Bruno Scarpellini

Comments on `Two Undecidable Problems of Analysis'

We first discuss some technical questions which arise in connection with the construction of undecidable propositions in analysis, in particular in connection with the notion of the normal form of a function representing a predicate. Then it is stressed that while a function f(x) may be computable in the sense of recursive function theory, it may nevertheless have undecidable properties in the realm of Fourier analysis. This has an implication for a conjecture of Penroses which states that classical physics is computable.

Center manifolds of infinite dimensions I: Main results and applications

In this paper, the first of a bipartite work, we consider an abstract, nonautonomous system of evolution equations of hyperbolic type, related to semilinear wave equations. Theorem 1 states that under certain assumptions the system admits a global center manifold, or equivalently a global decoupling function which is continuously differentiable with respect to its arguments, among which timet occurs. The difficult proof is presented in part II, i.e. the continuation of the present paper. For purposes of applications a local version of Theorem 1 is proved, i.e. the local center manifold Theorem 2. We obtain a series of applications both to abstract, nonautonomous wave equations and to concrete nonautonomous, semilinear wave equations subject to Neumann and Dirichlet boundary conditions.

Two Undecidable Problems of Analysis

In this article two undecidable problems belonging to the domain of analysis will be constructed. The basic idea is sketched as follows: Let us imagine an area B of functions (rational functions, trigonometric and exponential functions) and certain operations (addition, multiplication, integration over finite or infinite domains, etc.) and consider the smallest quantity M of functions which contains B and is closed with regard to the selected operations. The question will then be examined whether there is in M a function f( x)for which the predicate P( n)≡ � f( x)cos nxdx > 0 is not recursive. It will be shown that by suitably choosing the area B and the operations, the answer comes out positively. We will deal in general with complex functions of real variables, although one could with somewhat more effort carry out all considerations in the real domain. In the first example, new functions will be generated by means of the following operations: addition, multiplication, integration over finite intervals and the solution of Fredholm integral equations of the second kind. Following this, it will be shown that certain logically characterised functions can be represented as limits of functions of the area M. In these constructions care will be taken that the number of integral equations to be solved remains as small as possible (namely two). In the second example, instead of the solution of Fredholm integral equations, we permit integration over infinite intervals, and then prove for this instance the same theorems as in the first example, in the context of which considerable use will be made of the result of M. Davis, H. Putnam and J. Robinson (cf. (1)) on the unsolvability of exponential diophantine equations.

Smooth manifolds for semilinear wave equations on R2: On the existence of almost-periodic breathers

Abstract In this paper, we prove several nonexistence and existence results for certain real solutions to semilinear wave equations in one space dimension. These solutions represent time almost-periodic free vibrations which decay toward a constant equilibrium solution as the spatial variable goes to infinity. Our nonexistence results are of two kinds. We first prove that the requirements of almost periodicity in time and of spatial decay at infinity force such solutions to be trivial, whenever their spectrum does not interact with the nonlinearity in a certain sense. We also prove non-existence results without requiring any spectral conditions, but this is at the expense of having to impose more stringent conditions on the nonlinearities, such as certain convexity properties in the vicinity of the equilibrium solution. Finally, we prove an existence result for time almost-periodic free vibrations whose profiles decay exponentially rapidly toward a constant equilibrium solution. To accomplish this, we convert the wave equation into a dynamical system in which the original spatial variable plays the role of time; we then embed that dynamical system into an appropriate Banach space of time almost-periodic functions, so that the one-parameter family of stable and unstable manifolds which we construct carry the solutions that we seek. The major difficulty to overcome in our construction is a small divisor problem; it is related to the fact that the spectrum of the infinitesimal generator for the linearized flow is a pure point spectrum without gaps. We discuss several examples and we also stress some important qualitative differences which distinguish the almost-periodic case from the purely periodic one.

Predicting the future of functions on flows

AbstractLet Ω be a topological space,St∈ R (R the reals) a homeomorphism group on Ω andμ a Borel measure invariant with respect toSt, (μ(Ω)=1); forP ∈Ω putSt(P)=Pt. Assumef ∈L2(Ω,μ); according to E. Hopf there is for almost everyP ∈ Ω a well-determined spectral function σ(P,λ),λ ∈ R with lim

Center manifolds of infinite dimensions II: Proofs of the main result

T^{ - 1} \int_0^T {f(P_{t + s} )\overline {f(P_t )} dt = \int_{ - \infty }^{ + \infty } {e^{i\lambda s} d\sigma (P\lambda )} }

Complex boolean networks obtained by diagonalization

. The question to be considered is:*) if for a fixedP ∈ Ω we know the “past”f(Pt), t ≦ 0, is it then possible to compute (or “predict”) the future valuesf(Pt), t > 0? By using ideas from linear prediction theory we show that if