Wednesday, December 6, 2017

Ernest William Brown: More Mathematician Than Astronomer?

Despite being more well-known for his astronomical work in lunar theory, Is Ernest William Brown more of a mathematician than an astronomer?

By Ringo Bones 

Born in November 29, 1 in Kingston upon Hull, UK, Earnest William Brown FRS was an English mathematician and astronomer, who spent the majority of his career working in the United States and thus became a naturalized American citizen in 1923. In 1907, he was appointed Professor of Mathematics at Yale University. 

His life’s work was the study of the Moon’s motion (lunar theory) and the compilation of extremely accurate lunar tables. Brown also studied the motion of the planets and calculated the orbits of Trojan asteroids. During the height of his professorship at Yale, Brown was also an active member of the American Mathematical Society as its president from 1915 to 1916. 

Since 1923, the Lunar Tables of Ernest William Brown have reduced the Moon’s complicated motion to a numerical theory that yielded serviceable tables, which only proves that his mathematical skills are way better than his mathematical skills. Brown’s Tables were adopted by nearly all of the national ephemerides in 1923 for their calculations of the Moon’s position and continued to be used with some modification until 1983. With the advent of programmable digital computers, Brown’s original trigonometric expressions, given in the introduction to his 1919 tables (and from which the tables had been compiled), began to be used for direct computation instead of the tables themselves. This also gained some improvement precision, since the tables had embodied some minor approximations, in a trade-off between accuracy and the amount of labor needed for computations in those days of manual calculation. 

By the middle of 20th Century, the difference between Universal and Ephemeris Time had been recognized and evaluated and the troublesome empirical terms were removed. Further adjustments to Brown’s theory were made, arising from improved observational values of the fundamental astronomical constants used in the theory and from reworking Brown’s original analytical expansions to gain more precise versions of the coefficients used in the theory. Eventually in 1984, Brown’s work was replaced by results gained from more modern observational data – including data from lunar laser ranging - and altogether new computational methods for calculating the Moon’s ephemeris. 

A heavy smoker, Brown suffered from bronchial trouble for much of his life. He was afflicted by ill-health during most of the six years of his retirement and died in New Haven, Connecticut in 1938. 

Saturday, August 13, 2016

Largest Known Prime Number Discovered in a University of Central Missouri Computer

It may be seen only as a mathematical curiosity to most of us, but did you know that very large prime numbers are indispensable in maintaining effective cyber security?

By: Ringo Bones 

Previously seen as a mere mathematical curiosity – and it still is by most of the population – but prime numbers – such as two, three, five and seven – numbers that are divisible only by themselves and one, play a vital role in computer data encryption. The latest prime number discovered so far back in January 20, 2016 is more than 22-million digits long – 22,338,618 digits long to be exact - five million digits longer than the previously discovered largest known prime number. Prime numbers this large could prove useful to computing in the future – which is sooner than you might think given the current rapidity of advances in hardware and software. 

The new prime number was found as part of the “endless mathematical quest” called the Great Internet Mersenne Prime Search or GIMPS, a global quest to find a particular type of large prime numbers. Mersenne Primes are named after a French monk, Marin Mersenne, who studied them in the 17th Century during his spare time. Given that modern programmable digital computers processes data in binary code, they can be configured to hunt for Mersenne Prime Numbers by multiplying two by itself a large number of times, then taking away one. It is a relatively manageable calculation for today’s computers, but not every result is a prime number. This year’s newly discovered prime number is written as 2^74,207,281-1, which denotes the number two, multiplied by itself 74,207,280 times with one subtracted afterwards. Since it began 29 years ago, the GIMPS project has calculated the 15 largest Mersenne Prime Numbers and it is possible that there could still be an infinite number of them to discover.  

Very large prime numbers are important in computer encryption and help make sure that online banking, shopping and private messaging services are secure, but current encryption typically use prime numbers that are only hundreds of digits long – not millions. But given our increasing reliance on computers for online commerce and private messaging, the search for very large prime numbers can be very important to maintain encryption with ever increasing processing power – although mathematicians involved in the GIMPS project admitted in a statement that this year’s newly discovered prime number is “too large to currently be of practical value”. 

However, searching for large prime numbers is intensive work for computer processors and can have unexpected benefits. “One prime project discovered that there was a problem in some computer processors that only showed up in certain circumstances.” said Dr. Steven Murdoch, cybersecurity expert at University College London. This year’s new large prime number – the 49th known Mersenne Prime Number, was discovered by Dr. Curtis Cooper at the University of Central Missouri. Although computers do most of the hard work, very large prime numbers are said to be discovered only after when a human operator takes note of the result. 

Sunday, January 31, 2016

Ancient Babylonians: First To Use Sophisticated Geometry?

Previously known for starting an order of astrologer-priests, are the Ancient Babylonians are also the first ones to use sophisticated geometry? 

By: Ringo Bones

Before the recent research findings were published back in January 29, 2016, Ancient Babylonians were more famous for establishing the first order of astrologer-priests that would later evolve into what we know as the science of astronomy. But that all changed when evidence were uncovered that Ancient Babylonians were using a branch of geometry that only got widespread use in the 14th Century. The new study is published in the journal Science. Its author, Prof. Mathieu Ossendrijver from the Humboldt University of Berlin, Germany said: “I wasn’t expecting this. It is completely fundamental to physics and all branches of science use this method.” The study suggests that sophisticated geometry – the branch of mathematics that deals with shapes – was being used at least 1,400 years earlier than previously thought. 

The possibility that Ancient Babylonians were using geometrical calculations to track the planet Jupiter across the night sky entered the realm of plausibility after Prof. Ossendrijver examined five Babylonian tablets that were excavated in the 19th Century and which are now held in the British Museum’s archives. The script reveals that the Babylonians were using four-sided shapes, called trapezoids, to calculate when Jupiter would appear in the night sky and also the speed and distance that it traveled. “This figure – a rectangle with a slanted top – describes how the velocity of a planet, which is Jupiter, changes with time,” he said. “We have a figure where one axis, the horizontal side, represents time, and the other axis, the vertical side, represents velocity.” “The area of the trapezoid gives you the distance traveled by Jupiter along its orbit.” “What is so special is that this type of graph is unknown from antiquity – so making figures of motion in this rather abstract space of velocity against time – this is something very, very new.” It has been previously thought that complex geometry was first used by scholars in Oxford and Paris in Medieval times.    

The Ancient Babylonians once lived in what is now Iraq and Syria. The civilization emerged in about 1,800 BC. Clay tablets engraved in their Cuneiform writing system have already shown these people were advanced in astronomy. “They wrote reports about what they saw in the sky,” Prof. Ossendrijver told the BBC World Service’s Science In Action programme. “And they did this over a very long period of time, over centuries,” he says.  

Wednesday, May 27, 2015

Farewell Dr. John Nash....

As the world mourns of his recent tragic car crash, will the world be a sadder place without mathematician Dr. John Nash?

By: Ringo Bones

He’s probably more famous to the world at large via the 2001 movie A Beautiful Mind as he’s portrayed by actor Russell Crowe than by his works on game theory during the height of the Cold War and his being a 1994 Nobel Economics Prize laureate, but back in Saturday, May 23, 2015, mathematician Dr. John Nash together with his wife Alicia tragically dies in a car crash in the New Jersey Turnpike. The whole world – and not just the mathematicians’ corner – will be a sadder place without him. 

His work on noncooperative games, published in 1950 and known as the Nash equilibrium is considered as his most influential work of the 20th Century. It provided a conceptually simple but powerful mathematical tool for analyzing a wide range of competitive situations, from cooperative rivalries to legislative decision making. His theories are used in economics, computing, evolutionary biology, artificial intelligence, accounting, politics and military theory. Dr. Nash also made contributions to pure mathematics that many mathematicians view as more significant than his Nobel-winning work on game theory, including solving an intractable problem in differential geometry derived from the work of the 19th century mathematician G.F.B. Riemann. His achievements were more remarkable, colleagues say, for being contained in a small handful of papers published before he was 30.  

Given his lifelong struggle with depression and paranoid schizophrenia, it is quite remarkable feat indeed that Dr. Nash managed to communicate his mathematical brilliance to the whole world and managed to get recognition for it. Looks like Russell Crowe’s Tweet back in Sunday, May 24, 2015 is indeed both a touching and fitting tribute of Dr. Nash’s mathematical legacy.

Tuesday, March 17, 2015

Homer Simpson: Mathematical Genius?

Even though the world-renowned patriarch of The Simpsons is a well-known bumbling oaf, but did you know that Homer Simpson, at one time, exhibited his “mathematical genius”?

By: Ringo Bones

Though he is more well-known as a dunce and a bumbling oaf, Homer Simpson – a world-renown animated character often used by its creators to assess the prevailing zeitgeist – once displayed his mathematical genius and even predicted the mass of the Higgs Boson to within more than 90-percent accuracy 14 years before it was confirmed by a team of particle physicists operating CERN’s Large Hadron Collider. To the curious, this was from an episode titled “The Wizard of Evergreen Terrace” where Homer Simpson got envious of Thomas Alva Edison and tries to out-invent the “Wizard of Menlo Park”.

The episode would have been forgotten and would have languished in some obscure footnote of 20th Century history if not for Dr. Simon Singh who wrote a book back in 2013 titled “The Simpsons And Their Mathematical Secrets” that included a spotlight on the 1998 episode “The Wizard of Evergreen Terrace” when Homer becomes “obsessed” with Thomas Alva Edison and decides to become an inventor. A scene in that particular The Simpsons episode script required a reading glasses-clad Homer to be placed in front of a chalkboard with complex mathematical equations. One of the writers on staff had a physicist friend who was researching the then-theoretical Higgs Boson particle and needed a “scientifically believable” illustration of Homer dabbling with a complex mathematical equation predicting the mass of the Higgs Boson particle – which is also known as the “God Particle”.

“That particular equation - as shown on TV on that particular 1998 The Simpsons episode – predicts the mass of the Higgs Boson” says Dr. Simon Singh. “If you work it out, you get the mass of the Higgs Boson that’s only a bit larger than the nano-mass of a Higgs Boson actually is. It is kind of amazing as Homer makes the prediction 14 years before it was discovered” (in the CERN’s Large Hadron Collider). For those super interested, the Higgs Boson particle was discovered to have a mass of 126 GeV.

The Higgs Boson particle is the “visible” that interacts with the Higgs Field – just like gravitons do with the gravitational field. The Higgs Field is an energy force that permeates across the universe that gives baryonic matter mass and allows the weak nuclear force and the electromagnetic force to co-exist in the “Standard Model” of how we think, so far, on how universal molecular physics work.
Even though Homer’s mathematical musings on the Higgs Boson somewhat reminds me of 1984 Nobel Physics Prize winner Carlo Rubbia’s mathematical musings that was pictured on a 1990 era Time magazine, the field of particle physics / quantum mechanics, mathematics can be a very useful tool in discovering and describing an “unknown particle” with better than 90-percent accuracy. Back in 1962, a then 32 year old Caltech physicist named Murray Gell-Mann proposed a search for a then theoretical particle called the Omega Minus. The particle’s existence was mathematically predicted by the Standard Model, Gell-Mann argued by a theory he formulated himself and by another physicist – a then 37 year old former Israeli Army officer named Yuval Ne’eman.

This theory which Gell-Mann called “The Eightfold Way” was based on an obscure mathematical system invented in the 19th Century in order to manipulate numbers in groups of eight since each interacting nuclear particle had eight quantum numbers how subatomic baryons and mesons are organized into octets. Independently, Ne’eman did the same. Eventually, Gell-Mann was awarded the 1969 Nobel Physics Prize for his work on elementary particles and by 1971 began work in search for a then unknown family of particles called “quarks” using "The Eightfold Way".

Tuesday, June 3, 2014

Career Mathematicians: America’s Most Lucrative Profession?

Given that in a 2013 survey shows that they now earn about the same as - or slightly higher than - a typical Beverly Hills plastic surgeon, are mathematicians now America’s most lucrative profession? 

By: Ringo Bones

An overwhelming majority of the American public view career mathematicians as lone researchers into the most abstruse of matters, but frequently, America’s career mathematicians frequently work with other scientists. A survey conducted back in 2013 has shown that the median annual salary of a career mathematician in the United States was about U.S. $101,360 – comparable to that of a typical Beverly Hills plastic surgeon. Given that career tenured mathematicians in the United States could turn out to be one of the best-paid jobs, could there be any prevailing trends that lead to this rather fortunate outcome? Though, if you ask me, one should not put a cheap price on brain power.  

Since the internet boom of the latter half of the 1990s, “big data” and the analytical mathematical models describing them had become a hot commodity for the top commercial internet firms. Remember how career statistician Nate Silver (full name Nathaniel Read Silver) who used mathematics to show an uncannily accurate Obama victory prediction for the 2012 U.S. Presidential Race weeks before the November election via the use of big data is a powerful proof of the power of mathematics. Though years before, Nate Silver’s powerful analytic mathematical contribution to Major League Baseball has been immortalized in the movie Money Ball. 

Will – if favorable trends continue – career mathematicians will soon be earning more money than investment bankers? Could be, given that the leading internet firms had been inexplicably quick in commoditizing and monetizing big data and are also very keen on using analytical mathematics to describe and predict trends via big data – or to use higher mathematics to manipulate big data for commercial gain.     

Monday, March 25, 2013

The Somewhat Secret Life of Pi

It became a secret by default but is the overtly pragmatic outlook of contemporary mathematics educators causing undue mysticism of the “life” of pi? 

By: Ringo Bones 

Most contemporary mathematics educators discuss pi only as far as the ratio of the circumference of a circle to its diameter. Though it may allow students to solve the area of a given circle and other related engineering problems, but should today’s mathematics educators be “more adventurous” to have an extensive discussion on the somewhat secret historic life of pi? 

The symbol of pi that used the Greek letter was first employed in its present meaning by William Jones in 1707 and its adoption by Leonhard Euler in 1737 brought it into general use. In far earlier times, the determination of the value of pi and the calculation of the area of a circle of a given diameter were recognized early as closely related problems. 

The mathematicians of antiquity had learned how to calculate precisely the areas of plane figures bounded by straight lines, such as the rectangle, parallelogram and triangles. The circle, however, eluded the mathematicians of antiquity’s methods, although many approximations for its area were developed. It remained for the mathematicians of the last 150 years to disclose the real nature of the problem. 

The ancient Chinese and the ancient Hebrews seem to have taken pi as equal to 3. In the Ahmes Papyrus of ancient Egypt back in 1650 BC, the area of a circle is found by subtracting from the diameter 1/9 of its length and squaring the remainder. This is equivalent to taking pi as 256/81 or 3.1605. 

In ancient Greece, the problem took the form of requiring the construction of a square, by straight edge and compass alone, whose area equals that of a given circle – i.e. squaring the circle. Back around 225 BC, Archimedes approximated the area of a circle by comparing the areas of regular inscribed and circumscribed polygons of a large number of sides. Using polygons of 96 sides, Archimedes found that the ratio of the circumference of a circle to its diameter lies between 3 1/7 and 3 10/71. After the time of Archimedes, 3 1/7 and the square root of 10 were taken as a satisfactory approximation of pi. An approximation equivalent to the now familiar decimal 3.1416 dates from before 200 AD. 

After mathematicians invented decimals and promoted its widespread use, the calculation of pi was carried out to 35 decimal places by Ludolph van Ceulen (1540 – 1610). In Germany, pi is sometimes called the Lundolphian Number in his honor. 

Toward the end of the 17th Century, new methods of analysis were developed that made it possible to express pi as the sum or product of an infinite number of terms. One of the earliest of such expressions was formulated by Franciscus Vieta (1540 – 1603). In 1655, John Wallis (1616 – 1703) and in 1658 William Brouncker and Leibniz in 1673 published their various expressions of series to compute the value of pi.
By the use of such series, the value of pi may be computed to any desired number of decimal places. By 1873, William Shanks had carried the computation of the value of pi to 707 decimal places; in 1946, however, it was discovered that Shanks’ vas incorrect after the 527th decimal place. With the advent of mechanical – and later electronic – computing machines, more extensive calculations of the value of pi became feasible. Even as far back as 1958, pi was already accurately calculated to 10,000 decimal places using the most advanced electronic computers of the time. 

In our more pragmatic contemporary engineering practices computing the value of pi beyond 10 decimal places offer little practical value. To ten places, pi is given as 3.1415926536 – this is of sufficient accuracy to permit computing the circumference of a sphere the size of the planet Earth with an error less than 1/8 of an inch. Out to about 40 decimal places, pi would give the circumference of the entire visible universe with an error imperceptible even with an electron microscope. Thus these extensive calculations have no practical value and serve chiefly to show the power of modern methods of computation. 

Though in the field of number theory, the value of pi manifest itself in an exiting way that can sometimes only appreciated by pure mathematics enthusiasts. In 1761, Johann Lambert showed that pi is irrational – i.e. cannot be expressed as the ratio of two integers. While in 1794 Andre Marie Legendre made the proof a rigorous one, thus finally establishing that the value of pi is not a repeating decimal. 

Joseph Liouville in 1884 proved the existence of transcendental numbers – i.e. numbers that cannot be expressed as roots of algebraic equations with rational coefficients. Charles Hermite in 1873 proved that e – the base of natural logarithms – was such a number. Using this information and Euler’s theorem that e raised to the power of pi times the square root of negative one is equal to zero, Ferdinand Lindermann proved in 1882 that pi is transcendental thus assuring the impossibility of squaring the circle. 

The number pi enters into the measurement of many more curved plane figures other than the circle. Thus the area of an ellipse is given in terms of half the length of its axes as A = pi (a) (b). It also appears in many of the relations of mathematics, physics and engineering other than those directly concerned with areas or lengths of arcs.