Some formulas in invariant theory

Some Problems in Invariant Theory

We present summaries of a number of research problems in invariant theory. These problems were posed during the organized problem sessions held during the April 2002, Kingston Ontario, Invariant Theory Workshop and Conference.

Some Formulas for Invariant Phases of Unitary Matrices by Jarlskog

CP violation is expected in the standard model of particle physics with three or more families [1], [2]. Therefore it is an important problem that what is the measure of CP violation with such families which is invariant under the action of phase factors. To construct invariants for matrix action, the determinant is a useful tool. In the previous paper [3], C. Jarlskog succeeded to define invar...

Subspace-Invariant AC Formulas

The n-variable PARITY function is computable (by a well-known recursive construction) by AC formulas of depth d+ 1 and leafsize n·2dn1/d . These formulas are seen to possess a certain symmetry: they are syntactically invariant under the subspace P of even-weight elements in {0, 1}, which acts (as a group) on formulas by toggling negations on input literals. In this paper, we prove a 2 −1) lower...

On Some Asymptotic Formulas in the Theory of Partitions

They also conjectured that for Jki<feâ£o, pk!(n)Spk2(n) and for ko<h<k2i pki(n)<pk2(n). They verify this conjecture for n^32. Recently Todd published a table of all the pkin) for n^lOO, and it is easy to verify the conjecture for w^glOO. I am unable to prove or disprove this conjecture. They also remark that pk0{n) differs from c-inm i0g n by i e s s than 1 for n^ 32; (2) shows that for large n...

Some Asymptotic Formulas in the Theory of Numbers(')