Let D be a convex domain in the n-dimensional real number space Rn, not containing any affine line and A(D) the group of all affine transformations of Rn leaving D invariant. If the group A(D) acts transitively on D, then the domain D is said to be homogeneous. From a homogeneous convex domain D in Rn, a homogeneous convex cone V = V(D) in Rn+1 = Rn × R is constructed as follows (cf. Vinberg [11]):

which is called the cone fitted on the convex domain D.

It is well known [c.f. Kas] that every elliptic surface, with geometric genus 0, is given by a Weierstrass equation of the form

(relative to a suitable parameter, u, for the base) where the a’s and b’s are constants. For sufficiently general choices of a’s and b’s, the Mordell-Weil group (i.e., the group of solutions (x, y), with x and y rational functions of u) has rank 8.

The existence and basic properties of what are now referred to as spinor exceptional integers for a genus of integral ternary quadratic forms were first observed in the 1950’s by Jones and Watson [7] and Kneser [8] in the context of indefinite forms. The study of these integers and their generalizations has been undertaken by a number of authors in recent years, and has contributed significantly to the understanding of representation properties unique to ternary forms. In this direction, the present author proved in a previous paper [4] that if c is a primitive spinor exceptional integer for a genus of integral ternary quadratic forms and f is some form in this genus, then a form in the spinor genus of f primitively represents c if and only if f primitively represents an integer of the type ct2, for some odd positive integer t, relatively prime to the discriminant d, which satisfies the condition that the Jacobi symbol (–cd/t) equals 1.

Let R be a (commutative) Noetherian ring, and let {n0, n1 …, nm} {k1 k2, …, km} be two sequences of integers satisfying m > 0, ki ≧ 0 (i = 1, 2, …, m) and ni ≧ ki ki + 1 (i = 1, 2, …, m) with k0 = km + 1 = 0.

Let K be a finite Galois extension of an algebraic number field k with G = Gal (K/k), and M be a Galois extension of k containing K. We denote by resp. the genus field resp. the central class field of K with respect to M/k. By definition, the field is the composite of K and the maximal abelian extension over k contained in M. The field is the maximal Galois extension of k contained in M satisfying the condition that the Galois group over K is contained in the center of that over k. Then it is well known that Gal is isomorphic to a factor group of the Schur multiplicator H-3(G, Z), and is isomorphic to H-3(G, Z) when M is sufficiently large. In this case we call M abundant for K/k (See Heider [3, § 4] and Miyake [6, Theorem 5]).

In this paper we are concerned with stochastic relaxed control problems of the following kind. Let X(t), t ≥ 0, denote the state of a process being controlled, Y(t), t ≥ 0, the observation process and p(t, ·) a relaxed control, that is a process with values probability measures on the control region Г. The state and observation processes are governed by stochastic differential equations

and

where B and W are independent Brownian motions with values in Rn and Rm respectively, (put m = 1 for simplicity).

The main purposes of this paper are to investigate the existence and the uniqueness of a non-local problem for a linear parabolic equation

in a cylinder D = Ω × (0, T].

Let k be an algebraic number field of finite degree, and K a finite Galois extension of k. A central extension L of K/k is an algebraic number field which contains K and is normal over k, and whose Galois group over K is contained in the center of the Galois group Gal(L/k). We denote the maximal abelian extensions of k and K in the algebraic closure of k by kab and Kab respectively, and the maximal central extension of K/k by MCK/k. Then we have Kab⊃MCK/k⊃kab·K.

Our purpose is to prove the following

Theorem. Let k be an even integer ≥ 6. Let

be a Siegel cusp form of degree two, weight k. Then we have

In this paper we will investigate integrability of derivations in rings of unequal characteristic. Generally speaking this case is more difficult than the case of characteristic p which was studied in [4], New interesting phenomena arise, cf. e.g. Theorem 1 and Example 3. Although we obtain useful sufficient conditions for integrability only in very restricted cases, we also give several examples to illuminate the situation.