Up-to-date Interval Arithmetic From Closed Intervals to Connected Sets of Real Numbers

We consider biperiodic integral equations of the second kind with weakly singular kernels such as they arise in boundary integral equation methods. The equations are solved numerically using a collocation scheme based on trigonometric polynomials. The weak singularity is removed by a local change to polar coordinates. The resulting operators have smooth kernels and are discretized using the tensor product composite trapezodial rule. We prove stability and convergence of the scheme under suitable parameter choices, achieving algebraic convergence of any order under appropriate regularity assumptions. The method can be applied to typical boundary value problems such as potential and scattering problems both for bounded obstacles and for periodic surfaces. We present numerical results demonstrating that the expected convergence rates can be observed in practice

Complete Interval Arithmetic and its Implementation

A Complete Interval Arithmetic and its Implementation is discussed

A Mathematical Basis for an Interval Arithmetic Standard

Basic concepts for an interval arithmetic standard are discussed in the paper. Interval arithmetic deals with closed and connected sets of real numbers. Unlike floating-point arithmetic it is free of exceptions. A complete set of formulas to approximate real interval arithmetic on the computer is displayed in section 3 of the paper. The essential comparison relations and lattice operations are discussed in section 6. Evaluation of functions for interval arguments is studied in section 7. The desirability of variable length interval arithmetic is also discussed in the paper. The requirement to adapt the digital computer to the needs of interval arithmetic is as old as interval arithmetic. An obvious, simple possible solution is shown in section 8

Begegnungen eines Mathematikers mit Informatik

Es handelt sich hier um die nachträgliche Niederschrift eines Vortrages, den der Autor während eines Festkolloquiums gehalten hat, das die Fakultät für Mathematik und das Institut für Angewandte und Numerische Mathematik anlässlich seines 75sten Geburtstages am 27. Juni 2008 veranstaltet haben. Der Vortrag skizziert zunächst kurz die Entwicklung der Rechengeschwindigkeit während der letzten zwanzig Jahre. Es wird dann auf die Entwicklung von Rechenanlagen in Deutschland und Europa in den 50ger und 60ger Jahren eingegangen. Schließlich wird anhand von 4 Fallbeispielen gezeigt, dass sich aus der Tatsache, dass es heute in Europa keine eigenständige Rechnerentwicklung mehr gibt, schwerwiegende Nachteile für zukunftsweisende Softwareentwicklungen ergeben

Una caracterización de la circunferencia

El objetivo de esta nota es demostrar que una condición necesaria y suficiente para que una curva de ecuación x(s) : I→ ℝ3       parametrizada por  la longuitud de arco, y que no es una recta(esto es, con curvatura R ǂO), o sea una circunferencia, es que [formula