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Roger's Equations

This blog is all about science and technology (with occasional math thrown in for fun). The goal of this blog is to try and pass on the sense of excitement and wonder I feel when I read about these topics. I hope you enjoy the posts.

Mathematical Structures in Classical Books

Posted January 28, 2016 10:09 AM by Bayes

I came across this very interesting article regarding fractal structures in classical literature and thought I'd pass it along. - R

Scientists find evidence of mathematical structures in classic books

James Joyce's Finnegans Wake has been described as many things, from a masterpiece to unreadable nonsense. But it is also, according to scientists at the Institute of Nuclear Physics in Poland, almost indistinguishable in its structure from a purely mathematical multifractal. The academics put more than 100 works of world literature, by authors from Charles Dickens to Shakespeare, Alexandre Dumas, Thomas Mann, Umberto Eco and Samuel Beckett, through a detailed statistical analysis. Looking at sentence lengths and how they varied, they found that in an "overwhelming majority" of the studied texts, the correlations in variations of sentence length were governed by the dynamics of a cascade - meaning that their construction is a fractal: a mathematical object in which each fragment, when expanded, has a structure resembling the whole.

Fractals are used in science to model structures that contain re-occurring patterns, including snowflakes and galaxies. "All of the examined works showed self-similarity in terms of organisation of the lengths of sentences. Some were more expressive - here The Ambassadors by Henry James stood out - others to far less of an extreme, as in the case of the French 17th-century romance Artamene ou le Grand Cyrus. However, correlations were evident, and therefore these texts were the construction of a fractal," said Dr Paweł Oświęcimka from the Institute of Nuclear Physics of the Polish Academy of Sciences, one of the authors of the new paper Quantifying Origin and Character of Long-range Correlations in Narrative Texts. Some works, however, were more mathematically complex than others, with stream-of-consciousness narratives the most complex, comparable to multifractals, or fractals of fractals. Finnegans Wake, the scientists found, was the most complex of all.

Article Continues Here

7 comments; last comment on 02/03/2016
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Failures in Hypothetical Planets - Vulcan

Posted January 24, 2016 3:07 PM by Bayes

Mercury's Perihelion Precession

In 1859, Urbain Le Verrier was the first to report that the slow precession of Mercury's orbit around the Sun could not be completely explained by Newtonian Mechanics and perturbations by the known planets. He suggested that it was possible another planet, or series of minor planets (asteroids) might exist in an orbit even closer to the Sun, perturbing Mercury.

This was significant, because Le Verrier had basically discovered Neptune the same way. Back in the 1840's Le Verrier was studying small but systematic discrepancies between Uranus's observed orbit and the one predicted by Newtonian gravity. He engaged for months in complex calculations and on August 31st, 1846 he predicted the location of a planet that could be responsible for perturbing Uranus' orbit. Johann Galle of the Berlin Observatory found the predicted planet, Neptune, within 1 degree or where Le Verrier predicted it should be. So, yeah, it was a big deal when Le Verrier predicted a planet closer to the sun than Mercury a decade later.

In 1860, Le Verrier, after a meeting with an amateur astronomer who detailed a transit he had observed that he thought might be Le Verrier's hypothetical planet, Le Verrier announced the discovery of a previously unknown planet, Vulcan, to a meeting of the Academie des Sciences in Paris. The amateur astronomer, Lescarbault, was awarded the Legion d'honneur and invited to appear before numerous learned societies.

In 1877 Le Verrier died, convinced of having discovered another planet, although it was never convincingly confirmed. It in fact remained a possibility until 1915. That year, Einstein's theory of relativity showed that Mercury's perihelion precession should be slightly faster than Newtonian gravity predicted, thus eliminating the need for a perturbing planet inside Mercury's orbit. Still, astronomers have persisted, though never found, a planet inside of Mercury's orbit.

Something to keep in mind when you read this:

3 comments; last comment on 01/25/2016
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The Tabby's Star Mystery Deepens

Posted January 19, 2016 9:49 AM by Bayes

Tabby's Star

KIC 8462852, known also as Tabby's star or the WTF star (Where's the Flux), is an F-type main-sequence star located in the constellation Cygnus, about 1,480 ly from Earth. Unusual light fluctuations of the star were discovered by citizen scientists as part of the Planet Hunters project. No suitable explanation has been put forth to date for the light fluctuations and new research has raised even more questions.

New research put forward by Bradley Schaefer shows an unusual and steady decline in output from Tabby's star since 1890. The paper can be found here.

Pretty cool mystery. I have a feeling this star will be studied pretty heavily over the next few years...

5 comments; last comment on 01/20/2016
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As Bright as Half a Trillion Suns

Posted January 15, 2016 9:33 AM by Bayes

Most Luminous Supernova Ever Detected

There's an interesting story going around the internet this morning regarding the most luminous Supernova ever detected. A peer reviewed paper in the journal Science is reporting the discovery of ASASSN-15lh (SN 2015L) in a large quiescent (inactive) galaxy. Usually super-luminous supernovae reside in active star-forming dwarf galaxies so this discovery is unique in many ways.

Here is the abstract from the Science Paper:

We report the discovery of ASASSN-15lh (SN 2015L), which we interpret as the most luminous supernova yet found. At redshift z = 0.2326, ASASSN-15lh reached an absolute magnitude of Mu,AB = -23.5 ± 0.1 and bolometric luminosity Lbol = (2.2 ± 0.2) × 1045 ergs s-1, which is more than twice as luminous as any previously known supernova. It has several major features characteristic of the hydrogen-poor super-luminous supernovae (SLSNe-I), whose energy sources and progenitors are currently poorly understood. In contrast to most previously known SLSNe-I that reside in star-forming dwarf galaxies, ASASSN-15lh appears to be hosted by a luminous galaxy (MK ≈ -25.5) with little star formation. In the 4 months since first detection, ASASSN-15lh radiated (1.1 ± 0.2) × 1052 ergs, challenging the magnetar model for its engine.

If you're interested in reading a less technical version, an article on the discovery can be found here:

3 comments; last comment on 01/18/2016
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Relativistic Effects in Chemistry

Posted January 14, 2016 10:24 AM by Bayes

Atomic Orbital Contraction

Atoms can get very large and electrons in the innermost orbitals of a large atom can see a very large charge. As a result, these electrons orbit the atomic nucleus very quickly. So quickly in fact that they achieve meaningful fractions of the speed of light.

As a result, relativistic effects come into play. Chemists and Physicists discovered this soon after the formulation of quantum mechanics in the early 20th century. The result is that, for example, the 1S orbital of a Uranium atom has a smaller radius than would be expected without considering relativistic effects. In other words, relativistic effects have led to atomic orbital contraction.

I recently found a neat video that explains this concept fairly concisely and wanted to share it. Usually when we think of relativity, or at least when I do, we tend to think about astrophysics, not chemistry, and yet it plays an important role in chemistry. It's nice to see it has more of an impact in the world around us than we realize.

6 comments; last comment on 01/18/2016
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Human Cells to Human Bacteria Cells Ratio

Posted January 10, 2016 12:44 PM by Bayes

The Human Menagerie

We tend to think of ourselves as a single organism made up of human cells with specific purposes. Liver cells, brain cells, red blood cells all coming together to, human beings. However, the truth is more complicated. Human beings consist of human cells, yes, but also what's known as the human microbiota, an aggregate of microorganisms such as bacteria and fungi, that live on and in the human body performing essential tasks.

It was believed that the human microbiota far outnumbered human cells with some estimates providing ratios of 10-1 or even 100-1 bacteria to human cells. However, more recently those ratios have been increasingly viewed with skepticism and now a study recently released is suggesting that ratio is probably closer to 1-1.

Here is an interesting article from ScienceNews detailing that study:

Body's bacteria don't outnumber human cells so much after all

Human bodies don't contain 10 times as many bacteria as human cells, new calculations suggest. A "standard man" weighing 70 kilograms has roughly the same number of bacteria and human cells in his body, researchers report online January 6 at This average guy would be composed of about 40 trillion bacteria and 30 trillion human cells, calculate researchers at the Weizmann Institute of Science in Rehovot, Israel, and the Hospital for Sick Children in Toronto. That's a ratio of 1.3 bacteria to every one human cell.

That estimate could be off by as much as 25 percent, with the average number of bacteria ranging from 30 trillion to 50 trillion. Among individual people, the bacterial count could vary as much as 52 percent, say Ron Sender, Shai Fuchs and Ron Milo. With a fudge factor of 10 trillion to 20 trillion bacteria, the number of microbes may pretty well match the number of human cells in the body, which also varies somewhat. "Indeed, the numbers are similar enough that each defecation event may flip the ratio to favor human cells over bacteria," the researchers write.

Article continues here

6 comments; last comment on 01/13/2016
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