Let G be a connected linear algebraic group and X an algebraic variety, both defined over Q, the field of rational numbers. Suppose that G acts on X transitively and the action is defined over Q. Suppose that the set of rational points X(Q) is non-empty. Choosing x ∈ X(Q) allows us to identify G/Gx and X as varieties over Q, there Gx is the stabilizer of x.

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}.

In this article we discuss three types of mean values of the Euler double zeta-function. To get the results, we introduce three approximate formulas for this function.

We prove a McShane-type identity: a series, expressed in terms of geodesic lengths, that sums to 2π for any closed hyperbolic surface with one distinguished point. To do so, we prove a generalized Birman-Series theorem showing that the set of complete geodesics on a hyperbolic surface with large cone angles is sparse.

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.

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.

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.

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.

Let p be an odd prime and d be a positive integer prime to p such that d ≢ 2 mod 4. For technical reasons, we also assume that . For each integer n ≥ 1, we choose a primitive nth root ζn of 1 so that whenever n | m. Let be its cyclotomic Zp-extension, where is the nth layer of this extension. For n ≤ 1, we denote the Galois group Ga\(Kn/K0) by Gn, the unit group of the ring of integers of Kn by En, and the group of cyclotomic units of Kn by Cn. For the definition and basic properties of cyclotomic units such as the index theorem, we refer [6] and [7]. In this paper we examine the injectivity of the homomorphism between the first cohomology groups induced by the inclusion Cn → En.

Segregated algebras have been nicely characterized by M. Ikeda [4]. In this paper §1, we consider segregated rings and study the structure of such rings in Theorems 1.1 and 1.2. In §2, we specialize to the case of segregated algebras of finite dimension over a field. Theorem 2.1 gives a new characterization of such algebras. Theorem 2.2 shows an interesting property of segregated algebras; two segregated algebras S and T, with radicals N and P respectively, are isomorphic if and only if S/N2 and T/P2 are isomorphic.

A distribution μ on R+ = [0, ∞) is said to be a distribution if there are an increasing (in the strict sense) sequence of positive real numbers such that, for each j = 0, …, m, there is at least one ak satisfying bj < ak < b+1 and the

Laplace transform of μ is represented as

Let L be a divergence form elliptic operator with complex bounded measurable coefficients, let ω be a positive Musielak-Orlicz function on (0, ∞) of uniformly strictly critical lower-type pω ∈ (0, 1], and let ρ(x,t) = t−1/ω−1 (x,t−1) for x ∈ ℝn, t ∊ (0, ∞). In this paper, we study the Musielak-Orlicz Hardy space Hω,L(ℝn) and its dual space BMOρ,L* (ℝ n), where L* denotes the adjoint operator of L in L2 (ℝ n). The ρ-Carleson measure characterization and the John-Nirenberg inequality for the space BMOρ,L (ℝn) are also established. Finally, as applications, we show that the Riesz transform ∇L−1/2 and the Littlewood–Paley g-function gL map Hω,L(ℝn) continuously into L(ω).

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.

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.

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.

We show that every Fricke-invariant meromorphic modular form for $\unicode[STIX]{x1D6E4}_{0}(N)$ whose divisor on $X_{0}(N)$ is defined over $\mathbb{Q}$ and supported on Heegner divisors and the cusps is a generalized Borcherds product associated to a harmonic Maass form of weight $1/2$. Further, we derive a criterion for the finiteness of the multiplier systems of generalized Borcherds products in terms of the vanishing of the central derivatives of $L$-functions of certain weight $2$ newforms. We also prove similar results for twisted Borcherds products.

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.

The aim of this paper is to investigate the behaviour of Green lines at Royden’s boundary Γ of a Riemann surface R with the Green function g(z, o) with the fixed pole o in R. We denote by the totality of Green lines L issuing from the fixed point o.