Let X be a projective Gorenstein variety, Y ⊂ X a proper closed subscheme such that X is smooth at all points of Y, so that the formal completion of X along Y is regular.

In the study of elliptic differential operators of second order, we found that the automorphism pseudogroups are of finite type ([4]). However this fact takes a complete change in elliptic differential operators of first order.

So as to make the objects which can be dealt with clear, we introduce the concept of pseudoellipticity of first order differential operators (Definition 1.1), which is naturally satisfied by first order elliptic differential operators.

The aim of this paper is to investigate some applications of a certain universal problem. The universal problem deals with categories C which for every object A have some ‘‘structure” on the set C(A, A) of endomorphisms of A and calls for a universal solution, relative to this structure, that is associated with C(A, A) and centralizes the set of automorphisms of A. The commutative version of this universal problem asks for a universal solution, relative to the said structure, that abelianizes the canonical monoid structure of C(A, A).

Let Ω be a bounded domain in Rn with the boundary of class C0,1 and E be a compact subset (resp. a compact subset on an (n —dimensional hypersurface of class C0,1) in Ω. We assume that the usual function spaces and are known.

Let w = f(z) be a normal meromorphic function defined in the upper half plane U = {Im(z) >0}. We recall that a meromorphic function f(z) is normal in U if the family {f(S(z))}, where z′ = S(z) is an arbitrary one-one conformal mapping of U onto U, is normal in the sense of Montel. It is the purpose of this paper to state some results on the behavior of f(z) on curves which approach a point x0 on the real axis R with a fixed (finite) order of contact q at x0.

1. Let F be an algebraic number field of finite degree, and let K/F be a normal extension of degree n. Denote by OK the ring of all integers in K. In we proved the following:

Improving and clarifying a construction of Horowitz and Shelah, we show how to construct (in $\mathsf {ZF}$, i.e., without using the Axiom of Choice) maximal cofinitary groups. Among the groups we construct, one is definable by a formula in second-order arithmetic with only a few natural number quantifiers.

Let $\unicode[STIX]{x1D6EC}$ be a connected hereditary artin algebra. We show that the set of functorially finite torsion classes of $\unicode[STIX]{x1D6EC}$-modules is a lattice if and only if $\unicode[STIX]{x1D6EC}$ is either representation-finite (thus a Dynkin algebra) or $\unicode[STIX]{x1D6EC}$ has only two simple modules. For the case of $\unicode[STIX]{x1D6EC}$ being the path algebra of a quiver, this result has recently been established by Iyama–Reiten–Thomas–Todorov and our proof follows closely some of their considerations.

The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient ${\mathcal {M}}^{\operatorname {GIT}}$, as a Baily–Borel compactification of a ball quotient ${(\mathcal {B}_4/\Gamma )^*}$, and as a compactified K-moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup ${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$, whereas from the ball quotient point of view, it is natural to consider the toroidal compactification ${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$. The spaces ${\mathcal {M}}^{\operatorname {K}}$ and ${\overline {\mathcal {B}_4/\Gamma }}$ have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in fact not the case. Indeed, we show the more refined statement that ${\mathcal {M}}^{\operatorname {K}}$ and ${\overline {\mathcal {B}_4/\Gamma }}$ are equivalent in the Grothendieck ring, but not K-equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients.

Let $K=\mathbb{F}_{q}(T)$ and $A=\mathbb{F}_{q}[T]$. Suppose that $\unicode[STIX]{x1D719}$ is a Drinfeld $A$-module of rank $2$ over $K$ which does not have complex multiplication. We obtain an explicit upper bound (dependent on $\unicode[STIX]{x1D719}$) on the degree of primes ${\wp}$ of $K$ such that the image of the Galois representation on the ${\wp}$-torsion points of $\unicode[STIX]{x1D719}$ is not surjective, in the case of $q$ odd. Our results are a Drinfeld module analogue of Serre’s explicit large image results for the Galois representations on $p$-torsion points of elliptic curves (Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331; Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Etudes Sci. Publ. Math. 54 (1981), 323–401.) and are unconditional because the generalized Riemann hypothesis for function fields holds. An explicit isogeny theorem for Drinfeld $A$-modules of rank $2$ over $K$ is also proven.

The purpose of this paper is to study threefolds X, with negative Kodaira dimension k(X) and positive irregularity q(X), defined over the complex field C.

We prove the vanishing and non-vanishing theorems for an intersection of a finite number of q-complete domains in a complex manifold of dimension n. When q does not divide n, it is stronger than the result naturally obtained by combining the approximation theorem of Diederich-Fornaess for q-convex functions with corners and the vanishing theorem of Andreotti-Grauert for q-complete domains. We also give an example which implies our result is best possible.

We characterize numbers having purely periodic β-expansions where β is a Pisot number satisfying a certain irreducible polynomial. The main tool of the proof is to construct a natural extension on a d-dimensional domain with a fractal boundary.

Let M be a compact non-singular real affine algebraic variety and let A, B be open disjoint semialgebraic subsets of M. Define (where —4 denotes the Zariski closure).

The primary purpose of this paper is to give an affirmative answer to a problem posed by Ohtsuka [13] whether there exists a p-harmonic measure on the unit disc in the 2-dimensional Euclidean space R2 with an infinite p-Dirichlet integral for the exponent 1 < p < 2.

Let Ω be a proper subdomain of ℝn, n ≥ 2, and let x0∈ Ω be fixed. By GΩ and KΩ we denote the Green function and the Martin kernel for Ω, respectively. Under a certain assumption on Ω near a boundary point ξ, we show that the product GΩ(x, x0)KΩ(x, ξ) is comparable to |x - ξ|2-n for x in a nontangential cone with vertex at ξ. We also give an estimate for the product KΩ(x, ξ)KΩ(x,η) in a uniform domain, where η is another boundary point.

Let P be a pseudodifferential operator of the form

where s, b ≥ 0 are even integers and is odd function with f′(t) > 0 (t ≠ 0). Here . We shall call P an operator of Egorov type because P with f(t) = tk, (k odd) is an important model of subelliptic operators studied by Egorov [1] and Hörmander [3], [4, Chapter 27]. Roughly speaking, any subelliptic operator can be reduced to this operator or Mizohata one after several steps of microlocalization arguments. In this paper we shall study the hypoellipticity of P and the local solvability of adjoint operator P* in the case where f(t) vanishes infinitely at the origin and moreover consider the case where ts and are replaced by functions with zero of infinite order.

1. Nous avons un théorème, ce qu’on appelle le théorème de Cartan-Dieudonné, sur les générateurs du groupe orthogonal: Toute transformation orthogonale à n variables sur un corps de caractéristique ≠2 est un produit de n symetries au plus, et sur un corps de caractéristique 2 elle est un produit au plus de n transvections orthogonales, sauf un seul cas, [1] ou [3]. C’est une généralisation d’un résultat obtenu par E. Cartan, relatif au corps des nombres réels ou au corps des nombres complexes [4].

Let k be a p-adic number field and o be the ring of all integers in k. Moreover, let K\k be a finite Galois extension with the Galois group G = G(K/k). Then the ring of all integers in K is an -module.

According to Kazhdan-Lusztig and Ginzburg, the Hecke algebra of an affine Weyl group is identified with the equivariant K-group of Steinberg’s triple variety. The K-group is equipped with a filtration indexed by closed G-stable subvarieties of the nilpotent variety, where G is the corresponding reductive algebraic group over ℂ. In this paper we will show in the case of type A that the filtration is compatible with the Kazhdan-Lusztig basis of the Hecke algebra.