Relations between Approximation Numbers and Entropy Numbers

Recent Developments concerning Entropy and Approximation Numbers

Width invariant approximation of fuzzy numbers

In this paper, we consider the width invariant trapezoidal and triangularapproximations of fuzzy numbers. The presented methods avoid the effortful computation of Karush-Kuhn-Tucker Theorem. Some properties of the new approximation methods are presented and the applicability of the methods is illustrated by examples. In addition, we show that the proposed approximations of fuzzy numbers preserv...

Approximation and entropy numbers in Besov spaces of generalized smoothness

Let X and Y be Banach (or quasi-Banach) spaces and let T ∈ L(X,Y ) be a compact operator. In order to measure “the degree of compactness of T” one can use the asymptotic decay of the sequence of approximation numbers of T or the decay of the sequence of entropy numbers of T. In general, the behaviour of these sequences is quite different. However, as we shall show, under certain assumptions the...

Quadratic relations between logarithms of algebraic numbers

A well known conjecture asserts that Q-linearly independent logarithms of algebraic numbers should be algebraically independent. So far it has only been proved that they are linearly independent over the field of algebraic numbers (Baker, 1966). It is not yet known that there exist two logarithms of algebraic numbers which are algebraically independent. The next step might be to consider quadra...

Zarankiewicz Numbers and Bipartite Ramsey Numbers

The Zarankiewicz number z(b; s) is the maximum size of a subgraph of Kb,b which does not contain Ks,s as a subgraph. The two-color bipartite Ramsey number b(s, t) is the smallest integer b such that any coloring of the edges of Kb,b with two colors contains a Ks,s in the rst color or a Kt,t in the second color.In this work, we design and exploit a computational method for bounding and computing...