A Chebyshev polynomial method for line integrals with singularities

In this paper we consider a Chebyshev polynomial method for the calculation of line integrals along curves with Cauchy principal value or Hadamard finite part singularities. The major point we address is how to reconstruct the value of the integral when the parametrization of the curve is unknown and only empirical data are available at some discrete set of nodes. We replace the curve by a near‐minimax parametric polynomial approximation, and express the integrand by means of a sum of Chebyshev polynomials. We make use of a mapping property of the Hadamard finite part operator to calculate the value of the integral. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotic properties of local polynomial regression with missing data and correlated errors

The main objective of this work is the nonparametric estimation of the regression function with correlated errors when observations are missing in the response variable. Two nonparametric estimators of the regression function are proposed. The asymptotic properties of these estimators are studied; expresions for the bias and the variance are obtained and the joint asymptotic normality is established. A simulation study is also included.     

Generation and removal of apparent singularities in linear ordinary differential equations with polynomial coefficients

We discuss several examples of generating apparent singular points as a result of differentiating particular homogeneous linear ordinary differential equations with polynomial coefficients and formulate two general conjectures on the generation and removal of apparent singularities in arbitrary Fuchsian differential equations with polynomial coefficients. We consider a model problem in polymer physics.     

Euler-Maruyama approximations for SDEs with non-Lipschitz coefficients and applications

In this paper Euler-Maruyama approximation for SDE with non-Lipschitz coefficients is proved to converge uniformly to the solution in Lp-space with respect to the time and starting points. As an application, we also study the existence of solution and large deviation principle for anticipative SDE with random initial condition.     

External rays for polynomial maps of two variables associated with Chebyshev maps

The dynamics of regular polynomial endomorphisms of two variables is investigated. Especially, the landing points of the external rays for maps associated with Chebyshev maps are completely characterized.     

Asymptotic error expansions for numerical solutions of one-dimensional problems with singularities

A systematic analysis is given on asymptotic error expansions for numerical solutions of one-dimensional problems whose solutions are singular. Numerical examples show a great improvement on the accuracy of numerical solutions by using the Richardson extrapolation technique.     

Asymptotic approximations for solutions to quasilinear and linear parabolic problems with different perturbed boundary conditions in perforated domains

We consider quasilinear and linear parabolic problems with rapidly oscillating coefficients in a domain Ω _ε that is ε-periodically perforated by small holes of order      

Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients

We show that all eigenfunctions of linear partial differential operators in Rn with polynomial coefficients of Shubin type are extended to entire functions in Cn of finite exponential type 2 and decay like exp(−²|z|) for |z|→∞ in conic neighbourhoods of the form |Imz|?γ|Rez|. We also show that under semilinear polynomial perturbations all nonzero homoclinics keep the super-exponential decay of the above type, whereas a loss of the holomorphicity occurs, namely we show holomorphic extension into a strip {z∈Cn||Imz|?T} for some T>0. The proofs are based on geometrical and perturbative methods in Gelfand-Shilov spaces. The results apply in particular to semilinear Schrödinger equations of the form

(∗)     

Asymptotic behavior and uniqueness for an ultrahyperbolic equation with variable coefficients

This paper describes the asymptotic behavior of solutions of a class of semilinear ultrahyperbolic equations with variable coefficients. One consequence of the general analysis is a uniqueness theorem for a mixed boundary-value problem. Another demonstrates unique continuation at infinity. These results extend previous work by M. H. Protter, [Asymptotic decay for ultrahyperbolic operators, in “Contributions to Analysis” (Lars Ahlfors et al., Eds.), Academic Press, New York, 1974], and A. C. Murray and M. M. Protter, [Indiana U. Math. J.24 (1974), 115–130], on a more restricted class of equations.     

Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients

^** Email: todor{at}math.ethz.ch^*** Corresponding author. Email: schwab{at}math.ethz.ch A scalar, elliptic boundary-value problem in divergence formwith stochastic diffusion coefficient a(x, ) in a bounded domainD ^d is reformulated as a deterministic, infinite-dimensional,parametric problem by separation of deterministic (x D) andstochastic ( ) variables in a(x, ) via Karhúnen–Loèveor Legendre expansions of the diffusion coefficient. Deterministic,approximate solvers are obtained by projection of this probleminto a product probability space of finite dimension M and sparsediscretizations of the resulting M-dimensional parametric problem.Both Galerkin and collocation approximations are considered.Under regularity assumptions on the fluctuation of a(x, ) inthe deterministic variable x, the convergence rate of the deterministicsolution algorithm is analysed in terms of the number N of deterministicproblems to be solved as both the chaos dimension M and themultiresolution level of the sparse discretization resp. thepolynomial degree of the chaos expansion increase simultaneously.     

On polynomial collocation for second kind integral equations with fixed singularities of Mellin type

Summary.  We consider a polynomial collocation for the numerical solution of a second kind integral equation with an integral kernel of Mellin convolution type. Using a stability result by Junghanns and one of the authors, we prove that the error of the approximate solution is less than a logarithmic factor times the best approximation and, using the asymptotics of the solution, we derive the rates of convergence. Finally, we describe an algorithm to compute the stiffness matrix based on simple Gau? quadratures and an alternative algorithm based on a recursion in the spirit of Monegato and Palamara Orsi. All together an almost best approximation to the solution of the integral equation can be computed with 𝒪(n ²[log n]²) resp. 𝒪(n ²) operations, where n is the dimension of the polynomial trial space. Received February 18, 2002 / Revised version received May 15, 2002 / Published online October 29, 2002 RID="⋆" ID="⋆" Correspondence to: A. Rathsfeld Mathematics Subject Classification (1991): 65R20     

Propagation of Gevrey singularities for solutions of linear differential operators with constant coefficients

Summary In [4]L. Hörmander has given sufficient conditions for propagation of C singularities for solutions of linear differential operators P with constant coefficients in terms of limit operators called «localization of P at infinity». In this paper a result (Theorem 1.2) of the same type concerning the propagation of Gevrey singularities is given.     

Asymptotic behavior of spectral functions for elliptic operators with non-smooth coefficients

We consider the asymptotic formula of spectral functions for elliptic operators with non-smooth coefficients of order 2m in . If the coefficients of top order are Hölder continuous of exponent τ∈(0,1], we can derive the remainder estimate of the form O(t^(n−θ)/2m) with any θ∈(0,τ). This result holds without the condition 2m>n, which was always assumed in many papers. We also show that the spectral function is differentiable up to order <m.     

Necessary second-order conditions for a weak local minimum in a problem with endpoint and control constraints

One of the first steps towards necessary second-order optimality conditions in problems with constraints was taken by Dubovitskii and Milyutin in 1965. They offered a scheme that was very effective in smooth optimization problems, but seemed to be not suitable for applications in problems with pointwise control constraints. In this article we consider a modification of the Dubovitskii–Milyutin scheme, which allows to derive necessary second-order conditions for a weak local minimum in optimal control problems with a finite number of endpoint constraints of equality and inequality type and with pointwise control constraints of inequality type given by smooth functions. Assuming that the gradients of active control constraints are linearly independent, we provide rather straightforward proof of these conditions for a measurable and essentially bounded optimal control.     

A method for computing all the zeros of a polynomial with real coefficients

The problem of finding all the zeros of a polynomialP _n (x)=x ⁿ +a _n–1 x ^n–1+...+a ₁ x+a ₀, where the coefficientsa _i are real, can be posed as a system ofn nonlinear equations. The structure of this system allows an efficient numerical solution using a damped Newton method; in particular it is possible to generate the triangular factors of the associated Jacobian matrix directly. This approach provides a natural generalisation of the well-known method of Durand and Kerner.     

Morrey space regularity for weak solutions of Stokes systems with VMO coefficients

We prove that for weak solutions (u, p) of Stokes system with symmetric elliptic coefficients matrix A whose entries are bounded and VMO functions and with right-hand side f in Morrey space L ^2,μ their symmetric gradients Eu and p belong to the same Morrey space L ^2,μ .