Summability and estimates for polynomials and multilinear mappings

In this paper we extend and generalize several known estimates for homogeneous polynomials and multilinear mappings on Banach spaces. Applying the theory of absolutely summing nonlinear mappings, we prove that estimates which are known for mappings on ?_p spaces in fact hold true for mappings on arbitrary Banach spaces.     

A convenient preparation of 8R,25-dihydroxy-9,10-seco-4,6,10(19)-cholestatrien-3-one

Summary The title compound was prepared in three steps from 25-hydroxyvitamin D₃. The keystep in this sequence is the transformation of3b to4 by means of the LiBr/HMPT complex.Dedicated to Prof. Dr. Karl Schlögl on the occasion of his 65th anniversary     

On the tensor rank of the multiplication in the finite fields

First, we prove the existence of certain types of non-special divisors of degree g−1 in the algebraic function fields of genus g defined over Fq. Then, it enables us to obtain upper bounds of the tensor rank of the multiplication in any extension of quadratic finite fields Fq by using Shimura and modular curves defined over Fq. From the preceding results, we obtain upper bounds of the tensor rank of the multiplication in any extension of certain non-quadratic finite fields Fq, notably in the case of F₂. These upper bounds attain the best asymptotic upper bounds of Shparlinski-Tsfasman-Vladut [I.E. Shparlinski, M.A. Tsfasman, S.G. Vladut, Curves with many points and multiplication in finite fields, in: Lecture Notes in Math., vol. 1518, Springer-Verlag, Berlin, 1992, pp. 145-169].     

Risk-neutral valuation of participating life insurance contracts in a stochastic interest rate environment

Over the last years, the valuation of life insurance contracts using concepts from financial mathematics has become a popular research area for actuaries as well as financial economists. In particular, several methods have been proposed of how to model and price participating policies, which are characterized by an annual interest rate guarantee and some bonus distribution rules. However, despite the long terms of life insurance products, most valuation models allowing for sophisticated bonus distribution rules and the inclusion of frequently offered options assume a simple Black–Scholes setup and, more specifically, deterministic or even constant interest rates.We present a framework in which participating life insurance contracts including predominant kinds of guarantees and options can be valuated and analyzed in a stochastic interest rate environment. In particular, the different option elements can be priced and analyzed separately. We use Monte Carlo and discretization methods to derive the respective values.The sensitivity of the contract and guarantee values with respect to multiple parameters is studied using the bonus distribution schemes as introduced in [Bauer, D., Kiesel, R., Kling, A., Ruß, J., 2006. Risk-neutral valuation of participating life insurance contracts. Insurance: Math. Econom. 39, 171–183]. Surprisingly, even though the value of the contract as a whole is only moderately affected by the stochasticity of the short rate of interest, the value of the different embedded options is altered considerably in comparison to the value under constant interest rates. Furthermore, using a simplified asset portfolio and empirical parameter estimations, we show that the proportion of stock within the insurer’s asset portfolio substantially affects the value of the contract.     

Some variants of Cauchy's method with accelerated fourth-order convergence

In this paper, we present some variants of Cauchy's method for solving non-linear equations. Analysis of convergence shows that the methods have fourth-order convergence. Per iteration the new methods cost almost the same as Cauchy's method. Numerical results show that the methods can compete with Cauchy's method.     

A generalized conjugate gradient,least square method

Summary A generalizeds-term truncated conjugate gradient method of least square type, proposed in [1a, b], is extended to a form more suitable for proving when the truncated version is identical to the full-term version. Advantages with keeping a control term in the truncated version is pointed out. A computationally efficient new algorithm, based on a special inner product with a small demand of storage is also presented.We also give simplified and slightly extended proofs of termination of the iterative sequence and of existence of ans-term recursion, identical to the full-term version. Important earlier results on this latter topic are found in [15, 16, 8 and 11].The research reported in this paper was partly supported by NATO Grant No. 648/83     

On the quantum Fermi accelerator and its relevance to ‘quantum chaos’

We discuss a quantum version of the Fermi acceleration model, which consists of a particle bouncing between a fixed and oscillating wall. The actual movement of the particle crucially depends on the boundary conditions of the Schrödinger equation. Under Dirichlet boundary conditions, the quantum system displays a regular behaviour, but its classical limit exhibits some unphysical attributes. Only for certain initial conditions does it correspond to the stable motion of a ball bouncing once for an integer number of wall oscillations. In the classical model that situation gives rise to regular islands imbedded in the chaotic sea.     

Certified error bounds for uncertain elliptic equations

In many applications, partial differential equations depend on parameters which are only approximately known. Using tools from functional analysis and global optimization, methods are presented for obtaining certificates for rigorous and realistic error bounds on the solution of linear elliptic partial differential equations in arbitrary domains, either in an energy norm, or of key functionals of the solutions, given an approximate solution. Uncertainty in the parameters specifying the partial differential equations can be taken into account, either in a worst case setting, or given limited probabilistic information in terms of clouds.     

A concept of convergence in geodesic spaces

A CAT(0) space is a geodesic space for which each geodesic triangle is at least as ‘thin’ as its comparison triangle in the Euclidean plane. A notion of convergence introduced independently several years ago by Lim and Kuczumow is shown in CAT(0) spaces to be very similar to the usual weak convergence in Banach spaces. In particular many Banach space results involving weak convergence have precise analogues in this setting. At the same time, many questions remain open.     

An explicit solution of the large deformation of a cantilever beam under point load at the free tip

The large deformation of a cantilever beam under point load at the free tip is investigated by an analytic method, namely the homotopy analysis method (HAM). The explicit analytic formulas for the rotation angle at the free tip are given, which provide a convenient and straightforward approach to calculate the vertical and horizontal displacements of a cantilever beam with large deformation. These explicit formulas are valid for most practical problems, thus providing a useful reference for engineering applications. The corresponding Mathematica code is given in the Appendix.