lower bounds for the condition number of vandermonde matrices we derive lower bounds for the condition number of then n vandermonde matrixvn x in the cases where the node vectorxt x1 x2 xn has positive elements or real elements located symmetrically with respect to the origin the bounds obtained grow exponentially inn witho 2n ando 2n 2 respectively we also compute the optimal spectral condition numbers ofvn x for the two node configurations including the optimal nodes and compare them with the bounds obtained

two species annihilation in three dimensions a numerical study two species diffusion limited annihilation in less than four dimensions is well known to have a concentration decaying as t d4 as opposed to the rate equation result of t 1 this result had not however been demonstrated numerically in three dimensions due to very strong transient effects a variant of the model is proposed here and is shown to reach the exactly known asymptotic behaviour in numerically accessible times this also allows one to investigate open questions in this system with some certainty that the results are indeed asymptotic in particular it is shown that the distance between two particles of the same species scales identically to the distance between particles of different species this is in contradistinction to recent numerical and scaling results in one and two dimensions some evidence is also presented that contrary to previous suggestions the domains formed in three dimensional two species annihilation are indeed regular objects with smooth interfaces

acceleration of euclidean algorithm and rational number reconstruction we accelerate the known algorithms for computing a selected entry of the extended euclidean algorithm for integers and consequently for the modular and numerical rational number reconstruction problems the acceleration is from quadratic to nearly linear time matching the known complexity bound for the integer gcd which our algorithm computes as a special case

inhomogeneous two species annihilation in the steady state the authors investigate steady state geometrical properties of a reaction interface in the two species annihilation process a b to 0 when a flux j of a and b particles is injected at opposite extremities of a finite domain by balancing the input flux with the number of reactions they determine that the width w of the reaction zone scales as j 1 3 in the large flux limit and that the concentration in this zone is proportional to j2 3 this same behaviour is deduced from the solution to the reaction diffusion equation in the low flux limit the concentration is almost independent of position and is proportional to j in the latter case the local reaction rate reaches maximum at the edges of the system rather than at the midpoint when the two species approach at a finite velocity there exists a critical velocity above which the reactants essentially pass through each other results similar to those in one dimension are found in two and three dimensional radial geometries finally they apply the quasistatic approximation to their steady state solution to recover the known time dependence for the reaction zone width for the case of initially separated components with no external input

scaling of reaction zones in the a b 0 diffusion limited reaction we study reaction zones in three different versions of the a b 0 system for a steady state formed by opposing currents of a and b particles we derive scaling behavior via renormalization group analysis by use of a previously developed analogy these results are extended to the time dependent case of an initially segregated system we also consider an initially mixed system which forms reaction zones for dimension d 4 in this case an extension of the steady state analogy gives scaling results characterized by new exponents

transformations of matrix structures work again abstract matrices with the structures of toeplitz hankel vandermonde and cauchy types are omnipresent in modern computations in sciences engineering and signal and image processing these four matrix classes have distinct features but in our 1990 paper in mathematics of computation we showed that vandermonde and hankel multipliers can be applied to transform each class into the others and then we demonstrated how by using these transforms we can readily extend any successful matrix inversion algorithm from one of these classes to all the others the power of this approach was widely recognized later when novel numerically stable algorithms solved nonsingular toeplitz linear systems of equations in quadratic versus classical cubic arithmetic time based on transforming toeplitz into cauchy matrix structures more recent papers combined the same transformation with a link of the cauchy matrices to the hierarchical semiseparable matrix structure which is a specialization of matrix representations employed by the fast multipole method this produced numerically stable algorithms that approximated the solution of a nonsingular toeplitz linear system of equations in nearly linear arithmetic time we first revisit the successful method of structure transformation covering it comprehensively then we analyze the latter approximation algorithms for toeplitz linear systems and extend them to approximate multiplication of vandermonde and cauchy matrices by a vector and to approximate solution of vandermonde and cauchy linear systems of equations provided that they are nonsingular and well conditioned we decrease the arithmetic cost of the known numerical approximation algorithms for these tasks from quadratic to nearly linear and similarly for the computations with the matrices having structures that generalize the structures of vandermonde and cauchy matrices and for polynomial and rational evaluation and interpolation we also accelerate a little further the known numerical approximation algorithms for a nonsingular toeplitz or toeplitz like linear system by employing distinct transformations of matrix structures and we briefly comment on some natural research challenges particularly some promising applications of our techniques to high precision computations

possibilities and limitations of gaussian closure approximations for phase ordering dynamics the nonlinear equations describing phase ordering dynamics can be closed by assuming the existence of an underlying gaussian stochastic field which is nonlinearly related to the observable order parameter field we discuss the relation between different implementations of the gaussian assumption and consider the limitations of this assumption for phase ordering dynamics the fact that the different approaches gives different results is a sign of the breakdown of the gaussian assumption we concentrate on the non conserved order parameter case but also touch on the conserved order parameter case we demonstrate that the gaussian assumption is fundamentally flawed in the latter case

solving structured linear systems with large displacement rank linear systems with structures such as toeplitz vandermonde or cauchy likeness can be solved in email protected a 2n operations where n is the matrix size a is its displacement rank and email protected denotes the omission of logarithmic factors we show that for such matrices this cost can be reduced to email protected a w 1n where w is a feasible exponent for matrix multiplication over the base field the best known estimate for w is w 2 38 resulting in costs of order email protected a 1 3 8n we present consequences for hermite pade approximation and bivariate interpolation

phase ordering dynamics of a vector order parameter mazenko s theory of phase ordering dynamics is generalized to an n component nonconserved vector order parameter the scaling functions for the equal time and two time correlation functions are calculated as well as the exponent characterizing the decay of autocorrelations the equal time correlation function is numerically quite close to that recently calculated by bray and puri 1991 and by toyoki 1991 and exhibits the same power law tail in its fourier transform the time dependent structure factor

a rasch model for partial credit scoring a unidimensional latent trait model for responses scored in two or more ordered categories is developed this partial credit model is a member of the family of latent trait models which share the property of parameter separability and so permit specifically objective comparisons of persons and items the model can be viewed as an extension of andrich s rating scale model to situations in which ordered response alternatives are free to vary in number and structure from item to item the difference between the parameters in this model and the category boundaries in samejima s graded response model is demonstrated an unconditional maximum likelihood procedure for estimating the model parameters is developed

solving sparse linear equations over finite fields a coordinate recurrence method for solving sparse systems of linear equations over finite fields is described the algorithms discussed all require o n 1 omega n 1 log k n 1 field operations where n 1 is the maximum dimension of the coefficient matrix omega is approximately the number of field operations required to apply the matrix to a test vector and the value of k depends on the algorithm a probabilistic algorithm is shown to exist for finding the determinant of a square matrix also probabilistic algorithms are shown to exist for finding the minimum polynomial and rank with some arbitrarily small possibility of error

hensel and newton methods in valuation rings we give a computational description of hensel s method for lifting approximate factorizations of polynomials the general setting of valuation rings provides the framework for this and the other results of the paper we describe a newton method for solving algebraic and differential equations finally we discuss a fast algorithm for factoring polynomials via computing short vectors in modules

how bad are hankel matrices considered are hankel vandermonde and krylov basis matrices it is proved that for any real positive definite hankel matrix of order n its spectral condition number is bounded from below by 3 cdot 2 n 6 also proved is that the spectral condition number of a krylov basis matrix is bounded from below by 3 frac 1 2 cdot 2 frac n 2 3 for v v x 1 ldots x n a vandermonde matrix with arbitrary but pairwise distinct nodes x 1 ldots x n we show that mbox rm cond 2 v geq 2 n 2 n frac 1 2 if either x j leq 1 or x j geq 1 for all j then mbox rm cond 2 v geq 2 n 2

spatial structure in diffusion limited two particle reactions we analyze the limiting behavior of the densities a t and b t and the random spatial structure r a t b t for the diffusion controlled chemical reaction a b inert for equal initial densities b 0 b 0 there is a change in behavior fromd 4 where a t b t c td 4 tod 4 where a t b t c t ast the termc depends on the initial densities and changes withd there is a corresponding change in the spatial structure ind 4 both particle types are after large times present locally in concentrations not depending on type or location ind 4 both particle types are present locally but with random concentrations and the process tends to a limit

marginal maximum likelihood estimation of item parameters application of an em algorithm maximum likelihood estimation of item parameters in the marginal distribution integrating over the distribution of ability becomes practical when computing procedures based on an em algorithm are used by characterizing the ability distribution empirically arbitrary assumptions about its form are avoided the em procedure is shown to apply to general item response models lacking simple sufficient statistics for ability this includes models with more than one latent dimension

a generalized speed accuracy response model for dichotomous items we propose a generalization of the speed accuracy response model sarm introduced by maris and van der maas psychometrika 77 615 633 2012 in these models the scores that result from a scoring rule that incorporates both the speed and accuracy of item responses are modeled our generalization is similar to that of the one parameter logistic or rasch model to the two parameter logistic or birnbaum model in item response theory an expectation maximization em algorithm for estimating model parameters and standard errors was developed furthermore methods to assess model fit are provided in the form of generalized residuals for item score functions and saddlepoint approximations to the density of the sum score the presented methods were evaluated in a small simulation study the results of which indicated good parameter recovery and reasonable type i error rates for the residuals finally the methods were applied to two real data sets it was found that the two parameter sarm showed improved fit compared to the one parameter sarm in both data sets

parallel computation of polynomial gcd and some related parallel computations over abstract fields several fundamental problems of computations with polynomials and structured matrices are well known for their resistance to effective parallel solution this includes computing the gcd the 1cm and the resultant of two polynomials as well as any selected entry of either the extended euclidean scheme for these polynomials or the pade approximation table associated to a fixed power series the solution of the berlekamp massey problem of recovering the n coefficients of a linear recurrence sequence from its first 2n terms the solution of a toeplitz linear system of equations computing the ranks and the null spaces of toeplitz matrices and several other computations with toeplitz hankel vandermonde and cauchy generalized hilbert matrices and with matrices having similar structures our algorithms enable us to reach new record estimates for randomized parallel arithmetic complexity of all these computations over any field of constants that is o log n h time and o n2 log log n log n h g arithmetic processors where n is the input size g varies from 1 to 2 and h varies from 2 to 4 depending on the computational problem and on the fixed field of constants the estimates include the cost of correctness tests for the solution the results have further applications to many related computations over abstract fields and rely on some new techniques in particular on recursive pade approximation and regularization of pade approximation via randomization which might be of some independent technical interest in particular an auxiliary algorithm computes over any field the coefficients of a polynomial from the power sums of its zeros which is an important problem of algebraic coding

complexity of parallel matrix computations abstract we estimate parallel complexity of several matrix computations under both boolean and arithmetic machine models using deterministic and probabilistic approaches those computations include the evaluation of the inverse the determinant and the characteristic polynomial of a matrix recently processor efficiency of the previous parallel algorithms for numerical matrix inversion has been substantially improved in pan and reif 1987 reaching optimum estimates up to within a logarithmic factor that work however applies neither to the evaluation of the determinant and the characteristic polynomial nor to exact matrix inversion nor to the numerical inversion of ill conditioned matrices we present four new approaches to the solution of those latter problems having several applications to combinatorial computations in order to extend the suboptimum time and processor bounds of pan and reif 1987 to the case of computing the inverse determinant and characteristic polynomial of an arbitrary integer input matrix in addition processor efficient algorithms using polylogarithmic parallel time are devised for some other matrix computations such as triangular and qr factorizations of a matrix and its reduction to hessenberg form

equation free implementation of statistical moment closures we present a general numerical scheme for the practical implementation of statistical moment closures suitable for modeling complex large scale nonlinear systems building on recently developed equation free methods this approach numerically integrates the closure dynamics the equations of which may not even be available in closed form although closure dynamics introduce statistical assumptions of unknown validity they can have significant computational advantages as they typically have fewer degrees of freedom and may be much less stiff than the original detailed model the numerical closure approach can in principle be applied to a wide class of nonlinear problems including strongly coupled systems either deterministic or stochastic for which there may be no scale separation we demonstrate the equation free approach for implementing entropy based eyink levermore closures on a nonlinear stochastic partial differential equation

the dutch identity a new tool for the study of item response models the dutch identity is a useful way to reexpress the basic equations of item response models that relate the manifest probabilities to the item response functions irfs and the latent trait distribution the identity may be exploited in several ways for example a to suggest how item response models behave for large numbers of items they are approximate submodels of second order loglinear models for 2 j tables b to suggest new ways to assess the dimensionality of the latent trait principle components analysis of matrices composed of second order interactions from loglinear models c to give insight into the structure of latent class models and d to illuminate the problem of identifying the irfs and the latent trait distribution from sample data

fast gaussian elimination with partial pivoting for matrices with displacement structure fast o n 2 implementation of gaussian elimination with partial pivoting is designed for matrices possessing cauchy like displacement structure we show how toeplitz like toeplitz plus hankel like and vandermonde like matrices can be transformed into cauchy like matrices by using discrete fourier cosine or sine transform matrices in particular this allows us to propose a new fast o n 2 toeplitz solver gko which incorporates partial pivoting a large set of numerical examples showed that gko demonstrated stable numerical behavior and can be recommended for solving linear systems especially with nonsymmetric indefinite and ill conditioned positive definite toeplitz matrices it is also useful for block toeplitz and mosaic toeplitz toeplitz block matrices the algorithms proposed in this paper suggest an attractive alternative to look ahead approaches where one has to jump over ill conditioned leading submatrices which in the worst case requires o n 3 operations

renormalization group study of thea b diffusion limited reaction thea b diffusion limited reaction with equal initial densitiesa 0 b 0 n0 is studied by means of a field theoretic renormalization group formulation of the problem for dimensiond 2 an effective theory is derived from which the density and correlation functions call be calculated we find the density decays in time as a b sim c sqrt delta dt d 4 ford 4 with n0 c n 0 d 2 wherec is a universal constant andc is nonuniversal the calculation is extended to the case of unequal diffusion constantsd a d b resulting in a new amplitude but the same exponent ford 2 a controlled calculation is not possible but a heuristic argument is presented that the results above give at least the leading term in an e 2 d expansion finally we address reaction zones formed in the steady state by opposing currents ofa andb particles and derive scaling properties

energy scaling approach to phase ordering growth laws we present a simple unified approach to determining the growth law for the characteristic length scale l t in the phase ordering kinetics of a system quenched from a disordered phase to within an ordered phase this approach based on a scaling assumption for pair correlations determines l t self consistently for purely dissipative dynamics by computig the time dependence of the energy in two ways we derive growth laws for conserved and nonconserved o n models including two dimensional xy models and systems with textures we demonstrate that the growth laws for other systems such as liquid crystals and potts models are determined by the type of topological defect in the order parameter field that dominates the energy we also obtain generalized porod laws for systems with topological textures

exact solution of linear equations usingp adic expansions a method is described for computing the exact rational solution to a regular systemax b of linear equations with integer coefficients the method involves i computing the inverse modp ofa for some primep ii using successive refinements to compute an integer vector bar x such that a bar x equiv b modp m for a suitably large integerm and iii deducing the rational solutionx from thep adic approximation bar x for matricesa andb with entries of bounded size and dimensionsn n andn 1 this method can be implemented in timeo n 3 logn 2 which is better than methods previously used

on practical algorithms for accelerated matrix multiplication abstract we present several bilinear algorithms for the acceleration of multiplication of n x n matrices that are superior to both the classical and strassen s algorithm for moderate n starting with n 20 the bini lotti result on the weak stability of bilinear algorithms applies to these algorithms we present them in two equivalent versions bilinear and trilinear we also apply one of these algorithms over finite fields or rings of constants surprisingly this enables us to decrease the bilinear complexity of n x n matrix multiplication for 20 n 1020 below the current record upper bound for the same computation over the infinite fields of complex real or rational numbers

doi peliti path integral methods for stochastic systems with partial exclusion abstract doi peliti methods are developed for stochastic models with finite maximum occupation numbers per site we provide a generalized framework for the different fock spaces reported in the literature paragrassmannian techniques are then utilized to construct path integral formulations of factorial moments we show that for many models of interest a magnus expansion is required to construct a suitable action meaning actions containing a finite number of terms are not always feasible however for such systems perturbative techniques are still viable and for some examples including carrying capacity population dynamics and diffusion with partial exclusion the expansions are exactly summable

kinetics of phase ordering in the o n model with a conserved order parameter we study the phase ordering dynamics of the o n model with a conserved order parameter for systems with topological defects we present results from both cell dynamical simulations and predictions of a gaussian auxiliary field gaf approximation for the xy n 2 model in two and three dimensions and the heisenberg n 3 model in three dimensions we describe the results for the structure factor s q and growth law l t from simulations the growth laws obtained are consistent with theoretical predictions based on energy scaling arguments the structure factor shows good dynamical scaling using a length extracted from its first moment the simulations are compared with the theoretical predictions of the gaf for the scaling functions our results show that the gaf gives a good qualitative description of most features of the structure factor however it overestimates the amplitude of the porod tail in the large q limit moreover for small q the structure factor exhibits a q2 behaviour instead of the expected generalized yeung result of q4

how can we speed up matrix multiplication due to the new algebraic methods of algorithm design recently it became possible to perform multiplication and inversion of n times n matrices using o n 2 496 rather than o n 3 arithmetical operations consequently algorithms for several other computational problems of linear algebra and combinatorics have been accelerated the major ideas and techniques that have led to that progress are surveyed

degeneration of structured integer matrices modulo an integer abstract hensel s lifting modulo a prime q is a customary means of the solution of an integer or rational linear system of equations in combination with some effective numerical algorithms this technique enables solution in nearly optimal time in the case of most popular structured inputs practically one can further benefit from choosing q 2 v for a proper positive integer v and performing binary computations within the computer precision if the input matrix becomes singular because of the reduction modulo q then the approach fails for larger integers q and random integer input matrices however such degeneration occurs rarely according to the analysis by brent and mckay 1987 based on distinct techniques we show that degeneration also occurs rarely for random integer matrices with all most popular structures such as the toeplitz hankel band and rank quasiseparable structures furthermore with random small rank modifications of an input matrix we have good chances to overcome degeneration safely solve the new linear system and recover the solution of the original one the results of our extensive tests support our formal analysis

nearly optimal solution of rational linear systems of equations with symbolic lifting and numerical initialization abstract hensel s symbolic lifting is a highly effective method for the solution of a general or structured e g toeplitz or hankel linear system of equations with integer or rational coefficients of bounded length it can handle ill conditioned inputs for which numerical methods become costly lifting amounts to recursive multiplications by vectors of the input coefficient matrices and its precomputed inverse modulo a fixed integer s such multiplications only involve small numbers of data movements and arithmetic operations with bounded precision the known methods for precomputation of the inverse are more costly however in particular they involve more data movements as our remedy for this bottleneck stage we create an auxiliary matrix sharing its inverse modulo s with the input matrix and we readily compute this inverse by applying numerical iterative refinement which is a numerical counterpart of lifting in the case of general unstructured as well as toeplitz hankel and other popular structured inputs our hybrid algorithms involve a small number of data movements and optimal number of boolean that is bitwise operations up to a logarithmic factor we extend the algorithms to nearly optimal computation of polynomial greatest common divisors gcds least common multiples lcms and pade approximations as well as the berlekamp massey reconstruction of linear recurrences we also cover newton s lifting for matrix inversion specialize it to the case of structured input and combine it with hensel s to enhance the overall efficiency our initialization techniques work for newton s lifting as well furthermore we extend all our lifting algorithms to allow their initialization modulo powers of two thus implementing them in the binary base

wilson renormalization of a reaction diffusion process abstract healthy and sick individuals a and b particles diffuse independently with diffusion constants da and db sick individuals upon encounter infect healthy ones at rate k but may also spontaneously recover at rate 1 the propagation of the epidemic therefore couples to the fluctuations in the total population density global extinction occurs below a critical value c of the spatially averaged total density the epidemic evolves as the diffusion reaction decay process a b 2b b a for which we write down the field theory the stationary state properties of this theory when da db were obtained by kree et al the critical behavior for da db remains unsolved

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

asymptotic behavior of densities for two particle annihilating random walks consider the system of particles on d where particles are of two types a andb and execute simple random walks in continuous time particles do not interact with their own type but when ana particle meets ab particle both disappear i e are annihilated this system serves as a model for the chemical reactiona b inert we analyze the limiting behavior of the densities a t and b t when the initial state is given by homogeneous poisson random fields we prove that for equal initial densities a 0 b 0 there is a change in behavior fromd 4 where a t b t c t d 4 tod 4 where a t b t c tast for unequal initial densities a 0 b 0 a t e c l ind 1 a t e ct logt ind 2 and a t e ct ind 3 the termc depends on the initial densities and changes withd techniques are from interacting particle systems the behavior for this two particle annihilation process has similarities to those for coalescing random walks a a a and annihilating random walks a a inert the analysis of the present process is made considerably more difficult by the lack of comparison with an attractive particle system

renormalisation of fluctuation effects in the a a to a reaction the field theoretical description of the a a to a reaction is exactly renormalised to all orders in perturbation theory it is confirmed that it belong to the same universality class as the a a to oe reaction and that the concentration decay like t d 2 for space dimensionality d smaller than two and like ln t t for d 2 are exact results

generalized residuals for general models for contingency tables with application to item response theory generalized residuals are a tool employed in the analysis of contingency tables to examine possible sources of model error they have typically been applied to log linear models and to latent class models a general approach to generalized residuals is developed for a very general class of models for contingency tables to illustrate their use generalized residuals are applied to models based on item response theory irt models such models are commonly applied to analysis of standardized achievement or aptitude tests to obtain a realistic perspective on application of generalized residuals actual testing data are employed

assessing item fit for unidimensional item response theory models using residuals from estimated item response functions residual analysis e g hambleton swaminathan item response theory principles and applications kluwer academic boston 1985 hambleton swaminathan rogers fundamentals of item response theory sage newbury park 1991 is a popular method to assess fit of item response theory irt models we suggest a form of residual analysis that may be applied to assess item fit for unidimensional irt models the residual analysis consists of a comparison of the maximum likelihood estimate of the item characteristic curve with an alternative ratio estimate of the item characteristic curve the large sample distribution of the residual is proved to be standardized normal when the irt model fits the data we compare the performance of our suggested residual to the standardized residual of hambleton et al fundamentals of item response theory sage newbury park 1991 in a detailed simulation study we then calculate our suggested residuals using data from an operational test the residuals appear to be useful in assessing the item fit for unidimensional irt models

complexity of multiplication with vectors for structured matrices abstract fast algorithms for computing the product with a vector are presented for a number of classes of matrices whose properties relate to the properties of toeplitz vandermonde or cauchy matrices these matrices are defined using the concept of displacement of a matrix and also for their inverses all the actions which are not dependent upon the coordinates of the input vector are singled out in a separate preprocessing stage the proposed algorithms are based on new representations of these matrices involving factor circulants

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

general framework of the non perturbative renormalization group for non equilibrium steady states this paper is devoted to presenting in detail the non perturbative renormalization group nprg formalism to investigate out of equilibrium systems and critical dynamics in statistical physics the general nprg framework for studying non equilibrium steady states in stochastic models is expounded and fundamental technicalities are stressed mainly regarding the role of causality and of it s discretization we analyze the consequences of it s prescription in the nprg framework and eventually provide an adequate regularization to encode them automatically besides we show how to build a supersymmetric nprg formalism with emphasis on time reversal symmetric problems whose supersymmetric structure allows for a particularly simple implementation of nprg in which causality issues are transparent we illustrate the two approaches on the example of model a within the derivative expansion approximation at order 2 and check that they yield identical results we stress though that the framework presented here also applies to genuinely out of equilibrium problems

finding the observed information matrix when using the em algorithm a procedure is derived for extracting the observed information matrix when the em algorithm is used to find maximum likelihood estimates in incomplete data problems the technique requires computation of a complete data gradient vector or second derivative matrix but not those associated with the incomplete data likelihood in addition a method useful in speeding up the convergence of the em algorithm is developed two examples are presented

the aggregation and cancellation techniques as a practical tool for faster matrix multiplication the main purpose of this paper is to present a fast matrix multiplication algorithm taken from the paper of laderman et al linear algebra appl 162 164 1992 557 in a refined compact analytical form and to demonstrate that it can be implemented as quite efficient computer code our improved presentation enables us to simplify substantially the analysis of the computational complexity and numerical stability of the algorithm as well as its computer implementation the algorithm multiplies two n n matrices using o n2 7760 arithmetic operations in the case where n 18 48k for a positive integer k the total number of flops required by the algorithm is 4 894n2 7760 16 165n2 which may be compared to a similar estimate for the winograd algorithm 3 732n2 8074 5n2 flops n 8 2k the latter being current record bound among all known practical algorithms moreover we present a pseudo code of the algorithm which demonstrates its very moderate working memory requirements much smaller than that of the best available implementations of strassen and winograd algorithms for matrices of medium large size say 2000 n 10 000 we consider one level algorithms and compare them with the multilevel strassen and winograd algorithms the results of numerical tests clearly indicate that our accelerated matrix multiplication routines implementing two or three disjoint product based algorithm are comparable in computational time with an implementation of winograd algorithm and clearly outperform it with respect to working space and especially numerical stability the tests were performed for the matrices of the order of up to 7000 both in double and single precision

a field theoretic approach to the wiener sausage the wiener sausage the volume traced out by a sphere attached to a brownian particle is a classical problem in statistics and mathematical physics initially motivated by a range of field theoretic technical questions we present a single loop renormalised perturbation theory of a stochastic process closely related to the wiener sausage which however proves to be exact for the exponents and some amplitudes the field theoretic approach is particularly elegant and very enjoyable to see at work on such a classic problem while we recover a number of known classical results the field theoretic techniques deployed provide a particularly versatile framework which allows easy calculation with different boundary conditions even of higher momenta and more complicated correlation functions at the same time we provide a highly instructive non trivial example for some of the technical particularities of the field theoretic description of stochastic processes such as excluded volume lack of translational invariance and immobile particles the aim of the present work is not to improve upon the well established results for the wiener sausage but to provide a field theoretic approach to it in order to gain a better understanding of the field theoretic obstacles to overcome