Brownian Motion in a Weyl Chamber, Non-Colliding Particles, and Random Matrices

Let $n$ particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is $A_{n-1}$, the symmetric group. For any starting positions, we compute a determinant formula for the density function for the particles to be at specified positions at time $t$ without having collided by time $t$. We show that the probability that there will be no collision up to time $t$ is asymptotic to a constant multiple of $t^{-n(n-1)/4}$ as $t$ goes to infinity, and compute the constant as a polynomial of the starting positions. We have analogous results for the other classical Weyl groups; for example, the hyperoctahedral group $B_n$ gives a model of $n$ independent particles with a wall at $x=0$. We can define Brownian motion on a Lie algebra, viewing it as a vector space; the eigenvalues of a point in the Lie algebra correspond to a point in the Weyl chamber, giving a Brownian motion conditioned never to exit the chamber. If there are $m$ roots in $n$ dimensions, this shows that the radial part of the conditioned process is the same as the $n+2m$-dimensional Bessel process. The conditioned process also gives physical models, generalizing Dyson's model for $A_{n-1}$ corresponding to ${\mathfrak s}{\mathfrak u}_n$ of $n$ particles moving in a diffusion with a repelling force between two particles proportional to the inverse of the distance between them

Random Walk in an Alcove of an Affine Weyl Group, and Non-Colliding Random Walks on an Interval

We use a reflection argument, introduced by Gessel and Zeilberger, to count the number of k-step walks between two points which stay within a chamber of a Weyl group. We apply this technique to walks in the alcoves of the classical affine Weyl groups. In all cases, we get determinant formulas for the number of k-step walks. One important example is the region m>x_1>x_2>...>x_n>0, which is a rescaled alcove of the affine Weyl group C_n. If each coordinate is considered to be an independent particle, this models n non-colliding random walks on the interval (0,m). Another case models n non-colliding random walks on the circle.Comment: v.2, 22 pages; correction in a definition led to changes in many formulas, also added more background, references, and example

Continued Fractions and Unique Additive Partitions

A partition of the positive integers into sets A and B avoids a set S ⊂ N if no two distinct elements in the same part have a sum in S. If the partition is unique, S is uniquely avoidable. For any irrational α > 1, Chow and Long constructed a partition which avoids the numerators of all convergents of the continued fraction for α, and conjectured that the set S α which this partition avoids is uniquely avoidable. We prove that the set of numerators of convergents is uniquely avoidable if and only if the continued fraction for α has infinitely many partial quotients equal to 1. We also construct the set S α and show that it is always uniquely avoidable.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47656/1/11139_2004_Article_200319.pd

Cauchy's infinitesimals, his sum theorem, and foundational paradigms

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy's proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy's proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy's proof closely and show that it finds closer proxies in a different modern framework. Keywords: Cauchy's infinitesimal; sum theorem; quantifier alternation; uniform convergence; foundational paradigms.Comment: 42 pages; to appear in Foundations of Scienc

Response and Acquired Resistance to Everolimus in Anaplastic Thyroid Cancer

Everolimus, an inhibitor of the mammalian target of rapamycin (mTOR), is effective in treating tumors harboring alterations in the mTOR pathway. Mechanisms of resistance to everolimus remain undefined. Resistance developed in a patient with metastatic anaplastic thyroid carcinoma after an extraordinary 18-month response. Whole-exome sequencing of pretreatment and drug-resistant tumors revealed a nonsense mutation in TSC2, a negative regulator of mTOR, suggesting a mechanism for exquisite sensitivity to everolimus. The resistant tumor also harbored a mutation in MTOR that confers resistance to allosteric mTOR inhibition. The mutation remains sensitive to mTOR kinase inhibitors

‘The price is different depending on whether you want a receipt or not’: examining the purchasing of goods and services from the informal economy in South-East Europe

Research on the informal economy has largely focussed on supply-side issues, addressing questions like what motivates individuals to work in the informal economy and how can governments tackle this phenomenon. To date, much less attention has been given to demand-side aspects, examining issues around who purchases goods and services from the informal economy, why, and to what extent there are variations according to demographic, socio-economic and geographic dimensions. This paper addresses this imbalance by examining the purchasing of goods and services from the informal economy in South-East Europe. Firstly, this paper identifies the prevalence of such informal purchasing in South-East Europe as well as who undertakes such purchasing. Next, it examines the relative significance of cost factors, social factors and failures in the formal economy, in motivating such purchasing. Finally, it explores variability in the significance of these motivators based on individual-level factors, within and across three South-East European countries

Activating mTOR Mutations in a Patient with an Extraordinary Response on a Phase I Trial of Everolimus and Pazopanib

Understanding the genetic mechanisms of sensitivity to targeted anticancer therapies may improve patient selection, response to therapy, and rational treatment designs. One approach to increase this understanding involves detailed studies of exceptional responders: rare patients with unexpected exquisite sensitivity or durable responses to therapy. We identified an exceptional responder in a phase I study of pazopanib and everolimus in advanced solid tumors. Whole-exome sequencing of a patient with a 14-month complete response on this trial revealed two concurrent mutations in mTOR, the target of everolimus. In vitro experiments demonstrate that both mutations are activating, suggesting a biologic mechanism for exquisite sensitivity to everolimus in this patient. The use of precision (or “personalized”) medicine approaches to screen patients with cancer for alterations in the mTOR pathway may help to identify subsets of patients who may benefit from targeted therapies directed against mTOR.National Human Genome Research Institute (U.S.) (5U54HG003067-11

Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond

Many historians of the calculus deny significant continuity between infinitesimal calculus of the 17th century and 20th century developments such as Robinson's theory. Robinson's hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, Robinson regards Berkeley's criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley's criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz's infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz's defense of infinitesimals is more firmly grounded than Berkeley's criticism thereof. We show, moreover, that Leibniz's system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz's strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity.Comment: 69 pages, 3 figure

Ten Misconceptions from the History of Analysis and Their Debunking

The widespread idea that infinitesimals were "eliminated" by the "great triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d'Alembert, Cauchy, and others.Comment: 46 pages, 4 figures; Foundations of Science (2012). arXiv admin note: text overlap with arXiv:1108.2885 and arXiv:1110.545