cluster expansion for abstract polymer models a new direct proof of convergence of cluster expansions for polymer contour models is given in an abstract setting it does not rely on kirkwood salsburg type equations or combinatorics of trees a distinctive feature is that at all steps the considered clusters contain every polymer at most once

the lee yang and polya schur programs i linear operators preserving stability in 1952 lee and yang proposed the program of analyzing phase transitions in terms of zeros of partition functions linear operators preserving non vanishing properties are essential in this program and various contexts in complex analysis probability theory combinatorics and matrix theory we characterize all linear operators on finite or infinite dimensional spaces of multivariate polynomials preserving the property of being non vanishing whenever the variables are in prescribed open circular domains in particular this solves the higher dimensional counterpart of a long standing classification problem originating from classical works of hermite laguerre hurwitz and polya schur on univariate polynomials with such properties

an asymptotic expansion for the incomplete beta function a new asymptotic expansion is derived for the incomplete beta function i a b x which is suitable for large a small b and x 0 5 this expansion is of the form formula math where q is the incomplete gamma function ratio and a b 1 2 this form has some advantages over previous asymptotic expansions in this region in which t n depends on a as well as on b and x

extension of the partial derivatives of the incomplete beta function for complex values in this paper by using the hypergeometric function and the neutrix limit we extend the definition of the partial derivatives of the incomplete beta function p q x p y q b z x y p q 0 1 2 to all complex values of x and y as complex number z satisfying 0 z 1 moreover we establish the recursive formula of p q x p y q b z x y for x q q 1 q 2 p q 0 1 2 in addition we pay our special attention to the closed forms of p q x p y q b z n m for n m 0 1 2 which can be expressed by the elementary function special constants and riemann zeta function

abstract polymer gas a simple inductive proof of the fernandez procacci criterion this note contains an alternative proof of the fernandez procacci criterion for the convergence of cluster expansion on the abstract polymer gas via a simple inductive argument a la dobrushin

numerical techniques for conformal mapping onto a rectangle abstract this paper is concerned with the problem of determining approximations to the function f which maps conformally a simply connected domain onto a rectangle r so that four specified points on are mapped respectively onto the four vertices of r in particular we study the following two classes of methods for the mapping of domains of the form z x iy 00 1 x 2 x i methods which approximate f r by f s f where f is an approximation to the conformal map of onto the unit disc and s is a simple schwarz christoffel transformation ii methods based on approximating the conformal map of a certain symmetric doubly connected domain onto a circular annulus

algorithmic pirogov sinai theory we develop an efficient algorithmic approach for approximate counting and sampling in the low temperature regime of a broad class of statistical physics models on finite subsets of the lattice mathbb z d and on the torus mathbb z n mathbb z d our approach is based on combining contour representations from pirogov sinai theory with barvinok s approach to approximate counting using truncated taylor series some consequences of our main results include an fptas for approximating the partition function of the hard core model at sufficiently high fugacity on subsets of mathbb z d with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic potts model on the discrete torus mathbb z n mathbb z d at sufficiently low temperature

the repulsive lattice gas the independent set polynomial and the lovasz local lemma we elucidate the close connection between the repulsive lattice gas in equi librium statistical mechanics and the lovasz local lemma in probabilistic com binatorics we show that the conclusion of the lovasz local lemma holds for dependency graph g and probabilities px if and only if the independent set polynomial for g is nonvanishing in the polydisc of radii px furthermore we show that the usual proof of the lovasz local lemma which provides a suffi cient condition for this to occur corresponds to a simple inductive argument for the nonvanishing of the independent set polynomial in a polydisc which was discovered implicitly by shearer 98 and explicitly by dobrushin 37 38 we also present some refinements and extensions of both arguments including a generalization of the lovasz local lemma that allows for soft dependencies in addition we prove some general properties of the partition function of a repulsive lattice gas most of which are consequences of the alternating sign property for the mayer coefficients we conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs

the accuracy of numerical conformal mapping methods a survey of examples and results this paper shows how the geometry of the region affects the conditioning and the accuracy of numerical conformal mapping methods for simply connected regions especially fourier series methods both explicit examples of popular test cases and more general estimates are discussed the severe ill conditioning that is known as the crowding phenomenon is discussed and its effect on a conformally transplanted boundary value problem is illustrated remarks on various numerical methods are included

on some conjectures by morriset al about zeros of an entire function abstract it is known that the entire function n 0 z n q n n 1 2 n q 0 1 has infinitely many positive but no other zeros in the complex plane a number of conjectures on these zeros have been made by morris feldstein and bowen and by iserles the main objective of this note is to establish some connection among these conjectures and to prove the following estimates formula where t 0 t 1 are the zeros in monotonic increasing order some identities involving those zeros are proved by using hadamard s factorisation theorem

stability and convergence properties of bergman kernel methods for numerical conformal mapping in this paper we study the stability and convergence properties of bergman kernel methods for the numerical conformal mapping of simply and doubly connected domains in particular by using certain wellknown results of carleman we establish a characterization of the level of instability in the methods in terms of the geometry of the domain under consideration we also explain how certain known convergence results can provide some theoretical justification of the observed improvement in accuracy which is achieved by the methods when the basis set used contains functions that reflect the main singular behaviour of the conformal map

when a local hamiltonian must be frustration free a broad range of quantum optimization problems can be phrased as the question of whether a specific system has a ground state at zero energy i e whether its hamiltonian is frustration free frustration free hamiltonians in turn play a central role for constructing and understanding new phases of matter in quantum many body physics unfortunately determining whether this is the case is known to be a complexity theoretically intractable problem this makes it highly desirable to search for efficient heuristics and algorithms to at least partially answer this question here we prove a general criterion a sufficient condition under which a local hamiltonian is guaranteed to be frustration free by lifting shearer s theorem from classical probability theory to the quantum world remarkably evaluating this condition proceeds via a fully classical analysis of a hardcore lattice gas at negative fugacity on the hamiltonian s interaction graph which as a statistical mechanics problem is of interest in its own right we concretely apply this criterion to local hamiltonians on various regular lattices while bringing to bear the tools of spin glass physics that permit us to obtain new bounds on the satisfiable to unsatisfiable transition in random quantum satisfiability we are then led to natural conjectures for when such bounds will be tight as well as to a novel notion of universality for these computer science problems besides providing concrete algorithms leading to detailed and quantitative insights this work underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory

a nonoverlapping domain decomposition method for symm s equation for conformal mapping symm s equation is a first kind integral equation for computing conformal maps of simply connected regions the package confpack solves symm s equation by an indirect boundary element method using an accurate corner representation this solution technique is extended here to include nonoverlapping domain decomposition degrees of freedom are introduced on one or more interfaces and different unknowns are used leading to a system of second kind equations the corresponding linear system can be expressed in schur complement form the accurate treatment of corners is preserved in the new formulation the results of serial and parallel matlab implementations of the new algorithm show significant speedups as the number of unknowns grows

extremal distance harmonic measure and numerical conformal mapping abstract delillo t k and j a pfaltzgraff extremal distance harmonic measure and numerical conformal mapping journal of computational and applied mathematics 46 1993 103 113 estimates of extremal distance and harmonic measure are used to show how the geometric properties of a simply connected domain influence the boundary distortion of a conformal map from the unit disk to the domain numerical examples and remarks on the conditioning of numerical conformal mapping methods are included a sharp estimate is given of the exponential ill conditioning known as the crowding phenomenon which occurs for slender regions

counting regions with bounded surface area define a cubical complex to be a collection of integer aligned unit cubes in d dimensions lebowitz and mazel 1998 proved that there are between c 1d n 2d and c 2d 64n d complexes containing a fixed cube with connected boundary of d 1 volume n in this paper we narrow these bounds to between c 3d n d and c 4d 2n d we also show that there are n n 2d d 1 o 1 connected complexes containing a fixed cube with not necessarily connected boundary of volume n

boundary fitted coordinate systems for numerical solution of partial differential equations a review a comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given some general mathematical framework and error analysis common to such coordinate systems is also included the general categories of generating systems are those based on conformal mapping orthogonal systems nearly orthogonal systems systems produced as the solution of elliptic and hyperbolic partial differential equations and systems generated algebraically by interpolation among the boundaries also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation dynamically adaptive coordinate systems coupled with the physical solution and time dependent systems that follow moving boundaries are treated references reporting experience using such coordinate systems are reviewed as well as those covering the system development

rogers ramanujan identities in the hard hexagon model the hard hexagon model in statistical mechanics is a special case of a solvable class of hard square type models in which certain special diagonal interactions are added the sublattice densities and order parameters of this class are obtained and it is shown that many rogers ramanujan type identities naturally enter the working

asymptotic inversion of the incomplete beta function abstract the normalized incomplete beta function ix a b is inverted for large values of the parameters a and b that is x solutions of the equation ix a b p p 0 1 are considered especially for large values of a and b the approximations are obtained by using uniform asymptotic expansions of the incomplete beta function in which an error function or an incomplete gamma function is the dominant term the inversion problem is started by inverting this dominant term further terms in the expansion are obtained by using standard perturbation methods which were recently introduced in a paper describing a method for asymptotic inversion of the incomplete gamma functions numerical results indicate that for obtaining an accuracy of four correct digits the asymptotic method can already be used for a b 5

curvilinear crosscuts of subdivision for a domain decomposition method in numerical conformal mapping abstract let q z 1 z 2 z 3 z 4 be a quadrilateral consisting of a jordan domain and four points z 1 z 2 z 3 z 4 in counterclockwise order on we consider a domain decomposition method for computing approximations to the conformal module m q of q in cases where q is long or equivalently m q is large this method is based on decomposing the original quadrilateral q into two or more component quadrilaterals q1 q2 and then approximating m q by the sum j m q j of the modules of the component quadrilaterals the purpose of this paper is to consider ways for determining appropriate crosscuts of subdivision so that the sum j m q j does indeed give a good approximation to m q and in particular to show that there are cases where the use of curved crosscuts is much more appropriate than the straight line crosscuts that have been used so far

the leading root of the partial theta function abstract i study the leading root x 0 y of the partial theta function 0 x y n 0 x n y n n 1 2 considered as a formal power series i prove that all the coefficients of x 0 y are strictly positive indeed i prove the stronger results that all the coefficients of 1 x 0 y after the constant term 1 are strictly negative and all the coefficients of 1 x 0 y 2 after the constant term 1 are strictly negative except for the vanishing coefficient of y 3

zeros of the deformed exponential function abstract let f x n 0 1 n q n n 1 2 x n 0 q 1 be the deformed exponential function it is known that the zeros of f x are real and form a negative decreasing sequence x k k 1 we investigate the complete asymptotic expansion for x k and prove that for any n 1 as k x k k q 1 k 1 i 1 n c i q k 1 i o k 1 n where c i q are some q series which can be determined recursively we show that each c i q q a 0 a 1 a 2 where a i m 1 m i m q m and m denotes the sum of positive divisors of m when writing c i as a polynomial in a 0 a 1 and a 2 we find explicit formulas for the coefficients of the linear terms by using bernoulli numbers moreover we also prove that c i q q e 2 e 4 e 6 where e 2 e 4 and e 6 are the classical eisenstein series of weight 2 4 and 6 respectively

mayer expansions and the hamilton jacobi equation we review the derivation of wilson s differential equation in infinitely many variables which describes the infinitesimal change in an effective potential of a statistical mechanical model or quantum field theory when an infinitesimal integration out is performed we show that this equation can be solved for short times by a very elementary method when the initial data are bounded and analytic the resulting series solutions are generalizations of the mayer expansion in statistical mechanics the differential equation approach gives a remarkable identity for connected parts and precise estimates which include criteria for convergence of iterated mayer expansions applications include the yukawa gas in two dimensions past the 4 threshold and another derivation of some earlier results of gopfert and mack

the lee yang and polya schur programs ii theory of stable polynomials and applications in the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains for such sets this completes the multivariate generalization of the classification program initiated by polya and schur for univariate real polynomials we build on these classification theorems to develop here a theory of multivariate stable polynomials applications and examples show that this theory provides a natural framework for dealing in a uniform way with lee yang type problems in statistical mechanics combinatorics and geometric function theory in one or several variables in particular we answer a question of hinkkanen on multivariate apolarity

on the convergence of cluster expansions abstract a new simple proof of the convergence of cluster expansions in statistical mechanics is presented in the case of an abstract polymer system

potts model on infinite graphs and the limit of chromatic polynomials given an infinite graph quasi transitive and amenable with maximum degree we show that reduced ground state degeneracy per site w r q of the q state antiferromagnetic potts model at zero temperature on is analytic in the variable 1 q whenever 2 e 3 q 1 this result proves in an even stronger formulation a conjecture originally sketched in 12 and explicitly formulated in 16 and 19 based on which a sufficient condition for w r q to be analytic at 1 q 0 is that is a regular lattice

partial theta functions i beyond the lost notebook it is shown how many of the partial theta function identities in ramanujan s lost notebook can be generalized to infinite families of such identities key in our construction is the bailey lemma and a new generalization of the jacobi triple product identity by computing residues around the poles of our identities we find a surprising connection between partial theta function identities and garrett ismail stanton type extensions of multisum rogers ramanujan identities

two numerical methods for the conformal mapping of simply connected domains abstract two numerical methods are considered for the conformai mapping of a bounded simply connected domain onto the unit disc the two methods are respectively the bergman kernel method which has been described in 17 and the so called ritz method in this paper we indicate the close theoretical relationship of the two methods compare their computational efficiencies and present a number of practical applications of the approximate conformal maps

the ising partition function zeros and deterministic approximation we study the problem of approximating the partition function of the ferromagnetic ising model with both pairwise as well as higher order interactions equivalently in graphs as well as hypergraphs our approach is based on the classical lee yang theory of phase transitions along with a new lee yang theorem for the ising model with higher order interactions and on an extension of ideas developed recently by barvinok and patel and regts that can be seen as an algorithmic realization of the lee yang theory our first result is a deterministic polynomial time approximation scheme an fptas for the partition function in bounded degree graphs that is valid over the entire range of parameters beta the interaction and lambda the external field except for the case left lambda right 1 the zero field case a polynomial time randomized approximation scheme fpras for all graphs and all beta lambda based on markov chain monte carlo simulation has long been known unlike most other deterministic approximation algorithms for problems in statistical physics and counting our algorithm does not rely on the decay of correlations property but as pointed out above on lee yang theory this approach extends to the more general setting of the ising model on hypergraphs of bounded degree and edge size where no previous algorithms even randomized were known for a wide range of parameters in order to achieve this latter extension we establish a tight version of the lee yang theorem for the ising model on hypergraphs improving a classical result of suzuki and fisher

stacking models of vesicles and compact clusters we investigate three simple lattice models of two dimensional vesicles these models differ in their behavior from the universality class of partially convex polygons which has been recently established they do not have the tricritical scaling of those models and furthermore display a surprising feature their perimeter free energy is discontinuous with an isolated value at zero pressure we give the full asymptotic descriptions of the generating functions in area and perimeter variables from theq series solutions and obtain the scaling functions where applicable

a simple inductive approach to the problem of convergence of cluster expansions of polymer models we explain a simple inductive method for the analysis of the convergence of cluster expansions taylor expansions mayer expansions for the partition functions of polymer models we give a very simple proof of the dobrushin koteck preiss criterion and formulate a generalization usable for situations where a successive expansion of the partition function has to be used

on the phase transitions of graph coloring and independent sets abstract we study combinatorial indicators related to the characteristic phase transitions associated with the optimization problems of coloring the nodes of a graph with the minimum number of colors and of finding an independent set of maximum cardinality in a graph in particular we investigate the role of the acyclic orientations of the graph in the hardness of finding the graph s chromatic number and independence number we provide empirical evidence that along a sequence of increasingly denser random graphs the fraction of acyclic orientations that are shortest peaks when the chromatic number increases and that such maxima tend to coincide with locally easiest instances of the problem similar evidence is provided concerning the widest acyclic orientations and the independence number

cluster expansion in the canonical ensemble we consider a system of particles confined in a box lambda subset mathbb r d interacting via a tempered and stable pair potential we prove the validity of the cluster expansion for the canonical partition function in the high temperature low density regime the convergence is uniform in the volume and in the thermodynamic limit it reproduces mayer s virial expansion providing an alternative and more direct derivation which avoids the deep combinatorial issues present in the original proof

polymer gas approach to n body lattice systems we give a simple proof based only on combinatorial arguments of the koteck preiss condition for the convergence of the cluster expansion then we consider spin systems with long range n body interactions we prove directly using the polymer gas representation that the pressure may be written in terms of an absolutely convergent series uniformly in the volume when the interaction is summable in a suitable sense we also give an estimate of this radius of convergence in order to get the proof we use a method introduced by cassandro and olivieri in the early 1980s we apply this method to various concrete examples

mayer expansions and the hamilton jacobi equation ii fermions dimensional reduction formulas we obtain estimates on effective actions for fermionic field theories by studying the flow of a continuous renormalization group transformation for bosonic theories and statistical mechanics we establish some new formulas for mayer coefficients which are consequences of dimensional reduction

comment on the exact location of partition function zeros a new method for statistical mechanics in a recent paper wood 1985 conjectured that the exact location of the branch point singularity for the hard square lattice gas on the negative activity z axis is at z 0 119392 a careful analysis of the available series data implies that this conjecture is almost certainly false the author finds the correct value to be z 0 1193388809 or 0 000000001 with the corresponding exponent indistinguishably close to 5 6 corresponding to a zero of the singular part of the function

efficient and accurate algorithms for the computation and inversion of the incomplete gamma function ratios algorithms for the numerical evaluation of the incomplete gamma function ratios p a x gamma a x gamma a and q a x gamma a x gamma a are described for positive values of a and x also inversion methods are given for solving the equations p a x p q a x q with 0 p q 1 both the direct computation and the inversion of the incomplete gamma function ratios are used in many problems in statistics and applied probability the analytical approach from earlier literature is summarized and new initial estimates are derived for starting the inversion algorithms the performance of the associated software to our algorithms the fortran 90 module incgamfi is analyzed and compared with earlier published algorithms

witness trees in the moser tardos algorithmic lovasz local lemma and penrose trees in the hard core lattice gas we point out a close connection between the moser tardos algorithmic version of the lovasz local lemma a central tool in probabilistic combinatorics and the cluster expansion of the hard core lattice gas in statistical mechanics we show that the notion of witness trees given by moser and tardos is essentially coincident with that of penrose trees in the cluster expansion scheme of the hard core gas such an identification implies that the moser tardos algorithm is successful in a polynomial time if the cluster expansion converges

the treatment of corner and pole type singularities in numerical conformal mapping techniques abstract this paper is a report of recent developments concerning the nature and the treatment of singularities that affect certain numerical conformal mapping techniques the paper also includes some new results on the nature of singularities that the mapping function may have in the complement of the closure of the domain under consideration

nonmonotonic behavior in hard core and widom rowlinson models we give two examples of nonmonotonic behavior in symmetric systems exhibiting more than one critical point at which spontaneous symmetry breaking appears or disappears the two systems are the hard core model and the widom rowlinson model and both examples take place on a variation of the cayley tree bethe lattice devised by schonmann and tanaka we obtain similar though less constructive examples of non monotonicity via certain local modifications of any graph e g the square lattice which is known to have a critical point for either model en route we prove that the widom rowlinson model does behave monotonically on the ordinary cayley tree and we obtain an explicit description of its critical behavior there we conclude with some results about monotonicity of the phase transition phenomenon relative to graph structure

asymptotic inversion of the erlang b formula the erlang b formula represents the steady state blocking probability in the erlang loss model or m m s s queue we derive asymptotic expansions for the offered load that matches for a given number of servers a certain blocking probability in addressing this inversion problem we make use of various asymptotic expansions for the incomplete gamma function a similar inversion problem is investigated for the erlang c formula

cluster expansion for abstract polymer models new bounds from an old approach we revisit the classical approach to cluster expansions based on tree graphs and establish a new convergence condition that improves those by koteck preiss and dobrushin as we show in some examples the two ingredients of our approach are i a careful consideration of the penrose identity for truncated functions and ii the use of iterated transformations to bound tree graph expansions

numerical conformal mapping methods based on function conjugation abstract a unifying treatment of methods for computing conformal maps from the unit disk onto a jordan region is presented integral and integro differential equations involving the conjugation operator for the boundary correspondence function are first derived using an arbitrary auxiliary function having certain properties then various iteration methods for solving these equations are described in this generality so that the basic ideas become manifest specific methods are then treated as examples of the general theory among them are in particular the successive conjugation methods of theodorsen melentiev and kulisch timman and friberg the projection method of bergstrom and the newton methods of vertgeim wegmann and hubner which make use of the easy construction of the solutions of riemann hilbert problems many of these methods are treated in greater generality than in the literature the connections with the methods of fornberg menikoff zemach chakravarthy anderson and challis burley are also outlined

on the entire functions from the laguerre polya i class having the increasing second quotients of taylor coefficients abstract we prove that if f x k 0 a k x k a k 0 is an entire function such that the sequence q a k 2 a k 1 a k 1 k 1 is non decreasing and a 1 2 a 0 a 2 2 2 3 then all but a finite number of zeros of f are real and simple we also present a criterion in terms of the closest to zero roots for such a function to have only real zeros in other words for belonging to the laguerre polya class of type i under additional assumption on the sequence q

on the theory and application of a domain decomposition method for computing conformal modules we consider the theory and application of a domain decomposition method for computing the conformal modules of long quadrilaterals the method has been studied already by us and also by gaier and hayman our main purpose here is to extend its area of application and in the same time improve some of our earlier error estimates

an iterative method for the conformal mapping of doubly connected regions an iterative method is described for the determination of the conformal mapping of a circular annulus rq z q z 1 onto a doubly connected region g with smooth boundary curves the method is analogous to the method of 7 for simply connected regions it requires in each step the solution of a riemann hilbert problem on rq this problem can be solved explicitly in terms of a generalized conjugation operator k a degeneracy of this problem leads in a natural way to an adjustment of the parameter q in each step if the boundary curves of g admit parametrizations by functions with second derivatives which are holder continuous with exponent 0 1 then the method converges locally in the sobolev space w of 2 periodic absolutely continuous functions with square integrable derivatives the order of convergence is at least 1 the convergence is quadratic if the have lipschitz continuous second derivatives for the numerical implementation of the method the operator k can be approximated by a wittich type method which can be performed very effectively using fft a calculation with n 2m grid points on the computer requires storage of the order o n and computing time o mn the results of some test calculations are reported and compared with results of calculations with the method of theodorsen garrick

a certain functional differential equation abstract we resolve some conjectures concerning the zeros t n of the solution f of the initial value problem f t f qt 0 q f 0 1 in particular lim n t n 1 t n 1 q

a conformal map formula for difficult cases abstract a simple approximate formula is derived for the boundary map function of a conformal map which provides a remarkably good fit for mappings to regions with highly distorted boundaries the approximation follows from the reduction of a nonlocal integral equation for the map function to a local equation based on the nature of the crowding associated with distorted regions numerical examples are given

asymptotic inversion of incomplete gamma functions the normalized incomplete gamma functions p a x and q a x are inverted for large values of the parameter a that is x solutions of the equations p a x p q a x q p e o 1 q i p are considered especially for large values of a the approximations are obtained by using uniform asymptotic expansions of the incomplete gamma functions in which an error function is the dominant term the inversion problem is started by inverting this error function term numerical results indicate that for obtaining an accuracy of four correct digits the method can already be used for a 2 although a is a large parameter it is indicated that the method can be applied to other cumulative distribution functions

compactness and the maximal gibbs state for random gibbs fields on a lattice we prove for a general class of gibbsian random field on that the set of tempered gibbs states is compact this class contains the euclidean random fields moreover if the interaction is attractive there is a unique minimal and maximal gibbs state and are unique translation invariant ant and have the global markov property we also prove that uniqueness of the tempered gibbs state is equivalent to the magnetizationsm qx being equal which is true if the pressure is differentiable

convergence of cluster and virial expansions for repulsive classical gases we study the convergence of cluster and virial expansions for systems of particles subject to positive two body interactions our results strengthen and generalize existing lower bounds on the radii of convergence and on the value of the pressure our treatment of the cluster coefficients is based on expressing the truncated weights in terms of trees and partition schemes and generalize to soft repulsions previous approaches for models with hard exclusions our main theorem holds in a very general framework that does not require translation invariance and is applicable to models in general measure spaces our virial results stated only for homogeneous single space systems rely on an approach due to ramawadh and tate the virial coefficients are computed using lagrange inversion techniques but only at the level of formal power series thereby yielding diagrammatic expressions in terms of trees rather than the doubly connected diagrams traditionally used we obtain a new criterion that strengthens for repulsive interactions the best criterion previously available proposed by groeneveld and proven by ramawadh and tate we illustrate our results with a few applications showing noticeable improvements in the lower bound of convergence radii

fisher zeros and correlation decay in the ising model we study the complex zeros of the partition function of the ising model viewed as a polynomial in the interaction parameter these are known as fisher zeros in light of their introduction by fisher in 1965 fisher m e the nature of critical points in lecture notes in theoretical physics edited by brittin w e university of colorado press 1965 vol 7c while the zeros of the partition function as a polynomial in the field parameter have been extensively studied since the classical work of lee and yang comparatively little is known about fisher zeros for general graphs our main result shows that the zero field ising model has no fisher zeros in a complex neighborhood of the entire region of parameters where the model exhibits correlation decay in addition to shedding light on fisher zeros themselves this result also establishes a formal connection between two distinct notions of phase transition for the ising model the absence of complex zeros analyticity of the free energy density or the normalized logarithm of the partition function and decay of correlations with distance we also discuss the consequences of our result for efficient deterministic approximation of the partition function our proof relies heavily on algorithmic techniques notably weitz s self avoiding walk tree and as such belongs to a growing body of work that uses algorithmic methods to resolve classical questions in statistical physics we study the complex zeros of the partition function of the ising model viewed as a polynomial in the interaction parameter these are known as fisher zeros in light of their introduction by fisher in 1965 fisher m e the nature of critical points in lecture notes in theoretical physics edited by brittin w e university of colorado press 1965 vol 7c while the zeros of the partition function as a polynomial in the field parameter have been extensively studied since the classical work of lee and yang comparatively little is known about fisher zeros for general graphs our main result shows that the zero field ising model has no fisher zeros in a complex neighborhood of the entire region of parameters where the model exhibits correlation decay in addition to shedding light on fisher zeros themselves this result also establishes a formal connection between two distinct notions of phase transition for the ising model the absence of complex zeros analyticity of the free energy density

finite size analysis of the hard square lattice gas we investigate the hard square lattice gas model by means of transfer matrix calculations and a finite size scaling analysis using a minimal set of assumptions we find that the spectrum of correction to scaling exponents is consistent with that of the exactly solved ising model and that the critical exponents and correlation length amplitudes closely follow the relation predicted by conformal invariance assuming that these spectra are exactly identical and conformal invariance we determine the critical point the conformal anomaly and the temperature and magnetic exponents with numerical margins of 10 11 or less these results are in a perfect agreement with the exactly known ising universal parameters in two dimensions in order to obtain this degree of precision we included system sizes as large as feasible and used extended precision floating point arithmetic the latter resource provided a substantial improvement of the analysis despite the fact that it restricted the transfer matrix calculations to finite sizes of at most 34 lattice units

a domain decomposition method for approximating the conformal modules of long quadrilaterals this paper is concerned with the study of a domain decomposition method for approximating the conformal modules of long quadrilaterals the method has been studied already by us and also by d gaier and w k hayman but only in connection with a special class of quadrilaterals viz quadrilaterals where a the defining domain is bounded by two parallel straight lines and two jordan arcs and b the four specified boundary points are the four corners where the arcs meet the straight lines our main purpose here is to explain how the method may be extended to a wider class of quadrilaterals than that indicated above

the use of splines and singular functions in an integral equation method for conformal mapping we consider the integral equation method of symm for the conformal mapping of simply connected domains for the numerical solution we examine the use of spline functions of various degrees for the approximation of the source density in particular we consider ways for overcoming the difficulties associated with corner singularities for this we modify the spline approximation and in the neighborhood of each corner where a boundary singularity occurs we approximate by a function which reflects the main singular behaviour of the source density the singular functions are then blended with the splines which approximate on the remainder of the boundary so that the global approximating function has continuity of appropriate order at the transition points between the two types of approximation we show by means of numerical examples that such approximations overcome the difficulties associated with corner singularities and lead to numerical results of high accuracy

on the convergence of cluster expansions for polymer gases we compare the different convergence criteria available for cluster expansions of polymer gases subjected to hard core exclusions with emphasis on polymers defined as finite subsets of a countable set e g contour expansions and more generally high and low temperature expansions in order of increasing strength these criteria are i dobrushin criterion obtained by a simple inductive argument ii gruber kunz criterion obtained through the use of kirkwood salzburg equations and iii a criterion obtained by two of us via a direct combinatorial handling of the terms of the expansion we show that for subset polymers our sharper criterion can be proven both by a suitable adaptation of dobrushin inductive argument and by an alternative in fact more elementary handling of the kirkwood salzburg equations in addition we show that for general abstract polymers this alternative treatment leads to the same convergence region as the inductive dobrushin argument and furthermore to a systematic way to improve bounds on correlations

ramanujan s lost notebook ix the partial theta function as an entire function abstract we establish the infinite product expansion for n 0 a n q n 2 this is a corrected version of the expansion stated in ramanujan s lost notebook the proof requires the study of a new family of polynomials which while not orthogonal have many similar properties

sufficient conditions for uniform bounds in abstract polymer systems and explorative partition schemes we present several new sufficient conditions for uniform boundedness of the reduced correlations and free energy of an abstract polymer system in a complex multidisc around zero fugacity they resolve a discrepancy between two incomparable and previously known extensions of dobrushin s classic condition all conditions arise from an extension of the tree operator approach introduced by fernandez and procacci combined with a novel family of partition schemes of the spanning subgraph complex of a cluster the key technique is the increased transfer of structural information from the partition scheme to a tree operator on an enhanced space

a new exact integration method for the drucker prager elastoplastic model with linear isotropic hardening abstract this paper presents the exact stress solution of the non associative drucker prager elastoplastic model governed by linear isotropic hardening rule the stress integration is performed under constant strain rate assumption and the derived formulas are valid in the setting of small strain elastoplasticity theory based on the time continuous stress solution a complete discretized stress updating algorithm is also presented providing the solutions for the special cases when the initial stress state is located in the apex and when the increment produces a stress path through the apex explicit expression for the algorithmically consistent tangent tensor is also derived in addition a fully analytical strain solution is also derived for the stress driven case using constant stress rate assumption in order to get a deeper understanding of the features of these solutions two numerical test examples are also presented

high dimensional lattice gases we investigate the critical behaviour of hard core lattice gases in four five and six dimensions by means of monte carlo simulations in order to suppress critical slowing down we use a geometrical cluster monte carlo algorithm in particular nearest neighbour exclusion lattice gases on simple hypercubic lattices are investigated these models undergo ising like ordering transitions where the majority of the lattice gas particles settle on one of two sublattices a finite size scaling analysis of the simulation data confirms that these lattice gases display classical critical behaviour the results agree with the renormalization predictions at and above the upper critical dimensionality in particular the predicted value of the binder cumulant is confirmed

numerical conformal mapping onto a rectangle with applications to the solution of laplacian problems abstract let f be the function which maps conformally a simply connected domain onto a rectangle r so that four specified points on are mapped respectively onto the four vertices of r in this paper we consider the problem of approximating the conformal map f and present a survey of the available numerical methods we also illustrate the practical significance of the conformal map by presenting a number of applications involving the solution of laplacian boundary value problems

an extension of the incomplete beta function for negative integers abstract the incomplete beta function b x a b is defined for a b 0 and 0 x 1 its definition can be extended by regularization to negative non integer values of a and b in this paper we define the incomplete beta function b x a b for negative integer values of a and b further we prove that the function m n a m b n b x a b exists for m n 0 1 2 and all a and b

on the spectrum of a partial theta function the bivariate series defines a partial theta function for fixed q q is an entire function we show that for q 0 108 the function q has no multiple zeros

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

hardy petrovitch hutchinson s problem and partial theta function in 1907 m petrovitch initiated the study of a class of entire functions all whose finite sections i e truncations are real rooted polynomials he was motivated by previous studies of e laguerre on uniform limits of sequences of real rooted polynomials and by an interesting result of g h hardy an explicit description of this class in terms of the coefficients of a series is impossible since it is determined by an infinite number of discriminant inequalities one for each degree however interesting necessary or sufficient conditions can be formulated in particular j i hutchinson has shown that an entire function p x a 0 a 1 x a n x n with strictly positive coefficients has the property that all of its finite segments a i x i a i 1 x i 1 a j x j have only real roots if and only if a i 2 a i 1 a i 1 4 for i 1 2 in the present paper we give sharp lower bounds on the ratios a i 2 a i 1 a i 1 i 1 2 for the class considered by m petrovitch in particular we show that the limit of these minima when i infinity equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the laguerre polya class l pi we also explain the relation between newton s and hutchinson s inequalities and the logarithmic image of the set of all real rooted polynomials with positive coefficients

an asymptotic formula for the zeros of the deformed exponential function abstract we study the asymptotic representation for the zeros of the deformed exponential function n 0 1 n q n n 1 2 x n q 0 1 indeed we obtain an asymptotic formula for these zeros x n n q 1 n 1 g q n 2 o n 2 n 1 where g q k 1 k q k is the generating function of the sum of divisors function k this improves earlier results by langley 3 and liu 4 the proof of this formula is reduced to estimating the sum of an alternating series where the jacobi s triple product identity plays a key role

on neutral functional differential equations with proportional delays abstract in this paper we develop a comprehensive theory on the well posedness of the initial value problem for the neutral functional differential equation y t ay t i 1 b i y q i t i 1 c i y p i t t 0 y 0 y 0 and the asymptotic behaviour of its solutions we prove that the existence and uniqueness of solutions depend mainly on the coefficientsci i 1 2 and on the smoothness of functions in the solution space as far as the asymptotic behaviour of analytic solutions is concerned thecihave little effect we prove that if re a 0 then the solutiony t either grows exponentially or is polynomial the most interesting result is that if re a 0 anda 0 then the asymptotic behaviour of the solution depends mainly on the characteristic equation a i 1 b i q i 0 these results can be generalized to systems of equations finally we present some examples to illustrate the change of asymptotic behaviour in response to the variation of some parameters the main idea used in this paper is to express the solution in either dirichlet or dirichlet taylor series form

some canonical sequences of integers abstract extending earlier work of r donaghey and p j cameron we investigate some canonical eigen sequences associated with transformations of integer sequences several known sequences appear in a new setting for instance the sequences such as 1 3 11 49 257 1531 studied by t tsuzuku h o foulkes and a kerber in connection with multiply transitive groups are eigen sequences for the binomial transform many interesting new sequences also arise such as 1 1 2 26 152 1144 which shifts one place left when transformed by the stirling numbers of the second kind and whose exponential generating function satisfies a x a ex 1 1

an integral equation method for the numerical conformal mapping of interior exterior and doubly connected domains a numerical method based on the integral equation formulation of symm is described for computing approximations to the mapping functions which accomplish the following conformal maps a the mapping of a domain interior to a closed jordan curve onto the interior of the unit disc b the mapping of a domain exterior to a closed jordan curve onto the exterior of the unit disc c the mapping of a doubly connected domain bounded by two closed jordan curves onto a circular annulus the numerical method is based on approximating the unknown source density by cubic splines and singular functions and is particularly suited for the mapping of difficult domains having sharp corners

uniform sampling through the lovasz local lemma we propose a new algorithmic framework called partial rejection sampling to draw samples exactly from a product distribution conditioned on none of a number of bad events occurring our framework builds new connections between the variable framework of the lovasz local lemma and some classical sampling algorithms such as the cycle popping algorithm for rooted spanning trees among other applications we discover new algorithms to sample satisfying assignments of k cnf formulas with bounded variable occurrences