A FORMALISM FOR SOME CLASS OF FORCING NOTIONS

We introduce a class of forcing notions, called forcing notions of type S, which contains among other Sacks forcing, Prikry-Silver forcing and their iterations and products with countable supports. We construct and investigate some formalism suitable for this forcing notions, which allows all standard tricks for iterations or products with countable supports of Sacks forcing. On the other hand it does not involve internal combinatorial structure of conditions of iterations or products. We prove that the class of forcing notions of type S is closed under products and certain iterations with countable supports.     

Uniform unfolding and analytic measurability

We generalize Solovay's unfolding technique for infinite games and use an Unfolding Theorem to give a uniform method to prove that all analytic sets are in the -algebras of measurability connected with well-known forcing notions. Received: 19 July 1996     

Measured creatures

We prove that two basic questions on outer measure are undecidable. First we show that consistently every sup-measurable functionf: ℝ² → ℝ is measurable. The interest in sup-measurable functions comes from differential equations and the question for which functionsf: ℝ² → ℝ the Cauchy problemy′=f(x,y), y(x₀)=y₀ has a unique almost-everywhere solution in the classAC _t(ℝ) of locally absolutely continuous functions on ℝ. Next we prove that consistently every functionf: ℝ → ℝ is continuous on some set of positive outer Lebesgue measure. This says that in a strong sense the family of continuous functions (from the reals to the reals) is dense in the space of arbitrary such functions. For the proofs we discover and investigate a new family of nicely definable forcing notions (so indirectly we deal with nice ideals of subsets of the reals—the two classical ones being the ideal of null sets and the ideal of meagre ones). Concerning the method, i.e., the development of a family of forcing notions, the point is that whereas there are many such objects close to the Cohen forcing (corresponding to the ideal of meagre sets), little has been known on the existence of relatives of the random real forcing (corresponding to the ideal of null sets), and we look exactly at such forcing notions. The first author thanks The Hebrew University of Jerusalem for support during his visits to Jerusalem and the KBN (Polish Committee of Scientific Research) for partial support through grant 2P03A03114. The research of the second author was partially supported by the Israel Science Foundation. Publication 736.     

Iteration of Souslin forcing,projective measurability and the Borel conjecture

From an inaccessible cardinal we construct a model of ZFC where the Borel Conjecture holds and all projective sets of reals are measurable. This continues the investigation of countable support iterations of Proper Souslin forcing notions, started in a paper of Judah and Shelah. The first author thanks the Sloan Foundation for supporting him as a dissertation fellow during the year 1989/90. The second author thanks Israel Foundation for Basic Research, Israel Academy of Science.     

Symmetries between two Ramsey properties

In this article we compare the well-known Ramsey property with a dual form of it, the so called dual-Ramsey property (which was suggested first by Carlson and Simpson). Even if the two properties are different, it can be shown that all classical results known for the Ramsey property also hold for the dual-Ramsey property. We will also show that the dual-Ramsey property is closed under a generalized Suslin operation (the similar result for the Ramsey property was proved by Matet). Further we compare two notions of forcing, the Mathias forcing and a dual form of it, and will give some symmetries between them. Finally we give some relationships between the dual-Mathias forcing and the dual-Ramsey property. Received: July 1, 1996     

Distributive proper forcing axiom and cardinal invariants

In this paper, we study the forcing axiom for the class of proper forcing notions which do not add ω sequence of ordinals. We study the relationship between this forcing axiom and many cardinal invariants. We use typical iterated forcing with large cardinals and analyse certain property being preserved in this process. Lastly, we apply the results to distinguish several forcing axioms.     

On forcing matching number of boron-nitrogen fullerene graphs

Let G be a graph that admits a perfect matching M. A forcing set S for a perfect matching M is a subset of M such that it is contained in no other perfect matchings of G. The smallest cardinality of forcing sets of M is called the forcing number of M. Computing the minimum forcing number of perfect matchings of a graph is an NP-complete problem. In this paper, we consider boron-nitrogen (BN) fullerene graphs, cubic 3-connected plane bipartite graphs with exactly six square faces and other hexagonal faces. We obtain the forcing spectrum of tubular BN-fullerene graphs with cyclic edge-connectivity 3. Then we show that all perfect matchings of any BN-fullerene graphs have the forcing number at least two. Furthermore, we mainly construct all seven BN-fullerene graphs with the minimum forcing number two.     

How much sweetness is there in the universe?

We continue investigations of forcing notions with strong ccc properties introducing new methods of building sweet forcing notions. We also show that quotients of topologically sweet forcing notions over Cohen reals are topologically sweet while the quotients over random reals do not have to be such. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Baumgartnerʼs conjecture and bounded forcing axioms

We study the spectrum of forcing notions between the iterations of σ-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of α-proper forcings for indecomposable countable ordinals α, the Axiom A forcings and forcings completely embeddable into an iteration of a σ-closed followed by a ccc forcing. For the latter class, we present an equivalent characterization in terms of Baumgartner?s Axiom A. This resolves a conjecture of Baumgartner from the 1980s.     

Notions of boundaries for function spaces

Sibony and the author independently defined a higher order generalization of the usual Shilov boundary of a function algebra which yielded extensions of results about analytic structure from one dimension to several dimensions. Tonev later obtained an alternative characterization of this generalized Shilov boundary by looking at closed subsets of the spectrum whose image under the spectral mapping contains the topological boundary of the joint spectrum. In this note we define two related notions of what it means to be a higher order/higher dimensional boundary for a space of functions without requiring that the boundary be a closed set. We look at the relationships between these two boundaries, and in the process we obtain an alternative proof of Tonev's result. We look at some examples, and we show how the same concepts apply to convex sets and linear functions.

     

Some weak fragments of Martin’s axiom related to the rectangle refining property

We introduce the anti-rectangle refining property for forcing notions and investigate fragments of Martin’s axiom for ℵ₁ dense sets related to the anti-rectangle refining property, which is close to some fragment of Martin’s axiom for ℵ₁ dense sets related to the rectangle refining property, and prove that they are really weaker fragments. T. Yorioka was partially supported by Grant-in-aids for Scientific Research No.16340022 and No.18840022.     

On Genericity and Ershov's Hierarchy

It is natural to wish to study miniaturisations of Cohen forcing suitable to sets of low arithmetic complexity. We consider extensions of the work of Schaeffer [9] and Jockusch and Posner [6] by looking at genericity notions within the Δ₂ sets. Different equivalent characterisations of 1‐genericity suggest different ways in which the definition might be generalised. There are two natural ways of casting the notion of 1‐genericity: in terms of sets of strings and in terms of density functions, as we will see here. While these definitions coincide at the first level of the difference hierarchy, they turn out to differ at other levels. Furthermore, these differences remain when the remainder of the Δ⁰₂ sets are considered. While the string characterization of 1‐genericity collapses at the second level of the difference hierarchy to 2‐genericity, the density function definition gives a very interesting hierarchy at level w and above. Both of these results point towards the deep similarities exhibited by the n‐c.e. degrees for n ≥ 2.     

Mathias–Prikry and Laver–Prikry type forcing

We study the Mathias–Prikry and Laver–Prikry forcings associated with filters on ω. We give a combinatorial characterization of Martin?s number for these forcing notions and present a general scheme for analyzing preservation properties for them. In particular, we give a combinatorial characterization of those filters for which the Mathias–Prikry forcing does not add a dominating real.     

Towards Martins minimum

We show that it is consistent with MA + ?CH that the Forcing Axiom fails for all forcing notions in the class of ω^ω–bounding forcing notions with norms of [17]. Received: 28 April 1999 / Published online: 12 December 2001     

Universal forcing notions and ideals

Our main result states that a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give some results concerning cardinal characteristics of the σ-ideals determined by those universality parameters. Both authors acknowledge support from the United States-Israel Binational Science Foundation (Grant no. 2002323). Also, we would like to thank the referee for valuable comments and suggestions concerning the exposition of the paper. This is publication 845 of the second author     

A Riemannian inexact Newton‐CG method for stochastic inverse singular value problems

In this article, we consider the stochastic inverse singular value problem (ISVP) of constructing a stochastic matrix from the prescribed realizable singular values. We propose a Riemannian inexact Newton‐CG method with various choices of forcing terms for solving the stochastic ISVP. We show the proposed method converges linearly or superlinearly for different forcing terms under some assumptions. We also extend the proposed method to the case of prescribed entries. Finally, we report some numerical results to demonstrate the effectiveness of the proposed method. MOS SUBJECT CLASSIFICATION 65F18; 65F15; 15A18; 65K05; 90C26; 90C48     

Divergence spectra and Morse boundaries of relatively hyperbolic groups

We introduce a new quasi-isometry invariant, called the divergence spectrum, to study finitely generated groups. We compare the concept of divergence spectrum with the other classical notions of divergence and we examine the divergence spectra of relatively hyperbolic groups. We show the existence of an infinite collection of right-angled Coxeter groups which all have exponential divergence but they all have different divergence spectra. We also study Morse boundaries of relatively hyperbolic groups and examine their connection with Bowditch boundaries.     

Computability in structures representing a Scott set

Continuing work begun in [10], we utilize a notion of forcing for which the generic objects are structures and which allows us to determine whether these “generic” structures compute certain sets and enumerations. The forcing conditions are bounded complexity types which are consistent with a given theory and are elements of a given Scott set. These generic structures will “represent” this given Scott set, in the sense that the structure has a certain weak saturation property with respect to bounded complexity types in the Scott set. For example, if ? is a nonstandard model of PA, then ? represents the Scott set ? = n∈ω | ?⊧“the nth prime divides a” | a∈?. The notion of forcing yields two main results. The first characterizes the sets of natural numbers computable in all models of a given theory representing a given Scott set. We show that the characteristic function of such a set must be enumeration reducible to a complete existential type which is consistent with the given theory and is an element of the given Scott set. The second provides a sufficient condition for the existence of a structure ? such that ? represents a countable jump ideal and ? does not compute an enumeration of a given family of sets ?. This second result is of particular interest when the family of sets which cannot be enumerated is ? = Rep[Th(?)]. Under this additional assumption, the second result generalizes a result on TA [6] and on certain other completions of PA [10]. For example, we show that there also exist models of completions of ZF from which one cannot enumerate the family of sets represented by the theory. Received: 8 October 1997 / Published online: 25 January 2001     

Krein–Langer Factorization and Related Topics in the Slice Hyperholomorphic Setting

We study various aspects of Schur analysis in the slice hyperholomorphic setting. We present two sets of results: first, we give new results on the functional calculus for slice hyperholomorphic functions. In particular, we introduce and study some properties of the Riesz projectors. Then we prove a Beurling–Lax type theorem, the so-called structure theorem. A crucial fact which allows to prove our results is the fact that the right spectrum of a quaternionic linear operator and the point S-spectrum coincide. Finally, we study the Krein–Langer factorization for slice hyperholomorphic generalized Schur functions. Both the Beurling–Lax type theorem and the Krein–Langer factorization are far-reaching results which have not been proved in the quaternionic setting using notions of hyperholomorphy other than slice hyperholomorphy.