163 research outputs found
Similarity Analysis of Nonlinear Equations and Bases of Finite Wavelength Solitons
We introduce a generalized similarity analysis which grants a qualitative
description of the localised solutions of any nonlinear differential equation.
This procedure provides relations between amplitude, width, and velocity of the
solutions, and it is shown to be useful in analysing nonlinear structures like
solitons, dublets, triplets, compact supported solitons and other patterns. We
also introduce kink-antikink compact solutions for a nonlinear-nonlinear
dispersion equation, and we construct a basis of finite wavelength functions
having self-similar properties.Comment: 18 pages Latex, 6 figures ep
On effective compactness and sigma-compactness
Using the Gandy -- Harrington topology and other methods of effective
descriptive set theory, we prove several theorems on compact and sigma-compact
pointsets. In particular we show that any set of the Baire
space either is covered by a countable union of compact
sets, or contains a subset closed in and homeomorphic to (and
then is not covered by a sigma-compact set, of course)
The combinatorics of the Baer-Specker group
Denote the integers by Z and the positive integers by N.
The groups Z^k (k a natural number) are discrete, and the classification up
to isomorphism of their (topological) subgroups is trivial. But already for the
countably infinite power Z^N of Z, the situation is different. Here the product
topology is nontrivial, and the subgroups of Z^N make a rich source of examples
of non-isomorphic topological groups. Z^N is the Baer-Specker group.
We study subgroups of the Baer-Specker group which possess group theoretic
properties analogous to properties introduced by Menger (1924), Hurewicz
(1925), Rothberger (1938), and Scheepers (1996). The studied properties were
introduced independently by Ko\v{c}inac and Okunev. We obtain purely
combinatorial characterizations of these properties, and combine them with
other techniques to solve several questions of Babinkostova, Ko\v{c}inac, and
Scheepers.Comment: To appear in IJ
Some new directions in infinite-combinatorial topology
We give a light introduction to selection principles in topology, a young
subfield of infinite-combinatorial topology. Emphasis is put on the modern
approach to the problems it deals with. Recent results are described, and open
problems are stated. Some results which do not appear elsewhere are also
included, with proofs.Comment: Small update
Nonlinear analysis of spacecraft thermal models
We study the differential equations of lumped-parameter models of spacecraft
thermal control. Firstly, we consider a satellite model consisting of two
isothermal parts (nodes): an outer part that absorbs heat from the environment
as radiation of various types and radiates heat as a black-body, and an inner
part that just dissipates heat at a constant rate. The resulting system of two
nonlinear ordinary differential equations for the satellite's temperatures is
analyzed with various methods, which prove that the temperatures approach a
steady state if the heat input is constant, whereas they approach a limit cycle
if it varies periodically. Secondly, we generalize those methods to study a
many-node thermal model of a spacecraft: this model also has a stable steady
state under constant heat inputs that becomes a limit cycle if the inputs vary
periodically. Finally, we propose new numerical analyses of spacecraft thermal
models based on our results, to complement the analyses normally carried out
with commercial software packages.Comment: 29 pages, 4 figure
Singularity Structure and Stability Analysis of the Dirac Equation on the Boundary of the Nutku Helicoid Solution
Dirac equation written on the boundary of the Nutku helicoid space consists
of a system of ordinary differential equations. We tried to analyze this system
and we found that it has a higher singularity than those of the Heun's
equations which give the solutions of the Dirac equation in the bulk. We also
lose an independent integral of motion on the boundary. This facts explain why
we could not find the solution of the system on the boundary in terms of known
functions. We make the stability analysis of the helicoid and catenoid cases
and end up with an appendix which gives a new example where one encounters a
form of the Heun equation.Comment: Version to appear in JM
Operator renewal theory and mixing rates for dynamical systems with infinite measure
We develop a theory of operator renewal sequences in the context of infinite
ergodic theory. For large classes of dynamical systems preserving an infinite
measure, we determine the asymptotic behaviour of iterates of the
transfer operator. This was previously an intractable problem.
Examples of systems covered by our results include (i) parabolic rational
maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly
expanding interval maps with indifferent fixed points.
In addition, we give a particularly simple proof of pointwise dual ergodicity
(asymptotic behaviour of ) for the class of systems under
consideration.
In certain situations, including Pomeau-Manneville intermittency maps, we
obtain higher order expansions for and rates of mixing. Also, we obtain
error estimates in the associated Dynkin-Lamperti arcsine laws.Comment: Preprint, August 2010. Revised August 2011. After publication, a
minor error was pointed out by Kautzsch et al, arXiv:1404.5857. The updated
version includes minor corrections in Sections 10 and 11, and corresponding
modifications of certain statements in Section 1. All main results are
unaffected. In particular, Sections 2-9 are unchanged from the published
versio
On two-dimensional surface attractors and repellers on 3-manifolds
We show that if is an -diffeomorphism with a surface
two-dimensional attractor or repeller and is a
supporting surface for , then and
there is such that: 1) is a union
of disjoint tame surfaces such that every is
homeomorphic to the 2-torus . 2) the restriction of to
is conjugate to Anosov automorphism of
Combinatorics of Open Covers VI: Selectors for Sequences of Dense Sets
We consider the following two selection principles for topological spaces:
[Principle 1:] { For each sequence of dense subsets, there is a sequence of points from the space, the n-th point coming from the n-th dense set, such that this set of points is dense in the space;
[Principle 2:]{ For each sequence of dense subsets, there is a sequence of finite sets, the n-th a subset of the n-th dense set, such that the union of these finite sets is dense in the space.
We show that for separable metric space X one of these principles holds for the space C_p(X) of realvalued continuous functions equipped with the pointwise convergence topology if, and only if, a corresponding principle holds for a special family of open covers of X. An example is given to show that these equivalences do not hold in general for Tychonoff spaces. It is further shown that these two principles give characterizations for two popular cardinal numbers, and that these two principles are intimately related to an infinite game that was studied by Berner and Juhasz
Gauging and symplectic blowing up in nonlinear sigma-models: I. point singularities
In this paper a two dimensional non-linear sigma model with a general
symplectic manifold with isometry as target space is used to study symplectic
blowing up of a point singularity on the zero level set of the moment map
associated with a quasi-free Hamiltonian action. We discuss in general the
relation between symplectic reduction and gauging of the symplectic isometries
of the sigma model action. In the case of singular reduction, gauging has the
same effect as blowing up the singular point by a small amount. Using the
exponential mapping of the underlying metric, we are able to construct
symplectic diffeomorphisms needed to glue the blow-up to the global reduced
space which is regular, thus providing a transition from one symplectic sigma
model to another one free of singularities.Comment: 32 pages, LaTex, THEP 93/24 (corrected and expanded(about 5 pages)
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