10,562 research outputs found
The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation
The paper is to reveal the direct links between the well known Sylvester
equation in matrix theory and some integrable systems. Using the Sylvester
equation we introduce a scalar
function
which is defined as same as in discrete case. satisfy some
recurrence relations which can be viewed as discrete equations and play
indispensable roles in deriving continuous integrable equations. By imposing
dispersion relations on and , we find the
Korteweg-de Vries equation, modified Korteweg-de Vries equation, Schwarzian
Korteweg-de Vries equation and sine-Gordon equation can be expressed by some
discrete equations of defined on certain points. Some special
matrices are used to solve the Sylvester equation and prove symmetry property
. The solution provides function
by . We hope our results can not only
unify the Cauchy matrix approach in both continuous and discrete cases, but
also bring more links for integrable systems and variety of areas where the
Sylvester equation appears frequently.Comment: 23 page
Solution of polynomial Lyapunov and Sylvester equations
A two-variable polynomial approach to solve the one-variable polynomial Lyapunov and Sylvester equations is proposed. Lifting the problem from the one-variable to the two-variable context gives rise to associated lifted equations which live on finite-dimensional vector spaces. This allows for the design of an iterative solution method which is inspired by the method of Faddeev for the computation of matrix resolvents. The resulting algorithms are especially suitable for applications requiring symbolic or exact computation
On hyperbolic systems with time dependent H\"older characteristics
In this paper we study the well-posedness of weakly hyperbolic systems with
time dependent coefficients. We assume that the eigenvalues are low regular, in
the sense that they are H\"older with respect to . In the past these kind of
systems have been investigated by Yuzawa \cite{Yu:05} and Kajitani \cite{KY:06}
by employing semigroup techniques (Tanabe-Sobolevski method). Here, under a
certain uniform property of the eigenvalues, we improve the Gevrey
well-posedness result of \cite{Yu:05} and we obtain well-posedness in spaces of
ultradistributions as well. Our main idea is a reduction of the system to block
Sylvester form and then the formulation of suitable energy estimates inspired
by the treatment of scalar equations in \cite{GR:11
New and Old Results in Resultant Theory
Resultants are getting increasingly important in modern theoretical physics:
they appear whenever one deals with non-linear (polynomial) equations, with
non-quadratic forms or with non-Gaussian integrals. Being a subject of more
than three-hundred-year research, resultants are of course rather well studied:
a lot of explicit formulas, beautiful properties and intriguing relationships
are known in this field. We present a brief overview of these results,
including both recent and already classical. Emphasis is made on explicit
formulas for resultants, which could be practically useful in a future physics
research.Comment: 50 pages, 15 figure
Solvability and uniqueness criteria for generalized Sylvester-type equations
We provide necessary and sufficient conditions for the generalized
-Sylvester matrix equation, , to have exactly one
solution for any right-hand side E. These conditions are given for arbitrary
coefficient matrices (either square or rectangular) and generalize
existing results for the same equation with square coefficients. We also review
the known results regarding the existence and uniqueness of solution for
generalized Sylvester and -Sylvester equations.Comment: This new version corrects some inaccuracies in corollaries 7 and
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