We prove that the category of $\mathbb{Z}_2 ^n$-manifolds has all finite products. Further, we show a ^n$-manifold (resp., ^n$-morphism) can be reconstructed from its algebra global ^n$-functions morphism between ^n$-function algebras). These results are importance in study ^n$ Lie groups. The investigation is more challenging, since completed tensor product structure sheafs two not sheaf. rely...

For a semistable degeneration of surfaces without triple point, we show that two models Hilbert scheme points the family, Gulbrandsen-Halle-Hulek given in [GHH] and one by author [N], are actually isomorphic.

Simon (Theoret. Comput. Sci. 72 (1990) 65–94) has proved that every morphism from a free semigroup to a $nite semigroup S admits a Ramseyan factorization forest of height at most 9|S|. In this paper, we prove the same result of Simon with an improved bound of 7|S|. We provide a simple algorithm for constructing a factorization forest. In addition, we show that the algorithm cannot be improved s...

We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are either classical patterns 1-2, 2-1, 1-1· · · -1, or arbitrary generalized patterns without internal dashes, in which repetitions of letters are allowed. In p...

We first provide here a very short proof of a refinement of a theorem of Kodiyalam and Cutkosky, Herzog and Trung on the regularity of powers of ideals. This result implies a conjecture of Hà and generalizes a result of Eisenbud and Harris concerning the case of ideals primary for the graded maximal ideal in a standard graded algebra over a field. It also implies a new result on the regularitie...

Let G be a connected, semisimple algebraic group defined over an algebraically closed field k of characteristic p > 0. We assume that G is defined and split over the prime field k0. In general, for any positive integer r, and any affine fc-group scheme H defined over k0, the r-th infinitesimal subgroup scheme Hr of H is defined to be the (scheme-theoretic) kernel of the r-th power of the Froben...