We find some new expressions of density for straight lines and planes in euclidean spaces of dimension two and three. Also, we show the density for straight lines in surfaces which is expressed in terms of the coefficient of the first fundamental form. Finally, the density for tangent lines to a differentiable plane curve and for tangent plane of a differentiable surface are presented.

We have generalized the fermionic coherent states to pseudo-fer\-mion oscillator system. The system of coherent states constructed consist of two subsets, which are bi-normalized and bi-overcomplete. The two subsets are built up as eigenstates of two annihilation operators $b$ and $\tilde{b}= \eta b\eta^{-1}$ of respectively $H$ and $H^+$ where $\eta$ is the Hermitian and invertible operator that ensures the pseudo-Hermiticity of the Hamiltonian $H=\eta^{-1}H^+\eta$.

We classify all proper-biharmonic Legendre curves in a Sasakian space form and point out some of their geometric properties. Then we provide a method for constructing anti-invariant proper-biharmonic submanifolds in the Sasakian space forms. Finally, using the Boothby-Wang fibration, we determine all proper-biharmonic Hopf cylinders over homogeneous real hypersurfaces in complex projective spaces.

We give one parameter deformation of level $k$ free field realization of the screening current of the elliptic algebra $\mathrm{U}_{q,p}(\widehat{\mathfrak{sl}_2})$. By means of these free field realizations, we construct infinitely many commutative operators, which are called the nonlocal integrals of motion associated with the elliptic algebra $\mathrm{U}_{q,p}(\widehat{\mathfrak{sl}_2})$ for level $k$. They are given as integrals involving a product of the screening current and elliptic theta functions. This paper give level $k$ generalization of the nonlocal integrals of motion given in [1].

The special interest is devoted to such situations when the material space of object with affine degrees of freedom has generally lower dimension than the one of the physical space. In other words when we have $m$-dimensional affinely-rigid body moving in the $n$-dimensional physical space, $m\lt n$. We mainly concentrate on the physical situation $m=2$, $n=3$ when “thickness” of flat bodies performs one-dimensional oscillations orthogonal to the two-dimensional central plane of the body. For the isotropic case in two “flat” dimensions some special solutions, namely, the stationary ellipses, which are analogous to the ellipsoidal figures of equilibrium well known in astro- and geophysics, e.g., in the theory of the Earth's shape, are obtained.

Dynamical Systems theory is a valuable tool in the profound study of Cosmology, as qualitative methods allow to characterize cosmological solutions on the basis of their relevant physical features (e.g. stability and asymptotic behaviour). Here we briefly review some well established results for cosmological models in the framework of General Relativity. Then we consider a family of modified cosmological models which are gaining more and more relevance, namely Loop Quantum Cosmologies. In particular we analyze the geometrical structure and dynamical properties of the model presented in earlier.

We study noncommutative (NC) instantons and vortexes. At first, we construct instanton solutions which are deformations of instanton solutions on commutative Euclidean four-space. We show that the instanton numbers of these NC instanton solutions coincide with the commutative solutions. Next, we also deform vortex solutions similarly and we show that their vortex numbers are unchanged under the NC deformation.