spherical codes generated by binary partitions of symmetric pointsets several constructions are presented by which spherical codes are generated from groups of binary codes in the main family of constructions the codes are generated from equally spaced symmetric pointsets on the real line the main ideas are code concatenation and set partitioning extensive tables are presented for spherical codes in dimension n spl les 24

approximate travelling waves for generalized kpp equations and classical mechanics we consider the existence of approximate travelling waves of generalized kpp equations in which the initial distribution can depend on a small parameter which in the limit 0 is the sum of some functions or a step function using the method of elworthy truman 1982 we construct a classical path which is the backward flow of a classical newtonian mechanics with given initial position and velocity before the time at which the caustic appears by the feynman kac formula and the maruyama girsanov cameron martin transformation we obtain an identity from which with a late caustic assumption we see the propagation of the global wave front and the shape of the trough our theory shows clearly how the initial distribution contributes to the propagation of the travelling wave finally we prove a huygens principle for kpp equations on complete riemannian manifolds without cut locus with some bounds on their volume element in particular cartan hadamard manifolds

stabilized multilevel monte carlo method for stiff stochastic differential equations a multilevel monte carlo mlmc method for mean square stable stochastic differential equations with multiple scales is proposed for such problems that we call stiff the performance of mlmc methods based on classical explicit methods deteriorates because of the time step restriction to resolve the fastest scales that prevents to exploit all the levels of the mlmc approach we show that by switching to explicit stabilized stochastic methods and balancing the stabilization procedure simultaneously with the hierarchical sampling strategy of mlmc methods the computational cost for stiff systems is significantly reduced while keeping the computational algorithm fully explicit and easy to implement numerical experiments on linear and nonlinear stochastic differential equations and on a stochastic partial differential equation illustrate the performance of the stabilized mlmc method and corroborate our theoretical findings

defect free global minima in thomson s problem of charges on a sphere given n unit points charges on the surface of a unit conducting sphere what configuration of charges minimizes the coulombic energy ensuremath sum igj 1 n 1 r ij due to an exponential rise in good local minima finding global minima for this problem or even approaches to do so has proven extremely difficult for n 10 h 2 hk k 2 2 recent theoretical work based on elasticity theory and subsequent numerical work has shown that for n ensuremath gtrsim 500 char21 1000 adding dislocation defects to a symmetric icosadeltahedral lattice lowers the energy here we show that in fact this approach holds for all n and we give a complete or near complete catalogue of defect free global minima

numerical minimization of dirichlet laplacian eigenvalues of four dimensional geometries we develop the first numerical study in four dimensions of optimal eigenmodes associated with the dirichlet laplacian we describe an extension of the method of fundamental solutions adapted to the four dimensional context based on our numerical simulation and a postprocessing adapted to the identification of relevant symmetries we provide and discuss the numerical description of the eighth first optimal domains

streamline topologies for integrable vortex motion on a sphere we describe the instantaneous streamline patterns that occur on the surface of a two dimensional sphere in the presence of point vortices of general strength for cases in which the system is completely integrable after stating some general results based on the spherical topology we categorize all possible instantaneous streamline patterns and describe their stagnation point structure for the cases of two and three vortices it is found that for the case of two vortices the only non degenerate topologies that can arise are a figure eight lemniscate or a limacon which are homotopically equivalent for the case of three vortices there are 12 topologically distinct primitives from which an additional 23 patterns can be produced via continuous deformations on the sphere homotopies all possible streamline patterns that arise from three vortex motion can be obtained via linear superposition of the primitive topologies and their homotopic equivalents in this sense the primitives can be viewed as the building blocks for the general flow patterns the equations of motion and streamline patterns in the stereographic plane are obtained using the projected equations of motion in hamiltonian form we describe streamline patterns for three vortex fixed equilibria and relative equilibria as seen in both a fixed and a rotating frame of reference non equilibrium streamline patterns for a collapsing and special periodic solution are studied although to generally understand and classify the many bifurcations associated with non equilibrium patterns would require more extensive computation we conclude with a discussion of the relevance of the three vortex topological classification scheme and the bifurcation of streamline topologies for understanding global atmospheric weather patterns spherical isobars and large scale mixing phenomena 2000 elsevier science b v all rights reserved

finiteness of stationary configurations of the four vortex problem we show that the number of relative equilibria equilibria and rigidly translating configurations in the problem ot tour point vortices is hnite the proof is based on symbolic and exact integer computations which are carried out by computer we also provide upper bounds for these classes of stationary configurations

distributions of singular values for some random matrices the singular value decomposition is a matrix decomposition technique widely used in the analysis of multivariate data such as complex space time images obtained in both physical and biological systems in this paper we examine the distribution of singular values of low rank matrices corrupted by additive noise past studies have been limited to uniform uncorrelated noise using diagrammatic and saddle point integration techniques we extend these results to heterogeneous and correlated noise sources we also provide perturbative estimates of error bars on the reconstructed low rank matrix obtained by truncating a singular value decomposition 1999 the american physical society

averaging sets a generalization of mean values and spherical designs 0 1 r one cannot of course expect there to be a single point x at which every one assumes its mean value but there is an easy generalization in terms of weighted averages there exist at most m points xi and positive weights oi summing to 1 so that aifj xi i fi dt for j 1 2 m

packings of a charged line on a sphere we find equilibrium configurations of open and closed lines of charge on a sphere and track them with respect to varying sphere radius closed lines transition from a circle to a spiral like shape through two low wave number bifurcations char22 baseball seam and twist char22 which minimize coulomb energy the spiral shape is the unique stable equilibrium of the closed line other unstable equilibria arise through tip splitting events an open line transitions smoothly from an arc of a great circle to a spiral as the sphere radius decreases under repulsive potentials with faster than coulomb power law decay the spiral is tighter in initial stages of sphere shrinkage but at later stages of shrinkage the equilibria for all repulsive potentials converge on a spiral with uniform spacing between turns multiple stable equilibria of the open line are observed

the n vortex problem on a rotating sphere ii heterogeneous platonic solid equilibria we describe a new method of constructing point vortex equilibria on a sphere made up of n vortices with different strengths such equilibria called heterogeneous equilibria are obtained for the five platonic solid configurations hence for graphic 1 the method is based on calculating a basis set for the nullspace of a matrix obtained by enforcing the necessary and sufficient condition that the mutual distances between each pair of vortices remain constant by symmetries inherent in the platonic solid configurations this matrix is reduced for each case and we call the dimension of the nullspace the degree of heterogeneity of the structure for the tetrahedron n 4 and octahedron n 6 the degree of heterogeneity is 4 and 6 respectively hence we are free to choose each of the vortex strengths independently for the cube n 8 the degree of heterogeneity is 5 for the icosahedron n 12 it is 7 while for the dodecahedron n 20 it is 4 thus the entire set of equilibria based on the platonic solid configurations is obtained including substructures associated with each configuration constructed by taking different linear combinations of the basis elements 1 embed inline graphic 1 gif

equidistribution on the sphere a concept of generalized discrepancy which involves pseudodifferential operators to give a criterion of equidistributed pointsets is developed on the sphere a simply structured formula in terms of elementary functions is established for the computation of the generalized discrepancy with the help of this formula five kinds of point systems on the sphere namely lattices in polar coordinates transformed two dimensional sequences rotations on the sphere triangulations and sum of three squares sequence are investigated quantitative tests are done and the results are compared with one another our calculations exhibit different orders of convergence of the generalized discrepancy for different types of point systems

an iterative technique for the numerical solution of nonlinear stochastic ito volterra integral equations abstract the main aim of this study is to propose a numerical iterative approach for obtaining approximate solutions of nonlinear stochastic ito volterra integral equations the method is based on a combination of the successive approximations method the linear spline interpolation and ito approximation moreover the convergence analysis of this method is investigated in detail to confirm the validity and applicability of the new method several illustrative examples and some applications are presented

construction of point vortex equilibria via brownian ratchets a theory capable of producing equilibrium configurations of point vortices in the plane along with a numerical scheme to compute them is described the theory is formulated as a problem in linear algebra where one must find solutions to the matrix equation agz 0 where a is the 1 2 n nk1 n non normal configuration matrix obtained by requiring that all intervortical distances remain fixed and g2r n are the n vortex strengths for existence of an equilibrium a must have a non trivial nullspace we consider the singular values of a when this has one or more zero singular values the nullspace of a is non empty and an equilibrium exists for some choice of g new equilibrium configurations are found numerically by randomly depositing n points in the plane which generically gives rise to a configuration matrix a with empty nullspace using the sum of squares of the k smallest singular values of a as a ratchet we thermally fluctuate the configuration allowing each point to execute a random walk in the plane retaining only those configurations which reduce this quantity at the next step the configuration is thus driven to one with nullspace a zko0 these converged states are not necessarily nearby their initial configurations typically they are asymmetric and often we can drive the same initial state to several different equilibria a reverse ratchet method is also described which can produce initial conditions that would evolve to a specified equilibrium state once a converged final state is achieved the full singular value decomposition of a is used to calculate an optimal basis set for the nullspace of a and thus all allowable g the distribution of the singular values gives important information on the size of each equilibrium state as measured by frobenius norm their distance from each other spacing and density and how far a randomly chosen system of n points in the plane is from the nearest equilibrium configuration with a specified rank as well as its shannon entropy

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

saturated configuration and new large construction of equiangular lines abstract a set of lines through the origin in euclidean space is called equiangular when any pair of lines from the set intersects with each other at a common angle we study the maximum size of equiangular lines in euclidean space and use a graph theoretic approach to prove that all the currently known constructions for maximum equiangular lines in r d cannot be added by any more lines to form a larger equiangular set of lines when d 14 16 17 18 19 and 20 we give new constructions of large equiangular lines which are 248 equiangular lines in r 42 200 equiangular lines in r 41 168 equiangular lines in r 40 152 equiangular lines in r 39 with angle 1 7 and 56 equiangular lines in r 18 with angle 1 5

minimum energy point charge configurations on a circular disk stochastic global optimization methods are used together with problem specific heuristics to find configurations of n equal point charges on a circular disk such that the potential energy is minimized previous such optimal or conjecturally optimal configurations have been published only for and n 29 30 50 earlier results are examined and the best currently known configurations most of them new for are presented

a numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis functions abstract in this paper we use hat basis functions to solve the system of fredholm integral equations sfies and the system of volterra integral equations svies of the second kind this method converts the system of integral equations into a linear or nonlinear system of algebraic equations also we consider the order of convergence of the method and show that it is o h 2 application of the method on some examples show its accuracy and efficiency

stochastic volterra equations in banach spaces and stochastic partial differential equation abstract in this paper we study the existence uniqueness and large deviation estimate for stochastic volterra integral equations with singular kernels in 2 smooth banach spaces then we apply them to a large class of semilinear stochastic partial differential equations spde and obtain the existence of unique maximal strong solutions in the sense of sde and pde under local lipschitz conditions moreover stochastic navier stokes equations are also investigated

density of spherically embedded stiefel and grassmann codes the density of a code is the fraction of the coding space covered by packing balls centered around the codewords a high density indicates that a code performs well when used as a uniform point wise discretization of an ambient space this paper investigates the density of codes in the complex stiefel and grassmann manifolds equipped with the chordal distance arising from an euclidean embedding including the unitary group as a special case the choice of distance enables the treatment of the manifolds as subspaces of euclidean hyperspheres in this geometry the densest packings are not necessarily equivalent to maximum minimum distance codes computing a code s density follows from computing 1 the normalized volume of a metric ball and 2 the kissing radius the radius of the largest balls one can pack around the codewords without overlapping first the normalized volume of a metric ball is evaluated by asymptotic approximations the volume of a small ball can be well approximated by the volume of a locally equivalent tangential ball in order to properly normalize this approximation the precise volumes of the manifolds induced by their spherical embedding are computed for larger balls a hyperspherical cap approximation is used which is justified by a volume comparison theorem showing that the normalized volume of a ball in the stiefel or grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity then bounds on the kissing radius are derived alongside corresponding bounds on the density unlike spherical codes or codes in flat spaces the kissing radius of grassmann or stiefel codes cannot be exactly determined from its minimum distance it is nonetheless possible to derive bounds on density as functions of the minimum distance stiefel and grassmann codes have larger density than their image spherical codes when dimensions tend to infinity finally the bounds on density lead to refinements of the standard hamming bounds for stiefel and grassmann codes

the damped pendulum random differential equation a comprehensive stochastic analysis via the computation of the probability density function this work has been supported by the spanish ministerio de economia y competitividad grant mtm2017 89664 p marc jornet acknowledges the doctorate scholarship granted by programa de ayudas de investigacion y desarrollo paid universitat politecnica de valencia the authors are grateful for the valuable comments raised by the reviewers that have improved the final version of the paper

mean square numerical solution of random differential equations facts and possibilities this paper deals with the construction of numerical solutions of random initial value differential problems the random euler method is presented and the conditions for the mean square convergence are established numerical examples show that random euler method gives good results even if the sufficient convergence conditions are not satisfied

vortex lattice theory a particle interaction perspective recent experiments on the formation of vortex lattices in bose einstein condensates has produced the need for a mathematical theory that is capable of predicting a broader class of lattice patterns ones that are free of discrete symmetries and can form in a random environment we give an overview of an n particle based hamiltonian theory which if formulated in terms of the o n 2 interparticle distances leads to the analysis of a nonnormal configuration matrix whose nullspace structure determines the existence or nonexistence of a lattice the singular value decomposition of this matrix leads to a method in which all lattice patterns and the associated particle strengths in principle can be classified and calculated by a random walk scheme which systematically uses the m smallest singular values as a ratchet mechanism to home in on lattices with m dimensional nullspaces where 0 m le n the resulting distribution of singular values encodes detailed geometric properties of the lattice and allows us to identify and calculate important quantitative measures associated with each lattice including its size as measured by the frobenius or 2 norms distance between the lattices hence lattice density robustness and shannon entropy as a quantitative measure of its level of disorder this article gives an overview of vortex lattice theory from 1957 to the present highlighting recent experiments in bose einstein condensate systems and formulating questions that can be addressed by understanding the singular value decomposition of the configuration matrix we discuss some of the computational challenges associated with producing large n lattices the subtleties associated with understanding and exploiting complicated hamiltonian energy surfaces in high dimensions and we highlight ten important directions for future research in this area

point vortex equilibria on the sphere via brownian ratchets we describe a brownian ratchet scheme that we use to calculate relative equilibrium configurations of n point vortices of mixed strength on the surface of a unit sphere we formulate it as a proble

green s function for the laplace beltrami operator on a toroidal surface green s function for the laplacebeltrami operator on the surface of a three dimensional ring torus is constructed an integral ingredient of our approach is the stereographic projection of the toru

singular solutions of some nonlinear elliptic equations abstract we give a classification of isolated singularities of any solution of the equation u u u q 1 0 if 0 is the singular point we prove that there exist two types of singularities when 1 n n 2 and that we have essentially the following in 0 i either u x l q n x 2 q 1 with l q n 2 q 1 2q q 1 n 1 q 1 ii or u x c x where c is any constant and the fundamental solution of the laplacian in r n

complex ginzburg landau equations and dynamics of vortices filaments and codimension 2 submanifolds here we study the asymptotic behavior of solutions to the complex ginzburg landau equations and their associated heat flows in arbitrary dimensions when the ginzburg landau parameter tends to infinity we prove that the energies of solutions in the flow are concentrated at vortices in two dimensions filaments in three dimensions and codimension 2 submanifolds in higher dimensions moreover we show the dynamical laws for the motion of these vortices filaments and codimension 2 submanifolds away from the energy concentration sets we use some measure theoretic arguments to show the strong convergence of solutions in both static and heat flow cases 1998 john wiley sons inc

euler schemes and large deviations for stochastic volterra equations with singular kernels in this paper an euler type approximation is constructed for stochastic volterra equation with singular kernels which provides an algorithm for numerical calculation then the large deviation estimates of small perturbation to equations of this type are obtained we finally apply them to sdes with the kernel of fractional brownian motion with hurst parameter h 0 1

rotating 2n vortex solutions to the gross pitaevskii equation on s2 we establish the existence of rotating solutions to the gross pitaevskii equation iut u 1 2 1 u 2 uposed on s2 that is for u s2 r c these solutions possess vortices that for all time follow the vortex paths of known relative equilibria to the point vortex problem on the two sphere in the asymptotic regime 1 the approach is variational based on minimization of the ginzburg landau energy subject to a momentum constraint we also establish orbital stability within a class of symmetric initial data

the n vortex problem on a rotating sphere i multi frequency configurations the problem of n point vortices moving on a rotating unit sphere is considered through a sequence of linear coordinate transformations which takes into account the orientation of the centre of vorticity vector with respect to the axis of rotation we show how to reduce the problem to that on a non rotating sphere where the centre of vorticity vector is aligned with the z axis as a consequence we prove that integrability on the rotating sphere is the same as on the non rotating sphere namely the three vortex problem on the rotating sphere is integrable for all vortex strengths while the four vortex problem is integrable in the special case where the centre of vorticity is zero rigid multi frequency configurations that retain their shape while rotating about two independent axes with two independent frequencies are obtained and necessary conditions for one and two frequency motions are derived examples including dipoles which exhibit global wobbling and tumbling dynamics rings and platonic solid configurations are shown to undergo either periodic or quasi periodic evolution on the rotating sphere depending on the ratio of the solid body rotational frequency to the rotational frequency associated with the rigid structure

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

a computational method for solving stochastic ito volterra integral equations based on stochastic operational matrix for generalized hat basis functions abstract in this paper a new computational method based on the generalized hat basis functions is proposed for solving stochastic ito volterra integral equations in this way a new stochastic operational matrix for generalized hat functions on the finite interval 0 t is obtained by using these basis functions and their stochastic operational matrix such problems can be transformed into linear lower triangular systems of algebraic equations which can be directly solved by forward substitution also the rate of convergence of the proposed method is considered and it has been shown that it is o 1 n 2 further in order to show the accuracy and reliability of the proposed method the new approach is compared with the block pulse functions method by some examples the obtained results reveal that the proposed method is more accurate and efficient in comparison with the block pule functions method

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

point vortex equilibria and optimal packings of circles on a sphere we answer the question of whether optimal packings of circles on a sphere are equilibrium solutions to the logarithmic particle interaction problem for values of n 3 12 and 24 the only values of n for which the optimal packing problem also known as the tammes problem has rigorously known solutions we also address the cases n 13 23 where optimal packing solutions have been computed but not proven analytically as in jamaloodeen newton jamaloodeen newton 2006 proc r soc lond ser a 462 3277 3299 doi 10 1098 rspa 2006 1731 1 a logarithmic or point vortex equilibrium is determined by formulating the problem as the one in linear algebra graphic 2 where a is a n n 1 2 n non normal configuration matrix obtained by requiring that all interparticle distances remain constant if a has a kernel the strength vector graphic 3 is then determined as a right singular vector associated with the zero singular value or a vector that lies in the nullspace of a where the kernel is multi dimensional first we determine if the known optimal packing solution for a given value of n has a configuration matrix a with a non empty nullspace the answer is yes for n 3 9 11 14 16 and no for n 10 15 17 24 we then determine a basis set for the nullspace of a associated with the optimally packed state answer the question of whether n equal strength particles graphic 4 form an equilibrium for this configuration and describe what is special about the icosahedral configuration from this point of view we also find new equilibria by implementing two versions of a random walk algorithm first we cluster sub groups of particles into patterns during the packing process and grow a packed state using a version of the yin yang algorithm of longuet higgins longuet higgins 2008 proc r soc a doi 10 1098 rspa 2008 0219 we also implement a version of our brownian ratchet algorithm to find new equilibria near the optimally packed state for n 10 15 17 24 1 http dx doi org 10 1098 rspa 2006 1731 2 embed inline graphic 1 gif 3 embed inline graphic 2 gif 4 embed inline graphic 3 gif

snub polyhedra and organic growth this paper describes a new application of polyhedral theory to the growth of the outer sheath of certain viruses such structures are often modular consisting of one or two types of units arranged in a symmetric pattern in particular the polyoma virus has a structure apparently related to the snub dodecahedron here we consider the problem of how such patterns might grow in time starting from a given number n of randomly placed circles on the surface of a sphere the circles are first jostled by random perturbations then their radii are enlarged then they are jostled again and so on this yin yang method of growth can result in some very close packings when n 12 the closest packing corresponds to the snub tetrahedron and when n 24 the closest packing corresponds to the snub cube however when n 60 the closest packing does not correspond to the snub dodecahedron but to a less symmetric arrangement special attention is given to the structure of the human polyoma virus for which n 72 it is shown that the yin yang procedure successfully assembles the observed structure provided that the 72 circles are pre assembled in clusters of six each cluster consists of five circles arranged symmetrically around a sixth at the centre as in a flower with five petals this has implications for the assembly of the capsomeres in a polyoma virus

vector stabilizers and subgroups of leech lattice groups the leech lattice is a 2bdimensional lattice whose automo hism group is the double cover 2 co of conway s group co see 3 p 1801 from it the other conway groups co and co mclaughlin s group mcl and the higman sims group hs may be obtained by fixing certain sets of vectors the lattice played a central role in the determination of the maximal subgroups of co see 10 and co l see 1 i the crucial observation is that if a group fixes a non zero vector in a rational representation then it fixes a non zero vector in any reduction of this representation modulo 2 one of the aims of the present paper is to generalise this result and put it in its proper context in modular representation theory the other main aim is to apply this result to other leech lattice groups and so give a uniform classification of the maximal non local subgroups of mcl and co originally determined by finkelstein in s and hs originally determined by magliveras in s and co see lo as well as the automorphism groups mcl 2 of mcl and hs 2 of hs originally determined in 127 in section a we study the geometry of the leech lattice in detail section b contains the representation theory we need in section c to determine the maximal non local subgroups of the various groups finally in section d we show how to obtain various graphs from the leech lattice and in particular deduce that the groups we are studying really are isomorphic to mcl hs and u 2 it should perhaps be stressed that few if any of the results in this paper are new although a number of them have not appeared in print before my intention has been rather to present a unified picture of the relation between the subgroup structure of the various groups involved and the geometric structure of the leech lattice

finiteness of relative equilibria of the four body problem we show that the number of relative equilibria of the newtonian four body problem is finite up to symmetry in fact we show that this number is always between 32 and 8472 the proof is based on symbolic and exact integer computations which are carried out by computer

three ways to solve the poisson equation on a sphere with gaussian forcing motivated by the needs of vortex methods we describe three different exact or approximate solutions to the poisson equation on the surface of a sphere when the forcing is a gaussian of the three dimensional distance 2 j exp 2 e 2 1 cos q c g a u s s e more precisely the forcing is a gaussian minus the gauss constraint constant c g a u s s this subtraction is necessary because j is bounded for any type of forcing only if the integral of the forcing over the sphere is zero y kimura h okamoto vortex on a sphere j phys soc jpn 56 1987 4203 4206 d g dritschel contour dynamics surgery on the sphere j comput phys 79 1988 477 483 the legendre polynomial series is simple and yields the exact value of the gauss constraint constant but converges slowly for large e the analytic solution involves nothing more exotic than the exponential integral but all four terms are singular at one or the other pole cancelling in pairs so that j is everywhere nice the method of matched asymptotic expansions yields simpler uniformly valid approximations as series of inverse even powers of e that converge very rapidly for the large values of e e 40 appropriate for geophysical vortex computations the series converges to a nonzero o exp 4 e 2 error everywhere except at the south pole where it diverges linearly with order instead of the usual factorial order

symmetric four dimensional polytope and visualization method in four eight and sixteen dimensions using hopf maps inspired by and using methods of optimization derived from classical three dimensional electrostatics we have found a four dimensional polytope new to our knowledge with a high degree of symmetry in terms of the lengths of sides 64 of the 80 vertices have twelve nearest neighbors with the same four nearest neighbor distances and the other 16 vertices have ten nearest neighbors with distances that are two of the four nearest neighbor distances for the set of 64 vertices we give and illustrate a simple geometric method to visualize this configuration and other configurations in four eight and sixteen dimensions

a numerical method for solving m dimensional stochastic ito volterra integral equations by stochastic operational matrix the multidimensional ito volterra integral equations arise in many problems such as an exponential population growth model with several independent white noise sources in this paper we obtain a stochastic operational matrix of block pulse functions on interval 0 1 to solve m dimensional stochastic ito volterra integral equations by using block pulse functions and their stochastic operational matrix of integration m dimensional stochastic ito volterra integral equations can be reduced to a linear lower triangular system which can be directly solved by forward substitution we prove that the rate of convergence is o h furthermore a 95 confidence interval of the errors mean is made the results shows that the approximate solutions have a credible degree of accuracy

motion of three point vortices on a sphere abstract we solve the equations governing the relative motion of three point vortices of arbitrary strength moving on the surface of a sphere of radius r the system is more general than the corresponding one in the plane 1 3 5 19 28 33 34 37 38 and reduces to it in the limit r as long as the three vortices remain sufficiently close to each other during the course of their motion instead of using spherical coordinates we use cartesian coordinates in which the vector i points from the center of the sphere to the vortex with strength i an important conserved quantity is the center of vorticity vector c i i i which with no loss of generality we align with the z axis based on the size of this vector relative to the radius of the sphere we classify the motion into one of five types super radial radial sub radial degenerate or a limiting super radial case this categorization allows us to draw several conclusions about the qualitative motion of the vortices we then fully characterize all fixed and relative equilibria on the sphere for fixed equilibria the vortices must lie on great circles geodesics if the strengths are equal they form an equilateral triangle otherwise the triangle shape is specified once the strength of the vortices is given the relative equilibria are classified as either degenerate c 0 or non degenerate c 0 for each type the shape of the vortex triangle is described and the frequency of rotation is computed as in the planar problem it is possible to introduce trilinear coordinates and study the motion in a phase plane which allows us to locate all the equilibria as well as to characterize more complex relative dynamics we then describe self similar vortex collapse on the sphere stating necessary and sufficient conditions for collapse to occur computing the collapse times and vortex trajectories on the route towards collapse comparisons are made with the collapse formulas in the plane derived in 1 30

arrangement of 23 points on a sphere on a conjecture of r m robinson the problem of determining the largest angular diameter d n of n equal circles which can be packed on the surface of a sphere without overlapping is investigated it is known that the best packing of 5 11 circles on a sphere is obtained if one circle is removed from the best packing of 6 12 circles robinson has suggested that perhaps there are some other cases also where this property holds possibly n 24 48 60 120 are the circle numbers for which d n 1 d n in this paper it is proved that this property does not hold for n 24 thus d 23 d 24 and it is conjectured that d n 1 d n for n 48 60 120 a new packing construction is presented for 23 circles on a sphere with circle diameter 43 709964 and for 29 circles with circle diameter 38 677079

transition from spherical circle packing to covering geometrical analogues of chemical isomerization how must n equal circles spherical caps of given angular radius r be arranged on the surface of a sphere so that the area covered by the circles will be as large as possible in this paper conjectured solutions of this problem for n 5 7 8 9 10 11 are given when r varies from the maximum packing radius to the minimum covering radius the relation of the results of this problem to the extremal configurations of n mutually repulsive equal point charges on the surface of a sphere is discussed variation of the solution polyhedra with r gives an abstract model of chemical isomerization processes for example for n 5 where the packing covering transition models the berry pseudo rotation of a trigonal bipyramid

multiple folding and packing in dna modeling in this paper we introduce and analyze a set of equations to study geometric and energetic aspects associated with the kinematics of multiple folding and coiling of closed filaments for dna modeling by these equations we demonstrate that a high degree of coiling may be achieved at relatively low energy costs through appropriate writhe and twist distribution and independently from the number of coils formed for sufficiently high twist we show that coiling is actually favoured by elastic energy relaxation when the deformation energy is due to curvature and the mean twist of the filament we give estimates for the writhing process based purely on curvature information and the number of coils produced we also determine the packing rate associated with filament compaction in the case of a hierarchical helical coiling these results find useful applications in dna biology especially in modeling dna wrapping in proteic regions where there is a strong connection between high coiling efficient compaction and energy localization

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

well posedness and em approximations for non lipschitz stochastic fractional integro differential equations abstract this paper considers the nonlinear stochastic fractional integro differential equations sfides under the non lipschitz conditions which are general and include many stochastic fractional integro differential equations discussed in literature an important connection between sfides and stochastic volterra integral equations svies is derived in detail by the fubini theorem using the euler maruyama em approximation we prove the existence uniqueness and stability results of the solution to sfides moreover it is shown that the modified em solution of sfides shares strong first order sharp convergence the numerical examples are performed to show the accuracy and effectiveness of the numerical scheme and verify the correctness of our theoretical analysis

approximate solution of the trust region problem by minimization over two dimensional subspaces the trust region problem minimization of a quadratic function subject to a spherical trust region constraint occurs in many optimization algorithms in a previous paper the authors introduced an inexpensive approximate solution technique for this problem that involves the solution of a two dimensional trust region problem they showed that using this approximation in an unconstrained optimization algorithm leads to the same theoretical global and local convergence properties as are obtained using the exact solution to the trust region problem this paper reports computational results showing that the two dimensional minimization approach gives nearly optimal reductions in then dimension quadratic model over a wide range of test cases we also show that there is very little difference in efficiency and reliability between using the approximate or exact trust region step in solving standard test problems for unconstrained optimization these results may encourage the application of similar approximate trust region techniques in other contexts

motion of vortex lines in the ginzburg landau model equations of motion for vortex lines in the ginzburg landau theory are derived we construct asymptotic approximations for the complex order parameter which are valid in the core region and in the far field using the method of matched asymptotic expansions we show that to leading order the line moves in the binormal direction with a curvature dependent velocity we also consider the contribution of remote parts of the line interaction between several vortex lines and interaction with external fields

upper bounds on the minimum distance of spherical codes we use linear programming techniques to obtain new upper bounds on the maximal squared minimum distance of spherical codes with fixed cardinality functions q sub j n s are introduced with the property that q sub j n s m if and only if the levenshtein bound l sub m n s on a n s max w w is an n w s code can be improved by a polynomial of degree at least m 1 general conditions on the existence of new bounds are presented we prove that for fixed dimension n spl ges 5 there exists a constant k k n such that all levenshtein bounds l sub m n s for m spl ges 2k 1 can be improved an algorithm for obtaining new bounds is proposed and discussed

on a discrete variational problem involving interacting particles we consider the minimization problem quad min cal f bf z equiv sum k j log z k z j qquad k j 1 2 dots n for bf z z 1 z 2 dots z n in mathbb c n satisfying sum k 1 n z k 2 1 and z k ne z j for 1 le k j le n this problem arises in different contexts in several physical models such as incompressible euler equations and the ginzburg landau model in superconductivity we study the stability properties of some symmetric critical points such as regular polygons and configurations that consist of a regular polygon plus the origin we also establish the existence of an infinite number of critical points enjoying different kinds of symmetry we show that when the number of particles n exceeds a critical value the global minimizer cannot be a regular polygon n ge6 and if n ge 11 it cannot be a star configuration i e an n 1 sides regular polygon plus the origin

a class of extremal functions for the fourier transform we determine a class of real valued integrable functions f x and corresponding functions m x such that f x 1 and the value of mfo is miimi several applications of these functions to number theory and analysis are given

explicit parallel poisson integration of point vortices on the sphere point vortex models are frequently encountered in conceptual studies in geophysical fluid dynamics but also in practical applications for instance in aeronautics in spherical geometry the motion of vortex centres is governed by a dynamical system with a known poisson structure we construct poisson integration methods for these dynamics by splitting the hamiltonian into its constituent vortex pair terms from backward error analysis the method is formally known to provide solutions to a modified poisson system with the correct bracket but with a modified hamiltonian function different orderings of the pairwise interactions are considered and also used for the construction of higher order methods the energy and momentum conservation of the splitting schemes is demonstrated for several test cases for particular orderings of the pairwise interactions the schemes allow scalable parallelization this results in a linear as opposed to quadratic scaling of computation time with system size when scaling the number of processors accordingly

the closest packing of equal circles on a sphere the closest packing of x circles on the surface of a sphere is examined with the use of techniques that have been developed to determine the stereochemical arrangement of atoms packed around a central atom the technique is based on the concept that the centres of the circles repel one another and minimizing the total repulsion energy leads to an approximate structure from which exact solutions can be determined a number of improved packings have been discovered for values of x in the range 20 40 many different types of structure are found that are of lower symmetries than those previously described the packing density p defined as the fraction of the spherical surface that is enclosed by the circles is found to increase as the number of circles increases in contrast to the conclusion from previous studies however this increase in p is very slight and the values remain substantially below that for an infinite number of circles or a close packed plane

cubic b spline approximation for linear stochastic integro differential equation of fractional order abstract in this paper the cubic b spline collocation method is used for solving the stochastic integro differential equation of fractional order we show that stochastic integro differential equation of fractional order is equivalent to a modified stochastic integral equation then we apply the proposed method to obtain a numerical scheme of the modified stochastic integral equation using this method the problem solving turns into a linear system solution of equations also the convergence analysis of this numerical approach has been discussed in the end examples are given to test the accuracy and the implementation of the method the results are compared with the results obtained by other methods to verify that this method is accurate and efficient

many particle jumps algorithm and thomson s problem we study thomson s problem using a new numerical algorithm valid for any interacting complex system based on the consideration of simultaneous many particle transitions to reduce the characteristic slowing down of numerical algorithms when applied to critical or complex systems we improve or reproduce all previous results on the thomson problem using much less computer time than the other numerical algorithms we report ground state energies for and study the stability of the ground state as a function of the number of charges considered we associate this stability with how well defined are the charges surrounded by five nearest neighbours whose number always seems to be equal to 12

numerical approach for solving stochastic volterra fredholm integral equations by stochastic operational matrix in this paper we obtain stochastic operational matrix of block pulse functions on interval 0 1 to solve stochastic volterra fredholm integral equations by using block pulse functions and their stochastic operational matrix of integration the stochastic volterra fredholm integral equation can be reduced to a linear lower triangular system which can be directly solved by forward substitution we prove that the rate of convergence is o h furthermore the results show that the approximate solutions have a good degree of accuracy

an iterative numerical method for fredholm volterra integral equations of the second kind in this paper we propose a simple numerical method for approximating solutions of fredholm volterra integral equations of the second kind the method is based on picard iteration and uses a suitable quadrature formula under certain conditions we prove the existence and uniqueness of the solution and give error estimates for our approximations the paper concludes with numerical examples and a discussion of the approximations proposed

narrow escape problem for the unit sphere homogenization limit optimal arrangements of large numbers of traps and the n 2 conjecture a narrow escape problem is considered to calculate the mean first passage time mfpt needed for a brownian particle to leave a unit sphere through one of its n small boundary windows traps a procedure is established to calculate optimal arrangements of n 1 equal small boundary traps that minimize the asymptotic mfpt based on observed characteristics of such arrangements a remarkable property is discovered that is the sum of squared pairwise distances between optimally arranged n traps on a unit sphere is integer equal to n 2 it is observed for 2 n 1004 with high precision it is conjectured that this is the case for such optimal arrangements for all n a dilute trap limit of homogenization theory when n can be used to replace the strongly heterogeneous dirichlet neumann mfpt problem with a spherically symmetric robin problem for which an exact solution is readily found parameters of the robin homogenization problem are computed that capture the first four terms of the asymptotic mfpt close agreement of asymptotic and homogenization mfpt values is demonstrated the homogenization approach provides a radically faster way to estimate the mfpt since it is given by a simple formula and does not involve expensive global optimization to determine locations of n 1 boundary traps

new upper bounds on sphere packings i we continue the study of the linear programming bounds for sphere packing introduced by cohn and elkies we use theta series to give another proof of the principal theorem and present some related results and conjectures this article is in the arxiv as arxiv math mg 0110010

euler polynomial solutions of nonlinear stochastic ito volterra integral equations abstract in this study a practical matrix method based on operational matrices of integration and collocation points is presented to find the approximate solution of nonlinear stochastic ito volterra integral equations for this aim first we compute the stochastic operational matrix of euler polynomials then by using this operational matrix nonlinear stochastic ito volterra integral equation is reduced to a system of nonlinear algebraic equations with unknown euler coefficients which can be solved by using a convenient numerical method such as newton s method also convergence analysis and error estimation of the approach is discussed and an upper error bound is provided under mild conditions finally some numerical examples are provided to confirm the accuracy and efficiency of the proposed method all of the numerical computations have been performed on a pc by running some programs written in matlab software

beamforming codebooks for two transmit antenna systems based on optimum grassmannian packings precoding codebook design for limited feedback mimo systems is known to reduce to a discretization problem on a grassmann manifold the case of two antenna beamforming is special in that it is equivalent to quantizing the real sphere the isometry between the grassmannian g2 1 and the real sphere s2 shows that discretization problems in the grassmannian g2 1 are directly solved by corresponding spherical codes notably the grassmannian line packing problem in 2 namely maximizing the minimum distance is equivalent to the tammes problem on the real sphere so that optimum spherical packings give optimum grassmannian packings moreover a simple isomorphism between g2 1 and s2 enables to analytically derive simple codebooks in closed form having low implementation complexity using the simple geometry of some of these codebooks we derive closed form expressions of the probability density function of the relative snr loss due to limited feedback we also investigate codebooks based on other spherical arrangements such as solutions maximizing the harmonic mean of the mutual distances among the codewords which is known as the thomson problem we find that in some special cases grassmannian codebooks based on these other spherical arrangements outperform codebooks from grassmannian packing

a subspace interior and conjugate gradient method for large scale bound constrained minimization problems a subspace adaptation of the coleman li trust region and interior method is proposed for solving large scale bound constrained minimization problems this method can be implemented with either sparse cholesky factorization or conjugate gradient computation under reasonable conditions the convergence properties of this subspace trust region method are as strong as those of its full space version computational performance on various large test problems is reported advantages of our approach are demonstrated our experience indicates that our proposed method represents an efficient way to solve large bound constrained minimization problems

relative equilibria of point vortices and the fundamental theorem of algebra relative equilibria of identical point vortices may be associated with a generating polynomial that has the vortex positions as its roots a formula is derived that relates the first and second derivatives of this polynomial evaluated at a vortex position using this formula along with the fundamental theorem of algebra one can sometimes write a general polynomial equation in this way results about relative equilibria of point vortices may be proved in a compact and elegant way for example the classical result of stieltjes that if the vortices are on a line they must be situated at the zeros of the n th hermite polynomial follows easily it is also shown that if in a relative equilibrium the vortices are all situated on a circle they must form a regular n gon several other results are proved using this approach an ordinary differential equation for the generating polynomial when the vortices are situated on two perpendicular lines is derived the method is extended to vortex systems where all the vortices have the same magnitude but may be of either sign derivations of the equation of tkachenko for completely stationary configurations and its extension to translating relative equilibria are given

point vortex dynamics a classical mathematics playground the idealization of a two dimensional ideal flow as a collection of point vortices embedded in otherwise irrotational flow yields a surprisingly large number of mathematical insights and connects to a large number of areas of classical mathematics several examples are given including the integrability of the three vortex problem the interplay of relative equilibria of identical vortices and the roots of certain polynomials addition formulas for the cotangent and the weierstras function projective geometry and other topics the hope and intent of the article is to garner further participation in the exploration of this intriguing dynamical system from the mathematical physics community

icosahedral discretization of the two sphere we present an almost uniform triangulation of the two sphere derived from the icosahedron and describe a procedure for discretization of a partial differential equation using this triangular grid the accuracy of our procedure is described by a strong theoretical estimate and verified by large scale numerical experiments we also describe a data structure for this spherical discretization that allows fast computation on either a vector computer or an asynchronous parallel computer

an efficient computational method for solving nonlinear stochastic ito integral equations because of the nonlinearity closed form solutions of many important stochastic functional equations are virtually impossible to obtain thus numerical solutions are a viable alternative in this paper a new computational method based on the generalized hat basis functions together with their stochastic operational matrix of ito integration is proposed for solving nonlinear stochastic ito integral equations in large intervals in the proposed method a new technique for computing nonlinear terms in such problems is presented the main advantage of the proposed method is that it transforms problems under consideration into nonlinear systems of algebraic equations which can be simply solved error analysis of the proposed method is investigated and also the efficiency of this method is shown on some concrete examples the obtained results reveal that the proposed method is very accurate and efficient as two useful applications the proposed method is applied to obtain approximate solutions of the stochastic population growth models and stochastic pendulum problem

relative equilibria of point vortices on the sphere we prove the existence of many different symmetry types of relative equilibria for systems of identical point vortices on a non rotating sphere the proofs use the rotational symmetry group so 3 and the resulting conservation laws the time reversing reflectional symmetries in o 3 and the finite symmetry group of permutations of identical vortices results include both global existence theorems and local results on bifurcations from equilibria a more detailed study is made of relative equilibria which consist of two parallel rings with n vortices in each rotating about a common axis the paper ends with discussions of the bifurcation diagrams for systems of 3 6 identical vortices

differential systems with strongly indefinite variational structure abstract the euler lagrange equations of lagrangians with a strongly indefinite quadratic part are studied by means of the direct min max method of benci and rabinowitz the simplest case is a system of two coupled semilinear poisson equations using an analytic framework of a suitable family of products of fractional sobolev spaces existence results are obtained for all nonlinearities with subcritical growth rates

completely integrable gradient flows in this paper we exhibit the toda lattice equations in a double bracket form which shows they are gradient flow equations on their isospectral set on an adjoint orbit of a compact lie group representations for the flows are given and a convexity result associated with a momentum map is proved some general properties of the double bracket equations are demonstrated including a discussion of their invariant subspaces and their function as a lie algebraic sorter

application of operational matrices for solving system of linear stratonovich volterra integral equation abstract in this paper we give a brief review on block pulse functions bpfs and obtain operational matrix and stochastic operational matrix of integration based on bpfs then these operational matrices are used to solve system of linear stratonovich volterra integral equations by applying proposed method the system of linear stratonovich volterra integral equations reduce to system of linear algebraic equations which can be solved by a convenient numerical method also the error analysis is proved under several mild conditions we show the rate of convergence is o h finally by applying this method on two examples we demonstrate accuracy and efficiency of the proposed method all of the numerical calculation are performed on computer using a program written in matlab

force enhancing vortex equilibria for two parallel plates in uniform flow a two dimensional potential flow in an unbounded domain with two parallel plates is considered we examine whether two free point vortices can be trapped near the two plates in the presence of a uniform flow and observe whether these stationary point vortices enhance the force on the plates the present study is an extension of previously published work in which a free point vortex over a single plate is investigated the flow problem is motivated by an airfoil design problem for the double wings moreover it also contributes to a design problem for an efficient wind turbine with vertical blades in order to obtain the point vortex equilibria numerically we make use of a linear algebraic algorithm combined with a stochastic process called the brownian ratchet scheme the ratchet scheme allows us to capture a family of stationary point vortices in multiply connected domains with ease as a result we find that stationary point vortices exist around the two plates and they enhance the downward force and the counter clockwise rotational force acting on the two plates

spectral gradient flow and equilibrium configurations of point vortices we formulate the problem of finding equilibrium configurations of n point vortices in the plane in terms of a gradient flow on the smallest singular value of a skew symmetric matrix m whose nullspa