17,559 research outputs found
Modular elliptic curves over real abelian fields and the generalized Fermat equation
Using a combination of several powerful modularity theorems and class field
theory we derive a new modularity theorem for semistable elliptic curves over
certain real abelian fields. We deduce that if is a real abelian field of
conductor , with and , , , then every
semistable elliptic curve over is modular.
Let , , be prime, with , and .To a
putative non-trivial primitive solution of the generalized Fermat
we associate a Frey elliptic curve defined over
, and study its mod representation with the help
of level lowering and our modularity result. We deduce the non-existence of
non-trivial primitive solutions if , or if and , .Comment: Introduction rewritten to emphasise the new modularity theorem. Paper
revised in the light of referees' comment
The Shimura curve of discriminant 15 and topological automorphic forms
We find defining equations for the Shimura curve of discriminant 15 over
Z[1/15]. We then determine the graded ring of automorphic forms over the 2-adic
integers, as well as the higher cohomology. We apply this to calculate the
homotopy groups of a spectrum of "topological automorphic forms" associated to
this curve, as well as one associated to a quotient by an Atkin-Lehner
involution.Comment: 36 pages, 5 figures, updated with corrections and new introduction
(this version corrects image issues in the previous
SamACO: variable sampling ant colony optimization algorithm for continuous optimization
An ant colony optimization (ACO) algorithm offers
algorithmic techniques for optimization by simulating the foraging behavior of a group of ants to perform incremental solution
constructions and to realize a pheromone laying-and-following
mechanism. Although ACO is first designed for solving discrete
(combinatorial) optimization problems, the ACO procedure is
also applicable to continuous optimization. This paper presents
a new way of extending ACO to solving continuous optimization
problems by focusing on continuous variable sampling as a key
to transforming ACO from discrete optimization to continuous
optimization. The proposed SamACO algorithm consists of three
major steps, i.e., the generation of candidate variable values for
selection, the ants’ solution construction, and the pheromone
update process. The distinct characteristics of SamACO are the
cooperation of a novel sampling method for discretizing the
continuous search space and an efficient incremental solution
construction method based on the sampled values. The performance
of SamACO is tested using continuous numerical functions
with unimodal and multimodal features. Compared with some
state-of-the-art algorithms, including traditional ant-based algorithms
and representative computational intelligence algorithms
for continuous optimization, the performance of SamACO is seen
competitive and promising
Orthogonal methods based ant colony search for solving continuous optimization problems
Research into ant colony algorithms for solving continuous optimization problems forms one of the most
significant and promising areas in swarm computation. Although traditional ant algorithms are designed for combinatorial
optimization, they have shown great potential in solving a wide range of optimization problems, including continuous
optimization. Aimed at solving continuous problems effectively, this paper develops a novel ant algorithm termed "continuous orthogonal ant colony" (COAC), whose pheromone deposit mechanisms would enable ants to search for
solutions collaboratively and effectively. By using the orthogonal design method, ants in the feasible domain can explore
their chosen regions rapidly and e±ciently. By implementing an "adaptive regional radius" method, the proposed
algorithm can reduce the probability of being trapped in local optima and therefore enhance the global search capability and accuracy. An elitist strategy is also employed to reserve the most valuable points. The performance of the COAC is
compared with two other ant algorithms for continuous optimization of API and CACO by testing seventeen functions
in the continuous domain. The results demonstrate that the proposed COAC algorithm outperforms the others
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