On the Arithmetic of Errors

An approximate number is an ordered pair consisting of a (real) number and an error bound, briefly error, which is a (real) non-negative number. To compute with approximate numbers the arithmetic operations on errors should be well-known. To model computations with errors one should suitably define and study arithmetic operations and order relations over the set of non-negative numbers. In this work we discuss the algebraic properties of non-negative numbers starting from familiar properties of real numbers. We focus on certain operations of errors which seem not to have been sufficiently studied algebraically. In this work we restrict ourselves to arithmetic operations for errors related to addition and multiplication by scalars. We pay special attention to subtractability-like properties of errors and the induced “distance-like” operation. This operation is implicitly used under different names in several contemporary fields of applied mathematics (inner subtraction and inner addition in interval analysis, generalized Hukuhara difference in fuzzy set theory, etc.) Here we present some new results related to algebraic properties of this operation.

Approximation Classes for Adaptive Methods

* This work has been supported by the Office of Naval Research Contract Nr. N0014-91-J1343, the Army Research Office Contract Nr. DAAD 19-02-1-0028, the National Science Foundation grants DMS-0221642 and DMS-0200665, the Deutsche Forschungsgemeinschaft grant SFB 401, the IHP Network “Breaking Complexity” funded by the European Commission and the Alexan- der von Humboldt Foundation.

Large Distinct Part Sizes in a Random Integer Partition

A partition of a positive integer n is a way of writing it as the sum of positive integers without regard to order; the summands are called parts. The number of partitions of n, usually denoted by p(n), is determined asymptotically by the famous partition formula of Hardy and Ramanujan [5]. We shall introduce the uniform probability measure P on the set of all partitions of n assuming that the probability 1/p(n) is assigned to each n-partition. The symbols E and V ar will be further used to denote the expectation and variance with respect to the measure P . Thus, each conceivable numerical characteristic of the parts in a partition can be regarded as a random variable.