The aim of this paper is to make progress towards a geometric model theory for non first order theories. The main difficulty is to work in an environment where the compactness theorem fails. This paper continues the work started in [GrLel]. The main result is an axiomatic approach to the Hrushovski-Zilber group configuration theorem.

After introducing many different types of prefilter convergence, we introduce an universal method to define various notions of compactness using cluster point and convergence of a prefilter and to prove the Tychonoff theorem using characterizations of ultra(maximal) prefilters. : prefilter convergence, universal method, Tychonoff theorem, ultra prefilter, good extension

G. Žitković defined the notion of a convexly compact set in a topological space and, among other things, used it to give an extension of the Walrasian excess-demand theorem. We continue the study of convexly compactness in LCS spaces and prove a Krein-Milman theorem in this setting.

We prove the following results for toric Deligne–Mumford stacks, under minimal compactness hypotheses: the Localization Theorem in equivariant K-theory; the equivariant Hirzebruch–Riemann–Roch theorem; the Fourier–Mukai transformation associated to a crepant toric wall-crossing gives an equivariant derived equivalence.

In this paper, the well-known Egoroff’s theorem in classical measure theory is established on monotone non-additive measure spaces. Taylor’s theorem, which concerns almost everywhere convergence of measurable function sequence in classical measure theory, is also generalized. The converse problem of the theorems are discussed, and a necessary and sufficient condition for the Egoroff’s theorem i...

We establish two new Easton theorems for the least supercompact cardinal that are consistent with the level by level equivalence between strong compactness and supercompactness. These theorems generalize [1, Theorem 1]. In both our ground model and the model witnessing the conclusions of our theorem, there are no restrictions on the structure of the class of supercompact cardinals.

We prove a Model Existence Theorem for a fully infinitary logic LA for metric structures. This result is based on a generalization of the notions of approximate formulas and approximate truth in normed structures introduced by Henson ([7]) and studied in different forms by Anderson ([1]) and Fajardo and Keisler ([2]). This theorem extends Henson’s Compactness Theorem for approximate truth in no...

In this paper, the well-known Egoroff’s theorem in classical measure theory is established on Sugeno fuzzy measure spaces. Taylor’s theorem, which concerns almost everywhere convergence of measurable function sequence in classical measure theory, is also generalized. The converse problem of the theorems are discussed, and a necessary and sufficient condition for the Egoroff’s theorem is obtaine...

Let Ω ⊂ RN be the upper half strip with a hole. In this paper, we show there exists a positive higher energy solution of semilinear elliptic equations in Ω and describe the dynamic systems of solutions of equation (1) in various Ω. We also show there exist at least two positive solutions of perturbed semilinear elliptic equations in Ω.

We consider the following nonlinear singular elliptic equation −div (|x| −2a ∇u) = K(x)|x| −bp |u| p−2 u + λg(x) in R N , where g belongs to an appropriate weighted Sobolev space, and p denotes the Caffarelli–Kohn– Nirenberg critical exponent associated to a, b, and N. Under some natural assumptions on the positive potential K(x) we establish the existence of some λ 0 > 0 such that the above pr...