Projections and Dyadic Parseval Frame MRA Wavelets

A classical theorem attributed to Naimark states that, given a Parseval frame $\mathcal{B}$ in a Hilbert space $\mathcal{H}$, one can embed $\mathcal{H}$ in a larger Hilbert space $\mathcal{K}$ so that the image of $\mathcal{B}$ is the projection of an orthonormal basis for $\mathcal{K}$. In the present work, we revisit the notion of Parseval frame MRA wavelets from two papers of Paluszy\'nski, \v{S}iki\'c, Weiss, and Xiao (PSWX) and produce an analog of Naimark's theorem for these wavelets at the level of their scaling functions. We aim to make this discussion as self-contained as possible and provide a different point of view on Parseval frame MRA wavelets than that of PSWX.Comment: 19 page

The Radio Emission from the Ultra-Luminous Far-Infrared Galaxy NGC 6240

We present new radio observations of the ``prototypical'' ultra-luminous far- infrared galaxy NGC~6240, obtained using the VLA at 20~cm in `B' configuration and at 3.6~cm in `A' configuration. These data, along with those from four previous VLA observations, are used to perform a comprehensive study of the radio emission from NGC~6240. Approximately 70\% (~3 x 10$^{23}$ W~Hz$^{-1}$) of the total radio power at 20 cm originates from the nuclear region ( <~ 1.5 kpc), of which half is emitted by two unresolved ($R <~ 36 pc) cores and half by a diffuse component. The radio spectrum of the nuclear emission is relatively flat ($\alpha$=~ 0.6;$S_{\nu} \propto \nu^{-\alpha}). The supernova rate required to power the diffuse component is consistent with that predicted by the stellar evolution models of Rieke \etal (1985). If the radio emission from the two compact cores is powered by supernova remnants, then either the remnants overlap and form hot bubbles in the cores, or they are very young ( <~ 100 yr). Nearly all of the other 30\% of the total radio power comes from an ``arm-like'' region extending westward from the nuclear region. The western arm emission has a steep spectrum (\alpha$=~ 1.0), suggestive of aging effects from synchrotron or inverse-Compton losses, and is not correlated with starlight; we suggest that it is synchrotron emission from a shell of material driven by a galactic superwind. Inverse-Compton scattering of far-infrared photons in the radio sources is expected to produce an X-ray flux of ~2-6 x 10$^{-14}$ erg/s/cm2 in the 2-10 keV band. No significant radio emission is detected from or near the possible ultra-massive ``dark core'' hypothesized by Bland-Hawthorn, Wilson \& Tully (1991).Comment: 36 pages (text and tables) as an uuencoded compressed postscript file (figures available upon request), accepted for the ApJ (20 Nov issue), STScI preprint no. ?? -- May 199

Discrete Symmetries in Covariant LQG

We study time-reversal and parity ---on the physical manifold and in internal space--- in covariant loop gravity. We consider a minor modification of the Holst action which makes it transform coherently under such transformations. The classical theory is not affected but the quantum theory is slightly different. In particular, the simplicity constraints are slightly modified and this restricts orientation flips in a spinfoam to occur only across degenerate regions, thus reducing the sources of potential divergences.Comment: 8 pages, v2: Minor change

Loop quantum cosmology of Bianchi type IX models

The loop quantum cosmology "improved dynamics" of the Bianchi type IX model are studied. The action of the Hamiltonian constraint operator is obtained via techniques developed for the Bianchi type I and type II models, no new input is required. It is shown that the big bang and big crunch singularities are resolved by quantum gravity effects. We also present the effective equations which provide modifications to the classical equations of motion due to quantum geometry effects.Comment: 20 page

Power-Ups in Digital Games: The Rewarding Effect of Phantom Game Elementson Player Experience

Power-ups are a type of game reward that allow the player tocustomise their experience by altering gameplay for a shortperiod of time. Despite the wide use of power-ups in videogames, little is known about their effect on gaming experiences.To explore this, we conducted an experimental study that compares the experiences of players depending on their exposureto power-ups in a recreational video game. The results show that players who collected power-ups felt significantly more immersed in the game, experienced more autonomy, but didnot feel more competent or challenged than those who played the game without these collectables. Interestingly, a similareffect was observed for those players who picked up ‘placebo ’power-ups, despite the items having no effect on the gameplay. We provide a discussion of these results and their implications both for games user researchers and game designers

Hybrid Quantization: From Bianchi I to the Gowdy Model

The Gowdy cosmologies are vacuum solutions to the Einstein equations which possess two space-like Killing vectors and whose spatial sections are compact. We consider the simplest of these cosmological models: the case where the spatial topology is that of a three-torus and the gravitational waves are linearly polarized. The subset of homogeneous solutions to this Gowdy model are vacuum Bianchi I spacetimes with a three-torus topology. We deepen the analysis of the loop quantization of these Bianchi I universes adopting the improved dynamics scheme put forward recently by Ashtekar and Wilson-Ewing. Then, we revisit the hybrid quantization of the Gowdy $T^3$ cosmologies by combining this loop quantum cosmology description with a Fock quantization of the inhomogeneities over the homogeneous Bianchi I background. We show that, in vacuo, the Hamiltonian constraint of both the Bianchi I and the Gowdy models can be regarded as an evolution equation with respect to the volume of the Bianchi I universe. This evolution variable turns out to be discrete, with a strictly positive minimum. Furthermore, we argue that this evolution is well-defined inasmuch as the associated initial value problem is well posed: physical solutions are completely determined by the data on an initial section of constant Bianchi I volume. This fact allows us to carry out to completion the quantization of these two cosmological models.Comment: 20 pages, version accepted for publication in Physical Review