# 数式導入 / Mathematical expression insertion

\begin{align} \overline{x} &= \frac{x_1 + x_2 + \cdots + x_n}{n} \\ &= \frac{1}{n} \sum_{k = 1}^n x_k \end{align}

このように「ビジュアル」→「LaTex数式記入(赤枠)」によりHP上で綺麗な数式を表示することができます。

***　確認サイト　***
aTeX – コマンド一覧(外部サイト)
http://www1.kiy.jp/~yoka/LaTeX/latex.html
MathJaxによる数式表示(外部サイト)
https://oku.edu.mie-u.ac.jp/~okumura/javascript/mathjax.html
MathJaxの使い方(外部サイト)
http://gilbert.ninja-web.net/math/mathjax1.html

My publication list in MathSciNet

○Some consequences from Proper Forcing Axiom together with large continuum and the negation of Martin’s Axiom,Journal of the Mathematical Society of Japan, accepted, Nov. 2015.

Recently, David Asperó and Miguel Angel Mota discovered a new method of iterated forcing using models as side conditions [Forcing consequences of PFA together with the continuum large, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6103-6129]. he side condition method with modelswas introduced by Stevo Todorčević in the 1980s. The Asperó-Mota iteration enables us to force some $\Pi_2$-statements over $H(\aleph_2)$ with the continuum greater than $\aleph_2$. In this article, by using the Asperó-Mota iteration, we prove that it is consistent that $\mho$ fails, there are no weak club guessing ladder systems, $\mathfrak p=\mathrm{add}({\mathcal N})=2^{\aleph_0}>\aleph_2$ and $\mathsf{MA}_{\aleph_1}$ fails.

○Keeping the covering number of the null ideal small,Fund. Math. 231 (2015), 139-159.

It is proved that ideal-based forcings with the side condition method of Todorčević add no random reals. By applying Judah-Repický’s preservation theorem, it is consistent with the covering number of the null ideal equal to $\aleph_1$ that there are no $S$-spaces, every poset of uniform density $\aleph_1$ adds $\aleph_1$ Cohen reals, there are only five cofinal types of directed posets of size $\aleph_1$, and so on. This extends the previous work due to Zapletal　[Keeping additivity of the null ideal small, Proc. Amer. Math. Soc.125 (1997), no. 6, 2443-2451].
Chodounský and Zapletal developed this result in

Club-isomorphisms of Aronszajn trees in $\mathsf{PFA}(S)[S]$ Notre Dame Journal of Formal Logic, accepted, October 2014.

○$\mathbb{P}_{\rm max}$ variations related to slaloms, Math. Log. Q. 52 (2006), no. 2, 203-216.

We prove the iteration lemmata, which are the key lemmata to show that extensions by $\mathbb{P}_{\rm max}$ variations satisfy absoluteness for $\Pi_2$-statements in the structure for some set of reals in $L(\mathbb R)$, for the following statements: (1) The cofinality of the null ideal is $\aleph_1$. (2) There exists a good basis of the strong measure zero ideal.

$\mathcal{P}(\omega)/{\rm fin}$　，　$(\omega_1,\mathfrak c)$　，　$\dot{C}$　 ，　$\mathsf R_{1,\aleph_1}$　，　$NS_{\omega_{1}}$

***　おまけ　***

$x = \frac{-b \pm \sqrt{b^2-4ac} }{2a}$ , $\frac{\pi}{2} = \left( \int_{0}^{\infty} \frac{\sin x}{\sqrt{x}} dx \right)^2 = \sum_{k=0}^{\infty} \frac{(2k)!}{2^{2k}(k!)^2} \frac{1}{2k+1} = \prod_{k=1}^{\infty} \frac{4k^2}{4k^2 – 1}$

$e^{i\theta} = \cos\theta + i\sin\theta$ , $J_\alpha(x) = \sum\limits_{m=0}^\infty \frac{(-1)^m}{m! \, \Gamma(m + \alpha + 1)}{\left({\frac{x}{2}}\right)}^{2 m + \alpha}$

$\begin{eqnarray} A =\left[ \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{array} \right] \end{eqnarray}$