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ItemAsymptotic Almost Periodicity of Scalar Parabolic Equations with Almost Periodic Time Dependence(Georgia Institute of Technology, 2009-12-07) Shen, Wenxian ; Yi, Yingfei
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ItemDynamics of Almost Periodic Scalar Parabolic Equations(Georgia Institute of Technology, 2009-12-07) Shen, Wenxian ; Yi, Yingfei
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ItemRandom Restarts in Global Optimization(Georgia Institute of Technology, 2009-12-07) Hu, X. ; Shonkwiler, Ronald W. ; Spruill, Marcus C.In this article we study stochastic multistart methods for global optimization, which combine local search with random initialization, and their parallel implementations. It is shown that in a minimax sense the optimal restart distribution is uniform. We further establish the rate of decrease of the ensemble probability that the global minimum has not been found by the nth iteration. Turning to parallelization issues, we show that under independent identical processing (iip), exponential speedup in the time to hit the goal bin normally results. Our numerical studies are in close agreement with these finndings.
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ItemA Curious Binomial Identity(Georgia Institute of Technology, 2009-12-07) Calkin, Neil J.In this note we shall prove the following curious identity of sums of powers of the partial sum of binomial coefficients.
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ItemConverse Poincaré Type Inequalities for Convex Functions(Georgia Institute of Technology, 2009-12-07) Bobkov, S. G. ; Houdré, ChristianConverse Poincaré type inequalities are obtained within the class of smooth convex functions. This is, in particular, applied to the double exponential distribution.
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ItemHierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws(Georgia Institute of Technology, 2009-07-31) Liu, Yingjie ; Shu, Chi-Wang ; Xu, ZhiliangThe hierarchical reconstruction (HR) [Liu, Shu, Tadmor and Zhang, SINUM ’07] can effectively reduce spurious oscillations without local characteristic decomposition for numerical capturing of discontinuous solutions. However, there are still small re- maining overshoots/undershoots in the vicinity of discontinuities. HR with partial neighboring cells [Xu, Liu and Shu, JCP ’09] essentially overcomes this drawback for the third order case, and in the mean time further improves the resolution of the numer- ical solution. Extending the technique to higher order cases we observe the returning of overshoots/undershoots. In this paper, we introduce a new technique to work with HR on partial neighboring cells, which lowers the order of the remainder while maintaining the theoretical order of accuracy, essentially eliminates overshoots/undershoots for the fourth and fifth order cases (in one dimensional numerical examples) and reduces the numerical cost.
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ItemWKB and Turning Point Theory for Second Order Difference Equations: External Fields and Strong Asymptotics for Orthogonal Polynomials(Georgia Institute of Technology, 2009-05) Geronimo, Jeffrey S.A LG-WKB and Turning point theory is developed for three term recurrence formulas associated with monotonic recurrence coefficients. This is used to find strong asymptotics for certain classical orthogonal polynomials including Wilson polynomials.
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ItemSensitive dependence of the motion of a legged robot on granular media(Georgia Institute of Technology, 2009-03-03) Li, Chen ; Umbanhowar, Paul B. ; Komsuoglu, Haldun ; Koditschek, Daniel E. ; Goldman, Daniel I.Legged locomotion on flowing ground (e.g., granular media) is unlike locomotion on hard ground because feet experience both solid- and fluid-like forces during surface penetration. Recent bioinspired legged robots display speed relative to body size on hard ground comparable with high-performing organisms like cockroaches but suffer significant performance loss on flowing materials like sand. In laboratory experiments, we study the performance (speed) of a small (2.3 kg) 6-legged robot, SandBot, as it runs on a bed of granular media (1-mm poppy seeds). For an alternating tripod gait on the granular bed, standard gait control parameters achieve speeds at best 2 orders of magnitude smaller than the 2 body lengths/s (≈60 cm/s) for motion on hard ground. However, empirical adjustment of these control parameters away from the hard ground settings restores good performance, yielding top speeds of 30 cm/s. Robot speed depends sensitively on the packing fraction φ and the limb frequency ω, and a dramatic transition from rotary walking to slow swimming occurs when φ becomes small enough and/or ω large enough. We propose a kinematic model of the rotary walking mode based on generic features of penetration and slip of a curved limb in granular media. The model captures the dependence of robot speed on limb frequency and the transition between walking and swimming modes but highlights the need for a deeper understanding of the physics of granular media.
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ItemNonlinear Oscillations and Multiscale Dynamics in a Closed Chemical Reaction System(Georgia Institute of Technology, 2009) Li, Yongfeng ; Qian, Hong ; Yi, YingfeiWe investigate the oscillatory chemical dynamics in a closed isothermal reaction system described by the reversible Lotka-Volterra model. This is a three-dimensional, dissipative, singular perturbation to the conservative Lotka-Volterra model, with the free energy serving as a global Lyapunov function. We will show that there is a natural distinction between oscillatory and non-oscillatory regions in the phase space, that is, while orbits ultimately reach the equilibrium in a non-oscillatory fashion, they exhibit damped, oscillatory behaviors as interesting intermediate dynamics.
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ItemIndefinite Quadratic Forms and the Invariance of the Interval in Special Relativity(Georgia Institute of Technology, 2009) Elton, John H.In this note, a simple theorem on proportionality of indefinite real quadratic forms is proved, and is used to clarify the proof of the invariance of the interval in special relativity from Einstein's postulate on the universality of the speed of light; students are often rightfully confused by the incomplete or incorrect proofs given in many texts. The result is illuminated and generalized using Hilbert's Nullstellensatz, allowing one form to be a homogeneous polynomial which is not necessarily quadratic. Also a condition for simultaneous diagonalizability of semi-definite real quadratic forms is given.