tricriticality and the blume capel model a monte carlo study within the microcanonical ensemble the microcanonical partition function ensuremath omega ensuremath alpha n e of a three dimensional ferromagnetic simple cubic blume capel model is calculated for several coupling ratios ensuremath alpha close to and on the first order side of the tricritical point to this end a single monte carlo simulation of a suitably extended partition function for systems of l ifmmode times else texttimes fi l ifmmode times else texttimes fi l spins with l ensuremath in 8 10 12 14 18 was performed a finite system analog of the latent heat is introduced in order to define a tricritical point in finite systems as well an empirical scaling of its coordinates to the thermodynamic limit yields ensuremath alpha t 2 84479 ifmmode pm else textpm fi 0 00030 for the tricritical coupling ratio and k b t t j 1 4182 ifmmode pm else textpm fi 0 0055 for the tricritical transition temperature the results are compared with values obtained on fcc lattices

rayleigh ritz based substructure synthesis for flexible multibody systems this paper is concerned with the modeling of flexible multibody systems by a rayleigh ritz based substructure synthesis method so that certain advantages can be accrued by using the variational approach to derive the eigenvalue problem as with the classical rayleigh ritz method if the admissible functions used to represent the motion of the substructures are not chosen properly convergence can suffer this paper presents a new substructure synthesis method with superior convergence characteristics achieved by representing the motion by means of a recently developed class of functions namely the class of quasi compari son functions this improved convergence is shown to be related to improved approximation of both the differential equations and the natural boundary conditions the theory is demonstrated by means of a numerical example i introduction t his investigation is concerned with the modeling of flexible multibody systems many structures such as fixedwing aircraft helicopters flexible spacecraft flexible robots etc can be modeled as assemblages of interacting flexible bodies a method proposed by hurty1 2 in the early 1960s and known as component mode synthesis consists of modeling the motion of the individual substructures referred to as components by means of component modes he uses three types of modes to represent the motion of a substructure rigid body modes constraint modes and normal modes to define constraint modes hurty divides the constraints into two classes statically determinate and redundant the constraint modes are equal in number to the number of redundant constraints and are defined by producing a unit displacement on each redundant constraint in turn with all other constraints fixed the normal modes represent the modes of vibration of the components with all constraints fixed an approach by craig and hampton3 differs from that of hurty 1 2 mainly in the selection of the component modes indeed craig and hampton suggest two types of component modes boundary modes providing for displacements and rotations at points along the substructure boundaries and related to the constraint modes of hurty and substructure modes corresponding to completely restrained boundaries the substructures modeled individually in refs 1 3 are made to act together as a single structure by eliminating the redundant generalized coordinates arising from the fact that displacements at points common to two adjacent substructures are included twice in the overall problem formulation once for each substructure the elimination process is based on the use of constraint equations resulting from the enforcement of compatibility conditions which amounts to saying that the displacement of a boundary point shared by two substructures is the same in another approach to the problem proposed by

new construction of solvable lattice models including an ising model in a field in this letter we report a new construction to obtain restricted solid on solid rsos models out of loop models the method is a generalization of ideas developed by owczarek and baxter and by pasquier in particular we consider a solvable o n model and point out that some of the rsos models thus obtained admit an off critical extension among these models we find a spin 1 ising model which is solvable not only at the critical point but also in a fieldlike deviation away from it we calculate the critical exponent ensuremath delta 15 directly from the relation between the free energy and the field this is the first determination of this exponent without the use of scaling relations

25th order high temperature expansion results for three dimensional ising like systems on the simple cubic lattice 25th order high temperature series are computed for a general nearest neighbor three dimensional ising model with arbitrary potential on the simple cubic lattice in particular we consider three improved potentials characterized by suppressed leading scaling corrections critical exponents are extracted from high temperature series specialized to improved potentials obtaining gamma 1 2373 2 nu 0 63012 16 alpha 0 1096 5 eta 0 03639 15 beta 0 32653 10 delta 4 7893 8 moreover biased analyses of the 25th order series of the standard ising model provide the estimate delta 0 52 3 for the exponent associated with the leading scaling corrections by the same technique we study the small magnetization expansion of the helmholtz free energy the results are then applied to the construction of parametric representations of the critical equation of state using a systematic approach based on a global stationarity condition accurate estimates of several universal amplitude ratios are also presented

reference motion in deformable bodies under rigid body motion and vibration part ii evaluation of the coefficient of restitution for impacts abstract the dynamic analysis of rigid body impacts is usually performed by using the coefficient of restitution as a measure of the mechanical energy lost in the process the coefficient of restitution provides an algebraic equation that allows impulse momentum balance equations to be solved this paper reports a method for calculating the kinematic coefficient of restitution for the impact of deformable bodies by using a numerical simulation procedure calculations were done within the framework of the floating reference frame approach as shown here for the first time discriminating between reference velocities and rigid body equivalent velocities is essential with a view to accurately calculating the kinematic coefficient of restitution thus the velocities at which the contact points approach and depart must be calculated as rigid body equivalent velocities if references velocities are used instead the resulting coefficient of restitution lacks physical significance the proposed method is illustrated with two applications namely an axial impact of a rigid body on a deformable rod and a transverse impact of a beam on a fixed stop

integrable boundaries conformal boundary conditions and a d e fusion rules the sl 2 minimal theories are classified by a lie algebra pair where g is of a d e type for these theories on a cylinder we propose a complete set of conformal boundary conditions labelled by the nodes of the tensor product graph the cylinder partition functions are given by fusion rules arising from the graph fusion algebra of we further conjecture that for each conformal boundary condition an integrable boundary condition exists as a solution of the boundary yang baxter equation for the associated lattice model the theory is illustrated using the or three state potts model

geometric properties of two dimensional critical and tricritical potts models we investigate geometric properties of the general q state potts model in two dimensions and define geometric clusters as sets of lattice sites in the same potts state connected by nearest neighbor bonds with variable probability p we find that besides the random cluster fixed point both the critical and the tricritical potts models have another fixed point in the p direction for the critical model the random cluster fixed point p r is unstable and the other point p g or p r is stable while p r is stable and p g or p r is unstable at tricriticality moreover we show that the fixed point p g of a critical and tricritical q state potts models can be regarded to correspond to p r of a tricritical and critical q state potts models respectively in terms of the coupling constant of the coulomb gas g these two models are related as gg 16 by means of monte carlo simulations we obtain p g 0 6227 2 and 0 6395 2 for the tricritical blume capel and the q 3 potts model respectively and confirm the predicted values of the magnetic and bond dilution exponents near p g

consistent superconformal boundary states we propose a supersymmetric generalization of cardy s equation for consistent n 1 superconformal boundary states we solve this equation for the superconformal minimal models p p 2 with p odd and thereby provide a classification of the possible superconformal boundary conditions in addition to the neveu schwarz ns and ramond boundary states there are states the ns and boundary states are related by a z2 spin reversal transformation we treat the tricritical ising model as an example and in an appendix we discuss the non superconformal case of the ising model

on the faber transform and efficient numerical rational approximation algorithms are presented for numerical type m n rational approximation m geqq n 1 which can be effectively applied either in general simply connected regions of the complex plane or as a special case on intervals of the real line in the former case particularly the methods are much more efficient than those previously proposed but they are also competitive even for intervals the approximations are based on the observation that the faber transform of a rational function is itself rational and are of two types a class of pade like approximants and approximations of rational caratheodory fejer type based on the singular value decomposition of a hankel matrix of faber coefficients which generalise those recently introduced for the unit disc by trefethen numer math 37 1981 pp 297 320 these latter particularly give rise to approximations which are sufficiently near to best for practical purposes

computations of the complex error function rational expansions for computing the complex error function w z e z 2 text erfc iz are presented these expansions have the following attractive properties 1 they can be evaluated using a polynomial evaluation routine such as homer s method 2 the polynomial coefficients can be computed once and for all by a single fast fourier transform fft and 3 high accuracy is achieved uniformly in the complex plane with only a small number of terms comparisons reveal that in some parts of the complex plane certain competitors may be more efficient however the difference in efficiency is never great and the new algorithms are simpler than existing ones a complete program takes eight lines of matlab code

complex chebyshev polynomials on circular sectors with degree six or less let tna denote the nth chebyshev polynomial on the circular sector sa z i z i this paper contains numerical values of ii tnaii o and the corresponding coefficients of t7 for n 1 1 6 and a 0 5 180 also all critical angles for t7 n 1 1 6 are listed where an angle is called critical when the number of absolute maxima of i t a changes at that angle all figures are given to six places the positions and hence the number of extremal points of t7a n 1 1 6 are presented graphically the method consists of a combination of semi infinite linear programming finite linear programming and newton s method

chebyshev series approximations for the bessel function y n z of complex argument we employ the truncated chebyshev series to approximate the bessel function of the second kind y n z for z 8 detailed manipulations and discussions for y 0 z and y 1 z are given results of numerical experiments are presents are presented to demonstrate the computed accuracy by using the chebyshev series approximations advantages and disadvantages of the chebyshev series approximation compared with other polynomial approximation methods e g the tau method approximations are discussed

monte carlo study of the ising model phase transition in terms of the percolation transition of physical clusters finite squarel l ising lattices with ferromagnetic nearest neighbor interaction are simulated using the swendsen wang cluster algorithm both thermal properties internal energyu specific heatc magnetization m susceptibility and percolation cluster properties relating to the physical clusters namely the fortuin kasteleyn clusters percolation probability p percolation susceptibility p cluster size distributionnl are evaluated paying particular attention to finite size effects it is shown that thermal properties can be expressed entirely in terms of cluster properties p being identical to m in the thermodynamic limit while finite size corrections differ in contrast p differs from even in the thermodynamic limit since a fluctuation in the size of the percolating net contributes to but not to p neartc the cluster size distribution has the scaling properties as hypothesized by earlier phenomenological theories we also present a generalization of the swendsen wang algorithm allowing one to cross over continuously to the glauber dynamics

red bond exponents of the critical and the tricritical ising model in three dimensions using the wolff and geometric cluster algorithms and finite size scaling analysis we investigate the critical ising and the tricritical blume capel models with nearest neighbor interactions on the simple cubic lattice the sampling procedure involves the decomposition of the ising configuration into geometric clusters each of which consists of a set of nearest neighboring spins of the same sign connected with bond probability p these clusters include the well known kasteleyn fortuin clusters as a special case for p 1 exp 2k where k is the ising spin spin coupling along the critical line k kc the size distribution of geometric clusters is investigated as a function of p we observe that unlike in the case of two dimensional tricriticality the percolation threshold in both models lies at pc 1 exp 2kc further we determine the corresponding red bond exponents as yr 0 757 2 and 0 501 5 for the critical ising and the tricritical blume capel models respectively on this basis we conjecture yr 1 2 for the latter model

green s functions in lossy layered media integration along the imaginary axis and asymptotic behavior this paper presents an efficient technique for evaluating green s functions associated to layered media when formulated as sommerfeld integrals in the space domain the key step in the formulation is that sommerfeld integrals are computed choosing a suitable integration path which is closed through the imaginary axis of the complex spectral plane it is shown that with this original choice of the integration contour the numerical effort usually involved in the evaluation of sommerfeld integrals can be greatly reduced specially when large source observer distances are involved one asset of this technique is that it can be easily incorporated into integral equation based cad packages for the efficient analysis of complex printed microwave circuit and antennas in addition the theoretical developments needed to set up the numerical algorithm throw a new light on the asymptotic behavior of the layered media green s functions for large source observer distances

a note on the tau method approximations for the bessel functions y0 z and y1 z abstract this paper is to complete and improve the work reported in 1 2 using the lanczos method in coleman s version to approximate the bessel functions y 0 z and y 1 z we introduce symbolic representations of the scaled faber polynomials on any fan shaped section of the complex plane these faber polynomials are used as the perturbation terms in the method numerical comparison among the power series the chebyshev series and the method are conducted to show the accuracy improvement achieved by this new version of the method some concluding remarks and suggestions on future research are given

discretely holomorphic parafermions and integrable loop models we define parafermionic observables in various lattice loop models including examples where no kramers wannier duality holds for a particular rhombic embedding of the lattice in the plane and a value of the parafermionic spin these variables are discretely holomorphic they satisfy a lattice version of the cauchy riemann equations as long as the boltzmann weights satisfy certain linear constraints in the cases considered the weights then also satisfy the critical yang baxter equations with the spectral parameter being related linearly to the angle of the elementary rhombus

critical points in coupled potts models and critical phases in coupled loop models we show how to couple two critical q state potts models to yield a new self dual critical point we also present strong evidence of a dense critical phase near this critical point when the potts models are defined in their completely packed loop representations in the continuum limit the new critical point is described by an su 2 coset conformal field theory while in this limit of the critical phase the two loop models decouple using a combination of exact results and numerics we also obtain the phase diagram in the presence of vacancies we generalize these results to coupling two potts models at different q

further exact solutions of the eight vertex sos model and generalizations of the rogers ramanujan identities the restricted eight vertex solid on solid sos model is an exactly solvable class of two dimensional lattice models to each sitei of the lattice there is associated an integer heightli restricted to the range 1 li r 1 the boltzmann weights of the model are expressed in terms of elliptic functions of period 2k and involve a parameter in an earlier paper we considered the case k r here we generalize those considerations to the case sk r s an integer relatively prime tor we are again led to generalizations of the rogers ramanujan identities

polynomial approximations in the complex plane abstract good polynomial approximations for analytic functions are potentially useful but are in short supply a new approach introduced here involves the lanczos method with perturbations proportional to faber or chebyshev polynomials for specific regions of the complex plane the results show that suitable forms of the method which are easy to use can produce near minimax polynomial approximations for functions which satisfy linear differential equations with polynomial coefficients in particular some accurate approximations of low degree for bessel functions are presented an appendix describes a simple algorithm which generates polynomial approximations for the bessel function j z of any given order

loop models for conformal field theories by interpreting the fusion matrix as an adjacency matrix we associate a loop model with every primary operator of a generic conformal field theory the weight of these loop models is given by the quantum dimension of the corresponding primary operator using the known results for the o n models we establish a relationship between these models and sles the method is applied to wzw c 1 minimal conformal field theories and other coset models

lattice derivation of modular invariant partition functions on the torus the author expresses the partition functions of the ade lattice models on the torus in terms of partition function of six vertex models and which can be computed in the continuum limit thus the author recovers the partition function of minimal models with c 1 and new ones with c 1

a hybrid arnoldi faber iterative method for nonsymmetric systems of linear equations we present here a new hybrid method for the iterative solution of large sparse nonsymmetric systems of linear equations say of the formax b wherea n n witha nonsingular andb n are given this hybrid method begins with a limited number of steps of the arnoldi method to obtain some information on the location of the spectrum ofa and then switches to a richardson iterative method based on faber polynomials for a polygonal domain the faber polynomials can be constructed recursively from the parameters in the schwarz christoffel mapping function in four specific numerical examples of non normal matrices we show that this hybrid algorithm converges quite well and is approximately as fast or faster than the hybrid gmres or restarted versions of the gmres algorithm it is however sensitive as other hybrid methods also are to the amount of information on the spectrum ofa acquired during the first arnoldi phase of this procedure

backbone and elastic backbone of percolation clusters obtained by the new method of burning we introduce a method of obtaining information about percolation clusters in which the clusters are burned in different ways this method yields the backbone and the elastic backbone their number of loops shortest path and other quantities it works in any dimension and can be applied also to other clusters we obtain accurate estimates for the backbone exponents in two and three dimensions

computation of the sommerfeld integral tails using the matrix pencil method the oscillating infinite domain sommerfeld integrals si are difficult to integrate using a numerical procedure when dealing with structures in a layered media even though several researchers have attempted to do that generally integration along the real axis is used to compute the si however significant computational effort is required to integrate the oscillating and slowly decaying function along the tail extrapolation methods are generally applied to accelerate the rate of convergence of these integrals however there are difficulties with the extrapolation methods such as locations for the breakpoints in this paper we illustrate a simplified approach for accurate and efficient calculation of the integrals dealing with the tails of the si in this paper we fit the tail by a sum of finite usually 10 to 20 complex exponentials using the matrix pencil method mpm the integral of the tail of the si is then simply calculated by summing some complex numbers no numerical integration is needed in this process as the integrals can be done analytically good accuracy is achieved with a small number of evaluations for the integral kernel 60 points for the mpm as compared with hundreds or thousands of functional evaluations using the traditional extrapolation methods along the tails of the si simulation results show that to obtain the similar accuracy in the evaluation of the si the mpm is approximately 10 times faster than the traditional extrapolation methods moreover since the mpm is robust to the effects of noise this method is more stable especially for large values of the horizontal distances the method proposed in this paper is thus a new and better technique to obtain accurate results for the computation of the green s function for a layered media in the spatial domain

rigorous lower bound on the dynamic critical exponents of the swendsen wang algorithm we prove the rigorous lower bound z sw v for the dynamic critical exponent of the swendsen wang algorithm for two dimensional q state potts models with q 2 3 4 this implies z sw 0 2 5 1 we present numerical data indicating that z sw 0 55 0 03 0 89 0 05 for q 3 4 95 confidence limits statistical errors only the discrepancy for q 4 appears to be caused by multiplicative logarithmic corrections

supplement to the faber polynomials for circular sectors degree of faber polynomial 6 p 0 6 u p l 3 sa2 2 s u 7 ua2 4 2 p 2 4 sa3 11 sa2 u 14 s ua2 16 s 11 ua 3 32 u 2 p 3 39 sa4 76 s 3 u 74 s 2 ua2 192 sa2 44 s ua3 256 s u 7 ua4 128 ua2 144 16 p 4 46 sa5 31 s 4 u 34 sa3 ua2 248 s 3 22 sa2 ua3 120 sa2u 20 s u 4 136 s ua2 256 s 7 ua5 24 ua3 104 u 20 p 5 21 s 6 34 sa5 u 65 s 4 u 2 1o sa4 20 sa3 ua3 112 s 3 u 65 s 2 u 4 312 s 2 ua 2 72 sa2 70 s ta5 304 s u 3 144 s u 21 u 6 100 u 4 72 u 2 32 16

new expansions of bessel functions of first kind and complex argument we present accurate trigonometric expansions of bessel functions of first kind and integer order for complex arguments of the form j v z sum k alpha k s beta k z where alpha k and beta k are constants and s is a sinusoidal function using the new expansions varying levels of accuracy and range of applicability can be achieved by varying the number of terms in the expansions for example a four term expansion of j 0 z yields an average relative error of for vert z vert leq 2 pi and same accuracy is achieved for an eight term expansion for an extended range vert z vert leq 5 pi further a phase and amplitude corrected large argument asymptotic formula is studied such that the lower limit of its usage is reduced to medium magnitude ranges of arguments the new set of formulas can not only be incorporated into math libraries very easily but also be useful for treatment of radiation and scattering problems involving bessel functions

a new method for chebyshev approximation of complex valued functions problems of this type appear in many connections and have been treated by various authors cf 2 12 13 considering the computation of best rational chebyshev approximations to complex valued functions by descent algorithms ellacott and williams 6 report that the main portion of computer time is spent in the solution of the linear subproblems they have applied the very slowly convergent lawson algorithm to these subproblems and they remark that a fast linear algorithm would bring about a significant improvement in efficiency in our paper we describe a new algorithm which seems to be the first one which works for the continuous case where t is not a finite set too in our method we transform the complex approximation problem into a real linear optimization problem with infinitely many constraints a so called semi infinite program sip cf 5 8 10 this is solved by a two step procedure in the first step we apply the stabilized simplex algorithm to a discrete approximation of the sip and in the second we make use of the newton raphson method in order to obtain very rapidly the solution of the sip and thus of our original approximation problem

periodic solutions of a nonlinear parabolic equation associated with the penetration of a magnetic field into a substance abstract we prove the existence and uniqueness of a t periodic weak solution of the nonlinear parabolic equation u t 1 r r a 0 t u r r s 2 ds r u r f u 0 r t 0 1 x o t u r 0 t u r 1 t h t u 1 t 0 in a sobolev space with weight in the proof the galerkin method is employed numerical results are given

symbolic and numerical computation on bessel functions of complex argument and large magnitude the lanczos t method with perturbations proportional to faber polynomials is employed to approximate the bessel functions of the first kind j v z and the second kind y v z the hankel functions of the first kind h v 1 z and the second kind h v 2 z of integer order v for specific outer regions of the complex plane i e z r for some r the scaled symbolic representation of the faber polynomials and the appropriate automated t method approximation are introduced both symbolic and numerical computation are discussed in addition numerical experiments are employed to test the proposed t method computed accuracy for j 0 z and y 0 z for z 8 are presented the results are compared with those obtained from the truncated chebyshev series approximations and with those of the t method approximations on the inner disk z 8 some concluding remarks and suggestions on future research are given

parallel computation of polynomials with minimal uniform norm and its application to large eigenproblems we investigate the parallelization of an algorithm that computes the polynomial with minimal uniform norm on polygonal domains the obtained polynomial is used to accelerate large non hermitian eigenvalue problems we report the performance results obtained on the machine paragon and compare this method with the one based on chebyshev acceleration techniques

numerical evaluation of airy functions with complex arguments abstract we present two methods for the evaluation of airy functions of complex argument the first method is accurate to any desired precision but is slow and unsuitable for fixed precision languages the second method is accurate to double precision 12 digits and is suitable for programming in a fixed precision language such as fortran the first method uses the symbolic manipulation language maple to evaluate either the taylor series expansion or an asymptotic expansion of each function the second method extends an idea of j c p miller to the complex plane it uses the first method to obtain a grid of points in the complex plane where the functions are known to high precision and then uses taylor series from these base points the resulting algorithm is accurate and efficient

error analysis of the tau method dependence of the approximation error on the choice of perturbation term we consider a system of ordinary differential equations with constant coefficients and deduce asymptotic estimates for the tau method approximation error vector per step for different choices of the perturbation term hn x the cases considered are legendre polynomials chebyshev polynomials powers of x and polynomials of the form x2 r2 n r x r the first two are standard choices for the tau method for chebyshev and legendre series expansion techniques and also for collocation the third one realizes the classical power series expansion techniques in the framework of the tau method and the last is related to the trial functions used in weighted residuals methods we shall refer to it as the weighted residuals choice we show that the resulting tau method implementations can be arranged into the following scale of increasing error estimates at the end point x r legendre chebyshev power series weighted residuals for the interesting case of legendre tau approximations we offer upper and lower error bounds for the end point of the interval of approximation in particular this last estimates solve a conjecture on increased accuracy at the end point of the interval of approximation formulated by lanczos in 1956 such conjecture has equivalent forms for other polynomial methods for the numerical solution of differential equations although formulated in the convenient framework of the recursive tau method see ortiz 1 the results given here apply without essential modifications to chebyshev or legendre series expansion techniques for differential equations collocation and spectral methods we give numerical examples which confirm the sharpness of the lemmas and theorems given in this paper finally we discuss in an example the application of our results to the analysis of singularly perturbed differential equations

approximations for the bessel and struve functions on presente des approximations polynomiales et fractions rationnelles a precision minimax pour les fonctions de bessel de premiere et seconde espece d ordre zero et un et pour les fonctions de struve du meme ordre

solving complex approximation problems by semiinfinite finite optimization techniques a study on convergence we transform a complex approximation problem into an equivalent semiinfinite optimization problem whose constraints are described in terms of a quantity 0 2 i we study the effect of disturbing the problem by replacingi by a compact subsetm i which includes as special case the discrete case wherem consists only of finitely many points we introduce a measure for the deviation ofm fromi and show that in any complex approximation problem the minimal distance of the disturbed problem converges quadratically with 0 to the minimal distance of the undisturbed problem which is a generalization of a result by streit and nuttall we also show that in a linear finite dimensional approximation problem the convergence of the coefficients of the disturbed problem is in general at most linear there are some graphical representations of best complex approximations computed with the described method

solvable lattice models labelled by dynkin diagrams an equivalence between generalized restricted solid on solid models associated with sets of graphs and multi colour loop models is established as an application the authors consider solvable loop models and in this way obtain new solvable families of critical rsos models these families can all be classified by the dynkin diagrams of the simply laced lie algebras for one of the rsos models labelled by the lie algebra pair al al and related to the c2 1 vertex model they give an off critical extension which breaks the z2 symmetry of the dynkin diagrams

chebyshev expansions for the bessel function _ in the complex plane polynomial based approximations for jo z and j1 z are presented the first quadrant of the complex plane is divided into six sectors and separate approximations are given for i z i 8 can also be used to evaluate the bessel functions yo z and y z and the hankel functions of the first and second kinds

boundary critical phenomena in the three state potts model boundary critical phenomena are studied in the three state potts model in two dimensions using conformal field theory duality and renormalization group methods a presumably complete set of boundary conditions is obtained using both fusion and orbifold methods besides the previously known free fixed and mixed boundary conditions a new one is obtained this illustrates the necessity of considering fusion with operators that do not occur in the bulk spectrum to obtain all boundary conditions it is shown that this new boundary condition is dual to the mixed ones the phase diagram for the quantum chain version of the potts model is analysed using duality and renormalization group arguments

computer simulation studies of critical phenomena abstract the recent strides which have been made in the use of computer simulations to study phase transitions and critical phenomena are reviewed we describe how the use of a combination of recently developed histogram techniques and sophisticated monte carlo simulation algorithms have allowed the determination of the static critical behavior of the d 3 ising and heisenberg models with unprecedented precision we also discuss the study of dynamic critical behavior in simple spin models by both monte carlo and spin dynamics methods recent estimates for dynamic critical exponents are reviewed including the first high precision estimate for true dynamics

polynomial approximations to bessel functions a polynomial approximation to bessel functions that arises from an electromagnetic scattering problem is examined the approximation is extended to bessel functions of any integer order and the relationship to the taylor series is derived numerical calculations show that the polynomial approximation and the taylor series truncated to the same order have similar accuracies

the o n model on the annulus we use coulomb gas methods to derive an explicit form for the scaling limit of the partition function of the critical o n model on an annulus with free boundary conditions as a function of its modulus this correctly takes into account the magnetic charge asymmetry and the decoupling of the null states it agrees with an earlier conjecture based on bethe ansatz and quantum group symmetry and with all known results for special values of n it gives new formulae for percolation the probability that a cluster connects the two opposite boundaries and for self avoiding loops the partition function for a single loop wrapping non trivially around the annulus the limit n 0 also gives explicit examples of partition functions in logarithmic conformal field theory

critical behaviour and conformal anomaly of the o n model on the square lattice finite size scaling and transfer matrix techniques are used to determine the conformal anomaly and critical exponents of o n models on the square lattice these calculations were performed on five branches of critical points parametrised by n the results for two of the branches agree with the known universal properties of the o n model as derived for the honeycomb lattice finite size convergence is poor for most of the third branch so that in this case the classification in terms of known models is uncertain the authors data show that the fourth branch can be interpreted in terms of a combination of a low temperature o n and a critical ising model finally the numerical results for the o n model on the remaining branch agree accurately with the known results for the low temperature o n 1 model they prove this equivalence by means of an exact mapping

the numerical computation of bessel functions of the first and second kind for integer orders and complex arguments a rigorous algorithm for computing integer order bessel functions of the first and second kind with complex arguments is discussed the algorithm makes use of backward recurrence for computing bessel functions of the first kind where applicable and of hankel s asymptotic expansion for large arguments the argument is limited only by the ability of the computer to handle large numbers orders up to a few thousand have been computed to 13 digit accuracy

analytical expressions for spatial domain green s functions in layered media accurate and efficient formulas for computing spatial domain green s functions are presented in this paper to apply the formulas we only require a set of sample points of the spectral domain green s function sdgf on an integration path avoiding its singularities and an infinite domain tail path since the sampling is carried out for the integrand excluding the bessel function the number of sample points required is much smaller than the numerical integration method this aspect of the proposed method proves to be very advantageous for evaluating closed form green s functions at large source observer distances it is shown that the proposed formulas provide accurate results for both the near and far field regions as well as a wide range of material parameters such as lossless lossy left handed materials as well as multilayered substrates

geometric properties of two dimensional o n loop configurations we study the fractal geometry of o n loop configurations in two dimensions by means of scaling and a monte carlo method and compare the results with predictions based on the coulomb gas technique the monte carlo algorithm is applicable to models with noninteger n and uses local updates although these updates typically lead to nonlocal modifications of loop connectivities the number of operations required per update is only of order 1 the monte carlo algorithm is applied to the honeycomb o n model for several values of n including noninteger ones we thus determine scaling exponents that describe the fractal nature of o n loops at criticality the results of the numerical analysis agree with the theoretical predictions

application of the absolute nodal co ordinate formulation to multibody system dynamics abstract the floating frame of reference formulation is currently the most widely used approach in flexible multibody simulations the use of this approach however has been limited to small deformation problems in this investigation the computer implementation of the newabsolute nodal co ordinate formulationand its use in the small and large deformation analysis of flexible multibody systems that consist of interconnected bodies are discussed while in the floating frame of reference formulation a mixed set of absolute reference and local elastic co ordinates are used in the absolute nodal co ordinate formulation only absolute co ordinates are used in the absolute nodal co ordinate formulation new interpretation of the nodal co ordinates of the finite elements is used no infinitesimal or finite rotations are used as nodal co ordinates from beams and plates instead global slopes are used to define the element nodal co ordinates using this interpretation of the element co ordinates beams and plates can be considered as isoparametric elements and as a result exact modelling of the rigid body dynamics can be obtained using the element shape function and the absolute nodal co ordinates unlike the floating frame of reference approach no co ordinate transformation is required in order to determine the element inertia the mass matrix of the finite elements is a constant matrix and therefore the centrifugal and coriolis forces are equal to zero when the absolute nodal co ordinate formulation is used another advantage of using the absolute nodal co ordinate formulation in the dynamic simulation of multibody systems is its simplicity in imposing some of the joint constraints and also its simplicity in formulating the generalized forces due to spring damper elements the results obtained in this investigation show an excellent agreement with the results obtained using the floating frame of reference formulation when large rotation small deformation problems are considered

the faber polynomials for annular sectors a conformai mapping of the exterior of the unit circle to the exterior of a region of the complex plane determines the faber polynomials for that region these polynomials are of interest in providing near optimal polynomial approximations in a variety of contexts including the construction of semi iterative methods for linear equations the relevant conformai map for an annular sector z r z 1 0 argz it with 0 9 it is derived here and a recurrence relation is established for the coefficients of its laurent expansion about the point at infinity the recursive evaluation of scaled faber polynomials is formulated in such a way that an algebraic manipulation package may be used to generate explicit expressions for their coefficients in terms of two parameters which are determined by the interior angle of the annular sector and the ratio of its radii properties of the coefficients of the scaled faber polynomials are established and those for polynomials of degree 15 are tabulated in a supplement at the end of this issue a simple closed form is obtained for the coefficients of the faber series for 1 z known results for an interval a circular arc and a circular sector are reproduced as special cases

a study of semiiterative methods for nonsymmetric systems of linear equations given a nonsingular linear systema x b a splittinga m n leads to the one step iteration 1 x m t x m 1 c witht m 1n andc m 1 b we investigate semiiterative methods sim s with respect to 1 under the assumption that the eigenvalues oft are contained in some compact set of with 1 there exist sim s which are optimal with respect to but except for some special sets such optimal methods are not explicitly known in general using results about maximal convergence of polynomials and uniformly distributed nodes from approximation and function theory we describe here sim s which are asymptotically optimal with respect to it is shown that euler methods extensively studied by niethammer varga nv are special sim s various algorithms for sim s are also derived here a 1 1 correspondence between euler methods and sim s generated by generalized faber polynomials is further established here this correspondence gives that asymptotically optimal euler methods are quite near the optimal sim s

computation of faber series with application to numerical polynomial approximation in the complex plane kovari and pommerenke 19 and elliott 8 have shown that the truncated faber series gives a polynomial approximation which for practical values of the degree of the polynomial is very close to the best approximation in this paper we discuss efficient fast fourier transform fft and recursive methods for the computation of faber polynomials and point out that the fft method described by geddes 13 for computing chebyshev coefficients can be generalized to compute faber coefficients we also give a corrected bound for the norm of the faber projection that given in elliott 8 being unfortunately slightly in error and very briefly discuss a possible extension of the method to the case when the mapping function which is required to compute the faber series is not known explicitly

on clenshaws s method and a generalisation to faber series the method of expanding the solution of linear ordinary differential equations in power or laurent series is classical and is usually associated with the name of frobenius in the early days of electronic computation it was appreciated that expansions in chebyshev series are often of more practical use and the necessary techniques developed by clenshaw this is usually carried out in order to approximate special functions defined by ordinary differential equations rather than as a technique for actually solving such equations in general for which finite difference methods are to be preferred in this paper we show that by means of a conformal map the problem of expansion in chebyshev series can in fact be reduced to that of expansion in a laurent series yielding a method which is usually computationally simpler moreoveer the method can be generalised to the case of faber expansions on regions of the complex plane a non trivial example is explored in order to illustrate the method and we also use the technique to generalise an identity relating chebyshev polynomials to the faber case

reference motion in deformable bodies under rigid body motion and vibration part i theory abstract this paper examines the motions of reference systems linked to deformable bodies under simultaneously vibration and large translations and rotations these motions depend on the particular type of linkage between the moving reference system and the deformable body which is defined by the so called reference conditions when using the rayleigh ritz method the reference conditions also dictate the boundary conditions to be fulfilled by the shape functions used to describe the body s elasticity this paper analyses three different types of reference conditions namely free linkage rigid linkage and two point linkage it is shown that moving reference frames only evolve at a constant velocity in the absence of external forces when the free linkage is used the reference velocities for systems with a free linkage are designated rigid body equivalent velocities for the deformable body here such velocities can also be calculated under other types of reference conditions and are usually functions of the elastic and reference co ordinates and also of their derivatives rigid body equivalent velocities are useful for purposes such as estimating the trajectory of deformable bodies moving freely in space without the need to examine the deformations they undergo also their calculation is required with a view to determining the kinematic restitution coefficient for deformable body collisions which is dealt within part ii of this series

tau method approximations for the bessel function y1 z abstract we present method approximations for the bessel function of the second kind of order one y 1 z with some numerical results

tau method approximations for the bessel function y0 z we exploit the lanczos method with perturbations proportional to faber polynomials to approximate the bessel function of the second kind y0 z for specific regions of the complex plane accuracy comparisons are made among the power series direct and integrated forms of the method and the chebyshev series

new approximations to j sub 0 and j sub 1 bessel functions new approximate solutions to the 0th and 1st order bessel functions of the first kind are derived the formulations are based upon using a new integral with no previously known solution the new integral in the limiting case is identical to the 0th order bessel function integral it is solved in closed form and the solution is expressed as a simple even order polynomial with integer coefficients the polynomial coefficients are all of integer value the 1st order bessel function approximation can then be found through a simple derivative comparisons are made between the exact solution classic solutions and the new approximation the new approximation proves to be much more accurate than the classic small argument approximation it is also sufficiently accurate to bridge the gap between the classic large and small argument approximations and has potential applications in allowing one to analytically evaluate integrals containing bessel functions

new frequency dependent edge mode current density approximations for tm scattering from a conducting strip grating two new frequency dependent strip grating current densities are derived for tm mode incidence on a conducting strip grating the current densities are assumed to be the product of a spatial function and a frequency function one spatial function is the traditional frequency independent edge mode current density the other is found by the use of conformal mapping the frequency function is derived by use of the radiation integral and the periodic green s function the new resultant current densities are both exact for small strip widths the conformally mapped solution is also valid for wide strips and is exact in the limit as the strip width equals the grating period although the solutions are not as accurate as the moment method at all frequencies they produce computationally faster solutions and offer further insight into strip grating behavior

the simple cubic lattice gas with nearest neighbour exclusion ising universality abstract the lattice gas with nearest neighbour exclusion on the simple cubic lattice is studied by means of statistically accurate monte carlo simulations with an efficient cluster algorithm our results for critical exponents are y h 2 47 1 and y t 1 60 2 these results agree well with the three dimensional ising universality class we explain the discrepancy with an earlier study the critical activity is z c 1 0559 1 the size distribution of the clusters indicates that the percolation threshold of the cluster formation process of the monte carlo algorithm coincides with the critical point

the faber polynomials for circular lunes abstract we study the faber polynomials f n z generated by a circular lune symmetric about both axes with vertices at the points z 0 2 and exterior angle an explicit expression of f n z was obtained by computing the coefficients via a cauchy integral formula we also illustrate the location of the zeros of faber polynomial and of its derivative our results include a circle and a segment as special cases when 1 2 respectively

ising universality in three dimensions a monte carlo study we investigate three ising models on the simple cubic lattice by means of monte carlo methods and finite size scaling these models are the spin 1 2 ising model with nearest neighbour interactions a spin 1 2 model with nearest neighbour and third neighbour interactions and a spin 1 model with nearest neighbour interactions the results are in accurate agreement with the hypothesis of universality analysis of the finite size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field we find that the correction to scaling amplitudes are strongly dependent on the introduction of further neighbour interactions or a third spin state in a spin 1 ising model these corrections appear to be very small this is very helpful for the determination of the universal constants of the ising model the renormalization exponents of the ising model are determined as yt 1 587 2 yh 2 4815 15 and yi 0 82 6 the universal ratio q m2 2 m4 is equal to 0 6233 4 for periodic systems with cubic symmetry the critical point of the nearest neighbour spin 1 2 model is kc 0 2216546 10

on semiiterative methods generated by faber polynomials consider a nonsingular system of linear algebraic equationsa x b or in fixed point formx tx c where the eigenvalues oft are contained in some compact subset of the complex plane using faber polynomials iteration methods of the form y m mu m 0 left c ty m 1 right mu m 1 y m 1 ldots mu m k y m k m 0 m j 1 for allm 1 are constructed which include the chebyshev iterative method as well as the stationaryk step methods when is a rectangle we also develop four step methods which yield fast convergence in cases when the speed of convergence of the optimized chebyshev method is poor

a faber series approach to cardinal interpolation for a compactly supported function 9 in rd we study quasiinterpolants based on point evaluations at the integer lattice we restrict ourselves to the case where the coefficient sequence af for given data f is computed by applying a univariate polynomial q to the sequence i zd and then convolving with the data flzd such operators appear in the well known neumann series formulation of quasi interpolation a criterion for the polynomial q is given such that the corresponding operator defines a quasi interpolant since our main application is cardinal interpolation which is well defined if the symbol of p does not vanish we choose q as the partial sum of a certain faber series this series can be computed recursively by this approach we avoid the restriction that the range of the symbol of p must be contained in a disk of the complex plane excluding the origin which is necessary for convergence of the neumann series furthermore for symmetric a we prove that the rate of convergence to the cardinal interpolant is superior to the one obtainable from the neumann series

on a novel iterative method to compute polynomial approximations to bessel functions of the first kind and its connection to the solution of fractional diffusion diffusion wave problems we present an iterative method to obtain approximations to bessel functions of the first kind jp x p 1 via the repeated application of an integral operator to an initial seed function f0 x the class of seed functions f0 x leading to sets of increasingly accurate approximations fn x is considerably large and includes any polynomial when the operator is applied once to a polynomial of degree s it yields a polynomial of degree s 2 and so the iteration of this operator generates sets of increasingly better polynomial approximations of increasing degree we focus on the set of polynomial approximations generated from the seed function f0 x 1 this set of polynomials is useful not only for the computation of jp x but also from a physical point of view as it describes the long time decay modes of certain fractional diffusion and diffusion wave problems

thermodynamically self consistent theory for the blume capel model we use a self consistent ornstein zernike approximation to study the blume capel ferromagnet on three dimensional lattices the correlation functions and the thermodynamics are obtained from the solution of two coupled partial differential equations the theory provides a comprehensive and accurate description of the phase diagram in all regions including the wing boundaries in a nonzero magnetic field in particular the coordinates of the tricritical point are in very good agreement with the best estimates from simulation or series expansion numerical and analytical analysis strongly suggest that the theory predicts a universal ising like critical behavior along the lambda line and the wing critical lines and a tricritical behavior governed by mean field exponents

complex polynomial approximation by the lanczos t method dawson s integral abstract the lanczos method with perturbations proportional to faber polynomials is used to obtain polynomial approximations for dawson s integral on circular sectors an upper bound on the truncation error is established for one form of the method which gives near minimax polynomial approximations and it is found that this bound provides a useful estimate of the truncation error the faber series for dawson s integral on a circular sector is also investigated numerical results show that the method can produce polynomial approximations as accurate as the truncated faber series with much less effort than is involved in computing the faber coefficients

boundary conformal field theories and loop models we propose a systematic method to extract conformal loop models for rational conformal field theories cft the method is based on defining an ade model for boundary primary operators by using the fusion matrices of these operators as adjacency matrices these loop models respect the conformal boundary conditions we discuss the loop models that can be extracted by this method for minimal cfts and then we give dilute o n loop models on the square lattice as examples for these loop models we also give some proposals for wzw su 2 models

percolation between vacancies in the two dimensional blume capel model using suitable monte carlo methods and finite size scaling we investigate the blume capel model on the square lattice we construct percolation clusters by placing nearest neighbor bonds between vacancies with a variable bond probability pb at the tricritical point we locate the percolation threshold of these vacancy clusters at pbc 0 706 336 at this point we determine the fractal dimension of the vacancy clusters as xf 0 13085 21 160 and the exponent governing the renormalization flow in the pb direction as yp 0 4262 17 40 for bond probability pbpbc the vacancy clusters maintain strong critical correlations the fractal dimension is xf 0 07502 3 40 and the leading correction exponent is yp 0 452 19 40 the above values fit well in the kac table for the tricritical ising model these vacancy clusters have much analogy with those consisting of ising spins of the same sign although the associated quantities and magnetization m are energylike and magnetic quantities respectively however along the critical line of the blume capel model the vacancies are more or less uniformly distributed over the whole lattice in this case no critical percolation correlations are observed in the vacancy clusters at least in the physical region pb 1

fractal dimensions of potts clusters potts clusters are connected sets of nearest neighbour sites for which the potts variable is in the same state at criticality there exists a fractal cluster using arguments from the renormalization group conformal invariance and numerical simulations the author determines the bulk surface hull and red bond dimension of these clusters as a function of the number of potts states

clusters and droplets in the q state potts model a potts correlated polychromatic percolation is studied the clusters are made of sites corresponding to a given value of the q state potts variables connected by bonds being active with probability pb to treat this problem an s state potts hamiltonian diluted with q state potts variables instead of lattice gas variables is introduced to which the the migdal kadanoff renormalisation group is applied it is found for a particular choice of pb 1 e k where k is the potts coupling constant divided by the boltzmann factor that these clusters called droplets diverge at the potts critical point with potts exponents