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Shuffling the sheep

\(\def\xx{\mathbf{x}}\def\Real{\mathbb{R}} \)

We consider the problem of flock control: what are the natural constraints on the steering agents in reconfiguring an ensemble of agents?

Our basic setup is following: the agents are modeled as points \(\xx=(x_1,\ldots,x_n), x_k\in X, k=1,\ldots, N\), with \(X\) some terrain (for simplicity, we constrain ourselves here to Euclidean plane, but in general it can be some reasonably tame subset of a manifold). Quasi-static behavior is assumed, under which the system, for each value of the controlling parameters quickly settles into (locally minimizing) the equilibrium whose basin of attraction contains current position.…

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Protected: Curvilinear Origami

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Fact of the day: Brownian centroids

\(\def\Real{\mathbb{R}}\def\Z{\mathcal{Z}}\def\sg{\mathfrak{S}}\def\B{\mathbf{B}}\def\xx{\mathbf{x}}\def\ex{\mathbb{E}}
\)

Consider \(n\) points in Euclidean space, \(\xx=\{x_1,\ldots, x_n\}, x_k\in \Real^d, n\leq d+1\).

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 \(o(x_1,\ldots,x_n)\)) lies at the intersection of the bisectors \(H_{kl}, 1\leq k\lt l\leq n\), hyperplanes of points equidistant from \(x_k, x_l\).

Assume now that the points \(x_k,k=1,\ldots,n\) are independent standard \(d\)-dimensional Brownian motions. What is the law of the centroid?

It quite easy to see that it is a continuous martingale.…

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Fact of the day: Condorset domains, tiling permutations and contractibility

\(\def\Real{\mathbb{R}}\def\Z{\mathcal{Z}}\def\sg{\mathfrak{S}}\def\B{\mathbf{B}}
\)

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 are the Minkowski sums \(\Z(k,l)=s_k\oplus s_l, 1\leq k\lt l\leq n\) of pairs of the segments. Tilings of \(\Z\) by \(n\choose 2\) rhombuses \(\Z(k,l)\) define also an oriented graph (formed by the edges of the rhombuses, and oriented up in the plane). An increasing path from the bottom to the top defines then a permutations in \(\sg_n\): each of the vectors \(e_k\) appears in an increasing path once, and the permutation then is the order of these vectors.…

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Topologically constrained models of statistical physics.

\(\def\Real{\mathbb{R}}
\def\Int{\mathbb{Z}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\def\blob{\mathcal{B}}
\)

The Blob

Consider the following planar “spin model”: the state of the system is a function from \(\Int^2\) into \(\{0,1\}\) (on and off states). We interpret the site \((i,j), i,j\in\Int\) as the plaque, i.e. the (closed) square given by the inequalities \(s_{ij}:=i-1/2\leq x\leq i+1/2; j-1/2\leq y\leq j+1/2\).

To any configuration \(\eta\) we associate the corresponding active domain,
\[
A_\eta=\bigcup_{(i,j): \eta(i,j)=1} s_{ij}.
\]

We are interested in the statistical ensembles supported by the finite contractible active domains – let’s refer to such domains as blobs.…

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hyperbolic geometry of Google maps

\(\def\Real{\mathbb{R}}
\def\views{\mathbb{G}}
\def\earth{\mathbb{E}}
\def\hsp{\mathbb{H}}
\)

Google Maps and the Space of Views

Let’s consider the geometry hidden behind one of the best user interfaces in mobile apps ever, – the smartphone maps, the ones which you can swipe, flick and pinch. We are not talking here about the incredible engine that generates the maps on the fly as your finger waddles over the screen – just about the interface, which in a very intuitive and remarkably efficient way allows you to move between various maps.…

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unimodal category

We say that a continuous function is unimodal if all the upper excursion sets
\(e_f(c):=\{f≥c\}\) are contractible. For a nonnegative function
\(f:\mathbb{R}^d\to\mathbb{R}_+\) with compact support its unimodal category UCat(f) as the least number \(u\) of summands in a decomposition \(f=\sum_{i=1}^u f_i\), where all \(f_i\) are unimodal. Unimodal category is a “soft” version of Lyusternik-Schnirelman category, hence the name. We know how to compute UCat only in one-dimensional situation. In other dimensions, even the simplest questions seem hard, for example:

Find the unimodal category of the suspension of univariate Morse functions, i.e.

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linear search

triangle_rich
Linear search problems deals with locating an object hidden on a half line (“long road”) by an absent-minded robot which gather everything it sees (see, e.g. here for more details). A program for the robot is a sequence \(0<x_1<x_2<\ldots\) of excursions it takes until the object is found.

In the situation when the object is hidden at a random location with a known distribution function \(F\), the problem of finding an optimal trajectory can be translated into understanding the dynamics of certain area preserving mappings.…

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