We consider countable edge colourings of complete graphs and investigate the existence of monochromatic paths. We show how this depends on the sizes of the complete graphs whose edges are being coloured.

We prove two theorems showing connections between multipliers for the symmetrized Jacobi expansions and multipliers for the Dunkl transform on \(\mathbb{R}\). These results can be seen as generalizations of the classical Igari’s result relating Jacobi and Hankel multipliers. Moreover, we show a transference theorem relating Jacobi and Dunkl transplantations.

We study integral operators with kernels $$K(x,y)= k_{1}( x- A_1y) \cdots k_{m}( x-A_my),$$ \(k_{i}(x)=\frac{\Omega_{i}(x)}{|x|^{n/q_i}}\) where \(\Omega_{i} \colon \mathbb{R}^{n} \to \mathbb{R}\) are homogeneous functions of degree zero, satisfying a size and a Dini condition, Ai are certain invertible matrices, and \(\frac n{q_1}+\cdots+ \frac n{q_m} = n - \alpha, 0 \leq \alpha

We introduce the concepts of p-proximal contraction and p-proximal contractive mappings on metric spaces. Then we give some best proximity point results for such mappings. Also we provide some illustrative examples to compare our results with some earliers.

We find all the solutions of the title Diophantine equation \(P_1^p+2P_2^p + \cdots +kP_k^p=P_n^q\) in positive integer variables \((k, n)\), where \(P_i\) is the \(i^{th}\) term of the Pell sequence if the exponents p, q are included in the set \(\{1,2\}\).

We introduce a definition of \({\pi}\) being injective with respect to a generalized topology and a hereditary class where \({\pi}\) is a generalized quotient map between generalized topological spaces. This definition is mainly a sufficient condition to show several relations about a generalized topology and its induced generalized quotient topology when either is extended by a hereditary class or

We prove that if the sequence \(\langle k_n:1 \le n < \omega\rangle\) contains a so-called gap then the sequence \(\langle S^{\aleph_n}_{\aleph_{k_n}}:1 \le n < \omega\rangle\) of stationary sets is not mutually stationary, provided that \(k_n

We give a regular extension of the Menon-type identity to residually finite Dedekind domains.

We extend the notion of Aczél’s iterations of a mean to three-variable means and prove that these iterations can be used to define a continuous and scale invariant weigthing procedure on an extensive class of means.

Let n, k and a be positive integers. The Stirling numbers of the first kind, denoted by s(n, k), count the number of permutations of n elements with k disjoint cycles. Let p be a prime. Lengyel, Komatsu and Young, Leonetti and Sanna, Adelberg, Hong and Qiu made some progress in the study of the p-adic valuations of s(n, k). In this paper, by using Washington’s congruence on the generalized harmonic

Let \(\beta >1\) be a non-integer. First we show that Lebesgue almost every number has a \(\beta \)-expansion of a given frequency if and only if Lebesgue almost every number has infinitely many \(\beta \)-expansions of the same given frequency. Then we deduce that Lebesgue almost every number has infinitely many balanced \(\beta \)-expansions, where an infinite sequence on the finite alphabet \(\{0

This birthday note gives a “non-asymptotic” version of our earlier result with G. N. Sárközy and Szemerédi [3], in which Endre had the lion’s share. A hypergraph H with vertex set V defines the shadow graph G(H) whose vertex set is V and whose edge set is the set of pairs of V that are covered by some hyperedge of H. An edge coloring C of H defines a multicoloring, the shadow coloring\(C^{\prime}\)

As it was introduced by Tkachuk and Wilson in [7], a topological space X is cellular-compact if for any cellular, i.e. disjoint, family \(\mathcal{U}\) of non-empty open subsets of X there is a compact subspace \(K \subset X\) such that \(K \cap U \ne \emptyset\) for each \(U \in \mathcal{U}\). In this note we answer several questions raised in [7] by showing that (1) any first countable cellular-compact

We give an explicit simple construction for classifying spaces of maps obtained as hyperplane projections of immersions. We prove structure theorems for these classifying spaces.

In [3], we derived three results in additive combinatorics for function fields. The proofs of these results depended on a recent bound for the large sieve with sparse sets of moduli for function fields by the first and third-named authors in [1]. Unfortunately, they discovered an error in this paper and demonstrated in [2] that this result cannot hold in full generality. In the present paper, we formulate

We show that, under natural conditions, the minimal, respectively maximal tensor product of a pro-C*-algebra associated to a pro-C*-correspondence and a pro-C*-algebra is isomorphic to the pro-C*-algebra associated with a tensor product pro-C*-correspondence.

A positive operator on a cone is a linear operator that maps the cone to a subcone of itself. The extended second order cones were introduced by Németh and Zhang [17] as a working tool to solve mixed complementarity problems. Sznajder [23] determined the automorphism group and the Lyapunov (or bilinearity) ranks of these cones. Ferreira and Németh [9] reduced the problem of projecting onto the second

We introduce and investigate a class \(\mathcal {P}\) of continuous and periodic functions on \(\mathbb {R}\). The class \(\mathcal {P}\) is defined so that second-order central differences of a function satisfy some concavity-type estimate. Although this definition seems to be independent of nowhere differentiable character, it turns out that each function in \(\mathcal {P}\) is nowhere differentiable

Given two continuous functions \(f,g \colon I \to\mathbb{R}\) such that g is positive and f/g is strictly monotone, a measurable space \((T,\mathcal{A})\), a measurable family of d-variable means \(m: I^{d} \times T \to I\), and a probability measure μ on the measurable sets \(\mathcal{A}\), the d-variable mean \(M_{f,g,m;\mu} \colon I^{d} \to I\) is defined by $$M_{f,g,m;\mu}({\bf x}) :=\Bigl(\fr

In 1826 Abel started the study of the polynomial Pell equationx2 − g(u)y2 = 1. Its solvability in polynomials x(u), y(u) depends on a certain torsion point on the Jacobian of the hyperelliptic curve v2 = g(u). In this paper we study the affine surfaces defined by the Pell equations in 3-space with coordinatesx, y, u, and aim to describe all affine lines on it. These are polynomial solutions of the

The notions of α-normality and β-normality were introduced by Arhangel’skii and L. Ludwig [4]. In the present paper, we have investigated these concepts in relative sense, proved their dependency and independency on other versions of relative normality. Some relative versions of β normality are defined, their relations with various version of relative normality are discussed and it is observed that

A positive integer n is called weakly prime-additive if n has at least two distinct prime divisors and there exist distinct prime divisors \(p_{1},\ldots, p_{t}\) of n and positive integers \(\alpha_{1}, \ldots , \alpha_{t}\) such that \(n = p_{1}^{\alpha_{1}}+ \cdots + p_{t}^{\alpha_{t}}\). Erdős and Hegyvári [2] proved that, for any prime p, there exists a weakly prime-additive number which is divisible

A positive integer n is called practical if every positive integer \(m \leq n\) can be written as a sum of distinct divisors of n. For any integers \(a, b, k > 0\), we show that if \(2 \nmid a\), then there are infinitely many nonnegative integers m such that \(am^{k} + bm^{k-1}\) is practical. Let qn denote the n-th practical number. Further, when \(n \geq 7\), we prove that \(\sqrt{q_{n}+1} - \sqrt{q_n}

This survey paper summarizes the more important recent applications of the eigenpairs formulas for a family of tridiagonal matrices based on Losonczi’s seminal work of almost thirty years ago, which not only seems to have been largely ignored, but has also been re-cast or re-discovered in alternative guises by various authors since. In the course of presenting these applications, we also make contact

We prove the consistency of \(\mathfrak{u}_{\aleph_\omega} < 2^{\aleph_\omega}\). We also show that the consistency strength of this statement is the existence of a measurable cardinal \(\kappa\) with \(o(\kappa) = \kappa^{++}\) .

It is shown that under fairly weak conditions on the measure the orthonormal polynomials have almost everywhere oscillatory behavior. A simple lower bound for the amplitude of oscillation is also given in terms of the measure and the equilibrium density of the support. This bound is also shown to be exact in some situations.

We introduce and study in detail a special class of backward continued fractions that represents a generalization of Rényi continued fractions. We investigate the main metrical properties of the digits occurring in these expansions and we construct the natural extension for the transformation that generates the Rényi-type expansion. We also define the random system with complete connections associated

We examine the Borel version of the \(\sigma \)-finite chain condition of Horn and Tarski for a class of posets T(X) which have been used in the solution of their well-known problem. More precisely, we show that the poset \(T(\pi \mathbb{Q} \it )\) does not have the \(\sigma \)-finite chain condition witnessed by Borel pieces. More precisely, we define a condition on the topological spaces X under

Let \(n\) and \(k\) be positive integers such that \(n\ge k+1\) and let \(\{a_i\}_{i=1}^n\) be an arbitrary given strictly increasing sequence of positive integers. Let \(S_{n, k}:=\sum _{i=1}^{n-k} \frac{1}{ 1{\rm cm} (a_{i},a_{i+k})}\). Borwein [3] proved a conjecture of Erdős stating that if \(n\ge 2\), then \(S_{n,1}\le 1-\frac{1}{2^{n-1}}\), with the equality holding if and only if \(a_{i}=2^{i-1}\)

We prove that $$\begin{aligned} \frac{5}{6} + \sum_{k=1}^{n} \frac{\cos k \theta}{k} \ge \frac{1}{4} (1+\cos \theta )^2 \quad (n=2,3,\ldots;\ \theta \in (0, \pi )), \end{aligned}$$ where equality holds if and only if \(n = 2\) and \(\theta = \pi - \cos^{-1} \frac{1}{3}\). This refines a result of Brown and Koumandos.