Rank 2 Local Systems, Barsotti-Tate Groups, and Shimura Curves

We develop a descent criterion for $K$-linear abelian categories. Using recent advances in the Langlands correspondence due to Abe, we build a correspondence between certain rank 2 local systems and certain Barsotti-Tate groups on complete curves over a finite field. We conjecture that such Barsotti-Tate groups "come from" a family of fake elliptic curves. As an application of these ideas, we provide a criterion for being a Shimura curve over $\mathbb{F}_q$. Along the way, we formulate a conjecture on the field-of-coefficients of certain compatible systems.Comment: 30 pages. Part of author's PhD thesis. Comments welcome

Alternation, Sparsity and Sensitivity : Bounds and Exponential Gaps

$\newcommand{\sp}{\mathsf{sparsity}}\newcommand{\s}{\mathsf{s}}\newcommand{\al}{\mathsf{alt}}$ The well-known Sensitivity Conjecture states that for any Boolean function $f$, block sensitivity of $f$ is at most polynomial in sensitivity of $f$ (denoted by \s(f)). The XOR Log-Rank Conjecture states that for any $n$ bit Boolean function, $f$ the communication complexity of a related function $f^{\oplus}$ on $2n$ bits, (defined as $f^{\oplus}(x,y)=f(x\oplus y)$) is at most polynomial in logarithm of the sparsity of $f$ (denoted by \sp(f)). A recent result of Lin and Zhang (2017) implies that to confirm the above conjectures it suffices to upper bound alternation of $f$ (denoted \al(f)) for all Boolean functions $f$ by polynomial in \s(f) and logarithm of \sp(f), respectively. In this context, we show the following : * There exists a family of Boolean functions for which \al(f) is at least exponential in \s(f) and \al(f) is at least exponential in \log \sp(f). En route to the proof, we also show an exponential gap between \al(f) and the decision tree complexity of $f$, which might be of independent interest. * As our main result, we show that, despite the above gap between \al(f) and \log \sp(f), the XOR Log-Rank Conjecture is true for functions with the alternation upper bounded by $poly(\log n)$. It is easy to observe that the Sensitivity Conjecture is also true for this class of functions. * The starting point for the above result is the observation (derived from Lin and Zhang (2017)) that for any Boolean function $f$ and $m \ge 2$, deg(f)\le \al(f)deg_2(f)deg_m(f) where $deg(f)$, $deg_2(f)$ and $deg_m(f)$ are the degrees of $f$ over $\mathbb{R}$, $\mathbb{F}_2$ and $\mathbb{Z}_m$ respectively. We also show three further applications of this observation.Comment: 19 pages, 1 figure, Journal versio

An analysis of correlating parameters relating to hot gas ingestion characteristics of jet VTOL aircraft

Jet VTOL fighter-type model inlet-air temperature rise analysis with various exhaust pressure ratios and gas temperatures and surface wind velocities for correlating parameter