Volume 28, Issue 2

In this study, given two crossed complexes 𝒞{\mathcal{C}} and 𝒟{\mathcal{D}} of associative R -algebras and a crossed complex morphism f:𝒞→𝒟{f\colon\mathcal{C}\to\mathcal{D}}, we construct a homotopy as a pair (H,f){(H,f\/)}, where H=(Hn){H=(H_{n})} is a sequence of R -linear maps Hn:Cn→Dn+1{H_{n}\colon C_{n}\to D_{n+1}}. Then we show that for a fixed pair 𝒞{\mathcal{C}} and 𝒟{\mathcal{D}} of crossed complexes of associative R -algebras, the family of all homotopies between crossed complex morphisms from 𝒞{\mathcal{C}} to 𝒟{\mathcal{D}} has a groupoid structure with crossed complex morphisms as objects and homotopies as morphisms.

A quasilinearization algorithm is developed for boundary value problems at resonance. To do so, a standard monotonicity condition is assumed to obtain the uniqueness of solutions for the boundary value problem at resonance. Then the method of upper and lower solutions and the shift method are applied to obtain the existence of solutions. A quasilinearization algorithm is developed and sequences of approximate solutions are constructed, which converge monotonically and quadratically to the unique solution of the boundary value problem at resonance. Two examples are provided in which explicit upper and lower solutions are exhibited.

For certain classes of functions of Λ-bounded variation on [-π,π]{[-\pi,\pi]}, Waterman and D’Antonio constructed a sequence of summability kernels such that the corresponding integral means converge for any function from the class, but not for any function from a broader Λ′⁢BV{\Lambda^{\prime}\mathrm{BV}} class. In our paper, we extend this result to the case of functions of two variables.

Let ℛ{\mathcal{R}} be a prime ring with center Z⁢(ℛ){Z(\mathcal{R})} and *{*} an involution of ℛ{\mathcal{R}}. Suppose that ℛ{\mathcal{R}} admits generalized derivations F , G and H associated with a nonzero derivation f , g and h of ℛ{\mathcal{R}}, respectively. In the present paper, we investigate the commutativity of a prime ring ℛ{\mathcal{R}} satisfying any of the following identities: (i) [F⁢(x),F⁢(x*)]=0[F(x),F(x^{*})]=\nobreak 0, (ii) [F⁢(x),F⁢(x*)]=±[x,x*][F(x),F(x^{*})]=\pm[x,x^{*}], (iii) F⁢(x)∘F⁢(x*)=0F(x)\circ\nobreak F(x^{*})=0, (iv) F⁢(x)∘F⁢(x*)=±(x∘x*)F(x)\circ\nobreak F(x^{*})=\pm(x\circ\nobreak x^{*}), (v) [F⁢(x),x*]±[x,G⁢(x*)]=0[F(x),x^{*}]\pm[x,G(x^{*})]=0, (vi) F⁢(x⁢x*)∈Z⁢(ℛ)F(xx^{*})\in Z(\mathcal{R}), (vii) F⁢(x)⁢G⁢(x*)±H⁢(x)⁢x*∈Z⁢(ℛ)F(x)G(x^{*})\pm H(x)x^{*}\in Z(\mathcal{R}), (viii) F⁢([x,x*])±[x,x*]∈Z⁢(ℛ)F([x,x^{*}])\pm[x,x^{*}]\in Z(\mathcal{R}), (ix) F⁢(x∘x*)±x∘x*∈Z⁢(ℛ)F(x\circ\nobreak x^{*})\pm x\circ x^{*}\in Z(\mathcal{R}), (x) [F⁢(x),x*]±[x,G⁢(x*)]∈Z⁢(ℛ)[F(x),x^{*}]\pm[x,G(x^{*})]\in Z(\mathcal{R}), (xi) F⁢(x)∘x*±x∘G⁢(x*)∈Z⁢(ℛ)F(x)\circ\nobreak x^{*}\pm x\circ\nobreak G(x^{*})\in Z(\mathcal{R}) for all x∈ℛ{x\in\mathcal{R}}. Finally, the restrictions imposed on the hypotheses have been justified by an example.

We consider explicit sufficient conditions for all solutions of a first-order linear difference equation with several variable delays and non-negative coefficients to be oscillatory. The conditions have the form of inequalities bounding below the upper and lower limits of the sums of coefficients over a subset of the discrete semiaxis. Our main results are oscillation tests based on a new principle for composing the estimated sums of coefficients. We also give some results in the form of examples, including a counterexample to a wrong oscillation test cited in several recent papers.

The purpose of the present research is to investigate a general mixed type boundary value problem for the Laplace–Beltrami equation on a surface with the Lipschitz boundary 𝒞{\mathcal{C}} in the non-classical setting when solutions are sought in the Bessel potential spaces ℍps⁢(𝒞){\mathbb{H}^{s}_{p}(\mathcal{C})}, 1p<s<1+1p{\frac{1}{p}<s<1+\frac{1}{p}}, 1<p<∞{1<p<\infty}. Fredholm criteria and unique solvability criteria are found. By the localization, the problem is reduced to the investigation of model Dirichlet, Neumann and mixed boundary value problems for the Laplace equation in a planar angular domain Ωα⊂ℝ2{\Omega_{\alpha}\subset\mathbb{R}^{2}} of magnitude α. The model mixed BVP is investigated in the earlier paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain, Georgian Math. J. 27 2020, 2, 211–231], and the model Dirichlet and Neumann boundary value problems are studied in the non-classical setting. The problems are investigated by the potential method and reduction to locally equivalent 2×2{2\times 2} systems of Mellin convolution equations with meromorphic kernels on the semi-infinite axes ℝ+{\mathbb{R}^{+}} in the Bessel potential spaces. Such equations were recently studied by R. Duduchava [Mellin convolution operators in Bessel potential spaces with admissible meromorphic kernels, Mem. Differ. Equ. Math. Phys. 60 2013, 135–177] and V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 2016, 2, 707–731].

In this paper, by means of the generalized Olsen type inequality, related to the Riesz potential and its commutators, the author establishes the interior estimates on the generalized Morrey space for Schrödinger type elliptic equations with potentials satisfying the reverse Hölder condition.

Based on a variational principle for smooth functionals defined on reflexive Banach spaces, the existence of at least one weak solution for a non-homogeneous Neumann problem in an appropriate Orlicz–Sobolev space is discussed.

We extend some numerical radius inequalities for adjointable operators on Hilbert C*{C^{*}}-modules. A new refinement of a numerical radius inequality for some Hilbert space operators is given. More precisely, we prove that if T∈ℬ⁢(ℋ){T\in\mathcal{B}(\mathcal{H})} is an invertible operator, then ∥T∥2≤∥T∥2+1∥T-1∥22≤ω⁢(T).\frac{\|T\|}{2}\leq\frac{\sqrt{\|T\|^{2}+\frac{1}{\|T^{-1}\|^{2}}}}{2}\leq% \omega(T).

In this article, an operational definition, generating function, explicit summation formula, determinant definition and recurrence relations of the generalized families of Hermite–Appell polynomials are derived by using integral transforms and some known operational rules. An analogous study of these results is also carried out for the generalized forms of the Hermite–Bernoulli and Hermite–Euler polynomials.

We study the capability property of Leibniz algebras via the non-abelian exterior product.

In this paper we first classify left-invariant generalized Ricci solitons on four-dimensional hypercomplex Lie groups equipped with three families of left-invariant Lorentzian metrics. Then, on these Lorentzian spaces, we explicitly calculate the energy of an arbitrary left-invariant vector field X and determine the exact form of all left-invariant unit time-like vector fields which are spatially harmonic. Furthermore, we give a complete and explicit description of all homogeneous structures on these spaces in both Riemannian and Lorentzian cases and determine some of their types. The existence of Einstein four-dimensional hypercomplex Lorentzian Lie groups is proved and it is shown that although the results concerning Einstein-like metrics, conformally flatness and some equations in the Riemannian case are much richer than their Lorentzian analogues, in the Lorentzian case, there exist some new critical points of energy functionals, homogeneous structures and geodesic vectors which do not exist in the Riemannian case.

It is known that if f⁢(z)=zp+∑n=p+1∞an⁢zn{f(z)=z^{p}+\sum_{n=p+1}^{\infty}a_{n}z^{n}} and it is analytic in a convex domain D⊂ℂ{D\subset\mathbb{C}} and, for some real α, we have ℜ⁢𝔢⁡{exp⁡(i⁢α)⁢f(p)⁢(z)}>0{\operatorname{\mathfrak{Re}}\{\exp(i\alpha)f^{(p)}(z)\}>0}, z∈D{z\in D}, then f⁢(z){f(z)} is at most p -valent in D . This Ozaki condition is a generalization of the well-known Noshiro–Warschawski univalence condition. In this paper, we consider the radius of univalence of functions g⁢(z)=z+∑n=1∞bn⁢zn{g(z)=z+\sum_{n=1}^{\infty}b_{n}z^{n}} such that g′⁢(z)≺[(1+z)2/(1-z)2]{g^{\prime}(z)\prec[(1+z)^{2}/(1-z)^{2}]} and some related problems.

The procedure of positioning of pipes, for example for the automobile exhaust construction, by the use of 3D CAD systems is difficult and time consuming because of obstacles in an engine compartment. There are also further strong technical requirements and restrictions. An automatic generation procedure based on the level set method and the numerical solution of the Eikonal equation is proposed. The positions of pipes which satisfy the technical requirements are obtained using spline functions.

In this paper, we study the nonlinear Volterra integral equation satisfied by the early exercise boundary of the American put option in the one-dimensional diffusion model for a stock price with constant interest rate and constant dividend yield and with a local volatility depending on the current time t and the current stock price S . In the classical Black–Sholes model for a stock price, Theorem 4.3 of [S. D. Jacka, Optimal stopping and the American put, Math. Finance 1 1991, 2, 1–14] states that if the family of integral equations (parametrized by the variable S ) holds for all S≤b⁢(t){S\leq b(t)} with a candidate function b⁢(t){b(t)}, then this b⁢(t){b(t)} must coincide with the American put early exercise boundary c⁢(t){c(t)}. We generalize Peskir’s result [G. Peskir, On the American option problem, Math. Finance 15 2005, 1, 169–181] to state that if the candidate function b⁢(t){b(t)} satisfies one particular integral equation (which corresponds to the upper limit S=b⁢(t){S=b(t)}), then all other integral equations (corresponding to S , S≤b⁢(t){S\leq b(t)}) will be automatically satisfied by the same function b⁢(t){b(t)}.

In this paper, we propose the concept of a weighted pseudo δ-almost automorphic function under the matched space for time scales and we present some properties. Also, we obtain sufficient conditions for the existence of weighted pseudo δ-almost automorphic mild solutions to a class of semilinear dynamic equations under the matched spaces for time scales.