finitized conformal spectrum of the ising model on the cylinder and torus abstract the spectrum of the critical ising model on a lattice with cylindrical and toroidal boundary conditions is calculated by commuting transfer matrix methods using a simple truncation procedure we obtain the natural finitizations of the conformal spectra recently proposed by melzer these finitizations imply polynomial identities which in the large lattice limit give rise to the rogers ramanujan identities for the c 1 2 virasoro characters

fermionic solution of the andrews baxter forrester model i unification of tba and ctm methods the problem of computing the one dimensional configuration sums of the abf model in regime iii is mapped onto the problem of evaluating the grandcanonical partition function of a gas of charged particles obeying certain fermionic exclusion rules we thus obtain a newfermionic method to compute the local height probabilities of the model combined with the originalbosonic approach of andrews baxter and forrester we obtain a new proof of some of melzer s polynomial identities in the infinite limit these identities yield rogers ramanujan type identities for the virasoro characters l 1 r l r q as conjectured by the stony brook group as a result of our work the corner transfer matrix and thermodynamic bethe ansatz approaches to solvable lattice models are unified

te tm scheme for computation of electromagnetic fields in accelerators we propose a new two level economical conservative scheme for short range wake field calculation in three dimensions the scheme does not have dispersion in the longitudinal direction and is staircase free second order convergent unlike the finite difference time domain method fdtd it is based on a te tm like splitting of the field components in time additionally it uses an enhanced alternating direction splitting of the transverse space operator that makes the scheme computationally as effective as the conventional fdtd method unlike the fdtd adi and low order strang methods the splitting error in our scheme is only of fourth order as numerical examples show the new scheme is much more accurate on the long time scale than the conventional fdtd approach

a 3 d enlarged cell technique ect for the conformal fdtd method in this paper we present an enlarged cell technique ect to avoid the time step reduction encountered in the conformal finite difference time domain cfdtd method due to small irregular cells truncated by metallic boundaries we focus our efforts on the discussion of the accuracy and stability of the ect and its comparison with other conformal methods especially the one called the uniformly stable conformal usc method we also provide a simplified ect which is much easier to implement in the ect a stability criterion is first constructed to identify instable irregular cells i e those having so small an area to cause instability those instable cells are then enlarged into their adjacent cells to obtain a large stable area careful treatment is performed on the communication between the intruding and intruded cells in terms of electromotive force by keeping the total electromotive force conservative this technique is verified by several 3 d numerical experiments results show that the ect is second order accurate and numerically stable at the regular courant time step limit

rogers ramanujan identities in the hard hexagon model the hard hexagon model in statistical mechanics is a special case of a solvable class of hard square type models in which certain special diagonal interactions are added the sublattice densities and order parameters of this class are obtained and it is shown that many rogers ramanujan type identities naturally enter the working

a theorem about representation of palais smale sequence and its applications abstract we proved a theorem about representation of palais smale sequence to a semilinear elliptic equation then use it to obtain some new existence results of nontrivial solutions for some semilinear elliptic equations on unbounded domain

fermionic solution of the andrews baxter forrester model ii proof of melzer s polynomial identities we compute the one dimensional configuration sums of the abf model using the fermionic technique introduced in part i of this paper combined with the results of andrews baxter and forrester we prove polynomial identities for finitizations of the virasoro characters chi b a r 1 r q as conjectured by melzer in the thermodynamic limit these identities reproduce rogers ramanujan type identities for the unitary minimal virasoro characters conjectured by the stony brook group we also present a list of additional virasoro character identities which follow from our proof of melzer s identities and application of bailey s lemma

locally conformal fdtd for anisotropic conductive interfaces locally conformal discretizations have been developed for the finite difference time domain fdtd method to mitigate staircasing errors that arise in the modeling of arbitrary e g curved slanted interfaces by regular orthogonal grids in particular locally conformal algorithms based on the computation of effective material parameters in partially filled fdtd cells are attractive because they are simple to implement and maintain the basic conditional stability of fdtd with only minor changes on the courant limit in this paper we propose a new locally conformal implementation for fdtd tailored for anisotropic conductive interfaces instead of employing arithmetic or geometrical averages we invoke the quasi static relationship between the current density and the electric field to derive the effective conductivities of partially filled fdtd cells

particles in rsos paths we introduce a new representation of the paths of the forrester baxter restricted solid on solid rsos models which represents the states of the irreducible modules of the minimal models this representation is obtained by transforming the rsos paths for the cases p 2p 1 to new paths for which horizontal edges are allowed at certain heights these new paths are much simpler in that their weight is nothing but the sum of the position of the peaks this description paves the way for the interpretation of the rsos paths in terms of fermi type charged particles out of which the fermionic characters could be obtained constructively the derivation of the fermionic character for p 2 and p kp 1 is outlined finally the particles of the rsos paths are put in relation to the kinks and the breathers of the restricted sine gordon model

development of a novel fdtd 2 4 compatible conformal scheme for electromagnetic computations of complex curved pec objects we present a modified fourth order finite difference time domain fdtd 2 4 conformal scheme for accurately computing electromagnetic characteristics of some complex three dimensional 3d perfectly conducting pec objects it has higher accuracy and efficiency than those of conventional fdtd and fdtd 2 4 methods as coarse meshes are employed with staircase errors reduced effectively during its implementation two integration loops of the faraday s law are used for updating magnetic field components while the updating equations of electric field ones are the same as those in the normal fdtd method a rigorous analysis of its global stability based on the conventional high order fdtd stability criterion and the fourier method is also performed in order to obtain stable and accurate numerical results a user defined geometric precision technique which gives a criterion for determining the time step is employed for our computations it is numerically demonstrated that using our proposed fdtd 2 4 compatible conformal scheme high accuracy and low dispersion errors can be achieved for fast predicting radar cross sections as well as induced current distribution of some complex 3 d structures

finite difference time domain modeling of curved surfaces em scattering the finite difference time domain fdtd method is generalized to include the accurate modeling of curved surfaces this generalization the contour path cp method accurately models the illumination of bodies with curved surfaces yet retains the ability to model corners and edges cp modeling of two dimensional electromagnetic wave scattering from objects of various shapes and compositions is presented

a fast multigrid based electromagnetic eigensolver for curved metal boundaries on the yee mesh for embedded boundary electromagnetics using the dey mittra dey and mittra 1997 1 algorithm a special grad div matrix constructed in this work allows use of multigrid methods for efficient inversion of maxwell s curl curl matrix efficient curl curl inversions are demonstrated within a shift and invert krylov subspace eigensolver open sourced at ofortt https github com bauerca maxwell cfortt on the spherical cavity and the 9 cell tesla superconducting accelerator cavity the accuracy of the dey mittra algorithm is also examined frequencies converge with second order error and surface fields are found to converge with nearly second order error in agreement with previous work nieter et al 2009 2 neglecting some boundary cut cell faces as is required in the time domain for numerical stability reduces frequency convergence to first order and surface field convergence to zeroth order i e surface fields do not converge additionally and importantly neglecting faces can reduce accuracy by an order of magnitude at low resolutions

exceptional structure of the dilute a 3 model e 8 and e 7 rogers ramanujan identities the dilute a3 lattice model in regime 2 is in the universality class of the ising model in a magnetic field here we directly establish the existence of an e8 structure in the dilute a3 model in this regime by expressing the one dimensional configuration sums in terms of fermionic sums which involve the e8 root system explicitly in the thermodynamic limit these polynomial identities yield a proof of the e8 rogers ramanujan identity recently conjectured by kedem et al 1993 the polynomial identities also apply to regime 3 which is obtained by transforming the modular parameter by q to 1 q in this case we find an a1xe7 structure and prove a rogers ramanujan identity of a1xe7 type finally in the critical q to 1 limit we give some intriguing expressions for the number of l step paths on the a3 dynkin diagram with tadpoles in terms of the e8 cartan matrix all our findings confirm the e8 and e7 structure of the dilute a3 model found recently by means of the thermodynamic bethe ansatz

on the existence of positive solution for a semilinear elliptic equation on esteban lions domain with hole abstract we prove the existence of nontrivial solution for the equation u u u p 2 u in some unbounded domain in r n

conformal fdtd modeling of imperfect conductors at millimeter wave bands the dey mittra conformal finite difference time domain cfdtd algorithm for perfect electrical conductors is modified for the analysis of finite conductivity conductors at millimeter wave frequencies formulas are derived for cfdtd coefficients using voltage state variables and a constant surface impedance boundary condition sibc the approach permits a fast implementation suitable for cuda type gpu hardware accuracy and stability are investigated with respect to the stability constraints on intersection areas introduced by dey mittra and benkler as well as the distance stability constraints of zagorodnov that permits 100 courant temporal sampling a relaxation of the zagorodnov distance constraints permits increased accuracy with respect to all alternative area constraints analytic solutions are used to judge the performance of the proposed cfdtd modifications for millimeter wave band applications

the finite difference time domain fdtd and the finite volume time domain fvtd methods in solving maxwell s equations the finite difference time domain fdtd and its current generalizations have been demonstrated to be useful and powerful tools for the calculation of the radar cross section rcs of complicated objects the radiation of antennas in the presence of other structures and other applications the mathematical techniques for conformal fdtd have matured the primary impediments to its implementation are the complex geometries and material properties associated with the problem even under these circumstances fdtd is more flexible to implement because it is based on first principles instead of a clever mathematical trick this paper gives an account of some new results on conformal fdtd obtained by the authors and their associates at lockheed martin space company since 1988 the emphasis is on nonsmooth boundary condition simulation

conformal fdtd methods to avoid time step reduction with and without cell enlargement during the last decades there have been considerable efforts to develop accurate and yet simple conformal methods for modelling curved boundaries within the finite difference time domain fdtd algorithm in an earlier publication we proposed the uniformly stable conformal usc approach as a general three dimensional extension of fdtd without the need to reduce the maximum stable time step the main idea of usc is the usage of virtually enlarged cells near to the boundary leading to an increased implementation effort in this paper we review the usc method and introduce a new simple and accurate conformal scheme which does not use such enlarged cells this simplified conformal sc scheme has the same number of operations and algorithmic logic as the standard staircase method and thus is easily realizable in existing fdtd codes like usc it leads to accurate results without time step reduction showing a nearly second order convergence in practice the method is verified and compared to other approaches by means of several numerical 2d and 3d examples

convergent cartesian grid methods for maxwell s equations in complex geometries a convergent second order cartesian grid finite difference scheme for the solution of maxwell s equations is presented the scheme employs a staggered grid in space and represents the physical location of the material and metallic boundaries correctly hence eliminating problems caused by staircasing and contrary to the popular yee scheme enforces the correct jump conditions on the field components across material interfaces a detailed analysis of the accuracy of the new embedding scheme is presented confirming its second order global accuracy furthermore the scheme is proven to be a bounded error scheme and thus convergent conditions for fully discrete stability is furthermore established this enables the derivation of bounds for fully discrete stability with cfl restrictions being almost identical to those of the much simpler yee scheme the analysis exposes that the effects of staircasing as well as a lack of properly enforced jump conditions on the field components have significant consequences for the global accuracy it is among other things shown that for cases in which a field component is discontinuous along a grid line as happens at general two and three dimensional material interfaces the yee scheme may exhibit local divergence and loss of global convergence to validate the analysis several one and two dimensional test cases are presented showing an improvement of typically 1 to 2 orders of accuracy at little or no additional computational cost over the yee scheme which in most cases exhibits first order accuracy c 2001 academic press

long time numerical computation of electromagnetic fields in the vicinity of a relativistic source we propose an implicit scheme for the calculation of electromagnetic fields in the vicinity of short electron bunches moving in a transversally bounded domain as it has place in particle accelerators the scheme is able to accurately model curved boundaries and does not suffer from dispersion in direction of motion it is based on splitting the discrete curl operator in its transversal and longitudinal parts unlike previous conformal schemes the new method has a second order convergence without the need to reduce the maximal stable time step of the conventional staircase approach this feature allows the usage of a moving mesh easily several numerical examples are presented and the algorithm is compared to other approaches

q trinomial coefficients and the dilute a model we express the configuration sums of the dilute model in regimes and in terms of q trinomial coefficients we then extend previous results to give expressions for all phases and regimes for odd l for finite system size

local height probabilities in a composite andrews baxter forrester model we study the local height probabilities in a composite height model derived from the restricted solid on solid model introduced by andrews baxter and forrester and their connection with conformal field theory characters the obtained conformal field theories also describe the critical behavior of the model at two different critical points in addition at criticality the model is equivalent to a one dimensional chain of anyons subject to competing two and three body interactions the anyonic chain interpretation provided the original motivation to introduce the composite height model and by obtaining the critical behavior of the composite height model the critical behavior of the anyonic chains is established as well depending on the overall sign of the hamiltonian this critical behavior is described by a diagonal coset model generalizing the minimal models for one sign and by fateev zamolodchikov parafermions for the other