Category Archives: Math

Prayer Book Reveals Lost Archimedes Work Studying Ideas at Heart of Calculus

A Prayer for Archimedes by Julie J. Rehmeyer

The top layer of writing in this 700-year-old book describes Christian prayers. But underneath, almost obliterated, are the only surviving copies of many of the works of the ancient Greek mathematician Archimedes.
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An intensive research effort over the last nine years has led to the decoding of much of the almost-obliterated Greek text. The results were more revolutionary than anyone had expected. The researchers have discovered that Archimedes was working out principles that, centuries later, would form the heart of calculus and that he had a more sophisticated understanding of the concept of infinity than anyone had realized.

Archimedes wrote his manuscript on a papyrus scroll 2,200 years ago. At an unknown later time, someone copied the text from papyrus to animal-skin parchment. Then, 700 years ago, a monk needed parchment for a new prayer book. He pulled the copy of Archimedes’ book off the shelf, cut the pages in half
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“The interesting breakthrough is that he is completely willing to operate with actual infinity,” Netz says, but he adds that “the argument is definitely not completely valid. He just had a strong intuition that it should work.” In this case, it did work, but it remained for Newton and Leibniz to figure out how to make the argument mathematically rigorous.

Related: Archimedes Palimpsest project – The Archimedes Codex (a book on the discovery) – PoincarÃ© Conjecture

Sexy Math

Mean Girls:

British researchers, stated that men had 12.7 heterosexual partners in their lifetimes and women had 6.5.” These numbers, though Kolata doesn’t say so, are means, not medians. In this case, it’s indeed mathematically impossible that the numbers are correct. The medians in the British sample? Seven and four, same as in the American study—so you can stop worrying about a transatlantic promiscuity gap.

“It is logically impossible for heterosexual men to have more partners on average than heterosexual women,” she explains. “Those survey results cannot be correct.” Kolata even quotes a theorem to this effect, backed up by mathematician David Gale of Berkeley: The average number of partners has to be the same for men and women.

This article makes several good mathematical and scientific points, including the dangers of trusting reports by participants. Also I can see if this page is more popular than some of the other math posts. For awhile now I have noticed “sex 100” showing up as one of the terms guiding the most visitors to this site. I wondered what that could be – I just took a look: Bdelloid Rotifers Abandoned Sex 100 Million Years Ago. I think maybe those searchers didn’t exactly find what they wanted.

Textbook Revolution

Textbook Revolution is a resource on free textbooks and free related course materials. In general, I must say the prices of textbooks seem crazy. This is another tool great open access resource.

At Textbook Revolution, you’ll find links to textbooks and select educational resources of all kinds. Some of the books are PDF files, others are viewable only online as e-books. Most books are aimed at undergraduates, but there are at least a few resources at every level, from kindergarten to post-doc. All of the books are offered for free by their respective copyright holders for online viewing. Beyond that, each book is as individual as the author behind it.

Some examples: The Scientist and Engineer’s Guide to Digital Signal Processing by Steven W. Smith; Light and Matter by Benjamin Crowell; A First Course in Linear Algebra by Robert A Beeze; Programming Ruby: The Pragmatic Programmer’s Guide by
Dave Thomas and Introduction to Statistical Thought by Michael Lavine.

While on the subject of textbooks, I will plug my father’s book: Statistics for Experimenters – it is my blog so I get to do what I want 🙂

Time

Newsflash: Time May Not Exist

Planck time—the smallest unit of time that has any physical meaning—is 10-43 second, less than a trillionth of a trillionth of an attosecond. Beyond that? Tempus incognito. At least for now. Efforts to understand time below the Planck scale have led to an exceedingly strange juncture in physics. The problem, in brief, is that time may not exist at the most fundamental level of physical reality.
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Einstein’s theories also opened a rift in physics because the rules of general relativity (which describe gravity and the large-scale structure of the cosmos) seem incompatible with those of quantum physics (which govern the realm of the tiny). Some four decades ago, the renowned physicist John Wheeler, then at Princeton, and the late Bryce DeWitt, then at the University of North Carolina, developed an extraordinary equation that provides a possible framework for unifying relativity and quantum mechanics. But the Wheeler-ÂDeWitt equation has always been controversial, in part because it adds yet another, even more baffling twist to our understanding of time.

“One finds that time just disappears from the Wheeler-DeWitt equation,” says Carlo Rovelli, a physicist at the University of the Mediterranean in Marseille, France. “It is an issue that many theorists have puzzled about. It may be that the best way to think about quantum reality is to give up the notion of time—that the fundamental description of the universe must be timeless.”

Interesting. As usual, quantum actions seem bizarre. Related: Quantum Mechanics Made Relatively Simple Podcasts – Physicists Observe New Property of Matter – Particles and Waves – Quantum Theory Fails Reality Checks – Physics Concepts in 60 Seconds

Quantum Random Number Generator

Quantum Random Bit Generator Service:

The work on QRBG Service has been motivated by scientific necessity (primarily of local scientific community) of running various simulations (in cluster/Grid environments), whose results are often greatly affected by quality (distribution, nondeterminism, entropy, etc.) of used random numbers. Since true random numbers are impossible to generate with a finite state machine (such as today’s computers), scientists are forced to either use specialized expensive hardware number generators, or, more frequently, to content themselves with suboptimal solutions (like pseudo-random numbers generators).
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To achieve high availability of the service, several network access modes are developed, or shall be developed. These include transparent acquisition of random numbers using C/C++ libraries, web services (access over the SOAP protocol), and Mathematica/MATLAB client add-ons.
To enable high security, in future, SSL protocol shall be supported, i.e. all data shall be encrypted, at users request, with user/service certificates.
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We use ‘Quantum Random Bit Generator’ (QRBG121), which is a fast non-deterministic random bit (number) generator whose randomness relies on intrinsic randomness of the quantum physical process of photonic emission in semiconductors and subsequent detection by photoelectric effect. In this process photons are detected at random, one by one independently of each other. Timing information of detected photons is used to generate random binary digits – bits.

1=2: A Proof

1=2: A Proof using Complex Numbers

See if you can figure out in which step the fallacy lies. When you think you’ve figured it out, click on that step and the computer will tell you whether you are correct or not, and will give an additional explanation of why that step is or isn’t valid.

See how many tries it takes you to correctly identify the fallacious step!

This is a nice quick math refresher but the real reason I posted it is so I can say I got it on the first try 🙂

Shaw Laureates 2007

Image of the Shaw Prize Medal

The Shaw Prize awards $1 million in each of 3 areas: Astronomy, Life Science and Medicine, and Mathematical Sciences. The award was created in 2004 by Run Run Shaw who was born in China and made his money in the movie industry. According to wikipedia he has “donated billions of dollars to charity, schools and hospitals.” The prize is administered in Hong Kong and awards those “who have achieved significant breakthrough in academic and scientific research or application and whose work has resulted in a positive and profound impact on mankind.”

2007 Laureates:
Astronomy: Professor Peter Goldreich of the School of Natural Sciences at the Institute for Advanced Study, Princeton and the Lee A. DuBridge Professor of Astrophysics and Planetary Physics at the California Institute of Technology, USA, in recognition of his lifetime achievements in theoretical astrophysics and planetary sciences.

Life Science and Medicine: Professor Robert Lefkowitz, an Investigator of the Howard Hughes Medical Institute, the James B. Duke Professor of Medicine and a Professor of Biochemistry at the Duke University Medical Center in Durham, North Carolina, USA, for his relentless elucidation of the major receptor system that mediates the response of cells and organs to drugs and hormones.

Mathematical Sciences: Professor Robert Langlands of the Institute for Advanced Study in Princeton and Professor Richard Taylor, Herchel Smith Professor of Mathematics at Harvard University, USA, for initiating and developing a grand unifying vision of mathematics that connects prime numbers with symmetry.

Freeware Math Programs

3 awesome free Math programs

Maxima – A general purpose CAS (Computer Algebra System) is a program that’s able to perform symbolic manipulation for the resolution of common problems. As a matter of fact, modern CAS covers an extremely wide range of functionalities
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Scilab – Matlab is the standard for numerical computing, but there are a few clones and valid alternatives that are entirely free. Scilab is the closest that you can get to Matlab without spending a penny. It’s very compatible with Matlab (albeit not 100%) but it’s really flexible, powerful and comes with a Matlab converter and Scicos which is a block diagram modeler and simulator.

R – For statistical computing and analysis in the Open Source world, it doesn’t get any better than R. It is a programming language and environment that enables you to do pretty much anything that the commercial software (S-Plus) does. It is so widely adopted that it can be considered a standard in the field.

lpsolve is another one that I like for linear (integer) programming. LaTeX is also a great tool – a typesetting system; it includes features designed for the production of technical and scientific documentation (many will already be familiar with it but if you are not, check it out).

Engineering, Math and Physics Applets

Java applets illustrating concepts in math, physics and engineering including: Coupled Oscillations, Digital Filters, 3-D Magnetostatic Fields and Ordinary Differential Equations.

Proof – Movie

It is a stretch to say this post is related to math but the plot of the movie involves a mathematical proof so… I just watched the Proof DVD and thought is was excellent, and I get to write about what I want, and therefore this post exists. It really is a great movie but doesn’t have much at all to do with math other than as a backdrop.

The main mathematician characters flirt with (or have) mental illness – which seems to be a common combination in movies: Pi, A Beautiful Mind and Good Will Hunting (all worth watching by the way).