Ali Belabbas proved the following clever result. Consider a Riemannian manifold \(M^m\), and the gradient flow of a generic function \(f\). Then the \(\omega\)-limit of a trajectory starting near a local maxima (i.e., with the starting point drawn from a density \(\lambda^mf(\lambda x)dx\)), consists, with asymptotic certainty as \(\lambda\to\infty\), of at most two local minima […]
a tale of two whiskers Read More »
A few years ago, I suggested to Daniel Feshbach at Cynthia Sung‘s lab to play with a curved origami design for a self-folding walking robot. He took off with that raw idea, and brought an extraordinary amount of engineering insight and ingenuity… (Here’s one of the first tries.) Now, the CurveQuad (I like to call
news of the day: CurveQuad (a.k.a. FlatBot) Read More »
2. Around Arrow published, – a riveting tale of cold war planning, trans-Atlantic rivalries, RAND corporation characters, lack of social networks in 1950-ies, and the ensuing non-emergence of applied topology half a century ago, – all there. No, really.
news of the day: LaTeX waste; around Arrow Read More »
This visual continues the exploration started here. I used the dataset of MathSciNet records of the publications by department members, collected by the IGL participants. Each article reviewed by MSN carries a few MSC codes, roughly classifying them. I used these codes to create a sankey diagram, showing who published in what area. A few
math department and its research Read More »
\(\def\Real{\mathbb{R}}\def\tree{\mathcal{T}}\) Consider a path metric space \(M\) and a continuous function \(f:M\to \Real\). For simplicity, all such pairs here will have sublevel sets of \(f\) compact (so that the function is bounded from below). Then one can form the so-called merge tree (see, e.g. here, for motivation, and some history), a new topological space (let’s
\(\def\Real{\mathbb{R}}\def\Z{\mathcal{Z}}\def\sg{\mathfrak{S}}\def\B{\mathbf{B}}\def\xx{\mathbf{x}}\def\ex{\mathbb{E}}\) Consider $latex n$ points in Euclidean space, $latex \xx={x_1,\ldots, x_n}, x_k\in \Real^d, n\leq d+1$. Generically, there is a unique sphere in the affine space spanned by those points, containing all of them. This centroid (which we will denote as $latex o(x_1,\ldots,x_n)$) lies at the intersection of the bisectors $latex H_{kl}, 1\leq k\lt l\leq n$,
Brownian centroids Read More »
\(\def\Real{\mathbb{R}}\def\Z{\mathcal{Z}}\def\sg{\mathfrak{S}}\def\B{\mathbf{B}}\) Consider a collection of vectors \(e_1,\ldots,e_n\) in the upper half-plane, such that \( e_k=(x_k,1)\) and \( x_1>x_2\gt \ldots \gt x_n\). Minkowski sum of the segments ( s_k:=[0,e_k]) is a zonotope ( \Z). Rhombus in this context is the Minkowski sums \( \Z(k,l)=s_k\oplus s_l, 1\leq k\lt l\leq n\) of a pair of the segments, perhaps
domain restrictions and topology Read More »
There is no excerpt because this is a protected post.
Protected: kod*fest Read More »
