The product-limit estimator and the bootstrap: Some asymptotic representations

Summary The product-limit estimator and its quantile process are represented as i.i.d. mean processes, with a remainder of ordern ^–3/4(logn)^3/4 a.s. Corresponding bootstrap versions of these representations are given, which can help one visualize how the bootstrap procedure operates in this set up.Research supported by NSF grants MCS-81-02341 and MCS 83-01082     

Cutting and stacking,interval exchanges and geometric models

Every aperiodic measure-preserving transformation can be obtained by a cutting and stacking construction. It follows that all such transformations are infinite interval exchanges. This in turn is used to represent any ergodic measure-preserving flow as aC ^∞-flow on an open 2-manifold. Several additional applications of the basic theorems are also given. Partial support for this work was given by the National Science Foundation under grant number MCS81-07092.     

Global minimum test problem construction

A method is presented for the construction of test problems for which the global minimum point is known.Given a bounded convex polyhedron inR ⁿ , and a selected vertex, a concave quadratic function is constructed which attains its global minimum at the selected vertex. In general, this function will also have many other local minima.This research was supported in part by NSF Grant MCS 81-01214.     

Pseudo-free energies and large deviations for non gibbsian FKG measures

Summary A large deviation theorem for the invariant measures of translation invariant attractive interacting particle systems on {0, 1{ ^{Z d} is proven. In this way a pseudo-free energy and pressure is defined. For ergodic systems the large deviations property holds with the usual scaling. The case of non ergodic systems is also discussed. A similar result holds for occupation times. The perturbation by an external field is treated.Work partially supported by NSF-DMR81-14726 (USA) and CNPq (Brazil)Also Department of Physics     

On triangulations of the convex hull ofn points

A setS ofn points in Euclideand-space determines a convex hull which can be triangulated into some numberm of simplices using the points ofS as vertices. We characterize those setsS for which all triangulations minimizem. This is used to characterize sets of points maximizing the volume of the smallest non-trivial simplex. This work was supported in part by NSF Grants MCS 81-02519 and MCS 82-03347. This work supported in part by NSF Grants MCS 81-02519 and MCS 82-03347 Dedicated to Paul Erdős on his seventieth birthday     

A note on a theorem of Yamamuro

Conditions are presented which are necessary for the existence of a regular fixed point of aC ¹ map.Work supported in part by NSF Grant No. MCS 77-15509.Work supported in part by ONR Grant No. N0014-75-C-0495 and NSF Grant No. Eng. 76-81058.     

Horocycle flows are loosely Bernoulli

It is proved that horocycle flows associated with transitiveC ²-Anosov flows are loosely Bernoulli with respect to their unique ergodic measures. Partially supported by the Sloan Foundation and NSF Grant MCS74-19388.     

The Arveson Extension Theorem and coanalytic models

We develop techniques which allow one to describe in simple terms the set of operators on Hilbert space of the form M^{* ()} |M, where M is multiplication by z on a Hilbert space of analytic functions satisfying certain technical assumptions, M^{* ()} is the direct sum of a countably infinite number of copies of M^*, andM is invariant for M^{* ()}. One of the main ingredients in our technique is the Arveson Extension Theorem and this paper illustrates the great power and tractability of that theorem in a concrete setting.Research partially supported by NSF grant MCS 81-02518     

A converse to the P. A. Smith theorem for nonunitary homology spheres

If p is an odd prime and F is the fixed point set of a smooth Z_p action on Sⁿ or Dⁿ, then F is a smooth manifold with a unitary structure. Conversely; most Z_p homology disks or spheres with unitary structures are fixed point sets of smooth Z_p actions on Dⁿ or Sⁿ for suitable n. The results of this paper show that an arbitrary oriented mod p homology disk or sphere is the fixed point set of a smooth Z_p action on some Z[l/2]-homology disk or sphere. This result is in general the best possible.Partially supported by NSF Grants MCS 81-04852 and MCS 83-00669     

Bernoulli maps of the interval

The ergodic properties of expanding piecewiseC ² maps of the interval are studied. It is shown that such a map is Bernoulli if it is weak-mixing. Conditions are given that imply weak-mixing (and hence Bernoulliness). Partially supported by NSF Grant MCS74-19388 and the Sloan Foundation.     

Large deviations for Gibbs random fields

A large deviation principle for Gibbs random fields on Z^d is proven and a corresponding large deviations proof of the Gibbs variational formula is given. A generalization of the Lanford theory of large deviations is also obtained.This work was partially supported by NSF-DMR81-14726     

An upper bound for the cardinality of ans-distance subset in real euclidean space,II

It is shown that ifX is ans-distance subset inR ^d , then |X|≦( _s ^d+s ). Supported in part by NSF grant MCS7903128 A01. Supported in part by NSF grant MCS.     

Asymptotics for Euclidean minimal spanning trees on random points

Summary Asymptotic results for the Euclidean minimal spanning tree onn random vertices inR ^d can be obtained from consideration of a limiting infinite forest whose vertices form a Poisson process in allR ^d. In particular we prove a conjecture of Robert Bland: the sum of thed'th powers of the edge-lengths of the minimal spanning tree of a random sample ofn points from the uniform distribution in the unit cube ofR ^d tends to a constant asn.Whether the limit forest is in fact a single tree is a hard open problem, relating to continuum percolation.Research supported by N.S.F. Grants MCS87-11426 and MCS 90-01710Research supported in part by N.S.F. Grant DMS88-12868, A.F.O.S.R. Grant 89-0301, ARO Grant DAAL03-89-G-0092 and NSA Grant MDA-904-H-2034     

On the central limit theorem in R p when p→∞

Summary Let X ₁ , X ₂ , ..., X _n be i.i.d. random vectors in R ^p where p tends to infinity. A theorem is presented showing that the Central Limit Theorem should hold if p ²/n tends to zero. Furthermore, an example is presented with X _i having a mixed multivariate normal distribution (with finite moment generating function) for which a uniform normal approximation to the distribution of the sample mean can not hold if p ²/n does not tend to zero. Research supported in part by National Science Foundation Grants MCS 80-02340, MCS 83-01834, and DMS 85-03785     

Graph theoretic techniques in algebraic geometry II: construction of singular complex surfaces of the rational cohomology type of CP2

Methods of graph theory are used to obtain rational projective surfaces with only rational double points as singularities and with rational cohomology rings isomorphic to that of the complex projective plane. Uniqueness results for such cohomologyCP ²'s and for rational and integral homologyCP ²'s are given in terms of the typesA _k,D _k, orE _k of singularities allowed by the construction. Supported in part by National Science Foundation grant no. MCS 77-03540.     

A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces

The following theorem is proven:if E is a uniformly rotund Banach space with a Fréchet differentiable norm, C is a bounded nonempty closed convex subset of E, and T: C→C is a contraction, then the iterates {T ⁿx} are weakly almost-convergent to a fixed-point of T. Supported by NSF Grant MCS 76-08217.     

r-Maximal major subsets

The question of which r.e. setsA possess major subsetsB which are alsor-maximal inA (A⊂_rm B) arose in attempts to extend Lachlan’s decision procedure for the αε-theory of ℰ^*, the lattice of r.e. sets modulo finite sets, and Soare’s theorem thatA andB are automorphic if their lattice of supersets ℒ^*(A) and ℒ^*(B) are isomorphic finite Boolean algebras. We characterize the r.e. setsA with someB⊂_rm A as those with a Δ₃ function that for each recursiveR _i specifiesR _i or as infinite on and to be preferred in the construction ofB. There are r.e.A andB with ℒ^*(A) and ℒ^*(B) isomorphic to the atomless Boolean algebra such thatA has anrm subset andB does not. Thus 〈ℰ^*,A〉 and 〈ℰ^*,B〉 are not even elementarily equivalent. In every non-zero r.e. degree there are r.e. sets with and withoutrm subsets. However the classF of degrees of simple sets with norm subsets satisfies . The authors were partially supported by NSF Grants MCS 76-07258, MCS 77-04013 and MCS 77-01965 respectively.     

Equivariant Surgery Theory: Deleting-Inserting Theorems of Fixed Point Manifolds on Spheres and Disks

The paper gives a tool to delete and insert fixed point manifolds for smooth actions of finite Oliver groups on spheres and disks. A similar result was already given in a joint article with E. Laitinen and K. Pawaowski for those of finite nonsolvable groups on spheres. It is useful in classifying smooth actions on spheres from the view point of fixed point data. The methods employed in the present paper are equivariant surgery and equivariant connected sum associated with elements in the Burnside ring. The idea of killing surgery obstructions is as follows: Let G be a finite group not of prime power order, C a contractible, finiteG CW complex, and an element in a K theoretic group arising as an obstruction class of geometric object f. It often holds that (1-[C])^m becomes trivial for large integers m where [C] is the element represented byC in the Burnside ring (G). One expects that the algebraic object (1 - [C])^m is realizable as the obstruction class of G connected sum of f's related to (1 - [C])^m. Since it is true for the case here, we can kill the obstruction by taking G connected sum off's     

Peak points for algebras on circled sets

Summary LetK be a compact, connected, circled subset of ⁿ . It is shown that each nonpeak point forH(K) lies on an analytic disc in the rational hull ofK. The peak points forH(K) that are disjoint from the coordinate axes are characterized in terms of the logarithmically extreme points of |K|.Supported in part by an NSF Grant No. MCS 77-02213     

A Large Deviation Principle for Stochastic Integrals

Assuming that {(X _n ,Y _n )} satisfies the large deviation principle with good rate function I ^♯ , conditions are given under which the sequence of triples {(X _n ,Y _n ,X _n ⋅Y _n )} satisfies the large deviation principle. An ε-approximation to the stochastic integral is proven to be almost compact. As is well known from the contraction principle, we can derive the large deviation principle when applying continuous functions to sequences that satisfy the large deviation principle; the method showed here skips the contraction principle, uses almost compactness and can be used to derive a generalization of the work of Dembo and Zeitouni on exponential approximations. An application of the main result to stochastic differential equations is given, namely, a Freidlin-Wentzell theorem is obtained for a sequence of solutions of SDE’s.