6,398 research outputs found
Particle-based and Meshless Methods with Aboria
Aboria is a powerful and flexible C++ library for the implementation of
particle-based numerical methods. The particles in such methods can represent
actual particles (e.g. Molecular Dynamics) or abstract particles used to
discretise a continuous function over a domain (e.g. Radial Basis Functions).
Aboria provides a particle container, compatible with the Standard Template
Library, spatial search data structures, and a Domain Specific Language to
specify non-linear operators on the particle set. This paper gives an overview
of Aboria's design, an example of use, and a performance benchmark
On the statistical distribution of first--return times of balls and cylinders in chaotic systems
We study returns in dynamical systems: when a set of points, initially
populating a prescribed region, swarms around phase space according to a
deterministic rule of motion, we say that the return of the set occurs at the
earliest moment when one of these points comes back to the original region. We
describe the statistical distribution of these "first--return times" in various
settings: when phase space is composed of sequences of symbols from a finite
alphabet (with application for instance to biological problems) and when phase
space is a one and a two-dimensional manifold. Specifically, we consider
Bernoulli shifts, expanding maps of the interval and linear automorphisms of
the two dimensional torus. We derive relations linking these statistics with
Renyi entropies and Lyapunov exponents.Comment: submitted to Int. J. Bifurcations and Chao
Determinant Sums for Undirected Hamiltonicity
We present a Monte Carlo algorithm for Hamiltonicity detection in an
-vertex undirected graph running in time. To the best of
our knowledge, this is the first superpolynomial improvement on the worst case
runtime for the problem since the bound established for TSP almost
fifty years ago (Bellman 1962, Held and Karp 1962). It answers in part the
first open problem in Woeginger's 2003 survey on exact algorithms for NP-hard
problems.
For bipartite graphs, we improve the bound to time. Both the
bipartite and the general algorithm can be implemented to use space polynomial
in .
We combine several recently resurrected ideas to get the results. Our main
technical contribution is a new reduction inspired by the algebraic sieving
method for -Path (Koutis ICALP 2008, Williams IPL 2009). We introduce the
Labeled Cycle Cover Sum in which we are set to count weighted arc labeled cycle
covers over a finite field of characteristic two. We reduce Hamiltonicity to
Labeled Cycle Cover Sum and apply the determinant summation technique for Exact
Set Covers (Bj\"orklund STACS 2010) to evaluate it.Comment: To appear at IEEE FOCS 201
Numerical Evidence that the Perturbation Expansion for a Non-Hermitian -Symmetric Hamiltonian is Stieltjes
Recently, several studies of non-Hermitian Hamiltonians having
symmetry have been conducted. Most striking about these complex Hamiltonians is
how closely their properties resemble those of conventional Hermitian
Hamiltonians. This paper presents further evidence of the similarity of these
Hamiltonians to Hermitian Hamiltonians by examining the summation of the
divergent weak-coupling perturbation series for the ground-state energy of the
-symmetric Hamiltonian recently
studied by Bender and Dunne. For this purpose the first 193 (nonzero)
coefficients of the Rayleigh-Schr\"odinger perturbation series in powers of
for the ground-state energy were calculated. Pad\'e-summation and
Pad\'e-prediction techniques recently described by Weniger are applied to this
perturbation series. The qualitative features of the results obtained in this
way are indistinguishable from those obtained in the case of the perturbation
series for the quartic anharmonic oscillator, which is known to be a Stieltjes
series.Comment: 20 pages, 0 figure
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