An Introduction to the Method of Refinements 2 Jonathan Hickman December 21 , 2014 1 Main Theorem

An Introduction to the Method of Refinements

We begin with an intuitive description of the kind of operators we wish to study. The most general set-up is as follows: let M1 and M2 be manifolds of dimension n1 and n2, respectively and 1 ≤ k < n1. Suppose there exists a pair of smooth assignments x2 7→ γx2 and x2 7→ dσx2 , the former attaching a k-dimensional submanifold γx2 ⊂ M1 and the latter a smooth measure dσx2 on γx2 to each point x2 ...

Szemerédi’s Theorem via Ergodic Theory

This essay investigates Furstenberg’s proof of Szemerédi’s Theorem. The necessary concepts and results from ergodic theory are introduced, including the Poincaré and Mean Ergodic Theorems which are proved in full. The Ergodic Decomposition Theorem is also discussed. Furstenberg’s Multiple Recurrence Theorem is then stated and shown to imply Szemerédi’s Theorem. The remainder of the essay concen...

African Green Revolution, No, and Ozone 1 1 Impacts of African Agricultural Intensification on Nitric Oxide Fluxes and Tropospheric 2 Ozone 3

1 channels identified in rodent myelinated axons ,

KV1 channels identified in rodent myelinated axons, linked to Cx29 in innermost myelin: 1 Support for electrically-active myelin in mammalian saltatory conduction 2 3 John E. Rash*, Kimberly G. Vanderpool, Thomas Yasumura, Jordan Hickman, Jonathan T. 4 Beatty, and James I. Nagy 5 6 Department of Biomedical Sciences, and Program in Molecular, Cellular and Integrative 7 Neurosciences, Colorado St...

Feyz Kashani and Jonathan Edwards on the Place of Heart in Religious Knowledge

This essay seeks to summarize and compare the theological ideas of Jonathan Edwards, a western philosopher, with those of Feyz Kashani, an eastern philosopher. The reason for this study is the numerous conjunctions that are seen in their ideas. These include the harmony of heart and reason in respect to belief, the priority of the heart in the reflection of truth, the belief in centrality of th...