In this paper, we consider the spatially homogeneous Boltzmann equation without angular cutoff. We prove that every $L^1$-weak solution to the Cauchy problem with finite moments of all orders acquires the $C^{\infty }$-regularity in the velocity variable for all positive time.

As a continuation of the work of Bourgade, Fujita, and Yor, we show how to recover the extension of the Euler formulae concerning some special values of the Hurwitz zeta function from the product of two, and then $N$, independent generalized Cauchy variables. Meanwhile, we consider the ratio of two independent generalized Cauchy variables and give another proof of the partial fraction expansion of the cotangent function.

We herein investigate the homotopy types of the independence complexes of square grid graphs $C_{m,n}$ with cylindrical identification for $m=3$ and $n=6,7$. We show that they are all homotopy equivalent to a wedge of spheres.

A Hartogs-type extension theorem is proved for the complement of certain effective divisors on compact Kähler manifolds. A method employed in the proof yields a result supplementing some solutions of the Levi problem on two-dimensional spaces.

This paper stresses the strong link between the existence of partial holomorphic connections on the normal bundle of a foliation seen as a quotient of the ambient tangent bundle and the extendability of a foliation to an infinitesimal neighborhood of a submanifold. We find the obstructions to extendability, and thanks to the theory developed we obtain some new Khanedani–Lehmann–Suwa type index theorems.

We study the Aharonov–Bohm (AB) effect through resonances for magnetic scattering in two dimensions. The scattering system consists of three scatterers, one bounded obstacle, and two scalar potentials with compact supports at large separation, where the obstacle is placed between two supports and the support of the magnetic field is completely shielded by the obstacle. The field does not influence particles from a classical mechanical point of view, but quantum particles are influenced by the corresponding vector potential which does not necessarily vanish outside the obstacle. This quantum phenomenon is called the AB effect. The resonances are shown to be generated near the real axis by the trajectories oscillating between two supports of the scalar potentials as the distances between the three scatterers go to infinity. The location is described in terms of the backward amplitudes for scattering by each of the scalar potentials and by the obstacle, and it depends heavily on the magnetic flux of the field.

We investigate toric varieties defined by arrangements of hyperplanes and call them strongly symmetric. The smoothness of such a toric variety translates to the fact that the arrangement is crystallographic. As a result, we obtain a complete classification of this class of toric varieties. Further, we show that these varieties are projective and describe associated toric arrangements in these varieties.

In this, the third paper of the series, we construct a large family of representations of the quantum toroidal ${\mathfrak{gl}}_1$-algebra whose bases are parameterized by plane partitions with various boundary conditions and restrictions. We study the corresponding formal characters. As an application we obtain a Gelfand–Zetlin-type basis for a class of irreducible lowest weight ${\mathfrak{gl}}_{\infty }$-modules.