Lipschitz regularity for elliptic equations with random coefficients

We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale $L^\infty$-type estimate for the gradient of a solution. The estimate is proved with optimal stochastic integrability under a one-parameter family of mixing assumptions, allowing for very weak mixing with non-integrable correlations to very strong mixing (e.g., finite range of dependence). We also prove a quenched $L^2$ estimate for the error in homogenization of Dirichlet problems. The approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence-form equations.Comment: 85 pages, minor revisio

The additive structure of elliptic homogenization

One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this paper, we address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the weak convergence of the gradients, fluxes and energy densities of the first-order correctors (under blow-down) which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument, completing the program initiated in \cite{AKM}: using the regularity theory recently developed for stochastic homogenization, we reduce the error in additivity as we pass to larger and larger length scales. In the second part of the paper, we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables. In particular, we prove that the first-order correctors converge, in the large-scale limit, to a variant of the Gaussian free field.Comment: 118 pages, to appear in Invent. Math. This version is a merger of v2 and arXiv:1603.03388 and supersedes the latter. Other changes in v3 are mino

Mesoscopic higher regularity and subadditivity in elliptic homogenization

We introduce a new method for obtaining quantitative results in stochastic homogenization for linear elliptic equations in divergence form. Unlike previous works on the topic, our method does not use concentration inequalities (such as Poincar\'e or logarithmic Sobolev inequalities in the probability space) and relies instead on a higher ($C^{k}$, $k \geq 1$) regularity theory for solutions of the heterogeneous equation, which is valid on length scales larger than a certain specified mesoscopic scale. This regularity theory, which is of independent interest, allows us to, in effect, localize the dependence of the solutions on the coefficients and thereby accelerate the rate of convergence of the expected energy of the cell problem by a bootstrap argument. The fluctuations of the energy are then tightly controlled using subadditivity. The convergence of the energy gives control of the scaling of the spatial averages of gradients and fluxes (that is, it quantifies the weak convergence of these quantities) which yields, by a new "multiscale" Poincar\'e inequality, quantitative estimates on the sublinearity of the corrector.Comment: 44 pages, revised version, to appear in Comm. Math. Phy

Mitigation of Side-Effect Modulation in Optical OFDM VLC Systems

Side-effect modulation (SEM) has the potential to be a significant source of interference in future visible light communication (VLC) systems. SEM is a variation in the intensity of the light emitted by a luminaire and is usually a side-effect caused by the power supply used to drive the luminaires. For LED luminaires powered by a switched mode power supply, the SEM can be at much higher frequencies than that emitted by conventional incandescent or fluorescent lighting. It has been shown that the SEM caused by commercially available LED luminaires is often periodic and of low power. In this paper, we investigate the impact of typical forms of SEM on the performance of optical OFDM VLC systems; both ACO-OFDM and DCO-OFDM are considered. Our results show that even low levels of SEM power can significantly degrade the bit-error-rate performance. To solve this problem, an SEM mitigation scheme is described. The mitigation scheme is decision-directed and is based on estimating and subtracting the fundamental component of the SEM from the received signal. We describe two forms of the algorithm; one uses blind estimation while the other uses pilot-assisted estimation based on a training sequence. Decision errors, resulting in decision noise, limit the performance of the blind estimator even when estimation is based on very long signals. However, the pilot system can achieve more accurate estimations, thus better performance. Results are first presented for typical SEM waveforms for the case where the fundamental frequency of the SEM is known. The algorithms are then extended to include a frequency estimation step and the mitigation algorithm is shown also to be effective in this case

A Comparison of CP-OFDM, PCC-OFDM and UFMC for 5G Uplink Communications

Polynomial-cancellation-coded orthogonal frequency division multiplexing (PCC-OFDM) is a form of OFDM that has waveforms which are very well localized in both the time and frequency domains and so it is ideally suited for use in the 5G network. This paper analyzes the performance of PCC-OFDM in the uplink of a multiuser system using orthogonal frequency division multiple access (OFDMA) and compares it with conventional cyclic prefix OFDM (CP-OFDM), and universal filtered multicarrier (UFMC). PCC-OFDM is shown to be much less sensitive than either CP-OFDM or UFMC to time and frequency offsets. For a given constellation size, PCC-OFDM in additive white Gaussian noise (AWGN) requires 3dB lower signal-to-noise ratio (SNR) for a given bit-error-rate, and the SNR advantage of PCC-OFDM increases rapidly when there are timing and/or frequency offsets. For PCC-OFDM no frequency guard band is required between different OFDMA users. PCC-OFDM is completely compatible with CP-OFDM and adds negligible complexity and latency, as it uses a simple mapping of data onto pairs of subcarriers at the transmitter, and a simple weighting-and-adding of pairs of subcarriers at the receiver. The weighting and adding step, which has been omitted in some of the literature, is shown to contribute substantially to the SNR advantage of PCC-OFDM. A disadvantage of PCC-OFDM (without overlapping) is the potential reduction in spectral efficiency because subcarriers are modulated in pairs, but this reduction is more than regained because no guard band or cyclic prefix is required and because, for a given channel, larger constellations can be used

Pathways and mechanisms of aeration in Phragmites australis

This thesis describes an investigation into the aeration pathways, resistances to gas-movement, mechanisms of internal aeration and the locations and quantities of oxygen efflux from the underground parts of the common reed Phragmites australis. The "ventilating pressure concept" was also tested and reappraised.Well-developed interconnecting gas-spaces within the culm, rhizome and roots offered relatively small resistance to either diffusive or convective gas transport, and the porosity of root-rhizome junctions was unusually high. Radial channels located at the nodes proved to be the only connecting points between cortex and pith in culms and rhizomes. Rhizome and old adventitious root surfaces were impermeable to oxygen but the young parts of adventitious roots and the numerous laterals readily released oxygen to anaerobic agar media and soil, markedly raising the redox potential of the rhizosphere.It was discovered that callus readily forms in Phragmites in response to wounding and senescence, blocking the gas-spaces of root-rhizome junctions, leaf-sheath-culm junctions, and rhizome nodal diaphragms. However, the culm-rhizome junctions normally remain callus-free despite senescence of the culms, and gaseous connexion between the underground parts and the atmosphere is thus maintained thoughout the year.A major discovery was that pressurised gas-flows are an important feature in Phragmites' aeration: Venturi- and/or Humidity-induced convections produced much higher rhizome oxygen concentrations and radial oxygen loss from the roots than when rhizome aeration was chiefly diffusive. Both experiments and mathematical modelling demonstrated that comparatively slow rates of convection are sufficient to achieve this. The humidity-induced convection, the first reported in a grass, was shown to be initiated chiefly in living leaf sheaths, the convected gases being transmitted via gas-spaces in the culm to those of the underground rhizome, and vented via old broken culms. The flows are particularly rapid at low atmospheric humidities and increase with increasing PAR. The major mechanism promoting the convection appears to be a humidity- induced diffusion or transitional Knudsen diffusion of atmospheric gases into the plant, the concentration gradient being maintained by the difference in humidity between the interior of the plant and the outer air. The process was mimicked and further investigated using Nuclepore membranes providing important insights into the mechanism and its modelling, and it was shown that pore diameters within the Knudsen regime are not essential to produce the static pressure differentials and convective flows found in Phragmites.The Venturi-induced convection, the first reported example in a plant, is created by the action of winds blowing across tall, dead, broken culms; air is drawn into the rhizome system via short broken culms in more sheltered positions.Mathematical models and polarographic measurements of radial oxygen loss from roots were used to estimate the quantities of oxygen which might be released by Phragmites in the root-zone process of sewage treatment. It was concluded that 5 to 12 g 02 m⁻² day⁻¹ would be a conservative estimate but that the amount could be greater or less depending upon root numbers and their physiological condition, as well as upon soil oxygen demand and diffusivities

Quantitative analysis of boundary layers in periodic homogenization

We prove quantitative estimates on the rate of convergence for the oscillating Dirichlet problem in periodic homogenization of divergence-form uniformly elliptic systems. The estimates are optimal in dimensions larger than three and new in every dimension. We also prove a regularity estimate on the homogenized boundary condition.Comment: 41 pages; updated to comment on results of arXiv:1610.0527

From Mill Town to Mill Town: The Transition of a New England Town from a Textile to a High-Technology Economy

This article describes and analyzes the economic transformation of Maynard, Massachusetts. Located twenty-six miles west of Boston, Maynard was the home of the world\u27s largest woolen mill. In 1950 the mill closed and 1,200 jobs were lost. During the next ten years, the town gradually recovered as the mill became the home of several innovative companies. Ultimately, the mill became the headquarters for the world\u27s largest manufacturer of minicomputers. The circle is now complete: the mill is full and houses a company that claims a world\u27s largest designation. The article analyses the rise and fall of the American Woolen Company; examines the steps taken to stimulate recovery, and their results; identifies and examines the important elements that led to the economic transformation; and offers potential lessons for planners in towns undergoing similar changes

Programmable Multimode Quantum Networks

Entanglement between large numbers of quantum modes is the quintessential resource for future technologies such as the quantum internet. Conventionally the generation of multimode entanglement in optics requires complex layouts of beam-splitters and phase shifters in order to transform the input modes in to entangled modes. These networks need substantial modification for every new set of entangled modes to be generated. Here we report on the highly versatile and efficient generation of various multimode entangled states with the ability to switch between different linear optics networks in real time. By defining our modes to be combinations of different spatial regions of one beam, we may use just one pair of multi-pixel detectors each with M photodiodes in order to measure N entangled modes, with a maximum number of N=M modes. We program virtual networks that are fully equivalent to the physical linear optics networks they are emulating. We present results for N=2 up to N=8 entangled modes here, including N=2,3,4 cluster states. Our approach introduces flexibility and scalability to multimode entanglement, two important attributes that are highly sought after in state of the art devices.Comment: 10 pages, 5 figures, 2 tables, comments welcome