arXiv research

Study on generalized derivations in polynomial vector fields Lie algebras.

Vector fields invariant under Lie group action are finitely generated by polynomial fields.

In this paper we address the following questions: (i) Let $C\subset \mathbb C^2$ be an orbit of a polynomial vector field which has finite total Gaussian curvature. Is $C$ contained in an algebraic curve? (ii) What can be said of a polynomial vector field which has a finitely curved transcendent orbit? We give a positi…

Characterizes values at infinity for real polynomial maps with 2D fibers.

The paper proves that certain modified conformal vector fields are trivial on compact and non-compact manifolds.

Proves a Gel'fand-Kolmogoroff type result for vector bundles and polynomial functions.

Exact universal interpolation property for landmark configurations in Euclidean space.

Researchers found indecomposable Killing tensor fields on quaternionic projective spaces.

We state some generalizations of a theorem due to G. Darboux, which originally states that a polynomial vector field in the complex plane exhibits a rational first integral and has all its orbits algebraic provided that it exhibits infinitely many algebraic orbits. In this paper, we give an interpretation of this resul…

Study examines null vector fields on Lorentzian manifolds.

Infinitesimal conformal transformations of $R^n$ are always polynomial and finitely generated when $n>2$. Here we prove that the Lie algebra of infinitesimal conformal polynomial transformations over $R^n$, $n>1$, is maximal in the Lie algebra of polynomial vector fields. When $n$ is greater than 2 and $p,q$ are such t…

In this text we give a decomposition result on polynomial poly-vector fields generalizing a result on the decomposition of homogeneous Poisson structures. We discuss consequences of this decomposition result in particular for low dimensions and low degrees. We provide the tools to calculate simple cubic Poisson structu…

We analyse the asymptotical growth of Vassiliev invariants on non-periodic flow lines of ergodic vector fields on domains of $\R^3$. More precisely, we show that the asymptotics of Vassiliev invariants is completely determined by the helicity of the vector field. As an application, we determine the asymptotic Alexander…

Researchers found non-Killing tensor fields on certain symmetric spaces.

Quadratic Killing tensors on Lie groups are always decomposable.

The Complex Axis theorem states that any endomorphism of a finite-dimensional complex vector space affords an eigen-vector (or "invariant axis"). A geometric proof of this geometric result was given by A. de Medeiros, transforming the endomorphism into a topological self-map with Lefschetz number not equal to zero. We …

The elaboration of new quantization methods has recently developed the interest in the study of subalgebras of the Lie algebra of polynomial vector fields over a Euclidean space. In this framework, these subalgebras define maximal equivariance conditions that one can impose on a linear bijection between observables tha…

The note answers a question about Betti numbers for 1D Euclidean space.

The paper generalizes polynomial functions on Lie groups and their properties.

Researchers prove Lie algebras of differential operators and Grothendieck constructions coincide.

Study the spectrum of Poincaré operator in triaxial ellipsoids.

We examine the existence of one parameter groups of diffeomorphisms whose infinitesimal generators annihilate all scalar polynomial curvature invariants through the application of the Lie derivative, known as $\mathcal{I}$-preserving diffeomorphisms. Such mappings are a generalization of isometries and appear to be rel…

Transitive local Lie algebras of vector fields can be easily constructed from dilations of $\mathbb{R}^n$ associating with coordinates positive weights (give me a sequence of $n$ positive integers and I will give you a transitive nilpotent Lie algebra of vector fields on $\mathbb{R}^n$). It is interesting that all tran…

We show that Killing tensors on conformally flat $n$-dimensional tori whose conformal factor only depends on one variable, are polynomials in the metric and in the Killing vector fields. In other words, every first integral of the geodesic flow polynomial in the momenta on the sphere bundle of such a torus is linear in…

A 3-dimensional vector field $B$ is said to be Beltrami vector field (force free-magnetic vector field in physics), if $B\times(\nabla\times B)=0$. Motivated by our investigations on projective an polynomial superflows, and as an important side result, in the first paper on this topic we constructed two unique Beltrami…

Using the Lie derivative of the metric we define a class of Lie algebras of vector fields by generalising the concept of Killing vectors. As a Lie algebra they define locally a group action on the pseudo-Riemannian manifold through exponentiation. The motivation behind studying these infinitesimal group actions is the …

Let $M$ be a smooth manifold, $\cal S$ the space of polynomial on fibers functions on $T^*M$ (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, $Vect(M)$, of vector fields on $M$ with coefficients in the space of linear differential operators on $\cal S$. This co…

We classify nontrivial deformations of the standard embedding of the Lie algebra $\Vect(S^1)$ of smooth vector fields on the circle, into the Lie algebra~$\PD(S^1)$ of pseudodifferential symbols on $S^1$. This approach leads to deformations of the central charge induced on $\Vect(S^1)$ by the canonical central extensio…

The study reveals the efficiency of sampling from tilted distributions.

Simplified KR polynomial for bipartite links reduces to tensor products of vector spaces.

We consider planar vector field without zeroes X and study the image of the associated Lie derivative operator LX acting on the space of smooth functions. We show that the cokernel of LX is infinite-dimensional as soon as X is not topologically conjugate to a constant vector field and that, if the topology of the integ…

The paper classifies Killing tensor fields on Riemannian symmetric spaces.

In this paper, we derive a new form of maximum principle for smooth functions on a complete noncompact Riemannian manifold $M$ for which there exists a bounded vector field $X$ such that $\langle\nabla f,X\rangle\geq 0$ on $M$ and $\mathrm{div} X\geq af$ outside a suitable compact subset} of $M$, for some constant $a>0…

In this paper we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle equations. In this approach, we have an integer coefficient monic polynomial $P_K(u)$ f…

For integrable Hamiltonian systems with two degrees of freedom whose Hamiltonian vector fields have incomplete flows, an analogue of the Liouville theorem is established. A canonical Liouville fibration is defined by means of an "exact" 2-parameter family of flat polygons equipped with certain pairing of sides. For the…

Cryptocurrency prices predicted using LSTM, SVM, and polynomial regression.

RFM uses tangent vector fields to match data on manifolds, analyzing TV convergence for Euler discretization.

The study explores vector flows on manifolds, focusing on polynomial constraints and equivalence relations.

We introduce a novel systematic construction for integrable (3+1)-dimensional dispersionless systems using nonisospectral Lax pairs that involve contact vector fields. In particular, we present new large classes of (3+1)-dimensional integrable dispersionless systems associated to the Lax pairs which are polynomial and …

We study a natural Lie algebra structure on the free vector space generated by all rooted planar trees as the associated Lie algebra of the nonsymmetric operad (non-$Σ$ operad, preoperad) of rooted planar trees. We determine whether the Lie algebra and some related Lie algebras are finitely generated or not, and prove …

Extends elliptic operator regularity to maximally hypoelliptic operators.

Let $X$ be a compact smooth manifold with boundary. In this article, we study the spaces $\mathcal V^\dagger(X)$ and $\mathcal V^\ddagger(X)$ of so called boundary generic and traversally generic vector fields on $X$ and the place they occupy in the space $\mathcal V(X)$ of all fields (see Theorems \ref{th3.4} and Theo…

Algorithm creates polynomials for knotted surfaces, with bounds on degree.

New concept of regular separation for ODEs leads to improved Hardy field results.

Study examines boundedness of oscillating singular integrals on specific Lie groups.

From the paper "Formality Conjecture" (Ascona 1996): "I am aware of only one such a class, it corresponds to simplest good graph, the complete graph with $4$ vertices $($and $6$ edges$)$. This class gives a remarkable vector field on the space of bi-vector fields on $\mathbb{R}^{d}$. The evolution with respect to the t…

We study the Hamiltonian vector field $v=(-\partial f/\partial w,\partial f/\partial z)$ on $\mathbb C^2$, where $f=f(z,w)$ is a polynomial in two complex variables, which is non-degenerate with respect to its Newton's polygon. We introduce coordinates in four-dimensional neighbourhoods of the "points at infinity", in …

The study computes trace fields and minimal polynomials for specific knots and links.