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
Math: Human Discovery or Human Invention? So just what, in essence, is this thing called math? In developing these numbers and systems of numbers, did we discover the hidden coding of the universe? Is mathematics, in the words of Galileo, the language of God? Or is math just a human-created system that happens to correspond with natural laws and structures? There is no definitive answer to this question, but mathematicians tend to side with one of several compelling theories. First, there is the Platonic theory. The opposing argument, therefore, is that math is a man-made tool -- an abstraction free of time and space that merely corresponds with the universe. Several theories expand on this idea. The logistic theory, for instance, holds that math is an extension of human reasoning and logic.The intuitionist theory defines math as a system of purely mental constructs that are internally consistent.The formalist theory argues that mathematics boils down to the manipulation of man-made symbols. Who's right?
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.
Front Page, Interactive Mathematics Miscellany and Puzzles Since May 6, 1997You are visitor number 27862784 in base 11 The heart of mathematics consists of concrete examples and concrete problems. P.R.Halmos, How to Write MathematicsAMS, 1973 Also, some pages are organized into series while others, especially the older ones, are accessible individually. Throughout the discussions at this site I refer to various titles I love and find useful enough to have them in my own library. First off, you may want to look at the page that explains to the curious the origin and nature of my logo. There is also a page where I offer a beautiful geometric problem. One page currently presents 118 different proofs of the Pythagorean Theorem which was a great fun putting together. Another page looks into different ways a specific statement may be related to a more general one. Other pages have educational content. To me Internet is one of life's wonders. Recently I discovered the source of the 4 Travelers problem. |Contents||Store|
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.
Lucia de B. is innocent This internet site has been created by a group of individuals who form the “Committee for Lucia”. None of its members are related to Lucia, nor are they from her circle of friends. They are merely individuals driven by the certainty that Lucia is the victim of a gross miscarriage of justice. # News: BBC 23-7-2010 – Can chance make you a killer? Richard Gill 18-4-2010 – Bureau of Lost Causes This organisation has been set up inspired by the self-less efforts by so many people over the last six years, which only just now led to the extraordinary and total rehabilitation of Lucia de Berk. Guardian 14-4-2010 – Dutch nurse acquitted of being a mass murderer The ruling ended a bizarre legal odyssey during which the country's judicial system – right up to the Supreme Court – interpreted evidence wrongly, including statistics, autopsy results, and the nurse's diaries. The Independent 10-4-2010 – Nigel Hawkes: Did statistics damn Lucia de Berk? Google "Lucia" Trends # A gross miscarriage link or
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 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 The expected unicity distance is accordingly:
Can Math Make a Better Marathon? Less than two miles from the finish of last month’s Chicago Marathon, with the race finally narrowed to a neck-and-neck duel between two Kenyan runners, the NBC television commentators began placing their bets. It wasn’t a hard call. Abel Kirui, who hadn’t won a marathon since 2011, was visibly struggling, repeatedly falling behind and then clawing his way back. Such moments are what make marathons fascinating to watch and their results impossible to predict. In theory, the pace that a runner can sustain in a marathon is a straightforward function of physiology—how well the heart can pump fresh blood through the arteries, how much energy the leg muscles burn, and so on. Noakes is now most famous for having abandoned the idea of physiological limits entirely. Real-time measurements during a race, as T.C.S. plans for this weekend’s marathon, are a slightly different proposition from pre-race lab tests.