sigma delta modulation with i i d gaussian inputs the response of a single loop sigma delta modulator to an independent identically distributed i i d gaussian input signal is analyzed a continuous time stochastic model is developed and the connection of the model to the system is described a condition is given so that the difference between the behavior of the model and that of the true system can be made arbitrarily small theories from renewal and wiener processes are applied to show the convergence and mixing properties of the output sequence also derived is the power spectrum of the quantization noise compared to the spectrum when the input is dc the i i d gaussian random process smears the discrete spectrum into band structures

approximate analytical solutions of systems of pdes by homotopy analysis method in this paper the homotopy analysis method ham is applied to obtain series solutions to linear and nonlinear systems of first and second order partial differential equations pdes the ham solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions it is shown in particular that the solutions obtained by the variational iteration method vim are only special cases of the ham solutions

stability analysis for pitchfork bifurcations of solitary waves in generalized nonlinear schrodinger equations abstract linear stability of both sign definite positive and sign indefinite solitary waves near pitchfork bifurcations is analyzed for the generalized nonlinear schrodinger equations with arbitrary forms of nonlinearity and external potentials in arbitrary spatial dimensions bifurcations of linear stability eigenvalues associated with pitchfork bifurcations are analytically calculated based on these eigenvalue bifurcation formulae linear stability of solitary waves near pitchfork bifurcations is then determined it is shown that the base solution branch switches stability at the bifurcation point in addition the two bifurcated solution branches and the base branch have the opposite same stability when their power slopes have the same opposite sign furthermore the stability of these solution branches can be determined almost exclusively from their power diagram especially for positive solitary waves these stability results are also compared with the hamiltonian krein index theory and they are shown to be consistent with each other lastly various numerical examples are presented and the numerical results confirm the analytical predictions both qualitatively and quantitatively

structural stability of the korteweg de vries solitons under a singular perturbation abstract we investigate the stability of a solitary wave solution of the korteweg de vries equation when a fifth order spatial derivative term is added we show that the solution ceases to be strictly localized but develops an infinite oscillating tail and we compute the amplitude of the latter

trees tight extensions of metric spaces and the cohomological dimension of certain groups a note on combinatorial properties of metric spaces abstract the concept of tight extensions of a metric space is introduced the existence of an essentially unique maximal tight extension t x the tight span being an abstract analogon of the convex hull is established for any given metric space x and its properties are studied applications with respect to 1 the existence of embeddings of a metric space into trees 2 optimal graphs realizing a metric space and 3 the cohomological dimension of groups with specific length functions are discussed

sigma delta modulation with leaky integration and constant input exact descriptions of the behavior of quantization noise for single loop multistage cascade and multiloop sigma delta modulators for a variety of input signals have been found during recent years under the assumption of ideal integration the techniques used to solve the ideal integrator case do not easily extend to the more realistic model of a leaky integrator sigma delta in this paper a dynamical system representation for the leaky integrator sigma delta with a constant input is developed several properties of the resulting piecewise monotone and piecewise linear transformation t on the interval

two dimensional legendre wavelet method for travelling wave solutions of time fractional generalized seventh order kdv equation in this paper a new method based on the legendre wavelets expansion together with operational matrices of fractional integration and derivative of these basis functions is proposed to solve time fractional seventh order kdv skdv equation two dimensional legendre wavelet method is applied to compute the numerical solution of nonlinear time fractional skdv equation the approximate solutions of nonlinear time fractional skdv equation thus obtained by legendre wavelet method are compared with the exact solutions as well as homotopy analysis method ham the present scheme is very simple effective and convenient for obtaining numerical solution of the skdv equation

subtree prune and regraft a reversible real tree valued markov process we use dirichlet form methods to construct and analyze a reversible markov process the stationary distribution of which is the brownian continuum random tree this process is inspired by the subtree prune and regraft spr markov chains that appear in phylogenetic analysis a key technical ingredient in this work is the use of a novel gromov hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure also the investigation of the dirichlet form hinges on a new path decomposition of the brownian excursion

a comparative study on the analytic solutions of fractional coupled sine gordon equations by using two reliable methods new modified homotopy analysis method has been proposed the proposed novel method is well suited for present fractional s g equation it is obvious to see that the mham is more accurate easy and efficient than hptm fractional coupled s g equations have been first time solved by mham and hptm in this paper modified homotopy analysis method mham and homotopy perturbation transform method hptm have been implemented for solving time fractional coupled sine gordon equations we consider fractional coupled sine gordon equations which models one dimensional nonlinear wave processes in two component media the results obtained by modified homotopy analysis method mham and homotopy perturbation transform method hptm are then compared with the modified decomposition method mdm by using an initial value system the numerical solutions of coupled sine gordon equations have been represented graphically here we obtain the solution of fractional coupled sine gordon s g equations which is obtained by replacing the time derivatives with a fractional derivatives of order 1 2 and 1 2 the fractional derivatives here are described in caputo sense

snakes and spiders brownian motion on ℝ trees we consider diffusion processes on a class of trees the processes are defined in a manner similar to that of le gall s brownian snake each point in the tree has a real valued height or generation and the height of the diffusion process evolves as a brownian motion when the height process decreases the diffusion retreats back along a lineage whereas when the height process increases the diffusion chooses among branching lineages according to relative weights given by a possibly infinite measure on the family of lineages the class of trees we consider can have branch points with countably infinite branching and lineages along which the branch points have points of accumulation

quantization noise in single loop sigma delta modulation with sinusoidal inputs an exact nonlinear difference equation is derived and solved for a simple sigma delta modulator consisting of a discrete time integrator and a binary quantizer inside a single feedback loop and an arbitrary input signal it is shown that the system can be represented as an affine operation discrete time integration of a biased input followed by a memoryless nonlinearity an extension of the transform method for the analysis of nonlinear systems is applied to obtain formulas for first and second order time average moments of the binary quantization noise including the sample mean energy and autocorrelation the results are applied to the special case of a sinusoidal input signal to evaluate these time averages and the power spectrum in the limit of large oversampling ratios the marginal moments behave as if the quantization noise had a uniform distribution the spectrum is neither white nor continuous however even in the limit of large oversampling ratios

non existence of truly solitary waves in water with small surface tension this paper considers permanent capillary gravity waves on the free surface of a two dimensional incompressible inviscid fluid of finite depth it is shown that there are no solitary wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity if the surface tension coefficient is less than its critical value and lies in some intervals the proof is based upon an estimate of a constant that is related to the approximation of the solution if it exists near its singularity the approximation satisfies a fourth order nonlinear ordinary differential equation when the solution is extended to the complex plane then the non existence of truly solitary waves is obtained by using a contradiction on this constant

solving the inverse problem of identifying an unknown source term in a parabolic equation the inverse problem of determining an unknown source term in a parabolic equation with the leading coefficient depending on time and space variables under a final overdetermination condition is considered the series pattern solution of the problem is obtained by using the weighted homotopy analysis method wham a description of the method for solving the problem and identifying the unknown source term is derived finally several examples are presented to illustrate this method

some windows with very good sidelobe behavior some of the windows presented by harris 1 are not correct in terms of their reported peak sidelobes and optimal behavior we present corrected plots of harris windows and also derive additional windows with very good sidelobes and optimal behavior under several different constraints the temporal weightings are characterized as a sum of weighted cosines over a finite duration the plots enable the reader to select a window to suit his requirements in terms of bias due to nearby sidelobes and bias due to distant sidelobes

exit times for an increasing levy tree valued process we give an explicit construction of the increasing tree valued process introduced by abraham and delmas using a random point process of trees and a grafting procedure this random point process will be used in companion papers to study record processes on levy trees we use the poissonian structure of the jumps of the increasing tree valued process to describe its behavior at the first time the tree grows higher than a given height using a spinal decomposition of the tree similar to the classical bismut and williams decompositions we also give the joint distribution of this exit time and the ascension time which corresponds to the first infinite jump of the tree valued process

radiating solitary waves of a model evolution equation in fluids of finite depth abstract the evolution equation proposed by kubota et al j hydronautics 12 1978 157 165 and joseph j phys a 10 1977 225 227 for long waves in stratified fluids of finite depth is considered under a small third order derivative dispersive perturbation solitary wave solutions to this model equation become weakly nonlocal they develop oscillatory tails of infinite extent and exponentially small amplitude here these tails are calculated asymptotically following a perturbation procedure in the wavenumber domain

rayleigh processes real trees and root growth with re grafting the real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree aldous s brownian continuum random tree the random tree like object naturally associated with a standard brownian excursion may be thought of as a random compact real tree the continuum random tree is a scaling limit as n of both a critical galton watson tree conditioned to have total population size n as well as a uniform random rooted combinatorial tree with n vertices the aldous broder algorithm is a markov chain on the space of rooted combinatorial trees with n vertices that has the uniform tree as its stationary distribution we construct and study a markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as n of the aldous broder chain a key technical ingredient in this work is the use of a pointed gromov hausdorff distance to metrize the space of rooted compact real trees berkeley statistics technical report no 654 february 2004 revised october 2004 to appear in probability theory and related fields

analysis of reaction diffusion system via a new fractional derivative with non singular kernel abstract in this paper we obtain analytical solutions for the fractional cubic isothermal auto catalytic chemical system with caputo fabrizio and atangana baleanu fractional time derivatives in liouville caputo sense we utilize the q homotopy analysis transform method to compute the approximate solutions we find the optimal values of h so we assure the convergence of the approximate solutions finally we compare our results numerically with the finite difference method and excellent agreement is found

branching processes in levy processes laplace functionals of snakes and superprocesses we use the exploration process introduced in a previous work to develop a new construction of superprocesses with a general branching mechanism this construction depends on a path valued process called the levy snake which is of independent interest our method of proof involves a calculation of the laplace functional of the occupation field of the levy snake this calculation relies on an evaluation of the corresponding moment functionals which requires precise information about the underlying genealogical structure

an investigation with hermite wavelets for accurate solution of fractional jaulent miodek equation associated with energy dependent schrodinger potential in the present paper a wavelet method based on the hermite wavelet expansion along with operational matrices of fractional derivative and integration is proposed for finding the numerical solution to a coupled system of nonlinear time fractional jaulent miodek jm equations consequently the approximate solutions of fractional jaulent miodek equations acquired by using hermite wavelet technique were compared with those derived by using optimal homotopy asymptotic method oham and exact solutions the present proposed numerical technique is easy expedient and powerful in computing the numerical solution of coupled system of nonlinear fractional differential equations like jaulent miodek equations

weakly non local solutions for capillary gravity waves fifth degree korteweg de vries equation hunter and scheurle have shown that capillary gravity water waves in the vicinity of bond number bo l 3 are consistently modelled by the korteweg de vries equation with the addition of a fifth derivative term this wave equation does not have strict soliton solutions for bo l 3 because the near solitons have oscillatory wings that extend indefinitely from the core of the wave however these solutions are arbitrarily small perturbations of solitary waves because the amplitude of the wings is exponentially small in the amplitude 6 of the core pomeau ramani and grammaticos have calculated the amplitude of the wings by applying matched asymptotics in the complex plane in the limit e 0 in this article we describe a mixed chebyshev radiation function pseudospectral method which is able to calculate the weakly non local solitons for all e we show that for fixed phase speed the solitons form a three parameter family because the linearized wave equation has three eigensolutions we also show that one may repeat the soliton with even spacing to create a three parameter of periodic solutions which we also compute because the amplitude of the wings is exponentially small these non local capillary gravity solitons are as interesting as the classical localized solitons that solve the korteweg de vries equation

spines skeletons and the strong law of large numbers for superdiffusions consider a supercritical superdiffusion x t t 0 on a domain d r d with branching mechanism x z x z x z 2 0 e zy 1 zy x dy the skeleton decomposition provides a pathwise description of the process in terms of immigration along a branching particle diffusion we use this decomposition to derive the strong law of large numbers slln for a wide class of superdiffusions from the corresponding result for branching particle diffusions that is we show that for suitable test functions f and starting measures f x t p f x t w p almost surely as t where w is a finite non deterministic random variable characterized as a martingale limit our method is based on skeleton and spine techniques and offers structural insights into the driving force behind the slln for superdiffusions the result covers many of the key examples of interest and in particular proves a conjecture by fleischmann and swart stochastic process appl 106 2003 141 165 for the super wright fisher diffusion

an explicit analytic solution to the thomas fermi equation a new kind of analytic technique namely the homotopy analysis method is employed to give an explicit analytic solution of the thomas fermi equation and the related recurrence formulae of constant coefficients this solution can be regarded as the definition of the exact solution of the thomas fermi equation

an adaptive wavelet collocation method for solving optimal control of elliptic variational inequalities of the obstacle type abstract this paper presents a fast computational technique based on the wavelet collocation method for the numerical solution of an optimal control problem governed by elliptic variational inequalities of obstacle type in this problem the solution divides the domain into contact and noncontact sets the boundary between the contact and noncontact sets is a free boundary which is not known a priori and the solution is not smooth on it accordingly a very fine grid is needed in order to obtain a solution with a reasonable accuracy in this paper our aim is to propose an adaptive scheme in order to generate an appropriate and economic irregular dyadic mesh for finding the optimal control and state functions the irregular mesh will be generated such that its density around the free boundary is higher than in other places and high resolution computations are focused on these zones to this aim we use an adaptive wavelet collocation method and take advantage of the fast wavelet transform of compact supported interpolating wavelets to develop a multi level algorithm which generates an adaptive computational grid using this adaptive grid takes less cpu time than using a full regular mesh at each step of the algorithm the active set method is used for solving the optimality system of the obstacle problem on the adapted mesh finally the numerical examples are presented to show the validity and efficiency of the technique

application of wavelet collocation method for hyperbolic partial differential equations via matrices abstract in this work we developed an efficient computational method based on legendre and chebyshev wavelets to find an approximate solution of one dimensional hyperbolic partial differential equations hpdes with the given initial conditions the operational matrices of integration for legendre and chebyshev wavelets are derived and utilized to transform the given pde into the linear system of equations by combining collocation method convergence analysis and error estimation associated to the presented idea are also investigated under several mild conditions numerical experiments confirm that the proposed method has good accuracy and efficiency moreover the use of legendre and chebyshev wavelets are found to be accurate simple and fast

branching brownian motion in a strip survival near criticality we consider a branching brownian motion with linear drift in which particles are killed on exiting the interval 0 k and study the evolution of the process on the event of survival as the width of the interval shrinks to the critical value at which survival is no longer possible we combine spine techniques and a backbone decomposition to obtain exact asymptotics for the near critical survival probability this allows us to deduce the existence of a quasi stationary limit result for the process conditioned on survival which reveals that the backbone thins down to a spine as we approach criticality this paper is motivated by recent work on survival of near critical branching brownian motion with absorption at the origin by aidekon and harris near critical survival probability of branching brownian motion with an absorbing barrier 2010 unpublished manuscript as well as the work of berestycki et al ann probab 41 2013 527 618 j stat phys 143 2011 833 854

growth of the brownian forest trees in brownian excursions have been studied since the late 1980s forests in excursions of brownian motion above its past minimum are a natural extension of this notion in this paper we study a forest valued markov process which describes the growth of the brownian forest the key result is a composition rule for binary galton watson forests with i i d exponential branch lengths we give elementary proofs of this composition rule and explain how it is intimately linked with williams decomposition for brownian motion with drift

two stage sigma delta modulation an exact analysis of the nonlinear difference equation describing a discrete time two stage sigma delta quantizer with dc and sinusoidal inputs is presented it is shown that for dc inputs the binary quantizer error of the second quantizer which appears at the output of the modulator as a second order difference is asymptotically white uniformly distributed and uncorrelated with the input for sinusoidal inputs this result holds with one exception the autocorrelation which is a sum of weighted bessel functions does not correspond to that of a white noise process at large oversampling ratios the autocorrelation is approximately white and hence the white additive noise model is a good approximation for evaluating the first and second order moments it is found that for both a sinc sup 3 decoder filter and an ideal low pass filter decoder the average quantization noise power is inversely proportional to the fifth power of the oversampling ratio r for both types of inputs this represents an improvement over the single loop sigma delta quantizer by a factor of r sup 2 simulation results that agree with the theoretical analysis are presented

multistage sigma delta modulation a theoretical basis is provided for multistage sigma delta modulation msm which is a cascade realization of several single loop sigma delta modulators with a linear combinatorial network equations are derived describing the output and the quantization noise of msm for an arbitrary input signal and the noise shaping characteristic of msm is investigated the spectral characteristics of an m stage sigma delta modulator with both dc and sinusoidal inputs are developed for both types of inputs the binary quantizer noise of the mth m or 3 quantizer which appears at the output as an mth order difference is asymptotically white uniformly distributed and uncorrelated with the input level it is also found that for an m stage sigma delta quantizer with either an ideal low pass filter or a sinc sup m 1 filter decoder the average quantization noise of the system is inversely proportional to the 2m 1 th power of the oversampling ratio this implies that the high order systems are favourable in terms of the trade off between the quantization noise and oversampling ratio simulation results are presented to support the theoretical analysis

on the application of the homotopy analysis method for a nonlocal mixed problem with bessel operator abstract in this research work we give analytic approximate solutions for a nonlocal initial boundary value problem where one of the classical conditions is replaced by an integral condition by employing the homotopy analysis method ham the series solution can be obtained by either using initial conditions partial t solution or by using the boundary conditions partial x solution this method allows us to control the convergence region of the series solution

branching processes in levy processes the exploration process the main idea of the present work is to associate with a general continuous branching process an exploration process that contains the desirable information about the genealogical structure the exploration process appears as a simple local time functional of a levy process with no negative jumps whose laplace exponent coincides with the branching mechanism function this new relation between spectrally positive levy processes and continuous branching processes provides a unified perspective on both theories in particular we derive the adequate formulation of the classical ray knight theorem for such levy processes as a consequence of this theorem we show that the path continuity of the exploration process is equivalent to the almost sure extinction of the branching process

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

exponential asymptotics and stokes lines in nonlinear ordinary differential equations a technique for calculating exponentially small terms beyond all orders in singularly perturbed ordinary differential equations is presented the approach is based on the application of a wkbj type ansatz to the late terms in the naive asymptotic expansion and the identification of stokes lines and is closely related to the well known stokes line smoothing phenomenon in linear ordinary differential equations the method is illustrated by application to examples

exact travelling wave solutions to seventh order and ninth order kdv like equations new ansatze that involve hyperbolic functions are introduced to handle seventh order and ninth order kdv like equations a set of entirely new exact travelling wave solutions is established the work introduces useful guidance for other related nonlinear evolution equations

solitary waves with damped oscillatory tails an analysis of the fifth order korteweg de vries equation abstract we construct oscillatory solitary wave solutions of a fifth order korteweg de vries equation where the oscillations decay at infinity these waves arise as a bifurcation from the linear dispersion curve at that wavenumber where the linear phase speed and group velocity coincide our approach is a wave packet analysis about this wavenumber which leads in the first instance to a higher order nonlinear schrodinger equation from which we then obtain the steady solitary wave solution we then describe a complementary normal form analysis which leads to the same result in addition we derive the nonlinear schrodinger equation for all wavenumbers and list all the various anomalous cases

a new iterative technique for a fractional model of nonlinear zakharov kuznetsov equations via sumudu transform abstract the main objective of this paper is to suggest a new computational technique namely new iterative sumudu transform method nistm to solve numerically nonlinear time fractional zakharov kuznetsov fzk equation in two dimensions we implemented the proposed technique on two test examples plotted the solution and compared the absolute error with the variational iterative technique vim and homotopy perturbation transform method hptm

oversampled sigma delta modulation oversampled sigma delta modulation has been proposed as a practical implementation for high rate analog to digital conversion because of its simplicity and its robustness against circuit imperfections to date mathematical developments of the basic properties of such systems have been based either on simplified continuous time approximate models or on linearized discrete time models where the quantizer is replaced by an additive white uniform noise source in this paper we rigorously derive several basic properties of a simple discrete time single integrator loop sigma delta modulator with an accumulate and dump demodulator the derivation does not require any assumptions on the correlation or distribution of the quantizer error and hence involves no linearization of the nonlinear system but it does show that when the input is constant the state sequence of the integrator in the encoder loop can be modeled exactly as a linear system in an appropriate space two basic properties are developed 1 the behavior of the sigma delta quantizer when driven by a constant input and its relation to uniform quantization and 2 the rate distortion tradeoffs between the oversampling ratio and the average mean squared quantization error

probabilistic and fractal aspects of levy trees we investigate the random continuous trees called levy trees which are obtained as scaling limits of discrete galton watson trees we give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted trees which is equipped with the gromov hausdorff distance to construct levy trees we make use of the coding by the height process which was studied in detail in previous work we then investigate various probabilistic properties of levy trees in particular we establish a branching property analogous to the well known property for galton watson trees conditionally given the tree below level a the subtrees originating from that level are distributed as the atoms of a poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root we study regularity properties of local times in the space variable and prove that the support of local time is the full level set except for certain exceptional values of a corresponding to local extinctions we also compute several fractal dimensions of levy trees including hausdorff and packing dimensions in terms of lower and upper indices for the branching mechanism function which characterizes the distribution of the tree we finally discuss some applications to super brownian motion with a general branching mechanism

shape selection of saffman taylor fingers among all problems of pattern selection the one posed by the saffman taylor finger is yet unsolved although numerics and experiments indicate the very simple property that the relative width of the finger tends to 1 2 in the low surface tension limit we explain this by performing an expansion beyond all orders leading to the formulation of a nonlinear eigenvalue problem with a discrete set of solutions in particular we predict that the relative width of the finger tends to 1 2 as the surface tension to the power 2 3 pacs numbers 47 10 g 03 40 gc 68 10 m

the homotopy analysis method for solving the fornberg whitham equation and comparison with adomian s decomposition method in this work an analytical technique namely the homotopy analysis method ham is applied to obtain an approximate analytical solution of the fornberg whitham equation a comparison is made between the ham results and the adomian s decomposition method adm and the homotopy perturbation method hpm the results reveal that ham is very simple and effective the ham contains the auxiliary parameter h which provides us with a simple way to adjust and control the convergence region of solution series

spectral analysis of quantization noise in a single loop sigma delta modulator with dc input an exact discrete time analysis of the moments and spectra of the quantization noise of a discrete time single loop sigma delta modulator with a dc input is presented an exact difference equation for the discrete time nonlinear system is used to evaluate the first and second order moments and power spectrum of the binary quantizer noise and the binary quantizer output for a single loop sigma delta encoder with a dc input it is shown that the sample mean and power of the binary quantization noise are consistent with the common uniform distribution assumption but that the autocorrelation and power spectrum are not consistent with the white noise assumption the results are used to evaluate the overall sample average mean squared quantization error as a function of the decimation filter used

the generation of radiating waves in a singularly perturbed korteweg de vries equation abstract we consider a fifth order kdv equation where the fifth order derivative term is multiplied by a small parameter it has been conjectured that this equation admits a non local solitary wave solution which has a central core and an oscillatory tail either behind or in front of the core we prove that this solution cannot be exactly steady and instead the amplitude of the central core decays due to the energy flux generated in the oscillatory tail the decay rate is calculated in the limit as the parameter tends to zero in order to verify the analytical results we have developed a high precision spectral method for numerical integration of this equation the analytical and numerical result show good agreement

on the stability of sigma delta modulators a framework for stability analysis of sigma delta modulators is presented it is argued that limit cycles for constant inputs are natural objects to investigate in this context a number of analytical and approximate techniques to aid the stability analysis of the double loop and interpolative modulators are discussed and lead to ways of improving the design that explicitly take stability into account

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

a crank nicolson adi galerkin legendre spectral method for the two dimensional riesz space distributed order advection diffusion equation abstract in the paper a crank nicolson alternating direction implicit adi galerkin legendre spectral scheme is presented for the two dimensional riesz space distributed order advection diffusion equation the gauss quadrature has a higher computational accuracy than the mid point quadrature rule which is proposed to approximate the distributed order riesz space derivative so that the considered equation is transformed into a multi term fractional equation moreover the transformed equation is solved by discretizing in space by the adi galerkin legendre spectral scheme and in time using the crank nicolson difference method stability and convergence analysis are verified for the numerical approximation a lot of numerical results are demonstrated to justify the theoretical analysis

analysis of dc input response for a class of one bit feedback encoders a class of one bit feedback encoders that includes sigma delta modulation encoders data modulation encoders and some dpcm encoders is defined for any encoder from the class it is shown that the encoder output sequence in response to a dc input sequence is a typical sequence for a certain strict sense stationary ergodic process whose power spectral density is determined the result is applied to give the performance of an analog to digital conversion system operated at high sampling rates

computing exact solutions to a generalized lax seventh order forced kdv equation kdv7 abstract in this paper the cole hopf transform is used to construct exact solutions to a generalization of both the seventh order lax kdv equation lax kdv7 and the seventh order sawada kotera ito kdv equation sawada kotera ito kdv7 with forcing term

a note on the interaction between solitary waves in a singularly perturbed korteweg de vries equation for the fifth order korteweg de vries equation with a small parameter multiplying the highest derivative term it is known that solitary waves are non local and are accompanied by co propagating oscillatory waves of small amplitude and short wavelength here the authors report on some preliminary results for mutual interactions of these waves first they impose a quantization condition upon a chain of solitary waves to obtain a periodic solution then they compute the interaction force between two neighbouring waves and hence estimate the conditions for a bound state to occur

the real tree in this note it is shown that there exists a natural metric on the set formula of bounded subsets of r containing their infimum which endowstwith the structure of an r tree so that for everyt t the number of connected components oft t coincides with the cardinality p r of the set of subsets of r in addition the set of ends oftis explicitly determined and various further features oftare discussed too

growth of levy trees we construct random locally compact real trees called levy trees that are the genealogical trees associated with continuous state branching processes more precisely we define a growing family of discrete galton watson trees with i i d exponential branch lengths that is consistent under bernoulli percolation on leaves we define the levy tree as the limit of this growing family with respect to the gromov hausdorff topology on metric spaces this elementary approach notably includes supercritical trees and does not make use of the height process introduced by le gall and le jan to code the genealogy of sub critical continuous state branching processes we construct the mass measure of levy trees and we give a decomposition along the ancestral subtree of a poisson sampling directed by the mass measure

on the homotopy analysis method for nonlinear problems a powerful easy to use analytic tool for nonlinear problems in general namely the homotopy analysis method is further improved and systematically described through a typical nonlinear problem i e the algebraically decaying viscous boundary layer flow due to a moving sheet two rules the rule of solution expression and the rule of coefficient ergodicity are proposed which play important roles in the frame of the homotopy analysis method and simplify its applications in science and engineering an explicit analytic solution is given for the first time with recursive formulas for coefficients this analytic solution agrees well with numerical results and can be regarded as a definition of the solution of the considered nonlinear problem

double loop sigma delta modulation with dc input a discrete time model having a two bit 2 b quantizer is analyzed exactly and an analytical expression for the quantizer noise sequence is found rigorous answers are then provided to two fundamental questions for a double loop sigma delta modulation system with dc input 1 what is the long term statistical behavior of the internal quantizer noise 2 how does the asymptotic mean square sigma delta quantization error vary as a function of the oversampling ratio

exponential asymptotics and capillary waves recently developed techniques in exponential asymptotics beyond all orders are employed on the problem of potential flows with a free surface and small surface tension in the absence of gravity exponentially small capillary waves are found to be generated on the free surface where the equipotentials from singularities in the flow for example stagnation points and corners meet it the amplitude of these waves is determined and the implications are considered for many quite general flows the asymptotic results are compared to numerical simulations of the full problem for flow over a polygonal plough and for flow round a right angled corner and they show remarkably good agreement even for quite large values of the surface tension parameter

a modified spectral method for solving operator equations in this paper we introduce a modified spectral method for solving the linear operator equation l u f l d l h 1 h 2 where h 1 and h 2 are normed vector spaces with norms 1 and respectively and d l is the domain of l also for each h h 2 h 2 h h where is an inner product on h 2 in this method we make a new set n n 0 for h 1 using l and two sets in h 1 and h 2 then using the new set n n 0 we solve this linear operator equation we show that this method does not have some shortcomings of spectral method also we prove the stability and convergence of the new method after introducing the method we give some conditions that under them the nonlinear operator equation l u n u f can be solved some examples are considered to show the efficiency of method

weakly nonlocal solitary waves in a singularly perturbed korteweg de vries equation a fifth order korteweg de vries equation is considered where the fifth order derivative term is multiplied by a small parameter it is known that solitary wave solutions of this model equation are nonlocal in that the central core of the wave is accompanied by copropagating trailing oscillations here using the techniques of exponential asymptotics these solutions are reexamined and it is established that they form a one parameter family characterized by the phase shift of the trailing oscillations explicit asymptotic formula relating the oscillation amplitude to the phase shift are obtained

regenerative tree growth binary self similar continuum random trees and poisson dirichlet compositions we use a natural ordered extension of the chinese restaurant process to grow a twoparameter family of binary self similar continuum fragmentation trees we provide an explicit embedding of ford s sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of ford s trees in general the markov branching trees induced by the two parameter growth rule are not sampling consistent so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case we develop here a new approach to establish such limits based on regenerative interval partitions and the urn model description of sampling from dirichlet random distributions ams 2000 subject classifications 60j80

the optimal homotopy asymptotic method for solving blasius equation starting from the reality that many known methods fail in the attempt to obtain analytic solutions of blasius type equations in this work a new procedure namely optimal homotopy asymptotic method oham is proposed to obtain an explicit analytical solution of the blasius problem comparison with howarth s numerical solution as well as the obtained residual reveals that the proposed method is highly accurate this proves the validity and great potential of the proposed procedure oham as a new kind of powerful analytical tool for nonlinear problems

strichartz estimates for dispersive equations and solvability of the kawahara equation abstract we first establish a series of strichartz estimates for a general class of linear dispersive equations by applying the theory of oscillatory integrals established by kenig ponce and vega next we use such estimates to study solvability of the cauchy problem of the kawahara equation t u a u x u x 3 u x 5 u 0 in the class c r h s r local existence is proved for s 1 4 and global existence is proved for s 2

analytical and numerical studies of a singularly perturbed boussinesq equation we study the singularly perturbed sixth order boussinesq equation recently introduced by daripa and hua appl math comput 101 1999 159 motivated by their work we formally derive this equation from two dimensional potential flow equations governing the small amplitude long capillary gravity waves on the surface of shallow water for bond number very close to but less than 1 3 on the basis of far field analyses and heuristic arguments we show that the traveling wave solutions of this equation are weakly non local solitary waves characterized by small amplitude fast oscillations in the far field we review various analytical and numerical methods originally devised to obtain this type of weakly non local solitary wave solutions of the singularly perturbed fifth order kdv equation using these methods we obtain weakly non local solitary wave solutions of the singularly perturbed sixth order boussinesq equation and provide estimates of the amplitude of oscillations which persist in the far field

energy decay for the modified kawahara equation posed in a bounded domain abstract studied here is the eventual dissipation of solutions to initial boundary value problems for the modified kawahara equation with and without a localized damping term included it is shown that solutions of undamped problems posed on a bounded interval may not decay if the length of the interval is critical in contrast the energy associated to the locally damped problems is shown to be exponentially decreased independently of the interval length

multi pulse embedded solitons as bound states of quasi solitons abstract embedded solitons ess is a name given to localised solutions of multi component dispersive wave models that occur in resonance with the linear spectrum they exist at isolated codimension one parameter values within the steady state equations which have linearisation with eigenvalues and i at the same time quasi solitons qss or generalised solitary waves with non decaying radiation tails are known to be endemic we consider the general question of when two qss may be glued together to form a two humped bound state bs es in the limit 0 a generalisation of the method of gorshkov and ostrovsky is used within the framework of general normal forms for hamiltonian reversible systems a simple asymptotic formula is derived that governs the existence symmetry and accumulation rate of the bss this generalises earlier ad hoc calculations for specific examples the formula depends on few simple ingredients the eigenvalues of the equilibrium an asymptotic estimate for the tail amplitude of the qss the birkhoff signature and symmetry properties of the hamiltonian only the tail amplitude estimate is difficult to calculate requiring exponential asymptotics but it is shown that this only affects the third order term in the asymptotic formula the calculation is worked out in detail for the steady states of several example pde systems taken from nonlinear optics and fluid mechanics and excellent agreement with numerical results is found

the continuum random tree iii let mathscr r k k geq 1 be random trees with k leaves satisfying a consistency condition removing a random leaf from mathscr r k gives mathscr r k 1 then under an extra condition this family determines a random continuum tree mathscr l which it is convenient to represent as a random subset of l 1 this leads to an abstract notion of convergence in distribution as n rightarrow infty of rescaled random trees mathscr j n on n vertices to a limit continuum random tree mathscr l the notion is based upon the assumption that for fixed k the subtrees of mathscr j n determined by k randomly chosen vertices converge to mathscr r k as our main example under mild conditions on the offspring distribution the family tree of a galton watson branching process conditioned on total population size equal to n can be rescaled to converge to a limit continuum random tree which can be constructed from brownian excursion