Nosimpler

The Pentagon wants to give away 13,000 mine-resistant, ambush-protected trucks because they have outlived their original purpose.

Although the trucks' armored bodies are credited with protecting U.S. troops from roadside bombs in Iraq and Afghanistan, military planners want more-versatile vehicles that can be deployed quickly as troop levels decrease. A full-size MRAP (pronounced EM-rap, of course) weighs about 40,000 pounds, stands 10 feet tall and costs the Pentagon about $500,000 new...

Interest from foreign militaries has been tepid. But they are a hit with stateside police agencies. Almost 200 trucks have been distributed to police departments since August and requests are pending for an additional 750 trucks. The vehicles, many of which feature machine-gun turrets, are off-limits to private citizens and businesses.

Lucky recipients run from the Ohio State University campus police force to Florence County, S.C., which replaced an armored vehicle from the 1970s that the sheriff department's SWAT team had used for about 15 years. A new armored truck would have cost at least $188,000...

"Nobody will want them," says Dean Lockwood, an analyst with Forecast International Inc. "The Afghan terrain is hell on vehicles. It's eating them alive."

For police, though, the bulky trucks project a show of force at hostage incidents, civil disturbances and other situations where SWAT officers with military-grade weapons, uniforms and helmets are deployed.

I was looking at a collection of perfectly looped gifs on Buzzfeed and thinking about how they remind me of sample-based electronic music. In both cases, you’re taking a piece of a linear recording and making it cyclical. Do it wrong and it’s extremely irritating. Do it right and it’s mesmerizing. I’ve given a lot of thought to how looping a segment of audio changes its meaning, but am only just starting to think about the visual equivalent.

Like samples in hip-hop and techno, good animated gifs take something familiar and make it strange, or they take something strange and make it familiar. And like samples, gifs are especially expressive when they come from pop culture. Unlike hip-hop and techno producers, gif creators are quite anonymous; I can’t name a single one, and their work is almost always unattributed on the web.

Usually gifs are silly, but sometimes they can be moving. I like this one of a kid riding a dirtbike through the desert from Breaking Bad. No spoilers, but like every character on the show, the kid meets with tragedy; it’s nice to imagine him riding through the desert forever unharmed.

Pairing video loops with sound has so far been mostly super irritating. There’s a TV commercial in rotation right now that does that, and it just looks like an epileptic seizure. Maybe because it’s very difficult to align the optimally satisfying video loop points with the optimally satisfying audio ones. There are some wonderful video remixes based on the idea of looping short segments, but there the priority is sound; the video edits are totally nonsensical, though still satisfying in their own way.

Maybe the only way to pair audio and video loops is to have one of them necessarily be meaningless. Or maybe we should just enjoy the sample and the gif on their own terms.

Author(s): Jianyong Cen, Ping Yuan, and Simin Xue

Ball lightning (BL) has been observed with two slitless spectrographs at a distance of 0.9 km. The BL is generated by a cloud-to-ground lightning strike. It moves horizontally during the luminous duration. The evolution of size, color, and light intensity is reported in detail. The spectral analysis...

[Phys. Rev. Lett. 112, 035001] Published Fri Jan 17, 2014

We investigate the problem of factorizing a matrix into several sparse matrices and propose an algorithm for this under randomness and sparsity assumptions. This problem can be viewed as a simplification of the deep learning problem where finding a factorization corresponds to finding edges in different layers and values of hidden units. We prove that under certain assumptions for a sparse linear deep network with n nodes in each layer, our algorithm is able to recover the structure of the network and values of top layer hidden units for depths up to O~(n1/6). We further discuss the relation among sparse matrix factorization, deep learning, sparse recovery and dictionary learning.

We give algorithms with provable guarantees that learn a class of deep nets in the generative model view popularized by Hinton and others. Our generative model is an n node multilayer neural net that has degree at most n for some \gamma less than 1 and each edge has a random edge weight in [-1; 1]. Our algorithm learns almost all networks in this class with polynomial running time. The sample complexity is quadratic or cubic depending upon the details of the model. The algorithm uses layerwise learning. It is based upon a novel idea of observing correlations among features and using these to infer the underlying edge structure via a global graph recovery procedure. The analysis of the algorithm reveals interesting structure of neural networks with random edge weights.

• Sunday Morning Insight: Randomization is not a dirty word
• Sunday Morning Insight: Sharp Phase Transitions in Machine Learning ?

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A "no-knowledge" measurement of an open quantum system yields no information about any system observable; it only returns noise input from the environment. Surprisingly, performing such a no-knowledge measurement can be advantageous. We prove that a system undergoing no-knowledge monitoring has reversible noise, which can be cancelled by directly feeding back the measurement signal. We show how no-knowledge feedback control can be used to cancel decoherence in an arbitrary quantum system coupled to a Markovian reservoir that is being monitored. Since no-knowledge feedback does not depend on the system state or Hamiltonian, such decoherence cancellation is guaranteed to be general, robust and can operate in conjunction with any other quantum control protocol. As an application, we show that no-knowledge feedback could be used to improve the performance of dissipative quantum computers subjected to local loss.

A former Halliburton manager was sentenced Tuesday to one year of probation for destroying evidence in the aftermath of BP's massive 2010 oil spill in the Gulf of Mexico.

EMBO reports:

The new approach is possible due to recent advances in mathematics and  computer technologies. The system involves giving all scientists an annual, unconditional fixed amount of funding to conduct their research. All funded scientists are, however, obliged to donate a fixed percentage of all of the funding that they previously received to other researchers. As a result, the funding circulates through the community, converging on researchers that are expected to make the best use of it. “Our alternative funding system is inspired by the mathematical models used to search the internet for relevant information,” said Bollen. “The decentralized funding model uses the wisdom of the entire scientific community to determine a fair distribution of funding.”

The authors believe that this system can lead to sophisticated behavior at a global level. It would certainly liberate researchers from the time-consuming process of submitting and reviewing project proposals, but could also reduce the uncertainty associated with funding cycles, give researchers much greater flexibility, and allow the community to fund risky but high-reward projects that existing funding systems may overlook.

“You could think of it as a Google-inspired crowd-funding system that encourages all researchers to make autonomous, individual funding decisions towards people, not projects or proposals,” said Bollen. “All you need is a centralized web site where researchers could log-in, enter the names of the scientists they chose to donate to, and specify how much they each should receive.”

The authors emphasize that the system would require oversight to prevent misuse, such as conflicts of interests and collusion.

The (short) paper itself is here, by Johan Bollen, David Crandall, Damion Junk, Ying Ding, and Katy Börner.

For the pointer I thank Charles Klingman.

Cognitive psychologist Mary Czerwinski and her boyfriend were having a vigorous argument as they drove to Vancouver, B.C., from Seattle, where she works at Microsoft Research. She can’t remember the subject, but she does recall that suddenly, his phone went off, and he read out the text message: “Your friend Mary isn’t feeling well. You might want to give her a call.”

At the time, Czerwinski was wearing on her wrist a wireless device intended to monitor her emotional ups and downs. Similar to the technology used in lie detector tests, it interprets signals such as heart rate and electrical changes in the skin. The argument may have been trivial, but Czerwinski’s internal response was not. That prompted the device to send a distress message to her cellphone, which broadcast it to a network of her friends. Including the one with whom she was arguing, right beside her.

There is more here.

Interneuron cell types are fit to function

Nature 505, 7483 (2014). doi:10.1038/nature12983

Authors: Adam Kepecs & Gordon Fishell

Understanding brain circuits begins with an appreciation of their component parts — the cells. Although GABAergic interneurons are a minority population within the brain, they are crucial for the control of inhibition. Determining the diversity of these interneurons has been a central goal of neurobiologists,

A Deventer scribe, writing around 1420, found his manuscript ruined by a urine stain left there by a cat the night before. He was forced to leave the rest of the page empty, drew a picture of a cat and cursed the creature with the following words:

“Hic non defectus est, sed cattus minxit desuper nocte quadam. Confundatur pessimus cattus qui minxit super librum istum in nocte Daventrie, et consimiliter omnes alii propter illum. Et cavendum valde ne permittantur libri aperti per noctem ubi cattie venire possunt.”

[Here is nothing missing, but a cat urinated on this during a certain night. Cursed be the pesty cat that urinated over this book during the night in Deventer and because of it many others [other cats] too. And beware well not to leave open books at night where cats can come.]

Collective violence in direct confrontations between two opposing groups happens in short bursts wherein small subgroups briefly attack small numbers of opponents, while the others form a non-fighting audience. The mechanism is fighters' synchronization of intentionalities during preliminary interactions, by which they feel one and overcome their fear. To explain these bursts, subgroups' small sizes and leaders' role, a social influence model and a synchronization model are compared.

The observed cosmic acceleration was attributed to an exotic dark energy in the framework of classical general relativity. The dark energy behaves very similar with vacuum energy in quantum mechanics. However, once the quantum effects are seriously taken into account, it predicts a completely wrong result and leads to a severe fine-tuning. To solve the problem, the exact meaning of time in quantum mechanics is reexamined. We abandon the standard interpretation of time in quantum mechanics that time is just a global parameter, replace it by a quantum dynamical variable playing the role of physical clock. We find that synchronization of two spatially separated clocks can not be precisely realized at quantum level. There is an intrinsic quantum uncertainty of distant clock time, which implies an apparent vacuum energy fluctuation and gives an observed dark energy density $\rho_{de}=\frac{6}{\pi}L_{P}^{-2}L_{H}^{-2}$ at tree level approximation, where $L_{P}$ and $L_{H}$ are the Planck and Hubble scale cutoffs. The fraction of the dark energy is given by $\Omega_{de}=\frac{2}{\pi}$, which does not evolve with the internal clock time. The "dark energy" as a quantum cosmic variance is always seen comparable with the matter energy density by an observer using the internal clock time. The corrected distance-redshift relation of cosmic observations due to the distant clock effect are also discussed, which again gives a redshift independent fraction $\Omega_{de}=\frac{2}{\pi}$. The theory is consistent with current cosmic observations.

Neural system prediction and identification challenge.

Front Neuroinform. 2013;7:43

Authors: Vlachos I, Zaytsev YV, Spreizer S, Aertsen A, Kumar A

Abstract
Can we infer the function of a biological neural network (BNN) if we know the connectivity and activity of all its constituent neurons?This question is at the core of neuroscience and, accordingly, various methods have been developed to record the activity and connectivity of as many neurons as possible. Surprisingly, there is no theoretical or computational demonstration that neuronal activity and connectivity are indeed sufficient to infer the function of a BNN. Therefore, we pose the Neural Systems Identification and Prediction Challenge (nuSPIC). We provide the connectivity and activity of all neurons and invite participants (1) to infer the functions implemented (hard-wired) in spiking neural networks (SNNs) by stimulating and recording the activity of neurons and, (2) to implement predefined mathematical/biological functions using SNNs. The nuSPICs can be accessed via a web-interface to the NEST simulator and the user is not required to know any specific programming language. Furthermore, the nuSPICs can be used as a teaching tool. Finally, nuSPICs use the crowd-sourcing model to address scientific issues. With this computational approach we aim to identify which functions can be inferred by systematic recordings of neuronal activity and connectivity. In addition, nuSPICs will help the design and application of new experimental paradigms based on the structure of the SNN and the presumed function which is to be discovered.

PMID: 24399966 [PubMed]

United States District Judge Jed S. Rakoff is already kind of a hero to me, given that he’s the guy who rejected a “do not admit wrongdoing” settlement between Citigroup and the SEC over mortgage-backed securities fraud because, according to Rakoff, the proposed settlement was “neither fair, nor reasonable, nor adequate, nor in the public interest.”

More recently Rakoff has written a fine essay in the New York Review of Books entitled The Financial Crisis: Why Have No High-Level Executives Been Prosecuted? which I will summarize below but is well worth your time to read.

Rakoff’s essay

First Rakoff made the point that if there was no intentional fraud we should not scapegoat people and put them to jail. But on the other hand, if there was intentional fraud, then it’s a reflection on a dysfunctional justice system that nobody has gone to jail.

Then he examined that first possibility and found it unlikely, given that “… the Financial Crisis Inquiry Commission, in its final report, uses variants of the word “fraud” no fewer than 157 times in describing what led to the crisis…” In fact, fraud permeated at every level.

The Department of Justice (DOJ) has focused on explaining why nobody has gone to jail in spite of the existence of fraud. They have three reasons.

First, the DOJ claims it’s hard to prove intent for high-level management. But Rakoff demurs on this point, explaining that in cases of accounting fraud, “willful blindness” or “conscious disregard” is a well-established basis on which federal prosecutors have asked juries to infer intent.

Second, since many counterparties were “sophisticated,” it’s difficult to prove “reliance“. Again Rakoff demurs, pointing out that “In actuality, in a criminal fraud case the government is never required to prove—ever—that one party to a transaction relied on the word of another.”

Third, because of the “Too Big To Jail” problem, namely that prosecuting fraud would kill the economy. To this, Rakoff points out what that means in terms of class: that poor people can be prosecuted but the rich are protected.

Next, Rakoff says what he thinks is actually happening. First he discounts the revolving door: he thinks lawyers are thoroughly incentivized to make a name for themselves. Then what? He’s got three reasons.

Well, first, people were distracted. The FBI was distracted by terrorists, and the SEC was focused on Ponzi schemes and insider trading. The DOJ was inexperienced and the Southern District US Attorney’s Office was also focused on insider trading. And given the complexity and incentives, it’s hard for a given lawyer to decide to go with an MBS case instead of insider trading.

Second, the government had direct conflict in the fraud, given that the Fed and the regulators had deregulated everything in sight and then kept interest rates low to keep the mortgage machine going. They also meddled a lot during the crisis, deciding which failing bank should be taken over by whom. It made it hard for them to admit shit went wrong.

Finally, it’s because it’s now in vogue to prosecute corporations instead of people, but that really doesn’t work. Here’s Rakoff on this prosecutorial method:

Although it is supposedly justified because it prevents future crimes, I suggest that the future deterrent value of successfully prosecuting individuals far outweighs the prophylactic benefits of imposing internal compliance measures that are often little more than window-dressing. Just going after the company is also both technically and morally suspect. It is technically suspect because, under the law, you should not indict or threaten to indict a company unless you can prove beyond a reasonable doubt that some managerial agent of the company committed the alleged crime; and if you can prove that, why not indict the manager? And from a moral standpoint, punishing a company and its many innocent employees and shareholders for the crimes committed by some unprosecuted individuals seems contrary to elementary notions of moral responsibility.

And then his final conclusion:

So you don’t go after the companies, at least not criminally, because they are too big to jail; and you don’t go after the individuals, because that would involve the kind of years-long investigations that you no longer have the experience or the resources to pursue.

Comments

First, I am super grateful for Judge Rakoff’s essay, because as an experienced lawyer he has way more ammunition than I do to explain this stuff from the perspective of what is actually done in law. The “willful blindness” issue is particularly ridiculous. I’m glad to hear that courts have a way to deal with that problem, even if they aren’t using their tools against Jamie Dimon.

I am also grateful to hear him make the point that widespread fraud, unprosecuted, is not simply a theoretical issue. It exposes the dysfunctionality of our justice system and it exposes basic unfairness in society, where depending on how rich you are and how complicated your crime is, you can avoid going to jail. Personally, in the past few months I’ve gone from being angry at the bankers to being angry at the prosecutors.

Finally, I disagree with Rakoff on one point. Namely, his argument against the negative effect of the revolving door. His argument, I stipulate, only works for lawyers in a US Attorney’s office. I don’t think the average SEC lawyer or economist, or for that matter an employee at any captured regulator, has that much incentive to take on a big MBS case and be hard-assed. I think we would have seen more cases if that were true.

A new variational method, the principle of least radix economy, is formulated. The mathematical and physical relevance of the radix economy, also called digit capacity, is established, showing how physical laws can be derived from this concept in a unified way. The principle reinterprets and generalizes the principle of least action yielding two classes of physical solutions: least action paths and quantum wavefunctions. A new physical foundation of the Hilbert space of quantum mechanics is then accomplished and it is used to derive the Schr\"odinger and Dirac equations and the breaking of the commutativity of spacetime geometry. The formulation provides an explanation of how determinism and random statistical behavior coexist in spacetime and a framework is developed that allows dynamical processes to be formulated in terms of chains of digits. These methods lead to a new (pre-geometrical) foundation for Lorentz transformations and special relativity. The Parker-Rhodes combinatorial hierarchy is encompassed within our approach and this leads to an estimate of the interaction strength of the electromagnetic and gravitational forces that agrees with the experimental values to an error of less than one thousandth. Finally, it is shown how the principle of least-radix economy naturally gives rise to Boltzmann's principle of classical statistical thermodynamics. A new expression for a general (path-dependent) nonequilibrium entropy is proposed satisfying the Second Law of Thermodynamics.

The title of this post is half-serious: it is designed to catch attention, but I think there are interesting things lurking here which deserve more exploration.

Species

A species describes a class of objects built on finite sets, in a “functorial” way, in the sense that any bijection between finite sets induces a bijection between the objects constructed on those sets. In category language, it is a functor from the category of finite sets and bijections to itself.

Two examples to start with: the species Set, which is the identity; and the species Graph, which constructs all the graphs on the input set.

Substitution

I will be concerned with the important object of substitution of species. If A and B are species, where B builds nothing on the empty set, then A[B] is the species for which an object on a set X is constructed as follows: partition X into non-empty parts in any manner; put a B-structure on each part; and put an A-structure on the set of parts.

For example, if CGraph is the species of connected graphs, then Set[CGraph] = Graph, where = is interpreted rather loosely. This simply says that an arbitrary graph is a disjoint union of connected graphs, with no structure on the set of components.

Substitution into Set is commonly referred to as the exponential principle. The exponential generating function for the labelled objects in Set is just the exponential function, so substitution of a species into Set corresponds to taking the exponential of its (labelled) counting series.

Groups

My main interest is that an oligomorphic permutation group G gives rise to a species: it preserves a canonical relational structure (consisting of all the G-invariant relations made up of tuples of distinct elements); and on a finite set, the species builds all the relational structures embeddable in this canonical structure.

In the group case, substitution of species corresponds to wreath product of permutation groups.

In order to arise from the group case, we require two conditions on the species:

• It should be hereditary, that is, closed under taking substructures: the structure induced on a subset by any object in the species should also belong to the species. Many important species satisfy this.
• It should have the amalgamation property: two structures can be amalgamated over (at least) a common substructure. This is a much more restrictive condition.

Then Fraïssé’s Theorem guarantees the existence of a countable homogeneous structure whose substructures are precisely those in the given class. All numerical data (numbers of labelled and unlabelled structures, cycle index) are then associated with a group, the automorphism group of the countable homogeneous structure.

The groups associated with the species Set and Graph are the symmetric group of countable degree and the automorphism group of the random graph, respectively.

Since connected graphs are not hereditary, the equation relating graphs to connected graphs is not mirrored by a group.

An example

I am interested in equations like A[B] ≈ 2B. So A is thought of as an operator, and B an “eigenspecies” for it. The 2 can be interpreted in the sense that the counting function for labelled objects in the species (or the cycle index of the species, in Joyal’s sense) is doubled; or, more elaborately, that objects in the substituted species are “doubled” copies of objects in B.

The approximation comes because very often there is only one A-structure on a 1-element set, in which case the A[B]-structures and B-structures on a 1-element set are equinumerous. So I will require the doubling only for sets with more than one element.

For example, what would Set[B] ≈ 2B (in this sense) mean? We would need a class of objects so that, on any set of size larger than 1, exactly half of the objects are connected. So the exponential generating function B(x) for labelled structures in B will satisfy exp(B(x)) = 2B(x)−1.

There is such a class, namely Nfree, the class of N-free graphs, those which do not contain a path of length 3 as induced subgraph. This important class has many names (for example, cographs), and many characterisations. In particular,

• it is the smallest class containing the 1-vertex graph and closed under the operations of complementation and disjoint union;
• an N-free graph with more than 1 vertex is connected if and only if its complement is disconnected.

A better example

The reason this example is unsatisfying is that it doesn’t come from a group. There is no countable homogeneous N-free graph, or (said another way) the class of finite N-free graphs does not have the amalgamation property.

But this can be rectified, as was done by Jacinta Covington in her thesis.

To see what is involved, let us see how amalgamation fails. Let a,b,c be three non-adjacent vertices, x a vertex joined to a and b (but not c), and y a vertex joined to b and c (but not a). If we try to amalgamate {a,b,c,x} with {a,b,c,y}, we cannot identify x and y because of their different adjacencies within {a,b,c}; and either joining or not joining a and b results in a path of length 3.

So we need a rule which says that, in any independent set of size 3, external vertices joined to two vertices in the independent set must always be joined to the same pair. So the third vertex must be “distinguished” in some way. Dually, in a triangle, we must distinguish a vertex as the possible single neighbour of an external vertex.

Consider 3-vertex subsets of an N-free graph. In such a set containing one or two edges, there is clearly a distinguished vertex. If the set has no edges, or three edges, there is no vertex obviously distinguished, but as we have seen, there must be a rule which does distinguish one if amalgamation is to be restored. Thus, we need to add a ternary relation which distinguishes one point from each triple of distinct points, so that if the 3-set contains one or two edges, the relation distinguishes the same point as the graph structure does.

One type of structure in which every 3-set has a distinguished vertex is LeavesBTree, the set of leaves of a binary tree. Of any three leaves, two are related to one another more closely than either is related to the third. (So, for example, among {human, chimpanzee, gorilla}, it is the gorilla which is distinguished.)

Now Covington’s work shows that this is exactly the kind of relation required to restore the amalgamation property for N-free graphs. We have:

• LeavesBTree has the amalgamation property;
• The class Nfree⁺ of N-free graphs possessing also the ternary structure of leaves in a binary tree, so that the vertex distinguished by the ternary relation on a 3-set containing one or two edges agrees with the one distinguished by the graph structure, has the amalgamation property.

In particular, there is a countable universal N-free graph which is homogeneous when the ternary relation is added. (We call such a structure homogenizable.) This graph is connected with diameter 2. Paradoxically, although the complement of a finite connected N-free graph is disconnected, the complement of Covington’s graph is connected; indeed, the graph is isomorphic to its complement.

Let CNfree⁺ denote the species of connected N-free graphs enriched with the ternary relation. In a disconnected N-free graph, the ternary relation corresponding to possible added points in an extension is unchanged if points are replaced by others in the same connected component. So there is a LeavesBTree relation on the set of connected components. This with a little further argument shows that LeavesBTree[CNfree⁺] = Nfree⁺ ≈ 2CNfree⁺. So CNfree⁺ is an eigenspecies for LeavesBTree.

However, connected graphs are not hereditary, so this situation is not yet supported by a group. Something different is required.

Since no finite N-free graph is self-complementary, the orbits on finite subgraphs (with more than one vertex) of Covington’s graph come in complementary pairs. The two objects in a pair have the same ternary relation, and make identical contributions to the enumeration or cycle index of the species. So if Nfree* denotes the species of complementary pairs of finite N-free graphs, then we have

LeavesBTree[Nfree*] = Nfree⁺ ≈ 2Nfree*,

where all three species in the equation are realised by oligomorphic groups. (The group of the pair of graphs contains the group of Covington’s graph as a subgroup of index 2.)

The automorphism group of Covington’s graph is primitive, while the wreath product is imprimitive. So the orbit counts and cycle indices associated with oligomorphic groups are not able to determine primitivity in general.

And finally

There is a nice description of the set of N-free graphs homogenenized by the leaves of a given binary tree. Colour the non-leaves of the binary tree black and white in any manner; then join two leaves if their common ancestor is coloured black.

Jan Hubička and Jarik Nešetřil discovered that Covington’s construction is not an accident: under rather mild conditions a structure can be homogenized. (However, we do not know conditions to ensure that a finite number of additional relations suffice; this is one of the big open problems in the area!) So there must be many more interesting phenomena to discover here.

Also, much less is known about Covington’s graph than about the analogous cases of the random graph and Henson’s K_n-free graphs. A topic surely worth exploring.

Questions like those in this post can be explored at the level of counting functions, by asking what it means for a formal power series to be doubled (apart from the constant term) by substitution into another given series, for example. Having worked out which series has this property, one can attempt to identify it using the On-line Encyclopedia of Integer Sequences. In this way, web browsing could become serious mathematical research!

Time travel has captured the public imagination for much of the past century, but little has been done to actually search for time travelers. Here, three implementations of Internet searches for time travelers are described, all seeking a prescient mention of information not previously available. The first search covered prescient content placed on the Internet, highlighted by a comprehensive search for specific terms in tweets on Twitter. The second search examined prescient inquiries submitted to a search engine, highlighted by a comprehensive search for specific search terms submitted to a popular astronomy web site. The third search involved a request for a direct Internet communication, either by email or tweet, pre-dating to the time of the inquiry. Given practical verifiability concerns, only time travelers from the future were investigated. No time travelers were discovered. Although these negative results do not disprove time travel, given the great reach of the Internet, this search is perhaps the most comprehensive to date.

One of the best discoveries I made while researching my thesis is the mathematician Godfried Toussaint. While the bookshelves groan with mathematical analyses of western harmony, Toussaint is the rare scholar who uses the same tools to understand Afro-Cuban rhythms. He’s especially interested in the rhythm known to Latin musicians as 3-2 son clave, to Ghanaians as the kpanlogo bell pattern, and to rock musicians as the Bo Diddley beat. Toussaint calls it “The Rhythm that Conquered the World” in his paper of the same name. Here are eight different representations of it as rendered by Toussaint:

And here it is in my preferred circular notation:

Son clave probably traveled from West Africa to Cuba with the slave trade. It may have arrived in its present form, or it could have evolved from a similar 12/8 pattern, fume-fume. It has a long history; I was delighted to learn from Toussaint that son clave appears under the name “al-thaqil al-awwal” in the Kitāb al-Adwār, a manuscript written in Baghdad in the middle of the Thirteenth Century by the music scholar Safi al-Din al-Urmawi. The whole book is extraordinarily beautiful; click through the image below to see many more.

There’s no way to know how old son clave is, but I would guess that it’s probably very ancient. A forty-thousand-year-old bone flute was found in Germany that plays the major pentatonic scale. Rhythm is probably vastly older than harmony, and for all we know, hominids were chipping away at their stone axes to a son clave beat millions of years ago.

Wherever son clave came from, it’s incredibly popular. Toussaint observes that the beat “is heard in all corners of the world, in almost any type of music, including rhythm and blues, salsa, rockabilly, rock, soukous, jazz, house, and the fusion pop music of scores of countries.” So what makes this rhythm so special? Toussaint has a series of solid mathematical explanations.

By and large, we prefer rhythms that are “maximally even,” meaning that they’re spaced more or less equally in time. Son clave is one of many widely-used beats consisting of five hits per sixteen-step cycle (two bars of 4/4 time counted in eighth notes, in western theory terms.) Think of the sixteen steps as sixteen cubbyholes, each of which can hold one object, one drum hit. Sixteen doesn’t divide by five evenly, so there are several different possible ways to distribute the five hits among the sixteen cubbyholes to make a maximally even beat. Toussaint lists them all, and labels the ones that are in common usage.

Several of these rhythms are rotations of each other, like different modes of the same scale. Rhythms 5 and 11 are “modes” of son clave; rhythms 1, 9, 15, and 16 are “modes” of bossa nova; and rhythm 3 is a “mode” of the rumba. Cool!

Toussaint asks why this combination of five beats distributed across a sixteen-step cycle should be so popular:

Why not eleven [beats], thirteen, or seventeen for example? And what is it that is so singular about five onsets? Why not four, six, or nine? These two numbers, the number of pulses in the cycle of a timeline, and the number of these pulses that are sounded, vary widely among different cultures around the world. It is quite common for the number of pulses in the cycle to be as little as four. In Bulgarian music it may go as high as 33, and in the talas of Indian classical art music it may be as long as 128. The answers to these questions are essentially physiological and psychological; they lie to a large extent in the nature of the mental and physical constraints imposed by the human brain and body. Fundamentally, to be popular a rhythm should not be so complex that it becomes difficult to grasp by the masses, and at the same time it should not be so simple that it quickly becomes boring. Furthermore, to serve well as a timeline for dancing, its realization should not take much more than about two seconds, the duration of our conscious sense of the present. Rhythms with an even number of pulses that is also a power of two are for most people of the world, easier to assimilate than other rhythms. These constraints are already sufficient to bring the workable number of pulses down to small values that are powers of two, such as eight or sixteen. As for the number of onsets, for a timeline to afford a rich enough structure, five appears to be a good choice. However, a cycle of eight pulses does not provide enough room (in the sense of time) for five onsets to be distributed so as to create interesting patterns. Thus we are left with sixteen pulses and five onsets as the most feasible candidates for creating a timeline that has a sufficiently rich structure.

So why, out of the sixteen patterns above, is son clave so much more popular than the others? Toussaint attributes it to son clave’s “rhythmic oddity.” There are no pairs of hits located directly across from each other across the circle. If there were, the pair would tend to divide the pattern in half, making you hear two simpler eight-beat patterns rather than one more complex sixteen-beat pattern. Son clave can’t be broken down into smaller symmetrical pieces.

Okay, so son clave has desirable rhythmic oddity. But so do many other beats. What else does son clave have? Toussaint points to some special symmetries hidden in the beat. Any rhythm comes with a “shadow rhythm” with an implicit hit in between each of the actual ones. When you’re drumming, your hands or sticks reach their maximum height at the onsets of the shadow rhythm, so while you may not hear it, you feel it, and both you and your listeners can see it.

So far I’ve been talking exclusively about the so-called three-side version of the clave. There’s also the two-side version, where the two hit pattern comes first, followed by the three-hit pattern. You can switch from one to the other by moving the downbeat from the top of the circle to the bottom. You might notice that the shadow rhythm of three-side clave bears a strong resemblance to two-side clave, and conversely, the shadow rhythm of two-side clave resembles three-side. (Thanks to Roberto Thais for this observation.)

So here’s where it gets interesting. We tend to hear rhythms as patterns of short-long time intervals, rather than perceiving the exact length of the time intervals directly. Toussaint calls the pattern of long and short intervals the “rhythmic contour.” Son clave and fume-fume feel like “the same” rhythm because they have the same rhythmic contour, even though they’re in two different time signatures.

If you take son clave’s “shadow” and rotate it 180 degrees around the circle, it has the same rhythmic contour as son clave itself. In other words, son clave sounds “the same” as its own shadow backwards. It’s the only one of the sixteen-step rhythms listed above to have this property. We might not be able to perceive this bit of symmetry consciously, but it must act on us somehow or we wouldn’t be so wild about the beat.

Son clave shares its special qualities with all of its own rotations, the beats you get treating each of the five onset as the downbeat. So why do we prefer the downbeat that we do? Toussaint thinks it has to do with son clave’s metrical ambiguity. Those first three hits strongly imply triple meter, which is at odds with the underlying 4/4. The last two hits confirm the 4/4 feeling, but without hitting the second downbeat, the one at the bottom of the circle. You, the listener, have to involve your musical intelligence to make sense of all this ambiguity. It’s this invitation to your own imaginative participation in the beat that ultimately makes son clave so much more popular than all of its close rhythmic cousins. Math! Who says it has to be boring?

[With special thanks to the Up-Goer Five Text Editor, which was inspired by this xkcd]

I study computers that would work in a different way than any computer that we have today.  These computers would be very small, and they would use facts about the world that are not well known to us from day to day life.  No one has built one of these computers yet—at least, we don’t think they have!—but we can still reason about what they could do for us if we did build them.

How would these new computers work? Well, when you go small enough, you find that, in order to figure out what the chance is that something will happen, you need to both add and take away a whole lot of numbers—one number for each possible way that the thing could happen, in fact. What’s interesting is, this means that the different ways a thing could happen can “kill each other out,” so that the thing never happens at all! I know it sounds weird, but the world of very small things has been known to work that way for almost a hundred years.

So, with the new kind of computer, the idea is to make the different ways each wrong answer could be reached kill each other out (with some of them “pointing” in one direction, some “pointing” in another direction), while the different ways that the right answer could be reached all point in more or less the same direction. If you can get that to happen, then when you finally look at the computer, you’ll find that there’s a very good chance that you’ll see the right answer. And if you don’t see the right answer, then you can just run the computer again until you do.

For some problems—like breaking a big number into its smallest parts (say, 43259 = 181 × 239)—we’ve learned that the new computers would be much, much faster than we think any of today’s computers could ever be. For other problems, however, the new computers don’t look like they’d be faster at all. So a big part of my work is trying to figure out for which problems the new computers would be faster, and for which problems they wouldn’t be.

You might wonder, why is it so hard to build these new computers? Why don’t we have them already? This part is a little hard to explain using the words I’m allowed, but let me try. It turns out that the new computers would very easily break. In fact, if the bits in such a computer were to “get out” in any way—that is, to work themselves into the air in the surrounding room, or whatever—then you could quickly lose everything about the new computer that makes it faster than today’s computers. For this reason, if you’re building the new kind of computer, you have to keep it very, very carefully away from anything that could cause it to lose its state—but then at the same time, you do have to touch the computer, to make it do the steps that will eventually give you the right answer. And no one knows how to do all of this yet. So far, people have only been able to use the new computers for very small checks, like breaking 15 into 3 × 5. But people are working very hard today on figuring out how to do bigger things with the new kind of computer.

In fact, building the new kind of computer is so hard, that some people even believe it won’t be possible! But my answer to them is simple. If it’s not possible, then that’s even more interesting to me than if it is possible! And either way, the only way I know to find out the truth is to try it and see what happens.

Sometimes, people pretend that they already built one of these computers even though they didn’t. Or they say things about what the computers could do that aren’t true. I have to admit that, even though I don’t really enjoy it, I do spend a lot of my time these days writing about why those people are wrong.

Oh, one other thing. Not long from now, it might be possible to build computers that don’t do everything that the new computers could eventually do, but that at least do some of it. Like, maybe we could use nothing but light and mirrors to answer questions that, while not important in and of themselves, are still hard to answer using today’s computers. That would at least show that we can do something that’s hard for today’s computers, and it could be a step along the way to the new computers. Anyway, that’s what a lot of my own work has been about for the past four years or so.

Besides the new kind of computers, I’m also interested in understanding what today’s computers can and can’t do. The biggest open problem about today’s computers could be put this way: if a computer can check an answer to a problem in a short time, then can a computer also find an answer in a short time? Almost all of us think that the answer is no, but no one knows how to show it. Six years ago, another guy and I figured out one of the reasons why this question is so hard to answer: that is, why the ideas that we already know don’t work.

Anyway, I have to go to dinner now. I hope you enjoyed this little piece about the kind of stuff that I work on.

That is from 2007 to 2012, the link is here.

At the Hollywood-style awards ceremony last night for $3 million string theory and biomedical research prizes, it was announced that Yuri Milner and Mark Zuckerberg will now start funding something similar in mathematics, called the Breakthrough Prize in Mathematics. According to the New York Times:

Yuri Milner, the Russian entrepreneur, philanthropist and self-described “failed physicist” who made a splash two years ago when he began handing out lavish cash awards to scientists, announced Thursday that he was expanding the universe of his largess again: This time, he will begin handing out $3 million awards to mathematicians…

For the new math award, Mr. Milner and Mr. Zuckerberg, the co-sponsor of the math prize, will decide who gets the money, in consultation with experts. Mr. Milner declined to say how many mathematicians would be chosen, but there could be quite a number of windfalls in store: for the physics price, there were nine inaugural winners, and for the life sciences prize, there were 11.

I’ve written extensively about the “Fundamental Physics Prize” and what I see as the worst problem with it (heavily rewarding and propping up a failed research program). While many physicists are privately unhappy about this prize and its effects, few prominent ones are willing to speak publicly with their name attached, since this kind of mouthing-off could turn out to be personally extremely expensive. Ian Sample at the Guardian has a story today, which quotes a “prominent physicist who did not wish to be named”:

One prominent physicist who did not wish to be named said the huge sums of money could be used better: “The great philanthropists of the 19th and 20th centuries, like the Rockefellers and the Carnegies, did not create prizes – they created universities and research institutes that have enabled thousands of scientists to make great breakthroughs over the succeeding decades.

“By contrast, giving a prize has a negligible effect on the progress of science. A few already well-recognised people get enriched, but there is little value added in terms of the progress of science compared to the multiplier effect of creating new institutions for scientific research.”

The Guardian does quote one critic by name, but it’s just the usual one.

The physics prize has turned out to be extremely narrowly targeted at one particular subfield of physics, and from what little I know of the life sciences, the prizes in that area seem to be also narrowly targeted (US biomedical research aimed at curing diseases that most afflict those in the developed world). I’m highly ignorant about life sciences research, but it seems striking that the 6 $3 million winners in this field were all men.

I have no idea how Milner and Zuckerberg will go about choosing the $3 million winners in mathematics, and whether this new prize will end up being narrowly targeted to a certain sort of mathematics research. If so, it may have very significant effects on what kinds of mathematics get done. Based on the other prizes, it seems likely that the winners will be mostly prominent US academics, people already well-rewarded by the current academic star system. I don’t see any reason to believe that these kinds of financial awards will allow such mathematicians to do work they wouldn’t otherwise do, so the main argument for the prizes is that the money (and Academy Awards-style ceremonies) will help make them celebrities, and that this is a good thing. One can predict that public criticism from prominent US academics may be rather muted once the checks start coming.

Even if the Milner-Zuckerberg prize does end up focused on the best mathematics research, I still think the whole concept is problematic. The US today is increasingly dominated by a grotesque winner-take-all culture that values wealth and celebrity above all else. While mathematics research, like the rest of academia, has been affected as a star system has become increasingly part of the picture, this field has been somewhat immune to celebrity culture. While people typically think that what mathematicians do is perfectly respectable, they don’t understand much about it and aren’t especially interested. Milner and Zuckerberg want to change this by turning mathematicians into celebrities, but I don’t see any reason to believe this is going to lead to better mathematics.

Update: Here’s the statement from Milner about the planned mathematics prize:

Yuri Milner said: “Einstein said, Pure mathematics is the poetry of logical ideas. It is in this spirit that Mark and myself are announcing a new Breakthrough Prize in Mathematics. The work that the Prize recognizes could be the foundation for genetic engineering, quantum computing or Artificial Intelligence; but above all, for human knowledge itself.”

Retrograde tracer injections in 29 of the 91 areas of the macaque cerebral cortex revealed 1,615 interareal pathways, a third of which have not previously been reported. A weight index (extrinsic fraction of labeled neurons [FLNe]) was determined for each area-to-area pathway. Newly found projections were weaker on average compared with the known projections; nevertheless, the 2 sets of pathways had extensively overlapping weight distributions. Repeat injections across individuals revealed modest FLNe variability given the range of FLNe values (standard deviation <1 log unit, range 5 log units). The connectivity profile for each area conformed to a lognormal distribution, where a majority of projections are moderate or weak in strength. In the G_{29 x 29} interareal subgraph, two-thirds of the connections that can exist do exist. Analysis of the smallest set of areas that collects links from all 91 nodes of the G_{29 x 91} subgraph (dominating set analysis) confirms the dense (66%) structure of the cortical matrix. The G_{29 x 29} subgraph suggests an unexpectedly high incidence of unidirectional links. The directed and weighted G_{29 x 91} connectivity matrix for the macaque will be valuable for comparison with connectivity analyses in other species, including humans. It will also inform future modeling studies that explore the regularities of cortical networks.

Many realistic networks are scale-free, with small characteristic path lengths, high clustering, and power law in their degree distribution. They can be obtained by dynamical networks in which a preferential attachment process takes place. However, this mechanism is non-local, in the sense that it requires knowledge of the whole graph in order for the graph to be updated. Instead, if preferential attachment and realistic networks occur in physical systems, these features need to emerge from a local model. In this paper, we propose a local model and show that a possible ingredient (which is often underrated) for obtaining scale-free networks with local rules is memory. Such a model can be realised in solid-state circuits, using non-linear passive elements with memory such as memristors, and thus can be tested experimentally.