Let $\Gamma$ be a minimal chart with exactly seven white vertices. In this paper, we show that $\Gamma$ is a chart of type $(7),(5,2),(4,3),(3,2,2)$ or $(2,3,2)$ if necessary we change the labels. We investigate minimal charts with loops or lenses.

We will introduce some combinatorics for given words. Such combinatorics can essentially determine the exact information of letters as well as the patterns of words. This method can induce a characterization of the so-called local indistinguishability for one dimensional tilings, which allows us to have a new development for tiling bialgebras. Using those combinatorics associated with words and one dimensional tilings, we can obtain their combinatorial spectra as certain sets of functions or positive real numbers. We will also discuss higher dimensional tilings. Furthermore, we will try to compute some genome examples.

In the present paper, we develop the theory of log homotopy exact sequences associated to proper log smooth morphisms and morphisms whose characteristic sheaves are locally constant with stalks isomorphic to the monoid of natural numbers. In the process of developing this theory, we also show the existence of a logarithmic version of the Stein factorization and develop the theory of algebraization of log formal schemes.

We prove Singer's formulas and unstable acyclic relations for the mod $2$ Steenrod algebra, which are important tools for the computations of products. These are given by analogy of the corresponding dual results for the lambda algebra.

We consider a weak convergence of the power divergence family of statistics $\{T_{\lambda}(\boldsymbol{Y}),\lambda\in\mathbb{R}\}$ constructed from the multinomial distribution of degree $k$, to chi-squared distribution with $k-1$ degrees of freedom. We show that

\Pr(T_{\lambda}(\boldsymbol{Y})<c)=G_{k-1}(c)+ O(n^{-1+ 1/k})

where $G_r(c)$ is the distribution function of a chi-squared variable with $r$ degrees of freedom. In the proof we use E. Hlawka's theorem (1950) on the approximation of a number of integer points in a convex set with a closed smooth boundary by a volume of the set.