infinite dimensional integration and the multivariate decomposition method abstract we further develop the multivariate decomposition method mdm for the lebesgue integration of functions of infinitely many variables x 1 x 2 x 3 with respect to a corresponding product of a one dimensional probability measure the method is designed for functions that admit a dominantly convergent decomposition f u f u where u runs over all finite subsets of positive integers and for each u i 1 i k the function f u depends only on x i 1 x i k although a number of concepts of infinite dimensional integrals have been used in the literature questions of uniqueness and compatibility have mostly not been studied we show that under appropriate convergence conditions the lebesgue integral equals the anchored integral independently of the anchor for approximating the integral the mdm assumes that point values of f u are available for important subsets u at some known cost in this paper we introduce a new setting in which it is assumed that each f u belongs to a normed space f u and that bounds b u on f u f u are known this contrasts with the assumption in many papers that weights u appearing in the norm of the infinite dimensional function space are somehow known often such weights u were determined by minimizing an error bound depending on the b u the u and the chosen algorithm resulting in weights that depend on the algorithm in contrast in this paper only the bounds b u are assumed to be known we give two examples in which we specialize the mdm in the first case f u is the u fold tensor product of an anchored reproducing kernel hilbert space in the second case it is a particular non hilbert space for integration over an unbounded domain

stability in the energy space of the sum of n peakons for a modified camassa holm equation with higher order nonlinearity considered herein is the orbital stability in the energy space h1 r of a decoupled sum of n peakons for a modified camassa holm equation with higher order nonlinearity the equation studied here admits single peakon and multi peakons based on our obtained result of the stability of a single peakon and then combining modulation argument with monotonicity of the local energy h1 norm we get the stability of the sum of n peakons considered herein is the orbital stability in the energy space h1 r of a decoupled sum of n peakons for a modified camassa holm equation with higher order nonlinearity the equation studied here admits single peakon and multi peakons based on our obtained result of the stability of a single peakon and then combining modulation argument with monotonicity of the local energy h1 norm we get the stability of the sum of n peakons

blow up and global existence for a general class of nonlocal nonlinear coupled wave equations abstract we study the initial value problem for a general class of nonlinear nonlocal coupled wave equations the problem involves convolution operators with kernel functions whose fourier transforms are nonnegative some well known examples of nonlinear wave equations such as coupled boussinesq type equations arising in elasticity and in quasi continuum approximation of dense lattices follow from the present model for suitable choices of the kernel functions we establish local existence and sufficient conditions for finite time blow up and as well as global existence of solutions of the problem

the simulation of aerial movement i the determination of orientation angles from film data abstract quantitative mechanical analyses of human movement require the time histories of the angles which specify body configuration and orientation when these angles are obtained from a filmed performance they may be used to evaluate the accuracy of a simulation model this paper presents a method of determining orientation angles and their rates of change from film data the stages used comprise the synchronization of data obtained from two camera views the determination of three dimensional coordinates of joint centres the calculation of an angle from a sequence of sine and cosine values and the curve fitting of angles using quintic splines for each state other possible approaches are discussed original procedures are presented for obtaining individual error estimates of both the film data and the calculated angles to permit the automatic fitting of quintic splines for interpolation and differentiation and for deriving the time history of an angle as a continuous function from a sequence of sine and cosine values the method is applied to a forward somersault with 1 1 2 twists and the average error estimate of 17 orientation angles is obtained as 2 1 degrees

non uniform dependence on initial data for the two component novikov system considered herein is the initial value problem for the two component novikov system it is shown that the solution map of this problem is not uniformly continuous in sobolev spaces hs hs 1 for s 5 2 on the line based on the well posedness result and the lifespan for this problem the method of approximate solutions is utilized

the rigorous approximation of long wavelength capillary gravity waves in a previous paper we proved that long wavelength solutions of the water wave problem in the case of zero surface tension split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries kdv equation in this paper we examine the effect of surface tension on this scenario we find that we obtain three different physical regimes depending on the strength of the surface tension for weak surface tension the propagation of the wave packets is very similar to that in the zero surface tension case for strong surface tension the evolution is again governed by a pair of kdv equations but the coefficients in these equations have changed in such a way that the kdv soliton now represents a wave of depression on the fluid surface finally at a special intermediate value of the surface tension where the bond number equals the kdv description breaks down and it is necessary to introduce a new approximating equation the kawahara equation which is a fifth order nonlinear partial differential equation in each of these regimes we give rigorous estimates of the difference between the solution of the appropriate modulation equation and the solution of the true water wave problem

the effect of increasing strength and approach velocity on triple jump performance the triple jump is an athletic event comprising three phases in which the optimal phase ratio the proportion of each phase to the total distance jumped is unknown this study used a planar whole body torque driven computer simulation model of the ground contact parts of all three phases of the triple jump to investigate the effect of strength and approach velocity on optimal performance the strength and approach velocity of the simulation model were each increased by up to 30 in 10 increments from baseline data collected from a national standard triple jumper increasing strength always resulted in an increased overall jump distance increasing approach velocity also typically resulted in an increased overall jump distance but there was a point past which increasing approach velocity without increasing strength did not lead to an increase in overall jump distance increasing both strength and approach velocity by 10 20 and 30 led to roughly equivalent increases in overall jump distances distances ranged from 14 05m with baseline strength and approach velocity up to 18 49m with 30 increases in both optimal phase ratios were either hop dominated or balanced and typically became more balanced when the strength of the model was increased by a greater percentage than its approach velocity the range of triple jump distances that resulted from the optimisation process suggests that strength and approach velocity are of great importance for triple jump performance

an optimal monte carlo algorithm for multivariate feynman kac path integrals we study a monte carlo algorithm for approximating multivariate feynman kac path integrals parameterized by initial value and potential d variate functions this problem suffers from the curse of dimensionality in the worst case deterministic setting that is why we study the randomized setting in which the functions are sampled at randomized points and the algorithm uses a finite number of such samples we achieve optimality of the randomized algorithm due to variance reduction obtained by smolyak s algorithm for approximating tensor product functions the optimal convergence depends on the smoothness of initial value and potential functions when the initial value and potential functions are r times continuously differentiable we obtain the optimal convergence of order m 1 2 r d for m function samples thus if r d is negligible we do not gain much over the commonly used classical monte carlo algorithm whose convergence is of order m 1 2 hence the classical monte carlo algorithm turns out to be almost

global existence and blow up of solutions for a general class of doubly dispersive nonlocal nonlinear wave equations this study deals with the analysis of the cauchy problem of a general class of nonlocal nonlinear equations modeling the bi directional propagation of dispersive waves in various contexts the nonlocal nature of the problem is reflected by two different elliptic pseudodifferential operators acting on linear and nonlinear functions of the dependent variable respectively the well known doubly dispersive nonlinear wave equation that incorporates two types of dispersive effects originated from two different dispersion operators falls into the category studied here the class of nonlocal nonlinear wave equations also covers a variety of well known wave equations such as various forms of the boussinesq equation local existence of solutions of the cauchy problem with initial data in suitable sobolev spaces is proven and the conditions for global existence and finite time blow up of solutions are established

global existence and asymptotic behaviour of solutions for the generalized boussinesq equation abstract in this paper we prove the existence and the uniqueness of global solution for the cauchy problem for the generalized boussinesq equation under some assumptions we also show that the l norm of small solution of the cauchy problem for the generalized boussinesq equation decays to zero as t tends to the infinity

global existence and asymptotic behavior of solution for the rosenau equation with hydrodynamical damped term abstract we consider the cauchy problem for rosenau equation with hydrodynamical damped term in n dimensional space under smallness condition on the initial data we prove the global existence and asymptotic behavior of the solution in the time weighted sobolev spaces by the contraction mapping principle

differentiability of the blow up curve for one dimensional nonlinear wave equations consider the cauchy problem rac artial 2 u artial t 2 umimits i 1 n rac artial 2 u artial x i 2 f u x n athbb r n t 0 ueft x 0 ight feft x ight eft x n athbb r n ight u t eft x 0 ight geft x ight eft x n athbb r n ight with n 3 it is well known that if f u is superlinear i e feft u ight a eft u ight p ifeft u ight o nfty eft a 0 p 1 ight 0 1 then solutions generally blow up in finite time see 2 6

zero curvature formulations of dual hierarchies zero curvature formulations are given for the dual hierarchies of standard soliton equation hierarchies recently introduced by olver and rosenau including the physically interesting fuchssteiner fokas camassa holm hierarchy

the effects of initial conditions and takeoff technique on running jumps for height and distance this study used a subject specific model with eight segments driven by joint torques for forward dynamics simulation to investigate the effects of initial conditions and takeoff technique on the performance of running jumps for height and distance the torque activation profiles were varied in order to obtain matching simulations for two jumping performances one for height and one for distance by an elite male high jumper resulting in a simulated peak height of 1 98 m and a simulated horizontal distance of 4 38 m the peak height reached horizontal distance travelled by the mass centre for the same corresponding initial conditions were then maximised by varying the activation timings resulting in a peak height of 2 09 m and a horizontal distance of 4 67 m in a further two optimizations the initial conditions were interchanged giving a peak height of 1 82 m and a horizontal distance of 4 04 m the four optimised simulations show that even with similar approach speeds the initial conditions at touchdown have a substantial effect on the resulting performance whilst the takeoff phase is clearly important unless the approach phase and the subsequent touchdown conditions are close to optimal then a jumper will be unable to compensate for touchdown condition shortcomings during the short takeoff phase to achieve a performance close to optimum

blow up and scattering of solution for a generalized boussinesq equation in this paper we consider the cauchy problem for a class of boussinesq equation we obtain the existence and uniqueness of the local solutions for a class of nonlinearity of the perturbation blow up solutions are obtained furthermore the global existence and nonlinear scattering for small amplitude solutions are established

blow up phenomena and global existence for a two component camassa holm system with an arbitrary smooth function abstract in the paper we mainly investigate blow up phenomena and global existence of strong solutions to a two component camassa holm systems with an arbitrary smooth function h for three types of smooth functions h by using a conservation law and the sign preserving property of strong solutions we obtain two new blow up results and a new global existence result for the two component system our obtained results generalize and cover the recent results in yan et al 2015 zhang and yin 2015 2016

global existence and asymptotic behavior of solution for the sixth order boussinesq equation with damped term abstract we investigate the small data cauchy problem for the sixth order boussinesq equation with damped term the decay estimates of solutions to the corresponding linear equation are given by using the dyadic decomposition and some properties of bessel functions then we prove the global existence and asymptotic of the small amplitude solution in the time weighted sobolev space by the contraction mapping principle

existence and stability of traveling waves for a class of nonlocal nonlinear equations in this article we are concerned with the existence and orbital stability of traveling wave solutions of a general class of nonlocal wave equations utt luxx b u p 1 u xx p 1 the main characteristic of this class of equations is the existence of two sources of dispersion characterized by two coercive pseudo differential operatorsl and b members of the class arise as mathematical models for the propagation of dispersive waves in a wide variety of situations for instance all boussinesq type equations and the so called double dispersion equation are members of the class we first establish the existence of traveling wave solutions to the nonlocal wave equations considered we then obtain results on the orbital stability or instability of traveling waves for the case l i corresponding to a class of klein gordon type equations we give an almost complete characterization of the values of the wave velocity for which the traveling waves are orbitally stable or unstable by blow up

a note on the camassa holm equation abstract we consider a generalized camassa holm equation we improve the compactness argument of 5 6 7 17 establishing the existence of global weak solutions in h 1 without the use of an ole nik type estimate the potential interest of this improvement is linked to convergence analysis of numerical schemes for which it may not be easy to verify an ole nik type estimate

determination of subject specific model parameters for visco elastic elements the determination of subject specific model parameter values is necessary in order for a computer simulation model of human motion to be evaluated quantitatively this study used an optimisation procedure along with a kinematically driven simulation model of the contact phase in running jumps to determine the elastic parameters of segmental wobbling masses and the foot ground interface kinetic and kinematic data were obtained on running jumps for height and distance performed by an elite male high jumper stiffness and damping coefficients of the visco elastic elements in the model were varied until the difference between simulation and performance was minimised percentage differences of 6 and 9 between the simulated and recorded performances were obtained in the jumps for height and distance respectively when the parameters obtained from the jump for height were used in a simulation of the jump for distance and vice versa there was poor agreement with the recorded jump on the other hand a common set of visco elastic parameters were obtained using the data from both recorded jumps resulting in a mean difference of only 8 made up of 7 and 10 between simulation and performance that was almost as good as the individual matches simulations were not overly sensitive to perturbations of the common set of visco elastic parameters it is concluded that subject specific elastic parameters should be calculated from more than a single jump in order to provide a robust set of values that can be used in different simulations

existence and scattering of small solutions to a boussinesq type equation of sixth order abstract in this paper we consider the existence and uniqueness of the global small solution as well as the small data scattering result to the cauchy problem for a boussinesq type equation of sixth order with the nonlinear term f u behaving as u p p 1 as u 0 in r n n 1 the main method and techniques used in our paper are the littlewood paley dyadic decomposition the stationary phase estimate and some properties of bessel function

on the cost of uniform and nonuniform algorithms we compare the costs of uniform and nonuniform algorithms for approximate solutions of continuous problems assuming the real number model we show that in general there is no relation between these costs that is the class of uniform algorithms may be empty moreover even if this class is nonempty then the cost of any uniform algorithm may be arbitrarily larger than the minimal cost of nonuniform algorithms we also provide conditions under which there exist uniform algorithms whose cost is basically the same as the minimal cost of nonuniform algorithms

peakons and coshoidal waves traveling wave solutions of the camassa holm equation abstract camassa and holm 1 have recently derived a new integrable wave equation for the special case 0 they showed it has solitary waves of the form c exp x ct which they named peakons in this work we derive a perturbation series for general which converges even at the peakon limit we also give three analytical representations for the spatially periodic generalization of the peakon the coshoidal wave the three representations are i a closed form analytical solution ii a fourier series with coefficients that are explicit rational functions and iii an imbricate fourier series which is the superposition of an infinite number of peakons each separated from its neighbors by distance p where p is the spatial period lastly we have numerically tested the soliton superposition principle although the camassa holm equation is integrable for general it appears that imbricating solitary waves generates an exact spatially periodic solution only for the special cases 0 c 1 2 however the imbricate soliton series is a very good approximate solution for general even when the spatial period is small and the solution resembles a sine wave more than a solitary wave

global weak solutions for a new periodic integrable equation with peakon solutions abstract we prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

infinite dimensional integration on weighted hilbert spaces we study the numerical integration problem for functions with infinitely many variables the functions we want to integrate are from a reproducing kernel hilbert space which is endowed with a weighted norm we study the worst case e complexity which is defined as the minimal cost among all algorithms whose worst case error over the hilbert space unit ball is at most e here we assume that the cost of evaluating a function depends polynomially on the number of active variables the infinite dimensional integration problem is polynomially tractable if the e complexity is bounded by a constant times a power of 1 e the smallest such power is called the exponent of tractability first we study finite order weights we provide improved lower bounds for the exponent of tractability for general finite order weights and improved upper bounds for three newly defined classes of finite order weights the constructive upper bounds are obtained by multilevel algorithms that use for each level quasimonte carlo integration points whose projections onto specific sets of coordinates exhibit a small discrepancy the newly defined finite intersection weights model the situation where each group of variables interacts with at most other groups of variables where is some fixed number for these weights we obtain a sharp upper bound this is the first class of weights for which the exact exponent of tractability is known for any possible decay of the weights and for any polynomial degree of the cost function for the other two classes of finite order weights our upper bounds are sharp if e g the decay of the weights is fast or slow enough we extend our analysis to the case of arbitrary weights in particular from our results for finite order weights we conclude a lower bound on the exponent of tractability for arbitrary weights and a constructive upper bound for product weights

some properties of solutions to the periodic two component hyperelastic rod wave equation abstract in this paper we study the cauchy problem of the periodic two component hyperelastic rod wave equation we first establish the local well posedness and precise blow up scenario then we obtain several blow up results and the blowup rate of strong solutions moreover we present two global existence results for strong solutions finally we consider analyticity of solutions and the initial boundary value problem

kawahara dynamics in dispersive media abstract in a previous paper it was shown that in the long wave limit the water wave problem without surface tension can be described approximately by two decoupled kdv equations here we study a model problem and prove that in a degenerate case which occurs for the water wave problem with surface tension and near other codimension two points at which the coefficient in front of the leading order dispersive term in the equation of motion vanishes the long wave limit can be rigorously approximated by two decoupled kawahara equations

global solutions to a special case of the generalized weakly dissipative periodic two component camassa holm system abstract we establish global well posedness to the generalized weakly dissipative periodic two component camassa holm system in some special case

on tractability of path integration many applications require approximate values of path integrals a typical approach is to approximate the path integral by a high dimensional integral and apply a monte carlo randomized algorithm however monte carlo algorithm requires roughly e 2 integrand evaluations to provide an e approximation moreover the error bound of e is guaranteed only in a stochastic sense do we really need to use randomized algorithms for path integrals perhaps we can find a deterministic algorithm that is more effective even in the worst case setting to answer this question we study the worst case complexity of path integration which roughly speaking is defined as the minimal number of the integrand evaluations needed to compute an approximation with error at most e we consider path integration with respect to a gaussian measure and for various classes of integrands tractability of path integration means that the complexity depends polynomially on 1 e we show that for the class of r times frechet differentiab

a path integration formulation of stochastic lagrangian models of turbulent flow abstract in analogy with the feynman kac path integration formulation of quantum mechanics a general path integration formulation of stochastic lagrangian models of turbulent flow is developed with respect to the equivalence with the usual eulerian and stochastic lagrangian descriptions we then discuss efficient numerical methods for the realisation of the lagrangian solutions via path integration which provide an effective alternative to the simpler monte carlo approach

well posedness and analyticity for the cauchy problem for the generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem for the generalized camassa holm equation which was proposed by hakkaev and kirchev s hakkaev k kirchev local well posedness and orbital stability of solitary wave solutions for the generalized camassa holm equation comm partial differential equations 30 2005 761 781 we establish the local well posedness for a range of besov spaces moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time which extends some results of danchin r danchin a few remarks on the camassa holm equation differential integral equations 14 2001 953 988 and himonas and misiolek a himonas g misiolek analyticity of the cauchy problem for an integrable evolution equation math ann 327 2003 575 584 to more general equations

approximation of wiener integrals we present a new proof of chorin s quadrature formula for weiner integrals numerical experiments show that the rate of convergence may not be quadratic if the functional is not twice differentiable copyright 1987 academic press inc

solitonic solutions for a variable coefficient variant boussinesq system in the long gravity waves variable coefficient variant boussinesq vcvb system is able to describe the nonlinear and dispersive long gravity waves traveling in two horizontal directions with varying depth in this paper with symbolic computation a lax pair associated with the vcvb system under some constraints for variable coefficients is derived and based on the lax pair two sorts of basic darboux transformations are presented by applying the darboux transformations some solitonic solutions are obtained with the relevant constraints given in the text in addition the vcvb system is transformed to a variable coefficient broer kaup system solitonic solutions and procedure of getting them could be helpful to solve the nonlinear and dispersive problems in fluid dynamics

a completely integrable hamiltonian system the dynamical system characterized by the hamiltonian h q p j k 1 n p jp kf q dj q k with f x cos x sin x is completely integrable here n is an arbitrary positive integer and are 4 arbitrary constants and real and both real or both imaginary

local well posedness of a new integrable equation in this article we focus on the local in time well posedness of the cauchy problem for a new integrable equation we proved the local in time existence and uniqueness of the entropy solutions by using the method of the vanishing viscosity and l1 contraction property

decay and scattering of solutions for a generalized boussinesq equation abstract in this paper we consider the long time behavior of small solutions of the cauchy problem for a generalized boussinesq equation a scattering operator and the nonlinear scattering for small amplitude solutions of the boussinesq equation are established under certain hypotheses

formation of singularities in solutions to nonlinear hyperbolic equations on etudie des systemes de lois de conservation hyperboliques et d equations d ondes completement non lineaires

the cauchy problem for a one dimensional nonlinear elastic peridynamic model this paper studies the cauchy problem for a one dimensional nonlinear peridynamic model describing the dynamic response of an infinitely long elastic bar the issues of local well posedness and smoothness of the solutions are discussed the existence of a global solution is proved first in the sublinear case and then for nonlinearities of degree at most three the conditions for finite time blow up of solutions are established

analytical behavior of two dimensional incompressible flow in porous media in this paper we study the analytic structure of a two dimensional mass balance equation of an incompressible fluid in a porous medium given by darcy s law we obtain local existence and uniqueness by the particle trajectory method and we present different global existence criterions these analytical results with numerical simulations are used to indicate nonformation of singularities moreover blow up results are shown in a class of solutions with infinite energy

existence of perturbed solitary wave solutions to a model equation for water waves abstract we prove the existence of travelling wave solutions to a fifth order partial differential equation which is a formal asymptotic approximation for water waves with surface tension these travelling waves are arbitrarily small perturbations of solitary waves but are not solitary waves themselves because they approach small amplitude oscillations for large values of the independent variable this result suggests that for bond numbers less than one third there are branches of travelling wave solutions to the water wave equations which are perturbations of supercritical elevation solitary waves and which bifurcate from froude number one and bond number one third

conditioning analysis of nonlocal integral operators in fractional sobolev spaces we study the conditioning of nonlocal integral operators with singular and integrable kernels in fractional sobolev spaces these operators are used for instance in peridynamics formulation and nonlocal diffusion in one dimension 1d we present sharp quantification of the extremal eigenvalues in all three parameters size of nonlocality mesh size and regularity of the fractional sobolev space we accomplish sharpness both rigorously and numerically for the minimal eigenvalue we obtain sharpness analytically by using a nonlocal characterization of sobolev spaces we verify this estimate by exploiting the toeplitz property of the stiffness matrix however the analytical approach fails to give sharp quantification of the maximal eigenvalue hence in 1d we take an algebraic approach by directly working with the stiffness matrix entries which have complicated expressions due to all three parameters we systematically characterize the nonzero entries and dramatically simplify their expressions by u

large time existence for 3d water waves and asymptotics we rigorously justify in 3d the main asymptotic models used in coastal oceanography including shallow water equations boussinesq systems kadomtsev petviashvili kp approximation green naghdi equations serre approximation and full dispersion model we first introduce a variable nondimensionalized version of the water waves equations which vary from shallow to deep water and which involves four dimensionless parameters using a nonlocal energy adapted to the equations we can prove a well posedness theorem uniformly with respect to all the parameters its validity ranges therefore from shallow to deep water from small to large surface and bottom variations and from fully to weakly transverse waves

the time dependent amplitude equation for the swift hohenberg problem precise estimates for the validity of the amplitude approximation for the swift hohenberg equation are given in a fully time dependent framework it is shown that small solutions of ordero which are modulated like stationary solutions have an evolution which is well described in the amplitude approximation for a time of ordero 2 for the proofs we use techniques for nonlinear semigroups and oscillatory integrals

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

finite propagation speed for the camassa holm equation we prove that any classical solution of the camassa holm equation will have compact support if its initial data has this property

traveling wave solutions of the degasperis procesi equation we classify all weak traveling wave solutions of the degasperis procesi equation in addition to smooth and peaked solutions the equation is shown to admit more exotic traveling waves such as cuspons stumpons and composite waves

on a class of nonlocal wave equations from applications we study equations from the area of peridynamics which is a nonlocal extension of elasticity the governing equations form a system of nonlocal wave equations we take a novel approach by applying operator theory methods in a systematic way on the unbounded domain n we present three main results as main result 1 we find that the governing operator is a bounded function of the governing operator of classical elasticity as main result 2 a consequence of main result 1 we prove that the peridynamic solutions strongly converge to the classical solutions by utilizing for the first time strong resolvent convergence in addition main result 1 allows us to incorporate local boundary conditions in particular into peridynamics this avenue of research is developed in companion papers providing a remedy for boundary effects as main result 3 employing spherical bessel functions we give a new practical series representation of the solution which allows straightforward numerical treatment with symbolic computation

the long wave limit for the water wave problem i the case of zero surface tension the korteweg de vries equation boussinesq equation and many other equations can be formally derived as approximate equations for the two dimensional water wave problem in the limit of long waves here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension we prove that the solutions of the water wave problem in the long wave limit split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries equation our result allows us to describe the nonlinear interaction of solitary waves 2000 john wiley sons inc

a new algorithm and worst case complexity for feynman kac path integration abstract we study algorithms for approximation of feynman kac path integrals in the worst case setting the algorithms use a finite number of samples of the initial condition and potential functions we present a new algorithm and an explicit bound on its cost to compute an e approximation to the feynman kac path integral we also establish bounds on the worst case complexity of feynman kac path integration the upper bound is equal to the cost of the new algorithm and is given in terms of the complexity of a certain function approximation problem the lower bound is given in terms of the complexity of a certain weighted integration problem for some classes of functions these two bounds coincide modulo a multiplicative factor in this case the new algorithm is almost optimal the new algorithm requires precomputation of some real coefficients that are combinations of multivariate integrals with special weights this precomputation is difficult and limits the application of the new algorithm we report the results of the precomputation for specific cases

the simulation of aerial movement ii a mathematical inertia model of the human body a mathematical inertia model which permits the determination of personalized segmental inertia parameter values from anthropometric measurements is described the human body is modelled using 40 geometric solids which are specified by 95 anthropometric measurements a stadium solid is introduced for modelling the torso segments using perimeter and width measurements this procedure is more accurate than the use of elliptical discs of given width and depth and permits a smaller number of such solids to be used inertia parameter values may be obtained for body models of up to 20 segments errors in total body mass estimates from this and other models are discussed with reference to the unknown lung volumes

global solution for a generalized boussinesq equation this paper considers the existence and uniqueness of the solution to the cauchy problem for a class of boussinesq equation proves the global existence and finite time blowup of the solution to the problem by the potential well method

some tricks from the symmetry toolbox for nonlinear equations generalizations of the camassa holm equation abstract the main subject of the paper is to give a survey and to present new methods on how integrability results i e results for symmetry groups inverse scattering formulations action angle transformations and the like can be transferred from one equation to others in case the equations are not related by backlund transformations as a main example the so called camassa holm equation is chosen for which the relevant results are obtained by having a look on the korteweg de vries kdv equation the camassa holm equation turns out to be a different factorization equation of the kdv it describes shallow water waves and reconciles the properties which were known for different orders of shallow water wave approximations we follow here an old method already marginally mentioned in fuchssteiner and fokas 1981 and fuchssteiner 1983 and recently applied by others olver and rosenau 1995 the method allows an immediate recovery of the recursion operator for the camassa holm equation from the invariance structure of the kdv although both equations are not related by backlund transformations however in addition and different from other approaches from there by use of the squared eigenfunction relation for the kdv equation the lax pair formulation for the different factorization equation is derived for the example under consideration it is of course the one obtained in camassa and holm 1993 since the methods proposed here can be transferred to any compatible factorization of recursion operators its application even in the special case which was chosen for illustration leads to a large class of integrable equations among which the camassa holm equation can be found as well as a three parameter family of generalizations of the equation the advantage of the general approach to the lax pair presentation is that direct transformations between action and angle variables are obtained so using this lax pair formulation as a novel result a direct transformation between action and angle variables for the camassa holm equation is derived further novel results in the paper are a hodograph link back from a backlund transformation of the camassa holm equation to a particular member of the kdv hierarchy additional symmetries and the construction of the conformal algebra for the hierarchy of the camassa holm equation the methods involved include hereditary operators bi hamiltonian formulations nilpotent flows and scaling symmetries

well posedness of a class of solutions to an integrable two component camassa holm system abstract in this paper we study a class of solutions to the cauchy problem for an integrable two component camassa holm system with the initial data u 0 0 1 h 1 r w 1 r l 2 r l r based on characteristics we study a corresponding ode and obtain a unique local solution by applying the contraction mapping principle we prove local existence and uniqueness of the solution to the camassa holm system by constructing a solution obtained from the ode and studying the regularity of the solution finally we show continuous dependence of the solution on the initial data in some weak sense

the long wave limit for a boussinesq equation we consider the long wave limit for a boussinesq equation it is shown that for spatially localized initial conditions the dynamics are under a korteweg de vries kdv regime on a large time interval i e in this limit the solutions split up into two counterpropagating wave packets where each of the wave packets evolve independently and approximately as a solution of a kdv equation for the proof exact estimates between the long wave solutions of the boussinesq equation and the approximations obtained via the decoupled set of kdv equations are demonstrated we expect that this result holds for the water wave problem too