Volume 29, Issue 3

In this paper, we establish some spectral analysis in Fredholm theory involving the concept of a polynomially Riesz operators perturbation. Moreover, we apply the obtained results to analyze the incidence of some essential spectra of a perturbed operator via this kind of perturbation. Our approach allows us to investigate a new description in the theory of unbounded operator matrices defined with maximal domain via this kind of perturbation. Finally, we conclude by presenting some progress in the theory of polynomially operators and the theory of semi-Fredholm and semi-Browder operators.

The estimate for the Bernoulli regression function is constructed using the Bernstein polynomial for group observations. The question of its consistency and asymptotic normality is studied. A testing hypothesis is constructed on the form of the Bernoulli regression function. The consistency of the constructed tests is investigated.

There is a Mazurkiewicz set in the Cohen–Halpern–Levy model.

We show that if the Hardy–Littewood maximal operator M is bounded on a reflexive variable exponent space L p ⁢ ( ⋅ ) ⁢ ( ℝ d ) {L^{p(\,\cdot\,)}(\mathbb{R}^{d})} , then for every q ∈ ( 1 , ∞ ) {q\in(1,\infty)} , the exponent p ⁢ ( ⋅ ) {p(\,\cdot\,)} admits, for all sufficiently small θ > 0 {\theta>0} , the representation 1 p ⁢ ( x ) = θ q + 1 - θ r ⁢ ( x ) {\frac{1}{p(x)}=\frac{\theta}{q}+\frac{1-\theta}{r(x)}} , x ∈ ℝ d {x\in\mathbb{R}^{d}} , such that the operator M is bounded on the variable Lebesgue space L r ⁢ ( ⋅ ) ⁢ ( ℝ d ) {L^{r(\,\cdot\,)}(\mathbb{R}^{d})} . This result can be applied for transferring properties like compactness of linear operators from standard Lebesgue spaces to variable Lebesgue spaces by using interpolation techniques.

A key ingredient of Janashia–Lagvilava matrix spectral factorization method, which is a theorem from complex function theory, is generalized to pure algebraic form.

The Robin problem for the diffusion equation in non-homogeneous media partial differential equation is reduced to a system of direct segregated parametrix-based Boundary-Domain Integral Equations (BDIEs). We use a parametrix different from the one applied by Chkadua, Mikhailov, Natroshvili. in [4]. We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain integral operators are also analysed.

In this paper, we have derived certain classical inequalities, namely Young’s, Hölder’s, Minkowski’s and the Hermite–Hadamard inequalities for a pseudo-integral (also known as g -integral). For Young’s, Hölder’s and Minkowski’s inequalities, the cases p > 1 {p>1} and p < 1 {p<1} , p ≠ 0 {p\neq 0} , have been covered. Moreover, in the case of the Hermite–Hadamard inequality, the refinement has also been proved and, as a special case, a g -analogue of a geometric-logarithmic-arithmetic inequality has been deduced.

In this paper, we consider the boundary value problem for one class of higher-order nonlinear partial differential equations. We prove theorems on existence, uniqueness and nonexistence of solutions of this problem.

In this paper, we introduce and study the notion of an 𝑆-weakly prime ideal which is a generalization of a weakly prime ideal. Let 𝐴 be a commutative ring and 𝑆 a multiplicative set of 𝐴. We define a proper ideal 𝑃 of 𝐴 to be 𝑆-weakly prime if there exists s ∈ S s\in S such that, for all a , b ∈ A a,b\in A with 0 ≠ a ⁢ b ∈ P 0\neq ab\in P , then s ⁢ a ∈ P sa\in P or s ⁢ b ∈ P sb\in P . We describe the behavior of an 𝑆-weakly prime property across various ring-theoretic constructions such as direct product, homomorphic image, localization, trivial ring extensions and amalgamation rings.

The present paper studies the optimization of the Bolza problem with a system of convex and nonconvex, discrete and differential state variable second-order inequality constraints by deriving necessary and sufficient conditions of optimality. The problem with a system of discrete-approximation inequalities is investigated using the proposed method of discretization and equivalence theorems for subdifferential inclusions, which greatly contributes to the derivation of adjoint discrete inclusions generated by a given system of nonlinear inequality constraints. Furthermore, we formulate sufficient conditions of optimality for the continuous problem by passing to the case of limit. A numerical example is provided to illustrate the theoretical approach’s effectiveness.

We investigate perturbations of the Moore–Penrose inverse and forward order law for the Moore–Penrose inverse in rings with involution, and thus we extend some results of Castro–Gonzalez and Hartwig to more general settings.

We prove that if there exists a continuous surjection from a metric compact space 𝑋 onto a product X × T X\times T , where 𝑇 is a T 1 T_{1} second countable topological space which has the cardinality of the continuum, then there exists a surjection from 𝑋 onto the product X × [ 0 , 1 ] X\times[0,1] , where the interval [ 0 , 1 ] [0,1] is equipped with the usual Euclidean topology.

The purpose of this paper is to solve a kind of the Riemann–Hilbert boundary value problem for ( φ , ψ ) {(\varphi,\psi)} -harmonic functions, which are linked with the use of two orthogonal bases of the Euclidean space ℝ m {\mathbb{R}^{m}} . We approach this problem using the language of Clifford analysis for obtaining an explicit expression of the solution of the problem in a Jordan domain Ω ⊂ ℝ m {\Omega\subset\mathbb{R}^{m}} with fractal boundary. Since our study is concerned with a second order differential operator, the boundary data are restricted to involve the higher order Lipschitz class Lip ⁡ ( 1 + α , Γ ) {\operatorname{Lip}(1+\alpha,\Gamma)} .

Let [ ⋅ ] {[\,\cdot\,]} be the floor function. In this paper, we prove that if 1 < c < 16559 15276 {1<c<\frac{16559}{15276}} , then every sufficiently large positive integer N can be represented in the form N = [ p 1 c ] + [ p 2 c ] + [ p 3 c ] , N=[p^{c}_{1}]+[p^{c}_{2}]+[p^{c}_{3}], where p 1 , p 2 , p 3 {p_{1},p_{2},p_{3}} are primes, such that p 1 = x 2 + y 2 + 1 {p_{1}=x^{2}+y^{2}+1} .

In the paper we consider the properties of Fourier coefficients of functions that possess derivatives of bounded variation. We investigate the convergence of the special series of Fourier coefficients with respect to general orthonormal systems (ONS). The obtained results are the best possible. We also describe the behavior of subsequences of general ONS.