Levi-Civita connections for conformally deformed metrics on tame differential calculi

Given a tame differential calculus over a noncommutative algebra [Formula: see text] and an [Formula: see text]-bilinear metric [Formula: see text] consider the conformal deformation [Formula: see text] [Formula: see text] being an invertible element of [Formula: see text] We prove that there exists a unique connection [Formula: see text] on the bimodule of one-forms of the differential calculus which is torsionless and compatible with [Formula: see text] We derive a concrete formula connecting [Formula: see text] and the Levi-Civita connection for the metric [Formula: see text] As an application, we compute the Ricci and scalar curvatures for a general conformal perturbation of the canonical metric on the noncommutative [Formula: see text]-torus as well as for a natural metric on the quantum Heisenberg manifold. For the latter, the scalar curvature turns out to be a negative constant.

A description of Rasmussen’s invariant from the divisibility of Lee’s canonical class

We give a description of Rasmussen’s [Formula: see text]-invariant from the divisibility of Lee’s canonical class. More precisely, given any link diagram [Formula: see text], for any choice of an integral domain [Formula: see text] and a non-zero, non-invertible element [Formula: see text], we define the [Formula: see text]-divisibility [Formula: see text] of Lee’s canonical class of [Formula: see text], and prove that a combination of [Formula: see text] and some elementary properties of [Formula: see text] yields a link invariant [Formula: see text]. Each [Formula: see text] possesses properties similar to [Formula: see text], which in particular reproves the Milnor conjecture. If we restrict to knots and take [Formula: see text], then our invariant coincides with [Formula: see text].

A class of linear codes of length 2 over finite chain rings

Let [Formula: see text] be a finite field of cardinality [Formula: see text], where [Formula: see text] is an odd prime, [Formula: see text] be positive integers satisfying [Formula: see text], and denote [Formula: see text], where [Formula: see text] is an irreducible polynomial in [Formula: see text]. In this note, for any fixed invertible element [Formula: see text], we present all distinct linear codes [Formula: see text] over [Formula: see text] of length [Formula: see text] satisfying the condition: [Formula: see text] for all [Formula: see text]. This conclusion can be used to determine the structure of [Formula: see text]-constacyclic codes over the finite chain ring [Formula: see text] of length [Formula: see text] for any positive integer [Formula: see text] satisfying [Formula: see text].

Behavior of the Auslander condition with respect to regradings

We show that a noetherian ring graded by an abelian group of finite rank satisfies the Auslander condition if and only if it satisfies the graded Auslander condition. In addition, we also study the injective dimension, the global dimension and the Cohen–Macaulay property from the same perspective as that for the Auslander condition. A key step of our approach is to establish homological relations between a graded ring [Formula: see text], its quotient ring modulo the ideal [Formula: see text] and its localization ring with respect to the Ore set [Formula: see text], where [Formula: see text] is a homogeneous regular normal non-invertible element of [Formula: see text].

The nilpotence degree of quantum Lie nilpotent algebras

We consider the quantum analog of the Lie commutator [Formula: see text] for an invertible element [Formula: see text] of the ground field and prove lower and upper bounds for the nilpotence degree of an associative algebra satisfying an identity of the form [Formula: see text].

Identification and Characterization of a Phase-Variable Element That Regulates the Autotransporter UpaE in UropathogenicEscherichia coli

ABSTRACTUropathogenicEscherichia coli(UPEC) is the most common etiologic agent of uncomplicated urinary tract infection (UTI). An important mechanism of gene regulation in UPEC is phase variation that involves inversion of a promoter-containing DNA element via enzymatic activity of tyrosine recombinases, resulting in biphasic, ON or OFF expression of target genes. The UPEC reference strain CFT073 has five tyrosine site-specific recombinases that function at two previously characterized promoter inversion systems,fimSandhyxS. Three of the five recombinases are located proximally to their cognate target elements, which is typical of promoter inversion systems. The genes for the other two recombinases, IpuA and IpuB, are located distal from these sites. Here, we identified and characterized a third phase-variable invertible element in CFT073,ipuS, located proximal toipuAandipuB. The inversion ofipuSis catalyzed by four of the five CFT073 recombinases. Orientation of the element drives transcription of a two-gene operon containingipuR, a predicted LuxR-type regulator, andupaE, a predicted autotransporter. We show that the predicted autotransporter UpaE is surface located and facilitates biofilm formation as well as adhesion to extracellular matrix proteins in a K-12 recombinant background. Consistent with this phenotype, theipuSON condition in CFT073 results in defective swimming motility, increased adherence to human kidney epithelial cells, and a positive competitive kidney colonization advantage in experimental mouse UTIs. Overall, the identification of a third phase switch in UPEC that is regulated by a shared set of recombinases describes a complex phase-variable virulence network in UPEC.IMPORTANCEUropathogenicEscherichia coli(UPEC) is the most common cause of urinary tract infection (UTI). ON versus OFF phase switching by inversion of small DNA elements at two chromosome sites in UPEC regulates the expression of important virulence factors, including the type 1 fimbria adhesion organelle. In this report, we describe a third invertible element,ipuS, in the UPEC reference strain CFT073. The inversion ofipuScontrols the phase-variable expression ofupaE, an autotransporter gene that encodes a surface protein involved in adherence to extracellular matrix proteins and colonization of the kidneys in a murine model of UTI.

Identification and characterization of a phase-variable element that regulates the autotransporter UpaE in uropathogenicEscherichia coli

UropathogenicEscherichia coli(UPEC) are the most common etiological agent of uncomplicated urinary tract infection (UTI). An important mechanism of gene regulation in UPEC is phase variation that involves inversion of a promoter-containing DNA element via enzymatic activity of tyrosine recombinases, resulting in biphasic, ON or OFF expression of target genes. The UPEC reference strain CFT073 has five tyrosine site-specific recombinases that function at two previously characterized promoter inversion systems,fimSandhyxS. Three of the five recombinases are located proximally to their cognate target elements, which is typical of promoter inversion systems. The genes for the other two recombinases, IpuA and IpuB are located distal from these sites. Here, we identified and characterized a third phase variable invertible element in CFT073,ipuSlocated proximal toipuAandipuB. The inversion ofipuSis catalyzed by four of the five CFT073 recombinases. Orientation of the element drives transcription of a two-gene operon containingipuR, a predicted LuxR-type regulator, andupaE, a predicted autotransporter. We show that the predicted autotransporter UpaE is surface-located and facilitates biofilm formation as well as adhesion to extracellular matrix proteins in a K-12 recombinant background. Consistent with this phenotype, theipuSON condition in CFT073 results in defective swimming motility, increased adherence to human kidney epithelial cells, and a positive competitive kidney colonization advantage in experimental mouse UTI infections. Overall, the identification of a third phase-switch in UPEC that is regulated by a shared set of recombinases describes a complex phase-variable virulence network in UPEC.ImportanceUropathogenicEscherichia coli(UPEC) is the most common cause of urinary tract infection (UTI). ON versus OFF phase-switching by inversion of small DNA elements at two chromosome sites in UPEC regulates the expression of important virulence factors, including the type 1 fimbriae adhesion organelle. In this report, we describe a third invertible element,ipuS, in the UPEC reference strain CFT073. The inversion ofipuScontrols the phase variable expression ofupaE, an autotransporter gene that encodes a surface protein involved in adherence to extracellular matrix proteins and colonization of the kidneys in a murine model of UTI.

On some deddens subspaces of Banach algebras

Let A be a Banach algebra with a unit e, and let a ? A be an invertible element. We define the following algebra: Bloca := n {x ? A :??anxa-n?? ? cxn?(x) for some ?(x) ? 0 and cx > 0}. In this articlewestudy some properties of this algebra; in particular, weprove that Bloce+p = {x ? A : px (e-p) = 0}, where p is an idempotent in A. We also investigate the following Deddens subspace. Let a,b ? A be two elements. Fix any number ?,0 ? ? < 1, and consider the following subspace of A : D? a,b := {x ? A : ??anxbn?? = O(n?) as n ? ?}. Here we study some properties of the subspaces D?a,b and D?b,a.