metastability of the cahn hilliard equation in one space dimension abstract we establish metastability of the one dimensional cahn hilliard equation for initial data that is order one in energy and order one in h 1 away from a point on the so called slow manifold with n well separated layers specifically we show that for such initial data on a system of lengthscale there are three phases of evolution 1 the solution is drawn after a time of order 2 into an algebraically small neighborhood of the n layer branch of the slow manifold 2 the solution is drawn after a time of order 3 into an exponentially small neighborhood of the n layer branch of the slow manifold 3 the solution is trapped for an exponentially long time exponentially close to the n layer branch of the slow manifold the timescale in phase 3 is obtained with the sharp constant in the exponential

slow motion manifolds far from the attractor in multistable reaction diffusion equations abstract we consider a scalar reaction diffusion equation with multistable nonlinearity with a particular symmetry by reduction to a family of transmission problems in r and by contraction arguments a manifold close to an invariant manifold formed by functions exhibiting a pattern of transition layers is constructed an approximation for the associated vector field is also provided this shows that the motion on those manifolds is exponentially slow as in the well known case of the bistable equation however in opposition to the bistable case some of these manifolds are far from the attractor since these manifolds correspond to metastable patterns this shows the importance of the transient motion toward the attractor and the importance of these manifolds in organizing that motion it is also shown that by a suitable perturbation we can obtain new equilibria on those manifolds

on the complexity of computing the volume of a polyhedron we show that computing the volume of a polyhedron given either as a list of facets or as a list of vertices is as hard as computing the permanent of a matrix

polytope volume computation a combinatorial form of gram s relation for convex polytopes can be adapted for use in computing polytope volume we present an algorithm for volume computation based on this observation this algorithm is useful in finding the volume of a polytope given as the solution set of a system of linear inequalities p x e r ax b as an illustration we compute a formula for the volume of a projective image of the n cube from this formula we deduce that when a and b have rational entries so that the volume of p is also a rational number the number of binary digits in the denominator of the volume cannot be bounded by a polynomial in the total number of digits in the numerators and denominators of entries of a and b this settles a question posed by dyer and frieze

fast mcmc sampling algorithms on polytopes we propose and analyze two new mcmc sampling algorithms the vaidya walk and the john walk for generating samples from the uniform distribution over a polytope both random walks are sampling algorithms derived from interior point methods the former is based on volumetric logarithmic barrier introduced by vaidya whereas the latter uses john s ellipsoids we show that the vaidya walk mixes in significantly fewer steps than the logarithmic barrier based dikin walk studied in past work for a polytope in d defined by n d linear constraints we show that the mixing time from a warm start is bounded as o n0 5d1 5 compared to the o nd mixing time bound for the dikin walk the cost of each step of the vaidya walk is of the same order as the dikin walk and at most twice as large in terms of constant pre factors for the john walk we prove an o d2 5 log4 n d bound on its mixing time and conjecture that an improved variant of it could achieve a mixing time of o d2 poly log n d additionally we propose variants of the vaidya and john walks that mix in polynomial time from a deterministic starting point the speed up of the vaidya walk over the dikin walk are illustrated in numerical examples

computing volumes of polyhedra in this note we give two simple methods for calculating the volume of any closed bounded polyhedron in r having an orientable boundary which is triangulated into a set of n 1 dimensional simplices the formulas given require only coordinates of the vertices of the polyhedron 1 introduction the purpose of this note is to give two simple methods for calculating the volume of any closed bounded polyhedron p in r having an orientable boundary ap which is triangulated into a set t of n 1 dimensional simplices following hadwiger 2 we define a polyhedron to be the union of pairwise disjoint convex polyhedra each of which is the convex hull of a finite number of points in 1 we have described an algorithm for obtaining a piecewise linear manifold which closely approximates an implicitly defined manifold if p has been given in such a way then the affine pieces of ap are in general easy to triangulate with an inherited orientation for polyhedra p which are determined by a given system of inequalities methods and programs for triangulating p have been given in 5 6 for such polyhedra a triangulation of the boundary is not easily available so our method is inappropriate of course our approach would also be unnecessary for computing the volume of a parallelotope practical applications of the methods given here may be made to the approxima tion of an area bounded by an implicitly defined curve or to approximation of the volume of a solid which is bounded by an implicitly defined surface although some formulas for the volume of convex polyhedra in rn appear in the literature e g 3 these formulas generally require the computation of the n 1 volume of the facets and additionally they involve extra computations of certain distances the volume formulas we give here involve only the coordinates of the vertices of p

incomplete partial fractions for parallel evaluation of rational matrix functions abstract frequently one needs to evaluate expressions of the form p a 1q a b where a r n n b r n and p and q are polynomials with degree q degree p and such that no zero of p is an eigenvalue of a algorithms based on the partial fraction representation of q p when evaluating p a 1 q a b lend themselves well to implementation on a parallel computer but might yield poor accuracy we discuss how to determine an incomplete partial fraction representation of q p which allows parallel computation while retaining high accuracy

algorithmic and modeling insights via volumetric comparison of polyhedral relaxations this is mostly a survey on some mathematical results concerning volumes of polytopes of interest in non convex optimization our motivation is in geometrically comparing relaxations in the context of mixed integer linear and nonlinear optimization with the goal of gaining algorithmic and modeling insights we consider relaxations of fixed charge formulations vertex packing polytopes boolean quadric polytopes and relaxations of graphs of monomials on box domains besides surveying the area we do give a few new results and we provide many directions for further work

dynamics of localized structures abstract we describe qualitative behaviour of solutions of the gross pitaevskii equation in 2d in terms of motion of vortices and radiation to this end we introduce the notion of the intervortex energy we develop a rather general adiabatic theory of motion of well separated vortices and present the method of effective action which gives a fairly straightforward justification of this theory finally we mention briefly two special situations where we are able to obtain a rather detailed picture of the vortex dynamics our approach is rather general and is applicable to a wide class of evolution nonlinear equation which exhibit localized stable static solutions it yields description of general time dependent solutions in terms of dynamics of those static solutions glued together

on the spectral side of arthur s trace formula combinatorial setup in arthur s trace formula a ubiquitous role is played by certain limiting expressions arising from piecewise smooth functions with respect to projections of the coxeter fan g m families these include terms resulting from intertwining operators on the spectral side and volumes of polytopes on the geometric side we introduce the combinatorial concept of a compatible family with respect to an arbitrary polyhedral fan and obtain new formulas for the corresponding limiting expressions in this general framework our formulas can be regarded as algebraic generalizations of certain volume formulas for convex polytopes in a companion paper the results are used to study the spectral side of the trace formula

bayesian analysis of binary sequences this manuscript details bayesian methodology for learning by example with binary n sequences encoding the objects under consideration priors prove influential conformable priors are described laplace approximation of bayes integrals yields posterior likelihoods for all n sequences this involves the optimization of a definite function over a convex domain efficiently effectuated by the sequential application of the quadratic program

a sweep plane algorithm for generating random tuples in simple polytopes a sweep plane algorithm by lawrence for convex polytope computation is adapted to generate random tuples on simple polytopes in our method an affine hyperplane is swept through the given polytope until a random fraction sampled from a proper univariate distribution of the volume of the polytope is covered then the intersection of the plane with the polytope is a simple polytope with smaller dimension in the second part we apply this method to construct a black box algorithm for log concave and t concave multivariate distributions by means of transformed density rejection author s abstract

slow dynamics for the cahn hilliard equation in higher space dimensions the motion of bubbles it is known that the van der waals cahn hilliard w c h dynamics can be approximated by a quasi static stefan problem with surface tension it turns out that the stefan problem has a manifold of equilibria equal in dimension to that of the domain any sphere of fixed radius with interface contained in the domain is an equilibrium indistinguishable from the point of view of the perimeter functional we resolve this degeneracy by showing that at the w c h level this manifold is replaced by a quasi invariant stable manifold on which the typical solution preserves its bubble like shape until it reaches the boundary moreover we show that the bubble moves superslowly we also obtain an equation that determines those special spheres that correspond to equilibria at the w c h level our work establishes the phenomenon of superslow motion in higher space dimensions in the class of single interface solutions

metastability and stability of patterns in a convolution model for phase transitions where e is a small positive parameter and f is bistable having only three zeros 1 a 1 and is balanced i e r 1 1 f u du 1 4 0 see fig 1 for a typical initial datum j which crosses a several times at the beginning the solution evolves as if the diffusion term euxx were not there and in the time scale o ej n ej the solution gets close to 1 or 1 in intervals separated by thin layers of thickness at the order of o ej n ej 17 28 at this moment a spatial pattern is generated it persists on a time scale o ece but is eventually destroyed typically u x t 1 u x t 1 uniformly for x on 1 20 1 as t 1 it is possible that u x t converges to a or a nonconstant steady state as t 1 but this is an unlikely event because all these steady states are unstable 16 27 research supported in part by the national science foundation and bor leqsf

slow motion in higher order systems and g convergence in one space dimension the slow motion in higher order systems and convergence in one space dimension was studied some of the theorems and computations that support and proves the stability of the equations are presented

spectral methods in time for parabolic problems a pseudospectral explicit scheme for solving linear periodic parabolic problems is described it has infinite accuracy both in time and in space the high accuracy is achieved while the time resolution parameter m m o 1 delta t for time marching algorithm and the space resolution parameter n n o 1 delta x must satisfy m o n 1 varepsilon varepsilon 0 compared to the common stability condition m o n 2 which must be satisfied in any explicit finite order time algorithm

an efficient algorithm for obtaining the volume of a special kind of pyramid and application to convex polyhedra an efficient method is given for obtaining the volume of a pyramid of species n 2 whose base is a convex polygon the pyramid is transformed into a simplex whose volume is computed directly a refinement is provided to the cohen hickey method for determining volumes of convex polyhedra

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

metastable bubble solutions for the allen cahn equation with mass conservation in a multidimensional domain the slow motion behavior of internal layer solutions with spherical interfaces referred to as bubble solutions is analyzed for the nonlocal allen cahn equation with mass conservation this problem represents the simplest model for the phase separation of a binary mixture in the presence of a mass constraint the bubble is shown to drift exponentially slowly across the domain without change of shape toward the closest point on the boundary of the domain an explicit ordinary differential equation for the motion of the center of the bubble is derived by extending to a multidimensional setting the asymptotic projection method developed previously by the author to treat metastable problems in one spatial dimension an asymptotic formula for the time of collapse of the bubble against the boundary of the domain is derived in terms of the principal radii of curvature of the boundary at the initial contact point an analogy between slow bubble motion and the classical exit prob

generalized exponential time differencing methods for fractional order problems the main aim of this paper is to discuss the generalization of exponential integrators to differential equations of non integer orders two methods of this kind are devised and the accuracy and stability are investigated some numerical experiments are presented to validate the theoretical findings

on some explicit adams multistep methods for fractional differential equations in this paper we present a family of explicit formulas for the numerical solution of differential equations of fractional order the proposed methods are obtained by modifying in a suitable way fractional adams moulton methods and they represent a way for extending classical adams bashforth multistep methods to the fractional case the attention is hence focused on the investigation of stability properties intervals of stability for k step methods k 1 5 are computed and plots of stability regions in the complex plane are presented

dispersive and friction induced stabilization of the cahn hilliard inverse cascade we discuss the stabilization of the inverse cascade in the large scale instability of the kolmogorov flow described by the complete cahn hilliard equation with inclusion of effect large scale friction and deformation radius the friction and the values halting the inverse cascade at the various possible intermediate states are calculated by means of singular perturbation techniques and compared to the values resulting from numerical simulation of the complete cahn hilliard equation the excellent agreement validates the theory our main result is that the critical values of friction or halting the inverse cascade scale exponentially as a function of the jet separation in the final flow contrary to previous theories and phenomenological approach 2002 elsevier science b v all rights reserved pacs 47 10 g 92 60 ek 47 27 i 47 27 ak 47 35 i

metastable patterns for the cahn hilliard equation part i abstract in this paper we study the dynamics of the one dimensional cahn hilliard equation in a neighborhood of an equilibrium having n 1 transition layers an approximation for an n dimensional invariant manifold and slow channel are defined and it is proved that ff a solution of the cahn hilliard equation starts outside but close to the slow channel around the approximate manifold then it will approach the channel at an exponentially large speed after it enters the slow channel it will follow the approximate manifold with speed o e c and stay in the channel for exponentially large time

a class of explicit multistep exponential integrators for semilinear problems a class of explicit multistep exponential methods for abstract semilinear equations is introduced and analyzed it is shown that the k step method achieves order k for appropriate starting values which can be computed by auxiliary routines or by one strategy proposed in the paper together with some implementation issues numerical illustrations are also provided

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

time splitting schemes for fractional differential equations i smooth solutions we propose three time splitting schemes for nonlinear time fractional differential equations with smooth solutions where the order of the fractional derivative is 0 alpha 1 while one of the schemes is of order alpha the other two schemes are of order 1 alpha and 2 alpha and thus they can be combined to provide flexible numerical methods with convergence order no less than 3 2 we prove the convergence and stability of the proposed schemes numerical examples illustrate the flexibility and the efficiency of these time splitting schemes and show that they work for multirate and stiff time fractional differential systems effectively

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

new polytope decompositions and euler maclaurin formulas for simple integral polytopes abstract we give new weighted decompositions for simple polytopes generalizing previous results of lawrence varchenko and brianchon gram we start with witten s non abelian localization principle in equivariant cohomology for the norm square of the moment map in the context of toric varieties to obtain a decomposition for delzant polytopes then by a purely combinatorial argument we show that this formula holds for any simple polytope as an application we study euler maclaurin formulas

robust multi criteria ranking with additive value models and holistic pair wise preference statements we consider a problem of ranking alternatives based on their deterministic performance evaluations on multiple criteria we apply additive value theory and assume the decision maker s dm preferences to be representable with general additive monotone value functions the dm provides indirect preference information in form of pair wise comparisons of reference alternatives and we use this to derive the set of compatible value functions then this set is analyzed to describe 1 the possible and necessary preference relations 2 probabilities of the possible relations 3 ranges of ranks the alternatives may obtain and 4 the distributions of these ranks our work combines previous results from robust ordinal regression extreme ranking analysis and stochastic multicriteria acceptability analysis under a unified decision support framework we show how the four different results complement each other discuss extensions of the main proposal and demonstrate practical use of the approach by considering a problem of ranking 20 european countries in terms of 4 criteria reflecting the quality of their universities

how to integrate a polynomial over a simplex this paper starts by settling the computational complexity of the problem of integrating a polynomial function f over a rational simplex we prove that the problem is np hard for arbitrary polynomials via a generalization of a theorem of motzkin and straus on the other hand if the polynomial depends only on a fixed number of variables while its degree and the dimension of the simplex are allowed to vary we prove that integration can be done in polynomial time as a consequence for polynomials of fixed total degree there is a polynomial time algorithm as well we explore our algorithms with some experiments we conclude the article with extensions to other polytopes and discussion of other available methods 1

approximate volume and integration for basic semialgebraic sets given a basic compact semialgebraic set mathbf k subset mathbb r n we introduce a methodology that generates a sequence converging to the volume of mathbf k this sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on mathbf k can be approximated as closely as desired which permits the approximation of the integral on mathbf k of any given polynomial the extension to integration against some weight functions is also provided finally some numerical issues associated with the algorithms involved are briefly discussed

the caratheodory fejer method for real rational approximation a caratheodory fejer method is presented for near best real rational approximation on intervals based on the eigenvalue or singular value analysis of a hankel matrix of chebyshev coefficients in approximation of a smooth function f the cf approximant r cf frequently differs from the best approximation r by only one part in millions or billions to account for this we show here under weak assumptions that if f is approximated on varepsilon varepsilon then as varepsilon to 0 f r o varepsilon m n 1 while r cf r o varepsilon 3m 2n 3 in contrast the latter figure would be o varepsilon m n 2 for the chebyshev economization approximant of maehly or the chebyshev pade approximant of gragg it follows that as varepsilon to 0 best approximation error curves approach the real parts of m n 1 winding rational functions of constant modulus to within o varepsilon 3m 2n 3 numerical examples are given

on the convergence of krylov subspace methods for matrix mittag leffler functions in this paper we analyze the convergence of some commonly used krylov subspace methods for computing the action of matrix mittag leffler functions as is well known such functions find application in the solution of fractional differential equations we illustrate the theoretical results by some numerical experiments

euler maclaurin with remainder for a simple integral polytope we give an euler maclaurin formula with remainder for the sum of the values of a smooth function on the integral points in a simple integral polytope we prove this formula by elementary methods

the complexity of vertex enumeration methods the computational complexity of problems relating to the enumeration of all the vertices of a convex polyhedron defined by linear inequalities is examined several published approaches are evaluated in this light a new algorithm is described and shown to have a better complexity estimate than existing methods empirical evidence supporting the theoretical superiority is presented finally vertex enumeration is discussed when the space containing the polyhedra is of fixed dimension and only the size of the inequality system is permitted to vary

spinodal decomposition for the cahn hilliard equation in higher dimensions part i probability and wavelength estimate this paper is the first in a series of two papers addressing the phenomenon of spinodal decomposition for the cahn hilliard equation where is a bounded domain with sufficiently smooth boundary and f is cubic like for example f u u u3 we will present the main ideas of our approach and explain in what way our method differs from known results in one space dimension due to grant 26 furthermore we derive certain probability and wavelength estimates the probability estimate is needed to understand why in a neighborhood of a homogeneous equilibrium u0 of the cahn hilliard equation with mass in the spinodal region a strongly unstable manifold has dominating effects this is demonstrated for the linearized equation but will be essential for the nonlinear setting in the second paper 37 as well moreover we introduce the notion of a characteristic wavelength for the strongly unstable directions

coarsening dynamics of the one dimensional cahn hilliard model the dynamics of one dimensional cahn hilliard model is studied the stationary and particle type solutions the bubbles are perused as a function of initial conditions boundary conditions and system size we characterize the bubble solutions which are involved in the coarsening dynamics and establish the bifurcation scenarios of the system a set of ordinary differential equation permits us to describe the coarsening dynamics in very good agreement with numerical simulations we also compare these dynamics with the bubble dynamics deduced from the classical kink interaction computation where our model seems to be more appropriated in the case of two bubbles we deduce analytical expressions for the bubble s position and the bubble s width besides a simple description of the ulterior dynamics is presented

the euler maclaurin formula for simple integral polytopes we give a euler maclaurin formula with remainder for the sum of a smooth function on the integral points in a simple integral lattice polytope our proof uses elementary methods

a family of adams exponential integrators for fractional linear systems the numerical solution of linear time invariant systems of fractional order is investigated we construct a family of exponential integrators of adams type possessing good convergence and stability properties the methods are devised in order to keep at a suitable level the computational effort necessary to solve problems of large size numerical experiments are provided to validate the theoretical results the effectiveness of the proposed approach is tested and compared to some other classical methods

computing the matrix exponential and other matrix functions abstract in this note we present an algorithm for the computation of matrix functions especially the matrix exponential it is a new application of the so called elimination method which was presented for the scalar case in previous papers e g 5

on accurate product integration rules for linear fractional differential equations this paper addresses the numerical solution of linear fractional differential equations with a forcing term competitive and highly accurate product integration rules are derived by starting from an equivalent formulation in terms of a volterra integral equation with a generalized mittag leffler function in the kernel the error analysis is reported and aspects related to the computational complexity are treated numerical tests confirming the theoretical findings are presented

shadowing matching errors for wave front like solutions abstract consider a singularly perturbed system formula assume that the system has a sequence of regular and internal layers occurring alternatively along the x direction these multiple wave solutions can formally be constructed by matched asymptotic expansions to obtain a genuine solution we derive a spatial shadowing lemma which assures the existence of an exact solution that is near the formal asymptotic series provided 1 the residual errors are small in all the layers and 2 the matching errors are small along the lateral boundaries of the adjacent layers the method should work on some other systems like eu t e 2 d xx m u

valuative invariants for polymatroids abstract many important invariants for matroids and polymatroids such as the tutte polynomial the billera jia reiner quasi symmetric function and the invariant g introduced by the first author are valuative in this paper we construct the z modules of all z valued valuative functions for labeled matroids and polymatroids on a fixed ground set and their unlabeled counterparts the z modules of valuative invariants we give explicit bases for these modules and for their dual modules generated by indicator functions of polytopes and explicit formulas for their ranks our results confirm a conjecture of the first author that g is universal for valuative invariants

dynamics as a mechanism preventing the formation of finer and finer microstructure we study the dynamics of pattern formation in the one dimensional partial differential equation u u w u x x u xxt u 0 text u u x t text x in 0 1 text t 0 proposed recently by ball holmes james pego swart bhjps as a mathematical cartoon for the dynamic formation of microstructures observed in various crystalline solids here w is a double well potential like 1 4 ux 2 1 2 what makes this equation interesting and unusual is that it possesses as a lyapunov function a free energy consisting of kinetic energy plus a nonconvex elastic energy but no interfacial energy contribution which does not attain a minimum but favours the formation of finer and finer phase mixtures e u u t int limits 0 1 frac u t 2 2 w u x frac u 2 2 dx our analysis of the dynamics confirms the following surprising and striking difference between statics and dynamics conjectured in bhjps on the basis of numerical simulations of swart holmes sh while minimizing the above energy predicts infinitely fine patterns mathematically weak but not strong convergence of all minimizing sequences unvn of e u v in the sobolev space w1p 0 1 l2 0 1 solutions to the evolution equation of ball et al typically develop patterns of small but finite length scale mathematically strong convergence in w1p 0 1 l2 0 1 of all solutions u t ut t with low initial energy as time t

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

a statistical approach to the analytic hierarchy process with interval judgements i distributions on feasible regions abstract this paper addresses the problem of extracting preferences for alternatives from interval judgement matrices in the analytic hierarchy process ahp the method of arbel for extracting such preferences which is based on the assumption that the interval judgements specify a feasible region in the weight space of the alternatives is critically appraised from a statistical perspective and some new ideas emanating from this approach are developed and discussed in particular it is proposed that a distribution for the weights on the feasible region which is both tractable and meaningful be adopted the mean of the distribution can then be used as an assessment of the overall ranking of the alternatives and quantities of interest such as probabilities and marginal distributions can immediately be quantified two specific distributions on the feasible region the uniform distribution and the distribution of random convex combinations with coefficients which are uniform spacings are examined in some detail and the ideas which emerge are illustrated by means of selected examples

finding simplicial subdivisions of polytopes subdivisions of polytopes are described which place any chosen vertex in a near minimal number of simplices ancillary procedures to find volumes and centroids of polytopes are indicated

the keller segel model with logistic sensitivity function and small diffusivity the keller segel model is the classical model for chemotaxis of cell populations it consists of a drift diffusion equation for the cell density coupled to an equation for the chemoattractant here a variant of this model is studied in one dimensional position space where the chemotactic drift is turned off for a limiting cell density by a logistic term and where the chemoattractant density solves an elliptic equation modeling a quasi stationary balance of reaction and diffusion with production of the chemoattractant by the cells the case of small cell diffusivity is studied by asymptotic and numerical methods on a time scale characteristic for the convective effects convergence of solutions to weak entropy solutions of the limiting nonlinear hyperbolic conservation law is proven numerical and analytic evidence indicates that solutions of this problem converge to irregular patterns of cell aggregates separated by entropic shocks from vacuum regions as time tends to infinity close to each of these p

on the complexity of computing mixed volumes this paper gives various positive and negative results on the complexity of the problem of computing and approximating mixed volumes of polytopes and more general convex bodies in arbitrary dimension on the negative side we present several p hardness results that focus on the difference of computing mixed volumes versus computing the volume of polytopes we show that computing the volume of zonotopes is p hard while each corresponding mixed volume can be computed easily but also give examples showing that computing mixed volumes is hard even when computing the volume is easy on the positive side we derive a randomized algorithm for computing the mixed volumes v overbrace k 1 ld k 1 m 1 overbrace k 2 dots k 2 m 2 dots overbrace k s dots k s m s of well presented convex bodies k 1 dots k s where m 1 dots m s in n 0 and m 1 geq n psi n with psi n o frac log n log log n the algorithm is an interpolation method based on polynomial time randomized algorithms for computing the volume of convex bodies this paper concludes with applications of our results to various problems in discrete mathematics combinatorics computational convexity algebraic geometry geometry of numbers and operations research

a laplace transform algorithm for the volume of a convex polytope we provide two algorithms for computing the volume of the convex polytope x n ax b for a m n b n the computational complexity of both algorithms is essentially described by nm which makes them especially attractive for large n and relatively small m when the other methods with o mn complexity fail the methodology which differs from previous existing methods uses a laplace transform technique that is well suited to the half space representation of

on conjugate gradient type methods and polynomial preconditioners for a class of complex non hermitian matrices we consider conjugate gradient type methods for the solution of large linear systemsa x b with complex coefficient matrices of the typea t i i wheret is hermitian and a real scalar three different conjugate gradient type approaches with iterates defined by a minimal residual property a galerkin type condition and an euclidean error minimization respectively are investigated in particular we propose numerically stable implementations based on the ideas behind paige and saunder s symmlq and minres for real symmetric matrices and derive error bounds for all three methods it is shown how the special shift structure ofa can be preserved by using polynomial preconditioning and results on the optimal choice of the polynomial preconditioner are given also we report on some numerical experiments for matrices arising from finite difference approximations to the complex helmholtz equation

on mittag leffler type functions in fractional evolution processes abstract we review a variety of fractional evolution processes so defined being governed by equations of fractional order whose solutions turn out to be related to mittag leffler type functions the chosen equations are the simplest of the fractional calculus and include the abel integral equations of the second kind which are relevant in typical inverse problems and the fractional differential equations which govern generalized relaxation and oscillation phenomena

analysis of projection methods for rational function approximation to the matrix exponential krylov subspace methods for approximating the action of the matrix exponential exp a on a vector v are analyzed with a hermitian and negative semidefinite our approach is based on approximating the exponential with the commonly employed diagonal pade and chebyshev rational functions which yield a system of equations with a polynomial coefficient matrix we derive optimality properties and error bounds for the convergence of a galerkin type approximation and of a computationally feasible and extensively used alternative as complementary results we theoretically justify the use of a popular a posteriori error estimate and we provide upper bounds for the components of the solution vector our theoretical and numerical results show that this methodology may provide an appropriate framework to devise new strategies such as more powerful acceleration schemes

linear programming system identification we define a version of the inverse linear programming problem that we call linear programming system identification this version of the problem seeks to identify both the objective function coefficient vector and the constraint matrix of a linear programming problem that best fits a set of observed vector pairs one vector is that of actual decisions that we call outputs these are regarded as approximations of optimal decision vectors the other vector consists of the inputs or resources actually used to produce the corresponding outputs we propose an algorithm for approximating the maximum likelihood solution the major limitation of the method is the computation of exact volumes of convex polytopes a numerical illustration is given for simulated data