Game theory, maximum entropy, minimum discrepancy and robust Bayesian decision theory

We describe and develop a close relationship between two problems that have customarily been regarded as distinct: that of maximizing entropy, and that of minimizing worst-case expected loss. Using a formulation grounded in the equilibrium theory of zero-sum games between Decision Maker and Nature, these two problems are shown to be dual to each other, the solution to each providing that to the other. Although Tops\oe described this connection for the Shannon entropy over 20 years ago, it does not appear to be widely known even in that important special case. We here generalize this theory to apply to arbitrary decision problems and loss functions. We indicate how an appropriate generalized definition of entropy can be associated with such a problem, and we show that, subject to certain regularity conditions, the above-mentioned duality continues to apply in this extended context. This simultaneously provides a possible rationale for maximizing entropy and a tool for finding robust Bayes acts. We also describe the essential identity between the problem of maximizing entropy and that of minimizing a related discrepancy or divergence between distributions. This leads to an extension, to arbitrary discrepancies, of a well-known minimax theorem for the case of Kullback-Leibler divergence (the ``redundancy-capacity theorem'' of information theory). For the important case of families of distributions having certain mean values specified, we develop simple sufficient conditions and methods for identifying the desired solutions.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Statistics (http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/00905360400000055

Global petrologic variations of the Moon: A ternary-diagram approach

A ternary-diagram approach is used to show on a single map as much detailed geochemical information concerning petrologic variations within the lunar crust as is possible. The classification map shows the global spatial distributions of end-member compositions, the transitional spatial relations between end-member compositions, and quantitative estimates of relative proportions of each end member at each pixel location within the orbital groundtracks. The use of elemental ratios in this analysis, instead of the commonly used elemental bivariate diagrams, shows geologic information that is otherwise hidden in individual elemental databases

A chemical and petrological model of the lunar crust

Information is given on the composition and structure of the lunar crust. A lunar model is illustrated, indicating that it has essentially two layers, anorthositic mixed rocks overlaying a generally noritic crystalline basement. Implications relative to lunar evolution are discussed

The University of Michigan Centimeter-Band All Stokes Blazar Monitoring Program: Single-Dish Polarimetry as a Probe of Parsec-Scale Magnetic Fields

The University of Michigan 26-m paraboloid was dedicated to obtaining linear polarization and total flux density observations of blazars from the mid-1960s until June 2012 providing an unprecedented record tracking centimeter-band variability over decades at 14.5, 8.0, and 4.8 GHz for both targeted objects and members of flux-limited samples. In the mid-1970s through the mid-1980s, and during the last decade of the program, observations were additionally obtained of circular polarization for a small sample of radio-bright (S>5Jy), active sources. Key program results include evidence supporting class-dependent differences in the magnetic field geometry of BL Lac and QSO jets, identification of linear polarization changes temporally associated with flux outbursts supporting a shock-in-jet scenario, and determination of the spectral evolution of the Stokes V amplitude and polarity for testing proposed models. Recent radiative transfer modeling during large flares supports a jet scenario with a kinetically-dominated, relativistic flow at parsec scales with embedded turbulent magnetic fields and dynamically-weak ordered components which may be helical; the circular polarization observations are consistent with linear-to-circular mode conversion within this turbulent jet environment.Comment: 8 pages, 4 figures, Proceedings of the conference "Polarised Emission from Astrophysical Jets", June 12-16, 2017, Ierapetra, Greece, eds. E. Angelakis, M. Boettcher, and J.-L. Gome

Dirac Quantization of the Pais-Uhlenbeck Fourth Order Oscillator

As a model, the Pais-Uhlenbeck fourth order oscillator with equation of motion $d^4q/dt^4+(\omega_1^2+\omega_2^2)d^2q/dt^2 +\omega_1^2\omega_2^2 q=0$ is a quantum-mechanical prototype of a field theory containing both second and fourth order derivative terms. With its dynamical degrees of freedom obeying constraints due to the presence of higher order time derivatives, the model cannot be quantized canonically. We thus quantize it using the method of Dirac constraints to construct the correct quantum-mechanical Hamiltonian for the system, and find that the Hamiltonian diagonalizes in the positive and negative norm states that are characteristic of higher derivative field theories. However, we also find that the oscillator commutation relations become singular in the $\omega_1 \to \omega_2$ limit, a limit which corresponds to a prototype of a pure fourth order theory. Thus the particle content of the $\omega_1 =\omega_2$ theory cannot be inferred from that of the $\omega_1 \neq \omega_2$ theory; and in fact in the $\omega_1 \to \omega_2$ limit we find that all of the $\omega_1 \neq \omega_2$ negative norm states move off shell, with the spectrum of asymptotic in and out states of the equal frequency theory being found to be completely devoid of states with either negative energy or negative norm. As a byproduct of our work we find a Pais-Uhlenbeck analog of the zero energy theorem of Boulware, Horowitz and Strominger, and show how in the equal frequency Pais-Uhlenbeck theory the theorem can be transformed into a positive energy theorem instead.Comment: RevTeX4, 20 pages. Final version, to appear in Phys. Rev.

The Smallest Particles in Saturn's A and C Rings

Radio occultations of Saturn's main rings by spacecraft suggest a power law particle size-distribution down to sizes of the order of 1 cm (Marouf et al., 1983), (Zebker et al., 1985). The lack of optical depth variations between ultraviolet and near-IR wavelengths indicate a lack of micron-sized particles. Between these two regimes, the particle-size distribution is largely unknown. A cutoff where the particle-size distribution turns over must exist, but the position and shape of it is not clear from existing studies. Using a series of solar occultations performed by the VIMS instrument on-board Cassini in the near-infrared, we are able to measure light forward scattered by particles in the A and C rings. With a model of diffraction by ring particles, and the previous radio work as a constraint on the slope of the particle size distribution, we estimate the minimum particle size using a truncated power-law size distribution. The C Ring shows a minimum particle size of $4.1^{+3.8}_{-1.3}$ mm, with an assumed power law index of q=3.1 and a maximum particle size of 10 m. The A Ring signal shows a similar level of scattered flux, but modeling is complicated by the presence of self-gravity wakes and higher optical depths. If q<3, our A Ring model requires a minimum particle size below one millimeter (< 0.34 mm for an assumed q=2.75, or $0.56^{+0.35}_{-0.16}$ mm for a steeper q=2.9) to be consistent with VIMS observations. These results might seem to contradict previous optical(Dones et al., 1993) and infrared (French and Nicholson, 2000) work, which implied that there were few particles in the A Ring smaller than 1 cm. But, because of the shallow power law, relatively little optical depth (between 0.03 and 0.16 in extinction, or 0.015 - 0.08 in absorption) is provided by these particles.Comment: 47 pages, 16 figures, 3 Table

Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart

In a recent paper Bender and Mannheim showed that the unequal-frequency fourth-order derivative Pais-Uhlenbeck oscillator model has a realization in which the energy eigenvalues are real and bounded below, the Hilbert-space inner product is positive definite, and time evolution is unitary. Central to that analysis was the recognition that the Hamiltonian $H_{\rm PU}$ of the model is PT symmetric. This Hamiltonian was mapped to a conventional Dirac-Hermitian Hamiltonian via a similarity transformation whose form was found exactly. The present paper explores the equal-frequency limit of the same model. It is shown that in this limit the similarity transform that was used for the unequal-frequency case becomes singular and that $H_{\rm PU}$ becomes a Jordan-block operator, which is nondiagonalizable and has fewer energy eigenstates than eigenvalues. Such a Hamiltonian has no Hermitian counterpart. Thus, the equal-frequency PT theory emerges as a distinct realization of quantum mechanics. The quantum mechanics associated with this Jordan-block Hamiltonian can be treated exactly. It is shown that the Hilbert space is complete with a set of nonstationary solutions to the Schr\"odinger equation replacing the missing stationary ones. These nonstationary states are needed to establish that the Jordan-block Hamiltonian of the equal-frequency Pais-Uhlenbeck model generates unitary time evolution.Comment: 39 pages, 0 figure