statistics of shocks in solutions of inviscid burgers equation the purpose of this paper is to analyze statistical properties of discontinuities of solutions of the inviscid burgers equation having a typical realizationb y of the brownian motion as an initial datum this case was proposed and studied numerically in the companion paper by she aurell and frisch the description of the statistics is given in terms of the behavior of the convex hull of the random process w y int limits 0 y b eta eta d eta the hausdorff dimension of the closed set of thosey where the convex hull coincides withw is also studied

statistical properties of shocks in burgers turbulence we consider the statistical properties of solutions of burgers equation in the limit of vanishing viscosity frac partial partial t u left x t right frac partial partial x left frac 1 2 u left x t right 2 right 0 with gaussian whitenoise initial data this system was originally proposed by burgers 1 as a crude model of hydrodynamic turbulence and more recently by zel dovichet al 12 to describe the evolution of gravitational matter at large spatio temporal scales with shocks playing the role of mass clusters we present here a rigorous proof of the scaling relationp s s 1 2 s 1 wherep s is the cumulative probability distribution of shock strengths we also show that the set of spatial locations of shocks is discrete i e has no accumulation points and establish an upper bound on the tails of the shock strength distribution namely 1 p s exp cs 3 fors 1 our method draws on a remarkable connection existing between the structure of burgers turbulence and classical probabilistic work on the convex envelope of brownian motion and related diffusion processes

accurate solution of dense linear systems part i algorithms in rounding to nearest we investigate how extra precise accumulation of dot products can be used to solve ill conditioned linear systems accurately for a given p bit working precision extra precise evaluation of a dot product means that the products and summation are executed in 2p bit precision and that the final result is rounded into the p bit working precision denote by u 2 p the relative rounding error unit in a given working precision we treat two types of matrices first up to condition number u 1 and second up to condition number u 2 for both types of matrices we present two types of methods first for calculating an approximate solution and second for calculating rigorous error bounds for the solution together with the proof of non singularity of the matrix of the linear system in the first part of this paper we present algorithms using only rounding to nearest in part ii we use directed rounding to obtain better results all algorithms are given in executable matlab code and are available from my homepage

lattice basis reduction improved practical algorithms and solving subset sum problems we report on improved practical algorithms for lattice basis reduction we present a variant of the l3 algorithm with deep insertions and a practical algorithm for blockwise korkine zolotarev reduction a concept extending l3 reduction that has been introduced by schnorr 1987 empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 58 random weights of arbitrary bit length within at most a few hours on a unisys 6000 70 or within a couple of minutes on a sparc 2 computer

the effect of lumping and expanding on kinetic differential equations let us consider the differential equation dot bf y t bf f bf y t with an bf f from bf r n to bf r n and suppose that there exists a transformation bf h from bf r n to bf r hat n hat n le n such that hat bf y bf h circ bf y obeys a differential equation dot hat bf y t hat bf f hat bf y t with some function hat bf f then the first equation is said to be lumpable to the second by bf h here mainly the case is investigated when the original differential equation has been induced by a complex chemical reaction we provided a series of necessary and sufficient conditions for the existence of such functions bf h and hat bf f some of them are formulated in terms of bf h and bf f only beyond these conditions our main concern here is how lumping changes those properties of the solutions which are either interesting from the point of view of the qualitative theory of differential equations or from the point

structure of shocks in burgers turbulence with stable noise initial data burgers equation can be used as a simplified model for hydrodynamic turbulence the purpose of this paper is to study the structure of the shocks for the inviscid equation in dimension 1 when the initial velocity is given by a stable levy noise with index 1 2 2 we prove that lagrangian regular points exist i e there are fluid particles that have not participated in shocks at any time between 0 and t if and only if 1 and the noise is not completely asymmetric and that otherwise the shock structure is discrete moreover in the cauchy case 1 we show that there are no rarefaction intervals i e at time t 0 there are fluid particles in any non empty open interval

a new error free floating point summation algorithm summation is a basic operation in scientific computing furthermore division free arithmetic computations can be boiled down to summation we propose a new summation algorithm which consists of double precision floating point operations and outputs the error free sums the computational time is proportional to the condition number of the problem is low according to both our estimates and extensive experiments and further decreases for producing faithful rounding of the sum rather than its error free value

computation of the monodromy matrix in floating point arithmetic with the wilkinson model in this study results have been obtained that compute the monodromy matrix in floating point arithmetic using the wilkinson model these results have been applied to the asymptotic stability of systems of linear difference equations with periodic coefficients also the effect of floating point arithmetic has been investigated on numerical examples

computing singular values of diagonally dominant matrices to high relative accuracy for a row diagonally dominant matrix if all of its off diagonal entries and its diagonally dominant parts which are defined for each row as the absolute value of the diagonal entry subtracted by the sum of the absolute values of off diagonal entries in that row are accurately known we develop an algorithm that computes all the singular values including zero ones if any with relative errors in the order of the machine precision when the matrix is also symmetric with positive diagonals i e a symmetric positive semi definite diagonally dominant matrix our algorithm computes all eigenvalues to high relative accuracy rounding error analysis will be given and numerical examples will be presented to demonstrate the high relative accuracy of the algorithm

accurate floating point summation part i faithful rounding given a vector of floating point numbers with exact sum s we present an algorithm for calculating a faithful rounding of s i e the result is one of the immediate floating point neighbors of s if the sum s is a floating point number we prove that this is the result of our algorithm the algorithm adapts to the condition number of the sum i e it is fast for mildly conditioned sums with slowly increasing computing time proportional to the logarithm of the condition number all statements are also true in the presence of underflow the algorithm does not depend on the exponent range our algorithm is fast in terms of measured computing time because it allows good instruction level parallelism it neither requires special operations such as access to mantissa or exponent it contains no branch in the inner loop nor does it require some extra precision the only operations used are standard floating point addition subtraction and multiplication in one working precision for example double precision certain constants used in the algorithm are proved to be optimal

structured population models conservation laws and delay equations abstract the general principles by which certain structured population models or renewal equations can be reduced to ordinary differential equations or delay equations are systematically investigated with particular attention to retarded and neutral delay differential equations approriate state spaces are defined and their relations are studied as well as the positivity of the resulting evolutionary systems

the accuracy of floating point summation the usual recursive summation technique is just one of several ways of computing the sum of n floating point numbers five summation methods and their variations are analyzed here the accuracy of the methods is compared using rounding error analysis and numerical experiments four of the methods are shown to be special cases of a general class of methods and an error analysis is given for this class no one method is uniformly more accurate than the others but some guidelines are given on the choice of method in particular cases

model selection and the principle of minimum description length this article reviews the principle of minimum description length mdl for problems of model selection by viewing statistical modeling as a means of generating descriptions of observed data the mdl framework discriminates between competing models based on the complexity of each description this approach began with kolmogorov s theory of algorithmic complexity matured in the literature on information theory and has recently received renewed attention within the statistics community here we review both the practical and the theoretical aspects of mdl as a tool for model selection emphasizing the rich connections between information theory and statistics at the boundary between these two disciplines we find many interesting interpretations of popular frequentist and bayesian procedures as we show mdl provides an objective umbrella under which rather disparate approaches to statistical modeling can coexist and be compared we illustrate the mdl principle by considering problems in regression nonpar

lipschitz minorants of brownian motion and levy processes for alpha 0 the alpha lipschitz minorant of a function f mathbb r rightarrow mathbb r is the greatest function m mathbb r rightarrow mathbb r such that m le f and m s m t le alpha s t for all s t in mathbb r should such a function exist if x x t t in mathbb r is a real valued levy process that is not pure linear drift with slope pm alpha then the sample paths of x have an alpha lipschitz minorant almost surely if and only if mathbb e x 1 alpha denoting the minorant by m we investigate properties of the random closed set fancyscript z t in mathbb r m t x t wedge x t which since it is regenerative and stationary has the distribution of the closed range of some subordinator made stationary in a suitable sense we give conditions for the contact set fancyscript z to be countable or to have zero lebesgue measure and we obtain formulas that characterize the levy measure of the associated subordinator we study the limit of fancyscript z as alpha rightarrow infty and find for the so called abrupt levy processes introduced by vigon that this limit is the set of local infima of x when x is a brownian motion with drift beta such that beta alpha we calculate explicitly the densities of various random variables related to the minorant

markov modelling of changes in hiv specific cytotoxic t lymphocyte responses with time in untreated hiv 1 infected patients hiv specific cytotoxic cd8 t lymphocytes ctl appear to be the cornerstone of the immune response to hiv infection recent studies show that ctl activity reflects patients anti hiv immune status and slows disease progression however the dynamics of the diversity of this response also appears as a key parameter for immune control but the dynamics of this diversity is largely undocumented we modelled changes in ctl responses against the seven principal hiv proteins over time we also studied the influence of plasma viral load on temporal changes in hiv protein recognition by memory ctl the generic model we developed is based on a continuous time homogeneous markov process with reversible states those states are defined by the number of proteins recognized by memory ctl in a given patient at a given time this approach was developed within a bayesian framework full bayesian inference is implemented using markov chain monte carlo simulations mcmc the gibbs sampling algorithm was used to estimate the marginal posterior distributions of the transition intensities between stages of ctl responses we applied our model to data of 152 hiv infected patients included in the immunoco cohort the model suggested that the diversity of hiv protein recognition by memory ctl in treatment naive patients decreases as the disease progresses namely the loss of t cytotoxic responses is globally faster than their acquisition indeed these patients t cytotoxic responses were characterized by marked individual turnover and a gradual loss of multiple protein recognition over time this loss accelerating as viral load increased copyright 2003 john wiley sons ltd

numerical modelling in biosciences using delay differential equations our principal purposes here are i to consider from the perspective of applied mathematics models of phenomena in the biosciences that are based on delay differential equations and for which numerical approaches are a major tool in understanding their dynamics ii to review the application of numerical techniques to investigate these models we show that there are prima facie reasons for using such models i they have a richer mathematical framework compared with ordinary differential equations for the analysis of biosystem dynamics ii they display better consistency with the nature of certain biological processes and predictive results we analyze both the qualitative and quantitative role that delays play in basic time lag models proposed in population dynamics epidemiology physiology immunology neural networks and cell kinetics we then indicate suitable computational techniques for the numerical treatment of mathematical problems emerging in the biosciences comparing them with those implemented by the bio modellers

the convex minorant of a levy process we offer a unified approach to the theory of convex minorants of levy processes with continuous distributions new results include simple explicit constructions of the convex minorant of a levy process on both finite and infinite time intervals and of a poisson point process of excursions above the convex minorant up to an independent exponential time the poisson dirichlet distribution of parameter 1 is shown to be the universal law of ranked lengths of excursions of a levy process with continuous distributions above its convex minorant on the interval 0 1

the statistics of burgers turbulence initialized with fractional brownian noise data the statistics of the solution to the inviscid burgers equation are investigated when the initial velocity potential is fractional brownian motion using the theory of large deviations for gaussian processes we characterize the tails of the probability distribution functions pdfs of the velocity the distance between shocks and the shock strength these pdfs are shown to decay like stretched exponentials of the form our method of proof can also be used to extend these results to a much larger class of gaussian potentials this work generalizes the results of avellaneda and e 2 3 on the inviscid burgers equation with white noise initial data

combinatorial characterization of the null spaces of symmetric h matrices abstract we characterize the structure of null spaces of symmetric diagonally dominant sdd matrices and symmetric h matrices with non negative diagonal entries we show that the structure of the null space of a symmetric sdd matrix or h matrix a depends on the structure of the connected components of its underlying graph each connected component contributes at most one vector to the null space this paper provides a combinatorial characterization of the rank of each connected component and a combinatorial characterization of a null vector if one exists for sdd matrices we also present an efficient combinatorial algorithm for constructing an orthonormal basis for the null space the paper also shows a close connection between gain graphs and h matrices which extends known results regarding the connection between undirected graphs and laplacian matrices and between signed graphs and sdd matrices we show how to exploit these combinatorial algorithms to reliably solve certain singular linear systems in finite precision arithmetic

matrix computations and polynomial root finding with preprocessing solution of homogeneous linear systems of equations is a basic operation of matrix computations the customary algorithms rely on pivoting orthogonalization and svd but we employ randomized preprocessing instead this enables us to accelerate the solution dramatically both in terms of the estimated arithmetic cost and the observed cpu time the approach is effective in the cases of both general and structured input matrices and we extend it and its computational advantages to the solution of nonhomogeneous linear systems of equations matrix eigen solving the solution of polynomial and secular equations and approximation of a matrix by a nearby matrix that has a smaller rank or a fixed structure e g of the toeplitz or hankel type our analysis and extensive experiments show the power of the presented algorithms

additive preconditioning and aggregation in matrix computations we combine our novel svd free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations our analysis and experiments show the power of our algorithms guide us in selecting most effective policies of preconditioning and aggregation and provide some new insights into these and related subjects compared to the popular svd based multiplicative preconditioners our additive preconditioners are generated more readily and for a much larger class of matrices furthermore they better preserve matrix structure and sparseness and have a wider range of applications e g they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem

rival approaches to mathematical modelling in immunology in order to formulate quantitatively correct mathematical models of the immune system one requires an understanding of immune processes and familiarity with a range of mathematical techniques selection of an appropriate model requires a number of decisions to be made including a choice of the modelling objectives strategies and techniques and the types of model considered as candidate models the authors adopt a multidisciplinary perspective

reducing floating point error in dot product using the superblock family of algorithms this paper discusses both the theoretical and statistical errors obtained by various well known dot products from the canonical to pairwise algorithms and introduces a new and more general framework that we have named superblock which subsumes them and permits a practitioner to make trade offs between computational performance memory usage and error behavior we show that algorithms with lower error bounds tend to behave noticeably better in practice unlike many such error reducing algorithms superblock requires no additional floating point operations and should be implementable with little to no performance loss making it suitable for use as a performance critical building block of a linear algebra kernel

john von neumann s analysis of gaussian elimination and the origins of modern numerical analysis just when modern computers digital electronic and programmable were being invented john von neumann and herman goldstine wrote a paper to illustrate the mathematical analyses that they believed would be needed to use the new machines effectively and to guide the development of still faster computers their foresight and the congruence of historical events made their work the first modern paper in numerical analysis von neumann once remarked that to found a mathematical theory one had to prove the first theorem which he and goldstine did for the accuracy of mechanized gaussian elimination but their paper was about more than that von neumann and goldstine described what they surmized would be the significant questions once computers became available for computational science and they suggested enduring ways to answer them

hausdorff dimension of regular points in stochastic burgers flows with levy α stable initial data this paper studies statistical properties of shocks for the inviscid burgers equation with an stable levy motion initial data in the absence of analytic results numerical and computer simulation tools are utilized qualitative and quantitative information on the scaling properties of lagrangian regular points of solutions is obtained and in particular their hausdorff dimension is estimated to be 1 this suggestsa possible extension of ya sinai s result for brownian initial data

accurate summation dot product and polynomial evaluation in complex floating point arithmetic several different techniques and softwares intend to improve the accuracy of results computed in a fixed finite precision here we focus on methods to improve the accuracy of summation dot product and polynomial evaluation such algorithms exist real floating point numbers in this paper we provide new algorithms which deal with complex floating point numbers we show that the computed results are as accurate as if computed in twice the working precision the algorithms are simple since they only require addition subtraction and multiplication of floating point numbers in the same working precision as the given data

on ergodic and mixing properties of the triangle map abstract in this paper we study in detail both analytically and numerically the dynamical properties of the triangle map a piecewise parabolic automorphism of the two dimensional torus for different values of the two independent parameters defining the map the dynamics is studied numerically by means of two different symbolic encoding schemes both relying on the fact that it maps polygons to polygons in the first scheme we consider dynamically generated partitions made out of suitable sets of disjoint polygons in the second we consider the standard binary partition of the torus induced by the discontinuity set these encoding schemes are studied in detail and shown to be compatible although not equivalent the ergodic properties of the triangle map are then investigated in terms of the markov transition matrices associated to the above schemes and furthermore compared to the spectral properties of the koopman operator in l 2 t 2 finally a stochastic version of the triangle map is introduced and studied a simple heuristic analysis of the latter yields the correct statistical and scaling behaviours of the correlation functions of the original map

points de croissance des processus de levy et theorie generale des processus we prove a conjecture of j bertoin a levy process has increase times if and only if the integral is finite where g and h are the distribution functions of the minimum and the maximum of the levy process killed at an independent exponential time the if part of the statement had been obtained before by r doney our proof uses different techniques from potential theory and the general theory of processes and is self contained our results also show that if p x t 0 1 2 for all t small enough then the process does not have increase times

kinetic theory and lax equations for shock clustering and burgers turbulence we study shock statistics in the scalar conservation law t u x f u 0 x t 0 with a convex flux f and spatially random initial data we show that the markov property in x is preserved for a large class of random initial data markov processes with downward jumps and derivatives of levy processes with downward jumps the kinetics of shock clustering is then described completely by an evolution equation for the generator of the markov process u x t x we present four distinct derivations for this evolution equation and show that it takes the form of a lax pair the lax equation admits a spectral parameter as in manakov funct anal appl 10 328 329 1976 and has remarkable exact solutions for burgers equation f u u 2 2 this suggests the kinetic equations of shock clustering are completely integrable

a distillation algorithm for floating point summation the addition of two or more floating point numbers is fundamental to numerical computations this paper describes an efficient distillation style algorithm which produces a precise sum by exploiting the natural accuracy of compensated cancellation the algorithm is applicable to all sets of data but is particularly appropriate for ill conditioned data where standard methods fail due to the accumulation of rounding error and its subsequent exposure by cancellation the method uses only standard floating point arithmetic and does not rely on the radix used by the arithmetic model the architecture of specific machines or the use of accumulators

limit clusters in the inviscid burgers turbulence with certain random initial velocities we study the infinite time shock limits given certain markovian initial velocities to the inviscid burgers turbulence specifically we consider the one sided case where initial velocities are zero on the negative half line and follow a time homogeneous nice markov process x on the positive half line finite shock limits occur if the markov process is transient tending to infinity they form a poisson point process if x is spectrally negative we give an explicit description when x is furthermore spatially homogeneous a levy process or a self similar process on 0 we also consider the two sided case where we suppose an independent dual process in the negative spatial direction both spatial homogeneity and an exponential levy condition lead to stationary shock limits

computation and storage of the bianisotropic scalar green s function and its derivatives a set of algorithms is proposed for the accurate and efficient computation and storage of the bianisotropic scalar green s function the computation is based on an expansion of the green s function into chebyshev polynomials the analytical properties of these polynomials are exploited to allow the accurate computation of the derivatives of the green s function as well as the green s function itself for lossy materials the proposed computation strategy is provably robust in addition a multilevel storage scheme with a favorable complexity based on the chebyshev polynomial expansion is proposed for the storage of the expansion coefficients numerical results showcase the accuracy and computational complexity of the proposed algorithms

the inviscid burgers equation with initial data of brownian type the solutions to burgers equation in the limit of vanishing viscosity are investigated when the initial velocity is a brownian motion or fractional brownian motion function i e a gaussian process with scaling exponent 0 h 1 typea or the derivative thereof with scaling exponent 1 h 0 typeb largesize numerical experiments are performed helped by the fact that the solution is essentially obtained by performing a legendre transform the main result is obtained for typea and concerns the lagrangian functionx a which gives the location at timet 1 of the fluid particle which started at the locationa it is found to be a complete devil s staircase the cumulative probability of lagrangian shock intervals a also the distribution of shock amplitudes follows a a h law for small a the remaining regular lagrangian locations form a cantor set of dimensionh in eulerian coordinates the shock locations are everywhere dense the scaling properties of various statistical quantities are also found heuristic interpretations are provided for some of these results rigorous results for the case of brownian motion are established in a companion paper by ya sinai for typeb initial velocities e g white noise there are very few small shocks and shock locations appear to be isolated finally it is shown that there are universality classes of random but smooth non scaling initial velocities such that the long time large scale behavior is after rescaling the same as for typea orb

parallel algorithms to evaluate orthogonal polynomial series in this paper four parallel algorithms for the evaluation of finite series of orthogonal polynomials are introduced the algorithms are based on the forsythe and clenshaw sequential algorithms several tests carried out on a cray t3d are presented

a new distillation algorithm for floating point summation the summation of n floating point numbers is ubiquitous in numerical computations we present a new distillation algorithm for floating point summation which is stable efficient and accurate the algorithm iteratively distills the summands without discarding any significant digit until the partial sums cannot change the whole sum it uses standard floating point arithmetic and does not rely on the choice of radix or any other specific assumption furthermore the error bound of our algorithm is independent of n and less than 1 ulp

some aspects of causal neutral equations used in modelling the problem that motivates the considerations here is the construction of mathematical models of natural phenomena that depend upon past states the paper divides naturally into two parts in the first we expound the inter connection between ordinary differential equations delay differential equations neutral delay differential equations and integral equations with emphasis on certain linear cases as we show this leads to a natural hierarchy of model complexity when such equations are used in mathematical and computational modelling and to the possibility of reformulating problems either to facilitate their numerical solution or to provide mathematical insight or both volterra integral equations include as special cases the others we consider in the second part we develop some practical and theoretical consequences of results given in the first part in particular we consider various approaches to the definition of an adjoint we establish notably in the context of sensitivity analysis for neutral delay differential equations roles for well defined adjoints and quasi adjoints and we explore relationships between sensitivity analysis the variation of parameters formulae the fundamental solution and adjoints

ultimately fast accurate summation we present two new algorithms fastaccsum and fastprecsum one to compute a faithful rounding of the sum of floating point numbers and the other for a result as if computed in k fold precision faithful rounding means the computed result either is one of the immediate floating point neighbors of the exact result or is equal to the exact sum if this is a floating point number the algorithms are based on our previous algorithms accsum and precsum and improve them by up to 25 the first algorithm adapts to the condition number of the sum i e the computing time is proportional to the difficulty of the problem the second algorithm does not need extra memory and the computing time depends only on the number of summands and k both algorithms are the fastest known in terms of flops they allow good instruction level parallelism so that they are also fast in terms of measured computing time the algorithms require only standard floating point addition subtraction and multiplication in one working precision for example double precision

randomized preprocessing of homogeneous linear systems of equations abstract our randomized preprocessing enables pivoting free and orthogonalization free solution of homogeneous linear systems of equations in the case of toeplitz inputs we decrease the estimated solution time from quadratic to nearly linear and our tests show dramatic decrease of the cpu time as well we prove numerical stability of our approach and extend it to solving nonsingular linear systems inversion and generalized moore penrose inversion of general and structured matrices by means of newton s iteration approximation of a matrix by a nearby matrix that has a smaller rank or a smaller displacement rank matrix eigen solving and root finding for polynomial and secular equations and for polynomial systems of equations some by products and extensions of our study can be of independent technical intersest e g our extensions of the sherman morrison woodbury formula for matrix inversion our estimates for the condition number of randomized matrix products and preprocessing via augmentation

estimating division and death rates from cfse data the division tracking dye carboxyfluorescin diacetate succinimidyl ester cfse is currently the most informative labeling technique for characterizing the division history of cells in the immune system gett and hodgkin nat immunol 1 2000 239 244 have proposed to normalize cfse data by the 2 fold expansion that is associated with each division and have argued that the mean of the normalized data increases linearly with time t with a slope reflecting the division rate p we develop a number of mathematical models for the clonal expansion of quiescent cells after stimulation and show within the context of these models under which conditions this approach is valid we compare three means of the distribution of cells over the cfse profile at time t the mean t the mean of the normalized distribution 2 t and the mean of the normalized distribution excluding nondivided cells 2 t in the simplest models which deal with homogeneous populations of cells with constant division and death rates the normalized frequency distribution of the cells over the respective division numbers is a poisson distribution with mean 2 t pt where p is the division rate the fact that in the data these distributions seem gaussian is therefore insufficient to establish that the times at which cells are recruited into the first division have a gaussian variation because the poisson distribution approaches the gaussian distribution for large pt excluding nondivided cells complicates the data analysis because 2 t pt and only approaches a slope p after an initial transient in models where the first division of the quiescent cells takes longer than later divisions all three means have an initial transient before they approach an asymptotic regime which is the expected t 2pt and 2 t 2 t pt such a transient markedly complicates the data analysis after the same initial transients the normalized cell numbers tend to decrease at a rate e dt where d is the death rate nonlinear parameter fitting of cfse data obtained from gett and hodgkin to ordinary differential equation ode models with first order terms for cell proliferation and death gave poor fits to the data the smith martin model with an explicit time delay for the deterministic phase of the cell cycle performed much better nevertheless the insights gained from analysis of the odes proved useful as we showed by generating virtual cfse data with a simulation model where cell cylce times were drawn from various distributions and then computing the various mean division numbers

passage of levy processes across power law boundaries at small times we wish to characterise when a l e vy process x t crosses boundaries like t kappa kappa 0 in a one or two sided sense for small times t thus we enquire when limsup t downarrow 0 x t t kappa limsup t downarrow 0 x t t kappa and or liminf t downarrow 0 x t t kappa are almost surely a s finite or infinite necessary and sufficient conditions are given for these possibilities for all values of kappa 0 often for many values of kappa when the limsups are finite a s they are in fact zero as we show but the limsups may in some circumstances take finite nonzero values a s in general the process crosses one or two sided boundaries in quite different ways but surprisingly this is not so for the case kappa 1 2 an integral test is given to distinguish the possibilities in that case some results relating to other norming sequences for x and when x is centered at a nonstochastic function are also given

computational approaches to parameter estimation and model selection in immunology one of the significant challenges in biomathematics and other areas of science is to formulate meaningful mathematical models our problem is to decide on a parametrized model which is in some sense most likely to represent the information in a set of observed data in this paper we illustrate the computational implementation of an information theoretic approach associated with a maximum likelihood treatment to modelling in immunology the approach is illustrated by modelling lcmv infection using a family of models based on systems of ordinary differential and delay differential equations the models which use parameters that have a scientific interpretation are chosen to fit data arising from experimental studies of virus cytotoxic t lymphocyte kinetics the parametrized models that result are arranged in a hierarchy by the computation of akaike indices the practical illustration is used to convey more general insight because the mathematical equations that comprise the models are solved numerically the accuracy in the computation has a bearing on the outcome and we address this and other practical details in our discussion

improvement of error free splitting for accurate matrix multiplication recently new algorithms for accurate matrix multiplication have been developed by the authors a characteristic of the algorithms is a high dependency on level 3 blas routines which are highly optimized for several architectures an error free splitting for floating point matrices is a key technique in the algorithms in this paper an improvement of the error free splitting is focused on it is shown by numerical examples that the accuracy of computed results of matrix products can be improved by the modified error free splitting compared to that by the previous algorithms

schur aggregation for linear systems and determinants we apply our recent preconditioning techniques to the solution of linear systems of equations and computing determinants we combine these techniques with the sherman morrison woodbury formula its new variations aggregation iterative refinement and advanced algorithms that rapidly compute sums and products either error free or with the desired high accuracy our theoretical and experimental study shows the power of this approach

estimating the number of stable configurations for the generalized thomson problem given a natural number n one may ask what configuration of n points on the two sphere minimizes the discrete generalized coulomb energy if one applies a gradient based numerical optimization to this problem one encounters many configurations that are stable but not globally minimal this led the authors of this manuscript to the question how many stable configurations are there in this manuscript we report methods for identifying and counting observed stable configurations and estimating the actual number of stable configurations these estimates indicate that for n approaching two hundred there are at least tens of thousands of stable configurations

large deviations estimates in burgers turbulence with stable noise initial data we consider the inviscid burgers equation where the initial datum is given by a stable levy noise the asymptotic behavior of the tail distribution of the solution is described the decay is much faster in the case when the stable noise is completely skewed to the left

accurate and efficient floating point summation we present and analyze several simple algorithms for accurately computing the sum of n floating point numbers using a wider accumulator let f and f be the number of significant bits in the summands and the accumulator respectively then assuming gradual underflow no overflow and round to nearest arithmetic up to approximately 2f f numbers can be added accurately by simply summing the terms in decreasing order of exponents yielding a sum correct to within about 1 5 units in the last place ulps we apply this result to the floating point formats in the ieee floating point standard for example a dot product of single precision vectors of length at most 33 computed using double precision and sorting is guaranteed correct to nearly 1 5 ulps if double extended precision is used the vector length can be as large as 65 537 we also investigate how the cost of sorting can be reduced or eliminated while retaining accuracy

what shape is your conjugate a survey of computational convex analysis and its applications computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields to further communications between practitioners we review the field of computational convex analysis which focuses on the numerical computation of fundamental transforms arising from convex analysis current models use symbolic numeric and hybrid symbolic numeric algorithms our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing computing the distance transform the generalized distance transform and mathematical morphology operators partial differential equations solving hamilton jacobi equations and using differential equations numerical schemes to compute the convex envelope max plus algebra computing the equivalent of the fast fourier transform multifractal analysis etc the fields of applications include among others computer vision robot navigation thermodynamics electrical networks medical imaging and network communication

structure of shocks in burgers turbulence with levy noise initial data we study the structure of the shocks for the inviscid burgers equation in dimension 1 when the initial velocity is given by levy noise or equivalently when the initial potential is a two sided levy process 0 when 0 is abrupt in the sense of vigon or has bounded variation with lim sup h 0h 2 0 h we prove that the set of points with zero velocity is regenerative and that in the latter case this set is equal to the set of lagrangian regular points which is non empty when 0 is abrupt we show that the shock structure is discrete when 0 is eroded we show that there are no rarefaction intervals

statistical properties of shocks in burgers turbulence ii tail probabilities for velocities shock strengths and rarefaction intervals this paper studies the structure of the random sawtooth profile corresponding to the solution of the inviscid burgers equation with white noise initial data this function consists of a countable sequence of rarefaction waves separated by shocks we are concerned here with calculating the probabilities of rare events associated with the occurrence of very large values of the normalized velocity shock strength and rarefaction intervals we find that these quantities have tail probabilities of the form exp cx3 x 1 this cubic exponential decay of probabilities was conjectured in the companion paper 1 the calculations are done using a representation of the shock strength and length of rarefaction intervals in terms of the statistics of certain conditional diffusion processes

reducing round off errors in rigid body dynamics in several recent publications numerical integrators based on jacobi elliptic functions are proposed for solving the equations of motion of the rigid body although this approach yields theoretically the exact solution a standard implementation shows an unexpected linear propagation of round off errors we explain how deterministic error contribution can be avoided so that round off behaves like a random walk