Luis Javier Hernández Paricio

Universidad de La Rioja (Spain) | UNIRIOJA · Mathematics and Computation

In this work we introduce a new notion of covering projection $E\longrightarrow X$ of a topological space $X$ such that if $X$ is a locally connected space we have the usual notion of covering projection. We use locally constant presheaves and covering reduced sieves to find a pro-groupoid $\pi \, crs(X)$ and an induced category pro$(\pi \, crs(X)...

For each n > 1 and each multiplicative closed set of integers S, we study closed model category structures on the pointed category of topological spaces, where the classes of weak equivalences are classes of maps inducing isomorphism on homotopy groups with coefficients in determined torsion abelian groups, in degrees higher than or equal to n. We...

In this paper we analyze some applications of the category of exterior spaces to the study of dynamical systems (flows). We study the notion of an absorbing open subset of a dynamical system; i.e., an open subset that contains the "future part" of all the trajectories. The family of all absorbing open subsets is a quasi-filter which gives the struc...

In this work, we analyze the combinatorial properties of cylinders and subdivisions of augmented semi-simplicial sets. These constructions are obtained as particular cases of a certain action from a co-semi-simplicial set on an augmented semi-simplicial set. We also consider cylinders and subdivision operators in the algebraic setting of augmented...

In this work, we analyze the combinatorial properties of the category of augmented semi- simplicial sets. We consider various monoidal structures induced by the co-product, the product, and the join operator in this category. In addition, we also consider monoidal structures on augmented sequences of integers induced by the sum and product of integ...

The aim of this paper is to study, from a topological and geometrical point of view, the iteration map obtained by the application of iterative methods (Newton or relaxed Newton’s method) to a polynomial equation. In fact, we present a collection of algorithms that avoid the problem of overflows caused by denominators close to zero and the problem...

Este libro contiene un desarrollo axiomático de la geometría plana. El árbol deducctivo se construye a partir de los axiomas de Hilbert que reformulan de modo más riguroso los postulados de Euclides. Se llama geometría neutral a la teoría que incluye los teoremas que se pueden probar sin considerar el quinto postulado de Euclides. Los descubrimient...

The goal of this manual is to present a new package "Plotting Basins of Univariate Rational Functions" (PBURF.jl) written in Julia v1.1.x which allows us to visualize the attraction basins associated to the end points of a discrete semi-flow induced by a rational function on the Riemann sphere by using its geometry and complex structure. The main a...

The goal of this paper is to present a new project "A Julia Implementation for the Iteration of Univariate Rational Functions" which allows us to iterate a rational function on the Riemann sphere by using its geometry and complex structure. We have developed and implemented in Julia language a collection of algorithms for the iteration of a rationa...

An exterior space is a topological space equipped with a distinguished quasi-filter of open subsets (closed by finite intersections) that we call externology. For an exterior space one can consider limits, bar-limits and different sets of end points (Steenrod, Čech, Brown-Grossman). In this work we analyze relations between exterior spaces and disc...

In memory of Sibe Mardešić, our friend. Sibe Mardešić has enriched algebraic topology developing shape and strong shape theories with important constructions and theorems. This paper relates computational topology to shape theory. We have developed some algorithms and implementations that under some conditions give a shape resolution of some Julia...

When a semi-flow is induced by a d-fold branched covering \( f:M \rightarrow M \) defined on a Riemannian manifold M, the associated Julia set J(f) is a compact invariant subset of M and, therefore, there exists an induced restriction \( f | _ {J (f)} :J (f) \rightarrow J (f) \). In order to construct an inverse system of regular sub-complexes whos...

In this paper we analyse some applications of the category of exterior spaces to the study of dynamical systems (flows). The limit space and end space of an exterior space are used to construct different types of limit spaces and end spaces of a dynamical system. In this work we analyse the relationships between the notions and constructions given...

Autonomous differential equations induced by continuous vector fields usually appear in non-smooth mechanics and other scientific contexts. For these type of equations, given an initial condition, one has existence theorems but, in general, the uniqueness of the solution can not be ensured. For continuous vector fields, the equation solutions do no...

The relaxed Newton's method modifies the classical Newton's method with a parameter h in such a way that when it is applied to a polynomial with multiple roots and we take as parameter one of these multiplicities, it is increased the order of convergence to the related multiple root. For polynomials of degree three or higher, the relaxed Newton's m...

When the numerical Newton-Raphson method is applied to find the intersections of two algebraic curves (that is, the roots of a pair of bivariate polynomials), some difficulties appear when the value of a denominator of the corresponding bivariate rational functions is zero. In this paper we give a solution to these problems by using adequate homo...

In this communication we propose a new and effecfive strategy to apply Newton's method to the problem of finding the intersections of two real algebraic curves, that is, the roots of a pair of real bivariate polynomials. The use of adequate homogeneous coordinates and the extension of the domain where the iteration function is defined allow us to a...

In this paper we propose a new and effective strategy to apply Newton's method to the problem of finding the intersections of two real algebraic curves, that is, the roots of a pair of real bivariate polynomials. The use of adequate homogeneous coordinates and the extension of the domain where the iteration function is defined allow us to avoid som...

In this talk we present geometric models for studying the iteration of the function obtained by applying the Newton-Raphson method to find the roots of two bivariate polynomial equations. We consider the iteration of a function of rational type defined on a product of two augmented (real or complex) projective lines. This technique solves the pro...

RESUMEN Este trabajo contiene algunos comentarios sobre las principales líneas de in-vestigación y algunos avances que el Grupo de investigación de Topología de la Universidad de La Rioja ha realizado durante los comienzos del siglo XXI. Se destacan las principales aportaciones teóricas y también algunas aplicaciones prác-ticas. Palabras clave: Teo...

In this work, we develop and implement two algorithms for plotting and computing the measure of the basins of attraction of rational maps defined on the Riemann sphere. These algorithms are based on the subdivisions of a cubical decomposition of a sphere and they have been made by using different computational environments. As an application, we st...

Las soluciones de ecuaciones diferenciales autónomas tienen la estructura de un flujo continuo aunque no es así para el caso de funciones continuas no Lipchizianas, ya que algunas veces se obtienen resultados de existencia pero no de unicidad. Sin embargo, cuando sólo se obtiene unicidad para valores positivos de la variable tiempo las soluciones a...

In this work we use the theory of exterior spaces to construct a (Formula Presented)-completion and a (Formula Presented)-completion of a dynamical system. If X is a flow, we construct canonical maps (Formula Presented) and (Formula Presented) and when these maps are homeomorphisms we have the class of (Formula Presented)-complete and (Formula Pres...

In this work we use the theory of exterior spaces to construct a ˇ C r 0-completion and a ˇ C l 0-completion of a dynamical system. If X is a flow, we construct canonical maps X → ˇ C r 0 (X) and X → ˇ C l 0 (X) and when these maps are homeomorphism we have the class ofČofˇofČ r 0-complete andČandˇandČ l 0-complete flows, respectively. In this stud...

An exterior space is a topological space provided with a quasi-filter of open subsets (closed by finite intersections). In this work, we analyze some relations between the notion of an exterior space and the notion of a discrete semi-flow. On the one hand, for an exterior space, one can consider limits, bar-limits and different sets of end points (...

The notion of closure finite complexes with weak topology introduced by J.H.C. Whitehead determines an adequate category for the study of homotopy theory. Nevertheless a noncompact space which can be described as a $CW$-complex always needs an infinite number of cells. In the present paper we develop a new notion that we call proper $CW$-complex wh...

The aim of this work is to present a new program written in Sage which allows us to visualize the attraction basins associated to the end points of a discrete semi-flow induced by a rational function on the Riemann sphere by using its geometry and complex structure. One interesting novelty brought by the developed program is that it is able to plot...

In this work we construct the $\Co^{\r}$-completion and $\Co^{\l}$-completion of a dynamical system. If $X$ is a flow, we construct canonical maps $X\to \Co^{\r}(X)$ and $X\to \Co^{\l}(X)$ and when these maps are homeomorphism we have the class of $\Co^{\r}$-complete and $\Co^{\l}$-complete flows, respectively. In this study we find out many relati...

This paper presents a new procedure to construct families of spatial approximation-prediction functions which depend on several parameters. The method is based on partitions of the unity. In order to find optimal functions in these families we introduce a Vietoris simplicial set associated to an influence radius. We consider error estimators induce...

For every closed model category with zero object, Quillen gave the construction of Eckman-Hilton and Puppe sequences. In this paper, we remove the hypothesis of the existence of zero object and construct (using the category over the initial object or the category under the final object) these sequences for unpointed model categories. We illustrate...

It is well known that for a connected locally path-connected semi-locally 1-connected space X, there exists a bi-unique correspondence between the pointed d-fold connected coverings and the transitive representations of the fundamental group of X in the symmetric group Σd of degree d.The classification problem becomes more difficult if X is a more...

The closed model category of exterior spaces, that contains the proper category, is a useful tool for the study of non compact spaces and manifolds. The notion of exterior weak ℕ-S-equivalences is given by exterior maps which induce isomorphisms on the k-th ℕ-exterior homotopy groups pk\mathbbN\pi_k^{\mathbb{N}} for k ∈ S, where S is a set of non...

We have developed Postnikov sections for Brown–Grossman homotopy groups and for Steenrod homotopy groups in the category of exterior spaces, which is an extension of the proper category. The homotopy fibre of a fibration in the factorization associated with Brown–Grossman groups is an Eilenberg–Mac Lane exterior space for this type of groups and it...

La belleza, una noción difícil de precisar, relacionada con la armonía, bondad, simplicidad y orden, está presente en todas las culturas a lo largo del tiempo y puede constatarse que de alguna manera hay un sentido universal de lo que es bello. El objetivo de crear algo bello, que en cierto modo representaría un reflejo de la perfección divina, apa...

The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a ‘system of open neighbourhoods at infinity’ while an exterior map is a continuous map which is ‘continuous at infinity’. The category of spaces and proper maps is a subcategory of the category of exterior spac...

For each integer n>1 and a multiplicative system S of non-zero integers, we give a distinct closed model category structure to the category of pointed spaces Top★ and we prove that the corresponding localized category Ho(Top★(S,n)), obtained by inverting the weak equivalences, is equivalent to the standard homotopy category of uniquely (S,n)-divisi...

The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a 'system of open neighborhoods at infinity'. An exterior map is a continuous map which is 'continuous at infinity'. In this paper we present and develop the category of exterior spaces as a good framework for p...

Taking into account the simplicial models given in the category of exterior spaces we define and develop homology invariants for this category: the M-homology and the R-homology, as well as the tubular and the closed tubular homologies. As an application we give a description of the reduced Steenrod homology for compact metric spaces, X, in terms o...

The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a 'system of open neighborhoods at infinity'. An exterior map is a continuous map which is 'continuous at infinity'. In this paper we present and develop the category of exterior spaces as a good framework for p...

The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a ‘system of open neighborhoods at infinity’. An exterior map is a continuous map which is ‘continuous at infinity’. A strongly locally finite CW-complex X, whose skeletons are provided with the family of the co...

POLIEDROS JOSÉJOS´JOSÉ IGNACIO EXTREMIANA ALDANA, LUIS JAVIER HERN´ANDEZHERN´ HERN´ANDEZ PARICIO Y MAR´IAMAR´MAR´IA TERESA RIVAS RODR´IGUEZRODR´RODR´IGUEZ Gracias, Chicho, te queremos Abstract. In this paper we have tried to explore the presence of polyhedra in some areas of the human activity. Taking as started point a reflection about the platoni...

In this paper, for a given space X, a structural category C, and a faithful functor η from C to the category of spaces, we introduce a notion of (C, η)-bundle which contains as particular cases, the notions of covering space, of overlaying space (introduced by Fox), of suspension foliation and other well-known topological structures.The new notion...

Given an integern>1 and any setP of positive integers, one can assign to each topological spaceX a homotopy universal mapX (P,n) →X whereX (P,n) is an (n−1)-connected CW-complex whose homotopy groups areP-torsion. We analyze this construction and its properties by means of a suitable closed model category structure on the pointed category of top...

In this paper, we introduce the notion of exterior space and give a full embedding of the category P of spaces and proper maps into the category E of exterior spaces. We show that the category E admits the structure of a closed simplicial model category. This technique solves the problem of using homotopy constructions available in the localised ca...

For m n > 0, a map f between pointed spaces is said to be a weak [n, m]-equivalence if f induces isomorphisms of the homotopy groups π k for n k m. Associated with this notion we give two different closed model category structures to the category of pointed spaces. Both structures have the same class of weak equivalences but different classes of fi...

For each integer n>0, we give a distinct closed model category structure to the category of pointed spaces, Top * such that the corresponding localized category Ho(Top * n ) is equivalent to the standard homotopy category of (n-1)-connected CW-complexes. The structure of closed model category given by D. Quillen to Top * [Ann. Math., II. Ser. 90, 2...

For each integer n ≥ 0, we give a distinct closed model category structure to the categories of spaces and of simplicial sets. Recall that a non-empty map is said to be a weak equivalence if it induces isomorphisms on the homotopy groups for any choice of base point. Putting the condition on dimensions ≥ n, we have the notion of a weak n-equivalenc...

In this paper we have tried to reduce the classical classification problems for spaces and maps of the proper category and of the strong shape category to similar problems in the homotopy category of simplicial sets or in the homotopy category of simplicial Af-sets, which M is the monoid of proper selfmaps of the discrete space N of nonnegative int...

In 1975 E. M. Brown constructed a functor P which carries the tower of fundamental groups of the end of a (nice) space to the Brown-Grossman fundamental group. In this work, we study this functor and its extensions and analogues defined for pro-sets, pro-pointed sets, pro-groups and pro-abelian groups. The new versions of the P functor are provided...

We extend the Freudenthal suspension theorem to the category of towers of spaces and as consequences we obtain suspension theorems for proper homotopy and strong shape theories.

In this paper we have tried to reduce the classical classification problems for spaces and maps of the proper category and of the strong shape category to similar problems in the homotopy category of simplicial sets or in the homotopy category of simplicial M-sets, which M is the monoid of proper selfmaps of the discrete space N of nonnegative inte...

This paper is centred around an embedding theorem for the proper n-homotopy category at infinity of Σ-compact simplicial complexes into the n-homotopy category of prospaces. There is a corresponding global version. This enables one to prove proper analogues of various classical results of Whitehead on n-types, Jn-complexes, etc.

As part of a program to study Pro n-types and Proper n-types of locally finite simplicial complexes, in this paper we give notions of n-fibrations, n-cofibrations and weak n-equivalences in the category of crossed complexes Crs that satisfy the axioms for a closed model structure in the sense of Quillen. The category obtained by formal inverting th...

The notion of closure finite complexes with weak topology introduced by J. H. C. Whitehead determines an adequate category for the study of homotopy theory. Nevertheless a noncompact space which can be described as a CW-complex always needs an infinite number of cells. In the present paper we develop a new notion that we call proper CW-complex whic...

The Brown–Grossman proper homotopy groups at an end of a locally compact Hausdorff space, X, were introduced by E. M. Brown in 1974. In this note we define and study ‘global’ versions of these groups and compare these global groups with the groups ‘at infinity’. We also obtain several interlocking exact sequences relating these groups with the Hure...

In this paper we study the group of pointed proper homotopy classes of proper pointed maps from (n+1, 0) to a pointed σ-compact space (X, x) and prove the existence of a diagram of exact sequences linking the groups with the Brown—Grossman groups , the Steenrod groups πn(X) and the classical Hurewicz homotopy groups πn(X).

In this paper, we study the extension problem in the category of topological spaces and proper maps. To attack this problem a new proper cohomology theory and a new obstruction cocycle are defined. This cohomology theory has coefficients in a morphism π′ → π where π′ is a pro-abelian group and π is an abelian group.Let Kn be the n-skeleton of a sec...

For a commutative unitary ring R, we have developed a new homotopy theory in the category of abelian groups. The homotopy category of this theory has the following property: if an abelian group A admits the structrure of an R-module, then A has the homotopy type of the zero abelian group. As a consequence of this fact, this theory is a useful tool...

Resumen Se introduce una nueva técnica para el estudio y clasificación de muestras de datos. Las ideas básicas están basadas en la llamada Teoría de la Forma introducida por Borsuk y que aproxima espacios localmente patológicos por una sucesión de poliedros. Aquí modificamos el modelo anterior tomando una aproximación poliedral (o un número finito)...