combined sequence of markov parameters and moments in linear systems the combined sequence of markov parameters and moments in linear systems is discussed this sequence is shown to be generated by a linear feedback shift register application to the minimal degree partial realization from a mixture of any number of markov parameters and moments is considered central in this particular realization problem is the extension of a toeplitz partial behavior matrix this is accomplished on the basis of the algebraic theory of iohvidov indexes for toeplitz matrices

vicious walkers and hook young tableaux we consider a generalization of the vicious walker model using a bijection map between the path configuration of the non intersecting random walkers and the hook young diagram we compute the probability concerning the number of movements of the walker applying the saddle point method we reveal that the scaling limit gives the tracy widom distribution which is the same with the limit distribution of the largest eigenvalues of the gaussian unitary ensemble

non intersecting paths random tilings and random matrices we investigate certain measures induced by families of non intersecting paths in domino tilings of the aztec diamond rhombus tilings of an abc hexagon a dimer model on a cylindrical brick lattice and a growth model the measures obtained e g the krawtchouk and hahn ensembles have the same structure as the eigenvalue measures in random matrix theory like gue which can in fact can be obtained from non intersecting brownian motions the derivations of the measures are based on the karlin mcgregor or lindstrom gessel viennot method we use the measures to show some asymptotic results for the models

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

fast inversion of triangular toeplitz matrices an algorithm requiring 0 n log n operations is proposed for the inversion of triangular toeplitz matrices this algorithm is faster than existing techniques even for relatively small values of n and can easily be written in fortran

multidomain spectral solution of the euler gas dynamics equations abstract we present interface treatments for computing the euler gas dynamics equations by a multidomain chebyshev spectral collocation method these treatments are suitable for use at the corners of subdomains and for patched or overlapping subdomains they can be applied to interfaces placed in subsonic supersonic or transonic regions of a flow computed results indicate that the solutions are spectrally accurate can be more accurate and more efficient than a single domain calculation and that reflections of waves at interfaces are not significant

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

a new algorithm for solving toeplitz systems of equations abstract we present some recurrences that are the basis for an algorithm to invert an n n toeplitz system of linear equations with computational complexity o n log 2 n the recurrences used are closely related to the zohar trench and bareiss algorithms but do not have any obvious connection with other asymptotically fast algorithms for the inversion of toeplitz systems

a generalization of the levinson algorithm for hermitian toeplitz matrices with any rank profile the paper describes a recursive algorithm for solving hermitian toeplitz systems of linear equations without any restriction on the ranks of their nested toeplitz subsystems such a general algorithm is needed e g to obtain the eigenfilters for signal processing applications or to compute the inverse of a nondefinite toeplitz matrix the regular portion of the algorithm is made of the classical levinson recursion the singular portion requires solving some well defined systems of linear equations with gradient structure the dimension of each of these sytems equals the amplitude of the corresponding singularity

accuracy resolution and stability properties of a modified chebyshev method while the chebyshev pseudospectral method provides a spectrally accurate method integration of partial differential equations with spatial derivatives of order m requires time steps of approximately o n 2m for stable explicit solvers theoretically time steps may be increased to o n m with the use of a parameter alpha dependent mapped method introduced by kosloff and tal ezer em j comput phys 104 1993 pp 457 469 our analysis focuses on the utilization of this method for reasonable practical choices for n namely n lesssim 30 as may be needed for two or three dimensional modeling results presented confirm that spectral accuracy with increasing n is possible both for constant alpha hesthaven dinesen and lynov em j comput phys 155 1999 pp 287 306 and for alpha scaled with n alpha sufficiently different from 1 don and solomonoff em siam j sci comput 18 1997 pp 1040 1055 theoretical bounds however show that any realistic choice for alpha in which both resolution and accuracy considerations are imposed permits no more than a doubling of the time step for a stable explicit integrator in time much less than the o n improvement claimed by kosloff and tal ezer on the other hand by choosing alpha carefully it is possible to improve on the resolution of the chebyshev method in particular one may achieve satisfactory resolution with fewer than pi points per wavelength moreover this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about 8 points per wavelength our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one dimensional wave equations specifically while alpha can be chosen such that the mapped method improves the accuracy and resolution of the chebyshev method for practical choices of n it is not possible to achieve both single precision accuracy and gain the advantage of an o n m time step

score statistics of global sequence alignment from the energy distribution of a modified directed polymer and directed percolation problem sequence alignment is one of the most important bioinformatics tools for modern molecular biology the statistical characterization of gapped alignment scores has been a long standing problem in sequence alignment research using a variant of the directed path in random media model we investigate the score statistics of global sequence alignment taking into account in particular the compositional bias of the sequences compared such statistics are used to distinguish accidental similarity due to compositional similarity from biologically significant similarity to accommodate the compositional bias we introduce an extra parameter p indicating the probability for positive matching scores to occur when p is small a high scoring alignment obviously cannot come from compositional similarity when p is large the highest scoring point within a global alignment tends to be close to the end of both sequences in which case we say the system percolates by applying finite size scaling theory on percolating probability functions of various sizes sequence lengths the critical p at infinite size is obtained for alignment of length t the fact that the score fluctuation grows as chi t 1 3 is confirmed upon investigating the scaling form of the alignment score using the kolmogorov smirnov statistics test we show that the random variable if properly scaled follows the tracy widom distributions gaussian orthogonal ensemble for p slightly larger than pc and gaussian unitary ensemble for larger p although these results deepen our understanding of the distribution of alignment scores the use of these results in practical applications remains somewhat heuristic and needs to be further developed nevertheless the possibility of characterizing score statistics for modest system size sequence lengths via proper reparametrization of alignment scores is illustrated

the asymptotics of monotone subsequences of involutions we compute the limiting distributions of the lengths of the longest monotone subsequences of random signed involutions with or without conditions on the number of fixed points and negated points as the sizes of the involutions tend to infinity the resulting distributions are depending on the number of fixed points 1 the tracy widom distributions for the largest eigenvalues of random goe gue gse matrices 2 the normal distribution or 3 new classes of distributions which interpolate between pairs of the tracy widom distributions we also consider the second rows of the corresponding young diagrams in each case the convergence of moments is also shown the proof is based on the algebraic work of j baik and e rains in 7 which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the riemann hilbert analysis for the orthogonal polynomials by p deift k johansson and baik in 3

stable multi domain spectral penalty methods for fractional partial differential equations we propose stable multi domain spectral penalty methods suitable for solving fractional partial differential equations with fractional derivatives of any order first a high order discretization is proposed to approximate fractional derivatives of any order on any given grids based on orthogonal polynomials the approximation order is analyzed and verified through numerical examples based on the discrete fractional derivative we introduce stable multi domain spectral penalty methods for solving fractional advection and diffusion equations the equations are discretized in each sub domain separately and the global schemes are obtained by weakly imposed boundary and interface conditions through a penalty term stability of the schemes are analyzed and numerical examples based on both uniform and nonuniform grids are considered to highlight the flexibility and high accuracy of the proposed schemes

scale invariance of the png droplet and the airy process we establish that the static height fluctuations of a particular growth model the png droplet converges upon proper rescaling to a limit process which we call the airy process a y the airy process is stationary it has continuous sample paths its single time fixed y distribution is the tracy widom distribution of the largest eigenvalue of a gue random matrix and the airy process has a slow decay of correlations as y 2 roughly the airy process describes the last line of dyson s brownian motion model for random matrices our construction uses a multi layer version of the png model which can be analyzed through fermionic techniques specializing our result to a fixed value of y one reobtains the celebrated result of baik deift and johansson on the length of the longest increasing subsequence of a random permutation

pade techniques for model reduction in linear system theory a survey abstract techniques of pade approximation and continued fractions have been used often in model reduction problems an extensive bibliography on this topic is given the ideas are explained for the simple situation of a scalar function where no singular blocks appear in the pade table extensions to the matrix case and the multivariable case are not explained in detail

recursive partial realization from the combined sequence of markov parameters and moments abstract in this paper we develop the recursive structure of partial realization from a two sided sequence of markov parameters and moments the specific feature of this recursive structure is that it accommodates sequences growing in two directions like in the two point pade approximation we present a recursive algorithm for the minimal partial realization from a sequence of markov parameters and moments that amounts to recursive lu factorization of a toeplitz behavior matrix leaping over nongeneric sections it turns out that this algorithm reveals a new continued fraction expansion which extends the classical cauer third form as a corollary various results related to toeplitz matrices two point pade approximations continued fractions etc are unified within this partial realization setting

formally biorthogonal polynomials and a look ahead levinson algorithm for general teoplitz systems abstract systems of linear equations with toeplitz coefficient matrices arise in many important applications the classical levinson algorithm computes solutions of toeplitz systems with only o n 2 arithmetic operations as compared to o n 3 operations that are needed for solving general linear systems however the levinson algorithm in its original form requires that all leading principal submatrices be nonsingular in this paper an extension of the levinson algorithm to general toeplitz systems is presented the algorithm uses look ahead to skip over exactly singular as well as ill conditioned leading submatrices and at the same time it still fully exploits the toeplitz structure in our derivation of this algorithm we make use of the intimate connection of toeplitz matrices with formally biorthogonal polynomials in particular the occurence of singular or ill conditioned submatrices corresponds to a breakdown or near breakdown in the standard recurrence relations for biorthogonal polynomials we present new general recurrence relations that connect successive pairs in any given subsequence of all existing formally biorthogonal polynomials these recurrences then immediately lead to the proposed look ahead levinson algorithm for solving toeplitz systems implementation details for this algorithm and operation counts are given numerical experiments for toeplitz systems with ill conditioned submatrices are reported

exact scaling functions for one dimensional stationary kpz growth we determine the stationary two point correlation function of the one dimensional kpz equation through the scaling limit of a solvable microscopic model the polynuclear growth model the equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a riemann hilbert problem related to the painleve ii equation we solve these equations numerically with very high precision and compare our up to numerical rounding exact result with the prediction of colaiori and moore 1 obtained from the mode coupling approximation

fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains a time dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered by applying the energy method well posed boundary conditions for the continuous problem are derived summation by parts sbp operators for the space and time discretization together with a weak imposition of boundary and initial conditions using simultaneously approximation terms sats lead to a provable fully discrete energy stable conservative finite difference scheme we show how to construct a time dependent sat formulation that automatically imposes boundary conditions when and where they are required we also prove that a uniform flow field is preserved i e the numerical geometric conservation law ngcl holds automatically by using sbp sat in time and space the developed technique is illustrated by considering an application using the linearized euler equations the sound generated by moving boundaries numerical calculations corroborate the stability and accuracy of the new fully discrete approximations

longest increasing subsequence as expectation of a simple nonlinear stochastic partial differential equation with a low noise intensity we report some observations concerning the statistics of longest increasing subsequences lis we argue that the expectation of lis its variance and apparently the full distribution function appears in statistical analysis of some simple nonlinear stochastic partial differential equation in the limit of very low noise intensity

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

asymmetric exclusion process and extremal statistics of random sequences a mapping is established between sequence alignment one of the most commonly used tools of computational biology at a certain choice of scoring parameters and the asymmetric exclusion process one of the few exactly solvable models of nonequilibrium physics the statistical significance of sequence alignments is characterized through studying the total hopping current of the discrete time and space version of the asymmetric exclusion process

limit theorems for height fluctuations in a class of discrete space and time growth models we introduce a class of one dimensional discrete space discrete time stochastic growth models described by a height function ht x with corner initialization we prove with one exception that the limiting distribution function of ht x suitably centered and normalized equals a fredholm determinant previously encountered in random matrix theory in particular in the universal regime of large x and large t the limiting distribution is the fredholm determinant with airy kernel in the exceptional case called the critical regime the limiting distribution seems not to have previously occurred the proofs use the dual rsk algorithm gessel s theorem the borodin okounkov identity and a novel rigorous saddle point analysis in the fixed x large t regime we find a brownian motion representation this model is equilvalent to the seppalainen johansson model hence some of our results are not new but the proofs are

spectral methods on arbitrary grids stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations these new methods are equivalent to conventional spectral methods but do not rely on specific grid distributions specifically we show how to implement legendre galerkin legendre collocation and laguerre galerkin methodology on arbitrary grids

height distributions in competitive one dimensional kardar parisi zhang systems we study the competitive rsos bd model focusing on the validity of the kardar parisi zhang kpz ansatz h t v ensuremath infty t ensuremath gamma t ensuremath beta ensuremath chi and the universality of the height distributions hds near the point where the model has edwards wilkinson ew scaling using numerical simulations for long times we show that the system is asymptotically kpz as expected for values of the probability of the rsos component very close to p p c ensuremath approx 0 83 namely the growth exponents converge to ensuremath beta mathrm kpz 1 3 and the hds converge to the goe tracy widom distribution however the convergence seems to be faster in the last ones while the ew kpz crossover appears in the roughness scaling in a broad range of probabilities p around p c a gaussian goe crossover is not observed in the hds into the same interval possibly being restricted to values of p very close to p c these results improve recent ones reported by de assis et al phys rev e 86 051607 2012 where based on smaller simulations the kpz scaling was argued to break down in a broad range of p around p c

probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension d 1 2 3 via a monte carlo procedure in the disorder in order to probe with high precision the tails of the ground state energy distribution of disordered spin systems k orner katzgraber and hartmann have recently proposed an importance sampling monte carlo markov chain in the disorder in this paper we combine their monte carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground state energy distribution p d e 0 for the directed polymer in a random medium of dimension d 1 2 3 in d 1 we check the validity of the algorithm by a direct comparison with the exact result namely the tracy widom distribution in dimensions d 2 and d 3 we measure the negative tail up to ten standard deviations which correspond to probabilities of order p d e 0 ensuremath sim 10 ensuremath 22 our results are in agreement with zhang s argument stating that the negative tail exponent ensuremath eta d of the asymptotic behavior mathrm ln phantom rule 0 2em 0ex p d e 0 ensuremath sim ensuremath ensuremath mid e 0 ensuremath mid ensuremath eta d as e 0 ensuremath rightarrow ensuremath ensuremath infty is directly related to the fluctuation exponent ensuremath theta d which governs the fluctuations ensuremath delta e 0 l ensuremath sim l ensuremath theta d of the ground state energy e 0 for polymers of length l via the simple formula ensuremath eta d 1 1 ensuremath ensuremath theta d throughout the paper we comment on the similarities and differences with spin glasses

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

a least squares approach to reduce stable discrete linear systems preserving their stability abstract a new stability preserving model reduction algorithm for discrete linear siso systems based on a least squares approach is proposed similar to the pade approximation an equation system for the markov parameters involving a high dimensional hankel matrix is considered it is proved that approximate solutions computed via the moore penrose pseudo inverse give rise to a stability preserving reduction scheme furthermore the proposed algorithm is compared to the balanced truncation method showing comparable performance of the reduced systems

scale invariant dynamics of surface growth we describe in detail and extend a recently introduced nonperturbative renormalization group rg method for surface growth the scale invariant dynamics which is the key ingredient of the calculation is obtained as the fixed point of a rg transformation relating the representation of the microscopic process at two different coarse grained scales we review the rg calculation for systems in the kardar parisi zhang kpz universality class and compute the roughness exponent for the strong coupling phase in dimensions from 1 to 9 discussions of the approximations involved and possible improvements are also presented moreover very strong evidence of the absence of a finite upper critical dimension for kpz growth is presented finally we apply the method to the linear edwards wilkinson dynamics where we reproduce the known exact results proving the ability of the method to capture qualitatively different behaviors

fluctuations of the one dimensional polynuclear growth model in half space we consider the multi point equal time height fluctuations of the one dimensional polynuclear growth model in half space for special values of the nucleation rate at the origin the multi layer version of the model is reduced to a process with a determinantal weight for which the asymptotics can be analyzed in the scaling limit the fluctuations near the origin are shown to be equivalent to those of the largest eigenvalue of the orthogonal symplectic to unitary transition ensemble at soft edge in random matrix theory

longest increasing subsequences from patience sorting to the baik deift johansson theorem we describe a simple one person card game patience sorting its analysis leads to a broad circle of ideas linking young tableaux with the longest increasing subsequence of a random permutation via the schensted correspondence a recent highlight of this area is the work of baik deift johansson which yields limiting probability laws via hard analysis of toeplitz determinants

a modified procedure for the realization of the transfer function matrix from a mixture of markov parameters and moments a procedure for the realization of a transfer function matrix using a mixture of markov parameters and moments is given the procedure results in the determination of a b and c and does not involve the determination of a 1and a k 1 etc as needed in existing techniques

the chebyshev legendre method implementing legendre methods on chebyshev points a new collocation method for the numerical solution of partial differential equations is presented this method uses the chebyshev collocation points but because of the way the boundary conditions are implemented it has all the advantages of the legendre methods in particular l 2 estimates can be easily obtained for hyperbolic and parabolic problems

an isoparametric spectral element method for solution of the navier stokes equations in complex geometry abstract an isoparametric spectral element technique is presented for solution of the two dimensional navier stokes equations in arbitrary curvy geometries the spatial discretization is described and the method is illustrated by solution of poisson s equation in a cylindrical annulus a fractional step navier stokes solver based on a discretely consistent poisson equation for the pressure is then presented and exponential convergence in space is demonstrated for the case of separated flow between eccentric rotating cylinders the technique is made efficient by the use of explicit collocation rather than galerkin operators for the convective terms and use of static condensation to solve the elliptic equations resulting from the stokes problem the algorithm is amenable to a high degree of parallelism in both multiprocessor and vector pipeline environments

exact solutions for kpz type growth processes random matrices and equilibrium shapes of crystals abstract three models from statistical physics can be analyzed by employing space time determinantal processes 1 crystal facets in particular the statistical properties of the facet edge and equivalently tilings of the plane 2 one dimensional growth processes in the kardar parisi zhang universality class and directed last passage percolation 3 random matrices multi matrix models and dyson s brownian motion we explain the method and survey results of physical interest

inversion of toeplitz operators levinson equations and gohberg krein factorization a simple and unified approach for the rational case the problem of the inversion of the t oeplitz operator r associated with the operator valued function 0 defined on the unit circle is known to involve the associated levinson system of equations and the gohberg krein factorization of 0 a simplified and self contained approach making clear the connections betireen these three problems is presented in the case nhere is matrix valued and rational the key idea consists in looking at the levinson system of equations associated with 2 rather than that associated with z j as a consequence a new invertibility criterion for toeplitz operators with rational matrix lalued symbols is derived

non equilibrium behaviour of a many particle process density profile and local equilibria one considers a simple exclusion particle jump process on where the underlying one particle motion is a degenerate random walk that moves only to the right one starts with the configuration in which the left halfline is completely occupied and the right one free it is shown that the number of particles at time t between site u t and v t divided by t converges a s to int limits u nu f w dw where f might be called the density profile it is explicitely determined and shown to be an affine function secondly we prove that the distribution of the process looked at by an observer travelling at constant speed u converges weakly to the bernoulli measure with density f u as the time tends to infinity

mean field theory of polynuclear surface growth we study statistical properties of a continuum model of polynuclear surface growth on an infinite substrate we develop a self consistent mean field theory which is solved to deduce the growth velocity and the extremal behaviour of the coverage numerical simulations show that this theory gives an improved approximation for the coverage compared to the previous linear recursion relations approach furthermore these two approximations provide useful upper and lower bounds for a number of characteristics including the coverage growth velocity and the width exponent

exact asymptotic results for the bernoulli matching model of sequence alignment finding analytically the statistics of the longest common subsequence lcs of a pair of random sequences drawn from c alphabets is a challenging problem in computational evolutionary biology we present exact asymptotic results for the distribution of the lcs in a simpler yet nontrivial variant of the original model called the bernoulli matching bm model we show that in the bm model for all c the distribution of the asymptotic length of the lcs suitably scaled is identical to the tracy widom distribution of the largest eigenvalue of a random matrix whose entries are drawn from a gaussian unitary ensemble

mean field theory for skewed height profiles in kpz growth processes we propose a mean field theory for interfaces growing according to the kardar parisi zhang kpz equation in 1 1 dimensions the mean field equations are formulated in terms of densities at different heights taking surface tension and the influence of the nonlinear term in the kpz equation into account although spatial correlations are neglected the mean field equations still reflect the spatial dimensionality of the system in the special case of edwards wilkinson growth our mean field theory correctly reproduces all features in the presence of a nonlinear term one observes a crossover to a kpz like behaviour with the correct dynamical exponent z 3 2 in particular we compute the skewed interface profile during roughening and we study the influence of a co moving reflecting wall which has been discussed recently in the context of nonequilibrium wetting and synchronization transitions also here the mean field approximation reproduces all qualitative features of the full kpz equation although with different values of the surface exponents

an algorithm for solving discrete time wiener hopf equations based upon euclid s algorithm an algorithm for solving a discrete time wiener hopf equation is presented based upon euclid s algorithm the discrete time wiener hopf equation is a system of linear inhomogeneous equations with a given toeplitz matrix m a given vector b and an unknown vector lambda such that m lambda b the algorithm is able to find a solution of the discrete time wiener hopf equation for any type of toeplitz matrices except for the all zero matrix while the levinson algorithm and the trench algorithm are not available when at least one of the principal submatrices of the toeplitz matrix m is singular the algorithm gives a solution if one exists even when the toeplitz matrix m is singular while the brent gustavson yun algorithm only states that the toeplitz matrix m is singular the algorithm requires o t 2 arithmetic operations for t unknowns in the sense that the number of multiplications or divisions is directly proportional to t 2 like the levinson and trench algorithms furthermore a faster algorithm is also presented based upon the half greatest common divisor algorithm and hence it requires o t log 2 t arithmetic operations like the brent gustavson yun algorithm

universal fluctuations in kardar parisi zhang growth on one dimensional flat substrates we present a numerical study of the evolution of height distributions hds obtained in interface growth models belonging to the kardar parisi zhang kpz universality class the growth is done on an initially flat substrate the hds obtained for all investigated models are very well fitted by the theoretically predicted gaussian orthogonal ensemble goe distribution the first cumulant has a shift that vanishes as t 1 3 while the cumulants of order 2 n 4 converge to goe as t 2 3 or faster behaviors previously observed in other kpz systems these results yield evidences for the universality of the goe distribution in kpz growth on flat substrates finally we further show that the surfaces are described by the airy 1 process

the tau method and a new preconditioner in relations to the order of linear ordinary differential equations using a modified form of the chebyshev or legendre and gegenbauer polynomials some particular integral operators are introduced these are used to give a factorization of the operators arising from the application of the chebyshev or legendre tau method the new tau method presented in this article is then compared with the standard tau method and preconditioned method of cabos the new tau method shows a superior performance an analysis of error and a bound for condition number is given numerical examples applying iterative solvers show dramatic reduction in condition number and improved convergence for the tau method with the new preconditioner

remarks on the inversion of hankel matrices the paper continues the investigation into the links between algebraic system theory more specifically the partial realization problem and the problem of hankel matrix inversion the representation of the inverse of a hankel matrix as a bezoutiant and the bezout equation play a principal role

functional renormalization group and the field theory of disordered elastic systems we study elastic systems such as interfaces or lattices pinned by quenched disorder to escape triviality as a result of dimensional reduction we use the functional renormalization group difficulties arise in the calculation of the renormalization group functions beyond one loop order even worse observables such as the two point correlation function exhibit the same problem already at one loop order these difficulties are due to the nonanalyticity of the renormalized disorder correlator at zero temperature which is inherent to the physics beyond the larkin length characterized by many metastable states as a result two loop diagrams which involve derivatives of the disorder correlator at the nonanalytic point are naively ambiguous we examine several routes out of this dilemma which lead to a unique renormalizable field theory at two loop order it is also the only theory consistent with the potentiality of the problem the beta function differs from previous work and the one at depinning by novel anomalous terms for interfaces and random bond disorder we find a roughness exponent zeta 0 208 298 04epsilon 0 006 858epsilon 2 epsilon 4 d for random field disorder we find zeta epsilon 3 and compute universal amplitudes to order o epsilon 2 for periodic systems we evaluate the universal amplitude of the two point function we also clarify the dependence of universal amplitudes on the boundary conditions at large scale all predictions are in good agreement with numerical and exact results and are an improvement over one loop finally we calculate higher correlation functions which turn out to be equivalent to those at depinning to leading order in epsilon

vicious walkers friendly walkers and young tableaux ii with a wall we derive new results for the number of star and watermelon configurations of vicious walkers in the presence of an impenetrable wall by showing that these follow from standard results in the theory of young tableaux and combinatorial descriptions of symmetric functions for the problem of n friendly walkers we derive exact asymptotics for the number of stars and watermelons both in the absence of a wall and in the presence of a wall

matrix kernels for measures on partitions we consider the problem of computation of the correlation functions for the z measures with the deformation jack parameters 2 or 1 2 such measures on partitions are originated from the representation theory of the infinite symmetric group and in many ways are similar to the ensembles of random matrix theory of 4 or 1 symmetry types for a certain class of such measures we show that correlation functions can be represented as pfaffians including 2 2 matrix valued kernels and compute these kernels explicitly we also give contour integral representations for correlation kernels of closely connected measures on partitions

summation by parts operators with minimal dispersion error for coarse grid flow calculations we present a procedure for constructing summation by parts operators with minimal dispersion error both near and far from numerical interfaces examples of such operators are constructed and compar

high order implicit time marching methods based on generalized summation by parts operators this article extends the theory of classical finite difference summation by parts fd sbp time marching methods to the generalized summation by parts gsbp framework dual consistent gsbp time marching methods are shown to retain a and l stability as well as superconvergence of integral functionals when integrated with the quadrature associated with the discretization this also implies that the solution approximated at the end of each time step is superconvergent in addition gsbp time marching methods constructed with a diagonal norm are bn stable this article also formalizes the connection between fd sbp gsbp time marching methods and implicit runge kutta methods through this connection the minimum accuracy of the solution approximated at the end of a time step is extended for nonlinear problems it is also exploited to derive conditions under which nonlinearly stable gsbp time marching methods can be constructed the gsbp approach to time marching can simplify the construction of high order f

unitary matrix models as exactly solvable string theories models of unitary matrices are solved exactly in a double scaling limit using orthogonal polynomials on a circle exact differential equations are found for the scaling functions of these models for the simplest model k 1 the painlev e ii equation with constant 0 is obtained there are possible nonperturbative phase transitions in these models the scaling function is of the form mathit n mathrm ensuremath 1 2 mathit k 1 ifmmode times else texttimes fi f mathit n 2 mathit k 2 mathit k 1 ensuremath lambda mathit x ensuremath lambda for the kth multicritical point the specific heat is mathit f 2 and is therefore manifestly positive equations are given for k 2 and 3 with a discussion of asymptotic behavior

a preconditioned implementation of pseudospectral methods on arbitrary grids in this article a preconditioner for pseudospectral methods in terms of expansion coefficients on arbitrary grids is presented the preconditioner is first developed for application in ordinary differential equations and then is generalized for partial differential equations a fully discrete pseudospectral approximation is then used to solve bivariate problems this preconditioner in effect makes the pseudospectral methods insensitive to specific grid points the numerical results show the efficiency of the method

convergence analysis for a second order elliptic equation by a collocation method using scattered points a collocation method using scattered points applied to a second order elliptic differential equation is analyzed by establishing a new quadrature formula for the space of the polynomials we show that a polynomial solution possesses stability and preserves a similar convergence property occurred in the classical high order collocation method

large time asymptotics of growth models on space like paths ii png and parallel tasep we consider the polynuclear growth png model in 1 1 dimension with flat initial condition and no extra constraints the joint distributions of surface height at finitely many points at a fixed time moment are given as marginals of a signed determinantal point process the long time scaling limit of the surface height is shown to coincide with the airy1 process this result holds more generally for the observation points located along any space like path in the space time plane we also obtain the corresponding results for the discrete time tasep totally asymmetric simple exclusion process with parallel update

statistical self similarity of one dimensional growth processes abstract for one dimensional growth processes we consider the distribution of the height above a given point of the substrate and study its scale invariance in the limit of large times we argue that for self similar growth from a single seed the universal distribution is the tracy widom distribution from the theory of random matrices and that for the growth from a flat substrate it is some other only numerically determined distribution in particular for the polynuclear growth model in the droplet geometry the height maps onto the longest increasing subsequence of a random permutation from which the height distribution is identified as the tracy widom distribution

kardar parisi zhang universality class in 2 1 dimensions universal geometry dependent distributions and finite time corrections the dynamical regimes of models belonging to the kardar parisi zhang kpz universality class are investigated in d 2 1 by extensive simulations considering flat and curved geometries geometry dependent universal distributions different from their tracy widom counterpart in one dimension were found distributions exhibit finite time corrections hallmarked by a shift in the mean decaying as t where is the growth exponent our results support a generalization of the ansatz h v t t t to higher dimensions where v and are nonuniversal quantities whereas and are universal and the last one depends on the surface geometry generalized gumbel distributions provide very good fits of the distributions in at least four orders of magnitude around the peak which can be used for comparisons with experiments our numerical results call for analytical approaches and experimental realizations of the kpz class in two dimensional systems

magnetic fluctuations in the classical xy model the origin of an exponential tail in a complex system we study the probability density function for the fluctuations of the magnetic order parameter in the low temperature phase of the mathrm xy model of finite size in two dimensions this system is critical over the whole of the low temperature phase it is shown analytically and without recourse to the scaling hypothesis that in this case the distribution is non gaussian and of universal form independent of both system size and critical exponent ensuremath eta an exact expression for the generating function of the distribution is obtained which is transformed and compared with numerical data from high resolution molecular dynamics and monte carlo simulations the asymptotes of the distribution are calculated and found to be of exponential and double exponential form the calculated distribution is fitted to three standard functions a generalization of gumbel s first asymptote distribution from the theory of extremal statistics a generalized log normal distribution and a ensuremath chi 2 distribution the calculation is extended to general dimension and an exponential tail is found in all dimensions less than 4 despite the fact that critical fluctuations are limited to d 2 these results are discussed in the light of similar behavior observed in models of interface growth and for dissipative systems driven into a nonequilibrium steady state

an algebraic approach to the partial realization problem abstract the partial realization problem is treated algebraically using polynomial models the main idea is the kernels of certain projections of the hankel map which give rise to orthogonal direct sum decomposition of the state space minimality of partial realizations is based on minimal elements of these subspaces which arise as a natural consequence of the structural properties

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

integration matrix based on arbitrary grids with a preconditioner for pseudospectral method a matrix representation of integration for arbitrary grids is introduced suitable results are then obtained to be used along with differentiation matrix preconditioner to implement pseudospectral method on integro differential equations using arbitrary grids numerical examples are given to clarify the efficiency of the new method

integration preconditioning of pseudospectral operators i basic linear operators this paper develops a family of preconditioners for pseudospectral approximations of pth order linear differential operators subject to various types of boundary conditions the approximations are based on ultraspherical polynomials with special attention being paid to legendre and chebyshev polynomial methods based on gauss lobatto quadrature points the eigenvalue spectrum of the preconditioned operators are obtained in closed analytic form and the weakly enforced boundary conditions are shown to result in a rank 2p perturbation of the identity operator i e the majority of the preconditioned eigenvalues are unity the spectrum of the preconditioned advective operator is shown to be bounded independent of the order of the approximation n however the preconditioned diffusive operator is in general indefinite with four real eigenvalues for dirichlet boundary conditions the spectral radius grows as sqrt n while it scales as n for the case of neumann boundary conditions these results are shown to be asymptotically optimal within the present framework generalizations to higher order differential operators general boundary conditions and arbitrary polynomial basis and quadrature nodes are discussed

hammersley s interacting particle process and longest increasing subsequences in a famous paper 8 hammersley investigated the lengthl n of the longest increasing subsequence of a randomn permutation implicit in that paper is a certain one dimensional continuous space interacting particle process by studying a hydrodynamical limit for hammersley s process we show by fairly soft arguments that limn 1 2 el n 2 this is a known result but previous proofs 14 11 relied on hard analysis of combinatorial asymptotics

corner corrected diagonal norm summation by parts operators for the first derivative with increased order of accuracy combined with simultaneous approximation terms summation by parts sbp operators offer a versatile and efficient methodology that leads to consistent conservative and provably stable discretizations however diagonal norm operators with a repeating interior point operator that have thus far been constructed suffer from a loss of accuracy while on the interior these operators are of degree 2p at a number of nodes near the boundaries they are of degree p and therefore of global degree p meaning the highest degree monomial for which the operators are exact at all nodes this implies that for hyperbolic problems and operators of degree greater than unity they lead to solutions with a global order of accuracy lower than the degree of the interior point operator in this paper we develop a procedure to construct diagonal norm first derivative sbp operators that are of degree 2p at all nodes and therefore can lead to solutions of hyperbolic problems of order 2 p 1 this is accomplished by adding nonzero entries in the upper right and lower left corners of sbp operator matrices with a repeating interior point operator this modification necessitates treating these new operators as elements where mesh refinement is accomplished by increasing the number of elements in the mesh rather than increasing the number of nodes the significant improvements in accuracy of this new family for the same repeating interior point operator are demonstrated in the context of the linear convection equation

on recursiveness and related topics in linear systems a decomposition of matrix formal power series linear systems is developed the interconnections between formal power series realization pade approximation recursiveness berlekamp massey algorithm feedback interconnections matrix linear fractional representations matrix continued fractions and the matrix euclidean algorithm are thereby revealed

vicious walkers friendly walkers and young tableaux iii between two walls we derive exact and asymptotic results for the number of star and watermelon configurations of vicious walkers confined to lie between two impenetrable walls as well as corresponding results for the analogous problem of friendly walkers our proofs make use of results from symmetric function theory and the theory of basic hypergeometric series

universal distributions for growth processes in 1 1 dimensions and random matrices we develop a scaling theory for kardar parisi zhang growth in one dimension by a detailed study of the polynuclear growth model in particular we identify three universal distributions for shape fluctuations and their dependence on the macroscopic shape these distribution functions are computed using the partition function of gaussian random matrices in a cosine potential

the spatial operator in the incompressible navier stokes oseen and stokes equations abstract we investigate the spatial operator in the incompressible navier stokes oseen and stokes equations and show how to avoid singularities associated with null spaces by choosing specific boundary conditions the theoretical results are derived for a general form of energy stable boundary conditions and applied to a few commonly used ones the analysis is done on a system that simultaneously covers the nonlinear incompressible navier stokes the oseen and the stokes equations when the spectrum of the spatial operator is investigated we restrict the analysis to the oseen and stokes equations the continuous analysis carries over to the discrete setting by using the summation by parts framework

a simple hankel interpretation of the berlekamp massey algorithm abstract a simple interpretation of the berlekamp massey algorithm in the light of the hankel matrix is presented the salient result is that the jump of the linear feedback shift register lfsr length is derived almost trivially from the so called iohvidov index of the hankel matrix prior to making any reference to the berlekamp massey algorithm itself next the hankel system of equations that yields the updated connection polynomial is solved via the natural lu factorization of the hankel matrix which itself leads to the berlekamp massey algorithm in a simple and transparent manner

optimal gegenbauer quadrature over arbitrary integration nodes this paper treats definite integrations numerically using gegenbauer quadratures the novel numerical scheme introduces the idea of exploiting the strengths of the chebyshev legendre and gegenbauer polynomials through a unified approach and using a unique numerical quadrature in particular the developed numerical scheme employs the gegenbauer polynomials to achieve rapid rates of convergence of the quadrature for the small range of the spectral expansion terms for a large scale number of expansion terms the numerical quadrature has the advantage of converging to the optimal chebyshev and legendre quadratures in the l norm and l 2 norm respectively the key idea is to construct the gegenbauer quadrature through discretizations at some optimal sets of points of the gegenbauer gauss gg type in a certain optimality sense we show that the gegenbauer polynomial expansions can produce higher order approximations to the definite integrals 1 x if x dx of a smooth function f x c 1 1 for the small range by minimizing the quadrature error at each integration point x i through a pointwise approach the developed gegenbauer quadrature can be applied for approximating integrals with any arbitrary sets of integration nodes exact integrations are obtained for polynomials of any arbitrary degree n if the number of columns in the developed gegenbauer integration matrix gim is greater than or equal to n the error formula for the gegenbauer quadrature is derived moreover a study on the error bounds and the convergence rate shows that the optimal gegenbauer quadrature exhibits very rapid convergence rates faster than any finite power of the number of gegenbauer expansion terms two efficient computational algorithms are presented for optimally constructing the gegenbauer quadrature we illustrate the high order approximations of the optimal gegenbauer quadrature through extensive numerical experiments including comparisons with conventional chebyshev legendre and gegenbauer polynomial expansion methods the present method is broadly applicable and represents a strong addition to the arsenal of numerical quadrature methods

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

polynuclear growth model with external source and random matrix model with deterministic source we present a random matrix interpretation of the distribution functions which have appeared in the study of the one dimensional polynuclear growth png model with external sources it is shown that the distribution goe2 which is defined as the square of the gaussian orthogonal ensemble goe tracy widom distribution can be obtained as the scaled largest eigenvalue distribution of a special case of a random matrix model with a deterministic source which have been studied in a different context previously compared to the original interpretation of the goe2 as the square of goe ours has an advantage in that it can also describe the transition from the gaussian unitary ensemble gue tracy widom distribution to the goe2 we further demonstrate that our random matrix interpretation can be obtained naturally by noting the similarity of the topology between a certain noncolliding brownian motion model and the multilayer png model with an external source this provides us with a multimatrix model interpretation of the multipoint height distributions of the png model with an external source

limiting distributions for a polynuclear growth model with external sources the purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources which was considered by prahofer and spohn depending on the strength of the sources the limiting distribution functions are either the tracy widom functions of random matrix theory or a new explicit function which has the special property that its mean is zero moreover we obtain transition functions between pairs of the above distribution functions in suitably scaled limits there are also similar results for a discrete totally asymmetric exclusion process