In locally compact spaces, (Borel-)measurable sets can be approximated by compact sets. Ulam extended this result to complete separable metric spaces. We give a constructive proof of Ulam’s theorem. It is first proved intuitionistically and then, using a logical ‘trick’ due to Ishihara, a proof acceptable in Bishop-style mathematics is obtained. We feel this proof provides some insight into Ish...

Feldman and Karlin conjectured that the number of isolated fixed points for deterministic models of viability selection and recombination among n possible haplotypes has an upper bound of 2(n)-1. Here a proof is provided. The upper bound of 3(n-1) obtained by Lyubich et al. (2001) using Bézout's Theorem (1779) is reduced here to 2(n) through a change of representation that reduces the third-ord...

Kantorovitch's theorem gives su cient conditions for the convergence of Newton's method. We presenta full formalization of this theorem in the case of a real function. The work is accomplished inside the Coqproof assistant and it is based on the Reals library provided by the theorem prover. For the general case ofthe theorem we rst describe a way to represent concepts from multivari...

We show that under well-defined conditions the Hohenberg-Kohn theorem (HKT) that provides the foundation of ground-state density functional theory (DFT) can be extended to the lowest-energy resonance of unbound electronic systems. The extended version of the HKT provides an adequate framework to carry out DFT calculations of negative electron affinities.

We give a new, elementary, purely analytical development of Descartes’ theorem that a smooth connected surface is a perfect focusing lens if and only if it is a connected subset of the ovoid obtained by revolving a cartesian oval around its axis of symmetry.

The. development of automated assistance for medical diagnosis and decision making is an area of both theoretical and practical interest. Of methods for utilizing evidence to select diagnoses or decisions, probability theory has the firmest appeal. Probability theory in the form of Bayes’ Theorem has been used by a number of" workers (Ross, 1972). Notable among recent developments are those of ...

In locally compact spaces, (Borel-)measurable sets can be approximated by compact sets. Ulam extended this result to complete separable metric spaces. We give a constructive proof of Ulam’s theorem. It is first proved intuitionistically and then, using a logical ‘trick’ due to Ishihara, a proof acceptable in Bishop-style mathematics is obtained. We feel this proof provides some insight into Ish...

where <p is Euler's function and n(x) counts all primes up to x. Indeed, (1.1) is known to hold, but only under the proviso that q grow more slowly than some fixed power of logx (Siegel-Walfisz theorem). Yet, for many applications, it is important to weaken this restriction. The assumption of the Generalized Riemann Hypothesis (GRH) yields (1.1) in the much wider range q < X I / 2/ log2+e x . I...

We introduce a new notion of computable function on R and prove some basic properties. We give two applications, first a short proof of Yoshinaga’s theorem that periods are elementary (they are actually lower elementary). We also show that the lower elementary complex numbers form an algebraically closed field closed under exponentiation and some other special functions.

where <p is Euler's function and n(x) counts all primes up to x. Indeed, (1.1) is known to hold, but only under the proviso that q grow more slowly than some fixed power of logx (Siegel-Walfisz theorem). Yet, for many applications, it is important to weaken this restriction. The assumption of the Generalized Riemann Hypothesis (GRH) yields (1.1) in the much wider range q < X I / 2/ log2+e x . I...