Smooth Vector Fields Research Articles

On improving sound source localization in the wind tunnel

When traveling in an open-jet wind tunnel, the path of an acoustic wave is affected by the flow causing a shift of source positions in acoustical maps of phased arrays outside the flow. In this paper, we start by comparing several well-known approaches to correct travel times between microphones and assumed sources, used, for example, by beamforming algorithms in such an environment. The methods under consideration include the original 1D-Amiet/Bahr formulation, a standard 2.5D-planar approach that assumes the constancy of angular frequency and wave-number-vector components along a plane, and finally, a 3D-ray-tracing method. Common to the former two of these methods is the assumption that the boundary layer of the flow, the so-called shear layer, is infinitely thin and refracts the acoustical ray, whereas, in principle, the latter algorithm allows for an arbitrary (but sufficiently smooth) vector field modeling the flow. Going through these methods, we discuss travel-time results relative to each other and in terms of differences in source localization using beamforming maps. In particular, we model the flow of an existing wind tunnel and discuss results of real-world measurements comparing the 2.5D-planar approach with the 3D-ray-tracing method.

Local models for smooth vector fields of the line

We present the local classification of singularities of smooth vector fields on the line, with respect to the equivalence relation of C1-conjugacy. Along the way, we recall the analogous classification, up to C0 and C∞ conjugacy. We also give the transversal unfoldings of the corresponding normal forms and treat the case where the changes of coordinates are tangent to the identity. Thus, a fairly complete description of the 1-d case is achieved.

Reactive optimal motion planning to anywhere in the presence of moving obstacles

In this paper, a novel optimal motion planning framework that enables navigating optimally from any initial, to any final position within confined workspaces with convex, moving obstacles is presented. Our method outputs a smooth velocity vector field, which is then employed as a reference controller in order to sub-optimally avoid moving obstacles. The proposed approach leverages and extends desirable properties of reactive methods in order to provide a provably convergent and safe solution. Our algorithm is evaluated with both static and moving obstacles in synthetic environments and is compared against a variety of existing methods. The efficacy and applicability of the proposed scheme is finally validated in a high-fidelity simulation environment.

Improving hybrid image and structure-based deformable image registration for large internal deformations

Objective. Deformable image registration (DIR) is a widely used technique in radiotherapy. Complex deformations, resulting from large anatomical changes, are a regular challenge. DIR algorithms generally seek a balance between capturing large deformations and preserving a smooth deformation vector field (DVF). We propose a novel structure-based term that can enhance the registration efficacy while ensuring a smooth DVF. Approach. The proposed novel similarity metric for controlling structures was introduced as a new term into a commercially available algorithm. Its performance was compared to the original algorithm using a dataset of 46 patients who received pelvic re-irradiation, many of which exhibited complex deformations. Main results. The mean Dice Similarity Coefficient (DSC) under the improved algorithm was 0.96, 0.94, 0.76, and 0.91 for bladder, rectum, colon, and bone respectively, compared to 0.69, 0.89, 0.62, and 0.88 for the original algorithm. The improvement was more pronounced for complex deformations. Significance. With this work, we have demonstrated that the proposed term is able to improve registration accuracy for complex cases while maintaining realistic deformations.

Integrable 2D and 3D piecewise smooth vector fields with chaotic behavior and preserving energy or not

In this paper we exhibit 2D and 3D piecewise smooth vector fields that have, at the same time, first integrals and chaotic behavior. The fact of existing first integrals means that the trajectories are tangent to foliations. Also, in some of these examples the trajectories live in an unique level of the first integrals, and so the system preserves energy, and in other cases the trajectories visit distinct levels of the first integrals and the system does not preserves the energy.

Planar quartic–quadratic fold–fold singularity of Filippov systems and its bifurcation

In this paper it is exhibited the bifurcation scenario concerning a typical singularity of planar piecewise smooth vector fields in two zones. This singularity is characterized by a quadratic contact of one vector field and a quartic contact of another vector field at the same point of the switching manifold. By means of a three parameter perturbation, we observe the presence of bifurcations like: Hopf Bifurcation, Sliding Hopf Bifurcation, the birth and annihilation of chaotic scenarios, among others. The two and three dimensional bifurcation diagrams are included.

Density of weak solutions of the fractional Navier-Stokes equations in the smooth divergence-free vector fields

Density of weak solutions of the fractional Navier-Stokes equations in the smooth divergence-free vector fields

3-dimensional piecewise linear and quadratic vector fields with invariant spheres

We consider the class X of 3 -dimensional piecewise smooth vector fields that admit a first integral which leaves invariant any sphere centered at the origin. In this class, we prove that a linear vector field does not admit isolated invariant cones. Moreover, we provide the existence of at least ten 1 -parameter families of crossing closed trajectories for quadratic vector fields in X .

S-Embedding of Lie Superalgebras and Its Implications for Fuzzy Lie Algebras

This paper performed an investigation into the s-embedding of the Lie superalgebra (→S1∣1), a representation of smooth vector fields on a (1,1)-dimensional super-circle. Our primary objective was to establish a precise definition of the s-embedding, effectively dissecting the Lie superalgebra into the superalgebra of super-pseudodifferential operators ( SψD⊙) residing on the super-circle S1|1. We also introduce and rigorously define the central charge within the framework of (→S1∣1), leveraging the canonical central extension of SψD⊙. Moreover, we expanded the scope of our inquiry to encompass the domain of fuzzy Lie algebras, seeking to elucidate potential connections and parallels between these ostensibly distinct mathematical constructs. Our exploration spanned various facets, including non-commutative structures, representation theory, central extensions, and central charges, as we aimed to bridge the gap between Lie superalgebras and fuzzy Lie algebras. To summarize, this paper is a pioneering work with two pivotal contributions. Initially, a meticulous definition of the s-embedding of the Lie superalgebra (→S1|1) is provided, emphasizing the representationof smooth vector fields on the (1,1)-dimensional super-circle, thereby enriching a fundamental comprehension of the topic. Moreover, an investigation of the realm of fuzzy Lie algebras was undertaken, probing associations with conventional Lie superalgebras. Capitalizing on these discoveries, we expound upon the nexus between central extensions and provide a novel deformed representation of the central charge.

Algebraic Structures on Smooth Vector Fields

The aim of this work is to investigate some algebraic structures of objects which are defined and related to a manifold. Consider L to be a smooth manifold and Γ∞(TL) to be the module of smooth vector fields over the ring of smooth functions C∞(L). We prove that the module Γ∞(TL) is projective and finitely generated, but it is not semisimple. Therefore, it has a proper socle and nonzero Jacobson radical. Furthermore, we prove that this module is reflexive by showing that it is isomorphic to its bidual. Additionally, we investigate the structure of the Lie algebra of smooth vector fields. We give some questions and open problems at the end of the paper. We believe that our results are important because they link two different disciplines in modern pure mathematics.

A layer potential approach to inverse problems in brain imaging

Abstract We study the inverse source localisation problem using the electric potential measured point-wise inside the head with stereo-ElectroEncephaloGraphy (sEEG), the electric potential measured point-wise on the scalp with ElectroEncephaloGraphy (EEG) or the magnetic flux density measured point-wise outside the head with MagnetoEncephaloGraphy (MEG). We present a method that works on a wide range of models of primary currents; in particular, we give details for primary currents that are assumed to be smooth vector fields that are supported on and normally oriented to the grey/white matter interface. Irrespective of the data used, we also solve the transmission problem of the electric potential associated with a recovered source; hence we solve the cortical mapping problem. To ensure that the electric potential and normal currents are continuous in the head, the electric potential is expressed as a linear combination of double layer potentials and the magnetic flux density is expressed as a linear combination of single layer potentials. Numerically, we solve the problems on meshed surfaces of the grey/white matter interface, cortical surface, skull and scalp. A main feature of the numerical approach we take is that, on the meshed surfaces, we can compute the double and single layer potentials exactly and at arbitrary points. Because we explicitly study the transmission of the electric potential in head when using magnetic data, the coupling of electric and magnetic data in the source recovery problem is made explicit and shows the advantage of using simultaneous electric and magnetic data. We provide numerical examples of the source recovery and inverse cortical mapping using synthetic data.

Rigidity and nonexistence of complete hypersurfaces via Liouville type results and other maximum principles, with applications to entire graphs

We investigate complete hypersurfaces with some positive higher order mean curvature in a semi-Riemannian warped product space. Under standard curvature conditions on the ambient space and appropriate constraints on the higher order mean curvatures, we establish rigidity and nonexistence results via Liouville type results and suitable maximum principles related to the divergence of smooth vector fields on a complete noncompact Riemannian manifold. Applications to standard warped product models, like the Schwarzschild, Reissner-Nordström and pseudo-hyperbolic spaces, as well as steady state type spacetimes, are given and a particular study of entire graphs is also presented.

Asymptotic symmetries in Bondi gauge and the sub-subleading soft graviton theorem

We investigate asymptotic symmetries which preserve the Bondi gauge conditions but do not preserve the asymptotic falloff conditions for the metric near the null boundary, and their connection to soft graviton theorems for scattering amplitudes. These include generalized superrotation symmetries parameterized by a smooth vector field Y A obeying , for which we show that the associated conserved charge can be derived by applying the Noether procedure to the Einstein–Katz action. We also discuss the connection between asymptotic symmetries and the conserved charge associated with the sub-subleading soft theorem, and we find that in Bondi gauge this charge is generated by the combination of a diffeomorphism together with an extra transformation of the metric.

Global Dynamics of Planar Piecewise Linear Inelastic systems having straight lines as switching manifolds

In this paper we classify the phase portrait and the limit sets of a special class of piecewise smooth vector fields in the plane with different configurations of straight lines as switching manifolds. We study the behavior at infinity through a Poincaré compactification as well as the relations between canonical regions and vector fields which are defined over the switching regions. Results addressing the global behavior of trajectories, tangency points, vector fields over the switching manifolds, and equilibrium and pseudo-equilibrium points at the finite and the infinite part are stated. In particular, we prove the existence of 123 distinct phase portraits for the class of piecewise smooth vector fields that we consider.

Integration of path planning and following control for the stratospheric airship with forecasted wind field data

The absence of real-time airspeed sensors, which was more often ignored in previous studies, and low dynamic characteristics render stratospheric airship control challenging. This study creatively overcomes the aforementioned problems in an integrated path planning and following control scheme using forecasted wind field data. Herein, an efficient and practicable path planning algorithm is designed. Further, a smooth vector field guidance law is proposed for solving the problem of complex path following. Subsequently, an event-triggered neural network-based adaptive tracking controller is designed, considering the wind forecast error influence. Finally, these three parts are organically integrated to achieve autonomous flight. The stability of the closed-loop system and the exclusion of Zeno behavior are rigorously proved. The simulation results reveal that the convergence rate is 63.8% improved, essentially exhibiting better optimization, the tracking errors are eliminated within 80 s, and 99.4% control input updating times are saved.

Structural Stability for 2-Dimensional Piecewise Smooth Vector Fields Where the Switching Manifold is a Double Discontinuity

Structural Stability for 2-Dimensional Piecewise Smooth Vector Fields Where the Switching Manifold is a Double Discontinuity

Modal Projection for Quasi-Homogeneous Anisotropic Turbulence

This article, or essay, addresses the anisotropic structure and the dynamics of quasi-homogeneous, incompressible turbulence. Modal projection and expansions in terms of spherical harmonics in three-dimensional Fourier space are in line with a seminal study by Jack Herring, around the so-called Craya–Herring frame of reference, with a large review of the related approaches to date. The research part is focused on structure and dynamics of rotating sheared turbulence, including a description of both directional and polarization anisotropy with a minimal number of modes. Effort is made to generalize expansions in terms of scalar spherical harmonics (SSHs) to vector spherical harmonics (VSHs). Looking at stochastic fields, for possibly intermittent vector fields, some directions are explored to reconcile modal projection, firstly used for smooth vector fields, and multifractal approaches for internal intermittency but far beyond scalar correlations, such as structure functions. In order to illustrate turbulence from Earth to planets, stars, and galaxies, applications to geophysics and astrophysics are touched upon, with generalization to coupled vector fields (for kinetic, magnetic, and potential energies), possibly dominated by waves (Coriolis, gravity, and Alfvén).

Slow-Fast Normal Forms Arising from Piecewise Smooth Vector Fields

Slow-Fast Normal Forms Arising from Piecewise Smooth Vector Fields

Weighted Radon transforms of vector fields, with applications to magnetoacoustoelectric tomography

Currently, theory of ray transforms of vector and tensor fields is well developed, but the Radon transforms of such fields have not been fully analyzed. We thus consider linearly weighted and unweighted longitudinal and transversal Radon transforms of vector fields. As usual, we use the standard Helmholtz decomposition of smooth and fast decreasing vector fields over the whole space. We show that such a decomposition produces potential and solenoidal components decreasing at infinity fast enough to guarantee the existence of the unweighted longitudinal and transversal Radon transforms of these components. It is known that reconstruction of an arbitrary vector field from only longitudinal or only transversal transforms is impossible. However, for the cases when both linearly weighted and unweighted transforms of either one of the types are known, we derive explicit inversion formulas for the full reconstruction of the field. Our interest in the inversion of such transforms stems from a certain inverse problem arising in magnetoacoustoelectric tomography (MAET). The connection between the weighted Radon transforms and MAET is exhibited in the paper. Finally, we demonstrate performance and noise sensitivity of the new inversion formulas in numerical simulations.

The Geometry of Vector Fields and Two Dimensional Heat Equation

The geometry of orbits of families of smooth vector fields was studied by many mathematicians due to its importance in applications in the theory of control systems, in dynamic systems, in geometry and in the theory of foliations. In this paper it is studied geometry of orbits of vector fields in four dimensional Euclidean space. It is shown that orbits generate singular foliation every regular leaf of which is a surface of negative Gauss curvature and zero normal torsion. In addition, the invariant functions of the considered vector fields are used to find solutions of the two-dimensional heat equation that are invariant under the groups of transformations generated by these vector fields.