As an application of the method used to obtain the result above, we will obtain the consistency of b = ω1 < s = κ where b is the bounding number, s is the splitting number and κ is an arbitrary regular cardinal. Suppose ν ≥ δ. Begin with a model of GCH and adjoin δ-many Cohen reals 〈rα : α ∈ δ〉 followed by ν-many Random reals 〈sξ : ξ ∈ ν〉. That is, if Vδ is the model obtained after the first δ ...

This is an expository note on the Lascar group. We also study the Lascar group over hyperimaginaries, and make some new observations on the strong types over those. In particular we show that in a simple theory, Ltp ≡ stp in real context implies that for hyperimaginary context. The Lascar group introduced by Lascar [8], and related subjects have been studied by many authors ([9][1][7][6][3] and...

Suppose κ is λ-supercompact witnessed by an elementary embedding j : V →M with critical point κ, and further suppose that F is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton’s theorem: (1) ∀α α < cf(F (α)) and (2) α < β =⇒ F (α) ≤ F (β). In this article we address the question: assuming GCH, what additional assumptions are necessar...

In this paper we examine the cardinal structure of inner models that satisfy GCH but do not contain 0. We show, assuming that 0 exists, that such models necessarily contain Mahlo cardinals of high order, but without further assumptions need not contain a cardinal κ which is κ-Mahlo. The principal tools are the Covering Theorem for L and the technique of reverse Easton iteration. Let I denote th...

Several approaches have been proposed for the computation of solutions to the phase and chemical equilibrium problem when the problem is posed as the minimization of the Gibbs free energy function. None of them can guarantee convergence to the true optimal solution, and are highly dependent on the supplied initial point. Convergence to local solutions often occurs, yielding incorrect phase and ...

New experimental data on liquid-liquid equilibria (LLE), solubility and critical states in quaternary system propionic acid – 1-propanol n-propyl propionate water at 303.15 K atmospheric pressure are presented. LLE was studied by Gas Chromatography (GC). Solubility (plait) points were investigated “cloud-point technique” method. Binodal curves surfaces presented concentration tetrahedron. The c...