Dual banach lattices and Banach lattices with the Radon-Nikodym property

We construct a separable dual Banach latticeE such that no non-trivial order interval of its dual is weakly compact. HenceE has the Radon-Nikodym property without being in some sense a dual in a natural way. The final draft of this paper was written while the author held a grant from NATO to visit the Ohio State University.     

Weakly self-injective semilattices

In this note we characterize weakly self-injective semilattices as Brouwerian semilattices which are compact in the residuated interval topology. We also characterize weakly self-injective semigroups which are semilattices of groups with trivial multiplication. Research of first author supported in part by a research grant from the Faculty Research Committee of Bowling Green State University. Research of second author supported in part by a postdoctoral fellowship in the Biomathematics Program at North Carolina State University under PHS Grant #GM-678 from NIGMS.     

A foliated metric rigidity theorem for higher rank irreducible symmetric spaces

LetF be a foliation of a compact manifold with a transverse invariant measure of finite total mass. We prove that ifF admits a leafwise metric such that every leaf is an irreducible symmetric space of noncompact type and higher rank, then any other leafwise metric of nonpositive curvature is also symmetric along any leaf in the support of the transverse measure. A rank one version of this result is also exposed.The second author is partially supported, by a Seed Grant from The Ohio State University.     

A new countably determined Banach space

We construct a Banach space which is weak^*-countably determined in its second dual, but which is notK-analytic for its weak topology. This paper was written while the author was visiting The Ohio State University.     

Mixing automorphisms of compact groups and a theorem of Schlickewei

Summary We prove that every mixing ^d by automorphisms of a compact, connected, abelian group is mixing of all orders.Oblatum 5-II-1992The second author gratefully acknowledges support from NSF grant DMS-91-03056 at the Ohio State University     

Optimal control of a class of strongly nonlinear parabolic systems

This paper considers some typical optimal control problems for a class of strongly nonlinear parabolic systems. After some necessary preparation, it is shown that the family of admissible trajectories is a weakly closed and weakly sequentially compact subset of a reflexive Banach space and that the set of attainable states at any given time is a weakly compact subset of a Hilbert space. Using these basic results, proofs of existence of optimal controls are presented. A terminal control problem, a special Bolza problem, and a time optimal control problem are solved, and the necessary conditions of optimality for the corresponding control problems are given.     

The Law of the Iterated Logarithm for Weakly Dependent Hilbert Space Valued Random Variables

We consider the bounded and compact laws of the iterated logarithm for weakly dependent Hilbert space valued random variables. Under optimal moment conditions, we prove the bounded and compact laws of the iterated logarithm for sequences of identically distributed Hilbert space valued random variables satisfying the uniform strong mixing condition.     

Remarks on convex cones

We point out in this note that the class of cones in a locally convex topological vector space satisfying property () or piecewise relatively weakly compact cones is exactly the class of cones admitting weakly compact bases or the class of cones whose closures admit weakly compact bases.This work was supported by a Monash University Postdoctoral Fellowship.     

On the banach spaces with the property (V *) of Pelczynski. — II

Summary We present a converse of a result contained in our paper «On the Banach spaces with the property (V^*) of Pelczynski» so obtaining a characterization of that family of Banach spaces. Hence some extensions of other results from that note are presented. At the end we study property (V^*) in spaces of operators with compact range.The results of this paper were obtained during the permanence of the author at Kent State University, Kent, Ohio 44242, U.S.A. thanks to a fellowship of the Italian C.N.R.     

Existence theorems in vector optimization

In this paper, existence theorems for minimal, weakly minimal, and properly minimal elements of a subset of a partially-ordered, real linear space are presented.This paper was written when the author was a visitor at the Department of Mathematics, North Carolina State University, Raleigh, North Carolina.     

On parametric optimization in hilbert spaces

We consider parametric optimization problems of the following type; Minimize p(x,t) subject to x?C, for all t?T, where C is a weakly locally compact, closed, and convex subset of a Hilbert space H, T is a topological space, and p is a real-valued function on H×T lower semicontinuous if H is provided with the weak topology, and continuous in the second variable for all x?H. We investigate how the optimal value, and the set of optimal solutions depend on tET, We apply our results to show that the discrete eigenvalues, and the corresponding normed eigenelements of certain self-adjoint operators in Hilbert spaces depend (the latter strongly) continuous on parameters under reasonable conditions.     

THE FINITE DIMENSION BEHAVIOR OF THE SCHRDINGER EQUATION WITH NONLOCAL INTEGRAL NONLINEARITY

1.IntroductionInthispaperweconsiderthefollowinginitialproblem,lUt box OIUlpU 7U["III'dX IkU~gi(1.l)j--colul'dx iku~g,(1.l)nit=0~u',(1.2)whereP,7andkarerealnumber,k>0,theexponentpsatisfiesPZ0,ifP50,05P<4,ifP>0.(l.3)ForthenonlinearSchrodingerequation.lat aam P(lul')u ikm=g,(1.4)itslongtimebehaviorhasbeenobservedbynumericalmethodandphysicalargUmefits,(1)chaoticattractorexists;(2)afinitedimensional"space"confinestheattractors(see[l,2]),ajndithasalsobeenprovedbymathematicalmethodthatthereex…     

Bases of convex cones and Borwein's proper efficiency

In this note, we establish some interesting relationships between the existence of Borwein's proper efficient points and the existence of bases for convex ordering cones in normed linear spaces. We show that, if the closed unit ball in a smooth normed space ordered by a convex cone possesses a proper efficient point in the sense of Borwein, then the ordering cone is based. In particular, a convex ordering cone in a reflexive space is based if the closed unit ball possesses a proper efficient point. Conversely, we show that, in any ordered normed space, if the ordering cone has a base, then every weakly compact set possesses a proper efficient point.The research was conducted while the author was working on his PhD Degree under the supervision of Professor J. M. Borwein, whose guidance and valuable suggestions are gratefully appreciated. The author would like to thank two anonymous referees for their constructive comments and suggestions. This research was supported by an NSERC grant and a Mount Saint Vincent University Research Grant.     

Primariness of spaces of continuous functions on ordinals

Necessary and sufficient conditions on an ordinal α are given, such thatC(α) is primary, and that the general linear group ofC(α) is contractible. In particularC(α) possesses both of these properties if α is countable. This is a part of the first author's Ph. D. dissertation, prepared at The Ohio State University under the supervision of Professor W. B. Johnson. The first author was supported by a University Fellowship of The Ohio State University. This author's research was partially supported by NSF Grant MPS-74-2449.     

A viability theorem for set-valued states in a Hilbert space

Applications in robust control problems and shape evolution motivate the mathematical interest in control problems whose states are compact (possibly non-convex) sets rather than vectors. This leads to evolutions in a basic set which can be supplied with a metric (like the well-established Pompeiu–Hausdorff distance), but it does not have an obvious linear structure. This article extends differential inclusions with state constraints to compact-valued states in a separable Hilbert space H. The focus is on sufficient conditions such that a given constraint set (of compact subsets) is viable a.k.a. weakly invariant. Our main result extends the tangential criterion in the well-known viability theorem (usually for differential inclusions in a vector space) to the metric space of non-empty compact subsets of H.     

Operators that are points of spectral continuity

In this paper a characterization is obtained of those bounded operators on a Hilbert space at which the spectrum is continuous, where the spectrum is considered as a function whose domain is the set of all operators with the norm topology and whose range is the set of compact subsets of the plane with the Hausdorff metric. Similar characterizations of the points of continuity of the Weyl spectrum, the spectral radius, and the essential spectral radius are also obtained.The first author was supported by National Science Foundation Grant MCS 77-28396.     

The bernoulli property for expansive ℤ2 actions on compact groups

We show that an expansive ℤ² action on a compact abelian group is measurably isomorphic to a two-dimensional Bernoulli shift if and only if it has completely positive entropy. The proof uses the algebraic structure of such actions described by Kitchens and Schmidt and an algebraic characterisation of theK property due to Lind, Schmidt and the author. As a corollary, we note that an expansive ℤ² action on a compact abelian group is measurably isomorphic to a Bernoulli shift relative to the Pinsker algebra. A further corollary applies an argument of Lind to show that an expansiveK action of ℤ² on a compact abelian group is exponentially recurrent. Finally an example is given of measurable isomorphism without topological conjugacy for ℤ² actions. Supported in part by N.S.F. grant No. DMS-88-02593 at the University of Maryland and by N.S.F. grant No. DMS-91-03056 at Ohio State University.     

Subspaces of sequential spaces

It is proved that the space of continuous functions on the ordinary closed interval with the topology of pointwise convergence is not subsequential. In sequential spaces satisfying certain conditions, subspaces dense-in-themselves without convergent sequences are found; such subspaces are constructed in certain sequential compact spaces and semitopological groups. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 407–413, September, 1998. The author thanks the participants of Arkhangel'skii's seminar held at Moscow State University for useful discussions of this work.     

Quotients ofC[0, 1] with separable dual

A necessary and sufficient condition for an operator fromC(K),K compact metric, into a Banach spaceX to be an isomorphism on a subspace ofC(K) isometric toC ₀(ω ^ω ) is given. This is part of the author’s Ph.D. dissertation being prepared at the Ohio State University under the supervision of Professor W. B. Johnson. This research was supported in part by NSF grant MPS 72-04634-A03 and a University Fellowship of the Ohio State University.     

Eberlein mulian定理在局部凸空间中的推广

设（Ｅ，Ｔ）是一个分离的局部凸空间．本文主要结果是：（１）如果（Ｅ，Ｔ）是次可分的，那么ＡＥ是相对弱紧集Ａ是相对弱可数紧集Ａ是相对弱序列紧集．（２）如果（Ｅ，Ｔ）是强次可分且（Ｅ，β（Ｅ，Ｅ））是桶型空间，那么ＡＥ是相对弱紧集Ａ是相对弱可数紧集Ａ是相对弱序列紧集．