Mathematics

Recently Rattle posted a wonderful mathy poem by Diana Rosen entitled "Mathematics and Molé." Here's the first stanza: Mathematics and Molé by Diana Rosen Numbers flicker in front of my eyes as I give him my full attention. Differential geometry explains the black hole, he says. It’s very obvious. I lean forward to catch his words, my chin in cupped hand, eyes intent on his, yet thinking of Mexican food. Mathematics is the language of science, he says. . . .Rozen's complete poem is here.

I've dealt with numbers all my life, of course, and after a while you begin to feel that each number has a personality of its own. A twelve is very different from a thirteen, for example. Twelve is upright, conscientious, intelligent, whereas thirteen is a loner, a shady character who won't think twice about breaking the law to get what he wants. Eleven is tough, an outdoorsman who likes tramping through woods and scaling mountains; ten is rather simpleminded, a bland figure who always does what he's told; nine is deep and mystical, a Buddha of contemplation.... ~Paul Auster, The Music of Chance The 138th day of the year; 138 has three prime factors, the concatenation of the first two is the third. *Prime Cuiros EVENTS1825 Faraday isolates benzine. In 1825, Faraday started work on a sample of oil that had been sent to him for analysis by the Portable Oil Company of London. He subjected this oil to fractional distillation, a process that proved to be extremely difficult, and it took him some time to resolve the oil into its pure components. By repeated fractional distillation followed by selective fractional freezing, each stage monitored by analysis, he produced a fairly pure sample of what he called bicarburet of hydrogen. Faraday’s notebook records these procedures, which he carried out on 18 and 19 May 1825. Auguste Laurent suggested the name benzene. *Jennifer Wilson, Celebrating Michael Faraday’s Discovery of Benzene, Ambix,Volume 59, Issue 3 1852 Massachusetts becomes the ?rst state to pass a compulsory attendance law for school children. *VFR 1896, the Supreme Court ruled separate-but-equal facilities constitutional on intrastate railroads. For some fifty years, the Plessy v. Ferguson decision upheld the principle of racial segregation. Across the country, laws mandated separate accommodations on buses and trains, and in hotels, theaters, and schools. In a speech delivered in the Ohio House of Representatives in 1886 and later published as The Black Laws, legislator Benjamin W. Arnett described life in segregated Ohio: "This foe of my race stands at the school house door and separates the children, by reason of 'color,' and denies to those who have a visible admixture of African blood in them the blessings of a graded school and equal privileges... We call upon all friends of 'Equal Rights' to assist in this struggle to secure the blessings of untrammeled liberty for ourselves and posterity. " After hearing arguments by NAACP lawyer Thurgood Marshall, the Supreme Court overruled the Plessy decision on May 17, 1954. In Brown v. the Board of Education, a unanimous Court adopted Justice Harlan's position that segregation violated the Thirteenth and Fourteenth Amendments to the Constitution. *Library of Congress 1901 Charles Sanders Peirce writes George A. Plimpton, head of Ginn and Company and famous collector of rare mathematical books, describing what the contents of a newly acquired book must be were it indeed the great Liber Abaci (1202) of Fibonacci. In 1949 Carolyn Eisele’s discovery of this letter—still tucked into the back cover of the volume—began her career as a Peirce scholar. [HM 9, 335] *VFR In 1910, Halley's Comet was visible from Earth, moving across the face of the sun. *TIS 1910 Halley's comet was big news during its visible period in New York City. Beginning with the Saturday edition of May 14 and continuing on through the Sunday edition of May 22, the comet was given top billing in the New York Times. This was the period when the comet was at the height of its brilliance and activity and the coverage clearly reflected this.May 18: Earth to pass through come tail for 6 hours; C.B. Harmon invites college deans to join him in viewing comet from balloon. *Joseph M. Laufer, Halley's Comet Society - USA 1933 John Kieran’s Sports of the Times column in the New York Times is entitled “The Coordinate Clash, or Block that Abscissa.” The column was a humorous analogy between football and the upcoming

There’s going to be a “thematic trimester” in Paris starting next spring: Semantics of proofs and certified mathematics, Institut Henri Poincar?, April 22nd - July 11th, 2014, organized by Pierre-Louis Curien, Hugo Herbelin, Paul-Andr? Melli?s. If you like applications of category theory to logic and computer science, there should be a lot for you here! The basic layout is this: Week 1 — Kick-off: Formalisation in mathematics and in computer science Week 3 — Workshop 1: Formalization of mathematics in proof assistants, organized by Georges Gonthier and Vladimir Voevodsky. Week 6 — Workshop 2: Constructive mathematics and models of type theory, organized by Thierry Coquand and Thomas Streicher. Week 8 — Workshop 3: Semantics of proofs and programs, organized by Thomas Ehrhard and Alex Simpson. Week 10 — Workshop 4: Abstraction and verification in semantics, organized by Paul-Andr? Melli?s and Luke Ong. Week 12 — Workshop 5, Certification of high-level and low-level programs organized by Christine Paulin and Zhong Shao. A lot of people I know will attend parts of this, such as Jean Benabou, Marcelo Fiore, Dan Ghica, André Joyal, Samuel Mimram, and Bas Spitters. And that makes me happy, because Paul-Andr? Melli?s has invited me to spend up to a month attending this series of workshops, perhaps in two 2-week stretches. With a little luck I’ll be able to actually do this. (My wife Lisa Raphals has gotten invited to Erlangen for the spring of 2014, meaning roughly April 1 - June 1. If she and I succeed in getting leaves of absence, I’ll go with her, and then take some trips to nearby places. Since live in the Wild West, Paris seems nearby to Erlangen to me. I also have vague invitations to IHES, Prague and Berlin which I might try to take advantage of.)

May 17 was the 137th day of the year, and as usual on my "On This Day in Math" post I added a note that "The 137th day of the year; 137 is the sum of the squares of the first seven digits of pi, 32+ 12 + 42 + 12 + 52 + 92 + 22 = 137." (Plus a bit more)When I posted a tweet of the above, I got a note from jignesh rathod ?@engineer_rathod to inform me that 137 was also unique in that if you drop any one of the digits, the remaining digits form a prime. I pushed this to the point that if we allow the old-fashioned idea that 1 was a prime then 137 was a prime number which would still be prime if any number was eliminated, and each digit is prime (2nd thoughts about this, more later) . In fact, when any digit is removed, the remaining digits may be placed in any order and would still be prime. (take out 3 and we have 17 and 71). I wrote and asked if the idea had a name and there seemed to be no formal (or much of an informal) consences . I had thought of ultra-prime right off the bat, but then I got a note from Chris Maslanka to suggest the idea of a Knockout Prime, and I knew I had found the term I wanted. There was still a problem about how to distinguish special cases. Was one acceptable as a prime, what about those that were reversible. My next epipheny came in a tweet from @mathematicsprof who suggested the notation C(3,3) is Combination Prime for the fact that the original 3-digit number (137) was prime, and C(3,2) is combination Prime for if all two digit pairs were prime ( 13, 17, 37) . I worried about using the C(m,n) notation when it might be confused with the use in combinations, so I decided to use CP(m,n) for Combination Prime... then I changed my mind again and decided to use Chris' name as part of the symbol and opted for KP(m,n) as in Knockout Prime. So 137 is KP(3,3) and KP(3,2) but not KP(3,1) (using the modern approach that 1 is not considered prime). Eliminating one may well mean that there exists no KP(3,1) numbers as conjectured by the Math Prof. Suddently it hit me, a lesser symbol for those allowing 1 as prime, Kp(3,1) . So 137 is Kp(3,1) but not KP(3,1). I toyed with a symbol to differentiate 23 and 37 since both are KP(2,2) but 23 is not a prime when reversed and 37 is. I decided to settle for just using reversible KP(2,2). So a number like 137 would be reversible (3,2) since any two digits remaining are KP in either order. I haven't taken the time to do a computer search of all the 2,3,4 digit numbers to see how they qualify. If anybody jumps on that before I get to it, send me a copy. I expect I will update this page as better ideas are contributed by others. Thanks for comments

Sylvain Gigan (also in Science) let me know of the following paper that is also making the rounds in Science. If you recall, it was shown earlier that compressive sensing was speeding up the process of acquiring the data. The physics itself is not changed and revolves around this formula: 3D Computational Ghost Imaging by Baoqing Sun, Matthew P. Edgar, Richard Bowman, Liberty E. Vittert,Stephen S. Welsh, Ardrian Bowman, Miles J. Padgett Computational ghost imaging retrieves the spatial information of a scene using a single pixel detector. By projecting a series of known random patterns and measuring the back reflected intensity for each one, it is possible to reconstruct a 2D image of the scene. In this work we overcome previous limitations of computational ghost imaging and capture the 3D spatial form of an object by using several single pixel detectors in different locations. From each detector we derive a 2D image of the object that appears to be illuminated from a different direction, using only a single digital projector as illumination. Comparing the shading of the images allows the surface gradient and hence the 3D form of the object to be reconstructed. We compare our result to that obtained from a stereo- photogrammetric system utilizing multiple high resolution cameras. Our low cost approach is compatible with consumer applications and can readily be extended to non-visible wavebands. The 3D of this paper has little to do with the 2D extraction from Three-dimensional ghost imaging ladar and we measure reflectance difference here as opposed to time of flight information. Here is the multispectral paper: Multi-wavelength compressive computational ghost imaging by Stephen S. Welsh, Matthew P. Edgar, Phillip Jonathan, Baoqing Sun, Miles. J. Padgett The field of ghost imaging encompasses systems which can retrieve the spatial information of an object through correlated measurements of a projected light eld, having spatial resolution, and the associated reflected or transmitted light intensity measured by a photodetector. By employing a digital light projector in a computational ghost imaging system with multiple spectrally fi ltered photodetectors we obtain high-quality multi-wavelength reconstructions of real macroscopic objects. We compare di erent reconstruction algorithms and reveal the use of compressive sensing techniques for achieving sub-Nyquist performance. Furthermore, we demonstrate the use of this technology in non-visible and fluorescence imaging applications. Thanks Sylvain ! Join the CompressiveSensing subreddit or the Google+ Community and post there ! Liked this entry ? subscribe to Nuit Blanche's feed, there's more where that came from. You can also subscribe to Nuit Blanche by Email, explore the Big Picture in Compressive Sensing or the Matrix Factorization Jungle and join the conversations on compressive sensing, advanced matrix factorization and calibration issues on Linkedin.

Here at Wolfram Research and at Wolfram|Alpha we love mathematics and computations. Our favorite topics are algorithms, followed by formulas and equations. For instance, Mathematica can calculate millions of (more precisely, for all practical purposes, infinitely many) integrals, and Wolfram|Alpha knows hundreds of thousands of mathematical formulas (from Euler’s formula and BBP-type formulas for pi to complicated definite integrals containing sin(x)) and plenty of physics formulas (e.g from Poiseuille’s law to the classical mechanics solutions of a point particle in a rectangle to the inverse-distance potential in 4D in hyperspherical coordinates), as well as lesser-known formulas, such as formulas for the shaking frequency of a wet dog, the maximal height of a sandcastle, or the cooking time of a turkey. Recently we added formulas for a variety of shapes and forms, and the Wolfram|Alpha Blog showed some examples of shapes that were represented through mathematical equations and inequalities. These included fictional character curves: Shape curves: And, most popular among our users, person curves: While these are curves in a mathematical sense, similar to say a lemniscate or a folium of Descartes, they are interesting less for their mathematical properties than for their visual meaning to humans. After Richard’s blog post was published, a coworker of mine asked me, “How can you make an equation for Stephen Wolfram’s face?” After a moment of reflection about this question, I realized that the really surprising issue is not that there is a formula: a digital image (assume a grayscale image, for simplicity) is a rectangular array of gray values. From such an array, you could build an interpolating function, even a polynomial. But such an explicit function would be very large, hundreds of pages in size, and not useful for any practical application. The real question is how you can make a formula that resembles a person’s face that fits on a single page and is simple in structure. The formula for the curve that depicts Stephen Wolfram’s face, about one page in length, is about the size of a complicated physics formula, such as the gravitational potential of a cube. In this post, I want to show how to generate such equations. As a “how to calculate…”, the post will not surprisingly contain a fair bit of Mathematica code, but I’ll start with some simple introductory explanations. Assume you make a line drawing with a pencil on a piece of paper, and assume you draw only lines; no shading and no filling is done. Then the drawing is made from a set of curve segments. The mathematical concept of Fourier series allows us to write down a finite mathematical formula for each of these line segments that is as close as wanted to a drawn curve. As a simple example, consider the series of functions yn(x), which is a sum of sine functions of various frequencies and amplitudes. Here are the first few members of this sequence of functions: Plotting this sequence of functions suggests that as n increases, yn(x) approaches a triangular function. The sine function is an odd function, and as a result all of the sums of terms sin(k x) are also odd functions. If we use the cosine function instead, we obtain even functions. A mixture of sine and cosine terms allows us to approximate more general curve shapes. Generalizing the above (-1)(k – 1)/2) k-2 prefactor in front of the sine function to the following even or odd functions, allows us to model a wider variety of shapes: It turns out that any smooth curve y(x) can be approximated arbitrarily well over any interval [x1, x2] by a Fourier series. And for smooth curves, the coefficients of the sin(k x) and cos(k x) terms approach zero for large k. Now given a parametrized curve ?(t) = {?x(t), ?y(t)}, we can use such superpositions of sine and cosine functions independently for the horizontal component ?x(t) and f

I imagine many readers of this blog are familiar with the fact that you can knot a circle in 3-space, but not in 4-space.    If you enjoy thinking about why that is true, please read on! Think of euclidean 3-space, as a linear subspace of euclidean 4-space, .  So if you have a knotted circle in 3-space, you can consider it as an embedded circle in 4-space.  And you can unknot it! I think one of the simplest explanations of of this would be the idea to push the knot up into the 4-th dimension every time a strand is close to being an overcrossing (in a planar diagram).   At this stage you could in effect change the crossing to be anything you want, after you’re done modifying the crossings, you could push the knot back into 3-space to get a different knot.  But there’s a more uniform version of this construction.  I think I first learned of it from Rolfsen’s textbook.  It sits most naturally in a slightly different formalism.  is a smooth embedding and   whenever 1 \}' title='|x| > 1 \}' class='latex' />. is called the space of long knots in .  There is a natural inclusion map .  I claim it is a null homotopic map.  The idea is pretty simple. When you perform the inclusion, there is a new direction in orthogonal to the original . Let’s call the unit vector in that direction . The idea is to take a little bump function , and add to . Choose so that it is strictly increasing along the interval , and then have it decrease to zero quickly near . There is a straight-line isotopy from to , and also from to where is the standard inclusion, i.e. always. Similarly there is a straight-line isotopy from to . Assemble these three maps together and you have your null-homotopy of . Once you see this null-homotopy, notice that there actually two such null-homotopies! If instead of choosing to be increasing along , we could have chosen it to be decreasing along . If you assemble those two null-homotopies together, you get a map , or equivalently . My question for people here is, is this map null-homotopic? I don’t know the answer to this question. From the perspective of the Vassiliev spectral sequence, or the Goodwillie embedding calculus, this is a difficult map to understand. You can check that this map is zero on rational homology and rational homotopy groups. Not enough is known about the torsion in homotopy or homology to say what’s going on there — in a way that’s part of why I love this question. It’s also possible one of you will notice there’s a naive null-homotopy of this map. I just don’t see it.

Dmitry Savostyanov pointed me to the location of an implementation of the other super Fast FFT we mentioned last September. It is on GitHub here as part of the very promising TT-Toolbox. Let us note that the sparse FFT "looks" slower than MIT's sFFT and that the MIT sFFT has currently only version 1 and 2 while Piotr mentioned results for version 3 and 4 on Wednsday at the "Big data: theoretical and practical challenges" workshop. We are waiting for the Berkeley implementation mentioned previously for the end of the summer. The Ann Arbor FFT (AAFFT) is here. With regards to the comparison between sFFT [2] and the TT_toolbox version [1] It looks like sFFT version 3.0 scales better for K sparse signals than the TT_toolvox one which scales as K^3. But I wouldn't mind seeing comparison for compressible signals (I think it is version 4.0 for sFFT) and the TT_toolbox one. An interesting comparison should eventually entail not integer frequencies and higher dimensions as well. [1] Superfast Fourier Transform Using QTT Approximation by Sergey Dolgov, Boris Khoromskij and Dmitry Savostyanov.[2] http://groups.csail.mit.edu/netmit/sFFT/results.html Join the CompressiveSensing subreddit or the Google+ Community and post there ! Liked this entry ? subscribe to Nuit Blanche's feed, there's more where that came from. You can also subscribe to Nuit Blanche by Email, explore the Big Picture in Compressive Sensing or the Matrix Factorization Jungle and join the conversations on compressive sensing, advanced matrix factorization and calibration issues on Linkedin.

A commentator named Oz proposed the following question: You have a box with n red balls and n blue balls. You take out each time a ball at random but, if the ball was red, you put it back in the box and take out a blue ball. If the ball was blue, you put it back in the box and take out a red ball. You keep doing it until left only with balls of the same color. How many balls will be left (as a function of n)? Peter Shor wrote in a comment “I’m fairly sure that there is not enough bias to get , but it intuitively seems far too much bias to still be . I want to say . At a wild guess, it’s either or , since those are the simplest exponents between  and .”  The comment followed by a heuristic argument of Kevin Kostelo and computer experiments by Lior Silberman that supported the answer . This is correct! Good intuition, Peter! In our student probability day a couple of weeks ago, Yuval Peres told me the origin of this problem. The way it was originally posed by D. Williams and P. McIlroy used roughly the following story (I modified it a little): There are two groups of n gunmen that shoot at each other. Once a gunman is hit he stops shooting, and leaves the place happily and peacefully. How many gunmen will be left after all gunmen in one team had left. The problem was solved by Kingman and Volkov in this paper. J. F. C. Kingman, and S. E. Volkov,  Solution to the OK Corral model via decoupling of Friedman’s urn, J. Theoret. Probab. 16 (2003), no. 1, 267–276. A nice presentation of the result entitled: Internal erosion and the exponent 3/4 was given by Lionel Levine and Yuval Peres.

Even stranger things have happened; and perhaps the strangest of all is the marvel that mathematics should be possible to a race akin to the apes.~Eric T. Bell, The Development of MathematicsThe 137th day of the year; 137 is the sum of the squares of the first seven digits of pi, 32+ 12 + 42 + 12 + 52 + 92 + 22 = 137. *Prime Curios (Can you find other such primes from sums of squares of Pi?) 137 is the third term in a sequence of primes that can be created by staring with 7 and creating a new term by adding a single digit to the front of the previous term; 7, 37, 137 ... It is possible to create a sequence of 15 Prime numbers in this way. OEISEVENTS1630 Belts on Jupiter ?rst recognized. According to Rogers, the first known mention of belts on Jupiter is that by Niccolo Zucchi in 1630.( J. H. Rogers, The Giant Planet Jupiter (Cambridge University Press, 1995).)1719 “The learned Dr. Halley is of opinion that the comet seen in 1680 is the same which appeared in Julius Caesar’s time. This shows more than any other that comets are hard, opaque bodies; for it descended so near to the sun, as to come within a sixth part of the diameter of this planet from it, and consequently might have contracted a degree of heat two thousand times stronger than that of red-hot iron; and would have been soon dispersed in vapour, had it not been a ?rm, dense body. The guessing the course of comets began then to be very much in vogue. The celebrated [Johann] Bernoulli concluded by his system than the famous comet of 1680 would appear again the 17th of May, 1719. Not a single astronomer in Europe went to bed that night. However, they needed not to have broke their rest, for the famous comet never appeared.” So wrote Voltaire (1694-1778) in his Letters on the English or Lettres Philosophiques, c. 1778. *VFR 1749 Oops.... In 1747 at a public session in the French Academy of Sciences Clairaut stated that Newton's theory of gravity was wrong. Euler and d’Alembert had simultaneously came to the same conclusion as all had been working on the motion of the moon as a special case of the three body problem. Clairaut suggested that the strength of gravity was proportional not to 1/r^2 , but the more complicated 1/r^2 +c/r^4 for some constant c. Over large distances, the c/r^4 term would effectively disappear, accounting for the utility of the inverse square law over large distances. He then began trying to find a value of c which could account for the moon's motion. He would continue to pursue this idea until May 17, 1749, when he made an equally dramatic announcement in which he claimed that Newton was right after all. (See Deaths below)1861 James Clerk Maxwell exhibited a three-color photographic process before the Royal Institution of Great Britain on May 17, 1861. Maxwell photographed a colored ribbon on photographic plates. He made three exposures: one through a red filter, one through a green filter, and one through a blue filter. He probably then re-exposed those images onto other plates, or somehow processed them into positive rather than negative images; the published paper is unclear on the process. Then, he used magic lanterns to project his transparencies, superimposed the three images, and filtered the projectors as he had filtered the original images—with red, green & blue filters. He produced a colored image, ".a coloured image was seen, which, if the red and green images had been as fully photographed as the blue, would have been a truly-coloured image of the ribbon." (The Muser)1882 A comet is discovered and photographed by Sir Arthur Schuster (1851-1934), Germany/UK, during an eclipse in Egypt: first time a comet discovered in this way has been photographed. The Total Solar Eclipse had been observed by Sir Joseph Norman Lockyer (1836-1920), Ranard and Schuster from England, Tacchini from Italy, Trépied, Thollon and Puiseux from France.Observation from Sohag at the Nile. *NASA Eclipse Calendar 1910 Halley's comet was big news during its visible