Effects of an intermediate scale in SUSY grand unification

We discuss the production of lepton flavor violation and EDMs and the viability of the $b-\tau$ unification hypothesis in SUSY grand unification with an intermediate gauge symmetry breaking scale.Comment: 3 pages (Latex, esprc2.sty used), talk given at 4th International Conference on Supersymmetry (SUSY '96, College Park, MD, May 29 - June 1, 1996

On the role of F\"ollmer-Schweizer minimal martingale measure in Risk Sensitive control Asset Management

Kuroda and Nagai \cite{KN} state that the factor process in the Risk Sensitive control Asset Management (RSCAM) is stable under the F\"ollmer-Schweizer minimal martingale measure . Fleming and Sheu \cite{FS} and more recently F\"ollmer and Schweizer \cite{FoS} have observed that the role of the minimal martingale measure in this portfolio optimization is yet to be established. In this article we aim to address this question by explicitly connecting the optimal wealth allocation to the minimal martingale measure. We achieve this by using a "trick" of observing this problem in the context of model uncertainty via a two person zero sum stochastic differential game between the investor and an antagonistic market that provides a probability measure. We obtain some startling insights. Firstly, if short-selling is not permitted and if the factor process evolves under the minimal martingale measure then the investor's optimal strategy can only be to invest in the riskless asset (i.e. the no-regret strategy). Secondly, if the factor process and the stock price process have independent noise, then even if the market allows short selling, the optimal strategy for the investor must be the no-regret strategy while the factor process will evolve under the minimal martingale measure .Comment: A.Deshpande (2015), On the role of F\"ollmer-Schweizer minimal martingale measure in Risk Sensitive control Asset Management,Vol. 52, No. 3, Journal of Applied Probabilit

Crossed S-matrices and Character Sheaves on Unipotent Groups

Let $\mathtt{k}$ be an algebraic closure of a finite field $\mathbb{F}_{q}$ of characteristic $p$. Let $G$ be a connected unipotent group over $\mathtt{k}$ equipped with an $\mathbb{F}_q$-structure given by a Frobenius map $F:G\to G$. We will denote the corresponding algebraic group defined over $\mathbb{F}_q$ by $G_0$. Character sheaves on $G$ are certain objects in the triangulated braided monoidal category $\mathscr{D}_G(G)$ of bounded conjugation equivariant $\bar{\mathbb{Q}}_l$-complexes (where $l\neq p$ is a prime number) on $G$. Boyarchenko has proved that the "trace of Frobenius" functions associated with $F$-stable character sheaves on $G$ form an orthonormal basis of the space of class functions on $G_0(\mathbb{F}_q)$ and that the matrix relating this basis to the basis formed by the irreducible characters of $G_0(\mathbb{F}_q)$ is block diagonal with "small" blocks. In this paper we describe these block matrices and interpret them as certain "crossed $S$-matrices". We also derive a formula for the dimensions of the irreducible representations of $G_0(\mathbb{F}_q)$ that correspond to one such block in terms of certain modular categorical data associated with that block. In fact we will formulate and prove more general results which hold for possibly disconnected groups $G$ such that $G^\circ$ is unipotent. To prove our results, we will establish a formula (which holds for any algebraic group $G$) which expresses the inner product of the "trace of Frobenius" function of any $F$-stable object of $\mathscr{D}_G(G)$ with any character of $G_0(\mathbb{F}_q)$ (or of any of its pure inner forms) in terms of certain categorical operations.Comment: 37 pages. Added a section about certain Grothendieck rings. Added some example

Natural convection in a cubical cavity: Case of multiple solutions

The purpose of this report is to study an interesting case of natural convection13; in a cubical cavity. A set of boundary conditions has been applied on the six walls and13; the buoyancy driven xB0;ow inside the cavity is due to the heating of the bottom wall.13; Interestingly two families of distinct steady solutions have been observed. Both the13; families have the same critical Rayleigh number (Ra)crit below which there is no xB0;ow13; and heat transfer is by pure conduction