Saharon Shelah. Middle Diamond. Archive for Mathematical Logic, vol. 44 (2005), pp. 527–560. Saharon Shelah. Diamonds. Proceedings of the American Mathematical Society, vol. 138 no. 6 (2010), pp. 2151–2161. Martin Zeman. Diamond, GCH and Weak Square. Proceedings of the American Mathematical Society, vol. 138 no. 5 (2010), pp. 1853–1859. The Continuum Hypothesis (CH) implies, and is in fact equi...

We show how to connect together the loops of a simple Conformal Loop Ensemble (CLE) in order to construct samples of chordal SLEκ processes and their SLEκ(ρ) variants, and we discuss some consequences of this construction.

The symbol (XI)κ (with κ ≥ ω) denotes the space XI := Πi∈I Xi with the κ-box topology; this has as base all sets of the form U = Πi∈I Ui with Ui open in Xi and with |{i ∈ I : Ui 6= Xi}| < κ. The symbols w, d and S denote respectively weight, density character, and Souslin number. Generalizing familiar results for the usual product space (the case κ = ω), the authors show inter alia: Theorem 3.1...

We prove that o(κ) = κ is sufficient to construct a model V [C] in which κ is measurable and C is a closed and unbounded subset of κ containing only inaccessible cardinals of V . Gitik proved that o(κ) = κ is necessary. We also calculate the consistency strength of the existence of such a set C together with the assumption that κ is Mahlo, weakly compact, or Ramsey. In addition we consider the ...

A wide variety of environmental cues, including inflammatory cytokines, ligands for pattern recognition receptors and endogenous danger signals, activate the inducible transcription factor nuclear factorκ B (NFκ B), which is a central regulator of inflammatory and immune responses. Excessive activation of NFκ B results in the development of severe diseases, such as chronic inflammatory disorder...

Let φ1 stand for the statement V = HOD and φ2 stand for the Ground Axiom. Suppose Ti for i = 1, . . . , 4 are the theories “ZFC + φ1 + φ2”, “ZFC + ¬φ1 + φ2”, “ZFC + φ1 + ¬φ2”, and “ZFC + ¬φ1 + ¬φ2” respectively. We show that if κ is indestructibly supercompact and λ > κ is inaccessible, then for i = 1, . . . , 4, Ai =df {δ < κ | δ is an inaccessible cardinal which is not a limit of inaccessible...

The tree property at κ states that there are no Aronszajn trees on κ, or, equivalently, that every κ tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ and failure of the singular cardinal hypothesis at κ; the former is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible...

We study the approachability ideal I[κ+] in the context of large cardinals and properties of the regular cardinals below a singular κ. As a guiding example consider the approachability ideal I[אω+1] assuming that אω is a strong limit. In this case we obtain that club many points in אω+1 of cofinality אn for some n > 1 are approachable assuming the joint reflection of countable families of stati...