Skip to main content
main-content

Über dieses Buch

This book is a collection of papers from the 9th International ISAAC Congress held in 2013 in Kraków, Poland. The papers are devoted to recent results in mathematics, focused on analysis and a wide range of its applications. These include up-to-date findings of the following topics:

- Differential Equations: Complex and Functional Analytic Methods

- Nonlinear PDE

- Qualitative Properties of Evolution Models

- Differential and Difference Equations

- Toeplitz Operators

- Wavelet Theory

- Topological and Geometrical Methods of Analysis

- Queueing Theory and Performance Evaluation of Computer Networks

- Clifford and Quaternion Analysis

- Fixed Point Theory

- M-Frame Constructions

- Spaces of Differentiable Functions of Several Real Variables

Generalized Functions

- Analytic Methods in Complex Geometry

- Topological and Geometrical Methods of Analysis

- Integral Transforms and Reproducing Kernels

- Didactical Approaches to Mathematical Thinking

Their wide applications in biomathematics, mechanics, queueing models, scattering, geomechanics etc. are presented in a concise, but comprehensible way, such that further ramifications and future directions can be immediately seen.

Inhaltsverzeichnis

Frontmatter

Tributes

Frontmatter

A Tribute to the 70th Birthday of Prof Saburou Saitoh

Tsutomu Matsuura

Sergei Rogosin: Achievements so Far and Further Plans

Maryna Dubatovskaya, Anna Koroleva, Gennady Mishuris

Differential Equations: Complex and Functional Analytic Methods, Applications

Frontmatter

The Degenerate Second-Order Elliptic Oblique Derivative Problem in a Domain with a Conical Boundary Point

We have investigated the behaviour of strong solutions to the degenerate oblique derivative problem for linear second-order elliptic equation in a neighborhood of a conical boundary point of an

n

-dimensional bounded domain (

n

≥2).

Mariusz Bodzioch, Mikhail Borsuk

On Locally Differentiable Solutions of the Stationary Schrödinger Equation with Discontinuous Coefficients

In this paper we study the binomial second order elliptic equation. The coefficient and the right-hand side of this equation belong to some space

M

of type

F

. We find necessary and sufficient conditions on

M

under which a generalized solution of this equation is continuously differentiable. We find necessary and sufficient conditions on

M

for continuous differentiability of the solution to the above equation, when

M

is a symmetric space, or one of the Lorentz spaces, a Sobolev or a Besov space.

K. N. Ospanov

Riemann Problems for Single-Periodic Polyanalytic Functions II

In this article, we obtain the explicit expression of the solution and the conditions of solvability for the Riemann problem of single-periodic polyanalytic function.

Wang Yufeng, Wang Yanjin

Riemann–Hilbert Problem for Multiply Connected Domains

We discuss scalar Riemann–Hilbert problems for circular multiply connected domains considered by Mityushev (Functional Equations in Mathematical Analysis, pp. 599–632,

2012

). The main attention is paid to the

$\mathbb{R}$

-linear and the Schwarz problems. Some details concerning applications of the metod of functional equation, outlined in Functional Equations in Mathematical Analysis, pp. 599–632,

2012

are extended in the present paper.

Anna Tytuła

The Schottky–Klein Prime Function

This paper describes the Schottky–Klein prime function. The classical Schottky groups and the Poincaré

α

-series are used to construct the Schottky–Klein prime function for arbitrary multiply connected circular domains.

Inez Badecka

Green Function of the Dirichlet Problem for the Laplacian and Inhomogeneous Boundary Value Problems for the Poisson Equation in a Punctured Domain

The aim of this work is to present a new definition of the Green function of the Dirichlet problem for the Laplace equation prompted by the theory of ordinary differential equations and investigate correctly solvable boundary value problems for the Poisson equation in a punctured domain.

Baltabek Kanguzhin, Niyaz Tokmagambetov

Boundary Value Problems and Method of Reflection for Quarter Ring and Half Hexagon

On basis of the reflection principle, the boundary value problems of Schwarz, Dirichlet and Neumann type are explicitly solved for two irregular domains.

B. Shupeyeva

Harmonic Dirichlet Problem in a Ring Sector

In this paper, we construct a harmonic Green function by reflection method in a general ring sector with angle

$\theta=\frac{\pi}{\alpha}$

and

$\alpha\geq \frac{1}{2}$

, then the related harmonic Dirichlet problem for the Poisson equation is discussed explicitly.

Ying Wang, Jinyuan Du

The Parqueting-Reflection Principle

For certain plane domains with boundaries composed by arcs from circles and straight lines the parqueting-reflection principle is used to construct the Schwarz, Green, and Neumann kernels for solving the Schwarz, Dirichlet, and Neumann boundary value problems for the inhomogeneous Cauchy–Riemann and the Poisson equation, respectively.

Heinrich Begehr

On Existence of the Resolvent and Discreteness of the Spectrum of a Class of Differential Operators of Hyperbolic Type

The existence and compactness of the resolvent and discreteness of the spectrum of some hyperbolic differential operators are studied in this paper. One of the main results is the criterion of discreteness of the spectrum of a hyperbolic singular differential operator.

M. B. Muratbekov, M. M. Muratbekov

On the Singularities of the Emden–Fowler Type Equations

We study the Emden–Fowler type equations and their analytic solutions at the origin. We explain the structure of movable singularities of these solutions and visualize them numerically.

Radosław Antoni Kycia, Galina Filipuk

Differential Equations with Degenerated Variable Operator at the Derivative

The theory of Jordan chains for multiparameter operator-functions

A

(

λ

):

E

1

E

2

,

λ

Λ

,

$\operatorname{dim}\varLambda=k$

,

$\operatorname{dim} E_{1}=\operatorname{dim} E_{2}=n$

is developed. Here

A

0

=

A

(0) is a degenerated operator,

$\operatorname{dim}\operatorname{Ker}A_{0}=1$

,

$\operatorname{Ker}A_{0}=\{\varphi\}$

,

$\operatorname{Ker}A_{0}^{*}=\{\psi\}$

and the operator-function

A

(

λ

) is supposed to be linear in

λ

. Applications to degenerate differential equations of the form [

A

0

+

R

(⋅,

x

)]

x

′=

Bx

are given.

B. V. Loginov, Y. B. Rousak, L. R. Kim-Tyan

Cauchy Problem for a First Order Ordinary Differential System with Variable Coefficients

In this article the general solution of the first order ordinary differential systems is found. The Cauchy problem is solved.

A. Tungatarov

General Solution of an n-th Order Linear Ordinary Differential Equation with Variable Coefficients

In this article the general solution of an

n

-th order ordinary differential equations is found. The Cauchy problem for this equation is solved.

A. Tungatarov, B. Omarbayeva

About a Class of Two Dimensional Volterra Type Integral Equations with Singular Boundary Lines

We examine 2-dimensional integral equations of Volterra type with two singular boundary lines corresponding to

x

=

a

and

y

=

b

. The non-homogeneous integral equation that we can consider involves constants

A

1

,

A

2

,

B

1

,

B

2

,

C

1

,

C

2

,

C

3

,

C

4

. Given certain inequalities for

A

1

,

A

2

,

B

1

,

B

2

, it always has solutions on suitable domains that contain arbitrary functions of one variable. With other hypotheses, the equation has a unique solution in some domain.

Lutfya Rajabova

Optimal Control Problem on Optimization of Resources Productivity

The paper is devoted to the optimal control problem which is based on the model of optimization of resources productivity. Model analysis is implemented within the framework of Pontryagin maximum principle for the problems with infinite time horizon. Qualitative analysis of the Hamiltonian system allows to formulate necessary and sufficient conditions of existence of a steady state in terms of the model parameters. Under these conditions and an assumption on the saddle character of the steady state, we construct a nonlinear regulator which allows to approximate optimal trajectories by the solutions of the stabilized Hamiltonian system at a vicinity of the steady state. Finally, comparative analysis of results of numerical simulations is carried out.

Anastasia A. Usova, Alexander M. Tarasyev

Spaces of Differentiable Functions of Several Real Variables and Applications

Frontmatter

The Amalgam Spaces $W( L^{p( x) },\ell^{\{ p_{n}\} })$ and Boundedness of Hardy–Littlewood Maximal Operators

Let

$L^{q ( x ) } (\mathbb{R} ) $

be variable exponent Lebesgue space and

$\ell^{ \{ q_{n} \} }$

be discrete analog of this space. In this work we define the amalgam spaces

W

(

L

p

(

x

)

,

L

q

(

x

)

) and

$W ( L^{p ( x ) },\ell^{ \{ q_{n} \} } ) $

, and discuss some basic properties of these spaces. Since the global components

$L^{q ( x ) } (\mathbb{R} ) $

and

$\ell^{ \{ q_{n} \} }$

are not translation invariant, these spaces are not a Wiener amalgam space. But we show that there are similar properties of these spaces to the Wiener amalgam spaces. We also show that there is a variable exponent

q

(

x

) such that the sequence space

$\ell^{ \{ q_{n} \}}$

is the discrete space of

$L^{q ( x ) } (\mathbb{R} )$

. By using this result we prove that

$W ( L^{p ( x ) },\ell ^{ \{ p_{n} \} } ) =L^{p ( x ) } (\mathbb{R} ) $

. We also study the frame expansion in

$L^{p ( x ) } ( \mathbb{R} )$

. At the end of this work we prove that the Hardy–Littlewood maximal operator from

$W ( L^{s ( x ) },\ell^{ \{t_{n} \} } ) $

into

$W ( L^{u ( x ) },\ell^{ \{v_{n} \} } ) $

is bounded under some assumptions.

A. Turan Gürkanlı

Spaces of Generalised Smoothness in Summability Problems for Φ-Means of Spectral Decomposition

We establish conditions for localization of generalised Riesz means of spectral decomposition by system of fundamental functions of Laplace operator in terms of belongingness of the decomposing function to the spaces of generalised smoothness.

Tsegaye G. Ayele, Mikhail L. Goldman

Viewing the Steklov Eigenvalues of the Laplace Operator as Critical Neumann Eigenvalues

We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of boundary mass concentration. We discuss the asymptotic behavior of the Neumann eigenvalues in a ball and we deduce that the Steklov eigenvalues minimize the Neumann eigenvalues. Moreover, we study the dependence of the eigenvalues of the Steklov problem upon perturbation of the mass density and show that the Steklov eigenvalues violates a maximum principle in spectral optimization problems.

Pier Domenico Lamberti, Luigi Provenzano

Generalized Fractional Integrals on Central Morrey Spaces and Generalized σ-Lipschitz Spaces

For the generalized fractional integrals

${\tilde{I}}_{\alpha,d}$

, which were defined in Function Spaces X, to appear, when

n

/

α

p

<∞, we will consider their boundedness from the central Morrey spaces

$B^{p,\lambda}(\mathbb{R}^{n})$

to the generalized

σ

-Lipschitz spaces

$\mathrm{Lip}^{(d)}_{\beta,\sigma }(\mathbb{R}^{n})$

.

Katsuo Matsuoka

Qualitative Properties of Evolution Models

Frontmatter

Well-Posedness for a Generalized Boussinesq Equation

We consider a generalization of the Boussinesq equation obtained by adding a term of the form

$a(t,x,u)\partial_{x}^{3}u$

. We prove local in time well-posedness of the Cauchy problem in Sobolev spaces under a suitable decay condition on the real part of the coefficient

a

(

t

,

x

,

u

), as

x

→∞.

Alessia Ascanelli, Chiara Boiti

Energy Solutions for Nonlinear Klein–Gordon Equations in de Sitter Spacetime

The Cauchy problem for nonlinear Klein–Gordon equations is considered in de Sitter spacetime. The nonlinear terms are power type or exponential type. The local and global solutions are shown in the energy class.

Makoto Nakamura

A Benefit from the L ∞ Smallness of Initial Data for the Semilinear Wave Equation with Structural Damping

In this note, we prove the global existence of small data solutions for a semilinear wave equation with

structural

damping,

$$u_{tt}-\Delta u + \mu(-\Delta)^{\frac{1}{2}} u_t = |u|^p, $$

for any

n

≥2 and

p

>1+2/(

n

−1). The damping term allows us to derive linear

$L^{q_{1}}-L^{q_{2}}$

estimates, for 1≤

q

1

q

2

≤∞, without loss of regularity, in any space dimension. These estimates provide the basic tool to state our result, in which we assume initial data to be small in (

L

1

H

1

L

)×(

L

1

L

n

).

Marcello D’Abbicco

A Regularity Criterion for the Schrödinger Map

We prove a regularity criterion

$$\nabla u\in L^2\bigl(0,T;\mathit{BMO}\bigl(\mathbb{R}^n\bigr) \bigr) $$

with 2≤

n

≤5 for the Schrödinger map. Here

BMO

is the space of functions with bounded mean oscillations.

Jishan Fan, Tohru Ozawa

Microlocal Analysis for Hyperbolic Equations in Einstein-de Sitter Spacetime

We consider the waves propagating in the Einstein-de Sitter spacetime, which obey the covariant d’Alembert’s equation. We construct the parametrixes in the terms of Fourier integral operators and discuss the propagation and reflection of the singularities phenomena.

Anahit Galstian

Nonlinear Evolution Equations with Strong Dissipation and Proliferation

We investigate the global existence in time and asymptotic profile of the solution of some nonlinear evolution equations with strong dissipation and proliferation arising in mathematical biology. We apply our result to mathematical models of tumour angiogenesis and invasion with proliferation of tumour cells.

Akisato Kubo, Hiroki Hoshino

A Note on Real Powers of Time Differentiation

A Hilbert space framework for fractional calculus is presented. The utility of the approach is exemplified by applications to abstract ordinary fractional differential equations with or without delay.

Rainer Picard

A Stationary Approach to the Scattering on Noncompact Star Graphs Containing Finite Rays

In this paper we consider Schrödinger operators on noncompact star-shaped graphs including some finite rays. We show that our spectral representation formula provides the time dependent formulation of the scattering theory. The scattering operator

S

is constructed in the configuration space, and then is related to the scattering matrix

S

(

λ

) in the momentum space. Corresponding inverse scattering problem is investigated.

Kiyoshi Mochizuki, Igor Trooshin

Integral Transform Approach to the Cauchy Problem for the Evolution Equations

In this note we describe some integral transform that allows to write solutions of the Cauchy problem for one partial differential equation via solution of another one. It was suggested by author in J. Differ. Equ. 206:227–252,

2004

in the case when the last equation is a wave equation, and then used in the series of articles (see, e.g., Yagdjian in J. Differ. Equ. 206:227–252,

2004

, Yagdjian and Galstian in J. Math. Anal. Appl. 346(2):501–520,

2008

, Yagdjian and Galstian in Commun. Math. Phys. 285:293–344,

2009

, Yagdjian in Rend. Ist. Mat. Univ. Trieste 42:221–243,

2010

, Yagdjian in J. Math. Anal. Appl. 396(1):323–344,

2012

, Yagdjian in Commun. Partial Differ. Equ. 37(3):447–478,

2012

, Yagdjian in Semilinear Hyperbolic Equations in Curved Spacetimepp, pp. 391–415,

2014

and Yagdjian in J. Math. Phys. 54(9):091503,

2013

) to investigate several well-known equations such as Tricomi-type equation, the Klein–Gordon equation in the de Sitter and Einstein–de Sitter spacetimes. The generalization given in this note allows us to consider also evolution equations with

x

-dependent coefficients.

Karen Yagdjian

Nonlinear PDE and Control Theory

Frontmatter

On Some Solutions of Certain Versions of “Sigma” Model and Some Skyrme-Like Models

Some results concerning certain versions of “sigma” model and some Skyrme-like models, are presented.

Łukasz T. Stȩpień

Sharp Sobolev–Strichartz Estimates for the Free Schrödinger Propagator

We consider gaussian extremisability of sharp linear Sobolev–Strichartz estimates and closely related sharp bilinear Ozawa–Tsutsumi estimates for the free Schrödinger equation.

Neal Bez, Chris Jeavons, Nikolaos Pattakos

Nonlinear PDE as Immersions

Investigating of the nonlinear PDE including their geometric nature is one of the topical problems. With geometric point of view the nonlinear PDE are considered as immersions. We consider some aspects of the simplest soliton immersions in multidimensional space in Fokas–Gelfand’s sense (Ceyhan et al. in J. Math. Phys. 41:2551–2270,

2000

). In (1+1)-dimensional case nonlinear PDE are given in compatibility condition some system of linear equations (Lakshmanan and Myrzakulov in J. Math. Phys. 39:3765–3771,

1998

). In this case there is a surface with immersion function. We find the second quadratic form in Fokas–Gelfand’s sense associated to one soliton solution of nonlinear Schrödinger equation.

Zhanat Zhunussova

Blow-Up for Nonlinear Inequalities with Singularities on Unbounded Sets

Many physical phenomena are described by nonlinear equations and inequalities with singular coefficients, for which blow-up situation occurs. In this paper we establish sufficient conditions of blow-up situation for some classes of nonlinear differential inequalities with singularities on unbounded sets.

Evgeny Galakhov, Olga Salieva

Well-Posedness and Stability of a Mindlin–Timoshenko Plate Model with Damping and Sources

This note gives a concise summary of results concerning the well-posedness and long-time behavior of (Reissner)–Mindlin–Timoshenko plate equations as presented in Pei et al. (Local and global well-posedness for semilinear Reissner–Mindlin–Timoshenko plate equations,

2013

and Global well-posedness and stability of semilinear Mindlin–Timoshenko system,

2013

). The main feature of the considered model is the interplay between nonlinear viscous interior damping and nonlinear source terms. The results include Hadamard local well-posedness, global existence, blow-up theorems, as well as estimates on the uniform energy decay rates.

Pei Pei, Mohammad A. Rammaha, Daniel Toundykov

On Deterministic and Stochastic Linear Quadratic Control Problems

The numerical treatment of linear quadratic regulator (LQR), linear quadratic Gaussian (LQG) design and stochastic control problems of certain type require solving Riccati equations. In the finite time horizon case, the Riccati differential equation (RDE) arises. We show that within a Galerkin projection framework the solutions of the finite-dimensional RDEs converge in the strong operator topology to the solutions of the infinite-dimensional RDEs. A discussion about LQG design in the context of receding horizon control for nonlinear problems as well as a brief discussion about stochastic control is also addressed. Numerical experiments validate the proposed convergence result.

Tijana Levajković, Hermann Mena

Topological and Geometrical Methods

Frontmatter

Ergodic Theory, Boole Type Transformations, Dynamical Systems Theory

The arithmetic properties of generalized one-dimensional ergodic Boole type transformations are studied in the framework of the operator-theoretic approach. Some invariant measure statements and ergodicity conjectures concerning generalized multi-dimensional Boole-type transformations are formulated.

Anatolij K. Prykarpatski

Fixed Points Theorems for Multivalued Mappings

In this article we discuss the solvability of some class of multivalued inclusions in Euclidean spaces based on a generalization of the “conditions of an acute angle”. As corollary we receive fixed-point theorems for multivalued mappings (continuous and non continuous).

Yuri Zelinskii

Parametric Continuity of Choquet and Sugeno Integrals

For a probabilistic space (

$T, \mathcal{T}, \mu$

), a fixed measurable set

A

and a fixed positive measurable function

f

, the continuity with respect to the real parameter

λ

of the Choquet or Sugeno integral ∫

A

dm

(

λ

,

μ

) is proved. Here

m

(

λ

,

μ

) are all possible

λ

-Sugeno measures generated by

μ

. Asymptotical properties are studied too.

Ion Chiţescu

Nonlinear Operators, Fixed-Point Theorems, Nonlinear Equations

In this article we discuss the solvability of some class of fully nonlinear equations, and equations with

p

-Laplacian in more general conditions by using a new approach given by Soltanov in Nonlinear Anal. 72:164–175,

2010

for studying the nonlinear continuous operator. Moreover we reduce certain general results for the continuous operators acting on Banach spaces, and investigate their image. Here we also consider the existence of a fixed-point of the continuous operators under various conditions.

Kamal N. Soltanov

Some Remarks About Chow, Hilbert and K-stability of Ruled Threefolds

Given a rank 2 holomorphic vector bundle

E

over a projective surface, we explain some relationships between the Gieseker stability of

E

and the Chow, Hilbert and K-stability of the polarized ruled manifold

$\mathbb{P}E$

with respect to polarizations that make fibres sufficiently small.

Julien Keller

Atiyah Classes of Lie Algebroids

Given a smooth morphism of analytic spaces

π

:

X

Y

, we introduce the notion of a relative Lie algebroid

$(\mathcal {A},\sharp)$

over

X

. By replacing the relative tangent sheaf

$\mathcal{T}_{X/Y}$

with the Lie algebroid

$\mathcal{A}$

, we define the notion of a relative

$(\mathcal{A},\sharp)$

-connection on a quasi-coherent

$\mathcal{O}_{X}$

-module

$\mathcal{E}$

. Then, we define the

$(\mathcal{A},\sharp)$

-Atiyah class of

$\mathcal{E}$

as the obstruction to the existence of a holomorphic

$(\mathcal{A},\sharp)$

-connection on

$\mathcal{E}$

. Many results of the classical theory of connections can be restated in the more general setting of Lie algebroid connections. As an application we prove the following result.

Let

X

be a complex manifold and (

A

,♯) a Lie algebroid over

X

. For any quasi-coherent sheaf of commutative

$\mathcal{O}_{X}$

-algebras

$\mathcal{F}$

, let us write

$\mathfrak{g}_{i} = H^{i-1}(X, A \otimes \mathcal{F})$

. The (

A

,♯)-Atiyah class of

A

yields maps

$\mathfrak{g}_{i} \otimes\mathfrak{g}_{j} \to\mathfrak{g}_{i+j}$

. These maps define a graded Lie algebra structure on the graded vector space

$\mathfrak{g}^{\bullet} = \bigoplus_{i} \mathfrak{g}_{i}$

. In a similar way, for any holomorphic vector bundle

E

over

X

, let us write

$V_{j} = H^{j-1}(X, E \otimes\mathcal{F})$

. Then, for any

i

and

j

, the (

A

,♯)-Atiyah class of

E

yields a map

$\mathfrak{g}_{i} \otimes V_{j} \to V_{i+j}$

, and these maps define a structure of graded module on the graded vector space

V

=⨁

j

V

j

, over the graded Lie algebra

$\mathfrak{g}^{\bullet}$

. This generalizes a similar result proved by Kapranov in Compos. Math. 115:71–113,

1999

. Similar results have been obtained by Chen, Stiénon and Xu in From Atiyah classes to homotopy Leibniz algebras

2012

, by using different techniques.

Francesco Bottacin

Kähler Metrics with Cone Singularities and Uniqueness Problem

The Kähler metric with cone singularities has been the main subject which is being studied recently. In this expository note, we focus on the modular space of the Kähler metric with cone singularities. We first summary our work on the construction of the geodesic of the cone singularities. Then we apply the cone geodesic to obtain a uniqueness theorem of the constant scalar curvature Kähler metrics with cone singularities.

Kai Zheng

Didactics and Education

Frontmatter

Teaching of Mathematics in Vocational Schools Upon 1951 Reorganisation

The radical reform in 1951 of organisation and policy in the vocational schooling in Poland is analyzed. The conclusive points for such a deep transformation of the vocational schooling were political and economical reasons discussed in the paper.

Ryszard Ślęczka

Arithmetic in Polish Parish Schools in the Period of the Commission of National Education

The paper is devoted to historical description and analysis of the Polish primers in 16th–18th centuries and arithmetic in Polish primers until 1795. Conceptions of various schools and famous educators are discussed.

Ryszard Ślęczka, Jan Ryś

Life as an Example. S.M. Nikolskij

This paper is devoted to the outstanding Russian mathematician Sergey Mikhailovich Nikolskij. His biography and his famous results are outlined. A list of his pupils with last known positions is given.

Alexandr Rusakov

The Area Method and Proving Plane Geometry Theorems

The process of proving, deriving and discovering theorems is important in mathematics investigation. In this paper, we will use the elimination technique which is based on the theory of the area method. The main idea of this method will be illustrated through an example from plane geometry. In addition, we look at the application possibilities of using GCLC geometry system with built-in theorem prover in verification and proving constructive geometric statements.

Martin Billich

Clifford, Quaternion and Wavelet Analysis

Frontmatter

Redundant Multiscale Haar Wavelet Transforms

We consider a redundant lifting scheme for Haar wavelet transform that does not use the polyphase decomposition. We also extend the method to a two-dimensional triangular lattice, and define a nonseparable two-dimensional redundant Haar wavelet transform on the lattice.

Kensuke Fujinoki

Gabor Transform of Analytic Functional on the Sphere

We studied the Fourier–Borel transform of analytic functional on the complex sphere. In this paper, we will consider the Gabor transformation which is a windowed Fourier transformation whose window function is the Gaussian function. Following our previous results we will represent the Gabor transform of analytic functional on the sphere using a series expansion by means of the Bessel functions.

Keiko Fujita

On the Interpolation of Orthonormal Wavelets with Compact Support

The

N

-th order Daubechies wavelet is obtained with the spectral decomposition method from modulus of associated low-pass filters

M

(

ξ

)=|

m

0

(

ξ

)|

2

. Meanwhile, we can denote

M

(

ξ

) with the integration. In this paper, we focus on integrands and construct some wavelets by changing them. Moreover, we construct some kind of fractional order wavelets and give regularity estimates of them.

Naohiro Fukuda, Tamotu Kinoshita

An Estimation Method of Shift Parameters in Image Separation Problem

The simplest spatio-temporal mixing model of blind source separation for images is discussed. Shift parameters are estimated by total correlation functions of continuous wavelet transforms. An image separation algorithm using an annular sector multiwavelet is proposed.

Ryuichi Ashino, Takeshi Mandai, Akira Morimoto

Slice Functional Calculus in Quaternionic Hilbert Spaces

We propose a continuous functional calculus in quaternionic Hilbert spaces. The class of continuous functions considered is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen a generalization to quaternions of holomorphic functions of one complex variable. The notion of slice function allows to introduce suitable classes of real, complex and quaternionic

C

-algebras and to define, on each of these

C

-algebras, a functional calculus for quaternionic normal operators.

R. Ghiloni, V. Moretti, A. Perotti

Recent Progress on Spheroidal Monogenic Functions

Monogenic function theories are considered as generalizations of the holomorphic function theory in the complex plane to higher dimensions and are refinements of the harmonic analysis based on the Laplace operator’s factorizations. The construction of spherical monogenic functions has been studied for decades with different methods. Recently, orthogonal monogenic bases are developed for spheroidal reference domains, first by J. Morais and later by others. This survey will go through the construction of spheroidal monogenic functions and discuss up-to-date results.

Hung Manh Nguyen

Clifford Algebras with Induced (Semi)-Riemannian Structures and Their Compactifications

Identifying the Clifford algebra

Cℓ

r

,

s

with the semi-Riemannian manifold

$\mathbb{R}^{p,q}$

, one is afforded an opportunity to examine the conformal geometry of the associated compact manifold, in a manner similar to the case of the Riemann sphere in complex analysis. In this work we consider some low-dimensional examples and provide conjectures to inspire further research.

Craig A. Nolder, John A. Emanuello

Integral Transforms and Reproducing Kernels

Frontmatter

Generalized Shift Operators Generated by Convolutions of Integral Transforms

In this work we discuss a problem of the equivalence of two main approaches to introducing of generalized convolution operators. The first of them is based on the constructing of a generalized shift operator. The idea of the second approach is based on the works by Valentin Kakichev. On this problem we demonstrate on the examples of classical and nonclassical convolution constructions of integral transforms. In particular, we consider the shift operators defined by the convolutions for Hankel integral transform with the function

j

ν

(

xt

)=(2

xt

)

ν

Γ

(

ν

+1)

J

ν

(

xt

) in the kernel. Here

J

ν

(

xt

) is the Bessel function of the first kind of order

ν

,

$\operatorname{Re} \nu>-1/2$

.

Lyubov Y. Britvina

An Approach for Developing Fourier Convolutions and Applications

Based on the papers published recently, this talk presents a concept of convolution so-called pair-convolution which is a generalization of known convolutions, and considers applications for solving integral equations.

Nguyen Minh Tuan

Whittaker Differential Equation Associated to the Initial Heat Problem

In this paper, by using the theory of reproducing kernels, we investigate integral transforms with kernels related to the solutions of the initial Whittaker heat problem.

M. M. Rodrigues, S. Saitoh

Recovery of Holomorphic Functions and Taylor Coefficients by Sampling

We introduce an interpolation formula for holomorphic functions and prove its convergence pointwise under very general condition. We obtain also a recovery formula for Taylor coefficients from discrete samples.

Vu Kim Tuan, Amin Boumenir

On Approximation of Lebedev Type Transforms

New modification of

Kontorovitch–Lebedev

and

Lebedev–Skalskaya

integral transforms was introduced by

Yakubovich

. These transforms contain modified

Bessel

functions

$K_{\frac{1}{4}+i\tau}(x) $

and

$K_{\frac{3}{4}+i\tau}(x) $

and their real and imaginary parts as kernels. The vector Tau method approach is used for the approximation and calculation of these functions. This approach is based on the general Tau method’s computational scheme and canonical vector-polynomial notion. We obtain the system of two differential equations and then the system of two Volterra integral equations for the determination of the polynomial approximation of the kernels. These results may be used for the application of

Yakubovich

transforms to the solution of boundary value problems of mathematical physics.

Juri Rappoport

Reproducing Kernels and Discretization

We give a short survey of a general discretization method based on the theory of reproducing kernels. We believe our method will become the next generation method for solving analytical problems by computers. For example, for solving linear PDEs with general boundary or initial value conditions, independently of the domains. Furthermore, we give an ultimate sampling formula and a realization of reproducing kernel Hilbert spaces.

L. P. Castro, H. Fujiwara, M. M. Rodrigues, S. Saitoh, V. K. Tuan

Dirichlet’s Problem by Using Computers with the Theory of Reproducing Kernels

In this paper we shall give practical and numerical solutions of the Laplace equation on multidimensional spaces and show their numerical experiments by using computers. Our method is based on the Dirichlet principle by combinations with generalized inverses, Tikhonov’s regularization and the theory of reproducing kernels.

Tsutomu Matsuura, Saburo Saitoh

Toeplitz Operators and Their Applications

Frontmatter

C ∗-Algebras of Two-Dimensional Singular Integral Operators with Shifts

We construct a Fredholm symbol calculus for the

C

-algebra

${\mathfrak{B}}$

generated by the

C

-algebra

${\mathfrak{A}}$

of two-dimensional singular integral operators with continuous coefficients on a bounded closed simply connected domain

$\overline{U}\subset\mathbb{R}^{2}$

with Liapunov boundary and by all unitary shift operators

W

g

where

g

runs through a discrete solvable group

G

=

F

H

of diffeomorphisms of

$\overline{U}$

onto itself, where

F

is a commutative group of conformal mappings,

H

={

e

,

γ

} and

γ

is similar to the shift

$z\mapsto \overline{z}$

. As a result, we establish a Fredholm criterion for the operators

$B\in{\mathfrak{B}}$

.

Y. I. Karlovich, V. A. Mozel

Uncertainty and Analyticity

We describe a connection between minimal uncertainty states and holomorphy-type conditions on the images of the respective wavelet transforms. The most familiar example is the Fock–Segal–Bargmann transform generated by the Gaussian, however, this also occurs under more general assumptions.

Vladimir V. Kisil

Toeplitz Operators on the Harmonic Bergman Space with Pseudodifferential Defining Symbols

We study the

C

-algebra

$\mathcal{T}(\mathcal{R}(C(\overline {\mathbb {D}});S_{\mathbb{D}}, S_{\mathbb{D}}^{*}))$

generated by Toeplitz operators acting on the harmonic Bergman space on the unit disk whose pseudodifferential defining symbols belong to the algebra

$\mathcal{R}=\mathcal{R}(C(\overline{\mathbb {D}});S_{\mathbb {D}}, S_{\mathbb{D}}^{*})$

. The algebra

$\mathcal{R}$

is generated by the multiplication operators

aI

, where

$a\in C(\overline{\mathbb{D}})$

, and the following two operators

$$S_{\mathbb{D}}(\varphi) (z)=\frac{-1}{\pi}\int_{\mathbb{D}} \frac{\varphi(\zeta)}{(\zeta-z)^2}d\nu(\zeta) \quad \text {and}\quad S_{\mathbb{D}}^*(\varphi) (z)=\frac{-1}{\pi}\int_{\mathbb{D}}\frac{\varphi(\zeta)}{(\overline{\zeta} -\overline{z})^2}d\nu(\zeta). $$

We describe the Fredholm symbol algebra of

$\mathcal{T}(\mathcal{R}(C(\overline{\mathbb{D}});S_{\mathbb{D}}, S_{\mathbb{D}}^{*}))$

and the index formula for its Fredholm elements.

Maribel Loaiza, Nikolai Vasilevski

Theorems of Paley–Wiener Type for Spaces of Polyanalytic Functions

We prove Paley–Wiener theorems for the true poly-Bergman and poly-Bergman spaces based on properties of the compression of the Beurling–Ahlfors transform to the upper half-plane. An isometric isomorphism between

j

copies of the Hardy space and the poly-Bergman space of order

j

is constructed.

Luís V. Pessoa, Ana Moura Santos

Fredholm Theory of Pseudodifferential Operators Acting in Variable Exponent Spaces of Bessel Potentials on Smooth Manifolds

The paper is devoted to the of Fredholm property of pseudodifferential operators acting in the spaces of Bessel potentials connected with variable exponent Lebesgue spaces on smooth compact manifolds and non compact manifolds with conical structure at infinity.

Vladimir Rabinovich

Differential and Difference Equations with Applications

Frontmatter

Boundary Value Problems for the Radiative Transfer Equation with Reflection and Refraction Conditions

The paper contains a brief description of some new results about the boundary value problems for the radiative transfer equation with the reflection and refraction conditions.

A. A. Amosov

Asymptotic Approximations of a Thin Elastic Beam with Auxiliary Coupled 1D System due to Robin Boundary Condition

In Bare et al. (Appl. Anal.,

2013

, doi:

10.1080/00036811.2013.823481

), the dimension of a 3D linear elasticity boundary value problem with Robin boundary condition is asymptotically reduced. Assumption 1.4 in Bare et al. (Appl. Anal.,

2013

, doi:

10.1080/00036811.2013.823481

), leads to a 1D system in which the bending and tensile components are decoupled. With a generalization in this contribution, we obtain a coupled 1D system. We prove that the asymptotic error estimate in Bare et al. (Appl. Anal.,

2013

, doi:

10.1080/00036811.2013.823481

) remains true and illustrate the influence of the tension and torsion on the bending by a numerical example.

Z. Bare, J. Orlik, G. Panasenko

Smooth Solution of an Initial Value Problem for a Mixed-Type Differential Difference Equation

In this paper, we show the construction of the solution to the mixed type differential difference equation:

$$\begin{aligned} x'(t)= A x(t+ a)+ Bx(t-a)+Cx(t), \end{aligned}$$

where

$A,B,C\in\mathbb{C}\setminus\{0\}$

,

a

>0 and

$t \in\mathbb{R}$

. We use a step derivative method and a certain condition on the initial function

φ

C

[−

a

,

a

] to assure the existence, uniqueness and smoothness of the solution in

$\mathbb{R}$

.

Valentina Iakovleva, Judith Vanegas

On the Classical Lorenz System

The classical Lorenz system is considered. For many years, this system has been the subject of study by numerous authors. However, until now the structure of the Lorenz attractor is not clear completely yet, and the most important question at present is to understand the bifurcation scenario of chaos transition in this system. Using some numerical results and our bifurcational geometric approach, we present a new scenario of chaos transition in the classical Lorenz system.

Valery A. Gaiko

Discrete Singular Integrals in a Half-Space

We consider Calderon–Zygmund singular integral in the discrete half-space

$h\mathbf{Z}^{m}_{+}$

, where

Z

m

is entire lattice (

h

>0) in

R

m

, and prove, that the discrete singular integral operator is invertible in

$L_{2}(h\mathbf{Z}^{m}_{+})$

iff such is its continual analogue. The key point for this consideration takes solvability theory of so-called periodic Riemann boundary problem, which is constructed by authors.

Alexander V. Vasilyev, Vladimir B. Vasilyev

Geometrical Features of the Soliton Solution

It is well known, that integrable equations are solvable by the inverse scattering method (Ablowitz and Clarkson in Solitons, Non-linear Evolution Equations and Inverse Scattering,

1992

). Investigating of the integrable spin equations in (1+1), (2+1) dimensions are topical both from the mathematical and physical points of view (Lakshmanan and Myrzakulov in J. Math. Phys. 39:3765–3771,

1998

; Gardner et al. in Phys. Rev. Lett. 19(19):1095–1097,

1967

). Integrable equations admit different kinds of physically interesting solutions as solitons, vortices, dromions etc. We consider an integrable spin M-I equation (Myrzakulov and Vijayalakshmi in Phys. Lett. A 233:391–396,

1997

). There is a corresponding Lax representation. And the equation allows an infinite number of integrals of motion. We construct a surface corresponding to soliton solution of the equation. Further, we investigate some geometrical features of the surface.

Zhanat Zhunussova

M-Frame Constructions

Frontmatter

Approximate Dual M-Frames Constructions: The Gabor Case

The aim of this work is to provide an efficient method to realize constructively approximate duals of Gabor frames with multivariate atoms. The proposed method is independent of the number of atoms needed and it is applicable also in the case of non-separable atoms. Due to the small number of atoms used in the construction the method is computationally inexpensive.

Darian M. Onchis, Anna Grybos

Matrices of Operators on Some Function Spaces

Some matrix representations for Hilbert space operators are considered. The corresponding matrices are related to frames and appear in a quite natural way especially in the case of reproducing kernel spaces. The membership in Schatten classes is discussed in terms of a discrete set of points, where the corresponding symbols are evaluated.

Wojciech Mikołajczyk, Krzysztof Rudol

Rank-M Frame Multipliers and Optimality Criterions for Density Operators of Rank M

Ever since the introduction of frames in Duffin and Schaeffer (Trans. Am. Math. Soc. 72:341–366,

1952

), the connection between frame theory and decompositions of certain operators, particularly the identity operator, into rank-ones began to be elaborated. Abandoning the idea of restricting to tight frame-like expansions, with respect to systems arising from a single template function, one is led to the concept of resolutions of the identity, with respect to more general systems than the usual rank-one expansions of the identity.

In this study, we will investigate various notions of possible generalizations of optimality criterions for rank-M frames and corresponding multipliers. Explicitly, we will lay stress on continuous M-frames, arising from irreducible group representations of locally compact groups, have a look at its connection to time-frequency analysis and comment on adequate notions of optimality.

Daniel Lantzberg

Audio Inpainting Using M-Frames

Classical short-time Fourier constructions lead to a signal decomposition with a fixed time-frequency resolution. However, having signals with varying features, such time-frequency decompositions are very restrictive. A more flexible and adaptive sampling of the time-frequency plane is achieved by the nonstationary Gabor transform. Here, the resolution can evolve over time or frequency, respectively, by using different windows for the different sampling positions in the time or frequency domain (Multiwindow-frames). This adaptivity in the time-frequency plane leads to a sparser signal representation.

In terms of audio inpainting, i.e., filling in blanks of a depleted audio signal, sparsity in some representation space profoundly influences the quality of the reconstructed signal. We will compare this quality using different nonstationary Gabor transforms and the regular Gabor transform with different types of audio signals.

Florian Lieb

Wavelet Frames to Optimally Learn Functions on Diffusion Measure Spaces

Based on the theory of wavelets on data defined manifolds we study the Kolmogorov metric entropy and related measures of complexity of certain function spaces. We also develop constructive algorithms to represent those functions within a prescribed accuracy that is asymptotically optimal up to a logarithmic factor.

Martin Ehler, Frank Filbir

Applications of Queueing Theory in Modelling and Performance Evaluation of Computer Networks

Frontmatter

A Study on Ateb Transform as a Generalization of Fourier Transform

The aim of this work is to construct the Ateb transforms based on Ateb-functions as a generalization of orthogonal Fourier transform. It was proved that these transforms satisfy the properties of linearity, symmetry and similarity. The Hartley transform is a real linear operator, and symmetric and self-inverse properties for Hartley Ateb-transform were proved. The one-dimensional discrete and two-dimensional discrete Ateb transforms were represented. Discrete transforms were used for construction digital watermark for the information security aim in the computer networks.

Ivanna Dronjuk, Maria Nazarkevich

Queue-Size Distribution in Energy-Saving Model Based on Multiple Vacation Policy

An energy-saving model based on the

M

/

G

/1/

N

-type finite-buffer queue with independent and generally distributed repeated vacations is considered. Using the formula of total probability and the idea of embedded Markov chain, a system of integral equations for conditional transient queue-size distributions is found. A closed-form representation for the solution of the corresponding system built for Laplace transforms is obtained. Numerical example is attached as well.

Wojciech M. Kempa

Automobile System Safety Based on the Model for Stochastic Networks with Dependent Service Times

Broadband wireless data transmission network for providing of automobile transport system safety is considered. The network operates under IEEE802.11n-2012 protocol that guarantees high-speed transmission of multimedia information from stationary and mobile automatic systems of traffic control. The model of stochastic network with dependent service time and processor sharing discipline for the problem solution is used. Product-form representation for the model steady-state probabilities is presented.

Vladimir Vishnevsky, Vladimir Rykov

Applied Mathematics

Frontmatter

Damage Prediction of the Femur with Postresection Defect

The aim of this study is to develop an approach to assessing the strength of the femur after sectoral resection in cases of benign bone tumors, tumor-like and metastatic lesions. The proposed approach is based on the finite element calculation of dangerous volumes in the area of bone defect. Load is static and equivalent to average human weight. Model of the femur is based on tomographic data. Postresection defect is localized in the middle third of the lateral side of the femur. As a conditions for the selection of dangerous volume fracture criterion Coulomb–Mohr is used. The analysis of damage near the concentrators of the bone defect is carried out for different loads. The domain of the bone defect with the largest damage is determined. For concentrators of the postresection hole the emergence and growth of crack is considered as a change of the dangerous volume with taking account the removal of the damaged finite elements. The ranges of the load corresponding to the various cases of damage development are determined. Three options to compensate for bone strength and the prevention the pathological bone fracture after sectoral resection are suggested.

S. Bosiakov, D. Alekseev, I. Shpileuski

Representative Elements for Polydispersed Composites

Effective properties of random 2D composites are discussed in the framework of the representative volume element (RVE) theory proposed by Mityushev (Complex Var. Elliptic Equ. 51:1033–1045

2006

). This theory is extended to 2D fiber composites with sections perpendicular to fibers of different radii. The RVE theory is applied to the mixture problem arisen in technological processes.

Natalia Rylko

Random Non-overlapping Walks of Disks on the Plane

The initial locations of disks on the plane form regular lattice. Random non-overlapping walks of disks approach to the uniform non-overlapping distribution of disks on the plane. The basic

e

-sums are computed in time to describe such a dynamic process within random walks.

Wojciech Nawalaniec

Symbolic Computation of Conformal Mappings onto Slit Domains

A present paper, the symbolic implementation of conformal mappings of arbitrary circular multiply connected domains onto the complex plane with slits of prescribed inclinations is applied to examine of distribution of lengths of slits.

Roman Czapla

On One Approach to the Simulation of the Periodontal Ligament Takes into Account Its Viscoelastic Properties

The mathematical model of the viscoelastic periodontal ligament is presented. The relaxation kernel corresponds to the Maxwell model. Model describes the viscoelastic deformations of the periodontal membrane and the tooth movements. Analysis of premolar root movements in the form of an elliptical hyperboloid under the vertical load is performed.

K. Yurkevich, S. Bosiakov

Biomechanical Effects of Maxillary Expansion in Cross-Bite Patients During Orthodontic Treatment with Hyrax Screw

The aim of this study was finite element analysis of stress-strain state of the human maxillary complex with and without cleft palate. Loading the skull is carried out by activating orthodontic device HYRAX. Model of the skull and supporting teeth of upper jaw obtained on the basis of tomographic data for dry intact skull of an adult. Design of orthodontic device differ position of screws and rods relative to the palate. Equivalent stresses in the bones of the craniofacial complex are assessed. It is shown that large stresses occur in the maxillary complex, if the screw and rods of orthodontic devices are located in a horizontal plane for skull with and without cleft. Also in the intact skull big stresses appear in the bone of the upper jaw with location of the screw and rods of orthodontic device in a horizontal plane. In the rest of the skull bones stresses are insignificant. By moving the device screw to the palate the values of maximum stresses are reduced, but the region of big stresses displaced to the pterygoid plate and pharyngeal tubercle. In the skull with cleft for different positions of screws and rods orthodontic device the upper jaw is loaded fragmentary. High stresses are observed in the region of the maxilla near the zygomatic arches and along the edges of eye-sockets. When placing screw of orthodontic device close to palate the stresses decreases, but are observed in most part of the zygomatic arches.

S. Bosiakov, A. Vinokurova, A. Dosta

Others

Frontmatter

On the Solvability of a Nonlinear Optimal Control Problem for the Thermal Processes Described by Fredholm Integro-Differential Equations

The problem of nonlinear optimal control of the thermal process described by Fredholm integro-differential equation was investigated. The concept of a weak generalized solution of the boundary problem was introduced and the algorithm for its construction was indicated. It was established that the optimal control is defined as a solution of a nonlinear integral equation satisfying the additional condition in the form of inequality. Sufficient conditions for unique solvability of nonlinear optimization were found and the algorithm for constructing approximate solutions was developed. The convergence of approximate solutions with respect to control, optimal process and functional was investigated.

Akylbek Kerimbekov

Exact Null-Controllability of Evolution Equations by Smooth Controls and Applications to Controllability of Interconnected Systems

The aim of this work is to establish exact null-controllability conditions for a linear evolution equation in the class of smooth controls. Applications to the controllability of system consisting of two serially connected abstract control systems are considered.

B. Shklyar

On the Sum of Contractive Type of Mappings I: Maps on the Same Class

In this paper, we will show that under some conditions the sum of two mappings belonging to a contractive class of maps is a mapping on the same class (but with different contractive parameters).

J. R. Morales, E. M. Rojas

On (α,ψ) Contractions of Integral Type on Generalized Metric Spaces

In this paper, we investigate the existence and uniqueness of fixed points of (

α

,

ψ

)-contractive mappings of integral type in complete generalized metric spaces, introduced by Branciari. Our results generalize and improve several results in literature.

Erdal Karapınar

Infinite Dimensional Stochastic Cauchy Problems in Ito and Differential Forms: Comparison of Solutions

We consider three types of solutions to the infinite dimensional stochastic Cauchy problem

$X'(t)=AX(t)+B{\mathbb{W}}(t)$

,

t

≥0,

X

(0)=

ζ

, with

A

being the generator of a regularized semigroup in a Hilbert space and a white noise

${\mathbb{W}}$

in another Hilbert space: weak, generalized in

t

, and generalized in a random variable. It is proved coincidence of the solutions under the conditions they exist.

Irina V. Melnikova, Olga Starkova

Biomechanical Model of the Human Eye on the Base of Nonlinear Shell Theory

The goal of this work is the development of a biomechanical model of the human eye and to prove software simulation systems of measuring the intraocular pressure (IOP) by an optical analyzer. We numerically simulate the eye deformation when the IOP is measured using the Ocular Response Analyzer developed by the USA company Reichert. The biomechanical model includes a cornea and a sclera, which are considered as axisymmetrically deformable shells of revolution with fixed boundaries; the space between these shells is filled with incompressible fluid. Nonlinear shell theory is used to describe the stressed and strained state of the cornea and sclera. The optical system is calculated from the viewpoint of the geometrical optics. Dependences between the pressure in the air jet and the area of the surface reflecting the light into a photo detector for the different thickness of the cornea were obtained. Three problems with different boundary conditions were considered. The shapes of the regions on the cornea surface were found from which the reflected light falls on the photo detector. First, the light is reflected from the center of the cornea, but then, as the cornea deforms, the light is reflected from its periphery. The numerical results make it possible to better interpret the measurement data. This work was supported by a grant No. 13-01-00801 from the Russian Foundation for Basic Research.

Vladimir Yakushev

Backmatter

Weitere Informationen

Premium Partner

BranchenIndex Online

Die B2B-Firmensuche für Industrie und Wirtschaft: Kostenfrei in Firmenprofilen nach Lieferanten, Herstellern, Dienstleistern und Händlern recherchieren.

Whitepaper

- ANZEIGE -

Best Practices für die Mitarbeiter-Partizipation in der Produktentwicklung

Unternehmen haben das Innovationspotenzial der eigenen Mitarbeiter auch außerhalb der F&E-Abteilung erkannt. Viele Initiativen zur Partizipation scheitern in der Praxis jedoch häufig. Lesen Sie hier  - basierend auf einer qualitativ-explorativen Expertenstudie - mehr über die wesentlichen Problemfelder der mitarbeiterzentrierten Produktentwicklung und profitieren Sie von konkreten Handlungsempfehlungen aus der Praxis.
Jetzt gratis downloaden!

Bildnachweise