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25 posts tagged maths

Proofs without words - Math Overflow

The cardinality of the real number line is the same as a finite open interval of the real number line.

Engineers prefer linear systems because they’re much easier to work with mathematically, but unfortunately, we live in a largely nonlinear world. So a lot of research is aimed at finding linear characterizations of the behavior of nonlinear systems. That research usually requires a great deal of mathematical insight and trial and error, and even when it’s successful, the results may be impossible to generalize to other cases.

Pablo Parrilo, the Finmeccanica Career Development Professor at MIT’s Laboratory for Information and Decision Systems, has developed a new set of techniques that make it easier to get a handle on nonlinear systems. Moreover, in many cases, his techniques provide algorithms — step-by-step instructions — for analyzing those systems, taking away much of the guesswork.

More good news!~  Great news, actually.  Nonlinear dynamics is/was one of my favorite subjects/classes, despite having to churn through perturbations and such.  I kind of enjoyed the handwork, a bit like Prof. Gallagher being addicted to long division.  =)

xkcd

sagatrope:

Flatland : Edwin Abbott Abbott (1884)

FULL TEXT

excerpt:

A complex form of mathematical symmetry linked to string theory has been glimpsed in the real world for the first time, in laboratory experiments on exotic crystals.

Mathematicians discovered a complex 248-dimensional symmetry called E8 in the late 1800s. The dimensions in the structure are not necessarily spatial, like the three dimensions we live in, but they correspond to mathematical degrees of freedom, where each dimension represents a different variable.

In the 1970s, the symmetrical form turned up in calculations related to string theory, a candidate for the “theory of everything” that might explain all the forces in the universe. But string theory still awaits experimental proof.

The structure is also the basis for another proposed theory of everything advancedMovie Camera in 2007 by surfer-physicist Garrett Lisi, who refers to E8 as “perhaps the most beautiful structure in mathematics”.

Now, physicists have detected the signature of E8 in a very different realm – experiments on super-chilled crystals.

Journal reference: Science (DOI: 10.1126/science.1180085)

(via lidel)

Structure+Strangeness: Irony, tinged with truth

During the G-20 protests in Pittsburgh held September 22-25, CMU machine learning students took to the streets to support their causes. “Support vector machines!” and “Ban genetic algorithms!”, they demanded. “Bayesians against discrimination!”, they cried. …

Tip to Jake Hofman and Arthur Gretton (whose photos these are).

xkcd - A Webcomic - Apocalypse

i lol’d all over myself.

A team of education, economics and public policy scholars has built a new tool that can quickly assess how a particular school finance reform proposal might impact individual California school districts.

The tool can be used to assess any formula that consolidates so-called “categorical” or restricted, special-purpose state and federal funds. It will be discussed at the annual meeting of the American Educational Research Association in San Diego on Tuesday, April 14.

"California currently allocates more than $40 billion of tax revenue — more than $1,000 per resident — through a school finance system that is not rational or transparent," said Heather Rose, assistant professor of education at UC Davis. "Reform is vital, and we hope this model will be a useful tool for legislators and others who are serious about achieving it."

The model appears as an Excel appendix to a 2008 PPIC-sponsored report, “Funding Formulas for California Schools II: An Analysis of a Proposal by the Governor’s Committee on Education Excellence.” It can be seen at http://www.ppic.org/main/publication.asp?i=830.

AT LAST! Now, how to propogate …

[The] process begins by taking the derivatives of every variable observed with respect to every other – a mathematical way of measuring how one quantity changes as another changes. Then the computer creates equations at random using various constants and variables from the data. It tests these against the known derivatives, keeps the equations that come closest to predicting correctly, modifies them at random and tests again, repeating until it literally evolves a set of equations that accurately describe the behavior of the real system.

Technically, the computer does not output equations, but finds “invariants” – mathematical expressions that remain true all the time.

billydalto:

This is one of the interactive graphics Eigenfactor.org has published as part of their effort to draw more meaningful conclusions from the citation network of published research. Their analysis, described here and in this PDF, uses an eigenvector centrality measure similar to PageRank to evaluate the relationships between articles.

Incidentally, Jack July recently posted a similarly intended graphic from Nature (full text available free here). Its authors use clickstream logs to create a map of greater resolution and timeliness than is possible with citation data.

A team of researchers from Perimeter Institute, Cambridge University, and Texas A&M has for the first time estimated, from mathematical symmetry arguments, the size of a fundamental imbalance pervading the subatomic world. This imbalance, called the CP violation, distinguishes matter from antimatter and is essential to understanding why matter predominates over antimatter in the natural world.

Applying a new statistical approach, researchers showed how random matrices can be used to estimate the size of the CP violation to be expected in nature. To their surprise, their results tallied well with experimentally observed data about quarks.

The team also showed how this approach could be applied to judge whether or not there are likely to be more than three subatomic particle families in nature, and to anticipate the properties of exotic particles called neutrinos. The work also provides clues about the physical mechanism which caused the imbalance between matter and antimatter in the Universe.

billydalto:

Jer Thorp, using a NYT API and Processing, created this graphic tracking the number of occurrences of ‘organic’ since 1981. The names of the organizations associated with the articles were used to create roots. This is just one of the several visualizations in this set, including one comparing frequencies of ‘communism’ and ‘terrorism’ and another with ‘acid rain’, ‘global warming’, and ‘climate change’. A few of these are even available as prints in his Etsy store.

Special Function Invocation

O Hive Mind, o Lazy Web, Urania’s child, I invoke thee! Is there a name for the function
\[  f_n(\theta) = \sum_{k=0}^{n}{{n \choose k} \theta^k {(1-\theta)}^{n-k} \log{k!}}   \]
i.e., for $ \mathbb{E}[\log{X!}] $ when X is binomially distributed?

Posted by crshalizi at March 22, 2009 21:52 | see also Enigmas of Chance

=) =) =)

♥ ♥ ♥

lol!

via www.italianacademy.columbia.edu

*sigh*

i miss nyc.

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