We investigate Riemann--Liouville processes $R_H$, with $H > 0$, and fractional Brownian motions $B_H$, for $0 < H < 1$, and study their small deviation properties in the spaces $L_q([0, 1], \mu)$. Of special interest here are thin (fractal) measures $\mu$, that is, those that are singular with respect to the Lebesgue measure. We describe the behavior of small deviation probabilities by numerical quantities of $\mu$, called mixed entropy numbers, characterizing size and regularity of the underlying measure. For the particularly interesting case of self-similar measures, the asymptotic behavior of the mixed entropy is evaluated explicitly. We also provide two-sided estimates for this quantity in the case of random measures generated by subordinators.

While the upper asymptotic bound for the small deviation probability is proved by purely probabilistic methods, the lower bound is verified by analytic tools concerning entropy and Kolmogorov numbers of Riemann--Liouville operators.

Let $\varphi:A\to B$ be a homomorphism of finite-dimensional algebras over an algebraically closed field and $\varphi^{(c)}:{\rm mod}_B^c\to{\rm mod}_A^c$ the induced morphism of the associated module schemes for any integer $c\geq 1$. We prove that if the induced functor ${\rm mod\,} B\to{\rm mod\,} A$ is hom-controlled then the restriction of $\varphi^{(c)}$ to any connected component of ${\rm mod}_B^c$ is a composition of a smooth morphism followed by an immersion. Some new results on the types of singularities in the orbit closures of module schemes are also proved.

2000 Mathematical Subject Classification: 14B05, 14L30, 16G10.

Consider the group scheme

$$ R \mapsto G(R) := R \rtimes R^* = \left( \begin{array}{cc} 1 & R \\ 0 & R^* \end{array} \right) = \left( \begin{array}{cc} 1 & *\\ 0 & * \end{array} \right) \cap \mathrm{GL}_2(R) $$

where $R$ is an arbitrary commutative ring with $1 \neq 0$ and a unit $x \in R^*$ acts on $R$ by multiplication.

We will study the finiteness properties of subgroups of $G(\Oka_S)$ where $\Oka_S$ is an $S$-arithmetic subring of a global function field. The subgroups we are interested in are of the form $\Oka_S \rtimes Q$ where $Q$ is a subgroup of $\Oka_S^*$. The finiteness properties of these metabelian groups can be expressed in terms of the $\Sigma$-invariant due to R. Bieri and R. Strebel.

{\sc Theorem A.} {\it Let $S$ be a finite set of places of a global function field (regarded as normalized discrete valuations) and $\Oka_S$ the corresponding $S$-arithmetic ring. Let $Q$ be a subgroup of $\Oka_S^*$. Then $Q$ is finitely generated and for all integers $n \geq 1$ the following are equivalent:

\begin{enumerate} \item[\rm (1)] $\Oka_S \rtimes Q$ is of type FP$_n$;

\item[\rm (2)] $\Oka_S$ is $n$-tame as a $\ZZZ Q$-module;

\item[\rm (3)] each $p \in S$ restricts to a non-trivial homomorphism $p|_{Q} : Q \rightarrow \RRR$ and the set $\{ [p|_Q] \mid p \in S \} \subseteq \SSS(Q)$ is $n$-tame. \end{enumerate}

If these conditions hold for at least one $n \geq 1$ then the identity

$$ \Sigma_{\Oka_S}^c(Q,\ZZZ) = \{ [p|_Q] \mid p \in S \} $$

holds.} {\sc Theorem B.} {\it Let $r$ denote the rank of $Q$. Then the following hold: \begin{enumerate} \item[\rm (1)] the group $\Oka_S \rtimes Q$ is not of type FP$_{r+1}$;

\item[\rm (2)] if $Q$ has maximum rank $r=|S|-1$ then the group $\Oka_S \rtimes Q$ is of type FP$_r$.

\end{enumerate} In particular, $G(\Oka_S) = \Oka_S \rtimes \Oka_S^*$ is of type FP$_{|S|-1}$ but not of type FP$_{|S|}$.}

1991 Mathematics Subject Classification: 20E08, 20F16, 20G30, 52A20.

We classify the finite-dimensional quotient Hopf algebras of the deformed algebra of functions of the general linear group (over an algebraically closed field of zero characteristic). This gives an interesting class of Hopf algebras if the deformation parameter is a root of unity (of odd order). We investigate the properties of these Hopf algebras and construct a new counterexample to Kaplansky's tenth conjecture. E-mail: eric.mueller@akdb.de 1991 Mathematics Subject Classification: 81R50, 16W30, 17B37.

In this paper, we apply Padé approximation methods to derive completely explicit measures of irrationality for certain classes of algebraic numbers. Our approach is similar to that taken previously by G.V. Chudnovsky but has some fundamental advantages with regards to determining implicit constants. Our general results may be applied to produce specific bounds of the flavour of

$$ \left| \sqrt[3]{2} - \frac{p}{q} \right| > \frac{1}{4}~ q^{-2.45} \hskip2ex \mbox{and} \hskip2ex \left| \sqrt[7]{5} - \frac{p}{q} \right| > \frac{1}{4}~ q^{-4.43} $$

which we show to hold for any nonzero integers $p$ and $q$. Further examples are tabulated and applications to Diophantine equations are briefly discussed as are other topics of related interest.

1991 Mathematics Subject Classification: primary 11J68, 11J82; secondary 11D41.

Methods of Lie group analysis of differential equations are extended to weak solutions of (linear and non-linear) partial differential equations, where the term `weak solution' comprises the following settings: distributional solutions; solutions in generalized function algebras; solutions in the sense of association (corresponding to a number of weak or integral solution concepts in classical analysis). Factorization properties and infinitesimal criteria that allow the treatment of all three settings simultaneously are developed, thereby unifying and extending previous work in this area.

2000 Mathematical Subject Classification: 46F30, 22E70, 35Dxx, 35A30.

We employ results from KK-theory, along with quasidiagonality techniques, to obtain wide-ranging classification results for nuclear C*-algebras. Using a new realization of the Cuntz picture of the Kasparov groups we show that two morphisms inducing equal KK-elements are approximately stably unitarily equivalent. Using K-theory with coefficients to associate a partial KK-element to an approximate morphism, our result is generalized to cover such maps. Conversely, we study the problem of lifting a (positive) partial KK-element to an approximate morphism. These results are employed to obtain classification results for certain classes of quasidiagonal C*-algebras introduced by H. Lin, and to reprove the classification of purely infinite simple nuclear C*-algebras of Kirchberg and Phillips. It is our hope that this work can be the starting point of a unified approach to the classification of nuclear C*-algebras.

2000 Mathematical Subject Classification: primary 46L35; secondary 19K14, 19K35, 46L80.

This paper is a comprehensive study of the nest representations for the free semigroupoid algebra ${\mathfrak{L}}_G$ of a countable directed graph $G$ as well as its norm-closed counterpart, the tensor algebra ${\mathcal{T}}^{+}(G)$.

We prove that the finite-dimensional nest representations separate the points in ${\mathfrak{L}}_G$, and a fortiori, in ${\mathcal{T}}^{+}(G)$. The irreducible finite-dimensional representations separate the points in ${\mathfrak{L}}_G$ if and only if $G$ is transitive in components (which is equivalent to being semisimple). Also the upper triangular nest representations separate points if and only if for every vertex $x \in {\mathcal{T}}(G)$ supporting a cycle, $x$ also supports at least one loop edge.

We also study faithful nest representations. We prove that ${\mathfrak{L}}_G$ (or ${\mathcal{T}}^{+}(G)$) admits a faithful irreducible representation if and only if $G$ is strongly transitive as a directed graph. More generally, we obtain a condition on $G$ which is equivalent to the existence of a faithful nest representation. We also give a condition that determines the existence of a faithful nest representation for a maximal type ${\mathbb{N}}$ nest.

Let $S(x,y)$ be the set $S(x,y)=\{ 1 \leq n \leq x : P(n)\leq y\}$, where $P(n)$ denotes the largest prime factor of $n$. We study $E_f(x,y;\theta)=\sum_{n\inS(x,y)}f(n)e^{2\pi in\theta}$, where $f$ is a multiplicative function. When $f=1$ and when $f=\mu$, we widen the domain of uniform approximation using the method of Fouvry and Tenenbaum and making explicit the contribution of the Siegel zero.

Soit $S(x,y)$ l'ensemble $S(x,y)=\{ 1 \leq n \leq x : P(n)\leq y\}$, o\`u$P(n)$ d\'esigne le plus grand facteur premier de $n$. Nous étudions $E_f(x,y;\theta)=\sum_{n\in S(x,y)}f(n)e^{2\pi in\theta}$, lorsque $f$ est une fonction multiplicative. Quand $f=1$ et quand $f=\mu$, nous élargissons le domaine d'approximation uniforme enutilisant la méthode d\'eveloppée par Fouvry et Tenenbaum et en explicitant la contribution du zéro de Siegel.

1991 Mathematics Subject Classification: 11N25, 11N99.

Let $Y \rightrightarrows X$ be a finite flat groupoid scheme with $X$ a quasi-projective variety and let $S$ be its coarse moduli scheme. We associate to the groupoid scheme a coherent sheaf of algebras $\mathcal{O}_{X / Y}$ on $S$ which we call the non-commutative coordinate ring of the groupoid scheme. We show that when $X$ is a smooth curve and the groupoid action is generically free, the non-commutative coordinate rings which can occur are, up to Morita equivalence, the hereditary orders on smooth curves. This gives a bijective correspondence between smooth one-dimensional Deligne–Mumford stacks of finite type and Morita equivalence classes of hereditary orders on smooth curves.

The Bieri-Neumann-Strebel invariant of a finitely generated group $G$ determines, among other things, whether or not a given normal subgroup $N$, with $G/N$ abelian, is finitely generated. We examine the BNS-invariants of “Pride groups”, a large class of groups containing the Artin groups; in particular we establish a criterion which implies that a character $\chi$ of a Pride group $G{\cal G}$ is in the BNS-invariant $\Sigma^1(G{\cal G})$.

This restricts in the case of an Artin group $A{\cal G}$ to a simple condition which implies a character is in $\Sigma^1(A{\cal G})$. As an application this can be used to compute the unit ball in the Thurston semi-norm for the complement of iterated connected sums of $(2,n)$-torus links.

1991 Mathematics Subject Classification: 20F36, 20E07.

A Mackey functor $M$ is a structure analogous to the representation ring functor $H \mapsto R(H)$ encoding good formal behaviour under induction and restriction. More explicitly, $M$ associates an abelian group $M(H)$ to each closed subgroup $H$ of a fixed compact Lie group $G$, and to each inclusion $K \subseteq H$ it associates a restriction map ${\rm res}^H_K:M(H) \rightarrow M(K)$ and an induction map${\rm ind}^H_K:M(K) \rightarrow M(H)$. This paper gives an analysis of the category of Mackey functors $M$ whose values are rational vector spaces: such a Mackey functor may be specified by giving a suitably continuous family consisting of a ${\Bbb Q} \pi_0(W_G(H))$-module $V(H)$ for each closed subgroup $H$ with restriction maps $V(\hat{K}) \rightarrow V(K)$ whenever $K$ is normal in $\hat{K}$ and $\hat{K}/K$ is a torus (a ‘continuous Weyl-toral module’). We show that the category of rational Mackey functors is equivalent to the category of rational continuous Weyl-toral modules. In Part II this will be used to give an algebraic analysis of the category of rational Mackey functors, showing in particular that it has homological dimension equal to the rank of the group.

1991 Mathematics Subject Classification: 19A22, 20C99, 22E15, 55N91, 55P42, 55P91.

A precise description of a subset $\mathcal{A}$ of $\mathbb{Z} / n \mathbb{Z}$ satisfying

$$ | \mathcal{A} + \mathcal{A} | \leq 2.04 | \mathcal{A} | $$

is given. Basically, there exists a subgroup $\mathcal{H}$ of $\mathbb{Z} / n \mathbb{Z}$ such that $\mathcal{A}$ is included in an arithmetic progression of $\ell$ cosets modulo $\mathcal{H}$ and

$$(\ell - 1) | \mathcal{H} | \leq | \mathcal{A} + \mathcal{A} | - | \mathcal{A} |.$$

2000 Mathematical Subject Classification: 11B50, 11B83, 20E34.

The Novikov-Landweber algebra and the Steenrod algebra are set up in terms of the primitive differential operators $D_k=\sum_i x_i^{k+1}\frac{\partial}{\partial x_i}$ acting in the usual way on the integral polynomial ring ${\bf Z}[x_1,\ldots ,x_n,\ldots]$. A commutative wedge product $\vee$ for differential operators is introduced and it is shown that the iterated wedge product $D_k^{\vee r}$ is divisible by $r!$ as an integral operator. The divided differential operator algebra $\cal D$ is generated over the integers by the divided operators $\frac{D_k^{\vee r}}{r!}$ under the wedge product. $\cal D$ is additively isomorphic to the abelian group of symmetric functions in the variables $x_i$. Furthermore $\cal D$ is closed under composition of operators and admits a natural coproduct which makes it a Hopf algebra in two ways, with respect to the composition and wedge products. Under composition $\cal D$ is isomorphic to the Landweber-Novikov algebra. A Hopf sub-algebra is generated under composition by the integral Steenrod squares $SQ^r = \frac{D_1^{\vee r}}{r!}$ and reduces mod 2 to the Steenrod algebra. An explicit product formula for two wedge expressions is developed and used to derive Milnor's product formula for his basis elements in the Steenrod algebra. The hit problem in the Steenrod algebra is reformulated in terms of partial differential operators.

1991 Mathematics Subject Classification 55S10.

In this paper we define a large class of regularity conditions on a stratified set, which generalise the $(t)$ condition introduced by Thom in 1964. According to the first author's 1977 generalisation, a pair of adjacent strata $(X,Y)$ is said to be $(t^k )$-regular at $y_o$ in $Y$ if for every $C^k$-submanifold $S$ transverse to $Y$ at $y_o \in Y \cap S$, there is a neighbourhood of $y_o$ in which $S$ is transverse to $X$.

The very general microlocal conditions of this paper include $(t^k)$ for transversals of arbitrary h\‘olderian continuity and differentiability classes. Special cases of our conditions are equivalent to Verdier's $(w)$ condition, Whitney's $(a)$ condition, and their relative versions $(w_f)$ and $(a_f)$, respectively.

We analyse how the new conditions transform under pushforward and pullback by maps defining parametrised families of transversals to a stratum, and we deduce many consequences. In particular, whenever the pullback verifies $(w)$, the Verdier isotopy theorem (of which we prove a strengthened version in \S 4) implies stratified topological triviality of the family of transversals. This enables us to prove a conjecture made by the first author in 1981 about bounding the number of topological types of germs of transverse intersections with a subanalytic set, and also to give new and unified proofs, with stronger statements, of most of the known results in topological sufficiency theory for $V$-equivalence and right-equivalence.

The alternating group of degree 6 is located at the junction of three series of simple non-commutative groups: simple sporadic groups, alternating groups and simple groups of Lie type. It plays a very special role in the theory of finite groups. We shall study its new roles both in a finite geometry of a certain pentagon in the Leech lattice and also in the complex algebraic geometry of K3 surfaces.

We study the operator $(\mathcal{L} u)(t) := u'(t) - A(t) u(t)$ on $L^p (\mathbb{R}; X)$ for sectorial operators $A(t)$, with $t \in \mathbb{R}$, on a Banach space $X$ that are asymptotically hyperbolic, satisfy the Acquistapace–Terreni conditions, and have the property of maximal $L^p$-regularity. We establish optimal regularity on the time interval $\mathbb{R}$ showing that $\mathcal{L}$ is closed on its minimal domain. We further give conditions for ensuring that $\mathcal{L}$ is a semi-Fredholm operator. The Fredholm property is shown to persist under $A(t)$-bounded perturbations, provided they are compact or have small $A(t)$-bounds. We apply our results to parabolic systems and to generalized Ornstein–Uhlenbeck operators.

The controlled wild algebra is introduced, a covering criterion for a finite-dimensional algebra to be controlled wild is given, and this criterion is applied to the algebras with radical square zero, algebras with zero relations, local algebras and finite p-group algebras.

2000 Mathematical Subject Classification: 16G20, 16G60.

This paper considers various extensions of results of Johnson and Williams and of Fong on the range inclusion of normal derivations. Let $C_p$, with $p \in [1,\infty)$, be the Schatten ideals of compact operators on a Hilbert space $H$ with norms $|\ |_p$, $C_\infty$ be the ideal of all compact operators, and $C_b$ the algebra $B(H)$ of all bounded operators. Any $S \in B(H)$ defines a bounded derivation $\delta_S$ on all $C_p$: $\delta_S(X) = SX - XS$, for $X \in C_p$. Johnson and Williams proved that, for normal $S$, the inclusion of ranges $\delta_T(C_b) \subseteq \delta_S (C_b)$ implies $T = g(S)$, where $g$ is a Lipschitz, differentiable function on $\sigma(S)$. Fong showed that the condition $T = g(S)$ is equivalent to the range inclusion $\delta_T(C_1) \subseteq \delta_S (C_b)$. This paper studies the range inclusion \begin{equation} \delta_T(C_p) \subseteq \delta_S (C_p) \end{equation} for normal $S$ and $p \in [1,\infty] \cup b$, and the classes of functions for which $T = g(S)$. Set $$ p_- = \min\bigg(p, \frac{p}{p - 1}\bigg),\quad p_+ = \max\bigg(p, \frac{p}{p - 1}\bigg), \quad \text{for } p \in (1, \infty), $$ and $$ p_- = 1,\quad p_+ = b, \quad \text{for } p \in \{1,\infty,b\}. $$ This paper shows that condition (1) implies: \begin{enumerate} \item[(i)] the range inclusions $\delta_T(C_r) \subseteq \delta_S (C_r)$, for $r \in [p_-,p_+]$; \item[(ii)] that there exists $D > 0$ such that $|\delta_T(X)|_p \leq D|\delta_S (X)|_p$, for $X \in B(H)$ ($|X|_p = \infty$ if $X \notin C_p$); \item[(iii)]the range inclusions $\delta_{g(A)}(C_p) \subseteq \delta_A (C_p)$ for any normal operator $A$ with $\sigma(A) \subseteq \sigma(S)$. \end{enumerate} It establishes that (1) implies that the function $g$ (in $T = g(S)$) is $C_p$-Lipschitzian on $\sigma(S)$, that is, there is $D > 0$ such that $|g(A) - g(B)|_p \leq D|A - B|_p$ for all normal $A$ and $B$ with spectra in $\sigma(S)$. Conversely, it is proved that, for any selfadjoint $S$ and $C_p$-Lipschitz function $g$ on $\sigma(S)$, $\delta_{g(S)}(C_p) \subseteq \delta_S(C_p)$. The paper also extends the above results of Johnson and Williams to bounded derivations of C$^*$-algebras. 2000 Mathematics Subject Classification: 46L57, 47B47, 58C07.