I thought it would be interesting to check what the 2017′ Tax Cuts and Jobs Act did to Mankiw’s productivity.…

]]>I thought it would be interesting to check what the 2017′ Tax Cuts and Jobs Act did to Mankiw’s productivity. The effective tax rates for savvy operators (I am sure he can and does hire good accountants) went way down, and thus the incentives to work, as he surmised, increased a lot.

The picture on the right illustrate what (I admit, just a part, albeit arguably most visible part of) Mankiw’s output looks like, since 2010. Height of the list for each year is a graphic representation of Mankiw’s productivity. The red line indicates when tax cuts became the law; the timeline goes from bottom to the top. If you mistrust your eye, the average number of his columns went down from 7.7 per year pre tax cut times to mere 5 per year thereafter.

This is surely just one datapoint, and who knows what are the idiosyncratic challenges Mankiw faced over the last 3 years. But it is a fair metric to look at, as he himself offered it as a measure of social efficacy of a certain policy.

Judging by this metric, tax cut was not a success. Perhaps, those who diligently follow Mankiw’s output should insist that his tax rates, should go up, so they enjoy more of their opinions, insights, revelations etc.

]]>Still, important to understand where your department is on the map, if one wants to steer it in some particular direction, or to keep it where it is.

I wrote a short python script, using BeautifulSoup (thank you!) and campus-wide subscription to MathSciNet (thank you too!) – and downloaded the raw numbers: how many items (whichever MathSciNet is indexing: articles, books, theses,…) are authored by folks from ‘1-IL’ (our code) vs.…

]]>Still, important to understand where your department is on the map, if one wants to steer it in some particular direction, or to keep it where it is.

I wrote a short python script, using BeautifulSoup (thank you!) and campus-wide subscription to MathSciNet (thank you too!) – and downloaded the raw numbers: how many items (whichever MathSciNet is indexing: articles, books, theses,…) are authored by folks from ‘1-IL’ (our code) vs. the total output, split by the rough areas as defined by the Mathematical Subject Classification scheme.

Here are the results. The left display shows the *fraction* of the world’s math output that folks at University of Illinois produced since (rather arbitrarily chosen cutoff) 2013… The right display shows the total number of published works in each area across all institutions.

The usual caveats comparing publication numbers in different areas apply: some publish at a much higher rate than others; the disparities somewhat reflected in the right table. Still, the tables below invite some deeper level of introspection.

Say, the number of works in *K*-theory – worldwide – is small. It is small not just because it is harder to write a competent paper there than, say, in game theory, but also because the sum total of people working in the area is low. So a department hosting a high fraction of the world output is akin to being home to an endangered species. Shall we protect our *K*-theorists from extinction? Or let them being taken over by operator algebra and algebraic topology? To exacerbate these existential worries, are our stats correct in the first place? – when typical number of papers in a subject area is *globally* in below a hundred a year, fluctuations start to be very pronounced.

ON the other end, at the highest output areas in math proper (PDEs, Combinatorics, Probability, Number theory…) it is important to remember that within each of them there are broad subareas with *very* different rates of typical output. A lot of publishable papers in, say, combinatorics could have been written pretty fast, but many are deep and use highly technical tools, and would clearly require years of gestation. Perhaps the top level rubrics of MSC provide too coarse a partition to use these numbers algorithmically.

Still, these numbers do tell us something. Judge yourself.

]]>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 $latex \xx=(x_1,\ldots,x_n), x_k\in X, k=1,\ldots, N$, with $latex X$ 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.…

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 $latex \xx=(x_1,\ldots,x_n), x_k\in X, k=1,\ldots, N$, with $latex X$ 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.

We will assume that the mutual interactions between the agents in the flock are governed by the interaction term of the energy

$$

R(\xx)=\sum_{k\neq l} R(x_k, x_l).

$$

Here $latex R$ is a repulsive term forcing the agents to avoid each other. For the case of planar terrain, Coulomb interactions are especially attractive options, thanks to connections to potential theory and related holomorphic parameterizations:

$$

R(x_k,x_l)=-\log |x_k-x_l|.

$$

To keep the agent confined, we impose a global term,

$$

U_0(\xx)=\sum_k U_0(x_k).

$$

Here $latex U$ is a coercive function ($latex U(x_k):=\alpha |x_k|^2$ is an obvious choice).

Last element of the setup, is the controller term, – or, rather, an explicit dependence of the global energy sector on some control parameters.

We consider here two versions:

- The steering agent is herself embedded into the terrain, and her contribution to the potential also has the repulsive force structure:

$$

U_p(x_k)=U_0(x_k)-K\log|p-x_k), p\in X.

$$

We will be referring to this setting as the “shepherd” setting. - The confining potential function $latex U_p$ is made explicitly dependent on the parameter $latex p\in P$, some space of parameters. As a non-example, we can consider $latex U_p(x)=U_0(x)+\langle p, x\rangle$. More interesting would be to make the quadratic form $latex U$ dependent on the parameter running through positive definite forms.

As we indicated above, the overall dynamics is assumed to be fast on the agents, settling to their equilibrium state corresponding to the fixed values of the steering parameter $latex p$, and slow for the parameter $latex p$. This leads one to the following formulation:

$$

\dot{x}_k=-\frac{1}{\epsilon}\frac{\partial U_p(\xx)}{\partial x_k},\\

U_p(\xx)=\sum_k \left( U_p(x_k)+\sum_{l\neq k} R(x_k,x_l)\right),\\

\dot{p}=u, u\in U.

$$

Intuitively, the model corresponds to Ginibre ensemble (eigenvalues of Gaussian complex matrices), with a perturbative term (corresponding to “shepherd”).

Agents positions $latex x(t)=(x_k(t))_{k=1}^M$, “shepherd” (steering agent) trajectory $latex p(t)$.

Potential function:

$$U(x,p) = \frac{\alpha}{2}\|x\|^2 + \frac{1}{2}\sum_{k\neq j} \log\|x_k-x_j\|^2 + \frac{1}{2}\sum_k\log\|x_k-p\|^2.$$

Dog trajectory initial and final condition: $latex p(0)=p(T)=p_0$

Agents initial condition at equilibrium: $latex \nabla U(x(0),p(0))=0$

Temporal evolution:

$$\dot x(t) = -\nabla U(x(t),p(t))$$

The gradient (for each agent agent) is:

$$

\begin{align}

\nabla_{x_k} U(x,p) &= \nabla \frac{\alpha}{2}\|x_k\|^2 + \frac{1}{2}\sum_{j\neq k} \nabla_{x_k}\log\|x_k-x_j\|^2 + \frac{1}{2} \nabla_{x_k}\log\|x_k-p\|^2\\

&=

\alpha x + \sum_{k\neq j} \frac{x_k-x_j}{\|x_k-x_j\|^2}

+ \frac{x_k-p}{\|x_k-p\|^2}

\end{align}

$$

Example of simulation with herd of 12:

One can see starting (dot) and ending (cross) positions of the agents. The trajectory of the steering agent is solid blue. The agents start and end in an equilibrium position; n3o braiding of the agents observed.

- Assume the first, “shepherd”, model. Fix a (smooth) trajectory $latex p:[0,T]\to X\equiv \Real^2$, starting and ending at the same point $latex p_0$ far away from the origin. This loop would generate a movement among the charges $latex x_k, k=1,\ldots,n$.Do the point return to the same positions reshuffled?
- When they do, one generates so called braid, an element in the fundamental group of the configuration space of $latex n$ (indistinguishable) points in plane. What elements can be generated?
- What elements can be generated if the movements of the shepherd are constrained, for example to stay outside of the convex hull of the agents?

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]]>

\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}}

\)

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**.…

\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}}

\)

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**. In other words, blobs are finite collections of plaques that are topologically equivalent to a disk. We will denote the collection of blobs as \(\blob\).

It is worth remarking that the contractibility is a natural condition if one is interested in local models of statistical physics – such that the transition rates between the states can be defined in terms of local factors, – unlike, say, connectivity. More precisely, one can define a (symmetric) collection of transitions between states \(\eta,\eta’\) differing at just one site \(\sigma\) such that the feasibility of the transitions depends only on the intersections of \(\eta,\eta’\) with a \(3\times 3\) vicinity of \(\sigma\), such that \(\eta’\) is a blob iff \(\eta\) is a blob, and turning \(\blob\) into a connected graph (which we still refer to as \(\blob\)).

Given a *fugacity* \(\rho\), on can turn \(\blob\) into a Markov chain, by setting the jump rate adding a site as \(\rho\), and the jump rate removing a site as \(1\). We will denote this Markov chain as \(\blob(rho)\).

There are several connections of the blob to the staple models of statistical physics. In essence, we are looking at the Ising model with just one contour. The standard constructions pertaining to, say, Wulff shapes, or the general framework of abstract polymer models. Alternatively, one can consider the blob as a self-avoiding loop weighted by the area it encloses.

Whichever the connection, the model seems to be new, and deserves some attention.

Consider the subcritical fugacity \(\rho<1\). In this case, the size of the blob is a.s. finite. What is the tail distribution of its diameter (or area)? If \(\rho=1-\epsilon\) is close to one, what is the expected size of the blob?

Ir seems that the shape of the blob is fractal like, but at a large scale resembling a round disk. Is that true?

Macroscopically, one can expect the blob to drift, converging, upon proper rescaling to Brownian motion. What are the correct scales?

Consider the situation where some of the nodes are stuck in the active state (equivalently, their rate of switching from ON to OFF is zero). In the limit of small fugacity \(\rho\to 0\) the resulting blob will be approximating a certain one-dimensional complex minimizing a functional – i.e. behaving like a Steiner tree.

Can we make this intuitive picture precise?

More on the sites stuck in the active state: assume one starts with the (infinite) configuration where the whole halfplane (say, the upper halfplane) is active. Assume that the site at the origin is always ON, and all other sites perform the usual dynamics for some small fugacity. On every compact subset, the resulting blob will look like a tail starting at the origin and meandering off to infinity. What are the properties of this blob? How many sites are typically active in a box of size \(N\times N\)?

]]>Below are links to Javascript based visualizations

(designed and implemented by Yuriy Mileyko)

of the Rips complexes derived from

Below are links to Javascript based visualizations

(designed and implemented by Yuriy Mileyko)

of the Rips complexes derived from

The sliders allow one to filter on several parameters.

**Warning**: the data takes some time to download and to pre-process; give it a minute or so. Firefox or Chrome are browsers of preference.

*Color-coded by the rank of first local homology, the (spring-embedded) graph of a snapshot of ASN relationships.*