Showing 71 - 80 of 94 Items
Damped-driven granular chains: An ideal playground for dark breathers and multibreathers
Date: 2014-03-31
Creator: C. Chong, F. Li, J. Yang, M. O. Williams, I. G., Kevrekidis, P. G. Kevrekidis
Access: Open access
- By applying an out-of-phase actuation at the boundaries of a uniform chain of granular particles, we demonstrate experimentally that time-periodic and spatially localized structures with a nonzero background (so-called dark breathers) emerge for a wide range of parameter values and initial conditions. We demonstrate a remarkable control over the number of breathers within the multibreather pattern that can be "dialed in" by varying the frequency or amplitude of the actuation. The values of the frequency (or amplitude) where the transition between different multibreather states occurs are predicted accurately by the proposed theoretical model, which is numerically shown to support exact dark breather and multibreather solutions. Moreover, we visualize detailed temporal and spatial profiles of breathers and, especially, of multibreathers using a full-field probing technology and enable a systematic favorable comparison among theory, computation, and experiments. A detailed bifurcation analysis reveals that the dark and multibreather families are connected in a "snaking" pattern, providing a roadmap for the identification of such fundamental states and their bistability in the laboratory. © 2014 American Physical Society.
Mechanical Autonomous Stochastic Heat Engine
Date: 2016-06-28
Creator: Marc Serra-Garcia, André Foehr, Miguel Molerón, Joseph Lydon, Christopher, Chong, Chiara Daraio
Access: Open access
- Stochastic heat engines are devices that generate work from random thermal motion using a small number of highly fluctuating degrees of freedom. Proposals for such devices have existed for more than a century and include the Maxwell demon and the Feynman ratchet. Only recently have they been demonstrated experimentally, using, e.g., thermal cycles implemented in optical traps. However, recent experimental demonstrations of classical stochastic heat engines are nonautonomous, since they require an external control system that prescribes a heating and cooling cycle and consume more energy than they produce. We present a heat engine consisting of three coupled mechanical resonators (two ribbons and a cantilever) subject to a stochastic drive. The engine uses geometric nonlinearities in the resonating ribbons to autonomously convert a random excitation into a low-entropy, nonpassive oscillation of the cantilever. The engine presents the anomalous heat transport property of negative thermal conductivity, consisting in the ability to passively transfer energy from a cold reservoir to a hot reservoir.
Module structure of cells in unequal-parameter Hecke algebras
Date: 2010-09-06
Creator: Thomas Pietraho
Access: Open access
- A conjecture of Bonnafé, Geck, Iancu, and Lam parametrizes Kazhdan-Lusztig left cells for unequal-parameter Hecke algebras in type Bn by families of standard domino tableaux of arbitrary rank. Relying on a family of properties outlined by Lusztig and the recent work of Bonnafé, we verify the conjecture and describe the structure of each cell as a module for the underlying Weyl group. © 2010 by The Editorial Board of the Nagoya Mathematical Journal.
Automorphisms of higher rank lamplighter groups
Date: 2015-12-01
Creator: Melanie Stein, Jennifer Taback, Peter Wong
Access: Open access
- Let τd(q) denote the group whose Cayley graph with respect to a particular generating set is the Diestel-Leader graph DLd(q), as described by Bartholdi, Neuhauser and Woess. We compute both Aut(τd(q)) and Out(τd(q)) for d ≥ 2, and apply our results to count twisted conjugacy classes in these groups when d ≥ 3. Specifically, we show that when d ≥ 3, the groups τd(q) have property R∞, that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when d = 2 the lamplighter groups τ2(q) = Lq = Zq Z have property R∞ if and only if (q, 6)≠1.
Free limits of Thompson's group F
Date: 2011-12-01
Creator: Azer Akhmedov, Melanie Stein, Jennifer Taback
Access: Open access
- We produce a sequence of markings Sk of Thompson's group F within the space Gn of all marked n-generator groups so that the sequence (F, Sk) converges to the free group on n generators, for n ≥ 3. In addition, we give presentations for the limits of some other natural (convergent) sequences of markings to consider on F within G3, including (F, {x0, x1, xn}) and (F, {x0, x1, x0n}) © 2011 Springer Science+Business Media B.V.
Bounding right-arm rotation distances
Date: 2007-03-01
Creator: Sean Cleary, Jennifer Taback
Access: Open access
- Rotation distance measures the difference in shape between binary trees of the same size by counting the minimum number of rotations needed to transform one tree to the other. We describe several types of rotation distance where restrictions are put on the locations where rotations are permitted, and provide upper bounds on distances between trees with a fixed number of nodes with respect to several families of these restrictions. These bounds are sharp in a certain asymptotic sense and are obtained by relating each restricted rotation distance to the word length of elements of Thompson's group F with respect to different generating sets, including both finite and infinite generating sets. © World Scientific Publishing Company.
Dead end words in lamplighter groups and other wreath products
Date: 2005-09-22
Creator: Sean Cleary, Jennifer Taback
Access: Open access
- We explore the geometry of the Cayley graphs of the lamplighter groups and a wide range of wreath products. We show that these groups have dead end elements of arbitrary depth with respect to their natural generating sets. An element w in a group G with finite generating set X is a dead end element if no geodesic ray from the identity to w in the Cayley graph Γ(G, X) can be extended past w. Additionally, we describe some non-convex behaviour of paths between elements in these Cayley graphs and seesaw words, which are potential obstructions to these graphs satisfying the k-fellow traveller property. © The Author 2005. Published by Oxford University Press. All rights reserved.
Nonlinear coherent structures in granular crystals
Date: 2017-09-06
Creator: C. Chong, Mason A. Porter, P. G. Kevrekidis, C. Daraio
Access: Open access
- The study of granular crystals, which are nonlinear metamaterials that consist of closely packed arrays of particles that interact elastically, is a vibrant area of research that combines ideas from disciplines such as materials science, nonlinear dynamics, and condensed-matter physics. Granular crystals exploit geometrical nonlinearities in their constitutive microstructure to produce properties (such as tunability and energy localization) that are not conventional to engineering materials and linear devices. In this topical review, we focus on recent experimental, computational, and theoretical results on nonlinear coherent structures in granular crystals. Such structures - which include traveling solitary waves, dispersive shock waves, and discrete breathers - have fascinating dynamics, including a diversity of both transient features and robust, long-lived patterns that emerge from broad classes of initial data. In our review, we primarily discuss phenomena in one-dimensional crystals, as most research to date has focused on such scenarios, but we also present some extensions to two-dimensional settings. Throughout the review, we highlight open problems and discuss a variety of potential engineering applications that arise from the rich dynamic response of granular crystals.
Sensitivity Analysis of Basins of Attraction for Gradient-Based Optimization Methods
Date: 2022-01-01
Creator: Gillian King
Access: Open access
- This project is an analysis of the effectiveness of five distinct optimization methods in their ability in producing clear images of the basins of attraction, which is the set of initial points that approach the same minimum for a given function. Basin images are similar to contour plots, except that they depict the distinct regions of points--in unique colors--that approach the same minimum. Though distinct in goal, contour plots are useful to basin research in that idealized basin images can be inferred from the steepness levels and location of extrema they depict. Effectiveness of the method changes slightly depending on the function, but is generally defined as how closely the basin image models contour information on where the true minima are located, and by the clarity of the resulting image in depicting well-defined regions. The methods are tested on four distinct functions which were chosen to assess how each method performs in the presence of various challenges. This project ranks the five methods for their overall effectiveness and consistency across the four functions, and also analyzes the sensitivity of the methods when small changes are made to the function. In general, less sensitive and consistently effective methods are more applicable and reliable in applied optimization research.
The Structure and Unitary Representations of SU(2,1)
Date: 2015-05-01
Creator: Andrew J Pryhuber
Access: Open access