We call a topological space κ-compact if every subset of size κ has a complete accumulation point in it. Let Φ(μ, κ, λ) denote the following statement: μ < κ < λ = cf(λ) and there is {Sξ : ξ < λ} ⊂ [κ] μ such that |{ξ : |Sξ ∩ A| = μ}| < λ whenever A ∈ [κ]. We show that if Φ(μ, κ, λ) holds and the space X is both μ-compact and λ-compact then X is κ-compact as well. Moreover, from PCF theory we d...

Let κ be a regular cardinal. Consider the Baire numbers of the spaces (2)κ for various θ ≥ κ. Let l be the number of such different Baire numbers. Models of set theory with l = 1 or l = 2 are known and it is also known that l is finite. We show here that if κ > ω, then l could be any given finite number. The Baire number of a topological space with no isolated points is the minimal cardinality ...

The terminology and notation used in this paper are introduced in the following papers: [22], [3], [20], [9], [5], [12], [10], [11], [15], [2], [18], [4], [1], [21], [16], [17], [14], [13], [19], [6], [7], and [8]. For simplicity, we follow the rules: X denotes a complex unitary space, s1, s2, s3 denote sequences of X, R1 denotes a sequence of real numbers, C1, C2, C3 denote complex sequences, ...

The noncommutative spacetimes associated to the κ-Poincaré relativistic symmetries and their “non-relativistic” (Galilei) “ultra-relativistic” (Carroll) limits are indistinguishable, since coordinates satisfy same algebra. In this work, we show that three quantum kinematical models can be differentiated when looking at spaces of time-like worldlines. Specifically, construct geodesics with κ-Gal...

We consider the equilibrium perturbations for two stochastic systems: d-dimensional generalized exclusion process and one-dimensional chain of anharmonic oscillators. add a perturbation order N−α to profile speed up by N1+κ parameters 0<κ≤α. Under some additional constraints on κ α, we show perturbed quantities evolve according Burgers equation in process, decoupled equations chain, both smooth...

Let κ be an arbitrary regular infinite cardinal and let C denote the set of κ-maximal cofinitary groups. We show that if GCH holds and C is a closed set of cardinals such that 1. κ ∈ C, ∀ν ∈ C(ν ≥ κ), 2. if |C| ≥ κ then [κ, |C|] ⊆ C, 3. ∀ν ∈ C(cof(ν) ≤ κ→ ν ∈ C), then there is a generic extension in which cofinalities have not been changed and such that C = {|G| : G ∈ C}. The theorem generalize...

We prove that the following two statements are equiconsistent: there exists a greatly Mahlo cardinal; there exists a regular uncountable cardinal κ such that no stationary subset of κ+ ∩ cof(κ) carries a partial square. A famous theorem in set theory is the result that the failure of the square principle κ, for a regular uncountable cardinal κ, is equiconsistent with a Mahlo cardinal. Solovay p...