A FUNCTIONAL MODEL FOR PURE Gamma-CONTRACTIONS

A pair of commuting operators (S,P) defined on a Hilbert space H for which the closed symmetrized bidisc Gamma = {(z(1) + z(2), z(1)z(2)) : vertical bar z(1)vertical bar <= 1, vertical bar z(2)vertical bar <= 1} subset of C-2 is a spectral set is called a Gamma-contraction in the literature. A Gamma-contraction (S, P) is said to be pure if P is a pure contraction, i.e., P*(n) -> 0 strongly as n -> infinity Here we construct a functional model and produce a set of unitary invariants for a pure Gamma-contraction. The key ingredient in these constructions is an operator, which is the unique solution of the operator equation S - S*P = DpXDp, where X is an element of B(D-p), and is called the fundamental operator of the Gamma-contraction (S, P). We also discuss some important properties of the fundamental operator.

Sensitivity of simulated climate to latitudinal distribution of solar insolation reduction in solar radiation management

Solar radiation management (SRM) geoengineering has been proposed as a potential option to counteract climate change. We perform a set of idealized geoengineering simulations using Community Atmosphere Model version 3.1 developed at the National Center for Atmospheric Research to investigate the global hydrological implications of varying the latitudinal distribution of solar insolation reduction in SRM methods. To reduce the solar insolation we have prescribed sulfate aerosols in the stratosphere. The radiative forcing in the geoengineering simulations is the net forcing from a doubling of CO2 and the prescribed stratospheric aerosols. We find that for a fixed total mass of sulfate aerosols (12.6 Mt of SO4), relative to a uniform distribution which nearly offsets changes in global mean temperature from a doubling of CO2, global mean radiative forcing is larger when aerosol concentration is maximum at the poles leading to a warmer global mean climate and consequently an intensified hydrological cycle. Opposite changes are simulated when aerosol concentration is maximized in the tropics. We obtain a range of 1 K in global mean temperature and 3% in precipitation changes by varying the distribution pattern in our simulations: this range is about 50% of the climate change from a doubling of CO2. Hence, our study demonstrates that a range of global mean climate states, determined by the global mean radiative forcing, are possible for a fixed total amount of aerosols but with differing latitudinal distribution. However, it is important to note that this is an idealized study and thus not all important realistic climate processes are modeled.

Effects of Patterned Substrate on Thin Films Simulated by Family Model

Patterned substrate growth has been a subject of much interest. In this work, characteristics of some statistical properties of a film grown on triangular and vicinal substrates using the Family model are studied. Substrate size and tilt angle are varied. It is found that the interface width and the correlation function increase as the roughness of the pattern is increased. The new scaling exponents are calculated and anomalous scaling is obtained. The transient persistence probability does not show a power law relation when the initial surface is sufficiently rough. The initial rough surface also causes multifractal behavior in the model.

A note on tetrablock contractions

A commuting triple of operators (A, B, P) on a Hilbert space H is called a tetrablock contraction if the closure of the set E = {(a(11),a(22),detA) : A = GRAPHICS] with parallel to A parallel to <1} is a spectral set. In this paper, we construct a functional model and produce a set of complete unitary invariants for a pure tetrablock contraction. In this construction, the fundamental operators, which are the unique solutions of the operator equations A - B* P = DPX1DP and B - A* P = DPX2DP where X-1, X-2 is an element of B(D-P) play a pivotal role. As a result of the functional model, we show that every pure tetrablock isometry (A, B, P) on an abstract Hilbert space H is unitarily equivalent to the tetrablock contraction (MG1*+G2z, MG2*+G1z, M-z) on H-DP*(2). (D), where G(1) and G(2) are the fundamental operators of (A*, B*, P*). We prove a Beurling Lax Halmos type theorem for a triple of operators (MF1*+F2z, MF2*+F1z, M-z), where epsilon is a Hilbert space and F-1, F-2 is an element of B(epsilon). We also deal with a natural example of tetrablock contraction on a functions space to find out its fundamental operators.

Simulated long-term climate response to idealized solar geoengineering

Solar geoengineering has been proposed as a potential means to counteract anthropogenic climate change, yet it is unknown how such climate intervention might affect the Earth's climate on the millennial time scale. Here we use the HadCM3L model to conduct a 1000year sunshade geoengineering simulation in which solar irradiance is uniformly reduced by 4% to approximately offset global mean warming from an abrupt quadrupling of atmospheric CO2. During the 1000year period, modeled global climate, including temperature, hydrological cycle, and ocean circulation of the high-CO2 simulation departs substantially from that of the control preindustrial simulation, whereas the climate of the geoengineering simulation remains much closer to that of the preindustrial state with little drift. The results of our study do not support the hypothesis that nonlinearities in the climate system would cause substantial drift in the climate system if solar geoengineering was to be deployed on the timescale of a millennium.

Rigorous finite-time guarantees for optimization on continuous domains by simulated annealing

Simulated annealing is a popular method for approaching the solution of a global optimization problem. Existing results on its performance apply to discrete combinatorial optimization where the optimization variables can assume only a finite set of possible values. We introduce a new general formulation of simulated annealing which allows one to guarantee finite-time performance in the optimization of functions of continuous variables. The results hold universally for any optimization problem on a bounded domain and establish a connection between simulated annealing and up-to-date theory of convergence of Markov chain Monte Carlo methods on continuous domains. This work is inspired by the concept of finite-time learning with known accuracy and confidence developed in statistical learning theory.