Abstract: Abstract Every finite non-nilpotent group can be extended by a term operation such that solving equations in the resulting algebra is NP-complete and checking identities is co-NP-complete. This result was firstly proven by Horváth and Szabó; the term constructed in their proof depends on the underlying group. In this paper we provide a uniform term extension that induces hard problems. In doing so we also characterize a big class of solvable, non-nilpotent groups for which extending by the commutator operation suffices. PubDate: 2019-03-13

Abstract: Abstract The classifying spaces of cobordisms of singular maps have two fairly different constructions. We expose a homotopy theoretical connection between them. As a corollary we show that the classifying spaces in some cases have a simple product structure. PubDate: 2019-02-28

Abstract: Abstract Let the finite soluble group \({G = G_{1}G_{2} \cdots G_{r}}\) be the product of pairwise mutually permutable subgroups \({G_{1}, G_{2}, \ldots, G_{r}}\) , let h(G) and \({\ell_{p}(G)}\) be respectively the Fitting length and the p-length of G. The aim of this paper is to prove that \({h(G) \leq {\rm max} \{h(G_{i}) \mid i = 1, 2, \ldots, r\}+1}\) and \({\ell_{p}(G) \leq {\rm max} \{\ell_{p}(G_{i}) \mid i = 1, 2, \ldots, r\}+1}\) . PubDate: 2019-02-26

Abstract: Abstract We present estimations of the roots of r-Dowling, r-Lah and r-Dowling–Lah polynomials. It is known that these polynomials have simple, real and non-positive roots. We give bounds for them and we also compute the real magnitude of the roots via computational methods. PubDate: 2019-02-26

Abstract: Abstract The order dual \({[{\rm Fil}{R_{R}}]^{{\rm du}}}\) of the set \({{\rm Fil}{R_{R}}}\) of all right topologizing filters on a fixed but arbitrary ring R is a complete lattice ordered monoid with respect to the (order dual) of inclusion and a monoid operation ‘ \({:}\) ’ that is, in general, noncommutative. It is known that \({[{\rm Fil}{R_{R}}]^{{\rm du}}}\) is always left residuated, meaning, for each pair \({\mathfrak{F}, \mathfrak{G} \in {\rm Fil}{R_{R}}}\) there exists a smallest \({\mathfrak{H} \in {\rm Fil}{R_{R}}}\) such that \({\mathfrak{H}: \mathfrak{G} \supseteq \mathfrak{F}}\) , but is not, in general, right residuated (there exists a smallest \({\mathfrak {H}}\) such that \({\mathfrak{G} : \mathfrak{H} \supseteq \mathfrak{F}}\) ). Rings R for which \({[{\rm Fil}{R_{R}}]^{{\rm du}}}\) is both left and right residuated are shown to satisfy the DCC on left annihilator ideals and possess only finitely many minimal prime ideals. It is shown that every maximal ideal P of a commutative ring R gives rise to an onto homomorphism of lattice ordered monoids \({\hat{\varphi}_{P}}\) from \({[{\rm Fil}{R}]^{{\rm du}}}\) to \({[{\rm Fil}{R_{P}}]^{{\rm du}}}\) where RP denotes the localization of R at P. The kernel \({\equiv_{\hat{\varphi}_{P}}}\) of \({\hat{\varphi}_{P}}\) is a congruence on \({[{\rm Fil}{R}]^{{\rm du}}}\) whose properties we explore. Defining \({{\rm Rad}({\rm Fil}{R})}\) to be the intersection of all congruences \({\equiv_{\hat{\varphi}_{P}}}\) as P ranges through all maximal ideals of R, we show that for commutative VNR rings R, \({{\rm Rad}({\rm Fil}{R})}\) is trivial (the identity congruence) precisely if R is noetherian (and thus a finite product of fields). It is shown further that for arbitrary commutative rings R, \({{\rm Rad}({\rm Fil}{R})}\) is trivial whenever \({{\rm Fil}{R}}\) is commutative (meaning, the monoid operation ‘ \({:}\) ’ on \({{\rm Fil}{R}}\) is commutative). This yields, for such rings R, a subdirect embedding of \(... PubDate: 2019-02-26

Abstract: Abstract Exceptional orthogonal polynomials fulfil recurrence relations with constant, and with variable dependent coefficients. Considering the second type relations we can define multi-indexed polynomials of the second kind. In some cases they are also exceptional orthogonal polynomials. The other types of multi-indexed polynomials of the second kind are investigated in case of 2-step Darboux transform. PubDate: 2019-02-26

Abstract: Abstract Let A be a sequence of positive integers and P(A) be the set of all integers which can be represented as the finite sum of distinct terms of A. By improving a result of Hegyvári, Chen and Fang [2] proved that, for a sequence of integers \({B = \{b_{1} < b_{2} < \cdots \}}\) , if \({b_{1} \in \{4, 7, 8\} \cup \{b : b \geq 11\}}\) and \({b_{n+1} \geq 3b_{n} + 5}\) for all \({n \geq 1}\) , then there exists an infinite sequence A of positive integers for which \({P(A) = \mathbb{N} \setminus B}\) ; on the other hand, if b2 = 3b1 + 4, then such A does not exist. In this paper, for b2 = 3b1 + 5, we determine the critical value for b3 such that there exists an infinite sequence A of positive integers for which \({P(A) = \mathbb{N} \setminus B}\) . PubDate: 2019-02-22

Abstract: Abstract This paper studies the extrema of some affine invariant functionals related to the volume of the Orlicz–Lorentz centroid body introduced by Nguyen. We obtain some variants of the Orlicz–Lorentz Busemann–Petty centroid inequality, and also prove the reverse form of these inequalities in the two-dimensional case. PubDate: 2019-02-22

Abstract: Abstract We find all positive integer solutions in x, y and n of \({x^{2}+19^{2k+1}=4y^{n}}\) for any non-negative integer k. PubDate: 2019-02-15

Abstract: Abstract We prove a p-nilpotency criterion for finite groups in terms of the element orders of its p′-reduced sections that extends a nilpotency criterion by Tărnăuceanu. PubDate: 2019-02-01

Abstract: Abstract We consider the Diophantine equation $$y^{p} = \frac{f(x)}{g(x)},$$ where \({x \in \mathbb{Z}}\) and \({y \in \mathbb{Q}}\) are unknowns, f(x) and g(x) are non-zero integer polynomials in variable x and p is prime. We give bounds for x, when \({(x, y) \in \mathbb{Z} \times \mathbb{Q}}\) is a solution of the equation. This improves the results of some recent papers. PubDate: 2019-02-01

Abstract: Abstract The main purpose of this paper is to investigate the distribution, simplicity and monotonicity of the roots of a class of trinomial equations that appears in certain financial mathematics problems. Furthermore, we show that it is possible to decompose some classes of trinomial equations into polynomials generated by a three term recurrence relation. PubDate: 2019-02-01

Abstract: Abstract We consider a problem of spectral synthesis in the topological vector space \({\mathcal{M}(G)}\) of tempered functions on a discrete Abelian group G. It is proved that the space of tempered solutions of a convolution system on discrete Abelian groups admits spectral synthesis, that is the space of tempered solutions of a convolution system coincides with the closed linear span in \({\mathcal{M}(G)}\) of all exponential monomial solutions of this system. PubDate: 2019-02-01

Abstract: Abstract Moments of probability measures on a group can be obtained from so called (generalized) moment functions of a given order. Characterization theorems for moment function sequences on different types of groups and hypergroups have been obtained. We study spherical and moment functions on affine groups and describe spherical and moment functions on the affine group of the SU(n) group. PubDate: 2019-02-01

Abstract: Abstract Motivated by the density condition in the sense of Heinrich for Fréchet spaces and by some results of Schlüchtermann and Wheeler for Banach spaces, we characterize in terms of certain weakly compact resolutions those Fréchet spaces enjoying the property that each bounded subset of its Mackey* dual is metrizable. We also characterize those Köthe echelon Fréchet spaces \({\lambda _{p}(A)}\) as well as those Fréchet spaces Ck (X) of real-valued continuous functions equipped with the compact-open topology that enjoy this property. PubDate: 2019-02-01

Abstract: Abstract In the setting of convex metric spaces, we introduce the two geometric notions of uniform convexity in every direction as well as sequential convexity. They are used to study a concept of proximal normal structure. We also consider the class of noncyclic relatively nonexpansive mappings and analyze the min-max property for such mappings. As an application of our main results we conclude with some best proximity pair theorems for noncyclic mappings. PubDate: 2019-02-01

Abstract: Abstract Let \({K_{w^{*}}(X^{*},Y)}\) denote the set of all w*−w continuous compact operators from X* to Y. We investigate whether the space \({K_{w^{*}}(X^{*},Y)}\) has property RDP p * ( \({1\le p < \infty}\) ) when X and \({Y}\) have the same property. Suppose X and Y are Banach spaces, K is a compact Hausdorff space, \({\Sigma}\) is the \({\sigma}\) -algebra of Borel subsets of K, C(K,X) is the Banach space of all continuous X-valued functions (with the supremum norm), and \({T: C(K,X)\to Y}\) is a strongly bounded operator with representing measure \({m: \Sigma \to L(X,Y)}\) . We show that if T is a strongly bounded operator and \({\hat{T}: B(K, X) \to Y}\) is its extension, then T* is p-convergent if and only if \({\hat{T}^{*}}\) is p-convergent, for \({1\le p < \infty}\) . PubDate: 2019-02-01

Abstract: Abstract Let \({\mathcal M}\) be a Hilbert \({C^{*}}\) -module over a \({C^{*}}\) -algebra \({\mathcal A}\) . Suppose that \({\mathcal{K}(\mathcal{M})}\) is the space of compact operators on \({\mathcal M}\) and the bounded anti-homomorphism \({\rho \colon \mathcal{A}\rightarrow \mathcal{B}(\mathcal{M})}\) defined by \({\rho(a)(m)=ma}\) for all \({a\in\mathcal{A}}\) and \({m\in\mathcal{M}}\) . In this paper, we first provide some characterizations of module maps on Banach modules over Banach algebras by several local conditions (some of our results are a generalization of previous results) and then apply them to characterize the reflexive closure of \({\mathcal{K}(\mathcal{M})}\) and \({\rho(\mathcal{A})}\) , i.e., \({{\rm Alg}{\rm Lat} \mathcal{K}(\mathcal{M})}\) and \({{\rm Alg}{\rm Lat} \rho(\mathcal{A})}\) , where we think of \({\mathcal{K}(\mathcal{M})}\) and \({\rho(\mathcal{A})}\) as operator algebras acting on \({\mathcal M}\) . As an application of our results on reflexive closure of \({\mathcal{K}(\mathcal{M})}\) and \({\rho(\mathcal{A})}\) , a characterization of commutativity for \({C^{*}}\) -algebras is given. PubDate: 2019-02-01

Abstract: Abstract We provide a characterization of coefficients of a general orthogonal double-index series which allows existence of an a.e. convergent rearrangement. Namely, we show that for any sequence of numbers \({(a_{\mathbf{n}})_{\mathbf{n}\in\mathbb{N}^d}}\) the condition \({\sum_{\mathbf{n}\in\mathbb{N}^d} a_{\mathbf{n}}^{2}{\rm log}^{2} a_{\mathbf{n}}^{2} < \infty}\) is equivalent to the existence of an injective map \({\varrho\colon\mathbb{N}^{d} \rightarrow \mathbb{N}^{d}}\) such that the multiple series \({\sum_{\mathbf{n}\in\mathbb{N}^d} a_{\varrho(\mathbf{n})}\Phi_{\varrho(\mathbf{n})}}\) converges a.e. for any orthonormal system of functions \({(\Phi_{\mathbf{n}})_{\mathbf{n}\in\mathbb{N}^d}}\) . To this end, a Menshov-type lemma for arbitrary subsets of \({\mathbb{N}^d}\) is proved. PubDate: 2019-02-01

Abstract: Abstract Let \({A = \{a_{1},a_{2},\dots{} \}}\) \({(a_{1} < a_{2} < \cdots )}\) be an infinite sequence of nonnegative integers, and let \({R_{A,2}(n)}\) denote the number of solutions of \({a_{x}+a_{y}=n}\) \({(a_{x},a_{y} \in A)}\) . P. Erdős, A. Sárközy and V. T. Sós proved that if \({\lim_{N\to\infty}\frac{B(A,N)}{\sqrt{N}}=+\infty}\) then \({ \Delta_{1}(R_{A,2}(n)) }\) cannot be bounded, where \({B(A,N)}\) denotes the number of blocks formed by consecutive integers in A up to N and \({\Delta_{l}}\) denotes the l-th difference. Their result was extended to \({\Delta_{l}(R_{A,2}(n))}\) for any fixed \({l\ge2}\) . In this paper we give further generalizations of this problem. PubDate: 2019-02-01