Numerical application of Euler's series transformation and its generalizations

Summary A nonlinear generalizationÊ _z of Euler's series transformation is compared with the (linear) Euler-Knopp transformationE _z and a twoparametric methodE . It is shown how to applyE orE _, to compute the valuef(z_o) of a functionf from the power series at 0 iff is holomorphic in a half plane or in the cut plane. BothE andE _, are superior toÊ _z . A compact recursive algorithm is given for computingE andE _,.     

Effective computation of periodic orbits and bifurcation diagrams in delay equations

Summary By employing a numerical method which uses only rather classical tools of Numerical Analysis such as Newton's method and routines for ordinary differential equations, unstable periodic solutions of differential-difference equations can be computed. The method is applied to determine bifurcation diagrams with backward bifurcation.Dedicated to Professor Lothar Collatz on the occasion of his 70th birthdayThis paper has been read at the Conference of Numerical Mathematicians, Zeist, The Netherlands, October 12–15, 1979     

A mixed finite element method for solving the nonstationary stokes equation

Summary This paper deals with a mixed finite element method for approximating a fourth order initial value problem arising from the nonstationary Stokes problem. For piecewise linear shape functions error estimates are given with convergence rates similar to the elliptic case. Some numerical computations will illustrate the theoretical results.     

Some new domains with complete Kähler metrics of negative curvature

We add to the known examples of complete Kähler manifolds with negative sectional curvature by showing that the following three classes of domains in euclidean spaces also belong: perturbations of ellipsoidal domains in ?ⁿ, intersections of complex-ellipsoidal domains in ?², and intersections of fractional linear transforms of the unit ball in ?². In the process, we prove the following theorem in differential geometry: in the intersection of two complex-ellipsoidal domains in ?², the sum of the Bergman metrics is a Kähler metric with negative curvature operator.     

Peripheral and ADI hopscotch methods for two dimensional parabolic and elliptic equations with a mixed term

The peripheral and ADI hopscotch methods are extended to solve problems in two space dimensions with a mixed derivative term. The method is compared numerically with existing hopscotch methods.     

On the order of composite multistep methods for ordinary differential equations

Summary In this paper, a general class ofk-step methods for the numerical solution of ordinary differential equations is discussed. It is shown that methods with order of consistencyq have order of convergence (q+1) if a very simple condition is satisfied. This result gives a new aspect to previous results of Spijker; it also serves as a starting point for a new theory of cyclick-step methods, completing an approach of Donelson and Hansen. It facilitates the practical determination of high-order cyclick-step methods, especially of stiffly stable,k-step methods.     

Family of symplectic implicit Runge-Kutta formulae

A family of formulae for the sympletic IRK method is investigated. Specifically, focus is given to general solutions for formula parameters of IRK under the symplectic and the order conditions. Examples of such formulae are constructed for up to three stages.     

Page usage in a quadtree index

This paper provides a characterization of the storage needs of a quadtree when used as an index to access large volumes of 2-dimensional data. It is shown that the page occupancy for data in random order approaches 33%. A precise mathematical analysis that involves a modicum of hypergeometric functions and dilogarithms, together with some computer algebra is presented.A brief survey of the analysis of storage usage in tree structures is included. The 33% ratio for quadtrees is to be compared to the figures for binary search trees (50%), tries (69%), and quadtries (46%).The research of this author was done while visiting INRIA, Rocquencourt, France under support from the Ministry of Education of Japanese Government.Work of this author was supported in part by the Basic Research Action of the E.C. under contract No. 3075 (Project ALCOM).     

Sample path Large Deviations and optimal importance sampling for stochastic volatility models

Sample path Large Deviation Principles (LDP) of the Freidlin–Wentzell type are derived for a class of diffusions, which govern the price dynamics in common stochastic volatility models from Mathematical Finance. LDP are obtained by relaxing the non-degeneracy requirement on the diffusion matrix in the standard theory of Freidlin and Wentzell. As an application, a sample path LDP is proved for the price process in the Heston stochastic volatility model.     

Loop group decompositions in almost split real forms and applications to soliton theory and geometry

We prove a global Birkhoff decomposition for almost split real forms of loop groups, when an underlying finite dimensional Lie group is compact. Among applications, this shows that the dressing action–by the whole subgroup of loops which extend holomorphically to the exterior disc–on the U$U$-hierarchy of the ZS-AKNS systems, on curved flats and on various other integrable systems, is global for compact cases. It also implies a global infinite dimensional Weierstrass-type representation for Lorentzian harmonic maps (1+1$1+1$ wave maps) from surfaces into compact symmetric spaces. An “Iwasawa-type” decomposition of the same type of real form, with respect to a fixed point subgroup of an involution of the second kind, is also proved, and an application given.