density of states potts zeros and fisher zeros of the q state potts model for continuous q the q state potts model can be extended to noninteger and even complex q by expressing the partition function in the fortuin kasteleyn f k representation in the f k representation the partition function z q a is a polynomial in q and v a ensuremath 1 a e ensuremath beta j and the coefficients of this polynomial ensuremath phi b c are the number of graphs on the lattice consisting of b bonds and c connected clusters we introduce the random cluster transfer matrix to compute ensuremath phi b c exactly on finite square lattices with several types of boundary conditions given the f k representation of the partition function we begin by studying the critical potts model z mathrm cp z q a c q where a c q 1 sqrt q we find a set of zeros in the complex w sqrt q plane that map to or close to the beraha numbers for real positive q we also identify q c l the value of q for a lattice of width l above which the locus of zeros in the complex p v sqrt q plane lies on the unit circle by finite size scaling we find that 1 q c l ensuremath rightarrow 0 as stackrel ensuremath rightarrow l ensuremath infty we then study zeros of the antiferromagnetic af potts model in the complex q plane and determine q c a the largest value of q for a fixed value of a below which there is af order we find excellent agreement with baxter s conjecture q c mathrm af a 1 ensuremath a a 3 we also investigate the locus of zeros of the ferromagnetic potts model in the complex q plane and confirm that q c mathrm fm a a ensuremath 1 2 we show that the edge singularity in the complex q plane approaches q c as q c l ensuremath sim q c al ensuremath y q and determine the scaling exponent y q for several values of q finally by finite size scaling of the fisher zeros near the antiferromagnetic critical point we determine the thermal exponent y t as a function of q in the range 2l ql 3 using data for lattices of size 3l ll 8 we find that y t is a smooth function of q and is well fitted by y t 1 au bu 2 c du where u ensuremath 2 ensuremath pi mathrm cos ensuremath 1 sqrt q 2 for q 3 we find y t ensuremath simeq 0 6 however if we include lattices up to l 12 we find y t ensuremath simeq 0 50 8 in rough agreement with a recent result of ferreira and sokal j stat phys 96 461 1999

exact partition functions for potts antiferromagnets on cyclic lattice strips abstract we present exact calculations of the zero temperature partition function of the q state potts antiferromagnet on arbitrarily long strips of the square triangular and kagome lattices with width l y 2 or 3 vertices and with periodic longitudinal boundary conditions from these in the limit of infinite length we obtain the exact ground state entropy s 0 k b ln w these results are of interest since this model exhibits nonzero ground state entropy s 0 0 for sufficiently large q and hence is an exception to the third law of thermodynamics we also include results for homeomorphic expansions of the square lattice strip the analytic properties of w q are determined and related to zeros of the chromatic polynomial for long finite strips

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

partition function zeros of the one dimensional potts model the recursive method the yang lee fisher and potts zeros of the one dimensional q state potts model are studied using the theory of dynamical systems an exact recurrence relation for the partition function is derived it is shown that zeros of the partition function may be associated with neutral fixed points of the recurrence relation further a general equation for zeros of the partition function is found and a classification of the yang lee fisher and potts zeros is given it is shown that the fisher zeros in a nonzero magnetic field are located on several lines in the complex temperature plane and that the number of these lines depends on the value of the magnetic field analytical expressions for the densities of the yang lee fisher and potts zeros are derived it is shown that densities of all types of zeros of the partition function are singular at the edge singularity points with the same critical exponent

exact results for the zeros of the partition function of the potts model on finite lattices the yang lee zeros of the q state potts model are investigated in one two and three dimensions analytical results derived from the transfer matrix for the one dimensional model reveal a systematic behavior of the locus of zeros as a function of q for 1 2 they lie outside the unit circle for finite temperature in the special case q 2 the zeros lie exactly on the unit circle as proved by lee and yang in two and three dimensions the zeros are calculated numerically and behave in the same way results are also presented for the critical line of the potts model in an external field as determined from the zeros of the partition function in the complex temperature plane

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

transfer matrices and partition function zeros for antiferromagnetic potts models iii triangular lattice chromatic polynomial we study the chromatic polynomial pg q for m n triangular lattice strips of widths m 12p 9f with periodic or free transverse boundary conditions respectively and arbitrary lengths n with free longitudinal boundary conditions the chromatic polynomial gives the zero temperature limit of the partition function for the q state potts antiferromagnet we compute the transfer matrix for such strips in the fortuin kasteleyn representation and obtain the corresponding accumulation sets of chromatic zeros in the complex q plane in the limit n we recompute the limiting curve obtained by baxter in the thermodynamic limit m n and find new interesting features with possible physical consequences finally we analyze the isolated limiting points and their relation with the beraha numbers

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

numerical results for the three state critical potts model on finite rectangular lattices partition functions for the three state critical potts model on finite square lattices and for a variety of boundary conditions are presented the distribution of their zeros in the complex plane of the spectral variable is examined and is compared to the expected infinite lattice result the partition functions are then used to test the finite size scaling predictions of conformal and modular invariance

the stationary bootstrap abstract this article introduces a resampling procedure called the stationary bootstrap as a means of calculating standard errors of estimators and constructing confidence regions for parameters based on weakly dependent stationary observations previously a technique based on resampling blocks of consecutive observations was introduced to construct confidence intervals for a parameter of the m dimensional joint distribution of m consecutive observations where m is fixed this procedure has been generalized by constructing a blocks of blocks resampling scheme that yields asymptotically valid procedures even for a multivariate parameter of the whole i e infinite dimensional joint distribution of the stationary sequence of observations these methods share the construction of resampling blocks of observations to form a pseudo time series so that the statistic of interest may be recalculated based on the resampled data set but in the context of applying this method to stationary data it is natural

the exact location of partition function zeros a new method for statistical mechanics a new mathematical mechanism is proposed for locating exactly sections of the locus of the limiting distribution of partition function zeros for model systems in statistical mechanics illustrations and applications to the two dimensional ising model and the hard square lattice gas are given the exact location of the branch point singularity for the hard square lattice gas on the negative activity z axis is conjectured to be the negative root of the polynomial 2z6 9z5 42z4 90z3 96z2 27z 2

maximum norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter

statistical inference for max stable processes in space and time max stable processes have proved to be useful for the statistical modelling of spatial extremes several families of max stable random fields have been proposed in the literature one such representation is based on a limit of normalized and rescaled pointwise maxima of stationary gaussian processes that was first introduced by kabluchko and co workers this paper deals with statistical inference for max stable space time processes that are defined in an analogous fashion we describe pairwise likelihood estimation where the pairwise density of the process is used to estimate the model parameters for regular grid observations we prove strong consistency and asymptotic normality of the parameter estimates as the joint number of spatial locations and time points tends to furthermore we discuss extensions to irregularly spaced locations a simulation study shows that the method proposed works well for these models

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

transfer matrices and partition function zeros for antiferromagnetic potts models vi square lattice with extra vertex boundary conditions we study using transfer matrix methods the partition function zeros of the square lattice q state potts antiferromagnet at zero temperature square lattice chromatic polynomial for the boundary conditions that are obtained from an m n grid with free boundary conditions by adjoining one new vertex adjacent to all the sites in the leftmost column and a second new vertex adjacent to all the sites in the rightmost column we provide numerical evidence that the partition function zeros are becoming dense everywhere in the complex q plane outside the limiting curve mathcal b infty mathrm sq for this model with ordinary e g free or cylindrical boundary conditions despite this the infinite volume free energy is perfectly analytic in this region

maximum norm a posteriori error estimate for a 2d singularly perturbed semilinear reaction diffusion problem a singularly perturbed semilinear reaction diffusion equation posed in the unit square is discretized on arbitrary nonuniform tensor product meshes we establish a second order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter no mesh aspect ratio assumption is made numerical results are presented that support our theoretical estimate

the phase diagram of the o n model we investigate the properties of the zero field o n model on the honeycomb lattice outside the range of presently available exact solutions to this purpose we make use of finite size scaling and transfer matrix techniques our results confirm the expected phase diagram of the o n model in the range 0 n 2 it consists of a disordered phase separated by a second order transition from a low temperature phase with algebraically decaying correlations we confirm that in this low temperature phase the exponents are independent of the temperature more significantly we find evidence for this same universal behaviour even in the case n 2 this contrasts with the temperature dependent exponents of the generic xy model our numerical evidence in this case is supported by arguments based on a mapping from the o n 2 model to the ashkin teller model on the triangular lattice furthermore our results agree with the absence of a phase transition for n 2 for negative values of n we observe unphysical singularities in the free energies of finite systems

a simple approach to reliable and robust a posteriori error estimation for singularly perturbed problems abstract a simple flux reconstruction for finite element solutions of reaction diffusion problems is shown to yield fully computable upper bounds on the energy norm of error in an approximation of singularly perturbed reaction diffusion problem the flux reconstruction is based on simple independent post processing operations over patches of elements in conjunction with standard raviart thomas vector fields we prove that the corresponding local error indicators are locally efficient and robust up to a nonstandard oscillation term with respect to any mesh size and any size of the reaction coefficient including the singularly perturbed limit

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

series expansion of the percolation probability for the directed square lattice by extrapolation from finite lattices the authors extend the known series for the percolation probability on the directed square lattice from eight terms to 41 analysing the series they obtain the estimates qc 0 355 299 or 0 000 001 beta 0 2764 or 0 0001 for the critical probability and the critical exponent from this together with scaling relations and previous results on the moments of the pair connectedness function the authors conjecture that beta may be exactly 199 720

first order transition in the one dimensional three state potts model with long range interactions the first order phase transition in the three state potts model with long range interactions decaying as 1 r 1 ensuremath sigma has been examined by numerical simulations using the recently proposed luijten bl ote algorithm by applying scaling arguments to the interface free energy the binder s fourth order cumulant and the specific heat maximum the change in the character of the transition through variation of parameter ensuremath sigma was studied

the algebraic construction of partition function zeros universality and algebraic cycles the author considers the one parameter lattice models of classical statistical mechanics from a simple algebraic viewpoint the way in which the limiting locus of partition function zeros emerges through a sequence of semi infinite m infinity lattice sections is considered convergence to the limiting locus cinfinity is obtained through two sequences of algebraic curves for all arbitrarily large but finite m the partition function per site is a branch of an algebraic function lambda 1 defined by an irreducible polynomial and possesses only a finite number of algebraic singular points in the limit of m to infinity infinitely many branch points accumulate on cinfinity an algebraic basis to the universality hypothesis is presented in terms of the accumulation of branch points at the critical point it is argued that for many two dimensional lattice models the critical exponents alpha and nu will be of the form alpha 2 s and nu s 1 s where s is an integer or 3

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

two dimensional o n model in a staggered field nienhuis truncated o n model gives rise to a model of self avoiding loops on the hexagonal lattice each loop having a fugacity of n we study such loops subjected to a particular kind of staggered field w which for n has the geometrical effect of breaking the three phase coexistence linked to the three colourability of the lattice faces we show that at t 0 for w 1 the model flows to the ferromagnetic potts model with q n2 states with an associated fragmentation of the target space of the coulomb gas for t 0 there is a competition between t and w which gives rise to multicritical versions of the dense and dilute loop universality classes via an exact mapping and numerical results we establish that the latter two critical branches coincide with those found earlier in the o n model on the triangular lattice using transfer matrix studies we have found the renormalization group flows in the full phase diagram in the t w plane with fixed n superposing three copies of such hexagonal lattice loop models with staggered fields produces a variety of one or three species fully packed loop models on the triangular lattice with certain geometrical constraints possessing integer central charges 0 c 6 in particular we show that benjamini and schramm s rgb loops have fractal dimension df 3 2

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

an a posteriori error estimator for anisotropic refinement besides an algorithm for local refinement an a posteriori error estimator is the basic tool of every adaptive finite element method using information generated by such an error estimator the refinement of the grid is controlled for 2nd order elliptic problems we present an error estimator for anisotropically refined grids like n d cuboidal and 3 d prismatic grids that gives correct information about the size of the error additionally it generates information about the direction into which some element has to be refined to reduce the error in a proper way numerical examples are presented for 2 d rectangular and 3 d prismatic grids

long range one dimensional potts models a cluster mean field and extrapolation approach we obtain approximations for the critical temperature tc of q state ferromagnetic potts models on one dimensional lattices with algebraically decaying ferromagnetic pair interactions i e decaying as 1 r with 1 2 initially we use a cluster mean field method to get approximations to the critical temperature and we look at increasingly large clusters this gives us a sequence of ever more accurate approximate values which can be used as input to various extrapolation algorithms we see that as the interaction decreases more slowly and the value of q increases we obtain more and more accurate tc estimates our best estimates we believe to be correct to four figure accuracy

pointwise a posteriori error estimates for monotone semi linear equations we derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi linear equations the estimates hold for lagrange elements of any fixed order non smooth nonlinearities and take numerical integration into account the proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately and then applying the continuous maximum principle no geometric mesh constraints are thus required numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature

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

a numerical method to compute exactly the partition function with application to z n theories in two dimensions i present a new method to exactly compute the partition function of a class of discrete models in arbitrary dimensions the time for the computation for ann state model on anl d lattice scales like n l d 1 nl d i show examples of the use of this method by computing the partition function of the 2d ising and 3 state potts models for maximum lattice sizes 10 10 and 8 8 respectively the critical exponentsv and and the critical temperature one obtains from these are very near the exactly known values the distribution of zeros of the partition function of the potts model leads to the conjecture that the ratio of the amplitudes of the specific heat below and above the critical temperature is unity

towards estimating extremal serial dependence via the bootstrapped extremogram abstract davis and mikosch 2009a introduced the extremogram as a flexible quantitative tool for measuring various types of extremal dependence in a stationary time series there we showed some standard statistical properties of the sample extremogram a major difficulty was the construction of credible confidence bands for the extremogram in this paper we employ the stationary bootstrap to overcome this problem the use of the stationary bootstrap for the extremogram and the resulting interpretations are illustrated with several financial time series

locus of the tricritical transition in a two dimensional q state potts model a simple two dimensional potts model with vacancies is introduced the phase transition between paramagnetic and ferromagnetic is located by simple symmetry arguments it is shown that the transition has a continuous and a first order segment separated by a tricritical transition the locus of the tricritical point is determined by a transformation of the partition sum into that of an o n model at the tricritical point and at some of the critical points this potts model is soluble

critical behaviour of the 1d q state potts model with long range interactions the critical behaviour of the one dimensional q state potts model with long range interactions decaying with distance r as r 1 sigma has been studied in the wide range of parameters 0 sigma or 1 and 1 16 or q or 64 a transfer matrix has been constructed for a truncated range of interactions for integer and continuous q and finite range scaling has been applied results for the phase diagram and the correlation length critical exponent are presented

inference on the tail process with application to financial time series modeling abstract to draw inference on serial extremal dependence within heavy tailed markov chains drees et al 2015 proposed nonparametric estimators of the spectral tail process the methodology can be extended to the more general setting of a stationary regularly varying time series the large sample distribution of the estimators is derived via empirical process theory for cluster functionals the finite sample performance of these estimators is evaluated via monte carlo simulations moreover two different bootstrap schemes are employed which yield confidence intervals for the pre asymptotic spectral tail process the stationary bootstrap and the multiplier block bootstrap the estimators are applied to stock price data to study the persistence of positive and negative shocks

an a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes a new a posteriori residual error estimator is defined and rigorously analysed for anisotropic tetrahedral finite element meshes all considerations carry over to anisotropic triangular meshes with minor changes only the lower error bound is obtained by means of bubble functions and the corresponding anisotropic inverse inequalities in order to prove the upper error bound it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration to measure this correspondence a so called matching function is defined and its discussion shows it to be a useful tool with its help anisotropic interpolation estimates and subsequently the upper error bound are proven additionally it is pointed out how to treat robin boundary conditions in a posteriori error analysis on isotropic and anisotropic meshes a numerical example supports the anisotropic error analysis

robust a posteriori error estimators for a singularly perturbed reaction diffusion equation we derive robust a posteriori error estimators for a singularly perturbed reaction diffusion equation here robust means that the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds is bounded from below and from above by constants which do neither depend on any meshsize nor on the perturbation parameter the estimators are based either on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

finite size scaling of the density of zeros of the partition function in first and second order phase transitions the finite size scaling form for the density of zeros of the partition function in first and second order phase transitions is derived using the finite size scaling of the density of zeros the order of a phase transition can be easily determined and the order parameter calculated from finite size data we illustrate the scaling theory using exact values for the zeros of the partition function of the two dimensional ising model in the complex magnetic field plane

finite lattice extrapolation algorithms two algorithms for sequence extrapolation due to van den broeck and schwartz 1979 and bulirsch and stoer 1964 are reviewed and critically compared applications to three state and six state quantum chains and to the 2 1 d ising model show that the algorithm of bulirsch and stoer is superior in particular if only very few finite lattice data are available

ground state degeneracy of potts antiferromagnets homeomorphic classes with noncompact w boundaries abstract we present exact calculations of the zero temperature partition function z g q t 0 and ground state degeneracy w g q for the q state potts antiferromagnet on a number of families of graphs g for which generalizing q from z to c the boundary b of regions of analyticity of w in the complex q plane is noncompact passing through z 1 q 0 for these types of graphs since the reduced function w red q 1 w is nonanalytic at z 0 there is no large q taylor series expansion of w red the study of these graphs thus gives insight into the conditions for the validity of the large q expansions it is shown how such families of graphs can be generated from known families by homeomorphic expansion

a non parametric entropy based approach to detect changes in climate extremes summary the paper focuses primarily on temperature extremes measured at 24 european stations with at least 90 years of data here the term extremes refers to rare excesses of daily maxima and minima as mean temperatures in this region have been warming over the last century it is automatic that this positive shift can be detected also in extremes after removing this warming trend we focus on the question of determining whether other changes are still detectable in such extreme events as we do not want to hypothesize any parametric form of such possible changes we propose a new non parametric estimator based on the kullback leibler divergence tailored for extreme events the properties of our estimator are studied theoretically and tested with a simulation study our approach is also applied to seasonal extremes of daily maxima and minima for our 24 selected stations

chromatic polynomials potts models and all that the q state potts model can be defined on an arbitrary finite graph and its partition function encodes much important information about that graph including its chromatic polynomial flow polynomial and reliability polynomial the complex zeros of the potts partition function are of interest both to statistical mechanicians and to combinatorists i give a pedagogical introduction to all these problems and then sketch two recent results a construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q plane except possibly for the disc q 1 1 b proof of a universal upper bound on the q plane zeros of the chromatic polynomial or antiferromagnetic potts model partition function in terms of the graph s maximum degree

zeros of the partition function using theorems of ruelle theorems of ruelle which provide a technique for finding regions of the relevant complex planes free of zeros of the partition function are used to study certain ising spin systems of particular interest is the antiferromagnetic triangle lattice system with h 0 and systems having three body interactions

integrability versus non integrability hard hexagons and hard squares compared in this paper we compare the integrable hard hexagon model with the nonintegrable hard squares model by means of partition function roots and transfer matrix eigenvalues we consider partition functions for toroidal cylindrical and free free boundary conditions up to sizes 40 40 and transfer matrices up to 30 sites for all boundary conditions the hard squares roots are seen to lie in a bounded area of the complex fugacity plane along with the universal hard core line segment on the negative real fugacity axis the density of roots on this line segment matches the derivative of the phase difference between the eigenvalues of largest and equal moduli and exhibits much greater structure than the corresponding density of hard hexagons we also study the special point z 1 of hard squares where all eigenvalues have unit modulus and we give several conjectures for the value at z 1 of the partition functions

maximum norm error analysis of a 2d singularly perturbed semilinear reaction diffusion problem a semilinear reaction diffusion equation with multiple solutions is considered in a smooth two dimensional domain its diffusion parameter e 2 is arbitrarily small which induces boundary layers constructing discrete sub and super solutions we prove existence and investigate the accuracy of multiple discrete solutions on layer adapted meshes of bakhvalov and shishkin types it is shown that one gets second order convergence with in the case of the shishkin mesh a logarithmic factor in the discrete maximum norm uniformly in e for 0 is the maximum side length of mesh elements while the number of mesh nodes does not exceed ch 2 numerical experiments are performed to support the theoretical results

the cross quantilogram measuring quantile dependence and testing directional predictability between time series this paper proposes the cross quantilogram to measure the quantile dependence between two time series we apply it to test the hypothesis that one time series has no directional predictability to another time series we establish the asymptotic distribution of the cross quantilogram and the corresponding test statistic the limiting distributions depend on nuisance parameters to construct consistent confidence intervals we employ the stationary bootstrap procedure we show the consistency of this bootstrap also we consider the self normalized approach which is shown to be asymptotically pivotal under the null hypothesis of no predictability we provide simulation studies and two empirical applications first we use the cross quantilogram to detect predictability from stock variance to excess stock return compared to existing tools used in the literature of stock return predictability our method provides a more complete relationship between a predictor and stock return second we investigate the systemic risk of individual financial institutions such as jp morgan chase goldman sachs and aig this article has supplementary materials online

partition function zeros of the q state potts model for non integer q abstract the distribution of the zeros of the partition function in the complex temperature plane fisher zeros of the two dimensional q state potts model is studied for non integer q on l l self dual lattices studied l 8 no fisher zero lies on the unit circle p0 ei in the complex p e j 1 q plane for q 1 and the number of such zeros increases with increasing q the ferromagnetic and antiferromagnetic properties of the potts model are investigated using the distribution of the fisher zeros for the potts ferromagnet we verify the den nijs formula for the thermal exponent yt for the potts antiferromagnet we also verify the baxter conjecture for the critical temperature and present new results for the thermal exponents in the range 0

complex q zeros of the partition function of the potts model with long range interactions abstract the zeros of the partition function of the ferromagnetic q state potts model with long range interactions in the complex q plane are studied in the mean field case while preliminary numerical results are reported for the finite 1 d chains with power law decaying interactions in both cases at any fixed temperature the zeros lie on the arc shaped contours which cross the positive real axis at the value for which the given temperature is transition temperature for finite number of spins the positive real axis is free of zeros which approach to it in the thermodynamic limit the convergence exponent of the zero closest to the positive real q axis is found to have the same value as the temperature critical exponent 1

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 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

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 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

spanning forests and the q state potts model in the limit q 0 we study the q state potts model with nearest neighbor coupling v e j 1 in the limit q v 0 with the ratio w v q held fixed combinatorially this limit gives rise to the generating polynomial of spanning forests physically it provides information about the potts model phase diagram in the neighborhood of q v 0 0 we have studied this model on the square and triangular lattices using a transfer matrix approach at both real and complex values of w for both lattices we have computed the symbolic transfer matrices for cylindrical strips of widths 2 l 10 as well as the limiting curves b of partition function zeros in the complex w plane for real w we find two distinct phases separated by a transition point w w0 where w0 1 4 resp w0 0 1753 0 0002 for the square resp triangular lattice for w w0 we find a non critical disordered phase that is compatible with the predicted asymptotic freedom as w for w w0 our results are compatible with a massless berker kadanoff phase with central charge c 2 and leading thermal scaling dimension xt 1 2 marginally irrelevant operator at w w0 we find a first order critical point the first derivative of the free energy is discontinuous at w0 while the correlation length diverges as w w0 and is infinite at w w0 the critical ehavior at w w0 seems to be the same for both lattices and it differs from that of the berker kadanoff phase our results suggest that the central charge is c 1 the leading thermal scaling dimension is xt 1 0 and the critical exponents are 1 d 1 2 and 1

ground state degeneracy of potts antiferromagnets cases with noncompact w boundaries having multiple points at we present exact calculations of the zero temperature partition function and ground state degeneracy per site for the q state potts antiferromagnet on a number of families of graphs for which the boundary of regions of analyticity of w in the complex q plane is noncompact and has the properties that i in the plane the point z 0 is a multiple point on and ii includes support for these families are generated by the method of homeomorphic expansion our results give further insight into the conditions for the validity of large q series expansions for the reduced function

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

zeroes of the jones polynomial abstract we study the distribution of zeroes of the jones polynomial v k t for a knot k we have computed numerically the roots of the jones polynomial for all prime knots with n 10 crossings and found the zeroes scattered about the unit circle t 1 with the average distance to the circle approaching a nonzero value as n increases for torus knots of the type m n we show that all zeroes lie on the unit circle with a uniform density in the limit of either m or n a fact confirmed by our numerical findings we have also elucidated the relation connecting the jones polynomial with the potts model and used this relation to derive the jones polynomial for a repeating chain knot with 3 n crossings for general n it is found that zeroes of its jones polynomial lie on three closed curves centered about the points 1 i and i in addition there are two isolated zeroes located one each near the points t e 2 i 3 at a distance of the order of 3 n 2 2 closed form expressions are deduced for the closed curves in the limit of n

lower a posteriori error estimates on anisotropic meshes lower a posteriori error bounds obtained using the standard bubble function approach are reviewed in the context of anisotropic meshes a numerical example is given that clearly demonstrates that the short edge jump residual terms in such bounds are not sharp hence for linear finite element approximations of the laplace equation in polygonal domains a new approach is employed to obtain essentially sharper lower a posteriori error bounds and thus to show that the upper error estimator in the recent paper kopteva in numer math 137 607 642 2017 is efficient on partially structured anisotropic meshes

linear finite elements may be only first order pointwise accurate on anisotropic triangulations we give a counterexample of an anisotropic triangulation on which the exact solution has a second order error of linear interpolation while the computed solution obtained using linear finite elements is only first order pointwise accurate our example is given in the context of a singularly perturbed reaction diffusion equation whose exact solution exhibits a sharp boundary layer furthermore we give a theoretical justification of the observed numerical phenomena using a finite difference representation of the considered finite element methods both standard and lumped mass cases are addressed

edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes summary both for the h 1 and l 2 norms we prove that up to higher order perturbation terms edge residuals yield global upper and local lower bounds on the error of linear finite element methods on anisotropic triangular or tetrahedral meshes we also show that with a correct scaling edge residuals yield a robust error estimator for a singularly perturbed reaction diffusion equation

new anisotropic a priori error estimates we prove a priori anisotropic estimates for the l 2 and h 1 interpolation error on linear finite elements the full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions this new h 1 error estimate does not require the maximal angle condition the analysis has been carried out for the 2d case but may be extended to three dimensions numerical experiments have been carried out to test our theoretical results

energy norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the energy norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter to deal with anisotropic triangulations a special quasi interpolation operator is employed that may be of independent interest

optimized energy calculation in lattice systems with long range interactions we discuss an efficient approach to the calculation of the internal energy in numerical simulations of spin systems with long range interactions although since the introduction of the luijten blote algorithm monte carlo simulations of these systems no longer pose a fundamental problem the energy calculation is still an o n 2 problem for systems of size n we show how this can be reduced to an o n log n problem with a break even point that is already reached for very small systems this allows the study of a variety of until now hardly accessible physical aspects of these systems in particular we combine the optimized energy calcula tion with histogram interpolation methods to investigate the specific heat of the ising model and the first order regime of the three state potts model with long range interactions

transfer matrices and partition function zeros for antiferromagnetic potts models i general theory and square lattice chromatic polynomial we study the chromatic polynomials zero temperature antiferromagnetic potts model partition functions p g q for m n rectangular subsets of the square lattice with m 8 free or periodic transverse boundary conditions and n arbitrary free longitudinal boundary conditions using a transfer matrix in the fortuin kasteleyn representation in particular we extract the limiting curves of partition function zeros when n which arise from the crossing in modulus of dominant eigenvalues beraha kahane weiss theorem we also provide evidence that the beraha numbers b2 b3 b4 b5 are limiting points of partition function zeros as n whenever the strip width m is 7 periodic transverse b c or 8 free transverse b c along the way we prove that a noninteger beraha number except perhaps b10 cannot be a chromatic root of any graph

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

maximum norm a posteriori error estimates for singularly perturbed elliptic reaction diffusion problems residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polyhedral domains standard finite element approximations are considered the error constants are independent of the diameters of mesh elements and the small perturbation parameter in our analysis we employ sharp bounds on the green s function of the linearized differential operator numerical results are presented that support our theoretical findings

a structural break test for extremal dependence in β mixing random vectors summarywe derive a structural break test for extremal dependence in beta mixing possibly high dimensional random vectors with either asymptotically dependent or asymptotically independent components existing tests require serially independent observations with asymptotically dependent components to avoid estimating a long run variance we use self normalization which obviates the need to estimate the coefficient of tail dependence when components are asymptotically independent simulations show favourable empirical size and power of the test which we apply to s p 500 and dax log returns we find evidence for one break in the coefficient of tail dependence for the upper and lower joint tail at the beginning of the 2007 08 financial crisis leading to more extremal co movement

families of graphs with chromatic zeros lying on circles we define an infinite set of families of graphs which we call p wheels and denote wh p n that generalize the wheel p 1 and biwheel p 2 graphs the chromatic polynomial for wh p n is calculated and remarkably simple properties of the chromatic zeros are found i the real zeros occur at q 0 1 p 1 for n p even and q 0 1 p 2 for n p odd and ii the complex zeros all lie equally spaced on the unit circle q p 1 1 in the complex q plane in the n to infty limit the zeros on this circle merge to form a boundary curve separating two regions where the limiting function w wh p q is analytic viz the exterior and interior of the above circle connections with statistical mechanics are noted