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In a recent paper Robinson has obtained an explicit formula for the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices. The object of the present note is to show that this formula may be deduced from known properties of Schur functions, with the aid of a result which the author has proved elsewhere.

A finitely presented 1-ended group $G$ has semistable fundamental group at infinity if $G$ acts geometrically on a simply connected and locally compact ANR $Y$ having the property that any two proper rays in $Y$ are properly homotopic. This property of $Y$ captures a notion of connectivity at infinity stronger than “1-ended”, and is in fact a feature of $G$, being independent of choices. It is a fundamental property in the homotopical study of finitely presented groups. While many important classes of groups have been shown to have semistable fundamental group at infinity, the question of whether every $G$ has this property has been a recognized open question for nearly forty years. In this paper we attack the problem by considering a proper but non-cocompact action of a group $J$ on such an $Y$. This $J$ would typically be a subgroup of infinite index in the geometrically acting over-group $G$; for example $J$ might be infinite cyclic or some other subgroup whose semistability properties are known. We divide the semistability property of $G$ into a $J$-part and a “perpendicular to $J$” part, and we analyze how these two parts fit together. Among other things, this analysis leads to a proof (in a companion paper) that a class of groups previously considered to be likely counter examples do in fact have the semistability property.

This paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions that are measurable in ($x,t$)-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John–Nirenberg space. The results are even new when the drifts are identically zero, because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón–Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a “double-scaling parameter” technique and the maximal function free approach introduced by Acerbi and Mingione.

Ostrowski's theorem implies that $\log(x),\log(x+1),\ldots$ are algebraically independent over $\mathbb{C}(x)$. More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution $y$ and particular transformations of $y$, such as derivatives of $y$ with respect to parameters, shifts of the arguments, rescaling, etc. In the present paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply it to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are finite free products of semigroups of the form $\mathbb{N}^n\times \mathbb{Z}/n_1\mathbb{Z}\times\ldots\times\mathbb{Z}/n_r\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications., Comment: minor revision

The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitely as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$., Comment: 22 pages; Minor corrections

J. Bellaïche and M. Dimitrov showed that the $p$-adic eigencurve is smooth but not étale over the weight space at $p$-regular theta series attached to a character of a real quadratic field $F$ in which $p$ splits. In this paper we prove the existence of an isomorphism between the subring fixed by the Atkin–Lehner involution of the completed local ring of the eigencurve at these points and a universal ring representing a pseudo-deformation problem. Additionally, we give a precise criterion for which the ramification index is exactly 2. We finish this paper by proving the smoothness of the nearly ordinary and ordinary Hecke algebras for Hilbert modular forms over $F$ at the overconvergent cuspidal Eisenstein points, being the base change lift for $\text{GL}(2)_{/F}$ of these theta series. Our approach uses deformations and pseudo-deformations of reducible Galois representations.

In this paper, we introduce weighted Carleson measure spaces associated with different homogeneities and prove that these spaces are the dual spaces of weighted Hardy spaces studied in a forthcoming paper. As an application, we establish the boundedness of composition of two Calderón–Zygmund operators with different homogeneities on the weighted Carleson measure spaces; this, in particular, provides the weighted endpoint estimates for the operators studied by Phong–Stein.

In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson-Schensted-Knuth correspondence. Then we use this result to generalize the mirabolic Robinson-Schensted-Knuth correspondence defined by Travkin, to the case of two partial flags and a line., Comment: 27 pages, slightly rewritten to combine two papers into one and clarify some sections

In this paper, we give several pictorial fractal representations of the ample or Kahler cone for surfaces in a certain class of K3 surfaces. The class includes surfaces described by smooth (2, 2, 2) forms in P×P×P defined over a sufficiently large number field K, which have a line parallel to one of the axes, and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface’s group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be 1.296± .010. The ample cone or Kahler cone for a surface is a significant and often complicated geometric object. Though much is known about the ample cone, particularly for K3 surfaces, only a few non-trivial examples have been explicitly described. These include the ample cones with a finite number of sides (see [N1] for n = 3, and [N2, N3] for n ≥ 5; the case n = 4 is attributed to Vinberg in an unpublished work [N1]); the ample cone for a class of K3 surfaces with n = 3 [Ba3]; and the ample cones for several Kummer surfaces, which are K3 surfaces with n = 20 [V, K-K, Kon]. Though the complexity of the problem generically increases with n, the problem for K3 surfaces with maximal Picard number (n = 20) appear to be tractable because of the small size of the transcendental lattice. In this paper, we introduce accurate pictorial representations of the ample cone and the associated fractal for surfaces within a class of K3 surfaces with Picard number n = 4 (see Figures 1, 3, 4, and 5). As far as the author is aware, the associated fractal has not been studied in any great depth for any ample cone for which the fractal has a non-integer dimension, except the one in [Ba3]. The fractal in that case is Cantor-like (it is a subset of S) and rigorous bounds on its Hausdorff dimension are calculated in [Ba1]. The Hausdorff dimension of the fractal of this paper is estimated to be 1.296± .010. Our second main result is to relate the Hausdorff dimension of the fractal to the growth of the height of curves for an orbit of curves on a surface in this class. Precisely, let V be a surface within our class of K3 surfaces and let A = Aut(V/K) be its group of automorphisms over a sufficiently large number field K. Let D be an ample divisor on V and let C be a curve on V . Define NA(C)(t,D) = #{C′ ∈ A(C) : C′ ·D < t}. Here we have abused notation by letting C′ also represent the divisor class that contains C′. The intersection C′ ·D should be thought of as a logarithmic height of 2000 Mathematics Subject Classification. 14J28, 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05.

Let f be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer n has a factor common with the n-th Fourier coefficient of f. It has been shown in previous papers that this happens very often. In this paper, we give an asymptotic formula for the number of integers n for which (n, a(n)) = 1, where a(n) is the n-th Fourier coefficient of a normalized Hecke eigenform f of weight 2 with rational integer Fourier coefficients and having complex multiplication.

In a previous paper the authors showed that, under some technical conditions, the local Galois representations attached to the members of a non-CM family of ordinary cusp forms are indecomposable for all except possibly finitely many members of the family. In this paper we use deformation theoretic methods to give examples of non-CM families for which every classical member of weight at least two has a locally indecomposable Galois representation. School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400005, India. e-mail: eghate@math.tifr.res.in Department of Mathematics, University of British Columbia, Vancouver, BC e-mail: vatsal@math.ubc.ca Received by the editors August 5, 2008. Published electronically December 29, 2010. AMS subject classification: 11F80. 1

This paper is concerned with the residual spectrum of the hermitian quaternionic classical groups G′n and H′n defined as algebraic groups for a quaternion algebra over an algebraic number field. Groups G′n and H′n are not quasi-split. They are inner forms of the split groups SO4n and Sp4n. Hence, the parts of the residual spectrum of G′n and H′n obtained in this paper are compared to the corresponding parts for the split groups SO4n and Sp4n.

In his seminal papers, Koblitz proposed curves for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix-τ expansion of integers in the number fields and . The (window) nonadjacent form of τ -expansion of integers in was first investigated by Solinas. For integers in , the nonadjacent form and the window nonadjacent form of the τ -expansion were studied. These are used for efficient point multiplications on Koblitz curves. In this paper, we complete the picture by producing the (window) nonadjacent radix-τ expansions for integers in all Euclidean imaginary quadratic number fields.

This paper is a continuation of three recent articles concerning the structure of hyperinvariant subspace lattices of operators on a (separable, infinite dimensional) Hilbert space . We show herein, in particular, that there exists a “universal” fixed block-diagonal operator B on such that if ε > 0 is given and T is an arbitrary nonalgebraic operator on , then there exists a compact operator K of norm less than ε such that (i) Hlat(T) is isomorphic as a complete lattice to Hlat(B + K) and (ii) B + K is a quasidiagonal, C00, (BCP)-operator with spectrum and left essential spectrum the unit disc. In the last four sections of the paper, we investigate the possible structures of the hyperlattice of an arbitrary algebraic operator. Contrary to existing conjectures, Hlat(T) need not be generated by the ranges and kernels of the powers of T in the nilpotent case. In fact, this lattice can be infinite.

We study the cardinal invariants of analytic P-ideals, concentrating on the ideal Zof asymptotic density zero. Among other results we prove min{b, cov (N)} ≤ cov∗(Z) ≤ max{b, non(N)}. Introduction Analytic P-ideals and their quotients have been extensively studied in recent years. The first step to better understanding the structure of the quotient forcings P(ω)/I is to understand the structure of the ideal itself. Significant progress in understanding the way in which the structure of an ideal affects the structure of its quotient has been done by I. Farah [Fa1, Fa2, Fa3, Fa4, Fa5]. Typically (but not always) the quotients P(ω)/I, where I is an analytic P-ideal, are proper and weakly distributive. For some special ideals these quotients have been identified: P(ω)/Z is as forcing notion equivalent to P(ω)/fin ∗ B(2) [Fa5], and P(ω)/ tr(N) [HZ1] is as forcing notion equivalent to the iteration of B(ω) followed by an א0-distributive forcing (see the definitions below). A secondary motivation comes from the problem of which ideals can be destroyed by a weakly distributive forcing. Even for the class of analytic P-ideals only partial results are known (see Section 3). In this note we contribute to this line of research by investigating cardinal invariants of analytic P-ideals, comparing them to other, standard, cardinal invariants of the continuum. In the first section we introduce cardinal invariants of ideals on ω, along the lines of the cardinal invariants contained in the Cichon’s diagram. We also recall the definitions of standard orderings on ideals on ω (Rudin–Keisler, Tukey, Katětov) and their impact on the cardinal invariants of the ideals. Basic theory of analytic P-ideals on ω and examples are also reviewed here. Known results on additivity and cofinality of analytic P-ideals are summarized in the second section. The main part of the paper is contained in the third section. There we study the order of Katětov restricted to analytic P-ideals, giving a detailed description of how the summable and density ideals are placed in the Katětov order. For the rest of the section, we focus on the ideal of asymptotic density zero and compare its covering number to standard cardinal invariants of the continuum. We prove that min{b, cov(N)} ≤ cov(Z) ≤ max{b, non(N)} and mention some consistency results. We introduce the notion of a totally bounded analytic P-ideal and show that Received by the editors October 5, 2004; revised September 19, 2005. The authors gratefully acknowledge support from PAPIIT grant IN106705. The second author’s research was also partially supported by GA CR grant 201-03-0933 and CONACYT grant 40057-F AMS subject classification: 03E17, 03E40. c ©Canadian Mathematical Society 2007. 575 576 F. Hernandez-Hernandez and M. Hrusak all analytic P-ideals which are not totally bounded can be destroyed by a weakly distributive forcing. In the last section we study the separating number of analytic P-ideals, an invariant closely related to the Laver and Mathias–Prikry type forcings associated with the ideal. Two major problems remain open here: (1) Is add(I) = add(N) for every tall analytic P-ideal I? (2) Can every analytic P-ideal be destroyed by a weakly distributive forcing? What about Z? We assume knowledge of the method of forcing as well as the basic theory of cardinal invariants of the continuum as covered in [BJ]. Our notation is standard and follows [Ku, Je, BJ]. In particular, c0, l1 and l∞ denote the standard Banach spaces of sequences of reals. For A,B infinite subsets of ω, we say that A is almost contained in B (A ⊆ B) if A \ B is finite. The symbol A = Bmeans that A ⊆ B and B ⊆ A. For functions f , g ∈ ω we write f ≤ g to mean that there is some m ∈ ω such that f (n) ≤ g(n) for all n ≥ m. The bounding number b is the least cardinal of an ≤-unbounded family of functions in ω . The dominating number d is the least cardinal of a ≤-cofinal family of functions in ω . Recall that a family of subsets of ω has the strong finite intersection property if any finite subfamily has infinite intersection. The pseudointersection number p is the minimal size of a family of subsets of ω with the strong finite intersection property but without an infinite pseudointersection (i.e., without a common lower bound in the ⊆ order). A family S ⊆ P(ω) is a splitting family if for every infinite A ⊆ ω there is an S ∈ S such that S ∩ A and A \ S are infinite. The splitting number s is the minimal size of a splitting family in P(ω). The set 2 is equipped with the product topology, that is, the topology with basic open sets of the form [s] = {x ∈ 2 : s ⊆ x}, where s ∈ 2 . The topology of P(ω) is that obtained via the identification of each subset of ω with its characteristic function. An ideal on X is a family of subsets of X closed under taking finite unions and subsets of its members. We assume throughout the paper that all ideals contain all singletons {x} for x ∈ X. An ideal I on ω is called P-ideal if for any sequence Xn ∈ I, n ∈ ω, there exists X ∈ I such that Xn ⊆ ∗ X for all n ∈ ω. An ideal I on ω is analytic if it is analytic as a subspace of P(ω) with the above topology. Recall that an ideal on ω is tall (or dense) if every infinite set of ω contains an infinite set from the ideal. If I is an ideal on ω and Y ⊆ ω is an infinite set, then we denote by I↾Y the ideal {I ∩ Y : I ∈ I}; note that the underlying set of the ideal I↾Y is not the underlying set of I but Y . For an ideal I on ω, I denotes the dual filter, M denotes the ideal of meager subsets of R, and N the ideal of Lebesgue null subsets of R. Given an ideal I on a set X, the following are standard cardinal invariants associated with I: add(I) = min{|A| : A ⊆ I ∧ ⋃ A / ∈ I}, cov(I) = min{|A| : A ⊆ I ∧ ⋃ A = X}, cof(I) = min{|A| : A ⊆ I ∧ (∀I ∈ I)(∃A ∈ A)(I ⊆ A)}, non(I) = min{|Y | : Y ⊆ X ∧Y / ∈ I}. The provable relationships between the cardinal invariants of M and N are Cardinal Invariants of Analytic P-Ideals 577 summed up in the following diagram: cov(N) // non(M) // cof(M) // cof(N)

Let $N/K$ be a finite Galois extension of p-adic number fields, and let $\rho ^{\mathrm {nr}} \colon G_K \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ be an r-dimensional unramified representation of the absolute Galois group $G_K$ , which is the restriction of an unramified representation $\rho ^{\mathrm {nr}}_{{{\mathbb Q}}_{p}} \colon G_{{\mathbb Q}_{p}} \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ . In this paper, we consider the $\mathrm {Gal}(N/K)$ -equivariant local $\varepsilon $ -conjecture for the p-adic representation $T = \mathbb Z_p^r(1)(\rho ^{\mathrm {nr}})$ . For example, if A is an abelian variety of dimension r defined over ${{\mathbb Q}_{p}}$ with good ordinary reduction, then the Tate module $T = T_p\hat A$ associated to the formal group $\hat A$ of A is a p-adic representation of this form. We prove the conjecture for all tame extensions $N/K$ and a certain family of weakly and wildly ramified extensions $N/K$ . This generalizes previous work of Izychev and Venjakob in the tame case and of the authors in the weakly and wildly ramified case.

The main problem that is solved in this paper has the following simple formulation (which is not used in its solution). The group K = Op(C) × Oq(C) acts on the space Mp, q of p × q complex matrices by (a; b) · x = axb–1, and so does its identity component K0 = SOp(C)×SOq(C). A K-orbit (or K0-orbit) in Mp,q is said to be nilpotent if its closure contains the zero matrix. The closure, , of a nilpotent K-orbit (resp. K0-orbit) in Mp,q is a union of and some nilpotent K-orbits (resp. K0-orbits) of smaller dimensions. The description of the closure of nilpotent K-orbits has been known for some time, but not so for the nilpotent K0-orbits. A conjecture describing the closure of nilpotent K0-orbits was proposed in [11] and verièd when min(p, q) ≤ 7. In this paper we prove the conjecture. The proof is based on a study of two prehomogeneous vector spaces attached to and determination of the basic relative invariants of these spaces.The above problem is equivalent to the problem of describing the closure of nilpotent orbits in the real Lie algebra so(p, q) under the adjoint action of the identity component of the real orthogonal group O(p, q).

For fixed$\alpha \in [0,1]$, consider the set$S_{\alpha ,N}$of dilated squares$\alpha , 4\alpha , 9\alpha , \dots , N^2\alpha \, $modulo$1$. Rudnick and Sarnak conjectured that, for Lebesgue, almost all such$\alpha $the gap-distribution of$S_{\alpha ,N}$is consistent with the Poisson model (in the limit asNtends to infinity). In this paper, we prove a new estimate for the triple correlations associated with this problem, establishing an asymptotic expression for the third moment of the number of elements of$S_{\alpha ,N}$in a random interval of length$L/N$, provided that$L> N^{1/4+\varepsilon }$. The threshold of$\tfrac {1}{4}$is substantially smaller than the threshold of$\tfrac {1}{2}$(which is the threshold that would be given by a naïve discrepancy estimate).Unlike the theory of pair correlations, rather little is known about triple correlations of the dilations$(\alpha a_n \, \text {mod } 1)_{n=1}^{\infty } $for a nonlacunary sequence$(a_n)_{n=1}^{\infty } $of increasing integers. This is partially due to the fact that the second moment of the triple correlation function is difficult to control, and thus standard techniques involving variance bounds are not applicable. We circumvent this impasse by using an argument inspired by works of Rudnick, Sarnak, and Zaharescu, and Heath-Brown, which connects the triple correlation function to some modular counting problems.In Appendix B, we comment on the relationship between discrepancy and correlation functions, answering a question of Steinerberger.

In a 1987 paper, Gross introduced certain curves associated to a definite quaternion algebra B over Q; he then proved an analog of his result with Zagier for these curves. In Gross’ paper, the curves were defined in a somewhat ad hoc manner. In this article, we present an interpretation of these curves as projective varieties arising from graded rings of automorphic forms on B×, analogously to the construction in the Satake compactification. To define such graded rings, one needs to introduce a “multiplication” of automorphic forms that arises from the representation ring of B×. The resulting curves are unions of projective lines equipped with a collection of Hecke correspondences. They parametrize two-dimensional complex tori with quaternionic multiplication. In general, these complex tori are not abelian varieties; they are algebraic precisely when they correspond to CM points on these curves, and are thus isogenous to a product E × E, where E is an elliptic curve with complex multiplication. For these CM points one can make a relation between the action of the p-th Hecke operator and Frobenius at p, similar to the well-known congruence relation of Eichler and Shimura.

In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is links whose Khovanov homology is supported on two adjacent diagonals, are known to only contain $\mathbb{Z}_2$ torsion. In this paper, we prove a local version of this result. If the Khovanov homology of a link is supported in two adjacent diagonals over a range of homological gradings and the Khovanov homology satisfies some other mild restrictions, then the Khovanov homology of that link has only $\mathbb{Z}_2$ torsion over that range of homological gradings. These conditions are then shown to be met by an infinite family of 3-braids, strictly containing all 3-strand torus links, thus giving a partial answer to Sazdanovic and Przytycki's conjecture that 3-braids have only $\mathbb{Z}_2$ torsion in Khovanov homology. We also give explicit computations of integral Khovanov homology for all links in this family., Comment: 20 pages, 11 figures. Section 4 has been simplified

Let M be a closed, connected, orientable topological 4-manifold, and G be a finite group acting topologically and locally linearly on M. In this paper we investigate the Borel spectral sequence for the G-equivariant cohomology of M, and establish new bounds on the rank of G for homologically trivial actions with discrete singular set., 22 pages (v2). Accepted for publication in the Canadian Journal of Mathematics

This paper expresses the character of certain depth-zero supercuspidal representations of the rank-2 symplectic group as the Fourier transform of a finite linear combination of regular elliptic orbital integrals—an expression which is ideally suited for the study of the stability of those characters. Building on work of F. Murnaghan, our proof involves Lusztig’s Generalised Springer Correspondence in a fundamental way, and also makes use of some results on elliptic orbital integrals proved elsewhere by the author using Moy-Prasad filtrations of p-adic Lie algebras. Two applications of the main result are considered toward the end of the paper.

In an earlier paper [10], we studied a generalized Rao bound for ordered orthogonal arrays and (T, M, S)-nets. In this paper, we extend this to a coding-theoretic approach to ordered orthogonal arrays. Using a certain association scheme, we prove a MacWilliams-type theorem for linear ordered orthogonal arrays and linear ordered codes as well as a linear programming bound for the general case. We include some tables which compare this bound against two previously known bounds for ordered orthogonal arrays. Finally we show that, for even strength, the LP bound is always at least as strong as the generalized Rao bound.

In a previous paper, we gave a correspondence between certain exact solutions to a (2 + 1)-dimensional integrable Chiral Model and holomorphic bundles on a compact surface. In this paper, we use algebraic geometry to derive a closed-form expression for those solutions and show by way of examples how the algebraic data which parametrise the solution space dictates the behaviour of the solutions.

A. R. Martínez Fernández obtained upper bounds for quermassintegrals of the p-inner parallel bodies: an extension of the classical inner parallel body to the $L_p$ -Brunn-Minkowski theory. In this paper, we establish (sharp) upper and lower bounds for quermassintegrals of p-inner parallel bodies. Moreover, the sufficient and necessary conditions of the equality case for the main inequality are obtained, which characterize the so-called tangential bodies.

In this paper we consider the following problem: Given a unital C*- algebra A and a collection of elements S in the identity component of the invertible group of A, denoted inv0(A), characterize the group of finite products of elements of S. The particular C*-algebras studied in this paper are either unital purely infinite simple or of the form (A ⊗ K)+, where A is any C*-algebra and K is the compact operators on an infinite dimensional separable Hilbert space. The types of elements used in the factorizations are unipotents (1+ nilpotent), positive invertibles and symmetries (s2 = 1). First we determine the groups of finite products for each collection of elements in (A ⊗ K)+. Then we give upper bounds on the number of factors needed in these cases. The main result, which uses results for (A ⊗ K)+, is that for A unital purely infinite and simple, inv0(A) is generated by each of these collections of elements.

In a recent paper [2] we defined four classes of infinite dimensional simple Lie algebras over a field of characteristic 0 which we called W*, S*, H*, and K*. As the names suggest, these classes generalize the Lie algebras of Cartan type. A second paper [3] investigates the derivations of the algebras W* and S*, and the possible isomorphisms between these algebras and the algebras defined by Block [1]. In the present paper we investigate the automorphisms of the algebras of type W*.

Positive, copositive, onesided and intertwining (co-onesided) polyno- mial and spline approximations of functions f W k( 1 1) are considered. Both uniform and pointwise estimates, which are exact in some sense, are obtained. 1. Introduction and main results. We start by recalling some of the notations and definitions used throughout this paper. Let C(a b )a nd C k ( a b) be, respectively, the sets of all continuous and k-times continuously differentiable functions on (a b), and let Lp(a b), 0 p , be the set of measurable functions on (a b) such that f Lp(a b) ,w here f L p ( a b ) := b a f (x) p dx 1 p Throughout this paper L (a b) is understood as C(a b) with the usual uniform norm, to simplify the notation. We also denote by W k (a b), p 1, the set of all functions f on (a b) such that f (k 1) are absolutely continuous and f (k) Lp, and by Pn the set of all m=0 m i ( 1) m i f (x mh 2 + ih) if x mh 2 (a b), 0 otherwise. Then the m-th (usual) modulus of smoothness of f Lp(a b )i s def ined by m ( f t ( a b))p := sup 0 h t Δ m (f ( a b)) Lp (a b) We will also use the so-called -modulus, an averaged modulus of smoothness, defined for all bounded measurable functions on (a b )b y m ( f t ( a b))p := m (f t ) L p ( a b )

The aim of this paper is to study circular units in the compositum K of t cyclic extensions of ${\mathbb {Q}}$ ( $t\ge 2$ ) of the same odd prime degree $\ell $ . If these fields are pairwise arithmetically orthogonal and the number s of primes ramifying in $K/{\mathbb {Q}}$ is larger than $t,$ then a nontrivial root $\varepsilon $ of the top generator $\eta $ of the group of circular units of K is constructed. This explicit unit $\varepsilon $ is used to define an enlarged group of circular units of K, to show that $\ell ^{(s-t)\ell ^{t-1}}$ divides the class number of K, and to prove an annihilation statement for the ideal class group of K.

In this paper, we are interested in obtaining large values of Dirichlet L-functions evaluated at zeros of a class of L-functions, that is, $$ \begin{align*}\max_{\substack{F(\rho)=0\\ T\leq \Im \rho \leq 2T}}L(\rho,\chi), \end{align*} $$ where $\chi $ is a primitive Dirichlet character and F belongs to a class of L-functions. The class we consider includes L-functions associated with automorphic representations of $GL(n)$ over ${\mathbb {Q}}$ .

In this paper, we extend to the non-Hille–Yosida case a variation of constants formula for a nonautonomous and nonhomogeneous Cauchy problems first obtained by Gühring and Räbiger. By using this variation of constants formula, we derive a necessary and sufficient condition for the existence of an exponential dichotomy for the evolution family generated by the associated nonautonomous homogeneous problem. We also prove a persistence result of the exponential dichotomy for small perturbations. Finally, we illustrate our results by considering two examples. The first example is a parabolic equation with nonlocal and nonautonomous boundary conditions, and the second example is an age-structured model that is a hyperbolic equation.

We define a limiting $\mathfrak{sl}_N$ Khovanov-Rozansky homology for semi-infinite positive multi-colored braids, and we show that this limiting homology categorifies a highest-weight projector for a large class of such braids. This effectively completes the extension of Cautis' similar result for infinite twist braids, begun in our earlier papers with Islambouli and Abel. We also present several similar results for other families of semi-infinite and bi-infinite multi-colored braids., 37 pages, 13 figures

We formulate a conjecture analogous to Gross' refinement of the Stark conjectures on special values of abelian L-series at s = 0. Some evidence for the conjecture can be obtained, thanks to the fundamental ideas of F. Thaine. 1. Introduction. This paper formulates a refined analogue of the usual class number formula for a real quadratic extension of Q, using circular units. The statement of this conjecture is inspired by an analogous conjecture of Gross (Gr). Strong evidence for this conjecture can be given thanks to F. Thaine's powerful method (Th) for generating relations in ideal class groups using circular units. The first two sections briefly recall Dirichlet's analytic class number formula and Gross's refinement of it; they are there mainly to fix notations and provide motivation. Section 4 states the new conjecture. The remaining sections are devoted to proving various results that support it. ACKNOWLEDGEMENTS. I wish to thank Massimo Bertolini and Benedict Gross for many stimulating conversations on the topics of this paper. NOTATIONS. If K is a number field and w is a place of K lying above a prime v of Q, we denote by Kw the localization of K at w, and let Nw be the order of its residue field. The w-adic norm || || w is normalized so that it is equal to Nw" 1 on uniformizing elements.

We answer a 1975 question of G. R. Gordh by showing that if X is a homogeneous compactum which is the continuous image of a compact ordered space then at least one of the following holds: (i) X is metrizable, (ii) dimX = 0 or (iii) X is a union of finitely many pairwise disjoint generalized simple closed curves. We begin to examine the structure of homogeneous 0-dimensional spaces which are continuous images of ordered compacta. 1. Introduction. The aim of this paper is to investigate homogeneous spaces which are continuous images of ordered compacta. In 1975, G. R. Gordh proved that if a homo­ geneous and hereditarily unicoherent continuum is the continuous image of an ordered compactum, then it is metrizable, and so indecomposable (7, Theorem 3). Further, he asked if, in general, every homogeneous continuum which is the continuous image of an ordered compactum must be either metrizable or a generalized simple closed curve. Our Theorem 1 provides an affirmative answer to Gordh's question. Moreover, in Theorem 2, we prove that a homogeneous space which is not 0-dimensional and which is the continuous image of an ordered compactum is either metrizable or a union of finitely many pairwise disjoint generalized simple closed curves. Our methods of proof involve characterizations of continuous images of arcs obtained in ( 16) in terms of cyclic elements and T-sets. When dealing with the class A of all homogeneous and 0-dimensional spaces which are the continuous images of ordered compacta, the situation becomes less clear. By a recent theorem of M. Bell, each member of A is first countable. Moreover, by a result of (18), each member of A can be embedded into a dendron. We give a rather simple construction leading to a wide subclass of A. In particular, we show that not all members of A are orderable, and that there exists a strongly homogeneous space X which is the continuous image of an ordered compactum and which is not first countable. It follows that X $ A. Our investigations of the class A led to some natural questions which are stated at the end of the paper. All spaces considered in this paper are Hausdorff.

Let $A$ be a symmetrisable generalised Cartan matrix, and let $\mathfrak g(A)$ be the corresponding Kac-Moody algebra. In this paper, we address the following fundamental question on the structure of $\mathfrak g(A)$: given two homogeneous elements $x,y \in \mathfrak g(A)$, when is their bracket $[x,y]$ a nonzero element? As an application of our results, we give a description of the solvable and nilpotent graded subalgebras of $\mathfrak g(A)$., 32 pages. Final version, to appear in Canadian Journal of Mathematics

In this paper we analyze the convergence of the following type of series $$\begin{eqnarray}T_{N}^{{\mathcal{L}}}f(x)=\mathop{\sum }_{j=N_{1}}^{N_{2}}v_{j}\big(e^{-a_{j+1}{\mathcal{L}}}f(x)-e^{-a_{j}{\mathcal{L}}}f(x)\big),\quad x\in \mathbb{R}^{n},\end{eqnarray}$$ where ${\{e^{-t{\mathcal{L}}}\}}_{t>0}$ is the heat semigroup of the operator ${\mathcal{L}}=-\unicode[STIX]{x1D6E5}+V$ with $\unicode[STIX]{x1D6E5}$ being the classical laplacian, the nonnegative potential $V$ belonging to the reverse Hölder class $RH_{q}$ with $q>n/2$ and $n\geqslant 3$, $N=(N_{1},N_{2})\in \mathbb{Z}^{2}$ with $N_{1}, ${\{v_{j}\}}_{j\in \mathbb{Z}}$ is a bounded real sequences, and ${\{a_{j}\}}_{j\in \mathbb{Z}}$ is an increasing real sequence.Our analysis will consist in the boundedness, in $L^{p}(\mathbb{R}^{n})$ and in $BMO(\mathbb{R}^{n})$, of the operators $T_{N}^{{\mathcal{L}}}$ and its maximal operator $T^{\ast }f(x)=\sup _{N}T_{N}^{{\mathcal{L}}}f(x)$.It is also shown that the local size of the maximal differential transform operators (with $V=0$) is the same with the order of a singular integral for functions $f$ having local support. Moreover, if ${\{v_{j}\}}_{j\in \mathbb{Z}}\in \ell ^{p}(\mathbb{Z})$, we get an intermediate size between the local size of singular integrals and Hardy–Littlewood maximal operator.

In the classical setting, a convex polytope is said to be semiregular if its facets are regular and its symmetry group is transitive on vertices. This paper continues our study of alternating semiregular abstract polytopes, which have abstract regular facets, still with combinatorial automorphism group transitive on vertices and with two kinds of regular facets occurring in an alternating fashion.Our main concern here is the universal polytope ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$, an alternating semiregular $(n+1)$-polytope defined for any pair of regular $n$-polytopes ${\mathcal{P}},{\mathcal{Q}}$ with isomorphic facets. After a careful look at the local structure of these objects, we develop the combinatorial machinery needed to explain how ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ can be constructed by “freely assembling” unlimited copies of ${\mathcal{P}}$, ${\mathcal{Q}}$ along their facets in alternating fashion. We then examine the connection group of ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$, and from that prove that ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ covers any $(n+1)$-polytope ${\mathcal{B}}$ whose facets alternate in any way between various quotients of ${\mathcal{P}}$ or ${\mathcal{Q}}$.

The generalized Soulé character was introduced by H. Nakamura and Z. Wojtkowiak and is a generalization of Soulé’s cyclotomic character. In this paper, we prove that certain linear sums of generalized Soulé characters essentially coincide with the image of generalized Beilinson elements in K-groups under Soulé’s higher regulator maps. This result generalizes Huber–Wildeshaus’ theorem, which is a cyclotomic field case of our results, to an arbitrary number fields.

In this paper, we investigate the weighted multilinear boundedness properties of the maximal higher order Calder\'on commutator for the dimensions larger than two. We establish all weighted multilinear estimates on the product of the $L^p(\mathbb{R}^d,w)$ space, including some peculiar endpoint estimates of the higher dimensional Calder\'on commutator., Comment: 36 pages, Canadian Journal of Mathematics, to appear. arXiv admin note: text overlap with arXiv:1712.09020

In this paper we prove some one-level density results for the low-lying zeros of families of quadratic and quartic Hecke $L$-functions of the Gaussian field. As corollaries, we deduce that at least 94.27% and 5%, respectively, of the members of the quadratic family and the quartic family do not vanish at the central point.

Archdeacon and Grable (1995) proved that the genus of the random graph $G\in\mathcal{G}_{n,p}$ is almost surely close to $pn^2/12$ if $p=p(n)\geq3(\ln n)^2n^{-1/2}$. In this paper we prove an analogous result for random bipartite graphs in $\mathcal{G}_{n_1,n_2,p}$. If $n_1\ge n_2 \gg 1$, phase transitions occur for every positive integer $i$ when $p=\Theta((n_1n_2)^{-\frac{i}{2i+1}})$. A different behaviour is exhibited when one of the bipartite parts has constant size, $n_1\gg1$ and $n_2$ is a constant. In that case, phase transitions occur when $p=\Theta(n_1^{-1/2})$ and when $p=\Theta(n_1^{-1/3})$., Comment: 19 pages

This paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous $f$-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the $f$-divergence is proposed, and related functional affine isoperimetric inequalities are established.

Root systems and Coxeter groups are important tools in multivariable analysis. This paper is concerned with differential-difference and integral operators, and orthogonality structures for polynomials associated to Coxeter groups. For each such group, the structures allow as many parameters as the number of conjugacy classes of reflections. The classical orthogonal polynomials of Gegenbauer and Jacobi type appear in this theory as two-dimensional cases. For each Coxeter group and admissible choice of parameters there is a structure analogous to spherical harmonics which relies on the connection between a Laplacian operator and orthogonality on the unit sphere with respect to a group-invariant measure. The theory has been developed in several papers of the author [4,5,6,7]. In this paper, the emphasis is on the study of an intertwining operator which allows the transfer of certain results about ordinary harmonic polynomials to those associated to Coxeter groups. In particular, a formula and a bound are obtained for the Poisson kernel.

Strong convergence has been investigated in summability theory and Fourier analysis. This paper extends strong convergence to a topological property of sequence spaces E. The more general property of strong boundedness is also defined and examined. One of the main results shows that for an FK-space E which contains all finite sequences, strong convergence is equivalent to the invariance property E = ℓ ν0. E with respect to coordinatewise multiplication by sequences in the space ℓν0 defined in the paper. Similarly, strong boundedness is equivalent to another invariance E = ℓν.E. The results of the paper are applied to summability fields and spaces of Fourier series.

The purpose of this paper is to introduce the notion of a CW complex over a topological category. The main theorem of this paper gives an equivalence between the homotopy theory of diagrams of spaces based on a topological category and the homotopy theory of CW complexes over the same base category.A brief description of the paper goes as follows: in Section 1 we introduce the homotopy category of diagrams of spaces based on a fixed topological category. In Section 2 homotopy groups for diagrams are defined. These are used to define the concept of weak equivalence and J-n equivalence that generalize the classical definition. In Section 3 we adapt the classical theory of CW complexes to develop a cellular theory for diagrams. In Section 4 we use sheaf theory to define a reasonable cohomology theory of diagrams and compare it to previously defined theories. In Section 5 we define a closed model category structure for the homotopy theory of diagrams. We show this Quillen type homotopy theory is equivalent to the homotopy theory of J-CW complexes. In Section 6 we apply our constructions and results to prove a useful result in equivariant homotopy theory originally proved by Elmendorf by a different method.

Introduction. The concept of a mitotic group was introduced in [3] by Baumslag, Dyer and Heller who showed that mitotic groups were acyclic. In [8] one of the authors introduced the concept of a pseudo-mitotic group, a concept weaker than that of a mitotic group, and showed that pseudo-mitotic groups were acyclic and that the group Gn of homeomorphisms of R n with compact support is pseudo-mitotic. In our present paper we develop techniques to prove pseudomitoticity of certain other homeomorphism groups. In [5] Kan and Thurston observed that the group of set theoretic bijections of Q with bounded support is acyclic. A natural question is to decide whether the group of homeomorphisms of Q (resp. the irrationals / ) with bounded support is acyclic or not. In the present paper we develop techniques to answer this question in the affirmative. Also the techniques developed here enable us to show that the group of homeomorphisms of the Cantor set which are identity in a neighbourhood of 0 and 1 is pseudo-mitotic and hence acyclic. It is worth noticing the contrast between our results and the results of R. D. Anderson in [1] and [2]. Anderson, using his techniques shows that the group of all homeomorphisms of Q, / or the Cantor set is a simple group. He also shows that the group of orientation preserving homeomorphisms of S or S is simple.

This paper is concerned with a combinatorial construction which mysteriously “mimics” or “models” the decomposition of certain reducible representations of orthogonal groups. Although no knowledge of representation theory is needed to understand the body of this paper, a little familiarity is necessary to understand the representation theoretic motivation given in the introduction. Details of the proofs will most easily be understood by people who have had some exposure to Schensted's algorithm or jeu de tacquin.

In the present paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to the computation of the splitting number. Finally, we use the slice-torus link invariants, and the Whitehead doubling to define new strong concordance invariants for links, which are proven to be independent from the corresponding slice-torus link invariant., 31 pages, 19 figures, 4 tables. Improved exposition, typos fixed, slight improvement of Propositions 2.10 and 3.5, and added a comment on a result of A. Conway related to Theorem 1.4. Comments are welcome!

A ring is called right annelidan if the right annihilator of any subset of the ring is comparable with every other right ideal. In this paper we develop the connections between this class of rings and the classes of right Bézout rings and rings whose right ideals form a distributive lattice. We obtain results on localization of right annelidan rings at prime ideals, chain conditions that entail left-right symmetry of the annelidan condition, and construction of completely prime ideals.

This paper is concerned with the maximisation of the k'th eigenvalue of the Laplacian amongst flat tori of unit volume in dimension d as k goes to infinity. We show that in any dimension maximisers exist for any given k, but that any sequence of maximisers degenerates as k goes to infinity when the dimension is at most 10. Furthermore, we obtain specific upper and lower bounds for the injectivity radius of any sequence of maximisers. We also prove that flat Klein bottles maximising the k'th eigenvalue of the Laplacian exhibit the same behaviour. These results contrast with those obtained recently by Gittins and Larson, stating that sequences of optimal cuboids for either Dirichlet or Neumann boundary conditions converge to the cube no matter the dimension. We obtain these results via Weyl asymptotics with explicit control of the remainder in terms of the injectivity radius. We reduce the problem at hand to counting lattice points inside anisotropically expanding domains, where we generalise methods of Yu. Kordyukov and A. Yakovlev by considering domains that expand at different rates in various directions., Comment: 20 pages