THE INDUCED REPRESENTATION OF C^＊-GROUPOID DYNAMIC SYSTEMS

1.IntroductionandDefinitionsTheinducedrepresentationofgroupC*-algebrasfirstwasalgebraicallyintroducedbyfueffelin[3].P.Greenin[6]similiaxlyintroducedtheoneforcovariantdynamicsystem.TheinducedrepresentationofcrossedproductsbycoactionswasintroducedbyK.Mansfieldin[9].Using[8]wecanobtaintheinducedrepresentationofgrollpoidC*-algebrasin[1Ol.TherichapplicationofthesecanbefoundamongmanypapersrelativetotheC*-analysisofgroupoidandgroup.TheC*-groupoiddynamicsystemanditsreducedcrossedprod-uctwereintro…     

Soap films spanning rectangular prisms

In this paper, we consider soap films spanning rectangular prisms with regular n-gon bases. As the number of edges n varies, we show that there are significant changes in the qualitative properties of the spanning soap films as well as a change in the number of spanning soap films whose existence we can prove: We can find two nontrivial soap films for n = 3, 4, 5 but only one for n ≥ 6. We also prove some results concerning the interval of aspect ratios through which the soap films exist: The interval is finite if n = 3, 4, 5 and infinite if n ≥ 6. Furthermore, for n > 6, we have that the spanning soap film converges to a soap film spanning the vertical lines through the vertices of a regular n-gon as the aspect ratio goes to infinity. We can also make sense of the case n = ∞. Here, we discover some interesting singly and triply periodic soap films spanning singly and doubly periodic sets of vertical lines or spanning singly periodic sets of vertical line segments connected by pairs of parallel, horizontal lines. Finally, for n = 3, 4, 5, 6, we can derive parameterizations for the spanning soap films, and these parameterizations are explicit up to knowing the aspect ratio.      

On a class of stopping times for M-estimators

For a given score function ψ = ψ(x, θ), let θ_n be Huber's M-estimator for an unknown population parameter θ. Under some mild smoothness assumptions it is known that $\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(θ}{}_{\mathrm{n}}\text{? θ)}$ is asymptotically normal. In this paper the stopping times $\text{τ}{}_{\mathrm{c}}\text{(m) =}\text{inf}\text{{n ≥ m: n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{|θ}{}_{\mathrm{n}}\text{? θ | > c }}$ associated with the sequence of confidence intervals for θ are investigated. A useful representation of M-estimators is derived, which is also appropriate for proving laws of the iterated logarithm and Donskertype invariance principles for (π_n)_n.     

Irreducible * representations of realC*-algebras

In this paper, we show that a topologically irreducible * representation of a realC*-algebra is also algebraically irreducible. Moreover, the properties of pure real states on a realC*-algebra and their left kernels are discussed.Partially supported by the National Natural Science Foundation of China.     

Equivariant cobordism of Grassmann and flag manifolds

We consider certain natural (ℤ₂)ⁿ actions on real Grassmann and flag manifolds andS ¹ actions on complex Grassmann manifolds with finite stationary point sets and determine completely which of them bound equivariantly.     

SEMI-LINEAR SYSTEMS OF BACKWARD STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN $R^n$

This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs.The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions.     

Solving the Simple Plant Location Problem using a Data Correcting Approach

The Data Correcting Algorithm is a branch and bound type algorithm in which the data of a given problem instance is `corrected' at each branching in such a way that the new instance will be as close as possible to a polynomially solvable instance and the result satisfies an acceptable accuracy (the difference between optimal and current solution). In this paper the data correcting algorithm is applied to determining exact and approximate optimal solutions to the simple plant location problem. Implementations of the algorithm are based on a pseudo-Boolean representation of the goal function of this problem, and a new reduction rule. We study the efficiency of the data correcting approach using two different bounds, the Khachaturov-Minoux bound and the Erlenkotter bound. We present computational results on several benchmark instances, which confirm the efficiency of the data-correcting approach.     

Problem of invariant subspaces in C*-algebras

Let A be a separable simple C*-algebra. For each a(≠0) in A, there exists a separable faithful and irreducible * representation (π, Hπ) on A such that π(a) has a non-trivial invariant subspace in Hπ.     

关于H．Yao－D．Knuth定理

本文研究一类有限连分数展开的部分商数之平均和。所得结果改进了这方面已有的最好工作。     

r-Matrix Structure for a Restricted Flow with Bargmann Constraint

This paper deals with the integrability of a finite-dimensional Hamiltonian system linked with the generalized coupled KdV hierarchy. For this purpose the associated Lax representation is presented after an elementary calculation. It is shown that the Lax representation enjoys a dynamical r-matrix formula instead of a classical one in the Poisson bracket on R^2N. Consequently the resulting system is proved to be completely integrable in view of its r-matrix structure.