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Method Of Steepest Descent
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals. The integral to be estimated is often of the form :\int_Cf(z)e^\,dz, where ''C'' is a contour, and λ is large. One version of the method of steepest descent deforms the contour of integration ''C'' into a new path integration ''C′'' so that the following conditions hold: # ''C′'' passes through one or more zeros of the derivative ''g''′(''z''), # the imaginary part of ''g''(''z'') is constant on ''C′''. The method of steepest descent was first published by , who used it to estimate Bessel functions and pointed out that it occurred in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace's Method
In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form :\int_a^b e^ \, dx, where f is a twice-differentiable function, M is a large number, and the endpoints a and b could be infinite. This technique was originally presented in the book by . In Bayesian statistics, Laplace's approximation can refer to either approximating the posterior normalizing constant with Laplace's method or approximating the posterior distribution with a Gaussian centered at the maximum a posteriori estimate. Laplace approximations are used in the integrated nested Laplace approximations method for fast approximations of Bayesian inference. Concept Let the function f(x) have a unique global maximum at x_0. M>0 is a constant here. The following two functions are considered: :\begin g(x) &= Mf(x), \\ h(x) &= e^. \end Then, x_0 is the global maximum of g and h as well. Hence: :\begin \frac &= \frac = \frac, \\ pt\frac &= \frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan Normal Form
\begin \lambda_1 1\hphantom\hphantom\\ \hphantom\lambda_1 1\hphantom\\ \hphantom\lambda_1\hphantom\\ \hphantom\lambda_2 1\hphantom\hphantom\\ \hphantom\hphantom\lambda_2\hphantom\\ \hphantom\lambda_3\hphantom\\ \hphantom\ddots\hphantom\\ \hphantom\lambda_n 1\hphantom\\ \hphantom\hphantom\lambda_n \end Example of a matrix in Jordan normal form. All matrix entries not shown are zero. The outlined squares are known as "Jordan blocks". Each Jordan block contains one number ''λi'' on its main diagonal, and 1s directly above the main diagonal. The ''λi''s are the eigenvalues of the matrix; they need not be distinct. In linear algebra, a Jordan normal form, also known as a Jordan canonical form, is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to some basis. Such a matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Caustic (optics)
In optics, a caustic or caustic network is the Envelope (mathematics), envelope of Ray (optics), light rays which have been Reflection (physics), reflected or refraction, refracted by a curved surface or object, or the Projection (mathematics), projection of that envelope of rays on another surface. The caustic is a curve or Surface (mathematics), surface to which each of the light rays is tangent, defining a boundary of an envelope of rays as a curve of concentrated light. In some cases caustics can be seen as patches of light or their bright edges, shapes which often have cusp (singularity), cusp singularities. Explanation Concentration of light, especially sunlight, can burn. The word ''caustic'', in fact, comes from the Greek καυστός, burnt, via the Latin ''causticus'', burning. A common situation where caustics are visible is when light shines on a drinking glass. The glass casts a shadow, but also produces a curved region of bright light. In ideal circumstances ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catastrophe Theory
In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry. Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analysing how the qualitative nature of equation solutions depends on the parameters that appear in the equation. This may lead to sudden and dramatic changes, for example the unpredictable timing and magnitude of a landslide. Catastrophe theory originated with the work of the French mathematician René Thom in the 1960s, and became very popular due to the efforts of Christopher Zeeman in the 1970s. It considers the special case where the long-run stable equilibrium can be identified as the minimum of a smooth, well-defined potential function (Lyapunov function). Small changes in certain parameters of a nonlinear system can cause equilibria to appear or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Of Unity
In mathematics, a partition of unity on a topological space is a Set (mathematics), set of continuous function (topology), continuous functions from to the unit interval [0,1] such that for every point x\in X: * there is a neighbourhood (mathematics), neighbourhood of where all but a finite set, finite number of the functions of are non zero, and * the sum of all the function values at is 1, i.e., \sum_ \rho(x) = 1. Partitions of unity are useful because they often allow one to extend local constructions to the whole space. They are also important in the interpolation of data, in signal processing, and the theory of spline functions. Existence The existence of partitions of unity assumes two distinct forms: # Given any open cover \_ of a space, there exists a partition \_ indexed ''over the same set'' such that Support (mathematics), supp \rho_i \subseteq U_i. Such a partition is said to be subordinate to the open cover \_i. # If the space is locally compact, given an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\subset X (indexed by the set A), then C is a cover of X if \bigcup_U_ = X. Thus the collection \lbrace U_\alpha : \alpha \in A \rbrace is a cover of X if each element of X belongs to at least one of the subsets U_. Definition Covers are commonly used in the context of topology. If the set X is a topological space, then a cover C of X is a collection of subsets \_ of X whose union is the whole space X = \bigcup_U_. In this case C is said to cover X, or that the sets U_\alpha cover X. If Y is a (topological) subspace of X, then a cover of Y is a collection of subsets C = \_ of X whose union contains Y. That is, C is a cover of Y if Y \subseteq \bigcup_U_. Here, Y may be covered with either sets in Y itself or sets in the parent spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maslov Index
In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space ''V''. Its dimension is ''n''(''n'' + 1) (where the dimension of ''V'' is ''2n''). It may be identified with the homogeneous space :, where is the unitary group and the orthogonal group. Following Vladimir Arnold it is denoted by Λ(''n''). The Lagrangian Grassmannian is a submanifold of the ordinary Grassmannian of V. A complex Lagrangian Grassmannian is the homogeneous space, complex homogeneous manifold of Lagrangian subspaces of a complex symplectic vector space ''V'' of dimension 2''n''. It may be identified with the homogeneous space of complex dimension ''n''(''n'' + 1) :, where is the symplectic group, compact symplectic group. As a homogeneous space To see that the Lagrangian Grassmannian Λ(''n'') can be identified with , note that \mathbb^n is a 2''n''-dimensional real vector space, with the imaginary part of its usual inner product making ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylvester's Law Of Inertia
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if A is a symmetric matrix, then for any invertible matrix S, the number of positive, negative and zero eigenvalues (called the inertia of the matrix) of D=SAS^\mathrm is constant. This result is particularly useful when D is diagonal, as the inertia of a diagonal matrix can easily be obtained by looking at the sign of its diagonal elements. This property is named after James Joseph Sylvester who published its proof in 1852. Statement Let A be a symmetric square matrix of order n with real entries. Any non-singular matrix S of the same size is said to transform A into another symmetric matrix , also of order , where S^\mathrm is the transpose of . It is also said that matrices A and B are congruent. If A is the coefficient matrix of some quadratic form of , then B is the matrix fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Definite Bilinear Form
In mathematics, a definite quadratic form is a quadratic form over some real vector space that has the same sign (always positive or always negative) for every non-zero vector of . According to that sign, the quadratic form is called positive-definite or negative-definite. A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "never negative" and "never positive", respectively. In other words, it may take on zero values for some non-zero vectors of . An indefinite quadratic form takes on both positive and negative values and is called an isotropic quadratic form. More generally, these definitions apply to any vector space over an ordered field. Associated symmetric bilinear form Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space.This is true only over a field of characteristic other than 2, but here we consider only ordered fields, which necessaril ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Method Of Steepest Descent
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals. The integral to be estimated is often of the form :\int_Cf(z)e^\,dz, where ''C'' is a contour, and λ is large. One version of the method of steepest descent deforms the contour of integration ''C'' into a new path integration ''C′'' so that the following conditions hold: # ''C′'' passes through one or more zeros of the derivative ''g''′(''z''), # the imaginary part of ''g''(''z'') is constant on ''C′''. The method of steepest descent was first published by , who used it to estimate Bessel functions and pointed out that it occurred in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Illustration To Derivation Of Asymptotic For Saddle Point Integration
An illustration is a decoration, interpretation, or visual explanation of a text, concept, or process, designed for integration in print and digitally published media, such as posters, flyers, magazines, books, teaching materials, animations, video games and films. An illustration is typically created by an illustrator. Digital illustrations are often used to make websites and apps more user-friendly, such as the use of emojis to accompany digital type. Illustration also means providing an example; either in writing or in picture form. The origin of the word "illustration" is late Middle English (in the sense ‘illumination; spiritual or intellectual enlightenment’): via Old French from Latin">-4; we might wonder whether there's a point at which it's appropriate to talk of the beginnings of French, that is, when it wa ... from Latin ''illustratio''(n-), from the verb ''illustrare''. Illustration styles Contemporary illustration uses a wide range of styles and technique ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a Subspace topology, subspace of X. Some related but stronger conditions are #Path connectedness, path connected, Simply connected space, simply connected, and N-connected space, n-connected. Another related notion is Locally connected space, locally connected, which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
