Let D denote the unit disk | z | < 1, and C the unit circle | z | = 1. Corresponding to any function f meromorphic in D we denote by f* the spherical derivative

Let D denote the unit disc | z | <1, C the unit circle | z | = 1 and Cr the circle | z| = r. Corresponding to any function w(z) meromorphic in D we denote by w*(z) the spherical derivative

There are a number of methods to find minimal two-level forms for a given Boolean function, e g. Harvard’s group [1], Veitch [2], Quine [3], [4], Karnaugh [5], Nelson [6], [7] etc,. This paper presents an approach which is suitable for mechanical or automatic computation, as the Harvard method and the Quine method are so. On the other hand, it shares the same property as the Veitch method in the sense that some of essential prime implicants may be found before all prime implicants are computed. It also adopts the procedure to reduce the necessary steps for computation which is shown in Lawler [8]. The method described is applicable to the interval of Boolean functions f, g such that f implies g where for simplification of sum form the variables occurring in g also occur in f and for product form the variables in f also occur in g.

One of the authors discussed finite dimensional approximations to a white noise and a periodic Brownian motion with period 2 π on the projective limit space of spheres ([2]). The group of unitary operators derived from the periodic white noise has a pure point spectrum which consists of all integers with countably infinite multiplicity. We also have much interest in the investigation of a band limited white noise which is another typical example having quite different spectral type. Indeed, the corresponding group of unitary operators has a continuous spectrum with countably infinite multiplicity.

In this paper, I should like to add some remarks to my previous note titled “Some results in the Fourier analysis” (Nagoya Math. Journal, Vol. 27, 1966). At first, we shall show that the orthnormal set exp [2 πi(m1x1+ … mnxn is complete in the Hilbert space L2 over the unit cube E = {(x1, …, xn); 0≧xj≦1 (1≦j≦n)}, where the inner product ‹f, g› for f, g ∊ L2 over E is defined by

Let be the class of analytic functions Ф(z) which are regular and of positive real part in the unit circle | z | <1 and normalized by Ф(0) = 1. Several distortion theorems have been obtained on various functionals in this class. In a previous paper [4] we have dealt with mean distortion which generalizes a classical theorem of Rogosinski [6].

In this paper, we shall consider the logarithmic potential

where μ is a positive measure in the plane, P and Q are any points and PQ denotes the distance from P to Q. In general, consider the potential

of a positive measure μ taken with respect to a kernel K(P, Q) which is a continuous function in P and Q and may be + ∞ for P = Q. A kernel K (P, Q) is said to satisfy the balayage principle if, given any compact set F and any positive measure μ with compact support, there exists a positive measure μ′ supported by F such that K(P, μ′) = K(P, μ) on F with a possible exception of a set of k-capacity zero and K(P, μ′)≦K(P, μ) everywhere. A kernel K(P, Q) is said to satisfy the equilibrium principle if, given any compact set F there exists a positive measure λ supported by F such that K(P, λ) = V (a constant) on F with a possible exception of a set of K-capacity zero and K(p, λ)≦V everywhere.

Several years ago, K. Oikawa, my colleague, investigated the properties of Schottky coverings of closed Riemann surfaces, leaving an interesting problem as open [3]:

Does exist a Schottky covering between the basic surface and a given planar covering?

The tangent bundle of a differentiable manifold is an important invariant of a differentiable structure. It is determined neither by the topological structure nor by the homotopy type of a manifold. But in some cases tangent bundles depend only on the homotopy types of manifolds.

In this note we will improve the dual EHP sequence which has been constructed in [6] by showing that that can be extended by one term. We then observe that this can be used to deduce a result which has been announced by T. Ganea in [4]. As another application we will establish a theorem which asserts that, under certain conditions, a principal fibration with a loop-space as fibre is principally equivalent to the one induced by some map.

Let k be an algebraic number field and K be its normal extension of finite degree. Then the genus field K* of K over k is defined as the maximal unramified extension of K which is obtained from K by composing an abelian extension over k2). We call the degree (K*: K) the genus number of K over k.

In a Green space Ω we can introduce Martin’s topology and make it the Martin space Ω, Ω is a dense open subset of and the kernel

can be extended continuously to , where G(p, x) is a Green function in Ω and y0 the fixed point of Ω. is a metric space. is divided into two disjoint subsets Δ0, Δ1 and s ∊ Δ1 is characterized by the fact that K(s, x) is a minimal positive harmonic function in x∊Ω.