Axisymmetric Solutions in the Geomagnetic Direction Problem

The magnetic field outside the earth is in good approximation a harmonic vector field determined by its values at the earth’s surface. The direction problem seeks to determine harmonic vector fields vanishing at infinity and with the prescribed direction of the field vector at the surface. In general this type of data neither guarantees the existence nor the uniqueness of solutions of the corresponding nonlinear boundary value problem. To determine conditions for existence, to specify the non-uniqueness and to identify cases of uniqueness is of particular interest when modeling the earth’s (or any other celestial body’s) magnetic field from these data. Here we consider the case of axisymmetric harmonic fields $$\mathbf{B}$$ B outside the sphere $$S^2 \subset {{\mathbb {R}}}^3$$ S 2 ⊂ R 3 . We introduce a rotation number $${r\!o}\in {{\mathbb {Z}}}$$ r o ∈ Z along a meridian of $$S^2$$ S 2 for any axisymmetric Hölder continuous direction field $$\mathbf{D}\ne 0$$ D ≠ 0 on $$S^2$$ S 2 and, moreover, the (exact) decay order $$3 \le \delta \in {{\mathbb {Z}}}$$ 3 ≤ δ ∈ Z of any axisymmetric harmonic field $$\mathbf{B}$$ B at infinity. Fixing a meridional plane and in this plane $${r\!o}- \delta +1 \geqq 0$$ r o - δ + 1 ≧ 0 points $$z_n$$ z n (symmetric with respect to the symmetry axis and with $$|z_n| > 1$$ | z n | > 1 , $$n = 1,\ldots ,{r\!o}-\delta +1$$ n = 1 , … , r o - δ + 1 ), we prove the existence of an (up to a positive constant factor) unique harmonic field $$\mathbf{B}$$ B vanishing at $$z_n$$ z n and nowhere else, with decay order $$\delta $$ δ at infinity, and with direction $$\mathbf{D}$$ D at $$S^2$$ S 2 . The proof is based on the global solution of a nonlinear elliptic boundary value problem, which arises from a complex analytic ansatz for the axisymmetric harmonic field in the meridional plane. The coefficients of the elliptic equation are discontinuous and singular at the symmetry axis, and this requires solution techniques that are adapted to this special situation.

SOME VECTORS FIELDS ON THE TANGENT BUNDLE WITH A SEMI-SYMMETRIC METRIC CONNECTION

Let $M$ is a (pseudo-)Riemannian manifold and $TM$ be its tangent bundlewith the semi-symmetric metric connection $\overline{\nabla }$. In thispaper, we examine some special vector fields, such as incompressible vectorfields, harmonic vector fields, concurrent vector fields, conformal vectorfields and projective vector fields on $TM$ with respect to thesemi-symmetric metric connection $\overline{\nabla }$ and obtain someproperties related to them.

Twisted Sasakian Metric on the Tangent Bundle and Harmonicity

In the present paper, we introduce a new class of natural metrics on the tangent bundle $TM$ of the Riemannian manifold $(M,g)$ denoted by $G^{f,h}$ which is named a twisted Sasakian metric. A necessary and sufficient conditions under which a vector field is harmonic with respect to the twisted Sasakian metric are established. Some examples of harmonic vector fields are presented as well.

Five-Phase Converter Control Algorithms for Implementing Space-Vector Modulation

Multiphase (five-phase) motors can be considered as an alternative to three-phase motors in implementing electric traction with vector control. The subject of the study is the five-phase winding voltage vector space formed by various control algorithms of a five-phase converter. Thirty logical states of the converter form three types of the vector voltage space of a symmetrical five-phase winding. Each of the vector space types forms the resulting vector of a certain value. The resulting (generalized) voltage vector can take three different values. With the phase time sequence ABCDE of a symmetrical five-phase winding ABCDE with the spatial phase shift equal to 72 electrical degrees, the fundamental harmonic vector is the resulting working voltage vector. With the phase time sequence ACEBD of a symmetrical five-phase winding ABCDE with the spatial phase shift equal to 72 electrical degrees, the resulting voltage vector of the third harmonic is the resulting working vector. The rotation frequency of the resulting third harmonic vector is three times the rotation frequency of the fundamental harmonic vector, and the rotation direction is opposite to the rotation direction of the fundamental harmonic vector. With one of the phase voltage forms and the phase time sequence ACEBD, the modulus of the resulting voltage vector of the third harmonic component is equal to the modulus of the resulting vector of the fundamental harmonic component. There are two converter control algorithms that form the resulting voltage vectors that are equal in modulus, but with different rotation speeds multiple of three. An approach to studying the multiphase phase winding voltage vector space has been elaborated, which can be applied in implementing vector control.