Search

We identify the maximal real interval on which $\sin_{p,n}$ is real-analytic for any real number $p>1$ and any integer $n>1$. We achieve this by first proving that $\sin_{p,n}$ is analytic at $(1/2)π_{p,n}$ iff $p=m/(m-1)$ for some integer $m>1$, in which case we determine the radius of convergence of the Taylor series at $(1/2)π_{p,n}$., 7 pages

Linear functions are arguably the most mundane among all functions. However, the basic fact that a multi-variable linear function has a constant gradient field can provide simple geometric insights into several familiar results such as the Cauchy-Schwarz inequality, the GM-AM inequality, and some distance formulae, as we shall show.

Via a unified geometric approach, certain generalized trigonometric functions with two parameters are analytically extended to maximal domains on which they are univalent. Some consequences are deduced concerning radius of convergence for the Maclaurin series, commutation with rotation, continuation beyond the domain of univalence, and periodicity.

Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of a harmonic function along the gradient flow is determined by the mean curvature of the level sets that the flow intersects.

This note establishes several integral identities relating certain metric properties of level hypersurfaces of Morse functions.

8. M. Queff?lec, Transcendance des fractions continues de Thue-Morse, /. Number Theory 73 (1998) 201? 211. 9. -, Irrational numbers with automaton-generated continued fraction expansion, in Dynamical sys tems (Luminy-Marseille, 1998), World Scientific, River Edge, NJ, 2000, pp. 190-198. 10. W. M. Schmidt, On simultaneous approximations of two algebraic numbers by rationals, Acta Math. 119 (1967)27-50. 11. J. O. Shallit, Real numbers with bounded partial quotients, Enseign. Math. 38 (1992) 151-187. 12. A. Thue, ?ber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Norske vid. Selsk. Skr. Mat. Nat. Kl. 1 (1912) 1-67; reprinted in Selected Mathematical Papers of Axel Thue, T. Nagell, ed., Univer sitetsforlaget, Oslo, 1977, pp. 413-478. 13. M. Waldschmidt, Un demi-si?cle de transcendance, in Development of Mathematics 1950-2000, Birkh?user, Basel, 2000, pp. 1121-1186.

Given a smooth closed S 1 -manifold M, this article studies the extent to which certain numbers of the form (f* (x) P · C) [M] are determined by the fixed-point set M S1 , where f: M → K(π 1 (M), 1) classifies the universal cover of M, x ∈ H* (π 1 (M); Q), P is a polynomial in the Pontrjagin classes of M, and C is in the subalgebra of H* (M; Q) generated by H 2 (M;Q). When M S1 = O, various vanishing theorems follow, giving obstructions to certain fixed-point-free actions. For example, if a fixed-point-free S 1 -action extends to an action by some semisimple compact Lie group G, then (f(x) · P · C) [M] = 0. Similar vanishing results are obtained for spin manifolds admitting certain S 1 -actions.