Random ultrametric trees and applications

Ultrametric trees are trees whose leaves lie at the same distance from the root. They are used to model the genealogy of a population of particles co-existing at the same point in time. We show how the boundary of an ultrametric tree, like any compact ultrametric space, can be represented in a simple way via the so-called comb metric. We display a variety of examples of random combs and explain how they can be used in applications. In particular, we review some old and recent results regarding the genetic structure of the population when throwing neutral mutations on the skeleton of the tree.Comment: 20 pages, 7 figures, proceedings of MAS 2016, Grenoble, France (Stochastic modeling and Statistics Conference, French Society for Applied and Industrial Math, SMAI

Species abundance distributions in neutral models with immigration or mutation and general lifetimes

We consider a general, neutral, dynamical model of biodiversity. Individuals have i.i.d. lifetime durations, which are not necessarily exponentially distributed, and each individual gives birth independently at constant rate \lambda. We assume that types are clonally inherited. We consider two classes of speciation models in this setting. In the immigration model, new individuals of an entirely new species singly enter the population at constant rate \mu (e.g., from the mainland into the island). In the mutation model, each individual independently experiences point mutations in its germ line, at constant rate \theta. We are interested in the species abundance distribution, i.e., in the numbers, denoted I_n(k) in the immigration model and A_n(k) in the mutation model, of species represented by k individuals, k=1,2,...,n, when there are n individuals in the total population. In the immigration model, we prove that the numbers (I_t(k);k\ge 1) of species represented by k individuals at time t, are independent Poisson variables with parameters as in Fisher's log-series. When conditioning on the total size of the population to equal n, this results in species abundance distributions given by Ewens' sampling formula. In particular, I_n(k) converges as n\to\infty to a Poisson r.v. with mean \gamma /k, where \gamma:=\mu/\lambda. In the mutation model, as n\to\infty, we obtain the almost sure convergence of n^{-1}A_n(k) to a nonrandom explicit constant. In the case of a critical, linear birth--death process, this constant is given by Fisher's log-series, namely n^{-1}A_n(k) converges to \alpha^{k}/k, where \alpha :=\lambda/(\lambda+\theta). In both models, the abundances of the most abundant species are briefly discussed.Comment: 16 pages, 4 figures. To appear in Journal of Mathematical Biology. The final publication is available at http://www.springerlink.co

Impartiality, Close Friendships and the Confucian Tradition

This article explores the relationship between friendship and morality. Two ideas have been influential in the history of moral philosophy: the impartial standpoint and close friendship. These two perspectives on thought and action can conflict, however, and such a case is presented here. In an attempt to resolve these tensions, and understand the assumption that gives rise to it, I explore an alternative conception of moral conduct and friendship suggested by early Confucian thought. Within this account, moral conduct is that which aims at harmony, understood as the appropriate blending of different elements. This suggests a conception of friendship that realizes harmony through a focus on shared activities, and the quality of interaction achieved between people as they participate in shared social events. This account offers a novel way of conceptualizing friendship, which also avoids the tension between the impartial standpoint and close friendship

A Maskless Photolithography Apparatus for the Microfabrication of Electrical Leads

Graphene is a new and exciting, two-dimensional material. Particularly interesting are the electrical features of graphene. The small size of graphene used in this experiment (on the scale of microns) presents the need for small electrical leads. Photolithography can be used to make appropriately sized leads by depositing metal onto substrates in specific patterns. The technique uses light to transfer geometric patterns onto a light sensitive photoresist on the surface of a substrate. We have built a low cost, maskless photolithography apparatus assembled from a computer, a consumer grade projector, and a microscope. With multiple exposures, we can make features ranging from approximately 1 μm to 785 μm. The 1 μm feature size is near the theoretical minimum for the wavelength of blue light used, and will be more than sufficient for contacting the flakes of graphene, which average 50 μm in size

Measurements of the polarisation amplitudes and triple product asymmetries in B_s^0 to phi phi

Using 1fb^{-1} of pp collision data collected at center of mass energy sqrt{s} = 7 TeV during 2011 by the LHCb detector. Measurements of the triple product asymmetries, polarisation amplitudes and strong phase difference in the decay B_s^0 to phi phi are presented.Comment: Written for Moriond 2012 Electroweak Session. 4 pages, 3 figure

The Real Debate over the Senate’s Role in the Confirmation Process

The five Supreme Court nominations between 2005 and 2010 brought renewed attention to the Senate’s role in the confirmation process. This Note explores the debate over the Senate’s proper role in that process. First, this Note summarizes and clarifies the two traditional views of the Senate’s role, classifying them as the assertive view and the deferential view, and offers a new framework for understanding these views. This Note then traces the traditional arguments made by proponents of these views. It first examines the historical arguments, both from original understanding and historical practice; it then turns to pragmatic arguments about which view better accomplishes the purposes of the Senate’s participation in the confirmation process. Neither the historical arguments nor the pragmatic arguments resolve the issue of which approach to the confirmation process is better. By recounting these arguments, however, this Note reveals the underlying—and unspoken—difference between adherents of the assertive view and adherents of the deferential view: their conceptions of the relationship between law and politics differ widely. Adherents of the assertive view can fall on either end of a spectrum in understanding the relationship between law and politics. For some adherents of the assertive view, law is completely distinct from politics, so they believe senators should carefully ensure that judicial nominees understand this distinction and should vote only for those nominees who do and will respect it. For other adherents of the assertive view, law and politics are two sides of the same coin, so they think senators should aggressively inquire into the views of judicial nominees and should vote only for those nominees whose views comport with their own. Either way, the assertive view results in the same role for the Senate in the confirmation process. Adherents of the deferential view, by contrast, fall somewhere in the middle of the spectrum, believing that law is underdetermined and is shaped, but not totally controlled, by politics. Adherents of this view make certain that nominees have reasonable legal views, but they are more willing to vote to confirm nominees whose views differ from their own. This Note brings this important difference to the forefront in hopes of promoting more meaningful discussions about the Senate’s role in the confirmation process