pseudospectral method based on prolate spheroidal wave functions for frequency domain electromagnetic simulations we apply prolate spheroidal wave functions of order zero as basis functions in the pseudospectral method for frequency domain electromagnetic simulation problems like the traditional pseudospectral frequency domain psfd methods based on chebyshev and legendre polynomial series the prolate psfd method yields exponential order of accuracy in terms of the number of samples utilized per wavelength the prolate expansion is superior to the chebyshev and legendre polynomial series by a factor of spl pi 2 in addition the prolate psfd method employs a more uniform spatial grid achieving better resolution near the center of the domain

on hp convergence of prolate spheroidal wave functions and a new well conditioned prolate collocation scheme the first purpose of this paper is to provide further illustrations from both theoretical and numerical perspectives for the nonconvergence of h refinement in hp approximation by the prolate spheroidal wave functions pswfs a surprising convergence property that was first discovered by boyd et al 2013 3 the second purpose is to offer a new basis that leads to prolate collocation systems with condition numbers independent of c n the intrinsic bandwidth parameter and the number of collocation points we highlight that the collocation scheme together with a very practical rule for pairing up c n significantly outperforms the legendre polynomial based method and likewise other jacobi polynomial based methods in approximating highly oscillatory bandlimited functions

iterative method for the study of the thermodynamic behavior of hard sphere yukawa fluids abstract semi analytic expressions for the calculation of the thermodynamic properties of the hard sphere yukawa fluid are derived from the solution of the ornstein zernike equation according to the mean spherical approximation these expressions are written in terms of a cumulative parameter that becomes the physically meaningful root of a fourth or larger order polynomial which can be determined directly by a proposed iterative method the method is illustrated by comparison against well known results including predictions from perturbation theories and its limitations are discussed

mixed spectral boundary element embedding algorithms for the navier stokes equations in the vorticity stream function formulation an embedding approach based on fourier expansions and boundary integral equations is applied to the vorticity stream function formulation of the navier stokes equations the algorithm only requires efficient solvers of scalar elliptic equations and in an asymptotic version the boundary element method is only needed to solve the laplace equation the capabilities of this embedding method in both its full and asymptotic versions are pointed out by considering the classical problem of the flow between two eccentric cylinders

cluster expansion for abstract polymer models a new direct proof of convergence of cluster expansions for polymer contour models is given in an abstract setting it does not rely on kirkwood salsburg type equations or combinatorics of trees a distinctive feature is that at all steps the considered clusters contain every polymer at most once

a comparison of numerical algorithms for fourier extension of the first second and third kinds the range of fourier methods can be significantly increased by extending a nonperiodic function f x to a periodic function f on a larger interval when f x is analytically known on the extended interval the extension is straightforward when f x is unknown outside the physical interval there is no standard recipe worse still like a radarless aircraft groping through fog the algorithm may wreck on the mountain in fog problem a function f x which is perfectly well behaved on the physical interval may very well have singularities in the extended domain in this article we compare several algorithms for successfully extending a function f x into the fog even when the analytic extension is singular the best third kind extension requires singular value decomposition with iterative refinement but achieves accuracy close to machine precision

some comments on fourier analysis uncertainty and modeling investigation of the problem of simultaneously concentrating a function and its fourier transform has led to some interesting special functions that have widespread applications in engineering they provide a means of proving a rigorous version of an engineering folk theorem called the 2wt theorem many generalizations of these ideas seem to possess a similar elegant mathematical structure a brief descriptive review is given of these developments

eigenfunctions of the airyʼs integral transform properties numerical computations and asymptotic behaviors abstract this paper is devoted to the study of the spectrum of the integral operator with airy s kernel we provide the reader with some efficient methods of computing the eigenfunctions and the eigenvalues of this operator the first method is based on the use of a differential operator which commutes with this integral operator the second method is based on a gaussian quadrature method applied to the finite airy s transform the asymptotic behavior of the eigenfunctions is studied by the use of a wkb method some numerical examples are given to illustrate results of this work

primitive model for highly asymmetric electrolytes associative mean spherical approximation abstract we study the primitive model of highly asymmetric electrolytes consiting of a mixture of m n sticky charged hard spheres where m of these components represent polyions and n represent counterions it is assumed that the species representing polyions are highly charged and large in size while the counterions are small in size and have at most two charges a two density version of the mean spherical approximation associative msa explicitly accounting for the effects of the asymmetry is discussed and solved analytically the approximation is based on the recently developed multidensity integral equation theory for highly asymmetric electrolyte solutions kalyuzhnyi et al chem phys lett 215 1993 518 j chem phys 102 1995 5770 the analytic solution and the thermodynamic excess functions are given in terms of a single screening parameter t the excess entropy has the same remarkably simple form smsa t 3 3 sticky terms as found in previous work

adaptive radial basis function and hermite function pseudospectral methods for computing eigenvalues of the prolate spheroidal wave equation for very large bandwidth parameter asymptotic approximations show that the lowest modes of the prolate spheroidal wave equation are concentrated with an o 1 c length scale where c is the bandwidth parameter of the prolate differential equation accurate computation of the ground state eigenvalue by the long known legendre galerkin method requires roughly 3 8 c legendre polynomials some studies have therefore applied a grid with 20 000 points in conjunction with high order finite differences to achieve c 10 7 here we show that by adaptively applying either hermite functions or gaussian radial basis functions rbfs it is never necessary to use more than eighty degrees of freedom to calculate the lowest dozen eigenvalues of each symmetry class for small c the eigenmodes are not confined to a small portion of the domain 2 2 in latitude but are global we show that by periodizing the basis functions via imbricate series it is possible to apply hermite and rbfs even in the limit c 0 the legendre method is probably a little more efficient in this limit since the prolate functions tend to legendre polynomials in this limit a sideband truncation restricts the discretization to a small block taken from the large hermite galerkin matrix we show that sideband truncation with blocks as small as 5 5 is a very efficient way to compute high order modes in an appendix we prove a rigorous convergence theorem for the periodized hermite expansion

the repulsive lattice gas the independent set polynomial and the lovasz local lemma we elucidate the close connection between the repulsive lattice gas in equi librium statistical mechanics and the lovasz local lemma in probabilistic com binatorics we show that the conclusion of the lovasz local lemma holds for dependency graph g and probabilities px if and only if the independent set polynomial for g is nonvanishing in the polydisc of radii px furthermore we show that the usual proof of the lovasz local lemma which provides a suffi cient condition for this to occur corresponds to a simple inductive argument for the nonvanishing of the independent set polynomial in a polydisc which was discovered implicitly by shearer 98 and explicitly by dobrushin 37 38 we also present some refinements and extensions of both arguments including a generalization of the lovasz local lemma that allows for soft dependencies in addition we prove some general properties of the partition function of a repulsive lattice gas most of which are consequences of the alternating sign property for the mayer coefficients we conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs

counting regions with bounded surface area define a cubical complex to be a collection of integer aligned unit cubes in d dimensions lebowitz and mazel 1998 proved that there are between c 1d n 2d and c 2d 64n d complexes containing a fixed cube with connected boundary of d 1 volume n in this paper we narrow these bounds to between c 3d n d and c 4d 2n d we also show that there are n n 2d d 1 o 1 connected complexes containing a fixed cube with not necessarily connected boundary of volume n

an extension of the analytical solution of the ornstein zernike equation with the yukawa closure the analytical solution of the ornstein zernike equation with one yukawa closure of the factorizable coefficient case is extended from the scalar factorization case to the vector factorization case as a result the scaling parameter is extended from a scalar quantity to a matrix quantity and the scaling matrix

asymptotic behaviors and numerical computations of the eigenfunctions and eigenvalues associated with the classical and circular prolate spheroidal wave functions abstract for a fixed bandwidth c the classical prolate spheroidal wave functions pswfs n c and the circular ones cpswfs n c form remarkable hilbertian bases for the spaces of fourier and hankel band limited functions with bandwidth c respectively the prolate spheroidal wave functions have already found many applications from various scientific area such as signal processing numerical analysis physics and random matrix theory these concrete applications of the pswfs and cpswfs require the accurate computation of these special functions to this end we develop in this work two approximate procedures the first procedure is based on pushing forward the wkb method and provide accurate approximations of the pswfs and the cpswfs in terms of some classical special functions the second procedure is based on efficient and accurate quadrature methods for the computation of the values and the eigenvalues associated with pswfs and the cpswfs both procedures are valid for small as well as for large values of the frequency band width c also we provide the reader with some numerical examples that illustrate the different results of this work

accuracy resolution and stability properties of a modified chebyshev method while the chebyshev pseudospectral method provides a spectrally accurate method integration of partial differential equations with spatial derivatives of order m requires time steps of approximately o n 2m for stable explicit solvers theoretically time steps may be increased to o n m with the use of a parameter alpha dependent mapped method introduced by kosloff and tal ezer em j comput phys 104 1993 pp 457 469 our analysis focuses on the utilization of this method for reasonable practical choices for n namely n lesssim 30 as may be needed for two or three dimensional modeling results presented confirm that spectral accuracy with increasing n is possible both for constant alpha hesthaven dinesen and lynov em j comput phys 155 1999 pp 287 306 and for alpha scaled with n alpha sufficiently different from 1 don and solomonoff em siam j sci comput 18 1997 pp 1040 1055 theoretical bounds however show that any realistic choice for alpha in which both resolution and accuracy considerations are imposed permits no more than a doubling of the time step for a stable explicit integrator in time much less than the o n improvement claimed by kosloff and tal ezer on the other hand by choosing alpha carefully it is possible to improve on the resolution of the chebyshev method in particular one may achieve satisfactory resolution with fewer than pi points per wavelength moreover this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about 8 points per wavelength our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one dimensional wave equations specifically while alpha can be chosen such that the mapped method improves the accuracy and resolution of the chebyshev method for practical choices of n it is not possible to achieve both single precision accuracy and gain the advantage of an o n m time step

hybrid eulerian lagrangian semi implicit time integrators hybrid eulerian lagrangian semi implicit time integrators helsi are presented which use the standard semi implicit formulation as their starting point the advantage of such an approach is that existing models which employ eulerian semi implicit time integrators can easily be augmented to use the helsi approach that is provided that either a series of advection problems can be solved efficiently as in the operator integration factor splitting oifs method or that the grid and underlying numerics allow the construction of efficient search and interpolation stencils typically required of semi lagrangian methods once the advection problem is solved the approach then follows the standard semi implicit approach an important caveat of this approach is that the semi implicit time integrator must be constructed around the implicit backward difference formulas

network forming fluids yukawa square well m point model thermal and connectivity properties of the yukawa square well m point yswmp model of the network forming fluid are studied using solution of the multidensity ornstein zernike and connectedness ornstein zernike equations supplemented by the associative mean spherical approximation amsa the model is represented by the multicomponent mixture of yukawa hard spheres with m s a square well sites located on the surface of each hard sphere to validate the accuracy of the theory computer simulation is used to calculate the structure thermodynamic and connectivity properties of the one component ysw4p version of the model which is compared against corresponding theoretical data in addition connectivity properties of the model were studied using flory stockmayer fs theory predictions of the amsa for the thermal properties of the model radial distribution functions rdf internal energy pressure fractions of the particles in different bonding states are in good agreement with computer simulation predictions similarly good agreement was found for the connectedness rdf crdf except for the statepoints located close to the percolation threshold where the theory fails to reproduce the long range behavior of the crdf results of both theories amsa and fs for the mean cluster size are reasonably accurate only at low degrees of association predictions of the fs theory for the percolation lines are in a good agreement with computer simulation predictions amsa predictions of percolation are much less accurate where corresponding percolation lines are located at a temperatures approximately 25 lower then those calculated using computer simulation

variational extensions of the mean spherical approximation abstract in a previous work we have proposed a method to study complex systems with objects of arbitrary size for certain specific forms of the atomic and molecular interactions surprisingly simple and accurate theories the variational mean spherical scaling approximation vmssa velazquez blum j chem phys 110 1990 10 931 blum velazquez j quantum chem theochem in press can be obtained the basic idea is that if the interactions can be expressed in a rapidly converging sum of complex exponentials then the ornstein zernike equation oz has an analytical solution this analytical solution is used to construct a robust interpolation scheme the variation mean spherical scaling approximation vmssa the helmholtz excess free energy a e t s is then written as a function of a scaling matrix both the excess energy e and the excess entropy s will be functionals of in previous work of this series the form of this functional was found for the two blum herrera mol phys 96 1999 821 and three exponential closures of the oz equation blum j stat phys submitted for publication in this paper we extend this to m yukawas a complete basis set we obtain a solution for the one component case and give a closed form expression for the msa excess entropy which is also the vmssa entropy

an instability phenomenon in spectral methods the eigenvalues of chebyshev and legendre spectral differentiation matrices which determine the allowable time step in an explicit time integration are extraordinarily sensitive to rounding errors and other perturbations on a grid of n points per space dimension machine rounding leads to errors in the eigenvalues of size o n 2 this phenomenon may lead to inconsistency between predicted and observed time step restrictions one consequence of it is that spectral differentiation by interpolation in legendre points which has a favorable o n 1 time step restriction for the model problem u t u x in theory is subject to an o n 2 restriction in practice the same effect occurs with chebyshev points for the model problem u t xu x another consequence is that a spectral calculation with a fixed time step may be stable in double precision but unstable in single precision we know of no other examples in numerical computation of this kind of precision dependent stability

accuracy enhancement for higher derivatives using chebyshev collocation and a mapping technique a new method is investigated to reduce the roundoff error in computing derivatives using chebyshev collocation methods by using a grid mapping derived by kosloff and tal ezer and the proper choice of the parameter alpha the roundoff error of the k th derivative can be reduced from o n 2k to o n lge k where epsilon is the machine precision and n is the number of collocation points this drastic reduction of roundoff error makes mapped chebyshev methods competitive with any other algorithm in computing second or higher derivatives with large n several other aspects of the mapped chebyshev differentiation matrix are also studied revealing that the mapped chebyshev methods require much less than pi points to resolve a wave the eigenvalues are less sensitive to perturbation by roundoff error and larger time steps can be used for solving pdes all these advantages of the mapped chebyshev methods can be achieved while maintaining spectral accuracy

traps and snares in eigenvalue calculations with application to pseudospectral computations of ocean tides in a basin bounded by meridians we make several observations about eigenvalue problems using as examples laplace s tidal equations and the differential equation satisfied by the associated legendre functions whatever the discretization only some of the eigenvalues of then dimensional matrix eigenvalue problem will be good approximations to those of the differential equation usuallythen 2 eigenvalues of smallest magnitude for the tidal problem however the good eigenvalues are scattered so our first point is it is important to plot the drift of eigenvalues with changes in resolution we suggest plotting the difference between a low resolution eigenvalue and the nearest high resolution eigenvalue divided by the magnitude of the eigenvalue or the intermodal separation whichever is smaller second as a final safeguard it is important to look at the chebyshev coefficients of the mode we show a numerically computed anti kelvin wave which has little eigenvalue drift but is completely spurious as is obvious from its spectral series third inverting the roles of parameters can drastically modify the spectrum legendre s equation may have either an infinite number of discrete modes or only a handful depending on which parameter is the eigenvalue fourth when the modes are singular but decay to zero at the endpoints as is true of tides a tanh mapping can retrieve the usual exponential accuracy of spectral methods fifth the pseudospectral method is more reliable than deriving a banded galerkin matrix by means of recurrence relations the pseudospectral code is simple to check whereas it is easy to make an untestable mistake with the intricate algebra required for the galerkin method sixth we offer a brief cautionary tale about overlooked modes all these cautions are applicable toallforms of spatial discretization including finite difference and finite element methods however we limit our illustrations to spectral schemes where these difficulties are most easily resolved with a bit of care the pseudospectral method is a very robust and efficient method for solving differential eigenproblems even with singular eigenmodes

spectral methods based on prolate spheroidal wave functions for hyperbolic pdes we examine the merits of using prolate spheroidal wave functions pswfs as basis functions when solving hyperbolic pdes using pseudospectral methods the relevant approximation theory is reviewed and some new approximation results in sobolev spaces are established an optimal choice of the band limit parameter for pswfs is derived for single mode functions our conclusion is that one might gain from using the pswfs over the traditional chebyshev or legendre methods in terms of accuracy and efficiency for marginally resolved broadband solutions

series study of random percolation in three dimensions new high density series data for the mean number percolation probability and susceptibility of finite clusters are presented for bond and site percolation on four standard three dimensional lattices a pade approximant analysis of both high and low density series makes particular use of the rather precise unbiased estimates of the percolation threshold pc obtained recently by heermann and stauffer 1981 for the simple cubic lattice for this lattice the authors obtain the biased estimate gamma 1 73 or 0 03 for the site problem and a similar estimate but with larger uncertainties for the bond problem such a value is significantly larger than earlier series estimates assuming gamma to be universal they obtain precise although biased estimates of pc for both bond and site percolation on all four lattices using the bond estimates of pc they find an overall biased estimate of beta 0 454 or 0 008 for bond percolation on all three dimensional lattices the corresponding site problem requires further study scaling estimates of other critical exponents are alpha 0 64 or 0 05 delta 4 81 or 0 14 delta 2 18 or 0 04 nu 0 88 or 0 02 and nu 0 03 or 0 03

parallel algorithms with local fourier basis abstract we present a nonoverlapping domain decomposition method with local fourier basis applied to a model problem in liquid flames the introduction of domain decomposition techniques in this paper is for numerical and parallel efficiency purposes when one requires a large number of grid points to catch complex structures we obtain then a high order accurate domain decomposition method that allows us to generalize our previous work on the use of local fourier basis to solve combustion problems with nonperiodic boundary conditions m garbey and d tromeur dervout j comput phys 145 316 1998 local fourier basis methodology fully uses the superposition principle to split the searched solution in a numerically computed part and an analytically computed part our present methodology generalizes the israeli et al 1993 j sci comput 8 135 method which applies domain decomposition with local fourier basis to the helmholtz s problem in the present work several new difficulties occur first the problem is unsteady and nonlinear which makes the periodic extension delicate to construct in terms of stability and accuracy second we use a streamfunction biharmonic formulation of the incompressible navier stokes equation in two space dimensions the application of domain decomposition with local fourier basis to a fourth order operator is more difficult to achieve than for a second order operator a systematic investigation of the influence of the method s parameters on the accuracy is done a detail parallel mimd implementation is given we give an a priori estimate that allows the relaxation of the communication between processors for the interface problem treatment results on nonquasi planar complex frontal polymerization illustrate the capability of the method

spectral methods on arbitrary grids stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations these new methods are equivalent to conventional spectral methods but do not rely on specific grid distributions specifically we show how to implement legendre galerkin legendre collocation and laguerre galerkin methodology on arbitrary grids

large mode number eigenvalues of the prolate spheroidal differential equation we show that the eigenvalues of the prolate spheroidal differential equation of zeroth order g n c where c is the so called bandwidth or ellipticity parameter are well approximated for large mode number n by a single function w in the form g n c email protected c c where c p 2 n 1 2 w is defined implicitly as the root of an algebraic equation e min 1 m 1 2 m 1 2 1 w where e is the usual incomplete integral of the second kind and email protected 2 w with c c c w has a weak singularity at c 1 proportional to c 1 log c 1 plus iterated logarithms we give chebyshev series for w accurate for email protected 0

the optimization of convergence for chebyshev polynomial methods in an unbounded domain by using the method of steepest descent i have compared the suitability of three different methods for solving problems in a semi infinite or infinite domain using chebyshev polynomials exponential mappings are uniformyly bad domain truncation and algebraic mapping both work well but each is superior for a different category of problems when the solution is an entire function then domain truncation is best it is always simpler to apply than algebraic mapping and always at least as accurate much more accurate for functions which decay very rapidly for singular functions on the other hand algebraic mapping is better because it is less sensitive to the mapping scale factor l permitting a better compromise in resolving both the singularity and the exponential decay for both types of problems i give simple explicit estimates of both the optimum choise of domain size or mapping factor and of the attainable accuracy for the model entire function exp azk on a semi infinite interval 0 the optimum domain size is l 0 896 na 1k and the smallest attainable error is roughly e n 0 896 k where n is the number of chebyshev polynomals used similar formulas for singular functions handled via algebraic mapping are given in 3 41 to 3 45 below

the general solution of the binding mean spherical approximation for pairing ions the mean spherical approximation msa for an arbitrary mixture of charged hard spheres with saturating bonds is solved in the wertheim formalism only pairs are allowed it is shown that the general solution is given in terms of two scaling parameters and the set of binding fractions one of the scales is the screening parameter and the other is a coupling parameter analogous to that of the simple msa but that now is found solving a cubic equation therefore the full solution requires solvingm 2 nonlinear algebraic equations for a system withm components a brief discussion of the thermodynamics is given

the prolate spheroidal wave functions as invariants of the time reversal operator for an extended scatterer in the fraunhofer approximation the decomposition of the time reversal operator known by the french acronym dort is widely used to detect locate and focus on scatterers in various domains such as underwater acoustics medical ultrasound and nondestructive evaluation in the case of point scatterers the theory is well understood the number of nonzero eigenvalues is equal to the number of scatterers and the eigenvectors correspond to the scatterers green s function in the case of extended objects however the formalism is not as simple it is shown here that in the fraunhofer approximation analytical solutions can be found and that the solutions are functions called prolate spheroidal wave functions these functions have been studied in information theory as a basis of band limited and time limited signals they also arise in optics the theoretical solutions are compared to simulation results most importantly the intuition that for an extended objects the number of nonzero eigenvalues is proportional to the number of resol

high dimensional lipschitz functions are typically flat a homomorphism height function on the d dimensional torus mathbb z n d is a function on the vertices of the torus taking integer values and constrained to have adjacent vertices take adjacent integer values a lipschitz height function is defined similarly but may also take equal values on adjacent vertices for each of these models we consider the uniform distribution over all such functions with predetermined values at some fixed vertices boundary conditions our main result is that in high dimensions and with zero boundary values the random function obtained is typically very flat having bounded variance at any fixed vertex and taking at most c log n 1 d values with high probability this result matches up to constants a lower bound of benjamini yadin and yehudayoff our results extend to any dimension d ge2 if one replaces the torus mathbb z n d by an enhanced version of it the torus mathbb z n d times mathbb z 2 d 0 for some fixed d 0 consequently we establish one side of a conjectured roughening transition in two dimensions the full transition is established for a class of tori with nonequal side lengths including for example the n times lfloor frac 1 10 log n rfloor torus in another case of interest we find that when the dimension d is taken to infinity while n remains fixed the random function takes at most r values with high probability where r 5 for the homomorphism model and r 4 for the lipschitz model suitable generalizations are obtained when n grows with d our results have consequences also for the related model of uniform 3 coloring and establish that for certain boundary conditions a uniformly sampled proper 3 coloring of mathbb z n d will be nearly constant on either the even or odd sublattice our proofs are based on the construction of a combinatorial transformation suitable to the homomorphism model and on a careful analysis of the properties of a class of cutsets which we term odd cutsets for the lipschitz model our results rely also on a bijection of yadin this work generalizes results of galvin and kahn refutes a conjecture of benjamini yadin and yehudayoff and answers a question of benjamini haggstrom and mossel

numerical perturbative and chebyshev inversion of the incomplete elliptic integral of the second kind abstract the incomplete elliptic integral of the second kind e sin t m 0 t 1 m sin 2 t dt where m 0 1 is the elliptic modulus can be inverted with respect to angle t by solving the transcendental equation e sin t m z 0 we show that newton s iteration t n 1 t n e sin t m z 1 m sin 2 t always converges to t z m e 1 z m within a relative error of less than 10 10 in three iterations or less from the first guess t 0 z m 2 r 2 where defining 1 z e 1 m r 1 m 2 2 and atan 1 m we briefly discuss three alternative initialization strategies homotopy initialization t 0 z m 1 m z 0 mt z m m 1 perturbation series in powers of m and inversion of the chebyshev interpolant of the incomplete elliptic integral although all work well and are general strategies applicable to a very wide range of problems none of these three alternatives is as efficient as the empirical initialization which is completely problem specific this illustrates tai tsun wu s maxim usefulness is often inversely proportional to generality

continuum percolation of the four bonding site associating fluids wertheim s integral equation theory for associating fluids is reformulated for the study of the connectedness properties of associating hard spheres with four bonding sites the association interaction is described as a square well saturable attraction between these sites the connectedness version of the ornstein zernike oz integral equation is supplemented by the py like closure relation and solved analytically within an ideal network approximation in which the network is represented as resulting from the crossing of ideal polymer chains the pair connectedness functions and the mean cluster size are calculated and discussed the condition for the percolation transition and the analytical form of the percolation threshold are derived the connection of the percolation with the gas liquid phase transition is discussed

mappings and accuracy for chebyshev pseudo spectral approximations abstract the effect of mappings on the approximation by chebyshev collocation of functions which exhibit localized regions of rapid variation is studied a general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials a particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in applications it is shown that the mapped function can be approximated much more accurately by chebyshev polynomial approximations than the original function the effect on the approximation of introducing subdomains is also studied the accuracy of the pseudo spectral approximation is very sensitive to the location of the interface although this sensitivity is reduced when mappings are employed within the subdomains

nonlinear optimization quadrature and interpolation we present a nonlinear optimization procedure for the design of generalized gaussian quadratures for a fairly broad class of functions while some of the components of the algorithm have been published previously we introduce an improved procedure for the determination of an acceptable initial point for the continuation scheme that stabilizes the newton type process used to find the quadratures the resulting procedure never failed when applied to chebyshev systems for which the existence and uniqueness of generalized gaussian quadratures are well known it also worked for many non chebyshev systems for which the generalized gaussian quadratures are not guaranteed to exist the performance of the algorithm is illustrated with several numerical examples some of the presented quadratures integrate efficiently large classes of singular functions

an improved estimate of pswf approximation and approximation by mathieu functions abstract in this paper an error estimate of spectral approximations by prolate spheroidal wave functions pswfs with explicit dependence on the bandwidth parameter and optimal order of convergence is derived which improves the existing result in chen et al spectral methods based on prolate spheroidal wave functions for hyperbolic pdes siam j numer anal 43 5 2005 1912 1933 the underlying argument is applied to analyze spectral approximations of periodic functions by mathieu functions which leads to new estimates featured with explicit dependence on the intrinsic parameter

on the analytical solution of the ornstein zernike equation with yukawa closure we discuss the solution of the ornstein zernike equation for most general closure consisting of a sum ofm yukawa type exponentials a formal solution for the factored case is bound for an arbitrary mixture of hard spheres introducing a general scaling matrix of dimensions m m a sufficient number of equations for this matrix is obtained from symmetry considerations and the boundary condition we discuss also restricted and semirestricted case for which explicit solutions in terms of the scaling parameters and input parameters are found

pseudospectral methods on a semi infinite interval with application to the hydrogen atom a comparison of the mapped fourier sine method with laguerre series and rational chebyshev expansions the fourier sine with mapping pseudospectral algorithm of fattal et al phys rev e 53 1996 1217 has been applied in several quantum physics problems here we compare it with pseudospectral methods using laguerre functions and rational chebyshev functions we show that laguerre and chebyshev expansions are better suited for solving problems in the interval r 0 for example the coulomb schrodinger equation than the fourier sinemapping scheme all three methods give similar accuracy for the hydrogen atom when the scaling parameter l is optimum but the laguerre and chebyshev methods are less sensitive to variations in l we introduce a new variant of rational chebyshev functions which has a more uniform spacing of grid points for large r and gives somewhat better results than the rational chebyshev functions of boyd j comp phys 70 1987 63

improved peierls argument for high dimensional ising models we consider the low temperature expansion for the ising model on mathbb z d d geqslant 2 with ferromagnetic nearest neighbor interactions in terms of peierls contours we prove that the expansion converges for all temperatures smaller than cd log d 1 which is the correct order in d

rapid prolate pseudospectral differentiation and interpolation with the fast multipole method pseudospectral methods utilizing prolate spheroidal wave functions as basis functions have been shown to possess advantages over the conventional pseudospectral methods based on trigonometric and orthogonal polynomials however the spectral differentiation and interpolation steps of the prolate pseudospectral method involve matrix vector products which if evaluated directly entail o n2 memory requirement and computational complexity where n is the number of unknowns utilized for discretization and interpolation in this work we show that the fast multipole method fmm can be used to reduce the memory requirement and computational complexity of the prolate pseudospectral method to o n example simulation results demonstrate the speed and accuracy of the resulting fast prolate pseudospectral solver

a new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations a new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested this method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions stability and convergence results are proven for the chebyshev approximation of linear scalar hyperbolic equations the eigenvalues of this method applied to parabolic equations are shown to be real and negative

fluids with highly directional attractive forces i statistical thermodynamics a new formulation of statistical thermodynamics is derived for classical fluids of molecules that tend to associate into dimers and possibly highers mers due to highly directional attraction a breakup of the pair potential into repulsive and highly directionally attractive parts is introduced into the expansion of the logarithm of the grand partition function in fugacity graphs the bonding by the directional attraction is used to classify the graphs and to introduce a topological reduction which results in the replacement of the fugacity by two variables singlet density and monomer density 0 results for the thermodynamic functions as functionals of and 0 are given in the form of graph sums pair correlations are analyzed in terms of a new matrix analog of the direct correlation function it is shown that the low density limit is treated exactly while major difficulties arise when the mayer expansion which employs onlyp is used the intricate resummations required for the mayer expansion are illustrated for the case where dimers are the only association products

orthogonal rational functions on a semi infinite interval abstract by applying a mapping to the chebyshev polynomials we define a new spectral basis the rational chebyshev functions on the semi infinite interval denoted by tl n y continuing earlier work by the author and by grosch and orszag we show that these rational functions inherit most of the good numerical characteristics of the chebyshev polynomials orthogonality completeness exponential or infinite order convergence matrix sparsity for equations with polynomial coefficients and simplicity seven numerical examples illustrate their versatility the charney stability problem of meteorology for example is solved to show the feasibility of applying spectral methods to a semi infinite atmosphere for functions that are singular at both endpoints such as k 1 y one may combine rational chebyshev functions with a preliminary mapping to obtain a single exponentially convergent expansion on y 0 finally we successfully generalize the wkb method to obtain for the j 0 bessel function an amplitude phase approximation which is convergent rather than asymptotic and is accurate not merely for large y but for all y even the origin

generalized holomorphic szego kernel in 3d spheroids monogenic orthogonal polynomials over 3d prolate spheroids were previously introduced and shown to have some remarkable properties in particular the underlying functions take values in the quaternions identified with r 4 and are generally assumed to be nullsolutions of the well known moisil theodoresco system in this paper we show that these polynomial functions play an important role in defining the szego kernel function over the surface of 3d prolate spheroids as a concrete application we prove an explicit expression of the monogenic szego kernel function over 3d prolate spheroids and present two numerical examples

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

analysis of spectral approximations using prolate spheroidal wave functions in this paper the approximation properties of the prolate spheroidal wave functions of order zero pswfs are studied and a set of optimal error estimates are derived for the pswf approximation of non periodic functions in sobolev spaces these results serve as an indispensable tool for the analysis of pswf spectral methods a pswf spectral galerkin method is proposed and analyzed for elliptic type equations illustrative numerical results consistent with the theoretical analysis are also presented

new quadrature formulas from conformal maps gauss and clenshaw curtis quadrature like legendre and chebyshev spectral methods make use of grids strongly clustered at boundaries from the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal nevertheless such methods may waste a factor of pi 2 with respect to each space dimension we propose new nonpolynomial quadrature methods that avoid this effect by conformally mapping the usual ellipse of convergence to an infinite strip or another approximately straight sided domain the new methods are compared with related ideas of bakhvalov kosloff and tal ezer rokhlin and alpert and others an advantage of the conformal mapping approach is that it leads to theorems guaranteeing geometric rates of convergence for analytic integrands for example one of the formulas presented is proved to converge 50 faster than gauss quadrature for functions analytic in an varepsilon neighborhood of 1 1

local distortion of m conformal mappings a conformal mapping in a plane domain locally maps circles to circles more generally quasiconformal mappings locally map circles to ellipses of bounded distortion in this work we study the corresponding situation for solutions to stein weiss systems in the n 1 d euclidean space this class of solutions is a transformation of a subset of monogenic locally quasiconformal mappings with nonvanishing jacobian in the theoretical part of this work we prove that an m conformal mapping locally maps the unit hypersphere onto explicitly characterized hyperellipsoids and vice versa then we discuss quasiconformal radial mappings and their relations with the cauchy kernel and p monogenic mappings this is followed by the consideration of quadratic m conformal mappings in the applications part of this work we provide the reader with some plot examples that demonstrate the effectiveness of our approach

comparison of interpolating functions and interpolating points in full wave multilevel green s function interpolation method the difficulty for the solution of full wave electromagnetic problems using multilevel green s function interpolation method mlgfim lies in applying interpolating approaches to efficiently and accurately approximate green s function with rapidly changing phase term we compare various interpolating schemes when radial basis function rbf is employed for the interpolation of scattered data of green s function we show that the infinitely smooth gaussian ga rbf has the best interpolation accuracy in order to improve the interpolation efficiency a new kind of staggered tartan grid is proposed a good calculation method for the shape parameter in ga rbf is given to solve its sensitivity to the group size and the number of interpolation points based on the analysis of variation of the number of interpolation points with electric length of the group adaptive choice of the types of interpolation functions and interpolation points are employed numerical examples show that the computational efficiency of this new interpolation scheme is much improved over the previously reported ones

inverse methods for reconstruction of channel taps in ofdm systems we describe a novel pilot aided method for estimation of doubly selective wireless channels in ofdm systems we compute the first few fourier coefficients of each channel tap from the pilot information we then estimate the bem coefficients of the channel taps from their respective fourier coefficients using a recently developed inverse reconstruction method for a system with l channel taps the proposed method uses o l log l operations and o l memory per ofdm symbol we validate our method by simulating a system conforming to the ieee 802 16e standard

a numerical comparison of seven grids for polynomial interpolation on the interval abstract seven types of chebyshev like grids in one dimension are compared according to four different criteria for accuracy the grid which minimizes the lebesgue constant is the best because it performs fairly well by all four criteria for the same reason minimizing the lebesgue constant seems to be the most useful measure of optimality because grids that are good according to this criterion are good when measured by other criteria too cfm optimality which is the property that all cardinal functions lagrange fundamental polynomials have maxima at the interpolation points seems to be the least discriminating criterion because all seven grids generate cardinal functions that have maxima at or very near the interpolation points the difference between these grids on all four criteria always less than a factor of two and usually much smaller are sufficiently modest so that the final choice between grids should probably not be made because of accuracy but rather based on other criteria such as ease of programming analytical simplicity conformality with other approximations and timestep in applications to partial differential equations to explore such nonaccuracy issues we also compared six of the grids on the basis of maximum allowed timestep when the grid is used to discretize the spatial coordinate and an explicit scheme is used for time marching the lebesgue optimal grid is also nearly optimal in the sense of allowing the longest timestep with a pseudospectral collocation algorithm the lebesgue grid allows a timestep three halves as long as that of the legendre lobatto spectral element method

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

regular article spectral collocation time domain modeling of diffractive optical elements a spectral collocation multi domain scheme is developed for the accurate and efficient time domain solution of maxwell s equations within multi layered diffractive optical elements special attention is being paid to the modeling of out of plane waveguide couplers emphasis is given to the proper construction of high order schemes with the ability to handle very general problems of considerable geometric and material complexity central questions regarding efficient absorbing boundary conditions and time stepping issues are also addressed the efficacy of the overall scheme for the time domain modeling of electrically large and computationally challenging problems is illustrated by solving a number of plane as well as non plane waveguide problems

prolate spheroidal wavefunctions as an alternative to chebyshev and legendre polynomials for spectral element and pseudospectral algorithms prolate spheroidal functions of order zero are generalizations of legendre polynomials which when the bandwidth parameter c 0 oscillate more uniformly on x 1 1 than either chebyshev or legendre polynomials this suggests that compared to these polynomials prolate functions give more uniform spatial resolution further when used as the spatial discretization for time dependent partial differential equations in combination with explicit time marching prolate functions allow a longer stable timestep than legendre polynomials we show that these advantages are real and further that it is almost trivial to modify existing pseudospectral and spectral element codes to use the prolate basis however improvements in spatial resolution are at most a factor of 2 approached slowly as n the timestep can be lengthened by several times but not by a factor that grows rapidly with n the prolate basis is not likely to radically expand the range of problems that can be done on a workstation however for production runs on the bleeding edge edge of arithmurgy such as numerical weather prediction the rewards for switching to a prolate basis may be considerable

the eps method a new method for constructing pseudospectral derivative operators we develop a new type of derivative matrix for pseudospectral methods the norm of these matrices grows at the optimal rate o n 2 for n by n matrices in contrast to standard pseudospectral constructions that result in o n 4 growth of the norm the smaller norm has a big advantage when using the derivative matrix for solving time dependent problems such as wave propagation the construction is based on representing the derivative operator as an integral kernel and does not rely on the interpolating polynomials in particular we construct second derivative matrices that incorporate dirichlet or neumann boundary conditions on an interval and on the disk but the method can be used to construct a wide variety of commonly used operators for solving pdes and integral equations the construction can be used with any quadrature including traditional gauss legendre quadratures but we have found that by using quadratures based on prolate spheroidal wave functions we can achieve a near optimal sampling rate close to two points per wavelength even for non periodic problems we provide numerical results for the new construction and demonstrate that the construction achieves similar or better accuracy than traditional pseudospectral derivative matrices while resulting in a norm that is orders of magnitude smaller than the standard construction to demonstrate the advantage of the new construction we apply the method for solving the wave equation in constant and discontinuous media and for solving pdes on the unit disk we also present two compression algorithms for applying the derivative matrices in o nlogn operations

static structure factor and thermodynamic properties of a binary yukawa mixture abstract we use the solution of the ornstein zernike equation in the mean spherical approximation to find the static structure factor for a hard spheres yukawa fluid the thermodynamic and the structure properties of this fluid are given in terms of an accumulative parameter which satisfies a polynomial equation of degree n 4 this parameter is obtained mumerically by an iterative method we study binary mixtures with a factored interaction for which the classical lorentz berthelot rules are satisfied our result for the static structure factor and thermodynamics properties are in good agreement with the computer simulations and former numerical solutions

spectral analysis of the finite hankel transform and circular prolate spheroidal wave functions in this paper we develop two practical methods for the computation of the eigenvalues as well as the eigenfunctions of the finite hankel transform operator these different eigenfunctions are called circular prolate spheroidal wave functions cpswfs this work is motivated by the potential applications of the cpswfs as well as the development of practical methods for computing their values also in this work we should prove that the cpswfs form an orthonormal basis of the space of hankel band limited functions an orthogonal basis of l 2 0 1 and an orthonormal system of l 2 0 our computation of the cpswfs and their associated eigenvalues is done by the use of two different methods the first method is based on a suitable matrix representation of the finite hankel transform operator the second method is based on the use of an efficient quadrature method based on a special family of orthogonal polynomials also we give two maple programs that implement the previous two methods finally we present some numerical results that illustrate the results of this work

an adaptive algorithm for spectral computations on unbounded domains abstract we consider spectral methods for solving differential equations on unbounded domains in particular we consider a spectral rational function method and a method based on domain truncation combined with fourier series both methods contain a free parameter which determines the accuracy of the spectral method when solving evolution equations the optimal parameter may change with time resulting in a significant loss of accuracy if the parameter is kept fixed we develop an automatic algorithm which adapts the parameter at regular time levels to maintain optimum accuracy in numerical tests we found that the proposed method is robust and efficient