conformal invariance and finite one dimensional quantum chains based on previous work of cardy 1984 the authors show in a systematic way how using conformal invariance one can determine the anomalous dimensions of various operators from finite quantum chains with different boundary conditions the method is illustrated in the case of the three and four state potts models where the anomalous dimensions of the para fermionic operators are found

finite size scaling of the three state potts model on a simple cubic lattice finite size scaling is studied for the three state potts model on a simple cubic lattice we show that the specific heat and the magnetic susceptibility scale accurately as the volume the correlation length exhibits behaviors expected for a genuine first order transition the one extracted from the unsubtracted correlation function shows a characteristic finite size behavior whereas the physical correlation length that characterizes the first excited state stays at a finite value and is discontinuous at the transition point

the three dimensional ising model revisited numerically several basic universal amplitude ratios are studied afresh for three dimensional nearest neighbor ising models in revising earlier work modern estimates of the critical temperature and exponents are used in conjunction with biased inhomogeneous differential approximants to extrapolate the longest available series expansions to find the critical amplitudes c for the susceptibility 1 for the correlation length 1 a for the specific heat c t and b for the spontaneous magnetization m0 we find c c 4 95 0 15 1 1 1 960 0 01 a a 0 523 0 009 a c b 0 0581 0 0010 while a 1 3 verifies hyperuniversality to within 0 8 a method for calculating amplitude ratios which allows for corrections to scaling yields estimates for c c and 1 1 in excellent agreement with those derived from the individual amplitudes finally explicit formulae are given for the numerical evaluation of t 1 t c t and m0 t over the full temperature range from criticality to t 0 and corresponding plots and convenient near critical representations are also presented

knuth s balanced codes revisited in 1986 don knuth published a very simple algorithm for constructing sets of bipolar codewords with equal numbers of 1 s and 1 s called balanced codes knuth s algorithm is well suited for use with large codewords the redundancy of knuth s balanced codes is a factor of two larger than that of a code comprising the full set of balanced codewords in this paper we will present results of our attempts to improve the performance of knuth s balanced codes

series analysis of the 3 state potts model in 2 1 dimensions both high temperature and low temperature series are used to locate and characterize the first order transition in the 3 state potts model in 2 1 dimensions on both square and triangular lattices estimates are presented for the vacuum energy latent heat magnetization susceptibility and mass gap at the transition the spontaneous magnetization and latent heat appear to display an approximate universality at this weak first order transition

the 3 state potts model in 2 1 dimensions the 3 state potts model in 2 1 dimensions is studied using series and monte carlo methods high temperature series for the vacuum energy mass gap and susceptibility are computed on the square and triangular lattices using linked cluster expansions monte carlo estimates are obtained by the stochastic truncation method the model is found to undergo a weak first order transition as in the euclidean version of the model

balancing sets of vectors for n 0 d or 0 n identical to d mod 2 let k n d denote the minimal cardinality of a family v of or 1 vectors of dimension n such that for any or 1 vector w of dimension n there is a v in v such that mod v w mod

aubry mather theory and periodic solutions of the forced burgers equation consider a hamiltonian system with hamiltonian of the form h x t p where h is convex in p and periodic in x a ndt and x2 r 1 it is well known that its smooth invariant curves correspond to smoothz 2 periodic solutions of the pde ut h x t u x 0 in this paper we establish a connection between the aubry mather theory of invariant sets of the hamiltonian system andz 2 periodic weak solutions of this pde by realizing the aubry mather sets as closed subsets of the graphs of these weak solutions we show that the complement of the aubry mather set on the graph can be viewed as a subset of the generalized unstable manifold of the aubry mather set defined in 2 24 the graph itself is a backward invariant set of the hamiltonian system the basic idea is to embed the globally minimizing orbits used in the aubry mather theory into the characteristic fields of the above pde this is done by making use of one and two sided minimizers a notion introduced in 12 and inspired by the work of morse on geodesics of type a 26 the asymptotic slope of the minimizers also known as the rotation number is given by the derivative of the homogenized hamiltonian defined in 21 as an application we prove that thez 2 periodic weak solution of the above pde with given irrational asymptotic slope is unique a similar connection also exists in multidimensional problems with the convex hamiltonian except that in higher dimensions two sided minimizers with a specified asymptotic slope may not exist c 1999 john wiley sons inc

finite size effects and phase transition in the three dimensional three state potts model the three state potts model in three dimensions is studied by monte carlo and finite size scaling techniques using a histogram method recently proposed by ferrenberg and swendsen the finite size dependence for the maximum of the specific heat is found to scale with the volume of the system indicating that the phase transition is of first order the value of the latent heat per spin and the correlation length at the transition are estimated

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

the euler equation and absolute minimizers of l functionals the aronsson euler equation for the functional on w g 1 m i e w 1 with boundary data g is this equation has been derived for smooth absolute minimizers i e a function which minimizes f on every subdomain we prove in this paper that for m 1 n 1 or n 1 m 1 an absolute minimizer of f exists in w g 1 m and for m 1 n 1 any absolute minimizer of f must be a viscosity solution of the aronsson euler equation

m ary balanced codes with parallel decoding an m ary block code m 2 3 4 ldots of length n in mathbf i mathbf i mskip 7mu mathbf n is called balanced if and only if every codeword is balanced that is the real sum of the codeword components or weight is equal to left lfloor m 1 n 2 right rfloor this paper presents efficient encoding schemes to m ary balanced codes with parallel hence fast decoding in fact the decoding time complexity is o 1 digit operations these schemes are a generalization to the m ary alphabet of knuth s complementation method with parallel decoding let binom n w m indicate the number of m ary words of length n and weight w in 0 1 ldots m 1 n for any m in mathbf i mathbf i mskip 7mu mathbf n m geq 2 a simple implementation of the method is given which uses r in mathbf i mathbf i mskip 7mu mathbf n check digits to balance k leq binom r left lfloor m 1 r 2 right rfloor vphantom r r m m bmod 2 m 1 k bmod 2 m 1 information digits with an encoding time complexity of o mk log m k digit operations a refined implementation of the parallel decoding method is also given with r check digits and k leq m r 1 m 1 information digits where the encoding time complexity is o k sqrt log m k thus the proposed codes are less redundant than the m ary balanced codes with parallel decoding found in the literature and yet maintain the same complexity

computing the effective hamiltonian using a variational approach a numerical method for homogenization of hamilton jacobi equations is presented and implemented as an l infty calculus of variations problem solutions are found by solving a nonlinear convex optimization problem the numerical method is shown to be convergent and error estimates are provided one and two dimensional examples are worked in detail comparing known results with the numerical ones and computing new examples the cases of nonstrictly convex hamiltonians and hamiltonians for which the cell problem has no solution are treated

effective hamiltonians and averaging for hamiltonian dynamics i this paper building upon ideas of mather moser fathi e and others applies pde partial differential equation methods to understand the structure of certain hamiltonian flows the main point is that the cell or corrector pde introduced and solved in a weak sense by lions papanicolaou and varadhan in their study of periodic homogenization for hamilton jacobi equations formally induces a canonical change of variables in terms of which the dynamics are trivial we investigate to what extent this observation can be made rigorous in the case that the hamiltonian is strictly convex in the momenta given that the relevant pde does not usually in fact admit a smooth solution

capacity approaching non binary balanced codes using auxiliary data it is known that for large user word lengths the auxiliary data can be used to recover most of the redundancy losses of knuth s simple balancing method compared with the optimal redundancy of balanced codes for the binary case here this important result is extended in a number of ways first an upper bound for the amount of auxiliary data is derived that is valid for all codeword lengths this result is primarily of theoretical interest as it defines the probability distribution of the number of balancing indices that results in optimal redundancy this result is equally valid for particular non binary generalizations of knuth s balancing method second an asymptotically exact expression for the amount of auxiliary data for the ternary case of a variable length realization of the modified balanced code construction is derived that in all respects is the analogue of the result obtained for the binary case the derivation is based on a generalization of the binary random walk to the ternary case and a simple modification of an existing generalization of knuth s method for the non binary balanced codes finally a conjecture is proposed regarding the probability distribution of the number of balancing indices for any alphabet size

the hamilton jacobi bellman equation for time optimal control in this paper several assertions concerning viscosity solutions of the hamilton jacobi bellman equation for the optimal control problem of steering a system to zero in minimal time are proved first two rather general uniqueness theorems are established asserting that any positive viscosity solution of the hjb equation must in fact agree with the minimal time function near zero if also a boundary condition introduced by bardi siam j control optim 27 1988 pp 776 785 is satisfied then the agreement is global additionally the holder continuity of any subsolution of the hjb equation is proved in the case where the related dynamics satisfy a hormander type hypothesis this last assertion amounts to a half derivative analogue of a theorem of crandall and lions traps amer math soc 277 1 983 pp 1 42 concerning lipschitz viscosity solutions

correlation length of the three state potts model in three dimensions a monte carlo study is carried out for the three state potts model in three dimensions the correlation length is shown to remain finite and to be discontinuous at the transition point it is also shown that the transition exhibits genuine first order characteristics in all aspects

approximate universality at a first order transition the three state potts model in 3 1 dimensions both high and low temperature perturbation series are used to locate and characterize the first order transition in the three state potts model in 3 1 dimensions on the simple cubic body centred cubic and face centred cubic lattices estimates are presented for the vacuum energy latent heat magnetization susceptibility and mass gap at the transition the results show a remarkable degree of universality between the different lattices and a law of corresponding states is followed very closely

duality in linear programming problems related to deterministic long run average problems of optimal control it has been established recently that under mild conditions deterministic long run average problems of optimal control are asymptotically equivalent to infinite dimensional linear programming problems lpps and that these lpps can be approximated by finite dimensional lpps in this paper we introduce the corresponding infinite and finite dimensional dual problems and study duality relationships we also investigate the possibility of using solutions of finite dimensional lpps and their duals for numerical construction of the optimal controls in periodic optimization problems the construction is illustrated with a numerical example

on balanced codes in a balanced code each codeword contains equally many 1 s and 0 s parallel decoding balanced codes with 2 sup r or 2 sup r 1 information bits are presented where r is the number of check bits the 2 sup 2 r 1 construction given by d e knuth ibid vol 32 no 1 p 51 3 1986 is improved the new codes are shown to be optimal when knuth s complementation method is used

on efficient balanced codes over the mth roots of unity let spl phi sub m spl sube spl copf be the set of all mth roots of unity m spl isin in a balanced code over spl phi sub m is a block code over the alphabet spl phi sub m such that each code word is balanced that is the complex sum of its components or weight is equal to 0 let b sub m n be the set of all balanced words of length n over spl phi sub m in this correspondence it is shown that when m is a prime number the set b sub m n is not empty if and only if m divides n in this case the minimum redundancy for a balanced code over spl phi sub m of length n is on the other hand it is shown that when m 4 the set b sub 4 n is not empty if and only if n is even and in this case the minimum redundancy for a balanced code over spl phi sub 4 of length n is further this correspondence completely solves the problem of designing efficient coding methods for balanced codes over spl phi sub m when m 4 in fact it reduces the problem of designing efficient coding schemes for balanced codes over spl phi sub 4 to the design of efficient balanced codes over the usual bipolar alphabet spl phi sub 2 1 1

variable length unordered codes in an unordered code no code word is contained in any other code word unordered codes are all unidirectional error detecting aued codes in the binary case it is well known that among all systematic codes with k information bits berger codes are optimal unordered codes with r log2 k 1 log2k check bits this paper gives some new theory on variable length unordered codes and introduces a new class of systematic instantaneous unordered codes with variable length check symbols the average redundancy of the new codes presented here is r 1 2 log2k c where c 1 0470 1 1332 ir and k in is the number of information bits when k is large it is shown that such redundancy is at most 0 6069 bits off the redundancy of an optimal systematic unordered code design with fixed length information symbols and variable length check symbols and at most 2 8075 bits off the redundancy of an optimal variable length unordered code design the generalization is also given for the nonbinary case and it is shown that similar results hold true

design of some new efficient balanced codes a balanced code with r check bits and k information bits is a binary code of length k r and cardinality 2 sup k such that each codeword is balanced that is it has k r 2 1 s and k r 2 0 s this paper contains new methods to construct efficient balanced codes to design a balanced code an information word with a low number of 1 s or 0 s is compressed and then balanced using the saved space on the other hand an information word having almost the same number of 1 s and 0 s is encoded using the single maps defined by knuth s 1986 complementation method three different constructions are presented balanced codes with r check bits and k information bits with k spl les 2 sup r 1 2 k spl les 3 spl times 2 sup r 8 and k spl les 5 spl times 2 sup r 10r c r c r spl isin 15 10 5 0 5 are given improving the constructions found in the literature in some cases the first two constructions have a parallel coding scheme