327 research outputs found
Canonical matrices of isometric operators on indefinite inner product spaces
We give canonical matrices of a pair (A,B) consisting of a nondegenerate form
B and a linear operator A satisfying B(Ax,Ay)=B(x,y) on a vector space over F
in the following cases:
(i) F is an algebraically closed field of characteristic different from 2 or
a real closed field, and B is symmetric or skew-symmetric;
(ii) F is an algebraically closed field or the skew field of quaternions over
a real closed field, and B is Hermitian or skew-Hermitian with respect to any
nonidentity involution on F.
We use a method that admits to reduce the problem of classifying an arbitrary
system of forms and linear mappings to the problem of classifying
representations of some quiver. This method was described in [V.V. Sergeichuk,
Math. USSR-Izv. 31 (1988) 481-501].Comment: 57 page
Infinitesimal semisimple symplectic extrinsic symmetric spaces
We study infinitesimal semi-simple extrinsic symmetric spaces and give a
classification in the symplectic case
The inverse spectral problem for indefinite strings
Motivated by the study of certain nonlinear wave equations (in particular,
the Camassa-Holm equation), we introduce a new class of generalized indefinite
strings associated with differential equations of the form
on an interval , where is a
real-valued distribution in , is a
non-negative Borel measure on and is a complex spectral parameter.
Apart from developing basic spectral theory for these kinds of spectral
problems, our main result is an indefinite analogue of M. G. Krein's celebrated
solution of the inverse spectral problem for inhomogeneous vibrating strings.Comment: 27 page
Quantum measure and integration theory
This article begins with a review of quantum measure spaces. Quantum forms
and indefinite inner-product spaces are then discussed. The main part of the
paper introduces a quantum integral and derives some of its properties. The
quantum integral's form for simple functions is characterized and it is shown
that the quantum integral generalizes the Lebesgue integral. A bounded,
monotone convergence theorem for quantum integrals is obtained and it is shown
that a Radon-Nikodym type theorem does not hold for quantum measures. As an
example, a quantum-Lebesgue integral on the real line is considered.Comment: 28 page
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