26,619 research outputs found
Cooperation between Top-Down and Bottom-Up Theorem Provers
Top-down and bottom-up theorem proving approaches each have specific
advantages and disadvantages. Bottom-up provers profit from strong redundancy
control but suffer from the lack of goal-orientation, whereas top-down provers
are goal-oriented but often have weak calculi when their proof lengths are
considered. In order to integrate both approaches, we try to achieve
cooperation between a top-down and a bottom-up prover in two different ways:
The first technique aims at supporting a bottom-up with a top-down prover. A
top-down prover generates subgoal clauses, they are then processed by a
bottom-up prover. The second technique deals with the use of bottom-up
generated lemmas in a top-down prover. We apply our concept to the areas of
model elimination and superposition. We discuss the ability of our techniques
to shorten proofs as well as to reorder the search space in an appropriate
manner. Furthermore, in order to identify subgoal clauses and lemmas which are
actually relevant for the proof task, we develop methods for a relevancy-based
filtering. Experiments with the provers SETHEO and SPASS performed in the
problem library TPTP reveal the high potential of our cooperation approaches
On Lagrangian tangent sweeps and Lagrangian outer billiards
Given a Lagrangian submanifold in linear symplectic space, its tangent sweep
is the union of its (affine) tangent spaces, and its tangent cluster is the
result of parallel translating these spaces so that the foot point of each
tangent space becomes the origin. This defines a multivalued map from the
tangent sweep to the tangent cluster, and we show that this map is a local
symplectomorphism (a well known fact, in dimension two).
We define and study the outer billiard correspondence associated with a
Lagrangian submanifold. Two points are in this correspondence if they belong to
the same tangent space and are symmetric with respect to its foot pointe. We
show that this outer billiard correspondence is symplectic and establish the
existence of its periodic orbits. This generalizes the well studied outer
billiard map in dimension two.Comment: revision as requested by the refere
Formal rigidity of the Witt and Virasoro Algebra
The formal rigidity of the Witt and Virasoro algebras was first established
by the author in [4]. The proof was based on some earlier results of the author
and Goncharowa, and was not presented there. In this paper we give an
elementary proof of these facts.Comment: 5 page
Castaing's instability in a trapped ultra-cold gas
We consider a trapped ultra-cold gas of (non-condensed) bosons with two
internal states (described by a pseudo spin) and study the stability of a
longitudinal pseudo spin polarization gradient. For this purpose, we
numerically solve a kinetic equation corresponding to a situation close to an
experiment at JILA. It shows the presence of Castaing's instability of
transverse spin polarization fluctuations at long wavelengths. This phenomenon
could be used to create spontaneous transverse spin waves.Comment: 5 pages, 3 figures; equation (8) corrected; submitted to EPJ
Large amplitude spin waves in ultra-cold gases
We discuss the theory of spin waves in non-degenerate ultra-cold gases, and
compare various methods which can be used to obtain appropriate kinetic
equations. We then study non-hydrodynamic situations, where the amplitude of
spin waves is sufficiently large to bring the system far from local
equilibrium. In the first part of the article, we compare two general methods
which can be used to derive a kinetic equation for a dilute gas of atoms
(bosons or fermions) with two internal states (treated as a pseudo-spin 1/2).
The collisional methods are in the spirit of Boltzmann's original derivation of
his kinetic equation where, at each point of space, the effects of all sorts of
possible binary collisions are added. We discuss two different versions of
collisional methods, the Yvon-Snider approach and the S matrix approach. The
second method uses the notion of mean field, which modifies the drift term of
the kinetic equation, in the line of the Landau theory of transport in quantum
liquids. For a dilute cold gas, it turns out that all these derivations lead to
the same drift terms in the transport equation, but differ in the precise
expression of the collision integral and in higher order gradient terms. In the
second part of the article, the kinetic equation is applied to spin waves in
trapped ultra-cold gases. Numerical simulations are used to illustrate the
strongly non-hydrodynamic character of the spin waves recently observed with
trapped Rb87 atoms. The decay of the phenomenon, which takes place when the
system relaxes back towards equilibrium, is also discussed, with a short
comment on decoherence.Comment: To appear in Eur. Phys. J.
Group theoretic, Lie algebraic and Jordan algebraic formulations of the SIC existence problem
Although symmetric informationally complete positive operator valued measures
(SIC POVMs, or SICs for short) have been constructed in every dimension up to
67, a general existence proof remains elusive. The purpose of this paper is to
show that the SIC existence problem is equivalent to three other, on the face
of it quite different problems. Although it is still not clear whether these
reformulations of the problem will make it more tractable, we believe that the
fact that SICs have these connections to other areas of mathematics is of some
intrinsic interest. Specifically, we reformulate the SIC problem in terms of
(1) Lie groups, (2) Lie algebras and (3) Jordan algebras (the second result
being a greatly strengthened version of one previously obtained by Appleby,
Flammia and Fuchs). The connection between these three reformulations is
non-trivial: It is not easy to demonstrate their equivalence directly, without
appealing to their common equivalence to SIC existence. In the course of our
analysis we obtain a number of other results which may be of some independent
interest.Comment: 36 pages, to appear in Quantum Inf. Compu
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