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The Modified Szpiro Conjecture, equivalent to the $abc$ Conjecture, states that for each $epsilon>0$, there are finitely many rational elliptic curves satisfying $N_{E}^{6+epsilon}<max!left{ leftvert c_{4}^{3}rightvert,c_{6}^{2}right} $ where $c_{4}$ and $c_{6}$ are the invariants associated to a minimal model of $E$ and $N_{E}$ is the conductor of $E$. We say $E$ is a good elliptic curve if $N_{E}^{6}<max!left{ leftvert c_{4}^{3}rightvert,c_{6}^{2}right} $. Masser showed that there are infinitely many good Frey curves. Here we give a constructive proof of this assertion.
Let $Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $fcolonmathbb{N}tomathbb{C}$ one has [ frac{1}{N}sum_{n=1}^N, f(Omega(n)+1)=frac{1}{N}sum_{n=1}^N, f(Omega(n))+mathrm{o}_{Ntoinfty}(1). ] This yields a new elementary proof of the Prime Number Theorem.
We give a general method of extending unital completely positive maps to amalgamated free products of C*-algebras. As an application we give a dilation theoretic proof of Bocas Theorem.
Greenberg proved that every countable group $A$ is isomorphic to the automorphism group of a Riemann surface, which can be taken to be compact if $A$ is finite. We give a short and explicit algebraic proof of this for finitely generated groups $A$.
We outline a simple proof of Hulanickis theorem, that a locally compact group is amenable if and only if the left regular representation weakly contains all unitary representations. This combines some elements of the literature which have not appeared together, before.
This article contains a complete proof of Gabrielovs rank Theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of t