We classify all edge-to-edge spherical isohedral 4-gonal tilings such that the skeletons are pseudo-double wheels. For this, we characterize these spherical tilings by a quadratic equation for the cosine of an edge-length. By the classification, we see: there are indeed two non-congruent, edge-to-edge spherical isohedral 4-gonal tilings such that the skeletons are the same pseudo-double wheel and the cyclic list of the four inner angles of the tiles are the same. This contrasts with that every edge-to-edge spherical tiling by congruent 3-gons is determined by the skeleton and the inner angles of the skeleton. We show that for a particular spherical isohedral tiling over the pseudodouble wheel of twelve faces, the quadratic equation has a double solution and the copies of the tile also organize a spherical non-isohedral tiling over the same skeleton.

We classify jets of Monge forms of generic surfaces in 4-space via projective transformations, which is an extension of Platonova’s result for surfaces in 3-space.

We give a necessary and su‰cient condition for orbits of commutative Hermann actions and actions of the direct product of two symmetric subgroups on compact Lie groups to be biharmonic in terms of symmetric triad with multiplicities. By this criterion, we determine all the proper biharmonic orbits of these Lie group actions under some additional settings. As a consequence, we obtain many examples of proper biharmonic homogeneous submanifolds in compact symmetric spaces and compact Lie groups.

Using the rank of the Mordell-Weil group $E(\mathbb {Q})$ of an elliptic curve $E$ over $\mathbb Q$, we give a lower bound of the class number of the number field $\mathbb {Q}(E[p^{n}])$ generated by $p^n$-division points of $E$ when the curve $E$ does not possess a $p$-adic point of order $p: E(\mathbb {Q}_p)[p]=0$.

This paper proves a new regularity criterion for a density-dependent incompressible liquid crystals model with vacuum but without any compatibility condition.

For a positive integer $d$, the $d$-dimensional Chebyshev-Frolov lattice is the $\mathbb Z$-lattice in $\mathbb {R}^d$ generated by the Vandermonde matrix associated to the roots of the $d$-dimensional Chebyshev polynomial. It is important to enumerate the points from the Chebyshev-Frolov lattices in axis-parallel boxes when $d=2^n$ for a non-negative integer $n$, since the points are used as the nodes of Frolov’s cubature formula, which achieves the optimal rate of convergence for many spaces of functions with bounded mixed derivatives and compact support. Kacwin, Oettershagen and Ullrich suggested an enumeration algorithm for such points and later Kacwin improved it, which are claimed to be e‰cient up to dimension $d = 16$. In this paper we suggest a new algorithm which enumerates such points in realistic time for $d = 2^n$, up to $d = 32$. Our algorithm is faster than theirs by a constant factor.

We study the set $R$ of nonplanar rational curves of degree $d \lt q + 2$ on a smooth Hermitian surface $X$ of degree $q + 1$ defined over an algebraically closed field of characteristic $p > 0$, where $q$ is a power of $p$. We prove that $R$ is the empty set when $d \lt q + 1$. In the case where $d = q + 1$, we count the number of elements of $R$ by showing that the group of projective automorphisms of $X$ acts transitively on $R$ and by determining the stabilizer subgroup. In the special case where $X$ is the Fermat surface, we present an element of $R$ explicitly.