exact critical exponent for the shortest path scaling function in percolation it is shown that the critical exponentg1 related to pair connectiveness and shortest path or chemical distance scaling recently studied by porto et al dokholyan et al and grassberger can be found exactly in two dimensions by using a crossing probability result of cardy with the outcomeg1d 25 24 this prediction is consistent with existing simulation results

on the power of finite automata with both nondeterministic and probabilistic states we study finite automata with both nondeterministic and random states npfa s we restrict our attention to those npfa s that accept their languages with a small probability of error and run in polynomial expected time equivalently we study arthur merlin games where arthur is limited to polynomial time and constant space dwork and stockmeyer siam j comput 19 1990 pp 1011 1023 asked whether these npfa s accept only the regular languages this was known if the automaton has only randomness or only nondeterminism we show that the answer is yes in the case of npfa s with a 1 way input head we also show that if l is a nonregular language then either l or bar l is not accepted by any npfa with a 2 way input head toward this end we define a new measure of the complexity of a language l called its 1 tiling complexity for each n this is the number of tiles needed to cover the 1 s in the characteristic matrix of l namely the binary matrix with a row and column for each string of length le n where entry x y 1 if and only if the string xy in l we show that a language has constant 1 tiling complexity if and only if it is regular from which the result on 1 way input follows our main result regarding the general 2 way input tape follows by contrasting two bounds an upper bound of polylog n on the 1 tiling complexity of every language computed by our model and a lower bound stating that the 1 tiling complexity of a nonregular language or its complement exceeds a function in 2 omega sqrt log n infinitely often the last lower bound follows by proving that the characteristic matrix of every nonregular language has rank n for infinitely many n this is our main technical result and its proof extends techniques of frobenius and iohvidov developed for hankel matrices sitzungsber der konigl preuss akad der wiss 1894 pp 407 431 hankel and toeplitz matrices and forms algebraic theory birkhauser boston 1982

conformal invariance in two dimensional percolation the word percolation borrowed from the latin refers to the seeping or oozing of a liquid through a porous medium usually to be strained in this and related senses it has been in use since the seventeenth century it was introduced more recently into mathematics by s r broadbent and j m hammersley bh and is a branch of probability theory that is especially close to statistical mechanics broadbent and hammersley distinguish between two types of spreading of a fluid through a medium or between two aspects of the probabilistic models of such processes diffusion processes in which the random mechanism is ascribed to the fluid and percolation processes in which it is ascribed to the medium a percolation process typically depends on one or more probabilistic parameters for example if molecules of a gas are absorbed at the surface of a porous solid as in a gas mask then their ability to penetrate the solid depends on the sizes of the pores in it and their positions both conceived to be distributed in some random manner a simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner that could be determined by a periodic graph and to be open thus very large or closed thus smaller than the molecules with probabilities p and 1 p as p increases the probability of deeper penetration of the gas into the interior of the solid grows

on the universality of distribution of ranked cluster masses at critical percolation the distribution of masses of clusters smaller than the infinite cluster is evaluated at the percolation threshold the clusters are ranked according to their masses and the distribution p m ld r of the scaled masses m for any rank r shows a universal behaviour for different lattice sizes l d is the fractal dimension for different ranks however there is a universal distribution function only in the large rank limit i e p m ld r r y g mry ld y and are defined in the text where the universal scaling function g is found to be gaussian in nature

scaling of the distribution of shortest paths in percolation we present a scaling hypothesis for the distribution function of the shortest paths connecting any two points on a percolating cluster which accounts for i the effect of the finite size of the system and ii the dependence of this distribution on the site occupancy probability p we test the hypothesis for the case of two dimensional percolation

parameter range reduction from partial data in systems of differential algebraic equations abstract this paper presents an algorithm for parameter range reduction from partial data in systems of differential algebraic equations interval arithmetic and linear multistep discretizations are used to prune regions of parameter space that are inconsistent with available time series data methods of splitting parameter space to ensure further progress are discussed the algorithm outputs a reduced range and an approximation for each parameter value these outputs are shown to have beneficial effect as input to traditional parameter estimation schemes the range reduction algorithm is tested on three systems with partial data sets

the fractal dimension of the minimum path in two and three dimensional percolation the authors calculate the fractal dimension d min of the shortest path l between two points on a percolation cluster where l approximately rd min and r is the pythagorean distance between the points they find d min 1 130 or 0 002 for d 2 and 1 34 or 0 01 for d 3

a critical exponent for shortest path scaling in continuum percolation we carry out monte carlo experiments to study the scaling behavior of shortest path lengths in continuum percolation these studies suggest that the critical exponent governing this scaling is the same for both continuum and lattice percolation we use splitting a technique that has not yet been fully exploited in the physics literature to increase the speed of our simulations this technique can also be applied to other models where clusters are grown sequentially

sharp bounds on davenport schinzel sequences of every order one of the longest standing open problems in computational geometry is bounding the complexity of the lower envelope of n univariate functions each pair of which crosses at most s times for some fixed s this problem is known to be equivalent to bounding the length of an order s davenport schinzel sequence namely a sequence over an n letter alphabet that avoids alternating subsequences of the form a b a b with length s 2 these sequences were introduced by davenport and schinzel in 1965 to model a certain problem in differential equations and have since been applied to bound the running times of geometric algorithms data structures and the combinatorial complexity of geometric arrangements let s n be the maximum length of an order s ds sequence over n letters what is s asymptotically this question has been answered satisfactorily hart and sharir 1986 agarwal et al 1989 klazar 1999 nivasch 2010 when s is even or s 3 however since the work of agarwal et al in the mid 1980s there has been a persistent gap in our understanding of the odd orders in this work we effectively close the problem by establishing sharp bounds on davenport schinzel sequences of every order s our results reveal that contrary to one s intuition s n behaves essentially like s 1 n when s is odd this refutes conjectures by alon et al 2008 and nivasch 2010

the integration of stiff initial value problems in odes using modified extended backward differentiation formulae abstract a class of modified extended backward differentiation formulae mebdf suitable for the approximate numerical integration of first order systems of stiff ordinary differential equations is introduced the computational aspects of this new class of formulae are examined in detail in particular algorithms for varying both order and stepsize are given and this leads to a variable step variable order process using highly stable formulae of order 2 8 extensive numerical results for the well known detest set are given and on this basis a comparison is made between a code incorporating mebdf a code due to hindmarsh based on conventional backward differentiation formulae and a code due to skeel and kong based on blended linear multistep methods it is shown that on this test set the mebdf code is very reliable and is competitive with the other two codes on a significant class of problems a computer listing of the mebdf code used to obtain the results presented in this paper is available from the author

a bdf a generalization of the backward differentiation formulae a one parameter family of generalized bdf the so called a bdf is proposed and analyzed formulae of this new family are a0 stable up to order 7 especially a stable in orders 1 and 2 and a alpha stable in orders 3 to 7 respectively the nonexistence result of a0 stable formulae of order larger than 7 given by c w cryer bit 12 1972 pp 17 25 d m creedon and j j h miller bit 15 1975 pp 244 249 is extended to the a bdf after implementing these formulae in lsode a considerably larger set of stiff problems with oscillatory modes can be solved efficiently and accurately

distribution of shortest paths in percolation we present a scaling ansatz for the distribution function of the shortest paths connecting any two points on a percolating cluster which accounts for i the eect of the nite size of the system and ii the dependence of this distribution on the site occupancy probability p we present evidence supporting the scaling ansatz for the case of two dimensional percolation c 1999 elsevier science b v all rights reserved

pair connectedness and shortest path scaling in critical percolation we present high statistics data on the distribution of shortest path lengths between two near by points on the same cluster at the percolation threshold our data are based on a new and very efficient algorithm for d 2 they clearly disprove a recent conjecture by m porto et al 1998 phys rev e 58 r5205 our data also provide upper bounds on the probability that two near by points are on different infinite clusters

postbreakthrough behavior in flow through porous media we numerically simulate the traveling time of a tracer in convective flow between two points injection and extraction separated by a distance r in a model of porous media d 2 percolation we calculate and analyze the traveling time probability density function for two values of the fraction of connecting bonds p the homogeneous case p 1 and the inhomogeneous critical threshold case p p c we analyze both constant current and constant pressure conditions at p p c the homogeneous p 1 case serves as a comparison base for the more complicated p p c situation we find several regions in the probability density of the traveling times for the homogeneous case p 1 and also for the critical case p p c for both constant pressure and constant current conditions for constant pressure the first region i p corresponds to the short times before the flow breakthrough occurs when the probability distribution is strictly zero the second region ii p corresponds to numerous fast flow lines reaching the extraction point with the probability distribution reaching its maximum the third region iii p corresponds to intermediate times and is characterized by a power law decay the fourth region iv p corresponds to very long traveling times and is characterized by a different power law decaying tail the power law characterizing region iv p is related to the multifractal properties of flow in percolation and an expression for its dependence on the system size l is presented the constant current behavior is different from the constant pressure behavior and can be related analytically to the constant pressure case we present theoretical arguments for the values of the exponents characterizing each region and crossover times our results are summarized in two scaling assumptions for the traveling time probability density one for constant pressure and one for constant current we also present the production curve associated with the probability of traveling times which is of interest to oil recovery

brief paper rigorous parameter reconstruction for differential equations with noisy data we present a method that given a data set a finitely parametrized system of ordinary differential equations odes and a search space of parameters discards portions of the search space that are inconsistent with the model ode and data the method is completely rigorous as it is based on validated integration of the vector field as a consequence no consistent parameters can be lost during the pruning phase for data sets with moderate levels of noise this yields a good reconstruction of the underlying parameters several examples are included to illustrate the merits of the method

improved bounds and new techniques for davenport schinzel sequences and their generalizations we present several new results regarding s n the maximum length of a davenport schinzel sequence of order s on n distinct symbols first we prove that s n n 2 1 t n t o n t 1 for s 4 even and s n n 2 1 t n t log2 n o n t for s 3 odd where t s 2 2 and n denotes the inverse ackermann function the previous upper bounds by agarwal et al 1989 had a leading coefficient of 1 instead of 1 t in the exponent the bounds for even s are now tight up to lower order terms in the exponent these new bounds result from a small improvement on the technique of agarwal et al more importantly we also present a new technique for deriving upper bounds for s n this new technique is very similar to the one we applied to the problem of stabbing interval chains alon et al 2008 with this new technique we 1 re derive the upper bound of 3 n 2n n o n n first shown by klazar 1999 2 re derive our own new upper bounds for general s and 3 obtain improved upper bounds for the generalized davenport schinzel sequences considered by adamec et al 1992 regarding lower bounds we show that 3 n 2n n o n the previous lower bound sharir and agarwal 1995 had a coefficient of 1 2 so the coefficient 2 is tight we also present a simpler variant of the construction of agarwal et al 1989 that achieves the known lower bounds of s n n 2 1 t n t o n t 1 for s 4 even

parameter range reduction for ode models using monotonic discretizations this paper analyzes the effectiveness of various monotonic discretizations of an ode in a parameter range reduction algorithm several properties of discretizations are given and five classes of discretizations are defined for various step numbers s the range reduction algorithm that employs these discretizations is described using both analytical results based on the prototypical model x email protected and empirical results based on two more complicated models it is shown that one particular class of discretizations the a1out class results in the tightest bounds on the parameters this result is shown to be attributed to a certain characteristic value a 0 of the discretization accumulation of these discretizations is also defined and its usefulness in the range reduction algorithm is described

numerical studies of critical percolation in three dimensions presents results of high statistics simulations of the spreading of 3d percolation close to the critical point the main results are i a more precise estimate of the spreading resp chemical distance dimension ii a novel heuristic scaling theory for percolation near surfaces and iii more precise values for critical surface and edge exponents in addition the author verifies the very precise results of ziff and stell 1988 for the percolation thresholds and studies in detail corrections to scaling the latter is made possible by simulating bond and site percolation in a number of different geometries in addition in most cases one could measure more than one observable during each simulation which provided quite severe internal consistency constraints in some cases the author found quite important deviations from scaling which are not easily described by a single power or logarithmic correction term

on capacity of line networks we consider communication through a cascade of discrete memoryless channels dmcs the source and destination node of this cascade are allowed to use coding schemes of arbitrary complexity but the intermediate relay nodes are restricted to process only blocks of a fixed length we investigate how the processing at the relays must be chosen in order to maximize the capacity of the cascade that is the maximum achievable end to end rate between the source and the destination for infinite cascades with fixed intermediate processing length at the relays we prove that this intermediate processing can be chosen to be identical without loss of optimality and that the capacity of the cascade coincides with the rate of the best zero error code of length equal to the block length of the intermediate processing we further show that for fixed and identical intermediate processing at all relays convergence of capacity as the length of the cascade goes to infinity is exponentially fast finally we characterize how the block length of the intermediate processing must scale with the length of the cascade to guarantee a constant end to end rate we prove that it is sufficient that the block length scales logarithmically with the network length in order to achieve any rate above the zero error capacity we show that in many cases of interest logarithmic growth is also necessary

weak e nets and interval chains we construct weak e nets of almost linear size forcertain types of point sets specifically for planar point sets inconvex position we construct weak 1 r nets of size o r r where r denotes the inverse ackermann function for pointsets along the moment curve in d we constructweak 1 r nets of size r 2poly r where the degree of the polynomial in the exponent depends quadratically on d our constructions result from a reduction to a new problem which we call stabbing interval chains with j tuples given therange of integers n 1 n an interval chain of length k is asequence of k consecutive disjoint nonempty intervals containedin n a j tuple bar p p1 pj is said to stab aninterval chain c i1 ik if eachpi falls on a different interval of c the problem is toconstruct a small size family z of j tuples that stabs allk interval chains in n let z j k n denote the minimum size ofsuch a family z we derive almost tight upper and lower bounds forz j k n for every fixed j our boundsinvolve functions m n of the inverse ackermannhierarchy specifically we show that for j 3 we havez 3 k n n lfloor k 2 rfloor n for all k 6 for each j 4 we construct a pair of functionsp j m q j m almost equalasymptotically such that z j p j m n o n m n andz j q j m n n m n

a limit theorem for sets of stochastic matrices abstract the following fact about row stochastic matrices is an easy consequence of well known results for each positive integer n 1 there is a positive integer q q n with the property that if a is any n n stochastic matrix then the sequence of matrices aq a2q a3q converges we prove a generalization of this for sets of stochastic matrices under the hausdorff metric let d be any metric inducing the standard topology on the set of n n real matrices for a matrix a and set of matrices b define d a b to be the infimum of d a b over all b b for two sets of matrices a and b define d a b to be the supremum of d a b over all a a and define d a b to be the maximum of d a b and d b a this is the hausdorff metric on the set of subsets of n n stochastic matrices if a is a set of stochastic matrices and k is a positive integer define a k to be the set of all matrices expressible as a product of a sequence of k matrices from a we prove for each positive integer n there is a positive integer p p n such that if a is any subset of n n stochastic matrices then the sequence of subsets a p a 2p a 3p converges with respect to the hausdorff metric

parameter range reduction for ode models using cumulative backward differentiation formulas we consider fitting an ode model to time series data of the system variables we assume that the parameters of the model have some initial range of possible values and the goal is to reduce these ranges to produce a smaller parameter region from which to start a global nonlinear optimization algorithm we introduce the class of cumulative backward differentiation formulas cbdfs and show that they inherit the accuracy and stability properties of their generating backward differentiation formulas bdfs discretizing the system with these cbdfs and applying consistency conditions results in reductions of the parameter ranges we show that these reductions are better than can be obtained simply using bdfs in addition cbdfs inherit any monotonicity properties with respect to the parameters that the vector field possesses and we exploit these properties to make the consistency checking more efficient we illustrate with several examples analyze some of the behavior of our range reduction method and discuss how the method could be extended and improved

probability distribution of the shortest path on the percolation cluster its backbone and skeleton we consider the mean distribution functions ensuremath phi r mathcal l ensuremath phi mathrm b r mathcal l and ensuremath phi mathrm s r mathcal l giving the probability that two sites on the incipient percolation cluster on its backbone and on its skeleton respectively connected by a shortest path of length l are separated by an euclidean distance r following a scaling argument due to de gennes for self avoiding walks we derive analytical expressions for the exponents g 1 d f d mathrm min ensuremath d and g 1 mathrm b g 1 mathrm s 3d mathrm min ensuremath d which determine the scaling behavior of the distribution functions in the limit x ensuremath equiv r mathcal l stackrel ifmmode tilde else fi ensuremath nu ensuremath ll 1 i e ensuremath phi r mathcal l ensuremath propto mathcal l ensuremath stackrel ifmmode tilde else fi ensuremath nu d x g 1 ensuremath phi mathrm b r mathcal l ensuremath propto mathcal l ensuremath stackrel ifmmode tilde else fi ensuremath nu d x g 1 mathrm b and ensuremath phi mathrm s r mathcal l ensuremath propto mathcal l ensuremath stackrel ifmmode tilde else fi ensuremath nu d x g 1 mathrm s with stackrel ifmmode tilde else fi ensuremath nu ensuremath equiv 1 d mathrm min where d f and d mathrm min are the fractal dimensions of the percolation cluster and the shortest path respectively the theoretical predictions for g 1 g 1 mathrm b and g 1 mathrm s are in very good agreement with our numerical results

efficient simulation of the random cluster model the simulation of spin models close to critical points of continuous phase transitions is heavily impeded by the occurrence of critical slowing down a number of cluster algorithms usually based on the fortuin kasteleyn representation of the potts model and suitable generalizations for continuous spin models have been used to increase simulation efficiency the first algorithm making use of this representation suggested by sweeny in 1983 has not found widespread adoption due to problems in its implementation however it has been recently shown that it is indeed more efficient in reducing critical slowing down than the more well known algorithm due to swendsen and wang here we present an efficient implementation of sweeny s approach for the random cluster model using recent algorithmic advances in dynamic connectivity algorithms

a multidimensional filter algorithm for nonlinear equations and nonlinear least squares we introduce a new algorithm for the solution of systems of nonlinear equations and nonlinear least squares problems that attempts to combine the efficiency of filter techniques and the robustness of trust region methods the algorithm is shown under reasonable assumptions to globally converge to zeros of the system or to first order stationary points of the euclidean norm of its residual preliminary numerical experience is presented that shows substantial gains in efficiency over the traditional monotone trust region approach

shortest path fractal dimension for percolation in two and three dimensions we carry out a high precision monte carlo study of the shortest path fractal dimension d mathrm min for percolation in two and three dimensions using the leath alexandrowicz method which grows a cluster from an active seed site a variety of quantities are sampled as a function of the chemical distance including the number of activated sites a measure of the radius and the survival probability by finite size scaling we determine d mathrm min 1 130 phantom rule 0 16em 0ex 77 2 and 1 3756 6 in two and three dimensions respectively the result in two dimensions rules out the recently conjectured value d mathrm min 217 192 deng et al phys rev e 81 020102 r 2010

the number of incipient spanning clusters in two dimensional percolation using methods of conformal field theory we conjecture an exact form for the probability that n distinct clusters span a large rectangle or open cylinder of aspect ratio k i n the limit when k is large the study of the structure of large clusters at the percolation threshold continues to pose interesting problems whose solution sheds light on the nature of the critical state in general recently some attention has been paid to incipient spanning clusters iscs these are clusters which connect two disjoint segments of the boundary of a macroscopically large region langlands et al 1 conjectured that the probability that at least one such cluster exists that is that the segments are connected is invariant under conformal transformations this statement was placed in the context of conformal field theory in 2 where an explicit formula was given for this crossing probability for a review of the status of conformal invariance in critical percolation see 3 more recently aizenman 4 has considered among other things the probability that there exist n distinct iscs connecting the two segmentsy in the case of a rectangular region 0 kl 0 l he has proved that the probability p n k l that the strip is traversed in the direction of length kl b y nindependent clusters satisfies the bounds a e n 2 k

on the stability of exponential fitting bdf algorithms we present bdf type formulas capable of the exact integration with only round off errors of differential equations whose solutions are linear combinations of an exponential with parameter l and ordinary polynomials for l 0 new formulas reduces to the classical bdf formulas plots of their 0 stability regions in terms of l are provided plots of their regions absolute stability that include all the negative real axis are provided numerical examples shows the efficiency of the proposed codes specially when we are integrating stiff oscillatory problems