markov chains of nonlinear markov processes and an application to a winner takes all model for social conformity we discuss nonlinear markov processes defined on discrete time points and discrete state spaces using markov chains in this context special attention is paid to the distinction between linear and nonlinear markov processes we illustrate that the chapman kolmogorov equation holds for nonlinear markov processes by a winner takes all model for social conformity

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

a product decomposition bound for bezout numbers most polynomial systems that arise in practice are not completely general but have special structures a common form is that each equation must be a sum of products where each factor has an identifiable generic type a theorem is proven for such systems which offers a method for obtaining a tighter upper bound on the number of nonsingular solutions than is generally available at the same time this theorem provides an approach for solving such systems via polynomial continuation which results in less computational work to illustrate the practical usefulness of these ideas we show that a significant design of mechanisms problem can be solved with an order of magnitude less work than the published solution

homotopy techniques for tensor decomposition and perfect identifiability let t be a general complex tensor of format n1 nd when the fraction i ni 1 i ni 1 is an integer and a natural inequality called balancedness is satisfied it is expected that t has finitely many minimal decomposition as a sum of decomposable tensors we show how homotopy techniques allow us to find all the decompositions of t starting from a given one in particular this gives a guess true with high probability about the total number of these decompositions this guess matches exactly with all cases previously known and predicts several unknown cases some surprising experiments yielded two new cases of generic identifiability formats 3 4 5 and 2 2 2 3 which have a unique decomposition as the sum of 6 and 4 decomposable tensors respectively we conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds i e the only identifiable cases building on the computational experiments we also use basic tools from algebraic geometry to prove these two new cases are indeed generically identifiable

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

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

on real parameter stability margins and their computation a stability margin for linear control systems subject to real parameter variations can be computed from the constraints arising from the conventional routh hurwitz criteria the proposed stability margin is similar to the real criterion but with the bound on the real parameter variations given by a differentiable function if the uncertain parameters are gaussian random variates then an upper bound on the probability of instability is constructed a computational procedure is proposed based on polynomial continuation

split 3 lie algebras in order to begin an approach to the structure of 3 lie algebras with restrictions neither on the dimension nor on the base field we introduce the class of split 3 lie algebras as the natural extension of the class of split lie algebras by developing techniques of connections of roots for this kind of ternary algebras we show that any of such split 3 lie algebras t is of the form t u jij with u a subspace of the 0 root space t0 and any ij a well described ideal of t satisfying ij t ik 0 if j k under certain conditions the simplicity of t is characterized and it is shown that t is the direct sum of the family of its minimal ideals each one being a simple split 3 lie algebra

3 lie algebras with an ideal n we define the hypo nilpotent ideal in n lie algebras and obtain all solvable 3 lie algebras with an m dimensional simplest filiform 3 lie algebra as a maximal hypo nilpotent ideal we prove that the dimension of such solvable 3 lie algebras is at most m 2 and there is no solvable 3 lie algebra with the simplest filiform 3 lie algebra as the nilradical

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 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

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

multivariate ornstein uhlenbeck processes with mean field dependent coefficients application to postural sway we study the transient and stationary behavior of many particle systems in terms of multivariate ornstein uhlenbeck processes with friction and diffusion coefficients that depend nonlinearly on process mean fields mean field approximations of this kind of system are derived in terms of fokker planck equations in such systems multiple stationary solutions as well as bifurcations of stationary solutions may occur in addition strictly monotonically decreasing steady state autocorrelation functions that decay faster than exponential functions are found which are used to describe the erratic motion of the center of pressure during quiet standing

on the solutions to polynomial systems obtained by homotopy methods a natural class of homotopy methods for solving polynomial systems is considered it is shown that at least one solution from each connected component of the solution set is obtained this generalizes the results of previous papers which concentrated on isolated solutions i e connected components with one single point the number of solution paths ending in a connected component is independent of the particular homotopy in use and defines in a natural way the multiplicity of the connected component a few numerical experiments illustrate the obtained results

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

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

centroid structures of n lie algebras abstract we concern the centroid l of an n lie algebra l over a field f of characteristic p 0 let c l be the central derivations of l we obtain the following results 1 if p is not a factor of n 1 then c l is the intersection of l and the derivation algebra of l 2 let b be a nonzero ideal of l and invariant under l then the vanishing ideal of b is isomorphic to a subspace of hom l b z l b where z l b is the centralizer of b 3 suppose l l 1 l 2 with l 1 l 2 ideals of l then l 1 and l 2 are components of l 4 if l is a heisenberg n lie algebra over an algebraically closed field of characteristic 0 then l is generated by central derivations and scalars and c l is made up of all the inner derivations of l 5 if dim l 2 and l consists of scalars then the centroid of the tensor product of an associative algebra and l is the same as the tensor product of the centroids of the two algebras

algebraic methods for image processing and computer vision many important problems in image processing and computer vision can be formulated as the solution of a system of simultaneous polynomial equations crucial issues include the uniqueness of solution and the number of solutions if not unique and how to find numerically all the solutions the goal of this paper is to introduce to engineers and scientists some mathematical tools from algebraic geometry which are very useful in resolving these issues three dimensional motion structure estimation is used as the context however these tools should also be helpful in other areas including surface intersection in computer aided design and inverse position problems in kinematics robotics the tools to be described are bezout numbers grobner bases homotopy methods and a powerful theorem which states that under rather general conditions one can draw general conclusions on the number of solutions of a polynomial system from a single numerical example

stochastic resonance in a mean field model of cooperative behavior we study the long time response of a stochastic system formed by very many interacting subsystems coupled by a mean field interaction and subject to a time periodic external field in the absence of a driving field the system shows an order disorder phase transition and its time evolution is well described by a fokker planck equation which is nonlinear in the probability density we carry out an analysis of the dynamics in the case of a weak driving field by means of a perturbation analysis linear response theory the response of the system is then given in terms of a generalized susceptibility its evaluation shows that the phenomenon of stochastic resonance typical of driven bistable systems is greatly enhanced by the dynamical feedback induced by mean field coupling

role of fluctuations in the response of coupled bistable units to weak time periodic driving forces we analyze the stochastic response of a finite set of globally coupled noisy bistable units driven by rather weak time periodic forces we focus on the stochastic resonance and phase frequency synchronization of the collective variable defined as the arithmetic mean of the variable characterizing each element of the array for single unit systems stochastic resonance can be understood with the powerful tools of linear response theory proper noise induced phase frequency synchronization for a single unit system in this linear response regime does not exist for coupled arrays our numerical simulations indicate an enhancement of the stochastic resonance effects leading to gains larger than unity as well as genuine phase frequency synchronization the nonmonotonicity of the response with the strength of the coupling strength is investigated comparison with simplifying schemes proposed in the literature to describe the random response of the collective variable is carried out

computing dynamic output feedback laws the pole placement problem asks to find laws to feed the output of a plant governed by a linear system of differential equations back to the input of the plant so that the resulting closed loop system has a desired set of eigenvalues converting this problem into a question of enumerative geometry efficient numerical homotopy algorithms to solve this problem for general multiple input multiple output systems have been proposed recently despite the wider application range of dynamic feedback laws the realization of the output of the numerical homotopies as a machine to control the plant in the time domain has not been addressed before in this note we present symbolic numeric algorithms to turn the solution to the question of enumerative geometry into a useful control feedback machine we report on numerical experiments with our publicly available software phcpack and illustrate its application on various control problems from the literature

the classification of six dimensional 4 lie algebras this paper concerns properties of n 2 dimensional n lie algebras and shows the classification of six dimensional 4 lie algebras over an algebraically closed field of characteristic zero

synchronization induced by periodic inputs in finite n unit bistable langevin models the augmented moment method we have studied the synchronization induced by periodic inputs applied to a finite n unit coupled bistable langevin model which is subjected to cross correlated additive and multiplicative noises the effects on the synchronization of the system size n the coupling strength and the cross correlation between additive and multiplicative noise have been investigated using the semi analytical augmented moment method amm which is a second order moment approximation for local and global variables h hasegawa phys rev e 67 2003 041903 a linear analysis of the stationary solution of the amm equations shows that the stability is improved degraded by positive negative couplings results of the nonlinear bistable langevin model are compared to those of the linear langevin model

enhancement of neuronal coherence by diversity in coupled rulkov map models abstract we study the influence of diversity on the dynamics of spiking in spatially extended systems with global or diffusive couplings which is locally modeled by a map based neuron the variance of the system s parameter d is used to measure the diversity it is found that the regularity of spikes reaches a maximum with a variation in d indicating the occurrence of diversity induced coherence resonance moreover diversity induced coherence resonance in this system can be greatly enhanced by increasing the coupling strength our analysis shows that these phenomena have inherent relevance to both the bifurcation feature of a single neuron and interaction through couplings these results may exhibit that diversity plays a constructive role in enhancing the time precision of firing in a neural network

a homotopy method for eigenvalue assignment using decentralized state feedback this paper addresses the following problem given an interconnected system m composed of n subsystems of the form a i b i k i i 1 n a i b i a controllable pair and where the off diagonal blocks of m lie in the image of the appropriate b i then is it possible to arbitrarily assign the characteristic polynomial of m by a suitable selection of the characteristic polynomials of a i b i k i moreover is it possible to compute the appropriate characteristic polynomials of the a i b i k i or equivalently construct the k i needed to do so the first question is answered by constructing a mapping f r n rightarrow r n which maps a prescribed set of n of the feedback gains elements of k i i 1 n to the n coefficients of the characteristic polynomial of m the question then becomes given a p in r n does f x p have a solution the answer is found by constructing a homotopy h r n x o 1 rightarrow r n where h x 1 f x and h x 0 is some trivial function degree theory is then applied to guarantee that there exists an x t such that h x t t p for all t in 0 1 the parameterized sard s theorem is then utilized to prove that with probability 1 x t is a smooth curve and hence can be followed numerically from x 0 to x 1 by the solution of a differential equation davidenko s method

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

membership tests for images of algebraic sets by linear projections given a witness set for an irreducible variety v and a linear map p we describe membership tests for both the constructible algebraic set p v and the algebraic set p v we also provide applications and examples of these new tests including computing the codimension one components of p v p v additionally we also describe computing the geometric genus of a curve section of an irreducible component of the solution set of a polynomial system and a test for deciding whether a plane quartic curve is a luroth quartic

determining articulated motion from perspective views a decomposition approach abstract we consider the problem of estimating the 3d motion of an articulated object such as a robot arm or a human body from a monocular sequence of 2d perspective views we advocatte an approach of decomposition the object under analysis is decomposed into simpler parts each containing a small number of links we first estimate the motion of the simplest part s and then propagate the analysis to the remaining parts of the object human gait is used as an example however the approach is general to use this decomposition approach we need a repertoire of results for the motion of simple articulated objects motion estimation algorithms uniqueness and number of solutions especially how many views are needed for uniqueness with the help of techniques in algebraic geometry we have results for a number of cases which are particularly useful for human gait analysis

stationary configurations of point vortices the motion of point vortices in a plane of fluid is an old problem of fluid mechanics which was given a hamiltonian formulation by kirchhoff stationary configurations are those which remain self similar throughout the motion results of two types are presented configurations which are in equilibrium or which translate uniformly are counted using methods of algebraic geometry which establish necessary and sufficient conditions for existence relative equilibria rigidly rotating configurations which lie on a line are studied using a topological construction applicable to other power law systems upper and lower bounds for such configurations are found for vortices with mixed circulations arrangements of three vortices which collide in finite time are well known one dimensional families of such configurations are shown to exist for more than three vortices stationary configurations of four vortices are examined in detail the motion of point vortices in a plane of fluid is an old problem of fluid mechanics which was given a hamiltonian formulation by kirchhoff stationary configurations are those which remain self similar throughout the motion and are of considerable physical interest it is known that a configuration of vortices in equilibrium must have total vortex angular momentum 0 a converse is proved namely that for almost every choice of circulations with zero vortex angular momentum there are exactly n 2 equilibrium configurations a similar statement is proved for rigidly translating configurations with total circulation zero the proofs involve ideas from algebraic geometry relative equilibria rigidly rotating configurations were studied by palmore in the case of positive circulations upper and lower bounds for the number of collinear relative equilibria for arbitrary circulations are obtained by means of a topological construction which is applicable to other power law systems arrangements of three vortices which collide in finite time are well known one dimensional families of such configurations are shown to exist for n vortices n 3 stationary configurations of four vortices are examined in detail chapter 0 introduction the motion of point vortices in the plane is an old problem in fluid mechanics it was first given a hamiltonian formulation by kirchhoff 1876 who proved that n vortices with distinct positions xi yi i 1 n in the x y plane and received by the editors july 31 1985 and in revised form june 13 1986 1980 mathematics subject classification 1985 revision primary 76c05 secondary 70h05 t1987 american mathematical society 0002 9947 87 1 00 25 per page

evaluation of stochastic resonance based detectors of weak harmonic signals in additive white gaussian noise a thorough evaluation of stochastic resonance in the framework of statistical detection theory is presented both as a nonlinear signal preprocessor and as a detector the pertinent receiver operating characteristics are compared with those of the known statistically optimum detector using extensive monte carlo simulations parameter optimization and computational budget aspects are discussed

a langevin approach for the microscopic dynamics of nonlinear fokker planck equations abstract for stochastic processes defined by nonlinear fokker planck equations a microscopic dynamics is proposed in terms of generalized langevin equations these langevin equations can for example be used to model stochastic processes with mean field interactions and random walks related to the generalized thermostatistics suggested by tsallis

coefficient parameter polynomial continuation we establish the theory for a very general approach to constructing homotopies for computing all geometrically isolated solutions to polynomial systems we assume that the coefficients for a polynomial system are given by parameters in engineering applications these would be derived from physical parameters since the coefficients themselves can be the parameters this is completely general we show that by considering the space of parameters instead of the space of coefficients we can exploit any special structure the coefficient parameter formulas impose on the solution set in particular we prove that we can ignore generic positive dimensional solution sets we also show that any side conditions additional equations can be used to reduce the number of paths these results unify and generalize all current approaches to polynimial continuation the effect of these new results is to greatly increase the efficiency of polynomial continuation in several engineering areas for example we establish a 36 path homotopy for a electrocardiographic application that needed 193 paths under the previous most appropriate theory also we show that a 16 path homotopy suffices for the inverse kinematics problem for 6 revolute joint manipulators a problem previously requiring 32 paths

nonlinear homotopies for solving deficient polynomial systems with parameters by a deficient polynomial system of n polynomial equation in n unknowns a system is meant that has fewer solutions than that predicted by the total degree or the bezout number of the system in practice a deficient polynomial system is often associated with a set of parameters q q 1 cdots q s in this paper nonlinear homotopies for numerically determining all isolated solutions of deficient m homogeneous systems are introduced the procedure is that when a system p q x 0 with particular parameter q 0 is solved it can be used to solve p q x 0 for any parameter q it is assumed that for a generic q p q x shares the same kind of deficiency at infinity thus far fewer paths need to be followed as an application it is shown how to compute the 32 solutions of the tsai morgan manipulator problem with 26 parameters by following only 32 solution paths

the continuation method for rational and deficient systems deficient polynomial systems those with fewer than the full number of solutions cannot be solved efficiently with the usual continuation methods this paper describes a continuation technique which allows some deficient and rational systems of equations to be solved with little wasted computation several examples from the literature are discussed

classification of n 3 dimensional metric n lie algebras in this paper we focus on n 3 dimensional metric n lie algebras to begin with we give some properties on n 3 dimensional n lie algebras then based on the properties we obtain the classification of n 3 dimensional metric n lie algebras

perturbed homotopies for finding all isolated solutions of polynomial systems abstract given a polynomial system f c n c n the methods of numerical algebraic geometry produce numerical approximations of the isolated solutions of f z 0 as well as points on any positive dimensional components of the solution set v f some of these methods are guaranteed to find all isolated solutions nonsingular and singular alike while others may not find singular solutions one of the most recent advances in this field is regeneration an equation by equation solver that is often more efficient than other methods however the basic form of regeneration will not necessarily find all isolated singular solutions of a polynomial system without the additional cost of using deflation the aim of this article is two fold first more generally we consider the use of perturbed homotopies for solving polynomial systems in particular we propose first solving a perturbed version of the polynomial system followed by a parameter homotopy to remove the perturbation such perturbed homotopies are sometimes more efficient than regular homotopies though they can also be less efficient second a useful consequence is that the application of this perturbation to regeneration will yield all isolated solutions including all singular isolated solutions this version of regeneration perturbed regeneration can decrease the efficiency of regeneration but increases its applicability

adaptive multiprecision path tracking this article treats numerical methods for tracking an implicitly defined path the numerical precision required to successfully track such a path is difficult to predict a priori and indeed it may change dramatically through the course of the path in current practice one must either choose a conservatively large numerical precision at the outset or rerun paths multiple times in successively higher precision until success is achieved to avoid unnecessary computational cost it would be preferable to adaptively adjust the precision as the tracking proceeds in response to the local conditioning of the path we present an algorithm that can be set to either reactively adjust precision in response to step failure or proactively set the precision using error estimates we then test the relative merits of reactive and proactive adaptation on several examples arising as homotopies for solving systems of polynomial equations

algebraic c actions and the inverse kinematics of a general 6r manipulator abstract let x be a smooth quadric of dimension 2 m in p c 2 m 1 and let y z x be subvarieties both of dimension m which intersect transversely in this paper we give an algorithm for computing the intersection points of y z based on a homotopy method the homotopy is constructed using a c action on x whose fixed points are isolated which induces bialynicki birula decompositions of x into locally closed invariant subsets as an application we present a new solution to the inverse kinematics problem of a general six revolute serial link manipulator

stochastic resonance of collective variables in finite sets of interacting identical subsystems we explore stochastic resonance effects in the response of a complex stochastic system formed by a finite number of interacting identical subunits driven by a time periodic force the driving force alone cannot induce sustained oscillations between the different attractors of the dynamics in the absence of noise we focus on a global stochastic variable defined as the arithmetic mean of the relevant stochastic variable of each subunit we construct numerical approximations to its first two long time cumulant moments and its long time correlation function we also compute the output signal to noise ratio and the stochastic resonance gain for a wide range of parameter values and several types of driving forces the coupling between the subsystems leads within adequate ranges of the parameter values to global outputs with very large signal to noise ratios we have also observed gains larger than unity in the global response to subthreshold sinusoidal driving forces

folds in solutions of two parameter systems and their calculation part i this paper is concerned with paths of turning points in solutions of nonlinear systems having two parameters it is well known that these paths are solutions of a particular extended system of nonlinear equations in this paper both regular points and simple turning points in the extended system are related to the local geometry of the solution surface of the original nonlinear system a description is given of numerical methods both for solving the extended system and for calculating certain quantities which determine the local geometry of the solution surface applications to perturbed bifurcation to the formation of isolas and to the calculation of the multiplicity of solutions are also discussed numerical examples are given

green functions and langevin equations for nonlinear diffusion equations a comment on markov processes hurst exponents and nonlinear diffusion equations by bassler et al abstract we discuss two central claims made in the study by bassler et al k e bassler g h gunaratne j l mccauley physica a 369 2006 343 bassler et al claimed that green functions and langevin equations cannot be defined for nonlinear diffusion equations in addition they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones we review bottom up and top down approaches that have been used in the literature to derive green functions for nonlinear diffusion equations and in doing so show that the first claim needs to be revised we show that the second claim as well needs to be revised to this end we point out similarities and differences between non autonomous linear fokker planck equations and autonomous nonlinear fokker planck equations in this context we raise the question whether bassler et al s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes such external entities can easily be eliminated when taking self organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear fokker planck equations

complete stagnation of gmres we study problems for which the iterative method gmres for solving linear systems of equations makes no progress in its initial iterations our tool for analysis is a nonlinear system of equations the stagnation system that characterizes this behavior we focus on complete stagnation for which there is no progress until the last iteration we give necessary and sufficient conditions for complete stagnation of systems involving unitary matrices and show that if a normal matrix completely stagnates then so does an entire family of nonnormal matrices with the same eigenvalues finally we show that there are real matrices for which complete stagnation occurs for certain complex right hand sides but not for real ones 2003 elsevier science inc all rights reserved

structure of low dimensional n lie algebras over a field of characteristic 2 in this paper we classify n 1 dimensional n lie algebras over a field f of characteristic 2 and prove that there are no simple n 2 dimensional n lie algebras

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

uniqueness of solutions to three perspective views of four points we show that there is in general a unique solution for the relative orientation of three cameras simultaneously photographing four feature points on a fixed object however multiple solutions are possible in rare cases even when the four feature points are not coplanar

incomplete grobner basis as a preconditioner for polynomial systems precondition plays a critical role in the numerical methods for large and sparse linear systems it is also true for nonlinear algebraic systems in this paper incomplete grobner basis igb is proposed as a preconditioner of homotopy methods for polynomial systems of equations which transforms a deficient system into a system with the same finite solutions but smaller degree the reduced system can thus be solved faster numerical results show the efficiency of the preconditioner

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