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Showing papers in "Communications on Pure and Applied Mathematics in 1978"


Journal ArticleDOI
TL;DR: In this paper, the Ricci form of some Kahler metric is shown to be closed and its cohomology class must represent the first Chern class of M. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation.
Abstract: Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the first Chern class of M. More than twenty years ago, E. Calabi [3] conjectured that the above necessary condition is in fact sufficient. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation.

2,903 citations


Journal ArticleDOI
TL;DR: In this paper, the existence of periodic solutions of Hamiltonian systems of ordinary differential equations is proved in various settings, including free and forced vibration problems, where the period is fixed, and the proofs involve finite dimensional approximation arguments, variational methods, and appropriate estimates.
Abstract: : The existence of periodic solutions of Hamiltonian systems of ordinary differential equations is proved in various settings. A case in which energy is prescribed is treated in Section 1. Both free and forced vibration problems, where the period is fixed, are studied in Section 2. The proofs involve finite dimensional approximation arguments, variational methods, and appropriate estimates. (Author)

681 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the position of any fixed percentile of the maximal displacement of standard branching Brownian motion in one dimension is 21/2t−3 · 2−3/2 log t + O(1) at time t, the second-order term having been previously unknown.
Abstract: It is shown that the position of any fixed percentile of the maximal displacement of standard branching Brownian motion in one dimension is 21/2t–3 · 2−3/2 log t + O(1) at time t, the second-order term having been previously unknown. This determines (to within O(1)) the position of the travelling wave of the semilinear heat equation, ut =1/2uxx +f(u), in the classic paper by Kolmogorov-Petrovsky-Piscounov, “Etude de l'equations de la diffusion avec croissance de la quantite de la matiere et son application a un probleme biologique”, 1937.

570 citations




Journal ArticleDOI
TL;DR: Product formulas constitute one of several bridges between numerical and functional analysis as discussed by the authors, and they represent algorithms intended to approximate some evolution equation and, in functional analysis, they are used to prove estimates, existence and representation theorems.
Abstract: Product formulas constitute one of several bridges between numerical and functional analysis. In numerical analysis, they represent algorithms intended to approximate some evolution equation and, in functional analysis, they are used to prove estimates, existence and representation theorems. Our aim is to survey the setting for product formulas and to discuss some recent results. Needless to say, we do not attempt to accommodate all the complex variations which occur in practical algorithms, nor the sharpest possible theoretical results. Nevertheless, we hope that our middle ground approach and some of the examples will be of interest to both groups. Because of its survey nature, we have not hesitated to include some well-known examples which are important for understanding the ideas. The general idea of product formulas is the following. Suppose one is interested in an initial value problem

338 citations



Journal ArticleDOI
TL;DR: In this paper, it was shown that the moments of a linear hyperbolic operator can be approximated by a solution of L/sub h/U = 0 with an error O(h/sup v-delta/, delta as small as desired, at all points, no matter how close to the discontinuity.
Abstract: Let L be a linear hyperbolic operator in any number of variables with C/sup infinity/ coefficients. A solution u of Lu - 0 which has C/sup infinity/ initial data is C/sup infinity/ for all time. Let L/sub h/ be a difference approximation to L that is stable, and accurate of order v. Denote by U the solution of L/sub h/U = 0 the initial values of which agree with those of u on the lattice points, h denoting the mesh width of the lattice. At all times t at which U is available, and in any fixed range 0 less than or equal to t less than or equal to T, the absolute value of (u(t) - U(t)) = O(h/sup v/). Consider piecewise C/sup infinity/ initial data whose discontinuities occur across C/sup infinity/ surfaces. It is known that solutions u of Lu = 0 with such initial data are themselves piecewise C/sup infinity/, and their discontinuities occur across characteristic surfaces issuing from the discontinuity surface of the initial data. What happens when such a solution is approximated by a solution of L/sub h/U = 0 is discussed. It is shown that, for a scheme of any order v, the momentsmore » of U approximate those of u with accuracy O(h/sup v/), provided that the initial data of U are pre-processed appropriately near the discontinuities; this is true for equations in any number of variables, and with variable coefficients. It is further shown that post-processing the approximate solution can recover the exact solution, as well as its derivatives, with an error O(h/sup v-delta/, delta as small as desired, at all points, no matter how close to the discontinuity. (RWR)« less

84 citations





Journal ArticleDOI
TL;DR: Solovay and Strassen, and Miller and Rabin have discovered fast algorithms for testing primality which use coin-flipping and whose coefficients do not depend on the value of the coin.
Abstract: Solovay and Strassen, and Miller and Rabin have discovered fast algorithms for testing primality which use coin-flipping and whose con




Journal ArticleDOI
TL;DR: In this article, it was shown that for systems with constant principal part P near P° the notions of weak and strong hyperbolicity coincide and that the corresponding differential equations (1) stay hyperbola if we add any low-order terms with complex variable coefficients.
Abstract: In the paper [18]† we dealt with a certain type of homogeneous second-order systems of partial differential equations (1) with constant coefficients. Here the matrix form P = P(λ, ξ) was represented by a point in ℝ81. We studied in particular the forms P in the set K ɛ ∩H, that is those P corresponding to a hyperbolic system (1) which lie in an e-neighborhood of the special form P°. It was found that these P are never strictly hyperbolic and cannot even be approximated by strictly hyperbolic ones. We shall derive now a further property of the P∈K ɛ∩H for e sufficiently small, namely, that the corresponding differential equations (1) stay hyperbolic if we add any low-order terms with complex variable coefficients. This means that for systems with constant principal part P near P° the notions of weak and strong hyperbolicity coincide.6 The proof makes essential use of the fact proved earlier that the second derivatives of the function D*(ξ) for ξh near a singular point form a positive definite matrix.





Journal ArticleDOI
TL;DR: In this paper, it was shown that the operator Hs has a complete set of eigenfunctions and eigenvalues, which satisfy the Laguerre functions and spherical harmonics.
Abstract: We show that the operator Hs has a complete set of eigenfunctions and eigenvalues , which satisfy [2l(l + 1) - (3n2 + 3n + 1)]s + o(s) and lims0 = 0. The functions are given in spherical coordinates as a product of generalized Laguerre functions and spherical harmonics.

Journal ArticleDOI
TL;DR: In this paper, the nonlinear stabilization of the well-known linearly unstable m = 2 magnetohydrodynamic kink modes in a cylindrical plasma with a vacuum interface was studied from a perturbational viewpoint.
Abstract: The present work is concerned with the nonlinear stabilization of the well-known linearly unstable m = 2 magnetohydrodynamic kink modes in a cylindrical plasma with a vacuum interface. We shall study this problem from a perturbational viewpoint and obtain the nonlinear equation which governs the evolution of the mode amplitude. We adopt the method used by Stuart in deriving the Landau equation in fluid mechanics.