The

unit sphere in [F.sub.n] (E) is termed the inertia ellipsoid and denoted by

The approach is elementary and relies on the fact that in an infinite dimensional normed linear space there exists a retraction from the unit ball to the

unit sphere (note also in a normed linear space there exists a retraction from the unit ball (in a cone) to the

unit sphere (in a cone)).

The cubic texture map can equivalently be thought of as a projection of the

unit sphere onto the cube faces when they are arranged correctly in three dimensions.

Indeed, if we consider, for example, a 2-dimensional section of the unit ball, say of C(-1, 1), by a 2-dimensional subspace F := span{f, g}, where f([not equal to] 0) [member of] 0(-1, 1) and g([not equal to] 0) [member of] E (-1, 1), then Theorem 23 tells us that the two lines [L.sub.f] := span{f} and [L.sub.g] := span{g} divide the 2-dimensional

unit sphere of F into four identical quarters modulo reflections with respect to [L.sub.f] and [L.sub.g].

The 20 papers discuss such aspects as the curvature Veronese of a 3-manifold immersed in Euclidean space, topological formulas for closed semi-algebraic sets by Euler integration, the topological classification of simple Morse Bott functions on surfaces, critical points of the Gauss map and the exponential tangent map, and Legendre curves in the unit spherical bundle over the

unit sphere and evolutes.

Given a function f, continuous and compactly supported, we consider for each x [member of] [R.sup.n] and t > 0, the operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where d[sigma] is the normalized Lebesgue measure over the

unit sphere [S.sup.n-1].

So the average bond energy for per surface area of the

unit sphere (Fig.

Most often, the domain D is either the unit disc (planar parameterization), or the

unit sphere (spherical parameterization).

Let BX and SX denote its closed unit ball and its

unit sphere, respectively.

The vector spherical harmonics constitute an orthogonal set of vector functions on the

unit sphereIn case of CSR it holds that D(r) = 1 - exp(-[lambda][[kappa].sub.d][r.sup.d]), where [lambda] is the mean number of sphere centers per unit volume, d the dimension and [[kappa].sub.d] the volume of the i-dimensional

unit sphere. Thus, in this case, D(r) is a strictly concave function, whereas the diagram shown in the middle of Fig.