computational aspects of cui freeden statistics for equidistribution on the sphere abstract in this paper we derive the asymptotic statistical properties of a class of generalized discrepancies introduced by cui and freeden siam j sci comput 1997 to test equidistribution on the sphere we show that they have highly desirable properties and encompass several statistics already proposed in the literature in particular it turns out that the limiting distribution is an infinite weighted sum of chi squared random variables issues concerning the approximation of this distribution are considered in detail and explicit bounds for the approximation error are given the statistics are then applied to assess the equidistribution of hammersley low discrepancy sequences on the sphere and the uniformity of a dataset concerning magnetic orientations

nonlinear image representation for efficient perceptual coding image compression systems commonly operate by transforming the input signal into a new representation whose elements are independently quantized the success of such a system depends on two properties of the representation first the coding rate is minimized only if the elements of the representation are statistically independent second the perceived coding distortion is minimized only if the errors in a reconstructed image arising from quantization of the different elements of the representation are perceptually independent we argue that linear transforms cannot achieve either of these goals and propose instead an adaptive nonlinear image representation in which each coefficient of a linear transform is divided by a weighted sum of coefficient amplitudes in a generalized neighborhood we then show that the divisive operation greatly reduces both the statistical and the perceptual redundancy amongst representation elements we develop an efficient method of inverting this transformation and we demonstrate through simulations that the dual reduction in dependency can greatly improve the visual quality of compressed images

symmetry reduction for nonlinear relativistically invariant equations symmetry reduction is studied for the relativistically invariant scalar partial differential equation h u u 2 u 0 in n 1 dimensional minkowski space m n 1 the introduction of k symmetry variables 1 k as invariants of a subgroup g of the poincare group p n 1 having generic orbits of codimension k n in m n 1 reduces the equation to a pde in k variables all codimension 1 symmetry variables in m n 1 n arbitrary reducing the equation studied to an ode are found as well as all codimension 2 and 3 variables for the low dimensional cases n 2 3 the type of equation studied includes many cases of physical interest in particular nonlinear klein gordon equations such as the sine gordon equation and hamilton jacobi equations

a note on classical ground state energies the pair specific ground state energy eg n g n n n 1 of newtonian n body systems grows monotonically in n this furnishes a whole family of simple new tests for minimality of putative ground state energies gx n obtained through computer experiments inspection of several publicly available lists of such computer experimentally obtained putative ground state energies gx n has yielded several dozen instances of gx n which failed one of these tests i e for those n one concludes that gx n g n strictly although the correct g n is not revealed by this method it does yield a better upper bound on g n than gx n whenever gx n fails a monotonicity test the surveyed n body systems include in particular n point charges with 2 or 3 dimensional coulomb pair interactions placed either on the unit 2 sphere or on a 2 torus a k a thomson fekete or riesz problems

modeling polymerization of microtubules a semi classical nonlinear field theory approach abstract in this paper for the first time a three dimensional treatment of microtubules polymerization is presented starting from fundamental biochemical reactions during microtubule s assembly and disassembly processes we systematically derive a nonlinear system of equations that determines the dynamics of microtubules in three dimensions we found that the dynamics of a microtubule is mathematically expressed via a cubic quintic nonlinear schrodinger nls equation we show that in 3d a vortex filament a generic solution of the nls equation exhibits linear growth shrinkage in time as well as temporal fluctuations about some mean value which is qualitatively similar to the dynamic instability of microtubules by solving equations numerically we have found spatio temporal patterns consistent with experimental observations

exact solutions of the multidimensional classical φ6 field equations obtained by symmetry reduction the often used 6 model of classical critical phenomena is studied in 3 1 dimensional minkowski and euclidean spaces the euler lagrange equations describing the kinetics of the scalar order parameter are in this case nonlinear klein gordon equations the method of symmetry reduction is systematically applied to derive all the solutions invariant under subgroups with generic orbits of codimension 1 whenever the obtained ordinary differential equations have the painleve property they can be transformed to one of two standard forms these are then solved in terms of elliptic functions or elementary ones this results in a large number of new exact solutions particularly interesting solutions are found in the immediate vicinity of the tricritical point our treatment of the 6 theory is complete only for the four dimensional spaces m 3 1 and e 4 but many of the results are given for the more general cases of m n 1 and e n 1

support of the logarithmic equilibrium measure on sets of revolution in r3 for sets of revolution b in r3 we investigate the limit distribution of minimum energy point masses on b that interact according to the logarithmic potential log 1 r where r is the euclidean distance between points we show that such limit distributions are supported only on the outmost portion of the surface e g for a torus only on that portion of the surface with positive curvature our analysis proceeds by reducing the problem to the complex plane where a nonsingular potential kernel arises the level lines of which are ellipses

swarming on random graphs ii we consider an individual based model where agents interact over a random network via first order dynamics that involve both attraction and repulsion in the case of all to all coupling of agents in mathbb r d this system has a lowest energy state in which an equal number of agents occupy the vertices of the d dimensional simplex the purpose of this paper is to sharpen and extend a line of work initiated in 56 which studies the behavior of this model when the interaction between the n agents occurs according to an erd s renyi random graph mathcal g n p instead of all to all coupling in particular we study the effect of randomness on the stability of these simplicial solutions and provide rigorous results to demonstrate that stability of these solutions persists for probabilities greater than np o log n in other words only a relatively small number of interactions are required to maintain stability of the state the results rely on basic probability arguments together with spectral properties of random graphs

minimizing the discrete logarithmic energy on the sphere the role of random polynomials we prove that points in the sphere associated with roots of random polynomials via the stereographic projection are surprisingly well suited with respect to the minimal logarithmic energy on the sphere that is roots of random polynomials provide a fairly good approximation to elliptic fekete points

coherent structures in strongly interacting many body systems ii classical solutions and quantum fluctuations for pt i see ibid vol 22 p 4877 1989 previously i the author considered a second quantised hamiltonian which can be used to model a large number of important strongly interacting many body systems he has shown that the dynamics of such systems can be described exactly using quantum fields the equations of motion derived in this way are highly nonlinear partial differential equations pde he now uses a standard field theoretical approach where initially the fields are treated as classical functions he studies the solutions of these equations which remarkably may be obtained making full use of very recent mathematical discoveries the symmetry reduction method he finds exact solutions of the equations of motion for the classical field for each of the four main cases separately found in the preceding paper these cases may correspond to different physical situations and the last the second order case corresponds to the most general physical situation subsequently he outlines possible quantisation procedures for these fields he then briefly discusses the type of boundary conditions which may be applied to both the classical field and the quantum fluctuations which arise

equivalence groupoids and group classification of multidimensional nonlinear schrodinger equations abstract we study admissible and equivalence point transformations between generalized multidimensional nonlinear schrodinger equations and classify lie symmetries of such equations we begin with a wide superclass of schrodinger type equations which includes all the other classes considered in the paper showing that this superclass is not normalized we partition it into two disjoint normalized subclasses which are not related by point transformations further constraining the arbitrary elements of the superclass we construct a hierarchy of normalized classes of schrodinger type equations this gives us an appropriate normalized superclass for the non normalized class of multidimensional nonlinear schrodinger equations with potentials and modular nonlinearities and allows us to partition the latter class into three families of normalized subclasses after a preliminary study of lie symmetries of nonlinear schrodinger equations with potentials and modular nonlinearities for an arbitrary space dimension we exhaustively solve the group classification problem for such equations in space dimension two

linearity inside nonlinearity exact solutions to the complex ginzburg landau equation abstract systems of two linear partial differential equations pdes with constant coefficients define a natural basis of elementary functions to build some exact solutions to nonlinear pdes for the one dimensional complex ginzburg landau equation the usual representations of the complex field re a or a are multivalued which makes difficult the search for exact solutions in this manner fortunately the painleve analysis naturally introduces an elementary single valued like representation of a by two fields z grad respectively complex abd real uniquely defined by an explicit expression for and a ze i another important feature is the invariance by parity on a which increases the class of expected solutions this allows to retrieve quite easily the four famous solutions of nozaki and bekki represented by the constant coefficients of two linear partial differential equations and a finite set of constants

asymptotics of greedy energy points for a symmetric kernel k x x x u on a locally compact metric space x we investigate the asymptotic behavior of greedy k energy points a i 1 for a compact subset a c x that are defined inductively by selecting a 1 a arbitrarily and a n 1 so that n i 1 k a n l1 a i inf x a n i 1 k x a i we give sufficient conditions under which these points also known as leja points are asymptotically energy minimizing i e have energy n i j k a i aj as n that is asymptotically the same as e a n min i j k x i x j x 1 x n a and have asymptotic distribution equal to the equilibrium measure for a for the case of riesz kernels k s x y x y s s 0 we show that if a is a rectifiable jordan arc or closed curve in rp and s 1 then greedy k s energy points are not asymptotically energy minimizing in contrast to the case s 1 additional results are obtained for greedy k s energy points on a sphere for greedy best packing points the case s and for weighted riesz kernels for greedy best packing points we provide a simple counterexample to a conjecture attributed to l bos

conditionally invariant solutions of the rotating shallow water wave equations this paper is devoted to the extension of the recently proposed conditional symmetry method to first order nonhomogeneous quasilinear systems which are equivalent to homogeneous systems through a locally invertible point transformation we perform a systematic analysis of the rank 1 and rank 2 solutions admitted by the shallow water wave equations in 2 1 dimensions and construct the corresponding solutions of the rotating shallow water wave equations these solutions involve in general arbitrary functions depending on riemann invariants which allow us to construct new interesting classes of solutions

improved lebesgue constants on the triangle new sets of points with improved lebesgue constants in the triangle are calculated starting with the fekete points a direct minimization process for the lebesgue constant leads to better results the points and corresponding quadrature weigths are explicitly given it is quite surprising that the optimal points are not symmetric the points along the boundary of the triangle are the 1d gauss lobatto points for all degrees our points yield the smallest lebesgue constants currently known numerical examples are presented which show the improved interpolation properties of our nodes

asymptotics for minimal discrete energy on the sphere we investigate the energy of arrangements of n points on the surface of the unit sphere sd in rd 1 that interact through a power law potential v 1 rs where s 0 and r is euclidean distance with ed s n denoting the minimal energy for such n point arrangements we obtain bounds valid for all n for ed s n in the cases when 0 s d and 2 d s for s d we determine the precise asymptotic behavior of ed d n as n as a corollary lower bounds are given for the separation of any pair of points in an n point minimal energy configuration when s d 2 for the unit sphere in r3 d 2 we present two conjectures concerning the asymptotic expansion of e2 s n that relate to the zeta function l s for a hexagonal lattice in the plane we prove an asymptotic upper bound that supports the first of these conjectures of related interest we derive an asymptotic formula for the partial sums of l s when 0 s 2 the divergent case

an algorithm for computing fekete points in the triangle on the line and its tensor products fekete points are known to be the gauss lobatto quadrature points but unlike high order quadrature fekete points generalize to non tensor product domains such as the triangle thus fekete points might serve as an alternative to the gauss lobatto points for certain applications in this work we present a new algorithm to compute fekete points and give results up to degree 19 for the triangle for degree d 10 these points have the smallest lebesgue constant currently known the computations validate a conjecture of bos j approx theory 64 1991 pp 271 280 that fekete points along the boundary of the triangle are the one dimensional gauss lobatto points

group classification of 1 1 dimensional schrodinger equations with potentials and power nonlinearities we perform the complete group classification in the class of nonlinear schrodinger equations of the form i t xx v t x 0 where v is an arbitrary complex valued potential depending on t and x is a real nonzero constant we construct all the possible inequivalent potentials for which these equations have nontrivial lie symmetries using a combination of algebraic and compatibility methods the proposed approach can be applied to solving group classification problems for a number of important classes of differential equations arising in mathematical physics

group invariant solutions of the 2 1 dimensional cubic schrodinger equation we use lie point symmetries of the 2 1 dimensional cubic schrodinger equation to obtain new analytic solutions in a systematic manner we present an analysis of the reduced odes and in particular show that although the original equation is not integrable they typically can belong to the class of painleve type equations

exact solutions of the multidimensional derivative nonlinear schrodinger equation for many body systems of criticality the authors investigate the problem of strongly interacting many body systems near criticality as recently described in terms of nonlinear dynamics by dixon and tuszynski 1989 they show that in the first order of approximation the equation of motion for the order parameter can be mapped on the derivative nonlinear schrodinger equation and thus can be solved exactly they solve the equations both in the absence and presence of current densities

asymptotics for discrete weighted minimal riesz energy problems on rectifiable sets given a closed d rectifiable set a embedded in euclidean space we investigate minimal weighted riesz energy points on a that is n points constrained to a and interacting via the weighted power law potential v w x y x y s where s 0 is a fixed parameter and w is an admissible weight in the unweighted case w 1 such points for n fixed tend to the solution of the best packing problem on a as the parameter s our main results concern the asymptotic behavior as n of the minimal energies as well as the corresponding equilibrium configurations given a distribution x with respect to d dimensional hausdorff measure on a our results provide a method for generating n point configurations on a that are well separated and have asymptotic distribution p x as n

vsnr a wavelet based visual signal to noise ratio for natural images this paper presents an efficient metric for quantifying the visual fidelity of natural images based on near threshold and suprathreshold properties of human vision the proposed metric the visual signal to noise ratio vsnr operates via a two stage approach in the first stage contrast thresholds for detection of distortions in the presence of natural images are computed via wavelet based models of visual masking and visual summation in order to determine whether the distortions in the distorted image are visible if the distortions are below the threshold of detection the distorted image is deemed to be of perfect visual fidelity vsnr infin and no further analysis is required if the distortions are suprathreshold a second stage is applied which operates based on the low level visual property of perceived contrast and the mid level visual property of global precedence these two properties are modeled as euclidean distances in distortion contrast space of a multiscale wavelet decomposition and vsnr is computed based on a simple linear sum of these distances the proposed vsnr metric is generally competitive with current metrics of visual fidelity it is efficient both in terms of its low computational complexity and in terms of its low memory requirements and it operates based on physical luminances and visual angle rather than on digital pixel values and pixel based dimensions to accommodate different viewing conditions

exact similarity solutions of ideal mhd equations for plane motions for the non linear two dimensional ideal mhd equations we use the known lie point symmetries and for the case of adiabatic exponent 2 we give local lie backlund symmetries we find exact solutions using similarity analysis among the calculated similarity solutions describing plasma expansion into vacuum shock waves and spiral flows with plasma sources

self similar solutions for a coupled system of nonlinear schrodinger equations this work is devoted to the study of self similar solutions of the 2 1 dimensional coupled nonlinear schrodinger equations which occur in nonlinear optics 1 psi 1 z psi 1 xx psi 1 yy eta mod psi 1 to 2 1 h mod psi 2 mod 2 psi 1 0 and square root 1 psi 2 psi 2 xx psi 2 yy eta mod psi 2 mod 2 1 h mod psi 1 psi 2 0 where psi 1 and psi 2 are complex functions h is a non vanishing real parameter eta or 1 and in or 1 the authors give the point symmetry properties of the model and calculate generic 2 1 dimensional symmetry reductions some exact and approximate solutions are obtained in particular they use a variational approach to determine and classify a set of physically relevant localized nonsingular self similar solutions

group theoretical analysis of a rotating shallow liquid in a rigid container a lie group analysis is undertaken for a nonlinear system which models the motion of a rotating shallow liquid in a rigid basin the lie algebra of the symmetry group is presented for elliptic and circular paraboloidal basins in the elliptic case the symmetry algebra is a six dimensional real lie algebra in the circular case the symmetry algebra is nine dimensional finite group transformations are constructed which in the circular paraboloidal case deliver a theorem concerning the time evolution of a key moment of inertia during the motion in the elliptic paraboloidal case a result concerning the motion of the centre of gravity of the liquid is retrieved the investigation ends with symmetry reduction of the original system and the generation of group invariant solutions which correspond to various initial data

lie symmetries of a generalised nonlinear schrodinger equation i the symmetry group and its subgroups the symmetry group of the generalised non linear schrodinger equation i psi t delta psi a0 psi a1 mod psi mod 2 psi a2 mod psi mod 4 psi in three space dimensions is shown to be the extended galilei group g 3 for a1a2 not 0 and the galilei similitude group gd 3 including a dilation for a1 oor a2 0 all lie subgroups of g 3 and gd 3 are found they will be used in a subsequent paper to obtain group invariant solutions of the equation

a generalization of the spherical ensemble to even dimensional spheres abstract in a recent article 1 alishahi and zamani discuss the spherical ensemble a rotationally invariant determinantal point process on s 2 in this paper we extend this process in a natural way to the 2d dimensional sphere s 2 d we prove that the expected value of the riesz s energy associated to this determinantal point process has a reasonably low value compared to the known asymptotic expansion of the minimal riesz s energy

optimal logarithmic energy points on the unit sphere we study minimum energy point charges on the unit sphere s d in r d 1 d 3 that interact according to the logarithmic potential log 1 r where r is the euclidean distance between points such optimal n point configurations are uniformly distributed as n we quantify this result by estimating the spherical cap discrepancy of optimal energy configurations the estimate is of order o n 1 d 2 essential is an improvement of the lower bound of the optimal logarithmic energy which yields the second term 1 d log n n in the asymptotical expansion of the optimal energy previously this was known for the unit sphere in r 3 only furthermore we present an upper bound for the error of integration for an equally weighted numerical integration rule q n with the n nodes forming an optimal logarithmic energy configuration for polynomials p of degree at most n this bound is cd n 1 d n d 2 p as n n 1 d 0 for continuous functions f of s d satisfying a lipschitz condition with constant c f the bound is 12dc f c d f o n 1 d 2 as n

are all the equations of the kadomtsev petviashvili hierarchy integrable the kadomtsev petviashvili kp hierarchy is an infinite set of nonlinear partial differential equations in which the number of independent variables increases indefinitely as one proceeds down the hierarchy since these equations were obtained as part of a group theoretical approach to soliton equations it would appear that the kp hierarchy provides integrable scalar equations with an arbitrary number of independent variables it is shown by investigating a specific equation in 3 1 dimensions that the higher equations in the kp hierarchy are only integrable in a conditional sense the equation under study taken in isolation does not pass certain well known and reliable integrability tests thus applying painleve analysis we find that solutions exist allowing movable critical points furthermore solitary wave solutions are shown to exist that do not behave like solitons in multiple collisions on the other hand if the dependence of a solution on the first 2 1 variables is restricted by the fact that it should also satisfy the kp equation itself then the integrability conditions in the other dimensions are satisfied conditional integrability thus means that linear techniques will provide only those solutions of equations in the hierarchy that simultaneously satisfy lower equations in the same hierarchy

on separation of minimal riesz energy points on spheres in euclidean spaces let sd denote the unit sphere in the euclidean space rd 1 d 1 let n be a natural number n 2 and let n x1 xn be a collection of n distinct points on sd on which the minimal riesz s energy is attained in this paper we show that the points x1 xn are well separated for the cases d 1 s d

a sequence of discrete minimal energy configurations that does not converge in the weak star topology abstract we demonstrate a set a and a value of s for which any sequence of n point discrete minimal riesz s energy configurations on a does not have an asymptotic distribution in the weak star sense as n tends to infinity

lie symmetries of a generalised non linear schrodinger equation ii exact solutions for pt i see ibid vol 21 p 1493 1988 the authors obtain group invariant solutions of the non linear equation i psi t delta psi a0 psi a1 psi mod psi mod 2 a2 psi mod psi mod 4 for which the symmetry group was previously shown to be the extended galilei group for a1a2 not 0 and the extended galilei similitude group for a1 0 or a2 0 they use the symmetry subgroups to reduce the equation to ordinary differential equations which are solved whenever possible with the help of a singularity analysis solutions are obtained in terms of elementary functions jacobi elliptic functions and painleve transcendents

minimal riesz energy point configurations for rectifiable d dimensional manifolds we investigate the energy of arrangements of n points on a rectifiable d dimensional manifold a r d 0 that interact through the power law riesz potential v 1 r s where s 0 and r is euclidean distance in r d 0 with es a n denoting the minimal energy for such n point configurations we determine the asymptotic behavior as n 1 of es a n for each fixed s d moreover if a has positive d dimensional hausdor measure we show that n point configurations on a that minimize the s energy are asymptotically uniformly distributed with respect to d dimensional hausdor measure on a when s d even for the unit sphere s d r d 1 these results are new

lie symmetries of a generalised non linear schrodinger equation iii reductions to third order ordinary differential equations for pt ii see ibid vol 22 p 469 1989 the study of group invariant solutions of the generalised non linear schrodinger equation gnlse is continued it is shown that eight types of subgroups of the symmetry group lead via symmetry reduction to third order real ordinary differential equations giving both the phase and the absolute value of the solution only two of the reductions provide to a painleve type equation and both of them only for the cubic gnlse this equation is solved in terms of the fourth painleve transcendent

random riesz energies on compact kahler manifolds the expected riesz energies e n h es of the zero sets of systems of independent gaussian random polynomials are determined asymptotically as the degree n in all dimensions and codimensions the asymptotics are proved for sections of any positive line bundle over any compact kahler manifold

symmetry reduction for the kadomtsev petviashvili equation using a loop algebra the kadomtsev petviashvili kp equation ut 3uux 2 1 4 uxxx x 3 uyy 4 0 allows an infinite dimensional lie group of symmetries i e a group transforming solutions amongst each other the lie algebra of this symmetry group depends on three arbitrary functions of time t and is shown to be related to a subalgebra of the loop algebra a 1 4 low dimensional subalgebras of the symmetry algebra are identified specifically all those of dimension n 3 and also a physically important six dimensional lie algebra containing translations dilations galilei transformations and quasirotations new solutions of the kp equation are obtained by symmetry reduction using the one dimensional subalgebras of the symmetry algebra these solutions contain up to three arbitrary functions of t

periodic discrete energy for long range potentials we consider periodic energy problems in euclidean space with a special emphasis on long range potentials that cannot be defined through the usual infinite sum one of our main results builds on more recent developments of ewald summation to define the periodic energy corresponding to a large class of long range potentials two particularly interesting examples are the logarithmic potential and the riesz potential when the riesz parameter is smaller than the dimension of the space for these examples we use analytic continuation methods to provide concise formulas for the periodic kernel in terms of the epstein hurwitz zeta function we apply our energy definition to deduce several properties of the minimal energy including the asymptotic order of growth and the distribution of points in energy minimizing configurations as the number of points becomes large we conclude with some detailed calculations in the case of one dimension which shows the utility of this approach

quadrature errors discrepancies and their relations to halftoning on the torus and the sphere this paper deals with continuous domain quantization which aims to create the illusion of a gray value image by appropriately distributing black dots for lack of notation we refer to the process as halftoning which is usually associated with the quantization on a discrete grid recently a framework for this task was proposed by minimizing an attraction repulsion functional consisting of the difference of two continuous convex functions the first one of these functions describes attracting forces caused by the image gray values the second one enforces repulsion between the dots in this paper we generalize this approach by considering quadrature error functionals on reproducing kernel hilbert spaces rkhss with respect to the quadrature nodes where we ask for optimal distributions of these nodes for special reproducing kernels these quadrature error functionals coincide with discrepancy functionals which leads to a geometric interpretation it turns out that the original attraction repulsion fu

qmc designs optimal order quasi monte carlo integration schemes on the sphere we study equal weight numerical integration or quasi monte carlo qmc rules for functions in a sobolev space hs sd with smoothness parameter s d 2 defined over the unit sphere sd in rd 1 focusing on n point configurations that achieve optimal order qmc error bounds as is the case for efficient spherical designs we are led to introduce the concept of qmc designs these are sequences of n point configurations xn on s d such that the worst case error satisfies sup f h s f hs 1 1 n x xn f x sd f x d d x o n s d n with an implied constant that depends on the hs sd norm but is independent of n here d is the normalized surface measure on s d we provide methods for generation and numerical testing of qmc designs an essential tool is an expression for the worst case error in terms of a reproducing kernel for the space hs sd with s d 2 as a consequence of this and a recent result of bondarenko et al on the existence of spherical designs with appropriate number of points we show that minimizers of the n point energy for this kernel form a sequence of qmc designs for hs sd furthermore without appealing to the bondarenko et al result we prove that point sets that maximize the sum of suitable powers of the euclidean distance between pairs of points form a sequence of qmc designs for hs sd with s in the interval d 2 d 2 1 for such spaces there exist reproducing kernels with simple closed forms that are useful for numerical testing of optimal order quasi monte carlo integration numerical experiments suggest that many familiar sequences of point sets on the sphere equal area points spiral points minimal coulomb or logarithmic energy points and fekete points are qmc designs for appropriate values of s for comparison purposes we show that configurations of random points that are independently and uniformly distributed on the sphere do not constitute qmc designs for any s d 2 if xn is a sequence of qmc designs for h s sd we prove that it is also a sequence of qmc designs for hs sd for all s d 2 s this leads to the question of determining the supremum of such s here called the qmc strength of the sequence for which we provide estimates based on computations for the aforementioned sequences received by the editor august 15 2012 and in revised form february 26 2013 2010 mathematics subject classification primary 65d30 65d32 secondary 11k38 41a55

riesz extremal measures on the sphere for axis supported external fields abstract we investigate the minimal riesz s energy problem for positive measures on the d dimensional unit sphere s d in the presence of an external field induced by a point charge and more generally by a line charge the model interaction is that of riesz potentials x y s with d 2 s d for a given axis supported external field the support and the density of the corresponding extremal measure on s d is determined the special case s d 2 yields interesting phenomena which we investigate in detail a weak asymptotic analysis is provided as s d 2

group analysis of the three wave resonant system in 2 1 dimensions the three wave resonant interaction equations 2d 3wr in two spatial and one temporal dimension within a group framework are analyzed the symmetry algebra of this system which turns out to be an infinite dimensional lie algebra whose subalgebra is of the kac moody type is found the one and two dimensional symmetry subalgebras are classified and the corresponding reduction equations are obtained from these the new invariant and the partially invariant solutions of the original 2d 3wr equations are obtained

compressed image quality metric based on perceptually weighted distortion objective quality assessment for compressed images is critical to various image compression systems that are essential in image delivery and storage although the mean squared error mse is computationally simple it may not be accurate to reflect the perceptual quality of compressed images which is also affected dramatically by the characteristics of human visual system hvs such as masking effect in this paper an image quality metric iqm is proposed based on perceptually weighted distortion in terms of the mse to capture the characteristics of hvs a randomness map is proposed to measure the masking effect and a preprocessing scheme is proposed to simulate the processing that occurs in the initial part of hvs since the masking effect highly depends on the structural randomness the prediction error from neighborhood with a statistical model is used to measure the significance of masking meanwhile the imperceptible signal with high frequency could be removed by preprocessing with low pass filters the relation is investigated between the distortions before and after masking effect and a masking modulation model is proposed to simulate the masking effect after preprocessing the performance of the proposed iqm is validated on six image databases with various compression distortions the experimental results show that the proposed algorithm outperforms other benchmark iqms

discrete weighted least squares method for the poisson and biharmonic problems on domains with smooth boundary in this article a discrete weighted least squares method for the numerical solution of elliptic partial differential equations exhibiting smooth solution is presented it is shown how to create well conditioned matrices of the resulting system of linear equations using algebraic polynomials carefully selected matching points and weight factors two simple algorithms generating suitable matching points the chebyshev matching points for standard two dimensional domains and the approximate fekete points of sommariva and vianello for general domains are described the efficiency of the presented method is demonstrated by solving the poisson and biharmonic problems with the homogeneous dirichlet boundary conditions defined on circular and annular domains using basis functions in the form satisfying and in the form not satisfying the prescribed boundary conditions

an octahedral equal area partition of the sphere and near optimal configurations of points we construct a new area preserving map from the unit sphere to the regular octahedron both centered at the origin its inverse map allows the construction of uniform and refinable grids on a sphere starting from any triangular uniform and refinable grid on the faces of the octahedron we prove that our new grids are diameter bounded and then for the resulting configurations of points we calculate some riesz s energies and we compare them with the optimal ones for some configurations we also calculate the point energies and we list the minimum and maximum values concluding that these values are very close finally we show how we can map a hemisphere of the earth onto a square using our new area preserving projection the simplicity and the symmetry of our formulas lead to fast computations

estimation of fekete points we aim here at presenting a new procedure to numerically estimate the fekete points of a wide variety of compact sets in r 3 we understand the fekete point problem in terms of the identification of near equilibrium configurations for a potential energy that depends on the relative position of n particles the compact sets for which our procedure works are basically the finite union of piecewise regular surfaces and curves in order to determine a good initial configuration to start the search of the fekete points of these objects we construct a sequence of approximating regular surfaces our algorithm is based on the concept of disequilibrium degree which is defined from a physical interpretation of the behavior of a system of particles when they search for a minimum energy configuration moreover the algorithm is efficient and robust independently of the considered compact set as well as of the kernel used to define the energy the numerical experimentation carried out suggests that a local minimum can be localized with a computational cost of order less than n 3

visibility of wavelet quantization noise the discrete wavelet transform dwt decomposes an image into bands that vary in spatial frequency and orientation it is widely used for image compression measures of the visibility of dwt quantization errors are required to achieve optimal compression uniform quantization of a single band of coefficients results in an artifact that we call dwt uniform quantization noise it is the sum of a lattice of random amplitude basis functions of the corresponding dwt synthesis filter we measured visual detection thresholds for samples of dwt uniform quantization noise in y cb and cr color channels the spatial frequency of a wavelet is r2 sup spl lambda where r is the display visual resolution in pixels degree and spl lambda is the wavelet level thresholds increase rapidly with wavelet spatial frequency thresholds also increase from y to cr to cb and with orientation from lowpass to horizontal vertical to diagonal we construct a mathematical model for dwt noise detection thresholds that is a function of level orientation and display visual resolution this allows calculation of a perceptually lossless quantization matrix for which all errors are in theory below the visual threshold the model may also be used as the basis for adaptive quantization schemes

on certain symmetries of the nonlinear schrodinger equation in this paper we reduce the problem of symmetries for the ginzburg landau equation to a specific problem of non classical symmetries for the linear free schrodinger equation with the help of an immersion result this new problem is related to the local u 1 invariance of the model and allows us to construct projective representations for arbitrary lie groups acting on the space time manifold of the related sigma model to achieve this result we propose an enlargement of the usual notion of non classical symmetry

energies and spacings of point charges on a sphere numerical methods are in general required for the determination of the stable configurations of n point charges on a sphere the stable configurations for n up to 50 have previously been ascertained and the authors extend the calculations for values up to 101 they report for the first time some remarkable global features of these configurations they show that the minimum energy accurately follows a simple half integral power law in 1 n over the full range they have investigated this power law is explicable in terms of the idealization of mapping a planar wigner lattice onto the surface of the unit sphere the pair distribution functions of the larger n configurations indicate predominant hexagonal coordination the coefficients of the observed power law are closely straddled by values calculated on the basis of hexagonal and square wigner lattices this highly accurate description of the energy permits them to remark on the detailed deviations of the individual structures from the general trend for n 30 they note that structures with n prime are relatively less stable while structures with n equal to 6 12 32 44 48 and 60 seem more stable

a visual model weighted cosine transform for image compression and quality assessment utilizing a cosine transform in image compression has several recognized performance benefits resulting in the ability to attain large compression ratios with small quality loss also incorporation of a model of the human visual system into an image compression or quality assessment technique intuitively should and has often proven to improve performance clearly then it should prove highly beneficial to combine the image cosine transform with a visual model in the past combining these two has been hindered by a fundamental problem resulting from the scene alteration that is necessary for proper cosine transform utilization a new analytical solution to this problem taking the form of a straightforward multiplicative weighting function is developed in this paper this solution is readily applicable to image compression and quality assessment in conjunction with a visual model and the image cosine transform in the development relevant aspects of a human visual system model are discussed and a refined version of the mean square error quality assessment measure is given which should increase this measure s utility

computational cost of the fekete problem i the forces method on the 2 sphere here we study the computational complexity of the fekete point problem namely we give an exhaustive description of the main properties of an algorithm for the minimization of the logarithmic potential energy on the 2 sphere and we characterize the probability distribution of the cost of the different minima in particular we show that a local minimum can be found with an average cost of about o n 2 8

optimal asymptotic bounds for spherical designs in this paper we prove the conjecture of korevaar and meyers for each n cdt d there exists a spherical t design in the sphere s d consisting of n points where cd is a constant depending only on d

a nodal triangle based spectral element method for the shallow water equations on the sphere a nodal triangle based spectral element se method for the shallow water equations on the sphere is presented the original se method uses quadrilateral elements and high order nodal lagrange polynomials constructed from a tensor product of the legendre gauss lobatto points in this work we construct the high order lagrange polynomials directly on the triangle using nodal sets obtained from the electrostatics principle j s hesthaven from electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex siam journal on numerical analysis 35 1998 655 676 and fekete points m a taylor b a wingate r e vincent an algorithm for computing fekete points in the triangle siam journal on numerical analysis 38 2000 1707 1720 these points have good approximation properties and far better lebesgue constants than any other nodal set derived for the triangle by employing triangular elements as the basic building blocks of the se method and the cartesian coordinate form of the equations we can use any grid imaginable including adaptive unstructured grids results for six test cases are presented to confirm the accuracy and stability of the method the results show that the triangle based se method yields the expected exponential convergence and that it can be more accurate than the quadrilateral based se method even while using 30 60 fewer grid points especially when adaptive grids are used to align the grid with the flow direction however at the moment the quadrilateral based se method is twice as fast as the triangle based se method because the latter does not yield a diagonal mass matrix

classes of hypercomplex polynomials of discrete variable based on the quasi monomiality principle with the aim of derive a quasi monomiality formulation in the context of discrete hypercomplex variables one will amalgamate through a clifford algebraic structure of signature 0 n the umbral calculus framework with lie algebraic symmetries the exponential generating function egf carrying the continuum dirac operator d j 1 n e j x j together with the lie algebraic representation of raising and lowering operators acting on the lattice h z n is used to derive the corresponding hypercomplex polynomials of discrete variable as appell sets with membership on the space clifford vector valued polynomials some particular examples concerning this construction such as the hypercomplex versions of falling factorials and the poisson charlier polynomials are introduced certain applications from the view of interpolation theory and integral transforms are also discussed

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

galilei invariant nonlinear systems of evolution equations all systems of n 1 dimensional quasilinear second order evolution equations invariant under chain of algebras ag 1 n contained in implied by ag1 1 n contained in implied by ag2 1 n are described the results obtained are illustrated by the examples of the nonlinear schrodinger equations hamilton jacobi type systems and reaction diffusion equations

on the classification of lie subalgebras and lie subgroups starting from a paper of patera et al j math phys 16 1597 1975 the classification of lie subalgebras is analyzed up to conjugation for a lie algebra having a solvable ideal here the hypothesis of semidirect product structure is not made this leads to some natural affine structure next a complete classification is made of closed lie subgroups of a given lie group up to conjugation supposing the corresponding classification for lie subalgebras is known this is illustrated in the case of the group se 3

group invariant solutions of semilinear schrodinger equations in multi dimensions symmetry group methods are applied to obtain all explicit group invariant radial solutions to a class of semilinear schrodinger equations in dimensions n 1 both focusing and defocusing cases of a power nonlinearity are considered including the special case of the pseudo conformal power p 4 n relevant for critical dynamics the methods involve first reduction of the schrodinger equations to group invariant semilinear complex 2nd order ordinary differential equations odes with respect to an optimal set of one dimensional point symmetry groups and second use of inherited symmetries hidden symmetries and conditional symmetries to solve each ode by quadratures through noether s theorem all conservation laws arising from these point symmetry groups are listed some group invariant solutions are found to exist for values of n other than just positive integers and in such cases an alternative two dimensional form of the schrodinger equations involving an extra modulation term with a parameter m

equidistributed icosahedral configurations on the sphere abstract we present and analyze a new sequence of equidistributed icosahedral configurations these configurations are created by combining the m n icosahedral nodes of caspar and klug and adapting the azimuthal projection method of snyder to create a low deformation equal area mapping from the regular icosahedron to s 2 we prove an upper bound on the mesh ratio of these configurations and analyze numerically the riesz potential energies of the point sets of size n 50 000

lie theoretical generalization and discretization of the pinney equation abstract a lie group approach is adopted to construct generalized pinney equations of two distinct types which admit nonlinear superposition principles the procedure also provides a route to discretizations of these pinney equations which preserves the property of admittance of a nonlinear superposition principle to conclude underlying linearizations are placed in the context of results forc integrable nonlinear schrodinger equations

energy of static electron lattices a new approach to evaluating the static lattice energy of any given wigner lattice is proposed the method is much simpler than the traditional one and the results are much faster to evaluate the new approach is applied to two dimensional triangular lattices where it is shown that the triangular lattice is more stable than the square lattice three dimensional hexagonal lattices are also investigated in addition the authors place many of their considerations on a rigorous mathematical footing

estimating the number of stable configurations for the generalized thomson problem given a natural number n one may ask what configuration of n points on the two sphere minimizes the discrete generalized coulomb energy if one applies a gradient based numerical optimization to this problem one encounters many configurations that are stable but not globally minimal this led the authors of this manuscript to the question how many stable configurations are there in this manuscript we report methods for identifying and counting observed stable configurations and estimating the actual number of stable configurations these estimates indicate that for n approaching two hundred there are at least tens of thousands of stable configurations

partially invariant solutions of nonlinear klein gordon and laplace equations the concept of partially invariant solutions is discussed in the framework of the group analysis of systems of partial differential equations using a general and systematic approach based on subgroup classification methods nontrivial partially invariant solutions for specific equations are explicitly obtained belonging to the nlkge and nlle classes in two dimensions it is proven that the obtained partially invariant solutions are distinct from the invariant ones and from solutions obtained by other methods of reducing pdes to odes

the random walk on the boundary method for calculating capacitance in this paper we present the random walk on the boundary method for the rapid solution of integral equations that arise in electrostatics and related areas this method is a monte carlo method based on the construction of a markov chain that is readily interpreted as a random walk along the boundary over which integration in the integral equation is taken to illustrate the usefulness of this technique we apply it to the computation of the capacitance of the unit cube obtaining the capacitance of the cube usually requires computing the charge density and this problem has been used as a benchmark by many in the field for algorithms of this kind here the random walk on the boundary method does not require charge density computation and obtains the capacitance of the cube within a statistical error of 2 7 10 7 the most accurate estimate to date

symmetry reductions of a generalized cylindrical nonlinear schrodinger equation in this work symmetry reductions for a generalized cylindrical nonlinear schrodinger equation are presented these are obtained using an extension of the direct method originally developed by clarkson and kruskal 1989 which involves no group theoretic techniques

adaptive cosine transform coding of images in perceptual domain an adaptive cosine transform coding scheme for color images which incorporates human visual properties into the coding scheme is described it employs adaptive quantization to exploit the statistical nature of the coefficients and adaptive block distortion equalization to reduce the block edge structures inherent in block transform coding schemes results show that the subjective quality of the reconstructed images at a bit rate of 0 4 bit pixel or a compression ratio of 60 1 is very good

group invariant solutions and optimal systems for multidimensional hydrodynamics the group properties of the three dimensional 3 d one temperature hydrodynamic equations including nonlinear conduction and a thermal source are presented a subgroup corresponding to axisymmetric geometry is chosen and the details of the construction of the one and two dimensional optimal systems are shown the two dimensional optimal system is used to generate 23 intrinsically different reductions of the 2 d partial differential equations to ordinary differential equations these ordinary differential equations can be solved to provide analytic solutions to the original partial differential equations two example analytic solutions are presented a 2 d axisymmetric flow with a p2 asymmetry and a 3 d spiraling flow

partially invariant solutions of a class of nonlinear schrodinger equations partially invariant solutions of a general class of nonlinear schrodinger equations involving four arbitrary functions of the modulus rho of the solution and its derivative rho x are obtained the modulus rho xi is assumed to depend on a symmetry variable xi whereas the phase omega x t depends on both independent variables both rho and omega are obtained explicitly as are the conditions on the coefficients in the equation necessary for such solutions to exist

computing multivariate fekete and leja points by numerical linear algebra we discuss and compare two greedy algorithms that compute discrete versions of fekete like points for multivariate compact sets by basic tools of numerical linear algebra the first gives the so called approximate fekete points by qr factorization with column pivoting of vandermonde like matrices the second computes discrete leja points by lu factorization with row pivoting moreover we study the asymptotic distribution of such points when they are extracted from weakly admissible meshes

solvable lie algebras with heisenberg ideals all finite dimensional indecomposable solvable lie algebras l n f having the heisenberg algebra h n as the nilradical are constructed the number of non nilpotent elements f that can be added to h n satisfies f or n 1 the casimir and generalized casimir operators of the algebras l n f are obtained

fast high dimensional node generation with variable density abstract we present an algorithm for producing discrete distributions with a prescribed nearest neighbor distance function our approach is a combination of quasi monte carlo q mc methods and weighted riesz energy minimization the initial distribution is a stratified q mc sequence with some modifications a suitable energy functional on the configuration space is then minimized to ensure local regularity the resulting node sets are good candidates for building meshless solvers and interpolants as well as for other purposes where a point cloud with a controlled separation covering ratio is required applications of a three dimensional implementation of the algorithm in particular to atmospheric modeling are also given

quasi monte carlo rules for numerical integration over the unit sphere mathbb s 2 we study numerical integration on the unit sphere mathbb s 2 subseteq mathbb r 3 using equal weight quadrature rules where the weights are such that constant functions are integrated exactly the quadrature points are constructed by lifting a 0 m 2 net given in the unit square 0 1 2 to the sphere mathbb s 2 by means of an area preserving map a similar approach has previously been suggested by cui and freeden siam j sci comput 18 2 595 609 1997 we prove three results the first one is that the construction is almost optimal with respect to discrepancies based on spherical rectangles further we prove that the point set is asymptotically uniformly distributed on mathbb s 2 and finally we prove an upper bound on the spherical cap l 2 discrepancy of order n 1 2 log n 1 2 where n denotes the number of points this improves upon the bound on the spherical cap l 2 discrepancy of the construction by lubotzky phillips and sarnak commun pure appl math 39 s suppl s149 s186 1986 by a factor of sqrt log n numerical results suggest that the 0 m 2 nets lifted to the sphere mathbb s 2 have spherical cap l 2 discrepancy converging with the optimal order of n 3 4