Showing 11 - 20 of 94 Items
Phononic rogue waves
Date: 2018-09-13
Creator: E. G. Charalampidis, J. Lee, P. G. Kevrekidis, C. Chong
Access: Open access
- We present a theoretical study of extreme events occurring in phononic lattices. In particular, we focus on the formation of rogue or freak waves, which are characterized by their localization in both spatial and temporal domains. We consider two examples. The first one is the prototypical nonlinear mass-spring system in the form of a homogeneous Fermi-Pasta-Ulam-Tsingou (FPUT) lattice with a polynomial potential. By deriving an approximation based on the nonlinear Schrödinger (NLS) equation, we are able to initialize the FPUT model using a suitably transformed Peregrine soliton solution of the NLS equation, obtaining dynamics that resembles a rogue wave on the FPUT lattice. We also show that Gaussian initial data can lead to dynamics featuring a rogue wave for sufficiently wide Gaussians. The second example is a diatomic granular crystal exhibiting rogue-wave-like dynamics, which we also obtain through an NLS reduction and numerical simulations. The granular crystal (a chain of particles that interact elastically) is a widely studied system that lends itself to experimental studies. This study serves to illustrate the potential of such dynamical lattices towards the experimental observation of acoustic rogue waves.
Animal-to-animal variability in the phasing of the crustacean cardiac motor pattern: An experimental and computational analysis
Date: 2013-01-01
Creator: Alex H. Williams, Molly A. Kwiatkowski, Adam L. Mortimer, Eve Marder, Mary Lou, Zeeman, Patsy S. Dickinson
Access: Open access
- The cardiac ganglion (CG) of Homarus americanus is a central pattern generator that consists of two oscillatory groups of neurons: "small cells" (SCs) and "large cells" (LCs). We have shown that SCs and LCs begin their bursts nearly simultaneously but end their bursts at variable phases. This variability contrasts with many other central pattern generator systems in which phase is well maintained. To determine both the consequences of this variability and how CG phasing is controlled, we modeled the CG as a pair of Morris-Lecar oscillators coupled by electrical and excitatory synapses and constructed a database of 15,000 simulated networks using random parameter sets. These simulations, like our experimental results, displayed variable phase relationships, with the bursts beginning together but ending at variable phases. The model suggests that the variable phasing of the pattern has important implications for the functional role of the excitatory synapses. In networks in which the two oscillators had similar duty cycles, the excitatory coupling functioned to increase cycle frequency. In networks with disparate duty cycles, it functioned to decrease network frequency. Overall, we suggest that the phasing of the CG may vary without compromising appropriate motor output and that this variability may critically determine how the network behaves in response to manipulations. © 2013 the American Physiological Society.
Transient phenomena in ecology
Date: 2018-09-07
Creator: Alan Hastings, Karen C. Abbott, Kim Cuddington, Tessa Francis, Gabriel, Gellner, Ying Cheng Lai
Access: Open access
- The importance of transient dynamics in ecological systems and in the models that describe them has become increasingly recognized. However, previous work has typically treated each instance of these dynamics separately. We review both empirical examples and model systems, and outline a classification of transient dynamics based on ideas and concepts from dynamical systems theory. This classification provides ways to understand the likelihood of transients for particular systems, and to guide investigations to determine the timing of sudden switches in dynamics and other characteristics of transients. Implications for both management and underlying ecological theories emerge.
Random subgroups of Thompson's group F
Date: 2010-01-13
Creator: Sean Cleary, Murray Elder, Andrew Rechnitzer, Jennifer Taback
Access: Open access
- We consider random subgroups of Thompson's group F with respect to two natural stratifications of the set of all k-generator subgroups. We find that the isomorphism classes of subgroups which occur with positive density are not the same for the two stratifications. We give the first known examples of persistent subgroups, whose isomorphism classes occur with positive density within the set of k-generator subgroups, for all sufficiently large k. Additionally, Thompson's group provides the first example of a group without a generic isomorphism class of subgroup. Elements of F are represented uniquely by reduced pairs of finite rooted binary trees. We compute the asymptotic growth rate and a generating function for the number of reduced pairs of trees, which we show is D-finite (short for differentiably finite) and not algebraic. We then use the asymptotic growth to prove our density results. © European Mathematical Society.
Determining hilbert modular forms by central values of rankin-selberg convolutions: The level aspect
Date: 2017-12-01
Creator: Alia Hamieh, Naomi Tanabe
Access: Open access
- In this paper, we prove that a primitive Hilbert cusp form g is uniquely determined by the central values of the Rankin-Selberg L-functions (formula presented), where f runs through all primitive Hilbert cusp forms of level q for infinitely many prime ideals q. This result is a generalization of the work of Luo (1999) to the setting of totally real number fields.
Wave mixing in coupled phononic crystals via a variable stiffness mechanism
Date: 2016-10-01
Creator: Gil Yong Lee, Christopher Chong, Panayotis G. Kevrekidis, Jinkyu Yang
Access: Open access
- We investigate wave mixing effects in a phononic crystal that couples the wave dynamics of two channels – primary and control ones – via a variable stiffness mechanism. We demonstrate analytically and numerically that the wave transmission in the primary channel can be manipulated by the control channel's signal. We show that the application of control waves allows the selection of a specific mode through the primary channel. We also demonstrate that the mixing of two wave modes is possible whereby a modulation effect is observed. A detailed study of the design parameters is also carried out to optimize the switching capabilities of the proposed system. Finally, we verify that the system can fulfill both switching and amplification functionalities, potentially enabling the realization of an acoustic transistor.
Cells and constructible representations in type B
Date: 2008-10-13
Creator: Thomas Pietraho
Access: Open access
- We examine the partition of a finite Coxeter group of type B into cells determined by a weight function L. The main objective of these notes is to reconcile Lusztig's description of constructible representations in this setting with conjectured combinatorial descriptions of cells.
Mathematically Modeling a Nonlinear, Passive Acoustic Filter This record is embargoed.
- Embargo End Date: 2026-05-18
Date: 2023-01-01
Creator: Bjorn Ludwig
Access: Embargoed
On L-functions and the 1-Level Density
Date: 2023-01-01
Creator: Arav Agarwal
Access: Open access
- We begin with the classical study of the Riemann zeta function and Dirichlet L-functions. This includes a full exposition on one of the most useful ways of exploiting their connection with primes, namely, explicit formulae. We then proceed to introduce statistics of low-lying zeros of Dirichlet L-functions, discussing prior results of Fiorilli and Miller (2015) on the 1-level density of Dirichlet L-functions and their achievement in surpassing the prediction of the powerful Ratios Conjecture. Finally, we present our original work partially generalizing these results to the case of Hecke L-functions over imaginary quadratic fields.
The structure of singularities in Π-minimizing networks in R2
Date: 1991-01-01
Creator: M. Alfaro, M. Conger, K. Hodges, A. Levy, R., Kochar, L. Kuklinski
Access: Open access