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| <title>MathJax Example</title> | |
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| tex: { | |
| inlineMath: [['$', '$'], ['\(', '\)']] | |
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| <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> | |
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| <body> | |
| <p> $T_{text}$ </p> | |
| <p> $p_{xy}$ </p> | |
| <p> $S_{text}$ </p> | |
| <p> $P(I_{0},I_{1},...,I_{F}|I_{0})=\prod_{i=1}^{F}P(I_{i}|I_{0}),$ </p> | |
| <p> $w_{p_{xy}}$ </p> | |
| <p> $L_{Transformer}=-\sum p(I_{pred})\log q(I_{GT}),$ </p> | |
| <p> $Dist(Z_{q_{ij}},Z_{\theta})=\left\|Z_{q_{ij}}-Z_{\theta}\right\|_{2},$ </p> | |
| <p> $S_{style}$ </p> | |
| <p> $q(I_{GT})$ </p> | |
| <p> $P_{i_{xy}}=\frac{\sum_{p_{xy}>0}w_{p_{xy}}p_{xy}}{\sum_{p_{xy}>0}w_{p_{xy}}},$ </p> | |
| <p> $\displaystyle\text{Queries}^{\mathcal{P}_{n}}_{A_{n}}(G)\leq c\cdot\mathrm{RAC% | |
| }(\mathcal{P}_{n},G)$ </p> | |
| <p> $\displaystyle\text{Queries}^{\mathcal{P}_{n}}_{A_{n}}(f)\leq c\cdot\mathrm{RAC% | |
| }(\mathcal{P}_{n},f)$ </p> | |
| <p> $\mathrm{RAC}(\mathcal{P}_{H},\Delta)=O(n^{1/2}\log n)$ </p> | |
| <p> $H=S_{k}$ </p> | |
| <p> $v_{\sqrt{n}}$ </p> | |
| <p> $H=([h],E)$ </p> | |
| <p> $O(1/n^{c})$ </p> | |
| <p> $n/{2.2^{i}}$ </p> | |
| <p> $\Pr(E_{j}|\neg E_{1}\wedge\neg E_{2}\wedge\ldots\wedge\neg E_{j-1})=\Theta% | |
| \left(\frac{2^{t}}{1.1^{t}}\cdot\frac{1}{n^{1/10}}\right)$ </p> | |
| <p> $\mathrm{RAC}(\mathcal{P}_{H},\Delta)=O(n^{1/2})$ </p> | |
| <p> $\displaystyle\mathrm{RAC}(\mathcal{P},G)=\min_{A\in\mathcal{A}_{\mathcal{P}}}% | |
| \max_{\pi\in\Gamma}\text{Queries}^{\mathcal{P}}_{A}(\pi(G)),$ </p> | |
| <p> $|H|/(|B|-i)=\Theta(1/n)$ </p> | |
| <p> $x\neq y\in[n]$ </p> | |
| <p> $n^{1-c}/{1.1^{t}}$ </p> | |
| <p> $|C|=\Theta(n/\log n)$ </p> | |
| <p> $1\leq i\leq\sqrt{n}-1$ </p> | |
| <p> $I\in\mathcal{I}_{i}\setminus\mathcal{L}_{i}$ </p> | |
| <p> $\mathop{\mathbb{E}}_{G\leftarrow\Delta}\text{Queries}_{A}(G)=\Omega(n^{1/10}% | |
| \log(n))$ </p> | |
| <p> $n^{0.9}$ </p> | |
| <p> $\Pr(E)=1-1/\Theta(n^{1/4})$ </p> | |
| <p> $C\sqrt{n}$ </p> | |
| <p> $(1+o(1))p$ </p> | |
| <p> $\mathop{\mathbb{E}}_{G\leftarrow\Delta}\text{Queries}_{A}(G)=\Omega(n)$ </p> | |
| <p> $|V_{t}|=n-O(t)\geq n/2$ </p> | |
| <p> $\alpha n^{1/4}$ </p> | |
| <p> $I\in\mathcal{L}_{i}$ </p> | |
| <p> $O(n^{0.1})$ </p> | |
| <p> $\mathop{\mathbb{E}}[I_{i}]\leq\alpha q_{i}/n^{0.1}$ </p> | |
| <p> $x_{1},\ldots x_{h}\in[n]$ </p> | |
| <p> $\{x_{1},\ldots,x_{h}\}$ </p> | |
| <p> $\Pr(E)=1-O(1/\sqrt{n})$ </p> | |
| <p> $b_{t}=n^{9/10}/1.1^{t}$ </p> | |
| <p> $\mathop{\mathbb{E}}_{G\leftarrow\Delta}\mathrm{RAC}(\mathcal{P},G)=O(n^{1/10})$ </p> | |
| <p> $|\mathcal{L}_{i}|\leq\frac{\kappa n^{0.1}}{2^{i-2}}.$ </p> | |
| <p> $I\in\mathcal{R}_{i}$ </p> | |
| <p> $O(1/p)=O(1/n^{1/4})$ </p> | |
| <p> $H_{a,b,c}$ </p> | |
| <p> $y_{i},y_{j},y_{l}$ </p> | |
| <p> $2^{t}\cdot n^{9/10}/1.1^{t}$ </p> | |
| <p> $\text{deg}(v)=1$ </p> | |
| <p> $H=P_{k}$ </p> | |
| <p> $\mathcal{A}=\{A_{n}\}_{n\in\mathbb{N}}$ </p> | |
| <p> $f\circ\pi$ </p> | |
| <p> $V_{i}=\bigcup_{I\in\mathcal{I}_{i}}V(I)$ </p> | |
| <p> $Cn^{0.1}\log n$ </p> | |
| <p> $Q\subset I$ </p> | |
| <p> $\mathcal{A}_{\mathcal{P}}$ </p> | |
| <p> $\{G_{n}\}_{n\in\mathbb{N}}$ </p> | |
| <p> $p\in\{p_{i_{1}},\ldots,p_{i_{k}}\}$ </p> | |
| <p> $f(x_{u})=x_{v}$ </p> | |
| <p> $2^{i}<2^{t}$ </p> | |
| <p> $B(t)/|V_{t}|\leq 1/n^{0.1+\Omega(1)}$ </p> | |
| <p> $\mathcal{P}_{S_{k}}$ </p> | |
| <p> $\alpha\kappa$ </p> | |
| <p> $n^{1-c}/1.1^{i}$ </p> | |
| <p> $B(n)\leq n^{9/10}/1.1^{\log(n)/1000}=n^{9/10-\Omega(1)}$ </p> | |
| <p> $G^{\pi}=(V,E^{\pi})$ </p> | |
| <p> $2^{i-2}$ </p> | |
| <p> $f(x_{1})=\ldots=f(x_{k})$ </p> | |
| <p> $2/2^{i-1}=4/2^{i}$ </p> | |
| <p> $O(n^{3/4}\cdot\log^{4}n/n^{1/4})=o(n^{3/4})$ </p> | |
| <p> $B=[n]\setminus C$ </p> | |
| <p> $\displaystyle\text{Queries}^{\mathcal{P}_{n}}_{A_{n}}(f_{n})\geq\omega(n)\cdot% | |
| \mathrm{RAC}(\mathcal{P}_{n},f_{n}).$ </p> | |
| <p> $b_{i}=n^{9/10}/1.1^{i}$ </p> | |
| <p> $f\in\Delta$ </p> | |
| <p> $\frac{2^{t}}{1.1^{t}}\cdot\frac{1.1^{t}\cdot n}{n^{1-c}\cdot 2^{t}}=n^{c}.$ </p> | |
| <p> $\mathop{\mathbb{E}}_{f\leftarrow\Delta}RAC(\mathcal{P}_{H},f)=O(n^{3/4})$ </p> | |
| <p> $\frac{n^{1-c}\cdot 2^{t}}{n\cdot 1.1^{t}}$ </p> | |
| <p> $\Omega(n^{0.1}\log n)$ </p> | |
| <p> $2^{t}+4$ </p> | |
| <p> $\mathop{\mathbb{E}}\left[X_{j}|\neg E_{1}\wedge\neg E_{2}\wedge\ldots\wedge% | |
| \neg E_{j-1}\right]=O\left(\frac{2^{t}}{1.1^{t}}\right)$ </p> | |
| <p> $c=1/10$ </p> | |
| <p> $\Pr({E_{\text{long}}})=O(\kappa)$ </p> | |
| <p> $\frac{a_{i}\cdot 2^{i}}{n}=\frac{1}{1.1^{i}}$ </p> | |
| <p> $\omega\colon\mathbb{N}\to\mathbb{N}$ </p> | |
| <p> $C_{i_{1}},\ldots,C_{i_{T}}$ </p> | |
| <p> $C=\bigcup_{i=1}^{N}C_{i}$ </p> | |
| <p> $\mathcal{P}_{H}$ </p> | |
| <p> $\mathop{\mathbb{E}}[\text{\# red vertices encountered}]\leq\sum_{\frac{1}{1000% | |
| }\log n\leq i\leq\frac{1}{100}\log n}\alpha\frac{q_{i}}{n^{0.1}}\leq\frac{% | |
| \alpha}{n^{0.1}}cn^{0.1}\log n\leq\alpha c\log n,$ </p> | |
| <p> $N=\alpha n^{1/4}/\log(n)$ </p> | |
| <p> $\pi\colon[n]\to[n]$ </p> | |
| <p> $\bigcup_{i=1}^{N}C_{i}$ </p> | |
| <p> $\Omega(n^{c}\log n)$ </p> | |
| <p> $\Pr({E_{\text{long}}})$ </p> | |
| <p> $\Pr({E_{\text{long}}})\leq\sum_{I\in\mathcal{L}_{i}}\Pr(I\in\mathcal{R}_{i})% | |
| \leq|\mathcal{L}_{i}|\cdot(1+o(1))p\leq(1+o(1))\cdot\frac{\kappa n^{0.1}}{2^{i% | |
| -2}}\cdot\frac{2^{i}}{n^{0.1}}=O(\kappa).$ </p> | |
| <p> $\pi\colon V\to V$ </p> | |
| <p> $G=(V,E)\in\mathcal{P}$ </p> | |
| <p> $i=1,...,\Theta(\sqrt{n}\log n)$ </p> | |
| <p> ${E_{\text{short}}}$ </p> | |
| <p> $n^{9/10}/1.1^{t}$ </p> | |
| </body> | |
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