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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 ratio of the curve decreases as the curve converges to a circular shape, before collapsing to a single point of singularity.
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https://en.wikipedia.org/wiki/Curve-shortening_flow
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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 infinite length keep their shape, including the grim reaper curve, an infinite curve that translates upwards, and spirals that rotate while remaining the same size and shape.
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https://en.wikipedia.org/wiki/Curve-shortening_flow
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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 polygon vertices and then resampling vertices on the resulting curve, or repeatedly applying a median filter to a digital image whose black and white pixels represent the inside and outside of the curve. The curve-shortening flow was originally studied as a model for annealing of metal sheets.
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https://en.wikipedia.org/wiki/Curve-shortening_flow
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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, and as a model for the behavior of higher-dimensional flows.
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https://en.wikipedia.org/wiki/Curve-shortening_flow
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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, and π(cn) = c1.The corresponding cycle of π is written as ( c1 c2 ... cn ); this expression is not unique since c1 can be chosen to be any element of the orbit. The size n of the orbit is called the length of the corresponding cycle; when n = 1, the single element in the orbit is called a fixed point of the permutation. A permutation is determined by giving an expression for each of its cycles, and one notation for permutations consist of writing such expressions one after another in some order.
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https://en.wikipedia.org/wiki/Cycles_and_fixed_points
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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 maps 1 to 2, 6 to 8, etc. Then one may write π = ( 1 2 4 3 ) ( 5 ) ( 6 8 ) (7) = (7) ( 1 2 4 3 ) ( 6 8 ) ( 5 ) = ( 4 3 1 2 ) ( 8 6 ) ( 5 ) (7) = ...Here 5 and 7 are fixed points of π, since π(5) = 5 and π(7)=7. It is typical, but not necessary, to not write the cycles of length one in such an expression. Thus, π = (1 2 4 3)(6 8), would be an appropriate way to express this permutation. There are different ways to write a permutation as a list of its cycles, but the number of cycles and their contents are given by the partition of S into orbits, and these are therefore the same for all such expressions.
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https://en.wikipedia.org/wiki/Cycles_and_fixed_points
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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).
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https://en.wikipedia.org/wiki/Cyclic_category
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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 ∑ d | n μ ( n d ) α d , {\displaystyle M(\alpha ,n)={1 \over n}\sum _{d\,|\,n}\mu \left({n \over d}\right)\alpha ^{d},} and μ is the classic Möbius function of number theory. The name comes from the denominator, 1 − z j, which is the product of cyclotomic polynomials. The left hand side of the cyclotomic identity is the generating function for the free associative algebra on α generators, and the right hand side is the generating function for the universal enveloping algebra of the free Lie algebra on α generators. The cyclotomic identity witnesses the fact that these two algebras are isomorphic. There is also a symmetric generalization of the cyclotomic identity found by Strehl: ∏ j = 1 ∞ ( 1 1 − α z j ) M ( β , j ) = ∏ j = 1 ∞ ( 1 1 − β z j ) M ( α , j ) {\displaystyle \prod _{j=1}^{\infty }\left({1 \over 1-\alpha z^{j}}\right)^{M(\beta ,j)}=\prod _{j=1}^{\infty }\left({1 \over 1-\beta z^{j}}\right)^{M(\alpha ,j)}}
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https://en.wikipedia.org/wiki/Cyclotomic_identity
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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.
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https://en.wikipedia.org/wiki/Cylinder_set
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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 (height). Each function Vn(k) is the product of three terms, each depending on one coordinate alone. The ρ-dependent term is given by Bessel functions (which occasionally are also called cylindrical harmonics).
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https://en.wikipedia.org/wiki/Cylindrical_harmonics
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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 Hurwitz's automorphisms theorem). More generally, the set of non-constant morphisms from X to Y is finite; fixing X, for all but a finite number of such Y, there is no non-constant morphism from X to Y.These results are named for Michele De Franchis (1875–1946). It is sometimes referenced as the De Franchis-Severi theorem. It was used in an important way by Gerd Faltings to prove the Mordell conjecture.
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https://en.wikipedia.org/wiki/De_Franchis_theorem
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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, and that below the x-axis subtracts from the total. The fundamental theorem of calculus establishes the relationship between indefinite and definite integrals and introduces a technique for evaluating definite integrals. If the interval is infinite the definite integral is called an improper integral and defined by using appropriate limiting procedures.
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https://en.wikipedia.org/wiki/List_of_definite_integrals
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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 is a list of some of the most common or interesting definite integrals. For a list of indefinite integrals see List of indefinite integrals.
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https://en.wikipedia.org/wiki/List_of_definite_integrals
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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 univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial.
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https://en.wikipedia.org/wiki/Polynomial_degree
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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 x 1 y 0 − 9 x 0 y 0 , {\displaystyle 7x^{2}y^{3}+4x^{1}y^{0}-9x^{0}y^{0},} has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0.
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https://en.wikipedia.org/wiki/Polynomial_degree
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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 products (by distributivity) and combining the like terms; for example, ( x + 1 ) 2 − ( x − 1 ) 2 = 4 x {\displaystyle (x+1)^{2}-(x-1)^{2}=4x} is of degree 1, even though each summand has degree 2. However, this is not needed when the polynomial is written as a product of polynomials in standard form, because the degree of a product is the sum of the degrees of the factors.
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https://en.wikipedia.org/wiki/Polynomial_degree
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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 intersection multiplicity, because of the possibility of multiple components. For (irreducible) varieties, if one takes into account the multiplicities and, in the affine case, the points at infinity, the hypothesis of general position may be replaced by the much weaker condition that the intersection of the variety has the dimension zero (that is, consists of a finite number of points).
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https://en.wikipedia.org/wiki/Degree_(algebraic_geometry)
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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 the variety in an affine or projective space. The degree of a hypersurface is equal to the total degree of its defining equation.
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https://en.wikipedia.org/wiki/Degree_(algebraic_geometry)
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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 evaluation at 1 of the numerator of the Hilbert series of its coordinate ring. It follows that, given the equations of the variety, the degree may be computed from a Gröbner basis of the ideal of these equations.
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https://en.wikipedia.org/wiki/Degree_(algebraic_geometry)
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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.
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https://en.wikipedia.org/wiki/Generalized_derivative
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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 velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
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https://en.wikipedia.org/wiki/First-order_derivative_expression
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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 independent variable. Derivatives can be generalized to functions of several real variables.
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https://en.wikipedia.org/wiki/First-order_derivative_expression
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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 transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables.
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https://en.wikipedia.org/wiki/First-order_derivative_expression
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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 antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.
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https://en.wikipedia.org/wiki/First-order_derivative_expression
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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) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences.
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https://en.wikipedia.org/wiki/Derived_category
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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 basic theory of Verdier was written down in his dissertation, published finally in 1996 in Astérisque (a summary had earlier appeared in SGA 4½). The axiomatics required an innovation, the concept of triangulated category, and the construction is based on localization of a category, a generalization of localization of a ring.
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https://en.wikipedia.org/wiki/Derived_category
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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 theory of D-modules and microlocal analysis. Recently derived categories have also become important in areas nearer to physics, such as D-branes and mirror symmetry.
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https://en.wikipedia.org/wiki/Derived_category
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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 matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism.
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https://en.wikipedia.org/wiki/Determinant_(mathematics)
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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 of a 3 × 3 matrix is | a b c d e f g h i | = a e i + b f g + c d h − c e g − b d i − a f h . {\displaystyle {\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=aei+bfg+cdh-ceg-bdi-afh.}
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https://en.wikipedia.org/wiki/Determinant_(mathematics)
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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.
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https://en.wikipedia.org/wiki/Determinant_(mathematics)
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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 a succession of elementary row operations. Determinants can also be defined by some of their properties: the determinant is the unique function defined on the n × n matrices that has the four following properties. The determinant of the identity matrix is 1; the exchange of two rows multiplies the determinant by −1; multiplying a row by a number multiplies the determinant by this number; and adding to a row a multiple of another row does not change the determinant.
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https://en.wikipedia.org/wiki/Determinant_(mathematics)
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(The above properties relating to rows may be replaced by the corresponding statements with respect to columns.) Determinants occur throughout mathematics.
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https://en.wikipedia.org/wiki/Determinant_(mathematics)
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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 defining the characteristic polynomial of a matrix, whose roots are the eigenvalues. In geometry, the signed n-dimensional volume of a n-dimensional parallelepiped is expressed by a determinant, and the determinant of (the matrix of) a linear transformation determines how the orientation and the n-dimensional volume are transformed. This is used in calculus with exterior differential forms and the Jacobian determinant, in particular for changes of variables in multiple integrals.
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https://en.wikipedia.org/wiki/Determinant_(mathematics)
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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 upper bound for the number of rational points of bounded height on or near algebraic varieties defined over the rational numbers. The main novelty of the determinant method is that in all incarnations, the estimates obtained are uniform with respect to the coefficients of the polynomials defining the variety and only depend on the degree and dimension of the variety.
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https://en.wikipedia.org/wiki/Determinant_method
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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 and λ(B)i are the eigenvalues of A and B, and σ is an element of the symmetric group Sn.
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https://en.wikipedia.org/wiki/Determinantal_conjecture
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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)(a-b)} in elementary algebra.
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https://en.wikipedia.org/wiki/Difference_of_two_squares
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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 their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves.
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https://en.wikipedia.org/wiki/Differentiable_surface
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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; they also provide an essential ingredient in the modern approach to intrinsic differential geometry through connections. On the other hand, extrinsic properties relying on an embedding of a surface in Euclidean space have also been extensively studied. This is well illustrated by the non-linear Euler–Lagrange equations in the calculus of variations: although Euler developed the one variable equations to understand geodesics, defined independently of an embedding, one of Lagrange's main applications of the two variable equations was to minimal surfaces, a concept that can only be defined in terms of an embedding.
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https://en.wikipedia.org/wiki/Differentiable_surface
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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)}}.} It is the first of the polygamma functions. This function is strictly increasing and strictly concave on ( 0 , ∞ ) {\displaystyle (0,\infty )} , and it asymptotically behaves as ψ ( z ) ∼ ln z − 1 2 z , {\displaystyle \psi (z)\sim \ln {z}-{\frac {1}{2z}},} for large arguments ( | z | → ∞ {\displaystyle |z|\rightarrow \infty } ) in the sector | arg z | < π − ε {\displaystyle |\arg z|<\pi -\varepsilon } with some infinitesimally small positive constant ε {\displaystyle \varepsilon } . The digamma function is often denoted as ψ 0 ( x ) , ψ ( 0 ) ( x ) {\displaystyle \psi _{0}(x),\psi ^{(0)}(x)} or Ϝ (the uppercase form of the archaic Greek consonant digamma meaning double-gamma).
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https://en.wikipedia.org/wiki/Gauss's_digamma_theorem
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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.}
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https://en.wikipedia.org/wiki/Digit_sum
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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. Dushnik & Miller (1941) first studied order dimension; for a more detailed treatment of this subject than provided here, see Trotter (1992).
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https://en.wikipedia.org/wiki/Order_dimension
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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 dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined.
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https://en.wikipedia.org/wiki/Infinite-dimensional_vector_space
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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 as dim F ( V ) {\displaystyle \dim _{F}(V)} or as , {\displaystyle ,} read "dimension of V {\displaystyle V} over F {\displaystyle F} ". When F {\displaystyle F} can be inferred from context, dim ( V ) {\displaystyle \dim(V)} is typically written.
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https://en.wikipedia.org/wiki/Infinite-dimensional_vector_space
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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 for defining the position of a point that is constrained to be on the object. For example, the dimension of a point is zero; the dimension of a line is one, as a point can move on a line in only one direction (or its opposite); the dimension of a plane is two etc. The dimension is an intrinsic property of an object, in the sense that it is independent of the dimension of the space in which the object is or can be embedded. For example, a curve, such as a circle, is of dimension one, because the position of a point on a curve is determined by its signed distance along the curve to a fixed point on the curve.
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https://en.wikipedia.org/wiki/Multidimensional_geometry
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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 is faced with the question "what makes En n-dimensional?" One answer is that to cover a fixed ball in En by small balls of radius ε, one needs on the order of ε−n such small balls.
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https://en.wikipedia.org/wiki/Multidimensional_geometry
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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 like En-1 and this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces.
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https://en.wikipedia.org/wiki/Multidimensional_geometry
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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: "The dimension of the tesseract is 4" or: 4D. Although the notion of higher dimensions goes back to René Descartes, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of Arthur Cayley, William Rowan Hamilton, Ludwig Schläfli and Bernhard Riemann. Riemann's 1854 Habilitationsschrift, Schläfli's 1852 Theorie der vielfachen Kontinuität, and Hamilton's discovery of the quaternions and John T. Graves' discovery of the octonions in 1843 marked the beginning of higher-dimensional geometry. The rest of this section examines some of the more important mathematical definitions of dimension.
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https://en.wikipedia.org/wiki/Multidimensional_geometry
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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 dimension of the vector space. Formally, the dimension theorem for vector spaces states that: As a basis is a generating set that is linearly independent, the theorem is a consequence of the following theorem, which is also useful: In particular if V is finitely generated, then all its bases are finite and have the same number of elements.
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https://en.wikipedia.org/wiki/Dimension_theorem_for_vector_spaces
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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, which is strictly weaker (the proof given below, however, assumes trichotomy, i.e., that all cardinal numbers are comparable, a statement which is also equivalent to the axiom of choice). The theorem can be generalized to arbitrary R-modules for rings R having invariant basis number. In the finitely generated case the proof uses only elementary arguments of algebra, and does not require the axiom of choice nor its weaker variants.
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https://en.wikipedia.org/wiki/Dimension_theorem_for_vector_spaces
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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 continuous map f: X → Y, we can define a new sheaf f∗F on Y, called the direct image sheaf or the pushforward sheaf of F along f, such that the global sections of f∗F is given by the global sections of F. This assignment gives rise to a functor f∗ from the category of sheaves on X to the category of sheaves on Y, which is known as the direct image functor. Similar constructions exist in many other algebraic and geometric contexts, including that of quasi-coherent sheaves and étale sheaves on a scheme.
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https://en.wikipedia.org/wiki/Direct_image
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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.
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https://en.wikipedia.org/wiki/Direct_image_with_compact_support
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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 method relies on methods of functional analysis and topology. As well as being used to prove the existence of a solution, direct methods may be used to compute the solution to desired accuracy.
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https://en.wikipedia.org/wiki/Direct_method_in_calculus_of_variations
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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 frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence.
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https://en.wikipedia.org/wiki/Discrete_Fourier_Transform
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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. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle.
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https://en.wikipedia.org/wiki/Discrete_Fourier_Transform
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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 signal, or daily temperature readings, sampled over a finite time interval (often defined by a window function). In image processing, the samples can be the values of pixels along a row or column of a raster image.
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https://en.wikipedia.org/wiki/Discrete_Fourier_Transform
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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 algorithms or even dedicated hardware. These implementations usually employ efficient fast Fourier transform (FFT) algorithms; so much so that the terms "FFT" and "DFT" are often used interchangeably. Prior to its current usage, the "FFT" initialism may have also been used for the ambiguous term "finite Fourier transform".
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https://en.wikipedia.org/wiki/Discrete_Fourier_Transform
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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.
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https://en.wikipedia.org/wiki/Number-theoretic_transform
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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 Laplace operator is more commonly called the Laplacian matrix. The discrete Laplace operator occurs in physics problems such as the Ising model and loop quantum gravity, as well as in the study of discrete dynamical systems. It is also used in numerical analysis as a stand-in for the continuous Laplace operator. Common applications include image processing, where it is known as the Laplace filter, and in machine learning for clustering and semi-supervised learning on neighborhood graphs.
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https://en.wikipedia.org/wiki/Discrete_Laplace_operator
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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 stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics.
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https://en.wikipedia.org/wiki/Discrete_Poisson_equation
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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 powerful in improving and analyzing finite element methods: for instance, DEC-based methods allow the use of highly non-uniform meshes to obtain accurate results. Non-uniform meshes are advantageous because they allow the use of large elements where the process to be simulated is relatively simple, as opposed to a fine resolution where the process may be complicated (e.g., near an obstruction to a fluid flow), while using less computational power than if a uniformly fine mesh were used.
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https://en.wikipedia.org/wiki/Discrete_exterior_calculus
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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. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. hn(x;q) is also called discrete q-Hermite I polynomials and ĥn(x;q) is also called discrete q-Hermite II polynomials.
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https://en.wikipedia.org/wiki/Discrete_q-Hermite_polynomials
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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 with odd symmetry (since the Fourier transform of a real and odd function is imaginary and odd), where in some variants the input and/or output data are shifted by half a sample. A family of transforms composed of sine and sine hyperbolic functions exists. These transforms are made based on the natural vibration of thin square plates with different boundary conditions.The DST is related to the discrete cosine transform (DCT), which is equivalent to a DFT of real and even functions.
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https://en.wikipedia.org/wiki/Discrete_sine_transform
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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 described by Nasir Ahmed, T. Natarajan, and K.R.
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https://en.wikipedia.org/wiki/Discrete_sine_transform
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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.
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https://en.wikipedia.org/wiki/Discrete_sine_transform
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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 term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function.
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https://en.wikipedia.org/wiki/Discrete-time_Fourier_transform
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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, but discrete samples of it can be readily calculated via the discrete Fourier transform (DFT) (see § Sampling the DTFT), which is by far the most common method of modern Fourier analysis. Both transforms are invertible.
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https://en.wikipedia.org/wiki/Discrete-time_Fourier_transform
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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 inverse DFT.
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https://en.wikipedia.org/wiki/Discrete-time_Fourier_transform
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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 discriminant is widely used in polynomial factoring, number theory, and algebraic geometry. The discriminant of the quadratic polynomial a x 2 + b x + c {\displaystyle ax^{2}+bx+c} is b 2 − 4 a c , {\displaystyle b^{2}-4ac,} the quantity which appears under the square root in the quadratic formula.
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https://en.wikipedia.org/wiki/Discriminant_of_a_polynomial
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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. Similarly, the discriminant of a cubic polynomial is zero if and only if the polynomial has a multiple root.
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https://en.wikipedia.org/wiki/Discriminant_of_a_polynomial
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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 positive degree is zero if and only if the polynomial has a multiple root. For real coefficients and no multiple roots, the discriminant is positive if the number of non-real roots is a multiple of 4 (including none), and negative otherwise. Several generalizations are also called discriminant: the discriminant of an algebraic number field; the discriminant of a quadratic form; and more generally, the discriminant of a form, of a homogeneous polynomial, or of a projective hypersurface (these three concepts are essentially equivalent).
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https://en.wikipedia.org/wiki/Discriminant_of_a_polynomial
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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 domain of the ring of integers, and it regulates which primes are ramified. The discriminant is one of the most basic invariants of a number field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K, and the analytic class number formula for K. A theorem of Hermite states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an open problem, and the subject of current research.The discriminant of K can be referred to as the absolute discriminant of K to distinguish it from the relative discriminant of an extension K/L of number fields. The latter is an ideal in the ring of integers of L, and like the absolute discriminant it indicates which primes are ramified in K/L. It is a generalization of the absolute discriminant allowing for L to be bigger than Q; in fact, when L = Q, the relative discriminant of K/Q is the principal ideal of Z generated by the absolute discriminant of K.
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https://en.wikipedia.org/wiki/Brill's_theorem
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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 smooth nowhere-zero vector fields can be constructed on a sphere in n {\displaystyle n} -dimensional Euclidean space. A definitive answer was provided in 1962 by Frank Adams. It was already known, by direct construction using Clifford algebras, that there were at least ρ ( n ) − 1 {\displaystyle \rho (n)-1} such fields (see definition below). Adams applied homotopy theory and topological K-theory to prove that no more independent vector fields could be found. Hence ρ ( n ) − 1 {\displaystyle \rho (n)-1} is the exact number of pointwise linearly independent vector fields that exist on an ( n − 1 {\displaystyle n-1} )-dimensional sphere.
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https://en.wikipedia.org/wiki/Radon–Hurwitz_number
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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 conditional probability measures. In a sense, "disintegration" is the opposite process to the construction of a product measure.
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https://en.wikipedia.org/wiki/Disintegration_theorem
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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 a homeomorphism, then it is quasiconformal. The distortion of a function ƒ of the plane is given by H ( z , f ) = lim sup r → 0 max | h | = r | f ( z + h ) − f ( z ) | min | h | = r | f ( z + h ) − f ( z ) | {\displaystyle H(z,f)=\limsup _{r\to 0}{\frac {\max _{|h|=r}|f(z+h)-f(z)|}{\min _{|h|=r}|f(z+h)-f(z)|}}} which is the limiting eccentricity of the ellipse produced by applying ƒ to small circles centered at z. This geometrical definition is often very difficult to work with, and the necessary analytical features can be extrapolated to the following definition. A mapping ƒ: Ω → R2 from an open domain in the plane to the plane has finite distortion at a point x ∈ Ω if ƒ is in the Sobolev space W1,1loc(Ω, R2), the Jacobian determinant J(x,ƒ) is locally integrable and does not change sign in Ω, and there is a measurable function K(x) ≥ 1 such that | D f ( x ) | 2 ≤ K ( x ) | J ( x , f ) | {\displaystyle |Df(x)|^{2}\leq K(x)|J(x,f)|} almost everywhere.
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https://en.wikipedia.org/wiki/Distortion_(mathematics)
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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.
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https://en.wikipedia.org/wiki/Distortion_(mathematics)
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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&{\text{if }}J(x,f)=0.\end{cases}}} The outer distortion KO and inner distortion KI are defined via the Rayleigh quotients K O ( x ) = sup ξ ≠ 0 ⟨ G ( x ) ξ , ξ ⟩ n / 2 | ξ | n , K O ( x ) = sup ξ ≠ 0 ⟨ G − 1 ( x ) ξ , ξ ⟩ n / 2 | ξ | n . {\displaystyle K_{O}(x)=\sup _{\xi \not =0}{\frac {\langle G(x)\xi ,\xi \rangle ^{n/2}}{|\xi |^{n}}},\quad K_{O}(x)=\sup _{\xi \not =0}{\frac {\langle G^{-1}(x)\xi ,\xi \rangle ^{n/2}}{|\xi |^{n}}}.} The outer distortion can also be characterized by means of an inequality similar to that given in the two-dimensional case. If Ω is an open set in Rn, then a function ƒ ∈ W1,1loc(Ω,Rn) has finite distortion if its Jacobian is locally integrable and does not change sign, and there is a measurable function KO (the outer distortion) such that | D f ( x ) | n ≤ K O ( x ) | J ( x , f ) | {\displaystyle |Df(x)|^{n}\leq K_{O}(x)|J(x,f)|} almost everywhere.
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https://en.wikipedia.org/wiki/Distortion_(mathematics)
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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 distributes over addition. This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted ∧ {\displaystyle \,\land \,} ) and the logical or (denoted ∨ {\displaystyle \,\lor \,} ) distributes over the other.
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https://en.wikipedia.org/wiki/Antidistributive
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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 a function can generally be thought of as "what x can be".More precisely, given a function f: X → Y {\displaystyle f\colon X\to Y} , the domain of f is X. In modern mathematical language, the domain is part of the definition of a function rather than a property of it. In the special case that X and Y are both subsets of R {\displaystyle \mathbb {R} } , the function f can be graphed in the Cartesian coordinate system.
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https://en.wikipedia.org/wiki/Domain_(function)
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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 is a subset of Y) is called its range or image. Any function can be restricted to a subset of its domain. The restriction of f: X → Y {\displaystyle f\colon X\to Y} to A {\displaystyle A} , where A ⊆ X {\displaystyle A\subseteq X} , is written as f | A: A → Y {\displaystyle \left.f\right|_{A}\colon A\to Y} .
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https://en.wikipedia.org/wiki/Domain_(function)
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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 used. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more). Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers.
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https://en.wikipedia.org/wiki/Vector_dot_product
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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, the dot product is used for defining lengths (the length of a vector is the square root of the dot product of the vector by itself) and angles (the cosine of the angle between two vectors is the quotient of their dot product by the product of their lengths). The name "dot product" is derived from the centered dot " · " that is often used to designate this operation; the alternative name "scalar product" emphasizes that the result is a scalar, rather than a vector (as with the vector product in three-dimensional space).
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https://en.wikipedia.org/wiki/Vector_dot_product
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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 directions.
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https://en.wikipedia.org/wiki/Double_Fourier_sphere_method
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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 × 7 × 5 × 3 × 1 = 945. The zero double factorial 0‼ = 1 as an empty product.The sequence of double factorials for even n = 0, 2, 4, 6, 8,... starts as The sequence of double factorials for odd n = 1, 3, 5, 7, 9,... starts as The term odd factorial is sometimes used for the double factorial of an odd number.
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https://en.wikipedia.org/wiki/Double_factorial
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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} th triangular number, then the doubly triangular numbers are the numbers of the form T T n {\displaystyle T_{T_{n}}} .
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https://en.wikipedia.org/wiki/Doubly_triangular_number
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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 , a , b ) = ( a − b + 1 ) n ( a + c + 1 ) n n ! 3 F 2 ( − n , a − s , a + s + 1 ; a − b + a , a + c + 1 ; 1 ) {\displaystyle w_{n}^{(c)}(s,a,b)={\frac {(a-b+1)_{n}(a+c+1)_{n}}{n!
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https://en.wikipedia.org/wiki/Dual_Hahn_polynomials
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}}{}_{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}}
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https://en.wikipedia.org/wiki/Dual_Hahn_polynomials
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In mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces.
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https://en.wikipedia.org/wiki/Dual_bundle
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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 dual module is a dual vector space. Every module has a canonical homomorphism to the dual of its dual (called the double dual).
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https://en.wikipedia.org/wiki/Dual_module
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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 Hopf algebra A over a commutative ring k, then the Cartier dual G D {\displaystyle G^{D}} is the Spec of the dual k-module of A. == References ==
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https://en.wikipedia.org/wiki/Dual_module
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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.
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https://en.wikipedia.org/wiki/Dual_q-Hahn_polynomials
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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.
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https://en.wikipedia.org/wiki/Dual_q-Krawtchouk_polynomials
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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 dual quaternion can be represented in the form A + εB, where A and B are ordinary quaternions and ε is the dual unit, which satisfies ε2 = 0 and commutes with every element of the algebra. Unlike quaternions, the dual quaternions do not form a division algebra.
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https://en.wikipedia.org/wiki/Dual_quaternion
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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 rotations, dual quaternions obeying two algebraic constraints are used in this application. Since unit quaternions are subject to two algebraic constraints, unit quaternions are standard to represent rigid transformations.Similar to the way that rotations in 3D space can be represented by quaternions of unit length, rigid motions in 3D space can be represented by dual quaternions of unit length. This fact is used in theoretical kinematics (see McCarthy), and in applications to 3D computer graphics, robotics and computer vision. Polynomials with coefficients given by (non-zero real norm) dual quaternions have also been used in the context of mechanical linkages design.
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https://en.wikipedia.org/wiki/Dual_quaternion
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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 (particularly harmonic analysis) as a way of discretizing objects in order to make computations or analysis easier. For example, to study an arbitrary subset of A of Euclidean space, one may instead replace it by a union of dyadic cubes of a particular size that cover the set. One can consider this set as a pixelized version of the original set, and as smaller cubes are used one gets a clearer image of the set A. Most notable appearances of dyadic cubes include the Whitney extension theorem and the Calderón–Zygmund lemma.
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https://en.wikipedia.org/wiki/Dyadic_cubes
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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 circle is 0. {\displaystyle 0.}
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https://en.wikipedia.org/wiki/Eccentricity_(geometry)
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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.
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https://en.wikipedia.org/wiki/Eccentricity_(geometry)
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{\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.
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https://en.wikipedia.org/wiki/Eccentricity_(geometry)
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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.
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https://en.wikipedia.org/wiki/Effective_topos
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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).
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https://en.wikipedia.org/wiki/Either–or_topology
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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 f ( x ) − f ( a ) x − a a f ( a ) = lim x → a f ( x ) − f ( a ) f ( a ) a x − a = lim x → a 1 − f ( x ) f ( a ) 1 − x a ≈ % Δ f ( a ) % Δ a {\displaystyle =\lim _{x\to a}{\frac {f(x)-f(a)}{x-a}}{\frac {a}{f(a)}}=\lim _{x\to a}{\frac {f(x)-f(a)}{f(a)}}{\frac {a}{x-a}}=\lim _{x\to a}{\frac {1-{\frac {f(x)}{f(a)}}}{1-{\frac {x}{a}}}}\approx {\frac {\%\Delta f(a)}{\%\Delta a}}} or equivalently E f ( x ) = d log f ( x ) d log x . {\displaystyle Ef(x)={\frac {d\log f(x)}{d\log x}}.} It is thus the ratio of the relative (percentage) change in the function's output f ( x ) {\displaystyle f(x)} with respect to the relative change in its input x {\displaystyle x} , for infinitesimal changes from a point ( a , f ( a ) ) {\displaystyle (a,f(a))} .
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https://en.wikipedia.org/wiki/Elasticity_of_a_function
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