Continuous Vector Fields Research Articles

Conformally natural extension of vector fields from 𝑆ⁿ⁻¹ to 𝐵ⁿ

Up to multiplication by a constant there is exactly one conformally natural continuous linear map from the space of continuous vector fields on S n − 1 {S^{n - 1}} to the space of continuous vector fields on B n {B^n} .

Weak solutions of differential equations in Banach spaces

We consider the Cauchy problem x (t) = f (t,x (t)), x (0) = x0 in a nonreflexive Banach space X and for f: T × X → X a weakly continuous vector field. Using a compactness hypothesis involving a weak measure of noncompactness we prove an existence result that generalizes earlier theorems by Chow-Shur, Kato and Cramer-Lakshmikantham-Mitchell.

A continuation principle for forced oscillations on differentiable manifolds

In the present paper we are concerned with the existence of Γ-periodic solutions for the differential equation x(t) = f(t, .x( r)), t e R, where / is a continuous time dependent Γ-periodic tangent vector field defined on an ^-dimensional differentiable manifold M possibly with boundary. We prove that if the Euler characteristic of the average vector field w(p) = (l/T)fίff(t,p)dt is defined and nonzero and if all the possible orbits of the parametrized equation x(t) = λf(ί,x(t)), /eR and λ £ (0,1], lie in a compact set and do not hit the boundary of M, then the given equation admits a Γ-periodic solution. 0. Introduction. Let M be an w-dimensional differentiable manifold, possibly with boundary, and let /: RxM->Γ(M) be a time dependent Γ-periodic tangent vector field on M. In this paper we give a topological result concerning the existence of Γ-periodic solutions for the differential equation

A characterization of Clarke’s strict tangent cone via nonlinear semigroups

Clarke’s strict tangent cone T X ↑ ( a ) T_X^ \uparrow (a) at a ∈ X a \in X to a closed subset of a Banach space E E is shown to contain the limit inferior of tangent cones T X ( x ) {T_X}(x) to X X at x x as x → a x \to a , x ∈ X x \in X . Several characterizations of T X ↑ ( a ) T_X^ \uparrow (a) are presented. As a consequence various tangential and subtangential conditions for continuous vector fields on X X are shown to be equivalent.

A general rice formula, palm measures, and horizontal-window conditioning for random fields

In the first main result the mean ( m−n)-dimensional Hausdorff measure of the set of crossing points of a level y ϵ R n by an m-dimensional continuous random vector field with values in R n , m⩾ n, is computed. The second one deals with horizontal-window conditional (Palm) distributions for such random fields. For this purpose, a general concept of Palm measures is introduced, which contains both the ‘stationary’ and the ‘nonstationary’ one.

Superconvergence phenomenon in the finite element method arising from averaging gradients

We study a superconvergence phenomenon which can be obtained when solving a 2nd order elliptic problem by the usual linear elements. The averaged gradient is a piecewise linear continuous vector field, the value of which at any nodal point is an average of gradients of linear elements on triangles incident with this nodal point. The convergence rate of the averaged gradient to an exact gradient in theL 2-norm can locally be higher even by one than that of the original piecewise constant discrete gradient.

Generic flows generated by continuous vector fields in Banach spaces

Soit f un champ vectoriel defini sur un sous-ensemble ouvert d'un espace de Banach reel. On considere le probleme de Cauchy x˙=f(t,x), x(0)=y, #7B-E un sous-ensemble non vide de points initiaux. On etudie les f∈Μ qui engendrent un flot pour y∈#7B-E

Generalized energy representations for current groups

The class of energy representations of the group D( X, G) of compactly supported smooth mappings from a manifold X into a compact semisimple Lie group G is substantially enlarged in the following two ways. One consists of when X has a Riemannian structure, to extend the work of Vershik, Gelfand, and Graev ( Compositio Math. 35 (1977), 299–334; 42 (1981), 217–243), see also (Albeverio, Høegh-Krohn, and Testard, J. Funct. Anal. 41 (1981), 378–396); it is shown that each pair ( ϱ, M), where ϱ is a strictly positive continuous density on X, and M a subbundle of the tangent bundle of X, gives rise to an energy representation U ϱ, M of D( X, G) which is irreducible if dim( X) ⩾ 3. The other, entirely new, does not require a Riemannian structure on X: a volume measure m on a smooth manifold X with Euler number e( X) = 0 being given, each pair ( ϱ, ξ), where ϱ is a strictly positive continuous density on X and ξ a nonvanishing continuous vectorfield on X, gives rise to a new energy representation Π ϱ dm ξ of D( X, G) which is irreductible if dim( X) ⩾ 3. Conditions of unitary equivalency of the U ϱ, M as well as the Π ϱ dm ξ are given.

Singularities of n-dimensional cell functions and the associated index theory

In [1,2] a new and promising method for global analysis of non-linear systems has been presented in the form of cell-to-cell mappings. It is seen to be a very powerful method indeed. However, many fundamental questions about this new kind of mapping remain to be investigated. In this paper we present a theory of singularities for cell functions by introducing a system of singular entities. The key steps in the development are to introduce a simplicial structure to the cell space and to use simplicial mappings to create an associated continuous vector field for a given cell function. The construction of the associated vector field also allows us to develop a theory of index for cell functions. This theory of index then serves very nicely as a unifying force for the whole development reported in this paper.

Bounds on the number of switchings for trajectories of piecewise analytic vector fields

We study the trajectories of systems x ̇ = X(x) , where X is a continuous “extendably piecewise analytic” vector field, i.e., a continuous vector field X such that the domain of ƒ admits a locally finite partition I into sets such that for each A ∈ I there is a vector field X A which is analytic on a neighborhood of the closure of A and whose restriction to A coincides with that of X. We prove that the trajectories are piecewise analytic, with a priori bounds on the number of switchings for all trajectories that stay in a fixed compact set and whose duration does not exceed a fixed number T. This result implies the existence of a regular synthesis for optimal control problems with a strictly convex Lagrangian, and a linear dynamics with polyhedral constraints on the controls.

Topology of Three-Dimensional Separated Flows

Based on the hypothesis that patterns of skin-friction lines and external streamlines reflect the properties of continuous vector fields, topology rules define a small number of singular points (nodes, saddle points, and foci) that characterize the patterns on the surface and on particular projections of the flow (e.g., the crossflow plane). The restricted number of singular points and the rules that they obey are considered as an organizing principle whose finite number of elements can be combined in various ways to connect together the properties common to all steady three dimensional viscous flows. Introduction of a distinction between local and global properties of the flow resolves an ambiguity in the proper definition of a three dimensional separated flow. Adoption of the notions of topological structure, structural stability, and bifurcation provides a framework to describe how three dimensional separated flows originate and succeed each other as the relevant parameters of the problem are varied.

On the spectrum of anosov diffeomorphisms

Any Anosov diffeomorphismf induces a linear operator in the Banach space of continuous vector fields, which spectrum is contained in two annuli both being disjoint with the unit circle. The main theorem states that, if these annuli are sufficiently “narrow”, thenf has a fixed point. It is also proved that, iff acts on an infranilmanifold (e.g., on a torus), then the spectrum annuli for the algebraic automorphism corresponding tof are always more “narrow” than forf.

Geometry of binocular vision and a model for stereopsis.

If a binocular observer looks at surfaces, the disparity is a continuous vector field defined on the manifold of cyclopean visual directions. We derive this field for the general case that the observer is presented with a curved surface and fixates an arbitrary point. We expand the disparity field in the neighbourhood of a visual direction. The first order approximation can be decomposed into congruences, similarities and deformations. The deformation component is described by the traceless part of the symmetric part of the gradient of the disparity. The deformation component carries all information concerning the slant of a surface element that is contained in the disparity field itself; it is invariant for changes of fixation, differential cyclotorsion and uniform aniseikonia. The deformation component can be found from a comparison of the orientation of surface details in the left and right retinal images. The theory provides a geometric explanation of the percepts obtained with uniform and oblique meridional aniseikonia. We utilize the geometric theory to construct a mechanistic model of stereopsis that obviates the need for internal zooming mechanisms, but nevertheless is insensitive to differential cyclotorsion or uniform aniseikonia.

Gradient approximation of vector fields

In this paper, we consider an approximation problem which arose in optimal control theory. We seek conditions on a compact subset K of Euclidean n-space such that every continuous vector field on K may be uniformly approximated (on K) by vector fields with strictly positive integrating factors. We prove that such approximation is possible for all K in a particular subclass of the compact sets with topological dimension less than or equal to 1.

The Numerical Evaluation of Fixed Points for Completely Continuous Operators

Given a completely continuous nonlinear operator $\mathcal{K}$ on a domain D contained in a Banach space $\mathcal{X}$, one common way of approximating $\mathcal{K}$ leads to a sequence $\{ \mathcal{K}_n \} $ which is collectively compact and pointwise convergent to $\mathcal{K}$ on D. If $\mathcal{K}$ is an integral operator, then $\mathcal{K}_n $ would usually be the result of replacing the integral with a numerical integral which depended on n. The results given here concern local existence and uniqueness of fixed points of $\mathcal{K}_n $ and convergence of them to fixed points of $\mathcal{K}$. The approach used is the invariance under homotopy of the rotation of a completely continuous vector field.

Smoothing derivatives of functions and applications

Thus the problem comes to approximating our given function by one which satisfies a Lipschitz condition, still preserving the derivative of course. This problem is solved in the second section by using an averaging technique which was developed by Kurzweil [3]. Unfortunately, while the convolution provides a close approximation for the derivative, the Kurzweil averaging only allows for a one-sided approximation of the derivative. However, this is enough for our purposes, and in the third section we apply these results to the converse Lyapunov problem. In the final section of this paper we apply these results to obtain a topological version of the Hirsch-Cairns smoothing theorem. We owe special gratitude to Charles C. Pugh for his careful reading of the manuscript, especially ?2. Because of his diligence, the reader will be spared many ambiguities and typographical errors, and in addition one rather annoying technical error which appeared in the original manuscript. 1. Smoothing Lipschitzian Functions. Let M be a paracompact Cco manifold, and let X be a continuous nonsingular vector field on M whose trajectories are defined for all time and depend uniquely upon their initial conditions. Suppose that f: M -- R is a given continuous function such that the derivative Xf is defined and

Principles of rotational invariance of a completely continuous vector field

Principles of rotational invariance of a completely continuous vector field

Integral representation of dominated operations on spaces of continuous vector fields

Integral representation of dominated operations on spaces of continuous vector fields

The singularities of continuous functions and vector fields.

The singularities of continuous functions and vector fields.

Vector Fields on the n -Sphere

Publisher Summary This chapter discusses vector fields on the n-sphere. Any set of 2 k continuous vector fields tangent to S n are somewhere dependent. The case k = 0 is the classical result that a vector field on a sphere of even dimension has at least one zero. The case k = 2 was based on the erroneous assertion of π 5 (S 3 ) =0. If n and k are with r > 0, then the fiber bundle h: V n+1, 2h+1 → S n does not have a cross-section. If n is not of the form 2 k — 1, then the fiber bundle R n+1 → S n does not have a cross-section.