On the weak Freese–Nation property of ?(ω)

Continuing [6], [8] and [16], we study the consequences of the weak Freese-Nation property of (?(ω),⊆). Under this assumption, we prove that most of the known cardinal invariants including all of those appearing in Cichoń's diagram take the same value as in the corresponding Cohen model. Using this principle we could also strengthen two results of W. Just about cardinal sequences of superatomic Boolean algebras in a Cohen model. These results show that the weak Freese-Nation property of (?(ω),⊆) captures many of the features of Cohen models and hence may be considered as a principle axiomatizing a good portion of the combinatorics available in Cohen models. Received: 7 June 1999 / Revised version: 17 October 1999 /?Published online: 15 June 2001     

Homogeneity and the disjoint arcs property

Some previous results of the author towards a classification of homogeneous metric continua are improved. The disjoint arcs property is fully revealed in this context. In particular, closed -manifolds, , are characterized as those homogeneous continua which do not have the disjoint arcs property.

     

Elliptic genera of toric varieties and applications to mirror symmetry

The paper contains a proof that elliptic genus of a Calabi-Yau manifold is a Jacobi form, finds in which dimensions the elliptic genus is determined by the Hodge numbers and shows that elliptic genera of a Calabi-Yau hypersurface in a toric variety and its mirror coincide up to sign. The proof of the mirror property is based on the extension of elliptic genus to Calabi-Yau hypersurfaces in toric varieties with Gorenstein singularities. Oblatum 12-V-1999 & 4-XI-1999?Published online: 21 February 2000     

Sharp thresholds of graph properties, and the -sat problem

Given a monotone graph property , consider , the probability that a random graph with edge probability will have . The function is the key to understanding the threshold behavior of the property . We show that if is small (corresponding to a non-sharp threshold), then there is a list of graphs of bounded size such that can be approximated by the property of having one of the graphs as a subgraph. One striking consequence of this result is that a coarse threshold for a random graph property can only happen when the value of the critical edge probability is a rational power of .

As an application of the main theorem we settle the question of the existence of a sharp threshold for the satisfiability of a random -CNF formula.

An appendix by Jean Bourgain was added after the first version of this paper was written. In this appendix some of the conjectures raised in this paper are proven, along with more general results.

     

Note on universal conditional consistency

This note clarifies the implication of universal conditional consistency (UCC) in Fudenberg and Levine (Games Econ Behav 29:104–130, 1999) from the viewpoint of the no-regret test. We first show that the UCC property implies (universally) passing any countable set of no-regret tests. Furthermore, we show that the UCC implies universally passing uncountable (particular) no-regret tests so that the UCC theorem and the wide-range no-regret (WRNR) theorem in Lehrer (Games Econ Behav 42:101–115, 2003) may differ, as opposed to the thought that the UCC property is a special case of the WRNR property. Y. Noguchi is very grateful to Drew Fudenberg for his encouragement and helpful comments. He also thank an anonymous referee and Tatsuro Ichiishi for their helpful comments.     

On tightness and depth in superatomic Boolean algebras

We introduce a large cardinal property which is consistent with and show that for every superatomic Boolean algebra and every cardinal with the large cardinal property, if tightness, then depth. This improves a theorem of Dow and Monk.

     

The diagonal subring and the Cohen-Macaulay property of a multigraded ring

Let be a multigraded ring defined over a local ring . This paper deals with the question how the Cohen-Macaulay property of is related to that of its diagonal subring . In the bigraded case we are able to give necessary and sufficient conditions for the Cohen-Macaulayness of . If are ideals of positive height, we can then compare the Cohen-Macaulay property of the multi-Rees algebra with the Cohen-Macaulay property of the usual Rees algebra . We also obtain a bound for the joint reduction numbers of two -primary ideals in the case the corresponding multi-Rees algebra is Cohen-Macaulay.

     

On Nehari disks and the inner radius

Let D be a simply connected plane domain and B the unit disk. The inner radius of D, , is defined by . Here S _f is the Schwarzian derivative of f, the hyperbolic density on D and . Domains for which the value of is known include disks, angular sectors and regular polygons, as well as certain classes of rectangles and equiangular hexagons. All of the mentioned domains except non-convex angular sectors have an interesting property in common, namely that , where h maps B conformally onto D. Because of the importance of this property for computing , we say that D is a Nehari disk if holds.?This paper is devoted to the problem of characterizing Nehari disks. We give a necessary and sufficient condition for a domain to be a Nehari disk provided it is a regulated domain with convex corners. Received: March 7, 1996; revised version: July 15, 1999     

A class of

Extending an example given by T. Kriete, we develop a class of measures each of which consists of a measure on along with a series of weighted point masses in . This class provides relatively simple examples of measures which have the property that the collection of analytic bounded point evaluations for varies with . The first known measures with this property were recently constructed by J. Thomson.

     

A discrete cost sharing model with technological cooperation

This article proposes a setting that allows for technological cooperation in the cost sharing model. Dealing with discrete demands, we study two properties: additivity and dummy. We show that these properties are insufficient to guarantee a unit-flow representation similar to that of Wang (Econ Lett 64:187–192, 1999). To obtain a characterization of unit flows, we strengthen the dummy axiom and introduce a property that requires the cost share of every agent to be non-decreasing in the incremental costs generated by their demand. Finally, a fairness requirement as to the compensation of technological cooperation is examined.     

Prime ideals on P ω (λ) with the partition property

 We use ideas of Fred Galvin to show that under Martin's axiom, there is a prime ideal on P _ω (λ) with the partition property for every . Received: 6 May 1999 / Revised version: 15 October 2001 Published online: 2 September 2002     

Upper Lipschitz behavior of solutions to perturbed. C1,1 programs

We analyze the local upper Lipschitz behavior of critical points, stationary solutions and local minimizers to parametric C ^1,1 programs. In particular, we derive a characterization of this property for the stationary solution set map without assuming the Mangasarian–Fromovitz CQ. Moreover, conditions which also ensure the persistence of solvability are given, and the special case of linear constraints is handled. The present paper takes pattern from [21] by continuing the approach via contingent derivatives of the Kojima function associated with the given optimization problem. Received: June 10, 1999 / Accepted: November 15, 1999?Published online July 20, 2000     

Modularity prevents tails

We establish a direct correspondence between two congruence properties for finite algebras. The first property is that minimal sets of type have empty tails. The second property is that congruence lattices omit pentagons of type .

     

Shelah's stability spectrum and homogeneity spectrum in finite diagrams

We present Saharon Shelah's Stability Spectrum and Homogeneity Spectrum theorems, as well as the equivalence between the order property and instability in the framework of Finite Diagrams. Finite Diagrams is a context which generalizes the first order case. Localized versions of these theorems are presented. Our presentation is based on several papers; the point of view is contemporary and some of the proofs are new. The treatment of local stability in Finite Diagrams is new. Received: 12 November 1999 / Revised version: 21 July 2000 / Published online: 3 October 2001     

Compressions of free products of von Neumann algebras

A reduction formula for compressions of von Neumann algebra II–factors arising as free products is proved. This shows that the fundamental group is for some such algebras. Additionally, by taking a sort of free product with an unbounded semicircular element, continuous one parameter groups of trace scaling automorphisms on II–factors are constructed; this produces type III factors with core , where can be a full II–factor without the Haagerup approximation property. Received: 26 October 1998 / in final form 18 March 1999     

A singularly perturbed spectral problem for a biharmonic operator with Neumann conditions

We study a mathematical model of a composite plate that consists of two components with similar elastic properties but different distributions of density. The area of the domain occupied by one of the components is infinitely small as ε → 0. We investigate the asymptotic behavior of the eigenvalues and eigenfunctions of the boundary-value problem for a biharmonic operator with Neumann conditions as ε → 0. We describe four different cases of the limiting behavior of the spectrum, depending on the ratio of densities of the medium components. In particular, we describe the so-called Sanches-Palensia effect of local vibrations: A vibrating system has a countable series of proper frequencies infinitely small as ε → 0 and associated with natural forms of vibrations localized in the domain of perturbation of density. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1467–1475, November, 1999.     

Probabilistic approach to the strong Feller property

A new probabilistic method, based on the Girsanov theorem, for establishing the strong Feller property of diffusion processes in both finite and infinite dimensional spaces is proposed. Applications to second order stochastic differential equations, stochastic delay equations and stochastic partial differential equations of parabolic type are discussed, with a twofold aim: both to extend some older results, usually by weakening the assumptions on the drift term, and to obtain simpler proofs of them. Received: 13 December 1999 / Published online: 5 September 2000