Convex Integer Optimization by Constantly Many Linear Counterparts

In this article we study convex integer maximization problems with composite objective functions of the form $f(Wx)$, where $f$ is a convex function on $\R^d$ and $W$ is a $d\times n$ matrix with small or binary entries, over finite sets $S\subset \Z^n$ of integer points presented by an oracle or by linear inequalities. Continuing the line of research advanced by Uri Rothblum and his colleagues on edge-directions, we introduce here the notion of {\em edge complexity} of $S$, and use it to establish polynomial and constant upper bounds on the number of vertices of the projection \conv(WS) and on the number of linear optimization counterparts needed to solve the above convex problem. Two typical consequences are the following. First, for any $d$, there is a constant $m(d)$ such that the maximum number of vertices of the projection of any matroid $S\subset\{0,1\}^n$ by any binary $d\times n$ matrix $W$ is $m(d)$ regardless of $n$ and $S$; and the convex matroid problem reduces to $m(d)$ greedily solvable linear counterparts. In particular, $m(2)=8$. Second, for any $d,l,m$, there is a constant $t(d;l,m)$ such that the maximum number of vertices of the projection of any three-index $l\times m\times n$ transportation polytope for any $n$ by any binary $d\times(l\times m\times n)$ matrix $W$ is $t(d;l,m)$; and the convex three-index transportation problem reduces to $t(d;l,m)$ linear counterparts solvable in polynomial time

Many lives in many worlds

I argue that accepting quantum mechanics to be universally true means that you should also believe in parallel universes. I give my assessment of Everett's theory as it celebrates its 50th anniversary.Comment: Nature version with better graphics at http://www.nature.com/nature/journal/v448/n7149/full/448023a.html Everett bio and other links at http://space.mit.edu/home/tegmark/quantum.htm

Summaries of the Presentations at the EGALS Seminar 2017 at Katowice University, Poland

Summaries of the presentations given at the seminar of the Educational Group of Animal Law Studies

Statistical Properties of Many Particle Eigenfunctions

Wavefunction correlations and density matrices for few or many particles are derived from the properties of semiclassical energy Green functions. Universal features of fixed energy (microcanonical) random wavefunction correlation functions appear which reflect the emergence of the canonical ensemble as the number of particles approaches infinity. This arises through a little known asymptotic limit of Bessel functions. Constraints due to symmetries, boundaries, and collisions between particles can be included.Comment: 13 pages, 4 figure

M is for the Many Things

My basic argument is this: Motherhood is a central but confusing icon within our social structure. It is at once domination and dominated, much as mothers are both revered and regulated. The reverence and regulation are not so much in conflict as in league. The rules remind women of how to behave in order to stay revered. This reverence is something more than a fan club for mothers. It matters in such practical and concrete ways as keeping one\u27s children, having credibility in court, getting promoted at work, and so on