A new four parameter q-series identity and its partition implications

We prove a new four parameter q-hypergeometric series identity from which the three parameter key identity for the Goellnitz theorem due to Alladi, Andrews, and Gordon, follows as a special case by setting one of the parameters equal to 0. The new identity is equivalent to a four parameter partition theorem which extends the deep theorem of Goellnitz and thereby settles a problem raised by Andrews thirty years ago. Some consequences including a quadruple product extension of Jacobi's triple product identity, and prospects of future research are briefly discussed.Comment: 25 pages, in Sec. 3 Table 1 is added, discussion is added at the end of Sec. 5, minor stylistic changes, typos eliminated. To appear in Inventiones Mathematica

An Economic Analysis of a Drug-Selling Gang's Finances

We analyze a unique data set detailing the financial activities of a drug-selling street gang on a monthly basis over a four-year period in the recent past. The data, originally compiled by the gang leader to aid in managing the organization, contain detailed information on both the sources of revenues (e.g. drug sales, extortion) and expenditrues (e.g. costs of drugs sold, weapons, tribute to the central gang organization, wages paid to various levels of the gang). Street-level drug dealing appears to be less lucrative than is generally though. We estimate the average wage in the organization to rise from roughly $6 per hour to$11 per hour over the time period studied. The distribution of wages, however, is extremely skewed. Gang leaders earn far more than they could in the legitimate sector, but the actual street-level dealers appear to earn less than the minimum wage throughout most of our sample, in spite of the substantial risks associated with such activities (the annual violent death rate in our sample is 0.07), There is some evidence consistent both with compensating differentials and efficiency wages. The markup on drugs suggests that the gang has substantial local market power. Gang wars appear to have an important strategic component: violence on another gang's turf shifts demand away from that area. The gang we observe responds to such attacks by pricing below marginal cost, suggesting either economic punishment for the rival gang or the presence of switching for users that makes market share maintenance valuable. We investigate a range of alternative methods for estimating the willingness of gang members to accept risks of death, all of which suggest that the implicit value that gang members place on their own lives is very low.

An analogue of the Wiener-Tauberian theorem for spherical transforms on semisimple Lie groups

Let G be a semi-simple connected noncompact Lie group with finite center and K a fixed maximal compact subgroup of G. Fix a Haar measure dx on G and let I1(G) denote those functions in L1(G,dx) which are biinvariant under K. The purpose of this paper is to prove that if f ∈ I1(G) is such that its spherical Fourier transform (i.e., Gelfand transform) f is nowhere vanishing on the maximal ideal space of I1(G) and f “does not vanish too fast at ∞”, then the ideal generated by f is dense in I1(G). This generalizes earlier results of Ehrenpreis-Mautner for G = SL(2,R) and R. Krier for G of real rank one