Restrictions of Quadratic Forms to Lagrangian Planes,Quadratic Matrix Equations,and Gyroscopic Stabilization

We discuss the symplectic geometry of linear Hamiltonian systems with nondegenerate Hamiltonians. These systems can be reduced to linear second-order differential equations characteristic of linear oscillation theory. This reduction is related to the problem on the signatures of restrictions of quadratic forms to Lagrangian planes. We study vortex symplectic planes invariant with respect to linear Hamiltonian systems. These planes are determined by the solutions of quadratic matrix equations of a special form. New conditions for gyroscopic stabilization are found.     

Cochran's statistical theorem for outer inverses of matrices and matrix quadratic forms

We extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix. As applications, we investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures.     

An inner product lemma

Given the operator product BA in which both A and B are symmetric positive‐definite operators, for which symmetric positive‐definite operators C is BA symmetric positive‐definite in the C inner product 〈x, y〉C? This question arises naturally in preconditioned iterative solution methods, and will be answered completely here. Copyright © 2004 John Wiley & Sons, Ltd.     

Pointwise stabilization of strings with non‐linear feedback

We consider an initial and boundary value problem for a homogenous string subject to an internal pointwise control. The solution resulting from a non‐linear feedback is studied. We give various explicit decay estimates depending on the control position and the feedback non‐linearity. Copyright © 2004 John Wiley & Sons, Ltd.     

The π-π-theorem for manifold pairs with boundaries

The surgery obstruction of a normal map to a simple Poincaré pair (X, Y) lies in the relative surgery obstruction group L _*(π ₁(Y) → π ₁(X)). A well-known result of Wall, the so-called π-π-theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincaré pair with π ₁(X) ? π ₁(Y) is normally bordant to a simple homotopy equivalence of pairs. In order to study normal maps to a manifold with a submanifold, Wall introduced the surgery obstruction groups LP _* for manifold pairs and splitting obstruction groups LS _*. In the present paper, we formulate and prove for manifold pairs with boundary results similar to the π-π-theorem. We give direct geometric proofs, which are based on the original statements of Wall’s results and apply obtained results to investigate surgery on filtered manifolds.     

纵横扩张的优化

得到了最小折数纵横扩张的判别准则．针对4-正则图的平面嵌入的纵横扩张的特殊性，给出了它的最小折数纵横扩张判别准则．     

关于算子方程AX=XAX与AX=XA=XAX

用A的不变子空间作参数，给出了算子方程AX＝XAX的全部解。当A是单射或稠值域时，或者当A是正规算子时，给出了算子方程AX＝XA＝XAX的全部解。我们还给出正规算子X是算子方程AX＝XZ＝XAX的解的充分必要条件。     

Conditional expectation of integrands and random sets

The conditional expectation of integrands and random sets is the main tool of stochastic optimization. This work wishes to make up for the lack of real synthesis about this subject. We improve the existing hypothesis and simplify the corresponding proofs. In the convex case we especially study the problem of the exchange of conditional expectation and subdifferential operators.