Showing 81 - 90 of 94 Items
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Unidirectional Transition Waves in Bistable Lattices
- We present a model system for strongly nonlinear transition waves generated in a periodic lattice of bistable members connected by magnetic links. The asymmetry of the on-site energy wells created by the bistable members produces a mechanical diode that supports only unidirectional transition wave propagation with constant wave velocity. We theoretically justify the cause of the unidirectionality of the transition wave and confirm these predictions by experiments and simulations. We further identify how the wave velocity and profile are uniquely linked to the double-well energy landscape, which serves as a blueprint for transition wave control.
2016
Computing word length in alternate presentations of thompson's group F
- We introduce a new method for computing the word length of an element of Thompson's group F with respect to a "consecutive" generating set of the form Xn = {x0,x1, ⋯, xn}, which is a subset of the standard infinite generating set for F. We use this method to show that (F, Xn) is not almost convex, and has pockets of increasing, though bounded, depth dependent on n. © 2009 World Scientific Publishing Company.
2009
Cone types and geodesic languages for lamplighter groups and Thompson's group F
- We study languages of geodesics in lamplighter groups and Thompson's group F. We show that the lamplighter groups Ln have infinitely many cone types, have no regular geodesic languages, and have 1-counter, context-free and counter geodesic languages with respect to certain generating sets. We show that the full language of geodesics with respect to one generating set for the lamplighter group is not counter but is context-free, while with respect to another generating set the full language of geodesics is counter and context-free. In Thompson's group F with respect to the standard finite generating set, we show there are infinitely many cone types and that there is no regular language of geodesics. We show that the existence of families of "seesaw" elements with respect to a given generating set in a finitely generated infinite group precludes a regular language of geodesics and guarantees infinitely many cone types with respect to that generating set. © 2005 Elsevier Inc. All rights reserved.
2006
On the nature of eγ and non-vanishing of derivatives of L-series at s=1/2
- In 2011, M.R. Murty and V.K. Murty [10] proved that if L(s, χD) is the Dirichlet L-series attached a quadratic character χD, and L'(1, χD)=0, then eγ is transcendental. This paper investigates such phenomena in wider collections of L-functions, with a special emphasis on Artin L-functions. Instead of s=1, we consider s=1/2. More precisely, we prove thatexp (L'(1/2,χ)L(1/2,χ)-αγ) is transcendental with some rational number α. In particular, if we have L(1/2, χ)≠0 and L'(1/2, χ)=0 for some Artin L-series, we deduce the transcendence of eγ.
2014
Resonance in the menstrual cycle: A new model of the LH surge
- In vertebrates, ovulation is triggered by a surge of LH from the pituitary. The precise mechanism by which rising oestradiol concentrations initiate the LH surge in the human menstrual cycle remains a fundamental open question of reproductive biology. It is well known that sampling of serum LH on a time scale of minutes reveals pulsatile release from the pituitary in response to pulses of gonadotrophin releasing hormone from the hypothalamus. The LH pulse frequency and amplitude vary considerably over the cycle, with the highest frequency and amplitude at the midcycle surge. Here a new mathematical model is presented of the pituitary as a damped oscillator (pulse generator) driven by the hypothalamus. The model LH surge is consistent with LH data on the time scales of both minutes and days. The model is used to explain the surprising pulse frequency characteristics required to treat human infertility disorders such as Kallmann's syndrome, and new experimental predictions are made.
2003
Extinction in nonautonomous competitive Lotka-Volterra Systems
- 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 nonautonomous systems of arbitrary finite dimension. That is, for the n species nonautonomous 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. The restriction of the system to the remaining axis is a nonautonomous logistic equation, which has a unique solution u(t) that is strictly positive and bounded for all time; see Coleman (Math. Biosci. 45 (1979), 159-173) and Ahmad (Proc. Amer. Math. Soc. 117 (1993), 199-205). We prove in addition that all solutions of the n-dimensional system with strictly positive initial conditions are asymptotic to u(t). © 1996 American Mathematical Society.
1996
Components of the Springer fiber and domino tableaux
- Consider a complex classical semisimple Lie group along with the set of its nilpotent coadjoint orbits. When the group is of type A, the set of orbital varieties contained in a given nilpotent orbit is described a set of standard Young tableaux. We parameterize both, the orbital varieties and the irreducible components of unipotent varieties in the other classical groups by sets of standard domino tableaux. The main tools are Spaltenstein's results on signed domino tableaux together with Garfinkle's operations on standard domino tableaux. © 2004 Elsevier Inc. All rights reserved.
2004
Classifying Flow-kick Equilibria: Reactivity and Transient Behavior in the Variational Equation
- In light of concerns about climate change, there is interest in how sustainable management can maintain the resilience of ecosystems. We use flow-kick dynamical systems to model ecosystems subject to a constant kick occurring every τ time units. We classify the stability of flow-kick equilibria to determine which management strategies result in desirable long-term characteristics. To classify the stability of a flow-kick equilibrium, we classify the linearization of the time-τ map given by the time-τ map of the variational equation about the equilibrium trajectory. Since the variational equation is a non-autonomous linear differential equation, we conjecture that the asymptotic stability classification of each instantaneous local linearization along the equilibrium trajectory indicates the stability of the variational time-τ map. In Chapter 3, we prove this conjecture holds when all of the asymptotic and transient behavior of the instantaneous local linearizations is the same. To explore whether the conjecture holds in general, we ask: To what degree can transient behavior differ from asymptotic behavior? Under what conditions can this transient behavior accumulate asymptotically? In Chapter 4, we develop the radial and tangential velocity framework to characterize transient behavior in autonomous linear systems. In Chapter 5, we use this framework to construct an example of a non-autonomous linear system whose time-τ map has asymptotic behavior that differs from the asymptotic behavior of each instantaneous linear system that composes it. Future work seeks to determine whether this constructed example can arise as a variational equation, and thus provide a counterexample for our conjecture.
2020