Abstract In the present paper a formula for calculation of the density function \(f_{\rho}(x)\) of the distance between two independent points randomly and uniformly chosen in a bounded convex body \(D\) is given. The formula permits to find an explicit form of density function \(f_{\rho}(x)\) for body with known chord length distributions. In particular, we obtain an explicit expression for \(f_{\rho}(x)\)

Abstract The paper continues the research to reconstruct a convex body in \({\mathbb{R}}^{n}\) from the distribution of characteristics of its \(k\)-dimensional sections (\(k

Abstract The paper considers measures in the space \(\mathbb{E}\) of planes in \(\mathbb{R}^{3}\), and combinatorial decompositions for their values on ‘‘Buffon sets’’ in \(\mathbb{E}\). These decompositions, written in terms of a ‘‘wedge function’’ depending on the measure, have been known since long in Combinatorial Integral Geometry, yet their explicit expressions have been well established only

Abstract In this paper, we derive an \(L^{p}\) version of the de Rham theorem. The key is an \(L^{p}\) version of the Nec̆as inequality. Using this result and the variational method, we show the existence of a solution to the Maxwell–Stokes type system.

Abstract It is proven that, if the Riemann sums of a trigonometric series are bounded everywhere, perhaps, except for some countable set and converge in measure to a bounded function \(f\), then this series is the Fourier series of the function \(f\). This implies the generalizations of the Cantor and Young theorems.

Abstract The integrable function universal for classes \(L^{p}[0,1]\), \(p\in\left(0,1\right)\) with respect to the Walsh system is constructed in terms of signs. The Fourier–Walsh series of this function converges everywhere in \((0,1)\) and in the metrics \(L^{p}[0,1]\), \(p\in(0,1)\), and its Fourier coefficients by the Walsh system are in decreasing order. After selecting the appropriate signs

Abstract In the paper the formulas are provided for the derivatives of Cauchy-type integral \(K[u]\) which are smooth on the skeleton of the polydisk of functions \(u\). These formulas express the derivatives of the order \(m\) of \(K[u]\) through the derivatives of lower order (Theorem 2.1). They are used for estimating the smoothness of the derivatives of the Cauchy-type integral in terms of Hölder

Abstract In this paper we generalize the concept of the Szegö kernel by putting the weight of integration in the definition of the inner product in the Szegö space. We give some sufficient conditions for the weight in order for the Szegö kernel of the correspoding space to exist. We give examples of weights on unit ball for which there is no Szegö kernel of the corresponding Szegö space. Then using

Abstract In this paper, at first we study boundedness and compactness criterions for generalized composition operator from the Lipschitz space into the Zygmund space. Then we estimate the essential norm of this operator.

Abstract Landesman-Lazer’s type efficient sufficient conditions are established for the solvability of the two-point boundary value problem \(u^{(4)}(t)=p(t)u(t)+f(t,u(t))+h(t)\) for \(a\leq t\leq b\), \(u^{(i)}(a)=0\), \(u^{(i)}(b)=0,\quad(i=0,1)\), where \(h,p\in L([a,b];R)\) and \(f\in K([a,b]\times R;R)\), in the case where the linear problem \(w^{(4)}(t)=p(t)w(t)\), \(w^{(i)}(a)=0\), \(w^{(i)}(b)=0\)

Abstract The complete description of weakly invertible elements in weighted \(L^{p}\) spaces of entire Fock-type functions is obtained in the paper.

erratum

Abstract \(\mathbb{I}_{n}\) (or \(\tilde{\mathbb{I}}_{n}\), respectively) denotes the set of polynomials in \(n\) variables \(P(\xi)=P(\xi_{1},\xi_{2},...,\xi_{n})\) with real coefficients such that \(|P(\xi)|\rightarrow\infty\) as \(|\xi|\rightarrow\infty\) (respectively, \(\tilde{P}(\xi):=\sum_{\nu}|D^{\nu}P(\xi)|\rightarrow\infty\) as \(|\xi|\rightarrow\infty\). The necessary and sufficient conditions

Abstract In this article, we study the existence results of large positive weak solution for nonlinear system with singular weights (1.4), where \(\Omega\) is a bounded domain of \(R^{n}\) with boundary \(\partial\Omega\), \(0\in\Omega\), \(10\). Here, there is no any sign-changing conditions on \(a\) or \(b\). The proof of the main results is based on the sub-supersolutions method. Application and

Abstract In this work we obtain a sufficient condition for analytic functions to belong to certain subclasses of p-valently starlike functions of order \(\beta\) and p-valently close-to-convex functions of order \(\beta\). Further, we get a generalization of some of the well-known results.

Abstract For weighted \(L^{p}\)-classes of holomorphic functions in the unit disc, a family of weighted integral representations with a weighted function of type \(|w|^{2\varphi}\cdot(1-|w|^{2\rho})^{\beta}\) and with reproducing Mittag–Leffler kernels are obtained.

Abstract In this paper, we prove some factorization theorems of Cesàro and Copson spaces on an arbitrary time scale \(\mathbb{T}\), which offer enhancements of dynamic Copson’s and Hardy’s inequalities. Our results enhance, among others, the best-known forms of dynamic Hardy’s inequality. The main results will be proved by employing the time scales Hölder’s inequality and the derived time scales power

Abstract In this paper, we introduce \(n\)-variables mappings which are mixed additive-quadratic in each variable. We show that such mappings can be described by a equation, namely, by a multi-mixed additive-quadratic functional equation. The main goal is to extend the applications of a fixed point method to establish the Hyers-Ulam stability for the multi-mixed additive-quadratic mappings.

Abstract This note is framed in the field of complex analysis and deals with some types of interpolating sequences for Lipschitz functions in the unit disk. We introduce recursion between each point of a sequence and the next. We also add interpolation by the derivative, linking its values to those that the function takes. On the supposition that the sequences are quite contractive and lie in a Stolz

Abstract In this work we introduce the notion of the excess risk in the setup of estimation of the Gaussian mean when the observations are corrupted by outliers. It is known that the sample mean loses its good properties in the presence of outliers [5, 6]. In addition, even the sample median is not minimax-rate-optimal in the multivariate setting. The optimal rate of the minimax risk in this setting

Abstract In this paper we have found the necessary and sufficient conditions for the power integrability with a weight of the sum of the sine and cosine series whose coefficients belong to a subclass of \(\gamma RBVS\) class.

Abstract In this paper, the existing results concerning difference operator sharing two sets have been extended up to the most general form, namely linear difference operator. Furthermore, we have been able to find out the specific form of the function. A considerable number of examples have been exhibited throughout the paper pertinent with different issues.

Abstract The Heun functions satisfy linear ordinary differential equations of second order with certain singularities in the complex plane. The first order derivatives of the Heun functions satisfy linear second order differential equations with one more singularity. In this paper we compare these equations with linear differential equations isomonodromy deformations of which are described by the Painlevé

Abstract In this paper we obtain, that if the partial sums \(\sigma_{q_{k}}(x)\) of a Franklin series converge in measure to a function \(f\), the ratio \(\frac{q_{k+1}}{q_{k}}\) is bounded and the majorant of partial sums satisfies to a necessary condition, then the coefficients of the series are restored by the function \(f\).

Abstract This paper deals with the problem of finding some upper bound estimates for the maximal modulus of the polar derivative of a complex polynomial on a disk under certain constraints on the zeros and on the functions involved. A variety of interesting results follow as special cases from our results.

Abstract In this paper we prove the Noether theorem with the multiplicities described by PD operators. Despite the known analog versions in this case the provided conditions are necessary and sufficient. We also prove the Cayley–Bacharach theorem with PD multiplicities. As far as we know this is the first generalization of this theorem for multiple intersections.

Abstract In the present paper a formula for calculation of the density function \(f_{\rho}(x)\) of the distance between two independent points randomly and uniformly chosen in a bounded convex body \(D\) is given. The formula permits to find an explicit form of density function \(f_{\rho}(x)\) for body \(D\) with known chord length distributions. In particular, we obtain an explicit expression for

Abstract This work is concerned with the solvability of multi-point boundary value problems for fractional differential equations with nonlinear growth at the resonance. Existence results are obtained with the use of the coincidence degree theory. As an application, we discuss an example to illustrate the obtained results.

Abstract For some large classes of differential equations of the first order we give bounds for Ahlfors-Shimizu characteristics of meromorphic solutions in the complex plane of these equations. The considered equations largely generalize algebraic ones for which the obtained results imply the known Goldberg theorem. Characteristics of meromorphic solutions in a given domain weren’t studied at al. We

Abstract One of the main problems in prediction theory of second-order stationary processes, called direct prediction problem, is to describe the asymptotic behavior of the best linear mean squared one-step ahead prediction error variance in predicting the value \(X(0)\) of a stationary process \(X(t)\) by the observed past of finite length \(n\) as \(n\) goes to infinity, depending on the regularity

Abstract In this paper, we investigate the following two Painlevé III equations: \(\overline{w}\underline{w}(w^{2}-1)=w^{2}+\mu\) and \(\overline{w}\underline{w}(w^{2}-1)=w^{2}-\lambda w\), where \(\overline{w}:=w(z+1)\), \(\underline{w}:=w(z-1)\) and \(\mu\) (\(\mu\neq-1\)) and \(\lambda\notin\{\pm 1\}\) are constants. We discuss the questions of existence of rational solutions, of Borel exceptional

Abstract By using lemmas of Dubinin and Osserman some results for rational functions with fixed poles and restricted zeros are proved. The obtained results strengthen some known results for rational functions and, in turn, produce refinements of some polynomial inequalities as well.

Abstract A finite group \(G\) is said to be a generalized Frobenius group with kernel\(F\), if \(F\) is a proper nontrivial normal subgroup of \(G\) and for every element \(Fx\) of prime order of the quotient group \(G/F\) the coset \(Fx\) of the group \(G\) over \(F\) has only \(p\)-elements for some prime \(p\) depending on \(x\). This article considers generalized Frobenius groups with insoluble

Abstract A group is called an \(n\)-torsion group if it has a system of defining relations of the form \(r^{n}=1\) for some elements \(r\), and for any of its finite order element \(a\) the defining relation \(a^{n}=1\) holds. In this paper, we prove that all the finite subgroups of the relatively free \(n\)-torsion groups are cyclic groups. Notice that for each rank \(m>1\) and for any odd \(n\geq

Abstract In this paper, we study the problem of normality of meromorphic functions with multiple values and obtain three normality criteria. Examples are given to illustrate that the conditions in our results are necessary.

Abstract For most of the cases of bounded measurement errors fuzzification of calculations can be used. In the case of reconstructing convex body by random line segments we introduce a fuzzy convex body concept and define orientation dependent distribution of the length of line segment. We consider several properties of the latter.

Abstract In this paper we prove that the quadratic partial sums of a multiple Franklin series with indices \(2^{\nu}\), \(\nu=1,2,...\), cannot converge to \(+\infty\) on a set of positive measure.

Abstract The paper is devoted to the study of a class of integral equations with a symmetric kernel and with convex nonlinearity on the positive semiaxis. Existence and uniqueness theorems for a nonnegative and bounded solution are proved. The qualitative properties of the constructed solution are investigated. At the end of the paper, some particular examples for the above mentioned class of equations

Abstract In this paper, by using variational methods and critical point theory we investigate the existence of solutions for the following fractional Kirchhoff-type equation: $$\begin{cases}S\left(\int\limits_{\mathbb{R}}{|{}_{-\infty}D_{t}^{\alpha}u(t)|}^{2}dt\right){}_{t}D_{\infty}^{\alpha}({}_{-\infty}D_{t}^{\alpha}u(t))+l(t)u(t)=f(t,u(t)),\quad t\in\mathbb{R},\\ u\in H^{\alpha}(\mathbb{R}),\end{cases}$$

In this paper we construct an integrable function of two variables for which the double Fourier-Walsh series converges both by rectangles and by spheres. Besides, we show that the coefficients of the series on the spectrum are positive and are arranged in decreasing order in all directions. Also, it is proved that after a suitable choice of signs for the Fourier coefficients of the series the spherical

In this paper we prove some strong convergence theorems for partial sums with respect to Vilenkin system.

The main purpose of the present paper is to investigate a number of useful properties such as sufficiency criteria, distortion bounds, coefficient estimates, radius of starlikness and radius of convexity for a new subclass of analytic functions, which are defined here by means of a newly defined q-linear differential operator.

A group is called an n-torsion group if it has a system of defining relations of the form rn = 1 for some elements r, and for any of its finite order element a the defining relation an = 1 holds. It is assumed that the group can contain elements of infinite order. In this paper, we show that for every odd n ≥ 665 for each n-torsion group can be constructed a theory similar to that of constructed in

Motivated by the ordinary Gabor frames in L2(ℝd) and operator-valued frames on abstract Hilbert spaces, we investigate operator-valued Gabor frames associated with locally compact Abelian groups. Necessary and sufficient conditions for an operator-valued Gabor frame to admit a Parseval/tight Gabor dual are given. In particular, we consider a special case, which includes the case of ordinary Gabor frames

In this paper, we prove that if {nk} is an arbitrary increasing sequence of natural numbers such that the ratio nk+1/nk is bounded, then the nk-th partial sum of a series by Franklin system cannot converge to +∞ on a set of positive measure. Also, we prove that if the ratio nk+1/nk is unbounded, then there exists a series by Franklin system, the nk-th partial sum of which converges to +∞ almost everywhere

It is known that for a wide class of discrete-time stationary processes possessing spectral densities f, the variance σ2n(f) of the best linear unbiased estimator for the mean depends asymptotically only on the behavior of the spectral density f near the origin, and behaves hyperbolically as n → ∞. In this paper, we obtain necessary as well as sufficient conditions for exponential rate of decrease

In this paper, we define a new concept of analytic Feynman integral on theWiener space, which is called the generalized analytic Feynman integral, to explain various physical circumstances. Furthermore, we evaluate the generalized analytic Feynman integrals for several important classes of functionals.We also establish various properties of these generalized analytic Feynman integrals. We conclude

In this paper, we focus on a conjecture concerning uniqueness problem of meromorphic functions sharing three distinct polynomials with their difference operators, which is mentioned in Chen and Yi (Result Math v. 63, pp. 557–565, 2013), and prove that it is true for meromorphic functions of finite order. Also, a result of Zhang and Liao, obtained for entire functions (Sci China Math v. 57, pp. 2143–2152

Different generalized Cesáro summation methods are compared with each other. Analogous of Hardy’s theorem, concerning the order of the partial sums of trigonometric Fourier series, for generalized Cesáro means are obtained.

The paper deals with the question of convergence of multiple Fourier-Haar series with partial sums taken over homothetic copies of a given convex bounded set \(W\subset\mathbb{R}_+^n\) containing the intersection of some neighborhood of the origin with \(\mathbb{R}_+^n\). It is proved that for this type sets W with symmetric structure it is guaranteed almost everywhere convergence of Fourier-Haar series

The paper is devoted to the solvability questions of the Wiener-Hopf integral equation in the case where the kernel K satisfies the conditions 0 ≤ K ∈ L1(ℝ), \(\int_{-\infty}^{\infty} K(t)dt>1\), K(±x) ∈ C(3)(ℝ+), (−1)nK(±x)(n)(x) ≥ 0, x ∈ ℝ+, n =1, 2, 3. Based on Volterra factorization of the Wiener-Hopf operator, and invoking the technique of nonlinear functional equations, we construct real-valued

In this paper, we study bivariate homogeneous interpolation polynomials. We show that the homogeneous Lagrange interpolation polynomial of a sufficiently smooth function converges to a homogeneous Hermite interpolation polynomial when the interpolation points coalesce.

The present paper is devoted to the study of existence and uniqueness of a solution and Ulam type stabilities for Volterra delay integro-differential equations on a finite interval. Our analysis is based on the Pachpatte’s inequality and Picard operator theory. Examples are provided to illustrate the stability results obtained in the case of a finite interval. Also, we give an example to illustrate

We derived a convergence result for a sequential procedure known as alternating maximization (minimization) to the maximum likelihood estimator for a pretty large family of models - Generalized Linear Models. Alternating procedure for linear regression becomes to the well-known algorithm of Alternating Least Squares, because of the quadraticity of log-likelihood function L(υ). In Generalized Linear

The paper considers some functions depending on the realizations of a random process with log-normal distribution and two types of expectations. The interpretations of these functions and expectations are given in terms of actuarial mathematics. The comparison of the behavior of these two types of expectations is given using the Black-Scholes formulas. Criteria for a random process to obey a stochastic

In this paper we study the Dirichlet problem in the half-space for regular hypoelliptic equations. Applying a special integral representation, we construct approximate solutions for this problem and thereby prove correct solvability of the problem.

We define sparse operators of strong type on abstract measure spaces with ball-bases. Weak and strong type inequalities for such operators are proved.

In this paper the concept of interassociativity via hyperidentities of associativity is extended and describe the set of semigroups which are {i, j}-interassociative to the free semigroup and the free commutative semigroup, where i, j = 1, 2, 3.

In 1987 Harris proved-among others that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ Lp such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space Lp(II2) (1 ≤ p < 2) with subsequence of triangular partial means \(S_{2^A}^\Delta(f)\) of the double Walsh-Fourier series convergent in measure

Let {X(t), t ∈ ℝ} be a centered real-valued stationary Gaussian process with spectral density f. The paper considers a question concerning asymptotic distribution of tapered Toeplitz type quadratic functional \(Q_T^h\) of the process X(t), generated by an integrable even function g and a taper function h. Sufficient conditions in terms of functions f, g and h ensuring central limit theorems for standard