We consider the problem of maximization of the product of inner radii of n disjoint domains symmetric about the unit circle and of the 𝛾 th power of the inner radius of the domain with respect to zero. We solve the problem for n = 2, n = 3, and some 𝛾 > 1.

We study multiplicity results for a perturbed fourth-order problem on the real line with a perturbed nonlinear term depending on one real parameter. Our approach is based on the variational methods and critical point theory proposed in [G. Bonanno, “ Nonlin. Anal., 75, 2992–3007 (2012)].

We investigate the existence of solutions for a fractional-order boundary-value problem by using some fixed-point theorems. As applications, we present examples illustrating our main results.

We establish new generalized trapezoid-type inequalities for complex functions defined on the unit circle via the functions of bounded variation and functions satisfying the Hölder-type condition. These results are also used to deduce the quadrature-rule formula.

It is shown that the nth quasitopological homotopy group of a topological space is isomorphic to the (n − 1)th quasitopological homotopy group of its loop space. By using this fact, we obtain some results about quasitopological homotopy groups. Finally, with the help of the long exact sequence of a based pair and a fibration in the qTop introduced by Brazas in 2013, we obtain some results in this field

We deduce a fractional integral equality for twice differentiable mappings defined on an m-invex set. By using this equality, we obtain new estimates generalizing trapezium-like inequalities for mappings whose second-order derivatives are generalized relative semi-(m, h1,h2)-preinvex via fractional integrals. We also discuss some new special cases that can be deduced from our main results.

We find periodic solutions of the Coulomb d-dimensional (d = 1, 2, 3) equation of motion for two identical negative point charges in the field of six identical positive point charges fixed at the vertices of a convex symmetric hexagon or of an octahedron. These systems possess an equilibrium configuration. Their periodic solutions are obtained with the help of the Lyapunov central limit theorem.

We consider the problem of finding asymptotic solutions to singular perturbed linear differential algebraic equations with simple turning points and propose an algorithm for the construction of these asymptotic solutions.

We present the definitions of two new Lyapunov functionals. These functionals are applied to establish sufficient conditions guaranteeing the asymptotic stability, uniform stability, and boundedness of solutions of certain nonlinear Volterra integrodifferential equations of the first order. The obtained results improve and extend known results available from literature. We also propose examples to

We study the growth of a Dirichlet series \( F(s)={\sum}_{n=1}^{\infty }{f}_n\exp \left\{s{\uplambda}_n\right\} \) with zero abscissa of absolute convergence with respect to the entire Dirichlet series \( G(s)={\sum}_{n=1}^{\infty }{g}_n\exp \left\{s{\uplambda}_n\right\} \) by using generalized quantities of an order \( {\upvarrho}_{\beta, \beta}^0{\left[F\right]}_G=\lim\ {\sup}_{\sigma \uparrow 0}\frac{\beta

We consider a quite general problem from the geometric theory of functions, namely, the problem of finding the maximal value of the product of inner radii of n nonoverlapping domains that contain points of the unit circle and are symmetric with respect to this circle and the γ power of the inner radius of a domain containing the origin. The posed problem is solved for n ≥ 20 and \( 1<\gamma \le {n

For a nonlinear kinetic Boltzmann equation that describes the model of rough spheres, we construct its approximate solution in the form of a continual distribution with global Maxwellians. We establish sufficient conditions for the coefficient functions and hydrodynamic parameters appearing in the distribution, which enable one to make the analyzed error as small as desired.

We consider a special configuration in which a finite group A acts by automorphisms on a finite group G and the semidirect product GA acts on the vector space V by linear transformations and discuss the existence of a regular A-module in VA.

We propose a new reciprocity theorem for the Hardy sum s5(h, p). In addition, a hybrid mean-value problem involving the Hardy sum s4(h, p) and a Kloosterman sum is studied and two exact computational formulas are obtained.

We propose an iterative method for solving the following iterative functional-differential equation: x ′′ (t) = ⋋1x(t) + ⋋2x[2](t) + … + ⋋nx[n](t) + f(t).

With the use of the theory of generalized inversion of operators and integral operators, we obtain a criterion of solvability and determine the general form of solutions of a linear boundary-value problem for the integrodifferential equation with degenerate kernel in a Banach space.

Two meromorphic functions are said to share a set S ⊂ ℂ∪{∞} ignoring multiplicities (IM) if S has the same preimages under both functions. If any two nonconstant meromorphic functions sharing a set IM are identical, then the set is called a “reduced unique-range set for meromorphic functions” [RURSM (or URSM-IM)]. From the existing literature, it is known that there exists a RURSM with 17 elements

By using variational methods and critical-point theorems, we prove the existence of nontrivial solutions for one-dimensional fourth-order equations. The multiplicity results are also presented.

For a class of linear descriptor systems, we establish new criteria for the existence of the regularities of control guaranteeing the asymptotic stability and satisfying the required estimate for the weighted level of decay of bounded disturbances. We propose a procedure of generalized H∞-optimization of the descriptor systems with controlled and observed outputs. The main computational procedures

We propose a divergent version of the Stokes formula for a Banach manifold with uniform atlas.

We prove that the idempotent barycenter map restricted to the points with nontrivial fibers is a trivial fibration with Hilbert cube fibers whenever it is open.

We establish new criteria of compatibility for a linear system of equations (equivalent to the Kronecker–Capelli theorem) or inequalities (equivalent to the S. Chernikov theorem) connected with the conditions of existence of linear interpolation polynomials in Euclidean spaces.

We study some new spaces of sequences arising from the notation of generalized de la Vallée-Poussin means and introduce the spaces of strongly λ-almost summable sequences. We also consider some topological results and the characterization of strongly λ-almost regular matrices.

We propose some refinements for the second inequality in \( \frac{1}{2}\left\Vert A\right\Vert \le w(A)\le \left\Vert A\right\Vert, \) where A ∈ B(H). In particular, if A is hyponormal, then, by refining the Young inequality with the Kantorovich constant K K(⋅, ⋅), we show that \( w(A)\le \frac{1}{2{\operatorname{inf}}_{\left\Vert x\right\Vert =1}\zeta (x)}\left|\left\Vert A\right.\right|++\left|\left

We obtain analogs of the results on equicontinuity of families of quasiregular mappings that take no values from a fixed continuum. We prove that these families are equicontinuous whenever the quasiconformality characteristics of the mappings have finite mean oscillation at every inner point. In this context, we also prove the equicontinuity of generalized quasiisometries of Riemannian manifolds.

Let ℙ be the set of all prime numbers. A subgroup H of a finite group G is called ℙ-subnormal if either H = G or there exists a chain of subgroups H = H0 ≤ H1 ≤ … ≤ Hn = G such that |Hi : Hi − 1| ∈ ℙ, 1 ≤ i ≤ n. We prove that any finite group with ℙ-subnormal Sylow p-subgroup of odd order is p-solvable and any group with ℙ-subnormal generalized Schmidt subgroups is metanilpotent.

We find periodic solutions of the Coulomb d-dimensional (d = 1, 2, 3) equations of motion for two identical negative point charges in the field of four identical positive point charges fixed at vertices of a rectangle. Systems of this kind have equilibrium configurations. Periodic solutions are obtained with the help of the Lyapunov center theorem.

Over the (1, 2)-dimensional supercircle, we investigate the second cohomology space associated with the Lie superalgebra\( \mathcal{K} \)(2) of vector fields on the supercircle S1|2 with coefficients in the space of weighted densities. We explicitly give the 2-cocycle spanning for these cohomology spaces.

We prove the following theorems: (1) every spherical convex body W of constant width \( \varDelta (W)\ge \frac{\uppi}{2} \) can be covered by a disk of radius \( \varDelta (W)+\arcsin \left(\frac{2\sqrt{3}}{3}\cos \frac{\varDelta (W)}{2}\right)-\frac{\uppi}{2}; \) (2) every reduced spherical convex body R of thickness \( \varDelta (R)<\frac{\uppi}{2} \) can be covered by a disk of radius arctan \(

We obtain new Kolmogorov-type sharp inequalities estimating the norm of the Marchaud fractional derivative \( {\left\Vert {D}_{-}^kf\right\Vert}_{\infty } \) of a function f defined on the positive half line in terms of ‖f‖p, 1 < p < ∞ , and ‖f〞‖1. We also solve the following related problems: the Stechkin problem of the best approximation of the operator \( {D}_{-}^k \) by linear bounded operators

Let R = ⊕j≥0Rj be a homogeneous Noetherian ring with semilocal base ring R0. Let R+ = ⊕j≥1Rj be the irrelevant ideal of R. For two finitely generated graded R-modules M and N, we investigate several results on the properties of vanishing, Artiniannes, and tameness of the graded R-modules \( {H}_{R+}^i\left(M,N\right). \)

We describe the direct and inverse spectral problems for the block Jacobi type matrices corresponding to the two-dimensional half-strong real moment problem. In particular, we obtain three matrices with three-diagonal block structure acting in an l2-type space as commuting self-adjoint operators; moreover, two of these matrices are mutually inverse.

We consider the Cauchy problem for an evolution equation used to model bidirectional surface waves in convecting fluids. We study the existence, uniqueness, and asymptotic properties of global solutions to the initial-value problem associated with this equation in ℝn. We also obtain some polynomial decay estimates for the energy.

To the article “On the Cardinality of Unique Range Sets with Weight One,” by B. Chakraborty and S. Chakraborty, Vol. 72, No. 7, pp. 1164–1174, December, 2020.

Given i.i.d. ℝd-valued stochastic processes X1(t), . . . ,Xp(t), p ≥ 2, with stationary increments, a minimal condition is provided for the occupation measure \( {\mu}_t(B)=\underset{\left[0,t\right]}{\int }1B\left({X}_1\left({s}_1\right)-{X}_2\left({s}_2\right),\dots, {X}_{p-1}\left({s}_{p-1}\right)-{X}_p\left({s}_p\right)\right){ds}_1\dots {ds}_p, \) B ⊂ ℝd(p−1), to be absolutely continuous with

We solve a selection problem for multidimensional SDE dX(t) = a(X𝜖(t)) dt + 𝜖σ(X𝜖(t)) dW(t), where the drift and diffusion are locally Lipschitz continuous outside a fixed hyperplane H. It is assumed that X𝜖 (0) = x0 𝜖 H, the drift a(x) has a Hölder asymptotics as x approaches H, and the limit ODE dX(t) = a(X(t)) dt does not have a unique solution. It is shown that if the drift pushes the solution

We prove the existence of a sticky-reflected solution to the heat equation on the space interval [0, 1] driven by colored noise. The process can be interpreted as an infinite-dimensional analog of the sticky-reflected Brownian motion on the real line but, in this case, the solution obeys the ordinary stochastic heat equation, except the points where it reaches zero. The solution has no noise at zero

We present a survey of the Skorohod differentiability of measures on linear spaces, which also gives new proofs of some key results in this area parallel with some new observations.

We study the distribution of a Brownian motion conditioned to start from the boundary of an open set G and to stay in G for a finite period of time. The characterizations of distributions of this kind in terms of certain singular stochastic differential equations are obtained. The accumulated results are applied to the study of boundaries of the clusters in some coalescing stochastic flows on ℝ.

We prove the solvability of Itô stochastic equations with uniformly nondegenerate bounded measurable diffusion and drift in Ld+1(Rd+1). Actually, the powers of summability of the drift in x and t could be different. Our results seem to be new even if the diffusion is constant. The method of proving solvability belongs to A.V. Skorokhod. Weak uniqueness of solutions is an open problem even if the diffusion

We establish the rate of weak convergence in the fractional step method for the Arratia flow in terms of the Wasserstein distance between the images of the Lebesgue measure under the action of the flow. We introduce finite-dimensional densities that describe sequences of collisions in the Arratia flow and derive an explicit expression for them. For the discretized initial interval, we also discuss

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 boundary-value problem for the nonlinear integrodifferential equation u ′ ′ ′ ′ − m ∫ 0 l u ′ 2 dx u ″ = f x u u ′ , m z ≥ α > 0 , 0 ≤ z < ∞ , $$ {u}^{\prime \prime \prime \prime }-m\left(\underset{0}{\overset{l}{\int }}{u}^{\prime 2} dx\right){u}^{{\prime\prime} }=f\left(x,u,{u}^{\prime}\right),\kern1em m(z)\ge \upalpha >0,\kern1em 0\le z<\infty, $$ simulating the static state of the

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

We first describe the conditions for being elastica or nonelastica for a lightlike elastic Cartan curve in the Minkowski space E 1 4 $$ {\mathbbm{E}}_1^4 $$ 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 E 1 4 $$ {\mathbbm{E}}_1^4 $$ and investigate its relationship

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 $$ {W}_p^{n+r} $$ , 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

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 K ⋅ p ⋅ , q ⋅ α ⋅ ℝ n $$ H{\overset{\cdot }{K}}_{p\left(\cdot \right),q\left(\cdot \right)}^{\alpha \left(\cdot \right)}\left({\mathrm{\mathbb{R}}}^n\right) $$ 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.

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

UDC 512.5 In this paper, we investigate the problem of covering the vertices of a graph associated to a finite vector space as introduced by Das [Commun. Algebra, 44, 3918 – 3926 (2016)], such that we can uniquely identify any vertex by examining the vertices that cover it. We use locating-dominating sets and identifying codes, which are closely related concepts for this purpose. We find the location-domination

UDC 517.9 Two meromorphic functions $f$ and $g$ are said to share the set $S\subset \mathbb{C}\cup\{\infty\}$ with weight $l\in\mathbb{N}\cup\{0\}\cup\{\infty\},$ if $E_{f}(S,l)=E_{g}(S,l),$ where $$$E_{f}(S,l)=\bigcup_{a \in S} \big \{(z,t) \in \mathbb{C}\times\mathbb{N} \bigm| f(z)=a \; \text{with multiplicity} \;p \big \},$$ where $t=p$ if $p\leq l$ and $t=p+1$ if $p>l.$ In this paper, we improve