6,390 research outputs found
Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology
We introduce some algebraic geometric models in cosmology related to the
"boundaries" of space-time: Big Bang, Mixmaster Universe, Penrose's crossovers
between aeons. We suggest to model the kinematics of Big Bang using the
algebraic geometric (or analytic) blow up of a point . This creates a
boundary which consists of the projective space of tangent directions to
and possibly of the light cone of . We argue that time on the boundary
undergoes the Wick rotation and becomes purely imaginary. The Mixmaster
(Bianchi IX) model of the early history of the universe is neatly explained in
this picture by postulating that the reverse Wick rotation follows a hyperbolic
geodesic connecting imaginary time axis to the real one. Penrose's idea to see
the Big Bang as a sign of crossover from "the end of previous aeon" of the
expanding and cooling Universe to the "beginning of the next aeon" is
interpreted as an identification of a natural boundary of Minkowski space at
infinity with the Big Bang boundary
Searching the solution space in constructive geometric constraint solving with genetic algorithms
Geometric problems defined by constraints have an exponential number
of solution instances in the number of geometric elements involved.
Generally, the user is only interested in one instance such that
besides fulfilling the geometric constraints, exhibits some additional
properties.
Selecting a solution instance amounts to selecting a given root every
time the geometric constraint solver needs to compute the zeros of a
multi valuated function. The problem of selecting a given root is
known as the Root Identification Problem.
In this paper we present a new technique to solve the root
identification problem. The technique is based on an automatic search
in the space of solutions performed by a genetic algorithm. The user
specifies the solution of interest by defining a set of additional
constraints on the geometric elements which drive the search of the
genetic algorithm. The method is extended with a sequential niche
technique to compute multiple solutions. A number of case studies
illustrate the performance of the method.Postprint (published version
Algebraic, geometric, and stochastic aspects of genetic operators
Genetic algorithms for function optimization employ genetic operators patterned after those observed in search strategies employed in natural adaptation. Two of these operators, crossover and inversion, are interpreted in terms of their algebraic and geometric properties. Stochastic models of the operators are developed which are employed in Monte Carlo simulations of their behavior
The Localization Transition of the Two-Dimensional Lorentz Model
We investigate the dynamics of a single tracer particle performing Brownian
motion in a two-dimensional course of randomly distributed hard obstacles. At a
certain critical obstacle density, the motion of the tracer becomes anomalous
over many decades in time, which is rationalized in terms of an underlying
percolation transition of the void space. In the vicinity of this critical
density the dynamics follows the anomalous one up to a crossover time scale
where the motion becomes either diffusive or localized. We analyze the scaling
behavior of the time-dependent diffusion coefficient D(t) including corrections
to scaling. Away from the critical density, D(t) exhibits universal
hydrodynamic long-time tails both in the diffusive as well as in the localized
phase.Comment: 13 pages, 7 figures
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