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.

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...

Given a supercompact cardinal κ and a regular cardinal λ < κ, we describe a type of forcing such that in the generic extension the cofinality of κ is λ, there is a very good scale at κ, a bad scale at κ, and SCH at κ fails. When creating our model we have great freedom in assigning the value of 2, and so we can make SCH hold or fail arbitrarily

We introduce an extension, indexed by a partially ordered set P and cardinal numbers κ, λ, denoted by (κ,<λ) ; P , of the classical relation (κ, n, λ) → ρ in infinite combinatorics. By definition, (κ, n, λ) → ρ holds if every map F : [κ] → [κ] has a ρ-element free set. For example, Kuratowski’s Free Set Theorem states that (κ, n, λ) → n + 1 holds iff κ ≥ λ+n, where λ+n denotes the n-th cardinal...

Given an uncountable regular cardinal κ, a partial order is κstationarily layered if the collection of regular suborders of P of cardinality less than κ is stationary in Pκ(P). We show that weak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal κ is weakly compact if and only if every partial order satisfying the κ-chain conditio...

Starting from a supercompact cardinal κ, we build a model, in which κ is singular string limit, the singular cardinal hypothesis fails at κ and there are no very good scales at κ. Moreover there is a bad scale at κ, and so weak square fails.