Bifurcation in Nirenberg's problem

In this paper, we discuss some bifurcation on Nirenberg's problem. We give some natural conditions such that the standard bifurcation or Thom—Mather theory can be used. Most interestingly, our conditions for the existence of solutions are very near to the Kazdan—Warner obstruction.     

On Bayes' theorem and the inverse Bernoulli theorem

A recent assertion by S. M. Stigler that Thomas Bayes was perhaps anticipated in the discovery of the result that today bears his name is exposed to further scrutiny here. The distinction between Bayes' theorem and the inverse Bernoulli theorem is examined, and pertinent early writings on this matter are discussed. A careful examination of the difference between these two theorems leads to the conclusion that a result given by David Hartley in 1749 is more in line with the inverse Bernoulli theorem than with Bayes' result, and it is suggested that there is not sufficient evidence to remove Bayes from his place as originator of the method adopted.     

An arithmetic proof of John’s ellipsoid theorem

Using an idea of Voronoi in the geometric theory of positive definite quadratic forms, we give a transparent proof of John’s characterization of the unique ellipsoid of maximum volume contained in a convex body. The same idea applies to the ‘hard part’ of a generalization of John’s theorem and shows the difficulties of the corresponding ‘easy part’.Received: 8 October 2004     

On the refinement of Cauchy's theorem and Pellet's theorem

Using the relationship of a polynomial and its associated polynomial, we derived a necessary and sufficient condition for determining all roots of a given polynomial on the circumference of a circle defined by its associated polynomial. By employing the technology of analytic inequality and the theory of distribution of zeros of meromorphic function, we refine two classical results of Cauchy and Pellet about bounds of modules of polynomial zeros. Sufficient conditions are obtained for the polynomial whose Cauchy's bound and Pellet's bounds are strict bounds. The characteristics is given for the polynomial whose Cauchy's bound or Pellet's bounds can be achieved by the modules of zeros of the polynomial.     

On Bloch's theorem

Mathematische Zeitschrift -     

关于Borg定理

Borg定理是判定周期系数二阶线性微分方程稳定的一个重要定理,这个定理在一定意义下是不可改进的.本文利用判别式的一个新形式,在弱的限制下,得到判定周期系数二阶线性微分方程稳定的定理,所得结论改进了Borg定理的判定结果.     

On Bertini's theorem

We study the singularities of a generic element of a linear system of divisors on a smooth projective variety inside the base locus of the linear system, and give a generalization of Bertini's theorem. Partially Supported by NSF grant DMS-9596097     

On Stark's theorem

A new proof is obtained for Gauss' hypothesis on class-one imaginary quadratic fields, based on the bound obtained by A. O. Gel'fond in 1939 for the modulus of linear forms of two logarithms of algebraic numbers.Translated from Matematicheskie Zametki, Vol. 11, No. 3, pp. 329–340, March, 1972.     

On Downing-Kirk's theorem

In this paper, using the notion of τ-distance, we prove a generalization of Downing-Kirk's theorem.     

On the Skitovich-Darmois theorem and Heyde theorem in a Banach space

According to the well-known Skitovich-Darmois theorem, the independence of two linear forms of independent random variables with nonzero coefficients implies that the random variables are Gaussian variables. This result was generalized by Krakowiak for random variables with values in a Banach space in the case where the coefficients of forms are continuous invertible operators. In the first part of the paper, we give a new proof of the Skitovich-Darmois theorem in a Banach space. Heyde proved another characterization theorem similar to the Skitovich-Darmois theorem, in which, instead of the independence of linear forms, it is supposed that the conditional distribution of one linear form is symmetric if the other form is fixed. In the second part of the paper, we prove an analog of the Heyde theorem in a Banach space. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1234–1242, September, 2008.     

On the Glivenko-Cantelli theorem

Summary Various generalizations of the classical Glivenko-Cantelli theorem are proved. In particular, we have strived for as general results as possible for theoretical distributions on euclidean spaces, which are absolutely continuous with respect to Lebesgue measure.