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School of Mathematics

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Now showing 1 - 10 of 59
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Persistence of Invariant Tori on Submanifolds in Hamiltonian Systems

1999 , Chow, Shui-Nee , Li, Yong , Yi, Yingfei

Generalizing the degenerate KAM theorem under the Rüssmann non-degeneracy and the isoenergetic KAM theorem, we employ a quasi-linear iterative scheme to study the persistence and frequency preservation of invariant tori on a smooth sub-manifold for a real analytic, nearly integrable Hamiltonian system. Under a nondegenerate condition of Rüssmann type on the sub-manifold, we shall show the following: a) the majority of the unperturbed tori on the sub-manifold will persist; b) the perturbed toral frequencies can be partially preserved according to the maximal degeneracy of the Hessian of the unperturbed system and be fully preserved if the Hessian is nondegenerate; c) the Hamiltonian admits normal forms near the perturbed tori of arbitrarily prescribed high order. Under a sub-isoenergetic nondegenerate condition on an energy surface, we shall show that the majority of unperturbed tori give rise to invariant tori of the perturbed system of the same energy which preserve the ratio of certain components of the respective frequencies.

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A Quasi-Periodic Poincaré's Theorem

1999 , Li, Yong , Yi, Yingfei

We study the persistence of invariant tori on resonant surfaces of a nearly integrable Hamiltonian system under the usual Kolmogorov non-degenerate condition. By introducing a quasi-linear iterative scheme to deal with small divisors, we generalize the Poincaré theorem on the maximal resonance case (i.e., the periodic case) to the general resonance case (i.e., the quasi-periodic case) by showing the persistence of majority of invariant tori associated to non-degenerate relative equilibria on any resonant surface.

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Topological Dynamics and Differential Equations

1997 , Sell, George R. , Shen, Wenxian , Yi, Yingfei

By reviewing our previous works on lifting dynamics in skew-product semi-flows and also the work of Johnson on almost periodic Floquet theory, we show several significant applications of the abstract theory of topological dynamics to the qualitative study of non-autonomous differential equations. The paper also contains some detailed discussions on a conjecture of Johnson.

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Sharp Constants in Some Multiplicative Sobolev Inequalities

1995-09-16 , Bobkov, S. G. , Houdré, Christian

The optimal constants in the multiplicative Sobolev inequalities where the gradient is estimated in the L_1-norm and the function in two different Lebesgue norms are found. With the optimal constants, these inequalities turn out to still be equivalent to the isoperimetric property of the balls in the Euclidean space. In the course of the proof, relations between Lorentz and Lebesgue spaces are studied (and also applied to some different measures, e.g., Riesz potentials).

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Persistence of Lower Dimensional Tori of General Types in Hamiltonian Systems

1999 , Li, Yong , Yi, Yingfei

The work is a generalization to [40] in which we study the persistence of lower dimensional tori of general type in Hamiltonian systems of general normal forms. By introducing a modified linear KAM iterative scheme to deal with small divisors, we shall prove a persistence result, under a Melnikov type of non-resonance condition, which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori.

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Persistence of Hyperbolic Tori in Hamiltonian Systems

1999 , Li, Yong , Yi, Yingfei

We generalize the well-known result of Graff and Zehnder on the persistence of hyperbolic invariant tori in Hamiltonian systems by considering non-Floquet, frequency varying normal forms and allowing the degeneracy of the unperturbed frequencies. The preservation of part or full frequency components associated to the degree of non-degeneracy is considered. As applications, we consider the persistence problem of hyperbolic tori on a submanifold of a nearly integrable Hamiltonian system and the persistence problem of a fixed invariant hyperbolic torus in a non-integrable Hamiltonian system.

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Convergence in Almost Periodic Fisher and Kolmogorov Models

1996 , Shen, Wenxian , Yi, Yingfei

We study convergence of positive solutions for almost periodic reaction diffusion equations of Fisher or Kolmogorov type. It is proved that under suitable conditions every positive solution is asymptotically almost periodic. Moreover, all positive almost periodic solutions are harmonic and uniformly stable, and if one of them is spatially homogeneous, then so are others. The existence of an almost periodic global attractor is also discussed.

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Center Manifolds for Invariant Sets

1999 , Chow, Shui-Nee , Liu, Weishi , Yi, Yingfei

We derive a general center manifolds theory for a class of compact invariant sets of flows generated by a smooth vector fields in R^n. By applying the Hadamard graph transform technique, it is shown that, associated to certain dynamical characteristics of the linearized flow along the invariant set, there exists an invariant manifold (called a center manifold) of the invariant set which contains every locally bounded solution (in particular, contains the invariant set) and is persistent under small perturbations.

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Center Manifolds for Invariant Manifolds

1997 , Chow, Shui-Nee , Liu, Weishi , Yi, Yingfei

We study dynamics of flows generated from smooth vector fields in R^n in the vicinity of an invariant and closed smooth manifold Y. By applying the Hadamard graph transform technique, we show that there exists an invariant manifold (called a center manifold of Y) based on the information of the linearization along Y, which contains every locally bounded solution and is persistent under small perturbations.

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Information Advantages in Hiring under a Budget Constraint: Weak Convergence Comparisons

1995-10-06 , Boshuizen, Frans A. , Kertz, Robert P.

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