On locally symmetric vector fields on Riemannian manifolds

It is shown that if ann-dimensional (n≧3) Riemannian manifold admitsr≧2 locally symmetric vector fields (LSVF's), then it is aV(k)-space. In particular, ifr=n−1 then the manifold is a space of constant curvature. In the case of a 3-dimensional Riemannian manifold a close connection between LSVF's and eigenvectors of the Ricci tensor is found.     

Affinely Prime Dynamical Systems

This paper deals with representations of groups by "affine" automorphisms of compact,convex spaces,with special focus on "irreducible" representations:equivalently "minimal" actions.When the group in question is PSL(2,R),the authors exhibit a oneone correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations.This analysis shows that,surprisingly,all these representations are equivalent.In fact,it is found that all irreducible affine representations of this group are equivalent.The key to this is a property called "linear Stone-Weierstrass"for group actions on compact spaces.If it holds for the "universal strongly proximal space"of the group (to be defined),then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.     

Imaging freely moving subjects using continuous interleaved orthogonal magnetic resonance imaging

Magnetic resonance imaging has shown increasing clinical utility for the diagnosis of abnormalities in fetal development. MRI is not yet as effective for fetal imaging as ultrasound because of the difficulty of imaging freely moving subjects. We describe a design approach to overcome this difficulty. By interleaving orthogonal images of a subject, it is possible to rapidly and interactively localize the scan plane in a moving subject and confirm image plane orientation relative to the subject. We derive the equations necessary to optimize the tip angles for the acquisition of the orthogonal images so as to minimize artifact in the main image despite the long T1 of a fluid environment (e.g., amniotic fluid). To fully utilize the orthogonal images for rapid localization, it is critical to minimize the delay between acquisition and display, and to avoid segmented reconstruction techniques that are commonly used in high frame rate imaging. We demonstrate that this approach can be used to perform interactive scan plane localization on a moving subject and can obtain high temporal resolution images while confirming the image plane orientation relative to the subject.     

The Number of Nodal Domains on Quantum Graphs as a Stability Index of Graph Partitions

The Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of a quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains ν _n of the n ^th eigenfunction satisfies n ≥ ν _n . Here, we provide a new interpretation for the Courant nodal deficiency d _n = n − ν _n in the case of quantum graphs. It equals the Morse index — at a critical point — of an energy functional on a suitably defined space of graph partitions. Thus, the nodal deficiency assumes a previously unknown and profound meaning — it is the number of unstable directions in the vicinity of the critical point corresponding to the n ^th eigenfunction. To demonstrate this connection, the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail.     

Inequalities for cohomology classes of Kaehler manifolds

A set of inequalities for deRham cohomology classes of Kaehler manifolds is proved. As an application, an inequality for Chern numbers of a Kaehler manifold is obtained.     

Connection colligations of the second order

The paper deals with a study of connection colligations of the second order, that is the colligations for which not only the connection, but also the curvature operator satisfies the colligation condition. The gyration operators in the coupling space and the coupling system of partial differential equations are introduced and used for investigations of the basic problem of finding all regular connection colligations with vanishing curvature.     

Germanium-Iodine System,Thermodynamic Study assessment and complementary experiments

Literature thermodynamic data (enthalpies, entropies, and heat capacities) of the condensed species GeI₄(s)(I) and GeI₂(s) are revised. Entropies and heat capacities of the gaseous species GeI₄(g), GeI₂(g), and GeI(g) are calculated by the R.R.H.O. method. Published experimental investigations of the equilibria: GeI₄(l) ? GeI₄(g), GeI₄(s) ? GeI₄(g), 2 GeI₂ ? Ge(s) + GeI₄(g), GeI₄ ? GeI₂(g) + I₂(g), GeI₂(s) ? GeI₂(g) and Ge(s) + GeI₄(g) ? 2 GeI₂(g) are discussed. In addition, new measurements of the total pressure above the saturation ranges of the GeI₄(l) and for the reaction: Ge(s) + GeI₄(l) ? 2 GeI₂(g) are performed. Enthalpies of formation of GeI₄(g) and GeI(g) are deduced. Enthalpy of formation of GeI(g) is estimated from the evolution of the dissociation energy D°(Ge? X) (X = Cl, Br, I) against the Z atomic number and from the spectroscopic constants of this molecule. In conclusion a self consistent set of thermodynamic data concerning the germanium-iodine system is proposed.     

Constructions of harmonic polynomial maps between spheres

The complexity of _q -eigenmaps, i.e. homogeneous degreeq harmonic polynomial mapsf:S ^m S ⁿ, increases fast with the degreeq and the source dimensionm. Here we introduce a variety of methods of manufacturing new eigenmaps out of old ones. They include degree and source dimension raising operators. As a byproduct, we get estimates on the possible range dimensions of full eigenmaps and obtain a geometric insight of the harmonic product of ₂-eigenmaps.