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In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, see Relative concreteness below). This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example the category of topological spaces and the category of groups, and trivially also the category of sets itself. On the other hand, the homotopy category of topological spaces is not concretizable, i.e. it does not admit a faithful functor to the category of sets. A concrete category, when defined without reference to the notion of a category, consists of a class of objects, each equipped with an underlying set; and for any two objects A and B a set of functions, called morphisms, from the underlying set of A to the underlying set of B. Furthermore, for every object A, the identity function on the underlying set of A must be a morphism from A to A, and the composition of a morphism from A to B followed by a morphism from B to C must be a morphism from A to C.
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https://en.wikipedia.org/wiki/Concrete_category
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In mathematics, a condensation point p of a subset S of a topological space is any point p such that every neighborhood of p contains uncountably many points of S. Thus "condensation point" is synonymous with " ℵ 1 {\displaystyle \aleph _{1}} -accumulation point".
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https://en.wikipedia.org/wiki/Condensation_point
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In mathematics, a conference matrix (also called a C-matrix) is a square matrix C with 0 on the diagonal and +1 and −1 off the diagonal, such that CTC is a multiple of the identity matrix I. Thus, if the matrix has order n, CTC = (n−1)I. Some authors use a more general definition, which requires there to be a single 0 in each row and column but not necessarily on the diagonal.Conference matrices first arose in connection with a problem in telephony. They were first described by Vitold Belevitch, who also gave them their name. Belevitch was interested in constructing ideal telephone conference networks from ideal transformers and discovered that such networks were represented by conference matrices, hence the name.
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https://en.wikipedia.org/wiki/Conference_matrix
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Other applications are in statistics, and another is in elliptic geometry.For n > 1, there are two kinds of conference matrix. Let us normalize C by, first (if the more general definition is used), rearranging the rows so that all the zeros are on the diagonal, and then negating any row or column whose first entry is negative. (These operations do not change whether a matrix is a conference matrix.)
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https://en.wikipedia.org/wiki/Conference_matrix
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Thus, a normalized conference matrix has all 1's in its first row and column, except for a 0 in the top left corner, and is 0 on the diagonal. Let S be the matrix that remains when the first row and column of C are removed. Then either n is evenly even (a multiple of 4), and S is antisymmetric (as is the normalized C if its first row is negated), or n is oddly even (congruent to 2 modulo 4) and S is symmetric (as is the normalized C).
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https://en.wikipedia.org/wiki/Conference_matrix
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In mathematics, a configuration space is a construction closely related to state spaces or phase spaces in physics. In physics, these are used to describe the state of a whole system as a single point in a high-dimensional space. In mathematics, they are used to describe assignments of a collection of points to positions in a topological space. More specifically, configuration spaces in mathematics are particular examples of configuration spaces in physics in the particular case of several non-colliding particles.
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https://en.wikipedia.org/wiki/Configuration_space_(mathematics)
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In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. The term confluent refers to the merging of singular points of families of differential equations; confluere is Latin for "to flow together". There are several common standard forms of confluent hypergeometric functions: Kummer's (confluent hypergeometric) function M(a, b, z), introduced by Kummer (1837), is a solution to Kummer's differential equation. This is also known as the confluent hypergeometric function of the first kind.
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https://en.wikipedia.org/wiki/Confluent_hypergeometric_series
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There is a different and unrelated Kummer's function bearing the same name. Tricomi's (confluent hypergeometric) function U(a, b, z) introduced by Francesco Tricomi (1947), sometimes denoted by Ψ(a; b; z), is another solution to Kummer's equation.
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https://en.wikipedia.org/wiki/Confluent_hypergeometric_series
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This is also known as the confluent hypergeometric function of the second kind. Whittaker functions (for Edmund Taylor Whittaker) are solutions to Whittaker's equation. Coulomb wave functions are solutions to the Coulomb wave equation.The Kummer functions, Whittaker functions, and Coulomb wave functions are essentially the same, and differ from each other only by elementary functions and change of variables.
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https://en.wikipedia.org/wiki/Confluent_hypergeometric_series
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In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U {\displaystyle U} and V {\displaystyle V} be open subsets of R n {\displaystyle \mathbb {R} ^{n}} . A function f: U → V {\displaystyle f:U\to V} is called conformal (or angle-preserving) at a point u 0 ∈ U {\displaystyle u_{0}\in U} if it preserves angles between directed curves through u 0 {\displaystyle u_{0}} , as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature.
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https://en.wikipedia.org/wiki/Conformal_transformation
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The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix.For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally invertible complex analytic functions. In three and higher dimensions, Liouville's theorem sharply limits the conformal mappings to a few types. The notion of conformality generalizes in a natural way to maps between Riemannian or semi-Riemannian manifolds.
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https://en.wikipedia.org/wiki/Conformal_transformation
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In mathematics, a congruence is an equivalence relation on the integers. The following sections list important or interesting prime-related congruences.
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https://en.wikipedia.org/wiki/Table_of_congruences
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In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example is the subgroup of invertible 2 × 2 integer matrices of determinant 1 in which the off-diagonal entries are even. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer. The existence of congruence subgroups in an arithmetic group provides it with a wealth of subgroups, in particular it shows that the group is residually finite. An important question regarding the algebraic structure of arithmetic groups is the congruence subgroup problem, which asks whether all subgroups of finite index are essentially congruence subgroups. Congruence subgroups of 2×2 matrices are fundamental objects in the classical theory of modular forms; the modern theory of automorphic forms makes a similar use of congruence subgroups in more general arithmetic groups.
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https://en.wikipedia.org/wiki/Congruence_subgroup
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In mathematics, a conical spiral, also known as a conical helix, is a space curve on a right circular cone, whose floor projection is a plane spiral. If the floor projection is a logarithmic spiral, it is called conchospiral (from conch).
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https://en.wikipedia.org/wiki/Conic_helix
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In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them.
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https://en.wikipedia.org/wiki/Mathematical_conjecture
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In mathematics, a connector is a map which can be defined for a linear connection and used to define the covariant derivative on a vector bundle from the linear connection.
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https://en.wikipedia.org/wiki/Connector_(mathematics)
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In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink over time. Precisely speaking, they are those dynamical systems that have a null wandering set: under time evolution, no portion of the phase space ever "wanders away", never to be returned to or revisited. Alternately, conservative systems are those to which the Poincaré recurrence theorem applies. An important special case of conservative systems are the measure-preserving dynamical systems.
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https://en.wikipedia.org/wiki/Conservative_system
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In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function y(x) = 4 is a constant function because the value of y(x) is 4 regardless of the input value x (see image).
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https://en.wikipedia.org/wiki/Constant_map
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In mathematics, a constant term (sometimes referred to as a free term) is a term in an algebraic expression that does not contain any variables and therefore is constant. For example, in the quadratic polynomial x 2 + 2 x + 3 , {\displaystyle x^{2}+2x+3,\ } the 3 is a constant term.After like terms are combined, an algebraic expression will have at most one constant term. Thus, it is common to speak of the quadratic polynomial a x 2 + b x + c , {\displaystyle ax^{2}+bx+c,\ } where x {\displaystyle x} is the variable, as having a constant term of c . {\displaystyle c.}
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https://en.wikipedia.org/wiki/Constant_term
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If the constant term is 0, then it will conventionally be omitted when the quadratic is written out. Any polynomial written in standard form has a unique constant term, which can be considered a coefficient of x 0 .
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https://en.wikipedia.org/wiki/Constant_term
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{\displaystyle x^{0}.} In particular, the constant term will always be the lowest degree term of the polynomial. This also applies to multivariate polynomials.
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https://en.wikipedia.org/wiki/Constant_term
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For example, the polynomial x 2 + 2 x y + y 2 − 2 x + 2 y − 4 {\displaystyle x^{2}+2xy+y^{2}-2x+2y-4\ } has a constant term of −4, which can be considered to be the coefficient of x 0 y 0 , {\displaystyle x^{0}y^{0},} where the variables are eliminated by being exponentiated to 0 (any non-zero number exponentiated to 0 becomes 1). For any polynomial, the constant term can be obtained by substituting in 0 instead of each variable; thus, eliminating each variable. The concept of exponentiation to 0 can be applied to power series and other types of series, for example in this power series: a 0 + a 1 x + a 2 x 2 + a 3 x 3 + ⋯ , {\displaystyle a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots ,} a 0 {\displaystyle a_{0}} is the constant term.
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https://en.wikipedia.org/wiki/Constant_term
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In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy. There are several types of constraints—primarily equality constraints, inequality constraints, and integer constraints. The set of candidate solutions that satisfy all constraints is called the feasible set.
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https://en.wikipedia.org/wiki/Non-binding_constraint
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In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinitely many constructible polygons, but only 31 with an odd number of sides are known.
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https://en.wikipedia.org/wiki/Constructible_polygon
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In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space X, such that X is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It has its origins in algebraic geometry, where in étale cohomology constructible sheaves are defined in a similar way (Artin, Grothendieck & Verdier 1972, Exposé IX § 2). For the derived category of constructible sheaves, see a section in ℓ-adic sheaf. The finiteness theorem in étale cohomology states that the higher direct images of a constructible sheaf are constructible.
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https://en.wikipedia.org/wiki/Constructible_sheaf
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In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof. A constructive proof may also refer to the stronger concept of a proof that is valid in constructive mathematics.
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https://en.wikipedia.org/wiki/Non-constructive_proof
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Constructivism is a mathematical philosophy that rejects all proof methods that involve the existence of objects that are not explicitly built. This excludes, in particular, the use of the law of the excluded middle, the axiom of infinity, and the axiom of choice, and induces a different meaning for some terminology (for example, the term "or" has a stronger meaning in constructive mathematics than in classical).Some non-constructive proofs show that if a certain proposition is false, a contradiction ensues; consequently the proposition must be true (proof by contradiction). However, the principle of explosion (ex falso quodlibet) has been accepted in some varieties of constructive mathematics, including intuitionism. Constructive proofs can be seen as defining certified mathematical algorithms: this idea is explored in the Brouwer–Heyting–Kolmogorov interpretation of constructive logic, the Curry–Howard correspondence between proofs and programs, and such logical systems as Per Martin-Löf's intuitionistic type theory, and Thierry Coquand and Gérard Huet's calculus of constructions.
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https://en.wikipedia.org/wiki/Non-constructive_proof
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In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive.
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https://en.wikipedia.org/wiki/Continued_fraction
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The integers a i {\displaystyle a_{i}} are called the coefficients or terms of the continued fraction.It is generally assumed that the numerator of all of the fractions is 1. If arbitrary values and/or functions are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a generalized continued fraction. When it is necessary to distinguish the first form from generalized continued fractions, the former may be called a simple or regular continued fraction, or said to be in canonical form.
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https://en.wikipedia.org/wiki/Continued_fraction
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Continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number p {\displaystyle p} / q {\displaystyle q} has two closely related expressions as a finite continued fraction, whose coefficients ai can be determined by applying the Euclidean algorithm to ( p , q ) {\displaystyle (p,q)} . The numerical value of an infinite continued fraction is irrational; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions.
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https://en.wikipedia.org/wiki/Continued_fraction
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Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number α {\displaystyle \alpha } is the value of a unique infinite regular continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values α {\displaystyle \alpha } and 1.
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https://en.wikipedia.org/wiki/Continued_fraction
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This way of expressing real numbers (rational and irrational) is called their continued fraction representation. The term continued fraction may also refer to representations of rational functions, arising in their analytic theory. For this use of the term, see Padé approximation and Chebyshev rational functions.
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https://en.wikipedia.org/wiki/Continued_fraction
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In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument.
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https://en.wikipedia.org/wiki/Continuous_(topology)
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A discontinuous function is a function that is not continuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.
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https://en.wikipedia.org/wiki/Continuous_(topology)
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Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their definition is the basis of topology.
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https://en.wikipedia.org/wiki/Continuous_(topology)
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A stronger form of continuity is uniform continuity. In order theory, especially in domain theory, a related concept of continuity is Scott continuity. As an example, the function H(t) denoting the height of a growing flower at time t would be considered continuous. In contrast, the function M(t) denoting the amount of money in a bank account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.
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https://en.wikipedia.org/wiki/Continuous_(topology)
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In mathematics, a continuous module is a module M such that every submodule of M is essential in a direct summand and every submodule of M isomorphic to a direct summand is itself a direct summand. The endomorphism ring of a continuous module is a clean ring. == References ==
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https://en.wikipedia.org/wiki/Continuous_module
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In mathematics, a continuous-time random walk (CTRW) is a generalization of a random walk where the wandering particle waits for a random time between jumps. It is a stochastic jump process with arbitrary distributions of jump lengths and waiting times. More generally it can be seen to be a special case of a Markov renewal process.
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https://en.wikipedia.org/wiki/Continuous-time_random_walk
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In mathematics, a continuum structure function (CSF) is defined by Laurence Baxter as a nondecreasing mapping from the unit hypercube to the unit interval. It is used by Baxter to help in the Mathematical modelling of the level of performance of a system in terms of the performance levels of its components.
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https://en.wikipedia.org/wiki/Continuum_structure_function
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In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some real number 0 ≤ k < 1 {\displaystyle 0\leq k<1} such that for all x and y in M, d ( f ( x ) , f ( y ) ) ≤ k d ( x , y ) . {\displaystyle d(f(x),f(y))\leq k\,d(x,y).} The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps. If the above condition is instead satisfied for k ≤ 1, then the mapping is said to be a non-expansive map.
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https://en.wikipedia.org/wiki/Contraction_mapping
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More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if (M, d) and (N, d') are two metric spaces, then f: M → N {\displaystyle f:M\rightarrow N} is a contractive mapping if there is a constant 0 ≤ k < 1 {\displaystyle 0\leq k<1} such that d ′ ( f ( x ) , f ( y ) ) ≤ k d ( x , y ) {\displaystyle d'(f(x),f(y))\leq k\,d(x,y)} for all x and y in M. Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant k is no longer necessarily less than 1). A contraction mapping has at most one fixed point.
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https://en.wikipedia.org/wiki/Contraction_mapping
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Moreover, the Banach fixed-point theorem states that every contraction mapping on a non-empty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point. This concept is very useful for iterated function systems where contraction mappings are often used. Banach's fixed-point theorem is also applied in proving the existence of solutions of ordinary differential equations, and is used in one proof of the inverse function theorem.Contraction mappings play an important role in dynamic programming problems.
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https://en.wikipedia.org/wiki/Contraction_mapping
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In mathematics, a contraharmonic mean is a function complementary to the harmonic mean. The contraharmonic mean is a special case of the Lehmer mean, L p {\displaystyle L_{p}} , where p = 2.
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https://en.wikipedia.org/wiki/Contraharmonic_mean
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In mathematics, a convergence group or a discrete convergence group is a group Γ {\displaystyle \Gamma } acting by homeomorphisms on a compact metrizable space M {\displaystyle M} in a way that generalizes the properties of the action of Kleinian group by Möbius transformations on the ideal boundary S 2 {\displaystyle \mathbb {S} ^{2}} of the hyperbolic 3-space H 3 {\displaystyle \mathbb {H} ^{3}} . The notion of a convergence group was introduced by Gehring and Martin (1987) and has since found wide applications in geometric topology, quasiconformal analysis, and geometric group theory.
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https://en.wikipedia.org/wiki/Convergence_group
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In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a convergence that satisfies certain properties relating elements of X with the family of filters on X. Convergence spaces generalize the notions of convergence that are found in point-set topology, including metric convergence and uniform convergence. Every topological space gives rise to a canonical convergence but there are convergences, known as non-topological convergences, that do not arise from any topological space. Examples of convergences that are in general non-topological include convergence in measure and almost everywhere convergence.
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https://en.wikipedia.org/wiki/Convergence_space
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Many topological properties have generalizations to convergence spaces. Besides its ability to describe notions of convergence that topologies are unable to, the category of convergence spaces has an important categorical property that the category of topological spaces lacks. The category of topological spaces is not an exponential category (or equivalently, it is not Cartesian closed) although it is contained in the exponential category of pseudotopological spaces, which is itself a subcategory of the (also exponential) category of convergence spaces.
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https://en.wikipedia.org/wiki/Convergence_space
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In mathematics, a convex body in n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is a compact convex set with non-empty interior. A convex body K {\displaystyle K} is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x {\displaystyle x} lies in K {\displaystyle K} if and only if its antipode, − x {\displaystyle -x} also lies in K . {\displaystyle K.}
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https://en.wikipedia.org/wiki/Convex_body
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Symmetric convex bodies are in a one-to-one correspondence with the unit balls of norms on R n . {\displaystyle \mathbb {R} ^{n}.} Important examples of convex bodies are the Euclidean ball, the hypercube and the cross-polytope.
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https://en.wikipedia.org/wiki/Convex_body
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In mathematics, a convex space (or barycentric algebra) is a space in which it is possible to take convex combinations of any sets of points.
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https://en.wikipedia.org/wiki/Convex_space
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In mathematics, a coordinate-induced basis is a basis for the tangent space or cotangent space of a manifold that is induced by a certain coordinate system. Given the coordinate system x a {\displaystyle x^{a}} , the coordinate-induced basis e a {\displaystyle e_{a}} of the tangent space is given by e a = ∂ ∂ x a {\displaystyle e_{a}={\frac {\partial }{\partial x^{a}}}} and the dual basis ω a {\displaystyle \omega ^{a}} of the cotangent space is ω a = d x a . {\displaystyle \omega ^{a}=dx^{a}.\,}
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https://en.wikipedia.org/wiki/Coordinate-induced_basis
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In mathematics, a corestriction of a function is a notion analogous to the notion of a restriction of a function. The duality prefix co- here denotes that while the restriction changes the domain to a subset, the corestriction changes the codomain to a subset. However, the notions are not categorically dual. Given any subset S ⊂ A , {\displaystyle S\subset A,} we can consider the corresponding inclusion of sets i S: S ↪ A {\displaystyle i_{S}:S\hookrightarrow A} as a function.
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https://en.wikipedia.org/wiki/Corestriction
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Then for any function f: A → B {\displaystyle f:A\to B} , the restriction f | S: S → B {\displaystyle f|_{S}:S\to B} of a function f {\displaystyle f} onto S {\displaystyle S} can be defined as the composition f | S = f ∘ i S {\displaystyle f|_{S}=f\circ i_{S}} . Analogously, for an inclusion i T: T ↪ B {\displaystyle i_{T}:T\hookrightarrow B} the corestriction f | T: A → T {\displaystyle f|^{T}:A\to T} of f {\displaystyle f} onto T {\displaystyle T} is the unique function f | T {\displaystyle f|^{T}} such that there is a decomposition f = i T ∘ f | T {\displaystyle f=i_{T}\circ f|^{T}} . The corestriction exists if and only if T {\displaystyle T} contains the image of f {\displaystyle f} .
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https://en.wikipedia.org/wiki/Corestriction
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In particular, the corestriction onto the image always exists and it is sometimes simply called the corestriction of f {\displaystyle f} . More generally, one can consider corestriction of a morphism in general categories with images.
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https://en.wikipedia.org/wiki/Corestriction
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The term is well known in category theory, while rarely used in print.Andreotti introduces the above notion under the name coastriction, while the name corestriction reserves to the notion categorically dual to the notion of a restriction. Namely, if p U: B → U {\displaystyle p^{U}:B\to U} is a surjection of sets (that is a quotient map) then Andreotti considers the composition p U ∘ f: A → U {\displaystyle p^{U}\circ f:A\to U} , which surely always exists. == References ==
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https://en.wikipedia.org/wiki/Corestriction
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In mathematics, a corollary is a theorem connected by a short proof to an existing theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective. More formally, proposition B is a corollary of proposition A, if B can be readily deduced from A or is self-evident from its proof. In many cases, a corollary corresponds to a special case of a larger theorem, which makes the theorem easier to use and apply, even though its importance is generally considered to be secondary to that of the theorem. In particular, B is unlikely to be termed a corollary if its mathematical consequences are as significant as those of A. A corollary might have a proof that explains its derivation, even though such a derivation might be considered rather self-evident in some occasions (e.g., the Pythagorean theorem as a corollary of law of cosines).
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https://en.wikipedia.org/wiki/Corollary
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In mathematics, a covering group of a topological group H is a covering space G of H such that G is a topological group and the covering map p: G → H is a continuous group homomorphism. The map p is called the covering homomorphism. A frequently occurring case is a double covering group, a topological double cover in which H has index 2 in G; examples include the spin groups, pin groups, and metaplectic groups. Roughly explained, saying that for example the metaplectic group Mp2n is a double cover of the symplectic group Sp2n means that there are always two elements in the metaplectic group representing one element in the symplectic group.
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https://en.wikipedia.org/wiki/Double_covering_group
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In mathematics, a covering number is the number of spherical balls of a given size needed to completely cover a given space, with possible overlaps. Two related concepts are the packing number, the number of disjoint balls that fit in a space, and the metric entropy, the number of points that fit in a space when constrained to lie at some fixed minimum distance apart.
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https://en.wikipedia.org/wiki/Covering_number
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In mathematics, a covering set for a sequence of integers refers to a set of prime numbers such that every term in the sequence is divisible by at least one member of the set. The term "covering set" is used only in conjunction with sequences possessing exponential growth.
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https://en.wikipedia.org/wiki/Covering_set
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In mathematics, a covering system (also called a complete residue system) is a collection { a 1 ( mod n 1 ) , … , a k ( mod n k ) } {\displaystyle \{a_{1}{\pmod {n_{1}}},\ \ldots ,\ a_{k}{\pmod {n_{k}}}\}} of finitely many residue classes a i ( mod n i ) = { a i + n i x: x ∈ Z } , {\displaystyle a_{i}{\pmod {n_{i}}}=\{a_{i}+n_{i}x:\ x\in \mathbb {Z} \},} whose union contains every integer.
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https://en.wikipedia.org/wiki/Covering_system
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In mathematics, a credal set is a set of probability distributions or, more generally, a set of (possibly only finitely additive) probability measures. A credal set is often assumed or constructed to be a closed convex set. It is intended to express uncertainty or doubt about the probability model that should be used, or to convey the beliefs of a Bayesian agent about the possible states of the world.If a credal set K ( X ) {\displaystyle K(X)} is closed and convex, then, by the Krein–Milman theorem, it can be equivalently described by its extreme points e x t {\displaystyle \mathrm {ext} } . In that case, the expectation for a function f {\displaystyle f} of X {\displaystyle X} with respect to the credal set K ( X ) {\displaystyle K(X)} forms a closed interval , E ¯ ] {\displaystyle ,{\overline {E}}]} , whose lower bound is called the lower prevision of f {\displaystyle f} , and whose upper bound is called the upper prevision of f {\displaystyle f}: E _ = min μ ∈ K ( X ) ∫ f d μ = min μ ∈ e x t ∫ f d μ {\displaystyle {\underline {E}}=\min _{\mu \in K(X)}\int f\,d\mu =\min _{\mu \in \mathrm {ext} }\int f\,d\mu } where μ {\displaystyle \mu } denotes a probability measure, and with a similar expression for E ¯ {\displaystyle {\overline {E}}} (just replace min {\displaystyle \min } by max {\displaystyle \max } in the above expression).
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https://en.wikipedia.org/wiki/Credal_set
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If X {\displaystyle X} is a categorical variable, then the credal set K ( X ) {\displaystyle K(X)} can be considered as a set of probability mass functions over X {\displaystyle X} . If additionally K ( X ) {\displaystyle K(X)} is also closed and convex, then the lower prevision of a function f {\displaystyle f} of X {\displaystyle X} can be simply evaluated as: E _ = min p ∈ e x t ∑ x f ( x ) p ( x ) {\displaystyle {\underline {E}}=\min _{p\in \mathrm {ext} }\sum _{x}f(x)p(x)} where p {\displaystyle p} denotes a probability mass function. It is easy to see that a credal set over a Boolean variable X {\displaystyle X} cannot have more than two extreme points (because the only closed convex sets in R {\displaystyle \mathbb {R} } are closed intervals), while credal sets over variables X {\displaystyle X} that can take three or more values can have any arbitrary number of extreme points.
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https://en.wikipedia.org/wiki/Credal_set
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In mathematics, a crunode (archaic) or node is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines at the point of intersection. A crunode is also known as an ordinary double point.For a plane curve, defined as the locus of points f (x, y) = 0, where f (x, y) is a smooth function of variables x and y ranging over the real numbers, a crunode of the curve is a singularity of the function f, where both partial derivatives ∂ f ∂ x {\displaystyle {\tfrac {\partial f}{\partial x}}} and ∂ f ∂ y {\displaystyle {\tfrac {\partial f}{\partial y}}} vanish. Further the Hessian matrix of second derivatives will have both positive and negative eigenvalues.
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https://en.wikipedia.org/wiki/Crunode
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In mathematics, a cube root of a number x is a number y such that y3 = x. All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted 8 3 {\displaystyle {\sqrt{8}}} , is 2, because 23 = 8, while the other cube roots of 8 are − 1 + i 3 {\displaystyle -1+i{\sqrt {3}}} and − 1 − i 3 {\displaystyle -1-i{\sqrt {3}}} . The three cube roots of −27i are 3 i , 3 3 2 − 3 2 i , and − 3 3 2 − 3 2 i . {\displaystyle 3i,\quad {\frac {3{\sqrt {3}}}{2}}-{\frac {3}{2}}i,\quad {\text{and}}\quad -{\frac {3{\sqrt {3}}}{2}}-{\frac {3}{2}}i.}
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https://en.wikipedia.org/wiki/Cubic_root
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In some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the principal cube root, denoted with the radical sign 3 . {\displaystyle {\sqrt{~^{~}}}.} The cube root is the inverse function of the cube function if considering only real numbers, but not if considering also complex numbers: although one has always ( x 3 ) 3 = x , {\displaystyle \left({\sqrt{x}}\right)^{3}=x,} the cube of a nonzero number has more than one complex cube root and its principal cube root may not be the number that was cubed. For example, ( − 1 + i 3 ) 3 = 8 {\displaystyle (-1+i{\sqrt {3}})^{3}=8} , but 8 3 = 2. {\displaystyle {\sqrt{8}}=2.}
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https://en.wikipedia.org/wiki/Cubic_root
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In mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form. In the case of a cubic form in three variables, the zero set is a cubic plane curve. In (Delone & Faddeev 1964), Boris Delone and Dmitry Faddeev showed that binary cubic forms with integer coefficients can be used to parametrize orders in cubic fields.
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https://en.wikipedia.org/wiki/Cubic_form
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Their work was generalized in (Gan, Gross & Savin 2002, §4) to include all cubic rings (a cubic ring is a ring that is isomorphic to Z3 as a Z-module), giving a discriminant-preserving bijection between orbits of a GL(2, Z)-action on the space of integral binary cubic forms and cubic rings up to isomorphism. The classification of real cubic forms a x 3 + 3 b x 2 y + 3 c x y 2 + d y 3 {\displaystyle ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3}} is linked to the classification of umbilical points of surfaces. The equivalence classes of such cubics form a three-dimensional real projective space and the subset of parabolic forms define a surface – the umbilic torus.
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https://en.wikipedia.org/wiki/Cubic_form
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In mathematics, a cubic function is a function of the form f ( x ) = a x 3 + b x 2 + c x + d , {\displaystyle f(x)=ax^{3}+bx^{2}+cx+d,} that is, a polynomial function of degree three. In many texts, the coefficients a, b, c, and d are supposed to be real numbers, and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to complex numbers. In other cases, the coefficients may be complex numbers, and the function is a complex function that has the set of the complex numbers as its codomain, even when the domain is restricted to the real numbers. Setting f(x) = 0 produces a cubic equation of the form a x 3 + b x 2 + c x + d = 0 , {\displaystyle ax^{3}+bx^{2}+cx+d=0,} whose solutions are called roots of the function.
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https://en.wikipedia.org/wiki/Cubic_polynomial
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A cubic function with real coefficients has either one or three real roots (which may not be distinct); all odd-degree polynomials with real coefficients have at least one real root. The graph of a cubic function always has a single inflection point. It may have two critical points, a local minimum and a local maximum.
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https://en.wikipedia.org/wiki/Cubic_polynomial
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Otherwise, a cubic function is monotonic. The graph of a cubic function is symmetric with respect to its inflection point; that is, it is invariant under a rotation of a half turn around this point. Up to an affine transformation, there are only three possible graphs for cubic functions. Cubic functions are fundamental for cubic interpolation.
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https://en.wikipedia.org/wiki/Cubic_polynomial
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In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation F ( x , y , z ) = 0 {\displaystyle F(x,y,z)=0} applied to homogeneous coordinates ( x: y: z ) {\displaystyle (x:y:z)} for the projective plane; or the inhomogeneous version for the affine space determined by setting z = 1 in such an equation. Here F is a non-zero linear combination of the third-degree monomials x 3 , y 3 , z 3 , x 2 y , x 2 z , y 2 x , y 2 z , z 2 x , z 2 y , x y z {\displaystyle x^{3},y^{3},z^{3},x^{2}y,x^{2}z,y^{2}x,y^{2}z,z^{2}x,z^{2}y,xyz} These are ten in number; therefore the cubic curves form a projective space of dimension 9, over any given field K. Each point P imposes a single linear condition on F, if we ask that C pass through P. Therefore, we can find some cubic curve through any nine given points, which may be degenerate, and may not be unique, but will be unique and non-degenerate if the points are in general position; compare to two points determining a line and how five points determine a conic. If two cubics pass through a given set of nine points, then in fact a pencil of cubics does, and the points satisfy additional properties; see Cayley–Bacharach theorem. A cubic curve may have a singular point, in which case it has a parametrization in terms of a projective line.
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https://en.wikipedia.org/wiki/Darboux_cubic
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Otherwise a non-singular cubic curve is known to have nine points of inflection, over an algebraically closed field such as the complex numbers. This can be shown by taking the homogeneous version of the Hessian matrix, which defines again a cubic, and intersecting it with C; the intersections are then counted by Bézout's theorem. However, only three of these points may be real, so that the others cannot be seen in the real projective plane by drawing the curve.
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https://en.wikipedia.org/wiki/Darboux_cubic
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The nine inflection points of a non-singular cubic have the property that every line passing through two of them contains exactly three inflection points. The real points of cubic curves were studied by Isaac Newton. The real points of a non-singular projective cubic fall into one or two 'ovals'.
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https://en.wikipedia.org/wiki/Darboux_cubic
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One of these ovals crosses every real projective line, and thus is never bounded when the cubic is drawn in the Euclidean plane; it appears as one or three infinite branches, containing the three real inflection points. The other oval, if it exists, does not contain any real inflection point and appears either as an oval or as two infinite branches.
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https://en.wikipedia.org/wiki/Darboux_cubic
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Like for conic sections, a line cuts this oval at, at most, two points. A non-singular plane cubic defines an elliptic curve, over any field K for which it has a point defined. Elliptic curves are now normally studied in some variant of Weierstrass's elliptic functions, defining a quadratic extension of the field of rational functions made by extracting the square root of a cubic.
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https://en.wikipedia.org/wiki/Darboux_cubic
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This does depend on having a K-rational point, which serves as the point at infinity in Weierstrass form. There are many cubic curves that have no such point, for example when K is the rational number field. The singular points of an irreducible plane cubic curve are quite limited: one double point, or one cusp. A reducible plane cubic curve is either a conic and a line or three lines, and accordingly have two double points or a tacnode (if a conic and a line), or up to three double points or a single triple point (concurrent lines) if three lines.
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https://en.wikipedia.org/wiki/Darboux_cubic
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In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space, and so cubic surfaces are generally considered in projective 3-space P 3 {\displaystyle \mathbf {P} ^{3}} .
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https://en.wikipedia.org/wiki/Cubic_surface
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The theory also becomes more uniform by focusing on surfaces over the complex numbers rather than the real numbers; note that a complex surface has real dimension 4. A simple example is the Fermat cubic surface x 3 + y 3 + z 3 + w 3 = 0 {\displaystyle x^{3}+y^{3}+z^{3}+w^{3}=0} in P 3 {\displaystyle \mathbf {P} ^{3}} . Many properties of cubic surfaces hold more generally for del Pezzo surfaces.
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https://en.wikipedia.org/wiki/Cubic_surface
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In mathematics, a cubical complex (also called cubical set and Cartesian complex) is a set composed of points, line segments, squares, cubes, and their n-dimensional counterparts. They are used analogously to simplicial complexes and CW complexes in the computation of the homology of topological spaces.
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https://en.wikipedia.org/wiki/Cubical_complex
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In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's Elements: "The line is the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which will leave from its imaginary moving some vestige in length, exempt of any width.
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https://en.wikipedia.org/wiki/Closed_curve
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"This definition of a curve has been formalized in modern mathematics as: A curve is the image of an interval to a topological space by a continuous function. In some contexts, the function that defines the curve is called a parametrization, and the curve is a parametric curve. In this article, these curves are sometimes called topological curves to distinguish them from more constrained curves such as differentiable curves.
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https://en.wikipedia.org/wiki/Closed_curve
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This definition encompasses most curves that are studied in mathematics; notable exceptions are level curves (which are unions of curves and isolated points), and algebraic curves (see below). Level curves and algebraic curves are sometimes called implicit curves, since they are generally defined by implicit equations. Nevertheless, the class of topological curves is very broad, and contains some curves that do not look as one may expect for a curve, or even cannot be drawn.
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https://en.wikipedia.org/wiki/Closed_curve
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This is the case of space-filling curves and fractal curves. For ensuring more regularity, the function that defines a curve is often supposed to be differentiable, and the curve is then said to be a differentiable curve. A plane algebraic curve is the zero set of a polynomial in two indeterminates.
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https://en.wikipedia.org/wiki/Closed_curve
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More generally, an algebraic curve is the zero set of a finite set of polynomials, which satisfies the further condition of being an algebraic variety of dimension one. If the coefficients of the polynomials belong to a field k, the curve is said to be defined over k. In the common case of a real algebraic curve, where k is the field of real numbers, an algebraic curve is a finite union of topological curves. When complex zeros are considered, one has a complex algebraic curve, which, from the topological point of view, is not a curve, but a surface, and is often called a Riemann surface. Although not being curves in the common sense, algebraic curves defined over other fields have been widely studied. In particular, algebraic curves over a finite field are widely used in modern cryptography.
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https://en.wikipedia.org/wiki/Closed_curve
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In mathematics, a cusp neighborhood is defined as a set of points near a cusp singularity.
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https://en.wikipedia.org/wiki/Cusp_neighborhood
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In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve.
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https://en.wikipedia.org/wiki/Cusp_(singularity)
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For a plane curve defined by an analytic, parametric equation x = f ( t ) y = g ( t ) , {\displaystyle {\begin{aligned}x&=f(t)\\y&=g(t),\end{aligned}}} a cusp is a point where both derivatives of f and g are zero, and the directional derivative, in the direction of the tangent, changes sign (the direction of the tangent is the direction of the slope lim ( g ′ ( t ) / f ′ ( t ) ) {\displaystyle \lim(g'(t)/f'(t))} ). Cusps are local singularities in the sense that they involve only one value of the parameter t, in contrast to self-intersection points that involve more than one value. In some contexts, the condition on the directional derivative may be omitted, although, in this case, the singularity may look like a regular point.
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https://en.wikipedia.org/wiki/Cusp_(singularity)
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For a curve defined by an implicit equation F ( x , y ) = 0 , {\displaystyle F(x,y)=0,} which is smooth, cusps are points where the terms of lowest degree of the Taylor expansion of F are a power of a linear polynomial; however, not all singular points that have this property are cusps. The theory of Puiseux series implies that, if F is an analytic function (for example a polynomial), a linear change of coordinates allows the curve to be parametrized, in a neighborhood of the cusp, as x = a t m y = S ( t ) , {\displaystyle {\begin{aligned}x&=at^{m}\\y&=S(t),\end{aligned}}} where a is a real number, m is a positive even integer, and S(t) is a power series of order k (degree of the nonzero term of the lowest degree) larger than m. The number m is sometimes called the order or the multiplicity of the cusp, and is equal to the degree of the nonzero part of lowest degree of F. In some contexts, the definition of a cusp is restricted to the case of cusps of order two—that is, the case where m = 2. The definitions for plane curves and implicitly-defined curves have been generalized by René Thom and Vladimir Arnold to curves defined by differentiable functions: a curve has a cusp at a point if there is a diffeomorphism of a neighborhood of the point in the ambient space, which maps the curve onto one of the above-defined cusps.
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https://en.wikipedia.org/wiki/Cusp_(singularity)
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In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form y 2 − a 2 x 3 = 0 {\displaystyle y^{2}-a^{2}x^{3}=0} (with a ≠ 0) in some Cartesian coordinate system. Solving for y leads to the explicit form y = ± a x 3 2 , {\displaystyle y=\pm ax^{\frac {3}{2}},} which imply that every real point satisfies x ≥ 0. The exponent explains the term semicubical parabola. (A parabola can be described by the equation y = ax2.)
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https://en.wikipedia.org/wiki/Cuspidal_cubic
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Solving the implicit equation for x yields a second explicit form x = ( y a ) 2 3 . {\displaystyle x=\left({\frac {y}{a}}\right)^{\frac {2}{3}}.} The parametric equation x = t 2 , y = a t 3 {\displaystyle \quad x=t^{2},\quad y=at^{3}} can also be deduced from the implicit equation by putting t = y a x .
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https://en.wikipedia.org/wiki/Cuspidal_cubic
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{\textstyle t={\frac {y}{ax}}.} The semicubical parabolas have a cuspidal singularity; hence the name of cuspidal cubic. The arc length of the curve was calculated by the English mathematician William Neile and published in 1657 (see section History).
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https://en.wikipedia.org/wiki/Cuspidal_cubic
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In mathematics, a cwatset is a set of bitstrings, all of the same length, which is closed with a twist. If each string in a cwatset, C, say, is of length n, then C will be a subset of Z 2 n {\displaystyle \mathbb {Z} _{2}^{n}} . Thus, two strings in C are added by adding the bits in the strings modulo 2 (that is, addition without carry, or exclusive disjunction). The symmetric group on n letters, Sym ( n ) {\displaystyle {\text{Sym}}(n)} , acts on Z 2 n {\displaystyle \mathbb {Z} _{2}^{n}} by bit permutation: p ( ( c 1 , … , c n ) ) = ( c p ( 1 ) , … , c p ( n ) ) , {\displaystyle p((c_{1},\ldots ,c_{n}))=(c_{p(1)},\ldots ,c_{p(n)}),} where c = ( c 1 , … , c n ) {\displaystyle c=(c_{1},\ldots ,c_{n})} is an element of Z 2 n {\displaystyle \mathbb {Z} _{2}^{n}} and p is an element of Sym ( n ) {\displaystyle {\text{Sym}}(n)} .
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https://en.wikipedia.org/wiki/Closure_with_a_twist
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Closure with a twist now means that for each element c in C, there exists some permutation p c {\displaystyle p_{c}} such that, when you add c to an arbitrary element e in the cwatset and then apply the permutation, the result will also be an element of C. That is, denoting addition without carry by + {\displaystyle +} , C will be a cwatset if and only if ∀ c ∈ C: ∃ p c ∈ Sym ( n ): ∀ e ∈ C: p c ( e + c ) ∈ C . {\displaystyle \forall c\in C:\exists p_{c}\in {\text{Sym}}(n):\forall e\in C:p_{c}(e+c)\in C.} This condition can also be written as ∀ c ∈ C: ∃ p c ∈ Sym ( n ): p c ( C + c ) = C . {\displaystyle \forall c\in C:\exists p_{c}\in {\text{Sym}}(n):p_{c}(C+c)=C.}
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https://en.wikipedia.org/wiki/Closure_with_a_twist
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In mathematics, a cyclic graph may mean a graph that contains a cycle, or a graph that is a cycle, with varying definitions of cycles. See: Cycle (graph theory), a cycle in a graph Forest (graph theory), an undirected graph with no cycles Biconnected graph, an undirected graph in which every edge belongs to a cycle Directed acyclic graph, a directed graph with no cycles Strongly connected graph, a directed graph in which every edge belongs to a cycle Aperiodic graph, a directed graph in which the cycle lengths have no nontrivial common divisor Pseudoforest, a directed or undirected graph in which every connected component includes at most one cycle Cycle graph, a graph that has the structure of a single cycle Pancyclic graph, a graph that has cycles of all possible lengths Cycle detection (graph theory), the algorithmic problem of finding cycles in graphsOther similarly-named concepts include Cycle graph (algebra), a graph that illustrates the cyclic subgroups of a group Circulant graph, a graph with an automorphism which permutes its vertices cyclically.
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https://en.wikipedia.org/wiki/Cyclic_graph
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In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "a < b". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defined as a ternary relation , meaning "after a, one reaches b before c".
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https://en.wikipedia.org/wiki/Cyclic_sequence
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For example, , but not , cf. picture. A ternary relation is called a cyclic order if it is cyclic, asymmetric, transitive, and connected.
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https://en.wikipedia.org/wiki/Cyclic_sequence
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Dropping the "connected" requirement results in a partial cyclic order. A set with a cyclic order is called a cyclically ordered set or simply a cycle. Some familiar cycles are discrete, having only a finite number of elements: there are seven days of the week, four cardinal directions, twelve notes in the chromatic scale, and three plays in rock-paper-scissors.
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https://en.wikipedia.org/wiki/Cyclic_sequence
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In a finite cycle, each element has a "next element" and a "previous element". There are also cyclic orders with infinitely many elements, such as the oriented unit circle in the plane. Cyclic orders are closely related to the more familiar linear orders, which arrange objects in a line.
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https://en.wikipedia.org/wiki/Cyclic_sequence
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Any linear order can be bent into a circle, and any cyclic order can be cut at a point, resulting in a line. These operations, along with the related constructions of intervals and covering maps, mean that questions about cyclic orders can often be transformed into questions about linear orders. Cycles have more symmetries than linear orders, and they often naturally occur as residues of linear structures, as in the finite cyclic groups or the real projective line.
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https://en.wikipedia.org/wiki/Cyclic_sequence
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In mathematics, a cyclic polytope, denoted C(n,d), is a convex polytope formed as a convex hull of n distinct points on a rational normal curve in Rd, where n is greater than d. These polytopes were studied by Constantin Carathéodory, David Gale, Theodore Motzkin, Victor Klee, and others. They play an important role in polyhedral combinatorics: according to the upper bound theorem, proved by Peter McMullen and Richard Stanley, the boundary Δ(n,d) of the cyclic polytope C(n,d) maximizes the number fi of i-dimensional faces among all simplicial spheres of dimension d − 1 with n vertices.
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https://en.wikipedia.org/wiki/Cyclic_polytope
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In mathematics, a cyclically ordered group is a set with both a group structure and a cyclic order, such that left and right multiplication both preserve the cyclic order. Cyclically ordered groups were first studied in depth by Ladislav Rieger in 1947. They are a generalization of cyclic groups: the infinite cyclic group Z and the finite cyclic groups Z/n. Since a linear order induces a cyclic order, cyclically ordered groups are also a generalization of linearly ordered groups: the rational numbers Q, the real numbers R, and so on. Some of the most important cyclically ordered groups fall into neither previous category: the circle group T and its subgroups, such as the subgroup of rational points.
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https://en.wikipedia.org/wiki/Cyclically_ordered_group
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