Flag varieties and interpretations of Young tableau algorithms

The conjugacy classes of nilpotent $n\times n$ matrices can be parametrised by partitions $\lambda$ of $n$, and for a nilpotent $\eta$ in the class parametrised by $\lambda$, the variety $F_\eta$ of $\eta$-stable flags has its irreducible components parametrised by the standard Young tableaux of shape $\lambda$. We indicate how several algorithmic constructions defined for Young tableaux have significance in this context, thus extending Steinberg's result that the relative position of flags generically chosen in the irreducible components of $F_\eta$ parametrised by tableaux $P$ and $Q$, is the permutation associated to $(P,Q)$ under the Robinson-Schensted correspondence. Other constructions for which we give interpretations are Sch\"utzenberger's involution of the set of Young tableaux, jeu de taquin (leading also to an interpretation of Littlewood-Richardson coefficients), and the transpose Robinson-Schensted correspondence (defined using column insertion). In each case we use a doubly indexed family of partitions, defined in terms of the flag (or pair of flags) determined by a point chosen in the variety under consideration. We show that for generic choices, the family satisfies certain combinatorial relations, whence the family describes the computation of the algorithmic operation being interpreted, as we described in a previous publication.Comment: 16 page

Bulk Viscosity of Interacting Hadrons

We show that first approximations to the bulk viscosity $\eta_v$ are expressible in terms of factors that depend on the sound speed $v_s$, the enthalpy, and the interaction (elastic and inelastic) cross section. The explicit dependence of $\eta_v$ on the factor $(\frac 13 - v_s^2)$ is demonstrated in the Chapman-Enskog approximation as well as the variational and relaxation time approaches. The interesting feature of bulk viscosity is that the dominant contributions at a given temperature arise from particles which are neither extremely nonrelativistic nor extremely relativistic. Numerical results for a model binary mixture are reported.Comment: 4 pages, 1 figure, Contribution to Quark Matter 2009, Knoxville, Tennessee, US

Development of non-equilibrium Green's functions for use with full interaction in complex systems

We present an ongoing development of an existing code for calculating ground-state, steady-state, and transient properties of many-particle systems. The development involves the addition of the full four-index two electron integrals, which allows for the calculation of transport systems, as well as the extension to multi-level electronic systems, such as atomic and molecular systems and other applications. The necessary derivations are shown, along with some preliminary results and a summary of future plans for the code

Learning by a nerual net in a noisy environment - The pseudo-inverse solution revisited

A recurrent neural net is described that learns a set of patterns in the presence of noise. The learning rule is of Hebbian type, and, if noise would be absent during the learning process, the resulting final values of the weights would correspond to the pseudo-inverse solution of the fixed point equation in question. For a non-vanishing noise parameter, an explicit expression for the expectation value of the weights is obtained. This result turns out to be unequal to the pseudo-inverse solution. Furthermore, the stability properties of the system are discussed.Comment: 16 pages, 3 figure

Probing the basins of attraction of a recurrent neural network

A recurrent neural network is considered that can retrieve a collection of patterns, as well as slightly perturbed versions of this `pure' set of patterns via fixed points of its dynamics. By replacing the set of dynamical constraints, i.e., the fixed point equations, by an extended collection of fixed-point-like equations, analytical expressions are found for the weights w_ij(b) of the net, which depend on a certain parameter b. This so-called basin parameter b is such that for b=0 there are, a priori, no perturbed patterns to be recognized by the net. It is shown by a numerical study, via probing sets, that a net constructed to recognize perturbed patterns, i.e., with values of the connections w_ij(b) with b unequal zero, possesses larger basins of attraction than a net made with the help of a pure set of patterns, i.e., with connections w_ij(b=0). The mathematical results obtained can, in principle, be realized by an actual, biological neural net.Comment: 17 pages, LaTeX, 2 figure