Related papers: Singular-value decomposition of solution operators…
The problem of specification of self-adjoint operators corresponding to singular bilinear forms is very important for applications, such as quantum field theory and theory of partial differential equations with coefficient functions being…
We consider the recursion operators with nonlocal terms of special form for evolution systems in (1+1) dimensions, and extend them to well-defined operators on the space of nonlocal symmetries associated with the so-called universal Abelian…
The cyclic evolutions and associated geometric phases induced by time-independent Hamiltonians are studied for the case when the evolution operator becomes the identity (those processes are called {\it evolution loops}). We make a detailed…
The continuous dependence of solutions to certain (non-autonomous, partial, integro-differential-algebraic, evolutionary) equations on the coefficients is addressed. We give criteria that guarantee that convergence of the coefficients in…
In this paper we consider evolution equations in the abstract Hilbert space under the special conditions imposed on the operator at the right-hand side of the equation. We establish the method that allows us to formulate the existence and…
This work is devoted to the geometric-theoretic analysis of the nodal set of solutions to degenerate or singular equations involving a class of operators including $$ L_a = \mbox{div}(\abs{y}^a \nabla), $$ with $a\in(-1,1)$ and their…
We associate to every central simple algebra with involution of orthogonal type in characteristic two a totally singular quadratic form which reflects certain anisotropy properties of the involution. It is shown that this quadratic form can…
The eigenvalue problem plays a central role in linear algebra and its applications in control and optimization methods. In particular, many matrix decompositions rely upon computation of eigenvalue-eigenvector pairs, such as diagonal or…
The relation between the spectral decomposition of a self-adjoint operator which is realizable as a higher order recurrence operator and matrix-valued orthogonal polynomials is investigated. A general construction of such operators from…
Allowing for the inclusion of the parity operator, it is possible to construct an oscillator model whose Hamiltonian admits an EXACT square root, which is different from the conventional approach based on creation and annihilation…
The notion of singular reduction modules, i.e., of singular modules of nonclassical (conditional) symmetry, of differential equations is introduced. It is shown that the derivation of nonclassical symmetries for differential equations can…
In evolution equations for a complex amplitude, the phase obeys a much more intricate equation than the amplitude. Nevertheless, general methods should be applicable to both variables. On the example of the traveling wave reduction of the…
Evolutionary algorithms are good general problem solver but suffer from a lack of domain specific knowledge. However, the problem specific knowledge can be added to evolutionary algorithms by hybridizing. Interestingly, all the elements of…
We develop a unified framework for constructing matrix approximations to the convolution operator of Volterra type defined by functions that are approximated using classical orthogonal polynomials on $[-1, 1]$. The numerically stable…
We study evolution algebras of arbitrary dimension. We analyze in deep the notions of evolution subalgebras, ideals and non-degeneracy and describe the ideals generated by one element and characterize the simple evolution algebras. We also…
The present paper is concerned a class of quasi-linear elliptic degenerate equations. The degenerate operator comes from the analysis of manifolds with corner singularity. Variational methods are applied to verify the existence of infinity…
We introduce the notion of a family of convolution operators associated with a given elliptic partial differential operator. Such a convolution structure is shown to exist for a general class of Laplace-Beltrami operators on two-dimensional…
For a class of non-selfadjoint semiclassical pseudodifferential operators with double characteristics, we study bounds for resolvents and estimates for low lying eigenvalues. Specifically, assuming that the quadratic approximations of the…
In this paper, we define an operator function as a series of operators corresponding to the Taylor series representing the function of the complex variable. In previous papers, we considered the case when a function has a decomposition in…
Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework…