These E(c, [mu])-spaces are 3-manifolds admitting a global unit Killing vector field whose integral curves
are the fibers of a certain Riemannian submersion over the simply connected constant Gaussian curvature surface M(c) and, therefore, determine Killing submersions [pi] : E(c, [mu]) [right arrow] M(c) (for more details, see ).
Observe that the integral curves
of the system do not depend on [eta], so without loss of generality we may assume that [eta] = 1.
Several authors studied the integral curves
by using matrix of a linear vector field in Euclidean and 3-dimensional Lorentz-Minkowski spaces.
In Section 3, we make the numerical simulation for bounded integral curves
. In Section 4, we derive the exact implicit or parameter expressions of the kink-like wave and compacton-like wave solutions.
Note that the periodic waves of (9) correspond to the periodic integral curves
of (40), and the periodic integral curves
correspond to the closed orbits of system (41).
The integral curves
for -1/(1 - t)F'[(x).sup.-1]F(x) are the paths implicitly determined by
Sometimes, M is a union of family of subsets (a plane as union of straight lines, an integral manifold of a Pfaff equation as union of integral curves
, and so on.
Key words: roughnes of the cut surface, differential equation, integral curves
Paper topics include studies of a singularly perturbed Riccati equation, asymptotic differential algebra, non- oscillating integral curves
and o-minimal structures, topological construction of transseries and an introduction of generalized Borel summability, higher-order norms for the Moivre-Laplace theorem, matching the similarities of canard values, and generalizing surreal numbers.
Our complete line of floors offers a wide variety of options, including performance characteristics, thickness, integral curves
, color and texture.
This paper considers the problem of minimizing 1/2 [integral of] [k.sub.2] between limits 0 and T (s) ds over all integral curves
g(t), satisfying the prescribed boundary con ditions g(0) = [g.sub.0], g(T) = [g.sub.l], of the following differential system:
of the vector fields [e.sub.1] and [e.sub.2], passing through the point ([t.sub.0], [x.sub.0]) [member of] [R.sup.2], are determined by solutions of the ordinary differential equations