The Beauty of Roots (Part 3)

 

Dan Christensen, Sam Derbyshire and I are writing a paper on a fun topic I’ve discussed here before: the beauty of roots. We’re getting help from Greg Egan, but he’s too busy writing his next novel to commit to being a coauthor! Anyway, I’m giving a talk about this stuff today, and I think you’ll at least enjoy the pretty pictures:

• The Beauty of Roots: easy fun version.

• The Beauty of Roots: version for mathematicians.

The version for mathematicians has some proofs; the easy fun version states a few theorems, but it’s mainly pictures.

For mathematicians, I think the coolest part is the close relation between our main object of interest:

namely the set of all roots of all polynomials whose coefficients are ±1, and the Cantor set, which you get by taking a closed interval and repeatedly chopping out the middle third of each piece, forever:

They’re related because a point in the Cantor set can be seen as an infinite string of 0’s and 1’s, while a power series with coefficients ±1 can be seen as an infinite string of 1’s and -1’s. The sets called ‘dragons’, like the one at the top of this post, are also images of the Cantor set under continuous maps to the complex plane. But to understand how these facts understand our set of roots of polynomials with coefficients ±1, read the mathematician’s version of the talk.

Azimuth on Google Plus (Part 6)

Lately the distribution of hits per hour on this blog has become very fat-tailed. In other words: the readership shoots up immensely now and then. I just noticed today’s statistics:

That spike on the right is what I’m talking about: 338 hits per hour, while before it was hovering in the low 80’s, as usual for the weekend. Why? Someone on Hacker News posted an item saying:

John Baez will give his Google Talk tomorrow in the form of a robot.

That’s true! If you’re near Silicon Valley on Monday the 13th and you want to see me in the form of a robot, come to the Google campus and listen to my talk Energy, the Environment and What We Can Do.

It starts at 4 pm in the Paramaribo Room (Building 42, Floor 2). You’ll need to check in 15 minutes before that at the main visitor’s lounge in Building 43, and someone will escort you to the talk.

But if you can’t attend, don’t worry! A video will appear on YouTube, and I’ll point you to it when it does.

I tested out the robot a few days ago from a hotel room in Australia—it’s a strange sensation! Suzanne Brocato showed me the ropes. To talk to me easily, she lowered my ‘head’ until I was just 4 feet tall. “You’re so short!” she laughed. I rolled around the offices of Anybot and met the receptionist, who was also in the form of a robot. Then we went to the office of the CEO, Trevor Blackwell, and planned out my talk a little. I need to practice more today.

But why did someone at Hacker News post that comment just then? I suspect it’s because I reminded people about my talk on Google+ last night.

The fat-tailed distribution of blog hits is also happening at the scale of days, not just hours:

The spikes happen when I talk about a ‘hot topic’. January 27th was my biggest day so far. Slashdot discovered my post about the Elsevier boycott, and send 3468 readers my way. But a total 6499 people viewed that post, so a bunch must have come from other sources.

January 31st was also big: 3271 people came to read about The Faculty of 1000. 2140 of them were sent over by Hacker News.

If I were trying to make money from advertising on this blog, I’d be pushed toward more posts about hot topics. Forget the mind-bending articles on quantropy, packed with complicated equations!

But as it is, I’m trying to do some mixture of having fun, figuring out stuff, and getting people to save the planet. (Open access publishing fits into that mandate: it’s tragic how climate crackpots post on popular blogs while experts on climate change publish their papers in journals hidden from public view!) So, I don’t want to maximize readership: what matters more is getting people to do good stuff.

Do you have any suggestions on how I could do this better, while still being me? I’m not going to get a personality transplant, so there are limits on what I’ll do.

One good idea would be to make sure every post on a ‘hot topic’ offers readers something they can do now.

Hmm, readership is still spiking:

But enough of this navel-gazing! Here are some recent Azimuth articles about energy on Google+.

Energy

1) In his State of the Union speech, Obama talked a lot about energy:

We’ve subsidized oil companies for a century. That’s long enough. It’s time to end the taxpayer giveaways to an industry that rarely has been more profitable, and double-down on a clean energy industry that never has been more promising.

He acknowledged that differences on Capitol Hill are “too deep right now” to pass a comprehensive climate bill, but he added that “there’s no reason why Congress shouldn’t at least set a clean-energy standard that creates a market for innovation.”

However, lest anyone think he actually wants to stop global warming, he also pledged “to open more than 75 percent of our potential offshore oil and gas resources.”

2) This paper claims a ‘phase change’ hit the oil markets around 2005:

• James Murray and David King, Climate policy: Oil’s tipping point has passed, Nature 481 (2011), 433–435.

They write:

In 2005, global production of regular crude oil reached about 72 million barrels per day. From then on, production capacity seems to have hit a ceiling at 75 million barrels per day. A plot of prices against production from 1998 to today shows this dramatic transition, from a time when supply could respond elastically to rising prices caused by increased demand, to when it could not (see ‘Phase shift’). As a result, prices swing wildly in response to small changes in demand. Other people have remarked on this step change in the economics of oil around the year 2005, but the point needs to be lodged more firmly in the minds of policy-makers.

3) Help out the famous climate blogger Joe Romm! He asks: What will the U.S. energy mix look like in 2050 if we cut CO₂ emissions 80%?

How much total energy is consumed in 2050… How much coal, oil, and natural gas is being consumed (with carbon capture and storage of some coal and gas if you want to consider that)? What’s the price of oil? How much of our power is provided by nuclear power? How much by solar PV and how much by concentrated solar thermal? How much from wind power? How much from biomass? How much from other forms of renewable energy? What is the vehicle fleet like? How much electric? How much next-generation biofuels?

As he notes, there are lots of studies on these issues. Point him to the best ones!

4) Due to plunging prices for components, solar power prices in Germany dropped by half in the last 5 years. Now solar generates electricity at levels only slightly above what consumers pay. The subsidies will disappear entirely within a few years, when solar will be as cheap as conventional fossil fuels. Germany has added 14,000 megawatts capacity in the last 2 years and now has 24,000 megawatts in total—enough green electricity to meet nearly 4% the country’s power demand. That is expected to rise to 10% by 2020. Germany now has almost 10 times more installed capacity than the United States.

That’s all great—but, umm, what about the other 90%? What’s their long-term plan? Will they keep using coal-fired power plants? Will they buy more nuclear power from France?

In May 2011, Britain claimed it would halve carbon emissions by 2025. Is Germany making equally bold claims or not? Of course what matters is deeds, not words, but I’m curious.

5) Stephen Lacey presents some interesting charts showing the progress and problems with sustainability in the US. For example, there’s been a striking drop in how much energy is being used per dollar of GNP:

Sorry for the archaic ‘British Thermal Units’: we no longer have a king, but for some reason the U.S. failed to throw off the old British system of measurement. A BTU is a bit more than a kilojoule.

Despite these dramatic changes, Lacey says “we waste around 85% of the energy produced in the U.S.” But he doesn’t say how that number was arrived at. Does anyone know?

6) The American Council for an Energy-Efficient Economy (ACEEE) has a new report called The Long-Term Energy Efficiency Potential: What the Evidence Suggests. It describes some scenarios, including one where the US encourages a greater level of productive investments in energy efficiency so that by the year 2050, it reduces overall energy consumption by 40 to 60 percent. I’m very interested in how much efficiency can help. Some, but not all, of the improvements will be eaten up by the rebound effect.

Quantropy (Part 2)

In my first post in this series, we saw that filling in a well-known analogy between statistical mechanics and quantum mechanics requires a new concept: ‘quantropy’. To get some feeling for this concept, we should look at some examples. But to do that, we need to develop some tools to compute quantropy. That’s what we’ll do today.

All these tools will be borrowed from statistical mechanics. So, let me remind you how to compute the entropy of a system in thermal equilibrium starting if we know the energy of every state. Then we’ll copy this and get a formula for the quantropy of a system if we know the action of every history.

Computing entropy

Everything in this section is bog-standard. In case you don’t know, that’s British slang for ‘extremely, perhaps even depressingly, familiar’. Apparently it rains so much in England that bogs are not only standard, they’re the standard of what counts as standard!

Let be a measure space: physically, the set of states of some system. In statistical mechanics we suppose the system occupies states with probabilities given by some probability distribution

where of course

The entropy of this probability distribution is

There’s a nice way to compute the entropy when our system is in thermal equilibrium. This idea makes sense when we have a function

saying the energy of each state. Our system is in thermal equilibrium when maximizes entropy subject to a constraint on the expected value of energy:

A famous calculation shows that thermal equilibrium occurs precisely when is the so-called Gibbs state:

for some real number , where is a normalization factor called the partition function:

The number is called the coolness, since physical considerations say that

where is the temperature in units where Boltzmann’s constant is 1.

There’s a famous way to compute the entropy of the Gibbs state; I don’t know who did it first, but it’s both straightforward and tremendously useful. We take the formula for entropy

and substitute the Gibbs state

getting

Reshuffling this a little bit, we obtain:

If we define the free energy by

then we’ve shown that

This justifies the term ‘free energy’: it’s the expected energy minus the energy in the form of heat, namely

It’s nice that we can compute the free energy purely in terms of the partition function and the temperature, or equivalently the coolness :

Can we also do this for the entropy? Yes! First we’ll do it for the expected energy:

This gives

So, if we know the partition function of a system in thermal equilibrium as a function of the temperature, we can work out its entropy, expected energy and free energy.

Computing quantropy

Now we’ll repeat everything for quantropy! The idea is simply to replace the energy by action and the temperature by where is Planck’s constant. It’s harder to get the integrals to converge in interesting examples. But we’ll worry about that next time, that when we actually do an example!

It’s annoying that in physics stands for both entropy and action, since in this article we need to think about both. People also use to stand for entropy, but that’s no better, since that letter also stands for ‘Hamiltonian’! To avoid this let’s use to stand for action. This letter is also used to mean ‘Helmholtz free energy’, but we’ll just have to live with that. It would be real bummer if we failed to unify physics just because we ran out of letters.

Let be a measure space: physically, the set of histories of some system. In quantum mechanics we suppose the system carries out histories with amplitudes given by some function

where perhaps surprisingly

The quantropy of this function is

There’s a nice way to compute the entropy in Feynman’s path integral formalism. This formalism makes sense when we have a function

saying the action of each history. Feynman proclaimed that in this case we have

where is Planck’s constant and is a normalization factor called the partition function:

Last time I showed that we obtain Feynman’s prescription for by demanding that it’s a stationary point for the quantropy

subject to a constraint on the expected action:

As I mentioned last time, the formula for quantropy is dangerous, since we’re taking the logarithm of a complex-valued function. There’s not really a ‘best’ logarithm for a complex number: if we have one choice we can add any multiple of and get another. So in general, to define quantropy we need to pick a choice of for each point That’s a lot of ambiguity!

Luckily, the ambiguity is much less when we use Feynman’s prescription for Why? Because then is defined in terms of an exponential, and it’s easy to take the logarithm of an exponential! So, we can declare that

Once we choose a logarithm for , this formula will let us define and thus the quantropy.

So let’s do this, and say the quantropy is

We can simplify this a bit, since the integral of is 1:

Reshuffling this a little bit, we obtain:

By analogy to free energy in statistical mechanics, let’s define the free action by

I’m using the same letter for free energy and free action, but they play exactly analogous roles, so it’s not so bad. Indeed we now have

which is the analogue of a formula we saw for free energy in thermodynamics.

It’s nice that we can compute the free action purely in terms of the partition function and Planck’s constant. Can we also do this for the quantropy? Yes!

It’ll be convenient to introduce a parameter

which is analogous to ‘coolness’. We could call it ‘quantum coolness’, but a better name might be classicality, since it’s big when our system is close to classical. Whatever we call it, the main thing is that unlike ordinary coolness, it’s imaginary!

In terms of classicality, we have

Now we can compute the expected action just as we computed the expected energy in thermodynamics:

This gives:

So, if we can compute the partition function in the path integral approach to quantum mechanics, we can also work out the quantropy, expected action and free action!

Next time I’ll use these formulas to compute quantropy in an example: the free particle. We’ll see some strange and interesting things.

Summary

Here’s where our analogy stands now:

Statistical Mechanics	Quantum Mechanics
states:	histories:
probabilities:	amplitudes:
energy:	action:
temperature:	Planck’s constant times i:
coolness:	classicality:
partition function:	partition function:
Boltzmann distribution:	Feynman sum over histories:
entropy:	quantropy:
expected energy:	expected action:
free energy:	free action:

I should also say a word about units and dimensional analysis. There’s enormous flexibility in how we do dimensional analysis. Amateurs often don’t realize this, because they’ve just learned one system, but experts take full advantage of this flexibility to pick a setup that’s convenient for what they’re doing. The fewer independent units you use, the fewer dimensionful constants like the speed of light, Planck’s constant and Boltzmann’s constant you see in your formulas. That’s often good. But here I don’t want to set Planck’s constant equal to 1 because I’m treating it as analogous to temperature—so it’s important, and I want to see it. I’m also finding dimensional analysis useful to check my formulas.

So, I’m using units where mass, length and time count as independent dimensions in the sense of dimensional analysis. On the other hand, I’m not treating temperature as an independent dimension: instead, I’m setting Boltzmann’s constant to 1 and using that to translate from temperature into energy. This is fairly common in some circles. And for me, treating temperature as an independent dimension would be analogous to treating Planck’s constant as having its own independent dimension! I don’t feel like doing that.

So, here’s how the dimensional analysis works in my setup:

Statistical Mechanics	Quantum Mechanics
probabilities: dimensionless	amplitudes: dimensionless
energy:	action:
temperature:	Planck’s constant:
coolness:	classicality:
partition function: dimensionless	partition function: dimensionless
entropy: dimensionless	quantropy: dimensionless
expected energy:	expected action:
free energy:	free action:

I like this setup because I often think of entropy as closely allied to information, measured in bits or nats depending on whether I’m using base 2 or base e. From this viewpoint, it should be dimensionless.

Of course, in thermodynamics it’s common to put a factor of Boltzmann’s constant in front of the formula for entropy. Then entropy has units of energy/temperature. But I’m using units where Boltzmann’s constant is 1 and temperature has the same units as energy! So for me, entropy is dimensionless.

The Federal Research Public Access Act

As of this minute, 5030 scholars have joined the Elsevier boycott. You should too! But now is the time to go further and take positive steps to develop new, better systems for refereeing and distributing scholarly papers.

Everyone I know is talking about this now. Today, quantum physicist Steve Flammia pointed out to me that U.S. Representative Mike Doyle has a good idea:

• The Federal Research Public Access Act.

It’s simple: we should get to see the research we paid for with our tax dollars. We shouldn’t have to pay for it twice: once to have it done, and once more to see the results.

As Doyle puts it:

Americans have the right to see the results of research funded with taxpayer dollars. Yet such research too often gets locked away behind a pay-wall, forcing those who want to learn from it to pay expensive subscription fees for access.

The Federal Research Public Access Act will encourage broader collaboration among scholars in the scientific community by permitting widespread dissemination of research findings. Promoting greater collaboration will inevitably lead to more innovative research outcomes and more effective solutions in the fields of biomedicine, energy, education, and health care.

But what does the bill actually do? It says this: any federal agency that spends more than $100 million per year funding research must make that research freely available in a public repository no later than 6 months after the research has been published in a peer-review journal.

This is already done by the National Institute of Health: the bill would expand this practice to the National Science Foundation, the Department of Energy, and other agencies.

What we should do

Someone with technical brains should make it easy for US citizens to contact Congress and support this bill. Google got 4.5 million people to sign their petition against SOPA, the so-called Stop Online Piracy Act. But we’ve been playing defense for too long. Let’s go on the offense and do something like this for a bill that’s good!

Emailing your congressperson incredibly easy, but telephone calls are even better, precisely because they’re a bit more work.

Here’s a sample of what you could write or say:

I am your constituent, and I urge you to support the Federal Research Public Access Act. As a taxpayer, I help support scientific research out of my own pocket. I deserve to see the results! The National Institute of Health already demands this for all the research they support, and the system works well. Broadening this policy will advance science and improve the lives and welfare of all Americans.

I believe an emphasis on ‘taxpayers getting their money worth’ and ‘improving the lives of all Americans’ may resonate well with the U.S. Congress: that’s why I’ve worded the message this way. Taxes and patriotism are hot-button issues. But of course you should feel free to modify this text!

Why it’s important

I think this bill is important: even if it doesn’t pass, it changes the debate and puts the publishers on the defensive.

Remember: the Association of American Publishers is still supporting the Research Works Act, a bill that would prevent federal agencies from requiring that the research they fund be made freely available online. It seems this bill would even roll back the existing requirement that research funded by the National Institute of Health be made freely available at PubMed Central!

There’s a built-in imbalance at work here. Publishers pays lobbyists to work full-time on advancing their agenda. Scientists and other scholars prefer to spend their time thinking about more interesting things. So, we’re usually reactive: we wait until something becomes intolerable before taking action. That’s why we’re fighting against a crisis of journal prices that bankrupt our libraries, and battling bad bills like the Research Works Act, when we should be developing better systems for communicating the results of our research, and supporting good bills…

… like the Federal Research Public Access Act!

For more

For more, see:

• David Dobbs, Open science revolt occupies Congress, Wired, 9 February 2012.

• Call to action: Tell Congress you support the Bipartisan Federal Research Public Access Act (FRPAA), Alliance for Taxpayer Access, 9 February 2012.

• Scholarly Publishing & Academic Resources Council, SPARC FAQ for university administrators and faculty: Federal Research Public Access Act (FRPAA).

The original sponsors of the Federal Research Public Access Act were Reps. Kevin Yoder (R-KS) and Wm. Lacy Clay (D-MO). Identical legislation is also being introduced in the U.S. Senate by Sens. John Cornyn (R-TX), Ron Wyden (D-OR), and Kay Bailey Hutchison (R-TX).

 

The Cost of Knowledge

As of this moment, 4760 scholars have joined a boycott of the publishing company Elsevier. Of these, only 20% are mathematicians. But since the boycott was started by a mathematician, 34 of us wrote and signed an official statement explaining the boycott:

• The Cost of Knowledge.

It’s also below. Please check it out and join the boycott! I’m sure more than 34 mathematicians would be happy to sign, but we wanted to get the statement out soon.

This is an attempt to describe some of the background to the current boycott of Elsevier by many mathematicians (and other academics) at http://thecostofknowledge.com, and to present some of the issues that confront the boycott movement. Although the movement is anything but monolithic, we believe that the points we make here will resonate with many of the signatories to the boycott.

The role of journals (1): dissemination of research.

The role of journals in professional mathematics has been under discussion for some time now.

Traditionally, while journals served several purposes, their primary purpose was the dissemination of research papers. The journal publishers were charging for the cost of typesetting (not a trivial matter in general before the advent of electronic typesetting, and particularly non-trivial for mathematics), the cost of physically publishing copies of the journals, and the cost of distributing the journals to subscribers (primarily academic libraries).

The editorial board of a journal is a group of professional
mathematicians. Their editorial work is undertaken as part of their scholarly duties, and so is paid for by their employer, typically a university. Thus, from the publisher’s viewpoint the editors are volunteers. (The editor in chief of a journal sometimes receives modest compensation from the publisher.) When a paper is submitted to the journal, by an author who is again typically a university-employed mathematician, the editors select the referee or referees for the paper, evaluate the referees’ reports, decide whether or not to accept the submission, and organize the submitted papers into volumes. These are passed on to the publisher, who then undertakes the job of actually publishing them. The publisher supplies some administrative assistance in handling the papers, as well as some copy-editing assistance, which is often quite minor but sometimes more substantial. The referees are again volunteers from the point of view of the publisher: as with editing, refereeing is regarded as part of the service component of a mathematician’s academic work. Authors are not paid by the publishers for their published papers, although they are usually asked to sign over the copyright to the publisher.

This system made sense when the publishing and dissemination of papers was a difficult and expensive undertaking. Publishers supplied a valuable service in this regard, for which they were paid by subscribers to the journals, which were mainly academic libraries. The academic institutions whose libraries subscribe to mathematics journals are broadly speaking the same institutions that employ the mathematicians who are writing for, refereeing for, and editing the journals. Therefore, the cost of the whole process of producing research papers is borne by these institutions (and the outside entities that partially fund them, such as the National Science Foundation in the United States): they pay for their academic mathematician employees to do research and to organize the publications of the results of their research in journals; and then (through their libraries) they pay the publishers to disseminate these results among all the world’s mathematicians. Since these institutions employ research faculty in order to foster research, it certainly used to make sense for them to pay for the dissemination of this research as well. After all, the sharing of scientific ideas and research results is unquestionably a key component for making progress in science.

Now, however, the world has changed in significant ways.
Authors typeset their own papers, using electronic typesetting. Publishing and distribution costs are not
as great as they once were. And most importantly,
dissemination of scientific ideas no longer takes place via the physical distribution of journal volumes. Rather, it takes place mainly electronically. While this means of dissemination is not free, it is much less expensive, and much of it happens quite independently of mathematical journals.

In conclusion, the cost of journal publishing has gone down
because the cost of typesetting has been shifted from
publishers to authors and the cost of publishing and distribution is significantly lower than it used to be.
By contrast, the amount of money being spent by university libraries on journals seems to be growing with no end in sight. Why do mathematicians contribute all this volunteer labor, and their employers pay all this money, for a service whose value no longer justifies its cost?

The role of journals (2): peer review and professional
evaluation

There are some important reasons that mathematicians haven’t just abandoned journal publishing. In particular, peer review plays an essential role in ensuring the correctness and readability of mathematical papers, and publishing papers in research journals is the main way of achieving professional recognition. Furthermore, not all journals count equally from this point of view: journals are (loosely) ranked, so that publications in top journals will often count more than publications in lower ranked ones. Professional mathematicians typically have a good sense of the relative prestige of the journals that publish papers in their area, and they will usually submit a paper to the highest ranked journal that they judge is likely to accept and publish it.

Because of this evaluative aspect of traditional journal publishing, the problem of switching to a different model
is much more difficult than it might appear at first. For
example, it is not easy just to begin a new journal (even an electronic one, which avoids the difficulties of printing and distribution), since mathematicians may not want to publish in it, preferring to submit to journals with known reputations. Secondly, although the reputation of various journals has been created through the efforts of the authors, referees, and editors who have worked (at no cost to the publishers) on it over the years, in many cases the name of the journal is owned by the publisher, making it difficult for the mathematical community to separate this valuable object that they have constructed from its present publisher.

The role of Elsevier

Elsevier, Springer, and a number of other commercial publishers (many of them large companies but less significant for their mathematics publishing, e.g., Wiley) all exploit our volunteer labor to extract very large profits from the academic community. They supply some value in the process, but nothing like enough to justify their prices.

Among these publishers, Elsevier may not be the most expensive, but in the light of other factors, such as scandals, lawsuits, lobbying, etc. (discussed further below), we consider them a good initial focus for our discontent. A boycott should be substantial enough to be meaningful, but not so broad that the choice of targets becomes controversial or the boycott becomes an unmanageable burden. Refusing to submit papers to all overpriced publishers is a reasonable further step, which some of us have taken, but the focus of this boycott is on Elsevier because of the widespread feeling among mathematicians that they are the worst offender.

Let us begin with the issue of journal costs. Unfortunately, it is difficult to make cost comparisons: journals differ greatly in quality, in number of pages per volume, and even in amount of text per page. As measured by list prices, Elsevier mathematics journals are amongst the most expensive. For instance, in the AMS mathematics journal price survey, seven of the ten most expensive journals (by 2007 volume list price) were published by Elsevier. (All prices are as of 2007 because both prices and page counts are easily available online.) However, that is primarily because Elsevier publishes the largest volumes. Price per page is a more meaningful measure that can be easily computed. By this standard, Elsevier is certainly not the worst publisher, but its prices do on the face of it look very high. The Annals of Mathematics, published by Princeton University Press, is one of the absolute top mathematics journals and quite affordably priced: $0.13/page as of 2007. By contrast, ten Elsevier journals (not including one that has since ceased publication) cost $1.30/page or more; they and three others cost more per page than any journal published by a university press or learned society. For comparison, three other top journals competing with the Annals are Acta Mathematica, published by the Institut Mittag Leffler for $0.65/page, Journal of the American Mathematical Society, published by the American Mathematical Society for $0.24/page, and Inventiones Mathematicae, published by Springer for $1.21/page. Note that none of Elsevier’s mathematics journals is generally considered comparable in quality to these journals.

However, there is an additional aspect which makes it hard to compute the true cost of mathematics journals. This is the widespread practice among large commercial publishers of “bundling” journals, which allows libraries to subscribe to large numbers of journals in order to avoid paying the exorbitant list prices for the ones they need. Although this means that the average price libraries pay per journal is less than the list prices might suggest, what really matters is the average price that they pay per journal (or page of journal) that they actually want, which is hard to assess, but clearly higher. We would very much like to be able to offer more concrete data regarding the actual costs to libraries of Elsevier journals compared with those of Springer or other publishers. Unfortunately, this is difficult, because publishers often make it a contractual requirement that their institutional customers should not disclose the financial details of their contracts. For example, Elsevier sued Washington State University to try to prevent release of this information. One common consequence of these arrangements, though, is that in many cases a library cannot actually save any money by cancelling a few Elsevier journals: at best the money can sometimes be diverted to pay for other Elsevier subscriptions.

One reason for focusing on Elsevier rather than, say, Springer is that Springer has had a rich and productive history with the mathematical community. As well as journals, it has published important series of textbooks, monographs, and lecture notes; one could perhaps regard the prices of its journals as a means of subsidizing these other, less profitable, types of publications. Although all these types of publications have become less important with the advent of the internet and the resulting electronic distribution of texts, the long and continuing presence of Springer in the mathematical world has resulted in a store of goodwill being built up in the mathematical community towards them. This store is being rapidly depleted, but has not yet reached zero. See for instance the recent petition to Springer by a number of French mathematicians and departments.

Elsevier does not have a comparable tradition of involvement in mathematics publishing. Many of the mathematics journals that it publishes have been acquired comparatively recently as it has bought up other, smaller publishers. Furthermore, in recent years it has been involved in various scandals regarding the scientific content, or lack thereof, of its journals. One in particular involved the journal Chaos, Solitons & Fractals, which, at the time the scandal broke in 2008–2009, was one of the highest impact factor mathematics journals that Elsevier published. (Elsevier currently reports the five-year impact factor of this journal at 1.729. For sake of comparison, Advances in Mathematics, also published by Elsevier, is reported as having a five-year impact factor of 1.575.) It turned out that the high impact factor was at least partly the result of the journal publishing many papers full of mutual citations. (See Arnold for more information on this and other troubling examples that show the limitations of bibliometric measures of scholarly quality.) Furthermore, Chaos, Solitons & Fractals published many papers that, in our professional judgement, have little or no scientific merit and should not have been published in any reputable journal.

In another notorious episode, this time in medicine, for at least five years Elsevier “published a series of sponsored article compilation publications, on behalf of pharmaceutical clients, that were made to look like journals and lacked the proper disclosures”, as noted by the CEO of Elsevier’s Health Sciences Division.

Recently, Elsevier has lobbied for the Research Works Act, a proposed U.S. law that would undo the National Institutes of Health’s public access policy, which guarantees public access to published research papers based on NIH funding within twelve months of publication (to give publishers time to make a profit). Although most lobbying occurs behind closed doors, Elsevier’s vocal support of this act shows their opposition to a popular and effective open access policy.

These scandals, taken together with the bundling practices, exorbitant prices, and lobbying activities, suggest a publisher motivated purely by profit, with no genuine interest in or commitment to mathematical knowledge and the community of academic mathematicians that generates it. Of course, many Elsevier employees are reasonable people doing their best to contribute to scholarly publishing, and we bear them no ill will. However, the organization as a whole does not seem to have the interests of the mathematical community at heart.

The boycott

Not surprisingly, many mathematicians have in recent years lost patience with being involved in a system in which commercial publishers make profits based on the free labor of mathematicians and subscription fees from their institutions’ libraries, for a service that has become largely unnecessary. (See Scott Aaronson’s scathing but all-too-true satirical description of the publishers’ business model.) Among all the commercial publishers, the behavior of Elsevier seemed to many to be the most egregious, and a number of mathematicians had made personal commitments to avoid any involvement with Elsevier journals. (Some journals were also successfully moved from Elsevier to other publishers; e.g., Annales Scientifiques de l’école Normale Supérieure which until recent years was published by Elsevier, is now published by the Société Mathématique de France.)

One of us (Timothy Gowers) decided that it might be useful to
publicize his own personal boycott of Elsevier, thus encouraging others to do the same. This led to the current boycott movement at http://thecostofknowledge.com, the success of which has far exceeded his initial expectations.

Each participant in the boycott can choose which activities they intend to avoid: submitting to Elsevier journals, refereeing for them, and serving on editorial boards. Of course, submitting papers and editing journals are purely voluntary activities, but refereeing is a more subtle issue. The entire peer review system depends on the availability of suitable referees, and its success is one of the great traditions of science: refereeing is felt to be both a burden and an honor, and practically every member of the community willingly takes part in it. However, while we respect and value this tradition, many of us do not wish to see our labor used to support Elsevier’s business model.

What next?

As suggested at the very beginning, different participants in the boycott have different goals, both in the short and long term. Some people would like to see the journal system eliminated completely and replaced by something else more adapted to the internet and the possibilities of electronic distribution. Others see journals as continuing to play a role, but with commercial publishing being replaced by open access models. Still others imagine a more modest change, in which commercial publishers are replaced by non-profit entities such as professional societies (e.g., the American Mathematical Society, the London Mathematical Society, and the Société Mathématique de France, all of which already publish a number of journals) or university presses; in this way the value generated by the work of authors, referees, and editors would be returned to the academic and scientific community. These goals need not be mutually exclusive: the world of mathematics journals, like the world of mathematics itself, is large, and open access journals can coexist with traditional journals, as well as with other, more novel means of dissemination and evaluation.

What all the signatories do agree on is that Elsevier is an exemplar of everything that is wrong with the current system of commercial publication of mathematics journals, and we will no longer acquiesce to Elsevier’s harvesting of the value of our and our colleagues’ work.

What future do we envisage for all the papers that would
otherwise be published in Elsevier journals? There are many
other journals being published; perhaps they can pick up at
least some of the slack. Many successful new journals have been founded in recent years, too, including several that are electronic (thus completely eliminating printing and physical distribution costs), and no doubt more will follow. Finally, we hope that the mathematical community will be able to reclaim for itself some of the value that it has given to Elsevier’s journals by moving some of these journals (in name, if possible, and otherwise in spirit) from Elsevier to other publishers. One notable example is the August 10, 2006 resignation of the entire editorial board of the Elsevier journal Topology and their founding of the Journal of Topology, owned by the London Mathematical Society.

None of these changes will be easy; editing a journal is hard work, and founding a new journal, or moving and relaunching an existing journal, is even harder. But the alternative is to continue with the status quo, in which Elsevier harvests ever larger profits from the work of us and our colleagues, and this is both unsustainable and unacceptable.

Signed by:

Scott Aaronson
Massachusetts Institute of Technology

Douglas N. Arnold
University of Minnesota

Artur Avila
IMPA and Institut de Mathématiques de Jussieu

John Baez
University of California, Riverside

Folkmar Bornemann
Technische Universität München

Danny Calegari
Caltech/Cambridge University

Henry Cohn
Microsoft Research New England

Jordan Ellenberg
University of Wisconsin, Madison

Matthew Emerton
University of Chicago

Marie Farge
École Normale Supérieure Paris

David Gabai
Princeton University

Timothy Gowers
Cambridge University

Ben Green
Cambridge University

Martin Grötschel
Technische Universität Berlin

Michael Harris
Université Paris-Diderot Paris 7

Frédéric Hélein
Institut de Mathéatiques de Jussieu

Rob Kirby
University of California, Berkeley

Vincent Lafforgue
CNRS and Université d’Orléans

Gregory F. Lawler
University of Chicago

Randall J. LeVeque
University of Washington

László Lovász
Eötvös Lor´nd University

Peter J. Olver
University of Minnesota

Olof Sisask
Queen Mary, University of London

Terence Tao
University of California, Los Angeles

Richard Taylor
Institute for Advanced Study

Bernard Teissier
Institut de Mathématiques de Jussieu

Burt Totaro
Cambridge University

Lloyd N. Trefethen
Oxford University

Takashi Tsuboi
University of Tokyo

Marie-France Vigneras
Institut de Mathématiques de Jussieu

Wendelin Werner
Université Paris-Sud

Amie Wilkinson
University of Chicago

Günter M. Ziegler
Freie Universität Berlin

Appendix: recommendations for mathematicians.

All mathematicians must decide for themselves whether, or to what extent, they wish to participate in the boycott. Senior
mathematicians who have signed the boycott bear some
responsibility towards junior colleagues who are forgoing the
option of publishing in Elsevier journals, and should do their
best to help minimize any negative career consequences.

Whether or not you decide to join the boycott, there are some
simple actions that everyone can take, which seem to us to be
uncontroversial:

1) Make sure that the final versions of all your papers, particularly new ones, are freely available online— ideally both on the arXiv. (Elsevier’ electronic preprint policy is unacceptable, because it explicitly does not allow authors to update their papers on the arXiv to incorporate changes made during peer review). When signing copyright transfer forms, we recommend amending them (if necessary) to reserve the right to make the author’s final version of the text available free online from servers such as the arXiv, and on your home page.

2) If you are submitting a paper and there is a choice between an expensive journal and a cheap (or free) journal of the same standard, then always submit to the cheap one.

Note

The PDF version of this statement has many useful references not included here.

Archimedean Tilings and Egyptian Fractions

Ever since I was a kid, I’ve loved Archimedean tilings of the plane: that is, tilings by regular polygons where all the edge lengths are the same and every vertex looks alike. Here’s my favorite:

There are also 11 others, two of which are mirror images of each other. But how do we know this? How do we list them all and be sure we haven’t left any out?

The interior angle of a regular -sided polygon is obviously

since it’s a bit less than 180 degrees, or and how much?— well, times a full turn, or But these ‘s are getting annoying: it’s easier to say ‘a full turn’ than write Then we can say the interior angle is

times a full turn.

Now suppose we have an Archimedean tiling where polygons meet: one with sides, one with sides, and so on up to one with sides. Their interior angles must add up to a full turn. So, we have

or

or

So: to get an Archimedean tiling you need n whole numbers whose reciprocals add up to one less than n/2.

Looking for numbers like this is a weird little math puzzle. The Egyptians liked writing numbers as sums of reciprocals, so they might have enjoyed this game if they’d known it. The tiling I showed you comes from this solution:

since it has 3 polygons meeting at each vertex: a 4-sided one, a 6-sided one and a 12-sided one.

Here’s another solution:

It gives us this tiling:

Hmm, now I think this one is my favorite, because my eye sees it as a bunch of linked 12-sided polygons, sort of like chain mail. Different tilings make my eyes move over them in different ways, and this one has a very pleasant effect.

Here’s another solution:

This gives two Archimedean tilings that are mirror images of each other!

Of course, whether you count these as two different Archimedean tilings or just one depends on what rules you choose. And by the way, people usually don’t say a tiling is Archimedean if all the polygons are the same, like this:

They instead say it’s regular. If modern mathematicians were inventing this subject, we’d say regular tilings are a special case of Archimedean tilings—but this math is all very old, and back then mathematicians treated special cases as not included in the general case. For example, the Greeks didn’t even consider the number 1 to be a number!

So here’s a fun puzzle: classify the Archimedean tilings! For starters, you need to find all ways to get whole numbers whose reciprocals add up to one less than . That sounds hard, but luckily it’s obvious that

since an equilateral triangle has the smallest interior angle, of any regular polygon, and you can only fit 6 of them around a vertex. If you think a bit, you’ll see this cuts the puzzle down to a finite search.

But you have to be careful, since there are some solutions that don’t give Archimedean tilings. As usual, the number 5 causes problems. We have

but there’s no way to tile the plane so that 2 regular pentagons and 1 regular decagon meet at each vertex! Kepler seems to have tried; here’s a picture from his book Harmonices Mundi:

It works beautifully at one vertex, but not for a tiling of the whole plane. To save the day he had to add some stars, and some of the decagons overlap! The Islamic tiling artists, and later Penrose, went further in this direction.

If you get stuck on this puzzle, you can find the answer here:

• Michal Krížek, Jakub Šolc, and Alena Šolcová, Is there a crystal lattice possessing five-fold symmetry?, AMS Notices 59 (January 2012), 22-30.

• Combinations of regular polygons that can meet at a vertex, Wikipedia.

Not enough?

In short, all Archimedean tilings of the plane arise from finding whole numbers whose reciprocals sum to But what if the total is not enough? Don’t feel bad: you might still get a tiling of the hyperbolic plane. For example,

so you can’t tile the plane with 3 heptagons meeting at each corner… but you still get this tiling of the hyperbolic plane:

which happens to be related to a wonderful thing called Klein’s quartic curve.

You don’t always win… but sometimes you do, so the game is worth playing. For example,

so you have a chance at a tiling of the hyperbolic plane where five equilateral triangles and a square meet at each vertex. And in this case, you luck out:

For more beautiful pictures like these, see:

• Uniform tilings in hyperbolic plane, Wikipedia.

• Don Hatch, Hyperbolic tesselations.

Too much?

Similarly, if you’ve got reciprocals that add up to more than , you’ve got a chance at tiling the sphere. For example,

and in this case we luck out and get the snub dodecahedron. I thought it was rude to snub a dodecahedron, but apparently not:

These tilings of the sphere are technically called Archimedean solids and (if all the polygons are the same) Platonic solids. Of these, only the snub dodecahedron and the ‘snub cube’ are different from their mirror images.

Fancier stuff

In short, adding up reciprocals of whole numbers is related to Archimedean tilings of the plane, the sphere and the hyperbolic plane. But this is also how Egyptians would write fractions! In fact they even demanded that all the reciprocals be distinct, so instead of writing 2/3 as , they’d write

It’s a lousy system—doubtless this is why King Tut died so young. But forget about the restriction that the reciprocals be distinct: that’s silly. If you can show that for every the number can be written as for whole numbers you’ll be famous! So far people have ‘only’ shown it’s true for up to a hundred trillion:

• Erdös–Straus conjecture, Wikipedia.

So, see if you can do better! But if you’re into fancy math, a less stressful activity might be to read about Egyptian fractions, tilings and ADE classifications:

• John Baez, This Week’s Finds in Mathematical Physics (Week 182).

This only gets into ‘Platonic’ or ‘regular’ tilings, not the more general ‘Archimedean’ or ‘semiregular’ ones I’m talking about today—so the arithmetic works a bit differently.

For the special magic arising from

see:

• John Baez, 42.

In another direction, my colleague Julie Bergner has talked about how they Egyptian fractions show up in the study of ‘groupoid cardinality’:

• Julie Bergner, Groupoids and Egyptian fractions.

So, while nobody uses Egyptian fractions much anymore, they have a kind of eerie afterlife. For more on what the Egyptians actually did, try these:

• Ron Knott, Egyptian fractions.

• Egyptian fractions, Wikipedia.

Quantizing Electrical Circuits

As you may know, there’s a wonderful and famous analogy between classical mechanics and electrical circuit theory. I explained it back in “week288”, so I won’t repeat that story now. If you don’t know what I’m talking about, take a look!

This analogy opens up the possibility of quantizing electrical circuits by straightforwardly copying the way we quantize classical mechanics problems. I’d often wondered if this would be useful.

It is, and people have done it:

• Michel H. Devoret, Quantum fluctuations in electrical circuits.

Michel Devoret, Rob Schoelkopf and others call this idea quantronics: the study of mesoscopic electronic effects in which collective degrees of freedom like currents and voltages behave quantum mechanically.

I just learned about this from a talk by Sean Barrett here in Coogee. There are lots of cool applications, but right now I’m mainly interested in how this extends the set of analogies between different physical theories.

One interesting thing is how they quantize circuits with resistors. Over in classical mechanics, this corresponds to systems with friction. These systems, called ‘dissipative’ systems, don’t have a conserved energy. More precisely, energy leaks out of the system under consideration and gets transferred to the environment in the form of heat. It’s hard to quantize systems where energy isn’t conserved, so people in quantronics model resistors as infinite chains of inductors and capacitors: see the ‘LC ladder circuit’ on page 15 of Devoret’s notes. This idea is also the basis of the Caldeira–Leggett model of a particle coupled to a heat bath made of harmonic oscillators: it amounts to including the environment as part of the system being studied.