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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 adjo... | https://en.wikipedia.org/wiki/Number_bond |
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 qu... | https://en.wikipedia.org/wiki/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 numb... | https://en.wikipedia.org/wiki/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: ... | https://en.wikipedia.org/wiki/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 ch... | https://en.wikipedia.org/wiki/Chunking_(division) |
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 fin... | https://en.wikipedia.org/wiki/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 br... | https://en.wikipedia.org/wiki/Finite_mathematics |
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... | https://en.wikipedia.org/wiki/Mathematical_manipulative |
The second and third steps are representational and abstract, respectively. Mathematical manipulatives can be purchased or constructed by the teacher. | https://en.wikipedia.org/wiki/Mathematical_manipulative |
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 count... | https://en.wikipedia.org/wiki/Mathematical_manipulative |
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... | https://en.wikipedia.org/wiki/Mathematical_manipulative |
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. | https://en.wikipedia.org/wiki/Number_sentence |
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... | https://en.wikipedia.org/wiki/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... | https://en.wikipedia.org/wiki/Procept |
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,... | https://en.wikipedia.org/wiki/Multiple_representations_(mathematics_education) |
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 c... | https://en.wikipedia.org/wiki/Differential_and_integral_calculus |
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 c... | https://en.wikipedia.org/wiki/History_of_calculus |
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,... | https://en.wikipedia.org/wiki/Concept_image_and_concept_definition |
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 ... | https://en.wikipedia.org/wiki/Ethnomathematics |
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. | https://en.wikipedia.org/wiki/Precalculus |
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 1960... | https://en.wikipedia.org/wiki/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 stra... | https://en.wikipedia.org/wiki/Van_Hiele_model |
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, ph... | https://en.wikipedia.org/wiki/Multiplication_and_repeated_addition |
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 divis... | https://en.wikipedia.org/wiki/Unit_fraction |
In mathematics graph theory a process graph or P-graph is a directed bipartite graph used in workflow modeling. | https://en.wikipedia.org/wiki/Process_graph |
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 ... | https://en.wikipedia.org/wiki/Single-entry_single-exit |
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. | https://en.wikipedia.org/wiki/Holomorphic_separability |
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 ... | https://en.wikipedia.org/wiki/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 )... | https://en.wikipedia.org/wiki/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 . | https://en.wikipedia.org/wiki/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 {\displaystyl... | https://en.wikipedia.org/wiki/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 ... | https://en.wikipedia.org/wiki/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 call... | https://en.wikipedia.org/wiki/Characterization_of_probability_distributions |
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. | https://en.wikipedia.org/wiki/Cocurvature |
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 variab... | https://en.wikipedia.org/wiki/Bracket_ring |
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. | https://en.wikipedia.org/wiki/Normal_convergence |
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 w... | https://en.wikipedia.org/wiki/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... | https://en.wikipedia.org/wiki/Nyström_method |
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 ... | https://en.wikipedia.org/wiki/Neuman–Sándor_mean |
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... | https://en.wikipedia.org/wiki/Parity_bit |
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 o... | https://en.wikipedia.org/wiki/Parity_bit |
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. | https://en.wikipedia.org/wiki/Parity_bit |
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, a... | https://en.wikipedia.org/wiki/Parity_bit |
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 permuta... | https://en.wikipedia.org/wiki/Cycle_shape |
{\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 ).} | https://en.wikipedia.org/wiki/Cycle_shape |
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... | https://en.wikipedia.org/wiki/Cycle_shape |
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 commutati... | https://en.wikipedia.org/wiki/Cycle_shape |
{\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. | https://en.wikipedia.org/wiki/Cycle_shape |
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 appli... | https://en.wikipedia.org/wiki/Cycle_shape |
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. | https://en.wikipedia.org/wiki/Cycle_shape |
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 blo... | https://en.wikipedia.org/wiki/Baum–Sweet_sequence |
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. | https://en.wikipedia.org/wiki/Function_field_sieve |
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 {\di... | https://en.wikipedia.org/wiki/Function_field_sieve |
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 ) ... | https://en.wikipedia.org/wiki/Goodwin–Staton_integral |
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 ... | https://en.wikipedia.org/wiki/Gould_polynomials |
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 ... | https://en.wikipedia.org/wiki/Jacobian_ideal |
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... | https://en.wikipedia.org/wiki/Split_idempotent |
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 Spl... | https://en.wikipedia.org/wiki/Split_idempotent |
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 ^ {\displaysty... | https://en.wikipedia.org/wiki/Split_idempotent |
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 examinati... | https://en.wikipedia.org/wiki/Korovkin_approximation |
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 representati... | https://en.wikipedia.org/wiki/Lawrence–Krammer_representation |
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 theor... | https://en.wikipedia.org/wiki/Markov_theorem |
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... | https://en.wikipedia.org/wiki/Markov_theorem |
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. | https://en.wikipedia.org/wiki/Montgomery_curve |
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 ! . | https://en.wikipedia.org/wiki/Mott_polynomials |
{\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 {\disp... | https://en.wikipedia.org/wiki/Mott_polynomials |
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... | https://en.wikipedia.org/wiki/Mott_polynomials |
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 . | https://en.wikipedia.org/wiki/Mott_polynomials |
{\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 ) ! | https://en.wikipedia.org/wiki/Mott_polynomials |
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 thi... | https://en.wikipedia.org/wiki/Padovan_cuboid_spiral |
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 th... | https://en.wikipedia.org/wiki/Padovan_cuboid_spiral |
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. | https://en.wikipedia.org/wiki/Petersson_inner_product |
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 f... | https://en.wikipedia.org/wiki/Vicsek_fractal |
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'... | https://en.wikipedia.org/wiki/Quintuple_product_identity |
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 {\displa... | https://en.wikipedia.org/wiki/Differential_calculus_over_commutative_algebras |
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 \Ga... | https://en.wikipedia.org/wiki/Differential_calculus_over_commutative_algebras |
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... | https://en.wikipedia.org/wiki/Differential_calculus_over_commutative_algebras |
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... | https://en.wikipedia.org/wiki/Discrete_least_squares_meshless_method |
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. | https://en.wikipedia.org/wiki/Division_polynomial |
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 o... | https://en.wikipedia.org/wiki/Elliptic_rational_function |
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 (... | https://en.wikipedia.org/wiki/Elliptic_rational_function |
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 ... | https://en.wikipedia.org/wiki/Estimation_lemma |
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, ... | https://en.wikipedia.org/wiki/Estimation_lemma |
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 ... | https://en.wikipedia.org/wiki/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. | https://en.wikipedia.org/wiki/Finite_Fourier_transform |
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. | https://en.wikipedia.org/wiki/Let_expression |
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. | https://en.wikipedia.org/wiki/Let_expression |
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... | https://en.wikipedia.org/wiki/Let_expression |
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 sim... | https://en.wikipedia.org/wiki/Let_expression |
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 cons... | https://en.wikipedia.org/wiki/Polynomial_basis |
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! | https://en.wikipedia.org/wiki/Central_binomial_coefficient |
)^{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 A0... | https://en.wikipedia.org/wiki/Central_binomial_coefficient |
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−. | https://en.wikipedia.org/wiki/Negative_sign |
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 pap... | https://en.wikipedia.org/wiki/Regular_paperfolding_sequence |
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 ... | https://en.wikipedia.org/wiki/Signal-to-noise_statistic |
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 {Sp... | https://en.wikipedia.org/wiki/Spin_group |
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 tran... | https://en.wikipedia.org/wiki/Spin_group |
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 ori... | https://en.wikipedia.org/wiki/Symmetrization_methods |
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 minim... | https://en.wikipedia.org/wiki/Symmetrization_methods |
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 modifie... | https://en.wikipedia.org/wiki/Synchrotron_function |
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