IMA Preprints Series
Persistent link for this collectionhttps://hdl.handle.net/11299/2061
The IMA Preprint Series is one of the IMA’s scientific publication channels, providing rapid exposure of the scholarly research conducted by IMA-supported mathematicians and scientists. These non-refereed publications date from 1982 until the present.
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Item Non-linear Stability of Asymptotic Suction(1982) Miklavcic, MilanA flow over a plane y = 0 in R3 given by U(x,y,z) = (1 - e-y, -1/R, 0) is called an asymptotic suction velocity profile [12]. R>0 is the Reynolds number. U satisfies the Navier-Stokes equation ðv / ðt + (v · ) v = - p0 + 1/R v div v = 0 with p0 = 0. In the present paper it is proved that the stability of U for small perturbations which initially decay exponentially in the y direction and are periodic in the x and z direction is governed by the eigenvalues of the classical Orr-Sommerfeld equation [1, 8, 12]. For precise statements see Theorems 4, 5, 9, and 15.Item A Simple System with a Continuum of Stable Inhomogeneous Steady States(1982) Weinberger, H.F.The system ut = {(1 + v)}xx + (R1 - au - bv)u vt = (R2 - bu - av)v {(1 + u)}xx = 0 at x = 0 and x = 1 with 1/2 (a/b + b/a) < R1/R2 < a/b and > a(a2 - b2) / 2abR1 - (a2 + b2) R2 was considered by M. Mimura [2] as a model for the population densities of two competing species, one of which increases its migration rate in response to crowding by the other species. It is a special case of the model of N. Shigesada, K. Kawasaki, and E. Teramoto [3].Item Period 3 Bifurcation for the Logistic Mapping(1982) Du, Bau-SenIn the context of continuous mappings of the interval, one of the most striking features may be Sharkovsky's theorem [6] which, among other thing, shows that the existence of a period 3 point implies the existence of periodic points of every period (see also [2, 5]). Therefore, for a one-parameter family of interval mappings, the determination of period 3 bifurcation points may be interesting. In recent years, the logistic mapping f(x) = 1 - x2 has been entensively studied ([1, 4]). By using computer simulation for this family f(x), as the parameter is increased from 0, we can observe the Feigenbaum "cascades" [3]. That is, stable periodic points of double periods accumulate in a geometric and universal way. As the parameter is approximately equal to 1.7498 ([1, p.129]), there seems to be a period 3 bifurcation. In this note, we show that this family f(x) does have a period 3 bifurcation exactly at = 7/4.Item On Copositive Matrices and Strong Ellipticity for Isotropic Elastic Materials(1982) Simpson, H.; Spector, S.In this paper we establish necessary and sufficient conditions for the strong-ellipticity of the equations governing an isotropic (compressible) nonlinerly elastic material at equilibrium. Our work extends results of Knowles and Sternberg [5] who obtained such conditions for both ordinary and strong ellipticity in the special case when the underlying deformations are plane.Item A Simple Proof of C. Siegel's Center Theorem(1982) Llave, R. de laWe give an elementary proof of a particular case of C. Siegel's center theorem, based on a method of M. Herman. Even if the proof has less generality than the standard one, it is simpler and provides sharper bounds.Item Vector Fields in the Vicinity of a Compact Invariant Manifold(1982) Sell, George R.Let us consider two vector fields (1) X' = F(X) (2) Y' = F(Y) defined on a give Euclidean space E where F and G are of class CN+1. Furthermore, assume that there is a smooth compact manifold M smoothly imbedded in E and that M is invariant for both vector fields. Also that F and G agree on M, i.e. F|M = G|M. We wish to study the question of CS-conjugacies between (1) and (2).Item The KAM Theory of Systems with Short Range Interactions II(1983) Wayne, C. EugeneThe proof of the results on the KAM theory of systems with short range interactions, stated in [4] is completed. Estimates on the decay of the interactions generated by the iterative procedure in the KAM theorem are proved, as well as the modification of the theorems of [1-3] needed for our results.Item Tori in Resonance(1983) Meyer, Kenneth R.This paper gives three examples of ordinary differential equations which depend on one or more parameters and which admit invariant tori for some values of the parameters. These examples illustrate how invariant tori evolve as the parameters are changed; in particular how they disappear, bifurcate and lose smoothness. The equations presented are choosen to be as simple as possible in order to clearly show the interesting phenomenon without unnecessary details. However, the theory of normal forms and unfoldings was used to select typical examples, but no attempt will be made to define precisely the universe of discourse where these examples are generic. The unfolding of invariant tori would consist of a mutitude of cases not all of which are that interesting.Item On the Renormalized Coupling Constant and the Susceptibility in f4 Field Theory and the Ising Model in Four Dimensions(1983) Aizenmann, MichaelWe discuss the Euclidean 44 field theory, and the critical behavior in ferromagnetic systems in four dimensions. It is rigorously shown that there are at most logarithmic corrections to the mean field law in the behavior of the magnetic susceptibility = 44 S2 (0, x). Furthermore, if any such corrections are present in a continuum limit which is used to construct a 44 field theory the limiting theory would be non-interacting. Our analysis extends to ferromagnetic systems of variables which belong to the Simon-Griffiths class.Item Manifolds of Global Solutions of Functional Differential Equations(1983) Magalhaes, LuisThis paper consider smooth invariant manifolds of global solutions of retarded Functional Differential Equations in Rn. The persistence, under small perturbations, of such manifolds where the flow is given by an Ordinary Differential Equation in Rn is studied. The novelty of the present approach lies on the use of the dynamics of the flow on the manifolds, instead of their attractivity properties.Item On Work and Constraints in Mixtures(1983) Pericak-Spector, K.A.; Williams, W.O.In recent years workers in mixture theory have become aware of the central role that volume-fraction, the parameter describing the relative proportion of the volume occupied by a constituent, must play in that theory. In particular, the rate of change of volume-fraction, which we here call the chority, must appear in a working term as contributing to the energy, in order to avoid various inconsistencies. This is true both in theories in which volume-fraction appears as a parameter of microstructure and in complete mixture theories.Item On the Thermodynamics of Interstitial Working(1983) Dunn, J.E.; Serrin, J.In order to model fluid capillarity effects, the Dutch physicist Korteweg formulated in 1901 a constitutive equation for the Cauchy stress that included density gradients. Specifically, Koretweg proposed for study a compressible fluid model in which the "elastic" or "equilibrium" portion of the Cauchy stress tensor T is given byItem Surfactant Diffusion; New Results and Interpretations(1983) Evans, D.F.; Mitchell, D.; Mukherjee, S.; Ninham, B.Data for surfactant diffusion are reproted for sodium dodecylsulfate at 25° and tetradecyltrimethylammonium bromide at 25°, 90°, and 135°C, as measured by Taylor tube dispersion. These data are analyzed in terms of two limiting forms of theory, one appropriate to "slow" reaction rates, the other to "fast" rates. It is shown that the usual extrapolation to the critical micelle concentration to infer intrinsic diffusion constants is not permissible. The data is explicable if transport occurs by a process wherein ionic micelles disassociate, diffuse as monomers and reassemble into micelles. This is directly contrary to current ideas on diffusion of surfactants.Item Optimal Numerical Approximation of a Linear Operator(1983) Weinberger, H.F.Many linear problems of numerical analysis can be formulated in the following way: One is given a set of n linear data Nu = and a bound for the norm ||u||B of an otherwise unknown element of u of a hilbert space B. One wishes to find a best approximation to the element Su, where S is a bounded linear operator from B to another Hilbert space . For example, Su may be the solution of an ordinary or partial differential equation with right-hand side, initial data, or boundary data u.Item On Stability and Uniqueness of Fluid Flow Through a Rigid Porous Medium(1983) Pericak-Spector, K.A.We study a set of equations describing the flow of an incompressible viscous fluid through a rigid porous medium. Existence, uniqueness and stability results are established for the case of a region impregnated with fluid, and uniqueness for an unsaturated region.Item Random Fluctuations of the Duration of Harvest(1983) Capasso, V.; Cooke, K.L.; Witten, M.In this report, we wish to discuss models of harvesting of a population when the duration of the harvest interval is subject to random fluctuations. This kind of situation arises, for example, if the harvestor or predator can harvest only when weather conditions are favorable. Clearly, the length of the favorable period will be subject to random variations.Item On the Mechanics of Modulated Structures(1983) Aifantis, Elias C.The purpose of this lecture is to illustrate the appropriateness and potential of the methods of continuum mechanics in modeling modulated structures. Modulations are viewed, in general, as occurrences which may involve one or more properties of a system and extend from a submicroscopic to a macroscopic scale. They are also viewed as capable of possessing wave lengths and amplitudes which may vary from very small to very large values.Item Passive Quasi-Free States of the Noninteracting Fermi Gas(1983) Canniere, Jean DeThe passive quasi-free states of the noninteracting Fermi gas with continuous one-particle Hamiltonian H are computed. They turn out to be the well known Fermi-Dirac states, or limits thereof. This still holds true if the spectrum of H has both a continuous and a discrete part, except for the appearance of a class of "ground state-like" states showing a local random excitation of the point spectrum in a neighborhood of the Fermi energy. When H has only pure point spectrum, the requirement that a state be passive and quasi-free is no longer sufficient to characterize the Fermi-Dirac distributions.Item The KAM Theory of Systems with Short Range Interactions I(1983) Wayne, C. EugeneThe Kolmogorov, Arnol'd, Moser (KAM) theory [15, 1, 16] proves that ``small" perturbations of integrable Hamiltonian systems possess ``large" sets of initial conditions for which the trajectories remain quasiperiodic. In this paper we discuss how the ``strength" of the allowed perturbation varies with the number of degrees of freedom, N, in the system.