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J.H. Conway, Contemporary Mathematics, Universal Quadratic Forms and the Fifteen Theorem, here.
The representation theory of quadratic forms has a long history, start- ing in the seventeenth century with Fermat’s assertions of 1640 about the numbers represented by x2 + y2. In the next century, Euler gave proofs of these and some similar assertions about other simple binary quadratics, and although these proofs had some gaps, they contributed greatly to setting the theory on a firm foundation. Lagrange started the theory of universal quadratic forms in 1770 by proving his celebrated Four Squares Theorem, which in current language is expressed by saying that the form x2+y2+z2+t2 is universal. The eighteenth century was closed by a considerably deeper statement – Legendre’s Three Squares Theorem of 1798; this found exactly which numbers needed all four squares. In his Theorie des Nombres of 1830, Legendre also created a very general theory of binary quadratics.
Navier Stokes looks like it’s gonna blow … up
Peter Woit, Not Even Wrong, More Quick Links, here. Tao on Navier Stokes story from Quanta and SCIgen papers in peer review journals.
First, a couple of examples of recent progress in mathematics
Damiano Brigo, Imperial College, Interest Rate Models with Credit Risk, Collateral, Funding Liquidity Risk and Multiple Curves, here. Nice comprehensive slides.
Check in with Otelbaev and Navier Stokes and SEFs
math.stackexchange, Has Prof. Otelbaev shown existence of strong solutions for Navier-Stokes equations? here. Looks like the same link Tao was pointing to.
A young guy in Russia seems to have found a concrete gap in the proof. This concerns Statement 6.3. In the ‘proof’, on p.56, the passage from (6.33) to (6.34) is made by saying ‘using this and that and also that’ . However no reasons are visible where does the extra ||z|| on the right hand side come from. At least some very detailed explanation for this is needed.
OTC Space, Article: SEF offshore push confirmed | ISDA analysis, here.
Table: Shift in Volume Share between D2D Counterparty Location from Q1-3 to Q4 2013
D2D counterparty Dealer IRS Ccy Europe USA Canada Europe EUR +16.2% -15.6% 0.0% USD +5.1% -4.7% -0.3% USA EUR -0.2% 0.2% 0.1% USD -1.9% 0.7% 0.7%
Check in with Navier Stokes, Otelbayev, and Tao
Andy Kiersz, BI, A Kazakh Mathematician Claims To Have Solved An Enormously Important Equation, here.
A huge mathematical breakthrough might have just been made, but a language barrier is slowing things down.
New Scientist reports that Kazakh mathematician Mukhtarbay Otelbayev may have solved an extremely difficult and useful mathematics problem: the Navier-Stokes equations.
This is another example of why you simply go directly to Tao, Gowers, and Lipton to see what is actually going on. The drama here is off the charts huge, you have a couple weeks until Tao presents his take on the Navier Stokes approach. He is thinking counterexample based on the non geometric approach. From Tao’s comment log:
Villatoro’s blog has some detailed analysis, and has recently raised some serious issues with the paper as well, in the crucial Section 6. (It’s in Spanish, but this is easy to translate online.)
I can’t read Russian either, so I am happy to defer the detailed checking to others, but my feeling is that this sort of abstract approach to the regularity problem, using only the energy identity and harmonic analysis estimates on the nonlinearity rather than more precise geometric information specific to the Navier-Stokes equation (e.g. the vorticity equation) is necessarily doomed to failure. I think I can formalise a specific obstruction in this regard and hope to present it here in a couple weeks.
Seriously, Netflix guys, you need something to replace CSI: Miami. This is way better – Clay Scene Investigation: Los Angeles, get Keanu to play Tao in the series. Get Goldblum to grow a beard to play Lipton. Jeremy Irons for Gowers? Get Shelby Lyman from Fischer Spassky 1 to be the announcer for the show. I’ll even spot you the elevator pitch – Reality TV meets Police Investigation Drama. Just have your writers point their feedlys to these blogs and the screenplays just pop out. You got Perelman, Twin Primes, the ABC conjecture, that Deolalikar P!=NP proof what else do you need to get started? There’s a bunch of guys working right now deep inside Fine Hall to save the proof that Peano Arithmetic is inconsistent. If they get it, all the math proofs in all the journals and math books have to be rechecked 1 by 1. A generation of young mathematics PhDs will have to be consigned to the old proof checking mines, like in that Indiana Jones movie Temple of Doom. The proposed show goes like that Vaya Con Dios scene in Point Break, with a little more math thrown in. I assume that the UCLA Math department is in Malibu, right? That is all.
Why the sum of all positive integers is -1/12
Cory Doctorow, Boing Boing, Why the sum of all positive integers is -1/12, here.
Here’s a brain-meltingly cool proof of the bizarre mathematical truth that the sum of all positive integers (1 + 2 + 3 + 4 + 5….) is -1/12. This is not only provably true, it’s also foundational to certain testable elements of physics. In other words: not just a logical curiosity, but also the bedrock of real-world, useful stuff.
Navier Stokes Solution Claimed and Twitter Algos
Soulskill, Slashdot, Kazakh Professor Claims Solution of Another Millennium Prize Problem, here. Did time as a first year grad student in the Navier Stokes mines, coding and debugging (mostly debugging) numerical PDE solver schemes for Wavy Vortex flow on an IBM mainframe. Can’t parse the paper in Russian though. I hope Tao can read Russian, otherwise confirmation could take a while.
An anonymous reader writes “Kazakh news site BNews.kz reports that Mukhtarbay Otelbaev, Director of the Eurasian Mathematical Institute of the Eurasian National University, is claiming to have found the solution to another Millennium Prize Problems. His paper, which is called ‘Existence of a strong solution of the Navier-Stokes equations’ and is freely available online (PDF in Russian), may present a solution to the fundamental partial differentials equations that describe the flow of incompressible fluids for which, until now, only a subset of specific solutions have been found.
Otelbaev M., EXISTENCE OF A STRONG SOLUTION OF THE NAVIER-STOKES EQUATION, here. 100+ pages and last page is in English. Confirmation complexity compared to ABC Conjecture should be a layup, unless of course the result depends on InterUniversal Teichmuller Theory. How do you say “InterUniversal Teichmuller Theory” in Russian? Here is an earlier publication M. Otelbaev et. al. 2006, in English – Existence Conditions for a Global Strong Solution to One Class of Nonlinear Evolution Equations in a Hilbert Space, here.
otelbaev.com, Activities, here. Work summary.
His main works are grouped around the following fields:
I. Spectral theory of differential operators.
M. Otelbaev developed new methods for studying the spectral properties of differential operators, which are the result of a consistent and skilled implementation of the general idea of the localization of the problems under consideration. In particular, he invented a construction of averaging coefficients well describing those features of their behaviour which influence the spectral properties of a differential operator. This construction known under the name made it possible to answer many of the hitherto open questions of the spectral theory of the Schrödinger operator and its generalizations.
The function and its different variants have a number of remarkable properties, which allowed to apply this function to a wide range of problems. Here we note some problems for the first time solved by M. Otelbaev by using the function on the basis of sophisticated analysis of the properties of differential operators.
1) A criterion for belonging of the resolvent of the Schrödinger type operator with a non-negative potential to the class was found (previously only a criterion for belonging to was known) and two-sided estimates for the eigenvalues of this operator were obtained with the minimal assumptions of the smoothness of the coefficients.
2) The general localization principle was proved for the problems of selfadjointness and of the maximal dissipativity (simultaneously with the American mathematician P. Chernov) which provided significant progress in this area.
3) Examples were given showing the classical Carleman-Titchmarsh formula for the distribution function of the eigenvalues of the Sturm-Liouville operator is not always correct even in the class of monotonic potentials and a new formula was found valid for all monotonic potentials .
4) The following result of M. Otelbaev is principally important: for there is no universal asymptotic formula.
5) From the time of Carleman, who found the asymptotics for and, by using it, the asymptotics of the eigenvalues themselves, all mathematicians started with finding the asymptotics for and as a result they could not get rid of the so-called Tauberian conditions. M. Otelbaev was the first who, when looking for the asymptotics of the eigenvalues, omitted the interim step of finding the asymptotics for , which allowed getting rid of all non-essential conditions for the problem including Tauberian conditions.
6) The two-sided asymptotics for for the Dirac operator was for the first time found when and are not equivalent.
The results of M. Otelbaev on the spectral theory were included as separate chapters in the monographs of B.M .Levitan and I.S. Sargsyan “ Sturm-Liouville and Dirac operators» (Moscow: Nauka, 1985), and of A.G. Kostyuchenko and I.S. Sargsyan «Distribution of eigenvalues» (Moscow: Nauka, 1979), which became classical
Tyler Durden, Zerohedge, How Twitter Algos Determine Who Is Market-Moving And Who Isn’t, here.
Now that even Bridgewater has joined the Twitter craze and is using user-generated content for real-time economic modelling, and who knows what else, the scramble to determine who has the most market-moving, and actionable, Twitter stream is on. Because with HFT algos having camped out at all the usual newswire sources: Bloomberg, Reuters, Dow Jones, etc. the scramble to find a “content edge” for market moving information has never been higher. However, that opens up a far trickier question: whose information on the fastest growing social network, one which many say may surpass Bloomberg in terms of news propagation and functionality, is credible and by implication: whose is not? Indeed, that is the $64K question. Luckily, there is an algo for that.
Check in with Yitang Zhang
Michael Segal, Nautilus, The Twin Prime Hero, here. Total Stud
What would you do with the money?
Maybe the best way would be to give the money to my wife. Let her deal with this issue.
The paper is quite large now (164 pages!) but it is fortunately rather modular, and thus hopefully somewhat readable (particularly regarding the first half of the paper, which does not need any of the advanced exponential sum estimates). The size should not be a major issue for the journal, so I would not seek to artificially shorten the paper at the expense of readability or content.
The new issue of Nautilus has a wonderful story about Yitang Zhang, called The Twin Prime Hero, which includes a long interview with him. Zhang’s remarkable mathematical career includes several years working at a Subway in Kentucky. His sucessful work on the twin prime conjecture (see here) was done over four years, working seven days a week without almost any breaks, while teaching two classes at a time.
For the next two semesters, I organized a seminar with five graduate students (including Yitang) on Prof. Hironaka’s monumental papers on the theory of resolutions of singularities. I believed that we doubled the world population of those who had studied the papers after we finished two semesters. Prof. Grothendick once described those papers as among the most complicated thesises in the human history.
Check in w RSA
John Leyden, The Register, RSA: That NSA crypto-algorithm we put in our products? Stop using that, here.
“Despite many valid concerns about this generator, RSA went ahead and made it the default generator used for all cryptography in its flagship cryptography library,” noted Green late last week. “The implications for RSA and RSA-based products are staggering. In a modestly bad but by no means worst case, the NSA may be able to intercept SSL/TLS connections made by products implemented with BSafe.”
EMC Corp, Yahoo Finance, here. EMC purchased RSA Securities in 2006?
Emin Gun Sirer, Hacking, Distributed, How the Snowden Saga will End, here.
We knew it because it’s in their DNA. Massive data collection is nothing new. It was only a few years ago that the Stasi archives were opened up to reveal “Geruchsproben,” carefully catalogued jars containing the smell of their citizens. The Stasi had field agents collect smell samples, sometimes by having the citizens sit on special chairs, or sometimes by breaking into people’s homes and literally stealing their underwear.
FFT in Finance
A. Cerny, Introduction to Fast Fourier Transform in Finance, here.
Abstract. The Fourier transform is an important tool in Financial Economics. It delivers real time pricing while allowing for a realistic structure of asset returns, taking into account excess kurtosis and stochastic volatility. Fourier transform is also rather abstract and therefore off-putting to many practitioners. The purpose of this paper is to explain the working of the fast Fourier transform in the familiar binomial option pricing model. We argue that a good understanding of FFT requires no more than some high school mathematics and familiarity with roulette, bicycle wheel, or a similar circular object divided into equally sized segments. The returns to such a small intellectual investment are overwhelming.
Carr and Madan, Option valuation using the fast Fourier transform, here.
The Black±Scholes model and its extensions comprise one of the major developments in modern ®nance. Much of the recent literature on option valuation has successfully applied Fourier analysis to determine option prices (see e.g. Bakshi and Chen 1997, Scott 1997, Bates 1996, Heston 1993, Chen and Scott 1992). These authors numerically solve for the delta and for the risk-neutral prob- ability of ®nishing in-the-money, which can be easily combined with the stock price and the strike price to generate the option value. Unfortunately, this approach is unable to harness the considerable computational power of the fast Fourier transform (FFT) (Walker 1996), which represents one of the most fundamental advances in scientific computing. Furthermore, though the decomposition of an option price into probability elements is theoretically attractive, as explained by Bakshi and Madan (1999), it is numerically undesirable owing to discontinuity of the payoffs.