7,136 research outputs found
Abelian Equations and Rank Problems for Planar Webs
We find an invariant characterization of planar webs of maximum rank. For
4-webs, we prove that a planar 4-web is of maximum rank three if and only if it
is linearizable and its curvature vanishes. This result leads to the direct
web-theoretical proof of the Poincar\'{e}'s theorem: a planar 4-web of maximum
rank is linearizable. We also find an invariant intrinsic characterization of
planar 4-webs of rank two and one and prove that in general such webs are not
linearizable. This solves the Blaschke problem ``to find invariant conditions
for a planar 4-web to be of rank 1 or 2 or 3''. Finally, we find invariant
characterization of planar 5-webs of maximum rank and prove than in general
such webs are not linearizable.Comment: 43 page
On a class of linearizable planar geodesic webs
We present a complete description of a class of linearizable planar geodesic
webs which contain a parallelizable 3-subweb.Comment: 7 page
Quantum Versus Classical Decay Laws in Open Chaotic Systems
We study analytically the time evolution in decaying chaotic systems and
discuss in detail the hierarchy of characteristic time scales that appeared in
the quasiclassical region. There exist two quantum time scales: the Heisenberg
time t_H and the time t_q=t_H/\sqrt{\kappa T} (with \kappa >> 1 and T being the
degree of resonance overlapping and the transmission coefficient respectively)
associated with the decay. If t_q < t_H the quantum deviation from the
classical decay law starts at the time t_q and are due to the openness of the
system. Under the opposite condition quantum effects in intrinsic evolution
begin to influence the decay at the time t_H. In this case we establish the
connection between quantities which describe the time evolution in an open
system and their closed counterparts.Comment: 3 pages, REVTeX, no figures, replaced with the published version
(misprints corrected, references updated
Geodesic Webs on a Two-Dimensional Manifold and Euler Equations
We prove that any planar 4-web defines a unique projective structure in the
plane in such a way that the leaves of the foliations are geodesics of this
projective structure. We also find conditions for the projective structure
mentioned above to contain an affine symmetric connection, and conditions for a
planar 4-web to be equivalent to a geodesic 4-web on an affine symmetric
surface. Similar results are obtained for planar d-webs, d > 4, provided that
additional d-4 second-order invariants vanish.Comment: 15 page
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