Uniform bounds under increment conditions

We apply a majorizing measure theorem of Talagrand to obtain uniform bounds for sums of random variables satisfying increment conditions of the type considered in Gál-Koksma Theorems. We give some applications.

     

Characterization of almost maximally almost-periodic groups

Let G be an Abelian group. We prove that a group G admits a Hausdorff group topology τ such that the von Neumann radical n(G,τ) of (G,τ) is non-trivial and finite iff G has a non-trivial finite subgroup. If G is a topological group, then n(n(G))≠n(G) if and only if n(G) is not dually embedded. In particular, n(n(Z,τ))=n(Z,τ) for any Hausdorff group topology τ on Z.     

Associative Cones and Integrable Systems

Abstract We identify ℝ⁷ as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S⁶. It is known that a cone over a surface M in S⁶ is an associative submanifold of ℝ⁷ if and only if M is almost complex in S⁶. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S⁶ are the equation for primitive maps associated to the 6-symmetric space G₂=T², and use this to explain some of the known results. Moreover, the equation for S¹-symmetric almost complex curves in S⁶ is the periodic Toda lattice, and a discussion of periodic solutions is given. (Dedicated to the memory of Shiing-Shen Chern) * Partially supported by NSF grant DMS-0529756.     

An Improved Result in Almost Sure Central Limit Theory for Products of Partial Sums with Stable Distribution

Consider a sequence of i.i.d. positive random variables with the underlying distribution in the domain of attraction of a stable distribution with an exponent in (1, 2]. A universal result in the almost sure limit theorem for products of partial sums is established. Our results significantly generalize and improve those on the almost sure central limit theory previously obtained by Gonchigdanzan and Rempale and by Gonchigdanzan. In a sense, our results reach the optimal form.     

一道Lebesgue积分题目的注记

设E为n维欧氏空间Rn的可测子集,m E+∞,f(x)为E上的非负可测函数,并记Ek=E{x|k≤f(x)k+1}(k=0,1,2,…),利用Lebesgue积分的性质,通过构造反例,指出"级数∑k km Ek收敛"并不是f(x)在E上Lebesgue可积的充分条件.进一步,通过增加条件"f在E上a.e.有限",得到相应的充分必要条件.     

On almost self-complementary graphs

A graph is called almost self-complementary if it is isomorphic to one of its almost complements X^c-I, where X^c denotes the complement of X and I a perfect matching (1-factor) in X^c. Almost self-complementary circulant graphs were first studied by Dobson and Šajna [Almost self-complementary circulant graphs, Discrete Math. 278 (2004) 23-44]. In this paper we investigate some of the properties and constructions of general almost self-complementary graphs. In particular, we give necessary and sufficient conditions on the order of an almost self-complementary regular graph, and construct infinite families of almost self-complementary regular graphs, almost self-complementary vertex-transitive graphs, and non-cyclically almost self-complementary circulant graphs.     

Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one —IV

This paper is the first of a series of papers in which we generalize our results in (Asian J. of Math. 4, 817–830 (2000); J. Geom. Anal. 12, 63–79 (2002); Intern. J. Math. 14, 259–287 (2003)) to the general complex compact almost homogeneous manifolds of real cohomogeneity one. In this paper we deal with the exceptional case of the G ₂ action (Cf. Intern. J. Math. 14, 259–287 (2003), p. 285). In particular, we prove the existence of Kähler-Einstein metric on this manifold.     

一类非齐次时滞微分方程概周期解的存在性

该文研究一类非齐次时滞微分方程的概周期解, 结合运用指数二分法, 给出了系统存在概周期解的一组充分条件.     

On Weyl almost periodic selections of multivalued maps

We prove that Weyl almost periodic multivalued maps R∋t→F(t)∈clU have Weyl almost periodic selections, where clU is the collection of non-empty closed sets of a complete metric space U.     

Massera type theorem for almost automorphic solutions of functional differential equations of neutral type

In this work, we study the existence of almost automorphic solutions for functional differential equations of neutral type. We prove that the existence of a bounded solution on R⁺ implies the existence of an almost automorphic solution.