In [section]2 we explain the interpretation of the Hecke operators on [M.sub.3m]([GAMMA]) and the Petersson inner product
on [S.sub.3]([GAMMA]) in terms of the pluricanonical forms.
The inner product
of incident wave [r.sub.s]'s unit vector and subreflector's unit normal vector is
By a number of authors [1-4], it has been recommended to enhance the flexibility of the formalism by making use of an ad hoc, quantum-system-adapted physical inner product
in H, that is, by an introduction of a nontrivial, stationary metric operator [THETA] [not equal to] [THETA](t).
In this section, we define a new inner product
to determine the solution which satisfies the initial boundary value problem.
These polynomials coincide with the orthogonal polynomials associated with the inner product
defined in (6) and verify
Obviously, the Fock-Sobolev type space [F.sup.p.sub.[alpha]] equipped with the natural inner product
Functional Encryption for Inner Product
. Although FE supports the computation of general circuits relying on a wide spectrum of assumptions, there are two major problems with the state-of-the-art general FE constructions.
Then for each block, a dissimilar dictionary is learned by using the inner product
In Section 2, we introduce the concept of a matrix-valued inner product
[<<x,x>>.sub.S] for block vectors with values in a *-algebra S [subset] [C.sup.s x s].
Since cochains are column vectors, we use an inner product
between two column vectors to represent the volume integration of their wedge product.
We will be able to use some specific mathematical tools as inner product
A solution for this problem is provided by conformal geometric algebra (CGA), which can represent the geometries of Euclidean space effectively by the outer product, inner product
, and geometric product.