An asymptotic formula for the number of non-negative integer matrices with prescribed row and column sums

We count mxn non-negative integer matrices (contingency tables) with prescribed row and column sums (margins). For a wide class of smooth margins we establish a computationally efficient asymptotic formula approximating the number of matrices within a relative error which approaches 0 as m and n grow.Comment: 57 pages, results strengthened, proofs simplified somewha

The number of graphs and a random graph with a given degree sequence

We consider the set of all graphs on n labeled vertices with prescribed degrees D=(d_1, ..., d_n). For a wide class of tame degree sequences D we prove a computationally efficient asymptotic formula approximating the number of graphs within a relative error which approaches 0 as n grows. As a corollary, we prove that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D. We also establish an asymptotic formula for the number of bipartite graphs with prescribed degrees of vertices, or, equivalently, for the number of 0-1 matrices with prescribed row and column sums.Comment: 52 pages, minor improvement

Laboratory Experiments, Numerical Simulations, and Astronomical Observations of Deflected Supersonic Jets: Application to HH 110

Collimated supersonic flows in laboratory experiments behave in a similar manner to astrophysical jets provided that radiation, viscosity, and thermal conductivity are unimportant in the laboratory jets, and that the experimental and astrophysical jets share similar dimensionless parameters such as the Mach number and the ratio of the density between the jet and the ambient medium. Laboratory jets can be studied for a variety of initial conditions, arbitrary viewing angles, and different times, attributes especially helpful for interpreting astronomical images where the viewing angle and initial conditions are fixed and the time domain is limited. Experiments are also a powerful way to test numerical fluid codes in a parameter range where the codes must perform well. In this paper we combine images from a series of laboratory experiments of deflected supersonic jets with numerical simulations and new spectral observations of an astrophysical example, the young stellar jet HH 110. The experiments provide key insights into how deflected jets evolve in 3-D, particularly within working surfaces where multiple subsonic shells and filaments form, and along the interface where shocked jet material penetrates into and destroys the obstacle along its path. The experiments also underscore the importance of the viewing angle in determining what an observer will see. The simulations match the experiments so well that we can use the simulated velocity maps to compare the dynamics in the experiment with those implied by the astronomical spectra. The experiments support a model where the observed shock structures in HH 110 form as a result of a pulsed driving source rather than from weak shocks that may arise in the supersonic shear layer between the Mach disk and bow shock of the jet's working surface.Comment: Full resolution figures available at http://sparky.rice.edu/~hartigan/pub.html To appear in Ap

Simulating Radiative Magnetohydrodynamical Flows with AstroBEAR: Implementation and Applications of Non-equilibrium Cooling

Radiative cooling plays a crucial role in the dynamics of many astrophysical flows, and is particularly important in the dense shocked gas within Herbig-Haro (HH) objects and stellar jets. Simulating cooling processes accurately is necessary to compare numerical simulations with existing and planned observations of HH objects, such as those from the Hubble Space Telescope and the James Webb Space Telescope. In this paper we discuss a new, non-equilibrium cooling scheme we have implemented into the 3-D magnetohydrodynamic (MHD) code AstroBEAR. The new cooling function includes ionization, recombination, and excitation of all the important atomic species that cool below 10000 K. We tested the routine by comparing its predictions with those from the well-tested 1-D Cox-Raymond shock code (Raymond 1979). The results show thatAstroBEAR accurately tracks the ionization fraction, temperature, and other MHD variables for all low-velocity (.90 km/s) magnetized radiative shock waves. The new routine allows us to predict synthetic emission maps in all the bright forbidden and permitted lines observed in stellar jets, including H{\alpha}, [NII], [OI], and [SII]. We present an example as to how these synthetic maps facilitate a direct comparison with narrowband images of HH objects.Comment: 8 figure

Surprised by Law

This year’s Association of American Law Schools convention provided a genuinely engaging panel discussion between Michael Sandel and Judge Stephen Reinhardt. Michael Sandel, Harvard philosopher of community and the “encumbered self,” delivered his defense of an ethics of appreciation which goes beyond mere toleration, arguing for honor for persons rather than mere dignity. Reinhardt countered by characterizing Sandel’s stuff as the sort of academic theorizing which has nothing much to do with the world, and raised with almost unconscious elegance the main issue, and a deeply troubling concrete dilemma. Reinhardt noted that Sandel’s portrait of the person did not work because we are not free and we are not determined, but rather are some mix. In one sense, this concedes much of Sandel’s point about the inadequacy of the Kantian, Rawlsian unencumbered rational self, yet at the same time it reveals Sandel’s fatal incompleteness. And it focuses the issue of religious freedom in a postmodern age; it highlights the concern of what religious liberty means in a world of fluid discourse. Public speech has been variously characterized as merging horizons, as an ongoing conversation, as the constitution of communities of discourse, as constructing a social world and as being constructed by a social world. What, in this dialogic constitutionalism, does religious freedom mean? Perhaps it is the right not only to exercise one’s religious beliefs, but also to offer something about that right, as a gift, to the public discourse