One of the many applications for the Jacobian matrix is to transfer mapping from one coordinate system to another, such as the transformation from a Cartesian to natural coordinate system, spherical to Cartesian coordinate system, polar to Cartesian coordinate system, and vice versa for each. In ...
The Jacobian Conjecture states that if for anyx∈R2, the eigenvalues of the Jacobian matrixD(x) have negative real parts, then the zero solution of the differential equationx= (x) is globally asymptotically stable. In this paper we prove that the conjecture is true.doi:10.1007/s101140000098...
differentiation when the dimension of the output is small in comparison to the dimension of the input. Forward differentiation is of interest when solving a non linear least squares for examples through Levenberg-Marquardt minimization where we want to compute the full jacobian matrix of the ...
In the present note we describe the invariant manifold of the integrable system (1) in the case when the leading term J of the Lax matrix...E. Previato, Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schr¨... E Previato - 《Duke Mathematical Journal》 被引量: 191...
I am used to the usual definition of the Jacobian (when the talk is about derivatives) as the Jacobian matrix for multi-valued functions. However, in the 1995 edition of the introductory book "Basic Training in Mathematics: A fitness program for science students" on page 45 , equations 2.2...
The jacobian(F,v) function computes the Jacobian matrix of F with respect to the variable(s) v. • Each row of the result corresponds to the partial derivatives of an entry in F with respect to the entries of v. Examples > withMTM: > M1≔Matrix2,3&comma...
However, the methods which underlie these algorithms do not appear to be easily adaptable for finding discrete logarithms in the groups as... LM Adleman,J Demarrais,MD Huang - Springer Berlin Heidelberg 被引量: 515发表: 1994年 Jacobians of Matrix Transformations and Functions of Matrix Arguments...
Dynamical Systems and Introduction to Chaos", the norm of the Jacobian matrix is defined to be: |DF_x| = sup |DF_x (U)|, where U is in R^n and F: R^n -> R^n and the |U| = 1 is under the sup. ...|U| = 1 DF_x (U) is the directional derivative of F in the di...
The Jacobian Conjecture states that if for any x∈R2, the eigenvalues of the Jacobian matrix Df(x) have negative real parts, then the zero solution of the differential equation x=f(x) is globally asymptotically stable. In this paper we prove that the conjecture is true. Key words Plane...
Gordan and Noether proved in their fundamental theorem that an hypersurface$X=V(F)\subseteq \mathbb{P}^n$ with $n\leq 3$ is a cone if and only if $F$ hasvanishing hessian (i.e. the determinant of the Hessian matrix). They alsoshowed that the statement is false if $n\geq 4$, ...