measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

bounds for the zeros of polynomials from eigenvalues and singular values of some companion matrices abstract let a monic polynomial pn x xn a1xn 1 an aj plane1d 49e j 1 2 n be given bounds for the eigenvalues and the computation of the singular values of some special companion matrices give bounds for the zeros of the polynomial pn which improve classical and recently found ones also some new bounds for the zeros are given

convergence analysis for operator splitting methods applied to conservation laws with stiff source terms we analyze the order of convergence for operator splitting methods applied to conservation laws with stiff source terms we suppose that the source term q u is dissipative it is proved that the l 1 error introduced by the time splitting can be bounded by o d t vert q u 0 vert l 1 which is an improvement of the o q d t upper bound where d t is the splitting time step q is the lipschitz constant of q or q max u vert q u vert in case q is smooth a generic model with a special form of stiff source is also investigated we propose a nonuniform temporal mesh to eliminate the effect of the initial layer introduced by the stiff source term our results are derived by using parabolic regularizations rather than using kuznetsov s approximation theory which has been employed as a standard approach for error analysis to the dimensional or time splitting methods numerical examples are presented to illustrate the theoretical results

hyperbolic conservation laws with relaxation the effect of relaxation is important in many physical situations it is present in the kinetic theory of gases elasticity with memory gas flow with thermo non equilibrium water waves etc the governing equations often take the form of hyperbolic conservation laws with lower order terms in this article we present and analyze a simple model of hyperbolic conservation laws with relaxation effects dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified stability criteria for diffusion waves expansion waves and traveling waves are found and justified nonlinearly time asymptotic expansion and the energy method are used in the analysis for dissipative waves the expansion is similar in spirit to the chapman enskog expansion in the kinetic theory for shock waves however a different approach is needed

bounds on polynomial roots using intercyclic companion matrices abstract the frobenius companion matrix and more recently the fiedler companion matrices have been used to provide lower and upper bounds on the modulus of any root of a polynomial p x in this paper we explore new bounds obtained from taking the 1 norm and norm of a matrix in the wider class of intercyclic companion matrices as is the case with fiedler matrices we observe that the new bounds from intercyclic companion matrices can improve those from the frobenius matrix by at most a factor of two by using the hessenberg form of an intercyclic companion matrix we describe how to determine the best upper bound when restricted to fiedler companion matrices using the norm we also obtain a new general bound by considering the polynomial x q p x for q 0 we end by considering upper bounds obtained from inverses of monic reversal polynomials of intercyclic companion matrices noting that these can make more significant improvements on the bounds from a frobenius companion matrix for certain polynomials

the extremal index hitting time statistics and periodicity abstract the extremal index appears as a parameter in extreme value laws for stochastic processes characterising the clustering of extreme events we apply this idea in a dynamical systems context to analyse the possible extreme value laws for the stochastic process generated by observations taken along dynamical orbits with respect to various measures we derive new easily checkable conditions which identify extreme value laws with particular extremal indices in the dynamical context we prove that the extremal index is associated with periodic behaviour the analogy of these laws in the context of hitting time statistics as studied in the authors previous works on this topic is explained and exploited extensively allowing us to prove for the first time the existence of hitting time statistics for balls around periodic points moreover for very well behaved systems uniformly expanding we completely characterise the extremal behaviour by proving that either we have an extremal index less than 1 at periodic points or equal to 1 at any other point this theory then also applies directly to general stochastic processes adding both useful tools to identify the extremal index and giving deeper insight into the periodic behaviour it suggests

asymptotical growth of a class of random trees we study three rules for the development of a sequence of finite subtrees t n of an infinite m ary tree t independent realizations omega n of a stationary ergodic process omega on m letters are used to trace out paths in t in the first rule t n is formed by adding a node to t n 1 at the first location where the path defined by omega n leaves t n 1 the second and third rules are similar but more complicated for each rule the height l n of the added node is shown to grow in probability as ln n divided by h the entropy per symbol of the generic process a typical retrieval time has the same behavior on the other hand lim inf nl n ln n sigma 1 lim sup n l n ln n sigma 2 a s where the constants sigma 1 sigma 2 are in general different depend on the rule in use and sigma 1 1 h sigma 2 it is proven along the way that the height of t n grows as sigma 2 ln n with probability one

local error estimates for discontinuous solutions of nonlinear hyperbolic equations let u x t be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data suppose u varepsilon x t is the solution of an approximate viscosity regularization where varepsilon 0 is the small viscosity amplitude it is shown that by post processing the small viscosity approximation u varepsilon pointwise values of u and its derivatives with an error as close to e as desired can be recovered the analysis relies on the adjoint problem of the forward error equation which in this case amounts to a backward linear transport equation with discontinuous coefficients the novelty of our approach is to use a generalized e condition of the forward problem in order to deduce a w 1 infty energy estimate for the discontinuous backward transport equation this in turn leads to e uniform estimate on moments of the error u varepsilon u the approach presented does not follow the characteristics and therefore applies mutatis mutandis to o

the convergence rate of approximate solutions for nonlinear scalar conservation laws let v varepsilon x t varepsilon 0 be a family of approximate solutions for the nonlinear scalar conservation law u t f u x 0 with c 0 1 initial data assume that v varepsilon x t are textit lip stable in the sense that they satisfy oleinik s e entropy condition it is shown that if these approximate solutions are textit lip consistent i e if v varepsilon cdot 0 u cdot 0 textit lip x v t varepsilon f v varepsilon x textit lip x t mathcal o varepsilon then they converge to the entropy solution and the convergence rate estimate v varepsilon cdot t u cdot t textit lip x mathcal o varepsilon holds consequently the familiar l p type and new pointwise error estimates are derived these convergence rate results are demonstrated in the context of entropy satisfying finite difference and glimm s schemes

from rates of mixing to recurrence times via large deviations abstract a classic approach in dynamical systems is to use particular geometric structures to deduce statistical properties for example the existence of invariant measures with stochastic like behaviour such as large deviations or decay of correlations such geometric structures are generally highly non trivial and thus a natural question is the extent to which this approach can be applied in this paper we show that in many cases stochastic like behaviour itself implies that the system has certain non trivial geometric properties which are therefore necessary as well as sufficient conditions for the occurrence of the statistical properties under consideration as a by product of our techniques we also obtain some new results on large deviations for certain classes of systems which include viana maps and multidimensional piecewise expanding maps

runge kutta methods for hyperbolic conservation laws with stiff relaxation terms underresolved numerical schemes for hyperbolic conservation laws with stiff relaxation terms may generate unphysical spurious numerical results or reduce to lower order if the small relaxation time is not temporally well resolved we design a second order runge kutta type splitting method that possesses the discrete analogue of the continuous asymptotic limit which thus is able to capture the correct physical behaviors with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved

characterization of a family of generalized companion matrices abstract matrices a of order n having entries in the field f x 1 x n of rational functions over a field f and characteristic polynomial det t i a t n x 1 t n 1 x n 1 t x n are studied it is known that such matrices are irreducible and have at least 2 n 1 nonzero entries such matrices with exactly 2 n 1 nonzero entries are called ma zhan matrices conditions are given that imply that a ma zhan matrix is similar via a monomial matrix to a generalized companion matrix that is a lower hessenberg matrix with ones on its superdiagonal and exactly one nonzero entry in each of its subdiagonals via the ax grothendieck theorem respectively its analog for the reals these conditions are shown to hold for a family of matrices whose entries are complex respectively real polynomials

the compound poisson limit ruling periodic extreme behaviour of non uniformly hyperbolic dynamics we prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of certain non uniformly hyperbolic dynamical systems is compound poisson the returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events or exceedances of high thresholds so that there is a connection between the laws of return times statistics and extreme value laws the fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average sizes is given by the extremal index which depends on the expansion of the system at the periodic point we recall that for generic points the exceedances in the limit are singular and occur at poisson times however around periodic points the picture is different the respective point processes of exceedances converge to a compound poisson process so instead of single exceedances we have entire clusters of exceedances occurring at poisson times with a geometric distribution ruling its multiplicity the systems to which our results apply include general piecewise expanding maps of the interval rychlik maps maps with indifferent fixed points manneville pomeau maps and benedicks carleson quadratic maps

convergence of a splitting scheme applied to the ruijgrok wu model of the boltzmann equation this paper deals with upwind splitting schemes for the ruijgrok wu model physica a 113 1982 401 416 of the kinetic theory of rare7ed gases in the 8uid dynamic scaling we prove the stability and the convergence for these schemes the relaxation limit is also investigated and the limit equation is proved to be a 7rst order quasi linear conservation law the loss of quasi monotonicity of the present model makes it necessary to give a more careful analysis of its structure we also obtain global error estimates in the spaces w s p for 16s61 p 16p6 and pointwise error estimates for the approximate solution the proof naturally uses the framework introduced by nessyahu and tadmor siam j numer anal 29 1992 1505 1519 due to the convexity of the 8ux function c

renyi entropies and large deviations for the first match function we define the first match function t n mathcal c n rightarrow 1 ldots n where mathcal c is a finite alphabet for two copies of x 1 n in mathcal c n this function gives the minimum number of steps one has to slide one copy of x 1 n to get a match with the other one for ergodic positive entropy processes saussol and coauthors proved the almost sure convergence of t n n we compute the large deviation properties of this function we prove that this limit is related to the renyi entropy function which is also proved to exist our results hold under a condition easy to check which defines a large class of processes we provide some examples

diophantine properties of iets and general systems quantitative proximality and connectivity three properties of dynamical systems recurrence connectivity and proximality are quantified by introducing and studying the gauges measurable functions corresponding to each of these properties the properties of the proximality gauge are related to the results in the active field of shrinking targets the emphasis in the present paper is on the iets interval exchange transformations mathcal i t mathcal i 0 1 in particular we prove that if an iet t is ergodic relative to the lebesgue measure then the equality a1 liminf n to infty n bigl t n x y bigr 0 holds for a a x y in mathcal i 2 the ergodicity assumption is essential the result does not extend to all minimal iets also the factor n in a1 is optimal e g it cannot be replaced by n ln ln lnn on the other hand for lebesgue almost all 3 iets mathcal i t we prove that for all 0 a2 liminf n to infty n epsilon bigl t n x t n y bigr infty quad text for lebesgue a a x y in mathcal i 2 this should be contrasted with the equality lim inf n t n x t n y 0 for a a x y in mathcal i 2 which holds since mathcal i 2 t times t is ergodic because generic 3 iets mathcal i t are weakly mixing we introduce the notion of entropy of an iet which is related to obtaining estimates of type a2 we also prove that no 3 iet is strongly topologically mixing

spectrally arbitrary patterns over rings with unity abstract a zero nonzero pattern a is a matrix with entries from the set 0 a square pattern a is spectrally arbitrary over r a commutative ring with unity if for each n th degree monic polynomial f x r x there exists a matrix a over r with pattern a such that the characteristic polynomial p a x f x a pattern a is relaxed spectrally arbitrary over r if for each n th degree monic polynomial f x r x there exists a matrix a over r with either pattern a or a subpattern of a such that the characteristic polynomial p a x f x we evaluate how the structure of rings affects how we determine if a pattern is spectrally arbitrary we consider whether a pattern a that is spectrally arbitrary over r is spectrally arbitrary or relaxed spectrally arbitrary over another commutative ring s with unity in particular we establish that a pattern that is spectrally arbitrary over z is relaxed spectrally arbitrary over z m for all m z and spectrally arbitrary over q similarly a pattern that is spectrally arbitrary over the p adic integers is spectrally arbitrary over the p adic numbers

extremal sparsity of the companion matrix of a polynomial abstract let c be the companion matrix of a monic polynomial p over a field f we prove that if a is a matrix whose entries are rational functions of the coefficients of p over f and whose characteristic polynomial is p then a has at least as many nonzero entries as c

a multifractal analysis of equilibrium measures for conformal expanding maps and moran like geometric constructions in this paper we establish the complete multifractal formalism for equilibrium measures for holder continuous conformal expanding maps andexpanding markov moran like geometric constructions examples include markov maps of an interval beta transformations of an interval rational maps with hyperbolic julia sets and conformal toral endomorphisms we also construct a holder continuous homeomorphism of a compact metric space with an ergodic invariant measure of positive entropy for which the dimension spectrum is not convex and hence the multifractal formalism fails

ergodic properties of invariant measures for piecewise monotonic transformations during the last decade a lot of research has been done on onedimensional dynamics different kinds of invariant measures for certain classes of piecewise monotonic transformations have been considered in most of these cases the perron frobenius operator plays an important role in this paper we try to unify these different examples which are discussed in detail below first we give a discription of the results proved in this paper

potential well spectrum and hitting time in renewal processes the potential well of a state can be interpreted physically as the energy that a stationary process needs to leave the state we prove that for discrete time renewal processes the potential well is the right scaling for the hitting and return time distributions of the state we further detail the potential well spectrum of these processes by giving a complete classification of the states according to their potential well

hitting time statistics and extreme value theory we consider discrete time dynamical systems and show the link between hitting time statistics the distribution of the first time points land in asymptotically small sets and extreme value theory distribution properties of the partial maximum of stochastic processes this relation allows to study hitting time statistics with tools from extreme value theory and vice versa we apply these results to non uniformly hyperbolic systems and prove that a multimodal map with an absolutely continuous invariant measure must satisfy the classical extreme value laws with no extra condition on the speed of mixing for example we also give applications of our theory to higher dimensional examples for which we also obtain classical extreme value laws and exponential hitting time statistics for balls we extend these ideas to the subsequent returns to asymptotically small sets linking the poisson statistics of both processes

the lip stability and error estimates for a relaxation scheme we show the discrete lip stability for a relaxation scheme proposed by jin and xin comm pure appl math 48 1995 pp 235 277 to approximate convex conservation laws equipped with the lip stability we obtain global error estimates in the spaces w s p for 1 leq s leq 1 p 1 leq p leq infty and pointwise error estimates for the approximate solution obtained by the relaxation scheme the proof uses the framework introduced by nessyahu and tadmor siam j numer anal 29 1992 pp 1505 1519 we also show a maximum principle for the relaxation scheme when the initial data are in an equilibrium state

fast computation of the zeros of a polynomial via factorization of the companion matrix a new fast algorithm for computing the zeros of a polynomial in o n 2 time using o n memory is developed the eigenvalues of the frobenius companion matrix are computed by applying a nonunitary analogue of francis s implicitly shifted qr algorithm to a factored form of the matrix the algorithm achieves high speed and low memory use by preserving the factored form it also provides a residual and an error estimate for each root numerical tests confirm the high speed of the algorithm

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

discrete velocity models of the boltzmann equation a survey on the mathematical aspects of the theory discrete velocity models of the boltzmann equation are of considerable conceptual interest in the kinetic theory of gases and at the same time a fascinating mathematical subject the last decade has seen a lot of work on this subject which is reviewed here we put an emphasis on mathematical aspects in particular the global existence question qualitative properties of solutions explicit solutions and the fluid dynamical limit the mechanical background is discussed more concisely in part i numerous contributions concerning other questions are outlined we try to give a complete reference list up to and including 1985

a note on companion matrices abstract we show that the usual companion matrix of a polynomial of degree n can be factored into a product of n matrices n 1 of them being the identity matrix in which a 2 2 identity submatrix in two consecutive rows and columns is replaced by an appropriate 2 2 matrix the remaining being the identity matrix with the last entry replaced by possibly different entry by a certain similarity transformation we obtain a simple new companion matrix in a pentadiagonal form some generalizations are also possible

a completely solvable model of the nonlinear boltzmann equation abstract in one space and one time dimension a model of the nonlinear boltzmann equation is presented that is exactly solvable for all initial conditions furthermore this model has the following desirable properties i conservation of the number of particles ii energy conservation iii nonlinearity iv positivity of distribution functions and v unique equilibrium state for any given density which is approached as t for most physically interesting initial conditions some of the simple properties of this model are studied

the erdos renyi strong law for pattern matching with a given proportion of mismatches 1 informal introduction the erdos bnyi law erdijs and r6nyi 1970 l for coin tossing is a strong law for the behavior of the length of long success rich runs for the length r of the longest run of pure heads in n tosses the result is that with probability l lin rjlog n 1 where p flheads 0 a more general result is quoted as formula 1 in section 2 for rz the longest head rich run in which the fraction of heads is at least a p note that r ri in many practical situations such as manufacturing or roulette observations are taken sequentially and each can be classified as success or failure for these cases it is possible to use the erdijs wnyi law to test the hypothesis that the success probability is p from another point of view the erdijs bnyi law can be used to recognize patterns of unusually long runs of succe88e8 or failures our interest is in the recognition of unusually long patterns or words common to two random sequences the patterns are unknown prior to an examination of the sequences the motivation for this work is the comparison of dna sequences which can be modeled as sequences of i i d or markov distributed letters evolution operates to conserve although imperfectly patterns important to biological function it is a task of biology to discover these patterns and their function in earlier work we generalized the erdiie bnyi law for pure head runs a 1 to exact matching patterns between two sequences in this paper we extend those results to include matchings of quality a between p and 1 the examples below illustrate the natural analogs of rz studied in this paper two words form a quality a matching if they have the same length and the

numerical schemes for hyperbolic conservation laws with stiff relaxation terms hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a long time behavior governed by reduced systems that are parabolic in nature in this article it is shown by asymptotic analysis and numerical examples that semidiscrete high resolution methods for hyperbolic conservation laws fail to capture this asymptotic behavior unless the small relaxation rate is resolved by a fine spatial grid we introduce a modification of higher order godunov methods that possesses the correct asymptotic behavior allowing the use of coarse grids large cell peclet numbers the idea is to build into the numerical scheme the asymptotic balances that lead to this behavior numerical experiments on 2 2 systems verify our analysis

on the shortest distance between orbits and the longest common substring problem abstract in this paper we study the behaviour of the shortest distance between orbits and show that under some rapidly mixing conditions the decay of the shortest distance depends on the correlation dimension for irrational rotations we prove a different behaviour depending on the irrational exponent of the angle of the rotation for random processes this problem corresponds to the longest common substring problem we extend the result of 6 on sequence matching to mixing processes with exponential decay

critical phenomena for sequence matching with scoring consider two independent sequences x 1 x n and y 1 y n suppose that x 1 x n are i i d x and y 1 y n are i i d y where x and y are distributions on finite alphabets x and y respectively a score f x x y r is assigned to each pair x i y j and the maximal nonaligned segment score is m n max 0 i j n 0 l 1 f x i l y j l our result is that m n log n x y a s with determined by a tractable variational formula moreover the pair empirical measure of x i l y j l during the segment where m n is achieved converges to a probability measure which is accessible by the same formula these results generalize to x i y j taking values in any polish space to intrasequence scores under shifts to long quality segments and to more than two sequences

corrigendum to companion matrix patterns linear algebra appl 463 2014 255 272 abstract we correct an error in the statements of corollary 2 5 and corollary 4 3 of 1

spectrally arbitrary pattern extensions abstract a matrix pattern is often either a sign pattern with entries in 0 or more simply a nonzero pattern with entries in 0 a matrix pattern a is spectrally arbitrary if for any choice of a real matrix spectrum there is a real matrix having the pattern a and the chosen spectrum we describe a graphical technique a triangle extension for constructing spectrally arbitrary patterns out of some known lower order spectrally arbitrary patterns these methods provide a new way of viewing some known spectrally arbitrary patterns as well as providing many new families of spectrally arbitrary patterns we also demonstrate how the technique can be applied to certain inertially arbitrary patterns to obtain larger inertially arbitrary patterns we then provide an additional extension method for zero nonzero patterns patterns with entries in 0

on the spectra of close to schwarz matrices several inverse eigenvalue problems for tridiagonal matrices are discussed in particular it is shown that for every set s of n complex numbers which is symmetric with respect to the real axis there exists a symmetric in modulus tridiagonal n n matrix a which has a nonnegative super diagonal a nonpositive subdiagonal all diagonal elements zero except for a11 which is nonnegative and ann which is nonpositive and with spectrum s 2002 elsevier science inc all rights reserved

spectrally arbitrary patterns abstract an n n sign pattern matrix s is an inertially arbitrary pattern iap if each nonnegative triple n 1 n 2 n 3 with n 1 n 2 n 3 n is the inertia of a matrix with sign pattern s analogously s is a spectrally arbitrary pattern sap if for any given real monic polynomial r x of order n there is a matrix with sign pattern s and characteristic polynomial r x focusing on tree sign patterns consider the n n tridiagonal sign pattern t n with each superdiagonal entry positive each subdiagonal entry negative the 1 1 entry negative the n n entry positive and every other entry zero it is conjectured that t n is an iap by constructing matrices a n with pattern t n it is proved that t n allows any inertia with n 3 0 1 2 n 1 n for all n 2 this leads to a proof of the conjecture for n 5 the truth of the conjecture is extended to n 7 by showing the stronger result that t n is a sap the proof of this latter statement involves finding a matrix a n with pattern t n that is nilpotent further questions about patterns that are saps and iaps are considered

maximal length of common words among random letter sequences on considere des suites de lettres aleatoires basees sur un alphabet fini genere par des processus stationnaires uniformement melangeants on etudie les proprietes des distributions asymptotiques de la longueur du plus grand mot commun

space localization and well balanced schemes for discrete kinetic models in diffusive regimes we derive and study well balanced schemes for quasi monotone discrete kinetic models by means of a rigorous localization procedure we reformulate the collision terms as nonconservative products and solve the resulting riemann problem whose solution is self similar the construction of an asymptotic preserving ap godunov scheme is straightforward and various compactness properties are established within different scalings finally some computational results are supplied to show that this approach is realizable and efficient on concrete 2 2 models

companion matrix patterns abstract companion matrices especially the frobenius companion matrices are used in algorithms for finding roots of polynomials and are also used to find bounds on eigenvalues of matrices in 2003 fiedler introduced a larger class of companion matrices that includes the frobenius companion matrices one property of the combinatorial pattern of these companion matrices is that up to diagonal similarity they uniquely realize every possible spectrum of a real matrix we characterize matrix patterns that have this property and consequently introduce more companion matrix patterns we observe that each fiedler companion matrix is permutationally similar to a unit hessenberg matrix we provide digraph characterizations of the classes of patterns described and in particular all sparse companion matrices noting that there are companion matrices that are not sparse

an extreme value theory for sequence matching on considere des suites finies x 1 x 2 x m et y 1 y 2 y m ou les lettres x i et y i sont choisies i i d sur un alphabet denombrable avec p p x 1 y 1 0 1 on etudie la distribution de la plus longue sequence contigue de couples entre les x et les y avec au plus k mauvais couplages

condition numbers for inversion of fiedler companion matrices abstract the fiedler matrices of a monic polynomial p z of degree n are n n matrices with characteristic polynomial equal to p z and whose nonzero entries are either 1 or minus the coefficients of p z fiedler matrices include as particular cases the classical frobenius companion forms of p z frobenius companion matrices appear frequently in the literature on control and signal processing but it is well known that they posses many properties that are undesirable numerically which limit their use in applications in particular as n increases frobenius companion matrices are often nearly singular i e their condition numbers for inversion are very large therefore it is natural to investigate whether other fiedler matrices are better conditioned than the frobenius companion matrices or not in this paper we present explicit expressions for the condition numbers for inversion of all fiedler matrices with respect the frobenius norm i e a f ij a ij 2 this allows us to get a very simple criterion for ordering all fiedler matrices according to increasing condition numbers and to provide lower and upper bounds on the ratio of the condition numbers of any pair of fiedler matrices these results establish that if p 0 1 then the frobenius companion matrices have the largest condition number among all fiedler matrices of p z and that if p 0 1 then the frobenius companion matrices have the smallest condition number we also provide families of polynomials where the ratio of the condition numbers of pairs of fiedler matrices can be arbitrarily large and prove that this can only happen when both fiedler matrices are very ill conditioned we finally study some properties of the singular values of fiedler matrices and determine how many of the singular values of a fiedler matrix are equal to one

an erdos renyi law with shifts abstract motivated by the comparison of dna sequences a generalization is given of the result of erdos and renyi on the length r n of the longest run of heads in the first n tosses of a coin consider two sequences x 1 x 2 x n and y 1 y 2 y n the length of the longest matching consecutive subsequence allowing shifts is m n max m x i k y j k for k 1 to m for some 0 i j n m suppose that all the letters are independent and identically distributed the length of the longest match without shifts has the same distribution as r n the length of the longest head run for a biased coin with p p x i y i described by the erdos renyi law p lim n r n log 1 p n 1 1 for matching with shifts our result is p lim n m n log 1 p n 2 1 loosely speaking allowing shifts doubles the length of the longest match the case of markov chains is also handled

laws of rare events for deterministic and random dynamical systems the object of this paper is twofold from one side we study the dichotomy in terms of the extremal index of the possible extreme value laws when the rare events are centred around periodic or non periodic points then we build a general theory of extreme value laws for randomly perturbed dynamical systems we also address in both situations the convergence of rare events point processes decay of correlations against l 1 observables will play a central role in our investigations

potentially nilpotent sign pattern matrices abstract a sign pattern is said to be potentially nilpotent if it allows nilpotence in this paper a number of qualitative necessary or sufficient conditions for a sign pattern to allow nilpotence are established the sign patterns that allow nilpotence of index 2 are investigated for orders up to three potentially nilpotent sign patterns are characterized potentially nilpotent tree sign patterns are also explored

the continuing influence of fiedler s work on companion matrices this is a reconstruction in article like form of a talk given at the minisymposium in honor of miroslav fiedler at the 17th ilas conference held at tu braunschweig germany on thurs 25 aug 2011

uniformly accurate schemes for hyperbolic systems with relaxation we develop high resolution shock capturing numerical schemes for hyperbolic systems with relaxation in such systems the relaxation time may vary from order 1 to much less than unity when the relaxation time is small the relaxation term becomes very strong and highly stiff and underresolved numerical schemes may produce spurious results usually one cannot decouple the problem into separate regimes and handle different regimes with different methods thus it is important to have a scheme that works uniformly with respect to the relaxation time using the broadwell model of the nonlinear boltzmann equation we develop a second order scheme that works effectively with a fixed spatial and temporal discretization for all ranges of the mean free path formal uniform consistency proof for a first order scheme and numerical convergence proof for the second order scheme are also presented we also make numerical comparisons of the new scheme with some other schemes this study is motivated by the reentry problem in hypersonic computations

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

convergence to equilibrium for the relaxation approximations of conservation laws we study the cauchy problem for 2 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term special comparison and compactness properties are established by assuming the subcharacteristic condition therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to 0 this research was strongly motivated by the recent numerical investigations of s jin and z xin on the relaxation schemes for conservation laws 1996 john wiley sons inc

critical phenomena in sequence matching we give a generalization of the result of erdos and renyi on the length r n of the longest head run in the first n tosses of a coin consider two independent sequences x 1 x 2 cdots x m and y 1y 2 cdots y n suppose that x 1 x 2 cdots are i i d mu and y 1 y 2 cdots are i i d nu where mu and nu are possibly different distributions on a common finite alphabet s let p equiv p x 1 y 1 in 0 1 the length of the longest matching consecutive subsequence is m m n equiv max k x i r y j r for r 1 to k for some 0 leq i leq m k 0 leq j leq n k for m and n rightarrow infty with log m log mn rightarrow lambda in 0 1 our result is that there is a constant k equiv k mu nu lambda in 0 1 rbrack such that p lim m m n log 1 p mn k 1 the proof uses large deviation methods the constant k is determined from a variational formula involving the kullback liebler distance or relative entropy a simple necessary and sufficient condition for k 1 is given for the case m n lambda 1 2 and mu nu k 1 the set of mu nu lambda for which k 1 has nonempty interior the boundary of this set is the location of a phase transition the results generalize to more than two sequences and to markov chains a strong law of large numbers is given for the proportion of letters within the longest matching word the limiting proportion exhibits critical behavior similar to that of k