Volume 28, Issue 6

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.

This paper is devoted to study the existence and uniqueness of solutions of a one parameter family of nonlinear Riemann–Liouville fractional differential equations with mixed boundary value conditions. An exhaustive study of the sign of the related Green’s function is carried out. Under suitable assumptions on the asymptotic behavior of the nonlinear part of the equation at zero and at infinity, and by application of the fixed point theory of compact operators defined in suitable cones, it is proved that there exists at least one solution of the considered problem. Moreover, the method of lower and upper solutions is developed and the existence of solutions is deduced by a combination of both techniques. In particular cases, the Banach contraction principle is used to ensure the uniqueness of solutions.

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

Let R and S be arbitrary rings and let C S R {{}_{R}C_{S}} be a semidualizing bimodule, and let 𝒜 C ⁢ ( R op ) {\mathcal{A}_{C}(R^{\mathrm{op}})} and ℬ C ⁢ ( R ) {\mathcal{B}_{C}(R)} be the Auslander and Bass classes, respectively. Then both pairs ( 𝒜 C ⁢ ( R op ) , ℬ C ⁢ ( R ) )   and   ( ℬ C ⁢ ( R ) , 𝒜 C ⁢ ( R op ) ) (\mathcal{A}_{C}(R^{\mathrm{op}}),\mathcal{B}_{C}(R))\quad\text{and}\quad(% \mathcal{B}_{C}(R),\mathcal{A}_{C}(R^{\mathrm{op}})) are coproduct-closed and product-closed duality pairs and both 𝒜 C ⁢ ( R op ) {\mathcal{A}_{C}(R^{\mathrm{op}})} and ℬ C ⁢ ( R ) {\mathcal{B}_{C}(R)} are covering and preenveloping; in particular, the former duality pair is perfect. Moreover, if ℬ C ⁢ ( R ) {\mathcal{B}_{C}(R)} is enveloping in Mod ⁡ R {\operatorname{Mod}R} , then 𝒜 C ⁢ ( S ) {\mathcal{A}_{C}(S)} is enveloping in Mod ⁡ S {\operatorname{Mod}S} . Also, some applications to the Auslander projective dimension of modules are given.

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 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.

The asymptotic behavior of extreme values of queueing systems is studied. For a system M / M / m {{M/M/m}} , 1 ≤ m < ∞ {1\leq m<\infty} , the weak convergence of extreme values of an actual waiting time and a statement of the type of the law of the iterated logarithm are established.

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.

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 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 this paper, we study the parameterized complexity of the induced matching problem in hamiltonian bipartite graphs and the inapproximability of the maximum induced matching problem in hamiltonian bipartite graphs. We show that, given a hamiltonian bipartite graph, the induced matching problem is W[1]-hard and cannot be solved in time n o ⁢ ( k ) {n^{o(\sqrt{k})}} , where n is the number of vertices in the graph, unless the 3SAT problem can be solved in subexponential time. In addition, we show that unless NP = P {\operatorname{NP}=\operatorname{P}} , a maximum induced matching in a hamiltonian bipartite graph cannot be approximated within a ratio of n 1 / 4 - ϵ {n^{1/4-\epsilon}} , where n is the number of vertices in the graph.

In this paper, we give a characterization of a hypercyclic weighted composition operator on ℓ p {\ell^{p}} ( 1 ≤ p < ∞ {1\leq p<\infty} ). 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<\infty} ).

Starting from n roots of a univariate polynomial of n -th degree, combining the first, the second, and up to the n -th class, new polynomials are generated. Each class of polynomials obtained in such a way is fully determined by its roots. The arithmetic means of corresponding coefficients are proportional to the values of the coefficients obtained by the derivation of the initial univariate polynomial. This gives the basis for the algebraic approach to the polynomial derivation.

We introduce interpolation sets for the Zygmund class 𝒵 {\mathcal{Z}} in the unit disc of the complex plane. This space lies between the Lipschitz classes of order α, 0 < α < 1 {0<\alpha<1} , and the class of order α = 1 {\alpha=1} , whose interpolation sets are given in a different way. We prove that the interpolation sets for 𝒵 {\mathcal{Z}} are interpolation sets for the Lipschitz classes of order α, 0 < α < 1 {0<\alpha<1} , and the latter are interpolation sets for a space slightly larger than 𝒵 {\mathcal{Z}} .