Note on a Curious Instance of Opposite Proper Motions

Structure of proper motions in a sunspot penumbra

A Note on Learning from Multiple - Instance

We describe a simple reduction from the problem of PAC-learning from multiple-instance examples to that of PAC-learning with one-sided random classi cation noise. Thus, all concept classes learnable with one-sided noise, which includes all concepts learnable in the usual 2-sided random noise model plus others such as the parity function, are learnable from multiple-instance examples. We also de...

A note on a curious formula for Euler’s constant

In this short note we will use the residue theorem to establish a formula for Euler’s constant. In particular, we offer a slightly generalized version of an interesting infinite series due to Flajolet, Gourdon, and Dumas. 1 Properties of γ Recall that γ is usually defined in terms of the partial sums of the harmonic series: γ := lim n→∞ 

A Note on Proper Poisson Actions

We show that the fixed point set of a proper action of a Lie group G on a Poisson manifold M by Poisson automorphisms has a natural induced Poisson structure and we give several applications.

A Note on De Concini and Procesi’s Curious Identity

We give a short, case-free and combinatorial proof of de Concini and Procesi’s formula from [1] for the volume of the simplicial cone spanned by the simple roots of any finite root system. The argument presented here also extends their formula to include the non-crystallographic root systems.