bluuebunny/crossref_metadata_2025 huggingface dataset:
Basic Boundary Value Problems of Thermoelasticity for Anisotropic Bodies with Cuts. II
Authors: Duduchava R. , Natroshvili D. , Shargorodsky E.
Published: February 2021

In the first part [Duduchava, Natroshvili and Shargorodsky, Georgian Math. J. 2: 123–140, 1985] of the paper the basic boundary value problems of the mathematical theory of elasticity for three-dimensional anisotropic bodies with cuts were formulated. It is assumed that the two-dimensional surface of a cut is a smooth manifold of an arbitrary configuration with a smooth boundary. The existence and uniqueness theorems for boundary value problems were formulated in the Besov () and Bessel-potential () spaces. In the present part we give the proofs of the main results (Theorems 7 and 8) using the classical potential theory and the nonclassical theory of pseudodifferential equations on manifolds with a boundary.
Necessary and Sufficient Conditions for Weighted Orlicz Class Inequalities for Maximal Functions and Singular Integrals. I
Authors: Gogatishvili A. , Kokilashvili V.
Published: February 2021

Criteria of various weak and strong type weighted inequalities are established for singular integrals and maximal functions defined on homogeneous type spaces in the Orlicz classes.
On the Solvability of A Spatial Problem of Darboux Type for the Wave Equation
Authors: Kharibegashvili S.
Published: February 2021

The question of the correct formulation of one spatial problem of Darboux type for the wave equation has been investigated. The correct formulation of that problem in the Sobolev space has been proved for surfaces having a quite definite orientation on which are given the boundary value conditions of the problem of Darboux type.
On A Darboux Problem for A Third Order Hyperbolic Equation with Multiple Characteristics
Authors: Jokhadze O.
Published: February 2021

A Darboux type problem for a model hyperbolic equation of the third order with multiple characteristics is considered in the case of two independent variables. The Banach space , α ≥ 0, is introduced where the problem under consideration is investigated. The real number α
0 is found such that for α > α
0 the problem is solved uniquely and for α < α
0 it is normally solvable in Hausdorff's sense. In the class of uniqueness an estimate of the solution of the problem is obtained which ensures stability of the solution.
On the Number of Representations of Integers by Some Quadratic Forms in Ten Variables
Authors: Lomadze G.
Published: February 2021

A method of finding the so-called Liouville's type formulas for the number of representations of integers by

quadratic forms is developed.
Commutativity for a Certain Class of Rings
Authors: Abujabal Hamza A. S.
Published: February 2021

We discuss the commutativity of certain rings with unity 1 and one-sided s-unital rings under each of the following conditions: xr
[xs, y] = ±[x, yt
]xn
, xr
[xs, y] = ±xn
[x, yt
], xr
[xs, y] = ±[x, yt
]ym
, and xr
[xs, y] = ±ym
[x, yt
], where r, n, and m are non-negative integers and t > 1, s are positive integers such that either s, t are relatively prime or s[x, y] = 0 implies [x, y] = 0. Further, we improve the result of [Jacobson, Structure of rings, Colloq. Publ., 1964, Theorem 3] and reprove several recent results.
On the Non-Commutative Neutrix Product
Authors: Kiliçman Adem , Fisher Brian
Published: February 2021

The non-commutative neutrix product of the distributions and is evaluated for r = 0, 1, 2, . . . and s = 1, 2, . . . . Further neutrix products are then deduced.
Criteria of Unitary Equivalence of Hermitian operators with a Degenerate Spectrum
Authors: Lomidze I.
Published: February 2021

Nonimprovable, in general, estimates of the number of necessary and sufficient conditions for two Hermitian operators to be unitarily equaivalent in a unitary space are obtained when the multiplicities of eigenvalues of operators can be more than 1. The explicit form of these conditions is given. In the Appendix the concept of conditionally functionally independent functions is given and the corresponding necessary and sufficient conditions are presented.
On a Darboux Type Multidimensional Problem for Second-Order Hyperbolic Systems
Authors: Kharibegashvili S.
Published: February 2021

The correct formulation of a Darboux type multidimensional problem for second-order hyperbolic systems is investigated. The correct formulation of such a problem in the Sobolev space is proved for temporal type surfaces on which the boundary conditions of a Darboux type problem are given.
On the Correctness of the Cauchy Problem for a Linear Differential System on an Infinite Interval
Authors: Kiguradze I.
Published: February 2021

The conditions ensuring the correctness of the Cauchy problem

on the nonnegative half-axis R
+ are found, where P : R
+ → R

n×n
and q : R
+ → Rn
are locally summable matrix and vector functions, respectively, t
0 ∈ R
+ and c
0 ∈ Rn
.
On the Correctness of Nonlinear Boundary Value Problems for Systems of Generalized Ordinary Differential Equations
Authors: Ashordia M.
Published: February 2021

The concept of a strongly isolated solution of the nonlinear boundary value problem

dx(t) = dA(t) · f (t, x(t)), h(x) = 0,

is introduced, where A : [a, b] → Rn×n
is a matrix-function of bounded variation, f : [a, b] × Rn
→ Rn
is a vector-function belonging to a Carathéodory class, and h is a continuous operator from the space of n-dimensional vector-functions of bounded variation into Rn
.

It is stated that the problems with strongly isolated solutions are correct. Sufficient conditions for the correctness of these problems are given.
Characterization of a Regular Family of Semimartingales by Line Integrals
Authors: Chitashvili R. , Mania M.
Published: February 2021

A characterization of a regular family of semimartingales as a maximal family of processes with respect to which one can define a stochastic line integral with natural continuity properties is given.
On Continuous Extensions
Authors: Narici Lawrence , Beckenstein Edward
Published: February 2021

We consider various possibilities concerning the continuous extension of continuous functions taking values in an ultrametric space. In Section 1 we consider Tietze-type extension theorems concerning continuous extendibility of continuous functions from compact and closed subsets to the whole space. In Sections 2 and 3 we consider extending “separated” continuous functions in such a way that certain continuous extensions remain separated. Functions taking values in a complete ultravalued field are dealt with in Section 2, and the real and complex cases in Section 3.
Weighted L
Φ Integral Inequalities for Maximal Operators
Authors: Rakotondratsimba Y.
Published: February 2021

Sufficient (almost necessary) conditions are given on the weight functions u(·), v(·) for

to hold when Φ1, Φ2 are -functions with subadditive , and Ms (0 ≤ s < n), is the usual fractional maximal operator.
Lyapunov Type Integral Inequalities for Certain Differential Equations
Authors: Pachpatte B. G.
Published: February 2021

In the present paper we establish Lyapunov type integral inequalities related to the zeros of solutions of certain second-order differential equations by using elementary analysis. We also present some immediate applications of our results to study the asymptotic behavior of solutions of the corresponding differential equations.
Parametrization of a Family of Minimal Surfaces Bounded by the Broken Lines in
Authors: Abdulaev R.
Published: February 2021

Consideration is given to a family of minimal surfaces bounded by the broken lines in which are locally injectively projected onto the coordinate plane. The considered family is bijectively mapped by means of the Enepper–Weierstrass representation onto a set of circular polygons of a certain type. The parametrization of this set is constructed, and the dimension of the parameter domain is established.
On Singular Functional Differential Inequalities
Authors: Kiguradze I. , Sokhadze Z.
Published: February 2021

Classical theorems on differential inequalities [Coddington and Levinson, Theory of ordinary differential equations, Mc Graw-Hill Book Company, Inc., 1995, Hartman, Ordinary differential equations, John Wiley & Sons, 1964, Walter, Differential and integral inequalities, Springer-Verlag, 1970] are generalized for initial value problems of the kind

and

where ƒ : C([a, b]; Rn
) → Lloc
(]a, b]; Rn
) is a singular Volterra operator, c
0 ∈ Rn
, h : [a, b] → [0, +∞[ is continuous and positive on ]a, b], ‖ · ‖ is a norm in Rn
, and [u]+ and [u]– are respectively the positive and the negative part of the vector u ∈ Rn
.
A Multidimensional Singular Boundary Value Problem of the Cauchy–Nicoletti Type
Authors: Diblík J.
Published: February 2021

A two-point singular boundary value problem of the Cauchy–Nicoletti type is studied by introducing a two-point boundary value set and using the topological principle. The results on the existence of solutions whose graph lies in this set are proved. Applications and comparisons to the known results are given, too.
On the Absolute Convergence of Fourier Series
Authors: Karchava T.
Published: February 2021

The necessary and sufficient conditions of the absolute convergence of a trigonometric Fourier series are established for continuous 2π-periodic functions which in [0, 2π] have a finite number of intervals of convexity, and whose 𝑛th Fourier coefficients are O(ω(1/𝑛; 𝑓)/𝑛), where ω(δ; 𝑓) is the continuity modulus of the function 𝑓.
On the Global Solvability of the Cauchy Problem for Singular Functional Differential Equations
Authors: Kiguradze I. , Sokhadze Z.
Published: February 2021

Sufficient conditions are found for the global solvability of the weighted Cauchy problem

where 𝑓 : 𝐶([𝑎, 𝑏]; 𝑅𝑛) → 𝐿𝑙𝑜𝑐(]𝑎, 𝑏]; 𝑅𝑛) is a singular Volterra operator, 𝑐0 ∈ 𝑅𝑛, 𝑕 : [𝑎, 𝑏] → [0, +∞[ is a function continuous and positive on ]𝑎, 𝑏], and ∥ · ∥ is the norm in 𝑅𝑛.
On the Boundedness of Classical Operators on Weighted Lorentz Spaces
Authors: Rakotondratsimba Y.
Published: February 2021

Conditions on weights 𝑢(·), υ(·) are given so that a classical operator T sends the weighted Lorentz space Lrs
(υd𝑥) into Lpq
(υd𝑥). Here T is either a fractional maximal operator Mα
or a fractional integral operator Iα
or a Calderón–Zygmund operator. A characterization of this boundedness is obtained for Mα
and Iα
when the weights have some usual properties and max(r, s) ≤ min(p, q).
Coverings and Ring-Groupoids
Authors: Mucuk Osman
Published: February 2021

We prove that the set of homotopy classes of the paths in a topological ring is a ring object (called ring groupoid). Using this concept we show that the ring structure of a topological ring lifts to a simply connected covering space.
Vertical Cohomologies and their Application to Completely Integrable Hamiltonian Systems
Authors: Tevdoradze Z.
Published: February 2021

Some functorial and topological properties of vertical cohomologies and their application to completely integrable Hamiltonian systems are studied.
A N. Bourbaki Type General Theory and the Properties of Contracting Symbols and Corresponding Contracted Forms
Authors: Pkhakadze SH.
Published: February 2021

A system of contracting symbols is introduced for a N. Bourbaki type general mathematical theory corresponding to a general classical mathematical theory .
Factorization Systems and Adjunctions
Authors: Zangurashvili D.
Published: February 2021

Several results on the translation of factorization systems along adjunctions are proved. The first of these results is the answer to the question posed by G.Janelidze.
On Vitali's Theorem for Groups of Motions of Euclidean Space
Authors: Kharazishvili A.
Published: February 2021

We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.
Tensor Products of Non-Archimedean Weighted Spaces of Continuous Functions
Authors: Katsaras A. K. , Beloyiannis A.
Published: February 2021

It is shown that the completion of the tensor product of two non-Archimedean weighted spaces of continuous functions is topologically isomorphic to another weighted space. Several applications of this result are given.
On Periodic Solutions of Nonlinear Functional Differential Equations
Authors: Kiguradze I. , Půža B.
Published: February 2021

Sufficient conditions are established for the existence and uniqueness of an ω-periodic solution of the functional differential equation

where f is a continuous operator acting from the space of n-dimensional ω-periodic continuous vector functions into the space of n-dimensional ω-periodic and summable on [0, ω] vector functions.
A Contact Problem of the Interaction of a Semi-Finite Inclusion with a Plate
Authors: Shavlakadze N.
Published: February 2021

A piecewise-homogeneous plane made up of twodifferent materials and reinforced by an elastic unclusion is considered on a semi-finite section where the different materials join. Vertical and horizontal forces are applied to the inclusion which haz a variable thichness and a variable elasticity modulus.

Under certain conditions the problem is reduced to integrodifferential equations of third order. The solution is constructed effectively by applying the methods of theory of analytic functions to a boundary value problem of the Carleman type for a strip. Asymptotic estimates of normal contact stress are obtained.
Properties 𝐴 and 𝐵 of 𝑛th Order Linear Differential Equations with Deviating Argument
Authors: Koplatadze R. , Kvinikadze G. , Stavroulakis I. P.
Published: February 2021

Sufficient conditions for the nth order linear differential equation

𝑢(𝑛) (𝑡) + 𝑝(𝑡)𝑢(τ(𝑡)) = 0, 𝑛 ≥ 2,

to have Property 𝐴 or Property 𝐵 are established in both the delayed and the advanced cases. These conditions essentially improve many known results not only for differential equations with deviating arguments but for ordinary differential equations as well.
Multiple Solutions of Generalized Multipoint Conjugate Boundary Value Problems
Authors: Wong Patricia J. Y. , Agarwal Ravi P.
Published: February 2021

We consider the boundary value problem

𝑦(𝑛) (𝑡) = 𝑃(𝑡, 𝑦), 𝑡 ∈ (0, 1)

𝑦(𝑗) (𝑡𝑖) = 0, 𝑗 = 0, . . . , 𝑛𝑖 – 1, 𝑖 = 1, . . . , 𝑟,

where 𝑟 ≥ 2, 𝑛𝑖 ≥ 1 for 𝑖 = 1, . . . , 𝑟, and 0 = 𝑡1 < 𝑡2 < ⋯ < 𝑡𝑟 = 1. Criteria are offered for the existence of double and triple ‘positive’ (in some sense) solutions of the boundary value problem. Further investigation on the upper and lower bounds for the norms of these solutions is carried out for special cases. We also include several examples to illustrate the importance of the results obtained.
Sufficient Conditions for the Oscillation of Bounded Solutions of A Class of Impulsive Differential Equations of Second Order with a Constant Delay
Authors: Bainov D. D. , Dimitrova M. B.
Published: February 2021

Sufficient conditions are found for oscillation of bounded solutions of a class of impulsive differential equations of second order with a constant delay. Some asymptotic properties are studied for the bounded solutions.
On the Number of Representations of Positive Integers by the Quadratic Form
Authors: Lomadze G.
Published: February 2021

An explicit exact (non asymptotic) formula is derived for the number of representations of positive integers by the quadratic form . The way by which this formula is derived, gives us a possibility to develop a method of finding the so-called Liouville type formulas for the number of representations of positive integers by positive diagonal quadratic forms in nine variables with integral coefficients.
Inequalities of Calderon–Zygmund Type for Trigonometric Polynomials
Authors: Runovski K. , Schmeisser H.-J.
Published: February 2021

We give a unified approach to inequalities of Calderon–Zygmund type for trigonometric polynomials of several variables based on the Fourier analytic methods. Sharp results are achieved for the full range of admissible parameters p, 0 < p ≤ +∞. The results obtained are applied to the problem of the image of the Fourier transform in the scale of Besov spaces.
On Stochastic Differential Equations in a Configuration Space
Authors: Skorokhod A.
Published: February 2021

Infinite systems of stochastic differential equations for randomly perturbed particle systems with pairwise interaction are considered. It is proved that under some reasonable assumption on the potential function there exists a local weak solution to the system and it is weakly locally unique for a wide class of initial conditions.
On the Rationality of Certain Strata of the Lange Stratification of Stable Vector Bundles on Curves
Authors: Ballico E.
Published: February 2021

Let 𝑋 be a smooth projective curve of genus 𝑔 ≥ 2 and 𝑆(𝑟, 𝑑) the moduli scheme of all rank 𝑟 stable vector bundles of degree 𝑑 on 𝑋. Fix an integer 𝑘 with 0 < 𝑘 < 𝑟. H. Lange introduced a natural stratification of 𝑆(𝑟, 𝑑) using the degree of a rank 𝑘 subbundle of any 𝐸 ∈ 𝑆(𝑟, 𝑑) with maximal degree. Every non-dense stratum, say 𝑊(𝑘, 𝑟 – 𝑘, 𝑎, 𝑑 – 𝑎), has in a natural way a fiber structure ℎ : 𝑊(𝑘, 𝑟 – 𝑘, 𝑎, 𝑑 – 𝑎) → Pic𝑎(𝑋) × Pic𝑏(𝑋) with ℎ dominant. Here we study the rationality or the unirationality of the generic fiber of ℎ.
Suspension and Loop Objects in Theories and Cohomology
Authors: Baues H.-J. , Jibladze M.
Published: February 2021

We introduce theories in which suspension, resp. loop objects, are defined and we describe the cohomology of such theories. Cohomology classes in degree 3 classify track theories.
Some Problems of Thermoelastic Equilibrium of a Rectangular Parallelepiped in Terms of Asymmetric Elasticity
Authors: Khomasuridze N.
Published: February 2021

An effective solution of a number of boundary value and boundary contact problems of thermoelastic equilibrium is constructed for a homogeneous isotropic rectangular parallelepiped in terms of asymmetric and pseudo-asymmetric elasticity (Cosserat's continuum and pseudo- continuum). Two opposite faces of a parallelepiped are affected by arbitrary surface disturbances and a stationary thermal field, while for the four remaining faces symmetry or anti-symmetry conditions (for a multilayer rectangular parallelepiped nonhomogeneous contact conditions are also defined) are given. The solutions are constructed in trigonometric series using the method of separation of variables.
A Remark on Polygonal Quasiconformal Maps
Authors: Krushkal S. L.
Published: February 2021

Given a quasisymmetric map , let 𝑓0 be an extremal quasiconformal extension of ℎ onto the upper half-plane 𝘜 = {𝑧 ∈ ℂ : 𝔍𝑧 > 0} whose dilatation 𝑘(𝑓0) = inf{𝑘(𝑓) : 𝑓|∂𝘜 = ℎ0} ≕ 𝑘(ℎ). Let 𝑘𝑛 be the minimal dilatation of polygonal quasiconformal maps 𝑓 : 𝘜 → 𝘜 satisfying 𝑓(𝑥𝑗) = ℎ(𝑥𝑗), 𝑗 = 1, 2, . . . , 𝑛, for any 𝑛 points of (vertices of 𝑛-gons). Already a long time ago, the question was posed whether 𝑘(ℎ) = sup 𝑘𝑛, where the supremum is taken over all possible 𝑛-gons of such kind. The answer is obtained (in the negative) for the case of quadrilaterals (𝑛 = 4).

We show that in the case of pentagons (𝑛 = 5) the answer is also negative, i.e., there are quasisymmetric ℎ with 𝑘(ℎ) > sup 𝑘5.
Asymptotic Behaviour and Hopf Bifurcation of a Three-Dimensional Nonlinear Autonomous System
Authors: Baráková Lenka
Published: February 2021

A three-dimensional real nonlinear autonomous system of a concrete type is studied. The Hopf bifurcation is analyzed and the existence of a limit cycle is proved. A positively invariant set, which is globally attractive, is found using a suitable Lyapunov-like function. Corollaries for a cubic system are presented. Also, a two-dimensional nonlinear system is studied as a restricted system. An application in economics to the Kodera's model of inflation is presented. In some sense, the model of inflation is an extension of the dynamic version of the neo-keynesian macroeconomic IS-LM model and the presented results correspond to the results for the IS-LM model.
AMD-Numbers, Compactness, Strict Singularity and the Essential Spectrum of Operators
Authors: Castejón A. , Corbacho E. , Tarieladze V.
Published: February 2021

For an operator 𝑇 acting from an infinite-dimensional Hilbert space 𝐻 to a normed space 𝑌 we define the upper AMD-number and the lower AMD-number as the upper and the lower limit of the net (δ(𝑇|𝐸))𝐸∈𝐹𝐷(𝐻), with respect to the family 𝐹𝐷(𝐻) of all finite-dimensional subspaces of 𝐻. When these numbers are equal, the operator is called AMD-regular.

It is shown that if an operator 𝑇 is compact, then and, conversely, this property implies the compactness of 𝑇 provided 𝑌 is of cotype 2, but without this requirement may not imply this. Moreover, it is shown that an operator 𝑇 has the property if and only if it is superstrictly singular. As a consequence, it is established that any superstrictly singular operator from a Hilbert space to a cotype 2 Banach space is compact.

For an operator 𝑇, acting between Hilbert spaces, it is shown that and are respectively the maximal and the minimal elements of the essential spectrum of , and that 𝑇 is AMD-regular if and only if the essential spectrum of |𝑇| consists of a single point.
A Problem of Linear Conjugation for Analytic Functions with Boundary Values from the Zygmund Class
Authors: Kokilashvili V. , Paatashvili V.
Published: February 2021

The solvability conditions are established for a problem of linear conjugation for analytic functions with boundary values from the Zygmund class 𝐿(ln+ 𝐿)
α
when the conjugation coefficient is piecewise-continuous in the Hölder sense. Solutions of the problem are constructed in explicit form.
On Obtaining Dual Sequences Via Quasi-Monomiality
Authors: Cheikh Youssèf Ben
Published: February 2021

In this paper, we introduce a method to obtain the dual sequence of a given polynomial set using the lowering operator associated with the involved polynomials. As application, we derive polynomial expansions of analytic functions. The particular case corresponding to Boas–Buck polynomials allows us to unify many polynomial expansions of analytic functions in the literature. This method can be useful in studying many problems arising in the theory of polynomials as the so-called connection and linearization problems.
Solvability and the Unique Solvability of A Periodic Type Boundary Value Problem for First Order Scalar Functional Differential Equations
Authors: Hakl R. , Lomtatidze A. , Šremr J.
Published: February 2021

Nonimprovable in a certain sense, sufficient conditions for the solvability and unique solvability of the problem

𝑢′ (𝑡) = 𝐹 (𝑢) (𝑡), 𝑢(𝑎) – λ𝑢(𝑏) = ℎ(𝑢)

are established, where 𝐹 : 𝐶([𝑎, 𝑏]; 𝑅) → 𝐿([𝑎, 𝑏];𝑅) is a continuous operator satisfying the Carathéodory condition, ℎ : 𝐶([𝑎, 𝑏]; 𝑅) → 𝑅 is a continuous functional, and λ ∈ 𝑅+.
Frame Characterizations of Besov and Triebel–Lizorkin Spaces on Spaces of Homogeneous Type and their Applications
Authors: Yang Dachun
Published: February 2021

The author first establishes the frame characterizations of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type. As applications, the author then obtains some estimates of entropy numbers for the compact embeddings between Besov spaces or between Triebel–Lizorkin spaces. Moreover, some real interpolation theorems on these spaces are also established by using these frame characterizations and the abstract interpolation method.
On the Existence of Continuous Modifications of Vector-Valued Random Fields
Authors: Mittmann Katrin , Steinwart Ingo
Published: February 2021

We consider vector-valued random fields on a general index set in and show the existence of continuous modifications if some Kolmogorov type condition is satisfied. Furthermore, we prove an extension result for Hölder continuous vector valued mappings.
Time-Dependent Barrier Options and Boundary Crossing Probabilities
Authors: Novikov A. , Frishling V. , Kordzakhia N.
Published: February 2021

The problem of pricing of time-dependent barrier options is considered in the case when interest rate and volatility are given functions in Black–Scholes framework. The calculation of the fair price reduces to the calculation of non-linear boundary crossing probabilities for a standard Brownian motion. The proposed method is based on a piecewise-linear approximation for the boundary and repeated integration. The numerical example provided draws attention to the performance of suggested method in comparison to some alternatives.
On Separation Properties for Families of Probability Measures
Authors: Pantsulaia G.
Published: February 2021

We consider the problem of transition from a weakly separated family of probability measures to a strictly separated family.
On the Mathematical Basis of the Linear Sampling Method
Authors: Cakoni Fioralba , Colton David
Published: February 2021

The linear sampling method is an algorithm for solving the inverse scattering problem for acoustic and electromagnetic waves. The method is based on showing that a linear integral equation of first kind has a solution that becomes unbounded as a parameter 𝑧 approaches the boundary of the scatterer 𝐷 from inside 𝐷. However, except for the case of the transmission problem, the case where z is in the exterior of 𝐷 is unresolved. Since for the inverse scattering problem 𝐷 is unknown, this step is crucial for the mathematical justification of the linear sampling method. In this paper we give a mathematical justification of the linear sampling method for arbitrary 𝑧 by using the theory of integral equations of first kind with singular kernels.
About Asymptotic and Oscillation Properties of the Dirichlet Problem for Delay Partial Differential Equations
Authors: Domoshnitsky Alexander
Published: February 2021

In this paper, oscillation and asymptotic properties of solutions of the Dirichlet boundary value problem for hyperbolic and parabolic equations are considered. We demonstrate that introducing an arbitrary constant delay essentially changes the above properties. For instance, the delay equation does not inherit the classical properties of the Dirichlet boundary value problem for the heat equation: the maximum principle is not valid, unbounded solutions appear while all solutions of the classical Dirichlet problem tend to zero at infinity, for “narrow enough zones” all solutions oscillate instead of being positive. We establish that the Dirichlet problem for the wave equation with delay can possess unbounded solutions. We estimate zones of positivity of solutions for hyperbolic equations.
A Bound for Reflections Across Jordan Curves
Authors: Krushkal Samuel L.
Published: February 2021

The properties of logarithmic derivative of the Riemann mapping function of a quasidisk are applied to quantitative estimation of quasiconformal reflections across its boundary.
On Two-Point Boundary Value Problems for Two-Dimensional Differential Systems with Singularities
Authors: Mukhigulashvili S.
Published: February 2021

For a differential system

where λ ∈]0, 1[ and ℎ𝑖 :]𝑎, 𝑏[×]0, +∞[×ℝ → [0, +∞[ (𝑖 = 0, 1, 2) are continuous functions, we have established sufficient conditions for the existence of at least one solution satisfying one of the two boundary conditions

and
Analytic Subsets of Products of Infinite-Dimensional Projective Spaces
Authors: Ballico E.
Published: February 2021

Let 𝑉𝑖, 1 ≤ 𝑖 ≤ 𝑠, be complex topological vector spaces with 𝑉1 infinite-dimensional and 𝑌 a closed analytic subset of finite codimension of 𝐏(𝑉1) × . . . × 𝐏(𝑉𝑠). Here we show that 𝑌 is algebraic (at least if each 𝑉𝑖 is a Banach space) and that any two points of 𝑌 may be connected by a chain of 𝑠 + 3 lines contained in 𝑌.
Simplicial Crossed Modules and Mapping Cones
Authors: Conduché D.
Published: February 2021

Given a bisimplicial group 𝐺∗∗ such that 𝑁(𝐺)∗𝑞 = {1} for 𝑞 ⩾ 2, a simplicial group is obtained whose Moore complex is a mapping cone of the chain morphism 𝑁(𝐺)∗1 → 𝑁(𝐺)∗0. This simplicial group is homotopy equivalent to the diagonal of 𝐺∗∗. In the last section a special case is considered.
Badora's Equation on Non-Abelian Locally Compact Groups
Authors: Elqorachi E. , Akkouchi M. , Bakali A. , Bouikhalene B.
Published: February 2021

This paper is mainly concerned with the following functional equation

where 𝐺 is a locally compact group, 𝐾 a compact subgroup of its morphisms, and μ is a generalized Gelfand measure. It is shown that continuous and bounded solutions of this equation can be expressed in terms of μ-spherical functions. This extends the previous results obtained by Badora (Aequationes Math. 43: 72–89, 1992) on locally compact abelian groups. In the case where 𝐺 is a connected Lie group, we characterize solutions of the equation in question as joint eigenfunctions of certain operators associated to the left invariant differential operators.
Convergence of Marcinkiewicz Means of Integrable Functions with Respect to Two-Dimensional Vilenkin Systems
Authors: Gát György
Published: February 2021

We prove that the maximal operator of the Marcinkiewicz mean of integrable two-variable functions is of weak type (1, 1) on bounded two-dimensional Vilenkin groups. Moreover, for any integrable function 𝑓 the Marcinkiewicz mean σ
𝑛𝑓 converges to 𝑓 almost everywhere.
On the Measurability of Functions with Respect to Certain Classes of Measures
Authors: Kharazishvili A. , Kirtadze A.
Published: February 2021

The concept of measurability of real-valued functions with respect to various classes of measures is introduced and the associated notion of an absolutely nonmeasurable function is investigated. A characterization of such functions is given. Also, it is shown that functions produced by the classical Vitali partition of the real line are measurable with respect to the class of all extensions of the Lebesgue measure on this line.
Complex Points of Two-Dimensional Surfaces
Authors: Aliashvili T.
Published: February 2021

We deal with complex points of two-dimensional surfaces. A short review of basic results about complex points of smooth surfaces in is presented at the beginning. For algebraic surfaces, a formula is proved which expresses the number of complex points as the local degree of an explicitly constructible polynomial endomorphism.
Equicontinuity and Quasi-Uniformities
Authors: Corbacho E. , Tarieladze V. , Vidal R.
Published: February 2021

For topological spaces 𝑋, 𝑌 with a fixed compatible quasi-uniformity 𝑄 in 𝑌 and for a family (𝑓𝑖)𝑖∈𝐼 of mappings from 𝑋 to 𝑌, the notions of even continuity in the sense of Kelley, topological equicontinuity in the sense of Royden and 𝑄-equicontinuity (i.e., equicontinuity with respect to the topology of 𝑋 and 𝑄) are compared. It is shown that 𝑄-equicontinuity implies even continuity, and if 𝑄 is locally symmetric, it implies topological equicontinuity too. It turns out that these notions are equivalent provided 𝑄 is a uniformity compatible with a compact topology, but the equivalence may fail even for a locally symmetric quasi-uniformity 𝑄 compatible with a compact metrizable topology.
Witt's Theorem for Groups with Action and Free Leibniz Algebras
Authors: Datuashvili T.
Published: February 2021

A subcategory of the category of groups with action is determined and it is proved that the functor defined in [Datuashvili, Georgian Math. J. 9: 671–681, 2002] takes free objects from this category to free Leibniz algebras. This result gives a solution to a problem stated by J.-L. Loday [Enseign. Math. 39: 269–293, 1993], [K-Theory, 30: 105–127, 2003] and is an analogue of Witt's theorem for groups and Lie algebras [Witt, J. Reine Angew. Math. 1: 152–160, 1937].
Linearization and Higher Order Nonlinear Oscillation Theorems Using Comparison Methods
Authors: Agarwal Ravi P. , Grace Said R. , O'Regan Donal
Published: February 2021

We establish some new oscillation criteria for higher order nonlinear differential equations. The main idea involves comparing our equation with the related linear second order differential equations so that the known oscillation theorems from the literature can be employed directly.
New Applications of Algebraic Formulae for Topological Invariants
Authors: Khimshiashvili G.
Published: February 2021

Some geometric results are presented which can be derived from algebraic formulae for topological degree and Euler characteristic. In particular, it is shown that Euler characteristics of configuration spaces and work spaces of mechanical linkages can be computed in an algorithmic way. We also find the expected gradient degree of rotation invariant Gaussian random polynomial on an even-dimensional space.
Two-Step Systems for G-H-Relaxed Pseudococoercive Nonlinear Variational Problems Based on Projection Methods
Authors: Agarwal Ravi P. , O'Regan Donal , Verma Ram U.
Published: February 2021

The approximation-solvability of a generalized system of nonlinear variational inequalities (SNVI) involving relaxed pseudococoercive mappings, based on the convergence of a system of projection methods, is presented. The class of relaxed pseudococoercive mappings is more general than classes of strongly monotone and relaxed cocoercive mappings. Let 𝐾1 and 𝐾2 be nonempty closed convex subsets of real Hilbert spaces 𝐻1 and 𝐻2, respectively. The two-step SNVI problem considered here is as follows: find an element (𝑥*, 𝑦*) ∈ 𝐻1 × 𝐻2 such that (𝑔(𝑥*), 𝑔(𝑦*)) ∈ 𝐾1 × 𝐾2 and

where 𝑆 : 𝐻1 × 𝐻2 → 𝐻1, 𝑇 : 𝐻1 × 𝐻2 → 𝐻2, 𝑔 : 𝐻1 → 𝐻1 and ℎ : 𝐻2 → 𝐻2 are nonlinear mappings.
Integrability of the Majorant of the Fourier Series Partial Sums with Respect to Bases
Authors: Tkebuchava George
Published: February 2021

The majorant of Fourier series partial sums with respect to the system of functions formed by the product of 𝐿([0, 1]) space bases is considered. It is proved that in any Orlicz space wider than 𝐿(log+ 𝐿)𝑑([0, 1]𝑑), 𝑑 ≥ 1, the set of functions with such a majorant is integrable on [0, 1]𝑑 and has the first Baire category.
Gelfand Pairs and Generalized d'Alembert's and Cauchy's Functional Equations
Authors: Bouikhalene Belaid , Kabbaj Samir
Published: February 2021

We show that Cauchy's, d'Alembert's functional equations and their generalizations are the functional equations for bounded spherical functions associated to some Gel'fand pairs. Our results are very close to the ones obtained by Stetkær in [Aequationes Math. 48: 220–227, 1994].
Interior and Exterior Problems of Couple-Stress and Classical Elastostatics with Given Friction
Authors: Gachechiladze Roland
Published: February 2021

We consider an interior problem of statics of couple-stress elasticity for anisotropic nonhomogeneous media with friction taken into account and the corresponding exterior problem for homogeneous isotropic media. The question of the existence and uniqueness of weak solutions of these problems is studied when friction forces occur along the boundary of an elastic medium or on some part of this boundary.
Some Basic 𝑑-Orthogonal Polynomial Sets
Authors: Zaghouani Ali
Published: February 2021

The purpose of this paper is to study the class of polynomial sets which are at the same time 𝑑-orthogonal and 𝑞-Appell. By a linear change of variable, the resulting set reduces to 𝑞-Al-Salam–Carlitz polynomials, for 𝑑 = 1. Various properties of the obtained polynomials are singled out: a generating function, a recurrence relation of order 𝑑 + 1. We also explicitly express a 𝑑-dimensional functional for which the d-orthogonality holds.
Convergence in Measure of Logarithmic Means of Double Walsh–Fourier Series
Authors: Gát György , Goginava Ushangi , Tkebuchava George
Published: February 2021

The main aim of this paper is to prove that the logarithmic means of the double Walsh–Fourier series do not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of 𝐿log 𝐿(𝐼2), the set of functions for which quadratic logarithmic means of the double Walsh–Fourier series converge in measure is of first Baire category.
On the Accuracy of Kraft Upper Bound for the 𝐿1-Distance between Gaussian Densities in 𝑅𝑘
Authors: Kvatadze Zurab , Shervashidze Tengiz
Published: February 2021

For a scaled family of centered nonsingular Gaussian measures in 𝑘𝑘 the 𝐿1-distance between any two densities is compared with its Kraft upper bound.
Continuity of Multilinear Marcinkiewicz Operators for the Extreme Cases of 𝑝
Authors: Liu Lanzhe
Published: February 2021

We prove the continuity properties of multilinear Marcinkiewicz operators for the extreme cases of 𝑝.
Optimal Control in Bombieri's and Tammi's Conjectures
Authors: Prokhorov Dmitri , Vasil'ev Alexander
Published: February 2021

Let 𝑆 stand for the usual class of univalent regular functions in the unit disk 𝑈 = {𝑧 : |𝑧| < 1} normalized by 𝑓(𝑧) = 𝑧 + 𝑎2𝑧2 + . . . in 𝑈, and let 𝑆𝑀 be its subclass defined by restricting |𝑓(𝑧)| < 𝑀 in 𝑈, 𝑀 ≥ 1. We consider two classical problems: Bombieri's coefficient problem for the class 𝑆 and the sharp estimate of the fourth coefficient of a function from 𝑆𝑀. Using Löwner's parametric representation and the optimal control method we propose a way to finding initial Bombieri's numbers and derive a sharp constant 𝑀0 such that for all 𝑀 ≥ 𝑀0 the Pick function gives the local maximum to |𝑎4|. Numerical approximation is given.
On One Estimate for Periodic Functions
Authors: Hakl Robert , Mukhigulashvili Sulkhan
Published: February 2021

For , the estimate

is derived, where

and 𝑑𝑛 are defined by a certain recurrent formula.
Almost Everywhere Convergence of (𝐶, α)-Means of Quadratical Partial Sums of Double Vilenkin–Fourier Series
Authors: Gát György , Goginava Ushangi
Published: February 2021

We prove that the maximal operator of the (𝐶, α)-means of quadratical partial sums of double Vilenkin–Fourier series is of weak type (1,1). Moreover, the (𝐶, α)-means of a function 𝑓 ∈ 𝐿1 converge a.e. to 𝑓 as 𝑛 → ∞.
On Systems of Linear Functional Differential Inequalities
Authors: Šremr Jiří
Published: February 2021

The Cauchy problem for 𝑛-dimensional systems of linear functional differential equations is studied. Some new theorems on systems of functional differential inequalities are established.
Oscillations of Higher Order Functional Differential Equations with Impulses
Authors: Zhang Chaolong , Feng Weizhen
Published: February 2021

A kind of higher order sub-and super-linear FDE with impulses is studied in this paper. Several criteria on the oscillations of solutions are given. In particular, in the case where the coefficients of equations are positive and continuous functions, we find some suitable impulse functions such that all solutions of the equation are oscillatory under the impulse control.
Triangulated Track Categories
Authors: Baues Hans-Joachim
Published: February 2021

The concept of a triangulated homotopy category can be canonically lifted to the level of a groupoid-enriched category. This way two natural axioms on track triangles replace the four somewhat obscure axioms of a triangulated category.
Completing the Operadic Butterfly
Authors: Loday Jean-Louis
Published: February 2021

We complete a certain diagram (the operadic butterfly) of the categories of algebras involving Com, As, and Lie by constructing a type of algebras which have 4 generating operations and 16 relations. The associated operad is self-dual for Koszul duality.
On a Double Series of Chan and Ong
Authors: Williams Kenneth S.
Published: February 2021

An arithmetic identity is used to prove a relation satisfied by the double series . As an application an explicit formula is given for the number of representations of the positive integer 𝑛 by the form .
The Dirichlet Problem for Harmonic Functions in the Smirnov Class with Variable Exponent
Authors: Kokilashvili Vakhtang , Paatashvili Vakhtang
Published: February 2021

A solution of the Dirichlet problem for harmonic functions from the Smirnov class is obtained in the framework of functional spaces with a nonstandard growth condition. It is found that the domain boundary geometry influences the character of a problem solution. In the case of solvability, solutions are constructed in explicit form.
Singular Dirichlet Boundary Value Problem for Second Order Ode
Authors: Rachůnková Irena , Stryja Jakub
Published: February 2021

This paper investigates the singular Dirichlet problem

–𝑢″ = 𝑓(𝑡, 𝑢, 𝑢′), 𝑢(0) = 0, 𝑢(𝑇) = 0,

where 𝑓 satisfies the Carathéodory conditions on the set and .

The function 𝑓(𝑡, 𝑥, 𝑦) can have time singularities at 𝑡 = 0 and 𝑡 = 𝑇 and space singularities at 𝑥 = 0 and 𝑦 = 0. The existence principle for the above problem is given and its application is presented here. The paper provides conditions which guarantee the existence of a solution which is positive on (0; T) and which has the absolutely continuous first derivative on [0, 𝑇].
The Robin Function and Its Eigenvalues
Authors: Dittmar Bodo , Hantke Maren
Published: February 2021

The paper deals with the Robin function and eigenvalue problems generated by the Robin operator. First we show that Green's function of an 𝑛-fold connected domain is the Robin function of an appropriate simply connected domain. The main part of the paper deals with eigenvalue problems for the Robin operator: the mixed Stekloff eigenvalue problem and the membrane problem with mixed boundary conditions. Isoperimetric inequalities are proved for the sum of reciprocal eigenvalues.
On a Priori Estimates of Solutions of Systems of Higher Order Nonlinear Functional-Differential Inequalities
Authors: Kiguradze Ivan , Partsvania Nino
Published: February 2021

Systems of higher order nonlinear functional-differential inequalities arising in the theory of boundary value problems as well as in the stability theory are considered on a finite and an infinite interval. On a finite interval, a priori estimates are obtained for solutions of these systems satisfying boundary conditions of periodic type, and on an infinite interval, a priori estimates are established for the derivatives of bounded solutions.
On Solvability of the Neumann Problem in an Energy Space for a Domain with Peak
Authors: Maz'ya Vladimir G. , Poborchi Sergei V.
Published: February 2021

We describe the dual space of the boundary trace space for functions with a finite Dirichlet integral for a domain with a vertex of an isolated cusp at the boundary. This leads to conditions of solvability of the Neumann problem for elliptic equations of second order. In particular, we give an explicit necessary and sufficient condition for 𝑞 such that the Neumann problem is solvable if the boundary function is in 𝐿𝑞 over the boundary of a domain with an outer peak.
On the Oscillation of Second Order Nonlinear Neutral Delay Dynamic Equations
Authors: Agwo Hassan A.
Published: February 2021

In this paper we obtain some new oscillation criteria for the second order nonlinear neutral delay dynamic equation

(𝑥(𝑡) – 𝑝(𝑡)𝑥(𝑡 – τ
1))ΔΔ + 𝑞(𝑡)𝑓(𝑥(𝑡 – τ
2)) = 0,

on a time scale 𝕋. Moreover, a new sufficient condition for the oscillation sublinear equation

(𝑥(𝑡) – 𝑝(𝑡)𝑥(𝑡 – τ
1))″ + 𝑞(𝑡)𝑓(𝑥(𝑡 – τ
2)) = 0,

is presented, which improves other conditions and an example is given to illustrate our result.
Vector Measures on Topological Spaces
Authors: Khurana Surjit Singh
Published: February 2021

Let 𝑋 be a completely regular Hausdorff space, 𝐸 a quasi-complete locally convex space, 𝐶(𝑋) (resp. 𝐶𝑏(𝑋)) the space of all (resp. all, bounded), scalar-valued continuous functions on 𝑋, and 𝐵(𝑋) and 𝐵0(𝑋) be the classes of Borel and Baire subsets of 𝑋. We study the spaces 𝑀𝑡(𝑋,𝐸), 𝑀
τ
(𝑋,𝐸), 𝑀
σ
(𝑋,𝐸) of tight, τ-smooth, σ-smooth, 𝐸-valued Borel and Baire measures on 𝑋. Using strict topologies, we prove some measure representation theorems of linear operators between 𝐶𝑏(𝑋) and 𝐸 and then prove some convergence theorems about integrable functions. Also, the Alexandrov's theorem is extended to the vector case and a representation theorem about the order-bounded, scalar-valued, linear maps from 𝐶(𝑋) is generalized to the vector-valued linear maps.
Elusive Examples of Non-Metrizable Continua which Admit a Whitney Map
Authors: Lončar Ivan
Published: February 2021

The main purpose of this paper is to prove that the class of near locally connected continua contains no non-metrizable continuum 𝑋 which admits a Whitney map for 𝐶(𝑋). In particular, each near locally connected continuum 𝑋 which admits a Whitney map for 𝐶(𝑋) is metrizable.
On the Absolute Convergence of Fourier Series of Functions of ∧𝐵𝑉(𝑝) and φ ∧𝐵𝑉
Authors: Vyas Rajendra G.
Published: February 2021

Let 𝑓 be a 2π periodic function in 𝐿1[0,2π] and , be its Fourier coefficients. Extending the classical result of Zygmund, Schramm and Waterman obtained the sufficiency conditions for the absolute convergence of Fourier series of functions of ∧𝐵𝑉(𝑝) and φ ∧𝐵𝑉. Here we have generalized these results by obtaining certain sufficiency conditions for the convergence of the series , where is a strictly increasing sequence of natural numbers and 𝑛–𝑘 = –𝑛𝑘 for all 𝑘, for such functions.
Existence of Multiple Positive Solutions for Even Order Multi-Point Boundary Value Problems
Authors: Wang Youyu , Ge Weigao
Published: February 2021

In this paper, we consider the existence of multiple positive solutions for the 2𝑛th order 𝑚-point boundary value problem:

where

(0,1), 0 < ξ
1 < ξ
2 < ⋯ < ξ
𝑚–2 < 1. Using the Leggett–Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The associated Green's function for the above problem is also given.
Functional Equations and μ-Spherical Functions
Authors: Akkouchi Mohamed , Bouikhalene Belaid , Elqorachi Elhoucien
Published: February 2021

We will study the properties of solutions 𝑓, {𝑔𝑖}, {ℎ𝑖} ∈ 𝐶𝑏(𝐺) of the functional equation

where 𝐺 is a Hausdorff locally compact topological group, 𝐾 a compact subgroup of morphisms of 𝐺, χ a character on 𝐾, and μ a 𝐾-invariant measure on 𝐺. This equation provides a common generalization of many functional equations (D'Alembert's, Badora's, Cauchy's, Gajda's, Stetkaer's, Wilson's equations) on groups.

First we obtain the solutions of Badora's equation [Aequationes Math. 43: 72–89, 1992] under the condition that (𝐺,𝐾) is a Gelfand pair. This result completes the one obtained in [Badora, Aequationes Math. 43: 72–89, 1992] and [Elqorachi, Akkouchi, Bakali and Bouikhalene, Georgian Math. J. 11: 449–466, 2004]. Then we point out some of the relations of the general equation to the matrix Badora functional equation and obtain explicit solution formulas of the equation in question for some particular cases. The results presented in this paper may be viewed as a continuation and a generalization of Stetkær's, Badora's, and the authors' works.
On a Generalization of Partial Isometries in Banach Spaces
Authors: Taoudi Mohamed Aziz
Published: February 2021

This paper is concerned with the definition and study of semipartial isometries on Banach spaces. This class of operators, which is a natural generalization of partial isometries from Hilbert to general Banach spaces, contains in particular the class of partial isometries recently introduced by M. Mbekhta [Acta Sci. Math. (Szeged) 70: 767–781, 2004]. First of all, we establish some basic properties of semi-partial isometries. Next, we introduce the notion of pseudo Moore–Penrose inverse as a natural generalization of the Moore–Penrose inverse from Hilbert spaces to arbitrary Banach spaces. This concept is used to carry out a classification for semi-partial isometries in Banach spaces and to provide a characterization for Hilbert spaces.
A New Approach to the Sawyer and Sinnamon Characterizations of Hardy's Inequality for Decreasing Functions
Authors: Johansson Maria , Persson Lars-Erik , Wedestig Anna
Published: February 2021

Some Hardy type inequalities for decreasing functions are characterized by one condition (Sinnamon), while others are described by two independent conditions (Sawyer). In this paper we make a new approach to deriving such results and prove a theorem, which covers both the Sinnamon result and the Sawyer result for the case where one weight is increasing. In all cases we point out that the characterizing condition is not unique and can even be chosen among some (infinite) scales of conditions.