Volume 31, Issue 4

In this paper, we introduce a new class of derivations that generalizes skew derivations and semi-derivations, and we call it skew semi-derivation. Furthermore, we present a study of the conditions under which this type of multiplicative derivation becomes additive.

The present study introduces the notions of statistical convergence of order α and strong p -Cesàro summability of order α in partial metric spaces. Also, we examine the inclusion relations between these concepts. In addition, we introduce the notion of λ-statistical convergence of order α in partial metric spaces while providing relations linked to these sequence spaces.

Let A and B be two associative rings, let I be an ideal of B and let f ∈ Hom ⁢ ( A , B ) {f\in\mathrm{Hom}(A,B)} . In this paper, we give a complete description of generalized derivations over A ⋈ f I {A\bowtie^{f}I} . Furthermore, when A is prime or semi-prime, we give several identities on generalized derivations which provide the commutativity of A ⋈ f I {A\bowtie^{f}I} .

Operator matrices have played a significant role in the study of properties of the numerical radii of Hilbert space operators. This paper presents several new sharp upper bounds for the numerical radii of operator matrices in terms of certain complex combinations. The obtained results reveal many interesting properties of the numerical radius.

Let ℛ {\mathcal{R}} be a prime ring of characteristic not equal to 2, let 𝒰 {\mathcal{U}} be Utumi quotient ring of ℛ {\mathcal{R}} and let 𝒞 {\mathcal{C}} be the extended centroid of ℛ {\mathcal{R}} . Let Δ be a generalized derivation on ℛ {\mathcal{R}} , and let δ 1 {\delta_{1}} and δ 2 {\delta_{2}} be derivations on ℛ {\mathcal{R}} . Let p ⁢ ( v ) {p(v)} be a multilinear polynomial on ℛ {\mathcal{R}} , which is non-central valued on ℛ {\mathcal{R}} . If δ 1 ⁢ ( Δ 2 ⁢ ( p ⁢ ( v ) ) ⁢ p ⁢ ( v ) ) = δ 2 ⁢ ( Δ ⁢ ( p ⁢ ( v ) 2 ) ) {\delta_{1}(\Delta^{2}(p(v))p(v))=\delta_{2}(\Delta(p(v)^{2}))} for all v ∈ ℛ n {v\in\mathcal{R}^{n}} , then we find the complete description of Δ, δ 1 {\delta_{1}} and δ 2 {\delta_{2}} .

Let Φ : [ 0 , ∞ ) → [ 0 , ∞ ) {\Phi:[0,\infty)\rightarrow[0,\infty)} be a Young’s function satisfying the Δ 2 {\Delta_{2}} -condition and let M ℬ {M_{\mathcal{B}}} be the geometric maximal operator associated to a homothecy invariant basis ℬ {\mathcal{B}} acting on measurable functions on ℝ n {\mathbb{R}^{n}} . Let Q be the unit cube in ℝ n {\mathbb{R}^{n}} and let L Φ ⁢ ( Q ) {L^{\Phi}(Q)} be the Orlicz space associated to Φ with the norm given by ∥ f ∥ L Φ ⁢ ( Q ) := inf ⁡ { c > 0 : ∫ Q Φ ⁢ ( | f | c ) ≤ 1 } . \|f\|_{L^{\Phi}(Q)}:=\inf\Biggl{\{}c>0:\int_{Q}\Phi\bigg{(}\frac{|f|}{c}\bigg{% )}\leq 1\Bigg{\}}. We show that M ℬ {M_{\mathcal{B}}} satisfies the weak type estimate | { x ∈ ℝ n : M ℬ ⁢ f ⁢ ( x ) > α } | ≤ C 1 ⁢ ∫ ℝ n Φ ⁢ ( | f | α ) |\{x\in\mathbb{R}^{n}:M_{\mathcal{B}}\kern 1.422638ptf(x)>\alpha\}|\leq C_{1}% \int_{\mathbb{R}^{n}}\Phi\bigg{(}\frac{|f|}{\alpha}\bigg{)} for all measurable functions f on ℝ n {\mathbb{R}^{n}} and α > 0 {\alpha>0} if and only if M ℬ {M_{\mathcal{B}}} satisfies the weak type estimate | { x ∈ Q : M ℬ ⁢ f ⁢ ( x ) > α } | ≤ C 2 ⁢ ∥ f ∥ L Φ ⁢ ( Q ) α |\{x\in Q:M_{\mathcal{B}}\kern 1.422638ptf(x)>\alpha\}|\leq C_{2}\frac{\|f\|_{% L^{\Phi}(Q)}}{\alpha} for all measurable functions f supported on Q and α > 0 {\alpha>0} . As a consequence of this equivalence, we prove that if Φ satisfies the above conditions and ℬ {\mathcal{B}} is a homothecy invariant basis differentiating integrals of all measurable functions f on ℝ n {\mathbb{R}^{n}} such that ∫ ℝ n Φ ⁢ ( | f | ) < ∞ {\int_{\mathbb{R}^{n}}\Phi(|f|)<\infty} , then the associated maximal operator M ℬ {M_{\mathcal{B}}} satisfies both of the above weak type estimates.

In the paper, we study a Balakrishnan–Taylor quasilinear wave equation | z t | α ⁢ z t ⁢ t - Δ ⁢ z t ⁢ t - ( ξ 1 + ξ 2 ⁢ ∥ ∇ ⁡ z ∥ 2 + σ ⁢ ( ∇ ⁡ z , ∇ ⁡ z t ) ) ⁢ Δ ⁢ z - Δ ⁢ z t + β ⁢ ( x ) ⁢ f ⁢ ( z t ) + g ⁢ ( z ) = 0 |z_{t}|^{\alpha}z_{tt}-\Delta z_{tt}-\bigl{(}\xi_{1}+\xi_{2}\|\nabla z\|^{2}+% \sigma(\nabla z,\nabla z_{t})\bigr{)}\Delta z-\Delta z_{t}+\beta(x)f(z_{t})+g(% z)=0 in a bounded domain of ℝ n {\mathbb{R}^{n}} with Dirichlet boundary conditions. By using Faedo–Galerkin method, we prove the existence of global weak solutions. By the help of the perturbed energy method, the exponential stability of solutions is also established.

Let 𝑅 be a prime ring and 𝐿 a non-central Lie ideal of 𝑅. In this paper, we aim to classify the generalized derivations of 𝑅 satisfying some algebraic identities with power values on 𝐿. Moreover, the same identities are studied locally on a two nonvoid open subsets of a prime Banach algebra.

Let ω be a doubling weight and 0 < p ≤ q < ∞ {0<p\leq q<\infty} . The essential norm of Riemann–Stieltjes operator T g {T_{g}} from the weighted Bergman space A ω p {A^{p}_{\omega}} to A ω q {A^{q}_{\omega}} was investigated in the unit ball of ℂ n {\mathbb{C}^{n}} .

In this paper, we thoroughly study the Nakayama property and some related concepts. Also, we describe multiplication modules that, among other things, satisfy the Nakayama property. Next, we show that a ring 𝑅 is a Max ring if and only if all modules that can be generated by a finite or countable set have the weak Nakayama property. We prove that a ring 𝑅 is a perfect ring if and only if every module that can be generated by a finite or countable set has the Nakayama property. Finally, we present some categorical results on the aforementioned properties.

Under some suitable conditions, we show that at least three weak solutions exist for a system of differential equations involving the ( p ⁢ ( x ) , q ⁢ ( x ) ) {(p(x),q(x))} Laplacian-like with indefinite weights. The proof is related to the Bonanno–Marano critical theorem ( Appl. Anal. 89 (2010), 1–10).

Let A and G be finite groups such that A acts coprimely on G by automorphisms. We prove that if every self-centralizing non-nilpotent A -invariant subgroup of G is a TI-subgroup or a subnormal subgroup, then every non-nilpotent A -invariant subgroup of G is subnormal and G is p -nilpotent or p -closed for any prime divisor p of | G | {|G|} . If every self-centralizing non-metacyclic A -invariant subgroup of G is a TI-subgroup or a subnormal subgroup, then every non-metacyclic A -invariant subgroup of G is subnormal and G is solvable.

In this paper, we study the problem with data on the boundary of the infinite layer { ( t , x ) : t ∈ ( 0 , h ) , x ∈ R s } , h > 0 , s ∈ N , \{(t,x):t\in(0,h),\,x\in\mathbb{R}^{s}\},\quad h>0,\,s\in\mathbb{N}, for the system of two differential equations of the second order in the time variable 𝑡 with shifts in the spatial variables x 1 , x 2 , … , x s x_{1},x_{2},\ldots,x_{s} . We propose a differential-symbol method of constructing a solution of the problem and identify a class of vector functions in which the obtained solution is unique. The method of solving the Dirichlet problem in the layer is illustrated by examples.

Let 𝑅 be a commutative ring with identity 1 ≠ 0 1\neq 0 and let Z ⁢ ( R ) ′ Z(R)^{\prime} be the set of all non-zero and non-unit elements of ring 𝑅. Further, Γ ′ ⁢ ( R ) \Gamma^{\prime}(R) denotes the cozero-divisor graph of 𝑅, is an undirected graph with vertex set Z ⁢ ( R ) ′ Z(R)^{\prime} , and w ∉ z ⁢ R w\notin zR and z ∉ w ⁢ R z\notin wR if and only if two distinct vertices 𝑤 and 𝑧 are adjacent, where q ⁢ R qR is the ideal generated by the element 𝑞 in 𝑅. In this paper, we find the signless Laplacian eigenvalues of the graphs Γ ′ ⁢ ( Z n ) \Gamma^{\prime}(\mathbb{Z}_{n}) for n = p 1 N ⁢ p 2 ⁢ p 3 n=p_{1}^{N}p_{2}p_{3} and p 1 N ⁢ p 2 M ⁢ p 3 p_{1}^{N}p_{2}^{M}p_{3} , where p 1 , p 2 , p 3 p_{1},p_{2},p_{3} are distinct primes and N , M N,M are positive integers. We also show that the cozero-divisor graph Γ ′ ⁢ ( Z p 1 ⁢ p 2 ) \Gamma^{\prime}(\mathbb{Z}_{p_{1}p_{2}}) is a signless Laplacian integral.

Route planning problems are among the activities that have the highest impact in emergency logistical planning, goods transportation and facility location-distribution because of their effects on efficiency in resource management, service levels and client satisfaction. In the extreme conditions, such as disaster-stricken zones, the difficulty of vehicle movement between nearest different affected areas (demand points) on planning routes cause the imprecision of time of movement and the uncertainty of feasibility of movement. In this paper, the imprecision is presented by triangular fuzzy numbers and the uncertainty is presented by a possibility measure. A new two-stage, fuzzy bi-criterion optimization approach for the vehicle routing problem (VRP) is considered. On the first stage, the sample of so-called “promising” closed routes are selected based on a “constructive” approach. On the second stage, triangular fuzzy valued Choquet aggregation (TFCA) operator is constructed for the selected closed routes. The evaluation of constructed routes, levels of failure and possibility of vehicle movement on the roads are aggregated by the TFCA operator by the new criterion – minimization of infeasibility of movement. The new criterion together with the classic criterion – minimization of the total distance traveled – creates a bi-criteria fuzzy VRP. The constructed VRP is reduced to the bi-criteria fuzzy partitioning problem, and an 𝜀-constraint approach is developed for solving it. For numerical experiments, a parallel algorithm is created on the basis of D. Knuth’s algorithm of Dancing Links (DLX). An example is presented with the results of our approach for the VRP, where all Pareto-optimal solutions are found from the set of promising routes. The optimal solutions tend to avoid roads that are problematic because of extreme situations.