ar X iv : m at h - ph / 0 20 60 17 v 1 1 2 Ju n 20 02 Z 3 - graded Grassmann Variables , Parafermions and their Coherent States

ar X iv : m at h - ph / 0 30 60 62 v 1 2 5 Ju n 20 03 VECTOR COHERENT STATES WITH AN UNBOUNDED INVERSE FRAME OPERATOR

We present a class of vector coherent states in the domain D × D × .... × D (n-copies), where D is the complex unit disc, using a specific class of hermitian matrices. Further, as an example, we build vector coherent states in the unit disc by considering the unit disc as the homogeneous space of the group SU (1, 1).

ar X iv : h ep - p h / 02 06 17 1 v 2 2 4 Ju n 20 02 IPHE 2002 - 009 June 24 , 2002 B 0 – B 0 MIXING

ar X iv : 0 70 4 . 17 02 v 2 [ m at h . A G ] 2 6 Ju n 20 08 THE CANONICAL VOLUME OF 3 - FOLDS OF GENERAL TYPE WITH χ ≤ 0

We prove that the canonical volume K ≥ 1 30 for all 3-folds of general type with χ(O) ≤ 0. This bound is sharp.

ar X iv : m at h - ph / 0 20 60 18 v 1 1 3 Ju n 20 02 ENTROPY AND HADAMARD MATRICES

The entropy of an orthogonal matrix is defined. It provides a new interpretation of Hadamard matrices as those that saturate the bound for entropy.It appears to be a useful Morse function on the group manifold. It has sharp maxima and other saddle points. The matrices corresponding to the maxima for 3 and 5 dimensions are presented. They are integer matrices (upto a rescaling.)

ar X iv : 0 70 4 . 17 02 v 3 [ m at h . A G ] 2 7 Ju n 20 08 THE CANONICAL VOLUME OF 3 - FOLDS OF GENERAL TYPE WITH χ ≤ 0

We prove that the canonical volume K ≥ 1 30 for all 3-folds of general type with χ(O) ≤ 0. This bound is sharp.