The multi-component nonlinear Schrödinger equations related to ${\bf C.I}\simeq {\rm Sp}(2p)/{\rm U}(p)$ and ${\bf D.III}\simeq {\rm SO}(2p)/{\rm U}(p)$-type symmetric spaces with non-vanishing boundary conditions are solvable by the inverse scattering method (ISM). We focus our attention on the single threshold case. We formulate the spectral properties of the Lax operator $L$, which is the generalized Zakharov-Shabat operator. Next we construct the corresponding fundamental analytic solutions (FAS) and adapt the Wronskian relations for the constant boundary conditions. They allow one to analyze the mapping from the class of allowed potentials $\mathcal{M}$ to each of the minimal sets of scattering data $\mathcal{T}_i$, $i=1,2$. The ISM for the Lax operator $L$ is interpreted as a nonlinear analog of the Fourier-transform method. As appropriate generalizations of the usual exponential functions we use the so-called “squared solutions” , which are constructed in terms of the FAS $\chi^{\pm}(x, \lambda)$ of $L$ and the Cartan-Weyl basis of the Lie algebra, relevant to the symmetric space. We derive the completeness relation for the “squared solution” which turns out to provide the map from $\mathcal{M}$ to each $\mathcal{T}_i$, $i=1,2$. Such decompositions allow one to derive all fundamental properties of the multi-component nonlinear Schrödinger equations.

Let ${\mathbb B} \subset {\mathbb C}^2$ be the unit ball and $\Gamma$ be a lattice of ${\rm SU} (2,1)$. Bearing in mind that all compact Riemann surfaces are discrete quotients of the unit disc $\Delta \subset {\mathbb C}$, Holzapfel conjectures that the discrete ball quotients ${\mathbb B} / \Gamma$ and their compactifications are widely spread among the smooth projective surfaces. There are known ball quotients ${\mathbb B} / \Gamma$ of general type, as well as rational, abelian, K3 and elliptic ones. The present note constructs three non-compact ball quotients, which are birational, respectively, to a hyperelliptic, Enriques or a ruled surface with an elliptic base. As a result, we establish that the ball quotient surfaces have representatives in any of the eight Enriques classification classes of smooth projective surfaces.

Known parametrizations of rotations are derived from the Lie group theoretical point of view considering the two groups ${\rm SO}(3)$ and ${\rm SU}(2)$. The concept of coordinates of the first and second kind for these groups is used to derive the axis and angle as well as the three-angle description of rotation matrices. With the homomorphism of the two groups the Euler parameter description arises from the axis and angle description of ${\rm SU}(2) $. Due to the topology of $ {\rm SO}(3)$ any three-angle description gives only a local parametrization like Euler angles such that the mapping from their time derivatives to the algebra $\frak{so}(3)$, i.e., to the angular velocity tensor, exhibits singularities. All these parametrizations are based on the generation of the respective group by the $\exp$ map from their algebras. Alternatively the Cayley transformation also maps algebra elements to group elements. This fact is well know on ${\rm SO}(3)$ and yields a representation of rotation matrices in terms for Rodrigues parameter, which is, however, not continuous. Generalizing this transformation to ${\rm SU}(2) $ allows for a singularity-free description of all rotations, which does not contain transcendental functions. While in the considered range the exponential map is of class $ C^\infty $ the $\mathrm{cay}$ map on ${\rm SU}(2) $ is only of class $ C^1$ and on ${\rm SO}(3)$ it is not even continuous. Simulation results exemplify the resultant numerical benefits for the simulation of rigid body dynamics. The problem caused by a lack of a continuous transformation from generalized accelerations to angular accelerations can be avoided for rigid body motions using the Bolzmann-Hamel equations.

We develop the Fourier transform interpretation of the inverse scattering method for nonlinear integrable evolution equations associated with a $\mathbb{Z}_3$ reduced Zakharov-Shabat system for the Lie algebra $\mathfrak{sl}(3,\mathbb{C})$. A simple representative of this integrable hierarchy is the well-known Kaup-Kupershmidt equation. Our results admit a natural extention for nonlinear equations connected to a deeply reduced Zakharov-Shabat system related to an arbitrary simple Lie algebra.

The gravitational interaction of light is analyzed considering its dual characteristic nature, i.e. as an (electromagnetic) wave or as a particle (photon). Considered as an electromagnetic wave, the light can be source of gravitational waves belonging to the larger class of exact solutions of Einstein field equations which are invariant for a non-Abelian two-dimensional Lie algebra of Killing fields. It is shown that in the would be quantum theory of gravity they correspond to spin$-1$ massless particles.

Here we extend the star products by means of complex symmetric matrices. In this way we obtain a family of star products. Next we consider the star exponentials with respect to these star products, and finally we obtain several interesting identities.