In the theory of modular representations of a finite group G in an algebraically closed field Ω of characteristic p, Brauer has proved a useful reduction theorem for blocks [2, §§11, 12], [5, (88.8)], which can be reformulated as follows:

THEOREM 1 (Brauer). Let P bean arbitraryp-subgroup of G; let N = NG(P) and W = PCG(P).

The device of representing a predicate by a universal sentence gives rise to logics of higher order. To represent an object constructed by a series of certain steps, we usually use terms. To represent a predicate constructed by a series of certain steps, we now use sentences in general. We can now substitute special predicates represented by sentences for predicate variables of a sentence just as we have been able to substitute special objects represented by terms for object variables. It is remarkable that we can always distinguish predicates from objects, proceeding in this way.

Let Ω and Ω′ be two locally compact, connected Hausdorff spaces having countable bases. On each of the spaces is defined a system of harmonic functions satisfying the axioms of M. Brelot [2]. The following is the description of such a system. To each open set of Ω is assigned a vector space of finite continuous functions, called the harmonic functions, on this set.

The equation y2 = x3 + k, k an integer, has been discussed by many authors. Mordell [1] has found many classes of k values for which the equation has no integral solutions. Fueter [2], Mordell [3] and Chang [4] have found classes of k values for which the equation has no rational solutions. The following two theorems exhibit two more sets of conditions which give rise to classes of k values for which the corresponding equations have no rational solutions.

A vast class of abstractions are proved self-contradictory by Russell-type paradoxes in the sense that the negation of any one of them can be proved tautologically. On the other hand, there are a vast class of abstractions, each being self-consistent. A simple criterion for abstractions to be self-consistent (a sufficient condition) can be given. However, even a fairly restricted class of abstractions, each satisfying the criterion to be self-consistent, may contradict to each other.

Let M = M(D π) be a (real analytic) fibered manifold of dimension n over a manifold D of dimension p, with projection π. We denote by M′ (D, α) the prolonged fibered manifold of M(D, π). Every point of M′ is a p-dimensional contact element of M, and a p-dimensional contact element X of M belongs to M′ if and only if π*X = Tπ(x)(D), where x is the origin of X. We write x = β(X) and α = π°β. We also denote by M″(D, α′) the prolonged fibered manifold of M′(D, α), where α′ = α°β′ and β′ is the projection of each X′ ∈ M″ to its origin in M′.

I have introduced in my paper [1] a formalism for lower order predicate logics in which any sequence of variables and brackets (‘[’ and ‘]’) can be regarded as a meaningful (well-formed) sentence. Also, in my paper [2], I have introduced another formalism of the same kind for a certain kind of predicate logics of higher order. Can we introduce a formalism of the same kind properly for sentence logics too? Since sentence logics can be regarded as a sub-system of predicate logics, we are apt to consider that our device introduced in [1] works well for sentence logics too.

In this paper, I would like to point out some characteristic features of formal systems having just one primitive notion. Most remarkable systems of this kind may be Zermelo’s and Fraenkel’s set-theories, both having just one primitive notion ∈.

It has been my program to develop fundamental theories of mathematics starting from TABOOS and standing on the primitive logic LO at first instead of starting from AXIOMS and standing on the fairly brought up logic, the lower classical logic LK. This was proposed in my work [1].

Introduction. Let S be the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, and let G denote the Galois group of the quotient field extension. Auslander and Rim have shown in [3] that the trivial crossed product Δ (1, S, G) is an hereditary order if and only if 5 is a tamely ramified extension of R. And the author has proved in [7] that if the extension S of R is tamely ramified then the crossed product Δ(f, 5, G) is a Π-principal hereditary order for each 2-cocycle f in Z2(G, U(S)). (See Section 1 for the definition of Π-principal hereditary order.) However, the author has exhibited in [8] an example of a crossed product Δ(f, S, G) which is a Π-principal hereditary order in the case when S is a wildly ramified extension of R.

Some important formal systems are really developable from a finite number of axioms in the lower classical predicate logic LK or in the intuitionistic predicate logic LJ. Any system of this kind can be developed in LK (or in LJ) from the single conjunction of all the axioms of the system.

The purpose of this paper is to investigate the moduli of compact Riemann surfaces with non-trivial automorphisms.

Let Ω be the projective limit space of a sequence of probability space Ωn which is a certain subset of (n − 1)-dimensional sphere with the usual uniform probability distribution on it. T. Hida [2], starting from a sequence of finite dimensional flows which are derived from some one-parameter subgroups of rotations of spheres, constructed a flow {Tt} on Ω as the limit of them. Observing his method, the concept of consistency of flows which approximate {Tt} seems to play an essential role in his work [2]. As will be made clear in the following sections, the concept of consistency is closely related to the projective limiting structure of our basic space Ω. The purpose of this paper is to determine all the flows on Ω which can be approximated in the sense of [2] by finite dimensional flows.

In this paper, I would like to show that there are many systems of abstractions, each abstraction being self-consistent, but each system as a whole leading to contradiction or to some pathological situation.

Let D be the open unit disc, and let C be the unit circle in the complex plane. Let f be a (real-valued) function that is harmonic in D. A simple continuous curve β: z(t) (0≦t<1) contained in D such that |z(t)|→1 as t→1 is a boundary path with end (the bar denotes closure).

In this short note, we would like to point out that the following property (called ASSUMPTION REMOVABILITY in the present paper) is characteristic of the positive logics, the primitive logic LO, the positive predicate logics LP (intuitionistic) and LQ (classical):

In my joint work with J. Ito [7], we have pointed out that the following property (called ASSUMPTION REMOVABILITY) is characteristic of positive logics LO, LP, and LQ:

ASSUMPTION REMOVABILITY. If → is provable for any pair of propositions andhaving no primitive notions (proposition-, predicate-, and relation-symbols) in common, thenis also provable.

Beurling and Deny [1], [2] introduced Dirichlet spaces by generalizing the notion of “energy” in potential theory. They showed the existences of balayaged measures and condensor measures in the theory of Dirichlet spaces. The purpose of this paper is to characterize the supports of those measures. First we obtain the following result.