Let \(q\ge3\) be a positive integer, and let \(D=\{0,a,b\}\) with \(\gcd(a,b)=1\). For the self-similar measure \(\mu_{q,D}\) which is generated by the iterated function system \(\{\tau_d(x)=\frac{x+d}{q}\}_{d\in D}\), it is well known that if \(q\) is divisible by 3 and \(\{a,b\}\equiv\{1,2\}\pmod 3\), \(\mu_{q,D}\) is a spectral measure with a spectrum $$\Lambda(q,C)=\Big\{\sum_{j=0}^{n}q^ja_j:a_j\in

We consider an extended version of Armacost’s problem of “description of topologically torsion elements” of the circle group and describe topologically \(s\)-torsion and topologically \(\alpha\)-torsion elements (which form the statistically characterized subgroups and \(\alpha\)-statistically characterized subgroups, recently developed in [16], [14] respectively) in terms of the support. Our main

We generalize the asymptotic estimates by Bubboloni, Luca and Spiga [2] on the number of \(k\)-compositions of n satisfying some coprimality conditions. We substantially refine the error term concerning the number of \(k\)-compositions of \(n\) with pairwise relatively prime summands. We use a different approach, based on properties of multiplicative arithmetic functions of \(k\) variables and on an

Let \(\mathcal{P}\) denote the set of primes. For a fixed dimension \(d\), Cook– Magyar–Titichetrakun, Tao–Ziegler and Fox–Zhao independently proved that any subset of positive relative density of \(\mathcal{P}^d\) contains an arbitrary linear configuration. In this paper, we prove that there exists such configuration with the step being a shifted prime (prime minus \(1\) or plus \(1\)).

We prove that if \(X^{*}\) is strictly convex, a convex function \(f\) is coercive and b-Lipschitzian iff there exist two convex function sequences \(\{f_{n}\}_{n=1}^{\infty}\) and \(\{g_{n}\}_{n=1}^{\infty}\) such that (1) \(f_{n}\leq f_{n+1}\leq f\) and \(f\leq g_{n+1}\leq g_{n}\) for all integers \(n \geq 1\); (2) \(f_{n}\) and \(g_{n}\) are continuous and Gâteaux differentiable on \(X\); (3) \(f_n

Very recently, I. Altun, M. Aslantas and H. Sahin [1] introduced the notion of \(p\)-proximal contractions and surveyed the existence of best proximity points for such class of non-self mappings in metric spaces. In this note we show that this existence result is a straightforward consequence of the same conclusion in fixed point theory.

We determine the maximum size of a graph of given order, diameter, and edge-connectivity \(\lambda\) for \(2\leq \lambda \leq 7\). This completes the determination of the maximum size of graphs with given order, diameter and edge-connectivity \(\lambda\) which had previously been done for \(\lambda=1\) and \(\lambda \geq 8\). We further prove that upper bounds on the size of a graph of given order

Let \(n > k > 1\) be integers, \([n] = \{1, \ldots, n\}\) the standard \(n\)-element set and \({[n]\choose k}\) the collection of all its \(k\)-subsets. The families \(\mathcal F_0, \ldots, \mathcal F_s \subset {[n]\choose k}\) are said to be cross-union if \(F_0 \cup \cdots \cup F_s \neq [n]\) for all choices of \(F_i \in \mathcal F_i\). It is known [13] that for \(n \leq k(s + 1)\) the geometric

We first prove some new weighted refinements of inequalities of Hardy’s type with negative powers. Next, we prove that any \(A_{\lambda }^{1}\) Muckenhoupt class with a weight \(\lambda \) belongs to some weighted Gehring class \(G_{\lambda }^{p}\) for \(p>1\). We also prove that the self-improving property of the weighted Muckenhoupt class \(A_{\lambda }^{q}\) holds. The main results give exact values

It is established that a space X with exponential \(\kappa\)-domination has density not exceeding \(\kappa\) if \(l(X) \leq\kappa\) and \(t(X) \leq\kappa\). We also introduce a weaker property called exponential \(\kappa\)-density and show that it behaves nicely under standard operations. It is proved that exponential \(\kappa\)-density is preserved by continuous images, open subspaces, arbitrary products

As generalizations of Ehresmann semigroups, P-Ehresmann semigroups were firstly introduced and investigated by Jones [12]. From a varietal perspective, P-Ehresmann semigroups are derived from reducts of regular ◦-semigroups. In this paper, inspired by the approach of Jones, P-Ehresmann semigroups are further extended to weakly P-Ehresmann semigroups derived instead from reducts of regular semigroups

This paper is devoted to the problem of where the critical points of a polynomial are relative to their zeros. Classical and new developments are surveyed along with illustrative examples. The paper finishes with a short proof of the sector theorem of Sendov and Sendov.

Using congruences, a Hoehnke radical can be defined for graphs in the same way as for universal algebras. This leads in a natural way to the connectednesses and disconnectednesses (= radical and semisimple classes) of graphs. It thus makes sense to talk about ideal-hereditary Hoehnke radicals for graphs (= hereditary torsion theories). All such radicals for the category of undirected graphs which allow

We prove that, consistently, there exists a weakly but not strongly inaccessible cardinal \(\lambda\) for which the sequence \(\langle 2^\theta:\theta<\lambda\rangle\) is not eventually constant and the weak diamond fails at \(\lambda\). We also prove that consistently diamond fails but a parametrized version of weak diamond holds at some strongly inaccessible \(\lambda\).

We continue to study one of the classic problems in general topology raised by P. S. Alexandrov: when a Hausdorff space X has a continuous bijection (a condensation) onto a compactum? We concentrate on the situation when not only X but also \(X\setminus Y\) can be condensed onto a compactum whenever the cardinality of Y does not exceed certain \(\tau\).

It was noticed that there is a misprint in the initial of the name of one of the authors in the original online version of the paper.

A subgroup \(H\) of a group G is called an \(IC\Phi\)-subgroup of G if \(H\cap [H,G]\leq \Phi(H)\). We investigate the structure of a finite group G under the assumption that some subgroups of G are \(IC\Phi\)-subgroups of G, and some results of p-nilpotency and supersolvability of finite groups are obtained.

Alon [1] proved that if \(p\) is an odd prime, \(1\le n < p\) and \(a_1,\ldots,a_n\) are distinct elements in \(Z_p\) and \(b_1,\ldots,b_n\) are arbitrary elements in \(Z_p\) then there exists a permutation of \(\sigma\) of the indices \(1,\ldots,n\) such that the elements \(a_1+b_{\sigma(1)},\ldots,a_n+b_{\sigma(n)}\) are distinct. In this paper we present a multiset variant of this result.

We show that squares of sides of length \( 1\), \(2^{-t}\), \(3^{-t}\), \(4^{-t}\),\(\ldots \) can be packed perfectly into a square, provided \( 1/2 < t \le 2/3\). Moreover, we prove that cubes of edges of length \( 1\), \(2^{-t}\), \(3^{-t}\), \(4^{-t}\), \( \ldots \) can be packed perfectly into a box, provided \( 2/3 < t \le 4/11\).

We study the half-linear advanced differential equation $$(r(t)| y'(t)|^{\alpha-1}y'(t))'+q(t)|y|^{\alpha-1}(\sigma(t))y(\sigma(t))= 0, \quad t \geq t_0 >0,$$ where \(\alpha>0, r(t)>0, q(t) >0, \sigma(t)\geq t\), and \(R(t):= \int_{t_0}^{t}r^{-1/\alpha}(s) \, {\rm d}s \to \infty\) as \(t\to \infty\). We prove that such an equation is oscillatory if $$ \lambda_*:= \liminf_{t\to \infty}\frac{R(\sigm

This paper deals with minimal reductions and core of ideals in various settings of pullback constructions with the aim of building original examples, where we explicitly compute the core. To this purpose, we use techniques and objects from multiplicative ideal theory to investigate the existence of minimal reductions in Section 2 and then develop explicit formulas for the core in Section 3. The last

A graph \(\Gamma\) is called homogenous if the group of automorphisms of \(\Gamma\) acts transitively on the set of vertices of \(\Gamma\). We will study the following properties for homogenous graphs: Removability: a vertex p is removable if the graph obtained by removing it is isomorphic to the original. Property (R): for every edge x there exists an edge y such that removing x and y disconnects

If \(\kappa > \aleph_0\) then \(\kappa \to(\kappa,\operatorname{Top}K_\kappa)^2\), i.e., every graph on \(\kappa\) vertices contains either an independent set of \(\kappa\) vertices, or a topological \(K_\kappa\), iff \(\kappa\) is regular and there is no \(\kappa\)-Suslin tree. Concerning the statement \(\omega_2\to(\operatorname{Top}K_{\omega_2})^2_\omega\), i.e., in every coloring of the edges of

Let \(D\) be a division ring and \(K\) a subfield of \(D\) which is not necessarily contained in the center \(F\) of \(D\). We study the structure of \(D\) under the condition of left algebraicity of certain subsets of \(D\) over \(K\). Among others, it is proved that if \(D^*\) contains a noncentral normal subgroup which is left algebraic over \(K\) of bounded degree \(d\), then \([D:F]\le d^2\).

We give an upper bound on the number of extensions of a triple to a quadruple for the Diophantine m-tuples with the property D(4). We also confirm the conjecture of the uniqueness of such an extension in some special cases.

Let \(E\) be the self-similar set generated by the iterated function system \(f_0(x)=\frac{x}{\beta},\quad f_1(x)=\frac{x+1}{\beta}, \quad f_{\beta+1}=\frac{x+\beta+1}{\beta}\) with \(\beta\ge 3\). Then \(E\) is a self-similar set with complete overlaps, i.e., \(f_{0}\circ f_{\beta+1}={f_{1}\circ f_1}\), but \(E\) is not totally self-similar. We investigate all of its generating iterated function systems

Given a family \(\varphi = (\varphi_1, \ldots, \varphi_d)\in \mathbb{Z}[T]^d\) of d distinct nonconstant polynomials, a positive integer \(k\le d\) and a real positive parameter \(\rho\), we consider the mean value $$M_{k, \rho} (\varphi, N) = \int_{{\rm x} \in [0,1]^k} \sup_{{\rm y} \in [0,1]^{d-k}} | S_{\varphi}({\rm x}, {\rm y}; N) |^\rho \,d{\rm x} $$ of exponential sums $$S_{\varphi}({\rm x},

We determine all pairs \((A_{f,g},A_{g,h})\) of generalized weighted quasi-arithmetic means being square iterative roots of \((A_{F,G},A_{G,H})\), that is, the equation \begin{equation*} ( A_{f,g},A_{g,h}) \circ ( A_{f,g},A_{g,h}) =(A_{F,G},A_{G,H}), \end{equation*} is solved under three times differentiability of the functions \(f, g, h, F, G, H\). As an application, some special cases are presented

We study preservation properties of relatively star-Hurewicz property under two mappings and in Alexandroff space. For a topological space X, the collection of relatively star-Hurewicz subspaces of X is an admissible \(\sigma\)-ideal in X. The characterizations of relatively star-Hurewicz spaces given using selection principles. For a paracompact space X, TWO has a winning strategy in the star-Hurewicz

We study the extension of a result of Loxton [5] on representation of algebraic integers as sums of roots of unity to Kummer extensions.

The classical density topology is the topology generated by the lower density operator connected with a density point of a set. One of the possible generalizations of the concept of a density point is replacing the Lebesgue measure by the outer Lebesgue measure. It turns out that the analogous family associated with such generalized density points is not a topology. In this case, one can prove that

A pair of surgeries on a knot is called purely cosmetic if the pair of resulting 3-manifolds are homeomorphic as oriented manifolds. An outstanding conjecture predicts that no nontrivial knots admit any purely cosmetic surgeries. Recent work of Hanselman [5] uses Heegaard Floer homology to obtain new obstructions for the existence of such surgeries. In this work, we apply those obstructions to show

We prove that there exists a continuous function on \([0,1]^2\), with a certain smoothness, whose double Fourier–Walsh series diverges on a set of positive measure by rectangles. Similar theorem is proved also for the double Walsh–Kaczmarz system.

We investigate the asymptotic distribution of perturbed geometric progression \(\{\theta^k x+ \gamma_k\}\) given by \(\theta\in (-\infty, -1)\cup (1, \infty)\) and \(\gamma_1, \gamma_2, \dots \in \mathbf {R}\). We prove that the discrepancy obeys the law of the iterated logarithm with limsup constant sensitively depending on \(\theta\) and \(\{\gamma_k \}\).

Minkowski’s classical existence theorem provides necessary and sufficient conditions for a Borel measure on the unit sphere of Euclidean space to be the surface area measure of a convex body. The solution is unique up to a translation. We deal with corresponding questions for unbounded convex sets, whose behavior at infinity is determined by a given closed convex cone. We provide an existence theorem

This paper is devoted to the studying of the weak Orlicz–Lorentz space \(\Lambda_X^{\varphi, \infty}(w)\), which can be regarded as an extension of weak Orlicz space \(L_X^{\varphi, \infty}\) and weak Lorentz space \(\Lambda_X^{p, \infty}(w)\). Results are obtained on the basic properties of convergence, normability and embedding relationships. Some interpolation theorems of operators are also given

A Sjölin–Soria–Antonov type extrapolation theorem for locally compact \(\sigma\)-compact non-discrete Hausdorff groups is proved. Applying this result it is shown that the Fourier series with respect to the Vilenkin orthonormal systems on the Vilenkin groups of bounded type converge almost everywhere for functions from the class \(L {\rm log}^{+} L {\rm log}^{+} {\rm log}^{+} {\rm log}^{+} L\).

Let \(f(x, k, d) = x(x + d)\cdots(x + (k - 1)d)\) be a polynomial with \(k \geq 2, d \geq 1\). We consider the Diophantine equation \(\prod_{i=1}^{r} f(x_i, k_i, d) = y^{2}, r \geq 1\). Using the theory of Pell equations, we affirm a conjecture of Bennett and van Luijk [3]; extend some results of this Diophantine equation for \(d=1\), and give a positive answer to Question 3.2 of Zhang [19].

The full root polytope of type A is the convex hull of all pairwise differences of the standard basis vectors which we represent by forward and backward arrows. We completely classify all flag triangulations of this polytope that are uniform in the sense that the edges may be described as a function of the relative order of the indices of the four basis vectors involved. These fifteen triangulations

Lovász proved that the chromatic number of the graph formed by the principal diagonals of an \(n\)-dimensional strongly self-dual polytope is greater than or equal to \(n+1\). There is equality if the length of the principal diagonals is greater than the Euclidean diameter of the monochromatic parts of that coloring of the unit sphere which is based on a partition of \(n+1\) congruent spherical regular

We prove that for an arbitrary real rational function r of degree n, a measure of the set \(\{x\in \mathbb{R}: |r'(x)/r(x)|\ge n\}\) is at most \(2\pi\Theta\) (\(\Theta\approx 1.347\) is the weak \((1,1)\)-norm of the Hilbert transform), and this bound is extremal. A problem of rational approximations on the whole real line is also considered.

Let \(x\to\infty\) be a parameter. Feng [5] proved that the Deshouillers–Dress–Tenenbaum’s arcsine law on divisors of the integers less than x also holds in arithmetic progressions for ``non-exceptional moduli" \(q\leqslant\exp\{(\frac{1}{4}-\varepsilon)(\log_2 x)^2\}\), where \(\varepsilon\) is an arbitrarily small positive number. We show that in the case of a prime-power modulus (\(q:=\mathfrak{p}^{\varpi}\)

Expansion of real numbers is a basic research topic in number theory. Usually we expand real numbers in one given base. In this paper, we begin to systematically study expansions in multiple given bases in a reasonable way, which is a generalization in the sense that if all the bases are taken to be the same, we return to the classical expansions in one base. In particular, we focus on greedy, quasi-greedy

We obtain some results on Ulam stability for the linear difference equation \(x_{n+1}=a_nx_n+b_n,\, n\geq0\), in a Banach space X. If there exists \(\lim_{n\to\infty}|a_n|=\lambda\), then the equation is Ulam stable if and only if \(\lambda\neq 1\). Moreover if \((|a_n|)_{n\geq 0}\) is a monotone sequence, then the best Ulam constant of the equation is \(\frac{1}{|\lambda-1|}\).

An integral basis of the simplest number fields of degrees 3, 4 and 6 over \(\mathbb{Q}\) is well-known, and widely investigated. We generalize the simplest number fields to any degree, and show that an integral basis of these fields is repeating periodically.

Let \(n, \ell\) be positive integers, \(n > 2\ell + 1\). Let X be an n-element set and \(\mathcal{F}\) an antichain, \(\mathcal{F} \subset 2^X\). Kiselev, Kupavskii and Patkós conjectured that if \(|F\cup G| \leq 2\ell + 1\) for all \(F, G \in \mathcal{F}\) then the minimum degree of \(\mathcal{F}\) is no more than \({n - 1\choose \ell - 1}\), the minimum degree of \({[n]\choose \ell}\). We prove this

For each cardinal \(\kappa\) we construct an infinite \(\kappa\)-bounded (and hence countably compact) regular space \(R_{\kappa}\) such that for any \(T_1\) space \(Y\) of pseudocharacter \(\leq\kappa\), each continuous function \(f : R_{\kappa}\rightarrow Y\) is constant. This result resolves two problems posted by Tzannes [13] and extends results of Ciesielski and Wojciechowski [4] and Herrlich

Stable basis algebras were introduced by Fountain and Gould and developed in a series of articles. They form a class of universal algebras, extending that of independence algebras, and reflecting the way in which free modules over well-behaved domains generalise vector spaces. If a stable basis algebra \(\mathbb{B}\) satisfies the distributivity condition (a condition satisfied by all the previously

Assume that \(\mathbb{K}\) is Gaussian field, and \({{a}_{\mathbb{K}}} (n) \) is the number of non-zero integral ideals in \(\mathbb{Z} [i] \) with norm \(n\). We establish an Erdős–Kac type theorem weighted by \({{a}_{\mathbb{K}}}( n^2 )^l (l\in \mathbb{Z}^{+})\) in short intervals. We also establish an asymptotic formula for the average behavior of \({{a}_{\mathbb{K}}}( n^2 )^l\) in short intervals

We show that if c is a positive integer satisfying \(2c-1=3p^l\ \hbox{or}\ 2c-1=5p^l\) with p prime and l positive integer, then the equation \({x^2 + (2c-1)^m}=c^n\) has only the positive integer solution \((x,m,n)=(c-1,1,2)\) without any congruence condition on a prime p.

We investigate Diophantine problems concerning linear combinations of polynomials of the shape \(a_0 x+a_1x(x+1)+ a_2x(x+1)(x+2)+ \cdots + a_n x(x+1)\ldots (x+n)\) with \(n\in \mathbb{N}\cup\{0\}\). We provide effective finiteness results for the power, shifted power, and quadratic polynomial values of these linear combinations, generalizing the analogous results of Hajdu, Laishram and Tengely [10]

We construct potentially new manifolds homeomorphic but not diffeomorphic to \(\mathbb{CP}^{2} \) # \(8\overline{{\mathbb{CP}}^{2}}\) and \(\mathbb{CP}^{2}\)# \(9\overline{{\mathbb{CP}}^{2}}\) via rational blowdown surgery along certain 4-valent plumbing graphs. This way all the graph classes from [5] have a representative which admits a rational blowdown leading to an exotic manifold. We emphasize

We study the sums of nondegenerate sine series with monotone coefficients and consider the sets of positivity of such functions. We obtain the sharp lower estimate of the measure of such a set on \([\pi/2, \pi]\) and a new lower bound on its measure on \([0,\pi]\). It is shown that the latter measure is at least \(\pi/2 + 0.24\) and in the case of fulfilling special conditions it is at least \(2\pi/3\)

Let A be a sequence of positive integers and P(A) be the set of all integers which are the finite sum of distinct terms of A. Let \(B=\{b_1

The generalized topology on the Cartesian product of sets can be defined by the generalized topology on the factors of the product. A G-method on a set can derive a generalized topology, which is called a G-generalized topology. On the one hand, we introduce the concept of product G-methods on sets which lead to a G-generalized topology on the Cartesian products that is different both from the Császár

The notion of \(\mu\)-monotonically T2 spaces is introduced, which is a generation of monotone T2 separation in general topological spaces. The characterization and some properties of \(\mu\)-monotonically T2 spaces are given. Besides, the definition of the box product of the topologies is extended to generalized topologies, and we prove that the box product of \(\mu\)-monotonically T2 spaces is \

An example is constructed of a subset \(A \subset \mathbb {Z}_N\) of density \({1\over 2}+o(1)\) such that all the non-trivial Fourier coefficients of the characteristic function of A are very small, but if \(x,d \in \mathbb {Z}_N\) are chosen uniformly at random, then the probability that \(x, x+d, x+2d\) and \(x+3d\) all belong to A is at most \({1\over 16}-c\), where \(c>0\) is an absolute constant

The set A is an asymptotic nonbasis of order h for an additive abelian semigroup X if there are infinitely many elements of X not in the h-fold sumset hA. For all \(h \geq 2\), this paper constructs new classes of asymptotic nonbases of order h for Z and for N0 that are not subsets of maximal asymptotic nonbases.

For \(k \in \mathbb {N}\), write S(k) for the largest natural number such that there is a k-colouring of \(\{1, \ldots ,S(k)\}\) with no monochromatic solution to \(x-y=z^2\). That S(k) exists is a result of Bergelson, and a simple example shows that \(S(k) \ge 2^{2^{k-1}}\). The purpose of this note is to show that \(S(k) \le 2^{2^{2^{O(k)}}}\) .

It is well known that for every even integer n, the complete graph \(K_{n}\) has a one-factorization, namely a proper edge coloring with \(n-1\) colors. Unfortunately, not much is known about the possible structure of large one-factorizations. Also, at present we have only woefully few explicit constructions of large one-factorizations. In particular, we know essentially nothing about the typical properties