rational cohomology of witt groups let k be an algebraically closed field of characteristic p and for each n 0 let w n denote the group of witt vectors of length n w n is a commutative algebraic group for reference see jacobson 2 serre 6 one of the important properties of the witt groups is the following every commutative algebraic k group whose underlying variety is an affine space is a homomorphic image of products of w n we compute the rational cohomology of w n for n 2 h w n k s v n 1 1 l e rn 1 l

on parallel integer merging the problem of merging two sorted arrays a a1 a2 an1 and b b1 b2 bn2 is considered for input elements that are drawn from a domain of integers 1 s we present an algorithm that runs in o log log log s time using n log log log s crew pram processors optimal speed up and o ns space where n n1 n2 for input elements that are drawn from a domain of integers 1 n we present a second algorithm that runs in o n time where n is the inverse of ackermann s function using n n crew pram processors and linear space this second algorithm is non uniform however it can be made uniform at a price of a certain loss of speed or by using a crcw pram

on three conjectures by k e shuler some fifteen years ago shuler formulated three conjectures relating to the large time asymptotic properties of a nearest neighbor random walk on 2 that is allowed to make horizontal steps everywhere but vertical steps only on a random fraction of the columns we give a proof of his conjectures for the situation where the column distribution is stationary and satisfies a certain mixing codition we also prove a strong form of scaling to anisotropic brownian motion as well as a local limit theorem the main ingredient of the proofs is a large deviation estimate for the number of visits to a random set made by a simple random walk on we briefly discuss extensions to higher dimension and to other types of random walk

asymptotic properties of multistate random walks ii applications to inhomogeneous periodic and random lattices the previously developed formalism for the calculation of asymptotic properties of multistate random walks is used to study random walks on several inhomogeneous periodic lattices where the periodically repeated unit cell contains a number of inequivalent sites as well as on lattices with a random distribution of inequivalent sites we concentrate on the question whether the random walk properties depend on the spatial arrangement of the sites in the unit cell or only on the number density of the different types of sites specifically we consider lattices with periodic and random arrangements of columns and lattices with periodic and random arrangements of anisotropic scatterers

two particle cluster integral in the expansion of the dielectric constant we study in detail the two particle cluster integral in the cluster expansion for the effective dielectric constant of a suspension of spherically symmetric polarizable inclusions embedded in a uniform medium although our form for the integrand differs from that derived earlier by finkel berg and by jeffrey we show that the integral is equivalent the two body dielectric problem for particles with an arbitrary radial dependence of the dielectric constant is solved by an expansion in spherical harmonics numerical results for some special models illustrate the importance of multipole contributions to the effective dielectric constant

improved deterministic parallel integer sorting abstract we consider the problem of deterministic sorting of integers on a parallel ram pram the best previous result t hagerup 1987 inform and comput 75 39 51 states that n integers of size polynomial in n can be sorted in time o log n on a priority crcw pram with o n log log n log n processors we prove that n integers drawn from a set 0 m 1 can be sorted on an arbitrary crcw pram in time o log n log log n log log m with a time processor product of o n log log m in particular if m n log n o 1 the time and number of processors used are o log n log log n and o n log log n 2 log n respectively this improves the previous result in several respects the new algorithm is faster it works on a weaker pram model and it is closer to optimality for input numbers of superpolynomial size if log log m o log n log log n the new algorithm is optimally fast for any polynomial number of processors and if log log m 1 1 log log n and log log m 0 log n it has optimal speedup relative to the fastest known sequential algorithm the space needed is o nme for arbitrary but fixed e 0 the sorting algorithm derives its speed from a fast solution to a special list ranking problem of possible independent interest the monotonic list ranking problem in monotonic list ranking each list element has an associated key and the keys are known to increase monotonically along the list we show that monotonic list ranking problems of size n can be solved optimally in time o log n log log n we also discuss and attempt to solve some of the problems arising in the precise description and implementation of parallel recursive algorithms as part of this effort we introduce a new pram variant the allocated pram

improved fast integer sorting in linear space we present a fast deterministic algorithm for integer sorting in linear space our algorithm sorts n integers in the range 0 1 2 m 1 in linear space in o n log log n log log log n time when log m log2 n 0 we can further achieve o n log log n time this improves the o n log log n 2 time bound given in m thorup 1998 in proc 1998 acm siam symp on discrete algorithms soda 98 pp 550 555 this result is obtained by combining our new technique with that of thorup s signature sorting a andersson t hagerup s nilsson and r raman 1995 in proc 1995 symposium on theory of computing pp 427 436 a andersson s result 1996 in proc 1996 ieee symp on foundations of computer science pp 135 141 r raman s result 1996 lecture notes in computer science vol 1136 pp 121 137 springer verlag berlin new york and our previous result y han and x shen 1999 in proc 1999 tenth annual acm siam symposium on discrete algorithms soda 99 baltimore md january pp 419 428 are also used for the design of our algorithms we provide an approach and techniques which are totally different from previous approaches and techniques for the problem as a consequence our technique can be extended to apply to nonconservative sorting and parallel sorting our nonconservative sorting algorithm sorts n integers in 0 1 m 1 in time o n log log n 2 log k log log log n using word length k log m n where k log n our erew parallel algorithm sorts n integers in 0 1 m 1 in o log n 2 time and o n log log n 2 log log log n operations provided log m log n 2

systematic density expansion for random resistor networks the authors present a systematic density expansion for calculating transport properties return probabilities and spectral densities for lattice hopping models with diluted randomness the method is applied to a disordered square lattice where the bonds carry conductances sigma 1 and sigma 2 in concentrations c and 1 c respectively and they calculate explicitly the o c and o c2 contributions to the static conductivity diffusion coefficient and to the coefficient beta c of the long time tail in the velocity autocorrelation function vacf upsilon x 0 upsilon x t approximately beta c t 2 to o c the vacf is calculated for all times and shows negative correlation cage effect for all times

series study of random percolation in three dimensions new high density series data for the mean number percolation probability and susceptibility of finite clusters are presented for bond and site percolation on four standard three dimensional lattices a pade approximant analysis of both high and low density series makes particular use of the rather precise unbiased estimates of the percolation threshold pc obtained recently by heermann and stauffer 1981 for the simple cubic lattice for this lattice the authors obtain the biased estimate gamma 1 73 or 0 03 for the site problem and a similar estimate but with larger uncertainties for the bond problem such a value is significantly larger than earlier series estimates assuming gamma to be universal they obtain precise although biased estimates of pc for both bond and site percolation on all four lattices using the bond estimates of pc they find an overall biased estimate of beta 0 454 or 0 008 for bond percolation on all three dimensional lattices the corresponding site problem requires further study scaling estimates of other critical exponents are alpha 0 64 or 0 05 delta 4 81 or 0 14 delta 2 18 or 0 04 nu 0 88 or 0 02 and nu 0 03 or 0 03

fast parallel space allocation estimation and integer sorting abstract the following problems are shown to be solvable in o log n time with optimal speedup with high probability on a randomized crcw pram using o n space space allocation given nonnegative integers allocate nonoverlapping blocks of consecutive memory cells of sizes from a base segment of consecutive memory cells estimation given integers in the range 1 compute good estimates of the number of occurrences of each value in the range 1 semisorting given integers in the range 1 store the integers 1 in an array of cells such that for all 1 all elements of 1 and occur together separated only by empty cells integer chain sorting given integers in the range 1 construct a linked list containing the integers 1 such that for all 1 if precedes in the list then moreover given slightly superlinear processor and space bounds these problems or variations of them can be solved in constant time with high probability as a corollary of the integer chain sorting result it follows that n integers in the range 1 n can be sorted in o log n log log n time with optimal speedup with high probability

upper bounds for sorting integers on random access machines two models of random access machines suitable for sorting integers are presented our main results show that i a ram with addition subtraction multiplication and integer division can sort n integers in the range 0 2 cn in o n log c n steps ii a ram with addition subtraction and left and right shifts can sort any n integers in linear time iii a ram with addition subtraction and left and right shifts can sort n integers in the range 0 n c in o n log c n steps where all intermediate results are bounded in value by the largest input

asymptotic properties of multistate random walks i theory a calculation is presented of the long time behavior of various random walk properties moments probability of return to the origin expected number of distinct sites visited for multistate random walks on periodic lattices in particular we consider inhomogeneous periodic lattices consisting of a periodically repeated unit cell which contains a finite number of internal states sites the results are identical to those for perfect lattices except for a renormalization of coefficients for walks without drift it is found that all the asymptotic random walk properties are determined by the diffusion coefficients for the multistate random walk the diffusion coefficients can be obtained by a simple matrix algorithm presented here both discrete and continuous time random walks are considered the results are not restricted to nearest neighbor random walks but apply as long as the single step probability distributions associated with each of the internal states have finite means and variances

dynamical properties of 2d systems with site disorder abstract dynamical properties are studied of particles which perform random walks on square lattices where a fraction c of the sites has been replaced by impurities the latter may be either better conducting than the host or worse conducting or even non conducting the quantities studied are the velocity autocorelation function the return probability the density of states and burnett functions the results include short time finite time and long time behavior prefactors of the long time behavior are calculated up to second order in c comparison with numerical simulations is made different definitions of the diffusion coefficient via the second moment of displacement the return probability or the density of states are shown to give different answers

on the calculation of random walk properties from lattice bond enumeration abstract we examine the range of validity of previously developed lattice bond enumeration methods for the calculation of asymptotic properties of random walks on inhomogeneous periodic lattices

improved parallel integer sorting without concurrent writing we show thatnintegers in the range 1 ncan be sorted stably on an erew pram usingo t time ando n lognloglogn logn 2 t operations for arbitrary givent lognloglogn and on a crew pram usingo t time ando n logn logn 2t logn operations for arbitrary givent logn in addition we are able to sortnarbitrary integers on a randomized crew pram within the same resource bounds with high probability in each case our algorithm is a factor of almost logn closer to optimality than all previous algorithms for the stated problem in the stated model and our third result matches the operation count of the best previous sequential algorithm we also show thatnintegers in the range 1 mcan be sorted ino logn 2 time witho n operations on an erew pram using a nonstandard word length ofo lognloglognlogm bits thereby greatly improving the upper bound on the word length necessary to sort integers with a linear time processor product even sequentially our algorithms were inspired by and in one case directly use the fusion trees of fredman and willard

delaunay triangulations in o sort n time and more we present several results about delaunay triangulations dts and convex hulls in transdichotomous and hereditary settings i the dt of a planar point set can be computed in expected time o sort n on a word ram where sort n is the time to sort n numbers we assume that the word ram supports the shuffle operation in constant time ii if we know the ordering of a planar point set in x and in y direction its dt can be found by a randomized algebraic computation tree of expected linear depth iii given a universe u of points in the plane we construct a data structure d for delaunay queries for any p s u d can find the dt of p in expected time o vpv log log vuv iv given a universe u of points in 3 space in general convex position there is a data structure d for convex hull queries for any p s u d can find the convex hull of p in expected time o vpv log log vuv 2 v given a convex polytope in 3 space with n vertices which are colored with g 2 colors we can split it into the convex hulls of the individual color classes in expected time o n log log n 2 the results i iii generalize to higher dimensions where the expected running time now also depends on the complexity of the resulting dt we need a wide range of techniques most prominently we describe a reduction from dts to nearest neighbor graphs that relies on a new variant of randomized incremental constructions using dependent sampling

density expansion of transport properties on 2d site disordered lattices i general theory a systematic kinetic theory is developed for calculating transport properties on 2d lattices with random site impurities in concentration c which can be modelled by hopping models the authors main results are expressions in terms of lattice sums for the static and frequency dependent conductivity and for the velocity autocorrelation function

deterministic lattice lorentz gas ii mirror model for pt i see ibid vol 24 p 787 806 1991 as a model for diffusion a lattice version of the lorentz gas is studied using specular collision rules a previously developed theory containing the molecular chaos approximation several ring approximations and the effective medium approximation ema is applied and compared with simulations the correction to the boltzmann equation result resulting from effective medium approximation ema is opposite to the simulation results this indicates that other diagrams than ring diagrams should be considered which is peculiar if one considers the successes of the ema for other models this is investigated by using variants of the model the diffusion tensor is in general nonisotropic the corresponding features are qualitatively described by theory

crossover dynamics in bond disordered lattices the asymptotic dynamics of the percolation model for a bond disordered lattice is studied the velocity autocorrelation function vacf is investigated for arbitrary concentration of disorder in two and three dimensions using an effective medium approximation ema corrections to the long time tails away from the percolation threshold and to the percolation tails at the threshold are calculated a characteristic time scale for the long time tails is identified and found to diverge at the threshold sufficiently close to the threshold the two types of asymptotic dynamics can be identified clearly for times greater than and less than this characteristic time respectively an approximate scaling of the ema equation is obtained near the threshold for investigation of the crossover region more generally the ema equation is solved numerically for arbitrary concentration in two dimensions to exhibit the complete time dependence of the vacf in all domains near and far from the threshold

optimal and sublogarithmic time randomized parallel sorting algorithms this paper assumes a parallel ram random access machine model which allows both concurrent reads and concurrent writes of a global memory the main result is an optimal randomized parallel algorithm for integer sort i e for sorting n integers in the range 1 n this algorithm costs only logarithmic time and is the first known that is optimal the product of its time and processor bounds is upper bounded by a linear function of the input size also given is a deterministic sublogarithmic time algorithm for prefix sum in addition this paper presents a sublogarithmic time algorithm for obtaining a random permutation of n elements in parallel and finally sublogarithmic time algorithms for general sort and integer sort are presented our sub logarithmic general sort algorithm is also optimal

a real space renormalisation group approach to electrical and noise properties of percolation clusters the migdal kadanoff renormalisation scheme which becomes exact on hierarchical lattices is used to investigate electrical properties of 2d and 3d percolation clusters the author studies in detail the frequency dependence of the conductivity and the loss angle and the amplification of resistance flicker noise this type of noise exhibits critical amplification with its own exponents x and y which are related to the exponent b recently defined by rammal et al 1985

density expansion of transport coefficients on a 2d site disordered lattice ii for pt i see ibid vol 20 p 1043 1987 for a hopping model on a square lattice with a fraction c of sites randomly replaced by impurities the static conductivity and diffusion coefficients can be expanded in powers of the concentration c the authors calculate the exact coefficients of o c and o c2 using analytical and numerical methods and the results are compared with computer simulations

renormalisation group study of a disordered model we study the potts model on a diamond hierarchical lattice with random interactions using weak disorder expansions we calculate analytically the position and the exponents of the random fixed point which appears when the specific heat exponent ap of the pure system becomes positive at ap 0 we find how the logarithmic singularity is modified by the disorder lastly we suggest that this model should present griffiths like singularities

the combinatorics of steenrod operations on the cohomology of grassmannians abstract the study of the action of the steenrod algebra on the mod p cohomology of spaces has many applications to the topological structure of those spaces in this paper we present combinatorial formulas for the action of steenrod operations on the cohomology of grassmannians both in the borel and the schubert picture we consider integral lifts of steenrod operations which lie in a certain hopf algebra of differential operators the latter has been considered recently as a realization of the landweber novikov algebra in complex cobordism theory it also has connections with the action of the virasoro algebra on the boson fock space our formulas for steenrod operations are based on methods which have not been used before in this area namely hammond operators and the combinatorics of schur functions we also discuss applications of our formulas to the geometry of grassmannians

undirected single source shortest paths with positive integer weights in linear time the single source shortest paths problem sssp is one of the classic problems in algorithmic graph theory given a positively weighted graph g with a source vertex s find the shortest path from s to all other vertices in the graph since 1959 all theoretical developments in sssp for general directed and undirected graphs have been based on dijkstra s algorithm visiting the vertices in order of increasing distance from s thus any implementation of dijkstra s algorithm sorts the vertices according to their distances from s however we do not know how to sort in linear time here a deterministic linear time and linear space algorithm is presented for the undirected single source shortest paths problem with positive integer weights the algorithm avoids the sorting bottleneck by building a hierarchical bucketing structure identifying vertex pairs that may be visited in any order