Abstract:
In this paper we will prove the nodal line $N$ of the second eigenfunction of the Laplacian over some simply connected concave domain $\Omega$ in $\mathbb{R}^2$ must intersect the boundary $\partial\Omega$ at exactly two points.

Abstract:
In this paper we obtain a new class of open sets, and we prove the class is compact under the Hausdorff distance, then we prove the existence of solutions of some shape optimization for elliptic equations.

Abstract:
In this study, a modeling method to analyze multidimensional time series based on complex networks is proposed. The rate of return sequence of the closing price and the trading volume fluctuation sequence of the Shanghai Composite Index, the Shenzhen Component Index, the S & P 500 index, and the Dow Jones Industrial Average are analyzed. The two-dimensional time series is transformed into a complex network. We analyze the spatial distribution characteristics of the network to determine the relationship between volume and price. It is found that the interaction of stock return and volume in China’ stock market is more obvious than that in the American market.

Abstract:
We establish a divergence free partition for vector-valued Sobolev functions with free divergence in ${\bf R}^n, n\geq 1$. We prove that for any domain $\om$ of class $\cal C$ in ${\bf R}^n,n=2,3$, the space $D_0^1(\om)\equiv\{{\mathbf{v}} \in H^1_0(\Omega)^n ; {div}{\mathbf{v}}=0\}$ and the space $H_{0,\sigma}^1(\om)\equiv \bar{\{{\mathbf{v}}\in C^{\infty}_0(\Omega)^n;{div}{\mathbf v}=0\}}^{\|\cdot\|_{H^1(\om)^n}}$, which is the completion of $\{{\mathbf{v}} \in C^{\infty}_0(\Omega)^n; {div}{\mathbf v}=0\}$ in the $H^1(\Omega)^n$-norm, are identical. We will also prove that $H_{0,\sigma}^1(D\setminus\bar\om)=\{{\mathbf v}\in H_{0,\sigma}^1(D); {\mathbf v}=0 {a.e. in} \om\}$, where $D$ is a bounded Lipschitz domain such that $\om\subset\subset D$. These results, together with properties for domains of class $\mathcal C$, are used to solve an existence problem in the shape optimization theory of the stationary Navier-Stokes equations.

Abstract:
Reaction resonance is a frontier topic in chemical dynamics research, and it is also essential to the understanding of mechanisms of elementary chemical reactions. This short article describes an important development in the frontier of research. Experimental evidence of reaction resonance has been detected in a full quantum state resolved reactive scattering study of the F+H2 reaction. Highly accurate full quantum scattering theoretical modeling shows that the reaction resonance is caused by two Feshbach resonance states. Further studies show that quantum interference is present between the two resonance states for the forward scattering product. This study is a significant step forward in our understanding of chemical reaction resonance in the benchmark F+H2 system. Further experimental studies on the effect of H2 rotational excitation on dynamical resonance have been carried out. Dynamical resonance in the F+H2 (j = 1) reaction has also been observed.

Abstract:
Reaction resonance is a frontier topic in chemical dynamics research,and it is also essential to the understanding of mechanisms of elementary chemical reactions.This short article describes an im- portant development in the frontier of research.Experimental evidence of reaction resonance has been detected in a full quantum state resolved reactive scattering study of the F H2 reaction.Highly accurate full quantum scattering theoretical modeling shows that the reaction resonance is caused by two Feshbach resonance states.Further studies show that quantum interference is present between the two resonance states for the forward scattering product.This study is a significant step forward in our understanding of chemical reaction resonance in the benchmark F H2 system.Further experimental studies on the effect of H2 rotational excitation on dynamical resonance have been carried out.Dy- namical resonance in the F H2(j=1)reaction has also been observed.

Abstract:
For the first time AZ91 (MgAl9Zn1) and AM60 (MgAl6) Mg alloy foams with homogeneous pore structures were prepared successfully via melt foaming method using CaCO3 as blowing agent. It is revealed that the blowing gas to foam the melt is not CO2 but CO, which comes from liquid-solid reaction between Mg melt. The reaction temperature is more than 100℃ lower than CaCO3 decomposition, which makes Mg alloy melts foam into cellular structure much more easily in the temperature range from 690℃ to 750℃.

Abstract:
Background and objective 32P-chromic phosphate-poly (L-lactic) acid (32P-CP-PLLA) microparticle is a novel potent brachytherapy implant, which has good biocompatibility and biodegradability. The aim of this study is to investigate the changes of pathology and PET/CT images in VX2 rabbit tumor after treatment with intratumorol administration of 32P-CP-PLLA microparticles, and to explore the effects and influence of tumor growth and apoptosis related proteins in VX2 lung tumor treatment with 32P-CP-PLLA microparticles. Methods Twenty-four tumor bearing rabbits were randomly divided into 4 groups (6 in each group). Group 1, 2 and 3 were treated groups; group 4 was the control. Under CT guidance, 32P-CPPLLA microparticles were implanted into tumors. Low, medium and high treatment doses were 93 MBq (group 1), 185 MBq (group 2) and 370 MBq (group 3), respectively. 18F-FDG PET/CT was performed at d0, d3, d7 and d14 after intratumoral administration. In the control group, 18F-FDG PET/CT images were acquired at the same time points but without treatment. The standardized uptake value (SUV) of tumor regions were calculated. After last PET/CT imaging, the rabbits were euthanized and the tumors were removed for histological and immunohistochemical examination. The pathology and the expression of apoptosis related proteins (bcl-2, bax) were compared. Results No significant difference of SUVmax was observed between the treatment groups and the control group at d0. At d14, the SUVmax values for group 1, 2 and 3 were 0.80±0.10, 1.1±0.19 and 2.85±0.15, respectively, and were significantly lower than that of the control group (5.61±0.50)(P < 0.05). Significant dose-response relationship was observed in SUVmax between group 1 and group 2, and the SUV values gradually decreased from d7 to d14 after treatment. In group 3, SUVmax gradually increased and reached a peak at d7 then significantly decreased. The SUVmax values of group 3 were significantly lower than those of the control at the same time point (P < 0.05). HE staining found degenerative necrosis at the site was nearby the microparticle. Necrosis became serious increasing with the radioactivity. Inflammatory cell infiltration was rarely seen in tumors treated with 93 MBq or 185 MBq 32P-CP-PLLA microparticles. In contrast, the necrotic area was surrounded by marked inflammatory cell infiltration in group 3. IHC analysis showed that the expression of bcl-2 in treated groups were lower than those in the control group, and the expression of bax in treated group was higher than those in the control group (P < 0.05). The

Abstract:
We elaborate on an alternative representation of
conditional probability to the usual tree diagram. We term the representation “turtleback
diagram” for its resemblance to the pattern on turtle shells. Adopting the set
theoretic view of events and the sample space, the turtleback diagram uses
elements from Venn diagrams—set intersection, complement and partition—for
conditioning, with the additional notion that the area of a set indicates
probability whereas the ratio of areas for conditional probability. Once parts
of the diagram are drawn and properly labeled, the calculation of conditional
probability involves only simple arithmetic on the area of relevant sets. We
discuss turtleback diagrams in relation to other visual representations of
conditional probability, and detail several scenarios in which turtleback
diagrams prove useful. By the equivalence of recursive space partition and the tree,
the turtleback diagram is seen to be equally expressive as the tree diagram for
abstract concepts. We also provide empirical data on the use of turtleback diagrams
with undergraduate students in elementary statistics or probability courses.

Abstract:
Flaky and nanospherical birnessite and birnessite-supported Pt catalysts were successfully prepared and characterized by means of X-ray diffraction (XRD), transmission electron microscopy (TEM), energy dispersive spectroscopy (EDS) and N2 adsorption-desorption. Effects of the birnessite morphology and Pt reduction method on the catalytic activity for the complete oxidation of formaldehyde (HCHO) were investigated. It was found that flaky birnessite exhibited higher catalytic activity than nanospherical birnessite. The promoting effect of Pt on the birnessite catalyst indicated that the reduction method of the Pt precursor greatly influenced the catalytic performance. Flaky birnessite-supported Pt nanoparticles reduced by KBH4 showed the highest catalytic activity and could completely oxidize HCHO into CO2 and H2O at 50℃, whereas the sample reduced using H2-plasma showed lower activity for HCHO oxidation. The differences in catalytic activity of these materials were jointly attributed to the effects of pore structure, surface active sites exposed to HCHO and the dispersion of Pt nanoparticles.