The theory of Schwarzschild geodesics is revisited. Using a theorem due to Weierstrass and Biermann, we derive concise formulas describing all timelike null trajectories in terms elliptic functions. formulation given this note uses an analogue the so-called Mino time.

We dedicate this paper to Ambikeshwar Sharma with esteem, admiration and gratitude for lighting our way. Abstract. This paper examines the set B(P) = fQ : P Q = 1 ; Q 2 R m g where P 2 R m is unimodular and R is either the algebra P R of algebraic polynomials which are real{valued on the cube I d or the algebra L R of Laurent polynomials which are real{valued on the torus T d : We sharpen previ...

Let K be an effective field of characteristic zero. An effective tribe is a subset of K[[z1, z2, ...]] =K ∪K[[z1]] ∪K[[z1, z2]] ∪ ··· which is effectively stable under the Kalgebra operations, restricted division, composition, the implicit function theorem, as well as restricted monomial transformations with arbitrary rational exponents. Given an effective tribe with an effective zero test, we ...

Takagi–Sugeno (TS) fuzzy systems have been employed as fuzzy controllers and fuzzy models in successfully solving difficult control and modeling problems in practice. Virtually all the TS fuzzy systems use linear rule consequent. At present, there exist no results (qualitative or quantitative) to answer the fundamentally important question that is especially critical to TS fuzzy systems as fuzz...

The object of this paper is to prove a new type of estimate for isogenies between elliptic curves. This has several diophantine applications (effective versions of Serre's Galois irreducibility theorem and Shafarevich's theorem, for example) which are presented in another paper [MW3]. Later articles will deal with the corresponding problems for abelian varieties of arbitrary dimension. Right at...

With the rapidly growing scales of statistical problems, subset based communicationfree parallel MCMC methods are a promising future for large scale Bayesian analysis. In this article, we propose a new Weierstrass sampler for parallel MCMC based on independent subsets. The new sampler approximates the full data posterior samples via combining the posterior draws from independent subset MCMC cha...