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.
Evaluation of CuAl2O4 as an Oxygen Carrier in Chemical-Looping Combustion
The chemical-looping combustion (CLC) process is a novel solution for efficient combustion with intrinsic separation of carbon dioxide. The process uses a metal oxide as an oxygen carrier to transfer oxygen from an air to a fuel reactor where the fuel, or gasification products of the fuel, reacts with the solid oxygen carrier. In this work, copper(II) aluminate (CuAl2O4) was assessed as a potential oxygen carrier using methane as fuel. The carrier particles were produced by freeze–granulation and calcined at 1050 °C for a duration of 6 h. The chemical-looping characteristics were evaluated in a laboratory-scale fluidized-bed reactor in the temperature range of 900–950 °C during 45 alternating redox cycles. The oxygen carrier exhibited reproducible and stable reactivity behavior in this temperature range. Neither agglomeration nor defluidization was noticed in any of the cycles carried out at 900–925 °C. However, after reactivity tests at 950 °C, soft agglomeration and particle fragmentation were observed. Systematic phase analysis of the Cu–Al–O system during the redox cycle was carried out as a function of duration of reduction and oxygen concentration during the oxidation period. It was found that the CuAl2O4 is reduced to copper(I) aluminate (CuAlO2; delafossite), Cu2O, and elemental Cu. The CuAlO2 phase is characterized by slow kinetics for oxidation into CuO and CuAl2O4. Despite this kinetic limitation, complete conversion of methane with reproducible reactivity of the oxygen carrier is achieved. Thus, CuAl2O4 could be a potential oxygen carrier for chemical-looping combustion.