Util on aiken's blog https://aikenh.cn/categories/util/ Recent content in Util on aiken's blog Hugo -- 0.137.0 en-us Tue, 15 Sep 2020 11:30:03 +0000 90080597450814464 72439538841050112 Latex Function Test https://aikenh.cn/posts/latex_test/ Tue, 15 Sep 2020 11:30:03 +0000 https://aikenh.cn/posts/latex_test/ test website's function is normal 该文档主要目的是用于测试Latex语法对应前端的渲染能力，主要用于测试Hexo站点是否能正常渲染Latex。

Example1: 2 inline in one sentence.

When $a \ne 0$, there are two solutions to $(ax^2 + bx + c = 0)$ and they are $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$ Example2: Matrix Example

$$ \begin{bmatrix} 1&x&x^2\ 1&y&y^2\ 1&z&z^2 \end{bmatrix} \ \begin{bmatrix} 1&x&x^2\\ 1&y&y^2\\ 1&z&z^2 \end{bmatrix} \ vmatrix ||、Bmatrix{}、pmatrix() $$

Example3: the Conditional Formula

$$ f(x)= \begin{cases} 0& \text{x=0}\\ 1& \text{x!=0} \end{cases} $$

Example4: Sprcial Symboy

$$ \lim_{\alpha \rightarrow +\infty} \frac{1}{\alpha(\beta+1)} $$

Example5: Complex Function Which Occurs Error in Much Situation

$$ \begin{gathered} \mathcal{L}_{POD-final} = \frac{\lambda_c}{L-1}\sum_{l=1}^{L-1} \mathcal{L}_{POD-spatial}(f_l^{t-1}(x),f_l^t(x)) + \\ \lambda_f \mathcal{L}_{POD-flat}(f_l^{t-1}(x),f_l^t(x)) \end{gathered} $$

Example6：Mathbb、Text、etc…

$$ \mathcal{L}_{\text {POD-pixel }}\left(\mathbf{h}_{\ell}^{t-1}, \mathbf{h}_{\ell}^{t}\right)=\sum_{c=1}^{C} \sum_{w=1}^{W} \sum_{h=1}^{H}\left\|\mathbf{h}_{\ell, c, w, h}^{t-1}-\mathbf{h}_{\ell, c, w, h}^{t}\right\|^{2} $$

Example7: Multiple Lines of Loss in Incremental Learning

$$ \begin{gathered} \mathcal{L}_{\text {POD-channel }}\left(\mathbf{h}_{\ell}^{t-1}, \mathbf{h}_{\ell}^{t}\right)=\sum_{w=1}^{W} \sum_{h=1}^{H}\left\|\sum_{c=1}^{C} \mathbf{h}_{\ell, c, w, h}^{t-1}-\sum_{c=1}^{C} \mathbf{h}_{\ell, c, w, h}^{t}\right\|^{2} \\ \mathcal{L}_{\text {POD-gap }}\left(\mathbf{h}_{\ell}^{t-1}, \mathbf{h}_{\ell}^{t}\right)=\sum_{c=1}^{C}\left\|\sum_{w=1}^{W} \sum_{h=1}^{H} \mathbf{h}_{\ell, c, w, h}^{t-1}-\sum_{w=1}^{W} \sum_{h=1}^{H} \mathbf{h}_{\ell, c, w, h}^{t}\right\|^{2} \\ \mathcal{L}_{\text {POD-width }}\left(\mathbf{h}_{\ell}^{t-1}, \mathbf{h}_{\ell}^{t}\right)=\sum_{c=1}^{C} \sum_{h=1}^{H}\left\|\sum_{w=1}^{W} \mathbf{h}_{\ell, c, w, h}^{t-1}-\sum_{w=1}^{W} \mathbf{h}_{\ell, c, w, h}^{t}\right\|^{2} \end{gathered} $$

如果这些都能正确渲染的话，基本整个文档中的Latex基本渲染应该都没问题，用该文档能验证当前本地渲染的版本是否是正确的。

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