spectral theory for the periodic sine gordon equation a concrete viewpoint a summary of the spectral theory for quasiperiodic sine and sinh gordon equations is given analogies with whole line solitons and scattering theory motivates the discussion the relation between the ingredients in the inverse spectral solution of the periodic sine gordon equation and physical characteristics of sine gordon waves is emphasized the explicit topics covered are summarized in the table of contents in the introduction

3d finite element forward modeling of electromagnetic data using vector and scalar potentials and unstructured grids abstractwe present a finite element solution to the 3d electromagnetic forward modeling problem in the frequency domain the method is based on decomposing the electric field into vector and scalar potentials in the helmholtz equation and in the equation of conservation of charge edge element and nodal element basis functions were used respectively for the vector and scalar potentials this decomposition was performed with the intention of satisfying the continuity of the tangential component of the electric field and the normal component of the current density across the interelement boundaries therefore finding an efficient solution to the problem the computational domain was subdivided into unstructured tetrahedral elements the system of equations was discretized using the galerkin variant of the weighted residuals method with the approximated vector and scalar potentials as the unknowns of a sparse linear system a generalized minimum residual solver with an incomplete lu preconditioner was use

transfer functions of regular linear systems part ii the system operator and the lax phillips semigroup this paper is a sequel to a paper by the second author on regular linear systems 1994 referred to here as part i we introduce the system operator of a well posed linear system which for a finite dimensional system described by x ax bu y cx du would be the s dependent matrix s s a si c b d in the general case s s is an unbounded operator and we show that it can be split into four blocks as in the finite dimensional case but the splitting is not unique the upper row consists of the uniquely determined blocks a si and b as in the finite dimensional case but the lower row is more problematic for weakly regular systems which are introduced and studied here there exists a special splitting of s s where the right lower block is the feedthrough operator of the system using s 0 we give representation theorems which generalize those from part i to well posed linear systems and also to the situation when the initial time is we also introduce the lax phillips semigroup t induced by a well posed linear system which is in fact an alternative representation of a system used in scattering theory our concept of a lax phillips semigroup differs in several respects from the classical one for example by allowing an index r which determines an exponential weight in the input and output spaces this index allows us to characterize the spectrum of a and also the points where s s is not invertible in terms of the spectrum of the generator of for various values of the system is dissipative if and only if with index zero is a contraction semigroup

backlund transformations for nonlinear sigma models with values in riemannian symmetric spaces this work deals with backlund transformations for the principal sl n sigma model together with all reduced models with values in riemannian symmetric spaces first the dressing method of zakharov mikhailov and shabat is shown for the case of a meromorphic dressing matrix to be equivalent to a backlund transformation for an associated linearly extended system comparison of this multi backlund transformation with the composition of ordinary ones leads to a new proof of the permutability theorem a new method of solution for such multi backlund transformations mbt is developed by the introduction of a soliton correlation matrix which satisfies a riccati system equivalent to the mbt using the geometric structure of this system a linearization is achieved leading to a nonlinear superposition formula expressing the solution explicitly in terms of solutions of a single backlund transformation through purely linear algebraic relations a systematic study of all reductions of the system by involutive automorphisms is made thereby defining the multi backlund transformations and their solution for all riemannian symmetric spaces

on the τ function of the painleve equations abstract concerning the hamiltonian structure associated with painleve equations we define the function related to a hamiltonian and show that the correlation function given in the study of the two dimensional ising model is a function for the third painleve equation the list of the hamiltonians which are polynomials in the two canonical variables is given and we prove that the function related to each of them is holomorphic

on the error of linear interpolation and the orientation aspect ratio and internal angles of a triangle in this paper we attempt to reveal the precise relation between the error of linear interpolation on a general triangle and the geometric characters of the triangle taking the model problem of interpolating quadratic functions we derive two exact formulas for the h 1 seminorm and l 2 norm of the interpolation error in terms of the area aspect ratio orientation and internal angles of the triangle these formulas indicate that 1 for highly anisotropic triangular meshes the h 1 seminorm of the interpolation error is almost a monotonically decreasing function of the angle between the orientations of the triangle and the function 2 maximum angle condition is not essential if the mesh is aligned with the function and the aspect ratio is of magnitude sqrt lambda 1 lambda 2 or less where lambda 1 and lambda 2 are the eigenvalues of the hessian matrix of the function with these formulas we identify the optimal triangles which produce the smallest h 1 seminorm of the interpolation error to be the acute isosceles aligned with the solution and of an aspect ratio about 0 8 frac lambda 1 lambda 2 the l 2 norm of the interpolation error depends on the orientation and the aspect ratio of the triangle but not directly on its maximum or minimum angles the optimal triangles for the l 2 norm are those aligned with the solution and of an aspect ratio sqrt lambda 1 lambda 2 these formulas can be used to formulate more accurate mesh quality measures and to derive tighter error bounds for interpolations

the c neumann problem as a completely integrable system on an adjoint orbit it is shown by purely lie algebraic methods that the c neumann problem the motion of a material point on a sphere under the influence of a quadratic potential is a completely integrable system of euler poisson equations on a minimal dimensional orbit of a semidirect product of lie algebras 1 the c neumann problem the motion of a point on the sphere s 1 under the influence of a quadratic potential u x ax x x e r a diag al an is a completely integrable hamiltonian system for n 3 this has been shown by c neumann in 1859 12 and for arbitrary n by k uhlenbeck 16 r devaney 3 j moser 10 11 m adler and p van moerbeke 2 in this paper we show how this problem fits naturally in the framework of euler poisson equations 41 5 14 17 proving that the c neumann problem is a hamiltonian system on a minimaldimensional adjoint orbit in a semidirect product of lie algebras thus its complete integrability will follow entirely from lie algebraic considerations the equations of motion are i ax1 axx i 1 n 1 1 where the lagrange multiplier a ax x 11x112 is chosen such that x e sn i during the motion set x y and get the equivalent system to 1 1 xi yi yi a1xi ax x 11y112 xi llxll 1 x y 0 1 2 the following crucial remark that motivated the present investigation is due to k uhlenbeck and can be verified without any difficulties lemma 1 1 put x xixj p yixj xiyj system 1 2 is equivalent to x p x p x a lixl 1 x y o 1 3 remark that if one replaces x and a by x id n and a tr a id n respectively where id is the n x n identity matrix equations 1 3 remain unchanged from now on we shall assume that in 1 3 this change has been made so that x p a e sl n the next section gives a lie algebraic interpretation to these equations received by the editors march 12 1980 1980 mathematics subject classification primary 58f07 70h05 secondary 53c15 17b99

h1 parametrizations of complex planar physical domains in isogeometric analysis abstract isogeometric analysis is a method for solving geometric partial differential equations pdes generating parametrizations of a pde s physical domain is a basic and important issue within isogeometric analysis framework in this paper we present a global h 1 parametrization method for a complex planar physical domain

stabilization of infinte dimensional systems with periodic feedback gains and sampled output abstract a novel control technique periodic output feedback is investigated in the context of infinite dimensional linear dynamic systems with unbounded input and output operators this control scheme differs from classical control schemes in that the output of the system in question need not be monitored continuously but only monitored at sampling instants it is shown that for systems satisfying an analog of detectability and stabilizability periodic output feedback provides a stabilization scheme which is robust under perturbations of the sampling time sequence and is easily implementable

the kowalewski and henon heiles motions as manakov geodesic flows on so 4 a two dimensional family of lax pairs the invariant surfaces for the kowalewski top the henon heiles system and the manakov geodesic flow onso 4 complete into abelian surfacesa by adjoining in each case a divisord of arithmetic genus 9 these divisors belong to the same linear system ona and they each define a polarization 2 4 therefore there are rational maps transforming the kowalewski top and the henon heiles system into manakov s geodesic flow onso 4 this paper deals with the precise geometric relationship between these three problems it is based on the splitting of the 8 dimensional space of sections ofd theta functions into an even and an odd part and also on a normal form for the six quadrics describinga as embedded in 7 as a byproduct we get a 2 dimensional family of lax pairs for both the kowalewski top and the henon heiles system

sampled data observers for semilinear damped wave equations under spatially sampled state measurements abstract sampled data observers controllers under the sampled in space and time measurements were suggested in the past for parabolic systems in the present paper for the first time a sampled data observer is constructed for a hyperbolic system governed by 1d semilinear wave equation with either viscous or boundary damping the measurements are sampled in space and time sufficient conditions for the exponential stability of the estimation error are derived by using the time delay approach to sampled data control and appropriate lyapunov krasovskii functionals the dual sampled data controller problems are formulated numerical examples including observer design for unstable damped sine gordon equation illustrate the efficiency of the method

isomonodromic deformations of a rational differential system and reconstruction with the topological recursion the sl2 case in this paper we show that it is always possible to deform a differential equation x x l x x with l x sl2 c x by introducing a small formal parameter in such a way that it satisfies the topological type properties of bergere borot and eynard annales henri poincare 16 12 2713 2782 2015 this is obtained by including the former differential equation in an isomonodromic system and using some homogeneity conditions to introduce the topological recursion is then proved to provide a formal series expansion of the corresponding tau function whose coefficients can thus be expressed in terms of intersections of tautological classes in the deligne mumford compactification of the moduli space of surfaces we present a few examples including any fuchsian system of sl2 c x as well as some elements of painleve hierarchies

dual isomonodromic deformations and moment maps to loop algebras the hamiltonian structure of the monodromy preserving deformation equations of jimboet al jmms is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras the nonautonomous hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps this construction is shown to lead to dual pairs of matrix differential operators whose monodromy is preserved under the same family of deformations as illustrative examples involving discrete and continuous reductions a higher rank generalization of the hamiltonian equations governing the correlation functions for an impenetrable bose gas is obtained as well as dual pairs of isomonodromy representations for the equations of the painleve transcendentspv andvi

sampled data boundary feedback control of 1 d parabolic pdes abstract the paper provides results for the application of boundary feedback control with zero order hold zoh to 1 d linear parabolic systems on bounded domains it is shown that the continuous time boundary feedback applied in a sample and hold fashion guarantees closed loop exponential stability provided that the sampling period is sufficiently small two different continuous time feedback designs are considered the reduced model design and the backstepping design the obtained results provide stability estimates for weighted 2 norms of the state and robustness with respect to perturbations of the sampling schedule is guaranteed

admissible input elements for systems in hilbert space and a carleson measure criterion we study the control system dot x ax bu quad x in x quad u text scalar where a generates a semigroup on the hilbert space x but in general the control input element b notin x many boundary value control systems point control force situations etc can be studied in this context we define and analyze admissible input elements b and develop sufficient conditions for b to be admissible in terms of the carleson measure theorem of h p theory

hyperelliptic quasi periodic and soliton solutions of the nonlinear schrodinger equation on considere la construction de solutions par des methodes de geometrie algebriques et en particulier par les fonctions theta hyperelliptiques

conditioning of finite element equations with arbitrary anisotropic meshes this is the published version also available here http dx doi org 10 1090 s0025 5718 2014 02822 6 first published in math comput in 2014 published by the american mathematical society

kac moody lie algebras and soliton equations iii stationary equations associated with a1 1 abstract the hamiltonian structure of stationary soliton equations associated with the akns eigenvalue problem is derived in two ways first it is shown to arise from the kostant kirillov symplectic structure on a coadjoint orbit in an infinite dimensional lie algebra second it is obtained as the restriction to a finite dimensional manifold of the infinite dimensional hamiltonian structure associated with a certain eigenvalue problem polynomial in the eigenvalue parameter

on balanced moving mesh methods moving mesh methods are a widely used approach in the numerical solution of pdes where the original pdes are transformed from a physical domain to a computational domain the objective is to utilize a uniform mesh in the computational domain to get a non uniform physical mesh that better captures the behavior of the solution the movement of the physical mesh points can be governed by a moving mesh pde associated with a corresponding monitor function and both the original pdes and the moving mesh pdes must be solved simultaneously the motivation for this paper is to study a balanced moving mesh method where the aim is to strike a balance between the properties of the solution of the original pde and that of the moving mesh pde we focus on particular choices of the monitor function that give both a well behaved mesh transformation and a well behaved solution in the computational domain both theoretical analysis and numerical experiments are presented as illustrations

admissibility in input elements for diagonal semigroups on l 2 abstract we characterize the admissible unbounded input elements for diagonal semigroups on l 2 the result verifies the necessity of a condition whose sufficience was previously established by l f ho and d l russell

bounds on spectral condition numbers of matrices arising in the p version of the finite element method we estimate condition numbers of p version matrices for tensor product elements with two choices of reference element degrees of freedom in one case lagrange elements the condition numbers grow exponentially in p whereas in the other hierarchical basis functions based on tchebycheff polynomials the condition numbers grow rapidly but only algebraically in p we conjecture that regardless of the choice of basis the condition numbers grow like p 4d or faster where d is the dimension of the spatial domain

periodic output feedback stabilization of neutral systems a new feedback control technique called periodic output feedback is investigated in the context of infinite dimensional linear systems modeled by neutral functional differential equations in this method discrete output samples are multiplied by a periodic gain function to generate a continuous feedback control this work focuses on stabilization of neutral systems with delayed control modeled in the state space w sub 2 sup 1 r 0 r sup n spl times l sup 2 r 0 r sup p where w sub 2 sup 1 r 0 r sup n denotes the sobolev space of r sup n valued absolutely continuous functions with square integrable derivatives on r 0 we show that a class of these systems can be stabilized by periodic output feedback even though their input operators are unbounded we overcome this difficulty by representing the system state using an abstract integral variation of constants formula an algorithm is presented at the end of this paper to construct a periodic output feedback gain function an example is provided to illustrate the construction of the gain function

isospectral hamiltonian flows in finite and infinite dimensions the approach to isospectral hamiltonian flow introduced in part i is further developed to include integration of flows with singular spectral curves the flow on finite dimensional ad invariant poisson submanifolds of the dual widetilde gl r of the positive part of the loop algebra widetilde gl r is obtained through a generalization of the standard method of linearization on the jacobi variety of the invariant spectral curves these curves are embedded in the total space of a line bundlet p1 c allowing an explicit analysis of singularities arising from the structure of the image of a moment map tilde j r m n r times m n r to widetilde gl r from the space of rank r deformations of a fixedn n matrixa it is shown how the linear flow of line bundles e t to tilde s over a suitably desingularized curve tilde s may be used to determine both the flow of matricial polynomialsl and the hamiltonian flow in the spacemn r mn r in terms of functions the resulting flows are proved to be completely integrable the reductions to subalgebras developed in part i are shown to correspond to invariance of the spectral curves and line bundles e t to tilde s under certain linear or anti linear involutions the integration of two examples from part i is given to illustrate the method the rosochatius system and the cnls coupled non linear schrodinger equation

mesh dependent stability for finite element approximations of parabolic equations with mass lumping in this paper based on some mesh dependent estimates on the extreme eigenvalues of a general finite element system defined on a simplicial mesh novel and sharp bounds on the permissible time step size are derived for the mass lumping finite element approximations of parabolic equations the bounds are dependent not only on the mesh size but also on the mesh shape these results provide guidance to the stability of numerical solutions of parabolic problems in relation to the unstructured geometric meshing numerical experiments on both uniform meshes and adaptive meshes are presented to validate the theoretical analysis

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

bounding the influence of domain parameterization and knot spacing on numerical stability in isogeometric analysis abstract isogeometric analysis iga was introduced by hughes et al 2005 1 as a new method to bridge the gap between the geometry description and numerical analysis similar to the finite element approach the iga concept to solve a partial differential equation leads to a linear system of equations the condition number of the coefficient matrix is a crucial factor for the stability of the system it depends strongly on the domain parameterization which provides the isogeometric discretization in this paper we derive a bound for the condition number of the stiffness matrix of the poisson equation in particular we investigate the influence of the domain parameterization and the knot spacing on the stability of the numerical system the factors appearing in our bound reflect the stability properties of a given domain parameterization

on the maximum angle condition for linear tetrahedral elements synge s maximum angle condition for triangular elements is generalized to tetrahedral elements for the generalized condition it is proved that tetrahedra may degenerate in a certain way and the error of the standard linear interpolation remains o h in the w p 1 omega norm for sufficiently smooth functions and p in 1 infty

transfer functions of regular linear systems part i characterizations of regularity we recall the main facts about the representation of regular linear systems essentially that they can be described by equations of the form x t ax t bu t y t cx t du t like finite dimensional systems but now a b and c are in general unbounded operators regular linear systems are a subclass of abstract linear systems we define transfer functions of abstract linear systems via a generalization of a theorem of foures and segal we prove a formula for the transfer function of a regular linear system which is similar to the formula in finite dimensions the main result is a simple to state but hard to prove necessary and sufficient condition for an abstract linear system to be regular in terms of its transfer function other conditions equivalent to regularity are also obtained the main result is a consequence of a new tauberian theorem which is of independent interest

mesh smoothing using a posteriori error estimates we develop a simple mesh smoothing algorithm for adaptively improving finite element triangulations the algorithm makes use of a posteriori error estimates which are now widely used in finite element calculations in this paper we derive the method present some numerical illustrations and give a brief analysis of the issue of uniqueness

adaptive finite element methods for elliptic pdes based on conforming centroidal voronoi delaunay triangulations a new triangular mesh adaptivity algorithm for elliptic pdes that combines a posteriori error estimation with centroidal voronoi delaunay tessellations of domains in two dimensions is proposed and tested the ability of the first ingredient to detect local regions of large error and the ability of the second ingredient to generate superior unstructured grids result in a mesh adaptivity algorithm that has several very desirable features including the following errors are very well equidistributed over the triangles at all levels of refinement the triangles remain very well shaped even if the grid size at any particular refinement level when viewed globally varies by several orders of magnitude and the convergence rates achieved are the best obtainable using piecewise linear finite elements this methodology can be easily extended to higher order finite element approximations or mixed finite element formulations although only the linear approximation is considered in this paper

an optimal triangulation for second order elliptic problems abstract let be a polygonal domain in r n h an associated triangulation and uh the finite element solution of a well posed second order elliptic problem on h let m mi p qi 1 be the set of nodes which defines the vertices of the triangulation h for each i m i x i l 1 l n in rn the object of this paper is to provide a computational tool to approximate the best set of positions m of the nodes and hence the best triangulation gt h which minimizes the solution error in the natural norm associated with the problem the main result of this paper are theorems which provide explicit expressions for the partial derivatives of the associated energy functional with respect to the coordinates xil 1 l n of each of the variable nodes mi i 1 p

mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions in this paper we study the linear bond based nonlocal peridynamic models with a particular focus on problems associated with nonstandard nonlocal displacement loading conditions both stationary and time dependent problems are considered for a one dimensional scalar equation defined on a finite bar and for a two dimensional system defined on a square the related peridynamic operators and associated functional spaces are analyzed applications to the numerical analysis of the finite dimensional approximations to peridynamic models are also illustrated

r matrix construction of electromagnetic models for the painleve transcendents the painleve transcendents pi pv and their representations as isomonodromic deformation equations are derived as nonautonomous hamiltonian systems from the classical r matrix poisson bracket structure on the dual space slr 2 of the loop algebra slr 2 the hamiltonians are obtained by composing elements of the poisson commuting ring of spectral invariant functions on slr 2 with a time dependent family of poisson maps whose images are four dimensional rational coadjoint orbits in slr 2 each system may be interpreted as describing a particle moving on a surface of zero curvature in the presence of a time varying electromagnetic field the painleve equations follow from reduction of these systems by the hamiltonian flow generated by a second commuting element in the ring of spectral invariants 1995 american institute of physics

stabilization of distributed parameter systems by piecewise polynomial control in this paper we consider the stabilization of continuous time distributed parameter systems by piecewise polynomial controls we obtain necessary and sufficient conditions for stabilization by piecewise polynomial controls

output feedback stabilization for heat equations with sampled data controls abstract in this paper we build up an output feedback law to stabilize a sampled data controlled heat equation with a potential in a bounded domain the feedback law abides the following rules first we divide equally the time interval 0 into infinitely many disjoint time periods and divide each time period into three disjoint subintervals second for each time period we observe a solution over an open subset of in the first subinterval take sample from outputs at one time point of the first subinterval add a time invariant output feedback control over another open subset of in the second subinterval let the equation evolve free in the last subinterval thus the corresponding feedback control is of sampled data our feedback law has the following advantages the sampling period which is the length of the above time period can be arbitrarily taken the feedback law has an explicit expression in terms of the sampling period the behaviors of the norm of the feedback law when the sampling period goes to zero or infinity are clear the construction of the feedback law is based on two kinds of approximate null controllability for heat equations one has time invariant controls while another has impulse controls the studies of the aforementioned controllability with time invariant controls need a new observability inequality for heat equations built up in the current work

on mesh geometry and stiffness matrix conditioning for general finite element spaces the performance of finite element computation depends strongly on the quality of the geometric mesh and the efficiency of the numerical solution of the linear systems resulting from the discretization of partial differential equation pde models it is common knowledge that mesh geometry affects not only the approximation error of the finite element solution but also the spectral properties of the corresponding stiffness matrix in this paper for typical second order elliptic problems some refined relationships between the spectral condition number of the stiffness matrix and the mesh geometry are established for general finite element spaces defined on simplicial meshes the derivation of such relations for general high order elements is based on a new trace formula for the element stiffness matrix it is shown that a few universal geometric quantities have the same dominant effect on the stiffness matrix conditioning for different finite element spaces these results provide guidance to the studies of both linear algebraic solvers and the unstructured geometric meshing

grid generation and optimization based on centroidal voronoi tessellations centroidal voronoi tessellations cvts are voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding voronoi regions such tessellations are of use in very diverse applications including data compression clustering analysis cell biology territorial behavior of animals and optimal allocation of resources in this paper we explore the use of cvts in grid generation in connection with finite element approximations of partial differential equations we being by describing these tessellations and methods for their determination we then discuss their application to mesh generation and finish with some examples of their use for the solution of partial differential equations

isospectral hamiltonian flows in finite and infinite dimensions i generalized moser systems and moment maps into loop algebras a moment map jr jia gl r is constructed from the poisson manifold j a of rank r perturbations of a fixed n xn matrix a to the dual r f e positive part of the formal loop algebra i r g r c 1 the adler kostant symes theorem is used to give hamiltonians which generate commutative isospectral flows on g r the pull back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in j a the latter may be identified with flows on finite dimensional coadjoint orbits in g and linearized on the jacobi variety of an invariant spectral curve xr which generically is an r sheeted riemann surface reductions of j a are derived corresponding to subalgebras of gl r c and sl r c determined as the fixed point set of automorphism groupes generated by involutions i e jdl the classical algebras as well as reductions to twisted subalgebras of s r c the theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of pde s

quadratic optimal control of stable well posed linear systems we consider the infinite horizon quadratic cost minimization problem for a stable time invariant well posed linear system in the sense of salamon and weiss and show that it can be reduced to a spectral factorization problem in the control space more precisely we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution if both the system and the spectral factor are regular then the feedback operator can be expressed in terms of the riccati operator and the riccati operator is a positive self adjoint solution of an algebraic riccati equation this riccati equation is similar to the usual algebraic riccati equation but one of its coefficients varies depending on the subspace in which the equation is posed similar results are true for unstable systems as we have proved elsewhere

generalized sampled data feedback control of distributed parameter systems abstract in the sampled data control literature there are necessary conditions and sufficient conditions for stabilizability of distributed parameter systems by generalized sampled data control for finite dimensional systems the necessary conditions are also known to be sufficient we show that this equivalence extends to the infinite dimensional case if the underlying semigroup is analytic however for general systems the necessary conditions are not sufficient nor are the sufficient conditions necessary we prove this by a single example with a free parameter one choice of parameter shows that the necessary conditions are too weak and another choice shows that the sufficient conditions are too strong

the soliton correlation matrix and the reduction problem for integrable systems integrable 1 1 dimensional systems associated to linear first order matrix equations meromorphic in a complex parameter as formulated by zakharov mikhailov and shabat 1 3 zms are analyzed by a new method based upon the soliton correlation matrix m matrix the multi backlund transformation which is equivalent to the introduction of an arbitrary number of poles in the zms dressing matrix is expressed by a pair of matrix riccati equations for them matrix through a geometrical interpretation based upon group actions on grassman manifolds the solution of this system is explicitly determined in terms of the solutions to the zms linear system reductions of the system corresponding to invariance under finite groups of automorphisms are also solved by reducing them matrix suitably so as to preserve the class of invariant solutions

stability and convergence properties of forced infinite dimensional discrete time lur e systems incremental stability and convergence properties for forced infinite dimensional discrete time lur e systems are addressed lur e systems have a linear and nonlinear component and arise as the fe

discrete time and sampled data low gain control of infinite dimensional linear systems in the presence of input hysteresis we introduce a general class of causal dynamic discrete time nonlinearities which have certain monotonicity and lipschitz continuity properties in particular the discretizations of a large class of continuous time hysteresis operators obtained by applying the standard sampling and hold operations belong to this class it is shown that closing the loop around a power stable linear infinite dimensional discrete time single input single output system subject to an input nonlinearity from the class under consideration and compensated by a discrete time integral controller guarantees asymptotic tracking of constant reference signals provided that a the positive integrator gain is sufficiently small and b the reference value is feasible in a very natural sense we apply this result in the development of sampled data low gain integral control for exponentially stable regular linear infinite dimensional continuous time systems subject to input hysteresis

optimal finite element mesh for elliptic equation of divergence form we consider an optimal triangular mesh minimizing the condition number of the finite element stiffness matrix for an elliptic equation email protected email protected email protected x i a i email protected email protected x j f u w g using a sharp bound for the condition number of the stiffness matrix it is shown that the element of the optimal uniform triangular mesh is equilateral with respect to the metric which is the inverse of the coefficient matrix in the equation it is verified by numerical examples that such a mesh is really effective in reducing the condition number of the stiffness matrix in addition we suggest an algorithm generating a mesh in which every element is almost equilateral with respect to a metric

tetrahedral mesh generation and optimization based on centroidal voronoi tessellations the centroidal voronoi tessellation based delaunay triangulation cvdt provides an optimal distribution of generating points with respect to a given density function and accordingly generates a high quality mesh in this paper we discuss algorithms for the construction of the constrained cvdt from an initial delaunay tetrahedral mesh of a three dimensional domain by establishing an appropriate relationship between the density function and the specified sizing field and applying the lloyd s iteration the constrained cvdt mesh is obtained as a natural global optimization of the initial mesh simple local operations such as edges faces flippings are also used to further improve the cvdt mesh several complex meshing examples and their element quality statistics are presented to demonstrate the effectiveness and efficiency of the proposed mesh generation and optimization method copyright 2003 john wiley sons ltd

multi hamiltonian structures for r matrix systems for the rational elliptic and trigonometric r matrices we exhibit the links between three levels of poisson spaces a some finite dimensional spaces of matrix valued meromorphic functions on the complex line b spaces of spectral curves and sheaves supported on them and c symmetric products of a surface we have at each level a linear space of compatible poisson structures and the maps relating the levels are poisson this leads in a natural way to the nijenhuis coordinates for these spaces at level b there are hamiltonian systems on these spaces which are integrable for each poisson structure in the family and which are such that the lagrangian leaves are the intersections of the symplectic leaves over the poisson structures in the family specific examples include many of the well known integrable systems

discrete time analogues of some nonlinear oscillators in the inverse square potential we propose a family of symplectic maps providing finite difference approximations for the wojciechowski and rosochatius systems the lax pair representations are found and complete integrability is proved

stability of infinite dimensional sampled data systems suppose that a static state feedback stabilizes a continuous time linear infinite dimensional control system we consider the following question if we construct a sampled data controller by applying an idealized sample and hold process to a continuous time stabilizing feedback will this sampled data controller stabilize the system for all sufficiently small sampling times here the state space x and the control space u are hilbert spaces the system is of the form x t ax t bu t where a is the generator of a strongly continuous semigroup on x and the continuous time feedback is u t fx t the answer to the above question is known to be yes if x and u are finite dimensional spaces in the infinite dimensional case if f is not compact then it is easy to find counterexamples therefore we restrict attention to compact feedback we show that the answer to the above question is yes if b is a bounded operator from u into x moreover if b is unbounded we show that the answer yes remains correct provided that the semigroup generated by a is analytic we use the theory developed for static state feedback to obtain analogous results for dynamic output feedback control

generalized sampled data stabilization of well posed linear infinite dimensional systems we consider well posed linear infinite dimensional systems the outputs of which are sampled in a generalized sense using a suitable weighting function under certain natural assump tions on the system the weighting function and the sampling period we show that there exists a generalized hold function such that unity sampled data feedback renders the closed loop system exponentially stable in the state space sense as well as l 2 stable in the input output sense to illustrate our main result we describe an application to a structurally damped euler bernoulli beam

infinite dimensional linear systems with unbounded control and observation a functional analytic approach abstract the object of these notes is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators on the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem the implications of the theory for large classes of functional and partial differential equations are discussed in detail keywords representation of infinite dimensional systems semigroups boundary control feedback linear quadratic control

sampled data boundary feedback control of 1 d linear transport pdes with non local terms abstract the paper provides results for the application of boundary feedback control with zero order hold zoh to 1 d inhomogeneous linear transport partial differential equations on bounded domains with constant velocity and non local terms it is shown that the emulation design based on the recently proposed continuous time boundary feedback designed by means of backstepping guarantees closed loop exponential stability provided that the sampling period is sufficiently small it is also shown that contrary to the parabolic case a smaller sampling period implies a faster convergence rate with no upper bound for the achieved convergence rate the obtained results provide stability estimates for the sup norm of the state and robustness with respect to perturbations of the sampling schedule is guaranteed

spectral determination for a cantilever beam the eigenvalue specification problem is discussed for a beam with one end clamped and the other end free which is allowed to have some structural damping and has a moment force at the free end acting as a control the spectrum and eigenvectors of the uncontrolled system are first analyzed and are then used to construct the feedback element which solves the problem it is shown how this feedback element can be numerically approximated and a spillover result is proved for the approximation in the case where there is no damping this method yields a bounded feedback law which exponentially stabilizes the beam

abelian integrals and the reduction method for an integrable hamiltonian system a classical finite dimensional integrable hamiltonian system corresponding to the motion of a particle constrained to an n dimensional sphere n 0 x2 1 with the hamiltonian h 1 2 y2 u2 x2 e x2 where u and e are constants and y are the momenta conjugate to x is integrated using several different methods these are the following 1 the projection of geodesic free flow on a larger space namely the sphere s2n 1 for e 0 the flow is obtained in terms of elementary functions 2 separation of variables in the hamilton jacobi equation in elliptic coordinates or alternatively the use of a complete set of integrals of motion in involution to reduce hamilton s equations to quadratures the flow is obtained in terms of abelian integrals which are then inverted in terms of generalized functions the relation between the different methods and results is clarified using methods of algebraic geometry in particular the geometry of quadrics

stability analysis of a text 2 times text 2 linear hyperbolic system with a sampled data controller via backstepping method and looped functionals this paper is concerned with the global exponential stability of a 2 times 2 linear hyperbolic system with a sampled data boundary feedback control designed by means of the backstepping method for a nominal continuous input we show that there exists a sufficiently small intersampling time that encompasses both periodic and aperiodic sampling for which the global exponential stability of the closed loop system is guaranteed in addition we provide easily tractable sufficient stability conditions that can be used to find an upper bound of the maximum intersampling time the results rely on the combination of the lyapunov method and looped functionals the effectiveness of the proposed results is illustrated with a numerical example

discrete time low gain control of uncertain infinite dimensional systems using a frequency domain analysis it is shown that the application of a feedback controller of the form k z 1 or kz z 1 where k spl isin r to a power stable infinite dimensional discrete time system with square transfer function matrix g z will result in a power stable closed loop system which achieves asymptotic tracking of arbitrary constant reference signals provided that i all the eigenvalues of g 1 have positive real parts and ii the gain parameter k is positive and sufficiently small moreover if g 1 is positive definite we show how the gain parameter gain x can be tuned adaptively the resulting adaptive tracking controllers are universal in the sense that they apply to any power stable system with g 1 0 in particular they are not based on system identification or plant parameter estimation algorithms nor is the injection of probing signals required finally we apply these discrete time results to obtain adaptive sampled data low gain controllers for the class of regular systems a rather general class of infinite dimensional continuous time systems for which convenient representations are known to exist both in state space and in frequency domain we emphasize that our results guarantee not only asymptotic tracking at the sampling instants but also in the sampling interval

kac moody lie algebras and soliton equations ii lax equations associated with a1 1 the soliton equations associated with sl 2 eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment they are shown to be generated by a single family of commuting hamiltonians on a subalgebra of the loop algebra of sl 2 the conserved densities and fluxes of the usual anks hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems the hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment

orthogonal polynomials and cubature formulae on balls simplices and spheres we report on recent developments on orthogonal polynomials and cubature formulae on the unit ball bd the standard simplex td and the unit sphere sd the main result shows that orthogonal structures and cubature formulae for these three regions are closely related this provides a way to study the structure of orthogonal polynomials for example it allows us to use the theory of h harmonics to study orthogonal polynomials on bd and on td it also provides a way to construct new cubature formulae on these regions

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments