21,847 research outputs found
Fixation in large populations: a continuous view of a discrete problem
We study fixation in large, but finite, populations with two types, and
dynamics governed by birth-death processes. By considering a restricted class
of such processes, we derive a continuous approximation for the probability of
fixation that is valid beyond the weak-selection (WS) limit. From the
continuous approximations, we then obtain asymptotic approximations for
evolutionary dynamics with at most one equilibrium, in the selection-driven
regime, that does not preclude a weak-selection regime. As an application, we
study the fixation pattern when the infinite population limit has an interior
Evolutionary Stable Strategy (ESS): (i) we show that the fixation pattern for
the Hawk and Dove game satisfies what we term the one-half law: if the
Evolutionary Stable Strategy (ESS) is outside a small interval around
\sfrac{1}{2}, the fixation is of dominance type; (ii) we also show that,
outside of the weak-selection regime, the long-term dynamics of large
populations can have very little resemblance to the infinite population case;
in addition, we also present results for the case of two equilibria. Finally,
we present continuous restatements valid for large populations of two classical
concepts naturally defined in the discrete case: (i) the definition of an
\textsc{ESS}_N strategy; (ii) the definition of a risk-dominant strategy. We
then present two applications of these restatements: (i) we obtain an
asymptotic definition valid in the quasi-neutral regime that recovers both the
one-third law under linear fitness and the generalised one-third law for
-player games; (ii) we extend the ideas behind the (generalised) one-third
law outside the quasi-neutral regime and, as a generalisation, we introduce the
concept of critical-frequency; (iii) we recover the classification of
risk-dominant strategies for -player games
A non-standard evolution problem arising in population genetics
We study the evolution of the probability density of an asexual, one locus
population under natural selection and random evolution. This evolution is
governed by a Fokker-Planck equation with degenerate coefficients on the
boundaries, supplemented by a pair of conservation laws. It is readily shown
that no classical or standard weak solution definition yields solvability of
the problem. We provide an appropriate definition of weak solution for the
problem, for which we show existence and uniqueness. The solution displays a
very distinctive structure and, for large time, we show convergence to a unique
stationary solution that turns out to be a singular measure supported at the
endpoints. An exponential rate of convergence to this steady state is also
proved.Comment: 16 pages. Proposition 4 statement and proof corrected. Also a number
of typos has been fixe
Discussion on the energy content of the galactic dark matter Bose-Einstein condensate halo in the Thomas-Fermi approximation
We show that the galactic dark matter halo, considered composed of an
axionlike particles Bose-Einstein condensate \cite{pir12} trapped by a
self-graviting potential \cite{boh07}, may be stable in the Thomas-Fermi
approximation since appropriate choices for the dark matter particle mass and
scattering length are made. The demonstration is performed by means of the
calculation of the potential, kinetic and self-interaction energy terms of a
galactic halo described by a Boehmer-Harko density profile. We discuss the
validity of the Thomas-Fermi approximation for the halo system, and show that
the kinetic energy contribution is indeed negligible.Comment: 10 pages, 2 figures; v.2: 11 pages, 3 figures, references added,
matches published version in JCA
The continuous limit of the Moran process and the diffusion of mutant genes in infinite populations
We consider the so called Moran process with frequency dependent fitness
given by a certain pay-off matrix. For finite populations, we show that the
final state must be homogeneous, and show how to compute the fixation
probabilities. Next, we consider the infinite population limit, and discuss the
appropriate scalings for the drift-diffusion limit. In this case, a degenerated
parabolic PDE is formally obtained that, in the special case of frequency
independent fitness, recovers the celebrated Kimura equation in population
genetics. We then show that the corresponding initial value problem is well
posed and that the discrete model converges to the PDE model as the population
size goes to infinity. We also study some game-theoretic aspects of the
dynamics and characterize the best strategies, in an appropriate sense.Comment: 40 pages, 14 figures, submitte
On the stochastic evolution of finite populations
This work is a systematic study of discrete Markov chains that are used to
describe the evolution of a two-types population. Motivated by results valid
for the well-known Moran (M) and Wright-Fisher (WF) processes, we define a
general class of Markov chains models which we term the Kimura class. It
comprises the majority of the models used in population genetics, and we show
that many well-known results valid for M and WF processes are still valid in
this class. In all Kimura processes, a mutant gene will either fixate or become
extinct, and we present a necessary and sufficient condition for such processes
to have the probability of fixation strictly increasing in the initial
frequency of mutants. This condition implies that there are WF processes with
decreasing fixation probability --- in contradistinction to M processes which
always have strictly increasing fixation probability. As a by-product, we show
that an increasing fixation probability defines uniquely an M or WF process
which realises it, and that any fixation probability with no state having
trivial fixation can be realised by at least some WF process. These results are
extended to a subclass of processes that are suitable for describing
time-inhomogeneous dynamics. We also discuss the traditional identification of
frequency dependent fitnesses and pay-offs, extensively used in evolutionary
game theory, the role of weak selection when the population is finite, and the
relations between jumps in evolutionary processes and frequency dependent
fitnesses
On the Classification of Finite Semigroups and RA-loop with the Hyperbolic Property
We classify the finite semigroups S, for which all the Z-orders O of the
rational Q-algebra QS, is such that the unit group U(O) is hyperbolic. We also
classify the RA-loops L, for which the unit loop U(ZL) does not contain any
free abelian subgroup of rank two.Comment: 5 pages; this is part of the second and third chapters of the second
author's PhD. Thesis. The last theorem was revised, as well as added some
definitions and reference
Free Groups in Quaternion Algebras
In \cite{jpsf} we constructed pairs of units in -orders of a
quaternion algebra over \Q (\sqrt{-d}), positive and
square free, such that is free for some . Here
we extend this result to any imaginary quadratic extension of ,
thus including matrix algebras. More precisely, we show that is a
free group for all and and for and all . The
units we use arise from Pell's and Gauss' equations. A criterion for a pair of
homeomorphisms to generate a free semigroup is also established and used to
prove that two certain units generate a free semigroup but that, in this case,
the Ping-Pong Lemma can not be applied to show that the group they generate is
free.Comment: 10 pages, article presented in conferences: Algebra School,
Brasilia-Brazil, Brasilia National University (july-2010); Summer 2009
Meeting of CMS in Groups and Hopf Algebras section, St. Jonh's-Canada,
Memorial University of Newfoundland (June-2009); Groups, Rings and Group
Rings, Ubatuba-Brazil (july-2008
Hyperbolicity of Orders of Quaternions Algebras and Group Rings
For a given divison algebra of the quaternions we construct two types of
units: Pell units and Gauss units. If K is a rational quadratic extension and G
is a finite group, we classify R and G, s.t., the unit group U(RG) of
augmentation one is hyperbolic. In particular we give necessary and sufficient
conditions when G is the quaternion group of order 8. In this case, the
hyperbolic boundary is isomorphic to the 2-dimensional sphere. As a result we
reach a class of groups of one end, that are not virtually free.Comment: This report is part of the first chapter of the second author's PhD.
Thesis. It was included a result in main theorem and a sketch of its proof
before it, as well as two new reference
Alternative algebras with the hyperbolic property
We investigate the structure of an alternative finite dimensional
\Q-algebra subject to the condition that for a -order
, and thus for every -order of ,
the loop of units of \U (\Gamma) does not contain a free abelian subgroup of
rank two. In particular, we prove that the radical of such an algebra
associates with the whole algebra. We also classify -loops for which
has this property. The classification for group rings is still an
open problem.Comment: Third author's Ph.D. (parcial); generalization of those results for
the non-associative case, 9 pps. Conf.: Geometry, Topology, Algebra and
Number Theory, Steklov Math. Inst., Moscow-Russia (Aug, 2010); XVIII Latin
American Colloquium of Algebra, S\~ao Paulo-Brazil (July, 2009); Algebras,
Representation and Applications, Ubatuba-Brazil (aug., 2007); Group and Group
Rings, Bedlewo-Poland (2005
Hyperbolicity of Semigroup Algebras II
In 1996 Jespers and Wang classified finite semigroups whose integral
semigroup ring has finitely many units. In a recent paper, Iwaki-Juriaans-Souza
Filho continued this line of research by partially classifying the finite
semigroups whose rational semigroup algebra %over a field of characteristic
zero, contains a -order with hyperbolic unit group. In this paper
we complete this classification by handling the case in which the semigroup is
semi-simple.Comment: In doi:10.1016/j.jalgebra.2008.03.015 it was, partially, classified
the finite semigroups S which the semigroup algebra QS has the hyperbolic
property. In this article we complete this classificatio
- β¦