Suppose that κ is indestructibly supercompact and there is a measurable cardinal λ > κ. It then follows that A1 = {δ < κ | δ is measurable, δ is not a limit of measurable cardinals, and δ is not δ+ supercompact} is unbounded in κ. If in addition λ is 2λ supercompact, then A2 = {δ < κ | δ is measurable, δ is not a limit of measurable cardinals, and δ is δ+ supercompact} is unbounded in κ as well...

We analyze Loewner traces driven by functions asymptotic to κ √ 1− t. We prove a stability result when κ 6= 4 and show that κ = 4 can lead to non locally connected hulls. As a consequence, we obtain a driving term λ(t) so that the hulls driven by κλ(t) are generated by a continuous curve for all κ > 0 with κ 6= 4 but not when κ = 4, so that the space of driving terms with continuous traces is n...

We analyze the effect of singularizing cardinals on square properties. An old theorem of Dzamonja-Shelah/Gitik says that if you singularize an inaccessible cardinal while preserving its successor, then κ,ω holds in the bigger model. We extend this to the situation where a finite interval of cardinals above κ is collapsed. More precisely, we show that if V ⊂ W , κ is inaccessible in V , cf (κ V ...

We give an application of our extender based Radin forcing to cardinal arithmetic. Assuming κ is a large enough cardinal we construct a model satisfying 2 = κ together with 2 = λ for each cardinal λ < κ, where 0 < n < ω. The cofinality of κ can be set arbitrarily or κ can remain inaccessible. When κ remains an inaccessible, Vκ is a model of ZFC satisfying 2 = λ for all cardinals λ.

The purpose of this note is to describe a framework which unifies radial, chordal and dipolar SLE. When the definition of SLE(κ; ρ) is extended to the setting where the force points can be in the interior of the domain, radial SLE(κ) becomes chordal SLE(κ; ρ), with ρ = κ− 6, and vice versa. We also write down the martingales describing the Radon–Nykodim derivative of SLE(κ; ρ1, . . . , ρn) with...

By using combinatorial properties of stationary sets, we give a simple proof of some generalization of Silver’s Theorem i.e. if κ is an uncountable regular cardinal such that אκ is a singular strong limit cardinal, then the following hold. (1). If {α < κ : א<κ α ≤ אα·2} is stationary, then 2אκ ≤ אκ·2. (2). If {α < κ : א<κ α ≤ אα+γ}, where 0 < γ < κ, is stationary, then 2אκ ≤ אκ+γ.

We explore the information geometric structures among the thermodynamic potentials in the κ-thermostatistics, which is a generalized thermostatistics based on the κ-deformed entropy. We show that there exists two different kinds of dualistic Hessian structures: one is associated with the κ-escort expectations and the other with the standard expectations. The associated κ-generalized metrics are...

The purpose of this note is to describe a framework which unifies radial, chordal and dipolar SLE. When the definition of SLE(κ; ρ) is extended to the setting where the force points can be in the interior of the domain, radial SLE(κ) becomes chordal SLE(κ; ρ), with ρ = κ − 6, and vice versa. We also write down the martingales describing the Radon-Nykodim derivative of SLE(κ; ρ1, . . . , ρn) wit...

We answer some question of [Gi]. The upper bound of [Gi] on the strength of N S µ + precipitous for a regular µ is proved to be exact. It is shown that saturatedness of N S ℵ 0 κ over inaccessible κ requires at least o(κ) = κ ++. The upper bounds on the strength of N S κ precipitous for inaccessible κ are reduced quite close to the lower bounds.

Let κ be a regular uncountable cardinal and λ ≥ κ. The principle of stationary reflection for Pκλ has been successful in settling problems of infinite combinatorics in the case κ = ω1. For a greater κ the principle is known to fail at some λ. This note shows that it fails at every λ if κ is the successor of a regular uncountable cardinal or κ is countably closed.