Abstract:
Furniture in the halls of Tibetan Buddhist temples is an important part and the product of materialization of Tibetan Buddhist culture in Northwest Sichuan. The decorative pattern of the furniture in the halls is the essence of Buddhist culture in Northwest Sichuan, which has unique regional and national characteristics. By excavating the contemporary value and significance of decorative patterns of furniture in the halls of Tibetan Buddhist temples, we can study the influence of the religious culture on the decorative patterns of furniture, and further explore the culture.

Abstract:
In this paper, we consider a network of energy constrained sensors deployed over a region. Each sensor node in such a network is systematically gathering and transmitting sensed data to a base station (via cluster- heads). This paper focuses on reducing the power consumption of wireless sensor networks. Firstly, we proposed an Energy-balanced Clustering Routing Algorithm called LEACH-L, which is suitable for a large scope wireless sensor network. Secondly, optimum hop-counts are deduced. Lastly, optimum position of transmitting node is estimated. Simulation results show that our modified scheme can extend the network lifetime by up to 80% before first node dies in the network. Through both theoretical analysis and numerical simulations, it is shown that the proposed algorithm achieves higher performance than the existing clustering algorithms such as LEACH, LEACH-M.

Abstract:
In this paper, we show the existence and nonexistence of entire positive solutions for a class of singular elliptic system
We have that entire large positive solutions fail to exist if f and g are sublinear and b and d have fast decay at infinity. However, if f and g satisfy some growth conditions at infinity, and b, d are of slow decay or fast decay at infinity, then the system has infinitely many entire solutions, which are large or bounded.

Abstract:
Recent advances in stem cell biology have established the feasibility of reprogramming human and murine fibroblast cells into induced pluripotent stem (iPS) cells [1-3]. The ectopic expression of four transcription factors (Oct4, Sox2, c-Myc, and Klf4) in fibroblasts was shown to be efficacious in the conversion of fibroblast cells into embryonic stem (ES) cell-like status. Generation of iPS cells ushers in a new era in reprogramming differentiated somatic cells, including fibroblasts, neural cells, liver cells, stomach cells, and blood cells, into ES cell-like stem cells [4]. With the progress of iPS technology, the concept of 'master regulators', defined as a group of major reprogramming factors playing a critical role in the management of cell status of 'pluripotent' versus 'differentiated', has been demonstrated. The group of master regulators for iPS cell generation is found to be effective with only three genes [5], which is far fewer than the hundreds or thousands of genes that were presumed to be involved in the determination of cell fate or status.Successful reprogramming of fibroblast cells into iPS cells raised the possibility of directly converting one somatic cell type into other cell types. By overexpressing Ngn3, Pdx1, and Mafa, Zhou and colleagues [6] reported the conversion of exocrine pancreas cells into cells closely resembling beta cells and having the function of secreting insulin. In 2010, successful reprogramming of fibroblast cells into functional neuron cells was reported with the enforced expression of Ascl1, Brn2, and Myt1l [7]. Recently, Srivastava's group [8] used the same strategy and reported another breakthrough in direct reprogramming of mouse fibroblast cells into beating cardiomyocyte-like cells: the transduction of a set of three cardiac master factors important in early heart development (Gata4, Mef2c, and Tbx5). This is the first paper to reveal the possibility of directly committing fibroblast cells into heart muscle cells that

Abstract:
We find that the first Hurwitz triplet possesses two distinct arithmetic structures. As Shimura curves $X_1$, $X_2$, $X_3$, whose levels are with norm 13. As non-congruence modular curves $Y_1$, $Y_2$, $Y_3$, whose levels are 7. Both of them are defined over ${\Bbb Q}(\cos \frac{2 \pi}{7})$. However, for the third non-congruence modular curve $Y_3$, there exist an "exotic" duality between the associated non-congruence modular forms and the Hilbert modular forms, both of them are related to ${\Bbb Q}(e^{\frac{2 \pi i}{13}})$! Our results have relations and applications to modular equations of degree fourteen (including Jacobian modular equation and "exotic" modular equation), "triality" of the representation of $PSL(2, 13)$, Haagerup subfactor, geometry of the exceptional Lie group $G_2$, and even the Monster finite simple group ${\Bbb M}$!

Abstract:
Let $H = \mathrm{SO}(n,1)$ and $A =\{a(t) : t \in \mathbb{R}\}$ be a maximal $\mathbb{R}$-split Cartan subgroup of $H$. Let $G$ be a Lie group containing $H$ and $\Gamma$ be a lattice of $G$. Let $x = g\Gamma \in G/\Gamma$ be a point of $G/\Gamma$ such that its $H$-orbit $Hx$ is dense in $G/\Gamma$. Let $\phi: I= [a,b] \rightarrow H$ be an analytic curve, then $\phi(I)x$ gives an analytic curve in $G/\Gamma$. In this article, we will prove the following result: if $\phi(I)$ satisfies some explicit geometric condition, then $a(t)\phi(I)x$ tends to be equidistributed in $G/\Gamma$ as $t \rightarrow \infty$. It answers the first question asked by Shah in ~\cite{Shah_1} and generalizes the main result of that paper.

Abstract:
In this article, we consider the product space of several non-compact finite volume hyperbolic spaces, $V_1, V_2, \dots , V_k$ of dimension $n$. Let $\mathrm{T}^1(V_i)$ denote the unit tangent bundle of $V_i$ for each $i=1,\dots , k$, then for every $(v_1, \dots , v_k) \in \mathrm{T}^1 (V_1) \times \cdots \times \mathrm{T}^1 (V_k)$, the diagonal geodesic flow $g_t$ is defined by $g_t (v_1, \dots , v_k) = (g_t v_1, \dots , g_t v_k)$. And we define $$\mathfrak{D}_k =\left\{ (v_1, \dots, v_k) \in \mathrm{T}^1 (V_1) \times \cdots \times \mathrm{T}^1 (V_k): g_t(v_1, \dots, v_k) \text{ divergent, as } t\rightarrow \infty\right\}. $$ We will prove that the Hausdorff dimension of $\mathfrak{D}_k$ is equal to $k(2n-1) - \frac{n-1}{2}$. This extends the result of Yitwah Cheung ~\cite{Cheung1}.

Abstract:
In this article, we study an analytic curve $\varphi: I=[a,b]\rightarrow \mathrm{M}(n\times n, \mathbb{R})$ in the space of $n$ by $n$ real matrices, and show that if $\varphi$ satisfies certain geometric conditions, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem is not improvable. To do this, we embed the curve into some homogeneous space $G/\Gamma$, and prove that under the action of some expanding diagonal flow $A= \{a(t): t \in \mathbb{R}\}$, the expanding curves tend to be equidistributed in $G/\Gamma$, as $t \rightarrow +\infty$. This solves a special case of a problem proposed by Nimish Shah in ~\cite{Shah_1}.

Abstract:
We solve a long-standing open problem with its own long history dating back to the celebrated works of Klein and Ramanujan. This problem concerns the invariant decomposition formulas of the Hauptmodul for $\Gamma_0(p)$ under the action of finite simple groups $PSL(2, p)$ with $p=5, 7, 13$. The cases of $p=5$ and $7$ were solved by Klein and Ramanujan. Little was known about this problem for $p=13$. Using our invariant theory for $PSL(2, 13)$, we solve this problem. This leads to a new expression of the classical elliptic modular function of Klein: $j$-function in terms of theta constants associated with $\Gamma(13)$. Moreover, we find an exotic modular equation, i.e., it has the same form as Ramanujan's modular equation of degree $13$, but with different kinds of modular parametrizations, which gives the geometry of the classical modular curve $X(13)$.

Abstract:
We find that the equation of $E_8$-singularity possesses two distinct symmetry groups and modular parametrizations. One is the classical icosahedral equation with icosahedral symmetry, the associated modular forms are theta constants of order five. The other is given by the group $\text{PSL}(2, 13)$, the associated modular forms are theta constants of order $13$. As a consequence, we show that $E_8$ is not uniquely determined by the icosahedron. This solves a problem of Brieskorn in his ICM 1970 talk on the mysterious relation between exotic spheres, the icosahedron and $E_8$. Simultaneously, it gives a counterexample to Arnold's $A, D, E$ problem, and this also solves the other related problem on the relation between simple Lie algebras and Platonic solids. Moreover, we give modular parametrizations for the exceptional singularities $Q_{18}$, $E_{20}$ and $x^7+x^2 y^3+z^2=0$ by theta constants of order $13$, the second singularity provides a new analytic construction of solutions for the Fermat-Catalan conjecture and gives an answer to a problem dating back to the works of Klein.