Wavelets and graph C∗C^*-algebras

Here we give an overview on the connection between wavelet theory and representation theory for graph $C^{\ast}$-algebras, including the higher-rank graph $C^*$-algebras of A. Kumjian and D. Pask. Many authors have studied different aspects of this connection over the last 20 years, and we begin this paper with a survey of the known results. We then discuss several new ways to generalize these results and obtain wavelets associated to representations of higher-rank graphs. In \cite{FGKP}, we introduced the "cubical wavelets" associated to a higher-rank graph. Here, we generalize this construction to build wavelets of arbitrary shapes. We also present a different but related construction of wavelets associated to a higher-rank graph, which we anticipate will have applications to traffic analysis on networks. Finally, we generalize the spectral graph wavelets of \cite{hammond} to higher-rank graphs, giving a third family of wavelets associated to higher-rank graphs

Almost-Euclidean subspaces of ℓ1N\ell_1^N via tensor products: a simple approach to randomness reduction

It has been known since 1970's that the N-dimensional $\ell_1$-space contains nearly Euclidean subspaces whose dimension is $\Omega(N)$. However, proofs of existence of such subspaces were probabilistic, hence non-constructive, which made the results not-quite-suitable for subsequently discovered applications to high-dimensional nearest neighbor search, error-correcting codes over the reals, compressive sensing and other computational problems. In this paper we present a "low-tech" scheme which, for any $a > 0$, allows to exhibit nearly Euclidean $\Omega(N)$-dimensional subspaces of $\ell_1^N$ while using only $N^a$ random bits. Our results extend and complement (particularly) recent work by Guruswami-Lee-Wigderson. Characteristic features of our approach include (1) simplicity (we use only tensor products) and (2) yielding "almost Euclidean" subspaces with arbitrarily small distortions.Comment: 11 pages; title change, abstract and references added, other minor change

Convex recovery of a structured signal from independent random linear measurements

This chapter develops a theoretical analysis of the convex programming method for recovering a structured signal from independent random linear measurements. This technique delivers bounds for the sampling complexity that are similar with recent results for standard Gaussian measurements, but the argument applies to a much wider class of measurement ensembles. To demonstrate the power of this approach, the paper presents a short analysis of phase retrieval by trace-norm minimization. The key technical tool is a framework, due to Mendelson and coauthors, for bounding a nonnegative empirical process.Comment: 18 pages, 1 figure. To appear in "Sampling Theory, a Renaissance." v2: minor corrections. v3: updated citations and increased emphasis on Mendelson's contribution

Bioprospecting the thermal waters of the Roman Baths:Isolation of oleaginous species and analysis of the FAME profile for biodiesel production

The extensive diversity of microalgae provides an opportunity to undertake bioprospecting for species possessing features suited to commercial scale cultivation. The outdoor cultivation of microalgae is subject to extreme temperature fluctuations; temperature tolerant microalgae would help mitigate this problem. The waters of the Roman Baths, which have a temperature range between 39°C and 46°C, were sampled for microalgae. A total of 3 green algae, 1 diatom and 4 cyanobacterial species were successfully isolated into ‘unialgal’ culture. Four isolates were filamentous, which could prove advantageous for low energy dewatering of cultures using filtration. Lipid content, profiles and growth rates of the isolates were examined at temperatures of 20, 30, 40°C, with and without nitrogen starvation and compared against the oil producing green algal species, Chlorella emersonii. Some isolates synthesized high levels of lipids, however, all were most productive at temperatures lower than those of the Roman Baths. The eukaryotic algae accumulated a range of saturated and polyunsaturated FAMEs and all isolates generally showed higher lipid accumulation under nitrogen deficient conditions (Klebsormidium sp. increasing from 1.9% to 16.0% and Hantzschia sp. from 31.9 to 40.5%). The cyanobacteria typically accumulated a narrower range of FAMEs that were mostly saturated, but were capable of accumulating a larger quantity of lipid as a proportion of dry weight (M. laminosus, 37.8% fully saturated FAMEs). The maximum productivity of all the isolates was not determined in the current work and will require further effort to optimise key variables such as light intensity and media composition

Response of CO2 and CH4 emissions from Arctic tundra soils to a multifactorial manipulation of water table, temperature and thaw depth

Significant uncertainties persist concerning how Arctic soil tundra carbon emission responds to environmental changes. In this study, 24 cores were sampled from drier (high centre polygons and rims) and wetter (low centre polygons and troughs) permafrost tundra ecosystems. We examined how soil CO2 and CH4 fluxes responded to laboratory-based manipulations of soil temperature (and associated thaw depth) and water table depth, representing current and projected conditions in the Arctic. Similar soil CO2 respiration rates occurred in both the drier and the wetter sites, suggesting that a significant proportion of soil CO2 emission occurs via anaerobic respiration under water-saturated conditions in these Arctic tundra ecosystems. In the absence of vegetation, soil CO2 respiration rates decreased sharply within the first 7 weeks of the experiment, while CH4 emissions remained stable for the entire 26 weeks of the experiment. These patterns suggest that soil CO2 emission is more related to plant input than CH4 production and emission. The stable and substantial CH4 emission observed over the entire course of the experiment suggests that temperature limitations, rather than labile carbon limitations, play a predominant role in CH4 production in deeper soil layers. This is likely due to the presence of a substantial source of labile carbon in these carbon-rich soils. The small soil temperature difference (a median difference of 1 °C) and a more substantial thaw depth difference (a median difference of 6 cm) between the high and low temperature treatments resulted in a non-significant difference between soil CO2 and CH4 emissions. Although hydrology continued to be the primary factor influencing CH4 emissions, these emissions remained low in the drier ecosystem, even with a water table at the surface. This result suggests the potential absence of a methanogenic microbial community in high-centre polygon and rim ecosystems. Overall, our results suggest that the temperature increases reported for these Arctic regions are not responsible for increases in carbon losses. Instead, it is the changes in hydrology that exert significant control over soil CO2 and CH4 emissions

The road to deterministic matrices with the restricted isometry property

The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.Comment: 24 page

Baryon number violation, baryogenesis and defects with extra dimensions

In generic models for grand unified theories(GUT), various types of baryon number violating processes are expected when quarks and leptons propagate in the background of GUT strings. On the other hand, in models with large extra dimensions, the baryon number violation in the background of a string is not trivial because it must depend on the mechanism of the proton stabilization. In this paper we argue that cosmic strings in models with extra dimensions can enhance the baryon number violation to a phenomenologically interesting level, if the proton decay is suppressed by the mechanism of localized wavefunctions. We also make some comments on baryogenesis mediated by cosmological defects. We show at least two scenarios will be successful in this direction. One is the scenario of leptogenesis where the required lepton number conversion is mediated by cosmic strings, and the other is the baryogenesis from the decaying cosmological domain wall. Both scenarios are new and have not been discussed in the past.Comment: 20pages, latex2e, comments and references added, to appear in PR

Cosmological bounds on large extra dimensions from non-thermal production of Kaluza-Klein modes

The existing cosmological constraints on theories with large extra dimensions rely on the thermal production of the Kaluza-Klein modes of gravitons and radions in the early Universe. Successful inflation and reheating, as well as baryogenesis, typically requires the existence of a TeV-scale field in the bulk, most notably the inflaton. The non-thermal production of KK modes with masses of order 100 GeV accompanying the inflaton decay sets the lower bounds on the fundamental scale M_*. For a 1 TeV inflaton, the late decay of these modes distort the successful predictions of Big Bang Nucleosynthesis unless M_*> 35, 13, 7, 5 and 3 TeV for 2, 3, 4, 5 and 6 extra dimensions, respectively. This improves the existing bounds from cosmology on M_* for 4, 5 and 6 extra dimensions. Even more stringent bounds are derived for a heavier inflaton.Comment: 17 pages, latex, 4 figure

User-friendly tail bounds for sums of random matrices

This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales. In other words, this paper provides noncommutative generalizations of the classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid. The matrix inequalities promise the same diversity of application, ease of use, and strength of conclusion that have made the scalar inequalities so valuable.Comment: Current paper is the version of record. The material on Freedman's inequality has been moved to a separate note; other martingale bounds are described in Caltech ACM Report 2011-0

Standard Model baryogenesis through four-fermion operators in braneworlds

We study a new baryogenesis scenario in a class of braneworld models with low fundamental scale, which typically have difficulty with baryogenesis. The scenario is characterized by its minimal nature: the field content is that of the Standard Model and all interactions consistent with the gauge symmetry are admitted. Baryon number is violated via a dimension-6 proton decay operator, suppressed today by the mechanism of quark-lepton separation in extra dimensions; we assume that this operator was unsuppressed in the early Universe due to a time-dependent quark-lepton separation. The source of CP violation is the CKM matrix, in combination with the dimension-6 operators. We find that almost independently of cosmology, sufficient baryogenesis is nearly impossible in such a scenario if the fundamental scale is above 100 TeV, as required by an unsuppressed neutron-antineutron oscillation operator. The only exception producing sufficient baryon asymmetry is a scenario involving out-of-equilibrium c quarks interacting with equilibrium b quarks.Comment: 39 pages, 5 figures v2: typos, presentational changes, references and acknowledgments adde