This paper develops the study of Fox pairings of a group $G$ from the viewpoint of group cohomology. We compute some cohomology groups of Fox pairings of $G$, where $G$ admits a Poincaré duality group pair. We also suggest fundamental Fox pairings and higher Fox pairings.

Let $\mathfrak A$ be a maximal subdiagonal algebra in a $\sigma$-finite von Neumann algebra $\mathcal M$ with respect to a faithful normal conditional expectation $\Phi$. We consider the characterizations for $\mathfrak A$ to be a type 1 subdiagonal algebra in the sense that every right invariant subspace in noncommutative $H^2$ space is of Beurling type. As an application, we give a necessary and sufficient condition that a nest subalgebra $\rm{Alg} \mathcal N$ with an injective nest $\mathcal N$ is a type 1 subdiagonal algebra in a factor von Neumann algebra $\mathcal M$.

We introduce two kinds of covariant derivatives defined on a real hypersurface in Kähler manifolds with respect to the Levi-Civita connection and the $k$-th generalized Tanaka-Webster connection (shortly, gTW-connection). Related to such two kinds of derivatives, we study a generalized parallelism of structure Jacobi operator on a real hypersurface in complex Grassmannians with rank two. And by using this property, we will give some classification results of real hypersurfaces in complex Grassmannians of rank two.

We establish Hardy-Sobolev inequalities in the unit ball $\mathbf{B}$ in the framework of general double phase functionals given by\[\varphi_p(x,t) = \varphi_1(t^p) + \varphi_2((b(x)t)^p),\quad x\in \mathbf{B}, t \ge 0,\]where $p>1$, $\varphi_1, \varphi_2$ are positive convex functions on $(0,\infty)$ and $b$ is a non-negative function on $\mathbf{B}$ which is radially Hölder continuous of order $\theta \in (0,1]$.

We prove finite field analogues of integral representations of Appell-Lauricella hypergeometric functions in many variables. We consider certain hypersurfaces having a group action and compute the numbers of rational points associated with characters of the group, which will be expressed in terms of Appell-Lauricella functions over finite fields.

A bifurcation model for a nonlinear equation is introduced. Under the non-degeneracy condition (Definition 2.1), our bifurcation model describes the bifurcation of solutions to the nonlinear equation. We also show how these models work for Dirichlet problem on the square. We observe a perturbation of rectangles to a square creates new bifurcation, which is not a limit of the bifurcations on rectangles.

Let $(u_{n})_{n\geq0}$ be an unbounded sequence of positive integers such that $u_{n+1}=\alpha u_{n}^{2}+O(u_{n}^{\gamma})$ for some positive rational number $\alpha$ and some $\gamma\in\left] 0,2\right[ .$ Let $(r_{n})_{n\geq0}$ be a sequence of rational numbers satisfying ``weak'' growth conditions. We give necessary and sufficient conditions for the series $\sum_{n=0}^{\infty}r_{n}/u_{n}$ and the infinite product $\prod_{n=0}^{\infty}\left(1+r_{n}/u_{n}\right)$ to be rational numbers. Moreover, in case of irrationality, we obtain an upper bound for their irrationality exponents.