numerical calculation of the multiplicity of a solution to algebraic equations a method to calculate numerically the multiplicity of solution to a system of algebraic equations is presented the method is an application of zeuthen s rule which gives the multiplicity of a solution as the multiplicity of a united point of an algebraic correspondence defined naturally by the system the numerical calculation is applicable to a large scale system of algebraic equations whicl may have a solution that we cannot calculate the multiplicity by a symbolic computation

on the stability of periodic solutions of the generalized benjamin bona mahony equation abstract we study the stability of a four parameter family of spatially periodic traveling wave solutions of the generalized benjamin bona mahony equation under two classes of perturbations periodic perturbations with the same periodic structure as the underlying wave and long wavelength localized perturbations in particular we derive necessary conditions for spectral instability under perturbation for both classes of perturbations by deriving appropriate asymptotic expansions of the periodic evans function and we outline a theory of nonlinear stability under periodic perturbations based on variational methods which effectively extends our periodic spectral stability results

computing isolated singular solutions of polynomial systems case of breadth one we present a symbolic numeric method to refine an approximate isolated singular solution hat mathbf x hat x 1 ldots hat x n of a polynomial system f f 1 ldots f n when the jacobian matrix of f evaluated at hat mathbf x has corank one approximately our new approach is based on the regularized newton iteration and the computation of differential conditions satisfied at the approximate singular solution the size of matrices involved in our algorithm is bounded by n times n the algorithm converges quadratically if hat mathbf x is close to the isolated exact singular solution

simple lie algebras of small characteristic ii exceptional roots abstract letlbe a finite dimensional simple lie algebra of absolute toral rank 2 over an algebraically closed field of characteristicp 3 andta 2 dimensional torus in the semisimplep envelope ofl suppose thatlis not isomorphic to a melikian algebra it is proved in this paper that for every root l t the subalgebrak generated by i f pki whereki x li x l i 0 acts triangulably onl in particular this implies that in the terminology of r e block and r l wilson 1988 j algebra114 115 259 all roots of l t are nonexceptional

simple lie algebras of small characteristic vi completion of the classification let l be a finite dimensional simple lie algebra over an algebraically closed field of characteristic p 3 it is proved in this paper that if the p envelope of adl in derl contains a torus of maximal dimension whose centralizer in adl acts nontriangulably on l then p 5 and l is isomorphic to one of the melikian algebras m m n in conjunction with a premet h strade simple lie algebras of small characteristic v the non melikian case j algebra 314 2007 664 692 theorem 1 2 this implies that up to isomorphism any finite dimensional simple lie algebra over an algebraically closed field of characteristic p 3 is either classical or a filtered lie algebra of cartan type or a melikian algebra of characteristic 5 this result finally settles the classification problem for finite dimensional simple lie algebras over algebraically closed fields of characteristic 2 3

simple lie algebras of small characteristic iv solvable and classical roots abstract let l be a finite dimensional simple lie algebra over an algebraically closed field f of characteristic p 3 and t a torus of maximal dimension in the p envelope of l in der l in this paper we describe the t semisimple quotients of the 2 sections of l relative to t and prove that if all 1 sections of l relative to t are compositionally classical or solvable then l is either classical or a block algebra or a filtered lie algebra of type s

determining dimension of the solution component that contains a computed zero of a polynomial system abstract when the jacobian of a computed numerical solution of a polynomial system in c n allows very small singular values the solution could be isolated with a multiple multiplicity or may belong to a solution component with positive dimension the algorithm constructed in this article intends to differentiate those cases by determining the dimension of the solution component m in which the solution lies of particular interest is the case when dim m 0 then the solution is of course isolated while the proposed algorithm is experimental it has been tested successfully on the class of problems with the solution in question belonging to a reduced component numerical results are provided to illustrate the accuracy of the algorithm

multidimensional realisation theory and polynomial system solving abstractmultidimensional systems are becoming increasingly important as they provide a promising tool for estimation simulation and control while going beyond the traditional setting of one dimen

multiple zeros of nonlinear systems as an attempt to bridge between numerical analysis and algebraic geometry this paper formulates the multiplicity for the general nonlinear system at an isolated zero presents an algorithm for computing the multiplicity strucuture proposes a depth deflation method for accurate computation of multiple zeros and introduces the basic algebraic theory of the multiplicity furthermore this paper elaborates and proves some fundamental theorems of the multiplicity including local finiteness consistency perturbation invarance and depth deflatability the proposed algorithms can accurately compute the multiplicity and the multiple zeros using floating point arithmetic even if the nonlinear system is perturbed

on multiplicities in polynomial system solving this paper deals with the description of the solutions of zero dimensional systems of polynomial equations based on different models for describing solutions we consider suitable representations of a multiple root or more precisely suitable descriptions of the primary component of the system at a root we analyse the complexity of finding the representations and of algorithms which perform transformations between the different representations

nonlinear stability of periodic traveling wave solutions to the schrodinger and the modified korteweg de vries equations abstract this work is concerned with stability properties of periodic traveling waves solutions of the focusing schrodinger equation i u t u x x u 2 u 0 posed in r and the modified korteweg de vries equation u t 3 u 2 u x u x x x 0 posed in r our principal goal in this paper is the study of positive periodic wave solutions of the equation 3 0 called dnoidal waves a proof of the existence of a smooth curve of solutions with a fixed fundamental period l 2 2 l 2 h per 0 l is given it is also shown that these solutions are nonlinearly stable in the energy space h per 1 0 l and unstable by perturbations with period 2l in the case of the schrodinger equation

sufficient conditions for orbital stability of periodic traveling waves abstract the present paper deals with sufficient conditions for orbital stability of periodic waves of a general class of evolution equations supporting nonlinear dispersive waves our method can be seen as an extension to spatially periodic waves of the theory of solitary waves recently developed in 25 firstly our main result do not depend on the parametrization of the periodic wave itself secondly motived by the well known orbital stability criterion for solitary waves we show that the same criterion holds for periodic waves in addition we show that the positiveness of the principal entries of the hessian matrix related to the energy surface function are also sufficient to obtain the stability consequently we establish the orbital stability of periodic waves for several nonlinear dispersive models we believe our method can be applied in a wide class of evolution equations in particular it can be extended to regularized dispersive wave equations

verified error bounds for isolated singular solutions of polynomial systems case of breadth one in this paper we describe how to improve the performance of the symbolic numeric method in 19 20 for computing the multiplicity structure and refining approximate isolated singular solutions in the breadth one case by introducing a parameterized deflated system with smoothing parameters we generalize the algorithm in 33 to compute verified error bounds such that a slightly perturbed polynomial system is guaranteed to have a breadth one multiple root within the computed bounds

newton s method with deflation for isolated singularities of polynomial systems we present a modification of newton s method to restore quadratic convergence for isolated singular solutions of polynomial systems our method is symbolic numeric we produce a new polynomial system which has the original multiple solution as a regular root using standard bases a tool for the symbolic computation of multiplicities we show that the number of deflation stages is bounded by the multiplicity of the isolated root our implementation performs well on a large class of applications

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

modified deflation algorithm for the solution of singular problems i a system of nonlinear algebraic equations abstract the behavior of the newton raphson method at the singular roots has been studied by a number of authors and the convergence of the newton raphson sequence has been shown to be linear in this paper a new method with symbolic and numerical manipulations termed the modified deflation algorithm is proposed for the singular root of a system of nonlinear algebraic equations the basic idea of the present method is to replace a part of the original equations by a set of new equations which pass through the singular root according to the method both convergency and accuracy can greatly be improved in addition it is often possible to obtain analytically the singular root from the new equations in order to show the effectiveness of the present method two illustrative examples are solved

a numerical local dimension test for points on the solution set of a system of polynomial equations the solution set v of a polynomial system i e the set of common zeroes of a set of multivariate polynomials with complex coefficients may contain several components e g points curves surfaces etc each component has attached to it a number of quantities one of which is its dimension given a numerical approximation to a point mathbf p on the set v this article presents an efficient algorithm to compute the maximum dimension of the irreducible components of v which pass through mathbf p i e a local dimension test such a test is a crucial element in the homotopy based numerical irreducible decomposition algorithms of sommese verschelde and wampler this article presents computational evidence to illustrate that the use of this new algorithm greatly reduces the cost of so called junk point filtering previously a significant bottleneck in the computation of a numerical irreducible decomposition for moderate size examples this results in well over an order of magnitude improvement in the computation of a numerical irreducible decomposition as the computation of a numerical irreducible decomposition is a fundamental backbone operation gains in efficiency in the irreducible decomposition algorithm carry over to the many computations which require this decomposition as an initial step another feature of a local dimension test is that one can now compute the irreducible components in a prescribed dimension without first computing the numerical irreducible decomposition of all higher dimensions for example one may compute the isolated solutions of a polynomial system without having to carry out the full numerical irreducible decomposition

a new deflation method for verifying the isolated singular zeros of polynomial systems abstract in this paper we develop a new deflation technique for refining or verifying the isolated singular zeros of polynomial systems starting from a polynomial system with an isolated singular zero by computing the derivatives of the input polynomials directly or the linear combinations of the related polynomials we construct a new system which can be used to refine or verify the isolated singular zero of the input system in order to preserve the accuracy in numerical computation as much as possible new variables are introduced to represent the coefficients of the linear combinations of the related polynomials to our knowledge it is the first time that considering the deflation problem of polynomial systems from the perspective of linear combinations some acceleration strategies are proposed to reduce the scale of the final system we also give some further analysis of the tolerances we use which can help us have a better understanding of our method the experiments show that our method is effective and efficient especially it works well for zeros with high multiplicities of large systems it also works for isolated singular zeros of non polynomial systems

classification of the restricted simple lie algebras soit l une algebre de lie simple restreinte de dimension finie sur un corps algebriquement clos f de caracteristique p 7 alors l est de type classique ou de cartan

tori in the melikian algebra this paper is concerned with the tori in the restricted melikian algebra g g 1 1 according to the latest classification result of premet and strade 16 g is the unique simple lie algebra of toral rank 2 over an algebraically closed field of characteristic p 5 distinct from the classical algebras and algebras of cartan type here we determine the one dimensional subtori in g up to conjugation by automorphisms and compute their centralizers in g one consequence of this investigation is that all maximal tori in g are two dimensional for each two dimensional torus we find the number of its roots of any possible type classical witt hamiltonian proper the answer depends only on the position of the torus relative to the invariant filtration of g the conjugacy classes of two dimensional tori are not determined in this paper it can be shown that their number is infinite the classification of simple lie algebras requires detailed information about the toral structure of low rank lie algebras see e g 3 19 one of the key points is the possibility of finding a torus of maximal dimension in the p envelope of a simple lie algebra l having all roots proper the classical argument is based on checking that when switching an improper root to a proper one the number of proper roots in every 2 section of l increases in the case of the melikian algebra we encounter a situation when the number of proper roots remains unchanged when this does occur we show that the number of hamiltonian roots becomes strictly less

contributions to the classification of simple modular lie algebras we develop results directed towards the problem of classifying the finite dimensional simple lie algebras over an algebraically closed field of characteristic p 7 a 1 section of such a lie algebra relative to a torus t of maximal absolute toral rank possesses a unique subalgebra maximal with respect to having a composition series with factors which are abelian or classical simple in this paper we show that the sum q of those compositionally classical subalgebras is a subalgebra this extends to the general case a crucial step in the classification by block and wilson of the restricted simple lie algebras we derive properties of the filtration which can be constructed using q and obtain structural information about the 1 sections and 2 sections of q relative to t we further classify all those algebras in which q is solvable

nonlocal models for nonlinear dispersive waves abstract considered herein are model equations for the unidirectional propagation of small amplitude nonlinear dispersive long waves such as those governed by the classical korteweg de vries equation of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial so that the operator modelling dispersive effect is nonlocal particular cases in view here are the benjamin ono equation and the intermediate long wave equation which arise in internal wave theory and smith s equation which governs certain types of continental shelf waves the initial value problem for these equations is shown to be globally well posed in the classical sense including continuous dependence upon the initial data and in certain cases upon the modelling of nonlinear and dispersive effects whilst the results are stated for the specific equations listed above the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear dispersive evolution equations particularly worthy of note is our theorem implying that solutions of the intermediate long wave equation converge strongly to solutions of the korteweg de vries equation or to solutions of the benjamin ono equation in appropriate asymptotic limits

a comment about the paper on the instability of elliptic traveling wave solutions of the modified camassa holm equation abstract we discuss the recent paper by a daros and l k arruda 2019 6 our intention is to correct some imperfections left by the authors and present the orbital stability of periodic snoidal waves in the energy space

the classification of the simple modular lie algebras ii the toral structure abstract let l be a simple lie algebra over an algebraically closed field of characteristic p 7 and t an optimal torus in some p envelope l p we determine the action of t on the two sections of l which have been described in st4 we also give some new and noncomputational proofs to determine the conjugacy classes of the tori in w n 1 and of the cartan subalgebras of w 1 n

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

computing multiple roots of inexact polynomials we present a combination of two algorithms that accurately calculate multiple roots of general polynomials algorithm i transforms the singular root finding into a regular nonlinear least squares problem on a pejorative manifold and it calculates multiple roots simultaneously from a given multiplicity structure and initial root approximations to fulfill the input requirement of algorithm i we develop a numerical gcd finder containing a successive singular value updating and an iterative gcd refinement as the main engine of algorithm ii that calculates the multiplicity structure and the initial root approximation while limitations exist in certain situations the combined method calculates multiple roots with high accuracy and consistency in practice without using multiprecision arithmetic even if the coefficients are inexact this is perhaps the first blackbox type root finder with such capabilities to measure the sensitivity of the multiple roots a structure preserving condition number is proposed and error bounds are established according to our computational experiments and error analysis a polynomial being ill conditioned in the conventional sense can be well conditioned with the multiplicity structure being preserved and its multiple roots can be computed with high accuracy

verified error bounds for isolated singular solutions of polynomial systems in this paper we generalize the algorithm described by rump and graillat as well as our previous work on verifying breadth one singular solutions of polynomial systems to compute verified and narrow error bounds such that a slightly perturbed system is guaranteed to possess an isolated singular solution within computed error bounds our verification method is based on deflation techniques using smoothing parameters we demonstrate the performance of the algorithm for polynomial systems with singular solutions of multiplicity up to hundreds

an index theorem for the stability of periodic travelling waves of korteweg de vries type we consider the stability of periodic travelling wave solutions to a generalized korteweg de vries gkdv equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling wave ordinary differential equation we illustrate this result with several examples including the integrable kdv and modified kdv equations the l 2 critical kdv 4 equation that arises in the study of blow up and the kdv equation which is an idealized model for plasmas

an instability index theory for quadratic pencils and applications primarily motivated by the stability analysis of nonlinear waves in second order in time hamiltonian systems in this paper we develop an instability index theory for quadratic operator pencils acting on a hilbert space in an extension of the known theory for linear pencils explicit connections are made between the number of eigenvalues of a given quadratic operator pencil with positive real parts to spectral information about the individual operators comprising the coefficients of the spectral parameter in the pencil as an application we apply the general theory developed here to yield spectral and nonlinear stability instability results for abstract second order in time wave equations more specifically we consider the problem of the existence and stability of spatially periodic waves for the good boussinesq equation in the analysis our instability index theory provides an explicit and somewhat surprising connection between the stability of a given periodic traveling wave solution of the good boussinesq equation and the stability of the same periodic profile but with different wavespeed in the nonlinear dynamics of a related generalized korteweg de vries equation

orbital stability of periodic traveling wave solutions for the kawahara equation in this paper we investigate the orbital stability of periodic traveling waves for the kawahara equation we prove that the periodic traveling wave under certain conditions minimizes a convenient functional by using an adaptation of the method developed by grillakis et al j funct anal 74 160 197 1987 the required spectral properties to ensure the orbital stability are obtained by knowing the positiveness of the fourier transform of the associated periodic wave established by angulo and natali siam j math anal 40 1123 1151 2008