This paper is a review of the classical Möbius-Lie geometry and recent works on its extension. The latter considers ensembles of cycles (qua\-drics), which are interconnected through conformal-invariant geometric relations (e.g. ``to be orthogonal'', ``to be tangent'', etc.), as new objects in an extended Möbius-Lie geometry. It is shown on examples, that such ensembles of cycles naturally parameterise many other conformally-invariant families of objects, two examples--the Poincar\'e extension and continued fractions are considered in detail. Further examples, e.g. loxodromes, wave fronts and integrable systems, are discussed elsewhere. The extended Möbius--Lie geometry is efficient due to a method, which reduces a collection of conformally in\-vari\-ant geometric relations to a system of linear equations, which may be accompanied by one fixed quadratic relation. The algorithmic nature of the method allows to implement it as a ${\rm C}$ library, which operates with numeric and symbolic data of cycles in spaces of arbitrary dimensionality and metrics with any signatures. Numeric calculations can be done in exact or approximate arithmetic. In the two- and three-dimensional cases illustrations and animations can be produced. An interactive {\Python} wrapper of the library is provided as well.

A discretization scheme provided by the noncommutativity of space is reviewed. In the representation chosen here the radial coordinate is rendered discrete, allowing fields to be put on a lattice in a natural way. Noncommutativity is traded for a controllable type of nonlocality of the field dynamics, which in turn allows fermions to be free of lattice artefacts. Exact, singularity-free solutions are found interpreted, and their continuum limit is well-defined.

This paper aims to introduce a comparison of variation iteration method and wavelet basis method for the numerical solution of the Kawahara equation. Test problem is used to compare between the two methods. The comparison shows that variation iteration method is efficient and easy to use. On the other hand, the wavelet method is more stable as time increases.

In the electrodynamics of special relativity (SR) or general relativity (GR), the energy tensors of the charged medium and its EM field add to give the total energy tensor that obeys the dynamical equation without external force. In the investigated scalar theory of gravitation (SET), this assumption leads to charge non-conservation, hence an additional, ``interaction'' energy tensor $\boldsymbol{T}_\mathrm{inter}$ has to be postulated. The present work aims at constraining this tensor. First we study the independent equations of electrodynamics and their number, beginning with SR and GR. As in SR and GR, the system of electrodynamics of SET is closed in the absence of $\boldsymbol{T}_\mathrm{inter}$. Hence, with $\boldsymbol{T}_\mathrm{inter}$, at least one additional equation must be provided. This is done by assuming that $\boldsymbol{T}_\mathrm{inter}$ is Lorentz-invariant in the situation of SR. We derive equations allowing one in principle to compute $\boldsymbol{T}_\mathrm{inter}$ in a given gravitational plus EM field. $\boldsymbol{T}_\mathrm{inter}$ may contribute to the dark matter.

Some aspects of the simple Lie algebra $\mathfrak{sl}(2,\mathbb{R})$ realized as a subalgebra of Noether point symmetries of Lagrangian systems and related inverse problems are discussed, specially in connection to Lagrangians of kinetic type and some geometric properties like sectional curvatures.

In the last decade topnotch experiments (LIGO and GP-B) have putted into evidence the viscoelastic nature of the space time. In the present work we have applied the viscoelastic constitutive equations for a spacetime model, based on the fractional Zener representation, which is the most general way of thinking about materials. Dispersion and dissipation are discussed in the frame of the spacetime, considered as a viscoelastic material.

The main goal of this paper is to present the possibility of application of some well known tools of Poisson geometry to classification of real low dimensional Lie algebras.

The concept of fundamental domain, as defined by Ahlfors, plays an important role in the study of different classes of analytic functions. For more than a century the Dirichlet functions have been intensely studied by mathematicians working in the field of number theory as well as by those interested in their analytic properties. The fundamental domains pertain to the last field, yet we found a lot of theoretic aspects which can be dealt with by knowing in detail those domains. We gathered together in this survey paper some recent advances in this field. Proofs are provided for some of the theorems, so that the reader can navigate easily through it.

The nine two-dimensional Cayley-Klein geometries are firstly reviewed by following a graded contraction approach. Each geometry is considered as a set of three symmetrical homogeneous spaces (of points and two kinds of lines), in such a manner that the graded contraction parameters determine their curvature and signature. Secondly, new Poisson homogeneous spaces are constructed by making use of certain Poisson-Lie structures on the corresponding motion groups. Therefore, the quantization of these spaces provides noncommutative analogues of the Cayley-Klein geometries. The kinematical interpretation for the semi-Riemannian and pseudo-Riemannian Cayley-Klein geometries is emphasized, since they are just Newtonian and Lorentzian spacetimes of constant curvature.

The Maslov quantization condition is a condition for Lagrangian submanifolds which is regarded as a mathematical extension of the Bohr-Sommerfeld quantization condition. In this survey note, we apply the Maslov quantization condition to several concrete Schr ödinger operators and quantize invariant Lagrangian submanifolds of their classical systems. We see the quasi-classical energy levels are equal to the quantum ones for these operators and also the number of Lagrangian submanifolds is equal to the multiplicities of eigenvalues for these operators.

Among nonlinear evolutionary equations integrable ones are of particular interest since only in this we case can theoretically study the model in detail and in-depth. In the present, we establish the geometric connection of the well-known integrable two-component Manakov system with a new two-layer spin system. This indicates that the latter system is also integrable. In this formalism, geometric invariants define some important conserved quantities associated with two interacting curves, and with the corresponding nonlinear evolution equations.

The appearance of a geometric flow in the conservation law of particle number in classical particle diffusion and in the conservation law of probability in quantum mechanics is discussed in the geometrical environment of a two-dimensional curved surface with thickness $\epsilon$ embedded in $\mathbb E^3$. In such a system, two-dimensional conservation law needs an additional term just like an anomaly. The additional term can be obtained by the $\epsilon$ expansion. This term has a Gaussian and a mean curvature dependence and can be written as the total divergence of geometric flow $J^i_{G}$. This fact holds in both classical and quantum mechanics.

We variationally characterize the profile curves of rotational linear Weingarten surfaces as planar p-elastic curves. Moreover, by evolving these planar p-elasticae under the binormal flow with prescribed velocity, we describe a procedure to construct all rotational linear Weingarten surfaces, locally. Finally, we apply our findings to two well-known family of surfaces.

The paper is devoted to various refinement strategies in the 4D Euclidean space. The red refinement strategy (RRS) have been widely used by researchers applying multigrid methods. This refinement method has a lot of advantages but it is not superior with respect to the degeneracy measure in the case of 3D quasi canonical domains, 3D ideal domains, 4D canonical domains and curved nonconvex domains. The RRS is compared with other subdivision methods for dividing ideal domains in the 4D Euclidean space. It is established that the RRS is superior than all other methods considered for all ideal domains.

Here we consider an interesting class of free of bending deformations of thin axial symmetric shells subjected to uniform normal pressure. The meridional $k_{\mu}$ and the parallel $k_{\pi}$ principal curvatures of the middle surfaces of such shells obey the non-linear relationship $k_{\mu}=2ak_{\pi}^2+3k_{\pi}$, $a=const$. These non-bending shells depend on two arbitrary parameters, which are the principal radii $r_{\mu}$ and $r_{\pi}$ of some fixed parallel of the shell. Besides, these surfaces have no closed form description in elementary functions. Our principle aim here is to present such a parameterization of the whole class of non-bending closed surfaces by making use of the canonical forms of the elliptic integrals. The obtained explicit formulas are then applied for the derivation of the basic geometrical characteristics of these surfaces.

In this paper, first we gave a summary of recent results on biconservative immersions. Then, we obtained a necessary and sufficient condition for the existence of biconservative quasi-minimal immersions into a four dimensional semi-Riemannian space form of index two with non-zero sectional curvatures. We also constructed an explicit example of biconservative quasi-minimal surface.

It is known that entangled particles involve Lorentz symmetry violation. Hence, we pay attention to Lorentz transformations of signature $(m,n)$ for all positive integers $m$ and $n$. We show that these form the symmetry groups by which systems of $m$ entangled $n$-dimensional particles can be understood, just as the common Lorentz group of signature $(1,3)$ forms the symmetry group by which Einstein's special theory of relativity is understood. A novel, unified parametric realization of the Lorentz transformations of any signature $(m,n)$ shakes down the underlying matrix algebra into elegant and transparent results.

In this report we consider a vector generalization of Heisenberg ferromagnet equation. That completely integrable system is related to a spectral problem in pole gauge for the Lie algebra $\mathfrak{sl}(n+1,\mathbb{C})$. We construct special solutions over constant background using dressing technique.

We consider the theory of the hierarchies of nonlinear evolution equations associated with two gauge-equivalent auxiliary systems. They are obtained from the Generalized Zakharov-Shabat system on $\mathfrak{sl}(3,\mathbb{C})$ in general position making a $\mathbb{Z}_2\times \mathbb{Z}_2$ reductions of Mikhailov type in canonical and in pole gauge respectively. Using the Recursion Operators approach we study the symplectic structures of the hierarchies of the nonlinear evolution equations associated with the above auxiliary systems and find the relation between them.