Large Deviations for Brownian Intersection Measures

We consider $p$ independent Brownian motions in $\R^d$. We assume that $p\geq 2$ and $p(d-2)<d$. Let $\ell_t$ denote the intersection measure of the $p$ paths by time $t$, i.e., the random measure on $\R^d$ that assigns to any measurable set $A\subset \R^d$ the amount of intersection local time of the motions spent in $A$ by time $t$. Earlier results of Chen \cite{Ch09} derived the logarithmic asymptotics of the upper tails of the total mass $\ell_t(\R^d)$ as $t\to\infty$. In this paper, we derive a large-deviation principle for the normalised intersection measure $t^{-p}\ell_t$ on the set of positive measures on some open bounded set $B\subset\R^d$ as $t\to\infty$ before exiting $B$. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalised occupation times measures of the $p$ motions. Our proof makes the classical Donsker-Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from $B$, conditional on a large total mass in some compact set $U\subset B$. This extends earlier studies on the intersection measure by K\"onig and M\"orters \cite{KM01,KM05}.Comment: To appear in "Communications on Pure and Applied Mathematics

Episodic Learning with Control Lyapunov Functions for Uncertain Robotic Systems

Many modern nonlinear control methods aim to endow systems with guaranteed properties, such as stability or safety, and have been successfully applied to the domain of robotics. However, model uncertainty remains a persistent challenge, weakening theoretical guarantees and causing implementation failures on physical systems. This paper develops a machine learning framework centered around Control Lyapunov Functions (CLFs) to adapt to parametric uncertainty and unmodeled dynamics in general robotic systems. Our proposed method proceeds by iteratively updating estimates of Lyapunov function derivatives and improving controllers, ultimately yielding a stabilizing quadratic program model-based controller. We validate our approach on a planar Segway simulation, demonstrating substantial performance improvements by iteratively refining on a base model-free controller

The Effects of Menthol Concentrations on Germination Rates of Arabidopsis thaliana

Faculty Research Day 2018: Undergraduate Student Poster 3rd PlaceMenthol is a toxin found in plants. Arabidopsis thaliana was tested with several different concentrations of menthol to observe whether or not germination and fungal growth would be inhibited due to the menthol

Response of clay soil to three-dimensional tunnelling simulation in centrifuge models

Tunnelling-induced ground movements are complicated and investigations into them normally require some simplifications. This paper provides a brief literature review which highlights the advantages of adopting simplifications in physical modelling and addresses some of the deficiencies in the assessment of soil deformation due to a simulated tunnel excavation. A set of centrifuge tests modelling a tunnel heading located at different depths in clay was carried out at 125g. The tunnel was modelled by a semi-circular cavity partly supported by a stiff lining. The unlined tunnel heading was supported by a thin rubber bag supplied with compressed air pressure. Tunnel excavation was simulated by reducing air pressure. The induced ground movements at the subsurface and surface were measured by a 2D image analysis and a new, novel 3D imaging system. The results show that the experiment successfully reproduced key aspects of tunnelling-induced soil deformation in practice. In addition, a new equation to predict horizontal displacements in the longitudinal direction is proposed

Potential automorphy over CM fields

Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over $F$ without any self-duality condition. We deduce that all elliptic curves $E$ over $F$ are potentially modular, and furthermore satisfy the Sato--Tate conjecture. As an application of a different sort, we also prove the Ramanujan Conjecture for weight zero cuspidal automorphic representations for $\mathrm{GL}_2(\mathbf{A}_F)$.Comment: A number of details have been included to address the concerns of the referees. The definition of decomposed generic (Def 4.3.1) has been weakened slightly to be in line with the current version of arxiv.org/abs/1909.01898, resulting in a strengthening of a number of our theorems. This is the accepted version of the pape

System to measure three-dimensional movements in physical models

A newly developed imaging system is presented, which measures three-dimensional (3D) deformations of a soil surface in geotechnical experiments involving physical modelling. The method adopts the computer vision technique ‘structure from motion and multi-view stereo’ delivered by an open-source software MicMac. Three, 2 megapixel industrial cameras were synchronised and used to capture images of a deforming soil surface. The images were used to reconstruct the observed scene to a high-density, accurate 3D point cloud. A new method has been developed to process the obtained 3D point clouds and images to determine the 3D displacement vectors. The procedure is highly automatic which allows large data sets to be processed with minimal manual intervention. Two series of quantification experiments were carried out to assess the performance of the system which has shown the overall accuracy to be within 0·05 mm over a field of view of 500 × 250 mm. An example application is presented to demonstrate the capabilities of the 3D imaging system

A study on performance of three-dimensional imaging system for physical models

A study by Le et al. (2017) reported the application of computer vision techniques structure from motion (SfM) and multi-view stereo (MVS) to measure three-dimensional soil displacements at the surface of physical models. However, little information exists on the significance of the camera resolution and the number of images to the measurement performance. This study assesses the measurement performance of the SfMMVS, provided by an open source software Micmac, with input images taken by two different types of camera including DSLR (18Mega-pixel) and mobile phone cameras (12Mega-pixel). Rigorous quantifications were carried out to examine the precision of the image analysis, in measuring vertical and horizontal displacements, over a region of interest of 420x200mm. The measurement precision, achieved by different numbers of images, ranged from 0.06mm to 0.03mm. The results from this paper can be useful for researchers to select appropriate camera that satisfies their measurement requirements

A study on the reinforcing capabilities of Forepoling Umbrella System in urban tunnelling

Adequate heading stability is crucial to the safe construction of any tunnel. Insufficient support will lead to ground movements which have the potential to cause damage to existing infrastructure. Congested urban environments have led to a requirement to minimise these tunnelling-induced deformations. Forepoling Umbrella Systems (FUS) have proved to be a beneficial soil reinforcement measure for controlling ground movements due to NATM tunnelling in urban areas. However, there is limited understanding of the influence of tunnel geometry and FUS parameters on its reinforcing efficiencies. A series of centrifuge tests has been conducted to investigate the benefits of FUS using different arrangements of steel pipes placed in a model tunnel heading at various depths. The results show the importance of the steel pipes near the tunnel spring-line and the embedded lengths on the reinforcing effects of forepoles. In addition, relative benefits of forepole location and embedded length are shown to vary as the soil cover above the tunnel changes

Time-dependent models of dense PDRs with complex molecules

We present a study of the chemistry of a dense photon-dominated region (PDR) using a time-dependent chemical model. Our major interest is to study the spatial distribution of complex molecules such as hydrocarbons and cyanopolyynes in the cool dense material bordering regions where star formation has taken place. Our standard model uses a homogeneous cloud of density 2x10e4 cm-3 and temperature T=40 K, which is irradiated by a far-ultraviolet radiation field of intermediate intensity, given by X=100. We find that over a range of times unsaturated hydrocarbons (e.g., C2H, C4H, C3H2) have relatively high fractional abundances in the more external layers of the PDR, whereas their abundances in the innermost layers are several orders of magnitudes lower. On the other hand, molecules that are typical of late-time chemistry are usually more abundant in the inner parts of the PDR. We also present results for models with different density, temperature, intensity of the radiation field and initial fractional abundances. Our results are compared with both high- and moderate-angular resolution observations of the Horsehead nebula. Our standard model is partially successful in reproducing the observations. Additional models run with different physical parameters are able to reproduce the abundance of many of the observed molecules, but we do not find a single model that fits all the observations at the same time. We discuss the suitability of a time-dependent model of a dense PDR such as ours as an estimator of the age of a PDR, provided that enough observational data exist.Comment: 15 pages, 12 figures, 8 tables, to be published in MNRA