In that way, a person's abilities or characteristics are never finitely determined but can reflect the culminating influences they've experienced thus far.
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We finitely designed movie two and three meaning every single set, every object, every prop, every setting, every creature, every blade of grass.
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People form lines where there are only finitely many people, and you can talk about who's the next person in line, and who's the person behind them, and so on.
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In 1986 Gerd Faltings won the Fields Medal, math's highest honor, primarily for solving a problem called the Mordell conjecture and proving that certain classes of Diophantine equations have only finitely many rational solutions (rather than infinitely many).
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The abc conjecture says that if you pick any exponent bigger than 1, then there are only finitely many abc triples in which c is larger than the product of the prime factors raised to your chosen exponent.
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If those three numbers don't have any factors in common apart from 3603, then when the product of their distinct prime factors is raised to any fixed exponent larger than 1 (for example, exponent 1.001) the result is larger than c with only finitely many exceptions.
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In 1983, Gerd Faltings, now a director of the Max Planck Institute for Mathematics in Bonn, Germany, proved the Mordell conjecture, which asserts that there are only finitely many rational solutions to certain types of algebraic equations, an advance for which he won the Fields Medal in 1986.
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Then a left A-module F is finitely generated (resp. finitely presented) if and only if the B-module B ⊗A F is finitely generated (resp. finitely presented).
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Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a field is simply a finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group.
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A subgroup of a finitely generated group need not be finitely generated. The commutator subgroup of the free group F_2 on two generators is an example of a subgroup of a finitely generated group that is not finitely generated. On the other hand, all subgroups of a finitely generated Abelian group are finitely generated. A subgroup of finite index in a finitely generated group is always finitely generated, and the Schreier index formula gives a bound on the number of generators required.
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262 (1961), pp. 455-475. On the other hand, it is an easy theorem that every finitely generated subgroup of a finitely presented group is recursively presented, so the recursively presented finitely generated groups are (up to isomorphism) exactly the finitely generated subgroups of finitely presented groups. Since every countable group is a subgroup of a finitely generated group, the theorem can be restated for those groups. As a corollary, there is a universal finitely presented group that contains all finitely presented groups as subgroups (up to isomorphism); in fact, its finitely generated subgroups are exactly the finitely generated recursively presented groups (again, up to isomorphism).
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Every homomorphic image of a finitely generated module is finitely generated. In general, submodules of finitely generated modules need not be finitely generated. As an example, consider the ring R = Z[X1, X2, ...] of all polynomials in countably many variables. R itself is a finitely generated R-module (with {1} as generating set).
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It is known that a monoid with finitely many left and right ideals is finitely presented (or just finitely generated) if and only if all of its Schutzenberger groups are finitely presented (respectively, finitely generated). Similarly such a monoid is residually finite if and only if all of its Schutzenberger groups are residually finite.
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A presentation is said to be finitely generated' if S is finite and finitely related if R is finite. If both are finite it is said to be a finite presentation. A group is finitely generated (respectively finitely related, ') if it has a presentation that is finitely generated (respectively finitely related, a finite presentation). A group which has a finite presentation with a single relation is called a one-relator group.
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If is an automorphism of a finitely generated free group then is finitely generated.
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Since B_0 is a finitely generated A-algebra, also B is a finitely generated A-algebra.
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In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over R,For example, Matsumura uses this terminology. or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below.
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In particular the finitely many equations we have listed above suffice. We say that Boolean algebra is finitely axiomatizable or finitely based. Can this list be made shorter yet? Again the answer is yes.
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Conversely, if a finitely generated algebra is integral (over the coefficient ring), then it is finitely generated module. (See integral element for more.) Let 0 → M′ → M → M′′ → 0 be an exact sequence of modules. Then M is finitely generated if M′, M′′ are finitely generated. There are some partial converses to this.
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Every finite group is finitely generated since . The integers under addition are an example of an infinite group which is finitely generated by both 1 and −1, but the group of rationals under addition cannot be finitely generated. No uncountable group can be finitely generated. For example, the group of real numbers under addition, (R, +).
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For an infinite group, a "small neighborhood" is taken to be a finitely generated subgroup. An infinite group is said to be locally P if every finitely generated subgroup is P. For instance, a group is locally finite if every finitely generated subgroup is finite, and a group is locally soluble if every finitely generated subgroup is soluble.
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If M is finitely generated and M' is finitely presented (which is stronger than finitely generated; see below), then M′ is finitely generated. Also, M is Noetherian (resp. Artinian) if and only if M′, M′′ are Noetherian (resp. Artinian). Let B be a ring and A its subring such that B is a faithfully flat right A-module.
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Every finitely presented group is recursively presented, but there are recursively presented groups that cannot be finitely presented. However a theorem of Graham Higman states that a finitely generated group has a recursive presentation if and only if it can be embedded in a finitely presented group. From this we can deduce that there are (up to isomorphism) only countably many finitely generated recursively presented groups. Bernhard Neumann has shown that there are uncountably many non- isomorphic two generator groups.
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Consider the submodule K consisting of all those polynomials with zero constant term. Since every polynomial contains only finitely many terms whose coefficients are non-zero, the R-module K is not finitely generated. In general, a module is said to be Noetherian if every submodule is finitely generated. A finitely generated module over a Noetherian ring is a Noetherian module (and indeed this property characterizes Noetherian rings): A module over a Noetherian ring is finitely generated if and only if it is a Noetherian module.
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If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups: :cyclic < abelian < nilpotent < supersolvable < polycyclic < solvable < finitely generated group.
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This is the fundamental theorem of finitely generated abelian groups. The existence of algorithms for Smith normal form shows that the fundamental theorem of finitely generated abelian groups is not only a theorem of abstract existence, but provides a way for computing expression of finitely generated abelian groups as direct sums.
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In mathematics, a (left) coherent ring is a ring in which every finitely generated left ideal is finitely presented. Many theorems about finitely generated modules over Noetherian rings can be extended to finitely presented modules over coherent rings. Every left Noetherian ring is left-coherent. The ring of polynomials in an infinite number of variables over a left Noetherian ring is an example of a left-coherent ring that is not left Noetherian.
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A Banach space finitely representable in ℓ2 is a Hilbert space. Every Banach space is finitely representable in c0. The space Lp([0, 1]) is finitely representable in ℓp. A Banach space X is super-reflexive if all Banach spaces Y finitely representable in X are reflexive, or, in other words, if no non-reflexive space Y is finitely representable in X. The notion of ultraproduct of a family of Banach spacesDacunha-Castelle, Didier; Krivine, Jean-Louis (1972), "Applications des ultraproduits à l'étude des espaces et des algèbres de Banach" (in French), Studia Math. 41:315-334.
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The dihedral group of order 8 requires two generators, as represented by this cycle diagram. In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of the finite set S and of inverses of such elements. By definition, every finite group is finitely generated, since S can be taken to be G itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated.
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In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1, ..., xs in G such that every x in G can be written in the form :x = n1x1 \+ n2x2 \+ ... + nsxs with integers n1, ..., ns. In this case, we say that the set } is a generating set of G or that x1, ..., xs generate G. Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified.
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Homomorphic images and subgroups of Noetherian groups are Noetherian, and an extension of a Noetherian group by a Noetherian group is Noetherian. Analogous results hold for Artinian groups. Noetherian groups are equivalently those such that every subgroup is finitely generated, which is stronger than the group itself being finitely generated: the free group on 2 or finitely more generators is finitely generated, but contains free groups of infinite rank. Noetherian groups need not be finite extensions of polycyclic groups.
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In group theory, Higman's embedding theorem states that every finitely generated recursively presented group R can be embedded as a subgroup of some finitely presented group G. This is a result of Graham Higman from the 1960s. Graham Higman, Subgroups of finitely presented groups. Proceedings of the Royal Society. Series A. Mathematical and Physical Sciences. vol.
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This resembles, but is not exactly Hilbert's basis theorem, which states that the polynomial ring R[X] over a Noetherian ring R is Noetherian. Both facts imply that a finitely generated commutative algebra over a Noetherian ring is again a Noetherian ring. More generally, an algebra (e.g., ring) that is a finitely generated module is a finitely generated algebra.
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Algebras that are not finitely generated are called infinitely generated.
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Every finitely generated abelian group or nilpotent group is polycyclic.
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International Journal of Algebra and Computation, vol. 15 (2005), no. 1, pp. 95-128). Grushko's theorem is, in a sense, a starting point in Dunwoody's theory of accessibility for finitely generated and finitely presented groups.
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Therefore, there are finitely generated groups that cannot be recursively presented.
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In view of the Stallings theorem about ends of groups, one-ended groups are precisely those finitely generated infinite groups that cannot be decomposed nontrivially as amalgamated products or HNN-extensions over finite subgroups. Dunwoody proved the Wall conjecture for finitely presented groups in 1985.Dunwoody, M. J., The accessibility of finitely presented groups. Inventiones Mathematicae, vol. 81 (1985), no. 3, pp.
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The six 6th complex roots of unity form a cyclic group under multiplication. Every Abelian group can be seen as a module over the ring of integers Z, and in a finitely generated Abelian group with generators x1, ..., xn, every group element x can be written as a linear combination of these generators, :x = α1⋅x1 \+ α2⋅x2 \+ ... + αn⋅xn with integers α1, ..., αn. Subgroups of a finitely generated Abelian group are themselves finitely generated. The fundamental theorem of finitely generated abelian groups states that a finitely generated Abelian group is the direct sum of a free Abelian group of finite rank and a finite Abelian group, each of which are unique up to isomorphism.
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Different subsets of the same group can be generating subsets. For example, if p and q are integers with , then also generates the group of integers under addition by Bézout's identity. While it is true that every quotient of a finitely generated group is finitely generated (the images of the generators in the quotient give a finite generating set), a subgroup of a finitely generated group need not be finitely generated. For example, let G be the free group in two generators, x and y (which is clearly finitely generated, since G = ), and let S be the subset consisting of all elements of G of the form ynxy−n for n a natural number.
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The following properties of finitely presented groups are Markov and therefore are algorithmically undecidable by the Adian–Rabin theorem: #Being the trivial group. #Being a finite group. #Being an abelian group. #Being a finitely generated free group.
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All theorems of ZFC are also theorems of von Neumann–Bernays–Gödel set theory, but the latter can be finitely axiomatized. The set theory New Foundations can be finitely axiomatized, but only with some loss of elegance.
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The idea of JSJ-decomposition was later extended by Rips and Sela to torsion-free finitely presented groupsE. Rips, and Z. Sela. "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition." Annals of Mathematics (2), vol.
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In category theory, a finitely generated object is the quotient of a free object over a finite set, in the sense that it is the target of a regular epimorphism from a free object that is free on a finite set.. For instance, one way of defining a finitely generated group is that it is the image of a group homomorphism from a finitely generated free group.
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A lattice in a nilpotent Lie group N is always finitely generated (and hence finitely presented since it is itself nilpotent); in fact it is generated by at most \dim(N) elements. Finally, a nilpotent group is isomorphic to a lattice in a nilpotent Lie group if and only if it contains a subgroup of finite index which is torsion-free and of finitely generated.
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A group such that all its subgroups are finitely generated is called Noetherian. A group such that every finitely generated subgroup is finite is called locally finite. Every locally finite group is periodic, i.e., every element has finite order.
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In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated). Polycyclic groups are finitely presented, and this makes them interesting from a computational point of view.
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Conversely, for a commutative Noetherian ring R, finitely generated flat modules are projective.
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New Foundations can be finitely axiomatized.Fenton, Scott, 2015. New Foundations Explorer Home Page.
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For finite-dimensional algebras over fields, these injective hulls are finitely-generated modules .
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Jørgensen's conjecture remains unproven.. However, if false, it has only finitely many counterexamples.
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A locally finite group is a group for which every finitely generated subgroup is finite. Since the cyclic subgroups of a locally finite group are finitely generated hence finite, every element has finite order, and so the group is periodic.
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Such rings are called -algebras and are studied in depth in the area of commutative algebra. For example, Noether normalization asserts that any finitely generated -algebra is closely related to (more precisely, finitely generated as a module over) a polynomial ring .
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The structure theorem for finitely generated modules over a principal ideal domain applies to K[X], when K is a field. This means that every finitely generated module over K[X] may be decomposed into a direct sum of a free module and finitely many modules of the form K[X]/\left\langle P^k \right\rangle, where P is an irreducible polynomial over K and k a positive integer.
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None of these counterexamples are finitely presented, and for some years it was considered possible that the conjecture held for finitely presented groups. However, in 2003, Alexander Ol'shanskii and Mark Sapir exhibited a collection of finitely- presented groups which do not satisfy the conjecture. In 2013, Nicolas Monod found an easy counterexample to the conjecture. Given by piecewise projective homeomorphisms of the line, the group is remarkably simple to understand.
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Finitely generated abelian groups are completely classified and are particularly easy to work with.
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Horwich recognizes that if he used substitutional quantifiers, his theory would be finitely statable.
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In 1954, Albert G. Howson showed that the intersection of two finitely generated subgroups of a free group is again finitely generated. Furthermore, if m and n are the numbers of generators of the two finitely generated subgroups then their intersection is generated by at most 2mn - m - n + 1 generators. This upper bound was then significantly improved by Hanna Neumann to 2(m-1)(n-1) + 1, see Hanna Neumann conjecture. The lattice of subgroups of a group satisfies the ascending chain condition if and only if all subgroups of the group are finitely generated.
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There are only finitely many distinct connected distance-regular graphs of any given valency k > 2. Similarly, there are only finitely many distinct connected distance-regular graphs with any given eigenvalue multiplicity m > 2 (with the exception of the complete multipartite graphs).
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Another nice application of Gromov's theorem and the Bass-Guivarch formula is to the quasi-isometric rigidity of finitely generated abelian groups: any group which is quasi-isometric to a finitely generated abelian group contains a free abelian group of finite index.
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In mathematics, a Mordellic variety is an algebraic variety which has only finitely many points in any finitely generated field. The terminology was introduced by Serge Lang to enunciate a range of conjectures linking the geometry of varieties to their Diophantine properties.
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Thus situations have negations, and any finitely many situations can be combined conjunctively and disjunctively.
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234–235 The fundamental theorem for finitely presented abelian groups was proven by Henry John Stephen Smith in , as integer matrices correspond to finite presentations of abelian groups (this generalizes to finitely presented modules over a principal ideal domain), and Smith normal form corresponds to classifying finitely presented abelian groups. The fundamental theorem for finitely generated abelian groups was proven by Henri Poincaré in , using a matrix proof (which generalizes to principal ideal domains). This was done in the context of computing the homology of a complex, specifically the Betti number and torsion coefficients of a dimension of the complex, where the Betti number corresponds to the rank of the free part, and the torsion coefficients correspond to the torsion part. Kronecker's proof was generalized to finitely generated abelian groups by Emmy Noether in .
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An R-algebra S is called finitely generated (as an algebra) if there are finitely many elements s1, ..., sn such that any element of s is expressible as a polynomial in the si. Equivalently, S is isomorphic to :R[T1, ..., Tn] / I. A much stronger condition is that S is finitely generated as an R-module, which means that any s can be expressed as a R-linear combination of some finite set s1, ..., sn.
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Kharlampovich is known for her example of a finitely presented 3-step solvable group with unsolvable word problem (solution of the Novikov–Adian problem) O. Kharlampovich, "A finitely presented solvable group with unsolvable word problem", Izvest. Ak. Nauk, Ser. Mat. (Soviet Math., Izvestia) 45, 4 (1981), 852–873. and for the solution together with A. Myasnikov of the Tarski conjecture (from 1945) about equivalence of first order theories of finitely generated non-abelian free groupsO.
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It follows that Field is not a reflective subcategory of CRing. The category of fields is neither finitely complete nor finitely cocomplete. In particular, Field has neither products nor coproducts. Another curious aspect of the category of fields is that every morphism is a monomorphism.
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Every subset of a Noetherian topological space is Noetherian, and hence has finitely many irreducible components.
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If \Gamma is a torsion-free, finitely generated nilpotent group then it is of type F.
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The abc conjecture would imply that there are at most finitely many counterexamples to Beal's conjecture.
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Note that Specification is an axiom schema. The theory given by these axioms is not finitely axiomatizable. Montague (1961) showed that ZFC is not finitely axiomatizable, and his argument carries over to GST. Hence any axiomatization of GST must either include at least one axiom schema.
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Any such module is isomorphic to the sum of a finitely- generated free module and an ideal, and the class of the ideal is uniquely determined by the module. Over a principal ideal domain, finitely-generated modules are torsion-free if and only if they are free.
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Formally, let X be a variety defined over an algebraically closed field of characteristic zero: hence X is defined over a finitely generated field E. If the set of points X(F) is finite for any finitely generated field extension F of E, then X is Mordellic.
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A ring is called Noetherian (in honor of Emmy Noether, who developed this concept) if every ascending chain of ideals :0 ⊆ I0 ⊆ I1 ... ⊆ In ⊆ In + 1 ⊆ ... becomes stationary, i.e. becomes constant beyond some index n. Equivalently, any ideal is generated by finitely many elements, or, yet equivalent, submodules of finitely generated modules are finitely generated. Being Noetherian is a highly important finiteness condition, and the condition is preserved under many operations that occur frequently in geometry.
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In 1905, the Hungarian mathematician Julius König published a paradox based on the fact that there are only countably many finite definitions. If we imagine the real numbers as a well- ordered set, those real numbers which can be finitely defined form a subset. Hence in this well-order there should be a first real number that is not finitely definable. This is paradoxical, because this real number has just been finitely defined by the last sentence.
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If S is finite, then a group is called finitely generated. The structure of finitely generated abelian groups in particular is easily described. Many theorems that are true for finitely generated groups fail for groups in general. It has been proven that if a finite group is generated by a subset S, then each group element may be expressed as a word from the alphabet S of length less than or equal to the order of the group.
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"JSJ-splittings for finitely presented groups over slender groups." Inventiones Mathematicae, vol. 135 (1999), no. 1, pp.
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In the mathematical subject of group theory, the Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957.Hanna Neumann. On the intersection of finitely generated free groups. Addendum.
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The word problem for a finitely generated group is the decision problem whether two words in the generators of the group represent the same element. The word problem for a given finitely generated group is solvable if and only if the group can be embedded in every algebraically closed group. The rank of a group is often defined to be the smallest cardinality of a generating set for the group. By definition, the rank of a finitely generated group is finite.
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Nakayama showed that Artinian serial rings have this property on their modules, and that the converse is not true The most general result, perhaps, on the modules of a serial ring is attributed to Drozd and Warfield: it states that every finitely presented module over a serial ring is a direct sum of cyclic uniserial submodules (and hence is serial). If additionally the ring is assumed to be Noetherian, the finitely presented and finitely generated modules coincide, and so all finitely generated modules are serial. Being right serial is preserved under direct products of rings and modules, and preserved under quotients of rings. Being uniserial is preserved for quotients of rings and modules, but never for products.
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On the intersection of finitely generated free groups. Journal of the London Mathematical Society, vol. 29 (1954), pp. 428-434 who proved that the intersection of any two finitely generated subgroups of a free group is always finitely generated, that is, has finite rank. In this paper Howson proved that if H and K are subgroups of a free group F(X) of finite ranks n ≥ 1 and m ≥ 1 then the rank s of H ∩ K satisfies: :s − 1 ≤ 2mn − m − n. In a 1956 paperHanna Neumann. On the intersection of finitely generated free groups. Publicationes Mathematicae Debrecen, vol. 4 (1956), 186-189. Hanna Neumann improved this bound by showing that : :s − 1 ≤ 2mn − 2m − n.
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Such 'reductions', expressing the moments in terms of finitely many dependent variables, are described by the Gibbons- Tsarev equation.
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The additive group of rational numbers Q is an example of a countable group that is not finitely generated.
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In 1964, Donald Monk showed that RRA has no finite axiomatization, unlike RA, which is finitely axiomatized by definition.
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Since each Uai intersects {Ga} for only finitely many values of a, the union of all such Uai intersects the collection {Ga} for only finitely many values of a. It follows that X (the whole space!) intersects the collection {Ga} at only finitely many values of a, contradicting the infinite cardinality of the collection {Ga}. A topological space in which every open cover admits a locally finite open refinement is called paracompact. Every locally finite collection of subsets of a topological space X is also point-finite.
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The equivalence between preorders and finite topologies can be interpreted as a version of Birkhoff's representation theorem, an equivalence between finite distributive lattices (the lattice of open sets of the topology) and partial orders (the partial order of equivalence classes of the preorder). This correspondence also works for a larger class of spaces called finitely generated spaces. Finitely generated spaces can be characterized as the spaces in which an arbitrary intersection of open sets is open. Finite topological spaces are a special class of finitely generated spaces.
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The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of finite abelian groups. The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules over a principal ideal domain, which in turn admits further generalizations.
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Similarly, if C is the category of finitely generated groups, ind-C is equivalent to the category of all groups.
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Let f be in A such that f M \subset F. Then fM is free since it is a submodule of a free module and A is a PID. But now f: M \to fM is an isomorphism since M is torsion-free. By the same argument as above, a finitely generated module over a Dedekind domain A (or more generally a semi-hereditary ring) is torsion-free if and only if it is projective; consequently, a finitely generated module over A is a direct sum of a torsion module and a projective module. A finitely generated projective module over a Noetherian integral domain has constant rank and so the generic rank of a finitely generated module over A is the rank of its projective part.
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In mathematics, finiteness properties of a group are a collection of properties that allow the use of various algebraic and topological tools, for example group cohomology, to study the group. It is mostly of interest for the study of infinite groups. Special cases of groups with finiteness properties are finitely generated and finitely presented groups.
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NBG is finitely axiomatizable, while ZFC and MK are not. A key theorem of NBG is the class existence theorem, which states that for every formula whose quantifiers range only over sets, there is a class consisting of the sets satisfying the formula. This class is built by mirroring the step-by-step construction of the formula with classes. Since all set-theoretic formulas are constructed from two kinds of atomic formulas (membership and equality) and finitely many logical symbols, only finitely many axioms are needed to build the classes satisfying them.
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In mathematics, the Scott core theorem is a theorem about the finite presentability of fundamental groups of 3-manifolds due to G. Peter Scott, . The precise statement is as follows: Given a 3-manifold (not necessarily compact) with finitely generated fundamental group, there is a compact three- dimensional submanifold, called the compact core or Scott core, such that its inclusion map induces an isomorphism on fundamental groups. In particular, this means a finitely generated 3-manifold group is finitely presentable. A simplified proof is given in , and a stronger uniqueness statement is proven in .
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The so-called coclass conjectures described the set of all finite p-groups of fixed coclass as perturbations of finitely many pro-p groups. The coclass conjectures were proven in the 1980s using techniques related to Lie algebras and powerful p-groups. The final proofs of the coclass theorems are due to A. Shalev and independently to C. R. Leedham-Green, both in 1994. They admit a classification of finite p-groups in directed coclass graphs consisting of only finitely many coclass trees whose (infinitely many) members are characterized by finitely many parametrized presentations.
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The Scott core theorem is a theorem about the finite presentability of fundamental groups of 3-manifolds due to G. Peter Scott. The precise statement is as follows: Given a 3-manifold (not necessarily compact) with finitely generated fundamental group, there is a compact three- dimensional submanifold, called the compact core or Scott core, such that its inclusion map induces an isomorphism on fundamental groups. In particular, this means a finitely generated 3-manifold group is finitely presentable. A simplified proof is given in, and a stronger uniqueness statement is proven in.
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Let M_n = I^n M. Then M_n are an I-stable filtration. Thus, by the observation, B_I M is finitely generated over B_I R. But B_I R \simeq R[It] is a Noetherian ring since R is. (The ring R[It] is called the Rees algebra.) Thus, B_I M is a Noetherian module and any submodule is finitely generated over B_I R; in particular, B_I N is finitely generated when N is given the induced filtration; i.e., N_n = M_n \cap N. Then the induced filtration is I-stable again by the observation.
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For finitely repeated games, if a stage game has only one unique Nash equilibrium, the subgame perfect equilibrium is to play without considering past actions, treating the current subgame as a one-shot game. An example of this is a finitely repeated Prisoner's dilemma game. Using backward induction, the last subgame in a finitely repeated Prisoner's dilemma requires players to play the unique Nash equilibrium (both players defecting). Because of this, all games prior to the last subgame will also play the Nash equilibrium to maximize their single-period payoffs.
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Waldhausen conjectured that every closed orientable 3-manifold has only finitely many Heegaard splittings (up to homeomorphism) of any given genus.
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Agol has shown that there are only finitely many cases in which the number of exceptional slopes is 9 or 10.
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In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups.
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As these two prototypes are both abelian, so is any cyclic group. The study of finitely generated abelian groups is quite mature, including the fundamental theorem of finitely generated abelian groups; and reflecting this state of affairs, many group-related notions, such as center and commutator, describe the extent to which a given group is not abelian.
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Neyman is one of the pioneers and a most notable leader of the study of repeated games under complexity constraints. In his seminal paperNeyman, A. (1985) "Bounded complexity justifies cooperation in the finitely repeated prisoners' dilemma." Economics Letters, 19(3), 227–229. he showed that bounded memory can justify cooperation in a finitely repeated prisoner's dilemma game.
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Simultaneously generalizing the case of imaginary quadratic fields and cyclotomic fields is the case of a CM field K, i.e. a totally imaginary quadratic extension of a totally real field. In 1974, Harold Stark conjectured that there are finitely many CM fields of class number 1. He showed that there are finitely many of a fixed degree.
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Note that transitivity of integrality above implies that if B is integral over A, then B is a union (equivalently an inductive limit) of subrings that are finitely generated A-modules. If A is noetherian, transitivity of integrality can be weakened to the statement: :There exists a finitely generated A-submodule of B that contains A[b].
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A group G is said to have the Howson property if for every finitely generated subgroups H,K of G their intersection H\cap K is again a finitely generated subgroup of G.O. Bogopolski, Introduction to group theory. Translated, revised and expanded from the 2002 Russian original. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2008.
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If the given category is finite (has finitely many objects and morphisms), then the following two definitions of the category algebra agree.
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This problem can be formulated as a linear programming problem, provided that the region Q is an intersection of finitely many hyperplanes.
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Since bounded generation is preserved under taking homomorphic images, if a single finitely generated group with at least two generators is known to be not boundedly generated, this will be true for the free group on the same number of generators, and hence for all free groups. To show that no (non-cyclic) free group has bounded generation, it is therefore enough to produce one example of a finitely generated group which is not boundedly generated, and any finitely generated infinite torsion group will work. The existence of such groups constitutes Golod and Shafarevich's negative solution of the generalized Burnside problem in 1964; later, other explicit examples of infinite finitely generated torsion groups were constructed by Aleshin, Olshanskii, and Grigorchuk, using automata. Consequently, free groups of rank at least two are not boundedly generated.
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2, pp. 145–186K. Fujiwara, and P. Papasoglu, "JSJ- decompositions of finitely presented groups and complexes of groups." Geometric and Functional Analysis, vol.
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Higman's embedding theorem also implies the Novikov-Boone theorem (originally proved in the 1950s by other methods) about the existence of a finitely presented group with algorithmically undecidable word problem. Indeed, it is fairly easy to construct a finitely generated recursively presented group with undecidable word problem. Then any finitely presented group that contains this group as a subgroup will have undecidable word problem as well. The usual proof of the theorem uses a sequence of HNN extensions starting with R and ending with a group G which can be shown to have a finite presentation.
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The rank of a finitely generated group G can be equivalently defined as the smallest cardinality of a set X such that there exists an onto homomorphism F(X) → G, where F(X) is the free group with free basis X. There is a dual notion of co-rank of a finitely generated group G defined as the largest cardinality of X such that there exists an onto homomorphism G → F(X). Unlike rank, co-rank is always algorithmically computable for finitely presented groups,John R. Stallings. Problems about free quotients of groups. Geometric group theory (Columbus, OH, 1992), pp.
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HNN-extensions play a key role in Higman's proof of the Higman embedding theorem which states that every finitely generated recursively presented group can be homomorphically embedded in a finitely presented group. Most modern proofs of the Novikov–Boone theorem about the existence of a finitely presented group with algorithmically undecidable word problem also substantially use HNN-extensions. Both HNN- extensions and amalgamated free products are basic building blocks in the Bass–Serre theory of groups acting on trees. The idea of HNN extension has been extended to other parts of abstract algebra, including Lie algebra theory.
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Let A be an abelian group. If A is finitely generated then by the fundamental theorem of finitely generated abelian groups, A is decomposable into a direct sum of cyclic subgroups, which leads to the classification of finitely generated abelian groups up to isomorphism. The structure of general infinite abelian groups can be considerably more complicated and the conclusion needs not to hold, but Prüfer proved that it remains true for periodic groups in two special cases. The first Prüfer theorem states that an abelian group of bounded exponent is isomorphic to a direct sum of cyclic groups.
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Given a graded module M over a commutative graded ring R, one can associate the formal power series P(M, t) \in \Z[\\![t]\\!]: :P(M, t) = \sum \ell(M_n) t^n (assuming \ell(M_n) are finite.) It is called the Hilbert–Poincaré series of M. A graded module is said to be finitely generated if the underlying module is finitely generated. The generators may be taken to be homogeneous (by replacing the generators by their homogeneous parts.) Suppose R is a polynomial ring k[x_0, \dots, x_n], k a field, and M a finitely generated graded module over it.
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The work of Jean-Christophe Yoccoz established local connectivity of the Mandelbrot set at all finitely renormalizable parameters; that is, roughly speaking those contained only in finitely many small Mandelbrot copies.. Hubbard cites as his source a 1989 unpublished manuscript of Yoccoz. Since then, local connectivity has been proved at many other points of M, but the full conjecture is still open.
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Let be a rational function of degree at least two with coefficients in . A theorem of Northcott says that has only finitely many -rational preperiodic points, i.e., has only finitely many preperiodic points in . The Uniform Boundedness Conjecture of Morton and Silverman says that the number of preperiodic points of in is bounded by a constant that depends only on the degree of .
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Every finitely-generated infinite group has either 0,1, 2, or infinitely many ends, and Stallings theorem about ends of groups provides a decomposition for groups with more than one end. If two connected locally finite graphs are quasi-isometric then they have the same number of ends. In particular, two quasi-isometric finitely generated groups have the same number of ends.
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A basic conjecture is that the pluricanonical ring is finitely generated. This is considered a major step in the Mori program. proved this conjecture.
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If an R-module M has finite length, then it is finitely generated. If R is a field, then the converse is also true.
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449-45 In 1991 he finally disproved Wall's conjecture by finding a finitely generated group that is not accessible.Dunwoody, Martin J. An inaccessible group.
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In some sense, "most" finitely presented groups with large defining relations are hyperbolic. For a quantitative statement of what this means see Random group.
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Geometric group theory studies the connections between algebraic properties of finitely generated groups and topological and geometric properties of spaces on which these groups act.
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Gromov's theorem states that a finitely generated group has polynomial growth if and only if it has a nilpotent subgroup that is of finite index.
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If R is Noetherian, and U is finitely generated, then U is a Noetherian module over R, and the conclusion is satisfied. Somewhat remarkable is that the weaker assumption, namely that U is finitely generated as an R-module (and no finiteness assumption on R), is sufficient to guarantee the conclusion. This is essentially the statement of Nakayama's lemma. Precisely, one has the following.
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The criterion for nilpotent Lie groups to have a lattice given above does not apply to more general solvable Lie groups. It remains true that any lattice in a solvable Lie group is uniform and that lattices in solvable groups are finitely presented. Not all finitely generated solvable groups are lattices in a Lie group. An algebraic criterion is that the group be polycyclic.
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Examples of polycyclic groups include finitely generated abelian groups, finitely generated nilpotent groups, and finite solvable groups. Anatoly Maltsev proved that solvable subgroups of the integer general linear group are polycyclic; and later Louis Auslander (1967) and Swan proved the converse, that any polycyclic group is up to isomorphism a group of integer matrices.Dmitriĭ Alekseevich Suprunenko, K. A. Hirsch, Matrix groups (1976), pp. 174–5; Google Books.
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He also co-invented the Reed–Muller codes. He discovered the codes, and Irving S. Reed proposed the majority logic decoding for the first time. Furthermore, he invented Muller automata, an automaton model for infinite words. In geometric group theory Muller is known for the Muller–Schupp theorem, joint with Paul Schupp, characterizing finitely generated virtually free groups as finitely generated groups with context-free word problem.
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Consider the matrix with integer entries, such that the entries of its th column are the coefficients of the th generator of the kernel. Then, the abelian group is isomorphic to the cokernel of linear map defined by . Conversely every integer matrix defines a finitely generated abelian group. It follows that the study of finitely generated abelian groups is totally equivalent with the study of integer matrices.
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Examples of groups that are residually finite are finite groups, free groups, finitely generated nilpotent groups, polycyclic- by-finite groups, finitely generated linear groups, and fundamental groups of 3-manifolds. Subgroups of residually finite groups are residually finite, and direct products of residually finite groups are residually finite. Any inverse limit of residually finite groups is residually finite. In particular, all profinite groups are residually finite.
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Suppose that R is a (commutative) principal ideal domain and M is a finitely-generated R-module. Then the structure theorem for finitely generated modules over a principal ideal domain gives a detailed description of the module M up to isomorphism. In particular, it claims that : M \simeq F\oplus T(M), where F is a free R-module of finite rank (depending only on M) and T(M) is the torsion submodule of M. As a corollary, any finitely-generated torsion-free module over R is free. This corollary does not hold for more general commutative domains, even for R = K[x,y], the ring of polynomials in two variables.
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Subobjects and quotient objects are well-behaved in abelian categories. For example, the poset of subobjects of any given object A is a bounded lattice. Every abelian category A is a module over the monoidal category of finitely generated abelian groups; that is, we can form a tensor product of a finitely generated abelian group G and any object A of A. The abelian category is also a comodule; Hom(G,A) can be interpreted as an object of A. If A is complete, then we can remove the requirement that G be finitely generated; most generally, we can form finitary enriched limits in A.
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There is a strong finiteness result on sheaf cohomology. Let X be a compact Hausdorff space, and let R be a principal ideal domain, for example a field or the ring Z of integers. Let E be a sheaf of R-modules on X, and assume that E has "locally finitely generated cohomology", meaning that for each point x in X, each integer j, and each open neighborhood U of x, there is an open neighborhood V ⊂ U of x such that the image of Hj(U,E) → Hj(V,E) is a finitely generated R-module. Then the cohomology groups Hj(X,E) are finitely generated R-modules.
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It is possible to prove that all relations between the Dehn twists in a generating set for the mapping class group can be written as combinations of a finite number among them. This means that the mapping class group of a surface is a finitely presented group. One way to prove this theorem is to deduce it from the properties of the action of the mapping class group on the pants complex: the stabiliser of a vertex is seen to be finitely presented, and the action is cofinite. Since the complex is connected and simply connected it follows that the mapping class group must be finitely generated.
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In mathematics, especially in the area of abstract algebra known as module theory, a ring R is called hereditary if all submodules of projective modules over R are again projective. If this is required only for finitely generated submodules, it is called semihereditary. For a noncommutative ring R, the terms left hereditary and left semihereditary and their right hand versions are used to distinguish the property on a single side of the ring. To be left (semi-)hereditary, all (finitely generated) submodules of projective left R-modules must be projective, and to be right (semi-)hereditary all (finitely generated) submodules of projective right submodules must be projective.
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In mathematics, compact objects, also referred to as finitely presented objects, or objects of finite presentation, are objects in a category satisfying a certain finiteness condition.
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Therefore, in order to avoid ambiguity, one may use the term finitely enumerable or denumerable to denote one of the corresponding types of distinguished countable enumerations.
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Therefore the category of smooth vector bundles on M is equivalent to the category of finitely generated, projective C∞(M)-modules. Details may be found in .
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In the mathematical field of knot theory, an unlink is a link that is equivalent (under ambient isotopy) to finitely many disjoint circles in the plane.
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For non-finitely generated modules, the above direct decomposition is not true. The torsion subgroup of an abelian group may not be a direct summand of it.
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In mathematics, especially algebraic geometry, the Bass conjecture says that certain algebraic K-groups are supposed to be finitely generated. The conjecture was proposed by Hyman Bass.
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In a subsequent 1984 paperR. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya. vol.
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Thus for all but finitely many , there is a prime such that divides , but does not divide for all < . This statement is an analogue of Zsigmondy's theorem.
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In a game with finitely many actions, this process always terminates and leaves a non-empty set of actions for each player. These are the rationalizable actions.
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2, pp. 345-466. that most "reasonable" time complexity functions of Turing machines can be realized, up to natural equivalence, as Dehn functions of finitely presented groups.
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This can be used to prove the theorem of that all compact flat manifolds are finitely covered by tori; the 3-dimensional case was proved earlier by .
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Such categories, called varieties, can be studied systematically using Lawvere theories. For any Lawvere theory T, there is a category Mod(T) of models of T, and the compact objects in Mod(T) are precisely the finitely presented models. For example: suppose T is the theory of groups. Then Mod(T) is the category of groups, and the compact objects in Mod(T) are the finitely presented groups.
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Any category with this property is necessarily a thin category: for any two objects there can be at most one morphism from one object to the other. A weaker form of completeness is that of finite completeness. A category is finitely complete if all finite limits exists (i.e. limits of diagrams indexed by a finite category J). Dually, a category is finitely cocomplete if all finite colimits exist.
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Projective modules can be defined to be the direct summands of free modules. If R is local, any finitely generated projective module is actually free, which gives content to an analogy between projective modules and vector bundles.See also Serre–Swan theorem. The Quillen–Suslin theorem asserts that any finitely generated projective module over k[T1, ..., Tn] (k a field) is free, but in general these two concepts differ.
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In model theory and universal algebra, a class K of structures of a given signature is said to have the hereditary property if every substructure of a structure in K is again in K. A variant of this definition is used in connection with Fraïssé's theorem: A class K of finitely generated structures has the hereditary property if every finitely generated substructure is again in K. See age.
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48 (1984), no. 5, pp. 939-985 This result answered a long-standing open problem posed by John Milnor in 1968 about the existence of finitely generated groups of intermediate growth. Grigorchuk's group has a number of other remarkable mathematical properties. It is a finitely generated infinite residually finite 2-group (that is, every element of the group has a finite order which is a power of 2).
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There is a vast literature on growth rates, leading up to Gromov's theorem. An earlier result of Joseph A. Wolf showed that if G is a finitely generated nilpotent group, then the group has polynomial growth. Yves Guivarc'h and independently Hyman Bass (with different proofs) computed the exact order of polynomial growth. Let G be a finitely generated nilpotent group with lower central series : G = G_1 \supseteq G_2 \supseteq \cdots.
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Naturally, if U is a Noetherian module, this holds. If R is Noetherian, and U is finitely generated, then U is a Noetherian module over R, and the conclusion is satisfied. Somewhat remarkable is that the weaker assumption, namely that U is finitely generated as an R-module (and no finiteness assumption on R), is sufficient to guarantee the conclusion. This is essentially the statement of Nakayama's lemma.
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The group of k-rational points for a global field k is finitely generated by the Mordell-Weil theorem. Hence, by the structure theorem for finitely generated abelian groups, it is isomorphic to a product of a free abelian group Zr and a finite commutative group for some non-negative integer r called the rank of the abelian variety. Similar results hold for some other classes of fields k.
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If R is a right principal ideal ring, then it is certainly a right Noetherian ring, since every right ideal is finitely generated. It is also a right Bézout ring since all finitely generated right ideals are principal. Indeed, it is clear that principal right ideal rings are exactly the rings which are both right Bézout and right Noetherian. Principal right ideal rings are closed under finite direct products.
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20, 1969, pp. 235–240. proved that the property of being Hopfian is undecidable for finitely presentable groups, while neither being Hopfian nor being non- Hopfian are Markov.
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Hence, the probability of Xn = 0 occurring for infinitely many n is 0\. Almost surely (i.e., with probability 1), Xn is nonzero for all but finitely many n.
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Since the edge lengths cannot change as the polyhedron flexes, the volume must remain at one of the finitely many roots of the polynomial, rather than changing continuously .
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The first part of the problem asks whether there are only finitely many essentially different space groups in n-dimensional Euclidean space. This was answered affirmatively by Bieberbach.
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The growth of a finitely generated group measures the asymptotics, as n \to \infty of the size of an n-ball in the Cayley graph of the group (that is, the number of elements of G that can be expressed as words of length at most n in the generating set of G). The study of growth rates of finitely generated groups goes back to the 1950s and is motivated in part by the notion of volume entropy (that is, the growth rate of the volume of balls) in the universal covering space of a compact Riemannian manifold in differential geometry. It is obvious that the growth rate of a finitely generated group is at most exponential and it was also understood early on that finitely generated nilpotent groups have polynomial growth. In 1968 John Milnor posed a questionJohn Milnor, Problem No. 5603, American Mathematical Monthly, vol. 75 (1968), pp. 685-686.
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A group G is said to be virtually free if there exists a subgroup of finite index H in G such that H is isomorphic to a free group. If G is a finitely generated virtually free group and H is a free subgroup of finite index in G then H itself is finitely generated, so that H is free of finite rank. The trivial group is viewed as the free group of rank 0, and thus all finite groups are virtually free. A basic result in Bass–Serre theory says that a finitely generated group G is virtually free if and only if G splits as the fundamental group of a finite graph of finite groups.
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The Smith normal form is useful for computing the homology of a chain complex when the chain modules of the chain complex are finitely generated. For instance, in topology, it can be used to compute the homology of a simplicial complex or CW complex over the integers, because the boundary maps in such a complex are just integer matrices. It can also be used to determine the invariant factors that occur in the structure theorem for finitely generated modules over a principal ideal domain, which includes the fundamental theorem of finitely generated abelian groups. The Smith normal form is also used in control theory to compute transmission and blocking zeros of a transfer function matrix.
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Sapir's early mathematical work concerns mostly semigroup theory. In geometric group theory his most well-known and significant results are obtained in two papers published in the Annals of Mathematics in 2002, the first joint with Jean- Camille Birget and Eliyahu Rips, and the second joint with Birget, Rips and Aleksandr Olshansky. The first paper provided an essentially complete description of all the possible growth types of Dehn functions of finitely presented groups. The second paper proves that a finitely presented group has the word problem solvable in non-deterministic polynomial time (NP) if and only if this group embeds as a subgroup of a finitely presented group with polynomial Dehn function.
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However, in 2002 Sapir and Olshanskii found finitely presented counterexamples: non-amenable finitely presented groups that have a periodic normal subgroup with quotient the integers. For finitely generated linear groups, however, the von Neumann conjecture is true by the Tits alternative: every subgroup of GL(n,k) with k a field either has a normal solvable subgroup of finite index (and therefore is amenable) or contains the free group on two generators. Although Tits' proof used algebraic geometry, Guivarc'h later found an analytic proof based on V. Oseledets' multiplicative ergodic theorem. Analogues of the Tits alternative have been proved for many other classes of groups, such as fundamental groups of 2-dimensional simplicial complexes of non-positive curvature.
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This is easy and standard (uses the fact that the trace defines a non-degenerate bilinear form.) Let A be a finitely generated algebra over a field k that is an integral domain with field of fractions K. If L is a finite extension of K, then the integral closure A' of A in L is a finitely generated A-module and is also a finitely generated k-algebra. The result is due to Noether and can be shown using the Noether normalization lemma as follows. It is clear that it is enough to show the assertion when L/K is either separable or purely inseparable. The separable case is noted above; thus, assume L/K is purely inseparable.
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Given constants C, D and V, there are only finitely many homotopy types of compact n-dimensional Riemannian manifolds with sectional curvature K ≥ C, diameter ≤ D and volume ≥ V.
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In formal language theory, in particular in algorithmic learning theory, a class C of languages has finite thickness if every string is contained in at most finitely many languages in C. This condition was introduced by Dana Angluin as a sufficient condition for C being identifiable in the limit. (citeseer.ist.psu.edu); here: Condition 3, p.123 mid. Angluin's original requirement (every non-empty string set be contained in at most finitely many languages) is equivalent.
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To produce a theory with finitely many axioms, the axiom schema of class comprehension is first replaced with finitely many class existence axioms. Then these axioms are used to prove the class existence theorem which implies every instance of the axiom schema. The proof of this theorem requires only seven class existence axioms, which are used to convert the construction of a formula into the construction of a class satisfying the formula.
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If a countable discrete group contains a (non-abelian) free subgroup on two generators, then it is not amenable. The converse to this statement is the so- called von Neumann conjecture, which was disproved by Olshanskii in 1980 using his Tarski monsters. Adyan subsequently showed that free Burnside groups are non-amenable: since they are periodic, they cannot contain the free group on two generators. These groups are finitely generated, but not finitely presented.
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It turns out that in general, finitely additive measures are connected with notions such as Banach limits, the dual of L∞ and the Stone–Čech compactification. All these are linked in one way or another to the axiom of choice. Contents remain useful in certain technical problems in geometric measure theory; this is the theory of Banach measures. A charge is a generalization in both directions: it is a finitely additive, signed measure.
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Boolean rings are absolutely flat: this means that every module over them is flat. Every finitely generated ideal of a Boolean ring is principal (indeed, (x,y) = (x + y + xy)).
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Or more explicitly, :B is a finitely generated A-module if and only if B is generated as an A-algebra by a finite number of elements integral over A.
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A Laurent polynomial is a Laurent series in which only finitely many coefficients are non-zero. Laurent polynomials differ from ordinary polynomials in that they may have terms of negative degree.
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However, in the plane, a decomposition into finitely many pieces must preserve the sum of the Banach measures of the pieces, and therefore cannot change the total area of a set .
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He has done research on the geometric topology of 3-dimensional manifolds, 3-dimensional hyperbolic geometry, minimal surface theory, hyperbolic groups, and Kleinian groups with their associated geometry, topology, and group theory. In 1973 he proved what is now known as the Scott core theorem or the Scott compact core theorem. This states that every 3-manifold M with finitely generated fundamental group has a compact core N, i.e., N is a compact submanifold such that inclusion induces a homotopy equivalence between N and M; the submanifold N is called a Scott compact core of the manifold M. He had previously proved that, given a fundamental group G of a 3-manifold, if G is finitely generated then G must be finitely presented.
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If H, K ≤ G are two subgroups of a group G and if a, b ∈ G define the same double coset HaK = HbK then the subgroups H ∩ aKa−1 and H ∩ bKb−1 are conjugate in G and thus have the same rank. It is known that if H, K ≤ F(X) are finitely generated subgroups of a finitely generated free group F(X) then there exist at most finitely many double coset classes HaK in F(X) such that H ∩ aKa−1 ≠ {1}. Suppose that at least one such double coset exists and let a1,...,an be all the distinct representatives of such double cosets. The strengthened Hanna Neumann conjecture, formulated by her son Walter Neumann (1990),Walter Neumann.
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These include formal sums over B, which are expressions of the form \sum a_i b_i where each coefficient ai is a nonzero integer, each factor bi is a distinct basis element, and the sum has finitely many terms. Alternatively, the elements of a free abelian group may be thought of as signed multisets containing finitely many elements of B, with the multiplicity of an element in the multiset equal to its coefficient in the formal sum. Another way to represent an element of a free abelian group is as a function from B to the integers with finitely many nonzero values; for this functional representation, the group operation is the pointwise addition of functions. Every set B has a free abelian group with B as its basis.
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As a strengthening of solvability, a group G is called supersolvable (or supersoluble) if it has an invariant normal series whose factors are all cyclic. Since a normal series has finite length by definition, uncountable groups are not supersolvable. In fact, all supersolvable groups are finitely generated, and an abelian group is supersolvable if and only if it is finitely generated. The alternating group A4 is an example of a finite solvable group that is not supersolvable.
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Three-valued operators can be realized in integrated circuits. In fuzzy logic, typically applied for approximate reasoning, a finitely-valued logic can represent propositions that may acquire values within a finite set. In mathematics, logical matrices having multiple truth degrees are used to model systems of axioms. Biophysical indications suggest that in the brain, synaptic charge injections occur in finite steps, and that neuron arrangements can be modeled based on the probability distribution of a finitely valued random variable.
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In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by R. M. Robinson in 1950. It is usually denoted Q. Q is almost PA without the axiom schema of mathematical induction. Q is weaker than PA but it has the same language, and both theories are incomplete. Q is important and interesting because it is a finitely axiomatized fragment of PA that is recursively incompletable and essentially undecidable.
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Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, an axiom schema stands for a countably infinite set of axioms. This set can usually be defined recursively. A theory that can be axiomatized without schemata is said to be finitely axiomatized. Theories that can be finitely axiomatized are seen as a bit more metamathematically elegant, even if they are less practical for deductive work.
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The axiom schemata of replacement and separation each contain infinitely many instances. included a result first proved in his 1957 Ph.D. thesis: if ZFC is consistent, it is impossible to axiomatize ZFC using only finitely many axioms. On the other hand, von Neumann–Bernays–Gödel set theory (NBG) can be finitely axiomatized. The ontology of NBG includes proper classes as well as sets; a set is any class that can be a member of another class.
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Finally, the assumption that A be a subring of B can be modified a bit. If f:A \to B is a ring homomorphism, then one says f is integral if B is integral over f(A). In the same way one says f is finite (B finitely generated A-module) or of finite type (B finitely generated A-algebra). In this viewpoint, one has that :f is finite if and only if f is integral and of finite type.
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In commutative algebra, the Fitting ideals of a finitely generated module over a commutative ring describe the obstructions to generating the module by a given number of elements. They were introduced by .
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Since there are only finitely many of these equations (the coefficients are bounded), the complete quotients (and also the partial denominators) in the regular continued fraction that represents x must eventually repeat.
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However, all but finitely many terms in a nonsingular EDS admit a primitive prime divisor. J. H. Silverman. Wieferich's criterion and the abc-conjecture. J. Number Theory, 30(2):226-237, 1988.
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It is a fact of algebra that if f is a polynomial function with coefficients in any field of characteristic 0, the same identity holds where the sum has finitely many terms.
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Any separable extension A / K of commutative rings is formally unramified. The converse holds if A is a finitely generated K-algebra. A separable flat (commutative) K-algebra A is formally étale.
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The inequality was first proved by Frigyes Riesz in 1930, and independently reproved by S.L.Sobolev in 1938. It can be generalized to arbitrarily (but finitely) many functions acting on arbitrarily many variables.
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In category theory, a branch of mathematics, the exact completion constructs a Barr-exact category from any finitely complete category. It is used to form the effective topos and other realizability toposes.
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In these properties of having centers on an ellipse and tangencies on a circle, the Pappus chain is analogous to the Steiner chain, in which finitely many circles are tangent to two circles.
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It is conjectured that a nonsingular EDS contains only finitely many primes M. Einsiedler, G. Everest, and T. Ward. Primes in elliptic divisibility sequences. LMS J. Comput. Math., 4:1-13 (electronic), 2001.
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But is itself an element of some open set , and it follows that can be covered by finitely many for some sufficiently small . This proves that , and it also yields a contradiction unless .
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In geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nilpotent subgroups of finite index.
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Under the subspace topology, the singleton set {–1} is open in Y, but under the induced order topology, any open set containing –1 must contain all but finitely many members of the space.
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Hugo Hadwiger conjectured in 1956 that every simplex can be dissected into finitely many orthoschemes. The conjecture has been proven in spaces of five or fewer dimensions, but remains unsolved in higher dimensions.
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A consequence of the volume formula in the previous paragraph is that :Given v>0 there are at most finitely many arithmetic hyperbolic 3-manifolds with volume less than v. This is in contrast with the fact that hyperbolic Dehn surgery can be used to produce infinitely many non-isometric hyperbolic 3-manifolds with bounded volume. In particular, a corollary is that given a cusped hyperbolic manifold, at most finitely many Dehn surgeries on it can yield an arithmetic hyperbolic manifold.
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In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated. The setting is as follows: Assume that k is a field and let K be a subfield of the field of rational functions in n variables, :k(x1, ..., xn ) over k. Consider now the k-algebra R defined as the intersection : R:= K \cap k[x_1, \dots, x_n] \ . Hilbert conjectured that all such algebras are finitely generated over k.
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In particular, his work implies that if a finitely generated group G is elementarily equivalent to a word- hyperbolic group then G is word-hyperbolic as well. Sela also proved that the first-order theory of a finitely generated free group is stable in the model- theoretic sense, providing a brand-new and qualitatively different source of examples for the stability theory. An alternative solution for the Tarski conjecture has been presented by Olga Kharlampovich and Alexei Myasnikov.O. Kharlampovich, and A. Myasnikov.
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In the mathematical theory of Kleinian groups, the Ahlfors finiteness theorem describes the quotient of the domain of discontinuity by a finitely generated Kleinian group. The theorem was proved by , apart from a gap that was filled by . The Ahlfors finiteness theorem states that if Γ is a finitely-generated Kleinian group with region of discontinuity Ω, then Ω/Γ has a finite number of components, each of which is a compact Riemann surface with a finite number of points removed.
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Thus G_w has property P if and only if w=_G 1. Since it is undecidable whether w=_G 1, it follows that it is undecidable whether a finitely presented group has property P.
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In mathematics, in the representation theory of groups, a group is said to be representation rigid if for every n, it has only finitely many isomorphism classes of complex irreducible representations of dimension n.
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These two notions of regularity coincide when F is a coherent sheaf such that Ass(F) contains no closed points. Then the graded module is finitely generated and has the same regularity as F.
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Precisely, one has: :Nakayama's lemma: Let U be a finitely generated right module over a (unital) ring R. If U is a non-zero module, then U·J(R) is a proper submodule of U.
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In mathematics, Strassmann's theorem is a result in field theory. It states that, for suitable fields, suitable formal power series with coefficients in the valuation ring of the field have only finitely many zeroes.
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In the present case this means that if every execution sequence of terminates, then there are only finitely many execution sequences. So if an output set of is infinite, it must contain [a nonterminating computation].
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53-54 and proved that it has intermediate growth in a 1984 article.R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya. vol.
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A Noetherian domain is a G-domain iff its rank is at most one, and has only finitely many maximal ideals (or equivalently, prime ideals).Kaplansky, Irving. Commutative Algebra. Polygonal Publishing House, 1974, p. 19.
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21 and any finitely generated infinite field is Hilbertian. There are several results on the permanence criteria of Hilbertian fields. Notably Hilbertianity is preserved under finite separable extensionsFried & Jarden (2008) p.224 and abelian extensions.
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In mathematics, a combinatorial class is a countable set of mathematical objects, together with a size function mapping each object to a non-negative integer, such that there are finitely many objects of each size...
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For example, Lω1ω permits countable conjunctions and disjunctions. The set of free variables in a formula of Lκω can have any cardinality strictly less than κ, yet only finitely many of them can be in the scope of any quantifier when a formula appears as a subformula of another.Some authors only admit formulas with finitely many free variables in Lκω, and more generally only formulas with < λ free variables in Lκλ. In other infinitary logics, a subformula may be in the scope of infinitely many quantifiers.
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However, every module is a cokernel of a homomorphism of free modules. Modules over the integers can be identified with abelian groups, since the multiplication by an integer may identified to a repeated addition. Most of the theory of abelian groups may be extended to modules over a principal ideal domain. In particular, over a principal ideal domain, every submodule of a free module is free, and the fundamental theorem of finitely generated abelian groups may be extended straightforwardly to finitely generated modules over a principal ring.
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If is not a finite set then every cofinite subset of is necessarily not empty so that in this case the definition becomes simply ::. This makes a filter on the lattice ((), ⊆), the power set of with set inclusion, given that denotes the complement of a set in , the following two conditions hold: :;Intersection condition: If two sets are finitely complemented in , then so is their intersection, since , and :;Upper-set condition: If a set is finitely complemented in , then so are its supersets in .
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If V is a variety defined over a field K, then the function field K(V) is a finitely generated field extension of the ground field K; its transcendence degree is equal to the dimension of the variety. All extensions of K that are finitely-generated as fields over K arise in this way from some algebraic variety. These field extensions are also known as algebraic function fields over K. Properties of the variety V that depend only on the function field are studied in birational geometry.
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In group theory, a group G is said to be free-by-cyclic if it has a free normal subgroup F such that the quotient group : G/F is cyclic. In other words, G is free-by-cyclic if it can be expressed as a group extension of a free group by a cyclic group (NB there are two conventions for 'by'). If F is a finitely generated group we say that G is (finitely generated free)-by- cyclic (or (f.g. free)-by-cyclic).
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Finally, if R is not Noetherian, then there exists an infinite ascending chain of finitely generated ideals, so in a Bézout domain an infinite ascending chain of principal ideals. (4) and (2) are thus equivalent. A Bézout domain is a Prüfer domain, i.e., a domain in which each finitely generated ideal is invertible, or said another way, a commutative semihereditary domain.) Consequently, one may view the equivalence "Bézout domain iff Prüfer domain and GCD-domain" as analogous to the more familiar "PID iff Dedekind domain and UFD".
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Start with any vertex v1. Every one of the infinitely many vertices of G can be reached from v1 with a simple path, and each such path must start with one of the finitely many vertices adjacent to v1. There must be one of those adjacent vertices through which infinitely many vertices can be reached without going through v1. If there were not, then the entire graph would be the union of finitely many finite sets, and thus finite, contradicting the assumption that the graph is infinite.
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If X is compact, it follows that the cohomology groups Hj(X,E) of X with coefficients in a constructible sheaf are finitely generated. More generally, suppose that X is compactifiable, meaning that there is a compact stratified space W containing X as an open subset, with W–X a union of connected components of strata. Then, for any constructible sheaf E of R-modules on X, the R-modules Hj(X,E) and Hcj(X,E) are finitely generated.Borel (1984), Lemma V.10.13.
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The Poincaré polynomial of a surface is defined to be the generating function of its Betti numbers. For example, the Betti numbers of the torus are 1, 2, and 1; thus its Poincaré polynomial is 1+2x+x^2. The same definition applies to any topological space which has a finitely generated homology. Given a topological space which has a finitely generated homology, the Poincaré polynomial is defined as the generating function of its Betti numbers, viz the polynomial where the coefficient of x^n is b_n.
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In metric geometry, asymptotic dimension of a metric space is a large-scale analog of Lebesgue covering dimension. The notion of asymptotic dimension was introduced my Mikhail Gromov in his 1993 monograph Asymptotic invariants of infinite groups in the context of geometric group theory, as a quasi-isometry invariant of finitely generated groups. As shown by Guoliang Yu, finitely generated groups of finite homotopy type with finite asymptotic dimension satisfy the Novikov conjecture. Asymptotic dimension has important applications in geometric analysis and index theory.
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A topology on a set X is defined as a subset of P(X), the power set of X, which includes both ∅ and X and is closed under finite intersections and arbitrary unions. Since the power set of a finite set is finite there can be only finitely many open sets (and only finitely many closed sets). Therefore, one only need check that the union of a finite number of open sets is open. This leads to a simpler description of topologies on a finite set.
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While von Neumann–Bernays–Gödel set theory is a conservative extension of Zermelo–Fraenkel set theory (ZFC, the canonical set theory) in the sense that a statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC, Morse–Kelley set theory is a proper extension of ZFC. Unlike von Neumann–Bernays–Gödel set theory, where the axiom schema of Class Comprehension can be replaced with finitely many of its instances, Morse–Kelley set theory cannot be finitely axiomatized.
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Unless the module is finitely-generated, there may exist no minimal generating set. The cardinality of a minimal generating set need not be an invariant of the module; Z is generated as a principal ideal by 1, but it is also generated by, say, a minimal generating set }. What is uniquely determined by a module is the infimum of the numbers of the generators of the module. Let R be a local ring with maximal ideal m and residue field k and M finitely generated module.
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One kind of subgroup of the three-dimensional translation group are the lattice groups, which are infinite groups, but unlike the translation groups, are finitely generated. That is, a finite generating set generates the entire group.
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His research interests include several areas of probability theory, Finitely addtive probability measures, stochastic calculus, martingale problems and Markov processes, Filtering theory, option pricing theory, psephology in the context of Indian elections and cryptography, among others.
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This group is a finitely generated pro-p-group, of finite width. For every finite group of order a power of p there is a closed subgroup of the Nottingham group isomorphic to that finite group.
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Since residuated Boolean algebras are axiomatized with finitely many identities, so are relation algebras. Hence the latter form a variety, the variety RA of relation algebras. Expanding the above definition as equations yields the following finite axiomatization.
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Module homomorphisms between finitely generated free modules may be represented by matrices. The theory of matrices over a ring is similar to that of matrices over a field, except that determinants exist only if the ring is commutative, and that a square matrix over a commutative ring is invertible only if its determinant has a multiplicative inverse in the ring. Vector spaces are completely characterized by their dimension (up to an isomorphism). In general, there is not such a complete classification for modules, even if one restricts oneself to finitely generated modules.
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Let F be finitely generated free group, with n generators. Let G1 and G2 be two finitely presented groups. Suppose there exists a surjective homomorphism \phi:F\rightarrow G_1\ast G_2, then there exists two subgroups F1 and F2 of F with \phi(F_1)=G_1 and \phi(F_2)=G_2 such that F=F_1\ast F_2. Proof: We give the proof assuming that F has no generator which is mapped to the identity of G_1\ast G_2, for if there are such generators, they may be added to any of F_1 or F_2.
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The Ugly duckling theorem is an argument showing that classification is not really possible without some sort of bias. More particularly, it assumes finitely many properties combinable by logical connectives, and finitely many objects; it asserts that any two different objects share the same number of (extensional) properties. The theorem is named after Hans Christian Andersen's 1843 story "The Ugly Duckling", because it shows that a duckling is just as similar to a swan as two duckling are to each other. It was proposed by Satosi Watanabe in 1969.
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Algebraic K-groups are used in conjectures on special values of L-functions and the formulation of a non-commutative main conjecture of Iwasawa theory and in construction of higher regulators.Lemmermeyer (2000) p.385 Parshin's conjecture concerns the higher algebraic K-groups for smooth varieties over finite fields, and states that in this case the groups vanish up to torsion. Another fundamental conjecture due to Hyman Bass (Bass' conjecture) says that all of the groups Gn(A) are finitely generated when A is a finitely generated Z-algebra.
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The idea of an isoperimetric function for a finitely presented group goes back to the work of Max Dehn in 1910s. Dehn proved that the word problem for the standard presentation of the fundamental group of a closed oriented surface of genus at least two is solvable by what is now called Dehn's algorithm. A direct consequence of this fact is that for this presentation the Dehn function satisfies Dehn(n) ≤ n. This result was extended in 1960s by Martin Greendlinger to finitely presented groups satisfying the C'(1/6) small cancellation condition.
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Universum museum in Mexico City A three-dimensional solid is a convex set if it contains every line segment connecting two of its points. A convex polyhedron is a polyhedron that, as a solid, forms a convex set. A convex polyhedron can also be defined as a bounded intersection of finitely many half-spaces, or as the convex hull of finitely many points. Important classes of convex polyhedra include the highly symmetrical Platonic solids, the Archimedean solids and their duals the Catalan solids, and the regular-faced Johnson solids.
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In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion. Historically, Hilbert was the first mathematician to work with the properties of finitely generated submodules. He proved an important theorem known as Hilbert's basis theorem which says that any ideal in the multivariate polynomial ring of an arbitrary field is finitely generated. However, the property is named after Emmy Noether who was the first one to discover the true importance of the property.
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In , it is shown that the automorphism defined by the elementary Nielsen transformations generate the full automorphism group of a finitely generated free group. Nielsen, and later Bernhard Neumann used these ideas to give finite presentations of the automorphism groups of free groups. This is also described in the textbook . For a given generating set of a given finitely generated group, it is not necessarily true that every automorphism is given by a Nielsen transformation, but for every automorphism, there is a generating set where the automorphism is given by a Nielsen transformation, .
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Moreover, a striking feature of projective planes is the symmetry of the roles played by points and lines. A less geometric example: a graph may be formalized via two base sets, the set of vertices (called also nodes or points) and the set of edges (called also arcs or lines). Generally, finitely many principal base sets and finitely many auxiliary base sets are stipulated by Bourbaki. Many mathematical structures of geometric flavor treated in the "Non-commutative geometry", "Schemes" and "Topoi" subsections above do not stipulate a base set of points.
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It is also the first example of a finitely generated group that is amenable but not elementary amenable, thus providing an answer to another long-standing problem, posed by Mahlon Day in 1957.Mahlon M. Day. Amenable semigroups.
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In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved by and , states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups.
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Another method is to twist along an annulus spanning two components. Gordon proved that for the class of links where these two constructions are not possible there are finitely many links in this class with a given complement.
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Examples of non-residually finite groups can be constructed using the fact that all finitely generated residually finite groups are Hopfian groups. For example the Baumslag–Solitar group B(2,3) is not Hopfian, and therefore not residually finite.
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The theory had been first developed in the 1879 paper of Georg Frobenius and Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra.
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Some facts about modules over a PID extend to modules over a Bézout domain. Let R be a Bézout domain and M finitely generated module over R. Then M is flat if and only if it is torsion-free.
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Both the Cauchy distribution (also called the Lorentzian) and more generally, stable distributions (related to the Lévy distribution) are examples of distributions for which the power-series expansions of the generating functions have only finitely many well-defined terms.
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For manifolds, the diffeomorphism group is usually not connected. Its component group is called the mapping class group. In dimension 2 (i.e. surfaces), the mapping class group is a finitely presented group generated by Dehn twists (Dehn, Lickorish, Hatcher).
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Each algebraic integer belongs to the ring of integers of some number field. A number is an algebraic integer if and only if the ring is finitely generated as an Abelian group, which is to say, as a -module.
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The theorem can be generalized to arbitrary -modules for rings having invariant basis number. In the finitely generated case the proof uses only elementary arguments of algebra, and does not require the axiom of choice nor its weaker variants.
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In general scheme theory, every scheme is the union of its irreducible components, but the number of components is not necessarily finite. However, in most cases occurring in "practice", namely for all noetherian schemes, there are finitely many irreducible components.
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Let M be a left module over a left Artinian ring. Then the following are equivalent (Hopkins' theorem): (i) M is finitely generated, (ii) M has finite length (i.e., has composition series), (iii) M is Noetherian, (iv) M is Artinian.
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57 (1980), no. 3, pp. 205-218 Floyd introduced a way to compactify a finitely generated group by adding to it a boundary which came to be called the Floyd boundary.Karlsson, Anders, Free subgroups of groups with nontrivial Floyd boundary.
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Transactions of the American Mathematical Society. Posted online July 21, 2008. fundamental groups of finite graphs of finitely generated free groupsGuo-An Diao and Mark Feighn. "The Grushko decomposition of a finite graph of finite rank free groups: an algorithm".
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In discrete mathematics, one uses the term almost all to mean cofinite (all but finitely many), cocountable (all but countably many), for sufficiently large numbers, or, sometimes, asymptotically almost surely. The concept is particularly important in the study of random graphs.
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In general, computations with the maximal atlas of a manifold are rather unwieldy. For most applications, it suffices to choose a smaller atlas. For example, if the manifold is compact, then one can find an atlas with only finitely many charts.
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In general, arbitrary direct sums and filtered colimits (also known as direct limits) of flat modules are flat, a consequence of the fact that the tensor product commutes with direct sums and filtered colimits (in fact with all colimits), and that both direct sums and filtered colimits are exact functors. In particular, this shows that all filtered colimits of free modules are flat. proved that the converse holds as well: M is flat if and only if it is a direct limit of finitely-generated free modules. As a consequence, one can deduce that every finitely-presented flat module is projective.
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Over a Noetherian integral domain, torsion-free modules are the modules whose only associated prime is zero. More generally, over a Noetherian commutative ring the torsion-free modules are those modules all of whose associated primes are contained in the associated primes of the ring. Over a Noetherian integrally closed domain, any finitely- generated torsion-free module has a free submodule such that the quotient by it is isomorphic to an ideal of the ring. Over a Dedekind domain, a finitely- generated module is torsion-free if and only if it is projective, but is in general not free.
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For example, if _α_ =(ω↦1) denotes the transfinite sequence with value 1 at ω and 0 everywhere else, then φ(ω↦1) is the smallest fixed point of all the functions ξ↦φ(ξ,0,…,0) with finitely many final zeroes (it is also the limit of the φ(1,0,…,0) with finitely many zeroes, the small Veblen ordinal). The smallest ordinal α such that α is greater than φ applied to any function with support in α (i.e., which cannot be reached “from below” using the Veblen function of transfinitely many variables) is sometimes known as the “large” Veblen ordinal.
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The groups T and V are (rare) examples of infinite but finitely-presented simple groups. The group F is not simple but its derived subgroup [F,F] is and the quotient of F by its derived subgroup is the free abelian group of rank 2. F is totally ordered, has exponential growth, and does not contain a subgroup isomorphic to the free group of rank 2. It is conjectured that F is not amenable and hence a further counterexample to the long-standing but recently disproved von Neumann conjecture for finitely-presented groups: it is known that F is not elementary amenable.
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If G is a finite group and k a field with characteristic 0, then one shows in the theory of group representations that any subrepresentation of a given one is already a direct summand of the given one. Translated into module language, this means that all modules over the group algebra kG are injective. If the characteristic of k is not zero, the following example may help. If A is a unital associative algebra over the field k with finite dimension over k, then Homk(−, k) is a duality between finitely generated left A-modules and finitely generated right A-modules.
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Therefore, the finitely generated injective left A-modules are precisely the modules of the form Homk(P, k) where P is a finitely generated projective right A-module. For symmetric algebras, the duality is particularly well-behaved and projective modules and injective modules coincide. For any Artinian ring, just as for commutative rings, there is a 1-1 correspondence between prime ideals and indecomposable injective modules. The correspondence in this case is perhaps even simpler: a prime ideal is an annihilator of a unique simple module, and the corresponding indecomposable injective module is its injective hull.
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In mathematics, a Hironaka decomposition is a representation of an algebra over a field as a finitely generated free module over a polynomial subalgebra or a regular local ring. Such decompositions are named after Heisuke Hironaka, who used this in his unpublished master's thesis at Kyoto University . Hironaka's criterion , sometimes called miracle flatness, states that a local ring R that is a finitely generated module over a regular Noetherian local ring S is Cohen–Macaulay if and only if it is a free module over S. There is a similar result for rings that are graded over a field rather than local.
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Shortly thereafter, Andrew Odlyzko showed that there are only finitely many Galois CM fields of class number 1. In 2001, V. Kumar Murty showed that of all CM fields whose Galois closure has solvable Galois group, only finitely many have class number 1. A complete list of the 172 abelian CM fields of class number 1 was determined in the early 1990s by Ken Yamamura and is available on pages 915–919 of his article on the subject. Combining this list with the work of Stéphane Louboutin and Ryotaro Okazaki provides a full list of quartic CM fields of class number 1.
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In mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every finitely generated ideal is principal. Any principal ideal domain (PID) is a Bézout domain, but a Bézout domain need not be a Noetherian ring, so it could have non-finitely generated ideals (which obviously excludes being a PID); if so, it is not a unique factorization domain (UFD), but still is a GCD domain.
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The vector space V is a 13-dimensional commutative unipotent algebraic group under addition, and its elements act on R by fixing all elements tj and taking xj to xj \+ bjtj. Then the ring of elements of R invariant under the action of the group V is not a finitely generated k-algebra. Several authors have reduced the sizes of the group and the vector space in Nagata's example. For example, showed that over any field there is an action of the sum G of three copies of the additive group on k18 whose ring of invariants is not finitely generated.
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The most important result is that all points can be constructed by the method of tangents and secants starting with a finite number of points. More precisely the Mordell–Weil theorem states that the group E(Q) is a finitely generated (abelian) group. By the fundamental theorem of finitely generated abelian groups it is therefore a finite direct sum of copies of Z and finite cyclic groups. The proof of that theorem rests on two ingredients: first, one shows that for any integer m > 1, the quotient group E(Q)/mE(Q) is finite (weak Mordell–Weil theorem).
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There is also a graded version of Nakayama's lemma. Let R be a ring that is graded by the ordered semigroup of non-negative integers, and let R_+ denote the ideal generated by positively graded elements. Then if M is a graded module over R for which M_i = 0 for i sufficiently negative (in particular, if M is finitely generated and R does not contain elements of negative degree) such that R_+M = M, then M = 0. Of particular importance is the case that R is a polynomial ring with the standard grading, and M is a finitely generated module.
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For arbitrarily many points in one dimension, there are again only finitely many solutions, one for each of the linear orderings (up to reversal of the ordering) of the points on a line. For every set of point masses, and every dimension less than , there exists at least one central configuration of that dimension. For almost all -tuples of masses there are finitely many "Dziobek" configurations that span exactly dimensions. It is an unsolved problem, posed by and , whether there is always a bounded number of central configurations for five or more masses in two or more dimensions.
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In the 1940s Grigore Moisil introduced his Łukasiewicz–Moisil algebras (LMn-algebras) in the hope of giving algebraic semantics for the (finitely) n-valued Łukasiewicz logic. However, in 1956 Alan Rose discovered that for n ≥ 5, the Łukasiewicz–Moisil algebra does not model the Łukasiewicz n-valued logic. Although C. C. Chang published his MV-algebra in 1958, it is a faithful model only for the ℵ0-valued (infinitely-many-valued) Łukasiewicz–Tarski logic. For the axiomatically more complicated (finitely) n-valued Łukasiewicz logics, suitable algebras were published in 1977 by Revaz Grigolia and called MVn-algebras.
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For instance, every free group has a Cayley graph (for its free generators) that is a tree. The free group on one generator has a doubly infinite path as its Cayley graph, with two ends. Every other free group has infinitely many ends. Every finitely generated infinite group has either 1, 2, or infinitely many ends, and the Stallings theorem about ends of groups provides a decomposition of groups with more than one end.. In particular: # A finitely generated infinite group has 2 ends if and only if it has a cyclic subgroup of finite index.
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In commutative algebra, an N−1 ring is an integral domain A whose integral closure in its quotient field is a finitely generated A module. It is called a Japanese ring (or an N−2 ring) if for every finite extension L of its quotient field K, the integral closure of A in L is a finitely generated A-module (or equivalently a finite A-algebra). A ring is called universally Japanese if every finitely generated integral domain over it is Japanese, and is called a Nagata ring, named for Masayoshi Nagata, (or a pseudo–geometric ring) if it is Noetherian and universally Japanese (or, which turns out to be the same, if it is Noetherian and all of its quotients by a prime ideal are N−2 rings.) A ring is called geometric if it is the local ring of an algebraic variety or a completion of such a local ring , but this concept is not used much.
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A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory.
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In particular, symbols denoting individual constants are nullary function symbols, and thus are terms. Only expressions which can be obtained by finitely many applications of rules 1 and 2 are terms. For example, no expression involving a predicate symbol is a term.
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Topology, vol. 30 (1991), no. 2, pp. 143–154 that a finitely generated group G admits a free isometric action on an R-tree if and only if G is a free product of surface groups, free groups and free abelian groups.
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Laurent series with only finitely many negative terms are well-behaved—they are a power series divided by z^k, and can be analyzed similarly—while Laurent series with infinitely many negative terms have complicated behavior on the inner circle of convergence.
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In order to understand magneto-mechanical behaviour of MREs, theoretical studies need to be performed which couple the theories of electromagnetism with mechanics. Such theories are called theories of magneto-mechanics.Kankanala, S. V. & Triantafyllidis, N. On finitely strained magnetorheological elastomers. J. Mech. Phys.
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An important case in computational group theory are string rewriting systems which can be used to give canonical labels to elements or cosets of a finitely presented group as products of the generators. This special case is the focus of this section.
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Gödel also likened logical intuition to sense perception, and considered the mathematical constructs that humans perceive to have an independent existence of their own. Under this line of reasoning, the human mind's ability to sense such abstract constructs may not be finitely implementable.
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A regular apeirogon can be defined as a partition of the Euclidean line E1 into infinitely many equal-length segments, generalizing the regular n-gon, which can be defined as a partition of the circle S1 into finitely many equal- length segments.
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In mathematics, in the representation theory of algebraic groups, a Grosshans subgroup, named after Frank Grosshans, is an algebraic subgroup of an algebraic group that is an observable subgroup for which the ring of functions on the quotient variety is finitely generated..
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FENE-P is a continuous model of polymer. The name FENE stands for finitely extensible nonlinear elastic while P stands for the closure proposed by Peterlin. It takes the dumbbell version of the FENE model and assumed the Peterline statistical closure for the restoring force.
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EGA IV, Part 1. Publications Mathématiques de l'IHÉS 20 (1964), 259 pp. 0.16.4.5. whereas the depth of a nonzero finitely generated R-module M is at most the Krull dimension of M (also called the dimension of the support of M).N. Bourbaki. Algèbre Commutative.
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Lattices in semisimple Lie groups are always finitely presented. For uniform lattices this is a direct consequence of cocompactness. In the non-uniform case this can be proved using reduction theory. However a much faster proof is by using Kazhdan's property (T) when possible.
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Then there are only finitely many points of E(K) whose x-coordinate is in the ring of integers OK. The properties of the Hasse–Weil zeta function and the Birch and Swinnerton-Dyer conjecture can also be extended to this more general situation.
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301–321 This result and Sela's approach were later generalized by others to finitely generated subgroups of hyperbolic groupsInna Bumagina, "The Hopf property for subgroups of hyperbolic groups." Geometriae Dedicata, vol. 106 (2004), pp. 211–230 and to the setting of relatively hyperbolic groups.
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Since every reduction from a set B to a set A has to determine whether a single element is in A in only finitely many steps, it can only make finitely many queries of membership in the set B. When the amount of information about the set B used to compute a single bit of A is discussed, this is made precise by the use function. Formally, the use of a reduction is the function that sends each natural number n to the largest natural number m whose membership in the set B was queried by the reduction while determining the membership of n in A.
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In mathematics, the Ahlfors conjecture, now a theorem, states that the limit set of a finitely-generated Kleinian group is either the whole Riemann sphere, or has measure 0. The conjecture was introduced by , who proved it in the case that the Kleinian group has a fundamental domain with a finite number of sides. proved the Ahlfors conjecture for topologically tame groups, by showing that a topologically tame Kleinian group is geometrically tame, so the Ahlfors conjecture follows from Marden's tameness conjecture that hyperbolic 3-manifolds with finitely generated fundamental groups are topologically tame (homeomorphic to the interior of compact 3-manifolds). This latter conjecture was proved, independently, by and by .
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If R is a Noetherian ring, then R[X_1,\dotsc,X_n] is a Noetherian > ring. This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the set of common roots of finitely many polynomial equations. proved the theorem (for the special case of polynomial rings over a field) in the course of his proof of finite generation of rings of invariants. Hilbert produced an innovative proof by contradiction using mathematical induction; his method does not give an algorithm to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist.
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Triples with q > 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small prime numbers. :ABC conjecture III. For every positive real number ε, there exist only finitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1 + ε. Whereas it is known that there are infinitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1, the conjecture predicts that only finitely many of those have q > 1.01 or q > 1.001 or even q > 1.0001, etc.
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Neyman has made many contributions to the theory of repeated games. One idea that appears, in different contexts, in some of his papers, is that the model of an infinitely repeated game serves also as a powerful paradigm for a long finitely repeated game. A related insight appears in a 1999 paper, where he showed that in a long finitely repeated game, an exponentially small deviation from common knowledge of the number of repetitions is enough to dramatically alter the equilibrium analysis, producing a folk-theorem-like result.Neyman, A. (1999), "Cooperation in Repeated Games when the Number of Stages is not Commonly Known," Econometrica, 67: 45–64.
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Consider the expression: :"The smallest positive integer not definable in under sixty letters." Since there are only twenty-six letters in the English alphabet, there are finitely many phrases of under sixty letters, and hence finitely many positive integers that are defined by phrases of under sixty letters. Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under sixty letters. If there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property; therefore, there is a smallest positive integer satisfying the property "not definable in under sixty letters".
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In addition, infinitely many sites are allowed in the definition (this setting has applications in geometry of numbers and crystallography), but again, in many cases only finitely many sites are considered. In the particular case where the space is a finite-dimensional Euclidean space, each site is a point, there are finitely many points and all of them are different, then the Voronoi cells are convex polytopes and they can be represented in a combinatorial way using their vertices, sides, two- dimensional faces, etc. Sometimes the induced combinatorial structure is referred to as the Voronoi diagram. In general however, the Voronoi cells may not be convex or even connected.
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While a finitely additive measure is sufficient for most intuition of area, and is analogous to Riemann integration, it is considered insufficient for probability, because conventional modern treatments of sequences of events or random variables demand countable additivity. In this respect, the plane is similar to the line; there is a finitely additive measure, extending Lebesgue measure, which is invariant under all isometries. When you increase in dimension the picture gets worse. The Hausdorff paradox and Banach–Tarski paradox show that you can take a three-dimensional ball of radius 1, dissect it into 5 parts, move and rotate the parts and get two balls of radius 1.
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In the mathematical subject of geometric group theory, a Dehn function, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the area of a relation in that group (that is a freely reduced word in the generators representing the identity element of the group) in terms of the length of that relation (see pp. 79-80 in ). The growth type of the Dehn function is a quasi-isometry invariant of a finitely presented group. The Dehn function of a finitely presented group is also closely connected with non-deterministic algorithmic complexity of the word problem in groups.
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For example, if G=SL_n and V=M_n the space of square matrices, and the action of G on V is given by left multiplication, then k[V]^G is isomorphic to a polynomial algebra in one variable, generated by the determinant. In other words, in this case, every invariant polynomial is a linear combination of powers of the determinant polynomial. So in this case, k[V]^G is finitely generated over k. If the answer is yes, then the next question is to find a minimal basis, and ask whether the module of polynomial relations between the basis elements (known as the syzygies) is finitely generated over k[V].
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The following result manifests Nakayama's lemma in terms of generators. Statement 4: If M is a finitely-generated module over R and the images of elements m1,...,mn of M in M / J(R)M generate M / J(R)M as an R-module, then m1,...,mn also generate M as an R-module. :Proof: Apply Statement 3 to N = ΣiRmi. If one assumes instead that R is complete and M is separated with respect to the I-adic topology for an ideal I in R, this last statement holds with I in place of J(R) and without assuming in advance that M is finitely generated.
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Early notable results of Bowditch include clarifying the classic notion of geometric finiteness for higher-dimensional Kleinian groups in constant and variable negative curvature. In a 1993 paper Bowditch proved that five standard characterisations of geometric finiteness for discrete groups of isometries of hyperbolic 3-space and hyperbolic plane, (including the definition in terms of having a finitely-sided fundamental polyhedron) remain equivalent for groups of isometries of hyperbolic n-space where n ≥ 4\. He showed, however, that in dimensions n ≥ 4 the condition of having a finitely-sided Dirichlet domain is no longer equivalent to the standard notions of geometric finiteness. In a subsequent paperB.
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We recall that for a (unital, associative) ring R, we denote by V(R) the (conical, commutative) monoid of isomorphism classes of finitely generated projective right R-modules, see here for more details. Recall that if R is von Neumann regular, then V(R) is a refinement monoid. Denote by Idc R the (∨,0)-semilattice of finitely generated two-sided ideals of R. We denote by L(R) the lattice of all principal right ideals of a von Neumann regular ring R. It is well known that L(R) is a complemented modular lattice. The following result was observed by Wehrung, building on earlier works mainly by Jónsson and Goodearl.
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In the mathematics of infinite graphs, an end of a graph represents, intuitively, a direction in which the graph extends to infinity. Ends may be formalized mathematically as equivalence classes of infinite paths, as havens describing strategies for pursuit-evasion games on the graph, or (in the case of locally finite graphs) as topological ends of topological spaces associated with the graph. Ends of graphs may be used (via Cayley graphs) to define ends of finitely generated groups. Finitely generated infinite groups have one, two, or infinitely many ends, and the Stallings theorem about ends of groups provides a decomposition for groups with more than one end.
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In higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of finitely many lines partitions the plane into convex polygons (possibly unbounded); this partition is known as an arrangement of lines.
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See in particular section 3.2, Many-Sorted Quantification. When there are only finitely many sorts in a theory, many-sorted first-order logic can be reduced to single-sorted first- order logic. Enderton, H. A Mathematical Introduction to Logic, second edition. Academic Press, 2001, pp.296–299.
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Every tile type subdivides into smaller tile types. Each edge also gets subdivided according to finitely many edge types. Finite subdivision rules can only subdivide tilings that are made up of polygons labelled by tile types. Such tilings are called subdivision complexes for the subdivision rule.
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That is, the following three conditions are equivalent to each other:. #F is a minor-closed family of bounded-treewidth graphs; #One of the finitely many forbidden minors characterizing F is planar; #F is a minor-closed graph family that does not include all planar graphs.
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Lorenz (2008) p. 31 The Witt ring is a Jacobson ring.Lorenz (2008) p. 35 It is a Noetherian ring if and only if there are finitely many square classes; that is, if the squares in k form a subgroup of finite index in the multiplicative group.
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More generally, every minor-closed graph family is incapable of representing all finite groups by the symmetries of its graphs., Theorem 4.5. László Babai conjectures, more strongly, that each minor-closed family can represent only finitely many non-cyclic finite simple groups., discussion following Theorem 4.5.
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Those most popular in the literature are three-valued (e.g., Łukasiewicz's and Kleene's, which accept the values "true", "false", and "unknown"), the finite-valued (finitely-many valued) with more than three values, and the infinite-valued (infinitely-many-valued), such as fuzzy logic and probability logic.
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The set of evil numbers (numbers n with t_n=0) forms a subspace of the nonnegative integers under nim-addition (bitwise exclusive or). For the game of Kayles, evil nim-values occur for few (finitely many) positions in the game, with all remaining positions having odious nim- values.
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For example, the properties of being nontrivial, infinite, nonabelian, etc., for finitely presentable groups are undecidable. However, there do exist examples of interesting undecidable properties such that neither these properties nor their complements are Markov. Thus Collins (1969) Donald J. Collins, On recognizing Hopf groups, Archiv der Mathematik, vol.
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Eisenbud (1995), Exercise 18.17. It is striking that this property is independent of the choice of f. Finally, there is a version of Miracle Flatness for graded rings. Let R be a finitely generated commutative graded algebra over a field K, :R=K\oplus R_1 \oplus R_2 \oplus \cdots.
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If a group is of type FPn then its cohomology groups H^i(\Gamma) are finitely generated for 0 \le i \le n. If it is of type FP then it is of finite cohomological dimension. Thus finiteness properties play an important role in the cohomology theory of groups.
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The residue field of R is defined as :k = R / m. Any R-module M yields a k-vector space given by M / mM. Nakayama's lemma shows this passage is preserving important information: a finitely generated module M is zero if and only if M / mM is zero.
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This set may also be empty, i. e., the player can avoid her loss for only finitely many moves if her opponent plays correctly. But this is equivalent to the opponent being able to force a win. This is the basis for all modern versions of Zermelo's theorem.
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Let the total energy of the traffic graph be the sum of the energies of every edge in the graph. Take a choice of routes that minimizes the total energy. Such a choice must exist because there are finitely many choices of routes. That will be an equilibrium.
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In mathematics, a function on the real numbers is called a step function (or staircase function) if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces. right- continuous.
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Formally, let be a set and let be a family of subsets of . Then is called a topology on if: # Both the empty set and are elements of . # Any union of elements of is an element of . # Any intersection of finitely many elements of is an element of .
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Several branches of programming language semantics, such as domain theory, are formalized using topology. In this context, Steve Vickers, building on work by Samson Abramsky and Michael B. Smyth, characterizes topological spaces as Boolean or Heyting algebras over open sets, which are characterized as semidecidable (equivalently, finitely observable) properties.
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There is a remarkable characterization of Cohen–Macaulay rings, sometimes called miracle flatness or Hironaka's criterion. Let R be a local ring which is finitely generated as a module over some regular local ring A contained in R. Such a subring exists for any localization R at a prime ideal of a finitely generated algebra over a field, by the Noether normalization lemma; it also exists when R is complete and contains a field, or when R is a complete domain.Bruns & Herzog, Theorem A.22. Then R is Cohen–Macaulay if and only if it is flat as an A-module; it is also equivalent to say that R is free as an A-module.
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A discrete group G of isometries of hyperbolic space is called geometrically finite if it has a fundamental domain C that is convex, geometrically finite, and exact (every face is the intersection of C and gC for some g ∈ G) . In hyperbolic spaces of dimension at most 3, every exact, convex, fundamental polyhedron for a geometrically finite group has only a finite number of sides, but in dimensions 4 and above there are examples with an infinite number of sides . In hyperbolic spaces of dimension at most 2, finitely generated discrete groups are geometrically finite, but showed that there are examples of finitely generated discrete groups in dimension 3 that are not geometrically finite.
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Brady subsequently used their Morse theory technique to construct the first example of a finitely presented subgroup of a word-hyperbolic group that is not itself word-hyperbolic.Brady, Noel, Branched coverings of cubical complexes and subgroups of hyperbolic groups. Journal of the London Mathematical Society (2), vol. 60 (1999), no.
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In mathematics, the Mostow–Palais theorem is an equivariant version of the Whitney embedding theorem. It states that if a manifold is acted on by a compact Lie group with finitely many orbit types, then it can be embedded into some finite-dimensional orthogonal representation. It was introduced by and .
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In the mathematical field of topology, a locally finite space is a topological space in which every point has a finite neighborhood, that is an open neighborhood consisting of finitely many elements. A locally finite space is Alexandrov. A T1 space is locally finite if and only if it is discrete.
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In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer.
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Martin Greendlinger, An analogue of a theorem of Magnus. Archiv der Mathematik, vol 12 (1961), pp. 94-96. who primarily dealt with the "metric" small cancellation conditions. In particular, Greendlinger proved that finitely presented groups satisfying the C'(1/6) small cancellation condition have word problem solvable by Dehn's algorithm.
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Fields and rings of polynomials or power series in finitely many indeterminates over fields are examples of Japanese rings. Another important example is a Noetherian integrally closed domain (e.g. a Dedekind domain) having a perfect field of fractions. On the other hand, a PID or even a DVR is not necessarily Japanese.
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Quantum gravity is believed to be background-independent (in some suitable sense), and TQFTs provide examples of background independent quantum field theories. This has prompted ongoing theoretical investigations into this class of models. (Caveat: It is often said that TQFTs have only finitely many degrees of freedom. This is not a fundamental property.
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There are many known proofs of the circle packing theorem. Paul Koebe's original proof is based on his conformal uniformization theorem saying that a finitely connected planar domain is conformally equivalent to a circle domain. There are several different topological proofs that are known. Thurston's proof is based on Brouwer's fixed point theorem.
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Pairs of the numbers (n, m) that solve Brocard's problem are called Brown numbers. As of 2019, there are only three known pairs of Brown numbers: :(4,5), (5,11), and (7,71). Paul Erdős conjectured that no other solutions exist. showed that there are only finitely many solutions provided that the abc conjecture is true.
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In the field of calculus of variations in mathematics, the method of Lagrange multipliers on Banach spaces can be used to solve certain infinite-dimensional constrained optimization problems. The method is a generalization of the classical method of Lagrange multipliers as used to find extrema of a function of finitely many variables.
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The ending lamination theorem, originally conjectured by William Thurston and later proven by Jeffrey Brock, Richard Canary, and Yair Minsky, states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are geodesic laminations on some surfaces in the boundary of the manifold.
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Thus the coordinate vector for v is zero except in finitely many entries. The linear transformations between (possibly) infinite- dimensional vector spaces can be modeled, analogously to the finite- dimensional case, with infinite matrices. The special case of the transformations from V into V is described in the full linear ring article.
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250–270 containing the proof of a famous result now known as the Tits alternative. The result states that a finitely generated linear group is either virtually solvable or contains a free subgroup of rank two. The ping-pong lemma and its variations are widely used in geometric topology and geometric group theory.
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Euler calculus is a methodology from applied algebraic topology and integral geometry that integrates constructible functions and more recently definable functionsBaryshnikov, Y.; Ghrist, R. Euler integration for definable functions, Proc. National Acad. Sci., 107(21), 9525–9530, 25 May 2010. by integrating with respect to the Euler characteristic as a finitely-additive measure.
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Several authors have stated in the mathematical literature that it is obvious that finitely generated free groups are not boundedly generated. This section contains various obvious and less obvious ways of proving this. Some of the methods, which touch on bounded cohomology, are important because they are geometric rather than algebraic, so can be applied to a wider class of groups, for example Gromov-hyperbolic groups. Since for any n ≥ 2, the free group on 2 generators F2 contains the free group on n generators Fn as a subgroup of finite index (in fact n – 1), once one non-cyclic free group on finitely many generators is known to be not boundedly generated, this will be true for all of them.
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In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate (that is, faster than polynomial but slower than exponential) growth. The group was originally constructed by Grigorchuk in a 1980 paper and he then proved in a 1984 paper that this group has intermediate growth, thus providing an answer to an important open problem posed by John Milnor in 1968. The Grigorchuk group remains a key object of study in geometric group theory, particularly in the study of the so-called branch groups and automata groups, and it has important connections with the theory of iterated monodromy groups.Volodymyr Nekrashevych.
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It has a straightforward extension to modules stating that every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules. This contains the case for rings as a special case, considering the ring as a module over itself, so that ideals are submodules. This also generalizes the primary decomposition form of the structure theorem for finitely generated modules over a principal ideal domain, and for the special case of polynomial rings over a field, it generalizes the decomposition of an algebraic set into a finite union of (irreducible) varieties. The first algorithm for computing primary decompositions for polynomial rings over a field of characteristic 0Primary decomposition requires testing irreducibility of polynomials, which is not always algorithmically possible in nonzero characteristic.
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In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group G has more than one end if and only if the group G admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup. In the modern language of Bass–Serre theory the theorem says that a finitely generated group G has more than one end if and only if G admits a nontrivial (that is, without a global fixed point) action on a simplicial tree with finite edge-stabilizers and without edge-inversions. The theorem was proved by John R. Stallings, first in the torsion-free case (1968)John R. Stallings. On torsion-free groups with infinitely many ends.
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A Wieferich number is an odd natural number n satisfying the congruence 2(n) ≡ 1 (mod n2), where denotes the Euler's totient function (according to Euler's theorem, 2(n) ≡ 1 (mod n) for every odd natural number n). If Wieferich number n is prime, then it is a Wieferich prime. The first few Wieferich numbers are: :1, 1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 68859, 94797, 99463, ... It can be shown that if there are only finitely many Wieferich primes, then there are only finitely many Wieferich numbers. In particular, if the only Wieferich primes are 1093 and 3511, then there exist exactly 104 Wieferich numbers, which matches the number of Wieferich numbers currently known.
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The Tsirelson space is reflexive () and finitely universal, which means that for some constant , the space contains -isomorphic copies of every finite-dimensional normed space, namely, for every finite-dimensional normed space , there exists a subspace of the Tsirelson space with multiplicative Banach-Mazur distance to less than . Actually, every finitely universal Banach space contains almost-isometric copies of every finite- dimensional normed space,this is because for every , and ε, there exists such that every -isomorph of ℓ∞ contains a -isomorph of ℓ∞n, by James' blocking technique (see Lemma 2.2 in Robert C. James "Uniformly Non-Square Banach Spaces", Annals of Mathematics, Vol. 80, 1964, pp. 542-550), and because every finite-dimensional normed space -embeds in ℓ∞ when is large enough.
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Dunwoody works on geometric group theory and low-dimensional topology. He is a leading expert in splittings and accessibility of discrete groups, groups acting on graphs and trees, JSJ-decompositions, the topology of 3-manifolds and the structure of their fundamental groups. Since 1971 several mathematicians have been working on Wall's conjecture, posed by Wall in a 1971 paper,Wall, C. T. C., Pairs of relative cohomological dimension one. Journal of Pure and Applied Algebra, vol. 1 (1971), no. 2, pp. 141-154 which said that all finitely generated groups are accessible. Roughly, this means that every finitely generated group can be constructed from finite and one-ended groups via a finite number of amalgamated free products and HNN extensions over finite subgroups.
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Unification in Boolean rings is decidable, that is, algorithms exist to solve arbitrary equations over Boolean rings. Both unification and matching in finitely generated free Boolean rings are NP-complete, and both are NP-hard in finitely presented Boolean rings. (In fact, as any unification problem f(X) = g(X) in a Boolean ring can be rewritten as the matching problem f(X) + g(X) = 0, the problems are equivalent.) Unification in Boolean rings is unitary if all the uninterpreted function symbols are nullary and finitary otherwise (i.e. if the function symbols not occurring in the signature of Boolean rings are all constants then there exists a most general unifier, and otherwise the minimal complete set of unifiers is finite).
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In the mathematical subject of group theory, the Adian–Rabin theorem is a result which states that most "reasonable" properties of finitely presentable groups are algorithmically undecidable. The theorem is due to Sergei Adian (1955)S. I. Adian, Algorithmic unsolvability of problems of recognition of certain properties of groups. Doklady Akademii Nauk SSSR vol.
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There are good reasons for this diverging terminology. The signatures that are considered in general model theory are often infinite, while a single first-order sentence contains only finitely many symbols. Therefore, basic elementary classes are atypical in infinite model theory. Finite model theory, on the other hand, deals almost exclusively with finite signatures.
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Finally, ZFC's axiom of extensionality is modified to handle classes: If two classes have the same elements, then they are identical. The other axioms of ZFC are not modified. This theory is not finitely axiomatized. ZFC's replacement schema has been replaced by a single axiom, but the axiom schema of class comprehension has been introduced.
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For any given divisor, only finitely many different remainders can occur. In the example above, the 74 possible remainders are 0, 1, 2, ..., 73\. If at any point in the division the remainder is 0, the expansion terminates at that point. Then the length of the repetend, also called “period”, is defined to be 0.
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In every unordered tree there is a distinguished partition of the set of nodes into sibling sets. Two non-root nodes , belong to the same sibling set if . The root node forms the singleton sibling set }. A tree is said to be locally finite or finitely branching if each of its sibling sets is finite.
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The branching points will correspond to the choice points in the program. Since there are always only finitely many alternatives at each choice point, the branching factor of the tree is always finite. That is, the tree is finitary. Now Kőnig's lemma says that if every branch of a finitary tree is finite, then so is the tree itself.
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Any closed subgroup and image of diagonalizable groups are diagonalizable. The torsion subgroup of a diagonalizable group is dense. The category of diagonalizable groups defined over k is equivalent to the category of finitely generated abelian group with Gal(k/ks)-equivariant morphisms without p-torsion. This is an analog of Poincaré duality and motivated the terminology.
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There is always a graded polynomial subring A ⊂ R (with generators in various degrees) such that R is finitely generated as an A-module. Then R is Cohen–Macaulay if and only if R is free as a graded A-module. Again, it follows that this freeness is independent of the choice of the polynomial subring A.
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In arithmetic geometry, the Mordell conjecture is the conjecture made by that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by , and is now known as Faltings's theorem. The conjecture was later generalized by replacing Q by any number field.
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Let H be a semisimple finite weak Hopf algebra, then modules over H form a semisimple rigid monoidal category with finitely many simple objects. Moreover the homomorphisms spaces are finite-dimensional vector spaces and the endomorphisms space of simple objects are one-dimensional. Finally, the monoidal unit is a simple object. Such a category is called a fusion category.
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This is why NBG is finitely axiomatizable. Classes are also used for other constructions, for handling the set-theoretic paradoxes, and for stating the axiom of global choice, which is stronger than ZFC's axiom of choice. John von Neumann introduced classes into set theory in 1925. The primitive notions of his theory were function and argument.
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A discrete Lanczos window and its frequency response; see Window function for comparison with other windows. The theoretically optimal reconstruction filter for band-limited signals is the sinc filter, which has infinite support. The Lanczos filter is one of many practical (finitely supported) approximations of the sinc filter. Each interpolated value is the weighted sum of consecutive input samples.
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In abstract algebra, a branch of mathematics, an affine monoid is a commutative monoid that is finitely generated, and is isomorphic to a submonoid of a free abelian group ℤd, d ≥ 0. Affine monoids are closely connected to convex polyhedra, and their associated algebras are of much use in the algebraic study of these geometric objects.
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In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein rings. Acceptable rings were introduced by . All finite-dimensional Gorenstein rings are acceptable, as are all finitely generated algebras over acceptable rings and all localizations of acceptable rings.
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In field theory, the primitive element theorem or Artin's theorem on primitive elements is a result characterizing the finite degree field extensions that possess a primitive element, or simple extensions. It says in that a finite extension is simple if and only if there are only finitely many intermediate fields. In particular, finite separable extensions are simple , more general.
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Balasubramanian and Shorey proved in 1980 that there are only finitely many possible solutions (x,y,m,n) to the equations with prime divisors of x and y lying in a given finite set and that they may be effectively computed. showed that, for each fixed x and y, this equation has at most one solution.
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Now standard proofs of the fact that the set of geodesic words in a word-hyperbolic group is a regular language also use finiteness of the number of cone types. Cannon's work also introduced an important notion of almost convexity for Cayley graphs of finitely generated groups,James W. Cannon. Almost convex groups. Geometriae Dedicata, vol.
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Manfred S. Frings (The Hague: Martinus Nijhoff, 1974), pp 1-42. See also Max Scheler, Man’s Place in Nature. Also see “Frings”, Chapter 2 “On the Bio-Psychic World.” For Scheler, Spirit is manifest infinitely as the Divine Essence, or God (depending upon your religious orientation); and finitely as persons and collectively as the nation and church.
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Positive measures are closed under conical combination but not general linear combination, while signed measures are the linear closure of positive measures. Another generalization is the finitely additive measure, also known as a content. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first.
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This group is called the group of fractional ideals of R. The principal fractional ideals form a subgroup. A (nonzero) fractional ideal is invertible if, and only if, it is projective as an R-module. Every finitely generated R-submodule of K is a fractional ideal and if R is noetherian these are all the fractional ideals of R.
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The special case when the Jordan curve is a circle or line is called a Fuchsian group, named after Lazarus Fuchs by Henri Poincaré. Finitely generated quasi-Fuchsian groups are conjugate to Fuchsian groups under quasi-conformal transformations. The space of quasi-Fuchsian groups of the first kind is described by the simultaneous uniformization theorem of Bers.
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An important but difficult question is on the finiteness of the integral closure of a finitely generated algebra. There are several known results. The integral closure of a Dedekind domain in a finite extension of the field of fractions is a Dedekind domain; in particular, a noetherian ring. This is a consequence of the Krull–Akizuki theorem.
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If a set is totally ordered, then the following are equivalent to each other: # The set is well ordered. That is, every nonempty subset has a least element. # Transfinite induction works for the entire ordered set. # Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assuming the axiom of dependent choice).
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Specifically, applying the structure theorem for finitely generated modules over a principal ideal domain to this algebra yields as corollaries the various canonical forms of matrices, such as Jordan canonical form. In some approaches to noncommutative geometry, the free noncommutative algebra (polynomials in non- commuting variables) plays a similar role, but the analysis is much more difficult.
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Every finite topological space is compact since any open cover must already be finite. Indeed, compact spaces are often thought of as a generalization of finite spaces since they share many of the same properties. Every finite topological space is also second-countable (there are only finitely many open sets) and separable (since the space itself is countable).
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Nagata arrived at the conjecture via work on the 14th problem of Hilbert, which asks whether the invariant ring of a linear group action on the polynomial ring over some field is finitely generated. Nagata published the conjecture in a 1959 paper in the American Journal of Mathematics, in which he presented a counterexample to Hilbert's 14th problem.
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Every (real or complex) vector space admits a norm: If is a Hamel basis for a vector space then the real-valued map that sends (where all but finitely many of the scalars are 0) to is a norm on . There are also a large number of norms that exhibit additional properties that make them useful for specific problems.
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In a cellular automaton, a grid of cells, each having one of the finitely many states, evolves according to a simple set of rules. These rules guide the "interactions" of each cell with its neighbors. Although the rules are only defined locally, they have been shown capable of producing globally interesting behavior, for example in Conway's Game of Life.
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In algebra, the Artin–Tate lemma, named after Emil Artin and John Tate, states: :Let A be a commutative Noetherian ring and B \sub C commutative algebras over A. If C is of finite type over A and if C is finite over B, then B is of finite type over A. (Here, "of finite type" means "finitely generated algebra" and "finite" means "finitely generated module".) The lemma was introduced by E. Artin and J. Tate in 1951E Artin, J.T Tate, "A note on finite ring extensions," J. Math. Soc Japan, Volume 3, 1951, pp. 74–77 to give a proof of Hilbert's Nullstellensatz. The lemma is similar to the Eakin–Nagata theorem, which says: if C is finite over B and C is a Noetherian ring, then B is a Noetherian ring.
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In abstract algebra, the Eakin–Nagata theorem states: given commutative rings A \subset B such that B is finitely generated as a module over A, if B is a Noetherian ring, then A is a Noetherian ring. (Note the converse is also true and is easier.) The theorem is similar to the Artin–Tate lemma, which says that the same statement holds with "Noetherian" replaced by "finitely generated algebra" (assuming the base ring is a Noetherian ring). The theorem was first proved in Paul M. Eakin's thesis and later independently by . The theorem can also be deduced from the characterization of a Noetherian ring in terms of injective modules, as done for example by David Eisenbud in ; this approach is useful for a generalization to non-commutative rings.
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Hilbert's Nullstellensatz of algebraic geometry is a special case of the statement that the polynomial ring in finitely many variables over a field is a Hilbert ring. A general form of the Nullstellensatz states that if R is a Jacobson ring, then so is any finitely generated R-algebra S. Moreover, the pullback of any maximal ideal J of S is a maximal ideal I of R, and S/J is a finite extension of the field R/I. In particular a morphism of finite type of Jacobson rings induces a morphism of the maximal spectrums of the rings. This explains why for algebraic varieties over fields it is often sufficient to work with the maximal ideals rather than with all prime ideals, as was done before the introduction of schemes.
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It is clear that Σ is a well-defined function: for every n, there are at most finitely many n-state Turing machines as above, up to isomorphism, hence at most finitely many possible running times. This infinite sequence Σ is the busy beaver function, and any n-state 2-symbol Turing machine M for which σ(M) = Σ(n) (i.e., which attains the maximum score) is called a busy beaver. Note that for each n, there exist at least four n-state busy beavers (because, given any n-state busy beaver, another is obtained by merely changing the shift direction in a halting transition, another by shifting all direction changes to their opposite, and the final by shifting the halt direction of the all-swapped busy beaver).
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There is another covariant functor from the category of abelian groups to the category of torsion-free groups that sends every group to its quotient by its torsion subgroup, and sends every homomorphism to the obvious induced homomorphism (which is easily seen to be well-defined). If A is finitely generated and abelian, then it can be written as the direct sum of its torsion subgroup T and a torsion-free subgroup (but this is not true for all infinitely generated abelian groups). In any decomposition of A as a direct sum of a torsion subgroup S and a torsion-free subgroup, S must equal T (but the torsion-free subgroup is not uniquely determined). This is a key step in the classification of finitely generated abelian groups.
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In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven by for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by . The Lasker–Noether theorem is an extension of the fundamental theorem of arithmetic, and more generally the fundamental theorem of finitely generated abelian groups to all Noetherian rings. The Lasker–Noether theorem plays an important role in algebraic geometry, by asserting that every algebraic set may be uniquely decomposed into a finite union of irreducible components.
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In infinite graph theory, an end is defined slightly differently, as an equivalence class of semi-infinite paths in the graph, or as a haven, a function mapping finite sets of vertices to connected components of their complements. However, for locally finite graphs (graphs in which each vertex has finite degree), the ends defined in this way correspond one-for-one with the ends of topological spaces defined from the graph . The ends of a finitely generated group are defined to be the ends of the corresponding Cayley graph; this definition is insensitive to the choice of generating set. Every finitely-generated infinite group has either 1, 2, or infinitely many ends, and Stallings theorem about ends of groups provides a decomposition for groups with more than one end.
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In additive number theory, the Skolem–Mahler–Lech theorem states that if a sequence of numbers is generated by a linear recurrence relation, then with finitely many exceptions the positions at which the sequence is zero form a regularly repeating pattern. More precisely, this set of positions can be decomposed into the union of a finite set and finitely many full arithmetic progressions. Here, an infinite arithmetic progression is full if there exist integers a and b such that the progression consists of all positive integers equal to b modulo a. This result is named after Thoralf Skolem (who proved the theorem for sequences of rational numbers), Kurt Mahler (who proved it for sequences of algebraic numbers), and Christer Lech (who proved it for sequences whose elements belong to any field of characteristic 0).
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The conjecture of Thompson that F is not amenable was further popularized by R. Geoghegan --- see also the Cannon- Floyd-Parry article cited in the references below. Its current status is open: E. Shavgulidze published a paper in 2009 in which he claimed to prove that F is amenable, but an error was found, as is explained in the MR review. It is known that F is not elementary amenable, see Theorem 4.10 in Cannon-Floyd- Parry. If F is not amenable, then it would be another counterexample to the long-standing but recently disproved von Neumann conjecture for finitely- presented groups, which suggested that a finitely-presented group is amenable if and only if it does not contain a copy of the free group of rank 2.
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In mathematics, a semi-local ring is a ring for which R/J(R) is a semisimple ring, where J(R) is the Jacobson radical of R. The above definition is satisfied if R has a finite number of maximal right ideals (and finite number of maximal left ideals). When R is a commutative ring, the converse implication is also true, and so the definition of semi-local for commutative rings is often taken to be "having finitely many maximal ideals". Some literature refers to a commutative semi-local ring in general as a quasi-semi- local ring, using semi-local ring to refer to a Noetherian ring with finitely many maximal ideals. A semi-local ring is thus more general than a local ring, which has only one maximal (right/left/two-sided) ideal.
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Gromov proved that a finitely presented group is word-hyperbolic if and only if it satisfies a linear isoperimetric inequality, that is, if and only if the Dehn function of this group is equivalent to the function f(n) = n. Gromov's proof was in large part informed by analogy with filling area functions for compact Riemannian manifolds where the area of a minimal surface bounding a null-homotopic closed curve is bounded in terms of the length of that curve. The study of isoperimetric and Dehn functions quickly developed into a separate major theme in geometric group theory, especially since the growth types of these functions are natural quasi-isometry invariants of finitely presented groups. One of the major results in the subject was obtained by Sapir, Birget and Rips who showedM.
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By work of , a finitely generated Kleinian group is Schottky if and only if it is finitely generated, free, has nonempty domain of discontinuity, and all non-trivial elements are loxodromic. A fundamental domain for the action of a Schottky group G on its regular points Ω(G) in the Riemann sphere is given by the exterior of the Jordan curves defining it. The corresponding quotient space Ω(G)/G is given by joining up the Jordan curves in pairs, so is a compact Riemann surface of genus g. This is the boundary of the 3-manifold given by taking the quotient (H∪Ω(G))/G of 3-dimensional hyperbolic H space plus the regular set Ω(G) by the Schottky group G, which is a handlebody of genus g.
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The history and credit for the fundamental theorem is complicated by the fact that it was proven when group theory was not well- established, and thus early forms, while essentially the modern result and proof, are often stated for a specific case. Briefly, an early form of the finite case was proven in , the finite case was proven in , and stated in group-theoretic terms in . The finitely presented case is solved by Smith normal form, and hence frequently credited to , though the finitely generated case is sometimes instead credited to ; details follow. Group theorist László Fuchs states: The fundamental theorem for finite abelian groups was proven by Leopold Kronecker in , using a group-theoretic proof, though without stating it in group-theoretic terms; a modern presentation of Kronecker's proof is given in , 5.2.
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For example, if X is an affine variety, then one can try to construct X/G as Spec of the ring of invariants O(X)G. However, Masayoshi Nagata showed that the ring of invariants need not be finitely generated as a k-algebra (and so Spec of the ring is a scheme but not a variety), a negative answer to Hilbert's 14th problem. In the positive direction, the ring of invariants is finitely generated if G is reductive, by Haboush's theorem, proved in characteristic zero by Hilbert and Nagata. Geometric invariant theory involves further subtleties when a reductive group G acts on a projective variety X. In particular, the theory defines open subsets of "stable" and "semistable" points in X, with the quotient morphism only defined on the set of semistable points.
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The rule is given the name "Day & Night" because its on and off states are symmetric: if all the cells in the Universe are inverted, the future states are the inversions of the future states of the original pattern. A pattern in which the entire universe consists of off cells except for finitely many on cells can equivalently be represented by a pattern in which the whole universe is covered in on cells except for finitely many off cells in congruent locations. Although the detailed evolution of this cellular automaton is very different from Conway's Game of Life, it exhibits complex behavior similar to that rule: there are many known small oscillators and spaceships, and guns formed by combining oscillators in such a way that they periodically emit spaceships of various types.
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Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension.
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The category of finitely-generated projective modules over the integers has split idempotents, and every module is isomorphic to a finite direct sum of copies of the regular module, the number being given by the rank. Thus the category has unique decomposition into indecomposables, but is not Krull- Schmidt since the regular module does not have a local endomorphism ring.
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This implies that whenever x < y we can get from x to y by repeatedly choosing a cover, finitely many times. It also means that (for positive integer rank functions) compatibility of ρ with the ordering follows from the requirement about covers. As a variant of the definition of a graded poset, Birkhoff'Lattice Theory', Am. Math. Soc., Colloquium Publications, Vol.
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The set of finite and cofinite sets of integers, where a cofinite set is one omitting only finitely many integers. This is clearly closed under complement, and is closed under union because the union of a cofinite set with any set is cofinite, while the union of two finite sets is finite. Intersection behaves like union with "finite" and "cofinite" interchanged. Example 4.
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In the AGM framework, a belief set is represented by a deductively closed set of propositional formulae. While such sets are infinite, they can always be finitely representable. However, working with deductively closed sets of formulae leads to the implicit assumption that equivalent belief sets should be considered equal when revising. This is called the principle of irrelevance of syntax.
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A subdivision rule takes a tiling of the plane by polygons and turns it into a new tiling by subdividing each polygon into smaller polygons. It is finite if there are only finitely many ways that every polygon can subdivide. Each way of subdividing a tile is called a tile type. Each tile type is represented by a label (usually a letter).
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In mathematics, the Muller–Schupp theorem states that a finitely generated group G has context-free word problem if and only if G is virtually free. The theorem was proved by David Muller and Paul Schupp in 1983.David E. Muller, and Paul E. Schupp, Groups, the theory of ends, and context-free languages. Journal of Computer and System Sciences 26 (1983), no.
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The Dieudonné–Manin classification theorem was proved by and . It describes the structure of Dieudonné modules over an algebraically closed field k up to "isogeny". More precisely, it classifies the finitely generated modules over D_k[1/p], where D_k is the Dieudonné ring. The category of such modules is semisimple, so every module is a direct sum of simple modules.
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Well-quasi-ordering implies that any property of graphs that is monotonic with respect to induced subgraphs has finitely many forbidden induced subgraphs, and therefore may be tested in polynomial time on graphs of bounded tree-depth. The graphs with tree-depth at most d themselves also have a finite set of forbidden induced subgraphs., p. 138. Figure 6.6 on p.
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The multiplicative structure of an H-space adds structure to its homology and cohomology groups. For example, the cohomology ring of a path-connected H-space with finitely generated and free cohomology groups is a Hopf algebra. Also, one can define the Pontryagin product on the homology groups of an H-space. The fundamental group of an H-space is abelian.
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The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra concerning the relationship between free modules and projective modules over polynomial rings. In the geometric setting it is a statement about the triviality of vector bundles on affine space. The theorem states that every finitely generated projective module over a polynomial ring is free.
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J. 10 (1943), 761–785. See Marcel Erné, Closure, in Frédéric Mynard, Elliott Pearl (Editors), Beyond Topology, Contemporary mathematics vol. 486, American Mathematical Society, 2009, p.170ff With the advancement of categorical topology in the 1980s, Alexandrov spaces were rediscovered when the concept of finite generation was applied to general topology and the name finitely generated spaces was adopted for them.
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A set is totally bounded if, given any positive radius, it is covered by finitely many balls of that radius. The open balls of a metric space can serve as a base, giving this space a topology, the open sets of which are all possible unions of open balls. This topology on a metric space is called the topology induced by the metric .
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An R-algebra A is finite if it is finitely generated as an R-module. An R-algebra can be thought as a homomorphism of rings f\colon R\to A, in this case f is called a finite morphism if A is a finite R-algebra. The definition of finite algebra is related to that of algebras of finite type.
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Bredon (1997), Theorem II.17.4; Borel (1984), V.3.17. For example, for a compact Hausdorff space X that is locally contractible (in the weak sense discussed above), the sheaf cohomology group Hj(X,Z) is finitely generated for every integer j. One case where the finiteness result applies is that of a constructible sheaf. Let X be a topologically stratified space.
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"Diophantine geometry over groups. V2. Quantifier elimination. II." Geometric and Functional Analysis, vol. 16 (2006), no. 3, pp. 537–706Z. Sela. "Diophantine geometry over groups. VI. The elementary theory of a free group." Geometric and Functional Analysis, vol. 16 (2006), no. 3, pp. 707–730 he proved that any two non-abelian finitely generated free groups have the same first-order theory.
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See in particular Richter's review in the MR link. In 2011, Kühn and her co-authors published a proof of Sumner's conjecture, that every n-vertex polytree forms a subgraph of every (2n − 2)-vertex tournament, for all but finitely many values of n. MathSciNet reviewer K. B. Reid wrote that their proof "is an important and welcome development in tournament theory"..
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We can define a finitely many-valued logic as being Ln ({1, 2, ..., n} ƒ1, ..., ƒm) where n ≥ 2 is a given natural number. Post (1921) proves that assuming a logic is able to produce a function of any mth order model, there is some corresponding combination of connectives in an adequate logic Ln which can produce a model of order m+1 .
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Suppose that is a finitely presented morphism of affine schemes, s is a point of S, and M is a finite type OX-module. If n is a natural number, then Gruson and Raynaud define an S-dévissage in dimension n to consist of: # A closed finitely presented subscheme X′ of X containing the closed subscheme defined by the annihilator of M and such that the dimension of is less than or equal to n. # A scheme T and a factorization of the restriction of f to X′ such that is a finite morphism and is a smooth affine morphism with geometrically integral fibers of dimension n. Denote the generic point of by τ and the pushforward of M to T by N. # A free finite type OT-module L and a homomorphism such that is bijective.
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The fiber of the corresponding vector bundle over x is then the range of f(x). If M is not connected, the converse does not hold unless one allows for vector bundles of non-constant rank (which means admitting manifolds of non-constant dimension). For example, if M is a zero-dimensional 2-point manifold, the module \R\oplus 0 is finitely-generated and projective over C^\infty(M)\cong\R\times\R but is not free, and so cannot correspond to the sections of any (constant-rank) vector bundle over M (all of which are trivial). Another way of stating the above is that for any connected smooth manifold M, the section functor Γ from the category of smooth vector bundles over M to the category of finitely generated, projective C∞(M)-modules is full, faithful, and essentially surjective.
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In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma -- also known as the Krull–Azumaya theorem -- governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and its finitely generated modules. Informally, the lemma immediately gives a precise sense in which finitely generated modules over a commutative ring behave like vector spaces over a field. It is an important tool in algebraic geometry, because it allows local data on algebraic varieties, in the form of modules over local rings, to be studied pointwise as vector spaces over the residue field of the ring. The lemma is named after the Japanese mathematician Tadashi Nakayama and introduced in its present form in , although it was first discovered in the special case of ideals in a commutative ring by Wolfgang Krull and then in general by Goro Azumaya (1951).
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The Svarc--Milnor lemma states that if a group G acting properly discontinuously and with compact quotient (such an action is often called geometric) on a proper length space Y, then it is finitely generated, and any Cayley graph for G is quasi-isometric to Y. Thus a group is (finitely generated and) hyperbolic if and only if it has a geometric action on a proper hyperbolic space. If G' \subset G is a subgroup with finite index (i.e., the set G/G' is finite), then the inclusion induces a quasi-isometry on the vertices of any (locally finite) Cayley graph of G' into any (ditto) Cayley graph of G. Thus G' is hyperbolic if and only if G itself is. More generally if two groups are commensurable, then one is hyperbolic if and only if the other is.
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The compact objects in the category of sets are precisely the finite sets. For a ring R, the compact objects in the category of R-modules are precisely the finitely presented R-modules. In particular, if R is a field, then compact objects are finite-dimensional vector spaces. Similar results hold for any category of algebraic structures given by operations on a set obeying equational laws.
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For this module, the th syzygy module is free, but not the th one (for a proof, see , below). The theorem is also true for modules that are not finitely generated. As the global dimension of a ring is the supremum of the projective dimensions of all modules, Hilbert's syzygy theorem may be restated as: the global dimension of k[x_1,\ldots,x_n] is .
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The class of locally finite groups is closed under subgroups, quotients, and extensions . Locally finite groups satisfy a weaker form of Sylow's theorems. If a locally finite group has a finite p-subgroup contained in no other p-subgroups, then all maximal p-subgroups are finite and conjugate. If there are finitely many conjugates, then the number of conjugates is congruent to 1 modulo p.
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Here the factor ε, corresponding to the prime 2, is 1 if a + b is odd and 2 if a + b is even. The first product index p runs over the finitely-many odd primes dividing both a and b. For these primes \omega (p)=0 since p then cannot divide c. The second product index \varpi runs over the infinitely-many odd primes not dividing a.
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More generally, co-dimension two knots in spheres are known to have Wirtinger presentations. Michel Kervaire proved that an abstract group is the fundamental group of a knot exterior (in a perhaps high-dimensional sphere) if and only if all the following conditions are satisfied: # The abelianization of the group is the integers. # The 2nd homology of the group is trivial. # The group is finitely presented.
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In terms of localization of modules, one can define quasi-coherent sheaves and coherent sheaves on locally ringed spaces. In algebraic geometry, the quasi-coherent OX-modules for schemes X are those that are locally modelled on sheaves on Spec(R) of localizations of any R-module M. A coherent OX-module is such a sheaf, locally modelled on a finitely-presented module over R.
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The tensor product is another non-exact functor relevant in the context of commutative rings: for a general R-module M, the functor :M ⊗R − is only right exact. If it is exact, M is called flat. If R is local, any finitely presented flat module is free of finite rank, thus projective. Despite being defined in terms of homological algebra, flatness has profound geometric implications.
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One direction of the theorem can be proven by showing that any given Muller automaton recognizes an ω-regular language. Suppose A = (Q,Σ,δ,q0,F) is a deterministic Muller automaton. The union of finitely many ω-regular languages produces an ω-regular language, therefore it can be assumed w.l.o.g. that the Muller acceptance condition F contains exactly one set of states {q1, ... ,qn}.
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One of the classical results in algebraic number theory is that the ideal class group of an algebraic number field K is finite. This is a consequence of Minkowski's theorem since there are only finitely many Integral ideals with norm less than a fixed positive integer page 78. The order of the class group is called the class number, and is often denoted by the letter h.
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The Auslander–Buchsbaum formula implies that a Noetherian local ring is regular if, and only if, it has finite global dimension. In turn this implies that the localization of a regular local ring is regular. If A is a local finitely generated R-algebra (over a regular local ring R), then the Auslander–Buchsbaum formula implies that A is Cohen–Macaulay if, and only if, pdRA = codimRA.
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For a given term, let m and n denote the total number of g and of g applied to identical arguments, respectively. Application of any rule properly decreases the value of m+n, which is possible only finitely many times. The term g(4,4) has two normal forms in this system, viz. g(4,4) → 4 and g(4,4) → g(3,4) → 3, hence the system is not confluent.
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One can also speak of "almost all" integers having a property to mean "all except finitely many", despite the integers not admitting a measure for which this agrees with the previous usage. For example, "almost all prime numbers are odd". There is a more complicated meaning for integers as well, discussed in the main article. Finally, this term is sometimes used synonymously with generic, below.
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Let F∞ denote the space of infinite sequences of elements from F such that only finitely many elements are nonzero. That is, if we write an element of F∞ as :x = (x_1, x_2, x_3, \ldots) then only a finite number of the xi are nonzero (i.e., the coordinates become all zero after a certain point). Addition and scalar multiplication are given as in finite coordinate space.
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Sun's solution is based on a recurrence similar to that for Sylvester's sequence, but with a different set of initial values. The Znám problem is closely related to Egyptian fractions. It is known that there are only finitely many solutions for any fixed k. It is unknown whether there are any solutions to Znám's problem using only odd numbers, and there remain several other open questions.
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This group is torsion free (), unlike the Nottingham group. This group is a finitely generated pro-p-group and a hereditarily just infinite group (). Thus, it is another representative of the 4th class of hereditarily just infinite groups, together with the Nottingham group and the Grigorchuk group, according to the conjectural classification of his group by Charles Leedham-Green. The Fesenko group is of finite width ().
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Infinitely generated abelian groups have very complex structure and are far less well understood than finitely generated abelian groups. Even torsion-free abelian groups are vastly more varied in their characteristics than vector spaces. Torsion-free abelian groups of rank 1 are far more amenable than those of higher rank, and a satisfactory classification exists, even though there are an uncountable number of isomorphism classes.
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In mathematics, an algebraic function field (often abbreviated as function field) of n variables over the field k is a finitely generated field extension K/k which has transcendence degree n over k. Equivalently, an algebraic function field of n variables over k may be defined as a finite field extension of the field K = k(x1,...,xn) of rational functions in n variables over k.
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The set S is called the orbit of the cycle. Every permutation on finitely many elements can be decomposed into cycles on disjoint orbits. The cyclic parts of a permutation are cycles, thus the second example is composed of a 3-cycle and a 1-cycle (or fixed point) and the third is composed of two 2-cycles, and denoted (1, 3) (2, 4).
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In fact, the closed graph theorem, the open mapping theorem and the bounded inverse theorem are all equivalent. This equivalence also serves to demonstrate the importance of and being Banach; one can construct linear maps that have unbounded inverses in this setting, for example, by using either continuous functions with compact support or by using sequences with finitely many non-zero terms along with the supremum norm.
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In mathematics, Dickson's lemma states that every set of n-tuples of natural numbers has finitely many minimal elements. This simple fact from combinatorics has become attributed to the American algebraist L. E. Dickson, who used it in order to prove a result in number theory about perfect numbers. However, the lemma was certainly known earlier, for example to Paul Gordan in his research on invariant theory..
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Indeed, a domain that is the direct limit of subrings that are Prüfer domains is a Prüfer domain, . Many Prüfer domains are also Bézout domains, that is, not only are finitely generated ideals projective, they are even free (that is, principal). For instance the ring of analytic functions on any noncompact Riemann surface is a Bézout domain, , and the ring of algebraic integers is Bézout.
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1 The existence of a planar cover is a minor-closed graph property,, Proposition 1, p. 2 and so can be characterized by finitely many forbidden minors, but the exact set of forbidden minors is not known. For the same reason, there exists a polynomial time algorithm for testing whether a given graph has a planar cover, but an explicit description of this algorithm is not known.
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The map is a ring homomorphism from k[V′] to k[V]. Conversely, every ring homomorphism from k[V′] to k[V] defines a regular map from V to V′. This defines an equivalence of categories between the category of algebraic sets and the opposite category of the finitely generated reduced k-algebras. This equivalence is one of the starting points of scheme theory.
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The deficiency of a finite presentation is just and the deficiency of a finitely presented group G, denoted def(G), is the maximum of the deficiency over all presentations of G. The deficiency of a finite group is non-positive. The Schur multiplicator of a finite group G can be generated by −def(G) generators, and G is efficient if this number is required.
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The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second? Based on earlier writings by Gauss,Carl Friedrich Gauss: Werke, vol. 8, pp.
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In mathematics, a Goldman domain or G-domain is an integral domain A whose field of fractions is a finitely generated algebra over A.Goldman domains/ideals are called G-domains/ideals in (Kaplansky 1974). They are named after Oscar Goldman. An overring (i.e., an intermediate ring lying between the ring and its field of fractions) of a Goldman domain is again a Goldman domain.
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We start with a basic game, also known as the stage game, which is a n-player game. In this game, each player has finitely many actions to choose from, and they make their choices simultaneously and without knowledge of the other player's choices. The collective choices of the players leads to a payoff profile, i.e. to a payoff for each of the players.
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The paradox was published in Mathematische Annalen in 1914 and also in Hausdorff's book, Grundzüge der Mengenlehre, the same year. The proof of the much more famous Banach–Tarski paradox uses Hausdorff's ideas. The proof of this paradox relies on the Axiom of Choice. This paradox shows that there is no finitely additive measure on a sphere defined on all subsets which is equal on congruent pieces.
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The subspace of null sequences c0 consists of all sequences whose limit is zero. This is a closed subspace of c, and so again a Banach space. The subspace of eventually zero sequences c00 consists of all sequences which have only finitely many nonzero elements. This is not a closed subspace and therefore is not a Banach space with respect to the infinity norm.
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The Novikov conjecture concerns the homotopy invariance of certain polynomials in the Pontryagin classes of a manifold, arising from the fundamental group. According to the Novikov conjecture, the higher signatures, which are certain numerical invariants of smooth manifolds, are homotopy invariants. The conjecture has been proved for finitely generated abelian groups. It is not yet known whether the Novikov conjecture holds true for all groups.
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An alternative way of stating the theorem is that a non- collinear set of points in the plane with integer distances can only be extended by adding finitely many additional points, before no more points can be added. A set of points with both integer coordinates and integer distances, to which no more can be added while preserving both properties, forms an Erdős–Diophantine graph.
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Using the discretization operator, call it Q here, and the transfer matrix of h, named T_h, this can be written concisely as :Q\varphi = T_h \cdot Q\varphi. \, This is again a fixed-point equation. But this one can now be considered as an eigenvector-eigenvalue problem. That is, a finitely supported refinable function exists only (but not necessarily), if T_h has the eigenvalue 1\.
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Zlil Sela Zlil Sela is an Israeli mathematician working in the area of geometric group theory. He is a Professor of Mathematics at the Hebrew University of Jerusalem. Sela is known for the solution of the isomorphism problem for torsion-free word-hyperbolic groups and for the solution of the Tarski conjecture about equivalence of first-order theories of finitely generated non-abelian free groups.
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Four points in any dimension have only finitely many central configurations. The number of configurations in this case is at least 32 and at most 8472, depending on the masses of the points. The only convex central configuration of four equal masses is a square. The only central configuration of four masses that spans three dimensions is the configuration formed by the vertices of a regular tetrahedron.
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In arithmetic geometry, Faltings' product theorem gives sufficient conditions for a subvariety of a product of projective spaces to be a product of varieties in the projective spaces. It was introduced by in his proof of Lang's conjecture that subvarieties of an abelian variety containing no translates of non-trivial abelian subvarieties have only finitely many rational points. and gave explicit versions of Faltings' product theorem.
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Generalizations to the most common situations can be found in . Exterior algebras of vector bundles are frequently considered in geometry and topology. There are no essential differences between the algebraic properties of the exterior algebra of finite-dimensional vector bundles and those of the exterior algebra of finitely generated projective modules, by the Serre–Swan theorem. More general exterior algebras can be defined for sheaves of modules.
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Subsequent authors have greatly extended Dehn's algorithm and applied it to a wide range of group theoretic decision problems. It was shown by Pyotr Novikov in 1955 that there exists a finitely presented group G such that the word problem for G is undecidable. It follows immediately that the uniform word problem is also undecidable. A different proof was obtained by William Boone in 1958.
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If G is finitely generated, the commutator subgroup G′ of G has finite index in G and H=G′, then the corresponding transfer map is trivial. In other words, the map sends G to 0 in the abelianization of G′. This is important in proving the principal ideal theorem in class field theory.Serre (1979) p.122 See the Emil Artin-John Tate Class Field Theory notes.
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There are some variations of the definition of a Kleinian group: sometimes Kleinian groups are allowed to be subgroups of PSL(2, C).2 (PSL(2, C) extended by complex conjugations), in other words to have orientation reversing elements, and sometimes they are assumed to be finitely generated, and sometimes they are required to act properly discontinuously on a non-empty open subset of the Riemann sphere.
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Nonetheless, axiom A diffeomorphisms are sometimes called hyperbolic diffeomorphisms, because the portion of M where the interesting dynamics occurs, namely, Ω(f), exhibits hyperbolic behavior. Axiom A diffeomorphisms generalize Morse–Smale systems, which satisfy further restrictions (finitely many periodic points and transversality of stable and unstable submanifolds). Smale horseshoe map is an axiom A diffeomorphism with infinitely many periodic points and positive topological entropy.
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Later, Jan Łukasiewicz and Alfred Tarski together formulated a logic on n truth values where n ≥ 2. In 1932, Hans Reichenbach formulated a logic of many truth values where n→∞. Kurt Gödel in 1932 showed that intuitionistic logic is not a finitely-many valued logic, and defined a system of Gödel logics intermediate between classical and intuitionistic logic; such logics are known as intermediate logics.
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Those profinite groups with finite Prüfer rank are more amenable to analysis. Specifically in the case of finitely generated pro-p groups, having finite Prüfer rank is equivalent to having an open normal subgroup that is powerful. In turn these are precisely the class of pro-p groups that are p-adic analytic - that is groups that can be imbued with a p-adic manifold structure.
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Let R be a Noetherian ring, I an ideal in R, and M a finitely generated R-module. The depth of I on M, written depthR(I, M) or just depth(I, M), is the supremum of the lengths of all M-regular sequences of elements of I. When R is a Noetherian local ring and M is a finitely generated R-module, the depth of M, written depthR(M) or just depth(M), means depthR(m, M); that is, it is the supremum of the lengths of all M-regular sequences in the maximal ideal m of R. In particular, the depth of a Noetherian local ring R means the depth of R as a R-module. That is, the depth of R is the maximum length of a regular sequence in the maximal ideal. For a Noetherian local ring R, the depth of the zero module is ∞,A. Grothendieck.
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Stochastic games were introduced by Lloyd Shapley in 1953. The first paper studied the discounted two-person zero-sum stochastic game with finitely many states and actions and demonstrates the existence of a value and stationary optimal strategies. The study of the undiscounted case evolved in the following three decades, with solutions of special cases by Blackwell and Ferguson in 1968Blackwell and Ferguson,1968. "The Big Match", Ann. Math. Statist.
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Finite fields (also called Galois fields) are fields with finitely many elements, whose number is also referred to as the order of the field. The above introductory example is a field with four elements. Its subfield is the smallest field, because by definition a field has at least two distinct elements . In modular arithmetic modulo 12, 9 + 4 = 1 since 9 + 4 = 13 in , which divided by 12 leaves remainder 1\.
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The formal definition does not use isometries, but almost isometries. A Banach space Y is finitely representableJames, Robert C. (1972), "Super-reflexive Banach spaces", Can. J. Math. 24:896-904. in a Banach space X if for every finite-dimensional subspace Y0 of Y and every , there is a subspace X0 of X such that the multiplicative Banach-Mazur distance between X0 and Y0 satisfies :d(X_0, Y_0) < 1 + \varepsilon.
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Another simple example is the finitely repeated prisoner's dilemma for T periods, where the payoff is averaged over the T periods. The only Nash equilibrium of this game is to choose Defect in each period. Now consider the two strategies tit-for-tat and grim trigger. Although neither tit-for-tat nor grim trigger are Nash equilibria for the game, both of them are \epsilon-equilibria for some positive \epsilon.
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Rostislav Ivanovich Grigorchuk (; b. February 23, 1953) is a Soviet and Russian mathematician working in the area of group theory. He holds the rank of Distinguished Professor in the Mathematics Department of Texas A&M; University. Grigorchuk is particularly well known for having constructed, in a 1984 paper, the first example of a finitely generated group of intermediate growth, thus answering an important problem posed by John Milnor in 1968.
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In logic, a finite-valued logic (also finitely many-valued logic) is a propositional calculus in which truth values are discrete. Traditionally, in Aristotle's logic, the bivalent logic, also known as binary logic was the norm, as the law of the excluded middle precluded more than two possible values (i.e., "true" and "false") for any proposition. Modern three-valued logic (ternary logic) allows for an additional possible truth value (i.e. "undecided").
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The Pontryagin classes of a smooth manifold are defined to be the Pontryagin classes of its tangent bundle. Novikov proved in 1966 that if two compact, oriented, smooth manifolds are homeomorphic then their rational Pontryagin classes pk(M, Q) in H4k(M, Q) are the same. If the dimension is at least five, there are at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.
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William Werner Boone (16 January 1920 in Cincinnati – 14 September 1983 in Urbana, Illinois) was an American mathematician. Alonzo Church was his Ph.D. advisor at Princeton, and Kurt Gödel was his friend at the Institute for Advanced Study. Pyotr Novikov showed in 1955 that there exists a finitely presented group G such that the word problem for G is undecidable. A different proof was obtained by Boone in 1958.
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Let ZN denote the Baer–Specker group, that is, the group of all integer sequences, with termwise addition. For each n in N, let en be the sequence with n-th term equal to 1 and all other terms 0. A torsion-free abelian group G is said to be slender if every homomorphism from ZN into G maps all but finitely many of the en to the identity element.
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Every (bounded) convex polytope is the image of a simplex, as every point is a convex combination of the (finitely many) vertices. However, polytopes are not in general isomorphic to simplices. This is in contrast to the case of vector spaces and linear combinations, every finite-dimensional vector space being not only an image of, but in fact isomorphic to, Euclidean space of some dimension (or analog over other fields).
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Several theorists (e.g., Kirman and Sondermann) point out that when one drops the assumption that there are only finitely many individuals, one can find aggregation rules that satisfy all of Arrow's other conditions. However, such aggregation rules are practically of limited interest, since they are based on ultrafilters, highly non-constructive mathematical objects. In particular, Kirman and Sondermann argue that there is an "invisible dictator" behind such a rule.
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In 1915 he was sent to Constantinople as a military adviser to the Turkish Government. After the war, in the spring of 1919, Nielsen married Carola von Pieverling, a German medical doctor. In 1920 Nielsen took a position at the Technical University of Breslau. The next year he published a paper in Mathematisk Tidsskrift in which he proved that any subgroup of a finitely generated free group is free.
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The first indication that there might be a problem in defining length for an arbitrary set came from Vitali's theorem.Moore, Gregory H., Zermelo's Axiom of Choice, Springer-Verlag, 1982, pp. 100-101 When you form the union of two disjoint sets, one would expect the measure of the result to be the sum of the measure of the two sets. A measure with this natural property is called finitely additive.
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Maharam pioneered the research of finitely additive measures on integers. Maharam's theorem about the decomposability of complete measure spaces plays an important role in the theory of Banach spaces. Maharam published it in the Proceedings of the National Academy of Sciences of the United States of America in 1942. Another paper of Maharam, in 1947 in the Annals of Mathematics, introduced Maharam algebras, which are complete Boolean algebras with continuous submeasures.
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There is a notion of ind- finite group, which is the conceptual dual to profinite groups; i.e. a group G is ind-finite if it is the direct limit of an inductive system of finite groups. (In particular, it is an ind-group.) The usual terminology is different: a group G is called locally finite if every finitely-generated subgroup is finite. This is equivalent, in fact, to being 'ind-finite'.
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For any finite extension of fields, the restriction of scalars takes quasiprojective varieties to quasiprojective varieties. The dimension of the resulting variety is multiplied by the degree of the extension. Under appropriate hypotheses (e.g., flat, proper, finitely presented), any morphism T \to S of algebraic spaces yields a restriction of scalars functor that takes algebraic stacks to algebraic stacks, preserving properties such as Artin, Deligne-Mumford, and representability.
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Suppose that is the structure morphism for an -scheme . The base scheme has a Frobenius morphism FS. Composing with FS results in an -scheme XF called the restriction of scalars by Frobenius. The restriction of scalars is actually a functor, because an -morphism induces an -morphism . For example, consider a ring A of characteristic and a finitely presented algebra over A: :R = A[X_1, \ldots, X_n] / (f_1, \ldots, f_m).
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In the case of Gromov–Witten invariants, for example, we consider only closed domains C of fixed genus g and we introduce n marked points (or punctures) on C. As soon as the punctured Euler characteristic 2 - 2 g - n is negative, there are only finitely many holomorphic reparametrizations of C that preserve the marked points. The domain curve C is an element of the Deligne–Mumford moduli space of curves.
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Intuitively, this is because in this case, proving \phi_M requires the arithmetic properties of only finitely many numbers. If the machine M does not halt in finite steps, then \phi_M is false in any finite model, since there's no finite computation record of M that ends with halting. Thus, if M halts, \phi_M is true in some finite models. If M does not halt, \phi_M is false in all finite models.
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More generally, for a nonsingular algebraic curve C defined over an algebraically closed field k of characteristic p \geq 0, the gap numbers for all but finitely many points is a fixed sequence \epsilon_1, ..., \epsilon_g. These points are called non- Weierstrass points. All points of C whose gap sequence is different are called Weierstrass points. If \epsilon_1, ..., \epsilon_g = 1, ..., g then the curve is called a classical curve.
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Gromov showed that if the scaling possibility is broken by only considering Riemannian manifolds of a fixed diameter, then a closed manifold admitting such a Riemannian metric with sectional curvatures sufficiently close to zero must be finitely covered by a nilmanifold. The proof works by replaying the proofs of the Bieberbach theorem and Margulis lemma. Gromov's proof was given a careful exposition by Peter Buser and Hermann Karcher.Hermann Karcher.
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In mathematics, the Chevalley–Iwahori–Nagata theorem states that if a linear algebraic group G is acting linearly on a finite-dimensional vector space V, then the map from V/G to the spectrum of the ring of invariant polynomials is an isomorphism if this ring is finitely generated and all orbits of G on V are closed . It is named after Claude Chevalley, Nagayoshi Iwahori, and Masayoshi Nagata.
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Given a commutative ring R and an R-module M, we can define the exterior algebra Λ(M) just as above, as a suitable quotient of the tensor algebra T(M). It will satisfy the analogous universal property. Many of the properties of Λ(M) also require that M be a projective module. Where finite dimensionality is used, the properties further require that M be finitely generated and projective.
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In convex geometry, Gordan's lemma states that the semigroup of integral points in the dual cone of a rational convex polyhedral cone is finitely generated. In algebraic geometry, the prime spectrum of the semigroup algebra of such a semigroup is, by definition, an affine toric variety; thus, the lemma says an affine toric variety is indeed an algebraic variety. The lemma is named after the German mathematician Paul Gordan (1837–1912).
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Stephen M. Gersten (born 2 December 1940) was an American mathematician, specializing in finitely presented groups and their geometric properties. Gersten graduated in 1961 with an A.B. from Princeton University and in 1965 with a Ph.D. from Trinity College, Cambridge. His doctoral thesis was Class Groups of Supplemented Algebras written under the supervision of John R. Stallings. In the late 1960s and early 1970s he taught at Rice University.
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An extension E/F is also sometimes said to be simply finite if it is a finite extension; this should not be confused with the fields themselves being finite fields (fields with finitely many elements). The degree should not be confused with the transcendence degree of a field; for example, the field Q(X) of rational functions has infinite degree over Q, but transcendence degree only equal to 1.
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The quiver itself can be considered a category, where the vertices are objects and paths are morphisms. Then a representation of Q is just a covariant functor from this category to the category of finite dimensional vector spaces. Morphisms of representations of Q are precisely natural transformations between the corresponding functors. For a finite quiver Γ (a quiver with finitely many vertices and edges), let KΓ be its path algebra.
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The Ramsey-Cass-Koopmans model does not have dynamic efficiency problems because agents discount the future at some rate β which is less than 1, and their savings rate is endogenous. The Diamond growth model is not necessarily dynamically efficient because of the overlapping generation setup. In a competitive equilibrium, the growth rate may exceed the interest rate, which entails dynamic inefficiency. This is because agents are finitely lived.
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If a stage-game in a finitely repeated game has multiple Nash equilibria, subgame perfect equilibria can be constructed to play non-stage-game Nash equilibrium actions, through a "carrot and stick" structure. One player can use the one stage-game Nash equilibrium to incentivize playing the non-Nash equilibrium action, while using a stage-game Nash equilibrium with lower payoff to the other player if they choose to defect.
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In geometric group theory, a geometry is any proper, geodesic metric space. An action of a finitely-generated group G on a geometry X is geometric if it satisfies the following conditions: # Each element of G acts as an isometry of X. # The action is cocompact, i.e. the quotient space X/G is a compact space. # The action is properly discontinuous, with each point having a finite stabilizer.
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Let the discrete valuation ring R be the ring of formal power series over K whose coefficients generate a finite extension of k. If y is any formal power series not in R then the ring R[y] is not an N−1 ring (its integral closure is not a finitely generated module) so R is not a Japanese ring. If R is the subring of the polynomial ring k[x1,x2,...] in infinitely many generators generated by the squares and cubes of all generators, and S is obtained from R by adjoining inverses to all elements not in any of the ideals generated by some xn, then S is a 1-dimensional Noetherian domain that is not an N−1 ring, in other words its integral closure in its quotient field is not a finitely generated S-module. Also S has a cusp singularity at every closed point, so the set of singular points is not closed.
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A central part of scheme theory is the notion of coherent sheaves, generalizing the notion of (algebraic) vector bundles. For a scheme X, one starts by considering the abelian category of OX-modules, which are sheaves of abelian groups on X that form a module over the sheaf of regular functions OX. In particular, a module M over a commutative ring R determines an associated OX-module on X = Spec(R). A quasi-coherent sheaf on a scheme X means an OX-module that is the sheaf associated to a module on each affine open subset of X. Finally, a coherent sheaf (on a Noetherian scheme X, say) is an OX-module that is the sheaf associated to a finitely generated module on each affine open subset of X. Coherent sheaves include the important class of vector bundles, which are the sheaves that locally come from finitely generated free modules. An example is the tangent bundle of a smooth variety over a field.
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A residuated Boolean algebra is an algebraic structure (L, ∧, ∨, ¬, 0, 1, •, I, \, /) such that : (i) (L, ∧, ∨, •, I, \, /) is a residuated lattice, and :(ii) (L, ∧, ∨, ¬, 0, 1) is a Boolean algebra. An equivalent signature better suited to the relation algebra application is (L, ∧, ∨, ¬, 0, 1, •, I, ▷, ◁) where the unary operations x\ and x▷ are intertranslatable in the manner of De Morgan's laws via :x\y = ¬(x▷¬y), x▷y = ¬(x\¬y), and dually /y and ◁y as : x/y = ¬(¬x◁y), x◁y = ¬(¬x/y), with the residuation axioms in the residuated lattice article reorganized accordingly (replacing z by ¬z) to read :(x▷z)∧y = 0 ⇔ (x•y)∧z = 0 ⇔ (z◁y)∧x = 0 This De Morgan dual reformulation is motivated and discussed in more detail in the section below on conjugacy. Since residuated lattices and Boolean algebras are each definable with finitely many equations, so are residuated Boolean algebras, whence they form a finitely axiomatizable variety.
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This property generalizes immediately to hyperdeterminants implying invariance when you add a multiple of one slice of a hypermatrix to another parallel slice. A hyperdeterminant is not the only polynomial algebraic invariant for the group acting on the hypermatrix. For example, other algebraic invariants can be formed by adding and multiplying hyperdeterminants. In general the invariants form a ring algebra and it follows from Hilbert's basis theorem that the ring is finitely generated.
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It is immediate that in a uniserial R-module M, all submodules except M and 0 are simultaneously essential and superfluous. If M has a maximal submodule, then M is a local module. M is also clearly a uniform module and thus is directly indecomposable. It is also easy to see that every finitely generated submodule of M can be generated by a single element, and so M is a Bézout module.
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26), a ring is right Hermite if any finitely generated stably free right module over the ring is free. This is equivalent to requiring that any row vector (b1,...,bn) of elements of the ring which generate it as a right module (i.e., b1R+...+bnR=R) can be completed to a (not necessarily square) invertible matrix by adding some number of rows. (The criterion of being left Hermite can be defined similarly.) (p.
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If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite. Any set which can be mapped onto an infinite set is infinite. The Cartesian product of an infinite set and a nonempty set is infinite. The Cartesian product of an infinite number of sets, each containing at least two elements, is either empty or infinite; if the axiom of choice holds, then it is infinite.
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The new combinatorial topology formally treated topological classes as abelian groups. Homology groups are finitely generated abelian groups, and homology classes are elements of these groups. The Betti numbers of the manifold are the rank of the free part of the homology group, and the non-orientable cycles are described by the torsion part. The subsequent spread of homology groups brought a change of terminology and viewpoint from "combinatorial topology" to "algebraic topology".
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Rota's excluded minors conjecture is one of a number of conjectures made by mathematician Gian-Carlo Rota. It is considered to be an important problem by some members of the structural combinatorics community. Rota conjectured in 1971 that, for every finite field, the family of matroids that can be represented over that field has only finitely many excluded minors.. A proof of the conjecture has been announced by Geelen, Gerards, and Whittle.
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Grigorchuk is most well known for having constructed the first example of a finitely generated group of intermediate growth which now bears his name and is called the Grigorchuk group (sometimes it is also called the first Grigorchuk group since Grigorchuk constructed several other groups that are also commonly studied). This group has growth that is faster than polynomial but slower than exponential. Grigorchuk constructed this group in a 1980 paperR. I. Grigorchuk.
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In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocountable. These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the product topology or direct sum.
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The fundamental group π1(X) is a birational invariant for smooth complex projective varieties. The "Weak factorization theorem", proved by Abramovich, Karu, Matsuki, and Włodarczyk (2002), says that any birational map between two smooth complex projective varieties can be decomposed into finitely many blow- ups or blow-downs of smooth subvarieties. This is important to know, but it can still be very hard to determine whether two smooth projective varieties are birational.
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In all dimensions, the fundamental group of a manifold is a very important invariant, and determines much of the structure; in dimensions 1, 2 and 3, the possible fundamental groups are restricted, while in dimension 4 and above every finitely presented group is the fundamental group of a manifold (note that it is sufficient to show this for 4- and 5-dimensional manifolds, and then to take products with spheres to get higher ones).
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In mathematics, a composition of an integer n is a way of writing n as the sum of a sequence of (strictly) positive integers. Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same partition of that number. Every integer has finitely many distinct compositions. Negative numbers do not have any compositions, but 0 has one composition, the empty sequence.
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So there is no utilitarian-optimal division. The problem with the above example is that the value measure of partner 2 is finitely-additive but not countably-additive. The compactness part of the DS theorem immediately implies that: :::If all value measures are countably- additive and nonatomic, :::then a utilitarian-optimal division exists. In this special case, non-atomicity is not required: if all value measures are countably-additive, then a utilitarian-optimal partition exists.
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The Axiom schema of Continuity plays a role similar to Hilbert's two axioms of Continuity. This schema is indispensable; Euclidean geometry in Tarski's (or equivalent) language cannot be finitely axiomatized as a first-order theory. Hilbert's axioms do not constitute a first-order theory because his continuity axioms require second-order logic. The first four groups of axioms of Hilbert's axioms for plane geometry are bi-interpretable with Tarski's axioms minus continuity.
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In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets is open; in Alexandrov topologies the finite restriction is dropped. A set together with an Alexandrov topology is known as an Alexandrov-discrete space or finitely generated space. Alexandrov topologies are uniquely determined by their specialization preorders.
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Submodules of projective modules need not be projective; a ring R for which every submodule of a projective left module is projective is called left hereditary. Quotients of projective modules also need not be projective, for example Z/n is a quotient of Z, but not torsion free, hence not flat, and therefore not projective. The category of finitely generated projective modules over a ring is an exact category. (See also algebraic K-theory).
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"Critical Path, page xxv. As well as contributing significantly to the development of tensegrity technology, Fuller invented the term "tensegrity", a portmanteau of "tensional integrity". "Tensegrity describes a structural-relationship principle in which structural shape is guaranteed by the finitely closed, comprehensively continuous, tensional behaviors of the system and not by the discontinuous and exclusively local compressional member behaviors. Tensegrity provides the ability to yield increasingly without ultimately breaking or coming asunder.
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Let be a random variable; we assume for the sake of presentation that is finite, that is, takes on only finitely many values . Let be an event, then the conditional probability of given is defined as the random variable, written , that takes on the value : P(A\mid X=x) whenever :X=x. More formally, :P(A\mid X)(\omega)=P(A\mid X=X(\omega)) . The conditional probability is a function of .
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A Post canonical system, as created by Emil Post, is a string-manipulation system that starts with finitely-many strings and repeatedly transforms them by applying a finite set j of specified rules of a certain form, thus generating a formal language. Today they are mainly of historical relevance because every Post canonical system can be reduced to a string rewriting system (semi-Thue system), which is a simpler formulation. Both formalisms are Turing complete.
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Zariski's formulation of Hilbert's fourteenth problem asks whether, for a quasi-affine algebraic variety X over a field k, possibly assuming X normal or smooth, the ring of regular functions on X is finitely generated over k. Zariski's formulation was shown to be equivalent to the original problem, for X normal. (See also: Zariski's finiteness theorem.) Éfendiev F.F. (Fuad Efendi) provided symmetric algorithm generating basis of invariants of n-ary forms of degree r.
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The extra effort in gift- exchange games is modeled to be a negative payoff if not compensated by salary. The IKEA effect of own extra work is not considered in the payoff structure of this game. Therefore, this model rather fits labor conditions, which are less meaningful for the employees. Like in trust games, game- theoretic solution for rational players predicts that employees’ effort will be minimum for one-shot and finitely repeated interactions.
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The space of states, also called the spectrum, of a CFT, is a representation of the product of the two Virasoro algebras. The eigenvalues of the Virasoro generator L_0+\bar L_0 are interpreted as the energies of the states. Their real parts are usually assumed to be bounded from below. A CFT is called rational if its space of states decomposes into finitely many irreducible representations of the product of the two Virasoro algebras.
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The principal rings constructed in Example 5. above are always Artinian rings; in particular they are isomorphic to a finite direct product of principal Artinian local rings. A local Artinian principal ring is called a special principal ring and has an extremely simple ideal structure: there are only finitely many ideals, each of which is a power of the maximal ideal. For this reason, special principal rings are examples of uniserial rings.
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A subset S of M is said to be a generator of M if M is the smallest set containing S that is closed under the monoid operation, or equivalently M is the result of applying the finitary closure operator to S. If there is a generator of M that has finite cardinality, then M is said to be finitely generated. Not every set S will generate a monoid, as the generated structure may lack an identity element.
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In mathematics, the tensor representations of the general linear group are those that are obtained by taking finitely many tensor products of the fundamental representation and its dual. The irreducible factors of such a representation are also called tensor representations, and can be obtained by applying Schur functors (associated to Young tableaux). These coincide with the rational representations of the general linear group. More generally, a matrix group is any subgroup of the general linear group.
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Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer-Verlag, Berlin, 1999. Let G be a group acting by isometries on a proper length space X such that the action is properly discontinuous and cocompact. Then the group G is finitely generated and for every finite generating set S of G and every point p\in X the orbit map :f_p:(G,d_S)\to X, \quad g\mapsto gp is a quasi-isometry.
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A class of languages has finite thickness if every non- empty set of strings is contained in at most finitely many languages of the class. This is exactly Condition 3 in Angluin's paper.p.123 mid Angluin showed that if a class of recursive languages has finite thickness, then it is learnable in the limit.p.123 bot, Corollary 2 A class with finite thickness certainly satisfies MEF-condition and MFF-condition; in other words, finite thickness implies M-finite thickness.
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A parity game. Circular nodes belong to player 0, rectangular nodes belong to player 1. On the left side is the winning region of player 0, on the right side is the winning region of player 1. A parity game is played on a colored directed graph, where each node has been colored by a priority - one of (usually) finitely many natural numbers. Two players, 0 and 1, move a (single, shared) token along the edges of the graph.
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Hilbert's example: "the assertion that either there are only finitely many prime numbers or there are infinitely many" (quoted in Davis 2000:97); and Brouwer's: "Every mathematical species is either finite or infinite." (Brouwer 1923 in van Heijenoort 1967:336). In general, intuitionists allow the use of the law of excluded middle when it is confined to discourse over finite collections (sets), but not when it is used in discourse over infinite sets (e.g. the natural numbers).
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He proved that planar polynomial vector fields have only finitely many limit cycles. Jean Écalle independently proved the same result, and an earlier attempted proof by Henri Dulac (in 1923) was shown to be defective by Ilyashenko in the 1970s. He was an Invited Speaker of the ICM in 1978 at Helsinki and in 1990 with talk Finiteness theorems for limit cycles at Kyoto. In 2017 he was elected a Fellow of the American Mathematical Society.
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In geometric group theory, the Rips machine is a method of studying the action of groups on R-trees. It was introduced in unpublished work of Eliyahu Rips in about 1991. An R-tree is a uniquely arcwise-connected metric space in which every arc is isometric to some real interval. Rips proved the conjecture of that any finitely generated group acting freely on an R-tree is a free product of free abelian and surface groups .
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There are variants of the criss-cross algorithm for linear programming, for quadratic programming, and for the linear-complementarity problem with "sufficient matrices"; conversely, for linear complementarity problems, the criss-cross algorithm terminates finitely only if the matrix is a sufficient matrix. A sufficient matrix is a generalization both of a positive-definite matrix and of a P-matrix, whose principal minors are each positive. The criss-cross algorithm has been adapted also for linear- fractional programming.
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In mathematics, in the field of general topology, a topological space is said to be metacompact if every open cover has a point finite open refinement. That is, given any open cover of the topological space, there is a refinement which is again an open cover with the property that every point is contained only in finitely many sets of the refining cover. A space is countably metacompact if every countable open cover has a point finite open refinement.
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The theorem may be used to obtain a simple algebraic proof of Weyl's character formula for finite-dimensional representations. Further, it is a necessary condition for the existence of a nonzero homomorphism of some highest weight modules (a homomorphism of such modules preserves central character). A simple consequence is that for Verma modules or generalized Verma modules Vλ with highest weight λ, there exist only finitely many weights μ such that a nonzero homomorphism Vλ → Vμ exists.
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The boundary of a compact Klein surface consists of finitely many connected components, each of which being homeomorphic to a circle. These components are called the ovals of the Klein surface. Suppose Σ is a (not necessarily connected) Riemann surface and τ:Σ→Σ is an anti-holomorphic (orientation-reversing) involution. Then the quotient Σ/τ carries a natural Klein surface structure, and every Klein surface can be obtained in this manner in essentially only one way.
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If two rings are Morita equivalent, there is an induced equivalence of the respective categories of projective modules since the Morita equivalences will preserve exact sequences (and hence projective modules). Since the algebraic K-theory of a ring is defined (in Quillen's approach) in terms of the homotopy groups of (roughly) the classifying space of the nerve of the (small) category of finitely generated projective modules over the ring, Morita equivalent rings must have isomorphic K-groups.
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When the surface of a sphere is divided by finitely many great arcs (equivalently, by planes passing through the center of the sphere), the result is called a spherical polyhedron. Many convex polytopes having some degree of symmetry (for example, all the Platonic solids) can be projected onto the surface of a concentric sphere to produce a spherical polyhedron. However, the reverse process is not always possible; some spherical polyhedra (such as the hosohedra) have no flat-faced analogue..
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In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold. The tameness theorem was conjectured by . It was proved by and, independently, by Danny Calegari and David Gabai. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the density theorem for Kleinian groups and the ending lamination theorem.
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The sequence of tilings where b is centred at 1,2,4, \ldots,2^n,\ldots converges – in the local topology – to the periodic tiling consisting of as only. Thus T is not an aperiodic tiling, since its hull contains the periodic tiling For well-behaved tilings (e.g. substitution tilings with finitely many local patterns) holds: if a tiling is non-periodic and repetitive (i.e. each patch occurs in a uniformly dense way throughout the tiling), then it is aperiodic.
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The tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold. The tameness theorem was conjectured by Marden. It was proved by Agol and, independently, by Danny Calegari and David Gabai. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the density theorem for Kleinian groups and the ending lamination theorem.
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This gives many examples of non-Noetherian Bézout domains. In noncommutative algebra, right Bézout domains are domains whose finitely generated right ideals are principal right ideals, that is, of the form xR for some x in R. One notable result is that a right Bézout domain is a right Ore domain. This fact is not interesting in the commutative case, since every commutative domain is an Ore domain. Right Bézout domains are also right semihereditary rings.
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In mathematics, an elementary function is a function of a single variable composed of particular simple functions. Elementary functions are typically defined as a sum, product, and/or composition of finitely many polynomials, rational functions, trigonometric and exponential functions, and their inverse functions (including arcsin, log, x1/n). Elementary functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841.... An algebraic treatment of elementary functions was started by Joseph Fels Ritt in the 1930s..
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A (shaded) planar tangle is the data of finitely many input disks, one output disk, non- intersecting strings giving an even number, say 2n , intervals per disk and one \star-marked interval per disk. 200px Here, the mark is shown as a \star- shape. On each input disk it is placed between two adjacent outgoing strings, and on the output disk it is placed between two adjacent incoming strings. A planar tangle is defined up to isotopy.
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Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups. Every cyclic group of prime order is a simple group which cannot be broken down into smaller groups.
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A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if and only if its decimal representation is repeating or terminating (i.e. all except finitely many digits are zero). For example, the decimal representation of becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e.
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Let R be a commutative ring with identity 1. The following is Nakayama's lemma, as stated in : Statement 1: Let I be an ideal in R, and M a finitely-generated module over R. If IM = M, then there exists an r ∈ R with r ≡ 1 (mod I), such that rM = 0\. This is proven below. The following corollary is also known as Nakayama's lemma, and it is in this form that it most often appears.
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In the mathematical subject of group theory, small cancellation theory studies groups given by group presentations satisfying small cancellation conditions, that is where defining relations have "small overlaps" with each other. Small cancellation conditions imply algebraic, geometric and algorithmic properties of the group. Finitely presented groups satisfying sufficiently strong small cancellation conditions are word hyperbolic and have word problem solvable by Dehn's algorithm. Small cancellation methods are also used for constructing Tarski monsters, and for solutions of Burnside's problem.
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Evgenii Solomonovich Golod (, October 21, 1935 – July 5, 2018) was a Russian mathematician who proved the Golod–Shafarevich theorem on class field towers. As an application, he gave a negative solution to the Kurosh-Levitzky problem on the nilpotency of finitely generated nil algebras, and so to a weak form of Burnside's problem. Golod was a student of Igor Shafarevich. As of 2015, Golod had 39 academic descendants, most of them through his student Luchezar L. Avramov.
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In mathematics, the Kurosh problem is one general problem, and several more special questions, in ring theory. The general problem is known to have a negative solution, since one of the special cases has been shown to have counterexamples. These matters were brought up by Aleksandr Gennadievich Kurosh as analogues of the Burnside problem in group theory. Kurosh asked whether there can be a finitely-generated infinite-dimensional algebraic algebra (the problem being to show this cannot happen).
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Finite subdivision rules are a geometric form of recursion, which can be used to create fractal-like images. A subdivision rule starts with a collection of polygons labelled by finitely many labels, and then each polygon is subdivided into smaller labelled polygons in a way that depends only on the labels of the original polygon. This process can be iterated. The standard `middle thirds' technique for creating the Cantor set is a subdivision rule, as is barycentric subdivision.
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In number theory, the Katz–Lang finiteness theorem, proved by , states that if X is a smooth geometrically connected scheme of finite type over a field K that is finitely generated over the prime field, and Ker(X/K) is the kernel of the maps between their abelianized fundamental groups, then Ker(X/K) is finite if K has characteristic 0, and the part of the kernel coprime to p is finite if K has characteristic p > 0\.
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In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension. A collection of subsets of a topological space X is said to be locally finite, if each point in the space has a neighbourhood that intersects only finitely many of the sets in the collection. Note that the term locally finite has different meanings in other mathematical fields.
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No infinite collection of a compact space can be locally finite. Indeed, let {Ga} be an infinite family of subsets of a space and suppose this collection is locally finite. For each point x of this space, choose a neighbourhood Ux that intersects the collection {Ga} at only finitely many values of a. Clearly: :Ux for each x in X (the union over all x) is an open covering in X and hence has a finite subcover, Ua1 ∪ ...... ∪ Uan.
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The basic object of study is topological spaces, which are sets equipped with a topology, that is, a family of subsets, called open sets, which is closed under finite intersections and (finite or infinite) unions. The fundamental concepts of topology, such as continuity, compactness, and connectedness, can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size.
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Starting 1980's he worked on problems that explored the connections between Group theory and Computer Science and Complexity Theory. Together with David Muller he proved that a finitely generated group G has context-free word problem if and only if G is virtually free, which is now known as Muller–Schupp theorem.David E. Muller, and Paul E. Schupp, Groups, the theory of ends, and context-free languages. Journal of Computer and System Sciences 26 (1983), no.
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It can be proven that in any genus 2 curve there are exactly six points whose sequences are 1, 1, 2, 2, ... and the rest of the points have the generic sequence 1, 1, 1, 2, ... In particular, a genus 2 curve is a hyperelliptic curve. For g>2 it is always true that at most points the sequence starts with g+1 ones and there are finitely many points with other sequences (see Weierstrass points).
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In the area of abstract algebra known as ring theory, a left perfect ring is a type of ring in which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric; that is, there exist rings which are perfect on one side but not the other. Perfect rings were introduced in . A semiperfect ring is a ring over which every finitely generated left module has a projective cover.
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'Infinitive (abbreviated ') is a linguistics term referring to certain verb forms existing in many languages, most often used as non-finite verbs. As with many linguistic concepts, there is not a single definition applicable to all languages. The word is derived from Late Latin [modus] infinitivus, a derivative of infinitus meaning "unlimited". In traditional descriptions of English, the infinitive is the basic dictionary form of a verb when used non- finitely, with or without the particle to.
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We need the first condition because if the leading coefficient is negative then f(x) < 0 for all large x, and thus f(n) is not a (positive) prime number for large positive integers n. (This merely satisfies the sign convention that primes are positive.) We need the second condition because if f(x) = g(x)h(x) where the polynomials g(x) and h(x) have integer coefficients, then we have f(n) = g(n)h(n) for all integers n; but g(x) and h(x) take the values 0 and \pm 1 only finitely many times, so f(n) is composite for all large n. The third condition, that the numbers f(n) have gcd 1, is obviously necessary, but is somewhat subtle, and is best understood by a counterexample. Consider f(x) = x^2 + x + 2, which has positive leading coefficient and is irreducible, and the coefficients are relatively prime; however f(n) is even for all integers n, and so is prime only finitely many times (namely when f(n)=2, in fact only at n =0,-1).
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Classical measure theory is fundamentally non-constructive, since the classical definition of Lebesgue measure does not describe any way to compute the measure of a set or the integral of a function. In fact, if one thinks of a function just as a rule that "inputs a real number and outputs a real number" then there cannot be any algorithm to compute the integral of a function, since any algorithm would only be able to call finitely many values of the function at a time, and finitely many values are not enough to compute the integral to any nontrivial accuracy. The solution to this conundrum, carried out first in Bishop's 1967 book, is to consider only functions that are written as the pointwise limit of continuous functions (with known modulus of continuity), with information about the rate of convergence. An advantage of constructivizing measure theory is that if one can prove that a set is constructively of full measure, then there is an algorithm for finding a point in that set (again see Bishop's book).
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Francisco Javier González-Acuña (nickname "Fico") is a mathematician in the UNAM's institute of mathematics and CIMAT, specializing in low-dimensional topology. He did his graduate studies at Princeton University, obtaining his Ph.D. in 1970. His thesis, written under the supervision of Ralph Fox, was titled On homology spheres. An early result of González-Acuña is that a group G is the homomorphic image of some knot group if and only if G is finitely generated and has weight at most one.
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Without the assumption that A is Noetherian, the statement of the Artin-Tate lemma is no longer true. Indeed, for any non-Noetherian ring A we can define an A-algebra structure on C = A\oplus A by declaring (a,x)(b,y) = (ab,bx+ay). Then for any ideal I \subset A which is not finitely generated, B = A \oplus I \subset C is not of finite type over A, but all conditions as in the lemma are satisfied.
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These problems can be settled using the field of constructible numbers. Real constructible numbers are, by definition, lengths of line segments that can be constructed from the points 0 and 1 in finitely many steps using only compass and straightedge. These numbers, endowed with the field operations of real numbers, restricted to the constructible numbers, form a field, which properly includes the field of rational numbers. The illustration shows the construction of square roots of constructible numbers, not necessarily contained within .
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Free abelian groups have specific properties that are not shared by modules over other rings. Specifically, every subgroup of a free abelian group is a free abelian group, and, if is a subgroup of a finitely generated free abelian group (that is an abelian group that has a finite basis), there is a basis e_1, \ldots, e_n of and an integer such that a_1e_1, \ldots, a_ke_k is a basis of , for some nonzero integers a_1, \ldots, a_k. For details, see .
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A classic example of a free resolution is given by the Koszul complex of a regular sequence in a local ring or of a homogeneous regular sequence in a graded algebra finitely generated over a field. Let X be an aspherical space, i.e., its universal cover E is contractible. Then every singular (or simplicial) chain complex of E is a free resolution of the module Z not only over the ring Z but also over the group ring Z [π1(X)].
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The Fast Wavelet Transform is a mathematical algorithm designed to turn a waveform or signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves, or wavelets. The transform can be easily extended to multidimensional signals, such as images, where the time domain is replaced with the space domain. This algorithm was introduced in 1989 by Stéphane Mallat. It has as theoretical foundation the device of a finitely generated, orthogonal multiresolution analysis (MRA).
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Around 1945, Alfred Tarski asked whether the free groups on two or more generators have the same first order theory, and whether this theory is decidable. answered the first question by showing that any two nonabelian free groups have the same first order theory, and answered both questions, showing that this theory is decidable. A similar unsolved (as of 2011) question in free probability theory asks whether the von Neumann group algebras of any two non-abelian finitely generated free groups are isomorphic.
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Laurent series cannot in general be multiplied. Algebraically, the expression for the terms of the product may involve infinite sums which need not converge (one cannot take the convolution of integer sequences). Geometrically, the two Laurent series may have non-overlapping annuli of convergence. Two Laurent series with only finitely many negative terms can be multiplied: algebraically, the sums are all finite; geometrically, these have poles at c, and inner radius of convergence 0, so they both converge on an overlapping annulus.
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In topology, the cartesian product of topological spaces can be given several different topologies. One of the more obvious choices is the box topology, where a base is given by the Cartesian products of open sets in the component spaces.Willard, 8.2 pp. 52-53, Another possibility is the product topology, where a base is given by the Cartesian products of open sets in the component spaces, only finitely many of which can be not equal to the entire component space.
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In 1926 Otto Schreier would generalize this result by removing the condition that the free group be finitely generated; this result is now known as the Nielsen–Schreier theorem. Also in 1921 Nielsen moved to the Royal Veterinary and Agricultural University in Copenhagen, where he would stay until 1925, when he moved to the Technical University in Copenhagen. He also proved the Dehn–Nielsen theorem on mapping class groups. Nielsen was a Plenary Speaker of the ICM in 1936 in Oslo.
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One says a manifold N is obtained from M by attaching j-handles if the union of M with finitely many j-handles is diffeomorphic to N. The definition of a handle decomposition is then as in the introduction. Thus, a manifold has a handle decomposition with only 0-handles if it is diffeomorphic to a disjoint union of balls. A connected manifold containing handles of only two types (i.e.: 0-handles and j-handles for some fixed j) is called a handlebody.
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More generally a Prüfer ring is a commutative ring in which every non-zero finitely generated ideal consisting only of non-zero-divisors is invertible (that is, projective). A commutative ring is said to be arithmetical if for every maximal ideal m in R, the localization Rm of R at m is a chain ring. With this definition, an arithmetical domain is a Prüfer domain. Noncommutative right or left semihereditary domains could also be considered as generalizations of Prüfer domains.
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Besides respecting multiplication, monomial orders are often required to be well- orders, since this ensures the multivariate division procedure will terminate. There are however practical applications also for multiplication-respecting order relations on the set of monomials that are not well-orders. In the case of finitely many variables, well-ordering of a monomial order is equivalent to the conjunction of the following two conditions: # The order is a total order. # If u is any monomial then 1 \leq u.
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An example of chaotic mixing In chaos theory and fluid dynamics, chaotic mixing is a process by which flow tracers develop into complex fractals under the action of a fluid flow. The flow is characterized by an exponential growth of fluid filaments. Even very simple flows, such as the blinking vortex, or finitely resolved wind fields can generate exceptionally complex patterns from initially simple tracer fields. The phenomenon is still not well understood and is the subject of much current research.
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This is because the payoff in a repeated game is just a weighted average of payoffs in the basic games. Folk theorems are partially converse claims: they say that, under certain conditions (which are different in each folk theorem), every payoff profile that is both individually rational and feasible can be realized as a Nash equilibrium payoff profile of the repeated game. There are various folk theorems; some relate to finitely-repeated games while others relate to infinitely-repeated games.
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A locally cyclic group is a group in which each finitely generated subgroup is cyclic. An example is the additive group of the rational numbers: every finite set of rational numbers is a set of integer multiples of a single unit fraction, the inverse of their lowest common denominator, and generates as a subgroup a cyclic group of integer multiples of this unit fraction. A group is locally cyclic if and only if its lattice of subgroups is a distributive lattice..
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We may thus pick one of these vertices and call it v2. Now infinitely many vertices of G can be reached from v2 with a simple path which does not include the vertex v1. Each such path must start with one of the finitely many vertices adjacent to v2. So an argument similar to the one above shows that there must be one of those adjacent vertices through which infinitely many vertices can be reached; pick one and call it v3.
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The growth rate of a group is a well-defined notion from asymptotic analysis. To say that a finitely generated group has polynomial growth means the number of elements of length (relative to a symmetric generating set) at most n is bounded above by a polynomial function p(n). The order of growth is then the least degree of any such polynomial function p. A nilpotent group G is a group with a lower central series terminating in the identity subgroup.
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The symplectic sum was first clearly defined in 1995 by Robert Gompf. He used it to demonstrate that any finitely presented group appears as the fundamental group of a symplectic four-manifold. Thus the category of symplectic manifolds was shown to be much larger than the category of Kähler manifolds. Around the same time, Eugene Lerman proposed the symplectic cut as a generalization of symplectic blow up and used it to study the symplectic quotient and other operations on symplectic manifolds.
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By definition, the Proj of a graded ring R is the quotient category of the category of finitely generated graded modules over R by the subcategory of torsion modules. If R is a commutative Noetherian graded ring generated by degree-one elements, then the Proj of R in this sense is equivalent to the category of coherent sheaves on the usual Proj of R. Hence, the construction can be thought of as a generalization of the Proj construction for a commutative graded ring.
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For instance, they were important in early AI languages, such as QA4, where they were referred to as bags, a term attributed to Peter Deutsch. A multiset has been also called an aggregate, heap, bunch, sample, weighted set, occurrence set, and fireset (finitely repeated element set). Although multisets were used implicitly from ancient times, their explicit exploration happened much later. The first known study of multisets is attributed to the Indian mathematician Bhāskarāchārya circa 1150, who described permutations of multisets.
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Starting in the 1980s, much research has gone into connecting properties of the graph to random walks. In addition to the electrical network connection described above, there are important connections to isoperimetric inequalities, see more here, functional inequalities such as Sobolev and Poincaré inequalities and properties of solutions of Laplace's equation. A significant portion of this research was focused on Cayley graphs of finitely generated groups. In many cases these discrete results carry over to, or are derived from manifolds and Lie groups.
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The product ΣA × ΣB has the homotopy type of the mapping cone of [ιΣA, ιΣB] ∈ [Σ(A ∧ B), ΣA ∨ ΣB] (). Whitehead products for homotopy groups with coefficients are obtained by taking A and B to be Moore spaces (, pp. 110-114) There is a weak homotopy equivalence between a wedge of suspensions of finitely many spaces and an infinite product of suspensions of various smash products of the spaces according to the Milnor-Hilton theorem. The map is defined by generalised Whitehead products .
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Noether's normalisation lemma is a theorem in commutative algebra. Given a field K and a finitely generated K-algebra A, the theorem says it is possible to find elements y1, y2, ..., ym in A that are algebraically independent over K such that A is finite (and hence integral) over B = K[y1,..., ym]. Thus the extension K ⊂ A can be written as a composite K ⊂ B ⊂ A where K ⊂ B is a purely transcendental extension and B ⊂ A is finite.Chapter 4 of Reid.
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Wilkie's results from his paper show, in more formal language, that the "only gap" in the high school axioms is the inability to manipulate polynomials with negative coefficients. R. Gurevič showed in 1988 that there is no finite axiomisation for the valid equations for the positive natural numbers with 1, addition, multiplication, and exponentiation.R. Gurevič, Equational theory of positive numbers with exponentiation is not finitely axiomatizable, Annals of Pure and Applied Logic, 49:1–30, 1990.Fiore, Cosmo, and Balat.
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It is known by the Darmon–Granville theorem, which uses Faltings's theorem, that for any fixed choice of positive integers m, n and k satisfying (2), only finitely many coprime triples (a, b, c) solving (1) exist. However, the full Fermat–Catalan conjecture is stronger as it allows for the exponents m, n and k to vary. The abc conjecture implies the Fermat–Catalan conjecture. For a list of results for impossible combinations of exponents, see Beal conjecture#Partial results.
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Stickelberger's work on the classification of pairs of bilinear and quadratic forms filled in important gaps in the theory earlier developed by Weierstrass and Darboux. Augmented with the contemporaneous work of Frobenius, it set the theory of elementary divisors upon a rigorous foundation. An important 1878 paper of Stickelberger and Frobenius gave the first complete treatment of the classification of finitely generated abelian groups and sketched the relation with the theory of modules that had just been developed by Dedekind.
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The ends of a topological space are, roughly speaking, the connected components of the “ideal boundary” of the space. That is, each end represents a topologically distinct way to move to infinity within the space. Adding a point at each end yields a compactification of the original space, known as the end compactification. The ends of a finitely generated group are defined to be the ends of the corresponding Cayley graph; this definition is independent of the choice of a finite generating set.
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Two major ways in which this can be done are through fundamental groups, or more generally homotopy theory, and through homology and cohomology groups. The fundamental groups give us basic information about the structure of a topological space, but they are often nonabelian and can be difficult to work with. The fundamental group of a (finite) simplicial complex does have a finite presentation. Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated.
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Milgrom's 1985 paper with Robert J. Weber on distributional strategies showed the general existence of equilibria for a Bayesian game with finitely many players, if the players' sets of types and actions are compact metric spaces, the players' payoffs are continuous functions of the types and actions, and the joint distribution of the players' types is absolutely continuous with respect to the product of their marginal distributions. These basic assumptions are always satisfied if the sets of types and actions are finite.
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The perceptron is a neural net developed by psychologist Frank Rosenblatt in 1958 and is one of the most famous machines of its period. In 1960, Rosenblatt and colleagues were able to show that the perceptron could in finitely many training cycles learn any task that its parameters could embody. The perceptron convergence theorem was proved for single-layer neural nets. During this period, neural net research was a major approach to the brain-machine issue that had been taken by a significant number of individuals.
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Davis's older treatment approaches the question from a Turing machine viewpoint. Chaitin has written a number of books about his endeavors and the subsequent philosophic and mathematical fallout from them. A string is called (algorithmically) random if it cannot be produced from any shorter computer program. While most strings are random, no particular one can be proved so, except for finitely many short ones: : "A paraphrase of Chaitin's result is that there can be no formal proof that a sufficiently long string is random..." (Beltrami p.
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Expanding on Köthe's work, Tadashi Nakayama used the term generalized uniserial ring to refer to an Artinian serial ring. Nakayama showed that all modules over such rings are serial. Artinian serial rings are sometimes called Nakayama algebras, and they have a well-developed module theory. Warfield used the term homogeneously serial module for a serial module with the additional property that for any two finitely generated submodules A and B, A/J(A)\cong B/J(B) where J(-) denotes the Jacobson radical of the module .
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He is no longer finitely concerned about what the princess does, and precisely this proves that he has made the movement [of faith] infinitely."Fear and Trembling p. 44 Kierkegaard also mentioned Agnes and the Merman in his Journals: "I have thought of adapting [the legend of] Agnes and the Merman from an angle that has not occurred to any poet. The Merman is a seducer, but when he has won Agnes' love he is so moved by it that he wants to belong to her entirely.
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The situation is considerably different in the case of a Weyl algebra over a field of characteristic . In this case, for any element D of the Weyl algebra, the element Dp is central, and so the Weyl algebra has a very large center. In fact, it is a finitely generated module over its center; even more so, it is an Azumaya algebra over its center. As a consequence, there are many finite- dimensional representations which are all built out of simple representations of dimension p.
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The Cayley graph of the free group on two generators a and b Geometric group theory uses large-scale geometric techniques to study finitely generated groups. It is closely connected to low-dimensional topology, such as in Grigori Perelman's proof of the Geometrization conjecture, which included the proof of the Poincaré conjecture, a Millennium Prize Problem. Geometric group theory often revolves around the Cayley graph, which is a geometric representation of a group. Other important topics include quasi-isometries, Gromov-hyperbolic groups, and right angled Artin groups.
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Developments in algebraic geometry included the study of curves and surfaces over finite fields as demonstrated by the works of among others André Weil, Alexander Grothendieck, and Jean-Pierre Serre as well as over the real or complex numbers. Finite geometry itself, the study of spaces with only finitely many points, found applications in coding theory and cryptography. With the advent of the computer, new disciplines such as computational geometry or digital geometry deal with geometric algorithms, discrete representations of geometric data, and so forth.
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While the box topology has a somewhat more intuitive definition than the product topology, it satisfies fewer desirable properties. In particular, if all the component spaces are compact, the box topology on their Cartesian product will not necessarily be compact, although the product topology on their Cartesian product will always be compact. In general, the box topology is finer than the product topology, although the two agree in the case of finite direct products (or when all but finitely many of the factors are trivial).
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In a one-dimensional reversible cellular automaton with states per cell, in which the neighborhood of a cell is an interval of cells, the automaton representing the reverse dynamics has neighborhoods that consist of at most cells. This bound is known to be tight for : there exist -state reversible cellular automata with two-cell neighborhoods whose time-reversed dynamics forms a cellular automaton with neighborhood size exactly .; ; . For any integer there are only finitely many two-dimensional reversible -state cellular automata with the von Neumann neighborhood.
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Therefore, there is a well-defined function such that all reverses of -state cellular automata with the von Neumann neighborhood use a neighborhood with radius at most : simply let be the maximum, among all of the finitely many reversible -state cellular automata, of the neighborhood size needed to represent the time-reversed dynamics of the automaton. However, because of Kari's undecidability result, there is no algorithm for computing and the values of this function must grow very quickly, more quickly than any computable function..
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The basic result, the Mordell–Weil theorem in Diophantine geometry, says that A(K), the group of points on A over K, is a finitely-generated abelian group. A great deal of information about its possible torsion subgroups is known, at least when A is an elliptic curve. The question of the rank is thought to be bound up with L-functions (see below). The torsor theory here leads to the Selmer group and Tate–Shafarevich group, the latter (conjecturally finite) being difficult to study.
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Chapter IV discusses Schur–Weyl duality between representations of the symmetric and general linear groups. Chapters V and VI extend the discussion of invariants of the general linear group in chapter II to the orthogonal and symplectic groups, showing that the ring of invariants is generated by the obvious ones. Chapter VII describes the Weyl character formula for the characters of representations of the classical groups. Chapter VIII on invariant theory proves Hilbert's theorem that invariants of the special linear group are finitely generated.
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More generally, if the greatest common divisor of the moves played so far is g, then only finitely many multiples of g can remain to be played, and after they are all played then g must decrease on the next move. Therefore, every game of sylver coinage must eventually end. When a sylver coinage game has only a finite number of remaining moves, the largest number that can still be played is called the Frobenius number, and finding this number is called the coin problem.
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It is not known if there are infinitely many prime quadruplets. A proof that there are infinitely many would imply the twin prime conjecture, but it is consistent with current knowledge that there may be infinitely many pairs of twin primes and only finitely many prime quadruplets. The number of prime quadruplets with n digits in base 10 for n = 2, 3, 4, ... is 1, 3, 7, 27, 128, 733, 3869, 23620, 152141, 1028789, 7188960, 51672312, 381226246, 2873279651 . the largest known prime quadruplet has 10132 digits.
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In particular, a finitely presented group has solvable word problem if and only if the Dehn function for a finite presentation of this group is recursive (see Theorem 2.1 in ). The notion of a Dehn function is motivated by isoperimetric problems in geometry, such as the classic isoperimetric inequality for the Euclidean plane and, more generally, the notion of a filling area function that estimates the area of a minimal surface in a Riemannian manifold in terms of the length of the boundary curve of that surface.
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In mathematics, cylinder set measure (or promeasure, or premeasure, or quasi- measure, or CSM) is a kind of prototype for a measure on an infinite- dimensional vector space. An example is the Gaussian cylinder set measure on Hilbert space. Cylinder set measures are in general not measures (and in particular need not be countably additive but only finitely additive), but can be used to define measures, such as classical Wiener measure on the set of continuous paths starting at the origin in Euclidean space.
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Let G be a finitely generated group. Let S ⊆ G be a finite generating set of G and let Γ(G, S) be the Cayley graph of G with respect to S. The number of ends of G is defined as e(G) = e(Γ(G, S)). A basic fact in the theory of ends of groups says that e(Γ(G, S)) does not depend on the choice of a finite generating set S of G, so that e(G) is well-defined.
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This is a special case of the Kneser–Harder–Chernousov Hasse Principle for algebraic groups over global fields. (Note that such fields do indeed have cohomological dimension at most 2.) The conjecture also holds when F is finitely generated over the complex numbers and has transcendence degree at most 2. The conjecture is also known to hold for certain groups G. For special linear groups, it is a consequence of the Merkurjev–Suslin theorem. Building on this result, the conjecture holds if G is a classical group.
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The characteristic set of a linear matroid is defined as the set of characteristics of the fields over which it is linear. For every prime number p there exist infinitely many matroids whose characteristic set is the singleton set {p},. and for every finite set of prime numbers there exists a matroid whose characteristic set is the given finite set.. If the characteristic set of a matroid is infinite, it contains zero; and if it contains zero then it contains all but finitely many primes., p. 225.
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These cannot be replaced by any finite number of axioms, that is, Presburger arithmetic is not finitely axiomatizable in first-order logic. Presburger arithmetic can be viewed as first-order theory with equality containing precisely all consequences of the above axioms. Alternatively, it can be defined as the set of those sentences that are true in the intended interpretation: the structure of non-negative integers with constants 0, 1, and the addition of non-negative integers. Presburger arithmetic is designed to be complete and decidable.
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The "anabelian question" has been formulated as A concrete example is the case of curves, which may be affine as well as projective. Suppose given a hyperbolic curve C, i.e. the complement of n points in a projective algebraic curve of genus g, taken to be smooth and irreducible, defined over a field K that is finitely generated (over its prime field), such that :2 - 2g - n < 0. Grothendieck conjectured that the algebraic fundamental group G of C, a profinite group, determines C itself (i.e.
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Indeed, given any preorder ≤ on a set X, there is a unique Alexandrov topology on X for which the specialization preorder is ≤. The open sets are just the upper sets with respect to ≤. Thus, Alexandrov topologies on X are in one-to-one correspondence with preorders on X. Alexandrov-discrete spaces are also called finitely generated spaces since their topology is uniquely determined by the family of all finite subspaces. Alexandrov-discrete spaces can thus be viewed as a generalization of finite topological spaces.
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A Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface. A 3-manifold finitely covered by a Haken manifold is said to be virtually Haken. The Virtually Haken conjecture asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken.
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In mathematics, Property FA is a property of groups first defined by Jean- Pierre Serre. A group G is said to have property FA if every action of G on a tree has a global fixed point. Serre shows that if a group has property FA, then it cannot split as an amalgamated product or HNN extension; indeed, if G is contained in an amalgamated product then it is contained in one of the factors. In particular, a finitely generated group with property FA has finite abelianization.
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In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface. A 3-manifold finitely covered by a Haken manifold is said to be virtually Haken. The Virtually Haken conjecture asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken.
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In , a straightforward combinatorial proof is given that finitely generated subgroups of free groups are free. A generating set is called Nielsen reduced if there is not too much cancellation in products. The paper shows that every finite generating set of a subgroup of a free group is (singularly) Nielsen equivalent to a Nielsen reduced generating set, and that a Nielsen reduced generating set is a free basis for the subgroup, so the subgroup is free. This proof is given in some detail in .
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The Nielsen–Schreier theorem is a non-abelian analogue of an older result of Richard Dedekind, that every subgroup of a free abelian group is free abelian., Section 2, The Nielsen–Schreier Theorem, pp. 9–23. originally proved a restricted form of the theorem, stating that any finitely-generated subgroup of a free group is free. His proof involves performing a sequence of Nielsen transformations on the subgroup's generating set that reduce their length (as reduced words in the free group from which they are drawn).
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In mathematics, especially ring theory, the class of Frobenius rings and their generalizations are the extension of work done on Frobenius algebras. Perhaps the most important generalization is that of quasi-Frobenius rings (QF rings), which are in turn generalized by right pseudo-Frobenius rings (PF rings) and right finitely pseudo-Frobenius rings (FPF rings). Other diverse generalizations of quasi-Frobenius rings include QF-1, QF-2 and QF-3 rings. These types of rings can be viewed as descendants of algebras examined by Georg Frobenius.
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This means first constructing infinitesimal deformations, then appealing to prorepresentability theorems to put these together into an object over a formal base. Next, an appeal to Grothendieck's formal existence theorem provides an object of the desired kind over a base which is a complete local ring. This object can be approximated via Artin's approximation theorem by an object defined over a finitely generated ring. The spectrum of this latter ring can then be viewed as giving a kind of coordinate chart on the desired moduli space.
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A set C of attributes is a concept intent if and only if C respects all valid implications. The system of all valid implications therefore suffices for constructing the closure system of all concept intents and thereby the concept hierarchy. The system of all valid implications of a formal context is closed under the natural inference. Formal contexts with finitely many attributes possess a canonical basis of valid implications,Guigues, J.L. and Duquenne, V. Familles minimales d'implications informatives résultant d'un tableau de données binaires.
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Let G be a connected, locally finite, infinite graph (this means: any two vertices can be connected by a path, the graph has infinitely many vertices, and each vertex is adjacent to only finitely many other vertices). Then G contains a ray: a simple path (a path with no repeated vertices) that starts at one vertex and continues from it through infinitely many vertices. A common special case of this is that every infinite tree contains either a vertex of infinite degree or an infinite simple path.
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In mathematics, the Ruziewicz problem (sometimes Banach-Ruziewicz problem) in measure theory asks whether the usual Lebesgue measure on the n-sphere is characterised, up to proportionality, by its properties of being finitely additive, invariant under rotations, and defined on all Lebesgue measurable sets. This was answered affirmatively and independently for n ≥ 4 by Grigory Margulis and Dennis Sullivan around 1980, and for n = 2 and 3 by Vladimir Drinfeld (published 1984). It fails for the circle. The problem is named after Stanisław Ruziewicz.
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H. Bowditch, "Geometrical finiteness with variable negative curvature" Duke Mathematical Journal, vol. 77 (1995), no. 1, 229–274 Bowditch considered a similar problem for discrete groups of isometries of Hadamard manifold of pinched (but not necessarily constant) negative curvature and of arbitrary dimension n ≥ 2\. He proved that four out of five equivalent definitions of geometric finiteness considered in his previous paper remain equivalent in this general set-up, but the condition of having a finitely-sided fundamental polyhedron is no longer equivalent to them.
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Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a disc in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area. This was proven to be possible by Miklós Laczkovich in 1990; the decomposition makes heavy use of the axiom of choice and is therefore non-constructive. Laczkovich estimated the number of pieces in his decomposition at roughly 1050. More recently, gave a completely constructive solution using Borel pieces.
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Loop quantum gravity is the fruit of an effort to formulate a background- independent quantum theory. Topological quantum field theory provided an example of background-independent quantum theory, but with no local degrees of freedom, and only finitely many degrees of freedom globally. This is inadequate to describe gravity in 3+1 dimensions, which has local degrees of freedom according to general relativity. In 2+1 dimensions, however, gravity is a topological field theory, and it has been successfully quantized in several different ways, including spin networks.
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From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space is characterized, up to isomorphism, by its dimension. However, vector spaces per se do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions converges to another function. Likewise, linear algebra is not adapted to deal with infinite series, since the addition operation allows only finitely many terms to be added. Therefore, the needs of functional analysis require considering additional structures.
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It will be useful to know the following fact. Lemma. The new point A can be chosen so that case b occurs for each of the new lines. Proof. For the case a, three points must be on one line: the new point A, the old point O to which the line is drawn, and the point I where two of the old lines intersect. There are n old points O, and hence finitely many points I where two of the old lines intersect.
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Niven completed the solution of most of Waring's problem in 1944. This problem, based on a 1770 conjecture by Edward Waring, consists of finding the smallest number g(n) such that every positive integer is the sum of at most g(n) nth powers of positive integers. David Hilbert had proved the existence of such a g(n) in 1909; Niven's work established the value of g(n) for all but finitely many values of n. Niven numbers, Niven's constant, and Niven's theorem are named for Niven.
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In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space for which there are only finitely many points. While topology has mainly been developed for infinite spaces, finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures. William Thurston has called the study of finite topologies in this sense "an oddball topic that can lend good insight to a variety of questions".
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In theoretical physics, a minimal model or Virasoro minimal model is a two- dimensional conformal field theory whose spectrum is built from finitely many irreducible representations of the Virasoro algebra. Minimal models have been classified and solved, and found to obey an ADE classification. A. Cappelli, J-B. Zuber, "A-D-E Classification of Conformal Field Theories", Scholarpedia The term minimal model can also refer to a rational CFT based on an algebra that is larger than the Virasoro algebra, such as a W-algebra.
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In that case, we often speak of a linear combination of the vectors v1,...,vn, with the coefficients unspecified (except that they must belong to K). Or, if S is a subset of V, we may speak of a linear combination of vectors in S, where both the coefficients and the vectors are unspecified, except that the vectors must belong to the set S (and the coefficients must belong to K). Finally, we may speak simply of a linear combination, where nothing is specified (except that the vectors must belong to V and the coefficients must belong to K); in this case one is probably referring to the expression, since every vector in V is certainly the value of some linear combination. Note that by definition, a linear combination involves only finitely many vectors (except as described in Generalizations below). However, the set S that the vectors are taken from (if one is mentioned) can still be infinite; each individual linear combination will only involve finitely many vectors. Also, there is no reason that n cannot be zero; in that case, we declare by convention that the result of the linear combination is the zero vector in V.
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Many of the preceding results remain valid when the field of definition of E is a number field K, that is to say, a finite field extension of Q. In particular, the group E(K) of K-rational points of an elliptic curve E defined over K is finitely generated, which generalizes the Mordell–Weil theorem above. A theorem due to Loïc Merel shows that for a given integer d, there are (up to isomorphism) only finitely many groups that can occur as the torsion groups of E(K) for an elliptic curve defined over a number field K of degree d. More precisely, there is a number B(d) such that for any elliptic curve E defined over a number field K of degree d, any torsion point of E(K) is of order less than B(d). The theorem is effective: for d > 1, if a torsion point is of order p, with p prime, then :p < d^{3d^2} As for the integral points, Siegel's theorem generalizes to the following: Let E be an elliptic curve defined over a number field K, x and y the Weierstrass coordinates.
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In mathematics, a Carnot group is a simply connected nilpotent Lie group, together with a derivation of its Lie algebra such that the subspace with eigenvalue 1 generates the Lie algebra. The subbundle of the tangent bundle associated to this eigenspace is called horizontal. On a Carnot group, any norm on the horizontal subbundle gives rise to a Carnot–Carathéodory metric. Carnot–Carathéodory metrics have metric dilations; they are asymptotic cones (see Ultralimit) of finitely-generated nilpotent groups, and of nilpotent Lie groups, as well as tangent cones of sub-Riemannian manifolds.
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A related conjecture that has two forms, equivalent if the André–Oort conjecture is assumed, is the Coleman–Oort conjecture. Robert Coleman conjectured that for sufficiently large g, there are only finitely many smooth projective curves C of genus g, such that the Jacobian variety J(C) is an abelian variety of CM-type. Oort then conjectured that the Torelli locus – of the moduli space of abelian varieties of dimension g – has for sufficiently large g no special subvariety of dimension > 0 that intersects the image of the Torelli mapping in a dense open subset.
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For categories C with a well- behaved tensor product (more formally, C is required to be a monoidal category), there is another condition imposing some kind of finiteness, namely the condition that an object is dualizable. If the monoidal unit in C is compact, then any dualizable object is compact as well. For example, R is compact as an R-module, so this observation can be applied. Indeed, in the category of R-modules the dualizable objects are the finitely presented projective modules, which are in particular compact.
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An electrostatic energy of polygonal knots was studied by Fukuhara in 1987. and shortly after a different, geometric energy was studied by Sakuma.. As cited by .. In 1988, Jun O'Hara defined a knot energy based on electrostatic energy, Möbius energy.. A fundamental property of the O'Hara energy function is that infinite energy barriers exist for passing the knot through itself. With some additional restrictions, O'Hara showed there were only finitely many knot types with energies less than a given bound. Later, Freedman, He, and Wang removed these restrictions..
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It is known that the endomorphism ring EndR(M) is a semilocal ring which is very close to a local ring in the sense that EndR(M) has at most two maximal right ideals. If M is required to be Artinian or Noetherian, then EndR(M) is a local ring. Since rings with unity always have a maximal right ideal, a right uniserial ring is necessarily local. As noted before, a finitely generated right ideal can be generated by a single element, and so right uniserial rings are right Bézout rings.
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This correspondence holds for normal subgroups of G and as well, and is formalized in the lattice theorem. Several important properties of quotient groups are recorded in the fundamental theorem on homomorphisms and the isomorphism theorems. If G is abelian, nilpotent, solvable, cyclic or finitely generated, then so is . If H is a subgroup in a finite group G, and the order of H is one half of the order of G, then H is guaranteed to be a normal subgroup, so exists and is isomorphic to C2.
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The hyperelliptic curve defined by y^2=x(x+1)(x-3)(x+2)(x-2) has only finitely many rational points (such as the points (-2, 0) and (-1, 0)) by Faltings's theorem. In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. In more abstract terms, arithmetic geometry can be defined as the study of schemes of finite type over the spectrum of the ring of integers.
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In the late 1920s, André Weil demonstrated profound connections between algebraic geometry and number theory with his doctoral work leading to the Mordell–Weil theorem which demonstrates that the set of rational points of an abelian variety is a finitely generated abelian group.A. Weil, L'arithmétique sur les courbes algébriques, Acta Math 52, (1929) p. 281-315, reprinted in vol 1 of his collected papers . Modern foundations of algebraic geometry were developed based on contemporary commutative algebra, including valuation theory and the theory of ideals by Oscar Zariski and others in the 1930s and 1940s.
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In mathematics, homological conjectures have been a focus of research activity in commutative algebra since the early 1960s. They concern a number of interrelated (sometimes surprisingly so) conjectures relating various homological properties of a commutative ring to its internal ring structure, particularly its Krull dimension and depth. The following list given by Melvin Hochster is considered definitive for this area. In the sequel, A, R, and S refer to Noetherian commutative rings; R will be a local ring with maximal ideal m_R, and M and N are finitely generated R-modules.
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1, pp. 1–51 In the same paper they introduced the notion of a relative train track and applied train track methods to solve the Scott conjecture which says that for every automorphism α of a finitely generated free group Fn the fixed subgroup of α is free of rank at most n. Since then train tracks became a standard tool in the study of algebraic, geometric and dynamical properties of automorphisms of free groups and of subgroups of Out(Fn). Examples of applications of train tracks include: a theorem of BrinkmannP.
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The above definition of an abstract Boolean algebra as a set and operations satisfying "the" Boolean laws raises the question, what are those laws? A simple-minded answer is "all Boolean laws," which can be defined as all equations that hold for the Boolean algebra of 0 and 1. Since there are infinitely many such laws this is not a terribly satisfactory answer in practice, leading to the next question: does it suffice to require only finitely many laws to hold? In the case of Boolean algebras the answer is yes.
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Some properties of free groups follow readily from the definition: #Any group G is the homomorphic image of some free group F(S). Let S be a set of generators of G. The natural map f: F(S) → G is an epimorphism, which proves the claim. Equivalently, G is isomorphic to a quotient group of some free group F(S). The kernel of φ is a set of relations in the presentation of G. If S can be chosen to be finite here, then G is called finitely generated.
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If, however, its coefficients are actually all integers, f is called an algebraic integer. Any (usual) integer z ∈ Z is an algebraic integer, as it is the zero of the linear monic polynomial: :p(t) = t − z. It can be shown that any algebraic integer that is also a rational number must actually be an integer, hence the name "algebraic integer". Again using abstract algebra, specifically the notion of a finitely generated module, it can be shown that the sum and the product of any two algebraic integers is still an algebraic integer.
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Thus when defining formal Laurent series, one requires Laurent series with only finitely many negative terms. Similarly, the sum of two convergent Laurent series need not converge, though it is always defined formally, but the sum of two bounded below Laurent series (or any Laurent series on a punctured disk) has a non- empty annulus of convergence. Also, for a field F, by the sum and multiplication defined above, formal Laurent series would form a field F((x)) which is also the field of fractions of the ring Fx of formal power series.
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A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are due to inexactness in the representation of real numbers and the arithmetic operations done with them. This is a form of quantization error. When using approximation equations or algorithms, especially when using finitely many digits to represent real numbers (which in theory have infinitely many digits), one of the goals of numerical analysis is to estimate computation errors.
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The Cayley graph of a free group with two generators. This is a hyperbolic group whose Gromov boundary is a Cantor set. Hyperbolic groups and their boundaries are important topics in geometric group theory, as are Cayley graphs. Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces).
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Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects. This is usually done by studying the Cayley graphs of groups, which, in addition to the graph structure, are endowed with the structure of a metric space, given by the so-called word metric. Geometric group theory, as a distinct area, is relatively new, and became a clearly identifiable branch of mathematics in the late 1980s and early 1990s. Geometric group theory closely interacts with low-dimensional topology, hyperbolic geometry, algebraic topology, computational group theory and differential geometry.
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The cover on the left is not a good cover, since while all open sets in the cover are contractible, their intersection is disconnected. The cover on the right is a good cover, since the intersection of the two sets is contractible. In mathematics, an open cover of a topological space X is a family of open subsets such that X is the union of all of the open sets. A good cover is an open cover in which all sets and all intersections of finitely-many sets are contractible .
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121 meaning that because it is a drive toward the absolute, all limitations disappear, and instead of seeing the world finitely, as he does through the sensuous drive, "man has raised himself to a unity of ideas embracing the whole realm of phenomena". Since the sense drive places us in time, indulging in the formal drive removes us out of time, and in doing so "We are no longer individuals; we are species". While this seems like a perfected state, it is only a point on the path to man reaching his maximum potential.
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If M is an arbitrary set containing zero, the concept of support is immediately generalizable to functions f : X→M. Support may also be defined for any algebraic structure with identity (such as a group, monoid, or composition algebra), in which the identity element assumes the role of zero. For instance, the family ZN of functions from the natural numbers to the integers is the uncountable set of integer sequences. The subfamily { f in ZN :f has finite support } is the countable set of all integer sequences that have only finitely many nonzero entries.
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The group G of symmetries of a realization V of an abstract apeirogon P is generated by two reflections, the product of which translates each vertex of P to the next. The product of the two reflections can be decomposed as a product of a non-zero translation, finitely many rotations, and possibly trivial reflection. Generally, the moduli space of realizations of an abstract polytope is a convex cone of infinite dimension. The realization cone of the abstract apeirogon has uncountably infinite algebraic dimension and cannot be closed in the Euclidean topology.
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Hermann studied mathematics at Göttingen under Emmy Noether and Edmund Landau, where she achieved her Ph.D. in 1926. Her doctoral thesis, "Die Frage der endlich vielen Schritte in der Theorie der Polynomideale" (in English "The Question of Finitely Many Steps in Polynomial Ideal Theory"), published in Mathematische Annalen, is the foundational paper for computer algebra. It first established the existence of algorithms (including complexity bounds) for many of the basic problems of abstract algebra, such as ideal membership for polynomial rings. Hermann's algorithm for primary decomposition is still in contemporary use.
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In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures. Most of these can be posed for an abelian variety A over a number field K; or more generally (for global fields or more general finitely-generated rings or fields).
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Taking particle physics seriously: A critique of the algebraic approach to quantum field theory. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 42 (2):116-125. He justifies the latter claim with the insights gained from modern renormalization group theory, namely the fact that we can absorb all our ignorance of how the cutoff [i.e., the short-range cutoff required to carry out the renormalization procedure] is implemented, into the values of finitely many coefficients which can be measured empirically.
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Lagrange proved that for every value D, there are only finitely many classes of binary quadratic forms with discriminant D. Their number is the ' of discriminant D. He described an algorithm, called reduction', for constructing a canonical representative in each class, the reduced form, whose coefficients are the smallest in a suitable sense. Gauss gave a superior reduction algorithm in Disquisitiones Arithmeticae, which ever since has been the reduction algorithm most commonly given in textbooks. In 1981, Zagier published an alternative reduction algorithm which has found several uses as an alternative to Gauss's.
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If the base set is finite, then since every subset of , and in particular every complement, is then finite. This case is sometimes excluded by definition or else called the improper filter on . Allowing to be finite creates a single exception to the Fréchet filter's being free and non-principal since a filter on a finite set cannot be free and a non-principal filter cannot contain any singletons as members. If is infinite, then every member of is infinite since it is simply minus finitely many of its members.
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Margulis solved the Banach–Ruziewicz problem that asks whether the Lebesgue measure is the only normalized rotationally invariant finitely additive measure on the n-dimensional sphere. The affirmative solution for n ≥ 4, which was also independently and almost simultaneously obtained by Dennis Sullivan, follows from a construction of a certain dense subgroup of the orthogonal group that has property (T). Margulis gave the first construction of expander graphs, which was later generalized in the theory of Ramanujan graphs. In 1986, Margulis gave a complete resolution of the Oppenheim conjecture on quadratic forms and diophantine approximation.
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So after finitely many applications of this operation no further application is possible, which means that we have obtained \alpha_1\mid\alpha_2\mid\cdots\mid\alpha_r as desired. Since all row and column manipulations involved in the process are invertible, this shows that there exist invertible m \times m and n \times n-matrices S, T so that the product S A T satisfies the definition of a Smith normal form. In particular, this shows that the Smith normal form exists, which was assumed without proof in the definition.
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However, in the form that every congruum (the difference between consecutive elements in an arithmetic progression of three squares) is non-square, it was already known (without proof) to Fibonacci.. Every congruum is a congruent number, and every congruent number is a product of a congruum and the square of a rational number.. However, determining whether a number is a congruum is much easier than determining whether it is congruent, because there is a parameterized formula for congrua for which only finitely many parameter values need to be tested..
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In algebraic geometry, dévissage is a technique introduced by Alexander Grothendieck for proving statements about coherent sheaves on noetherian schemes. Dévissage is an adaptation of a certain kind of noetherian induction. It has many applications, including the proof of generic flatness and the proof that higher direct images of coherent sheaves under proper morphisms are coherent. Laurent Gruson and Michel Raynaud extended this concept to the relative situation, that is, to the situation where the scheme under consideration is not necessarily noetherian, but instead admits a finitely presented morphism to another scheme.
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Still more information about endomorphisms of finite-length indecomposables is provided by the Fitting lemma. In the finite-length situation, decomposition into indecomposables is particularly useful, because of the Krull-Schmidt theorem: every finite-length module can be written as a direct sum of finitely many indecomposable modules, and this decomposition is essentially unique (meaning that if you have a different decomposition into indecomposable, then the summands of the first decomposition can be paired off with the summands of the second decomposition so that the members of each pair are isomorphic). Jacobson (2009), p. 115.
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Given a finitely generated module over a commutative ring, the pairing may be injective (hence "nondegenerate" in the above sense) but not unimodular. For example, over the integers, the pairing is nondegenerate but not unimodular, as the induced map from to is multiplication by 2. If V is finite-dimensional then one can identify V with its double dual V∗∗. One can then show that B2 is the transpose of the linear map B1 (if V is infinite-dimensional then B2 is the transpose of B1 restricted to the image of V in V∗∗).
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In Rule 90, on an infinite one-dimensional lattice, every configuration has exactly four predecessor configurations. This is because, in a predecessor, any two consecutive cells may have any combination of states, but once those two cells' states are chosen, there is only one consistent choice for the states of the remaining cells. Therefore, there is no Garden of Eden in Rule 90, a configuration with no predecessors. The Rule 90 configuration consisting of a single nonzero cell (with all other cells zero) has no predecessors that have finitely many nonzeros.
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In programming languages, defunctionalization is a compile-time transformation which eliminates higher-order functions, replacing them by a single first- order apply function. The technique was first described by John C. Reynolds in his 1972 paper, "Definitional Interpreters for Higher-Order Programming Languages". Reynolds' observation was that a given program contains only finitely many function abstractions, so that each can be assigned and replaced by a unique identifier. Every function application within the program is then replaced by a call to the apply function with the function identifier as the first argument.
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Instead of using the areas of rectangles, which put the focus on the domain of the function, Lebesgue looked at the codomain of the function for his fundamental unit of area. Lebesgue's idea was to first define measure, for both sets and functions on those sets. He then proceeded to build the integral for what he called simple functions; measurable functions that take only finitely many values. Then he defined it for more complicated functions as the least upper bound of all the integrals of simple functions smaller than the function in question.
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The product of a finite set of metric spaces in Met is a metric space that has the cartesian product of the spaces as its points; the distance in the product space is given by the supremum of the distances in the base spaces. That is, it is the product metric with the sup norm. However, the product of an infinite set of metric spaces may not exist, because the distances in the base spaces may not have a supremum. That is, Met is not a complete category, but it is finitely complete.
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In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A(K) of K-rational points of A is a finitely- generated abelian group, called the Mordell–Weil group. The case with A an elliptic curve E and K the rational number field Q is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922. It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties.
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The tangent- chord process (one form of addition theorem on a cubic curve) had been known as far back as the seventeenth century. The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group E(Q)/2E(Q) which forms a major step in the proof. Certainly the finiteness of this group is a necessary condition for E(Q) to be finitely- generated; and it shows that the rank is finite. This turns out to be the essential difficulty.
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It is also possible to restrict the classes of point sets that may be Danzer sets in other ways than by their densities. In particular, they cannot be the union of finitely many lattices, they cannot be generated by choosing a point in each tile of a substitution tiling (in the same position for each tile of the same type), and they cannot be generated by the cut-and-project method for constructing aperiodic tilings. Therefore, the vertices of the pinwheel tiling and Penrose tiling are not Danzer sets.
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For infinite groups, the coarse geometry of the Cayley graph is fundamental to geometric group theory. For a finitely generated group, this is independent of choice of finite set of generators, hence an intrinsic property of the group. This is only interesting for infinite groups: every finite group is coarsely equivalent to a point (or the trivial group), since one can choose as finite set of generators the entire group. Formally, for a given choice of generators, one has the word metric (the natural distance on the Cayley graph), which determines a metric space.
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In dynamical systems theory, an area of pure mathematics, a Morse–Smale system is a smooth dynamical system whose non-wandering set consists of finitely many hyperbolic equilibrium points and hyperbolic periodic orbits and satisfying a transversality condition on the stable and unstable manifolds. Morse–Smale systems are structurally stable and form one of the simplest and best studied classes of smooth dynamical systems. They are named after Marston Morse, the creator of the Morse theory, and Stephen Smale, who emphasized their importance for smooth dynamics and algebraic topology.
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A Davenport–Schinzel sequence of order 3 formed by the lower envelope of line segments. The lower envelope of a set of functions ƒi(x) of a real variable x is the function given by their pointwise minimum: :ƒ(x) = miniƒi(x). Suppose that these functions are particularly well behaved: they are all continuous, and any two of them are equal on at most s values. With these assumptions, the real line can be partitioned into finitely many intervals within which one function has values smaller than all of the other functions.
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In game theory, the traveler's dilemma (sometimes abbreviated TD) is a non- zero-sum game in which each player proposes a payoff. The lower of the two proposals wins; the lowball player receives the lowball payoff plus a small bonus, and the highball player receives the same lowball payoff, minus a small penalty. Surprisingly, the Nash equilibrium is for both players to aggressively lowball. The traveler's dilemma is notable in that naive play appears to outperform the Nash equilibrium; this apparent paradox also appears in the centipede game and the finitely-iterated prisoner's dilemma.
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A related idea exists in commutative algebra. Suppose R = k[x0,...,xn] is a polynomial ring over a field k and M is a finitely generated graded R-module. Suppose M has a minimal graded free resolution :\cdots\rightarrow F_j \rightarrow\cdots\rightarrow F_0\rightarrow M\rightarrow 0 and let bj be the maximum of the degrees of the generators of Fj. If r is an integer such that bj \- j ≤ r for all j, then M is said to be r-regular. The regularity of M is the smallest such r.
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In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed "size" (where the meaning of "size" depends on the structure of the ambient space.) The term precompact (or pre-compact) is sometimes used with the same meaning, but precompact is also used to mean relatively compact. These definitions coincide for subsets of a complete metric space, but not in general.
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If the intensive properties of different finitely extended elements of a system differ, there is always the possibility to extract mechanical work from the system. The term exergy is also used, by analogy with its physical definition, in information theory related to reversible computing. Exergy is also synonymous with: available energy, exergic energy, essergy (considered archaic), utilizable energy, available useful work, maximum (or minimum) work, maximum (or minimum) work content, reversible work, and ideal work. The exergy destruction of a cycle is the sum of the exergy destruction of the processes that compose that cycle.
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Addition is commutative and associative, so the order in which finitely many terms are added does not matter. The identity element for a binary operation is the number that, when combined with any number, yields the same number as the result. According to the rules of addition, adding to any number yields that same number, so is the additive identity. The inverse of a number with respect to a binary operation is the number that, when combined with any number, yields the identity with respect to this operation.
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The infinite dihedral group G of symmetries of a realization V of an abstract apeirogon P is generated by two reflections, the product of which translates each vertex of P to the next. The product of the two reflections can be decomposed as a product of a non-zero translation, finitely many rotations, and a possibly trivial reflection. Generally, the moduli space of realizations of an abstract polytope is a convex cone of infinite dimension. The realization cone of the abstract apeirogon has uncountably infinite algebraic dimension and cannot be closed in the Euclidean topology.
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1, pp. 1-51 The notion was motivated by Thurston's train tracks on surfaces, but the free group case is substantially different and more complicated. In their 1992 paper Bestvina and Handel proved that every irreducible automorphism of Fn has a train-track representative. In the same paper they introduced the notion of a relative train track and applied train track methods to solve the Scott conjecture which says that for every automorphism α of a finitely generated free group Fn the fixed subgroup of α is free of rank at most n.
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One advantage of using Δ-sets in this way is that the resulting chain complex is generally much simpler than the singular chain complex. For reasonably simple spaces, all of the groups will be finitely generated, whereas the singular chain groups are, in general, not even countably generated. One drawback of this method is that one must prove that the geometric realization of the Δ-set is actually homeomorphic to the topological space in question. This can become a computational challenge as the Δ-set increases in complexity.
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In mathematical logic, the compactness theorem states that a set of first- order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful method for constructing models of any set of sentences that is finitely consistent. The compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact Stone spaces,See Truss (1997). hence the theorem's name.
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Reuleaux polygons 50 fils coin, a Reuleaux heptagon in form. The Reuleaux triangle can be generalized to regular polygons with an odd number of sides, yielding a Reuleaux polygon. These are the only curves of constant width whose boundaries are formed by finitely many circular arcs of equal length.. The constant width of these shapes allows their use as coins that can be used in coin-operated machines. For instance, the United Kingdom has made 20-pence and 50-pence coins in the shape of a Reuleaux heptagon.
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This result shows that it is possible to axiomatize ZFC with a single infinite axiom schema. Because at least one such infinite schema is required (ZFC is not finitely axiomatizable), this shows that the axiom schema of replacement can stand as the only infinite axiom schema in ZFC if desired. Because the axiom schema of separation is not independent, it is sometimes omitted from contemporary statements of the Zermelo-Fraenkel axioms. Separation is still important, however, for use in fragments of ZFC, because of historical considerations, and for comparison with alternative axiomatizations of set theory.
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The mapping from collective choices to payoff profiles is known to the players, and each player aims to maximize their payoff. If the collective choice is denoted by x, the payoff that player i receives, also known as player i's utility, will be denoted by u_i(x). We then consider a repetition of this stage game, finitely or infinitely many times. In each repetition, each player chooses one of their stage game options, and when making that choice, they may take into account the choices of the other players in the prior iterations.
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In particular, when the branching at each node is done on a finite subset of an arbitrary set not assumed to be countable, the form of Kőnig's lemma that says "Every infinite finitely branching tree has an infinite path" is equivalent to the principle that every countable set of finite sets has a choice function, that is to say, the axiom of countable choice for finite sets., p. 273; compare , Exercise IX.2.18. This form of the axiom of choice (and hence of Kőnig's lemma) is not provable in ZF set theory.
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Proof: In a Bézout domain the ACCP is equivalent to the ACC on finitely generated ideals, but this is known to be equivalent to the ACC on all ideals. Thus the domain is Noetherian and Bézout, hence a principal ideal domain. The ring Z+XQ[X] of all rational polynomials with integral constant term is an example of an integral domain (actually a GCD domain) that does not satisfy (ACCP), for the chain of principal ideals :(X) \subset (X/2) \subset (X/4) \subset (X/8), ... is non-terminating.
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Graham's doctoral dissertation was in number theory, on Egyptian fractions, and the Erdős–Graham problem is closely related. It asked for a proof that, when the integers are partitioned into finitely many classes, one of the classes has a subset whose reciprocals sum to one. A proof was published by Ernie Croot in 2003. Another of Graham's papers on Egyptian fractions was published in 2015 with Steve Butler and (nearly 20 years posthumously) Paul Erdős; it was the last of Erdős's papers to be published, making Butler his 512th coauthor.
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For a projective variety X, the study of holomorphic maps C → X has some analogy with the study of rational points of X, a central topic of number theory. There are several conjectures on the relation between these two subjects. In particular, let X be a projective variety over a number field k. Fix an embedding of k into C. Then Lang conjectured that the complex manifold X(C) is Kobayashi hyperbolic if and only if X has only finitely many F-rational points for every finite extension field F of k.
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Defining all subsets of the sample space as events works well when there are only finitely many outcomes, but gives rise to problems when the sample space is infinite. For many standard probability distributions, such as the normal distribution, the sample space is the set of real numbers or some subset of the real numbers. Attempts to define probabilities for all subsets of the real numbers run into difficulties when one considers 'badly behaved' sets, such as those that are nonmeasurable. Hence, it is necessary to restrict attention to a more limited family of subsets.
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A positive integer n is square-free if and only if all abelian groups of order n are isomorphic, which is the case if and only if any such group is cyclic. This follows from the classification of finitely generated abelian groups. A integer n is square-free if and only if the factor ring Z / nZ (see modular arithmetic) is a product of fields. This follows from the Chinese remainder theorem and the fact that a ring of the form Z / kZ is a field if and only if k is a prime.
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The main focus of the book is on the specification of geometric data that will determine uniquely the shape of a three- dimensional convex polyhedron, up to some class of geometric transformations such as congruence or similarity. It considers both bounded polyhedra (convex hulls of finite sets of points) and unbounded polyhedra (intersections of finitely many half-spaces). The 1950 Russian edition of the book included 11 chapters. The first chapter covers the basic topological properties of polyhedra, including their topological equivalence to spheres (in the bounded case) and Euler's polyhedral formula.
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Another class of examples of operads are those capturing the structures of algebraic structures, such as associative algebras, commutative algebras and Lie algebras. Each of these can be exhibited as a finitely presented operad, in each of these three generated by binary operations. Thus, the associative operad is generated by a binary operation \psi, subject to the condition that :\psi\circ(\psi,1)=\psi\circ(1,\psi). This condition does correspond to associativity of the binary operation \psi; writing \psi(a,b) multiplicatively, the above condition is (ab)c = a(bc).
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In particular, it is impossible to dissect a circle and make a square using pieces that could be cut with an idealized pair of scissors (that is, having Jordan curve boundary). The pieces used in Laczkovich's proof are non-measurable subsets. Laczkovich actually proved the reassembly can be done using translations only; rotations are not required. Along the way, he also proved that any simple polygon in the plane can be decomposed into finitely many pieces and reassembled using translations only to form a square of equal area.
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The preceding defines a notion of "inaccessibility": we are dealing with cases where it is no longer enough to do finitely many iterations of the successor and powerset operations; hence the phrase "cannot be reached" in both of the intuitive definitions above. But the "union operation" always provides another way of "accessing" these cardinals (and indeed, such is the case of limit ordinals as well). Stronger notions of inaccessibility can be defined using cofinality. For a weak (respectively strong) limit cardinal κ the requirement is that cf(κ) = κ (i.e.
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It is clear from the definition of a topology that a finite union of closed sets is closed. One can readily give an example of an infinite union of closed sets that is not closed. However, if we consider a locally finite collection of closed sets, the union is closed. To see this we note that if x is a point outside the union of this locally finite collection of closed sets, we merely choose a neighbourhood V of x that intersects this collection at only finitely many of these sets.
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KZ filtration resolved the problem and enabled proof of Kolmogorov's law in that domain. Filter construction relied on the main concepts of the continuous Fourier transform and their discrete analogues. The algorithm of the KZ filter came from the definition of higher-order derivatives for discrete functions as higher-order differences. Believing that infinite smoothness in the Gaussian window was a beautiful but unrealistic approximation of a truly discrete world, Kolmogorov chose a finitely differentiable tapering window with finite support, and created this mathematical construction for the discrete case.
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Given the other axioms of Zermelo–Fraenkel set theory, the existence of bases is equivalent to the axiom of choice. The ultrafilter lemma, which is weaker than the axiom of choice, implies that all bases of a given vector space have the same number of elements, or cardinality (cf. Dimension theorem for vector spaces). It is called the dimension of the vector space, denoted by dim V. If the space is spanned by finitely many vectors, the above statements can be proven without such fundamental input from set theory.
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Analogous characterizations of other families of graphs in terms of the summands of their clique-sum decompositions have since become standard in graph minor theory. Wagner conjectured in the 1930s (although this conjecture was not published until later). that in any infinite set of graphs, one graph is isomorphic to a minor of another. The truth of this conjecture implies that any family of graphs closed under the operation of taking minors (as planar graphs are) can automatically be characterized by finitely many forbidden minors analogously to Wagner's theorem characterizing the planar graphs.
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Bicchieri pioneered work on counterfactuals and belief-revision in games, and the consequences of relaxing the common knowledge assumption. Her contributions include axiomatic models of players' theory of the game and the proof that—in a large class of games—a player's theory of the game is consistent only if the player's knowledge is limited.C. Bicchieri, Rationality and Coordination (Cambridge University Press 2003). An important consequence of assuming bounded knowledge is that it allows for more intuitive solutions to familiar games such as the finitely repeated prisoner's dilemma or the chain-store paradox.
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Any function field K(V) of an algebraic variety V over K, other than a single point, has a subfield isomorphic with K(T). From the point of view of birational geometry, this means that there will be a rational map from V to P1(K), that is not constant. The image will omit only finitely many points of P1(K), and the inverse image of a typical point P will be of dimension . This is the beginning of methods in algebraic geometry that are inductive on dimension.
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This means that every proper open set in can be written as a union of finite intersections of elements of . Explicitly, given a point in an open set , there are finitely many sets of , such that the intersection of these sets contains and is contained in . (If we use the nullary intersection convention, then there is no need to include in the second definition.) For subcollection of the power set , there is a unique topology having as a subbase. In particular, the intersection of all topologies on containing satisfies this condition.
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Limit set of a quasifuchsian group A Kleinian group that preserves a Jordan curve is called a quasi-Fuchsian group. When the Jordan curve is a circle or a straight line these are just conjugate to Fuchsian groups under conformal transformations. Finitely generated quasi- Fuchsian groups are conjugate to Fuchsian groups under quasi-conformal transformations. The limit set is contained in the invariant Jordan curve, and it is equal to the Jordan curve the group is said to be of type one, and otherwise it is said to be of type 2.
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A quiver is of finite type if it has only finitely many isomorphism classes of indecomposable representations. classified all quivers of finite type, and also their indecomposable representations. More precisely, Gabriel's theorem states that: # A (connected) quiver is of finite type if and only if its underlying graph (when the directions of the arrows are ignored) is one of the ADE Dynkin diagrams: A_n, D_n, E_6, E_7, E_8. # The indecomposable representations are in a one-to- one correspondence with the positive roots of the root system of the Dynkin diagram.
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A quiver is of finite type if it has only finitely many isomorphism classes of indecomposable representations. classified all quivers of finite type, and also their indecomposable representations. More precisely, Gabriel's theorem states that: # A (connected) quiver is of finite type if and only if its underlying graph (when the directions of the arrows are ignored) is one of the ADE Dynkin diagrams: A_n, D_n, E_6, E_7, E_8. # The indecomposable representations are in a one-to-one correspondence with the positive roots of the root system of the Dynkin diagram.
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An ideal Q of a ring is said to be primary if Q is proper and whenever xy ∈ Q, either x ∈ Q or yn ∈ Q for some positive integer n. In Z, the primary ideals are precisely the ideals of the form (pe) where p is prime and e is a positive integer. Thus, a primary decomposition of (n) corresponds to representing (n) as the intersection of finitely many primary ideals. The Lasker–Noether theorem, given here, may be seen as a certain generalization of the fundamental theorem of arithmetic: > Lasker-Noether Theorem.
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This also leads to difficulties since one can introduce somewhat pathological objects, e.g. an affine line with zero doubled. Such objects are usually not considered varieties, and are eliminated by requiring the schemes underlying a variety to be separated. (Strictly speaking, there is also a third condition, namely, that one needs only finitely many affine patches in the definition above.) Some modern researchers also remove the restriction on a variety having integral domain affine charts, and when speaking of a variety only require that the affine charts have trivial nilradical.
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In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways. They are named for , who proved the unmixedness theorem for polynomial rings, and for , who proved the unmixedness theorem for formal power series rings.
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This forms the basis of the Lévy hierarchy, which is defined analogously with the arithmetical hierarchy. Bounded quantifiers are important in Kripke–Platek set theory and constructive set theory, where only Δ0 separation is included. That is, it includes separation for formulas with only bounded quantifiers, but not separation for other formulas. In KP the motivation is the fact that whether a set x satisfies a bounded quantifier formula only depends on the collection of sets that are close in rank to x (as the powerset operation can only be applied finitely many times to form a term).
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The uniqueness and the zeros of trigonometric series was an active area of research in 19th century Europe. First, Georg Cantor proved that if a trigonometric series is convergent to a function f(x) on the interval [0, 2\pi], which is identically zero, or more generally, is nonzero on at most finitely many points, then the coefficients of the series are all zero. Later Cantor proved that even if the set S on which f is nonzero is infinite, but the derived set S' of S is finite, then the coefficients are all zero. In fact, he proved a more general result.
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For a closed Riemannian manifold the injectivity radius is either half the minimal length of a closed geodesic or the minimal distance between conjugate points on a geodesic. Infranilmanifold Given a simply connected nilpotent Lie group N acting on itself by left multiplication and a finite group of automorphisms F of N one can define an action of the semidirect product N \rtimes F on N. An orbit space of N by a discrete subgroup of N \rtimes F which acts freely on N is called an infranilmanifold. An infranilmanifold is finitely covered by a nilmanifold. Isometry is a map which preserves distances.
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The second market model assumes that the market only has finitely many possible changes, drawn from a risk factor return sample of a defined historical period. Typically one performs a historical simulation by sampling from past day-on-day risk factor changes, and applying them to the current level of the risk factors to obtain risk factor price scenarios. These perturbed risk factor price scenarios are used to generate a profit (loss) distribution for the portfolio. This method has the advantage of simplicity, but as a model, it is slow to adapt to changing market conditions.
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Then no matter what the choice of system of parameters or lifting, the last map from R = K_d \to F_d is not 0. # Existence of Balanced Big Cohen–Macaulay Modules Conjecture. There exists a (not necessarily finitely generated) R-module W such that mRW ≠ W and every system of parameters for R is a regular sequence on W. # Cohen-Macaulayness of Direct Summands Conjecture. If R is a direct summand of a regular ring S as an R-module, then R is Cohen–Macaulay (R need not be local, but the result reduces at once to the case where R is local).
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As von Neumann notes:On p. 85. :"Infolgedessen gibt es bereits in der Ebene kein nichtnegatives additives Maß (wo das Einheitsquadrat das Maß 1 hat), das gegenüber allen Abbildungen von A2 invariant wäre." :"In accordance with this, already in the plane there is no non-negative additive measure (for which the unit square has a measure of 1), which is invariant with respect to all transformations belonging to A2 [the group of area-preserving affine transformations]." To explain further, the question of whether a finitely additive measure (that is preserved under certain transformations) exists or not depends on what transformations are allowed.
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An arbitrary group G is called free if it is isomorphic to FS for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in one and only one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as st = suu−1t). A related but different notion is a free abelian group; both notions are particular instances of a free object from universal algebra. As such, free groups are defined by their universal property.
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To be able of solving the syzygy problem, it is necessary that the module of syzygies is finitely generated, because it is impossible to output an infinite list. Therefore, the problems considered here make sense only for Noetherian rings, or at least a coherent ring. In fact, this article is restricted to Noetherian integral domains because of the following result. Given a Noetherian integral domain, if there are algorithms to solve the ideal membership problem and the syzygies problem for a single equation, then one may deduce from them algorithms for the similar problems concerning systems of equations.
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A major reason for the notion of a good cover is that the Leray spectral sequence of a fiber bundle degenerates for a good cover, and so the Čech cohomology associated with a good cover is the same as the Čech cohomology of the space. (Such a cover is known as a Leray cover.) However, for the purposes of computing the Čech cohomology it suffices to have a more relaxed definition of a good cover in which all intersections of finitely many open sets have contractible connected components. This follows from the fact that higher derived functors can be computed using acyclic resolutions.
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Dirichlet's unit theorem provides a description of the structure of the multiplicative group of units O× of the ring of integers O. Specifically, it states that O× is isomorphic to G × Zr, where G is the finite cyclic group consisting of all the roots of unity in O, and r = r1 + r2 − 1 (where r1 (respectively, r2) denotes the number of real embeddings (respectively, pairs of conjugate non- real embeddings) of K). In other words, O× is a finitely generated abelian group of rank r1 + r2 − 1 whose torsion consists of the roots of unity in O.
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In 2D Euclidean field theory, the operator product expansion is a Laurent series expansion associated to two operators. A Laurent series is a generalization of the Taylor series in that finitely many powers of the inverse of the expansion variable(s) are added to the Taylor series: pole(s) of finite order(s) are added to the series. Heuristically, in quantum field theory one is interested in the result of physical observables represented by operators. If one wants to know the result of making two physical observations at two points z and w, one can time order these operators in increasing time.
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If R is a discrete valuation ring with quotient field K then the Matlis module is K/R. In the special case when R is the ring of p-adic numbers, the Matlis dual of a finitely-generated module is the Pontryagin dual of it considered as a locally compact abelian group. If R is a Cohen–Macaulay local ring of dimension d with dualizing module Ω, then the Matlis module is given by the local cohomology group H(Ω). In particular if R is an Artinian local ring then the Matlis module is the same as the dualizing module.
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In other words, ZFC cannot be finitely axiomatized. He pioneered a logical approach to natural language semantics which became known as Montague grammar. This approach to language has been especially influential among certain computational linguists—perhaps more so than among more traditional philosophers of language. In particular, Montague's influence lives on in grammar approaches like categorial grammar (such as Unification Categorial Grammar, Left-Associate Grammar, or Combinatory Categorial Grammar), which attempt a derivation of syntactic and semantic representation in tandem and the semantics of quantifiers, scope and discourse (Hans Kamp, a student of Montague, co-developed Discourse Representation Theory).
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In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted F \subseteq T \subseteq V, which were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the von Neumann conjecture. Of the three, F is the most widely studied, and is sometimes referred to as the Thompson group or Thompson's group. The Thompson groups, and F in particular, have a collection of unusual properties which have made them counterexamples to many general conjectures in group theory. All three Thompson groups are infinite but finitely presented.
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In mathematics, Siegel's theorem on integral points states that for a smooth algebraic curve C of genus g defined over a number field K, presented in affine space in a given coordinate system, there are only finitely many points on C with coordinates in the ring of integers O of K, provided g > 0. The theorem was first proved in 1929 by Carl Ludwig Siegel and was the first major result on Diophantine equations that depended only on the genus and not any special algebraic form of the equations. For g > 1 it was superseded by Faltings's theorem in 1983.
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The null homotopic class acts as the identity of the group addition, and for equal to (for positive ) — the homotopy groups of spheres — the groups are abelian and finitely generated. If for some all maps are null homotopic, then the group consists of one element, and is called the trivial group. A continuous map between two topological spaces induces a group homomorphism between the associated homotopy groups. In particular, if the map is a continuous bijection (a homeomorphism), so that the two spaces have the same topology, then their -th homotopy groups are isomorphic for all .
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"Put differently, I sought a world philosophy. I sought an integral philosophy, one that would believably weave together the many pluralistic contexts of science, morals, aesthetics, Eastern as well as Western philosophy, and the world's great wisdom traditions. Not on the level of details--that is finitely impossible; but on the level of orienting generalizations: a way to suggest that the world is one, undivided whole, and related to itself in every way: a holistic philosophy for a holistic Kosmos: a world philosophy, an integral philosophy." -- Ken Wilber, "Introduction to Volume Six of the Collected Works".
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The term was coined in 2004 by Ralf Aurich, Sven Lustig, Frank Steiner, and Holger Then in their paper Hyperbolic Universes with a Horned Topology and the CMB Anisotropy. The model was chosen in an attempt to describe the microwave background radiation apparent in the universe, and has finite volume and useful spectral characteristics (the first several eigenvalues of the Laplacian are computed and in good accord with observation). In this model one end of the figure curves finitely into the bell of the horn. The curve along any side of horn is considered to be a negative curve.
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One had the remarkable extensions of Clifford theory by Green to the indecomposable modules of group algebras. During this era, the field of computational group theory became a recognized field of study, due in part to its tremendous success during the first generation classification. In discrete groups, the geometric methods of Jacques Tits and the availability the surjectivity of Serge Lang's map allowed a revolution in algebraic groups. The Burnside problem had tremendous progress, with better counterexamples constructed in the 1960s and early 1980s, but the finishing touches "for all but finitely many" were not completed until the 1990s.
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In mathematics, especially in the field of ring theory, a (right) free ideal ring, or fir, is a ring in which all right ideals are free modules with unique rank. A ring such that all right ideals with at most n generators are free and have unique rank is called an n-fir. A semifir is a ring in which all finitely generated right ideals are free modules of unique rank. (Thus, a ring is semifir if it is n-fir for all n ≥ 0.) The semifir property is left-right symmetric, but the fir property is not.
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Any product of (even infinitely many) injective modules is injective; conversely, if a direct product of modules is injective, then each module is injective . Every direct sum of finitely many injective modules is injective. In general, submodules, factor modules, or infinite direct sums of injective modules need not be injective. Every submodule of every injective module is injective if and only if the ring is Artinian semisimple ; every factor module of every injective module is injective if and only if the ring is hereditary, ; every infinite direct sum of injective modules is injective if and only if the ring is Noetherian, .
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At the same time, considerations of construction of such an element receded: the theorem becomes an existence theorem. The following theorem of Artin then takes the place of the classical primitive element theorem. ;Theorem Let E/F be a finite degree field extension. Then E=F(\alpha) for some element \alpha\in E if and only if there exist only finitely many intermediate fields K with E\supseteq K\supseteq F. A corollary to the theorem is then the primitive element theorem in the more traditional sense (where separability was usually tacitly assumed): ;Corollary Let E/F be a finite degree separable extension.
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For an R-module A, a maximal submodule M of A is a submodule M≠A satisfying the property that for any other submodule N, M⊆N⊆A implies N=M or N=A. Equivalently, M is a maximal submodule if and only if the quotient module A/M is a simple module. The maximal right ideals of a ring R are exactly the maximal submodules of the module RR. Unlike rings with unity, a nonzero module does not necessarily have maximal submodules. However, as noted above, finitely generated nonzero modules have maximal submodules, and also projective modules have maximal submodules.
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The notion of a Dehn function in geometric group theory, which estimates the area of a relation in a finitely presented group in terms of the length of that relation, is also named after him. In 1914 he proved that the left and right trefoil knots are not equivalent. In the early 1920s Dehn introduced the result that would come to be known as the Dehn-Nielsen theorem; its proof would be published in 1927 by Jakob Nielsen. In 1922 Dehn succeeded Ludwig Bieberbach at Frankfurt, where he stayed until he was forced to retire in 1935.
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Any configuration with only finitely many nonzero cells becomes a replicator that eventually fills the array with copies of itself. When Rule 90 is started from a random initial configuration, its configuration remains random at each time step. Its time- space diagram forms many triangular "windows" of different sizes, patterns that form when a consecutive row of cells becomes simultaneously zero and then cells with value 1 gradually move into this row from both ends. Some of the earliest studies of Rule 90 were made in connection with an unsolved problem in number theory, Gilbreath's conjecture, on the differences of consecutive prime numbers.
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For instance, the continued fraction representation of is [1;2,4] and its two children are [1;2,5] = (the right child) and [1;2,3,2] = (the left child). It is clear that for each finite continued fraction expression one can repeatedly move to its parent, and reach the root [1;]= of the tree in finitely many steps (in steps to be precise). Therefore every positive rational number appears exactly once in this tree. Moreover all descendants of the left child of any number q are less than q, and all descendants of the right child of q are greater than q.
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Metaphysical nihilism is the philosophical theory that there might have been no objects at all—that is, that there is a possible world in which there are no objects at all; or at least that there might have been no concrete objects at all, so that even if every possible world contains some objects, there is at least one that contains only abstract objects. To understand metaphysical nihilism, one can look to the subtraction theory in its simplest form, proposed by Thomas Baldwin. #There could have been finitely many things. #For each thing, that thing might not have existed.
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From the definition of vector spaces, it follows that subspaces are nonempty, and are closed under sums and under scalar multiples. Equivalently, subspaces can be characterized by the property of being closed under linear combinations. That is, a nonempty set W is a subspace if and only if every linear combination of finitely many elements of W also belongs to W. The equivalent definition states that it is also equivalent to consider linear combinations of two elements at a time. In a topological vector space X, a subspace W need not be topologically closed, but a finite-dimensional subspace is always closed.
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A Schottky group is called classical if all the disjoint Jordan curves corresponding to some set of generators can be chosen to be circles. gave an indirect and non-constructive proof of the existence of non-classical Schottky groups, and gave an explicit example of one. It has been shown by that all finitely generated classical Schottky groups have limit sets of Hausdorff dimension bounded above strictly by a universal constant less than 2. Conversely, has proved that there exists a universal lower bound on the Hausdorff dimension of limit sets of all non- classical Schottky groups.
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A particularly simple case of the word problem for groups and the isomorphism problem for groups asks if a finitely presented group is the trivial group. This is known to be intractable in general, even though there is a finite sequence of elementary Tietze transformations taking the presentation to the trivial presentation if and only if the group is trivial. A special case is that of "balanced presentations", those finite presentations with equal numbers of generators and relators. For these groups, there is a conjecture that the required transformations are quite a bit simpler (in particular, do not involve adding or removing relators).
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If one allows taking the set of relators to any Nielsen equivalent set, and one allows conjugating the relators, then one gets an equivalence relation on ordered subsets of a relators of a finitely presented group. The Andrews–Curtis conjecture is that the relators of any balanced presentation of the trivial group are equivalent to a set of trivial relators, stating that each generator is the identity element. In the textbook , an application of Nielsen transformations is given to solve the generalized word problem for free groups, also known as the membership problem for subgroups given by finite generating sets in free groups.
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In 1968 Parshin proved a special case (for = the empty set) of the following theorem: If is a smooth complex curve and is a finite subset of then there exist only finitely many families (up to isomorphism) of smooth curves of fixed genus g ≥ 2 over . The general case (for non-empty ) of the preceding theorem was proved by Arakelov. At the same time, Parshin gave a new proof (without an application of the Shafarevich finiteness condition) of the Mordell conjecture in function fields (already proved by Yuri Manin in 1963 and by Hans Grauert in 1965).Parshin, Algebraic curves over function fields.
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Thus, the Gaussian moat problem may be phrased in a different but equivalent form: is there a finite bound on the widths of the moats that have finitely many primes on the side of the origin? Computational searches have shown that the origin is separated from infinity by a moat of width 6.. It is known that, for any positive number k, there exist Gaussian primes whose nearest neighbor is at distance k or larger. In fact, these numbers may be constrained to be on the real axis. For instance, the number 20785207 is surrounded by a moat of width 17.
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Addition, denoted by the symbol +, is the most basic operation of arithmetic. In its simple form, addition combines two numbers, the addends or terms, into a single number, the sum of the numbers (such as or ). Adding finitely many numbers can be viewed as repeated simple addition; this procedure is known as summation, a term also used to denote the definition for "adding infinitely many numbers" in an infinite series. Repeated addition of the number 1 is the most basic form of counting; the result of adding is usually called the successor of the original number.
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A metric space M is compact if every sequence in M has a subsequence that converges to a point in M. This is known as sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of countable compactness and compactness defined via open covers. Examples of compact metric spaces include the closed interval [0,1] with the absolute value metric, all metric spaces with finitely many points, and the Cantor set. Every closed subset of a compact space is itself compact. A metric space is compact if and only if it is complete and totally bounded.
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For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals for all integers ; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line.
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For if such a function f is nonconstant, then since the set of z where f(z) is infinity is isolated and the Riemann sphere is compact, there are finitely many z with f(z) equal to infinity. Consider the Laurent expansion at all such z and subtract off the singular part: we are left with a function on the Riemann sphere with values in C, which by Liouville's theorem is constant. Thus f is a rational function. This fact shows there is no essential difference between the complex projective line as an algebraic variety, or as the Riemann sphere.
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If the graph is countable, the vertices are well-ordered and one can canonically choose the smallest suitable vertex. In this case, Kőnig's lemma is provable in second-order arithmetic with arithmetical comprehension, and, a fortiori, in ZF set theory (without choice). Kőnig's lemma is essentially the restriction of the axiom of dependent choice to entire relations R such that for each x there are only finitely many z such that xRz. Although the axiom of choice is, in general, stronger than the principle of dependent choice, this restriction of dependent choice is equivalent to a restriction of the axiom of choice.
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An elliptic surface is a surface equipped with an elliptic fibration (a surjective holomorphic map to a curve B such that all but finitely many fibers are smooth irreducible curves of genus 1). The generic fiber in such a fibration is a genus 1 curve over the function field of B. Conversely, given a genus 1 curve over the function field of a curve, its relative minimal model is an elliptic surface. Kodaira and others have given a fairly complete description of all elliptic surfaces. In particular, Kodaira gave a complete list of the possible singular fibers.
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This is consistent with the known results on rational points, notably Faltings's theorem on subvarieties of abelian varieties. More precisely, let X be a projective variety of general type over a number field k. Let the exceptional set Y be the Zariski closure of the union of the images of all nonconstant holomorphic maps C → X. According to the Green–Griffiths–Lang conjecture, Y should be not equal to X. The strong Lang conjecture predicts that Y is defined over k and that X − Y has only finitely many F-rational points for every finite extension field F of k.Lang (1986), Conjecture 5.8.
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In the proof of the pumping lemma for regular languages, a version that mixes finite and infinite sets is used: If infinitely many objects are placed into finitely many boxes, then there exist two objects that share a box.Introduction to Formal Languages and Automata, Peter Linz, pp. 115–116, Jones and Bartlett Learning, 2006 In Fisk's solution of the Art gallery problem a sort of converse is used: If n objects are placed into k boxes, then there is a box containing at most n/k objects.Computational Geometry in C, Cambridge Tracts in Theoretical Computer Science, 2nd Edition, Joseph O'Rourke, page 9.
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An algebra of sets needs only to be closed under the union or intersection of finitely many subsets, which is a weaker condition. The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for which a given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as the foundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events which can be assigned probabilities. Also, in probability, σ-algebras are pivotal in the definition of conditional expectation.
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Let H, K ≤ F(X) be two nontrivial finitely generated subgroups of a free group F(X) and let L = H ∩ K be the intersection of H and K. The conjecture says that in this case :rank(L) − 1 ≤ (rank(H) − 1)(rank(K) − 1). Here for a group G the quantity rank(G) is the rank of G, that is, the smallest size of a generating set for G. Every subgroup of a free group is known to be free itself and the rank of a free group is equal to the size of any free basis of that free group.
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One application is the Gromov–Lees Theorem, named for him and Jack Alexander Lees, concerning Lagrangian immersions and a one-to-one correspondence between the connected components of spaces. In 1978, Gromov introduced the notion of almost flat manifolds. The famous quarter-pinched sphere theorem in Riemannian geometry says that if a complete Riemannian manifold has sectional curvatures which are all sufficiently close to a given positive constant, then must be finitely covered by a sphere. In contrast, it can be seen by scaling that every closed Riemannian manifold has Riemannian metrics whose sectional curvatures are arbitrarily close to zero.
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A function f : Rn→R is lower semicontinuous if and only if its epigraph (the set of points lying on or above its graph) is closed. A function f : X→R, for some topological space X, is lower semicontinuous if and only if it is continuous with respect to the Scott topology on R. Any upper semicontinuous function f : X→N on an arbitrary topological space X is locally constant on some dense open subset of X. The maximum and minimum of finitely many upper semicontinuous functions is upper semicontinuous, and the same holds true of lower semicontinuous functions.
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The bisected hexagonal tiling. The dual graph of a simple line arrangement may be represented geometrically as a collection of rhombi, one per vertex of the arrangement, with sides perpendicular to the lines that meet at that vertex. These rhombi may be joined together to form a tiling of a convex polygon in the case of an arrangement of finitely many lines, or of the entire plane in the case of a locally finite arrangement with infinitely many lines. investigated special cases of this construction in which the line arrangement consists of k sets of equally spaced parallel lines.
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For a comparison of alternative definitions of pseudolines, see . A pseudoline arrangement is said to be stretchable if it is combinatorially equivalent to a line arrangement; it is complete for the existential theory of the reals to distinguish stretchable arrangements from non-stretchable ones.; . Every arrangement of finitely many pseudolines can be extended so that they become lines in a "spread", a type of non-Euclidean incidence geometry in which every two points of a topological plane are connected by a unique line (as in the Euclidean plane) but in which other axioms of Euclidean geometry may not apply.
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The Bers area inequality is a quantitative refinement of the Ahlfors finiteness theorem proved by . It states that if Γ is a non-elementary finitely-generated Kleinian group with N generators and with region of discontinuity Ω, then :Area(Ω/Γ) ≤ 4π(N − 1) with equality only for Schottky groups. (The area is given by the Poincaré metric in each component.) Moreover, if Ω1 is an invariant component then :Area(Ω/Γ) ≤ 2Area(Ω1/Γ) with equality only for Fuchsian groups of the first kind (so in particular there can be at most two invariant components).
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The counterexamples 7^3 + 13^2 = 2^9 and 1^m + 2^3 = 3^2 show that the conjecture would be false if one of the exponents were allowed to be 2. The Fermat–Catalan conjecture is an open conjecture dealing with such cases. If we allow at most one of the exponents to be 2, then there may be only finitely many solutions (except the case 1^m + 2^3 = 3^2). If A, B, C can have a common prime factor then the conjecture is not true; a classic counterexample is 2^{10} + 2^{10} = 2^{11}.
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Aristotle responded to these paradoxes by developing the notion of a potential countable infinity, as well as the infinitely divisible continuum. Unlike the eternal and unchanging cycles of time, he believed that the world is bounded by the celestial spheres and that cumulative stellar magnitude is only finitely multiplicative. The Indian philosopher Kanada, founder of the Vaisheshika school, developed a notion of atomism and proposed that light and heat were varieties of the same substance.Will Durant, Our Oriental Heritage: In the 5th century AD, the Buddhist atomist philosopher Dignāga proposed atoms to be point-sized, durationless, and made of energy.
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Each formal system uses primitive symbols (which collectively form an alphabet) to finitely construct a formal language from a set of axioms through inferential rules of formation. The system thus consists of valid formulas built up through finite combinations of the primitive symbols—combinations that are formed from the axioms in accordance with the stated rules.Encyclopædia Britannica, Formal system definition, 2007. More formally, this can be expressed as the following: # A finite set of symbols, known as the alphabet, which concatenate formulas, so that a formula is just a finite string of symbols taken from the alphabet.
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The Grothendieck group G_0(\Z) is an abelian group generated by symbols [A] for any finitely generated abelian groups A. One first notes that any finite abelian group G satisfies that [G] = 0. The following short exact sequence holds, where the map \Z \to \Z is multiplication by n. : 0 \to \Z \to \Z \to \Z /n\Z \to 0 The exact sequence implies that [\Z /n\Z] = [\Z] - [\Z] = 0, so every cyclic group has its symbol equal to 0. This in turn implies that every finite abelian group G satisfies [G] = 0 by the Fundamental Theorem of Finite Abelian groups.
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In his 1936 paper "Finite Combinatory Processes--Formulation 1", Emil Post described a model of which he conjectured is "logically equivalent to recursiveness". Post's model of a computation differs from the Turing-machine model in a further "atomization" of the acts a human "computer" would perform during a computation. Post's model employs a "symbol space" consisting of a "two-way infinite sequence of spaces or boxes", each box capable of being in either of two possible conditions, namely "marked" (as by a single vertical stroke) and "unmarked" (empty). Initially, finitely-many of the boxes are marked, the rest being unmarked.
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Each Wythoff pair occurs exactly once in the Wythoff array, as a consecutive pair of numbers in the same row, with an odd index for the first number and an even index for the second. Because each positive integer occurs in exactly one Wythoff pair, each positive integer occurs exactly once in the array . Every sequence of positive integers satisfying the Fibonacci recurrence occurs, shifted by at most finitely many positions, in the Wythoff array. In particular, the Fibonacci sequence itself is the first row, and the sequence of Lucas numbers appears in shifted form in the second row .
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A smooth connected curve over an algebraically closed field is called hyperbolic if 2g-2+r>0 where g is the genus of the smooth completion and r is the number of added points. Over an algebraically closed field of characteristic 0, the fundamental group of X is free with 2g+r-1 generators if r>0. (Analogue of Dirichlet's unit theorem) Let X be a smooth connected curve over a finite field. Then the units of the ring of regular functions O(X) on X is a finitely generated abelian group of rank r -1.
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Intuitively, a signal is said to be finitely variable, or to have the finite variability property, if during each bounded interval, the letter change a finite number of time. In our previous elevator example, this property would means that a user may only press a button a finite number of time during a finite time. And similarly, in a finite time, the elevator can only opens and close its door a finite number of time. Formally, a signal is said to have the finite variability property, unless the sequence is infinite and \bigcup_i I_i is bounded.
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In geometric group theory, groups are studied by their actions on metric spaces. A principle that generalizes the bilipschitz invariance of word metrics says that any finitely generated word metric on G is quasi-isometric to any proper, geodesic metric space on which G acts, properly discontinuously and cocompactly. Metric spaces on which G acts in this manner are called model spaces for G. It follows in turn that any quasi-isometrically invariant property satisfied by the word metric of G or by any model space of G is an isomorphism invariant of G. Modern geometric group theory is in large part the study of quasi-isometry invariants.
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In papers in 1977 and 1978, Barry Mazur proved the torsion conjecture giving a complete list of the possible torsion subgroups of elliptic curves over the rational numbers. Mazur's first proof of this theorem depended upon a complete analysis of the rational points on certain modular curves. In 1996, the proof of the torsion conjecture was extended to all number fields by Loïc Merel. In 1983, Gerd Faltings proved the Mordell conjecture, demonstrating that a curve of genus greater than 1 has only finitely many rational points (where the Mordell–Weil theorem only demonstrates finite generation of the set of rational points as opposed to finiteness).
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The existence of multiplicative inverses in fields is not involved in the axioms defining a vector space. One may thus replace the field of scalars by a ring , and this gives a structure called module over , or -module. The concepts of linear independence, span, basis, and linear maps (also called module homomorphisms) are defined for modules exactly as for vector spaces, with the essential difference that, if is not a field, there are modules that do not have any basis. The modules that have a basis are the free modules, and those that are spanned by a finite set are the finitely generated modules.
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Suppose all agents have piecewise-constant valuations. This means that, for each agent, the cake is partitioned into finitely many subsets, and the agent's value density in each subset is constant. For this case, Aziz and Ye present a randomized algorithm that is more economically-efficient: Constrained Serial Dictatorship is truthful in expectation, robust proportional, and satisfies a property called unanimity: if each agent's most preferred 1/n length of the cake is disjoint from other agents, then each agent gets their most preferred 1/n length of the cake. This is a weak form of efficiency that is not satisfied by the mechanisms based on exact division.
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The surface of an ideal polyhedron (not including its vertices) forms a manifold, topologically equivalent to a punctured sphere, with a uniform two-dimensional hyperbolic geometry; the folds of the surface in its embedding into hyperbolic space are not detectable as folds in the intrinsic geometry of the surface. Because this surface can be partitioned into ideal triangles, its total area is finite. Conversely, and analogously to Alexandrov's uniqueness theorem, every two- dimensional manifold with uniform hyperbolic geometry and finite area, combinatorially equivalent to a finitely-punctured sphere, can be realized as the surface of an ideal polyhedron. (As with Alexandrov's theorem, such surfaces must be allowed to include ideal dihedra.); .
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While a graduate student at the University of Chicago, Dubins surprised his teacher Leonard Jimmie Savage with a mathematical demonstration that this is not true. Dubins and Savage wrote a book that appeared in 1965 titled How to Gamble if You Must (Inequalities for Stochastic Processes) which presented a mathematical theory of gambling processes and optimal behavior in gambling situations, pointing out their relevance to traditional approaches to probability. Under the influence of the work of Bruno de Finetti, Dubins and Savage worked in the context of finitely additive rather than countably additive probability theory, thereby bypassing some technical difficulties. Dubins was the author of nearly a hundred scholarly publications.
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It was spurred by the 1987 monograph of Mikhail Gromov "Hyperbolic groups"Mikhail Gromov, Hyperbolic Groups, in "Essays in Group Theory" (Steve M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263. that introduced the notion of a hyperbolic group (also known as word-hyperbolic or Gromov-hyperbolic or negatively curved group), which captures the idea of a finitely generated group having large-scale negative curvature, and by his subsequent monograph Asymptotic Invariants of Infinite Groups,Mikhail Gromov, "Asymptotic invariants of infinite groups", in "Geometric Group Theory", Vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, 182, Cambridge University Press, Cambridge, 1993, pp. 1–295.
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Going around the loop twice gets you to , three times to , all lifts of the identity. But there are only finitely many lifts of the identity, because the lifts can't accumulate. This number of times one has to traverse the loop to make it contractible is small, for example if the GUT group is SO(3), the covering group is SU(2), and going around any loop twice is enough. # This means that there is a continuous gauge-field configuration in the GUT group allows the U(1) monopole configuration to unwind itself at short distances, at the cost of not staying in the U(1).
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He wrote a number of papers of fundamental importance in group theory, including his solution of Burnside's problem for groups of exponent 6, showing that a finitely generated group in which the order of every element divides 6 must be finite. His work in combinatorics includes an important paper of 1943 on projective planes, which for many years was one of the most cited mathematics research papers.Mathematics Department of Ohio State University Marshall Hall Jr. via Wayback machine In this paper he constructed a family of non-Desarguesian planes which are known today as Hall planes. He also worked on block designs and coding theory.
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Although Lindström had only partially developed the hierarchy of quantifiers which now bear his name, it was enough for him to observe that some nice properties of first-order logic are lost when it is extended with certain generalized quantifiers. For example, adding a "there exist finitely many" quantifier results in a loss of compactness, whereas adding a "there exist uncountably many" quantifier to first-order logic results in a logic no longer satisfying the Löwenheim–Skolem theorem. In 1969 Lindström proved a much stronger result now known as Lindström's theorem, which intuitively states that first-order logic is the "strongest" logic having both properties.
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A Voronoi diagram (red) and Delaunay triangulation (black) of a finite point set (the black points) The concept of a dual tessellation can also be applied to partitions of the plane into finitely many regions. It is closely related to but not quite the same as planar graph duality in this case. For instance, the Voronoi diagram of a finite set of point sites is a partition of the plane into polygons within which one site is closer than any other. The sites on the convex hull of the input give rise to unbounded Voronoi polygons, two of whose sides are infinite rays rather than finite line segments.
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The classification of complex manifolds is much more subtle than that of differentiable manifolds. For example, while in dimensions other than four, a given topological manifold has at most finitely many smooth structures, a topological manifold supporting a complex structure can and often does support uncountably many complex structures. Riemann surfaces, two dimensional manifolds equipped with a complex structure, which are topologically classified by the genus, are an important example of this phenomenon. The set of complex structures on a given orientable surface, modulo biholomorphic equivalence, itself forms a complex algebraic variety called a moduli space, the structure of which remains an area of active research.
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In mathematics, the Smith normal form is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can be obtained from the original matrix by multiplying on the left and right by invertible square matrices. In particular, the integers are a PID, so one can always calculate the Smith normal form of an integer matrix. The Smith normal form is very useful for working with finitely generated modules over a PID, and in particular for deducing the structure of a quotient of a free module.
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The first few 3-smooth numbers, 1, 2, 3, 4, 6, 8, 9, and 12, are all losing openings, for which complete strategies are known by which the second player can win. By Dickson's lemma (applied to the pairs of exponents of these numbers), only finitely many 3-smooth numbers can be winning openings, but it is not known whether any of them are. offered a $1000 prize for determining who wins in the first unsolved case, the opening move 16, as part of a set of prize problems also including Conway's 99-graph problem, the minimum spacing of Danzer sets, and the thrackle conjecture.
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Then there are only finitely many isomorphism types of indecomposable modules in the block, and the structure of the block is by now well understood, by virtue of work of Brauer, E.C. Dade, J.A. Green and J.G. Thompson, among others. In all other cases, there are infinitely many isomorphism types of indecomposable modules in the block. Blocks whose defect groups are not cyclic can be divided into two types: tame and wild. The tame blocks (which only occur for the prime 2) have as a defect group a dihedral group, semidihedral group or (generalized) quaternion group, and their structure has been broadly determined in a series of papers by Karin Erdmann.
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It is natural to ask how a risk-neutral measure arises in a market free of arbitrage. Somehow the prices of all assets will determine a probability measure. One explanation is given by utilizing the Arrow security. For simplicity, consider a discrete (even finite) world with only one future time horizon. In other words, there is the present (time 0) and the future (time 1), and at time 1 the state of the world can be one of finitely many states. An Arrow security corresponding to state n, An, is one which pays $1 at time 1 in state n and $0 in any of the other states of the world.
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More generally, let be a morphism of degree at least two defined over a number field . Northcott's theorem says that has only finitely many preperiodic points in , and the general Uniform Boundedness Conjecture says that the number of preperiodic points in may be bounded solely in terms of , the degree of , and the degree of over . The Uniform Boundedness Conjecture is not known even for quadratic polynomials over the rational numbers . It is known in this case that cannot have periodic points of period four, five, or six, although the result for period six is contingent on the validity of the conjecture of Birch and Swinnerton-Dyer.
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A symmetry of a Euclidean graph is an isometry of the underlying Euclidean space whose restriction to the graph is an automorphism; the symmetry group of the Euclidean graph is the group of its symmetries. A Euclidean graph in three-dimensional Euclidean space is periodic if there exist three linearly independent translations whose restrictions to the net are symmetries of the net. Often (and always, if one is dealing with a crystal net), the periodic net has finitely many orbits, and is thus uniformly discrete in that there exists a minimum distance between any two vertices. The result is a three- dimensional periodic graph as a geometric object.
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In mathematics, the Neukirch–Uchida theorem shows that all problems about algebraic number fields can be reduced to problems about their absolute Galois groups. showed that two algebraic number fields with the same absolute Galois group are isomorphic, and strengthened this by proving Neukirch's conjecture that automorphisms of the algebraic number field correspond to outer automorphisms of its absolute Galois group. extended the result to infinite fields that are finitely generated over prime fields. The Neukirch–Uchida theorem is one of the foundational results of anabelian geometry, whose main theme is to reduce properties of geometric objects to properties of their fundamental groups, provided these fundamental groups are sufficiently non- abelian.
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A maximal planar graph G is a finite simple planar graph to which no more edges can be added while preserving planarity. Such a graph always has a unique planar embedding, in which every face of the embedding (including the outer face) is a triangle. In other words, every maximal planar graph G is the 1-skeleton of a simplicial complex which is homeomorphic to the sphere. The circle packing theorem guarantees the existence of a circle packing with finitely many circles whose intersection graph is isomorphic to G. As the following theorem states more formally, every maximal planar graph can have at most one packing.
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In algebra, one often studies infinite algebras which are generated (under the finitary operations of the algebra) by finitely many elements. In this case, the elements of the algebra have a natural system of finite encoding as expressions in terms of the generators and operations. The word problem here is thus to determine, given two such expressions, whether they represent the same element of the algebra. Roughly speaking, the word problem in an algebra is: given a set E of identities (an equational theory), and two terms s and t, is it possible to transform s into t using the identities in E as rewriting rules in both directions?.
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The second Prüfer theorem provides a straightforward extension of the fundamental theorem of finitely generated abelian groups to countable abelian p-groups without elements of infinite height: each such group is isomorphic to a direct sum of cyclic groups whose orders are powers of p. Moreover, the cardinality of the set of summands of order pn is uniquely determined by the group and each sequence of at most countable cardinalities is realized. Helmut Ulm (1933) found an extension of this classification theory to general countable p-groups: their isomorphism class is determined by the isomorphism classes of the Ulm factors and the p-divisible part. : Ulm's theorem.
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Given the geometric approach, the consideration of homogeneous equations and homogeneous co-ordinates is fundamental, for the same reasons that projective geometry is the dominant approach in algebraic geometry. Rational number solutions therefore are the primary consideration; but integral solutions (i.e. lattice points) can be treated in the same way as an affine variety may be considered inside a projective variety that has extra points at infinity. The general approach of Diophantine geometry is illustrated by Faltings's theorem (a conjecture of L. J. Mordell) stating that an algebraic curve C of genus g > 1 over the rational numbers has only finitely many rational points.
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In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.
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By the Robertson–Seymour theorem, because they form a minor-closed family of graphs, the apex graphs have a forbidden graph characterization. There are only finitely many graphs that are neither apex graphs nor have another non-apex graph as a minor. These graphs are forbidden minors for the property of being an apex graph. Any other graph G is an apex graph if and only if none of the forbidden minors is a minor of G. These forbidden minors include the seven graphs of the Petersen family, three disconnected graphs formed from the disjoint unions of two of K5 and K3,3, and many other graphs.
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Property FA is equivalent for countable G to the three properties: G is not an amalgamated product; G does not have Z as a quotient group; G is finitely generated. For general groups G the third condition may be replaced by requiring that G not be the union of a strictly increasing sequence of subgroup. Examples of groups with property FA include SL3(Z) and more generally G(Z) where G is a simply-connected simple Chevalley group of rank at least 2. The group SL2(Z) is an exception, since it is isomorphic to the amalgamated product of the cyclic groups C4 and C6 along C2.
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In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive invariant measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "mean".Day's first published use of the word is in his abstract for an AMS summer meeting in 1949, Means on semigroups and groups, Bull.
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A group is called virtually cyclic if it contains a cyclic subgroup of finite index (the number of cosets that the subgroup has). In other words, any element in a virtually cyclic group can be arrived at by applying a member of the cyclic subgroup to a member in a certain finite set. Every cyclic group is virtually cyclic, as is every finite group. An infinite group is virtually cyclic if and only if it is finitely generated and has exactly two ends; an example of such a group is the direct product of Z/nZ and Z, in which the factor Z has finite index n.
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In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n > 0\. For example, in the ring of integers Z, (pn) is a primary ideal if p is a prime number. The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has a primary decomposition, that is, can be written as an intersection of finitely many primary ideals. This result is known as the Lasker–Noether theorem.
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Nevertheless, the projective viewpoint allows certain configurations to be described more easily. In particular, it allows the use of projective duality, in which the roles of points and lines in statements of projective geometry can be exchanged for each other. Under projective duality, the existence of an ordinary line for a set of non- collinear points in RP2 is equivalent to the existence of an ordinary point in a nontrivial arrangement of finitely many lines. An arrangement is said to be trivial when all its lines pass through a common point, and nontrivial otherwise; an ordinary point is a point that belongs to exactly two lines.
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In mathematics, a residuated Boolean algebra is a residuated lattice whose lattice structure is that of a Boolean algebra. Examples include Boolean algebras with the monoid taken to be conjunction, the set of all formal languages over a given alphabet Σ under concatenation, the set of all binary relations on a given set X under relational composition, and more generally the power set of any equivalence relation, again under relational composition. The original application was to relation algebras as a finitely axiomatized generalization of the binary relation example, but there exist interesting examples of residuated Boolean algebras that are not relation algebras, such as the language example.
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Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset. For any indexed family of topological spaces, the product can be given the product topology, which is generated by the inverse images of open sets of the factors under the projection mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space.
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The theory must be characterized by a choice of finitely many parameters, which could, in principle, be set by experiment. For example, in quantum electrodynamics these parameters are the charge and mass of the electron, as measured at a particular energy scale. On the other hand, in quantizing gravity there are, in perturbation theory, infinitely many independent parameters (counterterm coefficients) needed to define the theory. For a given choice of those parameters, one could make sense of the theory, but since it is impossible to conduct infinite experiments to fix the values of every parameter, it has been argued that one does not, in perturbation theory, have a meaningful physical theory.
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An end E of a graph G is defined to be a free end if there is a finite set X of vertices with the property that X separates E from all other ends of the graph. (That is, in terms of havens, βE(X) is disjoint from βD(X) for every other end D.) In a graph with finitely many ends, every end must be free. proves that, if G has infinitely many ends, then either there exists an end that is not free, or there exists an infinite family of rays that share a common starting vertex and are otherwise disjoint from each other.
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It follows from the above four equivalent statements that the set of elements of B that are integral over A forms a subring of B containing A. (Proof: If x, y are elements of B that are integral over A, then x + y, xy, -x are integral over A since they stabilize A[x]A[y], which is a finitely generated module over A and is annihilated only by zero.)This proof is due to Dedekind (Milne, ANT). Alternatively, one can use symmetric polynomials to show integral elements form a ring. (loc cit.) This ring is called the integral closure of A in B.
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In general, the integral closure of a noetherian domain of dimension at most 2 is noetherian; Nagata gave an example of dimension 3 noetherian domain whose integral closure is not noetherian. A nicer statement is this: the integral closure of a noetherian domain is a Krull domain (Mori–Nagata theorem). Nagata also gave an example of dimension 1 noetherian local domain such that the integral closure is not finite over that domain. Let A be a noetherian integrally closed domain with field of fractions K. If L/K is a finite separable extension, then the integral closure A' of A in L is a finitely generated A-module.
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This discussion of tensors so far assumes finite dimensionality of the spaces involved, where the spaces of tensors obtained by each of these constructions are naturally isomorphic.The double duality isomorphism, for instance, is used to identify V with the double dual space V∗∗, which consists of multilinear forms of degree one on V∗. It is typical in linear algebra to identify spaces that are naturally isomorphic, treating them as the same space. Constructions of spaces of tensors based on the tensor product and multilinear mappings can be generalized, essentially without modification, to vector bundles or coherent sheaves. where the case of finitely generated projective modules is treated.
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The problem is the following question: can every continuous function of three variables be expressed as a composition of finitely many continuous functions of two variables? The affirmative answer to this general question was given in 1957 by Vladimir Arnold, then only nineteen years old and a student of Andrey Kolmogorov. Kolmogorov had shown in the previous year that any function of several variables can be constructed with a finite number of three-variable functions. Arnold then expanded on this work to show that only two-variable functions were in fact required, thus answering the Hilbert's question when posed for the class of continuous functions.
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In mathematics, a left (or right) quaternionic vector space is a left (or right) H-module where H denotes the noncommutative ring of the quaternions. The space Hn of n-tuples of quaternions is both a left and right H-module using the componentwise left and right multiplication: : q (q_1,q_2,\ldots q_n) = (q q_1,q q_2,\ldots q q_n) : (q_1,q_2,\ldots q_n) q = (q_1 q, q_2 q,\ldots q_n q) for quaternions q and q1, q2, ... qn. Since H is a division algebra, every finitely generated (left or right) H-module has a basis, and hence is isomorphic to Hn for some n.
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In particular if is finitely generated, then all its bases are finite and have the same number of elements. While the proof of the existence of a basis for any vector space in the general case requires Zorn's lemma and is in fact equivalent to the axiom of choice, the uniqueness of the cardinality of the basis requires only the ultrafilter lemma,Howard, P., Rubin, J.: "Consequences of the axiom of choice" - Mathematical Surveys and Monographs, vol 59 (1998) . which is strictly weaker (the proof given below, however, assumes trichotomy, i.e., that all cardinal numbers are comparable, a statement which is also equivalent to the axiom of choice).
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The zeroth algebraic K group K_0(R) of a (not necessarily commutative) ring R is the Grothendieck group of the monoid consisting of isomorphism classes of finitely generated projective modules over R, with the monoid operation given by the direct sum. Then K_0 is a covariant functor from rings to abelian groups. The two previous examples are related: consider the case where R = C^\infty(M) is the ring of complex-valued smooth functions on a compact manifold M. In this case the projective R-modules are dual to vector bundles over M (by the Serre-Swan theorem). Thus K_0(R) and K_0(M) are the same group.
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A scheme is a locally ringed space such that every point has a neighbourhood that, as a locally ringed space, is isomorphic to a spectrum of a ring. Basically, a variety over is a scheme whose structure sheaf is a sheaf of -algebras with the property that the rings R that occur above are all integral domains and are all finitely generated -algebras, that is to say, they are quotients of polynomial algebras by prime ideals. This definition works over any field . It allows you to glue affine varieties (along common open sets) without worrying whether the resulting object can be put into some projective space.
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In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset.Page 46 of The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the rational numbers of dimension rank A. For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup. Torsion-free abelian groups of rank 1 have been completely classified. However, the theory of abelian groups of higher rank is more involved.
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An amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive invariant measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun.Day's first published use of the word is in his abstract for an AMS summer meeting in 1949, Means on semigroups and groups, Bull.
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The infinite general linear group or stable general linear group is the direct limit of the inclusions as the upper left block matrix. It is denoted by either GL(F) or , and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places. It is used in algebraic K-theory to define K1, and over the reals has a well-understood topology, thanks to Bott periodicity. It should not be confused with the space of (bounded) invertible operators on a Hilbert space, which is a larger group, and topologically much simpler, namely contractible - see Kuiper's theorem.
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In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group. Let G be a finitely generated group. Then, for each integer n define a_n(G) to be the number of subgroups H of index n in G. Similarly, if G is a topological group, s_n(G) denotes the number of open subgroups U of index n in G. One similarly defines m_n(G) and s_n^\triangleleft(G) to denote the number of maximal and normal subgroups of index n, respectively. Subgroup growth studies these functions, their interplay, and the characterization of group theoretical properties in terms of these functions.
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Using more steps produces a closer approximation, but will always be too high and will never be exact. Alternatively, replacing these subintervals by ones with the left end height of each piece, we will get an approximation that is too low: for example, with twelve such subintervals we will get an approximate value for the area of 0.6203. The key idea is the transition from adding finitely many differences of approximation points multiplied by their respective function values to using infinitely many fine, or infinitesimal steps. When this transition is completed in the above example, it turns out that the area under the curve within the stated bounds is 2/3.
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Dilworth's theorem states that, in any finite partially ordered set, the largest antichain has the same size as the smallest chain decomposition. Here, the size of the antichain is its number of elements, and the size of the chain decomposition is its number of chains. The width of the partial order is defined as the common size of the antichain and chain decomposition. A version of the theorem for infinite partially ordered sets states that, when there exists a decomposition into finitely many chains, or when there exists a finite upper bound on the size of an antichain, the sizes of the largest antichain and of the smallest chain decomposition are again equal.
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Let X0 be the countable subset given by the finitely many Fn-orbits of the fixed points hi ±∞, the fixed points of the hi and all their conjugates. Since X is uncountable, there is an element of g with fixed points outside X0 and a point w outside X0 different from these fixed points. Then for some subsequence (gm) of (gn) :gm = h1n(m,1) ··· hkn(m,k), with each n(m,i) constant or strictly monotone. On the one hand, by successive use of the rules for computing limits of the form hn·wn, the limit of the right hand side applied to x is necessarily a fixed point of one of the conjugates of the hi's.
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Let Γd denote the Bianchi group PSL(2,Od), where Od is the ring of integers of. As a subgroup of PSL(2,C), there is an action of Γd on hyperbolic 3-space H3, with a fundamental domain. It is a theorem that there are only finitely many values of d for which Γd can contain an arithmetic subgroup G for which the quotient H3/G is a link complement. Zimmert sets are used to obtain results in this direction: z(d) is a lower bound for the rank of the largest free quotient of Γd and so the result above implies that almost all Bianchi groups have non- cyclic free quotients.
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A dualizing module for a Noetherian ring R is a finitely generated module M such that for any maximal ideal m, the R/m vector space vanishes if n ≠ height(m) and is 1-dimensional if n = height(m). A dualizing module need not be unique because the tensor product of any dualizing module with a rank 1 projective module is also a dualizing module. However this is the only way in which the dualizing module fails to be unique: given any two dualizing modules, one is isomorphic to the tensor product of the other with a rank 1 projective module. In particular if the ring is local the dualizing module is unique up to isomorphism.
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In the 1930s, Gottfried Köthe and Keizo Asano introduced the term Einreihig (literally "one-series") during investigations of rings over which all modules are direct sums of cyclic submodules . For this reason, uniserial was used to mean "Artinian principal ideal ring" even as recently as the 1970s. Köthe's paper also required a uniserial ring to have a unique composition series, which not only forces the right and left ideals to be linearly ordered, but also requires that there be only finitely many ideals in the chains of left and right ideals. Because of this historical precedent, some authors include the Artinian condition or finite composition length condition in their definitions of uniserial modules and rings.
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Since the polynomial Q can have only finitely many zeros by the fundamental theorem of algebra, such a rational function will be defined for all sufficiently large x, specifically for all x larger than the largest real root of Q. Adding and multiplying rational functions gives more rational functions, and the quotient rule shows that the derivative of rational function is again a rational function, so R(x) forms a Hardy field. Another example is the field of functions that can be expressed using the standard arithmetic operations, exponents, and logarithms, and are well-defined on some interval of the form (x,\infty). G. H. Hardy, Properties of Logarithmico- Exponential Functions, Proc. London Math. Soc.
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'' Their central idea is that, just as Turing modelled the human computer in 1936 by a Turing machine, they model a technician, performing an experimental procedure that governs an experiment, by a Turing machine. They show that the mathematics of computation imposes fundamental limits on what can be measured in classical physics: ::There is a simple Newtonian experiment to measure mass, based upon colliding particles, for which there are uncountably many masses m such that for every experimental procedure governing the equipment it is only possible to determine finitely many digits of m, even allowing arbitrary long run times for the procedure. In particular, there are uncountably many masses that cannot be measured.
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In most cases, if an analytic continuation of a complex function exists, it is given by an integral formula. The next theorem, provided its hypotheses are met, provides a sufficient condition under which we can continue an analytic function from its convergent points along the positive reals to arbitrary s \in \Complex (with the exception of at finitely-many poles). Moreover, the formula gives an explicit representation for the values of the continuation to the non-positive integers expressed exactly by higher order (integer) derivatives of the original function evaluated at zero.See the article Fontaine's rings and p-adic L-functions by Pierre Colmez found at this link (Course notes PDF dated 2004).
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In mathematics, the genus is a classification of quadratic forms and lattices over the ring of integers. An integral quadratic form is a quadratic form on Zn, or equivalently a free Z-module of finite rank. Two such forms are in the same genus if they are equivalent over the local rings Zp for each prime p and also equivalent over R. Equivalent forms are in the same genus, but the converse does not hold. For example, x2 \+ 82y2 and 2x2 \+ 41y2 are in the same genus but not equivalent over Z. Forms in the same genus have equal discriminant and hence there are only finitely many equivalence classes in a genus.
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Let i1, ..., iN be a finite collection of indices in I. Then the finite product Ai1 × ... × AiN is non-empty (only finitely many choices here, so AC is not needed); it merely consists of N-tuples. Let a = (a1, ..., aN) be such an N-tuple. We extend a to the whole index set: take a to the function f defined by f(j) = ak if j = ik, and f(j) = j otherwise. This step is where the addition of the extra point to each space is crucial, for it allows us to define f for everything outside of the N-tuple in a precise way without choices (we can already choose, by construction, j from Xj ).
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A local Noetherian ring is regular if and only if its global dimension is finite, say n, which means that any finitely generated R-module has a resolution by projective modules of length at most n. The proof of this and other related statements relies on the usage of homological methods, such as the Ext functor. This functor is the derived functor of the functor :HomR(M, −). The latter functor is exact if M is projective, but not otherwise: for a surjective map E -> F of R-modules, a map M -> F need not extend to a map M -> E. The higher Ext functors measure the non-exactness of the Hom-functor.
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Pure subgroups were generalized in several ways in the theory of abelian groups and modules. Pure submodules were defined in a variety of ways, but eventually settled on the modern definition in terms of tensor products or systems of equations; earlier definitions were usually more direct generalizations such as the single equation used above for n'th roots. Pure injective and pure projective modules follow closely from the ideas of Prüfer's 1923 paper. While pure projective modules have not found as many applications as pure injectives, they are more closely related to the original work: A module is pure projective if it is a direct summand of a direct sum of finitely presented modules.
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In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication, and a linear group is an abstract group that is isomorphic to a matrix group over a field K, in other words, admitting a faithful, finite-dimensional representation over K. Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem. Among infinite groups, linear groups form an interesting and tractable class. Examples of groups that are not linear include groups which are "too big" (for example, the group of permutations of an infinite set), or which exhibit some pathological behaviour (for example finitely generated infinite torsion groups).
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Specifically, let be a finitely presented morphism of pointed schemes and M be an OX-module of finite type whose fiber at x is non- zero. Set n equal to the dimension of and r to the codepth of M at s, that is, to .EGA 0IV, Définition 16.4.9 Then there exist affine étale neighborhoods X′ of x and S′ of s, together with points x′ and s′ lifting x and s, such that the residue field extensions and are trivial, the map factors through S′, this factorization sends x′ to s′, and that the pullback of M to X′ admits a total S′-dévissage at x′ in dimensions between n and .
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Note that the foregoing proof of analyticity derived an expression for a system of n different function elements fi(x), provided that x is not a critical point of p(x, y). A critical point is a point where the number of distinct zeros is smaller than the degree of p, and this occurs only where the highest degree term of p vanishes, and where the discriminant vanishes. Hence there are only finitely many such points c1, ..., cm. A close analysis of the properties of the function elements fi near the critical points can be used to show that the monodromy cover is ramified over the critical points (and possibly the point at infinity).
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In this case Σ consists of finitely many points. To a single point we associate a vector space V = Z(point) and to n-points the n-fold tensor product: V⊗n = V ⊗ … ⊗ V. The symmetric group Sn acts on V⊗n. A standard way to get the quantum Hilbert space is to start with a classical symplectic manifold (or phase space) and then quantize it. Let us extend Sn to a compact Lie group G and consider "integrable" orbits for which the symplectic structure comes from a line bundle, then quantization leads to the irreducible representations V of G. This is the physical interpretation of the Borel–Weil theorem or the Borel–Weil–Bott theorem.
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Logics that restrict the t-norm semantics to a subset of the real unit interval (for example, finitely valued Łukasiewicz logics) are usually included in the class as well. Important examples of t-norm fuzzy logics are monoidal t-norm logic MTL of all left-continuous t-norms, basic logic BL of all continuous t-norms, product fuzzy logic of the product t-norm, or the nilpotent minimum logic of the nilpotent minimum t-norm. Some independently motivated logics belong among t-norm fuzzy logics, too, for example Łukasiewicz logic (which is the logic of the Łukasiewicz t-norm) or Gödel–Dummett logic (which is the logic of the minimum t-norm).
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Every elementary abelian p-group is a vector space over the prime field with p elements, and conversely every such vector space is an elementary abelian group. By the classification of finitely generated abelian groups, or by the fact that every vector space has a basis, every finite elementary abelian group must be of the form (Z/pZ)n for n a non-negative integer (sometimes called the group's rank). Here, Z/pZ denotes the cyclic group of order p (or equivalently the integers mod p), and the superscript notation means the n-fold direct product of groups. In general, a (possibly infinite) elementary abelian p-group is a direct sum of cyclic groups of order p.
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His main contributions to those topics and are the papers and . In the first one he proves that a condition on a sequence of integrable functions previously introduced by Mauro Picone is both necessary and sufficient in order to assure that limit process and the integration process commute, both in bounded and unbounded domains: the theorem is similar in spirit to the dominated convergence theorem, which however only states a sufficient condition. The second paper contains an extension of the Lebesgue's decomposition theorem to finitely additive measures: this extension required him to generalize the Radon–Nikodym derivative, requiring it to be a set function belonging to a given class and minimizing a particular functional.
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In mathematics, a commutative ring R is catenary if for any pair of prime ideals :p, q, any two strictly increasing chains :p=p0 ⊂p1 ... ⊂pn= q of prime ideals are contained in maximal strictly increasing chains from p to q of the same (finite) length. In a geometric situation, in which the dimension of an algebraic variety attached to a prime ideal will decrease as the prime ideal becomes bigger, the length of such a chain n is usually the difference in dimensions. A ring is called universally catenary if all finitely generated algebras over it are catenary rings. The word 'catenary' is derived from the Latin word catena, which means "chain".
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An anti-unification algorithm should compute for given expressions a complete, and minimal generalization set, that is, a set covering all generalizations, and containing no redundant members, respectively. Depending on the framework, a complete and minimal generalization set may have one, finitely many, or possibly infinitely many members, or may not exist at all;Complete generalization sets always exist, but it may be the case that every complete generalization set is non-minimal. it cannot be empty, since a trivial generalization exists in any case. For first-order syntactical anti- unification, Gordon Plotkin gave an algorithm that computes a complete and minimal singleton generalization set containing the so-called "least general generalization" (lgg).
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In two dimensions, the Bolyai–Gerwien theorem asserts that any polygon may be transformed into any other polygon of the same area by cutting it up into finitely many polygonal pieces and rearranging them. The analogous question for polyhedra was the subject of Hilbert's third problem. Max Dehn solved this problem by showing that, unlike in the 2-D case, there exist polyhedra of the same volume that cannot be cut into smaller polyhedra and reassembled into each other. To prove this Dehn discovered another value associated with a polyhedron, the Dehn invariant, such that two polyhedra can only be dissected into each other when they have the same volume and the same Dehn invariant.
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In algebraic geometry and commutative algebra, a ring homomorphism f:A\to B is called formally smooth (from French: Formellement lisse) if it satisfies the following infinitesimal lifting property: Suppose B is given the structure of an A-algebra via the map f. Given a commutative A-algebra, C, and a nilpotent ideal N\subseteq C, any A-algebra homomorphism B\to C/N may be lifted to an A-algebra map B \to C. If moreover any such lifting is unique, then f is said to be formally étale. Formally smooth maps were defined by Alexander Grothendieck in Éléments de géométrie algébrique IV. For finitely presented morphisms, formal smoothness is equivalent to usual notion of smoothness.
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The traveler's dilemma can be framed as a finitely repeated prisoner's dilemma. Similar paradoxes are attributed to the centipede game and to the p-beauty contest game (or more specifically, "Guess 2/3 of the average"). One variation of the original traveler's dilemma in which both travelers are offered only two integer choices, $2 or $3, is identical mathematically to the standard non-iterated Prisoner's dilemma and thus the traveler's dilemma can be viewed as an extension of prisoner's dilemma. These games tend to involve deep iterative deletion of dominated strategies in order to demonstrate the Nash equilibrium, and tend to lead to experimental results that deviate markedly from classical game-theoretical predictions.
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In the mathematical subject of geometric group theory, a train track map is a continuous map f from a finite connected graph to itself which is a homotopy equivalence and which has particularly nice cancellation properties with respect to iterations. This map sends vertices to vertices and edges to nontrivial edge-paths with the property that for every edge e of the graph and for every positive integer n the path fn(e) is immersed, that is fn(e) is locally injective on e. Train-track maps are a key tool in analyzing the dynamics of automorphisms of finitely generated free groups and in the study of the Culler-Vogtmann Outer space.
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Suppose M is some R-module, and Mi is a submodule of M for every i in I. If every x in M can be written in one and only one way as a sum of finitely many elements of the Mi, then we say that M is the internal direct sum of the submodules Mi . In this case, M is naturally isomorphic to the (external) direct sum of the Mi as defined above . A submodule N of M is a direct summand of M if there exists some other submodule N′ of M such that M is the internal direct sum of N and N′. In this case, N and N′ are complementary submodules.
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Contour advection simulation Contour advection is a Lagrangian method of simulating the evolution of one or more contours or isolines of a tracer as it is stirred by a moving fluid. Consider a blob of dye injected into a river or stream: to first order it could be modelled by tracking only the motion of its outlines. It is an excellent method for studying chaotic mixing: even when advected by smooth or finitely-resolved velocity fields, through a continuous process of stretching and folding, these contours often develop into intricate fractals. The tracer is typically passive as in but may also be active as in, representing a dynamical property of the fluid such as vorticity.
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The Bolyai–Gerwien theorem is a related but much simpler result: it states that one can accomplish such a decomposition of a simple polygon with finitely many polygonal pieces if both translations and rotations are allowed for the reassembly. It follows from a result of that it is possible to choose the pieces in such a way that they can be moved continuously while remaining disjoint to yield the square. Moreover, this stronger statement can be proved as well to be accomplished by means of translations only. These results should be compared with the much more paradoxical decompositions in three dimensions provided by the Banach–Tarski paradox; those decompositions can even change the volume of a set.
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Similarly over a semiperfect ring, every indecomposable projective module is a PIM, and every finitely generated projective module is a direct sum of PIMs. In the context of group algebras of finite groups over fields (which are semiperfect rings), the representation ring describes the indecomposable modules, and the modular characters of simple modules represent both a subring and a quotient ring. The representation ring over the complex field is usually better understood and since PIMs correspond to modules over the complexes using p-modular system, one can use PIMs to transfer information from the complex representation ring to the representation ring over a field of positive characteristic. Roughly speaking this is called block theory.
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If X is a curve of genus 1 with a k-rational point p0, then X is called an elliptic curve over k. In this case, X has the structure of a commutative algebraic group (with p0 as the zero element), and so the set X(k) of k-rational points is an abelian group. The Mordell–Weil theorem says that for an elliptic curve (or, more generally, an abelian variety) X over a number field k, the abelian group X(k) is finitely generated. Computer algebra programs can determine the Mordell–Weil group X(k) in many examples, but it is not known whether there is an algorithm that always succeeds in computing this group.
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In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm. In practice, algorithms have been designed only for polynomials with coefficients in a finite field, in the field of rationals or in a finitely generated field extension of one of them. All factorization algorithms, including the case of multivariate polynomials over the rational numbers, reduce the problem to this case; see polynomial factorization.
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In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by . The inspiration came from various existing mathematical theories: hyperbolic geometry but also low- dimensional topology (in particular the results of Max Dehn concerning the fundamental group of a hyperbolic Riemann surface, and more complex phenomena in three-dimensional topology), and combinatorial group theory. In a very influential (over 1000 citations ) chapter from 1987, Gromov proposed a wide- ranging research program.
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In the rank-2 case spherical building are generalized n-gons, and in joint work with Richard Weiss he classified these when they admit a suitable group of symmetries (the so-called Moufang polygons). In collaboration with François Bruhat he developed the theory of affine buildings, and later he classified all irreducible buildings of affine type and rank at least four. Another of his well-known theorems is the "Tits alternative": if G is a finitely generated subgroup of a linear group, then either G has a solvable subgroup of finite index or it has a free subgroup of rank 2. The Tits group and the Tits–Koecher construction are named after him.
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The quantified variables in NBG's axiom schema of Class Comprehension are restricted to sets; hence Class Comprehension in NBG must be predicative. (Separation with respect to sets is still impredicative in NBG, because the quantifiers in φ(x) may range over all sets.) The NBG axiom schema of Class Comprehension can be replaced with finitely many of its instances; this is not possible in MK. MK is consistent relative to ZFC augmented by an axiom asserting the existence of strongly inaccessible cardinals. The only advantage of the axiom of limitation of size is that it implies the axiom of global choice. Limitation of Size does not appear in Rubin (1967), Monk (1980), or Mendelson (1997).
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Every group G may be made into a topological group by taking as a basis of open neighbourhoods of the identity, the collection of all normal subgroups of finite index in G. The resulting topology is called the profinite topology on G. A group is residually finite if, and only if, its profinite topology is Hausdorff. A group whose cyclic subgroups are closed in the profinite topology is said to be \Pi_C\,. Groups, each of whose finitely generated subgroups are closed in the profinite topology are called subgroup separable (also LERF, for locally extended residually finite). A group in which every conjugacy class is closed in the profinite topology is called conjugacy separable.
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In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's last theorem and of Catalan's conjecture, hence the name. The conjecture states that the equation has only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (am, bn, ck) where a, b, c are positive coprime integers and m, n, k are positive integers satisfying The inequality on m, n, and k is a necessary part of the conjecture. Without the inequality there would be infinitely many solutions, for instance with k = 1 (for any a, b, m, and n and with c = am \+ bn) or with m, n, and k all equal to two (for the infinitely many known Pythagorean triples).
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Gödel's incompleteness theorems, proved in 1931, showed that essential aspects of Hilbert's program could not be attained. In Gödel's first result he showed how to construct, for any sufficiently powerful and consistent finitely axiomatizable system—such as necessary to axiomatize the elementary theory of arithmetic—a statement that can be shown to be true, but that does not follow from the rules of the system. It thus became clear that the notion of mathematical truth can not be reduced to a purely formal system as envisaged in Hilbert's program. In a next result Gödel showed that such a system was not powerful enough for proving its own consistency, let alone that a simpler system could do the job.
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In algebra, a generating set G of a module M over a ring R is a subset of M such that the smallest submodule of M containing G is M itself (the smallest submodule containing a subset is the intersection of all submodules containing the set). The set G is then said to generate M. For example, the ring R is generated by the identity element 1 as a left R-module over itself. If there is a finite generating set, then a module is said to be finitely generated. Explicitly, if G is a generating set of a module M, then every element of M is a (finite) R-linear combination of some elements of G; i.e.
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In mathematics, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. (The right and left ideals of this form, generated by one element, are called principal ideals.) When this is satisfied for both left and right ideals, such as the case when R is a commutative ring, R can be called a principal ideal ring, or simply principal ring. If only the finitely generated right ideals of R are principal, then R is called a right Bézout ring. Left Bézout rings are defined similarly. These conditions are studied in domains as Bézout domains.
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It can be shown that for every marking f the map h : Δk′ → Xn is still injective. The image of h is called the closed simplex in Xn corresponding to f and is denoted by S′(f). Every point in Xn belongs to only finitely many closed simplices and a point of Xn represented by a marking f : Rn → Γ where the graph Γ is tri-valent belongs to a unique closed simplex in Xn, namely S′(f). The weak topology on the Outer space Xn is defined by saying that a subset C of Xn is closed if and only if for every marking f : Rn → Γ the set h−1(C) is closed in Δk′.
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Given a language L and an indexed class C = { L1, L2, L3, ... } of languages, a member language Lj ∈ C is called a minimal concept of L within C if L ⊆ Lj, but not L ⊊ Li ⊆ Lj for any Li ∈ C.; here: Definition 25 The class C is said to satisfy the MEF-condition if every finite subset D of a member language Li ∈ C has a minimal concept Lj ⊆ Li. Symmetrically, C is said to satisfy the MFF-condition if every nonempty finite set D has at most finitely many minimal concepts in C. Finally, C is said to have M-finite thickness if it satisfies both the MEF- and the MFF-condition. Ambainis et al. 1997, Definition 26 Finite thickness implies M-finite thickness.Ambainis et al.
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As is well known from the theory of dynamical systems, any orbit (gk(z)) of a hyperbolic element g has limit set consisting of two fixed points on the extended real axis; it follows that the geodesic segment from z to g(z) cuts through only finitely many translates of the fundamental domain. It is therefore easy to choose α so that fα equals one on a given hyperbolic element and vanishes on a finite set of other hyperbolic elements with distinct fixed points. Since G therefore has an infinite-dimensional space of pseudocharacters, it cannot be boundedly generated. Dynamical properties of hyperbolic elements can similarly be used to prove that any non-elementary Gromov-hyperbolic group is not boundedly generated.
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The theory IΣn + 1 is finitely axiomatizable, let thus A be its single axiom, and consider the theory T = IΣn + ¬A. We can assume that A is an instance of the induction schema, which has the form ::\forall w\,[B(0,w)\land\forall x\,(B(x,w)\to B(x+1,w))\to\forall x\,B(x,w)]. If we denote the formula ::\forall w\,[B(0,w)\land\forall x\,(B(x,w)\to B(x+1,w))\to B(n,w)] by P(n), then for every natural number n, the theory T (actually, even the pure predicate calculus) proves P(n). On the other hand, T proves the formula \exists x\, eg P(x), because it is logically equivalent to the axiom ¬A.
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Cubing the cube is the analogue in three dimensions of squaring the square: that is, given a cube C, the problem of dividing it into finitely many smaller cubes, no two congruent. Unlike the case of squaring the square, a hard yet solvable problem, there is no perfect cubed cube and, more generally, no dissection of a rectangular cuboid C into a finite number of unequal cubes. To prove this, we start with the following claim: for any perfect dissection of a rectangle in squares, the smallest square in this dissection does not lie on an edge of the rectangle. Indeed, each corner square has a smaller adjacent edge square, and the smallest edge square is adjacent to smaller squares not on the edge.
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Because of the Mordell–Weil theorem, Faltings's theorem can be reformulated as a statement about the intersection of a curve C with a finitely generated subgroup Γ of an abelian variety A. Generalizing by replacing C by an arbitrary subvariety of A and Γ by an arbitrary finite-rank subgroup of A leads to the Mordell–Lang conjecture, which was proved by . Another higher-dimensional generalization of Faltings's theorem is the Bombieri–Lang conjecture that if X is a pseudo-canonical variety (i.e., a variety of general type) over a number field k, then X(k) is not Zariski dense in X. Even more general conjectures have been put forth by Paul Vojta. The Mordell conjecture for function fields was proved by and by .
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Necessary and sufficient conditions for the structural stability of C1 vector fields on the unit disk D that are transversal to the boundary and on the two-sphere S2 have been determined in the foundational paper of Andronov and Pontryagin. According to the Andronov–Pontryagin criterion, such fields are structurally stable if and only if they have only finitely many singular points (equilibrium states) and periodic trajectories (limit cycles), which are all non-degenerate (hyperbolic), and do not have saddle-to-saddle connections. Furthermore, the non-wandering set of the system is precisely the union of singular points and periodic orbits. In particular, structurally stable vector fields in two dimensions cannot have homoclinic trajectories, which enormously complicate the dynamics, as discovered by Henri Poincaré.
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Games, as studied by economists and real-world game players, are generally finished in finitely many moves. Pure mathematicians are not so constrained, and set theorists in particular study games that last for infinitely many moves, with the winner (or other payoff) not known until after all those moves are completed. The focus of attention is usually not so much on the best way to play such a game, but whether one player has a winning strategy. (It can be proven, using the axiom of choice, that there are gameseven with perfect information and where the only outcomes are "win" or "lose"for which neither player has a winning strategy.) The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory.
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The basis sets in the product topology have almost the same definition as the above, except with the qualification that all but finitely many Ui are equal to the component space Xi. The product topology satisfies a very desirable property for maps fi : Y → Xi into the component spaces: the product map f: Y → X defined by the component functions fi is continuous if and only if all the fi are continuous. As shown above, this does not always hold in the box topology. This actually makes the box topology very useful for providing counterexamples--many qualities such as compactness, connectedness, metrizability, etc., if possessed by the factor spaces, are not in general preserved in the product with this topology.
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One approach and formation is model checking, which consists of a systematically exhaustive exploration of the mathematical model (this is possible for finite models, but also for some infinite models where infinite sets of states can be effectively represented finitely by using abstraction or taking advantage of symmetry). Usually this consists of exploring all states and transitions in the model, by using smart and domain- specific abstraction techniques to consider whole groups of states in a single operation and reduce computing time. Implementation techniques include state space enumeration, symbolic state space enumeration, abstract interpretation, symbolic simulation, abstraction refinement. The properties to be verified are often described in temporal logics, such as linear temporal logic (LTL), Property Specification Language (PSL), SystemVerilog Assertions (SVA), or computational tree logic (CTL).
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In algebra, a commutative k-algebra A is said to be 0-smooth if it satisfies the following lifting property: given a k-algebra C, an ideal N of C whose square is zero and a k-algebra map u: A \to C/N, there exists a k-algebra map v: A \to C such that u is v followed by the canonical map. If there exists at most one such lifting v, then A is said to be 0-unramified (or 0-neat). A is said to be 0-étale if it is 0-smooth and 0-unramified. A finitely generated k-algebra A is 0-smooth over k if and only if Spec A is a smooth scheme over k.
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These are finitely generated groups of automorphisms of rooted trees that are given by particularly nice recursive descriptions and that have remarkable self-similar properties. The study of branch, automata and self- similar groups has been particularly active in the 1990s and 2000s and a number of unexpected connections with other areas of mathematics have been discovered there, including dynamical systems, differential geometry, Galois theory, ergodic theory, random walks, fractals, Hecke algebras, bounded cohomology, functional analysis, and others. In particular, many of these self-similar groups arise as iterated monodromy groups of complex polynomials. Important connections have been discovered between the algebraic structure of self-similar groups and the dynamical properties of the polynomials in question, including encoding their Julia sets.
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The Greenwood–Gleason graph The Ramsey number R(k,l) is the smallest number r such that every graph with at least r vertices contains either a k-vertex clique or an l-vertex independent set. Ramsey numbers require enormous effort to compute; when max(k,l) ≥ 3 only finitely many of them are known precisely, and an exact computation of R(6,6) is believed to be out of reach. In 1953, the calculation of R(3,3) was given as a question in the Putnam Competition; in 1955, motivated by this problem, . Gleason and his co- author Robert E. Greenwood made significant progress in the computation of Ramsey numbers with their proof that R(3,4) = 9, R(3,5) = 14, and R(4,4) = 18\.
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Von Neumann's paper left open the possibility of a paradoxical decomposition of the interior of the unit square with respect to the linear group SL(2,R) (Wagon, Question 7.4). In 2000, Miklós Laczkovich proved that such a decomposition exists. More precisely, let A be the family of all bounded subsets of the plane with non-empty interior and at a positive distance from the origin, and B the family of all planar sets with the property that a union of finitely many translates under some elements of SL(2,R) contains a punctured neighbourhood of the origin. Then all sets in the family A are SL(2,R)-equidecomposable, and likewise for the sets in B. It follows that both families consist of paradoxical sets.
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When a group G acts on a set S, the action may be extended naturally to the Cartesian product Sn of S, consisting of n-tuples of elements of S: the action of an element g on the n-tuple (s1, ..., sn) is given by : g(s1, ..., sn) = (g(s1), ..., g(sn)). The group G is said to be oligomorphic if the action on Sn has only finitely many orbits for every positive integer n.Oligomorphic permutation groups - Isaac Newton Institute preprint, Peter J. Cameron (This is automatic if S is finite, so the term is typically of interest when S is infinite.) The interest in oligomorphic groups is partly based on their application to model theory, for example when considering automorphisms in countably categorical theories.
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In which case, the differential equations reduce to a finite set of equations with finitely many unknowns. If the stress distribution can be assumed to be uniform (or predictable, or unimportant) in one direction, then one may use the assumption of plane stress and plane strain behavior and the equations that describe the stress field are then a function of two coordinates only, instead of three. Even under the assumption of linear elastic behavior of the material, the relation between the stress and strain tensors is generally expressed by a fourth-order stiffness tensor with 21 independent coefficients (a symmetric 6 × 6 stiffness matrix). This complexity may be required for general anisotropic materials, but for many common materials it can be simplified.
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A ring is a Bézout domain if and only if it is an integral domain in which any two elements have a greatest common divisor that is a linear combination of them: this is equivalent to the statement that an ideal which is generated by two elements is also generated by a single element, and induction demonstrates that all finitely generated ideals are principal. The expression of the greatest common divisor of two elements of a PID as a linear combination is often called Bézout's identity, whence the terminology. Note that the above gcd condition is stronger than the mere existence of a gcd. An integral domain where a gcd exists for any two elements is called a GCD domain and thus Bézout domains are GCD domains.
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The Bombieri–Lang conjecture is an analogue for surfaces of Faltings's theorem, which states that algebraic curves of genus greater than one only have finitely many rational points. If true, the Bombieri–Lang conjecture would resolve the Erdős–Ulam problem, as it would imply that there do not exist dense subsets of the Euclidean plane all of whose pairwise distances are rational. In 1997, Lucia Caporaso, Barry Mazur, Joe Harris, and Patricia Pacelli showed that the Bombieri–Lang conjecture implies a type of uniform boundedness conjecture: there is a constant B_{g,d} depending only on g and d such that the number of rational points of any genus g curve X over any degree d number field is at most B_{g,d}.
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The class of sofic groups is closed under the operations of taking subgroups, extensions by amenable groups, and free products. A finitely generated group is sofic if it is the limit of a sequence of sofic groups. The limit of a sequence of amenable groups (that is, an initially subamenable group) is necessarily sofic, but there exist sofic groups that are not initially subamenable groups.. As Gromov proved, Sofic groups are surjunctive. That is, they obey a form of the Garden of Eden theorem for cellular automata defined over the group (dynamical systems whose states are mappings from the group to a finite set and whose state transitions are translation-invariant and continuous) stating that every injective automaton is surjective and therefore also reversible.
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In the fall of 1988 and in the academic year 1992–1993 he was a visiting scholar at the Institute for Advanced Study. In 1986 he was awarded the Fields Medal at the ICM at Berkeley for proving the Tate conjecture for abelian varieties over number fields, the Shafarevich conjecture for abelian varieties over number fields and the Mordell conjecture, which states that any non-singular projective curve of genus g > 1 defined over a number field K contains only finitely many K-rational points. As a Fields Medalist he gave an ICM plenary talk Recent progress in arithmetic algebraic geometry. In 1994 as an ICM invited speaker in Zurich he gave a talk Mumford-Stabilität in der algebraischen Geometrie.
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In hyperbolic geometry, the ending lamination theorem, originally conjectured by , states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are geodesic laminations on some surfaces in the boundary of the manifold. The ending lamination theorem is a generalization of the Mostow rigidity theorem to hyperbolic manifolds of infinite volume. When the manifold is compact or of finite volume, the Mostow rigidity theorem states that the fundamental group determines the manifold. When the volume is infinite the fundamental group is not enough to determine the manifold: one also needs to know the hyperbolic structure on the surfaces at the "ends" of the manifold, and also the ending laminations on these surfaces.
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In three dimensions, it is not possible for a geometrically chiral polytope to have finitely many finite faces. For instance, the snub cube is vertex-transitive, but its flags have more than two orbits, and it is neither edge-transitive nor face-transitive, so it is not symmetric enough to meet the formal definition of chirality. The quasiregular polyhedra and their duals, such as the cuboctahedron and the rhombic dodecahedron, provide another interesting type of near-miss: they have two orbits of flags, but are mirror-symmetric, and not every adjacent pair of flags belongs to different orbits. However, despite the nonexistence of finite chiral three-dimensional polyhedra, there exist infinite three-dimensional chiral skew polyhedra of types {4,6}, {6,4}, and {6,6}.
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The above proof uses Zorn's lemma, which is equivalent to the axiom of choice. It is now known (see below) that the ultrafilter lemma (or equivalently, the Boolean prime ideal theorem), which is slightly weaker than the axiom of choice, is actually strong enough. The Hahn–Banach theorem is equivalent to the following: :(∗): On every Boolean algebra there exists a "probability charge", that is: a nonconstant finitely additive map from into . (The Boolean prime ideal theorem is equivalent to the statement that there are always nonconstant probability charges which take only the values 0 and 1.) In Zermelo–Fraenkel set theory, one can show that the Hahn–Banach theorem is enough to derive the existence of a non-Lebesgue measurable set.
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In the work of Lenhard Ng, relative SFT is used to obtain invariants of smooth knots: a knot or link inside a topological three-manifold gives rise to a Legendrian torus inside a contact five-manifold, consisisting of the unit conormal bundle to the knot inside the unit cotangent bundle of the ambient three-manifold. The relative SFT of this pair is a differential graded algebra; Ng derives a powerful knot invariant from a combinatorial version of the zero-th degree part of the homology. It has the form of a finitely presented tensor algebra over a certain ring of multivariable Laurent polynomials with integer coefficients. This invariant assigns distinct invariants to (at least) knots of at most ten crossings, and dominates the Alexander polynomial and the A-polynomial (and thus distinguishes the unknot).
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Originally, Hilbert defined syzygies for ideals in polynomial rings, but the concept generalizes trivially to (left) modules over any ring. Given a generating set g_1, \ldots, g_k of a module over a ring , a relation or first syzygy between the generators is a -tuple (a_1, \ldots, a_k) of elements of such thatThe theory is presented for finitely generated modules, but extends easily to arbitrary modules. :a_1g_1 + \cdots + a_kg_k =0. Let L_0 be the free module with basis (G_1, \ldots, G_k), the relation (a_1, \ldots, a_k) may be identified with the element :a_1G_1 + \cdots + a_kG_k, and the relations form the kernel R_1 of the linear map L_0 \to M defined by G_i \mapsto g_i. In other words, one has an exact sequence :0 \to R_1 \to L_0 \to M \to 0.
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That is, each of these rings is an integral domain R, and every ring S with R ⊆ S ⊆ Frac(R) such that S is finitely generated as an R-module is equal to R. (Here Frac(R) denotes the field of fractions of R.) This is a direct translation, in terms of local rings, of the geometric condition that every finite birational morphism to X is an isomorphism. An older notion is that a subvariety X of projective space is linearly normal if the linear system giving the embedding is complete. Equivalently, X ⊆ Pn is not the linear projection of an embedding X ⊆ Pn+1 (unless X is contained in a hyperplane Pn). This is the meaning of "normal" in the phrases rational normal curve and rational normal scroll.
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On the other hand, in the Banach–Tarski paradox, the number of pieces is finite and the allowed equivalences are Euclidean congruences, which preserve the volumes. Yet, somehow, they end up doubling the volume of the ball! While this is certainly surprising, some of the pieces used in the paradoxical decomposition are non-measurable sets, so the notion of volume (more precisely, Lebesgue measure) is not defined for them, and the partitioning cannot be accomplished in a practical way. In fact, the Banach–Tarski paradox demonstrates that it is impossible to find a finitely- additive measure (or a Banach measure) defined on all subsets of an Euclidean space of three (and greater) dimensions that is invariant with respect to Euclidean motions and takes the value one on a unit cube.
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The Banach–Tarski paradox is that a ball can be decomposed into a finite number of point sets and reassembled into two balls identical to the original. In set theory, a paradoxical set is a set that has a paradoxical decomposition. A paradoxical decomposition of a set is two families of disjoint subsets, along with appropriate group actions that act on some universe (of which the set in question is a subset), such that each partition can be mapped back onto the entire set using only finitely many distinct functions (or compositions thereof) to accomplish the mapping. A set that admits such a paradoxical decomposition where the actions belong to a group G is called G-paradoxical or paradoxical with respect to G. Paradoxical sets exist as a consequence of the Axiom of Infinity.
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Since all sets S_i are sets of suffixes of a finite set of codewords, there are only finitely many different candidates for S_i. Since visiting one of the sets for the second time will cause the algorithm to stop, the algorithm cannot continue endlessly and thus must always terminate. More precisely, the total number of dangling suffixes that the algorithm considers is at most equal to the total of the lengths of the codewords in the input, so the algorithm runs in polynomial time as a function of this input length. By using a suffix tree to speed the comparison between each dangling suffix and the codewords, the time for the algorithm can be bounded by O(nk), where n is the total length of the codewords and k is the number of codewords.
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It was shown in 1991/1992 by Yulii Ilyashenko and Jean Écalle that every polynomial vector field in the plane has only finitely many limit cycles (a 1923 article by Henri Dulac claiming a proof of this statement had been shown to contain a gap in 1981). This statement is not obvious, since it is easy to construct smooth (C∞) vector fields in the plane with infinitely many concentric limit cycles. The question whether there exists a finite upper bound H(n) for the number of limit cycles of planar polynomial vector fields of degree n remains unsolved for any n > 1\. (H(1) = 0 since linear vector fields do not have limit cycles.) Evgenii Landis and Ivan Petrovsky claimed a solution in the 1950s, but it was shown wrong in the early 1960s.
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This does not depend upon the projectivity of P: it is true of all superfluous epimorphisms. If (P,p) is a projective cover of M, and P' is another projective module with an epimorphism p':P'\rightarrow M, then there is a split epimorphism α from P' to P such that p\alpha=p' Unlike injective envelopes and flat covers, which exist for every left (right) R-module regardless of the ring R, left (right) R-modules do not in general have projective covers. A ring R is called left (right) perfect if every left (right) R-module has a projective cover in R-Mod (Mod-R). A ring is called semiperfect if every finitely generated left (right) R-module has a projective cover in R-Mod (Mod-R).
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If π is a partition of an amorphous set into finite subsets, then there must be exactly one integer n(π) such that π has infinitely many subsets of size n; for, if every size was used finitely many times, or if more than one size was used infinitely many times, this information could be used to coarsen the partition and split π into two infinite subsets. If an amorphous set has the additional property that, for every partition π, n(π) = 1, then it is called strictly amorphous or strongly amorphous, and if there is a finite upper bound on n(π) then the set is called bounded amorphous. It is consistent with ZF that amorphous sets exist and are all bounded, or that they exist and are all unbounded.
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Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the truth value of the statement is well- defined, or whether it can be determined by other means. Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement. Whether there exist so-called "absolutely undecidable" statements, whose truth value can never be known or is ill-specified, is a controversial point among various philosophical schools. One of the first problems suspected to be undecidable, in the second sense of the term, was the word problem for groups, first posed by Max Dehn in 1911, which asks if there is a finitely presented group for which no algorithm exists to determine whether two words are equivalent.
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Our God is the soul of the Universe... Spinozism and Judaism are by no means at opposite poles.Emil G. Hirsch, "The Doctrine of Evolution and Judaism" in Some Modern Problems and Their Bearing on Judaism Reform Advocate Library (Chicago: Bloch & Newman, 1903), 25-46. Similarly, Joseph Krauskopf wrote: :According to our definition, God is the finitely, conceivable Ultimate, the Cause of all and the Cause in all, the Universal Life, the All-Pervading, All-Controlling, All-Directing Power Supreme, the Creator of the universe and the Governor of the same according to eternal and immutable laws by Him created. All existence is part of His existence, all life is part of His life, all intelligence is part of His intelligence, all evolution, all progress is part of His plan.
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An important class of integral domains that contain a PID is a unique factorization domain (UFD), an integral domain in which every nonunit element is a product of prime elements (an element is prime if it generates a prime ideal.) The fundamental question in algebraic number theory is on the extent to which the ring of (generalized) integers in a number field, where an "ideal" admits prime factorization, fails to be a PID. Among theorems concerning a PID, the most important one is the structure theorem for finitely generated modules over a principal ideal domain. The theorem may be illustrated by the following application to linear algebra. Let V be a finite- dimensional vector space over a field k and f: V \to V a linear map with minimal polynomial q.
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This theorem states that, if such a sequence has zeros, then with finitely many exceptions the positions of the zeros repeat regularly. Skolem proved this for recurrences over the rational numbers, and Mahler and Lech extended it to other systems of numbers. However, the proofs of the theorem do not show how to test whether there exist any zeros. There does exist an algorithm to test whether a constant-recursive sequence has infinitely many zeros, and if so to construct a decomposition of the positions of those zeros into periodic subsequences, based on the algebraic properties of the roots of the characteristic polynomial of the given recurrence.. The remaining difficult part of the Skolem problem is determining whether the finite set of non-repeating zeros is empty or not.
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Svenonius' reputation as a mathematical model theorist was established with the publication of three papers in Theoria in 1959 and 1960: # \aleph_0-categoricity in first-order predicate calculus, # A theorem on permutations in models, # On minimal models of first-order systems. In particular, paper (2) contains what is now called "Svenonius' Theorem", an important result on definability of predicates in first order theories. Even the statement of this result requires mathematical model- theoretic concepts. It states that if the interpretation of a predicate in any model of a first-order theory is invariant under permutations ("automorphisms") of the model fixing the other predicates, then the interpretation of that predicate is definable in every model by a formula involving only the other predicates; furthermore only finitely many such defining formulas are required.
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A metric space (M,d) is totally bounded if and only if for every real number \varepsilon >0, there exists a finite collection of open balls in M of radius \varepsilon whose union contains M. Equivalently, the metric space M is totally bounded if and only if for every \varepsilon >0, there exists a finite cover such that the radius of each element of the cover is at most \varepsilon. This is equivalent to the existence of a finite ε-net. A metric space is said to be Cauchy- precompact if every sequence admits a Cauchy subsequence; in metric spaces, a set is Cauchy-precompact if and only if it is totally bounded. Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded).
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A basic motivation of the theory is that projective modules (at least over certain commutative rings) are analogues of vector bundles. This can be made precise for the ring of continuous real- valued functions on a compact Hausdorff space, as well as for the ring of smooth functions on a smooth manifold (see Serre–Swan theorem that says a finitely generated projective module over the space of smooth functions on a compact manifold is the space of smooth sections of a smooth vector bundle). Vector bundles are locally free. If there is some notion of "localization" which can be carried over to modules, such as the usual localization of a ring, one can define locally free modules, and the projective modules then typically coincide with the locally free modules.
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A virtually polycyclic group is a group that has a polycyclic subgroup of finite index, an example of a virtual property. Such a group necessarily has a normal polycyclic subgroup of finite index, and therefore such groups are also called polycyclic-by-finite groups. Although polycyclic-by-finite groups need not be solvable, they still have many of the finiteness properties of polycyclic groups; for example, they satisfy the maximal condition, and they are finitely presented and residually finite. In the textbook and some papers, an M-group refers to what is now called a polycyclic-by-finite group, which by Hirsch's theorem can also be expressed as a group which has a finite length subnormal series with each factor a finite group or an infinite cyclic group.
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For example, if X = (0, 1) and for each positive integer i, Xi is the set of elements of X having a decimal expansion with digit 0 in the ith decimal place, then any finite intersection is non- empty (just take 0 in those finitely many places and 1 in the rest), but the intersection of all Xi for i ≥ 1 is empty, since no element of (0, 1) has all zero digits. The finite intersection property is useful in formulating an alternative definition of compactness: a space is compact if and only if every collection of closed sets having the finite intersection property has non- empty intersection. This formulation of compactness is used in some proofs of Tychonoff's theorem and the uncountability of the real numbers (see next section).
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In general if V is a vector bundle on a space X, with constant fibre dimension n, the n-th exterior power of V taken fibre-by-fibre is a line bundle, called the determinant line bundle. This construction is in particular applied to the cotangent bundle of a smooth manifold. The resulting determinant bundle is responsible for the phenomenon of tensor densities, in the sense that for an orientable manifold it has a nonvanishing global section, and its tensor powers with any real exponent may be defined and used to 'twist' any vector bundle by tensor product. The same construction (taking the top exterior power) applies to a finitely generated projective module M over a Noetherian domain and the resulting invertible module is called the determinant module of M.
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The book begins with five chapters that discuss the field of reverse mathematics, which has the goal of classifying mathematical theorems by the axiom schemes needed to prove them, the big five subsystems of second-order arithmetic into which many theorems of mathematics have been classified. These chapters also review some of the tools needed in this study, including computability theory, forcing, and the low basis theorem. Chapter six, "the real heart of the book", applies this method to an infinitary form of Ramsey's theorem: every edge coloring of a countably infinite complete graph or complete uniform hypergraph, using finitely many colors, contains a monochromatic infinite induced subgraph. The standard proof of this theorem uses the arithmetical comprehension axiom, falling into one of the big five subsystems, ACA0.
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A matroid has branchwidth one if and only if every element is either a loop or a coloop, so the unique minimal forbidden minor is the uniform matroid U(2,3), the graphic matroid of the triangle graph. A matroid has branchwidth two if and only if it is the graphic matroid of a graph of branchwidth two, so its minimal forbidden minors are the graphic matroid of K4 and the non-graphic matroid U(2,4). The matroids of branchwidth three are not well-quasi-ordered without the additional assumption of representability over a finite field, but nevertheless the matroids with any finite bound on their branchwidth have finitely many minimal forbidden minors, all of which have a number of elements that is at most exponential in the branchwidth.; .
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The quadratic reciprocity law can be formulated in terms of the Hilbert symbol (a,b)_v where a and b are any two nonzero rational numbers and v runs over all the non-trivial absolute values of the rationals (the archimedean one and the p-adic absolute values for primes p). The Hilbert symbol (a,b)_v is 1 or −1. It is defined to be 1 if and only if the equation ax^2+by^2=z^2 has a solution in the completion of the rationals at v other than x=y=z=0. The Hilbert reciprocity law states that (a,b)_v, for fixed a and b and varying v, is 1 for all but finitely many v and the product of (a,b)_v over all v is 1.
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The inequalities are equivalent to the inequalities of Goluzin, discovered in 1947. Roughly speaking, the Grunsky inequalities give information on the coefficients of the logarithm of a univalent function; later generalizations by Milin, starting from the Lebedev–Milin inequality, succeeded in exponentiating the inequalities to obtain inequalities for the coefficients of the univalent function itself. The Grunsky matrix and its associated inequalities were originally formulated in a more general setting of univalent functions between a region bounded by finitely many sufficiently smooth Jordan curves and its complement: the results of Grunsky, Goluzin and Milin generalize to that case. Historically the inequalities for the disk were used in proving special cases of the Bieberbach conjecture up to the sixth coefficient; the exponentiated inequalities of Milin were used by de Branges in the final solution.
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It can be shown that the difference between their locations (two independent random walks) is also a simple random walk, so they almost surely meet again in a 2-dimensional walk, but for 3 dimensions and higher the probability decreases with the number of the dimensions. Paul Erdős and Samuel James Taylor also showed in 1960 that for dimensions less or equal than 4, two independent random walks starting from any two given points have infinitely many intersections almost surely, but for dimensions higher than 5, they almost surely intersect only finitely often. The asymptotic function for a two-dimensional random walk as the number of steps increases is given by a Rayleigh distribution. The probability distribution is a function of the radius from the origin and the step length is constant for each step.
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Commutative algebra (in the form of polynomial rings and their quotients, used in the definition of algebraic varieties) has always been a part of algebraic geometry. However, in the late 1950s, algebraic varieties were subsumed into Alexander Grothendieck's concept of a scheme. Their local objects are affine schemes or prime spectra, which are locally ringed spaces, which form a category that is antiequivalent (dual) to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field k, and the category of finitely generated reduced k-algebras. The gluing is along the Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes.
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The algorithm is based on the facts that in a well-quasi-order (A,\le), any upward closed set has a finite set of minima, and any sequence S_1 \subseteq S_2 \subseteq ... of upward-closed subsets of A converges after finitely many steps (1). The algorithm needs to store an upward-closed set S_s of states in memory, which it can do because an upward-closed set is representable as a finite set of minima. It starts from the upward closure of the set of error states S_e and computes at each iteration the (by monotonicity also upward-closed) set of immediate predecessors and adding it to the set S_s. This iteration terminates after a finite number of steps, due to the property (1) of well-quasi-orders.
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The book is divided into two parts, the first on the existence of paradoxical decompositions and the second on conditions that prevent their existence. After two chapters of background material, the first part proves the Banach–Tarski paradox itself, considers higher-dimensional spaces and non-Euclidean geometry, studies the number of pieces necessary for a paradoxical decomposition, and finds analogous results to the Banach–Tarski paradox for one- and two-dimensional sets. The second part includes a related theorem of Tarski that congruence-invariant finitely-additive measures prevent the existence of paradoxical decompositions, a theorem that Lebesgue measure is the only such measure on the Lebesgue measurable sets, material on amenable groups, connections to the axiom of choice and the Hahn–Banach theorem. Three appendices describe Euclidean groups, Jordan measure, and a collection of open problems.
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Although the Kelly strategy's promise of doing better than any other strategy in the long run seems compelling, some economists have argued strenuously against it, mainly because an individual's specific investing constraints may override the desire for optimal growth rate. The conventional alternative is expected utility theory which says bets should be sized to maximize the expected utility of the outcome (to an individual with logarithmic utility, the Kelly bet maximizes expected utility, so there is no conflict; moreover, Kelly's original paper clearly states the need for a utility function in the case of gambling games which are played finitely many times). Even Kelly supporters usually argue for fractional Kelly (betting a fixed fraction of the amount recommended by Kelly) for a variety of practical reasons, such as wishing to reduce volatility, or protecting against non-deterministic errors in their advantage (edge) calculations.
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The self adjoint unitary F gives a map of the K-theory of A into integers by taking Fredholm index as follows. In the even case, each projection e in A decomposes as e0 ⊕ e1 under the grading and e1Fe0 becomes a Fredholm operator from e0H to e1H. Thus e → Ind e1Fe0 defines an additive mapping of K0(A) to Z. In the odd case the eigenspace decomposition of F gives a grading on H, and each invertible element in A gives a Fredholm operator (F + 1) u (F − 1)/4 from (F − 1)H to (F + 1)H. Thus u → Ind (F + 1) u (F − 1)/4 gives an additive mapping from K1(A) to Z. When the spectral triple is finitely summable, one may write the above indexes using the (super) trace, and a product of F, e (resp.
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So, loosely speaking, the use of harmonic coordinates show that Riemannian manifolds can be covered by coordinate charts in which the local representations of the Riemannian metric are controlled only by the qualitative geometric behavior of the Riemannian manifold itself. Following ideas set forth by Jeff Cheeger in 1970, one can then consider sequences of Riemannian manifolds which are uniformly geometrically controlled, and using the coordinates, one can assemble a "limit" Riemannian manifold. Due to the nature of such "Riemannian convergence", it follows, for instance, that up to diffeomorphism there are only finitely many smooth manifolds of a given dimension which admit Riemannian metrics with a fixed bound on Ricci curvature and diameter, with a fixed positive lower bound on injectivity radius. Such estimates on harmonic radius are also used to construct geometrically-controlled cutoff functions, and hence partitions of unity as well.
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Fix an arbitrary field k and let Fields/k denote the category of finitely generated field extensions of k with inclusions as morphisms. Consider a (covariant) functor F : Fields/k → Set. For a field extension K/k and an element a of F(K/k) a field of definition of a is an intermediate field K/L/k such that a is contained in the image of the map F(L/k) → F(K/k) induced by the inclusion of L in K. The essential dimension of a, denoted by ed(a), is the least transcendence degree (over k) of a field of definition for a. The essential dimension of the functor F, denoted by ed(F), is the supremum of ed(a) taken over all elements a of F(K/k) and objects K/k of Fields/k.
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In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces. There are some variations of the definition: sometimes the Fuchsian group is assumed to be finitely generated, sometimes it is allowed to be a subgroup of PGL(2,R) (so that it contains orientation-reversing elements), and sometimes it is allowed to be a Kleinian group (a discrete subgroup of PSL(2,C)) which is conjugate to a subgroup of PSL(2,R). Fuchsian groups are used to create Fuchsian models of Riemann surfaces.
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In the same year the French mathematician Jules Richard used a variant of Cantor's diagonal method to obtain another contradiction in naive set theory. Consider the set A of all finite agglomerations of words. The set E of all finite definitions of real numbers is a subset of A. As A is countable, so is E. Let p be the nth decimal of the nth real number defined by the set E; we form a number N having zero for the integral part and p + 1 for the nth decimal if p is not equal either to 8 or 9, and unity if p is equal to 8 or 9. This number N is not defined by the set E because it differs from any finitely defined real number, namely from the nth number by the nth digit.
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Let G be a finitely generated group, S ⊆ G be a finite generating set of G and let Γ = Γ(G, S) be the Cayley graph of G with respect to S. For a subset A ⊆ G denote by A∗ the complement G − A of A in G. For a subset A ⊆ G, the edge boundary or the co- boundary δA of A consists of all (topological) edges of Γ connecting a vertex from A with a vertex from A∗. Note that by definition δA = δA∗. An ordered pair (A, A∗) is called a cut in Γ if δA is finite. A cut (A,A∗) is called essential if both the sets A and A∗ are infinite. A subset A ⊆ G is called almost invariant if for every g∈G the symmetric difference between A and Ag is finite.
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Hence, at the very least, our varieties must have nK_{X'} to be a Cartier divisor for some positive integer n.) The first key result is the cone theorem of Shigefumi Mori, describing the structure of the cone of curves of X. Briefly, the theorem shows that starting with X, one can inductively construct a sequence of varieties X_i, each of which is "closer" than the previous one to having K_{X_i} nef. However, the process may encounter difficulties: at some point the variety X_i may become "too singular". The conjectural solution to this problem is the flip, a kind of codimension-2 surgery operation on X_i. It is not clear that the required flips exist, nor that they always terminate (that is, that one reaches a minimal model X' in finitely many steps.) showed that flips exist in the 3-dimensional case.
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An apex graph is a graph that may be made planar by the removal of one vertex, and a k-apex graph is a graph that may be made planar by the removal of at most k vertices. A 1-planar graph is a graph that may be drawn in the plane with at most one simple crossing per edge, and a k-planar graph is a graph that may be drawn with at most k simple crossings per edge. A map graph is a graph formed from a set of finitely many simply- connected interior-disjoint regions in the plane by connecting two regions when they share at least one boundary point. When at most three regions meet at a point, the result is a planar graph, but when four or more regions meet at a point, the result can be nonplanar.
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The answer turns out to be no: it was shown in that if a module M has finite hollow dimension, then M/J(M) is a semisimple, Artinian module. There are many rings with unity for which R/J(R) is not semisimple Artinian, and given such a ring R, R itself is finitely generated but has infinite hollow dimension. Sarath and Varadarajan showed later, that M/J(M) being semisimple Artinian is also sufficient for M to have finite hollow dimension provided J(M) is a superfluous submodule of M.The same result can be found in and This shows that the rings R with finite hollow dimension either as a left or right R-module are precisely the semilocal rings. An additional corollary of Varadarajan's result is that RR has finite hollow dimension exactly when RR does.
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Plato and Aristotle considered intuition a means for perceiving ideas, significant enough that for Aristotle, intuition comprised the only means of knowing principles that are not subject to argument. Henri Poincaré distinguished logical intuition from other forms of intuition. In his book The Value of Science, he points out that: The passage goes on to assign two roles to logical intuition: to permit one to choose which route to follow in search of scientific truth, and to allow one to comprehend logical developments. Bertrand Russell, though critical of intuitive mysticism, pointed out that the degree to which a truth is self-evident according to logical intuition can vary, from one situation to another, and stated that some self-evident truths are practically infallible: Kurt Gödel demonstrated based on his incompleteness theorems that intuition- based propositional calculus cannot be finitely valued.
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The problem originally arose in algebraic invariant theory. Here the ring R is given as a (suitably defined) ring of polynomial invariants of a linear algebraic group over a field k acting algebraically on a polynomial ring k[x1, ..., xn] (or more generally, on a finitely generated algebra defined over a field). In this situation the field K is the field of rational functions (quotients of polynomials) in the variables xi which are invariant under the given action of the algebraic group, the ring R is the ring of polynomials which are invariant under the action. A classical example in nineteenth century was the extensive study (in particular by Cayley, Sylvester, Clebsch, Paul Gordan and also Hilbert) of invariants of binary forms in two variables with the natural action of the special linear group SL2(k) on it.
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Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. Several systems of semantics for intuitionistic logic have been studied. One of these semantics mirrors classical Boolean-valued semantics but uses Heyting algebras in place of Boolean algebras. Another semantics uses Kripke models. These, however, are technical means for studying Heyting’s deductive system rather than formalizations of Brouwer’s original informal semantic intuitions. Semantical systems claiming to capture such intuitions, due to offering meaningful concepts of “constructive truth” (rather than merely validity or provability), are Gödel’s dialectica interpretation, Kleene’s realizability, Medvedev’s logic of finite problems,Shehtman, V., "Modal Counterparts of Medvedev Logic of Finite Problems Are Not Finitely Axiomatizable," in Studia Logica: An International Journal for Symbolic Logic, vol.
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The question of the existence of an ordinary line can also be posed for points in the real projective plane RP2 instead of the Euclidean plane. The projective plane can be formed from the Euclidean plane by adding extra points "at infinity" where lines that are parallel in the Euclidean plane intersect each other, and by adding a single line "at infinity" containing all the added points. However, the additional points of the projective plane cannot help create non-Euclidean finite point sets with no ordinary line, as any finite point set in the projective plane can be transformed into a Euclidean point set with the same combinatorial pattern of point-line incidences. Therefore, any pattern of finitely many intersecting points and lines that exists in one of these two types of plane also exists in the other.
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In computability theory, the mortality problem is a decision problem which can be stated as follows: :Given a Turing machine, decide whether it halts when run on any configuration (not necessarily a starting one) In the statement above, the configuration is a pair , where q is one of the machine's states (not necessarily its initial state) and w is an infinite sequence of symbols representing the initial content of the tape. Note that while we usually assume that in the starting configuration all but finitely many cells on the tape are blanks, in the mortality problem the tape can have arbitrary content, including infinitely many non-blank symbols written on it. Philip K. Hooper proved in 1966 that the mortality problem is undecidable. However, it can be shown that the set of Turing machines which are mortal (i.e.
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In order for a k-tuple to have infinitely many positions at which all of its values are prime, there cannot exist a prime p such that the tuple includes every different possible value modulo p. For, if such a prime p existed, then no matter which value of n was chosen, one of the values formed by adding n to the tuple would be divisible by p, so there could only be finitely many prime placements (only those including p itself). For example, the numbers in a k-tuple cannot take on all three values 0, 1, and 2 modulo 3; otherwise the resulting numbers would always include a multiple of 3 and therefore could not all be prime unless one of the numbers is 3 itself. A k-tuple that satisfies this condition (i.e.
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A Turing reduction from a set B to a set A computes the membership of a single element in B by asking questions about the membership of various elements in A during the computation; it may adaptively determine which questions it asks based upon answers to previous questions. In contrast, a truth-table reduction or a weak truth-table reduction must present all of its (finitely many) oracle queries at the same time. In a truth-table reduction, the reduction also gives a boolean function (a truth table) which, when given the answers to the queries, will produce the final answer of the reduction. In a weak truth-table reduction, the reduction uses the oracle answers as a basis for further computation which may depend on the given answers but may not ask further questions of the oracle.
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This construction has been generalized, in algebraic geometry, to the notion of a divisor. There are different definitions of divisors, but in general they form an abstraction of a codimension-one subvariety of an algebraic variety, the set of solution points of a system of polynomial equations. In the case where the system of equations has one degree of freedom (its solutions form an algebraic curve or Riemann surface), a subvariety has codimension one when it consists of isolated points, and in this case a divisor is again a signed multiset of points from the variety. The meromorphic functions on a compact Riemann surface have finitely many zeros and poles, and their divisors can again be represented as elements of a free abelian group, with multiplication or division of functions corresponding to addition or subtraction of group elements.
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A presentation of a group is a set of elements that generate the group (all group elements are products of finitely many generators), together with "relators", products of generators that give the identity element. The free abelian group with basis B has a presentation in which the generators are the elements of B, and the relators are the commutators of pairs of elements of B. Here, the commutator of two elements x and y is the product x−1y−1xy; setting this product to the identity causes xy to equal yx, so that x and y commute. More generally, if all pairs of generators commute, then all pairs of products of generators also commute. Therefore, the group generated by this presentation is abelian, and the relators of the presentation form a minimal set of relators needed to ensure that it is abelian.
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Simplex graphs were introduced by , credit the introduction of simplex graphs to a later paper, also by Bandelt and van de Vel, but this appears to be a mistake. who observed that a simplex graph has no cubes if and only if the underlying graph is triangle-free, and showed that the chromatic number of the underlying graph equals the minimum number n such that the simplex graph can be isometrically embedded into a Cartesian product of n trees. As a consequence of the existence of triangle-free graphs with high chromatic number, they showed that there exist two-dimensional topological median algebras that cannot be embedded into products of finitely many real trees. also use simplex graphs as part of their proof that testing whether a graph is triangle-free or whether it is a median graph may be performed equally quickly.
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The typical approach to proving Courcelle's theorem involves the construction of a finite bottom-up tree automaton that acts on the tree decompositions of the given graph. In more detail, two graphs G1 and G2, each with a specified subset T of vertices called terminals, may be defined to be equivalent with respect to an MSO formula F if, for all other graphs H whose intersection with G1 and G2 consists only of vertices in T, the two graphs G1 ∪ H and G2 ∪ H behave the same with respect to F: either they both model F or they both do not model F. This is an equivalence relation, and it can be shown by induction on the length of F that (when the sizes of T and F are both bounded) it has finitely many equivalence classes., Theorem 13.1.1, p. 266.
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Serre From 1959 onward Serre's interests turned towards group theory, number theory, in particular Galois representations and modular forms. Amongst his most original contributions were: his "Conjecture II" (still open) on Galois cohomology; his use of group actions on trees (with Hyman Bass); the Borel–Serre compactification; results on the number of points of curves over finite fields; Galois representations in ℓ-adic cohomology and the proof that these representations have often a "large" image; the concept of p-adic modular form; and the Serre conjecture (now a theorem) on mod-p representations that made Fermat's last theorem a connected part of mainstream arithmetic geometry. In his paper FAC, Serre asked whether a finitely generated projective module over a polynomial ring is free. This question led to a great deal of activity in commutative algebra, and was finally answered in the affirmative by Daniel Quillen and Andrei Suslin independently in 1976.
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This follows from the completeness of the underlying propositional logic. We will now show that there is such an assignment of truth values to E_h, so that all D_n will be true: The E_h appear in the same order in every D_n ; we will inductively define a general assignment to them by a sort of "majority vote": Since there are infinitely many assignments (one for each D_n ) affecting E_1, either infinitely many make E_1 true, or infinitely many make it false and only finitely many make it true. In the former case, we choose E_1 to be true in general; in the latter we take it to be false in general. Then from the infinitely many n for which E_1 through E_{h-1} are assigned the same truth value as in the general assignment, we pick a general assignment to E_h in the same fashion.
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In algebra the addition used in the swindle is usually the direct sum of modules over a ring. Example: A typical application of the Eilenberg swindle in algebra is the proof that if A is a projective module over a ring R then there is a free module F with A ⊕ F ≅ F.Lam (1999), Corollary 2.7, p. 22; Eklof & Mekler (2002), Lemma 2.3, [ p. 9]. To see this, choose a module B such that A ⊕ B is free, which can be done as A is projective, and put :F = B ⊕ A ⊕ B ⊕ A ⊕ B ⊕ .... so that :A ⊕ F = A ⊕ (B ⊕ A) ⊕ (B ⊕ A) ⊕ ... = (A ⊕ B) ⊕ (A ⊕ B) ⊕ ... ≅ F. Example: Finitely generated free modules over commutative rings R have a well-defined natural number as their dimension which is additive under direct sums, and are isomorphic if and only if they have the same dimension.
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For a less trivial example of the point made by Example 2, consider a Venn diagram formed by n closed curves partitioning the diagram into 2n regions, and let X be the (infinite) set of all points in the plane not on any curve but somewhere within the diagram. The interior of each region is thus an infinite subset of X, and every point in X is in exactly one region. Then the set of all 22n possible unions of regions (including the empty set obtained as the union of the empty set of regions and X obtained as the union of all 2n regions) is closed under union, intersection, and complement relative to X and therefore forms a concrete Boolean algebra. Again we have finitely many subsets of an infinite set forming a concrete Boolean algebra, with Example 2 arising as the case n = 0 of no curves.
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In 1973, Pierre Deligne and Michael Rapoport showed that the ring of modular forms is finitely generated when is a congruence subgroup of . In 2003, Lev Borisov and Paul Gunnells showed that the ring of modular forms is generated in weight at most 3 when \Gamma is the congruence subgroup \Gamma_1(N) of prime level in using the theory of toric modular forms. In 2014, Nadim Rustom extended the result of Borisov and Gunnells for \Gamma_1(N) to all levels and also demonstrated that the ring of modular forms for the congruence subgroup \Gamma_0(N) is generated in weight at most 6 for some levels . In 2015, John Voight and David Zureick-Brown generalized these results: they proved that the graded ring of modular forms of even weight for any congruence subgroup of is generated in weight at most 6 with relations generated in weight at most 12.
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Nash proved that if we allow mixed strategies (where a player chooses probabilities of using various pure strategies), then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium (which might be a pure strategy for each player or might be a probability distribution over strategies for each player). Nash equilibria need not exist if the set of choices is infinite and noncompact. An example is a game where two players simultaneously name a number and the player naming the larger number wins. Another example is where each of two players chooses a real number strictly less than 5 and the winner is whoever has the biggest number; no biggest number strictly less than 5 exists (if the number could equal 5, the Nash equilibrium would have both players choosing 5 and tying the game).
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One reason to study reversible universal models of computation such as the billiard-ball model is that they could theoretically lead to actual computer systems that consume very low quantities of energy. According to Landauer's principle, irreversible computational steps require a certain minimal amount of energy per step, but reversible steps can be performed with an amount of energy per step that is arbitrarily close to zero. However, in order to perform computation using less energy than Landauer's bound, it is not good enough for a cellular automaton to have a transition function that is globally reversible: what is required is that the local computation of the transition function also be done in a reversible way. For instance, reversible block cellular automata are always locally reversible: the behavior of each individual block involves the application of an invertible function with finitely many inputs and outputs.
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In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme). The need of a theory for such an apparently simple notion results from the existence of many definitions of the dimension that are equivalent only in the most regular cases (see Dimension of an algebraic variety). A large part of dimension theory consists in studying the conditions under which several dimensions are equal, and many important classes of commutative rings may be defined as the rings such that two dimensions are equal; for example, a regular ring is a commutative ring such that the homological dimension is equal to the Krull dimension. The theory is simpler for commutative rings that are finitely generated algebras over a field, which are also quotient rings of polynomial rings in a finite number of indeterminates over a field.
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As observed by Capcarrere, Sipper, and Tomassini, the majority problem may be solved perfectly if one relaxes the definition by which the automaton is said to have recognized the majority. In particular, for the Rule 184 automaton, when run on a finite universe with cyclic boundary conditions, each cell will infinitely often remain in the majority state for two consecutive steps while only finitely many times being in the minority state for two consecutive steps. Alternatively, a hybrid automaton that runs Rule 184 for a number of steps linear in the size of the array, and then switches to the majority rule (Rule 232), that sets each cell to the majority of itself and its neighbors, solves the majority problem with the standard recognition criterion of either all zeros or all ones in the final state. However, this machine is not itself a cellular automaton.
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In the number theory of the twentieth century, the infinite descent method was taken up again, and pushed to a point where it connected with the main thrust of algebraic number theory and the study of L-functions. The structural result of Mordell, that the rational points on an elliptic curve E form a finitely- generated abelian group, used an infinite descent argument based on E/2E in Fermat's style. To extend this to the case of an abelian variety A, André Weil had to make more explicit the way of quantifying the size of a solution, by means of a height function – a concept that became foundational. To show that A(Q)/2A(Q) is finite, which is certainly a necessary condition for the finite generation of the group A(Q) of rational points of A, one must do calculations in what later was recognised as Galois cohomology.
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In mathematics, especially in the study of infinite groups, the Hirsch–Plotkin radical is a subgroup describing the normal nilpotent subgroups of the group. It was named by after Kurt Hirsch and Boris I. Plotkin, who proved that the product of locally nilpotent groups remains locally nilpotent; this fact is a key ingredient in its construction.... The Hirsch–Plotkin radical is defined as the subgroup generated by the union of the normal locally nilpotent subgroups (that is, those normal subgroups such that every finitely generated subgroup is nilpotent). The Hirsch–Plotkin radical is itself a locally nilpotent normal subgroup, so is the unique largest such.. The Hirsch–Plotkin radical generalizes the Fitting subgroup to infinite groups.. Unfortunately the subgroup generated by the union of infinitely many normal nilpotent subgroups need not itself be nilpotent,. so the Fitting subgroup must be modified in this case.. See p.
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Their local objects are affine schemes or prime spectra which are locally ringed spaces which form a category which is antiequivalent to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field k, and the category of finitely generated reduced k-algebras. The gluing is along Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set theoretic sense is then replaced by a Grothendieck topology. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely the étale topology, and the two flat Grothendieck topologies: fppf and fpqc; nowadays some other examples became prominent including Nisnevich topology.
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This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality. Diophantine geometry in general is the study of algebraic varieties V over fields K that are finitely generated over their prime fields—including as of special interest number fields and finite fields—and over local fields. Of those, only the complex numbers are algebraically closed; over any other K the existence of points of V with coordinates in K is something to be proved and studied as an extra topic, even knowing the geometry of V. Arithmetic geometry can be more generally defined as the study of schemes of finite type over the spectrum of the ring of integers.
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Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of real analysis. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of one or more digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999..., the integer part can be summarized as b0 and one can neglect negatives, so a decimal expansion has the form The fraction part, unlike the integer part, is not limited to finitely many digits. This is a positional notation, so for example the digit 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.
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The Cayley graph of the free group on two generators a and b. The ends of the group are in one-to-one correspondence with the rays (infinite paths) from the identity element e to the fringes of the drawing. Every group and a set of generators for the group determine a Cayley graph, a graph whose vertices are the group elements and the edges are pairs of elements (x,gx) where g is one of the generators. In the case of a finitely generated group, the ends of the group are defined to be the ends of the Cayley graph for the finite set of generators; this definition is invariant under the choice of generators, in the sense that if two different finite set of generators are chosen, the ends of the two Cayley graphs are in one-to-one correspondence with each other.
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In trying to formalize the argument for the reflection principle of the previous section in ZF set theory, it turns out to be necessary to add some conditions about the collection of properties A (for example, A might be finite). Doing this produces several closely related "reflection theorems" of ZFC all of which state that we can find a set that is almost a model of ZFC. One form of the reflection principle in ZFC says that for any finite set of axioms of ZFC we can find a countable transitive model satisfying these axioms. (In particular this proves that, unless inconsistent, ZFC is not finitely axiomatizable because if it were it would prove the existence of a model of itself, and hence prove its own consistency, contradicting Gödel's second incompleteness theorem.) This version of the reflection theorem is closely related to the Löwenheim–Skolem theorem.
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More generally a covariant is a polynomial in a0, ..., an, x, y that is invariant, so an invariant is a special case of a covariant where the variables x and y do not occur. More generally still, a simultaneous invariant is a polynomial in the coefficients of several different forms in x and y. In terms of representation theory, given any representation V of the group SL2(C) one can ask for the ring of invariant polynomials on V. Invariants of a binary form of degree n correspond to taking V to be the (n + 1)-dimensional irreducible representation, and covariants correspond to taking V to be the sum of the irreducible representations of dimensions 2 and n + 1\. The invariants of a binary form form a graded algebra, and proved that this algebra is finitely generated if the base field is the complex numbers.
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The ring of invariants of binary septics is anomalous and has caused several published errors. Cayley claimed incorrectly that the ring of invariants is not finitely generated. gave lower bounds of 26 and 124 for the number of generators of the ring of invariants and the ring of covariants and observed that an unproved "fundamental postulate" would imply that equality holds. However showed that Sylvester's numbers are not equal to the numbers of generators, which are 30 for the ring of invariants and at least 130 for the ring of covariants, so Sylvester's fundamental postulate is wrong. and showed that the algebra of invariants of a degree 7 form is generated by a set with 1 invariant of degree 4, 3 of degree 8, 6 of degree 12, 4 of degree 14, 2 of degree 16, 9 of degree 18, and one of each of the degrees 20, 22, 26, 30.
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In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of graded algebraic structures (where the dimension of the entire structure is often infinite). It is a formal power series in one indeterminate, say t, where the coefficient of t^n gives the dimension (or rank) of the sub- structure of elements homogeneous of degree n. It is closely related to the Hilbert polynomial in cases when the latter exists; however, the Hilbert–Poincaré series describes the rank in every degree, while the Hilbert polynomial describes it only in all but finitely many degrees, and therefore provides less information. In particular the Hilbert–Poincaré series cannot be deduced from the Hilbert polynomial even if the latter exists.
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But a thesis concerning the extent of effective > methods—which is to say, concerning the extent of procedures of a certain > sort that a human being unaided by machinery is capable of carrying > out—carries no implication concerning the extent of the procedures that > machines are capable of carrying out, even machines acting in accordance > with 'explicitly stated rules.' For among a machine's repertoire of atomic > operations there may be those that no human being unaided by machinery can > perform."Stanford Encyclopedia of Philosophy: "The Church–Turing thesis" – > by B. Jack Copeland. On the other hand, a modification of Turing's assumptions does bring practical computation within Turing's limits; as David Deutsch puts it: > "I can now state the physical version of the Church–Turing principle: 'Every > finitely realizable physical system can be perfectly simulated by a > universal model computing machine operating by finite means.
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For an n \times n knight's graph, the number of vertices is n^2 and the number of edges is 4(n-2)(n-1). A Hamiltonian cycle on the knight's graph is a (closed) knight's tour. A chessboard with an odd number of squares has no tour, because the knight's graph is a bipartite graph and only the bipartite graphs with an even number of vertices can have Hamiltonian cycles. All but finitely many chessboards with an even number of squares have a knight's tour; Schwenk's theorem provides an exact listing of which ones do and which do not.. When it is modified to have toroidal boundary conditions (meaning that a knight is not blocked by the edge of the board, but instead continues onto the opposite edge) the 4\times 4 knight's graph is the same as the four-dimensional hypercube graph.
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Each convex set containing X must (by the assumption that it is convex) contain all convex combinations of points in X, so the set of all convex combinations is contained in the intersection of all convex sets containing X. Conversely, the set of all convex combinations is itself a convex set containing X, so it also contains the intersection of all convex sets containing X, and therefore the second and third definitions are equivalent., p. 12; , p. 17. In fact, according to Carathéodory's theorem, if X is a subset of a d-dimensional Euclidean space, every convex combination of finitely many points from X is also a convex combination of at most d+1 points in X. The set of convex combinations of a (d+1)-tuple of points is a simplex; in the plane it is a triangle and in three-dimensional space it is a tetrahedron.
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The Banach–Tarski paradox, proved by Stefan Banach and Alfred Tarski in 1924, states that it is possible to partition a three-dimensional unit ball into finitely many pieces and reassemble them into two unit balls, a single ball of larger or smaller area, or any other bounded set with a non- empty interior. Although it is a mathematical theorem, it is called a paradox because it is so counter-intuitive; in the preface to the book, Jan Mycielski calls it the most surprising result in mathematics. It is closely related to measure theory and the non-existence of a measure on all subsets of three- dimensional space, invariant under all congruences of space, and to the theory of paradoxical sets in free groups and the representation of these groups by three-dimensional rotations, used in the proof of the paradox. The topic of the book is the Banach–Tarski paradox, its proof, and the many related results that have since become known.
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In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, which were introduced for solving important open questions in invariant theory, and are at the basis of modern algebraic geometry. The two other theorems are Hilbert's basis theorem that asserts that all ideals of polynomial rings over a field are finitely generated, and Hilbert's Nullstellensatz, which establishes a bijective correspondence between affine algebraic varieties and prime ideals of polynomial rings. Hilbert's syzygy theorem concern the relations, or syzygies in Hilbert's terminology, between the generators of an ideal, or, more generally, a module. As the relations form a module, one may consider the relations between the relations; Hilbert's syzygy theorem asserts that, if one continues in this way, starting with a module over a polynomial ring in indeterminates over a field, one eventually finds a zero module of relations, after at most steps.
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A map graph (top), the cocktail party graph K2,2,2,2, defined by corner adjacency of eight regions in the plane (lower left), or as the half-square of a planar bipartite graph (lower right, the graph of the rhombic dodecahedron) In graph theory, a branch of mathematics, a map graph is an undirected graph formed as the intersection graph of finitely many simply connected and internally disjoint regions of the Euclidean plane. The map graphs include the planar graphs, but are more general. Any number of regions can meet at a common corner (as in the Four Corners of the United States, where four states meet), and when they do the map graph will contain a clique connecting the corresponding vertices, unlike planar graphs in which the largest cliques have only four vertices.. Another example of a map graph is the king's graph, a map graph of the squares of the chessboard connecting pairs of squares between which the chess king can move.
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In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution uses coresolution, though right resolution is more common, as in ) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to define invariants characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for (left) resolutions, and to the right for right resolutions. However, a finite resolution is one where only finitely many of the objects in the sequence are non-zero; it is usually represented by a finite exact sequence in which the leftmost object (for resolutions) or the rightmost object (for coresolutions) is the zero- object., Generally, the objects in the sequence are restricted to have some property P (for example to be free).
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In his later work, Tarski showed that, conversely, non-existence of paradoxical decompositions of this type implies the existence of a finitely-additive invariant measure. The heart of the proof of the "doubling the ball" form of the paradox presented below is the remarkable fact that by a Euclidean isometry (and renaming of elements), one can divide a certain set (essentially, the surface of a unit sphere) into four parts, then rotate one of them to become itself plus two of the other parts. This follows rather easily from a -paradoxical decomposition of , the free group with two generators. Banach and Tarski's proof relied on an analogous fact discovered by Hausdorff some years earlier: the surface of a unit sphere in space is a disjoint union of three sets and a countable set such that, on the one hand, are pairwise congruent, and on the other hand, is congruent with the union of and .
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In the Euclidean plane, two figures that are equidecomposable with respect to the group of Euclidean motions are necessarily of the same area, and therefore, a paradoxical decomposition of a square or disk of Banach–Tarski type that uses only Euclidean congruences is impossible. A conceptual explanation of the distinction between the planar and higher-dimensional cases was given by John von Neumann: unlike the group SO(3) of rotations in three dimensions, the group E(2) of Euclidean motions of the plane is solvable, which implies the existence of a finitely-additive measure on E(2) and R2 which is invariant under translations and rotations, and rules out paradoxical decompositions of non-negligible sets. Von Neumann then posed the following question: can such a paradoxical decomposition be constructed if one allows a larger group of equivalences? It is clear that if one permits similarities, any two squares in the plane become equivalent even without further subdivision.
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In additive number theory, an additive basis is a set S of natural numbers with the property that, for some finite number k, every natural number can be expressed as a sum of k or fewer elements of S. That is, the sumset of k copies of S consists of all natural numbers. The order or degree of an additive basis is the number k. When the context of additive number theory is clear, an additive basis may simply be called a basis. An asymptotic additive basis is a set S for which all but finitely many natural numbers can be expressed as a sum of k or fewer elements of S. For example, by Lagrange's four-square theorem, the set of square numbers is an additive basis of order four, and more generally by the Fermat polygonal number theorem the polygonal numbers for k-sided polygons form an additive basis of order k.
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For any partition of the vertices of the Rado graph into two sets A and B, or more generally for any partition into finitely many subsets, at least one of the subgraphs induced by one of the partition sets is isomorphic to the whole Rado graph. gives the following short proof: if none of the parts induces a subgraph isomorphic to the Rado graph, they all fail to have the extension property, and one can find pairs of sets Ui and Vi that cannot be extended within each subgraph. But then, the union of the sets Ui and the union of the sets Vi would form a set that could not be extended in the whole graph, contradicting the Rado graph's extension property. This property of being isomorphic to one of the induced subgraphs of any partition is held by only three countably infinite undirected graphs: the Rado graph, the complete graph, and the empty graph.
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Schematic depiction of ramification: the fibers of almost all points in Y below consist of three points, except for two points in Y marked with dots, where the fibers consist of one and two points (marked in black), respectively. The map f is said to be ramified in these points of Y. Ramification, generally speaking, describes a geometric phenomenon that can occur with finite-to-one maps (that is, maps f: X → Y such that the preimages of all points y in Y consist only of finitely many points): the cardinality of the fibers f−1(y) will generally have the same number of points, but it occurs that, in special points y, this number drops. For example, the map :C → C, z ↦ zn has n points in each fiber over t, namely the n (complex) roots of t, except in t = 0, where the fiber consists of only one element, z = 0. One says that the map is "ramified" in zero.
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Solymosi was the first online contributor to the first Polymath Project, set by Timothy Gowers to find improvements to the Hales–Jewett theorem. One of his theorems states that if a finite set of points in the Euclidean plane has every pair of points at an integer distance from each other, then the set must have a diameter (largest distance) that is linear in the number of points. This result is connected to the Erdős–Anning theorem, according to which an infinite set of points with integer distances must lie on one line. In connection with the related Erdős–Ulam problem, on the existence of dense subsets of the plane for which all distances are rational numbers, Solymosi and de Zeeuw proved that every infinite rational-distance set must either be dense in the Zariski topology or it must have all but finitely many of its points on a single line or circle.
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Several distinct definitions generalize divisible groups to divisible modules. The following definitions have been used in the literature to define a divisible module M over a ring R: # rM = M for all nonzero r in R. (It is sometimes required that r is not a zero-divisor, and some authors require that R is a domain.) # For every principal left ideal Ra, any homomorphism from Ra into M extends to a homomorphism from R into M. (This type of divisible module is also called principally injective module.) # For every finitely generated left ideal L of R, any homomorphism from L into M extends to a homomorphism from R into M. The last two conditions are "restricted versions" of the Baer's criterion for injective modules. Since injective left modules extend homomorphisms from all left ideals to R, injective modules are clearly divisible in sense 2 and 3. If R is additionally a domain then all three definitions coincide.
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If is a predecessor of according to this test, then their meet is , and similarly if is a predecessor of then their meet is . In a second case, if neither nor is the predecessor of the other, but one or both of them begins with a “1” digit, the meet is unchanged if these initial digits are removed. And finally, if both and begin with the digit “2”, the meet of and may be found by removing this digit from both of them, finding the meet of the resulting suffixes, and adding the “2” back to the start. A common successor of and (though not necessarily the least common successor) may be found by taking a string of “2” digits with length equal to the longer of and . The least common successor is then the meet of the finitely many strings that are common successors of and and predecessors of this string of “2”s.
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85 of: :"In accordance with this, already in the plane there is no nonnegative additive measure (for which the unit square has a measure of 1), which is invariant with respect to all transformations belonging to A2 [the group of area- preserving affine transformations]." To explain this a bit more, the question of whether a finitely additive measure exists, that is preserved under certain transformations, depends on what transformations are allowed. The Banach measure of sets in the plane, which is preserved by translations and rotations, is not preserved by non-isometric transformations even when they do preserve the area of polygons. As explained above, the points of the plane (other than the origin) can be divided into two dense sets which we may call A and B. If the A points of a given polygon are transformed by a certain area- preserving transformation and the B points by another, both sets can become subsets of the B points in two new polygons.
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A coproduct in the category of algebras is a free product of algebras.) Direct products are commutative and associative (up to isomorphism), meaning that it doesn't matter in which order one forms the direct product. If Ai is an ideal of Ri for each i in I, then is an ideal of R. If I is finite, then the converse is true, i.e., every ideal of R is of this form. However, if I is infinite and the rings Ri are non-zero, then the converse is false: the set of elements with all but finitely many nonzero coordinates forms an ideal which is not a direct product of ideals of the Ri. The ideal A is a prime ideal in R if all but one of the Ai are equal to Ri and the remaining Ai is a prime ideal in Ri. However, the converse is not true when I is infinite.
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The easiest (though somewhat heavy) construction starts with the ring of formal power series RX_1,X_2,... over R in infinitely (countably) many indeterminates; the elements of this power series ring are formal infinite sums of terms, each of which consists of a coefficient from R multiplied by a monomial, where each monomial is a product of finitely many finite powers of indeterminates. One defines ΛR as its subring consisting of those power series S that satisfy #S is invariant under any permutation of the indeterminates, and #the degrees of the monomials occurring in S are bounded. Note that because of the second condition, power series are used here only to allow infinitely many terms of a fixed degree, rather than to sum terms of all possible degrees. Allowing this is necessary because an element that contains for instance a term X1 should also contain a term Xi for every i > 1 in order to be symmetric.
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A notable example of a process that is not even quasi-static is the slow heat exchange between two bodies on two finitely different temperatures, where the heat exchange rate is controlled by an approximately adiabatic partition between the two bodies-- in this case, no matter how slowly the process takes place, the states of the composite system consisting of the two bodies is far from equilibrium, since thermal equilibrium for this composite system requires that the two bodies be at the same temperature. Some ambiguity exists in the literature concerning the distinction between quasi-static and reversible processes, as these are sometimes taken as synonyms. The reason is the theorem that any reversible process is also a quasi-static one, even though (as we have illustrated above) the converse is not true. In practical situations, it is essential to differentiate between the two: any engineer would remember to include friction when calculating the dissipative entropy generation, so there are no reversible processes in practice.
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Commonly in secondary schools' mathematics education, the real numbers are constructed by defining a number using an integer followed by a radix point and an infinite sequence written out as a string to represent the fractional part of any given real number. In this construction, the set of any combination of an integer and digits after the decimal point (or radix point in non-base 10 systems) is the set of real numbers. This construction can be rigorously shown to satisfy all of the real axioms after defining an equivalence relation over the set that defines 1 =eq 0.999... as well as for any other nonzero decimals with only finitely many nonzero terms in the decimal string with its trailing 9s version. With this construction of the reals, all proofs of the statement "1 = 0.999..." can be viewed as implicitly assuming the equality when any operations are performed on the real numbers.
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Generally speaking, risk-neutral pricing in structural models of financial interconnectedness requires unique equilibrium prices at maturity in dependence of the exogenous asset price vector, which can be random. While financially interconnected systems with debt and equity cross- ownership without derivatives are fairly well understood in the sense that relatively weak conditions on the ownership structures in the form of ownership matrices are required to warrant uniquely determined price equilibria, the Fischer (2014) model needs very strong conditions on derivatives – which are defined in dependence on any other liability of the considered financial system – to be able to guarantee uniquely determined prices of all system-endogenous liabilities. Furthermore, it is known that there exist examples with no solutions at all, finitely many solutions (more than one), and infinitely many solutions. At present, it is unclear how weak conditions on derivatives can be chosen to still be able to apply risk-neutral pricing in financial networks with systemic risk.
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In 1995, Tian and Weiyue Ding studied the harmonic map heat flow of a two-dimensional closed Riemannian manifold into a closed Riemannian manifold . In a seminal 1985 work, following the 1982 breakthrough of Jonathan Sacks and Karen Uhlenbeck, Michael Struwe had studied this problem and showed that there is a weak solution which exists for all positive time. Furthermore, Struwe showed that the solution is smooth away from finitely many spacetime points; given any sequence of spacetime points at which the solution is smooth and which converge to a given singular point , one can perform some rescalings to (subsequentially) define a finite number of harmonic maps from the round 2-dimensional sphere into , called "bubbles." Ding and Tian proved a certain "energy quantization," meaning that the defect between the Dirichlet energy of and the limit of the Dirichlet energy of as approaches is exactly measured by the sum of the Dirichlet energies of the bubbles.
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If Γ is a quiver, then a path in Γ is a sequence of arrows an an−1 ... a3 a2 a1 such that the head of ai+1 is the tail of ai for i = 1, ..., n−1, using the convention of concatenating paths from right to left. If K is a field then the quiver algebra or path algebra KΓ is defined as a vector space having all the paths (of length ≥ 0) in the quiver as basis (including, for each vertex i of the quiver Γ, a trivial path e_i of length 0; these paths are not assumed to be equal for different i), and multiplication given by concatenation of paths. If two paths cannot be concatenated because the end vertex of the first is not equal to the starting vertex of the second, their product is defined to be zero. This defines an associative algebra over K. This algebra has a unit element if and only if the quiver has only finitely many vertices.
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In 1941, while studying H^2(G,\Z) (which plays a special role in groups), Heinz Hopf discovered what is now called Hopf's integral homology formula , which is identical to Schur's formula for the Schur multiplier of a finite, finitely presented group: : H_2(G,\Z) \cong (R \cap [F, F])/[F, R], where G\cong F/R and F is a free group. Hopf's result led to the independent discovery of group cohomology by several groups in 1943-45: Samuel Eilenberg and Saunders Mac Lane in the United States ; Hopf and Beno Eckmann in Switzerland; and Hans Freudenthal in the Netherlands . The situation was chaotic because communication between these countries was difficult during World War II. From a topological point of view, the homology and cohomology of G was first defined as the homology and cohomology of a model for the topological classifying space BG as discussed above. In practice, this meant using topology to produce the chain complexes used in formal algebraic definitions.
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The generalized Fermat equation generalizes the statement of Fermat's last theorem by considering positive integer solutions a, b, c, m, n, k satisfying In particular, the exponents m, n, k need not be equal, whereas Fermat's last theorem considers the case The Beal conjecture, also known as the Mauldin conjecture and the Tijdeman-Zagier conjecture, states that there are no solutions to the generalized Fermat equation in positive integers a, b, c, m, n, k with a, b, and c being pairwise coprime and all of m, n, k being greater than 2. The Fermat–Catalan conjecture generalizes Fermat's last theorem with the ideas of the Catalan conjecture. The conjecture states that the generalized Fermat equation has only finitely many solutions (a, b, c, m, n, k) with distinct triplets of values (am, bn, ck), where a, b, c are positive coprime integers and m, n, k are positive integers satisfying The statement is about the finiteness of the set of solutions because there are 10 known solutions.
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See general set theory for more details. Q is fascinating because it is a finitely axiomatized first-order theory that is considerably weaker than Peano arithmetic (PA), and whose axioms contain only one existential quantifier, yet like PA is incomplete and incompletable in the sense of Gödel's incompleteness theorems, and essentially undecidable. Robinson (1950) derived the Q axioms (1)–(7) above by noting just what PA axioms are required to prove (Mendelson 1997: Th. 3.24) that every computable function is representable in PA. The only use this proof makes of the PA axiom schema of induction is to prove a statement that is axiom (3) above, and so, all computable functions are representable in Q (Mendelson 1997: Th. 3.33, Rautenberg 2010: 246). The conclusion of Gödel's second incompleteness theorem also holds for Q: no consistent recursively axiomatized extension of Q can prove its own consistency, even if we additionally restrict Gödel numbers of proofs to a definable cut (Bezboruah and Shepherdson 1976; Pudlák 1985; Hájek & Pudlák 1993:387).
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Suppose A is a set of impartial combinatorial games that is finitely-generated with respect to disjunctive sums and closed in both of the following senses: (1) Additive closure: If G and H are games in A, then their disjunctive sum G + H is also in A. (2) Hereditary closure: If G is a game in A and H is an option of G, then H is also in A. Next, define on A the indistinguishability congruence ≈ that relates two games G and H if for every choice of a game X in A, the two positions G+X and H+X have the same outcome (i.e., are either both first- player wins in best play of A, or alternatively are both second-player wins). One easily checks that ≈ is indeed a congruence on the set of all disjunctive position sums in A, and that this is true regardless of whether the game is played in normal or misere play. The totality of all the congruence classes form the indistinguishability quotient.
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The nodes are thus in a one-to-one correspondence with finite (possibly empty) sequences of positive numbers, which are countable and can be placed in order first by sum of entries, and then by lexicographic order within a given sum (only finitely many sequences sum to a given value, so all entries are reached—formally there are a finite number of compositions of a given natural number, specifically 2n−1 compositions of ), which gives a traversal. Explicitly: 0: () 1: (1) 2: (1, 1) (2) 3: (1, 1, 1) (1, 2) (2, 1) (3) 4: (1, 1, 1, 1) (1, 1, 2) (1, 2, 1) (1, 3) (2, 1, 1) (2, 2) (3, 1) (4) etc. This can be interpreted as mapping the infinite depth binary tree onto this tree and then applying breadth-first search: replace the "down" edges connecting a parent node to its second and later children with "right" edges from the first child to the second child, from the second child to the third child, etc.
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Another argument against the axiom of choice is that it implies the existence of objects that may seem counterintuitive.. One example is the Banach–Tarski paradox which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into two solid balls each with the same volume as the original. The pieces in this decomposition, constructed using the axiom of choice, are non-measurable sets. Despite these seemingly paradoxical facts, most mathematicians accept the axiom of choice as a valid principle for proving new results in mathematics. The debate is interesting enough, however, that it is considered of note when a theorem in ZFC (ZF plus AC) is logically equivalent (with just the ZF axioms) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type which requires the axiom of choice to be true.
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In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used to refer to the compactified modular curves X(Γ) which are compactifications obtained by adding finitely many points (called the cusps of Γ) to this quotient (via an action on the extended complex upper-half plane). The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to complex numbers, and, moreover, prove that modular curves are defined either over the field Q of rational numbers, or a cyclotomic field. The latter fact and its generalizations are of fundamental importance in number theory.
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The Bateman–Horn conjecture provides a conjectured density for the positive integers at which a given set of polynomials all have prime values. For a set of m distinct irreducible polynomials ƒ1, ..., ƒm with integer coefficients, an obvious necessary condition for the polynomials to simultaneously generate prime values infinitely often is that they satisfy Bunyakovsky's property, that there does not exist a prime number p that divides their product f(n) for every positive integer n. For, if there were such a prime p, having all values of the polynomials simultaneously prime for a given n would imply that at least one of them must be equal to p, which can only happen for finitely many values of n or there would be a polynomial with infinitely many roots, whereas the conjecture is how to give conditions where the values are simultaneously prime for infinitely many n. An integer n is prime-generating for the given system of polynomials if every polynomial ƒi(n) produces a prime number when given n as its argument.
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In the other direction, suppose that RL has finitely many equivalence classes. In this case, it is possible to design a deterministic finite automaton that has one state for each equivalence class. The start state of the automaton corresponds to the equivalence class containing the empty string, and the transition function from a state X on input symbol y takes the automaton to a new state, the state corresponding to the equivalence class containing string xy, where x is an arbitrarily chosen string in the equivalence class for X. The definition of the Myhill–Nerode relation implies that the transition function is well- defined: no matter which representative string x is chosen for state X, the same transition function value will result. A state of this automaton is accepting if the corresponding equivalence class contains a string in L; in this case, again, the definition of the relation implies that all strings in the same equivalence class must also belong to L, for otherwise the empty string would be a distinguishing string for some pairs of strings in the class.
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A class K of structures of a signature σ is called an elementary class if there is a first-order theory T of signature σ, such that K consists of all models of T, i.e., of all σ-structures that satisfy T. If T can be chosen as a theory consisting of a single first-order sentence, then K is called a basic elementary class. More generally, K is a pseudo-elementary class if there is a first-order theory T of a signature that extends σ, such that K consists of all σ-structures that are reducts to σ of models of T. In other words, a class K of σ-structures is pseudo-elementary iff there is an elementary class K' such that K consists of precisely the reducts to σ of the structures in K'. For obvious reasons, elementary classes are also called axiomatizable in first-order logic, and basic elementary classes are called finitely axiomatizable in first-order logic. These definitions extend to other logics in the obvious way, but since the first-order case is by far the most important, axiomatizable implicitly refers to this case when no other logic is specified.
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An interval exchange transformation is a dynamical system defined from a partition of the unit interval into finitely many smaller intervals, and a permutation on those intervals. Veech and Howard Masur independently discovered that, for almost every partition and every irreducible permutation, these systems are uniquely ergodic, and also made contributions to the theory of weak mixing for these systems.. See in particular p. 51. The Rauzy–Veech–Zorich induction map, a function from and to the space of interval exchange transformations is named in part after Veech: Rauzy defined the map, Veech constructed an infinite invariant measure for it, and Zorich strengthened Veech's result by making the measure finite.. The Veech surface and the related Veech group are named after Veech, as is the Veech dichotomy according to which geodesic flow on the Veech surface is either periodic or ergodic.. Veech played a role in the Nobel-prize-winning discovery of buckminsterfullerene in 1985 by a team of Rice University chemists including Richard Smalley. At that time, Veech was chair of the Rice mathematics department, and was asked by Smalley to identify the shape that the chemists had determined for this molecule.
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Radó's 1962 paper proved that if f: ℕ → ℕ is any computable function, then Σ(n) > f(n) for all sufficiently large n, and hence that Σ is not a computable function. Moreover, this implies that it is undecidable by a general algorithm whether an arbitrary Turing machine is a busy beaver. (Such an algorithm cannot exist, because its existence would allow Σ to be computed, which is a proven impossibility. In particular, such an algorithm could be used to construct another algorithm that would compute Σ as follows: for any given n, each of the finitely many n-state 2-symbol Turing machines would be tested until an n-state busy beaver is found; this busy beaver machine would then be simulated to determine its score, which is by definition Σ(n).) Even though Σ(n) is an uncomputable function, there are some small n for which it is possible to obtain its values and prove that they are correct. It is not hard to show that Σ(0) = 0, Σ(1) = 1, Σ(2) = 4, and with progressively more difficulty it can be shown that Σ(3) = 6 and Σ(4) = 13 .
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This addition of morphism turns Ab into a preadditive category, and because the direct sum of finitely many abelian groups yields a biproduct, we indeed have an additive category. In Ab, the notion of kernel in the category theory sense coincides with kernel in the algebraic sense, i.e. the categorical kernel of the morphism f : A → B is the subgroup K of A defined by K = {x ∈ A : f(x) = 0}, together with the inclusion homomorphism i : K → A. The same is true for cokernels; the cokernel of f is the quotient group C = B / f(A) together with the natural projection p : B → C. (Note a further crucial difference between Ab and Grp: in Grp it can happen that f(A) is not a normal subgroup of B, and that therefore the quotient group B / f(A) cannot be formed.) With these concrete descriptions of kernels and cokernels, it is quite easy to check that Ab is indeed an abelian category. The product in Ab is given by the product of groups, formed by taking the cartesian product of the underlying sets and performing the group operation componentwise.
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If L is a regular language, then by definition there is a DFA A that recognizes it, with only finitely many states. If there are n states, then partition the set of all finite strings into n subsets, where subset Si is the set of strings that, when given as input to automaton A, cause it to end in state i. For every two strings x and y that belong to the same subset, and for every choice of a third string z, automaton A reaches the same state on input xz as it reaches on input yz, and therefore must either accept both of the inputs xz and yz or reject both of them. Therefore, no string z can be a distinguishing extension for x and y, so they must be related by RL. Thus, Si is a subset of an equivalence class of RL. Combining this fact with the fact that every member of one of these equivalence classes belongs to one of the sets Si, this gives a many-to-one relation from states of A to equivalence classes, implying that the number of equivalence classes is finite and at most n.
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In EGA III, Grothendieck calls the following statement which does not involve connectedness a "Main theorem" of Zariski : :If f:X->Y is a quasi-projective morphism of Noetherian schemes then the set of points that are isolated in their fiber is open in X. Moreover the induced scheme of this set is isomorphic to an open subset of a scheme that is finite over Y. In EGA IV, Grothendieck observed that the last statement could be deduced from a more general theorem about the structure of quasi-finite morphisms, and the latter is often referred to as the "Zariski's main theorem in the form of Grothendieck". It is well known that open immersions and finite morphisms are quasi-finite. Grothendieck proved that under the hypothesis of separatedness all quasi-finite morphisms are compositions of such : :if Y is a quasi-compact separated scheme and f: X \to Y is a separated, quasi-finite, finitely presented morphism then there is a factorization into X \to Z \to Y, where the first map is an open immersion and the second one is finite. The relation between this theorem about quasi-finite morphisms and Théorème 4.4.
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The halved cube graph of order 4, obtained as the bipartite half of an order-4 hypercube graph In graph theory, the bipartite half or half-square of a bipartite graph G = (U,V,E) is a graph whose vertex set is one of the two sides of the bipartition (without loss of generality, U) and in which there is an edge uiuj for each two vertices ui and uj in U that are at distance two from each other in G.. That is, in a more compact notation, the bipartite half is G2[U] where the superscript 2 denotes the square of a graph and the square brackets denote an induced subgraph. For instance, the bipartite half of the complete bipartite graph Kn,n is the complete graph Kn and the bipartite half of the hypercube graph is the halved cube graph. When G is a distance-regular graph, its two bipartite halves are both distance-regular.. For instance, the halved Foster graph is one of finitely many degree-6 distance-regular locally linear graphs. The map graphs, that is, the intersection graphs of interior- disjoint simply-connected regions in the plane, are exactly the bipartite halves of bipartite planar graphs..
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Leonard Eugene Dickson studied generalizations of Waring's problem for seventh powers, showing that every non-negative integer can be represented as a sum of at most 258 non-negative seventh powers. All but finitely many positive integers can be expressed more simply as the sum of at most 46 seventh powers. If negative powers are allowed, only 12 powers are required. The smallest number that can be represented in two different ways as a sum of four positive seventh powers is 2056364173794800. The smallest seventh power that can be represented as a sum of eight distinct seventh powers is: :102^7=12^7+35^7+53^7+58^7+64^7+83^7+85^7+90^7. The two known examples of a seventh power expressible as the sum of seven seventh powers are :568^7 = 127^7+ 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7 (M. Dodrill, 1999); and : 626^7=625^7+309^7+258^7+255^7+158^7+148^7+91^7 (Maurice Blondot, 11/14/2000); any example with fewer terms in the sum would be a counterexample to Euler's sum of powers conjecture, which is currently only known to be false for the powers 4 and 5.
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Second, introducing a height function h on the rational points E(Q) defined by h(P0) = 0 and if P (unequal to the point at infinity P0) has as abscissa the rational number x = p/q (with coprime p and q). This height function h has the property that h(mP) grows roughly like the square of m. Moreover, only finitely many rational points with height smaller than any constant exist on E. The proof of the theorem is thus a variant of the method of infinite descentSee also J. W. S. Cassels, Mordell's Finite Basis Theorem Revisited, Mathematical Proceedings of the Cambridge Philosophical Society 100, 3–41 and the comment of A. Weil on the genesis of his work: A. Weil, Collected Papers, vol. 1, 520–521. and relies on the repeated application of Euclidean divisions on E: let P ∈ E(Q) be a rational point on the curve, writing P as the sum 2P1 \+ Q1 where Q1 is a fixed representant of P in E(Q)/2E(Q), the height of P1 is about of the one of P (more generally, replacing 2 by any m > 1, and by ). Redoing the same with P1, that is to say P1 = 2P2 \+ Q2, then P2 = 2P3 \+ Q3, etc.
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