We introduce the concept of mixed sumset and estimate the lower bound for its cardinality by means of the polynomial method. The result generalizes the well known Cauchy–Davenport Theorem and a theorem of Alon, Nathanson and Ruzsa regarding the lower bound for the cardinality of restricted sumsets of distinct sets in a field \(\mathbb{F}\). As a consequence of this result, we also obtain a new proof

We introduce a new class of Picard operators which includes the class of enriched contractions, enriched Kannan mappings, enriched Chatterjea mappings, and prove some fixed point theorems for these mappings. Some examples will illustrate the generality of our results.

Loomis and Whitney proved an inequality between volume and areas of projections of an open set in n-dimensional space related to the isoperimetric inequality. They reduced the problem to a combinatorial theorem proved by a repeated use of Hölder inequality. In this paper we prove a general inequality between real numbers which easily implies the combinatorial theorem of Loomis and Whitney.

For \(L/K\) a finite extension of algebraic number fields, L may or may not have a relative integral basis over K. We show the existence of relative integral basis of a biquadratic field \(L=\mathbb{Q}(\sqrt{m},\sqrt{n})\) over its quadratic subfield \(K=\mathbb{Q}(\sqrt{m})\) is equivalent to that K is a Pólya field, or equivalently all strongly ambiguous ideal classes of K are trivial.

The \(r\)-Lambert function is a generalization of the classical Lambert W function which has proven to be useful in physics and other disciplines. In this paper we construct the Riemann surface of this function. It turns out that this surface has some peculiar properties, therefore it might be useful for demonstration purposes, for those who would like to see non-standard examples of Riemann surfaces

The sine addition formula on a semigroup S is the functional equation \(f(xy) = f(x)g(y) + g(x)f(y)\) for all \(x,y \in S\). For some time the solutions have been known on groups, regular semigroups, and semigroups which are generated by their squares. The obstacle to finding the solution on all semigroups arose in the special case that g is a multiplicative function. We overcome this obstacle and

We give a construction of a set \(A \subset \mathbb N\) such that any subset \({A' \subset A}\) with \(|A'| \gg |A|^{2/3}\) is neither an additive nor multiplicative Sidon set. In doing so, we refute a conjecture of Klurman and Pohoata.

We consider the empirical two-sample U-statistic with \(\beta\)-mixing strictly stationary data and investigate its convergence in Skorohod spaces. We then provide an application of such convergence.

The present article is devoted to representations of rational numbers in terms of sign variable Cantor expansions. The main attention is given to one of the discussions given by J. Galambos in [4].

Let \(q \geq 2\) be an integer and let \(s_{q}(n)\) be the sum-of-digits function in base q of the positive integer n. In this paper we obtain asymptotic formulae for the distribution of \((s_q(n^2 {\rm mod} q^k))_{n 2\) is a prime. Furthermore, we give exact identities for the distribution of \((s_p(n^{d} {\rm mod} p^k))_{n

We discuss when an order in a Dedekind domain \(R\) is equal to \(R\) (is the maximal order in \(R\)). Every order in \(R\) is a subring of \(R\). This fact implies the existence of natural homomorphisms between objects related to orders such that the group of Cartier divisors, the Picard group, the group of Weil divisors, the Chow group and the Witt ring of an order. We examine the maximality of an

We study complete convergence and closely related Hsu–Robbins–Erdős-Spitzer–Baum–Katz series for sums whose terms are elements of linear autoregression sequences. We obtain criteria for convergence of these series expressed in terms of moment assumptions, which for “weakly dependent” sequences are the same as in classical results concerning the independent case.

A Furstenberg family \(\mathcal{F}\) is a collection of infinite subsets ofthe set of positive integers such that if \(A\subset B\) and \(A\in \mathcal{F}\), then \(B\in \mathcal{F}\). For aFurstenberg family \(\mathcal{F}\), an operator \(T\) on a topological vector space \(X\) is said tobe \(\mathcal{F}\)-transitive provided that for each non-empty open subsets \(U\), \(V\) of \(X\) the set\(\{n

Using the Littlewood–Paley decomposition technique, Fourier transform and inverse Fourier transform, the boundedness of bilinear Fourier multiplier operators in Besov spaces with variable smoothness and integrability are proved.

Let \(G\) be a non-abelian finite simple group. A famous result of Liebeck and Shalev is that there is an absolute constant \(c\) such that whenever \(S\) is a non-trivial normal subset in \(G\) then \(S^{k} = G\) for any integer \(k\) at least \(c \cdot (\log|G|/\log|S|)\). This result is generalized by showing that there exists an absolute constant \(c\) such that whenever \(S_{1}\), \(\ldots , \)

Let \(n\) be a positive integer and \(\mathcal{R}\) be a \(2\) and \(n!\)-torsion free commutative ring with unity. Let \(X\) be a locally finite pre-ordered set. If any two directed edges in each connected component of the complete Hasse diagram \((X,\mathfrak{D})\) are contained in one cycle, then every \(n\)-commuting map on the incidence algebra \(I(X,\mathcal{R})\) is proper.

Let K be an imaginary quadratic number field, and \( \mathcal{O}_K\) its ring of integers. In this article, we prove the non-existence of Diophantine \(m\)-tuples in \(\mathcal{O}_K\) with the property \(D(-1)\), for \(m > 36\).

An irregular vertex in a tiling by polygons is a vertex of one tile and belongs to the interior of an edge of another tile. In this paper we show that for any integer \(k\geq 3\), there exists a normal tiling of the Euclidean plane by convex hexagons of unit area with exactly \(k\) irregular vertices. Using the same approach we show that there are normal edge-to-edge tilings of the plane by hexagons

Let X, Y be Banach spaces and let \(A,B :X \rightarrow Y\) be two operators, where A is a linear homeomorphism and B is a \(C^{1} \)-compact operator. Sufficient weakly coerciveness conditions are provided to assert that the perturbed operator \(A+B\) is a \(C^{1} \)-diffeomorphism. The proof of our result is based on properties of Fredholm mappings as well as on local and global inverse mapping theorems

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

In this paper, we introduce vast generalizations of the Hardy-Berndt sums. They involve higher-order Euler and/or Bernoulli functions, in which the variables are affected by certain linear shifts. By employing the Fourier series technique we derive linear relations for these sums. In particular, these relations yield reciprocity formulas for Carlitz, Rademacher, Mikolas and Apostol type generalizations

Let $$K$$ be an imaginary quadratic number field and $$\mathcal{O}_K$$ be its ring of integers. We show that, if the arithmetic functions $$f, g\colon\mathcal{O}_K\rightarrow \mathbb{C}$$ both have level of distribution $$\vartheta$$ for some $$0<\vartheta\leq 1/2$$ then the Dirichlet convolution $$f*g$$ also has level of distribution $$\vartheta$$ . As an application we also obtain an analogue of

We prove existence and uniqueness of a fixed point for non-self mappings with nonlinear contractive condition defined in strictly convex Menger $$PM\mbox{-}$$ spaces.

We establish results concerning covers of spaces by compact and related sets. Several cardinality bounds follow as corollaries. Introducing the cardinal invariant $$\overline{\psi}_c(X)$$ , we show that $$|X|\leq \pi\chi(X)^{c(X)\overline{\psi}_c(X)}$$ for any topological space X. If X is Hausdorff then $$\overline{\psi}_c(X)\leq\psi_c(X)$$ ; this gives a strengthening of a theorem of Shu-Hao [24]

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].