Existence of (Dirac-)harmonic Maps from Degenerating (Spin) Surfaces

We study the existence of harmonic maps and Dirac-harmonic maps from degenerating surfaces to a nonpositive curved manifold via the scheme of Sacks and Uhlenbeck. By choosing a suitable sequence of $$\alpha $$ α -(Dirac-)harmonic maps from a sequence of suitable closed surfaces degenerating to a hyperbolic surface, we get the convergence and a cleaner energy identity under the uniformly bounded energy assumption. In this energy identity, there is no energy loss near the punctures. As an application, we obtain an existence result about (Dirac-)harmonic maps from degenerating (spin) surfaces. If the energies of the map parts also stay away from zero, which is a necessary condition, both the limiting harmonic map and Dirac-harmonic map are nontrivial.

The Magnetic Hooke-Newton Transmutation in Momentum Space

The magnetic Hooke-Newton transmutation, which emerges from the transformation design of the quadratic conformal mapping for the system of charged particles moving in the uniform magnetic field, is investigated in the momentum space. It is shown that there are two ways to turn the linear interaction force of the system into the inverse square interaction. The first one, which involves simply applying the mapping to the system, has the spectrum with the Landau levels of even angular momentum quantum number. The second one considers the geometric structure of the mapping as an effective potential which leads us to the transmuted Coulomb system with the novel quantum spectrum. The wave functions of momentum for the bound and scattering states of the transmutation system are given. It is also shown that the effective potential due to the geometric structure can be generalized to a general 2D surface, and the Schrödinger equation of a particle moving on the surface while under the action of the potential can be solved by the form-invariant Schrödinger equation of the free particle. The solution of a particle moving on the hyperbolic surface under the potential is given with the conclusion. The presentation manifests the transformation design of the quantum state, depending mainly on the geometric structure of the representation space, not on the action of the specific potential field. This characteristic makes it possible for us to use the geometric structure of different representation spaces to explore some novel behaviors of quantum particles.

Mathematical model of geolocation of a satellite communication station using one relay satellite

At present, multi-satellite geolocation systems based on the TDOA are actively used to localization of radio emission sources in satellite communication systems operating via relay satellites without on-board processing. In General, information about the location of the radio emission sources is contained in the difference of the inclined range from the multiple fixed points with known coordinates. Such points of space in the classical geolocation system are two or more relay satellites in geostationary orbit. It is not always possible to have two or more satellites retransmitting the same signal. Therefore, it is necessary to develop a mathematical model for geolocation using a single relay satellite. Single-satellite geolocation is based on the use of Doppler, TDOA, or phase direction finding methods. With this approach, it is desirable that a single satellite has the ability to move in a controlled manner, either in altitude or at different speeds relative to its standing point. Moving the satellite along the equator in position and along the meridian in height allows you to calculate several orthogonal bases of estimates of the inclined range to the radio source. In this case, the determination of coordinates is based on the increment of the distance of the object's signal runs between the end points of each base. This provides the construction of position lines (hyperballs), the intersection of which is the source location. If the movement of the satellite along the equator and the meridian is performed with a change in speed, then geolocation is based on measurements of several orthogonal components of the Doppler frequency shift of the radio source signals. The base will be called two, four or more pairwise taken orbital positions of the satellite at points with fixed coordinates; S x y z1 1 1 1( , , ) S x y z2 2 2 2( , , ); S x y z2 2 2 2( , , ) S x y z3 3 3 3( , , ); etc. in all possible combinations. An arbitrary inclined base formed in the spacecraft orbit has an extension of Бп (x2  x1)2  (y2  y1)2  (z2  z1)2 . Differential range Дд = Дн2 – Дн1. To geolocate the M-object, you must: 1. Measure the difranges between M over two or more different shifted Дн bases at multiple satellite drift positions – Дд1, Дд2, ..., Дд4, etc. 2. Calculate the parameters al, bl, cl of each l-th hyperbolic surface of the section of the conic equations of the geometric location of the points of position M with the measured Дд1, Дд2, ..., Дд4 and the known Дн. Construct a common point of intersection of several such hyperbolic surfaces of the cross-section of the conic equations of the geometric location of the points of the position of the object M(x, y, z). The resulting vector of linear coordinates M(x, y, z) of the object must be converted from geocentric to geographical coordinates of the spherical coordinate system of the object M (longitude, latitude, Position-vector).