unified convergence results on a minimax algorithm for finding multiple critical points in banach spaces a minimax method for finding multiple critical points in banach spaces is successfully developed in x yao and j zhou siam j sci comput 26 2005 pp 1796 1809 by using a projected pseudogradient as a search direction since several different techniques can be used to compute a projected pseudogradient the uniform stepsize and the continuity of a search direction two key properties for the convergence results in y li and j zhou siam j sci comput 24 2002 pp 865 885 get lost in this paper instead of proving convergence results of the algorithm for each technique unified convergence results are obtained with a weaker stepsize assumption an abstract existence convergence result is also established it is independent of the algorithm and explains why function values always converge faster than their gradients do the weaker stepsize assumption is then verified for several different cases as an illustration to the new results the banach space w 0 1 p omega is considered and the conditions posed in the new results are verified for a quasi linear elliptic pde

technical communique generalized discrete time pi control of output pdfs using square root b spline expansion a new control approach is proposed for the probability density function pdf control of non gaussian stochastic systems using pi controllers using the square root output pdf model and the weight dynamics the pdf tracking is transformed to a constrained dynamical tracking control problem for weight dynamics where lmi techniques are used to design a generalized pi controller such that stability state constraints and tracking performances can be guaranteed simultaneously

brief robust multivariable pid control for linear parameter varying systems practical control problems often deal with uncertain systems depending on parametric and possibly time varying uncertainties quadratic stability theory and recent advances in convex and nonconvex optimisation algorithms provide useful tools to cope with this kind of problems on the other hand pid is still the controller most commonly used for industrial problems in this paper we consider a certain class of multi input multi output linear parameter varying systems which are encountered in practical control problems and we propose a state space parameterisation which allows to convert pi and pid controller synthesis into an lmi or bmi optimisation problem the proposed technique guarantees uniform exponential stability of the closed loop system a desired rate of convergence and an h norm bound

vojta s refinement of the subspace theorem vojta s refinement of the subspace theorem says that given linearly independent linear forms lx ln in n variables with algebraic coeffi cients there is a finite union u of proper subspaces of q such that for any g 0 the points x e z 0 with 1 lx x l x 1 then x e c u in the present paper it is shown that x e is in fact constant when e lies outside a simply described finite set of rational numbers more generally let k be an algebraic number field and 5 a finite set of absolute values of k containing the archimedean ones for v 6 s let l lvm be linear forms with coefficients in k and for x e 0 with av x dv height hk x 1 define avi x by lv k jc hk x where the dv are the local degrees the approximation set a consists of tuples t dvi v s 1 i m such that for every neighborhood o of a the points x with avj x e o are dense in the subspace topology then a is a polyhedron whose vertices are rational points

energy dissipating flows for solving nonlinear eigenpair problems abstract this work is concerned with computing nonlinear eigenpairs which model solitary waves and various other physical phenomena we aim at solving nonlinear eigenvalue problems of the general form t u q u in our setting t is a variational derivative of a convex functional such as the laplacian operator with respect to the dirichlet energy q is an arbitrary bounded nonlinear operator and is an unknown real eigenvalue we introduce a flow that numerically generates an eigenpair solution by its steady state analysis for the general case is performed showing a monotone decrease in the convex functional throughout the flow when t is the laplacian operator a complete discretized version is presented and anlalyzed we implement our algorithm on korteweg and de vries kdv and nonlinear schrodinger nls equations in one and two dimensions the proposed approach is very general and can be applied to a large variety of models moreover it is highly robust to noise and to perturbations in the initial conditions compared to classical petiashvili based methods

adaptive statistic tracking control based on two step neural networks with time delays this paper presents a new type of control framework for dynamical stochastic systems called statistic tracking control stc the system considered is general and non gaussian and the tracking objective is the statistical information of a given target probability density function pdf rather than a deterministic signal the control aims at making the statistical information of the output pdfs to follow those of a target pdf for such a control framework a variable structure adaptive tracking control strategy is first established using two step neural network models following the b spline neural network approximation to the integrated performance function the concerned problem is transferred into the tracking of given weights the dynamic neural network dnn is employed to identify the unknown nonlinear dynamics between the control input and the weights related to the integrated function to achieve the required control objective an adaptive controller based on the proposed dnn is developed so as to track a reference trajectory stability analysis for both the identification and tracking errors is developed via the use of lyapunov stability criterion simulations are given to demonstrate the efficiency of the proposed approach

marked d0l systems and the 2n conjecture we show that to test the equivalence of two d0l sequences over an n letter alphabet generated by marked morphisms it suffices to compare the first 2n 1 initial terms of the sequences under an additional condition it is enough to consider the 2n initial terms

the presence of a zero in an integer linear recurrent sequence is np hard to decide we show that the problem of determining if a given integer linear recurrent sequence has a zero a problem that is known as pisot s problem is np hard with a similar argument we show that the problem of finding the minimal realization dimension of a one letter max plus rational series is np hard this last result answers a folklore question raised in the control literature on the max plus approach to discrete event systems our results are simple consequences of a construction due to stockmeyer and meyer c 2002 elsevier science inc all rights reserved

numerical methods for computing nonlinear eigenpairs part i iso homogeneous cases with a rayleigh quotient formulation a local minimax method is developed to solve a class of iso homogeneous nonlinear eigenpair problems for multiple solutions in banach spaces following their instability order the algorithm is implemented to compute weighted eigenpairs of the p laplacian numerical eigenfunctions are illustrated by their graphics several interesting phenomena have been observed and are open for further investigation mathematical analysis related to convergence and instability order of computed eigenfunctions etc is also presented

two classes of ljusternik schnirelman minimax algorithms and an application for finding multiple negative energy solutions of a class of p laplacian equations abstract in this paper two classes of ls minimax algorithms are presented they are applied to numerically find multiple negative energy solutions of the p laplacian equation p u u r 1 u u q 1 u x r l u 0 x where is an open bounded domain 0 r p 1 q p 1 0 and p is the sobolev exponent and mathematical justification and global convergence result for them are established by combining ls minimax algorithm with the finite element method it is verified that as element size goes to zero numerical solutions of p laplacian equation captured by ls minimax algorithm converge to solutions of p laplacian equation two ls minimax algorithms developed in 1 are two special algorithms in these two classes of algorithms

a local min orthogonal method for finding multiple saddle points abstract the purpose of this paper is twofold the first is to remove a possible ill posedness related to a local minimax method developed in siam j sci comput 23 2001 840 865 siam j sci comput 24 2002 840 865 and the second is to provide a local characterization for nonminimax type saddle points to do so a local l selection is defined and a necessary and sufficient condition for a saddle point is established which leads to a min orthogonal method those results exceed the scope of a minimax principle the most popular approach in critical point theory an example is given to illustrate the new theory with this local characterization the local minimax method in siam j sci comput 23 2001 840 865 siam j sci comput 24 2002 840 865 is generalized to a local min orthogonal method for finding multiple saddle points in a subsequent paper this approach is applied to define a modified pseudo gradient flow of a functional for finding multiple saddle points in banach spaces

a skolem mahler lech theorem in positive characteristic and finite automata lech proved in 1953 that the set of zeroes of a linear recurrence sequence in a field of characteristic 0 is the union of a finite set and finitely many infinite arithmetic progressions this result is known as the skolem mahler lech theorem lech gave a counterexample to a similar statement in positive characteristic we will present some more pathological examples we will state and prove a correct analog of the skolem mahler lech theorem in positive characteristic the zeroes of a recurrence sequence in positive characteristic can be described using finite automata

on the diophantine equation gn x gm p x higher order recurrences let k be a field of characteristic 0 and let g n x n 0 be a linear recurring sequence of degree d in k x defined by the initial terms g 0 c d 1 k x and by the difference equation g n d x a d 1 x g n d 1 x a 0 x g n x for n 0 with a 0 a d 1 k x finally let p x be an element of k x in this paper we are giving fairly general conditions depending only on g 0 g d 1 on p and on a 0 a d 1 under which the diophantine equation g n x g m p x has only finitely many solutions n m z 2 n m 0 moreover we are giving an upper bound for the number of solutions which depends only on d this paper is a continuation of the work of the authors on this equation in the case of second order linear recurring sequences

a quantitative version of the absolute subspace theorem the celebrated subspace theorem of w m schmidt 12 says the following subspace theorem let l1 ln be linearly independent linear forms in n variables with real or complex algebraic coefficients suppose 0 consider the inequality 1 1 l1 x ln x 0 put 1 5 q c x r li x qi 1 i n then the parametric subspace theorem can be stated as follows parametric subspace theorem let c be a fixed tuple satisfying 1 3 let 0 then there are finitely many proper linear subspaces t1 tb of q n such that for any q which is sufficiently large there exists a subspace ti t1 tb with 1 6 q q c z ti

explicit test sets for iterated morphisms in free monoids and metabelian groups for checking equivalence of any two word morphisms restricted on a subset language it suffices to do this for a finite subset of the language the so called finite test set a way of effectively obtaining a finite test set for a special class of languages the so called iterated morphisms is presented here together with an explicit upper bound for its size the method can be extended to free metabelian groups and to certain other metabelian groups

optimizing schrodinger functionals using sobolev gradients applications to quantum mechanics and nonlinear optics in this paper we study the application of the sobolev gradients technique to the problem of minimizing several schrodinger functionals related to timely and difficult nonlinear problems in quantum mechanics and nonlinear optics we show that these gradients act as preconditioners over traditional choices of descent directions in minimization methods and show a computationally inexpensive way to obtain them using a discrete fourier basis and a fast fourier transform we show that the sobolev preconditioning provides a great convergence improvement over traditional techniques for finding solutions with minimal energy as well as stationary states and suggest a generalization of the method using arbitrary linear operators

on polynomial exponential equations let m k be the set of absolute values of k normalized such that they extend the standard or a p adic absolute value of q let s c m k be the subset consisting of the archimedean absolute values as well as of the non archimedean absolute values v e m k having i ol 4 1 for some ej thus s is a finite set an s unit is an element e k having i 1 1 for every v s then in particular the o and therefore every a txe with x e z is an s unit therefore 1 2 is related to the s unit equation

on s unit equations in two unknowns j h evertse 1 k gy6ry 2 c l stewart 3 and r tijdeman 4 department of pure mathematics centre for mathematics and computer science kruislaan 413 nl 1098 sj amsterdam the netherlands 2 mathematical institute kossuth lajos university h 4010 debrecen hungary 3 department of pure mathematics university of waterloo waterloo ontario canada n2l 3gi 4 mathematical institute university of leiden nl 2333 ca leiden the netherlands

stable polynomials and crystalline measures explicit examples of positive crystalline measures and fourier quasicrystals are constructed using pairs of stable polynomials answering several open questions in the area

a minimax method for finding multiple critical points and its applications to semilinear pdes most minimax theorems in critical point theory require one to solve a two level global optimization problem and therefore are not for algorithm implementation the objective of this research is to develop numerical algorithms and corresponding mathematical theory for finding multiple saddle points in a stable way in this paper inspired by the numerical works of choi mckenna and ding costa chen and the idea to define a solution submanifold some local minimax theorems are established which require us to solve only a two level local optimization problem based on the local theory a new local numerical minimax method for finding multiple saddle points is developed the local theory is applied and the numerical method is implemented successfully to solve a class of semilinear elliptic boundary value problems for multiple solutions on some nonconvex non star shaped and multiconnected domains numerical solutions are illustrated by their graphics for visualization in a subsequent paper y li and j zhou convergence results of a minimax method for finding critical points in review we establish some convergence results for the algorithm

computing the ground state solution of bose einstein condensates by a normalized gradient flow in this paper we present a continuous normalized gradient flow cngf and prove its energy diminishing property which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of bose einstein condensates bec we also investigate the energy diminishing property for the discretization of the cngf two numerical methods are proposed for such discretizations one is the backward euler centered finite difference befd method the other is an explicit time splitting sine spectral tssp method energy diminishing for befd and tssp for the linear case and monotonicity for befd for both linear and nonlinear cases are proven comparison between the two methods and existing methods e g crank nicolson finite difference cnfd or forward euler finite difference fefd shows that befd and tssp are much better in terms of preserving the energy diminishing property of the cngf numerical results in one two and three dimensions with magnetic trap confinement potential as well as a potential of a stirrer corresponding to a far blue detuned gaussian laser beam are reported to demonstrate the effectiveness of befd and tssp methods furthermore we observe that the cngf and its befd discretization can also be applied directly to compute the first excited state solution in bec when the initial data is chosen as an odd function

soliton solutions for quasilinear schrodinger equations ii abstract for a class of quasilinear schrodinger equations we establish the existence of ground states of soliton type solutions by a variational method

robust control of the output probability density functions for multivariable stochastic systems with guaranteed stability presents two robust solutions to the control of the output probability density function for general multi input and multi output stochastic systems the control inputs of the system appear as a set of variables in the probability density functions of the system output and the signal available to the controller is the measured probability density function of the system output a type of dynamic probability density model is formulated by using a b spline neural network with all its weights dynamically related to the control input it has been shown that the so formed robust control algorithms can control the shape of the output probability density function and can guaranteed the closed loop stability when the system is subjected to a bounded unknown input an illustrative example is included to demonstrate the use of the developed control algorithms and desired results have been obtained

binary equality sets are generated by two words we show that the equality set eq g h of two non periodic binary morphisms g h a is generated by at most two words if the rank of eq g h is two then and begin and end with different letters this in particular implies that any binary language has a test set of cardinality at most two

computing electronic structures a new multiconfiguration approach for excited states we present a new method for the computation of electronic excited states of molecular systems this method is based upon a recent theoretical definition of multiconfiguration excited states due to one of us see m lewin solutions of the multiconfiguration equations in quantum chemistry arch rat mech anal 171 2004 83 114 our algorithm dedicated to the computation of the first excited state always converges to a stationary state of the multiconfiguration model which can be interpreted as an approximate excited state of the molecule the definition of this approximate excited state is variational an interesting feature is that it satisfies a non linear hylleraas undheim macdonald type principle the energy of the approximate excited state is an upper bound to the true excited state energy of the n body hamiltonian to compute the first excited state one has to deform paths on a manifold like this is usually done in the search for transition states between reactants and products on potential energy surfaces we propose here a general method for the deformation of paths which could also be useful in other settings we also compare our method to other approaches used in quantum chemistry and give some explanation of the unsatisfactory behaviours which are sometimes observed when using the latters numerical results for the special case of two electron systems are provided we compute the first singlet excited state potential energy surface of the h 2 molecule

a proof of ehrenfeucht s conjecture abstract ehrenfeucht s conjecture states that each subset s of a finitely generated free monoid has a finite subset t such that if two endomorphisms of the monoid agree on t then they agree on s it is the purpose of this note to verify the conjecture

resonance problems for the p laplacian abstract we consider resonance problems at an arbitrary eigenvalue of the p laplacian and prove the existence of weak solutions assuming a standard landesman lazer condition we use variational arguments to characterize certain eigenvalues and then to establish the solvability of the given boundary value problem

linear equations in variables which lie in a multiplicative group let k be a field of characteristic 0 and let n be a natural number let r be a subgroup of the multiplicative group k n of finite rank r given a 1 a n e k write a a 1 a n for the number of solutions x x 1 x n of the equation a 1 x 1 a n x n 1 such that no proper subsum of a 1 x 1 a n x n vanishes we derive an explicit upper bound for a a 1 a n r which depends only on the dimension n and on the rank r

an efficient and stable method for computing multiple saddle points with symmetries in this paper an efficient and stable numerical algorithm for computing multiple saddle points with symmetries is developed by modifying the local minimax method established in y li and j zhou siam j sci comput 23 2001 pp 840 865 y li and j zhou siam j sci comput 24 2002 pp 840 865 first an invariant space is defined in a more general sense and a principle of invariant criticality is proved for the generalization then the orthogonal projection to the invariant space is used to preserve the invariance and to reduce computational error across iterations simple averaging formulas are used for the orthogonal projections numerical computations of examples with various symmetries of which some can and others cannot be characterized by a compact group of linear isomorphisms are carried out to confirm the theory and to illustrate applications the mathematical features of various problems demonstrated in these examples fall into two categories nodal solutions of saddle point type with large morse indices and nonradial positive solutions via symmetry breaking in radially symmetric elliptic problems the new numerical algorithm generates these rather unstable solutions in an efficient and stable way the existence of many unstable solutions and their behavior found in this paper remain to be investigated

convergence results of a local minimax method for finding multiple critical points in y li and j zhou siam j sci comput 23 2001 pp 840 865 a new local minimax method that characterizes a saddle point as a solution to a local minimax problem was established based on the local characterization a numerical minimax algorithm was designed for finding multiple saddle points numerical computations of many examples in semilinear elliptic pde were successfully carried out to solve for multiple solutions one of the important issues which remains unsolved is the convergence of the numerical minimax method in this paper first step 5 in the algorithm is modified with the design of a new stepsize rule that is easier to implement practically and with which convergence results of the numerical minimax method are established for isolated and nonisolated critical points the convergence results show that the algorithm indeed exceeds the scope of a minimax principle in the last section numerical multiple solutions to the henon equation and a sublinear elliptic equation subject to zero dirichlet boundary condition are presented to show their numerical convergence and profiles

une demonstration simple du theoreme de skolem mahler lech abstract a simple proof of the skolem mahler lech theorem is given which does not make the use of the p adic number theory it mainly relies on the equivalence between rational and recognizable series in particular this equivalence allows to deduce easily the general case of a commutative field of characteristic zero from the rational case

the algorithmic theory of finitely generated metabelian groups algorithms are constructed which when an explicit presentation of a finitely generated metabelian group g in the variety x 2 is given produce finitary presentations for the derived subgroup g the centre z g the fitting subgroup fit g and the frattini subgroup 0 g additional algorithms of independent interest are developed for commutative algebra which construct the associated set of primes ass m of a finitely generated module m over a finitely generated commutative ring r and the intersection pr m of the maximal submodules of m

on the equivalence problem for binary dol systems it is shown that to test whether two dol sequences in the binary case coincide it is enough to test whether four first words of these sequences are the same the result is optimal

instability analysis of saddle points by a local minimax method the objective of this work is to develop some tools for local instability analysis of multiple critical points which can be computationally carried out the morse index can be used to measure local instability of a nondegenerate saddle point however it is very expensive to compute numerically and is ineffective for degenerate critical points a local weak linking index can also be defined to measure local instability of a degenerate saddle point but it is still too difficult to compute in this paper a local instability index called a local minimax index is defined by using a local minimax method this new instability index is known beforehand and can help in finding a saddle point numerically relations between the local minimax index and other local instability indices are established those relations also provide ways to numerically compute the morse local linking indices in particular the local minimax index can be used to define a local instability index of a saddle point relative to a reference trivial critical point even in a banach space while others failed to do so

a further improvement of the quantitative subspace theorem in 2002 evertse and schlickewei 11 obtained a quantitative version of the so called absolute parametric subspace theorem this result deals with a parametrized class of twisted heights one of the consequences of this result is a quantitative version of the absolute subspace theorem giving an explicit upper bound for the number of subspaces containing the solutions of the diophantine inequality under consideration in the present paper we further improve evertse s and schlickewei s quantitative version of the absolute parametric subspace theorem and deduce an improved quantitative version of the absolute subspace theorem we combine ideas from the proof of evertse and schlickewei which is basically a substantial refinement of schmidt s proof of his subspace theorem from 1972 22 with ideas from faltings and wustholz proof of the subspace theorem 14 a new feature is an interval result which gives more precise information on the distribution of the heights of the solutions of the system of inequalities considered in the subspace theorem

solitons in several space dimensions derrick s problem and infinitely many solutions in this paper we study a class of lorentz invariant nonlinear field equations in several space dimensions the main purpose is to obtain soliton like solutions these equations were essentially proposed by c h derrick in a celebrated paper in 1964 as a model for elementary particles however an existence theory was not developed

a high linking algorithm for sign changing solutions of semilinear elliptic equations where n is a bounded domain with regular boundary the operator can be replaced by a more general second order uniformly elliptical operator in divergence form the solutions we are interested in nding are of minimax type and sign changing nonlinear elliptic equations of type 1 1 arise naturally in physics engineering and mathematical biology it is known that many such equations have multiple solutions their study has attracted the attention of many pure and applied mathematicians during the past two decades let us assume that f satis es the following regularity and growth conditions a1 f x t is locally lipschitz continuous in