The notion of essential amenability of Banach algebras has been defined and investigated. We introduce this concept for Fr´echet algebras. Then numerous well-known results concerning the essential amenability of Banach algebras are generalized for Fréchet algebras. Moreover, related results for the Segal–Fréchet algebras are also provided. As the main result, it is proved that if (𝒜,pℯ) is an amenable

For any p ∈ (0, ∞], ω > 0, β ∈ (0, 2ω), and any measurable set B ⊂ Id ≔ [0, d], μB ≤ β, we deduce a sharp Remez-type inequality$$ {\left\Vert {x}_{\pm}\right\Vert}_{\infty}\le \frac{{\left\Vert {\left(\varphi +c\right)}_{\pm}\right\Vert}_{\infty }}{{\left\Vert \varphi +c\right\Vert}_{L_p\left({I}_{2\upomega}\backslash {B}_y^c\right)}}{\left\Vert x\right\Vert}_{L_p\left({I}_d\backslash B\right)} $$

We consider a semidirect product G = K ⋉ V where K is a connected compact Lie group acting by automorphisms on a finite-dimensional real vector space V equipped with an inner product <, >. By Ĝ we denote the unitary dual of G and by 𝔤‡/G we denote the space of admissible coadjoint orbits, where 𝔤 is the Lie algebra of G. It was indicated by Lipsman that the correspondence between Ĝ and 𝔤‡/G is bijective

We consider a boundary-value problem for the nonlinear integrodifferential equation$$ {u}^{\prime \prime \prime \prime }-m\left(\underset{0}{\overset{l}{\int }}{u}^{\prime 2} dx\right){u}^{{\prime\prime} }=f\left(x,u,{u}^{\prime}\right),\kern1em m(z)\ge \upalpha >0,\kern1em 0\le z<\infty, $$ simulating the static state of the Kirchhoff beam. The problem is reduced to a nonlinear integral equation,

The Shen connection cannot be obtained by using Matsumoto’s processes from the other well-known connections. Hence, Tayebi and Najafi introduced two new processes called Shen’s C and L-processes and showed that the Shen connection is obtained from the Chern connection by Shen’s C-process. We study the Shen’s C- and L-process on Berwald connection and introduce two new torsion-free connections in Finsler

We first describe the conditions for being elastica or nonelastica for a lightlike elastic Cartan curve in the Minkowski space \( {\mathbbm{E}}_1^4 \) by using the Bishop orthonormal vector frame and associated Bishop components. Then we compute the energy of the lightlike elastic and nonelastic Cartan curves in the Minkowski space \( {\mathbbm{E}}_1^4 \) and investigate its relationship with the energy

We generalize the relationship between the classical moment problem and the spectral theory of Jacobi matrices. We present the solution of the two-dimensional half-strong moment problem and suggest an analog of Jacobi-type matrices associated with the two-dimensional half-strong moment problem and the corresponding system of polynomials orthogonal with respect to a measure with compact support in the

A new decomposition for square matrices is constructed by using two known matrix decompositions. A new characterization of the core-EP order is obtained by using this new matrix decomposition. We also use this matrix decomposition to investigate the minus, star, sharp and core partial orders in the setting of complex matrices.

We consider a time-optimal problem for a set-valued linear control system in the case where a section of the solution of the system coincides with a target set. For this problem, we establish both the solvability conditions and the optimal time and optimal controls. The results are illustrated by model examples.

We consider the most general class of multipoint boundary-value problems for systems of linear ordinary differential equations of any order whose solutions belong to a given Sobolev space \( {W}_p^{n+r} \), with n ≥ 0, r ≥ 1, and 1 ≤ p ≤ ∞. We establish constructive sufficient conditions under which the solutions of the analyzed problems are continuous with respect to the parameter ε for ε = 0 in the

We establish constructive necessary and sufficient conditions of solvability and a scheme of construction of the solutions for a nonlinear boundary-value problem unsolved with respect to the derivative. We also suggest convergent iterative schemes for finding approximate solutions of this problem. As an example of application of the proposed iterative scheme, we find approximations to the solutions

We study the problem of boundedness of fractional integral operators on a variable Herz-type Hardy space \( H{\overset{\cdot }{K}}_{p\left(\cdot \right),q\left(\cdot \right)}^{\alpha \left(\cdot \right)}\left({\mathrm{\mathbb{R}}}^n\right) \) by using the atomic decomposition.

We establish the rate of convergence to the exponential distribution in the general limit theorem for the extreme values of regenerative processes. We also suggest some applications of this result to the birth and death processes and to the queue length processes.

For any partition 𝜎 of the set ℙ of all primes, it is proved that if every complete Hall set of type 𝜎 for a finite 3'-group G is reducible into a certain subgroup H of G, then H is 𝜎-subnormal in G.

Integral manifolds are very useful in studying the dynamics of nonlinear evolution equations. We consider a nondensely defined partial differential equation $$ \frac{du}{dt}=\left(A+B(t)\right)u(t)+f\left(t,{u}_t\right),\kern0.72em t\in \mathrm{\mathbb{R}}, $$(1) where (A,D(A)) satisfies the Hille–Yosida condition, (B(t))t∈R is a family of operators in L(D(A),X) satisfying certain measurability and

In the metric L2, we establish exact inequalities, which relate the best approximations by trigonometric “angles” for the functions f(x, y) differentiable and 2𝜋-periodic in each variable to the integrals containing the moduli of continuity of higher order for mixed derivatives of these functions. For some classes of functions defined by the moduli of continuity, we determine the Kolmogorov and linear

We study the problem of covering of the vertices of a graph associated with a finite vector space introduced by Das [Comm. Algebra, 44, 3918–3926 (2016)] such that any vertex can be uniquely identified by examining the vertices from the covering. For this purpose, we use the locating-dominating sets and the identifying codes, i.e., closely related concepts. We find the location-domination number and

Two meromorphic functions f and g are said to share a set S ⊂ ℂ∪ {∞} with weight l ∈ ℕ∪{0}∪{∞} if Ef (S, l) = Eg(S, l), where$$ {E}_f\left(S,l\right)=\underset{\alpha \in S}{\cup}\left\{\left(z,t\right)\in \mathrm{\mathbb{C}}\times \left.\mathrm{\mathbb{N}}\right|f(z)=a\kern0.5em \mathrm{with}\kern0.5em \mathrm{multiplicity}\kern1em p\right\}, $$ provided that t = p for p ≤ l and t = p + 1 for p >

We present a survey of the results on the exact rates of approximation of functions by linear means of Fourier series and Fourier integrals. The corresponding K-functionals are expressed via the special moduli of smoothness.

Within the framework of variable-exponent Morrey and Morrey–Herz spaces, we prove some boundedness results for the commutator of Marcinkiewicz integrals with rough kernels. The proposed approach is based on the theory of variable exponents and on the generalization of the BMO-norms.

We analyze the problem of estimation of the error of approximation of a branched continued fraction, which is a multidimensional generalization of a continued fraction. By the method of fundamental inequalities, we establish truncation error bounds for the branched continued fraction$$ {\sum}_{i_{1=1}}^N\frac{a_{i(1)}}{1}+{\sum}_{i_{2=1}}^{i_1}\frac{a_{i(2)}}{1}+{\sum}_{i_{3=1}}^{i_2}\frac{a_{i(3)}}{1}+\cdots

Let R be a ring and let ΩR be the set of maximal right ideals of R. An R-module M is called an sd-Rickart module if, for every nonzero endomorphism f of M, Imf is a fully invariant direct summand of M. We obtain a characterization for an arbitrary direct sum of sd-Rickart modules to be sd-Rickart. We also obtain a decomposition of an sd-Rickart R-module M provided that R is a commutative Noetherian

We establish the solvability of a nonlocal multipoint (in time) problem regarded as a generalization of the Cauchy problem for the evolution equation with pseudodifferential operator (differentiation operator of infinite order) with initial conditions from the space of generalized functions of the ultradistribution type.

We present some facts about the smoothness of solutions of the initial-boundary-value problems for the parabolic system of partial differential equations$$ {\displaystyle \begin{array}{c}{u}_t-{\left(-1\right)}^mP\left(x,t,{D}_x\right)u=f\left(x,t\right)\kern1em \mathrm{in}\kern1em \Omega \times \left(0,T\right),\\ {}\frac{\partial^ju}{\partial {v}^j}=0\kern1em \mathrm{on}\kern1em \left(\mathrm{\partial

In the present paper, we establish bounds for the sum of the moduli of right eigenvalues of a quaternionic matrix. As a consequence, we establish bounds for the right spectral radius of a quaternionic matrix. We also present a minimal ball in 4D spaces that contains all Geršgorin balls of a quaternionic matrix. As an application, we introduce the estimation for the right eigenvalues of quaternionic

We construct a Boehmian space of quaternion-valued functions by using the quaternionic fractional convolution. By applying the convolution theorem, the quaternionic fractional Fourier transform is extended to the case of Boehmians and its properties are established.

We consider a three-point boundary-value problem for a p-Laplacian dynamic equation on time scales. We prove the existence of at least three positive solutions of the boundary-value problem by using the Avery and Peterson fixed-point theorem. The conditions used in this case differ from the conditions used in the major part of available papers. As an interesting point, we can mention the fact that

In a Hilbert space defined as the image of a unit ball under the action of a compact operator, we solve the problems of optimal recovery of elements according to their first n Fourier coefficients known with errors. Similar problems are also solved for the scalar products of elements from two different classes.

We establish existence conditions and construct bounded solutions of a system of linear inhomogeneous differential equations of the first order with rectangular matrices.

We establish necessary and sufficient conditions for the exponential dichotomy of the solutions of linear difference equations with piecewise constant operator coefficients.

Let FG be the group algebra of a finite 2-group G over a finite field F of characteristic two and let ⊛ be an involution that arises from G. The ⊛-unitary subgroup of FG is denoted by V⊛(FG) and defined as the set of all normalized units u satisfying the property u⊛ = u−1. We establish the order of V⊛(FG) for all involutions ⊛ arising from G, where G is a finite cyclic 2-group, and show that all ⊛-unitary

The paper deals with the existence and multiplicity of nontrivial weak solutions for the p(x)-Kirchhofftype problem$$ {\displaystyle \begin{array}{c}-M\left(\underset{\Omega}{\int}\frac{1}{p(x)}{\left|\Delta u\right|}^{p(x)} dx\right){\Delta}_{p(x)}^2u=f\left(x,u\right)\kern0.6em \mathrm{in}\kern0.48em \Omega, \\ {}u=\Delta u=0\kern0.48em \mathrm{on}\kern0.48em \mathrm{\partial \Omega }.\end{array}}

Assume that a continuous 2𝜋 -periodic function f defined on the real axis changes its sign at 2s, s ∈ ℕ, points yi : −𝜋 ≤ y2s < y2s−1 < ... < y1 < 𝜋 and that the other points yi, i ∈ ℤ, are defined by periodicity. Then, for any natural n > N(k, yi), where N(k, yi) is a constant that depends only on k ∈ ℕ and mini=1,...,2s{yi − yi+1}, we construct a trigonometric polynomial Pn of order ≤ n, which

We describe the asymptotic behavior of the logarithms of entire functions of improved regular growth with zeros on a finite system of rays in the metric of Lq[0, 2𝜋].

We study translation-invariant Gibbs measures for the Blume–Capel model with wand on a Cayley tree of order k. We find the exact critical value 𝜃cr = 1 such that, for 𝜃 ≥ 𝜃cr , there exists a unique translation-invariant Gibbs measure and, for 0 < 𝜃 < 𝜃cr , there are exactly three translation-invariant Gibbs measures in the presence of a wand in the analyzed model. In addition, we study the problem

It is known that any number x ∈ (0; 1] ≡ Ω has a unique Engel expansion$$ x=\sum \limits_{n=1}^{\infty}\frac{1}{\left({p}_1(x)+1\right)\dots \left({p}_n(x)+1\right)}, $$ where pn(x) ∈ ℕ, pn+1(x) ≥ pn(x) for all n ∈ ℕ. This means that pn(x) is a well-defined measurable function on the probability space (Ω, ℱ, λ), where ℱ is the σ-algebra of Lebesgue-measurable subsets of Ω and λ is the Lebesgue measure

For the λ-model on a Cayley tree of order k ≥ 2, we describe the set of periodic and weakly periodic ground states corresponding to normal divisors of index 2 of the group representation of Cayley tree.

We establish upper estimates for the distortion of the modulus of families of curves under mappings from the Sobolev class whose dilatation is locally integrable. As a consequence, we prove theorems on the local and boundary behaviors of these mappings.

We prove that any Hermitian matrix whose trace is integer and all eigenvalues lie in the segment [1 + 1/(k − 3),k − 1 − 1/(k − 3)] can be represented as a sum of k orthogonal projections. For the sums of k orthogonal projections, it is shown that the ratio of the number of eigenvalues that do not exceed 1 to the number of eigenvalues that are not smaller than 1 (with regard for the multiplicities)

Let (M,g) be a Riemannian manifold and let TM be its tangent bundle equipped with a Riemannian (or pseudo-Riemannian) lift metric derived from g. We give a classification of infinitesimal fiber-preserving conformal transformations on the tangent bundle.

We consider a generalization, under weaker conditions, of the main theorem on quasi-σ-power increasing sequences applied to |A, θn|k summability factors of infinite series and Fourier series. We obtain some new and known results related to the basic summability methods.

We construct some Cheney–Sharma-type operators defined on a triangle with two or three curved edges, their product, and Boolean sum. We study the interpolation properties of these operators and the degree of exactness.

We give the full classification of left-invariant metrics of an arbitrary signature on the Lie group corresponding to the real hyperbolic space. It is shown that all metrics have constant sectional curvature and that they are geodesically complete only in the Riemannian case.

We consider symmetric Dirac operators on bounded time scales. Under general boundary conditions, we describe extensions (dissipative, accumulative, self-adjoint, etc.) of these symmetric operators. We construct a self-adjoint dilation of the dissipative operator. Hence, we determine the scattering matrix of dilation. Then we construct a functional model of this operator and define its characteristic

We deal with the solution of the integral equation with generalized Mittag-Leffler function \( {E}_{\alpha, \beta}^{\upgamma, \mathrm{q}}(z) \) specifying the kernel with the help of a fractional integral operator. The existence of the solution is justified and necessary conditions for the integral equation admitting a solution are discussed. In addition, the solution of the integral equation is obtained

Sharp Ostrowski-type inequalities are proved for multidimensional sets and functions of bounded variation.

In the case where either r = 2, k = 1 or r = 3, k = 1, 2, for any q, p ≥ 1, β ∈ [0, 2π), and a Lebesgue-measurable set B ⊂ I2π ≔ [−π/2, 3π/2], μB ≤ β, we prove a sharp Kolmogorov–Remez-type inequality \( {\left\Vert {f}^{(k)}\right\Vert}_q\le \frac{\left\Vert \varphi r-k\right\Vert q}{E_0{{\left(\varphi r\right)}_L^{\alpha}}_p\left({I}_{2\uppi}/{B}_{2m}\right)}{\left\Vert f\right\Vert}_{L_p}^{\alp

We prove that each totally disconnected closed subset E of a domain G in the complex plane is removable for analytic functions f(z) defined in G \ E and such that, for any point z0 𝜖 E, the real or imaginary part of f(z) vanishes at z0.

We study three-dimensional trans-Sasakian manifolds admitting η-Ricci solitons. Actually, we investigate manifolds whose Ricci tensor satisfy some special conditions, such as cyclic parallelity, Ricci semisymmetry, and 𝜙-Ricci semisymmetry, after reviewing the properties of the second-order parallel tensors on these manifolds. We determine the form of the Riemann curvature tensor for trans-Sasakian

We study the relationship between the class of entire functions of completely regular growth of order 𝜌 and the class of entire functions with bounded l-index, where l(z) = |z|𝜌−1 + 1 for |z| ≥ 1. Possible applications of these functions in the analytic theory of differential equations are considered. We formulate three new problems on the existence of functions with given properties that belong

We establish approximation properties of Cesàro (C,−α,−β) means with α, β 𝜖 (0, 1) for the Vilenkin–Fourier series. This result enables one to establish a condition sufficient for the convergence of the means \( {\sigma}_{n,m}^{-\alpha, -\beta } \) (x, y, f) to f(x, y) in the Lp-metric.

We approximate ε-isometries of the unit sphere in \( {\ell}_2^n \) and \( {\ell}_{\infty}^n \) by linear isometries.

We obtain vector analogs of the Abel and Dirichlet criteria.

We consider the Stark operator \( T=-\frac{d^2}{dx^2}+x+q(x) \) on the semiaxis 0 ≤ x < ∞ with Dirichlet boundary condition at the origin. By the method of transformation operators, we study the direct and inverse spectral problems, deduce the main integral equation for the inverse problem, and prove that this equation is uniquely solvable. We also propose an effective algorithm of reconstruction of

We introduce a (p, q)-analog of the poly-Euler polynomials and numbers by using the (p, q)-polylogarithm function. These new sequences are generalizations of the poly-Euler numbers and polynomials. We present several combinatorial identities and properties of these new polynomials and also show some relations with (p, q)-poly-Bernoulli polynomials and (p, q)-poly-Cauchy polynomials. The (p, q)-analogs

We prove some results on the existence and uniqueness of fixed points defined on a b-metric space endowed with an arbitrary binary relation. As applications, we obtain some statements on the coincidence of points involving a pair of mappings. Our results generalize, extend, modify and unify several well-known results and, especially, the results obtained by Alam and Imdad [J. Fixed Point Theory Appl

We consider a boundary-value problem for degenerate equations with discontinuous coefficients and establish the unique strong solvability (almost everywhere) of this problem in the corresponding weighted Sobolev space.

We deal with the existence result for nonlinear elliptic equations of the form Au + g(x, u,∇u) = f, where the term –div (a(x, u, ∇u)) is a Leray–Lions operator from a subset of \( {W}_0^1{L}_M\left(\Omega \right) \) into its dual. The growth and coercivity conditions on the monotone vector field a are prescribed by an N-function M, which not necessarily satisfies a Δ2-condition. Therefore, we use Orlicz–Sobolev

We study the solvability of the nonlocal boundary-value problem for a differential equation with weak nonlinearity. By using the Nash–Mozer iterative scheme, we establish the solvability conditions for the posed problem in the Hilbert H¨ormander spaces of functions of several real variables, which form a refined Sobolev scale.

Assume that G is a finite group, π(G) = {s} ∪ σ, s > 2, Σ is a set of Sylow σ-subgroups in which one subgroup is taken for each pi 2 σ, and R(G) is the largest normal soluble subgroup in G (soluble radical of G). Moreover, suppose that each Sylow pi-subgroup Gpi 𝜖 Σ normalizes the s-subgroup T(i) ≠ 1 of the group G. In this case, we establish the conditions under which s divides |R(G)|.