Preservation of splitting families and cardinal characteristics of the continuum

We show how to construct, via forcing, splitting families that are preserved by a certain type of finite support iterations. As an application, we construct a model where 15 classical characteristics of the continuum are pairwise different, concretely: the 10 (non-dependent) entries in Cichoń’s diagram, $$\mathfrak{m}$$ m (2-Knaster), $$\mathfrak{p}$$ p , $$\mathfrak{h}$$ h , the splitting number $$\mathfrak{s}$$ s and the reaping number $$\mathfrak{r}$$ r .

Atypical Parkinsonian Syndromes

The cardinal characteristics of parkinsonism are represented in the mnemonic TRAP: tremor at rest, rigidity, akinesia and bradykinesia, and postural instability. The parkinsonian phenotype encompasses a broad range of clinical and pathologic disorders; the most common (about 55% of cases) is idiopathic (sporadic) Parkinson disease. Rapid disease progression, poor initial response to dopaminergic therapy, or the early presence of certain other signs may suggest an atypical parkinsonian syndrome, sometimes called parkinsonism-plus syndrome.

THE SOUTH AFRICAN ACT REVISITED – SOME CONSTITUTIONAL AND POLITICAL REFLECTIONS ON LESSONS LEARNT FROM THE CENTENARY OF THE UNION OF SOUTH AFRICA IN 1910

Just over a hundred years ago South Africa became a politically united state governed by the South Africa Act of 1909, which constituted the first constitution for a territory comprised of the four erstwhile British colonies of the Cape of Good Hope, Natal, the Transvaal and the Orange River Colony. This article revisits this historic constitution and attempts to revaluate its cardinal characteristics in the light of our subsequent constitutional and political development. This article also examines the constitutional and political configuration of the South Africa Act and how it came into being through a so-called National Convention and the part played by the prominent politicians and role players of the day. It examines the constitutional precedents that were available at the time. The crucial issues relating to the nature of the state that was to be established and why a unitary model and not a federation was adopted,are explained. It also considers the vexed question of the franchise and how a compromise was reached in this regard. Other important issues on which decisions had to be taken such as, inter alia, language, native and Indians affairs, are examined and evaluated. The article attempts to address certain important constitutional and political lessons that can be learnt from such an evaluation.

Controlling cardinal characteristics without adding reals

We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new [Formula: see text]-sequences (for some regular [Formula: see text]). As an application, we show that consistently the following cardinal characteristics can be different: The (“independent”) characteristics in Cichoń’s diagram, plus [Formula: see text]. (So we get thirteen different values, including [Formula: see text] and continuum). We also give constructions to alternatively separate other MA-numbers (instead of [Formula: see text]), namely: MA for [Formula: see text]-Knaster from MA for [Formula: see text]-Knaster; and MA for the union of all [Formula: see text]-Knaster forcings from MA for precaliber.

σ-Continuous Functions and Related Cardinal Characteristics of the Continuum

A function f : X → Y between topological spaces is called σ-continuous (resp. ̄σ-continuous) if there exists a (closed) cover {Xn}n∈ω of X such that for every n ∈ ω the restriction f ↾ Xn is continuous. By 𝔠 σ (resp. 𝔠¯σ)we denote the largest cardinal κ ≤ 𝔠 such that every function f : X → ℝ defined on a subset X ⊂ ℝ of cardinality |X| <κ is σ-continuous (resp. ¯σ-continuous). It is clear that ω1 ≤ 𝔠¯σ ≤ 𝔠 σ ≤ 𝔠.We prove that 𝔭 ≤ 𝔮0 = 𝔠¯σ =min{𝔠 σ, 𝔟, 𝔮 }≤ 𝔠 σ ≤ min{non(ℳ), non(𝒩)}.