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(Some authors write it as fg. )such that the following axiom holds: (associativity) if f: A → B, g: B → C and h: C → D then h ∘ (g ∘ f) = (h ∘ g) ∘ f. == References == | https://en.wikipedia.org/wiki/Semigroupoid |
In mathematics, a semimodule over a semiring R is like a module over a ring except that it is only a commutative monoid rather than an abelian group. | https://en.wikipedia.org/wiki/Semimodule |
In mathematics, a semiorthogonal decomposition is a way to divide a triangulated category into simpler pieces. One way to produce a semiorthogonal decomposition is from an exceptional collection, a special sequence of objects in a triangulated category. For an algebraic variety X, it has been fruitful to study semiorth... | https://en.wikipedia.org/wiki/Semiorthogonal_decomposition |
In mathematics, a semiperfect magic cube is a magic cube that is not a perfect magic cube, i.e., a magic cube for which the cross section diagonals do not necessarily sum up to the cube's magic constant. | https://en.wikipedia.org/wiki/Semiperfect_magic_cube |
In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime numbers, there are also infinitely many semiprimes. Semiprimes are also ca... | https://en.wikipedia.org/wiki/Semiprime_number |
In mathematics, a semitopological group is a topological space with a group action that is continuous with respect to each variable considered separately. It is a weakening of the concept of a topological group; all topological groups are semitopological groups but the converse does not hold. | https://en.wikipedia.org/wiki/Semitopological_group |
In mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension. | https://en.wikipedia.org/wiki/Separability_idempotent |
In mathematics, a separation relation is a formal way to arrange a set of objects in an unoriented circle. It is defined as a quaternary relation S(a, b, c, d) satisfying certain axioms, which is interpreted as asserting that a and c separate b from d.Whereas a linear order endows a set with a positive end and a negati... | https://en.wikipedia.org/wiki/Separation_relation |
In mathematics, a separatrix is the boundary separating two modes of behaviour in a differential equation. | https://en.wikipedia.org/wiki/Separatrix_(dynamical_systems) |
In mathematics, a separoid is a binary relation between disjoint sets which is stable as an ideal in the canonical order induced by inclusion. Many mathematical objects which appear to be quite different, find a common generalisation in the framework of separoids; e.g., graphs, configurations of convex sets, oriented m... | https://en.wikipedia.org/wiki/Separoid |
In mathematics, a sequence (s1, s2, s3, ...) of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that subinterval. Such sequences are studied in Diophantine approximation theory and have applications to Monte Carlo... | https://en.wikipedia.org/wiki/Weyl's_equidistribution_criterion |
In mathematics, a sequence a = (a0, a1, ..., an) of nonnegative real numbers is called a logarithmically concave sequence, or a log-concave sequence for short, if ai2 ≥ ai−1ai+1 holds for 0 < i < n . Remark: some authors (explicitly or not) add two further conditions in the definition of log-concave sequences: a is non... | https://en.wikipedia.org/wiki/Logarithmically_concave_sequence |
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple tim... | https://en.wikipedia.org/wiki/Finite_sequence |
Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an arbitrary index set. For example, (M, A, R, Y) is a sequence of letters wit... | https://en.wikipedia.org/wiki/Finite_sequence |
This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6, ...). | https://en.wikipedia.org/wiki/Finite_sequence |
The position of an element in a sequence is its rank or index; it is the natural number for which the element is the image. The first element has index 0 or 1, depending on the context or a specific convention. In mathematical analysis, a sequence is often denoted by letters in the form of a n {\displaystyle a_{n}} , b... | https://en.wikipedia.org/wiki/Finite_sequence |
In mathematics, a sequence of discrete orthogonal polynomials is a sequence of polynomials that are pairwise orthogonal with respect to a discrete measure. Examples include the discrete Chebyshev polynomials, Charlier polynomials, Krawtchouk polynomials, Meixner polynomials, dual Hahn polynomials, Hahn polynomials, and... | https://en.wikipedia.org/wiki/Discrete_orthogonal_polynomials |
In mathematics, a sequence of functions { f n } {\displaystyle \{f_{n}\}} from a set S to a metric space M is said to be uniformly Cauchy if: For all ε > 0 {\displaystyle \varepsilon >0} , there exists N > 0 {\displaystyle N>0} such that for all x ∈ S {\displaystyle x\in S}: d ( f n ( x ) , f m ( x ) ) < ε {\displaysty... | https://en.wikipedia.org/wiki/Uniformly_Cauchy_sequence |
In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n = 7, the set of all such locations is called 7-dimensional space. Often such a space is studied as a vector space, without any notion of distance. Seven-dimensional Euclidean space is seven-dimensional space equi... | https://en.wikipedia.org/wiki/Seven-dimensional_space |
It may also refer to a seven-dimensional manifold such as a 7-sphere, or a variety of other geometric constructions. Seven-dimensional spaces have a number of special properties, many of them related to the octonions. | https://en.wikipedia.org/wiki/Seven-dimensional_space |
An especially distinctive property is that a cross product can be defined only in three or seven dimensions. This is related to Hurwitz's theorem, which prohibits the existence of algebraic structures like the quaternions and octonions in dimensions other than 2, 4, and 8. The first exotic spheres ever discovered were ... | https://en.wikipedia.org/wiki/Seven-dimensional_space |
In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n = 8, the set of all such locations is called 8-dimensional space. Often such spaces are studied as vector spaces, without any notion of distance. Eight-dimensional Euclidean space is eight-dimensional space equip... | https://en.wikipedia.org/wiki/Eight-dimensional_space |
In mathematics, a sequence of natural numbers is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once. For example, the sequence of powers of two (1, 2, 4, 8, ...), the basis of the binary numeral system, is a complete sequence; given an... | https://en.wikipedia.org/wiki/Complete_sequence |
In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals I n {\displaystyle I_{n}} on the real number line with natural numbers n = 1 , 2 , 3 , … {\displaystyle n=1,2,3,\dots } as an index. In order for a sequence of intervals to be considered nested intervals, ... | https://en.wikipedia.org/wiki/Nested_sequences_of_intervals |
Historically - long before anyone defined nested intervals in a textbook - people implicitly constructed such nestings for concrete calculation purposes. For example, the ancient Babylonians discovered a method for computing square roots of numbers. In contrast, the famed Archimedes constructed sequences of polygons, t... | https://en.wikipedia.org/wiki/Nested_sequences_of_intervals |
In mathematics, a sequence of positive integers an is called an irrationality sequence if it has the property that for every sequence xn of positive integers, the sum of the series ∑ n = 1 ∞ 1 a n x n {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{a_{n}x_{n}}}} exists (that is, it converges) and is an irrational number... | https://en.wikipedia.org/wiki/Irrationality_sequence |
In mathematics, a sequence of positive real numbers ( s 1 , s 2 , . . . ) {\displaystyle (s_{1},s_{2},...)} is called superincreasing if every element of the sequence is greater than the sum of all previous elements in the sequence.Formally, this condition can be written as s n + 1 > ∑ j = 1 n s j {\displaystyle s_{n+1... | https://en.wikipedia.org/wiki/Superincreasing_sequence |
In mathematics, a sequence of vectors (xn) in a Hilbert space ( H , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle (H,\langle \cdot ,\cdot \rangle )} is called a Riesz sequence if there exist constants 0 < c ≤ C < + ∞ {\displaystyle 0 | https://en.wikipedia.org/wiki/Riesz_sequence |
In mathematics, a sequence transformation is an operator acting on a given space of sequences (a sequence space). Sequence transformations include linear mappings such as convolution with another sequence, and resummation of a sequence and, more generally, are commonly used for series acceleration, that is, for improvi... | https://en.wikipedia.org/wiki/Linear_sequence_transformation |
In mathematics, a series expansion is a technique that expresses a function as an infinite sum, or series, of simpler functions. It is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division).The resulting so-called series often can b... | https://en.wikipedia.org/wiki/Series_expansion |
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence ( a 0 , a 1 , a 2 , … ) {\displaystyle (a_{0},a_{1},a_{2},\ldots )} defines a series S that is denoted S = a 0 + a 1 + a 2 + ⋯ = ∑ k = 0 ∞ a k . {\displaystyle S=a_{0}+a_{1}+a_{2}+\cdots =\sum _{k=0... | https://en.wikipedia.org/wiki/Convergent_series |
{\displaystyle S_{n}=\sum _{k=1}^{n}a_{k}.} A series is convergent (or converges) if the sequence ( S 1 , S 2 , S 3 , … ) {\displaystyle (S_{1},S_{2},S_{3},\dots )} of its partial sums tends to a limit; that means that, when adding one a k {\displaystyle a_{k}} after the other in the order given by the indices, one get... | https://en.wikipedia.org/wiki/Convergent_series |
{\displaystyle \left|S_{n}-\ell \right|<\varepsilon .} If the series is convergent, the (necessarily unique) number ℓ {\displaystyle \ell } is called the sum of the series. The same notation ∑ k = 1 ∞ a k {\displaystyle \sum _{k=1}^{\infty }a_{k}} is used for the series, and, if it is convergent, to its sum. This conve... | https://en.wikipedia.org/wiki/Convergent_series |
In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite stru... | https://en.wikipedia.org/wiki/Partial_sums |
In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. | https://en.wikipedia.org/wiki/Partial_sums |
This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached... | https://en.wikipedia.org/wiki/Partial_sums |
Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary fo... | https://en.wikipedia.org/wiki/Partial_sums |
To emphasize that there are an infinite number of terms, a series may be called an infinite series. Such a series is represented (or denoted) by an expression like or, using the summation sign, The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time).... | https://en.wikipedia.org/wiki/Partial_sums |
This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the series. That is, When this limit exists, one says that the series is convergent or summable, or that the sequence ( a 1 , a 2 , a 3 , … ) {\displaystyl... | https://en.wikipedia.org/wiki/Partial_sums |
Otherwise, the series is said to be divergent.The notation ∑ i = 1 ∞ a i {\textstyle \sum _{i=1}^{\infty }a_{i}} denotes both the series—that is the implicit process of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the result of the process. This is a generali... | https://en.wikipedia.org/wiki/Partial_sums |
In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely. | https://en.wikipedia.org/wiki/Conditionally_convergent |
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a semilinear manner, thus the nam... | https://en.wikipedia.org/wiki/Hermitian_product |
This case arises naturally in mathematical physics applications. Another important case allows the scalars to come from any field and the twist is provided by a field automorphism. An application in projective geometry requires that the scalars come from a division ring (skew field), K, and this means that the "vectors... | https://en.wikipedia.org/wiki/Hermitian_product |
In mathematics, a sesquipower or Zimin word is a string over an alphabet with identical prefix and suffix. Sesquipowers are unavoidable patterns, in the sense that all sufficiently long strings contain one. | https://en.wikipedia.org/wiki/Sesquipower |
In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A. Explicitly, this means that there exists a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no s... | https://en.wikipedia.org/wiki/Directly_finite_ring |
Until the foundational crisis of mathematics showed the need for a more careful treatment of set theory, most mathematicians assumed that a set is infinite if and only if it is Dedekind-infinite. In the early twentieth century, Zermelo–Fraenkel set theory, today the most commonly used form of axiomatic set theory, was ... | https://en.wikipedia.org/wiki/Directly_finite_ring |
However, there exists a model of Zermelo–Fraenkel set theory without the axiom of choice (ZF) in which there exists an infinite, Dedekind-finite set, showing that the axioms of ZF are not strong enough to prove that every set that is Dedekind-finite is finite. There are definitions of finiteness and infiniteness of set... | https://en.wikipedia.org/wiki/Directly_finite_ring |
In mathematics, a set A {\displaystyle A} is inhabited if there exists an element a ∈ A {\displaystyle a\in A} . In classical mathematics, the property of being inhabited is equivalent to being non-empty. However, this equivalence is not valid in constructive or intuitionistic logic, and so this separate terminology is... | https://en.wikipedia.org/wiki/Inhabited_set |
In mathematics, a set B of vectors in a vector space V is called a basis (PL: bases) if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B. The element... | https://en.wikipedia.org/wiki/Vector_decomposition |
In mathematics, a set S {\displaystyle S} of functions with domain D {\displaystyle D} is called a separating set for D {\displaystyle D} and is said to separate the points of D {\displaystyle D} (or just to separate points) if for any two distinct elements x {\displaystyle x} and y {\displaystyle y} of D , {\displayst... | https://en.wikipedia.org/wiki/Separating_set |
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural... | https://en.wikipedia.org/wiki/Denumerable_set |
In mathematics, a set of n functions f1, f2, ..., fn is unisolvent (meaning "uniquely solvable") on a domain Ω if the vectors , , … , {\displaystyle {\begin{bmatrix}f_{1}(x_{1})\\f_{1}(x_{2})\\\vdots \\f_{1}(x_{n})\end{bmatrix}},{\begin{bmatrix}f_{2}(x_{1})\\f_{2}(x_{2})\\\vdots \\f_{2}(x_{n})\end{bmatrix}},\dots ,{... | https://en.wikipedia.org/wiki/Unisolvent_functions |
In mathematics, a set of natural numbers is called a K-trivial set if its initial segments viewed as binary strings are easy to describe: the prefix-free Kolmogorov complexity is as low as possible, close to that of a computable set. Solovay proved in 1975 that a set can be K-trivial without being computable. The Schno... | https://en.wikipedia.org/wiki/K-trivial_set |
Thus the K-trivials are far from random. This is why these sets are studied in the field of algorithmic randomness, which is a subfield of Computability theory and related to algorithmic information theory in computer science. At the same time, K-trivial sets are close to computable. For instance, they are all superlow... | https://en.wikipedia.org/wiki/K-trivial_set |
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single equations, namely as a: System of linear equations, System of nonlinear eq... | https://en.wikipedia.org/wiki/Simultaneous_equation |
In mathematics, a set of uniqueness is a concept relevant to trigonometric expansions which are not necessarily Fourier series. Their study is a relatively pure branch of harmonic analysis. | https://en.wikipedia.org/wiki/Set_of_uniqueness |
In mathematics, a setoid (X, ~) is a set (or type) X equipped with an equivalence relation ~. A setoid may also be called E-set, Bishop set, or extensional set.Setoids are studied especially in proof theory and in type-theoretic foundations of mathematics. Often in mathematics, when one defines an equivalence relation ... | https://en.wikipedia.org/wiki/Extensional_set |
In mathematics, a sheaf (PL: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Su... | https://en.wikipedia.org/wiki/Sheaf_of_sets |
The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. | https://en.wikipedia.org/wiki/Sheaf_of_sets |
Their correct definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with ... | https://en.wikipedia.org/wiki/Sheaf_of_sets |
On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of she... | https://en.wikipedia.org/wiki/Sheaf_of_sets |
First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space. In such contexts, several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very... | https://en.wikipedia.org/wiki/Sheaf_of_sets |
Especially in algebraic geometry and the theory of complex manifolds, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of D-modules, which provide applications to the theory of differential equations. In addition, generalisat... | https://en.wikipedia.org/wiki/Sheaf_of_sets |
In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module and the restriction maps F(U) → F(V) are compatible with the restriction maps O(U) → O(V): the restriction of fs is the restriction of f times that of s for an... | https://en.wikipedia.org/wiki/Sheaf_of_a_module |
Similarly, if R is a graded ring and X is the Proj of R, then any graded module defines an OX-module in a natural way. O-modules arising in such a fashion are examples of quasi-coherent sheaves, and in fact, on affine or projective schemes, all quasi-coherent sheaves are obtained this way. Sheaves of modules over a rin... | https://en.wikipedia.org/wiki/Sheaf_of_a_module |
In mathematics, a sheaf of planes is the set of all planes that have the same common line. It may also be known as a fan of planes or a pencil of planes. When extending the concept of line to the line at infinity, a set of parallel planes can be seen as a sheaf of planes intersecting in a line at infinity. To distingui... | https://en.wikipedia.org/wiki/Sheaf_of_planes |
In mathematics, a shelling of a simplicial complex is a way of gluing it together from its maximal simplices (simplices that are not a face of another simplex) in a well-behaved way. A complex admitting a shelling is called shellable. | https://en.wikipedia.org/wiki/Shelling_(topology) |
In mathematics, a shift matrix is a binary matrix with ones only on the superdiagonal or subdiagonal, and zeroes elsewhere. A shift matrix U with ones on the superdiagonal is an upper shift matrix. The alternative subdiagonal matrix L is unsurprisingly known as a lower shift matrix. The (i,j):th component of U and L ar... | https://en.wikipedia.org/wiki/Shift_matrix |
For example, the 5×5 shift matrices are U 5 = ( 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 ) L 5 = ( 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 ) . {\displaystyle U_{5}={\begin{pmatrix}0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\end{pmatrix}}\quad L_{5}={\begin{pmatrix}0&0&0&0&0\\1&0&0&0&0\\0&1&... | https://en.wikipedia.org/wiki/Shift_matrix |
As a linear transformation, a lower shift matrix shifts the components of a column vector one position down, with a zero appearing in the first position. An upper shift matrix shifts the components of a column vector one position up, with a zero appearing in the last position.Premultiplying a matrix A by a lower shift ... | https://en.wikipedia.org/wiki/Shift_matrix |
Similar operations involving an upper shift matrix result in the opposite shift. Clearly all finite-dimensional shift matrices are nilpotent; an n by n shift matrix S becomes the null matrix when raised to the power of its dimension n. Shift matrices act on shift spaces. The infinite-dimensional shift matrices are part... | https://en.wikipedia.org/wiki/Shift_matrix |
In mathematics, a shrewd cardinal is a certain kind of large cardinal number introduced by (Rathjen 1995), extending the definition of indescribable cardinals. For an ordinal λ, a cardinal number κ is called λ-shrewd if for every proposition φ, and set A ⊆ Vκ with (Vκ+λ, ∈, A) ⊧ φ there exists an α, λ' < κ with (Vα+λ',... | https://en.wikipedia.org/wiki/Shrewd_cardinal |
(Corollary 4.3) Shrewdness was developed by Michael Rathjen as part of his ordinal analysis of Π12-comprehension. It is essentially the nonrecursive analog to the stability property for admissible ordinals. More generally, a cardinal number κ is called λ-Πm-shrewd if for every Πm proposition φ, and set A ⊆ Vκ with (Vκ+... | https://en.wikipedia.org/wiki/Shrewd_cardinal |
(Definition 4.1) Πm is one of the levels of the Lévy hierarchy, in short one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal. For finite n, an n-Πm-shrewd cardinals is the same thing as a Πmn-indescribable cardinal.If κ is a subtle cardinal, then the set of κ-shrewd ... | https://en.wikipedia.org/wiki/Shrewd_cardinal |
In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product X ⧢ Y of two words X, Y: the sum of all ways of interlacing them. The interlacing is given by the riffle shuffle permutation. The shuffle algebra on a finite set is the grad... | https://en.wikipedia.org/wiki/Shuffle_algebra |
Over the rational numbers, the shuffle algebra is isomorphic to the polynomial algebra in the Lyndon words. The shuffle product occurs in generic settings in non-commutative algebras; this is because it is able to preserve the relative order of factors being multiplied together - the riffle shuffle permutation. This ca... | https://en.wikipedia.org/wiki/Shuffle_algebra |
In mathematics, a sign sequence, or ±1–sequence or bipolar sequence, is a sequence of numbers, each of which is either 1 or −1. One example is the sequence (1, −1, 1, −1, ...). Such sequences are commonly studied in discrepancy theory. | https://en.wikipedia.org/wiki/Erdős_discrepancy_problem |
In mathematics, a signalizer functor gives the intersections of a potential subgroup of a finite group with the centralizers of nontrivial elements of an abelian group. The signalizer functor theorem gives conditions under which a signalizer functor comes from a subgroup. The idea is to try to construct a p ′ {\display... | https://en.wikipedia.org/wiki/Signalizer_functor |
{\displaystyle G.} The technique has origins in the Feit–Thompson theorem, and was subsequently developed by many people including Gorenstein (1969) who defined signalizer functors, Glauberman (1976) who proved the Solvable Signalizer Functor Theorem for solvable groups, and McBride (1982a, 1982b) who proved it for all... | https://en.wikipedia.org/wiki/Signalizer_functor |
In mathematics, a signature matrix is a diagonal matrix whose diagonal elements are plus or minus 1, that is, any matrix of the form: A = ( ± 1 0 ⋯ 0 0 0 ± 1 ⋯ 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ ± 1 0 0 0 ⋯ 0 ± 1 ) {\displaystyle A={\begin{pmatrix}\pm 1&0&\cdots &0&0\\0&\pm 1&\cdots &0&0\\\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&... | https://en.wikipedia.org/wiki/Signature_matrix |
Noting that signature matrices are both symmetric and involutory, they are thus orthogonal. Consequently, any linear transformation corresponding to a signature matrix constitutes an isometry. Geometrically, signature matrices represent a reflection in each of the axes corresponding to the negated rows or columns. | https://en.wikipedia.org/wiki/Signature_matrix |
In mathematics, a signed set is a set of elements together with an assignment of a sign (positive or negative) to each element of the set. | https://en.wikipedia.org/wiki/Signed_set |
In mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces. Together with the commutative Lie group of the real numbers, R {\dis... | https://en.wikipedia.org/wiki/Exceptional_Lie_group |
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one... | https://en.wikipedia.org/wiki/Simple_group |
In mathematics, a simple subcubic graph (SSCG) is a finite simple graph in which each vertex has a degree of at most three. Suppose we have a sequence of simple subcubic graphs G1, G2, ... such that each graph Gi has at most i + k vertices (for some integer k) and for no i < j is Gi homeomorphically embeddable into (i.... | https://en.wikipedia.org/wiki/Friedman's_SSCG_function |
The function SSCG(k) denotes that length for simple subcubic graphs. The function SCG(k) denotes that length for (general) subcubic graphs. The SCG sequence begins SCG(0) = 6, but then explodes to a value equivalent to fε2*2 in the fast-growing hierarchy. | https://en.wikipedia.org/wiki/Friedman's_SSCG_function |
The SSCG sequence begins slower than SCG, SSCG(0) = 2, SSCG(1) = 5, but then grows rapidly. SSCG(2) = 3 × 2(3 × 295) − 8 ≈ 3.241704 × 1035775080127201286522908640065. Its first and last 20 digits are 32417042291246009846...34057047399148290040. | https://en.wikipedia.org/wiki/Friedman's_SSCG_function |
SSCG(3) is much larger than both TREE(3) and TREETREE(3)(3), that is, the TREE function nested TREE(3) times with 3 at the bottom. Adam P. Goucher claims there is no qualitative difference between the asymptotic growth rates of SSCG and SCG. He writes "It's clear that SCG(n) ≥ SSCG(n), but I can also prove SSCG(4n + 3)... | https://en.wikipedia.org/wiki/Friedman's_SSCG_function |
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial coun... | https://en.wikipedia.org/wiki/Geometric_simplicial_complex |
In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets.... | https://en.wikipedia.org/wiki/Degeneracy_map |
This realization consists of geometric simplices, glued together according to the rules of the simplicial set. Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a "well-behaved" topological space for the purposes of homotopy theory. Specifically, the categor... | https://en.wikipedia.org/wiki/Degeneracy_map |
In mathematics, a simplicial space is a simplicial object in the category of topological spaces. In other words, it is a contravariant functor from the simplex category Δ to the category of topological spaces. == References == | https://en.wikipedia.org/wiki/Simplicial_space |
In mathematics, a simplicially enriched category, is a category enriched over the category of simplicial sets. Simplicially enriched categories are often also called, more ambiguously, simplicial categories; the latter term however also applies to simplicial objects in Cat (the category of small categories). Simplicial... | https://en.wikipedia.org/wiki/Simplicially_enriched_category |
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set { 0 } {\displaystyle \{0\}} is a singleton whose single element is 0 {\displaystyle 0} . | https://en.wikipedia.org/wiki/Unit_set |
In mathematics, a singular perturbation problem is a problem containing a small parameter that cannot be approximated by setting the parameter value to zero. More precisely, the solution cannot be uniformly approximated by an asymptotic expansion φ ( x ) ≈ ∑ n = 0 N δ n ( ε ) ψ n ( x ) {\displaystyle \varphi (x)\approx... | https://en.wikipedia.org/wiki/Singular_Perturbation |
Singularly perturbed problems are generally characterized by dynamics operating on multiple scales. Several classes of singular perturbations are outlined below. The term "singular perturbation" was coined in the 1940s by Kurt Otto Friedrichs and Wolfgang R. Wasow. | https://en.wikipedia.org/wiki/Singular_Perturbation |
In mathematics, a singular trace is a trace on a space of linear operators of a separable Hilbert space that vanishes on operators of finite rank. Singular traces are a feature of infinite-dimensional Hilbert spaces such as the space of square-summable sequences and spaces of square-integrable functions. Linear operato... | https://en.wikipedia.org/wiki/Singular_trace |
American mathematician Gary Weiss and, later, British mathematician Nigel Kalton observed in the infinite-dimensional case that there are non-trivial singular traces on the ideal of trace class operators. Therefore, in distinction to the finite-dimensional case, in infinite dimensions the canonical operator trace is no... | https://en.wikipedia.org/wiki/Singular_trace |
Singular traces measure the asymptotic spectral behaviour of operators and have found applications in the noncommutative geometry of French mathematician Alain Connes. In heuristic terms, a singular trace corresponds to a way of summing numbers a1, a2, a3, ... that is completely orthogonal or 'singular' with respect to... | https://en.wikipedia.org/wiki/Singular_trace |
In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity.For example, the function f ( x ) = 1 x {\displaystyle f(x)={\frac {1}{x}}} has a s... | https://en.wikipedia.org/wiki/Mathematical_singularities |
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