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Transvectant
In mathematical invariant theory, a transvectant is an invariant formed from ''n'' invariants in ''n'' variables using Cayley's Ω process. Definition If ''Q''1,...,''Q''''n'' are functions of ''n'' variables x = (''x''1,...,''x''''n'') and ''r'' ≥ 0 is an integer then the ''r''th transvectant of these functions is a function of ''n'' variables given by : tr \Omega^r(Q_1\otimes\cdots \otimes Q_n) where Ω is Cayley's Ω process, the tensor product means take a product of functions with different variables x1,..., x''n'', and tr means set all the vectors x''k'' equal. Examples The zeroth transvectant is the product of the ''n'' functions. The first transvectant is the Jacobian determinant In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ... of the ''n'' functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant (mathematics)
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley's Ω Process
In mathematics, Cayley's Ω process, introduced by , is a relatively invariant differential operator on the general linear group, that is used to construct invariants of a group action. As a partial differential operator acting on functions of ''n''2 variables ''x''''ij'', the omega operator is given by the determinant : \Omega = \begin \frac & \cdots &\frac \\ \vdots& \ddots & \vdots\\ \frac & \cdots &\frac \end. For binary forms ''f'' in ''x''1, ''y''1 and ''g'' in ''x''2, ''y''2 the Ω operator is \frac - \frac. The ''r''-fold Ω process Ω''r''(''f'', ''g'') on two forms ''f'' and ''g'' in the variables ''x'' and ''y'' is then # Convert ''f'' to a form in ''x''1, ''y''1 and ''g'' to a form in ''x''2, ''y''2 # Apply the Ω operator ''r'' times to the function ''fg'', that is, ''f'' times ''g'' in these four variables # Substitute ''x'' for ''x''1 and ''x''2, ''y'' for ''y''1 and ''y''2 in the result The result of the ''r''-fold Ω process Ω''r''(''f'', ''g'') on the two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. Suppose is a function such that each of its first-order partial derivatives exist on . This function takes a point as input and produces the vector as output. Then the Jacobian matrix of is defined to be an matrix, denoted by , whose th entry is \mathbf J_ = \frac, or explicitly :\mathbf J = \begin \dfrac & \cdots & \dfrac \end = \begin \nabla^ f_1 \\ \vdots \\ \nabla^ f_m \end = \begin \dfrac & \cdots & \dfrac\\ \vdots & \ddots & \vdots\\ \dfrac & \cdo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hessian Matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants". Definitions and properties Suppose f : \R^n \to \R is a function taking as input a vector \mathbf \in \R^n and outputting a scalar f(\mathbf) \in \R. If all second-order partial derivatives of f exist, then the Hessian matrix \mathbf of f is a square n \times n matrix, usually defined and arranged as follows: \mathbf H_f= \begin \dfrac & \dfrac & \cdots & \dfrac \\ .2ex \dfrac & \dfrac & \cdots & \dfrac \\ .2ex \vdots & \vdots & \ddots & \vdots \\ .2ex \dfrac & \dfrac & \cdots & \dfrac \end, or, by stating an equation for the coefficients using indices i and j, (\mathbf H_f)_ = \fra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
