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In mathematics, projective differential geometry is the study of differential geometry, from the point of view of properties of mathematical objects such as functions, diffeomorphisms, and submanifolds, that are invariant under transformations of the projective group. This is a mixture of the approaches from Riemannian geometry of studying invariances, and of the Erlangen program of characterizing geometries according to their group symmetries. The area was much studied by mathematicians from around 1890 for a generation (by J. G. Darboux, George Henri Halphen, Ernest Julius Wilczynski, E. Bompiani, G. Fubini, Eduard Čech, amongst others), without a comprehensive theory of differential invariants emerging.
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https://en.wikipedia.org/wiki/Projective_differential_geometry
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Élie Cartan formulated the idea of a general projective connection, as part of his method of moving frames; abstractly speaking, this is the level of generality at which the Erlangen program can be reconciled with differential geometry, while it also develops the oldest part of the theory (for the projective line), namely the Schwarzian derivative, the simplest projective differential invariant.Further work from the 1930s onwards was carried out by J. Kanitani, Shiing-Shen Chern, A. P. Norden, G. Bol, S. P. Finikov and G. F. Laptev. Even the basic results on osculation of curves, a manifestly projective-invariant topic, lack any comprehensive theory. The ideas of projective differential geometry recur in mathematics and its applications, but the formulations given are still rooted in the language of the early twentieth century.
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https://en.wikipedia.org/wiki/Projective_differential_geometry
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In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice-versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations).
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https://en.wikipedia.org/wiki/Axioms_of_projective_geometry
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The first issue for geometers is what kind of geometry is adequate for a novel situation. It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant with respect to projective transformations, as is seen in perspective drawing. One source for projective geometry was indeed the theory of perspective.
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https://en.wikipedia.org/wiki/Axioms_of_projective_geometry
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Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing.
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https://en.wikipedia.org/wiki/Axioms_of_projective_geometry
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See projective plane for the basics of projective geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory of complex projective space, the coordinates used (homogeneous coordinates) being complex numbers.
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https://en.wikipedia.org/wiki/Axioms_of_projective_geometry
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Several major types of more abstract mathematics (including invariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programme resulting in the study of the classical groups) were motivated by projective geometry. It was also a subject with many practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry. The topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry (the study of projective varieties) and projective differential geometry (the study of differential invariants of the projective transformations).
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https://en.wikipedia.org/wiki/Axioms_of_projective_geometry
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In mathematics, projectivization is a procedure which associates with a non-zero vector space V a projective space P ( V ) {\displaystyle {\mathbb {P} }(V)} , whose elements are one-dimensional subspaces of V. More generally, any subset S of V closed under scalar multiplication defines a subset of P ( V ) {\displaystyle {\mathbb {P} }(V)} formed by the lines contained in S and is called the projectivization of S.
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https://en.wikipedia.org/wiki/Projectivization
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In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic square matrix is invertible." As another example, a generic property of a space is a property that holds at "almost all" points of the space, as in the statement, "If f: M → N is a smooth function between smooth manifolds, then a generic point of N is not a critical value of f." (This is by Sard's theorem.)
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https://en.wikipedia.org/wiki/Generic_property
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There are many different notions of "generic" (what is meant by "almost all") in mathematics, with corresponding dual notions of "almost none" (negligible set); the two main classes are: In measure theory, a generic property is one that holds almost everywhere, with the dual concept being null set, meaning "with probability 0". In topology and algebraic geometry, a generic property is one that holds on a dense open set, or more generally on a residual set, with the dual concept being a nowhere dense set, or more generally a meagre set.There are several natural examples where those notions are not equal. For instance, the set of Liouville numbers is generic in the topological sense, but has Lebesgue measure zero.
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https://en.wikipedia.org/wiki/Generic_property
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In mathematics, pseudoanalytic functions are functions introduced by Lipman Bers (1950, 1951, 1953, 1956) that generalize analytic functions and satisfy a weakened form of the Cauchy–Riemann equations.
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https://en.wikipedia.org/wiki/Pseudoanalytic_function
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In mathematics, quadratic Jordan algebras are a generalization of Jordan algebras introduced by Kevin McCrimmon (1966). The fundamental identities of the quadratic representation of a linear Jordan algebra are used as axioms to define a quadratic Jordan algebra over a field of arbitrary characteristic. There is a uniform description of finite-dimensional simple quadratic Jordan algebras, independent of characteristic. If 2 is invertible in the field of coefficients, the theory of quadratic Jordan algebras reduces to that of linear Jordan algebras.
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https://en.wikipedia.org/wiki/Quadratic_Jordan_algebra
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In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process.
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https://en.wikipedia.org/wiki/Quadratic_variation
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In mathematics, quadrature is a historical term for the process of determining area. This term is still used in the context of differential equations, where "solving an equation by quadrature" or "reduction to quadrature" means expressing its solution in terms of integrals. Quadrature problems served as one of the main sources of problems in the development of calculus. They introduce important topics in mathematical analysis.
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https://en.wikipedia.org/wiki/Quadrature_(mathematics)
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In mathematics, quantales are certain partially ordered algebraic structures that generalize locales (point free topologies) as well as various multiplicative lattices of ideals from ring theory and functional analysis (C*-algebras, von Neumann algebras). Quantales are sometimes referred to as complete residuated semigroups.
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https://en.wikipedia.org/wiki/Quantale
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In mathematics, quasi-bialgebras are a generalization of bialgebras: they were first defined by the Ukrainian mathematician Vladimir Drinfeld in 1990. A quasi-bialgebra differs from a bialgebra by having coassociativity replaced by an invertible element Φ {\displaystyle \Phi } which controls the non-coassociativity. One of their key properties is that the corresponding category of modules forms a tensor category.
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https://en.wikipedia.org/wiki/Quasi-bialgebra
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In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called. As with complex and real analysis, it is possible to study the concepts of analyticity, holomorphy, harmonicity and conformality in the context of quaternions. Unlike the complex numbers and like the reals, the four notions do not coincide.
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https://en.wikipedia.org/wiki/Quaternionic_analysis
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In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H . {\displaystyle \mathbb {H} .} Quaternionic projective space of dimension n is usually denoted by H P n {\displaystyle \mathbb {HP} ^{n}} and is a closed manifold of (real) dimension 4n. It is a homogeneous space for a Lie group action, in more than one way. The quaternionic projective line H P 1 {\displaystyle \mathbb {HP} ^{1}} is homeomorphic to the 4-sphere.
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https://en.wikipedia.org/wiki/Quaternionic_projective_line
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In mathematics, quaternions are a non-commutative number system that extends the complex numbers. Quaternions and their applications to rotations were first described in print by Olinde Rodrigues in all but name in 1840, but independently discovered by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. They find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations.
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https://en.wikipedia.org/wiki/History_of_quaternions
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In mathematics, racks and quandles are sets with binary operations satisfying axioms analogous to the Reidemeister moves used to manipulate knot diagrams. While mainly used to obtain invariants of knots, they can be viewed as algebraic constructions in their own right. In particular, the definition of a quandle axiomatizes the properties of conjugation in a group.
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https://en.wikipedia.org/wiki/Racks_and_quandles
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In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs lies at the intersection between graph theory and probability theory.
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https://en.wikipedia.org/wiki/Random_graphs
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From a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs. Its practical applications are found in all areas in which complex networks need to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, random graph refers almost exclusively to the Erdős–Rényi random graph model. In other contexts, any graph model may be referred to as a random graph.
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https://en.wikipedia.org/wiki/Random_graphs
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In mathematics, random groups are certain groups obtained by a probabilistic construction. They were introduced by Misha Gromov to answer questions such as "What does a typical group look like?" It so happens that, once a precise definition is given, random groups satisfy some properties with very high probability, whereas other properties fail with very high probability. For instance, very probably random groups are hyperbolic groups. In this sense, one can say that "most groups are hyperbolic".
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https://en.wikipedia.org/wiki/Random_group
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In mathematics, rational reconstruction is a method that allows one to recover a rational number from its value modulo a sufficiently large integer.
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https://en.wikipedia.org/wiki/Rational_reconstruction_(mathematics)
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In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial mappings). Semialgebraic geometry is the study of semialgebraic sets, i.e. real-number solutions to algebraic inequalities with-real number coefficients, and mappings between them. The most natural mappings between semialgebraic sets are semialgebraic mappings, i.e., mappings whose graphs are semialgebraic sets.
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https://en.wikipedia.org/wiki/Real_algebraic_variety
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In mathematics, real projective space, denoted R P n {\displaystyle \mathbb {RP} ^{n}} or P n ( R ) , {\displaystyle \mathbb {P} _{n}(\mathbb {R} ),} is the topological space of lines passing through the origin 0 in the real space R n + 1 . {\displaystyle \mathbb {R} ^{n+1}.} It is a compact, smooth manifold of dimension n, and is a special case G r ( 1 , R n + 1 ) {\displaystyle \mathbf {Gr} (1,\mathbb {R} ^{n+1})} of a Grassmannian space.
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https://en.wikipedia.org/wiki/Real_projective_space
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In mathematics, real trees (also called R {\displaystyle \mathbb {R} } -trees) are a class of metric spaces generalising simplicial trees. They arise naturally in many mathematical contexts, in particular geometric group theory and probability theory. They are also the simplest examples of Gromov hyperbolic spaces.
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https://en.wikipedia.org/wiki/Real_tree
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In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise statements to be made (as in Alexander duality) and eliminates many exceptional cases (as in the homology groups of spheres). If P is a single-point space, then with the usual definitions the integral homology group H0(P)is isomorphic to Z {\displaystyle \mathbb {Z} } (an infinite cyclic group), while for i ≥ 1 we have Hi(P) = {0}.More generally if X is a simplicial complex or finite CW complex, then the group H0(X) is the free abelian group with the connected components of X as generators. The reduced homology should replace this group, of rank r say, by one of rank r − 1.
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https://en.wikipedia.org/wiki/Reduced_homology
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In mathematics, reduction refers to the rewriting of an expression into a simpler form. For example, the process of rewriting a fraction into one with the smallest whole-number denominator possible (while keeping the numerator a whole number) is called "reducing a fraction". Rewriting a radical (or "root") expression with the smallest possible whole number under the radical symbol is called "reducing a radical". Minimizing the number of radicals that appear underneath other radicals in an expression is called denesting radicals.
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https://en.wikipedia.org/wiki/Reduction_(mathematics)
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In mathematics, reductionism can be interpreted as the philosophy that all mathematics can (or ought to) be based on a common foundation, which for modern mathematics is usually axiomatic set theory. Ernst Zermelo was one of the major advocates of such an opinion; he also developed much of axiomatic set theory. It has been argued that the generally accepted method of justifying mathematical axioms by their usefulness in common practice can potentially weaken Zermelo's reductionist claim.Jouko Väänänen has argued for second-order logic as a foundation for mathematics instead of set theory, whereas others have argued for category theory as a foundation for certain aspects of mathematics.The incompleteness theorems of Kurt Gödel, published in 1931, caused doubt about the attainability of an axiomatic foundation for all of mathematics. Any such foundation would have to include axioms powerful enough to describe the arithmetic of the natural numbers (a subset of all mathematics).
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https://en.wikipedia.org/wiki/Scientific_reductionism
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Yet Gödel proved that, for any consistent recursively enumerable axiomatic system powerful enough to describe the arithmetic of the natural numbers, there are (model-theoretically) true propositions about the natural numbers that cannot be proved from the axioms. Such propositions are known as formally undecidable propositions. For example, the continuum hypothesis is undecidable in the Zermelo–Fraenkel set theory as shown by Cohen.
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https://en.wikipedia.org/wiki/Scientific_reductionism
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In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric. In conclusion, a line of symmetry splits the shape in half and those halves should be identical.
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https://en.wikipedia.org/wiki/Reflection_symmetry
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In mathematics, restricted root systems, sometimes called relative root systems, are the root systems associated with a symmetric space. The associated finite reflection group is called the restricted Weyl group. The restricted root system of a symmetric space and its dual can be identified. For symmetric spaces of noncompact type arising as homogeneous spaces of a semisimple Lie group, the restricted root system and its Weyl group are related to the Iwasawa decomposition of the Lie group.
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https://en.wikipedia.org/wiki/Relative_root_system
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In mathematics, restriction of scalars (also known as "Weil restriction") is a functor which, for any finite extension of fields L/k and any algebraic variety X over L, produces another variety ResL/kX, defined over k. It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields.
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https://en.wikipedia.org/wiki/Weil_descent
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In mathematics, rigid cohomology is a p-adic cohomology theory introduced by Berthelot (1986). It extends crystalline cohomology to schemes that need not be proper or smooth, and extends Monsky–Washnitzer cohomology to non-affine varieties. For a scheme X of finite type over a perfect field k, there are rigid cohomology groups Hirig(X/K) which are finite dimensional vector spaces over the field K of fractions of the ring of Witt vectors of k. More generally one can define rigid cohomology with compact supports, or with support on a closed subscheme, or with coefficients in an overconvergent isocrystal. If X is smooth and proper over k the rigid cohomology groups are the same as the crystalline cohomology groups. The name "rigid cohomology" comes from its relation to rigid analytic spaces. Kedlaya (2006) used rigid cohomology to give a new proof of the Weil conjectures.
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https://en.wikipedia.org/wiki/Rigid_cohomology
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In mathematics, rigidity of K-theory encompasses results relating algebraic K-theory of different rings.
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https://en.wikipedia.org/wiki/Suslin_rigidity
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In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ring is an abelian group whose operation is called addition, with a second binary operation called multiplication that is associative, is distributive over the addition operation, and has a multiplicative identity element.
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https://en.wikipedia.org/wiki/Unital_ring
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(Some authors use the term "rng" with a missing "i" to refer to the more general structure that omits this last requirement; see § Notes on the definition.) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has profound implications on its behavior. Commutative algebra, the theory of commutative rings, is a major branch of ring theory.
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https://en.wikipedia.org/wiki/Unital_ring
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Its development has been greatly influenced by problems and ideas of algebraic number theory and algebraic geometry. The simplest commutative rings are those that admit division by non-zero elements; such rings are called fields. Examples of commutative rings include the set of integers with their standard addition and multiplication, the set of polynomials with their addition and multiplication, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field.
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https://en.wikipedia.org/wiki/Unital_ring
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Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators, and cohomology rings in topology. The conceptualization of rings spanned the 1870s to the 1920s, with key contributions by Dedekind, Hilbert, Fraenkel, and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. They later proved useful in other branches of mathematics such as geometry and analysis.
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https://en.wikipedia.org/wiki/Unital_ring
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In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector—without changing its direction. The term "scalar" itself derives from this usage: a scalar is that which scales vectors. Scalar multiplication is the multiplication of a vector by a scalar (where the product is a vector), and is to be distinguished from inner product of two vectors (where the product is a scalar).
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https://en.wikipedia.org/wiki/Scalar_multiplication
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In mathematics, scattering theory deals with a more abstract formulation of the same set of concepts. For example, if a differential equation is known to have some simple, localized solutions, and the solutions are a function of a single parameter, that parameter can take the conceptual role of time. One then asks what might happen if two such solutions are set up far away from each other, in the "distant past", and are made to move towards each other, interact (under the constraint of the differential equation) and then move apart in the "future". The scattering matrix then pairs solutions in the "distant past" to those in the "distant future".
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https://en.wikipedia.org/wiki/Scattering_process
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Solutions to differential equations are often posed on manifolds. Frequently, the means to the solution requires the study of the spectrum of an operator on the manifold.
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https://en.wikipedia.org/wiki/Scattering_process
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As a result, the solutions often have a spectrum that can be identified with a Hilbert space, and scattering is described by a certain map, the S matrix, on Hilbert spaces. Spaces with a discrete spectrum correspond to bound states in quantum mechanics, while a continuous spectrum is associated with scattering states. The study of inelastic scattering then asks how discrete and continuous spectra are mixed together. An important, notable development is the inverse scattering transform, central to the solution of many exactly solvable models.
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https://en.wikipedia.org/wiki/Scattering_process
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In mathematics, secondary calculus is a proposed expansion of classical differential calculus on manifolds, to the "space" of solutions of a (nonlinear) partial differential equation. It is a sophisticated theory at the level of jet spaces and employing algebraic methods.
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https://en.wikipedia.org/wiki/Secondary_calculus
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In mathematics, self-affinity is a feature of a fractal whose pieces are scaled by different amounts in the x- and y-directions. This means that to appreciate the self similarity of these fractal objects, they have to be rescaled using an anisotropic affine transformation.
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https://en.wikipedia.org/wiki/Self_similarity
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In mathematics, semi-infinite objects are objects which are infinite or unbounded in some but not all possible ways.
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https://en.wikipedia.org/wiki/Semi-infinite
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In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of simple objects, and simple objects are those that do not contain non-trivial proper sub-objects. The precise definitions of these words depends on the context. For example, if G is a finite group, then a nontrivial finite-dimensional representation V over a field is said to be simple if the only subrepresentations it contains are either {0} or V (these are also called irreducible representations).
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https://en.wikipedia.org/wiki/Semi-simple_category
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Now Maschke's theorem says that any finite-dimensional representation of a finite group is a direct sum of simple representations (provided the characteristic of the base field does not divide the order of the group). So in the case of finite groups with this condition, every finite-dimensional representation is semi-simple. Especially in algebra and representation theory, "semi-simplicity" is also called complete reducibility.
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https://en.wikipedia.org/wiki/Semi-simple_category
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For example, Weyl's theorem on complete reducibility says a finite-dimensional representation of a semisimple compact Lie group is semisimple. A square matrix (in other words a linear operator T: V → V {\displaystyle T:V\to V} with V finite dimensional vector space) is said to be simple if its only invariant subspaces under T are {0} and V. If the field is algebraically closed (such as the complex numbers), then the only simple matrices are of size 1 by 1. A semi-simple matrix is one that is similar to a direct sum of simple matrices; if the field is algebraically closed, this is the same as being diagonalizable. These notions of semi-simplicity can be unified using the language of semi-simple modules, and generalized to semi-simple categories.
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https://en.wikipedia.org/wiki/Semi-simple_category
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In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.
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https://en.wikipedia.org/wiki/Separation_of_variables
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In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration. Series acceleration techniques may also be used, for example, to obtain a variety of identities on special functions. Thus, the Euler transform applied to the hypergeometric series gives some of the classic, well-known hypergeometric series identities.
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https://en.wikipedia.org/wiki/Series_acceleration
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In mathematics, set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements. The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation.
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https://en.wikipedia.org/wiki/Subset_inclusion
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In mathematics, set inversion is the problem of characterizing the preimage X of a set Y by a function f, i.e., X = f −1(Y ) = {x ∈ Rn | f(x) ∈ Y }. It can also be viewed as the problem of describing the solution set of the quantified constraint "Y(f (x))", where Y( y) is a constraint, e.g. an inequality, describing the set Y. In most applications, f is a function from Rn to Rp and the set Y is a box of Rp (i.e. a Cartesian product of p intervals of R). When f is nonlinear the set inversion problem can be solved using interval analysis combined with a branch-and-bound algorithm.The main idea consists in building a paving of Rp made with non-overlapping boxes. For each box , we perform the following tests: if f () ⊂ Y we conclude that ⊂ X; if f () ∩ Y = ∅ we conclude that ∩ X = ∅; Otherwise, the box the box is bisected except if its width is smaller than a given precision.To check the two first tests, we need an interval extension (or an inclusion function) for f. Classified boxes are stored into subpavings, i.e., union of non-overlapping boxes. The algorithm can be made more efficient by replacing the inclusion tests by contractors.
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https://en.wikipedia.org/wiki/Set_inversion
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In mathematics, set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of Zermelo–Fraenkel set theory (ZFC).
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https://en.wikipedia.org/wiki/Set-theoretic_topology
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In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper. Sheaves, sheaf cohomology, and spectral sequences were introduced by Jean Leray at the prisoner-of-war camp Oflag XVII-A in Austria.
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https://en.wikipedia.org/wiki/Sheaf_cohomology
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From 1940 to 1945, Leray and other prisoners organized a "université en captivité" in the camp. Leray's definitions were simplified and clarified in the 1950s.
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https://en.wikipedia.org/wiki/Sheaf_cohomology
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It became clear that sheaf cohomology was not only a new approach to cohomology in algebraic topology, but also a powerful method in complex analytic geometry and algebraic geometry. These subjects often involve constructing global functions with specified local properties, and sheaf cohomology is ideally suited to such problems. Many earlier results such as the Riemann–Roch theorem and the Hodge theorem have been generalized or understood better using sheaf cohomology.
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https://en.wikipedia.org/wiki/Sheaf_cohomology
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In mathematics, sieved Jacobi polynomials are a family of sieved orthogonal polynomials, introduced by Askey (1984). Their recurrence relations are a modified (or "sieved") version of the recurrence relations for Jacobi polynomials.
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https://en.wikipedia.org/wiki/Sieved_Jacobi_polynomials
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In mathematics, sieved Pollaczek polynomials are a family of sieved orthogonal polynomials, introduced by Ismail (1985). Their recurrence relations are a modified (or "sieved") version of the recurrence relations for Pollaczek polynomials.
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https://en.wikipedia.org/wiki/Sieved_Pollaczek_polynomials
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In mathematics, sieved orthogonal polynomials are orthogonal polynomials whose recurrence relations are formed by sieving the recurrence relations of another family; in other words, some of the recurrence relations are replaced by simpler ones. The first examples were the sieved ultraspherical polynomials introduced by Waleed Al-Salam, W. R. Allaway, and Richard Askey (1984). Mourad Ismail later studied sieved orthogonal polynomials in a long series of papers. Other families of sieved orthogonal polynomials that have been studied include sieved Pollaczek polynomials, and sieved Jacobi polynomials.
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https://en.wikipedia.org/wiki/Sieved_orthogonal_polynomials
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In mathematics, signal processing and control theory, a pole–zero plot is a graphical representation of a rational transfer function in the complex plane which helps to convey certain properties of the system such as: Stability Causal system / anticausal system Region of convergence (ROC) Minimum phase / non minimum phaseA pole-zero plot shows the location in the complex plane of the poles and zeros of the transfer function of a dynamic system, such as a controller, compensator, sensor, equalizer, filter, or communications channel. By convention, the poles of the system are indicated in the plot by an X while the zeros are indicated by a circle or O. A pole-zero plot is plotted in the plane of a complex frequency domain, which can represent either a continuous-time or a discrete-time system: Continuous-time systems use the Laplace transform and are plotted in the s-plane: s = σ + j ω {\displaystyle s=\sigma +j\omega } Real frequency components are along its vertical axis (the imaginary line s = j ω {\displaystyle s{=}j\omega } where σ = 0 {\displaystyle \sigma {=}0} ) Discrete-time systems use the Z-transform and are plotted in the z-plane: z = A e j ϕ {\displaystyle z=Ae^{j\phi }} Real frequency components are along its unit circle
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https://en.wikipedia.org/wiki/Pole–zero_plot
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In mathematics, signed frequency (negative and positive frequency) expands upon the concept of frequency, from just an absolute value representing how often some repeating event occurs, to also have a positive or negative sign representing one of two opposing orientations for occurrences of those events. The following examples help illustrate the concept: For a rotating object, the absolute value of its frequency of rotation indicates how many rotations the object completes per unit of time, while the sign could indicate whether it is rotating clockwise or counterclockwise. Mathematically speaking, the vector ( cos ( t ) , sin ( t ) ) {\displaystyle (\cos(t),\sin(t))} has a positive frequency of +1 radian per unit of time and rotates counterclockwise around the unit circle, while the vector ( cos ( − t ) , sin ( − t ) ) {\displaystyle (\cos(-t),\sin(-t))} has a negative frequency of -1 radian per unit of time, which rotates clockwise instead. For a harmonic oscillator such as a pendulum, the absolute value of its frequency indicates how many times it swings back and forth per unit of time, while the sign could indicate in which of the two opposite directions it started moving. For a periodic function represented in a Cartesian coordinate system, the absolute value of its frequency indicates how often in its domain it repeats its values, while changing the sign of its frequency could represent a reflection around its y-axis.
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https://en.wikipedia.org/wiki/Negative_frequency
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In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values, i.e., to aqcuire sign.
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https://en.wikipedia.org/wiki/Signed_measure
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In mathematics, simple homotopy theory is a homotopy theory (a branch of algebraic topology) that concerns with the simple-homotopy type of a space. It was originated by Whitehead in his 1950 paper "Simple homotopy type".
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https://en.wikipedia.org/wiki/Simple_homotopy_theory
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In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle θ {\displaystyle \theta } , the sine and cosine functions are denoted simply as sin θ {\displaystyle \sin \theta } and cos θ {\displaystyle \cos \theta } .More generally, the definitions of sine and cosine can be extended to any real value in terms of the lengths of certain line segments in a unit circle.
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https://en.wikipedia.org/wiki/Sine_and_cosine
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More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. The sine and cosine functions are commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. They can be traced to the jyā and koṭi-jyā functions used in Indian astronomy during the Gupta period.
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https://en.wikipedia.org/wiki/Sine_and_cosine
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In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rn and Tn through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in harmonic analysis are the harmonic conjugation operator on the circle, the Hilbert transform on the circle and the real line, the Beurling transform in the complex plane and the Riesz transforms in Euclidean space. The continuity of these operators on L2 is evident because the Fourier transform converts them into multiplication operators.
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https://en.wikipedia.org/wiki/Singular_integral_operators_of_convolution_type
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Continuity on Lp spaces was first established by Marcel Riesz. The classical techniques include the use of Poisson integrals, interpolation theory and the Hardy–Littlewood maximal function. For more general operators, fundamental new techniques, introduced by Alberto Calderón and Antoni Zygmund in 1952, were developed by a number of authors to give general criteria for continuity on Lp spaces. This article explains the theory for the classical operators and sketches the subsequent general theory.
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https://en.wikipedia.org/wiki/Singular_integral_operators_of_convolution_type
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In mathematics, singular integral operators on closed curves arise in problems in analysis, in particular complex analysis and harmonic analysis. The two main singular integral operators, the Hilbert transform and the Cauchy transform, can be defined for any smooth Jordan curve in the complex plane and are related by a simple algebraic formula. In the special case of Fourier series for the unit circle, the operators become the classical Cauchy transform, the orthogonal projection onto Hardy space, and the Hilbert transform a real orthogonal linear complex structure. In general the Cauchy transform is a non-self-adjoint idempotent and the Hilbert transform a non-orthogonal complex structure.
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https://en.wikipedia.org/wiki/Singular_integral_operators_on_closed_curves
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The range of the Cauchy transform is the Hardy space of the bounded region enclosed by the Jordan curve. The theory for the original curve can be deduced from that of the unit circle, where, because of rotational symmetry, both operators are classical singular integral operators of convolution type. The Hilbert transform satisfies the jump relations of Plemelj and Sokhotski, which express the original function as the difference between the boundary values of holomorphic functions on the region and its complement. Singular integral operators have been studied on various classes of functions, including Hölder spaces, Lp spaces and Sobolev spaces. In the case of L2 spaces—the case treated in detail below—other operators associated with the closed curve, such as the Szegő projection onto Hardy space and the Neumann–Poincaré operator, can be expressed in terms of the Cauchy transform and its adjoint.
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https://en.wikipedia.org/wiki/Singular_integral_operators_on_closed_curves
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In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator T ( f ) ( x ) = ∫ K ( x , y ) f ( y ) d y , {\displaystyle T(f)(x)=\int K(x,y)f(y)\,dy,} whose kernel function K: Rn×Rn → R is singular along the diagonal x = y. Specifically, the singularity is such that |K(x, y)| is of size |x − y|−n asymptotically as |x − y| → 0. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over |y − x| > ε as ε → 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on Lp(Rn).
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https://en.wikipedia.org/wiki/Singular_integral_operator
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In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. In some places the flat string will cross itself in an approximate "X" shape.
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https://en.wikipedia.org/wiki/Singularity_theory
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The points on the floor where it does this are one kind of singularity, the double point: one bit of the floor corresponds to more than one bit of string. Perhaps the string will also touch itself without crossing, like an underlined "U". This is another kind of singularity.
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https://en.wikipedia.org/wiki/Singularity_theory
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Unlike the double point, it is not stable, in the sense that a small push will lift the bottom of the "U" away from the "underline". Vladimir Arnold defines the main goal of singularity theory as describing how objects depend on parameters, particularly in cases where the properties undergo sudden change under a small variation of the parameters. These situations are called perestroika (Russian: перестройка), bifurcations or catastrophes. Classifying the types of changes and characterizing sets of parameters which give rise to these changes are some of the main mathematical goals. Singularities can occur in a wide range of mathematical objects, from matrices depending on parameters to wavefronts.
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https://en.wikipedia.org/wiki/Singularity_theory
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In mathematics, size theory studies the properties of topological spaces endowed with R k {\displaystyle \mathbb {R} ^{k}} -valued functions, with respect to the change of these functions. More formally, the subject of size theory is the study of the natural pseudodistance between size pairs. A survey of size theory can be found in .
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https://en.wikipedia.org/wiki/Size_theory
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In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, as demonstrated with the counterexample below. One of the most important applications of smooth functions with compact support is the construction of so-called mollifiers, which are important in theories of generalized functions, such as Laurent Schwartz's theory of distributions.
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https://en.wikipedia.org/wiki/Non-analytic_smooth_function
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The existence of smooth but non-analytic functions represents one of the main differences between differential geometry and analytic geometry. In terms of sheaf theory, this difference can be stated as follows: the sheaf of differentiable functions on a differentiable manifold is fine, in contrast with the analytic case. The functions below are generally used to build up partitions of unity on differentiable manifolds.
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https://en.wikipedia.org/wiki/Non-analytic_smooth_function
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In mathematics, sociable numbers are numbers whose aliquot sums form a periodic sequence. They are generalizations of the concepts of perfect numbers and amicable numbers. The first two sociable sequences, or sociable chains, were discovered and named by the Belgian mathematician Paul Poulet in 1918. In a sociable sequence, each number is the sum of the proper divisors of the preceding number, i.e., the sum excludes the preceding number itself.
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https://en.wikipedia.org/wiki/Sociable_number
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For the sequence to be sociable, the sequence must be cyclic and return to its starting point. The period of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle. If the period of the sequence is 1, the number is a sociable number of order 1, or a perfect number—for example, the proper divisors of 6 are 1, 2, and 3, whose sum is again 6. A pair of amicable numbers is a set of sociable numbers of order 2. There are no known sociable numbers of order 3, and searches for them have been made up to 5 × 10 7 {\displaystyle 5\times 10^{7}} as of 1970.It is an open question whether all numbers end up at either a sociable number or at a prime (and hence 1), or, equivalently, whether there exist numbers whose aliquot sequence never terminates, and hence grows without bound.
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https://en.wikipedia.org/wiki/Sociable_number
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In mathematics, solid partitions are natural generalizations of partitions and plane partitions defined by Percy Alexander MacMahon. A solid partition of n {\displaystyle n} is a three-dimensional array of non-negative integers n i , j , k {\displaystyle n_{i,j,k}} (with indices i , j , k ≥ 1 {\displaystyle i,j,k\geq 1} ) such that ∑ i , j , k n i , j , k = n {\displaystyle \sum _{i,j,k}n_{i,j,k}=n} and n i + 1 , j , k ≤ n i , j , k , n i , j + 1 , k ≤ n i , j , k and n i , j , k + 1 ≤ n i , j , k {\displaystyle n_{i+1,j,k}\leq n_{i,j,k},\quad n_{i,j+1,k}\leq n_{i,j,k}\quad {\text{and}}\quad n_{i,j,k+1}\leq n_{i,j,k}} for all i , j and k . {\displaystyle i,j{\text{ and }}k.}
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https://en.wikipedia.org/wiki/Solid_partition
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Let p 3 ( n ) {\displaystyle p_{3}(n)} denote the number of solid partitions of n {\displaystyle n} . As the definition of solid partitions involves three-dimensional arrays of numbers, they are also called three-dimensional partitions in notation where plane partitions are two-dimensional partitions and partitions are one-dimensional partitions. Solid partitions and their higher-dimensional generalizations are discussed in the book by Andrews.
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https://en.wikipedia.org/wiki/Solid_partition
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In mathematics, some boundary value problems can be solved using the methods of stochastic analysis. Perhaps the most celebrated example is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. However, it turns out that for a large class of semi-elliptic second-order partial differential equations the associated Dirichlet boundary value problem can be solved using an Itō process that solves an associated stochastic differential equation.
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https://en.wikipedia.org/wiki/Stochastic_processes_and_boundary_value_problems
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In mathematics, some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a large theory of special functions which developed out of statistics and mathematical physics. A modern, abstract point of view contrasts large function spaces, which are infinite-dimensional and within which most functions are 'anonymous', with special functions picked out by properties such as symmetry, or relationship to harmonic analysis and group representations. See also List of types of functions
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https://en.wikipedia.org/wiki/List_of_mathematical_functions
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In mathematics, spaces of non-positive curvature occur in many contexts and form a generalization of hyperbolic geometry. In the category of Riemannian manifolds, one can consider the sectional curvature of the manifold and require that this curvature be everywhere less than or equal to zero. The notion of curvature extends to the category of geodesic metric spaces, where one can use comparison triangles to quantify the curvature of a space; in this context, non-positively curved spaces are known as (locally) CAT(0) spaces.
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https://en.wikipedia.org/wiki/Non-positive_curvature
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That is d h ∘ d v + d v ∘ d h = 0. {\displaystyle d_{h}\circ d_{v}+d_{v}\circ d_{h}=0.} This eases the definition of Total Complexes. By setting f p , q = ( − 1 ) p d p , q v: C p , q → C p , q − 1 {\displaystyle f_{p,q}=(-1)^{p}d_{p,q}^{v}\colon C_{p,q}\to C_{p,q-1}} , we can switch between having commutativity and anticommutativity. If the commutative definition is used, this alternating sign will have to show up in the definition of Total Complexes.
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https://en.wikipedia.org/wiki/Double_complex
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In mathematics, specifically Riemannian geometry, Synge's theorem is a classical result relating the curvature of a Riemannian manifold to its topology. It is named for John Lighton Synge, who proved it in 1936.
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https://en.wikipedia.org/wiki/Synge_theorem
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In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant. This may have different meanings. We say that (G, ≤) is a: left-ordered group if ≤ is left-invariant, that is a ≤ b implies ca ≤ cb for all a, b, c in G, right-ordered group if ≤ is right-invariant, that is a ≤ b implies ac ≤ bc for all a, b, c in G, bi-ordered group if ≤ is bi-invariant, that is it is both left- and right-invariant.A group G is said to be left-orderable (or right-orderable, or bi-orderable) if there exists a left- (or right-, or bi-) invariant order on G. A simple necessary condition for a group to be left-orderable is to have no elements of finite order; however this is not a sufficient condition. It is equivalent for a group to be left- or right-orderable; however there exist left-orderable groups which are not bi-orderable.
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https://en.wikipedia.org/wiki/Totally_ordered_abelian_group
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In mathematics, specifically abstract algebra, a square class of a field F {\displaystyle F} is an element of the square class group, the quotient group F × / F × 2 {\displaystyle F^{\times }/F^{\times 2}} of the multiplicative group of nonzero elements in the field modulo the square elements of the field. Each square class is a subset of the nonzero elements (a coset of the multiplicative group) consisting of the elements of the form xy2 where x is some particular fixed element and y ranges over all nonzero field elements.For instance, if F = R {\displaystyle F=\mathbb {R} } , the field of real numbers, then F × {\displaystyle F^{\times }} is just the group of all nonzero real numbers (with the multiplication operation) and F × 2 {\displaystyle F^{\times 2}} is the subgroup of positive numbers (as every positive number has a real square root). The quotient of these two groups is a group with two elements, corresponding to two cosets: the set of positive numbers and the set of negative numbers. Thus, the real numbers have two square classes, the positive numbers and the negative numbers.Square classes are frequently studied in relation to the theory of quadratic forms.
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https://en.wikipedia.org/wiki/Square_class
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The reason is that if V {\displaystyle V} is an F {\displaystyle F} -vector space and q: V → F {\displaystyle q:V\to F} is a quadratic form and v {\displaystyle v} is an element of V {\displaystyle V} such that q ( v ) = a ∈ F × {\displaystyle q(v)=a\in F^{\times }} , then for all u ∈ F × {\displaystyle u\in F^{\times }} , q ( u v ) = a u 2 {\displaystyle q(uv)=au^{2}} and thus it is sometimes more convenient to talk about the square classes which the quadratic form represents. Every element of the square class group is an involution. It follows that, if the number of square classes of a field is finite, it must be a power of two. == References ==
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https://en.wikipedia.org/wiki/Square_class
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In mathematics, specifically abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself (with left or right multiplication). Both concepts are named for Emil Artin. In the presence of the axiom of (dependent) choice, the descending chain condition becomes equivalent to the minimum condition, and so that may be used in the definition instead.
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https://en.wikipedia.org/wiki/Artinian_module
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Like Noetherian modules, Artinian modules enjoy the following heredity property: If M is an Artinian R-module, then so is any submodule and any quotient of M.The converse also holds: If M is any R-module and N any Artinian submodule such that M/N is Artinian, then M is Artinian.As a consequence, any finitely-generated module over an Artinian ring is Artinian. Since an Artinian ring is also a Noetherian ring, and finitely-generated modules over a Noetherian ring are Noetherian, it is true that for an Artinian ring R, any finitely-generated R-module is both Noetherian and Artinian, and is said to be of finite length. It also follows that any finitely generated Artinian module is Noetherian even without the assumption of R being Artinian. However, if R is not Artinian and M is not finitely-generated, there are counterexamples.
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https://en.wikipedia.org/wiki/Artinian_module
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In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition. Precisely, a ring is left Artinian if it satisfies the descending chain condition on left ideals, right Artinian if it satisfies the descending chain condition on right ideals, and Artinian or two-sided Artinian if it is both left and right Artinian.
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https://en.wikipedia.org/wiki/Artinian_ring
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For commutative rings the left and right definitions coincide, but in general they are distinct from each other. The Wedderburn–Artin theorem characterizes every simple Artinian ring as a ring of matrices over a division ring.
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https://en.wikipedia.org/wiki/Artinian_ring
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This implies that a simple ring is left Artinian if and only if it is right Artinian. The same definition and terminology can be applied to modules, with ideals replaced by submodules. Although the descending chain condition appears dual to the ascending chain condition, in rings it is in fact the stronger condition.
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https://en.wikipedia.org/wiki/Artinian_ring
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Specifically, a consequence of the Akizuki–Hopkins–Levitzki theorem is that a left (resp. right) Artinian ring is automatically a left (resp. right) Noetherian ring. This is not true for general modules; that is, an Artinian module need not be a Noetherian module.
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https://en.wikipedia.org/wiki/Artinian_ring
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In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.
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https://en.wikipedia.org/wiki/Associate_(ring_theory)
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"Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity. Noncommutative integral domains are sometimes admitted. This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "domain" for the general case including noncommutative rings. Some sources, notably Lang, use the term entire ring for integral domain.Some specific kinds of integral domains are given with the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields
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https://en.wikipedia.org/wiki/Associate_(ring_theory)
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In mathematics, specifically abstract algebra, if ( G , + ) {\displaystyle (G,+)} is an (abelian) group with identity element e {\displaystyle e} then ν: G → R {\displaystyle \nu \colon G\to \mathbb {R} } is said to be a norm on ( G , + ) {\displaystyle (G,+)} if: Positive definiteness: ν ( g ) > 0 for all g ≠ e and ν ( e ) = 0 {\displaystyle \nu (g)>0{\text{ for all }}g\neq e{\text{ and }}\nu (e)=0} , Subadditivity: ν ( g + h ) ≤ ν ( g ) + ν ( h ) {\displaystyle \nu (g+h)\leq \nu (g)+\nu (h)} , Inversion (Symmetry): ν ( − g ) = ν ( g ) for all g ∈ G {\displaystyle \nu (-g)=\nu (g){\text{ for all }}g\in G} .An alternative, stronger definition of a norm on ( G , + ) {\displaystyle (G,+)} requires ν ( g ) > 0 for all g ≠ e {\displaystyle \nu (g)>0{\text{ for all }}g\neq e} , ν ( g + h ) ≤ ν ( g ) + ν ( h ) {\displaystyle \nu (g+h)\leq \nu (g)+\nu (h)} , ν ( m g ) = | m | ν ( g ) for all m ∈ Z {\displaystyle \nu (mg)=|m|\,\nu (g){\text{ for all }}m\in \mathbb {Z} } .The norm ν {\displaystyle \nu } is discrete if there is some real number ρ > 0 {\displaystyle \rho >0} such that ν ( g ) > ρ {\displaystyle \nu (g)>\rho } whenever g ≠ 0 {\displaystyle g\neq 0} .
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https://en.wikipedia.org/wiki/Norm_(abelian_group)
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In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.
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https://en.wikipedia.org/wiki/First_ring_isomorphism_theorem
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