An analysis of the fixation probability of a mutant on special classes of non-directed graphs

There is a growing interest in the study of evolutionary dynamics on populations with some non-homogeneous structure. In this paper we follow the model of Lieberman et al. (Lieberman et al. 2005 Nature 433, 312–316) of evolutionary dynamics on a graph. We investigate the case of non-directed equally weighted graphs and find solutions for the fixation probability of a single mutant in two classes of simple graphs. We further demonstrate that finding similar solutions on graphs outside these classes is far more complex. Finally, we investigate our chosen classes numerically and discuss a number of features of the graphs; for example, we find the fixation probabilities for different initial starting positions and observe that average fixation probabilities are always increased for advantageous mutants as compared with those of unstructured populations

On the Consensus Threshold for the Opinion Dynamics of Krause-Hegselmann

In the consensus model of Krause-Hegselmann, opinions are real numbers between 0 and 1 and two agents are compatible if the difference of their opinions is smaller than the confidence bound parameter \epsilon. A randomly chosen agent takes the average of the opinions of all neighbouring agents which are compatible with it. We propose a conjecture, based on numerical evidence, on the value of the consensus threshold \epsilon_c of this model. We claim that \epsilon_c can take only two possible values, depending on the behaviour of the average degree d of the graph representing the social relationships, when the population N goes to infinity: if d diverges when N goes to infinity, \epsilon_c equals the consensus threshold \epsilon_i ~ 0.2 on the complete graph; if instead d stays finite when N goes to infinity, \epsilon_c=1/2 as for the model of Deffuant et al.Comment: 15 pages, 7 figures, to appear in International Journal of Modern Physics C 16, issue 2 (2005

On the Rigorous Derivation of the 3D Cubic Nonlinear Schr\"odinger Equation with A Quadratic Trap

We consider the dynamics of the 3D N-body Schr\"{o}dinger equation in the presence of a quadratic trap. We assume the pair interaction potential is N^{3{\beta}-1}V(N^{{\beta}}x). We justify the mean-field approximation and offer a rigorous derivation of the 3D cubic NLS with a quadratic trap. We establish the space-time bound conjectured by Klainerman and Machedon [30] for {\beta} in (0,2/7] by adapting and simplifying an argument in Chen and Pavlovi\'c [7] which solves the problem for {\beta} in (0,1/4) in the absence of a trap.Comment: Revised according to the referee report. Accepted to appear in Archive for Rational Mechanics and Analysi

Pair excitations and the mean field approximation of interacting Bosons, I

In our previous work \cite{GMM1},\cite{GMM2} we introduced a correction to the mean field approximation of interacting Bosons. This correction describes the evolution of pairs of particles that leave the condensate and subsequently evolve on a background formed by the condensate. In \cite{GMM2} we carried out the analysis assuming that the interactions are independent of the number of particles $N$. Here we consider the case of stronger interactions. We offer a new transparent derivation for the evolution of pair excitations. Indeed, we obtain a pair of linear equations describing their evolution. Furthermore, we obtain apriory estimates independent of the number of particles and use these to compare the exact with the approximate dynamics

Sidon Sets in Groups and Induced Subgraphs of Cayley Graphs

Let S be a subset of a group G. We call S a Sidon subset of the first (second) kind, if for any x, y, z, w ∈ S of which at least 3 are different, xy ≠ zw (xy-1 ≠ zw-1, resp.). (For abelian groups, the two notions coincide.) If G has a Sidon subset of the second kind with n elements then every n-vertex graph is an induced subgraph of some Cayley graph of G. We prove that a sufficient condition for G to have a Sidon subset of order n (of either kind) is that (❘G❘ ⩾ cn3. For elementary Abelian groups of square order, ❘G❘ ⩾ n2 is sufficient. We prove that most graphs on n vertices are not induced subgraphs of any vertex transitive graph with <cn2/log2n vertices. We comment on embedding trees and, in particular, stars, as induced subgraphs of Cayley graphs, and on the related problem of product-free (sum-free) sets in groups. We summarize the known results on the cardinality of Sidon sets of infinite groups, and formulate a number of open problems.We warn the reader that the sets considered in this paper are different from the Sidon sets Fourier analysts investigate

Optimization of Robustness of Complex Networks

Networks with a given degree distribution may be very resilient to one type of failure or attack but not to another. The goal of this work is to determine network design guidelines which maximize the robustness of networks to both random failure and intentional attack while keeping the cost of the network (which we take to be the average number of links per node) constant. We find optimal parameters for: (i) scale free networks having degree distributions with a single power-law regime, (ii) networks having degree distributions with two power-law regimes, and (iii) networks described by degree distributions containing two peaks. Of these various kinds of distributions we find that the optimal network design is one in which all but one of the nodes have the same degree, $k_1$ (close to the average number of links per node), and one node is of very large degree, $k_2 \sim N^{2/3}$, where $N$ is the number of nodes in the network.Comment: Accepted for publication in European Physical Journal

Rate of Convergence in Nonlinear Hartree Dynamics with Factorized Initial Data

The mean field dynamics of an $N$-particle weekly interacting Boson system can be described by the nonlinear Hartree equation. In this paper, we present estimates on the 1/N rate of convergence of many-body Schr\"{o}dinger dynamics to the one-body nonlinear Hartree dynamics with factorized initial data with two-body interaction potential $V$ in $L^3 (\mathbb{R}^3)+ L^{\infty} (\mathbb{R}^3)$.Comment: AMS LaTex, 21 page

Microscopic analysis of multipole susceptibility of actinide dioxides: A scenario of multipole ordering in AmO2_2

By evaluating multipole susceptibility of a seven-orbital impurity Anderson model with the use of a numerical renormalization group method, we discuss possible multipole states of actinide dioxides at low temperatures. In particular, here we point out a possible scenario for multipole ordering in americium dioxide. For Am$^{4+}$ ion with five $5f$ electrons, it is considered that the ground state is $\Gamma_7^{-}$ doublet and the first excited state is $\Gamma_8^{-}$ quartet, but we remark that the $f^5$ ground state is easily converted due to the competition between spin-orbit coupling and Coulomb interactions. Then, we find that the $\Gamma_8^-$ quartet can be the ground state of AmO$_2$ even for the same crystalline electric field potential. In the case of $\Gamma_8^-$ quartet ground state, the numerical results suggest that high-order multipoles such as quadrupole and octupole can be relevant to AmO$_2$.Comment: 8 pages, 4 figures. To appear in Phys. Rev.

Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in d=3d=3 based on spacetime norms

We derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy in dimension $d=3$, from an $N$-body Schr\"{o}dinger equation describing a gas of interacting bosons in the GP scaling, in the limit $N\rightarrow\infty$. The main result of this paper is the proof of convergence of the corresponding BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work on the well-posedness of the Cauchy problem for GP hierarchies, \cite{chpa2,chpa3,chpa4}, which are inspired by the solutions spaces based on space-time norms introduced by Klainerman and Machedon in \cite{klma}. We note that in $d=3$, this has been a well-known open problem in the field. While our results do not assume factorization of the solutions, consideration of factorized solutions yields a new derivation of the cubic, defocusing nonlinear Schr\"odinger equation (NLS) in $d=3$.Comment: 44 pages, AMS Late

Boltzmann limit and quasifreeness for a homogenous Fermi gas in a weakly disordered random medium

We discuss some basic aspects of the dynamics of a homogenous Fermi gas in a weak random potential, under negligence of the particle pair interactions. We derive the kinetic scaling limit for the momentum distribution function with a translation invariant initial state and prove that it is determined by a linear Boltzmann equation. Moreover, we prove that if the initial state is quasifree, then the time evolved state, averaged over the randomness, has a quasifree kinetic limit. We show that the momentum distributions determined by the Gibbs states of a free fermion field are stationary solutions of the linear Boltzmann equation; this includes the limit of zero temperature.Comment: AMS Latex, 26 pages. 2 figures. References added, minor typos correcte