exact t 0 partition functions for potts antiferromagnets on sections of the simple cubic lattice we present exact solutions for the zero temperature partition function of the q state potts antiferromagnet equivalently the chromatic polynomial p on tube sections of the simple cubic lattice of fixed transverse size l x ifmmode times else texttimes fi l y and arbitrarily great length l z for sizes l x ifmmode times else texttimes fi l y 2 ifmmode times else texttimes fi 3 and 2 ifmmode times else texttimes fi 4 and boundary conditions a mathrm fbc x mathrm fbc y mathrm fbc z and b mathrm pbc x mathrm fbc y mathrm fbc z where fbc pbc denote free periodic boundary conditions in the limit of infinite length l z ensuremath rightarrow ensuremath infty we calculate the resultant ground state degeneracy per site w exponent of the ground state entropy generalizing q from z to c we determine the analytic structure of w and the related singular locus mathcal b which is the continuous accumulation set of zeros of the chromatic polynomial for the l z ensuremath rightarrow ensuremath infty limit of a given family of lattice sections w is analytic for real q down to a value q c we determine the values of q c for the lattice sections considered and address the question of the value of q c for a d dimensional cartesian lattice analogous results are presented for a tube of arbitrarily great length whose transverse cross section is formed from the complete bipartite graph k m m

ground state entropy of the potts antiferromagnet on strips of the square lattice abstract we present exact solutions for the zero temperature partition function chromatic polynomial p and the ground state degeneracy per site w exponent of the ground state entropy for the q state potts antiferromagnet on strips of the square lattice of width ly vertices and arbitrarily great length lx vertices the specific solutions are for a ly 4 fbcy pbcx cyclic b ly 4 fbcy tpbcx mobius c ly 5 6 pbcy fbcx cylindrical and d l y 5 fbc y fbc x open where fbc pbc and tpbc denote free periodic and twisted periodic boundary conditions respectively in the lx limit of each strip we discuss the analytic structure of w in the complex q plane the respective w functions are evaluated numerically for various values of q several inferences are presented for the chromatic polynomials and analytic structure of w for lattice strips with arbitrarily great ly the absence of a nonpathological lx limit for real nonintegral q in the interval 0 0 for strips of the square triangular lattice is discussed

a perron frobenius type of theorem for quantum operations we define a special class of quantum operations we call markovian and show that it has the same spectral properties as a corresponding markov chain we then consider a convex combination of a quantum operation and a markovian quantum operation and show that under a norm condition its spectrum has the same properties as in the conclusion of the perron frobenius theorem if its markovian part does moreover under a compatibility condition of the two operations we show that its limiting distribution is the same as the corresponding markov chain we apply our general results to partially decoherent quantum random walks with decoherence strength 0 le p le 1 we obtain a quantum ergodic theorem for partially decoherent processes we show that for 0 p le 1 the limiting distribution of a partially decoherent quantum random walk is the same as the limiting distribution for the classical random walk

linear invariants of complex manifolds and their plurisubharmonic variations abstract for a bounded domain d and a real number p 0 denote by a p d the linear space of l p integrable holomorphic functions on d equipped with the l p pseudonorm we prove that two bounded hyperconvex domains d 1 c n and d 2 c m are biholomorphic in particular n m if there is a linear isometry between a p d 1 and a p d 2 for some 0 p 2 the same result holds for p 2 p 2 4 provided that the p bergman kernels on d 1 and d 2 are exhaustive with this as a motivation we show that for all p 0 the p bergman kernel on a strongly pseudoconvex domain with c 2 boundary or a simply connected homogeneous regular domain is exhaustive these results show that spaces of pluricanonical sections of complex manifolds equipped with canonical pseudonorms are important linear invariants of complex manifolds the second part of the present work devotes to studying variations of these invariants we show that the direct image sheaf of the twisted relative pluricanonical bundle associated to a holomorphic family of stein manifolds or compact kahler manifolds is positively curved with respect to the canonical singular finsler metric

complex temperature phase diagrams for the q state potts model on self dual families of graphs and the nature of the q limit exact calculations of the potts model partition function z g q v have been presented for arbitrary q and temperature like variable v on self dual strip graphs g of the square lattice with fixed width l y and arbitrarily great length lx with two types of boundary conditions letting lx infinity the resultant free energy and complex temperature phase diagram have been computed including the locus b where the free energy is nonanalytic results are analyzed for widths l y 1 2 3 these results have been used to study the approach to the large q limit of b

chromatic polynomials for j h i strip graphs and their asymptotic limits abstract we calculate the chromatic polynomials p for n vertex strip graphs of the form j l 1 m h i where j and i are various subgraphs on the left and right ends of the strip whose bulk is comprised of m fold repetitions of a subgraph h the strips have free boundary conditions in the longitudinal direction and free or periodic boundary conditions in the transverse direction this extends our earlier calculations for strip graphs of the form l 1 m h i we use a generating function method from these results we compute the asymptotic limiting function w lim n p 1 n for q z this has physical significance as the ground state degeneracy per site exponent of the ground state entropy of the q state potts antiferromagnet on the given strip in the complex q plane w is an analytic function except on a certain continuous locus b in contrast to the l 1 m h i strip graphs where b i is independent of i and ii consists of arcs and possible line segments that do not enclose any regions in the q plane we find that for some j l 1 m h i strip graphs b i does depend on i and j and ii can enclose regions in the q plane our study elucidates the effects of different end subgraphs i and j and of boundary conditions on the infinite length limit of the strip graphs

asymptotic limits and zeros of chromatic polynomials and ground state entropy of potts antiferromagnets we study the asymptotic limiting function w g q lim n to infty p g q 1 n where p g q is the chromatic polynomial for a graph g with n vertices we first discuss a subtlety in the definition of w g q resulting from the fact that at certain special points q s the following limits do not commute lim n to infty lim q to q s p g q 1 n ne lim q to q s lim n to infty p g q 1 n we then present exact calculations of w g q and determine the corresponding analytic structure in the complex q plane for a number of families of graphs g including circuits wheels biwheels bipyramids and cyclic and twisted ladders we study the zeros of the corresponding chromatic polynomials and prove a theorem that for certain families of graphs all but a finite number of the zeros lie exactly on a unit circle whose position depends on the family using the connection of p g q with the zero temperature potts antiferromagnet we derive a theorem concerning the maximal finite real point of non analyticity in w g q denoted q c and apply this theorem to deduce that q c sq 3 and q c hc 3 sqrt 5 2 for the square and honeycomb lattices finally numerical calculations of w hc q and w sq q are presented and compared with series expansions and bounds

exact potts model partition functions on strips of the honeycomb lattice abstract we present exact calculations of the partition function of the q state potts model on i open ii cyclic and iii mobius strips of the honeycomb brick lattice of width l y 2 and arbitrarily great length in the infinite length limit the thermodynamic properties are discussed the continuous locus of singularities of the free energy is determined in the q plane for fixed temperature and in the complex temperature plane for fixed q values we also give exact calculations of the zero temperature partition function chromatic polynomial and w q the exponent of the ground state entropy for the potts antiferromagnet for honeycomb strips of type iv l y 3 cyclic v l y 3 mobius vi l y 4 cylindrical and vii l y 4 open in the infinite length limit we calculate w q and determine the continuous locus of points where it is nonanalytic we show that our exact calculation of the entropy for the l y 4 strip with cylindrical boundary conditions provides an extremely accurate approximation to a few parts in 10 5 for moderate q values to the entropy for the full 2d honeycomb lattice where the latter is determined by monte carlo measurements since no exact analytic form is known

kahler einstein submanifolds of the infinite dimensional projective space this paper consists of two main results in the first one we describe all kahler immersions of a bounded symmetric domain into the infinite dimensional complex projective space in terms of the wallach set of the domain in the second one we exhibit an example of complete and non homogeneous kahler einstein metric with negative scalar curvature which admits a kahler immersion into the infinite dimensional complex projective space

homogeneous open quantum random walks on a lattice we study open quantum random walks oqrws for which the underlying graph is a lattice and the generators of the walk are homogeneous in space using the results recently obtained in carbone and pautrat ann henri poincare 2015 we study the quantum trajectory associated with the oqrw which is described by a position process and a state process we obtain a central limit theorem and a large deviation principle for the position process we study in detail the case of homogeneous oqrws on the lattice mathbb z d with internal space mathfrak h mathbb c 2

zeros of jones polynomials for families of knots and links abstract we calculate jones polynomials v l t for several families of alternating knots and links by computing the tutte polynomials t g x y for the associated graphs g and then obtaining v l t as a special case of the tutte polynomial for each of these families we determine the zeros of the jones polynomial including the accumulation set in the limit of infinitely many crossings a discussion is also given of the calculation of jones polynomials for non alternating links

improved lower bounds on the ground state entropy of the antiferromagnetic potts model we present generalized methods for calculating lower bounds on the ground state entropy per site s 0 or equivalently the ground state degeneracy per site w e s 0 k b of the antiferromagnetic potts model we use these methods to derive improved lower bounds on w for several lattices

flow polynomials and their asymptotic limits for lattice strip graphs we present exact calculations of flow polynomials f g q for lattice strips of various fixed widths l y 4 and arbitrarily great lengths l x with several different boundary conditions square honeycomb and triangular lattice strips are considered we introduce the notion of flows per face fl in the infinite length limit we study the zeros of f g q in the complex q plane and determine exactly the asymptotic accumulation sets of these zeros in the infinite length limit for the various families of strips the function fl is nonanalytic on this locus the loci are found to be noncompact for many strip graphs with periodic or twisted periodic longitudinal boundary conditions and compact for strips with free longitudinal boundary conditions we also find the interesting feature that aside from the trivial case l y 1 the maximal point q cf where crosses the real axis is universal on cyclic and mobius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value q cf 3 for the infinite square lattice

ground state entropy of potts antiferromagnets on homeomorphic families of strip graphs abstract we present exact calculations of the zero temperature partition function and ground state degeneracy per site w for the q state potts antiferromagnet on a variety of homeomorphic families of planar strip graphs g ch k1 k2 k m where k1 k2 and k describe the homeomorphic structure and m denotes the length of the strip several different ways of taking the total number of vertices to infinity by sending i m with k1 k2 and k fixed ii k1 and or k2 with m and k fixed and iii k with m and p k1 k2 fixed are studied and the respective loci of points b where w is nonanalytic in the complex q plane are determined the b s for limit i are comprised of arcs which do not enclose regions in the q plane and for many values of p and k include support for re q b for limits ii and iii is the unit circle q 1 1

strictly pseudoconvex domains and smoothly varying peak functions abstract we answer affirmatively the question of deng guan and zhang 1 concerning the existence for k n k 2 of the family of c k 2 smoothly varying peak functions at the boundary points of strictly pseudoconvex domains varying c k smoothly

ground state entropy of the potts antiferromagnet on cyclic strip graphs we present exact calculations of the zero temperature partition function chromatic polynomial and the exponent of the ground state entropy s0 for the q state potts antiferromagnet on families of cyclic and twisted cyclic mobius strip graphs composed of p sided polygons our results suggest a general rule concerning the maximal region in the complex q plane to which one can analytically continue from the physical interval where s0 0 the chromatic zeros and their accumulation set exhibit the rather unusual property of including support for re q 0 and provide further evidence for a relevant conjecture

some properties of h extendible domains in cn 1 abstract the purpose of this article is twofold the first aim is to characterize h extendibility of smoothly bounded pseudoconvex domains in c n 1 by their noncompact automorphism groups our second goal is to show that if the squeezing function tends to 1 at an h extendible boundary point of a smooth pseudoconvex domain in c n 1 then this point must be strongly pseudoconvex

absence of phase transition for antiferromagnetic potts models via the dobrushin uniqueness theorem we prove that theq state potts antiferromagnet on a lattice of maximum coordination numberr exhibits exponential decay of correlations uniformly at all temperatures including zero temperature wheneverq 2r we also prove slightly better bounds for several two dimensional lattices square lattice exponential decay forq 7 triangular lattice q 11 hexagonal lattice q 4 and kagome lattice q 6 the proofs are based on the dobrushin uniqueness theorem

chromatic polynomials for lattice strips with cyclic boundary conditions abstract the zero temperature q state potts model partition function for a lattice strip of fixed width l y and arbitrary length l x has the form p g q j 1 n g c g j g j l x and is equivalent to the chromatic polynomial for this graph we present exact zero temperature partition functions for strips of several lattices with fbc y pbc x i e cyclic boundary conditions in particular the chromatic polynomial of a family of generalized dodecahedra graphs is calculated the coefficient c g j of degree d in q is c d u 2d q 2 where u n x is the chebyshev polynomial of the second kind we also present the chromatic polynomial for the strip of the square lattice with pbc y pbc x i e toroidal boundary conditions and width l y 4 with the property that each set of four vertical vertices forms a tetrahedron a number of interesting and novel features of the continuous accumulation set of the chromatic zeros b are found

ground state entropy in potts antiferromagnets abstract the q state potts antiferromagnet exhibits nonzero ground state entropy s0 g q 0 for sufficiently large q on a given n vertex lattice or graph g and its n limit g we present exact calculations of the zero temperature partition function z g q t 0 and w g q where s 0 k b ln w for this model on a number of families g these calculations have interesting connections with graph theory since z g q t 0 p g q where the chromatic polynomial p g q gives the number of ways of coloring the vertices of the graph g such that no adjacent vertices have the same color generalizing q from z to c we determine the accumulation set b of the zeros of p g q which constitute the continuous loci of points on which w is nonanalytic the potts antiferromagnet has a zero temperature critical point at the maximal value qc where b crosses the real q axis in particular exact solutions for w and b are given for infinitely long finite width strips of various lattices in addition to their intrinsic interest these yield insight into the approach to the 2d thermodynamic limit some corresponding results are presented for the exact finite temperature potts free energy on families of graphs finally we present rigorous upper and lower bounds on w for 2d lattices

some exact results on bond percolation we present some exact results on bond percolation we derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice by l bonds connecting the same adjacent vertices thereby yielding the lattice l this relation is used to calculate the bond percolation threshold on l we show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality d 2 but changes it on a one dimensional lattice and quasi one dimensional infinite length strips we also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the n limits of several families of n vertex graphs finally we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as n

exact potts model partition functions on ladder graphs abstract we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex ladder graphs i e strips of the square lattice with width ly 2 and arbitrary length lx with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of these ladder graphs and the thermodynamics is discussed by comparison with strip graphs of other widths we analyze how the singularities at the zero temperature critical point of the ferromagnet on infinite length finite width strips depend on the width we point out and study the following noncommutativity at certain special values q s lim n lim q q s z 1 n lim q q s lim n z 1 n it is shown that the potts antiferromagnet on both the infinite length line and ladder graphs with cyclic or mobius boundary conditions exhibits a phase transition at finite temperature if 0 b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros certain properties of the complex temperature phase diagrams are shown to exhibit close connections with those of the model on the square lattice showing that exact solutions on infinite length strips provide a way of gaining insight into these complex temperature phase diagrams

geometry of domains with the uniform squeezing property abstract we introduce the notion of domains with uniform squeezing property study various analytic and geometric properties of such domains and show that they cover many interesting examples including teichmuller spaces and hermitian symmetric spaces of non compact type the properties supported by such manifolds include pseudoconvexity hyperconvexity kahler hyperbolicity vanishing of cohomology groups and quasi isometry of various invariant metrics it also leads to nice geometric properties for manifolds covered by bounded domains and a simple criterion to provide positive examples to a problem of serre about stein properties of holomorphic fiber bundles

potts model partition functions for self dual families of strip graphs abstract we consider the q state potts model on families of self dual strip graphs g d of the square lattice of width l y and arbitrarily great length l x with periodic longitudinal boundary conditions the general partition function z and the t 0 antiferromagnetic special case p chromatic polynomial have the respective forms j 1 n f l y c f l y j f l y j l x with f z p for arbitrary l y we determine i the general coefficient c f l y j in terms of chebyshev polynomials ii the number n f l y d of terms with each type of coefficient and iii the total number of terms n f l y we point out interesting connections between the n z l y d and temperley lieb algebras and between the n f l y and enumerations of directed lattice animals exact calculations of p are presented for 2 l y 4 in the limit of infinite length we calculate the ground state degeneracy per site exponent of the ground state entropy w q generalizing q from z to c we determine the continuous locus b in the complex q plane where w q is singular we find the interesting result that for all l y values considered the maximal point at which b crosses the real q axis denoted q c is the same and is equal to the value for the infinite square lattice q c 3 this is the first family of strip graphs of which we are aware that exhibits this type of universality of q c

open quantum random walks a new model of quantum random walks is introduced on lattices as well as on finite graphs these quantum random walks take into account the behavior of open quantum systems they are the exact quantum analogues of classical markov chains we explore the quantum trajectory point of view on these quantum random walks that is we show that measuring the position of the particle after each time step gives rise to a classical markov chain on the lattice times the state space of the particle this quantum trajectory is a simulation of the master equation of the quantum random walk the physical pertinence of such quantum random walks and the way they can be concretely realized is discussed differences and connections with the already well known quantum random walks such as the hadamard random walk are established

comparison of invariant metrics and distances on strongly pseudoconvex domains and worm domains we prove that for a strongly pseudoconvex domain d subset mathbb c n the infinitesimal caratheodory metric g c z v and the infinitesimal kobayashi metric g k z v coincide if z is sufficiently close to bd and if v is sufficiently close to being tangential to bd also we show that every two close points of d sufficiently close to the boundary and whose difference is almost tangential to bd can be joined by a unique up to reparameterization complex geodesic of d which is also a holomorphic retract of d the same continues to hold if d is a worm domain as long as the points are sufficiently close to a strongly pseudoconvex boundary point we also show that a strongly pseudoconvex boundary point of a worm domain can be globally exposed this has consequences for the behavior of the squeezing function

canonical metrics on the moduli space of riemann surfaces i in this paper we continue our study of the canonical metrics on the moduli space of curves we first prove the bounded geometry of the ricci and perturbed ricci metrics by carefully choosing the pertubation constant and by studying the asymptotics we show that the ricci and holomorphic sectional curvatures of the perturbed ricci metric are bounded from above and below by negative constants based on our understanding of the kahler einstein metric we show that the logarithmic cotangent bundle over the deligne mumford moduli space is stable with respect to the canonical polarization finally in the last section we prove the strongly bounded geometry of the kahler einstein metric by using the kahler ricci flow and a priori estimates of the complex monge ampere equation

a vanishing theorem on generalized cartan hartogs domain of the second type abstract in this paper we consider the vanishing theorem of square integrable harmonic forms with respect to the bergman metric on the second type of generalized cartan hartogs domain by using three main tools the properties of bergman kernel function holomorphic invariant and holomorphic automorphism group

ground state entropy of the potts antiferromagnet with next nearest neighbor spin spin couplings on strips of the square lattice we present exact calculations of the zero temperature partition function chromatic polynomial and w q the exponent of the ground state entropy for the q state potts antiferromagnet with next nearest neighbor spin spin couplings on square lattice strips of width l y 3 and l y 4 vertices and arbitrarily great length l x vertices with both free and periodic boundary conditions the resultant values of w for a range of physical q values are compared with each other and with the values for the full two dimensional lattice these results give insight into the effect of such nonnearest neighbor couplings on the ground state entropy we show that the q 2 ising and q 4 potts antiferromagnets have zero temperature critical points on the l x ensuremath rightarrow ensuremath infty limits of the strips that we study with the generalization of q from z to c we determine the analytic structure of w q in the q plane for the various cases

estimate of the squeezing function for a class of bounded domains we construct a class of bounded domains on which the squeezing function is not uniformly bounded from below near a smooth and pseudoconvex boundary point

open quantum random walks with decoherence on coins with n degrees of freedom in this paper we define a new type of decoherent quantum random walk with parameter 0 p 1 which becomes a unitary quantum random walk uqrw when p 0 and an open quantum random walk oqrw when p 1 respectively we call this process a partially open quantum random walk poqrw we study the limiting distribution of a poqrw on z1 subject to decoherence on coins with n degrees of freedom the limiting distribution of the poqrw converges to a convex combination of normal distributions under an eigenvalue condition a perron frobenius type of theorem is established to determine whether or not a poqrw satisfies the eigenvalue condition moreover we explicitly compute the limiting distributions of characteristic equations of the position probability functions when n 2 and 3

properties of squeezing functions and global transformations of bounded domains the central purpose of the present paper is to study boundary behavior of squeezing functions on bounded domains we prove that the squeezing function of a strongly pseudoconvex domain tends to 1 near the boundary in fact such an estimate is proved for the squeezing function on any domain near its globally strongly convex boundary points we also study the stability of squeezing functions on a sequence of bounded domains and give comparisons of intrinsic measures and metrics on bounded domains in terms of squeezing functions as applications we give new and simple proofs of several well known results about geometry of strongly pseudoconvex do mains and prove that all cartan hartogs domains are homogenous regular finally some related problems that ask for further study are proposed

transfer matrices for the zero temperature potts antiferromagnet on cyclic and mobius lattice strips abstract we present transfer matrices for the zero temperature partition function of the q state potts antiferromagnet equivalently the chromatic polynomial on cyclic and mobius strips of the square triangular and honeycomb lattices of width l y and arbitrarily great length l x we relate these results to our earlier exact solutions for square lattice strips with l y 3 4 5 triangular lattice strips with l y 2 3 4 and honeycomb lattice strips with l y 2 3 and periodic or twisted periodic boundary conditions we give a general expression for the chromatic polynomial of a mobius strip of a lattice and exact results for a subset of honeycomb lattice transfer matrices both of which are valid for arbitrary strip width l y new results are presented for the l y 5 strip of the triangular lattice and the l y 4 and l y 5 strips of the honeycomb lattice using these results and taking the infinite length limit l x we determine the continuous accumulation locus of the zeros of the above partition function in the complex q plane including the maximal real point of nonanalyticity of the degeneracy per site w as a function of q

exact potts model partition functions for strips of the square lattice we present exact calculations of the potts model partition function z g q v for arbitrary q and temperature like variable v on n vertex square lattice strip graphs g for a variety of transverse widths l t and for arbitrarily great length l l with free longitudinal boundary conditions and free and periodic transverse boundary conditions these have the form z g q v sum j 1 n z g lambda c z g j lambda z g j l ell we give general formulas for n z g j and its specialization to v 1 for arbitrary l t for both types of boundary conditions as well as other general structural results on z the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed it is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the potts model on the infinite square lattice considering the full generalization to arbitrary complex q and v we determine the singular locus b arising as the accumulation set of partition function zeros as l l in the q plane for fixed v and in the v plane for fixed q

exact chromatic polynomials for toroidal chains of complete graphs abstract we present exact calculations of the partition function of the zero temperature potts antiferromagnet equivalently the chromatic polynomial for graphs of arbitrarily great length composed of repeated complete subgraphs kb with b 5 6 which have periodic or twisted periodic boundary condition in the longitudinal direction in the lx limit the continuous accumulation set of the chromatic zeros b is determined we give some results for arbitrary b including the extrema of the eigenvalues with coefficients of degree b 1 and the explicit forms of some classes of eigenvalues we prove that the maximal point where b crosses the real axis qc satisfies the inequality qc b for 2 b the minimum value of q at which b crosses the real q axis is q 0 and we make a conjecture concerning the structure of the chromatic polynomial for klein bottle strips

q plane zeros of the potts partition function on diamond hierarchical graphs we report exact results concerning the zeros of the partition function of the potts model in the complex q plane as a function of a temperature like boltzmann variable v for the m th iterate graphs dm of the diamond hierarchical lattice including the limit m in this limit we denote the continuous accumulation locus of zeros in the q planes at fixed v v0 as bq v0 we apply theorems from complex dynamics to establish the properties of bq v0 for v 1 the zero temperature potts antiferromagnet or equivalently chromatic polynomial we prove that bq 1 crosses the real q axis at i a minimal point q 0 ii a maximal point q 3 iii q 32 27 iv a cubic root that we give with the value q q1 1 638 896 9 and v an infinite number of points smaller than q1 converging to 32 27 from above similar results hold for bq v0 for any 1 0 potts ferromagnet we also provide the computer generated plots of bq v0 at various values of v0 in both the antiferromagnetic and ferromagnetic regimes and compare them to the numerically computed zeros of z d4 q v0

characterizing strong pseudoconvexity obstructions to biholomorphisms and lyapunov exponents in this paper we consider the following question for bounded domains with smooth boundary can strong pseudoconvexity be characterized in terms of the intrinsic complex geometry of the domain our approach to answering this question is based on understanding the dynamical behavior of real geodesics in the kobayashi metric and allows us to prove a number of results for domains with low regularity for instance we show that for convex domains with c 2 epsilon boundary strong pseudoconvexity can be characterized in terms of the behavior of the squeezing function near the boundary the behavior of the holomorphic sectional curvature of the bergman metric near the boundary or any other reasonable measure of the complex geometry near the boundary the first characterization gives a partial answer to a question of fornaess and wold as an application of these characterizations we show that a convex domain with c 2 epsilon boundary which is biholomorphic to a strongly pseudoconvex domain is also strongly pseudoconvex

chromatic polynomials of large triangular lattices evaluating the q colourings of a lattice is equivalent to solving the q state zero temperature antiferromagnetic potts model this has recently been done exactly for an infinite triangular lattice with q real here the results are extended to the full complex q plane giving the limiting distribution of the zeros of the chromatic polynomial the results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the beraha numbers is noted

exact potts model partition function on strips of the triangular lattice abstract in this paper we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex strip graphs of width ly 2 and arbitrary length of the triangular lattice with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed considering the full generalization to arbitrary complex q and temperature we determine the singular locus b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros comparisons are made with our previous exact calculation of potts model partition functions for the corresponding strips of the square lattice our present calculations yield as special cases several quantities of graph theoretic interest

the squeezing function on doubly connected domains via the loewner differential equation for any bounded domains varomega in mathbb c n deng guan and zhang introduced the squeezing function s varomega z which is a biholomorphic invariant of bounded domains we show that for n 1 the squeezing function on an annulus a r lbrace z in mathbb c r z 1 rbrace is given by s a r z max left z frac r z right for all 0 r 1 this disproves the conjectured formula for the squeezing function proposed by deng guan and zhang and establishes up to biholomorphisms the squeezing function for all doubly connected domains in mathbb c other than the punctured plane it provides the first non trivial formula for the squeezing function for a wide class of plane domains and answers a question of wold our main tools used to prove this result are the schottky klein prime function following the work of crowdy and a version of the loewner differential equation on annuli due to komatu we also show that these results can be used to obtain lower bounds on the squeezing function for certain product domains in mathbb c n

upper and lower bounds for the ground state entropy of antiferromagnetic potts models we derive rigorous upper and lower bounds for the ground state entropy of the q state potts antiferromagnet on the honeycomb and triangular lattices these bounds are quite restrictive especially for large q copyright ital 1997 ital the american physical society

reliability polynomials and their asymptotic limits for families of graphs we present exact calculations of reliability polynomials r g p for lattice strips g of fixed widths l y 4 and arbitrarily great length l x with various boundary conditions we introduce the notion of a reliability per vertex r g p lim v r g p 1 v where v denotes the number of vertices in g and g denotes the formal limit lim v g we calculate this exactly for various families of graphs we also study the zeros of r g p in the complex p plane and determine exactly the asymptotic accumulation set of these zeros b across which r g is nonanalytic

the structure of chromatic polynomials of planar triangulations and implications for chromatic zeros and asymptotic limiting quantities we present an analysis of the structure and properties of chromatic polynomials of one parameter and multi parameter families of planar triangulation graphs where is a vector of integer parameters we use these to study the ratio of to the tutte upper bound 1 n 5 where and n is the number of vertices in in particular we calculate limiting values of this ratio as n for various families of planar triangulations we also use our calculations to analyze zeros of these chromatic polynomials we study a large class of families with p 1 and p 2 and show that these have a structure of the form for p 1 where 1 q 2 2 q 3 and 3 1 and for p 2 we derive properties of the coefficients and show that has a real chromatic zero that approaches as one or more of the mi the generalization to p 3 is given further we present a one parameter family of planar triangulations with real zeros that approach 3 from below as m implications for the ground state entropy of the potts antiferromagnet are discussed

ground state entropy of potts antiferromagnets and the approach to the two dimensional thermodynamic limit we study the ground state degeneracy per site exponent of the ground state entropy w lambda l x infty times l y q for the q state potts antiferromagnet on infinitely long strips with width l y of 2d lattices lambda with free and periodic boundary conditions in the y direction denoted fbc y and pbc y we show that the approach of w to its 2d thermodynamic limit as l y increases is quite rapid for moderate values of q and l y simeq 4 w lambda l x infty times l y q is within about 5 and cal o 10 3 of the 2d value w lambda l x infty times l y infty q for fbc y and pbc y respectively the approach of w to the 2d thermodynamic limit is proved to be monotonic non monotonic for fbc y pbc y it is noted that ground state entropy determinations on infinite strips can be used to obtain the central charge for cases with critical ground states

t 0 partition functions for potts antiferromagnets on lattice strips with fully periodic boundary conditions abstract we present exact calculations of the zero temperature partition function for the q state potts antiferromagnet equivalently the chromatic polynomial for families of arbitrarily long strip graphs of the square and triangular lattices with width ly 4 and boundary conditions that are doubly periodic or doubly periodic with reversed orientation i e of torus or klein bottle type these boundary conditions have the advantage of removing edge effects in the limit of infinite length we calculate the exponent of the entropy w q and determine the continuous locus b where it is singular we also give results for toroidal strips involving crossing subgraphs these make possible a unified treatment of torus and klein bottle boundary conditions and enable us to prove that for a given strip the locus b is the same for these boundary conditions

the explicit forms and zeros of the bergman kernel function for hartogs type domains abstract we define the cartan hartogs domain which is the hartogs type domain constructed over the product of bounded hermitian symmetric domains and compute the explicit form of the bergman kernel for the cartan hartogs domain using the virtual bergman kernel as the main contribution of this paper we show that the main part of the explicit form of the bergman kernel is a polynomial whose coefficients are combinations of stirling numbers of the second kind using this observation as an application we give an algorithmic procedure to determine the condition that their bergman kernel functions have zeros

structural properties of potts model partition functions and chromatic polynomials for lattice strips abstract the q state potts model partition function equivalent to the tutte polynomial for a lattice strip of fixed width l y and arbitrary length l x has the form z g q v j 1 n z g c z g j z g j l x where v is a temperature dependent variable the special case of the zero temperature antiferromagnet v 1 is the chromatic polynomial p g q using coloring and transfer matrix methods we give general formulas for c x g j 1 n x g c x g j for x z p on cyclic and mobius strip graphs of the square and triangular lattice combining these with a general expression for the unique coefficient c z g j of degree d in q c d u 2d q 2 where u n x is the chebyshev polynomial of the second kind we determine the number of z g j s with coefficient c d in z g q v for these cyclic strips of width l y to be n z l y d 2d 1 l y d 1 1 2l y l y d for 0 d l y and zero otherwise for both cyclic and mobius strips of these lattices the total number of distinct eigenvalues z g j is calculated to be n z l y 2l y l y results are also presented for the analogous numbers n p l y d and n p l y for p g q we find that n p l y 0 n p l y 1 1 m l y 1 motzkin number n z l y 0 c l y the catalan number and give an exact expression for n p l y our results for n z l y and n p l y apply for both the cyclic and mobius strips of both the square and triangular lattices we also point out the interesting relations n z l y 2 n da tri l y and n p l y 2 n da sq l y where n da n denotes the number of directed lattice animals on the lattice we find the asymptotic growths n z l y l y 1 2 4 l y and n p l y l y 1 2 3 l y as l y some general geometric identities for potts model partition functions are also presented

limit theorems for open quantum random walks we consider the limit distributions of open quantum random walks on one dimensional lattice space we introduce a dual process to the original quantum walk process which is quite similar to the relation of schrodinger heisenberg representation in quantum mechanics by this we can compute the distribution of the open quantum random walks concretely for many examples and thereby we can also obtain the limit distributions of them in particular it is possible to get rid of the initial state when we consider the evolution of the walk it appears only in the last step of the computation

the summarizations on research of hua domain the hua domain makes the cartan domain and the egg domain into a unit and provides a new research field for several complex variables up to now a series of important results were obtained in this paper the author introduces the brief history of hua domain and introduces the computations for berman kernel functions and einstein k hler metrics on hua domains and discuss the equivalences between four classical metrics bergman caratheodory einstein k hler and kobayashi metrics and prove that the bergman metric is equivalent to the einstein kahler metric on cartan hartogs which is the special cases of hua domains that means the s t yau s conjecture is also true on cartan hartogs domains the author point out the new idea and new methods and give some open problems to expect the mathematicians interested in hua domains and to lucubrate the hua domains