Volume 71, Issue 2

The Tribonacci sequence is a generalization of the Fibonacci sequence which starts with 0,0,1 and each term afterwards is the sum of the three preceding terms. Here, we show that the only Tribonacci numbers that are concatenations of two repdigits are 13,24,44,81. This paper continues a previous work that searched for Fibonacci numbers which are concatenations of two repdigits.

Let ( P n ) n ≥0 be the sequence of Padovan numbers defined by P 0 = 0, P 1 = 1 = P 2 , and P n +3 = P n +1 + P n for all n ≥ 0. In this paper, we find all Padovan numbers that are concatenations of two distinct repdigits.

Let K = ℚ(pd24)$\begin{array}{} \displaystyle (\sqrt[4]{pd^{2}}) \end{array}$ be a real pure quartic number field and k = ℚ(p$\begin{array}{} \displaystyle \sqrt{p} \end{array}$) its real quadratic subfield, where p ≡ 5 (mod 8) is a prime integer and d an odd square-free integer coprime to p . In this work, we calculate r 2 ( K ), the 2-rank of the class group of K , in terms of the number of prime divisors of d that decompose or remain inert in ℚ(p$\begin{array}{} \displaystyle \sqrt{p} \end{array}$), then we will deduce forms of d satisfying r 2 ( K ) = 2. In the last case, the 4-rank of the class group of K is given too.

We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [ An Introduction to Combinatorial Identities , Wiley, 1968].

We establish several versions of the subcritical and critical Hardy type inequalities with Bessel pairs on the Euclidean space endowed with a Finsler norm.

In this paper we introduce some new subclasses of the p -valent analytic functions with higher-order derivatives that generalize some related subclasses of starlike and convex functions of a positive order. We found the order of ( p , q )-valent starlikeness and convexity for the products of functions that belong to these classes. The order of ( p , q )-valent starlikeness and convexity of certain integral operators for the product of functions of these classes were also obtained.

In this contribution, we study properties of block Hessenberg matrices associated with matrix orthonormal polynomials on the unit circle. We also consider the Uvarov and Christoffel spectral matrix transformations of the orthogonality measure, and obtain some relations between the associated Hessenberg matrices.

For the period T ( α ) of a simple pendulum with the length L and the amplitude (the initial elongation) α ∈ (0, π ), a strictly increasing sequence T n ( α ) is constructed such that the relations T1(α )=2Lgπ − 2+1ϵ ln1+ϵ 1− ϵ +π 4− 23ϵ 2,Tn+1(α )=Tn(α )+2Lgπ wn+12− 22n+3ϵ 2n+2,$$\begin{array}{c} \displaystyle T_1(\alpha)=2\sqrt{\frac{L}{g}}\left[\pi-2+\frac{1}{\epsilon} \ln\left(\frac{1+\epsilon}{1-\epsilon}\right)+\left(\frac{\pi}{4}-\frac{2}{3}\right)\epsilon^2\right],\\ \displaystyle T_{n+1}(\alpha)=T_n(\alpha)+2\sqrt{\frac{L}{g}}\left(\pi w_{n+1}^2 - \frac{2}{2n+3}\right)\epsilon^{2n+2}, \end{array}$$ and 0< T(α )− Tn(α )T(α )< 2ϵ 2n+2π (2n+1),$$\begin{array}{} \displaystyle 0 \lt \frac{T(\alpha)-T_n(\alpha)}{T(\alpha)} \lt \frac{2\epsilon^{2n+2}}{\pi(2n+1)}\,, \end{array}$$ holds true, for α ∈ (0, π ), n ∈ ℕ, wn:=∏ k=1n2k− 12k$\begin{array}{} \displaystyle w_n:=\prod_{k=1}^n\frac{2k-1}{2k} \end{array}$ (the n th Wallis’ ratio) and ϵ = sin( α /2).

In this paper, we study convolution operators on an Orlicz space L Φ ( G ) commuting with left translations, where Φ is an N-function and G is a locally compact group. We also present some basic properties of the Fourier transform of a Φ-convolution operator in the context of locally compact abelian groups.

In this paper, we obtain some characterizations on the Reeb vector field for a trans-Sasakian manifold to be proper.

The p -adic completion ℚ p of the rational numbers induces a different absolute value |⋅| p than the typical | ⋅| we have on the real numbers. In this paper we compare and contrast functions f : ℝ + → ℝ + , for which the composition with the p-adic metric d p generated by |⋅| p is still a metric on ℚ p , with the usual metric preserving functions and the functions that preserve the Euclidean metric on ℝ. In particular, it is shown that f ∘ d p is still an ultrametric on ℚ p if and only if there is a function g such that f ∘ d p = g ∘ d p and g ∘ d is still an ultrametric for every ultrametric d . Some general variants of the last statement are also proved.

In this paper, we do further investigations on the statistical inner and outer limits of sequences of closed sets in metric spaces, which were introduced by Nuray, Rhoades, and Talo, Sever, Başar, and generalize the conventional Painleve-Kuratowski inner and outer limits. Also, we provide criteria for checking statistical Wijsman and Hausdorff set convergences and we examine the relationship between Kuratowski and Wijsman statistical convergence. A closer look on the concept of statistical Cauchyness, with respect to the Hausdorff “extended” metric h , completes this research.

We characterize the uniform convergence points set of a pointwisely convergent sequence of real-valued functions defined on a perfectly normal space. We prove that if X is a perfectly normal space which can be covered by a disjoint sequence of dense subsets and A ⊆ X , then A is the set of points of the uniform convergence for some convergent sequence ( f n ) n ∈ ω of functions f n : X → ℝ if and only if A is G δ -set which contains all isolated points of X . This result generalizes a theorem of Ján Borsík published in 2019.

In this paper, we investigate the problem of the local linear estimation of the conditional ageing intensity function, when the variable of interest is subject to random right-censored. We establish under appropriate conditions the asymptotic normality of this estimator.

We prove an analogue of the Donsker theorem under the Lindeberg condition in a fuzzy setting. Specifically, we consider a certain triangular system of d -dimensional fuzzy random variables {Xn,i∗ },$\begin{array}{} \{X_{n,i}^*\}, \end{array}$ n ∈ ℕ and i = 1, 2, …, k n , which take as their values fuzzy vectors of compact and convex α -cuts. We show that an appropriately normalized and interpolated sequence of partial sums of the system may be associated with a time-continuous process defined on the unit interval t ∈ [0, 1] which, under the assumption of the Lindeberg condition, tends in distribution to a standard Brownian motion in the space of support functions.

Having only two parameters, the Gamma-Lindley distribution does not provide enough flexibility for analyzing different types of lifetime data. From this perspective, in order to further enhance its flexibility, we set forward in this paper a new class of distributions named Generalized Gamma-Lindley distribution with four parameters. Its construction is based on certain mixtures of Gamma and Lindley distributions. The truncated moment, as a characterization method, has drawn a little attention in the statistical literature over the great popularity of the classical methods. We attempt to prove that the Generalized Gamma-Lindley distribution is characterized by its truncated moment of the first order statistics. This method rests upon finding a survival function of a distribution, that is a solution of a first order differential equation. This characterization includes as special cases: Gamma, Lindley, Exponential, Gamma-Lindley and Weighted Lindley distributions. Finally, a simulation study is performed to help the reader check whether the available data follow the underlying distribution.

Matrix variate generalizations of Pareto distributions are proposed. Several properties of these distributions including cumulative distribution functions, characteristic functions and relationship to matrix variate beta type I and matrix variate type II distributions are studied.

In this paper, the global exponential stability and periodicity are investigated for delayed neural network models with continuous coefficients and piecewise constant delay of generalized type. The sufficient condition for the existence and uniqueness of periodic solutions of the model is established by applying Banach’s fixed point theorem and the successive approximations method. By constructing suitable differential inequalities with generalized piecewise constant delay, some sufficient conditions for the global exponential stability of the model are obtained. Typical numerical examples with simulations are utilized to illustrate the validity and improvement in less conservatism of the theoretical results. This paper ends with a brief conclusion.

In [ An optimal inequality for CR - warped products in complex space forms involving CR δ - invariant , Internat. J. Math. 23 (3) (2012)], B.-Y. Chen introduced the CR δ -invariant for CR-submanifolds. Then, in [ Two optimal inequalities for anti - holomorphic submanifolds and their applications , Taiwan. J. Math. 18 (2014), 199–217], F. R. Al-Solamy, B.-Y. Chen and S. Deshmukh proved two optimal inequalities for anti-holomorphic submanifolds in complex space forms involving the CR δ -invariant. In this paper, we obtain optimal inequalities for this invariant for contact CR -submanifolds in almost contact metric manifolds.