Connections to fixed points and Sil’nikov saddle-focus homoclinic orbits in singularly perturbed systems

We consider a singularly perturbed system depending on two parameters with two (possibly the same) normally hyperbolic centre manifolds. We assume that the unperturbed system has an orbit connecting a hyperbolic fixed point on one centre manifold to a hyperbolic fixed point on the other. Then we prove some old and new results concerning the persistence of these connecting orbits and apply the results to find examples of systems in dimensions greater than three which possess Sil’nikov saddle-focus homoclinic orbits.Розглянуто сингулярно збурену систему, що залежить вiд двох параметрiв та має два (можливо, однаковi) нормально гiперболiчнi центрованi многовиди. При цьому припускається, що незбурена система має орбiту, яка поєднує гiперболiчну нерухому точку на одному центрованому многовидi з гiперболiчною нерухомою точкою на iншому. Доведено деякi вiдомi та новi результати щодо збереження цих орбiт та наведено приклади систем розмiрностi бiльше, нiж три, що мають сiдловi фокуснi гомоклiнiчнi орбiти Сiльнiкова

Exponential dichotomy for noninvertible linear difference equations: block triangular systems

In this paper, block upper triangular systems of linear difference equations are considered, in which the coefficient matrices are not assumed invertible. The relationship between the exponential dichotomy properties of such a system and its associated block diagonal system is studied. The reason it is important to study triangular systems is that any system of linear difference equations is kinematically similar to an upper triangular system. In the bounded invertible case, it is known that for equations on the intervals J = Z(+) or Z(-), a block upper triangular system has an exponential dichotomy if and only if the associated block diagonal system has one. However, when J = Z, only the sufficiency holds. The sufficiency extends to the noninvertible case, provided the off-diagonal matrices are bounded. However, the necessity does not hold even when J = Z(+) or Z(-). Nevertheless, if certain conditions are added, then the necessity does hold and it is also shown that these conditions are needed since it turns out that if both the triangular and diagonal systems have dichotomies, then these extra conditions must hold

Coal dust alters β-naphthoflavone-induced aryl hydrocarbon receptor nuclear translocation in alveolar type II cells

<p>Abstract</p> <p>Background</p> <p>Many polycyclic aromatic hydrocarbons (PAHs) can cause DNA adducts and initiate carcinogenesis. Mixed exposures to coal dust (CD) and PAHs are common in occupational settings. In the CD and PAH-exposed lung, CD increases apoptosis and causes alveolar type II (AT-II) cell hyperplasia but reduces CYP1A1 induction. Inflammation, but not apoptosis, appears etiologically associated with reduced CYP1A1 induction in this mixed exposure model. Many AT-II cells in the CD-exposed lungs have no detectable CYP1A1 induction after PAH exposure. Although AT-II cells are a small subfraction of lung cells, they are believed to be a potential progenitor cell for some lung cancers. Because CYP1A1 is induced via ligand-mediated nuclear translocation of the aryl hydrocarbon receptor (AhR), we investigated the effect of CD on PAH-induced nuclear translocation of AhR in AT-II cells isolated from <it>in vivo</it>-exposed rats. Rats received CD or vehicle (saline) by intratracheal (IT) instillation. Three days before sacrifice, half of the rats in each group started daily intraperitoneal injections of the PAH, β-naphthoflavone (BNF).</p> <p>Results</p> <p>Fourteen days after IT CD exposure and 1 day after the last intraperitoneal BNF injection, AhR immunofluorescence indicated that proportional AhR nuclear expression and the percentage of cells with nuclear AhR were significantly increased in rats receiving IT saline and BNF injections compared to vehicle controls. However, in CD-exposed rats, BNF did not significantly alter the nuclear localization or cytosolic expression of AhR compared to rats receiving CD and oil.</p> <p>Conclusion</p> <p>Our findings suggest that during particle and PAH mixed exposures, CD alters the BNF-induced nuclear translocation of AhR in AT-II cells. This provides an explanation for the modification of CYP1A1 induction in these cells. Thus, this study suggests that mechanisms for reduced PAH-induced CYP1A1 activity in the CD exposed lung include not only the effects of inflammation on the lung as a whole, but also reduced PAH-associated nuclear translocation of AhR in an expanded population of AT-II cells.</p

Extrapulmonary transport of MWCNT following inhalation exposure

Background Inhalation exposure studies of mice were conducted to determine if multi-walled carbon nanotubes (MWCNT) distribute to the tracheobronchial lymphatics, parietal pleura, respiratory musculature and/or extrapulmonary organs. Male C57BL/6 J mice were exposed in a whole-body inhalation system to a 5 mg/m3 MWCNT aerosol for 5 hours/day for 12 days (4 times/week for 3 weeks, lung burden 28.1 ug/lung). At 1 day and 336 days after the 12 day exposure period, mice were anesthetized and lungs, lymph nodes and extrapulmonary tissues were preserved by whole body vascular perfusion of paraformaldehyde while the lungs were inflated with air. Separate, clean-air control groups were studied at 1 day and 336 days post-exposure. Sirius Red stained sections from lung, tracheobronchial lymph nodes, diaphragm, chest wall, heart, brain, kidney and liver were analyzed. Enhanced darkfield microscopy and morphometric methods were used to detect and count MWCNT in tissue sections. Counts in tissue sections were expressed as number of MWCNT per g of tissue and as a percentage of total lung burden (Mean ± S.E., N = 8 mice per group). MWCNT burden in tracheobronchial lymph nodes was determined separately based on the volume density in the lymph nodes relative to the volume density in the lungs. Field emission scanning electron microscopy (FESEM) was used to examine MWCNT structure in the various tissues. Results Tracheobronchial lymph nodes were found to contain 1.08 and 7.34 percent of the lung burden at 1 day and 336 days post-exposure, respectively. Although agglomerates account for approximately 54% of lung burden, only singlet MWCNT were observed in the diaphragm, chest wall, liver, kidney, heart and brain. At one day post exposure, the average length of singlet MWCNT in liver and kidney, was comparable to that of singlet MWCNT in the lungs 8.2 ± 0.3 versus 7.5 ± 0.4 um, respectively. On average, there were 15,371 and 109,885 fibers per gram in liver, kidney, heart and brain at 1 day and 336 days post-exposure, respectively. The burden of singlet MWCNT in the lymph nodes, diaphragm, chest wall and extrapulmonary organs at 336 days post-exposure was significantly higher than at 1 day post-exposure. Conclusions Inhaled MWCNT, which deposit in the lungs, are transported to the parietal pleura, the respiratory musculature, liver, kidney, heart and brain in a singlet form and accumulate with time following exposure. The tracheobronchial lymph nodes contain high levels of MWCNT following exposure and further accumulate over nearly a year to levels that are a significant fraction of the lung burden 1 day post-exposure