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I regret to announce that there is a serious error in my paper in these Transactions, volume 84, pp. 430-443. The error was discovered by Louis Kokoris who found that on line 8 of page 434 the expression given as 4[g(bz)](az) should have been 4[g(az)](bz). As a consequence the computation of P(z, g, az, b) yields nothing, the proof of formula (30) is not valid, and the important Lemma 9 is not proved. Thus the paper does not give a proof of its major result stated as Theorem 1. Nevertheless, the theorems of the paper are all correct and we shall provide a revision of the proof here. This revised proof has been checked by Louis Kokoris to whom the author wishes to express his great thanks. We observe first that the equation

The reading of Ayres' interesting paper suggested to me the following remarks: 1. The order of a subset of a set S in a point p4 cannot surpass the order of S in p. Hence if S2 denotes the set of all points of S of order 2, then S2 has in each point of S the order 2, the order 1, or the order 0, where the terms "order 0" and "0-dimensional" are used synonymously. SI(M), S2(1), S" may denote the set of all points of S in which S2 has the order 0, 1, 2, respectively. The points of order 2 of S are also called the ordinary points of S, and the set S2 of all ordinarv points of S may be called the ordinary part of S. The set S" of all ordinary points of the ordinary part of S may be designated the ordinary kernel of S. WVe have

The purpose of this note is to present a new proof of a generalized form of Day's fixed point theorem. The proof we give is suggested by the work of T. Mitchell in his paper, Function algebras, means, and fixed points, [2]. The version of Day's theorem which we present here has not appeared explicitly in the literature before, and seems especially well suited for application to questions concerning fixed point properties of topological semigroups. 1. Preliminaries. We adopt the terminology and notation of [2] except where otherwise specified. New terminology will be introduced as needed. Let y be a convex compactum (compact convex set in a real locally convex linear topological space E), and let A( Y) denote the Banach space of all (real) continuous affine functions on Y under the supremum norm. Observe that A(Y) contains every function of the form h=f\Y + r where fe E* and r is real; thus A(Y) separates points of Y.

It has been brought to my attention by Mr. N. E. Steenrod that the lemma of page 708 in the paper referred to is in error. The final assertion of the proof is false. It is therefore necessary to point out that the paper is not "disturbed" by this fault. For if one requires that the Pn, «= 1,2, • • •, of the lemma be arcs then the (altered) lemma does hold, since it is true that the connected sum of a perfect continuous curve and an arc is "perfect." One verifies that this restricted lemma is sufficient for the uses of the paper.

This gives the desired result. It might be pointed out that Theorem 4 shows that Ilfn-gn I-->O In ?4 it is tacitly assumed that the measurable set E can be partitioned into measurable sets E,,. This is always the case in separable spaces. To proceed without this assumption it will not be necessary to assume that J is metric. The class So(E) is defined as the class of functions finitely valued on E. Such a function is one for which there is a decomposition of E into a finite number of disjoint measurable subsets on each of which it is constant. This basis necessitates only a slight rewording in a few places. In Lemma 1 the set E should be taken as a set in A. In Theorem 2 the words "functions uniformly continuous" should be replaced by "functions finitely valued." In the proof of Theorem 11 the sentence "Fix . . . continuous on e" should be worded "Fix e with 18(E-e)

We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollob\'as and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees. In this paper, we lay the foundations of the $L^p$ theory of graphons, characterize convergence, and develop corresponding random graph models, while we prove the equivalence of several alternative metrics in a companion paper., Comment: 44 pages

The main purpose of this paper is to explore the structure of local and regular Dirichlet forms associated with symmetric one-dimensional diffusions, which are also called symmetric linear diffusions. Let ( E , F ) (\mathcal {E},\mathcal {F}) be a regular and local Dirichlet form on L 2 ( I , m ) L^2(I,m) , where I I is an interval and m m is a fully supported Radon measure on I I . We shall first present a complete representation for ( E , F ) (\mathcal {E},\mathcal {F}) , which shows that ( E , F ) (\mathcal {E},\mathcal {F}) lives on at most countable disjoint “effective" intervals with an “adapted" scale function on each interval, and any point outside these intervals is a trap of the one-dimensional diffusion. Furthermore, we shall give a necessary and sufficient condition for C c ∞ ( I ) C_c^\infty (I) being a special standard core of ( E , F ) (\mathcal {E},\mathcal {F}) and shall identify the closure of C c ∞ ( I ) C_c^\infty (I) in ( E , F ) (\mathcal {E},\mathcal {F}) when C c ∞ ( I ) C_c^\infty (I) is contained but not necessarily dense in F \mathcal {F} relative to the E 1 1 / 2 \mathcal {E}_1^{1/2} -norm. This paper is partly motivated by a result of Hamza’s that was stated in a theorem of Fukushima, Oshima, and Takeda’s and that provides a different point of view to this theorem. To illustrate our results, many examples are provided.

The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin’s axiom for σ \sigma -centered posets. In his 1973 paper he showed under this assumption that both ω 1 {\omega }_{1} and the reals can be embedded. Analogous results were obtained later for the coarser notion of Tukey reducibility. We prove in this paper that Martin’s axiom for σ \sigma -centered posets implies that the Boolean algebra P ( ω ) / FIN \mathcal {P}(\omega ) / \operatorname {FIN} equipped with its natural partial order can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility. Consequently, the continuum hypothesis implies that every partial order of size at most continuum embeds into the P-points under both notions of reducibility.

In this paper we investigate nilpotenct and probabilistically nilpotent Hopf algebras. We define nilpotency via a descending chain of commutators and give a criterion for nilpotency via a family of central invertible elements. These elements can be obtained from a commutator matrix A A which depends only on the Grothendieck ring of H . H. When H H is almost cocommutative we introduce a probabilistic method. We prove that every semisimple quasitriangular Hopf algebra is probabilistically nilpotent. In a sense we thereby answer the title of our paper Are we counting or measuring anything? by Yes, we are.

The aim of this paper is to define the notion of the Cohen- Macaulay cone of a Noetherian local domain R R and to present its applications to the theory of Hilbert-Kunz functions. It has been shown by the second author that with a mild condition on R R , the Grothendieck group G 0 ( R ) ¯ \overline {G_0(R)} of finitely generated R R -modules modulo numerical equivalence is a finitely generated torsion-free abelian group. The Cohen-Macaulay cone of R R is the cone in G 0 ( R ) ¯ R \overline {G_0(R)}_{\mathbb R} spanned by cycles represented by maximal Cohen-Macaulay modules. We study basic properties on the Cohen-Macaulay cone in this paper. As an application, various examples of Hilbert-Kunz functions in the polynomial type will be produced. Precisely, for any given integers ϵ i = 0 , ± 1 \epsilon _i = 0, \pm 1 ( d / 2 > i > d d/2 > i > d ), we shall construct a d d -dimensional Cohen-Macaulay local ring R R (of characteristic p p ) and a maximal primary ideal I I of R R such that the function ℓ R ( R / I [ p n ] ) \ell _R(R/I^{[p^n]}) is a polynomial in p n p^n of degree d d whose coefficient of ( p n ) i (p^n)^i is the product of ϵ i \epsilon _i and a positive rational number for d / 2 > i > d d/2 > i > d . The existence of such ring is proved by using Segre products to construct a Cohen-Macaulay ring such that the Chow group of the ring is of certain simplicity and that test modules exist for it.

In this paper, we discuss the equidistribution phenomena for holomorphic endomorphisms over $\mathbb{P}^k$ in the case of bidegree $(p,p)$ with $1, Comment: Corrected an error in the statement of the main theorem and readability has been improved. This paper is based on the work in arXiv:math/0703702 and arXiv:0901.3000 by other authors; for precise estimates, we go over the proofs with modification

We compute divisors class groups of singular surfaces. Most notably we produce an exact sequence that relates the Cartier divisors and almost Cartier divisors of a surface to the those of its normalization. This generalizes Hartshorne's theor em for the cubic ruled surface in P 3 . We apply these results to limit the possible curves that can be s et-theoretic complete intersection in P 3 in characteristic zero. On a nonsingular variety, the study of divisors and linear systems is classical. In fact the entire theory of curves and surfaces is dependent on this study of codimension one subvarieties and the linear and algebraic families in which they move. This theory has been generalized in two directions: the Weil divisors on a normal variety, taking codimension one subvarieties as prime divisors; and the Cartier divisors on an arbitrary scheme, based on locally principal codimension one subschemes. Most of the literature both in algebraic geometry and commutative algebra up to now has been limited to these kinds of divisors. More recently there have been good reasons to consider divisors on non-normal varieties. Jaffe (9) introduced the notion of an almost Cartier divisor, which is locally principal off a subset of codimension two. A theory of generalized divisors was proposed on curves in (14), and extended to any dimension in (15). The latter paper gave a complete description of the generalized divisors on the ruled cubic surface in P 3 . In this paper we extend that analysis to an arbitrary integra l surface X, explaining the group APicX of linear equivalence classes of almost Cartier divisors on X in terms of the Picard group of the normalization S of X and certain local data at the singular points of X. We apply these results to give limitations on the possible curves that can b e set-theoretic compete intersections in P 3 in characteristic zero In section 2 we explain our basic set-up, comparing divisors on a variety X to its normalization S. In Section 3 we prove a local isomorphism that computes the group of almost Cartier divisors at a singular point of X in terms of the Cartier divisors along the curve of singulari ties and its inverse image in the normalization. In Section 4 we derive some global exact sequences for the groups PicX, APicX, and PicS, which generalize the results of (15, §6) to arbitrary surfaces These results are particularly transparent for surfaces wi th ordinary singularities, meaning a double curve with a finite number of pinch points and triple points.

Let $\Omega$ be either $\mathbb{R}^n$ or a strongly Lipschitz domain of $\mathbb{R}^n$, and $\omega\in A_{\infty}(\mathbb{R}^n)$ (the class of Muckenhoupt weights). Let $L$ be a second order divergence form elliptic operator on $L^2 (\Omega)$ with the Dirichlet or Neumann boundary condition, and assume that the heat semigroup generated by $L$ has the Gaussian property $(G_1)$ with the regularity of their kernels measured by $\mu\in(0,1]$. Let $\Phi$ be a continuous, strictly increasing, subadditive, positive and concave function on $(0,\infty)$ of critical lower type index $p_{\Phi}^-\in(0,1]$. In this paper, the authors introduce the "geometrical" weighted local Orlicz-Hardy spaces $h^{\Phi}_{\omega,\,r}(\Omega)$ and $h^{\Phi}_{\omega,\,z}(\Omega)$ via the weighted local Orlicz-Hardy spaces $h^{\Phi}_{\omega}(\mathbb{R}^n)$, and obtain their two equivalent characterizations in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by $L$ when $p_{\Phi}^-\in(n/(n+\mu),1]$. As applications, the authors prove that the operators $\nabla^2{\mathbb G}_D$ are bounded from $h^{\Phi}_{\omega,\,r}(\Omega)$ to the weighted Orlicz space $L^{\Phi}_{\omega}(\Omega)$, and from $h^{\Phi}_{\omega,\,r}(\Omega)$ to itself when $\Omega$ is a bounded semiconvex domain in $\mathbb{R}^n$ and $p_{\Phi}^-\in(\frac{n}{n+1},1]$, and the operators $\nabla^2{\mathbb G}_N$ are bounded from $h^{\Phi}_{\omega,\,z}(\Omega)$ to $L^{\Phi}_{\omega}(\Omega)$, and from $h^{\Phi}_{\omega,\,z}(\Omega)$ to $h^{\Phi}_{\omega,\,r}(\Omega)$ when $\Omega$ is a bounded convex domain in $\mathbb{R}^n$ and $p_{\Phi}^-\in(\frac{n}{n+1},1]$, where ${\mathbb G}_D$ and ${\mathbb G}_N$ denote, respectively, the Dirichlet Green operator and the Neumann Green operator., Comment: This paper has been withdrawn by the authors

In 1945 Duffin and Schaeffer proved that a power series that is bounded in a sector and has coefficients from a finite subset of C is already a rational function. Their proof is relatively indirect. It is one purpose of this paper to give a shorter direct proof of this beautiful and surprising theorem. This will allow us to give an easy proof of a recent result of two of the authors stating that a sequence of polynomials with coefficients from a finite subset of C cannot tend to zero uniformly on an arc of the unit circle. Another main result of this paper gives explicit estimates for the number and location of zeros of polynomials with bounded coefficients. Let n be so large that v satisfies δn≤ 1. We show that any polynomial in has at least zeros in any disk with center on the unit circle and radius αn.

We classify all homomorphisms between Weyl modules for SL3(k) when k is an algebraically closed field of characteristic at least three, and show that the Hom-spaces are all at most one-dimensional. As a corollary we obtain all homomorphisms between Specht modules for the symmetric group when the labelling partitions have at most three parts and the prime is at least three. We conclude by showing how a result of Fayers and Lyle on Hom-spaces for Specht modules is related to earlier work of Donkin for algebraic groups. Let G be a reductive algebraic group over an algebraically closed field of characteristic p > 0. An important class of modules for such a group are the Weyl modules �(λ), labelled by dominant weights; these can be constructed (relatively) explicitly, and their heads provide a full set of simple modules for G. (Equivalently one can study the duals of these modules, denoted ∇(λ) which have the advantage of being induced from one-dimensional modules for a Borel subgroup). In determining the structure of such modules, or indeed their cohomology, the calculation of Hom- spaces between them is a useful tool. Relatively little is known in general about such Hom-spaces. In type A, when λ andare related by a (suitable) single reflection, explicit non-zero homomorphisms from �(λ) to �(� ) were constructed (with some restrictions) by Carter and Lusztig (3), and (more generally) by Carter and Payne (4). The corresponding cases in other types were considered by Franklin (14). While it is clear that there should be a hierarchy of families of homomorphisms corresponding to different powers of p, the only case where the above results provide a complete classification is when G is SL2, where it is relatively easy to determine all Hom-spaces exactly (6). The only other general results in this area, by Andersen (1) and Koppinen (18), concern homo- morphisms between modules labelled by weights which are 'close together'. Typically such results show that certain Hom-spaces are non-zero, or in some cases one-dimensional. For weights which are far apart and not related by a single reflection almost nothing is known. In this paper we will determine all homomorphisms between Weyl modules for SL3 when p ≥ 3, and provide a recursive procedure for determining the composition factors arising in the image (or kernel) of such maps in most cases. From these results we will also classify all homomorphisms between Specht modules for the symmetric groups corresponding to three part partitions, when p ≥ 3. After a section of preliminaries, we review the SL3 data concerning p-filtrations that we will need from (21). This describes certain filtrations of induced modules which will allow us to proceed by induction, together with the set of p-good homomorphisms which will be fundamental in our later constructions. We also recall a theorem of Carter and Payne (4) on the existence of certain homomorphisms. These will be the two key sets of data which we need to determine all possible homomorphisms. With the notation developed up to that point in place, in Section 4 we can give the strategy to be followed in the remainder of the paper, and in particular the translation functor arguments

Existing black box and other algorithms for explicitly recognising groups of Lie type over G F ( q ) \mathrm {GF}(q) have asymptotic running times which are polynomial in q q , whereas the input size involves only log ⁡ q \log q . This has represented a serious obstruction to the efficient recognition of such groups. Recently, Brooksbank and Kantor devised new explicit recognition algorithms for classical groups; these run in time that is polynomial in the size of the input, given an oracle that recognises P S L ( 2 , q ) \mathrm {PSL}(2,q) explicitly. The present paper, in conjunction with an earlier paper by the first two authors, provides such an oracle. The earlier paper produced an algorithm for explicitly recognising S L ( 2 , q ) \mathrm {SL}(2,q) in its natural representation in polynomial time, given a discrete logarithm oracle for G F ( q ) \mathrm {GF}(q) . The algorithm presented here takes as input a generating set for a subgroup G G of G L ( d , F ) \mathrm {GL}(d,F) that is isomorphic modulo scalars to P S L ( 2 , q ) \mathrm {PSL}(2,q) , where F F is a finite field of the same characteristic as G F ( q ) \mathrm {GF}(q) ; it returns the natural representation of G G modulo scalars. Since a faithful projective representation of P S L ( 2 , q ) \mathrm {PSL}(2,q) in cross characteristic, or a faithful permutation representation of this group, is necessarily of size that is polynomial in q q rather than in log ⁡ q \log q , elementary algorithms will recognise P S L ( 2 , q ) \mathrm {PSL} (2,q) explicitly in polynomial time in these cases. Given a discrete logarithm oracle for G F ( q ) \mathrm {GF}(q) , our algorithm thus provides the required polynomial time oracle for recognising P S L ( 2 , q ) \mathrm {PSL}(2,q) explicitly in the remaining case, namely for representations in the natural characteristic. This leads to a partial solution of a question posed by Babai and Shalev: if G G is a matrix group in characteristic p p , determine in polynomial time whether or not O p ( G ) O_p(G) is trivial.

Fix a prime p p , and let A A be the polynomial part of the dual Steenrod algebra. The Frobenius map on A A induces the Steenrod operation P ~ 0 \widetilde {\mathscr {P}}^{0} on cohomology, and in this paper, we investigate this operation. We point out that if p = 2 p=2 , then for any element in the cohomology of A A , if one applies P ~ 0 \widetilde {\mathscr {P}}^{0} enough times, the resulting element is nilpotent. We conjecture that the same is true at odd primes, and that “enough times” should be “once.” The bulk of the paper is a study of some quotients of A A in which the Frobenius is an isomorphism of order n n . We show that these quotients are dual to group algebras, the resulting groups are torsion-free, and hence every element in Ext over these quotients is nilpotent. We also try to relate these results to the questions about P ~ 0 \widetilde {\mathscr {P}}^{0} . The dual complete Steenrod algebra makes an appearance.

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic s r a n k ( π ) = O ( π ) − O ( π ′ ) , \begin{equation*} \mathrm {srank}(\pi ) = {\mathcal O}(\pi ) - {\mathcal O}(\pi ’), \end{equation*} where O ( π ) {\mathcal O}(\pi ) denotes the number of odd parts of the partition π \pi and π ′ \pi ’ is the conjugate of π \pi . In a forthcoming paper, Andrews proved the following refinement of Ramanujan’s partition congruence mod 5 5 : p 0 ( 5 n + 4 ) a m p ; ≡ p 2 ( 5 n + 4 ) ≡ 0 ( mod 5 ) , p ( n ) a m p ; = p 0 ( n ) + p 2 ( n ) , \begin{align*} p_0(5n+4) &\equiv p_2(5n+4) \equiv 0 \pmod {5}, p(n) &= p_0(n) + p_2(n), \end{align*} where p i ( n ) p_i(n) ( i = 0 , 2 i=0,2 ) denotes the number of partitions of n n with s r a n k ≡ i ( mod 4 ) \mathrm {srank}\equiv i\pmod {4} and p ( n ) p(n) is the number of unrestricted partitions of n n . Andrews asked for a partition statistic that would divide the partitions enumerated by p i ( 5 n + 4 ) p_i(5n+4) ( i = 0 , 2 i=0,2 ) into five equinumerous classes. In this paper we discuss three such statistics: the ST-crank, the 2 2 -quotient-rank and the 5 5 -core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the 2 2 -quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan’s congruence mod 5 5 . This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo 5 5 . Finally, we discuss some new formulas for partitions that are 5 5 -cores and discuss an intriguing relation between 3 3 -cores and the Andrews-Garvan crank.

In this paper we study first-order definability in the lattice of equational theories of commutative semigroups. In a series of papers, J. Ježek, solving problems posed by A. Tarski and R. McKenzie, has proved, in particular, that each equational theory is first-order definable in the lattice of equational theories of a given type, up to automorphism, and that such lattices have no automorphisms besides the obvious syntactically defined ones (with exceptions for special unary types). He has proved also that the most important classes of theories of a given type are so definable. In a later paper, Ježek and McKenzie have “almost proved" the same facts for the lattice of equational theories of semigroups. There were good reasons to believe that the same can be proved for the lattice of equational theories of commutative semigroups. In this paper, however, we show that the case of commutative semigroups is different.

Let K[?] denote the group algebra of an infinite locally finite group?. In recent years, the lattice of ideals of K[?] has been extensively studied under the assumption that? is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra K[V] stable under the action of an appropriate automorphism group of V. Specifically, in this paper, we let B be a quasi-simple group of Lie type defined over an infinite locally finite field F, and we let V be a finite-dimensional vector space over a field E of the same characteristic p. If B acts nontrivially on V by way of the homomorphism Φ: G → GL(V), and if V has no proper G-stable subgroups, then we show that the augmentation ideal ωK[V] is the unique proper G-stable ideal of K[V] when char K ¬= p. The proof of this result requires, among other things, that we study characteristic p division rings D, certain multiplicative subgroups G of D*, and the action of G on the group algebra K[A], where A is the additive group D + . In particular, properties of the quasi-simple group G come into play only in the final section of this paper.

The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type(S+)L(S_{+})_{L}acting from a real, reflexive and separable Banach spaceXXintoX∗.X^{*}.This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given inlp, p>2,l^{p},~p>2,illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type(S+)).(S_{+})).Applications to nonlinear Dirichlet problems have appeared elsewhere.

The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp’s conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.

To date, integral bases for the centre of the Iwahori-Hecke algebra of a finite Coxeter group have relied on character theoretical results and the isomorphism between the Iwahori-Hecke algebra when semisimple and the group algebra of the finite Coxeter group. In this paper, we generalize the minimal basis approach of an earlier paper, to provide a way of describing and calculating elements of the minimal basis for the centre of an Iwahori-Hecke algebra which is entirely combinatorial in nature, and independent of both the above mentioned theories. This opens the door to further generalization of the minimal basis approach to other cases. In particular, we show that generalizing it to centralizers of parabolic subalgebras requires only certain properties in the Coxeter group. We show here that these properties hold for groups of type A A and B B , giving us the minimal basis theory for centralizers of any parabolic subalgebra in these types of Iwahori-Hecke algebra.

This paper deals with upper bounds on arithmetic discriminants of algebraic points on curves over number fields. It is shown, via a result of Zhang, that the arithmetic discriminants of algebraic points that are not pullbacks of rational points on the projective line are smaller than the arithmetic discriminants of families of linearly equivalent algebraic points. It is also shown that bounds on the arithmetic discriminant yield information about how the fields of definition k(P) and k(f(P)) differ when P is an algebraic point on a curve C and f : C --C' is a nonconstant morphism of curves. In particular, it is demonstrated that k(P) 74 k(f(P)), with at most finitely many exceptions, whenever the degrees of P and f are sufficiently small, relative to the difference between the genera g(C) and g(C'). The paper concludes with a detailed analysis of the arithmetic discriminants of quadratic points on bi-elliptic curves of genus 2. Let C be a curve defined over a number field k, and let X be a regular model for C over the ring of integers R of k. In [V 3], Vojta proves that, for any c > 0, all points P E C(k) of bounded degree satisfy the inequality (0.0.1) hK(P) < da(P) + EhA(P) + 0(1), where A is any ample divisor on C, hA and hK are Weil heights associated with A and the canonical divisor K, respectively, and da(P) is the arithmetic discriminant of P. The arithmetic discriminant depends on the choice of a regular model X; it is defined as (Hp.(Wx/B+HP)) [k(P) : k] where Hp is the arithmetic curve on X corresponding to P and Wx/B is the sheaf of relative differentials of X over B = Spec R. The difference between arithmetic discriminants derived from different regular models is always bounded. Hence, (0.0.1) is not affected by one's choice of a regular model. The arithmetic discriminant is related to d(P), the normalized field discriminant of the field of definition of P, which is defined as d(P) '_ log IN|Dk(p)/k [k(P): ] One may obtain da(P) by adding to d(P) terms corresponding to singularities of Hp at finite places and v-adic distances between the conjugates of P at infinite places v (see [V 2, Section 3]). The precise nature of the relationship between Received by the editors November 30, 1999 and, in revised form, February 25, 2000. 2000 Mathematics Subject Classification. Primary 11G30, 11J25. ()2001 American Mathematical Society

The distance from the origin in the word metric for generalizations F (p) of Thompson’s group F is quasi-isometric to the number of carets in the reduced rooted tree diagrams representing the elements of F (p). This interpretation of the metric is used to prove that every F (p) admits a quasiisometric embedding into every F (q), and also to study the behavior of the shift maps under these embeddings. The first example of a finitely presented, infinite, simple group was discovered by R.J. Thompson in 1965. This group G arose in the study of logic and it has many interesting descriptions. In 1974, Higman [10] used a description as the group of automorphisms of the Jonsson-Tarski algebra to generalize G to an infinite family of groups. G has another description as a group of homeomorphisms of the Cantor set. It has subgroups which may be described as homeomorphism groups of different objects, for instance, the circle and the unit interval. Among the people studying these generalizations are Brown [3] and Bieri and Strebel, in a set of unpublished notes [1]. In this paper we will concentrate on the groups F (p), which correspond to the groups Fp,∞ in [3]. These families of groups have also been extensively studied by Brin and Guzman [2]. The group F (2), commonly known as Thompson’s group F , can be regarded as the group of piecewise-linear, orientation-preserving homeomorphisms of the interval [0, 1] which have breakpoints only at dyadic points and whose slopes, where defined, are powers of two. In 1984 Brown and Geoghegan [4] found it to be the first example of a finitely presented, infinite-dimensional, torsion-free FP∞ group. This fact has been extended to all F (p) in [3], and studied by Stein [11], where generalizations to more general groups of homeomorphisms with general rational breakpoint sets are considered. Cleary [7] has studied these properties for groups of piecewise-linear homeomorphisms with irrational breakpoints and slopes. In [5] Burillo established the relationship between the word metric of Thompson’s group F and an estimate of the distance derived from the unique normal form of the elements. This algebraic estimate is quasi-isometric to the word metric and was used to prove that some subgroups are nondistorted in F . In this paper we find a geometric estimate of the word metric in terms of rooted tree diagrams (see [6]), show that this is quasi-isometric to the word metric, and use this interpretation to prove that some embeddings of F (p) into F (q), including some of those outlined in [2], are quasi-isometric. The interpretations of these embeddings in terms of Received by the editors September 25, 1998 and, in revised form, August 11, 1999. 2000 Mathematics Subject Classification. Primary 20F65; Secondary 20F05, 20F38, 20E99, 05C25. c ©2000 American Mathematical Society

This paper is devoted to giving the theoretical background for an algorithm for computing homomorphisms induced by maps in homology. The principal idea is to insert the graph of a given continuous map f into ag raph of a multi-valued representable map F. The multi-valued representable maps have well developed continuity properties and admit a nite coding that per- mits treating them by combinatorial methods. We provide the construction of the homomorphism F induced by F such that F = f. The presented construction does not require subsequent barycentric subdivisions and simpli- cial approximations of f. The main motivation for this paper comes from the project of computing the Conley Index for discrete dynamical systems.

We study in this paper polarized infinite exponent partition relations. We apply our results to constructing a model for the theory “ZF+DC+ω1 is the only regular, uncountable cardinal ≤ ωω1+1.” This gives a partial answer to a question of Woodin. In 1994, J. Steel proved what had long been suspected: that assuming AD, the regular cardinals of L(R) below Θ are all measurable, where Θ is the least ordinal onto which the real numbers R cannot be mapped [St3]. This remarkable theorem motivated the work of [A], which showed essentially that the cofinality of any regular cardinal below Θ can be changed to ω without perturbing cardinal structure. Specifically, the following theorem was proved. Theorem 0.1. Assume V′ AD, and let V = L(R)V . Then for any subset A of the regular, uncountable cardinals below Θ, there is a partial ordering P ∈ V and a symmetric inner model N of ZF such that (1) V ⊆ N ⊆ VP , (2) N and V contain the same cardinals, (3) Θ = Θ, (4) for all κ, cof(κ) = ω if cof(κ) ∈ A and cof(κ) = cof(κ) otherwise, and (5) the measurable cardinals not in A remain measurable in N. What was left unanswered is whether similar facts can be proven if the cofinality of κ < Θ is to be made uncountable. Consideration of this question was the genesis of this paper. The principal tool, infinite–exponent partition sequences, proved as interesting as its application. We begin, in the first three sections, with background material on infinite– exponent partition properties of cardinals, their abundance in models of AD, and their use in “Magidor–like” forcing to change cofinalities. The second three sections deal with infinite–exponent partition properties of cardinal sequences. With appropriate notation, the basic theory of partition sequences and the associated Magidor–like forcing exactly parallels the cardinal theory. In “Finite Support,” we apply the theory to show the extent to which possibly uncountable cofinal sequences can be added if we sacrifice DC. In “Countable Support,” we preserve DC. Received by the editors September 30, 1997. 1991 Mathematics Subject Classification. Primary 03E15, 03E35, 03E60. The first author’s research was partially supported by PSC–CUNY grant 665337. c ©1999 American Mathematical Society

The main result of the paper is a description of the maximal ideals in the modular group algebras of the finitary symmetric and alternating groups (provided the characteristic p of the ground field is greater than 2). For the symmetric group there are exactly p 1 such ideals and for the alternating group there are (p 1)/2 of them. The description is obtained in terms of the annihilators of certain systems of the 'completely splittable' irreducible modular representations of the finite symmetric and alternating groups. The main tools used in the proofs are the modular branching rules (obtained earlier by the second author) and the 'Mullineux conjecture' proved recently by FordKleshchev and Bessenrodt-Olsson. The results obtained are relevant to the theory of PI-algebras. They are used in a later paper by the authors and A. E. Zalesskii on almost simple group algebras and asymptotic properties of modular representations of symmetric groups.

This paper completes the classification of Artin-Schelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree one. We show that there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism, just as in the case where the algebras are assumed to be generated in degree one. In particular, we study the elliptic algebras A ( + ) A(+) , A ( − ) A(-) , and A ( a ) A({\mathbf {a}}) , where a ∈ P 2 {\mathbf {a}}\in \mathbb {P}^{2} , which were first defined in an earlier paper. We omit a set S ⊂ P 2 S\subset \mathbb {P}^2 consisting of 11 specified points where the algebras A ( a ) A({\mathbf {a}}) become too degenerate to be regular. Theorem. Let A A represent A ( + ) A(+) , A ( − ) A(-) or A ( a ) A({\mathbf {a}}) , where a ∈ P 2 ∖ S {\mathbf {a}} \in \mathbb {P}^2\setminus S . Then A A is an Artin-Schelter regular algebra of global dimension three. Moreover, A A is a Noetherian domain with the same Hilbert series as the (appropriately graded) commutative polynomial ring in three variables. This, combined with our earlier results, completes the classification.

In his notebooks, Ramanujan gave the values of over 100 class invariants which he had calculated. Many had been previously calculated by Heinrich Weber, but approximately half of them had not been heretofore determined. G. N. Watson wrote several papers devoted to the calculation of class invariants, but his methods were not entirely rigorous. Up until the past few years, eighteen of Ramanujan’s class invariants remained to be verified. Five were verified by the authors in a recent paper. For the remaining class invariants, in each case, the associated imaginary quadratic field has class number 8, and moreover there are two classes per genus. The authors devised three methods to calculate these thirteen class invariants. The first depends upon Kronecker’s limit formula, the second employs modular equations, and the third uses class field theory to make Watson’s “empirical method”rigorous.

We study the dimension of the space of homomorphisms from a given representation X of a quiver to a general representation of dimension vector 3. We prove a theorem about this number, and derive two corollaries concerning its asymptotic behaviour as 3 increases. These results are related to work of A. Schofield on homological epimorphisms from the path algebra to a simple artinian ring. In this paper we prove a number of results about hom(X, 3), the dimension of the space of homomorphisms from a fixed representation X of a quiver to a general representation of dimension vector 3. Our basic result relates this number to the dimension of a subset of a Grassmannian of submodules of X. This result is in the spirit of A. Schofield's paper on general representations of quivers [S2], and generalizes one of his results. We derive two corollaries concerning the asymptotic behaviour of hom(X, r/3), of interest in themselves, and also because of their connection with another theorem of Schofield, that the homological epimorphisms from a path algebra to a simple artinian ring are in 1-1 correspondence with indivisible Schur roots. (A homological epimorphism is a ring homomorphism R -* S with S OR S -S and Tori(S, S) = 0 for i > 0. This correspondence was announced by Schofield in a lecture in March 1995 in Krippen, Germany, but actually dates from 1991.) In fact our second corollary is already known, proved by Schofield and used in the proof of the correspondence, but his proof of the corollary involves difficult results about semi-invariants of quivers, whereas the proof here is quite elementary. Let K be an algebraically closed field, Q a finite quiver with vertex set I, and let (-,-) be the Ringel form on Z'. If 3 E N' we write RePKQW() = 17 Hom(K8('), K:(j)) a:i-j for the configuration space of representations of Q of dimension vector 3, and if y E RePKQ(/) we write Ky for the corresponding left KQ-module. If X is a finitely generated left KQ-module and y E RePKQ (/3) then Hom(X, Ky) is a finite dimensional vector space. Moreover the function y F dim Hom(X, Ky) is an upper semicontinuous function on RePKQ(/3), and it follows that the minimum value of this function, denoted hom(X,/3), is also its general value. The rank of a homomorphism X -* Ky is the dimension vector of its image. For any Ol E N' the set of homomorphisms of rank at most Ol is a closed subset of Hom(X, Ky). It follows that there is a unique maximal rank Yx,y of homomorphisms from X to Ky, Received by the editors July 21, 1995. 1991 Mathematics Subject Classification. Primary 16G20; Secondary 14M15. ?1996 American Mathematical Society

Let H be a fixed digraph. We consider the H-colouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the absence in G of certain tree-like obstructions. Specifically, we say that H has tree duality if, for all digraphs G, G is not homomorphic to H if and only if there is an oriented tree which is homomorphic to G but not to H. We prove that if H has tree duality then the H-colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the X-property studied by Gutjahr, Welzl, and Woeginger. We then focus on the case when H itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads H for which the H-colouring problem is NP -complete. We contrast these with several families of oriented triads H which have tree duality, or bounded treewidth duality, and hence polynomial H-colouring problems. If P 6= NP , then no oriented triad H with an NP -complete H-colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad H. We prove that none of the oriented triads H with NP -complete Hcolouring problems given in the companion paper has tree duality.

This paper presents a survey and some (hopefully) new facts on germs of maps tangent to the identity (in R , C , \mathbb {R},\mathbb {C}, or R 2 {\mathbb {R}^2} ), (maps f f defined near 0 0 , such that f ( 0 ) = 0 f(0) = 0 , and f ′ ( 0 ) f’(0) is the identity). Proofs are mostly original. The paper is mostly oriented towards precise examples and the questions of descriptions of members in the conjugacy class, flows, k k th root. It happened that entire functions provide clear and easy examples. However they should be considered just as a tool, not as the main topic. For example in Proposition 2 2 the function z ↦ z + z 2 z \mapsto z + {z^2} should be better thought of as the map ( x , y ) → ( x + x 2 − y 2 , y + 2 x y ) (x,y) \to (x + {x^2} - {y^2},y + 2xy) .

In this paper we prove two conjectures on partitions with certain conditions. A motivation for this is given by a problem in the modular representation theory of the covering groups Sn, of the finite symmetric groups S,n in characteristic 5 . One of the conjectures (Conjecture B below) has been open since 1974, when it was stated by the first author in his memoir [A3]. Recently the second and third author (jointly with A. 0. Morris) arrived at essentially the same conjecture from a completely different direction. Their paper [BMO] was concerned with decomposition matrices of Sn, in characteristic 3. A basic difficulty for obtaining similar results in characteristic 5 (or larger) was the lack of a class of partitions which would be "natural" character labels for the modular characters of these groups. In this connection two conjectures were stated (Conjectures A and B * below), whose solutions would be helpful in the characteristic 5 case. One of them, Conjecture B *, is equivalent to the old Conjecture B mentioned above. Conjecture A is concerned with a possible inductive definition of the set of partitions which should serve as the required labels. In ? 1 we give a brief description of the groups Sn and their representations, leading up to Conjectures A and B * as they were formulated in [BMO]. That section also presents the background for Conjecture B as stated in [A3] and the equivalence of Conjectures B and B * is explained. Sections 2 and 3 are devoted to the proof of Conjecture B, and ?4 to the proof of Conjecture A. 1. THE CONJECTURES AND THEIR BACKGROUND For facts concerning the general representation theory of finite groups needed in the following, the reader is referred to [F, NT]. In 191 1 Schur [S1] proved that the finite symmetric groups S, have covering groups S, of order 2IS, I = 2 * n ! This means that there is an exact sequence

A unit index-class number formula is proved for subfields of cyclotomic function fields in analogy with similar results for subfields of cyclotomic number fields. Let m be a positive integer and let Cm = exp(27ri/m). Let Km = Q(4m) denote the mth cyclotomic field, and K+ its maximal real subfield. The ring of integers in Km (resp. K+) is Z[Cm] (resp. Z[4m + C-1]). In [2] Sinnott showed that the index of circular units in the full group of units of Q(Cm) equals the class number of Z[4m + Cm'] multiplied by a power of 2 which depends exclusively on the number of prime factors of m. Sinnott [3] subsequently generalized this result to an arbitrary abelian field. There is a parallel setup for function fields of characteristic p . Let Fq be the finite field with q elements, let RT = Fq[T] be the ring of polynomials over Fq (with T transcendental over Fq), and let Fq(T) be the field of rational functions over Fq . To each polynomial M E RT one can associate an extension KM, called the Mth cyclotomicfunctionfield, which enjoys properties analogous to those of the cyclotomic number field Km. In particular, Galovich and Rosen [1] proved the analogue of Sinnott's theorem in this setting. The purpose of this paper is to extend this unit index-class number formula to an arbitrary subfield of KM. Let k be any subfield of the Mth cyclotomic function field (M monic), G the Galois group of k over IFq(T), k+ the maximal subfield of k in which oo splits, Ok(Ok+) the integral closure of Fq[T] in k(k'), and Ok* the unit group of Ok. In ?3, we define a subgroup C of Ok*, which we call the circular units of k. Our main result is that C has finite index in Ok* and that this index may be written in the form [0k :C]=h(Ok+).c,k+ where h (Ok+) is the class number of Ok+, and ck+ is a rational number whose definition does not involve h(Ok+). We now briefly describe the contents of the rest of this paper. In ? 1, we present the relevant definitions and facts in the function field setting. We also state the analytic class number formula. In ?2, we review ordinary distributions on lFq(T)/RT, discuss an index notation, and obtain a preliminary result on the structure of a certain module. The circular units are introduced in ?3 and Received by the editors February 1, 1990 and, in revised form, November 27, 1991. 1991 Mathematics Subject Classification. Primary 1 lR58. (D 1994 American Mathematical Society 0002-9947/94 $1.00 + $.25 per page

A skew lattice is a noncommutative associative analogue of a lattice and as such may be viewed both as an algebraic object and as a geometric ob- ject. Whereas recent papers on skew lattices primarily treated algebraic aspects of skew lattices, this article investigates their intrinsic geometry. This geometry is obtained by considering how the coset geometries of the maximal primitive subalgebras combine to form a global geometry on the skew lattice. While this geometry is derived from the algebraic operations, it can be given a description that is independent of these operations, but which in turn induces them. Var- ious aspects of this geometry are investigated including: its general properties; algebraic and numerical consequences of these properties; connectedness; the geometry of skew lattices in rings; connections between primitive skew lattices and completely simple semigroups; and finally, this geometry is used to help classify symmetric skew lattices on two generators. Recall that a band is a semigroup satisfying the idempotent law: xx = x. Upon examining bands which are multiplicative subsemigroups of rings, one uncovers classes of bands which also possess an idempotent countermultipli- cation. This leads one to define a skew lattice to be an algebra with a pair of associative idempotent binary operations, the join and the meet, which are con- nected by a set of absorption laws (see 1.1). While skew lattices of idempotents in rings remain important sources of motivation, the results of (10-12) make it clear that skew lattices can sustain mathematical life on their own. Perhaps the most natural way to think about a skew lattice is as a noncommutative ana- logue of a lattice. As such, a skew lattice is not only an algebraic object, but also a geometric object. Thus far most of the research given in (10-12) has emphasized the algebraic side of skew lattices. The purpose of this paper is to investigate their geometric aspects and in particular the role of the natural partial order in determining their algebraic structure, much as a lattice is deter- mined by its natural partial ordering. It is not our goal, however, to reinvent lattice theory. Hence the geometry of a skew lattice will be studied relative to the fixed structure of its underlying lattice. Saying this entails an implicit refer- ence to the fundamental Clifford-McLean Theorem which in effect provides a first sketch of a skew lattice: a congruence is defined on each skew lattice (called natural equivalence) which induces its maximal lattice image and whose equiv

In his previous paper the author characterized free arrangements bythe vanishing of cohomology modules of a certain sheaf of graded modules overa polynomial ring. Thus the homogeneous components of these cohomologymodules can be viewed as obstructions to the freeness of an arrangement. Inthis paper the first two obstructions are studied in detail. In particular thecomponent of degree zero of the first nontrivial cohomology module has a closerelation to formal arrangements and to the operation of truncation. This enablesus to prove that in dimension greater than two every free arrangement is formal and not a proper truncation of an essential arrangement. IntroductionIn this paper we develop further the ideas from [Y] where certain sheaf co-homology was applied to the theory of arrangements.Let V be an /-dimensional (/ > 2) vector space over an arbitrary field Kand sf an arrangement of « (n > 2) hyperplanes in V. For each H £ sfwe fix a linear functional aH £ V* such that kera// = 77. Let S = ®d>0Sdbe the graded symmetric algebra of V* and Der S = Der^S, S) the gradedS-module of derivations of S. The following graded submodule of Deri' wasintroduced by H. Terao [TI, T3] and studied in many papers on arrangements:D = D(s/) = {6 £ Der(S)\0(aH) £ SaH , H £ sf}. If the module D is freethen the arrangement sf is called free.Let Lo be the lattice of all intersections of hyperplanes from sf except

Let A A be a regular local ring of positive characteristic. This paper is concerned with the local cohomology modules of A A itself, but with respect to an arbitrary ideal of A A . The results include that all the Bass numbers of all such local cohomology modules are finite, that each such local cohomology module has finite set of associated prime ideals, and that, whenever such a local cohomology module is Artinian, then it must be injective. (This last result had been proved earlier by Hartshorne and Speiser under the additional assumptions that A A is complete and contains its residue field which is perfect.) The paper ends with some low-dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 0 are true.

The growth series of certain finitely generated groups which are wreath products are investigated. These growth series are intimately related to the traveling salesman problem on certain graphs. A large class of these growth series is shown to consist of irrational algebraic functions. n=0 Since only one generating set will be associated with each group below, the generating set associated with fr(x) will be obvious. Let H be a group with a finite generating set Sh ■ These will be fixed for the rest of the paper. The Cayley graph of the pair (H, Sh) is, as usual, the directed graph whose vertices are the elements of H and there is an edge from a vertex hx Xoa vertex h2 if and only if h2 = hxh for some h in SH US^1. In particular, the edge from hx to h2 has an opposite edge from h2 to hx . Let C be the graph gotten from the Cayley graph of (H, Sh) simply by identifying opposite edges. In other words, C might be called the undirected Cayley graph of (H,SH). hex K also be a group with a finite generating set Sk ■ These will also be fixed for the rest of the paper. It is possible to form what might be called a restricted direct product group P, which consists of all functions p from the vertices of C to K such that there are only finitely many vertices v in C with p(v) t? 1. This is a subgroup of the direct product group whose elements consist of all functions from the vertex set of C to K. The group P admits H as a group of automorphisms by means of the action of H on C. The resulting semidirect product P x H is the restricted wreath product K\H.

In view of the well-known conjecture concerning the classification of lines in the affine plane in characteristic p > 0, it is desirable to understand how the characteristic pairs of an irreducible algebroid plane curve are affected by the relative Frobenius. This paper determines the relation between the char- acteristic sequences (x, y) and (x, f), where x and y are formal power series in one variable with coefficients in a field of characteristic p > 0 . Throughout this paper, k is an algebraically closed field of characteristic p > 0, t is an indeterminate over k and k((t)) is the field of fractions of the ring k((t)) of formal power series in t. An affine plane curve with one place at infinity may be thought of as an epimorphism e: k(X, Y) -> A, where A is a one dimensional domain contained in all but one (say A g R ) valuation rings of qt(A)/k. Note that if x = e(X) and y = e(Y) then e':k(X, Y) -, A1 = k(x, yp) defined by e'(X) = x and e'(Y) = yp is also an affine plane curve with one place at infinity. We say that e is obtained from e by performing the "relative Frobenius" operation. This paper is part of an attempt to answer: Question 1. How is the singularity at infinity affected by the relative Frobenius? As a motivation for this question we recall the well-known conjecture which asserts that every affine plane curve biregular to the affine line can be obtained from a line of degree one by performing a finite sequence of operations, each operation being either an automoprhism of the plane or the relative Frobenius, the latter being allowed only if A = A1. (See (1, 2, 4, 5).) Let the notation be as above. Then the completion of R may be identified with k((i)) and as a result x, y £ k((t)) have negative orders (unless x £ k or y £ k, which is a trivial case). Define x^, yoo£ tk((t\) by ' (1/x, y/x), if ordx < ordy

The main result in this paper is the following linearization theorem. For each open set U in a complex Banach space E, there is a complex Banach space Goo (U) and a bounded holomorphic mapping gu: U -G? (U) with the following universal property: For each complex Banach space F and each bounded holomorphic mapping f: U -* F , there is a unique continuous linear operator Tf: Goo (U) -F such that Tf ? gu = f . The correspondence f Tf is an isometric isomorphism between the space Ho (U; F) of all bounded holomorphic mappings from U into F , and the space L(G?? (U); F) of all continuous linear operators from G? (U) into F. These properties characterize G??'(U) uniquely up to an isometric isomorphism. The rest of the paper is devoted to the study of some aspects of the interplay between the spaces H?O(U;F) and L(G?(U);F). This paper consists of five sections. In ? 1 we establish our notation and terminology. In ?2 we prove the aforementioned linearization theorem. In ?3 we translate certain properties of a mapping f E H' (U; F) into properties of the corresponding operator Tf E L(G'(U); F). We show, for instance, that f has a relatively compact range if and only if Tf is a compact operator. In ?4 we give a seminorm characterization of the unique locally convex topology T y on H?(U; F) such that the correspondence f --* Tf is a topological isomorphism between the spaces (H (U; F), TY) and (L(G ( U); F), T,), where TC denotes the compact-open topology. Finally, in ? 5 we use the preceding results to establish necessary and sufficient conditions for the spaces G? (U) and H? (U) to have the approximation property. These are holomorphic analogues of classical results of A. Grothendieck [8], and complement results of R. Aron and M. Schottenloher [2]. We show, in particular, that if U is a balanced, bounded, open set in a complex Banach space E, then G? (U) has the approximation property if and only if E has the approximation property. We also show that if U is an arbitrary open set in a complex Banach space E, then H? (U) has the approximation property if and only if, for each complex Banach space F, each mapping in H?(U; F) with a relatively compact range can be uniformly approximated on U by mappings in H?(U; F) with finite-dimensional range. Since it is still unknown whether Received by the editors April 10, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46G20, 46E15; Secondary 46E10. (D 1991 American Mathematical Society 0002-9947/91 $1.00+ $.25 per page

We show that under certain assumptions, the measurable cohomology class of the linear holonomy cocycle of a foliation yields information about the characteristic classes of the foliation. Combined with the results of a previous paper, this yields vanishing theorems for characteristic classes of certain actions of lattices in higher rank semisimple Lie groups. Let F be a discrete group acting by diffeomorphisms on a smooth compact manifold M. Associated to this action are certain characteristic classes in H (F, R), which are constructed as characteristic classes of a natural foliation associated to the group action. The action of F on M induces an action of F on the principal frame bundle P(M) of M and the characteristic classes of the action can be interpreted as obstructions to the existence of invariant geometric structures on M, i.e., principal subbundles of P(M) invariant by the F-action. If F is a lattice in a higher rank semisimple Lie group, then F has strong rigidity properties (see, e.g., [M, Z1]). In an earlier paper [S], we showed, using techniques from ergodic theory, that for a certain class of F-actions there is always an invariant measurable reductive geometric structure, i.e., a measurable principal subbundle with reductive structure group, which is invariant by the F-action. Moreover, the noncompact semisimple part of this reductive group is locally isomorphic to a semisimple factor of the ambient Lie group of F [Zi]. Zimmer [Z3] recently proved this result for a large class of actions (which does not a priori include the class of actions considered in [S]). A natural question is whether these results remain true in the smooth category. The purpose of this paper is to show that the characteristic classes, which obstruct a smooth geometric structure, vanish in the presence of a measurable geometric structure. Explicitly, we have Main Theorem. Let (M, Y) be a codimension n, C 2-foliated manifold and suppose that the linear holonomy cocycle is measurably equivalent to a locally tempered cocycle taking values in a subgroup H c GL(n, R) which is stable under transpose. Then the Weil homomorphism X: H'(g[(n), 0(n)) -Hc(M, Yi) Received by the editors February 24, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 57R32; Secondary 57S20. Research partially supported by an Alfred P. Sloan Dissertation Fellowship. ? 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page

Let f f and g g be algebraically independent entire functions. We first give an estimate of the Nevanlinna counting function for the common zeros of f n − 1 f^n-1 and g n − 1 g^n-1 for sufficiently large n n . We then apply this estimate to study divisible sequences in the sense that f n − 1 f^n-1 is divisible by g n − 1 g^n-1 for infinitely many n n . For the first part of establishing our gcd estimate, we need to formulate a truncated second main theorem for effective divisors by modifying a theorem from a paper by Hussein and Ru and explicitly computing the constants involved for a blowup of P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 along a point with its canonical divisor and the pull-back of vertical and horizontal divisors of P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 .

The asymptotic behavior of exponential sums ∑ k = 1 N exp ⁡ ( 2 π i n k α ) \sum _{k=1}^N \exp ( 2\pi i n_k \alpha ) for Hadamard lacunary ( n k ) (n_k) is well known, but for general ( n k ) (n_k) very few precise results exist, due to number theoretic difficulties. It is therefore natural to consider random ( n k ) (n_k) , and in this paper we prove the law of the iterated logarithm for ∑ k = 1 N exp ⁡ ( 2 π i n k α ) \sum _{k=1}^N \exp (2\pi i n_k \alpha ) if the gaps n k + 1 − n k n_{k+1}-n_k are independent, identically distributed random variables. As a comparison, we give a lower bound for the discrepancy of { n k α } \{n_k \alpha \} under the same random model, exhibiting a completely different behavior.

In this paper we give a structure theorem for the module of vector valued modular forms in the case of a three dimensional ball with the action of the Picard modular group Γ 3 [ − 3 ] \Gamma _3 [\sqrt {-3}] . The corresponding modular variety of dimension 3 3 is a copy of the Segre cubic.

We use the progenerator constructed in our previous paper to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees., Comment: arXiv admin note: text overlap with arXiv:1611.07373

We say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module which decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains. In this paper, we characterize the UDI property for reduced Noetherian rings.

Let S S be an orientable surface with negative Euler characteristic. For k ∈ N k \in \mathbb {N} , let C k ( S ) \mathcal {C}_{k}(S) denote the k-curve graph, whose vertices are isotopy classes of essential simple closed curves on S S and whose edges correspond to pairs of curves that can be realized to intersect at most k k times. The theme of this paper is that the geometry of Teichmüller space and of the mapping class group captures local combinatorial properties of C k ( S ) \mathcal {C}_{k}(S) , for large k k . Using techniques for measuring distance in Teichmüller space, we obtain upper bounds on the following three quantities for large k k : the clique number of C k ( S ) \mathcal {C}_{k}(S) (exponential in k k , which improves on previous bounds of Juvan, Malnič, and Mobar and Przytycki); the maximum size of the intersection, whenever it is finite, of a pair of links in C k \mathcal {C}_{k} (quasi-polynomial in k k ); and the diameter in C 0 ( S ) \mathcal {C}_{0}(S) of a large clique of C k ( S ) \mathcal {C}_{k}(S) (uniformly bounded). As an application, we obtain quasi-polynomial upper bounds, depending only on the topology of S S , on the number of short simple closed geodesics on any unit-square tiled surface homeomorphic to S S .

In this paper, we study the Heegner points on more general modular curves other than $X_0(N)$, which generalizes Gross' work "Heegner points on $X_0(N)$". The explicit Gross-Zagier formula and the Euler system property are stated in this case. Using such kind of Heegner points, we construct certain families of quadratic twists of $X_0(36)$, with the ranks of Mordell-Weil groups being zero and one respectively, and show that the $2$-part of their BSD conjectures hold., comments are welcome