random constraint satisfaction easy generation of hard satisfiable instances in this paper we show that the models of random csp instances proposed by xu and li k xu w li exact phase transitions in random constraint satisfaction problems journal of artificial intelligence research 12 2000 93 103 k xu w li many hard examples in exact phase transitions with application to generating hard satisfiable instances technical report corr report cs cc 0302001 revised version in theoretical computer science 355 2006 291 302 are of theoretical and practical interest indeed these models called rb and rd present several nice features first it is quite easy to generate random instances of any arity since no particular structure has to be integrated or property enforced in such instances then the existence of an asymptotic phase transition can be guaranteed while applying a limited restriction on domain size and on constraint tightness in that case a threshold point can be precisely located and all instances have the guarantee to be hard at the threshold i e to have an exponential tree resolution complexity next a formal analysis shows that it is possible to generate forced satisfiable instances whose hardness is similar to unforced satisfiable ones this analysis is supported by some representative results taken from an intensive experimentation that we have carried out using complete and incomplete search methods

combinatorial sharpness criterion and phase transition classification for random csps we investigate the nature of the phase transition sharp or coarse for random constraint satisfaction problems we first give a sharp threshold criterion specified for csps which is derived from friedgut bourgain s one thus we get a complete and precise classification of the nature of the threshold for symmetric boolean csps in particular we show that it is governed by two local properties strongly related to the problems 1 sat and 2 xor sat

a small time solutions for the korteweg de vries equation using spline approximation in this paper the one dimensional nonlinear korteweg de vries kdv equation is numerically solved using second order spline approximation the test problems concerning the propagation of a solution and two solution interaction are used to validate the proposal scheme and it is found to be both accurate and efficient at small times also it is shown that the second order spline approximation may be used effectively at small times when the exact solution of the kdv equation is not known

eventual periodicity of the forced oscillations for a korteweg de vries type equation on a bounded domain using a sinc collocation method abstract we demonstrate numerically the eventual time periodicity of solutions u t to the korteweg de vries type equation with periodic forcing at one end using the sinc collocation method this method approximates the space dimension of the solution with a cardinal expansion of sinc functions thus allowing the avoidance of a costly finite difference grid for a third order boundary value problem the first order time derivative is approximated with a weighted finite difference method the sinc collocation method was found to be more robust and more efficient than other numerical schemes when applied to this problem

on the phase transitions of random k constraint satisfaction problems constraint satisfaction has received increasing attention over the years intense research has focused on solving all kinds of constraint satisfaction problems csps in this paper first we propose a random csp model named k csp that guarantees the existence of phase transitions under certain circumstances the exact location of the phase transition is quantified and experimental results are provided to illustrate the performance of the proposed model second we revise the model k csp to a random linear csp by incorporating certain linear structure to constraint relations we also prove the existence of the phase transition and exhibit its exact location for this random linear csp model

the galerkin method for the kdv equation using a new basis of smooth piecewise cubic polynomials abstract in this paper we introduce the galerkin method using a new basis of smooth piecewise cubic polynomials for the one dimensional nonlinear kdv equation the stability analysis shows that the present method is unconditionally stable in l 2 norm numerical experiments show that the proposed method preserves the conservation laws and is effective for simulating the motions and the interactions of solitary waves

classifying the complexity of constraints using finite algebras many natural combinatorial problems can be expressed as constraint satisfaction problems this class of problems is known to be np complete in general but certain restrictions on the form of the constraints can ensure tractability here we show that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and we explore how the computational complexity of the corresponding constraint satisfaction problem is connected to the properties of this algebra hence we completely translate the problem of classifying the complexity of restricted constraint satisfaction problems into the language of universal algebra we introduce a notion of tractable algebra and investigate how the tractability of an algebra relates to the tractability of the smaller algebras which may be derived from it including its subalgebras and homomorphic images this allows us to reduce significantly the types of algebras which need to be classified using our results we also show that if the decision problem associated with a given collection of constraint types can be solved efficiently then so can the corresponding search problem we then classify all finite strictly simple surjective algebras with respect to tractability obtaining a dichotomy theorem which generalizes schaefer s dichotomy for the generalized satisfiability problem finally we suggest a possible general algebraic criterion for distinguishing the tractable and intractable cases of the constraint satisfaction problem

optimal error estimates of the legendre petrov galerkin method for the korteweg de vries equation in this paper the legendre petrov galerkin method for the korteweg de vries equation with nonperiodic boundary conditions is analyzed the nonlinear term is computed with the legendre spectral method and some pseudospectral methods respectively optimal error estimates in l2 norm are obtained for both semidiscrete and fully discrete schemes the method is also applicable to some 2m 1 th order differential equations

a small time solutions for the korteweg de vries equation in this paper a heat balance integral hbi method is applied to the one dimensional non linear korteweg de vries kdv equation prescribed by appropriate homogenous boundary conditions and a set of initial conditions to obtain its approximate analytical solutions at small times it is shown that the hbi solutions obtained by the method may be used effectively at small times when the exact solution of the kdv equation is not known

locating the phase transition in binary constraint satisfaction problems abstract the phase transition in binary constraint satisfaction problems i e the transition from a region in which almost all problems have many solutions to a region in which almost all problems have no solutions as the constraints become tighter is investigated by examining the behaviour of samples of randomly generated problems in contrast to theoretical work which is concerned with the asymptotic behaviour of problems as the number of variables becomes larger this paper is concerned with the location of the phase transition in finite problems the accuracy of a prediction based on the expected number of solutions is discussed it is shown that the variance of the number of solutions can be used to set bounds on the phase transition and to indicate the accuracy of the prediction a class of sparse problems for which the prediction is known to be inaccurate is considered in detail it is shown that for these problems the phase transition depends on the topology of the constraint graph as well as on the tightness of the constraints

a hybrid numerical method for the kdv equation by finite difference and sinc collocation method abstract in this paper we propose a hybrid numerical method for the kdv equation more precisely we discretize the temporal derivative of kdv equation by a weighted scheme and treat the implicitly nonlinear term with the combination of finite difference and sinc collocation method the stability analysis is presented and numerical experiments illustrate the efficiency and stability of the proposed hybrid method the motions and the interactions of solitary waves relying on particular initial boundary conditions are also simulated

generalized satisfiability problems minimal elements and phase transitions we develop a probabilistic model on the generalized satisfiability problems defined by schaefer in proceedings of the 10th stoc san diego ca usa association for computing machinery new york 1978 pp 216 226 for which the arity of the constraints is fixed in order to study the associated phase transition we establish new results on minimal elements associated with such generalized satisfiability problems these results are the keys of the exploration we conduct on the location and on the nature of the phase transition for generalized satisfiability we first prove that the phase transition occurs at the same scale for every reasonable problem and we provide lower and upper bounds for the associated critical ratio our framework allows one to get these bounds in a uniform way in particular we obtain a lower bound proportional to the number of variables for k sat without analyzing any algorithm finally we reveal the seed of coarseness for the phase transition of generalized satisfiability 2 xor sat

solution of nonlinear initial boundary value problems by sinc collocation interpolation methods abstract this paper deals with the solution of initial boundary value problems for nonlinear evolution equations in one and two space dimensions the solution technique is based on collocation interpolation methods which use sinc functions the application refers to some direct and inverse type problems

constructing an asymptotic phase transition in random binary constraint satisfaction problems the standard models used to generate random binary constraint satisfaction problems are described at the problem sizes studied experimentally a phase transition is seen as the constraint tightness is varied however achlioptas et al showed that if the problem size number of variables increases while the remaining parameters are kept constant asymptotically almost all instances are unsatisfiable in this paper an alternative scheme for one of the standard models is proposed in which both the number of values in each variable s domain and the average degree of the constraint graph are increased with problem size it is shown that with this scheme there is asymptotically a range of values of the constraint tightness in which instances are trivially satis able with probability at least 0 5 and a range in which instances are almost all unsatisfiable hence there is a crossover point at some value of the constraint tightness between these two ranges this scheme is compared to a similar scheme due to xu and li

an empirical study of phase transitions in binary constraint satisfaction problems abstract an empirical study of randomly generated binary constraint satisfaction problems reveals that for problems with a given number of variables domain size and connectivity there is a critical level of constraint tightness at which a phase transition occurs at the phase transition problems change from being soluble to insoluble and the difficulty of problems increases dramatically a theory developed by williams and hogg 44 and independently developed by smith 37 predicts where the hardest problems should occur it is shown that the theory is in close agreement with the empirical results except when constraint graphs are sparse

sinc collocation interpolation method for the simulation of nonlinear waves abstract numerical simulation is of great importance in the study of nonlinear waves where several delicate aspects must be dealt with the presence of spatial and time gradients of complex nonlinearities and high order derivatives in fact generates various complex problems this paper develops a collocation interpolation method based on interpolation by the sine functions after a description of the method several examples are shown these show the good results that can be obtained the efficency and stability of the method and the low computing time

thresholds of random quasi abelian codes for a q ary random quasi abelian code with fixed coindex and constant rate r it is shown that the gilbert varshamov gv bound is a threshold point if r is less than the gv bound at 0 1 q 1 then the probability of the relative distance of the random code being greater than approaches 1 as the index goes to infinity whereas if r is bigger than the gv bound at then the probability approaches 0 as a corollary there exist numerous asymptotically good quasi abelian codes attaining the gv bound

an analytical numerical method for solving the korteweg de vries equation in this paper an analytical numerical method is applied to the one dimensional korteweg de vries equation with a variant of boundary and initial conditions to obtain its numerical solutions at small times two test problem with known exact solutions are studied to demonstrate the accuracy of the present method the obtained results are compared with the exact solution of each problem and are found to be in good agreement with each other the numerical scheme is also compared with earlier work and shown to be accurate and efficient

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

a legendre petrov galerkin and chebyshev collocation method for third order differential equations a legendre petrov galerkin lpg method for the third order differential equation is developed by choosing appropriate base functions the method can be implemented efficiently also this new approach enables us to derive an optimal rate of convergence in l2 norm the method is applied to some nonlinear problems such as the korteweg de vries kdv equation with the chebyshev collocation treatment for the nonlinear term it is a legendre petrov galerkin and chebyshev collocation lpg cc method numerical experiments are given to confirm the theoretical result

path decompositions for markov chains we present two path decompositions of markov chains with general state space by means of harmonic functions which are dual to each other they can be seen as generalization of williams decomposition of a brownian motion with drift 1973 the results may be illustrated by a multitude of examples we confine ourselves to dierent types of

reflected brownian motion on an orthant we consider a k dimensional diffusion process z whose state space is the nonnegative orthant on the interior of the orthant z behaves like a kdimensional brownian motion with arbitrary covariance matrix and drift vector at each of the k 1 dimensional hyperplanes that form the boundary of the orthant z reflects instantaneously in a direction that is constant over that hyperplane there is no extant theory of multidimensional diffusion that applies to this process because the boundary of its state space is not smooth we adopt an approach that requires a restriction on the directions of reflection but reiman has shown that this restriction is met by all diffusions z arising as heavy traffic limits in open k station queuing networks our process z is defined as a path to path mapping of k dimensional brownian motion from this construction it follows that z is a continuous markov process and a semimartingale using the latter property we obtain a change of variable formula from which one can develop a complete analytical theory for the process z

exponential finite difference method applied to korteweg de vries equation for small times the korteweg de vries equation is numerically solved by using the exponential finite difference technique the accuracy of computed solutions is examined by comparison with other numerical and analytical solutions using two examples the close results agreement between the current results and the exact solutions confirms that the proposed finite difference procedure is an effective technique for the solution of the korteweg de vries equation at the small times

a new dual petrov galerkin method for third and higher odd order differential equations application to the kdv equation a new dual petrov galerkin method is proposed analyzed and implemented for third and higher odd order equations using a spectral discretization the key idea is to use trial functions satisfying the underlying boundary conditions of the differential equations and test functions satisfying the dual boundary conditions the method leads to linear systems which are sparse for problems with constant coefficients and well conditioned for problems with variable coefficients our theoretical analysis and numerical results indicate that the proposed method is extremely accurate and efficient and most suitable for the study of complex dynamics of higher odd order equations

energy conserving hamiltonian boundary value methods for the numerical solution of the korteweg de vries equation abstract in this paper we study the efficient solution of the well known korteweg de vries equation equipped with periodic boundary conditions a fourier galerkin space semi discretization at first provides a large size hamiltonian ode problem whose solution in time is then carried out by means of energy conserving methods in the hbvm class hamiltonian boundary value methods the efficient implementation of the methods for the resulting problem is also considered and several numerical examples are reported

numerical solution of korteweg de vries equation by galerkin b spline finite element method a finite element solution of the kdv equation is presented to demonstrate the efficiency of the method two test problems are considered the numerical solutions of the kdv equation are compared with both the exact solutions and other numerical solutions in the literature the numerical solutions are found to be in good agreement with the exact solutions

the scaling window of the model d k csp abstract consider the model d k csp which consists of a set of n variables with the domain size d and a set of t r n ln d ln p constraints where each constraint binds k distinct variables and permits p d k tuples of values we prove that the exact phase transition of the model d k csp occurs at r c 1 as long as k ln d ln n where satisfies that 1 2 1 2 2 k 1 e 1 e 0 is a sufficiently small constant more importantly we establish the finite size scaling about this transition specifically let 0 be a given constant the probability of satisfiability of the model d k csp is greater than 1 if r 1 1 n 1 ln d is dependent on the control parameters and is less than if r 1 1 n ln d

quasi product forms for levy driven fluid networks we study stochastic tree fluid networks driven by a multidimensional levy process we are interested in the joint distribution of the steady state content in each of the buffers the busy periods and the idle periods to investigate these fluid networks we relate the above three quantities to fluctuations of the input levy process by solving a multidimensional skorokhod reflection problem this leads to the analysis of the distribution of the componentwise maximums the corresponding epochs at which they are attained and the beginning of the first last passage excursion using the notion of splitting times we are able to find their laplace transforms it turns out that if the components of the levy process are ordered the laplace transform has a so called quasi product form the theory is illustrated by working out special cases such as tandem networks and priority queues

many hard examples in exact phase transitions this paper analyzes the resolution complexity of two random constraint satisfaction problem csp models i e model rb rd for which we can establish the existence of phase transitions and identify the threshold points exactly by encoding csps into cnf formulas it is proved that almost all instances of model rb rd have no tree like resolution proofs of less than exponential size thus we not only introduce new families of csps and cnf formulas hard to solve which can be useful in the experimental evaluation of csp and sat algorithms but also propose models with both many hard instances and exact phase transitions finally conclusions are presented as well as a detailed comparison of model rb rd with the hamiltonian cycle problem and random 3 sat which respectively exhibit three different kinds of phase transition behavior in np complete problems