Showing 61 - 70 of 96 Items
Thompson's group F is not almost convex
Date: 2003-12-01
Creator: Sean Cleary, Jennifer Taback
Access: Open access
- We show that Thompson's group F does not satisfy Cannon's almost convexity condition AC(n) for any positive integer n with respect to the standard generating set with two elements. To accomplish this, we construct a family of pairs of elements at distance n from the identity and distance 2 from each other, which are not connected by a path lying inside the n-ball of length less than k for increasingly large k. Our techniques rely upon Fordham's method for calculating the length of a word in F and upon an analysis of the generators' geometric actions on the tree pair diagrams representing elements of F. © 2003 Elsevier Inc. All rights reserved.
Extinction in competitive lotka-volterra systems
Date: 1995-01-01
Creator: Mary Lou Zeeman
Access: Open access
- It is well known that for the two species autonomous competitive Lotka-Volterra model with no fixed point in the open positive quadrant, one of the species is driven to extinction, whilst the other population stabilises at its own carrying capacity. In this paper we prove a generalisation of this result to arbitrary finite dimension. That is, for the n-species autonomous competitive Lotka-Volterra model, we exhibit simple algebraic criteria on the parameters which guarantee that all but one of the species is driven to extinction, whilst the one remaining population stabilises at its own carrying capacity. © 1995 American Mathematical Society.
Nonlinear excitations in magnetic lattices with long-range interactions
Date: 2019-06-24
Creator: Miguel Molerón, C. Chong, Alejandro J. Martínez, Mason A. Porter, P. G., Kevrekidis, Chiara Daraio
Access: Open access
- We study - experimentally, theoretically, and numerically - nonlinear excitations in lattices of magnets with long-range interactions. We examine breather solutions, which are spatially localized and periodic in time, in a chain with algebraically-decaying interactions. It was established two decades ago (Flach 1998 Phys. Rev. E 58 R4116) that lattices with long-range interactions can have breather solutions in which the spatial decay of the tails has a crossover from exponential to algebraic decay. In this article, we revisit this problem in the setting of a chain of repelling magnets with a mass defect and verify, both numerically and experimentally, the existence of breathers with such a crossover.
Sign changes of Fourier coefficients of Hilbert modular forms
Date: 2014-01-01
Creator: Jaban Meher, Naomi Tanabe
Access: Open access
- Sign changes of Fourier coefficients of various modular forms have been studied. In this paper, we analyze some sign change properties of Fourier coefficients of Hilbert modular forms, under the assumption that all the coefficients are real. The quantitative results on the number of sign changes in short intervals are also discussed. © 2014 Elsevier Inc.
Combinatorial and metric properties of Thompson's group t
Date: 2009-02-01
Creator: José Burillo, Sean Cleary, Melanie Stein, Jennifer Taback
Access: Open access
- We discuss metric and combinatorial properties of Thompson's group T, including normal forms for elements and unique tree pair diagram representatives. We relate these properties to those of Thompson's group F when possible, and highlight combinatorial differences between the two groups. We define a set of unique normal forms for elements of T arising from minimal factorizations of elements into natural pieces. We show that the number of carets in a reduced representative of an element of T estimates the word length, that F is undistorted in T, and we describe how to recognize torsion elements in T. © 2008 American Mathematical Society Reverts to public domain 28 years from publication.
On the Dirichlet L-functions and the L-functions of Cusp Forms
Date: 2021-01-01
Creator: Nawapan Wattanawanichkul
Access: Open access
- The main objects of our study are L-functions, which are meromorphic functions on the complex plane that analytically continue from the series of the form \sum_{n=1}^{\infty} \frac{a_n}{n^s}, where {a_n} is a sequence of complex numbers. In particular, we are interested in two families of L-functions: ''The Dirichlet L-functions" and ''the L-functions of cusp forms." The former refers to the L-functions whose a_n's are determined by Dirichlet characters, whereas cusp forms determine the latter. We begin our study with the celebrated Riemann zeta function, the simplest Dirichlet L-function, and discuss some of its well-known properties: the Euler product, analytic continuation, functional equation, Riemann hypothesis, and Euler's formula for its critical values. Then, we generalize our exploration to the Dirichlet L-functions and point out some analogous properties to those of the Riemann zeta function. Moreover, we present our original work on computing the critical values of the Dirichlet L-function associated with the primitive character mod 4, or what is known as the Dirichlet beta function. Lastly, we establish some knowledge of the theory of modular forms and cusp forms, which are nicely-behaved modular forms, and discuss some properties of the L-functions of cusp forms.
Formation of rarefaction waves in origami-based metamaterials
Date: 2016-04-15
Creator: H. Yasuda, C. Chong, E. G. Charalampidis, P. G. Kevrekidis, J., Yang
Access: Open access
- We investigate the nonlinear wave dynamics of origami-based metamaterials composed of Tachi-Miura polyhedron (TMP) unit cells. These cells exhibit strain softening behavior under compression, which can be tuned by modifying their geometrical configurations or initial folded conditions. We assemble these TMP cells into a cluster of origami-based metamaterials, and we theoretically model and numerically analyze their wave transmission mechanism under external impact. Numerical simulations show that origami-based metamaterials can provide a prototypical platform for the formation of nonlinear coherent structures in the form of rarefaction waves, which feature a tensile wavefront upon the application of compression to the system. We also demonstrate the existence of numerically exact traveling rarefaction waves in an effective lumped-mass model. Origami-based metamaterials can be highly useful for mitigating shock waves, potentially enabling a wide variety of engineering applications.
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.
Discrete breathers in a mass-in-mass chain with Hertzian local resonators
Date: 2017-02-22
Creator: S. P. Wallen, J. Lee, D. Mei, C. Chong, P. G., Kevrekidis, N. Boechler
Access: Open access
- We report on the existence of discrete breathers in a one-dimensional, mass-in-mass chain with linear intersite coupling and nonlinear, precompressed Hertzian local resonators, which is motivated by recent studies of the dynamics of microspheres adhered to elastic substrates. After predicting theoretically the existence of discrete breathers in the continuum and anticontinuum limits of intersite coupling, we use numerical continuation to compute a family of breathers interpolating between the two regimes in a finite chain, where the displacement profiles of the breathers are localized around one lattice site. We then analyze the frequency-amplitude dependence of the breathers by performing numerical continuation on a linear eigenmode (vanishing amplitude) solution of the system near the upper band gap edge. Finally, we use direct numerical integration of the equations of motion to demonstrate the formation and evolution of the identified localized modes in energy-conserving and dissipative scenarios, including within settings that may be relevant to future experimental studies.
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.