on the extreme values of the riemann zeta function on random intervals of the critical line in the present paper we show that under the riemann hypothesis and for fixed h epsilon 0 the supremum of the real and the imaginary parts of log zeta 1 2 it for t in ut h ut h are in the interval 1 epsilon log log t 1 epsilon log log t with probability tending to 1 when t goes to infinity u being a uniform random variable in 0 1 this proves a weak version of a conjecture by fyodorov hiary and keating which has recently been intensively studied in the setting of random matrices we also unconditionally show that the supremum of mathfrak r log zeta 1 2 it is at most log log t g t with probability tending to 1 g being any function tending to infinity at infinity

the effect of the 2 d laplacian operator approximation on the performance of finite difference time domain schemes for maxwell s equations the behavior of the finite difference time domain method fdtd is investigated with respect to the approximation of the two dimensional laplacian associated with the curl curl operator our analysis begins from the observation that in a two dimensional space the yee algorithm approximates the laplacian operator via a strongly anisotropic 5 point approximation it is demonstrated that with the aid of a transversely extended curl operator any 9 point laplacian can be mapped onto fdtd update equations our analysis shows that the mapping of an isotropic laplacian approximation results in an isotropic and less dispersive fdtd scheme the properties of the extended curl are further explored and it is proved that a unity courant number can be achieved without the resulting scheme suffering from grid decoupling additionally the case of a 25 point isotropic laplacian is examined and it is shown that the corresponding scheme is fourth order accurate in space and exhibits isotropy up to sixth order representative numerical simulations are performed that validate the theoretically derived results

phase error control for fd td methods of second and fourth order accuracy for fd td methods used to solve maxwell s equations we determine the spatial resolution of the discretized domain in terms of the total computation time and the desired phase error it is shown that the spatial step should vary as spl delta x spl sim g e sub spl phi t sub c sup 1 s in order to maintain a prescribed phase error level e sub spl phi throughout the computation time t sub c where s 2 or 4 is the spatial order of accuracy of the scheme and g is a geometric factor significantly we show that the rule of thumb of using 10 20 points per wavelength to determine the spatial cell size for the standard scheme is not optimal our results are verified by numerical simulations in two dimensions with the yee 1966 scheme and a new fourth order accurate fd td scheme

comparison of the dispersion properties of higher order fdtd schemes and equivalent sized mrtd schemes the dispersion errors of higher order finite difference time domain ho fdtd algorithms are compared to those of multiresolution time domain mrtd algorithms that have equivalent spatial stencil sizes both scaling function based mrtd s mrtd and wavelet function based mrtd w mrtd schemes are considered in particular the mrtd schemes considered include the coifman scaling functions and the cohen daubechies feauveau cdf biorthogonal scaling and wavelet functions in general the ho fdtd schemes are more accurate than their mrtd counterparts

a block pseudospectral method for maxwell s equations a block pseudospectral bps method is proposed as a new way to couple pseudospectral discretizations across interfaces in computations for a linear hyperbolic system the coupling is achieved via discretized derivative matching conditions obtained from the system compared to the standard technique of imposing compatibility conditions based on characteristics of the system the bps method offers better stability and accuracy especially in the case where equation coefficients are discontinuous computational examples for maxwell s equations in nonhomogeneous media demonstrate that bps retains high accuracy over times that are orders of magnitude larger than those for not only low order methods such as yee s but also high order methods such as characteristic based spectral elements

construction and analysis of fourth order finite difference schemes for the acoustic wave equation in nonhomogeneous media in this article we construct and analyse a family of finite difference schemes for the acoustic wave equation with variable coefficients these schemes are fourth order accurate in space and time in the case of smooth media and are designed to remain stable and optimal for reflection transmission phenomena in the case of discontinuous coefficients together with a detailed mathematical study various numerical results are presented

ising model on nonorientable surfaces exact solution for the mobius strip and the klein bottle closed form expressions are obtained for the partition function of the ising model on an mxn simple quartic lattice embedded on a mobius strip and a klein bottle the solutions all lead to the same bulk free energy but for finite m and n the expressions are different depending on whether the strip width m is odd or even finite size corrections at criticality are analyzed and compared with those under cylindrical and toroidal boundary conditions our results are consistent with the conformal field prediction of a central charge c 1 2 provided that the twisted mobius boundary condition is regarded as a free or fixed boundary

equidistribution in shrinking sets and l 4 norm bounds for automorphic forms we study two closely related problems stemming from the random wave conjecture for maas forms the first problem is bounding the l 4 norm of a maas form in the large eigenvalue limit we complete the work of spinu to show that the l 4 norm of an eisenstein series e z 1 2 it g restricted to compact sets is bounded by sqrt log t g the second problem is quantum unique ergodicity in shrinking sets we show that by averaging over the centre of hyperbolic balls in gamma backslash mathbb h quantum unique ergodicity holds for almost every shrinking ball whose radius is larger than the planck scale this result is conditional on the generalised lindelof hypothesis for hecke maas eigenforms but is unconditional for eisenstein series we also show that equidistribution for hecke maas eigenforms need not hold at or below the planck scale finally we prove similar equidistribution results in shrinking sets for heegner points and closed geodesics associated to ideal classes of quadratic fields

dynamics of continuous time quantum walks in restricted geometries we study quantum transport on finite discrete structures and we model the process by means of continuous time quantum walks a direct and effective comparison between quantum and classical walks can be attained based on the average displacement of the walker as a function of time indeed a fast growth of the average displacement can be advantageously exploited to build up efficient search algorithms by means of analytical and numerical investigations we show that the finiteness and the inhomogeneity of the substrate jointly weaken the quantum walk performance we further highlight the interplay between the quantum walk dynamics and the underlying topology by studying the temporal evolution of the transfer probability distribution and the lower bound of long time averages

dissimilar behaviors of coherent exciton transport on scale free networks with identical degree sequence conventional wisdom is that the power law degree sequence has a profound impact on diverse dynamical processes taking place on scale free networks here we show that power law degree distribution alone is not sufficient to determine the behaviors of coherent exciton transport on scale free networks for this purpose we study coherent exciton transport on a class of scale free networks controlled by a tunable parameter q which have the same degree sequence independent of q but are very different in other structural properties determined by q such as average distance and fractality first through numerical determination of the time evolutions of return probabilities we compare in detail the dissimilarities in oscillation periods and oscillation amplitudes between two limiting cases of q 0 and q 1 second we demonstrate that in the network of q 0 up to the third generation the quantum transport problem between node groups partitioned by node degrees can be mapped onto a line furthermore we show that in the case of long time limit the transition probabilities in the networks of q 0 and q 1 exhibit different characteristic patterns with identical values finally we find that the degree averaged return probability decreases when the parameter q increases from 0 to 1 indicating that the stronger localization can be found in those networks with lower values of q our results suggest that it is far from enough to characterize the behaviors of coherent exciton transport on scale free networks merely by power law degree distribution

a hierarchy of explicit low dispersion fdtd methods for electrically large problems a framework for the construction of a hierarchy of explicit low dispersion low anisotropy fdtd algorithms is developed herein the framework allows for the separation of dispersive and anisotropic effects due to a given explicit fdtd model construction using an extended stencil on a yee type staggered grid isotropic low dispersion 2 4 and 4 4 fdtd algorithms are constructed by eliminating second and fourth order errors due to dispersion and anisotropy the dispersive and anisotropic performance of these algorithms is compared to previously published schemes that fit within this hierarchical structure the algorithms are examined for propagation of a wave pulse in an electrically large domain

toward the construction of a fourth order difference scheme for transient em wave simulation staggered grid approach a compact central difference approximation in conjunction with the yee 1966 grid is used to compute the spatial derivatives in maxwell s equations to advance the semi discrete equations the four stage runge kutta rk integrator is invoked this combination of spatial and temporal differencing leads to a scheme that is fourth order accurate conditionally stable and highly efficient moreover the use of compact differencing allows one to apply the compact operator in the vicinity of a perfect conductor an attribute not found in other higher order methods results are provided that quantify the spectral properties of the method simulations are conducted on problem spaces that span one and three dimensions and whose domains are of the open and closed type results from these simulations are compared with exact closed form solutions the agreement between these results is consistent with numerical analysis

an upwinding boundary condition capturing method for maxwell s equations in media with material interfaces by using ghost points on either side of the interfaces a global second order accurate upwinding boundary condition capturing method for time domain maxwell s equations in media with material interfaces is proposed the equations are discretized on a uniform cartesian grid and the interfaces are allowed to intersect the grid in an arbitrary fashion the method is then obtained by combining central finite difference schemes with applicable nodes being replaced by the ghost points and upwinding technique with jump conditions across the interfaces being captured in a manner that the upwind property is always satisfied the resulting discretization has the desirable property that the allowed time step size is independent of the locations and the shapes of the interfaces numerical examples are then given to demonstrate the second order accuracy as well as the stability of the method where it is used to study wave equations with various types of material interfaces including electromagnetic scattering of a plane incident wave by a dielectric circular cylinder

an explicit fourth order orthogonal curvilinear staggered grid fdtd method for maxwell s equations the explicit fourth order staggered finite difference time domain scheme previously proposed for a cartesian grid is extended to maxwell s equations in an orthogonal curvilinear coordinate system and applied to electromagnetic wave problems a simple technique is also presented for generating orthogonal curvilinear grids that conform to the material boundaries and interfaces of the problem numerical experiments are presented to illustrate the efficiency and accuracy of the method

relaxation properties of small world networks recently watts and strogatz introduced the so called small world networks in order to describe systems that combine simultaneously properties of regular and random lattices in this work we study diffusion processes defined on such structures by considering explicitly the probability for a random walker to be present at the origin the results are intermediate between the corresponding ones for fractals and cayley trees

reducing the phase error for finite difference methods without increasing the order the phase error in finite difference fd methods is related to the spatial resolution and thus limits the maximum grid size for a desired accuracy greater accuracy is typically achieved by defining finer resolutions or implementing higher order methods both these techniques require more memory and longer computation times in this paper new modified methods are presented which are optimized to problems of electromagnetics simple methods are presented that reduce numerical phase error without additional processing time or memory requirements furthermore these methods are applied to both the helmholtz equation in the frequency domain and the finite difference time domain fdtd method both analytical and numerical results are presented to demonstrate the accuracy of these new methods

continuous time quantum walks on one dimensional regular networks in this paper we consider continuous time quantum walks ctqws on a one dimensional ring lattice of n nodes in which every node is connected to its 2m nearest neighbors m on either side in the framework of the bloch function ansatz we calculate the space time transition probabilities between two nodes of the lattice we find that the transport of ctqws between two different nodes is faster than that of the classical continuous time random walks ctrws the transport speed which is defined by the ratio of the shortest path length and propagating time increases with the connectivity parameter m for both ctqws and ctrws for fixed parameter m the transport of ctrws gets slower with the increase of the shortest distance while the transport speed of ctqws turns out to be a constant value in the long time limit depending on the network size n and connectivity parameter m the limiting probability distributions of ctqws show various patterns when the network size n is an even number the probability of being at the original node differs from that of being at the opposite node which also depends on the precise value of parameter m

a highly efficient time space domain optimized method with lax wendroff type time discretization for the scalar wave equation abstract in this paper we propose an efficient time space domain optimized optts finite difference scheme to model 2d and 3d scalar wave propagation it adopts piecewise constant interpolation coefficients for several consecutive courant number ranges which avoids the extra time costs caused by loading the coefficients consecutively according to different wave velocities in heterogeneous media the new scheme is capable of suppressing numerical dispersion errors using a large time interval because we introduce a lax wendroff type time discretization which utilizes the wave field information of o m 1 m 2 neighboring grid points in each step with merely o m 1 m 2 computation where m 1 and m 2 are the stencil radii to generate the 2nd order and 4th order spatial derivatives the improved computational efficiency and accuracy of optts are validated by substantial experiments using both theoretical analysis and numerical modeling they show that in heterogeneous models the maximal courant numbers reach 1 0 and 0 85 in 2d and 3d cases respectively without severe temporal error and the scheme is computationally nearly three times faster than its high order counterpart providing a powerful tool for large scale modeling and high resolution imaging

dispersion relation preserving fdtd algorithms for large scale three dimensional problems we introduce dispersion relation preserving drp algorithms to minimize the numerical dispersion error in large scale three dimensional 3d finite difference time domain fdtd simulations the dispersion error is first expanded in spherical harmonics in terms of the propagation angle and the leading order terms of the series are made equal to zero frequency dependent fdtd coefficients are then obtained and subsequently expanded in a polynomial taylor series in the frequency variable an inverse fourier transformation is used to allow for the incorporation of the new coefficients into the fdtd updates butterworth or chebyshev filters are subsequently employed to fine tune the fdtd coefficients for a given narrowband or broadband range of frequencies of interest numerical results are used to compare the proposed 3d drp fdtd schemes against traditional high order fdtd schemes

an fdtd scheme on a face centered cubic fcc grid for the solution of the wave equation a method is proposed to improve the numerical dispersion characteristics for simulations of the scalar wave equation in 3d using the fdtd method the improvements are realized by choosing a face centered cubic fcc grid instead of the typical cartesian yee grid which exhibits non physical distortions of the wavefront due to the fd stencil fcc grids are the logical extension of hexagonal grids in 2d and have been shown previously to provide optimal sampling of space based on close packing of spheres highest density the difference equations are developed for the wave equation on this alternative grid and the dispersion relationship and stability for grids of equal and non equal aspect ratios are derived a comparison is made between fcc and cartesian formulations based upon having an equal volume density of gridpoints in each method i e the computational storage requirements of each method would be the same for the same simulated space the comparison shows that the fcc grid exhibits a much more isotropic dispersion relation than the cartesian grid of equivalent density furthermore for an equivalent density the fcc method has a more relaxed stability criterion by a factor of approximately 1 35 resulting in a further reduction in computational resources

decouplings for surfaces in r4 abstract we prove a sharp decoupling for nondegenerate surfaces in r 4 this puts the progress in 1 on the lindelof hypothesis into a more general perspective

fdtd on face centered cubic fcc grids for maxwell s equations the standard fdtd method for maxwell s equations is reformulated on face centered cubic fcc grids as an alternative to the standard cartesian yee grid dispersion and stability for the formulation are also presented and a comparison is made with the standard fdtd formulation using the metric of equal volume density of gridpoints although the new formulation has four times as many lattice points per unit voxel in comparison with the standard formulation the new formulation shows much better grid isotropy furthermore the new formulation has a stability criterion that is approximately 1 4 times more relaxed than that of the standard formulation

large greatest common divisor sums and extreme values of the riemann zeta function it is shown that the maximum of 1 2 it on the interval t1 2 t t is at least exp 1 2 o 1 logtlogloglogt loglogt our proof uses soundararajan s resonance method and a certain large greatest common divisor sum the method of proof shows that the absolute constant a in the inequality sup 1 n1 nn k l 1ngcd nk nl nknl nexp alognlogloglognloglogn established in a recent paper of ours cannot be taken smaller than 1

relative accuracy of several finite difference time domain methods in two and three dimensions a comparison of the accuracy of several orthogonal grid finite difference time domain fdtd schemes is made in both two and three dimensions the relative accuracy is determined from the dispersion error associated with each algorithm and the number of floating point operations required to obtain a desired accuracy level in general in both 2 d and 3 d fourth order algorithms are more efficient than second order schemes in terms of minimizing the number of computations for a given accuracy level in 2 d a second order approach proposed by z chen et al 1991 is much more accurate than the scheme of k s yee 1966 for a given amount of computation and can be as efficient as fourth order algorithms in 3 d yee s algorithm is slightly more efficient than the approach of chen et al in terms of operations but much more efficient in terms of memory requirements

maximum of the riemann zeta function on a short interval of the critical line we prove the leading order of a conjecture by fyodorov hiary and keating about the maximum of the riemann zeta function on random intervals along the critical line more precisely as t for a set of t t 2t of measure 1 o 1 t we have max t u 1log 12 iu 1 o 1 loglogt

the spatial fourth order energy conserved s fdtd scheme for maxwell s equations abstract in this paper we develop a new spatial fourth order energy conserved splitting finite difference time domain method for maxwell s equations based on the staggered grids the splitting technique is applied to lead to a three stage energy conserved splitting scheme at each stage using the spatial fourth order difference operators on the strict interior nodes by a linear combination of two central differences one with a spatial step and the other with three spatial steps we first propose the spatial high order near boundary differences on the near boundary nodes which ensure the scheme to preserve energy conservations and to have fourth order accuracy in space step the proposed scheme has the important properties energy conserved unconditionally stable non dissipative high order accurate and computationally efficient we first prove that the scheme satisfies energy conversations and is in unconditional stability we then prove the optimal error estimates of fourth order in spatial step and second order in time step for the electric and magnetic fields and obtain the convergence and error estimate of divergence free as well numerical dispersion analysis and numerical experiments are presented to confirm our theoretical results

power moments and value distribution of functions in this paper we study various abscissae which one can associate to a given function f or rather to the power moments of f these are motivated by long standing open problems in analytic number theory we show how these abscissae connect to the distribution of values of f in a very elegant way using convex conjugates this connection allows us to show which abscissae are realizable for both general and more specific arithmetical functions further it may give a new approach to for example dirichlet s divisor problem

an isotropy improved nonstandard finite difference time domain method by introducing extra degree of freedom into the space domain stencils an isotropy improved nonstandard finite differ ence nsfd 2 2 method is presented in this paper the technique is also extended to the higher order 2 4 stencils both analytical and numerical results are presented and compared to demonstrate the accuracy and the strengths of these methods index terms dispersion finite difference time domain fdtd methods isotropy nonstandard

the spectral order of accuracy a new unified tool in the design methodology of excitation adaptive wave equation fdtd schemes we define a new concept termed the spectral order of accuracy sooa which is the spectral domain analogue of the familiar order of accuracy ooa the sooa is pivotal in a refined version of a recently introduced methodology for formulating excitation adaptive wave equation fdtd we fdtd schemes described below this concept is the basis for a unified classification for both existing and new schemes both one and two dimensional cases are presented for boundless source free homogeneous isotropic and lossless media the 1 d and 2 d cases are developed in detail for the 3 2m 1 temporal spatial and 3 3 2 d stencils respectively stability analysis is built into the methodology in terms of either analytical conditions or stability maps defined herein the methodology is seen as a generalization of many existing schemes that also provides a unified tool for a systematical design of we fdtd schemes subject to specific requirements in terms of the spectral content of the excitation the computational efficiency for all schemes remains the same for a given stencil since the core of the fdtd code is unchanged between schemes the difference being only in the values of scheme coefficients

a staggered fourth order accurate explicit finite difference scheme for the time domain maxwell s equations abstract we consider a model explicit fourth order staggered finite difference method for the hyperbolic maxwell s equations appropriate fourth order accurate extrapolation and one sided difference operators are derived in order to complete the scheme near metal boundaries and dielectric interfaces an eigenvalue analysis of the overall scheme provides a necessary but not sufficient stability condition and indicates long time stability numerical results verify both the stability analysis and the scheme s fourth order convergence rate over complex domains that include dielectric interfaces and perfectly conducting surfaces for a fixed error level we find the fourth order scheme is computationally cheaper in comparison to the yee scheme by more than an order of magnitude some open problems encountered in the application of such high order schemes are also discussed

on the multilinear restriction and kakeya conjectures we prove d linear analogues of the classical restriction and kakeya conjectures in rd our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians closely related to heat flow we conclude by giving some applications to the corresponding variable coefficient problems and the so called joints problem as well as presenting some n linear analogues for n d

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

quantum transport on two dimensional regular graphs we study the quantum mechanical transport on two dimensional graphs by means of continuous time quantum walks and analyse the effect of different boundary conditions bcs for periodic bcs in both directions i e for tori the problem can be treated in a large measure analytically some of these results carry over to graphs which obey open boundary conditions obcs such as cylinders or rectangles under obcs the long time transition probabilities lps also display asymmetries for certain graphs as a function of their particular sizes interestingly these effects do not show up in the marginal distributions obtained by summing the lps along one direction

coherent exciton transport and trapping on long range interacting cycles we consider coherent exciton transport modeled by continuous time quantum walks on long range interacting cycles lrics which are constructed by connecting all the two nodes of distance m in the cycle graph lric has a symmetric structure and can be regarded as the extensions of the cycle graph nearest neighboring lattice for small values of m the classical and quantum return probabilities show power law behavior p t approximately t 0 5 and pi t approximately t 1 respectively however for large values of m the classical and quantum efficiency scales as p t approximately t 1 and pi t approximately t 2 we give a theoretical explanation of this transition using the method of stationary phase approximation in the long time limit depending on the network size n and parameter m the limiting probability distributions of quantum transport show various patterns when the network size n is an even number we find an asymmetric transition probability of quantum transport between the initial node and its opposite node this asymmetry depends on the precise values of n and m finally we study the transport processes in the presence of traps and find that the survival probability decays faster on networks of large m

the correct formulation of intermediate boundary conditions for runge kutta time integration of initial boundary value problems a disturbing feature of applying explicit high order runge kutta rk time integrators to initial boundary value problems solved by the method of lines is the loss of accuracy that results from wrong specifications of intermediate stage boundary conditions the solution proposed here is to prescribe analytically those values that would result from applying the rk solver at the boundaries and hence maintain the order of the rk method without a reduction in the step size limit the procedure is described in detail for nonlinear hyperbolic equations and both scalar and vector examples are presented

a second order 3d electromagnetics algorithm for curved interfaces between anisotropic dielectrics on a yee mesh a new frequency domain electromagnetics algorithm is developed for simulating curved interfaces between anisotropic dielectrics embedded in a yee mesh with second order error in resonant frequencies the algorithm is systematically derived using the finite integration formulation of maxwell s equations on the yee mesh second order convergence of the error in resonant frequencies is achieved by guaranteeing first order error on dielectric boundaries and second order error in bulk possibly anisotropic regions convergence studies conducted for an analytically solvable problem and for a photonic crystal of ellipsoids with anisotropic dielectric constant both show second order convergence of frequency error the convergence is sufficiently smooth that richardson extrapolation yields roughly third order convergence the convergence of electric fields near the dielectric interface for the analytic problem is also presented

a finite difference time domain algorithm optimized for arbitrary propagation angles we present a two dimensional higher order fdtd algorithm to minimize the numerical dispersion error at arbitrary preassigned angles filtering schemes are further employed to optimize the fdtd frequency response at a given frequency range a stability analysis of the resulting fdtd algorithms is included numerical dispersion properties are compared against yee s scheme and a higher order 2 4 fdtd scheme numerical results demonstrate that the dispersion error around a preassigned angle and with filtering around a frequency band of choice can be significantly reduced

optimal staggered grid finite difference schemes by combining taylor series expansion and sampling approximation for wave equation modeling high order staggered grid finite difference sfd schemes have been universally used to improve the accuracy of wave equation modeling however the high order sfd coefficients on spatial derivatives are usually determined by the taylor series expansion te method which just leads to great accuracy at small wavenumbers for wave equation modeling some conventional optimization methods can achieve high accuracy at large wavenumbers but they hardly guarantee the small numerical dispersion error at small wavenumbers in this paper we develop new optimal explicit sfd esfd and implicit sfd isfd schemes for wave equation modeling we first derive the optimal esfd and isfd coefficients for the first order spatial derivatives by applying the combination of the te and the sampling approximation to the dispersion relation and then analyze their numerical accuracy finally we perform elastic wave modeling with the esfd and isfd schemes based on the te method and the optimal method respectively when the appropriate number and interval for the sampling points are chosen these optimal schemes have extremely high accuracy at small wavenumbers and can also guarantee small numerical dispersion error at large wavenumbers numerical accuracy analyses and modeling results demonstrate the optimal esfd and isfd schemes can efficiently suppress the numerical dispersion and significantly improve the modeling accuracy compared to the te based esfd and isfd schemes

a low dispersion 3 d second order in time fourth order in space fdtd scheme m3d sub 24 a second order in time fourth order in space modified finite difference time domain fdtd scheme for three dimensional electromagnetic problems m3d sub 24 is presented the algorithm enables the numerical phase error to be minimized so that it leads to high accuracy with low resolution grids the advantage of this method is demonstrated by considering the long distance propagation of the wave radiated from a time harmonic elementary dipole using a low resolution grid and comparing the results with other fdtd schemes

dimer statistics on the klein bottle abstract the problem of enumerating close packed dimers or perfect matchings on a quadratic lattice embedded on the klein bottle is considered thomassen c thomassen tilings of the torus and the klein bottle and vertex transitive graphs on a fixed surface trans amer math soc 323 2 1991 605 635 characterized that there are six quadrilateral lattices embedded on the klein bottle lu and wu w t lu and f y wu close packed dimers on nonorientable surfaces phys lett a 293 2002 235 246 had obtained a expression for the number of close packed dimers on one of them in this paper we investigate four other embeddings and obtain explicit expressions of the numbers of close packed dimers and free energy per dimer by enumerating pfaffians

an explicit fourth order staggered finite difference time domain method for maxwell s equations a new explicit fourth order accurate staggered finite difference time domain fdtd scheme is proposed and applied to electromagnetic wave problems it is fourth order accurate in both space and time conditionally stable and highly efficient with respect to yee s scheme and still retains much of the original simplicity of yee s scheme both extension to perfectly matched layers and modification to deal with dielectric interfaces and perfectly conducting boundaries of the scheme have also been presented numerical examples are shown to illustrate the efficiency of the method

a comprehensive new methodology for formulating fdtd schemes with controlled order of accuracy and dispersion numerical dispersion errors in the wave equation finite difference time domain we fdtd method have been treated by higher order schemes coefficient modification schemes dispersion relation preserving and non standard schemes in this work a unified methodology is formulated for the systematic generation of we fdtd schemes tailored to the spectrum of the excitation the methodology enables the scheme designer to gradually trade order of accuracy ooa for lower dispersion errors in a controlled manner at the cost of sacrificing low frequency behavior that is not deemed critical for this type of excitation the methodology is shown to encompass both existing and new schemes stability analysis is carried out concurrently with the generation of each scheme using a stencil size of 3 and 5 temporal and spatial samples respectively long term errors of a scheme designed for a specific pulse are compared with the standard 4 4 scheme that has the same computational complexity via simulation of a modulated pulse that propagates over a million time steps

novel explicit 2 d fdtd scheme with isotropic dispersion and enhanced stability a two dimensional 2 d finite difference time domain fdtd method based on a novel finite difference scheme is developed to eliminate the numerical dispersion errors in this paper the numerical dispersion and stability analysis of the new scheme are given these show that the proposed method is nearly dispersionless and stable for a larger time step than the standard fdtd method for a monochromatic frequency the new algorithm is also verified for two cases

the convergence rate for difference approximations to general mixed initial boundary value problems the convergence rate is investigated for the solutions to difference schemes approximating the mixed initial boundary value problem for general systems of differential equations it is shown that if an energy estimate holds then the extra boundary conditions can be of one order lower accuracy without destroying the convergence rate expected from the approximation at inner points if the maximal order of the derivatives occurring in the boundary conditions is low enough then even lower accuracy can be permitted for the extra boundary conditions

a locking free finite difference method on staggered grids for linear elasticity problems abstract a finite difference method on staggered grids is constructed on general nonuniform rectangular partition for linear elasticity problems stability optimal order error estimates in discrete h 1 norms on general nonuniform grids and second order superconvergence on almost uniform grids have been obtained these theoretical results are uniform about the lame constant 0 so the finite difference method is locking free the method and theoretical results can be extended to three dimensional problems numerical experiments using the method show agreement of the numerical results with theoretical analysis

splitting multisymplectic integrators for maxwell s equations in the paper we describe a novel kind of multisymplectic method for three dimensional 3 d maxwell s equations splitting the 3 d maxwell s equations into three local one dimensional lod equations then applying a pair of symplectic runge kutta methods to discretize each resulting lod equation it leads to splitting multisymplectic integrators we say this kind of schemes to be lod multisymplectic scheme lod ms the discrete conservation laws convergence dispersive relation dissipation and stability are investigated for the schemes theoretical analysis shows that the schemes are unconditionally stable non dissipative and of first order accuracy in time and second order accuracy in space as a reduction we also consider the application of lod ms to 2 d maxwell s equations numerical experiments match the theoretical results well they illustrate that lod ms is not only efficient and simple in coding but also has almost all the nature of multisymplectic integrators

small world networks links with long tailed distributions small world networks swn obtained by randomly adding to a regular structure additional links al are of current interest in this paper we explore based on physical models a new variant of swn in which the probability of realizing an al depends on the chemical distance between the connected sites we assume a power law probability distribution and study random walkers on the network focusing especially on their probability of being at the origin we connect the results to l evy flights which follow from a mean field variant of our model

energy dissipation splitting finite difference time domain method for maxwell equations with perfectly matched layers abstract in this paper we develop a novel kind of energy dissipation splitting finite difference time domain scheme for solving two dimensional maxwell equations with perfectly matched layers the discrete energy dissipation law convergence dispersion relation and stability are investigated for the scheme theoretical analysis shows that the scheme is unconditionally stable and of second order accuracy both in time and space numerical experiments confirm the theoretical results

suppression of numerical anisotropy and dispersion with optimized finite difference time domain methods to reduce numerical dispersion in finite difference time domain fdtd methods large computational stencils are often used this paper proposes an optimized two dimensional method by weighting the 2 4 stencil and the neighborhood stencil after obtaining the amplification factor and the numerical dispersion relation the optimal value of the weight parameter is obtained to minimize the numerical dispersion at a designated frequency the anisotropy dispersion error and the accumulated phase errors are greatly reduced over a broad bandwidth both the maximum anisotropy and the maximum dispersion error are 8 9 spl times 10 sup 5 and the accumulated phase error is 0 002367 spl deg per cell respectively for a broad band of frequencies if optimized at 10 cells per wavelength numerical experiments are performed which show very good agreement with theoretical analysis the time step size bound is the same as for yee s fdtd

exponential sums and the riemann zeta function v a van der corput exponential sum is s sigma exp 2 pi i f m where m has size m the function f x has size t and alpha log m log t 1 there are different bounds for s in different ranges for alpha in the middle range where alpha is near 1 over 2 s o sqrt m t theta epsilon this theta bounds the exponent of growth of the riemann zeta function on its critical line rm re s 1 over 2 van der corput used an iteration which changed alpha at each step the bombieri iwaniec method whilst still based on mean squares introduces number theoretic ideas and problems the second spacing problem is to count the number of resonances between short intervals of the sum when two arcs of the graph of y f x coincide approximately after an automorphism of the integer lattice in the previous paper in this series proc london math soc 3 66 1993 1 40 and the monograph area lattice points and exponential sums we saw that coincidence implies that there is an integer point close to some resonance curve one of a family of curves in some dual space now calculated accurately in the paper resonance curves in the bombieri iwaniec method which is to appear in funct approx comment math we turn the whole bombieri iwaniec method into an axiomatised step an upper bound for the number of integer points close to a plane curve gives a bound in the second spacing problem and a small improvement in the bound for s ends and cusps of resonance curves are treated separately bounds for sums of type s lead to bounds for integer points close to curves and another branching iteration luckily swinnerton dyer s method is stronger we improve theta from 0 156140 in the previous paper and monograph to 0 156098 in fact 32 205 epsilon 269 410 epsilon is an exponent pair for every epsilon 0

coherent dynamics on hierarchical systems abstract we study the coherent transport modeled by continuous time quantum walks focussing on hierarchical structures for these we use husimi cacti lattices dual to the dendrimers we find that the transport depends strongly on the initial site of the excitation for systems of sizes n 21 we find that processes which start at central sites are nearly recurrent furthermore we compare the classical limiting probability distribution to the long time average of the quantum mechanical transition probability which shows characteristic patterns we succeed in finding a good lower bound for the space average of the quantum mechanical probability to be still or again at the initial site

a high order accurate embedded boundary method for first order hyperbolic equations abstract a stable and high order accurate embedded boundary method for first order hyperbolic equations is derived where the grid boundaries and the physical boundaries do not coincide high order interpolation is used the boundary stencils are based on a summation by parts framework and the boundary conditions are imposed by the sat penalty method which guarantees linear stability for one dimensional problems second fourth and sixth order finite difference schemes are considered the resulting schemes are fully explicit accuracy and numerical stability of the proposed schemes are demonstrated for both linear and nonlinear hyperbolic systems in one and two spatial dimensions

block pseudospectral methods for maxwell s equations ii two dimensional discontinuous coefficient case block pseudospectral bps methods are examined for maxwell s equations in two dimensional inhomogeneous media for the case of a rectangular strip with a straight line interface blocks may be coupled via fictitious points or a generalization of characteristic outflow conditions the bps methods generalize to curvilinear strips described by a change of variables such strips can conform to interfaces while overlapping with a high order free space grid to form a composite grid method numerical experiments on strips with dielectrics lossy materials perfect conductors and absorbing layers indicate that the two coupling methods are comparable and accurate with just 3 4 points per wavelength full composite examples are included to demonstrate high accuracy and geometric flexibility

continuous time quantum walks on nonorientable surfaces analytical solutions for mobius strips and klein bottles in this paper we present an extensive analytical study of continuous time quantum walks ctqws occurring on two dimensional lattices under various boundary conditions focusing on the m n lattice wrapped on mobius strips and klein bottles which are featured by the twisted boundary conditions we find that the eigenvalue spectra and transport efficiency both quantum mechanical and classical for the two structures show comparable results with those for other well known two dimensional lattices including rectangles cylinders and tori we also demonstrate that the behaviors of ctqws depend on the initial node of the excitation and the network size both on short and long timescales in addition we discover the asymmetric behaviors of limiting probabilities for mobius strips and klein bottles which are quite different from each other and are also compared to those discovered in other two dimensional networks our work provides a comprehensive understanding of recent results about ctqws on two dimensional lattices and sheds light on quantum dynamics on lattices especially those with different boundary conditions

the lobatto cell robust explicit higher order fdtd that handles inhomogeneous media to improve numerical dispersion and second order convergence behavior of the traditional yee finite difference time domain fdtd algorithm various higher order finite difference implementations have been proposed many have difficulty handling interfaces between distinct dielectric media or metallic sheets because the high order taylor expansions on which these codes are based are invalidated by field or derivative discontinuities one sided extensions extrapolations are a common approach to remedy interface discontinuities here another approach based upon whitney form mass lumped finite elements is pursued to develop the ldquolobatto cell rdquo a high order replacement for the yee cell with appropriate continuity behavior at contrasting media interfaces the lobatto cell retains many robust features of the yee method discrete conservation laws a guarantee of conditional stability and straightforward explicit updating but provides higher order interpolation and dispersion error convergence a notable drawback of the method is larger modeling granularity staircasing which is managed here by combining the lobatto cell method with preexisting robust subgridding techniques

high accuracy yee algorithm based on nonstandard finite differences new developments and verifications we previously described a high accuracy version of the yee algorithm that uses second order nonstandard finite differences nsfds and demonstrated its accuracy numerically we now prove that at fixed frequency and grid spacing h the leading error term is o h sup 6 versus o h sup 2 for the ordinary yee algorithm with standard finite differences sfds we numerically verify the superior accuracy of the nsfd algorithm by simulating near field mie scattering on a coarse grid and comparing with the sfd one and with analytical solutions we present an updated stability analysis and show that the maximum time step for the nsfd algorithm is 20 longer than the sfd time step in two dimensions and 36 longer in three dimensions finally parameters that were previously given numerically are now analytically defined

time space domain dispersion relation based finite difference method with arbitrary even order accuracy for the 2d acoustic wave equation high order finite difference fd methods have been widely used for numerical solution of acoustic wave equations it has been reported that the modeling accuracy is of 2nd order when the conventional 2m th order space domain fd and the 2nd order time domain fd stencils are directly used to solve the acoustic wave equation under the same discretization the present version of time space domain dispersion relation based fd method can improve the accuracy from 2nd order to 2m th order along eight directions for the 2d acoustic wave equation to increase the accuracy further we propose a new fd stencil for 2d acoustic wave equation modeling this new time space domain dispersion relation based fd stencil can reach the same arbitrary even order accuracy along all directions and is more accurate and more stable than the conventional one for the same m dispersion analysis and modeling examples demonstrate its advantages

the splitting finite difference time domain methods for maxwell s equations in two dimensions in this paper we consider splitting methods for maxwell s equations in two dimensions a new kind of splitting finite difference time domain methods on a staggered grid is developed the corresponding schemes consist of only two stages for each time step which are very simple in computation the rigorous analysis of the schemes is given by the energy method it is proved that the scheme is unconditionally stable and convergent for the problems with perfectly conducting boundary conditions numerical dispersion analysis and numerical experiments are presented to show the efficient performance of the proposed methods furthermore the methods are also applied to solve a scattering problem successfully

uniform dispersion reduction schemes for the one dimensional wave equation in isotropic media finite difference fd methods for the wave equation in general suffer from numerical dispersion although fd methods based on accuracy give good dispersion at low frequencies waves tend to disperse for higher wavenumber in this work we will give a unified methodology to derive dispersion reduction fd schemes for the one dimensional wave equation and this new method can reduce dispersion error uniformly for all wavenumbers up to the nyquist stability criteria are given and stability analysis is done for each generated scheme

finite difference time domain dispersion reduction schemes the finite difference time domain fdtd although recognized as a flexible robust and simple to implement method for solving complex electromagnetic problems is subject to numerical dispersion errors in addition to the traditional ways for reducing dispersion i e increasing sampling rate and using higher order degrees of accuracy a number of schemes have been proposed recently in this work an unified methodology for deriving new difference schemes is presented it is based on certain modifications of the characteristic equation that accompanies any given discretized version of the wave equation the method is duly compared with existing schemes and verified numerically