On rational extension of Heisenberg algebra

Construction of rational extension for Heisenberg algebra with one pair of generators is discussed.Comment: AmSTeX, 12 pages, amsppt styl

Normal shift in general Lagrangian dynamics

It is well known that Lagrangian dynamical systems naturally arise in describing wave front dynamics in the limit of short waves (which is called pseudoclassical limit or limit of geometrical optics). Wave fronts are the surfaces of constant phase, their points move along lines which are called rays. In non-homogeneous anisotropic media rays are not straight lines. Their shape is determined by modified Lagrange equations. An important observation is that for most usual cases propagating wave fronts are perpendicular to rays in the sense of some Riemannian metric. This happens when Lagrange function is quadratic with respect to components of velocity vector. The goal of paper is to study how this property transforms for the case of general (non-quadratic) Lagrange function.Comment: AmSTeX, 27 pages, amsppt styl

Gauge or not gauge?

The analogy of the nonlinear dislocation theory in crystals and the electromagnetism theory is studied. The nature of some quantities is discussed.Comment: AmSTeX, 12 pages, amsppt styl

Comparative analysis for pair of dynamical systems, one of which is Lagrangian

It is known that some equations of differential geometry are derived from variational principle in form of Euler-Lagrange equations. The equations of geodesic flow in Riemannian geometry is an example. Conversely, having Lagrangian dynamical system in a manifold, one can consider it as geometric equipment of this manifold. Then properties of other dynamical systems can be studied relatively as compared to this Lagrangian one. This gives fruitful analogies for generalization. In present paper theory of normal shift of hypersurfaces is generalized from Riemannian geometry to the geometry determined by Lagrangian dynamical system. Both weak and additional normality equations for this case are derived.Comment: AmSTeX, 40 pages, amsppt styl

A note on solutions of the cuboid factor equations

A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. It is described by a system of four quadratic equations with respect to six variables. The cuboid factor equations were derived from these four equations by symmetrization procedure. They constitute a system of eight polynomial equations. Recently two sets of formulas were derived providing two solutions for the cuboid factor equations. These two solutions are studied in the present paper. They are proved to coincide with each other up to a change of parameters in them.Comment: AmSTeX, 15 pages, amsppt style, 3 ancillary file

The Higgs field can be expressed through the lepton and quark fields

The Higgs field is a central point of the Standard Model supplying masses to other fields through the symmetry breaking mechanism. However, it is associated with an elementary particle which is not yet discovered experimentally. In this short note I suggest a way for expressing the Higgs field through other fields of the Standard Model. If this is the case, being not an independent field, the Higgs field does not require an elementary particle to be associated with it.Comment: AmSTeX, 4 pages, amsppt styl

Minimal tori in five-dimensional sphere in C3C^3

Special class of surfaces in five-dimensional sphere in $C^3$ is considered. Immersion equations for minimal tori of that class are shown to be reducible to the equation $u_{z\bar z}=e^u-e^{-2u}$ which is integrable by means of inverse scattering method. Finite-gap minimal tori are constructed.Comment: AmSTeX, 10 pages, amsppt styl

V-representation for normality equations in geometry of generalized Legendre transformation

Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. These equations were first derived in Euclidean geometry. Then very soon they were rederived in Riemannian and in Finslerian geometry. Recently I have found that normality equations can be derived in geometry given by classical and/or generalized Legendre transformation. However, in this case they appear to be written in p-representation, i. e. in terms of momentum covector and its components. The goal of present paper is to transform normality equations back to v-representation, which is more natural for Newtonian dynamical systems.Comment: AmSTeX, 32 pages, amsppt styl

Transfinite normal and composition series of groups

Normal and composition series of groups enumerated by ordinal numbers are studied. The Jordan-Holder theorem for them is proved.Comment: AmSTeX, 12 pages, amsppt styl

Algorithm for generating orthogonal matrices with rational elements

Special orthogonal matrices with rational elements form the group SO(n,Q), where Q is the field of rational numbers. A theorem describing the structure of an arbitrary matrix from this group is proved. This theorem yields an algorithm for generating such matrices by means of random number routines.Comment: AmSTeX, 7 pages, amsppt style, English wording is improved, references are transformed to hyperlinks, the fugure is incorporated into the PS and PDF file