The Restriction Principle and Commuting Families of Toeplitz Operators on the Unit Ball

On the unit ball B^n we consider the weighted Bergman spaces H_\lambda and their Toeplitz operators with bounded symbols. It is known from our previous work that if a closed subgroup H of \widetilde{\SU(n,1)} has a multiplicity-free restriction for the holomorphic discrete series of \widetilde{\SU(n,1)}, then the family of Toeplitz operators with H-invariant symbols pairwise commute. In this work we consider the case of maximal abelian subgroups of \widetilde{\SU(n,1)} and provide a detailed proof of the pairwise commutativity of the corresponding Toeplitz operators. To achieve this we explicitly develop the restriction principle for each (conjugacy class of) maximal abelian subgroup and obtain the corresponding Segal-Bargmann transform. In particular, we obtain a multiplicity one result for the restriction of the holomorphic discrete series to all maximal abelian subgroups. We also observe that the Segal-Bargman transform is (up to a unitary transformation) a convolution operator against a function that we write down explicitly for each case. This can be used to obtain the explicit simultaneous diagonalization of Toeplitz operators whose symbols are invariant by one of these maximal abelian subgroups

Toeplitz operators on the domain {Z∈M2×2(C)∣ZZ∗<I}\{Z\in M_{2\times2}(\mathbb{C}) \mid Z Z^* < I\} with U(2)×T2\mathrm{U}(2)\times\mathbb{T}^2-invariant symbols

Let $D$ be the irreducible bounded symmetric domain of $2\times2$ complex matrices that satisfy $ZZ^* < I_2$. The biholomorphism group of $D$ is realized by $\mathrm{U}(2,2)$ with isotropy at the origin given by $\mathrm{U}(2)\times\mathrm{U}(2)$. Denote by $\mathbb{T}^2$ the subgroup of diagonal matrices in $\mathrm{U}(2)$. We prove that the set of $\mathrm{U}(2)\times\mathbb{T}^2$-invariant essentially bounded symbols yield Toeplitz operators that generate commutative $C^*$-algebras on all weighted Bergman spaces over $D$. Using tools from representation theory, we also provide an integral formula for the spectra of these Toeplitz operators

Vanishing Cosmological Constant in Modified Gauss-Bonnet Gravity with Conformal Anomaly

We consider dark energy cosmology in a de Sitter universe filled with quantum conformal matter. Our model represents a Gauss-Bonnet model of gravity with contributions from quantum effects. To the General Relativity action an arbitrary function of the GB invariant, f(G), is added, and taking into account quantum effects from matter the cosmological constant is studied. For the considered model the conditions for a vanishing cosmological constant are considered. Creation of a de Sitter universe by quantum effects in a GB modified gravity is discussed.Comment: 8 pages latex, 1 figure. To appear in Int. J. Mod. Phys.

Tachyon fields with effects of quantum matter in an Anti-de Sitter Universe

We consider an Anti-de Sitter universe filled by quantum conformal matter with the contribution from the usual tachyon and a perfect fluid. The model represents the combination of a trace-anomaly annihilated and a tachyon driven Anti-de Sitter universe. The influence exerted by the quantum effects and by the tachyon on the AdS space is studied. The radius corresponding to this universe is calculated and the effect of the tachyon potential is discussed, in particular, concerning to the possibility to get an accelerated scale factor for the proposed model (implying an accelerated expansion of the AdS type of universe). Fulfillment of the cosmological energy conditions in the model is also investigatedComment: 14 Latex pages, no figure

Estructura de tallas y composición sexual de Eurypodius Latreillii Guérin, 1828 (Brachyura) y Pagurus Comptus White, 1847 (Anomura) en el extremo sur de Chile

The decapod crustaceans Eurypodius latreillii (Brachyura) and Pagurus comptus (Paguridae) were the most frequently occurring species in Agassiz trawl (AGT) samples obtained during the Chilean - German - Italian Victor Hensen Expedition in the Straits of Magellan and south of the Beagle Channel. Sex composition and length structure aspects were studied using the material from 15 stations in the two areas, which comprised 773 specimens of E. latreillii and 647 P. comptus. The size frequency of both species was characterized by the presence of small individuals in shallow waters. The sex composition varied with depth, indicating a dominance of E. latreillii females in deep waters, whereas no such pattern was observed in P. comptus. These differences may be caused by epizoites of E. latreillii and the availability of vacant gastropod shells for P. comptus, in addition to geographical aspects and life-cycle patterns of these speciesLos crustáceos decápodos Eurypodius latreillii (Brachyura) y Pagurus comptus (Anomura), fueron las especies encontradas con mayor frecuencia en las muestras obtenidas durante la Expedición Chilena - Alemana - Italiana “Victor Hensen” (Octubre/Noviembre de 1994) en el Estrecho de Magallanes y al sur del Canal del Beagle, en el extremo sur de Chile. En estas localidades se obtuvieron las muestras mediante una red de tipo Agassiz. Los ejemplares fueron medidos, pesados y sexados, posteriormente se analizó la composición sexual y la estructura de tallas de estas especies. Se recolectaron un total de 773 individuos de E. latreillii y 647 individuos de P. comptus. Este material corresponde a los especímenes capturados en 15 estaciones. La estructura de tallas para ambas especies se caracterizó por la presencia de ejemplares de menor tamaño hacia zonas someras. La composición sexual presentó variaciones respecto de la profundidad, indicando una dominancia de hembras en aguas profundas para E. latreillii, pero este patrón no se observó en P. comptus. Los patrones observados dependerían de la presencia de epibiontes sobre el caparazón de E. latreillii y la disponibilidad de conchas vacías de gastrópodos para P. comptus, además de aspectos geográficos y de los ciclos de vida que presentarían ambas especies en la región

A Tale of Two Animats: What does it take to have goals?

What does it take for a system, biological or not, to have goals? Here, this question is approached in the context of in silico artificial evolution. By examining the informational and causal properties of artificial organisms ('animats') controlled by small, adaptive neural networks (Markov Brains), this essay discusses necessary requirements for intrinsic information, autonomy, and meaning. The focus lies on comparing two types of Markov Brains that evolved in the same simple environment: one with purely feedforward connections between its elements, the other with an integrated set of elements that causally constrain each other. While both types of brains 'process' information about their environment and are equally fit, only the integrated one forms a causally autonomous entity above a background of external influences. This suggests that to assess whether goals are meaningful for a system itself, it is important to understand what the system is, rather than what it does.Comment: This article is a contribution to the FQXi 2016-2017 essay contest "Wandering Towards a Goal

Analytic results for NN particles with 1/r21/r^2 interaction in two dimensions and an external magnetic field

The $2N$-dimensional quantum problem of $N$ particles (e.g. electrons) with interaction $\beta/r^2$ in a two-dimensional parabolic potential $\omega_0$ (e.g. quantum dot) and magnetic field $B$, reduces exactly to solving a $(2N-4)$-dimensional problem which is independent of $B$ and $\omega_0$. An exact, infinite set of relative mode excitations are obtained for any $N$. The $N=3$ problem reduces to that of a ficticious particle in a two-dimensional, non-linear potential of strength $\beta$, subject to a ficticious magnetic field $B_{\rm fic}\propto J$, the relative angular momentum.Comment: To appear in Physical Review Letters (in press). RevTeX file. Two figures available from [email protected] or [email protected]

Reduced abrasion drilling fluid

A reduced abrasion drilling fluid system and method of drilling a borehole by circulating the reduced abrasion drilling fluid through the borehole is disclosed. The reduced abrasion drilling fluid comprises a drilling fluid, a first additive and a weighting agent, wherein the weighting agent has a particle size of at least 90% by volume less than 50 um

Interplay between excitation kinetics and reaction-center dynamics in purple bacteria

Photosynthesis is arguably the fundamental process of Life, since it enables energy from the Sun to enter the food-chain on Earth. It is a remarkable non-equilibrium process in which photons are converted to many-body excitations which traverse a complex biomolecular membrane, getting captured and fueling chemical reactions within a reaction-center in order to produce nutrients. The precise nature of these dynamical processes -- which lie at the interface between quantum and classical behaviour, and involve both noise and coordination -- are still being explored. Here we focus on a striking recent empirical finding concerning an illumination-driven transition in the biomolecular membrane architecture of {\it Rsp. Photometricum} purple bacteria. Using stochastic realisations to describe a hopping rate model for excitation transfer, we show numerically and analytically that this surprising shift in preferred architectures can be traced to the interplay between the excitation kinetics and the reaction center dynamics. The net effect is that the bacteria profit from efficient metabolism at low illumination intensities while using dissipation to avoid an oversupply of energy at high illumination intensities.Comment: 21 pages, 13 figures, accepted for publication in New Journal of Physic