Journal articles: 'Combinatorial dynamical systems'

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Wisniewski, Rafael. "Combinatorial Abstractions of Dynamical Systems." Electronic Proceedings in Theoretical Computer Science 124 (August 22, 2013): 5–8. http://dx.doi.org/10.4204/eptcs.124.2.

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Díaz, Rafael, and Sergio Villamarín. "Combinatorial micro–macro dynamical systems." São Paulo Journal of Mathematical Sciences 14, no. 1 (August 29, 2018): 66–122. http://dx.doi.org/10.1007/s40863-018-0103-2.

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NEKRASHEVYCH, VOLODYMYR. "Combinatorial models of expanding dynamical systems." Ergodic Theory and Dynamical Systems 34, no. 3 (January 24, 2013): 938–85. http://dx.doi.org/10.1017/etds.2012.163.

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AbstractWe prove homotopical rigidity of expanding dynamical systems, by showing that they are determined by a group-theoretic invariant. We use this to show that the Julia set of every expanding dynamical system is an inverse limit of simplicial complexes constructed by inductive cut-and-paste rules. Moreover, the cut-and-paste rules can be found algorithmically from the algebraic invariant.

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Forman, Robin. "Combinatorial vector fields and dynamical systems." Mathematische Zeitschrift 228, no. 4 (August 1998): 629–81. http://dx.doi.org/10.1007/pl00004638.

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Harmer, Russ, Vincent Danos, Jérôme Feret, Jean Krivine, and Walter Fontana. "Intrinsic information carriers in combinatorial dynamical systems." Chaos: An Interdisciplinary Journal of Nonlinear Science 20, no. 3 (September 2010): 037108. http://dx.doi.org/10.1063/1.3491100.

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Banzhaf, Wolfgang. "Artificial Chemistries – Towards Constructive Dynamical Systems." Solid State Phenomena 97-98 (April 2004): 43–50. http://dx.doi.org/10.4028/www.scientific.net/ssp.97-98.43.

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In this contribution we consider constructive dynamical systems, taking one particular Artificial Chemistry as an example. We argue that constructive dynamical systems are in fact widespread in combinatorial spaces of Artificial Chemistries.

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Suzuki, Tomoya. "Dynamical Combinatorial Optimization for Predicting Multivariate Complex Systems." Journal of Signal Processing 16, no. 6 (2012): 537–46. http://dx.doi.org/10.2299/jsp.16.537.

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Yang, Insoon, Samuel A. Burden, Ram Rajagopal, S. Shankar Sastry, and Claire J. Tomlin. "Approximation Algorithms for Optimization of Combinatorial Dynamical Systems." IEEE Transactions on Automatic Control 61, no. 9 (September 2016): 2644–49. http://dx.doi.org/10.1109/tac.2015.2504867.

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Dey, Tamal K., Marian Mrozek, and Ryan Slechta. "Persistence of Conley--Morse Graphs in Combinatorial Dynamical Systems." SIAM Journal on Applied Dynamical Systems 21, no. 2 (April 11, 2022): 817–39. http://dx.doi.org/10.1137/21m143162x.

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Morrison, Rebecca E., Eric J. Friedman, and Adam S. Landsberg. "Combinatorial games with a pass: A dynamical systems approach." Chaos: An Interdisciplinary Journal of Nonlinear Science 21, no. 4 (December 2011): 043108. http://dx.doi.org/10.1063/1.3650234.

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Murota, Kazuo. "Some recent results in combinatorial approaches to dynamical systems." Linear Algebra and its Applications 122-124 (September 1989): 725–59. http://dx.doi.org/10.1016/0024-3795(89)90674-5.

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Jaewook Lee. "Pseudobasin of attraction for combinatorial dynamical systems: theory and its application to combinatorial optimization." IEEE Transactions on Circuits and Systems II: Express Briefs 52, no. 4 (April 2005): 189–93. http://dx.doi.org/10.1109/tcsii.2004.842025.

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Silva, L., J. Leonel Rocha, and M. T. Silva. "Bifurcations of 2-Periodic Nonautonomous Stunted Tent Systems." International Journal of Bifurcation and Chaos 27, no. 06 (June 15, 2017): 1730020. http://dx.doi.org/10.1142/s0218127417300208.

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In this paper, we will consider a family of 2-periodic nonautonomous dynamical systems, generated by the alternate iteration of two stunted tent maps and study its bifurcation skeleton. We will describe the bifurcation phenomena along and around the bones accomplished with the combinatorial data furnished by the respective symbolic dynamics.

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Ovchinnikov, Igor V., and Kang L. Wang. "Stochastic dynamics and combinatorial optimization." Modern Physics Letters B 31, no. 31 (November 6, 2017): 1750285. http://dx.doi.org/10.1142/s0217984917502852.

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Natural dynamics is often dominated by sudden nonlinear processes such as neuroavalanches, gamma-ray bursts, solar flares, etc., that exhibit scale-free statistics much in the spirit of the logarithmic Ritcher scale for earthquake magnitudes. On phase diagrams, stochastic dynamical systems (DSs) exhibiting this type of dynamics belong to the finite-width phase (N-phase for brevity) that precedes ordinary chaotic behavior and that is known under such names as noise-induced chaos, self-organized criticality, dynamical complexity, etc. Within the recently proposed supersymmetric theory of stochastic dynamics, the N-phase can be roughly interpreted as the noise-induced “overlap” between integrable and chaotic deterministic dynamics. As a result, the N-phase dynamics inherits the properties of the both. Here, we analyze this unique set of properties and conclude that the N-phase DSs must naturally be the most efficient optimizers: on one hand, N-phase DSs have integrable flows with well-defined attractors that can be associated with candidate solutions and, on the other hand, the noise-induced attractor-to-attractor dynamics in the N-phase is effectively chaotic or aperiodic so that a DS must avoid revisiting solutions/attractors thus accelerating the search for the best solution. Based on this understanding, we propose a method for stochastic dynamical optimization using the N-phase DSs. This method can be viewed as a hybrid of the simulated and chaotic annealing methods. Our proposition can result in a new generation of hardware devices for efficient solution of various search and/or combinatorial optimization problems.

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Kopp, Stephan, and Christian M. Reidys. "Neutral networks: a combinatorial perspective." Advances in Complex Systems 02, no. 03 (September 1999): 283–301. http://dx.doi.org/10.1142/s0219525999000151.

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The existence of neutral networks in genotype-phenotype maps has provided significant insight in theoretical investigations of evolutionary change and combinatorial optimization. In this paper we will consider neutral networks of two particular genotype-phenotype maps from a combinatorial perspective. The first map occurs in the context of folding RNA molecules into their secondary structures and the second map occurs in the study of sequential dynamical systems, a new class of dynamical systems designed for the analysis of computer simulations. We will prove basic properties of neutral nets and present an error threshold phenomenon for evolving populations of simulation schedules.

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MISIUREWICZ, MICHAŁ. "THIRTY YEARS AFTER SHARKOVSKIĬ'S THEOREM." International Journal of Bifurcation and Chaos 05, no. 05 (October 1995): 1275–81. http://dx.doi.org/10.1142/s0218127495000946.

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BERTHÉ, VALÉRIE, JÉRÉMIE BOURDON, TIMO JOLIVET, and ANNE SIEGEL. "A combinatorial approach to products of Pisot substitutions." Ergodic Theory and Dynamical Systems 36, no. 6 (March 19, 2015): 1757–94. http://dx.doi.org/10.1017/etds.2014.141.

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We define a generic algorithmic framework to prove a pure discrete spectrum for the substitutive symbolic dynamical systems associated with some infinite families of Pisot substitutions. We focus on the families obtained as finite products of the three-letter substitutions associated with the multidimensional continued fraction algorithms of Brun and Jacobi–Perron. Our tools consist in a reformulation of some combinatorial criteria (coincidence conditions), in terms of properties of discrete plane generation using multidimensional (dual) substitutions. We also deduce some topological and dynamical properties of the Rauzy fractals, of the underlying symbolic dynamical systems, as well as some number-theoretical properties of the associated Pisot numbers.

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Ban, Hyunju, and William D. Kalies. "A Computational Approach to Conley’s Decomposition Theorem." Journal of Computational and Nonlinear Dynamics 1, no. 4 (May 14, 2006): 312–19. http://dx.doi.org/10.1115/1.2338651.

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Background. The discrete dynamics generated by a continuous map can be represented combinatorially by an appropriate multivalued map on a discretization of the phase space such as a cubical grid or triangulation. Method of approach. We describe explicit algorithms for computing dynamical structures for the combinatorial multivalued maps. Results. We provide computational complexity bounds and numerical examples. Specifically we focus on the computation attractor-repeller pairs and Lyapunov functions for Morse decompositions. Conclusions. The computed discrete Lyapunov functions are weak Lyapunov functions and well-approximate a continuous Lyapunov function for the underlying map.

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Glasscock, Daniel, Andreas Koutsogiannis, and Florian Karl Richter. "Multiplicative combinatorial properties of return time sets in minimal dynamical systems." Discrete & Continuous Dynamical Systems - A 39, no. 10 (2019): 5891–921. http://dx.doi.org/10.3934/dcds.2019258.

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Mischaikow, Konstantin. "Topological techniques for efficient rigorous computation in dynamics." Acta Numerica 11 (January 2002): 435–77. http://dx.doi.org/10.1017/s0962492902000065.

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We describe topological methods for the efficient, rigorous computation of dynamical systems. In particular, we indicate how Conley's Fundamental Decomposition Theorem is naturally related to combinatorial approximations of dynamical systems. Furthermore, we show that computations of Morse decompositions and isolating blocks can be performed efficiently. We conclude with examples indicating how these ideas can be applied to finite- and infinite-dimensional discrete and continuous dynamical systems.

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BOLLT, ERIK M. "COMBINATORIAL CONTROL OF GLOBAL DYNAMICS IN A CHAOTIC DIFFERENTIAL EQUATION." International Journal of Bifurcation and Chaos 11, no. 08 (August 2001): 2145–62. http://dx.doi.org/10.1142/s0218127401003401.

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Controlling chaos has been an extremely active area of research in applied dynamical systems, following the introduction of the Ott, Grebogi, Yorke (OGY) technique in 1990 [Ott et al., 1990], but most of this research based on parametric feedback control uses local techniques. Associated with a dynamical system which pushes forward initial conditions in time, transfer operators, including the Frobenius–Perron operator, are associated dynamical systems which push forward ensemble distributions of initial conditions. In [Bollt, 2000a, 2000b; Bollt & Kostelich, 1998], we have shown that such global representations of a discrete dynamical system are useful in controlling certain aspects of a chaotic dynamical system which could only be accessible through such a global representation. Such aspects include invariant measure targeting, as well as orbit targeting. In this paper, we develop techniques to show that our previously discrete time techniques are accessible also to a differential equation. We focus on the Duffing oscillator as an example. We also show that a recent extension of our techniques by Góra and Boyarsky [1999] can be further simplified and represented in a convenient and compact way by using a tensor product.

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Scott-Phillips, Thomas C., and Richard A. Blythe. "Why is combinatorial communication rare in the natural world, and why is language an exception to this trend?" Journal of The Royal Society Interface 10, no. 88 (November 6, 2013): 20130520. http://dx.doi.org/10.1098/rsif.2013.0520.

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In a combinatorial communication system, some signals consist of the combinations of other signals. Such systems are more efficient than equivalent, non-combinatorial systems, yet despite this they are rare in nature. Why? Previous explanations have focused on the adaptive limits of combinatorial communication, or on its purported cognitive difficulties, but neither of these explains the full distribution of combinatorial communication in the natural world. Here, we present a nonlinear dynamical model of the emergence of combinatorial communication that, unlike previous models, considers how initially non-communicative behaviour evolves to take on a communicative function. We derive three basic principles about the emergence of combinatorial communication. We hence show that the interdependence of signals and responses places significant constraints on the historical pathways by which combinatorial signals might emerge, to the extent that anything other than the most simple form of combinatorial communication is extremely unlikely. We also argue that these constraints can be bypassed if individuals have the socio-cognitive capacity to engage in ostensive communication. Humans, but probably no other species, have this ability. This may explain why language, which is massively combinatorial, is such an extreme exception to nature's general trend for non-combinatorial communication.

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Lu, Ke Bo. "Some combinatorial properties of words in discrete dynamical systems from antisymmetric cubic maps." Acta Mathematica Sinica, English Series 29, no. 11 (January 4, 2013): 2181–92. http://dx.doi.org/10.1007/s10114-013-1743-x.

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ALSEDÀ, LL, and A. FALCÓ. "ON THE STRUCTURE OF THE KNEADING SPACE OF BIMODAL DEGREE ONE CIRCLE MAPS." International Journal of Bifurcation and Chaos 20, no. 09 (September 2010): 2701–21. http://dx.doi.org/10.1142/s0218127410027301.

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In this paper, we introduce an index space and two ⋆-like operators that can be used to describe bifurcations for parametrized families of degree one circle maps. Using these topological tools, we give a description of the kneading space, that is, the set of all dynamical combinatorial types for the class of all bimodal degree one circle maps considered as dynamical systems.

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WANG, QIUDONG, and LAI-SANG YOUNG. "Dynamical profile of a class of rank-one attractors." Ergodic Theory and Dynamical Systems 33, no. 4 (May 8, 2012): 1221–64. http://dx.doi.org/10.1017/s014338571200020x.

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AbstractThis paper contains results on the geometric and ergodic properties of a class of strange attractors introduced by Wang and Young [Towards a theory of rank one attractors. Ann. of Math. (2) 167 (2008), 349–480]. These attractors can live in phase spaces of any dimension, and have been shown to arise naturally in differential equations that model several commonly occurring phenomena. Dynamically, such systems are chaotic; they have controlled non-uniform hyperbolicity with exactly one unstable direction, hence the name rank-one. In this paper we prove theorems on their Lyapunov exponents, Sinai–Ruelle–Bowen (SRB) measures, basins of attraction, and statistics of time series, including central limit theorems, exponential correlation decay and large deviations. We also present results on their global geometric and combinatorial structures, symbolic coding and periodic points. In short, we build a dynamical profile for this class of dynamical systems, proving that these systems exhibit many of the characteristics normally associated with ‘strange attractors’.

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Nishimoto, Ryu, and Jun Tani. "Learning to generate combinatorial action sequences utilizing the initial sensitivity of deterministic dynamical systems." Neural Networks 17, no. 7 (September 2004): 925–33. http://dx.doi.org/10.1016/j.neunet.2004.02.003.

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Tanevski, Jovan, Ljupčo Todorovski, and Sašo Džeroski. "Combinatorial search for selecting the structure of models of dynamical systems with equation discovery." Engineering Applications of Artificial Intelligence 89 (March 2020): 103423. http://dx.doi.org/10.1016/j.engappai.2019.103423.

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DRACH, KOSTIANTYN, YAUHEN MIKULICH, JOHANNES RÜCKERT, and DIERK SCHLEICHER. "A combinatorial classification of postcritically fixed Newton maps." Ergodic Theory and Dynamical Systems 39, no. 11 (March 13, 2018): 2983–3014. http://dx.doi.org/10.1017/etds.2018.2.

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We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials as dynamical systems. This lays the foundation for classification results of more general classes of Newton maps. A fundamental ingredient is the proof that for every Newton map (postcritically finite or not) every connected component of the basin of an attracting fixed point can be connected to$\infty$through a finite chain of such components.

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PIVATO, MARCUS, and REEM YASSAWI. "Embedding Bratteli–Vershik systems in cellular automata." Ergodic Theory and Dynamical Systems 30, no. 5 (October 15, 2009): 1561–72. http://dx.doi.org/10.1017/s0143385709000601.

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AbstractMany dynamical systems can be naturally represented as Bratteli–Vershik (or adic) systems, which provide an appealing combinatorial description of their dynamics. If an adic system X is linearly recurrent, then we show how to represent X using a two-dimensional subshift of finite type Y; each ‘row’ in a Y-admissible configuration corresponds to an infinite path in the Bratteli diagram of X, and the vertical shift on Y corresponds to the ‘successor’ map of X. Any Y-admissible configuration can then be recoded as the space-time diagram of a one-dimensional cellular automaton Φ; in this way X is embedded in Φ (i.e. X is conjugate to a subsystem of Φ). With this technique, we can embed many odometers, Toeplitz systems, and constant-length substitution systems in one-dimensional cellular automata.

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MANGANARO, GABRIELE, and JOSE PINEDA DE GYVEZ. "NONLINEAR COMPUTABILITY BASED ON CHAOS." International Journal of Bifurcation and Chaos 10, no. 02 (February 2000): 415–29. http://dx.doi.org/10.1142/s0218127400000268.

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Two new computing models based on information coding and chaotic dynamical systems are presented. The novelty of these models lies on the blending of chaos theory and information coding to solve complex combinatorial problems. A unique feature of our computing models is that despite the nonpredictability property of chaos, it is possible to solve any combinatorial problem in a systematic way, and with only one dynamical system. This is in sharp contrast to methods based on heuristics employing an array of chaotic cells. To prove the computing power and versatility of our models, we address the systematic solution of classical NP-complete problems such as the three colorability and the directed Hamiltonian path in addition to a new chaotic simulated annealing scheme.

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Zhukov, Vladislav, and Anastasia Smirnova. "Morphology images of visual-cognitive character of dynamical system information." E3S Web of Conferences 244 (2021): 05030. http://dx.doi.org/10.1051/e3sconf/202124405030.

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The study of images of bladed-edged weapons, which allowed to design a paryura, was carried out. To create images of design objects –parures represented by visual-symbolic cognitive information dynamic systems (VKIDS) with locally stable structures (LUS) in the development of cognitive technologies of plastic arts and design, the methods of linguistic-combinatorial and tabular modelling were used. RESULTS: the study of morphology, colourists, eidos, and concept in the creation of images of design objects-jewellery associated with the main symbolized elements of state and social policy management, represented by images of sacred, astral cosmogony, and cosmology.

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MORTVEIT, H. S., and C. M. REIDYS. "NEUTRAL EVOLUTION AND MUTATION RATES OF SEQUENTIAL DYNAMICAL SYSTEMS." Advances in Complex Systems 07, no. 03n04 (September 2004): 395–418. http://dx.doi.org/10.1142/s0219525904000275.

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In this paper we study the evolution of sequential dynamical systems [Formula: see text] as a result of the erroneous replication of the SDS words. An [Formula: see text] consists of (a) a finite, labeled graph Y in which each vertex has a state, (b) a vertex labeled sequence of functions (Fvi,Y), and (c) a word w, i.e. a sequence (w1,…,wk), where each wi is a Y-vertex. The function Fwi,Y updates the state of vertex wi as a function of the states of wi and its Y-neighbors and leaves the states of all other vertices fixed. The [Formula: see text] over the word w and Y is the composed map: [Formula: see text]. The word w represents the genotype of the [Formula: see text] in a natural way. We will randomly flip consecutive letters of w with independent probability q and study the resulting evolution of the [Formula: see text]. We introduce combinatorial properties of [Formula: see text] which allow us to construct a new distance measure [Formula: see text] for words. We show that [Formula: see text] captures the similarity of corresponding [Formula: see text]. We will use the distance measure [Formula: see text] to study neutrality and mutation rates in the evolution of words. We analyze the structure of neutral networks of words and the transition of word populations between them. Furthermore, we prove the existence of a critical mutation rate beyond which a population of words becomes essentially randomly distributed, and the existence of an optimal mutation rate at which a population maximizes its mutant offspring.

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HUA, JIANPING, CHAO SIMA, MILANA CYPERT, GERALD C. GOODEN, SONSOLES SHACK, LALITAMBA ALLA, EDWARD A. SMITH, JEFFREY M. TRENT, EDWARD R. DOUGHERTY, and MICHAEL L. BITTNER. "DYNAMICAL ANALYSIS OF DRUG EFFICACY AND MECHANISM OF ACTION USING GFP REPORTERS." Journal of Biological Systems 20, no. 04 (December 2012): 403–22. http://dx.doi.org/10.1142/s0218339012400049.

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Two issues are critical to the development of effective cancer-drug combinations. First, it is necessary to determine common combinations of alterations that exert strong control over proliferation and survival regulation for the general type of cancer being considered. Second, it is necessary to have a drug testing method that allows one to assess the variety of responses that can be provoked by drugs acting at key points in the cellular processes dictating proliferation and survival. Utilizing a previously reported GFP (green fluorescent protein) reporter-based technology that provides dynamic measurements of individual reporters in individual cells, the present paper proposes a dynamical systems approach to these issues. It involves a three-state experimental design: (1) formulate an oncologic pathway model of relevant processes; (2) perturb the pathways with the test drug and drugs with known effects on components of the pathways of interest; and (3) measure process activity indicators at various points on cell populations. This design addresses the fundamental problems in the design and analysis of combinatorial drug treatments. We apply the dynamical approach to three issues in the context of colon cancer cell lines: (1) identification of cell subpopulations possessing differing degrees of drug sensitivity; (2) the consequences of different drug dosing strategies on cellular processes; and (3) assessing the consequences of combinatorial versus monotherapy. Finally, we illustrate how the dynamical systems approach leads to a mechanistic hypothesis in the colon cancer HCT116 cell line.

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Diamond, Phil, Peter Kloeden, and Aleksej Pokrovskii. "Interval stochastic matrices: A combinatorial lemma and the computation of invariant measures of dynamical systems." Journal of Dynamics and Differential Equations 7, no. 2 (April 1995): 341–64. http://dx.doi.org/10.1007/bf02219360.

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Bagheri, Neda, Marisa Shiina, Douglas A. Lauffenburger, and W. Michael Korn. "A Dynamical Systems Model for Combinatorial Cancer Therapy Enhances Oncolytic Adenovirus Efficacy by MEK-Inhibition." PLoS Computational Biology 7, no. 2 (February 17, 2011): e1001085. http://dx.doi.org/10.1371/journal.pcbi.1001085.

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Buluç, Aydın, and John R. Gilbert. "The Combinatorial BLAS: design, implementation, and applications." International Journal of High Performance Computing Applications 25, no. 4 (May 19, 2011): 496–509. http://dx.doi.org/10.1177/1094342011403516.

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This paper presents a scalable high-performance software library to be used for graph analysis and data mining. Large combinatorial graphs appear in many applications of high-performance computing, including computational biology, informatics, analytics, web search, dynamical systems, and sparse matrix methods. Graph computations are difficult to parallelize using traditional approaches due to their irregular nature and low operational intensity. Many graph computations, however, contain sufficient coarse-grained parallelism for thousands of processors, which can be uncovered by using the right primitives. We describe the parallel Combinatorial BLAS, which consists of a small but powerful set of linear algebra primitives specifically targeting graph and data mining applications. We provide an extensible library interface and some guiding principles for future development. The library is evaluated using two important graph algorithms, in terms of both performance and ease-of-use. The scalability and raw performance of the example applications, using the Combinatorial BLAS, are unprecedented on distributed memory clusters.

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MORA, JUAN CARLOS SECK TUOH, MANUEL GONZÁLEZ HERNÁNDEZ, GENARO JUÁREZ MARTÍNEZ, SERGIO V. CHAPA VERGARA, and HAROLD V. McINTOSH. "UNCONVENTIONAL INVERTIBLE BEHAVIORS IN REVERSIBLE ONE-DIMENSIONAL CELLULAR AUTOMATA." International Journal of Bifurcation and Chaos 18, no. 12 (December 2008): 3625–32. http://dx.doi.org/10.1142/s0218127408022597.

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Reversible cellular automata are discrete invertible dynamical systems determined by local interactions among their components. For the one-dimensional case, there are classical references providing a complete characterization based on combinatorial properties. Using these results and the simulation of every automaton by another with neighborhood size 2, this paper describes other types of invertible behaviors embedded in these systems different from the classical one observed in the temporal evolution. In particular, spatial reversibility and diagonal surjectivity are studied, and the generation of macrocells in the evolution space is analyzed.

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Bauer, U., H. Edelsbrunner, G. Jabłoński, and M. Mrozek. "Čech–Delaunay gradient flow and homology inference for self-maps." Journal of Applied and Computational Topology 4, no. 4 (August 30, 2020): 455–80. http://dx.doi.org/10.1007/s41468-020-00058-8.

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Abstract We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior. We establish a sampling theorem to recover the eigenspaces of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for Čech and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive Čech complexes to the computationally efficient Delaunay triangulations. The fast chain map algorithm has applications beyond dynamical systems.

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Zhilinskii, Boris. "Interplay of symmetry and topology in science." Acta Crystallographica Section A Foundations and Advances 70, a1 (August 5, 2014): C1419. http://dx.doi.org/10.1107/s2053273314085805.

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Qualitative methods in natural science are based mainly on simultaneous use of symmetry and topology arguments. The idea of the present talk is to demonstrate how the corresponding mathematical tools (based on symmetry and topology arguments) initially applied to describe classification of different phases of matter and transitions between them are extended to construct qualitative theory of finite particle systems and more general dynamical systems. I start with reminding basic notions and tools associated with application of group action ideas to physics as initiated and developed by Louis Michel (1923-1999) [1,2]. Then geometric combinatorial and topological ideas are used to give qualitative description of singularities of dynamical integrable classical system and their quantum analogs. Quantum monodromy and its various generalizations as well as description of energy bands of isolated finite particle quantum systems in terms of topological invariant, Chern number [3], will be discussed on concrete molecular and atomic examples.

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GIESEN, JOACHIM, MATTHIAS JOHN, and MICHEL STÖCKLIN. "SYMMETRY OF FLOW DIAGRAMS DERIVED FROM WEIGHTED POINTS IN THE PLANE." International Journal of Computational Geometry & Applications 13, no. 04 (August 2003): 327–37. http://dx.doi.org/10.1142/s0218195903001219.

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We introduce a condition that establishes a symmetry for diagrams associated with a class of dynamical systems (flows) which are induced by weighted points in the plane. This symmetry is known from weighted Delaunay- and power diagrams where it always holds. Under this condition the combinatorial and worst case algorithmic complexities of the flow diagrams are lower than in the general case. The condition is natural in the sense that it is automatically fulfilled for sets of unweighted points.

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TOŠIĆ, PREDRAG T. "ON THE COMPLEXITY OF COUNTING FIXED POINTS AND GARDENS OF EDEN IN SEQUENTIAL DYNAMICAL SYSTEMS ON PLANAR BIPARTITE GRAPHS." International Journal of Foundations of Computer Science 17, no. 05 (October 2006): 1179–203. http://dx.doi.org/10.1142/s0129054106004339.

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We study counting various types of configurations in certain classes of graph automata viewed as discrete dynamical systems. The graph automata models of our interest are Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively). These models have been proposed as the mathematical foundation for a theory of large-scale simulations of complex multi-agent systems. Our emphasis in this paper is on the computational complexity of counting the fixed point and the garden of Eden configurations in Boolean SDSs and SyDSs. We show that counting these configurations is, in general, computationally intractable. We also show that this intractability still holds when both the underlying graphs and the node update rules of these SDSs and SyDSs are severely restricted. In particular, we prove that the problems of exactly counting fixed points, gardens of Eden and two other types of S(y)DS configurations are all #P-complete, even if the SDSs and SyDSs are defined over planar bipartite graphs, and each of their nodes updates its state according to a monotone update rule given as a Boolean formula. We thus add these discrete dynamical systems to the list of those problem domains where counting combinatorial structures of interest is intractable even when the related decision problems are known to be efficiently solvable.

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PASK, DAVID, IAIN RAEBURN, and NATASHA A. WEAVER. "A family of 2-graphs arising from two-dimensional subshifts." Ergodic Theory and Dynamical Systems 29, no. 5 (March 12, 2009): 1613–39. http://dx.doi.org/10.1017/s0143385708000795.

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AbstractHigher-rank graphs (ork-graphs) were introduced by Kumjian and Pask to provide combinatorial models for the higher-rank Cuntz–KriegerC*-algebras of Robertson and Steger. Here we consider a family of finite 2-graphs whose path spaces are dynamical systems of algebraic origin, as studied by Schmidt and others. We analyse theC*-algebras of these 2-graphs, find criteria under which they are simple and purely infinite, and compute theirK-theory. We find examples whoseC*-algebras satisfy the hypotheses of the classification theorem of Kirchberg and Phillips, but are not isomorphic to theC*-algebras of ordinary directed graphs.

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Zhang, Xiaoyan, and Chao Wang. "Solutions to All-Colors Problem on Graph Cellular Automata." Complexity 2019 (June 2, 2019): 1–11. http://dx.doi.org/10.1155/2019/3164692.

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The All-Ones Problem comes from the theory of σ+-automata, which is related to graph dynamical systems as well as the Odd Set Problem in linear decoding. In this paper, we further study and compute the solutions to the “All-Colors Problem,” a generalization of “All-Ones Problem,” on some interesting classes of graphs which can be divided into two subproblems: Strong-All-Colors Problem and Weak-All-Colors Problem, respectively. We also introduce a new kind of All-Colors Problem, k-Random Weak-All-Colors Problem, which is relevant to both combinatorial number theory and cellular automata theory.

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Dembo, Amir, and Reza Gheissari. "Diffusions interacting through a random matrix: universality via stochastic Taylor expansion." Probability Theory and Related Fields 180, no. 3-4 (February 4, 2021): 1057–97. http://dx.doi.org/10.1007/s00440-021-01027-7.

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AbstractConsider $$(X_{i}(t))$$ ( X i ( t ) ) solving a system of N stochastic differential equations interacting through a random matrix $${\mathbf {J}} = (J_{ij})$$ J = ( J ij ) with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $$(X_i(t))$$ ( X i ( t ) ) , initialized from some $$\mu $$ μ independent of $${\mathbf {J}}$$ J , are universal, i.e., only depend on the choice of the distribution $$\mathbf {J}$$ J through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks.

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Papadimitriou, Christos, and Georgios Piliouras. "From Nash Equilibria to Chain Recurrent Sets: An Algorithmic Solution Concept for Game Theory." Entropy 20, no. 10 (October 12, 2018): 782. http://dx.doi.org/10.3390/e20100782.

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In 1950, Nash proposed a natural equilibrium solution concept for games hence called Nash equilibrium, and proved that all finite games have at least one. The proof is through a simple yet ingenious application of Brouwer’s (or, in another version Kakutani’s) fixed point theorem, the most sophisticated result in his era’s topology—in fact, recent algorithmic work has established that Nash equilibria are computationally equivalent to fixed points. In this paper, we propose a new class of universal non-equilibrium solution concepts arising from an important theorem in the topology of dynamical systems that was unavailable to Nash. This approach starts with both a game and a learning dynamics, defined over mixed strategies. The Nash equilibria are fixpoints of the dynamics, but the system behavior is captured by an object far more general than the Nash equilibrium that is known in dynamical systems theory as chain recurrent set. Informally, once we focus on this solution concept—this notion of “the outcome of the game”—every game behaves like a potential game with the dynamics converging to these states. In other words, unlike Nash equilibria, this solution concept is algorithmic in the sense that it has a constructive proof of existence. We characterize this solution for simple benchmark games under replicator dynamics, arguably the best known evolutionary dynamics in game theory. For (weighted) potential games, the new concept coincides with the fixpoints/equilibria of the dynamics. However, in (variants of) zero-sum games with fully mixed (i.e., interior) Nash equilibria, it covers the whole state space, as the dynamics satisfy specific information theoretic constants of motion. We discuss numerous novel computational, as well as structural, combinatorial questions raised by this chain recurrence conception of games.

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Bullett, Shaun, and Marianne Freiberger. "Hecke Groups, Polynomial Maps and Matings." International Journal of Modern Physics B 17, no. 22n24 (September 30, 2003): 3922–31. http://dx.doi.org/10.1142/s0217979203021915.

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We show that there is a topological conjugacy between the action of the Hecke group Gn on the completed positive real axis and the action of the shift on base n-1 expressions of real numbers in the unit interval. This conjugacy is shown to occur in holomorphic dynamical systems: we construct (n-1:n-1) holomorphic correspondences (multivalued self-maps of the Riemann sphere) which are matings between Gn acting on the complex upper half plane, and polynomials of degree n-1 acting on their filled Julia sets. Certain of these correspondences split into unions of (2:2) and (1:1) correspondences: we present combinatorial descriptions of limit sets and the connectivity locus in the case n=4.

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KHOLODENKO, A. L. "NEW MODELS FOR VENEZIANO AMPLITUDES: COMBINATORIAL, SYMPLECTIC AND SUPERSYMMETRIC ASPECTS." International Journal of Geometric Methods in Modern Physics 02, no. 04 (August 2005): 563–84. http://dx.doi.org/10.1142/s0219887805000703.

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The bosonic string theory evolved as an attempt to find a physical/quantum mechanical model capable of reproducing Euler's beta function (Veneziano amplitude) and its multidimensional analogue. The multidimensional analogue of beta function was studied mathematically for some time from different angles by mathematicians such as Selberg, Weil and Deligne among many others. The results of their studies apparently were not taken into account in physics literature on string theory. In several recent publications, attempts were made to restore the missing links. As discussed in these publications, the existing mathematical interpretation of the multidimensional analogue of Euler's beta function as one of the periods associated with the corresponding differential form "living" on the Fermat-type (hyper) surface, happens to be crucial for restoration of the quantum/statistical mechanical models reproducing such generalized beta function. There is a number of nontraditional models — all interrelated — capable of reproducing the Veneziano amplitudes. In this work we would like to discuss two of such new models: symplectic and supersymmetric. The symplectic model is based on observation that the Veneziano amplitude is just the Laplace transform of the generating function for the Ehrhart polynomial. Such a polynomial counts the number of lattice points inside the rational polytope (i.e. polytope whose vertices are located at the nodes of a regular lattice) and at its boundaries. In the present case, the polytope is a regular simplex. It is a deformation retract for the Fermat-type (hyper) surface (perhaps inflated, as explained in the text). Using known connections between polytopes and dynamical systems, the quantum mechanical system associated with such a dynamical system is found. The ground state of this system is degenerate with degeneracy factor given by the Ehrhart polynomial. Using some ideas by Atiyah, Bott and Witten we argue that the supersymmetric model related to the symplectic can be recovered. While recovering this model, we demonstrate that the ground state of such a model is degenerate with the same degeneracy factor as for earlier obtained symplectic model. Since the wave functions of this model are in one to one correspondence with the Veneziano amplitudes, this exactly solvable supersymmetric (and, hence, also symplectic) model is sufficient for recovery of the partition function reproducing the Veneziano amplitudes thus providing the exact solution of the Veneziano model.

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Pelillo, Marcello. "Replicator Equations, Maximal Cliques, and Graph Isomorphism." Neural Computation 11, no. 8 (November 1, 1999): 1933–55. http://dx.doi.org/10.1162/089976699300016034.

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We present a new energy-minimization framework for the graph isomorphism problem that is based on an equivalent maximum clique formulation. The approach is centered around a fundamental result proved by Motzkin and Straus in the mid-1960s, and recently expanded in various ways, which allows us to formulate the maximum clique problem in terms of a standard quadratic program. The attractive feature of this formulation is that a clear one-to-one correspondence exists between the solutions of the quadratic program and those in the original, combinatorial problem. To solve the program we use the so-called replicator equations—a class of straightforward continuous- and discrete-time dynamical systems developed in various branches of theoretical biology. We show how, despite their inherent inability to escape from local solutions, they nevertheless provide experimental results that are competitive with those obtained using more elaborate mean-field annealing heuristics.

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Kaczynski, Tomasz, Marian Mrozek, and Anik Trahan. "Ideas from Zariski Topology in the Study of Cubical Homology." Canadian Journal of Mathematics 59, no. 5 (October 1, 2007): 1008–28. http://dx.doi.org/10.4153/cjm-2007-043-3.

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AbstractCubical sets and their homology have been used in dynamical systems as well as in digital imaging. We take a fresh look at this topic, following Zariski ideas from algebraic geometry. The cubical topology is defined to be a topology in ℝd in which a set is closed if and only if it is cubical. This concept is a convenient frame for describing a variety of important features of cubical sets. Separation axioms which, in general, are not satisfied here, characterize exactly those pairs of points which we want to distinguish. The noetherian property guarantees the correctness of the algorithms. Moreover, maps between cubical sets which are continuous and closed with respect to the cubical topology are precisely those for whom the homology map can be defined and computed without grid subdivisions. A combinatorial version of the Vietoris–Begle theorem is derived. This theorem plays the central role in an algorithm computing homology of maps which are continuous with respect to the Euclidean topology.

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Lituiev, Dmytro S., and Ueli Grossniklaus. "Patterning of the angiosperm female gametophyte through the prism of theoretical paradigms." Biochemical Society Transactions 42, no. 2 (March 20, 2014): 332–39. http://dx.doi.org/10.1042/bst20140036.

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The FG (female gametophyte) of flowering plants (angiosperms) is a simple highly polar structure composed of only a few cell types. The FG develops from a single cell through mitotic divisions to generate, depending on the species, four to 16 nuclei in a syncytium. These nuclei are then partitioned into three or four distinct cell types. The mechanisms underlying the specification of the nuclei in the FG has been a focus of research over the last decade. Nevertheless, we are far from understanding the patterning mechanisms that govern cell specification. Although some results were previously interpreted in terms of static positional information, several lines of evidence now show that local interactions are important. In the present article, we revisit the available data on developmental mutants and cell fate markers in the light of theoretical frameworks for biological patterning. We argue that a further dissection of the mechanisms may be impeded by the combinatorial and dynamical nature of developmental cues. However, accounting for these properties of developing systems is necessary to disentangle the diversity of the phenotypic manifestations of the underlying molecular interactions.

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Journal articles on the topic 'Combinatorial dynamical systems'

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