Terms Of Vector Fields Research Articles

Eigenvalue Estimates for Submanifolds with Locally Bounded Mean Curvature

We present a method to obtain lower bounds for firstDirichlet eigenvalue in terms of vector fields with positivedivergence. Applying this to the gradient of a distance functionwe obtain estimates of eigenvalue of balls inside the cut locus and of domains Ω ⊂ M ∩ BN(p, r) in submanifolds M ⊂ϕNwith locally bounded mean curvature. Forsubmanifolds of Hadamard manifolds with bounded mean curvaturethese lower bounds depend only on the dimension of the submanifold and the bound on its mean curvature.

Morse--Smale Vector Fields without Closed Trajectories on \(3 \)-Manifolds

We study Morse--Smale vector fields on \(3\)-manifolds. A topological classification of such vector fields in terms of homeomorphisms of surfaces with two sets of circles on them is given. An algorithm verifying the existence of such homeomorphisms is constructed.

A vector wavelet approach to iono- and magnetospheric geomagnetic satellite data

A multiscale method is introduced using spherical (vector) wavelets for the computation of the earth's magnetic field within source regions of ionospheric and magnetospheric currents. The considerations are essentially based on two geomathematical keystones, namely (i) the Mie representation of solenoidal vector fields in terms of toroidal and poloidal parts and (ii) the Helmholtz decomposition of spherical (tangential) vector fields. Vector wavelets are shown to provide adequate tools for multiscale geomagnetic modelling in form of a multiresolution analysis, thereby completely circumventing the numerical obstacles caused by vector spherical harmonics. The applicability and efficiency of the multiresolution technique is tested with real satellite data.

Nonlinear evolution equations invariant under the Schrödinger group in three-dimensional spacetime

A classification of all possible realizations of the Galilei, Galilei-similitude and Schrödinger Lie algebras in three-dimensional spacetime in terms of vector fields under the action of the group of local diffeomorphisms of the space is presented. Using this result a variety of second-order evolution equations invariant under the corresponding groups are constructed and their physical significance is discussed.

Computation of magnetic fields within source regions of ionospheric and magnetospheric currents

A general method of computing the magnetic effect caused by a predetermined three-dimensional external current density is presented. It takes advantage of the representation of solenoidal vector fields in terms of toroidal and poloidal modes expressed by two independent series of spherical harmonics. In order to test the method, it is applied to two ionospheric–magnetospheric model current densities. The first example assumes a special large-scale current system which has an analytical solution for the magnetic field everywhere. In the second example, the results for a model current distribution that is of small-scale at high latitudes are compared with a direct integration using Biot–Savarts law. It is demonstrated that the method provides a fast and numerically stable determination of the magnetic field both outside and inside the current density.

Characteristic relations for quantum matrices

General algebraic properties of the algebras of vector fields over quantum linear groups $GL_q(N)$ and $SL_q(N)$ are studied. These quantum algebras appears to be quite similar to the classical matrix algebra. In particular, quantum analogues of the characteristic polynomial and characteristic identity are obtained for them. The $q$-analogues of the Newton relations connecting two different generating sets of central elements of these algebras (the determinant-like and the trace-like ones) are derived. This allows one to express the $q$-determinant of quantized vector fields in terms of their $q$-traces.

Boundary condition transfer method for the solution of electromagnetic scattering by rotationally symmetric penetrable bodies

A novel differential formulation of electromagnetic scattering by rotationally symmetric penetrable bodies is presented. The formulation is essentially based on the representation of vector fields in terms of spherical vector wave functions. In regions where we have physical inhomogenity the representation is in terms of position‐dependent expansion coefficients. The differential equation system for these variable coefficients is deduced from the relevant wave equation. This system is numerically solved as an initial value problem, and the unknown constant expansion coefficients are found with a matrix inversion. The method was tested and validated on a number of problems, whose exact solutions are available, and the results obtained for scatterers for which no exact solutions are present were succesfully compared with the ones obtained by other methods.

Complex Power Series—a Vector Field Visualization

How can we visualize the behavior of complex power series and convey to students a clear understanding of convergence and divergence? In this note we describe a visualization technique based on the use of Polya vector fields, as introduced by Polya and Latta [3] and further developed by Braden [1], [2]. In [3], Polya introduced the notion of a vector field representation of a complex function of a complex variable: If f(x + iy) = u(x, y) + iv(x, y), then the associated Polya field is F(x, y) = (u(x, y), v(x, y)). A plot of the vector field can then be used to visualize the behavior of f. The complex conjugate is used rather than (u, v) itself because the Cauchy-Riemann equations for analyticity of f translate into zero divergence and zero curl for F; thus a nice interpretation of analyticity in vector field terms is available. Of course, the conjugate field preserves all important features of f. The vector field representation is ideally suited to depict the process of convergence of a sequence of complex functions. Each function f, is represented by a vector field Fn on the common domain of definition, and convergence of f, to a limiting function f can be envisioned as a stabilizing of the vector fields F11 to a limiting configuration, that of the vector field for f. Convergence at a point is thus represented by a sequence of arrows based at a common point tending toward a limiting arrow. Divergence appears as either an oscillation or a growth in magnitude without bound of a sequence of common-based arrows. The production and exploration of such sequences of vector fields is easy in an interactive computer system such as Mathernatica. We describe a program written in the Mathernatica language and implemented on the Macintosh II that allows construction of such fields for a complex power series. The program displays the partial sums of a series oo=a1a(z Z0) l in vector field form. The first input to the program is a domain, which can be a disc or an annulus with center z0 and any radii. The next input is a pair [a, b]; only vectors whose magnitudes are between a and b will be shown. Then the user gives the number of annuli into which the display will be divided, and the closed form expression for an. The program computes and displays the fields in succession; vectors are plotted as arrows with bases on circles determined by the subdivision annuli. The lengths of the vectors are scaled logarithmically to avoid messy overlaps; this does not alter the behavior of the sequence. Arrows whose lengths exceed the upper tolerance level are deleted and replaced by dots. Termination of the program leaves us with a field that may be taken as a representation of E,0a11(z Zo). The animation feature of Mathematica allows the fields to be presented in an animated sequence, showing the meaning of convergence and divergence dynamically.

Combinatorics of Realizations of Nilpotent Control Systems

This article gives a very simple algorithm which allows to immediately write down a canonical nonlinear representation of a free nilpotent Lie algebra. Specifically, it defines a set of local coordinates and gives a formula for the components of a set of system vector fields in terms of these coordinates. The components of iterated Lie brackets of the system vector fields can also be read off easily without any further differentiation. The formulae given here are very close to Sussmann’s product expansion of the Chen-Fliess series and to the chronological calculus introduced by Agrachev and Gamkrelidze.

Vector fields and transfers

For a smooth fibre bundle\(F\mathop \to \limits^i E\mathop \to \limits^p B\) whereF is a compact manifold with or without boundary, a vertical vector fieldV gives rise to a transfer τV as anS-map. Our goal is to show these transfers satisfy an equation analogous to one that the index of vector fields satisfy. This equation gives results involving equivariant vector fields as well as a characterization of those transfers defined by vector fields in terms of the ordinary Euler-Poincare transfers.

Geometry of collective motions

A general projection method for decomposing the kinetic energy of an N-particle system into collective and intrinsic parts defined respectively on the orbits and the orbit space of a Lie transformation group is given. Specific targets of the application of the method are the kinematical group GL+(3,R) and the quotient set GL+(3,R)/SO(3) for their importance in microscopic formulation of nuclear collective motions. For these two cases the orbit spaces in the particle configuration space are shown to be identifiable with the Grassman and Stiefel manifolds of 3-planes and 3-frames respectively. Some problems related to expressing the kinetic energy in terms of vector fields on these manifolds are resolved. In particular, non-integrable coordinates previously used by one of the present authors is shown to arise from the imposition of unacceptable conditions. Finally the authors consider the corresponding decomposition of the N-particle Hilbert space. It is proposed that an appropriate basis function for the GL+(3,R) collective model is provided by an irreducible representation of the boson SU(6) group.

Intrinsic Lipschitz classes on manifolds with applications to complex function theory and estimates for the $$\bar \partial $$ and $$\bar \partial _b $$ equations

An intrinsic definition of Lipschitz classes in terms of vector fields on man-ifolds is provided and it is shown that it is locally equivalent with a more classical definition. A finer result is then proved for strongly pseudo-convex CR manifolds and applications of the theorems are given to smoothness of holomorphic functions and estimates for the $$\bar \partial $$ and $$\bar \partial _b $$ . equations.

Ambiguities in the Lagrangian and Hamiltonian formalism: Transformation properties

It is demonstrated that the classical isotropic oscillator can be described by simple Lagrangians and Hamiltonians which do not have the symmetry properties usually associated with these functions. For instance, Noether's theorem need not connect angular momentum with rotations. By starting from this example, the relation between symmetries and constants of the motion is analysed in terms of vector fields on symplectic manifolds in the modern view of classical dynamics. This relation is shown to be far more ambiguous than is usually thought.

Eigenfunctions of the Curl Operator, Rotationally Invariant Helmholtz Theorem, and Applications to Electromagnetic Theory and Fluid Mechanics

In the present paper we introduce eigenfunctions of the curl operator. The expansion of vector fields in terms of these eigenfunctions leads to a decomposition of such fields into three modes, one of which corresponds to an irrotational vector field and two of which correspond to rotational circularly polarized vector fields of opposite signs of polarization. Under a rotation of coordinates, the three modes which are introduced in this fashion remain invariant. Hence we have introduced the Helmholtz decomposition of vector fields in an irreducible, rotationally invariant form.These expansions enable one to handle the curl and divergence operators simply. As illustrations of the use of the curl eigenfunctions, we solve four problems. The first problem that is solved is the initial value problem of electromagnetic theory with given time- and space-dependent sources and currents and we show that the radiation and longitudinal modes uncouple in a very simple way. In the second problem we show how fluid motion...

A class of self-sustaining dissipative spherical dynamos

The present paper treats rigorously the dynamo equations describing the effects of the internal motion of a bounded volume of incompressible fluid with nonzero ohmic resistivity on the magnetic field produced by electric currents in that fluid. The procedure involves representing an arbitrary solenoidal vector field in terms of two scalars, analogous to the representation of an arbitrary irrotational field as the gradient of a single scalar. The dynamo equations are reduced to scalar heat equations for the two field scalars, the coupling between them taking the form of a heat source term. Precise results about the magnetic field can be obtained from these heat equations with the help of several variational inequalities analogous to Rayleigh's variational estimate for the fundamental frequency of a vibrating system. The main result is the explicit construction of a large class of continuously differentiable fluid velocities capable of indefinitely maintaining or amplifying the dipole moment of the external magnetic field. These motions all involve periods of stasis in the fluid, and cannot, therefore, be expected to occur in the earth's core. It is believed that it will be possible eventually to obtain more exact bounds than those presented here for the magnetic field components with high wave number, thus eliminating the need for such periods of stasis. The fluid motions shown capable of dynamo maintenance are of this sort: a toroidal shear symmetric about the z axis proceeds long enough to produce from P 1 z1 , the lowest poloidal free-decay mode symmetric about that axis, a very large energy in T 1 z1 , the lowest toroidal free-decay mode with such symmetry. During a period of stasis, everything else almost dies out, leaving a field which is largely T 1 z1 . Then almost any velocity which has a radial component and is not axisymmetric about the z axis will regenerate P 1 z1 and the external dipole moment. A critique of some previous attempts to produce dissipative self-regenerative spherical dynamos is included. The techniques which lead to the existence of self-sustaining dynamos produce other results about the dynamo equations, most of which are to the author's knowledge either new or not previously precisely formulated. These results are listed below. 1. (i) Fluid motions in a sphere can be regarded as bounded linear operators on the Hilbert space of magnetic fields with finite total energy. 2. (ii) The free-decay modes in the rigid sphere are complete in that space. 3. (iii) The magnetic effect of a given fluid motion on a given initial field depends continuously on the resistivity ϱ of the fluid even at ϱ = 0. 4. (iv) The magnetic effect of any motion can be approximated with arbitrary accuracy by replacing it by a series of rapid jerks interpersed with periods of rest. 5. (v) The effect of a rigid rotation of the fluid is to rotate the magnetic field while that field decays as if the fluid were motionless. This effect can be approximated with arbitrary accuracy by rotating all but a sufficiently thin shell at the surface, even if every point on the surface remains fixed and a large shear develops in the shell. 6. (vi) If the fluid velocity has no radial component, the poloidal magnetic field decays as rapidly as if the fluid were at rest. If, further, no poloidal field is initially present, the toroidal field decays as rapidly as if the fluid were at rest. 7. (vii) Dynamo maintenance is impossible if the local strain-rate of the fluid is always and everywhere less than the decay rate of P 1 z1 when the velocity of the fluid is zero.

A Continuous Model of Transportation

Abstract : Interlocal commodity flows are considered in terms of vector fields. Efficient transportation then gives rise to a problem in the calculus of variations; the flow functions solving this problem contain price distribution functions as parameters. With net production functions dependent on prices, the model describes the flow and prices in a competitive regional market economy. (Author)

LXIV. Divergence and curl in a vector field in terms of curvature and tortuosity

LXIV. <i>Divergence and curl in a vector field in terms of curvature and tortuosity</i>