Showing 41 - 50 of 94 Items
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.
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.
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.
Bounding the number of cycles of O.D.E.S in Rn
Date: 2001-01-01
Creator: M. Farkas, P. Van Den Driessche, M. L. Zeeman
Access: Open access
- Criteria are given under which the boundary of an oriented surface does not consist entirely of trajectories of the C1 differential equation ẋ = f(x) in Rn. The special case of an annulus is further considered, and the criteria are used to deduce sufficient conditions for the differential equation to have at most one cycle. A bound on the number of cycles on surfaces of higher connectivity is given by similar conditions. ©2000 American Mathematical Society.
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.
Time-Periodic Solutions of Driven-Damped Trimer Granular Crystals
Date: 2015-01-01
Creator: E. G. Charalampidis, F. Li, C. Chong, J. Yang, P. G., Kevrekidis
Access: Open access
- We consider time-periodic structures of granular crystals consisting of alternate chrome steel (S) and tungsten carbide (W) spherical particles where each unit cell follows the pattern of a 2: 1 trimer: S-W-S. The configuration at the left boundary is driven by a harmonic in-time actuation with given amplitude and frequency while the right one is a fixed wall. Similar to the case of a dimer chain, the combination of dissipation, driving of the boundary, and intrinsic nonlinearity leads to complex dynamics. For fixed driving frequencies in each of the spectral gaps, we find that the nonlinear surface modes and the states dictated by the linear drive collide in a saddle-node bifurcation as the driving amplitude is increased, beyond which the dynamics of the system becomes chaotic. While the bifurcation structure is similar for solutions within the first and second gap, those in the first gap appear to be less robust. We also conduct a continuation in driving frequency, where it is apparent that the nonlinearity of the system results in a complex bifurcation diagram, involving an intricate set of loops of branches, especially within the spectral gap. The theoretical findings are qualitatively corroborated by the experimental full-field visualization of the time-periodic structures.
Conical wave propagation and diffraction in two-dimensional hexagonally packed granular lattices
Date: 2016-01-25
Creator: C. Chong, P. G. Kevrekidis, M. J. Ablowitz, Yi Ping Ma
Access: Open access
- Linear and nonlinear mechanisms for conical wave propagation in two-dimensional lattices are explored in the realm of phononic crystals. As a prototypical example, a statically compressed granular lattice of spherical particles arranged in a hexagonal packing configuration is analyzed. Upon identifying the dispersion relation of the underlying linear problem, the resulting diffraction properties are considered. Analysis both via a heuristic argument for the linear propagation of a wave packet and via asymptotic analysis leading to the derivation of a Dirac system suggests the occurrence of conical diffraction. This analysis is valid for strong precompression, i.e., near the linear regime. For weak precompression, conical wave propagation is still possible, but the resulting expanding circular wave front is of a nonoscillatory nature, resulting from the complex interplay among the discreteness, nonlinearity, and geometry of the packing. The transition between these two types of propagation is explored.
Multistable solitons in higher-dimensional cubic-quintic nonlinear Schrödinger lattices
Date: 2009-01-15
Creator: C. Chong, R. Carretero-González, B. A. Malomed, P. G. Kevrekidis
Access: Open access
- We study the existence, stability, and mobility of fundamental discrete solitons in two- and three-dimensional nonlinear Schrödinger lattices with a combination of cubic self-focusing and quintic self-defocusing onsite nonlinearities. Several species of stationary solutions are constructed, and bifurcations linking their families are investigated using parameter continuation starting from the anti-continuum limit, and also with the help of a variational approximation. In particular, a species of hybrid solitons, intermediate between the site- and bond-centered types of the localized states (with no counterpart in the 1D model), is analyzed in 2D and 3D lattices. We also discuss the mobility of multi-dimensional discrete solitons that can be set in motion by lending them kinetic energy exceeding the appropriately defined Peierls-Nabarro barrier; however, they eventually come to a halt, due to radiation loss. © 2008 Elsevier B.V. All rights reserved.
Traveling waves in 2D hexagonal granular crystal lattices
Date: 2014-01-01
Creator: A. Leonard, C. Chong, P. G. Kevrekidis, C. Daraio
Access: Open access
- This study describes the dynamic response of a two-dimensional hexagonal packing of uncompressed stainless steel spheres excited by localized impulsive loadings. The dynamics of the system are modeled using the Hertzian normal contact law. After the initial impact strikes the system, a characteristic wave structure emerges and continuously decays as it propagates through the lattice. Using an extension of the binary collision approximation for one-dimensional chains, we predict its decay rate, which compares well with numerical simulations and experimental data. While the hexagonal lattice does not support constant speed traveling waves, we provide scaling relations that characterize the directional power law decay of the wave velocity for various angles of impact. Lastly, we discuss the effects of weak disorder on the directional amplitude decay rates. © 2014 The Author(s).