Not just social consequence, but there's high emotional quotients, there's all kinds of things.
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Head-turning quotients: It is ugly, or cute, depending on who is looking, or their mood.
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At Labs in the 1930s, a roomful of human 'computers' figured complex number quotients and products using commercial mechanical calculators.
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He's sharp and alert adapting to different speeds and melody quotients; what he's less capable of modulating is vocal grain.
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I learned, among other things, that some of the greatest psychologists believed in the ability of pets to elevate our happiness quotients.
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Instead, it heightens the screenplay's sentimentality, tidy psychologizing and life-affirming messages by thickening their syrup and corn quotients in ways presumably deemed palatable to theatergoing children and their parents.
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The victory was seen a defining moment in Indian politics as it seemed to user in an alternative to the existing political establishment populated with dynastic, wealthy and people with 'winnability quotients'.
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As I slowly rubbed her back, counting each stroke and feeling my love and gratitude quotients rising, I knew that this was a perfect moment, a small but vital opportunity to cement a bond that would help to benefit the two of us now and forever.
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A basic result is that geometric quotients (e.g., G/H) and GIT quotients (e.g., X/\\!/G) are categorical quotients.
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DM stacks are limited to quotients by finite group actions. While this suffices for many problems in moduli theory, it is too restrictive for others, and Artin stacks permit more general quotients.
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Her mathematical research concerns symplectic geometry, including work on Hamiltonian actions and symplectic quotients.
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The definition of strict differentiability avoids this problem by imposing a condition directly on the difference quotients.
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The quotients by congruence subgroups are of significant interest. Other important quotients are the triangle groups, which correspond geometrically to descending to a cylinder, quotienting the coordinate modulo , as . is the group of icosahedral symmetry, and the triangle group (and associated tiling) is the cover for all Hurwitz surfaces.
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The nested interval model stores the position of the nodes as rational numbers expressed as quotients (n/d).
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In 1978 he was an Invited Speaker with talk The spectrum of compact quotients of semisimple Lie groupsWallach, Nolan R. "The spectrum of compact quotients of semisimple Lie groups." In Proceedings of the International Congress of Mathematicians (Helsinki, 1978 (Helsinki), pp. 715–719 (Acad. Sci. Fennica, 1980), (81f: 22021). 1980.
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In particular, for an integral domain, the injective hull of the ring (considered as a module over itself) is the field of fractions. The injective hulls of nonsingular rings provide an analogue of the ring of quotients for non-commutative rings, where the absence of the Ore condition may impede the formation of the classical ring of quotients. This type of "ring of quotients" (as these more general "fields of fractions" are called) was pioneered in , and the connection to injective hulls was recognized in .
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Of the 26 sporadic groups, 20 can be seen inside the Monster group as subgroups or quotients of subgroups (sections).
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If one drops "pseudo", one cannot take quotients. Approach spaces are a generalization of metric spaces that maintains these good categorical properties.
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It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums.
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The arithmetic rods included a set of Napier's Bones. They were capable of assisting with multiplication of multi-digit numbers, and producing quotients.
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The corresponding simple Lie groups with non-trivial center can be obtained as quotients of this universal cover by a subgroup of the center.
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In mathematics, a Zimmert set is a set of positive integers associated with the structure of quotients of hyperbolic three-space by a Bianchi group.
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While national input–output tables are commonly created by countries' statistics agencies, officially published regional input–output tables are rare. Therefore, economists often use location quotients to create regional multipliers starting from national data. A. T. Flegg , C. D. Webber & M. V. Elliott "On the Appropriate Use of Location Quotients in Generating Regional Input–Output Tables", 16 July 2007. Retrieved on 29 May 2019.
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Two dimensional log terminal singularities are analytically isomorphic to quotients of C2 by finite subgroups of GL2(C). Two dimensional log canonical singularities have been classified by .
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A right Goldie ring is a ring that has finite uniform dimension (also called finite rank) as a right module over itself, and satisfies the ascending chain condition on right annihilators of its subsets. Goldie's theorem states that the semiprime right Goldie rings are precisely those that have a semisimple Artinian right classical ring of quotients. The Artin–Wedderburn theorem then completely determines the structure of this ring of quotients.
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This implicitly gives all modular partitions of V. It is in this sense that the modular decomposition tree "subsumes" all other ways of recursively decomposing G into quotients.
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Kronheimer, P.B. The construction of ALE spaces as hyper-Kähler quotients. J. Differential Geom. 29 (1989), no. 3, 665–683.Joyce, Dominic D. Compact Riemannian 7-manifolds with holonomy G2.
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Strong approximation for semi- simple groups over function fields, Annals of Mathematics 105(1977), 553-572. [4]. Volumes of S-arithmetic quotients of semi-simple groups, Publ.Math.IHES 69(1989), 91-117. [5].
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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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These algebras include certain quotients of the group algebras of braid groups. The presence of this commutative operator algebra plays a significant role in the harmonic analysis of modular forms and generalisations.
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In addition its canonical slot is marked as occupied. The runs for quotients 1 and 2 now comprise a cluster. In state 3 element a has been added. Its quotient is 1.
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However, for quotients more generally, the blocks of the partition giving rise to the quotient do not need to form connected subgraphs. If G is a covering graph of another graph H, then H is a quotient graph of G. The blocks of the corresponding partition are the inverse images of the vertices of H under the covering map. However, covering maps have an additional requirement that is not true more generally of quotients, that the map be a local isomorphism..
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A Fitting chain' (or Fitting series or ') for a group is a subnormal series with nilpotent quotients. In other words, a finite sequence of subgroups including both the whole group and the trivial group, such that each is a normal subgroup of the previous one, and such that the quotients of successive terms are nilpotent groups. The Fitting length or nilpotent length of a group is defined to be the smallest possible length of a Fitting chain, if one exists.
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Let X and Y be sets equipped with equivalence relations (or PERs) \approx_X, \approx_Y. For f,g : X \to Y, define f \approx g to mean: : \forall x_0 \; x_1, \quad x_0 \approx_X x_1 \Rightarrow f(x_0) \approx_Y g(x_1) then f \approx f means that f induces a well-defined function of the quotients X / \approx_X \; \to \; Y / \approx_Y. Thus, the PER \approx captures both the idea of definedness on the quotients and of two functions inducing the same function on the quotient.
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Lehmer's algorithm is based on the observation that the initial quotients produced by Euclid's algorithm can be determined based on only the first few digits; this is useful for numbers that are larger than a computer word. In essence, one extracts initial digits, typically forming one or two computer words, and runs Euclid's algorithms on these smaller numbers, as long as it is guaranteed that the quotients are the same with those that would be obtained with the original numbers. Those quotients are collected into a small 2-by-2 transformation matrix (that is a matrix of single-word integers), for using them all at once for reducing the original numbers. This process is repeated until numbers have a size for which the binary algorithm (see below) is more efficient.
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Thomas Friedman's formula for CQ Friedman's claim is that Curiosity quotient plus Passion quotient is greater than Intelligence Quotient. There is no evidence that this inequality is true. Friedman may believe that curiosity and passion are 'greater' than intelligence, but there is no evidence to suggest that the sum of a person's curiosity and passion quotients will always exceed their IQ. Indeed, given the ordinal nature of psychometric quotients, it is not clear whether it makes sense to add the curiosity and passion quotients or even if they can have numerical values attributed to them. According to Friedman, curiosity and passion are key components for education in a world where information is readily available to everyone and where global markets reward those who have learned how to learn and are self-motivated to learn.
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Hopf surfaces are quotients of C2−(0,0) by a discrete group G acting freely, and have vanishing second Betti numbers. The simplest example is to take G to be the integers, acting as multiplication by powers of 2; the corresponding Hopf surface is diffeomorphic to S1×S3. Inoue surfaces are certain class VII surfaces whose universal cover is C×H where H is the upper half plane (so they are quotients of this by a group of automorphisms). They have vanishing second Betti numbers.
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A Melnikov formation is closed under taking quotients, normal subgroups and group extensions. Thus a Melnikov formation M has the property that for every short exact sequence :1 \rightarrow A \rightarrow B \rightarrow C \rightarrow 1\ A and C are in M if and only if B is in M.Fried & Jarden (2004) p.344 A full formation is a Melnikov formation which is also closed under taking subgroups. An almost full formation is one which is closed under quotients, direct products and subgroups, but not necessarily extensions.
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The subdirect representation theorem of universal algebra states that every algebra is subdirectly representable by its subdirectly irreducible quotients. An equivalent definition of "subdirect irreducible" therefore is any algebra A that is not subdirectly representable by those of its quotients not isomorphic to A. (This is not quite the same thing as "by its proper quotients" because a proper quotient of A may be isomorphic to A, for example the quotient of the semilattice (Z, min) obtained by identifying just the two elements 3 and 4.) An immediate corollary is that any variety, as a class closed under homomorphisms, subalgebras, and direct products, is determined by its subdirectly irreducible members, since every algebra A in the variety can be constructed as a subalgebra of a suitable direct product of the subdirectly irreducible quotients of A, all of which belong to the variety because A does. For this reason one often studies not the variety itself but just its subdirect irreducibles. An algebra A is subdirectly irreducible if and only if it contains two elements that are identified by every proper quotient, equivalently, if and only if its lattice Con A of congruences has a least nonidentity element.
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A voter who has given a dangler transfers a portion of the vote to the other party. So if there are 30 seats in the council, and a person votes Labor but writes in a personal vote to a person on the Conservative list, Labor gets 29/30 of a vote, while the Conservative list gets 1/30 of a vote. Quotients are determined by dividing the number of votes each party received by 1.4, then by 3, 5, 7, 9, and so on. The quotients are then ranked from largest to smallest.
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Using a counting argument one can show that there exist transcendental numbers which have bounded partial quotients and hence are not Liouville numbers. Using the explicit continued fraction expansion of e, one can show that e is not a Liouville number (although the partial quotients in its continued fraction expansion are unbounded). Kurt Mahler showed in 1953 that is also not a Liouville number. It is conjectured that all infinite continued fractions with bounded terms that are not eventually periodic are transcendental (eventually periodic continued fractions correspond to quadratic irrationals)..
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In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory. Geometric invariant theory studies an action of a group G on an algebraic variety (or scheme) X and provides techniques for forming the 'quotient' of X by G as a scheme with reasonable properties. One motivation was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects.
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There is a fruitful relation between infinite abstract groups and topological groups: whenever a group Γ can be realized as a lattice in a topological group G, the geometry and analysis pertaining to G yield important results about Γ. A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups (profinite groups): for example, a single p-adic analytic group G has a family of quotients which are finite p-groups of various orders, and properties of G translate into the properties of its finite quotients.
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The projective linear group and the projective special linear group are the quotients of and by their centers (which consist of the multiples of the identity matrix therein); they are the induced action on the associated projective space.
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The Odd-Man-Out RT test correlates with "Intelligent Quotients (IQ) in the range of 0.30 to 0.60, a reliable and substantial effect." This correlation range is typically higher than the correlations to IQ in other reaction time tests.
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Unlike virtually every other palaeognath, which are generally small-brained by bird standards, kiwi have proportionally large encephalisation quotients. Hemisphere proportions are even similar to those of parrots and songbirds, though there is no evidence of similarly complex behaviour.
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In a sample of 19 children, a 1997 study found that 3 died before the age of 3, and 2 never learned to walk. The children had various levels of delayed development with developmental quotients from 60 to 85.
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Other studies, such as those by Steve Brusatte, indicate the encephalization quotient of Tyrannosaurus was similar in range (2.0-2.4) to a chimpanzee (2.2-2.5), though this may be debatable as reptilian and mammalian encephalization quotients are not equivalent.
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Also, one may consider classical groups over a unital associative algebra R over F; where R = H (an algebra over reals) represents an important case. For the sake of generality the article will refer to groups over R, where R may be the ground field F itself. Considering their abstract group theory, many linear groups have a "special" subgroup, usually consisting of the elements of determinant 1 over the ground field, and most of them have associated "projective" quotients, which are the quotients by the center of the group. For orthogonal groups in characteristic 2 "S" has a different meaning.
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A local complete intersection ring is a Noetherian local ring whose completion is the quotient of a regular local ring by an ideal generated by a regular sequence. Taking the completion is a minor technical complication caused by the fact that not all local rings are quotients of regular ones. For rings that are quotients of regular local rings, which covers most local rings that occur in algebraic geometry, it is not necessary to take completions in the definition. There is an alternative intrinsic definition that does not depend on embedding the ring in a regular local ring.
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For example, one possible biomarker for prenatal testosterone's effect on the brain is a low ratio of the second to fourth finger (the 2D:4D ratio), which has been found to be associated with several male specific psychological factors. A significantly lower 2D:4D ratio than the general population has been found in autistic individuals, however there was no correlation between the empathizing and systemizing quotients and the 2D:4D ratio. The authors give many possible explanations for this finding which are contrary to the extreme male brain theory of autism, for example it is possible that the psychometric properties of the quotients are lacking or that the theory itself is incorrect and the difference in autistic brains is not an extreme of normal functioning but of a different structure altogether.Voracek, M., & Dressler, S. (2006). Lack of correlation between digit ratio (2D:4D) and baron-Cohen’s “Reading the mind in the eyes” test, empathy, systemising, and autism-spectrum quotients in a general population sample.
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The dorsal fluke is typically slightly hooked. The beak is well-defined and of moderate length. There are 26 to 36 pairs of teeth in the upper and lower jaws.. The tucuxi has one of the largest known encephalization quotients among mammals.
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Let Γ and Δ be discrete subgroups of the isometry group of hyperbolic n-space H, where n ≥ 3, whose quotients H/Γ and H/Δ have finite volume. If Γ and Δ are isomorphic as discrete groups then they are conjugate.
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There are two common ways to define algebraic spaces: they can be defined as either quotients of schemes by etale equivalence relations, or as sheaves on a big etale site that are locally isomorphic to schemes. These two definitions are essentially equivalent.
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A far-reaching generalization of algebraic spaces is given by the algebraic stacks. In the category of stacks we can form even more quotients by group actions than in the category of algebraic spaces (the resulting quotient is called a quotient stack).
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However, a procedure used in several textbooks to construct the continuous-time \Sigma as the limit of finite difference quotients of the discrete-time \Sigma, which does not depend on the control, is circular or a best incomplete; see Remark 4 in Georgiou and Lindquist.
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A secant approaches a tangent when \Delta x \to 0. The most common approach to turn this intuitive idea into a precise definition is to define the derivative as a limit of difference quotients of real numbers.Spivak 1994, chapter 10. This is the approach described below.
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Since that time, formulas and algorithms for sums have been generalized and extended to differences, products, quotients and other binary and unary functions under various dependence assumptions.Yager, R.R. (1986). Arithmetic and other operations on Dempster–Shafer structures. International Journal of Man-machine Studies 25: 357–366.
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These two honeycombs, and three others using the ideal cuboctahedron, triangular prism, and truncated tetrahedron, arise in the study of the Bianchi groups, and come from cusped manifolds formed as quotients of hyperbolic space by subgroups of Bianchi groups. The same manifolds can also be interpreted as link complements.
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Kirwan was educated at Oxford High School, and studied maths as an undergraduate at Clare College in the University of Cambridge. She took a D.Phil at Oxford in 1984, with the dissertation title The Cohomology of Quotients in Symplectic and Algebraic Geometry, which was supervised by Michael Atiyah.
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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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Quotients of flat modules are not in general flat. For example, for each integer n > 1, \Z/n\Z is not flat over \Z, because n: \Z \to \Z, x \mapsto nx is injective, but tensored with \Z/n\Z it is not. Similarly, \Q/\Z is not flat over \Z.
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However, the word fraction can also be used to describe mathematical expressions that are not rational numbers. Examples of these usages include algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as /2 (see square root of 2) and π/4 (see proof that π is irrational).
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Cauchy real numbers can't be represented without this.Altenkirch, Thorsten, Thomas Anberrée, and Nuo Li. "Definable Quotients in Type Theory." Homotopy type theory works on resolving this problem. It allows one to define higher inductive types, which not only define first order constructors (values or points), but higher order constructors, i.e.
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Let f : X → Y be a continuous function. Then for any x and y in X :x ≡ y implies f(x) ≡ f(y). The converse is generally false (There are quotients of T0 spaces which are trivial). The converse will hold if X has the initial topology induced by f.
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It is this consequence, for any two primes not just 2 and 3, that Alaoglu and Erdős desired in their paper as it would imply the conjecture that the quotient of two consecutive colossally abundant numbers is prime, extending Ramanujan's results on the quotients of consecutive superior highly composite number.Ramanujan, (1915), section IV.
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If a normal space is R0, then it is in fact completely regular. Thus, anything from "normal R0" to "normal completely regular" is the same as what we usually call normal regular. Taking Kolmogorov quotients, we see that all normal T1 spaces are Tychonoff. These are what we usually call normal Hausdorff spaces.
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In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups. They share many properties with their finite quotients: for example, both Lagrange's theorem and the Sylow theorems generalise well to profinite groups. A non-compact generalization of a profinite group is a locally profinite group.
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In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring R that has finite uniform dimension (also called "finite rank") as a right module over itself, and satisfies the ascending chain condition on right annihilators of subsets of R. Goldie's theorem states that the semiprime right Goldie rings are precisely those that have a semisimple Artinian right classical ring of quotients. The structure of this ring of quotients is then completely determined by the Artin–Wedderburn theorem. In particular, Goldie's theorem applies to semiprime right Noetherian rings, since by definition right Noetherian rings have the ascending chain condition on all right ideals.
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In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring R that has finite uniform dimension (="finite rank") as a right module over itself, and satisfies the ascending chain condition on right annihilators of subsets of R. Goldie's theorem states that the semiprime right Goldie rings are precisely those that have a semisimple Artinian right classical ring of quotients. The structure of this ring of quotients is then completely determined by the Artin–Wedderburn theorem. In particular, Goldie's theorem applies to semiprime right Noetherian rings, since by definition right Noetherian rings have the ascending chain condition on all right ideals.
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In mathematics, the Gabriel–Popescu theorem is an embedding theorem for certain abelian categories, introduced by . It characterizes certain abelian categories (the Grothendieck categories) as quotients of module categories. There are several generalizations and variations of the Gabriel–Popescu theorem, given by (for an AB5 category with a set of generators), , (for triangulated categories).
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This suggests that H_1(X) admits an increasing filtration : 0\subset W_0\subset W_1 \subset W_2=H^1(X),\, whose successive quotients Wn/Wn−1 originate from the cohomology of smooth complete varieties, hence admit (pure) Hodge structures, albeit of different weights. Further examples can be found in "A Naive Guide to Mixed Hodge Theory".
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To calculate the convergents of we may set , define and , and , . Continuing like this, one can determine the infinite continued fraction of as :[3;7,15,1,292,1,1,...] . The fourth convergent of is [3;7,15,1] = = 3.14159292035..., sometimes called Milü, which is fairly close to the true value of . Let us suppose that the quotients found are, as above, [3;7,15,1].
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The ring Z and its quotients Z/nZ have no subrings (with multiplicative identity) other than the full ring. Every ring has a unique smallest subring, isomorphic to some ring Z/nZ with n a nonnegative integer (see characteristic). The integers Z correspond to in this statement, since Z is isomorphic to Z/0Z.
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In combinatorial game theory, and particularly in the theory of impartial games in misère play, an indistinguishability quotient is a commutative monoid that generalizes and localizes the Sprague–Grundy theorem for a specific game's rule set. In the specific case of misere-play impartial games, such commutative monoids have become known as misere quotients.
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Many more complicated and intricate misere quotients have been calculated for various impartial games, and particularly for octal games. A general-purpose algorithm for computing the misere quotient monoid presentation of a given a finite set of misere impartial game positions has been devised by Aaron N. Siegel and is described in its Appendix C.
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Proceeding in this way (lifting the lower central series for each quotient of the Fitting series) yields a subnormal series: :G = G1 ⊵ G2 ⊵ ⋯ ⊵ F1 = F1,1 ⊵ F1,2 ⊵ ⋯ ⊵ F2 = F2,1 ⊵ ⋯ ⊵ Fn = 1, like the coarse and fine divisions on a ruler. The successive quotients are abelian, showing the equivalence between being solvable and having a Fitting series.
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Due to the fact that inverse images commute with arbitrary unions and intersections, the property of being an Alexandrov-discrete space is preserved under quotients. Alexandrov-discrete spaces are named after the Russian topologist Pavel Alexandrov. They should not be confused with the more geometrical Alexandrov spaces introduced by the Russian mathematician Aleksandr Danilovich Aleksandrov.
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But this is a contradiction, and thus it must be the case that at least one of the coefficients is transcendental. The non-computable numbers are a strict subset of the transcendental numbers. All Liouville numbers are transcendental, but not vice versa. Any Liouville number must have unbounded partial quotients in its continued fraction expansion.
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Difference quotients are used as approximations in numerical differentiation, but they have also been subject of criticism in this application. The difference quotient is sometimes also called the Newton quotient (after Isaac Newton) or Fermat's difference quotient (after Pierre de Fermat).Donald C. Benson, A Smoother Pebble: Mathematical Explorations, Oxford University Press, 2003, p. 176.
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Among his most notable contributions to the logical foundations of mathematics are his work on typical geometries, on the problem of the infinitude of space, the three-body problem, on difference quotients, and on mathematical induction. In psychology, he developed theories about the observation of the transparent and on the depth and observation of compound colours.
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Starting from this formula, the exponential function as well as all the trigonometric and hyperbolic functions can be expressed in terms of the gamma function. More functions yet, including the hypergeometric function and special cases thereof, can be represented by means of complex contour integrals of products and quotients of the gamma function, called Mellin–Barnes integrals.
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In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.
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It is natural to depict cyclically ordered groups as quotients: one has and . Even a once-linear group like , when bent into a circle, can be thought of as . showed that this picture is a generic phenomenon. For any ordered group and any central element that generates a cofinal subgroup of , the quotient group is a cyclically ordered group.
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Local complete intersection rings, and a fortiori, regular local rings are Cohen–Macaulay, but not conversely. Cohen–Macaulay combine desirable properties of regular rings (such as the property of being universally catenary rings, which means that the (co)dimension of primes is well-behaved), but are also more robust under taking quotients than regular local rings.
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However, they suffered losing streak and ended up in a four-way tie with Alaska, GlobalPort and Barako Bull. Since Barako Bull and GlobalPort have higher quotients, Meralco was forced to play a sudden death game with Alaska Aces to determine the eighth and final playoff spot. The Aces defeated them and thus they were eliminated from playoff contention.
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In mathematics, the 2 theorem of Gromov and Thurston states a sufficient condition for Dehn filling on a cusped hyperbolic 3-manifold to result in a negatively curved 3-manifold. Let M be a cusped hyperbolic 3-manifold. Disjoint horoball neighborhoods of each cusp can be selected. The boundaries of these neighborhoods are quotients of horospheres and thus have Euclidean metrics.
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In mathematics, a Hopfian group is a group G for which every epimorphism :G -> G is an isomorphism. Equivalently, a group is Hopfian if and only if it is not isomorphic to any of its proper quotients. A group G is co-Hopfian if every monomorphism :G -> G is an isomorphism. Equivalently, G is not isomorphic to any of its proper subgroups.
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There are several minor variations of these, given by taking derived subgroups or central quotients, the latter yielding projective linear groups. They can be constructed over finite fields (or any other field) in much the same way that they are constructed over the real numbers. They correspond to the series An, Bn, Cn, Dn,2An, 2Dn of Chevalley and Steinberg groups.
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The range of respiratory coefficients for organisms in metabolic balance usually ranges from 1.0 (representing the value expected for pure carbohydrate oxidation) to ~0.7 (the value expected for pure fat oxidation). In general, molecules that are more oxidized (e.g., glucose) require less oxygen to be fully metabolized and, therefore, have higher respiratory quotients. Conversely, molecules that are less oxidized (e.g.
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In algebraic geometry, a Mori Dream Space is a projective variety whose cone of effective divisors has a well-behaved decomposition into certain convex sets called "Mori chambers". showed that Mori dream spaces are quotients of affine varieties by torus actions. The notion is named so because it behaves nicely from the point of view of Mori's minimal model program.
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If for odd, and for even, then if it is calculated with the central difference scheme. This is particularly troublesome if the domain of is discrete. See also Symmetric derivative Authors for whom finite differences mean finite difference approximations define the forward/backward/central differences as the quotients given in this section (instead of employing the definitions given in the previous section).
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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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In algebraic geometry, a toric stack is a stacky generalization of a toric variety. More precisely, a toric stack is obtained by replacing in the construction of a toric variety a step of taking GIT quotients with that of taking quotient stacks. Consequently, a toric variety is a coarse approximation of a toric stack. A toric orbifold is an example of a toric stack.
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In mathematics, the Iwahori–Hecke algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a deformation of the group algebra of a Coxeter group. Hecke algebras are quotients of the group rings of Artin braid groups. This connection found a spectacular application in Vaughan Jones' construction of new invariants of knots. Representations of Hecke algebras led to discovery of quantum groups by Michio Jimbo.
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The Indian sub- continent became the company's largest delivery hub. Under his leadership, IGATE recorded revenues of $1.2 billion and grew to 34,000 employees. The company was featured among the five best companies to work at in India. CareerBliss, based on employee happiness and employee satisfaction quotients, identified IGATE among the twenty-five best companies to work for in the United States of America.
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The Zassenhaus lemma gives an isomorphism between certain combinations of quotients and products in the lattice of subgroups. In general, there is no restriction on the shape of the lattice of subgroups, in the sense that every lattice is isomorphic to a sublattice of the subgroup lattice of some group. Furthermore, every finite lattice is isomorphic to a sublattice of the subgroup lattice of some finite group .
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Best-possible bounds for the distribution of a sum—a problem of Kolmogorov. Probability Theory and Related Fields 74: 199–211. generalized the result of Makarov and expressed it in terms of copulas. Since that time, formulas and algorithms for sums have been generalized and extended to differences, products, quotients and other binary and unary functions under various dependence assumptions.Williamson, R.C., and T. Downs (1990).
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In Canada, the constitution mandates that redistribution occur "on the completion of each decennial census." District boundaries are based on electoral quotients, with some exceptions. When the census indicates that a population change has occurred, an independent boundary commission issues a report of recommended changes. Changes are only made if passed into law—by Parliament for national redistribution, by the provincial legislature for provincial redistribution.
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Göttingen 1863. In 1838 Peter Gustav Lejeune Dirichlet came up with his own approximating function, the logarithmic integral (under the slightly different form of a series, which he communicated to Gauss). Both Legendre's and Dirichlet's formulas imply the same conjectured asymptotic equivalence of and stated above, although it turned out that Dirichlet's approximation is considerably better if one considers the differences instead of quotients.
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In terms of cognitive function, intelligence and cognitive deficits are common amongst youths with conduct disorder, particularly those with early-onset and have intelligence quotients (IQ) one standard deviation below the meanLynham, D. & Henry, B. (2001). The role of neuropsychological deficits in conduct disorders. In J. Hill & B. Maughan (Eds.), Conduct disorders in childhood and adolescence (pp.235-263). New York: Cambridge University Press.
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He introduced Matsushima's formula for the Betti numbers of quotients of symmetric spaces. In 1967, he became an editor of the Journal of Differential Geometry and remained on the editorial board for the rest of his life. After 14 years at Notre Dame, he returned to Japan in 1980. A conference was organized in his honor in May 1980 before he left Notre Dame.
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Given a mathematical structure, there are very often associated structures which can be constructed as a quotient of part of the original structure via an equivalence relation. An important example is a quotient group of a group. One might say that to understand the full structure one must understand these quotients. When the equivalence relation is definable, we can give the previous sentence a precise meaning.
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When one quotients Spin(8) by one central Z2, breaking this symmetry and obtaining SO(8), the remaining outer automorphism group is only Z2. The triality symmetry acts again on the further quotient SO(8)/Z2. Sometimes Spin(8) appears naturally in an "enlarged" form, as the automorphism group of Spin(8), which breaks up as a semidirect product: Aut(Spin(8)) ≅ PSO (8) ⋊ S3.
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In 1838 Peter Gustav Lejeune Dirichlet came up with his own approximating function, the logarithmic integral li(x) (under the slightly different form of a series, which he communicated to Gauss). Both Legendre's and Dirichlet's formulas imply the same conjectured asymptotic equivalence of π(x) and x / ln(x) stated above, although it turned out that Dirichlet's approximation is considerably better if one considers the differences instead of quotients.
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Two dimensional terminal singularities are smooth. If a variety has terminal singularities, then its singular points have codimension at least 3, and in particular in dimensions 1 and 2 all terminal singularities are smooth. In 3 dimensions they are isolated and were classified by . Two dimensional canonical singularities are the same as du Val singularities, and are analytically isomorphic to quotients of C2 by finite subgroups of SL2(C).
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Philovenator is a troodontid, a group of small, bird-like, gracile maniraptorans. All troodontids have many unique features of the skull, such as closely spaced teeth in the lower jaw, and large numbers of teeth. Troodontids have sickle-claws and raptorial hands, and some of the highest non-avian encephalization quotients, meaning they were behaviourally advanced and had keen senses. Several distinguishing traits were established in the initial description.
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Bézout's identity states that the greatest common divisor g of two integers a and b can be represented as a linear sum of the original two numbers a and b. In other words, it is always possible to find integers s and t such that g = sa + tb. The integers s and t can be calculated from the quotients q0, q1, etc. by reversing the order of equations in Euclid's algorithm.
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These are the complex surfaces such that q = 0 and the canonical line bundle is non-trivial, but has trivial square. Enriques surfaces are all algebraic (and therefore Kähler). They are quotients of K3 surfaces by a group of order 2 and their theory is similar to that of algebraic K3 surfaces. Invariants: The plurigenera Pn are 1 if n is even and 0 if n is odd.
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We assume aR < bR so the remainders in slots 1 through 4 must be shifted. Slot 2 receives bR and is marked as a continuation and shifted. Slot 5 receives eR and is marked as shifted. The runs for quotients 1, 2 and 4 now comprise a cluster, and the presence of those three runs in the cluster is indicated by having slots 1, 2 and 4 being marked as occupied.
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Dosage quotient analysis is the usual method of interpreting MLPA data. If a and b are the signals from two amplicons in the patient sample, and A and B are the corresponding amplicons in the experimental control, then the dosage quotient DQ = (a/b) / (A/B). Although dosage quotients may be calculated for any pair of amplicons, it is usually the case that one of the pair is an internal reference probe.
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Arithmetic expressions involving operations such as additions, subtractions, multiplications, divisions, minima, maxima, powers, exponentials, logarithms, square roots, absolute values, etc., are commonly used in risk analyses and uncertainty modeling. Convolution is the operation of finding the probability distribution of a sum of independent random variables specified by probability distributions. We can extend the term to finding distributions of other mathematical functions (products, differences, quotients, and more complex functions) and other assumptions about the intervariable dependencies.
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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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The quotient can be identified with the real projective plane. It is non- orientable and can be described as the quotient of by the antipodal map (multiplication by −1). The sphere is simply connected, while the real projective plane has fundamental group . The finite subgroups of , corresponding to the finite subgroups of and the symmetry groups of the platonic solids, do not act freely on , so the corresponding quotients are not 2-manifolds, just orbifolds.
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In mathematics, in his "Tôhoku paper" introduced a sequence of axioms of various kinds of categories enriched over the symmetric monoidal category of abelian groups. Abelian categories are sometimes called AB2 categories, according to the axiom (AB2). AB3 categories are abelian categories possessing arbitrary coproducts (hence, by the existence of quotients in abelian categories, also all colimits). AB5 categories are the AB3 categories in which filtered colimits of exact sequences are exact.
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Over fields of characteristic 0 the Specht modules are irreducible, and form a complete set of irreducible representations of the symmetric group. A partition is called p-regular if it does not have p parts of the same (positive) size. Over fields of characteristic p>0 the Specht modules can be reducible. For p-regular partitions they have a unique irreducible quotient, and these irreducible quotients form a complete set of irreducible representations.
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Kassel received her PhD under the direction of Yves Benoist at the University of Paris-Sud in 2009. Her thesis was on "Compact quotients of real or p-adic homogeneous spaces". She then entered the CNRS and worked at the Paul-Painlevé Laboratory of the University of Lille I until 2016, when she joined the IHÉS as detached CNRS researcher."Fanny Kassel joins IHES as a CNRS Researcher", IHES website (20-09-2016).
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The company was featured among the five best companies to work at in India. CareerBliss - based on employee happiness and satisfaction quotients - featured IGATE among the twenty-five best companies to work for in the United States of America. By July 2015, French-IT company, Capgemini completed a $4 billion acquisition of IGATE, recorded as one of the largest deals in the Indian information technology sector. It saw the exit of both Sunil and Ashok Trivedi.
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The proof is straightforward. From the fraction itself, one can construct the quadratic equation with integral coefficients that x must satisfy. Lagrange proved the converse of Euler's theorem: if x is a quadratic irrational, then the regular continued fraction expansion of x is periodic. Given a quadratic irrational x one can construct m different quadratic equations, each with the same discriminant, that relate the successive complete quotients of the regular continued fraction expansion of x to one another.
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Patient VP is a woman who underwent a two-stage callosotomy in 1979 at the age of 27. Although the callosotomy was reported to be complete, follow-up MRI in 1984 revealed spared fibers in the rostrum and splenium. The spared rostral fibers constituted approximately 1.8% of the total cross-sectional area of the corpus callosum and the spared splenial fibers constituted approximately 1% of the area. VP's postsurgery intelligence and memory quotients were within normal limits.
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The main interest in approach spaces and their contractions is that they form a category with good properties, while still being quantitative like metric spaces. One can take arbitrary products, coproducts, and quotients, and the results appropriately generalize the corresponding results for topologies. One can even "distancize" such badly non-metrizable spaces like βN, the Stone–Čech compactification of the integers. Certain hyperspaces, measure spaces, and probabilistic metric spaces turn out to be naturally endowed with a distance.
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In mathematics, a Kodaira surface is a compact complex surface of Kodaira dimension 0 and odd first Betti number. The concept is named after Kunihiko Kodaira. These are never algebraic, though they have non-constant meromorphic functions. They are usually divided into two subtypes: primary Kodaira surfaces with trivial canonical bundle, and secondary Kodaira surfaces which are quotients of these by finite groups of orders 2, 3, 4, or 6, and which have non-trivial canonical bundles.
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The use of chlorpyrifos in agriculture can leave chemical residue on food commodities. The FFDCA requires EPA to set limits, known as tolerances, for pesticide residue in human food and animal feed products based on risk quotients for acute and chronic exposure from food in humans. These tolerances limit the amount of chlorpyrifos that can be applied to crops. FDA enforces EPA's pesticide tolerances and determines "action levels" for the unintended drift of pesticide residues onto crops without tolerances.
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Pectoral girdle and forelimb Linhevenator is a troodontid, a group of small, bird-like, gracile maniraptorans. All troodontids have many unique features of the skull, such as closely spaced teeth in the lower jaw, and large numbers of teeth. Troodontids have sickle-claws and raptorial hands, and some of the highest non-avian encephalization quotients, meaning they were behaviourally advanced and had keen senses. Linhevenator is a rather large troodontid with an estimated body weight of .
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Restoration and size comparison Gobivenator is a troodontid, a group of small, bird-like, gracile maniraptorans. All troodontids have many unique features of the skull, such as closely spaced teeth in the lower jaw, and large numbers of teeth. Troodontids have sickle-claws and raptorial hands, and some of the highest non-avian encephalization quotients, meaning they were behaviourally advanced and had keen senses. Gobivenator possesses two autapomorphies, unique traits, that differentiate it from all other currently known troodontids.
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In mathematics, more specifically in the area of abstract algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients (equivalently, its abelianization, which is the universal abelian quotient, is trivial). In symbols, a perfect group is one such that G(1) = G (the commutator subgroup equals the group), or equivalently one such that Gab = {1} (its abelianization is trivial).
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Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics. Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Specifically, although Verma modules themselves are infinite dimensional, quotients of them can be used to construct finite-dimensional representations with highest weight \lambda, where \lambda is dominant and integral.E.g., Chapter 9 Their homomorphisms correspond to invariant differential operators over flag manifolds.
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Heath, p. 112 The Pythagoreans' conception of number included only what would today be called rational numbers, casting doubt on the validity of the theory in geometry where, as the Pythagoreans also discovered, incommensurable ratios (corresponding to irrational numbers) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables.Heath, p. 113 The existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients. However, this is a comparatively recent development, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios and quotients. The reasons for this are twofold: first, there was the previously mentioned reluctance to accept irrational numbers as true numbers, and second, the lack of a widely used symbolism to replace the already established terminology of ratios delayed the full acceptance of fractions as alternative until the 16th century.
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7 This is a special feature of hereditary rings like the integers Z: the direct sum of injective modules is injective because the ring is Noetherian, and the quotients of injectives are injective because the ring is hereditary, so any submodule generated by injective modules is injective. The converse is a result of : if every module has a unique maximal injective submodule, then the ring is hereditary. A complete classification of countable reduced periodic abelian groups is given by Ulm's theorem.
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Restoration Byronosaurus is a troodontid, a group of small, bird-like, gracile maniraptorans. All known troodontids share unique features of the skull, such as closely spaced teeth in the lower jaw, and large numbers of teeth. Troodontids have sickle-claws and raptorial hands, and some of the highest non-avian encephalization quotients, meaning they were behaviourally advanced and had keen senses. Byronosaurus is one of few troodontids that have no serrations on its teeth, similar to its closest relative Xixiasaurus.
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Troodontids are a group of small, bird-like, gracile maniraptorans. All troodontids have unique features of the skull, such as large numbers of closely spaced teeth in the lower jaw. Troodontids have sickle-claws and raptorial hands, and some of the highest non-avian encephalization quotients, suggesting that they were behaviourally advanced and had keen senses. They had unusually long legs compared to other theropods, with a large, curved claw on their retractable second toes, similar to the "sickle-claw" of the dromaeosaurids.
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Their logarithms did not lead to irrational numbers, however Theon tackled this discussion head on. He acknowledged that “one can prove that” the tone of value 9/8 cannot be divided into equal parts and so it is a number in itself. Many Pythagoreans believed in the existence of irrational numbers, but did not believe in using them because they were unnatural and not positive integers. Theon also does an amazing job of relating quotients of integers and musical intervals.
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The quotients are generally found by rounding the real and complex parts of the exact ratio (such as the complex number ) to the nearest integers. The second difference lies in the necessity of defining how one complex remainder can be "smaller" than another. To do this, a norm function is defined, which converts every Gaussian integer into an ordinary integer. After each step of the Euclidean algorithm, the norm of the remainder is smaller than the norm of the preceding remainder, .
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Size of Sinornithoides, compared to a human Sinornithoides is a troodontid, a group of small, bird-like, gracile maniraptorans. All troodontids have many unique features of the skull, such as closely spaced teeth in the lower jaw, and large numbers of teeth. Troodontids have sickle- claws and raptorial hands, and some of the highest non-avian encephalization quotients, meaning they were behaviourally advanced and had keen senses. In 2010, Gregory S. Paul estimated its body length at 1.1 metres, its weight at 2.5 kilogrammes.
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This algorithm improves speed, because it reduces the number of operations on very large numbers, and can use the speed of hardware arithmetic for most operations. In fact, most of the quotients are very small, so a fair number of steps of the Euclidean algorithm can be collected in a 2-by-2 matrix of single-word integers. When Lehmer's algorithm encounters a quotient that is too large, it must fall back to one iteration of Euclidean algorithm, with a Euclidean division of large numbers.
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Reiterman's theorem states that the two definitions above are equivalent. A scheme of the proof is now given. Given a variety V of semigroups as in the algebraic definition, one can choose the set P of profinite identities to be the set of profinite identities satisfied by every semigroup of V. Reciprocally, given a profinite identity u = v, one can remark that the class of semigroups satisfying this profinite identity is closed under subsemigroups, quotients, and finite products. Thus this class is a variety of finite semigroups.
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San Beda, Mapua and San Sebastian wound up in a three-way tie at the end of the elimination round. However, only the Red Lions and the Cardinals will figure in a knockout match due to their superior tie-break quotients. San Beda made it to the final four cast for the first time in seven years as they relied on their tested leader Arjun Cordero, who shot nine of his 14 points in the final four minutes to lift the Red Lions past the Mapua Cardinals.
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The general linear group GLn(R) is the group of all R-linear automorphisms of Rn. There is a subgroup: the special linear group SLn(R), and their quotients: the projective general linear group PGLn(R) = GLn(R)/Z(GLn(R)) and the projective special linear group PSLn(R) = SLn(R)/Z(SLn(R)). The projective special linear group PSLn(F) over a field F is simple for n ≥ 2, except for the two cases when n = 2 and the field has order 2 or 3\.
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For any A, B, and C subgroups of a group with A ≤ C (A subgroup of C) then AB ∩ C = A(B ∩ C); the multiplication here is the product of subgroups. This property has been called the modular property of groups or (Dedekind's) modular law (, ). Since for two normal subgroups the product is actually the smallest subgroup containing the two, the normal subgroups form a modular lattice. The Lattice theorem establishes a Galois connection between the lattice of subgroups of a group and that of its quotients.
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Every projective algebraic set may be uniquely decomposed into a finite union of projective varieties. The only regular functions which may be defined properly on a projective variety are the constant functions. Thus this notion is not used in projective situations. On the other hand, the field of the rational functions or function field is a useful notion, which, similarly to the affine case, is defined as the set of the quotients of two homogeneous elements of the same degree in the homogeneous coordinate ring.
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Martin Roček is a professor of theoretical physics at the State University of New York at Stony Brook and a member of the C. N. Yang Institute for Theoretical Physics. He received A.B. and Ph.D. degrees from Harvard University in 1975 and 1979. He did post-doctoral research at the University of Cambridge and Caltech before becoming a professor at Stony Brook University. He was one of the co-inventors of hyperkähler quotients, a hyperkahler analogue of Marsden–Weinstein reduction and the structure of Bihermitian manifolds.
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Size of Sinovenator compared to a human Sinovenator is a troodontid, a group of small, bird-like, gracile maniraptorans. All troodontids have many unique features of the skull, such as closely spaced teeth in the lower jaw, and large numbers of teeth. Troodontids have sickle- claws and raptorial hands, and some of the highest non-avian encephalization quotients, meaning they were behaviourally advanced and had keen senses. The holotype individual of Sinovenator was about the size of a chicken, less than a metre long.
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Categorically, Kolmogorov spaces are a reflective subcategory of topological spaces, and the Kolmogorov quotient is the reflector. Topological spaces X and Y are Kolmogorov equivalent when their Kolmogorov quotients are homeomorphic. Many properties of topological spaces are preserved by this equivalence; that is, if X and Y are Kolmogorov equivalent, then X has such a property if and only if Y does. On the other hand, most of the other properties of topological spaces imply T0-ness; that is, if X has such a property, then X must be T0.
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Intelligence quotients are sometimes held to be distributed according to the bell-shaped curve. About 40% of the area under the curve is in the interval from 100 to 120; correspondingly, about 40% of the population scores between 100 and 120 on IQ tests. Nearly 9% of the area under the curve is in the interval from 120 to 140; correspondingly, about 9% of the population scores between 120 and 140 on IQ tests, etc. Similarly many other things are distributed according to the "bell-shaped curve", including measurement errors in many physical measurements.
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Parts-per notations are all dimensionless quantities: in mathematical expressions, the units of measurement always cancel. In fractions like "2 nanometers per meter" (2 n ~~m~~ / ~~m~~ = 2 nano = 2 × 10−9 = 2 ppb = 2 × ), so the quotients are pure-number coefficients with positive values less than or equal to 1\. When parts-per notations, including the percent symbol (%), are used in regular prose (as opposed to mathematical expressions), they are still pure-number dimensionless quantities. However, they generally take the literal "parts per" meaning of a comparative ratio (e.g.
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Waterborne manganese has a greater bioavailability than dietary manganese. According to results from a 2010 study, higher levels of exposure to manganese in drinking water are associated with increased intellectual impairment and reduced intelligence quotients in school-age children. It is hypothesized that long-term exposure due to inhaling the naturally occurring manganese in shower water puts up to 8.7 million Americans at risk. However, data indicates that the human body can recover from certain adverse effects of overexposure to manganese if the exposure is stopped and the body can clear the excess.
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By taking the appropriate limits, the equation can also be made to hold even when the left-hand product contains zeros or poles. By taking limits, certain rational products with infinitely many factors can be evaluated in terms of the gamma function as well. Due to the Weierstrass factorization theorem, analytic functions can be written as infinite products, and these can sometimes be represented as finite products or quotients of the gamma function. We have already seen one striking example: the reflection formula essentially represents the sine function as the product of two gamma functions.
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These numbers are always divisible by (because a cyclic permutation of a foldable stamp sequence is always itself foldable),. As cited by and the quotients of this division are :1, 1, 2, 4, 10, 24, 66, 174, 504, 1406, 4210, 12198, 37378, 111278, 346846, 1053874, ... , the number of topologically distinct ways that a half-infinite curve can make crossings with a line, called "semimeanders". In the 1960s, John E. Koehler and W. F. Lunnon implemented algorithms that, at that time, could calculate these numbers for up to 28 stamps.
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The Akhmim Wooden Tablet, the Egyptian Mathematical Leather Roll, the RMP 2/n table, the Rhind Mathematical Papyrus, the Ebers Papyrus and other mathematical texts reported expected and observed Egyptian fractions totals. Totals were written in quotients and scaled/unscaled remainder units. A meta context of the Egyptian Middle Kingdom weights and measures system had empowered one of the earliest Ancient Near East monetary systems. The Egyptian economy was able to double-check its management elements by using double entry accounting, and theoretical or abstract weights and measures units.
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Some PSL groups arise as automorphism groups of Hurwitz surfaces, i.e., as quotients of the (2,3,7) triangle group, which is the symmetries of the order-3 bisected heptagonal tiling. PSL groups arise as Hurwitz groups (automorphism groups of Hurwitz surfaces – algebraic curves of maximal possibly symmetry group). The Hurwitz surface of lowest genus, the Klein quartic (genus 3), has automorphism group isomorphic to PSL(2, 7) (equivalently GL(3, 2)), while the Hurwitz surface of second-lowest genus, the Macbeath surface (genus 7), has automorphism group isomorphic to PSL(2, 8).
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Today, this can be done by simply stating that ratios are equal when the quotients of the terms are equal, but Euclid did not accept the existence of so such a definition would have been meaningless to him. Thus, a more subtle definition is needed where quantities involved are not measured directly to one another. In modern notation, Euclid's definition of equality is that given quantities p, q, r and s, p∶q∷r∶s if and only if for any positive integers m and n, npmq according as nrms, respectively.Heath p.
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The simplest linear Diophantine equation takes the form , where , and are given integers. The solutions are described by the following theorem: :This Diophantine equation has a solution (where and are integers) if and only if is a multiple of the greatest common divisor of and . Moreover, if is a solution, then the other solutions have the form , where is an arbitrary integer, and and are the quotients of and (respectively) by the greatest common divisor of and . Proof: If is this greatest common divisor, Bézout's identity asserts the existence of integers and such that .
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The modern formulation of geometric invariant theory is due to David Mumford, and emphasizes the construction of a quotient by the group action that should capture invariant information through its coordinate ring. It is a subtle theory, in that success is obtained by excluding some 'bad' orbits and identifying others with 'good' orbits. In a separate development the symbolic method of invariant theory, an apparently heuristic combinatorial notation, has been rehabilitated. One motivation was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects.
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To convert integer decimals to octal, divide the original number by the largest possible power of 8 and divide the remainders by successively smaller powers of 8 until the power is 1. The octal representation is formed by the quotients, written in the order generated by the algorithm. For example, to convert 12510 to octal: :125 = 82 × 1 + 61 :61 = 81 × 7 + 5 :5 = 80 × 5 + 0 Therefore, 12510 = 1758. Another example: :900 = 83 × 1 + 388 :388 = 82 × 6 + 4 :4 = 81 × 0 + 4 :4 = 80 × 4 + 0 Therefore, 90010 = 16048.
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The maximal ring of quotients Q(R) (in the sense of Utumi and Lambek) of a Boolean ring R is a Boolean ring, since every partial endomorphism is idempotent. Corollary 2. Every prime ideal P in a Boolean ring R is maximal: the quotient ring R/P is an integral domain and also a Boolean ring, so it is isomorphic to the field F2, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings. Boolean rings are von Neumann regular rings.
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One of the hallmarks of Context-Based Sustainability is its use of Context-Based Metrics (CBMs). Unlike other metrics used to measure, manage and report the sustainability performance of organizations (e.g., absolute and relative/intensity metrics), some CBMs, but not all, take the form of quotients that have two parts to them: 1) denominators that express organization-specific norms for what their impacts on vital capitals must be in order to be sustainable (i.e., equivalent to "allocations" as explained above), and 2) numerators that express their actual impacts on the same capitals.
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A holonomic function, also called a D-finite function, is a function that is a solution of a homogeneous linear differential equation with polynomial coefficients. Most functions that are commonly considered in mathematics are holonomic or quotients of holonomic functions. In fact, holonomic functions include polynomials, algebraic functions, logarithm, exponential function, sine, cosine, hyperbolic sine, hyperbolic cosine, inverse trigonometric and inverse hyperbolic functions, and many special functions such as Bessel functions and hypergeometric functions. Holonomic functions have several closure properties; in particular, sums, products, derivative and integrals of holonomic functions are holonomic.
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The requirement that the transitive nilpotent group acts by isometries leads to the following rigid characterization: every homogeneous nilmanifold is isometric to a nilpotent Lie group with left-invariant metric (see Wilson). Nilmanifolds are important geometric objects and often arise as concrete examples with interesting properties; in Riemannian geometry these spaces always have mixed curvature, almost flat spaces arise as quotients of nilmanifolds, and compact nilmanifolds have been used to construct elementary examples of collapse of Riemannian metrics under the Ricci flow.Chow, Bennett; Knopf, Dan, The Ricci flow: an introduction. Mathematical Surveys and Monographs, 110.
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Gautam Chintamani's book Qayamat Se Qayamat Tak: The Film That Revived Hindi Cinema (2016) credits the film with revitalizing Hindi cinema. In the late 1980s, Hindi cinema was experiencing a decline in box office turnout, due to increasing violence, decline in musical melodic quality, and rise in video piracy, leading to middle-class family audiences abandoning theaters. Qayamat Se Qayamat Taks blend of youthfulness, wholesome entertainment, emotional quotients and strong melodies is credited with luring family audiences back to the big screen. The film is credited with having "reinvented the romantic musical genre" in Bollywood.
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This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function f. This limit is the derivative of the function f at x = a, denoted f ′(a). Using derivatives, the equation of the tangent line can be stated as follows: : y=f(a)+f'(a)(x-a).\, Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the power function, trigonometric functions, exponential function, logarithm, and their various combinations.
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An intuitive way to understand an ε-quadratic form is to think of it as a quadratic refinement of its associated ε-symmetric form. For instance, in defining a Clifford algebra over a general field or ring, one quotients the tensor algebra by relations coming from the symmetric form and the quadratic form: and v^2=Q(v). If 2 is invertible, this second relation follows from the first (as the quadratic form can be recovered from the associated bilinear form), but at 2 this additional refinement is necessary.
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If is field of complex numbers, the fundamental theorem of algebra implies that all have degree one, and all numerators a_{ij} are constants. When is the field of real numbers, some of the may be quadratic, so, in the partial fraction decomposition, quotients of linear polynomials by powers of quadratic polynomials may also occur. In the preceding theorem, one may replace "distinct irreducible polynomials" by "pairwise coprime polynomials that are coprime with their derivative". For example, the may be the factors of the square-free factorization of .
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Fortunately, there exists such a recursive decomposition of a graph that implicitly represents all ways of decomposing it; this is the modular decomposition. It is itself a way of decomposing a graph recursively into quotients, but it subsumes all others. The decomposition depicted in the figure below is this special decomposition for the given graph. A graph, its quotient where "bags" of vertices of the graph correspond to the children of the root of the modular decomposition tree, and its full modular decomposition tree: series nodes are labeled "s", parallel nodes "//" and prime nodes "p".
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In 1971 she emigrated from the Soviet Union to Israel and she taught at the Hebrew University from 1971 until 1975. She began to work with Rufus Bowen at Berkeley and later emigrated to the United States and became a professor of mathematics at Berkeley. Her work included proofs of conjectures dealing with unipotent flows on quotients of Lie groups made by S. G. Dani and M. S. Raghunathan. For this and other work, she won the John J. Carty Award for the Advancement of Science in 1994.
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Moreover, every minimal free and discrete isometric action of Fn on a real tree with the quotient being a metric graph of volume one arises in this fashion from some point x of Xn. This defines a bijective correspondence between Xn and the set of equivalence classes of minimal free and discrete isometric actions of Fn on a real trees with volume-one quotients. Here two such actions of Fn on real trees T1 and T2 are equivalent if there exists an Fn-equivariant isometry between T1 and T2.
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Today, the term "finite difference" is often taken as synonymous with finite difference approximations of derivatives, especially in the context of numerical methods. Finite difference approximations are finite difference quotients in the terminology employed above. Finite differences were introduced by Brook Taylor in 1715 and have also been studied as abstract self-standing mathematical objects in works by George Boole (1860), L. M. Milne-Thomson (1933), and Károly Jordan (1939). Finite differences trace their origins back to one of Jost Bürgi's algorithms () and work by others including Isaac Newton.
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In the early 19th century, Carl Friedrich Gauss observed that non- zero integer solutions to homogeneous polynomial equations with rational coefficients exist if non-zero rational solutions exist. In the 1850s, Leopold Kronecker formulated the Kronecker–Weber theorem, introduced the theory of divisors, and made numerous other connections between number theory and algebra. He then conjectured his "liebster Jugendtraum" ("dearest dream of youth"), a generalization that was later put forward by Hilbert in a modified form as his twelfth problem, which outlines a goal to have number theory operate only with rings that are quotients of polynomial rings over the integers.
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An unusual 18th-century set of Napier's bones in which the numbers are on rotating cylinders rather than rods of square cross-section Napier's bones is a manually-operated calculating device created by John Napier of Merchiston, Scotland for the calculation of products and quotients of numbers. The method was based on lattice multiplication, and also called 'rabdology', a word invented by Napier. Napier published his version in 1617 in Rabdologiæ, printed in Edinburgh, dedicated to his patron Alexander Seton. Using the multiplication tables embedded in the rods, multiplication can be reduced to addition operations and division to subtractions.
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Israel Kleiner is a Canadian mathematician and historian of mathematics. Kleiner earned an MA at Yale University (1963) and a PhD at McGill University (1967) under Joachim Lambek with a thesis Lie modules and rings of quotients. Before his retirement as professor emeritus, he spent his career as a mathematics professor at York University, where he was a member of the faculty since 1965 and where he coordinated the training program for mathematics teachers teaching at the secondary school level. He is noted for his work on the history of algebra and on the combination of the history of mathematics and mathematics education.
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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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The orthogonal group On(R) preserves a non-degenerate quadratic form on a module. There is a subgroup, the special orthogonal group SOn(R) and quotients, the projective orthogonal group POn(R), and the projective special orthogonal group PSOn(R). In characteristic 2 the determinant is always 1, so the special orthogonal group is often defined as the subgroup of elements of Dickson invariant 1. There is a nameless group often denoted by Ωn(R) consisting of the elements of the orthogonal group of elements of spinor norm 1, with corresponding subgroup and quotient groups SΩn(R), PΩn(R), PSΩn(R).
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Peter Benedict Kronheimer (born 1963) is a British mathematician, known for his work on gauge theory and its applications to 3- and 4-dimensional topology. He is William Caspar Graustein Professor of Mathematics at Harvard University. Kronheimer's early work was on gravitational instantons, in particular the classification of hyperkähler 4-manifolds with asymptotical locally Euclidean geometry (ALE spaces), leading to the papers "The construction of ALE spaces as hyper-Kähler quotients" and "A Torelli-type theorem for gravitational instantons." He and Hiraku Nakajima gave a construction of instantons on ALE spaces generalizing the Atiyah–Hitchin–Drinfeld–Manin construction.
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Clifford algebras exhibit a 2-fold periodicity over the complex numbers and an 8-fold periodicity over the real numbers, which is related to the same periodicities for homotopy groups of the stable unitary group and stable orthogonal group, and is called Bott periodicity. The connection is explained by the geometric model of loop spaces approach to Bott periodicity: their 2-fold/8-fold periodic embeddings of the classical groups in each other (corresponding to isomorphism groups of Clifford algebras), and their successive quotients are symmetric spaces which are homotopy equivalent to the loop spaces of the unitary/orthogonal group.
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The answers were written in binary Eye of Horus quotients and exact Egyptian fraction remainders, scaled to a 1/320 factor named ro. The second half of the document proved the correctness of the five division answers by multiplying the two-part quotient and remainder answer by its respective (3, 7, 10, 11 and 13) dividend that returned the ab initio hekat unity, 64/64. In 2002, Hana Vymazalová obtained a fresh copy of the text from the Cairo Museum, and confirmed that all five two-part answers were correctly checked for accuracy by the scribe that returned a 64/64 hekat unity.
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In most contexts, both numbers are restricted to be positive. A ratio may be specified either by giving both constituting numbers, written as "a to b" or "a∶b", or by giving just the value of their quotient Equal quotients correspond to equal ratios. Consequently, a ratio may be considered as an ordered pair of numbers, a fraction with the first number in the numerator and the second in the denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero) natural numbers, are rational numbers, and may sometimes be natural numbers.
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The evolution of encephalization in cetaceans is similar to that in primates. Though the general trend in their evolutionary history increased brain mass, body mass, and encephalization quotient, a few lineages actually underwent decephalization, although the selective pressures that caused this are still under debate. Among cetaceans, Odontoceti tend to have higher encephalization quotients than Mysticeti, which is at least partially due to the fact that Mysticeti have much larger body masses without a compensating increase in brain mass. As far as which selective pressures drove the encephalization (or decephalization) of cetacean brains, current research espouses a few main theories.
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Stacks were first defined by , and the term "stack" was introduced by for the original French term "champ" meaning "field". In this paper they also introduced Deligne–Mumford stacks, which they called algebraic stacks, though the term "algebraic stack" now usually refers to the more general Artin stacks introduced by . When defining quotients of schemes by group actions, it is often impossible for the quotient to be a scheme and still satisfy desirable properties for a quotient. For example, if a few points have non-trivial stabilisers, then the categorical quotient will not exist among schemes.
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This is only a small selection of the rather large number of examples of surfaces of general type that have been found. Many of the surfaces of general type that have been investigated lie on (or near) the edges of the region of possible Chern numbers. In particular Horikawa surfaces lie on or near the "Noether line", many of the surfaces listed below lie on the line c_1^2 + c_2 = 12 \chi = 12, the minimum possible value for general type, and surfaces on the line 3c_2 = c_1^2 are all quotients of the unit ball in C2 (and are particularly hard to find).
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P. lujiatunensis skull LHPV 1 from the left and above The brain of P. lujiatunensis is well known; a study on the anatomy and functionality of three specimens was published in 2007. Until the study, it was generally thought the brain of Psittacosaurus would have been similar to other ceratopsians with low Encephalisation Quotients. Russell and Zhao (1996) believed "the small brain size of psittacosaurs implies a very restrictive behavioural repertoire relative to that of modern mammals of similar body size". However, the 2007 study dispelled this theory when it found the brain to be more advanced.
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Left foot of the type specimen as seen from the inside A restoration of Saurornithoides mongoliensis A comparison between a Saurornithoides mongoliensis specimen and an average human male Saurornithoides is a member of the troodontids, a group of small, bird-like, gracile maniraptorans. All troodontids have many unique features of the skull, such as closely spaced teeth in the lower jaw, and large numbers of teeth. Troodontids have sickle-claws and raptorial hands, and some of the highest non-avian encephalization quotients, meaning they were behaviourally advanced and had keen senses. Saurornithoides was a rather small troodontid.
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A positively curved universe is described by elliptic geometry, and can be thought of as a three-dimensional hypersphere, or some other spherical 3-manifold (such as the Poincaré dodecahedral space), all of which are quotients of the 3-sphere. Poincaré dodecahedral space is a positively curved space, colloquially described as "soccerball-shaped", as it is the quotient of the 3-sphere by the binary icosahedral group, which is very close to icosahedral symmetry, the symmetry of a soccer ball. This was proposed by Jean-Pierre Luminet and colleagues in 2003"Is the universe a dodecahedron?", article at PhysicsWeb.
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One can more generally view the classification problem from a homotopy-theoretic point of view. There is a universal bundle for real line bundles, and a universal bundle for complex line bundles. According to general theory about classifying spaces, the heuristic is to look for contractible spaces on which there are group actions of the respective groups C2 and S1, that are free actions. Those spaces can serve as the universal principal bundles, and the quotients for the actions as the classifying spaces BG. In these cases we can find those explicitly, in the infinite-dimensional analogues of real and complex projective space.
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In general, the Euclidean algorithm is convenient in such applications, but not essential; for example, the theorems can often be proven by other arguments. The Euclidean algorithm developed for two Gaussian integers and is nearly the same as that for ordinary integers, but differs in two respects. As before, the task at each step is to identify a quotient and a remainder such that :r_k = r_{k-2} - q_k r_{k-1}, where , where , and where every remainder is strictly smaller than its predecessor: . The first difference is that the quotients and remainders are themselves Gaussian integers, and thus are complex numbers.
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For example, the subdirect irreducibles in the variety generated by a finite linearly ordered Heyting algebra H must be just the nondegenerate quotients of H, namely all smaller linearly ordered nondegenerate Heyting algebras. The conditions cannot be dropped in general: for example, the variety of all Heyting algebras is generated by the set of its finite subdirectly irreducible algebras, but there exist subdirectly irreducible Heyting algebras of arbitrary (infinite) cardinality. There also exists a single finite algebra generating a (non- congruence-distributive) variety with arbitrarily large subdirect irreducibles.R. McKenzie, The residual bounds of finite algebras, Int.
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2 and 10.1.1. In particular, simple groups over an algebraically closed field k are classified (up to quotients by finite central subgroup schemes) by their Dynkin diagrams. It is striking that this classification is independent of the characteristic of k. For example, the exceptional Lie groups G2, F4, E6, E7, and E8 can be defined in any characteristic (and even as group schemes over Z). The classification of finite simple groups says that most finite simple groups arise as the group of k-points of a simple algebraic group over a finite field k, or as minor variants of that construction.
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In the late 1980s, Hindi cinema experienced another period of stagnation, with a decline in box office turnout, due to increasing violence, decline in musical melodic quality, and rise in video piracy, leading to middle-class family audiences abandoning theaters. The turning point came with Qayamat Se Qayamat Tak (1988), directed by Mansoor Khan, written and produced by his father Nasir Hussain, and starring his cousin Aamir Khan with Juhi Chawla. Its blend of youthfulness, wholesome entertainment, emotional quotients and strong melodies lured family audiences back to the big screen. It set a new template for Bollywood musical romance films that defined Hindi cinema in the 1990s.
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In 1879, four years after the signing of the Treaty of the Metre, the CIPM proposed a range of symbols for the various metric units then under the auspices of the CGPM. Among these were the use of the symbol "km" for "kilometre". In 1948, as part of its preparatory work for the SI, the CGPM adopted symbols for many units of measure that did not have universally agreed symbols, one of which was the symbol "h" for "hours". At the same time the CGPM formalised the rules for combining units – quotients could be written in one of three formats resulting in , and being valid representations of "kilometres per hour".
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From 1995 Shalev developed and applied probabilistic methods to finite groups and (nonabelian) finite simple groups in particular. A formative result in this area shows that almost every pair of elements in a finite simple group generate the group. This result, like many others in the field, were proved by Shalev in collaboration with Martin Liebeck of Imperial College at the University of London. The probabilistic approach led to the solution of many classical problems whose formulation does not involve probability; these problems concern quotients of the modular group, conjectures of Babai and of Cameron on permutation groups, diameters of certain Cayley graphs, Fuchsian groups, random walks, etc.
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It follows a complicated pattern rather analogous to that found on large continents at the same latitude in the northern hemisphere. Rainfall increases from around at Ouse in the centre to at Cradle Valley in the northwestern highlands. Sunshine is also highly differentiated, with average quotients ranging from around 4 hours a day (under 1500 hours a year) in the South West of the island, up to around 7 hours daily (2550 hours annually) in the North East around the Launceston area. It shares a similar climate to places like the United Kingdom, New Zealand, and the Pacific Northwest region of the United States, and Canada.
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In the study of p-groups, it is often important to use longer central series. An important class of such central series are the exponent-p central series; that is, a central series whose quotients are elementary abelian groups, or what is the same, have exponent p. There is a unique most quickly descending such series, the lower exponent-p central series λ defined by: :λ1(G) = G, and :λn + 1(G) = [G, λn(G)] (λn(G))p The second term, λ2(G), is equal to [G, G]Gp = Φ(G), the Frattini subgroup. The lower exponent-p central series is sometimes simply called the p-central series.
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Roughly fifteen years later in England, Henry Goodwyn set out to create a much more ambitious version of Haros’ table. In particular, Goodwyn wanted to tabulate the decimal values for all irreducible fractions with denominators less than or equal to 1,024. There are 318,963 such fractions. As a warm up and a test of the commercial market for such a table in 1816 he published for private circulation The First Centenary of a Series of Concise and Useful Tables of all the Complete Decimal Quotients, which can arise from dividing a unit, or any whole Number less than each Divisor by all Integers from 1 to 1024.
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Judge Ellen James Morphonios (September 30, 1929 – December 22, 2002) was a Dade County, Florida Circuit Judge, remembered for having prosecuted rock star Jim Morrison (The Doors) for allegedly exposing himself in her early days as a prosecutor. She was nicknamed "Maximum Morphonios" for the long sentences she routinely handed down to violent criminals. A native of Hyde County, North Carolina, Morphonios was a former model and beauty queen who passed a Florida exam that allowed her to enter law school without an undergraduate degree. She was also a member of the international organization for people with intelligence quotients in the top 2%, Mensa.
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In 2008, researchers from Kingston University in London discovered red wineThe sample wine was declared to be "a Shiraz from Southeast Australia", although no specific vintage, producer or wine region was stated in the report. to contain high levels of toxic metals relative to other beverages in the sample. Although the metal ions, which included chromium, copper, iron, manganese, nickel, vanadium and zinc, were also present in other plant-based beverages, the sample wine tested significantly higher for all metal ions, especially vanadium. Risk assessment was calculated using "target hazard quotients" (THQ), a method of quantifying health concerns associated with lifetime exposure to chemical pollutants.
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Putting these together, the octahedral axiom asserts the "third isomorphism theorem": :(Z/X)/(Y/X)\cong Z/Y. If the triangulated category is the derived category D(A) of an abelian category A, and X, Y, Z are objects of A viewed as complexes concentrated in degree 0, and the maps X\to Y and Y\to Z are monomorphisms in A, then the cones of these morphisms in D(A) are actually isomorphic to the quotients above in A. Finally, formulates the octahedral axiom using a two-dimensional commutative diagram with 4 rows and 4 columns. also give generalizations of the octahedral axiom.
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The modern approaches to algebraic geometry redefine and effectively extend the range of basic objects in various levels of generality to schemes, formal schemes, ind-schemes, algebraic spaces, algebraic stacks and so on. The need for this arises already from the useful ideas within theory of varieties, e.g. the formal functions of Zariski can be accommodated by introducing nilpotent elements in structure rings; considering spaces of loops and arcs, constructing quotients by group actions and developing formal grounds for natural intersection theory and deformation theory lead to some of the further extensions. Most remarkably, in late 1950s, algebraic varieties were subsumed into Alexander Grothendieck's concept of a scheme.
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Any other closed Riemannian 2-manifold of constant Gaussian curvature, after scaling the metric by a constant factor if necessary, will have one of these three surfaces as its universal covering space. In the orientable case, the fundamental group of can be identified with a torsion-free uniform subgroup of and can then be identified with the double coset space . In the case of the sphere and the Euclidean plane, the only possible examples are the sphere itself and tori obtained as quotients of by discrete rank 2 subgroups. For closed surfaces of genus , the moduli space of Riemann surfaces obtained as varies over all such subgroups, has real dimension .
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Abelian surfaces give rise to Kummer surfaces as quotients. There remains the class of elliptic surfaces, which are fiber bundles over a curve with elliptic curves as fiber, having a finite number of modifications (so there is a bundle that is locally trivial actually over a curve less some points). The question of classification is to show that any surface, lying in projective space of any dimension, is in the birational sense (after blowing up and blowing down of some curves, that is) accounted for by the models already mentioned. No more than other work in the Italian school would the proofs by Enriques now be counted as complete and rigorous.
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Also unlike modern reptiles, the brains of the juveniles did not seem to correlate any closer to the brain wall than those of adults. It was cautioned, however, that very young individuals were not included in the study. As with previous studies, EQ values were investigated, although a wider number range was given to account for uncertainty in brain and body mass. The range for the adult Hypacrosaurus was 2.3 to 3.7; the lowest end of this range was still higher than modern reptiles and most non-maniraptoran dinosaurs (nearly all having EQs below two), but fell well short of maniraptorans themselves, which had quotients higher than four.
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Earnings in the Health Care and Social Assistance sector grew at an average of over five percent per year during that period. Despite a relatively small share of total earnings, farming, forestry and tourism are all important economic drivers in Hood River County. In 2013, Farm Earnings and Forestry, Fishing and Related Activities had location quotients of 8.57 percent and 12.09 percent, respectively, indicating an outsized concentration of these sectors within the County. The Arts, Entertainment, and Recreation industry, while comprising less than four percent of total earnings, had a location quotient of 3.27 percent, the highest of any county in the State of Oregon, indicating a highly concentrated tourism sector.
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In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional elliptic modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular forms are Siegel modular varieties, which are basic models for what a moduli space for abelian varieties (with some extra level structure) should be and are constructed as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups. Siegel modular forms are holomorphic functions on the set of symmetric n × n matrices with positive definite imaginary part; the forms must satisfy an automorphy condition.
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If a given manifold admits a geometric structure, then it admits one whose model is maximal. A 3-dimensional model geometry X is relevant to the geometrization conjecture if it is maximal and if there is at least one compact manifold with a geometric structure modelled on X. Thurston classified the 8 model geometries satisfying these conditions; they are listed below and are sometimes called Thurston geometries. (There are also uncountably many model geometries without compact quotients.) There is some connection with the Bianchi groups: the 3-dimensional Lie groups. Most Thurston geometries can be realized as a left invariant metric on a Bianchi 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, in the realm of group theory, a group is said to be parafree if its quotients by the terms of its lower central series are the same as those of a free group and if it is residually nilpotent (the intersection of the terms of its lower central series is trivial). Parafree groups share many properties with free groups, making it difficult to distinguish between these two types. Gilbert Baumslag was led to the study of parafree groups in attempts to resolve the conjecture that a group of cohomological dimension one is free. One of his fundamental results is that there exist parafree groups that are not free.
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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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Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients). For example, noncommutative algebraic geometry is supposed to extend a notion of an algebraic scheme by suitable gluing of spectra of noncommutative rings; depending on how literally and how generally this aim (and a notion of spectrum) is understood in noncommutative setting, this has been achieved in various level of success. The noncommutative ring generalizes here a commutative ring of regular functions on a commutative scheme.
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In algebraic geometry, a Du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is an isolated singularity of a complex surface which is modeled on a double branched cover of the plane, with minimal resolution obtained by replacing the singular point with a tree of smooth rational curves, with intersection pattern dual to a Dynkin diagram of A-D-E singularity type. They are the canonical singularities (or, equivalently, rational Gorenstein singularities) in dimension 2. They were studied by and Felix Klein. The Du Val singularities also appear as quotients of C2 by a finite subgroup of SL2(C); equivalently, a finite subgroup of SU(2), which are known as binary polyhedral groups.
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The Cartan-Ambrose- Hicks theorem implies that M is locally Riemannian symmetric if and only if its curvature tensor is covariantly constant, and furthermore that every simply connected, complete locally Riemannian symmetric space is actually Riemannian symmetric. Every Riemannian symmetric space M is complete and Riemannian homogeneous (meaning that the isometry group of M acts transitively on M). In fact, already the identity component of the isometry group acts transitively on M (because M is connected). Locally Riemannian symmetric spaces that are not Riemannian symmetric may be constructed as quotients of Riemannian symmetric spaces by discrete groups of isometries with no fixed points, and as open subsets of (locally) Riemannian symmetric spaces.
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Arithmetic Chow groups are an amalgamation of Chow groups of varieties over Q together with a component encoding Arakelov-theoretical information, that is, differential forms on the associated complex manifold. The theory of Chow groups of schemes of finite type over a field extends easily to that of algebraic spaces. The key advantage of this extension is that it is easier to form quotients in the latter category and thus it is more natural to consider equivariant Chow groups of algebraic spaces. A much more formidable extension is that of Chow group of a stack, which has been constructed only in some special case and which is needed in particular to make sense of a virtual fundamental class.
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In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that : ax + by = \gcd(a, b). This is a certifying algorithm, because the gcd is the only number that can simultaneously satisfy this equation and divide the inputs. It allows one to compute also, with almost no extra cost, the quotients of a and b by their greatest common divisor. also refers to a very similar algorithm for computing the polynomial greatest common divisor and the coefficients of Bézout's identity of two univariate polynomials.
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A necessary and sufficient condition for a Heyting algebra to be subdirectly irreducible is for there to be a greatest element strictly below 1. The witnessing pair is that element and 1, and identifying any other pair a, b of elements identifies both a→b and b→a with 1 thereby collapsing everything above those two implications to 1. Hence every finite chain of two or more elements as a Heyting algebra is subdirectly irreducible. By Jónsson's Lemma, subdirectly irreducible algebras of a congruence-distributive variety generated by a finite set of finite algebras are no larger than the generating algebras, since the quotients and subalgebras of an algebra A are never larger than A itself.
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These are never algebraic, though they have non-constant meromorphic functions. They are usually divided into two subtypes: primary Kodaira surfaces with trivial canonical bundle, and secondary Kodaira surfaces which are quotients of these by finite groups of orders 2, 3, 4, or 6, and which have non-trivial canonical bundles. The secondary Kodaira surfaces have the same relation to primary ones that Enriques surfaces have to K3 surfaces, or bielliptic surfaces have to abelian surfaces. Invariants: If the surface is the quotient of a primary Kodaira surface by a group of order k = 1, 2, 3, 4, 6, then the plurigenera Pn are 1 if n is divisible by k and 0 otherwise.
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In the Fox television show Malcolm in the Middle, Malcolm Wilkerson displays astounding feats of automatic mental calculation, which causes him to fear his family will see him as a "freak", and causes his brother to ask, "Is Malcolm a robot?". In the 1991 movie Little Man Tate, Fred Tate in the audience blurts out the answer during a mental calculation contest. In the 1990s NBC TV sitcom NewsRadio, reporter/producer Lisa Miller can mentally calculate products, quotients, and square roots effortlessly and almost instantaneosly, on demand. In the 1997 Sci-Fi thriller Cube, one of the prisoners, Kazan, appears to be mentally disabled, but is revealed later in the film to be an autistic savant who is able to calculate prime factors in his head.
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The following is a rule by which we can write down at once the convergent fractions which result from these quotients without developing the continued fraction. The first quotient, supposed divided by unity, will give the first fraction, which will be too small, namely, . Then, multiplying the numerator and denominator of this fraction by the second quotient and adding unity to the numerator, we shall have the second fraction, , which will be too large. Multiplying in like manner the numerator and denominator of this fraction by the third quotient, and adding to the numerator the numerator of the preceding fraction, and to the denominator the denominator of the preceding fraction, we shall have the third fraction, which will be too small.
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The chemiosmotic theory was radical at the time, and it was not immediately accepted by the scientific community. One experiment that Moyle and Mitchell conducted in 1967 was an investigation of ways to improve the precision at which the quotients of translocation of a proton to oxygen can be measured, and what the optimum conditions are for these measurements. Their goal was to improve the precision of that measurement in order to allow others to confirm their findings in the proposed chemiosmotic hypothesis. They conducted this experiment because the arrangement of the electron transport chain was only a hypothesis at the time, and it had been inferred from a quotient of protons translocated to oxygen in the chain (→H+/O).
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The loop space is dual to the suspension of the same space; this duality is sometimes called Eckmann–Hilton duality. The basic observation is that :[\Sigma Z,X] \approxeq [Z, \Omega X] where [A,B] is the set of homotopy classes of maps A \rightarrow B, and \Sigma A is the suspension of A, and \approxeq denotes the natural homeomorphism. This homeomorphism is essentially that of currying, modulo the quotients needed to convert the products to reduced products. In general, [A, B] does not have a group structure for arbitrary spaces A and B. However, it can be shown that [\Sigma Z,X] and [Z, \Omega X] do have natural group structures when Z and X are pointed, and the aforementioned isomorphism is of those groups.
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For X0(N) and X1(N), the level structure is, respectively, a cyclic subgroup of order N and a point of order N. These curves have been studied in great detail, and in particular, it is known that X0(N) can be defined over Q. The equations defining modular curves are the best-known examples of modular equations. The "best models" can be very different from those taken directly from elliptic function theory. Hecke operators may be studied geometrically, as correspondences connecting pairs of modular curves. Remark: quotients of H that are compact do occur for Fuchsian groups Γ other than subgroups of the modular group; a class of them constructed from quaternion algebras is also of interest in number theory.
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In mathematics, Enriques surfaces are algebraic surfaces such that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial square. Enriques surfaces are all projective (and therefore Kähler over the complex numbers) and are elliptic surfaces of genus 0. Over fields of characteristic not 2 they are quotients of K3 surfaces by a group of order 2 acting without fixed points and their theory is similar to that of algebraic K3 surfaces. Enriques surfaces were first studied in detail by as an answer to a question discussed by about whether a surface with q=pg = 0 is necessarily rational, though some of the Reye congruences introduced earlier by are also examples of Enriques surfaces.
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He noted, similar to Marsh, noted the small predicted size of the organ, but also that it was significantly developed. A number of similarities to the brains of modern reptiles were noted. A 1905 diagram showing the small size of an Edmontosaurus annectens brain (bottom; alongside that of Triceratops horridus, top) commented on in early sources James Hopson investigated the encephalization quotients (EQs) of various dinosaurs in 1977 study. Three ornithopods for which brain endocasts had previously been produced – Camptosaurus, Iguanodon, and Anatosaurus (now known as Edmontosaurus annectens) – were investigated. It was found that they had relatively high EQs compared to many other dinosaurs (ranging from 0.8 to 1.5), comparable to that of carnosaurian theropods and of modern crocodilians, but far lower than that of coelurosaurian theropods.
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A particular but important case is given if A is a subset of the power set P(E) of some set E, ordered by reverse inclusion (⊇). Given this ordering of A, a subset B of A is cofinal in A if for every a ∈ A there is a b ∈ B such that a ⊇ b. For example, let E be a group and let A be the set of normal subgroups of finite index. The profinite completion of E is defined to be the inverse limit of the inverse system of finite quotients of E (which are parametrized by the set A). In this situation, every cofinal subset of A is sufficient to construct and describe the profinite completion of E.
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Each run is associated with a specific quotient value, which provides the most significant portion of the fingerprint, the runs are stored in order and each slot in the run provides the least significant portion of the fingerprint. So, by working from left to right, one can reconstruct all the fingerprints and the resulting list of integers will be in sorted order. Merging two quotient filters is then a simple matter of converting each quotient filter into such a list, merging the two lists and using it to populate a new larger quotient filter. Similarly, we can halve or double the size of a quotient filter without rehashing the keys since the fingerprints can be recomputed using just the quotients and remainders.
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These fractions can be found by the method of continued fractions: this arithmetical technique provides a series of progressively better approximations of any real numeric value by proper fractions. Since there may be an eclipse every half draconic month, we need to find approximations for the number of half draconic months per synodic month: so the target ratio to approximate is: SM / (DM/2) = 29.530588853 / (27.212220817/2) = 2.170391682 The continued fractions expansion for this ratio is: 2.170391682 = [2;5,1,6,1,1,1,1,1,11,1,...]:2.170391682 = 2 + 0.170391682 ; 1/0.170391682 = 5 + 0.868831085... ; 1/0.868831085... = 1 + 0.15097171... ; 1/0.15097171 = 6 + 0.6237575... ; etc. ; Evaluating this 4th continued fraction: 1/6 + 1 = 7/6; 6/7 + 5 = 41/7 ; 7/41 + 2 = 89/41 Quotients Convergents half DM/SM decimal named cycle (if any) 2; 2/1 = 2 5 11/5 = 2.2 1 13/6 = 2.166666667 semester 6 89/41 = 2.170731707 hepton 1 102/47 = 2.170212766 octon 1 191/88 = 2.170454545 tzolkinex 1 293/135 = 2.170370370 tritos 1 484/223 = 2.170403587 saros 1 777/358 = 2.170391061 inex 11 9031/4161 = 2.170391732 1 9808/4519 = 2.170391679 ... The ratio of synodic months per half eclipse year yields the same series: 5.868831091 = [5;1,6,1,1,1,1,1,11,1,...] Quotients Convergents SM/half EY decimal SM/full EY named cycle 5; 5/1 = 5 1 6/1 = 6 12/1 semester 6 41/7 = 5.857142857 hepton 1 47/8 = 5.875 47/4 octon 1 88/15 = 5.866666667 tzolkinex 1 135/23 = 5.869565217 tritos 1 223/38 = 5.868421053 223/19 saros 1 358/61 = 5.868852459 716/61 inex 11 4161/709 = 5.868829337 1 4519/770 = 5.868831169 4519/385 ... Each of these is an eclipse cycle. Less accurate cycles may be constructed by combinations of these.
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The divisor is chosen as necessary so that the resulting quotients, disregarding any fractional remainders, sum to the required total; in other words, pick a number so that there is no need to examine the remainders. Any number in one range of quotas will accomplish this, with the highest number in the range always being the same as the lowest number used by the D'Hondt method to award a seat (if it is used rather than the Jefferson method), and the lowest number in the range being the smallest number larger than the next number which would award a seat in the D'Hondt calculations. Applied to the above example of party lists, this range extends as integers from 20,001 to 25,000. More precisely, any number n for which 20,000 < n ≤ 25,000 can be used.
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The ergodicity of the geodesic flow on compact Riemann surfaces of variable negative curvature and on compact manifolds of constant negative curvature of any dimension was proved by Eberhard Hopf in 1939, although special cases had been studied earlier: see for example, Hadamard's billiards (1898) and Artin billiard (1924). The relation between geodesic flows on Riemann surfaces and one-parameter subgroups on SL(2, R) was described in 1952 by S. V. Fomin and I. M. Gelfand. The article on Anosov flows provides an example of ergodic flows on SL(2, R) and on Riemann surfaces of negative curvature. Much of the development described there generalizes to hyperbolic manifolds, since they can be viewed as quotients of the hyperbolic space by the action of a lattice in the semisimple Lie group SO(n,1).
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In the field of combinatorial group theory, it is an important and early result that free groups are residually nilpotent. In fact the quotients of the lower central series are free abelian groups with a natural basis defined by basic commutators, . If Gω = Gn for some finite n, then Gω is the smallest normal subgroup of G with nilpotent quotient, and Gω is called the nilpotent residual of G. This is always the case for a finite group, and defines the F1(G) term in the lower Fitting series for G. If Gω ≠ Gn for all finite n, then G/Gω is not nilpotent, but it is residually nilpotent. There is no general term for the intersection of all terms of the transfinite lower central series, analogous to the hypercenter (below).
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Specifically, data suggests that T. rex heard best in the low-frequency range, and that low-frequency sounds were an important part of tyrannosaur behavior. A 2017 study by Thomas Carr and colleagues found that the snout of tyrannosaurids was highly sensitive, based on a high number of small openings in the facial bones of the related Daspletosaurus that contained sensory neurons. The study speculated that tyrannosaurs might have used their sensitive snouts to measure the temperature of their nests and to gently pick-up eggs and hatchlings, as seen in modern crocodylians. A study by Grant R. Hurlburt, Ryan C. Ridgely and Lawrence Witmer obtained estimates for Encephalization Quotients (EQs), based on reptiles and birds, as well as estimates for the ratio of cerebrum to brain mass.
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In algebraic geometry, the existence of canonical metrics as proposed by Calabi allows one to give equally canonical representatives of characteristic classes by differential forms. Due to Yau's initial efforts at disproving the Calabi conjecture by showing that it would lead to contradictions in such contexts, he was able to draw striking corollaries to his primary theorem. In particular, the Calabi conjecture implies the Miyaoka–Yau inequality on Chern numbers of surfaces, as well as homotopical characterizations of the complex structures of the complex projective plane and of quotients of the two-dimensional complex unit ball. In string theory, it was discovered in 1985 by Philip Candelas, Gary Horowitz, Andrew Strominger, and Edward Witten that Calabi-Yau manifolds, due to their special holonomy, are the appropriate configuration spaces for superstrings.
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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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He constructed a pair of flat tori of 16 dimension, using arithmetic lattices first studied by Ernst Witt. After this example, many isospectral pairs in dimension two and higher were constructed (for instance, by M. F. Vignéras, A. Ikeda, H. Urakawa, C. Gordon). In particular , based on the Selberg trace formula for PSL(2,R) and PSL(2,C), constructed examples of isospectral, non-isometric closed hyperbolic 2-manifolds and 3-manifolds as quotients of hyperbolic 2-space and 3-space by arithmetic subgroups, constructed using quaternion algebras associated with quadratic extensions of the rationals by class field theory. In this case Selberg's trace formula shows that the spectrum of the Laplacian fully determines the length spectrum, the set of lengths of closed geodesics in each free homotopy class, along with the twist along the geodesic in the 3-dimensional case.
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Beginning with the next-to-last equation, g can be expressed in terms of the quotient qN−1 and the two preceding remainders, rN−2 and rN−3: : Those two remainders can be likewise expressed in terms of their quotients and preceding remainders, : and : Substituting these formulae for rN−2 and rN−3 into the first equation yields g as a linear sum of the remainders rN−4 and rN−5. The process of substituting remainders by formulae involving their predecessors can be continued until the original numbers a and b are reached: : : : After all the remainders r0, r1, etc. have been substituted, the final equation expresses g as a linear sum of a and b: g = sa + tb. Bézout's identity, and therefore the previous algorithm, can both be generalized to the context of Euclidean domains.
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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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In the mathematical field of group theory, an Artin transfer is a certain homomorphism from an arbitrary finite or infinite group to the commutator quotient group of a subgroup of finite index. Originally, such mappings arose as group theoretic counterparts of class extension homomorphisms of abelian extensions of algebraic number fields by applying Artin's reciprocity maps to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups. However, independently of number theoretic applications, a partial order on the kernels and targets of Artin transfers has recently turned out to be compatible with parent-descendant relations between finite p-groups (with a prime number p), which can be visualized in descendant trees. Therefore, Artin transfers provide a valuable tool for the classification of finite p-groups and for searching and identifying particular groups in descendant trees by looking for patterns defined by the kernels and targets of Artin transfers.
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Based on the equilibrium partitioning approach (which accounts for the varying biological availability of chemicals in different sediments), the ESB for total PAH is the sum of the quotients of a minimum of each of the 34 individual PAHs in a specific sediment, divided by the final chronic value concentration for each specific PAH in sediment. According to the U.S. EPA, freshwater or saltwater sediments that contain less or equal to 1.0 toxic units of the mixture of the 34 PAHs or more PAHs are acceptable for the protection of benthic organisms. Sediments that are greater than 1.0 toxic units are not protective and potentially have adverse effects to benthic organisms. U.S. EPA ESBs do not consider antagonistic, additive, or synergistic effects of other sediment contaminants and have been criticized as an overly conservative estimate for pyrogenic PAHs (such as those from manufactured gas plant processes).
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Like the orthogonal group, the projective orthogonal group can be defined over any field and with varied quadratic forms, though, as with the ordinary orthogonal group, the main emphasis is on the real positive definite projective orthogonal group; other fields are elaborated in generalizations, below. Except when mentioned otherwise, in the sequel PO and PSO will refer to the real positive definite groups. Like the spin groups and pin groups, which are covers rather than quotients of the (special) orthogonal groups, the projective (special) orthogonal groups are of interest for (projective) geometric analogs of Euclidean geometry, as related Lie groups, and in representation theory. More intrinsically, the (real positive definite) projective orthogonal group PO can be defined as the isometries of real projective space, while PSO can be defined as the orientation-preserving isometries of real projective space (when the space is orientable; otherwise PSO = PO).
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A review in 2005 found a U-shaped relationship between paternal age and low intelligence quotients (IQs). The highest IQ was found at paternal ages of 25–44; fathers younger than 25 and older than 44 tended to have children with lower IQs. It also found that "at least a half dozen other studies ... have demonstrated significant associations between paternal age and human intelligence." A 2009 study examined children at 8 months, 4 years and 7 years and found that higher paternal age was associated with poorer scores in almost all neurocognitive tests used but that higher maternal age was associated with better scores on the same tests; this was a reverse effect to that observed in the 2005 review, which found that maternal age began to correlate with lower intelligence at a younger age than paternal age, however two other past studies were in agreement with the 2009 study's results.
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To determine the county that each party will receive its leveling seats in, the following process is done: #For each county and eligible party, determine the first unused quotient when the regular district seats were distributed. If the party has not yet won a seat from that county, the quotient is equal to the number of votes the party received there. If the party already has won one mandate from that seat, the quotient is the number of votes received in that county divided by 3, if the party has already won two seats from the county, the quotient is the number of votes divided by 5, and so on. #The quotients for each county and party are divided by the total number of votes for all parties in that county and multiplied by the number of regular non-leveling seats allocated to that county.
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One passes between the first two by "pivoting" about X, to the third by pivoting about Z, and to the fourth by pivoting about X′. All enclosures in this diagram are commutative (both trigons and the square) but the other commutative square, expressing the equality of the two paths from Y′ to Y, is not evident. All the arrows pointing "off the edge" are degree 1: :300px This last diagram also illustrates a useful intuitive interpretation of the octahedral axiom. In triangulated categories, triangles play the role of exact sequences, and so it is suggestive to think of these objects as "quotients", Z' = Y/X and Y' = Z/X. In those terms, the existence of the last triangle expresses on the one hand :X' = Z/Y\ (looking at the triangle Y \to Z \to X' \to ), and :X' = Y'/Z' (looking at the triangle Z' \to Y' \to X' \to ).
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The cosets of with respect to outer automorphisms are then the elements of ; this is an instance of the fact that quotients of groups are not, in general, (isomorphic to) subgroups. If the inner automorphism group is trivial (when a group is abelian), the automorphism group and outer automorphism group are naturally identified; that is, the outer automorphism group does act on the group. For example, for the alternating group, , the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering as a subgroup of the symmetric group, , conjugation by any odd permutation is an outer automorphism of or more precisely "represents the class of the (non-trivial) outer automorphism of ", but the outer automorphism does not correspond to conjugation by any particular odd element, and all conjugations by odd elements are equivalent up to conjugation by an even element.
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Langlands' Ph.D. thesis was on the analytical theory of Lie semigroups,For context, see the note by Derek Robinson at the IAS site but he soon moved into representation theory, adapting the methods of Harish-Chandra to the theory of automorphic forms. His first accomplishment in this field was a formula for the dimension of certain spaces of automorphic forms, in which particular types of Harish-Chandra's discrete series appeared. He next constructed an analytical theory of Eisenstein series for reductive groups of rank greater than one, thus extending work of Hans Maass, Walter Roelcke, and Atle Selberg from the early 1950s for rank one groups such as . This amounted to describing in general terms the continuous spectra of arithmetic quotients, and showing that all automorphic forms arise in terms of cusp forms and the residues of Eisenstein series induced from cusp forms on smaller subgroups.
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This work led in turn, in the winter of 1966–67, to the now well known conjectures making up what is often called the Langlands program. Very roughly speaking, they propose a huge generalization of previously known examples of reciprocity, including (a) classical class field theory, in which characters of local and arithmetic abelian Galois groups are identified with characters of local multiplicative groups and the idele quotient group, respectively; (b) earlier results of Martin Eichler and Goro Shimura in which the Hasse–Weil zeta functions of arithmetic quotients of the upper half plane are identified with -functions occurring in Hecke's theory of holomorphic automorphic forms. These conjectures were first posed in relatively complete form in a famous letter to Weil, written in January 1967. It was in this letter that he introduced what has since become known as the -group and along with it, the notion of functoriality.
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Prasad's early work was on discrete subgroups of real and p-adic semi-simple groups. He proved the "strong rigidity" of lattices in real semi-simple groups of rank 1 and also of lattices in p-adic groups, see [1] and [2]. He then tackled group-theoretic and arithmetic questions on semi-simple algebraic groups. He proved the "strong approximation" property for simply connected semi-simple groups over global function fields [3]. In collaboration with M. S. Raghunathan, Prasad determined the topological central extensions of these groups, and computed the "metaplectic kernel" for isotropic groups, see [11], [12] and [10]. Later, together with Andrei Rapinchuk, Prasad gave a precise computation of the metaplectic kernel for all simply connected semi-simple groups, see [14]. Prasad and Raghunathan have also obtained results on the Kneser-Tits problem, [13]. In 1987, Prasad found a formula for the volume of S-arithmetic quotients of semi-simple groups, [4].
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Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, that describe groups as quotients of free groups; this field was first systematically studied by Walther von Dyck, student of Felix Klein, in the early 1880s, while an early form is found in the 1856 icosian calculus of William Rowan Hamilton, where he studied the icosahedral symmetry group via the edge graph of the dodecahedron. Currently combinatorial group theory as an area is largely subsumed by geometric group theory. Moreover, the term "geometric group theory" came to often include studying discrete groups using probabilistic, measure-theoretic, arithmetic, analytic and other approaches that lie outside of the traditional combinatorial group theory arsenal. In the first half of the 20th century, pioneering work of Max Dehn, Jakob Nielsen, Kurt Reidemeister and Otto Schreier, J. H. C. Whitehead, Egbert van Kampen, amongst others, introduced some topological and geometric ideas into the study of discrete groups.
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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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As an example, this section develops the case of the Anosov flow on the tangent bundle of a Riemann surface of negative curvature. This flow can be understood in terms of the flow on the tangent bundle of the Poincaré half-plane model of hyperbolic geometry. Riemann surfaces of negative curvature may be defined as Fuchsian models, that is, as the quotients of the upper half-plane and a Fuchsian group. For the following, let H be the upper half-plane; let Γ be a Fuchsian group; let M = H/Γ be a Riemann surface of negative curvature as the quotient of "M" by the action of the group Γ, and let T^1 M be the tangent bundle of unit-length vectors on the manifold M, and let T^1 H be the tangent bundle of unit-length vectors on H. Note that a bundle of unit-length vectors on a surface is the principal bundle of a complex line bundle.
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The tiling of the quartic by reflection domains is a quotient of the 3-7 kisrhombille. The Klein quartic admits tilings connected with the symmetry group (a "regular map"), and these are used in understanding the symmetry group, dating back to Klein's original paper. Given a fundamental domain for the group action (for the full, orientation-reversing symmetry group, a (2,3,7) triangle), the reflection domains (images of this domain under the group) give a tiling of the quartic such that the automorphism group of the tiling equals the automorphism group of the surface – reflections in the lines of the tiling correspond to the reflections in the group (reflections in the lines of a given fundamental triangle give a set of 3 generating reflections). This tiling is a quotient of the order-3 bisected heptagonal tiling of the hyperbolic plane (the universal cover of the quartic), and all Hurwitz surfaces are tiled in the same way, as quotients.
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Van der Poorten was the author of approximately 180 publications in number theory, on subjects that included Baker's theorem, continued fractions, elliptic curves, regular languages, the integer sequences derived from recurrence relations, and transcendental numbers. Some of his significant results include the 1988 solution of Pisot's conjecture on the rationality of Hadamard quotients of rational functions, his 1992 work with Bernard Dwork on the Eisenstein constant, his work with Enrico Bombieri on Diophantine approximation of algebraic numbers, and his 1999 paper with Kenneth Stuart Williams on the Chowla–Selberg formula. He had many co- authors, the most frequent being his colleague John H. Loxton, who joined the UNSW faculty in 1972 and who later like van der Poorten moved to Macquarie. As well as publishing his own research, van der Poorten was noted for his expository writings, among them a paper on Apéry's theorem on the irrationality of ζ(3) and his book on Fermat's last theorem.
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In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one sided maximal ideal A is not necessarily two-sided, the quotient R/A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J(R).
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Half-coins have infinitely many sides numbered with 0,1,2,... and the positive even numbers are taken with negative probabilities. Two half-coins make a complete coin in the sense that if we flip two half-coins then the sum of the outcomes is 0 or 1 with probability 1/2 as if we simply flipped a fair coin. In Convolution quotients of nonnegative definite functions and Algebraic Probability Theory Imre Z. Ruzsa and Gábor J. Székely proved that if a random variable X has a signed or quasi distribution where some of the probabilities are negative then one can always find two random variables, Y and Z, with ordinary (not signed / not quasi) distributions such that X, Y are independent and X + Y = Z in distribution. Thus X can always be interpreted as the "difference" of two ordinary random variables, Z and Y. If Y is interpreted as a measurement error of X and the observed value is Z then the negative regions of the distribution of X are masked / shielded by the error Y. Another example known as the Wigner distribution in phase space, introduced by Eugene Wigner in 1932 to study quantum corrections, often leads to negative probabilities.
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Vieta's formulas are frequently used with polynomials with coefficients in any integral domain . Then, the quotients a_i/a_n belong to the ring of fractions of (and possibly are in itself if a_n happens to be invertible in ) and the roots r_i are taken in an algebraically closed extension. Typically, is the ring of the integers, the field of fractions is the field of the rational numbers and the algebraically closed field is the field of the complex numbers. Vieta's formulas are then useful because they provide relations between the roots without having to compute them. For polynomials over a commutative ring which is not an integral domain, Vieta's formulas are only valid when a_n is a non zero-divisor and P(x) factors as a_n(x-r_1)(x-r_2)\dots(x-r_n). For example, in the ring of the integers modulo 8, the polynomial P(x)=x^2-1 has four roots: 1, 3, 5, and 7. Vieta's formulas are not true if, say, r_1=1 and r_2=3, because P(x) eq (x-1)(x-3). However, P(x) does factor as (x-1)(x-7) and as (x-3)(x-5), and Vieta's formulas hold if we set either r_1=1 and r_2=7 or r_1=3 and r_2=5.
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Of those smaller whales, four males ranged from , four females ranged from , and one of unknown sex was .Yamada, T. K., T. Kakuda & Y. Tajima, 2008. "Middle sized balaenopterid whale specimens in the Philippines and Indonesia". Memoirs of the National Science Museum, Tokyo, 45: 75–83. Lone individuals seen off Madagascar were estimated to range between 8 and 12 m (26.2 to 39.4 ft), while calves were estimated to be between 3 and 5 m (9.8 to 16.4 ft). The identity of three mature specimens (two females and a male) examined by biologist Graham Chittleborough in 1958 at a whaling station in Western Australia, which ranged in length from , is uncertain – they may refer to Omura's whale or the smaller form of Bryde's whale (B. edeni). These three individuals were noted to have very small baleen plates – about 22 cm (8.8 in) by , about 22 cm (8.8 in) by 16 cm (6.3 in), and , respectively – with length- breadth quotients of 1.34 to 1.46, within the upper range (1.00-1.43) of the 9 specimens included in the formal description of Omura's whale, but also within the lower range of the Bryde's whale complex (1.2 to 1.33). The holotype of the smaller form of Bryde's whale (B.
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Regardless if ΔP is infinitesimal or finite, there is (at least—in the case of the derivative—theoretically) a point range, where the boundaries are P ± (0.5) ΔP (depending on the orientation—ΔF(P), δF(P) or ∇F(P)): :LB = Lower Boundary; UB = Upper Boundary; Derivatives can be regarded as functions themselves, harboring their own derivatives. Thus each function is home to sequential degrees ("higher orders") of derivation, or differentiation. This property can be generalized to all difference quotients. As this sequencing requires a corresponding boundary splintering, it is practical to break up the point range into smaller, equi-sized sections, with each section being marked by an intermediary point (Pi), where LB = P0 and UB = Pń, the nth point, equaling the degree/order: LB = P0 = P0 \+ 0Δ1P = Pń − (Ń-0)Δ1P; P1 = P0 \+ 1Δ1P = Pń − (Ń-1)Δ1P; P2 = P0 \+ 2Δ1P = Pń − (Ń-2)Δ1P; P3 = P0 \+ 3Δ1P = Pń − (Ń-3)Δ1P; ↓ ↓ ↓ ↓ Pń-3 = P0 \+ (Ń-3)Δ1P = Pń − 3Δ1P; Pń-2 = P0 \+ (Ń-2)Δ1P = Pń − 2Δ1P; Pń-1 = P0 \+ (Ń-1)Δ1P = Pń − 1Δ1P; UB = Pń-0 = P0 \+ (Ń-0)Δ1P = Pń − 0Δ1P = Pń; ΔP = Δ1P = P1 − P0 = P2 − P1 = P3 − P2 = ... = Pń − Pń-1; ΔB = UB − LB = Pń − P0 = ΔńP = ŃΔ1P.
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