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September 21, 2006

Symmetry and Fermat's Last Theorem

Jay Goldman writes: "Let me suggest 'Fearless Symmetry' by Ash and Gross. It is not really about symmetry but an attempt to explain Wiles' proof of Fermat's Last Theorem to at least a general audience of mathematicians; maybe a wider group."
Fearless Symmetry: Exposing the Hidden Patterns of Numbers, by Avner Ash and Robert Gross. Princeton: Princeton University Press, 2006. Mathematics Library QA241 .A84 2006 Link to MnCat Record

Nelder on generalized linear models, again

Generalized Linear Models with Random Effects: Unified Analysis via H-Likelihood, by Youngjo Lee, John A. Nelder, Yudi Pawitan. Monographs on statistics and applied probability 106. Boca Raton, FL: Chapman & Hall/CRC, 2006.
QA279 .L43 2006 New Book Shelf Link to MNCAT record.

John Nelder pioneered these topics in several highly-cited works:
“Generalized linear models� (with R. W. M. Wedderburn), Journal of the Royal Statistical Society, Series A: General, 135, 370-384; Link to article
Generalized linear models (with P. McCullagh). Monographs on Statistics and Applied Probability. Chapman & Hall, London, 1983. Link to MNCAT record.
“Hierarchical generalized linear models� (with Y. Lee), Journal of the Royal Statistical Society, Series B 58 (1996), no. 4, 619--678. Link to article
and others. The current monograph greatly extends the class of GLMs: “First, to the fixed effects may be added one or more sets of random effects on the same linear scale; secondly GLMs may be fitted simultaneously to both mean and dispersion; thirdly the random effects may themselves be correlated, allowing the expression of models for both temporal and spatial correlation; lastly random effects may appear in the model for the dispersion as well as that for the mean. To allow likelihood-based inferences for the new model class, the idea of h-likelihood is introduced as a criterion to be maximized. This allows a single algorithm, expressed as a set of interlinked GLMs, to be used for fitting all members of the class. The algorithm does not require the use of quadrature in the fitting, and neither are prior probabilities required. The result is that the algorithm is orders of magnitude faster than some existing alternatives.� Applications to survival data, financial data, and denoising signals are discussed.

September 7, 2006

"99 Points of Intersection"

Walser, Hans. 99 Points of Intersection: Examples, Pictures, Proofs. Washington, DC: Mathematical Association of America, 2006. Link to MnCat Record
A fascinating collection of intersections beyond the common ones. The diagrams are clearly displayed but mostly without discussion, so they serve as puzzles for the reader. "If three straight lines pass through a common point--three such straight lines are called concurrent--this may either be an accident or a special property of these three straight lines (for example, the three medians of a triangle), or a property that holds in every triangle. The question then naturally follows, how and why the property of concurrence holds for all triangles; in other words, how can this property be proved for an arbitrary triangle?" Proof strategies, such as affine invariance and dynamic geometry software, are discussed, but Walser frankly mentions occasional difficulties: "Point of intersection 79 [Propeller] It took the author several years to find a proof for this point of intersection. The proof he got is far from nice. It uses complex numbers, homogeneous coordinates, which are usually used in projective geometry, and a huge amount of calculation, executed with the help of a computer algebra system (Maple). The author would be very happy if a reader found an elementary geometrical proof."