the effects of rounding errors in the nodes on barycentric interpolation we analyze the effects of rounding errors in the nodes on polynomial barycentric interpolation these errors are particularly relevant for the first barycentric formula with the chebyshev points of the second kind here we propose a method for reducing them

a coalgebraic perspective on linear weighted automata weighted automata are a generalisation of non deterministic automata where each transition in addition to an input letter has also a quantity expressing the weight e g cost or probability of its execution as for non deterministic automata their behaviours can be expressed in terms of either weighted bisimilarity or weighted language equivalence coalgebras provide a categorical framework for the uniform study of state based systems and their behaviours in this work we show that coalgebras can suitably model weighted automata in two different ways coalgebras on set the category of sets and functions characterise weighted bisimilarity while coalgebras on vect the category of vector spaces and linear maps characterise weighted language equivalence relying on the second characterisation we show three different procedures for computing weighted language equivalence the first one consists in a generalisation of the usual partition refinement algorithm for ordinary automata the second one is the backward version of the first one the third procedure relies on a syntactic representation of rational weighted languages

barycentric interpolation collocation methods for solving linear and nonlinear high dimensional fredholm integral equations abstract in this article two barycentric interpolation collocation methods are proposed for solving linear and nonlinear high dimensional f r e d h o l m integral equations of the second kind the approaches respectively utilize the modified weighted l a g r a n g e functions and the novel rational functions as the interpolation basis functions they are effective schemes for evaluating the multidimensional undetermined function through the numerical strategies and some composite quadrature formulas the linear and nonlinear f r e d h o l m integral equations are transformed into the corresponding linear and nonlinear algebraic equations further we prove that the discrete collocation methods are equivalent to the n y s t r o m quadrature methods then the convergence analysis is established by the collectively compact theory moreover the error estimation of the approximate solution and the exact solution are also provided numerical examples are presented to illustrate the capability and efficiency of the techniques by compared with the classic l a g r a n g e interpolation collocation method and other methodologies

on varieties of meet automata eilenberg s variety theorem gives a bijective correspondence between varieties of languages and varieties of finite monoids the second author gave a similar relation between conjunctive varieties of languages and varieties of semiring homomorphisms in this paper we add a third component to this result by considering varieties of meet automata we consider three significant classes of languages two of them consisting of reversible languages we present conditions on meet automata and identities for semiring homomorphisms for their characterization

newton interpolation in fejer and chebyshev points soit une courbe de jordan dans le plan complexe et soit l ensemble compact borne par soit f une fonction analytique sur l approximation de f sur par un polynome p de degre inferieur a n et qui interpole f en n points de est etudiee

accurate computation of divided differences of the exponential function abstract the traditional recurrence for the computation of exponential divided differences along with a new method based on the properties of the exponential function are studied in detail in this paper our results show that it is possible to combine these two methods to compute exponential divided differences accurately a hybrid algorithm is presented for which our error bound grows quite slowly with the order of the divided difference author

on the lebesgue constant of barycentric rational interpolation at equidistant nodes recent results reveal that the family of barycentric rational interpolants introduced by floater and hormann is very well suited for the approximation of functions as well as their derivatives integrals and primitives especially in the case of equidistant interpolation nodes these infinitely smooth interpolants offer a much better choice than their polynomial analogue a natural and important question concerns the condition of this rational approximation method in this paper we extend a recent study of the lebesgue function and constant associated with berrut s rational interpolant at equidistant nodes to the family of floater hormann interpolants which includes the former as a special case

a rational interpolation scheme with superpolynomial rate of convergence the purpose of this study is to construct a high order interpolation scheme for arbitrary scattered datasets the resulting function approximation is an interpolation function when the dataset is exact or a regression if measurement errors are present we represent each datapoint with a taylor series and the approximation error as a combination of the derivatives of the target function a weighted sum of the square of the coefficient of each derivative term in the approximation error is minimized to obtain the interpolation approximation the resulting approximation function is a high order rational function with no poles when measurement errors are absent the interpolation approximation converges to the target function faster than any polynomial rate of convergence

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

fast and stable contour integration for high order divided differences via elliptic functions in this paper we will present a new method for evaluating high order divided differences for certain classes of analytic possibly operator values functions this is a classical problem in numerical mathematics but also arises in new applications such as e g the use of generalized convolution quadrature to solve retarded potential integral equations the functions which we will consider are allowed to grow exponentially to the left complex half plane and the interpolation points are scattered in a large real interval our approach is based on the representation of divided differences as contour integral and we will employ a subtle parameterization of the contour in combination with a quadrature approximation by the trapezoidal rule

stability of barycentric interpolation formulas for extrapolation the barycentric interpolation formula defines a stable algorithm for evaluation at points in 1 1 of polynomial interpolants through data on chebyshev grids here it is shown that for evaluation at points in the complex plane outside 1 1 the algorithm becomes unstable and should be replaced by the alternative modified lagrange or first barycentric formula dating to jacobi in 1825 this difference in stability confirms the theory published by n j higham in 2004 ima j numer anal 24 2004 pp 547 556 and has practical consequences for computation with rational functions

the stability of barycentric interpolation at the chebyshev points of the second kind we present a new analysis of the stability of the first and second barycentric formulae for interpolation at the chebyshev points of the second kind our theory shows that the second formula is more stable than previously thought and our experiments confirm its stability in practice we also extend our current understanding regarding the accuracy problems of the first barycentric formula

an improved upper bound on the lebesgue constant of berrut s rational interpolation operator in approximation theory the lebesgue constant of an interpolation operator plays an important role the lebesgue constant of berrut s interpolation operator has been studied recently in the present work an improved upper bound for equidistant nodes is given

orbital stability of spatially synchronized solitary waves of an m coupled nonlinear schrodinger system in this paper we investigate the orbital stability of solitary wave solutions for an m coupled nonlinear schrodinger system i tuj 2 x2uj i 1mbijui2uj 0 j 1 m where m 2 uj are complex valued functions of x t 2 bjj j 1 2 m and bij i j are positive coupling constants satisfying bij bji it will be shown that spatially synchronized solitary wave solutions of the m coupled nonlinear schrodinger system exist and are orbitally stable here by synchronized solutions we mean solutions in which the components are proportional to one another our results completely settle the question on the existence and stability of synchronized solitary waves for the m coupled system while only partial results were known in the literature for the cases of m 3 heretofore furthermore the conditions imposed on the symmetric matrix b bij satisfied here are both sufficient and necessary for the m coupled nonlinear schrodinger system to admit synchronized ground state solutions

the accuracy of floating point summation the usual recursive summation technique is just one of several ways of computing the sum of n floating point numbers five summation methods and their variations are analyzed here the accuracy of the methods is compared using rounding error analysis and numerical experiments four of the methods are shown to be special cases of a general class of methods and an error analysis is given for this class no one method is uniformly more accurate than the others but some guidelines are given on the choice of method in particular cases

barycentric hermite interpolation let z 1 ldots z k be distinct grid points if f k 0 is the prescribed value of a function at the grid point z k and f k r the prescribed value of the r th derivative for 1 leq r leq n k 1 the hermite interpolant is the unique polynomial of degree n 1 n n 1 cdots n k which interpolates the prescribed function values and function derivatives we obtain another derivation of a method for hermite interpolation recently proposed by butcher et al numer algorithms 56 2011 pp 319 347 one advantage of our derivation is that it leads to an efficient method for updating the barycentric weights if an additional derivative is prescribed at one of the interpolation points we show how to update the barycentric coefficients using only mathcal o n operations even in the context of confluent newton series a comparably efficient and general method to update the coefficients appears not to be known if the method is properly implemented it computes the barycentric wei

on the numerical stability of newton s formula for lagrange interpolation abstract it was previously noticed in the literature that the computation of newton s formula can be very sensitive to the ordering of the input data and it may be numerically unstable even for interpolation at the chebyshev nodes nevertheless to the best of our knowledge the effects of rounding errors in the computation of newton s formula have not been properly quantized yet in this note we make a full stability analysis of the propagation of rounding errors in the computation of newton s formula for lagrange interpolation we derive sharp upper bounds for the numerical error in the computations of three algorithms for computing polynomial interpolants in newton form that can be used to understand the instabilities mentioned above we throw light on and give a practical meaning to the condition number defined in carnicer et al 2019 by showing that it is the right thermometer for measuring the effects of rounding errors in the computation of newton s formula

coalgebraic constructions of canonical nondeterministic automata for each regular language l we describe a family of canonical nondeterministic acceptors nfas their construction follows a uniform recipe build the minimal dfa for l in a locally finite variety v and apply an equivalence between the category of finite v algebras and a suitable category of finite structured sets and relations by instantiating this to different varieties we recover three well studied canonical nfas v boolean algebras yields the atomaton of brzozowski and tamm v semilattices yields the jiromaton of denis lemay and terlutte and v z 2 vector spaces yields the minimal xor automaton of vuillemin and gama moreover we obtain a new canonical nfa called the distromaton by taking v distributive lattices each of these nfas is shown to be minimal relative to a suitable measure and we derive sufficient conditions for their state minimality our approach is coalgebraic exhibiting additional structure and universal properties of the canonical nfas

stable standing waves of nonlinear klein gordon equations in this paper we give sufficient conditions for the stability of the standing waves of least energy for nonlinear klein gordon equations

stability of standing waves for the fractional schrodinger choquard equation abstract in this paper we consider the stability of standing waves for the fractional schrodinger choquard equation with an l 2 critical nonlinearity by using the profile decomposition of bounded sequences in h s and variational methods we prove that the standing waves are orbitally stable we extend the study of bhattarai for a single equation bhattarai 2017 to the l 2 critical case

on the constrained mock chebyshev least squares the algebraic polynomial interpolation on n 1 uniformly distributed nodes can be affected by the runge phenomenon also when the function f to be interpolated is analytic among all techniques that have been proposed to defeat this phenomenon there is the mock chebyshev interpolation which produces a polynomial p that interpolates f on a subset of m 1 of the given nodes whose elements mimic as well as possible the chebyshev lobatto points of order m in this work we use the simultaneous approximation theory to produce a polynomial p of degree r greater than m which still interpolates f on the m 1 mock chebyshev nodes minimizing at the same time the approximation error in a least squares sense on the other points of the sampling grid we give indications on how to select the degree r in order to obtain polynomial approximant good in the uniform norm furthermore we provide a sufficient condition under which the accuracy of the mock chebyshev interpolation in the uniform norm is improved numerical results are provided

on the lebesgue constant of barycentric rational hermite interpolants at equidistant nodes abstract barycentric rational floater hormann interpolants compare favourably to classical polynomial interpolants in the case of equidistant nodes because the lebesgue constant associated with these interpolants grows logarithmically in this setting in contrast to the exponential growth experienced by polynomials in the hermite setting in which also the first derivatives of the interpolant are prescribed at the nodes the same exponential growth has been proven for polynomial interpolants and the main goal of this paper is to show that much better results can be obtained with a recent generalization of floater hormann interpolants after summarizing the construction of these barycentric rational hermite interpolants we study the behaviour of the corresponding lebesgue constant and prove that it is bounded from above by a constant several numerical examples confirm this result

asymptotic stability of small gap solitons in nonlinear dirac equations we prove dispersive decay estimates for the one dimensional dirac operator and use them to prove asymptotic stability of small gap solitons in the nonlinear dirac equations with quintic and higher order nonlinear terms

on the dynamics of solitons in the nonlinear schrodinger equation we study the behavior of the soliton solutions of the equation i frac partial psi partial t frac 1 2m delta psi frac 1 2 w varepsilon prime psi v x psi where w varepsilon prime is a suitable nonlinear term which is singular for varepsilon 0 we use the strong nonlinearity to obtain results on existence shape stability and dynamics of the soliton the main result of this paper theorem 1 shows that for varepsilon rightarrow 0 the orbit of our soliton approaches the orbit of a classical particle in a potential v x

a numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis functions abstract in this paper we use hat basis functions to solve the system of fredholm integral equations sfies and the system of volterra integral equations svies of the second kind this method converts the system of integral equations into a linear or nonlinear system of algebraic equations also we consider the order of convergence of the method and show that it is o h 2 application of the method on some examples show its accuracy and efficiency

stable solutions of the logarithmic schrodinger equation on considere l equation de schrodinger suivante u t u ulog u 2 0 dans r r n on donne des resultats sur le comportement asymptotique des solutions

sharp thresholds of global existence and blowup for a system of schrodinger equations with combined power type nonlinearities in this paper we consider the cauchy problem of a nonlinear schrodinger system through establishing a sharp weighted vector valued gagliardo nirenberg s inequality we find that the best constant in this inequality can be regarded as the criterion of blowup and global existence of the solutions when p 4 n and we prove that the solutions of this system will always exist globally if p 4 n the sharp thresholds for blowup and global existence are also obtained when 4 n p 4 n 2

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

convergence of linear barycentric rational interpolation for analytic functions polynomial interpolation to analytic functions can be very accurate depending on the distribution of the interpolation nodes however in equispaced nodes and the like besides being badly conditioned these interpolants fail to converge even in exact arithmetic in some cases linear barycentric rational interpolation with the weights presented by floater and hormann can be viewed as blended polynomial interpolation and often yields better approximation in such cases this has been proven for differentiable functions and indicated in several experiments for analytic functions so far these rational interpolants have been used mainly with a constant parameter usually denoted by d the degree of the blended polynomials which leads to small condition numbers but to merely algebraic convergence with the help of logarithmic potential theory we derive asymptotic convergence results for analytic functions when this parameter varies with the number of nodes moreover we present suggestions for how to choos

a method for directly finding the denominator values of rational interpolants in this paper we give an algorithm for directly finding the denominator values of rational interpolants at the nodes and present an expression for the corresponding rational interpolant when the latter exists with these denominator values our method also provides information concerning the existence of the interpolant and the presence of unattainable points and poles

numerical solution of second kind fredholm integral equations system by using a taylor series expansion method in this paper we present a taylor series expansion method for a second kind fredholm integral equations system with smooth or weakly singular kernels this method reduce the system of integral equations to a linear system of ordinary differential equation after constructing boundary conditions this system reduce to a system of equations that can be solved easily with any of the usual methods finally we show the efficiency of the method using some numerical examples

a numerical solution for solving system of fredholm integral equations by using homotopy perturbation method abstract homotopy perturbation method is applied to the numerical solution for solving system of fredholm integral equations comparison of the result obtained by the present method with that obtained by taylor series expansion method k maleknejad n aghazade m rabbani numerical solution of second kind fredholm integral equations system by using a taylor series expansion method appl math comput 175 2006 1229 1234 reveals that the present method is very effective and convenient

relaxation of solitons in nonlinear schrodinger equations with potential abstract in this paper we study dynamics of solitons in the generalized nonlinear schrodinger equation nls with an external potential in all dimensions except for 2 for a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space centered near different critical points of the potential we call those solutions which are centered near the minima of the potential and which minimize energy restricted to l 2 unit sphere trapped solitons or just solitons in this paper we prove under certain conditions on the potentials and initial conditions that trapped solitons are asymptotically stable moreover if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position the dynamical law of motion of the soliton i e effective equations of motion for the soliton s center and momentum is close to newton s equation but with a dissipative term due to radiation of the energy to infinity

optimal asymptotic lebesgue constant of berrut s rational interpolation operator for equidistant nodes in approximation theory the lebesgue constant of an interpolation operator plays an important role the lebesgue constant of berrut s interpolation operator has been extensive studied in the present work by introducing a new method we obtain an optimal asymptotic lebesgue constant of berrut s rational interpolant at equidistant nodes

numerical solutions for the system of fredholm integral equations of second kind by a new approach involving semiorthogonal b spline wavelet collocation method in this paper a new approach based on linear b spline wavelet method has been developed to approximate the solutions of system of linear fredholm integral equations of second kind compactly supported linear semi orthogonal b spline scaling functions and wavelet functions together with their dual functions are applied to approximate the solutions of linear fredholm integral equations system of second kind this method reduces the system of integral equations to a linear system of algebraic equations that can be solved easily with any of the usual methods the numerical results obtained by the present method have been compared with those obtained by adaptive method based on trapezoidal rule numerical examples are presented to illustrate the accuracy of the method

the bezout matrix for hermite interpolants abstract the aim of this paper is to introduce the construction of the bezout matrix of two univariate polynomials given by values in the hermite interpolation basis namely the confluent bezout matrix moreover if such polynomials have exactly one common simple zero we describe how to compute it from the null space of the confluent bezout matrix

polynomial interpolation via mapped bases without resampling abstract in this work we propose a new method for univariate polynomial interpolation based on what we call mapped bases as theoretically shown constructing the interpolating function via the mapped bases i e in the mapped space turns out to be equivalent to map the nodes and then construct the approximant in the classical form without the need of resampling in view of this we also refer to such mapped points as fake nodes numerical evidence confirms that such scheme can be applied to mitigate runge and gibbs phenomena

the linear rational pseudospectral method with iteratively optimized poles for two point boundary value problems an algorithm is proposed which improves upon the polynomial pseudospectral method for solving linear two point boundary value problems in the latter the collocation points are the vertical projection onto the interval of points equidistant or nearly equidistant on the circle and they therefore accumulate in the vicinity of the extremities of the interval thus the method is well suited for solving problems whose solutions have boundary layers but not as good at approximating solutions with large gradients shocks away from the extremities of their domain of definition our idea is to modify the polynomial ansatz by attaching a denominator so as to make it a rational interpolant the denominator is then successively optimized in an iterative procedure with each step consisting of the solution of two problems an optimization of the denominator for given values of the approximation to the solution u at the interpolation points and a collocation in the linear space of the rational interpolants with the just obtained fixed denominator to obtain new approximate values of u we show the efficiency of a galerkin version of the method and discuss the power of the collocation version with several numerical examples

moments in quadrature problems abstract an account is given of the role played by moments and modified moments in the construction of quadrature rules specifically weighted newton cotes and gaussian rules fast and slow lagrange interpolation algorithms combined with gaussian quadrature as well as linear algebra methods based on moment equations are described for generating newton cotes formulae the weaknesses and strengths of these methods are illustrated in concrete examples involving weight functions with and without singularities new conjectures are formulated concerning the positivity of certain newton cotes formulae for jacobi weight functions and for the logistics weight with numerical evidence being provided to support them finally an inherent limitation is pointed out in the use of moment information to construct gauss type quadrature rules for the hermite weight function on bounded or half infinite intervals

instability of nonlinear bound states we establish a sharp instability theorem for the bound states of lowest energy of the nonlinear klein gordon equation utt u f u 0 and the nonlinear schrodinger equation iut u f u 0

stability and instability of the kdv solitary wave under the kp i flow we consider the kp i and gkp i equations in mathbb r times mathbb r 2 pi mathbb z we prove that the kdv soliton with subcritical speed 0 c in the spirit of the work by duyckaerts and merle gafa 18 1787 1840 2009 we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity this last result also holds for the gkp i equation

polynomial interpolation lagrange versus newton we show that the lagrangian form of the interpolating polynomial can be calculated with the same number of arithmetic operations as the newtonian form

the stability of extended floater hormann interpolants we present a new analysis of the stability of extended floater hormann interpolants in which both noisy data and rounding errors are considered contrary to what is claimed in the current literature we show that the lebesgue constant of these interpolants can grow exponentially with the parameters that define them and we emphasize the importance of using the proper interpretation of the lebesgue constant in order to estimate correctly the effects of noise and rounding errors we also present a simple condition that implies the backward instability of the barycentric formula used to implement extended interpolants our experiments show that extended interpolants mentioned in the literature satisfy this condition and therefore the formula used to implement them is not backward stable finally we explain that the extrapolation step is a significant source of numerical instability for extended interpolants based on extrapolation

existence and orbital stability of standing waves for nonlinear schrodinger systems abstract in this paper we investigate the existence of solutions in h 1 r n h 1 r n for nonlinear schrodinger systems of the form u 1 1 u 1 1 u 1 p 1 2 u 1 r 1 u 1 r 1 2 u 1 u 2 r 2 u 2 2 u 2 2 u 2 p 2 2 u 2 r 2 u 1 r 1 u 2 r 2 2 u 2 under the constraints r n u 1 2 d x a 1 0 r n u 2 2 d x a 2 0 here n 1 0 i 0 r i 1 2 p i 2 4 n for i 1 2 and r 1 r 2 2 4 n this problem is motivated by the search of standing waves for an evolution problem appearing in several physical models our solutions are obtained as constrained global minimizers of an associated functional note that in the system 1 and 2 are unknown and will correspond to the lagrange multipliers our main result is the precompactness of the minimizing sequences up to translation assuming the local well posedness of the associated evolution problem we then obtain the orbital stability of the standing waves associated to the set of minimizers

lebesgue constant minimizing linear rational interpolation of continuous functions over the interval abstract polynomial interpolation between large numbers of arbitrary nodes does notoriously not in general yield useful approximations of continuous functions following 1 we suggest to determine for each set of n 1 nodes another denominator of degree n and then to interpolate every continuous function by a rational function with that same denominator so that the resulting interpolation process remains a linear projection the optimal denominator is chosen so as to minimize the lebesgue constant for the given nodes it has to be computed numerically for that purpose the barycentric representation of rational interpolants which displays the linearity of the interpolation and reduces the determination of the denominator to that of the barycentric weights is used the optimal weights can then be computed by solving an optimization problem with simple bounds which could not be solved accurately by the first author in 1 we show here how to do so and we present numerical results

global existence for a coupled system of schrodinger equations with power type nonlinearities in this manuscript we consider the cauchy problem for a schrodinger system with power type nonlinearitiesi tuj uj k 1majk uk p uj p 2uj 0 uj x 0 j0 x where uj rn r c j0 rn c for j 1 2 m and ajk akj are positive real numbers global existence for the cauchy problem is established for a certain range of p a sharp form of a vector valued gagliardo nirenberg inequality is deduced which yields the minimal embedding constant for the inequality using this minimal embedding constant global existence for small initial data is shown for the critical case p 1 2 n finite time blow up as well as stability of solutions in the critical case is discussed

on the lebesgue constant of berrut s rational interpolant at equidistant nodes it is well known that polynomial interpolation at equidistant nodes can give bad approximation results and that rational interpolation is a promising alternative in this setting in this paper we confirm this observation by proving that the lebesgue constant of berrut s rational interpolant grows only logarithmically in the number of interpolation nodes moreover the numerical results suggest that the lebesgue constant behaves similarly for interpolation at chebyshev as well as logarithmically distributed nodes

an extension of the floater hormann family of barycentric rational interpolants the barycentric rational interpolants introduced by floater and hormann in 2007 are blends of polynomial interpolants of fixed degree d in some cases these rational functions achieve approximation of much higher quality than the classical polynomial interpolants which e g are ill conditioned and lead to runge s phenomenon if the interpolation nodes are equispaced for such nodes however the condition of floater hormann interpolation deteriorates exponentially with increasing d in this paper an extension of the floater hormann family with improved condition at equispaced nodes is presented and investigated the efficiency of its applications such as the approximation of derivatives integrals and primitives of functions is compared to the corresponding results recently obtained with the original family of rational interpolants math subject classification 65d05 65l12 65d32 41a05 41a20 41a25

stability and instability of standing waves to a system of schrodinger equations with combined power type nonlinearities abstract we obtain the existence of standing wave solutions to a coupled nonlinear schrodinger system i u 1 t u 1 u 1 p k 1 m k u 1 p k 2 u 2 q k u 1 in r n 0 t i u 2 t u 2 u 2 p k 1 m k u 1 p k u 2 q k 2 u 2 in r n 0 t u 1 x 0 u 10 x u 2 x 0 u 20 x in r n where 0 p 4 n 2 and k k p k q k are real constants with 1 p k p 1 1 q k p 1 p k q k p 2 and k q k k p k k 1 m we also discuss the stability and instability of the standing wave solutions

terminal coalgebras in well founded set theory abstract this paper shows that in order to obtain the theorem of aczel and mendler on the existence of terminal coalgebras for an endofunctor on the category of sets it is entirely unnecessary to delve into such exotica as non well founded set theory in addition we discuss the canonical map from the initial algebra for an endofunctor on sets to the terminal coalgebra and show that in many cases it embeds the former as a dense subset of the latter in a certain natural topology by way of example we calculate the terminal coalgebra for various simple endofunctors

stability of solitary waves for the vector nonlinear schrodinger equation in higher order sobolev spaces abstract in this manuscript a sharp form of orbital stability for the schrodinger system also known as the vector nls i t u j 2 x x u j 2 i 1 m u i 2 u j 0 where u j are complex valued functions of x t r 2 j 1 2 m is established in l 2 based sobolev classes of arbitrarily high order the result means practically that not only does the bulk of what emanates from the perturbed solitary wave stay close in shape and propagation and phase speeds to the original solitary wave but emerging residual oscillations must also be very small and not only in the energy norm

erratum to sharp thresholds of global existence and blowup for a system of schrodinger equations with combined power type nonlinearities j math phys 51 033509 2010

ultimately fast accurate summation we present two new algorithms fastaccsum and fastprecsum one to compute a faithful rounding of the sum of floating point numbers and the other for a result as if computed in k fold precision faithful rounding means the computed result either is one of the immediate floating point neighbors of the exact result or is equal to the exact sum if this is a floating point number the algorithms are based on our previous algorithms accsum and precsum and improve them by up to 25 the first algorithm adapts to the condition number of the sum i e the computing time is proportional to the difficulty of the problem the second algorithm does not need extra memory and the computing time depends only on the number of summands and k both algorithms are the fastest known in terms of flops they allow good instruction level parallelism so that they are also fast in terms of measured computing time the algorithms require only standard floating point addition subtraction and multiplication in one working precision for example double precision

matrices for the direct determination of the barycentric weights of rational interpolation abstract let x0 xn be n 1 interpolation points nodes and 0 n be n 1 interpolation data then every rational function r with numerator and denominator degress n interpolating these values can be written in its barycentric form completely determined by a vector u of n 1 barycentric weights uk finding u is therefore an alternative to the determination of the coefficients in the canonical form of r it is advantageous inasmuch as u contains information about unattainable points and poles in classical rational interpolation the numerator and the denominator of r are made unique up to a constant factor by restricting their respective degrees we determine here the corresponding vectors u by applying a stabilized elimination algorithm to a matrix whose kernel is the space spanned by the u s the method is of complexity o n3 in terms of the denominator degree n it seems on the other hand to be among the most stable ones

orbital stability of solitary waves of a 3 coupled nonlinear schrodinger system abstract in this paper consideration is given to the 3 coupled nonlinear schrodinger system i t u j 2 x 2 u j i 1 3 b i j u i 2 u j 0 where u j are complex valued functions of x t r 2 j 1 2 3 and b i j are positive constants satisfying b i j b j i it will be shown first that if the symmetric matrix b b i j satisfies certain conditions then ground state solutions of the 3 coupled nonlinear schrodinger system exist and moreover they are orbitally stable the theory is then extended to include solitary waves as well in particular it will be shown that when a solitary wave is perturbed the perturbed solution must stay close to a solitary wave profile in which the translation and phase parameters are prescribed functions of time properties of these functions are then studied this is a continuous work of our previous paper where the 2 coupled nonlinear schrodinger system was considered

meshfree approach for solving multi dimensional systems of fredholm integral equations via barycentric lagrange interpolation abstract a highly accurate meshfree approach based on the barycentric lagrange basis functions is reported for solving the linear and nonlinear multi dimensional systems of fredholm integral equations fies of the second kind the method is an improved lagrange interpolation technique which possesses high precision and inexpensive procedure the systems of fies are handled with the barycentric formula and transformed into the corresponding linear and nonlinear systems of algebraic equations the stability of the solved integral equation systems is guaranteed by the numerical stability of the barycentric lagrange formula which is dominated by barycentric weights the convergence analysis and error estimation are included some computational results indicate that the method is simple and efficient

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

new numerical method for ordinary differential equations newton polynomial abstract the adams bashforth have been recognized to be a very efficient numerical method to solve linear and nonlinear differential equations including those with non integer orders this method is based on the well known lagrange interpolation however it is well known that the lagrange polynomial is less accurate than the newton polynomial in this paper we introduced a new numerical scheme based on two steps newton polynomial we present some applications and illustrative examples for both ordinary differential equations with classical and fractional derivative

explicit barycentric weights for polynomial interpolation in the roots or extrema of classical orthogonal polynomials barycentric interpolation is arguably the method of choice for numerical polynomial interpolation in this paper we show that barycentric weights for the roots or extrema of classical families of orthogonal polynomials are expressible explicitly in terms of the nodes and weights of the corresponding gaussian quadrature rule based on the glaser liu rokhlin algorithm for gaussian quadrature this leads to an o n computational scheme for computing barycentric weights for the jacobi case known results on the barycentric weights of the chebyshev points are recovered as special cases and some new results such as the barycentric weights associated with the jacobi points and the gauss jacobi lobatto points are obtained we also show that the interpolants in the roots or extrema of classical orthogonal polynomials can be computed rapidly and stably by using their barycentric representations

on the denominator values and barycentric weights of rational interpolants we improve upon the method of zhu and zhu a method for directly finding the denominator values of rational interpolants j comput appl math 148 2002 341 348 for finding the denominator values of rational interpolants reducing considerably the number of arithmetical operations required for their computation in a second stage we determine the points if existent which can be discarded from the rational interpolation problem furthermore when the interpolant has a linear denominator we obtain a formula for the barycentric weights which is simpler than the one found by berrut and mittelmann matrices for the direct determination of the barycentric weights of rational interpolation j comput appl math 78 1997 355 370 subsequently we give a necessary and sufficient condition for the rational interpolant to have a pole

on the solitary wave dynamics under slowly varying medium for nonlinear schrodinger equations we consider the problem of a solitary wave propagation in a slowly varying medium for a variable coefficients nonlinear schrodinger equation we prove global existence and uniqueness of solitary wave solutions for a large class of slowly varying media moreover we describe for all time the behavior of these solutions which include refracted and reflected solitary waves depending on the initial energy