a simple homotopy method for determining all isolated solutions to polynomial systems a new homotopy method for solving systems of polynomial equations is presented the homotopy equation is extremely simple it is linear with respect to the homotopy parameter and only one auxiliary parameter is needed to regularize the problem within some limits an arbitrary starting problem can be chosen as long as its solution set is known no restrictions on the polynomial systems are made a few numerical tests are reported which show the influence of the auxiliary parameter resp the starting problem upon the computa tional cost of the method

multiple objective programming with polynomial objectives and constraints abstract an approach to solving multiple objective programming problems with polynomial objectives and polynomial constraints is developed the notions of pre efficient solution and local efficient solution are investigated relative to the single objective problem of benson illustrative examples are presented in which the solutions are obtained by means of homotopy continuation

a globally convergent probability one homotopy for linear programs with linear complementarity constraints a solution of the standard formulation of a linear program with linear complementarity constraints lpcc does not satisfy a constraint qualification a family of relaxations of an lpcc associated with a probability one homotopy map proposed here is shown to have several desirable properties the homotopy map is nonlinear replacing all the constraints with nonlinear relaxations of ncp functions under mild existence and rank assumptions 1 the lpcc relaxations rlpcc lambda have a solution for 0 le lambda le1 2 rlpcc 1 is equivalent to lpcc 3 the kuhn tucker constraint qualification is satisfied at every local or global solution of rlpcc lambda for almost all 0 le lambda 1 4 a point is a local solution of rlpcc 1 and lpcc if and only if it is a kuhn tucker point for rlpcc 1 and 5 a homotopy algorithm can find a kuhn tucker point for rlpcc 1 since the homotopy map is a globally convergent probability one homotopy robust and efficient numerical algorithms exist to f

homotopy methods for solving the optimal projection equations for the h2 reduced order model problem the optimal projection approach to solving the h2 reduced order model problem produces two coupled highly nonlinear matrix equations with rank conditions as constraints due to the resemblance of these equations to standard matrix lyapunov equations they are called modified lyapunov equations the algorithms proposed herein utilize probability one homotopy theory as the main tool it is shown that there is a family of systems the homotopy that make a continuous transformation from some initial system to the final system with a carefully chosen initial problem a theorem guarantees that all the systems along the homotopy path will be asymptotically stable controllable and observable one method which solves the equations in their original form requires a decomposition of the projection matrix using the drazin inverse of a matrix it is shown that the appropriate inverse is a differentiable function an effective algorithm for computing the derivative of the projection matrix that involves solving a set of sylvester equations is given another class of methods considers the equations in a modified form using a decomposition of the pseudogramians based on a contragredient transformation some freedom is left in making an exact match between the number of equations and the number of unknowns thus effectively generating a family of methods

symmetric homotopy construction abstract homotopy continuation methods can be applied to compute all finite solutions to a given polynomial system computations will be performed more efficiently if the symmetric structure of the system can be exploited this paper presents the construction of a symmetric homotopy using this homotopy only the paths according to the generating solutions have to be traced during continuation

regularity results for solving systems of polynomials by homotopy method the homotopy method for solving systems of polynomial equations proposed by li also morgan is believed to be one of the simplest for computer implementation it is shown here that the parameters of the homotopy can be chosen from an open and dense set such that all isolated solutions of polynomial systems can be obtained

a transformation to avoid solutions at infinity for polynomial systems in finding all solutions to polynomial systems the existence of solutions at infinity makes the problem more difficult particularly when a continuation method is being used as the solution technique systems with solutions at infinity do arise in applications for example in computer graphics and geometric modeling in this paper a simple transformation of the system is given which eliminates solutions at infinity

local and global solvability of a class of semilinear parabolic equations pierre baras laborutoire imag tour des mathkmatiques b p 68 38402 saint martin d heres cedex france robert kersner computer and automation institute hungarian academy of sciences p 0 box 63 h 1502 budapest hungary received february 11 1986 revised september 5 1986

temporal monotonicity of the solutions of some semilinear parabolic equations with fractional diffusion abstract suppose that the functions g and are nonnegative and satisfy suitable regularity conditions then we prove in this work that the parabolic semilinear problem t u t x u t x g x f u t x x t 0 x r n u 0 x x x r n has a unique positive and time monotone solution here is the fractional laplacian with 0 2 and the source term f is a convex function with f 0 0 moreover using the temporal monotonicity we show that the elliptic equation v x g x f v x x x r d with boundary condition lim x v x 0 has a positive solution we provide also sufficient conditions for the integrability of both solutions

computation of critical boundaries on equilibrium manifolds the study of various equilibrium phenomena leads to nonlinear equations 1 f y u 0 where y in r n is a vector of behavior or state variables u in r p a vector of p geqq 2 parameters or controls and f d subset r n times r p to r n a sufficiently differentiable map the solution set of 1 in r n times r p is often called the equilibrium surface of the problem and in many applications it is of interest to determine the critical boundary that is the set of solutions of 1 where the derivative d y f x is singular for example in structural problems these points may represent buckling points after characterizing the various properties of the problem we present three different numerical methods which allow for a computational trace of paths in the critical boundary these methods represent extensions of the earlier developed locally parametrized continuation method for tracing regular paths on the equilibrium surface they permit for instance a direct computational deter

note on the homotopy perturbation method for multivariate vector value oscillatory integrals this paper considers a homotopy perturbation method for approximating multivariate vector value highly oscillatory integrals the asymptotic formulae of the integrals and the asymptotic order of the asymptotic method are presented numerical examples show the efficiency of the approximation method

theory of globally convergent probability one homotopies for nonlinear programming for many years globally convergent probability one homotopy methods have been remarkably successful on difficult realistic engineering optimization problems most of which were attacked by homotopy methods because other optimization algorithms failed or were ineffective convergence theory has been derived for a few particular problems and considerable fixed point theory exists but generally convergence theory for the homotopy maps used in practice for nonlinear constrained optimization has been lacking this paper derives some probability one homotopy convergence theorems for unconstrained and inequality constrained optimization for linear and nonlinear inequality constraints and with and without convexity some insight is provided into why the homotopies used in engineering practice are so successful and why this success is more than dumb luck by presenting the theory as variations on a prototype probability one homotopy convergence theorem the essence of such convergence theory is elucidated

a hybrid polynomial system solving method for mixed trigonometric polynomial systems mixed trigonometric polynomial systems arise in many fields of science and engineering commonly this class of systems is transformed into polynomial systems by variable substituting and adding some quadratic equations and then solved by some polynomial system solving method in this paper by exploiting the special structure of the additional quadratic equations an efficient hybrid method for solving polynomial systems coming from mixed trigonometric polynomial systems is presented it combines the homotopy method in which the homotopy is a combination of coefficient parameter homotopy and the random product homotopy with decomposition variable substitution and reduction techniques numerical tests are given to show its effectiveness and it is applied to solve a practical problem

bezout number calculations for multi homogeneous polynomial systems the bezout number of a system of multi homogeneous polynomial equations is the largest number of nonsingular solutions such a system can have and it is also the number of solution paths used to compute all geometrically isolated solutions of the system using multi homogeneous polynomial continuation any polynomial system can be homogenized in a variety of ways typically yielding different bezout numbers this paper presents an efficient algorithm for computing bezout numbers and describes a procedure for finding the minimal bezout number over all possible homogenizations

a homotopy for solving general polynomial systems that respects m homogeneous structures methods for finding all solutions to a polynomial system f z 0 of n equations in n unknowns using a generic homotopy have been presented by several researchers over the last ten years in this paper we describe a new method for defining a homotophy to find all solutions to a polynomial system a major feature of this new approach is that the homotopy need not to be generic or random the practical consequences are that the homotopy may be chosen to more closely resemble the system being solved with a consequent potential for reducing the arc length of paths and exploiting special structures in the original system in particular if the system is naturally m homogeneous then the number of homotopy paths that need to be tracked is reduced since many systems that arise in applications have such a structure this represents a significant advance over previous homotopies we also provide a proof that the homotopy parameter is strictly increasing on homotopy paths as a function of arc length a new result which has implications for the choice of numerical path tracking algorithms

probability one homotopies in computational science probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that under mild assumptions are globally convergent for a wide range of problems in science and engineering convergence theory robust numerical algorithms and production quality mathematical software exist for general nonlinear systems of equations and special cases such as brouwer fixed point problems polynomial systems and nonlinear constrained optimization using a sample of challenging scientific problems as motivation some pertinent homotopy theory and algorithms are presented the problems considered are analog circuit simulation for nonlinear systems reconfigurable space trusses for polynomial systems and fuel optimal orbital rendezvous for nonlinear constrained optimization the mathematical software packages hompack90 and polsys plp are also briefly described

numerical linear algebra aspects of globally convergent homotopy methods probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one these methods are theoretically powerful and if constructed and implemented properly are robust numerically stable accurate and practical the concomitant numerical linear algebra problems deal with rectangular matrices and good algorithms require a delicate balance not always achieved of accuracy robustness and efficiency in both space and time the author s experience with globally convergent homotopy algorithms is surveyed here and some of the linear algebra difficulties for dense and sparse problems are discussed

asymptotic expansions of bessel anger and weber transformations in this paper we explore higher order numerical quadrature for the integrations of systems containing bessel anger and weber functions the method is constructed by finding the approximate solution of the differential equation and truncating the asymptotic series numerical examples based on theoretical results are presented to demonstrate the efficiency of the method

probability one homotopy algorithms for solving the coupled lyapunov equations arising in reduced order h2 h modeling estimation and control optimal reduced order modeling estimation and control with respect to combined h 2 h criteria give rise to coupled lyapunov and riccati equations to develop reliable numerical algorithms for these problems this paper focuses on the coupled lyapunov equations which appear as a subset of the synthesis equations in particular this paper systematically examines the requirements of probability one homotopy algorithms to guarantee global convergence homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family the main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled lyapunov equations and variations based upon canonical forms these results are essential to the probability one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms

cosine law at the atomic scale toward realistic simulations of knudsen diffusion we propose to reconsider the diffusion of atoms in the knudsen regime in terms of a complex dynamical reflection process by means of molecular dynamics simulations we emphasize the asymptotic nature of the cosine law of reflection at the atomic scale and carefully analyze the resulting strong correlations in the reflection events a dynamical interpretation of the accommodation coefficient associated with the slip at the wall interface is also proposed finally we show that the first two moments of the stochastic process of reflection depend nonuniformly on the incident angle

homotopy method for generalized eigenvalue problems ax λbx abstract generalized eigenvalue problems can be considered as a system of polynomials the homotopy continuation method is used to find all the isolated zeros of the polynomial system which corresponds to the eigenpairs of the generalized eigenvalue problem a special homotopy is constructed in such a way that there are exactly n distinct smooth curves connecting trivial solutions to desired eigenpairs since the curves followed by general homotopy curve following scheme are computed independently of one another the algorithm is a likely candidate for exploiting the advantages of parallel processing to the generalized eigenvalue problems

existence of solutions for an inhomogeneous fractional semilinear heat equation abstract we obtain necessary conditions and sufficient conditions on the existence of solutions to the cauchy problem for a fractional semilinear heat equation with an inhomogeneous term we identify the strongest spatial singularity of the inhomogeneous term for the solvability of the cauchy problem

free rotation of a circular ring with an unbalanced mass large space structures are much more flexible than their terrestrial counterparts rotation of large space structures may be desirable for stability thermal or artificial gravity reasons previous literature includes the large deformations due to the free rotation of a slender rod and a ring about a diameter an important model of a space station is a ring rotating about its axis of symmetry if the ring is balanced it will be stable and remain circular the present note considers the case when the ring is unbalanced by a mass attached to a point on the ring

the spectrum of the billiard laplacian of a family of random billiards random billiards are billiard dynamical systems for which the reflection law giving the post collision direction of a billiard particle as a function of the pre collision direction is specified by a markov scattering operator p billiards with microstructure are random billiards whose markov operator is derived from a microscopic surface structure on the boundary of the billiard table the microstructure in turn is defined in terms of what we call a billiard cellq the shape of which completely determines the operator p this operator defined on an appropriate hilbert space is bounded self adjoint and for the examples considered here a hilbert schmidt operator a central problem in the statistical theory of such random billiards is to relate the geometric characteristics of q and the spectrum of p we show for a particular family of billiard cell shapes parametrized by a scale invariant curvature k fig 2 that the billiard laplacian p i is closely related to the ordinary spherical laplacian and indicate by partly analytical and partly numerical means how this provides asymptotic information about the spectrum of p for small values of k it is shown in particular that the second moment of scattering about the incidence angle closely approximates the spectral gap of p

on steplength algorithms for a class of continuation methods the continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form fx b f d subset r n 1 to r n for given b in r n while these methods are similar in structure to those used for ode solvers their errors are independent of the history of the process and are solely determined by the termination criterion of the corrector at the current step this suggests the use of a posteriors estimates of the convergence radii of the corrector it is proved here that such estimates cannot be obtained from the sequence of corrector iterates alone but that they require some global information about f however it is shown that a finite sequence of corrector iterates does allow for the computation of effective estimates of the convergence quality of certain types of correctors this is used for the design of various step algorithms for continuation processes two of them are based on a newton corrector while the third o

a new start system for solving deficient polynomial systems using continuation we are interested in all finite solutions of polynomial systems the start system normally used in the homotopy causes divergent continuation paths when applied to a deficient system our aim is to avoid this wasted computation by constructing a new start system the start system presented in this paper respects the m homogeneous structure of the system we want to solve

an efficient start system for multi homogeneous polynomial continuation when solving a system of polynomial equations using homotopy continuation the number of solution paths can often be significantly reduced by casting the equations in multi homogeneous form this requires a multi homogeneous start system having a full set of nonsingular solutions that can be easily computed we give a general form of such a start system that works for any multi homogeneous structure and that can be used efficiently in continuation

limit theorems in the stadium billiard we prove that the birkhoff sums for almost every relevant observable in the stadium billiard obey a non standard limit law more precisely the usual central limit theorem holds for an observable if and only if its integral along a one codimensional invariant set vanishes otherwise a open image in new window normalization is needed as one of the two key steps in the argument we obtain a limit theorem that holds in young towers with exponential return time statistics in general an abstract result that seems to be applicable to many other situations

preconditioned iterative methods for sparse linear algebra problems arising in circuit simulation the dc operating point of a circuit may be computed by tracking the zero curve of an associated artificial parameter homotopy homotopy algorithms exist that are globally convergent with probability one for the dc operating point problem these algorithms require computing the one dimensional kernel of the jacobian matrix of the homotopy mapping at each step along the zero curve and hence the solution of a linear system of equations at each step these linear systems are typically large highly sparse non symmetric and indefinite several iterative methods which are applicable to such problems including craig s method gmres k bicgstab qmr kacz and lsqr are applied to a suite of test problems derived from simulations of actual bipolar circuits preconditioning techniques considered include incomplete lu factorization ilu sparse submatrix ilu and ilu allowing restricted fill in bands or blocks timings and convergence statistics are given for each iterative method and preconditioner

numerical study of porous channel flow in a rotating system by a homotopy method abstract problems in lubrication transpiration cooling dialysis gaseous diffusion processes subterranean rivers fed by porous bedrock and flow under the polar icecap are examples of porous channel flow in a rotating system using similarity equations the navier stokes equations reduce to a 10th order two point boundary value problem for certain parameter ranges this problem is very difficult and is not amenable to simple shooting techniques or classical locally convergent iterative methods a recently developed globally convergent homotopy algorithm is applied to the boundary value problem and numerical results are reported for a wide range of parameter values

billiards in a general domain with random reflections we study stochastic billiards on general tables a particle moves according to its constant velocity inside some domain fancyscript d subset mathbb r d until it hits the boundary and bounces randomly inside according to some reflection law we assume that the boundary of the domain is locally lipschitz and almost everywhere continuously differentiable the angle of the outgoing velocity with the inner normal vector has a specified absolutely continuous density we construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location velocity we mainly focus on the case of bounded domains then we prove exponential ergodicity of these two markov processes we study their invariant distribution and their normal gaussian fluctuations of particular interest is the case of the cosine reflection law the stationary distributions for the two processes are uniform in this case the discrete time chain is reversible though the continuous time process is quasi reversible also in this case we give a natural construction of a chord picked at random in fancyscript d and we study the angle of intersection of the process with a d 1 dimensional manifold contained in fancyscript d

the random product homotopy and deficient polynomial systems most systems ofn polynomial equations inn unknowns arising in applications aredeficient in the sense that they have fewer solutions than that predicted by the total degree of the system we introduce the random product homotopy an efficient homotopy continuation method for numerically determining all isolated solutions of deficient systems in many cases the amount of computation required to find all solutions can be made roughly proportional to the number of solutions

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

existence of solutions to the slow diffusion equation with a nonlinear source abstract in this paper we consider the existence of solutions to cauchy problem for the slow diffusion equation with a source term t u u m u p x r n t 0 u x 0 x 0 where m 1 and p m our purpose is to obtain lower estimates of the existence time of solutions in the uniformly local lebesgue spaces furthermore we obtain the estimate of blow up time blow up rate of solutions and blow up of critical norm

regeneration homotopies for solving systems of polynomials we present a new technique based on polynomial continuation for solving systems of n polynomials in n complex variables the method allows equations to be introduced one by one or in groups obtaining at each stage a representation of the solution set that can be extended to the next stage until finally obtaining the solution set for the entire system at any stage where positive dimensional solution components must be found they are sliced down to isolated points by the introduction of hyperplanes by moving these hyperplanes one may build up the solution set to an intermediate system in which a union of hyperplanes regenerates the intersection of the component with the variety of the polynomial or system of polynomials brought in at the next stage the theory underlying the approach guarantees that homotopy paths lead to all isolated solutions and this capability can be used to generate witness supersets for solution components at any dimension the first step in computing an irreducible decomposition of the solution set of a system of polynomial equations the method is illustrated on several challenging problems where it proves advantageous over both the polyhedral homotopy method and the diagonal equation by equation method formerly the two leading approaches to solving sparse polynomial systems by numerical continuation

numerical approximations for highly oscillatory bessel transforms and applications abstract this paper presents an efficient numerical method for approximating highly oscillatory bessel transforms based on analytic continuation we transform the integrals into the problems of integrating the forms on 0 with the integrand that does not oscillate and decays exponentially fast which can be efficiently computed by using gauss laguerre quadrature rule we then derive the error of the method depending on the frequency and the node number moreover we apply the scheme for studying the approximations of the solutions of two kinds of highly oscillatory integral equations preliminary numerical results show the efficiency and accuracy of numerical approximations

least change secant update methods for undetermined systems least change secant updates for nonsquare matrices have been addressed recently in 6 here the use of these updates in iterative procedures for the numerical solution of underde retrained systems is considered the model method is the normal flow algorithm used in homotopy or continuation methods for determining points on an implicitly defined curve a kantorovich type local convergence analysis is given which supports the use of least change secant updates in this al gorithm this analysis also provides a kantorovich type local convergence analysis for least change secant update methods in the usual case of an equal number of equations and unknowns this in turn gives a local convergence analysis for augmented jacobian algorithms which use least change secant updates in conclusion the results of some numerical experiments are given

canonical coordinates method for equality constrained nonlinear optimization feasible points methods have several appealing advantages over the infeasible points methods for solving equality constrained nonlinear optimization problems the known feasible points methods however solve often large systems of nonlinear constraint equations in each step in order to maintain feasibility solving nonlinear equations in each step not only slows down the algorithms considerably but also the large amount of floating point computation involved introduces considerable numerical inaccuracy into the overall computation as a result the commercial software packages for equality constrained optimization are slow and not numerically robust we present a radically new approach to maintaining feasibility called the canonical coordinates method ccm the ccm unlike previous methods does not adhere to the coordinate system used in the problem specification rather as the algorithm progresses ccm dynamically chooses in each step a coordinate system that is most appropriate for describing the local geometry around the current iterate by dynamically changing the coordinate system to suit the local geometry the ccm is able to maintain feasibility in equality constrained nonlinear optimization without having to solve systems of nonlinear equations we describe the ccm and present a proof of its convergence we also present a few numerical examples which show that ccm can solve in very few iterations problems that cannot be solved using the commercial nlp solver in matlab 6 1

spectral gap for a class of random billiards a random billiard is a random dynamical system similar to an ordinary billiard system except that the standard specular reflection law is replaced with a more general stochastic operator specifying the post collision distribution of velocities for any given pre collision velocity we consider such collision operators for certain random billiards that we call billiards with microstructure collisions modeled by these operators can still be thought of as elastic and time reversible the operators are canonically determined by a second deterministic billiard system that models microscopic roughness on the billiard table boundary our main purpose here is to develop some general tools for the analysis of the collision operator of such random billiards among the main results we give geometric conditions for these operators to be hilbert schmidt and relate their spectrum and speed of convergence to stationary markov chains with geometric features of the microscopic billiard structure the relationship between spectral gap and the shape of the microstructure is illustrated with several simple examples

bifurcations via singular value decompositions we study the relation between bifurcation theory and singular value decomposition in numerical analysis given a nonlinear equation we give a local analysis in a neighborhood of a solution via the liapunov schmidt method and the singular value decomposition the theorem is applicable to regular turning or bifurcation points a numerical method for global bifurcation problems based on the above analysis is presented

existence of solutions for a fractional semilinear parabolic equation with singular initial data abstract in this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the cauchy problem t u 2 u u p x r n t 0 u 0 0 in r n where n 1 0 2 p 1 and is a radon measure or a measurable function in r n our conditions lead optimal estimates of the life span of the solution with behaving like x a a 0 at the space infinity as 0

on knudsen flows within thin tubes the dynamics of a test particle in an infinite tube is investigated it is proven that the evolution tends to that of a diffusion process as the radius of the tube decreases to zero this justifies the hypotheses underlying an experiment of clausing 1930

a globally convergent method for finding zeros of smooth functions computing a zero of a smooth function is an old and extensively researched problem in numerical computation while a large body of results and algorithms has been reported on this problem in the literature to the extent we are aware the published literature does not contain a globally convergent algorithm for finding a zero of an arbitrary smooth function in this paper we present the first globally convergent algorithm for computing a zero if one exists of a general smooth function after presenting the algorithm and a proof of global convergence we also clarify the connection between our algorithm and some known results in topological degree theory

finding all solutions to a system of polynomial equations given a polynomial equation of degree d over the complex domain the fundamen tal theorem of algebra tells us that there are d solutions assuming that the solutions are counted by multiplicity these solutions can be approximated by deforming a standard n th degree equation into the given equation and following the solutions through the deformation this is called the homotopy method the fundamental theorem of algebra can be proved by the same technique in this paper we extend these results and methods to a system of n polynomial equations in n complex variables we show that the number of solutions to such a system is the product of the degrees of the equations assuming that infinite solutions are included and solutions are counted by multiplicity the proof is based on a homotopy or deformation from a standard system of equations with the same degrees and known solutions this homotopy provides a computational method of approximating all solutions computational results demonstrating the feasibility of this method are also presented

solving nonlinear systems of equations with only one nonlinear variable abstract we describe a method for solving systems of n 1 nonlinear equations in n 1 unknowns y te r and z te r n of the form a y z b y 0 where the n 1 n matrix a y and vector b y are functions of y alone such equations arise in minimax approximation we reduce the problem to one equation in y only an efficient quadratically convergent numerical technique based on newton s method in one variable is used to solve this equation computational details and results are provided and two generalizations are discussed

combinatorial optimization in nash games abstract games in which at least one player must solve a combinatorial optimization problem are studied after a suitable transformation some cases may be solved by consideration of a polynomial system of equations by finding all the solutions of this system one compuutes a all the nash equiblibria and the pre equilibrium set which may be of independent interest the question of a lack of equilibrium is also resolved for these games

numerical complexiton solutions for the complex kdv equation by the homotopy perturbation method in this paper the homotopy perturbation method is extended to investigate the numerical complexiton solutions of the complex kdv equation by constructing special forms of initial conditions three new types of realistic numerical solutions are obtained numerical positon solution expressed by the trigonometric functions numerical negaton solution expressed by the hyperbolic functions and particularly the numerical analytical complexiton solutions expressed by combinations of the two kinds of functions all these numerical solutions obtained can rapidly converge to the exact solutions obtained by lou et al illustrative numerical figures are exhibited the efficiency of the proposed method

real time optimization algorithm for nonlinear receding horizon control abstract a real time optimization technique is discussed concerning state feedback control of general nonlinear systems an optimal state feedback law is determined so that a receding horizon performance index with a moving terminal time is minimized it is shown that the receding horizon control problem can be converted to an initial value problem for an ordinary differential equation that can be solved numerically without recourse to iterative methods the proposed solution technique is applied to a control experiment on a simplified space vehicle model the penalty function approach is employed in the performance index so that the space vehicle model attains the objective state avoiding an obstacle the model is controlled successfully at the sampling frequency of 30 hz in the hardware experiment

a homotopy for solving polynomial systems this paper presents a continuation method for finding all solutions to polynomial systems it features a simpler homotopy than has been previously published

numerical solution of a class of deficient polynomial systems most systems of polynomials which arise in applications have fewer than the expected number of solutions the amount of computation required to find all solutions of such a deficient system using current homotopy continuation methods is proportional to the expected number of solutions and roughly to the size of the system much time is wasted following paths which do not lead to solutions we suggest methods for solving some deficient polynomial systems for which the amount of computational effort is instead proportional to the number of solutions

multiple scattering in random mechanical systems and diffusion approximation this paper is concerned with stochastic processes that model multiple or iterated scattering in classical mechanical systems of billiard type defined below from a given deterministic system of billiard type a random process with transition probabilities operator p is introduced by assuming that some of the dynamical variables are random with prescribed probability distributions of particular interest are systems with weak scattering which are associated to parametric families of operators ph depending on a geometric or mechanical parameter h that approaches the identity as h goes to 0 it is shown that ph i h converges for small h to a second order elliptic differential operator mathcal l on compactly supported functions and that the markov chain process associated to ph converges to a diffusion with infinitesimal generator mathcal l both ph and mathcal l are self adjoint densely defined on the space l 2 mathbb h eta of square integrable functions over the lower half space mathbb h in mathbb r m where is a stationary measure this measure s density is either post collision maxwell boltzmann distribution or knudsen cosine law and the random processes with infinitesimal generator mathcal l respectively correspond to what we call mb diffusion and generalized legendre diffusion concrete examples of simple mechanical systems are given and illustrated by numerically simulating the random processes

a methodology for solving chemical equilibrium systems this paper describes a methodology for solving chemical equilibrium systems the methodology is especially appropriate when many systems with the same structure must be solved quickly as in finite difference models of fluid flow and combustion the approach features identifying a canonical form for such systems and exploiting this canonical form to effect a preliminary algebraic reduction then numerical scaling and iterative solution techniques are evoked both homotopy continuation and a variant of newton s method are effective solution methods this methodology is a significant advance over previous approaches to solving chemical equilibrium systems the problem of solving such systems has heretofore been regarded as challenging solution techniques have tended to emphasize case by case analyses and hit or miss numerical methods we show that the problem can be solved by a unified approach quickly and reliably

global existence large time behavior and life span of solutions of a semilinear parabolic cauchy problem we investigate the behavior of the solution u x t of where 1 1 n 2 xi 2 is the laplace operator p 1 is a constant t 0 and is a nonnegative bounded continuous function in r n the main results are for the case when the initial value has polynomial decay near x assuming 1 x 75a with a 0 various questions of global in time existence and nonexistence large time behavior or life span of the solution u x t are answered in terms of simple conditions on a p and the space dimension n

singular parabolic equations with measures as initial data abstract in this paper we study the cauchy problem for the singular evolution p laplacian equations with gradient term and source on the assumption of measures as initial conditions for the supercritical case q p 1 p n we obtain that for every nonnegative solution there exists a nonnegative radon measure as initial trace and has some local regularity

a globally convergent parallel algorithm for zeros of polynomial systems certain classes of nonlinear systems of equations such as polynomial systems have properties that make them particularly amenable to solution on distributed computing systems some algorithms considered unfavorably on a single processor serial computer may be excellent on a distributed system this paper considers the solution of polynomial systems of equations via a globally convergent homotopy algorithm on a hypercube some computational results are reported

compact structures in a class of nonlinearly dispersive equations with time fractional derivatives in this work variants of the kdv equation with time fractional derivatives which exhibits compactons solitons with compact support are investigated new solitary wave solutions with compact support are developed the homotopy perturbation method is employed to derive compact solutions for these variants the chosen initial solution or trial function plays a major role in changing the physical structure of the solution the proposed scheme can be used to construct compact and noncompact solutions for a wide class of nonlinear fractional evolution equations

supersolutions for a class of nonlinear parabolic systems abstract in this paper by using scalar nonlinear parabolic equations we construct supersolutions for a class of nonlinear parabolic systems including t u u v p x t 0 t v v u q x t 0 u v 0 x t 0 u x 0 v x 0 u 0 x v 0 x x where p 0 q 0 is a possibly unbounded smooth domain in r n and both u 0 and v 0 are nonnegative and locally integrable functions in the supersolutions enable us to obtain optimal sufficient conditions for the existence of the solutions and optimal lower estimates of blow up rate of the solutions

on the cauchy problem for the evolution p laplacian equations with gradient term and source and measures as initial data abstract in this paper we study the cauchy problem for the evolution p laplacian equations with gradient term and source on the assumption that measures as initial conditions

errata computing all solutions to polynomial systems using homotopy continuation in a previous paper we described a new method for defining homotopies for finding all solutions to polynomial systems a major feature of this new approach is that the start system for the homotopy need not be a random or generic system also homotopy paths are strictly increasing in the homotopy parameter in this paper we establish some principles of implementation and report on the performance of programs that use the new homotopies a feature of our implementation is that we eliminate divergent paths entirely we include performance statistics for homotopies derived from more traditional approaches for comparison generally the new approach is faster and more reliable

a homotopy method applied to elastica problems abstract the inverse problem in nonlinear incompressible elastica theory where the end positions and inclinations rather than the forces and moment are specified is considered based on the globally convergent chow yorke algorithm a new homotopy method which is simple accurate stable and efficient is developed for comparison numerical results of some other simple approaches e g newton s method based on shooting or finite differences standard embedding are presented the new homotopy method does not require a good initial estimate and is guaranteed to have no singular points the homotopy method is applied to the problem of a circular elastica ring subjected to n symmetrical point loads and numerical results are given for n 2 3 4