September 19, 2013 — Itai Seggev, Mathematica Algorithm R&D
I love Maxwell’s equations. As a freshman in college, while pondering whether to major in physics, computer science, or music, it was the beauty of these equations and the physical predictions that can be elegantly extracted from them that made me decide in favor of physics. On a more universal level, the hints in Maxwell’s equations led Einstein to write Zur Elektrodynamik bewegter Körper (“On the Electrodynamics of Moving Bodies”), more commonly known as Einstein’s first paper on the theory of relativity. The quantum version of the equations, quantum electrodynamics (QED), remains our most successful physical theory, with predictions verified to 12 decimal places. There are many reasons to love Maxwell’s equations. And with Mathematica 9′s new vector analysis functionality, exploring them has never been easier.
August 27, 2013 — Michael Trott, Chief Scientist
(This is the third post in a three-part series about electrostatic and magnetostatic problems involving sharp edges.)
In the first blog post of this series, we looked at magnetic field configurations of piecewise straight wires. In the second post, we discussed charged cubes and orbits of test particles in their electric field. Today we will look at magnetic systems, concretely, mainly at a rectangular bar magnet with uniform magnetization.
June 26, 2013 — Jon McLoone, International Business & Strategic Development
My government (I’m in the UK) recently said that children here should learn up to their 12 times table by the age of 9. Now, I always believed that the reason why I learned my 12 times table was because of the money system that the UK used to have—12 pennies in a shilling. Since that madness ended with decimalization the year after I was born, by the late 1970s when I had to learn my 12 times table, it already seemed to be an anachronistic waste of time.
June 3, 2013 — Oleksandr Pavlyk, Manager of Probability and Statistics, Mathematica Algorithm R&D
I am a junkie for a good math problem. Over the weekend, I encountered such a good problem on a favorite subject of mine–probability. It’s the last problem from the article “A Mathematical Trivium” by V. I. Arnol’d, Russian Mathematical Surveys 46(1), 1991, 271–278.
It’s short enough to reproduce in its entirety: “Find the mathematical expectation of the area of the projection of a cube with edge of length 1 onto a plane with an isotropically distributed random direction of projection.” In other words, what is the average area of a cube’s shadow over all possible orientations?
This blog post explores the use of Mathematica to understand and ultimately solve the problem. It recreates how I approached the problem.
A century ago, Srinivasa Ramanujan and G. H. Hardy started a famous correspondence about mathematics so amazing that Hardy described it as “scarcely possible to believe.” On May 1, 1913, Ramanujan was given a permanent position at the University of Cambridge. Five years and a day later, he became a Fellow of the Royal Society, then the most prestigious scientific group in the world at that time. In 1919 Ramanujan was deathly ill while on a long ride back to India, from February 27 to March 13 on the steamship Nagoya. All he had was a pen and pad of paper (no Mathematica at that time), and he wanted to write down his equations before he died. He claimed to have solutions for a particular function, but only had time to write down a few before moving on to other areas of mathematics. He wrote the following incomplete equation with 14 others, only 3 of them solved.
Within months, he passed away, probably from hepatic amoebiasis. His final notebook was sent by the University of Madras to G. H. Hardy, who in turn gave it to mathematician G. N. Watson. When Watson died in 1965, the college chancellor found the notebook in his office while looking through papers scheduled to be incinerated. George Andrews rediscovered the notebook in 1976, and it was finally published in 1987. Bruce Berndt and Andrews wrote about Ramanujan’s Lost Notebook in a series of books (Part 1, Part 2, and Part 3). Berndt said, “The discovery of this ‘Lost Notebook’ caused roughly as much stir in the mathematical world as the discovery of Beethoven’s tenth symphony would cause in the musical world.”
March 28, 2013 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project
RootApproximant can turn an approximate solution into a perfect solution, such as for a square divided into fifty 45°-60°-75° triangles.
A square can be divided into triangles, for example by connecting opposite corners. It’s possible to divide a square into seven similar but differently sized triangles or ten acute isosceles triangles. Classic puzzles involve cutting a square into eight acute triangles, or twenty 1 – 2 – √5 triangles. The last image uses 45°-60°-75° triangles, but one triangle has a flaw.
It’s easy to divide a square with similar right triangles. Can a square be divided into similar non-right triangles? In his paper “Tilings of Polygons with Similar Triangles” (Combinatorica, 10(3), 1990 pp. 281–306), Laczkovich proved exactly three triangles provided solutions, with angles 22.5°-45°-122.5°, 15°-45°-120°, and 45°-60°-75°. I read his paper to try to make an image for the 45°-60°-75° case, but his construction was complex, and seemed to require thousands of triangles, so I tried to find my own solutions. All my attempts had flaws, such as the last image above, so I made a contest out of it: $200, minus a dollar for every triangle in the solution.
March 21, 2013 — Devendra Kapadia, Mathematica Algorithm R&D
Waiting in line is a common, though not always pleasant, experience for us all. We wait patiently to be served by the next free teller at a bank, clear the security check at an airport, or be answered by technical support when we call a phone service provider. At a more abstract level, these waiting lines, or queues, are also encountered in computer and communication systems. For example, every email you send is broken up into a series of packets. Each packet is then sent off to its destination by the best available route to avoid the queues formed by other packets in the network. Hence, queues play an important role in our lives, and it seems worthwhile to spend some time understanding their dynamics, with a view to answering questions such as, “How many tellers does your bank need to provide good customer service?” or “How can you speed up the security check?” or “On average, how long will you have to wait for technical support?” My purpose in writing this post is to give a gentle introduction to queueing theory, which attempts to answer such questions, using new functions that are available in Mathematica 9.
Queueing theory has its origins in the research of the Danish mathematician A. K. Erlang (1878–1929). While working for the Copenhagen Telephone Company, Erlang was interested in determining how many circuits and switchboard operators were needed to provide an acceptable telephone service. This investigation resulted in his seminal paper “The Theory of Probabilities and Telephone Conversations,” which was published in 1909. Erlang proved that the arrivals for such queues can be modeled as a Poisson process, which immediately made the problem mathematically tractable. Another major advance was made by the American engineer and computer scientist Leonard Kleinrock (1934–), who used queueing theory to develop the mathematical framework for packet switching networks, the basic technology behind the internet. Queueing theory has continued to be an active area of research and finds applications in diverse fields such as traffic engineering and hospital emergency room management.
February 4, 2013 — Oleksandr Pavlyk, Manager of Probability and Statistics, Mathematica Algorithm R&D
On January 23, 1913 of the Julian calendar, Andrey A. Markov presented for the Royal Academy of Sciences in St. Petersburg his analysis of Pushkin’s Eugene Onegin. He found that the sequence of consonants and vowels in the text could be well described as a random sequence, where the likely category of a letter depended only on the category of the previous or previous two letters.
At the time, the Russian Empire was using the Julian calendar. The 100th anniversary of the celebrated presentation is actually February 5, 2013, in the now used Gregorian calendar.
To perform his analysis, Markov invented what are now known as “Markov chains,” which can be represented as probabilistic state diagrams where the transitions between states are labeled with the probabilities of their occurrences.
In this blog post, we want to report some work in progress that might interest users of probability and statistics and also those who wonder how we add new knowledge every day to Wolfram|Alpha.
Since the beginning in 1988, Mathematica knew not only elementary functions (sqrt, exp, log, etc.) but many special functions of mathematical physics (such as the Bessel function K and the Riemann Zeta function) and number theoretical functions. All together, Mathematica knows now more than 300 such functions. The Wolfram Functions Site lists 300,000+ formulas and identities for these functions. And, based on Mathematica‘s algorithmic computation capabilities and the Functions Site’s identities, most of this knowledge is now easily accessible in Wolfram|Alpha. For example, relation between sin(x) and cos(x), series representations of the Beta function, relation between BesselJ(n, x) and AiryAi(x), differential equation for ellipticF(phi, m), and examples of complicated indefinite integrals containing erf.
But Wolfram|Alpha also knows about many special functions that are not in Mathematica because they are less common or less general. For instance, haversine(x), double factorial binomial(2n, n), Dickman rho(10/3), BesselPolynomialY[6, x], Conway’s base 13 function(4003/371293), and Goldbach function(1000).
Mathematica 7 knew 42 probability distributions; Mathematica 9 knows over 130 (parametric) probability distributions. Based on Mathematica, Wolfram|Alpha can answer a lot of queries about these distributions, such as characteristic function of the hyperbolic distribution or variance of the binomial distribution with p = 1/3, and give general overview pages for queries such as Student’s t distribution or Gumbel distribution.