quadri tilings of the plane we introduce quadri tilings and show that they are in bijection with dimer models on a family of graphs r arising from rhombus tilings using two height functions we interpret a sub family of all quadri tilings called triangular quadri tilings as an interface model in dimension 2 2 assigning critical weights to edges of r we prove an explicit expression only depending on the local geometry of the graph r for the minimal free energy per fundamental domain gibbs measure this solves a conjecture of kenyon invent math 150 409 439 2002 we also show that when edges of r are asymptotically far apart the probability of their occurrence only depends on this set of edges finally we give an expression for a gibbs measure on the set of all triangular quadri tilings whose marginals are the above gibbs measures and conjecture it to be that of minimal free energy per fundamental domain

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

analyzing lattice networks through substructures abstract analyzing the topology of network structures is an important topic studied from many different aspects of science and mathematics the wiener polarity index number of unordered pairs of vertices at distance 3 from each other is one of the representative descriptors of graph structures it was computed for several lattice networks by chen et al 11 in an effort to understand the properties of these networks the wiener polarity index is a variation of the classic distance based graph invariant the wiener index sum of distances between all pairs of vertices which is known to be closely related to the number of substructures in this paper we examine the numbers of various subgraphs of order 4 for these lattice networks in addition to confirming their symmetric nature comparing the numbers of various substructures leads to insights on other less trivial characteristics of these network structures of common interest

spanning trees on graphs and lattices in d dimensions the problem of enumerating spanning trees on graphs and lattices is considered we obtain bounds on the number of spanning trees nst and establish inequalities relating the numbers of spanning trees of different graphs or lattices a general formulation is presented for the enumeration of spanning trees on lattices in d 2 dimensions and is applied to the hypercubic body centred cubic face centred cubic and specific planar lattices including the kagome diced 4 8 8 bathroom tile union jack and 3 12 12 lattices this leads to closed form expressions for nst for these lattices of finite sizes we prove a theorem concerning the classes of graphs and lattices with the property that nst exp nz as the number of vertices n where z is a finite non zero constant this includes the bulk limit of lattices in any spatial dimension and also sections of lattices whose lengths in some dimensions go to infinity while others are finite we evaluate z exactly for the lattices we consider and discuss the dependence of z on d and the lattice coordination number we also establish a relation connecting z to the free energy of the critical ising model for planar lattices

upper and lower bounds for the energy of bipartite graphs using lagrange s multiplier rule we find upper and lower bounds of the energy of a bipartite graph g in terms of the number of vertices edges and the spectral moment of fourth order moreover the upper bound is attained in a graph g if and only if g is the graph of a symmetric balanced incomplete block design bibd also we determine the graphs for which the lower bound is sharp

the emergence of the electrostatic field as a feynman sum in random tilings with holes we consider random lozenge tilings on the triangular lattice with holes q 1 q n in some fixed position for each unit triangle not in a hole consider the average orientation of the lozenge covering it we show that the scaling limit of this discrete field is the electrostatic field obtained when regarding each hole q i as an electrical charge of magnitude equal to the difference between the number of unit triangles of the two different orientations inside q i this is then restated in terms of random surfaces yielding the result that the average over surfaces with prescribed height at the union of the boundaries of the holes is in the scaling limit a sum of helicoids

arctic circles domino tilings and square young tableaux the arctic circle theorem of jockusch propp and shor asserts that uniformly random domino tilings of an aztec diamond of high order are frozen with asymptotically high probability outside the arctic circle inscribed within the diamond a similar arctic circle phenomenon has been observed in the limiting behavior of random square young tableaux in this paper we show that random domino tilings of the aztec diamond are asymptotically related to random square young tableaux in a more refined sense that looks also at the behavior inside the arctic circle this is done by giving a new derivation of the limiting shape of the height function of a random domino tiling of the aztec diamond that uses the large deviation techniques developed for the square young tableaux problem in a previous paper by pittel and the author the solution of the variational problem that arises for domino tilings is almost identical to the solution for the case of square young tableaux by pittel and the author the analytic techniques used to solve the variational problem provide a systematic guess free approach for solving problems of this type which have appeared in a number of related combinatorial probability models

left and right compatibility of order relations and fuzzy tolerance relations abstract in a recent paper de baets et al have studied the compatibility of a n strict order relation with a fuzzy relation and have characterized the fuzzy tolerance and in particular fuzzy equivalence relations that a given strict order relation is compatible with we extend this study by considering the left and right compatibility of a n strict order relation with a fuzzy tolerance relation and vice versa we characterize the fuzzy tolerance relations that are compatible with a given strict order relation conversely we provide a representation of the fuzzy tolerance relations that a given strict order relation is left or right compatible with specific attention is paid to the case of fuzzy equivalence relations we conclude by pointing out that the representation theorems in the above mentioned paper need some minor rectification

local statistics of realizable vertex models we study planar vertex models which are probability measures on edge subsets of a planar graph satisfying certain constraints at each vertex examples including the dimer model and 1 2 model which we will define we express the local statistics of a large class of vertex models on a finite hexagonal lattice as a linear combination of the local statistics of dimers on the corresponding fisher graph with the help of a generalized holographic algorithm using an n n torus to approximate the periodic infinite graph we give an explicit integral formula for the free energy and local statistics for configurations of the vertex model on an infinite bi periodic graph as an example we simulate the 1 2 model by the technique of glauber dynamics

algebraic arctic curves in the domain wall six vertex model the arctic curve i e the spatial curve separating ordered or frozen and disordered or temperate regions of the six vertex model with domain wall boundary conditions is discussed for the root of unity vertex weights in these cases the curve is described by algebraic equations which can be worked out explicitly from the parametric solution for this curve some interesting examples are discussed in detail the upper bound on the maximal degree of the equation in a generic root of unity case is obtained

conformal invariance of loops in the double dimer model the dimer model is the study of random dimer covers perfect matchings of a graph a double dimer configuration on a graph mathcal g is a union of two dimer covers of mathcal g we introduce quaternion weights in the dimer model and show how they can be used to study the homotopy classes relative to a fixed set of faces of loops in the double dimer model on a planar graph as an application we prove that in the scaling limit of the uniform double dimer model on mathbb z 2 or on any other bipartite planar graph conformally approximating mathbb c the loops are conformally invariant

asymptotic energy of lattices abstract the energy of a simple graph g arising in chemical physics denoted by e g is defined as the sum of the absolute values of eigenvalues of g as the dimer problem and spanning trees problem in statistical physics in this paper we propose the energy per vertex problem for lattice systems in general for a type of lattice in statistical physics to compute the entropy constant with toroidal cylindrical mobius band klein bottle and free boundary conditions are different tasks with different hardness and may have different solutions we show that the energy per vertex of plane lattices is independent of the toroidal cylindrical mobius band klein bottle and free boundary conditions in particular the asymptotic formulae of energies of the triangular 3 3 4 2 and hexagonal lattices with toroidal cylindrical mobius band klein bottle and free boundary conditions are obtained explicitly

algebraic genericity of frequently universal harmonic functions on trees abstract we show that the set of frequently universal harmonic functions on a tree t contains a vector space except 0 which is dense in the space of harmonic functions on t seen as subset of c t in order to prove this we replace the complex plane c by any separable frechet space e and we repeat all the theory

macroscopically separated gaps in dimer coverings of aztec rectangles in this paper we determine the interaction of diagonal defect clusters in regions of an aztec rectangle that scale to arbitrary points on its symmetry axis in earlier work we treated the case when this point was the center of the scaled aztec rectangle we use the resulting formulas to determine the asymptotics of the correlation of defects that are macroscopically separated from one another and feel the influence of the boundary in several of the treated situations this seems not to be accomplishable by previous methods our applications include the case of two long neutral strings which turn out to interact by an analog of the casimir force two families of neutral doublets that turn out to interact completely independently of one another a neutral doublet and a very long neutral string a general collection of macroscopically separated monomer and separation defects and the case of long strings consisting of consecutive monomers

dimers on the kagome lattice ii correlations and the grassmannian approach in this paper we continue our consideration of closed packed dimers on the kagome lattice using the pfaffian approach we evaluate the correlation between dimers on two lattice edges it is found that the correlation is extremely short ranged in the case of symmetric dimers weights explicit expressions for the nonvanishing correlations are obtained for two dimers in the interior of a large lattice we also describe a grassmannian functional integral approach and use it to evaluate the dimer generating function and correlation functions

a note on dimer models and mckay quivers we give one formulation of a procedure of hanany and vegh j high energy phys 0710 029 35 2007 which takes a lattice polygon as an input and produces a set of isoradial dimer models we study the case of lattice triangles in detail and discuss the relation with coamoebas following feng et al adv theor math phys 12 3 489 545 2008

dimers on two types of lattices on the klein bottle the problem of enumerating close packed dimers or perfect matchings on two types of lattices the so called 8 8 4 and 8 8 6 lattices embedded on the klein bottle is considered and we obtain the explicit expression of the number of close packed dimers and entropy our results imply that 8 8 4 lattices have the same entropy under three different boundary conditions cylindrical toroidal and klein bottle and 8 8 6 lattices have the same property this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

loop statistics in the toroidal honeycomb dimer model the dimer model on a graph embedded in the torus can be interpreted as a collection of random self avoiding loops in this paper we consider the uniform toroidal honeycomb dimer model we prove that when the mesh of the graph tends to zero and the aspect of the torus is fixed the winding number of the collection of loops converges in law to a two dimensional discrete gaussian distribution this is known to physicists in more generality from their analysis of toroidal two dimensional critical loop models and their mapping to the massless free field on the torus this paper contains the first mathematical proof of this more general physics result in the specific case of the loop model induced by a toroidal dimer model

spanning trees on lattices and integral identities for a lattice with n vertices and dimension d equal to or higher than 2 the number of spanning trees nst increases asymptotically as exp nz in the thermodynamic limit we present exact integral expressions for the asymptotic growth constant z for spanning trees on several lattices by taking different unit cells in the calculation many integral identities can be obtained we also give z p on the homeomorphic expansion of k regular lattices with p vertices inserted on each edge

universal properties of harmonic functions on trees abstract we consider an infinite locally finite tree t equipped with nearest neighbor transition coefficients giving rise to a space of harmonic functions we show that except for trivial cases the generic harmonic function on t has dense range in c by looking at forward only transition coefficients we show that the generic harmonic function induces a boundary martingale that approximates in probability all measurable functions on the boundary of t we also study algebraic genericity spaceability and frequent universality of these phenomena

non existence of non trivial l algebras containing field structure in the sense of belohlavek and an open question r email protected lohlavek has proposed the notion of l algebra in r email protected lohlavek fuzzy relational systems foundations and principles kluwer academic plenum press new york 2002 and a series of papers in this direction has been written by him and his colleague r email protected lohlavek v vychodil algebras with fuzzy equalities in proceedings of the 10th ifsa world congress june 29 july 2 2003 pp 1 4 algebras with fuzzy equalities fuzzy sets and systems accepted for publication v vychodil direct limits and reduced products of algebras with fuzzy equalities submitted for publication in this short note it will be shown that if the ordinary part of an l algebra contains two binary operations forming the field structure then the underlying l equality of the l algebra is constant trivial this means that l algebras containing field structures in their ordinary parts correspond to ordinary algebras in a one to one manner so all results in r email protected lohlavek fuzzy relational systems foundations and principles kluwer academic plenum press new york 2002 r email protected lohlavek v vychodil algebras with fuzzy equalities in proceedings of the 10th ifsa world congress june 29 july 2 2003 pp 1 4 algebras with fuzzy equalities fuzzy sets and systems accepted for publication v vychodil direct limits and reduced products of algebras with fuzzy equalities submitted for publication in their setting for l algebras containing field structure become trivial in addition to this observation other aim of this note is to draw attention to the natural question does there exist any l algebra with an l equality different from trivial l equalities in case the ordinary part of the l algebra includes two binary operations that define group ring module or vector space structure this question is equivalent to the problem of whether the notion of l algebra provides a meaningful generalization of ring module or vector space structure in the context of fuzzy equality

tilings of non convex polygons skew young tableaux and determinantal processes this paper studies random lozenge tilings of general non convex polygonal regions we show that the pairwise interaction of the non convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations the precise geometrical figure here consists of a hexagon with cuts along opposite edges for this model we take limits when the size of the hexagon and the cuts tend to infinity while keeping certain geometric data fixed in order to guarantee sufficient interaction between the cuts in the limit we show in this paper that the kernel for the finite tiling model can be expressed as a multiple integral where the number of integrations is related to the fixed geometric data above the limiting kernel is believed to be a universal master kernel

construction of finite l groups abstract in this paper we give some necessary and sufficient conditions for a fuzzy relation to be a compatible fuzzy equality of some l group in the sense of b lohlavek these characterizations enable practical applications for construction of l groups with a given skeleton the construction has two steps determination of the general form of fuzzy pre equalities on a group and then solving a system of inequalities in the given residuated lattice some concrete examples of construction of finite l groups with the given skeleton are presented

partition function zeroes of a self dual ising model abstract we consider the ising model on an m n rectangular lattice with an asymmetric self dual boundary condition and derive a closed form expression for its partition function we show that zeroes of the partition function are given by the roots of a polynomial equation of degree 2m 1 which trace out certain loci in the complex temperature plane particularly it is shown that a real solutions of the polynomial equations always lead to zeroes on the unit circle and a segment of the negative real axis and b all temperature zeroes lie on two circles in the limit of m for any n closed form expressions of the loci as well as the density of zero distributions in the limit of n are derived for m 1 and 2 in addition we explain the reason of and establish the criterion for partition function zeroes of any self dual spin model to reside precisely on the unit circle this elucidates a recent finding in the case of the self dual potts model

the triangular kagome lattices revisited abstract the dimer problem ising spins and bond percolation on the triangular kagome lattice have been studied extensively by physicists in this paper based on the fact the triangular kagome lattice with toroidal boundary condition can be regarded as the line graph of 3 12 12 lattice with toroidal boundary condition we derive the formulae of the number of spanning trees the energy and the kirchhoff index of the triangular kagome lattice with toroidal boundary condition

asymptotic incidence energy of lattices abstract the energy of a graph g arising in chemical physics denoted by e g is defined as the sum of the absolute values of the eigenvalues of g as an analogue to e g the incidence energy ie g defined as the sum of the singular values of the incidence matrix of g is a much studied quantity with well known applications in chemical physics in this paper based on the results by yan and zhang 2009 we propose the incidence energy per vertex problem for lattice systems and present the closed form formulae expressing the incidence energy of the hexagonal lattice triangular lattice and 3 3 4 2 lattice respectively moreover we show that the incidence energy per vertex of lattices is independent of the toroidal cylindrical and free boundary conditions in particular the explicit asymptotic values of the incidence energy in these lattices are obtained by utilizing the applications of analysis approach with the help of calculational software

degenerations of amoebae and berkovich spaces we prove a continuity result for the fibers of the berkovich analytification of a complex algebraic variety with respect to the maximum of the archimedean norm and the trivial norm as a consequence we obtain generalizations of a result of mikhalkin and rullgard about degenerations of amoebae onto tropical varieties

the spectrum and laplacian spectrum of the dice lattice the dice lattice is the dual lattice of kagome lattice many physical properties on the dice lattice have been studied by physicists such as ising model glassy dynamics of josephson arrays and lattice green s function in this paper we derive the spectrum and laplacian spectrum of the dice lattice with toroidal boundary condition in addition we apply our results to obtain the formulae of the number of spanning trees the kirchhoff index and the energy of the dice lattice with toroidal boundary condition

dimers on the 33 42 lattice in this work we obtain explicit expression of the number of close packed dimers perfect matchings of the 33 42 lattice with cylindrical boundary condition particularly we show that the entropy of 33 42 lattice is the same for cylindrical and toroidal boundary conditions

dimer problem on the cylinder and torus institute of mathematics academia sinica nsfc 10771086 10671162 fujian province university nsc nsc96 2115 m 001 005

a variational principle for domino tilings 1 1 description of results a domino is a 1 x 2 or 2 x 1 rectangle and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps in 1961 kasteleyn kal found a formula for the number of domino tilings of an m x n rectangle with mn even as shown in figure 1 for rm n 68 temperley and fisher tf used a differenit method and arrived at the same result at almost exactly the same time both lines of calculation showed that the logarithm of number of tilings divided by the number of dominos in a tiling that is mn 2 converges to 2g 7r 0 58 here g is catalan s constant on the other hand in 1992 elkies et al eklp studied domino tilings of regions they called aztec diamonds figure 2 shows an aztec diamond of order 48 and showed that the logarithm of the number of tilings divided by the number of dominos converges to the smaller number log 2 2 0 35 thus even though the region in figure 1 has slightly smaller area than the region in figure 2 the former has far more domino tilings for regions with other shapes neither of these asymptotic formulas may apply in the present paper we consider simply connected regions of arbitrary shape we give an exact formula for the limiting value of the logarithm of the number of tilings per unit area as a function of the shape of the boundary of the region as the size of the region goes to infinity in particular we show that computation of this limit is intimately linked with an understanding of long range variations in the local statistics of random domino tilings such variations can be seen by comparing figures 1 and 2 each of the two tilings is random in the sense that the algorithm pwi that was used to create it generates each of the possible tilings of the region being tiled with the same probability hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

dimers on the kagome lattice i finite lattices abstract we report exact results on the enumeration of close packed dimers on a finite kagome lattice with general asymmetric dimer weights under periodic and cylindrical boundary conditions for symmetric dimer weights the resulting dimer generating functions reduce to very simple expressions and we show how the simple expressions can be obtained from the consideration of a spin variable mapping

dimers and amoebae we study random surfaces which arise as height functions of random perfect matchings a k a dimer configurations on a weighted bipartite doubly periodic graph g embedded in the plane we derive explicit formulas for the surface tension and local gibbs measure probabilities of these models the answers involve a certain plane algebraic curve which is the spectral curve of the kasteleyn operator of the graph for example the surface tension is the legendre dual of the ronkin function of the spectral curve the amoeba of the spectral curve represents the phase diagram of the dimer model further we prove that the spectral curve of a dimer model is always a real curve of special type namely it is a harnack curve this implies many qualitative and quantitative statement about the behavior of the dimer model such as existence of smooth phases decay rate of correlations growth rate of height function fluctuations etc

corner percolation on ℤ2 and the square root of 17 we consider a four vertex model introduced by balint toth a dependent bond percolation model on 2 in which every edge is present with probability 1 2 and each vertex has exactly two incident edges perpendicular to each other we prove that all components are finite cycles almost surely but the expected diameter of the cycle containing the origin is infinite moreover we derive the following critical exponents the tail probability diameter of the cycle of the origin n n and the expectation mathbb e length of a typical cycle with diameter n n with gamma 5 sqrt 17 4 0 219 ldots and delta sqrt 17 1 4 1 28 ldots the value of comes from a singular sixth order ode while the relation 3 2 corresponds to the fact that the scaling limit of the natural height function in the model is the additive brownian motion whose level sets have hausdorff dimension 3 2 we also include many open problems for example on the conformal invariance of certain linear entropy models

arctic curves in path models from the tangent method recently colomo and sportiello introduced a powerful method known as the tangent method for computing the arctic curve in statistical models which have a non or weakly intersecting lattice path formulation we apply the tangent method to compute arctic curves in various models the domino tiling of the aztec diamond for which we recover the celebrated arctic circle a model of dyck paths equivalent to the rhombus tiling of a half hexagon for which we find an arctic half ellipse another rhombus tiling model with an arctic parabola the vertically symmetric alternating sign matrices where we find the same arctic curve as for unconstrained alternating sign matrices the latter case involves lattice paths that are non intersecting but that are allowed to have osculating contact points for which the tangent method was argued to still apply for each problem we estimate the large size asymptotics of a certain one point function using lu decomposition of the corresponding gessel viennot matrices and a reformulation of the result amenable to asymptotic analysis

a clone based representation of the fuzzy tolerance or equivalence relations a strict order relation is compatible with we show that although there exists no non trivial fuzzy tolerance relation a partial order relation is compatible with in the sense of blohlvek the situation is quite different when considering its strict part more specifically we provide a representation of all fuzzy tolerance and in particular all fuzzy equivalence relations a strict order relation is compatible with to that end we introduce the notion of clone relation associated with a partially ordered set and discuss its basic properties the mentioned representation is intimately connected with this clone relation

coupling functions for domino tilings of aztec diamonds the inverse kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the system such as local statistics which can be used to compute local and global asymptotics in this paper we consider three different weightings of domino tilings of the aztec diamond and show using recurrence relations that we can compute the inverse kasteleyn matrix these weights are the one periodic weighting where the horizontal edges have one weight and the vertical edges have another weight the qvol weighting which corresponds to multiplying the product of tile weights by q if we add a box to the height function and the two periodic weighting which exhibits a flat region with defects in the center

mixing times of monotone surfaces and sos interfaces a mean curvature approach we consider stochastic spin flip dynamics for i monotone discrete surfaces in mathbb z 3 with planar boundary height and ii the one dimensional discrete solid on solid sos model confined to a box in both cases we show almost optimal bounds o l2polylog l for the mixing time of the chain where l is the natural size of the system the dynamics at a macroscopic scale should be described by a deterministic mean curvature motion such that each point of the surface feels a drift which tends to minimize the local surface tension spohn in j stat phys 71 1081 1132 1993 inspired by this heuristics our approach consists in bounding the dynamics with an auxiliary one which with very high probability follows quite closely the deterministic mean curvature evolution key technical ingredients are monotonicity coupling and an argument due to wilson ann appl probab 14 274 325 2004 in the framework of lozenge tiling markov chains our approach works equally well for both models despite the fact that their equilibrium maximal height fluctuations occur on very different scales log l for monotone surfaces and sqrt l for the sos model finally combining techniques from kinetically constrained spin systems cancrini et al in probab th rel fields 140 459 504 2008 together with the above mixing time result we prove an almost diffusive lower bound of order 1 l2polylog l for the spectral gap of the sos model with horizontal size l and unbounded heights

topologically robust zero sum games and pfaffian orientation how network topology determines the long time dynamics of the antisymmetric lotka volterra equation to explore how the topology of interaction networks determines the robustness of dynamical systems we study the antisymmetric lotka volterra equation alve the alve is the replicator equation of zero sum games in evolutionary game theory in which the strengths of pairwise interactions between strategies are defined by an antisymmetric matrix such that typically some strategies go extinct over time here we show that there also exist topologically robust zero sum games such as the rock paper scissors game for which all strategies coexist for all choices of interaction strengths we refer to such zero sum games as coexistence networks and construct coexistence networks with an arbitrary number of strategies by mapping the long time dynamics of the alve to the algebra of antisymmetric matrices we identify simple graph theoretical rules by which coexistence networks are constructed examples are triangulations of cycles characterized by the golden ratio phi 1 6180 cycles with complete subnetworks and non hamiltonian networks in graph theoretical terms we extend the concept of a pfaffian orientation from even sized to odd sized networks our results show that the topology of interaction networks alone can determine the long time behavior of nonlinear dynamical systems and may help to identify robust network motifs arising for example in ecology

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

graphs and matrices with maximal energy abstract given a complex m n matrix a we index its singular values as 1 a 2 a and call the value e a 1 a 2 a the energy of a thereby extending the concept of graph energy introduced by gutman koolen and moulton proved that e g n 2 1 n for any graph g of order n and exhibited an infinite family of graphs with e g v g 2 1 v g we prove that for all sufficiently large n there exists a graph g g n with e g n 3 2 2 n 11 10 this implies a conjecture of koolen and moulton we also characterize all square nonnegative matrices and all graphs with energy close to the maximal one in particular such graphs are quasi random

fluctuations of an atomic ledge bordering a crystalline facet when a high symmetry facet joins the rounded part of a crystal the step line density vanishes as square root of r with r denoting the distance from the facet edge this means that the ledge bordering the facet has a lot of space to meander as caused by thermal activation we investigate the statistical properties of the border ledge fluctuations in the scaling regime they turn out to be non gaussian and related to the edge statistics of gaussian unitary ensemble multimatrix models

log gases on quadratic lattices via discrete loop equations and q boxed plane partitions abstract we study a general class of log gas ensembles on shifted quadratic lattices we prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are gaussian with a universal covariance we apply our general results to analyze the asymptotic behavior of a q boxed plane partition model introduced by borodin gorin and rains in particular we show that the global fluctuations of the height function on a fixed slice are described by a one dimensional section of a pullback of the two dimensional gaussian free field our approach is based on a q analogue of the schwinger dyson or loop equations which originate in the work of nekrasov and his collaborators and extends the methods developed by borodin gorin and guionnet to quadratic lattices

frequently hypercyclic operators we investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators t on separable complex f spaces t is frequently hypercyclic if there exists a vector x such that for every nonempty open subset u of x the set of integers n such that t n x belongs to u has positive lower density we give several criteria for frequent hypercyclicity and this leads us in particular to study linear transformations from the point of view of ergodic theory several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting

algebras with fuzzy equalities an algebra with fuzzy equality is a set with operations on it that is equipped with similarity i e a fuzzy equivalence relation such that each operation f is compatible with described verbally compatibility says that each f yields similar results if applied to pairwise similar arguments on the one hand algebras with fuzzy equalities are structures for the equational fragment of fuzzy logic and have been studied from this point of view before on the other hand they are the formal counterpart to the intuitive idea of having functions that are not allowed to map similar objects to dissimilar ones the present paper aims at developing fundamental points of algebras with fuzzy equalities we introduce the notion of an algebra with fuzzy equality present natural examples compare the notion with other approaches and introduce and develop basic structural notions subalgebras morphisms products direct unions in a follow up paper vychodil direct limits and reduced products of algebras with fuzzy equalities submitted for publication we deal with advanced topics in algebras with fuzzy equalities direct limits reduced products

graph energy change due to edge deletion the energy of a graph is the sum of the singular values of its adjacency matrix we are interested in how the energy of a graph changes when edges are deleted examples show that all cases are possible increased decreased unchanged our goal is to find possible graph theoretical descriptions and to provide an infinite family of graphs for each case the main tool is a singular value inequality for complementary submatrices and its equality case

the energy of graphs and matrices abstract given a complex m n matrix a we index its singular values as 1 a 2 a and call the value e a 1 a 2 a the energy of a thereby extending the concept of graph energy introduced by gutman let 2 m n a be an m n nonnegative matrix with maximum entry and a 1 n extending previous results of koolen and moulton for graphs we prove that e a a 1 m n m 1 a 2 2 a 1 2 m n n m m 2 furthermore if a is any nonconstant matrix then e a 1 a a 2 2 1 2 a 2 a finally we note that wigner s semicircle law implies that e g 4 3 o 1 n 3 2 for almost all graphs g

a unified approach to the asymptotic topological indices of various lattices we present a unified approach to the asymptotic topological indices of various lattices we propose the topological indices per vertex problem for lattice systems the explicit asymptotic values of laplacian energies for various lattices are obtained we deduce the laplacian energies of many types of lattices are independent of various boundary conditions in this paper we present a unified approach to the asymptotic topological indices of various lattices moreover we propose the various topological indices per vertex problem for lattice systems and show that the various topological indices per vertex of lattices are independent of the toroidal cylindrical and free boundary conditions our result is a generalization of some earlier results

asymptotic laplacian energy like invariant of lattices let 1 2 n denote the laplacian eigenvalues of a graph g with n vertices the laplacian energy like invariant denoted by lel g i 1 n 1 i is a novel topological index in this paper we show that the laplacian energy like per vertex of various lattices is independent of the toroidal cylindrical and free boundary conditions simultaneously the explicit asymptotic values of the laplacian energy like in these lattices are obtained moreover our approach implies that in general the laplacian energy like per vertex of other lattices is independent of the boundary conditions

thermodynamic limit of the six vertex model with domain wall boundary conditions we address the question of the dependence of the bulk free energy on boundary conditions for the six vertex model here we compare the bulk free energy for periodic and domain wall boundary conditions using a determinant representation for the partition function with domain wall boundary conditions we derive toda differential equations and solve them asymptotically in order to extract the bulk free energy we find that it is different and bears no simple relation to the free energy for periodic boundary conditions the six vertex model with domain wall boundary conditions is closely related to algebraic combinatorics alternating sign matrices this implies new results for the weighted counting for large size alternating sign matrices finally we comment on the interpretation of our results in particular in connection with domino tilings dimers on a square lattice

an answer to demirci s open question a clarification of his result and a correction of his interpretation of the result in his note this issue of this journal demirci shows that fields with fuzzy equalities have only trivial fuzzy equalities concludes that therefore in case l algebras contain field structure all results in r belohlavek fuzzy relational systems foundations and principles kluwer academic plenum publishers new york 2002 r belohlavek v vychodil algebras with fuzzy equalities in proceedings of the 10th ifsa world congress june 29 july 2 2003 pp 1 4 r belohlavek v vychodil algebras with fuzzy equalities fuzzy sets and systems accepted v vychodil direct limits and reduced products of algebras with fuzzy eqalities submitted are evident from their classical counterparts and asks a question does there exist any l algebra with an l equality different from trivial l equalities in case the ordinary part of the l algebra includes two binary operations that define group ring module or vector space structure in our reply we show the following first by presenting examples of group based l algebras with non trivial l equalities we show that the answer to demirci s question is positive second we clarify the meaning of demirci s result and show that it is in fact a natural generalization of the well known classical result saying that ordinary fields do not have non trivial congruences third we argue that demirci s interpretation of his result is mistaken and that it is not true that all results in r belohlavek fuzzy relational systems foundations and principles kluwer academic plenum publishers new york 2002 r belohlavek v vychodil algebras with fuzzy equalities fuzzy sets and systems to appear r belohlavek v vychodil fuzzy equational logic springer berlin to appear s burris h p sankappanavar a course in universal algebra springer new york 1981 are evident from their classical counterparts

planar dimers and harnack curves in this article we study the connection between dimers and harnack curves discovered in 15 we prove that every harnack curve arises as a spectral curve of some dimer model we also prove that the space of harnack curves of given degree is homeomorphic to a closed octant and that the areas of the amoeba holes and the distances between the amoeba tentacles give these global coordinates we characterize harnack curves of genus zero as spectral curves of isoradial dimers and also as minimizers of the volume under their ronkin function with given boundary conditions

exact solution of close packed dimers on the kagome lattice it is well known that exact enumerations of close packed dimers can be carried out for two dimensional lattices while details of results are now known for most lattices due to the unique nature of the lattice structure there has been no complete analysis for the kagom e lattice here we derive the close form expression 1 3 mathrm ln 4xyz for the free energy of close packed dimers on the kagom e lattice where x y and z are dimer weights we use two different approaches the kasteleyn method of evaluating a pfaffian and an alternative vertex model formulation both methods lead to the same final expression the correlation function between two dimers at a distance equal or greater than two lattice spacings is found to vanish identically

permanents pfaffian orientations and even directed circuits given a 0 1 square matrix a when can some of the 1 s be changed toi1 s in such a way that the permanent of a equals the determinant of the modifled matrix when does a real square matrix have the property that every real matrix with the same sign pattern that is the corresponding entries either have the same sign or are both zero is nonsingular when is a hypergraph with n vertices and n hyperedges minimally nonbipartite when does a bipartite graph have a pfa an orientation given a digraph does it have no directed circuit of even length given a digraph does it have a subdivision with no even directed circuit it is known that all of the above problems are equivalent we prove a structural characterization of the feasible instances which implies a polynomial time algorithm to solve all of the above problems the structural characterization says roughly speaking that a bipartite graph has a pfa an orientation if and only if it can be obtained by piecing together in a specifled way planar bipartite graphs and one sporadic nonplanar bipartite graph

dynamics of 2 1 dimensional sos surfaces above a wall slow mixing induced by entropic repulsion we study the glauber dynamics for the 2 1 d solid on solid model above a hard wall and below a far away ceiling on an l times l box of z 2 with zero boundary conditions at large inverse temperature beta it was shown by bricmont el mellouki and frohlich 1986 that the floor constraint induces an entropic repulsion effect which lifts the surface to an average height h asymp 1 beta log l as an essential step in understanding the effect of entropic repulsion on the glauber dynamics we determine the equilibrium height h to within an additive constant h 1 4 beta log l o 1 we then show that starting from zero initial conditions the surface rises to its final height h through a sequence of metastable transitions between consecutive levels the time for a transition from height h ah a in 0 1 to height h 1 is roughly exp c l a for some constant c 0 in particular the mixing time of the dynamics is exponentially large in l i e t mix geq e c l we also provide the matching upper bound t mix leq e c l requiring a challenging analysis of the statistics of height contours at low temperature and new coupling ideas and techniques finally to emphasize the role of entropic repulsion we show that without a floor constraint at height zero the mixing time is no longer exponentially large in l