Search

This paper relates to a recent trend in complex differential equations which studies solutions in a given domain. The classical settings in complex equations were widely studied for meromorphic solutions in the complex plane. For functions in the complex plane, we have a lot of results of general nature, in particular, the classical value distributions theory describing numbers of a-points. Many of these results do not work for functions in a given domain. A recent principle of derivatives permits us to study the numbers of Ahlfors simple islands for functions in a given domain; the islands play, to some extend, a role similar to that of the numbers of simple a-points. In this paper, we consider a large class of higher order differential equations admitting meromorphic solutions in a given domain. Applying the principle of derivatives, we get the upper bounds for the numbers of Ahlfors simple islands of similar solutions.

The subject of the paper is suggested by G. Janelidze and motivated by his earlier result giving a positive answer to the question posed by S. MacLane whether the Galois theory of homogeneous linear ordinary differential equations over a differential field (which is Kolchin–Ritt theory and an algebraic version of Picard–Vessiot theory) can be obtained as a particular case of G. Janelidze’s Galois theory in categories. One ground category in the Galois structure involved in this theory is dual to the category of commutative rings with unit, and another one is dual to the category of commutative differential rings with unit. In the present paper, we apply the general categorical construction, the particular case of which gives this Galois structure, by replacing “commutative rings with unit” by algebras from any variety V \mathscr{V} of universal algebras satisfying the amalgamation property and a certain condition (of the syntactical nature) for elements of amalgamated free products which was introduced earlier, and replacing “commutative differential rings with unit” by V \mathscr{V} -algebras equipped with additional unary operations which satisfy some special identities to construct a new Galois structure. It is proved that this Galois structure is admissible. Moreover, normal extensions with respect to it are characterized in the case where V \mathscr{V} is any of the following varieties: abelian groups, loops and quasigroups.

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers and several integer sequences related to them have been studied. In the present paper, new sets of Bell–Sheffer polynomials are introduced. Connections with Bell numbers are shown.

In the present paper, we discuss the approximation properties of Jain–Baskakov operators with parameter c. The paper deals with the modified forms of the Baskakov basis functions. Some direct results are established, which include the asymptotic formula, error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct the King modification of these operators, which preserves the test functions e 0 {e_{0}} and e 1 {e_{1}} . It is shown that these King type operators provide a better approximation order than some Baskakov–Durrmeyer operators for continuous functions defined on some closed intervals.

The “oldest quartic” functional equation f ⁢ ( x + 2 ⁢ y ) + f ⁢ ( x - 2 ⁢ y ) = 4 ⁢ [ f ⁢ ( x + y ) + f ⁢ ( x - y ) ] - 6 ⁢ f ⁢ ( x ) + 24 ⁢ f ⁢ ( y ) f(x+2y)+f(x-2y)=4[f(x+y)+f(x-y)]-6f(x)+24f(y) was introduced and solved by the second author of this paper (see J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glas. Mat. Ser. III 34(54) 1999, 2, 243–252). Similarly, an interesting “quintic” functional equation was introduced and investigated by I. G. Cho, D. Kang and H. Koh, Stability problems of quintic mappings in quasi-β-normed spaces, J. Inequal. Appl. 2010 2010, Article ID 368981, in the following form: 2 ⁢ f ⁢ ( 2 ⁢ x + y ) + 2 ⁢ f ⁢ ( 2 ⁢ x - y ) + f ⁢ ( x + 2 ⁢ y ) + f ⁢ ( x - 2 ⁢ y ) = 20 ⁢ [ f ⁢ ( x + y ) + f ⁢ ( x - y ) ] + 90 ⁢ f ⁢ ( x ) . 2f(2x+y)+2f(2x-y)+f(x+2y)+f(x-2y)=20[f(x+y)+f(x-y)]+90f(x). In this paper, we generalize this “Cho–Kang–Koh equation” by introducing pertinent Euler–Lagrange k-quintic functional equations, and investigate the “Ulam stability” of these new k-quintic functional mappings.

In the paper, we investigate the relation between the properties of functions and their Fourier–Haar coefficients. We show that for some classes of functions Fourier–Haar coefficients have constant signs and order of magnitude. In 1964, Golubov proved in [B. I. Golubov, On Fourier series of continuous functions with respect to a Haar system (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 28 1964, 1271–1296] that if f ⁢ ( x ) ∈ C ⁢ ( 0 , 1 ) {f(x)\in C(0,1)} , then its Fourier–Haar coefficients have constant signs when f ⁢ ( x ) {f(x)} is a nonincreasing function on [ 0 , 1 ] {[0,1]} , and in some cases those coefficients have a certain order of magnitude. In the present paper, we continue to investigate the properties of functions which follow from the behavior of their Fourier–Haar coefficients.

The value distribution and, in particular, the numbers of a-points, have not been studied for meromorphic functions which are solutions of some complex differential equations in a given domain. Instead, the numbers of good a-points and Ahlfors islands, which play to a certain extend a role similar to that of the numbers of a-points, have been considered in some recent papers. In this paper, we consider meromorphic functions in a given domain, which are the solutions of some higher order equations and largely generalize the solutions of Painlevé equations 3–6. We give the upper bounds for the numbers of good a-points and Ahlfors islands of similar solutions.

In the papers [1] and [6], for an inverse sequence of pointed topological spaces and fibrations preserving the base points E = E 1 ← p 1 E 2 ← p 2 ⋯ ← p m E m + 1 , E=E_{1}\xleftarrow{p_{1}}E_{2}\xleftarrow{p_{2}}\cdots\xleftarrow{p_{m}}E_{m+1}, there exists an exact sequence * → lim ← ( 1 ) [ X , Ω E m ] → [ X , lim ← E ] → lim ← ( 1 ) [ X , E m ] → * . *\rightarrow{\varprojlim}^{(1)}[X,\Omega E_{m}]\rightarrow[X,\varprojlim E]% \rightarrow{\varprojlim}^{(1)}[X,E_{m}]\rightarrow*. In the present paper, for an inverse sequence of pointed topological semi-simplicial sets and fibrations preserving base points E ¯ = E ¯ ← p 1 1 E ¯ ← p 2 2 ⋯ ← p m E ¯ ← m + 1 ⋯ , \underline{E}=\underline{E}{}_{1}\xleftarrow{p_{1}}\underline{E}{}_{2}% \xleftarrow{p_{2}}\cdots\xleftarrow{p_{m}}\underline{E}{}_{m+1}\xleftarrow{% \hphantom{p_{1}}}\cdots, an analogous formula is proved.

In this paper, we introduce some classes of R-modules that are closely related to the classes of Prüfer, Dedekind and Bezout modules. Let R be a commutative ring with identity and set ℍ = { M ∣ M ⁢ is an ⁢ R ⁢ -module and ⁢ Nil ⁢ ( M ) ⁢ is a divided prime submodule of ⁢ M } . \mathbb{H}=\bigl{\{}M\mid M\text{ is an }R\text{-module and }\mathrm{Nil}(M)% \text{ is a divided prime submodule of }M\bigr{\}}. For an R-module M ∈ ℍ {M\in\mathbb{H}} , set T = ( R ∖ Z ⁢ ( R ) ) ∩ ( R ∖ Z ⁢ ( M ) ) {T=(R\setminus Z(R))\cap(R\setminus Z(M))} , 𝔗 ⁢ ( M ) = T - 1 ⁢ M {\mathfrak{T}(M)=T^{-1}M} and P = ( Nil ( M ) : R M ) {P=(\mathrm{Nil}(M):_{R}M)} . In this case, the mapping Φ : 𝔗 ⁢ ( M ) → M P {\Phi:\mathfrak{T}(M)\to M_{P}} given by Φ ⁢ ( x / s ) = x / s {\Phi(x/s)=x/s} is an R-module homomorphism. The restriction of Φ to M is also an R-module homomorphism from M into M P {M_{P}} given by Φ ⁢ ( x ) = x / 1 {\Phi(x)=x/1} for every x ∈ M {x\in M} . A nonnil submodule N of M is said to be Φ-invertible if Φ ⁢ ( N ) {\Phi(N)} is an invertible submodule of Φ ⁢ ( M ) {\Phi(M)} . Moreover, M is called a Φ-Prüfer module if every finitely generated nonnil submodule of M is Φ-invertible. If every nonnil submodule of M is Φ-invertible, then we say that M is a Φ-Dedekind module. Furthermore, M is said to be a Φ-Bezout module if Φ ⁢ ( N ) {\Phi(N)} is a principal ideal of Φ ⁢ ( M ) {\Phi(M)} for every finitely generated submodule N of the R-module M. The paper is devoted to the study of the properties of Φ-Prüfer, Φ-Dedekind and Φ-Bezout R-modules.

This paper is devoted to the analysis of the boundary value problem ∂ t ⁡ u - Δ ⁢ u = f {\partial_{t}u-\Delta u=f} , with an N-dimensional space variable, subject to a Dirichlet–Robin type boundary condition on the lateral boundary of the domain. The problem is settled in a noncylindrical domain of the form Q = { ( t , x 1 ) ∈ ℝ 2 : 0 < t < T , φ 1 ( t ) < x 1 < φ 2 ( t ) } × ∏ i = 1 N - 1 ] 0 , b i [ Q=\{(t,x_{1})\in\mathbb{R}^{2}:0 , where φ 1 {\varphi_{1}} and φ 2 {\varphi_{2}} are smooth functions. One of the main issues of the paper is that the domain can possibly be non-regular; for instance, the significant case when φ 1 ⁢ ( 0 ) = φ 2 ⁢ ( 0 ) {\varphi_{1}(0)=\varphi_{2}(0)} is allowed. We prove well-posedness results for the problem in a number of different settings and under natural assumptions on the coefficients and on the geometrical properties of the domain. This work is an extension of the one-dimensional case studied in [4].

We study piecewise synergetic systems originating from Biochemical Systems Theory. In the first part of the paper, the emphasis is put on practical calculations with such systems. We consider four examples: calculation of trajectories and steady states, solution of an optimization problem and a method of estimation of parameters (kinetic orders), all examples being biologically motivated. In the second part of the paper, we study convergence of solutions, in particularly, steady states, of a sequence of piecewise synergetic systems approximating an arbitrary compartment model. This convergence analysis is then applied to the optimization problem and the method of estimating sensitivities (kinetic orders) in a generic compartment model. In this paper we put forward arguments for the importance of the theoretical and numerical analysis of piecewise synergetic systems.

In this paper, we establish several results about the existence of second-order impulsive differential inclusion with periodic conditions. By using critical point theory, several new existence results are obtained. We also provide an example in order to illustrate the main abstract results of this paper.

In this paper, we introduce a split general quasi-variational inequality problem which is a natural extension of a split variational inequality problem, quasivariational and variational inequality problems in Hilbert spaces. Using the projection method, we propose an iterative algorithm for a split general quasi-variational inequality problem and discuss some special cases. Further, we discuss the convergence criteria of these iterative algorithms. The results presented in this paper generalize, unify and improve many previously known results for quasi-variational and variational inequality problems.

In this paper, we establish some new Ostrowski type inequalities for s logarithmically convex functions. Some applications of our results to P.D.F.’s and in numerical integration are given. 1. INTRODUCTION Let f : I [0;1] ! R be a di¤erentiable mapping on I , the interior of the interval I, such that f 0 2 L [a; b] where a; b 2 I with a < b. If jf 0 (x)j M , then the following inequality holds (see [2]): (1.1) f(x) 1 b a Z b a f(u)du M b a " (x a) + (b x) 2 # : This inequality is well known in the literature as the Ostrowski inequality. For some results which generalize, improve and extend the inequality (1.1) see ([2]-[6]) and the references therein. Let us recall some known de…nitions and results which we will use in this paper. The function f : I R ! [0;1) is said to be log convex or multiplicatively convex if log t is convex, or, equivalenly, if for all x; y 2 I and t 2 [0; 1] ; one has the inequality (See [7], p.7): f (tx+ (1 t) y) [f (x)] + [f (y)] t) : In [1], Akdemir and Tunc were introduced the class of s logarithmically convex functions in the …rst sense as the following: De…nition 1. A function f : I R0 ! R+ is said to be s logarithmically convex in the …rst sense if (1.2) f ( x+ y) [f (x)] s [f (y)] s for some s 2 (0; 1], where x; y 2 I and s + s = 1: In [8], authors introduced the class of s logarithmically convex functions in the second sense as the following: De…nition 2. A function f : I R0 ! R+ is said to be s logarithmically convex in the second sense if (1.3) f (tx+ (1 t) y) [f (x)] s [f (y)] (1 t) 2000 Mathematics Subject Classi…cation. 26D10, 26A15, 26A16, 26A51. Key words and phrases. Hadamard’s inequality, s geometrically convex functions . 1 2 MEVLUT TUNC AND AHMET OCAK AKDEMIR for some s 2 (0; 1], where x; y 2 I and t 2 [0; 1]. Clearly, when taking s = 1 in De…nition 1.2 or De…nition 1.3, then f becomes the standard logarithmically convex function on I. The main purpose of this paper is to establish some new Ostrowski’s type inequalities for s logarithmically convex functions. We also give some applications to P.D.F.’s and to midpoint formula. 2. THE NEW RESULTS In order to prove our main results, we will use following Lemma which was used by Alomari and Darus (see [3]): Lemma 1. Let f : I R ! R, be a di¤erentiable mapping on I where a; b 2 I, with a < b. Let f 0 2 L[a; b]; then the following equality holds

In the present paper, it is indicated that the proof of the main lemma is not valid, which relates to the inverse scattering problem for the perturbed Stark operator on the semiaxis. A correct proof of the mentioned lemma is given.

In the paper, similarity solutions are constructed for a model equation with multiple characteristics of an arbitrary integer order. It is shown that the structure of similarity solutions depends on the mutual simplicity of the orders of derivatives with respect to the variable x and y, respectively. Frequent cases are considered in which they are shown as fundamental solutions of well-known equations, expressed in a linear way through the constructed similarity solutions.

The question whether every operator on infinite-dimensional Hilbert space 𝐻 has a nontrivial invariant subspace or a nontrivial hyperinvariant subspace is one of the most difficult problems in operator theory. This problem is open for more than half a century. A subnormal operator has a nontrivial invariant subspace, but the existence of nontrivial invariant subspace for a hyponormal operator 𝑇 still open. In this paper we give an affirmative answer of the existence of a nontrivial hyperinvariant subspace for a hyponormal operator. More generally, we show that a large classes of operators containing the class of hyponormal operators have nontrivial hyperinvariant subspaces. Finally, every generalized scalar operator on a Banach space 𝑋 has a nontrivial invariant subspace.

This paper deals with the existence and multiplicity of solutions for the p ⁢ ( x ) p(x) -Laplacian Robin problem without the well-known Ambrosetti–Rabinowitz type growth conditions. By means of the variational approach (with the Cerami condition), existence and multiplicity results of solutions are established under weaker conditions.

In this paper, we find a necessary and sufficient condition for the modulus of continuity for which subsequences of Fejér means with respect to Vilenkin systems are bounded from the Hardy space H p {H_{p}} to the Lebesgue space L p {L_{p}} for all 0 < p < 1 2 {0 .

The main aim of this paper is to investigate the sequences of positive numbers, for which multiplication with Fourier coefficients of functions f ∈ Lip ⁡ 1 {f\in\operatorname{Lip}1} class provides absolute convergence of Fourier series. In particular, we found special conditions for the functions of orthonormal system (ONS), for which the above sequences are absolute convergence factors of Fourier series of functions of Lip ⁡ 1 {\operatorname{Lip}1} class. It is established that the resulting conditions are best possible in certain sense.

Goginava proved that the maximal operator σ α , * {\sigma^{\alpha,*}} ( 0 < α < 1 {0 ) of two-dimensional Marcinkiewicz type ( C , α ) {(C,\alpha)} means is bounded from the two-dimensional dyadic martingale Hardy space H p ⁢ ( G 2 ) {H_{p}(G^{2})} to the space L p ⁢ ( G 2 ) {L^{p}(G^{2})} for p > 2 2 + α {p>\frac{2}{2+\alpha}} . Moreover, he showed that assumption p > 2 2 + α {p>\frac{2}{2+\alpha}} is essential for the boundedness of the maximal operator σ α , * {\sigma^{\alpha,*}} . It was shown that at the point p 0 = 2 2 + α {p_{0}=\frac{2}{2+\alpha}} the maximal operator σ α , * {\sigma^{\alpha,*}} is bounded from the dyadic Hardy space H 2 / ( 2 + α ) ⁢ ( G 2 ) {H_{2/(2+\alpha)}(G^{2})} to the space weak- L 2 / ( 2 + α ) ⁢ ( G 2 ) {L^{2/(2+\alpha)}(G^{2})} . The main aim of this paper is to investigate the behaviour of the maximal operators of weighted Marcinkiewicz type σ α , * {\sigma^{\alpha,*}} means ( 0 < α < 1 {0 ) in the endpoint case p 0 = 2 2 + α {p_{0}=\frac{2}{2+\alpha}} . In particular, the optimal condition on the weights is given which provides the boundedness from H 2 / ( 2 + α ) ⁢ ( G 2 ) {H_{2/(2+\alpha)}(G^{2})} to L 2 / ( 2 + α ) ⁢ ( G 2 ) {L^{2/(2+\alpha)}(G^{2})} . Furthermore, a strong summation theorem is stated for functions in the dyadic martingale Hardy space H 2 / ( 2 + α ) ⁢ ( G 2 ) {H_{2/(2+\alpha)}(G^{2})} .

The present work deals with the mathematical investigation of some generalizations of Bessel functions. The main motive of this paper is to show that the generating function can be employed efficiently to obtain certain results for special functions. The complex form of Bessel functions is introduced by means of the generating function. Certain enthralling properties for complex Bessel functions are investigated using the generating function method. By considering separately the real and the imaginary part of complex Bessel functions, we get respectively cosine-Bessel functions and sine-Bessel functions for which several novel identities and Jacobi–Anger expansions are established. Also, the generating function of degenerate Bessel functions is investigated and certain novel identities are obtained for them. A hybrid form of degenerate Bessel functions, namely, of degenerate Fubini–Bessel functions, is constructed using the replacement technique. Finally, the explicit forms of the real and the imaginary part of complex Bessel functions are established by a hypergeometric approach.

Various extensions of the beta function together with their associated extended hypergeometric and confluent hypergeometric functions have been introduced and investigated. In this paper, using the very recently contrived extended beta function, we aim to introduce an extension F v p , q ; λ ; σ , τ u {{}_{u}F_{v}^{p,q;\lambda;\sigma,\tau}} of the generalized hypergeometric function F v u {{}_{u}F_{v}} and investigate certain classes of transforms and several identities of a generalized probability distribution involving this extension. In fact, we present some interesting formulas of Jacobi, Gegenbauer, pathway, Laplace, and Legendre transforms of this extension multiplied by a polynomial. We also introduce a generalized probability distribution to investigate its several related properties. Further, we consider some special cases of our main results with an argument about the derived process of a known result.

In this paper, we study the oscillation of even-order nonlinear neutral differential equations with both retarded and advanced arguments. The arithmetic-geometric mean inequality is employed. Examples are provided to illustrate the main results.

In this paper, we present a general Banach space of absolutely k-summable series using a triangle matrix operator and prove that this is a BK-space isometrically isomorphic to the space ℓ k {\ell_{k}} . We also establish the α - {\alpha-} , β - {\beta-} , γ-duals and base of the new space. Finally, we qualify some matrix and compact operators on the new space making use of the Hausdorff measure of noncompactness. Our results include, as particular cases, a number of well-known results.

In the present paper, the cyclic homology functor from the category of A ∞ {A_{\infty}} -algebras over any commutative unital ring K to the category of graded K-modules is constructed. Further, it is shown that this functor sends homotopy equivalences of A ∞ {A_{\infty}} -algebras into isomorphisms of graded modules. As a corollary, it is stated that the cyclic homology of an A ∞ {A_{\infty}} -algebra over any field is isomorphic to the cyclic homology of the A ∞ {A_{\infty}} -algebra of homologies for the source A ∞ {A_{\infty}} -algebra.

In this paper, we use some methods of combinatorial set theory, in particular, the ones related to the construction of independent families of sets and also some modified version of the notion of small sets originally introduced by Riečan and Neubrunn, to give an abstract and generalized formulation of a remarkable theorem of Kakutani and Oxtoby related to a nonseparable invariant extension of the Lebesgue measure in spaces with transformation groups.

In this paper, we give a characterization of a hypercyclic weighted composition operator on ℓ p {\ell^{p}} ( 1 ≤ p < ∞ {1\leq p ). We also obtain some necessary conditions and sufficient conditions for the supercyclicity and cyclicity of a weighted composition operator on ℓ p {{\ell}^{p}} ( 1 ≤ p < ∞ {1\leq p ).

In this paper, necessary, sufficient, and necessary and sufficient conditions are given for the smoothness of functions of two variables.

The purpose of this paper is to give a degree of approximation of a function in the space H p ω {H^{\omega}_{p}} with norm ∥ ⋅ ∥ p ω {\lVert\,\cdot\,\rVert^{\omega}_{p}} by using even-type delayed arithmetic mean of its Fourier series.

In this paper, we study a conformally flat 3-space 𝔽 3 {\mathbb{F}_{3}} which is an Euclidean 3-space with a conformally flat metric with the conformal factor 1 F 2 {\frac{1}{F^{2}}} , where F ⁢ ( x ) = e - x 1 2 - x 2 2 {F(x)=e^{-x_{1}^{2}-x_{2}^{2}}} for x = ( x 1 , x 2 , x 3 ) ∈ ℝ 3 {x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}} . In particular, we construct all helicoidal surfaces in 𝔽 3 {\mathbb{F}_{3}} by solving the second-order non-linear ODE with extrinsic curvature and mean curvature functions. As a result, we give classification of minimal helicoidal surfaces as well as examples for helicoidal surfaces with some extrinsic curvature and mean curvature functions in 𝔽 3 {\mathbb{F}_{3}} .

The basic problem of satisfaction of boundary conditions is considered when the generalized stress vector is given on the surfaces of elastic plates and shells. This problem has so far remained open for both refined theories in a wide sense and hierarchic type models. In the linear case, it was formulated by I. N. Vekua for hierarchic models. In the nonlinear case, bending and compression-expansion processes do not split and in this context the exact structure is presented for the system of differential equations of von Kármán–Mindlin–Reisner (KMR) type, constructed without using a variety of ad hoc assumptions since one of the two relations of this system in the classical form is the compatibility condition, but not the equilibrium equation. In this paper, a unity mathematical theory is elaborated in both linear and nonlinear cases for anisotropic inhomogeneous elastic thin-walled structures. The theory approximately satisfies the corresponding system of partial differential equations and the boundary conditions on the surfaces of such structures. The problem is investigated and solved for hierarchic models too. The obtained results broaden the sphere of applications of complex analysis methods. The classical theory of finding a general solution of partial differential equations of complex analysis, which in the linear case was thoroughly developed in the works of Goursat, Weyl, Walsh, Bergman, Kolosov, Muskhelishvili, Bers, Vekua and others, is extended to the solution of basic nonlinear differential equations containing the nonlinear summand, which is a composition of Laplace and Monge–Ampére operators.

In this paper, by the Faà di Bruno formula and properties of Bell polynomials of the second kind, the authors reconsider the generating functions of Hermite polynomials and their squares, find an explicit formula for higher-order derivatives of the generating function of Hermite polynomials, and derive explicit formulas and recurrence relations for Hermite polynomials and their squares.

In this paper, we define consistent criteria for hypotheses testing, such that the probability of any kind of error is zero for a given criterion. The necessary and sufficient conditions for the existence of such criteria are given.

This paper is concerned with periodic boundary value problems involving first order differential equations with functional arguments. The main feature of the paper is that the existence of maximal and minimal solutions is obtained by constructing sequences of upper and lower solutions of the initial value problems and not by establishing the comparison principle.

The purpose of this paper is to investigate the behavior of prime rings involving skew derivations with m-potent commutators on Lie ideals. In addition, we provide an example that shows that we cannot expect the same conclusion in case of semiprime rings. Also, we prove some other related results and present some open problems.

We give a characterization of all those groups of isometric transformations of a Þnite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem (1) holds true. This charac- terization is formulated in purely geometrical terms. A well-known result due to Vitali (1) states that there are subsets of the real line R, nonmeasurable in the Lebesgue sense. Moreover, the argument of Vitali yields simultaneously that if X is an arbitrary Lebesgue measurable subset of R with strictly positive measure, then there exists a subset of X nonmeasurable in the Lebesgue sense. This classical result was generalized in many directions (see, e.g., (2)-(12)). In the present paper, we consider some questions relevant to the above-mentioned result of Vitali, for various groups of motions of Þnite-dimensional Euclidean spaces. The main goal of the paper is to describe those groups of motions for which an analogue of Vitali's result remains true (we recall that, in Vitali's theorem, a basic group of transformations is the group of all translations of the real line). Let E denote a Þnite-dimensional Euclidean space and let G be a sub- group of the group of all isometric transformations (i.e., motions) of E. Then the pair (E;G) can be regarded as a space equipped with a trans- formation group. We denote by K the open unit cube in E. Let n be a measure given on E and let dom(n) denote the domain of n. We shall say that n is a G-measure on E if the following two conditions are fulÞlled: (1) K 2 dom(n); n(K) = 1; (2) n is invariant with respect to G, i.e., dom(n) is a G-invariant o-ring of subsets of E and, for each set X 2 dom(n) and for each transformation g 2 G, we have n(g(X)) = n(X).

The paper deals with the contact problems of the theory of elasticity. The problems are reduced to Prandtl-type integral dif- ferential equations with a coecient at the singular operator which has higher-order zeros at the ends of the integration interval. In some concrete cases the solution is constructed eciently. Asymptotic rep- resentations are obtained. Investigation of the problem of stress concentration when contact be- tween dierent elastic media takes place remains one of the most topical tasks. Inclusions such as punches and cuts are stress concentrators and hence the study of the insuence of inclusions on the stress-strained state of solid deformable bodies, as well as the elaboration of methods allowing one to reduce stress concentration, are of great theoretical and practical importance. The problems of contact interaction between rigid inclusions of various geometrical forms and elastic bodies are considered in (1-5). In (6) some problems of contact interaction between a piecewise-homogeneous plane and a rigid inclusion, and also the anti-plane problem of an elastic half-space and problems of bending of plates with thin-shelled inclusions, are reduced to integral equations by means of the generalized method of integral transforms. As distinct from the previous papers, this paper deals with the problems of contact interaction between a thin elastic inclusion of varying rigidity and (a) an elastic half-space which is in the state of anti-plane deformation, and (b) an elastic piecewise-homogeneous plane.

In the paper, some properties of functions continuous with respect to a density type strong generalized topology are presented. In particular, it is proved that each real function is approximately continuous with respect to this generalized topology almost everywhere. Moreover, some separation axioms for this generalized topological space are investigated.

In this paper, for a discrete hypergroup K and its subhypergroup H, we initiate the related Hecke * {*} -algebra which is an extension of the classical one, and study its basic properties. Especially, we give a necessary and sufficient condition (named (β)) for this algebra to be associative. Also, we show that this new structure is an associative * {*} -algebra if and only if K is a locally compact group.

In this paper, we introduce an interesting kind of convergence for a double sequence called the uniform convergence at a point. We give an example and demonstrate a Korovkin-type approximation theorem for a double sequence of functions using the uniform convergence at a point. Then we show that our result is stronger than the Korovkin theorem given by Volkov and present several graphs. Finally, in the last section, we compute the rate of convergence.

The quantum groups nowadays attract a considerable interest of mathematicians and physicists. The theory of q-special functions has received a group-theoretic interpretation using the techniques of quantum groups and quantum algebras. This paper focuses on introducing the q-Tricomi functions and 2D q-Tricomi functions through the generating function and series expansion and for the first time establishing a connecting relation between the q-Tricomi and q-Bessel functions. The behavior of these functions is described through shapes, and the contrast between them is observed using mathematical software. Further, the problem of framing the q-Tricomi and 2D q-Tricomi functions in the context of the irreducible representation ( ω ) {(\omega)} of the two-dimensional quantum algebra ℰ q ⁢ ( 2 ) {\mathcal{E}_{q}(2)} is addressed, and certain relations involving these functions are obtained. 2-Variable 1-parameter q-Tricomi functions and their relationship with the 2-variable 1-parameter q-Bessel functions are also explored.

Sucient conditions are found for the bifurcation of sow of a viscous heat-conducting suid between two rotating permeable cylinders. This paper deals with second stationary sows generated in a heat-con- ducting suid contained between two permeable cylinders rotating in the same direction. Among other papers where similar problems are treated mention should be made of (1-4) for the case of a noncompressible suid and (5) for the case of a heat-conducting suid. The permeability of the cylinders changes the character of the obtained operator equations, which results in nonsymmetricity of the kernels of the corresponding integral equations. This fact necessitates to another method of investigation of this problem and this is what we do here. 1. Let a homogeneous viscous heat-conducting suid Þll up the hollow space between two rotating permeable cylinders heated up to dierent tem- peratures. The radii, angular velocities, and temperatures of the internal and outer cylinders are denoted by R1, S1, i1 and R2, S2, i2, respectively. It is assumed that there are no external mass forces, the velocity of the sow across the cross-section of the hollow space between the cylinders is zero, and the suid insow through one cylinder is equal to the suid outsow through the other. The scales of length, velocity, and temperature will be denoted by R1, S1R1, i1, while the density scale will be understood as the suid density at the temperature i1. Under these assumptions, if we write the Navier-Stokes equations and heat conductivity equation in terms of cylindrical coordinates (r,',z) with the axis z coinciding with the axis

The paper deals with the exploration of those subclasses of the variable exponent Lebesgue space L p ⁢ ( ⋅ ) {L^{p(\,\cdot\,)}} with min ⁡ p ⁢ ( ⋅ ) = 1 {\min p(\,\cdot\,)=1} , which are invariant with respect to Cauchy singular integral operators.

Let ℛ {\mathscr{R}} be a prime ring, 𝒬 r {\mathscr{Q}_{r}} the right Martindale quotient ring of ℛ {\mathscr{R}} and 𝒞 {\mathscr{C}} the extended centroid of ℛ {\mathscr{R}} . In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e., ( [ δ ⁢ ( f ⁢ ( x 1 , … , x n ) ) , f ⁢ ( x 1 , … , x n ) ] ) m = [ δ ⁢ ( f ⁢ ( x 1 , … , x n ) ) , f ⁢ ( x 1 , … , x n ) ] , \big{(}[\delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})]\big{)}^{m}=[% \delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})], where 1 < m ∈ ℤ + {1 , f ⁢ ( x 1 , x 2 , … , x n ) {f(x_{1},x_{2},\ldots,x_{n})} is a non-central multilinear polynomial over 𝒞 {\mathscr{C}} and δ is a skew derivation of ℛ {\mathscr{R}} .

Let ℜ {\mathfrak{R}} be a ring with center Z ⁢ ( ℜ ) {Z(\mathfrak{R})} . In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation 𝔉 {\mathfrak{F}} such that ( [ 𝔉 ⁢ ( s m ) ⁢ s n + s n ⁢ 𝔉 ⁢ ( s m ) , s r ] k ) l ∈ Z ⁢ ( ℜ ) {([\mathfrak{F}(s^{m})s^{n}+s^{n}\mathfrak{F}(s^{m}),s^{r}]_{k})^{l}\in Z(% \mathfrak{R})} for every s ∈ ℜ {s\in\mathfrak{R}} , then either 𝔉 ⁢ ( s ) = p ⁢ s {\mathfrak{F}(s)=ps} for every s ∈ ℜ {s\in\mathfrak{R}} or ℜ {\mathfrak{R}} satisfies s 4 {s_{4}} and 𝔉 ⁢ ( s ) = s ⁢ p {\mathfrak{F}(s)=sp} for every s ∈ ℜ {s\in\mathfrak{R}} and p ∈ 𝔘 {p\in\mathfrak{U}} , the Utumi quotient ring of ℜ {\mathfrak{R}} . As an application, we prove that any spectrally generalized derivation on a semisimple Banach algebra satisfying the above mentioned differential identity must be a left multiplication map.

In this paper, we consider a natural extension in the context of d-orthogonality for asymptotic analysis of orthogonal polynomials. We introduce, for several d-orthogonal polynomials, asymptotic expansions in terms of d-Hermite ones. From these expansions, several limits between d-orthogonal polynomials are obtained.

In this paper, we investigate the strong summability of two-dimensional Walsh–Fourier series obtained in [F. Weisz, Strong convergence theorems for two-parameter Walsh–Fourier and trigonometric-Fourier series, Studia Math. 117 1996, 2, 173–194] (see Theorem W) and prove the sharpness of this result.

In this paper, we establish weighted higher order exponential type inequalities in the geodesic space ( X , d , μ ) {({X,d,\mu})} by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given.

In this paper, we find the condition on a function ω and a weight v which ensures the equivalency of norms of the Riesz potential and the fractional maximal function in generalized weighted Morrey spaces ℳ p , ω ⁢ ( ℝ n , v ) {{\mathcal{M}}_{p,\omega}({\mathbb{R}}^{n},v)} and generalized weighted central Morrey spaces ℳ ˙ p , ω ⁢ ( ℝ n , v ) {\dot{\mathcal{M}}_{p,\omega}({\mathbb{R}}^{n},v)} , when v belongs to the Muckenhoupt A ∞ {A_{\infty}} -class.