Field of Science

Louis Nirenberg, the geometry and the Abel Prize

http://t.co/rTEYZLbVDZ by @ulaulaman about #AbelPrize #LouisNirenberg
Great news: John Nash and Louis Nirenberg win the Abel Prize for 2015:
The Norwegian Academy of Sciences and Letters has decided to award the Abel Prize for 2015 to the American mathematicians John F. Nash Jr. and Louis Nirenberg “for striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis.” The President of the Academy, Kirsti Strøm Bull, announced the new laureates today 25 March. They will receive the Abel Prize from His Majesty King Harald at a ceremony in Oslo on 19 May.
While Nash is known for his contribution to game theory with Nash equilibria, Nirenberg is
considered one of the outstanding analysts of the twentieth century. He has made fundamental contributions to linear and nonlinear partial differential equations and their application to complex analysis and geometry.
His first result was the completion of the solution of a problem in differential geometry, starting from the 1916 work of Weyl (read On the work of Louis Nirenberg by Simon Donaldson, pdf).
The statement is very simple: an abstract Riemannian metric on the 2-sphere with positive curvature can be realised by an isometric embedding in $R^3$.
In 1994 he received the Steele Prize by the American Mathematical Society. In that occasion, the Society writes (via MacTutor) a good summary of his activity:
Nirenberg is a master of the art and science of obtaining and applying a priori estimates in all fields of analysis. A minor such gem is the useful set of Garliardo-Nirenberg inequalities. A high point is his joint research with [Shmuel] Agmon and [Avron] Douglis on a priori estimates for general linear elliptic systems, one of the most widely quoted results in analysis. Another is his fundamental paper with Fritz John on functions of bounded mean oscillation which was crucial for the later work of [Charles] Fefferman on this function space. Nirenberg has been the centre of many major developments. His theorem with his student, Newlander, on almost complex structures has become a classic. In a paper building on earlier estimates of [Alberto] Calderón and [Antoni] Zygmund, he and [Joseph] Kohn introduced the notion of a pseudo-differential operator which helped to generate an enormous amount of later work. His research with [François] Trèves was an important contribution to the solvability of general linear PDEs. Some other highlights are his research on the regularity of free boundary problems with [David] Kinderlehrer and [Joel] Spuck, existence of smooth solutions of equations of Monge-Ampère type with [Luis] Caffarelli and Spuck, and singular sets for the Navier-Stokes equations with Caffarelli and [Robert] Kohn. His study of symmetric solutions of non-linear elliptic equations using moving plane methods with [Basilis] Gidas and [Wei Ming] Ni and later with [Henri] Berestycki, is an ingenious application of the maximum principle.
I hope that you can appreciate the list of collaboration performed by Nirenberg: indeed he is one of the most collaborative mathematician in the word. In particular for this reason I'm really happy for the award to Nirenberg.
Read also: Interview with Louis Nirenberg (pdf)

Pi day: Zilienski, Wallis and the square

http://t.co/SopVLuLJHM by @ulaulaman #piday
A good pi day to all readers! I hope that the following post, that I cannot review after the first writing, could be interesting to all!
The technique used by the ancient Greek for their geometric constructions was called "ruler and compass". In this way you can build a lot of regular polygons, for example, but there are three problems that are impossible unless you use different techniques: the angle trisection, doubling the cube, squaring the circle.
In particular for the squaring, it is easy to calculate the relation between the radius $r$ of the circke and the side $l$ of the square with the same area of the starting circle: \[L = \sqrt {\pi} r \] Now, since $\pi$ is a transcendental number, the formula above is the simplest representation of the impossibility of squaring the circle using ruler and compass: with these devices you can treat rational and irrational numbers, such as $\sqrt{2}$ (in this case simply draw a square of side 1).
So, using these two tools it is possible to obtain an approximate construction and, therefore, a corresponding approximate value for $\pi$: during the XX century there are produced a lot of approximations, for example by CD Olds (1963), Martin Gardner (1966), Benjamin Bold (1982). They are all variations of the geometric construction discovered by Srinivasa Ramanujan in 1913, that approached $\pi$ with the fraction \[\frac{355}{113} = 3.1415929203539823008 \dots\] right up to the sixth decimal place.
In 1914, Ramanujan discovered a more accurate approximation (to eight decimal places), always using ruler and compass: \[\left (9^2 + \frac{19}{22}^2 \right)^{1/4} = \sqrt[4]{\frac{2143}{22}} = 3.1415926525826461252 \dots\]

Riemann zeta function's fractal

#arXiv #abstract on #zetafunction and #fractals

via imgur
The quadratic Mandelbrot set has been referred to as the most complex and beautiful object in mathematics and the Riemann Zeta function takes the prize for the most complicated and enigmatic function. Here we elucidate the spectrum of Mandelbrot and Julia sets of Zeta, to unearth the geography of its chaotic and fractal diversities, combining these two extremes into one intrepid journey into the deepest abyss of complex function space.
Fractal Geography of the Riemann Zeta Function by Chris King

Rock, paper, scissors, lizard, Spock

http://t.co/a0l4FP6ftF Goodbye #LeonardNimoy, #Spock from #StarTrek
One popular five-weapon expansion is "rock-paper-scissors-lizard-Spock", invented by Sam Kass and Karen Bryla, which adds "Spock" and "lizard" to the standard three choices. "Spock" is signified with the Star Trek Vulcan salute, while "lizard" is shown by forming the hand into a sock-puppet-like mouth. Spock smashes scissors and vaporizes rock; he is poisoned by lizard and disproven by paper. Lizard poisons Spock and eats paper; it is crushed by rock and decapitated by scissors. This variant was mentioned in a 2005 article of The Times and was later the subject of an episode of the American sitcom The Big Bang Theory in 2008.
The majority of such proposed generalizations are isomorphic to a simple game of modulo arithmetic, where half the differences are wins for player one. For instance, rock-paper-scissors-Spock-lizard (note the different order of the last two moves) may be modeled as a game in which each player picks a number from one to five. Subtract the number chosen by player two from the number chosen by player one, and then take the remainder modulo 5 of the result. Player one is the victor if the difference is one or three, and player two is the victor if the difference is two or four. If the difference is zero, the game is a tie.
Alternatively, the rankings in rock-paper-scissors-Spock-lizard may be modeled by a comparison of the parity of the two choices. If it is the same (two odd-numbered moves or two even-numbered ones) then the lower number wins, while if they are different (one odd and one even) the higher wins. Using this algorithm, additional moves can easily be added two at a time while keeping the game balanced:
  • Declare a move N+1 (where N is the original total of moves) that beats all existing odd-numbered moves and loses to the others (for example, the rock (#1), scissors (#3), and lizard (#5) could fall into the German well (#6), while the paper (#2) covers it and Spock (#4) manipulates it).
  • Declare another move N+2 with the reverse property (such as a plant (#7) that grows through the paper (#2), poisons Spock (#4), and grows through the well (#6), while being damaged by the rock (#1), scissor (#3), and lizard(#5)).
(via en.wiki)

The dark side of the moon

#moon #astronomy #NASA #video #PinkFloyd

The first photo of the lunar far side taken by the Soviet spacecraft Luna 3 on Oct. 7, 1959 - via Universe Today

What Einstein thought about Galilei

about #AlbertEinsten #GalileoGalilei
Galileo's Dialogue Concerning the Two Chief World Systems is a mine of information for anyone interested in the cultural history of the Western world and its influence upon economic and political development.
(...) To begin with, the Dialogue gives an extremely lively and persuasive exposition of the then prevailing views on the structure of the cosmos in the large. The naïve picture of the earth as a flat disc, combined with obscure ideas about star-filled space and the motions of the celestial bodies, prevalent in the early Middle Ages, represented a deterioration of the much earlier conceptions of the Greeks, and in particular of Aristotle’s ideas and Ptolemy’s consistent spatial concept of the celestial bodies and their motions.
(...) In advocating and fighting for the Copernican theory Galileo was not only motivated by a striving to simplify the representation of the celestial motions. His aim was to substitute for a petrified and barren system of ideas the unbiased and strenuous quest for a deeper and more consistent comprehension of the physical and astronomical facts.
The form of dialogue used in his work may be partly due to Plato’s shining example; it enabled Galileo to apply his extraordinary literary talent to the sharp and vivid confrontation of opinion. To be sure, he wanted to avoid an open commitment in these controversial questions that would have delivered him to destruction by the Inquisition. Galileo had, in fact, been expressly forbidden to advocate the Copernican theory. Apart from its revolutionary factual content the Dialogue represents a down-right roguish attempt to comply with this order in appearance and yet in fact to disregard it. Unfortunately, it turned out that the Holy Inquisition was unable to appreciate adequately such subtle humor.
(...) It is difficult to us today to appreciate the imaginative power made manifest in the precise formulation of the concept of acceleration and in the recognition of its physical significance.
Once the conception of the center of the universe had, with good reason, been rejected, the idea of the immovable earth, and, generally, of an exceptional role of the earth, was deprived of its justification (...)
(...) Galileo takes great pains to demonstrate that the hypothesis of the rotation and revolution of the earth is not refuted by the fact that we do not observe any mechanical effects of these motions. Strictly speaking, such a demonstration was impossible because a complete theory of mechanics was lacking. I think it is just in the struggle with this problem that Galileo’s originality is demonstrated with particular force. Galileo is, of course, also concerned to show that the fixed stars are too remote for parallaxes produced by the yearly motion of the earth to be detectable with the measuring instruments of his time. This investigation also is ingenious, notwithstanding its primitiveness.
It was Galileo’s longing for a mechanical proof of the motion of the earth which misled him into formulating a wrong theory of the tides. The fascinating arguments in the last conversation would hardly have been accepted as proofs by Galileo, had his temperament not got the better of him. It is hard for me to resist the temptation to deal with this subject more fully.
The leitmotif which I recognize in Galileo’s work is the passionate fight against any kind of dogma based on authority. Only experience and careful reflection are accepted by him as criteria of truth. Nowadays it is hard for us to grasp how sinister and revolutionary such an attitude appeared at Galileo’s time, when merely to doubt the truth of opinions which had no basis but authority was considered a capital crime and punished accordingly. Actually we are by no means so far removed from such a situation even today as many of us would like to flatter ourselves; but in theory, at least, the principle of unbiased thought has won out, and most people are willing to pay lip service to this principle.
It has often been maintained that Galileo became the father of modern science by replacing the speculative, deductive method with the empirical, experimental method. I believe, however, that this interpretation would not stand close scrutiny. There is no empirical method without speculative concepts and systems; and there is no speculative thinking whose concepts do not reveal, on closer investigation, the empirical material from which they stem. To put into sharp contrast the empirical and the deductive attitude is misleading, and was entirely foreign to Galileo. Actually it was not until the nineteenth century that logical (mathematical) systems whose structures were completely independent of any empirical content had been cleanly extracted. Moreover, the experimental methods at Galileo’s disposal were so imperfect that only the boldest speculation could possibly bridge the gaps between the empirical data. (For example, there existed no means to measure times shorter than a second). The antithesis Empiricism vs. Rationalism does not appear as a controversial point in Galileo’s work. Galileo opposes the deductive methods of Aristotle and his adherents only when he considers their premises arbitrary or untenable, and he does not rebuke his opponents for the mere fact of using deductive methods. In the first dialogue, he emphasizes in several passages that according to Aristotle, too, even the most plausible deduction must be put aside if it is incompatible with empirical findings. And on the other hand, Galileo himself makes considerable use of logical deduction. His endeavors are not so much directed at "factual knowledge" as at "comprehension". But to comprehend is essentially to draw conclusions from an already accepted logical system.
(from the foreword to Dialogue Concerning the Two Chief World Systems: Ptolemaic and Copernican (1953), Einstein Archives 1-174 - via Open Parachute)
About the italian physicist, Galileo Galilei and the impossible biomechanics of giants is an interesting reading.

The mathematics of love

#ValentinesDay #mathematics
\[\left (x^2 + \frac{9}{4} y^2 + z^2 - 1 \right )^3 - x^2 z^3 - \frac{9}{200} y^2 z^3 = 0\]