```python
import pandas as pd
import numpy as np
from utils import read_file, tokenize_latex
df = read_file("data/1601/*")

```

    error with: data/1601/1601.06597.json

```python
import re
def clean_exp(exp):
    exp = re.sub("\\\\begin{equation[^}]*}","" ,exp , flags=re.IGNORECASE)
    exp = re.sub("\\\\end{equation[^}]*}","" ,exp , flags=re.IGNORECASE)
    exp = re.sub("\\\\begin{split[^}]*}","\\\\begin{aligned}" ,exp , flags=re.IGNORECASE)
    exp = re.sub("\\\\end{split[^}]*}","\\\\end{aligned}" ,exp , flags=re.IGNORECASE)
    exp = re.sub("\\\\begin{gather[^}]*}","\\\\begin{aligned}" ,exp , flags=re.IGNORECASE)
    exp = re.sub("\\\\end{gather[^}]*}","\\\\end{aligned}" ,exp , flags=re.IGNORECASE)
    exp = re.sub("\\\\begin{align[^}]*}","\\\\begin{aligned}" ,exp , flags=re.IGNORECASE)
    exp = re.sub("\\\\end{align[^}]*}","\\\\end{aligned}" ,exp, flags=re.IGNORECASE)
    exp = re.sub("\\\\label{[^}]*}","" ,exp , flags=re.IGNORECASE)
    exp = re.sub("\\\\n", "",exp, flags=re.IGNORECASE)
    exp = re.sub("\\$", "",exp , flags=re.IGNORECASE)
    return exp
```

```python
df["processed"] = df["text"].apply(lambda x: clean_exp(x)).apply(lambda x: tokenize_latex(x))
```

```python
from gensim import corpora
import logging
logging.basicConfig(format='%(asctime)s : %(levelname)s : %(message)s', level=logging.INFO)
equations = df["processed"].tolist()
dictionary = corpora.Dictionary(equations)

corpus = [dictionary.doc2bow(eq) for eq in equations]
corpora.MmCorpus.serialize('/tmp/equations.mm', corpus)
# our vector space model
print(corpus[1:5])
```

    2017-12-02 00:49:54,164 : INFO : adding document #0 to Dictionary(0 unique tokens: [])
    2017-12-02 00:49:54,691 : INFO : adding document #10000 to Dictionary(15769 unique tokens: [u'\\xi\\leq', u'4E+6F', u'0.000004', u'-P\\cdot\\frac', u'i|S']...)
    2017-12-02 00:49:55,213 : INFO : adding document #20000 to Dictionary(25840 unique tokens: [u'\\xi\\leq', u'eF', u'q-1', u'q-2', u'+i+1']...)
    2017-12-02 00:49:55,714 : INFO : adding document #30000 to Dictionary(35023 unique tokens: [u'\\xi\\leq', u'eF', u'-5AB\\sqrt', u'q-1', u'q-2']...)
    2017-12-02 00:49:56,251 : INFO : adding document #40000 to Dictionary(43511 unique tokens: [u'\\xi\\leq', u'eF', u'-5AB\\sqrt', u'Model~III.~~~~~~~c', u'q-1']...)
    2017-12-02 00:49:56,807 : INFO : adding document #50000 to Dictionary(53030 unique tokens: [u'\\xi\\leq', u'eF', u'-5AB\\sqrt', u'Model~III.~~~~~~~c', u'q-1']...)
    2017-12-02 00:49:57,126 : INFO : built Dictionary(57545 unique tokens: [u'\\xi\\leq', u'eF', u'-5AB\\sqrt', u'Model~III.~~~~~~~c', u'q-1']...) from 56379 documents (total 3317666 corpus positions)
    2017-12-02 00:49:59,982 : INFO : storing corpus in Matrix Market format to /tmp/equations.mm
    2017-12-02 00:49:59,984 : INFO : saving sparse matrix to /tmp/equations.mm
    2017-12-02 00:49:59,989 : INFO : PROGRESS: saving document #0
    2017-12-02 00:50:00,214 : INFO : PROGRESS: saving document #1000
    2017-12-02 00:50:00,492 : INFO : PROGRESS: saving document #2000
    2017-12-02 00:50:00,695 : INFO : PROGRESS: saving document #3000
    2017-12-02 00:50:00,897 : INFO : PROGRESS: saving document #4000
    2017-12-02 00:50:01,096 : INFO : PROGRESS: saving document #5000
    2017-12-02 00:50:01,327 : INFO : PROGRESS: saving document #6000
    2017-12-02 00:50:01,510 : INFO : PROGRESS: saving document #7000
    2017-12-02 00:50:01,744 : INFO : PROGRESS: saving document #8000
    2017-12-02 00:50:01,944 : INFO : PROGRESS: saving document #9000
    2017-12-02 00:50:02,147 : INFO : PROGRESS: saving document #10000
    2017-12-02 00:50:02,364 : INFO : PROGRESS: saving document #11000
    2017-12-02 00:50:02,544 : INFO : PROGRESS: saving document #12000
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    2017-12-02 00:50:03,622 : INFO : PROGRESS: saving document #19000
    2017-12-02 00:50:03,787 : INFO : PROGRESS: saving document #20000
    2017-12-02 00:50:04,009 : INFO : PROGRESS: saving document #21000
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    2017-12-02 00:50:05,071 : INFO : PROGRESS: saving document #27000
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    2017-12-02 00:50:05,350 : INFO : PROGRESS: saving document #29000
    2017-12-02 00:50:05,494 : INFO : PROGRESS: saving document #30000
    2017-12-02 00:50:05,629 : INFO : PROGRESS: saving document #31000
    2017-12-02 00:50:05,765 : INFO : PROGRESS: saving document #32000
    2017-12-02 00:50:05,912 : INFO : PROGRESS: saving document #33000
    2017-12-02 00:50:06,065 : INFO : PROGRESS: saving document #34000
    2017-12-02 00:50:06,231 : INFO : PROGRESS: saving document #35000
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    2017-12-02 00:50:07,132 : INFO : PROGRESS: saving document #40000
    2017-12-02 00:50:07,409 : INFO : PROGRESS: saving document #41000
    2017-12-02 00:50:07,598 : INFO : PROGRESS: saving document #42000
    2017-12-02 00:50:07,769 : INFO : PROGRESS: saving document #43000
    2017-12-02 00:50:07,915 : INFO : PROGRESS: saving document #44000
    2017-12-02 00:50:08,053 : INFO : PROGRESS: saving document #45000
    2017-12-02 00:50:08,210 : INFO : PROGRESS: saving document #46000
    2017-12-02 00:50:08,365 : INFO : PROGRESS: saving document #47000
    2017-12-02 00:50:08,495 : INFO : PROGRESS: saving document #48000
    2017-12-02 00:50:08,629 : INFO : PROGRESS: saving document #49000
    2017-12-02 00:50:08,771 : INFO : PROGRESS: saving document #50000
    2017-12-02 00:50:08,903 : INFO : PROGRESS: saving document #51000
    2017-12-02 00:50:09,041 : INFO : PROGRESS: saving document #52000
    2017-12-02 00:50:09,176 : INFO : PROGRESS: saving document #53000
    2017-12-02 00:50:09,391 : INFO : PROGRESS: saving document #54000
    2017-12-02 00:50:09,622 : INFO : PROGRESS: saving document #55000
    2017-12-02 00:50:09,792 : INFO : PROGRESS: saving document #56000
    2017-12-02 00:50:09,875 : INFO : saved 56379x57545 matrix, density=0.038% (1222940/3244329555)
    2017-12-02 00:50:09,895 : INFO : saving MmCorpus index to /tmp/equations.mm.index

    [[(5, 1), (6, 1), (8, 3), (12, 4), (13, 3), (14, 4), (15, 4), (17, 1), (18, 3), (19, 3), (21, 1), (22, 1), (23, 1)], [(0, 1), (3, 1), (5, 1), (6, 1), (8, 5), (9, 1), (10, 4), (12, 3), (13, 2), (14, 5), (15, 5), (17, 2), (18, 3), (19, 5), (20, 1), (22, 2), (24, 1), (25, 1), (26, 1)], [(2, 1), (4, 1), (5, 1), (6, 1), (12, 2), (14, 6), (15, 6), (17, 1), (19, 1), (27, 1), (28, 1), (29, 1), (30, 1), (31, 1), (32, 1), (33, 1), (34, 1), (35, 1), (36, 1), (37, 1), (38, 1), (39, 1)], [(1, 1), (2, 1), (4, 1), (5, 3), (6, 3), (7, 3), (8, 1), (14, 6), (15, 6), (17, 1), (19, 2), (20, 2), (29, 1), (40, 1), (41, 2), (42, 1), (43, 2)]]

```python
from gensim import corpora, models, similarities, matutils
tfidf = models.TfidfModel(corpus)
corpus_tfidf = tfidf[corpus]
from gensim import similarities
index = similarities.Similarity("index/", corpus_tfidf, len(dictionary.keys()), num_best = 100)
```

    2017-12-02 00:50:09,937 : INFO : collecting document frequencies
    2017-12-02 00:50:09,950 : INFO : PROGRESS: processing document #0
    2017-12-02 00:50:10,032 : INFO : PROGRESS: processing document #10000
    2017-12-02 00:50:10,099 : INFO : PROGRESS: processing document #20000
    2017-12-02 00:50:10,168 : INFO : PROGRESS: processing document #30000
    2017-12-02 00:50:10,247 : INFO : PROGRESS: processing document #40000
    2017-12-02 00:50:10,350 : INFO : PROGRESS: processing document #50000
    2017-12-02 00:50:10,406 : INFO : calculating IDF weights for 56379 documents and 57544 features (1222940 matrix non-zeros)
    2017-12-02 00:50:10,461 : INFO : starting similarity index under index/
    2017-12-02 00:50:17,575 : INFO : PROGRESS: fresh_shard size=10000
    2017-12-02 00:50:24,426 : INFO : PROGRESS: fresh_shard size=20000
    2017-12-02 00:50:28,874 : INFO : PROGRESS: fresh_shard size=30000
    2017-12-02 00:50:30,080 : INFO : creating sparse index
    2017-12-02 00:50:30,082 : INFO : creating sparse matrix from corpus
    2017-12-02 00:50:30,091 : INFO : PROGRESS: at document #0/32768
    2017-12-02 00:50:30,914 : INFO : PROGRESS: at document #10000/32768
    2017-12-02 00:50:31,633 : INFO : PROGRESS: at document #20000/32768
    2017-12-02 00:50:32,377 : INFO : PROGRESS: at document #30000/32768
    2017-12-02 00:50:32,615 : INFO : created <32768x57545 sparse matrix of type '<type 'numpy.float32'>'
    	with 707547 stored elements in Compressed Sparse Row format>
    2017-12-02 00:50:32,626 : INFO : creating sparse shard #0
    2017-12-02 00:50:32,627 : INFO : saving index shard to index/.0
    2017-12-02 00:50:32,629 : INFO : saving SparseMatrixSimilarity object under index/.0, separately None
    2017-12-02 00:50:32,661 : INFO : saved index/.0
    2017-12-02 00:50:32,663 : INFO : loading SparseMatrixSimilarity object from index/.0
    2017-12-02 00:50:32,747 : INFO : loaded index/.0
    2017-12-02 00:50:32,865 : INFO : PROGRESS: fresh_shard size=0
    2017-12-02 00:50:38,388 : INFO : PROGRESS: fresh_shard size=10000
    2017-12-02 00:50:45,378 : INFO : PROGRESS: fresh_shard size=20000

```python
docs = ["".join(eq) for eq in df["processed"].tolist()]
```

```python
docs
```

    [u'\\Phi_{z}(L)=\\sum_{i=1}^{N}\\frac{1}{C_{i}\\timesV_{\\rmmax,i}},',
     u'V_{\\rmmax}={\\rmmin}(V_{\\rmmax,radio},V_{\\rmmax,optical})',
     u'V_{\\rmmax,radio}=\\sum\\limits_{i=1}^{n}F_i\\timesV_{\\rmmax,radio,i}(z_{\\rmmax,i}),',
     u'C_{R}=0.5\\left[1+{\\rmerf}\\left(\\frac{{\\rmSNR}-\\mu}{\\sqrt{2}\\sigma}\\right)\\right]',
     u'\\Phi(L)=\\frac{C}{(L_{*}/L)^{A}+(L_{*}/L)^{B}},',
     u'\\Phi_{z}(L)=(1+z)^{k}\\Phi_{0}(L),',
     u'\\Phi_{z}(L)=\\Phi_{0}\\Bigg(\\frac{L}{(1+z)^{k}}\\Bigg),',
     u'\\log_{10}\\Phi_{z}=m\\log_{10}L+k\\log_{10}(1+z)+c,',
     u'\\log_{10}\\Phi_{z}=m(\\log_{10}(L)-k\\log_{10}(1+z))+c.',
     u'\\Gamma=\\gamma^{-1}(1-\\beta\\cos\\theta)^{-1},',
     u'\\sigma_{\\rmmNWA}=\\int_{M-\\Delta}^{M+\\Delta}dm_{{\\gamma\\gamma}}\\bigg[\\frac{d\\sigma_{\\rmsig}}{dm_{{\\gamma\\gamma}}}\\bigg]_{\\rmpeak},\\quad\\hbox{forabroadresonance}.',
     u'K_{\\rmintf}=\\frac{\\sigma_{\\rmmNWA}}{\\sigma_{\\rmprod}\\cdot{\\rmBr}_{{\\gamma\\gamma}}}',
     u'\\chi^2_{\\rmSM}=7.02,\\,4.93,\\,17.77,\\,1.52,\\,16.65,',
     u'{\\calL}\\,i\\,\\frac{1}{2}M_\\Phi^2\\Phi^2\\,+\\,\\sum_Q(s_Q\\Phi+M_Q)\\overline{Q}\\gamma_5Q\\,+\\,\\sum_L(s_L\\Phi+M_L)\\overline{L}\\gamma_5L,',
     u'M_Q=1\\,{\\rmTeV},\\,N_Q=2,\\,s_Q=0.2.',
     u'M_L=400\\,{\\rmGeV},\\,N_L=6,\\,s_L\\textrm{isvaried}.',
     u'\\Gamma_\\Phi=5\\,{\\rmGeV}.',
     u'\\phi\\simeq\\left\\{\\begin{array}{ll}8.3^\\circ&\\hbox{for}s_L>0;\\\\188.3^\\circ&\\hbox{for}s_L<0,\\end{array}\\right.',
     u'{\\calM}_{E}=\\left(\\begin{array}{cc}M&\\tfrac{1}{\\sqrt{2}}\\,Y_{E}v_{2}\\\\\\tfrac{1}{\\sqrt{2}}\\,Y_E^\\primev_2&M_E\\end{array}\\right)\\,.',
     u'\\phi\\simeq\\left\\{\\begin{array}{rl}90^\\circ&\\hbox{for}y_E>0;\\\\-90^\\circ&\\hbox{for}y_E<0.\\end{array}\\right.',
     u'da=a\\frac{|dm|}{m_1+m_2}\\.',
     u'\\frac{dN}{dVdt}=\\frac{dN_{\\rmgal}}{dV}\\dot{N}_{SF}f_{Z}f_{\\rmmass}f_{\\rmsep},',
     u'\\frac{dN}{dt}\\sim\\frac{0.01}{\\textrm{Mpc}^3}\\times\\frac{2}{\\textrm{yr}}\\times0.1\\times10^{-4}\\times0.1\\sim20\\\\textrm{Gpc}^{-3}\\textrm{yr}^{-1}.',
     u'\\begin{aligned}\\omega_c=&\\begin{cases}0.2+2.7\\times10^{-4}\\,\\left(\\frac{m}{M_\\odot}-50\\right)^2&\\text{for}m<50M_\\odot,\\\\0.2&\\text{for}m\\ge50M_\\odot.\\\\\\end{cases}\\end{aligned}',
     u'a_\\textrm{after}=\\frac{m_\\textrm{after}}{2m_\\textrm{after}-m_\\textrm{before}}a_\\textrm{before}\\,,',
     u'\\tau_\\textrm{GW}=\\frac{5}{256}\\frac{c^5}{G^3}\\frac{a^4}{m_1m_2(m_1+m_2)}\\,.',
     u'\\frac{d^2M_\\textrm{SFR}}{dtdV_c}=0.015\\frac{(1+z)^{2.7}}{[1+(1+z)/2.9]^{5.6}}\\,\\frac{M_\\odot}{\\textrm{Mpc}^{3}\\textrm{yr}}\\,.',
     u'\\textrm{CDF}(Z,z)=\\hat{\\Gamma}\\left(\\alpha+2,(Z/Z_\\odot)^\\beta10^{0.15\\betaz}\\right),',
     u'\\frac{d^4N_{\\rmmerge}}{dV_{\\rmc}\\,dt\\,dm_1\\,dm_2}(t_{\\rmm})=\\int_{P_{\\min}}^{P_{\\max}}dP\\int_0^1dZ\\int_0^{t_{\\rmm}}dt\\,p(t_{\\rmm};m_1,m_2,P,Z,t_{\\rmb})\\,\\,\\frac{d^2M_{\\rmSFR}}{dt\\,dV_{\\rmc}}(t_{\\rmb})\\,\\,\\frac{d^5N_\\textrm{binaries}}{dm_1\\,dm_2\\,dP\\,dZ\\,dM_{\\rmSFR}}(t_{\\rmb})\\,.',
     u'\\frac{dN_{k,i}^{\\rmbirth}}{dt\\,dV_{\\rmc}}=\\textrm{CDF}(Z,z_i)\\,\\,\\frac{d^2M_{\\rmSFR}}{dt\\,dV_{\\rmc}}\\,\\,\\frac{1}{dM_{\\rmSFR}},',
     u'\\frac{dN_{k,j}^{\\rmmerge}}{dt\\,dV_{\\rmc}}=\\sum\\frac{dN_{k,i}^{\\rmbirth}}{dt\\,dV_{\\rmc}}\\,\\delta_{t_i+\\tau_k,t_j}\\,\\frac{dt_i}{dt_j},',
     u'\\begin{aligned}\\omega_c=&\\begin{cases}0.25+3.2\\times10^{-4}\\,\\left(\\frac{m}{M_\\odot}-46\\right)^2&\\text{for}m<46M_\\odot,\\\\0.25&\\text{for}m\\ge46M_\\odot.\\\\\\end{cases}\\end{aligned}',
     u'u\\geq0;\\,\\,u(-\\Deltau-g)=0;\\,\\,-\\Deltau-g\\geq0\\,\\,in\\,\\,\\Omega;',
     u'-\\frac{\\partialu}{\\partialn}=\\alpha(u-b)\\,\\,on\\,\\,\\Gamma_1;\\,\\,-\\frac{\\partialu}{\\partialn}=q\\,\\,on\\,\\,\\Gamma_2;',
     u'a_{\\alpha}(u_{\\alphag},v-u_{\\alphag})\\geq(g,v-u_{\\alphag})_H-(q,v-u_{\\alphag})_Q+\\alpha(b,v-u_{\\alphag})_R,',
     u'V=H^1(\\Omega),\\hspace{.3cm}K_+=\\{v\\inV:\\,v\\geq0\\,\\,in\\,\\Omega\\},',
     u'\\[H=L^2(\\Omega),\\hspace{.2cm}Q=L^2(\\Gamma_2),\\hspace{.2cm}\\text{and}\\hspace{.2cm}\\,R=L^2(\\Gamma_1),\\]',
     u'\\[(u,v)_H=\\int_\\Omegau\\,v\\,\\,dx,\\hspace{.1cm}(u,v)_Q=\\int_{\\Gamma_2}u\\,v\\,\\,ds,\\hspace{.1cm}(u,v)_R=\\int_{\\Gamma_1}u\\,v\\,\\,ds,\\]',
     u'\\[a(u,v)=\\int_\\Omegaablau.ablav\\,\\,dx\\]',
     u'a_{\\alpha}(u,v)=a(u,v)+\\alpha(u,v)_R.',
     u'\\,\\,a_{\\alpha}(v,v)\\geq\\lambda_{\\alpha}\\,\\Arrowvertv\\Arrowvert^2_V\\hspace{.5cm}\\forall\\,\\,v\\inV.',
     u'J_\\alpha(g_{op_\\alpha})=\\min_{g\\inH}J_\\alpha(g)',
     u'J_\\alpha(g)=\\frac{1}{2}\\Arrowvertu_{\\alphag}\\Arrowvert^2_H+\\frac{M}{2}\\Arrowvertg\\Arrowvert^2_H',
     u'\\[V_h=\\{v_h\\inC^0(\\overline{\\Omega}):v_h/_T\\inP_1(T),\\,\\,\\forall\\,T\\,\\in\\tau_h\\},\\]',
     u'\\[K_{+h}=\\{v_h\\inV_{h}:v_h\\geq0\\,\\,in\\,\\Omega\\}\\]',
     u'\\Arrowvertv-\\Pi_h(v)\\Arrowvert_H\\,\\leq\\,c_0\\,h^r\\,\\Arrowvertv\\Arrowvert_r\\,\\,\\,\\forall\\,v\\inH^r(\\Omega),',
     u'\\Arrowvertv-\\Pi_h(v)\\Arrowvert_V\\,\\leq\\,c_0\\,h^{r-1}\\,\\Arrowvertv\\Arrowvert_r\\,\\,\\,\\forall\\,v\\inH^r(\\Omega).',
     u'a_\\alpha(u_{h\\alphag},v_h-u_{h\\alphag})\\geq(g,v_h-u_{h\\alphag})_H-(q,v_h-u_{h\\alphag})_Q+\\alpha(b,v_h-u_{h\\alphag})_R.',
     u'\\Arrowvertu_{h\\alphag_n}\\Arrowvert_V\\leqC;',
     u'lim_{n\\rightarrow\\infty}\\Arrowvertu_{h\\alphag_n}-u_{h\\alphag}\\Arrowvert_V=0.',
     u'\\[\\Arrowvertu_{h\\alphag_2}-u_{h\\alphag_1}\\Arrowvert_V\\leq\\frac{1}{\\lambda_\\alpha}\\Arrowvertg_2-g_1\\Arrowvert_H\\]',
     u'\\Arrowvertu_{h\\alphag}-u_{\\alphag}\\Arrowvert_V\\leqC(\\alpha)h^{(r-1)/2}.',
     u'\\[a_\\alpha(u_{h\\alphag}-u_{\\alphag},u_{h\\alphag}-u_{\\alphag})\\leqa_\\alpha(u_{h\\alphag},w)-(g,w)_H+(q,w)_Q-\\alpha(b,w)_R\\]',
     u'\\[\\Arrowvertu_{h\\alphag}-u_{\\alphag}\\Arrowvert_V^2\\,\\leq\\frac{C}{\\lambda_\\alpha}\\Arrowvert\\Pi_h(u_{\\alphag})-u_{\\alphag}\\Arrowvert_V\\leq\\frac{C}{\\lambda_\\alpha}h^{r-1}\\Arrowvertu_{\\alphag}\\Arrowvert_r\\]',
     u'\\,\\,J_{h\\alpha}(g)=\\frac{1}{2}\\Arrowvertu_{h\\alphag}\\Arrowvert^2_H+\\frac{M}{2}\\Arrowvertg\\Arrowvert^2_H',
     u'J_{h\\alpha}(g_{op_{h\\alpha}})=\\min_{g\\,\\in\\,H}J_{h\\alpha}(g).',
     u'\\[\\lim_{\\Arrowvertg\\Arrowvert_H\\rightarrow\\infty}J_{h\\alpha}(g)=\\infty.\\]',
     u'\\[\\liminf_{n\\rightarrow\\infty}J_{h\\alpha}(g_n)\\geq\\frac{1}{2}\\Arrowvertu_{h\\alphag}\\Arrowvert^2_H+\\frac{M}{2}\\Arrowvertg\\Arrowvert^2_H=J_{h\\alpha}(g).\\]',
     u'|J_{h\\alpha}(g)-J_\\alpha(g)|\\leqC(\\alpha)h^{\\frac{r-1}{2}}',
     u'J_{h\\alpha}(g)-J_{\\alpha}(g)=\\frac{1}{2}\\,(\\rVertu_{h\\alphag}\\lVert^2_H-\\rVertu_{\\alphag}\\lVert^2_H)=\\frac{1}{2}\\rVertu_{h\\alphag}-u_{\\alphag}\\lVert^2_H\\,+\\,(u_{\\alphag},u_{h\\alphag}-u_{\\alphag})_H',
     u'\\lvertJ_{h\\alpha}(g)-J{\\alpha}(g)\\rvert\\leq\\,(\\frac{1}{2}\\rVertu_{h\\alphag}-u_{\\alphag}\\lVert_H\\,+\\,\\rVertu_{\\alphag}\\lVert_H)\\,\\rVertu_{h\\alphag}-u_{\\alphag}\\lVert_H\\leqC(\\alpha)\\,h^{\\frac{r-1}{2}},',
     u'a_\\alpha(u_{h\\alphag},v_h-u_{h\\alphag})\\geq(g,v_h-u_{h\\alphag})_H-(q,v_h-u_{h\\alphag})_Q+\\alpha(b,v_h-u_{h\\alphag})_R',
     u'\\[a_\\alpha(\\eta_\\alpha,v-\\eta_\\alpha)\\geq(g,v-\\eta_\\alpha)_H-(q,v-\\eta_\\alpha)_Q+\\alpha(b,v-\\eta_\\alpha)_R,\\hspace{.4cm}\\forall\\,v\\inK_+\\]',
     u'\\[\\lim_{h\\rightarrow0^+}\\Arrowvertu_{h\\alphag}-u_{\\alphag}\\Arrowvert_V=0.\\]',
     u'g_3(\\mu)=\\mu\\,g_1+(1-\\mu)g_2\\,\\,\\inH,',
     u'u_{h\\alpha3}(\\mu)=\\mu\\,u_{h\\alphag_1}+(1-\\mu)u_{h\\alphag_2}\\,\\,\\,\\inK_{+h}',
     u'0\\lequ_{h\\alpha4}(\\mu)\\lequ_{h\\alpha3}(\\mu)\\hspace{0.1cm}in\\hspace{0.1cm}\\Omega,\\hspace{.5cm}\\forall\\,\\,\\mu\\in[0,1],\\,\\forallh>0,',
     u'\\rVertu_{h\\alpha4}(\\mu)\\lVert_H\\,\\leq\\,\\rVertu_{h\\alpha3}(\\mu)\\lVert_H\\hspace{.5cm}\\forall\\,\\,\\mu\\in[0,1],\\,\\forallh>0.',
     u'\\[\\rVertg_{3\\mu}\\lVert^2_H=\\mu\\rVertg_1\\lVert^2_H+(1-\\mu)\\rVertg_2\\lVert^2_H-\\mu(1-\\mu)\\rVertg_2-g_1\\lVert^2_H\\,\\,\\,\\forallg_1,g_2\\inH,\\forall\\mu\\in[0,1]\\]',
     u'\\[\\rVertu_{h\\alpha3}(\\mu)\\lVert^2_H=\\mu\\rVertu_{h\\alpha1}\\lVert^2_H+(1-\\mu)\\rVertu_{h\\alpha2}\\lVert^2_H-\\mu(1-\\mu)\\rVertu_{h\\alpha2}-u_{h\\alpha1}\\lVert^2_H\\]',
     u'\\muJ_{h\\alpha}(g_1)+(1-\\mu)J_{h\\alpha}(g_2)-J_{h\\alpha}(g_3(\\mu))',
     u'=\\frac{\\mu(1-\\mu)}{2}\\lVertu_{h\\alphag_2}-u_{h\\alphag_1}\\rVert^2_H+\\frac{M}{2}\\,\\mu(1-\\mu)\\,\\lVertg_2-g_1\\rVert^2_H+\\frac{1}{2}\\big[\\lVertu_{h\\alpha3}\\rVert^2_H-\\lVertu_{h\\alpha4}\\rVert^2_H\\big]',
     u'\\geq\\frac{\\mu(1-\\mu)}{2}\\lVertu_{h\\alphag_2}-u_{h\\alphag_1}\\rVert^2_H+\\frac{M}{2}\\,\\mu(1-\\mu)\\,\\lVertg_2-g_1\\rVert^2_H\\geq',
     u'\\frac{M}{2}\\,\\mu(1-\\mu)\\,\\lVertg_2-g_1\\rVert^2_H>0\\hspace{0.2cm}\\forall\\mu\\in(0,1),\\,\\,g_1eqg_2\\inH',
     u'u_{h\\alpha\\,g_{op_{h\\alpha}}}\\rightarrowu_{\\alphag_{op_\\alpha}}\\,\\,\\,in\\,\\,\\,V\\,\\,\\,strong\\,\\,\\,when\\,\\,\\,h\\rightarrow0^+,',
     u'g_{op_{h\\alpha}}\\rightarrowg_{op_\\alpha}\\,\\,\\,in\\,\\,\\,H\\,\\,\\,strong\\,\\,\\,when\\,\\,\\,h\\rightarrow0^+.',
     u'J_{h\\alpha}(g_{op_{h\\alpha}})=\\frac{1}{2}\\lVertu_{h\\alphag_{op_{h\\alpha}}}\\rVert^2_H+\\frac{M}{2}\\lVertg_{op_{h\\alpha}}\\rVert^2_H\\,\\leq\\,\\frac{1}{2}\\lVertu_{h\\alphag}\\rVert^2_H+\\frac{M}{2}\\lVertg\\rVert^2_H.',
     u'J_{h\\alpha}(g_{op_{h\\alpha}})=\\frac{1}{2}\\lVertu_{h\\alphag_{op_{h\\alpha}}}\\rVert^2_H+\\frac{M}{2}\\lVertg_{op_{h\\alpha}}\\rVert^2_H\\,\\leq\\,\\frac{1}{2}\\lVertu_{h\\alpha0}\\rVert^2_H.',
     u'\\lVertu_{h\\alphag_{op_{h\\alpha}}}\\rVert_H\\leqC\\hspace{0.4cm}\\forall\\hspace{0.2cm}h',
     u'\\lVertg_{op_{h\\alpha}}\\rVert_H\\,\\leq\\,\\frac{1}{\\sqrt{M}}\\,\\lVertu_{h\\alpha0}\\rVert_H\\leq\\frac{1}{\\sqrt{M}}\\,C\\hspace{0.4cm}\\forall\\hspace{0.2cm}h.',
     u'\\lVertu_{h\\alphag_{op_{h\\alpha}}}\\rVert_V\\leqC',
     u'a_\\alpha(\\eta_\\alpha,v-\\eta_\\alpha)\\geq(f_\\alpha,v-\\eta_\\alpha)-(q,v-\\eta_\\alpha)_Q+\\alpha(b,v-\\eta_\\alpha)_R,\\hspace{1cm}\\forall\\,\\,v\\inK_+',
     u'\\[a_\\alpha(w_h,w_h)\\leq-a_\\alpha(u_{h\\alphag_{op_{h\\alpha}}},\\Pi_h(u_{\\alphag_{op_\\alpha}})-u_{\\alpha{g_{op\\alpha}}})+(q,\\Pi_h(u_{\\alphag_{op_\\alpha}})-u_{\\alpha{g_{op\\alpha}}})_Q\\]',
     u'\\[-\\alpha(b,\\Pi_h(u_{\\alphag_{op_\\alpha}})-u_{\\alpha{g_{op\\alpha}}})_R+(g_{op_{h\\alpha}},\\Pi_h(u_{\\alphag_{op_\\alpha}})-u_{h\\alphag_{op_{h\\alpha}}})_H\\]',
     u'\\[-(g_{op_\\alpha},w_h)_H.\\]',
     u'\\,\\,\\,\\,u\\geq0;\\,\\,u(-\\Deltau-g)=0;\\,\\,-\\Deltau-g\\geq0\\,\\,in\\,\\,\\Omega;',
     u'u=b\\,\\,\\,on\\,\\,\\Gamma_1;\\,\\,\\,-\\frac{\\partialu}{\\partialn}=q\\,\\,\\,on\\,\\,\\,\\Gamma_2;',
     u'a(u,v-u_g)\\geq(g,v-u_g)_H-(q,v-u_g)_Q,\\,\\,\\,\\forall\\,v\\inK',
     u'K=\\{v\\inV:\\,v\\geq0\\,\\,in\\,\\,\\Omega,\\,v{/\\Gamma_1}=b\\}.',
     u'J(g_{op})=\\min_{g\\inH}J(g)',
     u'J(g)=\\frac{1}{2}\\Arrowvertu_g\\Arrowvert^2_H+\\frac{M}{2}\\Arrowvertg\\Arrowvert^2_H',
     u'a(u_{hg},v_h-u_{hg})\\geq(g,v_h-u_{hg})_H-(q,v_h-u_{hg})_Q\\,\\,\\,\\forallv_h\\inK_h.',
     u'\\[K_h=\\{v_h\\inV_h:v_h\\geq0\\,\\,in\\,\\,\\Omega,v_h/\\Gamma_1=b\\}\\]',
     u'J_h(g_{op_h})=\\min_{g\\inH}J_h(g)=\\frac{1}{2}\\Arrowvertu_{hg}\\Arrowvert^2_H+\\frac{M}{2}\\Arrowvertg\\Arrowvert^2_H,',
     u'\\[lim_{\\alpha\\rightarrow\\infty}\\Arrowvertu_{h\\alphag}-u_{hg}\\Arrowvert_V=0.\\]',
     u'\\[a_\\alpha(w,w)-a_1(w,w)=(\\alpha-1)\\lVertw\\rVert_R^2.\\]',
     u'a_1(w,w)\\leqa_1(w,w)+(\\alpha-1)\\lVertw\\rVert_R^2\\leq(g,w)_H-(q,w)_Q-a(u_{hg},w)',
     u'\\[\\lVertu_{h\\alphag}-u_{hg}\\rVert_R^2\\leq\\frac{C}{\\alpha-1}\\]',
     u'\\[u_{h\\alphag}\\rightharpoonup\\eta\\,\\,\\text{in}\\,V\\,\\,(\\text{inHstrong}).\\]',
     u'u_{h\\alpha\\,g_{op_{h\\alpha}}}\\rightarrowu_{hf_h}\\,\\,\\,in\\,\\,\\,V\\,\\,\\,when\\,\\,\\,\\alpha\\rightarrow\\infty,',
     u'g_{op_{h\\alpha}}\\rightarrowf_h\\,\\,\\,in\\,\\,\\,H\\,\\,\\,\\,when\\,\\,\\,\\alpha\\rightarrow\\infty.',
     u'\\[a_\\alpha(u_{h\\alpha\\,g_{op_{h\\alpha}}},w_h)\\geq(g_{op_{h\\alpha}},w_h)_H-(q,w_h)_Q+\\alpha(b,w_h)_R\\]',
     u'a_1(-w_h,w_h)+a_1(b,w_h)\\geq(g_{op_{h\\alpha}},w_h)_H-(q,w_h)_Q+(\\alpha-1)\\lVertw_h\\rVert_R.',
     u'\\lVertu_{h\\alpha\\,g_{op_{h\\alpha}}}\\rVert_V\\leqC\\,\\,\\,\\forall\\,\\alpha>0.',
     u'\\lVertu_{h\\alpha\\,g_{op_{h\\alpha}}}\\rVert_R\\leq\\frac{C}{\\alpha-1}\\,\\,\\,\\forall\\,\\alpha>0.',
     u'g_{op_{h\\alpha}}\\rightharpoonupf_h\\,\\,\\text{in}\\,\\,H',
     u'u_{h\\alpha\\,g_{op_{h\\alpha}}}\\rightharpoonup\\eta_h\\,\\,\\text{in}\\,\\,V\\,\\,\\text{(inHstrong)}',
     u'\\[a_\\alpha(u_{h\\alpha\\,g_{op_{h\\alpha}}},w_h)\\geq(g_{op_{h\\alpha}},w_h)_H-(q,w_h)_Q+\\alpha(b,w_h)_R\\]',
     u'\\[a(u_{h\\alpha\\,g_{op_{h\\alpha}}},w_h)\\geq(g_{op_{h\\alpha}},w_h)_H-(q,w_h)_Q+\\alpha(b-u_{h\\alpha\\,g_{op_{h\\alpha}}},w_h)_R\\]',
     u'\\[a_\\alpha(u_{h\\alpha\\,g_{op_{h\\alpha}}},w_h)\\geq(g_{op_{h\\alpha}},w_h)_H-(q,w_h)_Q+\\alpha(b,w_h)_R\\]',
     u'\\[a_1(w_h,w_h)\\leqa_1(u_{hf_h},w_h)-(g_{op_{h\\alpha}},w_h)_H+(q,w_h)_Q-\\alpha(b,w_h)_R\\,+\\]',
     u'\\[(\\alpha-1)(u_{h\\alphag_{op_{h\\alpha}}},w_h)_R.\\]',
     u'\\[J_h(f_h)\\leq\\lim_{\\alpha\\rightarrow\\infty}J_{h\\alpha}(g_{op_{h\\alpha}})\\leq\\lim_{\\alpha\\rightarrow\\infty}J_{h\\alpha}(g)\\,\\,\\\\forall\\,g\\inH\\]',
     u'\\[J_h(f_h)\\leqJ_h(g)\\,\\,\\\\forall\\,g\\inH.\\]',
     u'\\[J_h(f_h)\\leq\\lim_{\\alpha\\rightarrow\\infty}J_{h\\alpha}(g_{op_{h\\alpha}})\\leqJ_h(g)\\,\\,\\\\forall\\,g\\inH\\]',
     u'\\[\\lim_{\\alpha\\rightarrow\\infty}J_{h\\alpha}(g_{op_{h\\alpha}})=J_h(f_h)\\]',
     u'\\lim_{\\alpha\\rightarrow\\infty}\\lVertg_{op_{h\\alpha}}\\rVert_H=\\lVertf\\rVert_H.',
     u'u_{h\\alpha\\,g_{op_{h\\alpha}}}\\rightarrowu_{g_{op}}\\,\\,\\,in\\,\\,\\,V\\,\\,\\,when\\,\\,\\,(h,\\alpha)\\rightarrow(0^+,\\infty),',
     u'g_{op_{h\\alpha}}\\rightarrowg_{op}\\,\\,\\,in\\,\\,\\,H\\,\\,\\,\\,\\,when\\,\\,\\,(h,\\alpha)\\rightarrow(0^+,\\infty).',
     u'u_{h\\alphag_{op_{h\\alpha}}}\\rightharpoonupu^*\\hspace{1cm}(strong\\quadin\\quadH)',
     u'g_{op_{h\\alpha}}\\rightharpoonupg^*',
     u'\\[a(u_{h\\alphag_{op_{h\\alpha}}},w_h)\\geq(g_{op_{h\\alpha}},w_h)_H-(q,w_h)_Q.\\]',
     u'\\[a_\\alpha(u_{h\\alphag_{op_{h\\alpha}}}-u_{g^*},u_{h\\alphag_{op_{h\\alpha}}}-u_{g^*})\\leq(g_{op_{h\\alpha}},u_{h\\alphag_{op_{h\\alpha}}}-\\Pi(u_{g^*}))_H\\]',
     u'\\[+(q,u_{h\\alphag_{op_{h\\alpha}}}-\\Pi(u_{g^*}))_Q+\\alpha(b,u_{h\\alphag_{op_{h\\alpha}}}-\\Pi(u_{g^*}))_R-a_\\alpha(u_{g^*},u_{h\\alphag_{op_{h\\alpha}}}-\\Pi(u_{g^*}))\\]',
     u'\\[+a_\\alpha(u_{h\\alphag_{op_{h\\alpha}}},\\Pi(u_{g^*})-u_{g^*})-a_\\alpha(u_{g^*},\\Pi(u_{g^*})-u_{g^*}).\\]',
     u'R^s{\\stackrel{{u_{\\mathbf{a}}}}{\\longrightarrow}}R^\\ell{\\stackrel{{\\pi_{\\mathbf{a}}}}{\\longrightarrow}}M\\rightarrow0',
     u'0\\rightarrowK_{\\mathbf{a}}{\\stackrel{{j_{\\mathbf{a}}}}{\\longrightarrow}}R^\\ell{\\stackrel{{\\pi_{\\mathbf{a}}}}{\\longrightarrow}}M\\rightarrow0.',
     u'R^s/IR^s{\\stackrel{{u_{{\\mathbf{a}},I}}}{\\longrightarrow}}(R/I)^\\ell{\\stackrel{{\\pi_{{\\mathbf{a}},I}}}{\\longrightarrow}}M/IM\\rightarrow0.',
     u'0\\rightarrowK_{{\\mathbf{a}},I}{\\stackrel{{j_{{\\mathbf{a}},I}}}{\\longrightarrow}}(R/I)^\\ell{\\stackrel{{\\pi_{{\\mathbf{a}},I}}}{\\longrightarrow}}M/IM\\rightarrow0.',
     u'\\mu(M)\\cdot\\lambda_R(R/I)=\\lambda_R(K_{{\\mathbf{a}},I})+\\lambda_R(M/IM).',
     u'\\mu(J)\\cdot\\lambda_R(R/I)+\\lambda_R(R/J)=\\lambda_R(K_{{\\mathbf{a}},I})+\\lambda_R(R/IJ).',
     u'\\lambda_R(M/IM)\\leq\\mu(M)\\cdot\\lambda_R(R/I).',
     u'\\lambda_R(R/IJ)\\leq\\mu(J)\\cdot\\lambda_R(R/I)+\\lambda_R(R/J).',
     u'\\[\\lambda_R(R/IJ)\\leq\\min\\{\\mu(J)\\cdot\\lambda_R(R/I)+\\lambda_R(R/J),\\lambda_R(R/I)+\\mu(I)\\cdot\\lambda_R(R/J)\\}.\\]',
     u'\\[\\lambda_R(R/I^n)\\leq\\left(1+\\ell+\\cdots+\\ell^{n-1}\\right)\\cdot\\lambda_R(R/I).\\]',
     u'\\[v_{ij}({\\mathbf{a}}):=-a_je_i+a_ie_j,\\]',
     u'-a_je_i+a_ie_j\\inC_{ij}\\subseteq{\\mathrm{{im}}\\,}\\partial_2\\subseteqIR^\\ell=\\oplus_{h=1}^\\ellIe_h,',
     u'\\[K_{\\mathbf{a}}=\\ker\\partial_1={\\mathrm{{im}}\\,}\\partial_2=\\sum_{i<j}C_{ij}\\subseteqIR^\\ell,\\]',
     u'\\lambda_R(R/IJ)=\\mu(J)\\cdot\\lambda_R(R/I)+\\lambda_R(R/J)',
     u'\\[K_{{\\mathbf{a}},I}=p(j_{{\\mathbf{a}},I}(K_{{\\mathbf{a}},I}))\\subseteqp({\\mathfrak{{m}}}\\cdot(R/I)^\\ell)={\\mathfrak{{m}}}p((R/I)^\\ell)={\\mathfrak{{m}}}\\cdot{\\mathrm{{im}}\\,}p\\subseteq{\\mathfrak{{m}}}K_{{\\mathbf{a}},I}.\\]',
     u'\\[\\lambda_R(R/J^2)=(d+1)\\cdot\\lambda_R(R/J).\\]',
     u'\\[{e_{\\rmHK}}({\\mathfrak{{a}}}):=\\lim_{q\\rightarrow\\infty}\\frac{\\lambda_R(R/{\\mathfrak{{a}}}^{[q]})}{q^d}.\\]',
     u'\\[\\ell\\cdot\\lambda_R(R/I^{[q]})+\\lambda_R(R/J^{[q]})=\\lambda_R(K_{{\\mathbf{a}}^{q},I^{[q]}})+\\lambda_R(R/(IJ)^{[q]}).\\]',
     u'\\ell\\cdot{e_{\\rmHK}}(I)+{e_{\\rmHK}}(J)=\\lim_{q\\rightarrow\\infty}\\frac{\\lambda_R(K_{{\\mathbf{a}}^q,I^{[q]}})}{q^d}+{e_{\\rmHK}}(IJ).',
     u'{e_{\\rmHK}}(IJ)\\leq\\ell^*(J)\\cdot{e_{\\rmHK}}(I)+{e_{\\rmHK}}(J).',
     u'\\[{e_{\\rmHK}}(IJ)\\leq\\min\\{\\ell^*(J)\\cdot{e_{\\rmHK}}(I)+{e_{\\rmHK}}(J),\\{e_{\\rmHK}}(I)+\\ell^*(I)\\cdot{e_{\\rmHK}}(J)\\}.\\]',
     u'\\[{e_{\\rmHK}}(I^n)\\leq(1+\\ell+\\cdots+\\ell^{n-1})\\cdot{e_{\\rmHK}}(I).\\]',
     u'\\[{e_{\\rmHK}}(IJ)=\\ell^*(J)\\cdot{e_{\\rmHK}}(I)+{e_{\\rmHK}}(J)\\]',
     u'\\[\\lim_{q\\rightarrow\\infty}\\frac{\\lambda_R(K_{{\\mathbf{a}}^q,I^{[q]}})}{q^d}=0.\\]',
     u'\\[K_{{\\mathbf{a}}^q,I^{[q]}}=\\frac{K_{{\\mathbf{a}}^q}+I^{[q]}R^\\ell}{I^{[q]}R^\\ell}\\supseteq\\frac{C_{ijq}+I^{[q]}R^\\ell}{I^{[q]}R^\\ell}\\cong\\frac{R}{I^{[q]}R^\\ell:_Rv_{ij}({\\mathbf{a}}^q)}=\\frac{R}{(I^{[q]}:a_i^q)\\cap(I^{[q]}:a_j^q)},\\]',
     u'\\[{e_{\\rmHK}}(I)-{e_{\\rmHK}}(I+(a_i))=\\displaystyle\\lim_{q\\rightarrow\\infty}\\frac{\\lambda_R\\left(\\frac{I^{[q]}+(a_i^q)}{I^{[q]}}\\right)}{q^d}=\\displaystyle\\lim_{q\\rightarrow\\infty}\\frac{\\lambda_R(R/(I^{[q]}:a_i^q))}{q^d}\\leq\\displaystyle\\lim_{q\\rightarrow\\infty}\\frac{\\lambda_R(K_{{\\mathbf{a}}^q,I^{[q]}})}{q^d}=0.\\]',
     u'\\[K_{{\\mathbf{a}}^q,I^{[q]}}=\\frac{K_{{\\mathbf{a}}^q}+I^{[q]}R^d}{I^{[q]}R^d}\\subseteq\\frac{((J^{[q]})^*+I^{[q]})R^d}{I^{[q]}R^d}\\subseteq\\frac{((I+J)^{[q]})^*R^d}{I^{[q]}R^d}.\\]',
     u'\\begin{aligned}\\lim_{q\\rightarrow\\infty}\\frac{\\lambda_R(K_{{\\mathbf{a}}^q,I^{[q]}})}{q^d}&\\leq\\lim_{q\\rightarrow\\infty}\\frac{\\lambda_R\\left({\\dfrac{((I+J)^{[q]})^*R^d}{I^{[q]}R^d}}\\right)}{q^d}\\\\&=d\\cdot\\left(\\lim_{q\\rightarrow\\infty}\\frac{\\lambda_R(R/I^{[q]})}{q^d}-\\lim_{q\\rightarrow\\infty}\\frac{\\lambda_R(R/((I+J)^{[q]})^*)}{q^d}\\right).\\end{aligned}',
     u'\\[\\lim_{q\\rightarrow\\infty}\\frac{\\lambda_R(K_{{\\mathbf{a}}^q,I^{[q]}})}{q^d}\\leqd\\cdot\\left({e_{\\rmHK}}(I)-{e_{\\rmHK}}(I+J)\\right).\\]',
     u'{e_{\\rmHK}}(IJ)\\geqd\\cdot{e_{\\rmHK}}(I+J)+{e_{\\rmHK}}(J).',
     u'\\[{e_{\\rmHK}}(J^2)=(d+1)\\cdot{e_{\\rmHK}}(J)=(d+1)\\cdote(J).\\]',
     u'\\[{e_{\\rmHK}}(IJ^{[q]})=\\ell^*(J)\\cdot{e_{\\rmHK}}(I)+{e_{\\rmHK}}(J^{[q]}).\\]',
     u'\\[{e_{\\rmHK}}(IJ^{[q]})=d\\cdot{e_{\\rmHK}}(I)+{e_{\\rmHK}}(J^{[q]}).\\]',
     u'\\[{e_{\\rmHK}}(I)\\leq{e_{\\rmHK}}(R)\\cdot\\lambda_R(R/I).\\]',
     u'\\[\\lambda_R(M/IM)=\\lambda_R(R^{1/q}/IR^{1/q})=[k^{1/q}:k]\\cdot{\\operatorname{\\lambda}}_{R^{1/q}}(R^{1/q}/IR^{1/q})=[k:k^{1/q}]\\cdot\\lambda_R(R/I^{[q]}),\\]',
     u'\\begin{aligned}\\mu(M)&=\\lambda_R(M/{\\mathfrak{{m}}}M)=\\lambda_R(R^{1/q}/{\\mathfrak{{m}}}R^{1/q})\\\\&=[k^{1/q}:k]\\cdot{\\operatorname{\\lambda}}_{R^{1/q}}(R^{1/q}/{\\mathfrak{{m}}}R^{1/q})=[k^{1/q}:k]\\cdot\\lambda_R(R/{\\mathfrak{{m}}}^{[q]}).\\end{aligned}',
     u'\\[\\lambda_R(R/I^{[q]})\\leq\\lambda_R(R/{\\mathfrak{{m}}}^{[q]})\\cdot\\lambda_R(R/I).\\]',
     u'\\Delta(c)=:c_{(1)}\\otimesc_{(2)},\\quad\\delta_M(m)=:m_{(0)}\\otimesm_{(1)}\\,.',
     u"f_*\\colonK_0({\\mathcal{B}})\\longrightarrowK_0({\\mathcal{B}}')\\quad\\text{satisfies}\\quadf_*([{\\mathcal{A}}\\Box^{\\mathcal{H}}V])=[{\\mathcal{A}}'\\Box^{\\mathcal{H}}V].",
     u'{\\mathcal{B}}:=\\{b\\in\\mathcal{A}~|~\\forall\\,a\\in\\mathcal{A}\\colon\\delta_{\\mathcal{A}}(ba)=b\\delta_{\\mathcal{A}}(a)\\}',
     u'\\psi:\\mathcal{C}\\otimes\\mathcal{A}{\\longrightarrow}\\mathcal{A}\\otimes\\mathcal{C},\\quadc\\otimesa\\longmapstocan(can^{-1}(1\\otimesc)a).',
     u'{\\mathcal{B}}=\\{b\\in\\mathcal{A}~|~\\delta_{\\mathcal{A}}(b)=b\\otimese\\}=\\{b\\in\\mathcal{A}~|~\\psi(e\\otimesb)=b\\otimese\\}.',
     u"{\\mathcal{B}}'{\\otimes}_{{\\mathcal{B}}}(\\mathcal{A}\\Box^{\\mathcal{C}}V)\\cong\\mathcal{A'}\\Box^{\\mathcal{C}}V",
     u"\\widetilde{f}:=m_{\\mathcal{A'}}\\circ({\\mathrm{id}}_{\\mathcal{B'}}\\otimes_{{\\mathcal{B}}}f)\\colon{\\mathcal{B}}'\\otimes_{{\\mathcal{B}}}\\mathcal{A}\\longrightarrow\\mathcal{A'},\\quadm_{\\mathcal{A'}}(b'\\otimesa')=b'a',",
     u"{\\mathrm{id}}_{\\mathcal{A'}}\\otimes_{{\\mathcal{B}}'}\\widetilde{f}\\colon\\mathcal{A'}\\otimes_{{\\mathcal{B}}'}{\\mathcal{B}}'\\otimes_{{\\mathcal{B}}}\\mathcal{A}\\longrightarrow\\mathcal{A'}\\otimes_{{\\mathcal{B}}'}\\mathcal{A'}",
     u"{\\mathrm{id}}_{\\mathcal{A'}}\\otimes_{A}(f\\otimes_{{\\mathcal{B}}'}f)\\colon\\mathcal{A'}\\otimes_{\\mathcal{A}}\\mathcal{A}\\otimes_{{\\mathcal{B}}}\\mathcal{A}\\longrightarrow\\mathcal{A'}\\otimes_{{\\mathcal{B}}'}\\mathcal{A'}",
     u"\\xymatrixcolsep{5pc}\\xymatrix{\\mathcal{A'}\\otimes_{\\mathcal{A}}\\mathcal{A}\\otimes_{{\\mathcal{B}}}\\mathcal{A}\\quad\\ar[d]_-{{\\mathrm{id}}_{\\mathcal{A'}}\\otimes_{\\mathcal{A}}\\hspace{0.2em}can}\\ar[r]^-{{\\mathrm{id}}_{\\mathcal{A'}}\\otimes_{\\mathcal{A}}(f\\otimes_{{\\mathcal{B}}'}f)}&\\quad\\mathcal{A'}\\otimes_{{\\mathcal{B}}'}\\mathcal{A'}\\ar[d]^{can'}\\\\\\mathcal{A'}\\otimes_{\\mathcal{A}}\\mathcal{A}\\otimes\\mathcal{C}\\ar[r]^-\\cong&\\mathcal{A'}\\otimes\\mathcal{C}}",
     u"\\widetilde{f}\\hspace{0.2em}\\Box^{\\mathcal{C}}{\\mathrm{id}}_{V}:({\\mathcal{B}}'\\otimes_{{\\mathcal{B}}}\\mathcal{A})\\Box^{\\mathcal{C}}V\\longrightarrow\\mathcal{A'}\\Box^{\\mathcal{C}}V",
     u"{\\mathcal{B}}'{\\otimes}_{{\\mathcal{B}}}(\\mathcal{A}\\Box^{\\mathcal{C}}V)\\cong({\\mathcal{B}}'{\\otimes}_{{\\mathcal{B}}}\\mathcal{A})\\Box^{\\mathcal{C}}V\\cong\\mathcal{A'}\\Box^{\\mathcal{C}}V,",
     u'_\\mathcal{A}\\delta:\\mathcal{A}\\longrightarrow\\mathcal{C}\\otimes\\mathcal{A},\\quad_\\mathcal{A}\\delta(a)=\\psi^{-1}(a\\otimese).',
     u"(f\\otimes{\\mathrm{id}}_{{\\mathcal{C}}})\\circcan=can'\\circ(f\\otimes_{{\\mathcal{B}}}f).",
     u"(can')^{-1}\\circ(f\\otimes{\\mathrm{id}}_{{\\mathcal{C}}})=(f\\otimes_{{\\mathcal{B}}}f)\\circcan^{-1}.",
     u"\\begin{aligned}\\big((f\\otimes{\\mathrm{id}}_{{\\mathcal{C}}})\\circ\\psi\\big)(c\\otimesa)&=(f\\otimes{\\mathrm{id}}_{{\\mathcal{C}}})\\Big(can\\big((can^{-1}(1\\otimesc)a)\\big)\\Big)onumber\\\\&=\\big(can'\\circ(f\\otimes_{{\\mathcal{B}}}f)\\big)\\big(can^{-1}(1\\otimesc)a\\big)onumber\\\\&=can'\\big((f\\otimes_{{\\mathcal{B}}}f)\\big(can^{-1}(1\\otimesc)a\\big)\\big)onumber\\\\&=can'\\big((f\\otimes_{{\\mathcal{B}}}f)\\big(can^{-1}(1\\otimesc)\\big)f(a)\\big)onumber\\\\&=can'\\Big(\\big((can')^{-1}(f(1)\\otimesc)\\big)f(a)\\Big)onumber\\\\&=can'\\Big(\\big((can')^{-1}(1\\otimesc)\\big)f(a)\\Big)onumber\\\\&=\\psi'(c\\otimesf(a))onumber\\\\&=\\big(\\psi'\\circ({\\mathrm{id}}_{{\\mathcal{C}}}\\otimesf)\\big)(c\\otimesa).\\end{aligned}",
     u"(f\\otimes{\\mathrm{id}}_{{\\mathcal{C}}})\\circ\\psi=\\psi'\\circ({\\mathrm{id}}_{{\\mathcal{C}}}\\otimesf).",
     u"(\\psi')^{-1}\\circ(f\\otimes{\\mathrm{id}}_{{\\mathcal{C}}})=({\\mathrm{id}}_{{\\mathcal{C}}}\\otimesf)\\circ\\psi^{-1}.",
     u"\\big((\\psi')^{-1}\\circ(f\\otimes{\\mathrm{id}}_{{\\mathcal{C}}})\\big)(a\\otimese)=\\big(({\\mathrm{id}}_{{\\mathcal{C}}}\\otimesf)\\circ\\psi^{-1}\\big)(a\\otimese),",
     u"\\big(_{\\mathcal{A}'}\\delta\\circf\\big)(a)=\\big(({\\mathrm{id}}_{{\\mathcal{C}}}\\otimesf)\\circ_{\\mathcal{A}}\\!\\delta\\big)(a).",
     u"{\\mathcal{B}}'{\\otimes}_{{\\mathcal{B}}}(\\mathcal{A}\\Box^{\\mathcal{C}}\\mathcal{A}){\\otimes}_{{\\mathcal{B}}}{\\mathcal{B}}'\\cong\\mathcal{A'}\\Box^{\\mathcal{C}}\\mathcal{A'}.",
     u"{\\mathcal{B}}'{\\otimes}_{{\\mathcal{B}}}(\\mathcal{A}\\Box^{\\mathcal{C}}\\mathcal{A}){\\otimes}_{{\\mathcal{B}}}{\\mathcal{B}}'\\cong({\\mathcal{B}}'{\\otimes}_{{\\mathcal{B}}}(\\mathcal{A}\\Box^{\\mathcal{C}}\\mathcal{A})){\\otimes}_{{\\mathcal{B}}}{\\mathcal{B}}'\\cong(\\mathcal{A'}\\Box^{\\mathcal{C}}\\mathcal{A}){\\otimes}_{{\\mathcal{B}}}{\\mathcal{B}}'\\cong\\mathcal{A'}\\Box^{\\mathcal{C}}\\mathcal{A'}.\\qedhere",
     u"{\\mathcal{B}}'{\\otimes}_{{\\mathcal{B}}}(\\mathcal{A}\\Box^{\\mathcal{C}}V)\\cong\\mathcal{A'}\\Box^{\\mathcal{C}}V",
     u"f_*([{\\mathcal{A}}\\Box^{\\mathcal{C}}V])=[{\\mathcal{A}}'\\Box^{\\mathcal{C}}V].",
     u'\\ell:\\mathcal{C}\\longrightarrow\\mathcal{A}\\otimes\\mathcal{A},\\quad\\sum_{\\mu}a_{\\mu}\\otimesr_{\\mu}(c):=\\ell(c)=:\\ell\\left(c\\right)^{\\langle1\\rangle}\\otimes\\ell\\left(c\\right)^{\\langle2\\rangle}',
     u'\\begin{aligned}\\sigma(a):=a_{(0)}\\ell\\left(a_{(1)}\\right)^{\\langle1\\rangle}\\varphi\\left(\\ell\\left(a_{(1)}\\right)^{\\langle2\\rangle}\\right),\\end{aligned}',
     u'e_{(\\mu,i)(u,j)}:=\\sigma(r_{\\mu}(c_{ij})a_{u})\\in{\\mathcal{B}}',
     u"\\ell':=(f\\otimesf)\\circ\\ell:\\mathcal{C}\\longrightarrow\\mathcal{A}'\\otimes\\mathcal{A}'",
     u"\\{a'_{\\mu}=f(a_{\\mu})\\,\\mid\\,\\mu\\inI\\}\\text{isabasisof}f(\\mathcal{A})\\quad\\text{and}\\quad\\{a_{\\mu}\\,\\mid\\,\\muot\\inI\\}\\text{isabasisof}\\kerf.",
     u"\\sum_{\\mu}a'_{\\mu}\\otimesr'_{\\mu}(c):=\\ell'(c)=\\sum_{\\mu}f(a_{\\mu})\\otimesf(r_{\\mu}(c))=\\sum_{\\mu}a'_{\\mu}\\otimesf(r_{\\mu}(c)).",
     u"\\xymatrix@C=1.5pc@R=0.5pc{&\\mathbb{C}&\\\\&&\\\\\\mathcal{A}\\ar[ruu]^-{\\varphi}\\ar[rr]^-{f}\\ar@/_2em/[dd]_-{\\sigma}&&\\mathcal{A}'\\ar[luu]_-{\\varphi'}\\ar@/^2em/[dd]^-{\\sigma'}\\\\\\cup\\hspace{-0.31em}\\shortmid&&\\cup\\hspace{-0.31em}\\shortmid\\\\B\\ar[rr]^-{f\\mid_{B}}&&B'}",
     u"\\begin{aligned}\\sigma'(a')&=a'_{(0)}\\ell'(a'_{(1)})^{\\langle1\\rangle}\\varphi'\\big(\\ell'(a'_{(1)})^{\\langle2\\rangle}\\big)onumber\\\\&=a'_{(0)}f\\big(\\ell(a'_{(1)})^{\\langle1\\rangle}\\big)\\varphi'\\big(f(\\ell(a'_{(1)})^{\\langle2\\rangle})\\big)onumber\\\\&=a'_{(0)}f\\big(\\ell(a'_{(1)})^{\\langle1\\rangle}\\big)\\varphi\\big(\\ell(a'_{(1)})^{\\langle2\\rangle}\\big).\\end{aligned}",
     u"\\begin{aligned}\\sigma'(f(a))&=f(a)_{(0)}f\\left(\\ell\\left(f(a)_{(1)}\\right)^{\\langle1\\rangle}\\right)\\varphi\\left(\\ell(f(a)_{(1)})^{\\langle2\\rangle}\\right)onumber\\\\&=f\\left(a_{(0)}\\right)f\\left(\\ell\\left(a_{(1)}\\right)^{\\langle1\\rangle}\\right)\\varphi\\left(\\ell(a_{(1)})^{\\langle2\\rangle}\\right)onumber\\\\&=f\\left(a_{(0)}\\ell\\left(a_{(1)}\\right)^{\\langle1\\rangle}\\varphi\\left(\\ell(a_{(1)})^{\\langle2\\rangle}\\right)\\right)onumber\\\\&=f(\\sigma(a)).\\end{aligned}",
     u"\\begin{aligned}f\\left(e_{(\\mu,i)(u,j)}\\right)&=f\\left(\\sigma\\left(r_{\\mu}(c_{ij})a_{u}\\right)\\right)=\\sigma'\\left(f\\left(r_{\\mu}(c_{ij})a_{u}\\right)\\right)onumber\\\\&=\\sigma'\\left(f\\left(r_{\\mu}(c_{ij})\\right)f\\left(a_{u}\\right)\\right)=\\sigma'\\left(r'_{\\mu}(c_{ij})a'_{u}\\right).\\end{aligned}",
     u"\\sigma'\\left(r'_{\\mu}(c_{ij})a'_{u}\\right)=:e'_{(\\mu,i)(u,j)}\\,,",
     u"f(e)=\\left(\\begin{array}{cc}e'&0\\\\d&0\\end{array}\\right),",
     u"\\left(\\begin{array}{cc}e'&0\\\\d&0\\end{array}\\right)=\\left(\\begin{array}{cc}1&0\\\\d&1\\end{array}\\right)\\left(\\begin{array}{cc}e'&0\\\\0&0\\end{array}\\right)\\left(\\begin{array}{cc}1&0\\\\d&1\\end{array}\\right)^{-1}.",
     u"{\\mathcal{B}}'\\otimes_{\\mathcal{B}}(A\\Box^{\\mathcal{C}}V)\\cong{\\mathcal{B}}'\\otimes_{\\mathcal{B}}(B^Ne)\\cong{\\mathcal{B}}'^Nf(e)\\cong{\\mathcal{B}}'^{N'}e'\\cong{\\mathcal{A}}'\\Box^{\\mathcal{C}}V.",
     u'\\{a\\delta_{A}(\\tilde{a})\\;|\\;a,\\tilde{a}\\inA\\}^{\\mathrm{cls}}=A\\underset{\\mathrm{min}}{\\otimes}H.',
     u'{\\mathcal{P}}_H(A):=\\left\\{\\,a\\inA\\,|\\,\\delta_{A}(a)\\inA\\otimes{\\mathcal{O}}(H)\\,\\right\\}.',
     u'B:=\\{b\\inA\\;|\\;\\delta_{A}(b)=b\\otimes1\\}.',
     u"B'\\otimes_{B}\\big(\\mathcal{P}_H(A)\\Box^{\\mathcal{O}(H)}V\\big)\\cong\\mathcal{P}_H(A')\\Box^{\\mathcal{O}(H)}V.",
     u"f_*([\\mathcal{P}_H(A)\\Box^{\\mathcal{O}(H)}V])=[\\mathcal{P}_H(A')\\Box^{\\mathcal{O}(H)}V].",
     u'A\\circledast^{\\delta_A}H:=\\left\\{f\\inC([0,1],A)\\underset{\\mathrm{min}}{\\otimes}\\!H=C([0,1],A\\underset{\\mathrm{min}}{\\otimes}\\!H)\\;\\Big{|}\\;\tf(0)\\in\\mathbb{C}\\otimesH,\\;f(1)\\in\\delta_A(A)\\right\\}.',
     u'\\mathrm{id}\\!\\otimes\\!\\Delta\\colon\\;C([0,1],A)\\!\\underset{\\mathrm{min}}{\\otimes}\\!H\\;\\longrightarrow\\;C([0,1],A)\\!\\underset{\\mathrm{min}}{\\otimes}\\!H\\!\\underset{\\mathrm{min}}{\\otimes}\\!H',
     u'\\delta_\\Delta\\colonA\\circledast^{\\delta_A}H\\longrightarrow(A\\circledast^{\\delta_A}H)\\!\\underset{\\mathrm{min}}{\\otimes}\\!H',
     u"{\\mathrm{id}}\\otimesF\\otimes{\\mathrm{id}}:C([0,1])\\underset{\\mathrm{min}}{\\otimes}A\\underset{\\mathrm{min}}{\\otimes}H\\longrightarrowC([0,1])\\underset{\\mathrm{min}}{\\otimes}A'\\underset{\\mathrm{min}}{\\otimes}H",
     u"\\mathcal{P}_H(A'\\!\\circledast^{\\delta_{A'}}\\!H)\\Box^{{\\mathcal{O}}(H)}V\\congB'\\otimes_{B}\\big(\\mathcal{P}_H(A\\!\\circledast^{\\delta_{A}}\\!H)\\Box^{{\\mathcal{O}}(H)}V\\big)",
     u'\\mathrm{ev}_{\\frac{1}{2}}\\otimes\\chi\\otimes{\\mathrm{id}}:H\\!\\circledast^{\\Delta}\\!H\\longrightarrowH',
     u's(t^*)=\\frac{|\\{T_{a}|t^*\\subseteqT_{a},T_{a}\\inS\\}|}{m}',
     u'c(t^*\\rightarrowy)=\\frac{s(t^*\\capy)}{s(t^*)}.',
     u'\\begin{aligned}&c(t^*\\rightarrowy)\\geqc_{min}onumber\\\\&s(t^*)\\geqs_{min}onumber\\\\&t^*\\cap\\mathbf{I}eq\\emptysetonumber\\\\&t^*\\cap\\mathbf{C}eq\\emptyset.\\end{aligned}',
     u'w_{name}={\\operatorname{arg\\,max}}_{w\\in\\mathbf{V}}\\sum_{k\\inP}\\sum_{w_{kj}\\inW_k}\\mathbf{I}_p(w_{kj})*tfidf(w_{kj})',
     u'\\alpha(g_1g_2,x)=\\alpha(g_1,g_2x)\\alpha(g_2,x)\\quad\\forallg_1,g_2\\in\\Gamma,x\\inX.',
     u'\\int|\\alpha(g,x)|_G~d\\mu(x)<\\infty',
     u'\\#\\{g\\in\\Gamma:~\\alpha(g,x)=h\\}\\leC',
     u'\\alpha:\\Gamma\\timesX\\to\\Lambda,\\quad\\alpha(g,x)x=gx,',
     u'\\beta:\\Lambda\\timesX\\to\\Gamma,\\quad\\beta(h,x)x=hx',
     u'{\\operatorname{End}}(X):={\\mathop{\\varprojlim}\\limits}_K\\pi_0(X\\setminusK).',
     u'C\\cup\\{\\xi\\in{\\operatorname{End}}(X):~\\pi_K(\\xi)\\inC\\}\\subset{\\overlineX}',
     u'\\#\\{g\\in\\Gamma:~\\alpha(g,x)=h\\}\\leC',
     u"\\alpha'(g,x)=\\alpha(g^{-1},x)^{-1}.",
     u'\\[{\\operatorname{End}}(X)\\to\\pi_0(X\\setminusD)\\to\\pi_0(Y\\setminusK),\\]',
     u'\\gamma[h_1,h_2](n)^{-1}\\gamma[h_1,h_2](n+1)\\inS_G',
     u"F_{s,k}(x):=\\{g\\in\\Gamma:~k\\in\\gamma[\\alpha'_x(g),\\alpha'_x(gs)]\\}.",
     u'\\int\\#F_{s,k}(x)~d\\mu(x)\\leC\\int|\\alpha(s^{-1},x)^{-1}|_G~d\\mu(x)<\\infty',
     u'L_{s,k}=\\{(g,x)\\in\\Gamma\\timesX:~g^{-1}\\inF_{s,k}(x)\\}.',
     u'c_\\Gamma\\times\\mu(L_{s,k})=c_\\Gamma\\times\\mu(R_{s,k})',
     u"\\alpha'_{gx}(g)^{-1}k\\in\\gamma[e,\\alpha'_{gx}(g)^{-1}\\alpha'_{gx}(gs)].",
     u'F_K(x):=\\bigcup\\{F_{s,k}:~s\\inS_\\Gamma,k\\inK\\}.',
     u"K\\cap\\gamma[\\alpha'_x(g_1),\\alpha'_x(g_1s)]=\\emptyset.",
     u'\\phi(hx)=\\alpha(h,x)\\phi(x).',
     u'd(e,r_n(x))=n=(r_n(x)|\\phi(x))_e.',
     u'\\lim_{n\\to\\infty}\\int|F(n,x)|~d\\mu(x)=+\\infty,\\quad\\int|\\partialF(n,x)|~d\\mu(x)\\leM,',
     u'\\[f_j(\\mathbf{x})=\\mathbf{c}_j^\\textsf{T}\\cdot\\mathbf{x}\\]',
     u'\\smalll^*=\\arg\\max_jf_j(\\mathbf{x})',
     u'\\small\\begin{aligned}f_j({x})=&\\sum_{i=1}^{N}\\beta_j^ik({x},{x}_i)+b=\\sum_{i=1}^{N}\\beta_j^i\\varphi({x_i})^\\textsf{T}\\varphi({x})+b\\\\f_j({x})=&{\\boldsymbol{\\beta}_j}^\\textsf{T}\\cdot\\textbf{k}({x})=\\mathbf{c}_j^\\textsf{T}\\cdot[\\varphi(x);1],\\mathbf{c}_j=[\\sum_{i=1}^{N}\\beta_j^i\\varphi({x_i});b]\\\\\\end{aligned}',
     u'\\[f_{{t}_*}({x})=\\boldsymbol{\\beta}({t}_*)^\\textsf{T}\\cdot\\textbf{k}({x}),\\]',
     u'\\centering\\begin{aligned}{\\Phi(t_*)}=\\tilde{c}(\\mathbf{t}_*)=&\\underset{\\mathbf{c}}{\\operatorname{argmin}}[K_C(\\mathbf{c},\\mathbf{c})-2k_c(\\mathbf{c})^\\textsf{T}\\mathbf{u}-\\eta\\log(\\\\&K_C(\\mathbf{c},\\mathbf{c}-k_c(\\mathbf{c})^\\textsf{T}(\\mathbf{K}_C+\\lambda_c\\mathbf{I})^{-1}k_c(c))]\\end{aligned}',
     u'p_{reg}(\\mathbf{c}|\\mathbf{t}_*)=\\mathcal{N}(\\mu_c=\\tilde{c}(\\mathbf{t}_*),\\Sigma_c=\\mathbf{I})',
     u'\\begin{aligned}{\\Phi(t_*)}=\\small\\hat{c}(\\mathbf{t}_*)=\\underset{\\mathbf{c},\\zeta_i}{\\operatorname{argmin}}[\\mathbf{c}^\\textsf{T}\\mathbf{c}+\\alpha\\,\\psi(\\mathbf{c},\\tilde{c}(\\mathbf{t}_*))+C\\sum_{i=1}^N{\\zeta_i}]\\\\\\;s.t.:-\\mathbf{c}^\\textsf{T}{\\mathbf{x}}_{i}\\geq\\zeta_i,\\;\\zeta_i\\geq0,i=1,\\cdots,N\\end{aligned}',
     u'\\min_{\\mathbf{W}}r(\\mathbf{W})+\\lambda\\sum_ic_i(\\mathbf{T}\\mathbf{W}\\mathbf{X}^\\textsf{T})',
     u'\\underset{\\mathbf{L}}{\\operatorname{min}}[r(\\mathbf{L})+\\lambda\\sum_pc_p(\\mathbf{K}_{T}^{\\frac{1}{2}}\\mathbf{L}\\mathbf{K}_{X}^\\frac{1}{2})],',
     u'\\[\\small\\min_{\\mathbf{W}}r(\\mathbf{W})+\\lambda_1\\sum_ic_i(\\mathbf{T}\\mathbf{W}\\mathbf{X}^\\textsf{T})+\\lambda_2\\sum_k{(\\mathbf{c}_j-\\mathbf{t}^\\textsf{T}_j\\mathbf{W})^T(\\mathbf{c}_j-\\mathbf{t}^\\textsf{T}_j\\mathbf{W})}\\]',
     u'\\begin{aligned}{\\Phi(t_*)}=\\hat{c}(\\mathbf{t}_*)=&\\underset{\\mathbf{c},\\zeta_i}{\\operatorname{argmin}}\\big[\\mathbf{c}^\\textsf{T}\\mathbf{c}-\\alpha{\\mathbf{t}_*}^\\textsf{T}\\mathbf{W}\\mathbf{c}-\\gamma\\ln(p_{reg}(\\mathbf{c}|\\mathbf{t}_*))\\\\&+C\\sum{\\zeta_i}\\big]\\\\&s.t.:-(\\mathbf{c}^\\textsf{T}{\\mathbf{x}}_{i})\\geq\\zeta_i,\\,\\,\\zeta_i\\geq0,\\;\\;i=1\\cdotsN\\\\&\t\\,\\,{\\mathbf{t}_*}^\\textsf{T}\\mathbf{W}\\mathbf{c}\\geql\\\\&\\alpha,\\gamma,C,l\\,\\text{:hyperparameters}\\\\\\end{aligned}',
     u'\\begin{aligned}-\\lnp(\\mathbf{c}|\\mathbf{t}_*)\\propto(\\mathbf{c}-\\tilde{c}(\\mathbf{t}_*))^\\textsf{T}(\\mathbf{c}-\\tilde{c}(\\mathbf{t}_*))\\\\=\\mathbf{c}^\\textsf{T}\\mathbf{c}-2\\mathbf{c}^\\textsf{T}\\tilde{c}(\\mathbf{t}_*)+\\tilde{c}(\\mathbf{t}_*)^\\textsf{T}\\tilde{c}(\\mathbf{t}_*)\\end{aligned}',
     u'\\begin{aligned}{\\mathbf{\\Psi}^*}=\\arg\\min_{{\\mathbf{\\Psi}}}L({\\mathbf{\\Psi}})=[&\\frac{1}{2}r({\\mathbf{\\Psi}})+\\lambda_1\\sum_kc_k(\\mathbf{G}\\,{\\mathbf{\\Psi}}\\,\\mathbf{K})+\\\\&\\lambda_2\\sum_{i=1}^{N_{sc}}{\\|\\boldsymbol{\\beta}_i-{\\mathbf{\\Psi}}^\\textsf{T}\\,{\\textbf{g}(t_i)}\\|^2}\\end{aligned}',
     u'\\frac{\\deltaL({\\mathbf{\\Psi}})}{\\delta\\,{\\mathbf{\\Psi}}}={\\mathbf{\\Psi}}+\\lambda_1\\cdot\\sum_{i,j}{\\mathbf{g}(t_i)}{\\mathbf{k}(x_j)}^\\mathsf{T}v_{ij}+r\\cdot\\lambda_2\\cdot(\\mathbf{G}^2\\,\\,{\\mathbf{\\Psi}}-\\mathbf{G}\\textbf{B})',
     u'\\centering\\begin{aligned}\\Phi(t_*)=\\tilde{\\boldsymbol{\\beta}}_{DT}(t_*)={\\mathbf{\\Psi}^*}^\\textsf{T}\\,\\mathbf{g}(t_*)\\end{aligned}',
     u"\\small\\begin{aligned}{\\boldsymbol{\\beta}}^*_{+}=&\\underset{\\boldsymbol{\\beta}}{\\operatorname{argmin}}\\big[\\boldsymbol{\\beta}^\\textsf{T}\\mathbf{K^{'}}\\boldsymbol{\\beta}-\\boldsymbol{\\beta}^T\\mathbf{a}\\big]\\\\&s.t.:\\boldsymbol{\\beta}^T\\mathbf{1}=1,0\\le\\boldsymbol{\\beta}_i\\leC;i=1\\cdotsN\\\\\\end{aligned}",
     u"\\small\\begin{aligned}{\\boldsymbol{\\beta}}^*_{-}=&\\underset{\\boldsymbol{\\beta}}{\\operatorname{argmin}}\\big[\\boldsymbol{\\beta}^\\textsf{T}\\mathbf{K^{'}}\\boldsymbol{\\beta}+\\boldsymbol{\\beta}^T\\mathbf{a}\\big]\\\\&s.t.:\\boldsymbol{\\beta}^T\\mathbf{1}=-1,-C\\le\\boldsymbol{\\beta}_i\\le0;i=1\\cdotsN\\\\\\end{aligned}",
     u"\\small\\begin{aligned}\\Phi(t_*)=\\hat{\\boldsymbol{\\beta}}(t_*)=&\\underset{\\boldsymbol{\\beta}}{\\operatorname{argmin}}\\big[\\boldsymbol{\\beta}^\\textsf{T}\\mathbf{K^{'}}\\boldsymbol{\\beta}-\\zeta\\hat{\\boldsymbol{\\beta}}_{DT}(t_*)^\\textsf{T}\\mathbf{K^{'}}\\boldsymbol{\\beta}+\\boldsymbol{\\beta}^T\\mathbf{a}\\big]\\\\&s.t.:\\boldsymbol{\\beta}^T\\mathbf{1}=-1,{\\hat{\\boldsymbol{\\beta}}_{DT}}^{\\mathsf{T}}\\mathbf{K^{'}}\\boldsymbol{\\beta}>l,-C\\le\\boldsymbol{\\beta}_i\\le0;\\foralli\\\\&C,\\zeta,l\\,\\text{:hyper-parameters},\\\\\\end{aligned}",
     u'g_{DS}({D}_i,{D}_j)=\\textbf{F}_i^\\textsf{T}\\textbf{P}_i\\textbf{P}_j^\\textsf{T}\\textbf{F}_j',
     u'a(T)=a_0+a_1\\left(\\frac{T_0}T\\right)+a_2\\left(\\frac{T_0}T\\right)^2,\\quadb(T)=b_3\\left(\\frac{T_0}T\\right)^3.',
     u'f_0(\\xi_i;\\Phi,\\bar\\Phi)=\\frac{\\bar\\Phi\\xi_i^2+2\\Phi\\xi_i+1}{\\xi_i^3+3\\bar\\Phi\\xi_i^2+3\\Phi\\xi_i+1}',
     u'\\rho(t)=\\mathcalU(t,t_0)\\rho(t_0)\\mathcalU^\\dagger(t,t_0),',
     u'\\rho_0(t)=\\mathcalU_0(t,t_0)\\rho(t_0)\\mathcalU^\\dagger_0(t,t_0)',
     u'H^\\prime_I(t^\\prime)=U^\\dagger_0(t^\\prime,t_0)H^\\prime(t^\\prime)\\mathcalU_0(t^\\prime,t_0)',
     u'\\delta\\langleA(t)\\rangle\\approxi\\intd\\bart\\theta(t-\\bart)\\langle[H_I^\\prime(\\bart),A_I(t)]\\rangle_0,',
     u'\\widetilde\\chi(k)=\\intd^4x\\chi(x)e^{ikx}.',
     u'\\widetildeS_a(i\\omega_n,\\mathbfp)=-\\int\\frac{dp^0}{2\\pi}\\frac{\\widetilde{\\mathcalA}_a(p)}{i\\omega_n-p^0},',
     u'\\widetilde{\\mathcalA}_a(p)\\equiv\\intd^4x\\mathcalA_a(x)e^{ipx/\\hbar}\\equiv\\intd^4x\\{q_a(x),\\barq_a(0)\\}e^{ipx/\\hbar}.',
     u'\\widetilde{\\mathcalA}_a(p)=\\pi[\\Delta_+(\\mathbfp)\\delta(p^0+\\widetilde\\mu_a-E_{\\mathbfp})-\\Delta_-(\\mathbfp)\\delta(p^0+\\widetilde\\mu_a+E_{\\mathbfp})],',
     u'\\Delta_\\pm(\\mathbfp)=\\pm\\gamma^0-\\frac{\\mathbfp}{E_\\mathbf{p}}\\cdot\\boldsymbol\\gamma+\\fracM{E_\\mathbf{p}},',
     u'D[f_a]=C[f_z],',
     u'D[f_a]\\equiv\\partial_tf_a+\\mathbfv\\cdotabla_\\mathbf{r}f_a+\\frac{M_a}{E_a}abla_\\mathbf{r}V^Sabla_\\mathbfpf_a,',
     u'D[f_a]\\approx\\partial_t\\deltaf_a+\\mathbfv\\cdotabla_\\mathbf{r}\\deltaf_a+\\frac{M_a}{E_a}abla_\\mathbf{r}\\deltaV^Sabla_\\mathbfpf^0_a,',
     u'C[f_a]\\approx-\\frac1{\\tau_a}\\deltaf_a,',
     u'\\frac1{\\tau_a}=\\sum_{bcd}\\intd^3\\mathbfPd^3\\mathbfp\\frac{(2\\pi)^{-5}\\delta(E_a+E_b-E_c-E_d)}{(1+\\delta_{ab})2^32E_a2E_b2E_c2E_d}|\\mathcalM_{ab}|^2\\frac{f_b^0(1-f_c^0)(1-f_d^0)}{1-f_a^0}.',
     u'\\delta(E_a+E_b-E_c-E_d)=\\intdE\\delta(E_a+E_b-E)\\delta(E_c+E_d-E),',
     u'\\delta(E-E_a-E_b)=\\frac{E_b}{Pp_a}\\delta\\left(x-\\frac{p_a^2-p_b^2+P^2}{2Pp_a}\\right),',
     u'\\delta(E_c+E_d-E)=\\delta\\left(p-E\\sqrt{\\frac{s-2m^2}{E^2-P^2x^{\\prime2}}}\\right)\\frac{4EE_cE_d}{p(E^2-P^2x^{\\prime2})},',
     u'\\partial_t\\deltaf_a+\\mathbfv\\cdotabla_\\mathbf{r}\\deltaf_a++\\frac{M_a}{E_a}abla_\\mathbf{r}\\deltaV^S_aabla_\\mathbfpf^0_a+\\tau^{-1}_a\\deltaf_a=0.',
     u'(\\omega+i\\tau^{-1}_a-\\mathbfk\\cdot\\mathbfv_a)\\delta\\tildef_a+\\frac{M_a}{E_a}\\delta\\tildeV^S_a\\mathbfk\\cdotabla_\\mathbfpf^0_a=0.',
     u'\\left(\\begin{array}{cc}1-2(G_S+K\\langle\\barss\\rangle)(\\chi_q-\\xi_q)&-2K\\langle\\barqq\\rangle(\\chi_q-\\xi_q)\\\\-4K\\langle\\barqq\\rangle(\\chi_s-\\xi_s)&1-2G_S(\\chi_s-\\xi_s)\\end{array}\\right)\\left(\\begin{array}{c}\\delta\\langle\\barqq\\rangle\\\\\\delta\\langle\\barss\\rangle\\end{array}\\right)=0,',
     u'\\xi=\\frac{N_c}{\\pi^2}\\intdE\\frac{p^3}{E^2}(f^0+\\barf^0-1).',
     u'\\left|\\begin{array}{cc}1-2(G_S+K\\langle\\barss\\rangle)(\\chi_q-\\xi_q)&-2K\\langle\\barqq\\rangle(\\chi_q-\\xi_q)\\\\-4K\\langle\\barqq\\rangle(\\chi_s-\\xi_s)&1-2G_S(\\chi_s-\\xi_s)\\end{array}\\right|=0.',
     u'\\[R_i=\\{c_{x1},c_{y1},c_{x2},c_{y2}\\},\\quadc_i\\simU(x|0,w)\\]',
     u'f(s,R_1,R_2)=\\begin{cases}1&\\text{if}\\displaystyle\\sum_{i|x_i\\inR_1}\\frac{x_i}{n_{1}}<\\sum_{i|x_i\\inR_2}\\frac{x_i}{n_{2}}\\\\0&\\text{otherwise}\\end{cases}',
     u'\\hat{S}(t)=\\prod_{j:t_{(j)}\\leqt}{\\left(1-\\frac{d_j}{r_j}\\right)}',
     u'\\hat{\\chi^2}=\\frac{\\left(\\sum^m_{i=1}{(d_{1i}-\\hat{e}_{1i})}\\right)^2}{\\sum^m_{i=1}{\\hat{v}_{1i}}}',
     u'S(t)=1-p(T\\leqt_0)=1-\\int_0^{t_0}{p(t)dt},',
     u'h(t)=\\lim\\limits_{\\Deltat\\rightarrow0}\\frac{P(t<T\\leqt+\\Deltat|T>t)}{\\Deltat}=\\frac{p(T=t)}{S(t)}.',
     u'h(t|{\\boldsymbol{x}})=h_0(t)\\exp({\\boldsymbol{x}}^{T}{\\boldsymbol{\\beta}}),',
     u'p(t|\\alpha_w,\\lambda_w)=\\alpha_w\\frac{1}{\\lambda_w}t^{\\alpha_w-1}\\exp\\left(-\\frac{1}{\\lambda_w}t^{\\alpha_w}\\right),',
     u'p(\\left\\{t_i\\right\\}_{i=0}^{N}|\\alpha_w,\\lambda_w)=\\prod_{i=1}^N{\\left(\\frac{\\alpha_w}{\\lambda_w}t_i^{\\alpha_w-1}\\right)^{\\delta_i}\\exp\\left(-\\frac{1}{\\lambda_w}t_i^{\\alpha_w}\\right)},',
     u'\\eta={\\boldsymbol{x}}^{t}{\\boldsymbol{\\beta}}+\\epsilon,\\hspace{20pt}\\text{where\\}\\epsilon\\simN(0,\\sigma^2).',
     u'T_{ij}=\\frac{1}{N}\\sum_{k=1}^Nr_{ki}r_{kj}~,',
     u'R_p=\\left(\\frac{1}{N}\\sum_{i=1}^Nr_i^2\\right)^{1/2}=(\\Lambda_1+\\Lambda_2+\\Lambda_3)^{1/2}',
     u'R_g=\\sqrt{\\left\\langleR_p^2\\right\\rangle}=\\sqrt{\\left\\langle\\Lambda_1+\\Lambda_2+\\Lambda_3\\right\\rangle}~,',
     u'\\frac{x^2}{\\Lambda_1}+\\frac{y^2}{\\Lambda_2}+\\frac{z^2}{\\Lambda_3}=3~.',
     u'P_0(\\lambda_1,\\lambda_2,\\lambda_3)=P_1(\\lambda_1)P_2(\\lambda_2)P_3(\\lambda_3)~,',
     u'P_i(\\lambda_i)=\\frac{(a_id_i)^{n_i-1}\\lambda_i^{-n_i}}{2K_i}\\exp\\left(-\\frac{\\lambda_i}{a_i}-d_i^2\\frac{a_i}{\\lambda_i}\\right)~,',
     u'A(\\phi_n)=1-3\\frac{\\left\\langle\\lambda_1\\lambda_2+\\lambda_1\\lambda_3+\\lambda_2\\lambda_3\\right\\rangle}{\\left\\langle(\\lambda_1+\\lambda_2+\\lambda_3)^2\\right\\rangle}~.',
     u'\\frac{R_g(\\phi_n)}{R_g(0)}=\\sqrt{6\\left\\langle\\lambda_1+\\lambda_2+\\lambda_3\\right\\rangle}~,',
     u'R_i(\\phi_n)=R_g(0)\\sqrt{18\\lambda_i}~,\\quadi=1,2,3~.',
     u'f=k_BT~\\frac{4\\pin_pR_p^3}{q}\\left(1+\\frac{2}{\\sqrt{\\pi}q}+\\frac{1}{3q^2}\\right)~,',
     u'\\beta\\varepsilon=\\frac{4\\piR_p^3}{v_pq}\\left(1+\\frac{2}{\\sqrt{\\pi}q}+\\frac{1}{3q^2}\\right)~,',
     u'\\beta\\varepsilon=\\frac{3}{q}\\left(1+\\frac{2}{\\sqrt{\\pi}q}+\\frac{1}{3q^2}\\right)~,',
     u'v_p=\\frac{4\\pi}{3}\\intd\\lambda\\,P_0(\\lambda)R_1R_2R_3=1.8365~R_p^3~,',
     u'\\beta\\varepsilon=\\frac{6.8426}{q}\\left(1+\\frac{2}{\\sqrt{\\pi}q}+\\frac{1}{3q^2}\\right)~.',
     u'v_p(\\phi_n)=\\frac{4\\pi}{3}\\intd\\lambda\\,P(\\lambda;\\phi_n)R_1R_2R_3~.',
     u'{\\calP}_{\\rmconfig}({\\rmold}\\to{\\rmnew})=\\min\\left\\{e^{-\\beta\\DeltaF},~1\\right\\}~,',
     u"{\\bfu}'=\\frac{{\\bfu}+\\tau{\\bfv}}{|{\\bfu}+\\tau{\\bfv}|}~,",
     u"{\\calP}_{\\rmshape}(\\lambda\\to\\lambda')=\\min\\left\\{\\frac{P_0(\\lambda')}{P_0(\\lambda)}e^{-\\beta\\DeltaF},~1\\right\\}~,",
     u'v_{\\rmmf}(r)=\\Omega(r)-\\Omega(\\infty)~,',
     u"v_{\\rmmf}(r)=-\\Pi_p\\int_{\\infty}^rdr'\\,A_{\\rmov}(r')=-\\Pi_pV_{\\rmov}(r)~,",
     u'V_{\\rmov}(r)=\\frac{\\displaystyle4\\pi}{\\displaystyle3}\\left[\\left(R_n+R_p\\right)^3-\\frac{\\displaystyle3r}{\\displaystyle4}(R_n+R_p)^2+\\frac{\\displaystyler^3}{\\displaystyle16}\\right]~,',
     u'\\Xi(r)\\propto\\left\\langle\\exp[-\\betaU(r)]\\right\\rangle~,',
     u'\\betav_{\\rmmf}(r)=-\\ln\\left(\\frac{\\left\\langle\\exp[-\\betaU(r)]\\right\\rangle}{\\left\\langle\\exp[-\\betaU(\\infty)]\\right\\rangle}\\right)~,',
     u'\\betav_{\\rmmf}(r)=\\left\\langle\\exp[-\\betaU(\\infty)]\\right\\rangle-\\left\\langle\\exp[-\\betaU(r)]\\right\\rangle~.',
     u'\\left\\langle\\exp[-\\betaU(\\infty)]\\right\\rangle=1-2c(r)\\phi_n=2\\left\\langle\\exp(-\\betaU_0)\\right\\rangle-1~.',
     u'\\left\\langle\\exp(-\\betaU_0)\\right\\rangle=\\frac{V-\\frac{4\\pi}{3}(R_n+R_p)^3}{V}=1-\\phi_n(1+q)^3~.',
     u'\\left\\langle\\exp[-\\betaU(\\infty)]\\right\\rangle=1-2\\phi_n(1+q)^3~.',
     u'\\left\\langle\\exp[-\\betaU(r)]\\right\\rangle=\\frac{V-\\frac{8\\pi}{3}(R_n+R_p)^3+V_{\\rmov}(r)}{V}~.',
     u'\\left\\langle\\exp[-\\betaU(r)]\\right\\rangle=1-\\phi_n\\left[(1+q)^3+\\frac{3}{4}x(1+q)^2-\\frac{x^3}{16}\\right]~,',
     u'\\betav_{\\rmmf}(r)=-12\\phi_p\\left[\\frac{h(x/q)}{qx}+\\frac{g(x/q)}{q^2x}-\\frac{h(2x/q)}{2qx^2}\\right]~,',
     u'g(x)\\equiv\\frac{e^{-x^2}}{\\sqrt{\\pi}}-x+x{\\rmErf}(x)',
     u'h(x)\\equiv\\frac{1}{4}\\left[-\\frac{2}{\\sqrt{\\pi}}xe^{-x^2}+(1+2x^2){\\rmErfc}(x)\\right]~,',
     u'p^{(n)}(s)=\\frac{(n+1)^{n+1}}{\\Gamma(n+1)}\\frac{s^n}{{\\bars}^{n+1}}\\exp[-(n+1)s/{\\bars}]~,',
     u'P(\\lambda;\\phi_n)=P_0(\\lambda)\\frac{\\alpha(\\lambda;\\phi_n)}{\\alpha_{\\rmeff}(\\phi_n)}~,',
     u'\\alpha_{\\rmeff}(\\phi_n)\\equiv\\int_0^{\\infty}{\\rmd}\\lambda\\,P_0(\\lambda)\\alpha(\\lambda;\\phi_n)',
     u"\\alpha(\\lambda;\\phi_n)=(1-\\phi_n')\\exp[-\\beta(pv_p+\\gammaa_p+\\kappac_p)]~,",
     u'U_{i}(\\mathbf{x})=exp\\left(\\mathbf{|\\tildex_i|}\\right)^{\\beta}exp(i\\phi_{i})',
     u'\\phi_{p,i}(\\mathbf{x})=ka_{i}',
     u'\\phi_{t,i}(\\mathbf{x})=k\\mathbf{x}\\mathbf{b}_i',
     u'U(\\mathbf{x})=\\left[\\sum_{i=1}^{M}U_{i}(\\mathbf{x})\\right]exp\\left(\\dfrac{|\\mathbf{x}-\\mathbf{c}_0|_2}{R}\\right)^\\gamma',
     u'|\\mathbfc_{i}-\\mathbfc_{j}|>2r_s,ieqj',
     u'E^+(\\mathbf{v})=\\frac{1}{\\lambda}\\sum_{\\mathbf{x}}exp\\left(i\\frac{2\\pi}{\\lambda}\\mathbf{v}\\cdot\\mathbf{x}\\right)U(\\mathbf{x})exp(+ik_x\\mathbf{x})',
     u'E^-(\\mathbf{v})=\\frac{1}{\\lambda}\\sum_{\\mathbf{x}}exp\\left(i\\frac{2\\pi}{\\lambda}\\mathbf{v}\\cdot\\mathbf{x}\\right)U(\\mathbf{x})exp(-ik_x\\mathbf{x})',
     u'I(\\mathbf{x})=|E^+|^{2}+|E^-|^{2}',
     u'\\mathbf{d}_{r,i}=\\mathbf{d}_{0}-2\\left(\\mathbf{d}_{0}\\cdot\\mathbf{n}_{i}\\right)\\mathbf{n}_{i}',
     u'z=-(x^2+y^2)/4f+f',
     u'\\begin{aligned}\\theta_{x,i,0}=cos^{-1}(x/2pf)\\hspace{-4.8mm}\t\\\\\\theta_{y,i,0}=cos^{-1}(y/2pf)\\hspace{-4.2mm}\t\\\\\\theta_{z,i,0}=cos^{-1}(1/p)\t\\end{aligned}',
     u'\\mathbf{\\theta}_{i}=\\mathbf{\\theta}_{i,0}+\\mathbf{\\Delta\\theta_{i}}',
     u'\\mathbf{m}_{i}=\\mathbf{c}_{i}+\\alpha\\cost\\mathbf{e}_{1}+\\alpha\\cost\\mathbf{e}_{2}',
     u'\\mathbf{u}_{i}=\\mathbf{m}_{i}+\\mathbf{d}_{r}\\frac{(\\mathbf{q}-\\mathbf{m}_{i})\\cdot\\mathbf{n}_{d}}{\\mathbf{d}_{r}\\cdot\\mathbf{n}_{d}}',
     u'\\begin{aligned}E^+(\\mathbf{u}^+_{i})=A^+_iexp(ik\\mathbf{d}^+_{r,i}\\cdot\\mathbf{u}^+_{i})exp(ikL^+_{i})\\\\\\E^-(\\mathbf{u}^-_{i})=A^-_iexp(ik\\mathbf{d}^-_{r,i}\\cdot\\mathbf{u}^-_{i})exp(ikL^-_{i})\\end{aligned}',
     u'I(\\mathbf{x}_{d})=|\\sum_{i=1}^{M}E{_i}^+(\\mathbf{x}_{d})|^2+|\\sum_{i=1}^{M}E{_i}^-(\\mathbf{x}_{d})|^2',
     u'g_{n}=(f\\otimesh_{n})+q_{n}',
     u'\\{f,h_{n}\\}=\\min_{h_{n}\\geq0,f\\geq0}\\frac{1}{N^{2}}\\sum_{n=0}^{N}\\sum_{\\mathbfu}|{G_{n}}-\\tilde{F}\\tilde{H_{n}}|^{2}(\\mathbfu)',
     u'H_{i}=H_{i-1}\\odot\\frac{F_{i}^{T}G_{i}}{F_{i}^{T}(F_{i}H_{i-1})}',
     u'F_{i}=F_{i-1}\\odot\\frac{H_{i}^{T}G_{i}}{H_{i}^{T}(H_{i}F_{i-1})}',
     u'{\\frac{\\bar{\\rho}}{\\bar{\\rho}_{\\odot}}}\\simeq{\\left(\\frac{\\Deltau}{\\Deltau_{\\odot}}\\right)}^{2}',
     u'{\\frac{g}{g_{\\odot}}}\\simeq{\\left(\\frac{u_{\\rm{max}}}{u_{\\rm{max},\\\\odot}}\\right)}{\\left(\\frac{T_{\\rm{eff}}}{T_{\\rm{eff},\\\\odot}}\\right)}^{-1/2}.',
     u"I(t)\\equiv\\int_0^t\\left(\\frac{T_{\\rm{eff}}(t')}{4500\\rm{K}}\\right)^{4/3}\\left(\\frac{M_{\\rm{env}}(t')}{M_{\\odot}}\\right)^{2/3}\\left(\\frac{R(t')}{R_{\\odot}}\\right)^8dt'.onumber",
     u'\\begin{aligned}V_5&=V_5^L+\\chi_5V_2V_3\\,,\\\\V_6&=V_6^L+\\chi_{63}V_3^2+\\chi_{62}V_2^3\\,,\\\\V_7&=V_7^L+\\chi_7V_2^2V_3\\,.\\end{aligned}',
     u'\\begin{aligned}\\chi_4&=\\frac{{\\langle}V_4(V_2^*)^2{\\rangle}}{{\\langle}|V_2|^4{\\rangle}}=\\frac{v_4\\{\\Psi_2\\}}{\\sqrt{{\\langle}|V_2|^4{\\rangle}}}\\\\\\chi_5&=\\frac{{\\langle}V_5V_2^*V_3^*{\\rangle}}{{\\langle}|V_2|^2|V_3|^2{\\rangle}}=\\frac{v_5\\{\\Psi_{23}\\}}{\\sqrt{{\\langle}|V_2|^2|V_3|^2{\\rangle}}}\\\\\\chi_{62}&=\\frac{{\\langle}V_6(V_2^*)^3{\\rangle}}{{\\langle}|V_2|^6{\\rangle}}=\\frac{v_6\\{\\Psi_2\\}}{\\sqrt{{\\langle}|V_2|^6{\\rangle}}}\\,,\\qquad\\chi_{63}=\\frac{{\\langle}V_6(V_3^*)^2{\\rangle}}{{\\langle}|V_3|^4{\\rangle}}=\\frac{v_6\\{\\Psi_3\\}}{\\sqrt{{\\langle}|V_3|^4{\\rangle}}}\\\\\\chi_7&=\\frac{{\\langle}V_7(V_2^*)^2V_3^*{\\rangle}}{{\\langle}|V_2|^4|V_3|^2{\\rangle}}=\\frac{v_7\\{\\Psi_{23}\\}}{\\sqrt{{\\langle}|V_2|^4|V_3|^2{\\rangle}}}.\\end{aligned}',
     u'I(T,\\omega)=\\prod\\limits_{i\\inI}I({\\rmTh}(\\mathcal{A}_i),\\min\\{|A_i|,\\omega\\}).',
     u"\\Sigma=\\langle\\leq,\\leq',P_n,\\{c_n\\},\\{c'_n\\}\\rangle_{n\\in\\omega}",
     u'I_\\infty(T,\\lambda)=\\left\\{\\begin{array}{rl}\\sum\\limits_{i=0}^{{\\rmmin}\\{|J|,\\lambda\\}}C^i_{|J|},&\\mbox{if}J\\mbox{and}\\lambda\\mbox{arefinite;}\\\\|J|,&\\mbox{if}J\\mbox{isinfiniteand}|J|>\\lambda;\\\\2^{|J|},&\\mbox{if}J\\mbox{isinfiniteand}|J|\\leq\\lambda.\\end{array}\\right.',
     u'\\varphi^{E,\\sigma}(x_1,\\ldots,x_n)=\\existsx\\left(\\bigwedge\\limits_{i=1}^n(E(x,x_i)\\wedge\\existsy(E(x,y)\\wedge\\sigma(y))\\wedge\\psi^{E,\\sigma}(x,x_1,\\ldots,x_n)\\right);',
     u'\\varphi^{E,\\sigma}(x_1,\\ldots,x_n)=\\forallx\\left(\\bigwedge\\limits_{i=1}^nE(x,x_i)\\wedge\\existsy(E(x,y)\\wedge\\sigma(y))\\to\\psi^{E,\\sigma}(x,x_1,\\ldots,x_n)\\right).',
     u"{\\rmTH}(\\mathcal{A}_E)=\\bigcap\\limits_{\\mathcal{A}'\\equiv\\mathcal{A}_E}{\\rmTH}(\\mathcal{A}'),",
     u"{\\rmComLim}(\\mathcal{A}_{E},\\mathcal{B}_{E'})\\leq{\\rmmin}\\{e\\mbox{-}{\\rmSp}({\\rmTh}(\\mathcal{A}_{E})),e\\mbox{-}{\\rmSp}({\\rmTh}(\\mathcal{B}_{E'}))\\},",
     u"{\\rmmax}\\{e\\mbox{-}{\\rmSp}({\\rmTh}(\\mathcal{A}_{E})),e\\mbox{-}{\\rmSp}({\\rmTh}(\\mathcal{B}_{E'}))\\}\\leqe\\mbox{-}{\\rmSp}({\\rmTh}(\\mathcal{C}_{E''})),",
     u"e\\mbox{-}{\\rmSp}({\\rmTh}(\\mathcal{A}_{E}))+e\\mbox{-}{\\rmSp}({\\rmTh}(\\mathcal{B}_{E'}))=e\\mbox{-}{\\rmSp}({\\rmTh}(\\mathcal{C}_{E''}))+{\\rmComLim}(\\mathcal{A}_{E},\\mathcal{B}_{E'}).",
     u'\\models\\forallx,y\\:((x<y)\\to\\existsz\\:((x<z)\\wedge(z<y)\\wedgeR_i(z))),\\mbox{\\}i=0,\\ldots,n-2.',
     u'e\\mbox{-}{\\rmSp}(T)=\\sum\\limits_{l=1}^m(l+1)=\\frac{m(m+1)}{2}+m=\\frac{m^2+3m}{2}.',
     u"{\\rmCl}_E(\\mathcal{T})={\\rmTH}(\\mathcal{A}'_{E'}),",
     u'\\existsx,y(eg(x\\approxy)\\wedge\\forallz((z\\approxx)\\vee(z\\approxy)))',
     u'\\bigcup\\limits_{\\mathcal{T}\\subset\\mathcal{T}_{\\rmfin}}{\\rmCl}_E(\\mathcal{T})\\cup\\bigcup\\limits_{\\mathcal{T}\\subset\\mathcal{T}_{\\rmfin}}{\\rmCl}^d_P(\\mathcal{T}).',
     u"R'(x_1,\\ldots,x_{k_{n+1}})\\wedge(x_{k_{n+2}}\\approxx_{k_{n+2}})\\wedge(x_{r_{n+1}}\\approxx_{r_{n+1}}).",
     u'\\frac{d^2N}{dydp_T}=\\frac{gV}{(2\\pi)^2}p_T\\sqrt{p^2_T+m^2_0}\\coshy\\bigg[1+\\frac{q-1}{T}\\Big(\\sqrt{p^2_T+m^2_0}\\coshy-\\mu\\Big)\\bigg]^{-q/(q-1)},',
     u'f(p_T)=\\frac{1}{N}\\frac{dN}{dp_T}\\proptop_T\\sqrt{p^2_T+m^2_0}\\int^{y_{\\max}}_{y_{\\min}}\\coshy\\bigg[1+\\frac{q-1}{T}\\Big(\\sqrt{p^2_T+m^2_0}\\coshy-\\mu\\Big)\\bigg]^{-q/(q-1)}dy,',
     u'f(y)=\\frac{1}{N}\\frac{dN}{dy}\\propto\\coshy\\int^{p_{T\\max}}_0p_T\\sqrt{p^2_T+m^2_0}\\bigg[1+\\frac{q-1}{T}\\Big(\\sqrt{p^2_T+m^2_0}\\coshy-\\mu\\Big)\\bigg]^{-q/(q-1)}dp_T,',
     u'f(p)=\\frac{1}{N}\\frac{dN}{dp}\\proptop^2\\bigg[1+\\frac{q-1}{T}\\Big(\\sqrt{p^2+m^2_0}-\\mu\\Big)\\bigg]^{-q/(q-1)},',
     u'f(E)=\\frac{1}{N}\\frac{dN}{dE}\\proptoE\\sqrt{E^2-m_0^2}\\bigg[1+\\frac{q-1}{T}\\Big(E-\\mu\\Big)\\bigg]^{-q/(q-1)},',
     u'f(E_{kin})=\\frac{1}{N}\\frac{dN}{dE_{kin}}\\propto\\Big(E_{kin}+m_0\\Big)\\sqrt{E^2_{kin}+2m_0E_{kin}}\\bigg[1+\\frac{q-1}{T}\\Big(E_{kin}+m_0-\\mu\\Big)\\bigg]^{-q/(q-1)},',
     u'T=-(0.000050\\pm0.000002)C^2-(0.0015\\pm0.0001)C+(0.692\\pm0.001)',
     u'T=(0.155\\pm0.006)+(0.275\\pm0.010)m_0,',
     u'T=(0.160\\pm0.005)+(0.264\\pm0.005)m_0,',
     u'T=(0.199\\pm0.036)+(0.204\\pm0.022)m_0,',
     u'N_0=(0.016\\pm0.001)\\exp[-(0.030\\pm0.002)C]-(0.0010\\pm0.0001)',
     u'N_0=(547.849\\pm71.220)\\exp[-(5.809\\pm0.116)m_0]',
     u'\\langlep_T\\rangle=(0.305\\pm0.007)+(0.491\\pm0.011)m_0,',
     u'\\langlep_T\\rangle=(0.294\\pm0.009)+(0.517\\pm0.008)m_0,',
     u'\\langlep_T\\rangle=(0.347\\pm0.049)+(0.435\\pm0.031)m_0,',
     u'\\frac{\\partial\\mathbf{m}}{\\partialt}=-\\mathbf{m}\\times\\mathbf{h}_{\\rmeff}+\\mathbf{p},\\quad\\mathbf{m}:{\\mathbb{R}}^2\\times{\\mathbb{R}}\\to\\mathbb{S}^2,',
     u'\\mathbf{h}_{\\rmeff}=h_0\\mathbf{z}+abla^2\\mathbf{m}+m_z\\mathbf{z},',
     u'\\mathbf{p}=\\underbrace{-\\alpha\\mathbf{m}\\times(\\mathbf{m}\\times\\mathbf{h}_\\text{eff})}_\\text{damping}+\\underbrace{\\sigma\\mathcal{H}(\\mathbf{x})\\mathbf{m}\\times\\mathbf{m}\\times\\mathbf{m}_\\mathrm{p}}_\\text{NC-STO}-\\underbrace{\\mathbf{m}\\times\\mathbf{h}}_\\text{thermal},',
     u'\\mathcal{H}(\\mathbf{x})=\\begin{cases}1&|\\mathbf{x}|\\le\\rho_*\\\\0&\\mathrm{else}\\end{cases}',
     u'\tI_0=\\frac{4\\mu_0M_s^2(Q-1)e\\piR_*^2\\delta}{\\hbar\\epsilon}.',
     u'\\beta^2=\\frac{T}{T_0},\\quadT_0=\\frac{\\mu_0M_s^2V}{2\\alphak_B},',
     u'\\begin{aligned}\\cos\\Theta&=\\tanh\\left(\\rho-\\frac{1}{\\omega}\\right),\\\\\\Phi&=h_0t-\\frac{\\mathbf{v}\\cdot\\hat{\\boldsymbol{\\rho}}}{\\omega^2}+\\phi,\\quad\\phi=\\omegat+\\phi_0.\\end{aligned}',
     u'\\begin{aligned}d{\\phi}&=\\omega\\,dt-\\frac{\\sigma}{4\\pi}\\int_{\\lvert\\mathbf{x}\\rvert\\leq\\rho_*}(\\mathbf{v}\\cdot\\hat{\\boldsymbol{\\rho}})\\,\\sech^2\\left(\\rho-\\frac{1}{\\omega}\\right)\\hspc[2]d\\mathbf{x}dt+d\\mathcal{W}_\\phi,\\\\d{\\boldsymbol{\\xi}}&=\\mathbf{v}dt+\\frac{\\sigma\\omega}{2\\pi}\\int_{\\lvert\\mathbf{x}\\rvert\\leq\\rho_*}\\sech^2\\left(\\rho-\\frac{1}{\\omega}\\right)\\hat{\\boldsymbol{\\rho}}\\hspc[2]\\hspc[2]d\\mathbf{x}dt+d{\\boldsymbol{\\mathcal{W}}}_{{\\boldsymbol{\\xi}}},\\\\d\\omega&=\\alpha\\omega^2(\\omega+h_0)dt-\\frac{\\sigma\\omega^3}{4\\pi}\\int_{\\lvert\\mathbf{x}\\rvert\\leq\\rho_*}\\sech^2\\left(\\rho-\\frac{1}{\\omega}\\right)\\hspc[2]d\\mathbf{x}dt+d\\mathcal{W}_\\omega,\\\\d{\\mathbf{v}}&=\\alpha\\omega\\mathbf{v}(\\omega+2h_0)dt-\\frac{\\sigma\\omega^2}{2\\pi}\\int_{\\lvert\\mathbf{x}\\rvert\\leq\\rho_*}\\left(\\frac{3}{2}\\mathbf{v}-\\frac{(\\mathbf{v}\\cdot\\hat{\\boldsymbol{\\varphi}})}{\\rho\\omega}\\hat{\\boldsymbol{\\varphi}}\\right)\\sech^2\\left(\\rho-\\frac{1}{\\omega}\\right)\\hspc[2]d\\mathbf{x}dt+d{\\boldsymbol{\\mathcal{W}}}_{{\\mathbf{v}}},\t\t\\end{aligned}',
     u'\t\\text{E}[\\boldsymbol{\\mathcal{W}}\\boldsymbol{\\mathcal{W}}^T]=\\frac{\\beta^2t}{2\\pi}\\left[\\begin{array}{cc}\\begin{array}{cc}\\sigma_\\phi^2&\\mathbf{v}^T/2\\\\\\mathbf{v}/2&\\omega\\mathbf{I}\\end{array}&\\mathbf{0}\\\\\\mathbf{0}&\\begin{array}{cc}\\omega^5/2&\\omega^4\\mathbf{v}^T\\\\\\omega^4\\mathbf{v}&\\boldsymbol{\\sigma}_{\\mathbf{v}}^2\\end{array}\\end{array}\\right],',
     u'\\begin{aligned}\\sigma^2_\\phi&=\\frac{v^2}{4\\omega}+\\frac{\\omega}{2},\\\\\\boldsymbol{\\sigma}^2_{\\mathbf{v}}&=\\left[\\begin{array}{cc}\\omega^5+\\frac{\\omega^3}{4}\\left(9v_x^2+v_y^2\\right)&2\\omega^3v_xv_y\\\\2\\omega^3v_xv_y&\\omega^5+\\frac{\\omega^3}{4}\\left(v_x^2+9v_y^2\\right)\\end{array}\\right]\t\\end{aligned}',
     u'\\frac{\\sigma}{\\alpha}=\\frac{2(h_0+\\omega_*)}{1+\\omega_*\\left[\\log\\left(\\frac{1}{2}\\textrm{sech}\\left(\\rho_*-\\frac{1}{\\omega_*}\\right)\\right)+\\rho_*\\tanh\\left(\\rho_*-\\frac{1}{\\omega_*}\\right)\\right]}.',
     u'\\begin{aligned}\t\\dot{\\phi}&=\\omega,\\\\\t\\dot{\\boldsymbol{\\xi}}&=\\mathbf{v}+\\lambda_\\xi\\boldsymbol{\\xi},\\\\\t\\dot{\\omega}&=\\lambda_\\omega(\\omega-\\omega_*),\\\\\t\\dot{\\mathbf{v}}&=\\lambda_V\\mathbf{v},\t\\end{aligned}',
     u'\\begin{aligned}\t\\lambda_\\xi&=-\\frac{1}{2}\\sigma\\rho_*\\omega_*\\sech^2\\left(\\rho_*-\\frac{1}{\\omega_*}\\right),\\\\\t\\lambda_\\omega&=-h_0\\alpha\\omega_*+\\lambda_\\xi+\\frac{1}{2}\\sigma\\omega_*\\left(\\tanh\\left(\\rho_*-\\frac{1}{\\omega_*}\\right)+1\\right),\\\\\t\\lambda_v&=-2\\alpha\\omega_*^2+\\lambda_\\omega-\\lambda_\\xi.\t\\end{aligned}',
     u'\\alpha\\omega_*(2\\omega_*+h_0)>\\frac{1}{2}\\sigma\\omega_*\\left(\\tanh\\left(\\rho_*-\\frac{1}{\\omega_*}\\right)+1\\right).',
     u'\t\\zeta=\\frac{2\\alphah_0}{\\sigma}-1.',
     u'\\omega_*=\\rho_*^{-1}+\\mathrm{arctanh}\\,(\\zeta)\\rho_*^{-2}+\\mathcal{O}(\\rho_*^{-3}).',
     u'\\begin{aligned}\\lambda_\\xi&=-\\frac{\\sigma}{2}(1-\\zeta^2)+\\mathcal{O}(\\rho_*^{-1}),\\\\\\lambda_\\omega&=-\\frac{\\sigma}{2}(1-\\zeta^2)+\\mathcal{O}(\\rho_*^{-1}),\\\\\\lambda_v&=\\mathcal{O}(\\sigma\\rho_*^{-2}).\t\\end{aligned}',
     u'\\begin{aligned}\td\\phi&=\\omegadt+d\\mathcal{W}_\\phi^*,\\\\\td{\\boldsymbol{\\xi}}&={\\mathbf{v}}dt+\\lambda_\\xi\\boldsymbol{\\xi}dt+d{\\boldsymbol{\\mathcal{W}}}_{{\\boldsymbol{\\xi}}}^*,\\\\\td{\\omega}&=\\lambda_\\omega(\\omega-\\omega_*)dt+d\\mathcal{W}_{\\omega}^*,\\\\\td{\\mathbf{v}}&=\\lambda_v\\mathbf{v}dt+d{\\boldsymbol{\\mathcal{W}}}_{{\\mathbf{v}}}^*.\t\\end{aligned}',
     u'\\begin{aligned}\\text{E}[\\boldsymbol{\\mathcal{W^*}}\\boldsymbol{\\mathcal{W}^*}^T]&=\\beta^2t\\cdot\\text{Diag}\\left(\\frac{\\omega_*}{4\\pi},\\frac{\\omega_*}{2\\pi},\\frac{\\omega_*}{2\\pi},\\frac{\\omega_*^5}{4\\pi},\\frac{\\omega_*^5}{2\\pi},\\frac{\\omega_*^5}{2\\pi}\\right),onumber\\\\&\\equivt\\cdot\\text{Diag}\\left(\\beta_\\phi^2,\\beta_\\xi^2,\\beta_\\xi^2,\\beta_\\omega^2,\\beta_v^2,\\beta_v^2\\right),\\end{aligned}',
     u'\\begin{aligned}\\text{Var}[\\omega(t)]&=-\\frac{\\beta_\\omega^2}{2\\lambda_\\omega}\\left(1-\\exp{2\\lambda_\\omegat}\\right)onumber\\\\&\\rightarrow-\\frac{\\beta^2_\\omega}{2\\lambda_\\omega},\\quad\\text{as}\\,\\,t\\rightarrow\\infty.\\end{aligned}',
     u'\\phi(t)=\\int_0^t\\omega(s)ds+\\mathcal{W}^*_\\phi,',
     u'\\text{Var}[\\phi(t)]\\rightarrow\\left(\\frac{\\beta_\\omega^2}{\\lambda_\\omega^2}+\\beta_\\phi^2\\right)t\\quad\\text{as}\\,\\,t\\rightarrow\\infty,',
     u'\\begin{aligned}\\Deltaf&=\\left(\\frac{\\beta_\\omega^2}{\\lambda_\\omega^2}+\\beta_\\phi^2\\right)={\\beta^2}\\left(\\frac{\\omega_*^5}{4\\pi\\lambda_\\omega^2}+\\frac{\\omega_*}{2\\pi}\\right).\\end{aligned}',
     u'\\begin{aligned}s_{{\\boldsymbol{\\xi}}}^2(t)&=-\\frac{\\beta_{\\xi}^2}{2\\lambda_{\\xi}}\\left(1-e^{2\\lambda_{\\xi}t}\\right)-\\frac{\\beta_v^2}{2\\left(\\lambda_{\\xi}-\\lambda_v\\right){}^2}onumber\\\\&\\quad\\times\\left(\\frac{1-e^{2\\lambda_{\\xi}t}}{\\lambda_{\\xi}}+\\frac{4\\left(1-e^{\\left(\\lambda_{\\xi}+\\lambda_v\\right)t}\\right)}{\\lambda_{\\xi}+\\lambda_v}+\\frac{1-e^{2\\lambda_vt}}{\\lambda_v}\\right)onumber\\\\&\\rightarrow-\\frac{1}{2}\\frac{\\beta_v^2}{\\lambda_\\xi^2\\lambda_v}-\\frac{\\beta_\\xi^2}{\\lambda_\\xi}\\quad\\text{as}\\,\\,t\\rightarrow\\infty.\\end{aligned}',
     u'\\|\\boldsymbol{\\xi}_\\text{NL}(t)-\\boldsymbol{\\xi}_\\text{L}(t)\\|=\\mathcal{O}(s^2_{\\xi}).',
     u'\\begin{aligned}&abla\\cdot\\epsilon_r^{-1}abla\\left(H_{zs}+H_{zi}\\right)+k_0^2\\mu_r\\left(H_{zs}+H_{zi}\\right)=0,~\\mathrm{in}~\\Omega\\\\&\\epsilon_r^{-1}ablaH_{zs}\\cdot\\mathbf{n}+jk_0\\sqrt{\\epsilon_r^{-1}\\mu_r}H_{zs}=0,~\\mathrm{on}~\\Gamma_{a}\\\\&H_{zs}\\left(\\mathbf{x}+\\mathbf{a}\\right)=H_{zs}\\left(\\mathbf{x}\\right)e^{-j\\mathbf{k}\\cdot\\mathbf{a}},~\\mathbf{n}\\left(\\mathbf{x}+\\mathbf{a}\\right)\\cdotablaH_{zs}\\left(\\mathbf{x}+\\mathbf{a}\\right)=-e^{-j\\mathbf{k}\\cdot\\mathbf{a}}\\mathbf{n}\\left(\\mathbf{x}\\right)\\cdotablaH_{zs}\\left(\\mathbf{x}\\right),\\\\&\\mathrm{for}~\\forall\\mathbf{x}\\in\\Gamma_{ps},~\\mathbf{x}+\\mathbf{a}\\in\\Gamma_{pd}\\end{aligned}',
     u'\\epsilon_{rm}=\\epsilon_{r\\infty}-{{\\omega_p^2}\\over{\\omega\\left(\\omega-j\\gamma_c\\right)}}',
     u'\\epsilon_{rd}\\left(\\omega\\right)=10^{\\log{\\epsilon_{rm}\\left(\\omega\\right)}-{{1-\\rho^3}\\over{1+\\rho^3}}\\left[\\log{\\epsilon_{rm}\\left(\\omega\\right)}-\\log{\\epsilon_{rf}\\left(\\omega\\right)}\\right]}',
     u'\\begin{aligned}P_i&=\\int_{\\Gamma_i}{1\\over2}\\mathrm{Re}\\left(\\mathbf{E}_i\\times\\mathbf{H}_i^*\\right)\\cdot\\mathbf{n}_i\\,\\mathrm{d}\\Gamma\\\\&=\\int_\\Omega{1\\over2}\\mathrm{Re}\\left(\\mathbf{E}_i\\times\\mathbf{H}_i^*\\right)\\cdot\\mathbf{n}_i~\\delta\\left(dist\\left(\\mathbf{x},\\Gamma_i\\right)\\right)\\mathrm{d}\\Omega\\\\\\end{aligned}',
     u'\\begin{aligned}P_o&=\\int_{\\Gamma_o}{1\\over2}\\mathrm{Re}\\left(\\mathbf{E}\\times\\mathbf{H}^*\\right)\\cdot\\mathbf{n}_o\\,\\mathrm{d}\\Gamma\\\\&=\\int_\\Omega{1\\over2}\\mathrm{Re}\\left(\\mathbf{E}\\times\\mathbf{H}^*\\right)\\cdot\\mathbf{n}_o~\\delta\\left(dist\\left(\\mathbf{x},\\Gamma_o\\right)\\right)\\mathrm{d}\\Omega\\end{aligned}',
     u'T_r={P_o}/{P_i}',
     u'\\max_{\\rho\\in\\left[0,1\\right]}\\left\\{\\min_{\\lambda\\in\\left[\\lambda_0-{\\theta\\over2},\\lambda_0+{\\theta\\over2}\\right]}T_r\\left(\\lambda\\right)\\right\\}',
     u'(\\forallf:[{\\mathbb{N}}]^{n}\\tok)(\\existsY)(Y\\mbox{isinfinite}\\wedge\\Psi(f,Y))',
     u'{\\mathsf{WKL_0}}\\vdash{\\forall}X({\\exists}Y\\Theta(X,Y)\\leftrightarrow{\\exists}Z(Z\\mbox{isinfinite}\\wedge\\Psi(X,Z))).',
     u'\\begin{aligned}\\Psi(X,Z)\\equiv{\\forall}F\\subseteq_{{\\mathrm{fin}}}Z(Feq\\emptyset\\to(&{\\exists}\\sigma\\in2^{\\maxF})({\\forall}n<|F|)({\\exists}m<\\maxF)\\theta(X{\\upharpoonright}m,\\sigma{\\upharpoonright}m,n,m)).\\end{aligned}',
     u'\\begin{aligned}T=\\{\\sigma\\in2^{<{\\mathbb{N}}}\\mid{\\forall}F\\subseteqZ\\cap[0,|\\sigma|]_{{\\mathbb{N}}}(Feq\\emptyset\\to({\\forall}n<|F|)({\\exists}m<\\maxF)\\theta(X{\\upharpoonright}m,\\sigma{\\upharpoonright}m,k,n,m))\\}.\\end{aligned}',
     u'\\Gamma=(\\forallf:[{\\mathbb{N}}]^{n}\\tok)(\\existsY)(Y\\mbox{isinfinite}\\wedge\\Psi(f,Y)).',
     u'(\\forallf:[{\\mathbb{N}}]^{n}\\tok)(\\existsY)(Y\\mbox{isinfinite}\\wedge\\Psi(f,Y))',
     u'{\\forall}i_{1}<\\dots<i_{n}\\,{\\exists}c<k\\,{\\forall}x_{1}\\inF_{i_{1}},\\dots,{\\forall}x_{n}\\inF_{i_{n}}\\,f(x_{1},\\dots,x_{n})=c',
     u'\\begin{aligned}\\mathcal{D}^{1}_{\\theta,b}:=&\\left\\{(\\sigma,\\vec{Y})\\,\\Bigg|\\,\\begin{aligned}&\\quad{\\forall}\\tau\\supseteq\\sigma(\\tau\\text{isaminimal{\\mathsf{L}}-sequence\tconsistentwith}\\vec{Y}\\to{\\forall}n\\le|\\tau|eg\\theta(b,n,\\tau{\\upharpoonright}n))\\\\&\\vee{\\exists}c\\leb({\\forall}d<c{\\forall}\\tau\\supseteq\\sigma(\\tau\\text{isaminimal{\\mathsf{L}}-sequenceconsistentwith}\\vec{Y}\\\\&\\quad\\quad\\quad\\quad\\to{\\forall}n\\le|\\tau|eg\\theta(d,n,\\tau{\\upharpoonright}n))\t\\wedge{\\exists}n\\le|\\sigma|\\theta(c,n,\\sigma{\\upharpoonright}n))\\end{aligned}\\right\\},\\\\&\\mbox{whereb\\inMand\\theta(x,n,\\sigma)\\in\\Sigma^{0}_{0}withparametersfrom(M,S)},\\\\\\mathcal{D}^{2}_{e}:=&\\left\\{(\\sigma,\\vec{Y})\\mid\\|\\sigma\\|_{{\\mathsf{L}}}\\gee\\right\\},\\mbox{wheree\\inM}.\\end{aligned}',
     u'\\[{\\forall}m\\in{\\mathbb{N}}{\\exists}k\\in{\\mathbb{N}}({\\mathrm{I}\\Sigma^0_1}\\vdash\\mbox{Xis\\bbomega^{k}-large}\\to\\mbox{Xism-dense({\\mathrm{RT}}^{2}_{2})}).\\]',
     u'\\[{\\forall}m\\in{\\mathbb{N}}{\\exists}k\\leh(m)({\\mathrm{I}\\Sigma^0_1}\\vdash_{h(m)}\\mbox{Xis\\bbomega^{k}-large}\\to\\mbox{Xism-dense({\\mathrm{RT}}^{2}_{2})}),\\]',
     u"\\hat{C}_2|\\mu_L^{l}\\sigma_i\\sigma_{i'}\\mu_R^{l}\\rangle=(-1)^\\gamma|\\mu_R^{l}\\sigma_{i'}\\sigma_{i}\\mu_L^{l}\\rangle",
     u"\\begin{aligned}\\hat{C}_2|\\mu\\sigma\\sigma{'}\\mu{'}\\rangle=&(-1)^{\\gamma}|\\mu{'}\\sigma{'}\\sigma\\mu\\rangle\\\\\\gamma=n_{\\mu}(n_{\\sigma}+n_{\\sigma{'}}+n_{\\mu{'}}&)+n_{\\sigma}(n_{\\sigma{'}}+n_{\\mu{'}})+n_{\\sigma{'}}n_{\\mu{'}}\\end{aligned}",
     u'\\begin{aligned}\\hat{J}_i|0\\rangle&=|\\uparrow\\downarrow\\rangle\\\\\\hat{J}_i|\\uparrow\\downarrow\\rangle&=-|0\\rangle\\end{aligned}\\qquad\\qquad\\begin{aligned}\\hat{J}_i|\\uparrow\\rangle&=(-1)^{\\eta_i}|\\uparrow\\rangle\\\\\\hat{J}_i|\\downarrow\\rangle&=(-1)^{\\eta_i}|\\downarrow\\rangle\\end{aligned}',
     u'\\begin{aligned}\\hat{P}_i|0\\rangle&=|0\\rangle\\\\\\hat{P}_i|\\uparrow\\downarrow\\rangle&=-|\\uparrow\\downarrow\\rangle\\end{aligned}\\qquad\\qquad\\begin{aligned}\\hat{P}_i|\\uparrow\\rangle&=|\\downarrow\\rangle\\\\\\hat{P}_i|\\downarrow\\rangle&=|\\uparrow\\rangle\\end{aligned}',
     u'\\hat{J}=\\prod_i{}\\hat{J_i}\\qquad\\qquad\\qquad\\hat{P}=\\prod_i{}\\hat{P_i}',
     u"\\begin{aligned}\\langle\\mu'&\\sigma'\\sigma\\mu|{\\cal{\\hat{R}}}_{2l+2}|u\\tau\\tau'u'\\rangle\\\\&=\\langle\\mu|{\\cal{\\hat{R}}}_l|u\\rangle\\langle\\sigma|{\\cal{\\hat{R}}}_i|\\tau\\rangle\\langle\\sigma'|{\\cal{\\hat{R}}}_{i'}|\\tau'\\rangle\\langle\\mu'|{\\cal{\\hat{R}}}_l|u'\\rangle\\end{aligned}",
     u'(\\hat{C}_2)^2=1\\qquad\\hat{J}^{\\phantom{|}2}=(-1)^{(N_s+N_e)}\\qquad\\hat{P}^2=1',
     u'\\hat{J}_{L}\\mu_i(n_s>n_e)=\\mu_j(n_s<n_e)',
     u'\\hat{P}_{L}\\mu_i(M_S<0)=\\mu_j(M_S>0).',
     u'\\begin{aligned}\\resizebox{0.97\\hsize}{!}{\\hat{J}_{L}\\mbox{}\\mu_i(n_s>n_e,M_S<0)=\\mu_{i_J}(n_s<n_e,M_S<0)}\\\\\\resizebox{0.97\\hsize}{!}{\\hat{P}_{L}\\mbox{}\\mu_i(n_s>n_e,M_S<0)=\\mu_{i_P}(n_s>n_e,M_S>0)}\\\\\\resizebox{0.97\\hsize}{!}{\\hat{P}_{L}\\hat{J}_{L}\\mbox{}\\mu_i(n_s>n_e,M_S<0)=\\mu_{i_{PJ}}(n_s<n_e,M_S>0)}\\end{aligned}',
     u'(\\hat{J}_{L})_{i,j}=(-1)^{\\eta_J}\\delta_{i,i_J}\\qquad(\\hat{P}_{L})_{i,j}=(-1)^{\\eta_P}\\delta_{i,i_P}\\\\',
     u'(\\hat{P}_{L}\\hat{J}_{L})_{i,j}=(-1)^{\\eta_{PJ}}\\delta_{i,i_{PJ}}',
     u'{\\calP}(\\Gamma)=\\frac{1}{h}\\sum_i\\chi(\\Gamma,{\\cal{R}}_i){\\cal{\\hat{R}}}_i',
     u'\\tilde{H}_{2l+2}=S^{\\dagger}H_{2l+2}S',
     u'\\begin{aligned}H&=\\sum_{<i,j>,\\sigma}t_0(\\hatc_{i,\\sigma}^\\dagger\\hatc_{j,\\sigma}^{}+\\mbox{H.C.})+\\sum_i\\epsilon_i\\hatn_i\\\\&+\\sum_i\\frac{U}{2}\\hatn_i(\\hatn_i-1)+\\sum_{i>j}V_{ij}(\\hatn_i-z_i)(\\hatn_j-z_j)\\end{aligned}',
     u'2M\\rightarrowP^++P^-',
     u'E_C(N)=E_{P^+}(N)+E_{P^-}(N)-2E_M(N)',
     u'\\logL_{\\rm5~GHz}=(0.45\\pm0.11)\\logL_{\\rm[O\\,III]}+(5.6\\pm0.9),',
     u'\\alpha',
     u'S\\,=\\,\\int{d^4x}\\sqrt{-g}\\left[\\frac{R}{16\\piG}+{\\calL}_m\\right]\\,,',
     u'V(\\phi)\\,=\\,\\mu^4e^{\\phi/f}\\,,',
     u"{\\ddot\\phi}+3H{\\dot\\phi}+V'(\\phi)\\,=\\,\\frac{\\lambda}{8f}{\\vec{E}_{a}}\\cdot{\\vec{B}_{a}}\\,,",
     u"V'(\\phi)\\,\\simeq\\,\\frac{\\lambda}{8f}{\\vec{E}_{a}}\\cdot{\\vec{B}_{a}}\\,,",
     u'D^{ab}_\\alphaF^{b\\beta\\alpha}-\\frac{4\\lambda}{f}\\epsilon^{\\muu\\alpha\\beta}\\partial_\\alpha^{ab}(\\phiF_{\\muu}^b)=0\\,,',
     u'D_\\alpha^{ab}\\equiv\\delta^{ab}abla_\\alpha+gf^{acb}A^{c}_\\alpha',
     u'abla_{i}E^{i}_{a}=-\\frac{\\lambda}{f}\\vec{abla}\\phiabla_{j}B^{i}_{a},',
     u'abla\\timesE^{i}_{a}=-\\frac{\\partial}{\\partialt}B^{i}_{a}\\,',
     u'\\frac{\\partial}{\\partialt}E^{i}_{a}+2HE^{i}_{a}\\,=\\,0',
     u'\\frac{\\partial}{\\partialt}B^{i}_{a}+2HB^{i}_{a}\\,=\\,0\\,.',
     u'\\frac{\\partial}{\\partialt}E^{i}_{a}+2HE^{i}_{a}\\,=\\,-\\frac{\\lambda}{f}[\\dot{\\phi}B^{i}_{a}]',
     u'H(t_m)\\,\\simeq\\,m\\,',
     u'E(t),B(t)\\,\\sim\\,a(t)^{-3/2}\\,',
     u'{\\dotE}^{i}\\,+\\,2HE^{i}\\,=\\,S(t)^{i}\\,,',
     u'S^{i}(t)\\,=\\,\\frac{\\lambda}{f}|{\\dot\\phi}|a^2(t_i)a^{-2}(t)B(t_i)^{i}\\,,',
     u"E(t)\\,=\\,\\int_{t_i}^tdt'\\frac{t'}{t}S(t')+E(t_i)\\frac{t_i}{t}\\,,",
     u'E(t)\\,=\\,E(t_i)\\frac{t_i}{t}\\bigl[1+\\frac{\\lambda}{f}|{\\dot\\phi}|\\frac{B(t_i)}{E(t_i}(t-t_i)\\bigr]\\,',
     u'E(t)\\,=\\,E(t_m)\\bigl(\\frac{a(t_m)}{a(t)}\\bigr)^{3/2}\\bigl[1+\\frac{\\lambda}{f}|{\\dot\\phi}|\\frac{B(t_m)}{E(t_m)}(t-t_m)\\bigr]\\,',
     u"E(t)\\,=\\,\\int_{t_{eq}}^tdt'\\bigl(\\frac{a(t')}{a(t)}\\bigr)^{3/2}S(t')+E(t_{eq})\\bigl(\\frac{a(t_{eq)}}{a(t)}\\bigr)^{3/2}\\,,",
     u'S(t)\\,=\\,\\frac{\\lambda}{f}|{\\dot\\phi}|a^{-3/2}(t)B(t_{eq})\\,',
     u'E(t)\\,\\simeq\\,E(t_i)\\bigl(\\frac{a(t_{i})}{a(t_m)}\\bigr)^2\\bigl(\\frac{a(t_m)}{a(t)}\\bigr)^{3/2}\\bigl[1+\\frac{\\lambda}{f}|{\\dot\\phi}|\\frac{B(t_i)}{E(t_i)}t\\bigr]\\,.',
     u't_c\\,=\\,\\frac{f}{\\lambda}\\frac{E(t_i)}{B(t_i)}\\frac{1}{|{\\dot\\phi}|}\\,.',
     u'{\\vec{E}}\\cdot{\\vec{B}}\\,\\sim\\,a(t)^{-4}',
     u'{\\vec{E}}\\cdot{\\vec{B}}\\,\\sim\\,a(t)^{-3}',
     u'{\\vec{E}}\\cdot{\\vec{B}}\\,\\sim\\,a(t)^{-3}t',
     u"{\\ddot\\phi}+3H{\\dot\\phi}+V'(\\phi)\\,=\\,\\frac{\\lambda}{8f}{\\vec{E}}\\cdot{\\vec{B}}\\,,",
     u"V'(\\phi)\\,=\\,\\frac{\\lambda}{8f}{\\vec{E}}\\cdot{\\vec{B}}\\,",
     u'V(\\phi)\\,=\\mu^4e^{\\phi/f},',
     u'V(\\phi)\\,=\\,\\frac{\\lambda}{8}{\\vec{E}}\\cdot{\\vec{B}}\\,',
     u'\\phi(t)\\,\\simeq\\,f{\\rmlog}(\\kappat^{-1})',
     u'\\kappa\\,=\\,\\frac{\\lambda}{8}\\frac{({\\vec{E}}\\cdot{\\vec{B}})(t_i)}{\\mu^4}(\\frac{a(t_i)}{a(t_m)})^4\\frac{\\lambda}{f}|{\\dot\\phi}|\\frac{B(t_i)}{E(t_i)}t_0^2\\,.',
     u'{\\dot\\phi}^2(t)\\,\\simeq\\,\\frac{f^2}{t^2}',
     u'{\\ddot\\phi}\\,\\simeq\\,\\frac{f}{t^2}\\,.',
     u'f\\,\\ll\\,m_{pl}\\,.',
     u'\\Omega_{\\phi}(t_0)\\,\\simeq\\,\\frac{\\lambda}{8}\\frac{({\\vec{E}}\\cdot{\\vec{B}})(t_i)}{\\rho_r(t_i)}\\bigl(\\frac{a(t_{eq})}{a(t_m)}\\bigr)\\frac{\\lambda}{f}|{\\dot\\phi}|\\frac{B(t_i)}{E(t_i)}t_0\\,',
     u'\\frac{({\\vec{E}}\\cdot{\\vec{B}})(t_i)}{\\rho_r(t_i)}\\,\\sim\\,\\frac{n_B}{s}(t_i),\\,',
     u'\\bigl(\\frac{a(t_{eq})}{a(t_m)}\\bigr)\\,\\sim\\,\\bigl[\\frac{m}{T_{eq}}\\frac{m_{pl}}{T_{eq}}\\bigr]^{1/2}',
     u'\\lambda\\,\\sim\\,10^{-4}',
     u'\\partial_{\\mu}\\phi\\cdotJ_{5,B-L}^{\\mu}',
     u'\\phi(F\\wedgeF+\\frac{1}{f}\\sum_{j}m_{j}{\\bar{\\psi}}_{j}\\gamma_{5}\\psi_{j})\\,,',
     u'S\\,=\\,e^{-\\varphi}\\,+\\,ia\\,.',
     u'{\\tildeS}\\,=\\,e^{\\rho}\\,+\\,i{\\tildea}\\,.',
     u'\\[x=y\\iffx-y<\\text{eachfinitequantity}.\\]',
     u'|\\partial\\rho|_{\\omega_{0}}^{2},\\|\\sqrt{-1}\\partial\\bar{\\partial}\\rho|_{\\omega_{0}}\\leq\\frac{C}{r^{2}}',
     u'e^{-\\frac{\\delta}{t}}Q\\leq\\frac{C}{r^{2}}',
     u'\\frac{\\partial}{\\partialt}\\int_M\\omega_{\\varphi_{\\varepsilon}(t)}\\wedge\\eta=\\int_{M}\\frac{\\partial\\omega_{\\varphi_{\\varepsilon}(t)}}{\\partialt}\\wedge\\eta+\\int_M\\omega_{\\varphi_{\\varepsilon}(t)}\\wedge\\frac{\\partial\\eta}{\\partialt}',
     u't^2(R(g_{\\varepsilon,j}(t))-tr_{g_{\\varepsilon,j}(t)}\\theta_{\\varepsilon})\\geq-C',
     u'R(g_{\\varepsilon,j}(t))-tr_{g_{\\varepsilon,j}(t)}\\theta_{\\varepsilon}\\geq-C',
     u'R(g_{\\varepsilon}(t))-tr_{g_{\\varepsilon}(t)}\\theta_{\\varepsilon}\\geq-C',
     u'|a_{\\varepsilon}(t)|\\leqC',
     u'-Ric(\\omega_{\\varepsilon}(t))+\\beta\\omega_{\\varepsilon}(t)+\\theta_{\\varepsilon}=\\sqrt{-1}\\partial\\bar{\\partial}u_{\\varepsilon}(t)',
     u'\\|\\dot{\\psi}_{\\varepsilon}(t)\\|_{C^{0}}\\leqC',
     u"(1-\\delta)\\|x-x'\\|_2^2\\leq\\|\\mathcal{P}x-\\mathcal{P}x'\\|_2^2\\leq(1+\\delta)\\|x-x'\\|_2^2.",
     u"S(\\mathcal{X})=\\left\\{\\dfrac{x-x'}{\\|x-x'\\|_2},x,x'\\in\\mathcal{X},xeqx'\\right\\}",
     u'\\begin{aligned}(1-\\delta)\\|v_i\\|_2^2\\leq\\|\\psiv_i\\|_2^2&\\leq(1+\\delta)\\|v_i\\|_2^2\\\\|\\|\\psiv_i\\|_2^2-\\|v_i\\|_2^2|&\\leq\\delta\\\\|\\|\\psiv_i\\|_2^2-1|&\\leq\\delta\\\\|v_i^TPv_i-1|&\\leq\\delta\\end{aligned}',
     u'\\begin{aligned}&\\underset{P}{\\text{minimize}}&&\\operatorname{rank}(P)\\\\&\\text{subjectto}&&\\|\\mathcal{A}(P)-1_S\\|_\\infty\\leq\\delta\\\\&&&P\\succeq0\\end{aligned}',
     u'\\begin{aligned}&\\underset{P}{\\text{minimize}}&&\\|P\\|_*\\\\&\\text{subjectto}&&\\|\\mathcal{A}(P)-1_S\\|_\\infty\\leq\\delta\\\\&&&P\\succeq0\\end{aligned}',
     u'\\psi=\\Gamma_M^{1/2}U_M^T',
     u'\\begin{aligned}&\\underset{P,L,q}{\\text{minimize}}&&\\|P\\|_*\\\\&\\text{subjectto}&&P=L\\\\&&&\\mathcal{A}(L)=q,\\\\&&&\\|q-1_S\\|_\\infty\\leq\\delta\\\\&&&P\\succeq0\\end{aligned}',
     u'L_A(P,L,q;\\Gamma,\\omega)=\\|P\\|_*+\\frac{\\beta_1}{2}\\|P-L-\\Gamma\\|_F^2+\\frac{\\beta_2}{2}\\|\\mathcal{A}(L)-q-\\omega\\|_2^2',
     u'\\begin{aligned}&\\underset{P,L,q,\\Gamma,\\omega}{\\text{minimize}}&&L_A(P,L,q,\\Gamma,\\omega)\\\\&\\text{subjectto}&&\\|q-1_S\\|_\\infty\\leq\\delta\\\\&&&P\\succeq0\\end{aligned}',
     u'\\begin{aligned}&\\underset{W,H}{\\text{minimize}}&&\\|WH-V\\|_F^2\\\\&\\text{subjectto}&&W_{ij}\\geq0,H_{ij}\\geq0,\\forall~i,j\\\\\\end{aligned}',
     u'\\begin{aligned}&\\underset{P,\\psi}{\\text{minimize}}&&\\frac{1}{2}\\|P-\\psi^T\\psi\\|_F^2\\\\&\\text{subjectto}&&\\|\\mathcal{A}(P)-1_S\\|_\\infty\\leq\\delta\\end{aligned}',
     u'\\begin{aligned}&\\underset{P,X,Y,q}{\\text{minimize}}&&\\frac{1}{2}\\|P-XY\\|_F^2\\\\&\\text{subjectto}&&\\mathcal{A}(P)=q\\\\&&&Y=X^T\\\\&&&\\|q-1_S\\|_\\infty\\leq\\delta\\end{aligned}',
     u'\\begin{multline}L_A(X,Y,q,P)=\\frac{1}{2}\\|P-XY\\|_F^2+\\Gamma\\cdot(\\mathcal{A}(P)-q)+\\Pi\\cdot(Y-X^T)\\\\+\\frac{\\beta_1}{2}\\|\\mathcal{A}(P)-q\\|_2^2+\\frac{\\beta_2}{2}\\|Y-X^T\\|_F^2+\\iota_{\\{q:\\|q-1_S\\|_\\infty\\leq\\delta\\}}\\end{multline}',
     u'\\iota_{\\{q:\\|q-1_S\\|_\\infty\\leq\\delta\\}}=\\begin{cases}0&\\text{if}\\|q-1_S\\|_\\infty\\leq\\delta\\\\\\infty&\\text{otherwise}\\end{cases}',
     u'q_{k+1}\\leftarrow\\underset{q}{\\arg\\min}~\\Gamma\\cdot(\\mathcal{A}(P)-q)+\\frac{\\beta_1}{2}\\|\\mathcal{A}(P)-q\\|_2^2+\\iota_{\\{q:\\|q-\\textbf{1}_S\\|_\\infty\\leq\\delta\\}}',
     u'q_{k+1}=1_S+\\operatorname{sign}(z)\\cdot\\min(|z|,\\delta)',
     u'P_{k+1}\\leftarrow\\underset{P}{\\arg\\min}~\\frac{1}{2}\\|P-XY\\|_F^2+\\Gamma\\cdot(\\mathcal{A}(P)-q)+\\frac{\\beta_1}{2}\\|\\mathcal{A}(P)-q\\|_2^2',
     u'(P-XY)+\\sum_{j=1}^s\\Gamma_jv_jv_j^T+\\beta_1\\mathcal{A}^*(\\mathcal{A}(P)-q)=0',
     u'X_{k+1}\\leftarrow\\underset{X}{\\arg\\min}~\\frac{1}{2}\\|P-XY\\|_F^2+\\Pi\\cdot(Y-X^T)+\\frac{\\beta_2}{2}\\|Y-X^T\\|_F^2',
     u'Y_{k+1}\\leftarrow\\underset{Y}{\\arg\\min}~\\frac{1}{2}\\|P-XY\\|_F^2+\\Pi\\cdot(Y-X^T)+\\frac{\\beta_2}{2}\\|Y-X^T\\|_F^2',
     u'\\begin{aligned}\\Gamma_{k+1}\\leftarrow&\\Gamma_k+\\eta\\beta_1(\\mathcal{A}(P_{k+1})-q_{k+1})\\\\\\Pi_{k+1}\\leftarrow&\\Pi_k+\\eta\\beta_2(Y_{k+1}-X_{k+1}^T)\\end{aligned}',
     u'\\underset{\\psi}{\\text{minimize}}\\|\\mathcal{A}(\\psi^T\\psi)-1_S\\|_\\infty',
     u'\\begin{aligned}\\underset{q,\\psi}{\\text{minimize}}&\\|q-1_s\\|_\\infty\\\\\\text{subjectto}&q=\\mathcal{A}(\\psi^T\\psi)\\end{aligned}',
     u'{\\operatorname{prox}}_f(x)=\\underset{u}{\\arg\\min}(f(u)+\\frac{1}{2}\\|u-x\\|_2^2)',
     u'x={\\operatorname{prox}}_f(x)+{\\operatorname{prox}}_{f^*}(x)',
     u'L_A(\\psi,q;\\omega)=\\|q-1_S\\|_\\infty+\\frac{\\beta}{2}\\|\\mathcal{A}(\\psi^T\\psi)-q-\\omega\\|_2^2',
     u'\\psi_{k+1}\\leftarrow\\underset{\\psi}{\\arg\\min}~\\frac{\\beta}{2}\\|\\mathcal{A}(\\psi^T\\psi)-q-\\omega\\|_2^2',
     u'q_{k+1}\\leftarrow\\underset{q}{\\arg\\min}~\\|q-1_S\\|_\\infty+\\frac{\\beta}{2}\\|\\mathcal{A}(\\psi^T\\psi)-q-\\omega\\|_2^2',
     u'\\begin{aligned}X&=\\frac{1}{\\beta}(\\beta\\tau-{\\operatorname{prox}}_{(\\|X\\|_1\\leq1)}(\\beta\\tau))\\\\q&=\\frac{1}{\\beta}(\\beta\\tau-{\\operatorname{prox}}_{(\\|X\\|_1\\leq1)}(\\beta\\tau))+1_S\\end{aligned}',
     u'\\omega_{k+1}\\leftarrow\\omega_k-\\beta(\\mathcal{A}(\\psi^T_{k+1}\\psi)-q_{k+1})',
     u'\\widehat{S}\\leftarrow\\{v_i\\inS_0:|v_i^T\\widehat{\\psi}^T\\widehat{\\psi}v_i-1|\\geq\\delta\\}',
     u'\\widehat{S}\\leftarrow\\widehat{S}\\bigcup\\{v_i\\inS_1:|v_i^T\\widehat{\\psi}^T\\widehat{\\psi}v_i-1|\\geq\\delta\\}',
     u'\\begin{aligned}&012,015,023,034,045,124^{\\prime},153^{\\prime},13^{\\prime}4^{\\prime},235^{\\prime},24^{\\prime}5^{\\prime},341^{\\prime},onumber\\\\&31^{\\prime}5^{\\prime},452^{\\prime},41^{\\prime}2^{\\prime},52^{\\prime}3^{\\prime},0^{\\prime}1^{\\prime}2^{\\prime},0^{\\prime}1^{\\prime}5^{\\prime},0^{\\prime}2^{\\prime}3^{\\prime},0^{\\prime}3^{\\prime}4^{\\prime},0^{\\prime}4^{\\prime}5^{\\prime}.\\end{aligned}',
     u'\\begin{array}{cl}G=1,~R_\\oplus=1,~m_\\oplus=1\\end{array}',
     u'a,\\;e,\\;{i},\\;\\omega\\;,\\Omega,\\;M,',
     u'\\theta=\\lambda_2-\\lambda_1=M_2-M_1+\\omega_2-\\omega_1,',
     u'{\\bfe}={\\bfv}{{{\\bf\\times}}}({\\bfr}{{{\\bf\\times}}}{\\bfv})/m_\\oplus-{\\bfr}/r,',
     u'\\widetilde{{\\bfe}}={\\bfe}_2-{\\bfe}_1,',
     u'\\dot{{\\bfx}}={\\bfw},\\\\\\\\dot{{\\bfw}}=-m_\\oplus({\\bfx}/r^3-3{\\bfr}\\cdot{\\bfx}\\:{\\bfr}/r^5),',
     u'{\\bfx}=\\sum{\\frac{\\partial{{\\bfr}}}{\\partial{q_k}}}\\deltaq_k\\\\{\\rmand}\\{\\bfw}=\\sum{\\frac{\\partial{{\\bfv}}}{\\partial{q_k}}}\\deltaq_k.',
     u'{\\bfx}_a=2{\\bfr}-3\\:t\\:{\\bfv},',
     u'a(2e\\sin(M)+\\theta,1-e\\cos(M)),',
     u'(\\Deltax,\\Deltay)=a(2e\\sin(M)+\\theta,-e\\cos(M))+\\vecO(\\varepsilon^2),',
     u'\\Delta^2=a^2((\\theta+2e\\sin(M))^2+e^2\\cos^2(M))+O(\\varepsilon^2).',
     u'H=K_1+K_2-R_1-R_2-R_{12},',
     u'K_u={\\bfp}_u^2/2-\\frac{m_\\oplus}{r_u},',
     u'R=\\sum_{n=1}^{N_x}\\sum_{m=0}^n\\frac{P_{n}^m(\\cos(\\theta))}{r^{n+1}}(C_n^m\\cos(m\\psi)+S_n^m\\sin(m\\psi))+R_{ls}.',
     u'R_u=\\frac{m_\\oplus}{r_u}(\\frac{J_2R_\\oplus^2}{2r_u^2}(1-3(\\frac{z_u}{r_u})^2),',
     u'\\Delta=|{\\bfr}_2-{\\bfr}_1|.',
     u'\\langleK_u-R_u\\rangle=-\\frac{m_\\oplus}{2a_u}-\\frac{{J_2R_\\oplus^2}m_\\oplus\\left(2-3\\sin^2(i_u)\\right)}{4a_u^3\\left(1-e_u^2\\right)^{3/2}.}',
     u'R_{12}(\\Delta)={\\epsilon}\\Delta,',
     u'\\Delta^2\\approxa^2({\\widetildee}^2\\cos(w)^2+(\\theta+2{\\widetildee}\\sin(w))^2)',
     u'{\\widetildee}=|{\\bfe}_2-{\\bfe}_1|,\\\\\\w=M+{\\rmconstant}.',
     u'R=\\sum_{u=1}^2\\frac{{J_2}m_\\oplusR_\\oplus^2\\left(2-3\\sin^2(i_u)\\right)}{4a_u^3\\left(1-e_u^2\\right)^{3/2}}+{\\epsilon}<\\Delta>.',
     u'\\dot\\theta=\\dot\\lambda_2-\\dot\\lambda_1=\\dotM_2+\\dot\\omega_2-\\dotM_1-\\dot\\omega_1.',
     u'\\dot\\theta\\approxn_2-n_1\\approx-\\frac32n\\alpha',
     u'\\ddot\\theta\\approx-\\frac{3}{a^2}({\\frac{\\partial{R}}{\\partial{M_2}}}-{\\frac{\\partial{R}}{\\partial{M_1}}})=-\\frac{6}{a^2}{\\frac{\\partial{R}}{\\partial{\\theta}}},',
     u'\\langleR_{12}\\rangle=\\frac{1}{2\\pi}\\int_0^{2\\pi}R_{12}(\\Delta(w))\\:dw',
     u'\\langleR_{12}\\rangle=R_{12}^{(0)}+R_{12}^{(2)}\\theta^2/2+..,',
     u'\\ddot\\theta=-\\frac{6}{a^2}{\\frac{\\partial{R_{12}}}{\\partial{\\theta}}}\\approx\\frac{-1.711{\\epsilon}}{a{\\widetildee}}\\theta,',
     u'\\langleR\\rangle={\\epsilon}_n\\frac{1}{2\\pi}\\int_0^{2\\pi}\\Delta^n(w)\\:dw.',
     u'\\sigma=G_0\\gamma\\,,',
     u'\\vec{f}^{\\rmel}_{ij}=\\begin{cases}-k(\\delta_{ij})\\,\\delta_{ij}\\,\\hat{r}_{ij}\\,,&\\delta_{ij}>0\\\\{\\vec0},&\\delta_{ij}<0.\\end{cases}',
     u'k=k_0\\,\\delta^{\\alpha-2}\\,.',
     u'\\vec{f}^{\\rmvisc}_{ij}=-\\tau_0\\,k(\\delta_{ij})\\,\\Delta{\\vecv}^{\\,c}_{ij}\\,,',
     u'\\sigma_{\\alpha\\beta}=\\frac{1}{2V}\\sum_{ij}f_{ij,\\alpha}r_{ij,\\beta}-\\frac{1}{V}\\sum_im_iv_{i,\\alpha}v_{i,\\beta}\\,,',
     u'\\frac{p}{k}\\sim\\Delta\\phi\\sim\\Deltaz^2\\,.',
     u'G_0/k\\sim(p/k)^{\\mu}\\,,',
     u'\\gamma_{\\rmcc}^{(1)}\\sim\\frac{(p/k)^{1/2}}{N}\\,.',
     u'\\gamma(t)=\\left\\lbrace\\begin{array}{cl}0&t<0\\\\\\gamma_0&t\\ge0\\,.\\end{array}\\right.',
     u'G_r(t,\\gamma_0)\\equiv\\frac{\\sigma(t,\\gamma_0)}{\\gamma_0}\\,.',
     u'{\\tau^*}\\sim\\left(\\frac{k}{p}\\right)^\\lambda\\tau_0\\,\\,\\,{\\rmas}\\,\\,\\,N\\rightarrow\\infty\\,.',
     u'\\frac{G_r(t)}{k}\\sim\\left\\lbrace\\begin{array}{cc}\\left({\\tau_0}/{t}\\right)^{\\theta}&1\\llt/\\tau_0\\ll({k}/p)^\\lambda\\\\(p/k)^{\\mu}&({k}/p)^\\lambda\\llt/\\tau_0\\,.\\end{array}\\right.',
     u'\\gamma(t)=\\left\\lbrace\\begin{array}{cl}0&t<0\\\\\\dot\\gamma_0t&t\\ge0\\end{array}\\right.',
     u'\\gamma^\\dag\\sim\\frac{p}{k}\\simp^{1/(\\alpha-1)}\\,,',
     u'n_{\\rmcc}(\\gamma)\\simeq\\frac{1}{N}\\left(\\frac{\\gamma}{\\gamma_{\\rmcc}^{(1)}}\\right)\\,\\,\\,\\,\\,\\,{\\rmas}\\,\\,\\,\\,\\,\\,\\gamma\\rightarrow0\\,.',
     u'n_{\\rmcc}^\\dag\\equivn_{\\rmcc}(\\gamma^\\dag)\\sim\\Deltaz\\,.',
     u"\\frac{\\sigma}{\\gamma}=\\frac{1}{t}\\,\\int_0^{t}G_r(t')\\,{\\rmd}t'\\,.",
     u'\\frac{\\sigma}{\\gamma}\\simk\\left(\\frac{\\tau_0}{t}\\right)^{\\theta},\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\tau_0<t<\\tau^*\\,,',
     u'\\dot\\gamma^\\dag\\sim\\frac{(p/k)^{1+\\lambda}}{\\tau_0}\\,,',
     u'\\frac{d{\\bfv}}{dt}+f\\hat{\\bfk}\\times{\\bfv}+abla\\Phi=\\mathcal{F}_{\\mathrm{drag}}+\\mathcal{D}_{\\mathrm{S}}\\mathrm{,}',
     u'\\frac{\\partial\\Phi}{\\partialp}+\\frac{1}{\\rho}=0\\mathrm{,}',
     u'abla\\cdot{\\bfv}+\\frac{\\partial\\omega}{\\partialp}=0\\mathrm{,}',
     u'T\\frac{d(\\mathrm{ln}\\theta)}{dt}=\\frac{dT}{dt}-\\frac{\\omega}{\\rhoc_{p}}=\\frac{q}{c_{p}}+\\mathcal{E}_{\\mathrm{S}}\\mathrm{,}',
     u'p=\\rhoRT\\mathrm{.}',
     u'\\frac{q}{c_{p}}=\\frac{T_{\\mathrm{eq}}(\\lambda,\\phi,p)-T(\\lambda,\\phi,p,t)}{\\tau_{\\mathrm{rad}}(p)}\\mathrm{.}',
     u'\\everymath=\\expandafter{\\displaystyle}T_{\\mathrm{eq}}\\left(\\lambda,\\phi,p\\right)=\\begin{cases}T_{\\mathrm{night,eq}}(p)+\\DeltaT_{\\mathrm{eq}}(p)\\mathrm{cos}\\lambda\\mathrm{cos}\\phi\\\\mathrm{dayside,}\\\\T_{\\mathrm{night,eq}}(p)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathrm{nightside.}\\end{cases}',
     u'\\everymath=\\expandafter{\\displaystyle}\\DeltaT_{\\mathrm{eq}}(p)=\\begin{cases}\\DeltaT_{\\mathrm{eq,top}}\\hspace{3.2cm}p<p_{\\mathrm{eq,top}}\\\\\\DeltaT_{\\mathrm{eq,top}}\\frac{\\mathrm{ln}(p/p_{\\mathrm{bot}})}{\\mathrm{ln}(p_{\\mathrm{eq,top}}/p_{\\mathrm{bot}})}\\hspace{0.8cm}p_{\\mathrm{eq,top}}<p<p_{\\mathrm{bot}}\\\\0\\hspace{2pt}\\mathrm{Kelvin}\\hspace{3.2cm}p>p_{\\mathrm{bot}}\\mathrm{.}\\end{cases}',
     u'\\everymath=\\expandafter{\\displaystyle}\\tau_{\\mathrm{rad}}(p)=\\begin{cases}\\tau_{\\mathrm{rad,top}}\\hspace{2cm}p<p_{\\mathrm{rad,top}}\\\\\\tau_{\\mathrm{rad,bot}}\\left(\\frac{p}{p_{\\mathrm{bot}}}\\right)^\\alpha\\hspace{.5cm}p_{\\mathrm{rad,top}}<p<p_{\\mathrm{bot}}\\\\\\tau_{\\mathrm{rad,bot}}\\hspace{2cm}p>p_{\\mathrm{bot}}\\mathrm{,}\t\\end{cases}',
     u'\\alpha=\\frac{\\mathrm{ln}(\\tau_{\\mathrm{rad,top}}/\\tau_{\\mathrm{rad,bot}})}{\\mathrm{ln}(p_{\\mathrm{rad,top}}/p_{\\mathrm{bot}})}\\mathrm{.}',
     u'\\mathcal{F}_{\\mathrm{drag}}=-k_v(p){\\bfv}\\mathrm{,}',
     u'k_v(p)=\\max\\left[\\tau_{\\rmdrag}^{-1},\\tau_{\\mathrm{drag,bot}}^{-1}\\frac{(p-p_{\\mathrm{drag,top}})}{(p_{\\mathrm{drag,bot}}-p_{\\mathrm{drag,top}})}\\right]',
     u'U_{\\mathrm{rms}}(p)=\\sqrt{\\frac{\\int(u^2+v^2)dA}{A}}\\mathrm{.}',
     u'{\\bfv}\\cdotabla{\\bfv}+w^{\\star}\\frac{\\partial{\\bfv}}{\\partialz^{\\star}}+f{\\bfk}\\times{\\bfv}=-abla\\Phi-\\frac{{\\bfv}}{\\tau_{\\mathrm{drag}}}\\mathrm{,}',
     u'\\frac{\\partial\\Phi}{\\partialz^{\\star}}=RT\\mathrm{,}',
     u'abla\\cdot{\\bfv}+e^{z^\\star}\\frac{\\partial(e^{-z^\\star}w^{\\star})}{\\partialz^{\\star}}=0\\mathrm{,}',
     u'{\\bfv}\\cdotablaT+w^{\\star}\\frac{N^2H^2}{R}=\\frac{T_{\\mathrm{eq}}-T}{\\tau_{\\mathrm{rad}}}\\mathrm{.}',
     u'z^{\\star}\\equiv-\\mathrm{ln}\\frac{p}{p_{\\mathrm{00}}}\\mathrm{,}',
     u'\\frac{\\mathcal{U}}{\\mathcal{L}}\\sim\\max\\left[w^\\star,\\frac{w^\\star}{\\Deltaz^\\star}\\right].',
     u'\\frac{\\mathcal{U}}{\\mathcal{L}}\\sim\\frac{\\mathcal{W}}{H}.',
     u'\\left|T_{\\mathrm{eq}}-T\\right|_{\\mathrm{global}}\\equiv\\left|T_{\\mathrm{eq}}-T\\right|_{\\mathrm{day}}+\\left|T_{\\mathrm{eq}}-T\\right|_{\\mathrm{night}}\\mathrm{,}',
     u'\\DeltaT_{\\mathrm{eq}}-\\DeltaT\\sim\\left|T_{\\mathrm{eq}}-T\\right|_{\\mathrm{global}}\\mathrm{.}',
     u'\\frac{\\DeltaT_{\\mathrm{eq}}-\\DeltaT}{\\tau_{\\mathrm{rad}}}\\sim\\mathrm{max}\\left[\\frac{\\mathcal{U}\\DeltaT}{\\mathcal{L}}\\mathrm{,}\\frac{\\mathcal{W}N^2H}{R}\\right]\\mathrm{.}',
     u'\\frac{\\DeltaT_{\\mathrm{eq}}-\\DeltaT}{\\tau_{\\mathrm{rad}}}\\sim\\frac{\\mathcal{W}N^2H}{R}.',
     u"\\Delta\\Phi\\approxR\\int_p^{p_{\\mathrm{bot}}}\\DeltaTd\\mathrm{ln}p'\\mathrm{.}",
     u'\\DeltaT=\\DeltaT_{\\mathrm{top}}\\frac{\\mathrm{ln}(p/p_{\\mathrm{bot}})}{\\mathrm{ln}(p_{\\mathrm{eq,top}}/p_{\\mathrm{bot}})}\\mathrm{.}',
     u'\\Delta\\Phi\\approxR\\DeltaT\\mathrm{ln}\\left(\\frac{p_{\\mathrm{bot}}}{p}\\right)\\mathrm{.}',
     u'abla\\Phi\\sim\\mathrm{max}\\left[\\frac{\\mathcal{U}^2}{\\mathcal{L}}\\mathrm{,}\\frac{\\mathcal{U}\\mathcal{W}}{H}\\mathrm{,}f\\mathcal{U}\\mathrm{,}\\frac{\\mathcal{U}}{\\tau_{\\mathrm{drag}}}\\right]\\mathrm{.}',
     u'\\everymath=\\expandafter{\\displaystyle}\\mathcal{U}(p)\\sim\\begin{cases}\\frac{R\\DeltaT(p)\\Delta\\mathrm{ln}p\\\\tau_{\\mathrm{drag}}(p)}{\\mathcal{L}}\\hspace{0.8cm}\\mathrm{Drag}\\\\\\frac{R\\DeltaT(p)\\Delta\\mathrm{ln}p}{\\mathcal{L}f}\\hspace{2.cm}\\mathrm{Coriolis}\\\\\\sqrt{R\\DeltaT(p)\\Delta\\mathrm{ln}p}\\hspace{1.8cm}\\mathrm{Advection}\\mathrm{,}\\end{cases}',
     u'\\everymath=\\expandafter{\\displaystyle}\\mathcal{W}(p)\\sim\\begin{cases}\\frac{R\\DeltaT(p)H(p)\\Delta\\mathrm{ln}p\\\\tau_{\\mathrm{drag}}(p)}{\\mathcal{L}^2}\\hspace{0.8cm}\\mathrm{Drag}\\\\\\frac{R\\DeltaT(p)H(p)\\Delta\\mathrm{ln}p}{\\mathcal{L}^2f}\\hspace{2.cm}\\mathrm{Coriolis}\\\\{H(p)\\over\\mathcal{L}}\\sqrt{R\\DeltaT(p)\\Delta\\mathrm{ln}p}\\hspace{1.7cm}\\mathrm{Advection}\\mathrm{.}\\end{cases}',
     u'\\everymath=\\expandafter{\\displaystyle}\\frac{\\DeltaT(p)}{\\DeltaT_{\\mathrm{eq}}(p)}\\sim\t\\begin{cases}\\left(1+\\frac{\\tau_{\\mathrm{rad}}(p)\\tau_{\\mathrm{drag}}(p)}{\\tau_{\\mathrm{wave}}^2(p)}\\Delta\\mathrm{ln}p\\right)^{-1}\t\\hspace{.0cm}\\mathrm{Drag}\\\\\\left(1+\\frac{\\tau_{\\mathrm{rad}}(p)}{f\\tau_{\\mathrm{wave}}^2(p)}\\Delta\\mathrm{ln}p\\right)^{-1}\\hspace{0.7cm}\\mathrm{Coriolis}\\\\\\frac{\\sqrt{\\gamma(p)+4\\DeltaT_{\\mathrm{eq}}(p)}-\\sqrt{\\gamma(p)}}{\\sqrt{\\gamma(p)+4\\DeltaT_{\\mathrm{eq}}(p)}+\\sqrt{\\gamma(p)}}\\hspace{.3cm}\\mathrm{Advection}\\mathrm{,}\\end{cases}',
     u'\\begin{aligned}\\everymath=\\expandafter{\\displaystyle}\\mathcal{W}\\sim\t\\begin{cases}\\frac{RH(p)}{\\mathcal{L}^2}\\frac{\\tau_{\\mathrm{wave}}^2(p)}{\\tau_{\\mathrm{rad}}(p)}\\DeltaT_{\\mathrm{eq}}(p)\\\\\\times\\left[1-\\left(1+\\frac{\\tau_{\\mathrm{rad}}(p)\\tau_{\\mathrm{drag}}(p)}{\\tau_{\\mathrm{wave}}^2(p)}\\Delta\\mathrm{ln}p\\right)^{-1}\\right]\t\\hspace{.0cm}\\mathrm{Drag}\\\\\\\\\\frac{RH(p)}{\\mathcal{L}^2}\\frac{\\tau_{\\mathrm{wave}}^2(p)}{\\tau_{\\mathrm{rad}}(p)}\\DeltaT_{\\mathrm{eq}}(p)\\\\\\times\\left[1-\\left(1+\\frac{\\tau_{\\mathrm{rad}}(p)}{f\\tau_{\\mathrm{wave}}^2(p)}\\Delta\\mathrm{ln}p\\right)^{-1}\\right]\\hspace{0.7cm}\\mathrm{Coriolis}\\\\\\\\\\frac{RH(p)}{\\mathcal{L}^2}\\frac{\\tau_{\\mathrm{wave}}^2(p)}{\\tau_{\\mathrm{rad}}(p)}\\DeltaT_{\\mathrm{eq}}(p)\\\\\\times\\left[1-\\frac{\\sqrt{\\gamma(p)+4\\DeltaT_{\\mathrm{eq}}(p)}-\\sqrt{\\gamma(p)}}{\\sqrt{\\gamma(p)+4\\DeltaT_{\\mathrm{eq}}(p)}+\\sqrt{\\gamma(p)}}\\right]\\hspace{.3cm}\\mathrm{Advection}\\mathrm{.}\\end{cases}\\end{aligned}',
     u'\\begin{aligned}\\tau_{\\mathrm{adv}}(p)=&\\frac{\\mathcal{L}^2\\tau_{\\mathrm{rad}}(p)}{R\\tau^2_{\\mathrm{wave}}(p)\\DeltaT_{\\mathrm{eq}}(p)}\\\\&\\times\\left[1-\\frac{\\sqrt{\\gamma(p)+4\\DeltaT_{\\mathrm{eq}}(p)}-\\sqrt{\\gamma(p)}}{\\sqrt{\\gamma(p)+4\\DeltaT_{\\mathrm{eq}}(p)}+\\sqrt{\\gamma(p)}}\\right]^{-1}\\mathrm{.}\\end{aligned}',
     u'\\everymath=\\expandafter{\\displaystyle}A(p)\\sim\\begin{cases}\\left(1+\\frac{\\tau_{\\mathrm{rad}}(p)\\tau_{\\mathrm{drag}}(p)}{\\tau_{\\mathrm{wave}}^2(p)}\\Delta\\mathrm{ln}p\\right)^{-1}\\hspace{0.05cm}\\tau_{\\mathrm{drag}}<f^{-1}\\\\\\left(1+\\frac{\\tau_{\\mathrm{rad}}(p)}{f\\tau_{\\mathrm{wave}}^2(p)}\\Delta\\mathrm{ln}p\\right)^{-1}\\hspace{0.8cm}\\tau_{\\mathrm{drag}}>f^{-1}\\mathrm{.}\\end{cases}',
     u'\\everymath=\\expandafter{\\displaystyle}A_{\\mathrm{eq}}(p)\\sim\\begin{cases}\\\\\\left(1+\\frac{\\tau_{\\mathrm{rad}}(p)\\tau_{\\mathrm{drag}}(p)}{\\tau_{\\mathrm{wave}}^2(p)}\\Delta\\mathrm{ln}p\\right)^{-1}\\hspace{-0.3cm}\\tau_{\\mathrm{drag}}(p)<\\tau_{\\mathrm{adv}}(p)\\\\\\\\\\frac{\\sqrt{\\gamma(p)+4\\DeltaT_{\\mathrm{eq}}(p)}-\\sqrt{\\gamma(p)}}{\\sqrt{\\gamma(p)+4\\DeltaT_{\\mathrm{eq}}(p)}+\\sqrt{\\gamma(p)}}\\hspace{0cm}\\tau_{\\mathrm{drag}}(p)>\\tau_{\\mathrm{adv}}(p)\\mathrm{,}\\end{cases}',
     u'\\everymath=\\expandafter{\\displaystyle}\\tau_{\\mathrm{wave}}(p)\\sim\\begin{cases}\\sqrt{\\tau_{\\mathrm{drag}}(p)\\tau_{\\mathrm{rad}}(p)\\Delta\\mathrm{ln}p}\\hspace{0.5cm}\\tau_{\\mathrm{drag}}(p)<f^{-1}\\\\\\sqrt{f^{-1}\\tau_{\\mathrm{rad}}(p)\\Delta\\mathrm{ln}p}\\hspace{1.05cm}\\tau_{\\mathrm{drag}}(p)>f^{-1}\\mathrm{.}\\end{cases}',
     u'\\tau_{\\mathrm{vert}}\\sim\\frac{\\DeltaTR}{\\mathcal{W}N^2H}\\mathrm{.}',
     u'\\tau_{\\mathrm{vert}}\\sim\\frac{\\tau^2_{\\mathrm{wave}}}{\\tau_{\\mathrm{drag}}\\Delta\\mathrm{ln}p}\\mathrm{.}',
     u'\\tau_{\\mathrm{rad}}\\sim\\frac{\\tau^2_{\\mathrm{wave}}}{\\tau_{\\mathrm{drag}}\\Delta\\mathrm{ln}p}',
     u'\\tau_{\\mathrm{vert}}\\sim\\tau_{\\mathrm{rad}}\\sim\\frac{f\\tau^2_{\\mathrm{wave}}}{\\Delta\\mathrm{ln}p}\\mathrm{.}',
     u'A(\\phi,p)=\\left\\{\\frac{\\int_{0}^{2\\pi}\\left[T(\\lambda,\\phi,p)-\\bar{T}(\\phi,p)\\right]^2d\\lambda}{\\int_{0}^{2\\pi}\\left[T_{\\mathrm{eq}}(\\lambda,\\phi,p)-\\bar{T}(\\phi,p)\\right]^2d\\lambda}\t\\right\\}^{1/2}\\mathrm{.}',
     u'A(p)=\\frac{3}{2\\pi}\\int_{-\\pi/3}^{\\pi/3}A(\\phi,p)d\\phi{\\mathrm{.}}',
     u'W_{\\mathrm{rms}}(p)=\\sqrt{\\frac{\\intw^2dA}{A}}\\mathrm{,}',
     u'\\frac{E_v}{E_h}\\approx\\frac{W_{\\mathrm{rms}}N^2H}{R}\\left(\\frac{U_{\\mathrm{rms}}A\\DeltaT_{\\mathrm{eq}}}{\\mathcal{L}}\\right)^{-1}\\mathrm{.}',
     u'A_{\\mathrm{obs,ll}}=\\frac{F_{\\mathrm{day}}^{1/4}-F_{\\mathrm{night,max}}^{1/4}}{F_{\\mathrm{day}}^{1/4}}\\mathrm{.}\\vspace{0.01cm}',
     u'R=\\frac{\\theta_{int}}{t_{int}}=\\frac{d\\alpha}{dt}\\cos\\,\\delta_0',
     u'\\alpha_{i}=\\alpha_{0}+\\frac{R}{\\cos\\,\\delta_{0}}\\cdot(t_{i}-t_{0})',
     u'T_{A}=T_{sys}\\cdot\\frac{ON-OFF}{OFF}',
     u'\\small(i-\\gamma)\\psi_{t}=[-\\frac{1}{2}(\\partial_{x}^2+\\partial_{y}^2)+V(x,y)-\\mu+p|\\psi|^2-\\OmegaL_{z}]\\psi\\eqno{(1)}',
     u'\\largeJ_{i}=\\sum\\limits_{i=1}^N\\sum\\limits_{j=1}^cu_{ij}^m{||x_i-v_j||}^2\\\\',
     u'\\begin{aligned}\\large\\sum\\limits_{j=1}^cu_{ij}=1,1\\leqi\\leqn\\\\0<\\sum\\limits_{i=1}^nu_{ij}<n,1\\leqj\\leqc\\\\\\sum\\limits_{i=1}^c\\sum\\limits_{i=1}^nu_{ij}=n.\\end{aligned}',
     u'\\largev_{i}=\\frac{\\sum\\limits_{i=1}^Nu_{ij}^m.x_{i}}{\\sum\\limits_{i=1}^Nu_{ij}^m}',
     u'\\largeu_{ij}=\\frac{1}{\\sum\\limits_{k=1}^c\\bigg[{\\frac{||x_{i}-v_{j}||}{||x_{i}-v_{k}||}\\bigg]^\\frac{2}{m-1}}}',
     u'\\largeDSC=\\frac{2(PR\\capGT)}{PR+GT}',
     u'|H\\rangle=\\pi^{-}p-\\frac{1}{\\sqrt{2}}\\pi^{0}n+\\frac{1}{\\sqrt{3}}\\etan+\\sqrt{\\frac{2}{3}}K^{0}\\Lambda.',
     u'T=V_{p}[h_{K^{0}\\Lambda}+\\sum_{i}h_{i}G_{i}(M_{K^{0}\\Lambda})t_{i\\rightarrowK^{0}\\Lambda}(M_{K^0\\Lambda})],',
     u'h_{\\pi^{-}p}=1,h_{\\pi^{0}n}=-\\frac{1}{\\sqrt{2}},h_{\\etan}=\\frac{1}{\\sqrt{3}},h_{K^{0}\\Lambda}=\\sqrt{\\frac{2}{3}}.',
     u't_{J/\\psi\\Lambda\\rightarrowJ/\\psi\\Lambda}=\\frac{g_{J/\\psi\\Lambda}^2}{M_{J/\\psi\\Lambda}-M+i\\Gamma/2}.',
     u'\\frac{d^{2}\\Gamma}{dM_{J/\\psi\\Lambda}^{2}dM_{K^0\\Lambda}^{2}}=\\frac{1}{(2\\pi)^{3}}\\frac{4M_{\\Lambda_{b}}M_{\\Lambda}}{32M_{\\Lambda_{b}}^{3}}\\bar{\\sum}\\sum|\\mathcal{M}(M_{J/\\psi\\Lambda},M_{K^0\\Lambda})|^{2},',
     u'T_{N^{*}(1895)}=\\frac{\\alphaM_{N^{*}(1895)}}{M_{K\\Lambda}-M_{N^{*}(1895)}+i\\Gamma_{N^{*}(1895)}/2}',
     u'|\\mathcal{M}|^{2}\\rightarrow|\\mathcal{M}|^{2}+\\beta\\tilde{P_{K}}^{2}\\left|\\frac{1}{M_{K\\Lambda}-M_{N^{*}(1900)}+i\\Gamma_{N^{*}(1900)}/2}\\right|^{2},',
     u'ds_{6}^{2}=\\sigma(r)\\eta_{\\muu}dx^{\\mu}dx^{u}-dr^{2}-\\gamma(r)d\\theta^{2}\\,',
     u'M^2_P=2\\piM_6^4\\int_0^{\\infty}{\\sigma(r)\\sqrt{\\gamma(r)}dr}\\.',
     u'\\sigma_{_{SD}}(r)={\\operatorname{e}}^{-cr},\\quad\\gamma_{_{SD}}(r)=R_{0}^{2}\\sigma_{_{SD}}(r)\\,',
     u'\\sigma_{_{TA}}(r)=\\cosh^{-2\\delta}{\\left(\\frac{\\betar}{\\delta}\\right)},\\quad\\gamma_{_{TA}}(r)=R_{0}^{2}\\sigma_{_{TA}}(r)\\,',
     u'\\delta<\\frac{5\\beta}{4\\pi}q^2v^2\\.',
     u'\\frac{\\partial}{\\partial\\lambda}g_{MN}(\\lambda)=-2R_{MN}(\\lambda)\\,onumber',
     u'\\!\\!\\!\\!\\sigma_{_{HC}}(r)={\\operatorname{e}}^{-cr+\\tanh(cr)},\\;\\;\\gamma_{_{HC}}(r)=\\frac{\\tanh^2{cr}}{c^2}\\sigma_{_{HC}}(r)\\,.',
     u'S(\\tilde{f})=-\\int_{0}^{\\infty}{d\\omega}\\tilde{f}(\\omega)\\mbox{ln}\\left[\\tilde{f}(\\omega)\\right].',
     u't_{0}(r)=\\frac{1}{\\kappa}\\left(\\frac{5}{2}+\\frac{1}{\\beta}\\right)\\left[2\\beta{\\operatorname{sech}}{\\left(\\frac{\\betar}{\\delta}\\right)}\\right]^{2}.',
     u'\\mathcal{F}(\\omega)=\\sqrt{2\\pi}\\delta\\omega(5\\delta+2){\\operatorname{csch}}\\left(\\frac{\\pi\\delta\\omega}{2\\beta}\\right),',
     u'\\tilde{f}(\\omega)=\\left[\\frac{\\pi\\delta\\omega}{2\\beta}\\\\mbox{csch}\\left(\\frac{\\pi\\delta\\omega}{2\\beta}\\right)\\right]^{2}\\,.',
     u'\\beta=\\sqrt{\\frac{(-\\Lambda)\\kappa}{10}},\\quad\\text{with}\\qquad0<\\beta\\leq\\frac{1}{2}\\,',
     u'0.09\\beta<\\delta<0.40\\beta.\\,',
     u'\\!\\!\\!\\!\\!\\tilde{f}(\\omega)\\!=\\!\\frac{\\pi^{2}\\omega^{2}\\left(4096c^{4}\\!+\\!881c^{2}\\omega^{2}\\!+\\!25\\omega^{4}\\right)}{16384c^{6}}{\\operatorname{csch}}^{2}\\left(\\frac{\\pi\\omega}{2c}\\right).',
     u'c=\\sqrt{\\frac{2\\kappa}{5}\\left(-\\Lambda\\right)},\\quad\\text{with}\\qquad0<c\\leq1\\,',
     u'M^2_P\\approx\\frac{4\\piR_0}{3}\\frac{1}{c}M_6^4\\,',
     u'r_i({\\mathbbX}):=r_{i}(\\underbrace{{\\mathbbX},\\dots,{\\mathbbX}}_{i-times})=\\frac{d^i}{dt^i}\\Big|_{t=0}log(g(t)).',
     u'\\sum_{i=0}^\\infty\\frac{m_i}{i!}t^i=g(t)=exp\\Big(\\sum_{i=1}^\\infty\\frac{r_i}{i!}t^i\\Big)',
     u'r_1=m_1,\\qquadr_2=m_2-m_1^2=\\sigma,\\qquadr_3=m_3-3m_2m_1+2m_1^3.',
     u'r_{i_1+\\dots+i_n}(\\underbrace{{\\mathbbX}_1,\\dots,{\\mathbbX}_1}_{i_1-times},\\dots,\\underbrace{{\\mathbbX}_n,\\dots,{\\mathbbX}_n}_{i_n-times})=\\frac{d^{i_1+\\dots+i_n}}{dt_1^{i_1}\\dotsdt_n^{i_n}}\\Big|_{t=0}log(g(t_1,\\dots,t_n)),',
     u'\\begin{aligned}&h_k(a^2+b^2)=r_k(a{\\mathbbS}_1+{\\mathbbY}+b{\\mathbbS}_2+{\\mathbbZ})-r_k({\\mathbbY}+{\\mathbbZ})=r_k(a{\\mathbbS}_1+{\\mathbbY})-r_k({\\mathbbY})+r_k(b{\\mathbbS}_2+{\\mathbbZ})-r_k({\\mathbbZ}).\\end{aligned}',
     u'h_k(a^2+b^2)=h_k(a^2)+h_k(b^2).',
     u'\\varphi_k(\\alpha,\\beta)=h_k(\\alpha^2+\\beta^2)=r_k(\\alpha{\\mathbbS}_1+{\\mathbbY}+\\beta{\\mathbbS}_2+{\\mathbbZ})-r_k({\\mathbbY}+{\\mathbbZ}),',
     u'h_k(a^2)=a^2\\varphi_k(1,0),',
     u'\\begin{aligned}r_k(a{\\mathbbS}_1+{\\mathbbY})-r_k({\\mathbbY})=a^2(r_k({\\mathbbS}_1+{\\mathbbY})-r_k({\\mathbbY})).\\end{aligned}',
     u'2acov({\\mathbbS}_1,{\\mathbbY})+a^2Var({\\mathbbS}_1)=a^2(2cov({\\mathbbS}_1,{\\mathbbY})+Var({\\mathbbS}_1)),',
     u'\\begin{aligned}\\sum_{i=1}^ka^{i}{k\\choosei}r_k(\\underbrace{{\\mathbbS}_1,\\dots,{\\mathbbS}_1}_{i-times},\\underbrace{{\\mathbbY},\\dots,{\\mathbbY}}_{k-i-times})=a^2(r_k({\\mathbbS}_1+{\\mathbbY})-r_k({\\mathbbY})),\\end{aligned}',
     u'\\begin{aligned}{\\mathbbS}_1=\\frac{\\sum_{i=1}^ma_i{\\mathbbX}_i}{\\sqrt{\\sum_{i=1}^ma_i^2}}\\textrm{and}{\\mathbbS}_2=\\frac{\\sum_{i=m+1}^na_i{\\mathbbX}_i}{{\\sqrt{\\sum_{i=m+1}^na_i^2}}}\\end{aligned}',
     u'{\\mathbbS}_1a+{\\mathbbY}+{\\mathbbS}_2b+{\\mathbbZ}=\\sum_{i=1}^ma_i{\\mathbbX}_i+{\\mathbbY}+\\sum_{i=m+1}^na_i{\\mathbbX}_i+{\\mathbbZ},',
     u'a_1{\\mathbbX}_1+{\\mathbbX}_2+{\\mathbbY}+a_{m+1}{\\mathbbX}_{m+1}+{\\mathbbZ}',
     u'\\sum_{i=1}^ma_i{\\mathbbX}_i+{\\mathbbY}+\\sum_{i=m+1}^na_i{\\mathbbX}_i+{\\mathbbZ}=\\sum_{i=1}^n({\\mathbbX}_i+a_i/2)^2-\\frac{1}{4}\\times\\sum_{i=1}^na_i^2.',
     u'\\begin{aligned}\\sum_{i=1}^ka^{i}{k\\choosei}r_k(\\underbrace{{\\mathbbX},\\dots,{\\mathbbX}}_{i-times},\\underbrace{{\\mathbbY},\\dots,{\\mathbbY}}_{k-i-times})=a^2(r_k({\\mathbbX}+{\\mathbbY})-r_k({\\mathbbY})).\\end{aligned}',
     u'\\begin{aligned}r_k(\\underbrace{{\\mathbbX},\\dots,{\\mathbbX}}_{i-times},\\underbrace{{\\mathbbY},\\dots,{\\mathbbY}}_{l-times})=0\\end{aligned}',
     u'r_k(\\underbrace{{\\mathbbX},\\dots,{\\mathbbX}}_{l-times},\\underbrace{{\\mathbbY},\\dots,{\\mathbbY}}_{i-times})=0',
     u'S(\\Delta\\eta,\\Delta\\phi)=\\frac{1}{N_\\text{jets}}\\frac{\\rd^{2}N^\\text{same}}{\\rd\\Delta\\eta\\,\\rd\\Delta\\phi}.\\',
     u'ME(\\Delta\\eta,\\Delta\\phi)=\\frac{1}{N_\\text{jets}}\\frac{\\rd^{2}N^\\text{mix}}{\\rd\\Delta\\eta\\,\\rd\\Delta\\phi}.\\',
     u'\\frac{1}{N_\\text{jets}}\\frac{\\rd^{2}N}{\\rd\\Delta\\eta\\,\\rd\\Delta\\phi}=ME(0,0)\\,\\frac{S(\\Delta\\eta,\\Delta\\phi)}{ME(\\Delta\\eta,\\Delta\\phi).}',
     u'B(\\Delta\\phi)=B_{0}(1+2V_{1}\\cos{(\\Delta\\phi)}+2V_{2}\\cos{(2\\Delta\\phi)}+2V_{3}\\cos{(3\\Delta\\phi)})+A_\\mathrm{AS}\\exp{\\bigg[-\\bigg(\\frac{\\abs{\\Delta\\phi-\\pi}}{\\alpha}\\bigg)^{\\beta}\\bigg]},',
     u'\\Psi_{6D}=\\Psi_++\\Psi_-=\\begin{pmatrix}\\chi_+\\\\\\bar\\eta_+\\\\0\\\\0\\end{pmatrix}+\\begin{pmatrix}0\\\\0\\\\\\chi_-\\\\\\eta_-\\end{pmatrix}\\,.',
     u'{\\calL}_{\\Psi_+}=\\int\\sqrt{-g}dx^6\\i(\\chi_+^\\dagger)\\sigma_\\mu\\partial^\\mu\\chi_++i(\\bar\\eta_+^\\dagger)\\sigma_\\mu\\partial^\\mu\\bar\\eta_+i+\\left(\\chi^\\dagger\\bar\\Omega_+\\bar\\eta_++\\bar\\eta_+^\\dagger\\Omega_+\\chi_+\\right)\\,,',
     u'{\\ensuremath{\\mathbf{A}}}=\\frac{n}{2g}\\cos\\thetad\\phi,',
     u'R=\\frac{n\\sqrt{8\\piG_6}}{2g}',
     u'f:\\left\\{\\begin{array}{ccc}S^2\\timesS^1&\\rightarrow&\\mathbb{C}\\\\(\\Omega,{\\ensuremath{\\mathbf{\\hat{{v}}}}})&\\mapsto&f(\\Omega,{\\ensuremath{\\mathbf{\\hat{{v}}}}})\\end{array}\\right.',
     u'\\begin{array}{rclrcl}\\phantom{a\\hspace{-0.5em}}_{s\\hspace{-0.1em}}\\eth&=&-\\left(\\partial_\\theta+i\\frac{\\partial_\\phi}{\\sin\\theta}-s\\cot\\theta\\right),\\\\&\\phantom{a\\hspace{-0.5em}}_{s\\hspace{-0.1em}}\\bar\\eth&=&-\\left(\\partial_\\theta-i\\frac{\\partial_\\phi}{\\sin\\theta}+s\\cot\\theta\\right).\\end{array}',
     u'\\begin{array}{rclrcl}\\eth\\phantom{a\\hspace{-0.5em}}_{s\\hspace{-0.1em}}Y_{lm}&=&\\sqrt{l(l+1)-s(s+1)}\\phantom{a\\hspace{-0.2em}}_{s+1\\hspace{-0.1em}}Y_{lm},\\\\&\\bar\\eth\\phantom{a\\hspace{-0.5em}}_{s\\hspace{-0.1em}}Y_{lm}&=&-\\sqrt{l(l+1)-s(s-1)}\\phantom{a\\hspace{-0.2em}}_{s-1\\hspace{-0.1em}}Y_{lm}\\,.\\end{array}',
     u'\\begin{array}{rclrcl}\\eth\\bar\\eth\\phantom{a\\hspace{-0.5em}}_{s\\hspace{-0.1em}}Y_{lm}&=&-(l(l+1)-s(s-1))\\phantom{a\\hspace{-0.2em}}_{s\\hspace{-0.1em}}Y_{lm},\\\\&\\bar\\eth\\eth\\phantom{a\\hspace{-0.5em}}_{s\\hspace{-0.1em}}Y_{lm}&=&-(l(l+1)-s(s+1))\\phantom{a\\hspace{-0.2em}}_{s\\hspace{-0.1em}}Y_{lm}\\,,\\end{array}',
     u'\\begin{array}{c}\\phantom{a\\hspace{-0.2em}}_{s\\hspace{-0.1em}}Y_{lm}^*=(-1)^{m+s}\\phantom{a\\hspace{-0.2em}}_{s\\hspace{-0.1em}}Y_{lm}\\\\\\phantom{a\\hspace{-0.2em}}_{s\\hspace{-0.1em}}Y_{lm}(\\pi-\\theta,\\phi+\\pi)=(-1)^j\\phantom{a\\hspace{-0.2em}}_{-s\\hspace{-0.1em}}Y_{lm}(\\theta,\\phi)\\end{array}',
     u"\\begin{array}{rll}\\intd\\Omega\\phantom{a\\hspace{-0.2em}}_{s\\hspace{-0.1em}}Y_{lm}^*\\phantom{a\\hspace{-0.2em}}_{s\\hspace{-0.1em}}Y_{l'm'}=&\\delta_{ll'}\\delta_{mm'}&\\\\\\intd\\Omega\\phantom{a\\hspace{-0.2em}}_{s_1\\hspace{-0.1em}}Y_{l_1m_1}\\times\\phantom{a\\hspace{-0.2em}}_{s_2\\hspace{-0.1em}}Y_{l_2m_2}\\times\\phantom{a\\hspace{-0.2em}}_{s_3\\hspace{-0.1em}}Y_{l_3m_3}=&&\\\\\\sqrt{\\frac{l_1(l_1+1)l_2(l_2+1)l_3(l_3+1)}{4\\pi}}&\\left(\\begin{array}{ccc}l_1&l_2&l_3\\\\m_1&m_2&m_3\\end{array}\\right)\\left(\\begin{array}{ccc}l_1&l_2&l_3\\\\-s_1&-s_2&-s_3\\end{array}\\right)&\\end{array}",
     u'{\\calL}_\\Phi=\\Phi\\partial_\\mu\\partial^\\mu\\Phi+\\Phi\\Delta_{S^2}\\Phi-m_\\Phi^2\\Phi\\Phi\\,.',
     u'\\Phi(x^\\mu,\\theta,\\phi)=\\sum_{lm}\\frac{\\Phi_{lm}(x^\\mu)}{R}Y_{lm}(\\theta,\\phi)\\,,',
     u'\\begin{array}{rl}{\\calL}_{\\Phi4D}&=R^2\\intd\\Omega{\\calL}_\\Phi\\\\&=\\displaystyle\\sum_{lm}\\Phi_{lm}^\\dagger\\partial_\\mu\\partial^\\mu\\Phi_{lm}-\\left(m_\\Phi^2+l(l+1)\\right)\\Phi^\\dagger_{lm}\\Phi_{lm}\\,.\\end{array}',
     u'\\begin{aligned}\\tildeA_{\\theta}&=\\frac{A_{\\theta}}{R}\\,,&\\tilde{A}_\\phi&=\\frac{A_\\phi}{R\\sin\\theta}\\,,&&onumber\\\\A_\\pm&=\\frac{1}{\\sqrt{2}}(\\tildeA_\\theta\\pmi\\tilde{A_\\phi})\\,,&\\tildeA_\\theta&=\\frac{1}{\\sqrt{2}}(A_++A_-)\\,,&\\tildeA_\\phi&=\\frac{1}{\\sqrt{2}i}(A_+-A_-)\\,,onumber\\\\A_+&=A_-^\\dagger\\,,&A_+&=-i\\eth\\Xi^\\dagger\\,,&\\Xi&=\\frac{1}{\\sqrt{2}}(\\Phi+i\\Theta)\\,.\\end{aligned}',
     u'{\\calL}_{gf}=\\frac{1}{\\xi}tr(f^2)',
     u'f=\\eta^{\\muu}\\partial_\\muA^u+\\frac{\\xi}{R^2\\sin\\theta}\\partial_\\theta\\sin\\thetaA_\\theta+\\frac{\\xi}{R^2\\sin^2\\theta}\\partial_\\phiA_\\phi\\,.',
     u'{\\calL}_{A^\\mu}=A_\\mu\\left[\\eta^{\\muu}\\left(\\Box+\\frac{1}{R^2}\\phantom{a\\hspace{-0.5em}}_{1\\hspace{-0.1em}}\\bar\\eth\\phantom{a\\hspace{-0.5em}}_{0\\hspace{-0.1em}}\\eth\\right)-\\left(1-\\frac{1}{\\xi}\\right)\\partial^\\mu\\partial^u\\right]A_u',
     u'{\\calL}_{\\Phi,\\Theta}=\\Phi\\;\\phantom{a\\hspace{-0.5em}}_{1\\hspace{-0.1em}}\\bar\\eth\\phantom{a\\hspace{-0.5em}}_{0\\hspace{-0.1em}}\\eth\\left(\\Box-\\frac{1}{R^2}\\phantom{a\\hspace{-0.5em}}_{1\\hspace{-0.1em}}\\bar\\eth\\phantom{a\\hspace{-0.5em}}_{0\\hspace{-0.1em}}\\eth\\right)\\Phi+\\Theta\\;\\phantom{a\\hspace{-0.5em}}_{1\\hspace{-0.1em}}\\bar\\eth\\phantom{a\\hspace{-0.5em}}_{0\\hspace{-0.1em}}\\eth\\left(\\Box-\\frac{\\xi}{R^2}\\phantom{a\\hspace{-0.5em}}_{1\\hspace{-0.1em}}\\bar\\eth\\phantom{a\\hspace{-0.5em}}_{0\\hspace{-0.1em}}\\eth\\right)\\Theta\\;;',
     u'\\begin{array}{rl}A_\\mu&=\\displaystyle\\sum_{lm}\\frac{A_{\\mulm}}{R}Y_{lm}\\,,\\\\\\Phi&=\\displaystyle\\sum_{l>0,m}\\frac{\\Phi_{lm}}{R\\sqrt{l(l+1)}}Y_{lm}\\,,\\\\\\Theta&=\\displaystyle\\sum_{l>0,m}\\frac{\\Theta_{lm}}{R\\sqrt{l(l+1)}}Y_{lm}\\,.\\\\\\end{array}',
     u'G\\left\\{\\begin{aligned}\\theta&\\rightarrow\\pi-\\theta\\\\\\phi&\\rightarrow\\phi+\\pi\\left(1+\\frac{1}{n}\\right)\\end{aligned}\\right.',
     u'Y_{lm}(G\\Omega)=(-1)^lY_{lm}(\\Omega)e^{im\\pi/n}\\,.',
     u'\\begin{array}{lll}\\mathcal{L}&=&\\intR^2d\\Omega\\\\Phi\\;\\hspace{0.01em}_1\\bar\\eth_0\\eth\\left(\\Box-\\frac{\\hspace{0.01em}_1\\bar\\eth_0\\eth}{R^2}\\right)\\Phi\\i\\+\\Phi_{2,-2}\\left(\\Box-\\frac{6}{R^2}\\right)\\Phi_{2,2}+\\Phi_{2,2}\\left(\\Box-\\frac{6}{R^2}\\right)\\Phi_{2,-2}\\\\&=&\\Phi_{2,2}^\\dagger\\left(\\Box-\\frac{6}{R^2}\\right)\\Phi_{2,2}+\\Phi_{2,2}\\left(\\Box-\\frac{6}{R^2}\\right)\\Phi_{2,2}^\\dagger\\=\\2\\Phi_{2,2}^\\dagger\\left(\\Box-\\frac{6}{R^2}\\right)\\Phi_{2,2}\\mathrm{\\\\}\\\\&=&\\varphi^\\dagger\\left(\\Box-\\frac{6}{R^2}\\right)\\varphi\\\\\\end{array}',
     u'{\\calL}_H^{6D}=(D_M\\phi)^\\dagger(D^M\\phi)+\\mu_{6D}^2\\phi^\\dagger\\phi-\\lambda_{6D}(\\phi^\\dagger\\phi)^2',
     u'\\phi=\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{c}\\phi_1+i\\phi_2\\\\v_{6D}+H+i\\phi_3\\end{array}\\right)\\,.',
     u'\\phi(x^\\mu,\\Omega)=\\sum\\frac{Y_{lm}(\\Omega)}{R}\\phi_{lm}(x^\\mu)\\,.',
     u'v_{6D}=\\sqrt{\\frac{\\mu^2_{6D}}{2\\lambda_{6D}}}=\\frac{v_{4D}}{\\sqrt{4\\pi}R}\\,.',
     u'\\begin{aligned}{\\calL}_{gf}=&-\\frac{1}{\\xi}tr(f^af_a)\\\\f^a=&\\eta^{\\muu}\\partial_\\muA_u^a+\\frac{\\xi}{R^2\\sin\\theta}\\partial_\\theta\\left(\\sin\\thetaA_\\theta^a\\right)+\\frac{\\xi}{R^2\\sin\\theta}\\partial_\\phiA_\\phi^a\\\\&-\\xigF_i^a\\phi_i\\end{aligned}',
     u'M^2_{gf}=\\frac{\\xi}{R^2}\\begin{pmatrix}g^2\\zeta^2&g\\zeta\\sqrt{l\\left(l+1\\right)}\\\\g\\zeta\\sqrt{l\\left(l+1\\right)}&l\\left(l+1\\right)\\end{pmatrix}',
     u'\\begin{pmatrix}1\\\\-\\frac{g\\zeta}{\\sqrt{l\\left(l+1\\right)}}\\end{pmatrix}\\,.',
     u'{\\calL}_{m_{EW}}=-\\frac{1}{8}g^2_{4D}v_{4D}^2\\left[\\bar\\varphi_W^+\\varphi_W^-+\\bar\\varphi_W^-\\varphi_W^++\\frac{\\bar\\varphi_Z\\varphi_Z}{cos^2\\theta_W}\\right]',
     u'm_{\\varphi_W^+}^2=m_{\\varphi_W^-}^2=\\frac{6}{R^2}+m_W^2\\,,\\quadm_{\\varphi_Z}^2=\\frac{6}{R^2}+m_Z^2\\,,\\quadm_{\\varphi_A}^2=\\frac{6}{R^2}\\,.',
     u'{\\calL}_{m_\\varphi}=\\frac{-1}{4}\\left(f_\\text{bulk}(\\theta)F_{MN}F^{MN}+2f_\\text{loc}\\delta(\\theta)F_{\\theta\\phi}F^{\\theta\\phi}\\right)',
     u'\\frac{1}{R}>\\unit{1.6}{\\tera\\electronvolt}\\,,\\qquadM_{l=2}>{\\unit{{3.9}}{\\tera\\electronvolt}}\\,,',
     u'{\\frac{\\partial{n}}{\\partial{t}}}+3Hn=-\\left<{\\sigmav}\\right>\\left(n^2-n_{eq}^2\\right)',
     u'n_{eq}=g\\left(\\frac{mT}{2\\pi}\\right)^{\\frac{3}{2}}e^{-m/T}',
     u'\\Omega_{DM}h^2\\approx\\frac{1.04\\times10^9}{M_{pl}}\\frac{m}{T_f\\sqrt{g_*}}\\frac{1}{a+3mb/T_f}\\,.',
     u'{\\frac{\\partial{N}}{\\partial{t}}}+3HN=-\\left<{\\sigma_{eff}v}\\right>\\left(N^2-N_{eq}^2\\right)\\,,',
     u'\\sigma_{eff}=\\sum_i^K\\sum_j^K\\sigma_{ij}\\frac{g_ig_j}{g_{eff}^2}(1+\\Delta_i)^{3/2}(1+\\Delta_j)^{3/2}e^{-x(\\Delta_i+\\Delta_j)}\\,.',
     u'\\sum_i^Kg_i(1+\\Delta_i)^{3/2}e^{-x\\Delta_i}\\,.',
     u'{\\calL}_{SSV}=\\frac{g}{2}\\epsilon_{abc}A_\\mu^a\\left(\\eth\\Phi^b\\partial_\\mu\\bar\\eth\\Phi^c+\\bar\\eth\\Phi^b\\partial\\eth\\Phi^c\\right).',
     u'\\begin{array}{rl}{\\calL}^{\\sc{4D}}_{\\sc{SSV}_1}=\\left.g_{4D}\\frac{1}{2i}\\right[&\\left(\\partial_\\mu\\bar\\varphi^+_W\\varphi^-_W-\\partial_\\mu\\bar\\varphi^-_W\\varphi^+_W\\right)W^{3\\mu}_{\\sc{SM}}\\\\+&\\left(\\partial_\\mu\\varphi^+_W\\bar\\varphi^-_W-\\partial_\\mu\\varphi^-_W\\bar\\varphi^+_W\\right)W^{3\\mu}_{\\sc{SM}}\\\\-&\\left(W^{+\\mu}_{\\sc{SM}}\\varphi_W^--W^{-\\mu}_{\\sc{SM}}\\varphi_W^+\\right)\\partial_\\mu\\bar\\varphi_{W^3}\\\\-&\\left(W^{+\\mu}_{\\sc{SM}}\\bar\\varphi_W^--W^{-\\mu}_{\\sc{SM}}\\bar\\varphi_W^+\\right)\\partial_\\mu\\varphi_{W^3}\\\\+&\\left(W^{+\\mu}_{\\sc{SM}}\\partial_\\mu\\varphi_W^--W^{-\\mu}_{\\sc{SM}}\\partial_\\mu\\varphi_W^+\\right)\\bar\\varphi_{W^3}\\\\+&\\left.\\left(W^{+\\mu}_{\\sc{SM}}\\partial_\\mu\\bar\\varphi_W^--W^{-\\mu}_{\\sc{SM}}\\partial_\\mu\\bar\\varphi_W^+\\right)\\varphi_{W^3}\\right]\\\\\\end{array}',
     u'\\begin{array}{rl}{\\calL}^{\\sc{4D}}_{\\sc{SSV}_2}=\\left.g_{4D}\\frac{\\sqrt{5}}{14i}\\right[&\\left(\\partial_\\mu\\bar\\varphi^+_W\\varphi^-_W-\\partial_\\mu\\bar\\varphi^-_W\\varphi^+_W\\right)W^{3\\mu}_{(2,0)}\\\\+&\\left(\\partial_\\mu\\varphi^+_W\\bar\\varphi^-_W-\\partial_\\mu\\varphi^-_W\\bar\\varphi^+_W\\right)W^{3\\mu}_{(2,0)}\\\\-&\\left(W^{+\\mu}_{(2,0)}\\varphi_W^--W^{-\\mu}_{(2,0)}\\varphi_W^+\\right)\\partial_\\mu\\bar\\varphi_{W^3}\\\\-&\\left(W^{+\\mu}_{(2,0)}\\bar\\varphi_W^--W^{-\\mu}_{(2,0)}\\bar\\varphi_W^+\\right)\\partial_\\mu\\varphi_{W^3}\\\\+&\\left(W^{+\\mu}_{(2,0)}\\partial_\\mu\\varphi_W^--W^{-\\mu}_{(2,0)}\\partial_\\mu\\varphi_W^+\\right)\\bar\\varphi_{W^3}\\\\+&\\left.\\left(W^{+\\mu}_{(2,0)}\\partial_\\mu\\bar\\varphi_W^--W^{-\\mu}_{(2,0)}\\partial_\\mu\\bar\\varphi_W^+\\right)\\varphi_{W^3}\\right]\\\\\\end{array}',
     u'{\\calL}_{SSVV}=\\frac{1}{2}g^2\\left[A_\\mu^aA_a^\\mu\\eth\\Phi^b\\bar\\eth\\Phi_b-A^a_\\mu\\eth\\Phi_aA_b^\\mu\\bar\\eth\\Phi^b\\right],',
     u'{\\calL}_{V_sV_sWW}=g^2_{4D}W^+_{SM\\mu}W_{-\\mu}^{SM}\\bar\\varphi_{A^3}\\varphi_{A^3}',
     u'{\\calL}_{V_sW_sVW}=-\\frac{1}{2}g_{4D}A^3_\\mu\\left[\\varphi^3\\left(W^{+\\mu}\\bar\\varphi_W^-+W^{-\\mu}\\bar\\varphi_W^+\\right)+\\bar\\varphi^3\\left(W^{+\\mu}\\varphi_W^-+W^{-\\mu}\\varphi_W^+\\right)\\right]',
     u'\\begin{array}{rl}{\\calL}_{W_sW_sWW}&=\\frac{1}{2}g^2_{4D}\\left[W_+^{SM\\mu}W_{-\\mu}^{SM}\\left(\\varphi_W^+\\bar\\varphi_W^-+\\varphi_W^-\\bar\\varphi_W^+\\right)-W^{SM}_{+\\mu}W^{SM\\mu}_{+}\\bar\\varphi_W^-\\varphi_W^-\\right.\\\\&\\left.\\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}-W^{SM}_{-\\mu}W^{SM\\mu}_{-}\\bar\\varphi_W^+\\varphi_W^+\\right]\\end{array}',
     u'{\\calL}_{W_sW_sVV}=\\frac{1}{2}g^2_{4D}A^3_{SM\\mu}A^{SM\\mu_3}\\left[\\bar\\varphi_W^+\\varphi_W^-+\\bar\\varphi_W^-\\varphi_W^+\\right]',
     u'{\\calL}_{SSHH}=-\\frac{1}{8}H^\\daggerH\\bar\\eth\\phi^a\\eth\\phi',
     u'{\\calL}_{SSHH}^{4D}=-\\frac{1}{8}g^2_{4D}hh\\left[\\bar\\varphi_W^+\\varphi_W^-+\\bar\\varphi_W^-\\varphi_W^++\\frac{\\bar\\varphi_Z\\varphi_Z}{\\cos^2\\theta_W}\\right]',
     u'{\\calL}_{SSH}^{4D}=-\\frac{1}{4}g_{4D}^2vh\\left[\\bar\\varphi_W^+\\varphi_W^-+\\bar\\varphi_W^-\\varphi_W^++\\frac{\\bar\\varphi_Z\\varphi_Z}{cos^2\\theta_W}\\right]\\\\',
     u'tfidf-c=\\frac{tf(t_{k},d_{j})\\timeslog\\left(N/df(t_{k})\\right)}{\\sqrt{{\\displaystyle\\sum_{t\\ind_{j}}(tf(t_{k},d_{j})\\timeslog\\left(N/df(t_{k})\\right)^{2})}}}',
     u'sim(a,b)=\\frac{v}{|N(a)||N(b)|}\\sum_{i\\inN(a),,j\\inN(b)}sim(i,j)',
     u'Auth(w_{t})=(1-\\alpha)*e_{t}+\\alpha\\sum_{s\\inT(w_{t})}Hub(s)',
     u'Hub(w_{s})=(1-\\gamma)*e_{s}+\\alpha\\sum_{t\\inT(w_{s})}Auth(t)',
     u'\\tau=\\frac{N_{i}+1/2*N_{j}}{N}',
     u'S_{D}=\\sum_{i}^{n}p(S_{i})*W_{i}',
     u'\\frac{1}{min(i,N-i+1)}',
     u'tf-isf=\\sum_{t\\ins}tf(t,s)*log\\frac{N}{sf(t)}',
     u'\\sum_{i=1}^{5}\\lambda_{i}F_{i}',
     u'PW(w)=\\sum_{i\\inP}log_{2}(\\frac{p(w,i)}{p(w).p(i)})-\\sum_{i\\inN}log_{2}(\\frac{p(w,i)}{p(w).p(i)})',
     u'\\{-,-\\}:{\\mathcal{O}}_{X}\\times{\\mathcal{O}}_{X}\\longrightarrow{\\mathcal{O}}_{X}',
     u'\\{F,G\\}_{U}(m)=\\langled_{m}F\\wedged_{m}G,\\pi_{m}\\rangle',
     u'd(\\pi_{x}\\circ\\phi)|_{u}\\oplusd(\\pi_{y}\\circ\\phi)|_{u}:T_{u}U={\\mathbb{R}}^{n}\\longrightarrowT_{x}X\\oplusT_{y}Y',
     u'\\partial(X\\times_{f,Z,g}Y)\\cong(\\partialX\\times_{f\\circi_{X},Z,g}Y){\\scriptstyle\\coprod}(X\\times_{f,Z,g\\circi_{Y}}\\partialY).',
     u'p^{U}_{W}=p^{V}_{W}\\circp^{U}_{V}',
     u'\\{-,-\\}={\\mathcal{O}}\\times{\\mathcal{O}}\\longrightarrow{\\mathcal{O}}',
     u'f_{\\#}(U):{\\mathcal{O}}_{Y}(U)=C^{\\infty}(U)\\rightarrow{\\mathcal{O}}_{X}(f^{-1}(U))=C^{\\infty}(f^{-1}(U))',
     u'\\{f,gs\\}=\\{f,g\\}s+g.\\{f,s\\}',
     u'\\[\\theta=f_1\\frac{\\partial}{\\partialz_1}+\\ldots+f_n\\frac{\\partial}{\\partialz_n}\\]',
     u'\\[h\\cdot\\theta_{|V}=\\theta^\\prime_{|V}.\\]',
     u'\\[\\theta(w_1)=\\ldots=\\theta(w_{n-1})=0.\\]',
     u'\\[\\xymatrix{\\mathbb{C}^n&W\\ar@{^{(}->}[l]\\ar[d]&V\\capV^\\prime\\ar[l]_{\\cong}\\ar[ld]^{\\pi}\\ar[r]^{\\cong}\\ar[rd]_{\\pi^\\prime}&W^\\prime\\ar[d]\\ar@{^{(}->}[r]&\\mathbb{C}^n\\\\&\\mathbb{C}^{n-1}&&\\mathbb{C}^{n-1}&}\\]',
     u'\\[A=\\left(\\frac{\\partialf_j}{\\partialz_i}(0)\\right)_{i,j=1,\\ldots,n}\\]',
     u'\\[\\lambda_i=\\sum_{j=1}^nm_j\\lambda_j.\\]',
     u'\\[|\\sum_{i=1}^n\\lambda_it_i|\\geqc\\cdot\\sum_{i=1}^nt_i,\\]',
     u'\\[\\left(\\sum_{i=1}^n\\overline{p_i}dz_i+\\sum_{i=1}^np_id\\overline{z_i}\\right)\\left(\\sum_{i=1}^nf_i(p)\\frac{\\partial}{\\partialz_i}\\right)=\\sum_{i=1}^n\\overline{p_i}\\cdotf_i(p)=0,\\]',
     u'\\[\\left|\\sum_{j\\inJ}\\overline{p_j}\\frac{c}{2n}p_{j+1}\\right|\\leq|J|\\cdot\\frac{c}{2n}\\cdot\\epsilon^2\\leq\\frac{c}{2}\\cdot\\epsilon^2.\\]',
     u'\\[\\sup_{p\\inS^{2n-1}_{\\epsilon}}|\\sum_{i=1}^n\\overline{p_i}g_i(p)|\\leqC\\cdot\\epsilon^3.\\]',
     u'\\[\\left|\\left(\\sum_{i=1}^n\\overline{p_i}dz_i+\\sum_{i=1}^np_id\\overline{z_i}\\right)\\left(\\sum_{i=1}^nf_i(p)\\frac{\\partial}{\\partialz_i}\\right)\\right|=\\left|\\sum_{i=1}^n\\overline{p_i}\\cdotf_i(p)\\right|\\]',
     u'\\[\\Psi:B^{2n}_{\\epsilon_0}\\times[0,\\infty)\\rightarrowB^{2n}_{\\epsilon_0},\\x\\mapstoxe^{-t}\\]',
     u'\\[\\Psi(\\{\\mathrm{leaf\\of\\}\\mathcal{F}\\capS^{2n-1}_{\\epsilon_0}\\}\\times[0,\\infty))\\]',
     u'\\[H:B^{2n}_{\\epsilon_0}\\rightarrowB^{2n}_{\\epsilon_0},\\q\\mapsto\\Psi_{t_q}(\\Phi^{-1}_{t_q}(q)),\\\\qeq0,\\\\mathrm{and\\}H(0)=0,\\]',
     u'\\[h:S^{2n-1}_{\\epsilon^\\prime}\\stackrel{H_C}{\\hookrightarrow}B^{2n}_{\\epsilon}-\\{0\\}\\stackrel{r}{\\rightarrow}S^{2n-1}_{\\epsilon}\\]',
     u'\\[(h\\circ\\Phi_{\\mathcal{F}})(x,t)eq\\Phi_{\\mathcal{G}}(h(x),t).\\]',
     u'\\[\\Phi_{\\widetilde{\\mathcal{F}}}((x,t^\\prime),t):=(\\Phi_{\\mathcal{F}}(x,t),t+t^\\prime),x\\inS^{2n-1}_{\\epsilon^\\prime},t,t^\\prime\\in\\mathbb{R}.\\]',
     u'\\[\\Phi_{\\widetilde{\\mathcal{G}}}((y,s^\\prime),s):=(\\Phi_{\\mathcal{G}}(y,s),s+s^\\prime),y\\inS^{2n-1}_{\\epsilon},s,s^\\prime\\in\\mathbb{R}\\]',
     u'\\[\\widetilde{H}:S^{2n-1}_{\\epsilon^\\prime}\\times\\mathbb{R}\\rightarrowS^{2n-1}_{\\epsilon}\\times\\mathbb{R}\\]',
     u'\\[\\Phi_{\\widetilde{\\mathcal{G}}}((h(x),0),s)=\\widetilde{H}(\\Phi_{\\mathcal{F}}(x,t),t),\\]',
     u'\\[\\Phi_{\\mathcal{G}}((h(x),s),s)=(h(\\Phi_{\\mathcal{F}}(x,t)),(q_2\\circ\\widetilde{H})(\\Phi_{\\mathcal{F}}(x,t),t)).\\]',
     u'h(\\Phi_{\\mathcal{F}}(x,t))=\\Phi_{\\mathcal{G}}((h(x),\\tau(x,t)).',
     u'\\tau(x,0)=0.',
     u'\\tau(x,t+t^\\prime)=\\tau(x,t)+\\tau(\\Phi_{\\mathcal{F}}(x,t),t^\\prime).',
     u'\\[\\tau(x,t+t^\\prime)=\\tau(x,t)+\\tau(\\Phi_{\\mathcal{F}}(x,t),t^\\prime)+k(t,t^\\prime)\\cdotT_{\\mathcal{G},h(x)},\\]',
     u'\\[\\tau(x,t+T_{\\mathcal{F},x})=\\tau(x,t)+l\\cdotT_{\\mathcal{G},h(x)}.\\]',
     u'\\tau(x,t+T_{\\mathcal{F},x})=\\tau(x,t)+T_{\\mathcal{G},h(x)}.',
     u'\\lim_{t\\rightarrow\\infty}\\tau(x,t)=\\infty\\\\mathrm{and}\\\\lim_{t\\rightarrow-\\infty}\\tau(x,t)=-\\infty:',
     u'\\[y=h(\\Phi_{\\mathcal{F}}(x,t))=\\Phi_{\\mathcal{G}}(h(x),\\tau(x,t))\\]',
     u'\\[\\gamma_{\\delta}(x,t):=\\min_{t<t^\\prime}\\{t^\\prime:\\tau(x,t^\\prime)=\\tau(x,t)+\\delta\\}-t>0,\\]',
     u'\\[\\sigma(x,t):=\\frac{1}{\\gamma_\\delta(x,t)}\\int_t^{t+\\gamma_\\delta(x,t)}\\mu(x,t^\\prime)dt^\\prime\\]',
     u'\\sigma(x,t+t^\\prime)=\\sigma(\\Phi_{\\mathcal{F}}(x,t),t^\\prime)+\\tau(x,t).',
     u'\\[\\sigma(x,t)=\\sigma(x,0)+k\\cdotT_{\\mathcal{G},h(x)}=\\sigma(x,0)+k\\cdot\\tau(x,T_{\\mathcal{F},x})=\\sigma(x,k\\cdotT_{\\mathcal{F},x}),\\]',
     u'\\[\\lambdax\\frac{\\partial}{\\partialx}+y\\frac{\\partial}{\\partialy},\\\\lambda\\in\\mathbb{C}-\\mathbb{R}.\\]',
     u'\\[ye^{it_2}d(xe^{it_1})-\\lambdaxe^{it_1}d(ye^{it_2})=e^{i(t_1+t_2)}(ydx-\\lambdaxdy)\\]',
     u'\\[d(x\\overline{x})=\\overline{x}dx+xd\\overline{x}\\\\mathrm{and}\\d(y\\overline{y})=\\overline{y}dy+yd\\overline{y}.\\]',
     u'\\[(\\overline{y}dy+yd\\overline{y})(\\lambdax\\frac{\\partial}{\\partialx}+y\\frac{\\partial}{\\partialy})=\\overline{y}y\\in\\mathbb{R}\\\\mathrm{and}\\yeq0\\]',
     u'\\[(\\overline{x}dx+xd\\overline{x})(\\lambdax\\frac{\\partial}{\\partialx}+y\\frac{\\partial}{\\partialy})=\\lambda\\overline{x}x\\\\mathrm{and}\\xeq0\\]',
     u'\\[\\omega=d(y^2+|x|^2-1)=2ydy+\\overline{x}dx+xd\\overline{x}.\\]',
     u'\\[\\omega_{\\mathrm{Re}}=ydy+\\overline{y}d\\overline{y}+\\overline{x}dx+xd\\overline{x}\\\\mathrm{and}\\\\omega_{\\mathrm{Im}}=-i(ydy-\\overline{y}d\\overline{y}).\\]',
     u'\\[\\omega_{\\mathrm{Re}}(\\theta(x,y))=2y^2+\\lambdax\\overline{x}\\\\mathrm{and}\\d(\\overline{x}x+\\overline{y}y)(\\theta(x,y))=\\lambdax\\overline{x}+y\\overline{y},\\]',
     u'\\[\\Phi_x^{(i)}:T_{1/2}\\times(0,1/2]\\rightarrow\\{(x,y)\\inS^3:0<|x|\\leq1/2\\}\\]',
     u'\\[\\Phi_x^{(2)}\\circ\\left[\\mathrm{id}_{T^x{\\frac{1}{2}}}\\times((\\phi^{(2)}_x)^{-1}\\circ\\phi^{(1)}_x)\\right]\\circ(\\Phi_x^{(1)})^{-1}\\]',
     u'\\[(mx+y^m)\\frac{\\partial}{\\partialx}+y\\frac{\\partial}{\\partialy},\\m\\geq1.\\]',
     u'\\[\\overline{x}(mx+y^m)+\\overline{y}y=\\overline{x}x+\\overline{y}y+(m-1)\\overline{x}x+\\overline{x}y^meq0.\\]',
     u'\\[|\\overline{x}y^m|=|x|\\cdot|y|^m<\\epsilon^{m+1}\\leq\\epsilon^2,\\]',
     u'\\[\\lambda_{(a,b)}:\\mathbb{C}\\rightarrow\\mathbb{C}^2,\\t\\mapsto((a+b^mt)e^{mt},be^t)\\]',
     u'\\[1=e^{m(t+\\overline{t})}(a\\overline{a}+b^m\\overline{a}t+a\\overline{b}^m\\overline{t}+(b\\overline{b})^mt\\overline{t})+b\\overline{b}e^{t+\\overline{t}}.\\]',
     u'\\[\\tag{\\ast}(b\\overline{b})^mt_I^2+2\\mathrm{Im}(a\\overline{b}^m)t_I+a\\overline{a}+2\\mathrm{Re}(b^m\\overline{a})t_R+(b\\overline{b})^mt_R^2+b\\overline{b}e^{2(1-m)t_R}-e^{-2mt_R}=0.\\]',
     u'\\[2\\mathrm{Re}(b^m\\overline{a})+2(b\\overline{b})^mt_R+2(1-m)b\\overline{b}e^{2(1-m)t_R}+2me^{-2mt_R}\\]',
     u'\\[\\{(a,b)\\inS^3:\\mathrm{Im}(a\\overline{b}^m)=0,beq0\\}.\\]',
     u'\\[\\begin{array}{ccccc}V_1^\\prime&\\stackrel{\\Phi}{\\rightarrow}&\\Phi(V_1^\\prime)&\\subset&V_2^\\prime\\\\\\cap&&\\cap&&\\cap\\\\S^3-\\{y=0\\}&\\stackrel{\\Phi}{\\rightarrow}&S^3-\\{y=0\\}&=&S^3-\\{y=0\\}.\\\\\\end{array}\\]',
     u'\\[\\begin{array}{ccccc}\\rule{.5cm}{0cm}\\mathbb{Z}&\\stackrel{\\cdot\\pm1}{\\longrightarrow}&\\mathbb{Z}&\\rightarrow&\\mathbb{Z}\\rule{.5cm}{0cm}\\\\\\cdotm_1\\downarrow&&&&\\downarrow\\cdotm_2\\\\\\rule{.5cm}{0cm}\\mathbb{Z}&=&\\mathbb{Z}&=&\\mathbb{Z}\\rule{.5cm}{0cm}\\end{array}\\]',
     u'\\[t(1-m+t^mx^{m-1})dx-(mx+t^mx^m)dt\\\\mathrm{resp.\\}(mx+t^mx^m)\\frac{\\partial}{\\partialx}+t(1-m+t^mx^{m-1})\\frac{\\partial}{\\partialt}\\]',
     u'\\[yds+(s(1-m)+y^{m-1})dy\\\\mathrm{resp.\\}y\\frac{\\partial}{\\partialy}+((m-1)s+y^{m-1})\\frac{\\partial}{\\partials}\\]',
     u'\\[\\lambdax\\frac{\\partial}{\\partialx}+y\\frac{\\partial}{\\partialy},\\\\lambda\\in\\mathbb{R}_{>0}.\\]',
     u'\\[t\\cdot_\\lambda(e^{ia},e^{ib})=(e^{i(a+\\lambdat)},e^{i(b+t)}),\\t,a,b\\in\\mathbb{R}.\\]',
     u'\\[\\phi_Q:T^2\\rightarrowT^2,(e^{ix},e^{iy})\\mapsto(e^{i(ax+by)},e^{i(cx+dy)})\\]',
     u'\\[\\mu=Q\\cdot\\lambda=\\frac{a\\lambda+b}{c\\lambda+d}.\\]',
     u'\\[L_P^{(\\lambda)}:=\\{(e^{i(a+\\lambdat)},e^{i(b+t)})|t\\in\\mathbb{R}\\}\\subsetT^2\\]',
     u'\\[\\left[\\frac{\\lambdab_\\epsilon}{2\\pi}\\right]\\leq\\left|p([0,b_\\epsilon])\\capC_2\\right|\\leq\\left[\\frac{\\lambdab_\\epsilon}{2\\pi}\\right]+1\\\\mathrm{and}\\left[\\frac{b_\\epsilon}{2\\pi}\\right]\\leq\\left|p([0,b_\\epsilon])\\capC_1\\right|\\leq\\left[\\frac{b_\\epsilon}{2\\pi}\\right]+1,\\]',
     u'\\[\\lambda=\\lim_{\\epsilon\\rightarrow0}\\frac{[\\gamma_{P,\\epsilon}^{(\\lambda)}]\\cdot[C_2]}{[\\gamma_{P,\\epsilon}^{(\\lambda)}]\\cdot[C_1]}=\\lim_{\\epsilon\\rightarrow0}\\frac{[\\phi(\\gamma_{P,\\epsilon}^{(\\lambda)})]\\cdot[\\phi(C_2)]}{[\\phi(\\gamma_{P,\\epsilon}^{(\\lambda)})]\\cdot[\\phi(C_1)]}=\\lim_{\\epsilon\\rightarrow0}\\frac{[\\phi(\\gamma_{P,\\epsilon}^{(\\lambda)})]\\cdot[C_2]}{[\\phi(\\gamma_{P,\\epsilon}^{(\\lambda)})]\\cdot[C_1]},\\]',
     u'\\[\\lim_{\\epsilon\\rightarrow0}\\frac{[\\gamma_{\\phi(P),U_\\epsilon}^{(\\mu)}]\\cdot[C_2]}{[\\gamma_{\\phi(P),U_\\epsilon}^{(\\mu)}]\\cdot[C_1]}=\\mu,\\]',
     u'\\[0<\\mathrm{Re}\\lambda_1<\\cdots<\\mathrm{Re}\\lambda_n\\\\mathrm{resp.\\}0<\\mathrm{Re}\\mu_1<\\cdots<\\mathrm{Re}\\mu_n.\\]',
     u'\\[L_1:=\\left\\{z_l=0:leqi_1,\\ldots,i_k\\right\\},L_2:=\\left\\{z_m=0:meqj_1,\\ldots,j_k\\right\\}\\subset\\mathbb{C}^n\\]',
     u'\\mathscr{T}_{ij}\\equiv\\mathrm{H}(\\Phi)_{ij}=\\frac{\\partial^2\\Phi}{\\partial\\vec{x}_i\\partial\\vec{x}_j},',
     u'\\Delta\\Phi(\\vec{x})=\\delta(\\vec{x}),',
     u'\\vec{x}(\\vec{q})\\equiv\\vec{q}+\\vec{\\Psi}(\\vec{q}).',
     u'J(\\vec{q})\\equiv\\det\\left[\\frac{\\partial\\vec{x}}{\\partial\\vec{q}}\\right]=\\det\\mathscr{D}=\\det(\\mathscr{I}+\\mathscr{R}),',
     u'\\mathrm{T}_i(\\vec{q}_\\ell|\\xi)\\in\\{0,1\\}\\;\\mathrm{for}\\;i\\in\\llbracket0,3\\rrbracket\\quad\\mathrm{and}\\quad\\sum_{i=0}^{3}\\mathrm{T}_i(\\vec{q}_\\ell|\\xi)=1',
     u'\\mathcal{T}_i(\\vec{q}_\\ell|\\xi)\\in[0,1]\\;\\mathrm{for}\\;i\\in\\llbracket0,3\\rrbracket\\quad\\mathrm{and}\\quad\\sum_{i=0}^{3}\\mathcal{T}_i(\\vec{q}_\\ell|\\xi)=1.',
     u'\\left\\langle\\mathcal{P}(\\mathrm{T}_i(\\vec{q}_\\ell)|d,\\xi)\\right\\rangle=\\frac{1}{N}\\sum_{n=1}^{N}\\sum_{j=0}^{3}\\delta^{(\\mathrm{K})}_{\\mathrm{T}_i(\\vec{q}_\\ell)\\mathrm{T}^n_j(\\vec{q}_\\ell|\\xi)},',
     u'\\left\\langle\\mathcal{P}(\\mathrm{T}_i(\\vec{x}_k)|d,\\xi)\\right\\rangle=\\frac{1}{N_{\\vec{x}_k}}\\sum_{n=1}^N\\mathcal{P}_n(\\mathrm{T}_i(\\vec{x}_k)|d,\\xi),',
     u'N_{\\vec{x}_k}=\\sum_{n=1}^N\\delta^{\\mathrm{(K)}}_{\\mathcal{S}_n(\\vec{x}_k),1}.',
     u'{F=(c\\hbar/2\\pie)A},',
     u'{{R}_{T}}=\\frac{2{{\\pi}^{2}}{{k}_{B}}nT/\\hbar{{\\omega}_{c}}}{\\text{sh}(2{{\\pi}^{2}}{{k}_{B}}nT/\\hbar{{\\omega}_{c}})};',
     u'{{{R}_{\\tau}}=\\exp(-2{{\\pi}^{2}}{{k}_{B}}nx/\\hbar{{\\omega}_{c}}),}',
     u'{{{R}_{0}}=\\frac{16\\rho{{l}_{i}}}{3\\pi{{d}^{2}}}+\\beta({{l}_{i}}/d)\\frac{\\rho}{d}}',
     u'{{{N}_{\\omega}}(\\mathbf{r})=\\frac{eV-\\hbar\\omega}{2\\hbar\\omega}\\Theta(eV-\\hbar\\omega){{q}_{\\omega}}(\\mathbf{r}),}',
     u"{{{q}_{\\omega}}(\\mathbf{r})\\frac{1}{4\\pi\\Lambda_{\\omega}^{2}}\\int{d\\mathbf{r}\\frac{\\exp(-R/{{\\Lambda}_{\\omega}})}{R}}q(\\mathbf{{r}'});}",
     u"\\[\\begin{matrix}R=\\left|\\mathbf{r}-\\mathbf{{r}'}\\right|;\\\\q(r)=\\frac{1}{2}\\left[1-{{\\left(1-\\frac{\\Omega(\\textbf{r})}{2\\pi}\\right)}^{2}}\\right];\\\\{{\\Lambda}_{\\omega}}={{\\left(l_{i}^{ph}l_{\\varepsilon}^{ph}/3\\right)}^{1/2}}{{({{\\omega}_{D}}-\\omega)}^{{1}/{2}\\;}}\\\\\\end{matrix}\\]",
     u'\\[\\tau_{\\varepsilonD}^{-1}=\\frac{2\\pi}{\\hbar}\\int\\limits_{0}^{\\omegaD}{g(\\omega)d\\omega}\\]',
     u'{\\lambda(eV)=\\frac{{{\\lambda}_{0}}}{2}\\left[1+\\frac{{{(h{{u}_{0}})}^{2}}}{{{(h{{u}_{0}})}^{2}}-{{(eV)}^{2}}}\\right].}',
     u'{{{A}_{1}}/{{A}_{2}}=C\\cosh\\left(\\frac{2{{\\pi}^{2}}{{k}_{B}}T}{\\hbar{{\\omega}_{c}}}\\right)\\exp\\left(\\frac{\\pi}{{{\\omega}_{c}}\\tau}\\right).}',
     u'\\mathrm{F}_{ij}\\circ\\mathrm{F}_{st}=\\mathrm{F}_{it}^{\\oplus\\dim(e_jAe_s)}.',
     u'\\mathrm{F}\\circ\\mathrm{F}\\cong\\mathrm{F}^{\\oplus\\dim(A)}',
     u"\\mathrm{Hom}_Q(\\mathrm{K}P,{}_-)\\cong\\mathrm{Hom}_Q(P,\\mathrm{K}'{}_-).",
     u'V\\otimes_Q\\left(X\\otimesY\\right)\\cong\\left(V\\otimes_QX\\right)\\otimesY.',
     u'\\begin{array}{c||c|c|c|c}\\circ&\\mathrm{F}_{11}&\\mathrm{F}_{12}&\\mathrm{F}_{21}&\\mathrm{F}_{22}\\\\\\hline\\hline\\mathrm{F}_{11}&\\mathrm{F}_{11}&\\mathrm{F}_{12}&0&0\\\\\\hline\\mathrm{F}_{12}&\\mathrm{F}_{11}&\\mathrm{F}_{12}&\\mathrm{F}_{11}&\\mathrm{F}_{12}\\\\\\hline\\mathrm{F}_{21}&\\mathrm{F}_{21}&\\mathrm{F}_{22}&0&0\\\\\\hline\\mathrm{F}_{22}&\\mathrm{F}_{21}&\\mathrm{F}_{22}&\\mathrm{F}_{21}&\\mathrm{F}_{22}\\\\\\end{array}',
     u'\\mathrm{Hom}_{A\\text{-}A}(Ae_i\\otimese_jA,Ae_s\\otimese_tA)\\conge_iAe_s\\otimese_tAe_j,',
     u'\\begin{array}{c||c|c|c|c}X\\setminusY&\\mathrm{F}_{11}&\\mathrm{F}_{12}&\\mathrm{F}_{21}&\\mathrm{F}_{22}\\\\\\hline\\hline\\mathrm{F}_{11}&\\Bbbk&\\Bbbk&0&0\\\\\\hline\\mathrm{F}_{12}&0&\\Bbbk&0&0\\\\\\hline\\mathrm{F}_{21}&\\Bbbk&\\Bbbk&\\Bbbk&\\Bbbk\\\\\\hline\\mathrm{F}_{22}&0&\\Bbbk&0&\\Bbbk\\\\\\end{array}',
     u'\\begin{aligned}M_1:=\\left(3\\right),\\quadM_2:=\\left(\\begin{array}{cc}2&1\\\\2&1\\end{array}\\right),\\quadM_3:=\\left(\\begin{array}{cc}2&2\\\\1&1\\end{array}\\right),\\quadM_4:=\\left(\\begin{array}{cc}1&1\\\\2&2\\end{array}\\right),\\\\M_5:=\\left(\\begin{array}{cc}1&2\\\\1&2\\end{array}\\right),\\quadM_6:=\\left(\\begin{array}{ccc}1&1&1\\\\1&1&1\\\\1&1&1\\end{array}\\right).\\end{aligned}',
     u'\\begin{aligned}M_{11}=\\left(\\begin{array}{cc}1&0\\\\0&0\\end{array}\\right),\\,\\,M_{12}=\\left(\\begin{array}{cc}1&1\\\\0&0\\end{array}\\right),\\,\\,M_{21}=\\left(\\begin{array}{cc}0&0\\\\1&0\\end{array}\\right),\\,\\,M_{22}=\\left(\\begin{array}{cc}0&0\\\\1&1\\end{array}\\right).\\end{aligned}',
     u'\\begin{aligned}M_{11}=\\left(\\begin{array}{cc}0&1\\\\0&1\\end{array}\\right),\\,\\,M_{12}=\\left(\\begin{array}{cc}1&0\\\\1&0\\end{array}\\right),\\,\\,M_{21}=\\left(\\begin{array}{cc}0&1\\\\0&0\\end{array}\\right),\\,\\,M_{22}=\\left(\\begin{array}{cc}1&0\\\\0&0\\end{array}\\right).\\end{aligned}',
     u'\\begin{aligned}M_{11}=\\left(\\begin{array}{ccc}1&0&0\\\\1&0&0\\\\0&0&0\\end{array}\\right),\\,\\,M_{12}=\\left(\\begin{array}{ccc}0&1&1\\\\0&1&1\\\\0&0&0\\end{array}\\right),\\\\M_{21}=\\left(\\begin{array}{ccc}0&0&0\\\\0&0&0\\\\1&0&0\\end{array}\\right),\\,\\,M_{22}=\\left(\\begin{array}{ccc}0&0&0\\\\0&0&0\\\\0&1&1\\end{array}\\right).\\end{aligned}',
     u'\\begin{aligned}M_{11}=\\left(\\begin{array}{ccc}1&*&*\\\\*&0&*\\\\*&*&0\\end{array}\\right),\\,\\,M_{12}=\\left(\\begin{array}{ccc}0&*&*\\\\*&1&*\\\\*&*&0\\end{array}\\right),\\\\M_{21}=\\left(\\begin{array}{ccc}0&*&*\\\\*&0&*\\\\*&*&0\\end{array}\\right),\\,\\,M_{22}=\\left(\\begin{array}{ccc}0&*&*\\\\*&0&*\\\\*&*&1\\end{array}\\right).\\end{aligned}',
     u'\\begin{aligned}M_{11}=\\left(\\begin{array}{ccc}1&*&0\\\\*&0&0\\\\*&*&0\\end{array}\\right),\\,\\,M_{12}=\\left(\\begin{array}{ccc}0&*&1\\\\*&1&*\\\\*&*&0\\end{array}\\right),\\\\M_{21}=\\left(\\begin{array}{ccc}0&0&0\\\\*&0&*\\\\*&*&0\\end{array}\\right),\\,\\,M_{22}=\\left(\\begin{array}{ccc}0&0&0\\\\*&0&*\\\\*&*&1\\end{array}\\right).\\end{aligned}',
     u'\\begin{aligned}M_{11}=\\left(\\begin{array}{ccc}1&*&0\\\\1&0&0\\\\*&*&0\\end{array}\\right),\\,\\,M_{12}=\\left(\\begin{array}{ccc}0&*&1\\\\0&1&*\\\\*&*&0\\end{array}\\right),\\\\M_{21}=\\left(\\begin{array}{ccc}0&0&0\\\\0&0&*\\\\*&*&0\\end{array}\\right),\\,\\,M_{22}=\\left(\\begin{array}{ccc}0&0&0\\\\0&0&*\\\\*&*&1\\end{array}\\right).\\end{aligned}',
     u'\\begin{aligned}M_{11}=\\left(\\begin{array}{ccc}1&0&0\\\\1&0&0\\\\0&0&0\\end{array}\\right),\\,\\,M_{12}=\\left(\\begin{array}{ccc}0&1&1\\\\0&1&1\\\\0&0&0\\end{array}\\right),\\\\M_{21}=\\left(\\begin{array}{ccc}0&0&0\\\\0&0&0\\\\*&*&0\\end{array}\\right),\\,\\,M_{22}=\\left(\\begin{array}{ccc}0&0&0\\\\0&0&0\\\\*&*&1\\end{array}\\right).\\end{aligned}',
     u'\\begin{aligned}M_{11}=\\left(\\begin{array}{ccc}1&0&0\\\\1&0&0\\\\0&0&0\\end{array}\\right),\\,\\,M_{12}=\\left(\\begin{array}{ccc}0&1&1\\\\0&1&1\\\\0&0&0\\end{array}\\right),\\\\M_{21}=\\left(\\begin{array}{ccc}0&0&0\\\\0&0&0\\\\*&0&0\\end{array}\\right),\\,\\,M_{22}=\\left(\\begin{array}{ccc}0&0&0\\\\0&0&0\\\\*&1&1\\end{array}\\right).\\end{aligned}',
     u'\\begin{aligned}M_{11}=\\left(\\begin{array}{ccc}1&0&0\\\\1&0&0\\\\0&0&0\\end{array}\\right),\\,\\,M_{12}=\\left(\\begin{array}{ccc}0&1&1\\\\0&1&1\\\\0&0&0\\end{array}\\right),\\\\M_{21}=\\left(\\begin{array}{ccc}0&0&0\\\\0&0&0\\\\1&0&0\\end{array}\\right),\\,\\,M_{22}=\\left(\\begin{array}{ccc}0&0&0\\\\0&0&0\\\\0&1&1\\end{array}\\right).\\end{aligned}',
     u'\\begin{array}{c||c|c|c|c|c|c|c|c|c}\\circ&\\mathrm{F}_{11}&\\mathrm{F}_{12}&\\mathrm{F}_{13}&\\mathrm{F}_{21}&\\mathrm{F}_{22}&\\mathrm{F}_{23}&\\mathrm{F}_{31}&\\mathrm{F}_{32}&\\mathrm{F}_{33}\\\\\\hline\\hline\\mathrm{F}_{11}&\\mathrm{F}_{11}&\\mathrm{F}_{12}&\\mathrm{F}_{13}&0&0&0&0&0&0\\\\\\hline\\mathrm{F}_{12}&\\mathrm{F}_{11}&\\mathrm{F}_{12}&\\mathrm{F}_{13}&\\mathrm{F}_{11}&\\mathrm{F}_{12}&\\mathrm{F}_{13}&0&0&0\\\\\\hline\\mathrm{F}_{13}&0&0&0&\\mathrm{F}_{11}&\\mathrm{F}_{12}&\\mathrm{F}_{13}&\\mathrm{F}_{11}&\\mathrm{F}_{12}&\\mathrm{F}_{13}\\\\\\hline\\mathrm{F}_{21}&\\mathrm{F}_{21}&\\mathrm{F}_{22}&\\mathrm{F}_{23}&0&0&0&0&0&0\\\\\\hline\\mathrm{F}_{22}&\\mathrm{F}_{21}&\\mathrm{F}_{22}&\\mathrm{F}_{23}&\\mathrm{F}_{21}&\\mathrm{F}_{22}&\\mathrm{F}_{23}&0&0&0\\\\\\hline\\mathrm{F}_{23}&0&0&0&\\mathrm{F}_{21}&\\mathrm{F}_{22}&\\mathrm{F}_{23}&\\mathrm{F}_{21}&\\mathrm{F}_{22}&\\mathrm{F}_{23}\\\\\\hline\\mathrm{F}_{31}&\\mathrm{F}_{31}&\\mathrm{F}_{32}&\\mathrm{F}_{33}&0&0&0&0&0&0\\\\\\hline\\mathrm{F}_{32}&\\mathrm{F}_{31}&\\mathrm{F}_{32}&\\mathrm{F}_{33}&\\mathrm{F}_{31}&\\mathrm{F}_{32}&\\mathrm{F}_{33}&0&0&0\\\\\\hline\\mathrm{F}_{33}&0&0&0&\\mathrm{F}_{31}&\\mathrm{F}_{32}&\\mathrm{F}_{33}&\\mathrm{F}_{31}&\\mathrm{F}_{32}&\\mathrm{F}_{33}\\\\\\end{array}',
     u'\\begin{array}{c||c|c|c|c|c|c|c|c|c}X\\setminusY&\\mathrm{F}_{11}&\\mathrm{F}_{12}&\\mathrm{F}_{13}&\\mathrm{F}_{21}&\\mathrm{F}_{22}&\\mathrm{F}_{23}&\\mathrm{F}_{31}&\\mathrm{F}_{32}&\\mathrm{F}_{33}\\\\\\hline\\hline\\mathrm{F}_{11}&\\Bbbk&\\Bbbk&0&0&0&0&0&0&0\\\\\\hline\\mathrm{F}_{12}&0&\\Bbbk&\\Bbbk&0&0&0&0&0&0\\\\\\hline\\mathrm{F}_{13}&0&0&\\Bbbk&0&0&0&0&0&0\\\\\\hline\\mathrm{F}_{21}&\\Bbbk&\\Bbbk&0&\\Bbbk&\\Bbbk&0&0&0&0\\\\\\hline\\mathrm{F}_{22}&0&\\Bbbk&\\Bbbk&0&\\Bbbk&\\Bbbk&0&0&0\\\\\\hline\\mathrm{F}_{23}&0&0&\\Bbbk&0&0&\\Bbbk&0&0&0\\\\\\hline\\mathrm{F}_{31}&0&0&0&\\Bbbk&\\Bbbk&0&\\Bbbk&\\Bbbk&0\\\\\\hline\\mathrm{F}_{32}&0&0&0&0&\\Bbbk&\\Bbbk&0&\\Bbbk&\\Bbbk\\\\\\hline\\mathrm{F}_{33}&0&0&0&0&0&\\Bbbk&0&0&\\Bbbk\\end{array}',
     u'\\begin{array}{ccc}M_{11}&M_{12}&M_{13}\\\\M_{21}&M_{22}&M_{23}\\\\M_{31}&M_{32}&M_{33}.\\end{array}',
     u'\\Vert\\varphi\\Vert^2=\t\\int_{-h}^0\\Vert\\varphi(\\theta)\\Vert^2\\,d\\theta.',
     u'\\begin{aligned}x\\in\\mathcal{O}(R)&\\Leftrightarrow\\Vertx\\Vert^2\\leqR^2,\\\\(x,\\varphi)\\in\\mathcal{O}(R)&\\Leftrightarrow\\Vertx\\Vert^2+\\Vert\\varphi\\Vert^2\\leqR^2.\\end{aligned}',
     u'\\begin{multline}\\dot{x}(t)=f\\big(x(t)\\big)+B_0\\big(x(t)\\big)u(t)+B_1\\big(x(t)\\big)u(t-h)\\\\\t+\\int_{-h}^0B_\\text{int}\\big(\\theta,x(t)\\big)u(t+\\theta)\\,d\\theta,\\end{multline}',
     u'\\begin{aligned}\\Vertf(x)\\Vert&\\leqM_f\\Vertx\\Vert,\\\\f(x)&=f(x_0)+A(x_0)(x-x_0)+o(x-x_0),\\\\A&\\inC^0\\big(\\mathcal{O}(R),R^{n\\timesn}\\big);\\end{aligned}',
     u'\\begin{aligned}B_1&\\text{isboundedon}\\mathcal{O}(R),\\\\B_1(x)&=B_1(x_0)+\\Big(\\mathcal{B}_1^1(x_0)(x-x_0),\\dots,\\mathcal{B}_1^m(x_0)(x-x_0)\\Big)otag\\\\\t&\\phantom{=}\\:+o(x-x_0),\\\\\\mathcal{B}_1^i&\\inC^0\\big(\\mathcal{O}(R),R^{n\\timesn}\\big)\\quad\\foralli\\in\\overline{1,m};\\end{aligned}',
     u'\\begin{aligned}B_\\text{int}&\\text{isboundedon}[-h,0]\\times\\mathcal{O}(R),\\\\B_\\text{int}(\\cdot,x)&\\inPC\\big([-h,0],R^{n\\timesm}\\big),\\\\B_\\text{int}(\\theta,x)&=B_\\text{int}(\\theta,x_0)otag\\\\\t&\\phantom{=}\\:+\\Big(\\mathcal{B}_\\text{int}^1(\\theta,x_0)(x-x_0),\\dots,\\mathcal{B}_\\text{int}^m(\\theta,x_0)(x-x_0)\\Big)otag\\\\\t&\\phantom{=}\\:+o(x-x_0),\\\\\\mathcal{B}_\\text{int}^i(\\theta,\\cdot)&\\inC^0\\big(\\mathcal{O}(R),R^{n\\timesn}\\big)\\quad\\foralli\\in\\overline{1,m}.\\end{aligned}',
     u'\\dot{x}(t)=Ax(t)+B_1u(t-h).',
     u'Y\\big(x(t),u_t\\big)=e^{Ah}x(t)+\\int_{-h}^0e^{-A\\theta}B_1u(t+\\theta)\\,d\\theta;',
     u'y(t)=Y\\big(x(t),u_t\\big);',
     u'\\dot{y}(t)=Ay(t)+B_1u(t);',
     u'\\dot{x}(t)=Ax(t)+B_0u(t)+B_1u(t-h)+\t\\int_{-h}^0B_\\text{int}(\\theta)u(t+\\theta)\\,d\\theta.',
     u"\\begin{aligned}Y\\big(x(t),u_t\\big)&=\\xi(h),\\\\\\xi'(s)&=A\\xi(s)+B_1u(t+s-h)onumber\\\\\t&\\phantom{=}\\:+\\int_{-h}^{-s}B_\\text{int}(\\theta)u(t+s+\\theta)\\,d\\theta,\\\\\\xi(0)&=x(t)\\end{aligned}",
     u'Y\\big(x(t),u_t\\big)=e^{Ah}x(t)+\\int_{-h}^0Q(\\theta)u(t+\\theta)\\,d\\theta,',
     u'Q(\\theta)=e^{-A\\theta}B_1+\\int_{-h}^\\thetae^{A(h-\\theta+\\tau)}B_\\text{int}(\\tau)\\,d\\tau;',
     u'y(t)=Y\\big(x(t),u_t\\big);',
     u'\\dot{y}(t)=Ay(t)+\\big(e^{Ah}B_0+Q(0)\\big)u(t);',
     u'u(t)=KY\\big(x(t),u_t\\big).',
     u'v(x,\\varphi)=Y^T(x,\\varphi)VY(x,\\varphi)\t+\\int_{-h}^{\\,0}e^{\\sigma\\theta}\\Vert\\varphi(\\theta)\\Vert^2\\,d\\theta,',
     u'\\dot{x}(t)=f\\big(x(t),u(t-h)\\big).',
     u'\\dot{y}(t)=f\\big(y(t),u(t)\\big)',
     u'y(t)=Y\\big(x(t),u_t\\big)',
     u"\\begin{aligned}Y(x,\\varphi)&=\\xi(h),\\\\\\xi'(s)&=f\\big(\\xi(s)\\big)+B_1\\big(\\xi(s)\\big)\\varphi(s-h)otag\\\\\t&\\phantom{=}\\:+\\int_{-h}^{-s}B_\\textrm{int}\\big(\\theta,\\xi(s)\\big)\\varphi(s+\\theta)\\,d\\theta,\\\\\\xi(0)&=x.\\end{aligned}",
     u'(x,\\varphi)\\in\\mathcal{O}\\left(\\frac{R}{\\rho}\\right),',
     u'\\begin{aligned}\\Vertx\\Vert^2\\leq\t\\rho^2\\big(\\VertY(x,\\varphi)\\Vert^2+\\Vert\\varphi\\Vert^2\\big),\\\\\\VertY(x,\\varphi)\\Vert^2\\leq\t\\rho^2\\big(\\Vertx\\Vert^2+\\Vert\\varphi\\Vert^2\\big).\\end{aligned}',
     u'\\Vert\\xi(s)\\Vert^2\\leq\t\\rho^2\\big(\\Vertx\\Vert^2+\\Vert\\varphi\\Vert^2\\big)',
     u'\\dot{y}(t)=f\\big(y(t)\\big)+B\\big(y(t),u_t\\big)u(t)',
     u"\\begin{aligned}B(y,\\varphi)&=B_1\\big(\\xi(h)\\big)+\\beta(h),\\\\\\beta'(s)&=\\tilde{A}\\big(s,\\xi(s),\\varphi\\big)\\beta(s)+B_\\text{int}\\big(-s,\\xi(s)\\big),\\\\\\beta(0)&=B_0\\big(\\xi(0)\\big),\\\\\\xi'(s)&=f\\big(\\xi(s)\\big)+B_1\\big(\\xi(s)\\big)\\varphi(s-h)otag\\\\\t&\\phantom{=\\;}+\\int_{-h}^{-s}B_\\textrm{int}\\big(\\theta,\\xi(s)\\big)\\varphi(s+\\theta)\\,d\\theta,\\\\\\xi(h)&=y,\\\\\\tilde{A}(s,\\xi,\\varphi)&=A(\\xi)+\\sum_{i=1}^m\\mathcal{B}_1^i(\\xi)\\varphi_i(s-h)otag\\\\\t&\\phantom{=}\\:+\\sum_{i=1}^m\\int_{-h}^{-s}\\mathcal{B}_\\text{int}^i(\\theta,\\xi)\\varphi_i(s+\\theta)\\,d\\theta\\end{aligned}",
     u"\\begin{aligned}\\xi'(s)&=f\\big(\\xi(s)\\big)+B_1\\big(\\xi(s)\\big)u(t+s-h)onumber\\\\\t&\\phantom{=}\\:+\\int_{-h}^{-s}B_\\text{int}\\big(\\theta,\\xi(s)\\big)u(t+s+\\theta)\\,d\\theta,\\\\\\xi(0)&=x(t).\\end{aligned}",
     u"\\begin{aligned}\\tilde{\\xi}'(s)&=f\\big(\\tilde{\\xi}(s)\\big)+B_1\\big(\\tilde{\\xi}(s)\\big)u(t+\\Delta+s-h)onumber\\\\\t&\\phantom{=}\\:+\\int_{-h}^{-s}B_\\text{int}\\big(\\theta,\\tilde{\\xi}(s)\\big)u(t+\\Delta+s+\\theta)\\,d\\theta,\\\\\\tilde{\\xi}(0)&=x(t+\\Delta).\\end{aligned}",
     u'\\zeta(0)=x(t+\\Delta)-\\xi(\\Delta)=B_0\\big(\\xi(0)\\big)u(t)\\Delta+o(\\Delta),',
     u"\\begin{aligned}\\zeta'(s)&=\\tilde{\\xi}'(s)-\\xi'(s+\\Delta)onumber\\\\\t&=f\\big(\\tilde{\\xi}(s)\\big)-f\\big(\\xi(s+\\Delta)\\big)onumber\\\\\t&\\phantom{=}\\:+\\Big(B_1\\big(\\tilde{\\xi}(s)\\big)-B_1\\big(\\xi(s+\\Delta)\\big)\\Big)u(t+\\Delta+s-h)otag\\\\\t&\\phantom{=}\\:+\\int_{-h}^{-s}B_\\text{int}\\big(\\theta,\\tilde{\\xi}(s)\\big)u(t+\\Delta+s+\\theta)\\,d\\thetaotag\\\\\t&\\phantom{=}\\:-\\int_{-h}^{-s-\\Delta}B_\\text{int}\\big(\\theta,\\xi(s+\\Delta)\\big)u(t+\\Delta+s+\\theta)\\,d\\thetaonumber\\\\\t&=A\\big(\\xi(s)\\big)\\zeta(s)+\\sum_{i=1}^m\\mathcal{B}_1^i\\big(\\xi(s)\\big)\\zeta(s)u_i(t+s-h)otag\\\\\t&\\phantom{=}\\:+\\sum_{i=1}^m\\int_{-h}^{-s}\\mathcal{B}_\\text{int}^i\\big(\\theta,\\xi(s)\\big)\\zeta(s)u_i(t+s+\\theta)\\,d\\thetaotag\\\\\t&\\phantom{=}\\:+B_\\text{int}\\big(-s,\\xi(s)\\big)u(t)\\Delta+o(\\Delta)otag\\\\\t&=\\tilde{A}\\big(s,\\xi(s),u_t\\big)\\zeta(s)+B_\\text{int}\\big(-s,\\xi(s)\\big)u(t)\\Delta+o(\\Delta).\\end{aligned}",
     u'\\begin{aligned}\\zeta(h-\\Delta)&=\\beta(h)u(t)\\Delta+o(\\Delta).\\end{aligned}',
     u'\\begin{aligned}\\frac{y(t+\\Delta)-y(t)}{\\Delta}&=\\frac{\\tilde{\\xi}(h)-\\xi(h)}{\\Delta}onumber\\\\\t&=\\frac{\\tilde{\\xi}(h)-\\tilde{\\xi}(h-\\Delta)+\\zeta(h-\\Delta)}{\\Delta}onumber\\\\\t&=f\\big(y(t)\\big)+B\\big(y(t),u_t\\big)u(t)+\\frac{o(\\Delta)}{\\Delta},\\end{aligned}',
     u'\\begin{aligned}m_{v_0}\\Verty\\Vert^2\\leqv_0(y)\\leqM_{v_0}\\Verty\\Vert^2,\\\\w_0(y)\\geqm_{w_0}\\Verty\\Vert^2,\\\\\\Vert\\kappa(y,\\varphi)\\Vert\\leqM_\\kappa\\Verty\\Vert,\\\\\\big(f(y)+B(y,\\varphi)\\kappa(y,\\varphi)\\big)^Tablav_0(y)\\leq\t-w_0(y),\\end{aligned}',
     u'u(t)=\\kappa\\big(Y(x,u_t),u_t\\big);',
     u'\\big(x(t),u_t\\big)\\in\\mathcal{O}\\left(\\sqrt{\\frac{m_v}{M_v}}\\frac{R}{\\rho}\\right),',
     u'\\begin{aligned}m_v&=\\frac{1}{2\\rho^2}\\min\\big\\lbracem_{v_0},\\gammae^{-\\sigmah},\t\\rho^2\\gammae^{-\\sigmah}\\big\\rbrace,\\\\M_v&=\\gamma+\\rho^2M_{v_0},\\\\\\gamma&=\\frac{m_{w_0}}{2M_\\kappa^2},\\\\\\sigma&=\\frac{m_{w_0}}{2M_{v_0}};\\end{aligned}',
     u'\\Vertx(t)\\Vert^2+\\Vertu_t\\Vert^2\\leq\t\\frac{M_v}{m_v}e^{-\\sigmat}\\big(\t\t\\Vertx(0)\\Vert^2+\\Vertu_0\\Vert^2\t\\big).',
     u'v(x,\\varphi)=v_0\\big(Y(x,\\varphi)\\big)+\t\\gamma\\int_{-h}^0e^{\\sigma\\theta}\\Vert\\varphi(\\theta)\\Vert^2\\,d\\theta.',
     u'm_v\\big(\\Vertx\\Vert^2+\\Vert\\varphi\\Vert^2\\big)\\leq\tv(x,\\varphi)\\leq\tM_v\\big(\\Vertx\\Vert^2+\\Vert\\varphi\\Vert^2\\big)',
     u'v(t)=v\\big(x(t),u_t\\big).',
     u'\\begin{aligned}\\dot{v}(t)&\\leq\t\\big(-m_{w_0}+\\gammaM_\\kappa^2\\big)\\big\\VertY\\big(x(t),u_t\\big)\\big\\Vert^2onumber\\\\&\\phantom{\\leq}\\:-\\sigma\\gamma\\int_{-h}^0e^{\\sigma\\theta}\t\\Vertu(t+\\theta)\\Vert^2\\,d\\thetaonumber\\\\&\\leq-\\sigmav(t).\\end{aligned}',
     u'\\begin{aligned}\\Vertablav_0(y)\\Vert\\leqM_{ablav_0}\\Verty\\Vert,\\\\f^T(y)ablav_0(y)-k\\VertB^Tablav_0(y)\\Vert^2\\leq-w_0(y)\\end{aligned}',
     u'\\kappa(y,\\varphi)=-kB^T(y,\\varphi)ablav_0(y)',
     u'M_\\kappa=|k|\\max_{B\\in\\mathbb{B}}\\VertB\\VertM_{ablav_0}.',
     u'\\dot{x}(t)=f\\big(x(t)\\big)+b_0u(t)+b_1u(t-h)+\\int_{-h}^0b_\\text{int}(\\theta)u(t+\\theta)\\,d\\theta',
     u'\\kappa(y,\\varphi)=\\frac{-f(y)-y}{B(y,\\varphi)}.',
     u'\\left\\lbrace\\begin{array}{l}\t\\dot{x}_1(t)=x_2^2(t)+u(t-h),\\\\\t\\dot{x}_2(t)=x_2(t)+u(t)\\end{array}\\right.',
     u'\\begin{aligned}y_1&=x_1+\\frac{e^{2h}-1}{2}x_2^2+\\int_{-h}^0u(t+\\theta)\\,d\\theta,\\\\y_2&=e^hx_2\\end{aligned}',
     u'\\left\\lbrace\\begin{array}{l}\t\\dot{y}_1=y_2^2+\\big(1+\\big(e^h-e^{-h}\\big)y_2\\big)u,\\\\\t\\dot{y}_2=y_2+e^hu.\\end{array}\\right.',
     u'\\begin{aligned}z_1&=y_1-e^{-h}y_2+\\frac{e^{-2h}-1}{2}y_2^2,\\\\z_2&=e^{-h}y_2,\\\\u&=-2z_2+\\tilde{u}\\end{aligned}',
     u'\\left\\lbrace\\begin{array}{l}\t\\dot{z}_1=z_2^2-z_2,\\\\\t\\dot{z}_2=-z_2+\\tilde{u}.\\end{array}\\right.',
     u'V(z)=\\big(z_1+\\tfrac{1}{2}z_2(z_2-2)\\big)^2+z_2^2.',
     u'\\dot{V}=-2z_2^2+\\frac{\\partialV}{\\partialz_2}\\tilde{u}',
     u'\\tilde{u}=-\\frac{\\partialV}{\\partialz_2}',
     u'\\begin{aligned}z_1&=x_1-x_2+\\int_{-h}^0u(t+\\theta)\\,d\\theta,\\\\z_2&=x_2,\\\\u&=-2x_2+\\tilde{u}.\\end{aligned}',
     u'\\left\\lbrace\\begin{array}{l}\t\\dot{x}_1(t)=x_2(t),\\\\\t\\dot{x}_2(t)=\\sinx_1(t)+u(t)+u(t-h),\\quadh=\\pi/4.\\end{array}\\right.',
     u'f(x)=\\begin{pmatrix}\tx_2\\\\\t\\sinx_1\\end{pmatrix},\\quadB_0=B_1=\\begin{pmatrix}\t0\\\\\t1\\end{pmatrix}.',
     u'A(x)=\\begin{pmatrix}\t0&1\\\\\t\\cosx_1&0\\end{pmatrix},\\quadM_f=1,\\quadR=\\infty.',
     u"\\begin{aligned}\\beta'(s)&=A\\big(\\xi(s)\\big)\\beta(s),\\\\\\beta(0)&=\\begin{pmatrix}\t0\\\\1\\end{pmatrix}.\\end{aligned}",
     u'\\begin{aligned}\\frac{d}{ds}\\big(\\beta_1^2+\\beta_2^2\\big)&\\in\\big[0,2\\big(\\beta_1^2+\\beta_2^2\\big)\\big],\\\\\\frac{d}{ds}\\left(\\frac{\\beta_1}{\\beta_2}\\right)&\\in\\left[\t1-\\left(\\frac{\\beta_1}{\\beta_2}\\right)^2,\t1+\\left(\\frac{\\beta_1}{\\beta_2}\\right)^2\\right]\\end{aligned}',
     u'\\begin{aligned}\\beta_1^2(h)+\\beta_2^2(h)&\\in\\Big[1,e^{2h}\\Big],\\\\\\frac{\\beta_1(h)}{\\beta_2(h)}&\\in\\big[\\tanhh,\\tanh\\big],\\end{aligned}',
     u'v_0(y)=y^TVy',
     u'2f^T(y)Vy-4k(Vy)^TBB^TVy\\leq-w_0(y).',
     u'f(y)=F(y)y,',
     u'\\begin{aligned}F(y)=\\begin{pmatrix}\t0&1\\\\\t\\alpha(y)&0\\end{pmatrix},\\\\\\alpha(y)=\\frac{\\siny_1}{y_1}\\in[-0.22,1],\\end{aligned}',
     u'V^{-1}F^T(y)+F(y)V^{-1}-4kBB^T',
     u'\\kappa(y,\\varphi)=-B^T(y,\\varphi)y.',
     u'\\int_a^b{f^p(x)dx\\geqslant}\\left(\\int_a^b{f(x)dx}\\right)^{p-1}',
     u'\\int_a^b{f^{p+2}(x)dx\\geqslant}\\frac{1}{{(b-a)^{p-1}}}\\left(\\int_a^b{f(x)dx}\\right)^{p+1}.',
     u'\\int_a^b{f^{p+2}(x)d_qx\\geqslant}\\frac{1}{{(b-a)^{p-1}}}\\left(\\int_a^b{f(qx)d_qx}\\right)^{p+1}.',
     u'\\int_a^b{f^t(x)dx-}\\left({\\int_a^b{f(x)dx}}\\right)^{t-1}\\geqslantf^{t-1}(a)\\int_a^b{f(x)dx}.',
     u'\\int_a^b{f^{p+2}(x)dx-}\\frac{1}{{(b-a)^{p-1}}}\\left(\\int_a^b{f(x)dx}\\right)^{p+1}\\geqslantf^{p+1}(a)\\int_a^b{f(x)dx}.',
     u'\\sigma(t)=\\inf\\left\\{{s\\in{\\mathbb{T}}:s>t}\\right\\},',
     u'\\rho(t)=\\sup\\left\\{{s\\in{\\mathbb{T}}:s<t}\\right\\},',
     u'\\begin{aligned}|f(\\sigma\\left(t\\right))-f\\left(s\\right)-f^\\Delta(t)(\\sigma\\left(t\\right)-s)|<\\varepsilon|\\sigma\\left(t\\right)-s|\\hfill\\\\(|f(\\rho\\left(t\\right))-f\\left(s\\right)-f^abla(t)(\\rho\\left(t\\right)-s)|<\\varepsilon|\\rho\\left(t\\right)-s|)\\hfill\\\\\\end{aligned}',
     u'\\begin{aligned}(fg)^\\Delta=f^\\Deltag+f^\\sigmag^\\Delta=fg^\\Delta+f^\\Deltag^\\sigma\\hfill\\\\((fg)^abla=f^ablag+f^\\rhog^abla=fg^abla+f^ablag^\\rho)\\hfill\\\\\\end{aligned}',
     u'\\begin{aligned}\\left({\\frac{f}{g}}\\right)^\\Delta=\\frac{{f^\\Deltag-fg^\\Delta}}{{gg^\\sigma}}\\hfill\\\\\\biggl(\\left({\\frac{f}{g}}\\right)^abla=\\frac{{f^ablag-fg^abla}}{{gg^\\rho}}\\biggr)\\hfill\\\\\\end{aligned}',
     u'f^\\Delta(t)=\\frac{f(t+q)-f(t)}{q}',
     u'f^abla(t)=\\frac{f(t)-f(t-q)}{q}',
     u'f^\\Delta(t)=\\frac{f(qt)-f(t)}{(q-1)t},teq0',
     u'f^abla(t)=\\frac{f(t)-f(t/q)}{\\biggl(t-t/q\\biggr)},teq0',
     u'\\begin{aligned}\\int\\limits_b^c{f(t)\\Deltat=F(c)-F(b)}\\hfill\\\\\\biggl(\\int\\limits_b^c{f(t)ablat=F(c)-F(b)}\\biggr)\\hfill\\\\\\end{aligned}',
     u'\\sigma,\\rho,f^\\sigma(x),(f^\\sigma(x))^p,f^\\rho(x),(f^\\rho(x))^p~~~~~p\\in\\mathbb{N}',
     u'pg^{p-1}(x)g^\\Delta(x)\\leqslant(g^p(x))^\\Delta\\leqslantp(g^\\sigma(x))^{p-1}g^\\Delta(x).',
     u'\\int_t^{\\sigma(t)}{f(\\tau)\\Delta\\tau=}f(t)\\mu(t).',
     u'p(g^\\rho(x))^{p-1}g^abla(x)\\leqslant(g^p(x))^abla\\leqslantpg^{p-1}(x)g^abla(x).',
     u'\\left({g^2}\\right)^abla=\\left({g+g^\\rho}\\right)g^abla.',
     u'\\left({g^p}\\right)^abla=\\left({g^{p-1}+g^\\rhog^{p-2}+\\cdots+\\left({g^\\rho}\\right)^{p-1}}\\right)g^abla.',
     u'f^{t-2}(x)f^\\Delta(x)\\geqslant(t-2)(f^{\\sigma^2}(x))^{t-2}(\\sigma^2(x)-a)^{t-3}\\sigma^\\Delta(x)',
     u'\\begin{aligned}\\int_a^b{f^t(x)\\Deltax-\\left(\\int_a^b{f(x)\\Deltax}\\right)^{t-1}}\\hfill\\\\\\geqslantf^{t-2}(a)\\left[{f(a)-(t-1)\\mu^{t-2}(a)}\\right]\\int_a^b{f(x)\\Deltax}.\\end{aligned}',
     u'F(x)=\\int_a^x{f^t(u)\\Deltau-\\left(\\int_a^x{f(u)\\Deltau}\\right)^{t-1}}',
     u'\\begin{aligned}F^\\Delta(x)\\geqslantf^t(x)-(t-1)(g^\\sigma(x))^{t-2}g^\\Delta(x)\\hfill\\\\=f(x)F_1(x)\\hfill\\\\\\end{aligned}',
     u'F_1^\\Delta(x)\\geqslant(t-1)f^{t-2}(x)f^\\Delta(x)-(t-1)(t-2)(g^{\\sigma^2}(x))^{t-3}f^\\sigma(x)\\sigma^\\Delta(x).',
     u'g^{\\sigma^2}(x)=\\int_a^{\\sigma^2(x)}{f(u)\\Deltau}\\leqslantf^{\\sigma^2}(x)(\\sigma^2(x)-a).',
     u'\\begin{aligned}F_1^\\Delta(x)\\hfill\\\\\\geqslant(t-1)[f^{t-2}(x)f^\\Delta(x)-(t-2)(f^{\\sigma^2}(x))^{t-3}f^\\sigma(x)(\\sigma^2(x)-a)^{t-3}\\sigma^\\Delta(x)]\\hfill\\\\\\geqslant(t-1)[f^{t-2}(x)f^\\Delta(x)-(t-2)(f^{\\sigma^2}(x))^{t-2}(\\sigma^2(x)-a)^{t-3}\\sigma^\\Delta(x)]\\hfill\\\\\\geqslant0\\hfill\\\\\\end{aligned}',
     u'F_1(x)\\geqslantF_1(a)=f^{t-2}(a)[f(a)-(t-1)\\mu^{t-2}(a)]',
     u'F^{\\Delta}(x)\\geqslantf^{t-2}(a)[f(a)-(t-1)\\mu^{t-2}(a)]f(x)',
     u'\\biggl(F(x)-f^{t-2}(a)[f(a)-(t-1)\\mu^{t-2}(a)]g(x)\\biggr)^\\Delta\\geqslant0.',
     u'\\begin{aligned}F(b)-f^{t-2}(a)[f(a)-(t-1)\\mu^{t-2}(a)]g(b)\\hfill\\\\\\geqslantF(a)-f^{t-2}(a)[f(a)-(t-1)\\mu^{t-2}(a)]g(a)\\hfill\\\\=0.\\hfill\\\\\\end{aligned}',
     u'f^{p}(x)f^\\Delta(x)\\geqslant\\frac{p}{(b-a)^{p-1}}\\biggl(f^{\\sigma^2}(x)\\biggr)^{p}\\biggl(\\sigma^2(x)-a\\biggr)^{p-1}\\sigma^\\Delta(x).',
     u'\\begin{aligned}\\int_a^b{f^{p+2}(x)\\Deltax-\\frac{1}{(b-a)^{p-1}}\\left(\\int_a^b{f(x)\\Deltax}\\right)^{p+1}}\\hfill\\\\\\geqslantf^p(a)\\left[{f(a)-\\frac{{p+1}}{{(b-a)^{p-1}}}\\mu^p(a)}\\right]\\int_a^b{f(x)\\Deltax}.\\end{aligned}',
     u'G(x)=\\int_a^x{f^{p+2}(t)\\Deltat-\\frac{1}{(b-a)^{p-1}}\\left(\\int_a^x{f(t)\\Deltat}\\right)^{p+1}}',
     u'\\begin{aligned}G^\\Delta(x)=f^{p+2}(x)-\\frac{1}{(b-a)^{p-1}}(g^{p+1}(x))^\\Delta\\hfill\\\\\\geqslantf^{p+2}(x)-\\frac{p+1}{(b-a)^{p-1}}(g^\\sigma(x))^{p}g^\\Delta(x)\\hfill\\\\\\geqslantf(x)\\left[f^{p+1}(x)-\\frac{p+1}{(b-a)^{p-1}}(g^\\sigma(x))^{p}\\right]\\hfill\\\\=f(x)G_1(x)\\hfill\\\\\\end{aligned}',
     u'\\begin{aligned}G_1^\\Delta(x)\\hfill\\\\\\geqslant(p+1)f^{p}(x)f^\\Delta(x)-\\frac{p(p+1)}{(b-a)^{p-1}}\\biggl(g^{\\sigma^2}(x)\\biggr)^{p-1}f^{\\sigma}(x)\\sigma^\\Delta(x)\\hfill\\\\\\geqslant(p+1)\\left[f^{p}(x)f^\\Delta(x)-\\frac{p}{(b-a)^{p-1}}\\biggl(f^{\\sigma^2}(x)\\biggr)^{p}\\biggl(\\sigma^2(x)-a\\biggr)^{p-1}\\sigma^\\Delta(x)\\right]\\hfill\\\\\\geqslant0\\hfill\\\\\\end{aligned}.',
     u'\\begin{array}{l}G^\\Delta(x)\\gef(x)G_1(a)\\\\\\Leftrightarrow\\left({G(x)-g(x)G_1(a)}\\right)^\\Delta\\geqslant0\\\\\\end{array}',
     u'G(x)-g(x)G_1(a)\\geqslantG(a)-g(a)G_1(a)=0.',
     u'f^{p-3}(x)f^\\Delta(x)\\geqslant(p-2)\\biggl(f^{\\sigma^2}(x)\\biggr)^{p-3}\\biggl(\\sigma^2(x)-a\\biggr)^{p-3}\\sigma^\\Delta(x).',
     u'\\begin{aligned}\\int_a^b{f^p(x)\\Deltax}-\\left(\\int_a^b{f^{\\rho}(x)\\Deltax}\\right)^{p-1}\\hfill\\\\\\geqslant(f(a))^{p-2}[f(a)-(p-1)\\mu^{p-2}(a)]\\int_a^b{f(\\rho(x))\\Deltax}.\\hfill\\\\\\end{aligned}',
     u'H(x)=\\int_a^x{f^p(t)\\Deltat-\\left(\\int_a^x{f^{\\rho}(t)\\Deltat}\\right)^{p-1}}',
     u'\\begin{aligned}H^\\Delta(x)=f^p(x)-(g^{p-1}(x))^\\Delta\\hfill\\\\\\geqslantf^p(x)-(p-1)(g^\\sigma(x))^{p-2}g^\\Delta(x)\\hfill\\\\\\geqslantf(\\rho(x))H_1(x)\\hfill\\\\\\end{aligned}',
     u'\\begin{aligned}H_1^\\Delta(x)\\geqslant(p-1)f^{p-2}(x)f^\\Delta(x)-(p-1)(p-2)\\biggl(g^{\\sigma^2}(x)\\biggr)^{p-3}f(x)(\\sigma(x))^\\Delta\\hfill\\\\\\geqslant(p-1)f(x)\\biggl[f^{p-3}(x)f^\\Delta(x)-(p-2)\\biggl(f^{\\sigma^2}(x)\\biggr)^{p-3}\\biggl(\\sigma^2(x)-a\\biggr)^{p-3}\\sigma^\\Delta(x)\\biggr]\\geqslant0\\hfill\\\\\\end{aligned}',
     u'\\begin{aligned}H_1(x)\\geqslantH_1(a)=f^{p-1}(a)-(p-1)(g^\\sigma(a))^{p-2}\\hfill\\\\=(f(a))^{p-2}[f(a)-(p-1)\\mu^{p-2}(a)]\\hfill\\\\\\end{aligned}',
     u'(f^\\sigma(x))^\\Delta\\geqslantp\\sigma^\\Delta(x),',
     u'\\begin{aligned}\\int_a^b{(f^\\sigma(x))^{p+2}\\Deltax}-\\frac{1}{{(b-a)^{p-1}}}\\left(\\int_a^b{f^\\rho(x)\\Deltax}\\right)^{p+1}\\hfill\\\\\\geqslant\\biggl[(f^\\sigma(a))^{p+1}-\\frac{{p+1}}{{(b-a)^{p-1}}}(f^\\rho(a)\\mu(a))^p\\biggr]\\int_a^b{f(\\rho(x))\\Deltax}.\\hfill\\\\\\end{aligned}',
     u'W(x)=\\int_a^x{(f^{\\sigma}(t))^p\\Deltat-\\left(\\int_a^x{f^{\\rho}(t)\\Deltat}\\right)^{p-1}}',
     u'\\begin{aligned}W^\\Delta(x)\\geqslant(f^\\sigma(x))^{p+2}-\\frac{{p+1}}{{(b-a)^{p-1}}}(g^\\sigma(x))^pg^\\Delta(x)\\hfill\\\\\\geqslantf^\\sigma(x)\\left[f^\\sigma(x))^{p+1}-\\frac{{p+1}}{{(b-a)^{p-1}}}(g^\\sigma(x))^p\\right]\\hfill\\\\\\geqslantf(\\rho(x))W_1(x)\\hfill\\\\\\end{aligned}',
     u'\\begin{aligned}W_1^\\Delta(x)\\hfill\\\\\\geqslant(p+1)\\left[\\biggl(f^\\sigma(x)\\biggr)^p\\biggl(f^\\sigma(x)\\biggr)^\\Delta-\\frac{{p}}{{(b-a)^{p-1}}}\\biggl(g^{\\sigma^2}(x)\\biggr)^{p-1}f(x)(\\sigma(x))^\\Delta\\right]\\hfill\\\\\\end{aligned}',
     u'g^{\\sigma^2}(x)=\\int_a^{\\sigma^2(x)}{f^{\\rho}(t)\\Deltat}\\leqslantf^{\\rho\\sigma^2}(x)(\\sigma^2(x)-a)\\leqslantf^\\sigma(x)(b-a).',
     u'W_1^{\\Delta}(x)\\geqslant(p+1)\\biggl(f^\\sigma(x)\\biggr)^p\\biggl[\\biggl(f^\\sigma(x)\\biggr)^\\Delta-p(\\sigma(x))^\\Delta\\biggr].',
     u'\\begin{aligned}W_1(x)\\geqslantW_1(a)\\hfill\\\\=(f^\\sigma(a))^{p+1}-\\frac{{p+1}}{{(b-a)^{p-1}}}(g^\\sigma(a))^p\\hfill\\\\=(f^\\sigma(a))^{p+1}-\\frac{{p+1}}{{(b-a)^{p-1}}}(f^\\rho(a)\\mu(a))^p\\hfill\\\\\\end{aligned}',
     u'\\biggl(f^\\sigma(x)\\biggr)^\\Delta\\geqslant(p-2)\\biggl(\\sigma^2(x)-a\\biggr)^{p-3}\\sigma^\\Delta(x)',
     u'\\begin{aligned}\\int_a^b{f^{\\sigma}(x))^{p}\\Deltax}-\\left(\\int_a^b{f^{\\rho}(x)\\Deltax}\\right)^{p-1}\\hfill\\\\\\geqslant(f^\\sigma(a))^{p-2}\\biggl[f^\\sigma(a)-(p-1)\\mu^{p-2}(a)\\biggr]\\int_a^b{f(\\rho(x))\\Deltax}.\\hfill\\\\\\end{aligned}',
     u'Q(x)=\\int_a^x{(f^{\\sigma}(t))^p\\Deltat-\\left(\\int_a^x{f^{\\rho}(t)\\Deltat}\\right)^{p-1}}',
     u'\\begin{aligned}Q^\\Delta(x)=(f^\\sigma(x))^p-(g^{p-1}(x))^\\Delta\\hfill\\\\\\geqslant(f^\\sigma(x))^p-(p-1)(g^\\sigma(x))^{p-2}g^\\Delta(x)\\hfill\\\\\\geqslantf(\\rho(x))Q_1(x)\\hfill\\\\\\end{aligned}',
     u'\\begin{aligned}Q_1^\\Delta(x)\\hfill\\\\\\geqslant(p-1)[(f^\\sigma(x))^{p-2}(f^\\sigma(x))^\\Delta-(p-2)(g^{\\sigma^2}(x))^{p-3}(g^\\sigma(x))^\\Delta]\\hfill\\\\\\geqslant(p-1)\\biggl(f^\\sigma(x)\\biggr)^{p-2}\\biggl[(f^\\sigma(x))^\\Delta-(p-2)\\biggl(\\sigma^2(x)-a\\biggr)^{p-3}\\sigma^\\Delta(x)\\biggr]\\hfill\\\\\\geqslant0\\hfill\\\\\\end{aligned}.',
     u'\\begin{aligned}Q_1(x)\\geqslantQ_1(a)=(f^\\sigma(a))^{p-1}-(p-1)(g^\\sigma(a))^{p-2}\\hfill\\\\\\geqslant(f^\\sigma(a))^{p-2}\\biggl(f^\\sigma(a)-(p-1)\\mu^{p-2}(a)\\biggr)\\hfill\\\\\\end{aligned}',
     u'\\biggl(Q(x)-g(x)(f^\\sigma(a))^{p-2}\\biggl(f^\\sigma(a)-(p-1)\\mu^{p-2}(a)\\biggr)\\biggr)^\\Delta\\geqslant0.',
     u'\\biggl(f^\\rho(x)\\biggr)^{t-2}f^abla(x)\\geqslant(t-2)(f(x))^{t-2}(x-a)^{t-3}.',
     u'\\int_a^b{f^t(x)ablax-\\left(\\int_a^b{f(x)ablax}\\right)^{t-1}}\\geqslantf^{t-1}(a)\\int_a^b{f(x)ablax}.',
     u'F(x)=\\int_a^x{f^t(u)ablau-\\left(\\int_a^x{f(u)ablau}\\right)^{t-1}}',
     u'\\begin{aligned}F^abla(x)\\hfill\\\\\\geqslantf^t(x)-(t-1)(g(x))^{t-2}g^abla(x)\\hfill\\\\=f(x)F_1(x)\\hfill\\\\\\end{aligned}',
     u'F_1^abla(x)\\geqslant(t-1)(f^\\rho(x))^{t-2}(x)f^abla(x)-(t-1)(t-2)(g(x))^{t-3}f(x).',
     u'g(x)=\\int_a^{x}{f(t)ablat}\\leqslantf(x)(x-a).',
     u'\\begin{aligned}F_1^abla(x)\\hfill\\\\\\geqslant(t-1)\\biggl[(f^{\\rho}(x))^{t-2}f^abla(x)-(t-2)(f(x))^{t-2}(x-a)^{t-3}\\biggr]\\hfill\\\\\\geqslant0\\hfill\\\\\\end{aligned}',
     u'F_1(x)\\geqslantF_1(a)',
     u'F^{abla}(x)\\geqslantF_1(a)f(x).',
     u'\\biggl(F(x)-F_1(a)g(x)\\biggr)^abla\\geqslant0.',
     u'F(b)-f^{t-1}(a)g(b)\\geqslantF(a)-f^{t-1}(a)g(a)=0.',
     u'(f^{\\rho}(x))^{p}f^abla(x)\\geqslant\\frac{p}{(b-a)^{p-1}}(f^{p}(x))(x-a)^{p-1}.',
     u'\\int_a^b{f^{p+2}(x)ablax-\\frac{1}{(b-a)^{p-1}}\\left(\\int_a^b{f(x)ablax}\\right)^{p+1}}\\geqslantf^{p+1}(a)\\int_a^b{f(x)ablax}.',
     u'G(x)=\\int_a^x{f^{p+2}(t)ablat-\\frac{1}{(b-a)^{p-1}}\\left(\\int_a^x{f(t)ablat}\\right)^{p+1}}',
     u'\\begin{aligned}G^abla(x)=f^{p+2}(x)-\\frac{1}{(b-a)^{p-1}}(g^{p+1}(x))^abla\\hfill\\\\\\geqslantf^{p+2}(x)-\\frac{p+1}{(b-a)^{p-1}}g^{p}(x)g^abla(x)\\hfill\\\\\\geqslantf(x)\\left[f^{p+1}(x)-\\frac{p+1}{(b-a)^{p-1}}g^{p}(x)\\right]\\hfill\\\\=f(x)G_1(x)\\hfill\\\\\\end{aligned}',
     u'\\begin{aligned}G_1^abla(x)\\hfill\\\\\\geqslant(p+1)\\left[(f^{\\rho}(x))^{p}f^abla(x)-\\frac{p}{(b-a)^{p-1}}f^{p}(x)(x-a)^{p-1}\\right]\\hfill\\\\\\geqslant0\\hfill\\\\\\end{aligned}.',
     u'\\begin{array}{l}G^abla(x)\\gef(x)G_1(a)\\\\\\Leftrightarrow\\left({G(x)-g(x)G_1(a)}\\right)^abla\\geqslant0\\\\\\end{array}',
     u'G(x)-g(x)G_1(a)\\geqslantG(a)-g(a)G_1(a)=0.',
     u'\\begin{aligned}\\Pr([A\\in\\Gamma]\\&[B\\in\\Delta]\\&...)=\\left\\langle\\Psi(t_{1})|P^{A}(\\Gamma).P^{B}(\\Delta)...|\\Psi(t_{1})\\right\\rangle\\\\=\\mu_{\\Psi(t_{1})}(\\mathcal{M}_{\\Gamma}^{A}\\wedge\\mathcal{M}_{\\Delta}^{B}\\wedge...)otag\\end{aligned}',
     u'\\Psi^{-}=1/\\surd2(\\left\\vertHV\\right\\rangle-\\left\\vertVH\\right\\rangle).',
     u'\\left\\vert\\Psi_{12}^{-}\\right\\rangle\\left\\vert\\Psi_{34}^{-}\\right\\rangle=1/2(\\left\\vert\\Psi_{14}^{+}\\right\\rangle\\left\\vert\\Psi_{23}^{+}\\right\\rangle-\\left\\vert\\Psi_{14}^{-}\\right\\rangle\\left\\vert\\Psi_{23}^{-}\\right\\rangle-\\left\\vert\\Phi_{14}^{+}\\right\\rangle\\left\\vert\\Phi_{23}^{+}\\right\\rangle+\\left\\vert\\Phi_{14}^{-}\\right\\rangle\\left\\vert\\Phi_{23}^{-}\\right\\rangle.',
     u'{\\calH}=-\\sum\\limits_{\\langlei,j\\rangle}J_{ij}S_iS_j\\,,',
     u'{\\calH}=-\\frac{1}{\\sqrt{N}}\\sum\\limits_{i<j}J_{ij}S_iS_j\\,.',
     u'q_D=\\lim_{t\\to\\infty}\\overline{q_t}\\,.',
     u'q_t(N)=\\frac{1}{N}\\sum\\limits_{i=1}^{N}S_i^1(t)S_i^2(t)',
     u'Pr(A,B|\\Lambda,M,N)=Pr(A|\\Lambda,N).Pr(B|\\Lambda,M)',
     u'Pr(A,B|a,b,c,\\lambda)=Pr(A|a,c,\\lambda).Pr(B|b,c,\\lambda)\\tag{Factorizability}',
     u'Pr_{a,b}(A|B,\\lambda)=Pr_{a}(A|\\lambda)\\tag{4a}',
     ...]

```python
import json
json.dump(docs, open("formula_list.json", "w+"))
```

```python
class Query:
    '''
    The important thing here is that the labels on the data coincide with the labels on the columns.
    '''
    def __init__(self, index, docs, dictionary, tokenize_latex, cols):
        self.index = index
        self.docs = docs
        self.dictionary = dictionary
        self.tokenize_latex = tokenize_latex
        self.columns = cols
    def _convert_query(self, query):
        query = self.dictionary.doc2bow(self.tokenize_latex(query))
        sims = self.index[query]
        neighbors = sorted(sims, key=lambda item: -item[1])
        neighbors = {"neighbors":[{self.columns[0]: {"data": self.docs[n[0]], "fmt": "math"}, self.columns[1]: {"data": float(n[1])}} for n in neighbors]} if neighbors else {"neighbors": []}
        return neighbors
    def query(self, q):
        return self._convert_query(q)

```

```python
index
```

    <gensim.similarities.docsim.Similarity at 0x10e8bbad0>

```python
dictionary
```

    <gensim.corpora.dictionary.Dictionary at 0x110c47110>

```python
q = Query(index, docs, dictionary, tokenize_latex, ["neighbor", "similarity_score"])
q.query("\\frac")
```

    2017-12-01 11:17:28,732 : INFO : creating matrix with 4 documents and 44 features
    2017-12-01 11:17:28,736 : INFO : creating dense shard #0
    2017-12-01 11:17:28,741 : INFO : saving index shard to index/.0
    2017-12-01 11:17:28,745 : INFO : saving MatrixSimilarity object under index/.0, separately None
    2017-12-01 11:17:28,748 : INFO : saved index/.0
    2017-12-01 11:17:28,751 : INFO : loading MatrixSimilarity object from index/.0
    2017-12-01 11:17:28,752 : INFO : loaded index/.0

    {'neighbors': [{'neighbor': {'data': u'\\begin{aligned}\\min_{L,S}\\sum_{m=1}^{M}\\sum_{i=1}^{N_{m}}&\\frac{1}{2}[max(0,1-Y_{m}^{i}(Ls^{m})^{T}X_{m}^{i})]^{2}\\\\&+\\gamma\\|L\\|_{1}+\\lambda\\|L\\|^{2}_{F}\\\\\\end{aligned}',
        'fmt': 'math'},
       'similarity_score': {'data': 0.0750708281993866}},
      {'neighbor': {'data': u'\\begin{aligned}\\min_{L,S}\\sum_{m=1}^{M}\\sum_{i=1}^{N_{m}}&\\frac{1}{2}[max(0,1-Y_{m}^{i}(Ls^{m})^{T}X_{m}^{i})]^{2}\\\\&+\\mu\\sum_{k=1}^{K}\\sum_{g=1}^{G}\\|s_{k}^{g}\\|_{2}\\\\\\end{aligned}',
        'fmt': 'math'},
       'similarity_score': {'data': 0.061607733368873596}},
      {'neighbor': {'data': u'\\begin{aligned}\\min_{L,S}\\sum_{m=1}^{M}\\sum_{i=1}^{N_{m}}&\\frac{1}{2}[max(0,1-Y_{m}^{i}(Ls^{m})^{T}X_{m}^{i})]^{2}\\\\&+\\mu\\sum_{k=1}^{K}\\sum_{g=1}^{G}\\|s_{k}^{g}\\|_{2}+\\gamma\\|L\\|_{1}+\\lambda\\|L\\|^{2}_{F}\\\\\\end{aligned}',
        'fmt': 'math'},
       'similarity_score': {'data': 0.052263256162405014}}]}

```python
from mathviz_hopper.src.table import Table
t = Table(q)
t.print_ipython()

# TODO: lambda function to convert each column
# Schema:
# also, enable non-autocomplete selections (the models should be able to handle this)
# also, add a warning if the settings are still loading, and other notifications.
# Data: [{Column A: {name: A, index: [], filter_func: lambda x : f(x)}} , {Column B: {name: B, index: [], filter_func: lambda x : f(x)}}
```

    Bottle v0.13-dev server starting up (using MyWSGIRefServer())...
    Listening on http://localhost:8081/
    Hit Ctrl-C to quit.

<!DOCTYPE html><html lang="en"><head><meta charset="utf-8"><meta name="viewport" content="width=device-width,initial-scale=1,shrink-to-fit=no"><meta name="theme-color" content="#000000"><link rel="manifest" href="./viz/manifest.json"><link rel="stylesheet" href="./viz/katex.css"><title>React App</title><link href="./viz/static/css/main.2d3f6c1d.css" rel="stylesheet"></head><body><noscript>You need to enable JavaScript to run this app.</noscript><div id="root"></div></body></html>

    127.0.0.1 - - [01/Dec/2017 11:17:30] "GET /settings HTTP/1.1" 200 1007

```python
import requests
r = requests.post('http://localhost:8081/query', json={"query": "\\begin{equation}\n\\Phi_{z}(L) = \\sum_{i=1}^{N} \\frac{1}{C_{i} \\times V_{\\rm max, i}} ,\n\\label{EQ1}\n\\end{equation}\n"})
```

    127.0.0.1 - - [01/Dec/2017 11:17:30] "POST /query HTTP/1.1" 200 915

```python
r.json()
```

    {u'neighbors': [{u'neighbor': {u'data': u'\\begin{aligned}\\min_{L,S}\\sum_{m=1}^{M}\\sum_{i=1}^{N_{m}}&\\frac{1}{2}[max(0,1-Y_{m}^{i}(Ls^{m})^{T}X_{m}^{i})]^{2}\\\\&+\\gamma\\|L\\|_{1}+\\lambda\\|L\\|^{2}_{F}\\\\\\end{aligned}',
        u'fmt': u'math'},
       u'similarity_score': {u'data': 0.2967531681060791}},
      {u'neighbor': {u'data': u'\\begin{aligned}\\min_{L,S}\\sum_{m=1}^{M}\\sum_{i=1}^{N_{m}}&\\frac{1}{2}[max(0,1-Y_{m}^{i}(Ls^{m})^{T}X_{m}^{i})]^{2}\\\\&+\\mu\\sum_{k=1}^{K}\\sum_{g=1}^{G}\\|s_{k}^{g}\\|_{2}\\\\\\end{aligned}',
        u'fmt': u'math'},
       u'similarity_score': {u'data': 0.29066944122314453}},
      {u'neighbor': {u'data': u'\\begin{aligned}\\min_{L,S}\\sum_{m=1}^{M}\\sum_{i=1}^{N_{m}}&\\frac{1}{2}[max(0,1-Y_{m}^{i}(Ls^{m})^{T}X_{m}^{i})]^{2}\\\\&+\\mu\\sum_{k=1}^{K}\\sum_{g=1}^{G}\\|s_{k}^{g}\\|_{2}+\\gamma\\|L\\|_{1}+\\lambda\\|L\\|^{2}_{F}\\\\\\end{aligned}',
        u'fmt': u'math'},
       u'similarity_score': {u'data': 0.2799034118652344}}]}

```python
import requests
r = requests.get('http://localhost:8081/settings')
```

    127.0.0.1 - - [30/Nov/2017 14:03:54] "GET /settings HTTP/1.1" 200 1007

```python
r.json()
```

    {u'columns': [{u'Header': u'neighbor', u'accessor': u'neighbor'},
      {u'Header': u'similarity_score', u'accessor': u'similarity_score'}],
     u'docs': {u'\\': {u'b': {u'e': {u'g': {u'i': {u'n': {u'{': {u'a': {u'l': {u'i': {u'g': {u'n': {u'e': {u'd': {u'}': {u'\\': {u'm': {u'i': {u'n': {u'_': {u'{': {},
                          u'{L,S}\\sum_{m=1}^{M}\\sum_{i=1}^{N_{m}}&\\frac{1}{2}[max(0,1-Y_{m}^{i}(Ls^{m})^{T}X_{m}^{i})]^{2}\\\\&+\\gamma\\|L\\|_{1}+\\lambda\\|L\\|^{2}_{F}\\\\\\end{aligned}': {u'full_word': 1},
                          u'{L,S}\\sum_{m=1}^{M}\\sum_{i=1}^{N_{m}}&\\frac{1}{2}[max(0,1-Y_{m}^{i}(Ls^{m})^{T}X_{m}^{i})]^{2}\\\\&+\\mu\\sum_{k=1}^{K}\\sum_{g=1}^{G}\\|s_{k}^{g}\\|_{2}+\\gamma\\|L\\|_{1}+\\lambda\\|L\\|^{2}_{F}\\\\\\end{aligned}': {u'full_word': 1},
                          u'{L,S}\\sum_{m=1}^{M}\\sum_{i=1}^{N_{m}}&\\frac{1}{2}[max(0,1-Y_{m}^{i}(Ls^{m})^{T}X_{m}^{i})]^{2}\\\\&+\\mu\\sum_{k=1}^{K}\\sum_{g=1}^{G}\\|s_{k}^{g}\\|_{2}\\\\\\end{aligned}': {u'full_word': 1}}}}}},
                     u'w': {u'^': {u'{': {u'm': {u'}': {u'=': {},
                          u'=Ls^{m}\\end{aligned}': {u'full_word': 1}}}}}}}}}}}}}}}}}}}}}},
     u'port': 8081}

```python

def insert(st, trie):
    i = 0
    for s in st:
        if s not in trie.keys():
            trie[s] = {}
        if i == 20:
            break
        trie = trie[s]
        i+=1

    if i == 20:
        trie[st[i:]] = {}
        trie = trie[st[i:]]
    trie["full_word"] = 1

def construct_trie(list_of_str):
    trie = {}
    for st in list_of_str:
        insert(st, trie)
    return trie

```

```python
docs_trie = construct_trie(clean_docs)

```

```python

opts = {
    "columns": [
            {
                "Header": "Word",
                "accessor": "word"
            },
            {
                "Header": "Similarity",
                "accessor": "sim"
            }
   ],
    "port": "8081",
    "docs": docs_trie
}
```

```python
import json
with open("mathviz_hopper/webpage/mathviz-js-components/public/settings.json", "w+") as f:
    json.dump(opts, f)
```

```python
def word_exists(trie, word):

    for i, s in enumerate(word):
        if s in trie.keys():
            trie = trie[s]
        else: return False

        if i == 19:
            break

    if i == 19:
        s = word[i+1:]
        if s not in trie.keys(): return False
        trie = trie[s]

    return True if 'full_word' in trie.keys() else False

```

```python
word_exists(trie, "fdakfkajshdfjkahsdfkjhadfdfdfdffdfdsdjkfhaskj")
```

```python
docs = ["".join(eq) for eq in df["processed"].tolist()]
docs_trie = construct_trie(docs)
```

```python

```