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In mathematics, a hollow matrix may refer to one of several related classes of matrix: a sparse matrix; a matrix with a large block of zeroes; or a matrix with diagonal entries all zero.
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https://en.wikipedia.org/wiki/Hollow_matrix
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In mathematics, a holomorphic discrete series representation is a discrete series representation of a semisimple Lie group that can be represented in a natural way as a Hilbert space of holomorphic functions. The simple Lie groups with holomorphic discrete series are those whose symmetric space is Hermitian. Holomorphic discrete series representations are the easiest discrete series representations to study because they have highest or lowest weights, which makes their behavior similar to that of finite-dimensional representations of compact Lie groups. Bargmann (1947) found the first examples of holomorphic discrete series representations, and Harish-Chandra (1954, 1955a, 1955c, 1956a, 1956b) classified them for all semisimple Lie groups. Martens (1975) and Hecht (1976) described the characters of holomorphic discrete series representations.
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https://en.wikipedia.org/wiki/Holomorphic_discrete_series_representation
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In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space Cn. The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (analytic). Holomorphic functions are the central objects of study in complex analysis. Though the term analytic function is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain.
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https://en.wikipedia.org/wiki/Holomorphic_map
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That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis.Holomorphic functions are also sometimes referred to as regular functions. A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic at a point z0" means not just differentiable at z0, but differentiable everywhere within some neighbourhood of z0 in the complex plane.
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https://en.wikipedia.org/wiki/Holomorphic_map
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In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π: E → X is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle. By Serre's GAGA, the category of holomorphic vector bundles on a smooth complex projective variety X (viewed as a complex manifold) is equivalent to the category of algebraic vector bundles (i.e., locally free sheaves of finite rank) on X.
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https://en.wikipedia.org/wiki/Holomorphic_vector_bundle
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In mathematics, a homogeneous distribution is a distribution S on Euclidean space Rn or Rn \ {0} that is homogeneous in the sense that, roughly speaking, S ( t x ) = t m S ( x ) {\displaystyle S(tx)=t^{m}S(x)\,} for all t > 0. More precisely, let μ t: x ↦ x / t {\displaystyle \mu _{t}:x\mapsto x/t} be the scalar division operator on Rn. A distribution S on Rn or Rn \ {0} is homogeneous of degree m provided that S = t m S {\displaystyle S=t^{m}S} for all positive real t and all test functions φ. The additional factor of t−n is needed to reproduce the usual notion of homogeneity for locally integrable functions, and comes about from the Jacobian change of variables.
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https://en.wikipedia.org/wiki/Homogeneous_distribution
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The number m can be real or complex. It can be a non-trivial problem to extend a given homogeneous distribution from Rn \ {0} to a distribution on Rn, although this is necessary for many of the techniques of Fourier analysis, in particular the Fourier transform, to be brought to bear. Such an extension exists in most cases, however, although it may not be unique.
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https://en.wikipedia.org/wiki/Homogeneous_distribution
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In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the degree; that is, if k is an integer, a function f of n variables is homogeneous of degree k if f ( s x 1 , … , s x n ) = s k f ( x 1 , … , x n ) {\displaystyle f(sx_{1},\ldots ,sx_{n})=s^{k}f(x_{1},\ldots ,x_{n})} for every x 1 , … , x n , {\displaystyle x_{1},\ldots ,x_{n},} and s ≠ 0. {\displaystyle s\neq 0.} For example, a homogeneous polynomial of degree k defines a homogeneous function of degree k. The above definition extends to functions whose domain and codomain are vector spaces over a field F: a function f: V → W {\displaystyle f:V\to W} between two F-vector spaces is homogeneous of degree k {\displaystyle k} if for all nonzero s ∈ F {\displaystyle s\in F} and v ∈ V . {\displaystyle v\in V.}
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https://en.wikipedia.org/wiki/Homogenous_function
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This definition is often further generalized to functions whose domain is not V, but a cone in V, that is, a subset C of V such that v ∈ C {\displaystyle \mathbf {v} \in C} implies s v ∈ C {\displaystyle s\mathbf {v} \in C} for every nonzero scalar s. In the case of functions of several real variables and real vector spaces, a slightly more general form of homogeneity called positive homogeneity is often considered, by requiring only that the above identities hold for s > 0 , {\displaystyle s>0,} and allowing any real number k as a degree of homogeneity. Every homogeneous real function is positively homogeneous. The converse is not true, but is locally true in the sense that (for integer degrees) the two kinds of homogeneity cannot be distinguished by considering the behavior of a function near a given point.
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https://en.wikipedia.org/wiki/Homogenous_function
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A norm over a real vector space is an example of a positively homogeneous function that is not homogeneous. A special case is the absolute value of real numbers. The quotient of two homogeneous polynomials of the same degree gives an example of a homogeneous function of degree zero. This example is fundamental in the definition of projective schemes.
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https://en.wikipedia.org/wiki/Homogenous_function
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In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x 5 + 2 x 3 y 2 + 9 x y 4 {\displaystyle x^{5}+2x^{3}y^{2}+9xy^{4}} is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial x 3 + 3 x 2 y + z 7 {\displaystyle x^{3}+3x^{2}y+z^{7}} is not homogeneous, because the sum of exponents does not match from term to term. The function defined by a homogeneous polynomial is always a homogeneous function.
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https://en.wikipedia.org/wiki/Inhomogeneous_polynomial
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An algebraic form, or simply form, is a function defined by a homogeneous polynomial. A binary form is a form in two variables. A form is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis.
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https://en.wikipedia.org/wiki/Inhomogeneous_polynomial
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A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form. A form of degree 2 is a quadratic form.
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https://en.wikipedia.org/wiki/Inhomogeneous_polynomial
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In geometry, the Euclidean distance is the square root of a quadratic form. Homogeneous polynomials are ubiquitous in mathematics and physics. They play a fundamental role in algebraic geometry, as a projective algebraic variety is defined as the set of the common zeros of a set of homogeneous polynomials.
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https://en.wikipedia.org/wiki/Inhomogeneous_polynomial
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In mathematics, a homogeneous relation (also called endorelation) on a set X is a binary relation between X and itself, i.e. it is a subset of the Cartesian product X × X. This is commonly phrased as "a relation on X" or "a (binary) relation over X". An example of a homogeneous relation is the relation of kinship, where the relation is between people. Common types of endorelations include orders, graphs, and equivalences. Specialized studies of order theory and graph theory have developed understanding of endorelations. Terminology particular for graph theory is used for description, with an ordinary (undirected) graph presumed to correspond to a symmetric relation, and a general endorelation corresponding to a directed graph. An endorelation R corresponds to a logical matrix of 0s and 1s, where the expression xRy corresponds to an edge between x and y in the graph, and to a 1 in the square matrix of R. It is called an adjacency matrix in graph terminology.
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https://en.wikipedia.org/wiki/Identity_relation
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In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and topological groups. More precisely, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively.
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https://en.wikipedia.org/wiki/Homogeneous_space
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The elements of G are called the symmetries of X. A special case of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, X is homogeneous if intuitively X looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism (differential geometry), or homeomorphism (topology). Some authors insist that the action of G be faithful (non-identity elements act non-trivially), although the present article does not. Thus there is a group action of G on X which can be thought of as preserving some "geometric structure" on X, and making X into a single G-orbit.
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https://en.wikipedia.org/wiki/Homogeneous_space
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In mathematics, a homology manifold (or generalized manifold) is a locally compact topological space X that looks locally like a topological manifold from the point of view of homology theory.
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https://en.wikipedia.org/wiki/Homology_manifold
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In mathematics, a homology theory in algebraic topology is compactly supported if, in every degree n, the relative homology group Hn(X, A) of every pair of spaces (X, A)is naturally isomorphic to the direct limit of the nth relative homology groups of pairs (Y, B), where Y varies over compact subspaces of X and B varies over compact subspaces of A.Singular homology is compactly supported, since each singular chain is a finite sum of simplices, which are compactly supported. Strong homology is not compactly supported. If one has defined a homology theory over compact pairs, it is possible to extend it into a compactly supported homology theory in the wider category of Hausdorff pairs (X, A) with A closed in X, by defining that the homology of a Hausdorff pair (X, A) is the direct limit over pairs (Y, B), where Y, B are compact, Y is a subset of X, and B is a subset of A. == References ==
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https://en.wikipedia.org/wiki/Compactly-supported_homology
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In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number k {\displaystyle k} called its ratio, which sends point X {\displaystyle X} to a point X ′ {\displaystyle X'} by the rule S X ′ → = k S X → {\displaystyle {\overrightarrow {SX'}}=k{\overrightarrow {SX}}} for a fixed number k ≠ 0 {\displaystyle k\neq 0} .Using position vectors: x ′ = s + k ( x − s ) {\displaystyle \mathbf {x} '=\mathbf {s} +k(\mathbf {x} -\mathbf {s} )} .In case of S = O {\displaystyle S=O} (Origin): x ′ = k x {\displaystyle \mathbf {x} '=k\mathbf {x} } ,which is a uniform scaling and shows the meaning of special choices for k {\displaystyle k}: for k = 1 {\displaystyle k=1} one gets the identity mapping, for k = − 1 {\displaystyle k=-1} one gets the reflection at the center,For 1 / k {\displaystyle 1/k} one gets the inverse mapping defined by k {\displaystyle k} . In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if k > 0 {\displaystyle k>0} ) or reverse (if k < 0 {\displaystyle k<0} ) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line g is a line parallel to g. In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic involution) that leaves the line at infinity pointwise invariant.In Euclidean geometry, a homothety of ratio k {\displaystyle k} multiplies distances between points by | k | {\displaystyle |k|} , areas by k 2 {\displaystyle k^{2}} and volumes by | k | 3 {\displaystyle |k|^{3}} .
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https://en.wikipedia.org/wiki/Homothety
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Here k {\displaystyle k} is the ratio of magnification or dilation factor or scale factor or similitude ratio. Such a transformation can be called an enlargement if the scale factor exceeds 1. The above-mentioned fixed point S is called homothetic center or center of similarity or center of similitude.
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https://en.wikipedia.org/wiki/Homothety
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The term, coined by French mathematician Michel Chasles, is derived from two Greek elements: the prefix homo- (όμο), meaning "similar", and thesis (Θέσις), meaning "position". It describes the relationship between two figures of the same shape and orientation. For example, two Russian dolls looking in the same direction can be considered homothetic. Homotheties are used to scale the contents of computer screens; for example, smartphones, notebooks, and laptops.
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https://en.wikipedia.org/wiki/Homothety
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In mathematics, a horn angle, also called a cornicular angle, is a type of curvilinear angle defined as the angle formed between a circle and a straight line tangent to it, or, more generally, the angle formed between two curves at a point where they are tangent to each other.
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https://en.wikipedia.org/wiki/Horn_angle
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In mathematics, a hyper-finite field is an uncountable field similar in many ways to finite fields. More precisely a field F is called hyper-finite if it is uncountable and quasi-finite, and for every subfield E, every absolutely entire E-algebra (regular field extension of E) of smaller cardinality than F can be embedded in F. They were introduced by Ax (1968). Every hyper-finite field is a pseudo-finite field, and is in particular a model for the first-order theory of finite fields.
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https://en.wikipedia.org/wiki/Hyper-finite_field
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In mathematics, a hyperbola ( ; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows.
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https://en.wikipedia.org/wiki/Rectangular_hyperbola
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The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.)
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https://en.wikipedia.org/wiki/Rectangular_hyperbola
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If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola. Besides being a conic section, a hyperbola can arise as the locus of points whose difference of distances to two fixed foci is constant, as a curve for each point of which the rays to two fixed foci are reflections across the tangent line at that point, or as the solution of certain bivariate quadratic equations such as the reciprocal relationship x y = 1. {\displaystyle xy=1.}
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https://en.wikipedia.org/wiki/Rectangular_hyperbola
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In practical applications, a hyperbola can arise as the path followed by the shadow of the tip of a sundial's gnomon, the shape of an open orbit such as that of a celestial object exceeding the escape velocity of the nearest gravitational body, or the scattering trajectory of a subatomic particle, among others. Each branch of the hyperbola has two arms which become straighter (lower curvature) further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms.
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https://en.wikipedia.org/wiki/Rectangular_hyperbola
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So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the mirror point about which each branch reflects to form the other branch. In the case of the curve y ( x ) = 1 / x {\displaystyle y(x)=1/x} the asymptotes are the two coordinate axes.Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term. Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry (Lobachevsky's celebrated non-Euclidean geometry), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed for use in both relativity and quantum mechanics which is not Euclidean).
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https://en.wikipedia.org/wiki/Rectangular_hyperbola
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In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry. A hyperbolic knot is a hyperbolic link with one component. As a consequence of the work of William Thurston, it is known that every knot is precisely one of the following: hyperbolic, a torus knot, or a satellite knot. As a consequence, hyperbolic knots can be considered plentiful. A similar heuristic applies to hyperbolic links. As a consequence of Thurston's hyperbolic Dehn surgery theorem, performing Dehn surgeries on a hyperbolic link enables one to obtain many more hyperbolic 3-manifolds.
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https://en.wikipedia.org/wiki/Hyperbolic_knot
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In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman.
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https://en.wikipedia.org/wiki/Hyperbolic_manifolds
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In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by Mikhael Gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. Hyperbolicity is a large-scale property, and is very useful to the study of certain infinite groups called Gromov-hyperbolic groups.
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https://en.wikipedia.org/wiki/Gromov_hyperbolic_space
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In mathematics, a hyperbolic partial differential equation of order n {\displaystyle n} is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n − 1 {\displaystyle n-1} derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface. Many of the equations of mechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the wave equation.
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https://en.wikipedia.org/wiki/Hyperbolic_PDE
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In one spatial dimension, this is The equation has the property that, if u and its first time derivative are arbitrarily specified initial data on the line t = 0 (with sufficient smoothness properties), then there exists a solution for all time t. The solutions of hyperbolic equations are "wave-like". If a disturbance is made in the initial data of a hyperbolic differential equation, then not every point of space feels the disturbance at once. Relative to a fixed time coordinate, disturbances have a finite propagation speed.
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https://en.wikipedia.org/wiki/Hyperbolic_PDE
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They travel along the characteristics of the equation. This feature qualitatively distinguishes hyperbolic equations from elliptic partial differential equations and parabolic partial differential equations. A perturbation of the initial (or boundary) data of an elliptic or parabolic equation is felt at once by essentially all points in the domain.
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https://en.wikipedia.org/wiki/Hyperbolic_PDE
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Although the definition of hyperbolicity is fundamentally a qualitative one, there are precise criteria that depend on the particular kind of differential equation under consideration. There is a well-developed theory for linear differential operators, due to Lars Gårding, in the context of microlocal analysis. Nonlinear differential equations are hyperbolic if their linearizations are hyperbolic in the sense of Gårding. There is a somewhat different theory for first order systems of equations coming from systems of conservation laws.
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https://en.wikipedia.org/wiki/Hyperbolic_PDE
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In mathematics, a hyperelliptic surface, or bi-elliptic surface, is a surface whose Albanese morphism is an elliptic fibration. Any such surface can be written as the quotient of a product of two elliptic curves by a finite abelian group. Hyperelliptic surfaces form one of the classes of surfaces of Kodaira dimension 0 in the Enriques–Kodaira classification.
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https://en.wikipedia.org/wiki/Hyperelliptic_surface
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In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a directed hypergraph is a pair ( X , E ) {\displaystyle (X,E)} , where X {\displaystyle X} is a set of elements called nodes, vertices, points, or elements and E {\displaystyle E} is a set of pairs of subsets of X {\displaystyle X} . Each of these pairs ( D , C ) ∈ E {\displaystyle (D,C)\in E} is called an edge or hyperedge; the vertex subset D {\displaystyle D} is known as its tail or domain, and C {\displaystyle C} as its head or codomain.
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https://en.wikipedia.org/wiki/Dual_hypergraph
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The order of a hypergraph ( X , E ) {\displaystyle (X,E)} is the number of vertices in X {\displaystyle X} . The size of the hypergraph is the number of edges in E {\displaystyle E} . The order of an edge e = ( D , C ) {\displaystyle e=(D,C)} in a directed hypergraph is | e | = ( | D | , | C | ) {\displaystyle |e|=(|D|,|C|)}: that is, the number of vertices in its tail followed by the number of vertices in its head.
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https://en.wikipedia.org/wiki/Dual_hypergraph
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The definition above generalizes from a directed graph to a directed hypergraph by defining the head or tail of each edge as a set of vertices ( C ⊆ X {\displaystyle C\subseteq X} or D ⊆ X {\displaystyle D\subseteq X} ) rather than as a single vertex. A graph is then the special case where each of these sets contains only one element. Hence any standard graph theoretic concept that is independent of the edge orders | e | {\displaystyle |e|} will generalize to hypergraph theory.
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https://en.wikipedia.org/wiki/Dual_hypergraph
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Under one definition, an undirected hypergraph ( X , E ) {\displaystyle (X,E)} is a directed hypergraph which has a symmetric edge set: If ( D , C ) ∈ E {\displaystyle (D,C)\in E} then ( C , D ) ∈ E {\displaystyle (C,D)\in E} . For notational simplicity one can remove the "duplicate" hyperedges since the modifier "undirected" is precisely informing us that they exist: If ( D , C ) ∈ E {\displaystyle (D,C)\in E} then ( C , D ) ∈ → E {\displaystyle (C,D){\vec {\in }}E} where ∈ → {\displaystyle {\vec {\in }}} means implicitly in. While graph edges connect only 2 nodes, hyperedges connect an arbitrary number of nodes.
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https://en.wikipedia.org/wiki/Dual_hypergraph
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However, it is often desirable to study hypergraphs where all hyperedges have the same cardinality; a k-uniform hypergraph is a hypergraph such that all its hyperedges have size k. (In other words, one such hypergraph is a collection of sets, each such set a hyperedge connecting k nodes.)
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https://en.wikipedia.org/wiki/Dual_hypergraph
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So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. An undirected hypergraph is also called a set system or a family of sets drawn from the universal set.
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https://en.wikipedia.org/wiki/Dual_hypergraph
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Hypergraphs can be viewed as incidence structures. In particular, there is a bipartite "incidence graph" or "Levi graph" corresponding to every hypergraph, and conversely, every bipartite graph can be regarded as the incidence graph of a hypergraph when it is 2-colored and it is indicated which color class corresponds to hypergraph vertices and which to hypergraph edges. Hypergraphs have many other names.
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https://en.wikipedia.org/wiki/Dual_hypergraph
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In computational geometry, an undirected hypergraph may sometimes be called a range space and then the hyperedges are called ranges. In cooperative game theory, hypergraphs are called simple games (voting games); this notion is applied to solve problems in social choice theory. In some literature edges are referred to as hyperlinks or connectors.The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms.
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https://en.wikipedia.org/wiki/Dual_hypergraph
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In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a parameter n, the dimension of the hypercube. As a Coxeter group it is of type Bn = Cn, and as a Weyl group it is associated to the symplectic groups and with the orthogonal groups in odd dimensions.
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https://en.wikipedia.org/wiki/Signed_symmetric_group
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As a wreath product it is S 2 ≀ S n {\displaystyle S_{2}\wr S_{n}} where Sn is the symmetric group of degree n. As a permutation group, the group is the signed symmetric group of permutations π either of the set { − n , − n + 1 , ⋯ , − 1 , 1 , 2 , ⋯ , n } {\displaystyle \{-n,-n+1,\cdots ,-1,1,2,\cdots ,n\}} or of the set { − n , − n + 1 , ⋯ , n } {\displaystyle \{-n,-n+1,\cdots ,n\}} such that π ( i ) = − π ( − i ) {\displaystyle \pi (i)=-\pi (-i)} for all i. As a matrix group, it can be described as the group of n × n orthogonal matrices whose entries are all integers. Equivalently, this is the set of n × n matrices with entries only 0, 1, or –1, which are invertible, and which have exactly one non-zero entry in each row or column. The representation theory of the hyperoctahedral group was described by (Young 1930) according to (Kerber 1971, p.
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https://en.wikipedia.org/wiki/Signed_symmetric_group
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2). In three dimensions, the hyperoctahedral group is known as O × S2 where O ≅ S4 is the octahedral group, and S2 is a symmetric group (here a cyclic group) of order 2. Geometric figures in three dimensions with this symmetry group are said to have octahedral symmetry, named after the regular octahedron, or 3-orthoplex. In 4-dimensions it is called a hexadecachoric symmetry, after the regular 16-cell, or 4-orthoplex. In two dimensions, the hyperoctahedral group structure is the abstract dihedral group of order eight, describing the symmetry of a square, or 2-orthoplex.
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https://en.wikipedia.org/wiki/Signed_symmetric_group
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In mathematics, a hyperplane section of a subset X of projective space Pn is the intersection of X with some hyperplane H. In other words, we look at the subset XH of those elements x of X that satisfy the single linear condition L = 0 defining H as a linear subspace. Here L or H can range over the dual projective space of non-zero linear forms in the homogeneous coordinates, up to scalar multiplication. From a geometrical point of view, the most interesting case is when X is an algebraic subvariety; for more general cases, in mathematical analysis, some analogue of the Radon transform applies.
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https://en.wikipedia.org/wiki/Hyperplane_section
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In algebraic geometry, assuming therefore that X is V, a subvariety not lying completely in any H, the hyperplane sections are algebraic sets with irreducible components all of dimension dim(V) − 1. What more can be said is addressed by a collection of results known collectively as Bertini's theorem. The topology of hyperplane sections is studied in the topic of the Lefschetz hyperplane theorem and its refinements. Because the dimension drops by one in taking hyperplane sections, the process is potentially an inductive method for understanding varieties of higher dimension. A basic tool for that is the Lefschetz pencil.
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https://en.wikipedia.org/wiki/Hyperplane_section
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In mathematics, a hypertoric variety or toric hyperkähler variety is a quaternionic analog of a toric variety constructed by applying the hyper-Kähler quotient construction of N. J. Hitchin, A. Karlhede, and U. Lindström et al. (1987) to a torus acting on a quaternionic vector space. Roger Bielawski and Andrew S. Dancer (2000) gave a systematic description of hypertoric varieties.
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https://en.wikipedia.org/wiki/Hypertoric_variety
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In mathematics, a hypocontinuous is a condition on bilinear maps of topological vector spaces that is weaker than continuity but stronger than separate continuity. Many important bilinear maps that are not continuous are, in fact, hypocontinuous.
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https://en.wikipedia.org/wiki/Hypocontinuous_bilinear_map
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In mathematics, a jacket matrix is a square symmetric matrix A = ( a i j ) {\displaystyle A=(a_{ij})} of order n if its entries are non-zero and real, complex, or from a finite field, and A B = B A = I n {\displaystyle \ AB=BA=I_{n}} where In is the identity matrix, and B = 1 n ( a i j − 1 ) T . {\displaystyle \ B={1 \over n}(a_{ij}^{-1})^{T}.} where T denotes the transpose of the matrix. In other words, the inverse of a jacket matrix is determined its element-wise or block-wise inverse. The definition above may also be expressed as: ∀ u , v ∈ { 1 , 2 , … , n }: a i u , a i v ≠ 0 , ∑ i = 1 n a i u − 1 a i v = { n , u = v 0 , u ≠ v {\displaystyle \forall u,v\in \{1,2,\dots ,n\}:~a_{iu},a_{iv}\neq 0,~~~~\sum _{i=1}^{n}a_{iu}^{-1}\,a_{iv}={\begin{cases}n,&u=v\\0,&u\neq v\end{cases}}} The jacket matrix is a generalization of the Hadamard matrix; it is a diagonal block-wise inverse matrix.
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https://en.wikipedia.org/wiki/Jacket_matrix
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In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. A jet group is a group of jets that describes how a Taylor polynomial transforms under changes of coordinate systems (or, equivalently, diffeomorphisms).
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https://en.wikipedia.org/wiki/Jet_group
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In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a meet (or greatest lower bound) for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the inverse order and vice versa.
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https://en.wikipedia.org/wiki/Join_semilattice
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Semilattices can also be defined algebraically: join and meet are associative, commutative, idempotent binary operations, and any such operation induces a partial order (and the respective inverse order) such that the result of the operation for any two elements is the least upper bound (or greatest lower bound) of the elements with respect to this partial order. A lattice is a partially ordered set that is both a meet- and join-semilattice with respect to the same partial order. Algebraically, a lattice is a set with two associative, commutative idempotent binary operations linked by corresponding absorption laws.
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https://en.wikipedia.org/wiki/Join_semilattice
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In mathematics, a jumping line or exceptional line of a vector bundle over projective space is a projective line in projective space where the vector bundle has exceptional behavior, in other words the structure of its restriction to the line "jumps". Jumping lines were introduced by R. L. E. Schwarzenberger (1961). The jumping lines of a vector bundle form a proper closed subset of the Grassmannian of all lines of projective space.
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https://en.wikipedia.org/wiki/Jumping_line
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The Birkhoff–Grothendieck theorem classifies the n-dimensional vector bundles over a projective line as corresponding to unordered n-tuples of integers. This phenomenon cannot be generalized to higher dimensional projective spaces, namely, one cannot decompose an arbitrary bundle in terms of a Whitney sum of powers of the Tautological bundle, or in fact of line bundles in general. Still one can gain information of this type by using the following method.
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https://en.wikipedia.org/wiki/Jumping_line
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Given a bundle on C P n {\displaystyle \mathbb {CP} ^{n}} , E {\displaystyle {\mathcal {E}}} , we may take a line L {\displaystyle L} in C P n {\displaystyle \mathbb {CP} ^{n}} , or equivalently, a 2-dimensional subspace of C n + 1 {\displaystyle \mathbb {C} ^{n+1}} . This forms a variety equivalent to C P 1 {\displaystyle \mathbb {CP} ^{1}} embedded in C P n {\displaystyle \mathbb {CP} ^{n}} , so we can the restriction of E L {\displaystyle {\mathcal {E}}_{L}} to L {\displaystyle L} , and it will decompose by the Birkhoff–Grothendieck theorem as a sum of powers of the Tautological bundle. It can be shown that the unique tuple of integers specified by this splitting is the same for a 'generic' choice of line.
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https://en.wikipedia.org/wiki/Jumping_line
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More technically, there is a non-empty, open sub-variety of the Grassmannian of lines in C P n {\displaystyle \mathbb {CP} ^{n}} , with decomposition of the same type. Lines such that the decomposition differs from this generic type are called 'Jumping Lines'. If the bundle is generically trivial along lines, then the Jumping lines are precisely the lines such that the restriction is nontrivial.
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https://en.wikipedia.org/wiki/Jumping_line
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In mathematics, a k-Scorza variety is a smooth projective variety, of maximal dimension among those whose k–1 secant varieties are not the whole of projective space. Scorza varieties were introduced and classified by Zak (1993), who named them after Gaetano Scorza. The special case of 2-Scorza varieties are sometimes called Severi varieties, after Francesco Severi.
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https://en.wikipedia.org/wiki/Scorza_variety
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In mathematics, a k-hyperperfect number is a natural number n for which the equality n = 1 + k(σ(n) − n − 1) holds, where σ(n) is the divisor function (i.e., the sum of all positive divisors of n). A hyperperfect number is a k-hyperperfect number for some integer k. Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect.The first few numbers in the sequence of k-hyperperfect numbers are 6, 21, 28, 301, 325, 496, 697, ... (sequence A034897 in the OEIS), with the corresponding values of k being 1, 2, 1, 6, 3, 1, 12, ... (sequence A034898 in the OEIS). The first few k-hyperperfect numbers that are not perfect are 21, 301, 325, 697, 1333, ... (sequence A007592 in the OEIS).
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https://en.wikipedia.org/wiki/Hyperperfect_number
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In mathematics, a k-ultrahomogeneous graph is a graph in which every isomorphism between two of its induced subgraphs of at most k vertices can be extended to an automorphism of the whole graph. A k-homogeneous graph obeys a weakened version of the same property in which every isomorphism between two induced subgraphs implies the existence of an automorphism of the whole graph that maps one subgraph to the other (but does not necessarily extend the given isomorphism).A homogeneous graph is a graph that is k-homogeneous for every k, or equivalently k-ultrahomogeneous for every k.
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https://en.wikipedia.org/wiki/Homogeneous_graph
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In mathematics, a knee of a curve (or elbow of a curve) is a point where the curve visibly bends, specifically from high slope to low slope (flat or close to flat), or in the other direction. This is particularly used in optimization, where a knee point is the optimum point for some decision, for example when there is an increasing function and a trade-off between the benefit (vertical y axis) and the cost (horizontal x axis): the knee is where the benefit is no longer increasing rapidly, and is no longer worth the cost of further increases – a cutoff point of diminishing returns. In heuristic use, the term may be used informally, and a knee point identified visually, but in more formal use an explicit objective function is used, and depends on the particular optimization problem. A knee may also be defined purely geometrically, in terms of the curvature or the second derivative.
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https://en.wikipedia.org/wiki/Knee_of_a_curve
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In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot K is defined as the fundamental group of the knot complement of K in R3, π 1 ( R 3 ∖ K ) . {\displaystyle \pi _{1}(\mathbb {R} ^{3}\setminus K).} Other conventions consider knots to be embedded in the 3-sphere, in which case the knot group is the fundamental group of its complement in S 3 {\displaystyle S^{3}} .
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https://en.wikipedia.org/wiki/Knot_group
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In mathematics, a knot is an embedding of the circle S1 into three-dimensional Euclidean space, R3 (also known as E3). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of R3 which takes one knot to the other. A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed — there are no ends to tie or untie on a mathematical knot.
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https://en.wikipedia.org/wiki/Framed_knot
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Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term knot is also applied to embeddings of S j in Sn, especially in the case j = n − 2. The branch of mathematics that studies knots is known as knot theory and has many relations to graph theory.
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https://en.wikipedia.org/wiki/Framed_knot
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In mathematics, a lattice word (or lattice permutation) is a string composed of positive integers, in which every prefix contains at least as many positive integers i as integers i + 1. A reverse lattice word, or Yamanouchi word, is a string whose reversal is a lattice word.
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https://en.wikipedia.org/wiki/Lattice_word
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In mathematics, a layer group is a three-dimensional extension of a wallpaper group, with reflections in the third dimension. It is a space group with a two-dimensional lattice, meaning that it is symmetric over repeats in the two lattice directions. The symmetry group at each lattice point is an axial crystallographic point group with the main axis being perpendicular to the lattice plane. Table of the 80 layer groups, organized by crystal system or lattice type, and by their point groups:
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https://en.wikipedia.org/wiki/Layer_group
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In mathematics, a leaky integrator equation is a specific differential equation, used to describe a component or system that takes the integral of an input, but gradually leaks a small amount of input over time. It appears commonly in hydraulics, electronics, and neuroscience where it can represent either a single neuron or a local population of neurons.
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https://en.wikipedia.org/wiki/Leaky_integrator
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In mathematics, a left (or right) quaternionic vector space is a left (or right) H-module where H is the (non-commutative) division ring of quaternions. The space Hn of n-tuples of quaternions is both a left and right H-module using the componentwise left and right multiplication: q ( q 1 , q 2 , … q n ) = ( q q 1 , q q 2 , … q q n ) {\displaystyle q(q_{1},q_{2},\ldots q_{n})=(qq_{1},qq_{2},\ldots qq_{n})} ( q 1 , q 2 , … q n ) q = ( q 1 q , q 2 q , … q n q ) {\displaystyle (q_{1},q_{2},\ldots q_{n})q=(q_{1}q,q_{2}q,\ldots q_{n}q)} for quaternions q and q1, q2, ... qn. Since H is a division algebra, every finitely generated (left or right) H-module has a basis, and hence is isomorphic to Hn for some n.
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https://en.wikipedia.org/wiki/Quaternionic_vector_space
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In mathematics, a left derivative and a right derivative are derivatives (rates of change of a function) defined for movement in one direction only (left or right; that is, to lower or higher values) by the argument of a function.
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https://en.wikipedia.org/wiki/Semi-differentiability
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In mathematics, a lethargy theorem is a statement about the distance of points in a metric space from members of a sequence of subspaces; one application in numerical analysis is to approximation theory, where such theorems quantify the difficulty of approximating general functions by functions of special form, such as polynomials. In more recent work, the convergence of a sequence of operators is studied: these operators generalise the projections of the earlier work.
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https://en.wikipedia.org/wiki/Lethargy_theorem
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In mathematics, a level set of a real-valued function f of n real variables is a set where the function takes on a given constant value c, that is: L c ( f ) = { ( x 1 , … , x n ) ∣ f ( x 1 , … , x n ) = c } , {\displaystyle L_{c}(f)=\left\{(x_{1},\ldots ,x_{n})\mid f(x_{1},\ldots ,x_{n})=c\right\}~,} When the number of independent variables is two, a level set is called a level curve, also known as contour line or isoline; so a level curve is the set of all real-valued solutions of an equation in two variables x1 and x2. When n = 3, a level set is called a level surface (or isosurface); so a level surface is the set of all real-valued roots of an equation in three variables x1, x2 and x3. For higher values of n, the level set is a level hypersurface, the set of all real-valued roots of an equation in n > 3 variables. A level set is a special case of a fiber.
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https://en.wikipedia.org/wiki/Level_sets
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In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. In formulas, a limit of a function is usually written as lim x → c f ( x ) = L , {\displaystyle \lim _{x\to c}f(x)=L,} (although a few authors use "Lt" instead of "lim") and is read as "the limit of f of x as x approaches c equals L". The fact that a function f approaches the limit L as x approaches c is sometimes denoted by a right arrow (→ or → {\displaystyle \rightarrow } ), as in f ( x ) → L as x → c , {\displaystyle f(x)\to L{\text{ as }}x\to c,} which reads " f {\displaystyle f} of x {\displaystyle x} tends to L {\displaystyle L} as x {\displaystyle x} tends to c {\displaystyle c} ".
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https://en.wikipedia.org/wiki/Mathematical_limit
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In mathematics, a limit point, accumulation point, or cluster point of a set S {\displaystyle S} in a topological space X {\displaystyle X} is a point x {\displaystyle x} that can be "approximated" by points of S {\displaystyle S} in the sense that every neighbourhood of x {\displaystyle x} with respect to the topology on X {\displaystyle X} also contains a point of S {\displaystyle S} other than x {\displaystyle x} itself. A limit point of a set S {\displaystyle S} does not itself have to be an element of S . {\displaystyle S.} There is also a closely related concept for sequences.
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https://en.wikipedia.org/wiki/Limit_points
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A cluster point or accumulation point of a sequence ( x n ) n ∈ N {\displaystyle (x_{n})_{n\in \mathbb {N} }} in a topological space X {\displaystyle X} is a point x {\displaystyle x} such that, for every neighbourhood V {\displaystyle V} of x , {\displaystyle x,} there are infinitely many natural numbers n {\displaystyle n} such that x n ∈ V . {\displaystyle x_{n}\in V.} This definition of a cluster or accumulation point of a sequence generalizes to nets and filters.
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https://en.wikipedia.org/wiki/Limit_points
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The similarly named notion of a limit point of a sequence (respectively, a limit point of a filter, a limit point of a net) by definition refers to a point that the sequence converges to (respectively, the filter converges to, the net converges to). Importantly, although "limit point of a set" is synonymous with "cluster/accumulation point of a set", this is not true for sequences (nor nets or filters). That is, the term "limit point of a sequence" is not synonymous with "cluster/accumulation point of a sequence".
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https://en.wikipedia.org/wiki/Limit_points
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The limit points of a set should not be confused with adherent points (also called points of closure) for which every neighbourhood of x {\displaystyle x} contains another point of S {\displaystyle S} . Unlike for limit points, an adherent point x {\displaystyle x} of S {\displaystyle S} may have a neighbourhood not containing points other than x {\displaystyle x} itself. A limit point can be characterized as an adherent point that is not an isolated point.
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https://en.wikipedia.org/wiki/Limit_points
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Limit points of a set should also not be confused with boundary points. For example, 0 {\displaystyle 0} is a boundary point (but not a limit point) of the set { 0 } {\displaystyle \{0\}} in R {\displaystyle \mathbb {R} } with standard topology. However, 0.5 {\displaystyle 0.5} is a limit point (though not a boundary point) of interval {\displaystyle } in R {\displaystyle \mathbb {R} } with standard topology (for a less trivial example of a limit point, see the first caption).This concept profitably generalizes the notion of a limit and is the underpinning of concepts such as closed set and topological closure. Indeed, a set is closed if and only if it contains all of its limit points, and the topological closure operation can be thought of as an operation that enriches a set by uniting it with its limit points.
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https://en.wikipedia.org/wiki/Limit_points
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In mathematics, a limiting case of a mathematical object is a special case that arises when one or more components of the object take on their most extreme possible values. For example: In statistics, the limiting case of the binomial distribution is the Poisson distribution. As the number of events tends to infinity in the binomial distribution, the random variable changes from the binomial to the Poisson distribution. A circle is a limiting case of various other figures, including the Cartesian oval, the ellipse, the superellipse, and the Cassini oval.
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https://en.wikipedia.org/wiki/Limiting_case_(mathematics)
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Each type of figure is a circle for certain values of the defining parameters, and the generic figure appears more like a circle as the limiting values are approached. Archimedes calculated an approximate value of π by treating the circle as the limiting case of a regular polygon with 3 × 2n sides, as n gets large. In electricity and magnetism, the long wavelength limit is the limiting case when the wavelength is much larger than the system size.
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https://en.wikipedia.org/wiki/Limiting_case_(mathematics)
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In economics, two limiting cases of a demand curve or supply curve are those in which the elasticity is zero (the totally inelastic case) or infinity (the infinitely elastic case). In finance, continuous compounding is the limiting case of compound interest in which the compounding period becomes infinitesimally small, achieved by taking the limit as the number of compounding periods per year goes to infinity.A limiting case is sometimes a degenerate case in which some qualitative properties differ from the corresponding properties of the generic case. For example: A point is a degenerate circle, namely one with radius 0.
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https://en.wikipedia.org/wiki/Limiting_case_(mathematics)
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A parabola can degenerate into two distinct or coinciding parallel lines. An ellipse can degenerate into a single point or a line segment. A hyperbola can degenerate into two intersecting lines.
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https://en.wikipedia.org/wiki/Limiting_case_(mathematics)
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In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a vector bundle of rank 1.Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner. In topological applications, this vector space is usually real or complex.
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https://en.wikipedia.org/wiki/Complex_line_bundle
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The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 invertible real matrices, which is homotopy-equivalent to a discrete two-point space by contracting the positive and negative reals each to a point; whereas removing the origin from the complex plane yields the 1×1 invertible complex matrices, which have the homotopy type of a circle. From the perspective of homotopy theory, a real line bundle therefore behaves much the same as a fiber bundle with a two-point fiber, that is, like a double cover. A special case of this is the orientable double cover of a differentiable manifold, where the corresponding line bundle is the determinant bundle of the tangent bundle (see below).
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https://en.wikipedia.org/wiki/Complex_line_bundle
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The Möbius strip corresponds to a double cover of the circle (the θ → 2θ mapping) and by changing the fiber, can also be viewed as having a two-point fiber, the unit interval as a fiber, or the real line. Complex line bundles are closely related to circle bundles. There are some celebrated ones, for example the Hopf fibrations of spheres to spheres.
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https://en.wikipedia.org/wiki/Complex_line_bundle
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In algebraic geometry, an invertible sheaf (i.e., locally free sheaf of rank one) is often called a line bundle. Every line bundle arises from a divisor with the following conditions (I) If X is reduced and irreducible scheme, then every line bundle comes from a divisor. (II) If X is projective scheme then the same statement holds.
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https://en.wikipedia.org/wiki/Complex_line_bundle
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In mathematics, a line field on a manifold is a formation of a line being tangent to a manifold at each point, i.e. a section of the line bundle over the manifold. Line fields are of particular interest in the study of complex dynamical systems, where it is conventional to modify the definition slightly.
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https://en.wikipedia.org/wiki/Line_field
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In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane. The function to be integrated may be a scalar field or a vector field.
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https://en.wikipedia.org/wiki/Curve_integral
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The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as W = F ⋅ s {\displaystyle W=\mathbf {F} \cdot \mathbf {s} } , have natural continuous analogues in terms of line integrals, in this case W = ∫ L F ( s ) ⋅ d s {\textstyle W=\int _{L}\mathbf {F} (\mathbf {s} )\cdot d\mathbf {s} } , which computes the work done on an object moving through an electric or gravitational field F along a path L {\displaystyle L} .
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https://en.wikipedia.org/wiki/Curve_integral
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In mathematics, a linear algebraic group is a subgroup of the group of invertible n × n {\displaystyle n\times n} matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M T M = I n {\displaystyle M^{T}M=I_{n}} where M T {\displaystyle M^{T}} is the transpose of M {\displaystyle M} . Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(n,R).)
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https://en.wikipedia.org/wiki/Linear_algebraic_group_action
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The simple Lie groups were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include Maurer, Chevalley, and Kolchin (1948). In the 1950s, Armand Borel constructed much of the theory of algebraic groups as it exists today. One of the first uses for the theory was to define the Chevalley groups.
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https://en.wikipedia.org/wiki/Linear_algebraic_group_action
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In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.
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https://en.wikipedia.org/wiki/Tangent_line_approximation
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In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.
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https://en.wikipedia.org/wiki/Linear_combination
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In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a0(x), ..., an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y(n) are the successive derivatives of an unknown function y of the variable x. Such an equation is an ordinary differential equation (ODE). A linear differential equation may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives. A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-constant coefficients.
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https://en.wikipedia.org/wiki/First-order_linear_differential_equation
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An equation of order two or higher with non-constant coefficients cannot, in general, be solved by quadrature. For order two, Kovacic's algorithm allows deciding whether there are solutions in terms of integrals, and computing them if any.
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https://en.wikipedia.org/wiki/First-order_linear_differential_equation
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The solutions of homogeneous linear differential equations with polynomial coefficients are called holonomic functions. This class of functions is stable under sums, products, differentiation, integration, and contains many usual functions and special functions such as exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric functions. Their representation by the defining differential equation and initial conditions allows making algorithmic (on these functions) most operations of calculus, such as computation of antiderivatives, limits, asymptotic expansion, and numerical evaluation to any precision, with a certified error bound.
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https://en.wikipedia.org/wiki/First-order_linear_differential_equation
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In mathematics, a linear equation is an equation that may be put in the form a 1 x 1 + … + a n x n + b = 0 , {\displaystyle a_{1}x_{1}+\ldots +a_{n}x_{n}+b=0,} where x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are the variables (or unknowns), and b , a 1 , … , a n {\displaystyle b,a_{1},\ldots ,a_{n}} are the coefficients, which are often real numbers. The coefficients may be considered as parameters of the equation, and may be arbitrary expressions, provided they do not contain any of the variables. To yield a meaningful equation, the coefficients a 1 , … , a n {\displaystyle a_{1},\ldots ,a_{n}} are required to not all be zero.
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https://en.wikipedia.org/wiki/First_degree_equation
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Alternatively, a linear equation can be obtained by equating to zero a linear polynomial over some field, from which the coefficients are taken. The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true. In the case of just one variable, there is exactly one solution (provided that a 1 ≠ 0 {\displaystyle a_{1}\neq 0} ).
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https://en.wikipedia.org/wiki/First_degree_equation
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