Class Of Vector Fields Research Articles

Algebraic structures on parallelizable manifolds

In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold L, there exists a global trivialization of the tangent bundle, which defines a map ρp:l⟶TpL for each point p∈L, where l is some vector space. This allows us to define a particular class of vector fields, known as fundamental vector fields, that correspond to each element of l. Furthermore, flows of these vector fields give rise to a product between elements of l and L, which in turn induces a local loop structure (i.e. a non-associative analog of a group). Furthermore, we also define a generalization of a Lie algebra structure on l. We will describe the properties and examples of these constructions.

Classification of vector fields and soliton structures on a tangent bundle with a Ricci quarter-symmetric metric connection

Consider $TM$ as the tangent bundle of a (pseudo-)Riemannian manifold $M$, equipped with a Ricci quarter-symmetric metric connection $\overline{\nabla }$. This research article aims to accomplish two primary objectives. Firstly, the paper undertakes the classification of specific types of vector fields, including incompressible vector fields, harmonic vector fields, concurrent vector fields, conformal vector fields, projective vector fields, and $% \widetilde{\varphi }(Ric)$ vector fields, within the framework of $\overline{% \nabla }$ on $T\dot{M}$. Secondly, the paper establishes the necessary and sufficient conditions for the tangent bundle $TM$ to become as a Riemannian soliton and a generalized Ricci-Yamabe soliton with regard to the connection $\overline{\nabla }$.

Nonuniqueness of Solutions to the Euler Equations with Vorticity in a Lorentz Space

For the two dimensional Euler equations, a classical result by Yudovich states that solutions are unique in the class of bounded vorticity; it is a celebrated open problem whether this uniqueness result can be extended in other integrability spaces. We prove in this note that such uniqueness theorem fails in the class of vector fields u with uniformly bounded kinetic energy and vorticity in the Lorentz space L^{1, infty }.

Sensitivity analysis with respect to a stochastic stock price model with rough volatility via a Bismut–Elworthy–Li formula for singular SDEs

In this paper, we show the existence of unique Malliavin differentiable solutions to SDE‘s driven by a fractional Brownian motion with Hurst parameter H<12 and singular, unbounded drift vector fields, for which we also prove a stability result. Further, using the latter results, we propose a stock price model with rough and correlated volatility, which also allows for capturing regime switching effects. Finally, we derive a Bismut–Elworthy–Li formula with respect to our stock price model for certain classes of vector fields.

The Periodic Orbit Conjecture for Steady Euler Flows

The periodic orbit conjecture states that, on closed manifolds, the set of lengths of the orbits of a non-vanishing vector field all whose orbits are closed admits an upper bound. This conjecture is known to be false in general due to a counterexample by Sullivan. However, it is satisfied under the geometric condition of being geodesible. In this work, we use the recent characterization of Eulerisable flows (or more generally flows admitting a strongly adapted one-form) to prove that the conjecture remains true for this larger class of vector fields.

Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields

ABSTRACT In the present study, we consider planar piecewise linear vector fields with two zones separated by the straight line x = 0. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has centre, a real one for y<0 and a virtual one for y>0, and such that the real centre is a global centre. Then, working with a first-order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second-order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one adittional crossing limit cycle can appear. Furthermore, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov method, the Extended Chebyshev systems with positive accuracy, and the Bendixson transformation.

On the Correct Determination of Flow of a Discontinuous Solenoidal Vector Field

We prove inequalities connecting a flow through the (n−1)-dimensional surface S of a smooth solenoidal vector field with its Lp(U)-norm (U is an n-dimensional domain that contains S ). On the basis of these inequalities, we propose a correct definition of the flow through the surface S of a discontinuous solenoidal vector field f ∈ Lp(U) (or, more precisely, of the class of vector fields that are equal almost everywhere with respect to the Lebesgue measure).

Well-posedness for the continuity equation for vector fields with suitable modulus of continuity

We prove well-posedness of linear scalar conservation laws using only assumptions on the growth and the modulus of continuity of the velocity field, but not on its divergence. As an application, we obtain uniqueness of solutions in the atomic Hardy space, H1, for the scalar conservation law induced by a class of vector fields whose divergence is an unbounded BMO function.

Ricci Solitons on 3-Symmetric Lorentzian Manifolds

An important generalization of Einstein equations on (pseudo)Riemannian manifolds is the Ricci soliton equation which was first discussed by R. Hamilton. The solving of the Ricci soliton equation becomes possible when there are some restrictions on the structure of the manifold, or the dimension, or the class of metrics, or a class of vector fields, which appears in the Ricci soliton equation. If there is a special coordinate system, then the problem of solving the Ricci soliton equation reduces to solving a system of PDE’s. There are Brinkman coordinates on Lorentzian Walker manifolds, which are Lorentzian manifolds with a parallel (in terms of Levi-Civita) distribution of isotropic lines. This fact allows one to investigate the Ricci soliton equation on these manifolds. The geometry of Walker manifolds and Ricci solitons on them were studied by many mathematicians. In this paper, we investigate the Ricci soliton equation on 3-symmetric indecomposable Lorentzian manifolds. These manifolds have been studied byD.V. Alekseevskii and A.S. Galaev, who have built a special local coordinate system. This article continues the authors’ study and the study of K. Honda and B. Batat, who have investigated Ricci solitons on 2-symmetric Lorentzian manifolds.DOI 10.14258/izvasu(2018)1-21

Well posedness of ODE’s and continuity equations with nonsmooth vector fields, and applications

This paper provides an overview of the theory of flows associated to nonsmooth vector fields, describing the main developments from the seminal paper (DiPerna and Lions in Invent Math 98:511–547, 1989) till now. The problem of well posedness of ODE’s associated to vector fields arises in many fields, for instance conservation laws (via the theory of characteristics) and fluid mechanics (when looking for consistence between Eulerian and Lagrangian points of view). The theory developed so far covers many classes of vector fields and, besides uniqueness, also more quantitative aspects, as stability estimates and differentiability of the flow. Detailed lecture notes on this topic are given in Ambrosio (in: Dacorogna, Marcellini (eds) Lecture Notes in Mathematics “Calculus of variations and non-linear partial differential equations” (CIME Series, Cetraro, 2005), vol 1927, pp 2–41, 2008), Ambrosio and Crippa (Lect Notes UMI 5:3–54, 2008), Ambrosio and Crippa (Proc R Soc Edinb Sect A 144:1191–1244 2014), Ambrosio and Trevisan (Ann Fac Sci Toulouse, 2016).

Sub-subleading soft gravitons and large diffeomorphisms

We present strong evidence that the sub-subleading soft theorem in semiclassical (tree level) gravity discovered by Cachazo and Strominger is equivalent to the conservation of asymptotic charges associated to a new class of vector fields not contained within the previous extensions of BMS algebra. Our analysis crucially relies on analyzing the hitherto established equivalences between soft theorems and Ward identities from a new perspective. In this process we naturally (re)discover a class of ‘magnetic’ charges at null infinity that are associated to the dual of the Weyl tensor.

О солитонах Риччи на 2-симметрических лоренцевых многообразиях

Ricci solitons are an important generalization of Einstein metrics on (pseudo) Riemannian manifolds, and this notion was introduced by R.Hamilton. The problem of solving the Ricci soliton equation is quite difficult, therefore one can assume some restrictions either on a structure of the manifold or on the dimension or on a class of metrics, or on a class of vector fields, which are contained in the Ricci soliton equation. Walker manifolds are one of the most important examples of such restrictions, that is pseudo-Riemannian manifolds admitting a smooth parallel (in sense of Levi-Civita connection) isotropic distribution. The geometry of Walker manifolds and Ricci solitons on them were studied by many mathematicians. In this paper, we investigate the Ricci soliton equation on some Lorentzian manifolds. In particular, we study the Ricci solitons on 2-symmetric Lorentzian manifolds, which are Walker manifolds, as it was proven by D.V. Alekseevsky and A.S. Galaev. K. Onda and B. Batat investigated Ricci solitons on fourdimensional 2-symmetric Lorentzian manifolds, and proved local solvability of the Ricci soliton equation on such manifolds. In this paper we have obtained local solvability of the Ricci soliton equation on fivedimensional 2-symmetric Lorentzian manifolds.

Infinite-dimensional statistical manifolds based on a balanced chart

We develop a family of infinite-dimensional Banach manifolds of measures on an abstract measurable space, employing charts that are "balanced" between the density and log-density functions. The manifolds, $(\tilde{M}_{\lambda},\lambda\in [2,\infty))$, retain many of the features of finite-dimensional information geometry; in particular, the $\alpha$-divergences are of class $C^{\lceil\lambda\rceil-1}$, enabling the definition of the Fisher metric and $\alpha$-derivatives of particular classes of vector fields. Manifolds of probability measures, $(M_{\lambda},\lambda\in [2,\infty))$, based on centred versions of the charts are shown to be $C^{\lceil\lambda \rceil-1}$-embedded submanifolds of the $\tilde{M}_{\lambda}$. The Fisher metric is a pseudo-Riemannian metric on $\tilde{M}_{\lambda}$. However, when restricted to finite-dimensional embedded submanifolds it becomes a Riemannian metric, allowing the full development of the geometry of $\alpha$-covariant derivatives. $\tilde{M}_{\lambda}$ and $M_{\lambda}$ provide natural settings for the study and comparison of approximations to posterior distributions in problems of Bayesian estimation.

On a criterion for the complete continuity of the Fréchet derivative

In this paper we introduce, by means of the Hausdorff measure of noncompactness χ, two new classes of operators (not necessarily linear): operators locally strongly χ-condensing at a point and operators strongly χ-condensing at infinity (on spherical interlayers). These classes include all completely continuous operators and some noncondensing operators. Necessary and sufficient conditions for the complete continuity of a Frechet derivative at a point and of an asymptotic derivative (if they exist) are proved. M. A. Krasnosel′skii’s theorem on asymptotic bifurcation points for completely continuous vector fields is generalized to the class of vector fields strongly χ-condensing at infinity.

Description of a helical motion of an incompressible nonviscous fluid

We consider the problem of describing the motion of a fluid filling at any specific time t ≥ 0 a domain D ⊂ R3 in terms of velocity v and pressure p. We assume that the pair of variables (v, p) satisfies a system of equations that includes Euler’s equation and the incompressible fluid continuity equation. For the case of an axially symmetric cylindrical layer D, we find a general solution of this system of equations in the class of vector fields v whose lines for any t ≥ 0 coincide everywhere in D with their vortex lines and lie on axially symmetric cylindrical surfaces nested in D. The general solution is characterized in a theorem. As an example, we specify a family of solutions expressed in terms of cylindrical functions, which, for D = R3, includes a particular solution obtained for the first time by I.S. Gromeka in the case of steady-state helical cylindrical motions.

Mathematical modeling and vector fields of dynamical systems in chemical technology of organic substances

Some aspects of mathematical modeling used in chemical technology have been considered. The most attention has been concentrated on the mathematical models in differential form, for which the investigation of common regularities of multidimensional vector fields of dynamical systems, based on which the driving force of the process is formed, represents a necessary step. The classification of vector fields used in particular in the case of consideration of chemical reactions and reactors, separation processes and combined reaction-mass transfer processes has been performed. The studies of vector fields, which are typical of the chemical technology process under discussion make it possible to meaningfully proceed to the numerical mathematical modeling of this process on a computer.

Integrable systems on S3

We classify the links of basic periodic orbits of integrable vector fields on S3 generalizing results on two degree of freedom Hamiltonian systems. We also study the case of completely integrable systems and define invariants for the two classes of vector fields.

Topological and analytical classification of vector fields with only isochronous centres

We study vector fields on the plane having only isochronous centres. The most familiar examples are isochronous vector fields, they are the real parts of complex polynomial vector fields on having all their zeroes of centre type. We describe the number of topologically inequivalent isochronous (singular) foliations that can appear for degree s, up to orientation preserving homeomorphisms. For each s, there exists a real analytic variety parametrizing the isochronous vector fields of degree s, the group of complex automorphisms of the plane acts on it. Furthermore, if , then is a non-singular real analytic variety of dimension , and their number of connected components is bounded by . An explicit formula for the residues of the rational 1-form, canonically associated with a complex polynomial vector field with simple zeroes, is given. A collection of residues (i.e. periods) does not characterize an isochronous vector field, even up to complex automorphisms of . An exact bound for the number of isochronous vector fields, up to , having the same collection of residues (periods) is given. We develop several descriptions of the quotient space using residues, weighted s-trees and singular flat Riemannian metrics associated with isochronous vector fields.

Asymptotics of the monodromy transformation in certain classes of monodromy germs

We calculate the second term of the asymptotics of the monodromy transformation of a monodromic singular point of an analytic vector field on a plane whose Newton diagram consists of one or two edges. In the cases under consideration the principal term of the monodromy transformation coincides with the identity function. The case of two edges is characterized by the fact that, as a result of blowing up the singularity by the Newton diagram, a singular point emerges that is a degenerate saddle. The results obtained make it possible to state a sufficient condition for the existence of a focus and to construct the stability boundary in the classes of vector fields under consideration.

Quasispaces induced by vector fields measurable in ℝ3

We study some metric functions that are induced by a class of basis vector fields in ℝ3 with measurable coordinates. These functions are proved to be quasimetrics in the domain of definition of the vector fields. Under some natural constraints, the Rashevsky-Chow Theorem and the Ball-Box Theorem are established for the classes of vector fields we consider.