THREE POINT BOUNDARY VALUE PROBLEMS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, we study existence and uniqueness of solutions to nonlinear three point boundary value problems for fractional differential equation of the type c D δ 0＋ u（t） = f （t, u（t）, c D σ 0＋ u（t））, t ∈ [0, T ], u（0） = αu（η）, u（T ） = βu（η）, where 1 〈 δ 〈 2, 0 〈 σ 〈 1, α, β∈ R, η∈ （0, T ）, αη（1 -β） ＋ （1-α）（T βη） = 0 and c D δ 0＋ , c D σ 0＋ are the Caputo fractional derivatives. We use Schauder fixed point theorem and contraction mapping principle to obtain existence and uniqueness results. Examples are also included to show the applicability of our results.

Viscous Cosmology and Thermodynamics of Apparent Horizon

It is shown that the differential form of Friedmann equations of Friedman-Robertson-Walker （FRW） universe can be recast as a similar form of the first law ThdSh = dE＋ WdV of thermodynamics at the apparent horizon of FRW universe filled with the viscous fluid. It is also shown that by employing the general expression of temperature Th=｜k｜/2π=1/2π~rA（1-2~rA/2H^rA）associated with the apparent horizon of an FRW universe and assumed that the temperature Tm of the energy inside the apparent horizon is proportional to the horizon temperature Tm=bTh, we are able to show that the generalized second law of thermodynamics holds in the Einstein gravity provided Th-Tm/~rA≤（ρ＋~P）.

Foliation and the First Law of Black Hole Thermodynamics

There has been lots of interest in exploring the thermodynamic properties at the horizon of a black hole spacetime.It has been shown earlier that for different spacetimes,the Einstein field equations at the horizon can be expressed as the first law of black hole thermodynamics.Using the idea of foliation,we develop a simpler procedure toobtain such results.We considerγ=constant slices,for the Schwarzschild and Reissner-Nordstrom black hole spacetimes.The Einstein field equations for the induced 3-dimensional metrics of the hypersurfaces are expressed in thermodynamic quantities under the virtual displacements of the hypersurfaces.As expected,it is found that the field equations of the induced metric corresponding to the horizon can be written as afirst law of black hole thermodynamics.It is to be mentioned here that our procedure is much easier,to obtain such results,as here one has to essentially deal with(n-1)-dimensional induced metric for an n-dimensional spacetime.

Gauss-Bonnet and Lovelock Gravities and the Generalized Second Law of Thermodynamics

It has been shown [Chin. Phys. Lett.25 （2008） 4199] that the generalized second law of thermodynamics holds in Einstein gravity. Here we extend this procedure for Gauss-Bonnet and Lovelock gravities. It is shown that by employing the general expression for temperature Th =｜κ｜/2π= 1/2πτA （1-τA/2HτA） associated with the apparent horizon of a Friedman Robertson-Walker （FRW） universe and assuming Tm = bTh, we are able to construct conditions for which the generalized second law holds in Gauss Bonnet and Lovelock gravities, where Tm and Th are the temperatures of the source and the horizon respectively.

潜伏期和染病期均具有传染性的媒介传染病模型的全局稳定性分析(英文)

讨论潜伏期和染病期均具有传染性的媒介传染病模型.得到模型基本再生数的表达式,证明了当基本再生数小于1时,无病平衡点是全局渐近稳定的,此时疾病消亡;当基本再生数大于1时,无病平衡点是不稳定的,系统存在全局渐近稳定的地方病平衡点,此时,疾病将在人群中持续存在,数值模拟验证了理论结果.

ON CLASSES OF REGULAR GRAPHS WITH CONSTANT METRIC DIMENSION

In this paper, we are dealing with the study of the metric dimension of some classes of regular graphs by considering a class of bridgeless cubic graphs called the flower snarks Jn, a class of cubic convex polytopes considering the open problem raised in [M. Imran et al., families of plane graphs with constant metric dimension, Utilitas Math., in press] and finally Harary graphs H5,n by partially answering to an open problem proposed in Ⅱ. Javaid et al., Families of regular graphs with constant metric dimension, Utilitas Math., 2012, 88： 43-57]. We prove that these classes of regular graphs have constant metric dimension.

Ricci Symmetries of Static Space-Times with Maximal Symmetric Transverse Spaces

在纸[M。Akbar 和 R.G 蔡， Commun。Theor。Phys。45 (2006 ) 95 ] ，一个完全的分类提供向量地 V 的至少一个部件是零。在这篇论文，我与所有非零部件认为向量地是 V，有最大的对称的横向的空格的静态的空间时间根据他们的 Ricci 关口线条被分类。这些为非退化的 Ricci 张肌 det R ≠ 被调查0。结果唯一的关口线条由这些承认空格能是十，七，六或四。它也作为一个空间案例盖住我们的以前的结果。承认合适的 Ricci 关口线条的一些新度量标准也被调查。

Generalized Second Law of Thermodynamics in Wormhole Geometry with Logarithmic Correction

We construct various cases for validity of the generalized second law(GSL)of thermodynamics by assuming the logarithmic correction to the horizon entropy of an evolving wormhole.It is shown that the GSL is always respected forα0≤0,whereas forα0>0 the GSL is respected only ifπr2A+/ℏ<α.