Special Linear Combinations Research Articles

Conformal and non-conformal symmetries in 2D dilaton gravity

We study finite-dimensional extra symmetries of generic 2D dilaton gravity models. Using a non-linear sigma model formulation we show that the unique theories admitting an extra (conformal) symmetry are the models with an exponential potential $V \propto e^{\beta\phi}$ ($ S ={1\over2\pi} \int d^2 x \sqrt{-g} [ R \phi + 4 \lambda^2 e^{\beta\phi} ]$), which include the CGHS model as a particular though limiting ($\beta=0$) case. These models give rise to black hole solutions with a mass-dependent temperature. The underlying extra symmetry can be maintained in a natural way in the one-loop effective action, thus implying the exact solubility of the semiclassical theory including back-reaction. Moreover, we also introduce three different classes of (non-conformal) transformations which are extra symmetries for generic 2D dilaton gravity models. Special linear combinations of these transformations turn out to be the (conformal) symmetries of the CGHS and $V \propto e^{\beta\phi}$ models. We show that one of the non-conformal extra symmetries can be converted into a conformal one by means of adequate field redefinitions involving the metric and the derivatives of the dilaton. Finally, by expressing the Polyakov-Liouville effective action in terms of an invariant metric, we are able to provide semiclassical models which are also invariant. This generalizes the solvable semiclassical model of Bose, Parker and Peleg (BPP) for a generic 2D dilaton gravity model.

On the joint distribution of studentized order statistics

In this paper, the joint distribution of some special linear combinations of the (internally) studentized order statistics are derived for both normal and exponential populations; the exact relationship between their pdf's is also obtained. The exact sampling distributions of studentized extreme deviation statistic, which has been proposed by Pearson and Chandra Sekar (1936,Biometrika,28, 308–320), are derived for these two populations. An application to the most powerful location and scale invariant test is discussed briefly.

Spin polarization and magnetic dichroism in photoemission from core and valence states in localized magnetic systems. II. Emission from open shells.

With the use of many-particle theory general equations are derived for the six fundamental spectra, which are special linear combinations of the spin-polarized photoemission spectra measured with left-, Z-, and right-circularly polarized radiation. These fundamental spectra, which are the isotropic spectrum, spin spectrum, magnetic circular dichroism, spin-orbit spectrum, anisotropic spectrum, and anisotropic spin spectrum give the correlation between the spin, orbital, and quadrupole momenta in the ground state and spin and orbit of the hole created after photoemission. For emission from an incompletely filled shell the integrated intensities of the fundamental spectra are proportional to the expectation values of the number of electrons, spin magnetic moment, orbital magnetic moment, the alignment between orbital and spin magnetic moment, the quadrupole moment, and the correlation between quadrupole and spin magnetic moment, respectively. This can be used to study the magnetic anisotropy of localized magnetic systems. We calculated the fundamental spectra for the 4f photoemission of the rare-earth ions and show that the multiplet structure displays a strong polarization dependence in agreement with our analysis in terms of correlations of moments and the rules for the integrated intensities. Finally we present the relation between the polarization effects in inverse photoemission and normal photoemission and we discuss the magnetic dichroism in metallic iron, cobalt, and nickel.

Bayesian D-Optimal and Model Robust Designs in Linear Regression Models

We consider the Bayesian optimal design problem in the usual linear regression model. A version of Elfving’s Theorem is proved for a model robust Bayesian c-optimality criterion. The optimal design minimizes a weighted product where the factors are proportional to the expected posterior risks of the Bayesian estimators for the linear combinations of the parameters in different models. The geometric characterizations are used to state sufficient conditions which guarantee that the classical and the Bayesian optimal designs are supported at the same set of points or are identical. The Bayesian D-optimal design problem appears as a special case in this setup considering “nested” models and special linear combinations for the paramater of the “highest coefficients” in different models. Thus sufficient conditions on the precision matrices of the prior distribution are found that the Bayesian D-optimal and the classical optimal design are supported at the same set of points or are identical. The results are illustrated in several examples.

Adaptive robust estimation of location for stable distributions

An adaptive procedure based on using two selector statistics and ,where Xp is the pth sample quantile, is described and used to choose among some robust estimators of location with respect to a class V of symmetric stable distributions with a characteristic exponent a ranging from 0.5 to 2.0 with steps of 0.1 or 0.05. The estimators considered are special symmetrical linear combinations of order statistics. Asymptotic variances of these estimators are computed using the characteristic function. Optimum asymptotic efficiencies of these estimators for the entire class V as well as for several subclasses of it are given.

Inclusion of tidal parameters in satellite altimetry adjustment model

Abstract In the short‐arc algorithm used at AFGL, the data consist of altimeter observations. The smoothed surface approximating the geoid is described by a truncated set of spherical‐harmonic potential coefficients, and each satellite arc is described by six state vector components. Certain tidal effects are also included in the adjustment model. However, each tidal constituent considered is attributed to only two adjustable parameters: a global amplitude factor and a global phase angle correction. These parameters, representing very special linear combinations of spherical‐harmonic tidal coefficients, are essentially uncontaminated by geoidal errors or systematic orbital errors. Such an advantage would not exist if the adjustment were made in terms of the tidal coefficients themselves. Satellite altimetry adjustment with tidal parameters yields the geoidal residuals which can serve in a variety of tasks, such as studying ocean‐bottom phenomena, refining the geoidal resolution on a regional scale (which ...

Treatment of charge-divergence commutators and application to the πK x system

A technique is developed for treating matrix elements of charge-divergence commutators between the vacuum and one-particle states. it is found that the charge-charge algebra and the asymptotic behaviour of certain three-point functions determine special linear combinations of such matrix elements. We use this fact to obtain sum rules involving weak decay constants and form factors ofπ-, K- and the hypothetical к-meson. If we further assume pole dominated unsubtracted dispersion relations for the form factors, and use the experimental number, ƒ K /ƒ πƒ +(0) = 1.23 and the known masses as input, we can determine all the above-mentioned parameters in the one-particle approximation. Among other results, we find: ƒ +(0) = 1, ƒ K ƒ π = 1.23 and m к ∼ 1100 MeV . The commutators can be calculated in the frame of the model of Gell-Mann, Oakes and Renner. We then find almost all the previous results without assuming pole dominance in dispersion relations. We also determine the model parameter c to be c = 1.27.

Robust Estimation of Location

Abstract The problem of estimating a location parameter from a random sample when the form of distribution is unknown or there is contamination of the target distribution is attacked by deriving estimators which are efficient over a class of two or more forms (“pencils”) of continuous symmetric unimodal distributions. The pencils considered are the normal, double exponential, Cauchy, parabolic, triangular, and rectangular (a limiting case). The estimators considered are special symmetrical linear combinations of order statistics: trimmed means, Winsorized means, “linearly weighted” means, and a combination of the median and two other order statistics. These are also compared asymptotically with a Hodges-Lehmann estimator. The theory required for deriving asymptotic variances is outlined. Efficiences are tabulated for sample sizes of 4 or 5, 8 or 9, 16 or 17, and ∞. Asymptotic efficiences of at least 0.82 relative to the best estimator for any single pencil are achieved by using the best trimmed mean or linearly weighted mean over a range of pencils of distributions from the normal to the Cauchy. However, the combination of the median and two other order statistics is almost as efficient (0.80) over the same range.

Robust Estimation of Location

Abstract The problem of estimating a location parameter from a random sample when the form of distribution is unknown or there is contamination of the target distribution is attacked by deriving estimators which are efficient over a class of two or more forms (“pencils”) of continuous symmetric unimodal distributions. The pencils considered are the normal, double exponential, Cauchy, parabolic, triangular, and rectangular (a limiting case). The estimators considered are special symmetrical linear combinations of order statistics: trimmed means, Winsorized means, “linearly weighted” means, and a combination of the median and two other order statistics. These are also compared asymptotically with a Hodges-Lehmann estimator. The theory required for deriving asymptotic variances is outlined. Efficiences are tabulated for sample sizes of 4 or 5, 8 or 9, 16 or 17, and ∞. Asymptotic efficiences of at least 0.82 relative to the best estimator for any single pencil are achieved by using the best trimmed mean or linearly weighted mean over a range of pencils of distributions from the normal to the Cauchy. However, the combination of the median and two other order statistics is almost as efficient (0.80) over the same range.

The wave functions of electronically degenerate states

The dynamical coupling of nuclear and electronic motions in a degenerate electronic state is described by a complicated vibronic wave equation, because the adiabatic electronic functions vary rapidly during the vibrations. Hobey and McLachlan tried to show how special linear combinations of the electronic wave functions could be found, which vary slowly, and lead to a simpler wave equation. This paper points out a mistake in their argument, but corrects it, and shows that the simple wave equation is still justified. The new method is similar to Van Vleck's transformation in degenerate perturbation theory, and also applies to molecules which are almost degenerate.

Dynamical Jahn-Teller Effect in Hydrocarbon Radicals

The first part of this paper shows how the ordinary Born-Oppenheimer approximation for separating nuclear and electronic motion can be adapted to a degenerate electronic state. To set up equations of motion for the dynamical Jahn-Teller effect in their simplest form we use vibrational amplitudes associated with special linear combinations of the degenerate electronic wave functions, chosen to vary as slowly as possible with nuclear displacements. We also discuss briefly the symmetry-forbidden electronic transitions allowed by a Jahn-Teller distortion. In the second we make molecular orbital calculations of the energies and distorted shapes of some aromatic hydrocarbon molecules. The differences in energy between the distorted and symmetrical shapes (in kcal/mole) are cyclobutadiene 11.43; cyclopentadienyl 1.414, cycloheptatrienyl 0.859; benzene negative ion 1.077; triphenylene and coronene negative ions 0.385 and 0.299. In the last three each shape of minimum energy is separated from two equivalent ones by a small potential barrier, respectively, 0.000, 0.001, and 0.002. The ground state of each radical is doubly degenerate, and it can oscillate about a continuous series of distorted shapes. In the excited 1E1u+ and 3E1u+ states of benzene the distortions are much smaller, and the CC bonds probably bend rather than stretch.

Theory of theO+States ofO16

The usual shell-model interpretation of the 6.06-Mev ${0}^{+}$ level of ${\mathrm{O}}^{16}$ as two-nucleon excitation is replaced by one-nucleon excitation. The large excitation energy for an individual nucleon is greatly reduced for special linear combinations of the excited shell-model configurations, because of a resonance effect arising from the degeneracy and from the large off-diagonal matrix elements. The state of collective compressional-dilational oscillation of the nucleus is shown to be a particularly favorable such linear combination, and the results of an explicit calculation indicate that the resonance effect is strong enough to account for the 6.06-Mev level. It is also proposed to interpret the 6.91-Mev ${2}^{+}$ level as a superposition of one-nucleon excited configurations corresponding to spheroidal deformation.