Smooth Vector Fields Research Articles

Modified general relativity and dark matter

Modified General Relativity (MGR) is the natural extension of General Relativity (GR). MGR explicitly uses the smooth regular line element vector field [Formula: see text], which exists in all Lorentzian spacetimes, to construct a connection-independent symmetric tensor that represents the energy–momentum of the gravitational field. It solves the problem of the nonlocalization of gravitational energy–momentum in GR, preserves the ontology of the Einstein equation, and maintains the equivalence principle. The line element field provides MGR with the extra freedom required to describe dark energy and dark matter. An extended Schwarzschild solution for the matter-free Einstein equation of MGR is developed, from which the Tully–Fisher relation is derived, and the gravitational energy density is calculated. The mass of the invisible matter halo of galaxy NGC 3198 calculated with MGR is identical to the result obtained from GR using a dark matter profile. Although dark matter in MGR is described geometrically, it has an equivalent representation as a particle with the property of a vector boson or a pair of fermions; the geometry of spacetime and the quantum nature of matter are linked together by the unit line element covectors that belong to both the Lorentzian metric and the spin-1 Klein–Gordon wave equation. The three classic tests of GR provide a comparison of the theories in the solar system and several parts of the cosmos. MGR provides the flexibility to describe inflation after the Big Bang and galactic anisotropies.

On the indices of singular points for planar bounded piecewise smooth polynomial vector field

We prove that for any piecewise-smooth bounded polynomial vector field in R2 with finitely many finite H-singular points (which include singular points, hyperbolic pseudo-equilibria and two fold singularities), the sum of the indices of all its finite H-singular points is 1.

On topological entropy of piecewise smooth vector fields

Non-smooth vector fields do not have necessarily the property of uniqueness of solution passing through a point and this is responsible to enrich the behavior of the system. Even on the plane, non-smooth vector fields can be chaotic, a feature impossible for the smooth or continuous case. We propose a new approach to better understand chaos for non-smooth vector fields by using the notion of entropy of a system. We construct a metric space of all possible trajectories of a non-smooth vector field, where we define a flow inherited by the vector field and then define the topological entropy in this scenario. As a consequence, we are able to obtain some general results and give some examples of planar non-smooth vector fields with positive (finite and infinite) entropy.

Parallel Translations for a Left Invariant Spray

In this paper, we study the left invariant spray geometry on a connected Lie group. Using the technique of invariant frames, we find the ordinary differential equations on the Lie algebra describing for a left invariant spray structure the linearly parallel translations along a geodesic and the nonlinearly parallel translations along a smooth curve. In these equations, the connection operator plays an important role. Using parallel translations, we provide alternative interpretations or proofs for some homogeneous curvature formulae. In particular, the Riemannian curvture appears in both a double Lie derivative along the spray vector and brackets between smooth vector fields induced by the connection operator. We propose two questions in left invariant spray geometry. One question generalizes Landsberg Problem in Finsler geometry, and the other concerns the restricted holonomy group.

A class of exact solutions of the Navier–Stokes equations in three and four dimensions

A few basic, intuitive, properties of the Navier–Stokes system of equations for incompressible fluid flows are discussed in this paper. We present a rephrased interpretation of the Navier–Stokes equation in a space having an arbitrary number of dimensions. We then derive spatially periodic solutions for the velocity and pressure fields that span an unbounded domain in three and four dimensions, given a smooth solenoidal initial velocity vector field. In these solutions all velocity components depend non-trivially on all coordinate directions.

The Fourth-Linear aff(1)-Invariant Differential Operators and the First Cohomology of the Lie Algebra of Vector Fields on RP1

In this paper, we denote the Lie algebra of smooth vector fields on RP1 by V(RP1). This paper focuses on two parts. In the beginning, we determine the cohomology space of aff(1) with coefficients in Dτ,λ,μ;ν. Afterward, we classify aff(1)-invariant fourth-linear differential operators from V(RP1) to Dτ,λ,μ;ν vanishing on aff(1). This result enables us to compute the aff(1)-relative cohomology of V(RP1) with coefficients in Dτ,λ,μ;ν.

The local behavior around switching planes in a mathematical model to chemoimmunotherapy

In this paper we consider a piecewise smooth mathematical model to chemoimmunotherapy. The positive octant of the 3D space (N,I,Q) is partitioned into 4 regions separated by the planes I=I0 and N=N0. The plane I=I0 (respectively, N=N0) controls if the immunotherapy (respectively, chemotherapy) is being applied or not. As a consequence, there are 4 smooth vector fields modeling each one of the scenarios. By means of the theory developed by piecewise smooth vector fields we are able to show the kind of contact between the trajectories of the smooth vector fields with the switching planes, the sliding motion defined over the planes and even the sliding motion defined over the simultaneous tangency lines of two smooth vector fields and the switching planes. This description is very useful in the understanding of the behavior of the treatment proposed.

Helmholtz decomposition and potential functions for n-dimensional analytic vector fields

The Helmholtz decomposition splits a sufficiently smooth vector field into a gradient field and a divergence-free rotation field. Existing decomposition methods impose constraints on the behavior of vector fields at infinity and require solving convolution integrals over the entire coordinate space. To allow a Helmholtz decomposition in Rn, we replace the vector potential in R3 by the rotation potential, an n-dimensional, antisymmetric matrix-valued map describing n(n−1)/2 rotations within the coordinate planes. We provide three methods to derive the Helmholtz decomposition: (1) a numerical method for fields decaying at infinity by using an n-dimensional convolution integral, (2) closed-form solutions using line-integrals for several unboundedly growing fields including periodic and exponential functions, multivariate polynomials and their linear combinations, (3) an existence proof for all analytic vector fields. Examples include the Lorenz and Rössler attractor and the competitive Lotka–Volterra equations with n species.

Geometry of concircular curvature tensor of Nearly Kahler manifold

In this paper, we study the necessary condition where a nearly Kahler manifold of flat concircular tensor has been found. And the relationship between these invariants and additional properties of symmetry concircular tensor, as well as geometrical sense of the reference (manipulation) in zero of these invariants are studied.
 Let M - smooth manifold of dimension 2n; - algebra of smooth functions on M; - the module of smooth vector fields on manifold of M; g = <·, > - Riemannian metrics; Ñ – Riemannian connection of the metrics g on M; d - the operator of exterior differentiation. In the further all manifold, Tensor fields, etc. objects are assumed smooth a class .

Strongly mixing smooth planar vector field without asymptotic directions

We use a Voronoi-type tessellation based on a compound Poisson point process to construct a polynomially mixing stationary random smooth planar vector field with bounded nonnegative components such that, with probability one, none of the associated integral curves possess an asymptotic direction.

Periodic orbits in piecewise Riccati vector fields

In this paper a class of planar vector fields is characterized and their periodic orbits are studied. The class under study is formed by vector fields that can be written as Riccati ordinary differential equations. The characterization is stated in terms of some homogeneity conditions on the vector fields. The study of periodic orbits is done through Poincaré maps defined around monodromic points. In particular, some properties of the so-called Möbius transformations are considered. In the study, both smooth and discontinuous vector fields defined by sectors are studied.Theorem 2.2 characterizes the Riccati vector fields and Theorem 3.4 proves that the Poincaré maps of such vector fields are Möbius transformations. In Theorem 4.1 a sharp upper bound of two periodic orbits is stated. The polynomial case is considered in Theorems 4.3 and 4.4 for which existence, upper bounds, and hyperbolicity of periodic orbits are stated.

A nonexistence result of positive stable solutions to a weighted elliptic equation

We investigate the following weighted elliptic equation ??u + b(x) ? ?u = (1 + |x|2)?2 up in RN, where ? ? 0, p > 1, N ? 3 and the advection term b(x) is a smooth vector field satisfying certain decay condition. We establish a Liouville-type theorem for possible stable solutions of the equation above.

Gradient estimate of a Poisson equation under the almost Ricci solitons

In this paper, we consider an n-dimensional manifold M^{n} endowed with an almost Bakry–Émery Ricci curvature and study a special case of gradient estimate for the positive solutions of Delta u-X.u=f, for a smooth function f and a smooth vector field X under the almost Ricci solitons condition.

Linearization and Perturbations of Piecewise Smooth Vector Fields with a Boundary Equilibrium

Linearization and Perturbations of Piecewise Smooth Vector Fields with a Boundary Equilibrium

Vector fields and differential equations

The geometry of orbits of families of smooth vector fields is an important object of mathematics due to its importance in applications, in the theory of dynamic systems and in the foliation theory. The paper devoted to the applications of the geometry of orbits of vector fields in four dimensional Euclidean space in theory of differential equations. It is shown that orbits generate singular foliation ever 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. In this paper smoothness is smoothness of the class C ∞.

Connections between nonlocal operators: from vector calculus identities to a fractional Helmholtz decomposition

Nonlocal vector calculus, which is based on the nonlocal forms of gradient, divergence, and Laplace operators in multiple dimensions, has shown promising applications in fields such as hydrology, mechanics, and image processing. In this work, we study the analytical underpinnings of these operators. We rigorously treat compositions of nonlocal operators, prove nonlocal vector calculus identities, and connect weighted and unweighted variational frameworks. We combine these results to obtain a weighted fractional Helmholtz decomposition which is valid for sufficiently smooth vector fields. Our approach identifies the function spaces in which the stated identities and decompositions hold, providing a rigorous foundation to the nonlocal vector calculus identities that can serve as tools for nonlocal modeling in higher dimensions.

Self-supervised learning of neighborhood embedding for longitudinal MRI.

In recent years, several deep learning models recommend first to represent Magnetic Resonance Imaging (MRI) as latent features before performing a downstream task of interest (such as classification or regression). The performance of the downstream task generally improves when these latent representations are explicitly associated with factors of interest. For example, we derived such a representation for capturing brain aging by applying self-supervised learning to longitudinal MRIs and then used the resulting encoding to automatically identify diseases accelerating the aging of the brain. We now propose a refinement of this representation by replacing the linear modeling of brain aging with one that is consistent in local neighborhoods in the latent space. Called Longitudinal Neighborhood Embedding (LNE), we derive an encoding so that neighborhoods are age-consistent (i.e., brain MRIs of different subjects with similar brain ages are in close proximity of each other) and progression-consistent, i.e., the latent space is defined by a smooth trajectory field where each trajectory captures changes in brain ages between a pair of MRIs extracted from a longitudinal sequence. To make the problem computationally tractable, we further propose a strategy for mini-batch sampling so that the resulting local neighborhoods accurately approximate the ones that would be defined based on the whole cohort. We evaluate LNE on three different downstream tasks: (1) to predict chronological age from T1-w MRI of 274 healthy subjects participating in a study at SRI International; (2) to distinguish Normal Control (NC) from Alzheimer's Disease (AD) and stable Mild Cognitive Impairment (sMCI) from progressive Mild Cognitive Impairment (pMCI) based on T1-w MRI of 632 participants of the Alzheimer's Disease Neuroimaging Initiative (ADNI); and (3) to distinguish no-to-low from moderate-to-heavy alcohol drinkers based on fractional anisotropy derived from diffusion tensor MRIs of 764 adolescents recruited by the National Consortium on Alcohol and NeuroDevelopment in Adolescence (NCANDA). Across the three data sets, the visualization of the smooth trajectory vector fields and superior accuracy on downstream tasks demonstrate the strength of the proposed method over existing self-supervised methods in extracting information related to brain aging, which could help study the impact of substance use and neurodegenerative disorders. The code is available at https://github.com/ouyangjiahong/longitudinal-neighbourhood-embedding.

Frequency entrainment and suppression of stick–slip vibrations in a 3 DoF discontinuous disc brake model

Friction-induced stick–slip vibration has been identified as the main reason for various types of disc brake noises like creep groan, squeak and creak. In this work, we investigate the suppression of stick–slip vibrations in disc brakes due to the application of a small-amplitude normal harmonic force on the brake pad. The discontinuity induced bifurcations that lead to such suppression are studied in detail. The frequency range of the normal force in which suppression of stick–slip vibration occur is identified. A 3 DoF discontinuous model of the disc brake is considered. The discontinuous oscillator is modelled as a Filippov system with smooth vector fields separated by a discontinuity surface. It is shown that the normal harmonic force entrains the non-smooth limit cycle even for very small forcing amplitudes. Different frequency intervals exhibiting harmonic, super-harmonic, and sub-harmonic entrainments are identified. It is demonstrated that the initiation and termination of these entrainment intervals are associated with Neimark–Sacker bifurcations. It is also shown that discontinuity induced grazing-sliding mechanism brings about these bifurcations. Switching-sliding bifurcations happening in the entrainment region give rise to a sub-interval which is associated with suppression of stick–slip vibrations. We also report interesting discontinuity induced bifurcation scenarios in higher order entrainment regimes like grazing-sliding involving different orbit loops at lower and upper limits of the interval and splitting–merging of the sliding surface. • Stick–slip vibration suppression in disc brakes by using normal force is studied. • Various entrainment regions are detected and studied by Filippov’s method. • Discontinuity induced bifurcations causing these entrainments are analysed. • Entrainment sub-intervals with suppressed stick–slip vibration are identified.

Learning Stable Vector Fields on Lie Groups

Learning robot motions from demonstration requires models able to specify vector fields for the full robot pose when the task is defined in operational space. Recent advances in reactive motion generation have shown that learning adaptive, reactive, smooth, and stable vector fields is possible. However, these approaches define vector fields on a flat Euclidean manifold, while representing vector fields for orientations requires modeling the dynamics in non-Euclidean manifolds, such as Lie Groups. In this paper, we present a novel vector field model that can guarantee most of the properties of previous approaches i.e., stability, smoothness, and reactivity beyond the Euclidean space. In the experimental evaluation, we show the performance of our proposed vector field model to learn stable vector fields for full robot poses as SE(2) and SE(3) in both simulated and real robotics tasks.

Vorticity Gramian of Compact Riemannian Manifolds

The vorticity of a vector field on 3-dimensional Euclidean space is usually given by the curl of the vector field. In this paper, we extend this concept to n-dimensional compact and oriented Riemannian manifold. We analyse many properties of this operation. We prove that a vector field on a compact Riemannian manifold admits a unique Helmholtz decomposition and establish that every smooth vector field on an open neighbourhood of a compact Riemannian manifold admits a Stokes’ type identity.