Authors:Zahra Barati Abstract: Let G be a graph with n vertices and m edges and r connected components. The free rank of G, denoted by \(\text {frank}(G)\) , is the number of primitive cycles of G. Also, the cycle rank of G was defined as \(\text {rank}(G) := m-n+r\) . The family of graphs satisfying the equality \(\text {rank}(G) = \text {frank}(G)\) is called ring graphs. The full characterization of this family of graphs were given in Gitler et al. (J. Algebraic Comb. 38:721–744, 2013; Discret. Math. 310:430–441, 2010). In this paper, we consider the following problem: Find the minimum number n such that the n-iterated line graph of G is not a ring graph, i.e. \(\text {rank}(L^n(G)) \ne \text {frank}(L^n(G))\) . For this purpose, we define the ring index of G as the minimum number n such that the n-iterated line graph of G is not a ring graph. We show that the ring index of a graph is at most 3 or is \(\infty \) . Furthermore, we give a full characterization of graphs with respect to this index. PubDate: 2018-02-26 DOI: 10.1007/s40590-018-0195-9

Authors:Jesús Jerónimo-Castro Abstract: In this paper, we prove the following: Let \(\mathcal {F}=\{x_1+B,x_2+B,x_3+B,x_4+B,x_5+B\}\) be a family of translates of the unit diameter disc B such that every three members of \({\mathcal {F}}\) are intersected by some straight line, then there is a line intersecting every member of the family \({\mathcal {F}}'=\{x_1+\varphi B,x_2+\varphi B,x_3+ \varphi B,x_4+ \varphi B,x_5+ \varphi B\}\) , where \(\varphi =\frac{1+\sqrt{5}}{2}\) . PubDate: 2018-02-21 DOI: 10.1007/s40590-018-0193-y

Authors:Akbar Rezaei Abstract: In this paper we generalize the notion of an implication groupoid and introduce ei-groupoid, and investigate related properties. We study the notion of filters in this structure and we give the construction of implication algebra from ei-groupoid. Finally, we prove that for distributive ei-groupoids filters coincide with ideals. PubDate: 2018-02-19 DOI: 10.1007/s40590-018-0194-x

Authors:Nicolas Campanelli; Martín Eduardo Frías-Armenta; Jose Luis Martinez-Morales Pages: 527 - 535 Abstract: We formulate a group of graphs with graph union as operation. Out of the 256 possible graph products, only six can be used as means to define ring structures over such graph group. Likewise, using the graph join instead of the graph union, another set of graph products is available for defining ring structures. Unsurprisingly, both constructions lead to the same rings via an isomorphism. PubDate: 2017-10-01 DOI: 10.1007/s40590-015-0081-7 Issue No:Vol. 23, No. 2 (2017)

Authors:Francisco Escalona; Ruy Fabila-Monroy; Jorge Urrutia Pages: 537 - 547 Abstract: A tetrahedralization of a point set in three dimensional space is the analogue of a triangulation of a point set in the plane. The dual graph of a tetrahedralization is the graph having the tetrahedra as nodes, two of which are adjacent if they share a face. A tetrahedralization is Hamiltonian if its dual graph has a Hamiltonian path. Problem 29 of the “Open Problems Project” in Computational Geometry, asks whether every finite set of points in three dimensional space has a Hamiltonian tetrahedralization. Let S be a set of n points in general position in three dimensional space, m of which are convex hull vertices. In this paper we provide an \(O(m^\frac{3}{2}) + O(n \log n)\) time algorithm to compute a Hamiltonian tetrahedralization of S, by adding Steiner points. Our algorithm adds at most \(\left\lfloor \frac{m-2}{2} \right\rfloor -1\) Steiner points. If \(m \le 20\) , then no Steiner points are needed to find a Hamiltonian tetrahedralization of S. Finally, we construct a set of 84 points that does not admit a Hamiltonian tetrahedralization in which all tetrahedra share a common vertex. PubDate: 2017-10-01 DOI: 10.1007/s40590-015-0080-8 Issue No:Vol. 23, No. 2 (2017)

Authors:Ana Paulina Figueroa; Julián Fresán-Figueroa; Eduardo Rivera-Campo Pages: 549 - 556 Abstract: The perfect matching graph of a graph G, denoted by M(G), has one vertex for each perfect matching of G and two matchings are adjacent if their symmetric difference is a cycle of G. Let C be a family of cycles of G. The perfect matching graph defined by C is the spanning subgraph M(G, C) of M(G) in which two perfect matchings L and N are adjacent only if \(L \varDelta N\) lies in C. We give a necessary condition and a sufficient condition for M(G, C) to be connected. We also give examples of graphs and of families of cycles for which the sufficient condition is satisfied. PubDate: 2017-10-01 DOI: 10.1007/s40590-015-0079-1 Issue No:Vol. 23, No. 2 (2017)

Authors:Xavier Gómez-Mont; Luis Núñez-Betancourt Pages: 623 - 651 Abstract: Given a zero-dimensional Gorenstein algebra \(\mathbb {B}\) and two syzygies between two elements \(f_1,f_2\in {\mathbb B}\) , one constructs a double complex of \(\mathbb {B}\) -modules, \(\mathcal{G}_\mathbb {B},\) called the small Gobelin. We describe an inductive procedure to construct the even and odd hyperhomologies of this complex. For high degrees, the difference \(\dim \mathbb {H}_{j+2}(\mathcal{G}_\mathbb {B}) - \dim \mathbb {H}_j(\mathcal{G}_\mathbb {B})\) is constant, but possibly with a different value for even and odd degrees. We describe two flags of ideals in \(\mathbb {B}\) which codify the above differences of dimension. The motivation to study this double complex comes from understanding the tangency condition between a vector field and a complete intersection, and invariants constructed in the zero locus of the vector field \(\hbox {Spec}(\mathbb {B})\) . PubDate: 2017-10-01 DOI: 10.1007/s40590-015-0076-4 Issue No:Vol. 23, No. 2 (2017)

Authors:Ana González Pages: 667 - 689 Abstract: This paper presents a combinatorial proof of the existence of a Lie bialgebra structure over the vector space of reduced cyclic words. Any surface with non-empty boundary has an associated vector space determined by the corresponding surface symbol, this space is known as the space of reduced cyclic words. The Lie bialgebra structure over this space was introduced by Chas in the article Combinatorial Lie bialgebras of curves on surfaces, where a proof of the existence of this structure is given. This proof is based on the construction of an isomorphism between the space of reduced cyclic words and the space of curves on a surface. PubDate: 2017-10-01 DOI: 10.1007/s40590-016-0133-7 Issue No:Vol. 23, No. 2 (2017)

Authors:Zhuoyan Gao; Mengmeng Li; JinRong Wang Pages: 691 - 711 Abstract: In this paper, we establish two fractional integral equalities involving once and twice differential functions. Then, we apply such equalities to give some fractional Hermite–Hadamard inequalities via s-convex and s-Godunova–Levin functions. Some applications to special means of positive real numbers are also given. PubDate: 2017-10-01 DOI: 10.1007/s40590-016-0087-9 Issue No:Vol. 23, No. 2 (2017)

Authors:Martín-Eduardo Frías-Armenta; Jaume Llibre Pages: 737 - 758 Abstract: We characterize the 11 non-topological equivalent classes of phase portraits in the Poincaré disc of the new family of cubic polynomial Hamiltonian differential systems with a center at the origin and Hamiltonian $$\begin{aligned} H= \frac{1}{2} ( (x + a x^2 + b x y + c y^2)^2+y^2 ), \end{aligned}$$ with \(a^2+b^2+c^2\ne 0\) . PubDate: 2017-10-01 DOI: 10.1007/s40590-016-0126-6 Issue No:Vol. 23, No. 2 (2017)

Authors:Samad Mohseni Kolagar; Maryam Ramezani; Madjid Eshaghi Gordji Pages: 815 - 824 Abstract: In this paper, we introduce the concept of generalized contraction for multivalued operators defined on ordered complete metric spaces. We analyze the existence of fixed points for generalized multivalued operators. Moreover, as an application of our main theorem, we give an existence theorem for the solution of a hyperbolic differential inclusion problem. PubDate: 2017-10-01 DOI: 10.1007/s40590-015-0077-3 Issue No:Vol. 23, No. 2 (2017)

Authors:Yanlong Hao; Xiugui Liu Pages: 847 - 851 Abstract: Let Aut(p) denote the space of all self-fibre homotopy equivalences of a principal G-bundle \(p: E\rightarrow X\) of simply connected CW complexes with E finite. When G is a compact connected topological group, we show that there exists an inequality $$\begin{aligned} n-\mathrm{N}(p)\le \mathrm{Hnil}_{\mathbb {Q}}(\mathrm{{Aut}}(p)_0)\le n \end{aligned}$$ for any space X, where n is the number of non-trivial rational homotopy groups of G and \(\mathrm{N}(p)\) is defined in Sect. 2. In particular, \(\mathrm{Hnil}_{\mathbb {Q}}(\mathrm{{Aut}}(p)_{0})=n\) if p is a fibre homotopy trivial bundle and X is finite. PubDate: 2017-10-01 DOI: 10.1007/s40590-016-0098-6 Issue No:Vol. 23, No. 2 (2017)

Authors:J. A. López-Renteria; B. Aguirre-Hernández; F. Verduzco Pages: 869 - 889 Abstract: The aim of this work is to give a Hurwitz path (which is a family of polynomials) joining any two arbitrary stable polynomials in the set of monic Hurwitz polynomials with positive coefficients and fixed degree n, \(\mathcal {H}_{n}^{+}\) . This and the homotopy of paths allow to prove the existence of a dense trajectory in \(\mathcal {H}_{n}^{+}\) . It implies, by the Möbius transform and Viète’s map, that we can find a connecting-path in the set of the Schur polynomials, \(\mathcal {S}_{n}\) . Due to the form of the stable connecting-paths, a feedback control is designed whose structure can be used to stabilize continuous or discrete systems. PubDate: 2017-10-01 DOI: 10.1007/s40590-016-0086-x Issue No:Vol. 23, No. 2 (2017)

Authors:Abdon E. Choque-Rivero Pages: 891 - 918 Abstract: We obtain a new multiplicative decomposition of the resolvent matrix of the non-degenerate truncated Hausdorff matrix moment (THMM) problem in the case of odd and even number of moments with the help of Dyukarev–Stieltjes matrix parameters (DSMP). Our result generalizes the Dyukarev representation of the resolvent matrix of the truncated Stieltjes matrix moment problem published in (Math Notes 75(1–2):66–82, 2004). In the scalar case, these parameters appear in the celebrated Stieltjes’s (1894) work Recherches sur les fractions continues and are used to establish the determinateness of the moment problem. We also obtain explicit relations between four families of orthogonal matrix polynomials on [a, b] together with their matrix polynomials of the second kind and the DSMP of the THMM problem. Additionally, we derive new representations of the Christoffel–Darboux kernel. PubDate: 2017-10-01 DOI: 10.1007/s40590-015-0083-5 Issue No:Vol. 23, No. 2 (2017)

Authors:James F. Glazebrook; Alberto Verjovsky Abstract: In the context of orientable circuits and subcomplexes of these as representing certain singular spaces, we consider characteristic class formulas generalizing those classical results as seen for the Riemann–Hurwitz formula for regulating the topology of branched covering maps and that for monoidal transformations which include the standard blowing-up process. Here the results are presented as cap product pairings, which will be elements of a suitable homology theory, rather than characteristic numbers as would be the case when taking Kronecker products once Poincaré duality is defined. We further consider possible applications and examples including branched covering maps, singular varieties involving virtual tangent bundles, the Chern–Schwartz–MacPherson class, the homology L-class, generalized signature, and the cohomology signature class. PubDate: 2017-12-02 DOI: 10.1007/s40590-017-0191-5

Authors:Carlos Castano-Bernard Abstract: Let E be an elliptic curve defined over the rationals and let N be its conductor. Assume N is prime. In this paper, we prove that the index on E of the Heegner divisor of discriminant \(D=-~4N\) is even provided \(N\equiv 7\pmod {8}\) and discuss some conjectures on further parity properties for the indexes on E of Heegner divisors of discriminant D dividing 4N. One of these conjectures suggests a possible link between the parity of the eigenvalue \(a_A(2)\) and the parity of the Šafarevič-Tate group of certain elliptic curves A of square conductor. PubDate: 2017-11-30 DOI: 10.1007/s40590-017-0192-4

Authors:Baltazar Aguirre-Hernández; Francisco A. Carrillo; Jesús F. Espinoza; Horacio Leyva Abstract: The importance of the Hurwitz–Metzler matrices and the Hurwitz symmetric matrices can be appreciated in different applications: communication networks, biology and economics are some of them. In this paper, we use an approach of differential topology for studying such matrices. Our results are as follows: the space of the \(n\times n\) Hurwitz symmetric matrices has a product manifold structure given by the space of the \((n-1)\times (n-1)\) Hurwitz symmetric matrices and the Euclidean space. Additionally we study the space of Hurwitz–Metzler matrices and these ideas let us do an analysis of robustness of Hurwitz–Metzler matrices. In particular, we study the insulin model as an application. PubDate: 2017-11-22 DOI: 10.1007/s40590-017-0189-z

Authors:H. Özlem Güney; Sedat İlhan; Janusz Sokół Abstract: In this paper, we investigate the third Hankel determinant problem in some classes of analytic functions in the open unit disc connected with k-Fibonacci numbers \(F_{k,n}\) \((k>0)\) . For this, first, we prove a conjecture, posed in Güney et al. (2017), for sharp upper bound of the second Hankel determinant. In the sequel, we obtain another sharp coefficient bound which we apply in solving the problem of the third Hankel determinant for these functions. Finally, we give an upper bound for the third Hankel determinant in this class. The results presented in the present paper have been shown to generalize and improve some recent work of Sokół et al. (2017). PubDate: 2017-11-17 DOI: 10.1007/s40590-017-0190-6

Authors:Francisco G. Jimenez-Lopez Abstract: The length spectrum Teichmüller space \(T_{ls}(R)\) , based at hyperbolic surface of infinite type R, is the collection of all marked hyperbolic surfaces that are homeomorphic to R, satisfying that the quotients of the lengths of the corresponding geodesics are uniformly bounded from above and from below. Two points in \(T_{ls}(R)\) are called asymptotically length spectrum equivalent if the ratios of the lengths of geodesics outside compact sets is close to 1. The quotient space coming from this relation is called the asymptotic length spectrum Teichmüller space \(AT_{ls}(R)\) . In this paper we prove that if the base surface R admits a pair of pants decomposition that satisfies Shiga’s condition (i.e., that is upper and lower bounded), then \(AT_{ls}(R)\) is complete under the natural metric. We also prove that in this case, the space is homeomorphic to \(l^{\infty }/c_0\) , where \(l^{\infty }\) is the Banach space of bounded sequences and \(c_0\) is the subspace of sequences converging to zero. PubDate: 2017-11-16 DOI: 10.1007/s40590-017-0188-0

Authors:Tanuj Kumar; Akhilesh Prasad Abstract: In this work, we introduce translation and convolution for linear canonical Hankel transformations and studied some inequalities. For the particular values of linear canonical Hankel transformation (i.e., for Hankel–Clifford transformation), we investigate linear time-invariant filters. Furthermore, some applications of linear canonical Hankel transformation to a generalized non-linear parabolic equation and a canonical convolution integral equation are given. PubDate: 2017-10-23 DOI: 10.1007/s40590-017-0187-1