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c_dskmmn3jqeob
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Number bond
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Summary
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Number_bond
|
The term "number bond" is also used to refer to a pictorial representation of part-part-whole relationships, often found in the Singapore mathematics curriculum. Number bonds consist of a minimum of 3 circles that are connected by lines. The “whole” is written in the first circle and its “parts” are written in the adjoining circles. Number bonds are used to build deeper understanding of math facts.
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c_eq3gs4v6bfk9
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Chunking (division)
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Summary
|
Chunking_(division)
|
In mathematics education at the primary school level, chunking (sometimes also called the partial quotients method) is an elementary approach for solving simple division questions by repeated subtraction. It is also known as the hangman method with the addition of a line separating the divisor, dividend, and partial quotients. It has a counterpart in the grid method for multiplication as well. In general, chunking is more flexible than the traditional method in that the calculation of quotient is less dependent on the place values.
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c_y1jp5i30rqox
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Chunking (division)
|
Summary
|
Chunking_(division)
|
As a result, it is often considered to be a more intuitive, but a less systematic approach to divisions – where the efficiency is highly dependent upon one's numeracy skills. To calculate the whole number quotient of dividing a large number by a small number, the student repeatedly takes away "chunks" of the large number, where each "chunk" is an easy multiple (for example 100×, 10×, 5× 2×, etc.) of the small number, until the large number has been reduced to zero – or the remainder is less than the small number itself.
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c_pddrp3cnpgoa
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Chunking (division)
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Summary
|
Chunking_(division)
|
At the same time the student is generating a list of the multiples of the small number (i.e., partial quotients) that have so far been taken away, which when added up together would then become the whole number quotient itself. For example, to calculate 132 ÷ 8, one might successively subtract 80, 40 and 8 to leave 4: 132 80 (10 × 8) -- 52 40 ( 5 × 8) -- 12 8 ( 1 × 8) -- 4 -------- 132 = 16 × 8 + 4 Because 10 + 5 + 1 = 16, 132 ÷ 8 is 16 with 4 remaining. In the UK, this approach for elementary division sums has come into widespread classroom use in primary schools since the late 1990s, when the National Numeracy Strategy in its "numeracy hour" brought in a new emphasis on more free-form oral and mental strategies for calculations, rather than the rote learning of standard methods.Compared to the short division and long division methods that are traditionally taught, chunking may seem strange, unsystematic, and arbitrary.
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c_9dsz9cdphmz4
|
Chunking (division)
|
Summary
|
Chunking_(division)
|
However, it is argued that chunking, rather than moving straight to short division, gives a better introduction to division, in part because the focus is always holistic, focusing throughout on the whole calculation and its meaning, rather than just rules for generating successive digits. The more freeform nature of chunking also means that it requires more genuine understanding – rather than just the ability to follow a ritualised procedure – to be successful.An alternative way of performing chunking involves the use of the standard long division tableau – except that the partial quotients are stacked up on the top of each other above the long division sign, and that all numbers are spelled out in full. By allowing one to subtract more chunks than what one currently has, it is also possible to expand chunking into a fully bidirectional method as well.
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c_vdzxxa5e8r5l
|
Finite mathematics
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Summary
|
Finite_mathematics
|
In mathematics education, Finite Mathematics is a syllabus in college and university mathematics that is independent of calculus. A course in precalculus may be a prerequisite for Finite Mathematics. Contents of the course include an eclectic selection of topics often applied in social science and business, such as finite probability spaces, matrix multiplication, Markov processes, finite graphs, or mathematical models.
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c_1or9iu6ygd5x
|
Finite mathematics
|
Summary
|
Finite_mathematics
|
These topics were used in Finite Mathematics courses at Dartmouth College as developed by John G. Kemeny, Gerald L. Thompson, and J. Laurie Snell and published by Prentice-Hall. Other publishers followed with their own topics. With the arrival of software to facilitate computations, teaching and usage shifted from a broad-spectrum Finite Mathematics with paper and pen, into development and usage of software.
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c_mm9vc348q098
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Mathematical manipulative
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Summary
|
Manipulative_(mathematics_education)
|
In mathematics education, a manipulative is an object which is designed so that a learner can perceive some mathematical concept by manipulating it, hence its name. The use of manipulatives provides a way for children to learn concepts through developmentally appropriate hands-on experience. The use of manipulatives in mathematics classrooms throughout the world grew considerably in popularity throughout the second half of the 20th century. Mathematical manipulatives are frequently used in the first step of teaching mathematical concepts, that of concrete representation.
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c_fulfhkzzpf09
|
Mathematical manipulative
|
Summary
|
Manipulative_(mathematics_education)
|
The second and third steps are representational and abstract, respectively. Mathematical manipulatives can be purchased or constructed by the teacher.
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c_rhl6ukk3if1b
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Mathematical manipulative
|
Summary
|
Manipulative_(mathematics_education)
|
Examples of common manipulatives include number lines, Cuisenaire rods; fraction strips, blocks, or stacks; base ten blocks (also known as Dienes or multibase blocks); interlocking linking cubes (such as Unifix); construction sets (such as Polydron and Zometool); colored tiles or tangrams; pattern blocks; colored counting chips; numicon tiles; chainable links; abaci such as "rekenreks", and geoboards. Improvised teacher-made manipulatives used in teaching place value include beans and bean sticks, or single popsicle sticks and bundles of ten popsicle sticks. Virtual manipulatives for mathematics are computer models of these objects.
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c_qkj0wax3onnn
|
Mathematical manipulative
|
Summary
|
Manipulative_(mathematics_education)
|
Notable collections of virtual manipulatives include The National Library of Virtual Manipulatives and the Ubersketch. Multiple experiences with manipulatives provide children with the conceptual foundation to understand mathematics at a conceptual level and are recommended by the NCTM.Some of the manipulatives are now used in other subjects in addition to mathematics. For example, Cuisenaire rods are now used in language arts and grammar, and pattern blocks are used in fine arts.
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c_8jbgvpgj9lvk
|
Number sentence
|
Summary
|
Number_sentence
|
In mathematics education, a number sentence is an equation or inequality expressed using numbers and mathematical symbols. The term is used in primary level mathematics teaching in the US, Canada, UK, Australia, New Zealand and South Africa.
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c_afinqg62m2k9
|
Procept
|
Summary
|
Procept
|
In mathematics education, a procept is an amalgam of three components: a "process" which produces a mathematical "object" and a "symbol" which is used to represent either process or object. It derives from the work of Eddie Gray and David O. Tall. The notion was first published in a paper in the Journal for Research in Mathematics Education in 1994, and is part of the process-object literature. This body of literature suggests that mathematical objects are formed by encapsulating processes, that is to say that the mathematical object 3 is formed by an encapsulation of the process of counting: 1,2,3... Gray and Tall's notion of procept improved upon the existing literature by noting that mathematical notation is often ambiguous as to whether it refers to process or object.
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c_7dcirkabezuy
|
Procept
|
Summary
|
Procept
|
Examples of such notations are: 3 + 4 {\displaystyle 3+4}: refers to the process of adding as well as the outcome of the process. ∑ n = 0 ∞ ( a n ) {\displaystyle \sum _{n=0}^{\infty }(a_{n})}: refers to the process of summing an infinite sequence, and to the outcome of the process. f ( x ) = 3 x + 2 {\displaystyle f(x)=3x+2}: refers to the process of mapping x to 3x+2 as well as the outcome of that process, the function f {\displaystyle f} .
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c_2ykqas3pbag4
|
Multiple representations (mathematics education)
|
Summary
|
Multiple_representations_(mathematics_education)
|
In mathematics education, a representation is a way of encoding an idea or a relationship, and can be both internal (e.g., mental construct) and external (e.g., graph). Thus multiple representations are ways to symbolize, to describe and to refer to the same mathematical entity. They are used to understand, to develop, and to communicate different mathematical features of the same object or operation, as well as connections between different properties. Multiple representations include graphs and diagrams, tables and grids, formulas, symbols, words, gestures, software code, videos, concrete models, physical and virtual manipulatives, pictures, and sounds. Representations are thinking tools for doing mathematics.
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c_i8gfot03dszp
|
Differential and integral calculus
|
Etymology
|
Differential_and_integral_calculus > Etymology
|
In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word calculus is Latin for "small pebble" (the diminutive of calx, meaning "stone"), a meaning which still persists in medicine. Because such pebbles were used for counting out distances, tallying votes, and doing abacus arithmetic, the word came to mean a method of computation. In this sense, it was used in English at least as early as 1672, several years prior to the publications of Leibniz and Newton.In addition to the differential calculus and integral calculus, the term is also used for naming specific methods of calculation and related theories which seek to model a particular concept in terms of mathematics. Examples of this convention include propositional calculus, Ricci calculus, calculus of variations, lambda calculus, and process calculus. Furthermore, the term "calculus" has variously been applied in ethics and philosophy, for such systems as Bentham's felicific calculus, and the ethical calculus.
|
c_a1raklayzk0u
|
History of calculus
|
Etymology
|
History_of_calculus > Etymology
|
In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word calculus is Latin for "small pebble" (the diminutive of calx, meaning "stone"), a meaning which still persists in medicine. Because such pebbles were used for counting out distances, tallying votes, and doing abacus arithmetic, the word came to mean a method of computation. In this sense, it was used in English at least as early as 1672, several years prior to the publications of Leibniz and Newton.In addition to the differential calculus and integral calculus, the term is also used widely for naming specific methods of calculation. Examples of this include propositional calculus in logic, the calculus of variations in mathematics, process calculus in computing, and the felicific calculus in philosophy.
|
c_e4tpgdkl2g7n
|
Concept image and concept definition
|
Summary
|
Concept_image_and_concept_definition
|
In mathematics education, concept image and concept definition are two ways of understanding a mathematical concept. The terms were introduced by Tall & Vinner (1981). They define a concept image as such: "We shall use the term concept image to describe the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes. It is built up over the years through experiences of all kinds, changing as the individual meets new stimuli and matures. "A concept definition is similar to the usual notion of a definition in mathematics, with the distinction that it is personal to an individual: "a personal concept definition can differ from a formal concept definition, the latter being a concept definition which is accepted by the mathematical community at large."
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c_i4832w5xxosp
|
Ethnomathematics
|
Summary
|
Ethnomathematics
|
In mathematics education, ethnomathematics is the study of the relationship between mathematics and culture. Often associated with "cultures without written expression", it may also be defined as "the mathematics which is practised among identifiable cultural groups". It refers to a broad cluster of ideas ranging from distinct numerical and mathematical systems to multicultural mathematics education. The goal of ethnomathematics is to contribute both to the understanding of culture and the understanding of mathematics, and mainly to lead to an appreciation of the connections between the two.
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c_8hd2ds5cg86o
|
Precalculus
|
Summary
|
Precalculus
|
In mathematics education, precalculus is a course, or a set of courses, that includes algebra and trigonometry at a level which is designed to prepare students for the study of calculus, thus the name precalculus. Schools often distinguish between algebra and trigonometry as two separate parts of the coursework.
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c_tc7ab81fnjrn
|
Van Hiele model
|
Summary
|
Van_Hiele_model
|
In mathematics education, the Van Hiele model is a theory that describes how students learn geometry. The theory originated in 1957 in the doctoral dissertations of Dina van Hiele-Geldof and Pierre van Hiele (wife and husband) at Utrecht University, in the Netherlands. The Soviets did research on the theory in the 1960s and integrated their findings into their curricula. American researchers did several large studies on the van Hiele theory in the late 1970s and early 1980s, concluding that students' low van Hiele levels made it difficult to succeed in proof-oriented geometry courses and advising better preparation at earlier grade levels.
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c_56l8fggvp8n3
|
Van Hiele model
|
Summary
|
Van_Hiele_model
|
Pierre van Hiele published Structure and Insight in 1986, further describing his theory. The model has greatly influenced geometry curricula throughout the world through emphasis on analyzing properties and classification of shapes at early grade levels. In the United States, the theory has influenced the geometry strand of the Standards published by the National Council of Teachers of Mathematics and the Common Core Standards.
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c_kvwmgyn9hqa5
|
Multiplication and repeated addition
|
Summary
|
Multiplication_and_repeated_addition
|
In mathematics education, there was a debate on the issue of whether the operation of multiplication should be taught as being a form of repeated addition. Participants in the debate brought up multiple perspectives, including axioms of arithmetic, pedagogy, learning and instructional design, history of mathematics, philosophy of mathematics, and computer-based mathematics.
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c_l31gze3rmuaq
|
Unit fraction
|
Fair division and mathematics education
|
Unit_fractions > Applications > Fair division and mathematics education
|
In mathematics education, unit fractions are often introduced earlier than other kinds of fractions, because of the ease of explaining them visually as equal parts of a whole. A common practical use of unit fractions is to divide food equally among a number of people, and exercises in performing this sort of fair division are a standard classroom example in teaching students to work with unit fractions.
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c_i5dtrf9ak188
|
Process graph
|
Summary
|
Process_graph
|
In mathematics graph theory a process graph or P-graph is a directed bipartite graph used in workflow modeling.
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c_c0j96ebeqnso
|
Single-entry single-exit
|
Summary
|
Single-entry_single-exit
|
In mathematics graph theory, a single-entry single-exit (SESE) region in a given graph is an ordered edge pair. For example, with the ordered edge pair, (a, b) of distinct control-flow edges a and b where: a dominates b b postdominates a Every cycle containing a also contains b and vice versa.where a node x is said to dominate node y in a directed graph if every path from start to y includes x. A node x is said to postdominate a node y if every path from y to end includes x. So, a and b refer to the entry and exit edge, respectively. The first condition ensures that every path from start into the region passes through the region’s entry edge, a. The second condition ensures that every path from inside the region to end passes through the region’s exit edge, b. The first two conditions are necessary but not enough to characterize SESE regions: since backedges do not alter the dominance or postdominance relationships, the first two conditions alone do not prohibit backedges entering or exiting the region. The third condition encodes two constraints: every path from inside the region to a point 'above' a passed through b, and every path from a point 'below' b to a point inside the region passes through a. == References ==
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c_fsso98gaxvwj
|
Holomorphic separability
|
Summary
|
Holomorphic_separability
|
In mathematics in complex analysis, the concept of holomorphic separability is a measure of the richness of the set of holomorphic functions on a complex manifold or complex-analytic space.
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c_1nc9sju97vaj
|
Characterization of probability distributions
|
Summary
|
Characterization_of_probability_distributions
|
In mathematics in general, a characterization theorem says that a particular object – a function, a space, etc. – is the only one that possesses properties specified in the theorem. A characterization of a probability distribution accordingly states that it is the only probability distribution that satisfies specified conditions. More precisely, the model of characterization of probability distribution was described by V.M. Zolotarev in such manner.
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c_63vqxsblajfw
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Characterization of probability distributions
|
Summary
|
Characterization_of_probability_distributions
|
On the probability space we define the space X = { X } {\displaystyle {\mathcal {X}}=\{X\}} of random variables with values in measurable metric space ( U , d u ) {\displaystyle (U,d_{u})} and the space Y = { Y } {\displaystyle {\mathcal {Y}}=\{Y\}} of random variables with values in measurable metric space ( V , d v ) {\displaystyle (V,d_{v})} . By characterizations of probability distributions we understand general problems of description of some set C {\displaystyle {\mathcal {C}}} in the space X {\displaystyle {\mathcal {X}}} by extracting the sets A ⊆ X {\displaystyle {\mathcal {A}}\subseteq {\mathcal {X}}} and B ⊆ Y {\displaystyle {\mathcal {B}}\subseteq {\mathcal {Y}}} which describe the properties of random variables X ∈ A {\displaystyle X\in {\mathcal {A}}} and their images Y = F X ∈ B {\displaystyle Y=\mathbf {F} X\in {\mathcal {B}}} , obtained by means of a specially chosen mapping F: X → Y {\displaystyle \mathbf {F} :{\mathcal {X}}\to {\mathcal {Y}}} . The description of the properties of the random variables X {\displaystyle X} and of their images Y = F X {\displaystyle Y=\mathbf {F} X} is equivalent to the indication of the set A ⊆ X {\displaystyle {\mathcal {A}}\subseteq {\mathcal {X}}} from which X {\displaystyle X} must be taken and of the set B ⊆ Y {\displaystyle {\mathcal {B}}\subseteq {\mathcal {Y}}} into which its image must fall.
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c_def9nevko12x
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Characterization of probability distributions
|
Summary
|
Characterization_of_probability_distributions
|
So, the set which interests us appears therefore in the following form: X ∈ A , F X ∈ B ⇔ X ∈ C , i . e .
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c_61jxo2jn7zcv
|
Characterization of probability distributions
|
Summary
|
Characterization_of_probability_distributions
|
C = F − 1 B , {\displaystyle X\in {\mathcal {A}},\mathbf {F} X\in {\mathcal {B}}\Leftrightarrow X\in {\mathcal {C}},i.e.{\mathcal {C}}=\mathbf {F} ^{-1}{\mathcal {B}},} where F − 1 B {\displaystyle \mathbf {F} ^{-1}{\mathcal {B}}} denotes the complete inverse image of B {\displaystyle {\mathcal {B}}} in A {\displaystyle {\mathcal {A}}} . This is the general model of characterization of probability distribution. Some examples of characterization theorems: The assumption that two linear (or non-linear) statistics are identically distributed (or independent, or have a constancy regression and so on) can be used to characterize various populations.
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c_88mlu6xt4n0y
|
Characterization of probability distributions
|
Summary
|
Characterization_of_probability_distributions
|
"Memoryless" means that if X {\displaystyle X} is a random variable with such a distribution, then for any numbers 0 < y < x {\displaystyle 0 x ∣ X > y ) = Pr ( X > x − y ) {\displaystyle \Pr(X>x\mid X>y)=\Pr(X>x-y)} . Verification of conditions of characterization theorems in practice is possible only with some error ϵ {\displaystyle \epsilon } , i.e., only to a certain degree of accuracy. Such a situation is observed, for instance, in the cases where a sample of finite size is considered.
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c_e2435d05j7c2
|
Characterization of probability distributions
|
Summary
|
Characterization_of_probability_distributions
|
That is why there arises the following natural question. Suppose that the conditions of the characterization theorem are fulfilled not exactly but only approximately. May we assert that the conclusion of the theorem is also fulfilled approximately? The theorems in which the problems of this kind are considered are called stability characterizations of probability distributions.
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c_93w3iw7newhf
|
Cocurvature
|
Summary
|
Cocurvature
|
In mathematics in the branch of differential geometry, the cocurvature of a connection on a manifold is the obstruction to the integrability of the vertical bundle.
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c_jgl1j1mfmrsn
|
Bracket ring
|
Summary
|
Bracket_ring
|
In mathematics invariant theory, the bracket ring is the subring of the ring of polynomials k generated by the d-by-d minors of a generic d-by-n matrix (xij). The bracket ring may be regarded as the ring of polynomials on the image of a Grassmannian under the Plücker embedding.For given d ≤ n we define as formal variables the brackets with the λ taken from {1,...,n}, subject to = − and similarly for other transpositions. The set Λ(n,d) of size ( n d ) {\displaystyle {\binom {n}{d}}} generates a polynomial ring K over a field K. There is a homomorphism Φ(n,d) from K to the polynomial ring K in nd indeterminates given by mapping to the determinant of the d by d matrix consisting of the columns of the xi,j indexed by the λ. The bracket ring B(n,d) is the image of Φ. The kernel I(n,d) of Φ encodes the relations or syzygies that exist between the minors of a generic n by d matrix. The projective variety defined by the ideal I is the (n−d)d dimensional Grassmann variety whose points correspond to d-dimensional subspaces of an n-dimensional space.To compute with brackets it is necessary to determine when an expression lies in the ideal I(n,d). This is achieved by a straightening law due to Young (1928).
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c_xqh60vpdz87u
|
Normal convergence
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Summary
|
Normal_convergence
|
In mathematics normal convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is changed.
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c_x3s8xm7pjdgz
|
Nyström method
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Summary
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Nyström_method
|
In mathematics numerical analysis, the Nyström method or quadrature method seeks the numerical solution of an integral equation by replacing the integral with a representative weighted sum. The continuous problem is broken into n {\displaystyle n} discrete intervals; quadrature or numerical integration determines the weights and locations of representative points for the integral. The problem becomes a system of linear equations with n {\displaystyle n} equations and n {\displaystyle n} unknowns, and the underlying function is implicitly represented by an interpolation using the chosen quadrature rule.
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c_8bfo332vgcyf
|
Nyström method
|
Summary
|
Nyström_method
|
This discrete problem may be ill-conditioned, depending on the original problem and the chosen quadrature rule. Since the linear equations require O ( n 3 ) {\displaystyle O(n^{3})} operations to solve, high-order quadrature rules perform better because low-order quadrature rules require large n {\displaystyle n} for a given accuracy. Gaussian quadrature is normally a good choice for smooth, non-singular problems.
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c_tm7wxvo75mvu
|
Neuman–Sándor mean
|
Summary
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Neuman–Sándor_mean
|
In mathematics of special functions, the Neuman–Sándor mean M, of two positive and unequal numbers a and b, is defined as: M ( a , b ) = a − b 2 arsinh ( a − b a + b ) {\displaystyle M(a,b)={\frac {a-b}{2\operatorname {arsinh} \left({\frac {a-b}{a+b}}\right)}}} This mean interpolates the inequality of the unweighted arithmetic mean A = (a + b)/2) and of the second Seiffert mean T defined as: T ( a , b ) = a − b 2 arctan ( a − b a + b ) , {\displaystyle T(a,b)={\frac {a-b}{2\arctan \left({\frac {a-b}{a+b}}\right)}},} so that A < M < T. The M(a,b) mean, introduced by Edward Neuman and József Sándor, has recently been the subject of intensive research and many remarkable inequalities for this mean can be found in the literature. Several authors obtained sharp and optimal bounds for the Neuman–Sándor mean. Neuman and others utilized this mean to study other bivariate means and inequalities.
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c_qbws7lpt9enf
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Parity bit
|
Parity
|
Parity_check > Parity
|
In mathematics parity can refer to the evenness or oddness of an integer, which, when written in its binary form, can be determined just by examining only its least significant bit. In information technology parity refers to the evenness or oddness, given any set of binary digits, of the number of those bits with value one. Because parity is determined by the state of every one of the bits, this property of parity—being dependent upon all the bits and changing its value from even to odd parity if any one bit changes—allows for its use in error detection and correction schemes. In telecommunications the parity referred to by some protocols is for error-detection.
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c_tsv0gomh3a96
|
Parity bit
|
Parity
|
Parity_check > Parity
|
The transmission medium is preset, at both end points, to agree on either odd parity or even parity. For each string of bits ready to transmit (data packet) the sender calculates its parity bit, zero or one, to make it conform to the agreed parity, even or odd. The receiver of that packet first checks that the parity of the packet as a whole is in accordance with the preset agreement, then, if there was a parity error in that packet, requests a re-transmission of that packet.
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c_d57jwiy3rzjo
|
Parity bit
|
Parity
|
Parity_check > Parity
|
In computer science the parity stripe or parity disk in a RAID provides error-correction. Parity bits are written at the rate of one parity bit per n bits, where n is the number of disks in the array. When a read error occurs, each bit in the error region is recalculated from its set of n bits.
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c_qmup2gsxdfmd
|
Parity bit
|
Parity
|
Parity_check > Parity
|
In this way, using one parity bit creates "redundancy" for a region from the size of one bit to the size of one disk. See § Redundant Array of Independent Disks below. In electronics, transcoding data with parity can be very efficient, as XOR gates output what is equivalent to a check bit that creates an even parity, and XOR logic design easily scales to any number of inputs. XOR and AND structures comprise the bulk of most integrated circuitry.
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c_8ezuaertzupg
|
Cycle shape
|
Definition
|
Circular_notation > Definition
|
In mathematics texts it is customary to denote permutations using lowercase Greek letters. Commonly, either α {\displaystyle \alpha } and β , {\displaystyle \beta ,} or σ , τ {\displaystyle \sigma ,\tau } and π {\displaystyle \pi } are used.Permutations can be defined as bijections from a set S onto itself. All permutations of a set with n elements form a symmetric group, denoted S n {\displaystyle S_{n}} , where the group operation is function composition. Thus for two permutations, π {\displaystyle \pi } and σ {\displaystyle \sigma } in the group S n {\displaystyle S_{n}} , the four group axioms hold: Closure: If π {\displaystyle \pi } and σ {\displaystyle \sigma } are in S n {\displaystyle S_{n}} then so is π σ .
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c_9ys57f84hgbk
|
Cycle shape
|
Definition
|
Circular_notation > Definition
|
{\displaystyle \pi \sigma .} Associativity: For any three permutations π , σ , τ ∈ S n {\displaystyle \pi ,\sigma ,\tau \in S_{n}} , ( π σ ) τ = π ( σ τ ) . {\displaystyle (\pi \sigma )\tau =\pi (\sigma \tau ).}
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c_x3sr7la1whcd
|
Cycle shape
|
Definition
|
Circular_notation > Definition
|
Identity: There is an identity permutation, denoted id {\displaystyle \operatorname {id} } and defined by id ( x ) = x {\displaystyle \operatorname {id} (x)=x} for all x ∈ S {\displaystyle x\in S} . For any σ ∈ S n {\displaystyle \sigma \in S_{n}} , id σ = σ id = σ . {\displaystyle \operatorname {id} \sigma =\sigma \operatorname {id} =\sigma .}
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c_dw3og0rvjlx8
|
Cycle shape
|
Definition
|
Circular_notation > Definition
|
Invertibility: For every permutation π ∈ S n {\displaystyle \pi \in S_{n}} , there exists an inverse permutation π − 1 ∈ S n {\displaystyle \pi ^{-1}\in S_{n}} , so that π π − 1 = π − 1 π = id . {\displaystyle \pi \pi ^{-1}=\pi ^{-1}\pi =\operatorname {id} .} In general, composition of two permutations is not commutative, that is, π σ ≠ σ π .
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c_gv7a8c6y41qt
|
Cycle shape
|
Definition
|
Circular_notation > Definition
|
{\displaystyle \pi \sigma \neq \sigma \pi .} As a bijection from a set to itself, a permutation is a function that performs a rearrangement of a set, and is not an arrangement itself. An older and more elementary viewpoint is that permutations are the arrangements themselves.
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c_fxce5lx2wmeo
|
Cycle shape
|
Definition
|
Circular_notation > Definition
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To distinguish between these two, the identifiers active and passive are sometimes prefixed to the term permutation, whereas in older terminology substitutions and permutations are used.A permutation can be decomposed into one or more disjoint cycles, that is, the orbits, which are found by repeatedly tracing the application of the permutation on some elements. For example, the permutation σ {\displaystyle \sigma } defined by σ ( 7 ) = 7 {\displaystyle \sigma (7)=7} has a 1-cycle, ( 7 ) {\displaystyle (\,7\,)} while the permutation π {\displaystyle \pi } defined by π ( 2 ) = 3 {\displaystyle \pi (2)=3} and π ( 3 ) = 2 {\displaystyle \pi (3)=2} has a 2-cycle ( 2 3 ) {\displaystyle (\,2\,3\,)} (for details on the syntax, see § Cycle notation below). In general, a cycle of length k, that is, consisting of k elements, is called a k-cycle.
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c_1tmhayt4v23z
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Cycle shape
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Definition
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Circular_notation > Definition
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An element in a 1-cycle ( x ) {\displaystyle (\,x\,)} is called a fixed point of the permutation. A permutation with no fixed points is called a derangement. 2-cycles are called transpositions; such permutations merely exchange two elements, leaving the others fixed.
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c_r0h7iw9jj9xs
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Baum–Sweet sequence
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Summary
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Baum–Sweet_sequence
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In mathematics the Baum–Sweet sequence is an infinite automatic sequence of 0s and 1s defined by the rule: bn = 1 if the binary representation of n contains no block of consecutive 0s of odd length; bn = 0 otherwise;for n ≥ 0.For example, b4 = 1 because the binary representation of 4 is 100, which only contains one block of consecutive 0s of length 2; whereas b5 = 0 because the binary representation of 5 is 101, which contains a block of consecutive 0s of length 1. Starting at n = 0, the first few terms of the Baum–Sweet sequence are: 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1 ... (sequence A086747 in the OEIS)
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c_w61yz2ks4nme
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Function field sieve
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Summary
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Function_field_sieve
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In mathematics the Function Field Sieve is one of the most efficient algorithms to solve the Discrete Logarithm Problem (DLP) in a finite field. It has heuristic subexponential complexity. Leonard Adleman developed it in 1994 and then elaborated it together with M. D. Huang in 1999.
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c_7ooajd36nw42
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Function field sieve
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Summary
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Function_field_sieve
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Previous work includes the work of D. Coppersmith about the DLP in fields of characteristic two. The discrete logarithm problem in a finite field consists of solving the equation a x = b {\displaystyle a^{x}=b} for a , b ∈ F p n {\displaystyle a,b\in \mathbb {F} _{p^{n}}} , p {\displaystyle p} a prime number and n {\displaystyle n} an integer. The function f: F p n → F p n , x ↦ a x {\displaystyle f:\mathbb {F} _{p^{n}}\to \mathbb {F} _{p^{n}},x\mapsto a^{x}} for a fixed a ∈ F p n {\displaystyle a\in \mathbb {F} _{p^{n}}} is a one-way function used in cryptography. Several cryptographic methods are based on the DLP such as the Diffie-Hellman key exchange, the El Gamal cryptosystem and the Digital Signature Algorithm.
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c_5yq8czhbddys
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Goodwin–Staton integral
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Summary
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Goodwin–Staton_integral
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In mathematics the Goodwin–Staton integral is defined as: G ( z ) = ∫ 0 ∞ e − t 2 t + z d t {\displaystyle G(z)=\int _{0}^{\infty }{\frac {e^{-t^{2}}}{t+z}}\,dt} It satisfies the following third-order nonlinear differential equation： 4 w ( z ) + 8 z d d z w ( z ) + ( 2 + 2 z 2 ) d 2 d z 2 w ( z ) + z d 3 d z 3 w ( z ) = 0 {\displaystyle 4w(z)+8\,z{\frac {d}{dz}}w(z)+(2+2\,z^{2}){\frac {d^{2}}{dz^{2}}}w(z)+z{\frac {d^{3}}{dz^{3}}}w\left(z\right)=0}
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c_wpmqfn4lnmn9
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Gould polynomials
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Summary
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Gould_polynomials
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In mathematics the Gould polynomials Gn(x; a,b) are polynomials introduced by H. W. Gould and named by Roman in 1984. They are given by exp ( x f − 1 ( t ) ) = ∑ n = 0 ∞ G n ( x ; a , b ) t n n ! {\displaystyle \displaystyle \exp(xf^{-1}(t))=\sum _{n=0}^{\infty }G_{n}(x;a,b){\frac {t^{n}}{n!}}} where f ( t ) = e a t ( e b t − 1 ) {\displaystyle f(t)=e^{at}(e^{bt}-1)} so f − 1 ( t ) = 1 b ∑ k = 1 ∞ ( − ( b + a k ) / b k − 1 ) t k k {\displaystyle f^{-1}(t)={\frac {1}{b}}\sum _{k=1}^{\infty }{\binom {-(b+ak)/b}{k-1}}{\frac {t^{k}}{k}}} == References ==
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c_zd7ewgfje4md
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Jacobian ideal
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Summary
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Jacobian_ideal
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In mathematics the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let O ( x 1 , … , x n ) {\displaystyle {\mathcal {O}}(x_{1},\ldots ,x_{n})} denote the ring of smooth functions in n {\displaystyle n} variables and f {\displaystyle f} a function in the ring. The Jacobian ideal of f {\displaystyle f} is J f := ⟨ ∂ f ∂ x 1 , … , ∂ f ∂ x n ⟩ . {\displaystyle J_{f}:=\left\langle {\frac {\partial f}{\partial x_{1}}},\ldots ,{\frac {\partial f}{\partial x_{n}}}\right\rangle .}
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c_r5m9rghjnntz
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Split idempotent
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Summary
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Karoubi_envelope
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In mathematics the Karoubi envelope (or Cauchy completion or idempotent completion) of a category C is a classification of the idempotents of C, by means of an auxiliary category. Taking the Karoubi envelope of a preadditive category gives a pseudo-abelian category, hence the construction is sometimes called the pseudo-abelian completion. It is named for the French mathematician Max Karoubi.
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c_i0qzuzoa2nl4
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Split idempotent
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Summary
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Karoubi_envelope
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Given a category C, an idempotent of C is an endomorphism e: A → A {\displaystyle e:A\rightarrow A} with e ∘ e = e {\displaystyle e\circ e=e} .An idempotent e: A → A is said to split if there is an object B and morphisms f: A → B, g: B → A such that e = g f and 1B = f g. The Karoubi envelope of C, sometimes written Split(C), is the category whose objects are pairs of the form (A, e) where A is an object of C and e: A → A {\displaystyle e:A\rightarrow A} is an idempotent of C, and whose morphisms are the triples ( e , f , e ′ ): ( A , e ) → ( A ′ , e ′ ) {\displaystyle (e,f,e^{\prime }):(A,e)\rightarrow (A^{\prime },e^{\prime })} where f: A → A ′ {\displaystyle f:A\rightarrow A^{\prime }} is a morphism of C satisfying e ′ ∘ f = f = f ∘ e {\displaystyle e^{\prime }\circ f=f=f\circ e} (or equivalently f = e ′ ∘ f ∘ e {\displaystyle f=e'\circ f\circ e} ). Composition in Split(C) is as in C, but the identity morphism on ( A , e ) {\displaystyle (A,e)} in Split(C) is ( e , e , e ) {\displaystyle (e,e,e)} , rather than the identity on A {\displaystyle A} . The category C embeds fully and faithfully in Split(C).
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c_vrvvur86dg4w
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Split idempotent
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Summary
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Karoubi_envelope
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In Split(C) every idempotent splits, and Split(C) is the universal category with this property. The Karoubi envelope of a category C can therefore be considered as the "completion" of C which splits idempotents. The Karoubi envelope of a category C can equivalently be defined as the full subcategory of C ^ {\displaystyle {\hat {\mathbf {C} }}} (the presheaves over C) of retracts of representable functors. The category of presheaves on C is equivalent to the category of presheaves on Split(C).
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c_b54lblwlftdx
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Korovkin approximation
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Summary
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Korovkin_approximation
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In mathematics the Korovkin approximation is a convergence statement in which the approximation of a function is given by a certain sequence of functions. In practice a continuous function can be approximated by polynomials. With Korovkin approximations one comes a convergence for the whole approximation with examination of the convergence of the process at a finite number of functions. The Korovkin approximation is named after Pavel Korovkin. == References ==
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c_4br6x502b6pk
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Lawrence–Krammer representation
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Summary
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Lawrence–Krammer_representation
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In mathematics the Lawrence–Krammer representation is a representation of the braid groups. It fits into a family of representations called the Lawrence representations. The first Lawrence representation is the Burau representation and the second is the Lawrence–Krammer representation. The Lawrence–Krammer representation is named after Ruth Lawrence and Daan Krammer.
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c_24f34rdgq21w
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Markov theorem
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Summary
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Markov_theorem
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In mathematics the Markov theorem gives necessary and sufficient conditions for two braids to have closures that are equivalent knots or links. The conditions are stated in terms of the group structures on braids. Braids are algebraic objects described by diagrams; the relation to topology is given by Alexander's theorem which states that every knot or link in three-dimensional Euclidean space is the closure of a braid. The Markov theorem, proved by Russian mathematician Andrei Andreevich Markov Jr.
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c_syrgiv9f3kt6
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Markov theorem
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Summary
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Markov_theorem
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describes the elementary moves generating the equivalence relation on braids given by the equivalence of their closures. More precisely Markov's theorem can be stated as follows: given two braids represented by elements β n , β m ′ {\displaystyle \beta _{n},\beta _{m}'} in the braid groups B n , B m {\displaystyle B_{n},B_{m}} , their closures are equivalent links if and only if β m ′ {\displaystyle \beta _{m}'} can be obtained from applying to β n {\displaystyle \beta _{n}} a sequence of the following operations: conjugating β n {\displaystyle \beta _{n}} in B n {\displaystyle B_{n}} ; replacing β n {\displaystyle \beta _{n}} by β n σ n + 1 ± 1 ∈ B n + 1 {\displaystyle \beta _{n}\sigma _{n+1}^{\pm 1}\in B_{n+1}} (here σ i {\displaystyle \sigma _{i}} are the standard generators of the braid groups; geometrically this amounts to adding a strand to the right of the braid diagram and twisting it once with the (previously) last strand); the inverse of the previous operation (if β n = β n − 1 σ n ± 1 {\displaystyle \beta _{n}=\beta _{n-1}\sigma _{n}^{\pm 1}} with β n − 1 ∈ B n − 1 {\displaystyle \beta _{n-1}\in B_{n-1}} replace with β n − 1 {\displaystyle \beta _{n-1}} ). == References ==
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c_ago0kyxjtkkq
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Montgomery curve
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Summary
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Montgomery_curve
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In mathematics the Montgomery curve is a form of elliptic curve introduced by Peter L. Montgomery in 1987, different from the usual Weierstrass form. It is used for certain computations, and in particular in different cryptography applications.
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c_xsnlqwchbrkm
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Mott polynomials
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Summary
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Mott_polynomials
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In mathematics the Mott polynomials sn(x) are polynomials introduced by N. F. Mott (1932, p. 442) who applied them to a problem in the theory of electrons. They are given by the exponential generating function e x ( 1 − t 2 − 1 ) / t = ∑ n s n ( x ) t n / n ! .
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c_xtk52jtqc7by
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Mott polynomials
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Summary
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Mott_polynomials
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{\displaystyle e^{x({\sqrt {1-t^{2}}}-1)/t}=\sum _{n}s_{n}(x)t^{n}/n!.} Because the factor in the exponential has the power series 1 − t 2 − 1 t = − ∑ k ≥ 0 C k ( t 2 ) 2 k + 1 {\displaystyle {\frac {{\sqrt {1-t^{2}}}-1}{t}}=-\sum _{k\geq 0}C_{k}\left({\frac {t}{2}}\right)^{2k+1}} in terms of Catalan numbers C k {\displaystyle C_{k}} , the coefficient in front of x k {\displaystyle x^{k}} of the polynomial can be written as s n ( x ) = ( − 1 ) k n ! k !
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c_7hmymzf1s9hj
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Mott polynomials
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Summary
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Mott_polynomials
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2 n ∑ n = l 1 + l 2 + ⋯ + l k C ( l 1 − 1 ) / 2 C ( l 2 − 1 ) / 2 ⋯ C ( l k − 1 ) / 2 {\displaystyle s_{n}(x)=(-1)^{k}{\frac {n! }{k!2^{n}}}\sum _{n=l_{1}+l_{2}+\cdots +l_{k}}C_{(l_{1}-1)/2}C_{(l_{2}-1)/2}\cdots C_{(l_{k}-1)/2}} ,according to the general formula for generalized Appell polynomials, where the sum is over all compositions n = l 1 + l 2 + ⋯ + l k {\displaystyle n=l_{1}+l_{2}+\cdots +l_{k}} of n {\displaystyle n} into k {\displaystyle k} positive odd integers. The empty product appearing for k = n = 0 {\displaystyle k=n=0} equals 1.
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c_vaqb8i1sqnz2
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Mott polynomials
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Summary
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Mott_polynomials
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Special values, where all contributing Catalan numbers equal 1, are s n ( x ) = ( − 1 ) n 2 n . {\displaystyle s_{n}(x)={\frac {(-1)^{n}}{2^{n}}}.} s n ( x ) = ( − 1 ) n n ( n − 1 ) ( n − 2 ) 2 n .
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c_3m5836kp30qn
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Mott polynomials
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Summary
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Mott_polynomials
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{\displaystyle s_{n}(x)={\frac {(-1)^{n}n(n-1)(n-2)}{2^{n}}}.} By differentiation the recurrence for the first derivative becomes s ′ ( x ) = − ∑ k = 0 ⌊ ( n − 1 ) / 2 ⌋ n ! ( n − 1 − 2 k ) !
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c_rb58ux7h54cp
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Padovan cuboid spiral
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Summary
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Padovan_cuboid_spiral
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In mathematics the Padovan cuboid spiral is the spiral created by joining the diagonals of faces of successive cuboids added to a unit cube. The cuboids are added sequentially so that the resulting cuboid has dimensions that are successive Padovan numbers.The first cuboid is 1x1x1. The second is formed by adding to this a 1x1x1 cuboid to form a 1x1x2 cuboid. To this is added a 1x1x2 cuboid to form a 1x2x2 cuboid.
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c_sla5rj3ssjjo
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Padovan cuboid spiral
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Summary
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Padovan_cuboid_spiral
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This pattern continues, forming in succession a 2x2x3 cuboid, a 2x3x4 cuboid etc. Joining the diagonals of the exposed end of each new added cuboid creates a spiral (seen as the black line in the figure). The points on this spiral all lie in the same plane.The cuboids are added in a sequence that adds to the face in the positive y direction, then the positive x direction, then the positive z direction. This is followed by cuboids added in the negative y, negative x and negative z directions. Each new cuboid added has a length and width that matches the length and width of the face being added to. The height of the nth added cuboid is the nth Padovan number.Connecting alternate points where the spiral bends creates a series of triangles, where each triangle has two sides that are successive Padovan numbers and that has an obtuse angle of 120 degrees between these two sides.
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c_h0r7zqj1jlgf
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Petersson inner product
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Summary
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Petersson_inner_product
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In mathematics the Petersson inner product is an inner product defined on the space of entire modular forms. It was introduced by the German mathematician Hans Petersson.
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c_le3d0ekwnpmx
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Vicsek fractal
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Summary
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Vicsek_fractal
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In mathematics the Vicsek fractal, also known as Vicsek snowflake or box fractal, is a fractal arising from a construction similar to that of the Sierpinski carpet, proposed by Tamás Vicsek. It has applications including as compact antennas, particularly in cellular phones. Box fractal also refers to various iterated fractals created by a square or rectangular grid with various boxes removed or absent and, at each iteration, those present and/or those absent have the previous image scaled down and drawn within them. The Sierpinski triangle may be approximated by a 2 × 2 box fractal with one corner removed. The Sierpinski carpet is a 3 × 3 box fractal with the middle square removed.
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c_frljeuf8h2ee
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Quintuple product identity
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Summary
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Quintuple_product_identity
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In mathematics the Watson quintuple product identity is an infinite product identity introduced by Watson (1929) and rediscovered by Bailey (1951) and Gordon (1961). It is analogous to the Jacobi triple product identity, and is the Macdonald identity for a certain non-reduced affine root system. It is related to Euler's pentagonal number theorem.
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c_ha5774ueypju
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Differential calculus over commutative algebras
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Summary
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Differential_calculus_over_commutative_algebras
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In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this are: The whole topological information of a smooth manifold M {\displaystyle M} is encoded in the algebraic properties of its R {\displaystyle \mathbb {R} } -algebra of smooth functions A = C ∞ ( M ) , {\displaystyle A=C^{\infty }(M),} as in the Banach–Stone theorem. Vector bundles over M {\displaystyle M} correspond to projective finitely generated modules over A , {\displaystyle A,} via the functor Γ {\displaystyle \Gamma } which associates to a vector bundle its module of sections. Vector fields on M {\displaystyle M} are naturally identified with derivations of the algebra A {\displaystyle A} .
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c_dzpxq7j5zw3q
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Differential calculus over commutative algebras
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Summary
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Differential_calculus_over_commutative_algebras
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More generally, a linear differential operator of order k, sending sections of a vector bundle E → M {\displaystyle E\rightarrow M} to sections of another bundle F → M {\displaystyle F\rightarrow M} is seen to be an R {\displaystyle \mathbb {R} } -linear map Δ: Γ ( E ) → Γ ( F ) {\displaystyle \Delta :\Gamma (E)\to \Gamma (F)} between the associated modules, such that for any k + 1 {\displaystyle k+1} elements f 0 , … , f k ∈ A {\displaystyle f_{0},\ldots ,f_{k}\in A}: where the bracket : Γ ( E ) → Γ ( F ) {\displaystyle :\Gamma (E)\to \Gamma (F)} is defined as the commutator Denoting the set of k {\displaystyle k} th order linear differential operators from an A {\displaystyle A} -module P {\displaystyle P} to an A {\displaystyle A} -module Q {\displaystyle Q} with D i f f k ( P , Q ) {\displaystyle \mathrm {Diff} _{k}(P,Q)} we obtain a bi-functor with values in the category of A {\displaystyle A} -modules. Other natural concepts of calculus such as jet spaces, differential forms are then obtained as representing objects of the functors D i f f k {\displaystyle \mathrm {Diff} _{k}} and related functors. Seen from this point of view calculus may in fact be understood as the theory of these functors and their representing objects.
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c_fb5nzutuyw8u
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Differential calculus over commutative algebras
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Summary
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Differential_calculus_over_commutative_algebras
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Replacing the real numbers R {\displaystyle \mathbb {R} } with any commutative ring, and the algebra C ∞ ( M ) {\displaystyle C^{\infty }(M)} with any commutative algebra the above said remains meaningful, hence differential calculus can be developed for arbitrary commutative algebras. Many of these concepts are widely used in algebraic geometry, differential geometry and secondary calculus. Moreover, the theory generalizes naturally to the setting of graded commutative algebra, allowing for a natural foundation of calculus on supermanifolds, graded manifolds and associated concepts like the Berezin integral.
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c_3s9z5m3vf30v
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Discrete least squares meshless method
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Summary
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Discrete_least_squares_meshless_method
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In mathematics the discrete least squares meshless (DLSM) method is a meshless method based on the least squares concept. The method is based on the minimization of a least squares functional, defined as the weighted summation of the squared residual of the governing differential equation and its boundary conditions at nodal points used to discretize the domain and its boundaries.
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c_i6ykh3y5qvfp
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Division polynomial
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Summary
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Division_polynomial
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In mathematics the division polynomials provide a way to calculate multiples of points on elliptic curves and to study the fields generated by torsion points. They play a central role in the study of counting points on elliptic curves in Schoof's algorithm.
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c_q5bova2y1mhq
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Elliptic rational function
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Summary
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Elliptic_rational_function
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In mathematics the elliptic rational functions are a sequence of rational functions with real coefficients. Elliptic rational functions are extensively used in the design of elliptic electronic filters. (These functions are sometimes called Chebyshev rational functions, not to be confused with certain other functions of the same name).
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c_2z6rbvwpvpse
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Elliptic rational function
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Summary
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Elliptic_rational_function
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Rational elliptic functions are identified by a positive integer order n and include a parameter ξ ≥ 1 called the selectivity factor. A rational elliptic function of degree n in x with selectivity factor ξ is generally defined as: R n ( ξ , x ) ≡ c d ( n K ( 1 / L n ( ξ ) ) K ( 1 / ξ ) c d − 1 ( x , 1 / ξ ) , 1 / L n ( ξ ) ) {\displaystyle R_{n}(\xi ,x)\equiv \mathrm {cd} \left(n{\frac {K(1/L_{n}(\xi ))}{K(1/\xi )}}\,\mathrm {cd} ^{-1}(x,1/\xi ),1/L_{n}(\xi )\right)} where cd(u,k) is the Jacobi elliptic cosine function. K() is a complete elliptic integral of the first kind. L n ( ξ ) = R n ( ξ , ξ ) {\displaystyle L_{n}(\xi )=R_{n}(\xi ,\xi )} is the discrimination factor, equal to the minimum value of the magnitude of R n ( ξ , x ) {\displaystyle R_{n}(\xi ,x)} for | x | ≥ ξ {\displaystyle |x|\geq \xi } .For many cases, in particular for orders of the form n = 2a3b where a and b are integers, the elliptic rational functions can be expressed using algebraic functions alone. Elliptic rational functions are closely related to the Chebyshev polynomials: Just as the circular trigonometric functions are special cases of the Jacobi elliptic functions, so the Chebyshev polynomials are special cases of the elliptic rational functions.
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c_5ggp145qlu49
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Estimation lemma
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Summary
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Estimation_lemma
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In mathematics the estimation lemma, also known as the ML inequality, gives an upper bound for a contour integral. If f is a complex-valued, continuous function on the contour Γ and if its absolute value |f (z)| is bounded by a constant M for all z on Γ, then | ∫ Γ f ( z ) d z | ≤ M l ( Γ ) , {\displaystyle \left|\int _{\Gamma }f(z)\,dz\right|\leq M\,l(\Gamma ),} where l(Γ) is the arc length of Γ. In particular, we may take the maximum M := sup z ∈ Γ | f ( z ) | {\displaystyle M:=\sup _{z\in \Gamma }|f(z)|} as upper bound. Intuitively, the lemma is very simple to understand. If a contour is thought of as many smaller contour segments connected together, then there will be a maximum |f (z)| for each segment.
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c_y1nyafiyc769
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Estimation lemma
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Summary
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Estimation_lemma
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Out of all the maximum |f (z)|s for the segments, there will be an overall largest one. Hence, if the overall largest |f (z)| is summed over the entire path then the integral of f (z) over the path must be less than or equal to it. Formally, the inequality can be shown to hold using the definition of contour integral, the absolute value inequality for integrals and the formula for the length of a curve as follows: | ∫ Γ f ( z ) d z | = | ∫ α β f ( γ ( t ) ) γ ′ ( t ) d t | ≤ ∫ α β | f ( γ ( t ) ) | | γ ′ ( t ) | d t ≤ M ∫ α β | γ ′ ( t ) | d t = M l ( Γ ) {\displaystyle \left|\int _{\Gamma }f(z)\,dz\right|=\left|\int _{\alpha }^{\beta }f(\gamma (t))\gamma '(t)\,dt\right|\leq \int _{\alpha }^{\beta }\left|f(\gamma (t))\right|\left|\gamma '(t)\right|\,dt\leq M\int _{\alpha }^{\beta }\left|\gamma '(t)\right|\,dt=M\,l(\Gamma )} The estimation lemma is most commonly used as part of the methods of contour integration with the intent to show that the integral over part of a contour goes to zero as |z| goes to infinity. An example of such a case is shown below.
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c_vf5m2u80dw5h
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Finite Fourier transform
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Summary
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Finite_Fourier_transform
|
In mathematics the finite Fourier transform may refer to either another name for discrete-time Fourier transform (DTFT) of a finite-length series. E.g., F.J.Harris (pp. 52–53) describes the finite Fourier transform as a "continuous periodic function" and the discrete Fourier transform (DFT) as "a set of samples of the finite Fourier transform".
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c_ucwxak7tmg9q
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Finite Fourier transform
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Summary
|
Finite_Fourier_transform
|
In actual implementation, that is not two separate steps; the DFT replaces the DTFT. So J.Cooley (pp. 77–78) describes the implementation as discrete finite Fourier transform.or another name for the Fourier series coefficients.or another name for one snapshot of a short-time Fourier transform.
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c_vi8f2zmto0ve
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Let expression
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Let definition in mathematics
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Let_expression > Definition > Let definition in mathematics
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In mathematics the let expression is described as the conjunction of expressions. In functional languages the let expression is also used to limit scope. In mathematics scope is described by quantifiers.
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c_pckbora49ptt
|
Let expression
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Let definition in mathematics
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Let_expression > Definition > Let definition in mathematics
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The let expression is a conjunction within an existential quantifier. ( ∃ x E ∧ F ) ⟺ let x: E in F {\displaystyle (\exists xE\land F)\iff \operatorname {let} x:E\operatorname {in} F} where E and F are of type Boolean. The let expression allows the substitution to be applied to another expression.
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c_qwn5iv5im4wg
|
Let expression
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Let definition in mathematics
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Let_expression > Definition > Let definition in mathematics
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This substitution may be applied within a restricted scope, to a sub expression. The natural use of the let expression is in application to a restricted scope (called lambda dropping). These rules define how the scope may be restricted; { x ∉ FV ( E ) ∧ x ∈ FV ( F ) ⟹ let x: G in E F = E ( let x: G in F ) x ∈ FV ( E ) ∧ x ∉ FV ( F ) ⟹ let x: G in E F = ( let x: G in E ) F x ∉ FV ( E ) ∧ x ∉ FV ( F ) ⟹ let x: G in E F = E F {\displaystyle {\begin{cases}x\not \in \operatorname {FV} (E)\land x\in \operatorname {FV} (F)\implies \operatorname {let} x:G\operatorname {in} E\ F=E\ (\operatorname {let} x:G\operatorname {in} F)\\x\in \operatorname {FV} (E)\land x\not \in \operatorname {FV} (F)\implies \operatorname {let} x:G\operatorname {in} E\ F=(\operatorname {let} x:G\operatorname {in} E)\ F\\x\not \in \operatorname {FV} (E)\land x\not \in \operatorname {FV} (F)\implies \operatorname {let} x:G\operatorname {in} E\ F=E\ F\end{cases}}} where F is not of type Boolean.
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c_fvfqjb3c5rho
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Let expression
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Let definition in mathematics
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Let_expression > Definition > Let definition in mathematics
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From this definition the following standard definition of a let expression, as used in a functional language may be derived. x ∉ FV ( y ) ⟹ ( let x: x = y in z ) = z = ( λ x . z ) y {\displaystyle x\not \in \operatorname {FV} (y)\implies (\operatorname {let} x:x=y\operatorname {in} z)=z=(\lambda x.z)\ y} For simplicity the marker specifying the existential variable, x: {\displaystyle x:} , will be omitted from the expression where it is clear from the context. x ∉ FV ( y ) ⟹ ( let x = y in z ) = z = ( λ x . z ) y {\displaystyle x\not \in \operatorname {FV} (y)\implies (\operatorname {let} x=y\operatorname {in} z)=z=(\lambda x.z)\ y}
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c_r0zxo3f31vj0
|
Polynomial basis
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Summary
|
Monomial_form
|
In mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of all monomials. The monomials form a basis because every polynomial may be uniquely written as a finite linear combination of monomials (this is an immediate consequence of the definition of a polynomial).
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c_yvnlzqur1jov
|
Central binomial coefficient
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Summary
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Central_binomial_coefficient
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In mathematics the nth central binomial coefficient is the particular binomial coefficient ( 2 n n ) = ( 2 n ) ! ( n ! ) 2 = ∏ k = 1 n n + k k for all n ≥ 0. {\displaystyle {2n \choose n}={\frac {(2n)!}{(n!
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c_k2rs4jkuszn2
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Central binomial coefficient
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Summary
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Central_binomial_coefficient
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)^{2}}}=\prod \limits _{k=1}^{n}{\frac {n+k}{k}}{\text{ for all }}n\geq 0.} They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle. The first few central binomial coefficients starting at n = 0 are: 1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, ...; (sequence A000984 in the OEIS)
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c_67235qfzuwsb
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Negative sign
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Mathematics
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Positive_sign > Use as a qualifier > Mathematics
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In mathematics the one-sided limit x → a+ means x approaches a from the right (i.e., right-sided limit), and x → a− means x approaches a from the left (i.e., left-sided limit). For example, 1/x → + ∞ {\displaystyle \infty } as x → 0+ but 1/x → − ∞ {\displaystyle \infty } as x → 0−.
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c_m0ff2kaf9q6d
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Regular paperfolding sequence
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Summary
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Regular_paperfolding_sequence
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In mathematics the regular paperfolding sequence, also known as the dragon curve sequence, is an infinite sequence of 0s and 1s. It is obtained from the repeating partial sequence by filling in the question marks by another copy of the whole sequence. The first few terms of the resulting sequence are: If a strip of paper is folded repeatedly in half in the same direction, i {\displaystyle i} times, it will get 2 i − 1 {\displaystyle 2^{i}-1} folds, whose direction (left or right) is given by the pattern of 0's and 1's in the first 2 i − 1 {\displaystyle 2^{i}-1} terms of the regular paperfolding sequence. Opening out each fold to create a right-angled corner (or, equivalently, making a sequence of left and right turns through a regular grid, following the pattern of the paperfolding sequence) produces a sequence of polygonal chains that approaches the dragon curve fractal:
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c_97c9sf5dnkqf
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Signal-to-noise statistic
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Summary
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Signal-to-noise_statistic
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In mathematics the signal-to-noise statistic distance between two vectors a and b with mean values μ a {\displaystyle \mu _{a}} and μ b {\displaystyle \mu _{b}} and standard deviation σ a {\displaystyle \sigma _{a}} and σ b {\displaystyle \sigma _{b}} respectively is: D s n = ( μ a − μ b ) ( σ a + σ b ) {\displaystyle D_{sn}={(\mu _{a}-\mu _{b}) \over (\sigma _{a}+\sigma _{b})}} In the case of Gaussian-distributed data and unbiased class distributions, this statistic can be related to classification accuracy given an ideal linear discrimination, and a decision boundary can be derived.This distance is frequently used to identify vectors that have significant difference. One usage is in bioinformatics to locate genes that are differential expressed on microarray experiments.
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c_8olxmba1t1lc
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Spin group
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Summary
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Spinor_group
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In mathematics the spin group Spin(n) is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R), such that there exists a short exact sequence of Lie groups (when n ≠ 2) 1 → Z 2 → Spin ( n ) → SO ( n ) → 1. {\displaystyle 1\to \mathrm {Z} _{2}\to \operatorname {Spin} (n)\to \operatorname {SO} (n)\to 1.} The group multiplication law on the double cover is given by lifting the multiplication on SO ( n ) {\displaystyle \operatorname {SO} (n)} .
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c_7ni5sbyzy6yz
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Spin group
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Summary
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Spinor_group
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As a Lie group, Spin(n) therefore shares its dimension, n(n − 1)/2, and its Lie algebra with the special orthogonal group. For n > 2, Spin(n) is simply connected and so coincides with the universal cover of SO(n). The non-trivial element of the kernel is denoted −1, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted −I. Spin(n) can be constructed as a subgroup of the invertible elements in the Clifford algebra Cl(n). A distinct article discusses the spin representations.
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c_abk59mk6nef8
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Symmetrization methods
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Summary
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Symmetrization_methods
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In mathematics the symmetrization methods are algorithms of transforming a set A ⊂ R n {\displaystyle A\subset \mathbb {R} ^{n}} to a ball B ⊂ R n {\displaystyle B\subset \mathbb {R} ^{n}} with equal volume vol ( B ) = vol ( A ) {\displaystyle \operatorname {vol} (B)=\operatorname {vol} (A)} and centered at the origin. B is called the symmetrized version of A, usually denoted A ∗ {\displaystyle A^{*}} . These algorithms show up in solving the classical isoperimetric inequality problem, which asks: Given all two-dimensional shapes of a given area, which of them has the minimal perimeter (for details see Isoperimetric inequality). The conjectured answer was the disk and Steiner in 1838 showed this to be true using the Steiner symmetrization method (described below).
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c_2qq62c1y7nwt
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Symmetrization methods
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Summary
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Symmetrization_methods
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From this many other isoperimetric problems sprung and other symmetrization algorithms. For example, Rayleigh's conjecture is that the first eigenvalue of the Dirichlet problem is minimized for the ball (see Rayleigh–Faber–Krahn inequality for details). Another problem is that the Newtonian capacity of a set A is minimized by A ∗ {\displaystyle A^{*}} and this was proved by Polya and G. Szego (1951) using circular symmetrization (described below).
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c_qs84sdy6jbs8
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Synchrotron function
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Summary
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Synchrotron_function
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In mathematics the synchrotron functions are defined as follows (for x ≥ 0): First synchrotron function F ( x ) = x ∫ x ∞ K 5 3 ( t ) d t {\displaystyle F(x)=x\int _{x}^{\infty }K_{\frac {5}{3}}(t)\,dt} Second synchrotron function G ( x ) = x K 2 3 ( x ) {\displaystyle G(x)=xK_{\frac {2}{3}}(x)} where Kj is the modified Bessel function of the second kind.
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