5,562 research outputs found
Complexity vs Energy: Theory of Computation and Theoretical Physics
This paper is a survey dedicated to the analogy between the notions of {\it
complexity} in theoretical computer science and {\it energy} in physics. This
analogy is not metaphorical: I describe three precise mathematical contexts,
suggested recently, in which mathematics related to (un)computability is
inspired by and to a degree reproduces formalisms of statistical physics and
quantum field theory.Comment: 23 pages. Talk at the satellite conference to ECM 2012, "QQQ Algebra,
Geometry, Information", Tallinn, July 9-12, 201
On competitive discrete systems in the plane. I. Invariant Manifolds
Let be a competitive map on a rectangular region . The main results of this paper give conditions which guarantee
the existence of an invariant curve , which is the graph of a continuous
increasing function, emanating from a fixed point . We show that
is a subset of the basin of attraction of and that the set consisting
of the endpoints of the curve in the interior of is forward invariant.
The main results can be used to give an accurate picture of the basins of
attraction for many competitive maps.
We then apply the main results of this paper along with other techniques to
determine a near complete picture of the qualitative behavior for the following
two rational systems in the plane.
with
and arbitrary nonnegative initial conditions so
that the denominator is never zero.
with
and arbitrary nonnegative initial conditions.Comment: arXiv admin note: text overlap with arXiv:0905.1772 by other author
The Dynamics and Attractivity for a Rational Recursive Sequence of Order Three
This paper is concerned with the behavior of solution of the nonlinear difference equation
where the initial conditions are arbitrary positive real numbers and a,b,c,d,e are positive constants
The Margulis region and screw parabolic elements of bounded type
Given a discrete subgroup of the isometries of n-dimensional hyperbolic space
there is always a region kept precisely invariant under the stabilizer of a
parabolic fixed point, called the Margulis region. While in dimensions 2 and 3
this region is a horoball, it has in general a more complicated shape due to
the existence of screw parabolic elements in higher dimensions. In fact, P.
Susskind has shown that in a discrete group acting on hyperbolic 4-space
containing a screw parabolic element with irrational rotation, the
corresponding Margulis region does not contain a horoball. In this paper we
describe the asymptotic behavior of the boundary of the Margulis region when
the irrational screw parabolic is of bounded type. As a corollary we show that
the region is quasi-isometric to a horoball. Although Y. Kim has shown that two
screw parabolic isometries with irrational rotation are not conjugate by any
quasi-isometry of hyperbolic 4-space, this corollary implies that their
corresponding Margulis regions (in the bounded type case) are quasi-isometric
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