These notes describe some links between the group $SL_2(\mathbb{R})$, the Heisenberg group and hypercomplex numbers - complex, dual and double numbers. Relations between quantum and classical mechanics are clarified in this framework. In particular, classical mechanics can be obtained as a theory with noncommutative observables and a non-zero Planck constant if we replace complex numbers in quantum mechanics by dual numbers. Our consideration is based on induced representations which are build from complex-/dual/-double-valued characters. Dynamic equations, rules of additions of probabilities, ladder operators and uncertainty relations are also discussed. Finally, we prove a Calderón-Vaillancourt-type norm estimation for relative convolutions.

We explain rich geometric structures that appear in the quantisation of linear bosonic and fermionic systems. By contrasting with the quantisation of general curved phase spaces, we focus on results that shed light on one of the most basic problems in quantisation: the dependence of the quantum Hilbert space on auxiliary data such as the choice of polarisations that is necessary to define a quantum Hilbert space.

In this paper we study fundamental equations of geodesic mappings of manifolds with affine connection onto symmetric manifolds. We obtain fundamental equations of this problem. At the end of our paper we demonstrate example of studied mappings.

Here we develop a specific factorization technique for rotations in $\mathbb{R}^3$ into five factors about two or three fixed axes. Although not always providing the most efficient solution, the method allows for avoiding gimbal lock singularities and decouples the dependence on the invariant axis ${\bf n}$ and the angle $\phi$ of the compound rotation. In particular, the solutions in the classical Euler setting are given directly by the angle of rotation $\phi$ and the coordinates of the unit vector $\bf{n}$ without additional calculations. The immediate implementations in rigid body kinematics are also discussed and some generalizations and potential applications in other branches of science and technology are pointed out as well.

We studied complex and real hypersurfaces immersed in locally conformal Kähler manifolds. We have obtained conditions for the immersions, if the manifolds admit existence of such complex hypersurfaces that are orthogonal to both Lee and anti-Lee vector fields. Also we explore real hypersurfaces immersed in LCK-manifolds.

In this paper we will introduce a newly found knowledge above the existence and the uniqueness of isoperimetric extremals of rotation on two-dimensional (pseudo-)Riemannian manifolds and on surfaces on Euclidean space. We will obtain the fundamental equations of rotary diffeomorphisms from (pseudo-)Riemannian manifolds for twice-differentiable metric tensors onto manifolds with affine connections.

We study the existence of conservation laws in constrained systems described by quadratic Lagrangians; the type of which is encountered in mini-superspace cosmology. As is well known, variational symmetries lead to conserved quantities that can be used in the classical and quantum integration of a system. Additionally - and due to the parametrization invariance of such Lagrangians - conditional symmetries defined on phase space can lead to non-local integrals of motion. The latter may be of importance in various cosmological configurations. As an example we present the case of scalar field cosmology with an arbitrary potential.

The embeddings of the $\frak{so}(3)$ Lie algebra and the Lie group ${\rm SO}(3)$ in higher dimensions is an important construction from both mathematical and physical viewpoint. Here we present results based on a program package for building the generating matrices of the irreducible embeddings of the $\frak{so}(3)$ Lie algebra within $\frak{so}(n)$ in arbitrary dimension $n \geq 3, n \neq 4k+2, k \in \mathbb{N}$ relying on the algorithm developed recently by Campoamor-Strursberg [3]. For the remaining cases $n = 4k+2$ embeddings of $\frak{so}(3)$ into $\frak{so}(n)$ are also constructed. Besides, we investigate the characteristic polynomials of these $\frak{so}(n)$ elements. We show that the Cayley map applied to $\cal{C} \in \frak{so}(n)$ is well defined and generates a subset of ${\rm SO}(n)$. Furthermore, we obtain explicit formulas for the images of the Cayley map. The so obtained ${\rm SO}(n)$ matrices are expressed as polynomials of ${\cal C}$ whose coefficients are rational functions of the norm of the vector-parameter ${\bf c}$. The composition laws are extracted for the cases $n = 4, 6$ and for the first case it is shown that via the Cayley map the isomorphism ${\rm SU}(2) \cong {\rm im}{{\rm Cay}_{{\rm im}{j_4}}}\cup\{-\mathcal{I}_4\}$ holds. Also, for $n = 4$ explicit formulas for the the angular velocity matrices are derived. Comparisons between the results obtained via the exponential map and the Cayley map are made as well. In contrast to the case of the Cayley map, the results for the exponential map include either irrational or transcendental functions of the module of the vector-parameter.

In this paper we study fundamental equations of geodesic mappings of manifolds with affine connection onto (pseudo-)Riemannian manifolds. We proved that if a manifold with affine (or projective) connection of differentiability class $C^r (r\geq2)$ admits a geodesic mapping onto a (pseudo-)Riemannian manifold of class $C^1$, then this manifold belongs to the differentiability class $C^{r+1}$. From this result follows if an Einstein spaces admits non-trivial geodesic mappings onto (pseudo-)Riemannian manifolds of class $C^1$ then this manifold is an Einstein space, and there exists a common coordinate system in which the components of the metric of these Einstein manifolds are real analytic functions.

In this paper we describe semi-discrete isothermic constant mean curvature surfaces of revolution with smooth profile curves in Minkowski three-space. Unlike the case of semi-discrete constant mean curvature surfaces in Euclidean three-space, they might have certain types of singularities in a sense defined by the second author in a previous work. We analyze the singularities of such surfaces.

The plate-ball problem concerns the shortest trajectories traced by a rolling sphere on a horizontal plane between the prescribed initial and final states meaning the positions and orientations of the sphere. Here we present an explicit parametric representation of these trajectories in terms of the Jacobian elliptic functions and elliptic integrals.

In this paper we study fundamental equations of geodesics on surfaces of revolution. We obtain examples of existence of geodesic bifurcation.

We introduce the notion of twisted Fock representations of noncommutative Kähler manifolds and give their explicit expressions. The so-called twisted Fock representation is a representation of the Heisenberg like algebra whose states are constructed by acting creation operators on a vacuum state. “Twisted” means that creation operators are not Hermitian conjugate of annihilation operators. In deformation quantization of Kähler manifolds with separation of variables formulated by Karabegov, local complex coordinates and partial derivatives of the Kähler potential with respect to coordinates satisfy the commutation relations between the creation and annihilation operators. Based on these relations, the twisted Fock representation of noncommutative Kähler manifolds is constructed.

We consider a canonical transformation of parabolic coordinates on the plain associated with integrable Hénon-Heiles systems and suppose that this transformation together with some additional relations may be considered as a counterpart of the auto and hetero Bäcklund transformations.