General Vector Fields Research Articles

Second order parallel tensors on singular quasi-constant curvature manifolds

Extending the result for the regular case (i.e. Ricci curvature along the generating vector field does not vanish) to the singular case, we have presented a classification of second order parallel tensors on a non-flat manifold of quasi-constant curvature manifolds and applied it to classify parallel and affine Killing vector fields on those manifolds.

On the generalization of the Darboux theorem

Darboux theorem to more general context of Frechet manifolds we face an obstacle: in general vector fields do not have local flows. Recently, Fr\'{e}chet geometry has been developed in terms of projective limit of Banach manifolds. In this framework under an appropriate Lipchitz condition The Darboux theorem asserts that a symplectic manifold $(M^{2n},\omega)$ is locally symplectomorphic to $(R^{2n}, \omega_0)$, where $\omega_0$ is the standard symplectic form on $R^{2n}$. This theorem was proved by Moser in 1965, the idea of proof, known as the Moser’s trick, works in many situations. The Moser tricks is to construct an appropriate isotopy $ \ff_t $ generated by a time-dependent vector field $ X_t $ on $M$ such that $ \ff_1^{*} \omega = \omega_0$. Nevertheless, it was showed by Marsden that Darboux theorem is not valid for weak symplectic Banach manifolds. However, in 1999 Bambusi showed that if we associate to each point of a Banach manifold a suitable Banach space (classifying space) via a given symplectic form then the Moser trick can be applied to obtain the theorem if the classifying space does not depend on the point of the manifold and a suitable smoothness condition holds.
 If we want to try to generalize the local flows exist and with some restrictive conditions the Darboux theorem was proved by Kumar. In this paper we consider the category of so-called bounded Fr\'{e}chet manifolds and prove that in this category vector fields have local flows and following the idea of Bambusi we associate to each point of a manifold a Fr\'{e}chet space independent of the choice of the point and with the assumption of bounded smoothness on vector fields we prove the Darboux theorem.

Automorphisms of generic gradient vector fields with prescribed finite symmetries

Let $M$ be a compact and connected smooth manifold endowed with a smooth action of a finite group $\Gamma$, and let $f$ be a $\Gamma$-invariant Morse function on $M$. We prove that the space of $\Gamma$-invariant Riemannian metrics on $M$ contains a residual subset ${\mathcal M\mathrm{et}}\_f$ with the following property. Let $g\in\mathcal{M}\mathrm{et}\_f$ and let $\nabla^gf$ be the gradient vector field of $f$ with respect to $g$. For any diffeomorphism $\phi\in \mathrm {Diff}(M)$ preserving $\nabla^gf$ there exists some $t\in\mathbb R$ and some $\gamma\in\Gamma$ such that for every $x\in M$ we have $\phi(x)=\gamma,\Phi\_t^g(x)$, where $\Phi\_t^g$ is the time-$t$ flow of the vector field $\nabla^gf$.

Generalized symmetries and higher-order Conservation laws of the Camassa–Holm equation

In the present paper, we derive generalized symmetries of order three of the Camassa–Holm equation by infinite prolongation of a generalized vector field and applying infinitesimal symmetry criterion. In addition, one-dimensional optimal system of Lie subalgebras investigated by applying the adjoint representation. Furthermore, determining equation for multipliers and the 2- dimensional homotopy formula employed to construct higher–order conservation laws for the Camassa–Holm equation.

Weighted anisotropic Hardy and Rellich type inequalities for general vector fields

In this paper, we establish the weighted anisotropic Hardy and Rellich type inequalities with boundary terms for general (real-valued) vector fields. As consequences, we derive new as well as many of the fundamental Hardy and Rellich type inequalities which are known in different settings.

Inflation driven by massive vector fields with derivative self-interactions

Inspired by the Generalized Proca Theory, we study a vector–tensor model of inflation with massive vector fields and derivative self-interactions. The action under consideration contains a usual Maxwell-like kinetic term, a general potential term and a term with nonminimal derivative coupling between the vector field and gravity, via the dual Riemann tensor. In this theory, the last term contains a free parameter, [Formula: see text], which quantifies the nonminimal derivative coupling. In this scenario, taking into account a spatially flat Friedmann–Robertson–Walker (FRW) universe and a general vector field, we obtain the general expressions for the equation of motion and the total energy–momentum tensor. Choosing a Proca-type potential, a suitable inflationary regimen driven by massive vector fields is studied. In this model, the isotropy of expansion is guaranteed by considering a triplet of orthogonal vector fields. In order to obtain an inflationary solution with this model, the quasi de Sitter expansion was considered. In this case, the vector field behaves as a constant. Finally, slow-roll analysis is performed and slow-roll conditions are defined for this model, which, for suitable constraints of the model parameters, can give the required number of e-folds for sufficient inflation.

On the growth rate of periodic orbits for vector fields

We establish the relationship between the growth rate of periodic orbits and the topological entropy for C1 generic vector fields: this extends a classical result of Katok for C1+α(α>0) surface diffeomorphisms to C1 generic vector fields of any dimension. The main difficulty comes from the existence of singularities and the shear of the flow.

Characterization of tightly focused vector fields formed by off-axis parabolic mirror.

In this paper, we present a systematic investigation of the characterization of tightly focused vector fields formed by an off-axis parabolic mirror. Based on the Stratton-Chu integral of Green's theorem, the rigorous diffraction integrals that generate vector fields within and outside the focus arising from a collimated beam incident on an idealized parabolic mirror were derived in detail. In addition, explicit analytical expressions for the far-field vector diffraction electric and magnetic fields suitable for an off-axis parabolic mirror were also obtained. It is shown that there are significant differences in the vector diffraction characterizations between on- and off-axis parabolic mirrors. When the off-axis rate is greater than 4, the longitudinal field is predominant, and the maximum peak intensity ratio between the longitudinal field and the transverse field at the focus is approximately 103. This property is valuable for all applications in which a strong longitudinal field component is desirable. The effective focal length increases with increasing off-axis rate, and the depth of focus and the convergence angle of the focused beam are strongly dependent onf'/2ω: smaller values of f'/2ω lead to shorter focal depths and larger beam convergence (or tighter focusing).

Isotropic Realizability of Fields and Reconstruction of Invariant Measures under Positivity Properties. Asymptotics of the Flow by a Non-Ergodic Approach

The paper is devoted to the isotropic realizability of a regular gradient field u or a more general vector field b, namely the existence of a continuous positive function σ such that σb is divergence free in R d or in an open set of R d. First, we prove that under some suitable positivity condition satisfied by u, the isotropic realizability of u holds either in R d if u does not vanish, or in the open sets {c j < u < c j+1 } if the c j are the critical values of u (including inf R d u and sup R d u) which are assumed to be in finite number. It turns out that this positivity condition is not sufficient to ensure the existence of a continuous positive invariant measure σ on the torus when u is periodic. Then, we establish a new criterium of the existence of an invariant measure for the flow associated with a regular periodic vector field b, which is based on the equality b · v = 1 in R d. We show that this gradient invertibility is not related to the classical ergodic assumption, but it actually appears as an alternative to get the asymptotics of the flow.

Extremum seeking control with an adaptive dither signal

We present an output-feedback law that steers a fully actuated control-affine system with general drift vector field to a minimum of the output function. The proposed control law is based on a suitable approximation of descent directions by Lie brackets in the high-frequency limit. A novelty of our approach is an adaptive choice of the frequency parameter. In this way, the problem of determining a sufficiently large frequency parameter becomes obsolete. The adaptive choice of the frequency parameter also prevents finite escape times in the presence of a drift. The proposed control law does not only lead to convergence into a neighborhood of the set of minima, but leads to exact convergence. For the case of an output function with a global minimum and no other stationary points, we prove global convergence.

On Wick Polynomials of Boson Fields in Locally Covariant Algebraic QFT

This work presents some results about Wick polynomials of a vector field renormalization in locally covariant algebraic quantum field theory in curved spacetime. General vector fields are pictured as sections of natural vector bundles over globally hyperbolic spacetimes and quantized through the known functorial machinery in terms of local $^*$-algebras. These quantized fields may be defined on spacetimes with given classical background fields, also sections of natural vector bundles, in addition to the Lorentzian metric. The mass and the coupling constants are in particular viewed as background fields. Wick powers of the quantized vector field are axiomatically defined imposing in particular local covariance, scaling properties and smooth dependence on smooth perturbation of the background fields. A general classification theorem is established for finite renormalization terms (or counterterms) arising when comparing different solutions satisfying the defining axioms of Wick powers. The result is specialized to the case of general tensor fields. In particular, the case of a vector Klein-Gordon field and the case of a scalar field renormalized together with its derivatives are discussed as examples. In each case, a more precise statement about the structure of the counterterms is proved. The finite renormalization terms turn out to be finite-order polynomials tensorially and locally constructed with the backgrounds fields and their covariant derivatives whose coefficients are locally smooth functions of polynomial scalar invariants constructed from the so-called marginal subset of the background fields. The notion of local smooth dependence on polynomial scalar invariants is made precise in the text.

Renormalization of generalized vector field models in curved spacetime

We calculate the one-loop divergences for different vector field models in curved spacetime. We introduce a classification scheme based on their degeneracy structure, which encompasses the well-known models of the non-degenerate vector field, the Abelian gauge field and the Proca field. The renormalization of the generalized Proca model, which has important applications in cosmology, is more complicated. By extending standard heat-kernel techniques, we derive a closed form expression for the one-loop divergences of the generalized Proca model.

Modeling Tangential Vector Fields on a Sphere

ABSTRACTPhysical processes that manifest as tangential vector fields on a sphere are common in geophysical and environmental sciences. These naturally occurring vector fields are often subject to physical constraints, such as being curl-free or divergence-free. We start with constructing parametric models for curl-free and divergence-free vector fields that are tangential to the unit sphere through applying the surface gradient or the surface curl operator to a scalar random potential field on the unit sphere. Using the Helmholtz–Hodge decomposition, we then construct a class of simple but flexible parametric models for general tangential vector fields, which are represented as a sum of a curl-free and a divergence-free components. We propose a likelihood-based parameter estimation procedure, and show that fast computation is possible even for large datasets when the observations are on a regular latitude–longitude grid. Characteristics and practical utility of the proposed methodology are illustrated through extensive simulation studies and an application to a dataset of ocean surface wind velocities collected by satellite-based scatterometers. We also compare our model with a bivariate Matérn model and a non-stationary bivariate global model. Supplementary materials for this article are available online.

Investigating pedestrian navigation in indoor open space environments using big data

• Pedestrian navigation in indoor open space environments is modelled. • Several floor field specifications were calibrated using pedestrian trajectories. • Big data collected through video recordings were utilized to calibrate the pedestrian models. • Likelihood function optimisation method was used as calibration method. • A generalized floor field for pedestrian navigation is proposed. Reliable empirical data are required to understand and model realistic pedestrian navigation behaviours at public crowd-gathering places such as public transportation hubs and shopping malls. Naturalistic walking big data could provide reliable information for investigating realistic pedestrian behaviours and to overcome critical shortcomings of the data collected through controlled laboratory experiments. In this work, we investigate pedestrian navigation in indoor open spaces using naturalistic walking big data collected through video recordings. The extracted data include 299,082 trajectories of individual pedestrians who navigated through the atrium of the Informatics Forum building of the University of Edinburgh. We compare several pedestrian vector fields by calibrating several cellular automaton (CA) models and we finally identify a generalized vector field for pedestrians who are walking in indoor open space environments under normal walking conditions. The output of this study could be useful in enhancing CA-based pedestrian simulation models by representing pedestrian navigation as well as route-choice behaviours more realistically in those models. Simulation tools based on such enhanced models can facilitate practitioners, such as public building designers, to optimize designs considering naturalistic pedestrian behaviours in open spaces.

On a class of generating vector fields for the extremum seeking problem: Lie bracket approximation and stability properties

In this paper, we describe a broad class of control functions for extremum seeking problems. We show that it unifies and generalizes existing extremum seeking strategies which are based on Lie bracket approximations, and allows to design new controls with favorable properties in extremum seeking and vibrational stabilization tasks. The second result of this paper is a novel approach for studying the asymptotic behavior of extremum seeking systems. It provides a constructive procedure for defining frequencies of control functions to ensure the practical asymptotic and exponential stability. In contrast to many known results, we also prove asymptotic and exponential stability in the sense of Lyapunov for the proposed class of extremum seeking systems under appropriate assumptions on the vector fields.

A Type of the Shadowing Properties for Generic View Points

We show that if a C 1 generic diffeomorphism of a closed smooth two-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it is Anosov. Moreover, if a C 1 generic vector field of a closed smooth three-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it satisfies singular Axiom A without cycles.

Effective description of general extensions of the Standard Model: the complete tree-level dictionary

We compute all the tree-level contributions to the Wilson coefficients of the dimension-six Standard-Model effective theory in ultraviolet completions with general scalar, spinor and vector field content and arbitrary interactions. No assumption about the renormalizability of the high-energy theory is made. This provides a complete ultraviolet/infrared dictionary at the classical level, which can be used to study the low-energy implications of any model of interest, and also to look for explicit completions consistent with low-energy data.

Parametric active contour based on sparse decomposition for multi-objects extraction

Active contour model has been widely used over the past decade in image segmentation. For parametric active contour model, it is always used to segment objects because of its simplicity and fast evolution. However, it could not extract multi-objects with one initial contour because of topological invariance of parametric contour and existing equilibrium points between objects in vector fields. In this paper, a method of multi-objects extraction with parametric active contour model is proposed. Firstly, after an edge map is computed, the idea of sparse representation and decomposition for edge map is introduced in order to obtain new edge maps to better describe the objects; Secondly, the obtained edge maps are used to generate vector fields; Finally, one initial contour is evolved in every vector field to extract corresponding objects. Experimental results show that the proposed model could extract multi-target objects by using one initial contour. Quantitative evaluation of the segmentation results with tested methods also shows that the proposed method is more robust to noise and gets the better effect of segmentation accuracy. Furthermore, the proposed method allows parallel computation of algorithms, hence further reducing the computational time.

Multi-Hamiltonian formulations and stability of higher-derivative extensions of 3d Chern\u2013Simons

Most general third-order 3d linear gauge vector field theory is considered. The field equations involve, besides the mass, two dimensionless constant parameters. The theory admits two-parameter series of conserved tensors with the canonical energy-momentum being a particular representative of the series. For a certain range of the model parameters, the series of conserved tensors include bounded quantities. This makes the dynamics classically stable, though the canonical energy is unbounded in all the instances. The free third-order equations are shown to admit constrained multi-Hamiltonian form with the 00-components of conserved tensors playing the roles of corresponding Hamiltonians. The series of Hamiltonians includes the canonical Ostrogradski’s one, which is unbounded. The Hamiltonian formulations with different Hamiltonians are not connected by canonical transformations. This means, the theory admits inequivalent quantizations at the free level. Covariant interactions are included with spinor fields such that the higher-derivative dynamics remains stable at interacting level if the bounded conserved quantity exists in the free theory. In the first-order formalism, the interacting theory remains Hamiltonian and therefore it admits quantization, though the vertices are not necessarily Lagrangian in the third-order field equations.

&lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$C^1$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; weak Palis conjecture for nonsingular flows

This paper focuses on generic properties of continuous dynamical systems. We prove \begin{document}$C^1$\end{document} weak Palis conjecture for nonsingular flows: Morse-Smale vector fields and vector fields admitting horseshoes are open and dense among \begin{document}$C^1$\end{document} nonsingular vector fields. Our arguments contain three main ingredients: linear Poincare flow, Liao's selecting lemma and the adapting of Crovisier's central model. Firstly, by studying the linear Poincare flow, we prove for a \begin{document}$C^1$\end{document} generic vector field away from horseshoes, any non-trivial nonsingular chain recurrent class contains a minimal set which is partially hyperbolic with 1-dimensional center with respect to the linear Poincare flow. Secondly, to understand the neutral behaviour of the 1-dimensional center, we adapt Crovisier's central model. The difficulties are that we can not build invariant plaque family of any time, the periodic point of a flow is not periodic for the discrete time map. Through delicate analysis of the center manifold of a periodic orbit near the partially hyperbolic set, we manage to yield nice periodic points such that their stable manifolds and unstable manifolds are well-placed for transverse intersection.