After almost 20 years, math problem falls Mathematicians and engineers are often concerned with finding the minimum value of a particular mathematical function. That minimum could represent the optimal trade-off between competing criteria — between the surface area, weight and wind resistance of a car’s body design, for instance. In control theory, a minimum might represent a stable state of an electromechanical system, like an airplane in flight or a bipedal robot trying to keep itself balanced. There, the goal of a control algorithm might be to continuously steer the system back toward the minimum. For complex functions, finding global minima can be very hard. But it’s a lot easier if you know in advance that the function is convex, meaning that the graph of the function slopes everywhere toward the minimum. Almost 20 years later, researchers in MIT’s Laboratory for Information and Decision Systems have finally answered that question. Downhill from here On the first paper, Parrilo and Ahmadi were joined by John N. Squaring off
The Unreasonable Effectiveness of Mathematics in the Natural Sciences Reading Materials by R. W. HAMMING Reprinted From: The American Mathematical Monthly Volume 87 Number 2 February 1980 Prologue. Man, so far as we know, has always wondered about himself, the world around him, and what life is all about. Philosophy started when man began to wonder about the world outside of this theological framework. From these early attempts to explain things slowly came philosophy as well as our present science. Our main tool for carrying out the long chains of tight reasoning required by science is mathematics. Mathematicians working in the foundations of mathematics are concerned mainly with the self-consistency and limitations of the system. Once I had organized the main outline, I had then to consider how best to communicate my ideas and opinions to others. In some respects this discussion is highly theoretical. The inspiration for this article came from the similarly entitled article, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" [1.
The Unreasonable Effectiveness of Mathematics in the Natural Sciences Reading Materials by Eugene Wigner "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960). New York: John Wiley & Sons, Inc. Copyright © 1960 by John Wiley & Sons, Inc. Mathematics, rightly viewed, possesses not only truth, but supreme beautya beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. THERE IS A story about two friends, who were classmates in high school, talking about their jobs. Naturally, we are inclined to smile about the simplicity of the classmate's approach. The preceding two stories illustrate the two main points which are the subjects of the present discourse. The complex numbers provide a particularly striking example for the foregoing. Let me end on a more cheerful note. Merci W.
Unicity distance Consider an attack on the ciphertext string "WNAIW" encrypted using a Vigenère cipher with a five letter key. Conceivably, this string could be deciphered into any other string — RIVER and WATER are both possibilities for certain keys. This is a general rule of cryptanalysis: with no additional information it is impossible to decode this message. Of course, even in this case, only a certain number of five letter keys will result in English words. Relation with key size and possible plaintexts[edit] In general, given any particular assumptions about the size of the key and the number of possible messages, there is an average ciphertext length where there is only one key (on average) that will generate a readable message. A tremendous number of possible messages, N, can be generated using even this limited set of characters: N = 26L, where L is the length of the message. Relation with key entropy and plaintext redundancy[edit] The expected unicity distance is accordingly:
Polymathematics Proof claimed for deep connection between primes The usually quiet world of mathematics is abuzz with a claim that one of the most important problems in number theory has been solved. Mathematician Shinichi Mochizuki of Kyoto University in Japan has released a 500-page proof of the abc conjecture, which proposes a relationship between whole numbers — a 'Diophantine' problem. The abc conjecture, proposed independently by David Masser and Joseph Oesterle in 1985, might not be as familiar to the wider world as Fermat’s Last Theorem, but in some ways it is more significant. “The abc conjecture, if proved true, at one stroke solves many famous Diophantine problems, including Fermat's Last Theorem,” says Dorian Goldfeld, a mathematician at Columbia University in New York. “If Mochizuki’s proof is correct, it will be one of the most astounding achievements of mathematics of the twenty-first century.” Like Fermat’s theorem, the abc conjecture refers to equations of the form a+b=c. If you’ve got that, then you should get the abc conjecture.
Mathgen paper accepted! | That's Mathematics! I’m pleased to announce that Mathgen has had its first randomly-generated paper accepted by a reputable journal! On August 3, 2012, a certain Professor Marcie Rathke of the University of Southern North Dakota at Hoople submitted a very interesting article to Advances in Pure Mathematics, one of the many fine journals put out by Scientific Research Publishing. (Your inbox and/or spam trap very likely contains useful information about their publications at this very moment!) This mathematical tour de force was entitled “Independent, Negative, Canonically Turing Arrows of Equations and Problems in Applied Formal PDE”, and I quote here its intriguing abstract: Let \rho = A. Is it possible to extend isomorphisms? The full text was kindly provided by the author and is available as PDF. After a remarkable turnaround time of only 10 days, on August 13, 2012, the editors were pleased to inform Professor Rathke that her submission had been accepted for publication. has been accepted. Bummer.
Social Science Research Network (SSRN) Home Page