Some Remarks on Random Vectors and O(n)O(n)-Invariants

Computations involving invariant random vectors are directly related to the theory of invariants (cf. e.g \cite{Weing_1}). Some simple observations along these lines are presented in this paper. We note in particular that sum of elements of the standard basis of $O(n)$-invariants is equal to the expectation of a random Veronese tensor up to a known scalar multiplier

Do Moderate-Luminosity Active Galactic Nuclei Suppress Star Formation?

The growth of supermassive black holes and their host galaxies are thought to be linked, but the precise nature of this symbiotic relationship is still poorly understood. Both observations and simulations of galaxy formation suggest that the energy input from active galactic nuclei (AGN), as the central supermassive black hole accretes material and grows, heats the interstellar material and suppresses star formation. In this Letter, we show that most host galaxies of moderate-luminosity supermassive black holes in the local universe have intermediate optical colors that imply the host galaxies are transitioning from star formation to quiescence, the first time this has been shown to be true for all AGN independent of obscuration. The intermediate colors suggest that star formation in the host galaxies ceased roughly 100 Myr ago. This result indicates that either the AGN are very long-lived, accreting for more than 1 Gyr beyond the end of star formation, or there is a ~100 Myr time delay between the shutdown of star formation and detectable black hole growth. The first explanation is unlikely given current estimates for AGN lifetimes, so low-lumiosity AGN must shut down star formation before substantial black hole accretion activity is detected. The scarcity of AGN host galaxies in the blue cloud reported here challenges scenarios where significant star formation and black hole growth are coeval. Lastly, these observations also strongly support the `Unified Model' of AGN as the host galaxy colors are independent of obscuration towards the central engine.Comment: Accepted for publication in ApJ Letters. 7 pages, 4 figure

Geometric methods in degree theory for equivariant maps

The book introduces conceptually simple geometric ideas based on the existence of fundamental domains for metric G- spaces. A list of the problems discussed includes Borsuk-Ulam type theorems for degrees of equivariant maps in finite and infinite dimensional cases, extensions of equivariant maps and equivariant homotopy classification, genus and G-category, elliptic boundary value problem, equivalence of p-group representations. The new results and geometric clarification of several known theorems presented here will make it interesting and useful for specialists in equivariant topology and its applications to non-linear analysis and representation theory

Theoretical study of the distal-side steric and electrostatic effects on the vibrational characteristics of the FeCO unit of the carbonylheme proteins and their models.

The vibronic theory of activation and quantum chemical intermediate neglect of differential overlap (INDO) calculations are used to study the activation of carbon monoxide (change of the C-O bond index and force field constant) by the imidazole complex with heme in dependence on the distortion of the porphyrin ring, geometry of the CO coordination, iron-carbon and iron-imidazole distances, iron displacement out of the porphyrin plane, and presence of the charged groups in the heme environment. It is shown that the main contribution to the CO activation stems from the change in the sigma donation from the 5 sigma CO orbital to iron, and back-bonding from the iron to the 2 pi orbital of CO. It follows from the results that none of the studied distortions can explain, by itself, the wide variation of the C-O vibrational frequency in the experimentally studied model compounds and heme proteins. To study the dependence of the properties of the FeCO unit on the presence of charged groups in the heme environment, the latter are simulated by the homogeneous electric field and point charges of different magnitude and location. The results show that charged groups can strongly affect the strength of the C-O bond and its vibrational frequency. It is found that the charges located on the distal side of the heme plane can affect the Fe-C and C-O bond indexes (and, consequently, the Fe-C and C-O vibrational frequencies), both in the same and in opposite directions, depending on their position. The theoretical results allow us to understand the peculiarities of the effect of charged groups on the properties of the FeCO unit both in heme proteins and in their model compounds