We establish the existence of smooth critical sub-solutions of the HamiltonJacobi equation on compact manifolds for smooth convex Hamiltonians, that is in the context of weak KAM theory, under the assumption that the Aubry set is the union of finitely many hyperbolic periodic orbits or fixed points. Let M be a compact manifold without boundary. A function H(x, p) : T M −→ R is called a Tonelli ...

We elucidate the epistemological futility of using concepts such as unconscious thinking in research. Focusing on Newell & Shanks' (N&S's) use of the lens model as a framework, we clarify issues with regard to unconscious-thought theory (UTT) and self-insight studies. We examine these key points: Brunswikian psychology is absent in UTT; research on self-insight did not emerge to explore the unc...

In 1966, Shanks and Schmid investigated the asymptotic behavior of the number of positive integers less than or equal to x which are represented by the quadratic form X +nY . Based on some numerical computations, they observed that the constant occurring in the main term appears to be the largest for n = 2. In this paper, we prove that in fact this constant is unbounded as n runs through positi...

We present algorithms for computing the cube of an ideal in an imaginary quadratic number field or function field. In addition to a version that computes a non-reduced output, we present a variation based on Shanks’ NUCOMP algorithm that computes a reduced output and keeps the sizes of the intermediate operands small. Extensive numerical results are included demonstrating that in many cases our...