Abstract:
LetX t be the interaction measured-valued branchingα-symmetric stable process overR d (1< α ≤2) constructed by Meleard-Roelly[1]. Frist, it is shown thatX t is absolutely continuous with respect to the Lebesgue measure (onR) with a continuous density function which satisfies some SPDE. Second, it is proved that if the underlying process is a Brownian motion onR d(d≤3), the corresponding occupation-time processY t is also absolutely continuous with respect to the Lebesgue measure.

Abstract:
Porous SiC ceramic were prepared with silicon carbide powder as the aggregate, silicone resin as the binder and pore agent by the process of mixing, iso-static pressure molding, and calcination. The mechanical properties and microstructures of the samples were characterized with a universal testing machine, X-ray diffraction, scanning electron microscope, and mercury injection. Two main factors, molding pressures and silicone resin mass ratio were studied in the experiments. The thermal conductivity of the samples was tested. The compressive strength was up to 19.4 MPa, and the porosities up to 30%. The thermal conductivities, mainly influenced by porosities, increased from 0.68 W^{.}m^{-}^{1.}K^{-}^{1} to 1.03 W^{.}m^{-}^{1.}K^{-}^{1} with the porosity decreasing from 41.96% to 31.30%.

Abstract:
LetX t be the interaction measured-valued branchingα-symmetric stable process overR d (1< α ≤2) constructed by Meleard-Roelly1]. Frist, it is shown thatX t is absolutely continuous with respect to the Lebesgue measure (onR) with a continuous density function which satisfies some SPDE. Second, it is proved that if the underlying process is a Brownian motion onR d(d≤3), the corresponding occupation-time processY t is also absolutely continuous with respect to the Lebesgue measure.

Abstract:
In this paper concerns the third-order boundary-value problem $$displaylines{ u'''(t)+ f(t, u(t),u'(t), u''(t))=0, quad 0 < t < 1, cr r_1 u(0) - r_2 u' (0)= r_3 u(1) + r_4 u'(1)= u''(0)=0. }$$ By placing certain restrictions on the nonlinear term f, we prove the existence of at least one solution to the boundary-value problem with the use of lower and upper solution method and of Schauder fixed-point theorem. The construction of lower or upper solutions is also presented.

Abstract:
In this paper, a fixed point theorem in a cone, some inequalities of the associated Green's function and the concavity of solutions are applied to obtain the existence of positive solutions of third-order three-point boundary value problem with dependence on the first-order derivative $egin{cases}& x'''(t) = f(t, x(t), x'(t)), quad 0 < t < 1, & x(0) = x'(eta) = x''(1) = 0, end{cases}$where $f:[0, 1] imes[0, infty) imes R o [0,infty)$ is a nonnegative continuous function, $etain(1/2, 1).$

Abstract:
In this article, we establish the existence of a positive solution to a singular boundary-value problem of nonlinear fractional differential equation. Our analysis rely on nonlinear alternative of Leray-Schauder type and Krasnoselskii's fixed point theorem in a cone.

Abstract:
In this paper we describe a constructive method which yields two monotone sequences that converge uniformly to extremal solutions to the following boundary value problems $$ u^{(4)}(x)-Mu'(x) =f(x,u(x),u'(x),u'(x), u''(x)) , 0

Abstract:
We consider the boundary value problem: $ left(varphi_p(x'(t)) ight)'+ q(t)f(t, x(t), x'(t))=0, p>1, t in [0, 1] $, with $ x(0)=x(1)=0 $, or $ x(0)=x'(1)=0 $. Using a fixed point theorem due to Avery and Peterson, sufficient conditions are obtained that guarantee the existence of at least three positive solutions. The emphasis here is the nonlinear term $ f $ is involved with the first order derivative. An example is also included to illustrate the importance of the results obtained

Abstract:
We obtain sufficient conditions for the existence of at least three positive solutions for the equation $ x''(t) + q(t)f(t, x(t), x'(t)) = 0 $ subject to some boundary conditions. This is an application of a new fixed-point theorem introduced by Avery and Peterson [6].

Abstract:
Abstract: Objective] The study aims to investigate the phylogeny diversity of the denitrification bacteria communities in the sediments of the eutrophic East Lake, Wuhan based on nitrite reductase gene (nirS) restriction fragment length polymorphism (RFLP) method and sequencing analysis, and to analyse community variation according to the environment parameters. Methods] We collected the sediment samples from the four typical sub-lake of East Lake in Wuhan, Guozheng Lake, Tangling Lake, Tuan Lake and Miao Lake, and measured the environmental parameters appropriately. After extracted the genomic DNA from the sediment, four nirS gene clone libraries were successfully constructed. The operation taxonomy units (OTUs) were determined by RFLP method and the representative fragment of every OTU was sequenced. The diversity, richness and evenness statistics of the NirS-like communities were calculated by using DOTUR software. Neighbour-joining phylogenetic tree was constructed basing on the amino acid sequences of NirS from the East Lake sediments and reference sequences retrieved from the GenBank database. The relationship between tested NirS communities and the references from different environments was also discussed. Results] Environmental parameters showed that Miao Lake sediment contains the highest amount of total nitrogen (TN) and NH4+-N, while the Tuan Lake sediments contains the lowest amount. Among the four sub-lakes, Tuan Lake harbours the highest diversity and richness of NirS-like denitrifiers, while the denitrifiers in Miao Lake was the lowest. Phylogenic analyses suggested that sedimentary NirS-like denitrifiers in the East Lake could be distributed into three groups, Group I to III. Group I accounts for 67.7% of all tested communities. Eighty-one percent of sequences from Guozheng Lake were clustered into Group I, while 67.7% of sequences from Miao Lake were clustered into Group II. Comparative analysis of communities from East Lake and artificial wetland found there are phylogenetically related. Conclusion] There are diverse and abundant NirS-like denitrifiers inhabited in the sediments of East Lake, Wuhan. The diversity indices and spatial distribution of these communities are affected by the content of TN, NH4+ and NO3- nutrients in the sediments.