the scaling limit of the interface of the continuous space symbiotic branching model the continuous space symbiotic branching model describes the evolution of two interacting populations that can reproduce locally only in the simultaneous presence of each other if started with complementary heaviside initial conditions the interface where both populations coexist remains compact together with a diffusive scaling property this suggests the presence of an interesting scaling limit indeed in the present paper we show weak convergence of the diffusively rescaled populations as measure valued processes in the skorokhod respectively the meyer zheng topology for suitable parameter ranges the limit can be characterized as the unique solution to a martingale problem and satisfies a separation of types property this provides an important step toward an understanding of the scaling limit for the interface as a corollary we obtain an estimate on the moments of the width of an approximate interface

complementary cooperation minimal winning coalitions and power indices we introduce a new simple game which is referred to as the complementary weighted multiple majority game c wmmg for short c wmmg models a basic cooperation rule the complementary cooperation rule and can be taken as a sister model of the famous weighted majority game wmg for short in c wmmg each player is characterized by a nonnegative vector with a fixed dimension and players in the same coalition cooperate by producing a characteristic vector for this coalition each dimension of this vector equals the maximum of the corresponding dimensions of its members the value of a coalition is 1 if and only if the sum of its characteristic vector is larger than that of its complementary coalition in which case the coalition is called winning otherwise the coalitional value is 0 in this paper we concentrate on the two dimensional c wmmg an interesting property of this case is that there are at most n 1 minimal winning coalitions mwcs for short and they can be enumerated in time o nlogn where n is the number of players this property guarantees that the two dimensional c wmmg is more handleable than wmg in particular we prove that the main power indices i e the shapley shubik index the penrose banzhaf index the holler packel index and the deegan packel index are all polynomially computable to make a comparison with wmg we know that it may have exponentially many mwcs and none of the four power indices is polynomially computable unless p np still for the two dimensional case we show that local monotonicity holds for all of the four power indices in wmg this property is possessed by the shapley shubik index and the penrose banzhaf index but not by the holler packel index or the deegan packel index since our model fits very well the cooperation and competition in team sports we hope that it can be potentially applied in measuring the values of players in team sports say help people give more objective ranking of nba players and select mvps and consequently bring new insights into contest theory and the more general field of sports economics it may also provide some interesting enlightenments into the design of non additive voting mechanisms last but not least the threshold version of c wmmg is a generalization of wmg and natural variants of it are closely related with the famous airport game and the stable marriage roommates problem

approximate the dynamical behavior for stochastic systems by wong zakai approaching abstract random invariant manifolds and foliations play an important role in the study of the qualitative dynamical behaviors for nonlinear stochastic partial differential equations in a general way these random objects are difficult to be visualized geometrically or computed numerically the current work provides a perturbation approach to approximate these random invariant manifolds and foliations after briefly discussing the existence of random invariant manifolds and foliations for a class of stochastic systems driven by additive noises the corresponding wong zakai type of convergence result in path wise sense is established

np completeness for calculating power indices of weighted majority games in this paper we prove that both problems for calculating the banzhaf power index and the shapley shubik power index for weighted majority games are np complete

uniform approximation of eigenvalues in laguerre and hermite β ensembles by roots of orthogonal polynomials we derive strong uniform approximations for the eigenvalues in general laguerre and hermite ensembles by showing that the maximal discrepancy between the suitably scaled eigenvalues and roots of orthogonal polynomials converges almost surely to zero when the dimension converges to infinity we also provide estimates of the rate of convergence in the special case of a normalized real wishart matrix w in s s where n denotes the dimension and s the degrees of freedom the rate is logn s 1 4 if n s with n s and the rate is p logn n if n s with n s n k in the latter case we also show the a s convergence of the nt largest eigenvalue of w in s s to the corresponding quantile of the marucenko pastur law

the complexity of counting cuts and of computing the probability that a graph is connected several enumeration and reliability problems are shown to be p complete and hence at least as hard as np complete problems included are important problems in network reliability analysis namely computing the probability that a graph is connected and counting the number of minimum cardinality s t cuts or directed network cuts also shown to be p complete are counting vertex covers in a bipartite graph counting antichains in a partial order and approximating the probability that a graph is connected and the probability that a pair of vertices is connected

pp is as hard as the polynomial time hierarchy in this paper two interesting complexity classes pp and oplus text p are compared with ph the polynomial time hierarchy it is shown that every set in ph is polynomial time turing reducible to a set in pp and ph is included in text bp cdot oplus text p as a consequence of the results it follows that text pp subseteq text ph or oplus text p subseteq text ph implies a collapse of ph a stronger result is also shown every set in pp ph is polynomial time turing reducible to a set in pp

fluctuation properties of the tasep with periodic initial configuration we consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process tasep they are expressed as fredholm determinants with a kernel defining a signed determinantal point process we then consider certain periodic initial conditions and determine the kernel in the scaling limit this result has been announced first in a letter by one of us sasamoto in j phys a 38 l549 l556 2005 and here we provide a self contained derivation connections to last passage directed percolation and random matrices are also briefly discussed

on the approximation of stochastic differential equation and on stroock varadhan s support theorem on demontre un theoreme d approximation pour l equation differentielle stochastique dx t b t x t dt i t x t o dm i t dans le cas ou m t m 1 t m l t est une semi martingale continue approchee par des semi martingales continues il sert a generaliser le theroreme du suppport de stroock varadhan au cas ou les coefficients b et i sont non bornes

an impact of noise on invariant manifolds in nonlinear dynamical systems invariant manifolds provide geometric structures for understanding dynamical behavior of nonlinear systems however these nonlinear systems are often subject to random fluctuations or noises it is thus desirable to quantify the impact of noises on the invariant manifolds when the noise intensity is small this impact is estimated via asymptotic analysis in the context of liapunov perron formulation namely the random invariant manifold is represented as a perturbation of the deterministic invariant manifold with a well defined bound for the deviation

homogenization of linear and semilinear second order parabolic pdes with periodic coefficients a probabilistic approach we study the limit of the solution of linear and semilinear second order pdes of parabolic type with rapidly oscillating periodic coefficients singular drift and singular coefficient of the zeroth order term our method of proof is fully probabilistic and builds upon the arguments in earlier work in the linear case we use the feynman kac formula to represent the solution of the parabolic pde and in the semilinear case we use an associated backward stochastic differential equation

asymptotics and bounds for the zeros of laguerre polynomials a survey some of the work on the construction of inequalities and asymptotic approximations for the zeros n k k 1 2 n of the laguerre polynomial ln x as v 4n 2 2 is reviewed and discussed the cases when one or both parameters n and unrestrictedly diverge are considered two new uniform asymptotic representations are presented the first involves the positive zeros of the bessel function j x and the second is in terms of the zeros of the airy function ai x they hold for k 1 2 qn and for k pn pn 1 n respectively where p and q are fixed numbers in the interval 0 1 numerical results and comparisons are provided which favorably justify the consideration of the new approximations formulas

convergence in various topologies for stochastic integrals driven by semimartingales we generalize existing limit theory for stochastic integrals driven by semimartingales and with left continuous integrands joint skorohod convergence is replaced with joint finite dimensional convergence plus an assumption excluding the case when oscillations of the integrand appear immediately before oscillations of the integrator integrands may converge in a very weak topology it is also proved that convergence of integrators implies convergence of stochastic integrals with respect to the same topology

cross directional control on paper machines using gram polynomials a control algorithm for cross directional basis weight control based on orthogonal polynomials is presented instead of controlling the cross directional values at each point the parameters of the orthogonal polynomials are controlled taking into account the maximum bending moment of the slice lip and the relative stiffness of the slice with respect to the actuators the gram polynomials have inherent ability to limit bending of the slice lip in a way that conventional control does not the slice lip will have smooth deflection governed by the maximum order of the polynomial which can be lowered if the bending limits are exceeded by modelling the measurements and the actuator positions by polynomials the mapping between the actuators and the sensors which is the key element of the current control systems is effectively eliminated computational requirements are greatly reduced by avoiding inverses of large matrices or the inverses are found for a diagonal matrix the performance of the gram polynomial control method is shown to be superior to conventional control methods especially for badly conditioned matrices

classical orthogonal polynomials dependence on parameters most of the classical orthogonal polynomials continuous discrete and their q analogues can be considered as functions of several parameters ci a systematic study of the variation innitesimal and nite of these polynomials pn x ci with respect to the parameters ci is proposed a method to get recurrence relations for connection coecients linking r c r i pn x ci t opn x ci is given and in some situations explicit expressions are obtained this allows us to compute new integrals or sums of classical orthogonal polynomials using the digamma function a basic theorem on the zeros of ci pn x ci is also proved c 2000 elsevier science b v all rights reserved msc 33c25 42c05 33b15

random time changes and convergence in distribution under the meyer zheng conditions an analog of conditions of meyer and zheng for the relative compactness in the sense of convergence in distribution of a sequence of stochastic processes is formulated for general separable metric spaces and the corresponding notion of convergence is characterized in terms of the convergence in the skorohod topology of time changes of the original processes in addition convergence in distribution under the topology of convergence in measure is discussed and results of jacod memin and metivier on convergence under the skorohod topology are extended

planar formulae and their uses we define the set of planar boolean formulae and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is np complete using these results we are able to provide simple and nearly uniform proofs of np completeness for planar node cover planar hamiltonian circuit and line geometric connected dominating set and of polynomial space completeness for planar generalized geography the np completeness of planar node cover and planar hamiltonian circuit and line were first proved elsewhere m r garey and d s johnson the rectilinear steiner tree is np complete siam j appl math 32 1977 pp 826 834 and m r garey d s johnson and r e tarjan the planar hamilton circuit problem is np complete siam j comp 5 1976 pp 704 714

approximate calculation of sums i bounds for the zeros of gram polynomials let n be a positive integer and x j be n equidistant points we propose an algorithmic approach for approximate calculation of sums of the form sum j 1 n f x j the method is based on the gaussian type quadrature formula for sums sum j 1 n f x j approx sum k 1 n b n k f g n k n n ll n where g n k n are the zeros of the so called gram polynomials this allows the calculation of sums with very large number of terms n to be reduced to sums with a much smaller number of summands n the first task in constructing such a formula is to calculate its nodes g n k n in this paper we obtain precise lower and upper bounds for g n k n numerical experiments show that the estimates for the zeros g n k n are very sharp and that the proposed method for calculation of sums is efficient

errata for the paper predecessor existence problems for finite discrete dynamical systems theoret comput sci 386 1 2 2007 3 37 we point out some minor corrections to the proof of theorem 4 1 in the above paper as presented the proof of theorem 4 1 uses a reduction from a special version of the planar 3sat problem called rp3sat in which each clause has exactly three literals each variable appears in at most three clauses and the factor graph corresponding to the instance is planar it is incorrect to use a reduction from rp3sat to prove the hardness of the predecessor existence problems for sdss and sydss on grids since the rp3sat problem is efficiently solvable even without the planarity restriction as shown by tovey 2 however the result of theorem 4 1 can be established by a reduction from a restricted version of planar 3sat henceforth referred to as pl b3sat in which each clause has two or three literals and each variable appears in at most three clauses there is a local replacement based ultra efficient decision parsimonious reduction from 3sat to pl b3sat 1 thus pl b3sat is np complete pl b3sat is p complete unique pl b3sat is dp complete and ambiguous pl b3sat is np complete in the proof of theorem 4 1 when the reduction is carried out from pl b3sat the constructions of the underlying grids for both sdss and sydss remain the same the only change is that the local transition functions for each node corresponding to a two literal clause should be the 4 simple threshold function for each node corresponding to a three literal clause the local transition function remains the 3 simple threshold function as in the paper because of this change some minor modifications are also needed for a few sentences in the discussion of the construction these modifications are indicated below the page and line numbers used in the following discussion correspond to the corrected page proofs for the paper

a discrete approach to monotonicity of zeros of orthogonal polynomials we study the monotonicity with respect to a parameter of zeros of orthogonal polynomials our method uses the tridiagonal jacobi matrices arising from the three term recurrence relation for the polynomials we obtain new results on monotonicity of zeros of associated laguerre al salam carlitz meixner and pollaczek polynomials we also derive inequalities for the zeros of the al salam carlitz and meixner polynomials

approximation and support theorems in modulus spaces we prove an approximation theorem for stochastic differential equations under rather weak smoothness conditions on the coefficients when the driving semimartingales are approximated by continuous semimartingales in probability and the solutions are considered in several banach spaces defined in terms of different types of the modulus of continuity hence stroock varadhan s support theorem is obtained in these spaces in particular in appropriate besov and holder spaces

slow foliation of a slow fast stochastic evolutionary system abstract this work is concerned with the dynamics of a slow fast stochastic evolutionary system quantified with a scale parameter an invariant foliation decomposes the state space into geometric regions of different dynamical regimes and thus helps understand dynamics a slow invariant foliation is established for this system it is shown that the slow foliation converges to a critical foliation i e the scale parameter is zero in probability distribution as the scale parameter tends to zero the approximation of slow foliation is also constructed with error estimate in distribution furthermore the geometric structure of the slow foliation is investigated every fiber of the slow foliation parallels each other with the slow manifold as a special fiber in fact when an arbitrarily chosen point of a fiber falls in the slow manifold the fiber must be the slow manifold itself

approximation schemes for covering and packing problems in image processing and vlsi a unified and powerful approach is presented for devising polynomial approximation schemes for many strongly np complete problems such schemes consist of families of approximation algorithms for each desired performance bound on the relative error e o with running time that is polynomial when e is fixed though the polynomiality of these algorithms depends on the degree of approximation e being fixed they cannot be improved owing to a negative result stating that there are no fully polynomial approximation schemes for strongly np complete problems unless np p the unified technique that is introduced here referred to as the shifting strategy is applicable to numerous geometric covering and packing problems the method of using the technique and how it varies with problem parameters are illustrated a similar technique independently devised by b s baker was shown to be applicable for covering and packing problems on planar graphs

on unique graph 3 colorability and parsimonious reductions in the plane we prove that the satisfiability resp planar satisfiability problem is parsimoniously p time reducible to the 3 colorability resp planar 3 colorability problem that means that the exact number of solutions is preserved by the reduction provided that 3 colorings are counted modulo their six trivial color permutations in particular the uniqueness of solutions is preserved which implies that unique 3 colorability is exactly as hard as unique satisfiability in the general case as well as in the planar case a consequence of our result is the dp completeness of unique 3 colorability and unique planar 3 colorability under random p time reductions it also gives a finer and unified proof of the p completeness of 3 colorability that was first obtained by linial for the general case and later by hunt et al for the planar case previous authors reductions were either weakly parsimonious with a multiplication of the numbers of solutions by an exponential factor or involved p complete intermediate counting problems derived from trivial yes decision problems

decomposition of triebel lizorkin and besov spaces in the context of laguerre expansions abstract a pair of dual frames with almost exponentially localized elements needlets are constructed on r d based on laguerre functions it is shown that the triebel lizorkin and besov spaces induced by laguerre expansions can be characterized in terms of respective sequence spaces that involve the needlet coefficients

the complexity of the counting constraint satisfaction problem the counting constraint satisfaction problem csp h over a finite relational structure h can be expressed as follows given a relational structure g over the same vocabulary determine the number of homomorphisms from g to h in this article we characterize relational structures h for which csp h can be solved in polynomial time and prove that for all other structures the problem is p complete

the support of the solution to a hyperbolic spde in this paper we prove stroock varadhan type theorems for the topological support of a hyperbolic stochastic partial differential equation in the holder norm for 0 1 2 our approach is based on absolutely continuous transformations of defined using non homogeneous approximations of the brownian sheet

uniform asymptotics and zeros of a system of orthogonal polynomials defined via a difference equation abstract we study a class of sieved pollaczek polynomials defined by a second order difference equation three term recurrence relation the measure of orthogonality is determined by using the markov theorem and the perron stieltjes inversion formula and is shown consisting of an absolutely continuous part and a discrete part with infinitely many mass points uniform asymptotic approximations of these polynomials for large degree n are derived at a turning point n and a critical point n involving respectively the airy function ai and a 1 2 ai i 2 bi darboux s method the method of steepest descents and various uniform asymptotic techniques such as cubic transformations are used to derive the results asymptotic formulas for the least zeros the largest zeros and the zeros on both sides of n are also obtained several numerical examples are provided to compare the approximate zeros with the true values

on the construction of graphs with a planar bipartite double cover from boolean formulas and its application to counting satisfying solutions abstract we address the problem to construct a graph g with binary edge weights from the incidence graph of a boolean formula in a way that g has a planar bipartite double cover g this allows one to use the identity between the permanent of g s adjacency matrix a g and the number of perfect matchings of g i e perm a g pm g due to the algorithm of fisher kasteleyn and temperlay fkt the right hand side of the equation can be counted in polynomial time if g is planar we prove a theorem that describes necessary conditions for the gadgets i e small subgraphs with certain properties of g to allow a planar bipartite double cover g for arbitrary 3cnf formulas and the known approaches to build g our theorem shows that gadgets which enable a planar bipartite double cover do not exist however we show that for certain classes of boolean formulas e g 5 pl rtw 2cnf 7 pl rtw 2cnf and 3 forest 3cnf such gadgets exist hence these counting problems are in p the results probably extended to other moduli p and perhaps further counting classes

the stable manifold theorem for stochastic differential equations type equations are treated starting with the existence of a stochastic flow for a sde we introduce the notion of a hyperbolic stationary trajectory we prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution for stratonovich sdes the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde the proof of the stable manifold theorem is based on ruelle oseledec multiplicative ergodic theory

convergent numerical scheme for singular stochastic control with state constraints in a portfolio selection problem we consider a singular stochastic control problem with state constraints that arises in problems of optimal consumption and investment under transaction costs numerical approximations for the value function using the markov chain approximation method of kushner and dupuis are studied the main result of the paper shows that the value function of the markov decision problem mdp corresponding to the approximating controlled markov chain converges to that of the original stochastic control problem as various parameters in the approximation approach suitable limits all our convergence arguments are probabilistic the main assumption that we make is that the value function be finite and continuous in particular uniqueness of the solutions of the associated hjb equations is neither needed nor available in the generality under which the problem is considered specific features of the problem that make the convergence analysis nontrivial include unboundedness of the state and control space and the cost function degeneracies in the dynamics mixed boundary dirichlet neumann conditions and presence of both singular and absolutely continuous controls in the dynamics finally schemes for computing the value function and optimal control policies for the mdp are presented and illustrated with a numerical study

invariant manifolds for stochastic partial differential equations invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems the theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems and for stochastic ordinary differential equations is relatively mature in this paper we present a unified theory of invariant manifolds for infinite dimensional random dynamical systems generated by stochastic partial differential equations we first introduce a random graph transform and a fixed point theorem for nonautonomous systems then we show the existence of generalized fixed points which give the desired invariant manifolds

a wong zakai approximation for random invariant manifolds random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences but these random objects are difficult to be visualized geometrically or computed numerically the current work provides a perturbation approach to approximate these random invariant manifolds we first discuss the existence of a random invariant manifold for a class of stochastic evolutionary equations then we approximate the random invariant manifold by the invariant manifold of a new system with smooth colored noise i e integrated ornstein uhlenbeck processes the convergence in a pathwise wong zakai sense is shown

some recent results on the zeros of bessel functions and orthogonal polynomials several recent results are encountered on the zeros of bessel functions and of the classical orthogonal polynomials many of them are waiting for being published four problems are formulated as conjectures

the complexity of reliability computations in planar and acyclic graphs we show that the problem of computing source sink reliability is np hard in fact p complete even for undirected and acyclic directed source sink planar graphs having vertex degree at most three thus the source sink reliability problem is unlikely to have an efficient algorithm even when the graph can be laid out on a rectilinear grid

the complexity of counting self avoiding walks in subgraphs of two dimensional grids and hypercubes valiant siam j comput 8 1979 410 421 showed that the problem of computing the number of simple s t paths in graphs is p complete both in the case of directed graphs and in the case of undirected graphs welsh complexity knots colourings and counting cambridge university press cambridge 1993 p 17 asked whether the problem of computing the number of self avoiding walks of a given length in the complete two dimensional grid is complete for p1 the tally version of p this paper offers a partial answer to the question of welsh it is p complete to compute the number of self avoiding walks of a given length in a subgraph of a two dimensional grid several variations of the problem are also studied and shown to be p complete this paper also studies the problem of computing the number of self avoiding walks in a subgraph of a hypercube similar completeness results are shown for the problem by scaling the computation time to exponential it is shown that computing the number of self avoiding walks in hypercubes is a complete problem for exp in the case when a subgraph of a hypercube is specified by its dimension and a boolean circuit that accepts the nodes finally this paper studies the complexity of testing whether a given word over the four letter alphabet u d l r represents a self avoiding walk in a two dimensional grid a linear space lower bound is shown for nondeterministic turing machines with a 1 way input head to make this test

asymptotic distribution of zeros of polynomials satisfying difference equations we propose a way to find the asymptotic distribution of zeros of orthogonal polynomials pn x satisfying a difference equation of the form b x pn x c x n pn x d x pn x 0 we calculate the asymptotic distribution of zeros and asymptotics of extreme zeros of the meixner and meixner pollaczek polynomials the distribution of zeros of meixner polynomials shows some delicate features we indicate the relation of our technique to the approach based on the nevai dehesa ullman distribution

computational aspects of mining maximal frequent patterns in this paper we study the complexity theoretic aspects of mining maximal frequent patterns from the perspective of counting the number of all distinct solutions we present the first formal proof that the problem of counting the number of maximal frequent itemsets in a database of transactions given an arbitrary support threshold is p complete thereby providing theoretical evidence that the problem of mining maximal frequent itemsets is np hard we also extend our complexity analysis to other similar data mining problems that deal with complex data structures such as sequences trees and graphs we investigate several variants of these mining problems in which the patterns of interest are subsequences subtrees or subgraphs and show that the associated problems of counting the number of maximal frequent patterns are all either p complete or p hard

algorithms for four variants of the exact satisfiability problem we present four polynomial space and exponential time algorithms for variants of the exact satisfiability problem first an o 1 1120n where n is the number of variables time algorithm for the np complete decision problem of exact 3 satisfiability and then an o 1 1907n time algorithm for the general decision problem of exact satisfiability the best previous algorithms run in o 1 1193n and o 1 2299n time respectively for the p complete problem of counting the number of models for exact 3 satisfiability we present an o 1 1487n time algorithm we also present an o 1 2190n time algorithm for the general problem of counting the number of models for exact satisfiability presenting a simple reduction we show how this algorithm can be used for computing the permanent of a 0 1 matrix

modeling and analyzing social network dynamics using stochastic discrete graphical dynamical systems motivated by applications such as the spread of epidemics and the propagation of influence in social networks we propose a formal model for analyzing the dynamics of such networks our model is a stochastic version of discrete graphical dynamical systems using this model we formulate and study the computational complexity of two fundamental problems called reachability and predecessor existence problems which arise in the context of social networks we also address other problems that deal with the time evolution of such stochastic dynamical systems further we point out the implications of our results to problems for other computational models such as hopfield networks communicating finite state machines and systolic arrays in particular our polynomial time algorithms for the predecessor existence problem for stochastic dynamical systems imply similar results for one dimensional finite cellular automata

stochastic scalar conservation laws we introduce a notion of stochastic entropic solution a la kruzkov but with ito s calculus replacing deterministic calculus this results in a rich family of stochastic inequalities defining what we mean by a solution a uniqueness theory is then developed following a stochastic generalization of l1 contraction estimate an existence theory is also developed by adapting compensated compactness arguments to stochastic setting we use approximating models of vanishing viscosity solution type for the construction while the uniqueness result applies to any spatial dimensions the existence result in the absence of special structural assumptions is restricted to one spatial dimension only

a new asymptotic form for the laguerre polynomials a new asymptotic formula in terms of elementary functions is demonstrated for the laguerre polynomials l n x in the limit when all three parameters n x become large this asymptotic limit appears in the problem of obtaining the low field semiclassical approximation from the full quantum mechanical behavior of a two dimensional electron gas in a magnetic field

on the complexity of cooperative solution concepts we study from a complexity theoretic standpoint the various solution concepts arising in cooperative game theory we use as a vehicle for this study a game in which the players are nodes of a graph with weights on the edges and the value of a coalition is determined by the total weight of the edges contained in it the shapley value is always easy to compute the core is easy to characterize when the game is convex and is intractable np complete otherwise similar results are shown for the kernel the nucleolus the e core and the bargaining set as for the von neumann morgenstern solution we point out that its existence may not even be decidable many of these results generalize to the case in which the game is presented by a hypergraph with edges of size k 2

note the complexity of power index comparison abstract we study the complexity of the following problem given two weighted voting games g and g that each contain a player p in which of these games is p s power index value higher we study this problem with respect to both the shapley shubik power index and the banzhaf power index our main result is that for both of these power indices the problem is complete for probabilistic polynomial time i e is pp complete we apply our results to partially resolve some recently proposed problems regarding the complexity of weighted voting games we also study the complexity of the raw shapley shubik power index deng and papadimitriou showed that the raw shapley shubik power index is p metric complete we strengthen this by showing that the raw shapley shubik power index is many one complete for p and our strengthening cannot possibly be further improved to parsimonious completeness since we observe that in contrast with the raw banzhaf power index the raw shapley shubik power index is not p parsimonious complete

the complexity of counting in sparse regular and planar graphs we show that a number of graph theoretic counting problems remain cal np hard indeed cal p complete in very restricted classes of graphs in particular we prove that the problems of counting matchings vertex covers independent sets and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree we obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2 cnf formulae to achieve these results a new interpolation based reduction technique which preserves properties such as constant degree is introduced

invariant manifolds for stochastic wave equations abstract in this paper we consider a class of stochastic wave equations with nonlinear multiplicative noise we first show that these stochastic wave equations generate random dynamical systems or stochastic flows by transforming the stochastic wave equations to random wave equations through a stationary random homeomorphism then we establish the existence of random invariant manifolds for the random wave equations due to the temperedness of the nonlinearity we obtain only local invariant manifolds no matter how large the spectral gap is unlike the deterministic cases based on these random dynamical systems we prove the existence of random invariant manifolds in a tempered neighborhood of an equilibrium finally we show that the images of these invariant manifolds under the inverse stationary transformation give invariant manifolds for the stochastic wave equations

on the zeros of meixner polynomials we investigate the zeros of a family of hypergeometric polynomials m n x beta c beta n 2f 1 n x beta 1 frac 1 c n in mathbb n known as meixner polynomials that are orthogonal on 0 infty with respect to a discrete measure for beta 0 and 0 1 as well as the polynomials k n x p n with non hermitian orthogonality for 0 p 1 and n n 1 n 2 ldots we also show that the polynomials m n x beta c beta in mathbb r are real rooted when c rightarrow 0

discrete orthogonal polynomial ensembles and the plancherel measure we consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory these ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions the meixner ensemble is related to a two dimensional directed growth model and the charlier ensemble is related to the lengths of weakly increasing subsequences in random words the krawtchouk ensemble occurs in connection with zig zag paths in random domino tilings of the aztec diamond and also in a certain simplifled directed flrst passage percolation model we use the charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of tracy and widom as a limit of the meixner ensemble or the charlier ensemble we obtain the plancherel measure on partitions and using this we prove a conjecture of baik deift and johansson that under the plancherel measure the distribution of the lengths of the flrst k rows in the partition appropriately scaled converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the gaussian unitary ensemble in this problem a certain discrete kernel which we call the discrete bessel kernel plays an important role

predecessor existence problems for finite discrete dynamical systems we study the predecessor existence problem for finite discrete dynamical systems given a finite discrete dynamical system s and a configuration c the predecessor existence or pre problem is to determine whether there is a configuration c such that s has a transition from c to c in addition to the decision version we also study the following variants the predecessor existence or pre problem counting the number of predecessors the unique predecessor existence or upre problem deciding whether there is a unique predecessor and the ambiguous predecessor existence or apre problem given a configuration c and a predecessor c of c deciding whether there is a different predecessor c of c general techniques are presented for simultaneously characterizing the computational complexity of the pre problem and its three variants our hardness results are based on the concept of simultaneous reductions single transformations that can be used to simultaneously prove the hardness of the different variants of the pre problem for their respective complexity classes our easiness results are based on dynamic programming and they extend the previous results on pre problem for one dimensional cellular automata the hardness results together with the easiness results provide a tight separation between easy and hard instances further the results imply similar bounds for other classes of finite discrete dynamical systems including discrete hopfield and recurrent neural networks concurrent state machines systolic networks and one and two dimensional cellular automata our results extend the earlier results of green sutner and orponen on the complexity of the predecessor existence problem and its variants

approximation and support theorem in holder norm for parabolic stochastic partial differential equations the solution u t x of a parabolic stochastic partial differential equation is a random element of the space e of holder continuous functions on 0 t x 0 1 of order 1 4 e in the time variable and 1 2 e in the space variable for any e 0 we prove a support theorem in e for the law of u the proof is based on an approximation procedure in holder norm which should have its own interest using a space time polygonal interpolation for the brownian sheet driving the spde and a sequence of absolutely continuous transformations of the wiener space

on extreme zeros of classical orthogonal polynomials let x1 and xk be the least and the largest zeros of the laguerre or jacobi polynomial of degree k we shall establish sharp inequalities of the form x1 b which are uniform in all the parameters involved together with inequalities in the opposite direction recently obtained by the author this locates the extreme zeros of classical orthogonal polynomials with a high precision

reflected brownian motion in a cone semimartingale property in this paper the object of study is reflected brownian motion in a cone ind dimensions d 3 with nonconstant oblique reflection on each radial line emanating from the vertex of the cone the basic question considered here is when is this process a semimartingale conditions for the existence and uniqueness of the process for which the vertex is an instantaneous state were given by kwon which is resolved in terms of a real parameter depending on the cone and the direction of reflection it is shown that starting from any point of the cone the process is a semimartingale if 1 0 and not a semimartingale if 2

the complexity of planar counting problems we prove the p hardness of the counting problems associated with various satisfiability graph and combinatorial problems when restricted to planar instances these problems include 3sat 1 3sat 1 ex3sat minimum vertex cover minimum dominating set minimum feedback vertex set x3c partition into triangles and clique cover we also prove the np completeness of the ambiguous satisfiability problems j b saxe two papers on graph embedding problems tech report cmu cs 80 102 dept of computer science carnegie mellon univ pittsburgh pa 1980 and the dp completeness with respect to random polynomial reducibility of the unique satisfiability problems l g valiant and v v vazirani np is as easy as detecting unique solutions in proc 17th acm symp on theory of computing 1985 pp 458 463 associated with several of the above problems when restricted to planar instances previously very few p hardness results no sf np hardness results and no dp completeness results were known for counting problems ambiguous satisfiability problems and unique satisfiability problems respectively when restricted to planar instances assuming sf p neq np one corollary of the above results is that there are no epsilon approximation algorithms for the problems of maximizing or minimizing a linear objective function subject to a planar system of linear inequality constraints over the integers

power indices of simple games and vector weighted majority games by means of binary decision diagrams abstract a simple game is a pair consisting of a finite set n of players and a set w 2 n of winning coalitions vector weighted majority games v wmg are a special case of simple games in which an integer vector weight can be assigned to each player and there is a quota which a coalition has to achieve in order to win binary decision diagrams bdds are used as compact representations for boolean functions and sets of subsets this paper shows how a quasi reduced and ordered bdd qobdd of the winning coalitions of a v wmg can be build how one can compute the minimal winning coalitions and how one can easily compute the banzhaf shapley shubik holler packel and deegan packel indices of the players e g in case of weighted majority games it is shown that the banzhaf and holler packel indices of all players can be computed in expected time o nq and in general the banzhaf indices can be computed in time linear in the size of the qobdd representation of the winning coalitions other running times are proven as well the algorithms were tested on some real world games e g the international monetary fund and the eu treaty of nice

zeros of classical orthogonal polynomials of a discrete variable in this paper we obtain sharp bounds for the zeros of classical orthogonal polynomials of a discrete variable considered as functions of a parameter by using a theorem of a markov and the so called hellmannfeynman theorem comparisons with previous results for zeros of hahn meixner kravchuk and charlier polynomials are also presented

asymptotics of extreme zeros of the meixner pollaczek polynomials we study the asymptotics of the smallest and largest zeros of the symmetric and asymmetric meixner pollaczek polynomials using two techniques one is a coulomb fluid technique developed earlier where the primary input is the weight function the second uses the method of chain sequences which supplies inequalities for the largest zeros from the knowledge of the recurrence coefficients an upper bound for the largest zero of meixner polynomials is also given

zeros of gegenbauer and hermite polynomials and connection coefficients in this paper sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of obrechkoff and certain explicit connection coefficients for these polynomials as a consequence sharp bounds for the zeros of the hermite polynomials are obtained

on the complexity of some common geometric location problems given n demand points in the plane the p center problem is to find p supply points anywhere in the plane so as to minimize the maximum distance from a demand point to its respective nearest supply point the p median problem is to minimize the sum of distances from demand points to their respective nearest supply points we prove that the p center and the p median problems relative to both the euclidean and the rectilinear metrics are np hard in fact we prove that it is np hard even to approximate the p center problems sufficiently closely the reductions are from 3 satisfiability

monotonicity of zeros of laguerre polynomials denote by x n k a k 1 n the zeros of the laguerre polynomial l n a x we establish monotonicity with respect to the parameter a of certain functions involving x n k a as a consequence we obtain sharp upper bounds for the largest zero of l n a x