Rank weights for arbitrary finite field extensions

Galois Theory for Arbitrary Field Extensions

$G$-Weights and $p$-Local Rank

Let $k$ be field of characteristic $p$, andlet $G$ be any finite group with splitting field $k$. Assume that $B$ is a $p$-block of $G$.In this paper, we introduce the notion of radical $B$-chain $C_{B}$, and we show that the $p$-local rank of $B$ is equals the length of $C_{B}$. Moreover, we prove that the vertex of a simple $kG$-module $S$ is radical if and only if it has the same vertex of th...

Finite Rank Toeplitz Operators: Some Extensions of D.luecking’s Theorem

The recent theorem by D.Luecking about finite rank Bergman-Toeplitz operators is extended to weights being distributions with compact support and to the spaces of harmonic functions.

Coins with Arbitrary Weights

Given a set of m coins out of a collection of coins of k unknown distinct weights, we wish to decide if all the m given coins have the same weight or not using the minimum possible number of weighings in a regular balance beam. Let m(n, k) denote the maximum possible number of coins for which the above problem can be solved in n weighings. It is known that m(n, 2) = n 1 2 . Here we determine th...

Pencils of Higher Derivations of Arbitrary Field Extensions

Let L be a field of characteristic p ^ 0. A subfield K of L is Galois if A' is the field of constants of a group of pencils of higher derivations on L. Let F d K be Galois subfields of L. Then the group of L over F is a normal subgroup of the group of L over K if and only if F = K(W') for some nonnegative integer r. If L/K splits as the tensor product of a purely inseparable extension and a sep...