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Curve-shortening flow Summary Gage–Hamilton–Grayson_theorem It loses area at a constant rate, and its perimeter decreases as quickly as possible for any continuous curve evolution. If the curve is non-convex, its total absolute curvature decreases monotonically, until it becomes convex. Once convex, the isoperimetric r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Curve-shortening flow Summary Gage–Hamilton–Grayson_theorem If two disjoint simple smooth closed curves evolve, they remain disjoint until one of them collapses to a point. The circle is the only simple closed curve that maintains its shape under the curve-shortening flow, but some curves that cross themselves or have ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Curve-shortening flow Summary Gage–Hamilton–Grayson_theorem An approximation to the curve-shortening flow can be computed numerically, by approximating the curve as a polygon and using the finite difference method to calculate the motion of each polygon vertex. Alternative methods include computing a convolution of pol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Curve-shortening flow Summary Gage–Hamilton–Grayson_theorem Later, it was applied in image analysis to give a multi-scale representation of shapes. It can also model reaction–diffusion systems, and the behavior of cellular automata. The curve-shortening flow can be used to find closed geodesics on Riemannian manifolds,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cycles and fixed points Summary Cycles_and_fixed_points In mathematics, the cycles of a permutation π of a finite set S correspond bijectively to the orbits of the subgroup generated by π acting on S. These orbits are subsets of S that can be written as { c1, ..., cn }, such that π(ci) = ci + 1 for i = 1, ..., n − 1, a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cycles and fixed points Summary Cycles_and_fixed_points For example, let π = ( 1 6 7 2 5 4 8 3 2 8 7 4 5 3 6 1 ) = ( 1 2 3 4 5 6 7 8 2 4 1 3 5 8 7 6 ) {\displaystyle \pi ={\begin{pmatrix}1&6&7&2&5&4&8&3\\2&8&7&4&5&3&6&1\end{pmatrix}}={\begin{pmatrix}1&2&3&4&5&6&7&8\\2&4&1&3&5&8&7&6\end{pmatrix}}} be a permutation that ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cyclic category Summary Cyclic_category In mathematics, the cyclic category or cycle category or category of cycles is a category of finite cyclically ordered sets and degree-1 maps between them. It was introduced by Connes (1983). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cyclotomic identity Summary Cyclotomic_identity In mathematics, the cyclotomic identity states that 1 1 − α z = ∏ j = 1 ∞ ( 1 1 − z j ) M ( α , j ) {\displaystyle {1 \over 1-\alpha z}=\prod _{j=1}^{\infty }\left({1 \over 1-z^{j}}\right)^{M(\alpha ,j)}} where M is Moreau's necklace-counting function, M ( α , n ) = 1 n ∑... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cylinder set Summary Cylinder_set In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cylindrical harmonics Summary Cylindrical_harmonics In mathematics, the cylindrical harmonics are a set of linearly independent functions that are solutions to Laplace's differential equation, ∇ 2 V = 0 {\displaystyle \nabla ^{2}V=0} , expressed in cylindrical coordinates, ρ (radial coordinate), φ (polar angle), and z ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
De Franchis theorem Summary De_Franchis_theorem In mathematics, the de Franchis theorem is one of a number of closely related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, X and Y, in the case of genus g > 1. The simplest is that the automorphism group of X is finite (see though... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
List of definite integrals Summary List_of_definite_integrals In mathematics, the definite integral ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
List of definite integrals Summary List_of_definite_integrals for example: ∫ a ∞ f ( x ) d x = lim b → ∞ {\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{b\to \infty }\left} A constant, such pi, that may be defined by the integral of an algebraic function over an algebraic domain is known as a period. The following i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polynomial degree Summary Polynomial_degree In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polynomial degree Summary Polynomial_degree The term order has been used as a synonym of degree but, nowadays, may refer to several other concepts (see Order of a polynomial (disambiguation)). For example, the polynomial 7 x 2 y 3 + 4 x − 9 , {\displaystyle 7x^{2}y^{3}+4x-9,} which can also be written as 7 x 2 y 3 + 4 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polynomial degree Summary Polynomial_degree Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as ( x + 1 ) 2 − ( x − 1 ) 2 {\displaystyle (x+1)^{2}-(x-1)^{2}} , one can put it in standard form by expanding the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Degree (algebraic geometry) Summary Degree_of_an_algebraic_variety In mathematics, the degree of an affine or projective variety of dimension n is the number of intersection points of the variety with n hyperplanes in general position. For an algebraic set, the intersection points must be counted with their intersectio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Degree (algebraic geometry) Summary Degree_of_an_algebraic_variety This is a generalization of Bézout's theorem (For a proof, see Hilbert series and Hilbert polynomial § Degree of a projective variety and Bézout's theorem). The degree is not an intrinsic property of the variety, as it depends on a specific embedding of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Degree (algebraic geometry) Summary Degree_of_an_algebraic_variety A generalization of Bézout's theorem asserts that, if an intersection of n projective hypersurfaces has codimension n, then the degree of the intersection is the product of the degrees of the hypersurfaces. The degree of a projective variety is the eval... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generalized derivative Summary Derivative_(generalizations) In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
First-order derivative expression Summary Derivative In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's veloci... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
First-order derivative expression Summary Derivative The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
First-order derivative expression Summary Derivative In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear tra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
First-order derivative expression Summary Derivative For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector. The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidiff... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Derived category Summary Derived_category In mathematics, the derived category D(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects of D(A) shoul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Derived category Summary Derived_category The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had made remarkable strides. The ba... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Derived category Summary Derived_category The original impulse to develop the "derived" formalism came from the need to find a suitable formulation of Grothendieck's coherent duality theory. Derived categories have since become indispensable also outside of algebraic geometry, for example in the formulation of the theo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Determinant (mathematics) Summary Determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the linear map represented by the matri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Determinant (mathematics) Summary Determinant The determinant of a product of matrices is the product of their determinants (which follows directly from the above properties). The determinant of a 2 × 2 matrix is | a b c d | = a d − b c , {\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc,} and the determinant... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Determinant (mathematics) Summary Determinant The determinant of an n × n matrix can be defined in several equivalent ways, the most common being Leibniz formula, which expresses the determinant as a sum of n ! {\displaystyle n!} (the factorial of n) signed products of matrix entries. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Determinant (mathematics) Summary Determinant It can be computed by the Laplace expansion, which expresses the determinant as a linear combination of determinants of submatrices, or with Gaussian elimination, which expresses the determinant as the product of the diagonal entries of a diagonal matrix that is obtained by... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Determinant (mathematics) Summary Determinant (The above properties relating to rows may be replaced by the corresponding statements with respect to columns.) Determinants occur throughout mathematics. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Determinant (mathematics) Summary Determinant For example, a matrix is often used to represent the coefficients in a system of linear equations, and determinants can be used to solve these equations (Cramer's rule), although other methods of solution are computationally much more efficient. Determinants are used for de... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Determinant method Summary Determinant_method In mathematics, the determinant method is any of a family of techniques in analytic number theory. The name was coined by Roger Heath-Brown and comes from the fact that the center piece of the method is estimating a certain determinant. Its main application is to give an up... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Determinantal conjecture Summary Determinantal_conjecture In mathematics, the determinantal conjecture of Marcus (1972) and de Oliveira (1982) asks whether the determinant of a sum A + B of two n by n normal complex matrices A and B lies in the convex hull of the n! points Πi (λ(A)i + λ(B)σ(i)), where the numbers λ(A)i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Difference of two squares Summary Difference_of_squares In mathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity a 2 − b 2 = ( a + b ) ( a − b ) {\displaystyle a^{2}-b^{2}=(a+b)(... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Differentiable surface Summary Shape_operator In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Differentiable surface Summary Shape_operator An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of the Euclidean plane, the sphere and the hyperbolic plane. These Lie groups can be used to describe surfaces of constant Gaussian curvature; ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gauss's digamma theorem Summary Gauss's_digamma_theorem In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z ) . {\displaystyle \psi (z)={\frac {\mathrm {d} }{\mathrm {d} z}}\ln \Gamma (z)={\frac {\Gamma '(z)}{\Gamma (z)}}.} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Digit sum Summary Digit_sum In mathematics, the digit sum of a natural number in a given number base is the sum of all its digits. For example, the digit sum of the decimal number 9045 {\displaystyle 9045} would be 9 + 0 + 4 + 5 = 18. {\displaystyle 9+0+4+5=18.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Order dimension Summary Order_dimension In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order. This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order. Dushn... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinite-dimensional vector space Summary Finite_dimensional In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinite-dimensional vector space Summary Finite_dimensional We say V {\displaystyle V} is finite-dimensional if the dimension of V {\displaystyle V} is finite, and infinite-dimensional if its dimension is infinite. The dimension of the vector space V {\displaystyle V} over the field F {\displaystyle F} can be written ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multidimensional geometry In mathematics Higher-dimensional_space > In mathematics In mathematics, the dimension of an object is, roughly speaking, the number of degrees of freedom of a point that moves on this object. In other words, the dimension is the number of independent parameters or coordinates that are needed ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multidimensional geometry In mathematics Higher-dimensional_space > In mathematics This is independent from the fact that a curve cannot be embedded in a Euclidean space of dimension lower than two, unless it is a line. The dimension of Euclidean n-space En is n. When trying to generalize to other types of spaces, one ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multidimensional geometry In mathematics Higher-dimensional_space > In mathematics This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, but there are also other answers to that question. For example, the boundary of a ball in En looks locally l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multidimensional geometry In mathematics Higher-dimensional_space > In mathematics A tesseract is an example of a four-dimensional object. Whereas outside mathematics the use of the term "dimension" is as in: "A tesseract has four dimensions", mathematicians usually express this as: "The tesseract has dimension 4", or:... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dimension theorem for vector spaces Summary Dimension_theorem_for_vector_spaces In mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements. This number of elements may be finite or infinite (in the latter case, it is a cardinal number), and defines the dim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dimension theorem for vector spaces Summary Dimension_theorem_for_vector_spaces While the proof of the existence of a basis for any vector space in the general case requires Zorn's lemma and is in fact equivalent to the axiom of choice, the uniqueness of the cardinality of the basis requires only the ultrafilter lemma,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Direct image Summary Higher_direct_image_functor In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf F defined on a topological space X and a contin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Direct image with compact support Summary Direct_image_with_compact_support In mathematics, the direct image with compact (or proper) support is an image functor for sheaves that extends the compactly supported global sections functor to the relative setting. It is one of Grothendieck's six operations. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Direct method in calculus of variations Summary Direct_method_in_calculus_of_variations In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Stanisław Zaremba and David Hilbert around 1900. The ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discrete Fourier Transform Summary Circular_cross-correlation In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discrete Fourier Transform Summary Circular_cross-correlation The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discrete Fourier Transform Summary Circular_cross-correlation The DFT is the most important discrete transform, used to perform Fourier analysis in many practical applications. In digital signal processing, the function is any quantity or signal that varies over time, such as the pressure of a sound wave, a radio signa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discrete Fourier Transform Summary Circular_cross-correlation The DFT is also used to efficiently solve partial differential equations, and to perform other operations such as convolutions or multiplying large integers. Since it deals with a finite amount of data, it can be implemented in computers by numerical algorit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Number-theoretic transform Summary Discrete_weighted_transform In mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex numbers, over an arbitrary ring. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discrete Laplace operator Summary Discrete_Laplace_operator In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. For the case of a finite-dimensional graph (having a finite number of edges and vertices), the discrete... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discrete Poisson equation Summary Discrete_Poisson_equation In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place of the Laplace operator. The discrete Poisson equation is frequently used in numerical analysis as a sta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discrete exterior calculus Summary Discrete_exterior_calculus In mathematics, the discrete exterior calculus (DEC) is the extension of the exterior calculus to discrete spaces including graphs, finite element meshes, and lately also general polygonal meshes (non-flat and non-convex). DEC methods have proved to be very ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discrete q-Hermite polynomials Summary Discrete_q-Hermite_polynomials In mathematics, the discrete q-Hermite polynomials are two closely related families hn(x;q) and ĥn(x;q) of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Al-Salam and Carlitz (1965). Roelof Koekoek, Peter A. Lesk... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discrete sine transform Summary Discrete_sine_transform In mathematics, the discrete sine transform (DST) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using a purely real matrix. It is equivalent to the imaginary parts of a DFT of roughly twice the length, operating on real data w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discrete sine transform Summary Discrete_sine_transform See the DCT article for a general discussion of how the boundary conditions relate the various DCT and DST types. Generally, the DST is derived from the DCT by replacing the Neumann condition at x=0 with a Dirichlet condition. Both the DCT and the DST were describ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discrete sine transform Summary Discrete_sine_transform Rao in 1974. The type-I DST (DST-I) was later described by Anil K. Jain in 1976, and the type-II DST (DST-II) was then described by H.B. Kekra and J.K. Solanka in 1978. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discrete-time Fourier transform Summary Discrete-time_Fourier_transform In mathematics, the discrete-time Fourier transform (DTFT), also called the finite Fourier transform, is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discrete-time Fourier transform Summary Discrete-time_Fourier_transform Under certain theoretical conditions, described by the sampling theorem, the original continuous function can be recovered perfectly from the DTFT and thus from the original discrete samples. The DTFT itself is a continuous function of frequency, b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discrete-time Fourier transform Summary Discrete-time_Fourier_transform The inverse DTFT is the original sampled data sequence. The inverse DFT is a periodic summation of the original sequence. The fast Fourier transform (FFT) is an algorithm for computing one cycle of the DFT, and its inverse produces one cycle of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discriminant of a polynomial Summary Discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discriminant of a polynomial Summary Discriminant If a ≠ 0 , {\displaystyle a\neq 0,} this discriminant is zero if and only if the polynomial has a double root. In the case of real coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinct complex conjugate roots. Si... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Discriminant of a polynomial Summary Discriminant In the case of a cubic with real coefficients, the discriminant is positive if the polynomial has three distinct real roots, and negative if it has one real root and two distinct complex conjugate roots. More generally, the discriminant of a univariate polynomial of pos... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Brill's theorem Summary Discriminant_of_a_number_field In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field. More specifically, it is proportional to the squared volume of the fundamental ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Radon–Hurwitz number Summary Radon–Hurwitz_number In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras. Specifically, the question is how many linearly independent smo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Disintegration theorem Summary Disintegration_theorem In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of the measure space in question. It is related to the existence of c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Distortion (mathematics) Summary Distortion_(mathematics) In mathematics, the distortion is a measure of the amount by which a function from the Euclidean plane to itself distorts circles to ellipses. If the distortion of a function is equal to one, then it is conformal; if the distortion is bounded and the function is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Distortion (mathematics) Summary Distortion_(mathematics) Here Df is the weak derivative of ƒ, and |Df| is the Hilbert–Schmidt norm. For functions on a higher-dimensional Euclidean space Rn, there are more measures of distortion because there are more than two principal axes of a symmetric tensor. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Distortion (mathematics) Summary Distortion_(mathematics) The pointwise information is contained in the distortion tensor G ( x , f ) = { | J ( x , f ) | − 2 / n D T f ( x ) D f ( x ) if J ( x , f ) ≠ 0 I if J ( x , f ) = 0. {\displaystyle G(x,f)={\begin{cases}|J(x,f)|^{-2/n}D^{T}f(x)Df(x)&{\text{if }}J(x,f)\not =0\\I&... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Antidistributive Summary Antidistributive In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality is always true in elementary algebra. For example, in elementary arithmetic, one has Therefore, one would say that multiplication distribu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Domain (function) Summary Domain_(function) In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ( f ) {\displaystyle \operatorname {dom} (f)} or dom f {\displaystyle \operatorname {dom} f} , where f is the function. In layman's terms, the domain of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Domain (function) Summary Domain_(function) In this case, the domain is represented on the x-axis of the graph, as the projection of the graph of the function onto the x-axis. For a function f: X → Y {\displaystyle f\colon X\to Y} , the set Y is called the codomain, and the set of values attained by the function (which... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Vector dot product Summary Dot_products In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely use... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Vector dot product Summary Dot_products Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Double Fourier sphere method Summary Double_Fourier_sphere_method In mathematics, the double Fourier sphere (DFS) method is a simple technique that transforms a function defined on the surface of the sphere to a function defined on a rectangular domain while preserving periodicity in both the longitude and latitude dir... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Double factorial Summary Double_factorial In mathematics, the double factorial of a number n, denoted by n‼, is the product of all the positive integers up to n that have the same parity (odd or even) as n. That is, Restated, this says that for even n, the double factorial is while for odd n it is For example, 9‼ = 9 ×... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Doubly triangular number Summary Doubly_triangular_number In mathematics, the doubly triangular numbers are the numbers that appear within the sequence of triangular numbers, in positions that are also triangular numbers. That is, if T n = n ( n + 1 ) / 2 {\displaystyle T_{n}=n(n+1)/2} denotes the n {\displaystyle n} t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dual Hahn polynomials Summary Dual_Hahn_polynomials In mathematics, the dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined on a non-uniform lattice x ( s ) = s ( s + 1 ) {\displaystyle x(s)=s(s+1)} and are defined as w n ( c ) ( s ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dual Hahn polynomials Summary Dual_Hahn_polynomials }}{}_{3}F_{2}(-n,a-s,a+s+1;a-b+a,a+c+1;1)} for n = 0 , 1 , . . . , N − 1 {\displaystyle n=0,1,...,N-1} and the parameters a , b , c {\displaystyle a,b,c} are restricted to − 1 2 < a < b , | c | < 1 + a , b = a + N {\displaystyle -{\frac {1}{2}} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dual bundle Summary Dual_bundle In mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dual module Summary Dual_module In mathematics, the dual module of a left (respectively right) module M over a ring R is the set of module homomorphisms from M to R with the pointwise right (respectively left) module structure. The dual module is typically denoted M∗ or HomR(M, R). If the base ring R is a field, then a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dual module Summary Dual_module A reflexive module is one for which the canonical homomorphism is an isomorphism. A torsionless module is one for which the canonical homomorphism is injective. Example: If G = Spec ( A ) {\displaystyle G=\operatorname {Spec} (A)} is a finite commutative group scheme represented by a H... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dual q-Hahn polynomials Summary Dual_q-Hahn_polynomials In mathematics, the dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dual q-Krawtchouk polynomials Summary Dual_q-Krawtchouk_polynomials In mathematics, the dual q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dual quaternion Summary Dual_quaternion In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus, they may be constructed in the same way as the quaternions, except using dual numbers instead of real numbers as coefficients. A ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dual quaternion Summary Dual_quaternion In mechanics, the dual quaternions are applied as a number system to represent rigid transformations in three dimensions. Since the space of dual quaternions is 8-dimensional and a rigid transformation has six real degrees of freedom, three for translations and three for rotation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dyadic cubes Summary One-third_trick In mathematics, the dyadic cubes are a collection of cubes in Rn of different sizes or scales such that the set of cubes of each scale partition Rn and each cube in one scale may be written as a union of cubes of a smaller scale. These are frequently used in mathematics (particularl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Eccentricity (geometry) Summary Eccentricity_(geometry) In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular: The eccentricity of a c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Eccentricity (geometry) Summary Eccentricity_(geometry) The eccentricity of an ellipse which is not a circle is between 0 {\displaystyle 0} and 1. {\displaystyle 1.} The eccentricity of a parabola is 1. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Eccentricity (geometry) Summary Eccentricity_(geometry) {\displaystyle 1.} The eccentricity of a hyperbola is greater than 1. {\displaystyle 1.} The eccentricity of a pair of lines is ∞ {\displaystyle \infty } Two conic sections with the same eccentricity are similar. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Effective topos Summary Effective_topos In mathematics, the effective topos E f f {\displaystyle {\mathsf {Eff}}} introduced by Martin Hyland (1982) captures the mathematical idea of effectivity within the category theoretical framework. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Either–or topology Summary Either–or_topology In mathematics, the either–or topology is a topological structure defined on the closed interval by declaring a set open if it either does not contain {0} or does contain (−1, 1). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Elasticity of a function Summary Elasticity_of_a_function In mathematics, the elasticity or point elasticity of a positive differentiable function f of a positive variable (positive input, positive output) at point a is defined as E f ( a ) = a f ( a ) f ′ ( a ) {\displaystyle Ef(a)={\frac {a}{f(a)}}f'(a)} = lim x → a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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