The main purpose of this article is to establish the CKN-type inequalities for all $\alpha\in \mathbf{R}$ and to study the relating matters systematically. Roughly speaking, we shall discuss about the characterizations of the CKN-type inequalities for all $\alpha\in \mathbf{R}$ as the variational problems, the existence and nonexistence of the extremal solutions to these variational problems in proper spaces, the exact values and the assymptotic behaviors of the best constants $S(p,q,\alpha)$ and $C(p,q)$.

We consider an area minimizing scheme for anisotropic mean curvature flow originally due to Chambolle (2004). We show the convergence of the scheme to anisotropic mean curvature flow in the sense of Hausdorff distance by the level set method provided that no fattening occurs.

We observe properties of coefficients of certain basis elements for the space of weakly holomorphic modular forms of weight 2 for $SL_{2}(\mathbf{Z})$. Moreover we show that these coefficients are often highly divisible by the primes 2, 3, 5, 7, 11.

Let $p$ be a prime number with $p \equiv 3 \bmod 4$ and $q=(p-1)/2$. Let $k=\mathbf{Q}(\sqrt{-p})$ and $k_{\infty}/k$ be the cyclotomic $\mathbf{Z}_{p}$-extension. Denote by $h_{n}^{-}$ the relative class number of the $n$-th layer $k_{n}$. Let $\ell$ be a prime number with $\ell \neq p$. We show that, for any $n \geq 1$, $\ell$ does not divide $h_{n}^{-}/h_{n-1}^{-}$ (resp. $h_{1}^{-}/h_{0}^{-}$) if $\ell$ is a primitive root modulo $p^{2}$ (resp. $p$) and $\ell \geq q-2$ (resp. $\ell \geq q-6$). Further, we show with the help of computer that when $p < 10000$ and $n \leq 100$, $\ell$ does not divide $h_{n}^{-}/h_{n-1}^{-}$ (resp. $h_{1}^{-}/h_{0}^{-}$) for any prime $\ell$ which is a primitive root modulo $p^{2}$ (resp. $p$).

We give a simple proof for the fact that certain algebra generators of the $\mathrm{mod}\,2$ cohomology of classifying spaces of exceptional Lie groups are given by Chern classes and Stiefel-Whitney classes of certain representations.