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

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

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

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

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

exact potts model partition functions for strips of the triangular lattice we present exact calculations of the potts model partition function z g q v for arbitrary q and temperature like variable v on n vertex strip graphs g of the triangular lattice for a variety of transverse widths equal to l vertices and for arbitrarily great length equal to m vertices with free longitudinal boundary conditions and free and periodic transverse boundary conditions these partition functions have the form z g q v sum j 1 n z g lambda c z g j z g j m 1 we give general formulas for n z g j and its specialization to v 1 for arbitrary l the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed it is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the potts model on the infinite triangular lattice considering the full generalization to arbitrary complex q and v we determine the singular locus mathcal b arising as the accumulation set of partition function zeros as m in the q plane for fixed v and in the v plane for fixed q explicit results for partition functions are given in the text for l 3 free and l 3 4 cylindrical and plots of partition function zeros and their asymptotic accumulation sets are given for l up to 5 a new estimate for the phase transition temperature of the q 3 potts antiferromagnet on the 2d triangular lattice is given

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

shank s transformation revisited abstract a unified and self contained approach to the block structure of shank s table and its cross rules is presented wynn s regular and cordellier s singular cross rules are derived by the schur complement method in a unified manner without appealing to pade approximation moreover by extending the definition of shank s transformation to certain biinfinite sequences and by introducing a parameter it is possible to get more consistency with respect to mobius transformations it is well known that pade approximants in general don t have this property

free vibration analysis of symmetrically laminated thin composite plates by using discrete singular convolution dsc approach algorithm and verification this study presents a detailed procedure for the implementation of a discrete singular convolution dsc approach to the free vibration analysis of composite plates based on classical laminated plate theory clpt the approach performs a numerical solution of differential equation of motion by using a grid discretization based on distribution theory and wavelets in the paper firstly computational algorithm of the dsc method is presented then the accuracy of the computer code developed is verified by comparing dsc solutions with the exact results of simply supported isotropic thin beams fully simply supported one layer isotropic and specially orthotropic plates and also some symmetrically laminated thin composite plates orientated to become specially orthotropic besides dsc predictions for laminated composite plates with different boundary conditions and ply numbers for which there is no analytical solution are compared with those of several distinguished works available in the literature it is noteworthy that dsc results completely match with the exact solutions and are in perfect agreement with those of compared studies

transfer matrices for the partition function of the potts model on cyclic and mobius lattice strips abstract we present a method for calculating transfer matrices for the q state potts model partition functions z g q v for arbitrary q and temperature variable v on cyclic and mobius strip graphs g of the square sq triangular tri and honeycomb hc lattices of width l y vertices and of arbitrarily great length l x vertices for the cyclic case we express the partition function as z l y l x q v d 0 l y c d tr t z l y d m where denotes lattice type c d are specified polynomials of degree d in q t z l y d is the transfer matrix in the degree d subspace and m l x l x 2 for sq tri hc respectively an analogous formula is given for mobius strips we exhibit a method for calculating t z l y d for arbitrary l y explicit results for arbitrary l y are given for t z l y d with d l y and l y 1 in particular we find very simple formulas the determinant det t z l y d and trace tr t z l y corresponding results are given for the equivalent tutte polynomials for these lattice strips and illustrative examples are included we also present formulas for self dual cyclic strips of the square lattice

general structural results for potts model partition functions on lattice strips abstract we present a set of general results on structural features of the q state potts model partition function z g q v for arbitrary q and temperature boltzmann variable v for various lattice strips of arbitrarily great width ly vertices and length lx vertices including i cyclic and mobius strips of the square and triangular lattices and ii self dual cyclic strips of the square lattice we also present an exact solution for the chromatic polynomial for the cyclic and mobius strips of the square lattice with width ly 5 the greatest width for which an exact solution has been obtained so far for these families in the lx limit we calculate the ground state degeneracy per site w q and determine the boundary b across which w q is singular in the complex q plane

partition function zeros of a restricted potts model on lattice strips and effects of boundary conditions we calculate exactly the partition function z g q v of the q state potts model with temperature like boltzmann variable v for strip graphs g of the square and triangular lattices of various widths ly and arbitrarily great lengths lx with a variety of boundary conditions and with q and v restricted to satisfy conditions corresponding to the ferromagnetic phase transition on the associated two dimensional lattices from these calculations in the limit lx we determine the continuous accumulation loci of the partition function zeros in the v and q planes strips of the honeycomb lattice are also considered we discuss some general features of these loci

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

density of yang lee zeros for the ising ferromagnet the densities of yang lee zeros for the ising ferromagnet on the l x l square lattice are evaluated from the exact grand partition functions l 3 approximately 16 the properties of the density of yang lee zeros are discussed as a function of temperature t and system size l the three different classes of phase transitions for the ising ferromagnet first order phase transition second order phase transition and yang lee edge singularity are clearly distinguished by estimating the magnetic scaling exponent yh from the densities of zeros for finite size systems the divergence of the density of zeros at yang lee edge in high temperatures yang lee edge singularity which has been detected only by the series expansion until now for the square lattice ising ferromagnet is obtained from the finite size data the identification of the orders of phase transitions in small systems is also discussed using the density of yang lee zeros

vibratory characteristics of general laminates i symmetric trapezoids abstract this paper presents part i of investigations of free flexural vibration of multi layered symmetric and unsymmetric composite laminates with symmetric trapezoidal planform of arbitrary combinations of edge conditions the ritz procedure withpb 2shape functions as the admissible functions is used to arrive at the governing eivenvalue equation the set ofpb 2shape functions is intrinsically a product of a class of orthogonally generated two dimensional polynomial functions and a basic function which ensures the satisfaction of geometric boundary conditions at the outset different boundary conditions with various combinations of in plane constraints are considered the effects of wide ranges of chord ratios angles of fibre orientation and stacking sequences on the vibration response are investigated with the aim of increasing the existing database sets of previously unavailable vibration frequencies and mode shapes are presented as benchmarks for future reference

free vibration analysis of orthotropic rectangular plates with variable thickness and general boundary conditions a discrete method is developed for analyzing the free vibration problem of orthotropic rectangular plates with variable thickness the green function which is obtained by transforming the differential equations into integral equations and using numerical integration is used to establish the characteristic equation of the free vibration the effects of the aspect ratios boundary conditions the variation of the thickness on the frequencies are considered by comparing the numerical results obtained by the present method with the those previously published the efficiency and accuracy of the present method are investigated the frequency parameters are obtained for the orthotropic plates with general boundary conditions and variable thickness in one or two directions the model shapes are given for some of the square plates with three kinds of thickness variations

transfer matrices for the partition function of the potts model on lattice strips with toroidal and klein bottle boundary conditions abstract we present a method for calculating transfer matrices for the q state potts model partition functions z g q v for arbitrary q and temperature variable v on strip graphs g of the square sq triangular tri and honeycomb hc lattices of width l y vertices and of arbitrarily great length l x vertices subject to toroidal and klein bottle boundary conditions for the toroidal case we express the partition function as z l y l x q v d 0 l y j b j d z l y d j m where denotes lattice type b j d are specified polynomials of degree d in q z l y d j are eigenvalues of the transfer matrix t z l y d in the degree d subspace and m l x l x 2 for sq tri hc respectively an analogous formula is given for klein bottle strips we exhibit a method for calculating t z l y d for arbitrary l y in particular we find some very simple formulas for the determinant det t z l y d and trace tr t z l y illustrative examples of our general results are given including new calculations of transfer matrices for potts model partition functions on strips of the square triangular and honeycomb lattices with toroidal or klein bottle boundary conditions

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

a new proof of the cross rule for the e algorithm based on schur complements abstract matrices of the form h ge 1 f are called schur complements they are closely related to elimination procedures and to determinantal identities in this paper the quotient property for schur complements is used to establish the schur complement method a useful tool for proving certain formulas as an example a new proof of the cross rule for the algorithm is presented

recursive interpolation extrapolation and projection abstract the recursive projection algorithm derived in a previous paper is related to several well known methods of numerical analysis such as the conjugate gradient method rosen s method and henrici s it is connected with the general interpolation problem with extrapolation methods with orthogonal projection on a subspace and with fourier expansions several other connections and applications are presented

free vibration analysis of symmetric laminated skew plates by discrete singular convolution technique based on first order shear deformation theory in this study free vibration of laminated skew plates was investigated discrete singular convolution dsc method is used for numerical solution of vibration problems the straight sided quadrilateral domain is mapped into a square domain in the computational space using a four node element by using the geometric transformation typical results are presented for different geometric parameters and boundary conditions it is concluded from the results that the skew angle have considerable influence on the variations of the frequencies with fibre orientation angle and number of layers in the laminate the results obtained by dsc method are compared with those obtained by analytical and numerical approaches it is shown that reasonable accurate results are obtained present work also indicates that the method of dsc is a promising and potential approach for computational mechanics copyright 2009 john wiley sons ltd

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

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

t 0 partition functions for potts antiferromagnets on square lattice strips with twisted periodic boundary conditions we present exact calculations of the zero temperature partition function for the q state potts antiferromagnet or equivalently the chromatic polynomial for two families of arbitrarily long strip graphs of the square lattice with periodic boundary conditions in the transverse direction and i periodic and ii twisted periodic boundary conditions in the longitudinal direction so that the strip graphs are embedded on i a torus and ii a klein bottle in the limit of infinite length we calculate the exponent of the entropy w q show it to be the same for i and ii and determine its analytic structure

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

a mapping relating complex and physical temperatures in the 2d q state potts model and some applications we show an exact equivalence of the free energy of the q state potts antiferromagnet on a lattice for the full temperature interval and the free energy of the q state potts model on the dual lattice for a semi infinite interval of complex temperatures cts this implies the existence of two quite different types of ct singularities the generic kind which does not obey universality or various scaling relations and a special kind which does obey such properties and encodes information of direct physical relevance we apply this observation to characterize ct properties of the potts model on several lattices to rule out two existing conjectures and to determine the critical value of q above which the potts antiferromagnet on the diced lattice has no phase transition

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

complex temperature partition function zeros of the potts model on the honeycomb and kagome lattices we calculate complex temperature ct zeros of the partition function for the q state potts model on the honeycomb and kagom e lattices for several values of q these give information on the ct phase diagrams a comparison of results obtained for different boundary conditions and a discussion of some ct singularities are given among other results our findings show that the potts antiferromagnet with q 4 and q 5 on the kagom e lattice has no phase transition at either finite or zero temperature

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

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

zeros of the partition function for the triangular lattice three state potts model the authors obtain the zeros of the partition function for the isotropic triangular lattice three state potts model in a finite lattice the distribution exhibits new points on the real axis which are well fitted by an algebraic equation deduced from the inversion relation and the symmetries of the anisotropic model they identify a symmetry approximately relating all six transition points

statical and dynamical behaviour of thin fibre reinforced composite laminates with different shapes abstract based on the classical laminated plate theory a variational approach for the study of the statical and dynamical behaviour of arbitrary quadrilateral anisotropic plates with various boundary conditions is developed the analytical formulation uses the ritz method in conjunction with natural coordinates to express the geometry of general plates in a simple form the deflection of the plate is approximated by a set of beam characteristic orthogonal polynomials generated using the gram schmidt procedure the algorithm developed is quite general and can be used to study fibre reinforced composite laminates with symmetric lay ups which may have general anisotropy and any combinations of clamped simply supported and free edge support conditions various numerical applications are presented and some results are compared with existing values in the literature to demonstrate the accuracy and flexibility of the present method new results were also determined for plates with different geometrical shapes combinations of boundary conditions several stacking sequences and various angles of fibre orientation

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

on the frequency response of moderately thick simply supported rectangular plates with arbitrary lamination a here to fore unavailable analytical solution to the boundary value problem of free vibration response of arbitrarily laminated moderately thick rectangular plates subjected to an admissible boundary condition is presented a novel boundary continuous solution approach is employed to solve the highly coupled governing partial differential equations that arise from the implementation of the yang norris stavsky 1966 theory incorporating first order shear deformation and rotary and in plane inertias the numerical results presented for various parametric effects to study first lowest seven eigenvalues and corresponding eigenvectors can be capitalized as bench marks for a future comparison an eight node isoparametrics shear flexible element is considered to validate its performance with respect to the present solutions

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

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

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

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

study of the potts model on the honeycomb and triangular lattices low temperature series and partition function zeros we present and analyse low temperature series and complex temperature partition function zeros for the q state potts model with q 4 on the honeycomb lattice and q 3 4 on the triangular lattice a discussion is given on how the locations of the singularities obtained from the series analysis correlate with the complex temperature phase boundary extending our earlier work we include a similar discussion for the potts model with q 3 on the honeycomb lattice and with q 3 4 on the kagom lattice

determinantal identities gauss schur cauchy sylvester kronecker jacobi binet laplace muir and cayley abstract we give a common concise derivation of some important determinantal identities attributed to the mathematicians in the title we also give a formal treatment of determinantal identities of the minors of a matrix

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

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