X iv : m at h - ph / 0 30 70 65 v 2 6 A ug 2 00 3 Generalised definitions of certain functions and their uses

0 30 70 65 v 3 2 A pr 2 00 4 Generalised definitions of certain functions and their uses

Generalised definitions of exponential, trigonometric sine and cosine and hyperbolic sine and cosine functions are given. In the lowest order, these functions correspond to ordinary exponential, trigonometric sine etc. Some of the properties of the generalised functions are discussed. Importance of these functions and their possible applications are also considered. PACS No.: 02.30.Hq, 02.30.Gp...

ar X iv : m at h - ph / 0 40 80 51 v 1 2 6 A ug 2 00 4 Chern - Simons Integral as a Surface Term ∗

Under certain circumstances the Chern-Simons 3-form is exact (or is a sum of exact forms). Its volume integral can be written as a surface term, in a " holographic " representation.

ar X iv : m at h - ph / 0 60 70 40 v 1 2 0 Ju l 2 00 6 Invariant Form of BK - factorization and its Applications

ar X iv : 0 70 8 . 37 30 v 1 [ m at h . PR ] 2 8 A ug 2 00 7 DENSITIES FOR ROUGH DIFFERENTIAL EQUATIONS UNDER HÖRMANDER ’ S CONDITION

We consider stochastic differential equations dY = V (Y ) dX driven by a multidimensional Gaussian process X in the rough path sense. Using Malliavin Calculus we show that Yt admits a density for t ∈ (0, T ] provided (i) the vector fields V = (V1, ..., Vd) satisfy Hörmander’s condition and (ii) the Gaussian driving signal X satisfies certain conditions. Examples of driving signals include fract...

ar X iv : m at h - ph / 0 70 30 43 v 3 2 8 O ct 2 00 7 RANDOM MATRICES , NON - BACKTRACKING WALKS , AND ORTHOGONAL POLYNOMIALS

Several well-known results from the random matrix theory, such as Wigner’s law and the Marchenko–Pastur law, can be interpreted (and proved) in terms of non-backtracking walks on a certain graph. Orthogonal polynomials with respect to the limiting spectral measure play a rôle in this approach.