computation of homoclinic orbits in partial differential equations an application to cylindrical shell buckling this paper concerns numerical computation of localized solutions in partial differential equations pdes on unbounded domains the application is to the von karman donnell equations a coupled system of elliptic equations describing the equilibrium of an axially compressed cylindrical shell earlier work suggests that axially localized solutions are the physically preferred buckling modes hence the problem is posed on a cylindrical domain that is unbounded axially and solutions are sought which are homoclinic in the axial variable and periodic circumferentially the numerical method is based on a galerkin spectral decomposition circumferentially to pose ordinary differential equations odes in the unbounded coordinate methods for location and parameter continuation of homoclinic solutions of odes are then adapted making special use of the symmetry and reversibility properties of solutions observed experimentally thus a formally well posed continuation problem is reduced to a rotational subgroup circumferentially and posed over a truncation of the half interval axially the method for location of solutions makes use of asymptotic approximations where available more generally an adaptation of the successive continuation shooting method is used in the lowest possible number of circumferential modes followed by additional homotopies to add more modes by continuation in the strength of nonlinear mode interaction terms the method is illustrated step by step to produce a variety of homoclinic solutions of the equations and compute their bifurcation diagrams as the loading parameter varies good agreement with experimental data is found all computations are performed using auto the techniques illustrated here for the von karman donnell equations are applicable to a wider class of pdes

fuzzy vectors as a tool for modeling uncertain multidimensional quantities the paper deals with modeling uncertain multidimensional quantities by means of fuzzy vectors in connection with possible interactions among variables the separability of fuzzy vectors is discussed in detail for the case when interconnections among variables are given by a crisp relation a more general form of separability of fuzzy vectors the separability on a given relation e g on a probability simplex is introduced properties of fuzzy vectors separable on a given relation are studied and ways of convenient setting of such fuzzy vectors are proposed finally fuzzy extensions of real valued functions and real vector valued functions to general input fuzzy vectors are investigated

smoothness of bounded solutions of nonlinear evolution equations abstract it is shown that in many cases globally defined bounded solutions of evolution equations are as smooth in time as the corresponding operator even if a general solution of the initial value problem is much less smooth i e initial values for bounded solutions are selected in such a way that optimal smoothness is attained in particular solutions which bifurcate from certain steady states such as periodic orbits almost periodic orbits and also homo and heteroclinic orbits have this property as examples a neutral functional differential equation a slightly damped non linear wave equation and a heat equation are considered in the latter case the space variable is included into the discussion of smoothness finally generalized hopf bifurcation in infinite dimensions is considered here smoothness of the bifurcation function is discussed and known results on the order of a focus are generalized

trapezoidal approximations of fuzzy numbers the problem of the trapezoidal approximation of fuzzy numbers is discussed a set of criteria for approximation operators is formulated these constraints can be used for direct operator derivation a new nearest trapezoidal approximation operator preserving expected interval is suggested

an improved index of interactivity for fuzzy numbers in this paper we will introduce a new index of interactivity between marginal possibility distributions a and b of a joint possibility distribution c the starting point of our approach is to equip each c level set of c with a uniform probability distribution then the probabilistic correlation coefficient between its marginal probability distributions is interpreted as an index of interactivity between the c level sets of a and b then we define the index of interactivity between a and b as the weighted average of these indexes over the set of all membership grades this new index of interactivity is meaningful for the whole family of joint possibility distributions

stable and unstable manifolds for hyperbolic bi semigroups we show the existence of local lipschitzian stable and unstable manifolds for the ill posed problem of perturbations of hyperbolic bi semigroups we do not assume backward nor forward uniqueness of solutions we do not use cut off functions because we do not assume global smallness conditions on the nonlinearities we introduce what we call dichotomous flows which recovers the symmetry between the past and the future thus we need to prove only a stable manifold theorem we modify the conley mcgehee moeckel approach to avoid appealing to wazewski principle and brouwer degree theory hence we allow both the stable and unstable directions to be infinite dimensional we apply our theorem to the elliptic system u u g u u in an infinite cylinder r

on possibilistic correlation in 2004 fuller and majlender introduced the notion of covariance between fuzzy numbers by their joint possibility distribution to measure the degree to which they interact based on this approach in this paper we will present the concept of possibilistic correlation representing an average degree of interaction between marginal distributions of a joint possibility distribution as compared to their respective dispersions moreover we will formulate the classical cauchy schwarz inequality in this possibilistic environment and show that the measure of possibilistic correlation satisfies the same property as its probabilistic counterpart in particular applying the idea of transforming level sets of possibility distributions into uniform probability distributions we will point out a fundamental relationship between our proposed possibilistic approach and the classical probabilistic approach to measuring correlation

a new fuzzy programming approach for multi period portfolio optimization with return demand and risk control abstract this paper considers a multi period portfolio selection problem imposed by return demand and risk control in a fuzzy investment environment in which the returns of assets are characterized by fuzzy numbers a fuzzy lower semi deviation is originally defined as the risk control of portfolio a proportion entropy constraint is added as the divergence measure of portfolio based on the theories of possibility and necessity measures a new multi period portfolio optimization model with return demand and risk control is proposed and then the proposed model is transformed into a crisp nonlinear programming problem by using fuzzy programming approach furthermore an improved differential evolution algorithm is designed for obtaining the optimal strategy finally a numerical example is given to illustrate the practicality and efficiency of the proposed model and the corresponding algorithm

explicit inverse of a tridiagonal k toeplitz matrix we obtain explicit formulas for the entries of the inverse of a nonsingular and irreducible tridiagonal k toeplitz matrix a the proof is based on results from the theory of orthogonal polynomials and it is shown that the entries of the inverse of such a matrix are given in terms of chebyshev polynomials of the second kind we also compute the characteristic polynomial of a which enables us to state some conditions for the existence of a 1 our results also extend known results for the case when the residue mod k of the order of a is equal to 0 or k 1 numer math 10 1967 pp 153 161

necessary and sufficient conditions for the equality of interactive and non interactive extensions of continuous functions abstract in this contribution we find the class of n dimensional joint possibility distributions with the property that the interactive extension principle coincides with the non interactive extension principle as long as the interactive operations are determined by continuous functions strictly increasing in each argument this result completes recent studies by the authors where the particular case of interactive additions and multiplications versus non interactive additions and multiplications were investigated in addition this time we propose results that also cover the cases when we know the fuzzy numbers only from their membership functions it means that we eliminated the limitations that appear when we cannot pass from membership function representation to parametric representation of fuzzy numbers as important new applications we mention the study on the completely correlated fuzzy numbers also of note is that we propose two simple methods to extend bidimensional joint possibility distributions to n dimensional joint possibility distributions one method is based on an inductive construction while the other one is based on a pairwise construction

numerical approximation of relative equilibria for equivariant pdes we prove convergence results for the numerical approximation of relative equilibria of parabolic systems in one space dimension these systems are special examples of equivariant evolution equations we use finite differences on a large interval with appropriately chosen boundary conditions moreover we consider the approximation of isolated eigenvalues of finite multiplicity of the linear operator which arises from linearization at the equilibrium as well as the approximation of the corresponding invariant subspace the results in this paper which are a generalization of the results in v thummler numerical analysis of the method of freezing traveling waves ph d thesis bielefeld university 2005 are illustrated by numerical computations for the cubic quintic ginzburg landau equation

solutions of higher order linear fuzzy differential equations with interactive fuzzy values abstract in this study we consider higher order linear differential equations with additional conditions initial and or boundary given by interactive fuzzy numbers the concept of interactivity arises from the notion of a joint possibility distribution j the proposed method for solving fuzzy differential equations is based on an extension of the classical solution via sup j extension which is a generalization of zadeh s extension principle we prove that under certain conditions the solution via zadeh s extension principle is equal to the convex hull of the solutions produced by the sup j extension we also show that the solutions based on the frechet derivatives of fuzzy functions coincide with the solutions obtained via the sup j extension all of the results are illustrated based on a 3rd order fuzzy boundary value problem

relative growth of linear iterations and orthogonal polynomials on several intervals assume a c2 2 and y0 c2 let yn yn a y0 be defined by thelinear iteration yn ayn 1 n 1 2 then for every p 0 and for every norm in c2 there is a constant c depending only on p and such that sup a 1 y0 0 yl a y0 p lk 0 yk a y0 p cl l 1 2 where a denotes the spectral radius of a applications to orthogonal polynomials pn with respect to a positive borel measure on several intervals are studied in particular the growth of orthogonal polynomials relative to their sums satisfies limn supx supp pn x p n 1k 0 pk x p 0 provided that the three term recurrence coefficients of pn are asymptotically periodic

nonlinear stability of time periodic viscous shocks in order to understand the nonlinear stability of many types of time periodic travelling waves on unbounded domains one must overcome two main difficulties the presence of embedded neutral eigenvalues and the time dependence of the associated linear operator this problem is studied in the context of time periodic lax shocks in systems of viscous conservation laws using spatial dynamics and a decomposition into separate floquet eigenmodes it is shown that the linear evolution for the time dependent operator can be represented using a contour integral similar to that of the standard time independent case by decomposing the resulting green s distribution the leading order behavior associated with the embedded eigenvalues is extracted sharp pointwise bounds are then obtained which are used to prove that time periodic lax shocks are linearly and nonlinearly stable under the necessary conditions of spectral stability and minimal multiplicity of the translational eigenvalues the latter conditions hold for example for small oscillation time periodic waves that emerge through a supercritical hopf bifurcation from a family of time independent lax shocks of possibly large amplitude

orthogonal polynomials and cubic transformations abstract starting from a sequence pn of orthogonal polynomials with respect to a quasi definite or regular linear functional we construct a new sequence qn of orthogonal polynomials such that q3n x pn x3 the basic tool is the three term recurrence relation which such a family satisfies as an application we present some examples concerning chebyshev and hermite polynomials

modeling uncertain variables of the weighted average operation by fuzzy vectors the paper deals with the fuzzy extension of the weighted average operation first we study the convenient ways how uncertain weights and weighted values can be modeled by fuzzy vectors we show that in comparison to a tuple of fuzzy numbers that have been used for modeling uncertain values of particular weights and weighted values up to now fuzzy vectors extend the possibilities of utilizing the vague expert information concerning the weights and weighted values next we focus on computation of a fuzzy weighted average of a fuzzy vector of weighted values with a fuzzy vector of weights we derive a general formula and we study its special forms the advantage of the approach presented in the paper is that the resulting fuzzy weighted average is not overly imprecise since every available information about its variables is involved in computation this fact is illustrated by several examples finally we briefly discuss the problem of defuzzification of the resulting fuzzy weighted average

a parametrized sum of fuzzy numbers with applications to fuzzy initial value problems abstract this paper presents a new parametrized family of joint possibility distributions that can be used to control the pompeiu hausdorff norm of the sum resulting from the extension principle we are able to generate sums with norms covering the entire interval ranging from the smallest possible value to the largest one finally we apply our approach to numerical methods for solving first order fuzzy initial value problems

characterizations of generalized hermite and sieved ultraspherical polynomials a new characterization of the generalized hermite polynomials and of the orthogonal polynomials with respect to the measure ixy 1 x2 1 2dx is derived which is based on a reversing property of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of laguerre and jacobi polynomials a similar characterization can be obtained for a generalization of the sieved ultraspherical polynomials of the first and second kind these results are applied in order to determine the asymptotic limit distribution for the zeros when the degree and the parameters tend to infinity with the same order

evans functions for periodic waves on infinite cylindrical domains abstract using galerkin approximations an evans function for spatially periodic waves on infinite cylindrical domains is constructed it is also shown that the evans function can be used to define a parity index for periodic waves that detects whether the wave admits an odd number of real unstable eigenvalues this parity index depends only on local information for the existence problem of the wave in particular it uses information about the linear dispersion relation near zero and the orientability of the unstable and stable manifolds along the nonlinear wave the results are applied to small amplitude wave trains for a scalar equation on an infinite strip

numerical computation and continuation of invariant manifolds connecting fixed points a numerical method for the computation of an invariant manifold that connects two fixed points of a vector field in mathbb r n is given extending the results of an earlier paper comput appl math 26 1989 pp 159 170 by the authors basically a boundary value problem on the real line is truncated to a finite interval the method applies in particular to the computation of heteroclinic orbits the emphasis is on the systematic computation of such orbits by continuation using the fact that the linearized operator of our problem is fredholm in appropriate banach spaces the general theory of approximation of nonlinear problems is employed to show that the errors in the approximate solution decay exponentially with the length of the approximating interval several applications are considered including the computation of traveling wave solutions to reaction diffusion problems computations were done using the software package auto

sieved ultraspherical polynomials the continuous q ultraspherical polynomials contain a number of important examples as limiting or special cases one of these arose in allaway s ph d thesis in a previous paper we solved a characterization problem essentially equivalent to allaway s and showed that these polynomials arose from the q ultraspherical polynomials when q approached a root of unity a second class of such polynomials is found and the recurrence relation and orthogonality relation are found for each of these polynomials the orthogonality is interesting because the weight function has a finite number of zeros in 1 1 generating functions and other formulas are also found

small amplitude steady water waves with critical layers non symmetric waves abstract the problem for two dimensional steady water waves with vorticity is considered using methods of spatial dynamics we reduce the problem to a finite dimensional hamiltonian system the reduced system describes all small amplitude solutions of the problem and as an application we give a proof of the existence of non symmetric steady water waves whenever the number of roots of the dispersion equation is greater than one

existence of multidimensional travelling wave solutions of an initial boundary value problem on considere le probleme suivant u t u xx u yy f u u x 0 t u x l t 0 u x y 0 u 0 x y ou x r 1 0 y l et f u u u 1 u avec 0 1 2

population growth model via interactive fuzzy differential equation abstract in this manuscript we study a fuzzy initial value problem fivp which describes an autocorrelated evolutive process for this we use the f correlated derivative f derivative concept on fuzzy number valued functions this approach lets us establish a relationship between the diameter of the fivp solution and its derivative moreover we notice how the derivative influences the interactivity of the process by computing the measure of possibilistic interactivity of the fivp solution we analyze a population growth model namely the logistic model with our approach and compare the results with another approach that uses the generalized derivative of hukuhara gh derivative finally we conclude that in fuzzy differential equations describing locally f correlated fuzzy processes the use of the interactive derivative is more adequate than the gh derivative for a fuzzy number valued function

absolute and convective instabilities of waves on unbounded and large bounded domains abstract instabilities of nonlinear waves on unbounded domains manifest themselves in different ways an absolute instability occurs if the amplitude of localized wave packets grows in time at each fixed point in the domain in contrast convective instabilities are characterized by the fact that even though the overall norm of wave packets grows in time perturbations decay locally at each given point in the unbounded domain wave packets are convected towards infinity in experiments as well as in numerical simulations bounded domains are often more relevant we are interested in the effects that the truncation of the unbounded to a large but bounded domain has on the aforementioned in stability properties of a wave these effects depend upon the boundary conditions that are imposed on the bounded domain we compare the spectra of the linearized evolution operators on unbounded and bounded domains for two classes of boundary conditions it is proved that periodic boundary conditions reproduce the point and essential spectrum on the unbounded domain accurately spectra for separated boundary conditions behave in quite a different way firstly separated boundary conditions may generate additional isolated eigenvalues secondly the essential spectrum on the unbounded domain is in general not approximated by the spectrum on the bounded domain instead the so called absolute spectrum is approximated that corresponds to the essential spectrum on the unbounded domain calculated with certain optimally chosen exponential weights we interpret the difference between the absolute and the essential spectrum in terms of the convective behavior of the wave on the unbounded domain in particular it is demonstrated that the stability of the absolute spectrum implies convective instability of the wave but that convectively unstable waves can destabilize under domain truncation the theoretical predictions are compared with numerical computations

on interactive fuzzy numbers in this paper we will introduce a measure of interactivity between marginal distributions of a joint possibility distribution c as the expected value of the interactivity relation between the level sets of its marginal distributions

a note to the finite t sum of l r fuzzy numbers and further considerations abstract in the first part of this paper we consider the cut of a sum of two fuzzy sets we show that the cut of such a sum can be written as one closed interval determined by the minimum and maximum of some special functions over a set uniquely determined by the considered t norm in the second part of the paper we show that not only cut of a sum of two fuzzy intervals has this property but also cut of a continuous function of two variables strictly increasing in each argument as well two examples of the presented theory are considered

on weighted possibilistic mean and variance of fuzzy numbers dubois and prade defined an interval valued expectation of fuzzy numbers viewing them as consonant random sets carlsson and fuller defined an interval valued mean value of fuzzy numbers viewing them as possibility distributions in this paper we shall introduce the notation of weighted interval valued possibilistic mean value of fuzzy numbers and investigate its relationship to the interval valued probabilistic mean we shall also introduce the notations of crisp weighted possibilistic mean value variance and covariance of fuzzy numbers which are consistent with the extension principle furthermore we show that the weighted variance of linear combination of fuzzy numbers can be computed in a similar manner as in probability theory

zadeh s extension principle for continuous functions of non interactive variables a parallel optimization approach there is a growing interest in the use of fuzzy intervals in many engineering applications however a direct implementation of zadeh s extension principle which forms the basis for computing with fuzzy intervals is still computationally too demanding for practical use in the case of a continuous function and fuzzy intervals that describe non interactive variables as inputs the output is a fuzzy interval as well and can be determined for each cut separately the problem thus reduces to finding the endpoints of these cuts which amounts to a number of interwoven optimization problems in the case of a non monotone continuous function however these optimization problems are non trivial in this paper different optimization algorithms are applied for that purpose gradient descent based on sequential quadratic programming simplex simulated annealing particle swarm optimization and particle swarm optimization combined with gradient descent in addition two approaches are followed to determine a suitable number of cuts either a fixed predetermined number is used or an initially very small number is chosen that is subsequently increased according to a linearity criterion both a non parallel and a parallel implementation are designed the parallel version is restricted to work with particle swarm optimization and employs communication to optimize its internal performance by exploiting the dependence between the various optimization problems different configurations are evaluated on a set of benchmark functions in terms of the mean area under the output fuzzy interval and the number of function evaluations particle swarm optimization combined with gradient descent starting from a small number of cuts leads to the most accurate fuzzy intervals at the cost of a relatively large number of function evaluations

steady flows of inviscid fluids under localized perturbations on essaie d obtenir tous les ecoulements stationnaires dans un canal avec un petit obstacle pour des nombres de froude proches de la valeur critique fo

numerical computation of solitary waves in infinite cylindrical domains the numerical computation of solitary waves to semilinear elliptic equations in infinite cylindrical domains is investigated rather than solving on the infinite cylinder the equation is approximated by a boundary value problem on a finite cylinder convergence and stability results for this approach are given it is also shown that galerkin approximations can be used to compute solitary waves of the elliptic problem on the finite cylinder in addition it is demonstrated that the aforementioned procedures simplify in cases where the elliptic equation admits an additional reversibility structure finally the theoretical predictions are compared with numerical computations in particular post buckling of an infinitely long cylindrical shell under axial compression is considered it is shown numerically that for a fixed spatial truncation the error in the truncation scales with the length of the cylinder as predicted theoretically

on various approaches to normalization of interval and fuzzy weights abstract the paper deals with the problem of fuzzification of the procedure of normalization of weights first the existing methods for normalization of interval and fuzzy weights are reviewed second we study the problem of normalization of a fuzzy vector of weights that expresses the joint possibility distribution of initial weights we show that a correct way is to apply the extension principle proposed by zadeh since the result of such normalization is the fuzzy vector of normalized weights that expresses the true joint possibility distribution of normalized weights further we establish some properties of this approach to normalization that are important from the point of view of real applications finally since an n tuple of non interactive interval or fuzzy weights can be viewed as a fuzzy vector of weights of a special kind we investigate normalization of such kind of fuzzy vectors of weights according to the extension principle we show that from the point of view of the way of modelling uncertain normalized weights the result of this approach can be directly compared only with the result of normalization proposed by wang and elhag 2006 35 we find out that it is not sufficient to express the result of normalization only by an n tuple of normalized interval or fuzzy weights together with the constraint that the sum of the weights is equal to 1 since it can cause a false increase of uncertainty in the model this fact is illustrated by an example

on sieved orthogonal polynomials iii orthogonality on several intervals on introduit deux generalisations des polynomes de chelryshev dont le spectre continu est x 2 c 1 c t k x 2 c 1 c ou c est un parametre positif on definit les fonctions poids on obtient les fonctions generatrices et les formules explicites

necessary and sufficient conditions for the equality of the interactive and non interactive sums of two fuzzy numbers this study aims to answer the open question proposed by carlsson et al 2004 4 which asks whether it is possible to find the properties needed by a joint possibility distribution function such that the interactive sum with respect to this function and the standard sum of two fuzzy numbers will coincide we provide the necessary and sufficient conditions such that this equality would hold in addition we find the necessary and sufficient conditions such that the interactive and standard sums will coincide at some point in the domain in order to obtain an interactive addition that does not coincide with the standard addition we show that we need to search for a joint possibility distribution that coincides as little as possible with the independent joint possibility distribution throughout the so called diagonal of the two fuzzy numbers as simple consequences we deduce similar results for triangular norm based additions of fuzzy numbers

normal hyperbolicity of center manifolds and saint venant s principle the concept of normal hyperbolicity of center manifolds is generalized to infinite dimensional differential equations in particular to elliptic problems in cylindrical domains it is shown that all solutions u staying close to the center manifold for t l l satisfy an estimate of the form left u t tilde u t right leqslant ce alpha l t where c and are independent of l and is a solution on the center manifold these results are applied to saint venant s principle for the static deformation of nonlinearly elastic prismatic bodies the use of the center manifold permits the effective treatment of the general case of non zero resultant forces and moments acting on each cross section

an integrable system connected with the uvarov chihara problem for orthogonal polynomials a new non local integrable chain with continuous time and discrete space variable is considered in contrast to the case of toda and volterra chains the general solution can be presented in explicit form in terms of two arbitrary sequences it is shown that this solution is connected with the so called uvarov chihara problem inserting a discrete mass at the centre of the spectral interval of symmetric orthogonal polynomials the asymptotic behaviour of the recurrence coefficients of such polynomials is considered

on semiclassical orthogonal polynomials via polynomial mappings abstract we consider orthogonal polynomials via polynomial mappings in the framework of the semiclassical class we prove that this class is stable under polynomial transformations several consequences of this fact are deduced as an application we analyze in detail cubic transformations for semiclassical orthogonal polynomials of class at most 2 recovering and extending some results proved recently for class 1 and producing new examples of semiclassical orthogonal polynomials of class 2 in particular we show how to obtain integral representations for the regular functionals with respect to which these new semiclassical families are orthogonal

on the stability of periodic travelling waves with large spatial period abstract in many circumstances a pulse to a partial differential equation pde on the real line is accompanied by periodic wave trains that have arbitrarily large period it is then interesting to investigate the pde stability of the periodic wave trains given that the pulse is stable using the evans function gardner has demonstrated that every isolated eigenvalue of the linearization about the pulse generates a small circle of eigenvalues for the linearization about the periodic waves in this article the precise location of these circles is determined it is demonstrated that the stability properties of the periodic waves depend on certain decay and oscillation properties of the tails of the pulse as a consequence periodic waves with long wavelength typically destabilize at homoclinic bifurcation points at which multi hump pulses are created that is in contrast to the situation for the underlying pulses whose stability properties are not affected by these bifurcations the proof uses lyapunov schmidt reduction and relies on the existence of exponential dichotomies the approach is also applicable to periodic waves with large spatial period of elliptic problems on r n or on unbounded cylinders r with bounded

independence concepts in possibility theory part i abstract from both a theoretical and a practical point of view the study of the concept of independence has great importance in any formalism that manages uncertainty in independence concepts in possibility theory part i de campos and huete fuzzy sets and systems 103 1999 127 152 several independence relationships were proposed using different comparison criteria between conditional possibility measures and using hisdal conditioning as the conditioning operator in this paper we follow the same approach but considering possibility measures as particular cases of consonant plausibility measures and therefore using dempster conditioning instead of hisdal s we formalize several intuitive ideas to define independence relationships namely not to modify not to gain and to obtain similar information after conditioning and study their properties we also compare the results with the previous ones obtained in part i using hisdal conditioning finally the marginal problem i e how to obtain a joint possibility distribution from a set of marginals and the problem of factorizing large possibility distributions in terms of its conditionally independent components are considered

polynomial contractions and lyapunov regularity abstract the purpose of this note is twofold to introduce the notion of polynomial contraction for a linear nonautonomous dynamics with discrete time and to show that it persists under sufficiently small linear and nonlinear perturbations the notion of polynomial contraction mimics the notion of exponential contraction but with the exponential decay replaced by a polynomial decay we show that this behavior is exhibited by a large class of dynamics by giving necessary conditions in terms of polynomial lyapunov exponents finally we establish the persistence of the asymptotic stability of a polynomial contraction under sufficiently small linear and nonlinear perturbations we also consider the case of nonuniform polynomial contractions for which the lyapunov stability is not uniform

smooth conjugacy of centre manifolds in this paper we prove that for a system of ordinary differential equations of class c r 1 1 r 0 and two arbitrary c r 1 1 local centre manifolds of a given equilibrium point the equations when restricted to the centre manifolds are c r conjugate the same result is proved for similinear parabolic equations the method is based on the geometric theory of invariant foliations for centre stable and centre unstable manifolds

nonconvergent bounded solutions of semilinear heat equations on arbitrary domains abstract we consider the dirichlet problem for the semilinear heat equation 0 1 u t u g x u x where is an arbitrary bounded domain in r n n 2 with c 2 boundary we find a c function g x u such that 0 1 has a bounded solution whose limit set is a continuum of equilibria this extends and improves an earlier result of the first author with rybakowski in which is a disk in r 2 and g is of finite differentiability class we also show that 0 1 can have an infinite dimensional manifold of nonconvergent bounded trajectories

exponential dichotomies for solitary wave solutions of semilinear elliptic equations on infinite cylinders in applications solitary wave solutions of semilinear elliptic equations u g u u 0 x y r in infinite cylinders frequently arise as travelling waves of parabolic equations as such their bifurcations are an interesting issue interpreting elliptic equations on infinite cylinders as dynamical systems inxhas proved very useful still there are major obstacles in obtaining for instance bifurcation results similar to those for ordinary differential equations in this article persistence and continuation of exponential dichotomies for linear elliptic equations is proved with this technique at hands lyapunov schmidt reduction near solitary waves can be applied as an example existence of shift dynamics near solitary waves is shown if a perturbation h x u u periodic inxis added

orthogonal polynomials on several intervals via a polynomial mapping starting from a sequence pn x no of orthogonal polynomials with an orthogonality measure yurj supported on eo c 1 1 we construct a new sequence p x fi of orthogonal polynomials on t 1 eq t is a polynomial of degree tv with an orthogonality measure i that is related to no if eo 1 1 then e t 1 l l will in general consist of tv intervals we give explicit formulas relating pn x fi and pn x fio and show how the recurrence coefficients in the three term recurrence formulas for these orthogonal polynomials are related if one chooses t to be a chebyshev polynomial of the first kind then one gets sieved orthogonal polynomials

sieved orthogonal polynomials vii generalized polynomial mappings systems of symmetric orthogonal polynomials whose recurrence relations are given by compatible blocks of second order difference equations are studied in detail applications are given to the theory of the recently discovered sieved orthogonal polynomials the connection with polynomial mappings is examined an example of a family of orthogonal polynomials having discrete masses in the interior of the spectrum is included

an efficient shooting algorithm for evans function calculations in large systems abstract in evans function computations of the spectra of asymptotically constant coefficient linear operators a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ode for small systems a fast shooting algorithm may be obtained by representing subspaces as single exterior products j c alexander r sachs linear instability of solitary waves of a boussinesq type equation a computer assisted computation nonlinear world 2 4 1995 471 507 l q brin numerical testing of the stability of viscous shock waves ph d thesis indiana university bloomington 1998 l q brin numerical testing of the stability of viscous shock waves math comp 70 235 2001 1071 1088 l q brin k zumbrun analytically varying eigenvectors and the stability of viscous shock waves in seventh workshop on partial differential equations part i 2001 rio de janeiro mat contemp 22 2002 19 32 t j bridges g derks g gottwald stability and instability of solitary waves of the fifth order kdv equation a numerical framework physica d 172 1 4 2002 190 216 for large systems however the dimension of the exterior product space quickly becomes prohibitive growing as n k where n is the dimension of the system written as a first order ode and k typically n 2 is the dimension of the subspace we resolve this difficulty by the introduction of a simple polar coordinate algorithm representing pure monomial products as scalar multiples of orthonormal bases for which the angular equation is a numerically optimized version of the continuous orthogonalization method of drury davey a davey an automatic orthonormalization method for solving stiff boundary value problems j comput phys 51 2 1983 343 356 l o drury numerical solution of orr sommerfeld type equations j comput phys 37 1 1980 133 139 and the radial equation is evaluable by quadrature notably the polar coordinate method preserves the important property of analyticity with respect to parameters

efficient computation of analytic bases in evans function analysis of large systems in evans function computations of the spectra of asymptotically constant coefficient linearized operators of large systems a problem that becomes important is the efficient computation of global analytically varying bases for invariant subspaces of the limiting coefficient matrices in the case that the invariant subspace is spectrally separated from its complementary invariant subspace we propose an efficient numerical implementation of a standard projection based algorithm of kato for which the key step is the solution of an associated sylvester problem this may be recognized as the analytic cousin of a c k algorithm developed by dieci and collaborators based on orthogonal projection rather than eigenprojection as in our case for a one dimensional subspace it reduces essentially to an algorithm of bridges derks and gottwald based on path finding and continuation methods

on co recursive orthogonal polynomials and their application to potential scattering abstract let pn x n 0 be a system of polynomials orthogonal with respect to a positive definite moment functional and satisfying the recurrence relation pn x x cn pn 1 x npn 2 x n 1 2 where p0 x 1 and p 1 x 0 the corresponding co recursive orthogonal polynomials p n x n 0 satisfy the same recurrence relations except for n 1 where now p 1 x x c 1 0 and p 0 x 1 the p n are orthogonal with respect to a moment functional which is positive definite for 0 and quasi definite for p n x separation theorems true interval of orthogonality etc can be determined from those of the pn x these polynomials occur in the l2 solution of the radial schrodinger equation for a separable potential where pn x is the tchebichef polynomial of the second kind in the case of s waves

orthogonal polynomials on the unit circle via a polynomial mapping on the real line let f n n 0 be a sequence of monic orthogonal polynomials on the unit circle opuc with respect to a symmetric and finite positive borel measure email protected on 0 email protected and let 1 a 0 a 1 a 2 be the associated sequence of verblunsky coefficients in this paper we study the sequence email protected n n 0 of monic opuc whose sequence of verblunsky coefficients is 1 b 1 b 2 b n 1 a 0 b n 1 b 2 b 1 a 1 b 1 b 2 b n 1 a 2 b n 1 b 2 b 1 a 3 where b 1 b 2 b n 1 are n 1 fixed real numbers such that b email protected 1 1 for all j 1 2 n 1 so that email protected n n 0 is also orthogonal with respect to a symmetric and finite positive borel measure email protected on the unit circle we show that the sequences of monic orthogonal polynomials on the real line oprl corresponding to f n n 0 and email protected n n 0 by szego s transformation are related by some polynomial mapping giving rise to a one to one correspondence between the monic opuc email protected n n 0 on the unit circle and a pair of monic oprl on a subset of the interval 1 1 in particular we prove email protected q email protected n 1 q email protected q n q email protected q n q q n q supported on a subset of the union of 2n intervals contained in 0 email protected such that any two of these intervals have at most one common point and where up to an affine change in the variable z n 1 and email protected n are algebraic polynomials in email protected of degrees n 1 and n respectively defined only in terms of a 0 b 1 b n 1 this measure induces a measure on the unit circle supported on the union of 2n arcs pairwise symmetric with respect to the real axis the restriction to symmetric measures or real verblunsky coefficients is needed in order that szego s transformation may be applicable

the generalized fuzzy derivative is interactive abstract in this article we prove that the generalized difference between a b r f c i e fuzzy numbers with continuous endpoints is given by an interactive difference to be more precise we construct a certain joint possibility distribution i such that the generalized difference coincides with the sup i extension of the subtraction as an immediate consequence we have that every notion of difference between a b r f c that has so far appeared in the literature can be derived from a sup j extension for some particular choice of j moreover we show that both the generalized and the generalized hukuhara derivative of a function f r r f c at x r can be expressed as the limit for h 0 of a difference quotient where the difference is an interactive difference for each h for short we say that the generalized as well as the generalized hukuhara difference is interactive

spectra of certain jacobi operators we discuss the spectra and in particular the essential spectra of some bounded self adjoint jacobi operators associated with orthogonal polynomial sequences obtained via polynomial mappings