We prove the existence and uniqueness of geometric models local isometry classes locally homogeneous spaces with sectional curvature $|\operatorname{sec}|\leq 1$. Moreover, we show that set is compact in pointed $\mathcal{C}^{1,\alpha}$-topology.

We analyze the strength of Helly’s selection theorem (HST), which is the most important compactness theorem on the space of functions of bounded variation (BV ). For this we utilize a new representation of this space intermediate between L1 and the Sobolev space W , compatible with the—so called—weak∗ topology on BV . We obtain that HST is instance-wise equivalent to the Bolzano-Weierstraß prin...

In any model theoretic logic, Beth’s definability property together with Feferman-Vaught’s uniform reduction property for pairs imply recursive compactness, and the existence of models with infinitely many automorphisms for sentences having infinite models. The stronger Craig’s interpolation property plus the uniform reduction property for pairs yield a recursive version of Ehrenfeucht-Mostowsk...

It is well-known that the Palais-Smale Condition C does not hold for the YangMills functional on a principal bundle over a compact four-manifold. According to the Uhlenbeck weak compactness theorem, a Palais-Smale sequence will in general only have a subsequence that converges on the complement of a finite set of points. Moreover, even on the complement of these points, the convergence is not a...

We study the Tychonoff Compactness Theorem for several different definitions of a compact space.

  In this paper, an existence result for a class of infinite systems of functional-integral equations in the Banach sequence space $c_{0}$ is established via the well-known Schauder fixed-point theorem together with a criterion of compactness in the space $c_{0}$. Furthermore, we include some remarks to show the vastity of the class of infinite systems which can be covered by our result. The a...

This is a set of lecture notes used in a course on model theory at Virginia Commonwealth University (Math 591 Topics: Logic and Mathematical Structures), which I taught jointly with Sean Cox in the spring of 2014. Most of the material contained in these notes can be found in the following sources. References: • Chang and Keisler. Model Theory, Studies in Logic and the Foundations of Mathematics...