We consider the class of constant depth AND/OR circuits augmented with a layer of modular counting gates at the bottom layer, i.e AC◦MODm circuits. We show that the following holds for several types of gates G: by adding a gate of type G at the output, it is possible to obtain an equivalent probabilistic depth 2 circuit of quasipolynomial size consisting of a gate of type G at the output and a ...

We explicitly determine quasi-polynomials describing the weight multiplicities of the Lie algebra so5(C). This information entails immediate complete knowledge of the character of any simple representation as well as the asymptotic behavior of characters.

Smallest and greatest 1-Lipschitz aggregation operators with given diagonal section, opposite diagonal section, and with graphs passing through a single point of the unit cube, respectively, are determined. These results are used to find smallest and greatest copulas and quasi-copulas with these properties (provided they exist).

A d-space X = (X, %, μ) is a compact set X with respect to a quasi-metric % and a Borel measure μ such that the measure of a ball of radius r is equivalent to r, where d > 0. The paper deals with spaces B p(X;H) of Besov type where 1 < p < ∞ and s ∈ R. Here H is a bi-Lipschitzian map of the snowflaked version (X, %, μ) of X for some 0 < ε < 1, onto a fractal d/ε-set Γ = HX in some R, reducing t...

That acknowledged, I am of the opinion that mathematical philosophy matters more now than it has in nearly a century. The power of modern computers matched with that of modern mathematical software and the sophistication of current mathematics is changing the way we do mathematics. In my view it is now both necessary and possible to admit quasi-empirical inductive methods fully into mathematica...