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NEURONS ARE POISED NEAR THE EDGE OF CHAOS

INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 2012年第4期22卷 1250098-1250098页

作者： Chua, Leon Sbitnev, Valery Kim, Hyongsuk Univ Calif Berkeley Dept Elect Engn & Comp Sci Berkeley CA 94720 USA Tech Univ Munich Fak Elektrotech & Informationstech D-80333 Munich Germany NRC Kurchatov Inst St Petersburg Nucl Phys Inst St Petersburg 188350 Russia Chonbuk Natl Univ Div Elect Engn Jeonju 561756 Jeonbuk South Korea

This paper shows the action potential (spikes) generated from the Hodgkin-Huxley equations emerges near the edge of chaos consisting of a tiny subset of the locally active regime of the HH equations. The main result proves that the eigenvalues of the 4 x 4 Jacobian matrix associated with the mathematically intractable system of four nonlinear differential equations are identical to the zeros of a scalar complexity function from complexity theory. Moreover, we show the loci of a pair of complex-conjugate zeros migrate continuously as a function of an externally applied DC current excitation emulating the net synaptic excitation current input to the neuron. In particular, the pair of complex-conjugate zeros move from a subcritical Hopf bifurcation point at low excitation current to a super-critical Hopf bifurcation point at high excitation current. The spikes are generated as the excitation current approaches the vicinity of the edge of chaos, which leads to the onset of the subcritical Hopf bifurcation regime. It follows from this in-depth qualitative analysis that local activity is the origin of spikes.

关键词： Action potential spikes neurons axons Hodgkin-Huxley equations Hodgkin-Huxley axon memristor complexity function edge of chaos local activity Hopf bifurcation subcritical Hopf bifurcation super-critical Hopf bifurcation complexity limit cycles basin of attraction eigenvalues stability

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High performance computing for partial differential equations

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COMPUTERS & FLUIDS 2011年第1期43卷 68-73页

作者： Gruber, R. Ahusborde, E. Azaiez, M. Keller, V. Latt, J. Univ Bordeaux TREFLE UMR CNRS 8508 IPB F-33607 Pessac France Ecole Polytech Fed Lausanne CH-1015 Lausanne Switzerland Univ Zurich CH-8000 Zurich Switzerland

When representing realistic physical phenomena by partial differential equations (PDE), it is crucial to approximate the underlying physics correctly, to get precise results, and to efficiently use the computer architecture. Incorrect results can appear in incompressible Navier-Stokes or Stokes problems when the numerical approach couples into spurious modes. In Maxwell or magnetohydrodynamic (MHD) equations the so-called spectrum pollution effect can occur, and the numerical solution does not stably converge to the physical one. Problems coming from a mesh that is not adapted to the underlying physical problem, or from an inadequate choice of the dependent and independent variables can lead to low precision. Efficiency of a code implementation can be improved by well adapting the parallel computer to the application. A new monitoring system enables to detect poor implementations, to find best suited resources to execute the job, and to adapt the processor frequency during. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： COOL Incompressibility condition Mesh density function Adaptive mesh complexity function Processor frequency adaptation High performance computing methods Stokes problem Gyrotron simulation Linear ideal MHD stability Stellerator

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The complexity probabilistic quasi-metric space

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2011年第2期376卷 732-740页

作者： Romaguera, Salvador Tirado, Pedro Univ Politecn Valencia Inst Univ Matemat Pura & Aplicada E-46071 Valencia Spain

We introduce and study a probabilistic quasi-metric on the set of complexity functions, which provides an efficient framework to measure the distance from a complexity function f to another one g in the case that f is asymptotically more efficient than g. In this context we also obtain a version of the Banach fixed point theorem which allows us to show that the functionals associated both to Divide and Conquer algorithms and Quicksort algorithms have a unique fixed point. (C) 2010 Elsevier Inc. All rights reserved.

关键词： Probabilistic quasi-metric Quasi-Menger space complexity function Asymptotic Bicomplete Smyth complete Contractive Fixed point functional Algorithm

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On the Number of Balanced Words of Given Length and Height over a Two-Letter Alphabet

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DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE 2010年第3期12卷 41-62页

作者： Bedaride, Nicolas Domenjoud, Eric Jamet, Damien Remy, Jean-Luc Univ Aix Marseille 3 Marseille France Univ Nancy 1 CNRS LORIA Vandoeuvre Les Nancy France

We exhibit a recurrence on the number of discrete line segments joining two integer points in the plane using an encoding of such segments as balanced words of given length and height over the two- letter alphabet {0,1}. We give generating functions and study the asymptotic behaviour. As a particular case, we focus on the symmetrical discrete segments which are encoded by balanced palindromes.

关键词： Symbolic dynamics complexity function discrete geometry

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Local complexity functions of interval exchange transformations

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2009年第4期14卷 1454-1460页

作者： Afraimovich, V. Rechtman, R. Univ Autonoma San Luis Potosi Inst Invest Comunicac Opt San Luis Potosi Mexico Univ Nacl Autonoma Mexico Ctr Invest Energia Temixco 62580 Morelos Mexico

We studied numerically complexity functions for interval exchange transformations. We have shown that they grow linearly in time as well as the c-complexity function. Moreover, we found out that they depend also linearly on E where E is the Lebesgue measure of a set of initial points. This allows us to hypothesize that the dimension of the measure related to the E-complexity function could be determined by studying the dependence of local complexity functions on C. (c) 2008 Elsevier B.V. All rights reserved.

关键词： Separability complexity function Interval exchange transformation

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complexity CLASSES FOR MULTI-VARIABLE complexity functionS

COMPLEXITY CLASSES FOR MULTI-VARIABLE COMPLEXITY FUNCTIONS

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20th International Danube-Adria-Association-for-Automation-and-Manufacturing Symposium

作者： Mogos, Andrei-Horia Florea, Adina Magda

ISBN: (纸本)9783901509704

The complexity of an algorithm is usually expressed using complexity functions and complexity classes. In this paper we present a method to reduce the work with multi-variable complexity functions to the work with one-variable complexity functions. We define five complexity classes for multi-variable complexity functions and then we prove some properties for these classes.

关键词： algorithm complexity function complexity class

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complexity of finite-variable fragments of EXPTIME-complete logics *

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Journal of Applied Non-Classical Logics 2007年第3期17卷 359-382页

作者： Rybakov, Mikhail Department of Mathematics Tver State University Tver 170000 Zhelyabova Street 33 Russian Federation

The main result of the present paper is that the variable-free fragment of logic K*, the logic with a singleK-style modality and its "reflexive and transitive closure," is EXPTIMEcomplete. It is then shown that this immediately gives EXPTIME-completeness of variable-free fragments of a number of known EXPTIME-complete logics. Our proof contains a general idea of how to construct a polynomial-time reduction of a propositional logic to its n-variable- and even, in the cases of K*, PDL, CTL, ATL, and some others, variable-free-fragments. complexity of countermodels for such fragments is considered.

关键词： Alternating-time temporal logic complexity complexity function Computation tree logic Decision problems Finitevariable fragments Logics of knowledge Modal logic Non-classical logics Propositional dynamic logic Variable-free fragments

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On time complexity analysis of algorithms

On time complexity analysis of algorithms

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9th World Multi-Conference on Systemics, Cybernetics and Informatics

作者： Tarek, Ahmed Eastern Kentucky Univ Dept Comp Sci Richmond KY 40475 USA

ISBN: (纸本)9789806560550

There exists a variety of techniques for the time complexity analysis of algorithms and functions. This analysis is used to find out the upper-bound on time complexity in big-oh notation, which is denoted by O(g(n)) with g(n) is a function of n, and n is the size of the given problem. Besides the big-oh complexity, there axe other complexity notations, such as Omega, theta, and small o complexities. complexity analysis is used to select an appropriate algorithm for solving a given problem using computers. Unfortunately there does not exist any efficient, generalized framework for the time complexity analysis. Most of the existing techniques are complex, obsolete, and hard to use in practice. There is a misconception in some literature that 0(g(n)) is a function rather than a set. In this paper, it has been established that 0(g(n) is basically a set, which includes all algorithms with an upper bound on the order of g(n). Moreover, a generalized framework for analyzing the time-complexity function f (n) for algorithms in obtaining the big-oh notational complexity has been proposed. The method has been extended for functions f (n(1), n(2), ... , n(m)), m = 1, 2, ... involving multiple variables. Other complexity orders are discussed, and compared with the big-oh complexity.

关键词： time complexity complexity function big-oh notation complexity order algorithm analysis

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Balanced words

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BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY-SIMON STEVIN 2003年第5期10卷 787-805页

作者： Vuillon, L Univ Savoie Math Lab CNRS UMR 5127 F-73376 Le Bourget Du Lac France

This article presents a survey about balanced words. The balance property comes from combinatorics on words and is used as a characteristic property of the well-known Sturmian words. The main goal of this survey is to study various generalizations of this notion with applications and with open problems in number theory and in theoretical computer science. We also prove a new result about the generalized balance property of hypercubic billiard words.

关键词： balanced words sturmian words complexity function combinatorics on words Beatty sequences Arnoux-Rauzy words hypercubic billiard words heap of pieces optimization problems

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Construction of regular languages and recognizability of polynomials

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DISCRETE MATHEMATICS 2002年第1-3期254卷 485-496页

作者： Rigo, M Univ Liege Inst Math B-4000 Liege Belgium

A generalization of numeration systems in which NI is recognizable by finite automata can be obtained by describing a lexicographically ordered infinite regular language. We show that if P is an element of Q[x] is a polynomial such that P(N) subset of N then there exists a numeration system in which the set of representations of P(N) is regular. The main issue is to construct a regular language with a complexity function equals to P(n + 1) - P(n) for n large enough. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： numeration system recognizability regular language complexity function

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