A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow
In this paper, we give a complete proof of the Poincare and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow.
Graph Decomposition is NP-Complete: A Complete Proof of Holyer's Conjecture
An H-decomposition of a graph G=(V,E) is a partition of E into subgraphs isomorphic to H. Given a fixed graph H, the H-decomposition problem is to determine whether an input graph G admits an H-decomposition. In 1980, Holyer conjectured that H-decomposition is NP-complete whenever H is connected and has three edges or more. Some partial results have been obtained since then. A complete proof of Holyer's conjecture is the content of this paper. The characterization problem of all graphs H for which H-decomposition is NP-complete is hence reduced to graphs where every connected component contains at most two edges.
The Complete Proof Theory of Hybrid Systems
Hybrid systems are a fusion of continuous dynamical systems and discrete dynamical systems. They freely combine dynamical features from both worlds. For that reason, it has often been claimed that hybrid systems are more challenging than continuous dynamical systems and than discrete systems. We now show that, proof-theoretically, this is not the case. We present a complete proof-theoretical alignment that interreduces the discrete dynamics and the continuous dynamics of hybrid systems. We give a sound and complete axiomatization of hybrid systems relative to continuous dynamical systems and a sound and complete axiomatization of hybrid systems relative to discrete dynamical systems. Thanks to our axiomatization, proving properties of hybrid systems is exactly the same as proving properties of continuous dynamical systems and again, exactly the same as proving properties of discrete dynamical systems. This fundamental cornerstone sheds light on the nature of hybridness and enables flexible and provably perfect combinations of discrete reasoning with continuous reasoning that lift to all aspects of hybrid systems and their fragments.
A proof of the trace theorem of Sobolev spaces on Lipschitz domains
A complete proof of the trace theorem of Sobolev spaces on Lipschitz domains has not appeared in the literature yet. The purpose of this paper is to give a complete proof of the trace theorem of Sobolev spaces on Lipschitz domains by taking advantage of the intrinsic norm on Hs(∂Ω). It is proved that the trace operator is a linear bounded operator from Hs(Ω) to Hs− 1 2 (∂Ω) for 1 2 < s < 3 2 .
A Conceptual Proof of the Kesten-Stigum Theorem for Multi-Type Branching Processes
We give complete proofs of the theorem of convergence of types and the Kesten-Stigum theorem for multi-type branching processes. Very little analysis is used beyond the strong law of large numbers and some basic measure theory.
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