Distance Geometry in Quasihypermetric Spaces. III

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by $$I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y),$$ and set $M(X) = \sup I(\mu)$, where $\mu$ ranges over the collection of signed measures in $\mathcal{M}(X)$ of total mass 1. This paper, with two earlier papers [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and II], investigates the geometric constant $M(X)$ and its relationship to the metric properties of $X$ and the functional-analytic properties of a certain subspace of $\mathcal{M}(X)$ when equipped with a natural semi-inner product. Specifically, this paper explores links between the properties of $M(X)$ and metric embeddings of $X$, and the properties of $M(X)$ when $X$ is a finite metric space.Comment: 20 pages. References [10] and [11] are arXiv:0809.0740v1 [math.MG] and arXiv:0809.0744v1 [math.MG

On the topology of free paratopological groups

The result often known as Joiner's lemma is fundamental in understanding the topology of the free topological group $F(X)$ on a Tychonoff space$X$. In this paper, an analogue of Joiner's lemma for the free paratopological group \FP(X) on a $T_1$ space $X$ is proved. Using this, it is shown that the following conditions are equivalent for a space $X$: (1) $X$ is $T_1$; (2) \FP(X) is $T_1$; (3) the subspace $X$ of \FP(X) is closed; (4) the subspace $X^{-1}$ of \FP(X) is discrete; (5) the subspace $X^{-1}$ is $T_1$; (6) the subspace $X^{-1}$ is closed; and (7) the subspace \FP_n(X) is closed for all $n \in \N$, where \FP_n(X) denotes the subspace of \FP(X) consisting of all words of length at most $n$.Comment: http://blms.oxfordjournals.org/cgi/content/abstract/bds031?ijkey=9Su2bYV9e19JMxf&keytype=re

Finite Quasihypermetric Spaces

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by $I(mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y)$, and set $M(X) = \sup I(mu)$, where $\mu$ ranges over the collection of measures in $\mathcal{M}(X)$ of total mass 1. The space $(X, d)$ is \emph{quasihypermetric} if $I(\mu) \leq 0$ for all measures $\mu$ in $\mathcal{M}(X)$ of total mass 0 and is \emph{strictly quasihypermetric} if in addition the equality $I(\mu) = 0$ holds amongst measures $\mu$ of mass 0 only for the zero measure. This paper explores the constant $M(X)$ and other geometric aspects of $X$ in the case when the space $X$ is finite, focusing first on the significance of the maximal strictly quasihypermetric subspaces of a given finite quasihypermetric space and second on the class of finite metric spaces which are $L^1$-embeddable. While most of the results are for finite spaces, several apply also in the general compact case. The analysis builds upon earlier more general work of the authors [Peter Nickolas and Reinhard Wolf, \emph{Distance geometry in quasihypermetric spaces. I}, \emph{II} and \emph{III}].Comment: 21 pages. References [11], [12] and [13] are arXiv:0809.0740v1 [math.MG], arXiv:0809.0744v1 [math.MG] and arXiv:0809.0746v1 [math.MG], res

Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Clouds

We combine standard persistent homology with image persistent homology to define a novel way of characterizing shapes and interactions between them. In particular, we introduce: (1) a mixup barcode, which captures geometric-topological interactions (mixup) between two point sets in arbitrary dimension; (2) simple summary statistics, total mixup and total percentage mixup, which quantify the complexity of the interactions as a single number; (3) a software tool for playing with the above. As a proof of concept, we apply this tool to a problem arising from machine learning. In particular, we study the disentanglement in embeddings of different classes. The results suggest that topological mixup is a useful method for characterizing interactions for low and high-dimensional data. Compared to the typical usage of persistent homology, the new tool is sensitive to the geometric locations of the topological features, which is often desirable

Verification of a Distributed Ledger Protocol for Distributed Autonomous Systems Using Monterey Phoenix

Autonomous multi-vehicle systems are becoming increasingly relevant in military operations and have demonstrated potential applicability in civilian environments as well. A problem emerges, however, when logging data within these systems. In particular, potential loss of individual vehicles and inherently lossy and noisy communications environments can result in the loss of important mission data. This paper describes a novel distributed ledger protocol that can be used to ensure that the data in such a system survives and documents verification of the behavioral correctness of this protocol using informal verification methods and tools provided by the Monterey Phoenix project

Next Steps & Closing

Discuss next steps for NTAS & PS&DS, action items, participant feedback, etc

Microwave Devices Employing Magnetic Waves

Contains reports on six research projects.Joint Services Electronics Program (Contract DAAG29-78-C-0020)National Science Foundation (Grant ENG76-18359

Investigating Methodological Differences in the Assessment of Dendritic Morphology of Basolateral Amygdala Principal Neurons-A Comparison of Golgi-Cox and Neurobiotin Electroporation Techniques

Quantitative assessments of neuronal subtypes in numerous brain regions show large variations in dendritic arbor size. A critical experimental factor is the method used to visualize neurons. We chose to investigate quantitative differences in basolateral amygdala (BLA) principal neuron morphology using two of the most common visualization methods: Golgi-Cox staining and neurobiotin (NB) filling. We show in 8-week-old Wistar rats that NB-filling reveals significantly larger dendritic arbors and different spine densities, compared to Golgi-Cox-stained BLA neurons. Our results demonstrate important differences and provide methodological insights into quantitative disparities of BLA principal neuron morphology reported in the literature

Data Combination: Interferometry and Single-dish Imaging in Radio Astronomy

Modern interferometers routinely provide radio-astronomical images down to subarcsecond resolution. However, interferometers filter out spatial scales larger than those sampled by the shortest baselines, which affects the measurement of both spatial and spectral features. Complementary single-dish data are vital for recovering the true flux distribution of spatially resolved astronomical sources with such extended emission. In this work, we provide an overview of the prominent available methods to combine single-dish and interferometric observations. We test each of these methods in the framework of the CASA data analysis software package on both synthetic continuum and observed spectral data sets. We develop a set of new assessment tools that are generally applicable to all radio-astronomical cases of data combination. Applying these new assessment diagnostics, we evaluate the methods' performance and demonstrate the significant improvement of the combined results in comparison to purely interferometric reductions. We provide combination and assessment scripts as add-on material. Our results highlight the advantage of using data combination to ensure high-quality science images of spatially resolved objects.Comment: 29 pages, 20 figures. Accepted for publication in PASP. Code repository available at: github.com/teuben/DataCom