Abstract: 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. PubDate: Thu, 23 Dec 2021 00:00:00 GMT

Abstract: 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 satisfies 2𝒟(xn)=𝒟(xn-1)φ(x)+ϕ(xn-1)𝒟(x)+𝒟(x)φ(xn-1)+ϕ(x)𝒟(xn-1)2\mathcal{D}\left( {{x^n}} \right) = \mathcal{D}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right) 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 additive mappings 𝒟, 𝒢 : ℛ → ℛ satisfying the relations 2𝒟(xn)=𝒟(xn-1)φ(x)+ϕ(xn-1)𝒟(x)+𝒟(x)φ(xn-1)+ϕ(x)𝒟(xn-1)2\mathcal{D}\left( {{x^n}} \right) = \mathcal{D}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right)2𝒢(xn)=𝒢(xn- PubDate: Thu, 23 Dec 2021 00:00:00 GMT

Abstract: Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying A′n+1(x) = (n + 1)An(x) with A0(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. PubDate: Thu, 23 Dec 2021 00:00:00 GMT

Abstract: 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. PubDate: Thu, 23 Dec 2021 00:00:00 GMT

Abstract: In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that π(x2)<exlogxπ(xe)\pi \left( {{x^2}} \right) < {{ex} \over {\log x}}\pi \left( {{x \over e}} \right) for x sufficiently large. First, we study its sharpness by giving full asymptotic expansions of its left and right hand sides expressions. Then, we discuss the structure of Ramanujan’s inequality, by replacing the factor xlogx{x \over {\log x}} on its right hand side by the factor xlogx-h{x \over {\log x - h}} for a given h, and by replacing the numerical factor e by a given positive α. Finally, we introduce and study inequalities analogous to Ramanujan’s inequality. PubDate: Thu, 23 Dec 2021 00:00:00 GMT

Abstract: 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),\eqalign{& \Delta _{p\left( x \right)}^2u - {\Delta _{p\left( x \right)}}u = \lambda w\left( x \right){\left u \right ^{q\left( x \right) - 2}}u\,\,\,{\rm{in}}\,\,\Omega {\rm{,}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,u \in {W^{2,p\left( \cdot \right)}}\left( \Omega \right) \cap W_0^{ - 1,p\left( \cdot \right)}\left( \Omega \right), \cr} is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces Lp(·)(Ω) and Wm,p(·)(Ω). PubDate: Thu, 23 Dec 2021 00:00:00 GMT

Abstract: Let χ be a Banach space of dimension n > 1 and 𝔘 ⊂ 𝔅(χ) be a standard operator algebra. In the present paper it is shown that if a mapping d : 𝔘 → 𝔘 (not necessarily linear) satisfies d([[U,V],W])=[[d(U),V],W]+[[U,d(V),W]]+[[U,V],d(W)]d\left( {\left[ {\left[ {U,V} \right],W} \right]} \right) = \left[ {\left[ {d\left( U \right),V} \right],W} \right] + \left[ {\left[ {U,d\left( V \right),W} \right]} \right] + \left[ {\left[ {U,V} \right],d\left( W \right)} \right] for all U, V, W ∈ 𝔘, then d =ψ + τ, where ψ is an additive derivation of 𝔘 and τ : 𝔘 → 𝔽I vanishes at second commutator [[U, V ], W ] for all U, V, W ∈ 𝔘. Moreover, if d is linear and satisfies the above relation, then there exists an operator S ∈ 𝔘 and a linear mapping τ from 𝔘 into 𝔽I satisfying τ ([[U, V ], W ]) = 0 for all U, V, W ∈ 𝔘, such that d(U) = SU − US + τ (U) for all U ∈ 𝔘. PubDate: Thu, 23 Dec 2021 00:00:00 GMT

Abstract: Let b > a > 0. We prove the following asymptotic formula ∑n⩾0 {x/(n+a)}-{x/(n+b)} =2πζ(3/2)cx+O(c2/9x4/9),\sum\limits_{n \geqslant 0} {\left {\left\{ {x/\left( {n + a} \right)} \right\} - \left\{ {x/\left( {n + b} \right)} \right\}} \right = {2 \over \pi }\zeta \left( {3/2} \right)\sqrt {cx} + O\left( {{c^{2/9}}{x^{4/9}}} \right),} with c = b − a, uniformly for x ⩾ 40c−5(1 + b)27/2. PubDate: Thu, 23 Dec 2021 00:00:00 GMT

Abstract: The aim of the present note is to derive an integral transformI=∫0∞xs+1e-βx2+γxMk,v(2ζx2)Jμ(χx)dx,I = \int_0^\infty {{x^{s + 1}}{e^{ - \beta x}}^{2 + \gamma x}{M_{k,v}}} \left( {2\zeta {x^2}} \right)J\mu \left( {\chi x} \right)dx,involving the product of the Whittaker function Mk,ν and the Bessel function of the first kind Jµ of order µ. As a by-product, we also derive certain new integral transforms as particular cases for some special values of the parameters k and ν of the Whittaker function. Eventually, we show the application of the integral in the propagation of hollow higher-order circular Lorentz-cosh-Gaussian beams through an ABCD optical system (see, for details [13], [3]). PubDate: Fri, 01 Oct 2021 00:00:00 GMT

Abstract: 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. PubDate: Fri, 01 Oct 2021 00:00:00 GMT

Abstract: For X, Y ∈ Mn,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 Mn,m. PubDate: Fri, 01 Oct 2021 00:00:00 GMT

Abstract: 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. PubDate: Fri, 01 Oct 2021 00:00:00 GMT

Abstract: 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 ℍ2n+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 ℍ2n+1(−1) are equivalent. Further, it is proved that a (k, µ)′ -almost Kenmotsu manifold satisfying Q · R = 0 is locally isometric to ℍn+1(−4) × ℝn 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 ℍn+1(−4) × ℝn. Finally, an illustrative example is presented. PubDate: Fri, 01 Oct 2021 00:00:00 GMT

Abstract: 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. PubDate: Fri, 01 Oct 2021 00:00:00 GMT

Abstract: In the present paper we prove that every local and 2-local derivation on conservative algebras of 2-dimensional algebras are derivations. Also, we prove that every local and 2-local automorphism on conservative algebras of 2-dimensional algebras are automorphisms. PubDate: Thu, 15 Jul 2021 00:00:00 GMT

Abstract: We find examples of polynomials f ∈ D [t; σ, δ] whose eigenring ℰ(f) is a central simple algebra over the field F = C ∩ Fix(σ) ∩ Const(δ). PubDate: Thu, 15 Jul 2021 00:00:00 GMT

Abstract: We connect the theorems of Rentschler [18] and Dixmier [10] on locally nilpotent derivations and automorphisms of the polynomial ring A0 and of the Weyl algebra A1, both over a field of characteristic zero, by establishing the same type of results for the family of algebras Ah=〈x,y yx−xy=h(x)〉,{A_h} = \left\langle {x,y yx - xy = h\left( x \right)} \right\rangle ,, where h is an arbitrary polynomial in x. In the second part of the paper we consider a field 𝔽 of prime characteristic and study 𝔽[t]-comodule algebra structures on Ah. We also compute the Makar-Limanov invariant of absolute constants of Ah over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of Ah. PubDate: Thu, 15 Jul 2021 00:00:00 GMT

Abstract: In this paper we introduce the notion of existentially closed Leibniz algebras. Then we use HNN-extensions of Leibniz algebras in order to prove an embedding theorem. PubDate: Thu, 15 Jul 2021 00:00:00 GMT

Abstract: We give the description of Rota–Baxter operators, Reynolds operators, Nijenhuis operators and average operators on 3-dimensional nilpotent associative algebras over ℂ. PubDate: Thu, 15 Jul 2021 00:00:00 GMT

Abstract: In this note, we estimate the distance between two q-nomial coefficients (kn)q-(k′n′)q{\left( {_k^n} \right)_q} - {\left( {_{k'}^{n'}} \right)_q}, where (n, k) ≠ (n′, k′) and q ≥ 2 is an integer. PubDate: Mon, 10 Aug 2020 00:00:00 GMT