On the Commensurability of Directional Distance Functions

Shephard’s distance functions are widely used instruments for characterizing technology and for estimating efficiency in contemporary economic theory and practice. Recently, they have been generalized by the Luenberger shortage function, or Chambers-Chung-Färe directional distance function. In this study, we explore a very important property of an economic measure known as commensurability or independence of units of measurement up to scalar transformation. Our study discovers both negative and positive results for this property in the context of the directional distance function, which in turn helps us narrow down the most critical issue for this function in practice—the choice of direction of measurement.Directional distance functions, commensurability, efficiency

Quantifying the nonlinearity of a quantum oscillator

We address the quantification of nonlinearity for quantum oscillators and introduce two measures based on the properties of the ground state rather than on the form of the potential itself. The first measure is a fidelity-based one, and corresponds to the renormalized Bures distance between the ground state of the considered oscillator and the ground state of a reference harmonic oscillator. Then, in order to avoid the introduction of this auxiliary oscillator, we introduce a different measure based on the non-Gaussianity (nG) of the ground state. The two measures are evaluated for a sample of significant nonlinear potentials and their properties are discussed in some detail. We show that the two measures are monotone functions of each other in most cases, and this suggests that the nG-based measure is a suitable choice to capture the anharmonic nature of a quantum oscillator, and to quantify its nonlinearity independently on the specific features of the potential. We also provide examples of potentials where the Bures measure cannot be defined, due to the lack of a proper reference harmonic potential, while the nG-based measure properly quantify their nonlinear features. Our results may have implications in experimental applications where access to the effective potential is limited, e.g., in quantum control, and protocols rely on information about the ground or thermal state.Comment: 8 pages, 5 figures, published versio

Redshift Weights for Baryon Acoustic Oscillations : Application to Mock Galaxy Catalogs

Large redshift surveys capable of measuring the Baryon Acoustic Oscillation (BAO) signal have proven to be an effective way of measuring the distance-redshift relation in cosmology. Building off the work in Zhu et al. (2015), we develop a technique to directly constrain the distance-redshift relation from BAO measurements without splitting the sample into redshift bins. We parametrize the distance-redshift relation, relative to a fiducial model, as a quadratic expansion. We measure its coefficients and reconstruct the distance-redshift relation from the expansion. We apply the redshift weighting technique in Zhu et al. (2015) to the clustering of galaxies from 1000 QuickPM (QPM) mock simulations after reconstruction and achieve a 0.75% measurement of the angular diameter distance $D_A$ at $z=0.64$ and the same precision for Hubble parameter H at $z=0.29$. These QPM mock catalogs are designed to mimic the clustering and noise level of the Baryon Oscillation Spectroscopic Survey (BOSS) Data Release 12 (DR12). We compress the correlation functions in the redshift direction onto a set of weighted correlation functions. These estimators give unbiased $D_A$ and $H$ measurements at all redshifts within the range of the combined sample. We demonstrate the effectiveness of redshift weighting in improving the distance and Hubble parameter estimates. Instead of measuring at a single 'effective' redshift as in traditional analyses, we report our $D_A$ and $H$ measurements at all redshifts. The measured fractional error of $D_A$ ranges from 1.53% at $z=0.2$ to 0.75% at $z=0.64$. The fractional error of $H$ ranges from 0.75% at $z=0.29$ to 2.45% at $z = 0.7$. Our measurements are consistent with a Fisher forecast to within 10% to 20% depending on the pivot redshift. We further show the results are robust against the choice of fiducial cosmologies, galaxy bias models, and RSD streaming parameters.Comment: 13 pages, 8 figures, submitted to MNRA

Parameter estimation of coalescing supermassive black hole binaries with LISA

Laser Interferometer Space Antenna (LISA) will routinely observe coalescences of supermassive black hole (BH) binaries up to very high redshifts. LISA can measure mass parameters of such coalescences to a relative accuracy of $10^{-4}-10^{-6}$, for sources at a distance of 3 Gpc. The problem of parameter estimation of massive nonspinning binary black holes using post-Newtonian (PN) phasing formula is studied in the context of LISA. Specifically, the performance of the 3.5PN templates is contrasted against its 2PN counterpart using a waveform which is averaged over the LISA pattern functions. The improvement due to the higher order corrections to the phasing formula is examined by calculating the errors in the estimation of mass parameters at each order. The estimation of the mass parameters ${\cal M}$ and $\eta$ are significantly enhanced by using the 3.5PN waveform instead of the 2PN one. For an equal mass binary of $2\times10^6M_\odot$ at a luminosity distance of 3 Gpc, the improvement in chirp mass is $\sim 11$ and that of $\eta$ is $\sim 39$. Estimation of coalescence time $t_c$ worsens by 43%. The improvement is larger for the unequal mass binary mergers. These results are compared to the ones obtained using a non-pattern averaged waveform. The errors depend very much on the location and orientation of the source and general conclusions cannot be drawn without performing Monte Carlo simulations. Finally the effect of the choice of the lower frequency cut-off for LISA on the parameter estimation is studied.Comment: 12 pages, 5 figures (eps) significant revision, accepted for publication in Phys. Rev. D. Matches with the published versio

A Borda Measure for Social Choice Functions

The question addressed in this paper is the order of magnitude of the difference between the Borda rule and any given social choice function. A social choice function is a mapping that associates a subset of alternatives to any profile of individual preferences. The Borda rule consists in asking voters to order all alternatives, knowing that the last one in their ranking will receive a score of zero, the second lowest a score of 1, the third a score of 2 and so on. These scores are then weighted by the number of voters that support them to give the Borda score of each alternative. The rule then selects the alternatives with the highest Borda score. In this paper, a simple measure of the difference between the Borda rule and any given social choice function is proposed. It is given by the ratio of the best Borda score achieved by the social choice function under scrutiny over the Borda score of a Borda winner. More precisely, it is the minimum of this ratio over all possible profiles of preferences that is used. This "Borda measure" or at least bounds for this measure is also computed for well known social choice functions. Cet article se penche sur la distance entre la règle de Borda et n'importe quelle autre fonction de choix social. Ces dernières associent un sous-ensemble d'options possibles à tout profil ou configuration de préférences individuelles. La règle de Borda consiste à demander aux votants d'ordonner les options possibles, en leur disant que la dernière dans leur ordre recevra un score nul, l'avant-dernière un score égal à 1, celle qui vient au troisième pire rang un score égal à 2 et ainsi de suite. Ces scores sont ensuite pondérés par le nombre de votants qui les supportent pour donner le score de Borda de chaque option. La règle choisit les options qui ont reçu le score le plus élevé. Dans cet article, une mesure simple de la différence entre la règle de Borda et n'importe quelle autre fonction de choix social est proposée. Elle est donnée par le rapport du meilleur score de Borda obtenu par les options que sélectionne la fonction de choix social considérée sur le score de Borda d'un gagnant de Borda. De façon plus précise, c'est le minimum de ces rapports, sur l'ensemble des profils de préférences, qui est utilisé. Cette mesure de Borda ou, à tout le moins, un intervalle pour cette mesure est calculé pour un certain nombre de fonctions de choix social bien connues.

On the Commensurability of Directional Distance Functions

Shephard’s distance functions are widely used instruments for characterizing technology and for estimating efficiency in contemporary economic theory and practice. Recently, they have been generalized by the Luenberger shortage function, or Chambers-Chung-Färe directional distance function. In this study, we explore a very important property of an economic measure known as commensurability or independence of units of measurement up to scalar transformation. Our study discovers both negative and positive results for this property in the context of the directional distance function, which in turn helps us narrow down the most critical issue for this function in practice—the choice of direction of measurement