A Bernstein theorem for special Lagrangian graphs

We obtain a Bernstein theorem for special Lagrangian graphs in n-dimensional complex space for arbitrary n only assuming bounded slope, but no quantitative restriction.Comment: 17 page

Power Partial Isometry Index and Ascent of a Finite Matrix

We give a complete characterization of nonnegative integers $j$ and $k$ and a positive integer $n$ for which there is an $n$-by-$n$ matrix with its power partial isometry index equal to $j$ and its ascent equal to $k$. Recall that the power partial isometry index $p(A)$ of a matrix $A$ is the supremum, possibly infinity, of nonnegative integers $j$ such that $I, A, A^2, \ldots, A^j$ are all partial isometries while the ascent $a(A)$ of $A$ is the smallest integer $k\ge 0$ for which $\ker A^k$ equals $\ker A^{k+1}$. It was known before that, for any matrix $A$, either $p(A)\le\min\{a(A), n-1\}$ or $p(A)=\infty$. In this paper, we prove more precisely that there is an $n$-by-$n$ matrix $A$ such that $p(A)=j$ and $a(A)=k$ if and only if one of the following conditions holds: (a) $j=k\le n-1$, (b) $j\le k-1$ and $j+k\le n-1$, and (c) $j\le k-2$ and $j+k=n$. This answers a question we asked in a previous paper.Comment: 11 page

Functional Bandits

We introduce the functional bandit problem, where the objective is to find an arm that optimises a known functional of the unknown arm-reward distributions. These problems arise in many settings such as maximum entropy methods in natural language processing, and risk-averse decision-making, but current best-arm identification techniques fail in these domains. We propose a new approach, that combines functional estimation and arm elimination, to tackle this problem. This method achieves provably efficient performance guarantees. In addition, we illustrate this method on a number of important functionals in risk management and information theory, and refine our generic theoretical results in those cases

On the Erdos-Sos Conjecture for Graphs on n=k+4 Vertices

The Erd\H{o}s-S\'{o}s Conjecture states that if $G$ is a simple graph of order $n$ with average degree more than $k-2,$ then $G$ contains every tree of order $k$. In this paper, we prove that Erd\H{o}s-S\'{o}s Conjecture is true for $n=k+4$.Comment: 18 page