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A classical result of K. L. Chung and W. Feller deals with the partial sums S k S_k arising in a fair coin-tossing game. If N n N_n is the number of “positive” terms among S 1 S_1 , S 2 S_2 , …, S n S_n then the quantity P ( N 2 n = 2 r ) P(N_{2n} = 2r) takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for P ( N 2 n + 1 = r ) P(N_{2n+1} = r) , r = 0 r = 0 , 1 1 , 2 2 , …, 2 n + 1 2n+1 . We get to this ansatz by adaptating the Feynman–Kac methodology.

In a recent paper, Chao-Ping Chen and Feng Qi (2005) established sharp upper and lower bounds for the sequence P n := 1.3 … ( 2 n − 1 ) 2.4 … 2 n P_{n}:=\frac {1.3\ldots (2n-1)}{2.4\ldots 2n} . We show that their result follows easily from a theorem of G. N Watson published in 1959. We also show that the main result of Chen and Qi’s paper is a special case of a more general inequality which admits a very short proof.

Concerns have been voiced about the correctness of certain technical points in DiPerna’s paper (Comm. Math. Phys. 91 (1983), 1–30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods. In [Di], DiPerna found a global entropy solution of the isentropic Euler equations for the following exponents in the equation of state for the pressure: γ = 1 + 2/(2m+ 1), m ≥ 2 integer. (1) He divided his arguments into the following two steps. 1. Compactness framework Assume that a sequence of approximate solutions (ρ (x, t),m (x, t)), 0 ≤ t ≤ T , satisfies: (i). There exists a constant C(T ) > 0, independent of > 0, such that 0 ≤ ρ (x, t) ≤ C, |m (x, t)/ρ (x, t)| ≤ C; (ii). For all weak entropy pairs (η, q) of the isentropic Euler equations, the measure sequence η(ρ ,m )t + q(ρ ,m )x is contained in a compact subset of H −1 loc (R× [0, T ]). If γ satisfies (1), then the sequence (ρ (x, t),m (x, t)) is compact in Lloc(R× [0, T ]). The reason for the restriction on the number γ is that, in such a case, any weak entropy function is a polynomial function of the Riemann invariants (w, z). This is the key step in DiPerna’s arguments and is also his main contribution to the compensated compactness method in this aspect. Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 35K55, 35L65; Secondary 76N15, 35L60, 65M06.

We observe that a strictly singular operator is not necessarily condensing, so that the invariant subspace problem for strictly singular operators remains open.

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.

1. In a recent paper [2] Reich has made the observation that an example of Koebe given in 1918 [1] does not fulfill its asserted purpose. This example was to show that vanishing measure of the complement did not assure that a slit domain was minimal. Reich proceeded to fill the gap by carrying out the following somewhat more general construction. Let A be a compact perfect nowhere dense set on the x-axis in the z-plane (z=x+iy). Then there exists a compact set S in the z-plane with the properties: (i) A is the projection of S on the x-axis, (ii) S is composed of segments symmetric with respect to the x-axis and points on the x-axis, at least one segment being present, (iii) any point in S, not on the x-axis and not at the end of a segment in S, is the limit, both from the left and right, of points of S. Once this construction is performed the desired examples are easily given [2, ?4]. The object of the present paper is to give an alternative construction which is very explicit and direct.

Among other results, it is proved that if a sequence { μ n } \{ {\mu _n}\} of regular measures on a Hausdorff space, with values in a normed group, is convergent to zero for all σ \sigma -compact sets or all open sets, then there exists a maximal open set U such that μ ˙ n ( U ) → 0 , { μ ˙ n } {\dot \mu _n}(U) \to 0,\{ {\dot \mu _n}\} being the associated submeasures.

We give a simplified proof to the following theorem due to D. Quillen: if A is a commutative noetherian ring of global dimension ⩽ 1 \leqslant 1 , then finitely generated projective modules over A [ T 1 , … , T n ] A[{T_1}, \ldots ,{T_n}] are extended from A. We prove also that if A is a commutative noetherian ring of global dimension d, then finitely generated projective modules of rank > d > d over A [ T 1 , … , T n ] A[{T_1}, \ldots ,{T_n}] are extended from A.

J. L. Palacios claimed [2] that the author’s paper [1] contained errors. This note refutes those claims by showing that Palacios misunderstood [1] and adopted assumptions different from those of [1].

There is a mistake in each of [3] and [4]. The purpose of this note is to correct the error in [3] and to salvage what can be saved in [4]. In each case the notation will be that of the paper under discussion. Professor Homer Bechtell has kindly informed me of an error in the proof of the Theorem of [3] which occurs in the case MG1c G. The theorem is true however by altering the proof at this point. Assume that Mc: MG1=M1lc:G. M1 is an invariant subgroup of G, and since G1c: Soc(G), G1 is a direct sum of minimal invariant subgroups of G. By Clifford's theorem (p. 70 of [1]), each minimal invariant subgroup of G is either M1-central or M1-hypereccentric, and hence

The first part of the theorem of the above paper states that if ois a closed convex set in real X1 ... Xn-space which is invariant under permutations of coordinates, and if C(o-) denotes the set of real symmetric nXn matrices whose eigenvalues X1, X. form the coordinates of points in a, then C(o) is convex. I am obliged to Professor R. T. Rockafellar for pointing out that this statement is essentially contained in a theorem of Chandler Davis.2 The statement also follows from an earlier result of V. B. Lidskii,3 which was not known to me at the time of publication.

In this short paper we prove that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and 83 83 powers of 2 2 .

Wahl recently proved that the homology of the non-orientable mapping class group stabilizes as the genus increases. In this short paper we analyse the situation where the underlying non-orientable surfaces have marked points.

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.

This paper studies vanishing of Ext modules over Cohen–Macaulay local rings. The main result of this paper implies that the Auslander–Reiten conjecture holds for maximal Cohen–Macaulay modules of rank one over Cohen–Macaulay normal local rings. It also recovers a theorem of Avramov–Buchweitz–Şega and Hanes–Huneke, which shows that the Tachikawa conjecture holds for Cohen–Macaulay generically Gorenstein local rings.

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.

In this paper we consider Artin local K-algebras with maximal length in the class of Artin algebras with given embedding dimension and socle type. They have been widely studied by several authors, among others by Iarrobino, Froberg and Laksov. If the local K-algebra is Gorenstein of socle degree 3, then the authors proved that it is canonically graded, i.e. analytically isomorphic to its associated graded ring, see (6). This unexpected result has been extended to compressed level K-algebras of socle degree 3 in (4). In this paper we end the investigation proving that extremal Artin Gorenstein local K-algebras of socle degree s � 4 are canonically graded, but the result does not extend to extremal Artin Gorenstein local rings of socle degree 5 or to compressed level local rings of socle degree 4 and type > 1. As a consequence we present results on Artin compressed local K-algebras having a specified socle type.

We examine a certain type of abelian C*-subalgebra that allows one to give a unified treatment of two uniqueness theorems: for graph C*algebras and for certain reduced crossed products. This note is meant to complement the paper [NR], which provides one of the two main examples of our results, by offering a conceptual treatment of the Uniqueness Theorem for the C*-algebras associated with graphs satisfying condition (L) and its generalization found in [Sz]. As pointed out in many places in graph C*-algebra literature (see for instance [Re1]), for graphs satisfying condition (L), a natural abelian C*-subalgebra (which is referred to in [NR] as the “diagonal”) turns out to give substantial information about the ambient (graph) C*-algebra. A similar treatment was proposed by Kumijian in [Ku], where he introduced the notion of C*-diagonals. In this paper we explain how, by considerably weakening several hypotheses in Kumjian’s definitions (in particular by getting rid of normalizers entirely), one can still obtain several key results, the most significant one being an “abstract” uniqueness property (see Theorem 3.1 below). Another illustration of this general approach is given in the context of reduced crossed products of abelian C*-algebras by essentially free actions of discrete groups, where we recover another uniqueness result due to Archbold and Spielberg [AS]. Our treatment focuses on the unique state extension property, as discussed in [KS] and [An], by weakening the global requirement made in the so-called Extension Property discussed in [An]. 1. Notation and preliminaries Notation. For any C*-algebra A, we denote by S(A) its set of states and by P (A) the set of pure states. Given a C*-subalgebra B ⊂ A, using the Hahn-Banach Theorem, it follows that any φ ∈ S(B) has at least one extension to some ψ ∈ S(A), so the set Sφ(A) = {ψ ∈ S(A) : ψ ∣∣ B = φ} is non-empty, convex and weak*-compact. Remark that if φ ∈ P (B), then every extreme point in Sφ(A) is in fact an extreme point in S(A); thus the intersection Sφ(A) ∩ P (A) is non-empty. Received by the editors June 26, 2011 and, in revised form, March 8, 2012. 2010 Mathematics Subject Classification. Primary 46L10; Secondary 46L30.

In this paper, we prove that certain contravariant endofunctors of singularity categories characterize Gorenstein rings. Let Λ be a noetherian ring. Denote by Dsg(Λ) the singularity category of Λ, that is, the Verdier quotient of the bounded derived category D(Λ) of finitely generated (right) Λ-modules by the full subcategory consisting of bounded complexes of finitely generated projective Λ-modules. We are interested in the following question. Question 1. What contravariant endofunctor of Dsg(Λ) characterizes the Iwanaga– Gorenstein property of Λ? In this paper we shall consider this question in the case where Λ is commutative and Cohen–Macaulay. Let R be a commutative Cohen–Macaulay local ring of Krull dimension d. Denote by CM(R) the category of (maximal) Cohen–Macaulay R-modules and by CM(R) its stable category : the objects of CM(R) are the Cohen–Macaulay R-modules, and the hom-set HomCM(R)(M,N) is defined as HomR(M,N), the quotient module of HomR(M,N) by the submodule consisting homomorphisms factoring through finitely generated projective (or equivalently, free) R-modules. The natural full embedding functor CM(R) → D(R) induces an additive covariant functor η : CM(R) → Dsg(R). Furthermore, the assignment M 7→ ΩTrM , where Ω and Tr stand for the syzygy and transpose functors respectively (see [1, Chapter 2, §1] for details of the functors Ω and Tr), makes an additive contravariant functor

In a paper by Z. Lu and J. Rowlett, it is shown that the eigenvalues of the weighted Laplacian can be approximated by eigenvalues of a naturally associated family of narrow graphs. In this paper, we generalize this result to the p p -Laplacian. Our approach features overcoming the nonlinearity of the p p -Laplacian when p ≠ 2 p\neq 2 , which is different from the Laplacian case.

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

For a given endomorphism on a unitary k-algebra, A, with k in the center of A, there are definitions of twisted cyclic and Hochschild homology. This paper will show that the method used to define them can be used to define twisted de Rham homology. The main result is that twisted de Rham homology can be thought of as the kernel of the Connes map from twisted cyclic homology to twisted Hochschild homology. For a noncommutative algebra A, Connes [1] and Karoubi [4] define the module of n-forms by taking the iterated tensor product of the bimodule of 1-forms. Karoubi in [4, Chapter 2] defines noncommutative de Rham homology. For the case where A is an associative unitary algebra over a commutative ring, k, containing Q, he shows that the reduced noncommutative de Rham homology is isomorphic to the kernel of the Connes map B from HCn(A), reduced cyclic homology, to HHn+1(A), reduced Hochschild homology. The bimodule of 1-forms used comes from the derivation d : A → A⊗kA, with d(a) = 1⊗a−a⊗1. In this paper we will generalize that result to the case of twisted cyclic homology based on a given k-algebra endomorphism, h : A → A. Here the bimodule of 1-forms will come from the twisted derivation, d : A → A ⊗k A, with d(a) = 1 ⊗ a − h(a)⊗ 1. If h = id, then we have d = d. A twisted derivation is any k-linear map, d, from A to an A-bimodule such that d(ab) = d(a) · b + h(a) · d(b). The definitions for twisted Hochschild homology and twisted cyclic homology are given in [3]. As pointed out there, in the twisted case C∗(A) itself is a paracyclic object, and we need to take an appropiate quotient of C∗(A) to obtain a cyclic object. It is quite interesting that even though the intrinsic definitions of the h-twisted theories differ considerably from the classical nontwisted theories, still there is an appropriate extension of de Rahm homology to the twisted case and an extension of the maps between all the twisted theories that give us a generalization of [4, Theorem 2.15]. Theorem. Suppose A is a unitary k-algebra with Q ⊂ k. If h is a k-algebra endomorphism then the reduced twisted de Rham homology and the reduced twisted cyclic homology are related by the exact sequence 0 → HDR n(A) → HC n (A) → HH n+1(A). HH ∗ (A) is the reduced twisted Hochschild homology and HC h,λ ∗ (A) is the reduced twisted cyclic homology based on the Connes complex {C n (A), b}. Details of the Connes complex together with the definition of HDR n(A) will be given Received by the editors March 11, 2009 and, in revised form, March 15, 2011. 2010 Mathematics Subject Classification. Primary 16E40; Secondary 16T20. c ©2011 American Mathematical Society Reverts to public domain 28 years from publication

In this paper we present a probabilistic construction of countable abelian p p -groups with prescribed Ulm-sequence. This result provides a different proof for the existence theorem of abelian p p -groups with any given countable Ulm-sequence due to Ulm, which is sometimes called Zippin’s theorem. The basic idea, applying probabilistic arguments, comes from a result by Erdős and Rényi. They gave an amazing probabilistic construction of countable graphs which, with probability 1 1 , produces the universal homogeneous graph, therefore also called the random graph. P. J. Cameron says about this in his book Oligomorphic Permutation Groups [Cambridge University Press, 1990]: In 1963, Erdős and Rényi proved the following paradoxical result. … It is my contention that mathematics is unique among academic pursuits in that such an apparently outrageous claim can be made completely convincing by a short argument. The algebraic tool in the present paper needs methods developed in the 1970s, the theory of valuated abelian p p -groups. Valuated abelian p p -groups are natural generalizations of abelian p p -groups with the height valuation, investigated in detail by F. Richman and E. Walker, and others. We have to establish extensions of finite valuated abelian p p -groups dominated by a given Ulm-sequence. Probabilistic results of a similar nature have been established by A. Blass and G. Braun, and by M. Droste and D. Kuske.

This paper complements a recent result of Dilworth, Ferenczi, Kutzarova and Odell regarding the ergodicity of strongly asymptotic ℓ p \ell _p spaces. We state this result in a more general form, involving domination relations, and we show that every asymptotically Hilbertian space which is not isomorphic to ℓ 2 \ell _2 is ergodic. In particular, every weak Hilbert space which is not isomorphic to ℓ 2 \ell _2 must be ergodic. Throughout the paper we construct explicitly the maps which establish the fact that the relation E 0 E_0 is Borel reducible to isomorphism between subspaces of the Banach spaces involved.

This paper presents a curvature-free version of the Log ( 2 k − 1 ) \text {Log}(2k-1) Theorem of Anderson, Canary, Culler, and Shalen [J. Differential Geometry 44 (1996), pp. 738–782]. It generalizes a result by Hou [J. Differential Geometry 57 (2001), no. 1, pp. 173–193] and its proof is rather straightforward once we know the work by Lim [Trans. Amer. Math. Soc. 360 (2008), no. 10, pp. 5089–5100] on volume entropy for graphs. As a byproduct we obtain a curvature-free version of the Collar Lemma in all dimensions.

In this paper we give simple, sufficient conditions for the existence of the stochastic integral for vector-valued processes X with values in a Banach space E; namely, X is of class (LD), and the stochastic measure I X is bounded and strongly additive in L p E (in particular, if I X is bounded in L p E and c 0 ⊈ E) and has bounded semivariation. The result is then applied to martingales and processes with integrable variation or semivariation. For martingales the condition of being of class (LD) is superfluous. For a square-integrable martingale with values in a Hilbert space, all the conditions are superfluous. For processes with p-integrable semivariation or p-integrable variation, the conditions of I X to be bounded and have bounded semivariation are superfluous. For processes with 1-integrable variation, all conditions are superfluous. In a forthcoming paper, we shall extend these results to local summability. The extension needs additional nontrivial work.

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.

In this paper we study the nonexpansivity of the so-called relatively nonexpansive mappings. A relatively nonexpansive mapping with respect to a pair of subsets ( A , B ) (A,B) of a Banach space X X is a mapping defined from A ∪ B A\cup B into X X such that ‖ T x − T y ‖ ≤ ‖ x − y ‖ \|Tx-Ty\|\le \|x-y\| for x ∈ A x\in A and y ∈ B y\in B . These mappings were recently considered in a paper by Eldred et al. (Proximinal normal structure and relatively nonexpansive mappings, Studia Math. 171 (3) (2005), 283-293) to obtain a generalization of Kirk’s Fixed Point Theorem. In this work we show that, for certain proximinal pairs ( A , B ) (A,B) , there exists a natural semimetric for which any relatively nonexpansive mapping with respect to ( A , B ) (A,B) is nonexpansive. This fact will be used to improve one of the two main results from the aforementioned paper by Eldred et al. At that time we will also obtain several consequences regarding the strong continuity properties of relatively nonexpansive mappings and the relation between the two main results from the same work.

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.

We address the problem of interrelations between the properties of an action of a discrete group Γ \Gamma on a compact Hausdorff space X X and the algebraic and analytical properties of the module of all continuous functions C ( X ) C(X) over the algebra of invariant continuous functions C Γ ( X ) C_\Gamma (X) . The present paper is a continuation of our joint paper with M. Frank and V. Manuilov. Here we prove some statements inverse to the ones obtained in that paper: we deduce properties of actions from properties of modules. In particular, it is proved that if for a uniformly continuous action the module C ( X ) C(X) is finitely generated projective over C Γ ( X ) C_\Gamma (X) , then the cardinality of orbits of the action is finite and fixed. Sufficient conditions for existence of natural conditional expectations C ( X ) → C Γ ( X ) C(X)\to C_\Gamma (X) are obtained.

It is a well-known result of M. Brodmann that if a is an ideal of a commutative Noetherian ring A, then the set of associated primes Ass(A/α n ) of the n-th power of a is constant for all large n. This paper is concerned with the following question: given a prime ideal p of A which is known to be in Ass(A/a n ) for all large integers n, can one identify a term of the sequence (Ass(A/a n )) n ∈ N beyond which p will subsequently be an ever-present? This paper presents some results about convergence of sequences of sets of associated primes of graded components of finitely generated graded modules over a standard positively graded commutative Noetherian ring; those results are then applied to the above question.

This paper answers a question of Jan J. Dijkstra by giving a proof that all three-point sets are zero dimensional. It is known that all two-point sets are zero dimensional, and it is known that for all n > 3, there are n-point sets which are not zero dimensional, so this paper answers the question for the last remaining case.

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.

In this paper, we first characterize the rank one operators in the preannihilator U⊥ of a weakly closed T(N)-module U. Using this characterization for the rank one operators in U⊥, a complete description of the extreme points of the unit ball U 1 is given. Finally, we show how to apply the techniques of the present paper to other operator systems and characterize their extreme points.

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.

In this paper, we generalize Chirka's theorem on the extension of functions holomorphic in a neighbourhood of S U (oD x D) where D is the open unit disc in C and S is the graph of a continuous D-valued function on D to higher dimensions, for certain classes of graphs S C D x Dn, n > 1. In particular, we show that Chirka's extension theorem generalizes to configurations in Cn+1, n > 1, involving graphs of (non-holomorphic) polynomial maps with small coefficients. 1. THE MAIN THEOREM This paper is motivated by an article by Chirka [1] (also see [2]) in which he proves the following result (in what follows, D will denote the open unit disc in C, while Dr will denote the open disc of radius r, centered at 0 C C): Theorem 1.1 (Chirka). Let 0: D -> C be a continuous function having sup cIq5(z)I 1. Rosay [3] showed that the theorem fails in general for higher dimensions. A natural question that arises is whether holomorphic extension to Dn+1r n > 1, occurs when the component functions of the Dn -valued map defining our graph are small in some appropriate sense (for instance, when the graph is a sufficiently small perturbation of a holomorphic graph). We are able to answer Chirka's question in the affirmative for the class of graphs described in Theorem 1.3 below. Before stating that theorem, however, we state the following proposition, which is a special case of Theorem 1.3. We highlight this as a separate proposition because of the clarity of its statement. Received by the editors May 1, 2000. 2000 Mathematics Subject Classification. Primary 32D15.