Ten Must Read Books about Mathematics
Posted by Antonio Cangiano in Essential Math, Suggested Reading on July 17th, 2007 | 70 responses I love books with the ability to inspire readers. Many non-mathematicians consider mathematics as something abstruse and complicated, suitable only for ‘nerds’. Often I highlight the unfounded nature of this prejudice, but nothing is more effective at disproving this stigma than a good book. The Man Who Loved Only Numbers: an original biography of the genius Paul Erdős, who was arguably the most prolific mathematician of the last century , renowned for being just as much of an eccentric as a math whiz. In the comments below feel free to share your thoughts on these books (if you have read any of them) and add other to the list which are near and dear to your own mathematical heart. … No differential equations were harmed in the making of this post.
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Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (i.e., any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. Background[edit] First incompleteness theorem[edit] Diagonalization[edit] B.
New Pattern Found in Prime Numbers
(PhysOrg.com) -- Prime numbers have intrigued curious thinkers for centuries. On one hand, prime numbers seem to be randomly distributed among the natural numbers with no other law than that of chance. But on the other hand, the global distribution of primes reveals a remarkably smooth regularity. This combination of randomness and regularity has motivated researchers to search for patterns in the distribution of primes that may eventually shed light on their ultimate nature. In a recent study, Bartolo Luque and Lucas Lacasa of the Universidad Politécnica de Madrid in Spain have discovered a new pattern in primes that has surprisingly gone unnoticed until now. “Mathematicians have studied prime numbers for centuries,” Lacasa told PhysOrg.com. Benford’s law (BL), named after physicist Frank Benford in 1938, describes the distribution of the leading digits of the numbers in a wide variety of data sets and mathematical sequences. “BL is a specific case of GBL,” Lacasa explained.
A Non-Mathematical Introduction to Using Neural Networks
The goal of this article is to help you understand what a neural network is, and how it is used. Most people, even non-programmers, have heard of neural networks. There are many science fiction overtones associated with them. And like many things, sci-fi writers have created a vast, but somewhat inaccurate, public idea of what a neural network is. Most laypeople think of neural networks as a sort of artificial brain. Neural networks are one small part of AI. The human brain really should be called a biological neural network (BNN). There are some basic similarities between biological neural networks and artificial neural networks. Like I said, neural networks are designed to accomplish one small task. The task that neural networks accomplish very well is pattern recognition. Figure 1: A Typical Neural Network As you can see, the neural network above is accepting a pattern and returning a pattern. Neural Network Structure Neural networks are made of layers of similar neurons. Conclusion
Online texts
Professor Jim Herod and I have written Multivariable Calculus ,a book which we and a few others have used here at Georgia Tech for two years. We have also proposed that this be the first calculus course in the curriculum here, but that is another story.... Although it is still in print, Calculus,by Gilbert Strang is made available through MIT's OpenCourseWare electronic publishing initiative. Here is one that has also been used here at Georgia Tech. Linear Methods of Applied Mathematics, by Evans Harrell and James Herod. Yet another one produced at Georgia Tech is Linear Algebra, Infinite Dimensions, and Maple, by James Herod.
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Theorem of the Day
The Whole Jolly Lot (now enriched with The list is presented here in reverse chronological order, so that new additions will appear at the top. This is not the order in which the theorem of the day is picked which is more designed to mix up the different areas of mathematics and the level of abstractness or technicality involved. The way that the list of theorems is indexed is described here. Every theorem number is linked to its entry in the delightful 'Prime Curios!' All files are pdf , mostly between 100 and 300 Kbytes in size. A QED following a theorem indicates that the description includes a proof of the theorem. 211 Willans' Formula QED 210 The Basel Problem QED 209 The Erdős Discrepancy Conjecture QED ( a Theorem under construction!) 208 Toricelli's Trumpet QED 207 The Eratosthenes-Legendre Sieve QED 206 Euler's Formula QED 205 The Classification of the Semiregular Tilings 204 Singmaster's Binomial Multiplicity Bound QED ( 203 Euler's Continued Fraction Correspondence 191 L'Hospital's Rule
An Introduction to Neural Networks
Prof. Leslie Smith Centre for Cognitive and Computational Neuroscience Department of Computing and Mathematics University of Stirling. lss@cs.stir.ac.uk last major update: 25 October 1996: minor update 22 April 1998 and 12 Sept 2001: links updated (they were out of date) 12 Sept 2001; fix to math font (thanks Sietse Brouwer) 2 April 2003 This document is a roughly HTML-ised version of a talk given at the NSYN meeting in Edinburgh, Scotland, on 28 February 1996, then updated a few times in response to comments received. What is a neural network? Some algorithms and architectures. Where have they been applied? What new applications are likely? Some useful sources of information. Some comments added Sept 2001 NEW: questions and answers arising from this tutorial Why would anyone want a `new' sort of computer? What are (everyday) computer systems good at... .....and not so good at? Good at Not so good at Fast arithmetic Interacting with noisy data or data from the environment Massive parallelism Courses
