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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 ==
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https://en.wikipedia.org/wiki/Semigroupoid
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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.
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https://en.wikipedia.org/wiki/Semimodule
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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 semiorthogonal decompositions of the bounded derived category of coherent sheaves, D b ( X ) {\displaystyle {\text{D}}^{\text{b}}(X)} .
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https://en.wikipedia.org/wiki/Semiorthogonal_decomposition
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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.
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https://en.wikipedia.org/wiki/Semiperfect_magic_cube
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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 called biprimes.
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https://en.wikipedia.org/wiki/Semiprime_number
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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.
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https://en.wikipedia.org/wiki/Semitopological_group
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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.
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https://en.wikipedia.org/wiki/Separability_idempotent
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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 negative end, a separation relation forgets not only which end is which, but also where the ends are located. In this way it is a final, further weakening of the concepts of a betweenness relation and a cyclic order. There is nothing else that can be forgotten: up to the relevant sense of interdefinability, these three relations are the only nontrivial reducts of the ordered set of rational numbers.
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https://en.wikipedia.org/wiki/Separation_relation
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In mathematics, a separatrix is the boundary separating two modes of behaviour in a differential equation.
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https://en.wikipedia.org/wiki/Separatrix_(dynamical_systems)
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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 matroids, and polytopes. Any countable category is an induced subcategory of separoids when they are endowed with homomorphisms (viz., mappings that preserve the so-called minimal Radon partitions). In this general framework, some results and invariants of different categories turn out to be special cases of the same aspect; e.g., the pseudoachromatic number from graph theory and the Tverberg theorem from combinatorial convexity are simply two faces of the same aspect, namely, complete colouring of separoids.
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https://en.wikipedia.org/wiki/Separoid
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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 integration.
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https://en.wikipedia.org/wiki/Weyl's_equidistribution_criterion
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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-negative a has no internal zeros; in other words, the support of a is an interval of Z.These conditions mirror the ones required for log-concave functions. Sequences that fulfill the three conditions are also called Pólya Frequency sequences of order 2 (PF2 sequences). Refer to chapter 2 of for a discussion on the two notions. For instance, the sequence (1,1,0,0,1) satisfies the concavity inequalities but not the internal zeros condition. Examples of log-concave sequences are given by the binomial coefficients along any row of Pascal's triangle and the elementary symmetric means of a finite sequence of real numbers.
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https://en.wikipedia.org/wiki/Logarithmically_concave_sequence
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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 times at different positions in a sequence, and unlike a set, the order does matter.
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https://en.wikipedia.org/wiki/Finite_sequence
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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 with the letter 'M' first and 'Y' last.
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https://en.wikipedia.org/wiki/Finite_sequence
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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, ...).
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https://en.wikipedia.org/wiki/Finite_sequence
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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 n {\displaystyle b_{n}} and c n {\displaystyle c_{n}} , where the subscript n refers to the nth element of the sequence; for example, the nth element of the Fibonacci sequence F {\displaystyle F} is generally denoted as F n {\displaystyle F_{n}} . In computing and computer science, finite sequences are sometimes called strings, words or lists, the different names commonly corresponding to different ways to represent them in computer memory; infinite sequences are called streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.
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https://en.wikipedia.org/wiki/Finite_sequence
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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 Racah polynomials. If the measure has finite support, then the corresponding sequence of discrete orthogonal polynomials has only a finite number of elements. The Racah polynomials give an example of this.
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https://en.wikipedia.org/wiki/Discrete_orthogonal_polynomials
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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 {\displaystyle d(f_{n}(x),f_{m}(x))<\varepsilon } whenever m , n > N {\displaystyle m,n>N} .Another way of saying this is that d u ( f n , f m ) → 0 {\displaystyle d_{u}(f_{n},f_{m})\to 0} as m , n → ∞ {\displaystyle m,n\to \infty } , where the uniform distance d u {\displaystyle d_{u}} between two functions is defined by d u ( f , g ) := sup x ∈ S d ( f ( x ) , g ( x ) ) . {\displaystyle d_{u}(f,g):=\sup _{x\in S}d(f(x),g(x)).}
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https://en.wikipedia.org/wiki/Uniformly_Cauchy_sequence
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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 equipped with a Euclidean metric, which is defined by the dot product.More generally, the term may refer to a seven-dimensional vector space over any field, such as a seven-dimensional complex vector space, which has 14 real dimensions.
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https://en.wikipedia.org/wiki/Seven-dimensional_space
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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.
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https://en.wikipedia.org/wiki/Seven-dimensional_space
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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 seven-dimensional.
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https://en.wikipedia.org/wiki/Seven-dimensional_space
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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 equipped with the Euclidean metric. More generally the term may refer to an eight-dimensional vector space over any field, such as an eight-dimensional complex vector space, which has 16 real dimensions. It may also refer to an eight-dimensional manifold such as an 8-sphere, or a variety of other geometric constructions.
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https://en.wikipedia.org/wiki/Eight-dimensional_space
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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 any natural number, we can choose the values corresponding to the 1 bits in its binary representation and sum them to obtain that number (e.g. 37 = 1001012 = 1 + 4 + 32). This sequence is minimal, since no value can be removed from it without making some natural numbers impossible to represent. Simple examples of sequences that are not complete include the even numbers, since adding even numbers produces only even numbers—no odd number can be formed.
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https://en.wikipedia.org/wiki/Complete_sequence
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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, two conditions have to be met: Every interval in the sequence is contained in the previous one ( I n + 1 {\displaystyle I_{n+1}} is always a subset of I n {\displaystyle I_{n}} ). The length of the intervals get arbitrarily small (meaning the length falls below every possible threshold ε {\displaystyle \varepsilon } after a certain index N {\displaystyle N} ).In other words, the left bound of the interval I n {\displaystyle I_{n}} can only increase ( a n + 1 ≥ a n {\displaystyle a_{n+1}\geq a_{n}} ), and the right bound can only decrease ( b n + 1 ≤ b n {\displaystyle b_{n+1}\leq b_{n}} ).
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https://en.wikipedia.org/wiki/Nested_sequences_of_intervals
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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, that inscribed and surcumscribed a unit circle, in order to get a lower and upper bound for the circles circumference - which is the circle number Pi ( π {\displaystyle \pi } ). The central question to be posed is the nature of the intersection over all the natural numbers, or, put differently, the set of numbers, that are found in every Interval I n {\displaystyle I_{n}} (thus, for all n ∈ N {\displaystyle n\in \mathbb {N} } ). In modern mathematics, nested intervals are used as a construction method for the real numbers (in order to complete the field of rational numbers).
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https://en.wikipedia.org/wiki/Nested_sequences_of_intervals
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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. The problem of characterizing irrationality sequences was posed by Paul Erdős and Ernst G. Straus, who originally called the property of being an irrationality sequence "Property P".
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https://en.wikipedia.org/wiki/Irrationality_sequence
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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}>\sum _{j=1}^{n}s_{j}} for all n ≥ 1.
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https://en.wikipedia.org/wiki/Superincreasing_sequence
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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
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https://en.wikipedia.org/wiki/Riesz_sequence
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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 improving the rate of convergence of a slowly convergent sequence or series. Sequence transformations are also commonly used to compute the antilimit of a divergent series numerically, and are used in conjunction with extrapolation methods.
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https://en.wikipedia.org/wiki/Linear_sequence_transformation
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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 be limited to a finite number of terms, thus yielding an approximation of the function. The fewer terms of the sequence are used, the simpler this approximation will be. Often, the resulting inaccuracy (i.e., the partial sum of the omitted terms) can be described by an equation involving Big O notation (see also asymptotic expansion). The series expansion on an open interval will also be an approximation for non-analytic functions.
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https://en.wikipedia.org/wiki/Series_expansion
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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}^{\infty }a_{k}.} The nth partial sum Sn is the sum of the first n terms of the sequence; that is, S n = ∑ k = 1 n a k .
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https://en.wikipedia.org/wiki/Convergent_series
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{\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 gets partial sums that become closer and closer to a given number. More precisely, a series converges, if there exists a number ℓ {\displaystyle \ell } such that for every arbitrarily small positive number ε {\displaystyle \varepsilon } , there is a (sufficiently large) integer N {\displaystyle N} such that for all n ≥ N {\displaystyle n\geq N} , | S n − ℓ | < ε .
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https://en.wikipedia.org/wiki/Convergent_series
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{\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 convention is similar to that which is used for addition: a + b denotes the operation of adding a and b as well as the result of this addition, which is called the sum of a and b. Any series that is not convergent is said to be divergent or to diverge.
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https://en.wikipedia.org/wiki/Convergent_series
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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 structures (such as in combinatorics) through generating functions.
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https://en.wikipedia.org/wiki/Partial_sums
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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.
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https://en.wikipedia.org/wiki/Partial_sums
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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 a second position; when he reaches this second position, the tortoise is at a third position, and so on. Zeno concluded that Achilles could never reach the tortoise, and thus that movement does not exist.
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https://en.wikipedia.org/wiki/Partial_sums
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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 for Achilles to catch up with the tortoise. In modern terminology, any (ordered) infinite sequence ( a 1 , a 2 , a 3 , … ) {\displaystyle (a_{1},a_{2},a_{3},\ldots )} of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the ai one after the other.
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https://en.wikipedia.org/wiki/Partial_sums
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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). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series.
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https://en.wikipedia.org/wiki/Partial_sums
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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 , … ) {\displaystyle (a_{1},a_{2},a_{3},\ldots )} is summable. In this case, the limit is called the sum of the series.
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https://en.wikipedia.org/wiki/Partial_sums
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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 generalization of the similar convention of denoting by a + b {\displaystyle a+b} both the addition—the process of adding—and its result—the sum of a and b. Generally, the terms of a series come from a ring, often the field R {\displaystyle \mathbb {R} } of the real numbers or the field C {\displaystyle \mathbb {C} } of the complex numbers. In this case, the set of all series is itself a ring (and even an associative algebra), in which the addition consists of adding the series term by term, and the multiplication is the Cauchy product.
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https://en.wikipedia.org/wiki/Partial_sums
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In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.
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https://en.wikipedia.org/wiki/Conditionally_convergent
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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 name; which originates from the Latin numerical prefix sesqui- meaning "one and a half". The basic concept of the dot product – producing a scalar from a pair of vectors – can be generalized by allowing a broader range of scalar values and, perhaps simultaneously, by widening the definition of a vector. A motivating special case is a sesquilinear form on a complex vector space, V. This is a map V × V → C that is linear in one argument and "twists" the linearity of the other argument by complex conjugation (referred to as being antilinear in the other argument).
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https://en.wikipedia.org/wiki/Hermitian_product
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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" should be replaced by elements of a K-module. In a very general setting, sesquilinear forms can be defined over R-modules for arbitrary rings R.
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https://en.wikipedia.org/wiki/Hermitian_product
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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.
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https://en.wikipedia.org/wiki/Sesquipower
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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 such bijection exists). Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers.A simple example is N {\displaystyle \mathbb {N} } , the set of natural numbers. From Galileo's paradox, there exists a bijection that maps every natural number n to its square n2. Since the set of squares is a proper subset of N {\displaystyle \mathbb {N} } , N {\displaystyle \mathbb {N} } is Dedekind-infinite.
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https://en.wikipedia.org/wiki/Directly_finite_ring
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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 proposed as an axiomatic system to formulate a theory of sets free of paradoxes such as Russell's paradox. Using the axioms of Zermelo–Fraenkel set theory with the originally highly controversial axiom of choice included (ZFC) one can show that a set is Dedekind-finite if and only if it is finite in the usual sense.
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https://en.wikipedia.org/wiki/Directly_finite_ring
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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 sets besides the one given by Dedekind that do not depend on the axiom of choice. A vaguely related notion is that of a Dedekind-finite ring.
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https://en.wikipedia.org/wiki/Directly_finite_ring
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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 mostly used in the set theory of constructive mathematics.
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https://en.wikipedia.org/wiki/Inhabited_set
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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 elements of a basis are called basis vectors. Equivalently, a set B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B. In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.
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https://en.wikipedia.org/wiki/Vector_decomposition
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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 , {\displaystyle D,} there exists a function f ∈ S {\displaystyle f\in S} such that f ( x ) ≠ f ( y ) . {\displaystyle f(x)\neq f(y).} Separating sets can be used to formulate a version of the Stone–Weierstrass theorem for real-valued functions on a compact Hausdorff space X , {\displaystyle X,} with the topology of uniform convergence. It states that any subalgebra of this space of functions is dense if and only if it separates points. This is the version of the theorem originally proved by Marshall H. Stone.
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https://en.wikipedia.org/wiki/Separating_set
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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 number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is countable if its cardinality (the number of elements of the set) is not greater than that of the natural numbers. A countable set that is not finite is said to be countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers.
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https://en.wikipedia.org/wiki/Denumerable_set
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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 ,{\begin{bmatrix}f_{n}(x_{1})\\f_{n}(x_{2})\\\vdots \\f_{n}(x_{n})\end{bmatrix}}} are linearly independent for any choice of n distinct points x1, x2 ... xn in Ω. Equivalently, the collection is unisolvent if the matrix F with entries fi(xj) has nonzero determinant: det(F) ≠ 0 for any choice of distinct xj's in Ω. Unisolvency is a property of vector spaces, not just particular sets of functions. That is, a vector space of functions of dimension n is unisolvent if given any basis (equivalently, a linearly independent set of n functions), the basis is unisolvent (as a set of functions). This is because any two bases are related by an invertible matrix (the change of basis matrix), so one basis is unisolvent if and only if any other basis is unisolvent. Unisolvent systems of functions are widely used in interpolation since they guarantee a unique solution to the interpolation problem. The set of polynomials of degree at most d {\displaystyle d} (which form a vector space of dimension d + 1 {\displaystyle d+1} ) are unisolvent by the unisolvence theorem.
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https://en.wikipedia.org/wiki/Unisolvent_functions
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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 Schnorr–Levin theorem says that random sets have a high initial segment complexity.
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https://en.wikipedia.org/wiki/K-trivial_set
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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, i.e. sets whose Turing jump is computable from the Halting problem, and form a Turing ideal, i.e. class of sets closed under Turing join and closed downward under Turing reduction.
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https://en.wikipedia.org/wiki/K-trivial_set
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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 equations, System of bilinear equations, System of polynomial equations, System of differential equations, or a System of difference equations
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https://en.wikipedia.org/wiki/Simultaneous_equation
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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.
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https://en.wikipedia.org/wiki/Set_of_uniqueness
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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 on a set, one immediately forms the quotient set (turning equivalence into equality). In contrast, setoids may be used when a difference between identity and equivalence must be maintained, often with an interpretation of intensional equality (the equality on the original set) and extensional equality (the equivalence relation, or the equality on the quotient set).
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https://en.wikipedia.org/wiki/Extensional_set
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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. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts).
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https://en.wikipedia.org/wiki/Sheaf_of_sets
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The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects.
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https://en.wikipedia.org/wiki/Sheaf_of_sets
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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 their morphisms on a fixed topological space form a category.
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https://en.wikipedia.org/wiki/Sheaf_of_sets
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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 sheaf theory. Due to their general nature and versatility, sheaves have several applications in topology and especially in algebraic and differential geometry.
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https://en.wikipedia.org/wiki/Sheaf_of_sets
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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 general cohomology theory, which encompasses also the "usual" topological cohomology theories such as singular cohomology.
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https://en.wikipedia.org/wiki/Sheaf_of_sets
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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, generalisations of sheaves to more general settings than topological spaces, such as Grothendieck topology, have provided applications to mathematical logic and to number theory.
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https://en.wikipedia.org/wiki/Sheaf_of_sets
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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 any f in O(U) and s in F(U). The standard case is when X is a scheme and O its structure sheaf. If O is the constant sheaf Z _ {\displaystyle {\underline {\mathbf {Z} }}} , then a sheaf of O-modules is the same as a sheaf of abelian groups (i.e., an abelian sheaf). If X is the prime spectrum of a ring R, then any R-module defines an OX-module (called an associated sheaf) in a natural way.
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https://en.wikipedia.org/wiki/Sheaf_of_a_module
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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 ringed space form an abelian category. Moreover, this category has enough injectives, and consequently one can and does define the sheaf cohomology H i ( X , − ) {\displaystyle \operatorname {H} ^{i}(X,-)} as the i-th right derived functor of the global section functor Γ ( X , − ) {\displaystyle \Gamma (X,-)} .
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https://en.wikipedia.org/wiki/Sheaf_of_a_module
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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 distinguish it from the more general definition, the adjective parallel can be added to it, resulting in the expression parallel sheaf of planes.
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https://en.wikipedia.org/wiki/Sheaf_of_planes
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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.
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https://en.wikipedia.org/wiki/Shelling_(topology)
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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 are U i j = δ i + 1 , j , L i j = δ i , j + 1 , {\displaystyle U_{ij}=\delta _{i+1,j},\quad L_{ij}=\delta _{i,j+1},} where δ i j {\displaystyle \delta _{ij}} is the Kronecker delta symbol.
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https://en.wikipedia.org/wiki/Shift_matrix
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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&0&0&0\\0&0&1&0&0\\0&0&0&1&0\end{pmatrix}}.} Clearly, the transpose of a lower shift matrix is an upper shift matrix and vice versa.
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https://en.wikipedia.org/wiki/Shift_matrix
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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 matrix results in the elements of A being shifted downward by one position, with zeroes appearing in the top row. Postmultiplication by a lower shift matrix results in a shift left.
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https://en.wikipedia.org/wiki/Shift_matrix
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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 particularly important for the study of ergodic systems. Important examples of infinite-dimensional shifts are the Bernoulli shift, which acts as a shift on Cantor space, and the Gauss map, which acts as a shift on the space of continued fractions (that is, on Baire space.)
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https://en.wikipedia.org/wiki/Shift_matrix
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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', ∈, A ∩ Vα) ⊧ φ. It is called shrewd if it is λ-shrewd for every λ(Definition 4.1) (including λ > κ). This definition extends the concept of indescribability to transfinite levels. A λ-shrewd cardinal is also μ-shrewd for any ordinal μ < λ.
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https://en.wikipedia.org/wiki/Shrewd_cardinal
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(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+λ, ∈, A) ⊧ φ there exists an α, λ' < κ with (Vα+λ', ∈, A ∩ Vα) ⊧ φ.
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https://en.wikipedia.org/wiki/Shrewd_cardinal
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(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 cardinals is stationary in κ. (Lemma 4.6) A cardinal is strongly unfoldable iff it is shrewd.λ-shrewdness is an improved version of λ-indescribability, as defined in Drake; this cardinal property differs in that the reflected substructure must be (Vα+λ, ∈, A ∩ Vα), making it impossible for a cardinal κ to be κ-indescribable. Also, the monotonicity property is lost: a λ-indescribable cardinal may fail to be α-indescribable for some ordinal α < λ.
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https://en.wikipedia.org/wiki/Shrewd_cardinal
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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 graded dual of the universal enveloping algebra of the free Lie algebra on the set.
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https://en.wikipedia.org/wiki/Shuffle_algebra
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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 can be held in contrast to the divided power structure, which becomes appropriate when factors are commutative.
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https://en.wikipedia.org/wiki/Shuffle_algebra
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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.
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https://en.wikipedia.org/wiki/Erdős_discrepancy_problem
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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 ′ {\displaystyle p'} -subgroup of a finite group G {\displaystyle G} , which has a good chance of being normal in G {\displaystyle G} , by taking as generators certain p ′ {\displaystyle p'} -subgroups of the centralizers of nonidentity elements in one or several given noncyclic elementary abelian p {\displaystyle p} -subgroups of G .
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https://en.wikipedia.org/wiki/Signalizer_functor
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{\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 groups. This theorem is needed to prove the so-called "dichotomy" stating that a given nonabelian finite simple group either has local characteristic two, or is of component type. It thus plays a major role in the classification of finite simple groups.
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https://en.wikipedia.org/wiki/Signalizer_functor
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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&\cdots &\pm 1&0\\0&0&\cdots &0&\pm 1\end{pmatrix}}} Any such matrix is its own inverse, hence is an involutory matrix. It is consequently a square root of the identity matrix. Note however that not all square roots of the identity are signature matrices.
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https://en.wikipedia.org/wiki/Signature_matrix
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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.
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https://en.wikipedia.org/wiki/Signature_matrix
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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.
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https://en.wikipedia.org/wiki/Signed_set
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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 {\displaystyle \mathbb {R} } , and that of the unit-magnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension. Many commonly encountered Lie groups are either simple or 'close' to being simple: for example, the so-called "special linear group" SL(n) of n by n matrices with determinant equal to 1 is simple for all n > 1. The first classification of simple Lie groups was by Wilhelm Killing, and this work was later perfected by Élie Cartan. The final classification is often referred to as Killing-Cartan classification.
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https://en.wikipedia.org/wiki/Exceptional_Lie_group
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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 eventually arrives at uniquely determined simple groups, by the Jordan–Hölder theorem. The complete classification of finite simple groups, completed in 2004, is a major milestone in the history of mathematics.
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https://en.wikipedia.org/wiki/Simple_group
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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.e. is a graph minor of) Gj. The Robertson–Seymour theorem proves that subcubic graphs (simple or not) are well-founded by homeomorphic embeddability, implying such a sequence cannot be infinite. So, for each value of k, there is a sequence with maximal length.
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https://en.wikipedia.org/wiki/Friedman's_SSCG_function
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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.
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https://en.wikipedia.org/wiki/Friedman's_SSCG_function
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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.
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https://en.wikipedia.org/wiki/Friedman's_SSCG_function
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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) ≥ SCG(n)."
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https://en.wikipedia.org/wiki/Friedman's_SSCG_function
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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 counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex. : 7
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https://en.wikipedia.org/wiki/Geometric_simplicial_complex
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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. Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber.Every simplicial set gives rise to a "nice" topological space, known as its geometric realization.
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https://en.wikipedia.org/wiki/Degeneracy_map
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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 category of simplicial sets carries a natural model structure, and the corresponding homotopy category is equivalent to the familiar homotopy category of topological spaces. Simplicial sets are used to define quasi-categories, a basic notion of higher category theory. A construction analogous to that of simplicial sets can be carried out in any category, not just in the category of sets, yielding the notion of simplicial objects.
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https://en.wikipedia.org/wiki/Degeneracy_map
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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 ==
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https://en.wikipedia.org/wiki/Simplicial_space
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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). Simplicially enriched categories can, however, be identified with simplicial objects in Cat whose object part is constant, or more precisely, whose all face and degeneracy maps are bijective on objects. Simplicially enriched categories can model (∞, 1)-categories, but the dictionary has to be carefully built. Namely many notions, limits for example, are different from the limits in the sense of enriched category theory.
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https://en.wikipedia.org/wiki/Simplicially_enriched_category
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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} .
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https://en.wikipedia.org/wiki/Unit_set
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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 \sum _{n=0}^{N}\delta _{n}(\varepsilon )\psi _{n}(x)\,} as ε → 0 {\displaystyle \varepsilon \to 0} . Here ε {\displaystyle \varepsilon } is the small parameter of the problem and δ n ( ε ) {\displaystyle \delta _{n}(\varepsilon )} are a sequence of functions of ε {\displaystyle \varepsilon } of increasing order, such as δ n ( ε ) = ε n {\displaystyle \delta _{n}(\varepsilon )=\varepsilon ^{n}} . This is in contrast to regular perturbation problems, for which a uniform approximation of this form can be obtained.
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https://en.wikipedia.org/wiki/Singular_Perturbation
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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.
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https://en.wikipedia.org/wiki/Singular_Perturbation
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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 operators on a finite-dimensional Hilbert space have only the zero functional as a singular trace since all operators have finite rank. For example, matrix algebras have no non-trivial singular traces and the matrix trace is the unique trace up to scaling.
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https://en.wikipedia.org/wiki/Singular_trace
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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 not the unique trace up to scaling. The operator trace is the continuous extension of the matrix trace from finite rank operators to all trace class operators, and the term singular derives from the fact that a singular trace vanishes where the matrix trace is supported, analogous to a singular measure vanishing where Lebesgue measure is supported.
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https://en.wikipedia.org/wiki/Singular_trace
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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 the usual sum a1 + a2 + a3 + ... . This allows mathematicians to sum sequences like the harmonic sequence (and operators with similar spectral behaviour) that are divergent for the usual sum. In similar terms a (noncommutative) measure theory or probability theory can be built for distributions like the Cauchy distribution (and operators with similar spectral behaviour) that do not have finite expectation in the usual sense.
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https://en.wikipedia.org/wiki/Singular_trace
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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 singularity at x = 0 {\displaystyle x=0} , where the value of the function is not defined, as involving a division by zero. The absolute value function g ( x ) = | x | {\displaystyle g(x)=|x|} also has a singularity at x = 0 {\displaystyle x=0} , since it is not differentiable there.The algebraic curve defined by { ( x , y ): y 3 − x 2 = 0 } {\displaystyle \left\{(x,y):y^{3}-x^{2}=0\right\}} in the ( x , y ) {\displaystyle (x,y)} coordinate system has a singularity (called a cusp) at ( 0 , 0 ) {\displaystyle (0,0)} . For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory.
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https://en.wikipedia.org/wiki/Mathematical_singularities
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