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

What measurable zero point fluctuations can(not) tell us about dark energy

We show that laboratory experiments cannot measure the absolute value of dark energy. All known experiments rely on electromagnetic interactions. They are thus insensitive to particles and fields that interact only weakly with ordinary matter. In addition, Josephson junction experiments only measure differences in vacuum energy similar to Casimir force measurements. Gravity, however, couples to the absolute value. Finally we note that Casimir force measurements have tested zero point fluctuations up to energies of ~10 eV, well above the dark energy scale of ~0.01 eV. Hence, the proposed cut-off in the fluctuation spectrum is ruled out experimentally.Comment: 4 page

Commuting self-adjoint extensions of symmetric operators defined from the partial derivatives

We consider the problem of finding commuting self-adjoint extensions of the partial derivatives {(1/i)(\partial/\partial x_j):j=1,...,d} with domain C_c^\infty(\Omega) where the self-adjointness is defined relative to L^2(\Omega), and \Omega is a given open subset of R^d. The measure on \Omega is Lebesgue measure on R^d restricted to \Omega. The problem originates with I.E. Segal and B. Fuglede, and is difficult in general. In this paper, we provide a representation-theoretic answer in the special case when \Omega=I\times\Omega_2 and I is an open interval. We then apply the results to the case when \Omega is a d-cube, I^d, and we describe possible subsets \Lambda of R^d such that {e^(i2\pi\lambda \dot x) restricted to I^d:\lambda\in\Lambda} is an orthonormal basis in L^2(I^d).Comment: LaTeX2e amsart class, 18 pages, 2 figures; PACS numbers 02.20.Km, 02.30.Nw, 02.30.Tb, 02.60.-x, 03.65.-w, 03.65.Bz, 03.65.Db, 61.12.Bt, 61.44.B

The norm-1-property of a quantum observable

A normalized positive operator measure $X\mapsto E(X)$ has the norm-1-property if \no{E(X)}=1 whenever $E(X)\ne O$. This property reflects the fact that the measurement outcome probabilities for the values of such observables can be made arbitrary close to one with suitable state preparations. Some general implications of the norm-1-property are investigated. As case studies, localization observables, phase observables, and phase space observables are considered.Comment: 14 page

Dihedral symmetry of periodic chain: quantization and coherent states

Our previous work on quantum kinematics and coherent states over finite configuration spaces is extended: the configuration space is, as before, the cyclic group Z_n of arbitrary order n=2,3,..., but a larger group - the non-Abelian dihedral group D_n - is taken as its symmetry group. The corresponding group related coherent states are constructed and their overcompleteness proved. Our approach based on geometric symmetry can be used as a kinematic framework for matrix methods in quantum chemistry of ring molecules.Comment: 13 pages; minor changes of the tex

Zeroth Law compatibility of non-additive thermodynamics

Non-extensive thermodynamics was criticized among others by stating that the Zeroth Law cannot be satisfied with non-additive composition rules. In this paper we determine the general functional form of those non-additive composition rules which are compatible with the Zeroth Law of thermodynamics. We find that this general form is additive for the formal logarithms of the original quantities and the familiar relations of thermodynamics apply to these. Our result offers a possible solution to the longstanding problem about equilibrium between extensive and non-extensive systems or systems with different non-extensivity parameters.Comment: 18 pages, 1 figur

Stellar and Planetary Properties of K2 Campaign 1 Candidates and Validation of 17 Planets, Including a Planet Receiving Earth-like Insolation

The extended Kepler mission, K2, is now providing photometry of new fields every three months in a search for transiting planets. In a recent study, Foreman-Mackey and collaborators presented a list of 36 planet candidates orbiting 31 stars in K2 Campaign 1. In this contribution, we present stellar and planetary properties for all systems. We combine ground-based seeing-limited survey data and adaptive optics imaging with an automated transit analysis scheme to validate 21 candidates as planets, 17 for the first time, and identify 6 candidates as likely false positives. Of particular interest is K2-18 (EPIC 201912552), a bright (K=8.9) M2.8 dwarf hosting a 2.23 \pm 0.25 R_Earth planet with T_eq = 272 \pm 15 K and an orbital period of 33 days. We also present two new open-source software packages which enable this analysis. The first, isochrones, is a flexible tool for fitting theoretical stellar models to observational data to determine stellar properties using a nested sampling scheme to capture the multimodal nature of the posterior distributions of the physical parameters of stars that may plausibly be evolved. The second is vespa, a new general-purpose procedure to calculate false positive probabilities and statistically validate transiting exoplanets.Comment: 17 pages, 5 figures, 5 tables, accepted for publication in the Astrophysical Journal. Updated to closely reflect published version in ApJ (2015, 809, 25

The Outer Envelopes of Globular Clusters. I. NGC 7089 (M2)

We present the results of a wide-field imaging survey of the periphery of the Milky Way globular cluster NGC 7089 (M2). Data were obtained with MegaCam on the Magellan Clay Telescope, and the Dark Energy Camera on the Blanco Telescope. We find that M2 is embedded in a diffuse stellar envelope extending to a radial distance of at least $\sim 60^{\prime}$ ($\sim 210$ pc) -- five times the nominal tidal radius of the cluster. The envelope appears nearly circular in shape, has a radial density decline well described by a power law of index $\gamma = -2.2 \pm 0.2$, and contains approximately $1.6\%$ of the luminosity of the entire system. While the origin of the envelope cannot be robustly identified using the presently available data, the fact that M2 also hosts stellar populations exhibiting a broad dispersion in the abundances of both iron and a variety of neutron capture elements suggests that this object might plausibly constitute the stripped nucleus of a dwarf Galaxy that was long ago accreted and destroyed by the Milky Way.Comment: 15 pages, 12 figures, accepted for publishing in MNRA

At what time does a quantum experiment have a result?

This paper provides a general method for defining a generalized quantum observable (or POVM) that supplies properly normalized conditional probabilities for the time of occurrence (i.e., of detection). This method treats the time of occurrence as a probabilistic variable whose value is to be determined by experiment and predicted by the Born rule. This avoids the problematic assumption that a question about the time at which an event occurs must be answered through instantaneous measurements of a projector by an observer, common to both Rovelli (1998) and Oppenheim et al. (2000). I also address the interpretation of experiments purporting to demonstrate the quantum Zeno effect, used by Oppenheim et al. (2000) to justify an inherent uncertainty for measurements of times.Comment: To appear in proceedings of 2015 ETH Zurich Workshop on Time in Physic