arXiv subjects

Petr Zemánek

Publications and source records attributed to Petr Zemánek.

7 recordsLinked to original sources

arXiv2024

Eigenfunctions expansion for discrete symplectic systems with general linear dependence on spectral parameter

Eigenfunctions expansion for discrete symplectic systems on a finite discrete interval is established in the case of a general linear dependence on the spectral parameter as a significant generalization of the Expansion theorem given by Bohner, Došlý and Kratz in [Trans. Amer. Math. Soc. 361 (2009), 3109--3123]. Subsequently, an integral representation of the Weyl--Titchmarsh $M(λ)$-function is derived explicitly by using a suitable spectral function and a possible extension to the half-line case is discussed. The main results are illustrated by several examples.

math.SP↗

arXiv2024

Resolvent and spectrum for discrete symplectic systems in the limit point case

The spectrum of an arbitrary self-adjoint extension of the minimal linear relation associated with the discrete symplectic system in the limit point case is completely characterized by using the limiting Weyl--Titchmarsh $M_+(λ)$-function. Furthermore, a dependence of the spectrum on a boundary condition is investigated and, consequently, several results of the singular Sturmian theory are derived.

math.SP↗

arXiv2024

The Friedrichs extension of a class of discrete symplectic systems

The Friedrichs extension of minimal linear relation being bounded below and associated with the discrete symplectic system with a special linear dependence on the spectral parameter is characterized by using recessive solutions. This generalizes a similar result obtained by Došlý and Hasil for linear operators defined by infinite banded matrices corresponding to even-order Sturm--Liouville difference equations and, in a certain sense, also results of Marletta and Zettl or Šimon Hilscher and Zemánek for singular differential operators.

math.SP↗

arXiv2024

Findings of the IWSLT 2024 Evaluation Campaign

This paper reports on the shared tasks organized by the 21st IWSLT Conference. The shared tasks address 7 scientific challenges in spoken language translation: simultaneous and offline translation, automatic subtitling and dubbing, speech-to-speech translation, dialect and low-resource speech translation, and Indic languages. The shared tasks attracted 18 teams whose submissions are documented in 26 system papers. The growing interest towards spoken language translation is also witnessed by the constantly increasing number of shared task organizers and contributors to the overview paper, almost evenly distributed across industry and academia.

cs.CL↗

arXiv2016

On discrete symplectic systems: Associated maximal and minimal linear relations and nonhomogeneous problems

In this paper we characterize the definiteness of the discrete symplectic system, study a nonhomogeneous discrete symplectic system, and introduce the minimal and maximal linear relations associated with these systems. Fundamental properties of the corresponding deficiency indices, including a relationship between the number of square summable solutions and the dimension of the defect subspace, are also derived. Moreover, a sufficient condition for the existence of a densely defined operator associated with the symplectic system is provided.

math.SP↗

arXiv2016

Characterization of self-adjoint extensions for discrete symplectic systems

All self-adjoint extensions of minimal linear relation associated with the discrete symplectic system are characterized. Especially, for the scalar case on a finite discrete interval some equivalent forms and the uniqueness of the given expression are discussed and the Krein--von Neumann extension is described explicitly. In addition, a limit point criterion for symplectic systems is established. The result partially generalizes even a classical limit point criterion for the second order Sturm--Liouville difference equations.

math.SP↗

arXiv2016

Discrete trigonometric and hyperbolic systems: An overview

In this paper we present an overview of results for discrete trigonometric and hyperbolic systems. These systems are discrete analogues of trigonometric and hyperbolic linear Hamiltonian systems. We show results which can be viewed as discrete n-dimensional extensions of scalar continuous trigonometric and hyperbolic formulas

math.CA↗