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The culmination of the papers (arXiv:0905.0518, arXiv:0910.0909) was a proof of the norm convergence in $L^2(mu)$ of the quadratic nonconventional ergodic averages frac{1}{N}sum_{n=1}^N(f_1circ T_1^{n^2})(f_2circ T_1^{n^2}T_2^n)quadquad f_1,f_2in L^infty(mu) associated to an arbitrary probability-preserving bbZ^2-system (X,mu,T_1,T_2). This is a special case of the Bergelson-Leibman conjecture on the norm convergence of polynomial nonconventional ergodic averages. That proof relied on some new machinery for extending probability-preserving $bbZ^d$-systems to obtain simplified asymptotic behaviour for various nonconventional averages such as the above. The engine of this machinery is formed by some detailed structure theorems for the `characteristic factors that are available for some such averages after ascending to a suitably-extended system. However, these new structure theorems underwent two distinct phases of development, separated by the discovery of some new technical results in Moores cohomology theory for locally compact groups (arXiv:1004.4937). That discovery enabled a significant improvement to the main structure theorem (Theorem 1.1 in (arXiv:0905.0518)), which in turn afforded a much shortened proof of convergence. However, since the proof of convergence using the original structure theorem required some quite different ideas that are now absent from these other papers, I have recorded it here in case it has some independent interest.
Let $G$ be a countable group and $X$ be a totally regular curve. Suppose that $phi:Grightarrow {rm Homeo}(X)$ is a minimal action. Then we show an alternative: either the action is topologically conjugate to isometries on the circle $mathbb S^1$ (thi
Hindman and Leader first introduced the notion of semigroup of ultrafilters converging to zero for a dense subsemigroups of $((0,infty),+)$. Using the algebraic structure of the Stone-$breve{C}$ech compactification, Tootkabani and Vahed generalized a
In this paper we consider the classical problem of computing linear extensions of a given poset which is well known to be a difficult problem. However, in our setting the elements of the poset are multivariate polynomials, and only a small admissible
We consider dynamical systems $T: X to X$ that are extensions of a factor $S: Y to Y$ through a projection $pi: X to Y$ with shrinking fibers, i.e. such that $T$ is uniformly continuous along fibers $pi^{-1}(y)$ and the diameter of iterate images of
We consider an alternative to inflation for the generation of superhorizon perturbations in the universe in which the speed of sound is faster than the speed of light. We label such cosmologies, first proposed by Armendariz-Picon, {it tachyacoustic},