Optimal $\mathcal{L}^2$ error bounds for MITC3 type element

Summary. In this paper, we derive the optimal error bounds for the stabilized MITC3 element [3], the MIN3 type element [7] and the T3BL element [8]. In this way we have solved the problem proposed recently in [5] in a positive manner. Moreover, we estimate the difference between stabilized MITC3 and MIN3 and show it is of order uniform in the plate thickness. Received May 31, 2000 / Revised version received April 2, 2001 / Published online September 19, 2001     

The weak conjecture on spherical classes

Let be the mod 2 Steenrod algebra. We construct a chain-level representation of the dual of Singer's algebraic transfer, which maps Singer's invariant-theoretic model of the dual of the Lambda algebra, , to and is the inclusion of the Dickson algebra, , into . This chain-level representation allows us to confirm the weak conjecture on spherical classes (see [9]), assuming the truth of (1) either the conjecture that the Dickson invariants of at least k = 3 variables are homologically zero in }, (2) or a conjecture on ${mathcal{A}}$ -decomposability of the Dickson algebra in $Gamma_k^{wedge}$. We prove the conjecture in item (1) for k = 3 and also show a weak form of the conjecture in item (2). Received November 27, 1996; in final form March 6, 1998     

Ordinal arithmetic and \Sigma_{1}-elementarity

We will introduce a partial ordering on the class of ordinals which will serve as a foundation for an approach to ordinal notations for formal systems of set theory and second-order arithmetic. In this paper we use to provide a new characterization of the ubiquitous ordinal . Received: 18 August 1997     

Asymptotics for the zeros and poles of normalized Pad\'{e} approximants to{\rm e}^{z}

Summary. With denoting the -th partial sum of ${\rm e}^{z}$, the exact rate of convergence of the zeros of the normalized partial sums, , to the Szeg\"o curve was recently studied by Carpenter et al. (1991), where is defined by Here, the above results are generalized to the convergence of the zeros and poles of certain sequences of normalized Pad\'{e} approximants to , where is the associated Pad\'{e} rational approximation to . Received February 2, 1994     

An adaptive Chebyshev iterative method\newline for nonsymmetric linear systems based on modified moments

Summary. Large, sparse nonsymmetric systems of linear equations with a matrix whose eigenvalues lie in the right half plane may be solved by an iterative method based on Chebyshev polynomials for an interval in the complex plane. Knowledge of the convex hull of the spectrum of the matrix is required in order to choose parameters upon which the iteration depends. Adaptive Chebyshev algorithms, in which these parameters are determined by using eigenvalue estimates computed by the power method or modifications thereof, have been described by Manteuffel [18]. This paper presents an adaptive Chebyshev iterative method, in which eigenvalue estimates are computed from modified moments determined during the iterations. The computation of eigenvalue estimates from modified moments requires less computer storage than when eigenvalue estimates are computed by a power method and yields faster convergence for many problems. Received May 13, 1992/Revised version received May 13, 1993     

Upper critical field for superconductors with edges and corners

The effect of non-smoothness of sample surfaces on the value of the upper critical field and on the location of superconductivity nucleation is discussed. It is shown that, superconducting samples with edges and corners have higher value of comparing to samples with smooth surfaces. Superconductivity nucleates first at the top and bottom edges in a cylinder with a finite height, and nucleates first at vertices in a cuboid. Received: 10 September 2000 / Accepted: 11 May 2001 / Published online: 19 October 2001     

A characterization of the \Sigma_1-definable functions of KP\omega + (uniform\; AC)

The subject of this paper is a characterization of the -definable set functions of Kripke-Platek set theory with infinity and a uniform version of axiom of choice: . This class of functions is shown to coincide with the collection of set functionals of type 1 primitive recursive in a given choice functional and . This goal is achieved by a G?del Dialectica-style functional interpretation of and a computability proof for the involved functionals. Received October 9, 1996     

On the Monge mass transfer problem

The Monge mass transfer problem, as proposed by Monge in 1781, is to move points from one mass distribution to another so that a cost functional is minimized among all measure preserving maps. The existence of an optimal mapping was proved by Sudakov in 1979, using probability theory. A proof based on partial differential equations was recently found by Evans and Gangbo. In this paper we give a more elementary and shorter proof by constructing an optimal mapping directly from the potential functions of Monge and Kantorovich. Received May 23, 2000 / Accepted June 12, 2000 / Published online November 9, 2000     

Rational obstruction theory and rational homotopy sets

We develop an obstruction theory for homotopy of homomorphisms between minimal differential graded algebras. We assume that has an obstruction decomposition given by and that f and g are homotopic on . An obstruction is then obtained as a vector space homomorphism . We investigate the relationship between the condition that f and g are homotopic and the condition that the obstruction is zero. The obstruction theory is then applied to study the set of homotopy classes . This enables us to give a fairly complete answer to a conjecture of Copeland-Shar on the size of the homotopy set [A,B] whenA and B are rational spaces. In addition, we give examples of minimal algebras (and hence of rational spaces) that have few homotopy classes of self-maps. Received February 22, 1999; in final form July 7, 1999 / Published online September 14, 2000     

Computation of high order derivatives in optimal shape design

Summary. In shape optimization problems, each computation of the cost function by the finite element method leads to an expensive analysis. The use of the second order derivative can help to reduce the number of analyses. Fujii ([4], [10]) was the first to study this problem. J. Simon [19] gave the second order derivative for the Navier-Stokes problem, and the authors describe in [8], [11], a method which gives an intrinsic expression of the first and second order derivatives on the boundary of the involved domain. In this paper we study higher order derivatives. But one can ask the following questions: -- are they expensive to calculate? -- are they complicated to use? -- are they imprecise? -- are they useless? \medskip\noindent At first sight, the answer seems to be positive, but classical results of V. Strassen [20] and J. Morgenstern [13] tell us that the higher order derivatives are not expensive to calculate, and can be computed automatically. The purpose of this paper is to give an answer to the third question by proving that the higher order derivatives of a function can be computed with the same precision as the function itself. We prove also that the derivatives so computed are equal to the derivatives of the discrete problem (see Diagram 1). We call the discrete problem the finite dimensional problem processed by the computer. This result allows the use of automatic differentiation ([5], [6]), which works only on discrete problems. Furthermore, the computations of Taylor's expansions which are proposed at the end of this paper, could be a partial answer to the last question. Received January 27, 1993/Revised version received July 20, 1993     

Analysis of a part design procedure

Summary. In this paper we analyze and illustrate a new "ab initio" part design procedure, in which, given a cost function which reflects performance, materials, and manufacturing considerations, the topology and the geometry of the part are automatically produced. The analysis is based on demonstration of, first, the compactness of the metric space over which the cost function is defined, and, second, lower semi-continuity of the cost function. Examples include beams and elastic supports. Received November 15, 1993     

On the zeros of the partial sums of \cos(z) and \sin(z)

Summary. Let denote the -th partial sum of the exponential function. Carpenter et al. (1991) [1] studied the exact rate of convergence of the zeros of the normalized partial sums to the so-called Szeg?-curve Here we apply parts of the results found by Carpenter et al. to the zeros of the normalized partial sums of and . Received August 11, 1995     

A finite set of generators for the Kauffman bracket skein algebra

If F is a compact orientable surface it is known that the Kauffman bracket skein module of has a multiplicative structure. Our central result is the construction of a finite set of knots which generate the module as an algebra. We can then define an integer valued invariant of compact orientable 3-manifolds which characterizes . Received November 27, 1995; in final form September 29, 1997     

Classification of sequentially weakly continuous $JB^*$-triples

Let D be the open unit ball of a -triple A and let Aut(D) be the group of all biholomorphic automorphisms of D. It is shown that every element of Aut(D) is sequentially weakly continuous if and only if every primitive ideal of A is a maximal closed ideal and is a type I -triple without infinite-spin part. Implications for general structure theory are explored. In particular, it is deduced that every -triple A contains a smallest ideal J for which the sequentially weakly continuous biholomorphic automorphisms of the open unit ball of A/J are all linear. Received August 27, 1998; in final form February 10, 1999     

On the orthogonality of residual polynomials\newline of minimax polynomial preconditioning

Summary. This paper studies polynomials used in polynomial preconditioning for solving linear systems of equations. Optimum preconditioning polynomials are obtained by solving some constrained minimax approximation problems. The resulting residual polynomials are referred to as the de Boor-Rice and Grcar polynomials. It will be shown in this paper that the de Boor-Rice and Grcar polynomials are orthogonal polynomials over several intervals. More specifically, each de Boor-Rice or Grcar polynomial belongs to an orthogonal family, but the orthogonal family varies with the polynomial. This orthogonality property is important, because it enables one to generate the minimax preconditioning polynomials by three-term recursive relations. Some results on the convergence properties of certain preconditioning polynomials are also presented. Received February 1, 1992/Revised version received July 7, 1993