Let A be a finite-dimensional power-associative algebra over a field F, either R or C, and let S, a subset of A, be closed under scalar multiplication. A real-valued function f on S is called a subnorm if f(a) > 0 for all 0 = a ∈ S, and f(αa) = |α|f(a) for all a ∈ S and α ∈ F. If in addition, S is closed under raising to powers, then a subnorm f is said to be stable if there exists a positive c...

In his thesis Baire defined functions of Baire class 1. A function f is of Baire class 1 if it is the pointwise limit of a sequence of continuous functions. Baire proves the following theorem. A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity. We prove the automaton version of this theorem. An ω-rati...

We present fixed point theorems for a nonexpansive mapping from a closed convex subset of a uniformly convex Banach space into itself under some asymptotic contraction assumptions. They generalize results valid for bounded convex sets or asymptotically compact sets. In this note we generalize a famous result by Browder [3], Göhde [6] and Kirk [8], recently extended by Luc in [14], by using the ...

It is well-known that if T is an everywhere defined bounded operator on a Banach space X to a Banach space Y and T* is its adjoint, then the range R(T*) of T* is closed in X* if and only if the range R(T) of T is closed in Y (cf. [2, pp. 487-489]). The object of this note is to establish this result for closed but possibly unbounded operators. This result, for the unbounded case, is of great ut...

This is a common introduction to math.AG/0011176, math.RT/0101170, math.RT/0306333, math.RT/0506043, math.RT/0601028. In these papers one studies the automorphism group G of an extension F/k of algebraically closed fields, especially in the case of countable transcendence degree and characteristic zero, its smooth linear and semi-linear representations, and their relations to algebraic geometry...

In 1979, P. Frankl conjectured that in a finite union-closed family F of finite sets, F 6= {∅}, there has to be an element that belongs to at least half of the sets in F . We prove this when | ⋃

Let F be an algebraically closed field of characteristic p. We fashion an infinite dimensional basic algebra C ←− p(F ), with a transparent combinatorial structure, which we expect to control the rational representation theory of GL2(F ).

The energy form on a closed fractal curve F is constructed. As F is neither self-similar nor nested, it is regarded as a ”fractal manifold”. The energy is obtained by integrating the Lagrangian on F .

We consider the problem F = f(ν) for strictly convex, closed hypersurfaces in S and solve it for curvature functions F the inverses of which are of class (K).