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 defined by

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 approach.

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.