16,769 research outputs found
On existence and uniqueness of the carrying simplex for competitive dynamical systems
Certain dynamical models of competition have a unique invariant hypersurface
to whichevery nonzero tractory is asymptotic, having simple geometry and
topology.Comment: Submitted to Journal of Biological Dynamics. 13 page
Common zeros of inward vector fields on surfaces
A vector field X on a manifold M with possibly nonempty boundary is inward if
it generates a unique local semiflow . A compact relatively open set K
in the zero set of X is a block. The Poincar\'e-Hopf index is generalized to an
index for blocks that may meet the boundary. A block with nonzero index is
essential.
Let X, Y be inward vector fields on surface M such that and let K be an essential block of zeros for X. Among the main results are
that Y has a zero in K if X and are analytic, or Y is and
preserves area. Applications are made to actions of Lie algebras and groups
Zero sets of Lie algebras of analytic vector fields on real and complex 2-manifolds
Let X be an analytic vector field on a real or complex 2-manifold, and K a
compact set of zeros of X whose fixed point index is not zero. Let A denote the
Lie algebra of analytic vector fields Y on M such that at every point of M the
values of X and [X,Y] are linearly dependent. Then the vector fields in A have
a common zero in K.
Application: Let G be a connected Lie group having a 1-dimensional normal
subgroup. Then every action of G on M has a fixed point.Comment: 22 page
Meissner effect, Spin Meissner effect and charge expulsion in superconductors
The Meissner effect and the Spin Meissner effect are the spontaneous
generation of charge and spin current respectively near the surface of a metal
making a transition to the superconducting state. The Meissner effect is well
known but, I argue, not explained by the conventional theory, the Spin Meissner
effect has yet to be detected. I propose that both effects take place in all
superconductors, the first one in the presence of an applied magnetostatic
field, the second one even in the absence of applied external fields. Both
effects can be understood under the assumption that electrons expand their
orbits and thereby lower their quantum kinetic energy in the transition to
superconductivity. Associated with this process, the metal expels negative
charge from the interior to the surface and an electric field is generated in
the interior. The resulting charge current can be understood as arising from
the magnetic Lorentz force on radially outgoing electrons, and the resulting
spin current can be understood as arising from a spin Hall effect originating
in the Rashba-like coupling of the electron magnetic moment to the internal
electric field. The associated electrodynamics is qualitatively different from
London electrodynamics, yet can be described by a small modification of the
conventional London equations. The stability of the superconducting state and
its macroscopic phase coherence hinge on the fact that the orbital angular
momentum of the carriers of the spin current is found to be exactly ,
indicating a topological origin. The simplicity and universality of our theory
argue for its validity, and the occurrence of superconductivity in many classes
of materials can be understood within our theory.Comment: Submitted to SLAFES XX Proceeding
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