Functional Analysis and Its Applications最新文献

We consider the Schrödinger equation (ihpartial_tpsi=Hpsi), (psi=psi(cdot,t)in L^2(mathbb{T})). The operator (H=-partial^2_x+V(x,t)) includes a smooth potential (V), which is assumed to be time (T)-periodic. Let (W=W(t)) be the fundamental solution of this linear ODE system on (L^2(mathbb{T})). Then, according to the terminology from Lyapunov–Floquet theory, (mathcal M=W(T)) is the monodromy operator. We prove that (mathcal M) is unitarily conjugated to (D+mathcal C), where (D) is diagonal in the standard Fourier basis, while (mathcal C) is a compact operator with an arbitrarily small norm.

In this work, we study the interlace polynomial as a generalization of a graph invariant to delta-matroids. We prove that the interlace polynomial satisfies the four-term relation for delta-matroids and thus determines a finite type invariant of links in the (3)-sphere. Using the interlace polynomial, we give a lower bound for the size of the Hopf algebra of binary delta-matroids modulo the (4)-term relations.

This paper generalizes part of the author’s previous results. Let (L) be a multilinear differential operator with constant coefficients. The fundamental solution (phi) supported in a convex cone of a linear space (U) is piecewise polynomial. Choose a basis in the space (T) of polynomials and consider the corresponding set of convex cones in the space (U). We claim that (phi (x)) is equal to a sum of basis elements in (T), with the sum being taken over those elements for which the corresponding cones contain (x).

Sequences of elements from a Hilbert space with phase retrieval property are considered. We discuss a connection between the phase retrieval property, the complement property, and the full spark property (strongly complete systems). There are conditions on the reproducing kernels in the Hardy space of holomorphic functions which ensure phase recovery. We use not only Hilbert, but also Banach frames to construct such sequences of reproducing kernels.

Given a semitopological semigroup (S), let (operatorname{WAP}(S)) and (operatorname{AP}(S)) be the algebras of weakly and strongly almost periodic functions on (S), respectively. This paper centers around the study of the fixed point property ((mathbf{F}_{*,s})): whenever (picolon Stimes K to K) is a jointly (*)-weak continuous nonexpansive action on a non-empty norm separable (*)-weak compact convex set (K) in the dual (E^*) of a Banach space (E), then there is a common fixed point for (S) in (K). We are primarily interested in answering the following problems posed by Lau and Zhang. (1) Let (S) be a discrete semigroup. If the fixed point property ((mathbf{F}_{*,s})) holds, does (operatorname{WAP}(S)) have a left invariant mean? (2) Is the existence of a left invariant mean on (operatorname{WAP}(S)) a sufficient condition to ensure the fixed point property ((mathbf{F}_{*,s}))? (3) Do the bicyclic semigroups (S_2=langle e,a,b,c colon ab=ac=erangle) and (S_3=langle e,a,b,c,d colon ac=bd=erangle) have the fixed point property ((mathbf{F}_{*,s}))? Among other things, characterization theorems of the amenability property of the algebras (operatorname{WAP}(S)) and (operatorname{AP}(S)) are also given.

The paper discusses an applicability condition of a cutoff regularization to a fundamental solution of the Laplace operator in the coordinate representation in the Euclidean space of dimension greater than two. To regularize, we consider a deformation of the solution in a sufficiently small ball centered at the origin by cutting off a singular component, and further supplementing it with a continuous function. It is shown that a set of functions satisfying the applicability condition is not empty. As an example, a family of functions is constructed that can be represented by applying a set of averaging operators to the non-regularized solution, and some specific examples are given. Additionally, it is demonstrated that there exist functions that satisfy the condition in a more strict formulation.

Meromorphic differentials on Riemann surfaces are said to be real-normalized if all their periods are real. Moduli spaces of real-normalized differentials on Riemann surfaces of given genus with prescribed orders of their poles and residues admit a stratification by the orders of zeroes of the differentials. Subsets of real-normalized differentials with a fixed polarized module of periods compose isoperiodic subspaces, which also admit this stratification. In this work, we prove connectedness of the principal stratum for the isoperiodic subspaces in the space of real-normalized differentials with a single pole of order two when all the periods are incommesurable.

Let (X) be a smooth toric variety defined by the fan (Sigma). We consider (Sigma) as a finite set with topology and define a natural sheaf of graded algebras (mathcal{A}_Sigma) on (Sigma). The category of modules over (mathcal{A}_Sigma) is studied (together with other related categories). This leads to a certain combinatorial Koszul duality equivalence.

We describe the equivariant category of coherent sheaves (mathrm{coh}_{X,T}) and a related (slightly bigger) equivariant category (mathcal{O}_{X,T}text{-}mathrm{mod}) in terms of sheaves of modules over the sheaf of algebras (mathcal{A}_Sigma). Eventually (for a complete (X)), the combinatorial Koszul duality is interpreted in terms of the Serre functor on (D^b(mathrm{coh}_{X,T})).

We propose a mathematical model for a new phenomenon: multi-dimensional hyperbolic chaos. This model is a ring chain of (Nge 2) unidirectionally coupled maps of the two-dimensional torus (mathbb{T}^2), each of which is of Arnold’s cat map type. We provide sufficient conditions (independent of (N)) under which the chain gives rise to an Anosov diffeomorphism of (mathbb{T}^{2N}) for any (Nge 2).

In this paper, we investigate the spectrum of the differential operator (T) generated by an ordinary differential expression of order (n) with (mathrm{PT})-symmertic periodic (mtimes m) matrix coefficients. We prove that if (m) and (n) are odd numbers, then the spectrum of (T) contains all the real line. Note that in standard quantum theory, observable systems must be Hermitian operators, so as to ensure that the spectrum is real. Research on (mathrm{PT})-symmetric quantum theory is based on the observation that the spectrum of a (mathrm{PT})-symmetric non-self-adjoint operator can contain real numbers. In this paper, we discover a large class of (mathrm{PT})-symmetric operators whose spectrum contains all real axes. Moreover, the proof is very short.