Two orthonormal bases in dimension d are said “mutually unbiased” if the hermitian scalar products of any vector in one base with any vector in the other base are in modulus equal to the inverse square root of d. One important issue is to determine the maximum number of mutually unbiased bases (MUB) for d dimensions. If d is a prime number or a power of a prime number it is known that this maximum number is d+1. We revisit this problem in both cases using the simple notions of Discrete Fourier Transform, of unitary Hadamard matrices, and of circulant matrices (for which the successive rows are circulant permutations of each other)

Presently about 2000 nuclei both naturally occurring and man made are known; and different mass formulae predict the survival of another 5000 nuclei which could be synthesised in the Laboratory. Accordingly, Radioactive Ion Beam (RIB) facilities are under construction in the major laboratories around the world to explore the entire nuclear landscape. The three actinide nuclei U(233), U(235) and Pu(239) in the stability valley are thermally fissile which have been mainly used in the reactor for energy generation. The question – if other nuclei in particular heavier Uranium isotopes having such features could exist and if so what will be their fission decay properties – is of serious experimental and theoretical importance. Attempts will be made to answer these questions in the present talk. Studies reveal a number of new fission properties of such nuclei, the most prominent one being a new mode of fission decay in which along with the two heavy fragments simultaneously about 2/3 prompt neutrons will be released at the time of scission. This is termed as “Multifragmentation Fission”. Its implication for r-process nucleosynthesis in stellar evolution and energy generation will be discussed

String theory drastically modifies general relativity at small distances, and is believed to lead to a finite theory of quantum gravity. I will discuss higher derivative corrections to the Einstein-Hilbert action, in the context of type IIB superstring theory. Supersymmetry and string dualities play a crucial role in constructing these interactions in the low energy effective action