level spacing distributions and the bessel kernel scaling models of randomn n hermitian matrices and passing to the limitn leads to integral operators whose fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order for the gaussian unitary ensemble and for many others as well the kernel one obtains by scaling in the bulk of the spectrum is the sine kernel frac sin pi x y pi x y rescaling the gue at the edge of the spectrum leads to the kernel frac ai x ai y ai x ai y x y where ai is the airy function in previous work we found several analogies between properties of this airy kernel and known properties of the sine kernel a system of partial differential equations associated with the logarithmic differential of the fredholm determinant when the underlying domain is a union of intervals a representation of the fredholm determinant in terms of a painleve transcendent in the case of a single interval and also in this case asymptotic expansions for these determinants and related quantities achieved with the help of a differential operator which commutes with the integral operator in this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the laguerre or jacobi ensembles at the edge of the spectrum namely frac j alpha sqrt x sqrt y j alpha sqrt y sqrt x j alpha sqrt x j alpha sqrt y 2 x y wherej z is the bessel function of order in the cases 1 2 these become after a variable change the kernels which arise when taking scaling limits in the bulk of the spectrum for the gaussian orthogonal and symplectic ensembles in particular an asymptotic expansion we derive will generalize ones found by dyson for the fredholm determinants of these kernels

partially implicit schemes for the numerical solutions of some non linear differential equations as numerical methods are one of the main tools used to discover the non linear behaviour characteristics of continuous dynamical systems a study on the use of numerical schemes which delineate the true physics from numerical artefact is desirable as a step in that direction partially implicit rational schemes based on first order methods and a second order method are derived and analysed from the point of view of dynamical systems for a scalar autonomous ordinary differential equation the second order method is developed by taking a linear combination of first order methods though implicit in nature the methods are applied explicitly the approach adopted is extended to obtain numerical schemes for the solution of a class of reaction diffusion partial differential equations the methods are tested on differential equations with cubic and quadratic reaction terms it is found that for certain sets of initial conditions the first order partially implicit schemes converge monotonically to the correct fixed points for all time steps unlike the euler method and the second order method in other words for the same steady state solution the associated domain of attraction of the discrete problem and its continuous counterpart are exactly the same however for other initial conditions it is found that the domains of attraction of the second order method mimic more closely the domains of attraction of the continuous differential equations the comparative study of the various explicit runge kutta methods of orders 2 6 shows that it is the discretized form with its specific algebraic form recursive relation which are the source of spurious steady state solutions and numerical chaos from these methods it is found however that the use in general of fifth order runge kutta methods is seen to be preferred to other runge kutta methods as they have an extended stability range in particular the fifth order lawson method which has a far superior range of stability than the other runge kutta methods

upper semicontinuity of attractors for linear multistep methods approximating sectorial evolution equations this paper sets out a theoretical framework for approximating the attractor a of a semigroup s t defined on a banach space x by a q step semidiscretization in time with constant steplength k using the one step theory of hale lin and raugel sufficient conditions are established for the existence of a family of attractors a k x q for the discrete semigroups s k n defined by the numerical method the convergence properties of this family are also considered full details of the theory are exemplified in the context of strictly a stable linear multistep approximations of an abstract dissipative sectorial evolution equation

runge kutta solutions of a hyperbolic conservation law with source term spurious long term solutions of a finite difference method for a hyperbolic conservation law with a general nonlinear source term are studied results are contrasted with those that have been established for nonlinear ordinary differential equations various types of spurious behavior are examined including spatially uniform equilibria that exist for arbitrarily small time steps nonsmooth steady states with profiles that jump between fixed levels and solutions with oscillations that arise from nonnormality and exist only in finite precision arithmetic it appears that spurious behavior is associated in general with insufficient spatial resolution the potential for curbing spuriosity by using adaptivity in space or time is also considered

computational chaos a prelude to computational instability abstract chaotic behavior sometimes occurs when difference equations used as approximations to ordinary differential equations are solved numerically with an excessively large time increment in two simple examples we find that as increases chaos first sets in when attractor a acquires two distinct points that map to the same point this happens when a acquires slopes of the same sign in a rectifying coordinate system at two consecutive intersections with the critical curve chaotic and quasi periodic behavior may then alternate within a range of before computational instability finally prevails bifurcations to and from chaos and transitions to computational instability are highly scheme dependent even among differencing schemes of the same order systems exhibiting computational chaos can serve as illustrative examples in more general studies of noninvertible mappings

large deviation eigenvalue density for the soft edge laguerre and jacobi β ensembles we analyze the eigenvalue density for the laguerre and jacobi ensembles in the cases that the corresponding exponents are extensive in particular we obtain the asymptotic expansion up to terms o 1 in the large deviation regime outside the limiting interval of support as found in recent studies of the large deviation density for the gaussian ensemble and laguerre ensemble with fixed exponent there is a scaling from this asymptotic expansion to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge

equilibria of runge kutta methods it is known that certain runge kutta methods share the property that in a constant step implementation if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system such methods are calledregular in the present paper we provide a recursive test to check whether given method is regular moreover by examining solution trajectories of linear equations we prove that the order of ans stage regular method may not exceed 2 s 2 2 and that the maximal order of regular runge kutta method with an irreducible stability function is 4

universality of local eigenvalue statistics for some sample covariance matrices we consider random complex sample covariance matrices 1 x x where x is a p n random matrix with i i d entries of distribution it has been conjectured that both the distribution of the distance between nearest neighbor eigenva lues in the bulk and that of the smallest eigenvalues become in the limit n p n 1 the same as that identified for a complex gaussian distribution we prove these conjectures for a certain class of probability distributions c 2004 wiley periodicals inc

exact interpolation and iterative subdivision schemes we examine the circumstances under which a discrete time signal can be exactly interpolated given only every mth sample after pointing out the connection between designing an m fold interpolator and the construction of an m channel perfect reconstruction filter bank we derive necessary and sufficient conditions on the signal under which exact interpolation is possible bandlimited signals are one obvious example but numerous others exist we examine these and show how the interpolators may be constructed a main application is to iterative interpolation schemes used for the efficient generation of smooth curves we show that conventional bandlimited interpolators are not suitable in this context a better criterion is to use interpolators that are exact for polynomial functions we demonstrate that these interpolators converge when iterated and show how these may be designed for any polynomial degree n and any interpolation factor m this makes it possible to design interpolators for iterative schemes to make best use of the resolution available in a given display medium

regularity properties of one leg methods for delay differential equations abstract the main purpose of this paper is to investigate the asymptotic states of one leg methods for multidelay differential equations in particular the existence of spurious steady solutions and period 2 solutions in constant or variable timestep is studied and the concepts of r 1 regularity and r 2 regularity of one leg methods for multidelay differential equations are introduced and studied some conditions guaranteeing r 1 regularity and r 2 regularity of such methods applied to multidelay differential equations with some important structures are given

discrete hopf bifurcation for runge kutta methods an investigation into how well real bifurcations in the family of dynamical systems are approximated as the step size varies is carried out the preservation of bifurcation structures and stability under numerical simulations is discussed in addition the behaviour of numerical solutions generated by a runge kutta method applied to a dynamical system whose analytical solution undergoes a hopf bifurcation is investigated hopf bifurcation results for the numerical solution are presented and analysed

eigenvalue density of cross correlations in sri lankan financial market we apply the universal properties with gaussian orthogonal ensemble goe of random matrices namely spectral properties distribution of eigenvalues eigenvalue spacing predicted by random matrix theory rmt to compare cross correlation matrix estimators from emerging market data the daily stock prices of the sri lankan all share price index and milanka price index from august 2004 to march 2005 were analyzed most eigenvalues in the spectrum of the cross correlation matrix of stock price changes agree with the universal predictions of rmt we find that the cross correlation matrix satisfies the universal properties of the goe of real symmetric random matrices the eigen distribution follows the rmt predictions in the bulk but there are some deviations at the large eigenvalues the nearest neighbor spacing and the next nearest neighbor spacing of the eigenvalues were examined and found that they follow the universality of goe rmt with deterministic correlations found that each eigenvalue from deterministic correlations is observed at values which are repelled from the bulk distribution

theory of regular m band wavelet bases orthonormal m band wavelet bases have been constructed and applied by several authors this paper makes three main contributions first it generalizes the minimal length k regular 2 band wavelets of daubechies 1988 to the m band case by deriving explicit formulas for k regular m band scaling filters several equivalent characterizations of k regularity are given and their significance explained second two approaches to the construction of the m 1 wavelet filters and associated wavelet bases are described one relies on a state space characterization with a novel technique to obtain the unitary wavelet filters the other uses a factorization approach third this paper gives a set of necessary and sufficient condition on the m band scaling filter for it to generate an orthonormal wavelet basis the conditions are very similar to those obtained by cohen 1990 and lawton 1990 for 2 band wavelets

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

a non iterative algorithm based on richardson s extrapolation application to groundwater flow modelling in this report a promising third order algorithm based on richardson s extrapolation and crank nicholson s method is studied the algorithm having a wide scope for applications was initially intended to be used for time integration on non linear advection diffusion problems then stability low oscillations and low additional diffusion become important numerical properties it is shown that this method using three single steps for extrapolation is very close to be unconditionally stable under the previous requirements it behaves essentially as well as crank nicholson s scheme but with a better performance at least in relation to spurious oscillations and perhaps not so good for dispersion properties in addition the method is third order accurate and provides useful information related to time integration error to be used for time step control these features indicate that the algorithm is a good candidate to efficiently solve problems with significant variations in time additionally its non iterative version saves important amounts of cpu time specially for applied 3d modelling to illustrate the good performance of the algorithm we include several tests comparing it with crank nicholson s scheme and a 2d application to variably saturated groundwater flow modelling

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

nonnormality effects in a discretised nonlinear reaction convection diffusion equation what is the long time effect of adding convention to a discretised reaction diffusion equation for linear problems it is well known that convection may denormalise the process and in particular eigenvalue based stability predictions may be overoptimistic this work deals with a related issue with a nonlinear reaction term the nonnormality can greatly influence the long time dynamics for a nonlinear model problem with dirichlet boundary conditions it is shown that the basin of attraction of the correct steady state can be shrunk in a directionally biased manner a normwise analysis provides lower bounds on the basin of attraction and a more revealing picture is provided by pseudo eigenvalues in extreme cases the computed solution can converge to a spurious bounded steady state that exists only in finite precision arithmetic the impact of convection on the existence and stability of spurious periodic solutions is also quantified

the characterization of continuous four coefficient scaling functions and wavelets four coefficient dilation equations are examined and results converse to a theorem of daubechies lagarias are given these results complete the characterization of those four coefficient dilation equations having a continuous solution

induced measures in the space of mixed quantum states we analyse several product measures in the space of mixed quantum states in particular we study measures induced by the operation of partial tracing the natural rotationally invariant measure on the set of all pure states of a n k composite system induces a unique measure in the space of n n mixed states or in the space of k k mixed states if the reduction takes place with respect to the first subsystem for k n the induced measure is equal to the hilbert schmidt measure which is shown to coincide with the measure induced by singular values of non hermitian random gaussian matrices pertaining to the ginibre ensemble we compute several averages with respect to this measure and show that the mean entanglement of n n pure states behaves as lnn 1 2

necessary and sufficient conditions for constructing orthonormal wavelet bases this paper proves a previous conjecture of the author characterizing sequences h l2 z that yield orthonormal wavelet bases of l2 r in terms of the multiplicity of the eigenvalue 1 of an operator associated to h the proof utilizes a result of cohen characterizing these sequences in terms of the real zeros of their fourier transforms the mapping from sequences to wavelets is shown to define a continuous mapping from a subset of l2 z into l2 r related conjectures are discussed

does error control suppress spuriosity in the numerical solution of initial value ordinary differential equations to what extent does local error control confer global properties this work concentrates on global steady states or fixed points it is shown that for systems of equations spurious fixed points generally cease to exist when local error control is used for scalar problems on the other hand locally adaptive algorithms generally avoid spurious fixed points by an indirect method the stepsize selection process causes spurious fixed points to be unstable however problem classes exist where for arbitrarily small tolerances stable spurious fixed points persist with significant basins of attraction a technique is derived for generating such examples

the optimal coefficients in daubechies wavelets abstract the construction of wavelets begins with a dilation equation x ck 2x k the choice of the coefficients ck controls the accuracy and orthogonality of wavelet expansions we show how the daubechies coefficients which give pth order accuracy with 2p coefficients the minimum number follow from orthogonality and accuracy conditions applied to the polynomial ckeik

random matrix ensembles with an effective extensive external charge recent theoretical studies of chaotic scattering have encountered ensembles of random matrices in which the eigenvalue probability density function contains a one body factor with an exponent proportional to the number of eigenvalues two such ensembles have been encountered an ensemble of unitary matrices specified by the so called poisson kernel and the laguerre ensemble of positive definite matrices here we consider various properties of these ensembles jack polynomial theory is used to prove a reproducing property of the poisson kernel and a certain unimodular mapping is used to demonstrate that the variance of a linear statistic is the same as in the dyson circular ensemble for the laguerre ensemble the scaled global density is calculated exactly for all even values of the parameter while for random matrices with unitary symmetry the neighbourhood of the smallest eigenvalue is shown to be in the soft edge universality class

spurious behavior of a symplectic integrator in this article we study the existence and behavior of spurious solutions of symplectic euler method for some hamiltonian systems it is shown that the symplectic integrator applied to hamiltonian systems in general doesn t avoid spurious behavior even spurious period two solutions the numerical results are presented

contractibility of compact contractions in hilbert space for a finite set of compact contractions in a complex hilbert space h it is shown that r a 1 for all a in the multiplicative semigroup generated by if and only if there exists a positive integer n such that a 1 for all a in the multiplicative semigroup generated by with length greater than n here r a denotes the spectral radius of a as an application an answer is given to an infinite dimensional case of the finiteness conjecture for the generalized spectral radius attributed to j c lagarias and y wang linear algebra appl 214 1995 17

preservation of hopf bifurcation for neutral delay differential equations by θ methods this paper considers the preservation of hopf bifurcation of some neural delay differential equations by q method by analyzing the dynamics of the numerical discrete system derived by q method we show that q method could inherit the hopf bifurcation and the asymptotical stability for sufficiently small stepsize h 1 m where m is a positive integer in particular for q 1 2 the result holds for any stepsize h 1 m furthermore the stability of the closed invariant curve is established finally some numerical examples are illustrated to support the analytic results

new iterative improvement of a solution for an ill conditioned system of linear equations based on a linear dynamic system abstract in this paper the analysis of the dynamic system for iterative improvement of a solution is discussed and the new iterative improvement of a solution is proposed based on the dynamic system we have proved that the new iterative improvement of the solution is convergent unconditionally the numerical experiments illustrate that the new iterative improvement of the solution is more effective for an ill conditional system of linear equations

application of the τ function theory of painleve equations to random matrices pv piii the lue jue and cue with denoting an average with respect to the eigenvalue pdf for the laguerre unitary ensemble the object of our study is e n i a x l i l 0 i l for i 0 s and i s where x i l 1 for l i and x i l 0 other wise using okamoto s development of the theory of the painleve v equation it is shown that e n i a is a function associated with the hamiltonian therein and so can be characterized as the solution of a certain second order second degree differential equation or in terms of the solution of certain difference equations the cases 0 and 2 are of particular interest because they correspond to the cumulative distribution and density function respectively for the smallest and largest eigenvalue in the case i s e n i a is simply related to an average in the jacobi unitary ensemble and this in turn is simply related to certain averages over the orthogonal group the unitary symplectic group and the circular unitary ensemble the latter integrals are of interest for their combinatorial content also considered are the hard edge and soft edge scaled limits of e n i a in particular in the hard edge scaled limit it is shown that the limiting quantity e hard 0 s a can be evaluated as a function associated with the hamiltonian in okamoto s theory of the painleve iii equation

sets of matrices all infinite products of which converge an infinite product iit lmi of matrices converges on the right if limi m mi exists a set z ai i l of n x n matrices is called an rcp set rightconvergent product set if all infinite products with each element drawn from z converge such sets of matrices arise in constructing self similar objects like von koch s snowflake curve in various interpolation schemes in constructing wavelets of compact support and in studying nonhomogeneous markov chains this paper gives necessary conditions and also some sufficient conditions for a set x to be an rcp set these are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices necessary and sufficient conditions are given for a finite set z to be an rcp set having a limit function m d rit lad where d d d which is a continuous function on the space of all sequences d with the sequence topology finite rcp sets of column stochastic matrices are completely characterized some results are given on the problem of algorithmically deciding if a given set x is an rcp set

renormalizing rectangles and other topics in random matrix theory we consider random hermitian matrices made of complex or realm n rectangular blocks where the blocks are drawn from various ensembles these matrices haven pairs of opposite real nonvanishing eigenvalues as well asm n zero eigenvalues form n these zero eigenvalues are kinematical in the sense that they are independent of randomness we study the eigenvalue distribution of these matrices to leading order in the large n m limit in which the rectangularity r m n is held fixed we apply a variety of methods in our study we study gaussian ensembles by a simple diagrammatic method by the dyson gas approach and by a generalization of the kazakov method these methods make use of the invariance of such ensembles under the action of symmetry groups the more complicated wigner ensemble which does not enjoy such symmetry properties is studied by large n renormalization techniques in addition to the kinematical function spike in the eigenvalue density which corresponds to zero eigenvalues we find for both types of ensembles that if r 1 is held fixed asn then nonzero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin this separation arises because the nonzero eigenvalues are repelled macroscopically from the origin finally we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes a behavior governed by airy functions asr 1 the lobes come closer and the airy oscillatory behavior near the endpoints that are close to zero breaks down we interpret this breakdown as a signal thatr 1 drives a crossover to the oscillation governed by bessel functions near the origin for matrices made of square blocks

uniform refinement of curves abstract we propose and analyze a class of algorithms for the generation of curves and surfaces these algorithms encompass some well known methods of subdivision for bernstein bezier curves de casteljau s algorithm and b spline curves lane and riesenfeld s algorithm several results concerning properties of the limiting curves as well as related questions are discussed

the calogero sutherland model and generalized classical polynomials multivariable generalizations of the classical hermite laguerre and jacobi polynomials occur as the polynomial part of the eigenfunctions of certain schrodinger operators for calogero sutherland type quantum systems for the generalized hermite and laguerre polynomials the multidimensional analogues of many classical results regarding generating functions differentiation and integration formulas recurrence relations and summation theorems are obtained we use this and related theory to evaluate the global limit of the ground state density obtaining in the hermite case the wigner semi circle law and to give an explicit solution for an initial value problem in the hermite and laguerre case

string scattering from decaying branes we develop the general formalism of string scattering from decaying d branes in bosonic string theory in worldsheet perturbation theory amplitudes can be written as a sum of correlators in a grand canonical ensemble of unitary random matrix models with time setting the fugacity an approach employed in the past for computing amplitudes in this theory involves an unjustified analytic continuation from special integer momenta we give an alternative formulation which is well defined for general momenta we study the emission of closed strings from a decaying d brane with initial conditions perturbed by the addition of an open string vertex operator using an integral formula due to selberg the relevant amplitude is expressed in closed form in terms of zeta functions perturbing the initial state can suppress or enhance the emission of high energy closed strings for extended branes but enhances it for d0 branes the closed string two point function is expressed as a sum of toeplitz determinants of certain hypergeometric functions a large n limit theorem due to szego and its extension due to borodin and okounkov permits us to compute approximate results showing that previous naive analytic continuations amount to the large n approximation of the full result we also give a free fermion formulation of scattering from decaying d branes and describe the relation to a grand canonical ensemble for a 2d coulomb gas

c1 continuity of the generalized four point scheme abstract we show that the interpolatory four point scheme with tension parameter generates c 1 limit curves if and only if 0 where 0 19273 is the unique real solution of the cubic equation 32 3 4 1 0

the finiteness conjecture for the generalized spectral radius of a set of matrices the generalized spectral radius g9 of a set of n n matrices is g9 lim supk g9k 1k where g9k sup a1a2 ak each ai the joint spectral radius g9 is g9 lim supk g9k 1k where g9k sup a1 ak each ai it is known that g9 g9 holds for any finite set of n n matrices the finiteness conjecture asserts that for any finite set of real n n matrices there exists a finite k such that g9 g9 g9k 1k the normed finiteness conjecture for a given operator norm asserts that for any finite set a1 am having all ai op 1 either g9 1 or g9 g9 g9k 1k 1 for some finite k it is shown that the finiteness conjecture is true if and only if the normed finiteness conjecture is true for all operator norms the normed finiteness conjecture is proved for a large class of operator norms extending results of gurvits in particular for polytope norms and for the euclidean norm explicit upper bounds are given for the least k having g9 g9k 1k these results imply upper bounds for generalized critical exponents for these norms

unitary integrators and applications to continuous orthonormalization techniques in this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed such systems have skew hermitian coefficient matrices in the linear case and a

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

a unified approach to spurious solutions introduced by time discretization part i basic theory the asymptotic states of numerical methods for initial value problems are examined in particular spurious steady solutions solutions with period 2 in the timestep and spurious invariant curves

euler like discrete models of the logistic differential equation abstract to understand the behavior of difference schemes on nonlinear differential equations it seems desirable to extend the standard linear stability theory into a nonlinear theory as a step in that direction we investigate the stability properties of euler related integration algorithms by checking how they preserve and violate the dynamical structure of the logistic differential equation among the schemes considered are two linearly implicit nonstandard schemes which are adjoint to each other we find that these schemes are superior to explicit schemes when they are stable and the blow up time has not passed for these h values they are dynamically faithful when these schemes turn unstable however they have much less desirable properties than explicit or fully implicit schemes they become simultaneously superstable and unstable this is explained by the fact that these schemes are not self adjoint the linearly implicit self adjoint scheme is dynamically faithful in an euler typical range of step sizes and gives correct stability for all step sizes

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

computing the joint spectral radius this paper presents algorithms for finding an arbitrarily small interval that contains the joint spectral radius of a finite set of matrices it also presents a numerical criterion for verifying in certain cases that the joint spectral radius is the maximum of the spectral radii of the given matrices error bounds are derived for the case where calculations are done with finite precision and the matrices are not known exactly the algorithms are implemented and applied to estimate holder exponents of the orthonormal wavelets n constructed by daubechies for 3 n 8

nonlinear instability in dissipative finite difference schemes a unified analysis of reaction diffusion equations and their finite difference representations is presented the parallel treatment of the two problems shows clearly when and why the finite difference approximations break down the approach used provides a general framework for the analysis and interpretation of numerical instability in approximations of dissipative nonlinear partial differential equations continuous and discrete problems are studied from the perspective of bifurcation theory and numerical instability is shown to be associated with the bifurcation of periodic orbits in discrete systems an asymptotic approach due to newell siam j appl math 33 1977 133 160 is used to investigate the instability phenomenon further in particular equations are derived that describe the interaction of the dynamics of the partial differential equation with the artefacts of the discretization

on spurious fixed points of runge kutta methods in this paper we investigate the onset of spurious fixed points when runge kutta methods are applied to study the dynamics of differential equations it is shown computationally that the spurious equilibria of griffithset al 14 are connected at infinity with fixed points inherited from the differential equation we introduce and study the concept of b regularity which is in connection to the concept of regularity introduced by iserles principium cuius hinc nobis exordia sumet nullam rem e nilo gigni diuinitus umquam nam si de nilo fierent ex omnibus rebus omne genus nasci posset nil semine egeret de rerum natura liber primus 149 lucrecio

universality in invariant random matrix models existence near the soft edge we consider two non gaussian ensembles of large hermitian random matrices with strong level confinement and show that near the soft edge of the spectrum both scaled density of states and eigenvalue correlations follow so called airy laws inherent in gaussian unitary ensemble this suggests that the invariant one matrix models should display universal eigenvalue correlations in the soft edge scaling limit

exact results and universal asymptotics in the laguerre random matrix ensemble for various values of the parameters in the laguerre random matrix ensemble the distribution of the smallest eigenvalue and the scaled n level distribution function are calculated exactly in terms of generalized hypergeometric functions in certain cases these functions are expressed as multidimensional integrals from which asymptotic formulas are calculated and predictions of universal behavior verified

wavelets and quantum algebras wavelets known to be useful in nonlinear multiscale processes and in multiresolution analysis are shown to have a q deformed algebraic structure the translation and dilation operators associate to any scaling equation a nonlinear two parameter algebra this structure can be mapped onto the quantum group slq 2 in one limit and approaches a fourier series generating algebra in another limit a duality between any scaling function and its corresponding nonlinear algebra is obtained examples for the haar and b wavelets are worked out in detail

nonlinear diffusion and stable period 2 solutions of a discrete reaction diffusion model abstract a discrete form of a nonlinear parabolic equation is considered this equation is in fact a model for migrating populations and contains a quadratic reaction term and a diffusion term which can be either linear for nonlinear attention is focused on the case of nonlinera diffusion and in particular period 2 solutions of the discrete version conditions for the existence and stability of two types of period 2 solutions are determined analytically these analytical results which are local in the neighborhood of fixed points are compared with the results of some numerical experiments showing that one of these period 2 solutions is a considerably stronger attractor than the other