Center manifold approach to the control of a tethered satellite system

It is well known that in a linearized analysis the in-plane oscillation of a tethered satellite system about the radial earth pointing position decouples from the out-of-plane oscillation. By tension control, therefore, only the in-plane but not the out-of-plane oscillation can be affected. Hence, using tension control linearization of the equations of motion cannot be used and a nonlinear problem must be treated. For a simple mechanical model of a tethered satellite system we show by means of center manifold theory that for the nonlinear system the out-of-plane oscillations can be stabilized by tension control.     

Comparing the relative volume with a revolution manifold as a model

Given a pair (P, M), whereM is ann-dimensional connected compact Riemannian manifold andP is a connected compact hypersurface ofM, the relative volume of (P, M) is the quotient volume(P)/volume(M). In this paper we give a comparison theorem for the relative volume of such a pair, with some bounds on the Ricci curvature ofM and the mean curvature ofP, with respect to that of a model pair where ℳ is a revolution manifold and a “parallel” of ℳ. Work partially supported by a DGICYT Grant No. PB91-0324.     

Smooth maps of a foliated manifold in a symplectic manifold

Let M be a smooth manifold with a regular foliation $ \mathcal{F} $ \mathcal{F} and a 2-form ω which induces closed forms on the leaves of $ \mathcal{F} $ \mathcal{F} in the leaf topology. A smooth map f: (M, $ \mathcal{F} $ \mathcal{F} ) → (N, σ) in a symplectic manifold (N, σ) is called a foliated symplectic immersion if f restricts to an immersion on each leaf of the foliation and further, the restriction of f*σ is the same as the restriction of ω on each leaf of the foliation. If f is a foliated symplectic immersion then the derivative map Df gives rise to a bundle morphism F: TM → T N which restricts to a monomorphism on T $ \mathcal{F} $ \mathcal{F} ⊆ T M and satisfies the condition F*σ = ω on T $ \mathcal{F} $ \mathcal{F} . A natural question is whether the existence of such a bundle map F ensures the existence of a foliated symplectic immersion f. As we shall see in this paper, the obstruction to the existence of such an f is only topological in nature. The result is proved using the h-principle theory of Gromov.     

The Stasheff model of a simply-connected manifold and the string bracket

We revisit Stasheff's construction of a minimal Lie-Quillen model of a simply-connected closed manifold using the language of infinity-algebras. This model is then used to construct a graded Lie bracket on the equivariant homology of the free loop space of minus a point similar to the Chas-Sullivan string bracket.

     

A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

In the present paper, we consider a five-dimensional Riemannian manifold with an irreducible SO(3)-structure as an example of an abstract statistical manifold. We prove that if a five-dimensional Riemannian manifold with an irreducible SO(3)-structure is a statistical manifold of constant curvature, then the metric of the Riemannian manifold is an Einstein metric. In addition, we show that a five-dimensional Euclidean sphere with an irreducible SO(3)-structure cannot be a conjugate symmetric statistical manifold. Finally, we show some results for a five-dimensional Riemannian manifold with a nearly integrable SO(3)-structure. For example, we prove that the structure tensor of a nearly integrable SO(3)-structure on a five-dimensional Riemannian manifold is a harmonic symmetric tensor and it defines the first integral of third order of the equations of geodesics. Moreover, we consider some topological properties of five-dimensional compact and conformally flat Riemannian manifolds with irreducible SO(3)-structure.     

Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold

We consider biharmonic maps $\phi :(M,g)\rightarrow (N,h)$ from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $ p $ satisfies $ 2\le p <\infty $ . If for such a $ p $ , $\int _M|\tau (\phi )|^{ p }\,\mathrm{d}v_g<\infty $ and $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty ,$ where $\tau (\phi )$ is the tension field of $\phi $ , then we show that $\phi $ is harmonic. For a biharmonic submanifold, we obtain that the above assumption $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty $ is not necessary. These results give affirmative partial answers to the global version of generalized Chen’s conjecture.     

Fractional folding of a manifold

This paper introduces a new type of folding; it is the fractional folding or the folding by cut of a manifold. It gives us more facility for the folding of a manifold into itself. The limit of this type of folding is discussed. The connection between the folding and the folding by cut are deduced. Theorems governing this connection are established.     

Game Theory applied to gene expression analysis

This is a summary of the author’s Ph.D. thesis supervised by Fioravante Patrone and Stefano Bonassi and defended on 25 May 2006 at the Università degli Studi di Genova. The thesis in written in English and a copy is available from the author upon request. This work deals with the discussion and the application of a methodology based on Game Theory for the analysis of gene expression data. Nowadays, microarray technology is available for taking “pictures” of gene expressions. Within a single experiment of this sophisticated technology, the level of expression of thousands of genes can be estimated in a sample of cells under given conditions. Roughly speaking, the starting point is the observation of a “picture” of gene expressions in a sample of cells under a biological condition of interest, for example a tumor. Then, Game Theory plays a primary role to quantitatively evaluate the relevance of each gene in regulating or provoking the condition of interest, taking into account the observed relationships in all subgroups of genes.      

The metric geometry of the manifold of Riemannian metrics over a closed manifold

We prove that the L ² Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L ² metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Fréchet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finite-dimensional manifold or a strong Riemannian metric on a Hilbert manifold.     

Wavelet decompositions on a manifold

A general method for constructing chains of embedded spline spaces on a smooth (not necessarily compact) manifold is suggested. A wavelet decomposition is obtained for the case of an arbitrary vector space. The results are illustrated by constructing the wavelet decompositon of a chain of embedded spaces of the B _φ-splines of zero order on a smooth manifold. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 346, 2007, pp. 26–38.     

Obstructions to fibering a manifold

Given a map f : M → N of closed topological manifolds we define torsion obstructions whose vanishing is a necessary condition for f being homotopy equivalent to a projection of a locally trivial fiber bundle. If N = S ¹, these torsion obstructions are identified with the ones due to Farrell (Indiana Univ Math J 21:315–346, 1971/1972).     

Wavelet bases on a manifold

Using the decomposition method, we present in this paper constructions of multiresolution analyses on a compact Riemannian manifold M of dimension n(n∈N). These analyses are generated by a finite number of basic functions and are adapted to the study of the Sobolev spaces H¹(M) and .     

Immersions in a symplectic manifold

In this paper we give a homotopy classification of symplectic isometric immersions following Gromov’sh-principle theorem.     

Polypotentials on a Riemannian manifold

Polyharmonic functions are considered on open sets in a Riemannian manifold R and their potential-theoretic properties are studied using the notion of complete m-potentials. Also one obtains here some characterizations of domains in R on which such complete m-potentials exist.