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In mathematics, in the representation theory of groups, a group is said to be representation rigid if for every n {\displaystyle n} , it has only finitely many isomorphism classes of complex irreducible representations of dimension n {\displaystyle n} .
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https://en.wikipedia.org/wiki/Representation_rigid_group
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In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).
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https://en.wikipedia.org/wiki/Limit_cycle
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In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearisation theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that linearisation—a natural simplification of the system—is effective in predicting qualitative patterns of behaviour. The theorem owes its name to Philip Hartman and David M. Grobman. The theorem states that the behaviour of a dynamical system in a domain near a hyperbolic equilibrium point is qualitatively the same as the behaviour of its linearization near this equilibrium point, where hyperbolicity means that no eigenvalue of the linearization has real part equal to zero. Therefore, when dealing with such dynamical systems one can use the simpler linearization of the system to analyse its behaviour around equilibria.
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https://en.wikipedia.org/wiki/Hartman–Grobman_theorem
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In mathematics, in the study of fractals, a Hutchinson operator is the collective action of a set of contractions, called an iterated function system. The iteration of the operator converges to a unique attractor, which is the often self-similar fixed set of the operator.
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https://en.wikipedia.org/wiki/Hutchinson_operator
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In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.
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https://en.wikipedia.org/wiki/Periodic_mapping
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In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
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https://en.wikipedia.org/wiki/Mapping_class_group
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In mathematics, in the theory of Hopf algebras, a Hopf algebroid is a generalisation of weak Hopf algebras, certain skew Hopf algebras and commutative Hopf k-algebroids. If k is a field, a commutative k-algebroid is a cogroupoid object in the category of k-algebras; the category of such is hence dual to the category of groupoid k-schemes. This commutative version has been used in 1970-s in algebraic geometry and stable homotopy theory.
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https://en.wikipedia.org/wiki/Hopf_algebroid
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The generalization of Hopf algebroids and its main part of the structure, associative bialgebroids, to the noncommutative base algebra was introduced by J.-H. Lu in 1996 as a result on work on groupoids in Poisson geometry (later shown equivalent in nontrivial way to a construction of Takeuchi from the 1970s and another by Xu around the year 2000). They may be loosely thought of as Hopf algebras over a noncommutative base ring, where weak Hopf algebras become Hopf algebras over a separable algebra. It is a theorem that a Hopf algebroid satisfying a finite projectivity condition over a separable algebra is a weak Hopf algebra, and conversely a weak Hopf algebra H is a Hopf algebroid over its separable subalgebra HL. The antipode axioms have been changed by G. Böhm and K. Szlachányi (J. Algebra) in 2004 for tensor categorical reasons and to accommodate examples associated to depth two Frobenius algebra extensions.
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https://en.wikipedia.org/wiki/Hopf_algebroid
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In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form d r / d t = A r {\displaystyle dr/dt=Ar} , where r is the perturbation to the steady state, A is a linear operator whose spectrum contains eigenvalues with positive real part. If all the eigenvalues have negative real part, then the solution is called linearly stable. Other names for linear stability include exponential stability or stability in terms of first approximation. If there exist an eigenvalue with zero real part then the question about stability cannot be solved on the basis of the first approximation and we approach the so-called "centre and focus problem".
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https://en.wikipedia.org/wiki/Linear_stability
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In mathematics, in the theory of discrete groups, superrigidity is a concept designed to show how a linear representation ρ of a discrete group Γ inside an algebraic group G can, under some circumstances, be as good as a representation of G itself. That this phenomenon happens for certain broadly defined classes of lattices inside semisimple groups was the discovery of Grigory Margulis, who proved some fundamental results in this direction. There is more than one result that goes by the name of Margulis superrigidity.
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https://en.wikipedia.org/wiki/Superrigidity_theorem
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One simplified statement is this: take G to be a simply connected semisimple real algebraic group in GLn, such that the Lie group of its real points has real rank at least 2 and no compact factors. Suppose Γ is an irreducible lattice in G. For a local field F and ρ a linear representation of the lattice Γ of the Lie group, into GLn (F), assume the image ρ(Γ) is not relatively compact (in the topology arising from F) and such that its closure in the Zariski topology is connected. Then F is the real numbers or the complex numbers, and there is a rational representation of G giving rise to ρ by restriction.
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https://en.wikipedia.org/wiki/Superrigidity_theorem
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In mathematics, in the theory of finite groups, a Brauer tree is a tree that encodes the characters of a block with cyclic defect group of a finite group. In fact, the trees encode the group algebra up to Morita equivalence. Such algebras coming from Brauer trees are called Brauer tree algebras. Feit (1984) described the possibilities for Brauer trees.
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https://en.wikipedia.org/wiki/Brauer_tree
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In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be extended to a bigger domain. Formally, an open set Ω {\displaystyle \Omega } in the n-dimensional complex space C n {\displaystyle {\mathbb {C} }^{n}} is called a domain of holomorphy if there do not exist non-empty open sets U ⊂ Ω {\displaystyle U\subset \Omega } and V ⊂ C n {\displaystyle V\subset {\mathbb {C} }^{n}} where V {\displaystyle V} is connected, V ⊄ Ω {\displaystyle V\not \subset \Omega } and U ⊂ Ω ∩ V {\displaystyle U\subset \Omega \cap V} such that for every holomorphic function f {\displaystyle f} on Ω {\displaystyle \Omega } there exists a holomorphic function g {\displaystyle g} on V {\displaystyle V} with f = g {\displaystyle f=g} on U {\displaystyle U} In the n = 1 {\displaystyle n=1} case, every open set is a domain of holomorphy: we can define a holomorphic function with zeros accumulating everywhere on the boundary of the domain, which must then be a natural boundary for a domain of definition of its reciprocal. For n ≥ 2 {\displaystyle n\geq 2} this is no longer true, as it follows from Hartogs' lemma.
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https://en.wikipedia.org/wiki/Domain_of_holomorphy
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In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the Lax equation. Lax pairs were introduced by Peter Lax to discuss solitons in continuous media. The inverse scattering transform makes use of the Lax equations to solve such systems.
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https://en.wikipedia.org/wiki/Lax_form
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In mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways, it is the dual notion to that of the socle soc(M) of M.
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https://en.wikipedia.org/wiki/Radical_of_a_module
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In mathematics, in the theory of ordinary differential equations in the complex plane C {\displaystyle \mathbb {C} } , the points of C {\displaystyle \mathbb {C} } are classified into ordinary points, at which the equation's coefficients are analytic functions, and singular points, at which some coefficient has a singularity. Then amongst singular points, an important distinction is made between a regular singular point, where the growth of solutions is bounded (in any small sector) by an algebraic function, and an irregular singular point, where the full solution set requires functions with higher growth rates. This distinction occurs, for example, between the hypergeometric equation, with three regular singular points, and the Bessel equation which is in a sense a limiting case, but where the analytic properties are substantially different.
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https://en.wikipedia.org/wiki/Regular_singular_points
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In mathematics, in the theory of rewriting systems, Newman's lemma, also commonly called the diamond lemma, states that a terminating (or strongly normalizing) abstract rewriting system (ARS), that is, one in which there are no infinite reduction sequences, is confluent if it is locally confluent. In fact a terminating ARS is confluent precisely when it is locally confluent.Equivalently, for every binary relation with no decreasing infinite chains and satisfying a weak version of the diamond property, there is a unique minimal element in every connected component of the relation considered as a graph. Today, this is seen as a purely combinatorial result based on well-foundedness due to a proof of Gérard Huet in 1980. Newman's original proof was considerably more complicated.
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https://en.wikipedia.org/wiki/Newman's_lemma
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In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of n complex dimensions. They were introduced by and named after Karl Stein (1951). A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry.
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https://en.wikipedia.org/wiki/Levi_problem
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In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem. A simple and useful version of the loop theorem states that if for some 3-dimensional manifold M with boundary ∂M there is a map f: ( D 2 , ∂ D 2 ) → ( M , ∂ M ) {\displaystyle f\colon (D^{2},\partial D^{2})\to (M,\partial M)} with f | ∂ D 2 {\displaystyle f|\partial D^{2}} not nullhomotopic in ∂ M {\displaystyle \partial M} , then there is an embedding with the same property. The following version of the loop theorem, due to John Stallings, is given in the standard 3-manifold treatises (such as Hempel or Jaco): Let M {\displaystyle M} be a 3-manifold and let S {\displaystyle S} be a connected surface in ∂ M {\displaystyle \partial M} .
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https://en.wikipedia.org/wiki/Loop_theorem
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Let N ⊂ π 1 ( S ) {\displaystyle N\subset \pi _{1}(S)} be a normal subgroup such that k e r ( π 1 ( S ) → π 1 ( M ) ) − N ≠ ∅ {\displaystyle \mathop {\mathrm {ker} } (\pi _{1}(S)\to \pi _{1}(M))-N\neq \emptyset } . Let f: D 2 → M {\displaystyle f\colon D^{2}\to M} be a continuous map such that f ( ∂ D 2 ) ⊂ S {\displaystyle f(\partial D^{2})\subset S} and ∉ N . {\displaystyle \notin N.}
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https://en.wikipedia.org/wiki/Loop_theorem
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Then there exists an embedding g: D 2 → M {\displaystyle g\colon D^{2}\to M} such that g ( ∂ D 2 ) ⊂ S {\displaystyle g(\partial D^{2})\subset S} and ∉ N . {\displaystyle \notin N.}
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https://en.wikipedia.org/wiki/Loop_theorem
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Furthermore if one starts with a map f in general position, then for any neighborhood U of the singularity set of f, we can find such a g with image lying inside the union of image of f and U. Stalling's proof utilizes an adaptation, due to Whitehead and Shapiro, of Papakyriakopoulos' "tower construction". The "tower" refers to a special sequence of coverings designed to simplify lifts of the given map. The same tower construction was used by Papakyriakopoulos to prove the sphere theorem (3-manifolds), which states that a nontrivial map of a sphere into a 3-manifold implies the existence of a nontrivial embedding of a sphere.
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https://en.wikipedia.org/wiki/Loop_theorem
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There is also a version of Dehn's lemma for minimal discs due to Meeks and S.-T. Yau, which also crucially relies on the tower construction. A proof not utilizing the tower construction exists of the first version of the loop theorem.
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https://en.wikipedia.org/wiki/Loop_theorem
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This was essentially done 30 years ago by Friedhelm Waldhausen as part of his solution to the word problem for Haken manifolds; although he recognized this gave a proof of the loop theorem, he did not write up a detailed proof. The essential ingredient of this proof is the concept of Haken hierarchy. Proofs were later written up, by Klaus Johannson, Marc Lackenby, and Iain Aitchison with Hyam Rubinstein.
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https://en.wikipedia.org/wiki/Loop_theorem
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In mathematics, in the topology of 3-manifolds, the sphere theorem of Christos Papakyriakopoulos (1957) gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres. One example is the following: Let M {\displaystyle M} be an orientable 3-manifold such that π 2 ( M ) {\displaystyle \pi _{2}(M)} is not the trivial group. Then there exists a non-zero element of π 2 ( M ) {\displaystyle \pi _{2}(M)} having a representative that is an embedding S 2 → M {\displaystyle S^{2}\to M} . The proof of this version of the theorem can be based on transversality methods, see Jean-Loïc Batude (1971).
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https://en.wikipedia.org/wiki/Sphere_theorem_(3-manifolds)
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Another more general version (also called the projective plane theorem, and due to David B. A. Epstein) is: Let M {\displaystyle M} be any 3-manifold and N {\displaystyle N} a π 1 ( M ) {\displaystyle \pi _{1}(M)} -invariant subgroup of π 2 ( M ) {\displaystyle \pi _{2}(M)} . If f: S 2 → M {\displaystyle f\colon S^{2}\to M} is a general position map such that ∉ N {\displaystyle \notin N} and U {\displaystyle U} is any neighborhood of the singular set Σ ( f ) {\displaystyle \Sigma (f)} , then there is a map g: S 2 → M {\displaystyle g\colon S^{2}\to M} satisfying ∉ N {\displaystyle \notin N} , g ( S 2 ) ⊂ f ( S 2 ) ∪ U {\displaystyle g(S^{2})\subset f(S^{2})\cup U} , g: S 2 → g ( S 2 ) {\displaystyle g\colon S^{2}\to g(S^{2})} is a covering map, and g ( S 2 ) {\displaystyle g(S^{2})} is a 2-sided submanifold (2-sphere or projective plane) of M {\displaystyle M} .quoted in (Hempel 1976, p. 54).
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https://en.wikipedia.org/wiki/Sphere_theorem_(3-manifolds)
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In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An incidence structure is what is obtained when all other concepts are removed and all that remains is the data about which points lie on which lines. Even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure.
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https://en.wikipedia.org/wiki/Incidence_geometry
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Such fundamental results remain valid when additional concepts are added to form a richer geometry. It sometimes happens that authors blur the distinction between a study and the objects of that study, so it is not surprising to find that some authors refer to incidence structures as incidence geometries.Incidence structures arise naturally and have been studied in various areas of mathematics. Consequently, there are different terminologies to describe these objects.
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https://en.wikipedia.org/wiki/Incidence_geometry
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In graph theory they are called hypergraphs, and in combinatorial design theory they are called block designs. Besides the difference in terminology, each area approaches the subject differently and is interested in questions about these objects relevant to that discipline. Using geometric language, as is done in incidence geometry, shapes the topics and examples that are normally presented.
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https://en.wikipedia.org/wiki/Incidence_geometry
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It is, however, possible to translate the results from one discipline into the terminology of another, but this often leads to awkward and convoluted statements that do not appear to be natural outgrowths of the topics. In the examples selected for this article we use only those with a natural geometric flavor. A special case that has generated much interest deals with finite sets of points in the Euclidean plane and what can be said about the number and types of (straight) lines they determine. Some results of this situation can extend to more general settings since only incidence properties are considered.
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https://en.wikipedia.org/wiki/Incidence_geometry
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In mathematics, inertial manifolds are concerned with the long term behavior of the solutions of dissipative dynamical systems. Inertial manifolds are finite-dimensional, smooth, invariant manifolds that contain the global attractor and attract all solutions exponentially quickly. Since an inertial manifold is finite-dimensional even if the original system is infinite-dimensional, and because most of the dynamics for the system takes place on the inertial manifold, studying the dynamics on an inertial manifold produces a considerable simplification in the study of the dynamics of the original system.In many physical applications, inertial manifolds express an interaction law between the small and large wavelength structures.
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https://en.wikipedia.org/wiki/Inertial_manifold
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Some say that the small wavelengths are enslaved by the large (e.g. synergetics). Inertial manifolds may also appear as slow manifolds common in meteorology, or as the center manifold in any bifurcation. Computationally, numerical schemes for partial differential equations seek to capture the long term dynamics and so such numerical schemes form an approximate inertial manifold.
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https://en.wikipedia.org/wiki/Inertial_manifold
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In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom. Recent developments concern combinatorics of the continuum and combinatorics on successors of singular cardinals.
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https://en.wikipedia.org/wiki/Arrow_notation_(Ramsey_theory)
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In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function.
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https://en.wikipedia.org/wiki/Infinite_compositions_of_analytic_functions
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For infinite compositions of a single function see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system. Although the title of this article specifies analytic functions, there are results for more general functions of a complex variable as well.
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https://en.wikipedia.org/wiki/Infinite_compositions_of_analytic_functions
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In mathematics, infinite difference methods are numerical methods for solving differential equations by approximating them with difference equations, in which infinite differences approximate the derivatives.
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https://en.wikipedia.org/wiki/Infinite_difference_method
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In mathematics, infinite-dimensional holomorphy is a branch of functional analysis. It is concerned with generalizations of the concept of holomorphic function to functions defined and taking values in complex Banach spaces (or Fréchet spaces more generally), typically of infinite dimension. It is one aspect of nonlinear functional analysis.
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https://en.wikipedia.org/wiki/Infinite-dimensional_holomorphy
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In mathematics, infinitesimal cohomology is a cohomology theory for algebraic varieties introduced by Grothendieck (1966). In characteristic 0 it is essentially the same as crystalline cohomology. In nonzero characteristic p Ogus (1975) showed that it is closely related to etale cohomology with mod p coefficients, a theory known to have undesirable properties.
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https://en.wikipedia.org/wiki/Infinitesimal_cohomology
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In mathematics, infinity plus one is a concept which has a well-defined formal meaning in some number systems, and may refer to: Transfinite numbers, numbers that are larger than all finite numbers Cardinal numbers, representations of sizes (cardinalities) of abstract sets, which may be infinite Ordinal numbers, representations of order types of well-ordered sets, which may also be infinite Hyperreal numbers, an extension of the real number system that contains infinite and infinitesimal numbers Surreal numbers, another extension of the real numbers, that contain the hyperreal and all transfinite ordinal numbers
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https://en.wikipedia.org/wiki/Infinity_plus_one
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In mathematics, informal logic and argument mapping, a lemma (PL: lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however, a lemma can also turn out to be more important than originally thought.
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https://en.wikipedia.org/wiki/Lemma_(mathematics)
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In mathematics, informally speaking, Euclid's orchard is an array of one-dimensional "trees" of unit height planted at the lattice points in one quadrant of a square lattice. More formally, Euclid's orchard is the set of line segments from (x, y, 0) to (x, y, 1), where x and y are positive integers. The trees visible from the origin are those at lattice points (x, y, 0), where x and y are coprime, i.e., where the fraction x/y is in reduced form. The name Euclid's orchard is derived from the Euclidean algorithm.
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https://en.wikipedia.org/wiki/Euclid's_orchard
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If the orchard is projected relative to the origin onto the plane x + y = 1 (or, equivalently, drawn in perspective from a viewpoint at the origin) the tops of the trees form a graph of Thomae's function. The point (x, y, 1) projects to ( x x + y , y x + y , 1 x + y ) . {\displaystyle \left({\frac {x}{x+y}},{\frac {y}{x+y}},{\frac {1}{x+y}}\right).} The solution to the Basel problem can be used to show that the proportion of points in the n × n {\displaystyle n\times n} grid that have trees on them is approximately 6 π 2 {\displaystyle {\tfrac {6}{\pi ^{2}}}} and that the error of this approximation goes to zero in the limit as n goes to infinity.
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https://en.wikipedia.org/wiki/Euclid's_orchard
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In mathematics, injections, surjections, and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other. A function maps elements from its domain to elements in its codomain. Given a function f: X → Y {\displaystyle f\colon X\to Y}: The function is injective, or one-to-one, if each element of the codomain is mapped to by at most one element of the domain, or equivalently, if distinct elements of the domain map to distinct elements in the codomain. An injective function is also called an injection.
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https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection
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Notationally: ∀ x , x ′ ∈ X , f ( x ) = f ( x ′ ) ⟹ x = x ′ , {\displaystyle \forall x,x'\in X,f(x)=f(x')\implies x=x',} or, equivalently (using logical transposition), ∀ x , x ′ ∈ X , x ≠ x ′ ⟹ f ( x ) ≠ f ( x ′ ) . {\displaystyle \forall x,x'\in X,x\neq x'\implies f(x)\neq f(x').} The function is surjective, or onto, if each element of the codomain is mapped to by at least one element of the domain.
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https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection
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That is, the image and the codomain of the function are equal. A surjective function is a surjection. Notationally: ∀ y ∈ Y , ∃ x ∈ X such that y = f ( x ) .
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https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection
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{\displaystyle \forall y\in Y,\exists x\in X{\text{ such that }}y=f(x).} The function is bijective (one-to-one and onto, one-to-one correspondence, or invertible) if each element of the codomain is mapped to by exactly one element of the domain. That is, the function is both injective and surjective.
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https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection
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A bijective function is also called a bijection. That is, combining the definitions of injective and surjective, ∀ y ∈ Y , ∃ ! x ∈ X such that y = f ( x ) , {\displaystyle \forall y\in Y,\exists !x\in X{\text{ such that }}y=f(x),} where ∃ !
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https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection
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x {\displaystyle \exists !x} means "there exists exactly one x".In any case (for any function), the following holds: ∀ x ∈ X , ∃ ! y ∈ Y such that y = f ( x ) . {\displaystyle \forall x\in X,\exists !y\in Y{\text{ such that }}y=f(x).} An injective function need not be surjective (not all elements of the codomain may be associated with arguments), and a surjective function need not be injective (some images may be associated with more than one argument). The four possible combinations of injective and surjective features are illustrated in the adjacent diagrams.
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https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection
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In mathematics, injective sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext). There is a further group of related concepts applied to sheaves: flabby (flasque in French), fine, soft (mou in French), acyclic. In the history of the subject they were introduced before the 1957 "Tohoku paper" of Alexander Grothendieck, which showed that the abelian category notion of injective object sufficed to found the theory.
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https://en.wikipedia.org/wiki/Acyclic_sheaf
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The other classes of sheaves are historically older notions. The abstract framework for defining cohomology and derived functors does not need them. However, in most concrete situations, resolutions by acyclic sheaves are often easier to construct. Acyclic sheaves therefore serve for computational purposes, for example the Leray spectral sequence.
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https://en.wikipedia.org/wiki/Acyclic_sheaf
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In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals that its motion is confined to a submanifold of much smaller dimensionality than that of its phase space. Three features are often referred to as characterizing integrable systems: the existence of a maximal set of conserved quantities (the usual defining property of complete integrability) the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability)Integrable systems may be seen as very different in qualitative character from more generic dynamical systems, which are more typically chaotic systems.
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https://en.wikipedia.org/wiki/Completely_integrable_system
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The latter generally have no conserved quantities, and are asymptotically intractable, since an arbitrarily small perturbation in initial conditions may lead to arbitrarily large deviations in their trajectories over a sufficiently large time. Many systems studied in physics are completely integrable, in particular, in the Hamiltonian sense, the key example being multi-dimensional harmonic oscillators. Another standard example is planetary motion about either one fixed center (e.g., the sun) or two.
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https://en.wikipedia.org/wiki/Completely_integrable_system
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Other elementary examples include the motion of a rigid body about its center of mass (the Euler top) and the motion of an axially symmetric rigid body about a point in its axis of symmetry (the Lagrange top). In the late 1960's, it was realized that there are completely integrable systems in physics having an infinite number of degrees of freedom, such as some models of shallow water waves (Korteweg–de Vries equation), the Kerr effect in optical fibres, described by the nonlinear Schrödinger equation, and certain integrable many-body systems, such as the Toda lattice. The modern theory of integrable systems was revived with the numerical discovery of solitons by Martin Kruskal and Norman Zabusky in 1965, which led to the inverse scattering transform method in 1967.
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https://en.wikipedia.org/wiki/Completely_integrable_system
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In the special case of Hamiltonian systems, if there are enough independent Poisson commuting first integrals for the flow parameters to be able to serve as a coordinate system on the invariant level sets (the leaves of the Lagrangian foliation), and if the flows are complete and the energy level set is compact, this implies the Liouville-Arnold theorem; i.e., the existence of action-angle variables. General dynamical systems have no such conserved quantities; in the case of autonomous Hamiltonian systems, the energy is generally the only one, and on the energy level sets, the flows are typically chaotic. A key ingredient in characterizing integrable systems is the Frobenius theorem, which states that a system is Frobenius integrable (i.e., is generated by an integrable distribution) if, locally, it has a foliation by maximal integral manifolds. But integrability, in the sense of dynamical systems, is a global property, not a local one, since it requires that the foliation be a regular one, with the leaves embedded submanifolds. Integrability does not necessarily imply that generic solutions can be explicitly expressed in terms of some known set of special functions; it is an intrinsic property of the geometry and topology of the system, and the nature of the dynamics.
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https://en.wikipedia.org/wiki/Completely_integrable_system
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In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: where I i ( u ) {\displaystyle I^{i}(u)} is an integral operator acting on u. Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals. A direct comparison can be seen with the mathematical form of the general integral equation above with the general form of a differential equation which may be expressed as follows:where D i ( u ) {\displaystyle D^{i}(u)} may be viewed as a differential operator of order i. Due to this close connection between differential and integral equations, one can often convert between the two.
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https://en.wikipedia.org/wiki/Singular_integral_equations
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For example, one method of solving a boundary value problem is by converting the differential equation with its boundary conditions into an integral equation and solving the integral equation. In addition, because one can convert between the two, differential equations in physics such as Maxwell's equations often have an analog integral and differential form. See also, for example, Green's function and Fredholm theory.
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https://en.wikipedia.org/wiki/Singular_integral_equations
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In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformations from the space of functions on one geometrical space to the space of functions on another geometrical space. Such transformations often take the form of integral transforms such as the Radon transform and its generalizations.
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https://en.wikipedia.org/wiki/Integral_geometry
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In mathematics, integrals of inverse functions can be computed by means of a formula that expresses the antiderivatives of the inverse f − 1 {\displaystyle f^{-1}} of a continuous and invertible function f {\displaystyle f} , in terms of f − 1 {\displaystyle f^{-1}} and an antiderivative of f {\displaystyle f} . This formula was published in 1905 by Charles-Ange Laisant.
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https://en.wikipedia.org/wiki/Integral_of_inverse_functions
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In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form. There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, Gromov-Witten theory and the extension of intersection theory from schemes to stacks.
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https://en.wikipedia.org/wiki/Unlikely_intersections
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In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are not transitive relations. This may include any relation that is not transitive, or the stronger property of antitransitivity, which describes a relation that is never transitive.
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https://en.wikipedia.org/wiki/Intransitive_preference
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In mathematics, intuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" (L. E. J. Brouwer). From this springboard, intuitionists seek to reconstruct what they consider to be the corrigible portion of mathematics in accordance with Kantian concepts of being, becoming, intuition, and knowledge. Brouwer, the founder of the movement, held that mathematical objects arise from the a priori forms of the volitions that inform the perception of empirical objects.A major force behind intuitionism was L. E. J. Brouwer, who rejected the usefulness of formalized logic of any sort for mathematics. His student Arend Heyting postulated an intuitionistic logic, different from the classical Aristotelian logic; this logic does not contain the law of the excluded middle and therefore frowns upon proofs by contradiction.
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https://en.wikipedia.org/wiki/Mathematical_anti-realism
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The axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted. In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms. Attempts have been made to use the concepts of Turing machine or computable function to fill this gap, leading to the claim that only questions regarding the behavior of finite algorithms are meaningful and should be investigated in mathematics. This has led to the study of the computable numbers, first introduced by Alan Turing. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical computer science.
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https://en.wikipedia.org/wiki/Mathematical_anti-realism
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In mathematics, it can be shown that every function can be written as the composite of a surjective function followed by an injective function. Factorization systems are a generalization of this situation in category theory.
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https://en.wikipedia.org/wiki/Factorization_system
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In mathematics, it is common practice to chain relational operators, such as in 3 < x < y < 20 (meaning 3 < x and x < y and y < 20). The syntax is clear since these relational operators in mathematics are transitive. However, many recent programming languages would see an expression like 3 < x < y as consisting of two left (or right-) associative operators, interpreting it as something like (3 < x) < y. If we say that x=4, we then get (3 < 4) < y, and evaluation will give true < y which generally does not make sense.
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https://en.wikipedia.org/wiki/Comparison_operator
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However, it does compile in C/C++ and some other languages, yielding surprising result (as true would be represented by the number 1 here). It is possible to give the expression x < y < z its familiar mathematical meaning, and some programming languages such as Python and Raku do that. Others, such as C# and Java, do not, partly because it would differ from the way most other infix operators work in C-like languages. The D programming language does not do that since it maintains some compatibility with C, and "Allowing C expressions but with subtly different semantics (albeit arguably in the right direction) would add more confusion than convenience".Some languages, like Common Lisp, use multiple argument predicates for this. In Lisp (<= 1 x 10) is true when x is between 1 and 10.
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https://en.wikipedia.org/wiki/Comparison_operator
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In mathematics, iterated forcing is a method for constructing models of set theory by repeating Cohen's forcing method a transfinite number of times. Iterated forcing was introduced by Solovay and Tennenbaum (1971) in their construction of a model of set theory with no Suslin tree. They also showed that iterated forcing can construct models where Martin's axiom holds and the continuum is any given regular cardinal. In iterated forcing, one has a transfinite sequence Pα of forcing notions indexed by some ordinals α, which give a family of Boolean-valued models VPα.
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https://en.wikipedia.org/wiki/Iterated_forcing
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If α+1 is a successor ordinal then Pα+1 is often constructed from Pα using a forcing notion in VPα, while if α is a limit ordinal then Pα is often constructed as some sort of limit (such as the direct limit) of the Pβ for β<α. A key consideration is that, typically, it is necessary that ω 1 {\displaystyle \omega _{1}} is not collapsed.
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https://en.wikipedia.org/wiki/Iterated_forcing
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This is often accomplished by the use of a preservation theorem such as: Finite support iteration of c.c.c. forcings (see countable chain condition) are c.c.c. and thus preserve ω 1 {\displaystyle \omega _{1}} . Countable support iterations of proper forcings are proper (see Fundamental Theorem of Proper Forcing) and thus preserve ω 1 {\displaystyle \omega _{1}} . Revised countable support iterations of semi-proper forcings are semi-proper and thus preserve ω 1 {\displaystyle \omega _{1}} .Some non-semi-proper forcings, such as Namba forcing, can be iterated with appropriate cardinal collapses while preserving ω 1 {\displaystyle \omega _{1}} using methods developed by Saharon Shelah.
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https://en.wikipedia.org/wiki/Iterated_forcing
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In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry. They were introduced in 1981.
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https://en.wikipedia.org/wiki/Iterated_Function_Systems
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IFS fractals, as they are normally called, can be of any number of dimensions, but are commonly computed and drawn in 2D. The fractal is made up of the union of several copies of itself, each copy being transformed by a function (hence "function system"). The canonical example is the Sierpiński triangle.
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https://en.wikipedia.org/wiki/Iterated_Function_Systems
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The functions are normally contractive, which means they bring points closer together and make shapes smaller. Hence, the shape of an IFS fractal is made up of several possibly-overlapping smaller copies of itself, each of which is also made up of copies of itself, ad infinitum. This is the source of its self-similar fractal nature.
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https://en.wikipedia.org/wiki/Iterated_Function_Systems
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In mathematics, iteration may refer to the process of iterating a function, i.e. applying a function repeatedly, using the output from one iteration as the input to the next. Iteration of apparently simple functions can produce complex behaviors and difficult problems – for examples, see the Collatz conjecture and juggler sequences. Another use of iteration in mathematics is in iterative methods which are used to produce approximate numerical solutions to certain mathematical problems. Newton's method is an example of an iterative method. Manual calculation of a number's square root is a common use and a well-known example.
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https://en.wikipedia.org/wiki/Iteration
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In mathematics, k-Hessian equations (or Hessian equations for short) are partial differential equations (PDEs) based on the Hessian matrix. More specifically, a Hessian equation is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k ≥ 2, the k-Hessian equation is a fully nonlinear partial differential equation. It can be written as S k = f {\displaystyle {\cal {S}}_{k}=f} , where 1 ⩽ k ⩽ n {\displaystyle 1\leqslant k\leqslant n} , S k = σ k ( λ ( D 2 u ) ) {\displaystyle {\cal {S}}_{k}=\sigma _{k}(\lambda ({\cal {D}}^{2}u))} , and λ ( D 2 u ) = ( λ 1 , ⋯ , λ n ) {\displaystyle \lambda ({\cal {D}}^{2}u)=(\lambda _{1},\cdots ,\lambda _{n})} , are the eigenvalues of the Hessian matrix D 2 u = 1 ≤ i , j ≤ n {\displaystyle {\cal {D}}^{2}u=_{1\leq i,j\leq n}} and σ k ( λ ) = ∑ i 1 < ⋯ < i k λ i 1 ⋯ λ i k {\displaystyle \sigma _{k}(\lambda )=\sum _{i_{1}<\cdots
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https://en.wikipedia.org/wiki/Hessian_equation
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In mathematics, least squares function approximation applies the principle of least squares to function approximation, by means of a weighted sum of other functions. The best approximation can be defined as that which minimizes the difference between the original function and the approximation; for a least-squares approach the quality of the approximation is measured in terms of the squared differences between the two.
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https://en.wikipedia.org/wiki/Least-squares_function_approximation
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In mathematics, leximin order is a total preorder on finite-dimensional vectors. A more accurate, but less common term is leximin preorder. The leximin order is particularly important in social choice theory and fair division.
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https://en.wikipedia.org/wiki/Leximin_order
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In mathematics, lifting theory was first introduced by John von Neumann in a pioneering paper from 1931, in which he answered a question raised by Alfréd Haar. The theory was further developed by Dorothy Maharam (1958) and by Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea (1961). Lifting theory was motivated to a large extent by its striking applications. Its development up to 1969 was described in a monograph of the Ionescu Tulceas. Lifting theory continued to develop since then, yielding new results and applications.
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https://en.wikipedia.org/wiki/Lifting_theory
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In mathematics, like terms are summands in a sum that differ only by a numerical factor. Like terms can be regrouped by adding their coefficients. Typically, in a polynomial expression, like terms are those that contain the same variables to the same powers, possibly with different coefficients. More generally, when some variable are considered as parameters, like terms are defined similarly, but "numerical factors" must be replaced by "factors depending only on the parameters".
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https://en.wikipedia.org/wiki/Combining_like_terms
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For example, when considering a quadratic equation, one considers often the expression ( x − r ) ( x − s ) , {\displaystyle (x-r)(x-s),} where r {\displaystyle r} and s {\displaystyle s} are the roots of the equation and may be considered as parameters. Then, expanding the above product and regrouping the like terms gives x 2 − ( r + s ) x + r s . {\displaystyle x^{2}-(r+s)x+rs.}
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https://en.wikipedia.org/wiki/Combining_like_terms
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In mathematics, limit cardinals are certain cardinal numbers. A cardinal number λ is a weak limit cardinal if λ is neither a successor cardinal nor zero. This means that one cannot "reach" λ from another cardinal by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the context is clear. A cardinal λ is a strong limit cardinal if λ cannot be reached by repeated powerset operations. This means that λ is nonzero and, for all κ < λ, 2κ < λ. Every strong limit cardinal is also a weak limit cardinal, because κ+ ≤ 2κ for every cardinal κ, where κ+ denotes the successor cardinal of κ. The first infinite cardinal, ℵ 0 {\displaystyle \aleph _{0}} (aleph-naught), is a strong limit cardinal, and hence also a weak limit cardinal.
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https://en.wikipedia.org/wiki/Weak_limit_cardinal
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In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.
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https://en.wikipedia.org/wiki/Linear_interpolation
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In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see linear approximation). If the spaces involved are also topological spaces (that is, topological vector spaces), then it makes sense to ask whether all linear maps are continuous. It turns out that for maps defined on infinite-dimensional topological vector spaces (e.g., infinite-dimensional normed spaces), the answer is generally no: there exist discontinuous linear maps. If the domain of definition is complete, it is trickier; such maps can be proven to exist, but the proof relies on the axiom of choice and does not provide an explicit example.
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https://en.wikipedia.org/wiki/Discontinuous_linear_functional
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In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in fields such as engineering, physics, economics, and ecology.
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https://en.wikipedia.org/wiki/Linearization
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In mathematics, list edge-coloring is a type of graph coloring that combines list coloring and edge coloring. An instance of a list edge-coloring problem consists of a graph together with a list of allowed colors for each edge. A list edge-coloring is a choice of a color for each edge, from its list of allowed colors; a coloring is proper if no two adjacent edges receive the same color.
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https://en.wikipedia.org/wiki/List_chromatic_index
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A graph G is k-edge-choosable if every instance of list edge-coloring that has G as its underlying graph and that provides at least k allowed colors for each edge of G has a proper coloring. The edge choosability, or list edge colorability, list edge chromatic number, or list chromatic index, ch′(G) of graph G is the least number k such that G is k-edge-choosable. It is conjectured that it always equals the chromatic index.
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https://en.wikipedia.org/wiki/List_chromatic_index
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In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of the p-adic numbers Qp (where p is any prime number), or the field of formal Laurent series Fq((T)) over a finite field Fq.
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https://en.wikipedia.org/wiki/Local_class_field_theory
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In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before. In homotopy theory, for example, there are many examples of mappings that are invertible up to homotopy; and so large classes of homotopy equivalent spaces. Calculus of fractions is another name for working in a localized category.
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https://en.wikipedia.org/wiki/Serre_C-theory
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In mathematics, log-polar coordinates (or logarithmic polar coordinates) is a coordinate system in two dimensions, where a point is identified by two numbers, one for the logarithm of the distance to a certain point, and one for an angle. Log-polar coordinates are closely connected to polar coordinates, which are usually used to describe domains in the plane with some sort of rotational symmetry. In areas like harmonic and complex analysis, the log-polar coordinates are more canonical than polar coordinates.
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https://en.wikipedia.org/wiki/Log-polar_coordinates
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In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f, its logarithm, and its gradient ∇ f {\displaystyle \nabla f} . These inequalities were discovered and named by Leonard Gross, who established them in dimension-independent form, in the context of constructive quantum field theory. Similar results were discovered by other mathematicians before and many variations on such inequalities are known.
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https://en.wikipedia.org/wiki/Logarithmic_Sobolev_inequalities
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In mathematics, logarithmic growth describes a phenomenon whose size or cost can be described as a logarithm function of some input. e.g. y = C log (x). Any logarithm base can be used, since one can be converted to another by multiplying by a fixed constant. Logarithmic growth is the inverse of exponential growth and is very slow.A familiar example of logarithmic growth is a number, N, in positional notation, which grows as logb (N), where b is the base of the number system used, e.g. 10 for decimal arithmetic.
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https://en.wikipedia.org/wiki/Logarithmic_growth
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In more advanced mathematics, the partial sums of the harmonic series 1 + 1 2 + 1 3 + 1 4 + 1 5 + ⋯ {\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots } grow logarithmically. In the design of computer algorithms, logarithmic growth, and related variants, such as log-linear, or linearithmic, growth are very desirable indications of efficiency, and occur in the time complexity analysis of algorithms such as binary search.Logarithmic growth can lead to apparent paradoxes, as in the martingale roulette system, where the potential winnings before bankruptcy grow as the logarithm of the gambler's bankroll. It also plays a role in the St.
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https://en.wikipedia.org/wiki/Logarithmic_growth
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Petersburg paradox.In microbiology, the rapidly growing exponential growth phase of a cell culture is sometimes called logarithmic growth. During this bacterial growth phase, the number of new cells appearing is proportional to the population. This terminological confusion between logarithmic growth and exponential growth may be explained by the fact that exponential growth curves may be straightened by plotting them using a logarithmic scale for the growth axis.
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https://en.wikipedia.org/wiki/Logarithmic_growth
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In mathematics, logic and computer science, a formal language (a set of finite sequences of symbols taken from a fixed alphabet) is called recursive if it is a recursive subset of the set of all possible finite sequences over the alphabet of the language. Equivalently, a formal language is recursive if there exists a Turing machine that, when given a finite sequence of symbols as input, always halts and accepts it if it belongs to the language and halts and rejects it otherwise. In Theoretical computer science, such always-halting Turing machines are called total Turing machines or algorithms (Sipser 1997).
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https://en.wikipedia.org/wiki/Recursive_language
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Recursive languages are also called decidable. The concept of decidability may be extended to other models of computation. For example, one may speak of languages decidable on a non-deterministic Turing machine.
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https://en.wikipedia.org/wiki/Recursive_language
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Therefore, whenever an ambiguity is possible, the synonym used for "recursive language" is Turing-decidable language, rather than simply decidable. The class of all recursive languages is often called R, although this name is also used for the class RP. This type of language was not defined in the Chomsky hierarchy of (Chomsky 1959). All recursive languages are also recursively enumerable. All regular, context-free and context-sensitive languages are recursive.
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https://en.wikipedia.org/wiki/Recursive_language
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In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i.e., if there exists a Turing machine which will enumerate all valid strings of the language. Recursively enumerable languages are known as type-0 languages in the Chomsky hierarchy of formal languages. All regular, context-free, context-sensitive and recursive languages are recursively enumerable. The class of all recursively enumerable languages is called RE.
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https://en.wikipedia.org/wiki/Recognizable_language
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In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more commonly) another set that contains the thing being defined. There is no generally accepted precise definition of what it means to be predicative or impredicative. Authors have given different but related definitions.
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https://en.wikipedia.org/wiki/Impredicativity
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The opposite of impredicativity is predicativity, which essentially entails building stratified (or ramified) theories where quantification over lower levels results in variables of some new type, distinguished from the lower types that the variable ranges over. A prototypical example is intuitionistic type theory, which retains ramification so as to discard impredicativity. Russell's paradox is a famous example of an impredicative construction—namely the set of all sets that do not contain themselves.
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https://en.wikipedia.org/wiki/Impredicativity
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The paradox is that such a set cannot exist: If it would exist, the question could be asked whether it contains itself or not — if it does then by definition it should not, and if it does not then by definition it should. The greatest lower bound of a set X, glb(X), also has an impredicative definition: y = glb(X) if and only if for all elements x of X, y is less than or equal to x, and any z less than or equal to all elements of X is less than or equal to y. This definition quantifies over the set (potentially infinite, depending on the order in question) whose members are the lower bounds of X, one of which being the glb itself. Hence predicativism would reject this definition.
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https://en.wikipedia.org/wiki/Impredicativity
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In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general, type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of mathematics. Two influential type theories that were proposed as foundations are Alonzo Church's typed λ-calculus and Per Martin-Löf's intuitionistic type theory. Most computerized proof-writing systems use a type theory for their foundation, a common one is Thierry Coquand's Calculus of Inductive Constructions.
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https://en.wikipedia.org/wiki/System_of_types
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In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory, relations between primitive notions are restricted by axioms. Some authors refer to the latter as "defining" primitive notions by one or more axioms, but this can be misleading.
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https://en.wikipedia.org/wiki/Undefined_term
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