a tenth order a stable two step hybrid block method for solving initial value problems of odes in this article a new two step hybrid block method for the numerical integration of ordinary differential initial value systems is presented the method is obtained after considering two intermediate points and the approximation of the true solution by an adequate polynomial and imposing collocation conditions the proposed method has the tenth algebraic order of convergence and is a stable the numerical experiments considered revealed the superiority of the new method for solving this kind of problems in comparison with methods of similar characteristics appeared in the literature

variable stepsize implementation of multistep methods for y f x y y the falkner method is a multistep scheme intended for the numerical solution of second order initial value problems where the first derivative does appear explicitly in this paper we develop a procedure to obtain k step falkner methods explicit and implicit in their variable step size versions providing recurrence formulas to compute the coefficients efficiently considering a pair of explicit and implicit formulae these may be implemented in predictor corrector mode

a method based on the tsallis entropy for characterizing threshold channel bank profiles abstract as a stable channel forms a point is reached when particles shift to motion threshold threshold channel as such the most common challenges related to alluvial channels include the variation in the transverse bank slope s t and channel geometric profile formation the present study introduces a novel reliable method of estimating the transverse slope distribution of stable channel bank profiles in different hydraulic conditions by using the tsallis entropy concept accordingly an equation for calculating s t and subsequently the vertical boundary level of bank profiles is presented based on the principle of maximum entropy the threshold channel bank profile features are defined according to the probability density function and cumulative density function to adjust the general form of the tsallis entropy for this specific engineering problem an extensive detailed study of lagrange multipliers is carried out and a wide range of observational data are used to determine the tsallis entropy model accuracy in measuring the s t of stable channel banks the results indicate that the tsallis entropy model is capable of detecting the ascending trend governing the profiles of stable channel banks in different hydraulic conditions this model exhibits excellent predictive performance with a mean absolute relative error mare of 0 265 and a coefficient of determination r 2 of 0 975 the proposed tsallis entropy based equation can be used to predict river morphology changes for river management plans

third derivative modification of k step block falkner methods for the numerical solution of second order initial value problems abstract this paper is devoted to the development and analysis of a modified family of falkner type methods for solving differential systems of second order initial value problems the approaches of collocation and interpolation are adopted to derive the new methods these modified methods are implemented in block form to obtain the numerical solutions to the considered problems the study of the properties of the proposed block falkner type methods reveals that they are consistent and zero stable and thus convergent from the stability analysis it could be seen that the proposed falkner methods have non empty stability regions for k 2 3 4 some numerical test are presented to illustrate the efficiency of the proposed family

a new look at q exponential distributions via excess statistics q exponential distributions play an important role in nonextensive statistics they appear as the canonical distributions i e the maximum generalized q entropy distributions under mean constraint their relevance is also independently justified by their appearance in the theory of superstatistics introduced by beck and cohen in this paper we provide a third and independent rationale for these distributions we indicate that q exponentials are stable by a statistical normalization operation and that pickands extreme values theorem plays the role of a clt like theorem in this context this suggests that q exponentials can arise in many contexts if the system at hand or the measurement device introduces some threshold moreover we give an asymptotic connection between excess distributions and maximum q entropy we also highlight the role of generalized pareto distributions in many applications and present several methods for the practical estimation of q exponential parameters

evolution equation of phenotype distribution general formulation and application to error catastrophe an equation describing the evolution of phenotypic distribution is derived using methods developed in statistical physics the equation is solved by using the singular perturbation method and assuming that the number of bases in the genetic sequence is large applying the equation to the mutation selection model by eigen provides the critical mutation rate for the error catastrophe phenotypic fluctuation of clones individuals sharing the same gene is introduced into this evolution equation with this formalism it is found that the critical mutation rate is sometimes increased by the phenotypic fluctuations i e noise can enhance robustness of a fitted state to mutation our formalism is systematic and general while approximations to derive more tractable evolution equations are also discussed

stochastic feedback nonlinear families of markov processes and nonlinear fokker planck equations abstract we discuss two fundamental aspects of fokker planck equations that are nonlinear with respect to probability densities first we show that evolution equations of this kind describe processes involving stochastic feedback and interpret stochastic feedback processes in terms of hitchhiker processes and path integral solutions second we demonstrate that nonlinear fokker planck equations can be interpreted as linear fokker planck equations describing nonlinear families of markov diffusion processes we exploit this finding in order to derive complete hierarchies of probability densities from nonlinear fokker planck equations

solvable biological evolution models with general fitness functions and multiple mutations in parallel mutation selection scheme in a recent paper phys rev e 69 046121 2004 we used the suzuki trottere formalism to study a quasispecies biological evolution model in a parallel mutation selection scheme with a single peak fitness function and a point mutation in the present paper we extend such a study to evolution models with more general fitness functions or multiple mutations in the parallel mutation selection scheme we give some analytical equations to define the error thresholds for some general cases of mean field like or symmetric mutation schemes and fitness functions we derive some equations for the dynamics in the case of a point mutation and polynomial fitness functions we derive exact dynamics for two point mutations asymmetric mutations and the four value spin model with a single peak fitness function the same method is applied for the model with a royal road fitness function we derive the steady state distribution for the single peak fitness function

empirical regularities of opening call auction in chinese stock market abstract we study the statistical regularities of an opening call auction using the ultra high frequency data of 22 liquid stocks traded on the shenzhen stock exchange in 2003 the distribution of the relative price defined as the relative difference between the order price in the opening call auction and the closing price on the last trading day is asymmetric and that the distribution displays a sharp peak at the zero relative price and a relatively wide peak at the negative relative price the detrended fluctuation analysis dfa method is adopted to investigate the long term memory of relative order prices we further study the statistical regularities of order sizes in the opening call auction and observe a phenomenon of number preference known as order size clustering the probability density function pdf of order sizes could be well fitted by a q gamma function and the long term memory also exists in order sizes in addition both the average volume and the average number of orders decrease exponentially with the price level away from the best bid or ask price level in the limit order book lob established immediately after the opening call auction and a price clustering phenomenon is observed

quasispecies theory for multiple peak fitness landscapes we use a path integral representation to solve the eigen and crow kimura molecular evolution models for the case of multiple fitness peaks with arbitrary fitness and degradation functions in the general case we find that the solution to these molecular evolution models can be written as the optimum of a fitness function with constraints enforced by lagrange multipliers and with a term accounting for the entropy of the spreading population in sequence space the results for the eigen model are applied to consider virus or cancer proliferation under the control of drugs or the immune system

recombination in one and two dimensional fitness landscapes we consider many site mutation recombination models of molecular evolution where fitness is a function of a hamming distance from one one dimensional case or two two dimensional case sequences for the one dimensional case we calculate the population distribution dynamics for a model with zero fitness and an arbitrary symmetric initial distribution and find an error threshold transition point in the single peak fitness model for a given initial symmetric distribution we calculate the recombination period in the case of a single peak fitness function when the original population is located at one sequence at some hamming distance from the peak configuration steady state fitness is calculated with finite genome length corrections we derive analytical equations for the two dimensional mutation recombination model

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

monotonically equivalent entropies and solution of additivity equation generalized entropies are studied as lyapunov functions for the master equation markov chains three basic properties of these lyapunov functions are taken into consideration universality independence of the kinetic coefficients trace form the form of sum over the states and additivity for composition of independent subsystems all the entropies which have all three properties simultaneously and are defined for positive probabilities are found they form a one parametric family we consider also pairs of entropies s1 s2 which are connected by the monotonous transformation s2 f s1 equivalent entropies all classes of pairs of universal equivalent entropies one of which has a trace form and another is additive these entropies can be different one from another were found these classes consist of two one parametric families the family of entropies which are equivalent to the additive trace form entropies and the family of renyi tsallis entropies

solvable biological evolution model with a parallel mutation selection scheme based on the connection between a quantum spin model and an asexual biological evolution model with a single peak fitness function in parallel mutation selection scheme we solve exactly both static and dynamics of the evolution model we find that relaxation in such a parallel scheme is faster than that in a connected scheme of eigen model our method can also be extended to other fitness functions

extinction of metastable stochastic populations we investigate the phenomenon of extinction of a long lived self regulating stochastic population caused by intrinsic demographic noise extinction typically occurs via one of two scenarios depending on whether the absorbing state n 0 is a repelling scenario a or attracting scenario b point of the deterministic rate equation in scenario a the metastable stochastic population resides in the vicinity of an attracting fixed point next to the repelling point n 0 in scenario b there is an intermediate repelling point n n 1 between the attracting point n 0 and another attracting point n n 2 in the vicinity of which the metastable population resides the crux of the theory is a dissipative variant of wkb wentzel kramers brillouin approximation which assumes that the typical population size in the metastable state is large starting from the master equation we calculate the quasistationary probability distribution of the population sizes and the exponentially long mean time to extinction for each of the two scenarios when necessary the wkb approximation is complemented i by a recursive solution of the quasistationary master equation at small n and ii by the van kampen system size expansion valid near the fixed points of the deterministic rate equation the theory yields both entropic barriers to extinction and pre exponential factors and holds for a general set of multistep processes when detailed balance is broken the results simplify considerably for single step processes and near the characteristic bifurcations of scenarios a and b

fractal surfaces from simple arithmetic operations fractal surfaces patchwork quilts are shown to arise under most general circumstances involving simple bitwise operations between real numbers a theory is presented for all deterministic bitwise operations on a finite alphabet it is shown that these models give rise to a roughness exponent h that shapes the resulting spatial patterns larger values of the exponent leading to coarser surfaces

generalized kolmogorov entropy in the dynamics of multifractal generation abstract we point out that applying a maximization principle on a tsallis like generalized form of the kolmogorov entropy for iterated function systems naturally provides a canonical statistical frame for the description of the multifractal measures generated by such dynamical processes multifractal spectra can then be characterized by usual statistical parameters in particular the temperature we show that in the limit of zero temperature the multifractal measure collapses to a homogeneous distribution over a fractal support for finite temperatures multifractal spectra are studied numerically in an illustrative example

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

schwinger boson formulation and solution of the crow kimura and eigen models of quasispecies theory we express the crow kimura and eigen models of quasispecies theory in a functional integral representation we formulate the spin coherent state functional integrals using the schwinger boson method in this formulation we are able to deduce the long time behavior of these models for arbitrary replication and degradation functions we discuss the phase transitions that occur in these models as a function of mutation rate we derive for these models the leading order corrections to the infinite genome length limit

how many k step linear block methods exist and which of them is the most efficient and simplest one abstract there have appeared in the literature a lot of k step block methods for solving initial value problems the methods consist in a set of k simultaneous multistep formulas over k non overlapping intervals a feature of block methods is that there is no need of other procedures to provide starting approximations and thus the methods are self starting sharing this advantage of runge kutta methods all the formulas are usually obtained from a continuous approximation derived via interpolation and collocation at k 1 points nevertheless all the k step block methods thus obtained may be considered as different formulations of one of them which results to be the most efficient and simple formulation of all of them the theoretical analysis and the numerical experiments presented support this claim

statistical mechanics of eigen s evolution model the correspondence between eigen s model of macromolecular evolution and the equilibrium statistical mechanics of an inhomogeneous ising system is developed the free energy landscape of random ising systems with the hopfield hamiltonian as a special example is applied to the replication rate coefficient landscape the coupling constants are scaled with 1 l since the maxima of any landscape must not increase with the length of the macromolecules the calculated error threshold relation then agrees with eigen s expression which was derived in a different way it gives an explicit expression for the superiority parameter in terms of the parameters of the landscape the dynamics of selection and evolution is discussed

disequilibrium complexity the schottky effect and q entropies in paramagnetism abstract we investigate connections between statistical quantifiers and paramagnetism more concretely we apply the notions of i disequilibrium and ii statistical complexity to a paramagnetic system of non coupled dipoles interesting insights are thereby obtained in particular we encounter a kind of criticality not associated to the temperature but to the disequilibrium

error threshold in simple landscapes we consider the quasispecies description of a population evolving in both the master sequence landscape where a single sequence is evolutionarily preferred over all others and the rem landscape where the fitness of different sequences is an independent identically distributed random variable we show that in both cases the error threshold is analogous to a first order thermodynamical transition where the overlap between the average genotype and the optimal one drops discontinuously to zero

indices for testing neural codes one of the most critical challenges in systems neuroscience is determining the neural code a principled framework for addressing this can be found in information theory with this approach one can determine whether a proposed code can account for the stimulus response relationship specifically one can compare the transmitted information between the stimulus and the hypothesized neural code with the transmitted information between the stimulus and the behavioral response if the former is smaller than the latter i e if the code cannot account for the behavior the code can be ruled out the information theoretic index most widely used in this context is shannon s mutual information the shannon test however is not ideal for this purpose while the codes it will rule out are truly nonviable there will be some nonviable codes that it will fail to rule out here we describe a wide range of alternative indices that can be used for ruling codes out the range includes a continuum from shannon information to measures of the performance of a bayesian decoder we analyze the relationship of these indices to each other and their complementary strengths and weaknesses for addressing this problem

some properties of q logarithm and q exponential functions in tsallis statistics we present some formulae which q logarithmic and q exponential functions introduced in tsallis statistics satisfy exact expression of the mellin transform of q exponential function is given moreover the q generalization of the digamma function is presented

convergence in reaction diffusion systems an information theory approach abstract we have applied an information theory approach in order to study the problem of convergence to point like or extended attractors in reaction diffusion systems a distance between two states based on the kullback leibler relative information was defined different forms of the probability distribution some of them based on the knowledge of the nonequilibrium potential when accesible give the possibility to look for a faster and or more accurate convergence this approach offers the chance to estimate the attraction basins of the different attractors as well as detecting limit circles together with an easy evaluation of their periods

path integral approach to the nonextensive canonical density matrix abstract feynman s path integral is herein generalized to the nonextensive canonical density matrix based on tsallis entropy this generalization is done in two ways by using unnormalized and normalized constraints firstly we consider the path integral formulation with unnormalized constraints and this generalization is worked out through two different ways which are shown to be equivalent these formulations with unnormalized constraints are solutions to the two generalized bloch equations proposed in this work the first form of the generalized bloch equation is linear but with a temperature dependent effective hamiltonian the second form is nonlinear and resembles the anomalous correlated diffusion equation porous medium equation furthermore we can extend these results to the prescription of field theory using integral representations the second development is dedicated to analyzing the path integral formulation with normalized constraints to illustrate the methods introduced here we analyze the free particle case and a non interacting scalar field the results herein obtained are expected to be useful in the discussion of generic nonextensive contexts

finite genome length corrections for the mean fitness and gene probabilities in evolution models using the hamilton jacobi equation approach to study genomes of length l we obtain 1 l corrections for the steady state population distributions and mean fitness functions for horizontal gene transfer model as well as for the diploid evolution model with general fitness landscapes our numerical solutions confirm the obtained analytic equations our method could be applied to the general case of nonlinear markov models

student s t and r distributions unified derivation from an entropic variational principle abstract the optimization of the recently generalized entropy of the form s q 1 dx p x q q 1 with the constraints dx p x 1 and x 2 q dx x 2 p x q 1 yields the student s t distribution for q 1 and the r distribution for q

suspended sediment concentration distribution using tsallis entropy abstract assuming sediment movement in channel flow as steady tsallis entropy is employed for deriving the distribution of sediment concentration this distribution is found to be analogous to the distribution obtained from a random walk model the sediment concentration distribution parameters are obtained from physical considerations with these parameters the distribution is tested using experimental observations and its sensitivity is evaluated the agreement between entropy based distribution and laboratory observations is found to be close

zeroth law compatibility of nonadditive thermodynamics nonextensive thermodynamics is criticized by the statement that the zeroth law cannot be satisfied with nonadditive composition rules corrected in this paper we determine the general functional form of those nonadditive composition rules that are compatible with the zeroth law of thermodynamics we find that this general form is additive for the formal logarithms of the original quantities and the familiar relations of thermodynamics apply to these our result offers a possible solution to the long standing questions about equilibrium between extensive and nonextensive systems or systems with different nonextensivity parameters

a unified approach for the development of k step block falkner type methods for solving general second order initial value problems in odes a family of k step block multistep methods where the main formulas are of falkner type is proposed for the direct integration of the general second order initial value problem where the differential equation is of the general form y f x y y the two main falkner formulas and the additional ones to complete the block procedure are obtained from a continuous approximation derived via interpolation and collocation at k 1 points the main characteristics of the methods are discussed through their formulation in vector form each method is formulated as a group of 2 k simultaneous formulas over k non overlapping intervals in this way the method produces the approximation of the solution simultaneously at k points on these intervals as in other block methods there is no need of other procedures to provide starting approximations and thus the methods are self starting sharing this advantage of runge kutta methods the resulting family is efficient and competitive compared with other existing methods in the literature as may be seen from the numerical results

fractal structure with a typical scale in order to understand the characteristics common to distributions which have both fractal and non fractal scale regions in a unified framework we introduce the concept of a typical scale we employ a model of 2d gravity modified by the r2 term as a tool to understand such distributions through the typical scale this model is obtained by adding an interaction term with a typical scale to a scale invariant system a distribution derived in the model provides power law one in the large scale region but weibull like one in the small scale region as examples of distributions which have both fractal and non fractal regions we take those of personal income and citation number of scientific papers we show that these distributions are fitted fairly well by the distribution curves derived analytically in the r2 2d gravity model as a result we consider that the typical scale is a useful concept to understand various distributions observed in the real world in a unified way we also point out that the r2 2d gravity model provides us with an effective tool to read the typical scales of various distributions in a systematic way

multiwavelet packet entropy and its application in transmission line fault recognition and classification multiwavelets possess better properties than traditional wavelets multiwavelet packet transformation has more high frequency information spectral entropy can be applied as an analysis index to the complexity or uncertainty of a signal this paper tries to define four multiwavelet packet entropies to extract the features of different transmission line faults and uses a radial basis function rbf neural network to recognize and classify 10 fault types of power transmission lines first the preprocessing and postprocessing problems of multiwavelets are presented shannon entropy and tsallis entropy are introduced and their difference is discussed second multiwavelet packet energy entropy time entropy shannon singular entropy and tsallis singular entropy are defined as the feature extraction methods of transmission line fault signals third the plan of transmission line fault recognition using multiwavelet packet entropies and an rbf neural network is proposed finally the experimental results show that the plan with the four multiwavelet packet energy entropies defined in this paper achieves better performance in fault recognition the performance with sa4 symmetric antisymmetric multiwavelet packet tsallis singular entropy is the best among the combinations of different multiwavelet packets and the four multiwavelet packet entropies

eigen model as a quantum spin chain exact dynamics we map the eigen model of biological evolution naturwissenschaften 58 465 1971 into a quantum spin model with non hermitian hamiltonian based on such a connection we derive exact relaxation periods for the eigen model to approach static energy landscape from various initial conditions we also study a simple case of dynamic fitness function

control of local error stabilizes integrations codes for the solution of the initial value problem for a system of ordinary differential equations adjust their step sizes so as to keep the estimated local errors smaller than a given tolerance and at the same time solve the problem as efficiently as possible step sizes providing the desired accuracy may be too large for stability yet this does not cause the popular codes to become unstable the theory explaining why this is so holds only in very restricted circumstances this paper broadens and deepens the theory for explicit runge kutta methods

diploid biological evolution models with general smooth fitness landscapes and recombination using a hamilton jacobi equation approach we obtain analytic equations for steady state population distributions and mean fitness functions for crow kimura and eigen type diploid biological evolution models with general smooth hypergeometric fitness landscapes our numerical solutions of diploid biological evolution models confirm the analytic equations obtained we also study the parallel diploid model for the simple case of recombination and calculate the variance of distribution which is consistent with numerical results

on the derivation of power law distributions within classical statistical mechanics far from the thermodynamic limit we show that within classical statistical mechanics without taking the thermodynamic limit the most general boltzmann factor for the canonical ensemble is a q exponential function the only assumption here is that microcanonical distributions have to be separated from the total system energy which is the prerequisite for any sensible measurement we derive that all separable distributions are parametrized by a mathematical separation constant q which can be related to the non extensivity q parameter in tsallis distributions we further demonstrate that nature fixes the separation constant q to 1 for large dimensionality of gibbs phase space our results will be relevant for systems with a low dimensional space for example nanosystems comprised of a small number of particles or for systems with a dimensionally collapsed phase space which might be the case for a large class of complex systems

extracting features of gaussian self similar stochastic processes via the bandt pompe approach by recourse to appropriate information theory quantifiers normalized shannon entropy and martin plastino rosso intensive statistical complexity measure we revisit the characterization of gaussian self similar stochastic processes from a bandt pompe viewpoint we show that the ensuing approach exhibits considerable advantages with respect to other treatments in particular clear quantifiers gaps are found in the transition between the continuous processes and their associated noises

solving nonstiff higher order odes directly by the direct integration method the variable order and stepsize direct integration di method for solving systems of nonstiff higher order odes directly is reviewed mention is also made of the current inroads in solving stiff higher order odes directly the advantages of the di method over the corresponding adams method when used to solve the same problem as a first order system are pointed out numerical results are presented comparing the two methods in terms of execution times for each tolerance used and execution times for each solution of the same range of accuracy

a new method for the solution of models of biological evolution derivation of exact steady state distributions we investigate well known models of biological evolution and address the open problem of how construct a correct continuous analog of mutations in discrete sequence space we deal with models where the fitness is a function of a hamming distance from the reference sequence the mutation selection master equation in the discrete sequence space is replaced by a hamilton jacobi equation for the logarithm of relative frequencies of different sequences the steady state distribution mean fitness and the variance of fitness are derived all our results are asymptotic in the large genome limit a variety of important biological and biochemical models can be solved by this new approach