For fixed k ≧ 2, let Vk denote the class of normalized analytic functions

such that z ∈ E = {z; |z| <1} are regular and have f′(0) = l,f′(z) ≠ 0, and

1

Let Sk be the subclass of Vk whose members f(z) are univalent in E. It was pointed out by Paatero (4) that Vk coincides with Sk whenever 2 ≦ k ≦ 4. Later Rényi (5) showed that in this case, f(z) ∈ Sk is also convex in one direction in E. In (6) I showed that the Bieberbach conjecture

holds for functions convex in one direction.

We introduce and study several notions of amenability for unitary corepresentations and $*$-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantum groups. As a background for this study, we investigate the associated tensor ${{C}^{*}}$-categories.

The transformations we will discuss in this paper are the Hankel transformation Hυ defined for f ∊ C0, the collection of continuous functions compactly supported in (0, ∞), by

(1.1)

and the and transformations defined for such f by

(1.2)

and

(1.3)

where Jv >and Yv are the Bessel functions of the first and second kinds respectively, and Hv is the Struve function; for the theory of these functions see [1, Chapter VII].

These transformations were studied extensively by one of us in [5] and [6] on the spaces defined in [7; Sections 1 & 5]. In those papers the boundedness of the three transformations was fully given on the spaces for 1 < p < ∞, but not for p = 1. Also inversion formulae were given for the transformations only for portions of their respective ranges of boundedness.

If D is an integral domain with quotient field K, the group of divisibility G(D) of D is the partially ordered group of non-zero principal fractional ideals with aD ≦ bD if and only if aD contains bD. If K* denotes the multiplicative group of K and U(D) the group of units of D, then G(D) is order isomorphic to K*/U(D), where aU(D) ≦ bU(D) if and only if b/a ∊ D.

1.1. This paper is a continuation of the papers of the author [14], [15]. We begin by recapitulating the main definitions. If {n,} is an increasing sequence of positive integers, the value of the characteristic or the counting function ω(n) of {nv} is, for any , the number of n satisfying the inequality . Suppose that A is a linear method of summation corresponding to the transformation

In this paper we develop a variational method for the Loewner equation in higher dimensions. As a result we obtain a version of Pontryagin’s maximum principle from optimal control theory for the Loewner equation in several complex variables. Based on recent work of Arosio, Bracci, and Wold, we then apply our version of the Pontryagin maximum principle to obtain first-order necessary conditions for the extremal mappings for a wide class of extremal problems over the set of normalized biholomorphic mappings on the unit ball in ${{\mathbb{C}}^{n}}$.

In this paper we study subrings A of generated by monomials over . If A is normal and A ⊂ B integral, we can completely characterize A. If dim A = 2, we show that A is isomorphic to a subring A’ of B generated by monomials with A’ ⊂ B integral. The author became interested in these rings while studying projective modules over subrings of , For some applications, see [1].

Let $X$ be a nonsingular algebraic variety in characteristic zero. To an effective divisor on $X$ Kontsevich has associated a certain motivic integral, living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant motivic (Igusa) zeta function, associated to a regular function on $X$, which specializes to both the classical $p$-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant.

This paper treats a generalization to singular varieties. Batyrev already considered such a ‘Kontsevich invariant’ for log terminal varieties (on the level of Hodge polynomials of varieties instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any $\mathbb{Q}$-Gorenstein variety $X$ we associate a motivic zeta function and a ‘Kontsevich invariant’ to effective $\mathbb{Q}$-Cartier divisors on $X$ whose support contains the singular locus of $X$.

One of the fundamental results on graph coloring is the following classical theorem of Brooks.

BROOKS’ THEOREM. Suppose that k ≧ 3 and that G is a k-regular graph which does not induce a (k + 1)-clique. Then G is k-colorable.

Brooks proved his theorem in [1]; several more recent proofs have appeared in [3], [4] and [5]. All the proofs of this theorem have the common feature of applying only to finite graphs; the transition to infinite graphs can be accomplished by a very standard implementation of the Compactness Theorem (or some other equally noneffective device such as the theorem of deBruijn and Erdös [2] asserting that a graph is k-colorable if and only if each of its finite subgraphs is). Thus, it is not immediately apparent that an effective version of Brooks’ Theorem exists. It is our purpose to show, however, that the effective analogue of Brooks' Theorem is indeed true.

Besides stating the problem and the results, we shall give in this section a brief overview of the classical necessary and sufficient conditions in the calculus of variations, in order to clearly situate the contribution of this article.

1.1 The problem. We are given an interval [a, b], two points xa, xb in Rn, and a function L (the Lagrangian) mapping [a, b] × Rn × Rn to R. The basic problem in the calculus of variations, labeled (P), is that of minimizing the functional

over some class X of functions x and subject to the constraints x(a) = xa, x(b) = xb. Let us take for now the class X of functions to be the continuously differentiable mappings from [a, b] to Rn; we call such functions smooth arcs.

Let R be a hereditary noetherian prime ring ((hnp)-ring) with enough invertible ideals. Torsion modules over bounded (hnp)-rings were studied by the author in [10; 11]. All the results proved in [10; 11] also hold for torsion R-modules having no completely faithful submodules. In Section 2, indecomposable injective torsion R-modules which are not completely faithful are studied, and they are shown to have finite periodicities (Theorem (2.8) and Corollary (2.9)). These results are used to determine the structure of quasi-injective and quasi-projective modules over bounded (hnp)-rings (Theorems (2.13), (2.14) and (2.15)).

A simple finite dimensional module ${{V}_{\lambda }}$ of a simple complex algebraic group $G$ is naturally endowed with a filtration induced by the PBW-filtration of $U\,(\text{Lie}\,G)$. The associated graded space $\text{V}_{\text{ }\!\!\lambda\!\!\text{ }}^{a}$ is a module for the group ${{G}^{a}}$, which can be roughly described as a semi-direct product of a Borel subgroup of $G$ and a large commutative unipotent group $\mathbb{G}_{a}^{M}$. In analogy to the flag variety ${{\mathcal{F}}_{\lambda }}\,=\,G.[{{v}_{\lambda }}]\,\,\subset \,\,\mathbb{P}({{V}_{\lambda }})$, we call the closure $\overline{{{G}^{a}}\,.\,[{{v}_{\text{ }\!\!\lambda\!\!\text{ }}}]}\,\,\subset \,\,\mathbb{P}\,(V_{\text{ }\!\!\lambda\!\!\text{ }}^{a})$ of the ${{G}^{a}}$-orbit through the highest weight line the degenerate flag variety $\mathcal{F}_{\text{ }\!\!\lambda\!\!\text{ }}^{a}$. In general this is a singular variety, but we conjecture that it has many nice properties similar to that of Schubert varieties. In this paper we consider the case of $G$ being the symplectic group. The symplectic case is important for the conjecture because it is the first known case where, even for fundamental weights $\omega$, the varieties $\mathcal{F}_{\text{ }\!\!\omega\!\!\text{ }}^{a}$ differ from ${{\mathcal{F}}_{\text{ }\!\!\omega\!\!\text{ }}}$. We give an explicit construction of the varieties $\text{Sp}\,\mathcal{F}_{\text{ }\!\!\lambda\!\!\text{ }}^{a}$ and construct desingularizations, similar to the Bott–Samelson resolutions in the classical case. We prove that $\text{Sp}\,\mathcal{F}_{\text{ }\!\!\lambda\!\!\text{ }}^{a}$ are normal locally complete intersections with terminal and rational singularities. We also show that these varieties are Frobenius split. Using the above mentioned results, we prove an analogue of the Borel–Weil theorem and obtain a $q$-character formula for the characters of irreducible $\text{S}{{\text{p}}_{2\pi }}$-modules via the Atiyah–Bott–Lefschetz fixed points formula.

Non-commutative measure theory embraces measure theory on cr-fields of subsets of a set, on projection lattices of von Neumann algebras or JBW-algebras and on hypergraphs alike [20], [27], [33], [37], [39], [40], [41]. Due to the unifying structure of an orthoalgebra concepts can easily be transferred from one branch to the other. Additional conceptual inpetus is obtained from the logico-probabilistic foundations of quantum mechanics (see [6], [19], [21]).

In the late seventies the author studied the Jordan-Hahn decomposition of measures on orthomodular posets and certain graphs. These investigations revealed an interesting geometrical aspect of this decomposition in that the Jordan-Hahn property of the convex set of probability charges on a finite orthomodular poset can be characterized in terms of the extreme points of the unit ball of the Banach space dual of the base normed space of Jordan charges.

In 1913 L. Aubry [1] proved the following theorem:

If a and m are relatively prime, m > 0, and if is not an integer, then it is always possible to find integers x and y not both zero such that

We say that an abelian variety over a $p$-adic field $K$ has anisotropic reduction $(\text{AR})$ if the special fiber of its Néron minimal model does not contain a nontrivial split torus. This includes all abelian varieties with potentially good reduction and, in particular, those with complex or quaternionic multiplication. We give a bound for the size of the $K$-rational torsion subgroup of a $g$-dimensional $\text{AR}$ variety depending only on $g$ and the numerical invariants of $K$ (the absolute ramification index and the cardinality of the residue field). Applying these bounds to abelian varieties over a number field with everywhere locally anisotropic reduction, we get bounds which, as a function of $g$, are close to optimal. In particular, we determine the possible cardinalities of the torsion subgroup of an $\text{AR}$ abelian surface over the rational numbers, up to a set of 11 values which are not known to occur. The largest such value is 72.

The problem of classifying the normal subgroups of the general linear group over a field was solved in the general case by Dieudonné (see 2 and 3). If we consider the problem over a ring, it is trivial to see that there will be more normal subgroups than in the field case. Klingenberg (4) has investigated the situation over a local ring and has shown that they are classified by certain congruence groups which are determined by the ideals in the ring.

Klingenberg's solution roughly goes as follows. To a given ideal , attach certain congruence groups and . Next, assign a certain ideal (called the order) to a given subgroup G. The main result states that if G is normal with order a, then ≧ G ≧ , that is, G satisfies the so-called ladder relation at ; conversely, if G satisfies the ladder relation at , then G is normal and has order .

Let K be a finite normal extension of an algebraic number field k; let k2 be the compositum of all quadratic extensions of k which are contained in K. Let ζk(s), ζK(s) and ζk2(s) denote the Dedekind ζ-functions of these fields.

1.1 This paper gives a lattice theoretic investigation of “finiteness“ and “continuity of the lattice operations” in a complemented modular lattice. Although we usually assume that the lattice is -complete for some infinite ,3 we do not require completeness and continuity, as von Neumann does in his classical memoir on continuous geometry (3); nor do we assume orthocomplementation as Kaplansky does in his remarkable paper (1).

1.2. Our exposition is elementary in the sense that it can be read without reference to the literature. Our brief preliminary § 2 should enable the reader to read this paper independently.

1.3. Von Neumann's theory of independence (3, Part I, Chapter II) leans heavily on the assumption that the lattice is continuous, or at least upper continuous.

Fenchel's conjugate correspondence for convex functions may be viewed as a generalization of the classical Legendre correspondence, as indicated briefly in (6). Here the relationship between the two correspondences will be described in detail. Essentially, the conjugate reduces to the Legendre transform if and only if the subdifferential of the convex function is a one-to-one mapping. The one-to-oneness is equivalent to differentiability and strict convexity, plus a condition that the function become infinitely steep near boundary points of its effective domain. These conditions are shown to be the very ones under which the Legendre correspondence is well-defined and symmetric among convex functions. Facts about Legendre transforms may thus be deduced using the elegant, geometrically motivated methods of Fenchel. This has definite advantages over the more restrictive classical treatment of the Legendre transformation in terms of implicit functions, determinants, and the like.

Let there be given a parabolic differential operator

where A is a second order linear elliptic (<0) differential operator in an open set Ω ⊂ Rn, having coefficients depending on x ∈ Ω and t ∈ [0, ∞]. Recently, Protter (1) investigated the asymptotic behaviour of functions u(x, t) that satisfy the differential inequality

(1.1)

Under suitable restrictions on the functions ci(t) and the coefficients of A, he proved that any solution of (1.1), subject to certain homogeneous boundary conditions, that vanishes sufficiently fast, as t → ∞, must be identically zero in Ω × [0, ∞). For example, conditions are given under which no solution of (1.1) can vanish faster than e-λt, ∀ λ > 0, unless identically zero.