逆差商的一个紧凑行列式表示

本文给出了逆差商的一个紧凑行列式表示,从该表示式易知,一个函数在某ｎ＋１个点的ｎ阶逆差商与这些点的排序有关,但与前ｎ－１个点的局部换序无关．此外,还从另一角度定义倒差商,得出了倒差商与逆差商之间的关系以及倒差商的整体换序无关性．     

GCD和GCUD矩阵行列式的一个不等式（英）

Let S = {x₁, x₂,..., x_n} be a set of distinct positive integers. The n x n matrix (S) whose i, j-entry is the greatest common divisor (x_i, x_j) of x_i and x_j is called the GCD matrix on S. A divisor d of x is said to be a unitary divisor of x if (d, x/d) = 1. The greatest common unitary divisor (GCUD) matrix (S^**) is defined analogously. We show that if S is both GCD-closed and GCUD-closed, then det(S^**) ≥ det(S), where the equality holds if and Only if (S^**) = (S).     

Non-invertibility of certain almost Mathieu operators

It is shown that the almost Mathieu operators of the type where is real and is a rational multiple of and an orthonormal basis for a Hilbert space, is not invertible.

     

复矩阵的亚半正定性

复亚半正定矩阵是 Hermite正定阵的推广 ,研究了它的 Kronecker积、Hadamard积和行列式理论 ,将实对称阵的 Schur定理、华罗庚定理、Minkowski不等式、Ky-Fan不等式、Ostrowski-Taussky不等式推广到了一类非 Hermite复矩阵上 ,扩大了 Minkowski不等式的指数范围 ,削弱了华罗庚不等式的条件 .     

Rational curves on moduli spaces of vector bundles

We identify the spaces Hom_i(ℙ¹,M) fori = 1, 2, whereM is the moduli space of vector bundles of rank 2 and determinant isomorphic to ,x ₀ ∈X, on a compact Riemann surface of genusg ≥ 2.     

M-矩阵和H-矩阵在Fan积下的Oppenheim型不等式

M-矩阵和H-矩阵在数学物理、经济学、数学规划等领域中有广泛的应用.对于一般的M-矩阵,是否成立著名的O ppenheim型不等式,文[1]给出了答案.本文建立了一个M-矩阵和一个H-矩阵在Fan积下的O ppenheim型不等式.     

Determinant preserving maps: an infinite dimensional version of a theorem of Frobenius

In this paper we investigate the structure of maps on classes of Hilbert space operators leaving the determinant of linear combinations invariant. Our main result is an infinite dimensional version of the famous theorem of Frobenius about determinant preserving linear maps on matrix algebras. In this theorem of ours, we use the notion of (Fredholm) determinant of bounded Hilbert space operators which differ from the identity by an element of the trace class. The other result of the paper describes the structure of those transformations on sets of positive semidefinite matrices which preserve the determinant of linear combinations with fixed coefficients.     

Hankel determinants of linear combinations of consecutive Catalan-like numbers

Let ${\left(a_n\right)}_{n\ge 0}$ be a sequence of the Catalan-like numbers. We evaluate Hankel determinants $det{[\lambda a_{i+j}+\mu a_{i+j+1}]}_{0\le i,j\le n}$ and $det{[\lambda a_{i+j+1}+\mu a_{i+j+2}]}_{0\le i,j\le n}$ for arbitrary coefficients $\lambda$ and $\mu$. Our results unify many known results of Hankel determinant evaluations for classic combinatorial counting coefficients, including the Catalan, Motzkin and Schröder numbers.     

全纯曲线的例外超平面∗

文章讨论了到复射影空间P^N (C)的全纯曲线交超平面的问题,借助Vandermonde行列式, 构造了一些具有N+1个例外超平面的非线性退化的全纯曲线和具有2N个例外超平面的线性退化的非常映射全纯曲线,说明了 Nochka 的全纯曲线的第二基本定理是最优的.最后还构造了具有2N个例外值的N值非常数代数体函数.