We prove a conjecture of Gross regarding the “order of vanishing” of Stickelberger elements relative to an abelian tower of fields and give a cohomological construction of the conjectural Gross–Stark units. This is achieved by introducing an integral version of the Eisenstein cocycle.

Higher order coefficients of the inverse mass expansion of one–loop effective actions are obtained from a one–dimensional path integral representation. For the case of a massive scalar loop in the background of both a scalar potential and a (non Abelian) gauge field explicit results to O(T 5) in the proper time parameter are presented.

The cyclicity of period annuli of some classes of reversible and non-Hamiltonian quadratic systems under quadratic perturbations are studied. The argument principle method and the centroid curve method are combined to prove that the related Abelian integral has at most two zeros.

We introduce an integral version of the Eisenstein cocycle. As applications we prove a conjecture of Gross regarding the “order of vanishing” of Stickelberger elements relative to an abelian tower of fields and give a cohomological construction of the conjectural Gross–Stark units.

Integral formulae for form factors of a large family of charged local operators in SU(2) invariant Thirring model are given extending Smirnov’s construction of form factors of chargeless local operators in the sine-Gordon model. New abelian symmetry acting on this family of local operators is found. It creates Lukyanov’s operators which are not in the above family of local operators in general.

We classify the finite semigroups S, for which all Z-orders Γ of QS, the unit group U(Γ) is hyperbolic. We also classify the RA-loops L for which the unit loop of its integral loop ring does not contain any free abelian subgroup of rank two.

We present a new method of formulating the classical theory of SU(N + 1) non-Abelian Chern-Simons (NACS) particles for arbitrary N ≥ 1 using the symplectic reduction of CP (N) manifold from S2N+1. Quantizing the theory using BRST formulation and coherent state path integral method , we obtain a quantum mechanical model for SU(N + 1) NACS particles.

We compute the cohomology of the subgroup of the integral symplectic group of degree 2 consisting of matrices γ ≡ 1 mod 4. This is done by computing the cohomology of the moduli space of principally polarized abelian surfaces with a level 4 structure. This space has a projective smooth compactification whose topology was analyzed by Lee and Weintraub in a recent work.

In this paper, we complete the classification of those finite 3groups G whose integral group rings have the multiplicative Jordan decomposition property. If G is abelian, then it is clear that Z[G] satisfies MJD. In the nonabelian case, we show that Z[G] satisfies MJD if and only if G is one of the two nonabelian groups of order 33 = 27.

In this note we use a functional approach to the integral to obtain a special case of the Keisler-Fubini theorem; the general case can be obtained with a similar proof. An immediate appUcation is the standard Fubini theorem for products of Radon measures. Similar methods give the Weil formula for quotient groups of compact Abelian groups.