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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: dice are chiral

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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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JACT

coming soon.…

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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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Topology in Motion @ICERM Fall’16

If you are interested in Applied Topology, tend to plan far, far ahead and have some free time on your hands in Fall 2016, check this out, and let us know!

logo_tim

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more on humanitarian disaster

Bug in the system

Handling of all – not just Salaita’s – cases by U of I is deficient, with the final approval coming only after the move to campus is done. We all are culpable, registering this issue when hired, then happily forgetting about it, waiting for that ticking bomb to explode, relying on assumptions. Debugging this procedure should be done as soon as possible. It seems the administration acts on it.

Role of Trustees

in terms of procedure they were fully within their prerogative not confirming Salaita’s appointment, as

the Board of Trustees exercises final authority over the University.

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arrieregarde battles of postmodernism

The epicenter of the battle around Salaita’s offer withdrawal seems to be located at the question, whether or not political considerations are admissible when appointing a scholar. I checked what Dr. Salaita himself writes about the matter in a 2008 paper dealing specifically with academic freedom.

A general remark on the paper: most of it consists of unsolicited, unadulterated punditry, bloated with (intentionally, I guess) opaque and (sometimes comically) elevated prose, mainly arguing with a group of Internet writers expressing their views too noisily and/or too effectively, to author’s taste.…

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