Linear-Size Approximations to the Vietoris-Rips Filtration

The Vietoris-Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes useful information about the topology of the underlying metric space. This information is often extracted from its so-called persistence diagram. Unfortunately, this filtration is often too large to construct in full. We show how to construct an O(n)-size filtered simplicial complex on an $n$-point metric space such that its persistence diagram is a good approximation to that of the Vietoris-Rips filtration. This new filtration can be constructed in $O(n\log n)$ time. The constant factors in both the size and the running time depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets. We describe two different sparse filtrations. The first is a zigzag filtration that removes points as the scale increases. The second is a (non-zigzag) filtration that yields the same persistence diagram. Both methods are based on a hierarchical net-tree and yield the same guarantees

Learning in the Panopticon: ethical and social issues in building a virtual educational environment

This paper examines ethical and social issues which have proved important when initiating and creating educational spaces within a virtual environment. It focuses on one project, identifying the key decisions made, the barriers to new practice encountered and the impact these had on the project. It demonstrates the importance of the ‘backstage’ ethical and social issues involved in the creation of a virtual education community and offers conclusions, and questions, which will inform future research and practice in this area. These ethical issues are considered using Knobel’s framework of front-end, in-process and back-end concerns, and include establishing social practices for the islands, allocating access rights, considering personal safety and supporting researchers appropriately within this contex

A Fast Algorithm for Well-Spaced Points and Approximate Delaunay Graphs

We present a new algorithm that produces a well-spaced superset of points conforming to a given input set in any dimension with guaranteed optimal output size. We also provide an approximate Delaunay graph on the output points. Our algorithm runs in expected time $O(2^{O(d)}(n\log n + m))$, where $n$ is the input size, $m$ is the output point set size, and $d$ is the ambient dimension. The constants only depend on the desired element quality bounds. To gain this new efficiency, the algorithm approximately maintains the Voronoi diagram of the current set of points by storing a superset of the Delaunay neighbors of each point. By retaining quality of the Voronoi diagram and avoiding the storage of the full Voronoi diagram, a simple exponential dependence on $d$ is obtained in the running time. Thus, if one only wants the approximate neighbors structure of a refined Delaunay mesh conforming to a set of input points, the algorithm will return a size $2^{O(d)}m$ graph in $2^{O(d)}(n\log n + m)$ expected time. If $m$ is superlinear in $n$, then we can produce a hierarchically well-spaced superset of size $2^{O(d)}n$ in $2^{O(d)}n\log n$ expected time.Comment: Full versio

A Geometric Perspective on Sparse Filtrations

We present a geometric perspective on sparse filtrations used in topological data analysis. This new perspective leads to much simpler proofs, while also being more general, applying equally to Rips filtrations and Cech filtrations for any convex metric. We also give an algorithm for finding the simplices in such a filtration and prove that the vertex removal can be implemented as a sequence of elementary edge collapses

A self-consistent Hartree-Fock approach for interacting bosons in optical lattices

A theoretical study of interacting bosons in a periodic optical lattice is presented. Instead of the commonly used tight-binding approach (applicable near the Mott insulating regime of the phase diagram), the present work starts from the exact single-particle states of bosons in a cubic optical lattice, satisfying the Mathieu equation, an approach that can be particularly useful at large boson fillings. The effects of short-range interactions are incorporated using a self-consistent Hartree-Fock approximation, and predictions for experimental observables such as the superfluid transition temperature, condensate fraction, and boson momentum distribution are presented.Comment: 12 pages, 15 figure file

Cognitive Illusion, Lucid Dreaming, and the Psychology of Metaphor in Tibetan Buddhist Dzogchen Contemplative Practices

A classic set of eight similes of illusion (sgyu ma’i dpe brgyad) are employed recurrently throughout Indian and Tibetan Buddhist literature to illustrate the operations of cognition, its correlative perceptions, and experiences that emerge. To illustrate a Buddhist psychology of metaphor, the fourteenth century Tibetan scholar and synthesizer of the Dzogchen (rdzogs chen) or Great Perfection system, Longchen Rabjam Drimé Ödzer (1308-1363), composed his poetic text, Being at Ease with Illusion. This work on illusion is the third volume in Longchenpa’s Trilogy of Being at Ease (Ngal gso skor gsum) in which he presents a series of Dzogchen instructions on how to settle totally at ease. To complement each volume in his trilogy, Longchenpa composed auxiliary contemplative guidance instructions on their meaning (don khrid). This article contextualizes Longchenpa’s meditation manual on Being at Ease with Illusion, a translation of which is included in the appendix. Special attention is given to Dzogchen practices of lucid dreaming and working with cognitive illusions to spotlight underlying contemplative dynamics and correlative psychological effects. To analogically map these Tibetan language instructions in translation, this article interprets Buddhist psychological understandings of cognitive and perceptual processes in dialogue with current theories in the cognitive sciences

A Sparse Delaunay Filtration

We show how a filtration of Delaunay complexes can be used to approximate the persistence diagram of the distance to a point set in ?^d. Whereas the full Delaunay complex can be used to compute this persistence diagram exactly, it may have size O(n^?d/2?). In contrast, our construction uses only O(n) simplices. The central idea is to connect Delaunay complexes on progressively denser subsamples by considering the flips in an incremental construction as simplices in d+1 dimensions. This approach leads to a very simple and straightforward proof of correctness in geometric terms, because the final filtration is dual to a (d+1)-dimensional Voronoi construction similar to the standard Delaunay filtration. We also, show how this complex can be efficiently constructed

Induced p-wave superfluidity in strongly interacting imbalanced Fermi gases

The induced interaction among the majority spin species, due to the presence of the minority species, is computed for the case of a population-imbalanced resonantly-interacting Fermi gas. It is shown that this interaction leads to an instability, at low temperatures, of the recently observed polaron Fermi liquid phase of strongly imbalanced Fermi gases to a p-wave superfluid state. We find that the associated transition temperature, while quite small in the weakly interacting BCS regime, is experimentally accessible in the strongly interacting unitary regime.Comment: Published versio