The dynamical group associated with the Dirac equation with a radially symmetric potential in four space-time dimensions is represented in terms of integrals with respect to operator valued set functions associated with the free Dirac operator. In coordinates in which c = ~ = 1, the class of potentials treated includes Coulomb potentials ?a=r with jaj < 1.

Motivated by the need for new mathematical tools applicable to the study of fractal point-cloud distributions, expectations of complex-valued functions defined over general ‘deterministic’ fractal domains are considered, following the development of a measure-theoretic foundation for their analysis. In particular, we wish to understand and evaluate separation moments as given by integrals of th...

We give a new proof of the result, originally proved in unpublished work of Glenn Stevens [7], that every modular eigensymbol of non-critical slope lifts uniquely to a rigid-analytic distribution-valued eigensymbol. The proof is algorithmic and facilitates the efficient calculation of certain padic integrals.

We prove that a gerbe with a connection can be defined on classical phase space, taking the U(1)–valued phase of Feynman path integrals as Čech 2–cocycles. A quantisation condition on the corresponding 3–form field strength is derived and proved to be equivalent to Heisenberg’s uncertainty principle.

We prove the convexity and compactness of the closure of the lower partition range of an Rn-valued, nonatomic, supermodular capacity, employing a useful relationship between convex games and their Choquet integrals. The main result is applied to fair division problems, and the existence of Pareto optimal α-fair partitions is demonstrated for the case of nonadditive measures.

Principal value integrals are associated to multi–valued rational differential forms with normal crossings support on a non–singular algebraic variety. We prove their vanishing on rational surfaces in the context of a conjecture of Denef–Jacobs. As an application we obtain a strong vanishing result for candidate poles of p–adic and motivic Igusa zeta functions.

In this paper we study the concept of analyticity for complex-valued functions of a complex time scale variable, derive a time scale counterpart of the classical Cauchy–Riemann equations, introduce complex line delta and nabla integrals along time scales curves, and obtain a time scale version of the classical Cauchy integral theorem.

Studied is the Baxter equation for the quantum discrete Boussinesq equation. We explicitly construct the Baxter Q operator from a generating function of the local integrals of motion of the affine Toda lattice field theory, and show that it solves the third order operator-valued difference equation. nlin/0102021

By applying the vector-valued inequalities for the Littlewood-Paley operators and their commutators on Lebesgue spaces with variable exponent, the boundedness of the Littlewood-Paley operators, including the Lusin area integrals, the Littlewood-Paley g-functions and g μ *-functions, and their commutators generated by BMO functions, is obtained on the Morrey spaces with variable exponent.

Recent reverses for the discrete generalised triangle inequality and its continuous version for vector-valued integrals in Banach spaces are surveyed. New results are also obtained. Particular instances of interest in Hilbert spaces and for complex numbers and functions are pointed out as well.