For given group schemes $\mathcal{G}^{(\lambda_i)}\ (i=1,2,\ldots)$ deforming the additive group scheme $\mathbf{G}_a$ to the multiplicative group scheme $\mathbf{G}_m$, T. Sekiguchi and N. Suwa constructed extensions: $$0\rightarrow\mathcal{G}^{(\lambda_2)}\rightarrow\mathcal{E}^{(\lambda_1,\lambda_2)}\rightarrow\mathcal{G}^{(\lambda_1)}\rightarrow0,\ \ldots,\ 0\rightarrow\mathcal{G}^{(\lambda_n)}\rightarrow\mathcal{E}^{(\lambda_1,\ldots,\lambda_n)}\rightarrow\mathcal{E}^{(\lambda_1,\ldots,\lambda_{n-1})}\rightarrow0,\ldots$$ inductively, by calculating the group of extensions $\mathrm{Ext}^1(\mathcal{E}^{(\lambda_1,\ldots,\lambda_{n-1})},\mathcal{G}^{(\lambda_n)})$. Here we treat the group $\mathrm{Ext}^1(\mathcal{G}^{(\lambda_0)},\mathcal{E}^{(\lambda_1,\ldots,\lambda_n)})$ of extensions in the case of $n=2,\ 3$. The case of $n=2$ was studied by D. Horikawa.

Let $H(k)$ be the group of all homotopy classes of $k$-string links. It has been proved that $f,g\in H(k)$ have the same closure if and only if there is $\beta\in S_k(1)$ such that $\beta\cdot f=g$, where $S_k(1)$ is the stabilizer of 1 for a certain action of the group $H(2k)$ on the set $H(k)$. If $\beta\in S_k(1)$, Artin's automorphism $\overline{\beta}$, induced by $\beta$ on $RF(2k)$, the reduced free group in $2k$ generators, induces an automorphism $\overline{\overline{\beta}}\in A(RF(k))$, the group of all automorphisms of $RF(k)$ that send each generator to one of its conjugates. This can be used to compare the homotopy classes of links obtained by closing $f$ and $g$. The association $\beta\mapsto \overline{\overline{\beta}}$ is a homomorphism from $S_k(1)$ to $A(RF(k))$. In this paper we determine its kernel.

We give a formula for generalized Eulerian numbers, prove monotonicity of sequences of certain ratios of the Eulerian numbers, and apply these results to obtain a new proof that the natural symmetric measure for the Bratteli-Vershik {dynamical} system based on the Euler graph is the unique fully supported invariant ergodic Borel probability measure. Key ingredients of the proof are a two-dimensional induction argument and a one-to-one correspondence between most paths from two vertices at the same level to another vertex.

Let $S$ be a finitely generated standard multigraded algebra over an Artinian local ring $A$; $M$ a finitely generated multigraded $S$-module. This paper first investigates the relationship between the multiplicity and mixed multiplicities of $M$. Next, we give some applications to multigraded fiber cones.

We give an optimal upper bound for the eigenvalues of the Hill operator with a distributional coefficient.

O. Kobayashi [6] in 2007 proved that $C^1$-mappings preserving anharmonic ratio are Moebius transformations. We strengthen his theorem and prove that the requirement of differentiability and even of injectivity can be omitted.

Charts are oriented labeled graphs in a disk which correspond to surface braids. C-moves are local modifications of charts in a disk, which induces an ambient isotopy between the closures of the corresponding two surface braids. A chart is minimal if its complexity is minimal among the charts which are modified from the chart by C-moves. We investigate a disk whose boundary consists of edges of the same label, called a $k$-angled disk, for a minimal chart. In this paper we investigate 2-angled disks and 3-angled disks containing at most one white vertex in their interiors for a minimal chart.

In this paper, we will establish the correspondence between the tempered distributions supported on a regular closed set and the space of the solutions of Bloch equations with some conditions on its support.

We give a reduction theorem for the codimension of a compact $n$-dimensional minimal proper $CR$ submanifold $M$ immersed in a complex projective space $CP^m$ with complex structure $J$, under the assumption that the Ricci curvature of $M$ is equal to or greater than $n-1$. Moreover, we classify compact $n$-dimensional minimal $CR$ submanifolds whose Ricci tensor $S$ satisfies $S(X,X)\geq (n-1)g(X,X)+kg(PX,PX)$, $k=0,1,2$, for any vector field $X$ tangent to $M$, where $PX$ is the tangential part of $JX$.

In this paper we study Killing helices, which are helices generated by Killing vector fields, of order 4 on a complex projective space whose first and third geodesic curvatures coincide. We construct a natural foliation structure on the set of all congruence classes of these helices. Our study shows that the moduli space of circles is canonically embedded into the moduli space of Killing helices of proper order 4 on a complex projective space.

This paper studies new classes of infinitely divisible distributions on $\mathbf{R}^d$. Firstly, the connecting classes with a continuous parameter between the Jurek class and the class of selfdecomposable distributions are revisited. Secondly, the range of the parameter is extended to construct new classes and characterizations in terms of stochastic integrals with respect to Lévy processes are given. Finally, the nested subclasses of those classes are discussed and characterized in two ways: One is by stochastic integral representations and another is in terms of Lévy measures.

Chan ([2], [3]) considered some continued fraction expansions related to random Fibonacci-type sequences. A Wirsing-type approach to the Perron-Frobenius operator of the associated transformation under its invariant measure allows us to study the optimality of the convergence rate. Actually, we obtain upper and lower bounds of the convergence rate which provide a near-optimal solution to the Gauss-Kuzmin-Lévy problem.

A simple graph $G=(V,E)$ admits an $H$-covering if every edge in $E$ belongs to a subgraph of $G$ isomorphic to $H$. We say that $G$ is $H$-magic if there is a total labeling $f:V\cup E\rightarrow\{1,2,3,\ldots,|V|+|E|\}$ such that for each subgraph $H'=(V',E')$ of $G$ isomorphic to $H$, $s(f)=\sum_{v\in V'} f(v)+\sum_{e\in E'} f(e)$ is constant. When $f(V)=\{1,2,\ldots,|V|\}$, then $G$ is said to be $H$-supermagic. In this case, the $H$-supermagic strength of $G$ is defined as the minimum of all $s(f)$ where the minimum is taken over all $H$-supermagic labelings $f$ of $G$, and is denoted by $SM_H(G)$. In this paper we find the $C_k$-supermagic strength of k-polygonal snakes of any length and $H$-supermagic strength of a chain of an arbitrary 2-connected simple graph $H$. Also we make a conjecture regarding the $P_h$-supermagic strength of $P_n$ for $2 \leq h \leq n$.

We define braid presentation of edge-oriented spatial graphs as a natural generalization of braid presentation of oriented links. We show that every spatial graph has a braid presentation. For an oriented link, it is known that the braid index is equal to the minimal number of Seifert circles. We show that an analogy does not hold for spatial graphs.

We define the IH-complex on the set of spatial trivalent graphs by using the IH-move, which is a local spatial move appeared in a study of knotted handlebodies. The IH-distance between two spatial trivalent graphs is defined by the minimal number of IH-moves needed to transform one into the other. It gives a distance function on the IH-complex. We give a lower bound for the IH-distance, and evaluate it.

We compute the limit of the colored Jones invariant of a prime link, which gives the first evidence for Volume Conjecture of a link whose complement decomposes into two hyperbolic pieces.