Microwave Tube Research

Contains reports on three research projects

Finitary Boolean functions

We study functions on the infinite-dimensional Hamming cube (-1,1)**, in particu- lar Boolean functions into (-1,1), generalising results on analysis of Boolean functions on (-1,1); for n 2 N. The notion of noise sensitivity, first studied in [BKS99], is extended to this setting, and basic Fourier formulas are established. We also prove hypercontractivity estimates for these functions, and give a version of the Kahn-Kalai- Linial theorem giving a bound relating the total in uence to the maximal in uence. Particular attention is paid to so-called finitary functions, which are functions for which there exists an algorithm that almost surely queries only finitely many bits. Two versions of the Benjamini-Kalai-Schramm theorem characterizing noise sensitivity in terms of the sum of squared influences are given, under additional moment hypotheses on the amount of bits looked at by an algorithm. A version of the Kahn-Kalai-Linial theorem giving that the maximal influence is of order log(n) n is also given, replacing n with the expected number of bits looked at by an algorithm. Finally, we show that the result in [SS10] that revealments going to zero implies noise sensitivity also holds for finitary functions, and apply this to show noise sensitivity of a version of the voter model on suffciently sparse graphs

Universal lower bound for community structure of sparse graphs

We prove new lower bounds on the modularity of graphs. Specifically, the modularity of a graph $G$ with average degree $\bar d$ is $\Omega(\bar{d}^{-1/2})$, under some mild assumptions on the degree sequence of $G$. The lower bound $\Omega(\bar{d}^{-1/2})$ applies, for instance, to graphs with a power-law degree sequence or a near-regular degree sequence. It has been suggested that the relatively high modularity of the Erd\H{o}s-R\'enyi random graph $G_{n,p}$ stems from the random fluctuations in its edge distribution, however our results imply high modularity for any graph with a degree sequence matching that typically found in $G_{n,p}$. The proof of the new lower bound relies on certain weight-balanced bisections with few cross-edges, which build on ideas of Alon [Combinatorics, Probability and Computing (1997)] and may be of independent interest.Comment: 25 pages, 2 figure

Itererade slumpmässiga funktioner

Vi ger en kort introduktion till hur itererade slumpmässiga funktioner inducerar en markovkedja, samt till konvergens av sannolikhetsmått. Vi presenterar sedan Letacs sats, som ger förutsättningar för existensen hos en stationär fördelning i termer av Lipschitzkonstanterna för funktionerna. Vi studerar sedan till vilken grad satsen överlever utan Lipschitzkonstanter, och presenterar en generell sats som ger existens av en stationär fördelning, med andra förutsättningar. Vi studerar också huruvida satsen fortfarande håller om vi släpper på antaganden om oberoende och likafördelning, alltså släpper på att processen skall vara markovsk och tidshomogen. Vi ger en generalisering av Letacs sats som delvis innetäcker även detta fall

Nanoantenna enhancement for telecom-wavelength superconducting single photon detectors

Superconducting nanowire single photon detectors are rapidly emerging as a key infrared photon-counting technology. Two front-side-coupled silver dipole nanoantennas, simulated to have resonances at 1480 and 1525 nm, were fabricated in a two-step process. An enhancement of 50 to 130% in the system detection efficiency was observed when illuminating the antennas. This offers a pathway to increasing absorption into superconducting nanowires, creating larger active areas, and achieving more efficient detection at longer wavelengths

Itererade slumpmässiga funktioner

Vi ger en kort introduktion till hur itererade slumpmässiga funktioner inducerar en markovkedja, samt till konvergens av sannolikhetsmått. Vi presenterar sedan Letacs sats, som ger förutsättningar för existensen hos en stationär fördelning i termer av Lipschitzkonstanterna för funktionerna. Vi studerar sedan till vilken grad satsen överlever utan Lipschitzkonstanter, och presenterar en generell sats som ger existens av en stationär fördelning, med andra förutsättningar. Vi studerar också huruvida satsen fortfarande håller om vi släpper på antaganden om oberoende och likafördelning, alltså släpper på att processen skall vara markovsk och tidshomogen. Vi ger en generalisering av Letacs sats som delvis innetäcker även detta fall

Community Structure in Graphs : Algorithms and Theoretical Guarantees

This thesis consists of three papers in the area of community detection in graphs, together with an introductory section and summaries of the papers. In Paper I, we study the problem of finding overlapping clusterings of hypergraphs, continuing the line of research started by Carlsson and Mémoli (2013) of classifying clustering schemes as functors. We extend their notion of representability to the overlapping case, showing that any representable overlapping clustering scheme is excisive and functorial, and any excisive and functorial clustering scheme is isomorphic to a representable clustering scheme. In Paper II, we prove a new lower bound on the modularity of a graph, solely in terms of its degree sequence. We establish that, under mild assumptions on the tail of the degree sequence, a graph with average degree d has modularity of order d-1/2, and also use this result to give lower bounds for the modularity of preferential attachment graphs. In Paper III, we study the algorithmic problem of optimizing the modularity of temporal graphs, that is, graphs that may vary over time. We prove that computing a multiplicative approximation of this maximum temporal modularity is tractable parameterised by the underlying treewidth of the graph

A classification of overlapping clustering schemes for hypergraphs

Community detection in graphs is a problem that is likely to be relevant whenever network data appears, and consequently the problem has received much attention with many different methods and algorithms applied. However, many of these methods are hard to study theoretically, and they optimise for somewhat different goals. A general and rigorous account of the problem and possible methods remains elusive. We study the problem of finding overlapping clusterings of hypergraphs, continuing the line of research started by Carlsson and Mémoli (2013) of classifying clustering schemes as functors. We extend their notion of representability to the overlapping case, showing that any representable overlapping clustering scheme is excisive and functorial, and any excisive and functorial clustering scheme is isomorphic to a representable clustering scheme. We also note that, for simple graphs, any representable clustering scheme is computable in polynomial time on graphs of bounded expansion, with an exponent determined by the maximum independence number of a graph in the representing set. This result also applies to non-overlapping representable clustering schemes, and so may be of independent interest