# Volume 29 (2021), Issue 3

### 1. A note on the volume of ∇-Einstein manifolds with skew-torsion

We study the volume of compact Riemannian manifolds which are Einstein with respect to a metric connection with (parallel) skew-torsion. We provide a result for the sign of the first variation of the volume in terms of the corresponding scalar curvature. This generalizes a result of M. Ville [15] related with the first variation of the volume on a compact Einstein manifold.

### 2. On the gaps between q-binomial coefficients

In this note, we estimate the distance between two q-nomial coefficients ( k n ) q - ( k ′ […]

### 3. A Weighted Eigenvalue Problems Driven by both p(·)-Harmonic and p(·)-Biharmonic Operators

The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both p(·)-Harmonic and p(·)-biharmonic operatorsΔp(x)2u-Δp(x)u=λw(x)|u|q(x)-2u   in  Ω,            u∈W2,p(⋅)(Ω)∩W0-1,p(⋅)(Ω),is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces Lp(·)(Ω) and Wm,p(·)(Ω).

### 4. Sur la variation de certaines suites de parties fractionnaires

Let $b > a > 0$. We prove the following asymptotic formula :\begin{equation*}\sum_{n\ge 0} \big\lvert\{x/(n+a)\}-\{x/(n+b)\}\big\rvert=\frac{2}{\pi}\zeta(3/2)\sqrt{cx}+O(c^{2/9}x^{4/9}),\\\end{equation*}with $c=b-a$, uniformly for $x \ge 40 c^{-5}(1+b)^{27/2}$.

### 5. On Balancing and Lucas-balancing Quaternions

The aim of this article is to investigate two new classes of quaternions, namely, balancing and Lucas-balancing quaternions that are based on balancing and Lucas-balancing numbers, respectively. Further, some identities including Binet’s formulas, summation formulas, Catalan’s identity, etc. concerning these quaternions are also established.

### 6. Symmetric identity for polynomial sequences satisfying A′n+1(x) = (n + 1)An(x)

Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying A′ n +1(x) = (n + 1)An (x) with A 0(x) a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, Apostol--Euler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.

### 7. Multiplicative Lie triple derivations on standard operator algebras

Let X be a Banach space of dimension n > 1 and A ⊂ B(X )be a standard operator algebra. In the present paper it is shown that if amapping d : A → A (not necessarily linear) satisfiesd([[U, V ], W]) = [[d(U), V ], W] + [[U, d(V )], W] + [[U, V ], d(W)]for all U, V, W ∈ A, then d = ψ + τ , where ψ is an additive derivation of Aand τ : A → FI vanishes at second commutator [[U, V ], W] for all U, V, W ∈A. Moreover, if d is linear and satisfies the above relation, then thereexists an operator S ∈ A and a linear mapping τ from A into FI satisfyingτ ([[U, V ], W]) = 0 for all U, V, W ∈ A, such that d(U) = SU − US + τ (U)for all U ∈ A.

### 8. (ϕ, φ)-derivations on semiprime rings and Banach algebras

Let R be a semiprime ring with unity e and φ, ϕ be automorphisms of R. In this paper it is shown that if R satisfies2D(xn) = D(xn−1)φ(x) + ϕ(xn−1)D(x) + D(x)φ(xn−1) + ϕ(x)D(xn−1)for all x ∈ R and some fixed integer n ≥ 2, then D is an (φ, ϕ)-derivation.Moreover, this result makes it possible to prove that if R admits an additivemappings D, G : R → R satisfying the relations2D(xn) = D(xn−1)φ(x) + ϕ(xn−1)G(x) + G(x)φ(xn−1) + ϕ(x)G(xn−1),2G(xn) = G(xn−1)φ(x) + ϕ(xn−1)D(x) + D(x)φ(xn−1) + ϕ(x)D(xn−1),for all x ∈ R and some fixed integer n ≥ 2, then D and G are (φ, ϕ)--derivations under some torsion restriction. Finally, we apply these purelyring theoretic results to semi-simple Banach algebras.

### 9. Remarks on Ramanujan’s inequality concerning the prime counting function

In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that π ( x 2 ) &lt; e x log x π ( x e ) \pi […]

### 10. G-tridiagonal majorization on Mn,m

For X, Y ∈ M n,m , it is said that X is g-tridiagonal majorized by Y (and it is denoted by X ≺gt Y) if there exists a tridiagonal g-doubly stochastic matrix A such that X = AY. In this paper, the linear preservers and strong linear preservers of ≺gt are characterized on M n,m .

### 11. Some type of semisymmetry on two classes of almost Kenmotsu manifolds

The object of the present paper is to study some types of semisymmetry conditions on two classes of almost Kenmotsu manifolds. It is shown that a (k, µ)-almost Kenmotsu manifold satisfying the curvature condition Q · R = 0 is locally isometric to the hyperbolic space ℍ2 n +1(−1). Also in (k, µ)-almost Kenmotsu manifolds the following conditions: (1) local symmetry (∇R = 0), (2) semisymmetry (R·R = 0), (3) Q(S, R) = 0, (4) R·R = Q(S, R), (5) locally isometric to the hyperbolic space ℍ2 n +1(−1) are equivalent. Further, it is proved that a (k, µ) ′ -almost Kenmotsu manifold satisfying Q · R = 0 is locally isometric to ℍ n +1(−4) × ℝ n and a (k, µ) ′- -almost Kenmotsu manifold satisfying any one of the curvature conditions Q(S, R) = 0 or R · R = Q(S, R) is either an Einstein manifold or locally isometric to ℍ n +1(−4) × ℝ n . Finally, an illustrative example is presented.

### 12. An integral transform and its application in the propagation of Lorentz-Gaussian beams

The aim of the present note is to derive an integral transform I = ∫ 0 ∞ x s + 1 e - β x 2 + […]

### 13. On the completeness of total spaces of horizontally conformal submersions

In this paper, we address the completeness problem of certain classes of Riemannian metrics on vector bundles. We first establish a general result on the completeness of the total space of a vector bundle when the projection is a horizontally conformal submersion with a bound condition on the dilation function, and in particular when it is a Riemannian submersion. This allows us to give completeness results for spherically symmetric metrics on vector bundle manifolds and eventually for the class of Cheeger-Gromoll and generalized Cheeger-Gromoll metrics on vector bundle manifolds. Moreover, we study the completeness of a subclass of g-natural metrics on tangent bundles and we extend the results to the case of unit tangent sphere bundles. Our proofs are mainly based on techniques of metric topology and on the Hopf-Rinow theorem.

### 14. A non-linear discrete-time dynamical system related to epidemic SISI model

We consider SISI epidemic model with discrete-time. The crucial point of this model is that an individual can be infected twice. This non-linear evolution operator depends on seven parameters and we assume that the population size under consideration is constant, so death rate is the same with birth rate per unit time. Reducing to quadratic stochastic operator (QSO) we study the dynamical system of the SISI model.

### 15. A note on the solvability of homogeneous Riemann boundary problem with infinity index

In this note we establish a necessary and sufficient condition for solvability of the homogeneous Riemann boundary problem with infinity index on a rectifiable open curve. The index of the problem we deal with considers the influence of the requirement of the solutions of the problem, the degree of non-smoothness of the curve at the endpoints as well as the behavior of the coefficient at these points.