Showing 71 - 80 of 96 Items
Combinatorial properties of Thompson's group F
Date: 2004-07-01
Creator: Sean Cleary, Jennifer Taback
Access: Open access
- We study some combinatorial consequences of Blake Fordham's theorems on the word metric of Thompson's group F in the standard two generator presentation. We explore connections between the tree pair diagram representing an element w of F, its normal form in the infinite presentation, its word length, and minimal length representatives of it. We estimate word length in terms of the number and type of carets in the tree pair diagram and show sharpness of those estimates. In addition we explore some properties of the Cayley graph of F with respect to the two generator finite presentation. Namely, we exhibit the form of "dead end" elements in this Cayley graph, and show that it has no "deep pockets". Finally, we discuss a simple method for constructing minimal length representatives for strictly positive or negative words.
Surface symmetries and PSL2(p)
Date: 2007-05-01
Creator: Murad Özaydin, Charlotte Simmons, Jennifer Taback
Access: Open access
- We classify, up to conjugacy, all orientation-preserving actions of PSL2(p) on closed connected orientable surfaces with spherical quotients. This classification is valid in the topological, PL, smooth, conformal, geometric and algebraic categories and is related to the Inverse Galois Problem. © 2006 American Mathematical Society.
C-graph automatic groups
Date: 2014-09-01
Creator: Murray Elder, Jennifer Taback
Access: Open access
- We generalize the notion of a graph automatic group introduced by Kharlampovich, Khoussainov and Miasnikov by replacing the regular languages in their definition with more powerful language classes. For a fixed language class C, we call the resulting groups C-graph automatic. We prove that the class of C-graph automatic groups is closed under change of generating set, direct and free product for certain classes C. We show that for quasi-realtime counter-graph automatic groups where normal forms have length that is linear in the geodesic length, there is an algorithm to compute normal forms (and therefore solve the word problem) in polynomial time. The class of quasi-realtime counter-graph automatic groups includes all Baumslag-Solitar groups, and the free group of countably infinite rank. Context-sensitive-graph automatic groups are shown to be a very large class, which encompasses, for example, groups with unsolvable conjugacy problem, the Grigorchuk group, and Thompson's groups F, T and V. © 2014 Elsevier Inc.
Thompson's group F is 1-counter graph automatic
Date: 2016-05-01
Creator: Murray Elder, Jennifer Taback
Access: Open access
- It is not known whether Thompson's group F is automatic. With the recent extensions of the notion of an automatic group to graph automatic by Kharlampovich, Khoussainov and Miasnikov and then to C-graph automatic by the authors, a compelling question is whether F is graph automatic or C-graph automatic for an appropriate language class C. The extended definitions allow the use of a symbol alphabet for the normal form language, replacing the dependence on generating set. In this paper we construct a 1-counter graph automatic structure for F based on the standard infinite normal form for group elements.
Separation of internal and interaction dynamics for NLS-described wave packets with different carrier waves
Date: 2008-11-01
Creator: Martina Chirilus-Bruckner, Christopher Chong, Guido Schneider, Hannes Uecker
Access: Open access
- We give a detailed analysis of the interaction of two NLS-described wave packets with different carrier waves for a nonlinear wave equation. By separating the internal dynamics of each wave packet from the dynamics caused by the interaction we prove that there is almost no interaction of such wave packets. We also prove the validity of a formula for the envelope shift caused by the interaction of the wave packets. © 2008 Elsevier Inc. All rights reserved.
Origami-based impact mitigation via rarefaction solitary wave creation
Date: 2019-01-01
Creator: Hiromi Yasuda, Yasuhiro Miyazawa, Efstathios G. Charalampidis, Christopher Chong, Panayotis G., Kevrekidis, Jinkyu Yang
Access: Open access
- The principles underlying the art of origami paper folding can be applied to design sophisticated metamaterials with unique mechanical properties. By exploiting the flat crease patterns that determine the dynamic folding and unfolding motion of origami, we are able to design an origami-based metamaterial that can form rarefaction solitary waves. Our analytical, numerical, and experimental results demonstrate that this rarefaction solitary wave overtakes initial compressive strain waves, thereby causing the latter part of the origami structure to feel tension first instead of compression under impact. This counterintuitive dynamic mechanism can be used to create a highly efficient-yet reusable-impact mitigating system without relying on material damping, plasticity, or fracture.
Modeling Coupled Disease-Behavior Dynamics of SARS-CoV-2 Using Influence Networks
Date: 2021-01-01
Creator: Juliana C. Taube
Access: Open access
- SARS-CoV-2, the virus that causes COVID-19, has caused significant human morbidity and mortality since its emergence in late 2019. Not only have over three million people died, but humans have been forced to change their behavior in a variety of ways, including limiting their contacts, social distancing, and wearing masks. Early infectious disease models, like the classical SIR model by Kermack and McKendrick, do not account for differing contact structures and behavior. More recent work has demonstrated that contact structures and behavior can considerably impact disease dynamics. We construct a coupled disease-behavior dynamical model for SARS-CoV-2 by incorporating heterogeneous contact structures and decisions about masking. We use a contact network with household, work, and friend interactions to capture the variation in contact patterns. We allow decisions about masking to occur at a different time scale from disease spread which dramatically changes the masking dynamics. Drawing from the field of game theory, we construct an individual decision-making process that relies on perceived risk of infection, social influence, and individual resistance to masking. Through simulation, we find that social influence prevents masking, while perceived risk largely drives individuals to mask. Underlying contact structure also affects the number of people who mask. This model serves as a starting point for future work which could explore the relative importance of social influence and perceived risk in human decision-making.
Mathematical Modeling of the American Lobster Cardiac Muscle Cell: An Investigation of Calcium Ion Permeability and Force of Contractions
Date: 2014-05-01
Creator: Lauren A Skerritt
Access: Open access
- In the American lobster (Homarus americanus), neurogenic stimulation of the heart drives fluxes of calcium (Ca2+) into the cytoplasm of a muscle cell resulting in heart muscle contraction. The heartbeat is completed by the active transport of calcium out of the cytoplasm into extracellular and intracellular spaces. An increase in the frequency of calcium release is expected to increase amplitude and duration of muscle contraction. This makes sense because an increase in cytoplasmic calcium should increase the activation of the muscle contractile elements (actin and myosin). Since calcium cycling is a reaction-diffusion process, the extent to which calcium mediates contraction amplitude and frequency will depend on the specific diffusion relationships of calcium in this system. Despite the importance of understanding this relationship, it is difficult to obtain experimental information on the dynamics of cytoplasmic calcium. Thus, we developed a mathematical diffusion model of the myofibril (muscle cell) to simulate calcium cycling in the lobster cardiac muscle cell. The amplitude and duration of the force curves produced by the model empirically mirrored that of the experimental data over a range of calcium diffusion coefficients (1-16), nerve stimulation durations (1/6-1/3 of a contraction period), and frequencies (40-80 Hz). The characteristics that alter the response of the lobster cardiac muscle system are stimulation duration (i.e., burst duration), burst frequency, and the rate of calcium diffusion into the cell’s cytoplasm. For this reason, we developed protocols that allow parameters representing these characteristics in the calcium-force model to be determined from isolated whole muscle experiments on lobster hearts (Phillips et al., 2004). These parameters are used to predict variability in lobster heart muscle function consistent with data recorded in experiments. Within the physiological range of nerve stimulation parameters (burst duration and cycle period), calcium increased the cell’s force output for increased burst duration. For example, increased duration of stimulation increased the muscle contraction period and vice versa. In terms of diffusion, a slower rate of calcium diffusion out of the sarcoplasmic reticulum decreased both the calcium level and the contraction duration of the cell. Finally, changes in stimulation frequency did not produce changes in contraction amplitude and duration. When considered in conjunction with experimental stimulations using lobster heart muscle cells, these data illustrate the prominent role for calcium diffusion in governing contraction-relaxation cycles in lobster hearts.
Random Walks on Finite Groups: an
Application of Group Representations
Date: 2025-01-01
Creator: David Guan
Access: Open access
- Playing cards, a set of fifty-two cards with four suits and thirteen numbers, appear every- where in ourdaily lives. In particular, theyare commonly used ongambling tables in casinos. Before every game, the dealer needs to shuffle those cards to put them into a random order so that the game is fair. (Is it really the case in casinos?) One may wonder whether a shuffling technique is really efficient or not, i.e. whether it can turn the deck into a random configu- ration in a small number of rounds. Mathematically, this can be interpreted as a problem of random walks on the symmetric group of 52 elements S_{52} and we aim to determine how fast the random walk becomes (uniformly) random. This paper aims to explain an important application of group representations to this type of problems; in particular, techniques from group representations can provide a (roughly) accurate approximation. There will be six chapters in this paper. Chapter 1 is an introduction to problems of random walks. Chapter 2 and 3 discuss group representations in general, and a key lemma for application is discussed at the end of Chapter 3. Chapter 4 is an application to the easiest random walk one can encounter. Chapter 5 focuses on discussing properties distinctive to symmetric groups, and Chapter 6 discusses a card-shuffling example in details. Contents in Chapter 5 are mostly from the second chapter of [Sag01]; the key result in Chapter 6 along with the rest of the paper are mostly from [Dia88] with a few exceptions. This paper is written in a self-explanatory way, so anyone with necessary background of linear algebra, group theory, and probability will be able to follow the entirety of it.
Lipschitzian multifunctions and a Lipschitzian inverse mapping theorem
Date: 2001-01-01
Creator: A.B. Levy
Access: Open access