Search results for "vector [correlation function]"

The Lie algebra of polynomial vector fields and the Jacobian conjecture

1998

The Jacobian conjecture for polynomial maps ϕ:Kn→Kn is shown to be equivalent to a certain Lie algebra theoretic property of the Lie algebra\(\mathbb{D}\) of formal vector fields inn variables. To be precise, let\(\mathbb{D}_0 \) be the unique subalgebra of codimensionn (consisting of the singular vector fields),H a Cartan subalgebra of\(\mathbb{D}_0 \),Hλ the root spaces corresponding to linear forms λ onH and\(A = \oplus _{\lambda \in {\rm H}^ * } H_\lambda \). Then every polynomial map ϕ:Kn→Kn with invertible Jacobian matrix is an automorphism if and only if every automorphism Φ of\(\mathbb{D}\) with Φ(A)\( \subseteq A\) satisfies Φ(A)=A.

More limit cycles than expected in Liénard equations

2007

The paper deals with classical polynomial Lienard equations, i.e. planar vector fields associated to scalar second order differential equations x"+ f(x)x' + x = 0 where f is a polynomial. We prove that for a well-chosen polynomial f of degree 6, the equation exhibits 4 limit cycles. It induces that for n ≥ 3 there exist polynomials f of degree 2n such that the related equations exhibit more than n limit cycles. This contradicts the conjecture of Lins, de Melo and Pugh stating that for Lienard equations as above, with f of degree 2n, the maximum number of limit cycles is n. The limit cycles that we found are relaxation oscillations which appear in slow-fast systems at the boundary of classic…

Generalised power series solutions of sub-analytic differential equations

2006

Abstract We show that if a solution y ( x ) of a sub-analytic differential equation admits an asymptotic expansion ∑ i = 1 ∞ c i x μ i , μ i ∈ R + , then the exponents μ i belong to a finitely generated semi-group of R + . We deduce a similar result for the components of non-oscillating trajectories of real analytic vector fields in dimension n. To cite this article: M. Matusinski, J.-P. Rolin, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Missing value imputation in proximity extension assay-based targeted proteomics data

2020

Targeted proteomics utilizing antibody-based proximity extension assays provides sensitive and highly specific quantifications of plasma protein levels. Multivariate analysis of this data is hampered by frequent missing values (random or left censored), calling for imputation approaches. While appropriate missing-value imputation methods exist, benchmarks of their performance in targeted proteomics data are lacking. Here, we assessed the performance of two methods for imputation of values missing completely at random, the previously top-benchmarked ‘missForest’ and the recently published ‘GSimp’ method. Evaluation was accomplished by comparing imputed with remeasured relative concentrations…

Relative cohomology spaces for some osp($n|2$)-modules

2018

International audience; In this work, we describe the H-invariant, so(n)-relative cohomology of a natural class of osp(n|2)-modules M, for n ≠ 2. The Lie superalgebra osp(n|2) can be realized as a superalgebra of vector fields on the superline R1|n. This yields canonical actions on spaces of densities and differential operators on the superline. The above result gives the zero, first, and second cohomology spaces for these modules of densities and differential operators.

On lifting the approximation property from a Banach space to its dual

2014

Existence of common zeros for commuting vector fields on 3‐manifolds II. Solving global difficulties

2020

We address the following conjecture about the existence of common zeros for commuting vector fields in dimension three: if $X,Y$ are two $C^1$ commuting vector fields on a $3$-manifold $M$, and $U$ is a relatively compact open such that $X$ does not vanish on the boundary of $U$ and has a non vanishing Poincar\'e-Hopf index in $U$, then $X$ and $Y$ have a common zero inside $U$. We prove this conjecture when $X$ and $Y$ are of class $C^3$ and every periodic orbit of $Y$ along which $X$ and $Y$ are collinear is partially hyperbolic. We also prove the conjecture, still in the $C^3$ setting, assuming that the flow $Y$ leaves invariant a transverse plane field. These results shed new light on t…

Generic properties of singular trajectories

1997

Abstract Let M be a σ-compact C∞ manifold of dimension d ≥ 3. Consider on M a single-input control system : x (t) = F 0 (x(t)) + u(t) F 1 (x(t)) , where F0, F1 are C∞ vector fields on M and the set of admissible controls U is the set of bounded measurable mappings u : [0Tu]↦ R , Tu > 0. A singular trajectory is an output corresponding to a control such that the differential of the input-output mapping is not of maximal rank. In this article we show that for an open dense subset of the set of pairs of vector fields (F0, F1), endowed with the C∞-Whitney topology, all the singular trajectories are with minimal order and the corank of the singularity is one.

The first Chevalley–Eilenberg Cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations

2010

Abstract In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle V k to a decreasing family of k foliations F i on a manifold M . We have shown that there exists a ( 1 , 1 ) tensor J of V k such that J k ≠ 0 , J k + 1 = 0 and we defined by L J ( V k ) the Lie Algebra of vector fields X on V k such that, for each vector field Y on V k , [ X , J Y ] = J [ X , Y ] . In this note, we study the first Chevalley–Eilenberg Cohomology Group, i.e. the quotient space of derivations of L J ( V k ) by the subspace of inner derivations, denoted by H 1 ( L J ( V k ) ) .

Set-valued Brownian motion

2015

Brownian motions, martingales, and Wiener processes are introduced and studied for set valued functions taking values in the subfamily of compact convex subsets of arbitrary Banach space $X$. The present paper is an application of one the paper of the second author in which an embedding result is obtained which considers also the ordered structure of $ck(X)$ and f-algebras.