The purpose of this article is to construct a family of uncountably many non-periodic 3-Archimedean tilings with 6-fold rotational symmetry, which admit three types of vertex configurations by regular triangles and squares.

Generating functions of Box and Ball System (BBS) are defined and studied. When the number of balls is finite, we show that the generating function is a rational function. When there are infinitely many balls, we conjecture that the generating function is rational if and only if the BBS is semi-periodic. We prove the conjecture in a special case. We also study the generating function of the BBS with a limited cart, including semi-periodic cases.

The main result of this paper classifies two-point homogeneous quandles with cardinality of prime power. More precisely, such quandles are isomorphic to Alexander quandles defined by primitive roots over finite fields. This result classifies all two-point homogeneous finite quandles, by combining with the recent result of Vendramin.

The purpose of this paper is to clarify the conditions for consistency of the log-likelihood-based information criteria in canonical correlation analysis of q- and p-dimensional random vectors when the dimension p is large but does not exceed the sample size. Although the vector of observations is assumed to be normally distributed, we do not know whether the underlying distribution is actually normal. Therefore, conditions for consistency are evaluated in a high-dimensional asymptotic framework when the underlying distribution is not normal.

Let l be a prime number. In the paper, we study the outer Galois action on the profinite and the relative pro-l completions of mapping class groups of pointed orientable topological surfaces. In the profinite case, we prove that the outer Galois action is faithful. In the pro-l case, we prove that the kernel of the outer Galois action has certain stability properties with respect to the genus and the number of punctures. Also, we prove a variant of the above results for arbitrary families of curves.