A needle dropped on a plain with parallel and uniformly separated rulings, crosses exactly one of them, with finite probability. le Comte de Buffon, a French nobleman, posed the interesting problem of precisely determining this probability in 1777. This value, with much surprise, involves the number pi. E. Barbier in 1860 rigorously proved that the probability of a short needle of length l crossing exactly one line is 2l/\pi d if the uniform separation, d, between the lines is greater than l. In this seminar, the needle problem and Barbier’s proof are discussed. In addition to this, we attempt a classroom demonstration of needle problem and verify whether the ascertains are in agreement with the theoretical results

We study the localisation of Bochner Riesz means on sets of positive Hausdorff measure in R^d by making use of the decay of the spherical means of Fourier Transform of fractal measures. We study the localisation of Bochner Riesz means on these sets corresponding to the Torus T^d as well.

Let P(z) = \sum_{i=0}^m z^i A_i, where A_i are complex n cross n matrices for i= 0, . . . , m, be a regular matrix polynomial. The matrix polynomial P(z) is said to be structured if the matrices A_i, for i= 0, . . . , m belong to a special subset S of (C^{nxn})^{ m+1}. Structured matrix polynomials have occurred in many engineering applications and have been studied widely for the last two decades. Structured eigenvalue-eigenpair backward error analysis of structured matrix polynomials is important in order to know the backward stability of algorithms that compute them without losing the structure of the polynomial. In this talk, I will derive formulas for the structured eigenvalue backward errors of matrix polynomials that have Hermitian and related structures, like skew-Hermitian, ?-even, ?-odd. This involves a reformulation of the original problem of computing eigenvalue backward error into an equivalent problem of minimizing the maximum eigenvalue of a parameterized Hermitian matrix. Numerical experiments show that there is a significant difference between the backward errors with respect to perturbations that preserve structure and those with respect to arbitrary perturbations

Ramanujan's master theorem states that under suitable conditions, the Mellin transform of an alternating power series provides an interpolation formula for the coefficients of this power series. Ramanujan applied this theorem to compute several definite integrals and power series and this explains why it is referred as "Master Theorem". In this talk we shall try to explain its analogue for radial sections of line bundles over the real hyperbolic space. This a joint work (in progress) with Prof. Swagato K Ray.