2,454 research outputs found
On indecomposable normal matrices in spaces with indefinite scalar product
Finite dimensional linear spaces (both complex and real) with indefinite
scalar product [.,.] are considered. Upper and lower bounds are given for the
size of an indecomposable matrix that is normal with respect to this scalar
product in terms of specific functions of v = min{v-, v+}, where v-, (v+) is
the number of negative (positive) squares of the form [x,x]. All the bounds
except for one are proved to be strict.Comment: 9 page
Classification of normal operators in spaces with indefinite scalar product of rank 2
A finite-dimensional complex space with indefinite scalar product [.,.]
having v- = 2 negative squares and v+ >= 2 positive ones is considered. The
paper presents a classification of operators that are normal with respect to
this product. It is related to the study by Gohberg and Reichstein in which a
similar classification was obtained for the case v = min{v-, v+} = 1.Comment: 42 page
Discrete skew selfadjoint canonical systems and the isotropic Heisenberg magnet model
A discrete analog of a skew selfadjoint canonical (Zakharov-Shabat or AKNS)
system with a pseudo-exponential potential is introduced. For the corresponding
Weyl function the direct and inverse problem are solved explicitly in terms of
three parameter matrices. As an application explicit solutions are obtained for
the discrete integrable nonlinear equation corresponding to the isotropic
Heisenberg magnet model. State space techniques from mathematical system theory
play an important role in the proofs
PIETOOLS: A Matlab Toolbox for Manipulation and Optimization of Partial Integral Operators
In this paper, we present PIETOOLS, a MATLAB toolbox for the construction and
handling of Partial Integral (PI) operators. The toolbox introduces a new class
of MATLAB object, opvar, for which standard MATLAB matrix operation syntax
(e.g. +, *, ' e tc.) is defined. PI operators are a generalization of bounded
linear operators on infinite-dimensional spaces that form a *-subalgebra with
two binary operations (addition and composition) on the space RxL2. These
operators frequently appear in analysis and control of infinite-dimensional
systems such as Partial Differential equations (PDE) and Time-delay systems
(TDS). Furthermore, PIETOOLS can: declare opvar decision variables, add
operator positivity constraints, declare an objective function, and solve the
resulting optimization problem using a syntax similar to the sdpvar class in
YALMIP. Use of the resulting Linear Operator Inequalities (LOIs) are
demonstrated on several examples, including stability analysis of a PDE,
bounding operator norms, and verifying integral inequalities. The result is
that PIETOOLS, packaged with SOSTOOLS and MULTIPOLY, offers a scalable,
user-friendly and computationally efficient toolbox for parsing, performing
algebraic operations, setting up and solving convex optimization problems on PI
operators
Fredholm factorization of Wiener-Hopf scalar and matrix kernels
A general theory to factorize the Wiener-Hopf (W-H) kernel using Fredholm Integral Equations (FIE) of the second kind is presented. This technique, hereafter called Fredholm factorization, factorizes the W-H kernel using simple numerical quadrature. W-H kernels can be either of scalar form or of matrix form with arbitrary dimensions. The kernel spectrum can be continuous (with branch points), discrete (with poles), or mixed (with branch points and poles). In order to validate the proposed method, rational matrix kernels in particular are studied since they admit exact closed form factorization. In the appendix a new analytical method to factorize rational matrix kernels is also described. The Fredholm factorization is discussed in detail, supplying several numerical tests. Physical aspects are also illustrated in the framework of scattering problems: in particular, diffraction problems. Mathematical proofs are reported in the pape
Fidelity preserving maps on density operators
We prove that any bijective fidelity preserving transformation on the set of
all density operators on a Hilbert space is implemented by an either unitary or
antiunitary operator on the underlying Hilbert space.Comment: This is corrected version of the paper math.OA/0108060. The paper has
already appeared in ROMP (vol. 48 (2001), 299-303
On the convergence of second order spectra and multiplicity
Let A be a self-adjoint operator acting on a Hilbert space. The notion of
second order spectrum of A relative to a given finite-dimensional subspace L
has been studied recently in connection with the phenomenon of spectral
pollution in the Galerkin method. We establish in this paper a general
framework allowing us to determine how the second order spectrum encodes
precise information about the multiplicity of the isolated eigenvalues of A.
Our theoretical findings are supported by various numerical experiments on the
computation of inclusions for eigenvalues of benchmark differential operators
via finite element bases.Comment: 22 pages, 2 figures, 4 tables, research paper
Krein systems
In the present paper we extend results of M.G. Krein associated to the
spectral problem for Krein systems to systems with matrix valued accelerants
with a possible jump discontinuity at the origin. Explicit formulas for the
accelerant are given in terms of the matrizant of the system in question.
Recent developments in the theory of continuous analogs of the resultant
operator play an essential role
Analyticity and uniform stability in the inverse spectral problem for Dirac operators
We prove that the inverse spectral mapping reconstructing the square
integrable potentials on [0,1] of Dirac operators in the AKNS form from their
spectral data (two spectra or one spectrum and the corresponding norming
constants) is analytic and uniformly stable in a certain sense.Comment: 19 page
Strings in five-dimensional anti-de Sitter space with a symmetry
The equation of motion of an extended object in spacetime reduces to an
ordinary differential equation in the presence of symmetry. By properly
defining of the symmetry with notion of cohomogeneity, we discuss the method
for classifying all these extended objects. We carry out the classification for
the strings in the five-dimensional anti-de Sitter space by the effective use
of the local isomorphism between \SO(4,2) and \SU(2,2). We present a
general method for solving the trajectory of the Nambu-Goto string and apply to
a case obtained by the classification, thereby find a new solution which has
properties unique to odd-dimensional anti-de Sitter spaces. The geometry of the
solution is analized and found to be a timelike helicoid-like surface
- …