39 research outputs found
Generalized DPW method and an application to isometric immersions of space forms
Let be a complex Lie group and denote the group of maps from
the unit circle into , of a suitable class. A differentiable
map from a manifold into , is said to be of \emph{connection
order } if the Fourier expansion in the loop parameter of the
-family of Maurer-Cartan forms for , namely F_\lambda^{-1}
\dd F_\lambda, is of the form . Most
integrable systems in geometry are associated to such a map. Roughly speaking,
the DPW method used a Birkhoff type splitting to reduce a harmonic map into a
symmetric space, which can be represented by a certain order map,
into a pair of simpler maps of order and respectively.
Conversely, one could construct such a harmonic map from any pair of
and maps. This allowed a Weierstrass type description
of harmonic maps into symmetric spaces. We extend this method to show that, for
a large class of loop groups, a connection order map, for ,
splits uniquely into a pair of and maps. As an
application, we show that constant non-zero curvature submanifolds with flat
normal bundle of a sphere or hyperbolic space split into pairs of flat
submanifolds, reducing the problem (at least locally) to the flat case. To
extend the DPW method sufficiently to handle this problem requires a more
general Iwasawa type splitting of the loop group, which we prove always holds
at least locally.Comment: Some typographical correction
Curved Flats, Pluriharmonic Maps and Constant Curvature Immersions into Pseudo-Riemannian Space Forms
We study two aspects of the loop group formulation for isometric immersions
with flat normal bundle of space forms. The first aspect is to examine the loop
group maps along different ranges of the loop parameter. This leads to various
equivalences between global isometric immersion problems among different space
forms and pseudo-Riemannian space forms. As a corollary, we obtain a
non-immersibility theorem for spheres into certain pseudo-Riemannian spheres
and hyperbolic spaces.
The second aspect pursued is to clarify the relationship between the loop
group formulation of isometric immersions of space forms and that of
pluriharmonic maps into symmetric spaces. We show that the objects in the first
class are, in the real analytic case, extended pluriharmonic maps into certain
symmetric spaces which satisfy an extra reality condition along a totally real
submanifold. We show how to construct such pluriharmonic maps for general
symmetric spaces from curved flats, using a generalised DPW method.Comment: 21 Pages, reference adde
Tzitz\'eica transformation is a dressing action
We classify the simplest rational elements in a twisted loop group, and prove
that dressing actions of them on proper indefinite affine spheres give the
classical Tzitz\'eica transformation and its dual. We also give the group point
of view of the Permutability Theorem, construct complex Tzitz\'eica
transformations, and discuss the group structure for these transformations
Umbral Calculus, Discretization, and Quantum Mechanics on a Lattice
`Umbral calculus' deals with representations of the canonical commutation
relations. We present a short exposition of it and discuss how this calculus
can be used to discretize continuum models and to construct representations of
Lie algebras on a lattice. Related ideas appeared in recent publications and we
show that the examples treated there are special cases of umbral calculus. This
observation then suggests various generalizations of these examples. A special
umbral representation of the canonical commutation relations given in terms of
the position and momentum operator on a lattice is investigated in detail.Comment: 19 pages, Late
A Replication Study of GWAS-Derived Lipid Genes in Asian Indians: The Chromosomal Region 11q23.3 Harbors Loci Contributing to Triglycerides
Recent genome-wide association scans (GWAS) and meta-analysis studies on European populations have identified many genes previously implicated in lipid regulation. Validation of these loci on different global populations is important in determining their clinical relevance, particularly for development of novel drug targets for treating and preventing diabetic dyslipidemia and coronary artery disease (CAD). In an attempt to replicate GWAS findings on a non-European sample, we examined the role of six of these loci (CELSR2-PSRC1-SORT1 rs599839; CDKN2A-2B rs1333049; BUD13-ZNF259 rs964184; ZNF259 rs12286037; CETP rs3764261; APOE-C1-C4-C2 rs4420638) in our Asian Indian cohort from the Sikh Diabetes Study (SDS) comprising 3,781 individuals (2,902 from Punjab and 879 from the US). Two of the six SNPs examined showed convincing replication in these populations of Asian Indian origin. Our study confirmed a strong association of CETP rs3764261 with high-density lipoprotein cholesterol (HDL-C) (p = 2.03×10−26). Our results also showed significant associations of two GWAS SNPs (rs964184 and rs12286037) from BUD13-ZNF259 near the APOA5-A4-C3-A1 genes with triglyceride (TG) levels in this Asian Indian cohort (rs964184: p = 1.74×10−17; rs12286037: p = 1.58×10−2). We further explored 45 SNPs in a ∼195 kb region within the chromosomal region 11q23.3 (encompassing the BUD13-ZNF259, APOA5-A4-C3-A1, and SIK3 genes) in 8,530 Asian Indians from the London Life Sciences Population (LOLIPOP) (UK) and SDS cohorts. Five more SNPs revealed significant associations with TG in both cohorts individually as well as in a joint meta-analysis. However, the strongest signal for TG remained with BUD13-ZNF259 (rs964184: p = 1.06×10−39). Future targeted deep sequencing and functional studies should enhance our understanding of the clinical relevance of these genes in dyslipidemia and hypertriglyceridemia (HTG) and, consequently, diabetes and CAD
Bezlotoxumab for Prevention of Recurrent Clostridium difficile Infection
BACKGROUND Clostridium difficile is the most common cause of infectious diarrhea in hospitalized patients. Recurrences are common after antibiotic therapy. Actoxumab and bezlotoxumab are human monoclonal antibodies against C. difficile toxins A and B, respectively. METHODS We conducted two double-blind, randomized, placebo-controlled, phase 3 trials, MODIFY I and MODIFY II, involving 2655 adults receiving oral standard-of-care antibiotics for primary or recurrent C. difficile infection. Participants received an infusion of bezlotoxumab (10 mg per kilogram of body weight), actoxumab plus bezlotoxumab (10 mg per kilogram each), or placebo; actoxumab alone (10 mg per kilogram) was given in MODIFY I but discontinued after a planned interim analysis. The primary end point was recurrent infection (new episode after initial clinical cure) within 12 weeks after infusion in the modified intention-to-treat population. RESULTS In both trials, the rate of recurrent C. difficile infection was significantly lower with bezlotoxumab alone than with placebo (MODIFY I: 17% [67 of 386] vs. 28% [109 of 395]; adjusted difference, −10.1 percentage points; 95% confidence interval [CI], −15.9 to −4.3; P<0.001; MODIFY II: 16% [62 of 395] vs. 26% [97 of 378]; adjusted difference, −9.9 percentage points; 95% CI, −15.5 to −4.3; P<0.001) and was significantly lower with actoxumab plus bezlotoxumab than with placebo (MODIFY I: 16% [61 of 383] vs. 28% [109 of 395]; adjusted difference, −11.6 percentage points; 95% CI, −17.4 to −5.9; P<0.001; MODIFY II: 15% [58 of 390] vs. 26% [97 of 378]; adjusted difference, −10.7 percentage points; 95% CI, −16.4 to −5.1; P<0.001). In prespecified subgroup analyses (combined data set), rates of recurrent infection were lower in both groups that received bezlotoxumab than in the placebo group in subpopulations at high risk for recurrent infection or for an adverse outcome. The rates of initial clinical cure were 80% with bezlotoxumab alone, 73% with actoxumab plus bezlotoxumab, and 80% with placebo; the rates of sustained cure (initial clinical cure without recurrent infection in 12 weeks) were 64%, 58%, and 54%, respectively. The rates of adverse events were similar among these groups; the most common events were diarrhea and nausea. CONCLUSIONS Among participants receiving antibiotic treatment for primary or recurrent C. difficile infection, bezlotoxumab was associated with a substantially lower rate of recurrent infection than placebo and had a safety profile similar to that of placebo. The addition of actoxumab did not improve efficacy. (Funded by Merck; MODIFY I and MODIFY II ClinicalTrials.gov numbers, NCT01241552 and NCT01513239.
Complex surfaces of constant mean curvature fibered by minimal surfaces
We define complex constant mean curvature immersions in complex three space using a natural extension of constant mean curvature immersions in Euclidean three space via loop group techniques. We then discuss the fundamental properties of these complex constant mean curvature immersions. In particular, we prove that these immersions are doubly ruled by holomorphic null curves. We present a construction of minimal immersions from constant mean curvature immersions in Euclidean three space via the associated complex constant mean curvature immersions
Recommended from our members