• Volume 128, Issue 3

June 2018

• Some polynomials associated with the $r$-Whitney numbers

In the present article, we study three families of polynomials associated with the $r$-Whitney numbers of the second kind. They are the $r$-Dowling polynomials, $r$-Whitney–Fubini polynomials and the $r$-Eulerian–Fubini polynomials. Then we derive several combinatorial results by using algebraic arguments (Rota’s method), combinatorial arguments (set partitions) and asymptotic methods.

• Arithmetical Fourier and limit values of elliptic modular functions

Here, we shall use the first periodic Bernoulli polynomial $\bar{B}_{1}(x) = x-[x]-\frac{1}{2}$ to resurrect the instinctive direction of B Riemann in his posthumous fragment II on the limit values of elliptic modular functions à la C G J Jacobi, Fundamenta Nova $\S$40 (1829). In the spirit of Riemann who considered the odd part, we use a general Dirichlet–Abel theorem to condense Arias–de-Reyna’s theorems 8–15 into ‘a bigger theorem’ in Sect. 2 by choosing a suitable $R$-function in taking the radial limits. Wesupplement Wang (Ramanujan J. 24 (2011) 129–145). Furthermore, the same method is applied to obtain in Sect. 3 a correct representation for the ‘trigonometric series’, i.e., we prove that for every rational number $x$ the trigonometric series (3.5) is represented by $\sum^{\infty}_{n=1}(-1)^{n}\frac{\bar{B}_{1}(nx)}{n}$ as Dedekind suggested but not by $\sum^{\infty}_{n=1}\frac{\bar{B}_{1}(nx)}{n}$ as Riemann stated.

• Transcendence of some power series for Liouville number arguments

In this paper, we prove that some power series with rational coefficients take either values of rational numbers or transcendental numbers for the arguments from the set of Liouville numbers under certain conditions in the field of complex numbers. We then apply this result to an algebraic number field. In addition, we establish the $p$-adic analogues of these results and show that these results have analogues in the field of $p$-adic numbers.

• Subspace Lang conjecture and some remarks on a transcendental criterion

Let $b \geq 2$ be an integer and $\alpha$ is a non-zero real number written in $b$-ary expansion. Adamczewski et al. (C. R. Acad. Sci. Paris 339 (2004) 11–14) provided a criterion for an irrational number to be a transcendental number using $b$-ary expansion. In this paper, we make some remarks on this criterion and, under the assumption of Subspace Lang Conjecture, we extend this criterion for a much wider class of irrational numbers.

• $z$-Classes in finite groups of conjugate type ($n$, 1)

Two elements in a group $G$ are said to be $z$-equivalent or to be in the same $z$-class if their centralizers are conjugate in $G$. In a recent work, Kulkarni et al. (J. Algebra Appl., 15 (2016) 1650131) proved that a non-abelian $p$-group $G$ can have at most $\frac{p^{k}−1}{p−1} + 1$ number of $z$-classes, where $|G/Z(G)| = p^{k}$ . Here, we characterize the $p$-groups of conjugate type ($n$, 1) attaining this maximal number. As a corollary, we characterize $p$-groups having prime order commutator subgroup and maximal number of $z$-classes.

• Uniformly locally univalent harmonic mappings

The primary aim of this paper is to characterize the uniformly locally univalent harmonic mappings in the unit disk. Then, we obtain sharp distortion, growth and covering theorems for one parameter family $\mathcal{B}_{H}(\lambda)$ of uniformly locally univalent harmonic mappings. Finally, we show that the subclass of $k$-quasiconformal harmonic mappings in $\mathcal{B}_{H}(\lambda)$ and the class $\mathcal{B}_{H}(\lambda)$ are contained in the Hardy space of a specific exponent depending on $\lambda$, respectively, and we also discuss the growth of coefficients for harmonic mappings in $\mathcal{B}_{H}(\lambda)$.

• Analytic sets and extension of holomorphic maps of positive codimension

Let $D$, $D'$ be arbitrary domains in $\mathbb{C}^{n}$ and $\mathbb{C}^{N}$ respectively, 1 < $n \leq N$, both possibly unbounded and $M \subseteq \partial D$, $M' \subseteq \partial D'$ be open pieces of the boundaries. Suppose that $\partial D$ is smooth real-analytic and minimal in an open neighborhood of $\bar{M}$ and $\partial D'$ is smooth real-algebraic and minimal in an open neighborhood of $\bar{M}'$. Let $f : D \rightarrow D'$ be a holomorphic mapping such that the cluster set $\rm{cl}_{f}(M)$ does not intersect $D'$. It is proved that if the cluster set $\rm{cl}_{f}(p)$ of some point $p \in M$ contains some point $q \in M'$ and the graph of f extends as an analytic set to a neighborhood of $(p, q) \in \mathbb{C}^{n} \times \mathbb{C}^{N}$ , then $f$ extends as a holomorphic map to a dense subset of some neighborhood of $p$. If in addition, $M = \partial D$, $M' = \partial D'$ and $M'$ is compact, then $f$ extends holomorphically across an open dense subset of $\partial D$.

• Approximate controllability of a non-autonomous differential equation

In this paper, we establish the approximate controllability results for a non-autonomousfunctional differential equation using the theory of linear evolution system, Schauder fixed point theorem, and by making use of resolvent operators. The results obtained in this paper, improve the existing ones in this direction, to a considerable extent. An example is also given to illustrate the abstract results.

• Traveling wave solutions of a diffusive predator–prey model with modified Leslie–Gower and Holling-type II schemes

We study a diffusive predator–prey model with modified Leslie–Gower and Holling-II schemes with $D = 0$. We establish the existence of traveling wave solutions connecting a positive equilibrium and a boundary equilibrium via the ‘shooting method’, and the non-existence by the ‘eigenvalue method’. It should be emphasized that a threshold value $c^{\ast} = \sqrt{4\alpha}$ is found in our paper.

• Cross-product of Bessel functions: Monotonicity patterns and functional inequalities

In this paper, we study the Dini functions and the cross-product of Bessel functions. Moreover, we are interested on the monotonicity patterns for the cross-product of Bessel and modified Bessel functions. In addition, we deduce Redheffer-type inequalities, and the interlacing property of the zeros of Dini functions and the cross-product of Bessel and modified Bessel functions. Bounds for logarithmic derivatives ofthese functions are also derived. The key tools in our proofs are some recently developed infinite product representations for Dini functions and cross-product of Bessel functions.

• Some new estimates for the Helgason Fourier transform on rank 1 symmetric spaces

New estimates are proved for the Helgason Fourier transform in the space $L^{2}(X)$ on certain classes of functions characterized by the spherical modulus of continuity.

• Property of reflexivity for multiplication operators on Banach function spaces

In this paper, we give conditions under which the powers of the multiplication operator $M_{z}$ are reflexive on a Banach space of functions analytic on a plane domain.

• Stabilization of the higher order nonlinear Schrödinger equation with constant coefficients

We study the internal stabilization of the higher order nonlinear Schrödinger equation with constant coefficients. Combining multiplier techniques, a fixed point argument and nonlinear interpolation theory, we can obtain the well-posedness. Then, applying compactness arguments and a unique continuation property, we prove that the solution of the higher-order nonlinear Schrödinger equation with a damping term decays exponentially.