Congruences and Canonical Forms for a Positive Matrix: Application to the Schweinler-Wigner Extremum Principle

It is shown that a $N\times N$ real symmetric [complex hermitian] positive definite matrix $V$ is congruent to a diagonal matrix modulo a pseudo-orthogonal [pseudo-unitary] matrix in $SO(m,n)$ [ $SU(m,n)$], for any choice of partition $N=m+n$. It is further shown that the method of proof in this context can easily be adapted to obtain a rather simple proof of Williamson's theorem which states that if $N$ is even then $V$ is congruent also to a diagonal matrix modulo a symplectic matrix in $Sp(N,{\cal R})$ [$Sp(N,{\cal C})$]. Applications of these results considered include a generalization of the Schweinler-Wigner method of `orthogonalization based on an extremum principle' to construct pseudo-orthogonal and symplectic bases from a given set of linearly independent vectors.Comment: 7 pages, latex, no figure

Recursive parametrization of Quark flavour mixing matrices

We examine quark flavour mixing matrices for three and four generations using the recursive parametrization of $U(n)$ and $SU(n)$ matrices developed by some of us in Refs.[2] and [3]. After a brief summary of the recursive parametrization, we obtain expressions for the independent rephasing invariants and also the constraints on them that arise from the requirement of mod symmetry of the flavour mixing matrix

Grand canonical partition functions for multi level para Fermi systems of any order

A general formula for the grand canonical partition function for a para Fermi system of any order and of any number of levels is derived.Comment: 9 pages, latex, no figure

Bounds on quark mass matrices elements due to measured properties of the mixing matrix and present values of the quark masses

We obtain constraints on possible structures of mass matrices in the quark sector by using as experimental restrictions the determined values of the quark masses at the $M_Z$ energy scale, the magnitudes of the quark mixing matrix elements $V_{\rm ud}$, $V_{\rm us}$, $V_{\rm cd}$, and $V_{\rm cs}$, and the Jarlskog invariant $J(V)$. Different cases of specific mass matrices are examined in detail. The quality of the fits for the Fritzsch and Stech type mass matrices is about the same with $\chi^2/{\rm dof}=4.23/3=1.41$ and $\chi^2/{\rm dof}=9.10/4=2.28$, respectively. The fit for a simple generalization (one extra parameter) of the Fritzsch type matrices, in the physical basis, is much better with $\chi^2/{\rm dof}=1.89/4=0.47$. For comparison we also include the results using the quark masses at the 2 GeV energy scale. The fits obtained at this energy scale are similar to that at $M_Z$ energy scale, implying that our results are unaffected by the evolution of the quark masses from 2 to 91 GeV.Comment: Evolution effects include

Identities involving elementary symmetric functions

A systematic procedure for generating certain identities involving elementary symmetric functions is proposed. These identities, as particular cases, lead to a hierarchy of identities for q-binomial coefficients

Mapping of non-central potentials under point canonical transformations

Motivated by the observation that all known exactly solvable shape invariant central potentials are inter-related via point canonical transformations, we develop an algebraic framework to show that a similar mapping procedure is also exist between a class of non-central potentials. As an illustrative example, we discuss the inter-relation between the generalized Coulomb and oscillator systems.Comment: 11 pages article in LaTEX (uses standard article.sty). Please check http://www1.gantep.edu.tr/~gonul for other studies of Nuclear Physics Group at University of Gaziante