Reflection positive affine actions and stochastic processes

In this note we continue our investigations of the representation theoretic aspects of reflection positivity, also called Osterwalder--Schrader positivity. We explain how this concept relates to affine isometric actions on real Hilbert spaces and how this is connected with Gaussian processes with stationary increments

A spiral-like disk of ionized gas in IC 1459: Signature of a merging collision

The authors report the discovery of a large (15 kpc diameter) H alpha + (NII) emission-line disk in the elliptical galaxy IC 1459, showing weak spiral structure. The line flux peaks strongly at the nucleus and is more concentrated than the stellar continuum. The major axis of the disk of ionized gas coincides with that of the stellar body of the galaxy. The mass of the ionized gas is estimated to be approx. 1 times 10 (exp 5) solar mass, less than 1 percent of the total mass of gas present in IC 1459. The total gas mass of 4 times 10(exp 7) solar mass has been estimated from the dust mass derived from a broad-band color index image and the Infrared Astronomy Satellite (IRAS) data. The authors speculate that the presence of dust and gas in IC 1459 is a signature of a merger event

An extension of Wiener integration with the use of operator theory

With the use of tensor product of Hilbert space, and a diagonalization procedure from operator theory, we derive an approximation formula for a general class of stochastic integrals. Further we establish a generalized Fourier expansion for these stochastic integrals. In our extension, we circumvent some of the limitations of the more widely used stochastic integral due to Wiener and Ito, i.e., stochastic integration with respect to Brownian motion. Finally we discuss the connection between the two approaches, as well as a priori estimates and applications.Comment: 13 page

Asymptotic Behavior of Ext functors for modules of finite complete intersection dimension

Let $R$ be a local ring, and let $M$ and $N$ be finitely generated $R$-modules such that $M$ has finite complete intersection dimension. In this paper we define and study, under certain conditions, a pairing using the modules \Ext_R^i(M,N) which generalizes Buchweitz's notion of the Herbrand diference. We exploit this pairing to examine the number of consecutive vanishing of \Ext_R^i(M,N) needed to ensure that \Ext_R^i(M,N)=0 for all $i\gg 0$. Our results recover and improve on most of the known bounds in the literature, especially when $R$ has dimension at most two

Forming Galaxies with MOND

Beginning with a simple model for the growth of structure, I consider the dissipationless evolution of a MOND-dominated region in an expanding Universe by means of a spherically symmetric N-body code. I demonstrate that the final virialized objects resemble elliptical galaxies with well-defined relationships between the mass, radius, and velocity dispersion. These calculations suggest that, in the context of MOND, massive elliptical galaxies may be formed early (z > 10) as a result of monolithic dissipationless collapse. Then I reconsider the classic argument that a galaxy of stars results from cooling and fragmentation of a gas cloud on a time scale shorter than that of dynamical collapse. Qualitatively, the results are similar to that of the traditional picture; moreover, the existence, in MOND, of a density-temperature relation for virialized, near isothermal objects as well as a mass-temperature relation implies that there is a definite limit to the mass of a gas cloud where this condition can be met-- an upper limit corresponding to that of presently observed massive galaxies.Comment: 9 pages, 9 figures, revised in response to comments of referee. Table added, extended discussion, accepted MNRA

Analysis of unbounded operators and random motion

We study infinite weighted graphs with view to \textquotedblleft limits at infinity,\textquotedblright or boundaries at infinity. Examples of such weighted graphs arise in infinite (in practice, that means \textquotedblleft very\textquotedblright large) networks of resistors, or in statistical mechanics models for classical or quantum systems. But more generally our analysis includes reproducing kernel Hilbert spaces and associated operators on them. If $X$ is some infinite set of vertices or nodes, in applications the essential ingredient going into the definition is a reproducing kernel Hilbert space; it measures the differences of functions on $X$ evaluated on pairs of points in $X$. And the Hilbert norm-squared in $\mathcal{H}(X)$ will represent a suitable measure of energy. Associated unbounded operators will define a notion or dissipation, it can be a graph Laplacian, or a more abstract unbounded Hermitian operator defined from the reproducing kernel Hilbert space under study. We prove that there are two closed subspaces in reproducing kernel Hilbert space $\mathcal{H}(X)$ which measure quantitative notions of limits at infinity in $X$, one generalizes finite-energy harmonic functions in $\mathcal{H}(X)$, and the other a deficiency index of a natural operator in $\mathcal{H}(X)$ associated directly with the diffusion. We establish these results in the abstract, and we offer examples and applications. Our results are related to, but different from, potential theoretic notions of \textquotedblleft boundaries\textquotedblright in more standard random walk models. Comparisons are made.Comment: 38 pages, 4 tables, 3 figure

Essential selfadjointness of the graph-Laplacian

We study the operator theory associated with such infinite graphs $G$ as occur in electrical networks, in fractals, in statistical mechanics, and even in internet search engines. Our emphasis is on the determination of spectral data for a natural Laplace operator associated with the graph in question. This operator $\Delta$ will depend not only on $G$, but also on a prescribed positive real valued function $c$ defined on the edges in $G$. In electrical network models, this function $c$ will determine a conductance number for each edge. We show that the corresponding Laplace operator $\Delta$ is automatically essential selfadjoint. By this we mean that $\Delta$ is defined on the dense subspace $\mathcal{D}$ (of all the real valued functions on the set of vertices $G^{0}$ with finite support) in the Hilbert space $l^{2}$. The conclusion is that the closure of the operator $\Delta$ is selfadjoint in $l^{2}(G^{0})$, and so in particular that it has a unique spectral resolution, determined by a projection valued measure on the Borel subsets of the infinite half-line. We prove that generically our graph Laplace operator $\Delta=\Delta_{c}$ will have continuous spectrum. For a given infinite graph $G$ with conductance function $c$, we set up a system of finite graphs with periodic boundary conditions such the finite spectra, for an ascending family of finite graphs, will have the Laplace operator for $G$ as its limit.Comment: 50 pages with TOC and figure

A study of omega bands and Ps6 pulsations on the ground, at low altitude and at geostationary orbit

We investigate the electrodynamic coupling between auroral omega bands and the inner magnetosphere. The goal of this study is to determine the features to which omega bands map in the magnetosphere. To establish the auroral-magnetosphere connection, we appeal to the case study analysis of the data rich event of September 26, 1989. At 6 magnetic local time (MLT), two trains of Ps6 pulsations (ground magnetic signatures of omega bands) were observed to drift over the Canadian Auroral Network For the OPEN Program Unified Study (CANOPUS) chain. At the same time periodic ionospheric flow patterns moved through the collocated Bistatic Auroral Radar System (BARS) field of view. Similar coincident magnetic variations were observed by GOES 6, GOES 7 and SCATHA, all of which had magnetic foot points near the CANOPUS/BARS stations. SCATHA, which was located at 6 MLT, 0.5 RE earthward of GOES 7 observed the 10 min period pulsations, whereas GOES 7 did not. In addition, DMSP F6 and F8 were over-flying the region and observed characteristic precipitation and flow signatures. From this fortunate constellation of ground and space observations, we conclude that auroral omega bands are the electrodynamic signature of a corrugated current sheet (or some similar spatially localized magnetic structure) in the near-Earth geostationary magnetosphere

The history of mass assembly of faint red galaxies in 28 galaxy clusters since z=1.3

We measure the relative evolution of the number of bright and faint (as faint as 0.05 L*) red galaxies in a sample of 28 clusters, of which 16 are at 0.50<= z<=1.27, all observed through a pair of filters bracketing the 4000 Angstrom break rest-frame. The abundance of red galaxies, relative to bright ones, is constant over all the studied redshift range, 0<z<1.3, and rules out a differential evolution between bright and faint red galaxies as large as claimed in some past works. Faint red galaxies are largely assembled and in place at z=1.3 and their deficit does not depend on cluster mass, parametrized by velocity dispersion or X-ray luminosity. Our analysis, with respect to previous one, samples a wider redshift range, minimizes systematics and put a more attention to statistical issues, keeping at the same time a large number of clusters.Comment: MNRAS, 386, 1045. Half a single sentence (in sec 4.4) change