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Edwards Curves and Gaussian Hypergeometric Series
Let be an elliptic curve described by either an Edwards model or a
twisted Edwards model over , namely, is defined by one of the
following equations mod , or,
mod , respectively. We express the
number of rational points of over using the Gaussian
hypergeometric series where and are the trivial and
quadratic characters over respectively. This enables us to
evaluate for some elliptic curves , and prove the
existence of isogenies between and Legendre elliptic curves over
Gate recess engineering of pseudomorphic In0.30GaAs/GaAs HEMTs
The authors report how the performance of 0.12 μm GaAs pHEMTs is improved by controlling both the gate recess width, using selective dry etching, and the gate position in the source drain gap, using electron beam lithography. pHEMTs with a transconductance of 600 mS/mm, off state breakdown voltages >2 V, fτ of 120 GHz, f max of 180 GHz and MAG of 13.5 dB at 60 GHz are reported
The spectrum of kernel random matrices
We place ourselves in the setting of high-dimensional statistical inference
where the number of variables in a dataset of interest is of the same order
of magnitude as the number of observations . We consider the spectrum of
certain kernel random matrices, in particular matrices whose
th entry is or where is
the dimension of the data, and are independent data vectors. Here is
assumed to be a locally smooth function. The study is motivated by questions
arising in statistics and computer science where these matrices are used to
perform, among other things, nonlinear versions of principal component
analysis. Surprisingly, we show that in high-dimensions, and for the models we
analyze, the problem becomes essentially linear--which is at odds with
heuristics sometimes used to justify the usage of these methods. The analysis
also highlights certain peculiarities of models widely studied in random matrix
theory and raises some questions about their relevance as tools to model
high-dimensional data encountered in practice.Comment: Published in at http://dx.doi.org/10.1214/08-AOS648 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Positive linear functionals on BP*-algebras
Let A be a BP*-algebra with identity e, P_{1}(A) be the set of all positive
linear functionals f on A such that f(e) = 1, and let M_{s}(A) be the set of
all nonzero hermitian multiplicative linear functionals on A. We prove that
M_{s}(A) is the set of extreme points of P_{1}(A). We also prove that, if
M_{s}(A) is equicontinuous, then every positive linear functional on A is
continuous. Finally, we give an example of a BP*-algebra whose topological dual
is not included in the vector space generated by P_{1}(A), which gives a
negative answer to a question posed by M. A. Hennings.Comment: This is an English translation of the original article written in
Frenc
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