34,597 research outputs found
Least-Squares Approximation by Elements from Matrix Orbits Achieved by Gradient Flows on Compact Lie Groups
Let denote the orbit of a complex or real matrix 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 by the sum of matrices
in in the sense of finding the Euclidean least-squares
distance
Connections of the results to different pure and applied areas are discussed
On a new geometric homology theory
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
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
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