Modular Solutions to Equations of Generalized Halphen Type

Solutions to a class of differential systems that generalize the Halphen system are determined in terms of automorphic functions whose groups are commensurable with the modular group. These functions all uniformize Riemann surfaces of genus zero and have $q$--series with integral coefficients. Rational maps relating these functions are derived, implying subgroup relations between their automorphism groups, as well as symmetrization maps relating the associated differential systems.Comment: PlainTeX 36gs. (Formula for Hecke operator corrected.

Superwind-driven Intense H2_2 Emission in NGC 6240 II: Detailed Comparison of Kinematical and Morphological Structures of the Warm and Cold Molecular Gas

We report on our new analysis of the spatial and kinematical distribution of warm and cold molecular gas in NGC 6240, which was undertaken to explore the origin of its unusually luminous H$_2$ emission. By comparing three-dimensional emission-line data (in space and velocity) of CO (J=2-1) in the radio and H$_2$ in the near infrared, we are able to study the H$_2$ emitting efficiency, defined in terms of the intensity ratio of H$_2$ to CO [$I$(H$_2$)/$I$(CO)], as a function of velocity. The integrated H$_2$ emitting efficiency is calculated by integrating the velocity profile of H$_2$ emitting efficiency in blue, red, and total (blue + red) velocity regions of the profile. We find that (1) both the total H$_2$ emitting efficiency and the blue-to-red ratio of the efficiency are larger in regions surrounding the CO and H$_2$ intensity peaks, and (2) the H$_2$ emitting efficiency and the kinematical conditions in the warm molecular gas are closely related to each other. A collision between the molecular gas concentration and the external superwind outflow from the southern nucleus seems plausible to explain these characteristics, since it can reproduce the enhanced emitting efficiency of blueshifted H$_2$ around the molecular gas concentration, if we assume that the superwind blows from the southern nucleus toward us, hitting the entire gas concentration from behind. In this model, internal cloud-cloud collisions within the molecular gas concentration are enhanced by the interaction with the superwind outflow, and efficient and intense shock-excited H$_2$ emission is expected as a result of the cloud-crushing mechanism.Comment: 12 pages, 6 figures, accepted for publication in A

Sub-parsec-scale Accleration of the Radio Jet in the Powerful Radio Galaxy NGC 6251

In order to investigate the genesis of powerful radio jet, we have mapped the central 10 pc region of the nearby radio galaxy NGC 6251 with a 0.2 pc resolution using Very Long Baseline Interferometer (VLBI) at two radio frequencies, 5 GHz and 15 GHz, we have found the sub-parsec-scale counterjet for the first time in this radio galaxy. This discovery allows us to investigate the jet acceleration based on the relativistic beaming model.Comment: 7 pages with 7 figures. To appear in PASJ, 52, No. 5, Oct. 25, 200

Irregular conformal blocks, with an application to the fifth and fourth Painlev\'e equations

We develop the theory of irregular conformal blocks of the Virasoro algebra. In previous studies, expansions of irregular conformal blocks at regular singular points were obtained as degeneration limits of regular conformal blocks; however, such expansions at irregular singular points were not clearly understood. This is because precise definitions of irregular vertex operators had not been provided previously. In this paper, we present precise definitions of irregular vertex operators of two types and we prove that one of our vertex operators exists uniquely. Then, we define irregular conformal blocks with at most two irregular singular points as expectation values of given irregular vertex operators. Our definitions provide an understanding of expansions of irregular conformal blocks and enable us to obtain expansions at irregular singular points. As an application, we propose conjectural formulas of series expansions of the tau functions of the fifth and fourth Painlev\'e equations, using expansions of irregular conformal blocks at an irregular singular point.Comment: 26 page