Let G be a finite group, F a field and M an irreducible F[G]-module. By ^ we denote the F-linear involutary antiautomorphism of F[G], induced by inversion on group elements. Suppose that char (F) ≠. 2. We then show that M carries a non-singular G-invariant symmetric bilinear form with values in F if and only if there exists a ^-invariant idempotent e ∈ F [G] which generates the projective cover of M. This extends earlier results of W. Willems [Wi]. The assertion is not true if char(F) = 2.

The manifold in this paper is assumed to be connected differentiable of class C∞. Let Dr(M) and Ӿr(M) be the set of all diffeomorphisms and vector fields of class Cr on a manifold M with Whitney Cr topology, respectively. In [2], the concept of weak stability is defined. The definition is equivalent to the following ((2.1) of this paper); f∈Dr(M) or X ∈ Ӿr(M) is weakly (allowably) stable if and only if there is a neighborhood U of f or X in Dr(M) or Ӿr(M) such that for any (a suitable) g or Y ∈ U the set of all elements topologically equivalent to g or Y is dense in U, respectively. Here, f, g ∈ Dr(M) are said to be topologically equivalent if they are topologically conjugate and X, Y ∈ Ӿr(M) are said to be topologically equivalent if there is a homeomorphism mapping any trajectory of X onto a trajectory of Y preserving the orientations of the trajectories. Similarly, weak Ω-stability is defined for f and X.

It is the purpose of the present paper to give a natural method of realizing bounded symmetric domains as matrix-spaces. Our method yields, as special cases, the well-known bounded models of irreducible bounded symmetric domains of classical type (I)-(IV), as were already described in the original paper of E. Cartan [1] (see §3; we follow in this paper the classification table in [14], not in [1]). A direct application of this method will be to determine the canonical bounded models of the irreducible bounded symmetric domains of exceptional type; it will be published in another paper (see [6], [7] for the summary of the results).

Let D be a bounded domain of the complex n-space Cn(n≥2), or more generally a pair (M,D) a finite manifold (cf. Definition 2.1), and we assume the boundary ∂D is a smooth and connected submanifold. It is well known by Hartogs-Osgood’s theorem that every holomorphic function on a neighbourhood of ∂D can be continued holomorphically to D. Generalizing the above theorem we shall prove that if a differentiable function on ∂D satisfies certain conditions which are satisfied for the trace of a holomorphic function on a neighbourhood of ∂D, then it can be continued holomorphically to D (Theorem 2-5). The above conditions will be called the tangential Cauchy Riemann equations.

We are interested in the question how the spaces of solutions of elliptic equations vary according to the variations of underlying regions and coefficients of the equation. We will discuss this question for the case of equations Δu = Pu considered on noncompact Riemann surfaces R. Typically we ask the properties of mappings τx: (R,P) → dim PX(R) from the space Φ of pairs (R, P) of noncompact Riemann surfaces R and densities P on R, i.e. P(z)dxdy are 2-forms on R such that P(z)dxdy ≢ 0 and P(z) ≥ 0 are Hölder continuous with respect to local parameters z = x + iy, into cardinals, where PX(R) are the linear spaces of solutions of Δu = Pu on R with certain boundedness properties X.

Let A be a Weil algebra. The fibre bundle TAM of A-velocities over a manifold M was described by A. Morimoto [15] as another description of the bundle of near A-points by Weil [17]. In [4] for any tensor field τ of type (0,2) on M and any functional λ ∈ A* we have defined the so called λ-lift of τ to TAM. We recall this construction in Example 1.3. The λ-lift of τ is a naturally induced tensor field of type (0,2) on TAM.

A Willmore surface $y:M\rightarrow S^{n+2}$ has a natural harmonic oriented conformal Gauss map $Gr_{y}:M\rightarrow SO^{+}(1,n+3)/SO(1,3)\times SO(n)$, which maps each point $p\in M$ to its oriented mean curvature 2-sphere at $p$. An easy observation shows that all conformal Gauss maps of Willmore surfaces satisfy a restricted nilpotency condition, which will be called “strongly conformally harmonic.” The goal of this paper is to characterize those strongly conformally harmonic maps from a Riemann surface $M$ to $SO^{+}(1,n+3)/SO^{+}(1,3)\times SO(n)$, which are the conformal Gauss maps of some Willmore surface in $S^{n+2}.$ It turns out that generically, the condition of being strongly conformally harmonic suffices to be associated with a Willmore surface. The exceptional case will also be discussed.

On considère une variété différentielle M, paracompacte de classe C∞. Etant donné une algèbre locale A (algèbre commutative unitaire de dimension finie sur R dont l’idéal maximal m est de codimension 1 sur R), on rappelle qu’un point proche de x ∊ M d’espèce A est un homomorphisme d’algèbres ξ de C∞(M) [algèbre des fonctions numériques de classe C∞ sur M] dans A tel que ξ(f)≡f(x) mod m pour toute fonction f ∊ C∞(M) [9]. En notant l’ensemble des points proches de x d’espèce est une variété différentielle de dimension n × dim A où n = dim M. Si A = R[T1 … T3]/(T1, …, Ts)k+1, la variété MA s’identifie à la variété des jets, , des applications différentiables de classe C∞ de Rs dans M ayant 0 ∊ Rs pour source.

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

We shall discuss the sample path continuity of a stationary process assuming that the spectral distribution function F(λ) is given. Many kinds of sufficient conditions have been given in terms of the covariance function or the asymptotic behavior of the spectral distribution function.

Dans toute la suite Rn désignera l’espace euclidien à dimension n (≧ 1). Pour un point x = (x1, x2, · · ·, xn) de Rn, on notera |x| = .La coordonnée sphérique dans Rn s’écrira (r, σ). R+ désignera l’ensemble {t ∈ R1; t ≧ 0}.

Let f be a positive integer such that f ≢ 2 (mod 4). Let h0 be the class number of the maximal real subfield of fth cyclotomic field Q(ζf)- It is interesting to determine when h0 is even. Kummer [11] investigated this problem when f is a prime and showed that if h0 is even, then the relative class number h of the cyclotomic field is even (Satz III). Moreover he gave another necessary condition for h0 to be even (Satz IV). In [7] Hasse gave a necessary and sufficient condition for h to be even (Satz 45).

An R-module M is a generator (of the category of modules) provided every module is a homomorphic image of a suitable direct sum of copies of M. Equivalently, some M(k) has R as a summand. Except in the last section, all rings are assumed to be commutative, Noetherian domains, and modules are usually finitely generated. In this context generators are exactly those modules that have non-zero free summands locally. Of course, generators can fail to have free summands (e.g., over Dedekind domains), and we ask whether they necessarily have non-zero projective summands. The answer is “yes” for rings of dimension 1, as we point out in § 3, and for the polynomial ring in one variable over a Dedekind domain. In § 1 we show that for 2-dimensional rings the answer is intimately connected with the structure of projective modules. Our main result in the positive direction, Theorem 1.3, grew out of the attempt, in conversations with T. Stafford, to understand the case R = k[x, y]. In § 2 we give examples of rings having generators with no projective summands. The last section contains miscellaneous observations, some of them on rings without chain conditions.

Let S be a ring and d be a derivation of S. The Oreextension S(X,d) is the ring generated by S and an indeterminate X satisfying the ralation Xa — aX = da for all a in S.

Sario’s theory of principal functions fully discussed in his research monograph [3] with Rodin stems from the principal function problem which is to find a harmonic function p on an open Riemann surface R imitating the ideal boundary behavior of the given harmonic function s in a neighborhood A of the ideal boundary δ of R.

Es bezeichne Sn die n-dimensionale Sphäre, Eine solche ist der Rand einer (n + 1)-dimensionalen euklidischen Vollkugel vom Radius 1.

A class of exotic $_3F_2(1)$ -series is examined by integral representations, which enables the authors to present relatively easier proofs for a few remarkable formulae. By means of the linearization method, these $_3F_2(1)$ -series are further extended with two integer parameters. A general summation theorem is explicitly established for these extended series, and several sample summation identities are highlighted as consequences.

In his masterwork Disquisitiones Arithmeticae, Gauss stated an approximate formula for the average of the class number for negative discriminants. In this paper we improve the known estimates for the error term in Gauss approximate formula. Namely, our result can be written as for every ∊ > 0, where H(−n) is, in modern notation, h(−4n). We also consider the average of h(−n) itself obtaining the same type of result.

Proving this formula we transform firstly the problem in a lattice point problem (as probably Gauss did) and we use a functional equation due to Shintani and Dirichlet class number formula to express the error term as a sum of character and exponential sums that can be estimated with techniques introduced in a previous work on the sphere problem.

A radical of a field K is a non zero element of a given algebraic closure some positive power of which lies in K. The group R(K) of radicals reflects properties of the field K and is in turn easily determined as an extension of the multiplicative group K* of non zero elements of K. The elements of the quotient group R(K)/K* are then conveniently identified with certain subspaces of the algebraic closure, the radical spaces of K (cf. §1). What we are here concerned with is the corresponding arithmetic situation, in which we start with a Dedekind domain o with quotient field K. The role of the radicals is taken over by the radical modules. These form a group (o) which contains the group of fractional ideals of o (cf. §4).

Fix a positive integer m. Let Fm denote the Fermat curve over Q of degree m, given by the projective equation