5,346 research outputs found
Operator Ordering and Solution of Pseudo-Evolutionary Equations
The solution of pseudo initial value differential equations, either ordinary
or partial (including those of fractional nature), requires the development of
adequate analytical methods, complementing those well established in the
ordinary differential equation setting. A combination of techniques, involving
procedures of umbral and of operational nature, has been demonstrated to be a
very promising tool in order to approach within a unifying context
non-canonical evolution problems. This article covers the extension of this
approach to the solution of pseudo-evolutionary equations. We will comment on
the explicit formulation of the necessary techniques, which are based on
certain time- and operator ordering tools. We will in particular demonstrate
how Volterra-Neumann expansions, Feynman-Dyson series and other popular tools
can be profitably extended to obtain solutions of fractional differential
equations. We apply the method to a number of examples, in which fractional
calculus and a certain umbral image calculus play a role of central importance.Comment: 16 pages, 2 figure
Integrable viscous conservation laws
We propose an extension of the Dubrovin-Zhang perturbative approach to the study of normal forms for non-Hamiltonian integrable scalar conservation laws. The explicit computation of the first few corrections leads to the conjecture that such normal forms are parameterized by one single functional parameter, named viscous central invariant. A constant valued viscous central invariant corresponds to the well-known Burgers hierarchy. The case of a linear viscous central invariant provides a viscous analog of the Camassa-Holm equation, that formerly appeared as a reduction of a two-component Hamiltonian integrable systems. We write explicitly the negative and positive hierarchy associated with this equation and prove the integrability showing that they can be mapped respectively into the heat hierarchy and its negative counterpart, named the Klein-Gordon hierarchy. A local well-posedness theorem for periodic initial data is also proven.
We show how transport equations can be used to effectively construct asymptotic solutions via an extension of the quasi-Miura map that preserves the initial datum. The method is alternative to the method of the string equation for Hamiltonian conservation laws and naturally extends to the viscous case. Using these tools we derive the viscous analog of the Painlevé I2 equation that describes the universal behaviour of the solution at the critical point of gradient catastrophe
Elitism Levels Traverse Mechanism For The Derivation of Upper Bounds on Unimodal Functions
In this article we present an Elitism Levels Traverse Mechanism that we
designed to find bounds on population-based Evolutionary algorithms solving
unimodal functions. We prove its efficiency theoretically and test it on OneMax
function deriving bounds c{\mu}n log n - O({\mu} n). This analysis can be
generalized to any similar algorithm using variants of tournament selection and
genetic operators that flip or swap only 1 bit in each string.Comment: accepted to Congress on Evolutionary Computation (WCCI/CEC) 201
The problem of time and gauge invariance in the quantization of cosmological models. I. Canonical quantization methods
The paper is the first of two parts of a work reviewing some approaches to
the problem of time in quantum cosmology, which were put forward last decade,
and which demonstrated their relation to the problems of reparametrization and
gauge invariance of quantum gravity. In the present part we remind basic
features of quantum geometrodynamics and minisuperspace cosmological models,
and discuss fundamental problems of the Wheeler - DeWitt theory. Various
attempts to find a solution to the problem of time are considered in the
framework of the canonical approach. Possible solutions to the problem are
investigated making use of minisuperspace models, that is, systems with a
finite number of degrees of freedom. At the same time, in the last section of
the paper we expand our consideration beyond the minisuperspace approximation
and briefly review promising ideas by Brown and Kuchar, who propose that dust
interacting only gravitationally can be used for time measuring, and the
unitary approach by Barvinsky and collaborators. The latter approach admits
both the canonical and path integral formulations and anticipates the
consideration of recent developments in the path integral approach in the
second part of our work.Comment: 16 pages, to be published in Grav. Cosmo
Numerical Study of a Particle Method for Gradient Flows
We study the numerical behaviour of a particle method for gradient flows
involving linear and nonlinear diffusion. This method relies on the
discretisation of the energy via non-overlapping balls centred at the
particles. The resulting scheme preserves the gradient flow structure at the
particle level, and enables us to obtain a gradient descent formulation after
time discretisation. We give several simulations to illustrate the validity of
this method, as well as a detailed study of one-dimensional
aggregation-diffusion equations.Comment: 27 pages, 21 figure
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