exit problems associated with affine reflection groups we obtain a formula for the distribution of the first exit time of brownian motion from the alcove of an affine weyl group in most cases the formula is expressed compactly in terms of pfaffians expected exit times are derived in the type widetilde a case the results extend to other markov processes we also give formulas for the real eigenfunctions of the dirichlet and neumann laplacians on alcoves observing that the hot spots conjecture of j rauch is true for alcoves

a solution to an open problem for wave equations with generalized wentzell boundary conditions in this paper we solve an open problem put forward by a favini g r goldstein j a goldstein and s romanelli concerning the mixed problem for wave equations with generalized wentzell boundary conditions as a consequence we also develop the previous wellposedness result regarding the mixed problem for heat equations with generalized wentzell boundary conditions

stable computation of multiquadric interpolants for all values of the shape parameter abstract spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries since radial basis function rbf approxima ions permit this even for multivariate scattered data there has been much recent interest in practical algorithms to compute these approximations effectively several types of rbfs feature a free parameter e g c in the multiquadric mq case f r r 2 c 2 the limit of c increasingly flat basis functions has not received much attention because it leads to a severely ill conditioned problem we present here an algorithm which avoids this difficulty and which allows numerically stable computations of mq rbf interpolants for all parameter values we then find that the accuracy of the resulting approximations in some cases becomes orders of magnitude higher than was the case within the previously available parameter range our new method provides the first tool for the numerical exploration of mq rbf interpolants in the limit of c the method is in no way specific to mq basis functions and can without any change be applied to many other cases as well

rbf fd formulas and convergence properties the local rbf is becoming increasingly popular as an alternative to the global version that suffers from ill conditioning in this paper we study analytically the convergence behavior of the local rbf method as a function of the number of nodes employed in the scheme the nodal distance and the shape parameter we derive exact formulas for the first and second derivatives in one dimension and for the laplacian in two dimensions using these formulas we compute taylor expansions for the error from this analysis we find that there is an optimal value of the shape parameter for which the error is minimum this optimal parameter is independent of the nodal distance our theoretical results are corroborated by numerical experiments

simulation of incompressible viscous flows past a circular cylinder by hybrid fd scheme and meshless least square based finite difference method in this paper we describe an efficient method for simulating the two dimensional steady and unsteady incompressible flows the method is a hybrid approach which combines the conventional finite difference fd scheme and the meshfree least square based finite difference mlsfd method in this method the mlsfd method is adopted to deal with the complex geometry for its truly mesh free property and it is responsible for the spatial discretization in the region around the complex geometry which represents the stationary solid obstacle correspondingly conventional fd scheme is applied in the rest of the flow domain to take advantage of its high computational efficiency numerical simulation is carried out for the steady flow with reynolds numbers of 10 20 and 40 and unsteady flow with reynolds numbers of 100 and 200 the obtained numerical results agree very well with computational and experimental data available in the literature compared with the fully mlsfd method this new approach greatly improves the computational efficiency 2003 elsevier b v all rights reserved

the runge phenomenon and spatially variable shape parameters in rbf interpolation many studies mostly empirical have been devoted to finding an optimal shape parameter for radial basis functions rbf when exploring the underlying factors that determine what is a good such choice we are led to consider the runge phenomenon rp best known in cases of high order polynomial interpolation as a key error mechanism this observation suggests that it can be advantageous to let the shape parameter vary spatially rather than assigning a single value to it benefits typically include improvements in both accuracy and numerical conditioning still another benefit arises if one wishes to improve local accuracy by clustering nodes in selected areas this idea is routinely used when working with splines or finite element methods however local refinement with rbfs may cause rp type errors unless we use a spatially variable shape paremeter with this enhancement rbf approximations combine freedom from meshes with spectral accuracy on irregular domains and furthermore permit local node clustering to improve the resolution wherever this might be needed

a nonterminating 8 φ 7 summation for the root system c r using multiple q integrals and a determinant evaluation we establish a nonterminating 8 7 summation for the root system cr we also give some important specializations explicitly

observations on the behavior of radial basis function approximations near boundaries abstract rbf approximations would appear to be very attractive for approximating spatial derivatives in numerical simulations of pdes rbfs allow arbitrarily scattered data generalize easily to several space dimensions and can be spectrally accurate however accuracy degradations near boundaries in many cases severely limit the utility of this approach with that as motivation this study aims at gaining a better understanding of the properties of rbf approximations near the ends of an interval in 1 d and towards edges in 2 d

remarks on the schur complement abstract first a generalization of the schur determinantal formula is given using properties of quasidirect sums of matrices a new characterization of the schur complement is proved

double centralizing theorems for the alternating groups let v n be the n fold tensor product of a vector space v following i schur we consider the action of the symmetric group sn on v n by permuting coordinates in the super z2 graded case v v0 v1 a sign is added these actions give rise to the corresponding schur algebras s sn v here s sn v is compared with s an v the schur algebra corresponding to the alternating subgroup an sn while in the classical signless case these two schur algebras are the same for n large enough it is proved that in the super case where dim v0 dim v1 s an v is isomorphic to the cross product algebra s an v s sn v z2

error estimates of local multiquadric based differential quadrature lmqdq method through numerical experiments summary in this article we present an error estimate of the derivative approximation by the local multiquadricbased differential quadrature lmqdq method radial basis function is different from the polynomial approximation in which taylor series expansion is not applicable so the present analysis is performed through the numerical solution of poisson equation it is known that the approximation error of lmqdq method depends on three factors i e local density of knots h free shape parameter c and number of supporting knots ns by numerical experiments their contribution to the approximation error and correlation were studied and analysed in this paper an error estimate o h c n is thereafter proposed in which n is a positive constant and determined by the number of supporting knots ns copyright 2005 john wiley sons ltd

modular invariant of quantum tori the quantum modular invariant j qt of 2r is defined as a discontin uous pgl2 z invariant multi valued map using the distance to the nearest integer function k k for 2 q it is shown that j qt 1 and for quadratic irrationalities pari gp experiments suggest that j qt is a finite set in the case of the golden mean we produce explicit formulas involving weighted versions of the rogers ramanujan functions for the experimental supremum and infimum we then define a universal modular invariant j a mod c as a continuous and single valued map of ultrasolenoids such that 1 the classical modular invariant is a quotient of the re striction of j to a subsolenoid mod cl a mod fibering over the classical moduli space of elliptic curves and 2 the quantum modular invariant is a quotient of the restric tion of j to a subsolenoid mod qt a mod fibering over the moduli space of elliptic curves equipped with a kronecker foliation

numerical computation of three dimensional incompressible viscous flows in the primitive variable form by local multiquadric differential quadrature method abstract in this paper the local multiquadric differential quadrature lmqdq method is applied on three dimensional incompressible flow problems the lmqdq method is among the newly proposed mesh free methods unlike the traditional differential quadrature dq method the weighting coefficients of lmqdq method are determined by using the radial basis functions rbfs as the trial functions instead of high order polynomials the main concern of this paper is to discuss the effectiveness of using lmqdq method to solve 3 d incompressible navier stokes n s equations in the primitive variable form three dimensional lid driven cavity flow problem with reynolds numbers of 100 400 and 1000 was chosen as a test case to validate the lmqdq method the computed velocity profiles along the vertical and horizontal centerlines are given and compared with available data in the literature

optimal variable shape parameter for multiquadric based rbf fd method in this follow up paper to our previous study in bayona et al 2011 2 we present a new technique to compute the solution of pdes with the multiquadric based rbf finite difference method rbf fd using an optimal node dependent variable value of the shape parameter this optimal value is chosen so that to leading order the local approximation error of the rbf fd formulas is zero in our previous paper bayona et al 2011 2 we considered the case of an optimal constant value of the shape parameter for all the nodes our new results show that if one allows the shape parameter to be different at each grid point of the domain one may obtain very significant accuracy improvements with a simple and inexpensive numerical technique we analyze the same examples studied in bayona et al 2011 2 both with structured and unstructured grids and compare our new results with those obtained previously we also find that if there are a significant number of nodes for which no optimal value of the shape parameter exists then the improvement in accuracy deteriorates significantly in those cases we use generalized multiquadrics as rbfs and choose the exponent of the multiquadric at each node to assure the existence of an optimal variable shape parameter

vibration analysis of arbitrarily shaped membranes using local radial basis function based differential quadrature method in this study a recently developed local radial basis function based differential quadrature lrbfdq method is applied for the vibration analysis of arbitrarily shaped membranes lrbfdq method combines the good features of differential quadrature dq approximation of derivatives and mesh free nature of the radial basis functions rbfs in a local region the derivative at a reference point is approximated as a linear weighted sum of functional values at a set of scattered points in the local supporting region of the reference point the helmholtz equation governing membrane vibration is directly discretized into algebraic equations from which the wavenumbers natural frequencies and mode shapes of freely vibrating membranes are easily calculated owing to the properties of mesh free and local approximation of the lrbfdq method the problems with arbitrarily shaped domains can be solved readily and accurately in particular for highly concave shaped membranes and multi connected membranes with a hole very accurate numerical results can be easily obtained without the use of any domain decomposition technique it is also shown that the lrbfdq method can produce more accurate solutions than fem when the two methods use nearly the same number of points in a domain

a stable algorithm for flat radial basis functions on a sphere when radial basis functions rbfs are made increasingly flat the interpolation error typically decreases steadily until some point when runge type oscillations either halt or reverse this trend because the most obvious method to calculate an rbf interpolant becomes a numerically unstable algorithm for a stable problem in the case of near flat basis functions there will typically also be a separate point at which disastrous ill conditioning enters we introduce here a new method rbf qr which entirely eliminates such ill conditioning and we apply it in the special case when the data points are distributed over the surface of a sphere this algorithm works even for thousands of node points and it allows the rbf shape parameter to be optimized without the limitations imposed by stability concerns since interpolation in the flat rbf limit on a sphere is found to coincide with spherical harmonics interpolation new insights are gained as to why the rbf approach with nonflat basis functions often is the more accurate of the two methods

generation of localization in a discrete chain with periodic boundary conditions numerical and analytical results we study the generation of localization in a discrete chain composed of n subsystems and periodic boundary conditions strongly localized motions are studied that is time periodic motions where nearly all of the energy is spatially confined to a single subsystem for varying n numerical results indicate that the strongly localized solutions are generated through a bifurcation from an in phase spatially extended solution however in the limit as n rightarrow infty the bifurcation point tends to infinity and a smooth transition from localization to nonlocalization occurs we then present an analytic technique to complement the numerical results it is based on the matching local asymptotic expansions of a solution branch using pade approximants this leads to global analytic representations of the considered solutions valid over the entire range of the control parameter

interpolation in the limit of increasingly flat radial basis functions abstract many types of radial basis functions such as multiquadrics contain a free parameter in the limit where the basis functions become increasingly flat the linear system to solve becomes highly ill conditioned and the expansion coefficients diverge nevertheless we find in this study that limiting interpolants often exist and take the form of polynomials in the 1 d case we prove that with simple conditions on the basis function the interpolants converge to the lagrange interpolating polynomial hence differentiation of this limit is equivalent to the standard finite difference method we also summarize some preliminary observations regarding the limit in 2 d

seismic modeling with radial basis function generated finite differences rbf fd a simplified treatment of interfaces abstract in a previous study of seismic modeling with radial basis function generated finite differences rbf fd we outlined a numerical method for solving 2 d wave equations in domains with material interfaces between different regions the method was applicable on a mesh free set of data nodes it included all information about interfaces within the weights of the stencils allowing the use of traditional time integrators and was shown to solve problems of the 2 d elastic wave equation to 3rd order accuracy in the present paper we discuss a refinement of that method that makes it simpler to implement it can also improve accuracy for the case of smoothly variable model parameter values near interfaces we give several test cases that demonstrate the method solving 2 d elastic wave equation problems to 4th order accuracy even in the presence of smoothly curved interfaces with jump discontinuities in the model parameters

stable computations with gaussian radial basis functions radial basis function rbf approximation is an extremely powerful tool for representing smooth functions in nontrivial geometries since the method is mesh free and can be spectrally accurate a perceived practical obstacle is that the interpolation matrix becomes increasingly ill conditioned as the rbf shape parameter becomes small corresponding to flat rbfs two stable approaches that overcome this problem exist the contour pade method and the rbf qr method however the former is limited to small node sets and the latter has until now been formulated only for the surface of the sphere this paper focuses on an rbf qr formulation for node sets in one two and three dimensions the algorithm is stable for arbitrarily small shape parameters it can be used for thousands of node points in two dimensions and still more in three dimensions a sample matlab code for the two dimensional case is provided

on the role of polynomials in rbf fd approximations radial basis function generated finite difference rbf fd approximations generalize classical grid based finite differences fd from lattice based to scattered node layouts this greatly increases the geometric flexibility of the discretizations and makes it easier to carry out local refinement in critical areas many different types of radial functions have been considered in this rbf fd context in this study we find that i polyharmonic splines phs in conjunction with supplementary polynomials provide a very simple way to defeat stagnation also known as saturation error and ii give particularly good accuracy for the tasks of interpolation and derivative approximations without the hassle of determining a shape parameter in follow up studies we will focus on how to best use these hybrid rbf polynomial bases for fd approximations in the contexts of solving elliptic and hyperbolic type pdes

new combinatorial interpretations of ramanujan s partition congruences mod 5 7 and 11 soit p n le nombre de partitions non restreintes de n on donne de nouvelles interpretations combinatoires des congruences p 5n 4 p 7n 5 et p 11n 6 0 mod 5 7 et 11 respectivement

a de giorgi nash type theorem for time fractional diffusion equations we study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order alpha in 0 1 the coefficients are merely assumed to be bounded and measurable and they satisfy a uniform parabolicity condition our main result is a de giorgi nash type theorem which gives an interior holder estimate for bounded weak solutions in terms of the data and the l infty bound of the solution the proof relies on new a priori estimates for time fractional problems and uses de giorgi s technique and the method of non local growth lemmas which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional laplacian

some inequalities for the q beta and the q gamma functions via some q integral inequalities some new inequalities for the q gamma the q beta and the q analogue of the psi functions are established via some q integral inequalities

the x ray transform and its inversion for the series expansion basis functions in three dimensional tomography a formula for the three dimensional x ray transform and an expression for its inversion have been derived the spherical harmonics and the rotation matrices in terms of euler angles are used in the generalized fourier series expansion some properties of the transformed fourier components are discussed

some observations regarding interpolants in the limit of flat radial basis functions abstract radial basis functions rbfs form a primary tool for multivariate interpolation some of the most commonly used radial functions feature a shape parameter allowing them to vary from being nearly flat e small to sharply peaked e large the former limit can be particularly accurate when interpolating a smooth function based on scattered data this study discusses theoretical and computational aspects of the e 0 limit and includes the conjecture that gaussian rbf interpolants will never diverge in this limit

local radial basis function based differential quadrature method and its application to solve two dimensional incompressible navier stokes equations local radial basis function based differential quadrature method is presented in detail in this paper the method is a natural mesh free approach like the conventional differential quadrature dq method it discretizes any derivative at a knot by a weighted linear sum of functional values at its neighbouring knots which may be distributed randomly however different from the conventional dq method the weighting coefficients in present method are determined by taking the radial basis functions rbfs instead of high order polynomials as the test functions the method works in a similar fashion as conventional finite difference schemes but with truly mesh free property in this paper we mainly concentrate on the multiquadric rbfs since they have exponential convergence the effects of shape parameter c on the accuracy of numerical solution of linear and nonlinear partial differential equations are studied and how the value of optimal c varies with the number of local support knots is also numerically demonstrated the proposed method is validated by its application to the simulation of natural convection in a square cavity excellent numerical results are obtained on an irregular knot distribution

optimal constant shape parameter for multiquadric based rbf fd method radial basis functions rbfs have become a popular method for interpolation and solution of partial differential equations pdes many types of rbfs used in these problems contain a shape parameter and there is much experimental evidence showing that accuracy strongly depends on the value of this shape parameter in this paper we focus on pde problems solved with a multiquadric based rbf finite difference rbf fd method we propose an efficient algorithm to compute the optimal value of the shape parameter that minimizes the approximation error the algorithm is based on analytical approximations to the local rbf fd error derived in 1 we show through several examples in 1d and 2d both with structured and unstructured nodes that very accurate solutions compared to finite differences can be achieved using the optimal value of the constant shape parameter

stable evaluation of gaussian radial basis function interpolants we provide a new way to compute and evaluate gaussian radial basis function interpolants in a stable way with a special focus on small values of the shape parameter i e for flat kernels this work is motivated by the fundamental ideas proposed earlier by bengt fornberg and his coworkers however following mercer s theorem an l 2 mathbb r d rho orthonormal expansion of the gaussian kernel allows us to come up with an algorithm that is simpler than the one proposed by fornberg larsson and flyer and that is applicable in arbitrary space dimensions d in addition to obtaining an accurate approximation of the radial basis function interpolant using many terms in the series expansion of the kernel we also propose and investigate a highly accurate least squares approximation based on early truncation of the kernel expansion

on the role of polynomials in rbf fd approximations ii numerical solution of elliptic pdes rbf generated finite differences rbf fd have in the last decade emerged as a very powerful and flexible numerical approach for solving a wide range of pdes we find in the present study that combining polyharmonic splines phs with multivariate polynomials offers an outstanding combination of simplicity accuracy and geometric flexibility when solving elliptic equations in irregular or regular regions in particular the drawbacks on accuracy and stability due to runge s phenomenon are overcome once the rbf stencils exceed a certain size due to an underlying minimization property test problems include the classical 2 d driven cavity and also a 3 d global electric circuit problem with the earth s irregular topography as its bottom boundary the results we find are fully consistent with previous results for data interpolation

an improved sectional anosov closing lemma we prove that all nonwandering points of a sectional anosov flow on a compact 3 manifold can be approximated by periodic points or by points for which the omega limit set is a singularity this improves the closing lemma in morales mich math j 56 1 29 53 2008 we also describe a sectional anosov flow for which the recurrent points are not dense in the nonwandering set

enhancing finite differences with radial basis functions polynomials are used together with polyharmonic spline phs radial basis functions rbfs to create local rbf finite difference rbf fd weights on different node layouts for spatial discretizations that can be viewed as enhancements of the classical finite differences fd the presented method replicates the convergence properties of fd but for arbitrary node layouts it is tested on the 2d compressible navier stokes equations at low mach number relevant to atmospheric flows test cases are taken from the numerical weather prediction community and solved on bounded domains thus attention is given on how to handle boundaries with the rbf fd method as well as a novel implementation for hyperviscosity comparisons are done on cartesian hexagonal and quasi uniform node layouts consideration and guidelines are given on phs order polynomial degree and stencil size the main advantages of the present method are 1 capturing the basic physics of the problem surprisingly well even at very coarse resolutions 2 high order accuracy without the need of tuning a shape parameter and 3 the inclusion of polynomials eliminates stagnation saturation errors a matlab code is given to calculate the differentiation weights for this novel approach