83 Sentences With "boundedness" | Random Sentence Generator

By asking whether pay phones exist or not, he locates the photograph in time, recognizes his boundedness.

Yet these dual evocations (of a cave, and that of a particular starlit night) do not contradict each other; they each reminded me of our earth-boundedness, our relation to other critters here, and of both violent and beautiful histories that make up the black diaspora.

Rather, boundedness is an underlying semantic distinction that motivates countability.

Results that give sufficient conditions for boundedness are known as multiplier theorems. Three such results are given below.

The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.

We now introduce a number of important properties that lead to interesting special classes of lattices. One, boundedness, has already been discussed.

In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm. The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus, but it was also proven independently by Hans Hahn.

He proved several fundamental problems such as Shokurov's conjecture on boundedness of complements and Borisov–Alexeev–Borisov conjecture on boundedness of Fano varieties.C. Birkar, Anti-pluricanonical systems on Fano varieties. arXiv:1603.05765C. Birkar, Singularities of linear systems and boundedness of Fano varieties. arXiv:1609.05543. In 2018, Birkar was given the Fields Medal for his Fano varieties and his other contributions the minimal model problem. In a video made available by the Simons Foundation, Birkar expressed hope that his Fields Medal will put “just a little smile on the lips” of the world's estimated 40 million Kurds.

The (strong) torsion conjecture first posed by has been completely resolved in the case of elliptic curves. Barry Mazur proved uniform boundedness for elliptic curves over the rationals. His techniques were generalized by and , who obtained uniform boundedness for quadratic fields and number fields of degree at most 8 respectively. Finally, Loïc Merel () proved the conjecture for elliptic curves over any number field.

Local boundedness may also refer to a property of topological vector spaces, or of functions from a topological space into a topological vector space.

For verbs, certain grammatical aspects express boundedness. Boundedness is characteristic of perfective aspects such as the Ancient Greek aorist and the Spanish preterite. The simple past of English commonly expresses a bounded event ("I found out"), but sometimes expresses, for example, a stative ("I knew"). The perfective aspect often includes a contextual variation similar to an inchoative aspect or verb, and expresses the beginning of a state.

In mathematics, especially functional analysis, a bornology on a set X is a collection of subsets of X satisfying axioms that generalize the notion of boundedness.

In mathematics, Nemytskii operators are a class of nonlinear operators on Lp spaces with good continuity and boundedness properties. They take their name from the mathematician Viktor Vladimirovich Nemytskii.

The boundedness problem for Datalog asks, given a Datalog program, whether it is bounded, i.e., the maximal recursion depth reached when evaluating the program on an input database can be bounded by some constant. In other words, this question asks whether the Datalog program could be rewritten as a nonrecursive Datalog program. Solving the boundedness problem on arbitrary Datalog programs is undecidable, but it can be made decidable by restricting to some fragments of Datalog.

If X is a topological vector space (TVS) then the set of all boundedness subsets of X from a bornology (indeed, even a vector bornology) on X called the von Neumann bornology of X, the usual bornology, or simply the bornology of X and is referred to as natural boundedness. In any locally convex TVS X, the set of all closed bounded disks form a base for the usual bornology of X.

Let X be a topological space, Y a topological vector space, and f : X → Y a function. Then f is locally bounded if each point of X has a neighborhood whose image under f is bounded. The following theorem relates local boundedness of functions with the local boundedness of topological vector spaces: :Theorem. A topological vector space X is locally bounded if and only if the identity map idX: X → X is locally bounded.

In mathematics, particularly in functional analysis and convex analysis, the Ursescu theorem is a theorem that generalizes the closed graph theorem, the open mapping theorem, and the uniform boundedness principle.

One of the major problems in the subject is to determine, for any specified multiplier m, whether the corresponding Fourier multiplier operator continues to be well-defined when f has very low regularity, for instance if it is only assumed to lie in an Lp space. See the discussion on the "boundedness problem" below. As a bare minimum, one usually requires the multiplier m to be bounded and measurable; this is sufficient to establish boundedness on L^2 but is in general not strong enough to give boundedness on other spaces. One can view the multiplier operator T as the composition of three operators, namely the Fourier transform, the operation of pointwise multiplication by m, and then the inverse Fourier transform.

Since V_n \subset V, boundedness and ellipticity of the bilinear form apply to V_n. Therefore, the well- posedness of the Galerkin problem is actually inherited from the well- posedness of the original problem.

In fact, only single rules over extensional predicate symbols can be easily rewritten as an equivalent conjunctive query. The problem of deciding whether for a given Datalog program there is an equivalent nonrecursive program (corresponding to a positive relational algebra query, or, equivalently, a formula of positive existential first-order logic, or, as a special case, a conjunctive query) is known as the Datalog boundedness problem and is undecidable.Gerd G. Hillebrand, Paris C. Kanellakis, Harry G. Mairson, Moshe Y. Vardi: Undecidable Boundedness Problems for Datalog Programs. J. Log. Program.

In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.

It has been proved with the introduction of the concept itself that when two finite state machines communicate with only one type of messages, boundedness, deadlocks, and unspecified reception state can be decided and identified while such is not the case when the machines communicate with two or more types of messages. Later, it has been further proved that when only one finite state machine communicates with single type of message while the communication of its partner is unconstrained, we can still decide and identify boundedness, deadlocks, and unspecified reception state. It has been further proved that when the message priority relation is empty, boundedness, deadlocks and unspecified reception state can be decided even under the condition in which there are two or more types of messages in the communication between finite state machines.Gouda, Mohamed G; Rosier, Louis E. "Communicating finite state machines with priority channels," Automata, Languages and Programming.

Antwerp: ICALP, 1984 Boundedness, deadlocks, and unspecified reception state are all decidable in polynomial time (which means that a particular problem can be solved in tractable, not infinite, amount of time) since the decision problems regarding them are nondeterministic logspace complete.

If an algebra A is Koszul, there is an equivalence between certain subcategories of the derived categories of graded A\- and A^!-modules. These subcategories are defined by certain boundedness conditions on the grading vs. the cohomological degree of a complex.

He retired in 1989. His main research area was the theory of Jordan algebras, where he introduced the Kantor–Koecher–Tits construction and the Koecher–Vinberg theorem. He discovered the Koecher boundedness principle in the theory of Siegel modular forms.

The boundedness problem consists in deciding whether the set of reachable configuration is finite. I.e. the length of the content of each channel is bounded. This problem is trivially decidable over machine capable of insertion of errors. It is also decidable over counter machine.

He was elected as a Fellow of the American Mathematical Society in the 2020 Class, for "contributions to algebraic geometry, in particular his proof of the finite generation of the canonical ring, the existence of flips and the boundedness of varieties of log general type".

In the calculus of variations, where one is typically interested in infinite-dimensional function spaces, the condition is necessary because some extra notion of compactness beyond simple boundedness is needed. See, for example, the proof of the mountain pass theorem in section 8.5 of Evans.

More generally, let be a morphism of degree at least two defined over a number field . Northcott's theorem says that has only finitely many preperiodic points in , and the general Uniform Boundedness Conjecture says that the number of preperiodic points in may be bounded solely in terms of , the degree of , and the degree of over . The Uniform Boundedness Conjecture is not known even for quadratic polynomials over the rational numbers . It is known in this case that cannot have periodic points of period four, five, or six, although the result for period six is contingent on the validity of the conjecture of Birch and Swinnerton-Dyer.

The uniform boundedness principle (also known as the Banach–Steinhaus theorem) states that a set of linear maps between Banach spaces is equicontinuous if it is pointwise bounded; i.e., for each . The result can be generalized to a case when is locally convex and is a barreled space.

Standard Petri net properties like reachability, boundedness and liveness show a mixed picture. A paper Michael Köhler- Bußmeier: A Survey of Decidability Results for Elementary Object Systems: Fundamenta Informaticae, Vol. 130, No 1, pp. 99-123, 2014 of Köhler-Bußmeier gives a survey on decidability results for elementary object systems.

In topological vector spaces, a different definition for bounded sets exists which is sometimes called von Neumann boundedness. If the topology of the topological vector space is induced by a metric which is homogeneous, as in the case of a metric induced by the norm of normed vector spaces, then the two definitions coincide.

Because unbounded nouns refer to internally homogenous referents, any part of their expansive referent could be analyzed as an instance of that noun. Further, any removal of the expanse does not change the applicability of the noun to its referent. These two qualities are not possible of bounded nouns. Note that boundedness in nouns should not be thought of as synonymous with countability.

Let be a rational function of degree at least two with coefficients in . A theorem of Northcott says that has only finitely many -rational preperiodic points, i.e., has only finitely many preperiodic points in . The Uniform Boundedness Conjecture of Morton and Silverman says that the number of preperiodic points of in is bounded by a constant that depends only on the degree of .

Haddad's research in the area on nonlinear dynamical system theory is highlighted in his textbook on Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach , Princeton, NJ: Princeton University Press, 2008. This 1000-page "encyclopedic masterpiece" presents and develops an extensive treatment of stability analysis and control design of nonlinear dynamical systems, with an emphasis on Lyapunov-based methods. Topics include Lyapunov stability theory, partial stability, Lagrange stability, boundedness, ultimate boundedness, input-to-state stability, input-output stability, finite-time stability, semistability, stability of sets, stability of periodic orbits, and stability theorems via vector Lyapunov functions. In addition, a complete and thorough treatment of dissipativity theory, absolute stability theory, stability of feedback interconnections, optimal control, backstepping control, disturbance rejection control, and robust control via fixed and parameter-dependent Lyapunov functions for nonlinear continuous-time and discrete-time dynamical systems is also given.

Semantic properties of nouns/entities can be divided into eight classes: specificity, boundedness, animacy, gender, kinship, social status, physical properties, and function. Physical properties refer to how an entity exists in space. It can include shape, size, and material, for example. The function class of semantic properties refers to noun class markers that indicate the purpose of an entity or how humans utilize an entity.

Journal of the ACM, 30(2):323-342, 1983. and can be used as a model of concurrent processes like Petri nets. Communicating finite state machines are used frequently for modeling a communication protocol since they make it possible to detect major protocol design errors, including boundedness, deadlocks, and unspecified receptions.Rosier, Louis E; Gouda, Mohamed G. Deciding Progress for a Class of Communicating Finite State Machines.

In a Hilbert space , a sequence is weakly convergent to a vector when : \lim_n \langle x_n, v \rangle = \langle x, v \rangle for every . For example, any orthonormal sequence converges weakly to 0, as a consequence of Bessel's inequality. Every weakly convergent sequence is bounded, by the uniform boundedness principle. Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences (Alaoglu's theorem).

K Versteeg & W. Malalasekera . An introduction to Computational Fluid Dynamics.Chapter:5.Page 118(5.6.1.1). The upwind differencing scheme formulation is conservative. BoundednessH.K Versteeg & W. Malalasekera . An introduction to Computational Fluid Dynamics.Chapter:5.Page 118 (5.6.1.2). As the coefficients of the discretised equation are always positive hence satisfying the requirements for boundedness and also the coefficient matrix is diagonally dominant therefore no irregularities occur in the solution.

A semi-infinite integral is an improper integral over a semi-infinite interval. More generally, objects indexed or parametrised by semi-infinite sets may be described as semi-infinite.Cator, Pimentel, A shape theorem and semi-infinite geodesics for the Hammersley model with random weights, 2010. Most forms of semi-infiniteness are boundedness properties, not cardinality or measure properties: semi-infinite sets are typically infinite in cardinality and measure.

A space is called ω-bounded if the closure of every countable set is compact. For example, the long line and the closed long ray are ω-bounded but not compact. When restricted to a metric space ω-boundedness is equivalent to compactness. The bagpipe theorem states that every ω-bounded connected surface is the connected sum of a compact connected surface and a finite number of long pipes.

If the differencing scheme produces coefficients that satisfy the above criterion the resulting matrix of coefficients is diagonally dominant. To achieve diagonal dominance we need large values of net coefficient so the linearisation practice of source terms should ensure that SP is always negative. If this is the case –SP is always positive and adds to aP. Diagonal dominance is a desirable feature for satisfying the boundedness criterion.

While L^p to L^p bounds can be derived immediately from the L^1 to weak L^1 estimate by a clever change of variables, Marcinkiewicz interpolation is a more intuitive approach. Since the Hardy–Littlewood Maximal Function is trivially bounded from L^\infty to L^\infty, strong boundedness for all p>1 follows immediately from the weak (1,1) estimate and interpolation. The weak (1,1) estimate can be obtained from the Vitali covering lemma.

In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for abelian varieties states that the order of the torsion group of an abelian variety over a number field can be bounded in terms of the dimension of the variety and the number field. A stronger version of the conjecture is that the torsion is bounded in terms of the dimension of the variety and the degree of the number field.

The donor- acceptor scheme is based on two fundamental criteria, namely the boundedness criterion and the availability criterion. The first one states that the value of C has to be bounded between zero and one. The latter criterion ensures that the amount of fluid convected over a face during a time step is less than or equal to the amount available in the donor cell, i.e., the cell from which the fluid is flowing to the acceptor cell.

In 2019 Krieger was the Australian Mathematical Sciences Institute Mahler Lecturer, which involved giving a series of seminars and public lectures across Australia. In 2020 she won a Whitehead Prize of the London Mathematical Society "for her deep contributions to arithmetic dynamics, to equidistribution, to bifurcation loci in families of rational maps, and her recent proof (with DeMarco and Ye) of uniform boundedness results for numbers of torsion points on families of bielliptic genus two curves in their Jacobians".

In this context, it is still true that every continuous map is bounded, however the converse fails; a bounded operator need not be continuous. Clearly, this also means that boundedness is no longer equivalent to Lipschitz continuity in this context. If the domain is a bornological space (e.g. a pseudometrizable TVS, a Fréchet space, a normed space) then a linear operators into any other locally convex spaces is bounded if and only if it is continuous.

In any concrete category, especially for vector spaces, endomorphisms are maps from a set into itself, and may be interpreted as unary operators on that set, acting on the elements, and allowing to define the notion of orbits of elements, etc. Depending on the additional structure defined for the category at hand (topology, metric, ...), such operators can have properties like continuity, boundedness, and so on. More details should be found in the article about operator theory.

This is known as the Heine–Borel theorem. Note that compactness depends only on the topology, while boundedness depends on the metric. Lebesgue's number lemma states that for every open cover of a compact metric space M, there exists a "Lebesgue number" \delta such that every subset of M of diameter r<\delta is contained in some member of the cover. Every compact metric space is second countable, and is a continuous image of the Cantor set.

Since Fourier series have such good convergence properties, many are often surprised by some of the negative results. For example, the Fourier series of a continuous T-periodic function need not converge pointwise. The uniform boundedness principle yields a simple non-constructive proof of this fact. In 1922, Andrey Kolmogorov published an article titled Une série de Fourier- Lebesgue divergente presque partout in which he gave an example of a Lebesgue- integrable function whose Fourier series diverges almost everywhere.

Together with F. Fahroo, W. Kang and Q. Gong, Ross proved a series of results on the convergence of pseudospectral discretizations of optimal control problems.W. Kang, I. M. Ross, Q. Gong, Pseudospectral optimal control and its convergence theorems, Analysis and Design of Nonlinear Control Systems, Springer, pp. 109–124, 2008. Ross and his coworkers showed that the Legendre and Chebyshev pseudospectral discretizations converge to an optimal solution of a problem under the mild condition of boundedness of variations.

This clause boundedness somewhat restricts the QR. May also noticed a subject-object asymmetry with respect to the interaction of wh-words and quantifier phrases. A modified version of his past work that QR determines quantifier scope but does not disambiguate it was brought up. To regulate the interaction, The Scope Principle that if two operators govern each other, they can be interpreted in either scopal order was also brought up. However, this solution has eventually been abandoned.

The Lp boundedness problem (for any particular p) for a given group G is, stated simply, to identify the multipliers m such that the corresponding multiplier operator is bounded from Lp(G) to Lp(G). Such multipliers are usually simply referred to as "Lp multipliers". Note that as multiplier operators are always linear, such operators are bounded if and only if they are continuous. This problem is considered to be extremely difficult in general, but many special cases can be treated.

In this general case, necessary and sufficient conditions for boundedness have not been established, even for Euclidean space or the unit circle. However, several necessary conditions and several sufficient conditions are known. For instance it is known that in order for a multiplier operator to be bounded on even a single Lp space, the multiplier must be bounded and measurable (this follows from the characterisation of L2 multipliers above and the inclusion property). However, this is not sufficient except when p = 2.

If T is a closed operator (which includes the case that T is a bounded operator), boundedness of such inverses follows automatically if the inverse exists at all. The space of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra. Since the definition of the spectrum does not mention any properties of B(X) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.

If X is a topological vector space (TVS) then the set of all bounded subsets of X from a vector bornology on X called the von Neumann bornology of X, the usual bornology, or simply the bornology of X and is referred to as natural boundedness. In any locally convex TVS X, the set of all closed bounded disks form a base for the usual bornology of X. Unless indicated otherwise, it is always assumed that the real or complex numbers are endowed with the usual bornology.

In linguistics, boundedness is a semantic feature that relates to an understanding of the referential limits of a lexical item. Fundamentally, words that specify a spatio-temporal demarcation of their reference are considered bounded, while words that allow for a fluidly interpretable referent are considered unbounded. This distinction also relies on the divisibility of the lexical item's referent into distinct segments, or strata. Though this feature most often distinguishes countability in nouns and aspect in verbs, it applies more generally to any syntactic category.

A relatively refined theory is available for pseudocompact topological groups.See, for example, Mikhail Tkachenko, Topological Groups: Between Compactness and \aleph_0-boundedness, in Mirek Husek and Jan van Mill (eds.), Recent Progress in General Topology II, 2002 Elsevier Science B.V. In particular, W. W. Comfort and Kenneth A. Ross proved that a product of pseudocompact topological groups is still pseudocompact (this might fail for arbitrary topological spaces).Comfort, W. W. and Ross, K. A., Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16, 483-496, 1966.

In complex analysis and geometric function theory, the Grunsky matrices, or Grunsky operators, are infinite matrices introduced in 1939 by Helmut Grunsky. The matrices correspond to either a single holomorphic function on the unit disk or a pair of holomorphic functions on the unit disk and its complement. The Grunsky inequalities express boundedness properties of these matrices, which in general are contraction operators or in important special cases unitary operators. As Grunsky showed, these inequalities hold if and only if the holomorphic function is univalent.

Baptiste asserts that historical views of race and colonialism impact Trinidadian culture in such a way that are often excluded from Western feminist studies. "Caribbean gender theory has to wrangle with the boundedness of patriarchy at the same time as it tussles with the barnacles of colonialism and imperialism." Gender performances in Trinidad and Tobago occupy three distinct spaces: physical, social, and cultural. Baptiste argues that the physical, public spaces represent a "postcolonial essentialist collage" in which performances are gendered by the socialization of gender roles according to very essentialist views of men and women.

A classical particle has a definite position and momentum, and hence it is represented by a point in phase space. Given a collection (ensemble) of particles, the probability of finding a particle at a certain position in phase space is specified by a probability distribution, the Liouville density. This strict interpretation fails for a quantum particle, due to the uncertainty principle. Instead, the above quasiprobability Wigner distribution plays an analogous role, but does not satisfy all the properties of a conventional probability distribution; and, conversely, satisfies boundedness properties unavailable to classical distributions.

Also, Calderón insisted that the focus should be on algebras of singular integral operators with non-smooth kernels to solve actual problems arising in physics and engineering, where lack of smoothness is a natural feature. It led to what is now known as the "Calderón program", with major parts: Calderón's study of the Cauchy integral on Lipschitz curves,Calderón, A. P. (1977), Cauchy integrals on Lipschitz curves and related operators, Proc. Natl. Acad. Sci. U.S.A. 74, pp. 1324–1327 and his proof of the boundedness of the "first commutator".

In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed "size" (where the meaning of "size" depends on the structure of the ambient space.) The term precompact (or pre-compact) is sometimes used with the same meaning, but precompact is also used to mean relatively compact. These definitions coincide for subsets of a complete metric space, but not in general.

In mathematics, a Carleson measure is a type of measure on subsets of n-dimensional Euclidean space Rn. Roughly speaking, a Carleson measure on a domain Ω is a measure that does not vanish at the boundary of Ω when compared to the surface measure on the boundary of Ω. Carleson measures have many applications in harmonic analysis and the theory of partial differential equations, for instance in the solution of Dirichlet problems with "rough" boundary. The Carleson condition is closely related to the boundedness of the Poisson operator. Carleson measures are named after the Swedish mathematician Lennart Carleson.

A set of real numbers is bounded if and only if it has an upper and lower bound. This definition is extendable to subsets of any partially ordered set. Note that this more general concept of boundedness does not correspond to a notion of "size". A subset S of a partially ordered set P is called bounded above if there is an element k in P such that k ≥ s for all s in S. The element k is called an upper bound of S. The concepts of bounded below and lower bound are defined similarly.

The term is used widely with this definition that focuses on suprema and there is no common name for the dual property. However, bounded completeness can be expressed in terms of other completeness conditions that are easily dualized (see below). Although concepts with the names "complete" and "bounded" were already defined, confusion is unlikely to occur since one would rarely speak of a "bounded complete poset" when meaning a "bounded cpo" (which is just a "cpo with greatest element"). Likewise, "bounded complete lattice" is almost unambiguous, since one would not state the boundedness property for complete lattices, where it is implied anyway.

Rodolfo Humberto Torres (born November 20, 1960) is an Argentine mathematician specializing in harmonic analysis who works as a University Distinguished Professor of Mathematics at the University of Kansas. Torres did his undergraduate studies at the National University of Rosario in Argentina, completing a licenciatura there in 1984. He earned his doctorate in 1989 from Washington University in St. Louis, with a dissertation entitled On the Boundedness of Certain Operators with Singular Kernels on Distribution Spaces and supervised by Björn D. Jawerth. In 2012 he became one of the inaugural fellows of the American Mathematical Society.

In mathematicsspecifically, in the theory of stochastic processesDoob's martingale convergence theorems are a collection of results on the limits of supermartingales, named after the American mathematician Joseph L. Doob. Informally, the martingale convergence theorem typically refers to the result that any supermartingale satisfying a certain boundedness condition must converge. One may think of supermartingales as the random variable analogues of non-increasing sequences; from this perspective, the martingale convergence theorem is a random variable analogue of the monotone convergence theorem, which states that any bounded monotone sequence converges. There are symmetric results for submartingales, which are analogous to non-decreasing sequences.

In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by that property that a linear maps from a bornological space into any locally convex spaces is continuous if and only if it a bounded linear operator. Bornological spaces were first studied by Mackey. The name was coined by Bourbaki after , the French word for "bounded".

One of the presently unanswered questions about the universe is whether it is infinite or finite in extent. For intuition, it can be understood that a finite universe has a finite volume that, for example, could be in theory filled up with a finite amount of material, while an infinite universe is unbounded and no numerical volume could possibly fill it. Mathematically, the question of whether the universe is infinite or finite is referred to as boundedness. An infinite universe (unbounded metric space) means that there are points arbitrarily far apart: for any distance , there are points that are of a distance at least apart.

For example, compactness and connectedness are topological properties, whereas boundedness and completeness are not. Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces. ;Topological space: A topological space (X, T) is a set X equipped with a collection T of subsets of X satisfying the following axioms: :# The empty set and X are in T. :# The union of any collection of sets in T is also in T. :# The intersection of any pair of sets in T is also in T. :The collection T is a topology on X. ;Topological sum: See Coproduct topology. ;Topologically complete: Completely metrizable spaces (i. e.

The difference between a contract and a covenant is that a covenant is conceived under a deity and therefore has a spiritual context and boundedness that includes a higher power as not only a participant but also a guarantor. It was not surprising that the Indigenous people viewed Treaties as covenants based on their previous relationships with fur traders of the Hudson’s Bay Company and how their interactions were founded on a basis of religion and mutual respect for a higher power and their land. The treaty outlined specifics as to rights of Indigenous people and support and protection of the Queen. These included rights that Indigenous people could hunt and fish and had provisions on their land.

Galston (1999) Kavanaugh and Patterson (2001) did not find that increased Internet usage increased community involvement and attachment. According to Gilleard, C. et al. (2007), “ownership and use of domestic information and communication technology reduces the sense of attachment to the local neighborhood among individuals 50 and older in England.” But they continue that “domestic information and communication technology may be more liberating of neighborhood boundedness than destructive of social capital.”Kavanaugh and Patterson (2001) Anonymity is often mentioned in popular media as a possible cause for negative effects. But according to Bargh and McKenna (2004), anonymity also associated with positive effects: “research has found that the relative anonymity aspect encourages self-expression, and the relative absence of physical and nonverbal interaction cues (e.g.

Indeed, x can be any predetermined system metric and corresponding mathematical analysis would illustrate some hidden analytical behaviors of the system. The reciprocal-of-polynomial formulation is used for the same reason that computational boundedness is defined as polynomial running time: it has mathematical closure properties that make it tractable in the asymptotic setting (see #Closure properties). For example, if an attack succeeds in violating a security condition only with negligible probability, and the attack is repeated a polynomial number of times, the success probability of the overall attack still remains negligible. In practice one might want to have more concrete functions bounding the adversary's success probability and to choose the security parameter large enough that this probability is smaller than some threshold, say 2−128.

Any topological vector space is an abelian topological group under addition, so the above conditions apply (although note that the above are written multiplicatively). Historically, definition 1(b) was the first reformulation of total boundedness for topological vector spaces; it dates to a 1935 paper of John von Neumann. This definition has the appealing property that, in a locally convex space endowed with the weak topology, the precompact sets are exactly the bounded sets. For seperable Banach spaces, there is a nice characterization of the precompact sets (in the norm topology) in terms of weakly convergent sequences of functionals: if is a separable Banach space, then is precompact if and only if every weakly convergent sequence of functionals converges uniformly on .

The Bombieri–Lang conjecture is an analogue for surfaces of Faltings's theorem, which states that algebraic curves of genus greater than one only have finitely many rational points. If true, the Bombieri–Lang conjecture would resolve the Erdős–Ulam problem, as it would imply that there do not exist dense subsets of the Euclidean plane all of whose pairwise distances are rational. In 1997, Lucia Caporaso, Barry Mazur, Joe Harris, and Patricia Pacelli showed that the Bombieri–Lang conjecture implies a type of uniform boundedness conjecture: there is a constant B_{g,d} depending only on g and d such that the number of rational points of any genus g curve X over any degree d number field is at most B_{g,d}.

Stein worked primarily in the field of harmonic analysis, and made contributions in both extending and clarifying Calderón–Zygmund theory. These include Stein interpolation (a variable-parameter version of complex interpolation), the Stein maximal principle (showing that under many circumstances, almost everywhere convergence is equivalent to the boundedness of a maximal function), Stein complementary series representations, Nikishin–Pisier–Stein factorization in operator theory, the Tomas–Stein restriction theorem in Fourier analysis, the Kunze–Stein phenomenon in convolution on semisimple groups, the Cotlar–Stein lemma concerning the sum of almost orthogonal operators, and the Fefferman–Stein theory of the Hardy space H^1 and the space BMO of functions of bounded mean oscillation. He wrote numerous books on harmonic analysis (see e.g.

A metric appears in the definition of total boundedness only to ensure that each element of the finite cover is of comparable size, and can be weakened to that of a uniform structure. A subset S of a uniform space X is totally bounded if and only if, for any entourage E, there exists a finite cover of S by subsets of X each of whose Cartesian squares is a subset of E. (In other words, replaces the "size" , and a subset is of size E if its Cartesian square is a subset of E.) C.f. definition 39.7 and lemma 39.8. The definition can be extended still further, to any category of spaces with a notion of compactness and Cauchy completion: a space is totally bounded if and only if its (Cauchy) completion is compact.

In the mathematical fields of geometry and topology, a coarse structure on a set X is a collection of subsets of the cartesian product X × X with certain properties which allow the large-scale structure of metric spaces and topological spaces to be defined. The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space--such as boundedness, or the degrees of freedom of the space--do not depend on such features. Coarse geometry and coarse topology provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties.

After an initial section of the book, introducing computable analysis and leading up to an example of John Myhill of a computable continuously differentiable function whose derivative is not computable, the remaining two parts of the book concerns the authors' results. These include the results that, for a computable self-adjoint operator, the eigenvalues are individually computable, but their sequence is (in general) not; the existence of a computable self- adjoint operator for which 0 is an eigenvalue of multiplicity one with no computable eigenvectors; and the equivalence of computability and boundedness for operators. The authors' main tools include the notions of a computability structure, a pair of a Banach space and an axiomatically-characterized set of its sequences, and of an effective generating set, a member of the set of sequences whose linear span is dense in the space. The authors are motivated in part by the computability of solutions to differential equations.

Then the axiom schema is: : \forall w_1,\ldots,w_n \,[(\forall x\, \exists\, y \phi(x, y, w_1, \ldots, w_n)) \Rightarrow \forall A\, \exists B\, \forall x \in A\, \exists y \in B\, \phi(x, y, w_1, \ldots, w_n)] The axiom schema is sometimes stated without prior restrictions (apart from B not occurring free in \phi) on the predicate, \phi: : \forall w_1,\ldots,w_n \, \forall A\, \exists B\,\forall x \in A\, [ \exists y \phi(x, y, w_1, \ldots, w_n) \Rightarrow \exists y \in B\,\phi(x, y, w_1, \ldots, w_n)] In this case, there may be elements x in A that are not associated to any other sets by \phi. However, the axiom schema as stated requires that, if an element x of A is associated with at least one set y, then the image set B will contain at least one such y. The resulting axiom schema is also called the axiom schema of boundedness.

Connes, H. Moscovici; The Local Index Formula in Noncommutative Geometry expresses the pairing of the K-group of the manifold with this K-cycle in two ways: the 'analytic/global' side involves the usual trace on the Hilbert space and commutators of functions with the phase operator (which corresponds to the 'index' part of the index theorem), while the 'geometric/local' side involves the Dixmier trace and commutators with the Dirac operator (which corresponds to the 'characteristic class integration' part of the index theorem). Extensions of the index theorem can be considered in cases, typically when one has an action of a group on the manifold, or when the manifold is endowed with a foliation structure, among others. In those cases the algebraic system of the 'functions' which expresses the underlying geometric object is no longer commutative, but one may able to find the space of square integrable spinors (or, sections of a Clifford module) on which the algebra acts, and the corresponding 'Dirac' operator on it satisfying certain boundedness of commutators implied by the pseudo-differential calculus.

The Komar integral definition can also be generalized to non-stationary fields for which there is at least an asymptotic time translation symmetry; imposing a certain gauge condition, one can define the Bondi energy at null infinity. In a way, the ADM energy measures all of the energy contained in spacetime, while the Bondi energy excludes those parts carried off by gravitational waves to infinity. Great effort has been expended on proving positivity theorems for the masses just defined, not least because positivity, or at least the existence of a lower limit, has a bearing on the more fundamental question of boundedness from below: if there were no lower limit to the energy, then no isolated system would be absolutely stable; there would always be the possibility of a decay to a state of even lower total energy. Several kinds of proofs that both the ADM mass and the Bondi mass are indeed positive exist; in particular, this means that Minkowski space (for which both are zero) is indeed stable.