Abstract: We prove that the number of positive integers \(n\le x\) , such that \(\phi (1)+\phi (2)+\cdots +\phi (n)\) is a multiple of n, is less than \(x/(\log x)^{0.15742}\) for all sufficiently large x, where \(\phi \) stands for the Euler totient function. PubDate: 2017-03-27

Abstract: Recently we have shown that the distance trisector curve is a transcendental curve. Since from the computational point of view this implies that no closed expression to describe the curve in algebraic terms can be found, it is still of interest to know how to approximate it efficiently by means of polynomial or rational functions. We discuss here some of the remarkable properties of this curve, that among other things lead to very good approximations. PubDate: 2017-03-27

Authors:Ernesto Lupercio; Elias Micha Abstract: In his note we briefly analyze the rôle of the oeuvre of Mexican mathematician Samuel Gitler and his influence in twentieth and twenty-first century mathematics. PubDate: 2017-03-11 DOI: 10.1007/s40590-017-0163-9

Authors:M. Shirvani; S. Mirvakili Abstract: In this paper, we determine a family \({\mathfrak {U}}_{\mathfrak {R}}\) of subsets of a hyperring R and sufficient conditions, such that the geometric space \((R,{\mathfrak {U}}_{\mathfrak {R}})\) is strongly transitive. Finally, we prove that in any hyperfield or any hyperring \((R,+,\cdot )\) , such that \((R,+)\) has an identity element, \(\rho _\mathfrak {R}=\rho ^*_\mathfrak {R}\) . PubDate: 2017-02-18 DOI: 10.1007/s40590-017-0162-x

Authors:F. Javier Trigos-Arrieta Abstract: Denote by \({{\mathbb {T}}}\) the torus, i.e., the topological group consisting of the complex numbers of modulus 1 under multiplication. Every topological Abelian group (G, t) has associated a weaker topological group topology, denoted by \(t^+\) , defined as the weakest topology on G that makes the t-continuous homomorphisms (t-characters) \(\phi : G \rightarrow {{\mathbb {T}}}\) continuous. The topology \(t^+\) is called the Bohr topology on (G, t). Let \({{{\mathfrak {T}}}}\) denote the torsion subgroup of \({{\mathbb {T}}}\) . Then the weakest topology that makes the t-characters \(\phi : G \rightarrow {{\mathfrak {T}}}\) continuous is called the Bohr-torsion topology on (G, t) and is denoted by \(t^\oplus \) . When t is locally compact, we show that \(t^\oplus \) is Hausdorff if and only if (G, t) is zero dimensional, and if (G, t) is zero dimensional and H is a subgroup of G, then H is t-closed if and only if H is \(t^\oplus \) -closed. PubDate: 2017-02-17 DOI: 10.1007/s40590-017-0161-y

Authors:Zhanmin Zhu Abstract: Let R be a ring and n be a positive integer. R is called left n-coherent, if every n-presented left R-module is \((n+1)\) -presented. A left R-module M is called (n, 0)-injective if Ext \(^1_R(V, M)=0\) for every n-presented module V, a right R-module F is called (n, 0)-flat if Tor \(^R_1(F, V)=0\) for every n-presented module V, a left R-module M is called (n, 0)-projective if \(\mathrm{Ext}^1_R(M, N)=0\) for any (n, 0)-injective module N, and a right R-module M is called (n, 0)-cotorsion if \(\mathrm{Ext}^1_R(F, M)=0\) for any (n, 0)-flat module F. We give some characterizations and properties of (n, 0)-projective modules and (n, 0)-cotorsion modules. n-coherent rings, n-hereditary rings and n-regular rings are characterized by (n, 0)-projective modules, (n, 0)-cotorsion modules, (n, 0)-injective modules and (n, 0)-flat modules. PubDate: 2017-02-14 DOI: 10.1007/s40590-017-0160-z

Authors:M. Hamidi; A. Broumand Saeid Abstract: In this paper, we introduce a strongly regular relation on hyper EQ-algebra as a fundamental relation and study some of its properties. We construct the quotient of any hyper EQ-algebra via (regular relation) strongly regular relations. Finally, relationships between filter and EQ-filter are studied in detail. PubDate: 2017-01-10 DOI: 10.1007/s40590-016-0159-x

Authors:S. Mirvakili; S. M. Anvariyeh; B. Davvaz Abstract: This paper concerns a new relationship between hyperrings and rings. It is a continuation of ideas presented by Babaeia, Jafarpour, and Mousavi. We apply generalization of a complete part in a hyperring. We study the notion of strongly \(\mathcal {U}\) -regular relations on hyperrings and investigate some of their properties. In particular, we investigate the properties of rings derived from strongly \(\mathcal {U}\) -regular relations. PubDate: 2017-01-03 DOI: 10.1007/s40590-016-0157-z

Authors:M. R. Molaei Abstract: In this essay we consider topological cocycles via semibornology and group structures. Global and local attractors via semibornological nonautonomous sets are studied. We show that global semibornological pullback attractors are compact and positively invariant. We introduce the concept of global \((g,\beta )\) pullback attractors and we show that they are also compact and positively invariant. We prove that semibornological pullback attractors are invariant objects under topological equivalencies. We present a method for finding absorbing sets when the state space of the dynamical system of a cocycle is a convex semibornological vector space. PubDate: 2017-01-03 DOI: 10.1007/s40590-016-0158-y

Authors:Boris A. Kats; David B. Katz Pages: 361 - 366 Abstract: We study the problem to find a function holomorphic on a compact Riemann surface that is given by its jump on a non-rectifiable curve. We improve the known solvability criteria for this problem in terms of new metric characteristics of non-rectifiable curves. PubDate: 2016-10-01 DOI: 10.1007/s40590-016-0100-3 Issue No:Vol. 22, No. 2 (2016)

Authors:Thomas Pawlaschyk; Eduardo S. Zeron Pages: 367 - 388 Abstract: We study the properties of hulls of compact sets in \(\mathbb {C}^n\) that are generated by certain subfamilies of q-plurisubharmonic functions. We consider in particular those functions that are plurisubharmonic on the level sets of holomorphic mappings defined from \(\mathbb {C}^n\) into \(\mathbb {C}^q\) . We also compare the above-mentioned hulls against the generalised polynomially and rationally convex hulls already defined in the literature. Our main result yields that the hulls defined by q-plurisubharmonic functions or q-pseudoconvex sets are all q-maximum, so that their complement is relatively \((n{\text {-}}q{\text {-}}2)\) -pseudoconvex. PubDate: 2016-10-01 DOI: 10.1007/s40590-016-0123-9 Issue No:Vol. 22, No. 2 (2016)

Authors:Kevin Esmeral; Nikolai Vasilevski Pages: 567 - 582 Abstract: We introduce the so-called horizontal Toeplitz operators acting on the Fock space and give an explicit description of the C*-algebra generated by them. We show that any Toeplitz operator with \(L_{\infty }\) -symbol, which is invariant under imaginary translations, is unitarily equivalent to the multiplication operator by its “spectral function”. This result is also true for the Toeplitz operators whose defining symbols are invariant under translations over any Lagrangian plane. The main result of the paper states that the corresponding spectral functions form a dense subset in the C*-algebra of bounded uniformly continuous functions with respect to the standard metric on \(\mathbb {R}^{n}\) . PubDate: 2016-10-01 DOI: 10.1007/s40590-016-0110-1 Issue No:Vol. 22, No. 2 (2016)

Authors:Maribel Loaiza; Carmen Lozano Pages: 583 - 604 Abstract: We prove that Toeplitz operators, acting on the pluriharmonic Bergman space of the Siegel domain, generate commutative \(C^*\) - algebras in the following two cases: when the class of symbols are invariant functions under the action of the quasi-parabolic group of biholomorphisms of the Siegel domain and when the class of symbols consists of all invariant functions under the action of the nilpotent group of biholomorphisms of the Siegel domain. PubDate: 2016-10-01 DOI: 10.1007/s40590-016-0122-x Issue No:Vol. 22, No. 2 (2016)

Authors:Wolfram Bauer; Lewis A. Coburn Pages: 669 - 677 Abstract: With a family \((\mu _t)_{t>0}\) of Gaussian probability measures we consider the scale \((H_t^2)_{>0}\) of \(\mu _t\) -square integrable entire functions on \(\mathbb {C}^n\) . Here t plays the role of Planck’s constant. For f and g in the space \(\mathrm{BUC}(\mathbb {C}^n)\) of all bounded and uniformly continuous complex valued functions on \(\mathbb {C}^n\) we show the asymptotic composition formula 1 $$\begin{aligned} \lim _{t\downarrow 0} \Vert T_f^{(t)} T_g^{(t)} -T^{(t)}_{fg} \Vert _t =0, \end{aligned}$$ where \(\Vert \cdot \Vert _t\) denotes the norm in \(\mathcal {L}(H_t^2)\) and \(T_f^{(t)}\) is the Toeplitz operator with symbol f. Different from previously known results (e.g. Borthwick, Perspectives on quantization. Contemporary mathematics, vol 214. AMS, Providence, pp 23–37, 1998; Coburn, Commun Math Phys 149:415–424, 1992) neither differentiability nor compact support of the operator symbols is assumed. We provide an example which indicates that (1) in general fails for rapidly oscillating bounded symbols. PubDate: 2016-10-01 DOI: 10.1007/s40590-016-0108-8 Issue No:Vol. 22, No. 2 (2016)

Authors:Leo R. Ya. Doktorski Pages: 679 - 693 Abstract: The real interpolation method \( {\overline{X}}_{\theta ,q,b}=(X_0,X_1)_{\theta ,q,b} \) involving iterated logarithms with any number of iterations is considered. Reiteration relations of the types \( (X_0,{\overline{X}}_{0,q,a})_{\theta ,r,b} \) and \( ({\overline{X}}_{1,q,a}X_1)_{\theta ,r,b} \) \( (0\le \theta \le 1)\) are investigated. Using any number of iterations allows in particular obtaining effects where the resulting space includes three iterated logarithms although the initial scale includes only the uniterated logarithm. Application to Lorentz–Zygmund spaces is given. PubDate: 2016-10-01 DOI: 10.1007/s40590-016-0116-8 Issue No:Vol. 22, No. 2 (2016)

Authors:Abhijit Banerjee; Sujoy Majumder Abstract: With the notion of weighted sharing of values we investigate the possible relation between two meromorphic functions when \(f^{n}P(f)f^{(k)}\) and \(g^{n}P(g)g^{(k)}\) share a non-zero polynomial. The results obtained in this paper improve, generalize and rectify a recent one of Cao and Zhang (J Inequal Appl 1:100, 2012). PubDate: 2016-12-05 DOI: 10.1007/s40590-016-0156-0

Abstract: We determine the 2-primary components of the 32-stem homotopy groups of spheres. The method is based on the classical one including the Toda’s composition methods. PubDate: 2016-11-04 DOI: 10.1007/s40590-016-0154-2

Authors:Mouataz Billah Mesmouli; Abdelouaheb Ardjouni; Ahcene Djoudi Abstract: In this paper, we study the existence of periodic solutions of the nonlinear neutral system of differential equations $$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}x\left( t\right) =A\left( t\right) x\left( t\right) +\frac{\mathrm{d}}{\mathrm{d}t} Q\left( t,x\left( t-g\left( t\right) \right) \right) +\int _{-\infty }^{t}D\left( t,s\right) F\left( x\left( s\right) \right) \mathrm{d}s. \end{aligned}$$ Using Krasnoselskii’s fixed point theorem, we obtain the existence of periodic solution and by contraction mapping principle we obtain the uniqueness of periodic solution and stability of the zero solution. Our results extend some earlier publications. PubDate: 2016-10-28 DOI: 10.1007/s40590-016-0155-1

Authors:Gennadiy Gubreev; Anna Tarasenko Abstract: A quite general theorems on unconditional bases in separable Hilbert spaces, given in terms of values of entire operator valued vector-functions, were established in the papers [2–4]. In the present paper, we give a detailed analysis of the hypothesis of these theorems. We present examples of various classes of vector-functions that satisfy some of the hypothesis of the above theorems. On the other hand, we show that there exist natural classes of vector-functions that do not satisfy some of the hypothesis of Theorem A. PubDate: 2016-10-25 DOI: 10.1007/s40590-016-0153-3

Authors:I. M. Erusalimskiy Abstract: Graph lattice has vertices at points with non-negative integer coordinates. From each vertex we have two edges: horizontal and vertical neighboring vertices (right and top). In this paper, we considered the problem of random walk on the vertices of the graph lattice without limitation on the reachability and with two types of limitations on reachability—mixed and magnetic. The set of edges of the graph with mixed reachability U is a union of disjoint non-empty sets \(U_R\) and \(U_Z\) . Permitted path on the graph with mixed reachability is the path wherein the edges from the set \(U_Z\) are thinned by the edges from the set \(U_R\) , i.e. the edges from the set \(U_Z\) on the path are not adjacent. As a consequence of the consideration of this problem, we have obtained new proof of some combinatorial identities. PubDate: 2016-05-12 DOI: 10.1007/s40590-016-0115-9