This note concerns subgroups of the general linear group GL(n, F), where n is finite and F algebraically closed, g ∈ GL(n, F) is called a d-element if there exists an x ∈ GL(w, F) such that x–1gx is diagonal, and a u-element if (g — 1)n = 0. A subgroup G of GL(n, F) is called a d-group (or a u-group) if every element of G is a d-element (or a u-element). In view of the Jordan decomposition of the elements of GL(n, F) into products of d-elements and u-elements it is important to know the structure of d-groups and u-groups, u-groups present very little difficulty and their structure is well known (1, 19.4), but d-groups seem to have a more complicated structure.

Soit R un anneau et soit Gop un sous groupe de Aut (R). Formons l'anneau de groupe à gauche (“skew group ring”) RSG, en imposant, sur le module libre à gauche avec une base G, la multiplication rg = grg. (On écrit Gop au lieu de G car, si G < Inn (R) = le groupe d'automorphismes intérieurs, la condition g–rg = rg; implique que le homomorphisme naturel est vraiment un anti-homomorphisme).

Nous considérons le cas où R = MnF, F un corps, MnF l'anneau de matrices, d'ordre n, et Gop un sous-groupe (généralement) fini, de PGL(n, F) = Inn (R). Ce travail est consacré à l'étude de la simplicité de RSG.

We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini–Darmon, Longo, Nekovar, Pollack–Weston, and others. The construction has direct applications to Iwasawa's main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated $p$-adic $L$-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over $\text{CM}$ fields via the technique of Skinner and Urban.

The notions of monotone bases and bases of subspaces are well known in a normed linear space setting and have obvious extensions to pseudo-metrizable linear topological spaces. In this paper, these notions are extended to arbitrary linear topological spaces. The principal result gives a list of properties that are equivalent to a sequence (Mi) of complete subspaces being an e-Schauder basis of subspaces for the closed linear span of . A corollary of this theorem is the fact that an e-Schauder basis for a dense subspace of a linear topological space is an e-Schauder basis for the whole space.

In a previous work [6] it was shown that by imposing certain finiteness conditions on a nilpotent loop certain algebraic results yielded properties about [X, Y] where X is finite CW and Y is an H-Space. In this sequel we further restrict the category of nilpotent loops to a full subcategory called H-loops which still contains all loops of the form [X, Y], We prove that on this category there is a unique and universal P-localization if P ≠ ∅ which corresponds to topological localization. We also show that if the H-loop is a group then the two concepts of localization agree.

The first section of this paper is devoted to the definition and basic properties of H-loops. In the second section we develop the localization construction and prove uniqueness. Finally, in the third section we consider the topological and group theoretic situations.

In this paper, we prove that for $\ell \,=\,1$ or 2 the rate of best $\ell $ - monotone polynomial approximation in the ${{L}_{p}}$ norm $\left( 1\,\le \,p\,\le \,\infty \right)$ weighted by the Jacobi weight ${{w}_{\alpha ,\,\beta }}\left( x \right)\,:=\,{{\left( 1\,+\,x \right)}^{\alpha }}{{\left( 1\,-\,x \right)}^{\beta }}$ with $\alpha ,\,\beta \,>\,-1/p$ if $p\,<\,\infty $ , or $\alpha ,\,\beta \,\ge \,0$ if $p\,=\,\infty $ , is bounded by an appropriate $\left( \ell \,+\,1 \right)$ -st modulus of smoothness with the same weight, and that this rate cannot be bounded by the $\left( \ell \,+\,2 \right)$ -nd modulus. Related results on constrained weighted spline approximation and applications of our estimates are also given.

This note is an addendum to my earlier paper [8]. The class of adjoint abelian operators discussed there was small because the compatibility relation between the operator and the duality map was too restrictive. (In effect, the relation is appropriate for Hilbert space, but ill-suited for other Banach spaces where the unit ball is not round.) However, the techniques introduced in [8] permit us to readily obtain a spectral theory (of the Dunford type) for a wider class of operators on Banach spaces, as we shall show.

A duality system for the operator T is an ordered sextuple

(i) T is a bounded linear operator mapping the Banach space B into B,

(ii) ϕ is a duality map from B to B*. Thus, for x ∊ B, ϕ(x) = x* ∊ B*, where ‖x‖ = ‖x*‖ and x*(x) = ‖x‖2. The existence of ϕ follows easily from the Hahn-Banach Theorem.

In this paper, some contributions are made to the theory of algebraic equations over the rational field with conditions imposed on the Galois group. In § 1, for a given abstract group G all faithful permutation representations Ḡ are obtained, and it is shown that if one of them is the group of some equation with splitting field K, then any of them is the group of some equation, also with splitting field K. The method of proof enables us to construct an equation having as group a given faithful permutation representation Ḡ of a prescribed group G if we are given an equation having as group some faithful representation of G. In § 2, equations having nilpotent group are considered, non-normal extension fields are discussed, and a canonical form is obtained for the roots of non-normal irreducible equations; this form is used to characterize fields and equations with nilpotent groups.

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 contain only $\mathbb {Z}_2$ torsion. In this paper, we prove a local version of this result. If the Khovanov homology of a link is supported on 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 three-braids, strictly containing all three-strand torus links, thus giving a partial answer to Sazdanović and Przytycki’s conjecture that three-braids have only $\mathbb {Z}_2$ torsion in Khovanov homology. We use these computations and our main theorem to obtain the integral Khovanov homology for all links in this family.

We define a natural lattice structure on all subsets of a finite root system that extends the weak order on the elements of the corresponding Coxeter group. For crystallographic root systems, we show that the subposet of this lattice induced by antisymmetric closed subsets of roots is again a lattice. We then study further subposets of this lattice that naturally correspond to the elements, the intervals, and the faces of the permutahedron and the generalized associahedra of the corresponding Weyl group. These results extend to arbitrary finite crystallographic root systems the recent results of G. Chatel, V. Pilaud, and V. Pons on the weak order on posets and its induced subposets.

A class of graphs is bridge-addable if given a graph $G$ in the class, any graph obtained by adding an edge between two connected components of $G$ is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if ${\mathcal{G}}$ is bridge-addable and $G_{n}$ is a uniform $n$-vertex graph from ${\mathcal{G}}$, then $G_{n}$ is connected with probability at least $(1+o_{n}(1))e^{-1/2}$. The constant $e^{-1/2}$ is best possible, since it is reached for the class of all forests.

In this paper, we prove a form of uniqueness in this statement: if ${\mathcal{G}}$ is a bridge-addable class and the random graph $G_{n}$ is connected with probability close to $e^{-1/2}$, then $G_{n}$ is asymptotically close to a uniform $n$-vertex random forest in a local sense. For example, if the probability converges to $e^{-1/2}$, then $G_{n}$ converges in the sense of Benjamini–Schramm to the uniformly infinite random forest $F_{\infty }$. This result is reminiscent of so-called “stability results” in extremal graph theory, the difference being that here the stable extremum is not a graph but a graph class.

We study the number of ways of factoring elements in the complex reflection groups $G(r,s,n)$ as products of reflections. We prove a result that compares factorization numbers in $G(r,s,n)$ to those in the symmetric group $S_n$ , and we use this comparison, along with the Ekedahl, Lando, Shapiro, and Vainshtein (ELSV) formula, to deduce a polynomial structure for factorizations in $G(r,s,n)$ .

Given an action ${\varphi }$ of inverse semigroup S on a ring A (with domain of ${\varphi }(s)$ denoted by $D_{s^*}$ ), we show that if the ideals $D_e$ , with e an idempotent, are unital, then the skew inverse semigroup ring $A\rtimes S$ can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of (unital) rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.

Many authors have made interesting and important contributions to the study of vector fields or infinitesimal transformations on compact orientable Riemannian manifolds and Hermitian manifolds without boundary. Recently, Hsiung (6, 7, 8) has extended some of these results to compact orientable Riemannian manifolds with boundary. The purpose of this paper is to continue Hsiung's work by studying vector fields and infinitesimal transformations on almost-Hermitian manifolds with boundary.

In this note, we are concerned with the following generalization of a wellknown theorem of M. H. Stone; see (2, 8.2).

Theorem 1. Let L be a relatively complemented distributive lattice.

(I) If L has no least element, then L is isomorphic to the lattice of non-empty compact-open subsets of an anti-Hausdorff, nearly-Hausdorff, T1-space with a base of open sets consisting of compact-open sets.

(II) (3, Theorem 1) If L has a least element, then L is isomorphic to the lattice of all compact-open subsets of a locally compact totally disconnected space. Moreover, the spaces of (I) and (II) are compact if and only if L has a greatest element.

The space in question is the space of prime ideals of L with the hull-kernel topology.

The author is indebted to M. G. Stanley for several conversations concerning this note.

We study the range of the gradients of a ${{C}^{1,\alpha }}$ -smooth bump function defined on a Banach space. We find that this set must satisfy two geometrical conditions: It can not be too flat and it satisfies a strong compactness condition with respect to an appropriate distance. These notions are defined precisely below. With these results we illustrate the differences with the case of ${{C}^{1}}$-smooth bump functions. Finally, we give a sufficient condition on a subset of ${{X}^{*}}$ so that it is the set of the gradients of a ${{C}^{1,1}}$-smooth bump function. In particular, if $X$ is an infinite dimensional Banach space with a ${{C}^{1,1}}$-smooth bump function, then any convex open bounded subset of ${{X}^{*}}$ containing 0 is the set of the gradients of a ${{C}^{1,1}}$-smooth bump function.

We strengthen, in various directions, the theorem of Garnett that every $\unicode[STIX]{x1D70E}$-compact, completely regular space $X$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $X$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $X$ of a Euclidean space, there is a compact set $K$ in some $\mathbb{C}^{N}$ so that $\widehat{K}\backslash K$ contains a Gleason part homeomorphic to $X$, and $\widehat{K}$ contains no analytic discs.

It is proved that if $G\,=\,{{G}_{1}}\,{{*}_{{{G}_{3}}}}\,{{G}_{2}}$ is free product of probability measure preserving $s$–regular ergodic discrete groupoids amalgamated over an amenable subgroupoid ${{G}_{3}}$, then the sofic dimension $s(G)$ satisfies the equality

$$s(G)=\mathfrak{h}(G_{1}^{0})s({{G}_{1}})+\mathfrak{h}(G_{2}^{0})s\left( {{G}_{2}} \right)-\,\mathfrak{h}(G_{3}^{0})s({{G}_{3}}),$$

where $\mathfrak{h}$ is the normalized Haar measure on $G$.

The Kronecker modules, $\mathbb{V}\left( m,h,\alpha \right)$, where $m$ is a positive integer, $h$ is a height function, and $\alpha $ is a $K$-linear functional on the space $K(X)$ of rational functions in one variable $X$ over an algebraically closed field $K$, are models for the family of all torsion-free rank-2 modules that are extensions of finite-dimensional rank-1 modules. Every such module comes with a regulating polynomial $f$ in $K(X)[Y]$. When the endomorphism algebra of $\mathbb{V}\left( m,h,\alpha \right)$ is commutative and non-trivial, the regulator $f$ must be quadratic in $Y$. If $f$ has one repeated root in $K(X)$, the endomorphism algebra is the trivial extension $K\ltimes S$ for some vector space $S$. If $f$ has distinct roots in $K(X)$, then the endomorphisms form a structure that we call a bridge. These include the coordinate rings of some curves. Regardless of the number of roots in the regulator, those End $\mathbb{V}\left( m,h,\alpha \right)$ that are domains have zero radical. In addition, each semi-local End $\mathbb{V}\left( m,h,\alpha \right)$ must be either a trivial extension $K\ltimes S$ or the product $K\times K$.

If (Y, µ) is an H-Space (here all our spaces are assumed to be finitely generated) with homotopy associative multiplication µ. and X is a finite CW complex then [X, Y] has the structure of a nilpotent group. Using this and the relationship between the localizations of nilpotent groups and topological spaces one can demonstrate various properties of [X,Y] (see [1], [2], [6] for example). If µ is not homotopy associative then [X, Y] has the structure of a nilpotent loop [7], [9]. However this algebraic structure is not rich enough to reflect certain significant properties of [X, Y]. Indeed, we will show that there is no theory of localization for nilpotent loops which will correspond to topological localization or will restrict to the localization of nilpotent groups.