A theorem concerning the joint approximation of real numbers by rationals

In this note we prove a theorem concerning the joint approximation of irrational numbers by rationals.Translated from Matematicheskie Zametki, Vol. 11, No. 3, pp. 313–317, March, 1972.     

A seven-color theorem on the sphere

Ringel has shown that the set of vertices and regions of any normal map on the sphere can be admissibly colored by six colors. In this paper, it is shown that the set of vertices, edges and regions of any normal map on the sphere can be admissibly colored with seven colors.     

A Note on pinching sphere theorem

Let M²ⁿ be a 2n-dimensional compact, simply connected Riemannian manifold without boundary and S²ⁿ be the unit sphere of 2n+1 dimension Euclidean space $\text{R}{}^{\mathrm{2n+1}}$. We prove in this note that if the sectional curvature K_M varies in (0,1] and the volume V(M) is not larger than $\text{(}\text{3}\text{2}\text{+η)V(S}{}^{\mathrm{2n}}\text{)}$ for some positive number η depending only on n, then M²ⁿ is homeomorphic to S²ⁿ. To cite this article: Y. Wen, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A generalization of the classical sphere theorem

In this paper, we prove a sphere theorem for Riemannian manifolds with partially positive curvature which generalizes the classical sphere theorem.

     

A theorem concerning closed polygons

Sunto. Si assegna una condizione necessaria e sufficiente affinchè la somma algebrica delle distanze di un punto dan rette complanari orientate non muti al variare di quello sul piano di queste.      

A counterexample concerning smooth approximation

We answer a question of Smith, Stanoyevitch and Stegenga in the negative by constructing a simply connected planar domain with no two-sided boundary points and for which every point on is an -limit point of and such that is not dense in the Sobolev space .

     

A sphere theorem for the equations of elasticity

Zusammenfassung In dieser Arbeit werden Ausdrücke für das Verschiebungsfeld in einer elastischen Kugel angegeben, deren Oberfläche a) elastisch gelagert, b) eingespannt, c) frei ist. Die Verschiebungen werden durch diejenigen Verschiebungen und Spannungen ausgedrückt, die in einem unendlichen Körper unter derselben Verteilung von Singularitäten entstehen.     

A general poisson approximation theorem

We study the rate of convergence in a limit theorem due to Kabanov-Liptser-Shiryayev. We show how the probabilities P(N _t= k) can be computed from the compensator, when it is deterministic.     

A sphere theorem on locally conformally flat even-dimensional manifolds

In this paper, we prove that a closed even-dimensional manifold which is locally conformally flat with positive scalar curvature, positive Euler characteristic and which satisfies some additional condition on its curvature is diffeomorphic to the sphere or projective space.     

A sphere theorem for non-reversible Finsler metrics

For a non-reversible Finsler metric F on a compact smooth manifold M we introduce the reversibility = max {F(–X)|F(X)=1}1. We prove the following generalization of the classical sphere theorem in Riemannian geometry: A simply-connected and compact Finsler manifold of dimension n3 with reversibility and with flag curvature is homotopy equivalent to the n-sphere.Mathematics Subject Classification (2000):53C60; 53C20, 53C22     

On approximation of functions on sphere

Let f be an integrable function on the unit sphere Σ _n−1 of R ⁿ (n⩾3) and let σ _N ^δ be the Cesàro means of order σ of the Fourier-Laplace series of f. The special value λ:=n−2/2 of σ is known as the critical index. This paper proves that and where ω(f,t)_p is the 1st-order modulus of continuity of f in L^p-metric which is defined in a way different than in the classical case of n=2. In Memory of Professor M. T. Cheng Project supported by the NSF of China under the grans # 19771009.     

On approximation of functions on sphere

Let f be an integrable function on the unit sphere Σ _n−1 of R ⁿ (n⩾3) and let σ _N ^δ be the Cesàro means of order σ of the Fourier-Laplace series of f. The special value λ:=n−2/2 of σ is known as the critical index. This paper proves that      

A theorem on the Markov periodic approximation in ergodic theory

One presents a new variant of the theory of periodic approximations of dynamical systems and C^*-algebras, namely the construction for each automorphism of the Lebesgue space of a Markov tower (or adic model) of periodic automorphisms. One gives several examples.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 72–82, 1982.     

A differentiable sphere theorem on manifolds with reverse volume pinching

Based on the celebrated 1/4-pinching sphere theorem, we prove a differentiable sphere theorem on Riemannian manifolds with reverse volume pinching.