4,373 research outputs found
A heuristic for boundedness of ranks of elliptic curves
We present a heuristic that suggests that ranks of elliptic curves over the
rationals are bounded. In fact, it suggests that there are only finitely many
elliptic curves of rank greater than 21. Our heuristic is based on modeling the
ranks and Shafarevich-Tate groups of elliptic curves simultaneously, and relies
on a theorem counting alternating integer matrices of specified rank. We also
discuss analogues for elliptic curves over other global fields.Comment: 41 pages, typos fixed in torsion table in section
On the Relationship of Quantum Mechanics to Classical Electromagnetism and Classical Relativistic Mechanics
Some connections between quantum mechanics and classical physics are
explored. The Planck-Einstein and De Broglie relations, the wavefunction and
its probabilistic interpretation, the Canonical Commutation Relations and the
Maxwell--Lorentz Equation may be understood in a simple way by comparing
classical electromagnetism and the photonic description of light provided by
classical relativistic kinematics. The method used may be described as `inverse
correspondence' since quantum phenomena become apparent on considering the low
photon number density limit of classical electromagnetism. Generalisation to
massive particles leads to the Klein--Gordon and Schr\"{o}dinger Equations. The
difference between the quantum wavefunction of the photon and a classical
electromagnetic wave is discussed in some detail.Comment: 14 pages, no figures, no table
M\"{o}bius deconvolution on the hyperbolic plane with application to impedance density estimation
In this paper we consider a novel statistical inverse problem on the
Poincar\'{e}, or Lobachevsky, upper (complex) half plane. Here the Riemannian
structure is hyperbolic and a transitive group action comes from the space of
real matrices of determinant one via M\"{o}bius transformations. Our
approach is based on a deconvolution technique which relies on the
Helgason--Fourier calculus adapted to this hyperbolic space. This gives a
minimax nonparametric density estimator of a hyperbolic density that is
corrupted by a random M\"{o}bius transform. A motivation for this work comes
from the reconstruction of impedances of capacitors where the above scenario on
the Poincar\'{e} plane exactly describes the physical system that is of
statistical interest.Comment: Published in at http://dx.doi.org/10.1214/09-AOS783 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A Distributed Approach for the Optimal Power Flow Problem Based on ADMM and Sequential Convex Approximations
The optimal power flow (OPF) problem, which plays a central role in operating
electrical networks is considered. The problem is nonconvex and is in fact NP
hard. Therefore, designing efficient algorithms of practical relevance is
crucial, though their global optimality is not guaranteed. Existing
semi-definite programming relaxation based approaches are restricted to OPF
problems where zero duality holds. In this paper, an efficient novel method to
address the general nonconvex OPF problem is investigated. The proposed method
is based on alternating direction method of multipliers combined with
sequential convex approximations. The global OPF problem is decomposed into
smaller problems associated to each bus of the network, the solutions of which
are coordinated via a light communication protocol. Therefore, the proposed
method is highly scalable. The convergence properties of the proposed algorithm
are mathematically substantiated. Finally, the proposed algorithm is evaluated
on a number of test examples, where the convergence properties of the proposed
algorithm are numerically substantiated and the performance is compared with a
global optimal method.Comment: 14 page
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