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We consider positive classical solutions of $$ \left\{ \begin{array}{rcl} -{\Delta} u + \lambda a(x) \cdot \nabla u &=& u^{p} \qquad\text{in} {\Omega}, \\ u&=& 0 \qquad \quad \text{on} {\partial {\Omega}}, \end{array}\right. $$ where p > 1, a is a smooth divergence free vector field and λ > 0 is a large parameter. Under certain assumptions on a(x) and (or) assumptions on the existence of first integrals of a(x) we show there is a subsequence of smooth positive solutions which converge to a nonzero first integral of a(x) as $ \lambda \rightarrow \infty $ .

We show that graph map with zero topological entropy is Li-Yorke chaotic if and only if it has an NS-pair (a pair of non-separable points containing in a same solenoidal $\omega$-limit set), and a non-diagonal pair is an NS-pair if and only if it is an IN-pair if and only if it is an IT-pair. This completes characterization of zero topological sequence entropy for graph maps., Comment: 14 pages. Published online in Proc. Amer. Math. Soc

This article proposes a novel analysis method for the losses in a litz wire considering the circulating and eddy currents. In the proposed method, the macroscopic complex permeability is introduced to evaluate the eddy current loss owing to the proximity effect. Homogenization-based finite element analysis using the macroscopic complex permeability can effectively compute the proximity effect loss. The loss owing to the circulating currents is evaluated by solving the corresponding circuit equation. It is shown that the eddy current loss is dominant for the solenoidal and spiral coils without magnetic cores because of the uniform magnetic field distribution along the wire. Moreover, the magnetic reactors in which the circulating current loss cannot be ignored are analyzed using the proposed method. It is shown that the losses computed by the proposed method agree well with the measured results.

In the current energy scenario, due to the increment in power generation from renewable sources, the importance of electrical storage systems has increased significantly, and as a consequence, the study of the improvement of its efficiency and the design of new storage systems also increased. Superconducting material permits the design of Superconducting Magnetic Energy Storage (SMES). The main problem of SMEs is the low energy density they have, what make the optimization of design to be one of the keys for inclusion of this elements in the power grid and other specific applications as, for instance, flux pumps. As the only basic forms for SMES, and with the objective of its mathematical optimization, this work (i) evaluates the mathematical equations for a real solenoidal winding, and (ii) develops the equivalent equations for a toroidal winding, from the electromagnetic laws. Then, (iii) a practical structure formed by short solenoids connected in series along a circular axis (quasi-toroidal structure) is studied. Due to the large number of equations involved in this case, the finite-element method in used here. Finally (iv), in order to validate the results without building the complete solenoid (impossible at the time), one of the magnetic coupling between two solenoids in the quasi-toroidal winding was developed according with the theoretical method, and experimentally tested. The study was carried out by programming different dimensions in order to make conclusions for a further development of an optimization algorithm. These conclusions are presented. This work is the first stage for the optimized design of a SMES, and presents the complete equations of the real toroidal winding as the base of the outline dimensions of a practical quasi-toroidal SMES.

A two-mode cylindrical magnetic field is considered, the potential of which has a minimum. The object of this work is the study of the parameters of an electron beam when it moves in a solenoid field with the longitudinal trap formed by the magnetic field, and the construction of the computational model of the motion of an electron beam. The problem is posed of the stability of the motion of electrons in such solenoid magnetic field. The possibility of obtaining oscillatory modes of particle motion has been studied. It was found that for oscillations of particles with an energy of tens of kiloelectronvolts in the potential well in a well, the field with the amplitude of tens of thousands of Oersteds is required. For the solenoid magnetic field of the solenoid, the formation of electron beam with an energy of 55 keV in the longitudinal and radial directions during its transportation is studied. A section of a magnetron gun was used as the physical object. One possible direction is to combine the two matched magnetic systems of the gun to create the potential magnetic field well. It is shown that, for the chosen conditions, the motion of electrons can be associated with the model of three-dimensional oscillations. In this work, on the basis of the Hamiltonian formalism of the motion of electrons in a magnetic field and an algorithm for numerically finding solutions to the differential equations of dynamics, a software tool is constructed that allows one to obtain arrays of values of particle trajectories in the volume. The use of the software made it possible to simulate the main dependences of the motion of the electron beam in a given two-mode solenoid magnetic field. The results of numerical simulation of electron trajectories in the gradient magnetic field with the point secondary emission cathode located in the middle of the system are presented. The formation of the beam with energy of 55 keV in the radial and longitudinal directions during its transportation in a solenoid magnetic field with a large gradient is considered. For significant time intervals, the possibility of three-dimensional oscillations is shown and the operating modes of the magnetic system are obtained, in which the particle undergoes stable three-dimensional oscillations. The influence of the initial conditions during emission on the occurrence of the reciprocating oscillatory effect has been studied. It is shown that for a given electron energy and fixed magnetic field, the parameter that determines the reflection of a particle, is the polar angle of entry relative to the axis of the cylindrical magnetic field. The dependence of the formation of the final distribution of particles on the amplitude and gradient of the magnetic field along the axis of the system is investigated. The results of numerical simulation on the motion of the electron flow are presented. The characteristics of the resulting electron beam are considered on the basis of a model of electron flow motion. The obtained simulation results show that it is possible to establish the phenomenon of oscillatory-return longitudinal motion under experimental conditions. Keywords: electron beam, magnetron gun, three-dimensional oscillations, electron dynamics, gradient magnetic field, mathematical modeling.

The vacuum oscillators generally utilize high current annular relativistic electron beams to generate high-power microwaves. The annular electron beams are usually guided and focused by strong axial magnetic fields. A permanent magnet is developed for the application of annular electron beams. The configuration of the permanent magnet is based on the Halbach structure. A new method is proposed to design this solenoidal permanent magnet. The permanent magnet flux source can generate a 0.88 T axial magnetic field at the interaction region and a peak field of 0.94 T. The diameter of its clear bore is 50 mm. The total weight of the permanent magnet is 320 kg. The design, fabrication, and test results are presented in this article.

We consider almost Riemann solitons $$(V,\lambda )$$ in a Riemannian manifold and underline their relation to almost Ricci solitons. When V is of gradient type, using Bochner formula, we explicitly express the function $$\lambda $$ by means of the gradient vector field V and illustrate the result with suitable examples. Moreover, we deduce some properties for the particular cases when the potential vector field of the soliton is solenoidal or torse-forming, with a special view towards curvature.

In this note, we obtain a simpler expression for the constant number given by Costin–Maz’ya on the sharp Hardy–Leray inequality for a class of solenoidal (namely divergence-free) vector fields, with respect to any radial power-weighted measure. The dependence of the constant on the weight exponent will be clear.

We give an example of a bounded divergence free autonomous vector field in $${\mathbb {R}}^3$$ (and of a nonautonomous bounded divergence free vector field in $${\mathbb {R}}^2$$ ) and of a smooth initial data for which the Cauchy problem for the corresponding transport equation has 2 distinct solutions. We then show that both solutions are limits of classical solutions of transport equations for appropriate smoothings of the vector fields and of the initial data.

In this paper we prove the existence and uniqueness of time global mild solutions to the Navier-Stokes-Oseen equations, which describes dynamics of incompressible viscous fluid flows passing a translating and rotating obstacle, in the solenoidal Lorentz space L σ, w 3 . Besides, boundedness and polynomial stability of these solutions are also shown.

Experiments on proton and sodium MRI at 0.5 T are described to assess the sensitivity of various types of coils, namely solenoidal, surface, volumetric, and with specific complex geometry. Each type of coil, characterized by its geometric specificity, was built to record MR signal of protons (1H) and sodium (23Na) at 21.1 MHz and 5.6 MHz, respectively. Sodium coils were initially built for 1H MRI and modified for 23Na MRI. Measurements in a container phantom filled with a 12% salt solution showed that the quality factor and sensitivity of various type coils approximately coincide at both sodium and proton resonance frequencies. The possibility of using data on SNR for a highly concentrated sample and Q-factor measurements to predict MRI results for a weakly concentrated sample is considered. These measurements represent a guide for extension to multinuclear whole body low field MRI clinical platform that is usually designed to operate in proton mode only.

We show that for any given solenoidal initial data in $L^2$ and any solenoidal external force in $L_{\text{loc}}^q \bigcap L^{3/2}$ with $q>3$, there exist partially regular weak solutions of the Navier-Stokes equations in $\R^4 \times [0,\infty[$ which satisfy certain local energy inequalities and whose singular sets have locally finite $2$-dimensional parabolic Hausdorff measure. With the help of a parabolic concentration-compactness theorem we are able to overcome the possible lack of compactness arising in the spatially $4$-dimensional setting by using defect measures, which we then incorporate into the partial regularity theory., Various improvements

We prove a new decomposition result for rank $m$ symmetric tensor fields which generalizes the well-known solenoidal and potential decomposition of tensor fields. This decomposition is then used to...

New formulas are derived for once-differentiable 3-dimensional fields, using the operator . This new operator has a property similar to that of the Laplacian operator; however, unlike the Laplacian operator, the new operator requires only once-differentiability. A simpler formula is derived for the classical Helmholtz decomposition. Orthogonality of the solenoidal and irrotational parts of a vector field, the uniqueness of the familiar inverse-square laws, and the existence of solution of a system of first-order PDEs in 3 dimensions are proved. New proofs are given for the Helmholtz Decomposition Theorem and the Divergence theorem. The proofs use the relations between the rectangular-Cartesian and spherical-polar coordinate systems. Finally, an application is made to the study of Maxwell’s equations.

Magnetic fields are a dynamically important agent for regulating structure formation in the interstellar medium. The study of the relative orientation between the local magnetic field and gas (column-) density gradient has become a powerful tool to analyse the magnetic field's impact on the dense gas formation in the Galaxy. In this study, we perform numerical simulations of a non-gravitating, isothermal gas, where the turbulence is driven either solenoidally or compressively. We find that only simulations with an initially strong magnetic field (plasma-$��, accepted for publication by MNRAS

Beam breakup (BBU) in linear induction accelerators (LIA) is suppressed by the solenoidal magnetic-field focusing of the electron beam. Amplification of BBU instability can also be significantly reduced by using cells having different transverse impedances. This so-called “stagger tuning” occurs naturally on one of the LIAs used for flash radiography at the Los Alamos National Laboratory. In this article, we use computer simulations to demonstrate how the natural stagger tuning functions, and how it might be used on advanced LIAs under consideration. Other means for engineering stagger tuning into LIAs are also investigated. Results of these simulations show promise for mitigation of BBU by reduction of the effective impedance, thereby relieving the necessity for extraordinarily strong solenoidal fields for designs with a large number of cells.

Fluid flows around an obstacle generate vortices which, in turn, generate forces on the obstacle. This phenomenon is studied for planar viscous flows governed by the stationary Navier–Stokes equations with inhomogeneous Dirichlet boundary data in a (virtual) square containing an obstacle. In a symmetric framework the appearance of forces is strictly related to the multiplicity of solutions. Precise bounds on the data ensuring uniqueness are then sought and several functional inequalities (concerning relative capacity, Sobolev embedding, solenoidal extensions) are analyzed in detail: explicit bounds are obtained for constant boundary data. The case of “almost symmetric” frameworks is also considered. A universal threshold on the Reynolds number ensuring that the flow generates no lift is obtained regardless of the shape and the nature of the obstacle. Based on the asymmetry/multiplicity principle, the performance of different obstacle shapes is then compared numerically. Finally, connections of the results with elasticity and mechanics are emphasized.

We show the incompressible Navier–Stokes–Maxwell system with solenoidal Ohm's law can be derived from the two-fluid incompressible Navier–Stokes–Maxwell system when the momentum transfer coefficient tends to zero. The strategy is based on the decay and dissipative properties of the electromagnetic field.

Supersonic turbulence plays a pivotal role during the formation of molecular clouds and stars in galaxies. However, little is known about how the fraction of compressive and solenoidal modes in the velocity field evolves over time and how it depends on properties of the molecular cloud or the galactic environment. In this work, we carry out magnetohydrodynamical simulations of disc galaxies and study the time evolution of the turbulence driving parameter for an ensemble of clouds. We find that the time-averaged turbulence driving parameter is insensitive to the position of the cloud within the galaxy. The ensemble-averaged driving parameter is found to be rather compressive with $b\sim0.5-0.7$, indicating almost time-independent global star formation properties. However, each individual cloud shows a highly fluctuating driving parameter, which would strongly affect the cloud's star formation rate. We find that the mode of turbulence driving can rapidly change within only a few Myr, both from solenoidal to compressive and vice versa. We attribute these changes to cloud collisions and to tidal interactions with clouds or overdensities in the environment. Last, we find no significant differences in the average driving parameter between hydrodynamic and initially strongly magnetised galaxies. However, the magnetic field tends to reduce the overall fluctuation of the driving parameter. The average driving as well as its uncertainty are seen to be in agreement with recent constraints on the turbulence driving mode for solar neighbourhood clouds., Comment: Accepted for publication by MNRAS. Comments welcome

In this letter, we present a novel and efficient broadband spectral numerical Green's function (S-NGF) for inhomogeneous region. The proposed S-NGF is formulated in terms of the matrices that are obtained by the finite-element method (FEM). By performing modal analysis to FEM matrices, the proposed S-NGF is encapsulated by a series of resonant solenoidal modes where the operating frequency is embedded in the expansion coefficients. Besides, the convergence of the series of resonant solenoidal modes can be greatly accelerated by performing an extraction at one low wavenumber. The S-NGF can be rapidly reconstructed at different frequencies when it is integrated into the surface integral equation for inhomogeneous object modeling. The proposed algorithm is easy to implement, and well suited for the design and optimization of inhomogeneous electromagnetic structures, where fast solutions at massive frequencies are called for. The numerical examples demonstrate the efficiency and accuracy of the proposed scheme.

A novel design and fabrication of the gear reduction starter motor is proposed. In this proposed configuration, the conventional solenoid of the starter motor is eliminated, resulting in a reduction in the starting current required while cranking, and the weight of starter motor is reduced by making it more compact. Failed/dead cranking and the cost of starter motor are also decreased. The function of the solenoid in the starter motor is replaced with a spring-coupled drive mechanism, which makes the function of the starter motor an automatic process. This replacement will affect the state of charge (SoC) and depth of discharge (DoD) of the battery positively, thereby reducing the battery drain and making it last long.

The critical current of superconducting wires is measured at CERN in four test stations that use high-field superconducting solenoid magnets achieving up to 15 T. The reduction of the uncertainty in the critical-current measurements requires the field mapping of the solenoids in their operating condition. This paper presents the design and application of a solenoidal-field transducer based on a pair of nested induction coils, which are moved along the axis of the magnet. The system allows to extract the longitudinal and transverse field components as a function of the sensor's longitudinal position. The radial dependence of the magnetic flux density can be estimated directly from the nested coils of different diameters validated by applying the theory of pseudo-multipoles in solenoidal magnets. The results and uncertainties of the field mappings are presented.

The asymptotics from the two-fluid incompressible Navier-Stokes-Maxwell system with Ohm's law to solenoidal Ohm's law was pointed out in Arsenio-Saint-Raymond's work [3] . We justify rigorously this limit in the context of global-in-time classical solutions. The key is to derive the global-in-time uniform in e energy estimate of the rescaled system with Ohm's law by employing the decay properties of both the electric field E e and the wave equation with linear damping of the divergence free magnetic field B e , then take the limit as e → 0 to obtain the solutions of the system with solenoidal Ohm's law.

The experimental studies supported by the Russian Science Foundation, project no. 17-01525, have been briefly reviewed. Processes of formation and decay of vortex structures on a water surface have been studied. Vortex motions in the liquid have been excited by gravity waves propagating on the surface in different directions. It has been shown that the process of formation and decay of the vorticity on the surface can be described by the sum of two mechanisms-Stokes’ drag and generation of a solenoidal flow by interacting nonlinear surface waves. It has been shown that the processes of formation and decay of vortex structures after the beginning (or end) of the wave excitation at times smaller than the time of viscous damping of the wave are due to changes in the wave system. The process of formation at times much larger than the time of viscous damping is due only to the nonlinear mechanism of generation of the vorticity, whereas the process of decay is due to viscous losses. In the intermediate time interval, the intense interaction of vortices leads to the redistribution of energy between different scales.

An arbitrary diffeomorphism of an annular set of the form is considered, where is a ball in a Banach space and is a (finite- or infinite-dimensional) torus. A system of effective sufficient conditions is proposed which ensure that has a global attractor that can be represented as a generalized solenoid, that is, the inverse limit , where is an expanding linear endomorphism of the torus . Furthermore, the restriction is topologically conjugate to a shift map of the solenoid. Bibliography: 25 titles.

This paper presents for the first time implementation of zig–zag and solenoidal on-chip inductors in standard and cost effective SOI-MUMPs fabrication process. The solenoidal inductor offers high Q-factor and inductance as compared to planar zig–zag inductor. Bonding wires are used to construct the top conductor loop of the solenoidal inductor. The inductors are also characterized at elevated temperature for performance investigation in harsh environment. The mechanical resonance frequency of the inductors are characterized using the Finite element software COMSOL. The inductors are characterized using a vector network analyzer from 60 MHz to 3 GHz, and the results of both inductors are compared. The total size of the device is 2 × 0.7 mm2.

The study analyzes exact solutions to beam equations describing degenerate flows, generalized Brillouin states, and plane solenoidal flows unrelated to the class of invariant solutions. Previously unknown facts of the theory of solenoidal flows in an external inhomogeneous magnetic field and new interpretations of degenerate solutions are discussed.

In this article, discrete variants of several results from vector calculus are studied for classical finite difference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector fields cannot hold discretely because of grid oscillations, which are characterised explicitly. This results in a non-vanishing remainder associated with grid oscillations in the discrete Helmholtz Hodge decomposition. Nevertheless, iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposition via orthogonal projections are proposed and applied successfully. In numerical experiments, the discrete remainder vanishes and the potentials converge with the same order of accuracy as usual in other first-order partial differential equations. Motivated by the successful application of the Helmholtz Hodge decomposition in theoretical plasma physics, applications to the discrete analysis of magnetohydrodynamic (MHD) wave modes are presented and discussed.

A method using the coupling model combining the field and circuit equations is put forward to describe the wireless power transfer (WPT) system in this study. Hollow solenoidal resonant coils are usually excited by alternating currents in practice, doing studies related to the solutions to the solenoidal coils in alternating electromagnetic fields is important. In this study, the analytic expressions of the alternating magnetic vector potential are obtained from Maxwell's formulas for time-harmonic electromagnetic fields, and the mathematical expressions of the electric field intensity, the magnetic flux density, the induced voltage, and the mutual inductance are established for the coaxially symmetric solenoidal coils. The field-circuit coupling model is built for WPT, and the relationship between the input and output variables is obtained. The equations contain the dielectric information of the transmission space and the coils' radii and turn numbers, and can be used for analysing the dielectric effects of the space at lower or higher frequencies. The results are verified by experiments of 2-coil and 4-coil structures. The advantages of being able to analyse the distribution of the fields are illustrated by a comparison experiment with a laterally displaced coil, which is valuable to WPT design.

A general model of incompatible small-strain elasticity is presented and analyzed, based on the linearized strain and its associated incompatibility tensor field. Strain incompatibility accounts for the presence of dislocations, whose motion is ultimately responsible for the plastic behaviour of solids. The specific functional setting is built up, on which existence results are proved. Our solution strategy is essentially based on the projection of the governing equations on appropriate subspaces in the spirit of the Leray decomposition of solenoidal square-integrable velocity fields in hydrodynamics. It is also strongly related with the Beltrami decomposition of symmetric tensor fields in the wake of previous works by the authors. Moreover a novel model parameter is introduced, the incompatibility modulus, that measures the resistance of the elastic material to incompatible deformations. An important result of our study is that classical linearized elasticity is recovered as the limit case when the incompatibility modulus goes to infinity. Several examples are provided to illustrate this property and the physical meaning of the incompatibility modulus in connection with the dissipative nature of the processes under consideration.

In recent years, due to the digitization of the electrical devices electric load demand has been increased. Power security and stabilization are the other two challenges which necessitate the need of electrical energy storage. Along with the technological constraints, economical and environmental issues are the other challenges in the development of energy storage technologies. Fast response and high energy density features are the two key points due to which Superconducting Magnetic Energy Storage (SMES) Devices can work efficiently while stabilizing the power grid. Two types of geometrical combinations have been utilized in the expansion of SMES devices till today; solenoidal and toroidal. It has been found that the solenoidal arrangement is easier to fabricate and provides efficient approach to handle stresses produced by Lorentz forces. This work represents the development of 2.5MJ SMES solenoidal Magnet using 2G (SuperPower, YBCO having Tc=90K @ 0T) HTS tape. The effect of the operating current (passing through a single superconducting tape) has been evaluated. A reference magnetic flux of 3.5T has been considered in designing of HTS SMES. It has been found that the parallel fields for solenoidal magnet are large as compared to perpendicular fields which results into large magnitudes of Lorentz forces acting on HTS tape which may leading to structural instabilities and can cause failure. It has been concluded that higher currents can be used in order to lower down the total length of the superconductor as it can reduce the overall cost of the device

In this paper we present a generalization of Poincar\'e's Rotation Theory of homeomorphisms of the circle to the case of one-dimensional compact abelian groups which are solenoidal groups, {\it i.e.}, groups which fiber over the circle with fiber a Cantor abelian group. We define rotation elements, \emph{\`a la} Poincar\'e and discuss the dynamical properties of translations on these solenoidal groups. We also study the semiconjugation problem when the rotation element generates a dense subgroup of the solenoidal group. Finally, we comment on the relation between Rotation Theory and entropy for these homeomorphisms, since unlike the case of the circle, for the solenoids considered here there are homeomorphisms (not homotopic to the identity) with positive entropy.

This paper aims to study a degenerate parabolic problem for a solenoidal vector field in which the time derivative acts on a moving body. We propose a fully-discrete finite element scheme combined with backward Euler’s method for the saddle-point variational formulation. The convergence of this numerical scheme is proved and error estimates for some stable finite element pairs are also established.

We present a methodology to impose micromechanical constraints, i.e. stress equilibrium at grain and sub-grain scale, to an arbitrary (non-equilibrated) voxelized stress field obtained, for example, by means of synchrotron X-ray diffraction techniques. The method consists in finding the equilibrated stress field closest (in $L^2$-norm sense) to the measured non-equilibrated stress field, via the solution of an optimization problem. The extraction of the divergence-free (equilibrated) part of a general (non-equilibrated) field is performed using the Hodge decomposition of a symmetric matrix field, which is the generalization of the Helmholtz decomposition of a vector field into the sum of an irrotational field and a solenoidal field. The combination of: a) the Euler-Lagrange equations that solve the optimization problem, and b) the Hodge decomposition, gives a differential expression that contains the bi-harmonic operator and two times the curl operator acting on the measured stress field. These high-order derivatives can be efficiently performed in Fourier space. The method is applied to filter the non-equilibrated parts of a synthetic piecewise constant stress fields with a known ground truth, and stress fields in Gum Metal, a beta-Ti-based alloy measured in-situ using Diffraction Contrast Tomography (DCT). In both cases, the largest corrections were obtained near grain boundaries., Comment: 19 pages, 5 figures, submitted to mechanics of materials

We set up the analytical framework for studying the threshold for the appearance of a lift force exerted by a viscous steady fluid (the wind) on the deck of a bridge. We model this interaction as in a wind tunnel experiment, where at the inlet and outlet sections the velocity field of the fluid has a Poiseuille flow profile. Since in a symmetric configuration the appearance of lift forces is a consequence of non-uniqueness of solutions, we compute an explicit threshold on the incoming flow ensuring uniqueness. This requires building an explicit solenoidal extension of the prescribed Poiseuille flow and bounding some embedding and cutoff constants.

We consider a nonlinear autonomous system of $N\gg 1$ degrees of freedom randomly coupled by both relaxational ('gradient') and non-relaxational ('solenoidal') random interactions. We show that with increased interaction strength such systems generically undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically non-trivial regime of 'absolute instability' where equilibria are on average exponentially abundant, but typically all of them are unstable, unless the dynamics is purely gradient. When interactions increase even further the stable equilibria eventually become on average exponentially abundant unless the interaction is purely solenoidal. We further calculate the mean proportion of equilibria which have a fixed fraction of unstable directions., Comment: Main paper - 7 pages, Supplementary Information - 20 pages. Revised version - minor changes, including adding short discussion of model assumptions

Biehl et al (2020) present some interesting observations on an early formulation of the free energy principle in (Friston, 2013). We use these observations to scaffold a discussion of the technical arguments that underwrite the free energy principle. This discussion focuses on solenoidal coupling between various (subsets of) states in sparsely coupled systems that possess a Markov blanket - and the distinction between exact and approximate Bayesian inference, implied by the ensuing Bayesian mechanics., A response to a technical critique [arXiv:2001.06408] of the free energy principle as presented in "Life as we know it"

A novel source for broadband millimeter-wave RF is being designed at Los Alamos National Laboratory utilizing the two-stream instability. This RF source offers the potential for consistent output power over a large bandwidth on a single device. The source is designed to use two co-axial electron beams coupled into solenoidal magnetic fields. This is accomplished using two cathodes independently modulated nested in each other. The electron beam energy from the innermost cathode will fall in the range between 15 and 20 keV with the beam energy from the outer cathode being 75%–95% of the inner beam energy. In order to efficiently match the beam from the outer cathode into the solenoidal magnetic field, shields are used around the cathodes to shape the electric field in the gun region to extend the electrostatic focal length of the beams for better coupling into the magnetic field. We present the design of electron guns and expected performance using Trak simulations.

A combined superconducting magnetic system with a uniform field of up to 20 T has been developed. The system contains several solenoidal sections and a magnetic flux concentrator consisting of hydroextruded dysprosium. The sections are wound from NbTi, Nb3Sn wires and Nb3Sn ribbon. A magnetic shield made out of Nb3Sn ribbon is applied in order to increase the uniformity of the magnetic field. The magnetic field achieved in the gap between the ends of the concentrator has a high spatial uniformity ΔН/Н ∼ 5 × 10−5.

We show existence of a solenoidal vector field u belonging to $$ {W}^{2,q}\left(\Omega \right)\cap {W}_0^{1,s}\left(\Omega \right), $$ s ∈ (1,∞), q ∈ (1, n), such that (δ+|Du|)p−2, p ∈ (1, 2)∪(2,∞), δ ∈ [0, 1], does not belong to the Muckenhoupt class A∞(Ω). This is an extension of our result in [1] proved for δ = 1.

We prove inequalities connecting a flow through the (n−1)-dimensional surface S of a smooth solenoidal vector field with its Lp(U)-norm (U is an n-dimensional domain that contains S ). On the basis of these inequalities, we propose a correct definition of the flow through the surface S of a discontinuous solenoidal vector field f ∈ Lp(U) (or, more precisely, of the class of vector fields that are equal almost everywhere with respect to the Lebesgue measure).

The pseudopotential approach is a useful tool for the qualitative description of ion motion in inhomogeneous high-frequency electric fields, often used in mass spectrometric devices. However, in the theoretical study of the motion of ions in high-frequency electric fields with close frequencies, mathematical ambiguity arises, caused by the nonequivalence of different mathematical approaches. The paper considers and compares the time-dependent pseudopotential model and the model of solenoidal (vortex) pseudoforces, that is, vortex drift. The vortex drift model in high-frequency electric fields with close frequencies can be replaced by the model of ion motion in a pseudopotential varying in time. The theory of almost periodic time signals with two characteristic time scales, fast and slow, eliminates the ambiguity in selecting the correct mathematical method for describing ion motion in high-frequency electric fields.