Modeling Flocks and Prices: Jumping Particles with an Attractive Interaction

We introduce and investigate a new model of a finite number of particles jumping forward on the real line. The jump lengths are independent of everything, but the jump rate of each particle depends on the relative position of the particle compared to the center of mass of the system. The rates are higher for those left behind, and lower for those ahead of the center of mass, providing an attractive interaction keeping the particles together. We prove that in the fluid limit, as the number of particles goes to infinity, the evolution of the system is described by a mean field equation that exhibits traveling wave solutions. A connection to extreme value statistics is also provided.Comment: 35 pages, 9 figures. A shortened version appears as arXiv:1108.243

The measurable Kesten theorem

We give explicit estimates between the spectral radius and the densities of short cycles for finite d-regular graphs. This allows us to show that the essential girth of a finite d-regular Ramanujan graph G is at least c log log |G|. We prove that infinite d-regular Ramanujan unimodular random graphs are trees. Using Benjamini-Schramm convergence this leads to a rigidity result saying that if most eigenvalues of a d-regular finite graph G fall in the Alon-Boppana region, then the eigenvalue distribution of G is close to the spectral measure of the d-regular tree. Kesten showed that if a Cayley graph has the same spectral radius as its universal cover, then it must be a tree. We generalize this to unimodular random graphs.Comment: The previous, longer version 1 has been split in two parts: the present paper, and a more group-theoretic one with the title "Kesten's theorem for Invariant Random Subgroups

Kesten's theorem for Invariant Random Subgroups

An invariant random subgroup of the countable group {\Gamma} is a random subgroup of {\Gamma} whose distribution is invariant under conjugation by all elements of {\Gamma}. We prove that for a nonamenable invariant random subgroup H, the spectral radius of every finitely supported random walk on {\Gamma} is strictly less than the spectral radius of the corresponding random walk on {\Gamma}/H. This generalizes a result of Kesten who proved this for normal subgroups. As a byproduct, we show that for a Cayley graph G of a linear group with no amenable normal subgroups, any sequence of finite quotients of G that spectrally approximates G converges to G in Benjamini-Schramm convergence. In particular, this implies that infinite sequences of finite d-regular Ramanujan Schreier graphs have essentially large girth.Comment: 19 page

Discrete Scale Axis Representations for 3D Geometry

This paper addresses the fundamental problem of computing stable medial representations of 3D shapes. We propose a spatially adaptive classification of geometric features that yields a robust algorithm for generating medial representations at different levels of abstraction. The recently introduced continuous scale axis transform serves as the mathematical foundation of our algorithm. We show how geometric and topological properties of the continuous setting carry over to discrete shape representations. Our method combines scaling operations of medial balls for geometric simplification with filtrations of the medial axis and provably good conversion steps to and from union of balls, to enable efficient processing of a wide variety shape representations including polygon meshes, 3D images, implicit surfaces, and point clouds. We demonstrate the robustness and versatility of our algorithm with an extensive validation on hundreds of shapes including complex geometries consisting of millions of triangles

OCC: A Smart Reply System for Efficient In-App Communications

Smart reply systems have been developed for various messaging platforms. In this paper, we introduce Uber's smart reply system: one-click-chat (OCC), which is a key enhanced feature on top of the Uber in-app chat system. It enables driver-partners to quickly respond to rider messages using smart replies. The smart replies are dynamically selected according to conversation content using machine learning algorithms. Our system consists of two major components: intent detection and reply retrieval, which are very different from standard smart reply systems where the task is to directly predict a reply. It is designed specifically for mobile applications with short and non-canonical messages. Reply retrieval utilizes pairings between intent and reply based on their popularity in chat messages as derived from historical data. For intent detection, a set of embedding and classification techniques are experimented with, and we choose to deploy a solution using unsupervised distributed embedding and nearest-neighbor classifier. It has the advantage of only requiring a small amount of labeled training data, simplicity in developing and deploying to production, and fast inference during serving and hence highly scalable. At the same time, it performs comparably with deep learning architectures such as word-level convolutional neural network. Overall, the system achieves a high accuracy of 76% on intent detection. Currently, the system is deployed in production for English-speaking countries and 71% of in-app communications between riders and driver-partners adopted the smart replies to speedup the communication process.Comment: link to demo: https://www.youtube.com/watch?v=nOffUT7rS0A&t=32

Population dynamics and genetic changes of Picea abies in the South Carpathians revealed by pollen and ancient DNA analyses

Background: Studies on allele length polymorphism designate several glacial refugia for Norway spruce (Picea abies) in the South Carpathian Mountains, but infer only limited expansion from these refugia after the last glaciation. To better understand the genetic dynamics of a South Carpathian spruce lineage, we compared ancient DNA from 10,700 and 11,000-year-old spruce pollen and macrofossils retrieved from Holocene lake sediment in the Retezat Mountains with DNA extracted from extant material from the same site. We used eight primer pairs that amplified short and variable regions of the spruce cpDNA. In addition, from the same lake sediment we obtained a 15,000-years-long pollen accumulation rate (PAR) record for spruce that helped us to infer changes in population size at this site.Results: We obtained successful amplifications for Norway spruce from 17 out of 462 pollen grains tested, while the macrofossil material provided 22 DNA sequences. Two fossil sequences were found to be unique to the ancient material. Population genetic statistics showed higher genetic diversity in the ancient individuals compared to the extant ones. Similarly, statistically significant Ks and Kst values showed a considerable level of differentiation between extant and ancient populations at the same loci.Lateglacial and Holocene PAR values suggested that population size of the ancient population was small, in the range of 1/10 or 1/5 of the extant population. PAR analysis also detected two periods of rapid population growths (from ca. 11,100 and 3900 calibrated years before present (cal yr BP)) and three bottlenecks (around 9180, 7200 and 2200 cal yr BP), likely triggered by climatic change and human impact.Conclusion: Our results suggest that the paternal lineages observed today in the Retezat Mountains persisted at this site at least since the early Holocene. Combination of the results from the genetic and the PAR analyses furthermore suggests that the higher level of genetic variation found in the ancient populations and the loss of ancient allele types detected in the extant individuals were likely due to the repeated bottlenecks during the Holocene; however our limited sample size did not allow us to exclude sampling effect.This study demonstrates how past population size changes inferred from PAR records can be efficiently used in combination with ancient DNA studies. The joint application of palaeoecological and population genetics analyses proved to be a powerful tool to understand the influence of past population demographic changes on the haplotype diversity and genetic composition of forest tree species