Stark [9] has explicitly evaluated some symplectic Gauss sums when the “denominator” matrix has odd prime level. This result is useful in computing the exact tranformation formulas of multivariable theta functions (see Stark [10], Friedberg [3] and Styer [11]). It is particularly useful when considering theta functions with quadratic forms having an odd number of variables, often a troublesome case (see Eichler [2] and Andrianov-Maloletkin [1]).

One of the authors discussed the best non-linear predictor of the process X(t) = f(U(t)), which is obtained by transforming an Ornstein-Uhlenbeck process U(t) with a measurable function f(u) (see [5]).

Let R be a local ring with maximal ideal m and let us make the following definition according to the paper [NR] of Northcott and Rees, which is essentially due to F. S. Macaulay.

Let k be either a local or a global field, and K be a finite Galois extension of k with g = Gal (K/k). Let L be a Galois extension of K which is also Galois over k. Such an extension is called central if Gal(L/iT) lies inside the centre of Gal(L/K). Clearly L is abelian over K. Next set L* = L∩K · kab where kab is the maximal abelian extension of k in its algebraic closure. This is the genus field of L over K/k.

Let Rn denote the Euclidean space of dimension n ≧> 1 with the standard inner product ‹x, y› and the norm Nx = ‹x, x›. We shall denote by dωn-i the volume element of the unit sphere normalized so that the volume of Sn-1 is 1.

We are concerned with Dirichlet series which appear in the Fourier expansion of the non-analytic Eisenstein series on the Siegel upper half space Hm of degree m. In the case of m = 2 Kaufhold [1] evaluated them. Here we treat the general cases by a different method.

Let η(z) be Dedekind’s η-function. For any set of integer g = (k1 …, Ks), k1 ≥ k2 ≥ … ≥ks ≥ 1, put . In this paper, we shall prove McKay’s conjecture which gives some combinatorial conditions about ki on which ηg(z) is a primitive cusp form. As to McKay’s conjecture, we refer [5].

In this paper we wish to study the isometry invariant geodesic on a non-positively curved manifold from a point of view of the displacement function.

The paper with the above title (Vol. 90, 1983, 119-135) contains omissions. They are in the proof (p. 134) of uniform continuity and uniform convergence of maps hn stated in p. 133, l ↑ 3.

Let {(Sn, Tn), n = 0, 1, 2, …} be a two-dimensional random walk with stationary independent increments starting at the origin 0. Throughout the paper we always assume the following condition:

We prove a Bernstein-type property for maximal spacelike hypersurfaces in a Lorentzian manifold.

Let Q be the rational number field and h(m) be the class number of the real quadratic field with a positive square-free integer m. It is known that if h(m) = 1 holds, then m is one of the following four types with prime numbers p ≡ 1, pt ≡ 3 (mod 4) (1 昤 i ≥ 4) : i) m = p, ii) m = p1, iii) m = 2 or m = 2p2, iv) m = p3p4 (see Behrbohm and Rédei [1]). The sufficient conditions for h(m) > 1 with these m were given by several authors: in all cases by Hasse [2], in case i) by Ankeny, Chowla and Hasse [3] and by Lang [4], in case ii) by Takeuchi [5] and by Yokoi [6].

In his paper [2], [3], D. A. Hejhal investigated the variational theory of linear polynomic functions. In this paper we are concerned with the variational theory of higher-order differential equations. To be more precise, consider a compact Riemann surface having genus g > 1. As is well known, we can choose a projective coordinate covering U = (Ua, za). Fix this coordinate covering of X. We shall be concerned with linear ordinary differential operators of order n defined in each projective coordinate open set Ua

In [16] we studied the following problems which had been asked by H. Matsumura (cf. [11]):

• (I) What is the difference between smoothness and J-smoothness? In particular, concerning the characterization of smoothness,

• (II) When is a ring A[X1 …, Xn]/a smooth over A?

In this paper, according to these problems, we will study J-smoothness further when rings are noetherian as in [16].