Then the group G = O(p + 1, q + 1) acts as conformal diffeomorphisms on [S.sup.p] x [S.sup.q], and also on its quotient space X = ([S.sup.p] x [S.sup.q])/[Z.sub.2] by identifying the direct product of

antipodal points. By the general theory of conformal groups, one has a natural family of representations [[??].sub.[lambda]] on [C.sup.[infinity]](X) with parameter [lambda] [member of] C [12, Sect.

In Section 3, we build a framework of USTM constellation based on the

antipodal points. The optimal packing method of searching the orthogonal unitary matrices over Grassmannian manifold and the corresponding searching algorithm are investigated.

It will mean that, when he includes his previous challenges, Mark will have covered 20,000 miles and passed through two

antipodal points on the globe - making it a run around the world.

Another way of assessing the antipodal location of the Earth major landforms severity is the rectilinear correlation r coefficient calculation between absolute heights (depths) of the earth surface within

antipodal points. The calculation algorithm is the correlation coefficient for large groups is discussed in detail by the respective papers [8-10].

Definition (14) means that two different spin coherent states overlap unless they are directed into two

antipodal points on the sphere [17].

is a continuous function from S [equivalent] [S.sup.2] to [R.sup.2], by the Borsuk-Ulam theorem (see Corollary 9.3 in [18]) there exist two

antipodal points of S having the same image.

Proof: We choose a covering [([A.sub.i]).sub.i=0, ..., k+r] of [S.sup.k+r-1] by closed subsets such that no [A.sub.i] contains a pair of

antipodal points. Let D := [min.sub.i] dist([A.sub.i], -[A.sub.i]) > 0, 0 < [epsilon] < D and [epsilon]' := D - [epsilon].

Let S = [B.sub.1] [union] [B.sub.2] be a partition of S into two Borel pieces such that no

antipodal points of S lie in the same cell of the partition.

Due to symmetry, the ball [rB.sup.d] touches the boundary of Z in at least two

antipodal points rw and -rw, w [member of] [S.sup.d-1].

Paris was the starting point and the

antipodal points I had to pass through were Madrid and Wellington.

Plane elliptic geometry is closely related to spherical geometry, but it differs in that

antipodal points on the sphere are identified.

On the other hand, Borsuk's theorem only requires the mapping f to be continuous on the ball and to fulfill a non-colinearity condition for the function values at all pairs of

antipodal points on the boundary of the ball.