We define a total order, which we call rooted order, on minimal generating set of $J(P_n)^s$ where $J(P_n)$ is the cover ideal of a path graph on $n$ vertices. We show that each power of a cover ideal of a path has linear quotients with respect to the rooted order. Along the way, we characterize minimal generating set of $J(P_n)^s$ for $s\geq 3$ in terms of minimal generating set of $J(P_n)^2$. We also discuss the extension of the concept of rooted order to chordal graphs. Computational examples suggest that such order gives linear quotients for powers of cover ideals of chordal graphs as well.

We give an infinite presentation for the mapping class group of a non-orientable surface with boundary components. The presentation is a generalization of the presentation given by the second author [16]. We also give a finite presentation for the mapping class group to obtain the infinite presentation.

When we study degenerations of Riemann surfaces from a topological viewpoint, the topological monodromies play a very important role. In this paper, as an analogy, we introduce the concept of “topological monodromies of splitting families" for degenerations of Riemann surfaces, and their “monodromy assortments". We show that the monodromy assortments of barking families associated with tame simple crusts act as a pseudo-periodic homeomorphism of negative twist on each irreducible component of the main fibers. As an application of our results, we show an interesting example of two splitting families for one degeneration that have different topological monodromies, although they give the same splitting.

The present paper show that Conway-Coxeter friezes of zigzag type are characterized by (unoriented) rational links. As an application of this characterization Jones polynomial can be defined for Conway-Coxeter friezes of zigzag type. This gives a new method for computing the Jones polynomial for oriented rational links.

We construct a genus zero PALF structure on each of plugs introduced by Akbulut and Yasui and describe the monodromy as a positive factorization in the mapping class group of a fiber.

We describe all the trees with the property that the corresponding edge ideal of the square of the tree has a linear resolution. As a consequence, we give a complete characterization of those trees $T$ for which the square is co-chordal, that is the complement of the square, $(T^2)^c$, is a chordal graph. For particular classes of trees such as paths and double brooms, we determine the Krull dimension and the projective dimension.

For a symmetrizable GCM $C$ and its symmetrizer $D$, Geiss-Leclerc-Schröer [Invent. Math. 209 (2017)] has introduced a generalized preprojective algebra $\Pi$ associated to $C$ and $D$, that contains a class of modules, called locally free modules. We show that any basic support $\tau$-tilting $\Pi$-module is locally free and gives a classification theorem of torsion-free classes in $\operatorname{\mathbf{rep}}{\Pi}$ as the generalization of the work of Mizuno [Math. Z. 277 (2014)].

Let $f_{1} : (\Bbb{R}^{n}, \mathbf{0}_{n}) \rightarrow (\Bbb{R}^{p}, \mathbf{0}_{p})$ and $f_{2} : (\Bbb{R}^{m}, \mathbf{0}_{m}) \rightarrow (\Bbb{R}^{p}, \mathbf{0}_{p})$ be analytic germs of independent variables, where $n, m \geq p \geq 2$. In this paper, we assume that $f_{1}, f_{2}$ and $f = f_{1} + f_{2}$ satisfy $a_{f}$-condition. Then we show that the tubular Milnor fiber of $f$ is homotopy equivalent to the join of tubular Milnor fibers of $f_1$ and $f_2$. If $p = 2$, the monodromy of the tubular Milnor fibration of $f$ is equal to the join of the monodromies of the tubular Milnor fibrations of $f_1$ and $f_2$ up to homotopy.

The theory of Stark systems due to Burns, Sakamoto, and Sano is an important tool toward main conjectures in Iwasawa theory. In this paper, we propose a new perspective of their results, which produces more refined consequences. As a principal application, we prove one divisibility of the equivariant main conjecture for elliptic curves, under certain conditions without $\mu = 0$ hypothesis.

We show that the entropy of the $\alpha$-continued fraction map w.r.t the absolutely continuous invariant probability measure is strictly less than that of the nearest integer continued fraction map when $0 \lt \alpha \lt \frac{3 - \sqrt{5}}{2}$. This answers a question by C. Kraaikamp, T. A. Schmidt, and W. Steiner (2012). To prove this result we make use of the notion of the geodesic continued fractions introduced by A. F. Beardon, M. Hockman, and I. Short (2012).

In this paper, we consider the Cauchy problem of a class of higher order Schrödinger type equations with constant coefficients. By employing the energy inequality, we show the $L^2$ well-posedness, the parabolic smoothing and a breakdown of the persistence of regularity. We classify this class of equations into three types on the basis of their smoothing property.