Math in the Media

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Image of the month image of the month

"Penrose Pursuit, 2011," by Kerry Mitchell, is a tessellation of Penrose tiles. In this set, there are two different tile shapes, a fat rhombus and a thin rhombus. Read more about this image and see more in his Mathematically-Inspired Images gallery on AMS Mathematical Imagery.

Tony Phillips

Tony Phillips' Take on Math in the Media
A monthly survey of math news

This month's topics:

A "Rosetta Stone" from Linear Algebra

The September 23, 2019 press release from Fermilab was not very widely picked up (it did make Mashable India), perhaps because there were no catchy graphics. But the story is remarkable: as Stephen Parke tells it, "For years physicists have used important theorems in linear algebra to quickly calculate solutions to the most complicated problems. This August, three theoretical physicists ... turned the tables and, in the context of particle physics, discovered a fundamental identity in linear algebra." This was part of their work on neutrino oscillations, available here. Parke is one of the three; the others are Peter Denton (BNL) and Xining Zhang, a University of Chicago graduate student. Parke: "The identity relates eigenvectors and eigenvalues in a direct way that hadn't been previously recognized. Eigenvectors and eigenvalues are two important ways of reducing the properties of a matrix to their most basic components ... . The eigenvectors identify the directions in which a transformation occurs, and the eigenvalues specify the amount of stretching or compressing that occurs." [Note that the eigenvalues can be complex numbers]. The identity is in fact a formula giving the absolute value of the $j$th component of an eigenvector in terms of the eigenvalues of the matrix and the eigenvalues of a certain sub-matrix depending on $j$.

    Parke and his collaborators were convinced that such a basic formula must already exist in the literature, since eigenvalues and eigenvectors (so named by Hilbert in 1904) have been studied since Laplace, but could not find a reference. Finally they were put in touch with Terry Tao, who had done related work: "When we presented Tao with our result, he cheerfully declared that it was, in fact, the discovery of a new identity, and he provided several mathematical proofs, which have now been published online."

Progress in rapid integer factorization

Quantum computing is a work in progress, with the state of the art often measured by success in integer factorization. One method is Shor's algorithm (explained in these pages here) but there are others, mainly what is called Adiabatic Quantum Computing (AQC): see the article by Xinhua Peng et al. (published in 2008 as Phys. Rev. Lett 101 220405). The idea is to set up factorization of an integer $F$ as a minimization problem, where the function to be minimized is $E(X,Y) = (XY-F)^2$ for integers $X,Y$ in an appropriate range.

"Integer factorization using stochastic magnetic tunnel junctions" is a letter in Nature (September 19, 2019), by Shunsuke Fukami (Tohoku), Kerem Camsari (Purdue) and four collaborators, which proposes implementing the AQC factorization algorithm in a non-quantum way, using p-bits (probabilistic bits) instead of q-bits.

  • "Individual p-bits are stochastic building blocks with a normalized output $m_i$ that takes on the values 0 and 1 with probabilities $P_0$ and $P_1$, respectively. These probabilities are controlled by their normalized inputs $I_i$; for $I_i = 0$ they are equal ($P_0 = P_1 = 0.5$), large positive $I_i$ pins the output $m_i$ to 1 ($P_0 = 0, P_1 = 1$) and large negative $I_i$ pins $m_i$ to 0 ($P_0 = 1, P_1 = 0$). This is similar to the behaviour of a binary stochastic neuron, a well known concept in the field of stochastic neural networks and machine learning. These p-bits can be used to perform useful functions by interconnecting them so that the $i$th p-bit is driven by a synaptic input $I_i$ that is a function of all the other outputs {$m_1,\dots, m_n$}." For a minimization problem like the one at hand one takes $$I_i = -\frac{\partial E(m_1,\dots, m_n)}{\partial m_i}.$$

The p-bits are physically realized by stochastic magnetic tunnel junctions, which are "developed by modifying market-ready magnetoresistive random-access memory technology" and which can operate at room temperature.

In terms of the binary representations $(x_P,\dots,x_1,1)$ and $(y_Q,\dots,y_1,1)$ for the factors $X$ and $Y$ (which can be assumed to be odd), the "cost function" $E$ to be minimized is $$E( x_P,\dots,x_1;y_Q,\dots,y_1) = \left[(\sum_{p=0}^P 2^px_p)(\sum_{q=0}^Q 2^qy_q)-F\right]^2.$$ Note that the binary digits $x_i, y_j$ of $X$ and $Y$ take the role of $P+Q$ independent variables in this formulation.

The authors used this set-up to run what they describe as a "proof-of-concept experiment for probabilistic computing using spintronics technology." They used a four p-bit configuration to factorize 35, a six-bit configuration for $F=161$ (see Figure) and an eight-bit configuration for $F=945$.

factorization of 23x7

Factoring $F=161$. The uncorrelated (top) and correlated (bottom) state of the system when six p-bits are used to factorize 161. The $x$ and $y$ axes show the factors $X=\sum_{p=0}^P 2^px_p$ and $Y=\sum_{q=0}^Q 2^qy_q$. The ranges were chosen $1\leq X\leq 7$, so $(x_2, x_1, 1)$ uses two p-bits; and $1\leq Y\leq 31$, so $(y_4, y_3, y_2, y_1, 1)$ uses four p-bits. The authors show similar images for the factorization of 35 using four p-bits ($P=Q=2$) and 945 using eight p-bits ($P=3$ and $Q=5$). Image from Nature 573 390-393, used with permission.

Chaos in the Wall Street Journal

"The Chaos of Weather Forecasts," the latest installment in Eugenia Cheng's (paywall-protected) "Everyday Math" series in the WSJ ran on October 3, 2019. The sub-head says it all: "In principle, math should be able to tell us whether it will rain tomorrow—but in practice, things get complicated quickly." Cheng elaborates: "The weather is what mathematicians call a dynamical system. This means that if we input data about the system as it is right now, a set of mathematical equations is supposed to tell us what state the system will be in at any given time in the future. In some systems, a slight difference in starting data causes only a slight difference in outcome. ... With the weather, however, a very slight difference in starting conditions can yield a wildly different outcome further down the line. ... This is a feature of the field of mathematics called chaos theory, which is sometimes illustrated by the idea that a butterfly flapping its tiny wings in one part of the world can result in a storm somewhere else.

Chaotic systems have another feature: Not only can we not predict the future given our knowledge of the present, but once we reach the future, we can't retrace our steps to find out exactly what caused it. Things always happen for a reason, but we can't always tell what that reason is. A butterfly might have caused a storm.

The animated GIF at the top, by Tomasz Walenska, shows a fanciful butterfly spewing lighting and rain.

Heliknotons

"Three-dimensional crystals of adaptive knots" by Jung-Shen Tai and Ivan Smalyukh (University of Colorado, Boulder) appeared in Science, September 27, 2019. "We introduce energetically stable, micrometer-sized knots in helical fields of chiral liquid crystals. While spatially localized and freely diffusing in all directions, they resemble colloidal particles and atoms, self-assembling into crystalline lattices with open and closed structures. These knots are robust and topologically distinct from the host medium $\dots$ ." Tai and Smalyukh call such a localized 3D configuration a heliknoton.

gas

crystal

Top: a gas of heliknotons in a liquid crystal sample of thickness 30μm. Bottom: A 2D closed rhombic latice of heliknotons. The individual heliknotons are approximately $9\times 6$μm. Polarizing optical micrographs courtesy of Jung-Shen Tai.

In mathematical terms, a liquid crystal with rod-like molecules can be thought of as a 3-dimensional volume with a unit vector field (the molecules themselves may not be oriented but in a convex medium they can be coherently "decorated" with a direction); a helical field is described by a helical axis direction (perpendicular to the visual plane in the micrographs above), such that the vector field is perpendicular to that direction and rotates uniformly about it.

nematic

Mathematical idealization of a liquid crystal with rod-like molecules in a uniform helical field: a vector field spiraling around helical axes.

A heliknoton is topologically non-trivial in two related ways, illustrated here for the simplest example.

The vector field, which can be specified by a map ${\bf n}$ from the micro-region containing the heliknoton to the sphere of unit directions in 3-space, is a "topological soliton:" locally like the tangent field to the Hopf fibration, in that the inverse images of any two distinct directions form a Hopf link: two circles that link exactly once.

fibration

The vector field ${\bf n}$, in the micro-region containing the heliknoton, has the property that the locus ${\bf n}^{-1}({\bf a})$ of points where the vector field has direction ${\bf a}$ is a circle and links ${\bf n}^{-1}({\bf b})$ exactly once, for any other direction ${\bf b}$. Derived from an image in Science 365, 1449-1453.

A helical field can be described by three mutually perpendicular unit fields. One is ${\bf n}$, which points along the molecules; the other two are χ, the unit field along the helical axis, and τ, the unit field perpendicular to both χ and ${\bf n}$. When the field is deformed to make a heliknoton, the field ${\bf n}$ remains everywhere smooth and of unit modulus, but the other two become singular (i.e. their length goes to zero), along a set of curves called vortex lines. Those vortex lines are trefoil knots.

knot

A vortex line through the heliknoton. This knot is known as the trefoil, and also as the T(2,3) torus knot. Derived from an image in Science 365, 1449-1453.

The example just described is what Tai and Smalyukh call a "$Q=1$ heliknoton." But there is more: "In addition to the $Q=1$ heliknotons $\dots$, we also find larger $Q=2$ topological solitons. $\dots$ A $Q=2$ heliknoton contains a larger region of distorted helical background in both the lateral and axial directions. Preimages for two antiparallel vertical orientations of ${\bf n}$ form a pair of [i.e. two separate] Hopf links $\dots$, like all other preimage pairs. Singular vortex lines form closed cinquefoil T(2,5) torus knots." The article shows simulations of $Q=3$ heliknotons (3 sets of Hopf links, T(2,7) torus knots. Note the relation: T$(2,2Q+1)$ torus knot with a $Q$-heliknoton); Dr. Tai tells me "we are planning to publish more articles on heliknotons with higher Q in the near future." Stay tuned.

Tony Phillips
Stony Brook University
tony at math.sunysb.edu

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