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.

We provide results on differential stability of metric projection in Sobolev space onto convex set

(1.1)

where Ω ⊂ Rd is open, bounded domain.

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.

Classically, trace was defined as the sum of the diagonal entries of a square matrix with entries in a field. This notion played an important role in classical mathematics, e.g. in the theory of algebras over a field of characteristic zero, and in the theory of group characters (as in [1]). A generalization to endomorphisms of a finitely generated projective module over any ring R with unit is well-known. For such a module P the canonical homomorphism Ψ: P* ⊗ P→ EndR(P) is an isomorphism. Then the composite ε˚Ψ-1: EndR(P) → P* ↗ P→ R, where e denotes “evaluation”, is a homomorphism which coincides with the classical trace whenever P is free. This version of trace has been used by Hattori [3] and others to study projective modules. However, this approach to trace is limited to the finitely generated projective modules, since it can be shown that Ψ is an isomorphism if and only if P is finitely generated and projective.

Let V be a convex cone in a real vector space X, F: Y × Y → Xc a V-positive hermitian map on a complex vector space Y, and

the Siegel domain associated to V and F. D(V,F) is said to be symmetric, if for each point p ∈ D(V, F) there exists an involutive holomorphic automorphism σp of D(V,F) such that the fixed point set of σp consists of only the point p. Satake [6] showed that the symmetric Siegel domain D(V, F) is characterized by the following three conditions (i), (ii) and (iii).

The main goal of this paper is to show that Castelnuovo–Enriques’ $P_{12}$- theorem (a precise version of the rough classification of algebraic surfaces) also holds for algebraic surfaces $S$ defined over an algebraically closed field $k$ of positive characteristic ($\text{char}(k)=p>0$). The result relies on a main theorem describing the growth of the plurigenera for properly elliptic or properly quasielliptic surfaces (surfaces with Kodaira dimension equal to 1). We also discuss the limit cases, i.e., the families of surfaces which show that the result of the main theorem is sharp.

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.

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 theory of ordinal diagrams has been a most powerful means for consistency proofs of some systems of second order arithmetic. The last existing result in this line is the consistency proofs of the systems with the provably -comprehension axiom and the -comprehension axiom respectively (cf. [6]). In order to pursue the consistency problem further, one needs investigate the theory of ordinal diagrams in two directions—refinement and strengthening of the theory.

For this purpose we have begun to search for some properties concerning ordinal diagrams and some variations of the theory of ordinal diagrams. The reader is requested to refer to §26 of [7] for the basic knowledge of ordinal diagrams.

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.

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 T be an automorphism on a probability space (Ω, B, P). Given a finite partition α of Ω, into disjoint measurable sets A1, A2, · · ·, An, its entropy is

(D.1)

Let X be a locally compact and σ-compact abelian group and let denote the dual group of X. We denote by ξ a fixed Haar measure on X and by the Haar measure associated with ξ. In [2], we show the following

Theorem. Let (αt)t≧0 be a sub-Markov convolution semi-group on X and let ψ be the negative definite function associated with (αt)t≧0. Then (αt)t≧0 is transient if and only if Re (1/ψ) is locally -summable.

We prove that for $n\geqslant 4$, every knot has infinitely many conjugacy classes of $n$-braid representatives if and only if it has one admitting an exchange move.

A nonnegative locally Hölder continuous second order differential P = P(z)dxdy (z = x + iy) on a Riemann surface R is referred to as a density on R. A density P is said to be finite if P is integrable over R, i.e.

(1) ∫ R P(z)dxdy < ∞.

Let k be an algebraic number field of finite degree, and p be a fixed rational prime. We denote the set of all the non-Archimedian prime divisors of k by S0(k) and the set of all the real Archimedian ones by (k). Put and S = S0 ∪ S∞, and define a subgroup of the unit group (k) of k by

The analytic capacity γ(E) of a compact set E in the complex plane C is defined by γ(E) = sup , where — f′(∞) is the 1/z-coeffieient of f(ζ) at infinity and the supremum is taken over all bounded analytic functions f(ζ) outside E with supremum norm less than or equal to 1. Analytic capacity γ(·) plays various important roles in the theory of bounded analytic functions.

In the complex function theory, Picard’s Great Theorem plays an essential and important role. It is well-known as generalizations of this theorem that in a neighborhood of an isolated essential singularity, a meromorphic function cannot be exceptionally ramified (see W. Gross [2]) and that even it cannot be normal (see O. Lehto and K. I. Virtanen [7]). We are therefore interested in the behaviour of meromorphic functions with non-isolated essential singularities as well as in generalizations of the Gross’ result. Several approaches in this direction have been made by G. af Hällström [3], S. Kametani [4], K. Noshiro [13], K. Matsumoto [8], [9], [10], [11], [12], S. Toppila [15], etc..

In this paper we prove that a normal Gorenstein surface dominated by P2 is isomorphic to a quotient P2/G, where G is a finite group of automorphisms of P2 (except possibly for one surface ). We can completely classify all such quotients. Some natural conjectures when the surface is not Gorenstein are also stated.

Let K be a finite extension of (= the maximal unramified extension of Qp) of degree eK, its integer ring, p a rational prime and r a positive integer. If there exists a one parameter formal group defined over whose reduction is of height 2 with a cyclic subgroup V of order pr defined over , then .

We apply this result to a criterion for non-existence of Q-rational point of . (This criterion is Momose’s theorem in [14] except for the cases p = 5 and p = 13, but our new proof does not require defining equations of modular curves except for the case p = 2.)