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Let L be a nonnegative integer and ΓL the power set of the set {0, 1,…,L}. Then there is an adjacency relation in ΓL such that ΓL together with the relation forms a regular graph. In this paper, we propose a model of continuous-time magnetic quantum walk (MQW) on the graph ΓL, and investigate its properties from a viewpoint of probability and quantum information. We first introduce a magnetic Laplacian ΔL(휃) on the graph ΓL and examine its spectrum. And then, with ΔL(휃) as the Hamiltonian, we construct our model of continuous-time MQW on the graph ΓL. We find that the model has probability distributions that are completely independent of the magnetic potential at all times. And we show that it has perfect state transfer at time t = π 2 when the magnetic potential satisfies some mild conditions. Some other interesting results are also obtained. [ABSTRACT FROM AUTHOR]

We consider the following class of elliptic problems \[ - \Delta_A u + u = a_{\lambda}(x) |u|^{q-2}u+b_{\mu}(x) |u|^{p-2}u ,\quad x\in {\mathbb{R}}^N, \] − Δ A u + u = a λ (x) | u | q − 2 u + b μ (x) | u | p − 2 u , x ∈ R N , where $ 1 1 < q < 2 < p < 2 ∗ = 2 N N − 2 $ N\geq ~3 $ N ≥ 3 , $ a_{\lambda }(x) $ a λ (x) is a sign-changing weight function, $ b_{\mu }(x) $ b μ (x) is continuous, $ \lambda > 0 $ λ > 0 and $ \mu > 0 $ μ > 0 are real parameters, $ u \in H^1_A({\mathbb {R}}^N) $ u ∈ H A 1 ( R N) and $ A:{\mathbb {R}}^N \rightarrow {\mathbb {R}}^N $ A : R N → R N is a magnetic potential. Exploring the relationship between the Nehari manifold and fibering maps, we will discuss the existence, multiplicity and regularity of solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of multiple solutions of the following critical nonlocal elliptic equations with magnetic field:$\begin{align} \left\{\begin{aligned} (-i\nabla-A(x))^2u& = \lambda |u|^{p-2}u+\left(\int_{\Omega}\frac{|u(y)|^{2^*_\alpha}}{|x-y|^{\alpha}}dy\right)|u|^{2^*_\alpha-2}u\quad {\rm in}\quad \Omega,\\ &u = 0\quad \partial\Omega,\\ \end{aligned}\right. \end{align}\;\;\;\;(1)$where $ i $ is imaginary unit, $ N\geq4 $, $ 2^*_\alpha = \frac{2N-\alpha}{N-2} $ with $ 0<\alpha<4 $, $ \lambda>0 $ and $ 2\leq p<2^* = \frac{2N}{N-2} $. Suppose the magnetic vector potential $ A(x) = (A_1(x), A_2(x),..., A_N(x)) $ is real and local Hölder continuous, We show by the Ljusternik-Schnirelman theory that (1) has at least $ cat_\Omega(\Omega) $ nontrivial solutions for $ \lambda $ small. [ABSTRACT FROM AUTHOR]

We consider the nonlinear Schrödinger equation with magnetic field and the Neumann boundary condition: { − ∇ A 2 u + λ u = | u | p − 2 u in Ω , ν ⋅ ∇ A u = 0 on ∂ Ω , where Ω is a boundary domain in R n with a C 1 boundary, ν is the outward normal vector field at x ∈ ∂ Ω , n ≥ 3 , λ > − μ (A) , ( μ (A) is given by (3)), A ∈ C ∞ (Ω ¯ , R n) is a magnetic vector potential. When the exponent is subcritical, 2 < p < 2 ∗ = 2 n n − 2 we can obtain solutions by Nehari manifold. When the exponent is critical, p = 2 ∗ , we can obtain solutions by constrained minimization arguments. [ABSTRACT FROM AUTHOR]

We consider the elliptic problem −∆Au + u = aλ(x)|u|q−2u + bµ(x)|u|p−2u, for x ∈ RN, 1 < q < 2 < p < 2∗ = 2N/(N − 2), aλ(x) is a sign-changing weight function, bµ(x) satisfies some additional conditions, u ∈ H¹A(RN) and A : RN → RN is a magnetic potential. Exploring the Bahri-Li argument and some preliminary results we will discuss the existence of a four nontrivial solutions to the problem in question. [ABSTRACT FROM AUTHOR]

We derive conditional stability estimates for inverse scattering problems related to the time harmonic magnetic Schrödinger equation. We prove logarithmic-type estimates for retrieving the magnetic (up to a gradient) and electric potentials from near field or far field maps. Our approach combines techniques from similar results obtained in the literature for inhomogeneous inverse scattering problems based on the use of geometrical optics solutions. [ABSTRACT FROM AUTHOR]

The permanent magnet brushless DC motor (BLDCM) is typically controlled using the six-step commutation method, and the flux-weakening method is employed to enable the motor to operate at speeds higher than the base speed. Currently, it is considered that the weak magnetic angle range is 0-pi/3, while the range for deep weakening is pi/3-pi/2. In field-weakening control, a forward shift of the commutation point results in a circulating current flowing in the three-phase bridge of the inverter and the stator winding of the motor. This paper analyses the principle of the circulating current formed by the inverter. Through magnetic potential analysis and Simulink simulation, it is concluded that flux-weakening control generates a circulating current in the inverter and motor stator windings. The inverter's circulating current affects the motor's magnetic potential, causing it to shift towards the rotating direction of the motor rotor. When the forward shift angle of the inverter commutation point is within the range of 0-pi/6 electrical angle, the phase shift of the inverter circulating current remains below pi/6. This configuration weakens the magnetic field and provides the driving effect. However, when the forward shift angle falls within the range of pi/6-pi/3, the phase shift of the inverter circulating current exceeds pi/6, resulting in magnetic weakening and braking. During the braking effect, a reverse torque is generated, leading to a decrease in motor torque and efficiency. Therefore, the range of the weak magnetic angle should be between 0-pi/6. [ABSTRACT FROM AUTHOR]

This work presents an efficient method based on asymptotic models in order to solve numerically eddy current problems in a bi‐dimensional setting with a high contrast of linear relative magnetic permeability μr ≫ 1 between a conductor and a dielectric subdomain. We describe a magnetic skin effect by deriving a multiscale expansion for the magnetic potential in power series of a small parameter δ which represents the skin depth. We make explicit the first asymptotics up to the order three. Some numerical results based on the finite element method will be presented to illustrate the magnetic skin effect and to validate the performance of the proposed asymptotic models in the dielectric medium. We confirm that the proposed asymptotics provide reduced computational costs for a wide range of physical parameters introduced in our problem. [ABSTRACT FROM AUTHOR]

As the main contribution, a nonlocal strain gradient theory (NSGT) is developed in conjugation with the generalized differential quadrature method (GDQM) to analyze the free vibration of a transversely graded nanoshell subjected to magnetic and thermal loads derived by using the first-order shear deformation theory (FSDT) and Hamilton's principle. Comparisons with other common methods have shown the validity of the presented NSGT-GDQM hybrid approach. The effects of power-law exponent, nonlocal parameter, strain gradient parameter, magnetic potential, angular velocity, and temperature for both simply supported and clamped boundary conditions of a nanoshell composed of BaTiO3 and CoFe2O4 materials were investigated. Sensitivity analysis illustrates that more resolution will be available with clamped boundaries. With increasing the temperature gradient, the absolute value of the sensitivity increases and the maximum sensitivity value occurs always on the critical buckling point. Finally, comparing two studied boundary conditions revealed that if there is any temperature rise limit, the clamped boundary condition is suggested because of more sensitivity. Though the temperature gradient could be more than 400 K in the case of simply supported edges, the maximum absolute sensitivity could be obtained is about 0.12 wherein the case of clamped boundary condition it was about 0.17. [ABSTRACT FROM AUTHOR]

A new nanocomposite piezoelectromagnetic plate model is developed for studying free vibration based on a refined shear deformation theory (RDPT). The present model is composed of piezoelectromagnetic material reinforced with functionally graded graphene platelets (FG-GPLs). The nanocomposite panel rests on Winkler–Pasternak foundation and is subjected to external electric and magnetic potentials. It is assumed that the electric and magnetic properties of the GPLs are proportional to those of the electromagnetic materials. The effective material properties of the plate are estimated based on the modified Halpin–Tsai model. A refined graded rule is introduced to govern the variation in the volume fraction of graphene through the thickness of the plate. The basic partial differential equations are provided based on Hamilton's principle and then solved analytically to obtain the eigenfrequency for different boundary conditions. To check the accuracy of the present formulations, the depicted results are compared with the published ones. Moreover, impacts of the variation in elastic foundation stiffness, plate geometry, electric potential, magnetic potential, boundary conditions and GPLs weight fraction on the vibration of the smart plate are detailed and discussed. [ABSTRACT FROM AUTHOR]

This work is concerned with the time-harmonic eddy current problem in a bi-dimensional setting with a high contrast of magnetic permeabilities between a conducting medium and a dielectric medium. We describe a magnetic skin effect by deriving rigorously a multiscale expansion for the magnetic potential in power series of a small parameter ε which represents the inverse of the square root of a relative permeability. We make explicit the first asymptotics up to the order ε 3 . As an application, we obtain impedance conditions up to the fourth order of approximation for the magnetic potential. Finally, we measure this skin effect with a characteristic length that depends on the scalar curvature of the boundary of the conductor. [ABSTRACT FROM AUTHOR]

This study is about thermal enhancement in hybrid nano-ferrofluid. The conservation equations with thermo-correlations are solved numerically and computed solutions are used for parametric study related to flow of fluid and transfer of heat energy. The convergent solutions are derived via the finite element method (FEM). A mesh-free study is performed. Flow experiences a sufficient amount of resistive force by the porous medium. It is noted that Darcy porous is less resistive than Forchheirmer porous medium. It is also noted that convective heat transfer is compromised when the Forchheirmer parameter is increased. Dissipation effects are responsible for an increase in temperature and hence, an increase in thermal boundary layer thickness is noted. It is also observed that heat dissipation in a hybrid nanofluid is stronger than that in a nanofluid. The numerical values read the wall shear stress exerted by hybrid nanofluid is greater than wall shear stress by nano-ferrofluid. The wall shear stress increases as a function of the ferro-hydrodynamic parameter. However, the wall heat transfer rate (Nusselt number) decreases as the ferro-hydrodynamic parameter is increased. Similarly, wall shear stress increases versus Curie temperature number whereas Nusselt number decreases when the porosity parameter is increased. The porous medium is responsible for more wall shear stress on the surface. The transfer of heat in the presence of a porous medium in a fluid is greater than the rate of heat transfer in fluid in the absence of a porous medium. Viscous dissipation is responsible for the increase of the rate of heat transfer.

Micromagnetic tomography (MMT) is an emerging technique in rock and paleomagnetism to determine individual magnetic moments of tomographically defined magnetic source regions within a natural sample by means of surface scans of the magnetic field above the sample. MMT relies on combining large high-resolution data sets from X-ray tomography and magnetic scanning devices, like quantum diamond magnetometers, together with advanced inversion algorithms potentially capable to solve for millions of individual magnetic moment vectors. We here provide an overview of existing algorithms that have been developed to tackle different aspects of MMT-related problems and discuss recent advances and future challenges of MMT. • Micromagnetic Tomography MMT computes paleomagnetic data from rock magnetic grains. • Forward modeling and numerical inversions of MMT are coded in open Python libraries. • MMT can infer the internal state from single grains using micromagnetic modeling. • MMT codes are optimized for high performance computing using both CPUs and GPUs. • Proposed benchmark problems using experimental data support MMT results. [ABSTRACT FROM AUTHOR]

A new nanocomposite piezoelectromagnetic plate model is developed for studying free vibration based on a refined shear deformation theory (RDPT). The present model is composed of piezoelectromagnetic material reinforced with functionally graded graphene platelets (FG-GPLs). The nanocomposite panel rests on Winkler–Pasternak foundation and is subjected to external electric and magnetic potentials. It is assumed that the electric and magnetic properties of the GPLs are proportional to those of the electromagnetic materials. The effective material properties of the plate are estimated based on the modified Halpin–Tsai model. A refined graded rule is introduced to govern the variation in the volume fraction of graphene through the thickness of the plate. The basic partial differential equations are provided based on Hamilton’s principle and then solved analytically to obtain the eigenfrequency for different boundary conditions. To check the accuracy of the present formulations, the depicted results are compared with the published ones. Moreover, impacts of the variation in elastic foundation stiffness, plate geometry, electric potential, magnetic potential, boundary conditions and GPLs weight fraction on the vibration of the smart plate are detailed and discussed.

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An eccentric position diagnosis method of static eccentricity (SE) fault of external rotor permanent magnet synchronous motor (ER‐PMSM) is presented. Firstly, an analytical model of no‐load radial magnetic field of ER‐PMSM is established. Analytical models of no‐load Back‐EMF of both unit motors and the whole motor are carried out and are verified by finite element method (FEM) and experimental measurements. Then, the influences of SE ratio, SE circumferential angle, winding distribution mode and number of parallel branches on no‐load radial magnetic field and no‐load Back‐EMF are analyzed based on these analytical models. The results show that SE does not affect the frequency characteristics of no‐load radial magnetic field, but changes space order characteristics. On one hand, for ER‐PMSM, of which the number of unit motors is equal to 1, SE causes no‐load Back‐EMF distortion. On the other hand, for ER‐PMSM, of which the number of unit motors is greater than 1, SE does not affect no‐load Back‐EMF of the whole motor, but it still leads to no‐load Back‐EMF distortion of unit motors. Therefore, based on total harmonic distortion (THD) of no‐load Back‐EMF of unit motor, a projection method of intersection lines for SE fault diagnosis of ER‐PMSM is proposed finally. [ABSTRACT FROM AUTHOR]

In Magnetic Resonance Imaging there are several situations where, for simulation purposes, one wants to compute the magnetic field induced by a cluster of small metallic particles. Given the difficulty of the problem from a numerical point of view, the simplifying assumption that the field due to each particle interacts only with the main magnetic field but does not interact with the fields due to the other particles is usually made. In this paper we investigate from a mathematical point of view the relevancy of this assumption and provide error estimates for the scalar magnetic potential in terms of the key parameter that is the minimal distance between the particles. A special attention is paid to obtain explicit and relevant constants in the estimates. When the "non-interacting assumption" is deficient, we propose to compute a better approximation of the magnetic potential by taking into account pairwise magnetic field interactions between particles that enters in a general framework for computing the scalar magnetic potential as a series expansion. [ABSTRACT FROM AUTHOR]

A high-voltage AC double-circuit 400 kV overhead power transmission line runs from the city of Elk (Poland) to the city of Alytus (Lithuania). This international 400 kV power transmission line is potentially one of the strongest magnetic field-generating sources in the area. This 400 kV voltage double-circuit overhead transmission line and its surroundings were analyzed using the mathematical analytical methods of superposition and reflections. This research paper includes the calculation of the numerical values of the magnetic field and its distribution. The research showed that the values of the magnetic field strength near the international 400 kV power transmission line exceed the threshold values permitted by relevant standards. This overhead power line is connected to the general (50 Hz) power system and generates a highly intense magnetic field. It is suggested that experimental trials should be undertaken in order to determine the maximum values of the magnetic field strength. For the purpose of mitigating these values, it is suggested that the height of the support bars should be increased or that any individual and commercial activities near the object under investigation should be restricted. [ABSTRACT FROM AUTHOR]

Given a smooth closed oriented manifold M of dimension n embedded in R n + 2 , we study properties of the 'solid angle' function Φ : R n + 2 \ M → S 1 . It turns out that a non-critical level set of Φ is an explicit Seifert hypersurface for M. This gives an explicit analytic construction of a Seifert surface in higher dimensions. [ABSTRACT FROM AUTHOR]

We propose a simple situation in which the magnetic Aharonov–Bohm potential influences the values of the deficiency indices of the initial Schrödinger operator, so determining whether the particle interacts with the solenoid or not. Even with the particle excluded from the magnetic field, the number of self-adjoint extensions of the initial Hamiltonian depends on the magnetic flux. This is a new point of view of the Aharonov–Bohm effect. [ABSTRACT FROM AUTHOR]

In this article, we prove that the function Ψ(t, x) belongs to the domain of the definition of the operator H ̶̶̶̶̶̶̶̶. We discuss that the operator H and its powers is essentially self-adjoint in a region. Accordingly, we investigate that the operator H and its powers is essentially self-adjoint. Finally, we discuss the uniqueness solution of the. [ABSTRACT FROM AUTHOR]

One-dimensional Schrödinger operators with singular perturbed magnetic and electric potentials are considered. We study the strong resolvent convergence of two families of the operators with potentials shrinking to a point. Localized δ-like magnetic fields are combined with δ′-like perturbations of the electric potentials as well as localized rank-two perturbations. The limit results obtained heavily depend on zero-energy resonances of the electric potentials. In particular, the approximation for a wide class of point interactions in one dimension is obtained.

We study the problem of unique recovery of a nonsmooth one-form\scrA and a scalar function q from the Dirichlet to Neumann map,\Lambda\scrA,q, of a hyperbolic equation on a Riemannian manifold (M, g). We prove uniqueness of the one-form\scrA up to the natural gauge, under weak regularity conditions on\scrA, q and under the assumption that (M, g) is simple. Under an additional regularity assumption, we also derive uniqueness of the scalar function q. The proof is based on the geometric optic construction and inversion of the light ray transform extended as a Fourier integral operator to nonsmooth parameters and functions. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the Feynman-Kac Itô formula of the Schrödinger operator with electromagnetic ѱ(t; x) in equation (1) in [8] which defined as ... is differentiable of the variable t, and so establish that the infinitely differentiable in a region, therefore, investigate smoothness of this function. [ABSTRACT FROM AUTHOR]

Abstract We prove the existence of solutions to the nonlinear Schrödinger equation ε 2 (i ∇ + A) 2 u + V (y) u − | u | p − 1 u = 0 in R 2 with a magnetic potential A = (A 1 , A 2). Here V represents the electric potential, the index p is greater than 1. Along some sequence { ε n } tending to zero we exhibit complex-value solutions that concentrate along some closed curves. [ABSTRACT FROM AUTHOR]

In the paper, we consider different generalizations of the Dirichlet problem in an arbitrary domain of n-dimensional Euclidean space Rn for a magnetic Schrödinger operator. Their equivalence under different conditions on magnetic and electric potentials are proved. Magnetic Sobolev space of first order is introduced and it is proved that this space is topologically equivalent to the ordinary Sobolev space of first order. The interval of a real axis for a spectral parameter, where the Dirichlet first generalized problem has a unique solution, is shown. The Green operator for the Dirichlet first and second generalized problems is introduced and its boundedness from the first order conjugated Sobolev space to the first order Sobolev space is proved. [ABSTRACT FROM AUTHOR]

The paper presents a new method to solve the electromagnetic field in 2-D cylindrical geometries. The method is based on the analytical solution of the magnetic potential. The method can be used to compute inductor or transformer winding resistance or leakage inductance with high accuracy up to several megahertz. The proposed method assumes no simplification of the problem, and then achieves higher accuracy than the classical method such as homogenization techniques. The method requires no meshing unlike finite-element analysis (FEA) resulting in shorter computation time. The method is compared with FEA on the bifilar line. The method overperform existing methods in terms of accuracy and computation time. [ABSTRACT FROM AUTHOR]

In this article, we consider the existence of infinitely many nontrivial solutions for the following semilinear Schr\"odinger equation with electromagnetic potential $$ \big(-i\nabla+A(x)\big)^2u+V(x)u=f(x,|u|)u,\quad\text{in } \mathbb{R}^N $$ where i is the imaginary unit, V is the scalar (or electric) potential, A is the vector (or magnetic) potential. We establish the existence of infinitely many solutions via variational methods.

A high-voltage AC double-circuit 400 kV overhead power transmission line runs from the city of Elk (Poland) to the city of Alytus (Lithuania). This international 400 kV power transmission line is potentially one of the strongest magnetic field-generating sources in the area. This 400 kV voltage double-circuit overhead transmission line and its surroundings were analyzed using the mathematical analytical methods of superposition and reflections. This research paper includes the calculation of the numerical values of the magnetic field and its distribution. The research showed that the values of the magnetic field strength near the international 400 kV power transmission line exceed the threshold values permitted by relevant standards. This overhead power line is connected to the general (50 Hz) power system and generates a highly intense magnetic field. It is suggested that experimental trials should be undertaken in order to determine the maximum values of the magnetic field strength. For the purpose of mitigating these values, it is suggested that the height of the support bars should be increased or that any individual and commercial activities near the object under investigation should be restricted.