Precise evaluation of a polynomial at a point given in staggered correction format

The problem of the evaluation in floating-point arithmetic of a polynomial with floating-point coefficients at a point which is a finite sum of floating-point numbers is studied. The solution is obtained as an infinite convergent series of floating-point numbers. The algorithm requires a precise scalar product, but this can always be implemented by software in a high-level language without assembly language routines as we indicate. A convergence result is proved under a very weak restriction on the size of the degree of the polynomial in terms of the unit roundoff u; roughly speaking, the degree should not be larger than the square root of (1 + u)(2u). Even in the particular case when the point at which to evaluate the polynomial reduces to one floating-point number, we find a new simplified algorithm among the whole family that the preceding convergence result allows.

This problem occurs, among others, in the convergence of the Newton method to some real root of the given polynomial p. If we simply use the Horner scheme to evaluate the polynomial p in a neighbourhood of the root, in some cases the evaluation will contain no correct digits and will prevent us from getting convergence even to machine accuracy. The convergence of iterative methods, among which the Newton method, with added perturbations was the central theme of my talk given at the ICCAM'92. The second part will appear in a forthcoming paper. These added perturbations can represent for example forward or backward errors occurring in finite-precision computations.

The problem discussed here appears in validating some hypotheses of these general convergence results (see the forthcoming paper).     

On constraint controllability of linear systems in Banach spaces

In this paper, we give an addendum to a result of Dolecki and Russell (Ref. 1) related to the duality relationship between observation and control for linear systems in Banach spaces. Our results relate the controllability of a system to the constraint controllability of that system and to the observability of an adjoint system. The main tool used here is an extension of the classical open mapping theorem.     

Generalised technique for calculation of plane director profiles in bounded nematic liquid crystals

We propose an improved conformal mapping technique for analytical calculation of two-dimensional profiles of the nematic liquid crystal (NLC) director in a cell with a simply connected cross-sectional region. We consider the case of the strong anchoring and the piecewise constant director pretilt on the piecewise smooth curve which bounds the region. Obtained expressions for the director profile explicitly depend on the conformal mapping which maps the region onto the upper half plane of a complex plane. An advantage of our method in comparison with the standard conformal mapping technique is that it does not require the knowledge of the inverse mapping and the calculation of the integral in the Poisson formula. Proposed technique allows to take into account topological defects in the bulk of the NLC on the symmetry axis of the region. As an example of how the method can be used, we find an analytical expression for the director profile in a horizontal cylindrical groove partly filled with the NLC. We consider the case where a disclination line parallel to the axis of the groove occurs in the bulk of the NLC. The equilibrium position of the disclination line is found.     

负指数EmdenFowler方程奇异边值问题的正解

本文利用上下解方法和不动点理论给出了负指数ＥｍｄｅｎＦｏｗｌｅｒ方程奇异边值问题有Ｃ［０，１］和Ｃ１［０，１］正解存在的充分必要条件．     

A Concordance Extension Theorem

Let be a manifold approximate fibration between closed manifolds, where , and let be the mapping cylinder of . In this paper it is shown that if is any concordance on , then there exists a concordance such that and . As an application, if and are closed manifolds where is a locally flat submanifold of and and , then a concordance extends to a concordance on such that . This uses the fact that under these hypotheses there exists a manifold approximate fibration , where is a closed -manifold, such that the mapping cylinder is homeomorphic to a closed neighborhood of in by a homeomorphism which is the identity on .

     

On perturbations of M-accretive operators in Banach spaces

In this paper, we consider the solvability of nonlinear equations of the form

where is an m-accretive operator on a Banach space , is a mapping on and .

     

Classifying spaces and homotopy sets of axes of pairings

We consider the maps between classifying spaces of the form . The main theorem shows that if the restriction map on is a weak epimorphism, then the restriction on should factor through the classifying spaces of the center of the compact Lie group . An application implies that is an H-space (Hopf space) if and only if is abelian.

     

Fixed-Point Technique for Implicit Complementarity Problem in Hilbert Lattice

In this paper, we consider and study the implicit complementarity problem in the setting of a Hilbert lattice. It has been shown that this problem can be formulated as a fixed-point problem by using a suitable change of variables. Moreover, this formulation allows us to prove the existence and uniqueness of solutions of the implicit complementarity problem.     

Implicit functions and sensitivity of stationary points

We consider the spaceL(D) consisting of Lipschitz continuous mappings fromD to the Euclideann-space ⁿ ,D being an open bounded subset of ⁿ . LetF belong toL(D) and suppose that solves the equationF(x) = 0. In case that the generalized Jacobian ofF at is nonsingular (in the sense of Clarke, 1983), we show that forG nearF (with respect to a natural norm) the systemG(x) = 0 has a unique solution, sayx(G), in a neighborhood of Moreover, the mapping which sendsG tox(G) is shown to be Lipschitz continuous. The latter result is connected with the sensitivity of strongly stable stationary points in the sense of Kojima (1980); here, the linear independence constraint qualification is assumed to be satisfied.