We now come to the persistent field appearing in the 2nd rank tensor field
equations when matter is absent.
Generally, the vanishing of tensor field
h means that the Reeb foliation of [M.
Now we recall from a paper of Hall-da Costa  that the existence of a covariant constant second-order tensor field
, other than the metric tensor, must exclude some spacetimes, in particular, perfect fluids.
Then, we prove precise results regarding the definition of the Hilbert tensor field
where [LAMBDA] is a fourth order tensor, called the mobility tensor which is essentially the inverse of the relaxation time of the polymer fluids, c is the conformation tensor, [rho] is the fluid density, Q is anisotropic viscosity matrix that is related to viscous dissipation, L is coupling parameter between the velocity gradient field and the structural tensor field
, and A is Helmholtz free energy function represented by a combination of invariants of the conformation tensor, which can be written as follows based on the Hookean model:
An n-dimensional differentiable manifold M is said to admit an almost para-contact Riemannian structure ([phi], [xi], [eta], g), where [phi] is a (1, 1) tensor field
, [xi] is a vector field, [eta] is a 1-form and g is a Riemannian metric on M such that
tau]] of the vector field Z represents the controlled process (it is also possible to replace it by a transport of an arbitrary tensor field
along the flow [c.
where a, b, c, d are scalars of which b [not equal to] 0, c = 0, d [not equal to] 0 and A, B are two non-zero 1-forms such that A(X) = g(X,U) and B(X) = g(X, V); U, V being mutually orthogonal unit vector fields and D is a symmetric (0, 2) type tensor field
with zero trace which satisfies the condition D(X, U) = 0, [for all]X.
n] = (M, g) where g (the fundamental tensor) is a distinguished tensor field
denote the Christoffel symbols of the first type attached to tensor f and if we introduce the distinguished tensor field
In 1968 Yano and Sawaki (10) defined and studied a tensor field
W of type (1, 3) which includes both the conformal curvature tensor C and the concircular curvature tensor [~.
A sampling of paper topics includes real-time hand gesture recognition system based on Q6455 DSP board, a model of data reduction based on tensor field
, fast path searching in real time 3D game, and policy-gradient based actor-critic algorithms.