# 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 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 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 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 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 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 A Nonlinear Stability Analysis of a Model Equation for Alloy Solidification(1983) Wollkind, DavidControlled plane front solidification of alloys and other binary substances under an imposed temperature gradient is used in practice to grow single crystals, refine materials (e.g., zone refining), and obtain uniform or non-uniform composition within the material grown [1]. The most important industrial applications of this type of solidification are for growth of crystals for metal oxide semiconductors (MOS's) [1]. Growth of oxide crystals for jewels is another, much older commercial application of single crystal growth [1]. Another important application is in growth of oxides for laser systems and other optical devices [1]. Further industrial applications arise in ingot casting and in the steel and glass industries [2]. For all of these solidification situations involving binary materials, quantitative predictions of interfacial cellular morphology, including information on cell size and intracellular solute distribution, prove to be extremely valuable and are of a particular aid to industrial researchers.Item Maxwell and van der Waals Revisited(1983) Aifantis, Elias C.We utilize a modern continuum mechanic framework to reconsider an old problem for fluid interfaces, also addressed by Maxwell and van der Waals. We prove that their results need not be valid necessarily. This conclusion is arrived at as a consequence of questioning the existence of thermodynamic potentials and the validity of usual thermodynamic relations within unstable (spinodal) regions. One central result is that Maxwell's equal area rule needs not be valid and certain statistical models are shown to be internally inconsistent. Prescise conditions for the validity of Maxwell's rule and the variational theory of van der Waals established in terms of the coefficients defining the interfacial stress.Item The Duration of Transients(1983) Pelikan, StephanImagine a particle moving in a box and making elastic collisions with the sides. Suppose there is a small hole in one side of the box. For many initial conditions the particle will bounce around for a long time and then leave the box. These trajectories are examples of transients. In this paper we investigate the average duration of transients for a certain class of transformations T.Item Stability for Semilinear Parabolic Equations with Noninvertible Linear Operator(1983) Miklavcic, MilanSuppose that x'(t) + Ax(t) = f(t, x(t)), t ≥ 0 is a semilinear parabolic equation, e-At is bounded and f satisfies the usual continuity condition. If for some 0 < ≤ 1, 0 < < 1, p > 1, > 1 ||t Ae-At|| ≤ C, t ≥ 1 ||f(t, x)|| ≤ C(||A x|| p + (1 + t)-), t ≥ 0 whenever ||A x|| + ||x|| is small enough, then for small initial data there exist stable global solutions. Moreover, if the space is reflexive then their limit states exist. Some theorems that are useful for obtaining the above bounds and some examples are also presented.Item Homoclinic Orbits for Flow in R3(1983) Tresser, CharlesWe propose a rough classification for volume contracting flows in R3 with chaotic behavior. In the simplest cases, one looks at the nature of a homoclinic loop for the flow. Most configurations have been studied at length in the literature; here we examine briefly the ``forgotten" case.Item Perturbations of Geodesic Flows on Surfaces of Constant Negative Curvature and their Mixing Properties(1983) Collet, P.; Epstein, H.; Gallavotti, G.We consider one parameter analytic Hamiltonian perturbations of the geodesic flows on surfaces of constant negative curvature. We find two different necessary and sufficient conditions for the canonical equivalence of the perturbed flows and the non perturbed ones. One condition says that the "Hamilton-Jacobi" (introduced in this work) for the conjugation problem should admit a solution as a formal power series (not necessarily convergent) in the perturbation parameter. The alternative condition is based on the identification of a complete set of invariants for the canonical conjugation problem. The relation with the similar problems arising in the KAM theory of the perturbations of quasi periodic Hamiltonian motions is briefly discussed. As a byproduct of our analysis we obtain some results on the Livscic, Guillemin, Kazhdan equation and on the Fourier series for the SL(2, R) group. We also prove that the analytic functions on the phase space for the eodesic flow of unit speed have a mixing property (with respect to the geodesic flow and to the invariant volume measure) which is exponential with a universal exponent, independent on the particular function, equal to the curvature of the surface divided by 2. This result is contrasted with the slow mixing rates that the same functions show under the horocyclic flow: in this case we find that the decay rate is the inverse of the time ("up to logarithms").Item Phase Transitions in the Ising Model with Traverse Field(1983) Kirkwood, JamesThe Ising model perturbed by a small transverse field is shown to have a phase transition by two methods. With the first method, using a Peierls' contour argument, we are only able to show that spontaneous magnetization occurs with the transverse field goint to 0 as -1/3. With the second method, which used reflection positivity, long range order is shown to occur for a small transverse field independent of temperature.Item Invariant Manifolds for Functional Differential Equations Close to Ordinary Differential Equations(1983) Magalhaes, LuisThis paper considers invariant manifolds of global trajectories of retarded Functional Differential Equations in Rn. The persistence, smoothness and stability of such manifolds where the flow is given by an Ordinary Differential Equation (ODE) in Rn is studied for small perturbations of ODEs. The novelty of the present approach lies in the use of the dynamics of the flow on the manifolds, instead of their attractivity properties.Item Smooth Linearization Near a Fixed Point(1983) Sell, George R.In this paper we extend a theorem of Sternberg and Bileckii. We study a vector field, or a diffeomorphism, in the vicinity of a hyperbolic fixed point. We show that if the eigenvalues of the linear part (at the fixed point) satisfy 2N-algebraic conditions (where N > 1), then there is a CN-linearization in the vicinity of this fixed point. If the fixed point is stable, then the CN-linearization theorem follows when only (N + 1)-algebraic conditions are satisfied. Examples are given which show that the first of these results is sharp. An application to celestial mechanics is included.Item A Remark about the Final Aperiodic Regime for Maps on the Interval(1983) Arkeryd, LeifWe consider families of maps on the interval with one maximum, and prove the geometric convergence of the bifurcation parameter for the case of superstable periodic orbits converging towards the final aperiodic regime.Item Local Cinfinity Conjugacy on the Julia Set for some Holomorphic Perturbations of z to z2(1983) Collet, PierreWe consider holomorphic perturbations f of f0, f0(z) = z2, which are small in a neighborhood of the unit circle (the Julia set of f0). We show that if the C1 conjugacy invariants of f and f0 are identical, then f and f0 are conjugate on their part of the Julia set which remains near the unit circle.Item Some Remarks on Deformations of Minimal Surfaces(1983) Rosenberg, H.; Toubiana, E.We consider complete minimal surfaces (c.m.s.'s) in R3 and their deformations. M1 is an deformation of M0 if M1 is a graph over M0 in an tubular neighborhood of M1 and M1 is - C1 close to M0. A c.m.s. M0 is isolated if all minimal surfaces M1, which are sufficiently small deformations of M0, are congruent to M0. Many of the classical minimal surfaces in R3 are known to be isolated [2]; however, no example was known of a nonisolated minimal surface.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.