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In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form d x d t ( t ) ∈ F ( t , x ( t ) ) , {\displaystyle {\frac {dx}{dt}}(t)\in F(t,x(t)),} where F is a multivalued map, i.e. F(t, x) is a set rather than a single point in R d {\displaystyle \mathbb {R} ^{d}} . Differential inclusions arise in many situations including differential variational inequalities, projected dynamical systems, Moreau's sweeping process, linear and nonlinear complementarity dynamical systems, discontinuous ordinary differential equations, switching dynamical systems, and fuzzy set arithmetic.For example, the basic rule for Coulomb friction is that the friction force has magnitude μN in the direction opposite to the direction of slip, where N is the normal force and μ is a constant (the friction coefficient). However, if the slip is zero, the friction force can be any force in the correct plane with magnitude smaller than or equal to μN. Thus, writing the friction force as a function of position and velocity leads to a set-valued function.
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https://en.wikipedia.org/wiki/Differential_inclusion
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In differential inclusion, we not only take a set-valued map at the right hand side but also we can take a subset of a Euclidean space R N {\displaystyle \mathbb {R} ^{N}} for some N ∈ N {\displaystyle N\in \mathbb {N} } as following way. Let n ∈ N {\displaystyle n\in \mathbb {N} } and E ⊂ R n × n ∖ { 0 } . {\displaystyle E\subset \mathbb {R} ^{n\times n}\setminus \{0\}.} Our main purpose is to find a W 0 1 , ∞ ( Ω , R n ) {\displaystyle W_{0}^{1,\infty }(\Omega ,\mathbb {R} ^{n})} function u {\displaystyle u} satisfying the differential inclusion D u ∈ E {\displaystyle Du\in E} a.e. in Ω , {\displaystyle \Omega ,} where Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} is an open bounded set.
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https://en.wikipedia.org/wiki/Differential_inclusion
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In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1-forms. Given a complex manifold M, a differential of the first kind ω is therefore the same thing as a 1-form that is everywhere holomorphic; on an algebraic variety V that is non-singular it would be a global section of the coherent sheaf Ω1 of Kähler differentials. In either case the definition has its origins in the theory of abelian integrals. The dimension of the space of differentials of the first kind, by means of this identification, is the Hodge number h1,0.The differentials of the first kind, when integrated along paths, give rise to integrals that generalise the elliptic integrals to all curves over the complex numbers.
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https://en.wikipedia.org/wiki/Differential_of_the_first_kind
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They include for example the hyperelliptic integrals of type ∫ x k d x Q ( x ) {\displaystyle \int {\frac {x^{k}\,dx}{\sqrt {Q(x)}}}} where Q is a square-free polynomial of any given degree > 4. The allowable power k has to be determined by analysis of the possible pole at the point at infinity on the corresponding hyperelliptic curve. When this is done, one finds that the condition is k ≤ g − 1,or in other words, k at most 1 for degree of Q 5 or 6, at most 2 for degree 7 or 8, and so on (as g = ). Quite generally, as this example illustrates, for a compact Riemann surface or algebraic curve, the Hodge number is the genus g. For the case of algebraic surfaces, this is the quantity known classically as the irregularity q. It is also, in general, the dimension of the Albanese variety, which takes the place of the Jacobian variety.
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https://en.wikipedia.org/wiki/Differential_of_the_first_kind
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In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions.The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology.
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https://en.wikipedia.org/wiki/Differential_(mathematics)
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In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the geometric properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties, such as the number of holes in a manifold, its homotopy type, or the structure of its diffeomorphism group. Because many of these coarser properties may be captured algebraically, differential topology has strong links to algebraic topology.
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https://en.wikipedia.org/wiki/Differential_Topology
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The central goal of the field of differential topology is the classification of all smooth manifolds up to diffeomorphism. Since dimension is an invariant of smooth manifolds up to diffeomorphism type, this classification is often studied by classifying the (connected) manifolds in each dimension separately: In dimension 1, the only smooth manifolds up to diffeomorphism are the circle, the real number line, and allowing a boundary, the half-closed interval {\displaystyle } . In dimension 2, every closed surface is classified up to diffeomorphism by its genus, the number of holes (or equivalently its Euler characteristic), and whether or not it is orientable.
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https://en.wikipedia.org/wiki/Differential_Topology
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This is the famous classification of closed surfaces. Already in dimension two the classification of non-compact surfaces becomes difficult, due to the existence of exotic spaces such as Jacob's ladder. In dimension 3, William Thurston's geometrization conjecture, proven by Grigori Perelman, gives a partial classification of compact three-manifolds.
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https://en.wikipedia.org/wiki/Differential_Topology
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Included in this theorem is the Poincaré conjecture, which states that any closed, simply connected three-manifold is homeomorphic (and in fact diffeomorphic) to the 3-sphere.Beginning in dimension 4, the classification becomes much more difficult for two reasons. Firstly, every finitely presented group appears as the fundamental group of some 4-manifold, and since the fundamental group is a diffeomorphism invariant, this makes the classification of 4-manifolds at least as difficult as the classification of finitely presented groups.
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https://en.wikipedia.org/wiki/Differential_Topology
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By the word problem for groups, which is equivalent to the halting problem, it is impossible to classify such groups, so a full topological classification is impossible. Secondly, beginning in dimension four it is possible to have smooth manifolds that are homeomorphic, but with distinct, non-diffeomorphic smooth structures. This is true even for the Euclidean space R 4 {\displaystyle \mathbb {R} ^{4}} , which admits many exotic R 4 {\displaystyle \mathbb {R} ^{4}} structures.
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https://en.wikipedia.org/wiki/Differential_Topology
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This means that the study of differential topology in dimensions 4 and higher must use tools genuinely outside the realm of the regular continuous topology of topological manifolds. One of the central open problems in differential topology is the four-dimensional smooth Poincaré conjecture, which asks if every smooth 4-manifold that is homeomorphic to the 4-sphere, is also diffeomorphic to it. That is, does the 4-sphere admit only one smooth structure?
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https://en.wikipedia.org/wiki/Differential_Topology
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This conjecture is true in dimensions 1, 2, and 3, by the above classification results, but is known to be false in dimension 7 due to the Milnor spheres. Important tools in studying the differential topology of smooth manifolds include the construction of smooth topological invariants of such manifolds, such as de Rham cohomology or the intersection form, as well as smoothable topological constructions, such as smooth surgery theory or the construction of cobordisms. Morse theory is an important tool which studies smooth manifolds by considering the critical points of differentiable functions on the manifold, demonstrating how the smooth structure of the manifold enters into the set of tools available.
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https://en.wikipedia.org/wiki/Differential_Topology
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Oftentimes more geometric or analytical techniques may be used, by equipping a smooth manifold with a Riemannian metric or by studying a differential equation on it. Care must be taken to ensure that the resulting information is insensitive to this choice of extra structure, and so genuinely reflects only the topological properties of the underlying smooth manifold.
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https://en.wikipedia.org/wiki/Differential_Topology
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For example, the Hodge theorem provides a geometric and analytical interpretation of the de Rham cohomology, and gauge theory was used by Simon Donaldson to prove facts about the intersection form of simply connected 4-manifolds. In some cases techniques from contemporary physics may appear, such as topological quantum field theory, which can be used to compute topological invariants of smooth spaces. Famous theorems in differential topology include the Whitney embedding theorem, the hairy ball theorem, the Hopf theorem, the Poincaré–Hopf theorem, Donaldson's theorem, and the Poincaré conjecture.
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https://en.wikipedia.org/wiki/Differential_Topology
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In mathematics, digital Morse theory is a digital adaptation of continuum Morse theory for scalar volume data. This is not about the Samuel Morse's Morse code of long and short clicks or tones used in manual electric telegraphy. The term was first promulgated by DB Karron based on the work of JL Cox and DB Karron. The main utility of a digital Morse theory is that it serves to provide a theoretical basis for isosurfaces (a kind of embedded manifold submanifold) and perpendicular streamlines in a digital context. The intended main application of DMT is in the rapid semiautomatic segmentation objects such as organs and anatomic structures from stacks of medical images such as produced by three-dimensional computed tomography by CT or MRI technology.
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https://en.wikipedia.org/wiki/Digital_Morse_theory
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In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme). The need of a theory for such an apparently simple notion results from the existence of many definitions of dimension that are equivalent only in the most regular cases (see Dimension of an algebraic variety). A large part of dimension theory consists in studying the conditions under which several dimensions are equal, and many important classes of commutative rings may be defined as the rings such that two dimensions are equal; for example, a regular ring is a commutative ring such that the homological dimension is equal to the Krull dimension. The theory is simpler for commutative rings that are finitely generated algebras over a field, which are also quotient rings of polynomial rings in a finite number of indeterminates over a field.
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https://en.wikipedia.org/wiki/Dimension_theory_(algebra)
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In this case, which is the algebraic counterpart of the case of affine algebraic sets, most of the definitions of the dimension are equivalent. For general commutative rings, the lack of geometric interpretation is an obstacle to the development of the theory; in particular, very little is known for non-noetherian rings. (Kaplansky's Commutative rings gives a good account of the non-noetherian case.) Throughout the article, dim {\displaystyle \dim } denotes Krull dimension of a ring and ht {\displaystyle \operatorname {ht} } the height of a prime ideal (i.e., the Krull dimension of the localization at that prime ideal.) Rings are assumed to be commutative except in the last section on dimensions of non-commutative rings.
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https://en.wikipedia.org/wiki/Dimension_theory_(algebra)
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In mathematics, directed algebraic topology is a refinement of algebraic topology for directed spaces, topological spaces and their combinatorial counterparts equipped with some notion of direction. Some common examples of directed spaces are spacetimes and simplicial sets. The basic goal is to find algebraic invariants that classify directed spaces up to directed analogues of homotopy equivalence.
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https://en.wikipedia.org/wiki/Directed_algebraic_topology
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For example, homotopy groups and fundamental n-groupoids of spaces generalize to homotopy monoids and fundamental n-categories of directed spaces. Directed algebraic topology, like algebraic topology, is motivated by the need to describe qualitative properties of complex systems in terms of algebraic properties of state spaces, which are often directed by time. Thus directed algebraic topology finds applications in concurrency (computer science), network traffic control, general relativity, noncommutative geometry, rewriting theory, and biological systems.
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https://en.wikipedia.org/wiki/Directed_algebraic_topology
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In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the theme of classical discrepancy theory, namely distributing points in some space such that they are evenly distributed with respect to some (mostly geometrically defined) subsets.
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https://en.wikipedia.org/wiki/Discrepancy_theory
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The discrepancy (irregularity) measures how far a given distribution deviates from an ideal one. Discrepancy theory can be described as the study of inevitable irregularities of distributions, in measure-theoretic and combinatorial settings. Just as Ramsey theory elucidates the impossibility of total disorder, discrepancy theory studies the deviations from total uniformity. A significant event in the history of discrepancy theory was the 1916 paper of Weyl on the uniform distribution of sequences in the unit interval.
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https://en.wikipedia.org/wiki/Discrepancy_theory
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In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced by Pafnuty Chebyshev and rediscovered by Gram. They were later found to be applicable to various algebraic properties of spin angular momentum.
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https://en.wikipedia.org/wiki/Gram_polynomial
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In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation.Divided differences is a recursive division process. Given a sequence of data points ( x 0 , y 0 ) , … , ( x n , y n ) {\displaystyle (x_{0},y_{0}),\ldots ,(x_{n},y_{n})} , the method calculates the coefficients of the interpolation polynomial of these points in the Newton form.
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https://en.wikipedia.org/wiki/Divided_differences
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In mathematics, division by infinity is division where the divisor (denominator) is infinity. In ordinary arithmetic, this does not have a well-defined meaning, since infinity is a mathematical concept that does not correspond to a specific number, and moreover, there is no nonzero real number that, when added to itself an infinite number of times, gives a finite number. However, "dividing by infinity" can be given meaning as an informal way of expressing the limit of dividing a number by larger and larger divisors. : 201–204 Using mathematical structures that go beyond the real numbers, it is possible to define numbers that have infinite magnitude yet can still be manipulated in ways much like ordinary arithmetic.
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https://en.wikipedia.org/wiki/Division_by_infinity
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For example, on the extended real number line, dividing any real number by infinity yields zero, while in the surreal number system, dividing 1 by the infinite number ω {\displaystyle \omega } yields the infinitesimal number ϵ {\displaystyle \epsilon } . : 12 In floating-point arithmetic, any finite number divided by ± ∞ {\displaystyle \pm \infty } is equal to positive or negative zero if the numerator is finite. Otherwise, the result is NaN.
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https://en.wikipedia.org/wiki/Division_by_infinity
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The challenges of providing a rigorous meaning of "division by infinity" are analogous to those of defining division by zero. Within the domain of mathematical discourse, the contemplation of dividing infinity by itself gives rise to a proposition of interest. Specifically, the assertion that the result of dividing infinity by infinity ( ∞ ÷ ∞ = ∞ ) is tantamount to infinity itself merits exploration.
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https://en.wikipedia.org/wiki/Division_by_infinity
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A logical journey unveils the underpinnings of this concept and its mathematical validity. Consider a parameter denoted as "y," which, for the sake of analysis, is assigned the value 10.
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https://en.wikipedia.org/wiki/Division_by_infinity
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The crux of the matter rests in the equation ∞ ÷ y = ∞, where the introduction of y introduces an essential condition. To render the equation coherent, y must assume a magnitude that is sufficiently vast to accommodate the division operation involving infinity. This requirement reflects the conceptual intricacies associated with dealing with the concept of infinity in mathematical contexts.
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https://en.wikipedia.org/wiki/Division_by_infinity
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However, the narrative takes a noteworthy turn as we transition to the equation y × ∞ = ∞. This equation signifies a transformation of the division operation into one of multiplication. In essence, this transition underscores a relationship where division of infinity finds equivalence through multiplication with an appropriate value of y. This insight reinforces the notion that the division of infinity by itself materializes as an operation of multiplication, culminating in an outcome of infinity.
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https://en.wikipedia.org/wiki/Division_by_infinity
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Moreover, it's worth mentioning that if we carry forward the same line of thinking, something fascinating emerges. When we take infinity and divide it by a regular number like 10, the result still holds true: it's infinity. This adds another layer of insight to our mathematical journey, underscoring the depth of what we're uncovering here.
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https://en.wikipedia.org/wiki/Division_by_infinity
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In mathematics, division by two or halving has also been called mediation or dimidiation. The treatment of this as a different operation from multiplication and division by other numbers goes back to the ancient Egyptians, whose multiplication algorithm used division by two as one of its fundamental steps. Some mathematicians as late as the sixteenth century continued to view halving as a separate operation, and it often continues to be treated separately in modern computer programming. Performing this operation is simple in decimal arithmetic, in the binary numeral system used in computer programming, and in other even-numbered bases.
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https://en.wikipedia.org/wiki/Division_by_two
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In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as a 0 {\textstyle {\tfrac {a}{0}}} , where a is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number that, when multiplied by 0, gives a (assuming a ≠ 0 {\textstyle a\neq 0} ); thus, division by zero is undefined (a type of singularity).
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https://en.wikipedia.org/wiki/Division_by_zero
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Since any number multiplied by zero is zero, the expression 0 0 {\displaystyle {\tfrac {0}{0}}} is also undefined; when it is the form of a limit, it is an indeterminate form. Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to a 0 {\textstyle {\tfrac {a}{0}}} is contained in Anglo-Irish philosopher George Berkeley's criticism of infinitesimal calculus in 1734 in The Analyst ("ghosts of departed quantities").There are mathematical structures in which a 0 {\textstyle {\tfrac {a}{0}}} is defined for some a such as in the Riemann sphere (a model of the extended complex plane) and the projectively extended real line; however, such structures do not satisfy every ordinary rule of arithmetic (the field axioms). In computing, a program error may result from an attempt to divide by zero. Depending on the programming environment and the type of number (e.g., floating point, integer) being divided by zero, it may generate positive or negative infinity by the IEEE 754 floating-point standard, generate an exception, generate an error message, cause the program to terminate, result in a special not-a-number value, or crash.
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https://en.wikipedia.org/wiki/Division_by_zero
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In mathematics, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces. This duality, which is originally also due to Marshall H. Stone, generalizes the well-known Stone duality between Stone spaces and Boolean algebras. Let L be a bounded distributive lattice, and let X denote the set of prime filters of L. For each a ∈ L, let φ+(a) = {x∈ X: a ∈ x}. Then (X,τ+) is a spectral space, where the topology τ+ on X is generated by {φ+(a): a ∈ L}.
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https://en.wikipedia.org/wiki/Duality_theory_for_distributive_lattices
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The spectral space (X, τ+) is called the prime spectrum of L. The map φ+ is a lattice isomorphism from L onto the lattice of all compact open subsets of (X,τ+). In fact, each spectral space is homeomorphic to the prime spectrum of some bounded distributive lattice.Similarly, if φ−(a) = {x∈ X: a ∉ x} and τ− denotes the topology generated by {φ−(a): a∈ L}, then (X,τ−) is also a spectral space. Moreover, (X,τ+,τ−) is a pairwise Stone space.
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https://en.wikipedia.org/wiki/Duality_theory_for_distributive_lattices
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The pairwise Stone space (X,τ+,τ−) is called the bitopological dual of L. Each pairwise Stone space is bi-homeomorphic to the bitopological dual of some bounded distributive lattice.Finally, let ≤ be set-theoretic inclusion on the set of prime filters of L and let τ = τ+∨ τ−. Then (X,τ,≤) is a Priestley space. Moreover, φ+ is a lattice isomorphism from L onto the lattice of all clopen up-sets of (X,τ,≤).
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https://en.wikipedia.org/wiki/Duality_theory_for_distributive_lattices
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The Priestley space (X,τ,≤) is called the Priestley dual of L. Each Priestley space is isomorphic to the Priestley dual of some bounded distributive lattice.Let Dist denote the category of bounded distributive lattices and bounded lattice homomorphisms. Then the above three representations of bounded distributive lattices can be extended to dual equivalence between Dist and the categories Spec, PStone, and Pries of spectral spaces with spectral maps, of pairwise Stone spaces with bi-continuous maps, and of Priestley spaces with Priestley morphisms, respectively: Thus, there are three equivalent ways of representing bounded distributive lattices. Each one has its own motivation and advantages, but ultimately they all serve the same purpose of providing better understanding of bounded distributive lattices.
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https://en.wikipedia.org/wiki/Duality_theory_for_distributive_lattices
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In mathematics, dynamic equation can refer to: difference equation in discrete time differential equation in continuous time time scale calculus in combined discrete and continuous time
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https://en.wikipedia.org/wiki/Dynamic_equation
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In mathematics, economics and computer science, particularly in the fields of combinatorics, game theory and algorithms, the stable-roommate problem (SRP) is the problem of finding a stable matching for an even-sized set. A matching is a separation of the set into disjoint pairs ("roommates"). The matching is stable if there are no two elements which are not roommates and which both prefer each other to their roommate under the matching. This is distinct from the stable-marriage problem in that the stable-roommates problem allows matches between any two elements, not just between classes of "men" and "women".
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https://en.wikipedia.org/wiki/Stable_roommates_problem
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It is commonly stated as: In a given instance of the stable-roommates problem (SRP), each of 2n participants ranks the others in strict order of preference. A matching is a set of n disjoint pairs of participants. A matching M in an instance of SRP is stable if there are no two participants x and y, each of whom prefers the other to their partner in M. Such a pair is said to block M, or to be a blocking pair with respect to M.
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https://en.wikipedia.org/wiki/Stable_roommates_problem
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In mathematics, economics, and computer science, the Gale–Shapley algorithm (also known as the deferred acceptance algorithm or propose-and-reject algorithm) is an algorithm for finding a solution to the stable matching problem, named for David Gale and Lloyd Shapley. It takes polynomial time, and the time is linear in the size of the input to the algorithm. It is a truthful mechanism from the point of view of the proposing participants, for whom the solution will always be optimal.
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https://en.wikipedia.org/wiki/Gale–Shapley_algorithm
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In mathematics, economics, and computer science, the lattice of stable matchings is a distributive lattice whose elements are stable matchings. For a given instance of the stable matching problem, this lattice provides an algebraic description of the family of all solutions to the problem. It was originally described in the 1970s by John Horton Conway and Donald Knuth.By Birkhoff's representation theorem, this lattice can be represented as the lower sets of an underlying partially ordered set. The elements of this set can be given a concrete structure as rotations, with cycle graphs describing the changes between adjacent stable matchings in the lattice.
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https://en.wikipedia.org/wiki/Lattice_of_stable_matchings
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The family of all rotations and their partial order can be constructed in polynomial time, leading to polynomial time solutions for other problems on stable matching including the minimum or maximum weight stable matching. The Gale–Shapley algorithm can be used to construct two special lattice elements, its top and bottom element.
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https://en.wikipedia.org/wiki/Lattice_of_stable_matchings
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Every finite distributive lattice can be represented as a lattice of stable matchings. The number of elements in the lattice can vary from an average case of e − 1 n ln n {\displaystyle e^{-1}n\ln n} to a worst-case of exponential. Computing the number of elements is #P-complete.
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https://en.wikipedia.org/wiki/Lattice_of_stable_matchings
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In mathematics, economics, and computer science, the stable marriage problem (also stable matching problem or SMP) is the problem of finding a stable matching between two equally sized sets of elements given an ordering of preferences for each element. A matching is a bijection from the elements of one set to the elements of the other set. A matching is not stable if: In other words, a matching is stable when there does not exist any pair (A, B) which both prefer each other to their current partner under the matching.
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https://en.wikipedia.org/wiki/Stable_marriage_problem
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The stable marriage problem has been stated as follows: Given n men and n women, where each person has ranked all members of the opposite sex in order of preference, marry the men and women together such that there are no two people of opposite sex who would both rather have each other than their current partners. When there are no such pairs of people, the set of marriages is deemed stable. The existence of two classes that need to be paired with each other (heterosexual men and women in this example) distinguishes this problem from the stable roommates problem.
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https://en.wikipedia.org/wiki/Stable_marriage_problem
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In mathematics, economics, and computer science, the stable matching polytope or stable marriage polytope is a convex polytope derived from the solutions to an instance of the stable matching problem.
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https://en.wikipedia.org/wiki/Stable_matching_polytope
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In mathematics, effective dimension is a modification of Hausdorff dimension and other fractal dimensions that places it in a computability theory setting. There are several variations (various notions of effective dimension) of which the most common is effective Hausdorff dimension. Dimension, in mathematics, is a particular way of describing the size of an object (contrasting with measure and other, different, notions of size).
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https://en.wikipedia.org/wiki/Effective_dimension
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Hausdorff dimension generalizes the well-known integer dimensions assigned to points, lines, planes, etc. by allowing one to distinguish between objects of intermediate size between these integer-dimensional objects. For example, fractal subsets of the plane may have intermediate dimension between 1 and 2, as they are "larger" than lines or curves, and yet "smaller" than filled circles or rectangles. Effective dimension modifies Hausdorff dimension by requiring that objects with small effective dimension be not only small but also locatable (or partially locatable) in a computable sense. As such, objects with large Hausdorff dimension also have large effective dimension, and objects with small effective dimension have small Hausdorff dimension, but an object can have small Hausdorff but large effective dimension. An example is an algorithmically random point on a line, which has Hausdorff dimension 0 (since it is a point) but effective dimension 1 (because, roughly speaking, it can't be effectively localized any better than a small interval, which has Hausdorff dimension 1).
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https://en.wikipedia.org/wiki/Effective_dimension
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In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.
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https://en.wikipedia.org/wiki/Elliptic_cohomology
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In mathematics, elliptic curve primality testing techniques, or elliptic curve primality proving (ECPP), are among the quickest and most widely used methods in primality proving. It is an idea put forward by Shafi Goldwasser and Joe Kilian in 1986 and turned into an algorithm by A. O. L. Atkin the same year. The algorithm was altered and improved by several collaborators subsequently, and notably by Atkin and François Morain, in 1993. The concept of using elliptic curves in factorization had been developed by H. W. Lenstra in 1985, and the implications for its use in primality testing (and proving) followed quickly.
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https://en.wikipedia.org/wiki/Elliptic_curve_primality
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Primality testing is a field that has been around since the time of Fermat, in whose time most algorithms were based on factoring, which become unwieldy with large input; modern algorithms treat the problems of determining whether a number is prime and what its factors are separately. It became of practical importance with the advent of modern cryptography. Although many current tests result in a probabilistic output (N is either shown composite, or probably prime, such as with the Baillie–PSW primality test or the Miller–Rabin test), the elliptic curve test proves primality (or compositeness) with a quickly verifiable certificate.Previously-known prime-proving methods such as the Pocklington primality test required at least partial factorization of N ± 1 {\displaystyle N\pm 1} in order to prove that N {\displaystyle N} is prime. As a result, these methods required some luck and are generally slow in practice.
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https://en.wikipedia.org/wiki/Elliptic_curve_primality
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In mathematics, elliptic units are certain units of abelian extensions of imaginary quadratic fields constructed using singular values of modular functions, or division values of elliptic functions. They were introduced by Gilles Robert in 1973, and were used by John Coates and Andrew Wiles in their work on the Birch and Swinnerton-Dyer conjecture. Elliptic units are an analogue for imaginary quadratic fields of cyclotomic units. They form an example of an Euler system.
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https://en.wikipedia.org/wiki/Elliptic_unit
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In mathematics, endoscopic groups of reductive algebraic groups were introduced by Robert Langlands (1979, 1983) in his work on the stable trace formula. Roughly speaking, an endoscopic group H of G is a quasi-split group whose L-group is the connected component of the centralizer of a semisimple element of the L-group of G. In the stable trace formula, unstable orbital integrals on a group G correspond to stable orbital integrals on its endoscopic groups H. The relation between them is given by the fundamental lemma.
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https://en.wikipedia.org/wiki/Endoscopic_group
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In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.
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https://en.wikipedia.org/wiki/Clemens_conjecture
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In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. The equality between A and B is written A = B, and pronounced "A equals B". The symbol "=" is called an "equals sign". Two objects that are not equal are said to be distinct.
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https://en.wikipedia.org/wiki/Equality_(mathematics)
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For example: x = y {\displaystyle x=y} means that x and y denote the same object. The identity ( x + 1 ) 2 = x 2 + 2 x + 1 {\displaystyle (x+1)^{2}=x^{2}+2x+1} means that if x is any number, then the two expressions have the same value. This may also be interpreted as saying that the two sides of the equals sign represent the same function.
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https://en.wikipedia.org/wiki/Equality_(mathematics)
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{ x ∣ P ( x ) } = { x ∣ Q ( x ) } {\displaystyle \{x\mid P(x)\}=\{x\mid Q(x)\}} if and only if P ( x ) ⇔ Q ( x ) . {\displaystyle P(x)\Leftrightarrow Q(x).}
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https://en.wikipedia.org/wiki/Equality_(mathematics)
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This assertion, which uses set-builder notation, means that if the elements satisfying the property P ( x ) {\displaystyle P(x)} are the same as the elements satisfying Q ( x ) , {\displaystyle Q(x),} then the two uses of the set-builder notation define the same set. This property is often expressed as "two sets that have the same elements are equal." It is one of the usual axioms of set theory, called axiom of extensionality.
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https://en.wikipedia.org/wiki/Equality_(mathematics)
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In mathematics, equivalent definitions are used in two somewhat different ways. First, within a particular mathematical theory (for example, Euclidean geometry), a notion (for example, ellipse or minimal surface) may have more than one definition. These definitions are equivalent in the context of a given mathematical structure (Euclidean space, in this case). Second, a mathematical structure may have more than one definition (for example, topological space has at least seven definitions; ordered field has at least two definitions).
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https://en.wikipedia.org/wiki/Equivalent_definitions_of_mathematical_structures
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In the former case, equivalence of two definitions means that a mathematical object (for example, geometric body) satisfies one definition if and only if it satisfies the other definition. In the latter case, the meaning of equivalence (between two definitions of a structure) is more complicated, since a structure is more abstract than an object. Many different objects may implement the same structure.
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https://en.wikipedia.org/wiki/Equivalent_definitions_of_mathematical_structures
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In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation. Equivariant maps generalize the concept of invariants, functions whose value is unchanged by a symmetry transformation of their argument.
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https://en.wikipedia.org/wiki/Equivariant_morphism
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The value of an equivariant map is often (imprecisely) called an invariant. In statistical inference, equivariance under statistical transformations of data is an important property of various estimation methods; see invariant estimator for details. In pure mathematics, equivariance is a central object of study in equivariant topology and its subtopics equivariant cohomology and equivariant stable homotopy theory.
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https://en.wikipedia.org/wiki/Equivariant_morphism
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In mathematics, equivariant cohomology (or Borel cohomology) is a cohomology theory from algebraic topology which applies to topological spaces with a group action. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space X {\displaystyle X} with action of a topological group G {\displaystyle G} is defined as the ordinary cohomology ring with coefficient ring Λ {\displaystyle \Lambda } of the homotopy quotient E G × G X {\displaystyle EG\times _{G}X}: H G ∗ ( X ; Λ ) = H ∗ ( E G × G X ; Λ ) . {\displaystyle H_{G}^{*}(X;\Lambda )=H^{*}(EG\times _{G}X;\Lambda ).}
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https://en.wikipedia.org/wiki/Equivariant_cohomology_ring
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If G {\displaystyle G} is the trivial group, this is the ordinary cohomology ring of X {\displaystyle X} , whereas if X {\displaystyle X} is contractible, it reduces to the cohomology ring of the classifying space B G {\displaystyle BG} (that is, the group cohomology of G {\displaystyle G} when G is finite.) If G acts freely on X, then the canonical map E G × G X → X / G {\displaystyle EG\times _{G}X\to X/G} is a homotopy equivalence and so one gets: H G ∗ ( X ; Λ ) = H ∗ ( X / G ; Λ ) . {\displaystyle H_{G}^{*}(X;\Lambda )=H^{*}(X/G;\Lambda ).}
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https://en.wikipedia.org/wiki/Equivariant_cohomology_ring
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In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. In studying topological spaces, one often considers continuous maps f: X → Y {\displaystyle f:X\to Y} , and while equivariant topology also considers such maps, there is the additional constraint that each map "respects symmetry" in both its domain and target space. The notion of symmetry is usually captured by considering a group action of a group G {\displaystyle G} on X {\displaystyle X} and Y {\displaystyle Y} and requiring that f {\displaystyle f} is equivariant under this action, so that f ( g ⋅ x ) = g ⋅ f ( x ) {\displaystyle f(g\cdot x)=g\cdot f(x)} for all x ∈ X {\displaystyle x\in X} , a property usually denoted by f: X → G Y {\displaystyle f:X\to _{G}Y} . Heuristically speaking, standard topology views two spaces as equivalent "up to deformation," while equivariant topology considers spaces equivalent up to deformation so long as it pays attention to any symmetry possessed by both spaces. A famous theorem of equivariant topology is the Borsuk–Ulam theorem, which asserts that every Z 2 {\displaystyle \mathbf {Z} _{2}} -equivariant map f: S n → R n {\displaystyle f:S^{n}\to \mathbb {R} ^{n}} necessarily vanishes.
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https://en.wikipedia.org/wiki/Equivariant_algebraic_topoloy
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In mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been understood in terms of the theory of unitary representations of locally compact groups: if Γ is the fundamental group of a closed surface, regarded as a discrete subgroup of the Möbius group G = PSL(2,R), then the geodesic and horocycle flow can be identified with the natural actions of the subgroups A of real positive diagonal matrices and N of lower unitriangular matrices on the unit tangent bundle G / Γ. The Ambrose-Kakutani theorem expresses every ergodic flow as the flow built from an invertible ergodic transformation on a measure space using a ceiling function. In the case of geodesic flow, the ergodic transformation can be understood in terms of symbolic dynamics; and in terms of the ergodic actions of Γ on the boundary S1 = G / AN and G / A = S1 × S1 \ diag S1. Ergodic flows also arise naturally as invariants in the classification of von Neumann algebras: the flow of weights for a factor of type III0 is an ergodic flow on a measure space.
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https://en.wikipedia.org/wiki/Ergodic_flow
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In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored into smaller components.
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https://en.wikipedia.org/wiki/Unique_ergodicity
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Ergodic theory is the study of systems possessing ergodicity. Ergodic systems occur in a broad range of systems in physics and in geometry. This can be roughly understood to be due to a common phenomenon: the motion of particles, that is, geodesics on a hyperbolic manifold are divergent; when that manifold is compact, that is, of finite size, those orbits return to the same general area, eventually filling the entire space.
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https://en.wikipedia.org/wiki/Unique_ergodicity
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Ergodic systems capture the common-sense, every-day notions of randomness, such that smoke might come to fill all of a smoke-filled room, or that a block of metal might eventually come to have the same temperature throughout, or that flips of a fair coin may come up heads and tails half the time. A stronger concept than ergodicity is that of mixing, which aims to mathematically describe the common-sense notions of mixing, such as mixing drinks or mixing cooking ingredients. The proper mathematical formulation of ergodicity is founded on the formal definitions of measure theory and dynamical systems, and rather specifically on the notion of a measure-preserving dynamical system. The origins of ergodicity lie in statistical physics, where Ludwig Boltzmann formulated the ergodic hypothesis.
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https://en.wikipedia.org/wiki/Unique_ergodicity
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In mathematics, error analysis is the study of kind and quantity of error, or uncertainty, that may be present in the solution to a problem. This issue is particularly prominent in applied areas such as numerical analysis and statistics.
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https://en.wikipedia.org/wiki/Error_analysis_(mathematics)
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In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra.
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https://en.wikipedia.org/wiki/Categorical_Algebra
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In mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Many of the problems posed here first appeared in the Loops (Prague) conferences and the Mile High (Denver) conferences.
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https://en.wikipedia.org/wiki/List_of_problems_in_loop_theory_and_quasigroup_theory
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In mathematics, especially algebraic geometry, the Bass conjecture says that certain algebraic K-groups are supposed to be finitely generated. The conjecture was proposed by Hyman Bass.
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https://en.wikipedia.org/wiki/Bass_conjecture
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In mathematics, especially algebraic geometry, the decomposition theorem of Beilinson, Bernstein and Deligne or BBD decomposition theorem is a set of results concerning the cohomology of algebraic varieties. It was originally conjectured by Gelfand and MacPherson.
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https://en.wikipedia.org/wiki/Decomposition_theorem_of_Beilinson,_Bernstein_and_Deligne
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In mathematics, especially convex analysis, the recession cone of a set A {\displaystyle A} is a cone containing all vectors such that A {\displaystyle A} recedes in that direction. That is, the set extends outward in all the directions given by the recession cone.
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https://en.wikipedia.org/wiki/Recession_cone
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In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds, or (in the form of cotangent sheaf) algebraic varieties or schemes. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories.
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https://en.wikipedia.org/wiki/Cotangent_manifold
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In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x {\displaystyle x} in a Hilbert space with respect to an orthonormal sequence. The inequality was derived by F.W. Bessel in 1828.Let H {\displaystyle H} be a Hilbert space, and suppose that e 1 , e 2 , .
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https://en.wikipedia.org/wiki/Bessel's_inequality
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. . {\displaystyle e_{1},e_{2},...} is an orthonormal sequence in H {\displaystyle H} .
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https://en.wikipedia.org/wiki/Bessel's_inequality
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In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A {\displaystyle A} over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy This ensures that the multiplication operation is continuous. A Banach algebra is called unital if it has an identity element for the multiplication whose norm is 1 , {\displaystyle 1,} and commutative if its multiplication is commutative. Any Banach algebra A {\displaystyle A} (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra A e {\displaystyle A_{e}} so as to form a closed ideal of A e {\displaystyle A_{e}} .
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https://en.wikipedia.org/wiki/Banach_ring
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Often one assumes a priori that the algebra under consideration is unital: for one can develop much of the theory by considering A e {\displaystyle A_{e}} and then applying the outcome in the original algebra. However, this is not the case all the time. For example, one cannot define all the trigonometric functions in a Banach algebra without identity.
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https://en.wikipedia.org/wiki/Banach_ring
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The theory of real Banach algebras can be very different from the theory of complex Banach algebras. For example, the spectrum of an element of a nontrivial complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements. Banach algebras can also be defined over fields of p {\displaystyle p} -adic numbers. This is part of p {\displaystyle p} -adic analysis.
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https://en.wikipedia.org/wiki/Banach_ring
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In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra A {\displaystyle A} over the real or complex numbers that at the same time is also a (locally convex) Fréchet space. The multiplication operation ( a , b ) ↦ a ∗ b {\displaystyle (a,b)\mapsto a*b} for a , b ∈ A {\displaystyle a,b\in A} is required to be jointly continuous. If { ‖ ⋅ ‖ n } n = 0 ∞ {\displaystyle \{\|\cdot \|_{n}\}_{n=0}^{\infty }} is an increasing family of seminorms for the topology of A {\displaystyle A} , the joint continuity of multiplication is equivalent to there being a constant C n > 0 {\displaystyle C_{n}>0} and integer m ≥ n {\displaystyle m\geq n} for each n {\displaystyle n} such that ‖ a b ‖ n ≤ C n ‖ a ‖ m ‖ b ‖ m {\displaystyle \left\|ab\right\|_{n}\leq C_{n}\left\|a\right\|_{m}\left\|b\right\|_{m}} for all a , b ∈ A {\displaystyle a,b\in A} . Fréchet algebras are also called B0-algebras.A Fréchet algebra is m {\displaystyle m} -convex if there exists such a family of semi-norms for which m = n {\displaystyle m=n} .
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https://en.wikipedia.org/wiki/Fréchet_algebra
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In that case, by rescaling the seminorms, we may also take C n = 1 {\displaystyle C_{n}=1} for each n {\displaystyle n} and the seminorms are said to be submultiplicative: ‖ a b ‖ n ≤ ‖ a ‖ n ‖ b ‖ n {\displaystyle \|ab\|_{n}\leq \|a\|_{n}\|b\|_{n}} for all a , b ∈ A . {\displaystyle a,b\in A.} m {\displaystyle m} -convex Fréchet algebras may also be called Fréchet algebras.A Fréchet algebra may or may not have an identity element 1 A {\displaystyle 1_{A}} . If A {\displaystyle A} is unital, we do not require that ‖ 1 A ‖ n = 1 , {\displaystyle \|1_{A}\|_{n}=1,} as is often done for Banach algebras.
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https://en.wikipedia.org/wiki/Fréchet_algebra
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In mathematics, especially functional analysis, a bornology B {\displaystyle {\mathcal {B}}} on a vector space X {\displaystyle X} over a field K , {\displaystyle \mathbb {K} ,} where K {\displaystyle \mathbb {K} } has a bornology ℬ F {\displaystyle \mathbb {F} } , is called a vector bornology if B {\displaystyle {\mathcal {B}}} makes the vector space operations into bounded maps.
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https://en.wikipedia.org/wiki/Vector_bornology
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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. One of the key motivations behind bornologies and bornological analysis is the fact that bornological spaces provide a convenient setting for homological algebra in functional analysis. This is becausepg 9 the category of bornological spaces is additive, complete, cocomplete, and has a tensor product adjoint to an internal hom, all necessary components for homological algebra.
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https://en.wikipedia.org/wiki/Bornology
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In mathematics, especially functional analysis, a hypercyclic operator on a Banach space X is a bounded linear operator T: X → X such that there is a vector x ∈ X such that the sequence {Tn x: n = 0, 1, 2, …} is dense in the whole space X. In other words, the smallest closed invariant subset containing x is the whole space. Such an x is then called hypercyclic vector. There is no hypercyclic operator in finite-dimensional spaces, but the property of hypercyclicity in spaces of infinite dimension is not a rare phenomenon: many operators are hypercyclic.
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https://en.wikipedia.org/wiki/Hypercyclic_operator
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The hypercyclicity is a special case of broader notions of topological transitivity (see topological mixing), and universality. Universality in general involves a set of mappings from one topological space to another (instead of a sequence of powers of a single operator mapping from X to X), but has a similar meaning to hypercyclicity. Examples of universal objects were discovered already in 1914 by Julius Pál, in 1935 by Józef Marcinkiewicz, or MacLane in 1952. However, it was not until the 1980s when hypercyclic operators started to be more intensively studied.
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https://en.wikipedia.org/wiki/Hypercyclic_operator
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In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N: H → H that commutes with its hermitian adjoint N*, that is: NN* = N*N.Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood. Examples of normal operators are unitary operators: N* = N−1 Hermitian operators (i.e., self-adjoint operators): N* = N Skew-Hermitian operators: N* = −N positive operators: N = MM* for some M (so N is self-adjoint).A normal matrix is the matrix expression of a normal operator on the Hilbert space Cn.
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https://en.wikipedia.org/wiki/Normal_operator
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In mathematics, especially functional analysis, a quasitrace is a not necessarily additive tracial functional on a C*-algebra. An additive quasitrace is called a trace. It is a major open problem if every quasitrace is a trace.
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https://en.wikipedia.org/wiki/Quasitrace
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In mathematics, especially general topology and analysis, an exhaustion by compact sets of a topological space X {\displaystyle X} is a nested sequence of compact subsets K i {\displaystyle K_{i}} of X {\displaystyle X} (i.e. K 1 ⊆ K 2 ⊆ K 3 ⊆ ⋯ {\displaystyle K_{1}\subseteq K_{2}\subseteq K_{3}\subseteq \cdots } ), such that K i {\displaystyle K_{i}} is contained in the interior of K i + 1 {\displaystyle K_{i+1}} , i.e. K i ⊆ int ( K i + 1 ) {\displaystyle K_{i}\subseteq {\text{int}}(K_{i+1})} for each i {\displaystyle i} and X = ⋃ i = 1 ∞ K i {\displaystyle X=\bigcup _{i=1}^{\infty }K_{i}} . A space admitting an exhaustion by compact sets is called exhaustible by compact sets. For example, consider X = R n {\displaystyle X=\mathbb {R} ^{n}} and the sequence of closed balls K i = { x: | x | ≤ i } . {\displaystyle K_{i}=\{x:|x|\leq i\}.} Occasionally some authors drop the requirement that K i {\displaystyle K_{i}} is in the interior of K i + 1 {\displaystyle K_{i+1}} , but then the property becomes the same as the space being σ-compact, namely a countable union of compact subsets.
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https://en.wikipedia.org/wiki/Exhaustion_by_compact_sets
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In mathematics, especially group theory, the Zappa–Szép product (also known as the Zappa–Rédei–Szép product, general product, knit product, exact factorization or bicrossed product) describes a way in which a group can be constructed from two subgroups. It is a generalization of the direct and semidirect products. It is named after Guido Zappa (1940) and Jenő Szép (1950) although it was independently studied by others including B.H.
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https://en.wikipedia.org/wiki/Knit_product
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Neumann (1935), G.A. Miller (1935), and J.A. de Séguier (1904).
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https://en.wikipedia.org/wiki/Knit_product
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In mathematics, especially group theory, the centralizer (also called commutant) of a subset S in a group G is the set C G ( S ) {\displaystyle \operatorname {C} _{G}(S)} of elements of G that commute with every element of S, or equivalently, such that conjugation by g {\displaystyle g} leaves each element of S fixed. The normalizer of S in G is the set of elements N G ( S ) {\displaystyle \mathrm {N} _{G}(S)} of G that satisfy the weaker condition of leaving the set S ⊆ G {\displaystyle S\subseteq G} fixed under conjugation. The centralizer and normalizer of S are subgroups of G. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets S. Suitably formulated, the definitions also apply to semigroups.
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https://en.wikipedia.org/wiki/Normalizer_(group_theory)
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In ring theory, the centralizer of a subset of a ring is defined with respect to the semigroup (multiplication) operation of the ring. The centralizer of a subset of a ring R is a subring of R. This article also deals with centralizers and normalizers in a Lie algebra. The idealizer in a semigroup or ring is another construction that is in the same vein as the centralizer and normalizer.
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https://en.wikipedia.org/wiki/Normalizer_(group_theory)
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In mathematics, especially group theory, two elements a {\displaystyle a} and b {\displaystyle b} of a group are conjugate if there is an element g {\displaystyle g} in the group such that b = g a g − 1 . {\displaystyle b=gag^{-1}.} This is an equivalence relation whose equivalence classes are called conjugacy classes.
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https://en.wikipedia.org/wiki/Class_number_(group_theory)
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In other words, each conjugacy class is closed under b = g a g − 1 {\displaystyle b=gag^{-1}} for all elements g {\displaystyle g} in the group. Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element (singleton set). Functions that are constant for members of the same conjugacy class are called class functions.
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https://en.wikipedia.org/wiki/Class_number_(group_theory)
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In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is not only valid for abelian categories but also works in the category of groups, for example. The five lemma can be thought of as a combination of two other theorems, the four lemmas, which are dual to each other.
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https://en.wikipedia.org/wiki/Five_Lemma
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In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma. It states that for the following commutative diagram (in any abelian category, or in the category of groups), if the rows are short exact sequences, and if g and h are isomorphisms, then f is an isomorphism as well. It follows immediately from the five lemma. The essence of the lemma can be summarized as follows: if you have a homomorphism f from an object B to an object B′, and this homomorphism induces an isomorphism from a subobject A of B to a subobject A′ of B′ and also an isomorphism from the factor object B/A to B′/A′, then f itself is an isomorphism. Note however that the existence of f (such that the diagram commutes) has to be assumed from the start; two objects B and B′ that simply have isomorphic sub- and factor objects need not themselves be isomorphic (for example, in the category of abelian groups, B could be the cyclic group of order four and B′ the Klein four-group).
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https://en.wikipedia.org/wiki/Short_five_lemma
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In mathematics, especially homological algebra, a differential graded category, often shortened to dg-category or DG category, is a category whose morphism sets are endowed with the additional structure of a differential graded Z {\displaystyle \mathbb {Z} } -module. In detail, this means that Hom ( A , B ) {\displaystyle \operatorname {Hom} (A,B)} , the morphisms from any object A to another object B of the category is a direct sum ⨁ n ∈ Z Hom n ( A , B ) {\displaystyle \bigoplus _{n\in \mathbb {Z} }\operatorname {Hom} _{n}(A,B)} and there is a differential d on this graded group, i.e., for each n there is a linear map d: Hom n ( A , B ) → Hom n + 1 ( A , B ) {\displaystyle d\colon \operatorname {Hom} _{n}(A,B)\rightarrow \operatorname {Hom} _{n+1}(A,B)} ,which has to satisfy d ∘ d = 0 {\displaystyle d\circ d=0} . This is equivalent to saying that Hom ( A , B ) {\displaystyle \operatorname {Hom} (A,B)} is a cochain complex. Furthermore, the composition of morphisms Hom ( A , B ) ⊗ Hom ( B , C ) → Hom ( A , C ) {\displaystyle \operatorname {Hom} (A,B)\otimes \operatorname {Hom} (B,C)\rightarrow \operatorname {Hom} (A,C)} is required to be a map of complexes, and for all objects A of the category, one requires d ( id A ) = 0 {\displaystyle d(\operatorname {id} _{A})=0} .
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https://en.wikipedia.org/wiki/Differential_graded_category
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