1 1 General introduction 3 1.1 Combinatorial property testing . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Property testing in the dense graph (and hypergraph) model . . . . 4 1.1.2 Testing graph properties in other models . . . . . . . . . . . . . . . 5 1.1.3 Testing massively parameterized properties . . . . . . . . . . . . . . 5 1.2 Probabilistically checkable proofs of proximit...

We give a short proof of a lemma which generalizes both the main lemma from the original construction in the author’s thesis of a model with no ω2-Aronszajn trees, and also the “Key Lemma” in Hamkins’s gap forcing theorems. The new lemma directly yields Hamkins’s newer lemma stating that certain forcing notions have the approximation property. According to Hamkins [2], a partial ordering P sati...

n this paper, we consider continuity properties(especially, regularity, also viewed as an approximation property) for $%mathcal{p}_{0}(x)$-valued set multifunctions ($x$ being a linear,topological space), in order to obtain egoroff and lusin type theorems forset multifunctions in the vietoris hypertopology. some mathematicalapplications are established and several physical implications of thema...

Nonstandard finite difference schemes for the Black-Scholes partial differential equation preserving the positivity property are proposed. Computationally simple schemes are derived by using a nonlocal approximation in the reaction term of the Black-Scholes equation. Unlike the standard methods, the solutions of new proposed schemes are positive and free of the spurious oscillations.