Search results for "vector space"
showing 10 items of 287 documents
On Radon Transforms on Tori
2014
We show injectivity of the X-ray transform and the $d$-plane Radon transform for distributions on the $n$-torus, lowering the regularity assumption in the recent work by Abouelaz and Rouvi\`ere. We also show solenoidal injectivity of the X-ray transform on the $n$-torus for tensor fields of any order, allowing the tensors to have distribution valued coefficients. These imply new injectivity results for the periodic broken ray transform on cubes of any dimension.
Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms
2020
The graph complex acts on the spaces of Poisson bi-vectors $P$ by infinitesimal symmetries. We prove that whenever a Poisson structure is homogeneous, i.e. $P = L_{\vec{V}}(P)$ w.r.t. the Lie derivative along some vector field $\vec{V}$, but not quadratic (the coefficients of $P$ are not degree-two homogeneous polynomials), and whenever its velocity bi-vector $\dot{P}=Q(P)$, also homogeneous w.r.t. $\vec{V}$ by $L_{\vec{V}}(Q)=n\cdot Q$ whenever $Q(P)= Or(\gamma)(P^{\otimes^n})$ is obtained using the orientation morphism $Or$ from a graph cocycle $\gamma$ on $n$ vertices and $2n-2$ edges in each term, then the $1$-vector $\vec{X}=Or(\gamma)(\vec{V}\otimes P^{\otimes^{n-1}})$ is a Poisson co…
Symplectic Applicability of Lagrangian Surfaces
2009
We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with their integrability equa- tions. The invariant setup is applied to discuss the question of symplectic applicability for elliptic Lagrangian immersions. Explicit examples are considered.
Mass transportation on sub-Riemannian structures of rank two in dimension four
2017
International audience; This paper is concerned with the study of the Monge optimal transport problem in sub-Riemannian manifolds where the cost is given by the square of the sub-Riemannian distance. Our aim is to extend previous results on existence and uniqueness of optimal transport maps to cases of sub-Riemannian structures which admit many singular minimizing geodesics. We treat here the case of sub-Riemannian structures of rank two in dimension four.
Spectra for Semiclassical Operators with Periodic Bicharacteristics in Dimension Two
2014
We study the distribution of eigenvalues for selfadjoint $h$--pseudodifferential operators in dimension two, arising as perturbations of selfadjoint operators with a periodic classical flow. When the strength $\varepsilon$ of the perturbation is $\ll h$, the spectrum displays a cluster structure, and assuming that $\varepsilon \gg h^2$ (or sometimes $\gg h^{N_0}$, for $N_0 >1$ large), we obtain a complete asymptotic description of the individual eigenvalues inside subclusters, corresponding to the regular values of the leading symbol of the perturbation, averaged along the flow.
A symmetrization result for Monge–Ampère type equations
2007
In this paper we prove some comparison results for Monge–Ampere type equations in dimension two. We also consider the case of eigenfunctions and we derive a kind of “reverse” inequalities. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Asplund Operators on Locally Convex Spaces
2000
We study the relationship between the local Radon-Nikodým property, introduced by Defant [4] as a generalization of the Radon-Nikodým property to duals of locally convex spaces, and the Asplund operators, introduced by Robertson [7]. We also give a characterization of Asplund symmetric tensor products of Banach spaces in terms of Asplund maps.
Dimension gap under Sobolev mappings
2015
Abstract We prove an essentially sharp estimate in terms of generalized Hausdorff measures for the images of boundaries of Holder domains under continuous Sobolev mappings, satisfying suitable Orlicz–Sobolev conditions. This estimate marks a dimension gap, which was first observed in [2] for conformal mappings.
Some Open Problems
2009
We have extensively considered here the use of Stone's theorem on the paracompactness of metric spaces in order to build up new techniques to construct an equivalent locally uniformly rotund norm on a given normed space X. The discreetness of the basis for the metric topologies gives us the necessary rigidity condition that appears in all the known cases of existence of such a renorming property [Hay99, MOTV06]. Our approximation process is based on co-σ-continuous maps using that they have separable fibers, see Sect. 2.2. We present now some problems that remain open in this area. Some of them are classical and have been asked by different authors in conferences, papers and books. Others h…
On Discovering Low Order Models in Biochemical Reaction Kinetics
2007
We develop a method by which a large number of differential equations representing biochemical reaction kinetics may be represented by a smaller number of differential equations. The basis of our technique is a conjecture that the high dimension equations of biochemical kinetics, which involve reaction terms of specific forms, are actually implementing a low dimension system whose behavior requires right hand sides that can not be biochemically implemented. For systems that satisfy this conjecture, we develop a simple approximation scheme based on multilinear algebra that extracts the low dimensional system from simulations of the high dimension system. We demonstrate this technique on a st…