Gauss Equation Research Articles

MODELING OF THE EXTERNAL MAGNETIC FIELD OF ELECTRIC MACHINES

Powerful electric machines – such as electric generators, electric motors generate a magnetic field of great tension. These fields negatively effect on personnel and can to violate the stability of electronic equipment. To determine safe areas for workers, laying communication cables and placing sensitive electronic equipment, it is advisable to model the propagation of magnetic fields of electrical machines. This will make it possible to rationalize the placing of electrical equipment at design stages. The most common high-power electrical machines are four-pole electrical machines with a dipole-quadrupole structure of the external magnetic field. The purpose of the research is a development of models of the propagation of the external magnetic field generated by alternating current electric machines. Results of the research: it is substantiated that to simulate the propagation of the magnetic field of electric machines, it is advisable to use the Gauss equation for the magnetic scalar potential. This will make it possible to take into account the required number of spatial harmonics of the magnetic field to ensure an acceptable calculation error. An electric machine is considered in a spherical approximation. Calculations were carried out in spherical coordinates. The distances were determined in relative radii of the electric machine - the ratio of the radius of the machine to the definition of field strength. The calculations were made for two planes of spherical coordinates and a three-dimensional image was obtained. As a result of the simulation, it is possible to determine the magnetic field strength of a four-pole electric machine at selected distances and directions around the electric machine with the required accuracy. It has been established that there are points of zero external magnetic field strength around electrical machines. Verification of the simulation results was carried out using the method of full-scale measurements of the magnetic field strength around a real four-pole machine. The measurement results showed acceptable agreement with the calculated data. Conclusions: the chosen approach and the results of modeling the propagation of the external magnetic field of electrical machines can be used to design the placement of electrical equipment, taking into account the requirements for electromagnetic safety and electromagnetic compatibility of technical equipment.

C-R Immersions and Sub-Riemannian Geometry

On any strictly pseudoconvex CR manifold M, of CR dimension n, equipped with a positively oriented contact form θ, we consider natural ϵ-contractions, i.e., contractions gϵM of the Levi form Gθ, such that the norm of the Reeb vector field T of (M, θ) is of order O(ϵ−1). We study isopseudohermitian (i.e., f∗Θ=θ) Cauchy–Riemann immersions f:M→(A,Θ) between strictly pseudoconvex CR manifolds M and A, where Θ is a contact form on A. For every contraction gϵA of the Levi form GΘ, we write the embedding equations for the immersion f:M→A,gϵA. A pseudohermitan version of the Gauss equation for an isopseudohermitian C-R immersion is obtained by an elementary asymptotic analysis as ϵ→0+. For every isopseudohermitian immersion f:M→S2N+1 into a sphere S2N+1⊂CN+1, we show that Webster’s pseudohermitian scalar curvature R of (M, θ) satisfies the inequality R≤2n(f∗gΘ)(T,T)+n+1+12{∥H(f)∥gΘf2+∥traceGθΠH(M)∇⊤−∇∥f∗gΘ2} with equality if and only if B(f)=0 and ∇⊤=∇ on H(M)⊗H(M). This gives a pseudohermitian analog to a classical result by S-S. Chern on minimal isometric immersions into space forms.

A general inequality for warped product $CR$-submanifolds of Kähler manifolds

In this paper, warped product CRCR-submanifolds in Kahler manifolds and warped product contact CRCR-submanifolds in Sasakian, Kenmotsu and cosymplectic manifolds, are shown to possess a geometric property; namely DTDT-minimal. Taking benefit from this property, an optimal general inequality is established by means of the Gauss equation, we leave cosyplectic because it is an easy structure. Moreover, a rich geometry appears when the necessity and sufficiency are proved and discussed in the equality case. Applying this general inequality, the inequalities obtained by Munteanu are derived as particular cases. Up to now, the method used by Chen and Munteanu can not extended for general ambient manifolds, this is because many limitations in using Codazzi equation. Hence, Our method depends on the Gauss equation. The inequality is constructed to involve an intrinsic invariant (scalar curvature) controlled by an extrinsic one (the second fundamental form), which provides an answer for the well-know Chen's research problem (Problem 1.1???). As further research directions, we have addressed a couple of open problems arose naturally during this work and depending on its results.

MATHEMATICAL APPARATUS FOR MODELING OF THE PROPAGATION THE MAGNETIC FIELD ELECTRIC MACHINES WITH A GIVEN ACCURACY

. The problem of modeling the propagation local magnetic fields and spatially dispersed sources is large errors compared to field measurements. An important aspect of adequate modeling is the use of the correct mathematical apparatus. It is shown that in order to obtain reliable models of the propagation magnetic fields around electrical machines (generators, electric motors of different power, geometric dimensions and poles), it is advisable to apply the Gauss equation for a scalar potential. The solution of the equation in polar coordinates makes it possible to take into account not only the fundamental, but also other harmonics of the magnetic field (dipole, quadrupole, octupole). This allows, depending on the number of spatial harmonics taken into account, to obtain a model with the required accuracy (error) for predicting the magnetic field strength at any point around the machine. It is considered in the paper that an electronic machine is an object of base radius R0. The presented approach makes it possible to unambiguously determine the location of zero field points at a distance from the source (for a quadrupole source and zero field lines, for an octupole source). The results of modeling and their verification by full-scale measurements for the most common four-pole machines (quadrupole source) are presented. The main task of modeling the propagation the magnetic field of such sources is to ensure the required accuracy based on the goals of modeling. It is shown that the modeling accuracy and the presence of zero field points are due to different field levels near the electrical machine housing for different harmonics. The dipole harmonic at the cabinet is 20% of its own harmonic. But it falls more slowly with distance. This necessitates taking into account a different number of harmonics depending on the value of the ratio R0/R, R is the distance to the point of determining the field strength from the source. Therefore, with the ratio R0/R=2/3, the eighth harmonic is essential. At R0/R=1/5, already the fourth spatial harmonic can be neglected. Such data allow you to choose a rational number of harmonics. This reduces the amount of calculations and simplifies the process of modeling the propagation of the magnetic field around the source.

The holomorphic sectional curvature and “convex” real hypersurfaces in Kähler manifolds

We prove a sharp lower bound for the Tanaka-Webster holomorphic sectional curvature of strictly pseudoconvex real hypersurfaces that are "semi-isometrically" immersed in a K\"ahler manifold of nonnegative holomorphic sectional curvature under an appropriate convexity condition. This gives a partial answer to a question posed by Chanillo, Chiu, and Yang regarding the positivity of the Tanaka-Webster scalar curvature of the boundary of a strictly convex domain in $\mathbb{C}^2$ from 2012. In fact, the main result proves a stronger positivity property, namely the $\frac12$-positivity in the sense of Cao, Chang, and Chen, for compact "convex" real hypersurfaces in a K\"ahler manifold of nonnegative holomorphic sectional curvature. Our approach is rather simple and uses a version of the Gauss equation for semi-isometric CR immersions of pseudohermitian manifolds into K\"ahler manifolds.

On steady motions of an ideal fibre-reinforced fluid in a curved stratum. Geometry and integrability

It is shown that the kinematic equations governing steady motions of an ideal fibre-reinforced fluid in a curved stratum may be expressed entirely in terms of the intrinsic Gauss equation, which assumes the form of a partial differential equation of third order, for the surface representing the stratum. In particular, the approach adopted here leads to natural non-classical orthogonal coordinate systems on surfaces of constant Gaussian curvature with one family of coordinate lines representing the fibres. Integrable cases are isolated by requiring that the Gauss equation be compatible with another third-order hyperbolic differential equation. In particular, a variant of the integrable Tzitzéica equation is derived which encodes orthogonal coordinate systems on pseudospherical surfaces. This third-order equation is related to the Tzitzéica equation by an analogue of the Miura transformation for the (modified) Korteweg–de Vries equation. Finally, the formalism developed in this paper is illustrated by focussing on the simplest ‘fluid sheets’ of constant Gaussian curvature, namely the plane, sphere and pseudosphere.

Geometric Mechanics on Warped Product Semi-Slant Submanifold of Generalized Complex Space Forms

In this study, we develop a general inequality for warped product semi-slant submanifolds of type M n = N T n 1 × f N ϑ n 2 in a nearly Kaehler manifold and generalized complex space forms using the Gauss equation instead of the Codazzi equation. There are several applications that can be developed from this. It is also described how to classify warped product semi-slant submanifolds that satisfy the equality cases of inequalities (determined using boundary conditions). Several results for connected, compact warped product semi-slant submanifolds of nearly Kaehler manifolds are obtained, and they are derived in the context of the Hamiltonian, Dirichlet energy function, gradient Ricci curvature, and nonzero eigenvalue of the Laplacian of the warping functions.

Theory of perturbation of electric potential by a 3D object made of an anisotropic dielectric material

The extended boundary condition method (EBCM) was formulated for the perturbation of a source electric potential by a 3D object composed of a homogeneous anisotropic dielectric medium whose relative permittivity dyadic is positive definite. The formulation required the application of Green’s second identity to the exterior region to deduce the electrostatic counterpart of the Ewald–Oseen extinction theorem. The electric potential inside the object was represented using a basis obtained by implementing an affine bijective transformation of space to the Gauss equation for the electric field. The EBCM yields a transition matrix that depends on the geometry and the composition of the 3D object, but not on the source potential.

Research on Deep-Site Failure Mechanism of High-Steep Slope under Active Fault Creeping Dislocation

The reverse thrust in the deep site causes the upward propagation of stress and displacement in the overlying soil. The displacement field around the fault zone is maximum. As the spatial location becomes shallower, the soil displacement gradually becomes smaller. The deformation of the overlying soil is mainly affected by the vertical dislocation of the fracture zone. The monitoring curve showed no abrupt change value, indicating that the top surface of soil did not rupture, and only the influence of fault on the displacement transfer of the top surface of the soil. When a creeping dislocation occurs in the bottom fracture zone, the maximum principal stress of the upper boundary of the deep site is dominated by compressive stress. The maximum principal stress of the soil on both sides of the fracture zone has a maximum value, and the soil on the right side of the fracture zone has a significant compression effect. The maximum principal stress monitoring curve varies greatly, indicating the plastic failure development of soil, which is the same as the research results of the plastic failure zone in the following paper. When the bottom fracture zone starts to move, the plastic zone first appears at the junction area between the front end of the bottom fracture zone and the overlying soil. As the amount of dislocation of the fracture zone increases, the plastic zone continues to extend into the inner soil. The left and right sides of the fracture zone show tensile failure and compression failure, respectively. The development of the upper envelope curve in the plastic zone of the overlying soil satisfies the Boltzmann equation with a first-order exponential growth, while the development of the lower envelope curve satisfies the Gauss equation with a second-order exponential growth. The development curve equation of the plastic zone is verified according to the residual figures of the fitting result and the correlation parameters.

A General Inequality for CR-Warped Products in Generalized Sasakian Space Form and Its Applications

In the present paper, by considering the Gauss equation in place of the Codazzi equation, we derive new optimal inequality for the second fundamental form of CR-warped product submanifolds into a generalized Sasakian space form. Moreover, the inequality generalizes some inequalities for various ambient space forms.

Analytic continuation of Lauricella's function F D (N) for variables close to unit near hyperplanes {z j = z l }

For the Lauricella hypergeometric function with an arbitrary number of variables , we construct formulas for analytic continuation into the vicinity of hyperplanes and their intersections providing that all variables are close to unit. Such formulas represent the function near the point as linear combinations of N–multiple hypergeometric series that are solutions of the same system of partial differential equations as . Such series are the N–dimensional analog of the Kummer solutions known for the Gauss equation. The constructed analytical continuation formulas allow one to effectively calculate the function outside the unit polydisk.

Analytic continuation of Lauricella's function F D (N) for large in modulo variables near hyperplanes {z j = z l }

We consider the Lauricella hypergeometric function , depending on variables , and obtain formulas for its analytic continuation into the vicinity of a singular set which is an intersection of the hyperplanes . It is assumed that all N variables are large in modulo. This formulas represent the function outside of the unit polydisk in the form of linear combinations of other N-multiple hypergeometric series that are solutions of the same system of partial differential equations as . The derived hypergeometric series are N-dimensional analogues of the Kummer solutions that are well known in the theory of the classical hypergeometric Gauss equation.

On geometry of warped product pseudo-slant submanifolds on generalized Sasakian space forms

In the present paper, we consider warped product pseudo-slant submanifolds of generalized Sasakian space forms. We obtain an interesting inequality in terms of the second fundamental forms and the scalar curvature using Gauss equation. Next we compute to characterize the relation between the warping function and the squared norm of mean curvature of warped product pseudo-slant submanifolds of generalized Sasakian space forms under some certain conditions. Finally, we derive warped product pseudo-slant submanifolds in generalized Sasakian space forms to become Riemannian product with various mathematical and physical expressions.

Bifurcation for minimal surface equation in hyperbolic 3-manifolds

Initiated by the work of Uhlenbeck in late 1970s, we study existence, multiplicity and asymptotic behavior for minimal immersions of a closed surface in some hyperbolic three-manifold, with prescribed conformal structure on the surface and second fundamental form of the immersion. We prove several results in these directions, by analyzing the Gauss equation governing the immersion. We determine when existence holds, and obtain unique stable solutions for area minimizing immersions. Furthermore, we find exactly when other (unstable) solutions exist and study how they blow-up. We prove our class of unstable solutions exhibit different blow-up behaviors when the surface is of genus two or greater. We establish similar results for the blow-up behavior of any general family of unstable solutions. This information allows us to consider similar minimal immersion problems when the total extrinsic curvature is also prescribed.

Effects of mechanical fields on mobile charges in a composite beam of flexoelectric dielectrics and semiconductors

We study the interaction between mechanical fields and the motion of charge carriers in a composite beam of flexoelectric (nonpiezoelectric) and semiconductor layers. The macroscopic theory of flexoelectricity and the drift-diffusion theory of semiconduction are used. They are coupled by doping and mobile charges in Gauss's equation of electrostatics. A one-dimensional electromechanical model is established for the bending of the beam. Solutions from the model show that bending of the beam induces redistribution of charge carriers through the electric field produced by flexoelectric coupling. This provides a new means for constructing electromechanical semiconductor devices and extends piezotronic devices.

De Sitter cosmology on an expanding bubble

Constructing an explicit compactification yielding a metastable de Sitter (dS) vacuum in a UV consistent string theory is an incredibly difficult open problem. Motivated by this issue, as well as the conjecture that all non-supersymmetric AdS vacua must decay, we discuss the alternative possibility of realizing an effective four-dimensional dS cosmology on a codimension-one bubble wall separating two AdS5 vacua. The construction further elaborates on the scenario of arXiv:1807.01570, where the aforementioned cosmology arises due to a non-perturbative decay and is embedded in a five-dimensional bulk in a time­ dependent way. In this paper we discuss the relation between this scenario and the weak gravity conjecture and further develop the details of the four-dimensional cosmology. We provide a bulk interpretation for the dS temperature as the Unruh temperature experienced by an accelerated observer riding the bubble. A source of four-dimensional matter arises from a string cloud in the bulk, and we examine the consequences for the particle mass spectrum. Furthermore, we show how effective four-dimensional Einstein gravity on the bubble is obtained from the five-dimensional Gauss equation. We conclude by outlining some implications that this paradigm will have for holography, inflation, the standard model, and black holes.

On local isometric embeddings of three-dimensional Lie groups

Due to Janet–Cartan’s theorem, any analytic Riemannian manifolds can be locally isometrically embedded into a sufficiently high dimensional Euclidean space. However, for an individual Riemannian manifold (M, g), it is in general hard to determine the least dimensional Euclidean space into which (M, g) can be locally isometrically embedded, even in the case where (M, g) is homogeneous. In this paper, when the space (M, g) is locally isometric to a three-dimensional Lie group equipped with a left-invariant Riemannian metric, we classify all such spaces that can be locally isometrically embedded into the four-dimensional Euclidean space. Two types of algebraic equations, the Gauss equation and the derived Gauss equation, play an essential role in this classification.

Relationship between air temperature and horizontal sand-dust flux observed in the Taklimakan Desert, China

Over the hinterland of the Taklimakan Desert (TD), the diurnal and seasonal cycles of the air temperature (AT) and horizontal sand-dust flux (HSDF) were investigated, based on in situ data including sand saltation counts and meteorological parameters in the period 2008–2010. In particular, the relationship between AT and HSDF was determined. It was found that 74.7% of HSDF was observed between 10:00 and 20:00, and HSDF in spring and summer accounted for 92.3% of the annual flux. Further, the diurnal and annual cycles of HSDF and AT presented an exponential distribution, with correlation coefficients (R) of 0.97 and 0.93, respectively. Additionally, when AT was 0 °C or lower, the frequency of sand saltation was only 4.3%. The maximum sand saltation frequency was 70.1%, which occurred at 33–34 °C, while 67.6% of the total HSDF occurred at 24–34 °C. Finally, a Gauss equation was established over the TD: Q = 250.2 + 1368.5e−2 × [(T−30.8)/7.3]^2, with R = 0.90. Results revealed a strong correlation between HSDF and AT, indicating that AT should be noticeable in HSDF parameterization schemes.

Rigidity of holomorphic curves in a hyperquadric Q4

In this paper, firstly, we obtain the Gauss equation and Codazzi equations of a holomorphic curve in a hyperquadric Qn, and we also compute the Laplace of the square of the length of the second fundamental form. Secondly, we prove that any two linearly full holomorphic curves in Q4 are congruent if their first and second fundamental forms are the same. Finally, we determine a one-parameter family of homogeneous holomorphic curves in Q4 with constant curvature 2, but their second fundamental forms are different.

Warped product semi-slant submanifolds in locally conformal Kaehler manifolds II

In 1994 N.~Papaghiuc introduced the notion of semi-slant submanifold in a Hermitian manifold which is a generalization of $CR$- and slant-submanifolds, \cite{MR0353212}, \cite{MR760392}.
 In particular, he considered this submanifold in Kaehlerian manifolds, \cite{MR1328947}.
 Then, in 2007, V.~A.~Khan and M.~A.~Khan considered this submanifold in a nearly Kaehler manifold and obtained interesting results, \cite{MR2364904}.
 Recently, we considered semi-slant submanifolds in a locally conformal Kaehler manifold and we gave a necessary and sufficient conditions of the two distributions (holomorphic and slant) be integrable.
 Moreover, we considered these submanifolds in a locally conformal Kaehler space form.
 In the last paper, we defined $2$-kind warped product semi-slant submanifolds in almost hermitian manifolds and studied the first kind submanifold in a locally conformal Kaehler manifold.
 Using Gauss equation, we derived some properties of this submanifold in an locally conformal Kaehler space form, \cite{MR2077697}, \cite{MR3728534}.
 In this paper, we consider same submanifold with the parallel second fundamental form in a locally conformal Kaehler space form.
 Using Codazzi equation, we partially determine the tensor field $P$ which defined in~\eqref{1.3}, see Theorem~\ref{th4.1}.
 Finally, we show that, in the first type warped product semi-slant submanifold in a locally conformal space form, if it is normally flat, then the shape operators $A$
 satisfy some special equations, see Theorem~\ref{th5.2}.