On Lipschitz Retraction of Finite Subsets of Normed Spaces

If $X$ is a metric space, then its finite subset spaces $X(n)$ form a nested sequence under natural isometric embeddings $X = X(1)\subset X(2) \subset \cdots$. It was previously established, by Kovalev when $X$ is a Hilbert space and, by Ba\v{c}\'{a}k and Kovalev when $X$ is a CAT(0) space, that this sequence admits Lipschitz retractions $X(n)\rightarrow X(n-1)$ for all $n\geq 2$. We prove that when $X$ is a normed space, the above sequence admits Lipschitz retractions $X(n)\rightarrow X$, $X(n)\rightarrow X(2)$, as well as concrete retractions $X(n)\rightarrow X(n-1)$ that are Lipschitz if $n=2,3$ and H\"older-continuous on bounded sets if $n>3$. We also prove that if $X$ is a geodesic metric space, then each $X(n)$ is a $2$-quasiconvex metric space. These results are relevant to certain questions in the aforementioned previous work which asked whether Lipschitz retractions $X(n)\rightarrow X(n-1)$, $n\geq 2$, exist for $X$ in more general classes of Banach spaces.Comment: 20 pages, Isr. J. Math. (2019). "$\gamma$ is injective" added in Lemma 6.6(ii), Published in Israel Journal of Mathematic

Finite temperature field theory on the Moyal plane

10 pages, no figures.-- PACS nr.: 11.10.Wx.-- ArXiv pre-print available at: http://arxiv.org/abs/0907.0905v1.pdfIn this paper, we initiate the study of finite temperature quantum field theories on the Moyal plane. Such theories violate causality which influences the properties of these theories. In particular, causality influences the fluctuation-dissipation theorem: as we show, a disturbance in a space-time region M1 creates a response in a space-time region M2 spacelike with respect to M1 (M1×M2). The relativistic Kubo formula with and without noncommutativity is discussed in detail, and the modified properties of relaxation time and the dependence of mean square fluctuations on time are derived. In particular, the Sinha-Sorkin result [Phys. Rev. B 45, 8123 (1992)] on the logarithmic time dependence of the mean square fluctuations is discussed in our context. We derive an exact formula for the noncommutative susceptibility in terms of the susceptibility for the corresponding commutative case. It shows that noncommutative corrections in the four-momentum space have remarkable periodicity properties as a function of the four-momentum k. They have direction dependence as well and vanish for certain directions of the spatial momentum. These are striking observable signals for noncommutativity. The Lehmann representation is also generalized to any value of the noncommutativity parameter θ(μν) and finite temperatures.This work was supported by the US Department of Energy under grant number DE-FG02-85ER40231 and by the Universidad Carlos III de Madrid.Publicad

Causality and statistics on the Groenewold-Moyal plane

Quantum theories constructed on the noncommutative spacetime called the Groenewold-Moyal plane exhibit many interesting properties such as Lorentz and CPT noninvariance, causality violation and twisted statistics. We show that such violations lead to many striking features that may be tested experimentally. These theories predict Pauli forbidden transitions due to twisted statistics, anisotropies in the cosmic microwave background radiation due to correlations of observables in spacelike regions and Lorentz and CPT violations in scattering amplitudes.Comment: 12 pages, 1 figure. Based on the talk given by APB at the Workshop "Theoretical and Experimental Aspects of the Spin Statisics Connection and Related Symmetries", Stazione Marittima Conference Center, Trieste, Italy from the 21st to the 25th of October 200

Quantum Fields on the Groenewold-Moyal Plane

We give an introductory review of quantum physics on the noncommutative spacetime called the Groenewold-Moyal plane. Basic ideas like star products, twisted statistics, second quantized fields and discrete symmetries are discussed. We also outline some of the recent developments in these fields and mention where one can search for experimental signals.Comment: 50 pages, 3 figures. v2: published versio

On quasiconvexity of precompact-subset spaces

Suppose $X$ is a metric space and $BCl(X)$ the collection of its bounded closed subsets as a metric space with respect to Hausdorff distance (and call $BCl(X)$ the \emph{bounded-subset space} of $X$). The question of whether or not one can characterize (the existence of) a rectifiable path in some subspace $\mathcal{J}$ of $BCl(X)$ entirely in terms of rectifiable paths in $X$ does not seem to have been given serious consideration. In this paper, we make some progress with the case where $\mathcal{J}$ consists of precompact subsets of $X$ (with such a $\mathcal{J}$ called a \emph{precompact-subset space} of $X$). Specifically, in certain precompact-subset spaces $\mathcal{J}$ of $X$, we give a criterion to determine (the existence of) a rectifiable path in $\mathcal{J}$ using rectifiable paths in $X$. We then show that certain path connectivity properties, especially quasiconvexity, inherited from $X$ by such precompact-subset spaces of $X$ can be determined in an automatic way using our criterion. Meanwhile, we also give a concise review of our earlier work on quasiconvexity of \emph{finite-subset spaces} of $X$

Optimal Inference for Distributed Detection

In distributed detection, there does not exist an automatic way of generating optimal decision strategies for non-affine decision functions. Consequently, in a detection problem based on a non-affine decision function, establishing optimality of a given decision strategy, such as a generalized likelihood ratio test, is often difficult or even impossible. In this thesis we develop a novel detection network optimization technique that can be used to determine necessary and sufficient conditions for optimality in distributed detection for which the underlying objective function is monotonic and convex in probabilistic decision strategies. Our developed approach leverages on basic concepts of optimization and statistical inference which are provided in appendices in sufficient detail. These basic concepts are combined to form the basis of an optimal inference technique for signal detection. We prove a central theorem that characterizes optimality in a variety of distributed detection architectures. We discuss three applications of this result in distributed signal detection. These applications include interactive distributed detection, optimal tandem fusion architecture, and distributed detection by acyclic graph networks. In the conclusion we indicate several future research directions, which include possible generalizations of our optimization method and new research problems arising from each of the three applications considered