Instead of using white noise analysis, we use abstract Wiener measure to define the Chern-Simons path integral over $\mathbb{R}^3$. One rigorous and the other, not so rigorous, definitions will be given. The latter will be used to compute the Wilson Loop observable in the abelian case, which gives us the linking number of a link. .

Some new nonlinear dynamic integral inequalities of Gronwall type for retarded functions are established. These inequalities can be used as basic tools in the study of certain classes of functional dynamic equations as well as dynamic delay equations.

We study the structure of approximate solutions of a discrete-time control system with a compact metric space of states which arises in economic dynamics. We are interested in turnpike properties of the approximate solutions which are independent of the length of the interval, for all sufficiently large intervals and are stable under perturbations of an objective function.

In this paper we study an elliptic equation involving variable exponents and containing a singular lower order terms with $p(x)-$growth in the gradient. Through an approximation approach, we prove the existence of a nonnegative distributional solution in the whole space $\mathbb{R}^N. $

We introduce length dilation structures on metric spaces, tempered dilation structures and coherent projections and explore the relations between these objects and the Radon-Nikodym property and Gamma-convergence of length functionals. Then we show that the main properties of sub-riemannian spaces can be obtained from pairs of length dilation structures, the first being a tempered one and the second obtained via a coherent projection. Thus we get an intrinsic, synthetic, axiomatic description of sub-riemannian geometry, which transforms the classical construction of a Carnot-Carathéodory distance on a regular sub-riemannian manifold into a model for this abstract sub-riemannian geometry.

In this article we establish the global regularity of weak solutions of the Dirichlet problem for a class of degenerate elliptic equations.

Scattering theory between the fractional power $H_0=\kappa^{-1}(-\Delta)^{\kappa/2}$ $(\kappa\ge 1$) of negative Laplacian and the Hamiltonian $H=H_0+V$ perturbed by shortand long-range potentials considered in [14] is revisited and a new proof of the existence and asymptotic completeness of wave operators is given with utilizing the smooth operator technique.

The spaces $(L^{q},\;l^{p})^{\alpha}$ and $M^{p,\;\alpha}$ are closely related to classical problems in Harmonic Analysis: properties of multiplier and Fourier multiplier from a Lebesgue space to another, finite $(1,p)$-energy measures. We characterize the Fourier transforms of their elements and establish criteria of compactness in these spaces.