We give a detailed survey, known and new results on the domain and the range of the maximal operator. In particular we employ the grand Lp spaces and logarithmic Lebesgue spaces.

In [6], Mason reported some connections between sporadic simple group M24 and certain cusp forms which appear in the ‘denominator’ of Thompson series assigned to Fisher-Griess’s group F1. In this paper, we discuss the ‘numerator’ of these Thompson series.

One of the basic problems in the theory of dynamical systems is the characterization of stable systems.

Let M be a closed (i.e. compact without boundary) connected smooth manifold with a smooth Riemannian metric and Diffr (M) (r ≥ 1) denote the space of Cr diffeomorphisms on M with the uniform Cr topology.

Pontrjagin rings over the field of rational numbers of compact Lie groups are commutative in the sense of graded algebras (or anti-commutative in the classical terminology) [14]. Pontrjagin rings over the field Zp (p 0) of several compact simple Lie groups were studied by Borel [5]. The most examples are commutative. However, this is generally not true.

Let k be a real quadratic field and k, E the ring of integers and the group of units in k. Denoting by E() the subgroup represented by E of (k/)× for a prime ideal , we show that prime ideals for which the order of E() is theoretically maximal have a positive density under the Generalized Riemann Hypothesis.

Let (A, m, k) be a Noetherian local ring. We define

and call it the Rees algebra of A. Let X be an indeterminate over A, then R(A) can be identified with the A-subalgebra .

Since M.S. Baouendi and C. Goulaouic ([2], [3]) defined partial differential operators of Fuchs type and proved theorems of Cauchy-Kowalevskaya type and Holmgren type, many authors have investigated operators of Fuchs type in various categories, that is, real-analytic, C∞ and so on. (Cf. [1], [4], [6], [8], [9], [11], [12], [17], [18], [19], [20], [21] etc.)

Let F be a Siegel cusp form of integral weight k on Γ2: = Sp2(Z) and denote by a(T) (T a positive definite symmetric half-integral (2,2)-matrix) its Fourier coefficients. In [2] Kitaoka proved that

(1)

(the result is actually stated only under the assumption that k is even). In our previous paper [3] it was shown that one can attain

(2)

Polylogarithmic functions satisfy functional equations. The most famous equation is of course the functional equation of the logarithm

log x + log y = log(x · y).

The other well known equation is the Abel equation of the dilogarithm

A uniform upper bound for the Diederich-Fornaess index is given for weakly pseudoconvex domains whose Levi form of the boundary vanishes in l-directions everywhere.

Recently, systematic applications of the Scott-Solovay Boolean valued set theory were done by several authors; Takeuti [25, 26, 27, 28, 29, 30], Nishimura [13, 14] Jech [8] and Ozawa [15, 16, 17, 18, 19, 20] in analysis and Smith [23], Eda [2, 3] in algebra. This approach seems to be providing us with a new and powerful machinery in analysis and algebra. In the present paper, we shall study Banach space objects in the Scott-Solovay Boolean valued universe and provide some useful transfer principles from theorems of Banach spaces to theorems of Banach modules over commutative AW*-algebras. The obtained machinery will be applied to resolve some problems concerning the module structures of von Neumann algebras.

Let R be a perfect 𝔽-algebra equipped with the trivial norm. Let W(R) be the ring of p-typical Witt vectors over R equipped with the p-adic norm. At the level of nonarchimedean analytic spaces (in the sense of Berkovich), we demonstrate a close analogy between W(R) and the polynomial ring R[T] equipped with the Gauss norm, in which the role of the structure morphism from R to R[T] is played by the Teichmüller map. For instance, we show that the analytic space associated to R is a strong deformation retract of the space associated to W(R). We also show that each fiber forms a tree under the relation of pointwise comparison, and we classify the points of fibers in the manner of Berkovich’s classification of points of a nonarchimedean disk. Some results pertain to the study of p-adic representations of étale fundamental groups of nonarchimedean analytic spaces (i.e., relative p-adic Hodge theory).

We shall consider a semi-primary ring ۸ with radical N (i.e. N is nilpotent and ۸/N is semi-simple (with minimum condition)). All modules considered are left ۸-modules. We refer to [1] for all notions relevant to homological algebra.

The objective of this paper is to establish the following two theorems:

Theorem I. Let a be a two-sided ideal in A such that

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.

A conjecture concerning characterization of blocks of representations of symmetric groups by means of Young diagrams, given formerly by one of the writers, was affirmed recently by R. Brauer and G. de B. Robinson jointly. Namely, it was proved in Brauer’s paper, relying on his general, and profound, theory of blocks in finite groups, that the number of p-blocks in a symmetric group of degree n, is exactly equal to the number of p-cores, or p-kernels in the terminology of NI, with n- Ip (l= 0, 1, 2,…) nodes, and that certain statements designated as (A), (B) and (C) would secure that two irreducible representations with a same p-core belong to a same block (whence the conjecture itself when combined with the estimate of the number of blocks). Then these statements (A), (B), (C) were proved actually in Robinson’s paper.

This is a general study of twisted Calabi–Yau algebras that are $\mathbb {N}$ -graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi–Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin–Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi–Yau algebras of dimension 0 as separable k-algebras, and we similarly characterize graded twisted Calabi–Yau algebras of dimension 1 as tensor algebras of certain invertible bimodules over separable algebras. Finally, we prove that a graded twisted Calabi–Yau algebra of dimension 2 is noetherian if and only if it has finite GK dimension.

The group of cycles of codimension one algebraically equivalent to zero of a nonsingular projective variety modulo rational equivalence forms an abelian variety, i.e., the Picard variety. To the group of cycles of dimension zero and of degree zero, there corresponds an abelian variety, the Albanese variety. Similarly, Weil, Lieberman and Griffiths have attached complex tori to the cycles of intermediate dimension in the classical case. The aim of this article is to give a purely algebraic construction of such “intermediate Jacobian varieties.”

Let M be an n-dimensional (n≥2) connected C∞-manifold with a linear connection. For simplicity, tensor fields on M will simply be called tensors on M. A tensor S on M is said to be parallel if its covariant derivative is everywhere zero in M, i.e., if ▽S = 0. S is said to be recurrent if its covariant derivative is equal to the tensor product of a covector and S itself, i.e., if ▽S = W⊗S, where W is called the recurrence covector.

In this note, we introduce an approximation of harmonic maps via a sequence of exponentially harmonic maps. We then reestablish the existence theorem of harmonic maps due to Eells and Sampson.

Let K be an algebraically closed field of arbitrary characteristic. The term “variety” always means here an irreducible algebraic variety over K. The notations and the terminology are borrowed in general from EGA [4].