We discuss some well-known facts about Clifford algebras: matrix representations, Cartan’s periodicity of 8, double coverings of orthogonal groups by spin groups, Dirac equation in different formalisms, spinors in $n$ dimensions, etc. We also present our point of view on some problems. Namely, we discuss the generalization of the Pauli theorem, the basic ideas of the method of averaging in Clifford algebras, the notion of quaternion type of Clifford algebra elements, the classification of Lie subalgebras of specific type in Clifford algebra, etc.

According to the “gravitationally-modified” Maxwell equations that were proposed for an alternative scalar theory with an “ether”, electric charge would not be conserved in time-dependent gravitational field. We define an asymptotic expansion scheme for the electromagnetic field in a weak gravitational field. This allows us to assess the amounts of charge production or destruction which are thus predicted. These amounts seem high enough to discard that version of the gravitationally-modified Maxwell equations. We show that this failure is due to the former assumption of additivity of the energy tensors: an “interaction energy tensor” has to be added. Then the standard Maxwell equations in a curved spacetime become compatible with that scalar theory, and they predict charge conservation.

This paper aims to introduce the notion of hyper generalized weakly symmetric manifolds with a non-trivial example.

Computing Kauffman bracket grows exponentially with the number of crossings in the knot diagram. In this paper we illustrate how Kauffman bracket for rational tangles and rational knots can be computed so that it involves a low number of terms. Kauffman bracket and Jones polynomial are known to have connections with statistical mechanics, quantum theory and quantum field theory.

The paper provides a pedagogical study on vectorial parameterizations first proposed by O. Rodrigues for the rotation group in $\mathbb{R}^3$ by means of the so-called Rodrigue’s vector. Although his technique yields significant advantages in both theoretical and applied context, the vectorial interpretation is easily seen to be completely wrong and in order to benefit most from this otherwise fruitful approach, we put it in the proper perspective, namely, that of Clifford’s geometric algebras, spin groups and projective geometry. This allows for a natural generalization and straightforward implementations in various physical models, some of which are pointed out below in the text.

The present paper aims to emphasize the geometrical features of the quantum spacetime, considering gravity as an emergent feature similar to the elasticity of the solid state. A small scale structure is needed to explain the emergent gravity and how spacetime atoms are continuously created in the process of the expansion of the universe. A simple geometrical model has been introduced.

In this paper we construct holomorphically projective mappings of equidistant parabolic Kähler spaces. We discus fundamental equations of these mappings as well.

We introduce noncommutative deformations of locally symmetric Kähler manifolds. A Kähler manifold $M$ is said to be a locally symmetric Kähler manifold if the covariant derivative of the curvature tensor is vanishing . An algebraic derivation process to construct a locally symmetric Kähler manifold is given. As examples, star products for noncommutative Riemann surfaces and noncommutative $\mathbb {CP}^N$ are constructed.

In the present paper we prove non-existence theorems for conformal mappings of compact (pseudo-)Riemannian manifolds onto Einstein manifolds without boundary. We obtained certain conditions for which these mappings are only trivial.

Symplectic-Haantjes manifolds are constructed for several Hamiltonian systems following Tempesta-Tondo [5], which yields the complete integrability of systems.

In the present paper we investigate an alternative two-axes decomposition method for rotations that has been proposed in our earlier research. It is shown to provide a convenient parametrization for many important physical systems. As an example, the kinematics of a rotating rigid body is considered and a specific class of solutions to the Euler dynamical equations are obtained in the case of symmetric inertial ellipsoid. They turn out to be related to the Rabi oscillator in spin systems well known in quantum computation. The corresponding quantum mechanical angular momentum and Laplace operator are derived as well with the aid of infinitesimal variations. Curiously, the coefficients in this new representation happen to depend only on one of the angles, which simplifies the corresponding system of ODE's emerging from separation of variables. Some applications of the hyperbolic and complex analogues of this construction in quantum mechanics and relativity are considered in a different paper cited below.

For multiple Fourier series the convergence of partial sums essentially depends on the type of integer sets, to which the sequence numbers of their terms belong. The problem on the general form of such sets is studying in $u$-convergence theory ($u(K)$ - convergence) for multiple Fourier series. An alternative method of summation is based on the concept of the so-called correctly denumarable sets. In the paper some results describing the $u$-convergence relations and convergence on the system or correctly denumarable sets are presented. It is shown that the system of $U(K)$-sets containing a sphere of infinitely increasing radius for fixed $K$ is correctly denumarable. It is established that for the functions satisfying the Lipschitz condition and having a certain growthing $p-k$-variation, the coefficients of multiple Fourier series decrease at the average on the system of $U(K)$-sets faster than it is predicted by their ordinary estimations. It is shown the accurate estimation of the Fourier coefficients of functions of several variables is achieved at a very ``poor'' set of elements of the integer lattice.

We consider a class of linear Weingarten surfaces of revolution whose principal curvatures, meridional $k_{\mu}$ and parallel $k_{\pi}$, satisfy the relation $k_{\mu}=(n+1)k_{\pi}$, $n=0,\,1,\,2,\ldots\, .$ The first two members of this class of surfaces are the sphere $(n=0)$ and the Mylar balloon $(n=1)$. Elsewhere the Mylar balloon has been parameterized via the Jacobian and Weierstrassian elliptic functions and elliptic integrals. Here we derive six alternative parameterizations describing the third type of surfaces when $n=2$. The so obtained explicit formulas are applied for the derivation of the basic geometrical characteristics of this surface.

This paper is denoted to further study of geodesic bifurcation on surfaces of revolution. We demonstrate an example of bifurcation of closed geodesics on surfaces.

Four mathematical models of classical electrodynamics based on vector fields, tensor spaces, geometric algebras and differential forms are represented in parallel and compared.

In a new approach the graviton is defined as the field particle of spacetime rather than the mediator of gravity. The unification equation is derived and used to predict that for a freely falling body, the energy of incident gravitons is $6.12\times 10^{18}$ GeV. Redshift and scattering of gravitons should produce diffraction patterns, galactic halos and expansion of the Universe. The energy of incident gravitons remains constant as the Universe evolves because of the Doppler shift as bodies fall towards redshifted gravitons. Complex space is used to represent gravitons and explain Young’s two-slit interference. The approach is corroborated by empirical data and extends establish theory.

We derive the extended entanglement entropy and the Fisher information metric in the case of quantum models, described by time-independent diagonal quadratic Hamiltonians. Our research is conducted within the framework of Thermo field dynamics. We also study the properties of the Fisher metric invariants to identify the phase structure of the quasi-particle systems in equilibrium.

We introduce star products for certain function space containing polynomials, and then we obtain an associative algebra of functions. In this algebra we can consider exponential elements, which are called star exponentials. Using star exponentials we can define star functions in the star product algebra. We explain several examples.