Arevalo, I., Hernandez, R., Martin, M. J., & Vukotic, D. (2018). On weighted compositions preserving the Caratheodory class. Mon.heft. Math., 187(3), 459–477.
Abstract: We characterize in various ways the weighted composition transformations which preserve the class P of normalized analytic functions in the disk with positive real part. We analyze the meaning of the criteria obtained for various special cases of symbols and identify the fixed points of such transformations.
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Bravo, M., Cominetti, R., & Pavez-Signe, M. (2019). Rates of convergence for inexact Krasnosel'skii-Mann iterations in Banach spaces. Math. Program., 175(1-2), 241–262.
Abstract: We study the convergence of an inexact version of the classical Krasnosel'skii-Mann iteration for computing fixed points of nonexpansive maps. Our main result establishes a new metric bound for the fixed-point residuals, from which we derive their rate of convergence as well as the convergence of the iterates towards a fixed point. The results are applied to three variants of the basic iteration: infeasible iterations with approximate projections, the Ishikawa iteration, and diagonal Krasnosels'kii-Mann schemes. The results are also extended to continuous time in order to study the asymptotics of nonautonomous evolution equations governed by nonexpansive operators.
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Goles, E., & Noual, M. (2012). Disjunctive networks and update schedules. Adv. Appl. Math., 48(5), 646–662.
Abstract: In this paper, we present a study of the dynamics of disjunctive networks under all block-sequential update schedules. We also present an extension of this study to more general fair periodic update schedules, that is, periodic update schedules that do not update some elements much more often than some others. Our main aim is to classify disjunctive networks according to the robustness of their dynamics with respect to changes of their update schedules. To study this robustness, we focus on one property, that of being able to cycle dynamically. (C) 2012 Elsevier Inc. All rights reserved.
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Goles, E., & Ruz, G. A. (2015). Dynamics of neural networks over undirected graphs. Neural Netw., 63, 156–169.
Abstract: In this paper we study the dynamical behavior of neural networks such that their interconnections are the incidence matrix of an undirected finite graph G = (V, E) (i.e., the weights belong to {0, 1}). The network may be updated synchronously (every node is updated at the same time), sequentially (nodes are updated one by one in a prescribed order) or in a block-sequential way (a mixture of the previous schemes). We characterize completely the attractors (fixed points or cycles). More precisely, we establish the convergence to fixed points related to a parameter alpha(G), taking into account the number of loops, edges, vertices as well as the minimum number of edges to remove from E in order to obtain a maximum bipartite graph. Roughly, alpha(G') < 0 for any G' subgraph of G implies the convergence to fixed points. Otherwise, cycles appear. Actually, for very simple networks (majority functions updated in a block-sequential scheme such that each block is of minimum cardinality two) we exhibit cycles with nonpolynomial periods. (C) 2014 Elsevier Ltd. All rights reserved.
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Goles, E., & Salinas, L. (2008). Comparison between parallel and serial dynamics of Boolean networks. Theor. Comput. Sci., 396(1-3), 247–253.
Abstract: In this article we study some aspects about the graph associated with parallel and serial behavior of a Boolean network. We conclude that the structure of the associated graph can give some information about the attractors of the network. We show that the length of the attractors of Boolean networks with a graph by layers is a power of two and under certain conditions the only attractors are fixed points. Also, we show that, under certain conditions, dynamical cycles are not the same for parallel and serial updates of the same Boolean network. (C) 2007 Elsevier B.V. All rights reserved.
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Goles, E., Medina, P., Montealegre, P., & Santivanez, J. (2022). Majority networks and consensus dynamics. Chaos Solitons Fractals, 164, 112697.
Abstract: Consensus is an emergent property of many complex systems, considering this as an absolute majority phenomenon. In this work we study consensus dynamics in grids (in silicon), where individuals (the vertices) with two possible opinions (binary states) interact with the eight nearest neighbors (Moore’s neighborhood). Dynamics emerge once the majority rule drives the evolution of the system. In this work, we fully characterize the sub-neighborhoods on which the consensus may be reached or not. Given this, we study the quality of the consensus proposing two new measures, namely effectiveness and efficiency. We characterize attraction basins through the energy-like and magnetization-like functions similar to the Ising spin model.
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Goles, E., Montealegre, P., & Vera, J. (2016). Naming Game Automata Networks. J. Cell. Autom., 11(5-6), 497–521.
Abstract: In this paper we introduce automata networks to model some features of the emergence of a vocabulary related with the naming game model. We study the dynamical behaviour (attractors and convergence) of extremal and majority local functions.
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Oviedo, H., Lara, H., & Dalmau, O. (2023). SCALED FIXED POINT ALGORITHM FOR COMPUTING THE MATRIX SQUARE ROOT. Fixed Point Theory, 24(1), 295–308.
Abstract: This paper addresses the numerical solution of the matrix square root problem. Two fixed point iterations are proposed by rearranging the nonlinear matrix equation A – X2 = 0 and incorporating a positive scaling parameter. The proposals only need to compute one matrix inverse and at most two matrix multiplications per iteration. A global convergence result is established. The numerical comparisons versus some existing methods from the literature, on several test problems, demonstrate the efficiency and effectiveness of our proposals.
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