By exploiting Perelman’s pseudolocality theorem, we prove a new compactness theorem for Ricci flows. By optimising the theory in the two-dimensional case, and invoking the theory of quasiconformal maps, we establish a new existence theorem which generates a Ricci flow starting at an arbitrary incomplete metric, with Gauss curvature bounded above, on an arbitrary surface. The criterion we assert...

0. Introduction 1 1. Background on infinitary logic 2 1.1. Expressive power of Lω1ω 2 1.2. The back-and-forth construction 3 1.3. The Scott isomorphism theorem 4 1.4. ω-logic 7 1.5. Familiar theorems 8 1.6. Failure of compactness 9 2. Background on admissible sets 10 2.1. ∆0 formulas and Σ-formulas in set theory 10 2.2. Axioms of KP 11 2.3. Examples of admissible sets 12 2.4. The admissible set...

In this paper, we first establish a criterion based on contractive function for the existence of generalized polynomial attractors. This only involves some rather weak compactness associated with repeated inferior limit and requires no compactness, which makes it suitable critical cases. Then by abstract theorem, verify attractor estimate its attractive speed wave equation nonlocal damping, ant...

We explore the connection between the concepts “excess” and “duality gap” from epigraphical analysis and optimization, and the functional analytic concepts of weak* and weak compactness. We also discuss briefly the connection with R. C. James’s “sup theorem”.

We give a survey and a unified presentation of the defect theorem, its generalizations and recent aspects of interest. In its basic form the defect theorem states that if a set of n words satisfies a nontrivial relation, then these words can be expressed simultaneously as products of at most n 1 words. In other words, dependency of words causes a defect effect. There does not exist just one def...

Kochen and Specker’s theorem can be seen as a consequence of Gleason’s theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays in Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason’s theorem itself has a constructive proof, based on a generic, finite, effectively generated set of rays, on wh...

In this work, the authors study the existence of solutions for fractional differential inclusions in the sense of Almgren with Riemann-Liouville derivative. They also show the compactness of the solution set. A Peano type existence theorem is also proved.

We present a proof of Sklar’s Theorem that uses topological arguments, namely compactness (under the weak topology) of the class of copulas and some density properties of the class of distribution functions.

We study the compactness in a fuzzy metric space. We also give a generalization of Niemytzki–Tychonoff theorem for a fuzzy metric space. © 2013 Elsevier B.V. All rights reserved.

1 2 Contents 0. One-variable calculus 1. The derivative 2. Inverse function and implicit function theorem 3. Fundamental local existence theorem for ODE 4. The Riemann integral in n variables 5. Integration on surfaces 6. Differential forms 7. Products and exterior derivatives of forms 8. The general Stokes formula 9. The classical Gauss, Green, and Stokes formulas 10. Holomorphic functions and...