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List Of Things Named After Carl Gustav Jacob Jacobi ...
These are things named after Carl Gustav Jacob Jacobi (1804–1851), a German mathematician. Jacobi Jacobian {{columns-list, colwidth=30em, * Generalized Jacobian * Intermediate Jacobian * Jacobian conjecture * Jacobian curve * Jacobian matrix and determinant * Jacobian variety Jacobi Jacobi may refer to: * People with the surname Jacobi (surname), Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Gustav Jacob Jacobi
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasionally written as Carolus Gustavus Iacobus Iacobi in his Latin books, and his first name is sometimes given as Karl. Jacobi was the first Jewish mathematician to be appointed professor at a German university. Biography Jacobi was born of Ashkenazi Jewish parentage in Potsdam on 10 December 1804. He was the second of four children of banker Simon Jacobi. His elder brother Moritz von Jacobi would also become known later as an engineer and physicist. He was initially home schooled by his uncle Lehman, who instructed him in the classical languages and elements of mathematics. In 1816, the twelve-year-old Jacobi went to the Potsdam Gymnasium, where students were taught all the standard subjects: classical languages, history, philology, mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Map Projection Of The Triaxial Ellipsoid
In geodesy, a map projection of the triaxial ellipsoid maps Earth or some other astronomical body modeled as a triaxial ellipsoid to the plane. Such a model is called the reference ellipsoid. In most cases, reference ellipsoids are spheroids, and sometimes spheres. Massive objects have sufficient gravity to overcome their own rigidity and usually have an oblate ellipsoid shape. However, minor moons or small solar system bodies are not under hydrostatic equilibrium. Usually such bodies have irregular shapes. Furthermore, some of gravitationally rounded objects may have a tri-axial ellipsoid shape due to rapid rotation (such as Haumea) or unidirectional strong tidal forces (such as Io). Examples A triaxial equivalent of the Mercator projection was developed by John P. Snyder. Equidistant map projections of a triaxial ellipsoid were developed by Paweł Pędzich. Conic Projections of a triaxial ellipsoid were developed by Maxim Nyrtsov. Equal-area cylindrical and azimuthal projec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zech's Logarithm
Zech logarithms are used to implement addition in finite fields when elements are represented as powers of a generator \alpha. Zech logarithms are named after Julius Zech, and are also called Jacobi logarithms, after Carl G. J. Jacobi who used them for number theoretic investigations. Definition Given a primitive element \alpha of a finite field, the Zech logarithm relative to the base \alpha is defined by the equation :\alpha^ = 1 + \alpha^n, which is often rewritten as :Z_\alpha(n) = \log_\alpha(1 + \alpha^n). The choice of base \alpha is usually dropped from the notation when it is clear from the context. To be more precise, Z_\alpha is a function on the integers modulo the multiplicative order of \alpha, and takes values in the same set. In order to describe every element, it is convenient to formally add a new symbol -\infty, along with the definitions :\alpha^ = 0 :n + (-\infty) = -\infty :Z_\alpha(-\infty) = 0 :Z_\alpha(e) = -\infty where e is an integer satisfying \al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Integral
In celestial mechanics, Jacobi's integral (also known as the Jacobi integral or Jacobi constant) is the only known conserved quantity for the circular restricted three-body problem.Bibliothèque nationale de France Unlike in the two-body problem, the energy and momentum of the system are not conserved separately and a general analytical solution is not possible. The integral has been used to derive numerous solutions in special cases. It was named after German mathematician . Definition Synodic system One of the suitable coordinate systems used is the so-called ''syn ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associative property, any order of evaluation gives the same result (parentheses in a multiple product are not needed). The identity is named after the German mathematician Carl Gustav Jacob Jacobi. The cross product a\times b and the Lie bracket operation ,b/math> both satisfy the Jacobi identity. In analytical mechanics, the Jacobi identity is satisfied by the Poisson brackets. In quantum mechanics, it is satisfied by operator commutators on a Hilbert space and equivalently in the phase space formulation of quantum mechanics by the Moyal bracket. Definition Let + and \times be two binary operations, and let 0 be the neutral element for +. The is :x \times (y \times z) \ +\ y \times (z \times x) \ +\ z \times (x \times y)\ =\ 0. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Ideal
In mathematics the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let \mathcal(x_1,\ldots,x_n) denote the ring of smooth functions in n variables and f a function in the ring. The Jacobian ideal of f is : J_f := \left\langle \frac, \ldots, \frac \right\rangle. Relation to deformation theory In deformation theory, the deformations of a hypersurface given by a polynomial f is classified by the ring \frac This is shown using the Kodaira–Spencer map. Relation to Hodge theory In Hodge theory, there are objects called real Hodge structures which are the data of a real vector space H_\mathbb and an increasing filtration F^\bullet of H_\mathbb = H_\mathbb\otimes_\mathbb satisfying a list of compatibility structures. For a smooth projective variety X there is a canonical Hodge structure. Statement for degree d hypersurfaces In the special case X is defined by a homogeneous degree d polynomial f \in \Gamma(\mathbb^,\ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Group
In mathematics, the Jacobi group, introduced by , is the semidirect product of the symplectic group Sp2''n''(R) and the Heisenberg group R1+2''n''. The concept is named after Carl Gustav Jacob Jacobi. Automorphic forms on the Jacobi group are called Jacobi form In mathematics, a Jacobi form is an automorphic form on the Jacobi group, which is the semidirect product of the symplectic group Sp(n;R) and the Heisenberg group H^_R. The theory was first systematically studied by . Definition A Jacobi form of ...s. References * * Modular forms Lie groups {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi's Formula
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix ''A'' in terms of the adjugate of ''A'' and the derivative of ''A''., Part Three, Section 8.3 If is a differentiable map from the real numbers to matrices, then : \frac \det A(t) = \operatorname \left (\operatorname(A(t)) \, \frac\right ) = \left(\det A(t) \right) \cdot \operatorname \left (A(t)^ \cdot \, \frac\right ) where is the trace of the matrix . (The latter equality only holds if ''A''(''t'') is invertible.) As a special case, : = \operatorname(A)_. Equivalently, if stands for the differential of , the general formula is : d \det (A) = \operatorname (\operatorname(A) \, dA). The formula is named after the mathematician Carl Gustav Jacob Jacobi. Derivation Via Matrix Computation We first prove a preliminary lemma: Lemma. Let ''A'' and ''B'' be a pair of square matrices of the same dimension ''n''. Then :\sum_i \sum_j A_ B_ = \operatorname (A^ B). ''Proof.'' The product ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Form
In mathematics, a Jacobi form is an automorphic form on the Jacobi group, which is the semidirect product of the symplectic group Sp(n;R) and the Heisenberg group H^_R. The theory was first systematically studied by . Definition A Jacobi form of level 1, weight ''k'' and index ''m'' is a function \phi(\tau,z) of two complex variables (with τ in the upper half plane) such that *\phi\left(\frac,\frac\right) = (c\tau+d)^ke^\phi(\tau,z)\text\in \mathrm_2(\mathbb) *\phi(\tau,z+\lambda\tau+\mu) = e^\phi(\tau,z) for all integers λ, μ. *\phi has a Fourier expansion :: \phi(\tau,z) = \sum_ \sum_ C(n,r)e^. Examples Examples in two variables include Jacobi theta functions, the Weierstrass ℘ function, and Fourier–Jacobi coefficients of Siegel modular forms of genus 2. Examples with more than two variables include characters of some irreducible highest-weight representations of affine Kac–Moody algebras. Meromorphic Jacobi forms appear in the theory of Mock modular f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi's Four-square Theorem
Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer ''n'' can be represented as the sum of four squares. History The theorem was proved in 1834 by Carl Gustav Jakob Jacobi. Theorem Two representations are considered different if their terms are in different order or if the integer being squared (not just the square) is different; to illustrate, these are three of the eight different ways to represent 1: : \begin & 1^2 + 0^2 + 0^2 + 0^2 \\ & 0^2 + 1^2 + 0^2 + 0^2 \\ & (-1)^2 + 0^2 + 0^2 + 0^2. \end The number of ways to represent n as the sum of four squares is eight times the sum of the divisors of ''n'' if ''n'' is odd and 24 times the sum of the odd divisors of ''n'' if ''n'' is even (see divisor function), i.e. : r_4(n)=\begin8\sum\limits_m&\textn\text\\2pt24\sum\limits_m&\textn\text. \end Equivalently, it is eight times the sum of all its divisors which are not divisible by 4, i.e. :r_4(n)=8\sum_m. We may also write this as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Field
In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space of all geodesics. They are named after Carl Jacobi. Definitions and properties Jacobi fields can be obtained in the following way: Take a smooth one parameter family of geodesics \gamma_\tau with \gamma_0=\gamma, then :J(t)=\left.\frac\_ is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic \gamma. A vector field ''J'' along a geodesic \gamma is said to be a Jacobi field if it satisfies the Jacobi equation: :\fracJ(t)+R(J(t),\dot\gamma(t))\dot\gamma(t)=0, where ''D'' denotes the covariant derivative with respect to the Levi-Civita connection, ''R'' the Riemann curvature tensor, \dot\gamma(t)=d\gamma(t)/dt the tangent vector ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Elliptic Functions
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometry, trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation \operatorname for \sin. The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by . Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later. Overview There are twelve Jacob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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