Iterated function systems, representations, and Hilbert space

This paper studies a general class of Iterated Function Systems (IFS). No contractivity assumptions are made, other than the existence of some compact attractor. The possibility of escape to infinity is considered. Our present approach is based on Hilbert space, and the theory of representations of the Cuntz algebras O_n, n=2,3,.... While the more traditional approaches to IFS's start with some equilibrium measure, ours doesn't. Rather, we construct a Hilbert space directly from a given IFS; and our construction uses instead families of measures. Starting with a fixed IFS S_n, with n branches, we prove existence of an associated representation of O_n, and we show that the representation is universal in a certain sense. We further prove a theorem about a direct correspondence between a given system S_n, and an associated sub-representation of the universal representation of O_n.Comment: 22 pages, 3 figures containing 7 EPS graphics; LaTeX2e ("elsart" document class); v2 reflects change in Comments onl

Wavelets in mathematical physics: q-oscillators

We construct representations of a q-oscillator algebra by operators on Fock space on positive matrices. They emerge from a multiresolution scaling construction used in wavelet analysis. The representations of the Cuntz Algebra arising from this multiresolution analysis are contained as a special case in the Fock Space construction.Comment: (03/11/03):18 pages; LaTeX2e, "article" document class with "letterpaper" option An outline was added under the abstract (p.1), paragraphs added to Introduction (p.2), mat'l added to Proofs in Theorems 1 and 6 (pgs.5&17), material added to text for the conclusion (p.17), one add'l reference added [12]. (04/22/03):"number 1" replace with "term C" (p.9), single sentences reformed into a one paragraph (p.13), QED symbol moved up one paragraph and last paragraph labeled as "Concluding Remarks.

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

Time Matters: Temporally Enacted Frame-Works in Narrative Accounts of Mediation

Bateson\u27s (1979) method of double description is utilized to examine narrative accounts of participants\u27 mediation experiences, as a way to investigate significant change events. Comparing what changes to what remains more stable suggests that temporal differences are an indicator of contextualization, providing a framework for how meaning is made meaningful. Case studies of two of these structured interview transcripts are intensively analyzed, with triangulating measures of different logical type. Specifically, these include narrative analysis of key story points, temporal analysis of the frequency and distribution of in vivo codes to yield repetitive themes, and a modified lag analysis of codes in joint proximity to yield reliable thematic clusters. Results are integrated by means of grounded theory procedures of open and axial coding, arriving at semi-saturated categories dealing with temporal enactment of meaning-making. A lexicon of temporal devices for the social construction of common frames of reference between speaker and listener is developed. These are partitioned into three types of temporal progression (i.e., sequence, episodic structure, and co-occurrence) and three types of temporal duration (i.e., repetition, framing, and selection/deselection). Defining conditions and exemplars of each are provided, along with further permutations, including transposition, chained incidents, rival narratives, adjacency, inclusio, asymmetrical bracketing, and chiasm. These provide varied narrative solutions to address the limited attentional focus of a listener. An initial hypothesis—that longer duration meanings contextualize shorter—is given provisional support, in that it appears useful to construct and compare relative durations, with longer duration lying deeper in a hierarchy of logical types. A second hypothesis—that an increase in duration means an increase in perceived significance—is not sustained, in that deselection (and thereby decreasing a meaning\u27s duration) can nonetheless be a significant vehicle for therapeutic change. The study amounts to building a set of tautological linkages that “time matters,” and mapping descriptive territories such as narrative accounts onto it, with resulting increments in explanatory understanding. It is shown how participants shaped their accounts via temporality, by selecting themes, contextualizing, repeating, grouping, ordering, and weaving into stories. The tautology is reflexively applied to itself, and avenues for future theoretical sampling are suggested

Endomorphisms of B(H). II. Finitely Correlated States on On

AbstractWe identify sets of conjugacy classes of ergodic endomorphisms of B(H) where H is a fixed separable Hilbert space. They correspond to certain equivalence classes of pure states on the Cuntz algebras Onwherenis the Powers index. These states, called finitely correlated states, and strongly asymptotically shift invariant states, are defined and characterized. The subsets of these states defining shifts will in general be identified in a later work, but here an interesting cross section for the conjugacy classes of shifts called diagonalizable shifts is introduced and studied

Tuning of magnetic and electronic states by control of oxygen content in lanthanum strontium cobaltites

We report on the magnetic, resistive, and structural studies of perovskite La$_{1/3}$Sr$_{2/3}$CoO$_{3-\delta}$. By using the relation of synthesis temperature and oxygen partial pressure to oxygen stoichiometry obtained from thermogravimetric analysis, we have synthesized a series of samples with precisely controlled $\delta=0.00-0.49$. These samples show three structural phases at $\delta=0.00-0.15$, $\approx0.25$, $\approx0.5$, and two-phase behavior for other oxygen contents. The stoichiometric material with $\delta=0.00$ is a cubic ferromagnetic metal with the Curie temperature $T_{\rm C}=274$ K. The increase of $\delta$ to 0.15 is followed by a linear decrease of $T_{\rm C}$ to $\approx$ 160 K and a metal-insulator transition near the boundary of the cubic structure range. Further increase of $\delta$ results in formation of a tetragonal $2a_p\times 2a_p \times 4a_p$ phase for $\delta\approx 0.25$ and a brownmillerite phase for $\delta\approx0.5$. At low temperatures, these are weak ferromagnetic insulators (canted antiferromagnets) with magnetic transitions at $T_{\rm m}\approx230$ and 120 K, respectively. At higher temperatures, the $2a_p\times 2a_p \times 4a_p$ phase is $G$-type antiferromagnetic between 230 K and $\approx$360 K. Low temperature magnetic properties of this system for $\delta<1/3$ can be described in terms of a mixture of Co$^{3+}$ ions in the low-spin state and Co$^{4+}$ ions in the intermediate-spin state and a possible spin transition of Co$^{3+}$ to the intermediate-spin state above $T_{\rm C}$. For $\delta>1/3$, there appears to be a combination of Co$^{2+}$ and Co$^{3+}$ ions, both in the high-spin state with dominating antiferromagnetic interactions.Comment: RevTeX, 9 pages, 7 figures, to be published in Physical Review

Ages, metallicities and α\alpha-element enhancement for galaxies in Hickson compact groups

Central velocity dispersions and eight line-strength Lick indices have been determined from 1.3${\rm \AA}$ resolution long-slit spectra of 16 elliptical galaxies in Hickson compact groups. These data were used to determine galaxy properties (ages, metallicities and $\alpha$-element enhancements) and allowed a comparison with the parameters determined for a sample of galaxies in lower density environments, studied by Gonz\'alez (1993). The stellar population parameters were derived by comparison to single stellar population models of Thomas et al. (2003) and to a new set of SSP models for the indices Mg$_2$, Fe5270 and Fe5335 based on synthetic spetra. These models, based on an update version of the fitting functions presented in Barbuy et al. (2003), are fully described here. Our main results are: (1) the two samples have similar mean values for the metallicities and [$\alpha$/Fe] ratios, (2) the majority of the galaxies in compact groups seem to be old (median age of 14 Gyr for eight galaxies for which ages could be derived), in agreement with recent work by Proctor et al. (2004). These findings support two possible scenarios: compact groups are either young systems whose members have recently assembled and had not enough time to experience any merging yet or, instead, they are old systems that have avoided merging since their time of formation.Comment: Accepted for publication in A