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.

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].

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.

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 ).

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.

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.

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, 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.

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.

In this paper we study the boundedness of localization operators associated with the Stockwell transform with symbol in L p {L^{p}} acting on the Wiener amalgam space W ⁢ ( L p , L q ) ⁢ ( ℝ ) {W(L^{p},L^{q})(\mathbb{R})} .

The purpose of this paper is to study the problems of meromorphic functions sharing unique range sets with one or two elements, which can be seen as a new answer to a question due to Gross. The examples also illustrate the accuracy of all of the conditions.

In this paper, we investigate the absolute convergence of Fourier series of functions in several variables for an odd-dimensional space when these functions have continuous partial derivatives. It should be noted that similar properties for an even-dimensional space were given in [L. D. Gogoladze and V. S. Tsagareishvili, On absolute convergence of multiple Fourier series (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 2015, 9, 12–21; translation in Russian Math. (Iz. VUZ) 59 (2015), no. 9, 9–17]. The obtained results are the best possible in a certain sense.

In this paper we first classify left-invariant generalized Ricci solitons on four-dimensional hypercomplex Lie groups equipped with three families of left-invariant Lorentzian metrics. Then, on these Lorentzian spaces, we explicitly calculate the energy of an arbitrary left-invariant vector field X and determine the exact form of all left-invariant unit time-like vector fields which are spatially harmonic. Furthermore, we give a complete and explicit description of all homogeneous structures on these spaces in both Riemannian and Lorentzian cases and determine some of their types. The existence of Einstein four-dimensional hypercomplex Lorentzian Lie groups is proved and it is shown that although the results concerning Einstein-like metrics, conformally flatness and some equations in the Riemannian case are much richer than their Lorentzian analogues, in the Lorentzian case, there exist some new critical points of energy functionals, homogeneous structures and geodesic vectors which do not exist in the Riemannian case.

In Section 1 of the paper, we prove McCoy's property for the zero-divisors of polynomials in semirings. We also investigate zero-divisors of semimodules and prove that under suitable conditions, the monoid semimodule $M[G]$ has very few zero-divisors if and only if the $S$-semimodule $M$ does so. The concept of Auslander semimodules are introduced in this section as well. In Section 2, we introduce Ohm-Rush and McCoy semialgebras and prove some interesting results for prime ideals of monoid semirings. In Section 3, we investigate the set of zero-divisors of McCoy semialgebras. We also introduce strong Krull primes for semirings and investigate their extension in semialgebras., 20 pages, totally revised and extended. A couple of new results added

The paper is devoted to the construction and study of the additive averaged semi-discrete scheme for one nonlinear multi-dimensional integro-differential equation of parabolic type. The studied equation is based on the well-known Maxwell’s system arising in mathematical simulation of electromagnetic field penetration into a substance.

In this paper, by means of the generalized Olsen type inequality, related to the Riesz potential and its commutators, the author establishes the interior estimates on the generalized Morrey space for Schrödinger type elliptic equations with potentials satisfying the reverse Hölder condition.

We consider explicit sufficient conditions for all solutions of a first-order linear difference equation with several variable delays and non-negative coefficients to be oscillatory. The conditions have the form of inequalities bounding below the upper and lower limits of the sums of coefficients over a subset of the discrete semiaxis. Our main results are oscillation tests based on a new principle for composing the estimated sums of coefficients. We also give some results in the form of examples, including a counterexample to a wrong oscillation test cited in several recent papers.

For certain classes of functions of Λ-bounded variation on [ - π , π ] {[-\pi,\pi]} , Waterman and D’Antonio constructed a sequence of summability kernels such that the corresponding integral means converge for any function from the class, but not for any function from a broader Λ ′ ⁢ BV {\Lambda^{\prime}\mathrm{BV}} class. In our paper, we extend this result to the case of functions of two variables.

Let ℛ {\mathcal{R}} be a prime ring with center Z ⁢ ( ℛ ) {Z(\mathcal{R})} and * {*} an involution of ℛ {\mathcal{R}} . Suppose that ℛ {\mathcal{R}} admits generalized derivations F, G and H associated with a nonzero derivation f, g and h of ℛ {\mathcal{R}} , respectively. In the present paper, we investigate the commutativity of a prime ring ℛ {\mathcal{R}} satisfying any of the following identities: (i) [ F ⁢ ( x ) , F ⁢ ( x * ) ] = 0 [F(x),F(x^{*})]=\nobreak 0 , (ii) [ F ⁢ ( x ) , F ⁢ ( x * ) ] = ± [ x , x * ] [F(x),F(x^{*})]=\pm[x,x^{*}] , (iii) F ⁢ ( x ) ∘ F ⁢ ( x * ) = 0 F(x)\circ\nobreak F(x^{*})=0 , (iv) F ⁢ ( x ) ∘ F ⁢ ( x * ) = ± ( x ∘ x * ) F(x)\circ\nobreak F(x^{*})=\pm(x\circ\nobreak x^{*}) , (v) [ F ⁢ ( x ) , x * ] ± [ x , G ⁢ ( x * ) ] = 0 [F(x),x^{*}]\pm[x,G(x^{*})]=0 , (vi) F ⁢ ( x ⁢ x * ) ∈ Z ⁢ ( ℛ ) F(xx^{*})\in Z(\mathcal{R}) , (vii) F ⁢ ( x ) ⁢ G ⁢ ( x * ) ± H ⁢ ( x ) ⁢ x * ∈ Z ⁢ ( ℛ ) F(x)G(x^{*})\pm H(x)x^{*}\in Z(\mathcal{R}) , (viii) F ⁢ ( [ x , x * ] ) ± [ x , x * ] ∈ Z ⁢ ( ℛ ) F([x,x^{*}])\pm[x,x^{*}]\in Z(\mathcal{R}) , (ix) F ⁢ ( x ∘ x * ) ± x ∘ x * ∈ Z ⁢ ( ℛ ) F(x\circ\nobreak x^{*})\pm x\circ x^{*}\in Z(\mathcal{R}) , (x) [ F ⁢ ( x ) , x * ] ± [ x , G ⁢ ( x * ) ] ∈ Z ⁢ ( ℛ ) [F(x),x^{*}]\pm[x,G(x^{*})]\in Z(\mathcal{R}) , (xi) F ⁢ ( x ) ∘ x * ± x ∘ G ⁢ ( x * ) ∈ Z ⁢ ( ℛ ) F(x)\circ\nobreak x^{*}\pm x\circ\nobreak G(x^{*})\in Z(\mathcal{R}) for all x ∈ ℛ {x\in\mathcal{R}} . Finally, the restrictions imposed on the hypotheses have been justified by an example.

The well-known Gitik–Shelah theorem (1989) touches the problem of existence isomorphisms between some quotient algebras. In this paper, we study a relation between the existence of such isomorphisms and the existence of so-called Kuratowski partitions of adequate Baire spaces. For this purpose, we give strictly combinatorial methods.

In this paper, we study the question of the existence of Kuratowski partitions in tree structures and Ellentuck topology. On the examples of Marczewski and Laver structures and the structure of Ellentuck topology, we show that non-small sets of these structures do not admit Kuratowski partitions.

In the first part of the paper we study the sets of boundedness and of convergence and divergence to infinity of sequences of real closed-graph functions. Generalization on ideal convergence of such sequences is discussed. Limits and ideal-limits of sequences of functions with closed graphs are considered in the last part of the article.

In this paper, theorems are stated regarding geometrical realizations of finite families of sets, for a given countable family of sets the existence of families of point sets combinatorially strictly ω-isomorphic to this given family of sets, independent families of triangles in the Euclidean plane, also, the existence and extensions of ρ-at-sets, ρ-rt-sets, ρ-ot-sets and ot-sets. These theorems are a continuation of the corresponding results of [6, 7, 8, 5, 3, 4].

The paper deals with two ideals of subsets of ℕ {\mathbb{N}} defined with the help of density points and its category analogue. We present basic properties of these ideals and show that they are incomparable under inclusion.

In this paper we study (non-Abelian) extensions of a given hom-Lie color algebra and provide a geometrical interpretation of extensions. In particular, we characterize an extension of a hom-Lie algebra $\mathfrak{g}$ by another hom-Lie algebra $\mathfrak{h}$ and we discuss the case where $\mathfrak{h}$ has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, curvature and the Bianchi identity for the possible extensions in differential geometry. Moreover, we find a cohomological obstruction to the existence of extensions of hom-Lie color algebras, i. e. we show that in order to have an extendible hom-Lie color algebra, there should exist a trivial member of the third cohomology., Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv:1709.06164

In this paper, we define accessibility on an iterated function system (IFS) and show that it provides a sufficient condition for the transitivity of this system and its corresponding skew product. Then, by means of a certain tool, we obtain the topologically mixing property. We also give some results about the ergodicity and stability of accessibility and, further, illustrate accessibility by some examples.

In this paper, we show some refinements of generalized numerical radius inequalities involving the Young and Heinz inequality. In particular, we present w p p ⁢ ( A 1 * ⁢ T 1 ⁢ B 1 , … , A n * ⁢ T n ⁢ B n ) ≤ n 1 - 1 r 2 1 r ⁢ ∥ ∑ i = 1 n [ B i * ⁢ f 2 ⁢ ( | T i | ) ⁢ B i ] r ⁢ p + [ A i * ⁢ g 2 ⁢ ( | T i * | ) ⁢ A i ] r ⁢ p ∥ 1 r - inf ∥ x ∥ = 1 ⁡ η ⁢ ( x ) , w_{p}^{p}(A_{1}^{*}T_{1}B_{1},\dots,A_{n}^{*}T_{n}B_{n})\leq\frac{n^{1-\frac{1% }{r}}}{2^{\frac{1}{r}}}\bigg{\|}\sum_{i=1}^{n}[B_{i}^{*}f^{2}(|T_{i}|)B_{i}]^{% rp}+[A_{i}^{*}g^{2}(|T_{i}^{*}|)A_{i}]^{rp}\bigg{\|}^{\frac{1}{r}}-\inf_{\|x\|% =1}\eta(x), where T i , A i , B i ∈ 𝔹 ⁢ ( ℋ ) {T_{i},A_{i},B_{i}\in\mathbb{B}(\mathscr{H})} ( 1 ≤ i ≤ n ) {(1\leq i\leq n)} , f and g are nonnegative continuous functions on [ 0 , ∞ ) {[0,\infty)} satisfying f ⁢ ( t ) ⁢ g ⁢ ( t ) = t {f(t)g(t)=t} for all t ∈ [ 0 , ∞ ) {t\in[0,\infty)} , p , r ≥ 1 {p,r\geq 1} , N ∈ ℕ {N\in\mathbb{N}} , and η ( x ) = 1 2 ∑ i = 1 n ∑ j = 1 N ( 〈 ( A i * ⁢ g 2 ⁢ ( | T i * | ) ⁢ A i ) p ⁢ x , x 〉 2 j - 1 - k j ⁢ 〈 ( B i * ⁢ f 2 ⁢ ( | T i | ) ⁢ B i ) p ⁢ x , x 〉 k j 2 j \displaystyle\eta(x)=\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{N}\Bigl{(}\sqrt[2^{j% }]{\big{\langle}(A_{i}^{*}g^{2}(|T_{i}^{*}|)A_{i})^{p}x,x\big{\rangle}^{2^{j-1% }-k_{j}}\big{\langle}(B_{i}^{*}f^{2}(|T_{i}|)B_{i})^{p}x,x\big{\rangle}^{k_{j}}} - 〈 ( B i * ⁢ f 2 ⁢ ( | T i | ) ⁢ B i ) p ⁢ x , x 〉 k j + 1 ⁢ 〈 ( A i * ⁢ g 2 ⁢ ( | T i * | ) ⁢ A i ) p ⁢ x , x 〉 2 j - 1 - k j - 1 2 j ) 2 . \displaystyle -\sqrt[2^{j}]{\big{\langle}(B_{i}^{*}f^{2}(|T_{i}|)B_{i}% )^{p}x,x\big{\rangle}^{k_{j}+1}\big{\langle}(A_{i}^{*}g^{2}(|T_{i}^{*}|)A_{i})% ^{p}x,x\big{\rangle}^{2^{j-1}-k_{j}-1}}\,\Big{)}^{2}.

Let 𝒦 {\mathcal{K}} be an abelian category that has enough injective objects, let T : 𝒦 → A {T\colon\mathcal{K}\to A} be any left exact covariant additive functor to an abelian category A and let T ( i ) {T^{(i)}} be a right derived functor, u ≥ 1 {u\geq 1} , [S. Mardešić, Strong Shape and Homology, Springer Monogr. Math., Springer, Berlin, 2000]. If T ( i ) = 0 {T^{(i)}=0} for i ≥ 2 {i\geq 2} and T ( i ) ⁢ C n = 0 {T^{(i)}C_{n}=0} for all n ∈ ℤ {n\in\mathbb{Z}} , then there is an exact sequence 0 ⟶ T ( 1 ) ⁢ H n + 1 ⁢ ( C * ) ⟶ H n ⁢ ( T ⁢ C * ) ⟶ T ⁢ H n ⁢ ( C * ) ⟶ 0 , 0\longrightarrow T^{(1)}H_{n+1}(C_{*})\longrightarrow H_{n}(TC_{*})% \longrightarrow TH_{n}(C_{*})\longrightarrow 0, where C * = { C n } {C_{*}=\{C_{n}\}} is a chain complex in the category 𝒦 {\mathcal{K}} , H n ⁢ ( C * ) {H_{n}(C_{*})} is the homology of the chain complex C * {C_{*}} , T ⁢ C * {TC_{*}} is a chain complex in the category A, and H n ⁢ ( T ⁢ C * ) {H_{n}(TC_{*})} is the homology of the chain complex T ⁢ C * {TC_{*}} . This exact sequence is the well known Künneth’s correlation. In the present paper Künneth’s correlation is generalized. Namely, the conditions are found under which the infinite exact sequence ⋯ ⟶ T ( 2 ⁢ i + 1 ) ⁢ H n + i + 1 ⟶ ⋯ ⟶ T ( 3 ) ⁢ H n + 2 ⟶ T ( 1 ) ⁢ H n + 1 ⟶ H n ⁢ ( T ⁢ C * ) \displaystyle\cdots\longrightarrow T^{(2i+1)}H_{n+i+1}\longrightarrow\cdots% \longrightarrow T^{(3)}H_{n+2}\longrightarrow T^{(1)}H_{n+1}\longrightarrow H_% {n}(TC_{*}) ⟶ T ⁢ H n ⁢ ( C * ) ⟶ T ( 2 ) ⁢ H n + 1 ⟶ T ( 4 ) ⁢ H n + 2 ⟶ ⋯ ⟶ T ( 2 ⁢ i ) ⁢ H n + i ⟶ ⋯ \displaystyle\longrightarrow TH_{n}(C_{*})\longrightarrow T^{(2)}H_{n+1}% \longrightarrow T^{(4)}H_{n+2}\longrightarrow\cdots\longrightarrow T^{(2i)}H_{% n+i}\longrightarrow\cdots holds, where T ( 2 ⁢ i + 1 ) ⁢ H n + i + 1 = T ( 2 ⁢ i + 1 ) ⁢ H n + i + 1 ⁢ ( C * ) {T^{(2i+1)}H_{n+i+1}=T^{(2i+1)}H_{n+i+1}(C_{*})} , T ( 2 ⁢ i ) ⁢ H n + i = T ( 2 ⁢ i ) ⁢ H n + i ⁢ ( C * ) {T^{(2i)}H_{n+i}=T^{(2i)}H_{n+i}(C_{*})} . The formula makes it possible to generalize Milnor’s formula for the cohomologies of an arbitrary complex, relatively to the Kolmogorov homology to the Alexandroff–Čech homology for a compact space, to a generative result of Massey for a local compact Hausdorff space X and a direct system { U } {\{U\}} of open subsets U of X such that U ¯ {\overline{U}} is a compact subset of X.

The purpose of the present paper is to obtain the degree of approximation in terms of a Lipschitz type maximal function for the Kantorovich type modification of Jakimovski–Leviatan operators based on multiple Appell polynomials. Also, we study the rate of approximation of these operators in a weighted space of polynomial growth and for functions having a derivative of bounded variation. A Voronvskaja type theorem is obtained. Further, we illustrate the convergence of these operators for certain functions through tables and figures using the Maple algorithm and, by a numerical example, we show that our Kantorovich type operator involving multiple Appell polynomials yields a better rate of convergence than the Durrmeyer type Jakimovski Leviatan operators based on Appell polynomials introduced by Karaisa (2016).