Fusion and large cardinal preservation

In this paper we introduce some fusion properties of forcing notions which guarantee that an iteration with supports of size ?κ   not only does not collapse κ⁺${\kappa }^{+}$ but also preserves the strength of κ (after a suitable preparatory forcing). This provides a general theory covering the known cases of tree iterations which preserve large cardinals (cf. Dobrinen and Friedman (2010) [3], Friedman and Halilovi? (2011) [5], Friedman and Honzik (2008) [6], Friedman and Magidor (2009) [8], Friedman and Zdomskyy (2010) [10], Honzik (2010) [12]).     

A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees

We introduce a property of forcing notions, called the anti-, which comes from Aronszajn trees. This property canonically defines a new chain condition stronger than the countable chain condition, which is called the property .In this paper, we investigate the property . For example, we show that a forcing notion with the property does not add random reals. We prove that it is consistent that every forcing notion with the property has precaliber ℵ₁ and for forcing notions with the property fails. This negatively answers a part of one of the classical problems about implications between fragments of .     

Small embedding characterizations for large cardinals

We show that many large cardinal notions can be characterized in terms of the existence of certain elementary embeddings between transitive set-sized structures, that map their critical point to the large cardinal in question. As an application, we use such embeddings to provide new proofs of results of Christoph Weiß on the consistency strength of certain generalized tree properties. These new proofs eliminate problems contained in the original proofs provided by Weiß.     

Models of Cohen measurability

We show that in contrast with the Cohen version of Solovay's model, it is consistent for the continuum to be Cohen-measurable and for every function to be continuous on a non-meagre set.     

On guessing generalized clubs at the successors of regulars

König, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of a higher Souslin tree from the strong guessing principle.Complementary to the author’s work on the validity of diamond and non-saturation at the successor of singulars, we deal here with a successor of regulars. It is established that even the non-strong guessing principle entails non-saturation, and that, assuming the necessary cardinal arithmetic configuration, entails a diamond-type principle which suffices for the construction of a higher Souslin tree.We also establish the consistency of GCH with the failure of the weakest form of generalized club guessing. This, in particular, settles a question from the original paper.     

The approachability ideal without a maximal set

We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously adding partial square sequences on multiple stationary sets. We show that certain quotients of such forcings have the ${\omega }_1$-approximation property. We apply these ideas to prove, assuming the consistency of a greatly Mahlo cardinal, that it is consistent that the approachability ideal $I[{\omega }_2]$ does not have a maximal set modulo clubs.     

On the role of quadratic oscillations in nonlinear Schrödinger equations

We consider a nonlinear semi-classical Schrödinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations; the proof relies on a linearizability condition (which can be expressed in terms of Wigner measures). When the initial data is a sum of such quadratic oscillations, we prove that the associate solution is the superposition of the nonlinear evolution of each of them, up to a small remainder term. In an appendix, we transpose those results to the case of the nonlinear Schrödinger equation with harmonic potential.     

Lexicographically ordered trees

We characterize trees whose lexicographic ordering produces an order isomorphic copy of some sets of real numbers, or an order isomorphic copy of some set of ordinal numbers. We characterize trees whose lexicographic ordering is order complete, and we investigate lexicographically ordered ω-splitting trees that, under the open-interval topology of their lexicographic orders, are of the first Baire category. Finally we collect together some folklore results about the relation between Aronszajn trees and Aronszajn lines, and use earlier results of the paper to deduce some topological properties of Aronszajn lines.     

Generic embeddings associated to an indestructibly weakly compact cardinal

I use generic embeddings induced by generic normal measures on P_κ(λ) that can be forced to exist if κ is an indestructibly weakly compact cardinal. These embeddings can be applied in order to obtain the forcing axioms in forcing extensions. This has consequences in : The Singular Cardinal Hypothesis holds above κ, and κ has a useful Jónsson-like property. This in turn implies that the countable tower Q_<κ works much like it does when κ is a Woodin limit of Woodin cardinals. One consequence is that every set of reals in the Chang model is Lebesgue measurable and has the Baire Property, the Perfect Set Property and the Ramsey Property. So indestructible weak compactness has effects on cardinal arithmetic high up and also on the structure of sets of real numbers, down low, similar to supercompactness.     

A model of Cummings and Foreman revisited

This paper concerns the model of Cummings and Foreman where from ω   supercompact cardinals they obtain the tree property at each _ℵn${\mathrm{\aleph }}_n$ for 2≤n<ω$2\le n<\omega$. We prove some structural facts about this model. We show that the combinatorics at _ℵω+1${\mathrm{\aleph }}_{\omega +1}$ in this model depend strongly on the properties of _ω1${\omega }_1$ in the ground model. From different ground models for the Cummings–Foreman iteration we can obtain either _ℵω+1∈I[_ℵω+1]${\mathrm{\aleph }}_{\omega +1}\in I[{\mathrm{\aleph }}_{\omega +1}]$ and every stationary subset of _ℵω+1${\mathrm{\aleph }}_{\omega +1}$ reflects or there are a bad scale at _ℵω${\mathrm{\aleph }}_{\omega }$ and a non-reflecting stationary subset of _ℵω+1∩cof(_ω1)${\mathrm{\aleph }}_{\omega +1}\cap \mathrm{cof}({\omega }_1)$. We also prove that regardless of the ground model a strong generalization of the tree property holds at each _ℵn${\mathrm{\aleph }}_n$ for n≥2$n\ge 2$.     

Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain

In this paper we obtain the continuity of attractors for semilinear parabolic problems with Neumann boundary conditions relatively to perturbations of the domain. We show that, if the perturbations on the domain are such that the convergence of eigenvalues and eigenfunctions of the Neumann Laplacian is granted then, we obtain the upper semicontinuity of the attractors. If, moreover, every equilibrium of the unperturbed problem is hyperbolic we also obtain the continuity of attractors. We also give necessary and sufficient conditions for the spectral convergence of Neumann problems under perturbations of the domain.     

Large cardinals and basic sequences

The purpose of this paper is to present several applications of combinatorial principles, well-known in Set Theory, to the geometry of infinite dimensional Banach spaces, particularly to the existence of certain basic sequences. We mention also some open problems where set-theoretical techniques are relevant.     

Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations

We study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation

ut−diva(x,∇u)+f(x,u)=0     

Mob families and mad families

We show the consistency of where is the size of the smallest off-branch family, and is as usual the dominating number. We also prove the consistency of with large continuum. Here, is the unbounding number, and is the almost disjointness number. Received: September 12, 1996 / Revised version received: June 16, 1997     

Singular functions and relative index for elliptic corner operators

We show an explicit link between the nature of a singular point and the behaviour of the coefficients of the equation, under which formal asymptotic expansions are still available. We also derive a general relative index theorem for elliptic operators. To the memory of Lamberto Cattabriga     

Positional graphs and conditional structure of weakly null sequences

We prove that, unless assuming additional set theoretical axioms, there are no reflexive spaces without unconditional sequences of the density continuum. We show that for every integer n$n$ there are normalized weakly-null sequences of length _ωn${\omega }_n$ without unconditional subsequences. This together with a result of Dodos et al. (2011) [7] shows that _ωω${\omega }_{\omega }$ is the minimal cardinal κ$\kappa$ that could possibly have the property that every weakly null κ$\kappa$-sequence has an infinite unconditional basic subsequence. We also prove that for every cardinal number κ$\kappa$ which is smaller than the first ω$\omega$-Erd?s cardinal there is a normalized weakly-null sequence without subsymmetric subsequences. Finally, we prove that mixed Tsirelson spaces of uncountable densities must always contain isomorphic copies of either _c0$c_0$ or _?p${?}_p$, with p≥1$p\ge 1$.     

Compact spaces, elementary submodels, and the countable chain condition, II

Given a space 〈X,T〉 in an elementary submodel of H(θ), define XM to be X∩M with the topology generated by . It is established that if XM is compact and satisfies the countable chain condition, while X is not scattered and has cardinality less than the first inaccessible cardinal, then X=XM. If the character of XM is a member of M, then “inaccessible” may be replaced by “1-extendible”.     

Covering a bounded set of functions by an increasing chain of slaloms

A slalom is a sequence of finite sets of length ω. Slaloms are ordered by coordinatewise inclusion with finitely many exceptions. Improving earlier results of Mildenberger, Shelah and Tsaban, we prove consistency results concerning existence and non-existence of an increasing sequence of a certain type of slaloms which covers a bounded set of functions in ωω.     

A Boolean algebra and a Banach space obtained by push-out iteration

Under the assumption that c is a regular cardinal, we prove the existence and uniqueness of a Boolean algebra B of size c defined by sharing the main structural properties that P(ω)/fin has under CH and in the ℵ₂-Cohen model. We prove a similar result in the category of Banach spaces.