arXiv subjects
Publications and source records attributed to M. Fazeel Anwar.
Let $G$ be a non-trivial torsion free group and $t$ be an unknown. In this paper we consider three equations (over $G$) of arbitrary length and show that they have a solution (over $G$) provided two relations among their coefficients hold. Such equations appear for all lengths greater than or equal to eight and the results presented in this article can substantially simplify their solution.
We consider the one dimensional Schrödinger operator with properly connecting generalized point interaction at the origin. We derive a trace formula for trace of difference of resolvents of perturbed and unperturbed Schrödinger operators in terms of a Wronskian which results into an explicit expression for perturbation determinant. Using the estimate for large time real argument on the trace norm of the resolvent difference of the perturbed and unperturbed Schrödinger operators we express the spectral shift function in terms of perturbation determinant. Under certain integrability condition on the potential function, we calculate low energy asymptotics for the perturbation determinant and prove an analog of Levinson's formula.
Let $G$ be a non-trivial torsion free group and $s(t)=g_{1}t^{ε_{1}}g_{2}t^{ε_{2}} \cdots g_{n}t^{ε_{n}}=1 \; (g_{i} \in G,\ ε_i=\pm 1)$ be an equation over $G$ containing no blocks of the form $t^{-1}g_{i}t^{-1}, \; g_{i} \in G$. In this paper we show that $s(t)=1$ has a solution over $G$ provided a single relation on coefficients of $s(t)$ holds. We also generalize our results to equations containing higher powers of $t$. The later equations are also related to Kaplansky zero-divisor conjecture \cite{K}.
In his famous paper [2], Demazure introduced certain indecomposable modules and used them to give a short proof of Bott's theorem. In this paper we consider a generalization of these modules and give their cohomology.
Let G be a semisimple, simply connected linear algebraic group over an algebraically closed field k of characteristic p > 0. In a recent paper [4], Doty introduces the notion of r-minuscule weight and exhibits a tensor product factorization of a corresponding tilting module under the assumption p >= 2h-2, where h is the coxeter number. We remove this restriction and consider some variations involving the more general notion of (p,r)-minuscule weights.