Authors:Nicolas Campanelli; Martín Eduardo Frías-Armenta; Jose Luis Martinez-Morales Pages: 527 - 535 Abstract: 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: 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: 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:Raymundo Bautista; Ivon Dorado Pages: 557 - 609 Abstract: Abstract We introduce partially ordered sets (posets) with an additional structure given by a collection of vector subspaces of an algebra A. We call them algebraically equipped posets. Some particular cases of these, are generalized equipped posets and p-equipped posets, for a prime number p. We study their categories of representations and establish equivalences with some module categories, categories of morphisms and a subcategory of representations of a differential tensor algebra. Through this, we obtain matrix representations and its corresponding matrix classification problem. PubDate: 2017-10-01 DOI: 10.1007/s40590-016-0131-9 Issue No:Vol. 23, No. 2 (2017)

Authors:Leila Hamedi-Mobarra; Dawood Hassanzadeh-lelekaami; Hajar Roshan-Shekalgourabi Pages: 611 - 621 Abstract: Abstract In this paper, we introduce a new graph over a commutative ring R relative to an R-module M. We study the relationship between the algebraic properties of R and M and their associated graph, say G(R, M). More precisely, we consider the relationship between the completeness of subgraphs of the G(R, M) and algebraic properties of R and M. In addition, we study the bipartite and complete bipartite subgraphs of the associated graph. Finally, we completely characterize the diameter of the certain subgraph of G(R, M). PubDate: 2017-10-01 DOI: 10.1007/s40590-016-0132-8 Issue No:Vol. 23, No. 2 (2017)

Authors:Xavier Gómez-Mont; Luis Núñez-Betancourt Pages: 623 - 651 Abstract: 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:Sonia Trepode; Yadira Valdivieso-Díaz Pages: 653 - 666 Abstract: Abstract We show that Jacobian algebras arising from every tagged triangulation of a sphere with n-punctures, with \(n\ge 5\) , are finite dimensional algebras. We consider also a family of cyclically oriented quivers and we prove that, for any primitive potential, the associated Jacobian algebra is finite dimensional. PubDate: 2017-10-01 DOI: 10.1007/s40590-015-0082-6 Issue No:Vol. 23, No. 2 (2017)

Authors:Ana González Pages: 667 - 689 Abstract: 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: 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:Feng Qi; Xiao-Jing Zhang; Wen-Hui Li Pages: 713 - 736 Abstract: Abstract In the paper, the authors present by several approaches that both the harmonic mean and the geometric mean of two positive numbers are Bernstein functions and establish their integral representations. PubDate: 2017-10-01 DOI: 10.1007/s40590-016-0085-y Issue No:Vol. 23, No. 2 (2017)

Authors:Martín-Eduardo Frías-Armenta; Jaume Llibre Pages: 737 - 758 Abstract: 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:Valente Ramírez Pages: 759 - 813 Abstract: Abstract In this work, we consider holomorphic foliations of degree two on the complex projective plane \(\mathbb {P}^2\) having an invariant line. In a suitable choice of affine coordinates, these foliations are induced by a quadratic vector field over the affine part in such a way that the invariant line corresponds to the line at infinity. We say that two such foliations are topologically equivalent provided there exists a homeomorphism of \(\mathbb {P}^2\) which brings the leaves of one foliation onto the leaves of the other and preserves orientation both on the ambient space and on the leaves. The main result of this paper is that in the generic case, two such foliations may be topologically equivalent if and only if they are analytically equivalent. In fact, it is shown that the analytic conjugacy class of the holonomy group of the invariant line is the modulus of both topological and analytic classification. We obtain as a corollary that two generic orbitally topologically equivalent quadratic vector fields on \(\mathbb {C}^2\) must be orbitally affine equivalent. This result improves, in the case of quadratic foliations, a well-known result by Ilyashenko that claims that two generic and topologically equivalent foliations with an invariant line at infinity are affine equivalent, provided they are close enough in the space of foliations and the linking homeomorphism is close enough to the identity map of \(\mathbb {P}^2\) . PubDate: 2017-10-01 DOI: 10.1007/s40590-016-0127-5 Issue No:Vol. 23, No. 2 (2017)

Authors:Samad Mohseni Kolagar; Maryam Ramezani; Madjid Eshaghi Gordji Pages: 815 - 824 Abstract: 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:Jose Rosales-Ortega Pages: 825 - 845 Abstract: Abstract We give a new version of the Gromov’s centralizer theorem in the case of semisimple Lie group actions and arbitrary rigid geometric structures of algebraic type. PubDate: 2017-10-01 DOI: 10.1007/s40590-015-0084-4 Issue No:Vol. 23, No. 2 (2017)

Authors:Yanlong Hao; Xiugui Liu Pages: 847 - 851 Abstract: 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:G. Brumfiel; H. Hilden; M. T. Lozano; J. M. Montesinos-Amilibia; E. Ramírez-Losada; H. Short; D. Tejada; M. Toro Pages: 853 - 868 Abstract: Abstract In this paper, we show that the branched covering space map \(p: H^3\rightarrow E^3\) between the Riemannian manifolds \(H^ 3\) and \(E^3\) , the hyperbolic space and the Euclidean space, can be harmonic. PubDate: 2017-10-01 DOI: 10.1007/s40590-016-0114-x Issue No:Vol. 23, No. 2 (2017)

Authors:J. A. López-Renteria; B. Aguirre-Hernández; F. Verduzco Pages: 869 - 889 Abstract: 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: 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:Andrés Piedra Abstract: Abstract We study the components of the Chow variety \({\mathcal {C}}_{1,3}({\mathbb P}^3)\) of 1-cycles of degree 3 in \({\mathbb P}^3\) . To do this, we calculate explicit specializations at the components \(H(3,-1),\) and \(H(3,-2)\) of the Hilbert schemes Hilb \(^{3m+2}({\mathbb P}^3)\) and Hilb \(^{3m+3}({\mathbb P}^3)\) , respectively. This will give us a partial description of the stratifications of the components \(H(3,-1),\) and \(H(3,-2)\) and, therefore, the birational Hilbert–Chow morphism will give a partial description of the corresponding components of the Chow variety \({\mathcal {C}}_{1,3}({\mathbb P}^3)\) . PubDate: 2017-09-27 DOI: 10.1007/s40590-017-0182-6

Authors:M. E. Frías-Armenta; E. López-González Abstract: Abstract Geodesibility of vector fields was studied by Gluck and Sullivan in the 1970s. For the case of complex analytical vector fields, Jenkins shed light on the subject from the end of the 1950s. After the 1970s, multiple authors have studied the subject, such as K. Strebel, and Muciño-Raymundo and Valero-Valdéz. In this paper, we consider planar vector fields which are \(\mathbb {A}\) -algebrizable (differentiable in the sense of Lorch for some associative and commutative algebra \(\mathbb {A}\) with unit e). We give rectifications of these vector fields and metrics under which they are geodesible. PubDate: 2017-09-27 DOI: 10.1007/s40590-017-0186-2