Showing 21 - 30 of 94 Items
Tame combing and almost convexity conditions
Date: 2011-12-01
Creator: Sean Cleary, Susan Hermiller, Melanie Stein, Jennifer Taback
Access: Open access
- We give the first examples of groups which admit a tame combing with linear radial tameness function with respect to any choice of finite presentation, but which are not minimally almost convex on a standard generating set. Namely, we explicitly construct such combings for Thompson's group F and the Baumslag-Solitar groups BS(1, p) with p ≥ 3. In order to make this construction for Thompson's group F, we significantly expand the understanding of the Cayley complex of this group with respect to the standard finite presentation. In particular we describe a quasigeodesic set of normal forms and combinatorially classify the arrangements of 2-cells adjacent to edges that do not lie on normal form paths. © 2010 Springer-Verlag.
Effects of stochasticity on the length and behaviour of ecological transients
Date: 2021-07-01
Creator: Alan Hastings, Karen C. Abbott, Kim Cuddington, Tessa B. Francis, Ying Cheng, Lai, Andrew Morozov
Access: Open access
- There is a growing recognition that ecological systems can spend extended periods of time far away from an asymptotic state, and that ecological understanding will therefore require a deeper appreciation for how long ecological transients arise. Recent work has defined classes of deterministic mechanisms that can lead to long transients. Given the ubiquity of stochasticity in ecological systems, a similar systematic treatment of transients that includes the influence of stochasticity is important. Stochasticity can of course promote the appearance of transient dynamics by preventing systems from settling permanently near their asymptotic state, but stochasticity also interacts with deterministic features to create qualitatively new dynamics. As such, stochasticity may shorten, extend or fundamentally change a system's transient dynamics. Here, we describe a general framework that is developing for understanding the range of possible outcomes when random processes impact the dynamics of ecological systems over realistic time scales. We emphasize that we can understand the ways in which stochasticity can either extend or reduce the lifetime of transients by studying the interactions between the stochastic and deterministic processes present, and we summarize both the current state of knowledge and avenues for future advances.
The large-scale geometry of some metabelian groups
Date: 2003-01-01
Creator: Jennifer Taback, Kevin Whyte
Access: Open access
Convexity Properties of the Diestel-Leader Group Γ_3(2)
Date: 2014-05-01
Creator: Peter J Davids
Access: Open access
- The Diestel-Leader groups are a family of groups first introduced in 2001 by Diestel and Leader in [7]. In this paper, we demonstrate that the Diestel-Leader group Γ3(2) is not almost convex with respect to a particular generating set S. Almost convexity is a geometric property that has been shown by Cannon [3] to guarantee a solvable word problem (that is, in any almost convex group there is a finite-step algorithm to determine if two strings of generators, or “words”, represent the same group element). Our proof relies on the word length formula given by Stein and Taback in [10], and we construct a family of group elements X that contradicts the almost convexity condition. We then go on to show that Γ3(2) is minimally almost convex with respect to S.
Instability in a Time-Modulated Lattice This record is embargoed.
- Embargo End Date: 2025-05-19
Date: 2022-01-01
Creator: Evelyn Wallace
Access: Embargoed
Sensitivity Analysis of Basins of Attraction for Nelder-Mead
Date: 2022-01-01
Creator: Sonia K. Shah
Access: Open access
- The Nelder-Mead optimization method is a numerical method used to find the minimum of an objective function in a multidimensional space. In this paper, we use this method to study functions - specifically functions with three-dimensional graphs - and create images of the basin of attraction of the function. Three different methods are used to create these images named the systematic point method, randomized centroid method, and systemized centroid method. This paper applies these methods to different functions. The first function has two minima with an equivalent function value. The second function has one global minimum and one local minimum. The last function studied has several minima of different function values. The systematic point method is a reliable method in particular scenarios but is extremely sensitive to changes in the initial simplex. The randomized centroid method was not found to be useful as the basin of attraction images are difficult to understand. This made it particularly troublesome to know when the method was working effectively and when it was not. The systemized centroid method appears to be the most precise and effective method at creating the basin of attraction in most cases. This method rarely fails to find a minimum and is particularly adept at finding global minima more effectively compared to local minima. It is important to remember that these conclusions are simply based off the results of the methods and functions studied and that more effective methods may exist.
Demonstration of Dispersive Rarefaction Shocks in Hollow Elliptical Cylinder Chains
Date: 2018-05-11
Creator: H. Kim, E. Kim, C. Chong, P. G. Kevrekidis, J., Yang
Access: Open access
- We report an experimental and numerical demonstration of dispersive rarefaction shocks (DRS) in a 3D-printed soft chain of hollow elliptical cylinders. We find that, in contrast to conventional nonlinear waves, these DRS have their lower amplitude components travel faster, while the higher amplitude ones propagate slower. This results in the backward-tilted shape of the front of the wave (the rarefaction segment) and the breakage of wave tails into a modulated waveform (the dispersive shock segment). Examining the DRS under various impact conditions, we find the counterintuitive feature that the higher striker velocity causes the slower propagation of the DRS. These unique features can be useful for mitigating impact controllably and efficiently without relying on material damping or plasticity effects.
Wave transmission in time- and space-variant helicoidal phononic crystals
Date: 2014-11-04
Creator: F. Li, C. Chong, J. Yang, P. G. Kevrekidis, C., Daraio
Access: Open access
- We present a dynamically tunable mechanism of wave transmission in one-dimensional helicoidal phononic crystals in a shape similar to DNA structures. These helicoidal architectures allow slanted nonlinear contact among cylindrical constituents, and the relative torsional movements can dynamically tune the contact stiffness between neighboring cylinders. This results in cross-talking between in-plane torsional and out-of-plane longitudinal waves. We numerically demonstrate their versatile wave mixing and controllable dispersion behavior in both wavenumber and frequency domains. Based on this principle, a suggestion toward an acoustic configuration bearing parallels to a transistor is further proposed, in which longitudinal waves can be switched on and off through torsional waves.
The validity of the kdv approximation in case of resonances arising from periodic media
Date: 2011-11-15
Creator: Christopher Chong, Guido Schneider
Access: Open access
- It is the purpose of this short note to discuss some aspects of the validity question concerning the Korteweg-de Vries (KdV) approximation for periodic media. For a homogeneous model possessing the same resonance structure as it arises in periodic media we prove the validity of the KdV approximation with the help of energy estimates. © 2011 Elsevier Inc.
From local to global behavior in competitive Lotka-Volterra systems
Date: 2003-01-01
Creator: E. C. Zeeman, M. L. Zeeman
Access: Open access
- In this paper we exploit the linear, quadratic, monotone and geometric structures of competitive Lotka-Volterra systems of arbitrary dimension to give geometric, algebraic and computational hypotheses for ruling out nontrivial recurrence. We thus deduce the global dynamics of a system from its local dynamics. The geometric hypotheses rely on the introduction of a split Liapunov function. We show that if a system has a fixed point p ∈ int R+n and the carrying simplex of the system lies to one side of its tangent hyperplane at p, then there is no nontrivial recurrence, and the global dynamics are known. We translate the geometric hypotheses into algebraic hypotheses in terms of the definiteness of a certain quadratic function on the tangent hyperplane. Finally, we derive a computational algorithm for checking the algebraic hypotheses, and we compare this algorithm with the classical Volterra-Liapunov stability theorem for Lotka-Volterra systems.