论文标题
比较指数及其应用的环状总和
Cyclic sums of comparative indices and their applications
论文作者
论文摘要
在本文中，我们概括了一对拉格朗日子空间的比较指数的概念，该子空间具有基本应用在振荡理论中，并构成了线性差异系统和线性差异化的哈密顿系统。我们介绍了环状和$μ_c^{\ pm}（y_1，y_2，\ dots，y_m），\，\，m \ ge 2 $的比较指数的$ n- $ dimensional lagrangian子空间。我们尤其是循环总和的主要特性，尤其是我们指出了循环总和与kashiwara索引的连接。本文的主要结果将比较指数的环状和负征值的数量与$ Mn \ times mn $ s $对称矩阵的$ y_i^t \，j \，j \，j \，y_j，$ $ i，$ $ i，j = 1，j = 1，m。将其主要解决方案的焦点数与对称矩阵的负和正惯性连接起来的离散符号系统。
In this paper we generalize the notion of the comparative index for the pair of Lagrangian subspaces which has fundamental applications in oscillation theory of symplectic difference systems and linear differential Hamiltonian systems. We introduce cyclic sums $μ_c^{\pm}(Y_1,Y_2,\dots,Y_m),\,m\ge 2$ of the comparative indices for the set of $n-$ dimensional Lagrangian subspaces. We formulate and prove main properties of the cyclic sums, in particular, we state connections of the cyclic sums with the Kashiwara index. The main results of the paper connect the cyclic sums of the comparative indices with the number of positive and negative eigenvalues of $mn\times mn$ symmetric matrices defined in terms of the Wronskians $Y_i^T \,J\, Y_j,$ $i,j=1,\dots,m.$ We also present first applications of the cyclic sums of the comparative indices in the oscillation theory of the discrete symplectic systems connecting the number of focal points of their principal solutions with the negative and positive inertia of symmetric matrices.
