a connection between state space and doubly coprime fractional representations explicit formulas for a doubly coprime fractional representation of the transfer function of a lumped lti system are given in terms of a stabilizable and detectable state space realization of the transfer function these formulas allow existing computational algorithms to be utilized for the purpose of computing doubly coprime factorizations several additional implications of this result are briefly discussed as are some immediate extensions and future research directions

modeling and identification of symmetric noncausal impulse responces spectrally equivalent causal sec models for single input single output noncausal systems are derived with the objective of estimating the noncausal impulse response attention is restricted to symmetric noncausal impulse responses and the measurements of the system output are assumed to be noisy it is shown that by using structured sec models either state space or arma plus noise consistent estimates of the noncausal impulse response can be obtained by exploiting only the second order statistics of the measurements a new version of the correlation technique particularly suited to structured models is presented and analyzed computer simulation results are also presented to illustrate the performance of the proposed approach

frequency weighted controller reduction methods and loop transfer recovery abstract this paper shows that if one designs an lqg controller using the conventional technique of loop transfer recovery ltr then two frequency weighted controller reduction methods the enns frequency weighted balanced truncation enns 1984a ph d thesis stanford university ca 1984b proc 23rd cdc las vegas nv 127 132 and the bezout identity induced frequency weighted reduction method anderson and liu 1987 proc amer control conf mn will be equivalent under the condition that the plant transfer function is square nonsingular and minimum phase we also show that enns method is equivalent to the bezout identity induced frequency weighted reduction method if the controller itself is stable and a particular representation for the controller is assumed

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

percolation threshold parameters of fluids extensive monte carlo simulations on three qualitatively different model supercritical fluids square well fluid lennard jonesium and primitive water have been performed to examine percolation threshold parameters for continuum correlated models and their relation to general results valid for random lattice models random site percolation simple cubic lattice has therefore been considered as well two different bond criteria the configurational and self bound ones defining a cluster have been used in addition to the percolation threshold occupation probability p c and the percolation threshold fluid density ensuremath rho c the correlation length exponent ensuremath nu and the wrapping probability at the percolation threshold r w c have also been evaluated it is found that parameters ensuremath nu and r w c exhibit not only strong temperature dependence but also unlike the case of lattice systems dependence on the nature of the system considered and the employed definition of the cluster

frequency weighted optimal hankel norm approximation of stable transfer functions abstract this paper shows how to introduce arbitrary frequency weighting into the optimal hankel norm approximation problem for scalar finite dimensional linear time invariant systems the procedure has applications to the closed form design of reduced order controllers

an equation error method for the design of digital controllers the purpose of the proposed equation error technique is to design a digital controller such that the response of the resulting closed loop system matches approximately a specified response the technique is based on the minimization of the equation error in closed loop the necessary and sufficient condition for minimizing the equation error is given and is used in the design method

quasispecies theory for finite populations we present stochastic finite population formulations of the crow kimura and eigen models of quasispecies theory for fitness functions that depend in an arbitrary way on the number of mutations from the wild type we include back mutations in our description we show that the fluctuation of the population numbers about the average values is exceedingly large in these physical models of evolution we further show that horizontal gene transfer reduces by orders of magnitude the fluctuations in the population numbers and reduces the accumulation of deleterious mutations in the finite population due to muller s ratchet indeed the population sizes needed to converge to the infinite population limit are often larger than those found in nature for smooth fitness functions in the absence of horizontal gene transfer these analytical results are derived for the steady state by means of a field theoretic representation numerical results are presented that indicate horizontal gene transfer speeds up the dynamics of evolution as well

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

dynamics of the eigen and the crow kimura models for molecular evolution we introduce an alternative way to study molecular evolution within well established hamilton jacobi formalism showing that for a broad class of fitness landscapes it is possible to derive dynamics analytically within the 1n accuracy where n is the genome length for a smooth and monotonic fitness function this approach gives two dynamical phases smooth dynamics and discontinuous dynamics the latter phase arises naturally with no explicite singular fitness function counterintuitively the hamilton jacobi method yields straightforward analytical results for the models that utilize fitness as a function of hamming distance from a reference genome sequence we also show the way in which this method gives dynamical phase structure for multipeak fitness

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

almost sure convergence for stochastically biased random walks on trees we are interested in the biased random walk on a supercritical galton watson tree in the sense of lyons ann probab 18 931 958 1990 and lyons pemantle and peres probab theory relat fields 106 249 264 1996 and study a phenomenon of slow movement in order to observe such a slow movement the bias needs to be random the resulting random walk is then a tree valued random walk in random environment we investigate the recurrent case and prove under suitable general integrability assumptions that upon the system s non extinction the maximal displacement of the walk in the first n steps divided by log n 3 converges almost surely to a known positive constant

biased random walks on galton watson trees with leaves we consider a biased random walk xn on a galton watson tree with leaves in the sub ballistic regime we prove that there exists an explicit constant 0 1 depending on the bias such that xn is of order n denoting n the hitting time of level n we prove that n n1 is tight moreover we show that n n1 does not converge in law at least for large values of we prove that along the sequences n k k n n1 converges to certain infinitely divisible laws key tools for the proof are the classical harris decomposition for galton watson trees a new variant of regeneration times and the careful analysis of triangular arrays of i i d heavy tailed random variables

solution of the crow kimura model with a periodically changing two season fitness function since the origin of life both evolutionary dynamics and rhythms have played a key role in the functioning of living systems the crow kimura model of periodically changing fitness function has been solved exactly using integral equation with time ordered exponent we also found a simple approximate solution for the two season case the evolutionary dynamics accompanied by the rhythms provide important insights into the properties of certain biological systems and processes

explicit solutions of the discrete time lyapunov matrix equation and kalman yakubovich equations a general solution for the nonsquare nonsymmetric lyapunov matrix equation in a canonical form is presented the solution is shown to be a toeplitz matrix which may be calculated using the backwards levinson algorithm this solution is then applied to the kalman yakubovich equations to derive a method for generating strictly positive real functions via the positive real lemma this latter result has an application in system identification

a subdiffusive behaviour of recurrent random walk in random environment on a regular tree we are interested in the random walk in random environment on an infinite tree lyons and pemantle ann probab 20 125 136 1992 give a precise recurrence transience criterion our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk x n in random environment on a regular tree which is closely related to mandelbrot s c r acad sci paris 278 289 292 1974 multiplicative cascade we prove under some general assumptions upon the distribution of the environment the existence of a new exponent nu in big 0 1 over 2 big such that max 0 le i le n x i behaves asymptotically like n nu the value of is explicitly formulated in terms of the distribution of the environment

coprime factorization controller reduction with bezout identity induced frequency weighting abstract this paper introduces a frequency weighted controller reduction scheme which offers good closed loop stability under use of the reduced order controller by combining the use of a stable fractional representation of the controller with a frequency weighting in the reduction procedure derived by consideration of the closed loop stability margin we formulate a frequency weighted controller reduction procedure with the weight simplified by the bezout identity some examples are used to illustrate the reduction procedure

reduced order controller design using coprime factor model reduction two procedures for reduced order controller design that incorporate coprime factor model reduction techniques and a related robust stabilization problem are proposed it is shown that robust stability can be achieved with these controllers the results obtained show how to reduce the controller degree systematically they clearly indicate how the maximum stability margin influences allowable model reduction error

identification of linear stochastic systems via second and fourth order cumulant matching the identification problem for time invariant single input single output linear stochastic systems driven by non gaussian white noise is considered the system is not restricted to be minimum phase and it is allowed to contain all pass components a least squares criterion that involves matching the second and the fourth order cumulant functions of the noisy observations is proposed knowledge of the probability distribution of the driving noise is not required an order determination criterion that is a modification of the akaike information criterion is also proposed strong consistency of the proposed estimator is proved under certain sufficient conditions simulation results are presented to illustrate the method

random walks in random environments random walks in random environments and their diffusion analogues have been a source of surprising phenomena and challenging problems especially in the non reversible situation since they began to be studied in the 1970s we review the model available results and techniques and point out several gaps in the understanding of these processes

quenched invariance principle for simple random walk on percolation clusters we consider the simple random walk on the unique infinite cluster of super critical bond percolation in d with d 2 we prove that for almost every percolation configuration the path distribution of the walk converges weakly to that of non degenerate isotropic brownian motion our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square integrable martingale the size of the deformation expressed by the so called corrector is estimated by means of ergodicity arguments

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

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

an output error method for reduced order controller design an efficient computational technique is presented for the design of reduced order controllers for linear discrete time systems the technique is based on the minimization of the output error between the closed loop system and a specified reference model

effect of anisotropy on the scaling of connectivity and conductivity in continuum percolation theory we investigate the effects of anisotropy on the finite size scaling of connectivity and conductivity of continuum percolation in three dimensions we consider a system of size x ifmmode times else texttimes fi y ifmmode times else texttimes fi z in which cubic bodies of size a ifmmode times else texttimes fi b ifmmode times else texttimes fi c are placed randomly we define two aspect ratios to request anisotropy then we expect that the displacement of average connected fraction p averaged over the realizations about the isotropic universal curves will be a function of the two aspect ratios this is accounted by considering an apparent percolation threshold in each direction which leads to 50 of realizations connecting in that direction we find the aspect ratios dependency of the apparent threshold and investigate the finite size scaling transformations for the mean connected fraction and its associated fluctuations moreover we apply a single phase pressure solver to determine the conductivity of various realizations of the system finally we apply the same idea to account for the effect of anisotropy on the conductivity scaling

speed of the biased random walk on a galton watson tree we give an expression of the speed of the biased random walk on a galton watson tree in the particular case of the simple random walk we recover the result of lyons et al erg theory dyn syst 15 593 619 1995 the proof uses a description of the invariant distribution of the environment seen from the particle

controller reduction using canonical interactions a conceptual approach to controller reduction is proposed that is based on closed loop considerations the method assesses the interactions between the plant and a high order controller as described by canonical correlation coefficients a free parameter in the algorithm of the reduced controller makes it possible to trade stability robustness versus performance several examples demonstrate the efficacy of the method

convergence rate for an approximation approach to h norm optimization problems with an application to controller order reduction in a recent paper by fisher et al 1989 a constrained h smooth optimization technique proc 28th cdc florida u s a a smooth approximation technique is proposed to solve a general class of constrained h norm optimization problems the aim of this paper is to show that such approximation has a convergence rate of o in p2p furthermore this technique is applied to handle a general class of frequency weighted controller order reduction problems

conceptual proofs of l log l criteria for mean behavior of branching processes the kesten stigum theorem is a fundamental criterion for the rate of growth of a supercritical branching process showing that an l log l condition is decisive in critical and subcritical cases results of kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations we give conceptual proofs of these theorems based on comparisons of galton watson measure to another measure on the space of trees this approach also explains yaglom s exponential limit law for conditioned critical branching processes via a simple characterization of the exponential distribution

a model with darwinian dynamics on a rugged landscape we discuss the population dynamics with selection and random diffusion keeping the total population constant in a fitness landscape associated with constraint satisfaction a paradigm for difficult optimization problems we obtain a phase diagram in terms of the size of the population and the diffusion rate with a glass phase inside which the dynamics keeps searching for better configurations and outside which deleterious mutations spoil the performance the phase diagram is analogous to that of dense active matter in terms of temperature and drive

error threshold in optimal coding numerical criteria and classes of universalities for complexity the free energy of the random energy model at the transition point between the ferromagnetic and spin glass phases is calculated at this point equivalent to the decoding error threshold in optimal codes the free energy has finite size corrections proportional to the square root of the number of degrees the response of the magnetization to an external ferromagnetic phase is maximal at values of magnetization equal to one half we give several criteria of complexity and define different universality classes according to our classification at the lowest class of complexity are random graphs markov models and hidden markov models at the next level is the sherrington kirkpatrick spin glass connected to neuron network models on a higher level are critical theories the spin glass phase of the random energy model percolation and self organized criticality the top level class involves highly optimized tolerance design error thresholds in optimal coding language and maybe financial markets living systems are also related to the last class the concept of antiresonance is suggested for complex systems

robust stabilization of singularly perturbed systems abstract we discuss the problem of designing stabilizing controllers for singularly perturbed systems on the basis of simplified models in 1 it was shown that a constant gain output feedback controller designed on the basis of the simplified model need not stabilize the true system containing both fast and slow modes this phenomenon was then expanded to include the case where the simplified system is strictly proper in 2 the objectives of this note are threefold i to show that given any proper system and any stabilizing controller for it that is proper but not strictly proper there exists a singular perturbation of the system that is destabilized by that controller ii to show that any strictly proper controller for a singularly perturbed system designed on the basis of a reduced order model will stabilize the true system for sufficiently small values of the fast dynamics parameter and iii to provide a characterization in the same spirit as 3 4 of the set of all strictly proper controllers that stabilize a given proper plant by combining these results it is possible to generate the class of all robustly stabilizing controllers for a given singularly perturbed system

lqg control with an h sup infinity performance bound a riccati equation approach an lqg linear quadratic gaussian control design problem involving a constraint on h sup infinity disturbance attenuation is considered the h sup infinity performance constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on l sub 2 performance in contrast to the pair of separated riccati equations of standard lqg theory the h sup infinity constrained gains are given by a coupled system of three modified riccati equations the coupling illustrates the breakdown of the separation principle for the h sup infinity constrained problem both full and reduced order design problems are considered with an h sup infinity attenuation constraint involving both state and control variables an algorithm is developed for the full order design problem and illustrative numerical results are given

structure of principal eigenvectors and genetic diversity abstract the main concern of this paper is long term genotypic diversity genotypes are represented as finite sequences s 1 s 2 s n where the entries s i are drawn from a finite alphabet the mutation matrix is given in terms of hamming distances it is proved that the long time behavior of solutions for a class of genotype evolution models is governed by the principal eigenvectors of the sum of the mutation and fitness matrices it is proved that the components of principal eigenvectors are symmetric and monotonely decreasing in terms of hamming distances whenever the fitness matrix has those properties

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

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

a note on balanced controller reduction the standard linear quadratic gaussian lqg optimal controller is not minimal in general even if the plant is minimal hence the order of the optimal controller can be less than that of the plant this note shows that the original balancing method of moore 1 can be used to generate these reduced order optimal controllers but the newer balancing methods recently proposed for controller reduction might not generate the reduced order optimal controllers even when they exist

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

random walk on barely supercritical branching random walk let mathcal t be a supercritical galton watson tree with a bounded offspring distribution that has mean mu 1 conditioned to survive let varphi mathcal t be a random embedding of mathcal t into mathbb z d according to a simple random walk step distribution let mathcal t p be percolation on mathcal t with parameter p and let p c mu 1 be the critical percolation parameter we consider a random walk x n n ge 1 on mathcal t p and investigate the behavior of the embedded process varphi mathcal t p x n as n rightarrow infty and simultaneously mathcal t p becomes critical that is p p n searrow p c we show that when we scale time by n p n p c 3 and space by sqrt p n p c n the process varphi mathcal t p x n n ge 1 converges to a d dimensional brownian motion we argue that this scaling can be seen as an interpolation between the scaling of random walk on a static random tree and the anomalous scaling of processes in critical random environments

controller reduction via stable factorization and balancing abstract a controller reduction procedure based on a representation of a controller as a matrix function defined using stable proper transfer functions and employing a balancing technique is studied in this paper for a certain right coprime factorization of an lqg designed controller k s n s d 1 s we approximate using a balancing technique the pair d s n s t by a low order pair d1 s n1 s t defining a factorization of the reduced order controller k 1 s n1 s d1 1 s we show that reducing the controller order in this way is motivated in a natural way which leads to the expectation of both good stability properties and good accuracy of approximation of closed loop behaviour this is also demonstrated in some examples

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

random walks and percolation on trees there is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree it equals the exponential of the hausdorff dimension of the boundary in an appropriate metric its importance for probabilistic processes on a tree is shown in several ways including random walk and percolation where it provides points of phase transition

biased random walks on galton watson trees we consider random walks with a bias toward the root on the family tree t of a supercritical galton watson branching process and show that the speed is positive whenever the walk is transient the corresponding harmonic measures are carried by subsets of the boundary of dimension smaller than that of the whole boundary when the bias is directed away from the root and the extinction probability is positive the speed may be zero even though the walk is transient the critical bias for positive speed is determined

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

fractional balanced reduction model reduction via fractional representation a method for the model reduction of finite dimensional linear time invariant fdlti plants is proposed which uses normalized fractional representations is proposed the method dubbed fractional balanced reduction applies balance and truncate to a special representation of the graph operator of the plant this operation yields the graph operator of a reduced order plant the method has such properties as existence of an a priori error bound in the graph metric and preservation of sector containment coupling fractional representations with principal component analysis gives a model reduction method that is capable of producing in a numerically stable way a good reduced order model using the whole full order model sector properties are also preserved these are useful for deciding stability when nonlinearities are in the loop

anderson localization non linearity and stable genetic diversity in many models of genotypic evolution the vector of genotype populations satisfies a system of linear ordinary differential equations this system of equations models a competition between differential replication rates fitness and mutation mutation operates as a generalized diffusion process on genotype space in the large time asymptotics the replication term tends to produce a single dominant quasi species unless the mutation rate is too high in which case the asymptotic population becomes de localized we introduce a more macroscopic picture of genotypic evolution wherein a random fitness term in the linear model produces features analogous to anderson localization when coupled with density dependent non linearities which limit the population of any given genotype we obtain a model whose large time asymptotics display stable genotypic diversity

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

spin coherent state representation of the crow kimura and eigen models of quasispecies theory we present a spin coherent state representation of the crow kimura and eigen models of biological evolution we deal with quasispecies models where the fitness is a function of hamming distances from one or more reference sequences in the limit of large sequence length n we find exact expressions for the mean fitness and magnetization of the asymptotic quasispecies distribution in symmetric fitness landscapes the results are obtained by constructing a path integral for the propagator on the coset su 2 u 1 and taking the classical limit the classical limit gives a hamiltonian function on a circle for one reference sequence and on the product of 2m 1 circles for m reference sequences we apply our representation to study the schuster swetina phenomena where a wide lower peak is selected over a narrow higher peak the quadratic landscape with two reference sequences is also analyzed specifically and we present the phase diagram on the mutation fitness parameter phase space furthermore we use our method to investigate more biologically relevant system a model of escape from adaptive conflict through gene duplication and find three different phases for the asymptotic population distribution

another balanced controller reduction algorithm the problem of controller reduction has been addressed by a number of authors 1 4 using the concept of balanced coordinates originally proposed by moore 5 this note establishes a more direct link between the riccati balancing methods in 2 4 and moore s work and proposes yet another candidate method as a logical extension of 2 4

a central limit theorem for biased random walks on galton watson trees let mathcal t be a rooted galton watson tree with offspring distribution p k that has p 0 0 mean m kp k 1 and exponential tails consider the biased random walk x n n 0 on mathcal t this is the nearest neighbor random walk which when at a vertex v with d v offspring moves closer to the root with probability d v and moves to each of the offspring with probability 1 d v it is known that this walk has an a s constant speed tt v lim n x n n where x n is the distance of x n from the root with tt v 0 for 0 m and tt v 0 for m for all m we prove a quenched clt for x n n tt v for m the walk is positive recurrent and there is no clt the most interesting case by far is m where the clt has the following form for almost every mathcal t the ratio x nt sqrt n converges in law as n to a deterministic multiple of the absolute value of a brownian motion our approach to this case is based on an explicit description of an invariant measure for the walk from the point of view of the particle previously such a measure was explicitly known only for 1 and the construction of appropriate harmonic coordinates

biological evolution in a multidimensional fitness landscape we considered a multiblock molecular model of biological evolution in which fitness is a function of the mean types of alleles located at different parts blocks of the genome we formulated an infinite population model with selection and mutation and calculated the mean fitness for the case of recombination we formulated a model with a multidimensional fitness landscape the dimension of the space is equal to the number of blocks and derived a theorem about the dynamics of initially narrow distribution we also considered the case of lethal mutations we also formulated the finite population version of the model in the case of lethal mutations our models derived for the virus evolution are interesting also for the statistical mechanics and the hamilton jacobi equation as well

comments on ff pade method of model reduction in frequency domain with reply in a previously published paper see ibid vol ac 32 no 3 p 243 6 1987 h xiheng presented four examples for model reduction based on a modified pade approximation method it is pointed out that three of these contain errors a brief reply from h xiheng is also given

robust stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

universal finite size scaling functions for percolation on three dimensional lattices using a histogram monte carlo simulation method hmcsm hu lin and chen found that bond and site percolation models on planar lattices have universal finite size scaling functions for the existence probability e p the percolation probability p and the probability w n for the appearance of n percolating clusters in these models in this paper we extend above study to percolation on three dimensional lattices with various linear dimensions l using the hmcsm we calculate the existence probability e p and the percolation probability p for site and bond percolation on a simple cubic sc lattice and site percolation on body centered cubic and face centered cubic lattices each lattice has the same linear dimension in three dimensions using the data of e p and p in a percolation renormalization group method we find that the critical exponents obtained are quite consistent with the universality of critical exponents using a small number of nonuniversal metric factors we find that e p and p have universal finite size scaling functions this implies that the critical e p is a universal quantity which is 0 265 ifmmode pm else textpm fi 0 005 for free boundary conditions and 0 924 ifmmode pm else textpm fi 0 005 for periodic boundary conditions we also find that w n for site and bond percolation on sc lattices have universal finite size scaling functions

solution of the crow kimura and eigen models for alphabets of arbitrary size by schwinger spin coherent states to represent the evolution of nucleic acid and protein sequence we express the parallel and eigen models for molecular evolution in terms of a functional integral representation with an h letter alphabet lifting the two state purine pyrimidine assumption often made in quasi species theory for arbitrary h and a general mutation scheme we obtain the solution of this model in terms of a maximum principle euler s theorem for homogeneous functions is used to derive this thermodynamic formulation of evolution the general result for the parallel model reduces to known results for the purine pyrimidine h 2 alphabet and the nucleic acid h 4 alphabet for the kimura 3 st mutation scheme examples are presented for the h 4 and h 20 cases we also derive the maximum principle for the eigen model for general h the general result for the eigen model reduces to a known result for h 2 examples are presented for the nucleic acid h 4 and the amino acid h 20 alphabet an error catastrophe phase transition occurs in these models and the order of the phase transition changes from second to first order for smooth fitness functions when the alphabet size is increased beyond two letters to the generic case as examples we analyze the general analytic solution for sharp peak linear quadratic and quartic fitness functions

large deviations for random walks on galton watson trees averaging and uncertainty in the study of large deviations for random walks in random environment a key distinction has emerged between quenched asymptotics conditional on the environment and annealed asymptotics obtained from averaging over environments in this paper we consider a simple random walk x n on a galton watson tree t i e on the family tree arising from a supercritical branching process denote by x n the distance between the node x n and the root of t our main result is the almost sure equality of the large deviation rate function for x n n under the quenched measure conditional upon t and the rate function for the same ratio under the annealed measure averaging on t according to the galton watson distribution this equality hinges on a concentration of measure phenomenon for the momentum of the walk the momentum at level n for a specific tree t is the average over random walk paths of the forward drift at the hitting point of that level this concentration or certainty is a consequence of the uncertainty in the location of the hitting point we also obtain similar results when x n is a biased walk on a galton watson tree even though in that case there is no known formula for the asymptotic speed our arguments rely at several points on a ubiquity lemma for galton watson trees due to grimmett and kesten 1984

on direct versus indirect methods for reduced order controller design comparisons are obtained between direct parameter optimization and indirect controller reduction methods in reduced order controller design in particular the authors compare the controller canonical correlation coefficients method of c de villemagne and r e skelton 1988 to the optimal projection theory of d c hyland and d s bernstein 1984 it is noted that suboptimal methods may in certain cases be simpler to implement than optimal methods thus the choice of direct versus indirect methods can be viewed as a tradeoff between design simplicity and actual control system performance

phase diagram for the eigen quasispecies theory with a truncated fitness landscape using methods of statistical physics we present rigorous theoretical calculations of eigen s quasispecies theory with the truncated fitness landscape which dramatically limits the available sequence space of information carriers as the mutation rate is increased from small values to large values one can observe three phases the first i selective also known as ferromagnetic phase the second ii intermediate phase with some residual order and the third iii completely randomized also known as paramagnetic phase we calculate the phase diagram for these phases and the concentration of information carriers in the master sequence also known as peak configuration x 0 and other classes of information carriers as the phase point moves across the boundary between phase i and phase ii x 0 changes continuously as the phase point moves across the boundary between phase ii and phase iii x 0 has a large change our results are applicable for the general case of a fitness landscape

theorame central limite fonctionnel pour une marche au hasard en environnement aleatoire the simple random walk on a supercritical galton watson tree is transient when the tree is infinite moreover there exist regeneration times that is times when the walk crosses an edge for the first and the last time we prove that the distance and the range of the walk satisfy functional central limit theorems under the annealed law gp this result is a consequence of estimates of the law of the first regeneration time r we show that there exist positive c c and such that for all n 1 exp c n n exp c n la marche au hasard simple sur un arbre de galton watson super critique est transiente quand l arbre est infini de plus elle possede des instants de regeneration c est a dire des instants ou le marcheur traverse une arete pour la premiere et la derniere fois nous demontrons que la distance et le rang de la marche verifient un theoreme central limite fonctionnel sous la loi gp moyennee en l environnement ce resultat repose sur une estimation de la loi du premier instant de regeneration r nous montrons qu il existe des constantes c c et strictement positives telles que pour tout n 1 exp c n gp r n exp c n

galton watson trees with the same mean have the same polar sets evans defined a notion of what it means for a set b to be polar for a process indexed by a tree the main result herein is that a tree picked from a galton watson measure whose offspring distribution has mean m and finite variance will almost surely have precisely the same polar sets as a deterministic tree of the same growth rate this implies that deterministic and nondeterministic trees behave identically in a variety of probability models mapping subsets of euclidean space to trees and polar sets to capacity criteria it follows that certain random cantor sets are capacity equivalent to each other and to deterministic cantor sets an extension to branching processes in varying environment is also obtained

maximum minimum and optimal mutation rates in dynamic environments we analyze the dynamics of the parallel mutation selection quasispecies model with a changing environment for an environment with the sharp peak fitness function in which the most fit sequence changes by k spin flips every period t we find analytical expressions for the minimum and maximum mutation rates for which a quasispecies can survive valid in the limit of large sequence size we find an asymptotic solution in which the quasispecies population changes periodically according to the periodic environmental change in this state we compute the mutation rate that gives the optimal mean fitness over a period we find that the optimal mutation rate per genome k t is independent of genome size a relationship which is observed across broad groups of real organisms

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

structured coprime factor model reduction based on lmis in this paper we discuss dynamic model reduction methods which preserve a certain structure in the underlying system specifically we consider the situation where the reduction must be consistent with a partition of the system states this is motivated for instance in situations where state variables are associated with the topology of a networked system and the reduction should preserve this we build on the observation that imposing block structure to generalized controllability and observability gramians automatically yields such state partitioned model reduction the difficulty lies in ensuring feasibility of the resulting lyapunov inequalities which is in general very restrictive to overcome this we consider coprime factor model reduction we derive an lmi characterization of expansive and contractive coprime factorizations that preserve structure and use this to build a more flexible method for structured model reduction an example is given to illustrate the method

dependency of percolation critical exponents on the exponent of power law size distribution abstract the standard percolation theory uses objects of the same size moreover it has long been observed that the percolation properties of the systems with a finite distribution of sizes are controlled by an effective size and consequently the universality of the percolation theory is still valid in this study the effect of power law size distribution on the critical exponents of the percolation theory of the two dimensional models is investigated two different object shapes i e stick shaped and square are considered these two shapes are the representative of the fractures in fracture reservoirs and the sandbodies in clastic reservoirs the finite size scaling arguments are used for the connectivity to determine the dependency of the critical exponents on the power law exponent in particular the deviations of percolation exponents from their universal values as well as the connectivity behavior of such systems are investigated numerically as a result this extends the applicability of the conventional percolation approach to study the connectivity of systems with a very broad size distribution

computation of coprime factorizations of rational matrices abstract we propose numerically reliable state space algorithms for computing several coprime factorizations of rational matrices 1 factorizations with factors having poles in a given stability domain 2 factorizations with proper stable factors 3 factorizations with inner and j inner denominators the new algorithms are based on a recursive generalized schur algorithm for pole dislocation they are generally applicable whether the underlying descriptor state space representation is minimal or not and whether it is stabilizable detectable or not the proposed algorithms are useful in solving various computational problems for both standard and descriptor system representations

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

quasispecies model of evolution with migration we consider the model of asexual evolution with migration which was proposed by waclaw et al phys rev lett 105 268101 2010 prltao0031 900710 1103 physrevlett 105 268101 this model setting is based on the standard mutation scheme from the quasispecies theory but with replicators moving from one habitat to another the primary goal is to solve exactly the infinite population genome length version of the model for the independent random distribution of fitnesses considered in the original paper moreover we propose the analytical solution for the single peak and the symmetric fitness landscape our analytical solution is exact at the limit of large n we found two phases the correlated phase with the identical distributions of mutations in both habitats and the uncorrelated phase where the second habitat is choosing another peak of distribution in sequence space compared to the peak in the first habitat

quenched invariance principles for walks on clusters of percolation or among random conductances in this work we principally study random walk on the supercritical infinite cluster for bond percolation on d we prove a quenched functional central limit theorem for the walk when d 4 we also prove a similar result for random walk among i i d random conductances along nearest neighbor edges of d when d 1

controller reduction concepts and approaches the problem of passing from a linear time invariant high order controller designed for a linear time invariant plant of presumably high order to a low order approximation of the controller is discussed the approximation problem is often best posed as a frequency weighted l sub infinity approximation problem many different controller representations are possible giving different performances of the various reduction algorithms

performance bounds for coprime factor controller reductions this paper investigates the performance of coprime factor based controller reduction methods we show that under some mild conditions closed loop performance bounds can be derived for observer based controllers the results in this paper lay a mathematical foundation for using coprime factor controller reduction methods in certain classes of controller reduction problems

evolutionary dynamics of the most populated genotype on rugged fitness landscapes we consider an asexual population evolving on rugged fitness landscapes which are defined on the multidimensional genotypic space and have many local optima we track the most populated genotype as it changes when the population jumps from a fitness peak to a better one during the process of adaptation this is done using the dynamics of the shell model which is a simplified version of the quasispecies model for infinite populations and standard wright fisher dynamics for large finite populations we show that the population fraction of a genotype obtained within the quasispecies model and the shell model match for fit genotypes and at short times but the dynamics of the two models are identical for questions related to the most populated genotype we calculate exactly several properties of the jumps in infinite populations some of which were obtained numerically in previous works we also present our preliminary simulation results for finite populations in particular we measure the jump distribution in time and find that it decays as t ensuremath 2 as in the quasispecies problem

gramian based model reduction for data sparse systems model reduction is a common task within the simulation control and optimization of complex dynamical systems for instance in control and optimization problems for instationary partial differential equations pdes the associated large scale linear systems resulting from a semi discretization have to be solved repeatedly model reduction is a more and more accepted technique to attack these problems in reasonable time here we investigate model reduction for parabolic pdes based on balanced truncation bt the system theoretic properties of bt and the related method of singular perturbation approximation spa provide desirable features for the reduced order system the major computational task in bt is the solution of large scale lyapunov or stein equations depending on whether the semidiscretized or fully discretized system is considered therefore the method is often considered to be of limited use for really large scale applications i e problems that require gg1 gbyte of storage so that memory access becomes an issue here we develop effective implementations of bt and spa by exploiting the structure of the underlying pde problem this is achieved by a data sparse approximation of the semi discretized differential operator using the hierarchical matrix format furthermore we review how to integrate the corresponding formatted arithmetic in the sign function method for computing approximate solution factors of the lyapunov equations and suggest an iterative scheme for solving stein equations employing formatted arithmetic these approaches are well suited for a class of practically relevant problems and allow the application of bt and spa to systems coming from two dimensional 2d and three dimensional 3d finite element method fem and boundary element method bem discretizations the resulting methods are most beneficial in the context of bem discretizations for 3d problems where large scale dense matrices define the semi discretized control system we also analyze the additional approximation error in the reduced order model introduced by the hierarchical matrix approximation the efficiency of the methods is demonstrated for linear scalar convection diffusion reaction equations the numerical results confirm that the model reduction bt error and the hierarchical matrix approximation error can be balanced by choosing the tolerances and parameters involved appropriately

reduced order models and controllers for continuous time stochastic systems an information theory approach reduced order models and controllers for continuous time stochastic systems are described the reduced order models are chosen to minimize the kullback leibler information distance klid between the outputs of the actual and reduced systems an lqg controller based on a reduced order system model is described a second reduced order controller is found to minimize the klid between the closed loop system outputs with the full and reduced order controllers

scaling limit of the recurrent biased random walk on a galton watson tree we show that the trace of the null recurrent biased random walk on a galton watson tree properly renormalized converges to the brownian forest our result extends to the setting of the random walk in random environment on a galton watson tree

upper and lower bounds for approximation in the gap metric upper and lower bounds for the closest approximant of degree k n in the gap metric to a plant of degree n are obtained the bounds are expressed in terms of the singular values of two hankel operators defined in terms of the symbol of the graph of the plant the question of robust stability and performance of feedback systems is examined in the context of approximation of plant and controller in the gap metric

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

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

optimal mutation rates in dynamic environments the eigen model we consider the eigen quasispecies model with a dynamic environment for an environment with sharp peak fitness in which the most fit sequence moves by k spin flips each period t we find an asymptotic stationary state in which the quasispecies population changes regularly according to the regular environmental change from this stationary state we estimate the maximum and the minimum mutation rates for a quasispecies to survive under the changing environment and calculate the optimum mutation rate that maximizes the population growth interestingly we find that the optimum mutation rate in the eigen model is lower than that in the crow kimura model and at their optimum mutation rates the corresponding mean fitness in the eigen model is lower than that in the crow kimura model suggesting that the mutation process which occurs in parallel to the replication process as in the crow kimura model gives an adaptive advantage under changing environment