Normality and Short Exact Sequences of Hopf-Galois Structures

Every Hopf-Galois structure on a finite Galois extension K/k where G = Gal(K/k) corresponds uniquely to a regular subgroup N ≤ B = Perm(G), normalized by λ(G) ≤ B, in accordance with a theorem of Greither and Pareigis. The resulting Hopf algebra which acts on K/k is HN = (K[N])λ(G). For a given such N we consider the Hopf-Galois structure arising from a subgroup P ⊳ N that is also normalized by λ(G). This subgroup gives rise to a Hopf sub-algebra HP ⊆ HN with fixed field F = KHP . By the work of Chase and Sweedler, this yields a Hopf-Galois structure on the extension K/F where the action arises by base changing HP to F ⊗k HP which is an F-Hopf algebra. We examine this analogy with classical Galois theory, and also examine how the Hopf-Galois structure on K/F relates to that on K/k. We will also pay particular attention to how the Greither-Pareigis enumeration/construction of those HP acting on K/F relates to that of the HN which act on K/k. In the process we also examine short exact sequences of the Hopf algebras which act, whose exactness is directly tied to the descent theoretic description of these algebras

Isomorphism problems for Hopf-Galois structures on separable field extensions

Let L=K be a finite separable extension of fields whose Galois closure E=K has Galois group G. Greither and Pareigis use Galois descent to show that a Hopf algebra giving a Hopf-Galois structure on L=K has the form E[N]G for some group N of order [L : K]. We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as K-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and K-algebras that appear in the classification of Hopf-Galois structures on a cyclic extension of degree pn, for p an odd prime number

Tracing <i>Mycobacterium tuberculosi</i>s transmission by whole genome sequencing in a high incidence setting:A retrospective population-based study in East Greenland

In East Greenland, a dramatic increase of tuberculosis (TB) incidence has been observed in recent years. Classical genotyping suggests a genetically similar Mycobacterium tuberculosis (Mtb) strain population as cause, however, precise transmission patterns are unclear. We performed whole genome sequencing (WGS) of Mtb isolates from 98% of culture-positive TB cases through 21 years (n = 182) which revealed four genomic clusters of the Euro-American lineage (mainly sub-lineage 4.8 (n = 134)). The time to the most recent common ancestor of lineage 4.8 strains was found to be 100 years. This sub-lineage further diversified in the 1970s, and massively expanded in the 1990s, a period of lowered TB awareness in Greenland. Despite the low genetic strain diversity, WGS data revealed several recent short-term transmission events in line with the increasing incidence in the region. Thus, the isolated setting and the uniformity of circulating Mtb strains indicated that the majority of East Greenlandic TB cases originated from one or few strains introduced within the last century. Thereby, the study shows the consequences of even short interruptions in TB control efforts in previously TB high incidence areas and demonstrates the potential role of WGS in detecting ongoing micro epidemics, thus guiding public health efforts in the future

The Structure of Hopf Algebras Acting on Dihedral Extensions

We discuss isomorphism questions concerning the Hopf algebras that yield Hopf–Galois structures for a fixed separable field extension L/K. We study in detail the case where L/K is Galois with dihedral group Dp, p≥3 prime and give explicit descriptions of the Hopf algebras which act on L/K. We also determine when two such Hopf algebras are isomorphic, either as Hopf algebras or as algebras. For the case p=3 and a chosen L/K, we give the Wedderburn–Artin decompositions of the Hopf algebras

Self consistent and covariant propagation of pions, nucleon and isobar resonances in cold nuclear matter

We evaluate the in-medium spectral functions for pions, nucleon and isobar resonances in a self consistent and covariant manner. The calculations are based on a recently developed formulation which leads to predictions in terms of the pion-nucleon scattering phase shifts and a set of Migdal parameters describing important short range correlation effects. We do not observe significant softening of pion modes if we insist on reasonable isobar resonance properties but predict a considerable broadening of the N(1440) and N(1520) resonances in nuclear matter. Contrasted results are obtained for the s-wave N(1535) and N(1650) resonances which are affected by a nuclear environment very little. The properties of slowly moving isobar's in nuclear matter are found to depend very sensitively on a soft form factor in the piNN vertex, which is not controlled by the piN scattering data.Comment: 20 pages, 7 figure, revised manuscrip

Self energies of the pion and the delta isobar from the ^3He(e,e'pi^+)^3H reaction

In a kinematically complete experiment at the Mainz microtron MAMI, pion angular distributions of the $^3$He(e,e'$\pi^+)^3$H reaction have been measured in the excitation region of the $\Delta$ resonance to determine the longitudinal ($L$), transverse ($T$), and the $LT$ interference part of the differential cross section. The data are described only after introducing self-energy modifications of the pion and $\Delta$-isobar propagators. Using Chiral Perturbation Theory (ChPT) to extrapolate the pion self energy as inferred from the measurement on the mass shell, we deduce a reduction of the $\pi^+$ mass of $\Delta m_{\pi^+} = (-1.7^{+ 1.7}_{- 2.1})$ MeV/c$^2$ in the neutron-rich nuclear medium at a density of $\rho = (0.057^{+ 0.085}_{- 0.057})$ fm$^{-3}$. Our data are consistent with the $\Delta$ self energy determined from measurements of $\pi^0$ photoproduction from $^4$He and heavier nuclei.Comment: Elsart, 12 pages and 4 figures, Correspondent: Professor Dr. Dr. h.c. mult. Achim Richter, [email protected], submitted to Phys. Rev. Let

Informal Caregiving in Amyotrophic Lateral Sclerosis (ALS): A High Caregiver Burden and Drastic Consequences on Caregivers’ Lives

Amyotrophic lateral sclerosis (ALS) is a fatal neurodegenerative disease that causes progressive autonomy loss and need for care. This does not only affect patients themselves, but also the patients’ informal caregivers (CGs) in their health, personal and professional lives. The big efforts of this multi-center study were not only to evaluate the caregivers’ burden and to identify its predictors, but it also should provide a specific understanding of the needs of ALS patients’ CGs and fill the gap of knowledge on their personal and work lives. Using standardized questionnaires, primary data from patients and their main informal CGs (n = 249) were collected. Patients’ functional status and disease severity were evaluated using the Barthel Index, the revised Amyotrophic Lateral Sclerosis Functional Rating Scale (ALSFRS-R) and the King’s Stages for ALS. The caregivers’ burden was recorded by the Zarit Burden Interview (ZBI). Comorbid anxiety and depression of caregivers were assessed by the Hospital Anxiety and Depression Scale. Additionally, the EuroQol Five Dimension Five Level Scale evaluated their health-related quality of life. The caregivers’ burden was high (mean ZBI = 26/88, 0 = no burden, ≥24 = highly burdened) and correlated with patients’ functional status (rp = −0.555, p < 0.001, n = 242). It was influenced by the CGs’ own mental health issues due to caregiving (+11.36, 95% CI [6.84; 15.87], p < 0.001), patients’ wheelchair dependency (+9.30, 95% CI [5.94; 12.66], p < 0.001) and was interrelated with the CGs’ depression (rp = 0.627, p < 0.001, n = 234), anxiety (rp = 0.550, p < 0.001, n = 234), and poorer physical condition (rp = −0.362, p < 0.001, n = 237). Moreover, female CGs showed symptoms of anxiety more often, which also correlated with the patients’ impairment in daily routine (rs = −0.280, p < 0.001, n = 169). As increasing disease severity, along with decreasing autonomy, was the main predictor of caregiver burden and showed to create relevant (negative) implications on CGs’ lives, patient care and supportive therapies should address this issue. Moreover, in order to preserve the mental and physical health of the CGs, new concepts of care have to focus on both, on not only patients but also their CGs and gender-associated specific issues. As caregiving in ALS also significantly influences the socioeconomic status by restrictions in CGs’ work lives and income, and the main reported needs being lack of psychological support and a high bureaucracy, the situation of CGs needs more attention. Apart from their own multi-disciplinary medical and psychological care, more support in care and patient management issues is required

In Vivo Analysis of Disease-Associated Point Mutations Unveils Profound Differences in mRNA Splicing of Peripherin-2 in Rod and Cone Photoreceptors

Point mutations in peripherin-2 (PRPH2) are associated with severe retinal degenerative disorders affecting rod and/or cone photoreceptors. Various disease-causing mutations have been identified, but the exact contribution of a given mutation to the clinical phenotype remains unclear. Exonic point mutations are usually assumed to alter single amino acids, thereby influencing specific protein characteristics;however, they can also affect mRNA splicing. To examine the effects of distinct PRPH2 point mutations on mRNA splicing and protein expression in vivo, we designed PRPH2 minigenes containing the three coding exons and relevant intronic regions of human PRPH2. Minigenes carrying wild type PRPH2 or PRPH2 exon 2 mutations associated with rod or cone disorders were expressed in murine photoreceptors using recombinant adeno-associated virus (rAAV) vectors. We detect three PRPH2 splice isoforms in rods and cones: correctly spliced, intron 1 retention, and unspliced. In addition, we show that only the correctly spliced isoform results in detectable protein expression. Surprisingly, compared to rods, differential splicing leads to lower expression of correctly spliced and higher expression of unspliced PRPH2 in cones. These results were confirmed in qRT-PCR experiments from FAC-sorted murine rods and cones. Strikingly, three out of five cone disease-causing PRPH2 mutations profoundly enhanced correct splicing of PRPH2, which correlated with strong upregulation of mutant PRPH2 protein expression in cones. By contrast, four out of six PRPH2 mutants associated with rod disorders gave rise to a reduced PRPH2 protein expression via different mechanisms. These mechanisms include aberrant mRNA splicing, protein mislocalization, and protein degradation. Our data suggest that upregulation of PRPH2 levels in combination with defects in the PRPH2 function caused by the mutation might be an important mechanism leading to cone degeneration. By contrast, the pathology of rod-specific PRPH2 mutations is rather characterized by PRPH2 downregulation and impaired protein localization