Abstract:
"nTo date, there have been no curative drugs for inflammatory bowel disease (IBD). Conventional drugs and biologic agents are not always effective and may cause serious side effects. Therefore, it is still challenging to develop effective and safe novel drugs for IBD. Although the exact etiology of IBD remains elusive, it is generally accepted that the immune system of the gut plays a central role in the pathogenesis of IBD. Recently, the nuclear transcription factor kappa B (NF-κB) has been identified as the pivotal elements in the regulation of the increased inflammatory activity. Moreover, recent studies have shown that Cordyceps pruinosa extract is a inhibitor of NF- κB activation and can enhance weak immune functions. Based on these facts, I hypothesize that Cordyceps pruinosa extract may thus exert its therapeutic effect on IBD by regulating NF-κB activity and improving impaired immune functions.

Abstract:
Cropland soil is an important source of atmospheric nitric oxide (NO) and ammonia (NH3). Chinese croplands are characterized by intensive management, but limited information is available with regard to NO emissions from croplands in China and NH3 emissions in south China. In this study, a mesocosm experiment was conducted to measure NO and NH3 emissions from a typical vegetable-land soil in the Pearl River Delta following the applications of 150 kg N ha？1 as urea, ammonium nitrate (AN) and ammonium bicarbonate (ABC), respectively. Over the sampling period after fertilization (72 days for NO and 39 days for NH3), mean NO fluxes (± standard error of three replicates) in the control and urea, AN and ABC fertilized mesocosms were 10.9±0.9, 73.1±2.9, 63.9±1.8 and 66.0±4.0 ng N m？2 s？1, respectively; mean NH3 fluxes were 8.9±0.2, 493.6±4.4, 144.8±0.1 and 684.7±8.4 ng N m？2 s？1, respectively. The fertilizer-induced NO emission factors for urea, AN and ABC were 2.6±0.1%, 2.2±0.1% and 2.3±0.2%, respectively. The fertilizer-induced NH3 emission factors for the three fertilizers were 10.9±0.2%, 3.1±0.1% and 15.2±0.4%, respectively. From the perspective of air quality protection, it would be better to increase the proportion of AN application due to its lower emission factors for both NO and NH3.

Abstract:
We consider the It\^{o} SDE with non-degenerate diffusion coefficient and measurable drift coefficient. Under the condition that the gradient of the diffusion coefficient and the divergences of the diffusion and drift coefficients are exponentially integrable with respect to the Gaussian measure, we show that the stochastic flow leaves the reference measure absolutely continuous.

Abstract:
We consider stochastic differential equations on the group of volume-preserving homeomorphisms of the sphere $S^d\,(d\geq 2)$. The diffusion part is given by the divergence free eigenvector fields of the Laplacian acting on $L^2$-vector fields, while the drift is some other divergence free vector field. We show that the equation generates a unique flow of measure-preserving homeomorphisms when the drift has first order Sobolev regularity, and derive a formula for the distance between two Lagrangian flows. We also compute the rotation process of two particles on the sphere $S^2$ when they are close to each other.

Abstract:
This paper considers a mortgage contract where the borrower pays a fixed mortgage rate and has the choice of making prepayment. Assume the market interest follows the CIR model, a free boundary problem is formulated. Here we focus on the infinite horizon problem. Using variational method, we obtain an analytical solution to the problem, where the free boundary is implicitly given by a transcendental algebraic equation.

Abstract:
In this paper, some new characterizations on Gorenstein projective, injective and flat modules over commutative noetherian local ring are given. For instance, it is shown that an $R$-module $M$ is Gorenstein projective if and only if the Matlis dual $\text{Hom}_R(M,E(k))$ belongs to Auslander category $\mathcal{B}(\widehat{R})$ and $\text{Ext}^{i>0}_R(M,P)=0$ for all projective $R$-modules $P$.

Abstract:
Let $M$ be a compact Riemannian manifold without boundary and $V:M\to \mathbb R$ a smooth function. Denote by $P_t$ and ${\rm d}\mu=e^V\,{\rm d} x$ the semigroup and symmetric measure of the second order differential operator $L=\Delta+\nabla V\cdot\nabla$. For some suitable convex function $\Phi:{\mathcal I}\to\mathbb R$ defined on an interval $\mathcal I$, we consider the $\Phi$-entropy of $P_t f$ (with respect to $\mu$) for any $f\in C^\infty(M,\mathcal I)$. We show that an integral form curvature-dimension condition is equivalent to an estimate on the rate of change of the $\Phi$-entropy. We also generalize this result to bounded smooth domains of a complete Riemannian manifold.

Abstract:
We consider the It\^o SDE with partially Sobolev coefficients. Under some suitable conditions, we show the existence, uniqueness and stability of generalized stochastic flows associated to such an equation. As an application, we prove the weak differentiability of the stochastic flow generated by the It\^o SDE with Sobolev coefficients.