pagerank of integers we build up a directed network tracing links from a given integer to its divisors and analyze the properties of the google matrix of this network the pagerank vector of this matrix is computed numerically and it is shown that its probability is inversely proportional to the pagerank index thus being similar to the zipf law and the dependence established for the world wide web the spectrum of the google matrix of integers is characterized by a large gap and a relatively small number of nonzero eigenvalues a simple semi analytical expression for the pagerank of integers is derived that allows to find this vector for matrices of billion size this network provides a new pagerank order of integers

a mesh free numerical method for solution of the family of kuramoto sivashinsky equations in this paper radial basis function rbfs based mesh free method is implemented to find numerical solution of the kuramoto sivashinsky equations this approach has an edge over traditional methods such as finite difference and finite element methods because it does not require a mesh to discretize the problem domain and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method the accuracy of the method is assessed in terms of the error norms l 2 l number of nodes in the domain of influence free parameter dependent parameter rbfs and time step length numerical experiments demonstrate accuracy and robustness of the method for solving a class of nonlinear partial differential equations

families and clustering in a natural numbers network we develop a network in which the natural numbers are the vertices the decomposition of natural numbers by prime numbers is used to establish the connections we perform data collapse and show that the degree distribution of these networks scales linearly with the number of vertices we explore the families of vertices in connection with prime numbers decomposition we compare the average distance of the network and the clustering coefficient with the distance and clustering coefficient of the corresponding random graph in case we set connections among vertices each time the numbers share a common prime number the network has properties similar to a random graph if the criterion for establishing links becomes more selective only prime numbers greater than p l are used to establish links where the network has high clustering coefficient

comment on families and clustering in a natural numbers network corso phys rev e 69 036106 2004 constructs a family of graphs from subsets of the natural numbers and numerically estimates diameter degree and clustering we give exact asymptotic formulas for these quantities and thereby argue that number theory is a more appropriate tool than simulation

local discontinuous galerkin methods for the kuramoto sivashinsky equations and the ito type coupled kdv equations in this paper we develop a local discontinuous galerkin method to solve the kuramoto sivashinsky equations and the ito type coupled kdv equations the l2 stability of the schemes is obtained for both of these nonlinear equations we use both the traditional nonlinearly stable explicit high order runge kutta methods and the explicit exponential time differencing method for the time discretization the latter can achieve high order accuracy and maintain good stability while avoiding the very restrictive explicit stability limit of the former when the pde contains higher order spatial derivatives numerical examples are shown to demonstrate the accuracy and capability of these methods

volumes of balls in large riemannian manifolds we prove two lower bounds for the volumes of balls in a riemannian manifold if m n g is a complete riemannian manifold with lling radius at least r then it contains a ball of radius r and volume at least n r n if m n hyp is a closed hyperbolic manifold and if g is another metric on m with volume no greater than n vol m hyp then the universal cover of m g contains a unit ball with volume greater than the volume of a unit ball in hyperbolic n space

interpolation and approximation of 3 d and 4 d scattered data abstract given a set of 3 d or 4 d scattered data methods are presented that yield a bivariate or trivariate function that interpolate or approximate the given data the subroutine package includes hardy s multiquadric interpolant and multistage methods with many options available to the user thus several different interpolation and approximation methods can be generated the options available in the multistage methods can be effectively used if the data set is noisy rapidly varying or nonuniformly distributed the computational and storage requirements for the multistage methods are linear in the number of data points thus they are efficient on large data sets

hard and soft packing radius theorems singular spaces are playing an increasingly important role in riemannian geometry in particular this is true of the so called alexandrov spaces i e finite hausdorff dimensional complete inner metric spaces with a lower curvature bound in the local triangle comparison sense interest in alexandrov spaces is largely explained by the fact that the natural process of taking gromov hausdorff limits is closed in alexandrov geometry but not in riemannian geometry this simple observation has become a powerful tool in riemannian geometry see for example fy gp2 or gpw and pe in this paper we use a new means to obtain results in riemannian geometry by studying alexandrov spaces the operation of forming spherical suspensions of positively curved spaces an operation which is also closed in alexandrov geometry but not in riemannian geometry will be applied to prove a differentiable sphere theorem which is unlike all previous solutions to the following basic problem

a legendre petrov galerkin method for fourth order differential equations in this paper we present a legendre petrov galekin method for one dimensional linear fourth order differential equations a legendre petrov galerkin and chebyshev collocation method is developed for the nonlinear kuramoto sivashinsky equation numerical results are presented to demonstrate the efficiency of the proposed schemes and optimal rates of convergence in the l 2 norm are rigorously derived

an interval for the shape parameter in radial basis function approximation the radial basis functions rbfs depend on an auxiliary parameter called the shape parameter great theoretical and numerical efforts have been made to find the relationship between the accuracy of the rbf approximations and the value of the shape parameter in many cases the numerical approaches are based on minimization of an estimation of the error function such as rippa s approach and its modifications these approaches determine a value for the shape parameter in this paper we propose a practical approach to determine an interval instead of a value without any minimization and estimation of error function the idea is based on adding a loop on the shape parameter but not to minimize an error norm suitable values of the shape parameter are determined by take into account the practical convergence behavior of the problem the proposed method is applied to some illustrative examples including one and two dimensional interpolation and partial differential equations the results of the method are compared with those of some other approaches to confirm the reliability of the method furthermore numerical stability of the method of line respect to the shape parameter is considered for time dependent pdes the results show that the proposed method can be used as an independent method to find a shape parameter and also as an alternative for investigating the validity of the values of the other methods

rbf ps scheme for solving the equal width equation the equal width equation is solved by a rbf ps scheme this technique is meshless and there is no need to linearize the nonlinear terms the radial kernels are used to transform the pde into a system of odes a higher order ode solver is used to solve the ode system the method is tested for single solitary wave interaction of two and three solitary waves the development of an undular bore and the interaction of two solitary waves the results are compared with earlier work and other methods

an efficient numerical scheme for burgers equation this paper applies the multiquadric mq as a spatial approximation scheme for solving the nonlinear burgers equation for comparison purposes a low order explicit finite difference approximation of the time derivative is employed by decreasing the time step of the computation it is shown that the major numerical error is from the time integration instead of the mq spatial approximation the numerical results indicate that this mq offers an excellent approximation for all possible values of reynolds number an adaptive algorithm is also developed to adjust the mq interpolation points to the peak of the shock wave which is shown to provide an improved numerical result numerical comparisons are made with most of the existing numerical schemes for solving the burgers equation

on the filling radius of positively curved alexandrov spaces it was shown by f wilhelm that gromov s filling radius of any positively curved closed riemannian manifolds are less than that of the round sphere unless they are isometric to each other in this short paper we adapt his proof to see that the same is true for any positively curved closed alexandrov spaces as well

smooth interpolation of scattered data by local thin plate splines abstract an algorithm and the corresponding computer program for solution of the scattered data interpolation problem is described given points x k y k f k k 1 n a locally defined function f x y which has the property f x k y k f k k 1 n is constructed the algorithm is based on a weighted sum of locally defined thin plate splines and yields an interpolation function which is differentiable the program is available from the author

flowers on riemannian manifolds in this paper we will present two upper bounds for the length of a smallest flower shaped geodesic net in terms of the volume and the diameter of a manifold minimal geodesic nets are critical points of the length functional on the space of graphs immersed into a riemannian manifold let mn be a closed riemannian manifold of dimension n we prove that there exists a minimal geodesic net that consists of one vertex and at most 2n 1 geodesic loops based at that vertex of total length 2n d where d is the diameter of mn we also show that there exists a minimal geodesic net that consists of one vertex and at most 3 n 1 2 loops of total length leq2 n 1 2 3 n 1 3 fill rad m n leq2 n 1 frac 5 2 3 n 1 3 n 1 n n vol m n frac 1 n where fill rad mn denotes the filling radius and vol mn denotes the volume of mn

on traveling wave solutions of the kuramoto sivashinsky equation abstract previous numerical results indicate that the kuramoto sivashinsky equation admits three classes of non periodic traveling wave solutions namely regular shocks oscillatory shocks and solitary waves however it has been shown that regular monotonic shocks cease to exist in the weak shock limit due to the radiation of oscillatory waves of exponentially small with respect to the shock strength but growing in space amplitude here oscillatory shocks and solitary waves are constructed by asymptotic analysis it thus transpires that in the weak shock limit oscillatory shocks can only be antisymmetric otherwise oscillatory and monotonic waves of exponentially small but growing in space amplitude would inevitably be excited under certain conditions however the growing waves can link a nearly antisymmetric oscillatory shock with a radiating regular shock to form a solitary wave the predictions of the asymptotic theory are supported by numerical results

a method for interpolating scattered data based upon a minimum norm network a method for interpolating scattered data is described ctiven xi yi zi i 1 n a bivariate function s with continuous first order partial derivatives is defined which has the property that s xi yi zi i 1 n the method is based upon a triangulation of the domain and a curve network which has certain minimum pseudonorm properties algorithms and examples are included

gromov hyperbolic spaces and the sharp isoperimetric constant in this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is gromov hyperbolic as a particular consequence we obtain that euclidean space is a borderline case for gromov hyperbolicity in terms of the isoperimetric function we prove similar results for the linear filling radius inequality our results strengthen and generalize theorems of gromov papasoglu and others

iterative solution of systems originating from integral equations and surface interpolation a simple procedure for generating diagonal dominance in certain linear systems is presented and the iterative solution of the transformed systems is discussed the method is presented for systems encountered in the numerical solution of fredholm equations of the first kind with singular kernels and for systems arising in the interpolation of surfaces by certain sets of radial functions in the second case the method provides new sets of basis functions generalizing in some sense the univariate b splines

second order splitting combined with orthogonal cubic spline collocation method for the kuramoto sivashinsky equation in this paper a second order splitting method is applied to the kuramoto sivashinsky equation and then an orthogonal cubic spline collocation procedure is employed to the approximate resulting system this semidiscrete method yields a system of differential algebraic equations daes of index 1 error estimates in l 2 and l c norms are obtained for the semidiscrete approximation for the time discretization the time integrator radau5 is used the results of numerical experiments are presented to validate the theoretical findings

a least squares finite element scheme for the ew equation abstract the equal width ew equation is solved by a least squares technique using linear space time finite elements in simulations of the migration of a single solitary wave this algorithm is shown to have good accuracy and conservation the development of solitary waves from an arbitrary initial condition within an ew environment is examined

the kuramoto sivashinsky equation a bridge between pde s and dynamical systems abstract through extensive numerical simulation we have characterized the transition to chaos of the solutions to the kuramoto sivashinsky equation the attracting solution manifolds undergo a complex bifurcation sequence including multimodal fixed points invariant tori traveling wave trains and homoclinic orbits we relate this behavior to earlier work where the kuramoto sivashinsky equation was shown to behave as a finite dimensional dynamical system of ordinary differential equations

the riemann zeros and eigenvalue asymptotics comparison between formulae for the counting functions of the heights tn of the riemann zeros and of semiclassical quantum eigenvalues en suggests that the tn are eigenvalues of an unknown hermitean operator h obtained by quantizing a classical dynamical system with hamiltonian hcl many features of hcl are provided by the analogy for example the riemann dynamics should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers statistics of the tn have a similar structure to those of the semiclassical en in particular they display random matrix universality at short range and nonuniversal behaviour over longer ranges very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues the riemann siegel formula for the zeta function is described in detail its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations we speculate that the riemann dynamics is related to the trajectories generated by the classical hamiltonian hcl xp

meshless method of lines for the numerical solution of generalized kuramoto sivashinsky equation in this paper the numerical solution of the generalized kuramoto sivashinsky equation is presented by meshless method of lines mol in this method the spatial derivatives are approximated by radial basis functions rbfs giving an edge over finite difference method fdm and finite element method fem because no mesh is required for discretization of the problem domain only a set of scattered nodes is required to approximate the solution the numerical results in comparison with exact solution using different radial basis functions rbfs prove the efficiency and accuracy of the method

steady solutions of the kuramoto sivashinsky equation abstract steady solutions of the kuramoto sivashinsky equation are studied these solutions are defined on the whole x line and propagate with a constant speed c 2 in time for large c 2 it is shown that the solution is unique and has a conical form for small c 2 there is a periodic solution and an infinite set of quasi periodic solutions as asserted by moser s twist map theorem numerical computations for intermediate values of c 2 suggest that below c 1 6 of every speed there is a continuum of odd quasi periodic solutions or a cantor set of chaotic solutions wrapped by infinite sequences of conic solutions

lattice boltzmann model for high order nonlinear partial differential equations in this paper a general lattice boltzmann lb model is proposed for the high order nonlinear partial differential equation with the form ensuremath partial t ensuremath phi ensuremath sum k 1 m ensuremath alpha k ensuremath partial x k mathrm ensuremath pi k ensuremath phi 0 1 ensuremath le k ensuremath le m ensuremath le 6 ensuremath alpha k are constant coefficients mathrm ensuremath pi k ensuremath phi are some known differential functions of ensuremath phi as some special cases of the high order nonlinear partial differential equation the classical m kdv equation kdv burgers equation mathrm k n n burgers equation kuramoto sivashinsky equation and kawahara equation can be solved by the present lb model compared to the available lb models the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved in addition we also conducted a detailed chapman enskog analysis and found that the high order nonlinear partial differential equation can be correctly recovered from the proposed lb model finally a large number of simulations are performed and it is found that the numerical results agree with the analytical solutions and usually the present model is also more accurate than the existing lb models h lai and c ma sci china ser g 52 1053 2009 h lai and c ma phys a amsterdam 388 1405 2009 for high order nonlinear partial differential equations

improved accuracy of multiquadric interpolation using variable shape parameters given n scattered data points we examined the problem of finding n variable multiquadric mq shape parameters or r2 values because the problem of finding the optimal r2 values is a nonlinear one we optimized these parameters numerically by minimizing the root mean square rms errors the resulting r2 values varied over many orders of magnitude we have tested this approach on a number of univariate and bivariate franke s problems and found that the rms error reduction was substantial

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

on the spread of positively curved alexandrov spaces it was proved by f wilhelm that gromov s filling radius of closed positively curved manifolds with a uniform lower bound on sectional curvature attains the maximum with the round sphere recently the author proved that this is also the case for closed finite dimensional alexandrov spaces with a positive lower curvature bound these were proved as a corollary of a comparison theorem for the invariant called spread of those spaces in this paper we extend the latter result to infinite dimensional alexandrov spaces

a time adaptive finite volume method for the cahn hilliard and kuramoto sivashinsky equations this paper presents a complete finite volume method for the cahn hilliard and kuramoto sivashinsky type of equations the spatial discretization is high order accurate and suitable for general unstructured grids the time integration is addressed by means of implicit an implicit explicit fourth order runge kutta schemes with error control and adaptive time stepping the outcome is a practical accurate and efficient simulation tool which has been successfully applied to accuracy tests and representative simulations the use of adaptive time stepping is of paramount importance in problems governed by the cahn hilliard model an adaptive method may be several orders of magnitude more efficient than schemes using constant or heuristic time steps in addition to driving the simulations efficiently the time adaptive procedure provides a quantitative not just qualitative characterization of the rich temporal scales present in phase separation processes governed by the cahn hilliard phase field model

lower bounds on the energy of maps on demontre que pour la metrique standard sur rp n l application identite est minimisante de l energie dans sa classe d homotopie

a vectorized multigrid solver for the three dimensional poisson equation a vectorized version of the multigrid mg method has been implemented on the cdc cyber 205 for solving the three dimensional poisson equation on a rectangular domain with dirichlet boundary conditions this paper describes the techniques employed in vectorizing the mg processes of relaxation residual transfer and interpolation it discusses the data structures and programming concepts employed and presents resulting performance details it concludes with some suggestions for future applications efforts