Dimension formula of solution spaces of A-hypergeometric differential-difference systems
Let A = (a ij) be a matrix in M (d, n, Z ≥0). We suppose that the set of the column vectors of A spans Z d. We denote by S i : f (s i) → f (s i − 1) the difference operator with respect to a variable s i. Definition 1. We call the following differential-difference system H A an A-hypergeometric differential-difference system:   n j=1 a ij x j ∂ j − s i   • f = 0, ∂ j − d i=1 S a ij i • f = 0. Definition 2. Let I be a left ideal of the ring of differential-difference operators D = C We define the rank of I by rank(I) = dim C(x,s) D/I, where D/I is a vector space over C(x, s). Theorem 1. The rank of H A agrees with the normalized volume of A. Let a i be the i-th column vector of the matrix A and F (β, x) the integral F (β, x) = C exp n i=1 x i t a i t −β−1 dt, t = (t 1 ,. .. , t d).
HYPERGEOMETRIC FUNCTIONS
Given the hypergeometric function F (a, b; c; z) = ∞ P n=0 (a)n(b)n (c)n(1)n zn, we place conditions on a, b and c to guarante that zpF (a, b; c; z) will be in various subclasses of p-valent starlike and p-valent convex functions. Operators related to the hypergeometric function are also examined.
Rapid Spatially Coupled AVO Inversion
The Bayesian methodology yields a natural framework for many geophysical inverse problem, see e.g. (Tarantola and Valette, 1982; Tarantola, 1987; Duijndam, 1988a; Duijndam, 1988b). The solution of a Bayesian inverse problem is represented by the posterior distribution. From the posterior distribution the best estimate and the corresponding uncertainty can be extracted and a set of plausible solutions can be drawn.