exact large deviation bounds up to t c for the ising model in two dimensions we prove an upper large deviation bound for the block spin magnetization in the 2d ising model in the phase coexistence region the precise rate given by the wulff construction is shown to hold true for all c combined with the lower bounds derived in i those results yield an exact second order large deviation theory up to the critical temperature

limitedness theorem on finite automata with distance functions an algebraic proof a distance function on a finite automaton m is defined by assigning to each transition a distance of nonnegative integer value m is said to be limited in distance if there is a nonnegative integer k such that for each accepted string w in the language of m there is an accepting path p for w on which the sum of distances is bounded by k the limitedness theorem 6 9 on finite automata with distance functions states that it is decidable if an arbitrary finite automaton with a distance function is limited in distance in this paper we give an algebraic proof of the theorem and derive from it an exponential time decision algorithm in addition we prove that the decision problem is pspace hard

large deviations for the 2d ising model a lower bound without cluster expansions we show that a lower large deviation bound for the block spin magnetization in the 2d ising model can be pushed all the way forward toward its correct wulff value for all c

numerical solution of integral equations by means of the sinc collocation method based on the double exponential transformation numerical solution of linear integral equations by means of the sinc collocation method based on the double exponential transformation abbreviated the de transformation is considered we first apply the method to the volterra integral equation of the second kind and then to the volterra equation of the first kind this method is also applied to the fredholm integral equation of the second kind for the volterra equations we employed a formula for numerical indefinite integration developed by muhammad and mori obtained by applying the de transformation incorporated into the sinc expansion of the integrand while for the fredholm equation we employed the conventional de transformation for definite integrals an error analysis of the method is given and in every case a convergence rate of o exp cn logn for the error is established where n is a parameter representing the number of terms of the sinc expansion also the condition of the matrix of the main system of linear equations is watched through an estimate of the condition number returned by the program numerical examples show the convergence rate mentioned above and confirm the high efficiency of the present method

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

large deviations for the empirical field of a gibbs measure on applique le principe de contraction pour obtenir un principe de grande deviation pour la distribution des distributions empiriques on donne une description detaillee de la fonction de taux resultante

recognizable closures and submonoids of free partially commutative monoids abstract the main result of this paper is the presentation of two sufficient conditions for the closure x of x and the closure x of x respectively in any free partially commutative monoid for any regular language x to be recognizable several previously obtained sufficient conditions for x to be recognizable can be seen to imply one of those two conditions

an imt type quadrature formula with the same asymptotic performance as the de formula we propose an imt type quadrature formula which achieves the same asymptotic error estimate as the de formula the point of the idea is to optimize the parameters of the imt type transformation depending on the number of sampling points we also show the performance of our imt type quadrature formula by numerical examples

rational relations and rational series a rational relation is a rational subset of the direct product of two free monoids r a b consider r as a function of a into the family of subsets of b by posing for all u a r u v b u v r assume r u is a finite set for all u a we study how the cardinality of r u behaves as the length of u tends to infinity and we show that there exists an infinite hierachy of growth functions

second order large deviation estimates for ferromagnetic systems in the phase coexistence region we consider thed dimensional ising model with ferromagnetic nearest neighbor interaction at inverse temperature let m lambda lambda 1 sum limits i in lambda sigma i be the magnetization inside ad dimensional hyper cube be the gibbs state andm be the spontaneous magnetization for such thatm 0 we find a sufficient condition easily verified to hold for large for m a b to decay exponentially with d 1 d when m b m 1 a b ford 2 this sufficient condition is the exponential decay of a connectivity function we also prove a partial converse to this result obtain a sharper result for the magnetization ond 1 dimensional cross sections of the model and prove a similar result ford 2 m a b m and large when free boundary conditions are chosen outside

large deviations for gibbs random fields a large deviation principle for gibbs random fields on zd is proven and a corresponding large deviations proof of the gibbs variational formula is given a generalization of the lanford theory of large deviations is also obtained

an upper bound estimate and stability for the global error of numerical integration using double exponential transformation abstract the double exponential formula was introduced for calculating definite integrals with singular point oscillation functions and fourier integral the double exponential transformation is not only useful for numerical computations but it is also used in different methods of sinc theory in this paper we give an upper bound estimate for the error of double exponential transformation by improving integral estimates having singular final points in theorem 1 we prove that the method is convergent and the rate of convergence is o h 2 where h is a step size our main tool in the proof is de formula in sinc theory the advantage of our method is that the time and space complexity is drastically reduced furthermore we discovered an upper bound error in de formula independent of n truncated number as a matter of fact we proved stability numerical tests are presented to verify the theoretical predictions and confirm the convergence of the numerical solution

complete analyticity for 2d ising completed we study the behavior of the two dimensional nearest neighbor ferromagnetic ising model under an external magnetic fieldh we extend to every subcritical value of the temperature a result previously proven by martirosyan at low enough temperature and which roughly states that for finite systems with boundary conditions under a positive external field the boundary effect dominates in the bulk if the linear size of the system is of orderb h withb small enough while ifb is large enough then the external field dominates in the bulk as a consequence we are able to complete the proof that complete analyticity for nice sets holds for every value of the temperature and external field in the interior of the uniqueness region in the phase diagram of the model the main tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field and recently extended to all temperatures below the critical one by ioffe

wulff droplets and the metastable relaxation of kinetic ising models we consider the kinetic ising models glauber dynamics corresponding to the infinite volume ising model in dimension 2 with nearest neighbor ferromagnetic interaction and under a positive external magnetic field h minimal conditions on the flip rates are assumed so that all the common choices are being considered we study the relaxation towards equilibrium when the system is at an arbitrary subcritical temperature t and the evolution is started from a distribution which is stochastically lower than the phase we show that as h 0 the relaxation time blows up as exp c t h with lgr c t w t 2 12 t m t here m t is the spontaneous magnetization and w t is the integrated surface tension of the wulff body of unit volume moreover for 0 c the state of the process at time exp h is shown to be close when h is small to the phase the difference between this state and the phase can be described in terms of an asymptotic expansion in powers of the external field this expansion can be interpreted as describing a set of continuations in h of the family of gibbs distributions with the negative magnetic fields into the region of positive fields

surface order large deviations for ising potts and percolation models we derive uniform surface order large deviation estimates for the block magnetization in finite volume ising or potts models with plus or free or a combination of both boundary conditions in the phase coexistence regime ford 3 the results are valid up to a limit of slab thresholds conjectured to agree with the critical temperature our arguments are based on the renormalization of the random cluster model withq 1 andd 3 and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability the results are new even for the case of independent percolation q 1 as a byproduct of our methods we obtain further results in the fk model concerning semicontinuity inp andq of the percolation probability the second largest cluster in a box and the tail of the finite cluster size distribution

the double exponential transformation in numerical analysis the double exponential transformation was first proposed by takahasi and mori in 1974 for the efficient evaluation of integrals of an analytic function with end point singularity afterwards this transformation was improved for the evaluation of oscillatory functions like fourier integrals recently it turned out that the double exponential transformation is useful not only for numerical integration but also for various kinds of sinc numerical methods the purpose of the present paper is to review the double exponential transformation in numerical integration and in a variety of sinc numerical methods

on the decidability of some problems about rational subsets of free partially commutative monoids let i a b be a partially commutative alphabet such that two letters commute if one of them belongs to a and the other one belongs to b let m a b denote the free partially commutative monoid generated by i we consider the following six problems for rational given by regular expressions subsets x y of m q1 x y o q2 x y di x y q4 x m l25 m x finite x is recognizable it was proved by choffrut see 2 that all these problems are undecidable if card a 1 and card b 1 and they are decidable if card a card b 1 card u denotes the cardinality of u it was conjectured see 2 p 79 that these problems are decidable in the remaining cases where card a 1 and card b 1 in this paper we show that if card a 1 and card b 1 then the problem 01 is decidable and problems q2 06 are undecidable our paper is an application of results concerning reversal bounded nondeterministic multicounter machines and nondeterministic general sequential machines

function classes for successful de sinc approximations the de sinc formulas resulting from a combination of the sinc approximation formula with the double exponential de transformation provide a highly efficient method for function approximation in many cases they are more efficient than the se sinc formulas which are the sinc approximation formulas combined with the single exponential se transformations function classes suited to the se sinc formulas have already been investigated in the literature through rigorous mathematical analysis whereas this is not the case with the de sinc formulas this paper identifies function classes suited to the de sinc formulas in a way compatible with the existing theoretical results for the se sinc formulas furthermore we identify alternative function classes for the de sinc formulas as well as for the se sinc formulas which are more useful in applications in the sense that the conditions imposed on the functions are easier to verify

exponential convergence under mixing we show that for a mixing stationary sequence of bounded random variables the average of the firstn variables converges exponentially fast withn to the mean value of these random variables

fluctuations of shapes of large areas under paths of random walks we discuss statistical properties of random walks conditioned by fixing a large area under their paths we prove the functional central limit theorem invariance principle for these conditional distributions the limiting gaussian measure coincides with the conditional probability distribution of certain timenonhomogeneous gaussian random process obtained by an integral transformation of the white noise from the point of view of statistical mechanics the studied problem is the problem of describing the fluctuations of the phase boundary in the one dimensional sos model

optimality of the double exponential formula functional analysis approach in the light of the functional analysis theory we establish the optimality of the double exponential formula the argument consists of the following three ingredients 1 introduction of a number of spaces of functions analytic in a strip region about the real axis each space being characterized by the decay rate of their elements functions in the neighborhood of the infinity 2 proof of the near optimality of the trapezoidal formula in each space introduced in 1 by showing the near equality between an upper estimate for the error norm of the trapezoidal formula and a lower estimate for the minimum error norm of quadratures 3 nonexistence theorem for the spaces the characterizing decay rate of which is more rapid than the double exponential

string matching problems over free partially commutative monoids abstract this paper studies two string matching problems over free partially commutative monoids we analyze these two problems in detail and present two efficient polynomial time algorithms for solving them

the wulff construction and asymptotics of the finite cluster distribution for two dimensional bernoulli percolation we consider two dimensional bernoulli percolation at densityp p c and establish the following results

probabilistic estimation of the number of prefixes of a trace abstract we present a generating function technique to evaluate the number of strings of a given length recognized by a particular kind of finite state automaton using the method we determine some asymptotic estimations of the number of prefixes in free partially commutative monoids more precisely we prove that for every concurrent alphabet c assuming equiprobable strings all of length n the average number of prefixes of a trace of length n is n k o n k 1 and its variance is o n 2 k 1 where k is the number of components of the dependency graph c c and is a rational constant depending only on c these combinatorial results allow to determine the probabilistic behaviour of some algorithms for problems on trace languages including the membership problems for regular and context free trace languages