Let $\mathcal{M}\left(n,m;{\mathbb{F\mathbb{P}}}^n\right)$ be the configuration space of $m$-tuples of pairwise distinct points in ${\mathbb{F\mathbb{P}}}^n$, that is, the quotient of the set of $m$-tuples of pairwise distinct points in ${\mathbb{F\mathbb{P}}}^n$ with respect to the diagonal action of $\text{P}\text{U}(1,n;\mathbb{F})$ equipped with the quotient topology. In this paper, by mainly using the rotation-normalized and the block-normalized algorithms, we construct the parameter spaces of both $\mathcal{M}\left(n,m;\partial {\mathbf{H}}_{\mathrm{ℍ}}^n\right)$ and $\mathcal{M}\left(n,m;\mathrm{\mathbb{P}}\left(V_{+}\right)\right)$, respectively.

In this paper, we give a formula for the number of rational points on the Dwork hypersurfaces of degree six over finite fields by using Greene’s finite-field hypergeometric function, which is a generalization of Goodson’s formula for the Dwork hypersurfaces of degree four. Our formula is also a higher-dimensional and a finite field analogue of Matsumoto-Terasoma-Yamazaki’s formula. Furthermore, we also explain the relation between our formula and Miyatani’s formula.

We consider a parabolic problem with Robin boundary condition which arises when the edge of a micro-electro-mechanical-system (MEMS) device is connected with a flexible nonideal support. Then via a rigorous analysis we investigate the structure of the solution set of the corresponding steady-state problem. We show that a critical value (the pull-in voltage) exists so that the system has exactly two stationary solutions when the applied voltage is lower than this critical value, one stationary solution for applying this critical voltage, and no stationary solution above the critical voltage.

Let $X_n(n>1)$ be a finitistic space with cohomology type $\left(0,0\right)$. Let $\left(X_n,E,\pi ,B\right)$ be a fibre bundle and $\left({\mathrm{ℝ}}^k,E\prime ,\mathrm{\pi }\prime ,\mathrm{B}\right)$ be a $k$-dimensional real vector bundle with fibre preserving $G={\mathrm{\mathbb{Z}}}_p,p>2$ a prime, action such that $G$ acts freely on $E$ and $E\prime -\left\{0\right\}$, where $\left\{0\right\}$ is the zero section of the vector bundle. We determine a lower bound of the cohomological dimension of the set $A_{\mu }=\{x\in E|\mu \left(x\right)\cap \mu \left(gx\right)\cap \cdots \cap \mu \left(g^{P-1}x\right)\ne \varphi \}$ for an admissible multivalued fibre preserving map $\mu :E\to E\prime$.

It is known that a connected and simply-connected Lie group admits only one left-invariant Riemannian metric up to scaling and isometry if and only if it is isomorphic to the Euclidean space, the Lie group of the real hyperbolic space, or the direct product of the three dimensional Heisenberg group and the Euclidean space of dimension $n-3$. In this paper, we give a classification of left-invariant pseudo-Riemannian metrics of an arbitrary signature for the third Lie groups with $n\ge 4$ up to scaling and automorphisms. This completes the classifications of left-invariant pseudo-Riemannian metrics for the above three Lie groups up to scaling and automorphisms.

We are interested in the classification or finding conditions for the existence of left-invariant symplectic structures on Lie groups. Some classifications are known, especially in low dimensions. We approach this problem by studying the “moduli space of left-invariant nondegenerate $2$-forms”, which is a certain orbit space in the set of all nondegenerate $2$-forms on a Lie algebra. In this paper, using this approach, we give a classification of left-invariant symplectic structures on all almost abelian Lie algebras determined by diagonal matrices.