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School of Mathematics

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Asymptotic Almost Periodicity of Scalar Parabolic Equations with Almost Periodic Time Dependence

2009-12-07 , Shen, Wenxian , Yi, Yingfei

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A Curious Binomial Identity

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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WKB and Turning Point Theory for Second Order Difference Equations: External Fields and Strong Asymptotics for Orthogonal Polynomials

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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Invariant Tori in Hamiltonian Systems with High Order Proper Degeneracy

2009 , Han, Yuecai , Li, Yong , Yi, Yingfei

We study the existence of quasi-periodic, invariant tori in a nearly integrable Hamiltonian system of high order proper degeneracy, i.e., the integrable part of the Hamiltonian involves several time scales and at each time scale the corresponding Hamiltonian depends on only part of the action variables. Such a Hamiltonian system arises frequently in problems of celestial mechanics, for instance, in perturbed Kepler problems like the restricted and non-restricted 3-body problems and spatial lunar problems in which several bodies with very small masses are coupled with two massive bodies and the nearly integrable Hamiltonian systems naturally involve different time scales. Using KAM method, we will show under certain higher order nondegenerate conditions of Bruno-Rüssmann type that the majority of quasi-periodic, invariant tori associated with the integrable part will persist after the non-integrable perturbation. This actually concludes the KAM metric stability for such a properly degenerate Hamiltonian system.

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Dynamics of Almost Periodic Scalar Parabolic Equations

2009-12-07 , Shen, Wenxian , Yi, Yingfei

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Converse Poincaré Type Inequalities for Convex Functions

2009-12-07 , Bobkov, S. G. , Houdré, Christian

Converse 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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Nonlinear Oscillations and Multiscale Dynamics in a Closed Chemical Reaction System

2009 , Li, Yongfeng , Qian, Hong , Yi, Yingfei

We 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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Random Restarts in Global Optimization

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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Hierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws

2009-07-31 , Liu, Yingjie , Shu, Chi-Wang , Xu, Zhiliang

The 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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Indefinite Quadratic Forms and the Invariance of the Interval in Special Relativity

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.

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