holonomic functions of several complex variables and singularities of anisotropic ising n fold integrals focusing on examples associated with holonomic functions we try to bring new ideas on how to look at phase transitions for which the critical manifolds are not points but curves depending on a spectral variable or even fill higher dimensional submanifolds lattice statistical mechanics often provides a natural holonomic framework to perform singularity analysis with several complex variables that would in the most general mathematical framework be too complex or simply could not be defined in a learn by example approach considering several picard fuchs systems of two variables above calabi yau odes associated with double hypergeometric series we show that d finite holonomic functions are actually a good framework for finding properly the singular manifolds the singular manifolds are found to be genus zero curves we then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics the n fold integrals n corresponding to the n particle decomposition of the magnetic susceptibility of the anisotropic square ising model in this anisotropic case we revisit a set of so called nickelian singularities that turns out to be a two parameter family of elliptic curves we then find the first set of non nickelian singularities for 3 and 4 that also turns out to be rational or elliptic curves we underline the fact that these singular curves depend on the anisotropy of the ising model or equivalently that they depend on the spectral parameter of the model this has important consequences on the physical nature of the anisotropic n s which appear to be highly composite objects we address from a birational viewpoint the emergence of families of elliptic curves and that of calabi yau manifolds on such problems we also address the question of singularities of non holonomic functions with a discussion on the accumulation of these singular curves for the non holonomic anisotropic full susceptibility this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

optimal q ary error correcting all unidirectional error detecting codes codes that can correct up to t symmetric errors and detect all unidirectional errors known as t ec aued codes are studied in this paper given positive integers q a and t let n q a t 1 denote the length of the shortest q ary t ec aued code of size a we introduce combinatorial constructions for q ary t ec aued codes via one factorizations of complete graphs and concatenation of mds codes and codes from resolvable set systems consequently we determine the exact values of n q a t 1 for several new infinite families of q a and t

the potts model and flows iii standard and subgraph break collapse methods for pt ii see ibid vol 19 p 1655 1986 an algorithm is developed for the exact calculation of the many spin correlation functions of potts model clusters which is more efficient than the standard break collapse method successfully used in real space renormalisation group calculations the improved performance is based on a relationship which at any stage of the calculation allows the replacement of certain subgraphs by single effective edges thereby decreasing the number of iterations needed the authors method avoids as in the standard one the time consuming summation over spin states and may be used to extend series expansion and real space renormalisation group calculations on crystal lattices both methods are based on a number of combinatorial formulae the proofs of which are given in this paper

zeroes of the triangular potts model partition function a conjectured distribution abstract we consider the q state potts model on the triangular lattice with two and three site interactions in alternate triangular faces and determine zeroes of the partition function numerically in the case of pure three site interactions on the basis of a rigorous reciprocal symmetry and results on the zeroes for finite lattices we conjecture that zeroes of the partition function of the triangular potts model with pure three site interactions in alternate triangular faces lie on a circle and a segment of the negative real axis it is shown that the conjecture holds for q 2 and that it reproduces the known critical point for general q including the q 1 site percolation

disorder solutions for ising and potts models with a field the exact expressions of the disorder varieties and the corresponding values taken by the partition functions of the checkerboard ising and potts models with a magnetic field are obtained different specialisations and extensions of these exact results are examined

critical frontier of the potts and percolation models on triangular type and kagome type lattices i closed form expressions we consider the potts model and the related bond site and mixed site bond percolation problems on triangular type and kagome type lattices and derive closed form expressions for the critical frontier for triangular type lattices the critical frontier is known usually derived from a duality consideration in conjunction with the assumption of a unique transition our analysis however is rigorous and based on an established result without the need of a uniqueness assumption thus firmly establishing all derived results for kagome type lattices the exact critical frontier is not known we derive a closed form expression for the potts critical frontier by making use of a homogeneity assumption the closed form expression is unique and we apply it to a host of problems including site bond and mixed site bond percolations on various lattices it yields exact thresholds for site percolation on kagome martini and other lattices and is highly accurate numerically in other applications when compared to numerical determination

constant weight codes for correcting symmetric errors and detecting unidirectional errors proposes two classes of constant weight codes which can be used for correcting t symmetric errors and simultaneously detecting all unidirectional errors codes in the first class are in quasi systematic form and codes in the second class are in systematic form since each codeword of codes in both classes can be divided into a data part and a parity check part the proposed codes have the merit of easily mapping messages into codewords

a disorder solution for a cubic ising model the exact expression of the partition function of a three dimensional cubic ising model with nearest neighbour interactions is given when a certain relation between the coupling constants of the model is satisfied this disorder solution is then compared with a partially resummed high temperature expansion of the partition function of the model the constraints on this expansion which result from the existence of the disorder solution and from the inversion relation are discussed

on the singularity structure of the 2d ising model susceptibility some simplifications of the integrals 2n 1 derived by wu et al 1976 13 316 that contribute to the zero field susceptibility of the 2d square lattice ising model are reported in particular several alternate expressions for the integrands in 2n 1 are determined which greatly facilitate both the generation of high temperature series and analytical analysis one can show that as series 2n 1 22n s 2 4n n 1 1 o s where s is the high temperature variable sin 2k with k the conventional normalized inverse temperature analysis of the integrals near symmetry points of the integrands shows that 2n 1 s is singular on the unit circle at sk exp ik where 2cos k cos 2 k 2n 1 cos 2 2n 1 n k n the singularities k 0 excepted are logarithmic branch points of order 2n n 1 1ln with 1 s sk there is numerical evidence from series that these van hove points in addition to the known points at s 1 and i exhaust the singularities on the unit circle barring cancellation from extra unobserved singularities one can conclude that s 1 is a natural boundary for the susceptibility

generalized hypergeometric functions and rational curves on calabi yau complete intersections in toric varieties we formulate general conjectures about the relationship between the a model connection on the cohomology of ad dimensional calabi yau complete intersectionv ofr hypersurfacesv1 vr in a toric varietyp and the system of differential operators annihilating the special generalized hypergeometric series 0 constructed from the fan using this generalized hypergeometric series we propose conjectural mirrorsv ofv and the canonicalq coordinates on the moduli spaces of calabi yau manifolds

hyperbolic coxeter groups for triangular potts models the symmetry groups generated by the inversion relations of lattice models of statistical mechanics on triangular lattices are analysed for vertex models and for the standard scalar potts model with two and three site interactions these groups are generated by three inversion relations and are seen to be generically very large ones hyperbolic groups two situations for which the representations of these groups degenerate into smaller ones hopefully compatible with integrability are considered the first reduction for the vertex triangular model corresponds to the situation where the vertex of the triangular model coincides with the left or right hand side of a yang baxter relation in this case the representation of the group is isomorphic up to a semi direct product by a finite group to z z the second reduction for q state ports models occurs for particular values of q the so called tutte beraha numbers for this model algebraic varieties including the known ferromagnetic critical variety happen to be invariant under such large groups of symmetries as a byproduct this analysis provides nice birational representations of hyperbolic coxeter groups

on the phase diagram of the chiral potts model the authors study the phase diagram of the isotropic three state chiral potts model monte carlo simulations performed on twisted square lattices with different sizes up to 64 64 indicate different types of critical curves in particular a floating phase seems to occur finite size scaling analyses are performed at some specific critical points the phase diagram they propose is discussed using exact results

on unordered codes by extending the results obtained by d e knuth 1986 a parallel unordered coding scheme with 2 sup r information bits is described balanced codes in which each codeword contains equal amounts of zeros and ones with r check bits and up to 2 sup r 1 r 2 information bits are constructed unordered codes with r check bits and up to 2 sup r 2 sup r 1 1 information bits are designed codes capable of detecting 2 sup r 1 2 sup r 2 1 unidirectional errors using r check bits are also described a review of previous work is presented

ising model susceptibility the fuchsian differential equation for χ 4 and its factorization properties we give the fuchsian linear differential equation satisfied by x 4 the four particle contribution to the susceptibility of the isotropic square lattice ising model this fuchsian differential equation is deduced from a series expansion method introduced in two previous papers and is applied with some symmetries and tricks specific to x 4 the corresponding order ten linear differentia operator exhibits a large set of factorization properties among these factorizations one is highly remarkable it corresponds to the fact that the two particle contribution x 2 is actually a solution of this order ten linear differential operator this result together with a similar one for the order seven differential operator corresponding to the three particle contribution x 3 leads us to a conjecture on the structure of all the n particle contributions x n

differential algebra on lattice green functions and calabi yau operators we revisit miscellaneous linear differential operators mostly associated with lattice green functions in arbitrary dimensions but also calabi yau operators and order 7 operators corresponding to exceptional differential galois groups we show that these irreducible operators are not only globally nilpotent but are such that they are homomorphic to their formal adjoints considering these operators or sometimes equivalent operators we show that they are also such that either their symmetric square or their exterior square have a rational solution this is a general result an irreducible linear differential operator homomorphic to its formal adjoint is necessarily such that either its symmetric square or its exterior square has a rational solution and this situation corresponds to the occurrence of a special differential galois group we thus define the notion of being special geometry for a linear differential operator if it is irreducible globally nilpotent and such that it is homomorphic to its formal adjoint since many derived from geometry n fold integrals periods occurring in physics are seen to be diagonals of rational functions we address several examples of minimal order operators annihilating diagonals of rational functions and remark that they also seem to be systematically associated with irreducible factors homomorphic to their adjoint

criticality of the discrete n vector ferromagnet in planar self dual lattices the authors study an extended version of the discrete n vector or cubic ferromagnetic model within a real space renormalisation group approach which preserves the two spin correlation function the n evolution for real values of n of the wheatstone bridge hierarchical lattice phase diagram which presents paramagnetic intermediate nematic like and ferromagnetic phases as well as of the thermal nu and crossover phi critical exponents is presented the self avoiding walk problem is recovered in the n to o limit and the so called corner rule is re obtained in a larger context the ising n and 2n state potts ferromagnets are recovered as particular cases an interchange of stability occurs at n n approximately 6 9 in such a way that the 2n state potts special point where all three existing phases join is multicritical if n n but only critical if n n consistently phi n o for the cubic model nu n presents a maximum at n nmax approximately 1 5 the results are exact for all n for the wheatstone bridge hierarchical lattice and approximate for n or 2 for the square lattice last but not least they discuss the connection between the present approach and the phenomenological renormalisation group

duality relation for potts multispin correlation functions the duality relation for multispin correlation functions of the potts model on a plane graph is derived using the formulation of a percolation average it is shown that in simple cases the correlation functions are related to those of the dual graph but more generally the correlation functions are expressible as linear combinations of ratios of certain percolation averages the dual partitioned equivalent transmissivities and dual correlation functions an overall multiplication factor arises in these considerations and the author elucidate the graph theoretic meaning of this function

numerical experiments on yang lee zeros it is argued that since any real space renormalisation r is a transformation under which the partition function is invariant those r which are based upon various real space approximation schemes can in principle be extended into the complex field of the parameter space in this extension the julia set of r now extended to include r which are algebraic is an approximation to the locus of the yang lee zeros finite size scaling using onsager s solution of the two dimensional ising model is used to construct r which are highly accurate in real temperatures in the critical region these are extended into the complex temperature plane and results are displayed for both the anisotropic and isotropic cases

the susceptibility of the square lattice ising model new developments we have made substantial advances in elucidating the properties of the susceptibility of the square lattice ising model we discuss its analyticity properties certain closed form expressions for subsets of the coefficients and give an algorithm of complexity o n6 to determine its first n coefficients as a result we have generated and analyzed series with more than 300 terms in both the high and low temperature regime we quantify the effect of irrelevant variables to the scaling amplitude functions in particular we find and quantify the breakdown of simple scaling in the absence of irrelevant scaling fields arising first at order t tc 9 4 though high low temperature symmetry is still preserved at terms of order t tc 17 4 and beyond this symmetry is no longer present the short distance terms are shown to have the form t tc p log t tc q with p q2 conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments

location of zeros in the complex temperature plane absence of lee yang theorem in the yang lee theory 1952 the occurrence of a phase transition at real physical values of the magnetic field is related to the behaviour of the density of zeros of the partition function in the complex magnetic field plane as shown by lee and yang these zeros fall on lines for a wide class of ferromagnetic models in extensions of the yang lee theory to the complex temperature plane it has often been assumed that the zeros of the partition function would again fall on lines rather than in areas though this happens to be true for the isotropic ising model the authors point out that there is in general no basis for this assumption they discuss the location of the zeros of the partition function of the anisotropic ising model in the complex temperature plane and show that they indeed always fall in areas

z λ model and flows subgraph break collapse method a recursive algorithm previously developed for carrying out renormalisation calculations for the lambda state potts model is generalised to the z model the relations used are based on an expression the authors derive for the pair correlation function in terms of mod lambda flows which represents an extension of a similar result for the partition function previously obtained by biggs 1976 the use of flows enables them to prove and extend the formulae which appear in the break collapse method of mariz and co workers 1989 it is argued that the use of fixed flow bonds rather than the precollapsed bonds used by the latter authors leads to a more efficient algorithm

factorization of the ising model form factors we present a general method for analytically factorizing the n fold form factor integrals f n n n t for the correlation functions of the ising model on the diagonal in terms of the hypergeometric functions 2f1 1 2 n 1 2 n 1 t which appear in the form factor f 1 n n t new quadratic recursion and quartic identities are obtained for the form factors for n 2 3 for n 2 3 4 explicit results are given for the form factors these factorizations are proved for all n for n 2 3 these results yield the emergence of palindromic polynomials canonically associated with elliptic curves as a consequence understanding the form factors amounts to describing and understanding an infinite set of palindromic polynomials canonically associated with elliptic curves from an analytical viewpoint the relation of these palindromic polynomials with hypergeometric functions associated with elliptic curves is made very explicitly and from a differential algebra viewpoint this corresponds to the emergence of direct sums of differential operators homomorphic to symmetric powers of a second order operator associated with elliptic curve

fa yueh wu s contributions in physics abstract we sketch a subset of professor f y wu s contributions in lattice statistical mechanics solid state physics graph theory enumerative combinatorics and other domains of physics and mathematics we will recall some of f y wu s most important and well known classic results and we will also sketch some of his most recent researches dedicated to the connections of lattice statistical mechanical models with deep problems in pure mathematics

pure and dilute z n spin and generalised gauge lattice systems duality and criticality considers the pure z n spin systems including the standard ising and potts models as well as generalised gauge systems plaquettes or more complex simplex in d dimensional hypercubic lattices these models are self dual and the authors show how this duality can be thought of as a series parallel transformation the simplicity of the equations enables conjectures to be made on the approximate critical frontier of the diluted version of the above systems including some particular asymptotic behaviours which are exact as an illustration the d 2 diluted z 4 spin system is discussed in some detail for those regions where exact results are available the agreement is satisfactory

addendum to on the singularity structure of the 2d ising model susceptibility a remarkable product formula first derived by palmer and tracy 1981 adv appl math 2 329 for the integrand of the two dimensional ising model susceptibility expansion coefficients 2n for temperatures t less than the critical t c is shown to apply equally for 2n 1 for t t c and agrees with formulae derived by yamada 1984 prog theor phys 71 1416 this new representation simplifies the derivation of the results in the original paper of this title 1999 j phys a math gen 32 3889 to the extent that the leading series behaviour and the singularity structure can be deduced almost by inspection the derivation of series is also simplified and i show using extended series and knowledge of the singularity structure that there is now unambiguous evidence for correction to scaling terms in the susceptibility beyond those inferred from a nonlinear scaling field analysis

thermal transmissivity in discrete spin systems formulation and applications thermal transmissivity a notion which arises empirically and leads to considerable simplification of analyses of discrete spin systems is of fundamental interest in graphical and real space renormalization group approaches the authors present a formulation of the theory of thermal transmissivity they show that the situation is different depending on whether the interaction matrices commute in the commuting case the thermal transmissivity is given by the eigenvalues of the interaction matrix while in the non commuting case it is given by the eigenvalues as well as matrices which block diagonalize the interaction matrix the meaning of the thermal transmissivity in diagrammatic analyses of spin models is also elucidated and their results are illustrated by examples finally they present a general formulation of disorder solutions for spin models in terms of thermal transmissivities

phase diagram of a six state chiral potts model abstract we study the phase diagram of an isotropic six state chiral potts model on a square lattice by means of both exact and numerical methods the phase diagram of this model presents many similarities with the phase diagrams of the ashkin teller model or the models studied by zamolodchikov and monarstirskii a remarkable line globally invariant under a transformation generalizing the kramers wannier duality seems to correspond to a first order transition line up to a bifurcation point where this line splits into two second order lines all the numerical calculations are compared with exact results which can be performed using a canonical elliptic parametrization of this model the bifurcation point is found to correspond to the intersection of a generalized self dual line with an algebraic curve this curve corresponds to the set of points of the phase diagram for which a non trivial infinite symmetry group of the model degenerates into a finite group of order six the agreement between numerical and analytical results is very good

optimal unidirectional error detecting correcting codes in this correspondence a class of t error correcting and multiple unidirectional error detecting systematic codes is presented these codes are significantly more efficient than the earlier codes the efficiency of these codes approaches the efficiency of the bch codes asymptotically furthermore it is shown that these codes can be easily decoded also in this correspondence we have presented a generalization of berger codes over zq these new codes are also shown to be optimal

the failure distribution in percolation models of breakdown the probability of finding a largest defect cluster of size n in a percolation network is calculated analytically using a new distribution function scaling equation from this result and the stress voltage enhancement at the tip of the most critical defect in the network the probability of failure in percolation models of breakdown is calculated for defect fractions less than the percolation point this distribution is found to be of the form exponential of an exponential numerical simulations on the two dimensional random fuse network confirm the new distribution function and convincingly distinguish between it and the weibull 1951 form most often used in the fitting of breakdown data

singularities of n fold integrals of the ising class and the theory of elliptic curves we introduce some multiple integrals that are expected to have the same singularities as the singularities of the n particle contributions n to the susceptibility of the square lattice ising model we find the fuchsian linear differential equation satisfied by these multiple integrals for n 1 2 3 4 and only modulo some primes for n 5 and 6 thus providing a large set of possible new singularities of n we discuss the singularity structure for these multiple integrals by solving the landau conditions we find that the singularities of the associated odes identify up to n 6 with the leading pinch landau singularities the second remarkable obtained feature is that the singularities of the odes associated with the multiple integrals reduce to the singularities of the odes associated with a finite number of one dimensional integrals among the singularities found we underline the fact that the quadratic polynomial condition 1 3w 4w2 0 that occurs in the linear differential equation of 3 actually corresponds to a remarkable property of selected elliptic curves namely the occurrence of complex multiplication the interpretation of complex multiplication for elliptic curves as complex fixed points of the selected generators of the renormalization group namely isogenies of elliptic curves is sketched most of the other singularities occurring in our multiple integrals are not related to complex multiplication situations suggesting an interpretation in terms of motivic mathematical structures beyond the theory of elliptic curves

triangular lattice potts models in this paper we study the 3 state potts model on the triangular lattice which has two and three site interactions using a peierls argument we obtain a rigorous bound on the transition temperature thereby disproving a conjecture on the location of its critical point low temperature series are generated and analyzed for three particular choices of the coupling constants a phase diagram is then drawn on the basis of these considerations our analysis indicates that the antiferromagnetic transition and the transition along the coexistence line are of first order implying the existence of a multicritical point in the ferromagnetic region relation of the triangularq state potts model with other lattice statistical problems is also discussed in particular an ashkin teller model and the hard hexagon lattice gas solved by baxter emerge as special cases in appropriate limits

inversion relations and disorder solutions on potts models the inversion symmetry and the automorphy group generated by the latter when combined with the spatial symmetries are studied for the anisotropic potts models on the triangular and checkerboard lattices the exact expression of the partition function which is known on some particular disorder subvarieties of the triangular model is checked and a generalisation is proposed for the checkerboard lattice the automorphy group is then used to extend the disorder solutions to the infinity of transformed subvarieties

high order fuchsian equations for the square lattice ising model χ 6 this paper deals with the six particle contribution to the magnetic susceptibility of the square lattice ising model we have generated modulo a prime series coefficients for the length of the series is sufficient to produce the corresponding fuchsian linear differential equation modulo a prime we obtain the fuchsian linear differential equation that annihilates the depleted series the factorization of the corresponding differential operator is performed using a method of factorization modulo a prime introduced in a previous paper the depleted differential operator is shown to have a structure similar to the corresponding operator for it splits into factors of smaller orders with the left most factor of order 6 being equivalent to the symmetric fifth power of the linear differential operator corresponding to the elliptic integral e the right most factor has a direct sum structure and using series calculated modulo several primes all the factors in the direct sum have been reconstructed in exact arithmetics

on the density of partition function temperature zeros the density of partition function temperature zeros for the two dimensional spatially anisotropic ising model in the absence of a magnetic field satisfies a linear homogeneous partial differential equation a generalised form of this equation for a non zero specific heat exponent can be derived by applying non linear scaling to the expression for the specific heat as an integral over the density of zeros

globally nilpotent differential operators and the square ising model we recall various multiple integrals with one parameter related to the isotropic square ising model and corresponding respectively to the n particle contributions of the magnetic susceptibility to the lattice form factors to the two point correlation functions and to their extensions the univariate analytic functions defined by these integrals are holonomic and even g functions they satisfy fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties we recall the explicit forms found in previous work of these fuchsian equations as well as their russian doll and direct sum structures these differential operators are selected fuchsian linear differential operators and their remarkable properties have a deep geometrical origin they are all globally nilpotent or sometimes even have zero p curvature we also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known focusing on the factorized parts of all these operators we find out that the global nilpotence of the factors resp p curvature nullity corresponds to a set of selected structures of algebraic geometry elliptic curves modular curves curves of genus five six and even a remarkable weight 1 modular form emerging in the three particle contribution 3 of the magnetic susceptibility of the square ising model noticeably this associated weight 1 modular form is also seen in the factors of the differential operator for another n fold integral of the ising class 3 h for the staircase polygons counting and in ap ry s study of 3 g functions naturally occur as solutions of globally nilpotent operators in the case where we do not have g functions but hamburger functions one irregular singularity at 0 or that correspond to the confluence of singularities in the scaling limit the p curvature is also found to verify new structures associated with simple deformations of the nilpotent property

lattice green function at 0 for the 4d hypercubic lattice the generating function for recurrent polya walks on the four dimensional hypercubic lattice is expressed as a kampe de feriet function various properties of the associated walks are enumerated

inverse functional relations for lattice models inverse functional equations are studied for the partition functions and for the correlation functions of various models the validity of the inverse relation for the partition function is justified with the help of approximations by quasi one dimensional models defined on strips of increasing size the possibility of determining the partition function through the use of the inverse and other symmetry relations coupled to analyticity hypotheses is briefly investigated on the two dimensional ising model with a field the group generated by the inverse relation and the spatial symmetries of the model is studied in a three dimensional case inverse relations are also exhibited firstly for two point and then for n point correlation functions they are first put into evidence by a geometric approach and then verified on a particular high temperature expansion

landau singularities and singularities of holonomic integrals of the ising class we consider families of multiple and simple integrals of the ising class and the linear ordinary differential equations with polynomial coefficients they are solutions of we compare the full set of singularities given by the roots of the head polynomial of these linear odes and the subset of singularities occurring in the integrals with the singularities obtained from the landau conditions for these ising class integrals we show that the landau conditions can be worked out either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral the singular behaviour of these integrals is obtained in the self dual variable w s 2 1 s2 with s sinh 2k where k j kt is the usual ising model coupling constant switching to the variable s we show that the singularities of the analytic continuation of series expansions of these integrals actually break the kramers wannier duality we revisit the singular behaviour zenine et al 2005 j phys a math gen 38 9439 74 of the third contribution to the magnetic susceptibility of ising model 3 at the points 1 3w 4w2 0 and show that 3 s is not singular at the corresponding points inside the unit circle s 1 while its analytical continuation in the variable s is actually singular at the corresponding points 2 s s2 0 outside the unit circle s 1

on symmetric error correcting and all unidirectional error detecting codes the design of binary block codes that are capable of correcting up to t symmetric errors and detecting all unidirectional errors is discussed a class of systematic t symmetric error correcting all unidirectional error detecting t syec aued codes are proposed when t 0 the proposed codes become berger codes for t 1 the proposed codes are shown to be of asymptotically optimal order methods to construct nonsystematic t syec aued codes for t 2 and 3 are also presented

algebraic properties and symmetries of the symmetric ashkin teller model the authors investigate the automorphic properties of the partition function of an anisotropic but symmetric ashkin teller model on a square lattice they discuss the structure and the location of critical and solubility lines of the isotropic model in the present framework and comment on their algebraic and or non algebraic character

diagonal ising susceptibility elliptic integrals modular forms and calabi yau equations we give the exact expressions of the partial susceptibilities 3 d and 4 d for the diagonal susceptibility of the ising model in terms of modular forms and calabi yau odes and more specifically 3f2 1 3 2 3 3 2 1 1 z and 4f3 1 2 1 2 1 2 1 2 1 1 1 z hypergeometric functions by solving the connection problems we analytically compute the behavior at all finite singular points for 3 d and 4 d we also give new results for 5 d we see in particular the emergence of a remarkable order 6 operator which is such that its symmetric square has a rational solution these new exact results indicate that the linear differential operators occurring in the n fold integrals of the ising model are not only derived from geometry globally nilpotent but actually correspond to special geometry homomorphic to their formal adjoint this raises the question of seeing if these special geometry ising operators are special ones reducing in fact systematically to selected k balanced q 1fq hypergeometric functions or correspond to the more general solutions of calabi yau equations

partition function zeros for the two dimensional ising model iii abstract the density of the complex temperature zeros of the partition function of the two dimensional ising model on completely anisotropic triangular lattices is investigated near real and complex critical points the non uniform behaviour of the density of zeros at interior and boundary critical points is studied analytically and numerically for a lattice with interactions in the ratios 3 2 1 the limiting behaviour of the density at complex critical points depends on the direction of approach in the complex plane in the neighbourhood of interior critical points one finds only a single layer of zeros but generally there are two layers or superposed sets of zeros with different distributions and different limiting densities at boundary critical points on anisotropic quadratic and on partially anisotropic triangular lattices the two layers become identical by symmetry of the partition function the divergence of the two dimensional density of zeros at real critical points is discussed briefly in relation to scaling theory

the hyperdeterminant and triangulations of the 4 cube peter huggins bernd sturmfels josephine yu and debbie s yusterabstract the hyperdeterminant of format 2 2 2 2 is a polynomialof degree 24 in 16 unknowns which has 2894276 terms we compute thenewton polytope of this polynomial and the secondary polytope of the 4 cube the 87959448 regular triangulations of the 4 cube are classi ed into25448 d equivalence classes one for each vertex of the newton polytope the 4 cube has 80876 coarsest regular subdivisions one for each facet of thesecondary polytope but only 268 of them come from the hyperdeterminant

bounds and constructions for binary codes of length less than 24 and asymmetric distance less than 6 upper bounds to the maximum number of codewords in a binary code of length n and asymmetric distance delta are derived for some values of n and delta a method is given in which a code of length n m and asymmetric distance at least t 1 is constructed by expurgating and puncturing a code of length n and hamming distance at least 2t 1 novel asymmetric error correcting codes are constructed by applying this method to some celebrated symmetric error correcting codes a table is presented on the size of optimal asymmetric error correcting codes of length less than 24 and asymmetric distance less than 6

inversion relations and symmetry groups for potts models on the triangular lattice inversion relations are obtained for the standard scalar q state triangular potts model with two and three spin interactions generalizing previously known results for two spin interaction models it is shown that these inversion relations generate a group of symmetries of the model which is naturally represented in terms of birational transformations in a four dimensional parameter space this group of birational transformations is generically a very large one namely a hyperbolic coxeter group in this framework of very large groups of symmetries a remarkable situation pops out the one for which q corresponds to tutte beraha numbers

three state chiral potts models on the triangular lattice a monte carlo study abstract the phase transition of the three state chiral potts model is investigated by monte carlo simulations performed on two types of isotropic triangular lattices of various linear sizes up to 128 these two types of model are shown to have different phase diagrams in addition the critical antifferomagnetic point is analyzed in detail

systematic t error correcting all unidirectional error detecting codes in this paper we give methods for the construction of systematic t random error correcting and all unidirectional error detecting codes also we give the encoding decoding algorithms and discuss their implementation

algebraic approximations to the locus of partition function zeros applications of the mathematical formalism put forward in the previous paper see ibid p 3471 1987 are made to a number of well known lattice models in statistical mechanics sequences of algebraic curves are obtained from the branches of an algebraic function lambda 1 where lambda 1 at real temperatures is the branch of lambda 1 which is the partition function per site of an m infinity lattice section these sequences are viewed as approximations to parts of the limiting locus cinfinity of partition function zeros approximations to critical points can be obtained in a natural extension of these curves beyond the branch points of lambda 1

square lattice ising model susceptibility connection matrices and singular behaviour of χ 3 and χ 4 we present a simple but efficient way to calculate connection matrices between sets of independent local solutions defined at two neighbouring singular points of fuchsian differential equations of quite large orders such as those found for the third and fourth contribution 3 and 4 to the magnetic susceptibility of the square lattice ising model we deduce all the critical behaviours of the solutions 3 and 4 as well as the asymptotic behaviour of the coefficients in the corresponding series expansions we confirm that the newly found quadratic singularities of the fuchsian ode associated with 3 are not singularities of the particular solution 3 itself we use the previous connection matrices to get the exact expressions of all the monodromy matrices of the fuchsian differential equation for 3 and 4 expressed in the same basis of solutions these monodromy matrices are the generators of the differential galois group of the fuchsian differential equations for 3 and 4 whose analysis is just sketched here as far as the physics implications of the solutions are concerned we find challenging qualitative differences when comparing the corrections to scaling for the full susceptibility at high temperature respectively low temperature and the first two terms 1 and 3 respectively 2 and 4

holonomy of the ising model form factors we study the ising model two point diagonal correlation function c n n by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of wu mccoy tracy and barouch we extend this expansion weighting by powers of a variable the j particle contributions f j n n in the form factor expansion the corresponding extension of the two point diagonal correlation function c n n is shown for arbitrary to be a solution of the sigma form of the painleve vi equation introduced by jimbo and miwa in their isomonodromic approach to the ising model fuchsian linear differential equations for the form factors f j n n are obtained for j 9 and shown to have both a russian doll nesting and a decomposition of the corresponding linear differential operators as a direct sum of operators equivalent to symmetric powers of the second order linear differential operator associated with the elliptic integral e from this we show that each f j n n is unexpectedly simple being expressed polynomially in terms of the elliptic integrals e and k in contrast we exhibit some mathematical objects built from these form factors f j n n which break the direct sum of symmetric powers decomposition with its associated polynomial expressions first we show that the scaling limit of these differential operators and form factors breaks the direct sum structure but not the russian doll structure secondly we show that the previous extension of two point diagonal correlation functions c n n is for singled out values cos m n m n integers also solutions of fuchsian linear differential equations these solutions of painleve vi are not polynomial in e and k but are actually algebraic functions being associated with modular curves

zeros of the partition function for the two dimensional ising model the zeros of the partition function for the zero field two dimensional ising model in the complex temperature plane are represented in terms of the eigenvalues of the transfer matrix the boundaries of the limiting distribution have a particular simple representation in terms of the correlation range along the two principal axes of the quadratic lattice examples of the distribution in the complex plane of z exp 2k are given for the anisotropic model when the horizontal and vertical interactions are in integer ratio

fuchs versus painleve we briefly recall the fuchs painleve elliptic representation of painleve vi we then show that the polynomiality of the expressions of the correlation functions and form factors in terms of the complete elliptic integral of the first and second kinds k and e is a straight consequence of the fact that the differential operators corresponding to the entries of toeplitz like determinants are equivalent to the second order operator l e which has e as solution or for off diagonal correlations to the direct sum of l e and d dt we show that this can be generalized mutatis mutandis to the anisotropic ising model the singled out second order linear differential operator l e is replaced by an isomonodromic system of two third order linear partial differential operators associated with 1 the jacobi s form of the complete elliptic integral of the third kind or equivalently two second order linear partial differential operators associated with appell functions where one of these operators can be seen as a deformation of l e we finally explore the generalizations to the anisotropic ising models of the links we made in two previous papers between painleve nonlinear odes fuchsian linear odes and elliptic curves in particular the elliptic representation of painleve vi has to be generalized to an appellian representation of gamier systems

a heuristic principle for the phase boundaries of pure and quenched bond diluted spin models results from high temperature series for a class of discrete classical spin models on the square lattice suggest that the phase boundary is at least in part given by a simple heuristic principle it is shown how this principle can be extended to triangular and hexagonal lattices by making use of duality and the star triangle transformation for the potts model the exact phase transition is recovered not only for integer number of states but also in the percolation limit for more complicated models a number of phase diagrams are derived which are presumably exact except in regions where partially broken symmetry phases or massless phases intervene the principle can also be applied to quenched bond diluted spin models the results for the potts model are obtained for all three lattices and shown to be exact in certain limiting cases the m to 1 limit of the wreath product model s m s m which corresponds to double percolation is studied to provide an example for which the principle is completely incorrect

square lattice ising model tilde chi 5 ode in exact arithmetic we obtain in exact arithmetic the order 24 linear differential operator l24 and the right hand side e 5 of the inhomogeneous equation l24 5 e 5 where is a linear combination of n particle contributions to the susceptibility of the square lattice ising model in bostan et al 2009 j phys a math theor 42 275209 the operator l24 modulo a prime was shown to factorize into here we prove that no further factorization of the order 12 operator is possible we use the exact ode to obtain the behaviour of at the ferromagnetic critical point and to obtain a limited number of analytic continuations of beyond the principal disc defined by its high temperature series contrary to a speculation in boukraa et al 2008 j phys a math theor 41 455202 we find that is singular at w 1 2 on an infinite number of branches

a birational mapping with a strange attractor post critical set and covariant curves we consider some two dimensional birational transformations one of them is a birational deformation of the henon map for some of these birational mappings the post critical set i e the iterates of the critical set is infinite and we show that this gives straightforwardly the algebraic covariant curves of the transformation when they exist these covariant curves are used to build the preserved meromorphic 2 form one may also have an infinite post critical set yielding a covariant curve which is not algebraic transcendental for two of the birational mappings considered the post critical set is finite and we claim that there is no algebraic covariant curve and no preserved meromorphic 2 form for these two mappings with finite post critical sets attracting sets occur and we show that they pass the usual tests lyapunov exponents and the fractal dimension for being strange attractors the strange attractor of one of these two mappings is unbounded

theory of unidirectional error correcting detecting codes in this paper we present some basic theory on unidirectional error correcting detecting codes we define symmetric asymmetric and unidirectional error classes and proceed to derive the necessary and sufficient conditions for a binary code to be unidirectional error correcting detecting

extension of the kasteleyn fortuin formulas to directed percolation it is shown that the pair connectedness for percolation on an arbitrary directed graph may be obtained by setting 1 in the transmissivity function for a state chiral potts model the odd flow model and that the pair connectedness of the dual percolation model is expressible in terms of the correlation function of a second chiral potts model the odd potential difference model there is an important distinction between odd and even and only those models with odd exhibit chirality the duality relation between the transmissivity and correlation functions leads to a duality relation for pair connectedness of directed percolation theory which extends the result of dhar barna and phani for percolation probabilities

experimental mathematics on the magnetic susceptibility of the square lattice ising model we calculate very long low and high temperature series for the susceptibility of the square lattice ising model as well as very long series for the five particle contribution 5 and six particle contribution 6 these calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150 000 cpu hours on computer clusters the series for low and high temperature regimes 5 and 6 are now extended to 2000 terms in addition for 5 10 000 terms of the series are calculated modulo a single prime and have been used to find the linear ode satisfied by 5 modulo a prime a diff pade analysis of the 2000 terms series for 5 and 6 confirms to a very high degree of confidence previous conjectures about the location and strength of the singularities of the n particle components of the susceptibility up to a small set of additional singularities the exponents at all the singularities of the fuchsian linear ode of 5 and the as yet unknown ode of 6 are given they are all rational numbers we find the presence of singularities at w 1 2 for the linear ode of 5 and w2 1 8 for the ode of 6 which are not singularities of the physical 5 and 6 that is to say the series solutions of the ode s which are analytic at w 0 furthermore analysis of the long series for 5 and 6 combined with the corresponding long series for the full susceptibility yields previously conjectured singularities in some n n 7 the exponents at all these singularities are also seen to be rational numbers we also present a mechanism of resummation of the logarithmic singularities of the n leading to the known power law critical behaviour occurring in the full and perform a power spectrum analysis giving strong arguments in favour of the existence of a natural boundary for the full susceptibility

the potts model and flows i the pair correlation function it is shown that the partition function for the a state potts model with pair interactions is related to the expected number of integer mod a flows in a percolation model the relation is generalised to the pair correlation function the resulting high temperature expansion coefficients are shown to be the flow polynomials of graph theory we also prove an observation of tsallis and levy concerning the equivalent transmissivity of a cluster

random cluster models on the triangular lattice we study percolation and the random cluster model on the triangular lattice with 3 body interactions starting with percolation we generalize the star triangle transformation we introduce a new parameter the 3 body term and identify configurations on the triangles solely by their connectivity in this new setup necessary and sufficient conditions are found for positive correlations and this is used to establish regions of percolation and non percolation next we apply this set of ideas to the q 1 random cluster model we derive duality relations for the suitable random cluster measures prove necessary and sufficient conditions for them to have positive correlations and finally prove some rigorous theorems concerning phase transitions

rational mappings arborescent iterations and the symmetries of integrability we describe a class of nonlinear birational representations of groups generated by a finite number of involutions these groups are symmetries of the yang baxter equations and their higher dimensional generalizations they provide discrete dynamical systems with a variety of behaviors from chaotic to integrable according to the number of invariants of the representation

re examination of yang lee zeros of the anisotropic ising models on square triangular and honeycomb lattices we have determined the zeros of the partition functions of the anisotropic ising models on square triangular and honeycomb lattices in the absence of a magnetic field with arbitrary combinations of interactions it is found that the zeros are generally distributed over areas however the zeros near the positive real axis are distributed on the unit circle in a complex plane with the zero density g which leads to logarithmic singularity of the free energy our results generalize fisher s results for isotropic cases

on the boundaries and density of partition function temperature zeros for the two dimensional ising model explicit expressions which determine the boundaries of the regions containing temperature zeros of the partition function of the two dimensional ising model on an anisotropic triangular lattice are obtained along with a general formula for the density of zeros valid everywhere in regions containing zeros

the fuchsian differential equation of the square lattice ising model χ 3 susceptibility using an expansion method in the variables xi that appear in n 1 dimensional integrals representing the n particle contribution to the ising square lattice model susceptibility we generate a long series of coefficients for the three particle contribution 3 using an n4 polynomial time algorithm we give the fuchsian differential equation of order 7 for 3 that reproduces all the terms of our long series an analysis of the properties of this fuchsian differential equation is performed

critical point of a triangular potts model with two and three site interactions the q state potts model on the triangular lattice with nearest neighbour interactions and three site interactions in half of the triangular faces is considered the exact duality relation is re examined from the point of view of determining its critical point using the continuity and uniqueness arguments the authors determine the exact critical point in the ferromagnetic model it is argued that a transition exists in an antiferromagnetic model only for q 3 a conjecture is then made on the phase diagram for the q 3 isotropic model these results are used to determine the exact criticality of a dilute potts model on the honeycomb lattice

an exact critical frontier for the potts model on the 3 12 lattice an exact critical frontier for the potts model on the 3 12 lattice which includes the kagome lattice as a special case is determined in a special parameter space of the two and three site interactions the determination is made possible by the use of a star triangle transformation which converts the lattice into one whose exact critical point is known

the ising susceptibility scaling function we have dramatically extended the zero field susceptibility series at both high and low temperature of the ising model on the triangular and honeycomb lattices and used these data and newly available further terms for the square lattice to calculate a number of terms in the scaling function expansion around both the ferromagnetic and for the square and honeycomb lattices the antiferromagnetic critical point

three state chiral potts models in two dimensions integrability and symmetry abstract symmetry and integrability relations for the three state chiral potts model on two dimensional lattices are reviewed our detailed and systematic analysis leads to new integrability varieties for specific chiral potts models including new integrable points for the standard antiferromagnetic potts model we also describe a three state vertex model representation of the chiral potts model and write down the associated fundamental invariants for three coordinated lattices

automorphisms of algebraic varieties and yang baxter equations it is shown that commuting transfer matrix models in statistical mechanics are parametrized by algebraic varieties having a set of automorphisms deduced from the so called inversion relation in general this set of automorphisms is infinite this shows that for algebraic varieties of dimension 1 the models are parametrized by algebraic curves of genus 0 or 1

the n component cubic model and flows subgraph break collapse method we specialize to then component cubic model the subgraph break collapse method which we recently developed for thez model the cubic model has less symmetry than the potts model for which the method was originally developed but nevertheless it is still possible to reduce considerably the computational complexity of the generalz model our recursive algorithm is similar forn 2 to the break collapse method for thez 4 model proposed by mariz and co workers it allows the exact calculation for the partition function and correlation functions forn component cubic clusters withn as a variable without the need to examine all of the spin configurations an important application is therefore in real space renormalization group calculations

massless phases for the zp model on the union jack lattice the authors introduce and analyse zp symmetric models on the union jack lattice they show that these models have the same self dual structure already known for other zp systems by performing renormalisation group calculations as well as monte carlo simulations they analyse their phase diagrams showing in particular that for p or 5 they exhibit a disordered massless phase

the potts model on kagome and honeycomb lattices low temperature series have been derived for theq state potts model on the kagom e lattice qd 3 and 4 and for the 3 state honeycomb lattice the series are derived by the finite lattice method with many additional terms obtained by noting the structure of the correction terms accurate estimates of the critical points are found which confirm exact but not rigorous results in the case of the honeycomb lattice and refute previous conjectures in the case of the kagom e lattice in this paper we report on the derivation and analysis of low temperature series for the 3 state honeycomb and 3 and 4 state kagomlattice potts model from the series we obtain estimates for the critical points which for the honeycomb model confirms the validity of the exact but not rigorous result 7 and for the kagomlattice provides accurate results against which conjectures can be tested and found not to hold we comment on properties that the solution must have in an unsuccessful attempt to find an alternative conjecture in addition we obtain estimates for the location of various non physical singularities the remainder of the paper is organized as follows in section 2 we briefly describe the finite lattice method of series expansions give details regarding the specific implementations for the kagomand honeycomb lattices and show how the finite lattice method can be supplemented by an extension procedure allowing us to derive more series terms details regarding the analysis of the series are given in section 3 and the implications for possible conjectures for the critical point on the kagom

the ising model from elliptic curves to modular forms and calabi yau equations we show that almost all the linear differential operators factors obtained in the analysis of the n particle contributions s of the susceptibility of the ising model for n 6 are linear differential operators associated with elliptic curves beyond the simplest differential operators factors which are homomorphic to symmetric powers of the second order operator associated with the complete elliptic integral e the second and third order differential operators z2 f2 f3 can actually be interpreted as modular forms of the elliptic curve of the ising model a last order 4 globally nilpotent linear differential operator is not reducible to this elliptic curve modular form scheme this operator is shown to actually correspond to a natural generalization of this elliptic curve modular form scheme with the emergence of a calabi yau equation corresponding to a selected 4f3 hypergeometric function this hypergeometric function can also be seen as a hadamard product of the complete elliptic integral k with a remarkably simple algebraic pull back square root extension the corresponding calabi yau fourth order differential operator having a symplectic differential galois group the mirror maps and higher order schwarzian odes associated with this calabi yau ode present all the nice physical and mathematical ingredients we had with elliptic curves and modular forms in particular an exact isogenies representation of the generators of the renormalization group extending the modular group to a symmetry group