Main purpose of the present paper is to study formal deductions described along the line of my former work [1]. In the present paper, I restrict myself to the primitive logic. To extend this method to other logics such as the lower classical predicate logic or the intuitionistic predicaste logic, my way of practical discription has to undergo a certain extent of modification.

In this paper we shall study relations among the class B of the Bayes solutions in the strict sense, the class W of the Bayes solutions in the wide sense and the closure c(B), in a certain sense, of the class B. For abbreviation, we shall use the word “Bayes class” for the class B and “Wald class” for the class W.

We denote a permutation group H on a set Γ by (H, Γ). (H, Γ) is called a permutation group of rank 3 if (H, Γ) is transitive and (Ha, Γ), a ∈ Γ, has exactly three orbits, where Ha is the stabiliger of a point a, namely, {α ∈ H \aα = a}

A deformation theory for rings and algebras was introduced recently by M. Gerstenhaber [1]. Let K be an extension of a field k, and p denotes the characteristic. One of his results is that, if K is separable over k, then it is rigid. It was conjectured in [1] that, if K is not separable over k, then it is not rigid, and if it is further finitely generated, then an integrable element of (see [2]) will be found in the image of Sqp. In this note we shall study the above conjecture in certain special case.

Let R be a Riemann surface. Let G be a domain in R with relative boundary ∂G of positive capacity. Let U(z) be a positive superharmonic function in G such that the Dirichlet integral D(min(M,U(z))) < ∞ for every M. Let D be a compact domain in G.

Throughout this paper all functions are single-valued. Let R be a Riemann surface. We shall denote by φ∧ the least harmonic majorant of a function φ defined in R if it has the meaning.

Recently, S. Nagata gave an interesting series of rules beginning with Peirce’s rule introduced in [3], each rule in the series being really stronger than its successor in the intuitionistic logics. (See [1] Nagata.) Namely, let p0, p1, … be any series of mutually distinct propositional variables.

Main purpose of the present paper is to exhibit a class of truth-value evaluations of the primitive logic LO and its sentential part LOS.

In this paper we prove:

THEOREM: Let k be an algebraically closed field of characteristic p > 0, and let X and Y be abelian varieties over k. Then the group Ext2(X, Y) = 0.

Let E be a compact set of logarithmic capacity zero in the complex plane. Then the following is well-known as Evans-Selberg’s theorem [1] [8]: there is a measure with support contained in E such that its logarithmic potential is positively infinite at each point of E. But such a potential does not exist for E of logarithmic positive capacity. Now suppose that E is contained in the circumference of the unit disc |z| < 1 and is of linear measure zero.

In dieser Note soil die Struktur des Quasinormalteilers vom Exponenten p der endlichen p-Gruppe untersucht werden. Ferner sollen einige Beispiele sowohl im regularen Falle als auch im nicht regulären Falle angeführt werden, die zeigen, daß das erst gesagte Ergebnis im allgemeinen Quasinormalteiler nicht immer zutrifft.

Let G be a connected Lie group and H a closed subgroup, then the homogeneous space M = G/H is called reductive if there exists a decomposition (subspace direct sum) with where g (resp. ) is the Lie algebra of G (resp. H); in this case the pair (g,) is called a reductive pair.

At first, we define three relations ⊇, =, and ⊃ in connection with a pair of logics L and L* as follows:

• L ⊇ L*, if and only if every proposition provable in L* is also provable in L;

• L = L*, if and only if L ⊇ L* and L* ⊇ L;

• L ⊃ L*, if and only if L, ⊇ L* but not L* ⊇ L.

Let k denote the quotient field of a complete discrete rank one valuation ring R. The purpose of this paper is to establish a relationship between the Brauer group of k and the set of maximal orders over R which are equivalent to crossed products over tamely ramified extensions of R.

QF-3 algebras R are classified according to their second commutator algebras R′ with respect to the minimal faithful module, which satisfy dom.dim. R′ ≧ 2. The class C(S) of all QF-3 algebras whose second commutator is S, contains besides S only algebras R with dom.dim. R = 1. C(S) contains a unique (up to isomorphism) minimal algebra which can be represented as a subalgebra S0 of S describable in terms of the structure of S, and C(S) consists just of the algebras S0 ⊂ R ⊂ S (up to isomorphism).

Let V be a projectively embeddable complete non-singular variety of dimension n > 1. Let f be a projective embedding of V, U a non-singular variety, W a non-singular variety and φ a morphism of W onto U such that φ-1(u0) = f(V) for some point u0 of U. Denote by ∑(V) the set of all those complete non-singular fibres φ-1(u), u ∈ U, as we consider all possible (f, U, W). Suppose that we call members of ∑(V) (algebraic) deformations of V and propose to study ∑(V) from the stand point of algebraic geometry, as a generalization of the case of curves. This has been taken up at least locally by Kodaira, Spencer, Kuranishi and others in the case of characteristic 0 from a little more general point of view of complex manifolds (cf. [9] and references given in [16]).

Although tangent bundles of manifolds are not always homotopically invariant, but in some categories of the manifolds they can be homotopically invariant.

In this note, I show that tangent bundles of π-manifolds and almost parallelizable manifolds depend only on their homotopy types.

The main subject in the present article has the origin in the following quite primitive question: Linear systems of ordinary differential equations form a nice family. Then, from the projective point of view, what does correspond to linear systems?

The purpose of this paper is to extend the continuation theorem of holomorphic functions, especially the generalized Hartogs-Osgood’s theorem given in the previous papers [6] and [8], to the case of sections of torsion-free coherent analytic sheaves over a reduced complex space. In the following, we restrict ourselves only to reduced complex spaces.