In 1967, Heyde and Rohatgi obtained two results on the rate of convergence in the Marcinkiewicz–Zygmund strong law of large numbers. In the present work, we show that the independence requirement in their theorems is unnecessary when the common law satisfies a suitable integrability condition.

We study the quasi-nilpotency of generalized Volterra operators on spaces of power series with Taylor coefficients in weighted \(\ell ^p\) spaces \(1

We consider the pseudo-Euclidean space \((\mathbb {R}^n,g)\), \(n \ge 3\), with coordinates \(x=\left( x_1,...,x_n\right) \) and metric components \(g_{ij} = \delta _{ij}\epsilon _i\), \(1\le i, j\le n\), where \(\varepsilon _i=\pm 1\), with at least one \(\varepsilon _i=1\) and one diagonal (0,2)-tensors of the form \(T=\sum _i\epsilon _i{h_i(x)dx_i^2}\). We obtain necessary and sufficient conditions

We introduce the class of \(\alpha \)-firmly nonexpansive and quasi \(\alpha \)-firmly nonexpansive operators on r-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where \(\alpha \)-firmly nonexpansive operators coincide with so-called \(\alpha \)-averaged operators. For our more general setting, we show that \(\alpha \)-averaged operators form a subset of \(\alpha

We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a reaction which is parametric and exhibits the combined effects of a singular term and of a superdiffusive one. We prove an existence and nonexistence result for positive solutions depending on the value of the parameter \(\lambda \in \overset{\circ }{{\mathbb {R}}}_+=(0,+\infty )\).

In this paper generalized moment functions are considered. They are closely related to the well-known functions of binomial type which have been investigated on various abstract structures. The main purpose of this work is to prove characterization theorems for generalized moment functions on commutative groups. At the beginning a multivariate characterization of moment functions defined on a commutative

Explicit upper and lower bounds for the two-dimensional discrete Hardy–Littlewood maximal functions from \(\ell ^{1}({\mathbb {Z}}^{2}) \) to the space of functions of bounded variation \(\mathrm {BV}({\mathbb {Z}}^{2}) \) are obtained.

In this paper, we are interested in giving two characterizations for the so-called property L\(_{o,o}\), a local vector valued Bollobás type theorem. We say that (X, Y) has this property whenever given \(\varepsilon > 0\) and an operador \(T: X \rightarrow Y\), there is \(\eta = \eta (\varepsilon , T)\) such that if x satisfies \(\Vert T(x)\Vert > 1 - \eta \), then there exists \(x_0 \in S_X\) such

We point out to a connection between a problem of invariance of power series families of probability distributions under binomial thinning and functional equations which generalize both the Cauchy and an additive form of the Gołąb–Schinzel equation. We solve these equations in several settings with no or mild regularity assumptions imposed on unknown functions.

The subject of this note is the mixed Katugampola fractional integral of a bivariate function defined on a rectangular region in the Cartesian plane. This is a natural extension of the Katugampola fractional integral of a univariate function—a concept well-received in the recent literature on fractional calculus and its applications. It is shown that the mixed Katugampola fractional integral of a prescribed

We show that the formula of Faà di Bruno for the nth derivative of a composite function can be applied to deduce various combinatorial identities involving Lah numbers, Stirling numbers and harmonic numbers. One of our results states that for all integers m, n with \(m\ge n\ge 1\), we have $$\begin{aligned}&2^n {m\atopwithdelims ()n} (2 H_{2m-n}-H_{m-n})\\&\quad =\sum _{k=0}^{\left\lfloor n/2 \right\rfloor

In this paper, we consider the deformed Hankel transform \({\mathscr {F}}_{\kappa } \), which is a deformation of the Hankel transform by a parameter \(\kappa >\frac{1}{4}\). We introduce, via modulus of continuity, a function subspace of \(L^p(d\mu _{\kappa })\) that we call deformed Hankel Dini–Lipschitz spaces. In the case \(p = 2\), we provide equivalence theorem: we get a characterization of those

We continue our study of the mixed Einstein–Hilbert action as a functional of a pseudo-Riemannian metric and a linear connection. Its geometrical part is the total mixed scalar curvature on a smooth manifold endowed with a distribution or a foliation. We develop variational formulas for quantities of extrinsic geometry of a distribution on a metric-affine space and use them to derive Euler–Lagrange

Let \(\{U_n\}_{n\ge 0}\) and \(\{V_n\}_{n\ge 0}\) denote the sequence of ordinary ménage numbers and the sequence of straight ménage numbers respectively. In this paper, we mainly study the log-behavior of \(\{U_n\}_{n\ge 3}\) and \(\{V_n\}_{n\ge 2}\). We prove that \(\{U_n\}_{n\ge 6}\) and \(\{V_n\}_{n\ge 6}\) are log-balanced. In addition, we discuss the log-behavior of some sequences involving \(U_n\)

The main result of this paper states that for a given countable system of data \(\varDelta \), there exists a countable iterated function system consisting of Rakotch contractions, such that its attractor is the graph of a fractal interpolation function corresponding to \(\varDelta \). In this way, on the one hand, we generalize a result due to Secelean (see Univ Beograd Publ Elektrotehn Fak Ser Mat

In this work, we consider a class of analytic functions f defined in the unit disk for which the values of \(zf'/f\) lie in a parabolic region of the right-half plane. By using a well-known sufficient condition for functions to be in this class in terms of the Taylor coefficients of z/f, we introduce a subclass \(\mathcal {F}_{\alpha }\) of this class. The aim of the paper is to find the best approximation

We investigate the weighted approximation of functions in \(L_p\)-norm by Kantorovich modifications of the classical Szász–Mirakjan operator, with weights of type \((1+x)^{\alpha }\), \(\alpha \in \mathbb {R}\). By defining an appropriate K-functional we prove direct inequality for them.

In this paper, we explore a new concept of simmetry multilinear mappings and introduce a new approach to the concept of multipolynomials. We generalize several results of homogeneous polynomials and symmetric multilinear mappings, such the classic Polarization Formula. We also present a concept of coherence and compatibility for sequences of pairs formed by multipolynomial ideals and multi-ideals.

Let G be a graph of order \({\text {n}}(G)\) and vertex set V(G). Given a set \(S\subseteq V(G)\), we define the external neighbourhood of S as the set \(N_e(S)\) of all vertices in \(V(G){\setminus } S\) having at least one neighbour in S. The differential of S is defined to be \(\partial (S)=|N_e(S)|-|S|\). In this paper, we introduce the study of the 2-packing differential of a graph, which we define

In this paper, we initiate and study strictly and strongly X-positive definite functions on a hypergroup K, where X is a subset of K, and give some equivalent conditions.

We here deduce some supercongruence results for certain sums involving rising factorials using a method similar to the WZ-method. As particular cases, we confirm certain recent conjectural supercongruence of Guo.

When sectional curvatures of a Hadamard Kähler manifold are not greater than c, every trajectory half-line \(\gamma \) for a Kähler magnetic field of strength not greater than \(\sqrt{|c|}\) is unbounded and have limit point \(\gamma (\infty )\) in the ideal boundary. We take a geodesic half-line \(\sigma \) whose origin is the origin of \(\gamma \) and the limit point coincides with \(\gamma (\infty

Motivated by a recent paper of Leśniak and Snigireva [Iterated function systems enriched with symmetry, preprint], we investigate the properties of the semiattractor \(A_{\mathcal {F}\cup \mathcal {G}}^*\) of an IFS \(\mathcal {F}\) enriched by some other IFS \(\mathcal {G}\). We show that in natural cases, the semiattractor \(A_{\mathcal {F}\cup \mathcal {G}}^*\) is in fact the attractor of certain

We give formulas for the image Milnor number of a weighted-homogeneous map-germ \(({\mathbb {C}}^n,0)\rightarrow ({\mathbb {C}}^{n+1},0)\), for \(n=4\) and 5, in terms of weights and degrees. Our expressions are obtained by a purely interpolative method, applied to a result by Ohmoto. We use our approach to recover the formulas for \(n=2\) and 3 due to Mond and Ohmoto, respectively. For \(n\ge 6\)

We give a complete solution to the Borel–Ritt problem in non-uniform spaces \({\mathscr {A}}^-_{(M)}(S)\) of ultraholomorphic functions of Beurling type, where S is an unbounded sector of the Riemann surface of the logarithm and M is a strongly regular weight sequence. Namely, we characterize the surjectivity and the existence of a continuous linear right inverse of the asymptotic Borel map on \({\mathscr

Kovalevsky has developed an abstract cellular complex for representing digital images in the combinatorial grid. This paper initiates the notions of cellular homotopy and cellular path homotopy through the continuous connected preserving map and investigates some of their basic properties in the abstract cellular complex. Further, the concept of fundamental groups in the abstract cellular complex is

In this paper we present some relevant properties of Pečarić-type functions and of Mercer-type functions. We discuss both the real case (for convex functions) and the vector case (for delta-convex functions). As a result, we obtain an extension of the Hermite–Hadamard inequality. We also highlight some Schur-convexity properties. A series of recent results in the area are framed in a new vision and

We introduce three families of positive definite bivariate integral kernels whose eigenfunctions and eigenvalues can be explicitly derived in terms of classical orthogonal polynomials. All sequences of classical orthogonal polynomials are involved in at least one of our developments. We show that the well-known Karhunen–Loève expansions of the Wiener, Brownian bridge and Anderson–Darling Gaussian stochastic

Non-positive at infinity valuations are a class of real plane valuations which have a nice geometrical behavior. They are divided in three types. We study the dual graphs of non-positive at infinity valuations and give an algorithm for obtaining them. Moreover we compare these graphs attending the type of their corresponding valuation.

In this paper, we study the uniqueness of entire function and its differential-difference operators. We prove the following result: let f be a transcendental entire function of finite order, let \(\eta \) be a non-zero complex number, \(n\ge 1, k\ge 0\) two integers and let a and b be two distinct finite complex numbers. If f and \((\Delta _{\eta }^{n}f)^{(k)}\) share a CM and share b IM, then \(f\equiv

Let \({{\mathbb {K}}}\) be an algebraically closed field of characteristic zero and \({{\mathbb {G}}}_a\) be the additive group of \({{\mathbb {K}}}\). We say that an irreducible algebraic variety X of dimension n over the field \({{\mathbb {K}}}\) admits an additive action if there is a regular action of the group \({{\mathbb {G}}}_a^n = {{\mathbb {G}}}_a \times \cdots \times {{\mathbb {G}}}_a\) (n

In this paper, sharpening a theorem of Yamaguchi et al (J Diff Geom 11: 59–64, 1976), we give a complete classification of n-dimensional C-totally real minimal submanifolds with constant sectional curvature in the \((2n+1)\)-dimensional Sasakian space forms.

We prove the Hyers–Ulam stability of the functional equation $$\begin{aligned}&f(a_1x_1+a_2x_2,b_1y_1+b_2y_2)=C_{1}f(x_1,y_1)\nonumber \\ \nonumber \\&\quad +C_{2}f(x_1,y_2)+C_{3}f(x_2,y_1)+C_{4}f(x_2,y_2) \end{aligned}$$(*) in the class of functions from a real or complex linear space into a Banach space over the same field. We also study, using the fixed point method, the generalized stability of

We consider an analogue of the well-known Casson knot invariant for knotoids. We start with a direct analogue of the classical construction which gives two different integer-valued knotoid invariants and then focus on its homology extension. The value of the extension is a formal sum of subgroups of the first homology group \(H_1(\varSigma )\) where \(\varSigma \) is an oriented surface with (maybe)

On a compact connected manifold \(\mathbb {M}\), we obtain an \(L^{p}\) equivalence relation between the K-functional and the approximation of the Bochner–Riesz multiplier operator, under the sharp condition on its index. When \(\mathbb {M}\) is a compact Lie group, a similar result is obtained between the Bochner–Riesz multiplier operator and the K-functional on the Hardy space \(H^{p}\left( \mathbb

A generalization of the well–known Lucas sequence is the k–Lucas sequence with some fixed integer \(k\ge 2\). The first k terms of this sequence are \(0,\ldots ,0,2,1\), and each term afterwards is the sum of the preceding k terms. In this paper, we find all k–Lucas numbers that are concatenations of two repdigits. This generalizes a prior result which dealt with the above problem for the particular

We extend some approach to the family of symmetric means (i.e. symmetric functions \(\mathscr {M} :\bigcup _{n=1}^\infty I^n \rightarrow I\) with \(\min \le \mathscr {M}\le \max \); I is an interval). Namely, it is known that every symmetric mean can be written in a form \(\mathscr {M}(v_1,\dots ,v_n):=F(f(v_1)+\cdots +f(v_n))\), where \(f :I \rightarrow G\) and \(F :G \rightarrow I\) (G is a commutative

Two accelerated iterative methods for curves approximation are presented in this paper. These presented methods are used to reduce the degree of Bézier curves and approximate rational Bézier curves by polynomials. By employing the preconditioned progressive iterative approximation (PPIA), we approximate the points sampled from target curves, and generate polynomial approximations. The equi-parametric

Two different notions of approximate Birkhoff–James orthogonality in normed linear spaces have been introduced by Dragomir and Chmieliński. In the present paper we consider a global and a local approximate symmetry of the Birkhoff–James orthogonality related to each of the two definitions. We prove that the considered orthogonality is approximately symmetric in the sense of Dragomir in all finite-dimensional

New scales of grand variable exponent Hajłasz–Sobolev and Hölder spaces are introduced. Embeddings between these spaces are established under the log-Hölder continuity condition on exponent functions of spaces. Spaces are defined, generally speaking, on quasi-metric measure spaces with doubling condition (spaces of homogeneous type) but the results are new even for Euclidean domains with Lebesgue measure

Wefelscheid (Untersuchungen über Fastkörper und Fastbereiche, Habilitationsschrift, Hamburg, 1971) generalised the well-known Theorem of Artin/Schreier about the characterization of formally real fields and the fundamental result of Baer/Krull to near-fields. In the last fifty years arose from the Theorem of Baer/Krull a theory, which analyses the entirety of the orderings of a field (E. Becker, L

Over the past decades, the split inverse problem has been widely studied, and one of the objectives of those researches is to invent some efficient algorithms for approximating solutions. Most of those algorithms depend on the norms of bounded linear operators; however, the calculation of the operator norms is not an easy task in general practice. In this article, we study and investigate the split

In this research, we introduce the fractional Copson matrix and define the associated sequence spaces. We investigate the inclusion relations, dual spaces and matrix transformations of these new sequence spaces. Moreover, investigating the compactness of matrix operators and obtaining the norm of this matrix on the difference sequence spaces are another motive of this research.

The aim of this paper is to establish the existence and the nonexistence of solutions for double-phase problems depending on two parameters. This work improves and complements the existing ones in the literature. There seems to be no results on the nonexistence of solutions for double-phase problems.

The aim of the paper is to generalize decomposition theorems showed in Bagheri-Bardi et al. (Linear Algebra Appl 583:102–118, 2019; Linear Algebra Appl 539:117–133, 2018) by a unified approach. We show a general decomposition theorem with respect to a hereditary property. Then the vast majority of decompositions known in the algebra of Hilbert space operators is generalized to elements of Baer \(*\)-rings

In this note we formulate sufficient conditions under which a certain class of nonlinear and nonautonomous differential equations of second order is Hyers–Ulam stable. This class consists of equations obtained by perturbing Hill’s equation of the form \(x''=(\lambda ^2(t)-\lambda '(t))x\).

Let L be a finite-dimensional Lie algebra and I, J two ideals of L. Let \(\mathrm {Der}_J^I(L)\) denote the set of all derivations of L whose images are in I and send J to zero and let \(\mathrm {Der}^n_c(L)\) denote the set of all derivations \(\alpha \) of L for which \(\alpha (x)\in [x,L^n]\) for all \(x\in L\). In this paper, we have shown that if L and H are two n-isoclinic Lie algebras, then

Friedrich Engel and David Hilbert learned to know each other at Leipzig in 1885 and exchanged letters in particular during the next 15 years which contain interesting information on the academic life of mathematicians at the end of the 19th century. In the present article we will mainly discuss a statement by Hilbert himself on Moritz Pasch’s influence on his views of geometry, and on personnel politics

Let K be a body in the plane and let \(\lambda <1\) be a positive real number. Suppose that for every \(x\in \mathbb R^2\) such that \(\lambda K + x\) is exteriorly tangent to K, the common support cone of K and \(\lambda K + x\) has a constant angle \(\alpha \). Suppose that one of the following conditions is satisfied: (a) K is a body of constant width, (b) K is centrally symmetric. Then K is a Euclidean

S. Marcus showed that there is a quasicontinuous function from the interval [0, 1] to \(\mathbb {R}\) which is not Lebesgue measurable. We prove that if X is either an uncountable Polish space or a locally pathwise connected perfectly normal topological space with at least one non isolated point, then there is a quasicontinuous non Borel measurable function from X to [0, 1]. We also found new conditions

We use the concept of moduli of continuity to give some generalised characterisation theorems in some generalised Hölder type spaces of functions over \({\mathbb {R}}\), characterising certain Lipschitz type conditions on functions in terms of their Fourier transforms. Our main results are inspired by a theorem of Titchmarsh, combined with the notion of moduli of continuity, for characterising Jacobi–Lipschitz

According to the generalized Pólya theorem, the Gaussian distribution on the real line is characterized by the property of equidistribution of a monomial and a linear form of independent identically distributed random variables. We give a complete description of \({{\varvec{a}}}\)-adic solenoids for which an analog of this theorem is true. The proof of the main theorem is reduced to solving some functional

We study the Bishop–Phelps–Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that \(C_0(L)\) spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space L. To this end, on the one hand, we provide some techniques allowing to pass the BPBp-nu for compact operators from subspaces

The general position number \(\mathrm{gp}(G)\) of a connected graph G is the cardinality of a largest set S of vertices such that no three distinct vertices from S lie on a common geodesic; such sets are refereed to as gp-sets of G. The general position number of cylinders \(P_r\,\square \,C_s\) is deduced. It is proved that \(\mathrm{gp}(C_r\,\square \,C_s)\in \{6,7\}\) whenever \(r\ge s \ge 3\),

The existence of fundamental cardinal exponential B-splines of positive real order \(\sigma \) is established subject to two conditions on \(\sigma \) and their construction is implemented. A sampling result for these fundamental cardinal exponential B-splines is also presented.

Let \(\mathcal {H}\) be a complex Hilbert space and let A be a positive operator on \(\mathcal {H}\). We obtain new bounds for the A-numerical radius of operators in semi-Hilbertian space \(\mathcal {B}_A(\mathcal {H})\) that generalize and improve on the existing ones. Further, we estimate an upper bound for the \(\mathbb {A}\)-operator seminorm of \(2\times 2\) operator matrices, where \(\mathbb

In this note, we study discrete time majority dynamics over an inhomogeneous random graph G obtained by including each edge e in the complete graph \(K_n\) independently with probability \(p_n(e)\). Each vertex is independently assigned an initial state \(+1\) (with probability \(p_+\)) or \(-1\) (with probability \(1-p_+\)), updated at each time step following the majority of its neighbors’ states

In this article, we investigate the summability of the formal power series solutions in time of a certain class of inhomogeneous partial differential equations with a polynomial semilinearity, and with variable coefficients. In particular, we give necessary and sufficient conditions for the k-summability of the solutions in a given direction, where k is a positive rational number entirely determined

Motivated by the work of Frazier; and Gabardo and Nashed, we study \(P^{th}\)-stage nonuniform discrete wavelet frames (\(P^{th}\)-stage NUDW frames, in short) for \(\ell ^2(\Lambda )\), a nonuniform discrete space. In nonuniform discrete wavelet frames, the translation set is not necessary a group but a spectrum which is based on the theory of spectral pairs. We characterize first-stage nonuniform

This paper is concerned with the mathematical analysis of solutions to following class of magnetic Kirchhoff problems $$\begin{aligned} \left\{ \begin{array}{ll} -K\Big (\displaystyle \int _\Omega |\nabla _A u|^2dx\Big )\Delta _Au = g(x,|u|^2)u &{} \quad \mathrm{in} \ \Omega \\ {\,u|_{\partial \Omega }=0}\\ \end{array} \right\} , \end{aligned}$$ where \(\Omega \) is a smooth bounded domain in \({{\mathbb