Large conjugacy classes,projective Fraïssé limits and the pseudo-arc

We show that the automorphism group, Aut(?), of a projective Fraïssé limit ?, whose natural quotient is the pseudo-arc, has a comeager conjugacy class. This generalizes an unpublished result of Izhar Oppenheim that Aut(?) (and consequently, the group of all homeomorphisms of the pseudo-arc) has a dense conjugacy class. We also present a simple proof of the result of Oppenheim.     

Hausdorff dimension of some kind of alternating Oppenheim series expansion

For Oppenheim series epansions, the authors of [7] discussed the exceptional sets Bm={x∈（0,1]：1〈dj（x）/h（j-1）（d（j-1）（x））≤m for any j ≥2} In this paper, we investigate the Hausdorff dimension of a kind of exceptional sets occurring in alternating Oppenheim series expansion. As an application, we get the exact Hausdorff dimension of the-set in Luroth series expansion, also we give an estimate of such dimensional number.     

More results on Schur complements in Euclidean Jordan algebras

In a recent article Gowda and Sznajder (Linear Algebra Appl 432:1553–1559, 2010) studied the concept of Schur complement in Euclidean Jordan algebras and described Schur determinantal and Haynsworth inertia formulas. In this article, we establish some more results on the Schur complement. Specifically, we prove, in the setting of Euclidean Jordan algebras, an analogue of the Crabtree-Haynsworth quotient formula and show that any Schur complement of a strictly diagonally dominant element is strictly diagonally dominant. We also introduce the concept of Schur product of a real symmetric matrix and an element of a Euclidean Jordan algebra when its Peirce decomposition with respect to a Jordan frame is given. An Oppenheim type inequality is proved in this setting.     

On Exact Laws of Large Numbers for Oppenheim Expansions with Infinite Mean

Journal of Theoretical Probability - In this work, we investigate the asymptotic behaviour of weighted partial sums of a particular class of random variables related to Oppenheim series expansions....     

A quantitative Oppenheim theorem for generic diagonal quadratic forms

We establish a quantitative version of Oppenheim’s conjecture for one-parameter families of ternary indefinite quadratic forms using an analytic number-theory approach. The statements come with power gains and in some cases are essentially optimal.     

广义正定矩阵的Oppenheim不等式

建立了PDn类广义正定矩阵的广义Oppenheim不等式,推广了以前的结果.     

Arithmetic and metric properties of Oppenheim continued fraction expansions

We introduce a class of continued fraction expansions called Oppenheim continued fraction (OCF) expansions. Basic properties of these expansions are discussed and metric properties of the digits occurring in the OCF expansions are studied.     

一类Oppenheim展式的例外关系集

本文研究了Oppenheim展式中一类例外关系集的Hausdorff维数,作为其应用,我们得到了Lüroth级数展式中一些集合的Hausdorff维数的确切值,并给出了这些确切值的一个估计式     

Some inequalities involving determinants,eigenvalues, and Schur complements in Euclidean Jordan algebras

In this paper, using Schur complements, we prove various inequalities in Euclidean Jordan algebras. Specifically, we study analogues of the inequalities of Fischer, Hadamard, Bergstrom, Oppenheim, and other inequalities related to determinants, eigenvalues, and Schur complements.     

Prehomogeneous vector spaces and ergodic theory II

We apply M. Ratner's theorem on closures of unipotent orbits to the study of three families of prehomogeneous vector spaces. As a result, we prove analogues of the Oppenheim Conjecture for simultaneous approximation by values of certain alternating bilinear forms in an even number of variables and certain alternating trilinear forms in six and seven variables.

     

Schur’s theorem for a block Hadamard product

We define a generalized Kronecker product for block matrices, mention some of its properties, and apply it to the study of a block Hadamard product of positive semidefinite matrices, which was defined by Horn, Mathias, and Nakamura. Under strong commutation assumptions we obtain generalizations of Schur’s theorem and of Oppenheim’s inequality.     

New lower bound of the determinant for Hadamard product on some totally nonnegative matrices

Applying the properties of Hadamard core for totally nonnegative matrices, we give new lower bounds of the determinant for Hadamard product about matrices in Hadamard core and totally nonnegative matrices, the results improve Oppenheim inequality for tridiagonal oscillating matrices obtained by T. L. Markham.     

Local unimodularity of matrix-vector pairs

The main theorem establishes a close relationship between the seemingly separate concepts of balancedness and total unimodularity of {0,1} matrices. The result involves classes of Eulerian matrices, and it is presented using terms “local unimodularity” and “local total unimodularity” recently proposed by Hoffman and Oppenheim.     

Approximation orders of formal Laurent series by Oppenheim rational functions

We study formal Laurent series which are better approximated by their Oppenheim convergents. We calculate the Hausdorff dimensions of sets of Laurent series which have given polynomial or exponential approximation orders. Such approximations are faster than the approximation of typical Laurent series (with respect to the Haar measure).     

形式Laurent级数域上交错Oppenheim展开的研究

该文介绍了形式Laurent级数域上交错Oppenheim展开的算法,得到了该展开中数字的强(弱)大数定理、中心极限定理和重对数率,并且研究了这些级数部分和的逼近的度.     

Some distribution results of the Oppenheim continued fractions

We study some distribution properties of the Oppenheim continued fraction expansions. A Gauss–Kuzmin–Lévy type theorem is established. Based on this, a Fréchet law concerning the partial maxima of the growth rate of the digit sequence is derived, which extends previous work of Galambos and Philipp on the regular continued fraction expansion. Besides, a uniform distribution modulo 1 result is obtained.     

The equality cases for the inequalities of Oppenheim and Schur for positive semi-definite matrices

In this paper, necessary and sufficient conditions for equality in the inequalities of Oppenheim and Schur for positive semidefinite matrices are investigated. Supported by National Natural Science Foundation of China (No. 10531070), National Basic Research Program of China 973 Program (No. 2006CB805900), National Research Program of China 863 Program (No. 2006AA11Z209) and the Natural Science Foundation of Shanghai (Grant No. 06ZR14049).     

Metric Properties and Exceptional Sets of the Oppenheim Expansions over the Field of Laurent Series

We investigate metric properties of the polynomial digits occurring in a large class of Oppenheim expansions of Laurent series, including Lüroth, Engel, and Sylvester expansions of Laurent series and Cantor infinite products of Laurent series. The obtained results cover those for special cases of Lüroth and Engel expansions obtained by Grabner, A. Knopfmacher, and J. Knopfmacher. Our results applied in the cases of Sylvester expansions and Cantor infinite products are original. We also calculate the Hausdorff dimensions of different exceptional sets on which the above-mentioned metric properties fail to hold.     

Balanced 0, ±1-matrices,bicoloring and total dual integrality

A 0, ±1-matrixA is balanced if, in every submatrix with two nonzero entries per row and column, the sum of the entries is a multiple of four. This definition was introduced by Truemper (1978) and generalizes the notion of a balanced 0, 1-matrix introduced by Berge (1970). In this paper, we extend a bicoloring theorem of Berge (1970) and total dual integrality results of Fulkerson, Hoffman and Oppenheim (1974) to balanced 0, ±1-matrices.This work was supported in part by NSF grants DDM-8800281, ECS-8901495 and DDM-9001705.     

The generalized Oppenheim expansions for the direct product of non-Archimedean fields

The well-known Oppenheim expansion algorithm in the field ℚ _p is generalized to the ring ℚ _g , where g = p ₁ · ... · p _N . The metric properties of the digits of this expansion and also the metric properties of the coefficients of some expansions of polyadic numbers are studied.