We derive an asymptotic formula which counts the number of abelian extensions of prime degrees over rational function fields. Specifically, let l be a rational prime and K a rational function field F_q(t) with l\nmid q. Let Discf (F/K) denote the finite discriminant of F over K. Denote the number of abelian l-extensions F/K with deg (Discf (F/K)) = (l − 1)αn by a_l(n), where α = α(q, l) is the order of q in the multiplicative group (Z/lZ)^\times. We give a explicit asymptotic formula for a_l(n). In the case of cubic extensions with q ≡ 2 (mod 3), our formula gives an exact analogue of Cohn’s classical formula.

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In earlier work, Maire developed a class of cell-centered Lagrangian schemes for solving Euler equations of compressible gas dynamics in cylindrical coordinates. These schemes use a node-based discretization of the numerical fluxes. The control volume version has several distinguished properties, including the conservation of mass, momentum and total energy and compatibility with the geometric conservation law (GCL). However it also has a limitation in that it cannot preserve spherical symmetry for one-dimensional spherical flow. An alternative is also given to use the first order area-weighted approach which can ensure spherical symmetry, at the price of sacrificing conservation of momentum. In this paper, we apply the methodology proposed in our recent work to the first order control volume scheme of Maire to obtain the spherical symmetry property. The modified scheme can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid, and meanwhile it maintains its original good properties such as conservation and GCL. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of symmetry, non-oscillation and robustness properties.

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