Field of Science

Alan Turing's declassified papers

@ulaulaman via @MathisintheAir about #AlanTuring on #arXiv
Recently Ian Taylor has uploaded on arXiv a couple of declassified papers by Alan Turing about statistics, probability and cryptography:
1. The Statistics of Repetitions
In order to be able to obtain reliable estimates of the value of given repeats we need to have information about repetition in plain language. Suppose for example that we have placed two messages together and that we find repetitions consisting of a tetragramme, two bigrammes, and fifteen single letters, and that the total overlap was 105, i.e. that the maximum possible number of repetitions which could be obtained by altering letters of the messages is 105; suppose also that the lengths of the messages are 200 and 250; in such a case what is the probability of the fit being right, no other information about the day's traffic being taken into consideration, but information about the character of the enciphered text being available in considerable quantity?
2. The Applications of Probability to Cryptography
The theory of probability may be used in cryptography with most effect when the type of cipher used is already fully understood, and it only remains to find the actual keys. It is of rather less value when one is trying to diagnose the type of cipher, but if definite rival theories about the type of cipher are suggested it may be used to decide between them.

Inge Lehmann: the core of the Earth about #IngeLehmann #EarthCore #theCore #geophysics
Inge Lehmann was a Danish seismologist and geophysicist. Using seismic data, she discovered the inner solid core of the Earth with some physical properties distinct from the outer liquid core:
No rays emerged at epicentral distances between 112° and 154°. I then placed a smaller core inside the first core and let the velocity in it be larger so that a reflection would occur when the rays through the larger core met it. After a choice of velocities in the inner core was made, a time curve was obtained, part of which appeared in the interval where there had not been any rays before. The existence of a small solid core in the innermost part of the earth was seen to result in waves emerging at distances where it had not been possible to predict their presence.

Lehmann, I. (1987). Seismology in the days of old Eos, Transactions American Geophysical Union, 68 (3) DOI: 10.1029/EO068i003p00033-02 (full paper)
Read also: Bolt, B. (1994). Inge Lehmann Physics Today, 47 (1) DOI: 10.1063/1.2808386

The mathematics and geometry in John Hejduck about #JohnHejduck #mathematics #geometry #architecture
In the Diamond project he articulates his idea of the architect's plan as perpendicular to the observer's frontality. This makes the opposite positions between the architect and the imaginary observer equivalent, which, building on Mondrian's ideas, maximises the strength of the contrary oppositions. Hejduk's axes are no longer ordinary axes, not really comparable witb the axis of x,y and z of conventional descriptive geometry. As we turn around, the line becomes a plane. The plane becomes imagined as a wall, a wall corresponding to the bodily movement. It's the moment of passage. The outline is also a membrane, Hejduk's says, and as the relatioosbip between our present and future, the relationship between the architect and the observer, the moment of passage through the wall is a "moment of the hypotenuse", of moving from one condition to another, through an edge between two elements. The concept of the "hypotenuse" is like a cut between the equivalence, as an opening and a movement, moving across two apparently fixed conditions. Hejduk's moment of the hypotenuse, when you become physically inside, is the moment of thought appearing, memory, seeing and moving. It resembles the experience of reading a book; all of a sudden you are in it, on the inside, and it bas become a part of you. But tbere is a difference, Hejduk remarks: there is something special about the physical encounter.
from Architecture of the ineffable: on the work of John Hejduk by Einar Bjarki Malmquist

Snowden and the debate on surveillance versus privacy

In June 2013, NSA contractor Edward Snowden met with journalists Glenn Greenwald and Ewen Macaskill and film-maker Laura Poitras in Hong Kong. The whistleblower gave them documents which proved the existence of a massive scale surveillance system that allows the American NSA and other intelligence and security agencies to gather information on citizens without judicial supervision. While in the USA and in Germany the major media outlets reported extensively on the issue, in Italy there hasn't been a proper public debate on privacy and surveillance, as - except for the work of handful of journalists – the media chose not to cover the implications of such revelations. On the other hand, politics is quick to use the fight against terrorism to push for reforms that might limit people's privacy, a fundamental human right that is currently under attack all over the world. Without a proper balance between surveillance and privacy, the freedom of citizens is at risk. Without a proper public debate, it is hard to understand what is at stake. For the first time in Italy, such debate will take place including the voices of the people who made the information public: Edward Snowden, the whistleblower who revealed the scope of the NSA surveillance practices, will be joining the conversation, as well as the independent film-maker Laura Poitras. Poitras recently won an Academy Award for the documentary Citizenfour, where she shows the meetings between the whistleblower and the journalists, and a Pulitzer prize for her journalistic work on the story. The human rights implications will be explored by Ben Wizner (ACLU), Snowden's lawyer, and Andrea Menapace, who directs the newly-born Italian Coalition for Civil Rights and Freedoms. Organised in association with Italian Coalition for Civil Rights and Freedoms (CILD) and American Civil Liberties Union (ACLU) Speakers: Edward Snowden (via Skype) Laura Poitras (via Skype) Ben Wizner (ACLU) Andrea Menapace (CILD) Simon Davies (Privacy International) Introduction: Patrizio Gonnella (CILD) Moderator: Fabio Chiusi Con: Fabio Chiusi (journalist and author), Simon Davies (founder Privacy International), Patrizio Gonnella (president CILD), Andrea Menapace (director CILD), Laura Poitras (documentary film-maker (via Skype)), Edward Snowden (whistleblower (via Skype)), Ben Wizner (ACLU), Ben Wizner

Cassini ovals

#HappyEaster from @ulaulaman with #math

via commons
A Cassini oval is a quartic plane curve defined as the set (or locus) of points in the plane such that the product of the distances to two fixed points is constant. This may be contrasted to an ellipse, for which the sum of the distances is constant, rather than the product. Cassini ovals are the special case of polynomial lemniscates when the polynomial used has degree 2.
Cassini ovals are named after the astronomer Giovanni Domenico Cassini who studied them in 1680. Other names include Cassinian ovals, Cassinian curves and ovals of Cassini.
Read also: MathWorld, McTutor

Louis Nirenberg, the geometry and the Abel Prize 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)