on the degrees of bases of free modulesover a polynomial ring let k be an infinite field a the polynomial ring k x 1 x n and f in a n times m a matrix such that rm im f subset a n is a free in particular quillen suslin theorem implies that rm ker f is also free let d be the maximum of the degrees of the entries of f and s the rank of f we show that there exists a basis v 1 ldots v m of a m such that v 1 ldots v m s is a basis of rm ker f f v m s 1 ldots f v m is a basis of rm im f and the degrees of their coordinates are of order m s sd o n 4

new constructive methods in classical ideal theory abstract the recent development of computer algebra allows us to take up problems of classical ideal theory for which constructive solutions by nearly unfeasible algorithms exist like the problem of deciding whether a polynomial belongs to a given ideal g hermann math ann 95 1926 736 788 even macaulay s methods by using h bases are not satisfactory at least when considering modules the concept of grobner bases introduced in computer algebra by buchberger allows us to find feasible algorithms for many problems of classical ideal theory by transforming them into some low dimensional systems of linear equations our aim is to demonstrate the effectivity of this new concept considering only polynomial rings p k x 1 x n k a field we generalize in part i the notion of grobner bases to modules present some equivalent properties characterizing these bases and show their good computational qualities in part ii we construct starting with grobner bases of ideals explicit free solutions where all the bases of submodules under consideration are also grobner bases and present a reduction strategy for obtaining shorter resolutions where all the bases have also favourable properties t bases part iii deals with applications we propose a method for constructing minimal resolutions starting with the resolutions of part ii and present a convenient way for computing the hilbert function

on fitting the pareto levy distribution to stock market index data selecting a suitable cutoff value abstract the so called pareto levy or power law distribution has been successfully used as a model to describe probabilities associated to extreme variations of stock markets indexes worldwide the selection of the threshold parameter from empirical data and consequently the determination of the exponent of the distribution is often done using a simple graphical method based on a log log scale where a power law probability plot shows a straight line with slope equal to the exponent of the power law distribution this procedure can be considered subjective particularly with regard to the choice of the threshold or cutoff parameter in this work a more objective procedure based on a statistical measure of discrepancy between the empirical and the pareto levy distribution is presented the technique is illustrated for data sets from the new york stock exchange djia and the mexican stock market ipc

multi d wavelet filter bank design using quillen suslin theorem for laurent polynomials in this paper we present a new approach for constructing the wavelet filter bank our approach enables constructing nonseparable multidimensional non redundant wavelet filter banks with fir filters using the quillen suslin theorem for laurent polynomials our construction method presents some advantages over the traditional methods of multidimensional wavelet filter bank design first it works for any spatial dimension and for any sampling matrix second it does not require the initial lowpass filters to satisfy any additional assumption such as interpolatory condition third it provides an algorithm for constructing a wavelet filter bank from a single lowpass filter so that its vanishing moments are at least as many as the accuracy number of the lowpass filter

highly efficient strong stability preserving runge kutta methods with low storage implementations strong stability preserving ssp runge kutta methods were developed for time integration of semidiscretizations of partial differential equations ssp methods preserve stability properties satisfied by forward euler time integration under a modified time step restriction we consider the problem of finding explicit runge kutta methods with optimal ssp time step restrictions first for the case of linear autonomous ordinary differential equations and then for nonlinear or nonautonomous equations by using alternate formulations of the associated optimization problems and introducing a new more general class of low storage implementations of runge kutta methods new optimal low storage methods and new low storage implementations of known optimal methods are found the results include families of low storage second and third order methods that achieve the maximum theoretically achievable effective ssp coefficient independent of stage number as well as low storage fourth order methods that are more efficient than current full storage methods the theoretical properties of these methods are confirmed by numerical experiment

on some new low storage implementations of time advancing runge kutta methods in this paper explicit runge kutta rk schemes with minimum storage requirements for systems with very large dimension that arise in the spatial discretization of some partial differential equations are considered a complete study of all four stage fourth order schemes of the minimum storage families of williamson 1980 2 van der houwen 1977 8 and ketcheson 2010 12 that require only two storage locations per variable is carried out it is found that whereas there exist no schemes of this type in the williamson and van der houwen families there are two isolated schemes and a one parameter family of fourth order schemes in four stage ketcheson s family this available parameter is used to obtain the optimal scheme taking into account the 2 norm of the coefficients of the leading error term in addition a new alternative minimum storage family to the s stage ketcheson that depends also on 3 s 3 free parameters is proposed this family contains both the williamson and van der houwen schemes but it is not included in ketcheson s family finally the results of some numerical experiments are presented to show the behavior of fourth order optimal schemes for some nonlinear problems

multivariate mimo fir inverses the problem of computing exact finite impulse response fir inverses for multivariate multiple input multiple output mimo fir systems is considered necessary and sufficient conditions for invertibility are given along with computation techniques random systems and structured systems are defined necessary and sufficient conditions for the almost sure invertibility of structured random systems are derived bounds on the orders of the inverse filters are computed

internal error propagation in explicit runge kutta methods in practical computation with runge kutta methods the stage equations are not satisfied exactly due to roundoff errors algebraic solver errors and so forth we show by example that propagation of such errors within a single step can have catastrophic effects for otherwise practical and well known methods we perform a general analysis of internal error propagation emphasizing that it depends significantly on how the method is implemented we show that for a fixed method essentially any set of internal stability polynomials can be obtained by modifying the implementation details we provide bounds on the internal error amplification constants for some classes of methods with many stages including strong stability preserving methods and extrapolation methods these results are used to prove error bounds in the presence of roundoff or other internal errors

stepsize restrictions for the total variation diminishing property in general runge kutta methods much attention has been paid in the literature to total variation diminishing tvd numerical processes in the solution of nonlinear hyperbolic differential equations for special runge kutta methods conditions on the stepsize were derived that are sufficient for the tvd property see e g shu and osher j comput phys 77 1988 pp 439 471 and gottlieb and shu math comp 67 1998 pp 73 85 various basic questions are still open regarding the following issues 1 the extension of the above conditions to more general runge kutta methods 2 simple restrictions on the stepsize which are not only sufficient but at the same time necessary for the tvd property and 3 the determination of optimal runge kutta methods with the tvd property in this paper we propose a theory by means of which we are able to clarify the above questions moreover by applying our theory we settle analogous questions regarding the related strong stability preserving ssp property see e g gottlieb shu and

portfolio theory information theory and tsallis statistics abstract we developed a strategic of optimal portfolio based on information theory and tsallis statistics the growth rate of a stock market is defined by using q deformed functions and we find that the wealth after n days with the optimal portfolio is given by a q exponential function in this context the asymptotic optimality is investigated on causal portfolios showing advantages of the optimal portfolio over an arbitrary choice of causal portfolios finally we apply the formulation in a small number of stocks in brazilian stock market b 3 and analyzed the results

an algorithmic proof of suslin s stability theorem for polynomial rings abstract let k be a field then gaussian elimination over k and the euclidean division algorithm for the univariate polynomial ring k x allow us to write any matrix in sln k or sln k x n 2 as a product of elementary matrices suslin s stability theorem states that the same is true for sln k xl xm with n 3 and m 1 in this paper we present an algorithmic proof of suslin s stability theorem thus providing a method for finding an explicit factorization of a given polynomial matrix into elementary matrices grobner basis techniques may be used in the implementation of the algorithm

stock market return distributions from past to present abstract we show that recent stock market fluctuations are characterized by the cumulative distributions whose tails on short minute time scales exhibit power scaling with the scaling index 3 and this index tends to increase quickly with decreasing sampling frequency our study is based on high frequency recordings of the s p500 dax and wig20 indices over the interval may 2004 may 2006 our findings suggest that dynamics of the contemporary market may differ from the one observed in the past this effect indicates a constantly increasing efficiency of world markets

algorithms for the quillen suslin theorem abstract we give an algorithm for computing a free basis of a projective c x1 xn module which is presented as the image kernel or cokernel of a polynomial matrix our method can be implemented using grobner bases and it provides an elementary constructive proof of the quillen suslin theorem

an extension and analysis of the shu osher representation of runge kutta methods in the context of solving nonlinear partial differential equations shu and osher introduced representations of explicit runge kutta methods which lead to stepsize conditions under which the numerical process is total variation diminishing tvd much attention has been paid to these representations in the literature in general a shu osher representation of a given runge kutta method is not unique therefore of special importance are representations of a given method which are best possible with regard to the stepsize condition that can be derived from them several basic questions are still open notably regarding the following issues 1 the formulation of a simple and general strategy for finding a best possible shu osher representation for any given runge kutta method 2 the question of whether the tvd property of a given runge kutta method can still be guaranteed when the stepsize condition corresponding to a best possible shu osher representation of the method is violated 3 the generalization of the shu osher approach to general possibly implicit runge kutta methods in this paper we give an extension and analysis of the original shu osher representation by means of which the above questions can be settled moreover we clarify analogous questions regarding properties which are referred to in the literature by the terms monotonicity and strong stability preserving ssp

factoring and decomposing a class of linear functional systems abstract within a constructive homological algebra approach we study the factorization and decomposition problems for a class of linear functional determined over determined under determined systems using the concept of ore algebras of functional operators e g ordinary partial differential operators shift operators time delay operators we first concentrate on the computation of morphisms from a finitely presented left module m over an ore algebra to another one m where m resp m is a module intrinsically associated with the linear functional system ry 0 resp r z 0 these morphisms define applications sending solutions of the system r z 0 to solutions of r y 0 we explicitly characterize the kernel image cokernel and coimage of a general morphism we then show that the existence of a non injective endomorphism of the module m is equivalent to the existence of a non trivial factorization r r 2 r 1 of the system matrix r the corresponding system can then be integrated in cascade under certain conditions we also show that the system ry 0 is equivalent to a system r z 0 where r is a block triangular matrix of the same size as r we show that the existence of idempotents of the endomorphism ring of the module m allows us to reduce the integration of the system ry 0 to the integration of two independent systems r 1 y 1 0 and r 2 y 2 0 furthermore we prove that under certain conditions idempotents provide decompositions of the system ry 0 i e they allow us to compute an equivalent system r z 0 where r is a block diagonal matrix of the same size as r applications of these results in mathematical physics and control theory are given finally the different algorithms of the paper are implemented in the maple package m orphisms based on the library oremodules

representations of runge kutta methods and strong stability preserving methods over the last few years a great effort has been made to develop monotone high order explicit runge kutta methods by means of their shu osher representations in this context the stepsize restriction to obtain numerical monotonicity is normally computed using the optimal representation in this paper we extend the shu osher representations for any runge kutta method giving sufficient conditions for monotonicity we show how optimal shu osher representations can be constructed from the butcher tableau of a runge kutta method the optimum stepsize restriction for monotonicity is given by the radius of absolute monotonicity of the runge kutta method l ferracina and m n spijker siam j numer anal 42 2004 pp 1073 1093 and hence if this radius is zero the method is not monotone in the shu osher representation methods with zero radius require negative coefficients and to deal with them an extra associate problem is considered in this paper we interpret these schemes as representations of perturbed runge kutta methods we extend the concept of radius of absolute monotonicity and give sufficient conditions for monotonicity optimal representations can be constructed from the butcher tableau of a perturbed runge kutta method

semi discrete discontinuous galerkin methods and stage exceeding order strong stability preserving runge kutta time discretizations this paper investigates the use of a special class of strong stability preserving ssp runge kutta time discretization methods in conjunction with discontinuous galerkin dg finite element spatial discretizatons the class of ssp methods investigated here is defined by the property that the number of stages s is greater than the order k of the method from analysis cfl conditions for the linear l 2 stability of the methods defined using the s k ssp schemes are obtained that are less restrictive than those of the standard so called rkdg methods that use s k ssp runge kutta schemes the improvement in the cfl conditions for linear stability of the methods more than offsets the additional work introduced by the increased number of stages given that the cfl conditions for linear stability are what must be respected in practice in order to maintain high order accuracy the use of the s k ssp schemes results in rkdg methods that are more efficient than those previously defined furthermore with the application of a slope limiter the nonlinear stability properties of the forward euler method and the dg spatial discretization which have been previously proven are preserved with these methods under less restrictive cfl conditions than those required for linear stability thus more efficient rkdg methods that possess the same favorable accuracy and stability properties of the standard rkdg methods are obtained numerical results verify the cfl conditions for stability obtained from analysis and demonstrate the efficiency advantages of these new rkdg methods

sharp effective nullstellensatz the usual proofs of this result however give no information about the g1 s for instance they give no bound on their degrees this question was first considered by g hermann h she used elimination theory to get a bound on the degree of the gi s which was doubly exponential in the number of variables her results were later improved in mw and in th all these produce bounds that are doubly exponential in the number of variables a major breakthrough was achieved by brownawell b1 who proved the following result

fractional sylvester gallai theorems we prove fractional analogs of the classical sylvester gallai theorem our theorems translate local information about collinear triples in a set of points into global bounds on the dimension of the set specifically we show that if for every points v in a finite set there are at least v other points u v for which the line through v u contains a third point in v then the v resides in a 13 2 dimensional affine subspace of this result which is one of several variants we study is motivated by questions in theoretical computer science and in particular from the area of error correcting codes our proofs combine algebraic analytic and combinatorial arguments a key ingredient is a new lower bound for the rank of design matrices specified only by conditions on their zero non zero pattern

high order linear multistep methods with general monotonicity and boundedness properties we consider linear multistep methods that possess general monotonicity and boundedness properties strict monotonicity in terms of arbitrary starting values for the multistep schemes is only valid for a small class of methods under very stringent step size restrictions this makes them uncompetitive with the strong stability preserving ssp runge kutta methods by relaxing these strict monotonicity requirements a larger class of methods can be considered including many methods of practical interest in this paper we construct linear multistep methods of high order up to six that possess relaxed monotonicity or boundedness properties with optimal step size conditions numerical experiments show that the new schemes perform much better than the classical monotonicity preserving multistep schemes moreover there is a substantial gain in efficiency compared to recently constructed ssp runge kutta ssprk methods

an algorithm for unimodular completion over noetherian rings abstract we give an algorithm for the well known result asserting that if r is a polynomial ring in a finite number of variables over a noetherian ring a of krull dimension d then for n max 3 d 2 sl n r acts transitively on um n r for technical reasons we demand that the noetherian ring a has a theory of grobner bases and contains an infinite set e y 1 y 2 such that y i y j a for each i j the most important guiding examples are affine rings k x 1 x m i and localizations of polynomial rings s 1 k x 1 x m with k an infinite field moreover we give an algorithmic proof of suslin s stability theorem over these rings for the purpose to prepare the ground for this algorithmic generalizations of the quillen suslin theorem corresponding to the particular case a is a field we will give in the first section a constructive proof of an important lemma of suslin which is the only nonconstructive step in suslin s second elementary solution of serre s conjecture this lemma says that for a commutative ring a if v 1 x v n x a x where v 1 is monic and n 3 then there exist 1 l e n 1 a x such that res v 1 e 1 1 v 2 v n t res v 1 e 1 l v 2 v n t a thanks to this constructive proof suslin s second proof of serre s conjecture becomes fully constructive

global optimization of explicit strong stability preserving runge kutta methods strong stability preserving runge kutta ssprk methods are a type of time discretization method that are widely used especially for the time evolution of hyperbolic partial differential equations pdes under a suitable stepsize restriction these methods share a desirable nonlinear stability property with the underlying pde e g positivity or stability with respect to total variation this is of particular interest when the solution exhibits shock like or other nonsmooth behaviour a variety of optimality results have been proven for simple ssprk methods however the scope of these results has been limited to low order methods due to the detailed nature of the proofs in this article global optimization software baron is applied to an appropriate mathematical formulation to obtain optimality results for general explicit ssprk methods up to fifth order and explicit low storage ssprk methods up to fourth order throughout our studies allow for the possibility of negative coefficients which correspond to downwind biased spatial discretizations guarantees of optimality are obtained for a variety of third and fourth order schemes where optimality is impractical to guarantee specifically for fifth order methods and certain low storage methods extensive numerical optimizations are carried out to derive numerically optimal schemes as a part of these studies several new schemes arise which have theoretically improved time stepping restrictions over schemes appearing in the recent literature

a generic effective nullstellensatz kollar s sharp effective nullstellensatz which is independent of the number of polynomials is shown to be a generically pessimistic bound a generic effective nullstellensatz is proven the generic bound is approximately inversely proportional to the number of polynomials further the bound is linear rather than exponential in the degrees of the polynomials the generic effective nullstellensatz provides for minimum required equalization filter orders in signal and image processing

strong stability preserving transformed dimsims abstract in this paper we investigate the strong stability preserving ssp property of transformed diagonally implicit multistage integration methods dimsims within this class examples of ssp methods of order p 1 2 3 and 4 and stage order q p have been determined and also suitable starting and finishing procedure have been constructed the numerical experiments performed on a set of test problems have shown that ssp transformed dimsims can be more accurate and competitive with ssp runge kutta methods of the same order

sparse effective membership problems via residue currents we use residue currents on toric varieties to obtain bounds on the degrees of solutions to polynomial ideal membership problems our bounds depend on the volume of the newton polytope of the polynomial system and are therefore well adjusted to sparse polynomial systems we present sparse versions of max nother s af bg theorem macaulay s theorem and kollar s effective nullstellensatz as well as recent results by hickel and andersson gotmark

exact image deconvolution from multiple fir blurs we address the problem of restoring an image from its noisy convolutions with two or more blur functions channels deconvolution from multiple blurs is in general better conditioned than from a single blur and can be performed without regularization for moderate noise levels we characterize the problem of missing data at the image boundaries and show that perfect reconstruction is impossible even in the no noise case almost surely unless there are at least three channels conversely when there are at least three channels we show that perfect reconstruction is not only possible almost surely in the absence of noise but also that it can be accomplished by finite impulse response fir filtering such fir reconstruction is vastly more efficient computationally than the least squares solution and is suitable for low noise levels even in the high noise case the estimates obtained by fir filtering provide useful starting points for iterative least squares algorithms we present results on the minimum possible sizes of such deconvolver filters we derive expressions for the mean square errors in the fir reconstructions and show that performance comparable to that of the least squares reconstruction may be obtained with relatively small deconvolver filters finally we demonstrate the fir reconstruction on synthetic and real data

a comparative study of limiting strategies in discontinuous galerkin schemes for the m1 model of radiation transport abstract the m 1 minimum entropy moment system is a system of hyperbolic balance laws that approximates the radiation transport equation and has many desirable properties among them are symmetric hyperbolicity entropy decay moment realizability and correct behavior in the diffusion and free streaming limits however numerical difficulties arise when approximating the solution of the m 1 model by high order numerical schemes namely maintaining the realizability of the numerical solution and controlling spurious oscillations in this paper we extend a previously constructed one dimensional realizability limiting strategy to 2d in addition we perform a numerical study of various combinations of the realizability limiter and the tvbm local slope limiter on a third order discontinuous galerkin dg scheme on both triangular and rectangular meshes in several test cases we demonstrate that in general a combination of the realizability limiter and a tvbm limiter is necessary to obtain a robust and accurate numerical scheme our code is published so that all results can be reproduced by the reader

high order wave propagation algorithms for hyperbolic systems we present a finite volume method that is applicable to hyperbolic pdes including spatially varying and semilinear nonconservative systems the spatial discretization like that of the well known clawpack software is based on solving riemann problems and calculating fluctuations not fluxes the implementation employs weighted essentially nonoscillatory reconstruction in space and strong stability preserving runge kutta integration in time the method can be extended to arbitrarily high order of accuracy and allows a well balanced implementation for capturing solutions of balance laws near steady state this well balancing is achieved through the f wave riemann solver and a novel wave slope weno reconstruction procedure the wide applicability and advantageous properties of the method are demonstrated through numerical examples including problems in nonconservative form problems with spatially varying fluxes and problems involving near equilibrium solutions of balance laws

efficient low storage runge kutta schemes with optimized stability regions a variety of numerical calculations especially when considering wave propagation are based on the method of lines where time dependent partial differential equations pdes are first discretized in space for the remaining time integration low storage runge kutta schemes are particularly popular due to their efficiency and their reduced memory requirements in this work we present a numerical approach to generate new low storage runge kutta lsrk schemes with optimized stability regions for advection dominated problems adapted to the spectral shape of a given physical problem those methods are found to yield significant performance improvements over previously known lsrk schemes as a concrete example we present time domain calculations of maxwell s equations in fully three dimensional systems discretized by a discontinuous galerkin approach

a generalization of serre s conjecture and some related issues abstract several topics concerned with multivariate polynomial matrices like unimodular matrix completion matrix determinantal or primitive factorization matrix greatest common factor existence and subsequent extraction along with relevant primeness and coprimeness issues are related to a conjecture which may be viewed as a type of generalization of the original serre problem conjecture solved nonconstructively in 1976 and constructively more recently this generalized serre conjecture is proved to be equivalent to several other unsettled conjectures and therfore all these conjectures constitute a complete set in the sense that solution to any one also solves all the remaining

tight relaxations for polynomial optimization and lagrange multiplier expressions this paper proposes tight semidefinite relaxations for polynomial optimization the optimality conditions are investigated we show that generally lagrange multipliers can be expressed as polynomial functions in decision variables over the set of critical points the polynomial expressions are determined by linear equations based on these expressions new lasserre type semidefinite relaxations are constructed for solving the polynomial optimization we show that the hierarchy of new relaxations has finite convergence or equivalently the new relaxations are tight for a finite relaxation order

multidimensional multichannel fir deconvolution using grobner bases we present a new method for general multidimensional multichannel deconvolution with finite impulse response fir convolution and deconvolution filters using grobner bases previous work formulates the problem of multichannel fir deconvolution as the construction of a left inverse of the convolution matrix which is solved by numerical linear algebra however this approach requires the prior information of the support of deconvolution filters using algebraic geometry and grobner bases we find necessary and sufficient conditions for the existence of exact deconvolution fir filters and propose simple algorithms to find these deconvolution filters the main contribution of our work is to extend the previous grobner basis results on multidimensional multichannel deconvolution for polynomial or causal filters to general fir filters the proposed algorithms obtain a set of fir deconvolution filters with a small number of nonzero coefficients a desirable feature in the impulsive noise environment and do not require the prior information of the support moreover we provide a complete characterization of all exact deconvolution fir filters from which good fir deconvolution filters under the additive white noise environment are found simulation results show that our approaches achieve good results under different noise settings

nonextensive statistical features of the polish stock market fluctuations the statistics of return distributions on various time scales constitutes one of the most informative characteristics of the financial dynamics here we present a systematic study of such characteristics for the polish stock market index wig20 over the period 04 01 1999 31 10 2005 for the time lags ranging from 1min1min up to 1 h this market is commonly classified as emerging still on the shortest time scales studied we find that the tails of the return distributions are consistent with the inverse cubic power law as identified previously for majority of the mature markets within the time scales studied a quick and considerable departure from this law towards a gaussian can however be traced interestingly all the forms of the distributions observed can be comprised by the single q gaussians which provide a satisfactory and at the same time compact representation of the distribution of return fluctuations over all magnitudes of their variation the corresponding nonextensivity parameter q was found to systematically decrease when increasing the time scales the temporal correlations quantified here in terms of multifractality provide further arguments in favor of nonextensivity

contractivity in the numerical solution of initial value problems consider a linear autonomous system of ordinary differential equations with the property that the norm u t of each solutionu t satisfies u t u 0 t 0 we call a numerical process for solving such a system contractive if a discrete version of this property holds for the numerical approximations a givenk step method is said to be unconditionally contractive if for each stepsizeh 0 the numerical process is contractive in this paper a general theory is given which yields necessary and sufficient conditions for unconditional contractivity it turns out that unconditionally contractive methods are subject to an order barrierp 1 further the concept of a contractivity threshold is studied which makes it possible to compare the contractivity behaviour of methods with an orderp 1 as well most theoretical results in this paper are formulated for differential equations in arbitrary banach spaces applications are given to numerical methods for solving ordinary as well as partial differential equations

fir perfect signal reconstruction from multiple convolutions minimum deconvolver orders given p co prime finite impulse response fir filters h sub i spl isin r m sub h it well known that there exist q sub i spl isin r m sub q such that spl sigma sub i h sub i q sub i spl delta importantly this enables signal recovery from its convolutions with p spl ges 2 co prime fir filters by fir filtering we show that such q sub i exist almost surely if and only if m sub q spl ges m sub h 1 p 1 where x is the smallest integer greater than or equal to x the results also provide conditions for full rank of certain key matrices arising in the blind multichannel deconvolution problem

division formulas on projective varieties we introduce a division formula on a possibly singular projective subvariety x of complex projective space mathbb p n which e g provides explicit representations of solutions to various polynomial division problems on the affine part of x especially we consider a global effective version of the briancon skoda huneke theorem

the computation of the logarithmic cohomology for plane curves we will give algorithms of computing bases of logarithmic cohomology groups for square free polynomials in two variables

bound preserving modified exponential runge kutta discontinuous galerkin methods for scalar hyperbolic equations with stiff source terms abstract in this paper we develop bound preserving modified exponential runge kutta rk discontinuous galerkin dg schemes to solve scalar hyperbolic equations with stiff source terms by extending the idea in zhang and shu 43 exponential strong stability preserving ssp high order time discretizations are constructed and then modified to overcome the stiffness and preserve the bound of the numerical solutions it is also straightforward to extend the method to two dimensions on rectangular and triangular meshes even though we only discuss the bound preserving limiter for dg schemes it can also be applied to high order finite volume schemes such as weighted essentially non oscillatory weno finite volume schemes as well

generalised summation by parts operators and variable coefficients abstract high order methods for conservation laws can be highly efficient if their stability is ensured a suitable means mimicking estimates of the continuous level is provided by summation by parts sbp operators and the weak enforcement of boundary conditions recently there has been an increasing interest in generalised sbp operators both in the finite difference and the discontinuous galerkin spectral element framework however if generalised sbp operators are used the treatment of the boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely interpolating the product of two functions will in general result in a value different from the product of the interpolations thus desired properties such as conservation and stability are more difficult to obtain here new formulations are proposed allowing the creation of discretisations using general sbp operators that are both conservative and stable thus several shortcomings that might be attributed to generalised sbp operators are overcome cf nordstrom and ruggiu 2017 38 and manzanero et al 2017 39

polynomial automorphisms and grobner reductions abstract letpn k x1 xn be the polynomial algebra over a fieldkof characteristic 0 we show that applying an automorphism to a given polynomialp pnis mimicked by grobner transformations of a basis of the ideal ofpngenerated by partial derivatives of this polynomial in the case ofp2 this yields a miraculously simple algorithm for deciding whether or not a given polynomial fromp2is part of a basis another application is an algorithm which given a polynomialp p2that is part of a basis finds a sequence of elementary automorphisms that reducesptox1 we also speculate on how our method may be used for constructing a possible counterexample to the jacobian conjecture in higher dimensions

sharp estimates for the arithmetic nullstellensatz we present sharp estimates for the degree and the height of the polynomials in the nullstellensatz over the integer ring zz the result improves previous work of philippon berenstein yger and krick pardo we also present degree and height estimates of intrinsic type which depend mainly on the degree and the height of the input polynomial system as an application we derive an effective arithmetic nullstellensatz for sparse polynomial systems the proof of these results relies heavily on the notion of local height of an affine variety defined over a number field we introduce this notion and study its basic properties

scaling of the distribution of fluctuations of financial market indices up to more than one month we find that the distributions for dt 4d 1560 min are consistent with a power law asymptotic behavior characterized by an exponent a 3 well outside the stable levy regime 0 a 2 to test the robustness of the s p result we perform a parallel analysis on two other financial market indices database iv contains 3560 daily records of the nikkei index for the 14 year period 1984 1997 and database v contains 4649 daily records of the hang seng index for the 18 year period 1980 1997 we find estimates ofa consistent with those describing the distribution of s p 500 daily returns one possible reason for the scaling of these distributions is the long persistence of the autocorrelation function of the volatility for time scales longer than dt 3 4 d our results are consistent with a slow convergence to gaussian behavior s1063 651x 99 11211 x pacs number s 05 40 fb 05 45 tp 89 90 1n

pyclaw accessible extensible scalable tools for wave propagation problems development of scientific software involves tradeoffs between ease of use generality and performance we describe the design of a general hyperbolic pde solver that can be operated with the convenience of matlab yet achieves efficiency near that of hand coded fortran and scales to the largest supercomputers this is achieved by using python for most of the code while employing automatically wrapped fortran kernels for computationally intensive routines and using python bindings to interface with a parallel computing library and other numerical packages the software described here is pyclaw a python based structured grid solver for general systems of hyperbolic pdes k t mandli et al pyclaw software version 1 0 http numerics kaust edu sa pyclaw 2011 pyclaw provides a powerful and intuitive interface to the algorithms of the existing fortran codes clawpack and sharpclaw simplifying code development and use while providing massive parallelism and scalable solvers via the petsc library the

tail distribution of index fluctuations in world markets abstract we have investigated the tail distribution of the daily fluctuations in 202 different indices in the stock markets of 59 countries for the time span of the last 20 years power law log normal weibull exponential and power law with exponential cutoff distributions are considered as possible candidates for the tail distribution of the normalized returns it is found that the power exponent depends strongly on the choice of the tail threshold and a sizeable number of indices can be better fitted by a distribution function other than the power law at the region that has power law exponent of 3 also we have found that the power exponent is not an indicator of the maturity of the market

monotonicity preserving linear multistep methods in this paper we provide an analysis of monotonicity properties for linear multistep methods these monotonicity properties include positivity and the diminishing of total variation we also pay particular attention to related boundedness properties such as the total variation bounded tvb property in the analysis the multistep methods are considered in combination with suitable starting procedures this allows for monotonicity statements for classes of methods which are important and often used in practice but which were thus far not covered by theoretical results

making the use of maximal ideals constructive the purpose of this paper is to decipher constructively a lemma of suslin which played a central role in his second solution of serre s problem on projective modules over polynomial rings this lemma says that for a commutative ring a if a x where v 1 is monic and n 3 then there exist c 1 c e n 1 a x such that denoting by w i the first coordinate of c it v 2 v n we have a by the constructive proof we give suslin s proof of serre s problem becomes fully constructive moreover the new method with which we treat this academic example may be a model for miming constructively abstract proofs in which one works modulo a generic maximal ideal in order to prove that an ideal contains 1

absolute monotonicity of polynomials occurring in the numerical solution of initial value problems this paper deals with polynomial approximations o x to the exponential function exp x related to numerical procedures for solving initial value problems motivated by positivity and contractivity requirements imposed on these numerical procedures we study the smallest negative argument denoted by r o at which o is absolutely monotonic for given integersp 1 m 1 we determine the maximum ofr o when o varies over the class of all polynomials of a degree m with phi left x right exp left x right mathcal o left x p 1 right left for x to 0 right forx 0

runge kutta methods with minimum storage implementations solution of partial differential equations by the method of lines requires the integration of large numbers of ordinary differential equations odes in such computations storage requirements are typically one of the main considerations especially if a high order ode solver is required we investigate runge kutta methods that require only two storage locations per ode existing methods of this type require additional memory if an error estimate or the ability to restart a step is required we present a new more general class of methods that provide error estimates and or the ability to restart a step while still employing the minimum possible number of memory registers examples of such methods are found to have good properties

truncated levy walks and an emerging market economic index abstract in this paper we perform a statistical analysis of the major stock index in latin america the sao paulo stock exchange index in brazil ibovespa database contains daily records for the 15 year period 1986 2000 we find that the time evolution of the index of share prices is well described by an exponentially truncated levy flight etlf characterized by a levy exponent 1 6 1 7 and a cutoff exponent 1 7 the etlf statistics accounts for the observed short term large fluctuations of the financial data time series and describes the long term convergence to the gaussian regime we derive the characteristic crossover time scale n c dependence on and according to this model as well as the volatility dependence on and n c we find an uncorrelated behaviour of the historical data and n c 20 trading days which are in numerical agreement with the analytical results this dynamic model provides a framework within which it is possible to develop an efficient risk management and option pricing practice for emerging economies

computation of optimal monotonicity preserving general linear methods monotonicity preserving numerical methods for ordinary differential equations prevent the growth of propagated errors and preserve convex boundedness properties of the solution we formulate the problem of finding optimal monotonicity preserving general linear methods for linear autonomous equations and propose an efficient algorithm for its solution this algorithm reliably finds optimal methods even among classes involving very high order accuracy and that use many steps and or stages the optimality of some recently proposed methods is verified and many more efficient methods are found we use similar algorithms to find optimal strong stability preserving linear multistep methods of both explicit and implicit type including methods for hyperbolic pdes that use downwind biased operators

strong stability preserving explicit runge kutta methods of maximal effective order we apply the concept of effective order to strong stability preserving ssp explicit runge kutta methods relative to classical runge kutta methods methods with an effective order of accuracy are designed to satisfy a relaxed set of order conditions but yield higher order accuracy when composed with special starting and stopping methods we show that this allows the construc tion of four stage ssp methods with effective order four such methods cannot have classical order four however we also prove that effective order five methods like classical order five methods require the use of nonpositive weights and so cannot be ssp by numerical optimization we construct explicit ssp runge kutta methods up to effective order four and establish the optimality of many of them numerical experiments demonstrate the validity of these methods in practice

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem