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  1. output_mathjax_example_1.html +124 -0
  2. output_mathjax_example_10.html +137 -0
  3. output_mathjax_example_2.html +149 -0
  4. output_mathjax_example_3.html +146 -0
  5. output_mathjax_example_4.html +142 -0
  6. output_mathjax_example_5.html +136 -0
  7. output_mathjax_example_6.html +139 -0
  8. output_mathjax_example_7.html +146 -0
  9. output_mathjax_example_8.html +141 -0
  10. output_mathjax_example_9.html +123 -0
output_mathjax_example_1.html ADDED
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+ <!DOCTYPE html>
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+ <html>
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+ <head>
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+ <title>MathJax Example</title>
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+ <script>
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+ MathJax = {
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+ tex: {
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+ inlineMath: [['$', '$'], ['\(', '\)']]
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+ },
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+ svg: {
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+ fontCache: 'global'
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+ }
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+ };
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+ </script>
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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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+ </head>
17
+ <body>
18
+ <p> $O(n^{2})$ </p>
19
+ <p> $f$ </p>
20
+ <p> $n$ </p>
21
+ <p> $G(v)$ </p>
22
+ <p> $s_{o}\oplus s_{a}\in\mathbb{V}^{n+m}$ </p>
23
+ <p> $Z\in\mathbb{R}^{m\times d_{\text{token}}}$ </p>
24
+ <p> $E_{\psi}(s)$ </p>
25
+ <p> $\displaystyle=F^{i}(E_{\psi}(s_{o})\oplus\text{Proj}_{\psi}(Z)).$ </p>
26
+ <p> $\displaystyle\cos(v,v_{t}^{image})+\lambda\cos(v,v_{t}^{text})$ </p>
27
+ <p> $\cos(\psi_{i},\psi_{j})$ </p>
28
+ <p> ${}^{4}$ </p>
29
+ <p> $v_{t}^{text}=F^{t}(E_{\psi}(s^{\prime}))$ </p>
30
+ <p> ${}^{*}$ </p>
31
+ <p> $\displaystyle\text{argmax}_{Z}$ </p>
32
+ <p> $\rightarrow$ </p>
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+ <p> $\mathcal{A}(x,t,s_{o})$ </p>
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+ <p> $\displaystyle=F^{i}(E_{\psi}(s_{o}\oplus s_{a}))$ </p>
35
+ <p> ${}^{1}$ </p>
36
+ <p> $\text{Proj}_{\psi}(Z)_{i}=Z_{i}+\text{sg}(\psi_{j}-Z_{i})$ </p>
37
+ <p> $x_{t}$ </p>
38
+ <p> $500\times 20=10000$ </p>
39
+ <p> $w_{i},w_{j}$ </p>
40
+ <p> $v_{t}^{image}\leftarrow F^{i}(x_{t})$ </p>
41
+ <p> $m=4$ </p>
42
+ <p> $s_{a}=E_{\psi}^{-1}(\text{Proj}_{\psi}(Z))$ </p>
43
+ <p> ${}^{5}$ </p>
44
+ <p> $Z_{i}$ </p>
45
+ <p> ${}^{1,*}$ </p>
46
+ <p> $\text{Proj}_{\psi}(Z)$ </p>
47
+ <p> $s$ </p>
48
+ <p> $\displaystyle\text{argmax}_{s_{a}}$ </p>
49
+ <p> $t$ </p>
50
+ <p> $s^{\prime}\leftarrow$ </p>
51
+ <p> $v_{t}^{image}$ </p>
52
+ <p> $5\times 4\times 100=2000$ </p>
53
+ <p> ${}^{1,2}$ </p>
54
+ <p> $\psi\in\mathbb{R}^{|\mathbb{V}|\times d_{\text{token}}}$ </p>
55
+ <p> $bestloss\leftarrow\mathcal{L},bestZ\leftarrow Z$ </p>
56
+ <p> $G$ </p>
57
+ <p> $\lambda=0$ </p>
58
+ <p> $\text{Proj}_{\psi}:\mathbb{R}^{m\times d_{\text{token}}}\rightarrow\mathbb{R}^%
59
+ {m\times d_{\text{token}}}$ </p>
60
+ <p> $i\leftarrow 1$ </p>
61
+ <p> $s\in\mathbb{V}^{*}$ </p>
62
+ <p> $\displaystyle\text{argmax}_{s_{a}}\mathbb{E}_{x\sim G(F^{t}(E_{\psi}(s_{o}%
63
+ \oplus s_{a})))}\mathcal{A}(x,t,s_{o})~{},$ </p>
64
+ <p> $\displaystyle\cos(v,v_{t}^{image})+\lambda\cos(v,v_{t}^{text}),$ </p>
65
+ <p> $\cos(a,b)=\frac{a^{T}b}{\|a\|\|b\|}$ </p>
66
+ <p> $\eta$ </p>
67
+ <p> $512\times 512$ </p>
68
+ <p> $x$ </p>
69
+ <p> $E_{\psi}(s_{o}\oplus s_{a})=E_{\psi}(s_{o})\oplus E_{\psi}(s_{a})$ </p>
70
+ <p> $N$ </p>
71
+ <p> $bestloss>\mathcal{L}$ </p>
72
+ <p> $v_{t}^{image}=F^{i}(x_{t})$ </p>
73
+ <p> $d_{\text{emb}}$ </p>
74
+ <p> $\displaystyle\text{argmax}_{s_{a}}\cos(F^{i}(E_{\psi}(s_{o}\oplus s_{a})),v_{t%
75
+ }).$ </p>
76
+ <p> $s^{\prime}=$ </p>
77
+ <p> ${}^{3,*}$ </p>
78
+ <p> $Z\leftarrow Z-\eta\nabla_{Z}\mathcal{L}$ </p>
79
+ <p> $100$ </p>
80
+ <p> $s_{a}$ </p>
81
+ <p> $s_{o}\oplus s_{a}$ </p>
82
+ <p> $m$ </p>
83
+ <p> $v$ </p>
84
+ <p> $\displaystyle\text{s.t.}\quad v=F^{i}(E_{\psi}(s_{o}\oplus s_{a})),$ </p>
85
+ <p> $\mathbb{V}=\{w_{1},w_{2},\cdots,w_{|\mathbb{V}|}\}$ </p>
86
+ <p> $F^{i}$ </p>
87
+ <p> $\psi$ </p>
88
+ <p> $\displaystyle\text{s.t.}\quad v$ </p>
89
+ <p> $s_{o}$ </p>
90
+ <p> $F^{t}$ </p>
91
+ <p> ${}^{2}$ </p>
92
+ <p> $\oplus$ </p>
93
+ <p> $E_{\psi}(s)_{i}=\psi_{j}$ </p>
94
+ <p> $5\times 4=20$ </p>
95
+ <p> $3\times 100$ </p>
96
+ <p> ${}^{3}$ </p>
97
+ <p> $v\leftarrow F^{t}(E_{\psi}(s_{o})\oplus\text{Proj}_{\psi}(Z))$ </p>
98
+ <p> $\mathcal{L}=-\cos(v,v_{t}^{image})-\lambda\cos(v,v_{t}^{text})$ </p>
99
+ <p> $s_{o}\in\mathbb{V}^{n}$ </p>
100
+ <p> $s_{a}\leftarrow E_{\psi}^{-1}(\text{Proj}_{\psi}(bestZ))$ </p>
101
+ <p> $bestloss\leftarrow\infty,bestZ\leftarrow Z$ </p>
102
+ <p> $\displaystyle=F^{i}(E_{\psi}(s_{o}\oplus E_{\psi}^{-1}(\text{Proj}_{\psi}(Z))))$ </p>
103
+ <p> $t\in\mathbb{V}$ </p>
104
+ <p> $Z$ </p>
105
+ <p> $(\cdot)$ </p>
106
+ <p> $x\sim G(v)$ </p>
107
+ <p> $d_{\text{token}}$ </p>
108
+ <p> $s_{a}\in\mathbb{V}^{m}$ </p>
109
+ <p> $v_{t}$ </p>
110
+ <p> $\lambda$ </p>
111
+ <p> $\mathbb{V}$ </p>
112
+ <p> $w_{j}=s_{i}$ </p>
113
+ <p> $t\in\mathcal{V}$ </p>
114
+ <p> $x\sim G(F^{t}(E_{\psi}(s)))$ </p>
115
+ <p> $E_{\psi}$ </p>
116
+ <p> $j=\text{argmin}_{j^{\prime}}\|\psi_{j^{\prime}}-Z_{i}\|_{2}^{2}$ </p>
117
+ <p> $|s|\times d_{\text{token}}$ </p>
118
+ <p> $\displaystyle\text{argmax}_{v_{t}}\mathbb{E}_{x\sim G(v_{t})}\mathcal{A}(x,t,s%
119
+ _{o})~{}.$ </p>
120
+ <p> $E_{L}\cup E_{R}$ </p>
121
+ <p> $E_{L}=\{(u,w)|(u,w)\in E,w\neq v\}$ </p>
122
+
123
+ </body>
124
+ </html>
output_mathjax_example_10.html ADDED
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1
+ <!DOCTYPE html>
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+ <html>
3
+ <head>
4
+ <title>MathJax Example</title>
5
+ <script>
6
+ MathJax = {
7
+ tex: {
8
+ inlineMath: [['$', '$'], ['\(', '\)']]
9
+ },
10
+ svg: {
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+ fontCache: 'global'
12
+ }
13
+ };
14
+ </script>
15
+ <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
16
+ </head>
17
+ <body>
18
+ <p> $0.01$ </p>
19
+ <p> $\underset{\pm 0.10}{2.15}$ </p>
20
+ <p> $(\operatorname{\bm{\theta}}_{\text{client}}^{(t)}=\operatorname{\bm{\theta}}_{%
21
+ \text{client}}^{(0)}$ </p>
22
+ <p> $r=64$ </p>
23
+ <p> $297.78$ </p>
24
+ <p> $\mathbf{0.43}$ </p>
25
+ <p> $\operatorname{\mathbf{v}}_{i}\in\mathcal{V}$ </p>
26
+ <p> $0.76$ </p>
27
+ <p> $\underset{\pm 0.66}{45.89}$ </p>
28
+ <p> $0$ </p>
29
+ <p> $\rho$ </p>
30
+ <p> $0.26$ </p>
31
+ <p> $0.95$ </p>
32
+ <p> $p\approx 8.69\times 10^{-8}$ </p>
33
+ <p> $0.69$ </p>
34
+ <p> $47.32$ </p>
35
+ <p> $\mathbf{0.69}$ </p>
36
+ <p> $2.30$ </p>
37
+ <p> $\mathbf{A}\in\mathbb{R}^{d\times r}$ </p>
38
+ <p> $\operatorname{\bm{\theta}}_{\text{client}}^{(t)}$ </p>
39
+ <p> $500$ </p>
40
+ <p> $\operatorname{\mathbf{v}}_{i}$ </p>
41
+ <p> $0.78$ </p>
42
+ <p> $308$ </p>
43
+ <p> $\mathbf{0.36}$ </p>
44
+ <p> $-\frac{1}{|\mathcal{D}^{(t)}_{\bigtriangledown}|}\sum_{\operatorname{\mathbf{d%
45
+ }}^{(t)}\in\mathcal{D}^{(t)}_{\bigtriangledown}}\log p_{(\cdot)}\big{(}%
46
+ \operatorname{\mathbf{d}}^{(t)}\big{|}\operatorname{\bm{\theta}}_{(\cdot)}^{(t%
47
+ )}),$ </p>
48
+ <p> $\operatorname{\mathbf{pr}}_{\text{client}}$ </p>
49
+ <p> $p$ </p>
50
+ <p> $0.66$ </p>
51
+ <p> $2.65$ </p>
52
+ <p> $\operatorname{\bm{\theta}}_{\text{client}}^{(0)}$ </p>
53
+ <p> $25$ </p>
54
+ <p> $277.25$ </p>
55
+ <p> $\%$ </p>
56
+ <p> $57.16$ </p>
57
+ <p> $0.74$ </p>
58
+ <p> $0.77$ </p>
59
+ <p> $\tau=0.5$ </p>
60
+ <p> $256$ </p>
61
+ <p> $4.3$ </p>
62
+ <p> $\operatorname{\bm{\theta}}_{\text{agent}}^{(0)}$ </p>
63
+ <p> $0.90$ </p>
64
+ <p> $0.80$ </p>
65
+ <p> $4$ </p>
66
+ <p> $0.8$ </p>
67
+ <p> $9000$ </p>
68
+ <p> $\operatorname{\bm{\theta}}_{\text{client}}$ </p>
69
+ <p> $F_{1}$ </p>
70
+ <p> $\mathbf{55.25}$ </p>
71
+ <p> $0.60$ </p>
72
+ <p> $\operatorname{\mathbf{v}}_{\text{next}}\in\text{Children}(\operatorname{%
73
+ \mathbf{v}}_{i})$ </p>
74
+ <p> $\displaystyle\geq bx_{j}^{\prime}+\sum_{i>j}x_{i}^{\prime}+(b-1)\sum_{i>j}x_{i}$ </p>
75
+ <p> $x_{i}^{\prime}\geq x_{i}$ </p>
76
+ <p> $A_{1}$ </p>
77
+ <p> $\displaystyle(b^{k}-b^{k-1}-(b-1)^{k}+b^{k-1})x_{1}$ </p>
78
+ <p> $(u,w)$ </p>
79
+ <p> $C_{1},C_{2}$ </p>
80
+ <p> $P:(u,v)\cup P^{\prime}$ </p>
81
+ <p> $e_{>i}$ </p>
82
+ <p> $(u_{1},u_{2},\dots,u_{k},v)$ </p>
83
+ <p> $\displaystyle=\frac{(b-1)^{k-1}}{b^{k}-(b-1)^{k}}x.$ </p>
84
+ <p> $(v,z)\in N(u)\times N(u)$ </p>
85
+ <p> $1\leq i\leq k$ </p>
86
+ <p> $x_{\tau+1},\dots,x_{k}\mapsto\textsf{Chunk-Shortest-Edge}(k-\tau,x-\delta)$ </p>
87
+ <p> $\sum_{i}x_{i}<x$ </p>
88
+ <p> $\displaystyle=bx+c(v\to t)-c(u,w)-c(w\to t)=\delta+(b-1)x$ </p>
89
+ <p> $p(e_{k}^{O})<\beta$ </p>
90
+ <p> $i\geq\tau$ </p>
91
+ <p> $\displaystyle=\begin{cases*}c(u,y)+cost[y,y,i]&if $(y,z)\in\mathcal{P}^{\prime%
92
+ }(u,y)$\\
93
+ \infty&otherwise\end{cases*}$ </p>
94
+ <p> $y^{*}\leq\frac{x}{k-1}$ </p>
95
+ <p> $x-\delta$ </p>
96
+ <p> $p(e;b_{i})$ </p>
97
+ <p> $\displaystyle=\frac{b^{k-1}}{b^{k-1}-(b-1)^{k-1}}\left(\frac{b^{k}-(b-1)^{k-1}%
98
+ (b-1+1)}{b^{k}-(b-1)^{k}}\right)x+c(v\to t)$ </p>
99
+ <p> $(s,s_{1})$ </p>
100
+ <p> $i+j\leq k$ </p>
101
+ <p> $\displaystyle=\left(\frac{b^{k-1}}{b^{k-1}-(b-1)^{k-1}}-1\right)x$ </p>
102
+ <p> $\begin{array}[]{ll}p(e_{\tau})&=bx_{\tau}+c(u_{\tau+1}\to t)\\
103
+ &=bx_{\tau}+\sum_{i=\tau+1}^{k}x_{i}+c(v\to t)\hfill\mbox{(shortest path from
104
+ $u_{\tau+1}$ follows the chunking)}\\
105
+ &=bx_{\tau}+x-\sum_{i=1}^{\tau}x_{i}+c(v\to t)\\
106
+ &=b\cdot\frac{\delta}{\tau}-\tau\cdot\frac{\delta}{\tau}+x+c(v\to t)\hfill%
107
+ \mbox{(substituing $x_{i}=\delta/\tau$ for $i\leq\tau$)}\\
108
+ &=\frac{b\delta}{\tau}+c(u,w)+c(w\to t)\hfill\mbox{(since $\delta=x+c(v\to t)-%
109
+ c(u,w)-c(w\to t)$).}\end{array}$ </p>
110
+ <p> $(u_{3},z)$ </p>
111
+ <p> $\displaystyle\frac{(b-1)(b^{k-1}-(b-1)^{k-1})+b^{k-1}}{b^{k-1}-(b-1)^{k-1}}x_{1}$ </p>
112
+ <p> $\sum_{l>i}x_{l}\leq x$ </p>
113
+ <p> $\displaystyle(b-1)x_{1}+x$ </p>
114
+ <p> $c^{n}$ </p>
115
+ <p> $\delta/k$ </p>
116
+ <p> $p(e_{i})=bx_{i}+c(u,w)+c(w\to t)$ </p>
117
+ <p> $\alpha=\beta$ </p>
118
+ <p> $p(e_{i};b_{1})\leq\alpha_{u}^{(j)}$ </p>
119
+ <p> $p(e_{i}^{C})=bx_{i}^{C}+c(u_{i}^{C}\to t)$ </p>
120
+ <p> $x_{i}-x_{i}^{\prime}\geq 0$ </p>
121
+ <p> $p(e_{i})=\beta^{\prime}$ </p>
122
+ <p> $\displaystyle\min_{(v,z)\in\mathcal{P}(u,y)}\min(C_{1}(u,v,y),C_{2}(u,y,z),C_{%
123
+ 3}(u,v,y,z)).$ </p>
124
+ <p> $c(u,v)$ </p>
125
+ <p> $\beta^{*}=p(e_{i})$ </p>
126
+ <p> $p(e_{j};b_{1})\leq\alpha_{u}^{(1)}$ </p>
127
+ <p> $\displaystyle=\frac{x}{1-\left(\frac{b-1}{b}\right)^{k-\tau+1}}+c(v\to t)-c(w%
128
+ \to t)-c(u,w).$ </p>
129
+ <p> $x_{1}$ </p>
130
+ <p> $O(|E|^{2}k^{3}\log k+|V|)$ </p>
131
+ <p> $\beta\leftarrow\frac{x-\delta}{z_{\tau}}+c(v\to t)$ </p>
132
+ <p> $min\_bottleneck\leftarrow\min(\alpha,\beta)$ </p>
133
+ <p> $\delta/\tau$ </p>
134
+ <p> $(y,z)$ </p>
135
+
136
+ </body>
137
+ </html>
output_mathjax_example_2.html ADDED
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1
+ <!DOCTYPE html>
2
+ <html>
3
+ <head>
4
+ <title>MathJax Example</title>
5
+ <script>
6
+ MathJax = {
7
+ tex: {
8
+ inlineMath: [['$', '$'], ['\(', '\)']]
9
+ },
10
+ svg: {
11
+ fontCache: 'global'
12
+ }
13
+ };
14
+ </script>
15
+ <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
16
+ </head>
17
+ <body>
18
+ <p> $v_{1},v_{2}\in\overline{V_{m}}$ </p>
19
+ <p> $|\Delta E|^{\prime}=w^{*}$ </p>
20
+ <p> $1\leq j\leq k$ </p>
21
+ <p> ${\mathcal{R}}$ </p>
22
+ <p> $\pi\left(G^{\prime}\right)\leq\pi(G)-d$ </p>
23
+ <p> $\displaystyle\underbrace{n_{R}\times|y-C|\times k^{\prime}}_{\text{between the%
24
+ subpath of }w_{2}\text{ and the vertices in }u}+\underbrace{n_{R}\times|y-B-C%
25
+ |\times k}_{{\text{between the subpath of }w_{2}\text{ and the vertices in }v}}+$ </p>
26
+ <p> $\operatorname{min}(\mathcal{E}(M_{L}^{*}),\mathcal{E}(M_{R}^{*}))$ </p>
27
+ <p> $E_{m}=\{e^{*}\}$ </p>
28
+ <p> $\displaystyle n_{L}\times n_{R}\times|x+y-A-B-C|\xrightarrow{n_{R}=0}$ </p>
29
+ <p> $P$ </p>
30
+ <p> $v_{i}\in V_{L}$ </p>
31
+ <p> $M_{R}^{*}\xleftarrow[]{}\emptyset$ </p>
32
+ <p> $\alpha_{1}=A-x+x-A-B<B\xrightarrow{}0<2B$ </p>
33
+ <p> $M_{R}$ </p>
34
+ <p> $u,v\in V$ </p>
35
+ <p> $n_{L}=3$ </p>
36
+ <p> $\left|\sum_{k=i}^{j-2}(w_{k}+\epsilon_{k})-w^{*}-\sum_{k=i}^{j-2}w_{k}\right|=%
37
+ \left|w^{*}-\sum_{k=i}^{j-2}\epsilon_{k}\right|$ </p>
38
+ <p> $n_{L}(|x-A|+|x-A-B|)$ </p>
39
+ <p> $\mathcal{E}_{L}^{(i+1)}-\mathcal{E}_{L}^{(i)}=n_{L}\times\big{(}(i+1)\;w_{i+1}%
40
+ -(k-i)\;w_{i+1}\big{)}$ </p>
41
+ <p> $w^{\prime}(e_{i})=w(e_{i})+w(e^{*})$ </p>
42
+ <p> $S_{R}\xleftarrow{}\sum_{\forall e_{i}\in E_{R}}R_{i}$ </p>
43
+ <p> $u,v\in G_{1}$ </p>
44
+ <p> $x\geq A$ </p>
45
+ <p> $\Delta({\text{MARK\_RIGHT}})$ </p>
46
+ <p> $\operatorname{min}(\mathcal{E}(M_{R}^{*}),\mathcal{E}(M_{L}^{*}))\leq\mathcal{%
47
+ E}(M)$ </p>
48
+ <p> ${\mathcal{L}}-(\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}})%
49
+ =\frac{{\mathcal{L}}{n_{L}}-{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}$ </p>
50
+ <p> $u_{j+1}$ </p>
51
+ <p> $T_{j}^{R}$ </p>
52
+ <p> $\varphi(x)=x/\alpha-\beta$ </p>
53
+ <p> $\mathcal{E}=\mathcal{E}_{R}$ </p>
54
+ <p> $S_{R}-S_{L}\leq R_{i}$ </p>
55
+ <p> $e_{3}$ </p>
56
+ <p> $a\geq 0$ </p>
57
+ <p> $V_{L}$ </p>
58
+ <p> ${n_{L}}$ </p>
59
+ <p> $n_{R}$ </p>
60
+ <p> $n_{L}\geq 0$ </p>
61
+ <p> $c_{2}\geq\epsilon$ </p>
62
+ <p> ${n^{2}_{L}}\times((i+1)({\mathcal{L}}-i-1)-i({\mathcal{L}}-i))={n^{2}_{L}}%
63
+ \times(i{\mathcal{L}}-i^{2}-i+{\mathcal{L}}-i-1-i{\mathcal{L}}+i^{2})={n^{2}_{%
64
+ L}}\times({\mathcal{L}}-2i-1)$ </p>
65
+ <p> $e^{*}_{k}=(u_{j},u_{j+1})$ </p>
66
+ <p> $|\Delta E|=|\overline{V_{m}}|(w^{*}_{1}+\dots+w^{*}_{k})=(n-2k)(w^{*}_{1}+%
67
+ \dots+w^{*}_{k})$ </p>
68
+ <p> $e_{1},e_{2},\dots,e_{k}$ </p>
69
+ <p> $-\left|\sum_{k=n_{1}+1}^{j-2}\epsilon_{k}\right|$ </p>
70
+ <p> $\overline{E_{m}}$ </p>
71
+ <p> $e^{*}\in E$ </p>
72
+ <p> $\epsilon_{1}+\epsilon_{2}=B$ </p>
73
+ <p> $n_{L}+n_{R}=n-2$ </p>
74
+ <p> $c^{\prime}_{1}+c^{\prime}_{2}=1$ </p>
75
+ <p> $\mathcal{E}_{L}^{(j)},j\neq\frac{k}{2}$ </p>
76
+ <p> $L_{i}\geq S_{L}-S_{R}$ </p>
77
+ <p> $j>0$ </p>
78
+ <p> $\Delta_{1}({\text{MARK\_LEFT}})$ </p>
79
+ <p> $L_{i}\times R_{j}\times w^{*}$ </p>
80
+ <p> $e_{j}\neq e_{1}$ </p>
81
+ <p> $x-A$ </p>
82
+ <p> $j\leq\frac{{\mathcal{R}}}{2}+\frac{{n_{L}}(1-{\mathcal{L}})}{2{n_{R}}}$ </p>
83
+ <p> $|a|=a$ </p>
84
+ <p> $2\times L_{i}\times(S_{LM})$ </p>
85
+ <p> $n_{L}+n_{R}=|\overline{V_{m}}|=n-(k+k^{\prime})$ </p>
86
+ <p> $M_{L}^{*}\xleftarrow[]{}\emptyset$ </p>
87
+ <p> $\frac{\mathcal{E}_{L}}{n_{L}}$ </p>
88
+ <p> $\displaystyle(\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}})(%
89
+ {n^{2}_{L}}\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}({\mathcal{R}}))+\frac{{%
90
+ \mathcal{L}}{n_{L}}-{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}({n^{2}_{L}}\times({%
91
+ \mathcal{L}}-1)+{n_{L}}{n_{R}}(-{\mathcal{R}}))$ </p>
92
+ <p> $\displaystyle\mathcal{E}_{L}^{(0)}=n_{L}\times\big{(}w_{1}+w_{1}+w_{2}+w_{1}+w%
93
+ _{2}+w_{3}+\dots+w_{1}+w_{2}+\dots+w_{k}\big{)}$ </p>
94
+ <p> $\mathcal{W}^{\prime}(E^{(v_{1},u_{5})})=\epsilon_{1}+\epsilon_{2}+\epsilon_{2}%
95
+ +\epsilon_{3}+\epsilon_{4}$ </p>
96
+ <p> $v_{i}\in\overline{V_{m}}$ </p>
97
+ <p> $e_{4}=(v_{2},v_{6})$ </p>
98
+ <p> $u\in\overline{V_{m}}$ </p>
99
+ <p> ${T_{i}^{L}},i\in\{1,\dots,{\mathcal{L}}\}$ </p>
100
+ <p> $\operatorname{CONTRACTION}$ </p>
101
+ <p> ${{\mathcal{L}}\choose{2}}\times n_{L}\times n_{L}\times 2w^{*}={n^{2}_{L}}%
102
+ \times{\mathcal{L}}({\mathcal{L}}-1)\times w^{*}$ </p>
103
+ <p> $\displaystyle n_{L}(|x-A|+|x-A-B|)+n_{R}(|y-C|+|y-B-C|)+n_{L}n_{R}|x+y-A-B-C|$ </p>
104
+ <p> $e_{3}=(v_{2},v_{5})$ </p>
105
+ <p> $\mathcal{E}(M_{R}^{*})=\underbrace{\mathcal{E}(M_{0})}_{\text{The error %
106
+ associated with the empty marking}}+\underbrace{\sum_{e_{i}\in E_{R}}R_{i}%
107
+ \times w^{*}\times(S_{R}-R_{i}-S_{L})}_{\text{The sum of all }\Delta({\text{%
108
+ MARK\_RIGHT}})\text{'s that
109
+ transform }M_{0}\text{ into }M_{R}^{*}}$ </p>
110
+ <p> ${\mathcal{L}}=2$ </p>
111
+ <p> $\displaystyle\geq\mathcal{E}(M_{0})+\epsilon_{1}\times\sum_{e_{i}\in E_{L}}L_{%
112
+ i}\times w^{*}\times(S_{L}-L_{i}-S_{R})+\epsilon_{2}\times\sum_{e_{i}\in E_{L}%
113
+ }L_{i}\times w^{*}\times(S_{L}-L_{i}-S_{R})$ </p>
114
+ <p> $\mathcal{E}_{LR}=n_{L}\times n_{R}\times|x+y-w_{0}-w_{1}-\dots-w_{k+1}|$ </p>
115
+ <p> $w_{j}$ </p>
116
+ <p> $\displaystyle 0$ </p>
117
+ <p> $\Delta({\text{MARK\_LEFT}})$ </p>
118
+ <p> $j\xleftarrow{}j$ </p>
119
+ <p> $M_{0}$ </p>
120
+ <p> $x=A+\epsilon_{1}$ </p>
121
+ <p> $\Delta_{v_{i},u_{j}}+\Delta_{v_{i},u_{j+1}}\leq\left|w^{*}_{k}-\mathcal{W}^{%
122
+ \prime}(E^{(v_{i},u_{j})})+\mathcal{W}^{*}(E^{(v_{i},u_{j})})\right|-\left|w^{%
123
+ *}_{k}+\mathcal{W}^{*}(E^{(v_{i},u_{j})})-\mathcal{W}^{\prime}(E^{(v_{i},u_{j}%
124
+ )})\right|\leq 0$ </p>
125
+ <p> $L_{i}=|\{v|v\in{T_{i}^{L}}\}|$ </p>
126
+ <p> $R_{j}=|\{v|v\in{T_{j}^{R}}\}|$ </p>
127
+ <p> $E^{\prime}=E-e^{*}$ </p>
128
+ <p> $e^{\prime}=e_{1}$ </p>
129
+ <p> $n_{L}$ </p>
130
+ <p> $E_{R}=\{(v,w)|(v,w)\in E,w\neq u\}$ </p>
131
+ <p> $e^{*}_{k}=e^{*}_{3}=(u_{5},u_{6})$ </p>
132
+ <p> $(S_{LM}+(S_{LU}-L_{i})+S_{RM}-S_{RU})\leq 0\xrightarrow[]{S_{LM}+S_{LU}=S_{L}}%
133
+ S_{L}-L_{i}+S_{RM}-S_{RU}\leq 0$ </p>
134
+ <p> $S_{R}^{\prime}=S_{R}-R_{1}$ </p>
135
+ <p> $\mathcal{E}_{1}=|x-w_{0}|+\dots+|w-w_{0}-\dots-w_{k}|$ </p>
136
+ <p> $S_{LM}>0$ </p>
137
+ <p> $\Delta({\text{MARK\_LEFT}})\leq 0\text{ if }{n_{L}}\times({\mathcal{L}}-1)\leq%
138
+ {n_{R}}({\mathcal{R}}-2j)\xrightarrow[]{\text{Rearranging the terms}}j\leq%
139
+ \frac{{\mathcal{R}}}{2}+\frac{{n_{L}}(1-{\mathcal{L}})}{2{n_{R}}}$ </p>
140
+ <p> $\displaystyle\underbrace{\epsilon_{2}\times\sum_{e_{i}\in E_{R}}R_{i}\times w^%
141
+ {*}\times(S_{R}-R_{i}-S_{L})}_{\text{The sum of all }\Delta({\text{MARK\_RIGHT%
142
+ }})\text{'s by }\epsilon_{2}}$ </p>
143
+ <p> $k\geq k^{\prime}$ </p>
144
+ <p> $c^{\prime}_{i}=0$ </p>
145
+ <p> $w_{i+1}$ </p>
146
+ <p> $(v_{1},v_{4})$ </p>
147
+
148
+ </body>
149
+ </html>
output_mathjax_example_3.html ADDED
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1
+ <!DOCTYPE html>
2
+ <html>
3
+ <head>
4
+ <title>MathJax Example</title>
5
+ <script>
6
+ MathJax = {
7
+ tex: {
8
+ inlineMath: [['$', '$'], ['\(', '\)']]
9
+ },
10
+ svg: {
11
+ fontCache: 'global'
12
+ }
13
+ };
14
+ </script>
15
+ <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
16
+ </head>
17
+ <body>
18
+ <p> $c^{\prime}_{i}=c_{j}$ </p>
19
+ <p> $\mathcal{W}^{*}(E^{(v_{1},u_{5})})=w^{*}_{1}+w^{*}_{2}$ </p>
20
+ <p> $|w_{1}+w^{*}+w_{2}+w^{*}-w_{1}-w_{2}|=2w^{*}$ </p>
21
+ <p> $e_{i}\in E_{m},1\leq i\leq k$ </p>
22
+ <p> $|w^{*}|$ </p>
23
+ <p> $P^{{}^{\prime}}\subseteq P$ </p>
24
+ <p> $P^{\prime}\subseteq P$ </p>
25
+ <p> $k-i$ </p>
26
+ <p> $c_{1}+c_{2}>1$ </p>
27
+ <p> $c_{j}>0$ </p>
28
+ <p> $e_{1},e_{2}$ </p>
29
+ <p> $\displaystyle\pi^{\prime\prime}_{v_{i},u_{j}}$ </p>
30
+ <p> $n_{R}(|y-C|+|y-B-C|)$ </p>
31
+ <p> $\operatorname{min}(\mathcal{E}(M_{R}^{*}),\mathcal{E}(M_{L}^{*}))=\mathcal{E}(%
32
+ M_{L}^{*})$ </p>
33
+ <p> $i\xleftarrow{}i$ </p>
34
+ <p> $\displaystyle\leq$ </p>
35
+ <p> $\mathcal{E}_{L}=n_{L}\times\big{(}\underbrace{|x-w_{0}|}_{\text{between the %
36
+ vertices of }V_{L}\text{ and }v_{2}}+\underbrace{|x-w_{0}-w_{1}|}_{\text{%
37
+ between the vertices of }V_{L}\text{ and }v_{3}}+\dots+\underbrace{|x-w_{0}-w_%
38
+ {1}-\dots-w_{k}|}_{\text{between the vertices of }V_{L}\text{ and }v_{k+2}}%
39
+ \big{)}$ </p>
40
+ <p> ${n^{2}_{L}}\times{\mathcal{L}}({\mathcal{L}}-1)\leq{n^{2}_{R}}\times{\mathcal{%
41
+ R}}({\mathcal{R}}-1)$ </p>
42
+ <p> $M_{L}^{*}$ </p>
43
+ <p> ${n_{L}}\times{n_{L}}\times 2w^{*}$ </p>
44
+ <p> $M^{\prime\prime}$ </p>
45
+ <p> $f\in\mathcal{F}=\{{\text{MARK\_LEFT}},{\text{UNMARK\_LEFT}},{\text{MARK\_RIGHT%
46
+ }},{\text{UNMARK\_RIGHT}}\}$ </p>
47
+ <p> $7-21\leq 2=L_{2}$ </p>
48
+ <p> $\displaystyle n_{L}\times\big{(}\sum_{j=0}^{i}j\;w_{j}+\sum_{j=i+1}^{k}(k+1-j)%
49
+ \;w_{j}+(i+1)\;w_{i+1}-(k-i)\;w_{i+1}\big{)}$ </p>
50
+ <p> $x=w_{0}+w_{1}+\dots+w_{i}$ </p>
51
+ <p> $x>w_{0}+\dots+w_{k}$ </p>
52
+ <p> $L_{1}\times L_{2}\times w^{*}$ </p>
53
+ <p> $e_{2}=(v_{1},v_{4})$ </p>
54
+ <p> $e_{1}$ </p>
55
+ <p> $H$ </p>
56
+ <p> $S_{L}-L_{i}-S_{R}\leq 0\xrightarrow[]{}S_{L}-S_{R}\leq L_{i}$ </p>
57
+ <p> $j>\frac{{\mathcal{R}}}{2}+\frac{{n_{L}}(1-{\mathcal{L}})}{2{n_{R}}}$ </p>
58
+ <p> $\mathcal{E}(M)=\mathcal{E}(M^{\prime})+\Delta_{1}({\text{MARK\_LEFT}})$ </p>
59
+ <p> $w_{i}$ </p>
60
+ <p> $u,v$ </p>
61
+ <p> $v_{5}$ </p>
62
+ <p> $n_{2}=n_{R}$ </p>
63
+ <p> $u\in G_{2}$ </p>
64
+ <p> $e_{i}\in E$ </p>
65
+ <p> $\frac{\mathcal{E}^{(\frac{k}{2})}_{L}}{n_{L}}$ </p>
66
+ <p> $\alpha_{2}$ </p>
67
+ <p> $w(e^{\prime})\xleftarrow[]{}w(e^{\prime})+w(e)$ </p>
68
+ <p> $V_{L}=\{v_{i}|1\leq i\leq n_{1}+1\}$ </p>
69
+ <p> $\mathcal{E}_{L}^{(i)}=n_{L}\times\big{(}\sum_{j=0}^{i}j\;w_{j}+\sum_{j=i+1}^{k%
70
+ }(k+1-j)\;w_{j}\big{)}$ </p>
71
+ <p> $n_{L}\times\big{(}(k-i)\;w_{i+1}\big{)}$ </p>
72
+ <p> ${T_{i}^{L}},i\in\{1,2\}$ </p>
73
+ <p> $\mathcal{W}^{\prime}(E^{\prime})$ </p>
74
+ <p> $e^{\prime}\in E_{m}$ </p>
75
+ <p> $\displaystyle n_{L}\times n_{R}\times|x+y-A-B-C|$ </p>
76
+ <p> $M^{*}\xleftarrow[]{}\operatorname{argmin}(\mathcal{E}(M_{L}^{*}),\mathcal{E}(M%
77
+ _{R}^{*}))$ </p>
78
+ <p> $u,v\in G_{2}$ </p>
79
+ <p> $\Delta({\text{UNMARK\_LEFT}})=L_{i}\times\bigg{(}-(S_{LM}-L_{i})-S_{LU}-S_{RM}%
80
+ +S_{RU}\bigg{)}$ </p>
81
+ <p> $\mathcal{E}(M^{\prime\prime})<\mathcal{E}(M)$ </p>
82
+ <p> $L_{i}=n_{L},\;1\leq i\leq{\mathcal{L}}$ </p>
83
+ <p> $u\in G_{1}$ </p>
84
+ <p> $|w^{*}|-\left|\sum_{k=i}^{n_{1}}\epsilon_{k}\right|\leq\left|w^{*}-\sum_{k=i}^%
85
+ {n_{1}}\epsilon_{k}\right|$ </p>
86
+ <p> $\beta\geq 0$ </p>
87
+ <p> ${T_{j}^{R}},j\in\{1,\dots,{\mathcal{R}}\}$ </p>
88
+ <p> $\Delta({\text{UNMARK\_LEFT}})$ </p>
89
+ <p> $\displaystyle B\times k^{\prime}\times\underbrace{(n_{L}+n_{R})}_{=n-(k+k^{%
90
+ \prime})}=B\times k^{\prime}\times(n-(k+k^{\prime}))$ </p>
91
+ <p> $P^{\prime}\subset P$ </p>
92
+ <p> $0\leq i\leq k$ </p>
93
+ <p> $e_{i},e_{j}$ </p>
94
+ <p> $C\leq y\leq B+C$ </p>
95
+ <p> $c_{i}\neq c_{j}$ </p>
96
+ <p> $\alpha_{1}=B$ </p>
97
+ <p> $x-A-B$ </p>
98
+ <p> $\displaystyle\xrightarrow{\text{setting }x=A+B,y=C}=$ </p>
99
+ <p> $v\in V_{s}$ </p>
100
+ <p> $\mathcal{E}_{1}$ </p>
101
+ <p> $L_{1}$ </p>
102
+ <p> $X<0$ </p>
103
+ <p> $V_{s}$ </p>
104
+ <p> $|w^{*}-(c_{2}\times w^{*}+c_{j}\times w^{*}-\epsilon\times w^{*})|-|w^{*}-(c_{%
105
+ 2}\times w^{*}+c_{j}\times w^{*})|\leq\epsilon\times w^{*}$ </p>
106
+ <p> $e^{*}\in S$ </p>
107
+ <p> $c^{\prime}_{2}=c_{2}-\epsilon$ </p>
108
+ <p> $f\in\mathcal{F}=\{\text{MARK\_LEFT},\text{UNMARK\_RIGHT}\}$ </p>
109
+ <p> $S_{RM}$ </p>
110
+ <p> $|w_{1}+w^{*}+w_{3}+w^{*}-w_{1}-w^{*}-w_{3}|=w^{*}$ </p>
111
+ <p> $M_{L}^{*}\xleftarrow[]{}M_{L}^{*}\cup\{e_{i}\}$ </p>
112
+ <p> $\begin{array}[]{cc}\Delta_{1}({\text{UNMARK\_RIGHT}})\leq R_{1}\times\epsilon%
113
+ \times w^{*}\times\bigg{(}-S_{R}^{\prime}\underbrace{-L_{1}}_{<0}+S_{L}^{%
114
+ \prime}\bigg{)}&<R_{1}\times\epsilon\times w^{*}\times(\underbrace{-S_{R}^{%
115
+ \prime}+S_{L}^{\prime}}_{\leq 0})\leq 0\end{array}$ </p>
116
+ <p> $\mathcal{E}^{v_{i},u_{j}}_{1}=\left|\pi_{v_{i},u_{j}}-\pi^{\prime}_{v_{i},u_{j%
117
+ }}\right|=\left|\pi^{\prime}_{v_{i},u_{j}}-\pi_{v_{i},u_{j}}\right|=\left|%
118
+ \mathcal{W}^{\prime}(E^{(v_{i},u_{j})})-\mathcal{W}^{*}(E^{(v_{i},u_{j})})\right|$ </p>
119
+ <p> $\alpha_{1}=x-A+A+B-x<B\xrightarrow{}0<0$ </p>
120
+ <p> $(u,v),\;\;u\in V_{m},\;v\in\overline{V_{m}}$ </p>
121
+ <p> $w_{1}$ </p>
122
+ <p> $|\Delta E|\geq B(n_{L}+n_{R})+n_{L}n_{R}|x+y-A-B-C|$ </p>
123
+ <p> $v^{\prime}_{1}$ </p>
124
+ <p> $\operatorname{d}_{G^{\prime}}(u,v)\geq\varphi\left(\operatorname{d}_{G}(u,v)\right)$ </p>
125
+ <p> $S\xleftarrow[]{}E_{m}$ </p>
126
+ <p> $\mathcal{E}^{v_{i},u_{j+1}}_{1}=\left|\pi_{v_{i},u_{j+1}}-\pi^{\prime}_{v_{i},%
127
+ u_{j+1}}\right|=\left|\pi_{v_{i},u_{j}}+w^{*}_{k}-\pi^{\prime}_{v_{i},u_{j}}%
128
+ \right|=\left|w^{*}_{k}+\mathcal{W}^{*}(E^{(v_{i},u_{j})})-\mathcal{W}^{\prime%
129
+ }(E^{(v_{i},u_{j})})\right|$ </p>
130
+ <p> $7-21\leq 3=L_{3}$ </p>
131
+ <p> ${n^{2}_{R}}\times 2\big{(}{j-1\choose 2}-{j\choose 2}\big{)}={n^{2}_{R}}\times%
132
+ (-2(j-1))$ </p>
133
+ <p> $\mathcal{E}_{LR}=0$ </p>
134
+ <p> $w^{\prime}(e_{i})=w(e_{i})+\epsilon_{i},\epsilon_{i}\in[0,w^{*}]$ </p>
135
+ <p> $\displaystyle+$ </p>
136
+ <p> $i+1-k+i<0\xrightarrow[]{}2i<k-1\xrightarrow[]{}i<\frac{k}{2}-\frac{1}{2}%
137
+ \xrightarrow[\text{since }k\text{ is even}]{}i\leq\frac{k}{2}-1$ </p>
138
+ <p> $|x-A|$ </p>
139
+ <p> $\alpha_{1}$ </p>
140
+ <p> $V_{m}=\{v_{2},v_{3}\}$ </p>
141
+ <p> $\Delta({\text{MARK\_LEFT}})={n^{2}_{L}}\times(2i+{\mathcal{L}}-2i-1)+{n_{L}}{n%
142
+ _{R}}(j+j-{\mathcal{R}})={n^{2}_{L}}\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}(2j-%
143
+ {\mathcal{R}})$ </p>
144
+
145
+ </body>
146
+ </html>
output_mathjax_example_4.html ADDED
@@ -0,0 +1,142 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ <!DOCTYPE html>
2
+ <html>
3
+ <head>
4
+ <title>MathJax Example</title>
5
+ <script>
6
+ MathJax = {
7
+ tex: {
8
+ inlineMath: [['$', '$'], ['\(', '\)']]
9
+ },
10
+ svg: {
11
+ fontCache: 'global'
12
+ }
13
+ };
14
+ </script>
15
+ <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
16
+ </head>
17
+ <body>
18
+ <p> $E^{(v_{1},u_{5})}=\{e_{1},e^{*}_{1},e_{2},e_{3},e^{*}_{2},e_{4}\}$ </p>
19
+ <p> $x=\sum_{k=i}^{n_{1}}\epsilon_{k}$ </p>
20
+ <p> $v_{i},v_{j}\in V_{R}$ </p>
21
+ <p> $G^{\prime}$ </p>
22
+ <p> $c_{i+1},\dots,c_{k}$ </p>
23
+ <p> $S_{R}-R_{i}-S_{L}\leq 0$ </p>
24
+ <p> $n\geq 3$ </p>
25
+ <p> $\underbrace{|w^{*}-(c_{i}\times w^{*}+c_{j}\times w^{*}-\epsilon\times w^{*})|%
26
+ }_{=w^{*}-(c_{i}\times w^{*}+c_{j}\times w^{*}-\epsilon\times w^{*})\text{ %
27
+ because }c_{i}+c_{j}\leq 1}-\underbrace{|w^{*}-(c_{i}\times w^{*}+c_{j}\times w%
28
+ ^{*})|}_{=w^{*}-(c_{i}\times w^{*}+c_{j}\times w^{*})\text{ because }c_{i}+c_{%
29
+ j}\leq 1}=\epsilon\times w^{*}$ </p>
30
+ <p> $w_{i}=w(e_{i})\;\forall i\in\{0,\dots,k+1\}$ </p>
31
+ <p> $u_{1}\in T_{1}^{L},u_{2}\in T_{2}^{L}$ </p>
32
+ <p> $\pi$ </p>
33
+ <p> $\displaystyle\mathcal{E}_{L}^{(i+1)}=$ </p>
34
+ <p> $i\geq\frac{{\mathcal{L}}}{2}+\frac{{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}$ </p>
35
+ <p> $\displaystyle\mathcal{E}(M)=$ </p>
36
+ <p> $n_{L}n_{R}|x+y-A-B-C|$ </p>
37
+ <p> $V^{\prime}\subset V$ </p>
38
+ <p> $R_{i}={n_{R}},\;1\leq i\leq{\mathcal{R}}$ </p>
39
+ <p> $\pi_{v_{i},u_{j+1}}=\pi_{v_{i},u_{j}}+w^{*}_{k}\text{, }\pi^{\prime}_{v_{i},u_%
40
+ {j}}=\pi^{\prime}_{v_{i},u_{j+1}}\text{, and }\pi^{\prime\prime}_{v_{i},u_{j}}%
41
+ =\pi^{\prime\prime}_{v_{i},u_{j+1}}$ </p>
42
+ <p> $u_{1}\in T_{i}^{L},u_{2}\in T_{j}^{L}$ </p>
43
+ <p> $\epsilon=0.4$ </p>
44
+ <p> $e\in E_{m}$ </p>
45
+ <p> $c_{i}$ </p>
46
+ <p> $e_{i},w_{i}=w(e_{i})$ </p>
47
+ <p> $\displaystyle\mathcal{E}^{(x<w_{0})}_{L}$ </p>
48
+ <p> $V_{m}=\{u_{1},\dots,u_{2k}\}$ </p>
49
+ <p> $\displaystyle\underbrace{n_{L}\times n_{R}\times|x+y-A-B-C|}_{\text{between %
50
+ the subpath of }w_{1}\text{ and }w_{2}}$ </p>
51
+ <p> $\mathcal{E}^{v_{i},u_{j}}_{2}=\left|\pi^{\prime\prime}_{v_{i},u_{j}}-\pi_{v_{i%
52
+ },u_{j}}\right|=\left|w^{*}_{k}\right|$ </p>
53
+ <p> $c_{i}>0$ </p>
54
+ <p> $d$ </p>
55
+ <p> $x<A$ </p>
56
+ <p> $\frac{\mathcal{E}^{(\frac{k}{2})}_{L}}{n_{L}}-\frac{\mathcal{E}_{L}}{n_{L}}\leq
57
+ 0$ </p>
58
+ <p> $\mathcal{E}^{v_{i},u_{j+1}}_{2}=0$ </p>
59
+ <p> $\displaystyle\xrightarrow[]{}\mathcal{E}(M)\geq\mathcal{E}(M_{L}^{*})$ </p>
60
+ <p> $|x+y-A-B-C|$ </p>
61
+ <p> $E^{(u,v)}$ </p>
62
+ <p> $x=w_{0}+w_{1}+\dots+w_{\frac{k}{2}}$ </p>
63
+ <p> $w_{1}+w_{2}+w_{3}$ </p>
64
+ <p> $j\xleftarrow[]{}0$ </p>
65
+ <p> $S_{R}=S_{RU}+S_{RM}$ </p>
66
+ <p> $v^{*}$ </p>
67
+ <p> $c_{1}+c_{2}=0.6+0.8=1.4>1$ </p>
68
+ <p> $S_{L}=\sum_{i=1}^{{\mathcal{L}}}L_{i}$ </p>
69
+ <p> $c_{1}+c_{2}=1+\epsilon$ </p>
70
+ <p> $w^{\prime}(e_{i})=w(e_{i})+w^{*}$ </p>
71
+ <p> $\{e_{1},e_{3}\}$ </p>
72
+ <p> $G^{{}^{\prime}}$ </p>
73
+ <p> $i\xleftarrow{}i+1$ </p>
74
+ <p> $\Delta(f)$ </p>
75
+ <p> $\displaystyle n_{L}\times k^{\prime}\times\underbrace{\big{(}|x-A|+|x-A-B|\big%
76
+ {)}}_{\geq B\text{ (Lemma \ref{l12})}}+n_{R}\times k^{\prime}\times\underbrace%
77
+ {\big{(}|y-C|+|y-B-C|\big{)}}_{\geq B\text{ (Lemma \ref{l12})}}$ </p>
78
+ <p> $i<\frac{{\mathcal{L}}}{2}+\frac{{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}$ </p>
79
+ <p> $\epsilon_{2}\in\{0,w^{*}\}$ </p>
80
+ <p> $\displaystyle\mathcal{E}(M)$ </p>
81
+ <p> $|V|=n$ </p>
82
+ <p> $V_{L}=\{u|(u,v_{1})\in E^{\prime}\},{\mathcal{L}}=|V_{L}|$ </p>
83
+ <p> $e_{k+1}$ </p>
84
+ <p> $|x-A-B|$ </p>
85
+ <p> $n_{L}+n_{R}=n-2=\left|\overline{V_{m}}\right|$ </p>
86
+ <p> $T_{i}^{L}$ </p>
87
+ <p> $x,z$ </p>
88
+ <p> $i={\mathcal{L}}$ </p>
89
+ <p> $\epsilon>0$ </p>
90
+ <p> $G_{1}$ </p>
91
+ <p> $v_{1}$ </p>
92
+ <p> $E^{\prime}\subseteq E$ </p>
93
+ <p> $S_{R}^{\prime}\geq S_{L}^{\prime}$ </p>
94
+ <p> $M$ </p>
95
+ <p> $e^{*}$ </p>
96
+ <p> $u$ </p>
97
+ <p> $u_{1}\in T_{i}^{R},u_{2}\in T_{j}^{R}$ </p>
98
+ <p> $y$ </p>
99
+ <p> $\displaystyle|\Delta E|=$ </p>
100
+ <p> $v_{i}$ </p>
101
+ <p> $V_{L},V_{R}\subset V$ </p>
102
+ <p> $\displaystyle=\mathcal{E}(M_{0})+(\epsilon_{1}+\epsilon_{2})\times\sum_{e_{i}%
103
+ \in E_{L}}L_{i}\times w^{*}\times(S_{L}-L_{i}-S_{R})\xrightarrow[S_{L}-L_{i}-S%
104
+ _{R}\leq 0]{\epsilon_{1}+\epsilon_{2}\leq 1}$ </p>
105
+ <p> $\{u_{1},v_{1}\}$ </p>
106
+ <p> $\Delta({\text{UNMARK\_RIGHT}})=R_{i}\times\bigg{(}-(S_{RM}-R_{i})-S_{RU}-S_{LM%
107
+ }+S_{LU}\bigg{)}$ </p>
108
+ <p> $(u,v)$ </p>
109
+ <p> $\alpha_{1}=|x-A|+|x-A-B|$ </p>
110
+ <p> $|z|\leq|x|+|z-x|\xrightarrow[]{}|z|-|x|\leq|z-x|$ </p>
111
+ <p> $B<0$ </p>
112
+ <p> $n_{R}\geq 0$ </p>
113
+ <p> $21-7>1=R_{2}$ </p>
114
+ <p> $v^{*}=\{v_{2},v_{3},\dots,v_{k+2}\}$ </p>
115
+ <p> $w:E\rightarrow\mathbb{R}_{\geq 0}$ </p>
116
+ <p> $e\in E$ </p>
117
+ <p> $i=\{1,\dots,{\mathcal{L}}\}$ </p>
118
+ <p> $\Delta_{1}({\text{MARK\_LEFT}})<0$ </p>
119
+ <p> $0-\left|\sum_{k=i}^{j-1}\epsilon_{k}\right|$ </p>
120
+ <p> $u,v\in G$ </p>
121
+ <p> $\mathcal{E}_{R}=n_{R}\times\big{(}\underbrace{|y-w_{k+1}|}_{\text{between the %
122
+ vertices of }V_{R}\text{ and }v_{k+2}}+\underbrace{|y-w_{k+1}-w_{k}|}_{\text{%
123
+ between the vertices of }V_{R}\text{ and }v_{k+1}}+\dots+\underbrace{|y-w_{k+1%
124
+ }-w_{k}-\dots-w_{1}|}_{\text{between the vertices of }V_{R}\text{ and }v_{2}}%
125
+ \big{)}$ </p>
126
+ <p> $V_{m}=\{v_{1},v_{2}\}$ </p>
127
+ <p> $R_{i}\times R_{j}\times w^{*}$ </p>
128
+ <p> $L_{i}\times L_{j}\times w^{*}$ </p>
129
+ <p> $\displaystyle\underbrace{\mathcal{E}(M_{0})}_{\text{The error associated with %
130
+ the empty marking}}+\underbrace{\epsilon_{1}\times\sum_{e_{i}\in E_{L}}L_{i}%
131
+ \times w^{*}\times(S_{L}-L_{i}-S_{R})}_{\text{The sum of all }\Delta({\text{%
132
+ MARK\_LEFT}})\text{'s by
133
+ }\epsilon_{1}}$ </p>
134
+ <p> $(u_{2i-1},u_{2i})$ </p>
135
+ <p> $V_{R}=\{v_{i}|n_{1}+2\leq i\leq n_{2}+2\}$ </p>
136
+ <p> $E_{R}$ </p>
137
+ <p> $w:E\Rightarrow\mathbb{R}_{\geq 0}$ </p>
138
+ <p> $x=A$ </p>
139
+ <p> $\epsilon_{i}=0\;\;\forall i\neq n_{1}$ </p>
140
+
141
+ </body>
142
+ </html>
output_mathjax_example_5.html ADDED
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1
+ <!DOCTYPE html>
2
+ <html>
3
+ <head>
4
+ <title>MathJax Example</title>
5
+ <script>
6
+ MathJax = {
7
+ tex: {
8
+ inlineMath: [['$', '$'], ['\(', '\)']]
9
+ },
10
+ svg: {
11
+ fontCache: 'global'
12
+ }
13
+ };
14
+ </script>
15
+ <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
16
+ </head>
17
+ <body>
18
+ <p> $c_{j}=\epsilon_{2}$ </p>
19
+ <p> $\{e_{1},e_{2}\}$ </p>
20
+ <p> $\Delta_{2}({\text{MARK\_LEFT}})=(c_{1})\times X+(1-c_{1})\times X$ </p>
21
+ <p> $A$ </p>
22
+ <p> $S$ </p>
23
+ <p> $\alpha\geq 1$ </p>
24
+ <p> $|w_{3}+w^{*}+w_{4}-w_{3}-w_{4}|=w^{*}$ </p>
25
+ <p> $\displaystyle\pi_{v_{i},u_{j}}$ </p>
26
+ <p> $\{v^{\prime}_{4},v^{\prime}_{5},v^{\prime}_{6}\}$ </p>
27
+ <p> $x=w_{0}$ </p>
28
+ <p> $v_{n_{1}+1}$ </p>
29
+ <p> $\displaystyle=$ </p>
30
+ <p> $e^{*},$ </p>
31
+ <p> ${n_{L}}{n_{R}}\times(({\mathcal{L}}-i-1)({\mathcal{R}}-j)-({\mathcal{L}}-i)({%
32
+ \mathcal{R}}-j))={n_{L}}{n_{R}}\times({\mathcal{L}}{\mathcal{R}}-{\mathcal{L}}%
33
+ j-i{\mathcal{R}}+ij-{\mathcal{R}}+j-{\mathcal{L}}{\mathcal{R}}+{\mathcal{L}}j+%
34
+ i{\mathcal{R}}-ij)={n_{L}}{n_{R}}\times(j-{\mathcal{R}})$ </p>
35
+ <p> ${R}_{j}$ </p>
36
+ <p> $-2\times R_{i}\times(S_{RM}-R_{i})$ </p>
37
+ <p> $V_{m}=\{v|v,u\in V,\exists e=(u,v)\in E_{m}\}$ </p>
38
+ <p> $v_{2}$ </p>
39
+ <p> $\Delta_{v_{i},u_{j}}=\mathcal{E}^{v_{i},u_{j}}_{2}-\mathcal{E}^{v_{i},u_{j}}_{%
40
+ 1}=\left|w^{*}_{k}\right|-\left|\mathcal{W}^{\prime}(E^{(v_{i},u_{j})})-%
41
+ \mathcal{W}^{*}(E^{(v_{i},u_{j})})\right|\leq\left|w^{*}_{k}-\mathcal{W}^{%
42
+ \prime}(E^{(v_{i},u_{j})})+\mathcal{W}^{*}(E^{(v_{i},u_{j})})\right|$ </p>
43
+ <p> $k$ </p>
44
+ <p> $w^{\prime}(e_{i})=w(e_{i})+\epsilon_{i},\epsilon_{i}\in\{0,w^{*}\}$ </p>
45
+ <p> $c_{1}$ </p>
46
+ <p> $n_{R}=3$ </p>
47
+ <p> $\displaystyle=n_{L}\times(w_{0}-x+w_{0}+w_{1}-x+\dots+w_{0}+w_{1}+\dots+w_{k}-%
48
+ x)=n_{L}\times\big{(}\big{(}\sum_{j=0}^{k}(k+1-j)w_{j}\big{)}-(k+1)\times x%
49
+ \big{)}$ </p>
50
+ <p> $\mathcal{C}(v)=k$ </p>
51
+ <p> $\{v_{1},v_{2}\}$ </p>
52
+ <p> $\mathcal{E}_{L}^{(\frac{k}{2})}$ </p>
53
+ <p> $k^{\prime}$ </p>
54
+ <p> $c_{i}+c_{j}\leq 1$ </p>
55
+ <p> $S_{L}-L_{i}-S_{R}\leq 0$ </p>
56
+ <p> $\{v^{\prime}_{1},v^{\prime}_{2},v^{\prime}_{3}\}$ </p>
57
+ <p> $E^{\prime}\subset E$ </p>
58
+ <p> $v_{1},v_{3}\in\overline{V_{m}}$ </p>
59
+ <p> $w^{*}_{1},w^{*}_{2},\dots,w^{*}_{k}$ </p>
60
+ <p> $\mathcal{F}$ </p>
61
+ <p> $x<0$ </p>
62
+ <p> $C\leq x\leq B+C$ </p>
63
+ <p> $T_{j}^{L}$ </p>
64
+ <p> $\mathcal{E}(M_{L}^{*})\leq\mathcal{E}(M_{R}^{*})\xrightarrow[]{}\sum_{e_{i}\in
65
+ E%
66
+ _{L}}L_{i}\times w^{*}\times(S_{L}-L_{i}-S_{R})\leq\sum_{e_{i}\in E_{R}}R_{i}%
67
+ \times w^{*}\times(S_{R}-R_{i}-S_{L})$ </p>
68
+ <p> $x\geq A+B$ </p>
69
+ <p> $\mathcal{E}_{LR}$ </p>
70
+ <p> $\displaystyle\geq\mathcal{E}(M_{0})+\sum_{e_{i}\in E_{L}}L_{i}\times w^{*}%
71
+ \times(S_{L}-L_{i}-S_{R})=\mathcal{E}(M_{L}^{*})$ </p>
72
+ <p> $x<w_{0}$ </p>
73
+ <p> $S_{RM}>0$ </p>
74
+ <p> $\mathcal{E}(M)<\mathcal{E}(M^{\prime})$ </p>
75
+ <p> ${L}_{i}$ </p>
76
+ <p> ${n_{R}}$ </p>
77
+ <p> $e_{1}=(v_{1},v_{3})$ </p>
78
+ <p> ${\mathcal{L}}\xleftarrow[]{}|E_{L}|$ </p>
79
+ <p> $\mathcal{E}_{L}$ </p>
80
+ <p> $\mathcal{E}(M^{\prime\prime})\leq\mathcal{E}(M)$ </p>
81
+ <p> $w^{*}$ </p>
82
+ <p> $0<\epsilon<w_{\frac{k}{2}+1}$ </p>
83
+ <p> $|\sum_{k=i}^{n_{1}}\epsilon_{k}|$ </p>
84
+ <p> $|\Delta E|$ </p>
85
+ <p> $-R_{i}\times S_{LM}$ </p>
86
+ <p> $w^{\prime}(e)=w(e)+w(e^{*})$ </p>
87
+ <p> $u_{1}\in T_{i}^{L},u_{2}\in T_{j}^{R}$ </p>
88
+ <p> $S_{RM}=0$ </p>
89
+ <p> $\Delta({\text{MARK\_LEFT}})\leq 0$ </p>
90
+ <p> $C_{R}$ </p>
91
+ <p> $-L_{i}\times(S_{LM})+L_{i}\times(S_{LU}-L_{i})$ </p>
92
+ <p> $c_{0},c_{1},\dots,c_{i}$ </p>
93
+ <p> $k+1-j$ </p>
94
+ <p> $|\Delta E|\geq(n-2)B=|\overline{V_{m}}|B$ </p>
95
+ <p> $B\times k^{\prime}\times(n-(k+k^{\prime}))=B\times(n-2)$ </p>
96
+ <p> $P^{\prime}$ </p>
97
+ <p> $|\Delta E|^{\prime}\geq w^{*}$ </p>
98
+ <p> $C$ </p>
99
+ <p> $|y-C|+|y-B-C|$ </p>
100
+ <p> $E_{m}\subset E$ </p>
101
+ <p> $j<i+1$ </p>
102
+ <p> $\overline{V_{m}}=V-V_{m}$ </p>
103
+ <p> $v_{i}\in V_{L},v_{i}\neq v_{n_{1}+1}$ </p>
104
+ <p> ${\mathcal{L}}$ </p>
105
+ <p> $e^{*}=(v_{1},v_{2})$ </p>
106
+ <p> $y=C$ </p>
107
+ <p> $\mathcal{E}(M_{L}^{*})\leq\mathcal{E}(M_{R}^{*})$ </p>
108
+ <p> $u\in T_{2}^{L}$ </p>
109
+ <p> $\Delta({\text{UNMARK\_RIGHT}})$ </p>
110
+ <p> $v_{i},v_{j}\in V_{L}\;(i<j,\;j\neq n_{1}+1)$ </p>
111
+ <p> $c_{i}=1$ </p>
112
+ <p> ${n^{2}_{R}}\times((j-1)({\mathcal{R}}-j+1)-j({\mathcal{R}}-j))={n^{2}_{R}}%
113
+ \times(j{\mathcal{R}}-j^{2}+j-{\mathcal{R}}+j-1-j{\mathcal{R}}+j^{2})={n^{2}_{%
114
+ R}}\times(2j-{\mathcal{R}}-1)$ </p>
115
+ <p> $\mathcal{C}(u)=k^{\prime}$ </p>
116
+ <p> $E^{\prime}$ </p>
117
+ <p> $0.4$ </p>
118
+ <p> $\Delta({\text{UNMARK\_RIGHT}})=0$ </p>
119
+ <p> ${n_{L}}{n_{R}}((i+1)j-ij)={n_{L}}{n_{R}}\times j$ </p>
120
+ <p> $-\left|w^{*}-\sum_{k=i}^{n_{1}}\epsilon_{k}\right|$ </p>
121
+ <p> $c_{i}<c_{j}$ </p>
122
+ <p> $e\in E^{\prime}$ </p>
123
+ <p> $k-k^{\prime}\geq 0$ </p>
124
+ <p> $-R_{i}\times(S_{RU})+R_{i}\times(S_{RM}-R_{i})$ </p>
125
+ <p> $w^{*}_{k}$ </p>
126
+ <p> $\operatorname{CONTRACTION}(\pi)$ </p>
127
+ <p> $T-\{v_{1},v_{2}\}$ </p>
128
+ <p> $V_{R}$ </p>
129
+ <p> $\begin{array}[]{cc}\Delta&=R_{i}\times\bigg{(}\underbrace{-(S_{RM}-R_{i})}_{<0%
130
+ }-S_{RU}\underbrace{-S_{LM}}_{<0}+S_{LU}\bigg{)}<R_{i}(\underbrace{-S_{RU}+S_{%
131
+ LU}}_{\leq 0})<0\end{array}$ </p>
132
+ <p> $G=P_{n}$ </p>
133
+ <p> $\triangleright$ </p>
134
+
135
+ </body>
136
+ </html>
output_mathjax_example_6.html ADDED
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1
+ <!DOCTYPE html>
2
+ <html>
3
+ <head>
4
+ <title>MathJax Example</title>
5
+ <script>
6
+ MathJax = {
7
+ tex: {
8
+ inlineMath: [['$', '$'], ['\(', '\)']]
9
+ },
10
+ svg: {
11
+ fontCache: 'global'
12
+ }
13
+ };
14
+ </script>
15
+ <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
16
+ </head>
17
+ <body>
18
+ <p> $\alpha_{2}=|y-C|+|y-B-C|$ </p>
19
+ <p> $v_{j}\in V_{R}$ </p>
20
+ <p> $\displaystyle{n^{2}_{L}}\times{\mathcal{L}}({\mathcal{L}}-1)-{n^{2}_{R}}\times%
21
+ {\mathcal{R}}({\mathcal{R}}-1)$ </p>
22
+ <p> $u_{6}$ </p>
23
+ <p> $\alpha_{2}=B$ </p>
24
+ <p> $w_{1}+w_{2}=\sum_{k=1}^{2}w_{k}$ </p>
25
+ <p> $R_{i}$ </p>
26
+ <p> $k+1$ </p>
27
+ <p> $\mathcal{E}_{L}^{(\frac{k}{2})}+(\frac{k}{2}+1)\;\epsilon-(\frac{k}{2})\;%
28
+ \epsilon>\mathcal{E}_{L}^{(\frac{k}{2})}$ </p>
29
+ <p> $w^{\prime}(e)=w(e)$ </p>
30
+ <p> $i<{\mathcal{L}}$ </p>
31
+ <p> $\mathcal{C},\;\mathcal{C}:V_{s}\rightarrow\mathbb{N},$ </p>
32
+ <p> $v_{6}$ </p>
33
+ <p> $\Delta_{1}({\text{UNMARK\_LEFT}})\geq 0$ </p>
34
+ <p> $L_{1}\times R_{1}\times w^{*}$ </p>
35
+ <p> $v_{4}$ </p>
36
+ <p> $\mathcal{W}(E^{(v_{1},u_{5})})=w_{1}+w_{2}+w_{3}+w_{4}$ </p>
37
+ <p> $k=k^{\prime}=1$ </p>
38
+ <p> $n_{L}=0$ </p>
39
+ <p> $w(e_{i})$ </p>
40
+ <p> $\Delta(\text{UNMARK\_RIGHT})\leq 0$ </p>
41
+ <p> $\Delta({\text{MARK\_LEFT}})=L_{i}\times\bigg{(}S_{LM}+(S_{LU}-L_{i})+S_{RM}-S_%
42
+ {RU}\bigg{)}$ </p>
43
+ <p> $\alpha_{1}=A-x+A+B-x<B\xrightarrow{}A<x$ </p>
44
+ <p> $c_{1}+c_{2}<1+\epsilon=c_{1}+c_{2}$ </p>
45
+ <p> $M^{*}$ </p>
46
+ <p> $\mathcal{E}(M)$ </p>
47
+ <p> $n_{L}+n_{R}=n-(k+k^{\prime})$ </p>
48
+ <p> $\mathcal{O}(|V|)$ </p>
49
+ <p> $e^{\prime}$ </p>
50
+ <p> $S_{L}-S_{R}\leq L_{i}$ </p>
51
+ <p> $B\times k^{\prime}\times(n-(k+k^{\prime}))$ </p>
52
+ <p> $e\in S$ </p>
53
+ <p> $1\leq i\leq{\mathcal{L}}$ </p>
54
+ <p> $e_{2}$ </p>
55
+ <p> $\mathcal{E}(M^{\prime})\leq\mathcal{E}(M)$ </p>
56
+ <p> $-S_{L}+L_{i}+S_{R}\geq 0\xrightarrow{}L_{i}\geq S_{L}-S_{R}$ </p>
57
+ <p> $\displaystyle n_{L}\times k^{\prime}\times B$ </p>
58
+ <p> $e_{j}$ </p>
59
+ <p> $\frac{{\mathcal{L}}}{2}+\frac{{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}-i=\frac{{%
60
+ \mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}-i$ </p>
61
+ <p> $\displaystyle\pi^{\prime}_{v_{i},u_{j}}$ </p>
62
+ <p> $w^{\prime}\left(e_{i}\right)=w\left(e_{i}\right)+c_{i}w\left(e^{*}\right)$ </p>
63
+ <p> $i<j$ </p>
64
+ <p> $w^{\prime}$ </p>
65
+ <p> $u_{1}$ </p>
66
+ <p> $u,v\in V,$ </p>
67
+ <p> $E_{m}$ </p>
68
+ <p> $\displaystyle(k-k^{\prime})\times n_{L}\times|x-A|+n_{L}\times k^{\prime}%
69
+ \times\big{(}|x-A|+|x-A-B|\big{)}$ </p>
70
+ <p> $\displaystyle\underbrace{|w_{2}-(w_{2}+\epsilon_{2})|}_{\text{between }u\text{%
71
+ and }v_{1}}+\underbrace{|w_{2}+w^{*}-(w_{2}+\epsilon_{2})|}_{\text{between }u%
72
+ \text{ and }v_{2}}=|\epsilon_{2}|+|w^{*}-\epsilon_{2}|=|\epsilon_{2}|+|%
73
+ \epsilon_{2}-w^{*}|$ </p>
74
+ <p> $n_{1}=n_{L}$ </p>
75
+ <p> $T^{L}_{1}$ </p>
76
+ <p> $-\epsilon\times w^{*}$ </p>
77
+ <p> $V_{m}$ </p>
78
+ <p> $B$ </p>
79
+ <p> $V$ </p>
80
+ <p> $|\Delta E|\geq B(n-2)+n_{L}n_{R}|x+y-A-B-C|\geq B(n-2)=|\overline{V_{m}}|B$ </p>
81
+ <p> $e^{*}=(u_{1},v_{1})$ </p>
82
+ <p> $e=(u,w_{2})$ </p>
83
+ <p> $A\leq x\leq A+B$ </p>
84
+ <p> $T=(V,E)$ </p>
85
+ <p> $\epsilon_{1}$ </p>
86
+ <p> $|V|$ </p>
87
+ <p> $\mathcal{E}_{L}^{(i)}$ </p>
88
+ <p> $c^{\prime}_{i}=c_{i}+\epsilon$ </p>
89
+ <p> $R_{1}\times R_{2}\times w^{*}$ </p>
90
+ <p> $e^{*}=(u_{1},v_{1}),w^{*}=w(e^{*})$ </p>
91
+ <p> $|z|-|x|\leq|z-x|$ </p>
92
+ <p> $|x|=-x$ </p>
93
+ <p> ${\mathcal{R}}\xleftarrow[]{}|E_{R}|$ </p>
94
+ <p> $V_{L}=\{v_{3},v_{4}\}$ </p>
95
+ <p> $\displaystyle|\Delta E|=\underbrace{n_{L}\times|x-A|\times k}_{\text{between %
96
+ the subpath of }w_{1}\text{ and the vertices in }v}+\underbrace{n_{L}\times|x-%
97
+ A-B|\times k^{\prime}}_{\text{between the subpath of }w_{1}\text{ and the %
98
+ vertices in }u}+$ </p>
99
+ <p> $\displaystyle\frac{{\mathcal{L}}{n_{L}}-{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}({n%
100
+ ^{2}_{L}}\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}(-{\mathcal{R}}))$ </p>
101
+ <p> $|w^{*}|-\left|w^{*}-\sum_{k=n_{1}+1}^{j-2}\epsilon_{k}\right|\leq\left|\sum_{k%
102
+ =n_{1}+1}^{j-2}\epsilon_{k}\right|$ </p>
103
+ <p> $\left|w^{*}-\sum_{k=i}^{n_{1}}\epsilon_{k}\right|$ </p>
104
+ <p> $0\leq j\leq k$ </p>
105
+ <p> $e^{*}=(v_{n_{1}+1},v_{n_{1}+2}$ </p>
106
+ <p> $L_{i}\times L_{j}\times 2w^{*}$ </p>
107
+ <p> $i=\{1,\dots,{\mathcal{R}}\}$ </p>
108
+ <p> $e_{i}\in\overline{E_{m}}=E-E_{m}$ </p>
109
+ <p> $R_{i}\times S_{LU}$ </p>
110
+ <p> $\overline{V_{m}}$ </p>
111
+ <p> $c_{j}$ </p>
112
+ <p> $R_{1}$ </p>
113
+ <p> ${n^{2}_{L}}\times 2\bigg{(}{i+1\choose 2}-{i\choose 2}\bigg{)}={n^{2}_{L}}%
114
+ \times 2i$ </p>
115
+ <p> $i=\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}$ </p>
116
+ <p> $\begin{array}[]{cc}\Delta({\text{MARK\_LEFT}})\leq 0\xrightarrow[]{}{n^{2}_{L}%
117
+ }\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}(2j-{\mathcal{R}})&\leq 0\\
118
+ \end{array}$ </p>
119
+ <p> $\displaystyle n_{L}\times\big{(}\sum_{j=0}^{i+1}j\;w_{j}+\sum_{j=i+2}^{k}(k+1-%
120
+ j)\;w_{j}\big{)}$ </p>
121
+ <p> $\{e_{1},\dots,e_{k}\}$ </p>
122
+ <p> $e^{\prime}\neq e^{*}$ </p>
123
+ <p> $w_{1}+\epsilon_{1}+w_{2}+\epsilon_{2}=\sum_{k=1}^{2}(w_{k}+\epsilon_{k})$ </p>
124
+ <p> $e^{*}_{k}=(u_{j},u_{j+1})\in E_{m}$ </p>
125
+ <p> $n_{L}=n_{R}$ </p>
126
+ <p> $\Delta(\text{MARK\_LEFT})\leq 0$ </p>
127
+ <p> $e^{*}_{j}$ </p>
128
+ <p> $\alpha_{1}<B$ </p>
129
+ <p> $|w^{*}|-\left|\sum_{k=i}^{n_{1}}\epsilon_{k}\right|$ </p>
130
+ <p> $i+1$ </p>
131
+ <p> $\Delta_{v_{i},u_{j+1}}=\mathcal{E}^{v_{i},u_{j+1}}_{2}-\mathcal{E}^{v_{i},u_{j%
132
+ +1}}_{1}=-\left|w^{*}_{k}+\mathcal{W}^{*}(E^{(v_{i},u_{j})})-\mathcal{W}^{%
133
+ \prime}(E^{(v_{i},u_{j})})\right|$ </p>
134
+ <p> $v_{i},v_{j}\in V_{L}(i<j)$ </p>
135
+ <p> $w^{\prime}:E\rightarrow\mathbb{R}_{\geq 0}$ </p>
136
+ <p> $x=\epsilon+\sum_{j=0}^{\frac{k}{2}}w_{j}$ </p>
137
+
138
+ </body>
139
+ </html>
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1
+ <!DOCTYPE html>
2
+ <html>
3
+ <head>
4
+ <title>MathJax Example</title>
5
+ <script>
6
+ MathJax = {
7
+ tex: {
8
+ inlineMath: [['$', '$'], ['\(', '\)']]
9
+ },
10
+ svg: {
11
+ fontCache: 'global'
12
+ }
13
+ };
14
+ </script>
15
+ <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
16
+ </head>
17
+ <body>
18
+ <p> $\mathcal{E}_{L}^{(0)}$ </p>
19
+ <p> $\Delta({\text{UNMARK\_LEFT}})=L_{i}\times\bigg{(}-(S_{LM}-L_{i})-S_{LU}-S_{RM}%
20
+ +S_{RU}\bigg{)}=L_{i}\times\bigg{(}L_{i}-S_{L}+S_{R}\bigg{)}<0$ </p>
21
+ <p> $|\overline{V_{m}}|=n-2|E_{m}|=n-2k$ </p>
22
+ <p> $S_{LU}$ </p>
23
+ <p> $\mathcal{E}_{L}^{(0)}=n_{L}\times\big{(}\sum_{j=1}^{k}(k+1-j)\;w_{j}\big{)}$ </p>
24
+ <p> $k\leq\frac{n}{2}$ </p>
25
+ <p> $e_{i}$ </p>
26
+ <p> $\displaystyle=\mathcal{W}(E^{(v_{i},u_{j})})+\mathcal{W}^{*}(E^{(v_{i},u_{j})}%
27
+ )+w^{*}_{k}$ </p>
28
+ <p> $S_{RU}$ </p>
29
+ <p> ${n_{L}}{n_{R}}(i(j-1)-ij)={n_{L}}{n_{R}}\times(-i)$ </p>
30
+ <p> $w(e^{*})$ </p>
31
+ <p> $S_{R}^{\prime}=R_{2}\geq S_{L}^{\prime}=L_{2}+L_{3}$ </p>
32
+ <p> $\mathcal{W}(E^{\prime})=\sum_{e\in E^{\prime}\cap\overline{E_{m}}}w(e),\;\;%
33
+ \mathcal{W}^{*}(E^{\prime})=\sum_{e\in E^{\prime}\cap{E_{m}}}w(e),\;\;\mathcal%
34
+ {W}^{\prime}(E^{\prime})=\sum_{e_{i}\in\overline{E_{m}}\cap E^{\prime}}%
35
+ \epsilon_{i}$ </p>
36
+ <p> $|\Delta E|=n_{L}\times k^{\prime}\times B=(n-(k+k^{\prime}))\times k^{\prime}\times
37
+ B$ </p>
38
+ <p> $S_{R}=\sum_{i=1}^{{\mathcal{R}}}R_{i}$ </p>
39
+ <p> $\mathcal{E}$ </p>
40
+ <p> $|w_{1}+w^{*}+w_{3}-w_{1}-w^{*}-w_{3}|=0$ </p>
41
+ <p> $P_{n},n\geq 3$ </p>
42
+ <p> $\displaystyle>n_{L}\times\big{(}\sum_{j=1}^{k}(k+1-j)w_{j}\big{)}\xrightarrow{%
43
+ \text{See the proof of Lemma \ref{induction1}}}=\mathcal{E}^{(0)}_{L}>\mathcal%
44
+ {E}^{(\frac{k}{2})}_{L}$ </p>
45
+ <p> $e^{*}_{k}=(u_{j},u_{j+1})=e^{*}_{3}=(u_{5},u_{6})$ </p>
46
+ <p> $n_{R}=0$ </p>
47
+ <p> $\overline{V_{m}}=\{v^{\prime}_{1},v^{\prime}_{2},v^{\prime}_{3},v^{\prime}_{4}%
48
+ ,v^{\prime}_{5},v^{\prime}_{6}\}$ </p>
49
+ <p> ${n_{L}}{n_{R}}\times(({\mathcal{L}}-i)({\mathcal{R}}-j+1)-({\mathcal{L}}-i)({%
50
+ \mathcal{R}}-j))={n_{L}}{n_{R}}\times({\mathcal{L}}{\mathcal{R}}-{\mathcal{L}}%
51
+ j+{\mathcal{L}}-i{\mathcal{R}}+ij-i-{\mathcal{L}}{\mathcal{R}}+{\mathcal{L}}j+%
52
+ i{\mathcal{R}}-ij)={n_{L}}{n_{R}}\times({\mathcal{L}}-i)$ </p>
53
+ <p> $w^{\prime}(e_{i})=w(e_{i})+\epsilon_{i},\;\forall e_{i}\in E,\;\epsilon_{i}\in%
54
+ \mathbb{R}$ </p>
55
+ <p> $0<c_{i}<1$ </p>
56
+ <p> $e_{i}\in E_{R}$ </p>
57
+ <p> $\displaystyle|\Delta E|\geq$ </p>
58
+ <p> $\epsilon\leq c_{i}$ </p>
59
+ <p> $\Delta_{1}({\text{MARK\_LEFT}})=c_{1}\times(X)$ </p>
60
+ <p> $A,B,C,D,x,y$ </p>
61
+ <p> $(\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}-i)({n^{2}_{L}}%
62
+ \times({\mathcal{L}}-1)+{n_{L}}{n_{R}}(2j-{\mathcal{R}}))$ </p>
63
+ <p> $|w_{1}+w_{2}+w_{3}+w^{*}-w_{1}-w_{2}-w_{3}|=w^{*}$ </p>
64
+ <p> $v_{3}$ </p>
65
+ <p> $w^{\prime}(e_{i})=w(e_{i})$ </p>
66
+ <p> $v_{n_{1}+2}$ </p>
67
+ <p> $e^{*}=(u,v)$ </p>
68
+ <p> ${e^{*}}$ </p>
69
+ <p> $e_{i}\in E_{L}$ </p>
70
+ <p> $S_{RU}=S_{R}$ </p>
71
+ <p> $n_{L}\times\big{(}(i+1)\;w_{i+1}\big{)}$ </p>
72
+ <p> $c_{i}=\epsilon_{1}=0.4$ </p>
73
+ <p> $+L_{i}\times S_{RM}$ </p>
74
+ <p> $V_{m}=\{u_{1},v_{1}\}$ </p>
75
+ <p> $R_{i}\times R_{j}\times 2w^{*}$ </p>
76
+ <p> $\mathcal{E}^{(x<w_{0})}_{L}$ </p>
77
+ <p> $1\leq j\leq{\mathcal{R}}$ </p>
78
+ <p> $\mathcal{E}(M^{\prime\prime})=\underbrace{\mathcal{E}(M^{\prime})+c_{1}\times X%
79
+ }_{=\mathcal{E}(M)}+(c_{2}-c_{1})\times X$ </p>
80
+ <p> $X=L_{1}\times\bigg{(}S_{LM}+(S_{LU}-L_{1})+S_{RM}-S_{RU}\bigg{)}=L_{1}\times%
81
+ \bigg{(}S_{LM}+(S_{LU}-L_{1})-S_{R}\bigg{)}$ </p>
82
+ <p> $|x-A|+|x-A-B|$ </p>
83
+ <p> $\epsilon_{i}$ </p>
84
+ <p> ${e^{*}}=(u,v)$ </p>
85
+ <p> $c_{1}+c_{2}\leq 1$ </p>
86
+ <p> $|a|=-a$ </p>
87
+ <p> $v^{\prime}_{3}$ </p>
88
+ <p> $u_{1},u_{2}\in\overline{V_{m}}$ </p>
89
+ <p> $v_{j}\in V_{R},v_{j}\neq v_{n_{1}+2}$ </p>
90
+ <p> $u_{j}$ </p>
91
+ <p> $A+B+C$ </p>
92
+ <p> $\pi_{v,u}$ </p>
93
+ <p> $\pi^{\prime\prime}_{v,u}$ </p>
94
+ <p> $\Delta({\text{UNMARK\_RIGHT}})\leq 0\text{ if }{n_{L}}({\mathcal{L}}-2i)\leq{n%
95
+ _{R}}({\mathcal{R}}-1)\xrightarrow[]{\text{Rearranging the terms}}i\geq\frac{{%
96
+ \mathcal{L}}}{2}+\frac{{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}$ </p>
97
+ <p> $\epsilon$ </p>
98
+ <p> $-\left|\sum_{k=i}^{j-1}\epsilon_{k}\right|$ </p>
99
+ <p> $\displaystyle=\mathcal{W}(E^{(v_{i},u_{j})})+\mathcal{W}^{\prime}(E^{(v_{i},u_%
100
+ {j})})$ </p>
101
+ <p> $S_{L}^{\prime}=S_{L}-L_{1}$ </p>
102
+ <p> $w(e^{*}_{i})$ </p>
103
+ <p> $e_{0}$ </p>
104
+ <p> $y=w_{\frac{k}{2}+1}+w_{\frac{k}{2}+2}+\dots+w_{k+1}$ </p>
105
+ <p> $\pi^{\prime}_{v,u}$ </p>
106
+ <p> $\mathcal{E}_{LR}=E_{L}=0$ </p>
107
+ <p> $x-A+x-A-B<B\xrightarrow[]{}x<A+B$ </p>
108
+ <p> $\epsilon_{2}$ </p>
109
+ <p> $j\xleftarrow{}j-1$ </p>
110
+ <p> $\Delta({\text{MARK\_RIGHT}})=R_{i}\times\bigg{(}S_{RM}+(S_{RU}-R_{i})+S_{LM}-S%
111
+ _{LU}\bigg{)}$ </p>
112
+ <p> $H=G-\{e_{3}=(v^{\prime}_{3},u_{1}),e^{*}=(u_{1},v_{1}),e_{4}=(v_{1},v^{\prime}%
113
+ _{4})\}$ </p>
114
+ <p> $R_{j}$ </p>
115
+ <p> $S_{RU}<S_{LU}$ </p>
116
+ <p> $M_{R}^{*}$ </p>
117
+ <p> $\Delta_{1}({\text{UNMARK\_LEFT}})=L_{i}\times\epsilon\times w^{*}\times(-(S_{L%
118
+ }-L_{i})+S_{R})=L_{i}\times\epsilon\times w^{*}\times(-S_{L}+L_{i}+S_{R})$ </p>
119
+ <p> $P_{n}$ </p>
120
+ <p> $\displaystyle\mathcal{E}_{L}^{(i)}+n_{L}\times\big{(}(i+1)\;w_{i+1}-(k-i)\;w_{%
121
+ i+1}\big{)}$ </p>
122
+ <p> $T$ </p>
123
+ <p> $e^{*}_{k}\in E_{m}$ </p>
124
+ <p> $G_{2}$ </p>
125
+ <p> $S_{RU}>S_{LU}$ </p>
126
+ <p> $\Delta({\text{UNMARK\_RIGHT}})={n^{2}_{R}}(-2(j-1)+2j-{\mathcal{R}}-1)+{n_{L}}%
127
+ {n_{R}}({\mathcal{L}}-i-i)={n^{2}_{R}}(1-{\mathcal{R}})+{n_{L}}{n_{R}}({%
128
+ \mathcal{L}}-2i)$ </p>
129
+ <p> $w_{1}+w_{2}+w_{3}+w^{*}$ </p>
130
+ <p> $\left|\sum_{k=i}^{j-1}(w_{k}+\epsilon_{k})-\sum_{k=i}^{j-1}w_{k}\right|=\left|%
131
+ \sum_{k=i}^{j-1}\epsilon_{k}\right|$ </p>
132
+ <p> $v_{i}=v_{1}$ </p>
133
+ <p> $n_{L}=\left|\{v|v\in G_{1}\}\right|,n_{R}=\left|\{v|v\in G_{2}\}\right|$ </p>
134
+ <p> $\displaystyle=\mathcal{W}(E^{(v_{i},u_{j})})+\mathcal{W}^{*}(E^{(v_{i},u_{j})})$ </p>
135
+ <p> $\mathcal{E}_{L}^{(i+1)}-\mathcal{E}_{L}^{(i)}<0$ </p>
136
+ <p> $\displaystyle(n_{L}+n_{R})B=(n-2)B=|\overline{V_{m}}|B$ </p>
137
+ <p> $M_{L}$ </p>
138
+ <p> $M_{R}^{*}\xleftarrow[]{}M_{R}^{*}\cup\{e_{i}\}$ </p>
139
+ <p> $S_{LM}$ </p>
140
+ <p> $e=e_{3}$ </p>
141
+ <p> $x<A+B$ </p>
142
+ <p> $d_{G}(u,v)$ </p>
143
+ <p> $c_{j}=|x-w_{0}-w_{1}-\dots-w_{j}|$ </p>
144
+
145
+ </body>
146
+ </html>
output_mathjax_example_8.html ADDED
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1
+ <!DOCTYPE html>
2
+ <html>
3
+ <head>
4
+ <title>MathJax Example</title>
5
+ <script>
6
+ MathJax = {
7
+ tex: {
8
+ inlineMath: [['$', '$'], ['\(', '\)']]
9
+ },
10
+ svg: {
11
+ fontCache: 'global'
12
+ }
13
+ };
14
+ </script>
15
+ <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
16
+ </head>
17
+ <body>
18
+ <p> $w^{*}_{i}$ </p>
19
+ <p> ${T_{j}^{R}},j\in\{1,2\}$ </p>
20
+ <p> $G=(V,E)$ </p>
21
+ <p> $\displaystyle(k-k^{\prime})\times n_{L}\times|x-A|+n_{L}\times k^{\prime}%
22
+ \times\big{(}|x-A|+|x-A-B|\big{)}\xrightarrow{\text{no edge weights are %
23
+ changed, set }x=A}$ </p>
24
+ <p> $E_{L}$ </p>
25
+ <p> $\displaystyle(\underbrace{\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}%
26
+ {2{n_{L}}}-i}_{\leq\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L%
27
+ }}}})(\underbrace{{n^{2}_{L}}\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}(2j-{%
28
+ \mathcal{R}})}_{\leq{n^{2}_{L}}\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}({%
29
+ \mathcal{R}})})+j\underbrace{({n^{2}_{R}}(1-{\mathcal{R}})+{n_{L}}{n_{R}}({%
30
+ \mathcal{L}}-2\times\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{%
31
+ L}}})}_{=0})$ </p>
32
+ <p> $x=\sum_{j=0}^{i}w_{j}$ </p>
33
+ <p> $y=C+\epsilon_{2}$ </p>
34
+ <p> $E$ </p>
35
+ <p> $z$ </p>
36
+ <p> $\overline{E_{m}}=E-E_{m}$ </p>
37
+ <p> $\displaystyle\xrightarrow{0\leq x<w_{0}}$ </p>
38
+ <p> $\epsilon_{1}+\epsilon_{2}\leq 1$ </p>
39
+ <p> $C_{L}$ </p>
40
+ <p> $(v,u)$ </p>
41
+ <p> $w(e)$ </p>
42
+ <p> $-L_{i}\times S_{RU}$ </p>
43
+ <p> $V_{R}=\{w|(v_{2},w)\in E^{\prime}\},{\mathcal{R}}=|V_{R}|$ </p>
44
+ <p> $v^{\prime}_{4}$ </p>
45
+ <p> $S_{L}\xleftarrow{}\sum_{\forall e_{i}\in E_{L}}L_{i}$ </p>
46
+ <p> $\alpha_{2}\geq B$ </p>
47
+ <p> $w^{\prime}(e_{i})=w(e_{i})+\epsilon_{i}$ </p>
48
+ <p> $e^{*}=(v_{n_{1}+1},v_{n_{1}+2})$ </p>
49
+ <p> $i$ </p>
50
+ <p> $j=0$ </p>
51
+ <p> $w_{2}$ </p>
52
+ <p> $\displaystyle=\mathcal{E}(M_{0})+\epsilon_{1}\times\sum_{e_{i}\in E_{L}}L_{i}%
53
+ \times w^{*}\times(S_{L}-L_{i}-S_{R})+\epsilon_{2}\times\sum_{e_{i}\in E_{R}}R%
54
+ _{i}\times w^{*}\times(S_{R}-R_{i}-S_{L})$ </p>
55
+ <p> $S_{RU}\geq S_{LU}$ </p>
56
+ <p> $c^{\prime}_{i}=c_{i}-\epsilon$ </p>
57
+ <p> $n_{L}=n-(k+k^{\prime})$ </p>
58
+ <p> $T_{i}^{R}$ </p>
59
+ <p> $|\Delta E|=n_{L}(|x-A|+|x-A-B|)+n_{R}(|y-C|+|y-B-C|)+n_{L}n_{R}|x+y-A-B-C|=n_{%
60
+ L}\alpha_{1}+n_{R}\alpha_{2}+n_{L}n_{R}|x+y-A-B-C|$ </p>
61
+ <p> $\epsilon_{n_{1}}=w^{*}$ </p>
62
+ <p> $\displaystyle\geq$ </p>
63
+ <p> $|\Delta E|=\sum_{u\in V_{m},v\in\overline{V_{m}},\text{ or }u,v\in\overline{V_%
64
+ {m}},u\neq v}\left|d_{G}(u,v)-d_{G^{\prime}}(u,v)\right|$ </p>
65
+ <p> $n_{2}+1$ </p>
66
+ <p> $7-21\leq 2=L_{1}$ </p>
67
+ <p> $z=w^{*}$ </p>
68
+ <p> $x=\sum_{j=0}^{i+1}w_{j}$ </p>
69
+ <p> $M^{\prime}$ </p>
70
+ <p> $\displaystyle\Delta=$ </p>
71
+ <p> $u\in G_{1},v\in G_{2}$ </p>
72
+ <p> $\displaystyle|\Delta E|^{\prime}=$ </p>
73
+ <p> $x+y$ </p>
74
+ <p> $u_{1}\in T_{1}^{L},u_{2}\in T_{1}^{R}$ </p>
75
+ <p> $S_{L}=S_{LU}+S_{LM}$ </p>
76
+ <p> $S_{R}=S_{RU}$ </p>
77
+ <p> $\mathcal{E}_{R}$ </p>
78
+ <p> $w:E\xrightarrow[]{}\mathbb{R}_{\geq 0}$ </p>
79
+ <p> $c_{j}=\epsilon_{2}=0.5$ </p>
80
+ <p> $0.1$ </p>
81
+ <p> $e^{*}=(v_{2},v_{3})$ </p>
82
+ <p> ${\mathcal{L}}\choose 2$ </p>
83
+ <p> $\mathcal{E}(M^{\prime\prime})=\mathcal{E}(M^{\prime})+\Delta_{2}({\text{MARK\_%
84
+ LEFT}})=\underbrace{\mathcal{E}(M^{\prime})+(c_{1})\times X}_{=\mathcal{E}(M^{%
85
+ \prime})+\Delta_{1}({\text{MARK\_LEFT}})=\mathcal{E}(M)}+(1-c_{1})\times X<%
86
+ \mathcal{E}(M)$ </p>
87
+ <p> $|w^{*}|-\left|w^{*}-\sum_{k=n_{1}+1}^{j-2}\epsilon_{k}\right|$ </p>
88
+ <p> $V_{R}=\{v_{5},v_{6}\}$ </p>
89
+ <p> $u\in V_{L}$ </p>
90
+ <p> $L_{i}$ </p>
91
+ <p> $0<c_{i}\leq 1$ </p>
92
+ <p> $\overline{V_{m}}=V-\{v_{1},v_{2}\}$ </p>
93
+ <p> $(n-2k)w^{*}_{k}$ </p>
94
+ <p> $u_{1}\in T_{1}^{R},u_{2}\in T_{2}^{R}$ </p>
95
+ <p> $\alpha_{1}\geq B$ </p>
96
+ <p> $e$ </p>
97
+ <p> $L_{1}\times L_{2}\times 2w^{*}$ </p>
98
+ <p> $\mathcal{E}(M_{L}^{*})=\underbrace{\mathcal{E}(M_{0})}_{\text{The error %
99
+ associated with the empty marking}}+\underbrace{\sum_{e_{i}\in E_{L}}L_{i}%
100
+ \times w^{*}\times(S_{L}-L_{i}-S_{R})}_{\text{The sum of all }\Delta({\text{%
101
+ MARK\_LEFT}})\text{'s that
102
+ transform }M_{0}\text{ into }M_{L}^{*}}$ </p>
103
+ <p> $\mathcal{E}=\mathcal{E}_{L}+\mathcal{E}_{R}+\mathcal{E}_{LR}$ </p>
104
+ <p> $\epsilon_{i}\neq 0$ </p>
105
+ <p> $\epsilon\times w^{*}$ </p>
106
+ <p> $\pi:\mathcal{G}\xrightarrow{}\mathbb{R}$ </p>
107
+ <p> $j$ </p>
108
+ <p> $y=B+C$ </p>
109
+ <p> $S_{L}-S_{R}>L_{i}$ </p>
110
+ <p> $21-7\leq 20=R_{1}$ </p>
111
+ <p> $\mathcal{E}(.)$ </p>
112
+ <p> $e^{*}_{k}$ </p>
113
+ <p> ${\mathcal{L}}={\mathcal{R}}=2$ </p>
114
+ <p> $\displaystyle(k-k^{\prime})\times n_{R}\times|y-B-C|+n_{R}\times k^{\prime}%
115
+ \times\big{(}|y-C|+|y-B-C|\big{)}$ </p>
116
+ <p> $w(e_{i})+\epsilon_{i}$ </p>
117
+ <p> $w^{\prime}(e_{3})=w_{3}+w^{*}$ </p>
118
+ <p> $c_{i}=\epsilon_{1}$ </p>
119
+ <p> $0\leq\epsilon_{2}\leq w^{*}$ </p>
120
+ <p> $x=A+B$ </p>
121
+ <p> $T_{i}={t^{1}_{1},\dots,t^{1}_{k}}$ </p>
122
+ <p> $X=\{x_{1},x_{2},...,x_{n}\}$ </p>
123
+ <p> $T_{i}\sim p_{\theta}(t^{1}_{1\dots k}\mid x_{1}\dots,x_{n})$ </p>
124
+ <p> $t^{i}_{j}*\sim p^{vote}_{\theta}(t^{i}_{j}*|B)$ </p>
125
+ <p> $X$ </p>
126
+ <p> $n-1$ </p>
127
+ <p> $V(p_{\theta},T_{i})(t^{i}_{j})=1[t^{i}_{j}=t^{i}_{j}*]$ </p>
128
+ <p> $n_{i}\in N$ </p>
129
+ <p> $t^{n}_{1}\sim t^{n}_{k}$ </p>
130
+ <p> $G=F(X)$ </p>
131
+ <p> $X=\{x_{1},x_{2},...,x_{m}\}$ </p>
132
+ <p> $t^{i-1}_{j}$ </p>
133
+ <p> $x_{i}$ </p>
134
+ <p> $t^{i}_{j}$ </p>
135
+ <p> $F$ </p>
136
+ <p> $G=(N,E)$ </p>
137
+ <p> $\{t^{i}_{1},\dots,t^{i}_{k}\}\sim p_{\theta}(t^{i}_{1\dots k}\mid x_{1\dots n}%
138
+ ,t^{i-1}_{j})$ </p>
139
+
140
+ </body>
141
+ </html>
output_mathjax_example_9.html ADDED
@@ -0,0 +1,123 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ <!DOCTYPE html>
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+ <html>
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+ <head>
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+ <title>MathJax Example</title>
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+ <script>
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+ MathJax = {
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+ tex: {
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+ inlineMath: [['$', '$'], ['\(', '\)']]
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+ },
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+ svg: {
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+ fontCache: 'global'
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+ }
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+ };
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+ </script>
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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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+ </head>
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+ <body>
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+ <p> $T_{i}$ </p>
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+ <p> $\operatorname{\bm{\theta}}_{\text{agent}}$ </p>
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+ <p> $\operatorname{\bm{\theta}}_{(\cdot)}^{(t+1)}$ </p>
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+ <p> $2.41$ </p>
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+ <p> $5-20\%$ </p>
23
+ <p> $50$ </p>
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+ <p> $\mathbf{W}\in\mathbb{R}^{d\times d}$ </p>
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+ <p> $46.55$ </p>
26
+ <p> $16$ </p>
27
+ <p> $p(\operatorname{\mathbf{d}}|\operatorname{\bm{\theta}}_{\text{client}}^{(t)},%
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+ \operatorname{\bm{\theta}}_{\text{agent}}^{(t)},\operatorname{\mathbf{pr}}_{%
29
+ \text{agent}},\operatorname{\mathbf{pr}}_{\text{client}})$ </p>
30
+ <p> $\tau\approx 0.67$ </p>
31
+ <p> $\underset{\pm 6.21}{273.71}$ </p>
32
+ <p> $\alpha=0.9$ </p>
33
+ <p> $2.15$ </p>
34
+ <p> $0.67$ </p>
35
+ <p> $0.97$ </p>
36
+ <p> $40$ </p>
37
+ <p> $0.92$ </p>
38
+ <p> $5\%$ </p>
39
+ <p> $\mathbf{2.54}$ </p>
40
+ <p> $\bigstar$ </p>
41
+ <p> $0.29$ </p>
42
+ <p> $0.27$ </p>
43
+ <p> $0.87$ </p>
44
+ <p> $303.65$ </p>
45
+ <p> $\mathcal{V}$ </p>
46
+ <p> $46.62$ </p>
47
+ <p> $\tau$ </p>
48
+ <p> $0.93$ </p>
49
+ <p> $49.40$ </p>
50
+ <p> $3$ </p>
51
+ <p> $0.05$ </p>
52
+ <p> $0.33$ </p>
53
+ <p> $n=9$ </p>
54
+ <p> $p=0.95$ </p>
55
+ <p> $285.94$ </p>
56
+ <p> $5\%-20\%$ </p>
57
+ <p> $\mathbf{0.81}$ </p>
58
+ <p> $2.37$ </p>
59
+ <p> $343.07$ </p>
60
+ <p> $5\times 10^{-4}$ </p>
61
+ <p> $\underset{\pm 0.03}{0.24}$ </p>
62
+ <p> $200$ </p>
63
+ <p> $\Delta\mathbf{W}\in\mathbb{R}^{d\times d}$ </p>
64
+ <p> $0.22$ </p>
65
+ <p> $\underset{\pm 0.02}{0.37}$ </p>
66
+ <p> $0.30$ </p>
67
+ <p> $l_{1}$ </p>
68
+ <p> $5$ </p>
69
+ <p> $\varnothing$ </p>
70
+ <p> $r\ll d$ </p>
71
+ <p> $\underset{\pm 0.00}{0.77}$ </p>
72
+ <p> $0.63$ </p>
73
+ <p> $45.51$ </p>
74
+ <p> $\underset{\pm 0.01}{0.62}$ </p>
75
+ <p> $2.$ </p>
76
+ <p> $2.56$ </p>
77
+ <p> $\mathcal{D}^{(t)}_{\bigtriangledown}\subseteq\mathcal{D}^{(t)}$ </p>
78
+ <p> $373.87$ </p>
79
+ <p> $20$ </p>
80
+ <p> $37\%$ </p>
81
+ <p> $1$ </p>
82
+ <p> $280.53$ </p>
83
+ <p> $\approx 1.95\times 10^{-8}$ </p>
84
+ <p> $0.5$ </p>
85
+ <p> $\mathbf{B}\in\mathbb{R}^{r\times d}$ </p>
86
+ <p> $0.38$ </p>
87
+ <p> $266.77$ </p>
88
+ <p> $\operatorname{\mathbf{pr}}_{\text{agent}}$ </p>
89
+ <p> $\uparrow$ </p>
90
+ <p> $\operatorname{\bm{\theta}}_{\text{agent}}^{(t)}$ </p>
91
+ <p> $1-2\%$ </p>
92
+ <p> $1\%-5\%$ </p>
93
+ <p> $0.64$ </p>
94
+ <p> $0.98$ </p>
95
+ <p> $10042$ </p>
96
+ <p> $0.35$ </p>
97
+ <p> $2.22$ </p>
98
+ <p> $\mathbf{355.44}$ </p>
99
+ <p> $k=50$ </p>
100
+ <p> $6$ </p>
101
+ <p> $0.41$ </p>
102
+ <p> $2.31$ </p>
103
+ <p> $500k$ </p>
104
+ <p> $10\%$ </p>
105
+ <p> $54.14$ </p>
106
+ <p> $\mathcal{G}=\{\mathcal{V},\mathcal{E}\}$ </p>
107
+ <p> $28$ </p>
108
+ <p> $\mathcal{D}^{(t)}=\{\operatorname{\mathbf{d}}_{1}^{(t)},\ldots,\operatorname{%
109
+ \mathbf{d}}_{N}^{(t)}\}$ </p>
110
+ <p> $0.82$ </p>
111
+ <p> $5\times 10^{-5}$ </p>
112
+ <p> $\kappa\approx 0.52$ </p>
113
+ <p> $0.00$ </p>
114
+ <p> $0.39$ </p>
115
+ <p> $2.11$ </p>
116
+ <p> $0.32$ </p>
117
+ <p> $15$ </p>
118
+ <p> $0.99$ </p>
119
+ <p> $44.91$ </p>
120
+ <p> $\Delta\mathbf{W}=\mathbf{AB}$ </p>
121
+
122
+ </body>
123
+ </html>