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Why is Congruence Hard to Learn?

January 25, 2015

Gears and rings…

Flickr source:

Denys Fisher was a British engineer and maker of board games and other toys. In 1965 he invented the Spirograph toy. Some speculate that he was influenced by the designs of the artist, composer, and film animator John Whitney, whose opening sequence for Alfred Hitchcock’s 1958 film “Vertigo” is considered the first use of computer graphics in cinema. The Spirograph toy involves drawing with a pen guided by a gear with m teeth going around inside a ring or around a track or other gear with x teeth. The kind of design you get depends on how x relates to m.

Today Ken and I want to talk about a basic notion of mathematics and theory that is simple to define, very useful, and yet seems to be tough for some students to get.
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A Computer Chess Analysis Interchange Format

January 20, 2015

File formats are like watching paint dry…

Source: Chess-programming wiki on Edwards’s “Spector” program.

Steven Edwards is an American computer scientist and chess programmer. Two decades ago he spearheaded the development and adoption of three standards for communicating chess games and their moves.

Today Ken wishes to present a proposed new format for representing computer analysis of chess games and positions.
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Forgetting Results

January 14, 2015

Rejoining many who have forgotten or overlooked results

St. Andrews Mac Tutor biographies source

Henry Smith was a mathematician of the 19th century who worked mainly in number theory. He especially did important work on the representation of numbers by various quadratic forms. We have discussed how even in something seemingly settled, like Joseph Lagrange’s theorem that every natural number is representable as a sum of four squares, new questions are always around—especially when one considers complexity.

Today Ken and I want to discuss a private slip of forgetfulness, and how often others may have done the same.
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There Are Many Primes

January 8, 2015

Various senses of ‘many’ from proofs of the infinitude of primes

2008 Bowen Lectures source

Hillel Furstenberg is entering his 50th years as a professor of mathematics at Hebrew University in Jerusalem. He shared the 2006-07 Wolf Prize with Stephen Smale. He has shown many connections between analysis and combinatorics. One was showing how ergodic theory can prove Endre Szemerédi’s theorem that every positive-density subset of the integers includes arbitrarily-long arithmetic progressions {[a + bn]}. This led to a new multidimensional formulation with Yitzhak Katznelson, and the two later proved positive-density versions of the Hales-Jewett theorem.

Today Ken and I wish to discuss proofs of the infinitude of primes, and what they begin to say about analysis and combinatorics.
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Mathematics: It’s About the Future

January 4, 2015

How some predictions fared in 2020 and other years

Zimbio source

Simon Donaldson, Maxim Kontsevich, Terence Tao, Richard Taylor, and Jacob Lurie (photo order) won the 2015 Breakthrough Prize in Mathematics. This did not happen since Thursday; it happened last June. When Tao was asked to explain the 2015 prize date at the start of his 11/12/14 appearance on The Colbert Report, he said,

“It’s about the future.”

Today Dick and I salute the prize-winners, and preview a new book about advances that were made in the years 2015–2019.
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The New Chess World Champion

December 28, 2014

Available for $59.96, or—for free?

chessprogramming wiki source

Larry Kaufman is a Grandmaster of chess, and has teamed in the development of two champion computer chess programs, Rybka and Komodo. I have known him from chess tournaments since the 1970s. He earned the title of International Master (IM) from the World Chess Federation in 1980, a year before I did. He earned his GM title in 2008 by dint of winning the World Senior Chess Championship, equal with GM Mihai Suba.

Today we salute Komodo for winning the 7th Thoresen Chess Engines Competition (TCEC), which some regard as the de-facto world computer chess championship.
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Modulating the Permanent

December 21, 2014

When does it become hard to compute?


Thomas Muir coined the term “permanent” as a noun in his treatise on determinants in 1882. He took it from Augustin Cauchy’s distinction in 1815 between symmetric functions that alternate when rows of a matrix are interchanged versus ones that “stay permanent.” To emphasize that all terms of the permanent have positive sign, he modified the contemporary notation {\left| A \right|} for the determinant of a matrix {A} into

\displaystyle  \overset{+}{|} A \overset{+}{|}

for the permanent. Perhaps we should be glad that this notation did not become permanent.

Today Ken and I wish to highlight some interesting results on computing the permanent modulo some integer value.
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