Notes for a mini course at the University of Science and Technology of China in Hefei, China, June 23–July 12, 2014.

In this paper, we study extended modules for a special class of Ore extensions. We will assume that R is a ring and A will denote the Ore extension $A:=R[x_1,\ldots ,x_n;\sigma ]$ for which $\sigma $ is an automorphism of R, $x_ix_j=x_jx_i$ and $x_ir=\sigma (r)x_i$, for every $1\le i,j\le n$. With some extra conditions over the ring R, we will prove Vaserstein’s, Quillen’s patching, Horrocks’, and Quillen–Suslin’s theorems for this type of non-commutative rings.

In isogeometric analysis (IGA), parametrization is an important and difficult issue that greatly influences the numerical accuracy and efficiency of the numerical solution. One of the problems facing the parametrization in IGA is the existence of the singular points in the parametrization domain. To avoid producing singular points, boundary-mapping parametrization is given by mapping the computational domain to a polygon domain which may not be a square domain and mapping each segment of the boundary in computational domain to a corresponding boundary edge of the polygon. Two numerical examples in finite element analysis are presented to show the novel parametrization is efficient.

In the paper we first derive the evolution equation for eigenvalues of geometric operator $-\Delta _{\phi }+cR$ under the Ricci flow and the normalized Ricci flow on a closed Riemannian manifold M, where $\Delta _{\phi }$ is the Witten–Laplacian operator, $\phi \in C^{\infty }(M)$, and R is the scalar curvature. We then prove that the first eigenvalue of the geometric operator is nondecreasing along the Ricci flow on closed surfaces with certain curvature conditions when $0<c\le \frac{1}{2}$. As an application, we obtain some monotonicity formulae and estimates for the first eigenvalue on closed surfaces.

The sub-linear expectation or called G-expectation is a non-linear expectation having advantage of modeling non-additive probability problems and the volatility uncertainty in finance. Let $\{X_n;n\ge 1\}$ be a sequence of independent random variables in a sub-linear expectation space $(\Omega , \mathscr {H}, \widehat{\mathbb {E}})$. Denote $S_n=\sum _{k=1}^n X_k$ and $V_n^2=\sum _{k=1}^n X_k^2$. In this paper, a moderate deviation for self-normalized sums, that is, the asymptotic capacity of the event $\{S_n/V_n \ge x_n \}$ for $x_n=o(\sqrt{n})$, is found both for identically distributed random variables and independent but not necessarily identically distributed random variables. As an application, the self-normalized laws of the iterated logarithm are obtained. A Bernstein’s type inequality is also established for proving the law of the iterated logarithm.

In this article, we give an explicit way to construct representations of the fundamental group $\pi _1(X),$ where X is a hyperbolic curve over $\mathbb {C}.$ Our motivation is to study a special space in $M_\mathrm{DR}(X,\,\mathrm {SL}_2(\mathbb {C}))$ which is called the space of permissible connections in Faltings (Compos Math 48(2):223–269, 1983), or indigenous bundles in Gunning (Math Ann 170:67–86, 1967). We get representations by constructing Higgs bundles, and we show that the family we get intersects the space of permissible connections $\mathbf {PC}$ in a positive dimension. In this way, we actually get a deformation of the canonical representation in $\mathbf {PC},$ and all these deformations are given by explicit constructed Higgs bundles. We also estimate the dimension of this deformation space.