Search results for "Linear"
showing 10 items of 7165 documents
Expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta and the Appell generalized hypergeometric functions
2015
We construct several expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta functions and the Appell generalized hypergeometric functions of two variables of the first kind. The coefficients of different expansions obey four-, five-, or six-term recurrence relations that are reduced to ones involving less number of terms only in a few exceptional cases. The conditions for deriving finite-sum solutions via termination of the series are discussed.
Qualitative analysis of matrix splitting methods
2001
Abstract Qualitative properties of matrix splitting methods for linear systems with tridiagonal and block tridiagonal Stieltjes-Toeplitz matrices are studied. Two particular splittings, the so-called symmetric tridiagonal splittings and the bidiagonal splittings, are considered, and conditions for qualitative properties like nonnegativity and shape preservation are shown for them. Special attention is paid to their close relation to the well-known splitting techniques like regular and weak regular splitting methods. Extensions to block tridiagonal matrices are given, and their relation to algebraic representations of domain decomposition methods is discussed. The paper is concluded with ill…
Bessel sequences, Riesz-like bases and operators in Triplets of Hilbert spaces
2016
Riesz-like bases for a triplet of Hilbert spaces are investigated, in connection with an analogous study for more general rigged Hilbert spaces performed in a previous paper. It is shown, in particular, that every \(\omega \)-independent, complete (total) Bessel sequence is a (strict) Riesz-like basis in a convenient triplet of Hilbert spaces. An application to non self-adjoint Schrodinger-type operators is considered. Moreover, some of the simplest operators we can define by them and their dual bases are studied.
An isoperimetric type problem for primitive Pythagorean hodograph curves
2012
An isoperimetric type problem for primitive Pythagorean hodograph curves is studied. We show how to compute, for each possible degree, the Pythagorean hodograph curve of a given perimeter enclosing the greatest area. We also discuss the existence and construction of smooth solutions, obtaining a relationship with an interesting sequence of Appell polynomials.
Approximations of positive operators and continuity of the spectral radius III
1994
AbstractWe prove estimates on the speed of convergence of the ‘peripheral eigenvalues’ (and principal eigenvectors) of a sequence Tn of positive operators on a Banach lattice E to the peripheral eigenvalues of its limit operator T on E which is positive, irreducible and such that the spectral radius r(T) of T is a Riesz point of the spectrum of T (that is, a pole of the resolvent of T with a residuum of finite rank) under some conditions on the kind of approximation of Tn to T. These results sharpen results of convergence obtained by the authors in previous papers.
Some representation theorems for sesquilinear forms
2016
The possibility of getting a Radon-Nikodym type theorem and a Lebesgue-like decomposition for a non necessarily positive sesquilinear $\Omega$ form defined on a vector space $\mathcal D$, with respect to a given positive form $\Theta$ defined on $\D$, is explored. The main result consists in showing that a sesquilinear form $\Omega$ is $\Theta$-regular, in the sense that it has a Radon-Nikodym type representation, if and only if it satisfies a sort Cauchy-Schwarz inequality whose right hand side is implemented by a positive sesquilinear form which is $\Theta$-absolutely continuous. In the particular case where $\Theta$ is an inner product in $\mathcal D$, this class of sesquilinear form cov…
Local Spectral Theory
2018
In this chapter we shall introduce an important property, defined for bounded linear operators on complex Banach spaces, the so-called single-valued extension property (SVEP).
Sign-indefinite second order differential operators on finite metric graphs
2012
The question of self-adjoint realizations of sign-indefinite second order differential operators is discussed in terms of a model problem. Operators of the type $-\frac{d}{dx} \sgn (x) \frac{d}{dx}$ are generalized to finite, not necessarily compact, metric graphs. All self-adjoint realizations are parametrized using methods from extension theory. The spectral and scattering theory of the self-adjoint realizations are studied in detail.
Quantum moment maps and invariants for G-invariant star products
2002
We study a quantum moment map and propose an invariant for $G$-invariant star products on a $G$-transitive symplectic manifold. We start by describing a new method to construct a quantum moment map for $G$-invariant star products of Fedosov type. We use it to obtain an invariant that is invariant under $G$-equivalence. In the last section we give two simple examples of such invariants, which involve non-classical terms and provide new insights into the classification of $G$-invariant star products.
Closed star products and cyclic cohomology
1992
We define the notion of a closed star product. A (generalized) star product (deformation of the associative product of functions on a symplectic manifold W) is closed iff integration over W is a trace on the deformed algebra. We show that for these products the cyclic cohomology replaces the Hochschild cohomology in usual star products. We then define the character of a closed star product as the cohomology class (in the cyclic bicomplex) of a well-defined cocycle, and show that, in the case of pseudodifferential operators (standard ordering on the cotangent bundle to a compact Riemannian manifold), the character is defined and given by the Todd class, while in general it fails to satisfy t…