Polynomiality of monotone Hurwitz numbers in higher genera

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers, related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In previous work we gave an explicit formula for monotone Hurwitz numbers in genus zero. In this paper we consider monotone Hurwitz numbers in higher genera, and prove a number of results that are reminiscent of those for classical Hurwitz numbers. These include an explicit formula for monotone Hurwitz numbers in genus one, and an explicit form for the generating function in arbitrary positive genus. From the form of the generating function we are able to prove that monotone Hurwitz numbers exhibit a polynomiality that is reminiscent of that for the classical Hurwitz numbers, i.e., up to a specified combinatorial factor, the monotone Hurwitz number in genus g with ramification specified by a given partition is a polynomial indexed by g in the parts of the partition.Comment: 23 page

APPLICATION OF RECURSIVE PARTITIONING TO AGRICULTURAL CREDIT SCORING

Recursive Partitioning Algorithm (RPA) is introduced as a technique for credit scoring analysis, which allows direct incorporation of misclassification costs. This study corroborates nonagricultural credit studies, which indicate that RPA outperforms logistic regression based on within-sample observations. However, validation based on more appropriate out-of-sample observations indicates that logistic regression is superior under some conditions. Incorporation of misclassification costs can influence the creditworthiness decision.finance, credit scoring, misclassification, recursive partitioning algorithm, Agricultural Finance,

Radiative Transfer and Radiative driving of Outflows in AGN and Starbursts

To facilitate the study of black hole fueling, star formation, and feedback in galaxies, we outline a method for treating the radial forces on interstellar gas due to absorption of photons by dust grains. The method gives the correct behavior in all of the relevant limits (dominated by the central point source; dominated by the distributed isotropic source; optically thin; optically thick to UV/optical; optically thick to IR) and reasonably interpolates between the limits when necessary. The method is explicitly energy conserving so that UV/optical photons that are absorbed are not lost, but are rather redistributed to the IR where they may scatter out of the galaxy. We implement the radiative transfer algorithm in a two-dimensional hydrodynamical code designed to study feedback processes in the context of early-type galaxies. We find that the dynamics and final state of simulations are measurably but only moderately affected by radiative forces on dust, even when assumptions about the dust-to-gas ratio are varied from zero to a value appropriate for the Milky Way. In simulations with high gas densities designed to mimic ULIRGs with a star formation rate of several hundred solar masses per year, dust makes a more substantial contribution to the dynamics and outcome of the simulation. We find that, despite the large opacity of dust to UV radiation, the momentum input to the flow from radiation very rarely exceeds L/c due to two factors: the low opacity of dust to the re-radiated IR and the tendency for dust to be destroyed by sputtering in hot gas environments. We also develop a simplification of our radiative transfer algorithm that respects the essential physics but is much easier to implement and requires a fraction of the computational cost.Comment: 25 pages, 17 figures, submitted to MNRA

Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers

This article introduces mixed double Hurwitz numbers, which interpolate combinatorially between the classical double Hurwitz numbers studied by Okounkov and the monotone double Hurwitz numbers introduced recently by Goulden, Guay-Paquet and Novak. Generalizing a result of Okounkov, we prove that a certain generating series for the mixed double Hurwitz numbers solves the 2-Toda hierarchy of partial differential equations. We also prove that the mixed double Hurwitz numbers are piecewise polynomial, thereby generalizing a result of Goulden, Jackson and Vakil

Momentum Driving: which physical processes dominate AGN feedback?

The deposition of mechanical feedback from a supermassive black hole (SMBH) in an active galactic nucleus (AGN) into the surrounding galaxy occurs via broad-line winds which must carry mass and radial momentum as well as energy. The effect can be summarized by the dimensionless parameter $\eta=dot{M_outflow}/dot{M_accretion}= (2 \epsilon_w c^2)/v_w^2$ where (\epslion_w \equiv dot{E}_w/(dot{M_accretion} c^2)) is the efficiency by which accreted matter is turned into wind energy in the disc surrounding the central SMBH. The outflowing mass and omentum are proportional to $\eta$, and many prior treatments have essentially assumed that $\eta=0$. We perform one- and two-dimensional simulations and find that the growth of the central SMBH is very sensitive to the inclusion of the mass and momentum driving but is insensitive to the assumed mechanical efficiency. For example in representative calculations, the omission of momentum and mass feedback leads to an hundred fold increase in the mass of the SMBH to over 10^{10} \Msun. When allowance is made for momentum driving, the final SMBH mass is much lower and the wind efficiencies which lead to the most observationally acceptable results are relatively low with $\epsilon_w \lesssim 10^{-4}$.Comment: 10 pages, 8 figures, resubmitted to ApJ, added reference

AGN feedback in an isolated elliptical galaxy: the effect of strong radiative feedback in the kinetic mode

Based on two-dimensional high resolution hydrodynamic numerical simulation, we study the mechanical and radiative feedback effects from the central AGN on the cosmological evolution of an isolated elliptical galaxy. Physical processes such as star formation and supernovae are considered. The inner boundary of the simulation domain is carefully chosen so that the fiducial Bondi radius is resolved and the accretion rate of the black hole is determined self-consistently. In analogy to previous works, we assume that the specific angular momentum of the galaxy is low. It is well-known that when the accretion rates are high and low, the central AGNs will be in cold and hot accretion modes, which correspond to the radiative and kinetic feedback modes, respectively. The emitted spectrum from the hot accretion flows is harder than that from the cold accretion flows, which could result in a higher Compton temperature accompanied by a more efficient radiative heating, according to previous theoretical works. Such a difference of the Compton temperature between the two feedback modes, the focus of this study, has been neglected in previous works. Significant differences in the kinetic feedback mode are found as a result of the stronger Compton heating and accretion becomes more chaotic. More importantly, if we constrain models to correctly predict black hole growth and AGN duty cycle after cosmological evolution, we find that the favored model parameters are constrained: mechanical feedback efficiency diminishes with decreasing luminosity (the maximum efficiency being $\simeq 10^{-3.5}$) and X-ray Compton temperature increases with decreasing luminosity, although models with fixed mechanical efficiency and Compton temperature can be found that are satisfactory as well. We conclude that radiative feedback in the kinetic mode is much more important than previously thought.Comment: 35 pages, 7 figures, accepted by the Ap