Let $\rm Mod(\Sigma_{\textit{g,p}})$ be the mapping class group of a closed oriented surface $\Sigma_{g,p}$ of genus $g\geq 1$ with $p$ punctures. Wajnryb proved that $\rm Mod({\Sigma_{\textit{g},0}})$ is generated by two elements. Korkmaz proved that one of these generators may be taken to be a Dehn twist. Korkmaz also proved the same result in the case of $\rm Mod(\Sigma_{\textit{g},1})$. For $p\geq 2$, we prove that $\rm Mod(\Sigma_{\textit{g,p}})$ is generated by three elements.

We give a proof of McShane’s identity in "Weierstrass points and simple geodesics," Bull. London Math. Soc., 36 (2004), 181–187, based on the investigation on the arrangement of axes of simple hyperbolic elements in a once punctured torus group which are represented by palindromic words. Our argument includes a short proof of the fact that the linear measure of the infinitesimal Birman-Series set is zero.

Combining the finite form of Jacobi’s triple product identity with the $q$-Gauss summation theorem, we present a new and unified proof for the two transformation lemmas due to Andrews (1981). The same approach is then utilized to establish two further transformations from unilateral to bilateral series. They are employed to review forty identities of Rogers–Ramanujan type with quintuple products.

Li (1991) introduced the so called sliced inverse regression (SIR) which reduces the dimension of the input variable in regression analysis, without any modelfitting process. Akita et al. (2009) proposed an improvement of SIR, called SIR, which uses conditional covariate matrices. In this paper, developing the approach of PCASIR, we propose yet another improvement of SIR, which we call PCA-SIR2. Simulation results produced by SIR, PCA-SIR and PCA-SIR2 are compared.

The $L^{(\alpha)}$-harmonic function is the solution of the parabolic operator $L^{(\alpha)}= \partial_{t}+(-\Delta_{x})^{\alpha}$. We study a function space $\widetilde{{\cal B}}_{\alpha}(\sigma)$ consisting of $L^{(\alpha)}$-harmonic functions of parabolic Bloch type. In particular, we give a reproducing formula for functions in $\widetilde{{\cal B}}_{\alpha}(\sigma)$. Furthermore, we study the fractional calculus on $\widetilde{{\cal B}}_{\alpha}(\sigma)$. As an application, we also give a reproducing formula with fractional orders for functions in $\widetilde{{\cal B}}_{\alpha}(\sigma)$. Moreover, we investigate the dual and pre-dual spaces of function spaces of parabolic Bloch type.

In this paper, we give a complete classification of 3-bridge arborescent links.