This paper proves some results concerning the polar factorisation of an integrable vector-valued function u into the composition u = u ◦ s, where u = ∇ψ almost everywhere for some convex function ψ, and s is a measure-preserving mapping. Not every integrable function has a polar factorisation; we extend the class of counterexamples. We introduce a generalisation: u has a polar inclusion if u(x)...

We will also use Fubini’s theorem [2, p. 170–171] and Lebesgue’s dominated convergence theorem [2, p. 91]. Fubini’s theorem allows us to change the order of integration in a double integral of a function, when we know that the function is integrable in one of the orders (i.e., the integral of the absolute value of the function is finite in one of the orders). Lebesgue’s dominated convergence th...

Integrable optics is an innovation in particle accelerator design that potentially enables a greater betatron tune spread and damps collective instabilities. An integrable rapid-cycling synchrotron (RCS) would be an effective replacement for the Fermilab Booster, as part of a plan to reach multi-MW beam power at 120 GeV for the Fermilab high-energy neutrino program. We provide an example integr...

This paper has two aims. The first is to give a description of irreducible tempered representations of classical p-adic groups which follows naturally the classification of irreducible square integrable representations modulo cuspidal data obtained in [15] and [19]. The second aim of the paper is to give description of an invariant (partially defined function) of irreducible square integrable r...