On linearity of separating multi-particle differential Schr\"odinger operators for identical particles

We show that hierarchies of differential Schroedinger operators for identical particles which are separating for the usual (anti-)symmetric tensor product, are necessarily linear, and offer some speculations on the source of quantum linearity.Comment: As accepted by Journal of Mathematical Physics. Original title "Separating multi-particle differential Schroedinger operators for identical particles are necessarily linear". Some new discussion and references. Main result unchanged. Uses RevTeX 4, 9 page

The Non-Linear Dependence of Flux on Black Hole Mass and Accretion Rate in Core Dominated Jets

We derive the non-linear relation between the core flux F_{nu} of accretion powered jets at a given frequency and the mass M of the central compact object. For scale invariant jet models, the mathematical structure of the equations describing the synchrotron emission from jets enables us to cancel out the model dependent complications of jet dynamics, retaining only a simple, model independent algebraic relation between F_{nu} and M. This approach allows us to derive the F_{nu}-M relation for any accretion disk scenario that provides a set of input boundary conditions for the magnetic field and the relativistic particle pressure in the jet, such as standard and advection dominated accretion flow (ADAF) disk solutions. Surprisingly, the mass dependence of F_{nu} is very similar in different accretion scenarios. For typical flat-spectrum core dominated radio jets and standard accretion scenarios we find F_{nu}~M^{17/12}. The 7-9 orders of magnitude difference in black hole mass between microquasars and AGN jets imply that AGN jets must be about 3-4 orders of magnitude more radio loud than microquasars, i.e., the ratio of radio to bolometric luminosity is much smaller in microquasars than in AGN jets. Because of the generality of these results, measurements of this F_{nu}-M dependence are a powerful probe of jet and accretion physics. We show how our analysis can be extended to derive a similar scaling relation between the accretion rate mdot and F_{nu} for different accretion disk models. For radiatively inefficient accretion modes we find that the flat spectrum emission follows F_{nu}~(mdot*M)^{17/12}.Comment: Added key words and acknowledgements, minor editorial corrections. 6 pages, to appear in MNRAS 343, L59-L6

The predictive accuracy of credit ratings: measurement and statistical inference

Credit ratings are ordinal predictions for the default risk of an obligor. To evaluate the accuracy of such predictions commonly used measures are the Accuracy Ratio or, equivalently, the Area under the ROC curve. The disadvantage of these measures is that they treat default as a binary variable thereby neglecting the timing of the default events and also not using the full information from censored observations. We present an alternative measure that is related to the Accuracy Ratio but does not suffer from these drawbacks. As a second contribution, we study statistical inference for the Accuracy Ratio and the proposed measure in the case of multiple cohorts of obligors with overlapping lifetimes. We derive methods that use more sample information and lead to more powerful tests than alternatives that filter just the independent part of the dataset. All procedures are illustrated in the empirical section using a dataset of S&P Long Term Credit Ratings. --ratings,predictive accuracy,Accuracy Ratio,Harrell's C,overlapping lifetimes

Replica trick with real replicas: A way to build in thermodynamic homogeneity

We use real replicas to investigate stability of thermodynamic homogeneity of the free energy of the Sherrington-Kirkpatrick (SK) model of spin glasses. Within the replica trick with the replica symmetric ansatz we show that the averaged free energy at low temperatures is not thermodynamically homogeneous. The demand of minimization of the inhomogeneity of thermodynamic potentials leads in a natural way to the hierarchical solution of the Parisi type. Conditions for the global thermodynamic homogeneity are derived and evaluated for the SK and $p$-spin infinite range models.Comment: 6 pages, presented at SPDSA2004 Hayama (Japan), to appear in Progr. Theor. Phy

Singular values of the Dirac operator in dense QCD-like theories

We study the singular values of the Dirac operator in dense QCD-like theories at zero temperature. The Dirac singular values are real and nonnegative at any nonzero quark density. The scale of their spectrum is set by the diquark condensate, in contrast to the complex Dirac eigenvalues whose scale is set by the chiral condensate at low density and by the BCS gap at high density. We identify three different low-energy effective theories with diquark sources applicable at low, intermediate, and high density, together with their overlapping domains of validity. We derive a number of exact formulas for the Dirac singular values, including Banks-Casher-type relations for the diquark condensate, Smilga-Stern-type relations for the slope of the singular value density, and Leutwyler-Smilga-type sum rules for the inverse singular values. We construct random matrix theories and determine the form of the microscopic spectral correlation functions of the singular values for all nonzero quark densities. We also derive a rigorous index theorem for non-Hermitian Dirac operators. Our results can in principle be tested in lattice simulations.Comment: 3 references added, version published in JHE