Eta cocycles, relative pairings and the Godbillon-Vey index theorem

We prove a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary-foliation; this is a secondary invariant for longitudinal Dirac operators on type-III foliations. Moreover, employing the Godbillon-Vey index as a pivotal example, we explain a new approach to higher index theory on geometric structures with boundary. This is heavily based on the interplay between the absolute and relative pairings of K-theory and cyclic cohomology for an exact sequence of Banach algebras which in the present context takes the form $0\to J\to A\to B\to 0$, with J dense and holomorphically closed in the C^*-algebra of the foliation and B depending only on boundary data. Of particular importance is the definition of a relative cyclic cocycle $(\tau_{GV}^r,\sigma_{GV})$ for the pair $A\to B$; $\tau_{GV}^r$ is a cyclic cochain on A defined through a regularization, \`a la Melrose, of the usual Godbillon-Vey cyclic cocycle $\tau_{GV}$; $\sigma_{GV}$ is a cyclic cocycle on B, obtained through a suspension procedure involving $\tau_{GV}$ and a specific 1-cyclic cocycle (Roe's 1-cocycle). We call $\sigma_{GV}$ the eta cocycle associated to $\tau_{GV}$. The Atiyah-Patodi-Singer formula is obtained by defining a relative index class \Ind (D,D^\partial)\in K_* (A,B) and establishing the equality =<\Ind (D,D^\partial), [\tau^r_{GV}, \sigma_{GV}]>$. The Godbillon-Vey eta invariant$\eta_{GV}$is obtained through the eta cocycle$\sigma_{GV}$.Comment: 86 pages. This is the complete article corresponding to the announcement "Eta cocycles" by the same authors (arXiv:0907.0173

Pulmonary stretch receptor activity during partial liquid ventilation in cats with healthy lungs

Aim: To study whether pulmonary stretch receptor (PSR) activity in mechanically ventilated young cats with healthy lungs during partial liquid ventilation (PLV) is different from that during gas ventilation (GV). Methods: In 10 young cats (4.4 +/- 0.4 months, 2.3 +/- 0.3 kg; mean B SD), PSR instantaneous impulse frequency (PSR f(imp)) was recorded from single fibres in the vagal nerve during GV and PLV with perfluorocarbon (30 ml/kg) at increasing positive inspiratory pressures (PIP; 1.2, 1.8, 2.2 and 2.7 kPa), and at a positive end-expiratory pressure of 0.5 kPa. Results: All PSRs studied during GV maintained their phasic character with increased impulse frequency during inspiration during PLV. Peak PSR fimp was lower at PIP 1.2 kPa (p < 0.05) and at PIP 2.7 kPa (p = 0.10) during PLV than during GV, giving a lower number of PSR impulses at these two settings during PLV (p < 0.05). Conclusion: The phasic character of PSR activity is similar during GV and PLV. PSR activity is not higher during PLV than during GV in cats with healthy lungs, indicating no extensive stretching of the lung during PLV. Copyright (C) 2004 S. Karger AG, Basel

The relationship between glycaemic variability and cardiovascular autonomic dysfunction in patients with type 1 diabetes : a systematic review

Rigorous glycaemic control-reflected by low HbA1c goals-is of the utmost importance in the prevention and management of complications in patients with type 1 diabetes mellitus (T1DM). However, previous studies suggested that short-term glycaemic variability (GV) is also important to consider as excessive glucose fluctuations may have an additional impact on the development of diabetic complications. The potential relationship between GV and the risk of cardiovascular autonomic neuropathy (CAN), a clinical expression of cardiovascular autonomic dysfunction, is of increasing interest. This systematic review aimed to summarize existing evidence concerning the relationship between GV and cardiovascular autonomic dysfunction in T1DM. An electronic database search of Medline (PubMed), Web of Science and Embase was performed up to October 2019. There were no limits concerning year of publication. Methodological quality was evaluated using the Newcastle Ottawa Scale for observational studies. Six studies (four cross-sectional and two prospective cohorts) were included. Methodological quality of the studies varied from level C to A2. Two studies examined the association between GV and heart rate variability (HRV), and both found significant negative correlations. Regarding cardiovascular autonomic reflex tests (CARTs), two studies did not, while two other studies did find significant associations between GV parameters and CART scores. However, associations were attenuated after adjusting for covariates such as HbA1c, age and disease duration. In conclusion, this systematic review found some preliminary evidence supporting an association between GV and cardiovascular autonomic dysfunction in T1DM. Hence, uncertainty remains whether high GV can independently contribute to the onset or progression of CAN. The heterogeneity in the methodological approach made it difficult to compare different studies. Future studies should therefore use uniformly evaluated continuous glucose monitoring-derived parameters of GV, while standardized assessment of HRV, CARTs and other potential cardiac autonomic function parameters is needed for an unambiguous definition of CAN

Algebraic geometry codes with complementary duals exceed the asymptotic Gilbert-Varshamov bound

It was shown by Massey that linear complementary dual (LCD for short) codes are asymptotically good. In 2004, Sendrier proved that LCD codes meet the asymptotic Gilbert-Varshamov (GV for short) bound. Until now, the GV bound still remains to be the best asymptotical lower bound for LCD codes. In this paper, we show that an algebraic geometry code over a finite field of even characteristic is equivalent to an LCD code and consequently there exists a family of LCD codes that are equivalent to algebraic geometry codes and exceed the asymptotical GV bound