CARACTERIZACIÓN DE LA TRANSICIÓN AL CAOS EN ECONOMÍA

Básicamente, cualquier proceso que evoluciona con el tiempo es un sistema dinámico. Los sistemas dinámicos aparecen en todas las ramas de la ciencia y, virtualmente, en todos los aspectos de la vida. La Economía es un ejemplo de un sistema dinámico: las variaciones de precios en la Bolsa de Valores son un ejemplo simple de la evolución temporal de dicho sistema. El principal objetivo del estudio y análisis de un sistema dinámico es la posibilidad de predecir el resultado final de un proceso. Algunos sistemas dinámicos son predecibles y otros no lo son. Existen sistemas dinámicos muy simples que dependen de una sola variable y muestran un comportamiento sumamente no predecible, debido a la presencia del “caos”, esto es, poseen una dependencia sensible a los valores iniciales. El objetivo principal de este trabajo es investigar cuáles son los factores que producen caminos alternativos para pasar del orden al caos en problemas económicos. Palabras clave: caos, atractores y repulsores, bifurcación, atractor extraño, contornos fractales. Abstract Basically, any process evolving with time is a dynamical system. Dynamical systems appear at every branch of Science and virtually at every aspect of life. Economy is an example of a dynamical system: the prices variations at the Stock Exchange is a simple illustration of the temporal evolution of this system. The main objective of the study and analysis of a dynamical system is the possibility of predicting the final result of a process. Some dynamical systems are predictable and some are not. There are very simple dynamical systems depending only on one variable that show a highly non predictable behavior, due to the presence of “chaos”, that means they possess a sensitive dependence on the initial values. The main aim of this paper is to investigate which are the factors that produce alternative roads to pass from order to chaos in economic problems. Keywords: chaos, attractors and repellers, bifurcation, strange attractor, fractal basin boundaries

LA FAMILIA DE NUMEROS METALICOS

En este trabajo vamos a introducir una nueva familia de números irracionales cuadráticos positivos. Se llama familia de números metálicos [1], [2], [3], [4] y su miembro más importante es el número de oro φ . Entre sus parientes, podemos mencionar el número de plata, el número de bronce, el número de cobre, el número de níquel, etc. Los miembros de dicha familia gozan de propiedades matemáticas comunes que son fundamentales en la investigación actual sobre la estabilidad de macro- y microsistemas físicos, desde la estructura interna del ADN hasta las galaxias astronómicas. Los resultados más notables de esta nueva investigación son los siguientes: • Los miembros de la familia intervienen en la determinación del comportamiento cuasi-periódico de sistemas dinámicos no lineales, constituyendo una herramienta invalorable en la búsqueda de rutas universales al caos. • Las sucesiones numéricas basadas en los miembros de esta familia, satisfacen muchas propiedades aditivas y simultáneamente son sucesiones geométricas, por lo que han sido utilizadas con frecuencia como base de muchos sistemas de proporciones. Palabras clave: desarrollo en fracciones continuas, sucesiones de Fibonacci, caos, atractor extraño, ecuación logística. Abstract In this paper we introduce a new family of positive quadratic irrational numbers. It is called the Metallic Means family [1], [2], [3], [4] and its most renowned member is the Golden Mean. Among its relatives we may mention the Silver Mean, the Bronze Mean, the Cupper Mean, the Nickel Mean, etc. The members of such a family enjoy common mathematical properties that are fundamental in the present research about the stability of macro- and micro- physical systems, going from the internal structure of DNA up to the astronomical galaxies. The most important results of this new investigation are the following: ! The members of this family intervene in the determination of the quasi-periodic behavior of non linear dynamical systems, being essential tools in the search of universal routes to chaos. ! The numerical sequences based on the members of this family, satisfy many additive properties and simultaneously, are geometric sequences. This unique property has had as a consequence the use of some Metallic Means as a base for proportion systems. Keywords: Continued fraction expansion; Fibonacci sequences; Strange attractor; Logistic equation

Las redes y sus aplicaciones

La teoría de las redes es una rama de la Investigación Operativa que se aplica en el tratamiento de diversos problemas provenientes del campo económico, sociológico y tecnológico. Históricamente está comprobado que el hombre, ante el planteo de un problema, tiende a hacer una diagrama en el que los puntos representan individuos, localidades, actividades, etapas de un proyecto, etc., uniéndolos por medio de líneas que indican una cierta relación existente entre ellos. D. Konig fue el primero en proponer que tales diagramas recibieran el nombre de redes, haciendo un estudio sistemático de sus propiedades. Con todo rigor, la exposición de dichas propiedades incluiría un número de conceptos y teoremas, entre los cuales algunos son relativamente complicados. Dado que nuestro objetivo es presentar este tema de manera que resulte accesible a un gran número de lectores de distinto nivel científico, presentaremos los conceptos básicos del modo más simple posible, mostraremos como usarlos y daremos algunos métodos que pueden ser de uso fructífero en las aplicaciones

Teoría de las zonas alcanzables en sistemas bidimensionales

Fil:Winitzky de Spinadel, Vera Martha. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

VISUALIZACIÓN Y TECNOLOGÍA

El objetivo de este trabajo es mostrar cómo la visualización obtenida por medio de actuales herramientas matemático/informáticas tales como la gráfica computarizada, constituye un elemento indispensable en la aplicación en la investigación de conceptos matemáticos que van desde las estructuras fractales, los nudos y la transición al caos hasta las transformaciones topológicas más generales. Palabras clave: caos, atractor extraño, hipercubo, hiperestereograma. Abstract The main aim of this paper is to show how the visualization obtained through the use of sophisticated mathematical tools like the computerized graphics, is an essential element in the teaching of mathematical concepts that comprise from fractal structures, knots and the transition to chaos up to the more general topological transformations. Keywords: chaos, strange attractor, hypercube, hyper stereogra

Qubit-Qutrit Separability-Probability Ratios

Paralleling our recent computationally-intensive (quasi-Monte Carlo) work for the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional hyperarea of the (separable and nonseparable) N x N density matrices, based on the Bures (minimal monotone) metric -- and also their analogous formulas (quant-ph/0302197) for the (non-monotone) Hilbert-Schmidt metric. With the same seven billion well-distributed (``low-discrepancy'') sample points, we estimate the unknown volumes and hyperareas based on five additional (monotone) metrics of interest, including the Kubo-Mori and Wigner-Yanase. Further, we estimate all of these seven volume and seven hyperarea (unknown) quantities when restricted to the separable density matrices. The ratios of separable volumes (hyperareas) to separable plus nonseparable volumes (hyperareas) yield estimates of the separability probabilities of generically rank-six (rank-five) density matrices. The (rank-six) separability probabilities obtained based on the 35-dimensional volumes appear to be -- independently of the metric (each of the seven inducing Haar measure) employed -- twice as large as those (rank-five ones) based on the 34-dimensional hyperareas. Accepting such a relationship, we fit exact formulas to the estimates of the Bures and Hilbert-Schmidt separable volumes and hyperareas.(An additional estimate -- 33.9982 -- of the ratio of the rank-6 Hilbert-Schmidt separability probability to the rank-4 one is quite clearly close to integral too.) The doubling relationship also appears to hold for the N=4 case for the Hilbert-Schmidt metric, but not the others. We fit exact formulas for the Hilbert-Schmidt separable volumes and hyperareas.Comment: 36 pages, 15 figures, 11 tables, final PRA version, new last paragraph presenting qubit-qutrit probability ratios disaggregated by the two distinct forms of partial transpositio

Silver mean conjectures for 15-d volumes and 14-d hyperareas of the separable two-qubit systems

Extensive numerical integration results lead us to conjecture that the silver mean, that is, s = \sqrt{2}-1 = .414214 plays a fundamental role in certain geometries (those given by monotone metrics) imposable on the 15-dimensional convex set of two-qubit systems. For example, we hypothesize that the volume of separable two-qubit states, as measured in terms of (four times) the minimal monotone or Bures metric is s/3, and 10s in terms of (four times) the Kubo-Mori monotone metric. Also, we conjecture, in terms of (four times) the Bures metric, that that part of the 14-dimensional boundary of separable states consisting generically of rank-four 4 x 4 density matrices has volume (``hyperarea'') 55s/39 and that part composed of rank-three density matrices, 43s/39, so the total boundary hyperarea would be 98s/39. While the Bures probability of separability (0.07334) dominates that (0.050339) based on the Wigner-Yanase metric (and all other monotone metrics) for rank-four states, the Wigner-Yanase (0.18228) strongly dominates the Bures (0.03982) for the rank-three states.Comment: 30 pages, 6 tables, 17 figures; nine new figures and one new table in new section VII.B pertaining to 14-dimensional hyperareas associated with various monotone metric

Generalized Inverse Participation Numbers in Metallic-Mean Quasiperiodic Systems

From the quantum mechanical point of view, the electronic characteristics of quasicrystals are determined by the nature of their eigenstates. A practicable way to obtain information about the properties of these wave functions is studying the scaling behavior of the generalized inverse participation numbers $Z_q \sim N^{-D_q(q-1)}$ with the system size $N$. In particular, we investigate $d$-dimensional quasiperiodic models based on different metallic-mean quasiperiodic sequences. We obtain the eigenstates of the one-dimensional metallic-mean chains by numerical calculations for a tight-binding model. Higher dimensional solutions of the associated generalized labyrinth tiling are then constructed by a product approach from the one-dimensional solutions. Numerical results suggest that the relation $D_q^{d\mathrm{d}} = d D_q^\mathrm{1d}$ holds for these models. Using the product structure of the labyrinth tiling we prove that this relation is always satisfied for the silver-mean model and that the scaling exponents approach this relation for large system sizes also for the other metallic-mean systems.Comment: 7 pages, 3 figure