Abstract:
The biochemical indicators of wistar rat under low oxygen concentration, such as brain water content, necrosis, lactic acid and Na+-K+-ATPase, was detected to evaluate normobaric hypoxia-induced brain damage and to investigate the mechanism of wistar rat brain injury. Histopathological changes in brain tissue induced by hypoxia were investigated via hematoxylin and eosin stain (HE). Hypoxia induced factor-1α (HIF-1α) expression in brain was confirmed using immunohistochemistry. The results showed that the level of lactic acid was positively correlated with the degree of hypoxia, while concentration-dependent decrease in total Na+-K+-ATPase activity was observed. Compared with the control group, hypoxia group had a significant difference on brain water content under severe hypoxic conditions, the rate of brain necrosis increased obviously, followed by the increase of lactic acid level and the decrease of Na+-K+-ATPase activity. Histopathological analysis of brain confirmed that there was neuronal cell death in hippocampal gyrus. HIF-1α expression enhanced the hypoxia adaptation capability of the rat model through regulating the expressions of multiple genes. Lactic acid, Na+-K+-ATPase and HIF- 1α played an important role in brain injury as a possible mechanism.

Abstract:
The title compound, C15H17FO, was prepared directly from the aldol condensation of cyclooctanone with 4-fluorobenzaldehyde, catalysed by Pd(Ni,Ce) in the presence of trimethylsilyl chloride. The eight-membered ring adopts a boat-chair conformation.

Abstract:
In this paper, we give a precise description of the rescaling behaviors of global strong polynomial solutions to the reformulation of zero surface tension Hele-Shaw problem driven by injection, the Polubarinova-Galin equation, in terms of Richardson complex moments. From past results, we know that this set of solutions is large. This method can also be applied to zero surface tension Stokes flow driven by injection and a rescaling behavior is given in terms of many conserved quantities as well.

Abstract:
In this work, we give a perturbation theorem for strong polynomial solutions to the zero surface tension Hele-Shaw equation driven by injection or suction, so called the Polubarinova-Galin equation. This theorem enables us to explore properties of solutions with initial functions close to but are not polynomial. Applications of this theorem are given in the suction or injection case. In the former case, we show that if the initial domain is close to a disk, most of fluid will be sucked before the strong solution blows up. In the later case, we obtain precise large-time rescaling behaviors for large data to Hele-Shaw flows in terms of invariant Richardson complex moments. This rescaling behavior result generalizes a recent result regarding large-time rescaling behavior for small data in terms of moments. As a byproduct of a theorem in this paper, a short proof of existence and uniqueness of strong solutions to the Polubarinova-Galin equation is given.

Abstract:
The Polubarinova-Galin equation is derived from zero-surface-tension Hele-Shaw flow driven by injection. In this paper, we classify degree three polynomial solutions to the Polubarinova-Galin equation into three categories: global solutions, solutions which can be continued after blow-up and solutions which cannot be continued after blow-up. The coefficient region of the initial functions in each category is obtained.

Abstract:
This note gives two results neighboring on Fermat's last theorem that in principle induce elementary proofs of Fermat's last theorem; the first one gives a necessary condition for Fermat's last theorem to be false, and the second one divides Fermat's last theorem into exactly three cases and proves one of those.

Abstract:
We study non-univalent solutions of the Polubarinova-Galin equation, describing the time evolution of the conformal map from the unit disk onto a Hele-Shaw blob of fluid subject to injection at one point. In particular, we tackle the difficulties arising when the map is not even locally univalent, in which case one has to pass to weak solutions developing on a branched covering surface of the complex plane. One major concern is the construction of this Riemann surface, which is not given in advance but has to be constantly up-dated along with the solution. Once the Riemann surface is constructed the weak solution is automatically global in time, but we have had to leave open the question whether the weak solution can be kept simply connected all the time (as is necessary to connect to the Polubarinova-Galin equation). A certain crucial statement, a kind of stability statement for free boundaries, has therefore been left as a conjecture only. Another major part of the paper concerns the structure of rational solutions (as for the derivative of the mapping function). Here we have fairly complete results on the dynamics. Several examples are given.

Abstract:
We study the dynamics of roots of f'(z,t), where f(z,t) is a locally univalent polynomial solution of the Polubarinova-Galin equation for the evolution of the conformal map onto a Hele-Shaw blob subject to injection at one point. We give examples of the sometimes complicated motion of roots, but show also that the asymptotic behavior is simple. More generally we allow f'(z,t) to be a rational function and give sharp estimates for the motion of poles and for the decay of the Taylor coefficients. We also prove that any global in time locally univalent solution actually has to be univalent.