477 research outputs found
Irrationality proofs \`a la Hermite
As rewards of reading two great papers of Hermite from 1873, we trace the
historical origin of the integral Niven used in his well-known proof of the
irrationality of , uncover a rarely acknowledged simple proof by Hermite
of the irrationality of , give a new proof of the irrationality of
for nonzero rational , and generalize it to a proof of the
irrationality of certain ratios of Bessel functions.Comment: 8 page
Abel's Theorem in the Noncommutative Case
We define noncommutative binary forms. Using the typical representation of
Hermite we prove the fundamental theorem of algebra and we derive a
noncommutative Cardano formula for cubic forms. We define quantized elliptic
and hyperelliptic differentials of the first kind. Following Abel we prove
Abel's Theorem.Comment: 30 page
Functionals of exponential Brownian motion and divided differences
We provide a surprising new application of classical approximation theory to a fundamental asset-pricing model of mathematical finance. Specifically, we calculate an analytic value for the correlation coefficient between
exponential Brownian motion and its time average, and we find the use of divided differences greatly elucidates formulae, providing a path to several new results. As applications, we find that this correlation coefficient is always at least 1/p2 and, via the Hermite–Genocchi integral relation, demonstrate that all moments of the time average are certain divided differences of the exponential function. We also prove that these moments agree with the somewhat more complex formulae obtained by Oshanin and Yor
Differential analysis of matrix convex functions
We analyze matrix convex functions of a fixed order defined on a real
interval by differential methods as opposed to the characterization in terms of
divided differences given by Kraus. We obtain for each order conditions for
matrix convexity which are necessary and locally sufficient, and they allow us
to prove the existence of gaps between classes of matrix convex functions of
successive orders, and to give explicit examples of the type of functions
contained in each of these gaps. The given conditions are shown to be also
globally sufficient for matrix convexity of order two. We finally introduce a
fractional transformation which connects the set of matrix monotone functions
of each order n with the set of matrix convex functions of order n+1
Sur les racines de la fonction sphérique de seconde espèce. Extrait d'une lettre adressée à M. Lerch
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