Posts about Math

Mathematicians Critique Journal Rankings

Mathematicians Critique Journal Rankings

Three international math groups joined forces to issue a report last week decrying the use of citation statistics to evaluate scientific journals, research institutions and individual scientists. These statistics, sometimes called “bibliometrics,” measure how frequently a given journal’s articles are cited by other journals.

Read the report on Citation Statistics. This concern is justified. I do have some interest in some of these (and related) statistics but one must always remember their limitations.

Related: Country H-index Rank for Science PublicationsRanking Universities WorldwideBest Research University Rankings (2007)Don’t Forget the Proxy Nature of Data

Shaw Laureates 2008

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 established in 2002 by Run Run Shaw who was born in China and made his money in the movie industry. 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.” The 2008 Shaw Laureates have been selected.

Professor Reinhard Genzel, Managing Director of the Max Planck Institute for Extraterrestrial Physics, in recognition of his outstanding contribution in demonstrating that the Milky Way contains a supermassive black hole at its centre.

In 1969, Donald Lynden-Bell and Martin Rees suggested that the Milky Way might contain a supermassive black hole. But evidence for such an object was lacking at the time because the centre of the Milky Way is obscured by interstellar dust, and was detected only as a relatively faint radio source. Reinhard Genzel obtained compelling evidence for this conjecture by developing state-of-the-art astronomical instruments and carrying out a persistent programme of observing our Galactic Centre for many years, which ultimately led to the discovery of a black hole with a mass a few million times that of the Sun, in the centre of the Milky Way.

Supermassive black holes are now recognized to account for the luminous sources seen at the nuclei of galaxies and to play a fundamental role in the formation of galaxies.

Mathematical Sciences
Vladimir Arnold, together with Andrei Kolmogorov and Jurgen Moser, made fundamental contributions to the study of stability in dynamical systems, exemplified by the motion of the planets round the sun. This work laid the foundation for all subsequent developments right up to the present time.

Arnold also produced extremely fruitful ideas, relating classical mechanics to questions of topology. This includes the famous Arnold Conjecture which was only recently solved.

In classical hydrodynamics the basic equations of an ideal fluid were derived by Euler in 1757 and major steps towards understanding them were taken by Helmholtz in 1858, and Kelvin in 1869. The next significant breakthrough was made by Arnold a century later and this has provided the basis for more recent work.

Ludwig Faddeev has made many important contributions to quantum physics. Together with Boris Popov he showed the right way to quantize the famous non-Abelian theory which underlies all contemporary work on sub-atomic physics. This led in particular to the work of ”²t Hooft and Veltman which was recognized by the Nobel Prize for Physics of 1999.
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Presidential Award for Top Science and Math Teachers

Top Science and Math Teachers Receive Presidential Award

For the 2007 awards, 99 middle school and high school math and science teachers are receiving this recognition. In the citation from the president, winners are commended “for embodying excellence in teaching, for devotion to the learning needs of the students, and for upholding the high standards that exemplify American education at its finest.”

Each winner receives a $10,000 award from NSF, as well as a trip for two to Washington, D.C., for a week of celebratory events and professional development activities.

Among the activities during that week are a day with scientists and science educators at NSF; meetings with members of Congress and federal agency leadership; and a reception and dinner at the U.S. Department of State featuring guest speaker Dorothy Metcalf-Lindenburger, a NASA Astronaut-Mission Specialist.

Related: Presidential Award for Excellence in Mathematics and Science TeachingEinstein Fellowship for TeachersNSF Graduate Teaching Fellows in K-12 EducationThe Importance of Science EducationEducation Resources Directory for Science and Engineering

Aztec Math

Aztec Math Decoded, Reveals Woes of Ancient Tax Time

By reading Aztec records from the city-state of Tepetlaoztoc, a pair of scientists recently figured out the complicated equations and fractions that officials once used to determine the size of land on which tributes were paid. Two ancient codices, written from A.D. 1540 to 1544, survive from Tepetlaoztoc. They record each household and its number of members, the amount of land owned, and soil types such as stony, sandy, or “yellow earth.”

“The ancient texts were extremely detailed and well organized, because landowners often had to pay tribute according to the value of their holdings,” said co-author Maria del Carmen Jorge y Jorge at the National Autonomous University in Mexico City, Mexico. The Aztecs recorded only the total area of each parcel and the length of the four sides of its perimeter, Jorge y Jorge explained. Officials calculated the size of each parcel using a series of five algorithms—including one also employed by the ancient Sumerians—she added.

Aztec math finally adds up

That meant that some of the unknown symbols had to represent fractions of a rod, she said. By trial and error, she decoded the system. A hand equaled 3/5 of a rod, an arrow was 1/2 , a heart was 2/5 , an arm was 1/3 , and a bone was 1/5 .

A set of at least five formulas emerged showing how the Aztec surveyors determined the areas of irregular shapes. In some cases, the Aztecs averaged opposite sides and then multiplied. In others, they bisected the fields into triangles.

Related: Sexy MathPixar Is Inventing New Math1=2: A Mistaken Proof

Thompson and Tits share 2008 Abel Prize (Math)

Thompson and Tits share the Abel Prize for 2008

John Griggs Thompson, Graduate Research Professor, University of Florida, and Jacques Tits, Professor Emeritus, Collège de France, have been awarded the 2008 Abel Prize “for their profound achievements in algebra and in particular for shaping modern group theory.” In the prize citation, the Abel Committee writes that “Thompson revolutionized the theory of finite groups by proving extraordinarily deep theorems that laid the foundation for the complete classification of finite simple groups, one of the greatest achievements of twentieth century mathematics.”

In 1963, Thompson and Walter Feit proved that all nonabelian finite simple groups were of even order, work for which they both won the Frank Nelson Cole Prize in Algebra from the AMS in 1965. Thompson also won a Fields Medal in 1970. In the Abel citation for Tits, the committee writes that “Tits created a new and highly influential vision of groups as geometric objects. He introduced what is now known as a Tits building, which encodes in geometric terms the algebraic structure of linear groups.” The committee noted the link between the two winners’ work: “Tits’s geometric approach was essential in the study and realization of the sporadic groups, including the Monster.” Tits received the Grand Prix of the French Academy of Sciences in 1976, and the Wolf Prize in Mathematics in 1993.

The Abel Prize is awarded by the Norwegian Academy of Science and Letters for outstanding scientific work in the field of mathematics. The prize amount is 6,000,000 Norwegian kroner (over US$1,000,000).

Related: Professor Marcus du Sautoy on Thompson and TitsMath’s Architect of Beauty2007 Nobel Prize in PhysicsPoincaré Conjecture

Bigger Impact: 15 to 18 mpg or 50 to 100 mpg?

This is a pretty counter-intuitive statement, I believe:

You save more fuel switching from a 15 to 18 mpg car than switching from a 50 to 100 mpg car.

But some simple math shows it is true. If you drive 10,000 miles you would use: 667 gallons, 556 gallons, 200 gallons and 100 gallons. Amazing. I must admit, when I first read the quote I thought that it must be an wrong. But there is the math. You save 111 gallons improving from 15 mpg to 18 mpg and just 100 improving from 50 to 100 mpg. Other than those of you who automatically guess that whatever seems wrong must be the answer when you see a title like this I can’t believe anyone thinks 15 to 18 mpg is the change that has the bigger impact. It is great how a little understanding of math can help you see the errors in your initial beliefs. Via: 18 Is Enough.

It also illustrates that the way the data is presented makes a difference. You can also view 100 mpg as 1/100 gallon per mile, 2/100 gallons per mile, 5.6/100 gpm and 6.7 gpm. That way most everyone sees that the 6.7 to 5.6 gpm saves more fuel than 2 to 1 gpm does. Mathematics and scientific thinking are great – if you are willing to think you can learn to better understand the world we live in every day.

Related: Statistics Don’t Lie, But People Can be FooledUnderstanding DataSeeing Patterns Where None ExistsOptical Illusions and Other Illusions1=2: A Proof

Playing Dice and Children’s Numeracy

My father, Willaim Hunter, a professor of statistics and of Chemical Engineering at the University of Wisconsin, was a guest speaker for my second grade class (I think it was 2nd) to teach us about numbers – using dice. He gave every kid a die. I remember he asked all the kids what number do you think will show up when you roll the die. 6 was the answer from about 80% of them (which I knew was wrong – so I was feeling very smart).

Then he had the kids roll the die and he stood up at the front to create a frequency distribution of what was actually rolled. He was all ready for them to see how wrong they were and learn it was just as likely for any of the numbers on the die to be rolled. But as he asked each kid about what they rolled something like 5 out of the first 6 said they rolled a 6. He then modified the exercise a bit and had the kid come up to the front and roll the die on the teachers desk. Then my Dad read the number off the die and wrote on the chart 🙂

This nice blog post, reminded me of that story: Kids’ misconceptions about numbers — and how they fix them

in the real study, conducted by John Opfer and Rober Siegler, the kids used lines with just 0 and 1000 labeled. They were then given numbers within that range and asked to draw a vertical line through the number line where each number fell (they used a new, blank number line each time). The figure above represents (in red) the average results for a few of the numbers used in the study. As you can see, the second graders are way off, especially for lower numbers. They typically placed the number 150 almost halfway across the number line! Fourth graders perform nearly as well as adults on the task, putting all the numbers in just about the right spot.

But there’s a pattern to the second-graders’ responses. Nearly all the kids (93 were tested) understood that 750 was a larger number than 366; they just squeezed too many large numbers on the far-right side of the number line. In fact, their results show more of a logarithmic pattern than the proper linear pattern.