the eigenvalues of second order spectral differentiation matrices the eigenvalues of the pseudospectral second derivative matrix with homogeneous dirichlet boundary conditions are important in many applications of spectral methods this paper investigates some of their properties numerical results show that a certain fraction of the eigenvalues approximate the eigenvalues of the continuous operator very accurately but the errors in the remaining ones are large it is demonstrated that the inaccurate eigenvalues correspond to those eigenfunctions of the continuous operator that cannot be resolved by polynomial interpolation in the spectral grid in particular it is proved that 7r points on average per wavelength are sufficient for successful interpolation of the eigenfunctions of the continuous operator in a chebyshev distribution of nodes and six points per wavelength for a uniform distribution these results are in agreement with the observed fractions of accurate eigenvalues by using the characteristic polynomial a bound on the spectral radius of the differentiation matrix is derived that is accurate to 2 or better the effect of filtering on the eigenvalues is studied numerically

conformal maps to multiply slit domains and applications by exploiting conformal maps to vertically slit regions in the complex plane a recently developed rational spectral method t w tee and l n trefethen siam j sci comput 28 2006 pp 1798 1811 is able to solve pdes with interior layer like behavior using significantly fewer collocation points than traditional spectral methods the conformal maps are chosen to enlarge the region of analyticity in the solution an idea which can be extended to other numerical methods based upon global polynomial interpolation here we show how such maps can be rapidly computed in both periodic and nonperiodic geometries and apply them to some challenging differential equations

a conservative free energy dissipating and positivity preserving finite difference scheme for multi dimensional nonlocal fokker planck equation abstract in this work we design and analyze a conservative positivity preserving and free energy dissipating finite difference method for the multi dimensional nonlocal fokker planck fp equation based on a non logarithmic landau transformation a central differencing spatial discretization using harmonic mean approximations is developed both forward and backward euler discretizations in time are employed to derive an explicit scheme and a linearized semi implicit scheme respectively three desired properties that are possessed by analytical solutions i mass conservation ii free energy dissipation and iii positivity are proved to be maintained at discrete level remarkably numerical analysis demonstrates that the semi implicit time discretization ensures the property of positivity preserving unconditionally due to the advantages brought by the harmonic mean approximations an estimate on the upper bound of the condition number of the resulting coefficient matrix is further established for the semi implicit scheme extensive numerical tests are performed to validate aforementioned properties numerically

on a chemotaxis model with degenerate diffusion initial shrinking eventual smoothness and expanding abstract we investigate the propagating profiles of a degenerate chemotaxis model describing the bacteria chemotaxis and consumption of oxygen by aerobic bacteria in particular the effect of the initial attractant distribution on bacterial clustering we prove that the compact support of solutions may shrink if the signal concentration satisfies a special structure and show the finite speed propagating property without assuming the special structure on attractant concentration and obtain an explicit formula of the population spreading speed in terms of model parameters the presented results suggest that bacterial cluster formation can be affected by chemotactic attractants and density dependent dispersal

theoretical analysis of a sinc nystrom method for volterra integro differential equations and its improvement abstract a sinc nystrom method for volterra integro differential equations was developed by zarebnia 2010 the method is quite efficient in the sense that exponential convergence can be obtained even if the given problem has endpoint singularity however its exponential convergence has not been proved theoretically in addition to implement the method the regularity of the solution is required although the solution is an unknown function in practice this paper reinforces the method by presenting two theoretical results 1 the regularity of the solution is analyzed and 2 its convergence rate is rigorously analyzed moreover this paper improves the method so that a much higher convergence rate can be attained and theoretical results similar to those listed above are provided numerical comparisons are also provided

numerical simulation of nonlinear continuity equations by evolving diffeomorphisms abstract in this paper we present a numerical scheme for nonlinear continuity equations which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance it can be applied to a large class of nonlinear continuity equations whose dynamics are driven by internal energies given external potentials and or interaction energies the solver is based on its variational formulation as a gradient flow with respect to the wasserstein distance positivity of solutions as well as energy decrease of the semi discrete scheme are guaranteed by its construction we illustrate this property with various examples in spatial dimension one and two

a note on an identity based ring signature scheme with signer verifiability recently herranz presented an identity based ring signature scheme featuring signer verifiability where a signer can prove that he or she is the real signer by releasing an authorship proof in this paper we show that this scheme is vulnerable to a key recovery attack in which a user s secret signing key can be efficiently recovered through the use of two known ring signatures and their corresponding authorship proofs in addition we present a simple method to fix this security vulnerability by slightly modifying the authorship proof our modified scheme simplifies the original scheme and improves performance to show that the modified scheme is unforgeable we define two types of unforgeability notions for both signatures and authorship proofs in these notions an adversary has opening capability to confirm the real signers of ring signatures and thus can manipulate authorship proofs in an adaptive way we then prove that our modified scheme is secure in terms of these unforgeability notions

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

blow up dynamics for the aggregation equation with degenerate diffusion abstract we study radially symmetric finite time blow up dynamics for the aggregation equation with degenerate diffusion u t u m u k u in r d where the kernel k x is of power law form x depending on m d and the initial data the solution exhibits three kinds of blow up behavior self similar with no mass concentrated at the core imploding shock solution and near self similar blow up with a fixed amount of mass concentrated at the core computations are performed for different values of m d and using an arbitrary lagrangian eulerian method with adaptive mesh refinement

application of sinc collocation method for solving a class of nonlinear fredholm integral equations in this paper the numerical solution of nonlinear fredholm integral equations of the second kind is considered by two methods the methods are developed by means of the sinc approximation with the single exponential se and double exponential de transformations these numerical methods combine a sinc collocation method with the newton iterative process that involves solving a nonlinear system of equations we provide an error analysis for the methods so far approximate solutions with polynomial convergence have been reported for this equation these methods improve conventional results and achieve exponential convergence some numerical examples are given to confirm the accuracy and ease of implementation of the methods

a sinc function analogue of chebfun chebfun is an established software system for computing with functions of a real variable but its capabilities for handling functions with singularities are limited here an analogous system is described based on sinc function expansions instead of chebyshev series this experiment sheds light on the strengths and weaknesses of sinc function techniques it also serves as a review of some of the main features of sinc methods including construction evaluation zerofinding optimization integration and differentiation

error estimates with explicit constants for sinc approximation sinc quadrature and sinc indefinite integration error estimates with explicit constants are given for approximations of functions definite integrals and indefinite integrals by means of the sinc approximation although in the literature various error estimates have already been given for these approximations those estimates were basically for examining the rates of convergence and several constants were left unevaluated giving more explicit estimates i e evaluating these constants is of great practical importance since by this means we can reinforce the useful formulas with the concept of verified numerical computations in this paper we reveal the explicit form of all constants in a computable form under the same assumptions of the existing theorems the function to be approximated is analytic in a suitable region we also improve some formulas themselves to decrease their computational costs numerical examples that confirm the theory are also given

barycentric formulae for cardinal sinc interpolants we present a barycentric representation of cardinal interpolants as well as a weighted barycentric formula for their efficient evaluation we also propose a rational cardinal function which in some cases agrees with the corresponding cardinal interpolant and in other cases is even more accurate in numerical examples we compare the relative accuracy of those various interpolants with one another and with a rational interpolant proposed in former work

global existence versus blow up in a high dimensional keller segel equation with degenerate diffusion and nonlocal aggregation abstract in this paper we study the degenerate keller segel equation with nonlocal aggregation u t u m u b u in r d r where m 1 d 3 and b u 2 u with 2 d by analyzing the interaction between the degenerate diffusion and the nonlocal aggregation we determine the conditions for initial data under which weak solutions globally exist or blow up in finite time with m 1 d d d furthermore a sharper criterion is given for global existence and finite time blow up of weak solutions with m in the subrange 2 d 2 d d d 1 d d

new interior penalty discontinuous galerkin methods for the keller segel chemotaxis model we develop a family of new interior penalty discontinuous galerkin methods for the keller segel chemotaxis model this model is described by a system of two nonlinear pdes a convection diffusion equation for the cell density coupled with a reaction diffusion equation for the chemoattractant concentration it has been recently shown that the convective part of this system is of a mixed hyperbolic elliptic type which may cause severe instabilities when the studied system is solved by straightforward numerical methods therefore the first step in the derivation of our new methods is made by introducing the new variable for the gradient of the chemoattractant concentration and by reformulating the original keller segel model in the form of a convection diffusion reaction system with a hyperbolic convective part we then design interior penalty discontinuous galerkin methods for the rewritten keller segel system our methods employ the central upwind numerical fluxes originally developed in the context of finite volume methods for hyperbolic systems of conservation laws in this paper we consider cartesian grids and prove error estimates for the proposed high order discontinuous galerkin methods our proof is valid for pre blow up times since we assume boundedness of the exact solution we also show that the blow up time of the exact solution is bounded from above by the blow up time of our numerical solution in the numerical tests presented below we demonstrate that the obtained numerical solutions have no negative values and are oscillation free even though no slope limiting technique has been implemented

a finite volume scheme for the patlak keller segel chemotaxis model a finite volume method is presented to discretize the patlak keller segel pks modeling chemosensitive movements first we prove existence and uniqueness of a numerical solution to the proposed scheme then we give a priori estimates and establish a threshold on the initial mass for which we show that the numerical approximation converges to the solution to the pks system when the initial mass is lower than this threshold numerical simulations are performed to verify accuracy and the properties of the scheme finally in the last section we investigate blow up of the solution for large mass

notes on finite difference schemes to a parabolic elliptic system modelling chemotaxis we consider numerical solutions to a nonlinear parabolic elliptic system in mathematical biology an important property of solutions to the system is the conservation of l 1 norm applying the upwind technique we make a finite difference scheme satisfying the conservation of a discrete l 1 norm we also describe a remark on time discretization from the view point of lyapunov s property which is another important feature of the system several numerical examples clarify the nature of our scheme

de sinc methods have almost the same convergence property as se sinc methods even for a family of functions fitting the se sinc methods in this paper the theoretical convergence rate of the trapezoidal rule combined with the double exponential de transformation is given for a class of functions for which the single exponential se transformation is suitable it is well known that the de transformation enables the rule to achieve a much higher rate of convergence than the se transformation and the convergence rate has been analyzed and justified theoretically under a proper assumption here it should be emphasized that the assumption is more severe than the one for the se transformation and there actually exist some examples such that the trapezoidal rule with the se transformation achieves its usual rate whereas the rule with de does not such cases have been observed numerically but no theoretical analysis has been given thus far this paper reveals the theoretical rate of convergence in such cases and it turns out that the de s rate is almost the same as but slightly lower than that of the se by using the analysis technique developed here the theoretical convergence rate of the sinc approximation with the de transformation is also given for a class of functions for which the se transformation is suitable the result is quite similar to above the convergence rate in the de case is slightly lower than in the se case numerical examples which support those two theoretical results are also given

identity based ring signatures from rsa shamir proposed in 1984 the first identity based signature scheme whose security relies on the rsa problem a similar scheme was proposed by guillou and quisquater in 1988 formal security of these schemes was not argued and or proved until many years later d pointcheval j stern security arguments for digital signatures and blind signatures journal of cryptology 13 3 2000 361 396 y dodis j katz s xu m yung strong key insulated signature schemes in proceedings of pkc 03 in lncs vol 2567 springer verlag 2002 pp 130 144 m bellare c namprempre g neven security proofs for identity based identification and signature schemes in proceedings of eurocrypt 04 in lncs vol 3027 springer verlag 2004 pp 268 286 taking the guillou quisquater scheme as the starting point we design and analyze in this work ring signature schemes and distributed ring signature schemes for identity based scenarios whose security is based on the hardness of the rsa problem these are the first identity based ring signature schemes which do not employ bilinear pairings furthermore the resulting schemes satisfy an interesting property the real author s of a ring signature can later open the anonymity and prove that he is actually the person who signed the message

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

symmetric singularity formation in lubrication type equations for interface motion fourth order degenerate diffusion equations arise in a lubrication approximation of a thin film or neck driven by surface tension numerical studies of the lubrication equation le h t h n h xxx x 0 with various boundary conditions indicate that singularity formation in which h x t t to 0 occurs for small enough n with anomalous or second type scaling inconsistent with usual dimensional analysis this paper considers locally symmetric or even singularities in the le and in the modified lubrication equation mle h t h n h xxxx 0 both equations have the property that entropy bounds forbid finite time singularities when n is sufficiently large power series expansions for local symmetric similarity solutions are proposed for equation le with n 1 and mle for all n in mathbb r in the latter case special boundary conditions that force singularity formation are required to produce singularities when n is large matching conditions at higher order terms in the

precise computations of chemotactic collapse using moving mesh methods we consider the problem of computing blow up solutions of chemotaxis systems or the so called chemotactic collapse in two spatial dimensions such solutions can have approximate self similar behaviour which can be very challenging to verify in numerical simulations cf betterton and brenner collapsing bacterial cylinders phys rev e 64 2001 061904 we analyse a dynamic scale invariant remeshing method which performs spatial mesh movement based upon equidistribution using a suitably chosen monitor function the numerical solution resolves the fine detail in the asymptotic solution structure such that the computations are seen to be fully consistent with the asymptotic description of the collapse phenomenon given by herrero and velazquez singularity patterns in a chemotaxis model math ann 306 1996 583 623 we believe that the methods we construct are ideally suited to a large number of problems in mathematical biology for which collapse phenomena are expected

spectral methods for the navier stokes equations with one infinite and two periodic directions abstract two numerical methods were designed to solve the time dependent three dimensional incompressible navier stokes equations in boundary layers method a semi infinite domain and mixing layers or wakes method b fully infinite domain their originality lies in the use of rapidly decaying spectral basis functions to approximate the vertical dependence of the solutions combined with one method a or two method b slowly decaying extra functions for each wave vector that exactly represent the irrotational component of the solution at large distances both methods eliminate the pressure term as part of the formulation thus avoiding fractional step time integration they yield rapid convergence and are free of spurious modes in the orr sommerfeld spectra they are also efficient although the operation count is of order n 2 n is the number of modes in the infinite direction these methods have been used for extensive direct numerical simulations of transition and turbulence a new time integration scheme with low storage requirements and good stability properties is also described

an upper bound estimate and stability for the global error of numerical integration using double exponential transformation abstract the double exponential formula was introduced for calculating definite integrals with singular point oscillation functions and fourier integral the double exponential transformation is not only useful for numerical computations but it is also used in different methods of sinc theory in this paper we give an upper bound estimate for the error of double exponential transformation by improving integral estimates having singular final points in theorem 1 we prove that the method is convergent and the rate of convergence is o h 2 where h is a step size our main tool in the proof is de formula in sinc theory the advantage of our method is that the time and space complexity is drastically reduced furthermore we discovered an upper bound error in de formula independent of n truncated number as a matter of fact we proved stability numerical tests are presented to verify the theoretical predictions and confirm the convergence of the numerical solution

self similar blowup solutions to an aggregation equation in r n we present numerical simulations of radially symmetric finite time blowup for the aggregation equation ut u k u where the kernel k x x the dynamics of the blowup exhibits self similar behavior in which zero mass concentrates at the core at the blowup time computations are performed in r n for n between 2 and 10 using a method based on characteristics in all cases studied the self similarity exhibits second kind anomalous scaling

numerical indefinite integration by double exponential sinc method we present a numerical method for approximating an indefinite integral by the double exponential sinc method the approximation error of the proposed method with n integrand function evaluations is o exp c 1 n log c 2 n for a reasonably wide class of integrands including those with endpoint singularities the proposed method compares favorably with the existing formulas based on the ordinary sinc method computational results show the accordance of the actual convergence rates with the theoretical estimate

function classes for double exponential integration formulas the double exponential de formulas for numerical integration are known to be highly efficient more efficient than the single exponential se formulas in many cases function classes suited to the se formulas have already been investigated in the literature through rigorous mathematical analysis whereas this is not the case with the de formulas this paper identifies function classes suited to the de formulas in a way compatible with the existing theoretical results for the se formulas the de formulas are good for more restricted classes of functions but more efficient for such functions two concrete examples demonstrate the subtlety in the behavior of the de formulas that is revealed by our theoretical analysis

new exponential variable transform methods for functions with endpoint singularities the focus of this paper is the approximation of functions which are analytic on a compact interval except at the endpoints typical numerical methods for approximating such functions depend upon the use of particular conformal maps from the original interval to either a semi infinite or an infinite interval followed by an appropriate approximation procedure on the new region we first analyze the convergence of these existing methods and show that in a precisely defined sense they are suboptimal specifically they exhibit poor resolution properties by which we mean that many more degrees of freedom are required to resolve oscillatory functions than standard approximation schemes for analytic functions such as chebyshev interpolation to remedy this situation we introduce two new transforms one for each of the above settings we provide full convergence results for these new approximations and then demonstrate that for a particular choice of parameters these methods lead to substantially better re

near optimality of the sinc approximation near optimality of the sinc approximation is established in a variety of spaces of functions analytic in a strip region about the real axis each space being characterized by the decay rate of their elements functions in the neighborhood of the infinity

stochastic particle approximation for measure valued solutions of the 2d keller segel system we construct an approximation to the measure valued global in time solutions to the patlak keller segel model in 2d based on systems of stochastic interacting particles the advantage of our approach is that it reproduces the well known dichotomy in the qualitative behavior of the system and moreover captures the solution even after the possible blow up events we present a numerical method based on this approach and show some numerical results moreover we make a first step toward the convergence analysis of our scheme by proving the convergence of the stochastic particle approximation for the keller segel model with a regularized interaction potential the proof is based on a bbgky like approach for the corresponding particle distribution function

summary of sinc numerical methods sinc approximation methods excel for problems whose solutions may have singularities or infinite domains or boundary layers this article summarizes results obtained to date on sinc numerical methods of computation sinc methods provide procedures for function approximation over bounded or unbounded regions encompassing interpolation approximation of derivatives approximate definite and indefinite integration solving initial value ordinary differential equation problems approximation and inversion of fourier and laplace transforms approximation of hilbert transforms and approximation of indefinite convolutions the approximate solution of partial differential equations and the approximate solution of integral equations methods for constructing conformal maps and methods for analytic continuation indeed sinc are ubiquitous for approximating every operation of calculus

error estimates with explicit constants for the sinc approximation over infinite intervals abstract the sinc approximation is a function approximation formula that attains exponential convergence for rapidly decaying functions defined on the whole real axis even for other functions the sinc approximation works accurately when combined with a proper variable transformation the convergence rate has been analyzed for typical cases including finite semi infinite and infinite intervals recently for verified numerical computations a more explicit computable error bound has been given in the case of a finite interval in this paper such explicit error bounds are derived for other cases

discontinuous galerkin methods for the chemotaxis and haptotaxis models in this work first we formulate and compare three different discontinuous interior penalty galerkin methods for the 2d keller segel chemotaxis model keller segel chemotaxis model is the important starting step in the modeling of the real biological system we show in the numerical tests that two of the proposed methods fail to give accurate oscillation free solutions next we consider the application of the successful method for the keller segel model to the simulation of the more realistic and closely related haptotaxis model of tumor invasion into healthy tissues

rational scaled generalized laguerre function collocation method for solving the blasius equation in this paper we propose a collocation method for solving the blasius equation the blasius equation is a third order nonlinear ordinary differential equation this approach is based on a rational scaled generalized laguerre function collocation method we also present the comparison of this work with some well known results and show that the present solution is accurate

on the resolution power of fourier extensions for oscillatory functions functions that are smooth but non periodic on a certain interval possess fourier series that lack uniform convergence and suffer from the gibbs phenomenon however they can be represented accurately by a fourier series that is periodic on a larger interval this is commonly called a fourier extension when constructed in a particular manner fourier extensions share many of the same features of a standard fourier series in particular one can compute fourier extensions which converge spectrally fast whenever the function is smooth and geometrically fast if the function is analytic much the same as the fourier series of a smooth analytic and periodic function with this in mind the purpose of this paper is to describe analyse and explain the observation that fourier extensions much like classical fourier series also have excellent resolution properties for representing oscillatory functions the resolution power or required number of degrees of freedom per wavelength depends on a user controlled parameter and as we show it varies between 2 and p the former value is optimal and is achieved by classical fourier series for periodic functions for example the latter value is the resolution power of algebraic polynomial approximations thus fourier extensions with an appropriate choice of parameter are eminently suitable for problems with moderate to high degrees of oscillation

infinite time aggregation for the critical patlak keller segel model in ℝ2 we analyze the two dimensional parabolic elliptic patlak keller segel model in the whole euclidean space r2 under the hypotheses of integrable initial data with finite second moment and entropy we first show local in time existence for any mass of free energy solutions namely weak solutions with some free energy estimates we also prove that the solution exists as long as the entropy is controlled from above the main result of the paper is to show the global existence of free energy solutions with initial data as before for the critical mass 8 actually we prove that solutions blow up as a delta dirac at the center of mass when t keeping constant their second moment at any time furthermore all moments larger than 2 blow up as t if initially bounded

double exponential formulas for numerical indefinite integration in this paper we derive a formula for indefinite integration of analytic functions over 1 s where 1 s 1 by means of the double exponential transformation and the sinc method the integrand must be analytic on 1 x 1 but may have a singularity at the end points x 1 the error of the formula behaves approximately as exp c1n logc2n where n is the number of function evaluations of the integrand this error term shows a much faster convergence to zero when n becomes large than that of the known formula by haber also we derive efficient double exponential formulas for numerical evaluation of indefinite integrals over 0 s 0 s and over s s several numerical examples indicate high efficiency of the formulas

the rate of convergence of chebyshev polynomials for functions which have asymptotic power series about one endpoint the theorem proved here extends chebyshev theory into what has previously been no man s land functions which have an infinite number of bounded derivatives on the expansion interval a b but which are singular at one endpoint the chebyshev series in l x for all the familiar special functions fall into this category so this class of functions is very important indeed in words the theorem shows that the more slowly the asymptotic power series about the singular point converges the slower the convergence of the corresponding chebyshev series must be more formally if f x analytic on a b is singular at x b in such a way that it has an asymptotic power seriesf x e a x b about that endpoint then if loglanl nr logn r it is proved that the coefficients of the convergent chebyshev polynomial series on a bj f x i bn tn y wherey 2 x 0 5 b a l b a satisfy the inequality lim logj logjbn1 0 when a function has a singularity which is not on 1 1 it is known that bn ae for some a and x which depend received march 21 1980 revised august 21 1980 and october 29 1980 ams mos subject classifications 1970 primary 42a56 secondary 33a65 41a10

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

convergence of the mass transport steepest descent scheme for the subcritical patlak keller segel model variational steepest descent approximation schemes for the modified patlak keller segel equation with a logarithmic interaction kernel in any dimension are considered we prove the convergence of the suitably interpolated in time implicit euler scheme defined in terms of the euclidean wasserstein distance associated with this equation for subcritical masses as a consequence we recover the recent result about the global in time existence of weak solutions to the modified patlak keller segel equation for the logarithmic interaction kernel in any dimension in the subcritical case moreover we show how this method performs numerically in dimension one in this particular case this numerical scheme corresponds to a standard implicit euler method for the pseudoinverse of the cumulative distribution function we demonstrate its capabilities to reproduce the blow up of solutions for supercritical masses easily without the need of mesh refinement

chebyshev spectral collocation methods for laminar flow through a channel contraction abstract a chebyshev spectral element method is described for solving the navier stokes equations in a channel contraction the flow region is divided into two semi infinite elements the governing equation for the stream function is solved using a newton linearization the semi infinite elements are treated by means of algebraic type mappings to transform them onto finite domains the stream function is represented by a double chebyshev expansion in each element the coefficients are determined by collocating the linearized equation at each newton step and imposing c3 continuity conditions across the element interface in a collocation sense efficient direct methods based on capacitance matrix ideas are described which take advantage of the structure of the spectral element matrix results are presented for reynolds numbers in the range 0 200 which are in good agreement with previously published work but requiring fewer degrees of freedom only several steps of the newton process are required to achieve a converged solution for re 120 the method converges from a zero initial approximation and thereafter continuation in the reynolds number is used with increments of 10

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

recent developments of the sinc numerical methods abstract this paper gives a survey of recent developments of the sinc numerical methods a variety of sinc numerical methods have been developed by stenger and his school for a certain class of problems the sinc numerical methods have the convergence rates of o exp n with some 0 where n is the number of nodes or bases used in the methods recently it has turned out that the sinc numerical methods can achieve convergence rates of o exp n log n with some 0 for a smaller but still practically meaningful class of problems and that these convergence rates are best possible the present paper demonstrates these facts for two sinc numerical methods the sinc approximation and the sinc collocation method for two point boundary value problems

the double exponential transformation in numerical analysis the double exponential transformation was first proposed by takahasi and mori in 1974 for the efficient evaluation of integrals of an analytic function with end point singularity afterwards this transformation was improved for the evaluation of oscillatory functions like fourier integrals recently it turned out that the double exponential transformation is useful not only for numerical integration but also for various kinds of sinc numerical methods the purpose of the present paper is to review the double exponential transformation in numerical integration and in a variety of sinc numerical methods

numerical methods based on whittaker cardinal or sinc functions this paper summarizes the results known to date for using sine functions composed with other functions as bases for approximations in numerical analysis described in this paper are methods of interpolation and approximation of functions and their derivatives quadrature the approximate evaluation of transforms hilbert fourier laplace hankel and mellin and the approximate solution of differential and integral equations the methods have many advantages over classical methods which use polynomials as bases in addition all of the methods converge at an optimal rate if singularities on the boundary of approximation are ignored

how to leak a secret theory and applications of ring signatures in this work we formalize the notion of a ring signature which makes it possible to specify a set of possible signers without revealing which member actually produced the signature unlike group signatures ring signatures have no group managers no setup procedures no revocation procedures and no coordination any user can choose any set of possible signers that includes himself and sign any message by using his secret key and the others public keys without getting their approval or assistance ring signatures provide an elegant way to leak authoritative secrets in an anonymous way to sign casual email in a way that can only be verified by its intended recipient and to solve other problems in multiparty computations our main contribution lies in the presentation of efficient constructions of ring signatures the general concept itself under different terminology was first introduced by cramer et al cds94 our constructions of such signatures are unconditionally signer ambiguous secure in the random oracle model and exceptionally efficient adding each ring member increases the cost of signing or verifying by a single modular multiplication and a single symmetric encryption we also describe a large number of extensions modifications and applications of ring signatures which were published after the original version of this work in asiacrypt 2001

function classes for successful de sinc approximations the de sinc formulas resulting from a combination of the sinc approximation formula with the double exponential de transformation provide a highly efficient method for function approximation in many cases they are more efficient than the se sinc formulas which are the sinc approximation formulas combined with the single exponential se transformations function classes suited to the se sinc formulas have already been investigated in the literature through rigorous mathematical analysis whereas this is not the case with the de sinc formulas this paper identifies function classes suited to the de sinc formulas in a way compatible with the existing theoretical results for the se sinc formulas furthermore we identify alternative function classes for the de sinc formulas as well as for the se sinc formulas which are more useful in applications in the sense that the conditions imposed on the functions are easier to verify

an extension of matlab to continuous functions and operators an object oriented matlab system is described for performing numerical linear algebra on continuous functions and operators rather than the usual discrete vectors and matrices about eighty matlab functions from plot and sum to svd and cond have been overloaded so that one can work with our chebfun objects using almost exactly the usual matlab syntax all functions live on 1 1 and are represented by values at sufficiently many chebyshev points for the polynomial interpolant to be accurate to close to machine precision each of our overloaded operations raises questions about the proper generalization of familiar notions to the continuous context and about appropriate methods of interpolation differentiation integration zerofinding or transforms applications in approximation theory and numerical analysis are explored and possible extensions for more substantial problems of scientific computing are mentioned

optimality of the double exponential formula functional analysis approach in the light of the functional analysis theory we establish the optimality of the double exponential formula the argument consists of the following three ingredients 1 introduction of a number of spaces of functions analytic in a strip region about the real axis each space being characterized by the decay rate of their elements functions in the neighborhood of the infinity 2 proof of the near optimality of the trapezoidal formula in each space introduced in 1 by showing the near equality between an upper estimate for the error norm of the trapezoidal formula and a lower estimate for the minimum error norm of quadratures 3 nonexistence theorem for the spaces the characterizing decay rate of which is more rapid than the double exponential

computing the real roots of a fourier series plus linear polynomial a chebyshev companion matrix approach abstract fourier series often need to be generalized by appending a linear polynomial to the usual series of sines and cosines the integral of a trigonometric polynomial is one example another is a time series of climate data where the periodic oscillations of the diurnal and annual cycles are accompanied by a non periodic trend global warming stock market averages fluctuate about a generally upward trend such non periodic variations with time are commonly called secular trends we borrow secular to label a truncated fourier series plus a linear trend as a linear secular trigonometric polynomial standard fourier rootfinding methods are wrecked by the extra nonperiodic term therefore we introduce a new algorithm for computing the zeros of a fourier polynomial with secular trend first we expand the linear secular trigonometric polynomial f n t as a truncated chebyshev series because of the special structure it is easy to calculate a problem dependent truncation m such that the error of the truncated chebyshev series is guaranteed to be less than a user specified tolerance we then find the roots of the truncated chebyshev series as the eigenvalues of the chebyshev companion matrix this computes all roots but we explain why the method is not reliable for complex valued roots unless these are close to the real axis no a priori information is required of the user except the coefficients of the linear secular trigonometric polynomial numerical examples show that 13 decimal place accuracy for real roots is typical

eliminating the unbounded behavior of function derivative expansions in terms of sinc bases sinc methods are frequently used in treating mathematical physics problems such as in interpolation quadrature and solving integral and differential equations however for finite intervals the function derivatives become unbounded this unbounded behavior is addressed the choice of proper weights and the use of the sinc expansion of the function f x x are used to provide improved expansions and to eliminate the unbounded behavior of the derivatives at the terminal points

de sinc methods have almost the same convergence property as se sinc methods even for a family of functions fitting the se sinc methods part ii indefinite integration in this paper the theoretical convergence rate of the sinc indefinite integration combined with the double exponential de transformation is given for a class of functions for which the single exponential se transformation is suitable although the de transformation is considered as an enhanced version of the se transformation for sinc related methods the function space for which the de transformation is suitable is smaller than that for se and therefore there exist some examples such that the de transformation is not better than the se transformation even in such cases however some numerical observations in the literature suggest that there is almost no difference in the convergence rates of se and de in fact recently the observations have been theoretically explained for two explicit approximation formulas the sinc quadrature and the sinc approximation the conclusion is that in such cases the de s rate is slightly lower but almost the same as that of the se the contribution of this study is the derivation of the same conclusion for the sinc indefinite integration numerical examples that support the theoretical result are also provided

accurate function sinc interpolation and derivative estimations over finite intervals function expansion in terms of finite sinc bases is considered for finite intervals we considered proper transformation and weighted barycentric methods we adjusted the transformation function to comply with the truncation of the expansion this removes the unbounded behavior of the terminal derivatives and yields accurate results efficient weighted barycentric expansions are presented to improve the accuracies of function and derivative estimations a new expression for the estimation of the second derivative is derived the derived expressions are successfully applied to accurately locate the minimum of the solution and solve a singularly perturbed problem

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

on the rational second kind chebyshev pseudospectral method for the solution of the thomas fermi equation over an infinite interval in this paper we propose a pseudospectral method for solving the thomas fermi equation which is a nonlinear singular ordinary differential equation on a semi infinite interval this approach is based on the rational second kind chebyshev pseudospectral method that is indeed a combination of tau and collocation methods this method reduces the solution of this problem to the solution of a system of algebraic equations the slope at origin is provided with high accuracy comparison with some numerical solutions shows that the present solution is effective and highly accurate