The Measure of a Measurement

While finite non-commutative operator systems lie at the foundation of quantum measurement, they are also tools for understanding geometric iterations as used in the theory of iterated function systems (IFSs) and in wavelet analysis. Key is a certain splitting of the total Hilbert space and its recursive iterations to further iterated subdivisions. This paper explores some implications for associated probability measures (in the classical sense of measure theory), specifically their fractal components. We identify a fractal scale $s$ in a family of Borel probability measures $\mu$ on the unit interval which arises independently in quantum information theory and in wavelet analysis. The scales $s$ we find satisfy $s\in \mathbb{R}_{+}$ and $s\not =1$, some $s 1$. We identify these scales $s$ by considering the asymptotic properties of $\mu(J) /| J| ^{s}$ where $J$ are dyadic subintervals, and $| J| \to0$.Comment: 18 pages, 3 figures, and reference

Harmonic analysis of iterated function systems with overlap

In this paper we extend previous work on IFSs without overlap. Our method involves systems of operators generalizing the more familiar Cuntz relations from operator algebra theory, and from subband filter operators in signal processing.Comment: 37 page

Construction of Parseval wavelets from redundant filter systems

We consider wavelets in L^2(R^d) which have generalized multiresolutions. This means that the initial resolution subspace V_0 in L^2(R^d) is not singly generated. As a result, the representation of the integer lattice Z^d restricted to V_0 has a nontrivial multiplicity function. We show how the corresponding analysis and synthesis for these wavelets can be understood in terms of unitary-matrix-valued functions on a torus acting on a certain vector bundle. Specifically, we show how the wavelet functions on R^d can be constructed directly from the generalized wavelet filters.Comment: 34 pages, AMS-LaTeX ("amsproc" document class) v2 changes minor typos in Sections 1 and 4, v3 adds a number of references on GMRA theory and wavelet multiplicity analysis; v4 adds material on pages 2, 3, 5 and 10, and two more reference

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

Global energetic neutral atom (ENA) measurements and their association with the Dst index

We present a new global magnetospheric index that measures the intensity of the Earth\u27s ring current through energetic neutral atoms (ENAs). We have named it the Global Energetic Neutral Index (GENI), and it is derived from ENA measurements obtained by the Imaging Proton Spectrometer (IPS), part of the Comprehensive Energetic Particle and Pitch Angle Distribution (CEPPAD) experiment on the POLAR satellite. GENI provides a simple orbit-independent global sum of ENAs measured with IPS. Actual ENA measurements for the same magnetospheric state look different when seen from different points in the POLAR orbit. In addition, the instrument is sensitive to weak ion populations in the polar cap, as well as cosmic rays. We have devised a method for removing the effects of cosmic rays and weak ion fluxes, in order to produce an image of “pure” ENA counts. We then devised a method of normalizing the ENA measurements to remove the orbital bias effect. The normalized data were then used to produce the GENI. We show, both experimentally and theoretically the approximate proportionality between the GENI and the Dst index. In addition we discuss possible implications of this relation. Owing to the high sensitivity of IPS to ENAs, we can use these data to explore the ENA/Dst relationship not only during all phases of moderate geomagnetic storms, but also during quiescent ring current periods

The Mystery Deepens: Spitzer Observations of Cool White Dwarfs

We present 4.5$\mu$m and 8$\mu$m photometric observations of 18 cool white dwarfs obtained with the Spitzer Space Telescope. Our observations demonstrate that four white dwarfs with T_eff< 6000 K show slightly depressed mid-infrared fluxes relative to white dwarf models. In addition, another white dwarf with a peculiar optical and near-infrared spectral energy distribution (LHS 1126) is found to display significant flux deficits in Spitzer observations. These mid-infrared flux deficits are not predicted by the current white dwarf models including collision induced absorption due to molecular hydrogen. We postulate that either the collision induced absorption calculations are incomplete or there are other unrecognized physical processes occuring in cool white dwarf atmospheres. The spectral energy distribution of LHS 1126 surprisingly fits a Rayleigh-Jeans spectrum in the infrared, mimicking a hot white dwarf with effective temperature well in excess of 10$^5$ K. This implies that the source of this flux deficit is probably not molecular absorption but some other process.Comment: 17 pages, 4 figures, ApJ in press, 10 May 200

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