34,597 research outputs found

    Least-Squares Approximation by Elements from Matrix Orbits Achieved by Gradient Flows on Compact Lie Groups

    Full text link
    Let S(A)S(A) denote the orbit of a complex or real matrix AA under a certain equivalence relation such as unitary similarity, unitary equivalence, unitary congruences etc. Efficient gradient-flow algorithms are constructed to determine the best approximation of a given matrix A0A_0 by the sum of matrices in S(A1),...,S(AN)S(A_1), ..., S(A_N) in the sense of finding the Euclidean least-squares distance min⁑{βˆ₯X1+...+XNβˆ’A0βˆ₯:Xj∈S(Aj),j=1,>...,N}.\min \{\|X_1+ ... + X_N - A_0\|: X_j \in S(A_j), j = 1, >..., N\}. Connections of the results to different pure and applied areas are discussed

    On a new geometric homology theory

    Full text link
    In this note we present a new homology theory, we call it geometric homology theory (or GHT for brevity). We prove that the homology groups of GHT are isomorphic to the singular homology groups, which solves a Conjecture of Voronov. GHT has several nice properties compared with singular homology, which makes itself more suitable than singular homology in some situations, especially in chain-level theories. We will develop further of this theory in our sequel paper.Comment: Comments are appreciated !. arXiv admin note: text overlap with arXiv:0709.3874 by other author

    Reduction Operators of Linear Second-Order Parabolic Equations

    Full text link
    The reduction operators, i.e., the operators of nonclassical (conditional) symmetry, of (1+1)-dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described. This problem proves to be equivalent, in some sense, to solving the initial equations. The ``no-go'' result is extended to the investigation of point transformations (admissible transformations, equivalence transformations, Lie symmetries) and Lie reductions of the determining equations for the nonclassical symmetries. Transformations linearizing the determining equations are obtained in the general case and under different additional constraints. A nontrivial example illustrating applications of reduction operators to finding exact solutions of equations from the class under consideration is presented. An observed connection between reduction operators and Darboux transformations is discussed.Comment: 31 pages, minor misprints are correcte
    • …
    corecore