无穷维随机微分方程的最优宽松控制

W.H.Flcmng—M.Nisio在文[3]中讨论了n维空间上如下形式随机微分方程的最优宽松控制问题:本文利用A.Bensoussan—M.Nisiso在文[1]中引入的测度的殆紧性,将文[3]中最优宽松控制的存在性定理推广到无穷维情形.     

Indexes and Special Discretization Methods for Linear partial Differential Algebraic Equations

Linear partial differential algebraic equations (PDAEs) of the form Au _t(t, x) + Bu _xx(t, x) + Cu(t, x) = f(t, x) are studied where at least one of the matrices A, B R ^n×n is singular. For these systems we introduce a uniform differential time index and a differential space index. We show that in contrast to problems with regular matrices A and B the initial conditions and/or boundary conditions for problems with singular matrices A and B have to fulfill certain consistency conditions. Furthermore, two numerical methods for solving PDAEs are considered. In two theorems it is shown that there is a strong dependence of the order of convergence on these indexes. We present examples for the calculation of the order of convergence and give results of numerical calculations for several aspects encountered in the numerical solution of PDAEs.     

统计递归集的Hausdorff测度的上、下界

给出了由随机压缩算子生成的统计递归集的Ｈａｕｓｄｏｒｆ测度上、下界的估计．也就是说,在一些条件下,找出了Ｋ（ω）∩∞ｎ＝１∪ｉ∈Ｎｎ（ω）ｆ（ω）ｎ,ｉ（Ｅ）的α－维Ｈａｕｓｄｏｒｆ测度的上、下界,其中｛ｆｎ,ｉ,１≤ｉ,ｎ＜∞｝是一族随机压缩算子,｛Ｎｎ（ω）,ｎ≥１｝是一族可数的随机指标集．     

一阶时滞微分方程解的零点分布

Abstract. The paper gives two estimates of the distance between adjacent zeros of solutions     

Oscillatory Behaviour of Solutions of Two-Dimensional Differential Systems with Deviated Arguments

Sufficient conditions are established for the oscillation of proper solutions of the system

Gaussian estimates and regularized groups

We show that if a bounded analytic semigroup on satisfies a Gaussian estimate of order and is the generator of its consistent semigroup on , then generates a -regularized group on where . We obtain the estimate of () and the -independence of , and give applications to Schrödinger operators and elliptic operators of higher order.

     

Bifurcation of periodic solutions to variational inequalities in R κ based on Alexander-Yorke theorem

Variational inequalities are studied, where K is a closed convex cone in , 3, B is a × matrix, G is a small perturbation, a real parameter. The assumptions guaranteeing a Hopf bifurcation at some ₀ for the corresponding equation are considered and it is proved that then, in some situations, also a bifurcation of periodic solutions to our inequality occurs at some _{I 0}. Bifurcating solutions are obtained by the limiting process along branches of solutions to penalty problems starting at ₀ constructed on the basis of the Alexander-Yorke theorem as global bifurcation branches of a certain enlarged system.     

On the Infinitesimal Rigidity of Homogeneous Varieties

Let XP be a variety (respectively an open subset of an analytic submanifold) and let xX be a point where all integer valued differential invariants are locally constant. We show that if the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Segre P× P, n,m2, a Grassmaniann G(2,n+2), n4, or the Cayley plane OP², then X is the corresponding homogeneous variety (resp. an open subset of the corresponding homogeneous variety). The case of the Segre P²×P² had been conjectured by Griffiths and Harris in [GH]. If the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Veronese v₂(P) and the Fubini cubic form of X at x is zero, then X=v₂ (P) (resp. an open subset of v₂(P)). All these results are valid in the real or complex analytic categories and locally in the C category if one assumes the hypotheses hold in a neighborhood of any point x. As a byproduct, we show that the systems of quadrics I₂(P P) S²C, I₂(P¹× P) S²C and I₂(S₅) S²C¹⁶ are stable in the sense that if A S^* is an analytic family such that for t0,AA, then A₀A. We also make some observations related to the Fulton–:Hansen connectedness theorem.     

On oscillatory solutions of fourth order ordinary differential equations

The paper deals with the oscillation of a differential equation L ₄ y + P(t)L ₂ y + Q(t)y 0 as well as with the structure of its fundamental system of solutions.