In this survey paper we present some aspects of generalized inverses, which are related to inner and outer invertibility, Moore-Penrose inverse, the appropriate reverse order law, and Drazin inverse.

We extend the Feynman derivation of the Maxwell-Lorentz equations to the case in which coordinates do not commute, adding significantly to previous results. New dynamics is pinned down precisely both at the level of the homogeneous equations and for the Lorentz force, for which a complete derivation is given for the first time.

The problem of time – a foundational question in quantum gravity – is due to conceptual gaps between GR and physics’ other observationally-confirmed theories. Its multiple facets originated with Wheeler-DeWitt-Dirac over 50 years ago. They were subsequently classified by Kuchaˇr-Isham, who argued that most of the problem is facet interferences and posed the question of how to order the facets. We show the local classical level facets are two copies of Lie theory with a Wheelerian two-way route there-between. This solves facet ordering and facet interference. Closure by a Lie algorithm generalization of Dirac’s algorithm is central.

A quantum stochastic product is a binary operation on the space of quantum states preserving the convex structure. We describe a class of associative stochastic products, the twirled products, that have interesting connections with quantum measurement theory. Constructing such a product involves a square integrable group representation, a probability measure and a fiducial state. By extending a twirled product to the full space of trace class operators, one obtains a Banach algebra. This algebra is commutative if the underlying group is abelian. In the case of the group of translations on phase space, one gets a quantum convolution algebra, a quantum counterpart of the classical phase-space convolution algebra. The peculiar role of the fiducial state characterizing each quantum convolution product is highlighted.

In the paper we consider almost geodesic mappings of the first type of spaces with affine connections onto generalized 2-Ricci-symmetric spaces. The main equations for the mappings are obtained as a closed system of linear differential equations of Cauchy type in the covariant derivatives. The obtained result extends an amount of research produced by Sinyukov, Berezovski and Mikeš.

Various structural properties of semidirect sums of the rotation Lie algebra of rank one and an Abelian algebra described in terms of real representations with at most two irreducible constituents are obtained. The stability properties of these semidirect sums are studied by means of the co- homological and the Jacobi scheme methods.

In this paper, we review recent results on the interaction of the topological electromagnetic fields with matter, in particular with spinless and spin half charged particles obtained earlier. The problems discussed here are the generalized Finsler geometries and their dualities in the Trautman- Rañada backgrounds, the classical dynamics of the charged particles in the single non-null knot mode background and the quantization in the same background in the strong field approximation.

We investigate a version of Yang–Mills theory by means of general connections. In order to deduce a basic equation, which we regard as a version of Yang–Mills equation, we construct a self-action density using the curvature of general connections. The most different point from the usual theory is that the solutions are given in pairs of two general connections. This enables us to get nontrivial solutions as general connections. Especially, in the quaternionic Hopf fibration over four-sphere, we demonstrate that there certainly exist nontrivial solutions, which are made by twisting the well-known BPST anti-instanton.

In the paper we study the extremals and isoperimetric extremals of the rotations in the plane. We found that extremals of the rotations in the plane are arbitrary curves. By studying the Euler-Poisson equations for extended variational problems, we found that the isoperimetric extremals of the rotations in the Euclidian plane are straight lines.

A detailed derivation of the jet composition in local coordinates for jet (differential) groups is presented. A suitable faithful representation in matrix groups is demonstrated. Furthermore, Toupin subgroups which occur in continuum mechanics are demonstrated as an example in which representations can be used effectively.

Despite of their importance, the symplectic groups are not so popular like orthogonal ones as they deserve. The only explanation of this fact seems to be that their algebras can not be described so simply. While in the case of the orthogonal groups they are just the anti-symmetric matrices, those of the symplectic ones should be split in four blocks that have to be specified separately. It turns out however that in some sense they can be presented by the even dimensional symmetric matrices. Here, we present such a scheme and illustrate it in the lowest possible dimension via the Cayley map.

Besides, it is proved that by means of the exponential map all such matrices generate genuine symplectic matrices.

We developed the theory of superalgebraic spinors, which is based on the use of Grassmann densities and derivatives with respect to them in a pseudo- continuous space of momenta. The algebra that they form corresponds to the algebra of second quantization of fermions. We have con-structed a vacuum state vector and have shown that it is symmetric with respect to $P$, $CT$ and $CP T$ transformations. Operators $C$ and $T$ transforms the vacuum into an alternative one. Therefore, time inversion $T$ and charge conjugation $C$ cannot be exact symmetries of the spinors.

Noncommutative phase space of arbitrary dimension is discussed. We introduce momentum-momentum noncommutativity in addition to coordinate-coordinate noncommutativity. We find an exact form for the linear transformation which relates a noncommutative phase space to the corresponding ordinary one. By using this form, we show that a noncommutative phase space of arbitrary dimension can be represented by the direct sum of two-dimensional noncommutative ones. In two-dimension, we obtain the transformation which relates a noncommutative phase space to commutative one. The transformation has the Lorentz transformation-like forms and can also describe the Bopp’s shift.

Subject of present paper is the geometry of foliation defined by submersions on complete Riemannian manifold. It is proven foliation de- fined by Riemannian submersion on the complete manifold of zero sectional curvature is total geodesic foliation with isometric leaves. Also it is shown level surfaces of metric function are conformally equivalent.

In this work, we perform exact and concrete computations of star-product of functions on the Euclidean motion group in the plane, and list its $C$-star- algebra properties. The star-product of phase space functions is one of the main ingredients in phase space quantum mechanics, which includes Weyl quantization and the Wigner transform, and their generalizations. These methods have also found extensive use in signal and image analysis. Thus, the computations we provide here should prove very useful for phase space models where the Euclidean motion groups play the crucial role, for instance, in quantum optics.

As the title itself suggests here we are presenting extremely reach two/three parametric families of non-bending rotational surfaces in the three dimensional Euclidean space and provide the necessary details about their natural classifications and explicit parameterizations. Following the changes of the relevant parameters it is possible to trace out the “evolution” of these surfaces and even visualize them through their topological transformations.

Many, and more deeper questions about their metrical properties, mechanical applications, etc. are left for future explorations.

This work presents quantization of time of arrival functions using generalized Stratonovich-Weyl quantization. We take into account the ordering problems involved, mainly the Born-Jordan and the symmetric ordering schemes. We call attention to the combination of the group theoretic methods usually employed in Weyl quantization with the implementation of different ordering schemes via integral kernel factors. It is possible to, and we do, apply the Pegg-Barnett method to the quantization of time to address physical issues such as boundedness and self-adjointness.

Mathematicians have shown interest in manifolds endowed with several distributions, e.g., webs composed of different regular foliations and multiply warped products, as well as distributions having variable dimensions (e.g., singular Riemannian foliations). In this paper, we extend our previous study of the mixed scalar curvature of two orthogonal singular distributions for the case of $k > 2$ singular (or regular) pairwise orthogonal distributions, prove an integral formula with this kind of curvature, and illustrate it by characterizing auto parallel singular distributions.

Certain ways of characterizing integrable systems with (1, 1)-tensor field have been investigated, so far. For example, recursion operators and Haantjes operators are known. We show that geometrical examples of four- or six-dimensional symplectic Haantjes manifolds and recursion operators for several Hamiltonian systems. Through these examples, we consider the relation between recursion operators and Haantjes operators.

This work is dedicated to systems of matrix nonlinear evolution equations related to Hermitian symmetric spaces of the type A.III. The systems under consideration generalize the $1 + 1$ dimensional Heisenberg ferromagnet equation in the sense that their Lax pairs are linear bundles in pole gauge like for the original Heisenberg model. Here we present certain local and nonlocal reductions. A local integrable deformation and some of its reductions are discussed as well.

For given $k$ bodies of collinear central configuration of Newtonian $k$-body problem, we ask whether one can add another body on the line without changing the configuration and motion of the initial bodies so that the total $k + 1$ bodies provide a central configuration.

The case $k = 4$ is analyzed. We study the inverse problem of five bodies and obtain a global explicit formula. Then using the formula we find there are five possible positions of the added body and for each case the mass of the added body is zero. We further consider to deform the position of the added body without changing the positions of the initial four bodies so that the total five bodies are in a state of central configuration and the mass of the added body becomes positive. For each solution above, we find such a deformation of the position of the added body in an explicit manner starting from the solution.

Star product for functions of one variable is given. A deformation of the Mittag-Leffler functions is suggested by means of the star product.