harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{gathered}h^{-2\sigma}a^{-2k + \eta}= h^{-2\sigma}(h^{\frac{\sigma}{k}}(\log(h^{-1}))^{-T})^{2k - \eta}\le e^{\frac{\sigma}{k}}\log(h^{-1})^{-\frac{T}{3}}\le C \log(h^{-1})^{-\frac{T}{3}}, h^{-2\sigma}a^{-1 - 2\delta + k + \eta}\le h^{-2\sigma}a^{-2k + \eta}\le C \log(h^{-1})^{-\frac{T}{3}}. \end{gathered}
[\sigma^i(p')+\sigma^i(j(p'))+\sigma^{-i}(p')+\sigma^{-i}(j(p')) -\sigma^i(x')-\sigma^i(j(x'))-\sigma^{-i}(x')-\sigma^{-i}(j(x'))]= [p'_i+j(p'_{d-i})+p'_{d-i}+ j(p'_{i}) - x'_i-j(x'_{d-i})-x'_{d-i}-j(x'_{i})]= (1+j)([p'_i+p'_{d-i}-x'_i-x'_{d-i}]).
{\frac{\partial}{\partial r}}\left[ ( r^{2}+ A^{2}- 2 M r ){\frac{\partial G}{\partial r}}\right] +{\frac{1}{s i n \theta}}{\frac{\partial}{\partial \theta}}\left[ s i n \theta{\frac{\partial G}{\partial \theta}}\right] = - \delta ( r - r_{o}) \delta ( c o s \theta - c o s{\theta}_{o}) \delta ( \varphi -{\varphi}_{o})
{}\displaystyle\sum_{n=2}^{\infty}\Big|\Big(\frac{n-1}{n}\Big)^{\frac{\beta}{2}}\frac{A_n}{\sqrt{\lambda_{n}}}-\Big(\frac{n}{n-1}\Big)^{\frac{\beta}{2}}\frac{A_{n-1}}{\sqrt{\lambda_{n}}}\Big|^{2}\geq \displaystyle\sum_{n=1}^{\infty}(\widetilde{w_{n}}(\lambda,\beta)-w_{n}(\lambda,\beta))|A_{n}|^{2}\geq 0.
H(x)\triangleq[\begin{matrix}h_{1}(x)\\h_{2}(x)\\h_{3}(x)\\ \vdots\\h_{n}(x)\end{matrix}]\triangleq[\begin{matrix}h(x)\\L_{f}h(x)\\L_{f}^{2}h(x)\\ \vdots\\L_{f}^{n-1}h(x)\end{matrix}]
\begin{aligned}[b] |S_1^*|-|S_2^*|-\frac{2E_{S_1^*}-E_{S_1^*,S_3^*}+E_{S_2^*,V_1}-E_{S_1^*,V_1}}{\tau-1}&>|S_1^*|-1-|S_2^*|-\frac{2E_{S_1^*}-E_{S_1^*,S_3^*}-d_{i,S_1^*}-d_{i,S_3^*}+E_{S_2^*,V_1}-E_{S_1^*,V_1}+d_{i,V_1}}{\tau-1},\end{aligned}
( \hat{\nu}_{\epsilon}, a ) = \sum_{i , k}\hat{\nu}_{\epsilon \, i}( \delta_{i j}- \frac{1}{2}{\cal C}_{i j}) \, a_{j}= \sum_{i \in \Gamma_{\epsilon}^{( 0 )}}\hat{\nu}_{\epsilon \, i}\, a_{i}- c o s \frac{\pi}{h}\sum_{i \in \Gamma_{\epsilon + 1}^{( 0 )}}\hat{\nu}_{\epsilon + 1 \, i}\, a_{i}
T_{zz}^{\Psi}=T_{zz},\hspace{1em}T_{z\bar{z}}^{\Psi}={\cal O}_{0}\Psi\hspace{1em}T_{\bar{z}\bar{z}}^{\Psi}=T_{\bar{z}\bar{z}},\hspace{1em}T_{\bar{z}z}^{\Psi}=\overline{\cal O}_{0}\Psi .
e_{a_j}\left[t+1\right]= e_{a_j}\left[t\right]+2\mu\Re{\left\{\hat{\phi}^{\left(j,j\right)}\right\}}-4\mu\left(\sum_{l=1}^M\lvert\hat{\phi}^{\left(l,j\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2\right.\left.+\sum_{q=1}^{M-1}\sum_{r=q+1}^Mf_{q,r}^{\left(j\right)}\right)a_j\left[t\right].
\frac{d h_{p p^{\prime}}( l )}{d l}= \! \! - \sum_{q}\! \left( \frac{d h_{p q}( l )}{d l}\frac{1}{E_{p}( l ) - E_{q}( l )}h_{q p^{\prime}}( l ) + h_{p q}( l ) \frac{1}{E_{p^{\prime}}( l ) - E_{q}( l )}\frac{d h_{q p^{\prime}}( l )}{d l}\right) \, ,
\mathcal{B}_T =\bigg\{\theta \in L^{\infty}\left(0,T; L^{1}\left(\mathbb{T}^{n}\right)\right) \cap L^{\infty}\left([\tau,T); L^{\infty}\left(\mathbb{T}^{n}\right)\right): \sup_{\tau\leq t<T}\mathfrak{E}(\theta(t)) + \int^{T}_{\tau}\left\|\Lambda^{\gamma/2}\theta(t)\right\|^{2}_{L^{2}}dt<\infty \bigg\}.
\frac{1}{2}\sum_{k\in I_{a,b}}(k\pi)^{-2}\ge \frac{1}{2\pi}\int_{a+2\pi}^{b}k^{-2}\,dk = \frac{1}{2\pi}\big(\int_{a}^{b}k^{-2}\,dk - \int_{a}^{a+2\pi}k^{-2}\,dk \Big)\ge \frac{1}{2\pi}(a^{-1}- b^{-1}) - a^{-2}.
C_{n}= \prod_{m=1}^{n}(1- 2^{\frac{1}{2}-i(m\omega-\lambda_{*})}) \zeta (\frac{1}{2}+i(m\omega-\lambda_{*})), \tilde{C}_{n}= |(1-2^{\frac{1}{2}-i(n\omega -\lambda_{*})}) \zeta(\frac{1}{2}+i(\lambda_{*}-n\omega))|^{2}C_{n-1}.
Q_2(\lambda)=\begin{bmatrix}\lambda^2P_6+\lambda P_5+P_4 \lambda^2 P_{10}+\lambda P_9 0 \lambda^2 P_8+\lambda P_7 I_n \lambda P_1+P_0 0 -\lambda^2 I_n 0 0 0 -I_n 0 \lambda^2 I_n 0 \lambda P_3+P_2 0 I_n 0 -\lambda^2 I_n \lambda^2 I_n 0 0 -I_n 0\end{bmatrix}.
\left({\frac{\dot{R}}{R}}\right)^{2}={\frac{8 \pi}{3}}G_{N}( R ) \rho_{Q}( R ) +{\frac{8 \pi}{3}}G_{N}( R ) \rho ( R ) +{\frac{\kappa_{5}^{4}}{3 6}}\rho ( R )^{2}.
Q^+_{A_0}(e^{-iu}\psi) + \pi^{-}(a\cdot e^{-iu}\psi + idu\cdot e^{-iu}\psi) = e^{-iu}Q^+_{A_0}\psi -ie^{-iu}\pi^-(du\cdot \psi) + &+ e^{-iu}\pi^{-}(a\cdot \psi) + ie^{-iu}\pi^-(du \cdot \psi) = e^{-iu}(Q^+_{A_0}\psi + \pi^{-}(a\cdot \psi)).
t_{clear}=2\pi\sqrt{\frac{a^{3}}{GM}}\frac{C^{2}}{a^{2}}(\frac{m}{3M})^{2/3}\frac{a^{2}M^{2}}{100m^{2}}=\frac{2\pi}{100\sqrt{G}}\frac{C^{2}}{3^{2/3}}a^{3/2}M^{5/6}m^{-4/3}
b_R :={1\over |R|}\int_R b(x_1,x_2,x_3) dx_1dx_2dx_3={1\over a^4}\int_{a}^{a+a^2}\int_{a}^{2a}\int_{a}^{2a}x_1\,dx_1dx_2dx_3 &={1\over a}{x_1^2\over 2}\bigg|_a^{2a}={3a\over2}.
\sigma_{1}= \left\| \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right\| \quad \sigma_{2}= \left\| \begin{array}{c c}{0}&{- i}\\{i}&{0}\\ \end{array}\right\| \quad \sigma_{3}= \left\| \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right\|
d c-\sum_{i=1}^n\sum_{j=1}^n a_{ij}(x,t)D_{ij}cdt+\sum_{i=1}^n b_i(x,t)D_icdt+h(x,t)cdt=g(x,t)dt+c\sum_{k=0}^mh_kdB_k,~~(x,t)\in Q_T,
\frac{1}{n}\sum_{i=1}^n \sqrt{z_i}-\sqrt{\frac{-\widehat\alpha_{}-1}{L}}\frac{\Gamma ( -\widehat\alpha_{}-{\frac{1}{2}})}{\Gamma (-\widehat\alpha_{})}\frac{\Gamma (L+{\frac{1}{2}})}{\Gamma (L)}=0.
\iota(x):={}\pi I_0(x); \kappa(x):={}K_0(|x|); \iota_+(x):={}\iota(x)e^{- x}\mathbb1_{(0,\infty)}(x); \kappa_+(x):={}\kappa(x)e^{-x}; \iota_-(x):={}\iota(x)e^x\mathbb1_{(-\infty,0)}(x); \kappa_-(x):={}\kappa(x)e^{x},
| - \frac{1}{2}\rangle = - \frac{1}{\sqrt{6 ( 1 + 2 x^{2})}}[ ( 2 x + 2 ) | \downarrow \uparrow \downarrow \rangle + ( 2 x - 1 ) | \downarrow \downarrow \uparrow \rangle + ( 2 x - 1 ) | \uparrow \downarrow \downarrow \rangle ]
II_{t}=\frac{c_{b}}{4 \pi^{3/2}}\left(\int_{0}^{\infty}\left(1-\chi_{\leq 1}(r\xi)\right) \xi \left(e^{i(t-r)\xi}+e^{-i(t+r)\xi}\right)\frac{r \xi \Phi_{1}(r\xi) \chi_{\leq \frac{1}{4}}(\xi)}{\log^{b-1}(\frac{1}{\xi})}d\xi \right)
-\chi^A\epsilon_B \{Q_A,Q^B\}\,\equiv\,[\delta_{\chi},\delta_{\epsilon}]\,=\,\frac{i}{16}(\chi^{\top}C \Gamma_{MN}\epsilon) J^{MN}+\frac{i}{16}(\chi^{\top}C \Gamma_{M_1\ldots M_6}\epsilon)J^{M_1\ldots M_6}\;,
I_{n,q}^{r}= \frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}\mathrm{d}\binom{s-q+r-1}{r}= \frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}\sum_{n=0}^{r-1}\frac{1}{(s-q+n)}\binom{s-q+r-1}{r}\mathrm{d}s,
p_{1}^{\varphi_{t}}f(x_1-x_2)p_1^{\varphi_{t}}= p_{1}^{\varphi_{t}}\int dx_1 \, \frac{\varphi_t^*(x_1)}{\|\varphi_t\|_2}f(x_1-x_2) \frac{\varphi_t(x_1)}{\|\varphi_t\|_2}p_{1}^{\varphi_{t}}= \Lambda^{-1}f*|\varphi_{t}|^{2}(x_{2}) p_{1}^{\varphi_{t}}\;,
Z^{I}= \sum_{a = 1}^{a = 5}{\frac{q^{I}{}_{a}}{| \vec{x}- \vec{x}_{a}|}}\, \ , \qquad \sum_{a = 1}^{a = 5}q^{I}{}_{a}= 0 \ , \qquad I = 1 , 2 , 3 , 4 .
I \left( a \right) = \inf_{\gamma \in \left\lbrace \psi < \infty \right\rbrace}\left[ \ell \left( \gamma ; a \right) + \psi \left( \gamma \right) \right] &= \inf_{f \in H_{1}\left( \left[ 0,t \right] \right)}\left[ \ell \left( \phi_{t}\left( f \right) ; a \right) + \psi \left( \phi_{t}\left( f \right) \right) \right].
\frac{2 7 i}{2 0}\int \! \! d^{4}x \, \, \, \eta_{{\mathbf m}{\mathbf i}}[ D^{\mathbf m}{}_{{\mathbf j}}{\bar{D}}_{{\mathbf k}{\mathbf l}}+{\bar{D}}^{\mathbf m}{}_{{\mathbf j}}D_{{\mathbf k}{\mathbf l}}] T^{{\mathbf i}{\mathbf j}{\mathbf k}{\mathbf l}}| .
\begin{aligned}\begin{bmatrix}1-F_{j,j}-\Gamma_{j-1}0 -G_{j-1,j-1}-\widetilde{t}_{j}\rho^2 1-F_{j,j-1}^{T}0 0 0 1\end{bmatrix}-\begin{bmatrix}1-F_{j,j-1}1_{i-j}-f_{j,j}-\Gamma_{j}-G_{j-1,j-1}-\widetilde{t}_{j}\rho^2 -G_{j-1,j-1}-\widetilde{t}_{j}\rho^2 1-F_{j,j-1}^{T}x -g_{j,j}-f_{j,j}\end{bmatrix}\end{aligned}
\Big(\bar x_{\nu_n}^{(1)}(t,\omega ),\bar x_{\nu_n}^{(2)}(t,\omega),\dots ,\bar x_{\nu_n}^{(N)}(t,\omega )\Big)^{\top}\rightarrow\Big(\bar z(t,\omega ),\bar z(t,\omega ),\dots ,\bar z(t,\omega)\Big)^{\top}
A_{n}^{* i}= \mathrm{e}^{- i{\frac{\pi}{4}}}\Omega_{-}^{-{\frac{1}{2}}}\bar{\psi}_{n}^{i \dot{1}}, \qquad A_{n}^{i}= \mathrm{e}^{i{\frac{\pi}{4}}}\Omega_{-}^{{\frac{1}{2}}}\psi_{n 2}^{i}.
K_{\rm odd}:= \bigotimes_{}\big\{-(x_j^{a_j}+x_{j+1}^{a_{j+1}}x_{j+2}), x_{j+1}\big\},\mathrm{\ \and}K_{\rm even}:= \bigotimes_{}\big\{-(x_j^{a_j}+x_{j+1}^{a_{j+1}}x_{j+2}), x_{j+1}\big\}.
z_{1}= \frac{e^{i \psi / 2}}{\sqrt{2}}\left( \cos \left( \frac{\theta_{1}+ \theta_{2}}{2}\right) \cos \left( \frac{\phi_{1}+ \phi_{2}}{2}\right) + i \cos \left( \frac{\theta_{1}- \theta_{2}}{2}\right) \sin \left( \frac{\phi_{1}+ \phi_{2}}{2}\right) \right)
g ( u , y^{p}, \theta ) = \left( \begin{array}{c c c}{{- \frac{e}{2}u \, 1_{2}}}&{{\frac{i}{\sqrt{2}}y^{p}\, \sigma_{p}}}&{{\sqrt{e \,}\theta_{N}}}\\{{.}}&{{\frac{e}{2}u \, 1_{2}}}&{{.}}\\{{.}}&{{\sqrt{e}\, \vartheta^{\dagger N}}}&{{.}}\\ \end{array}\right) = u{\cal C}+ y^{p}{\cal N}_{p}+ \theta^{\dagger N}Q_{N}+ \overline{{{Q}}}^{N}\theta_{N}
\Omega(x,...,x,\Omega(x,...,x,y))=\Omega(x,...,x,y), \underbrace{tx+...+tx}_{m-1}+t\Omega(x,...,x,y) = \underbrace{tx+...+tx}_{m-1}+ty, t(\underbrace{tx+...+tx}_{m-1}+ty) = ty, \underbrace{tx+...+tx}_{m-1}+ty + tx = y+tx x+ty = y+tx \implies t=1.
\mbox{Prod}_{\mathbf{A}\otimes\mathbf{B}}\left(\left(\mathbf{x}_{1}\otimes\mathbf{y}_{1}\right)^{\top^{1}},\left(\mathbf{x}_{0}\otimes\mathbf{y}_{0}\right)^{\top^{0}}\right)\,:=\mbox{Prod}_{\mathbf{A}}\left(\mathbf{x}_{1}^{\top^{1}},\mathbf{x}_{0}^{\top^{0}}\right)\cdot\mbox{Prod}_{\mathbf{B}}\left(\mathbf{y}_{1}^{\top^{1}},\mathbf{y}_{0}^{\top^{0}}\right),
\begin{array}{l}XH=x^{k_2}z^{k_1}(1-x-y-z)(k_1 k_3 - k_2 k_4 ) XV=y^{k_3}(1-x-y-z)^{k_2}x(k_2 k_4 -k_1 k_3), X\widetilde{H}=x^{k_3}z^{k_4}(k_1 k_3 - k_2 k_4)y X\widetilde{V}=y^{k_4}(1-x-y-z)^{k_1}(k_2k_4 - k_1k_3)z\end{array}
\begin{gathered}\hat{R}_{01}=\hat{R}_{1001}+\hat{R}_{3031}=R_{1001}-g(A(l,k),A(k,l))+g(A(k,k),A(l,l)) +R_{3031}-g(A(m_3,m_3),A(k,l))+g(A(k,m_3),A(m_3,l))={R}_{1001}+{R}_{3031}-2\theta^2,\end{gathered}
x_{-}= \left( \begin{array}{c c c}{0}&{0}&{0}\\{X_{-}}&{0}&{0}\\{0}&{- X_{-}^{T}}&{0}\\ \end{array}\right) , \qquad x_{+}= \left( \begin{array}{c c c}{0}&{X_{+}}&{0}\\{0}&{0}&{- X_{+}^{T}}\\{0}&{0}&{0}\\ \end{array}\right) ,
\Gamma = \Gamma_{\min}\cup[ \tau_{\Gamma_{\min}}(x) , \tau_{\Gamma}(x) ], \Gamma_{\min}\cap [\tau_{\Gamma_{\min}}(x) , \tau_{\Gamma}(x)] = \{\tau_{\Gamma_{\min}}(x) \}& \tau_{\Gamma}(x) \not\in \Gamma_{\min}, \tau_{\Gamma}(x) \neq \tau_{\Gamma}(y).
\frac{\partial{u_{3}}}{\partial{t}}= \frac{\partial}{\partial{t}}\Gamma_{k}^{( 6 )}[ \varphi = 0 ] = \frac{1}{2}{T}{r}\frac{\delta^{6}}{\delta \varphi_{1}( q = 0 ) \cdots \delta \varphi_{6}( q = 0 )}\left\{( \Gamma_{k}^{( 2 )}[ \varphi ] +{R}_{k})^{- 1}\frac{\partial}{\partial{t}}{R}_{k}\right\}\Bigg \vert_{\varphi = 0}
|\sum_{k'}(R_{i'i'}R_{k'k'}-R_{i'k'}R_{k'i'})|\le R_{i'i'}R+\sum_{k'}R_{i'k'}R_{k'i'}\le R_{i'i'}R+R^2\le(1+\frac{1}{\epsilon})R_{i'i'}R=o(1)R_{i'i'}.
\ker B_{M^{(2)},M^{(1)}}(P^{(1)},P^{(2)}) =\mbox{Im}\ \!\! \begin{bmatrix}X_F\phi_{k-1}(T_F) \vdots X_F\phi_{0}(T_F) \end{bmatrix}\,\oplus \,\mbox{Im}\ \!\! \begin{bmatrix}X_\infty \phi_{0}(T_\infty) \vdots X_\infty \phi_{k-1}(T_\infty) \end{bmatrix}.
\psi(s_1, s_2) =s_1A_++s_2A_--s_1^{r/p}B_+-s_2^{r/p}B_--s_1^{q/p}C_+-s_2^{q/p}C_- &\quad+ \frac{2}{p}\int_{\Omega_+\times \Omega_-}\frac{|s_1^{1/p}u^+(x)-s_2^{1/p}u^-(y)|^p}{|x-y|^{N+ps}}\, dx\, dy.
f(t) = \begin{cases}1, \ \ \ \ \ \ \mbox{if}\ 0 \leq t \leq R + \frac{\delta}{3}, -\frac{3}{\delta}t + \frac{3R + 2 \delta}{3},\ \ \ \mbox{if}\R + \frac{\delta}{3}< t \leq R + \frac{2\delta}{3}, 0, \ \ \ \ \ \ \ \mbox{if}\t \geq R + \frac{2\delta}{3}, \end{cases}
B_3^{(n)}\subseteq\bigcup_{t=\lfloor p_n^{-1}\rfloor}^{\lfloor cn\rfloor}\{S_n'(t)\leq t\}\cup\bigcup_{t=\lfloor cn\rfloor}^{\lfloor n-p_n^{-1}\rfloor\wedge\lfloor n-\varepsilon f_1(n)\rfloor}\{S_n'(t)\leq t\}\cup\bigcup_{t=\lfloor n-p_n^{-1}\rfloor\wedge\lfloor n-\varepsilon f_1(n)\rfloor}^{\lfloor n-\varepsilon f_1(n)\rfloor}\{S_n'(t)\leq t\},
A_k= A_{k_0}\prod_{\ell = k_0}^{k - 1}\gamma_d(1 + s_\ell)\leq A_{k_0}\gamma_d^{k - k_0}\prod_{\ell = k_0}^{k - 1}(1 + C_3/\ell)= A_{k_0}\gamma_d^{k - k_0}\prod_{\ell = k_0}^{k - 1}\frac{\ell + C_3}{\ell}\leq C_4 \gamma_d^k k^n,
c_{n+1}(t)= - \frac{2(n+1)t}{1 + n(n+1)^3 t^2}\mbox{and}\mathfrak{m}_{n}(t) = \frac{n}{2(n+1)}\frac{1 + (n+1)^2 (n+2)^2 t^2}{1 + n (n+1)^2 (n+2)t^2}, n \geq 0.
P(^{(r,s)}S_t \ge x) = P\left(^{(r+1,s)}S_t + \Delta X_t^{(r+1)}/b_t \ge x \right) \ge P\left(^{(r+1,s)}S_t\ge -y,\ \Delta X_t^{(r+1)}\ge (x+y) b_t \right) =P\Big(\Delta X_t^{(r+1)}\ge (x+y)b_t\Big) - P\left(^{(r+1,s)}S_t < -y,\ \Delta X_t^{(r+1)}\ge (x+y) b_t\right) .
\bar{f}_{n, k}(t) \ := \ \frac{f_{n, k}(t) - \mathbb{E}[f_{n, k}(t)]}{\sqrt{{\rm{Var}}[f_{n, k}(t)]}},\quad\bar{\chi}_{n}(t) \ := \ \frac{\chi_{n}(t) - \mathbb{E}[\chi_{n}(t)]}{\sqrt{{{\rm{Var}}}[\chi_{n}(t)]}}.
\phi(s'\otimes a s\otimes m) =\phi(s's\otimes am)=\phi(a(s's\otimes m))=a\phi(f(r')s\otimes m)=a \phi(r'f(1_R)s\otimes m) &= r'a\phi(1_Ss\otimes m)=(1_S\otimes r'a)\phi(s\otimes m)=(f(1_Rr')\otimes a)\phi(s\otimes m)=s'\otimes a\phi(s\otimes m),
y_{jk}(0,\l) = i b_j \int_{0}^1 e^{i (b_k-b_j) \l t}q_{jk}(t) dt + i b_j \int_0^1 e^{i (b_k-b_j) \l t}\sum_{l=1}^n q_{jl}(t)\rho_{lk}(t,\l)dt.
\int_{a}^{b}e^{ig(t)}{\rm d}t = \int_{a}^{b}e^{ig(t)}\frac{g'(t)}{g'(t)}{\rm d}t = \left[e^{ig(t)}\frac{1}{g'(t)}\right]_{a}^{b}-\int_{a}^{b}e^{ig(t)}\left(\frac{1}{g'(t)}\right)'{\rm d}t.
{\rm pv}\int \cot\Big(\frac{z(e)-z(e')}{L/\pi}\Big) f(e')\d e' = \int \cot\Big(\frac{z(e)-z(e')}{L/\pi}\Big) \frac{f(e')z_{e}(e)-f(e)z_{e}(e')}{z_{e}(e)}\d e' \, ,
L^{2}({\cal A}_{\gamma}\times{\cal F}_{\gamma}) = \bigoplus_{\rho_{e}\in \mathrm{R e p}( G ) , \; \rho_{v}\in S}\quad \bigotimes_{v \in V}\bigl ( \bigotimes_{t ( e ) = v}\rho_{e}^{*}\otimes \bigotimes_{s ( e ) = v}\rho_{e}\otimes \rho_{v}\bigr )
\left\{\mathfrak{P}_{\ell}\chi F, \mathfrak{P}_{\ell}\chi G \right\}_{\ell}^{\widetilde{\Sigma}}[\widetilde{\psi}] = \left\langle E_{\widetilde{\Sigma}}, \left( \mathfrak{P}_{\ell}\chi F \right)^{(1)}[\widetilde{\psi}] \otimes \left( \mathfrak{P}_{\ell}\chi G \right)^{(1)}[\widetilde{\psi}] \right\rangle.
B_{\mu \nu}( t ) = \sum_{\rho = 0}^{N}\frac{\eta_{\mu \rho}\eta_{\nu \rho}}{\sqrt{8 \omega_{\mu}}}\left[ \left( 1 + \frac{\omega_{\mu}}{\Omega_{\rho}}\right) \mathrm{e}^{i \Omega_{\rho}t}+ \left( 1 - \frac{\omega_{\mu}}{\Omega_{\rho}}\right) \mathrm{e}^{- i \Omega_{\rho}t}\right] \; .
\varrho =\bigg[\mathrm{exp}\bigg( \frac{0.49\chi^2}{\left(1+\!0.18\xi^2+0.56\chi^{12/5}\right)^{7/6}}\bigg)-1\bigg]^{-1}, \beta =\bigg[\mathrm{exp}\bigg( \frac{0.51\chi^2\left(1\!+\!0.69\chi^{12/5}\right)^{-5/6}}{\left(1+\!0.9\xi^2+0.62\xi^2\chi^{12/5}\right)^{5/6}}\bigg)-1\bigg]^{-1},
\int_{0}^{t}e^{-|\xi|^{2\alpha}(t-s)}|H(\xi,s)|ds\leq C|\xi|\int_{0}^{t}(s+1)^{-\frac{5-4\alpha}{2\alpha}}ds \leq C\frac{2\alpha}{6\alpha-5}|\xi|((t+1)^{\frac{6\alpha-5}{2\alpha}}-1) \leq C\frac{2\alpha}{6\alpha-5}(t+1)^{-\frac{1}{2\alpha}}((t+1)^{\frac{6\alpha-5}{2\alpha}}-1) \leq C(t+1)^{\frac{3\alpha-3}{\alpha}}.
D =\Gamma(t_1t_2q^{n-1},t_3t_4q^{n-1},t_5t_6)^n\prod_{m=1}^n\prod_{\substack{1\leq j<k\leq 6 (j,k)\neq (1,2),\,(3,4),\,(5,6)}}\Gamma(t_jt_kq^{m-1}) &\quad\times\det_{1\leq j,k\leq n}\left((t_1t_3q^{k-1},t_1t_4q^{n-k})_{j-1}(t_2t_3q^{k-1},t_2t_4q^{n-k})_{n-j}\right).
l^{2}b^{2}=l^{2}\left( \frac{n_{1}a_{1}^{p}+n_{k}a_{k}^{p}}{2l}\right)^{2/p}\geq l^{2-\frac{2}{p}}\left( \sqrt{n_{1}a_{1}^{p}n_{k}a_{k}^{p}}\right)^{2/p}=\left( \frac{l^{2}}{n_{1}n_{k}}\right)^{1-1/p}n_{1}n_{k}a_{1}a_{k}& >n_{1}n_{k}a_{1}a_{k}.
\dot G_\pm +\frac{i}{2}\phi^\pm D^2_\pm G_\pm + \frac{i}{4}D_\pm\phi^\pm D_\pm G_\pm+ \frac{3}{4}i D^2_\pm \phi^\pm G_\pm= \frac{\kappa_0}{2}D^5_\pm \phi^\pm ~.
\min_{x}\ \sum_{i,j \in \mathcal{I}, \mathcal{J}}u_i q_j x_{i,j}, \mathrm{s.t.}\x \in \mathcal{X}\subseteq \left\{x \in \{0,1\}^{|\mathcal{I}|\cdot|\mathcal{J}|}:\sum_{j \in \mathcal{J}}x_{i,j}= 1 \ \forall i \in \mathcal{I}\right\}
N_{m_{1},m_{2},m_{3}}^{\l_{1},\l_{2},\l_{3}}=N_{m_{3},m_{1},m_{2}}^{\l_{3},\l_{1},\l_{2}}=(-1)^{l_{1}+l_{2}+l_{3}}N_{m_{2},m_{1},m_{3}}^{\l_{2},\l_{1},\l_{3}}& =(-1)^{l_{1}+l_{2}+l_{3}}N_{-m_{1},-m_{2},-m_{3}}^{\l_{1},\l_{2},\l_{3}}\ .
\frac{d+1}{k(d+1-k)}\left|\sigma - \frac{k^{2}}{d+1}\right| = \frac{d+1}{k(d+1-k)}\left[\frac{k^{2}}{d+1}-\sigma\right] &= \frac{k}{d+1-k}-\frac{d+1}{k(d+1-k)}[k-(r-\lambda)].
\delta^{2}(\varphi_{rest_{1}}^{1})=\left(x^{1}+\bar{u}^{1}\cos\left(\tfrac{x^{3}+v_{1}^{1}}{2}\right)\right)\sin\left(\tfrac{v_{1}^{1}+v_{1,[1]}^{1}}{2}\right)-\left(x^{2}+\bar{u}^{1}\sin\left(\tfrac{x^{3}+v_{1}^{1}}{2}\right)\right)\cos\left(\tfrac{v_{1}^{1}+v_{1,[1]}^{1}}{2}\right)\,.
\log\left[e^{-\bar\rho}(r + \bar\rho + 1)\right] = \frac{1}{2}(r - \rho) + \log\left(\frac{1}{2}(\rho+r)+1\right) \leq \frac{1}{2}(r - \rho) + \log\left(r+1\right) + \frac{\rho-r}{2r+ 2}= \frac{r}{2}-\frac{r(\rho+1)}{2(r+1)}+ \log(r+1) = \log \gamma_1 \epsilon,
\mathbf{s}\left( t\right) :=\underset{(\eta ,l^{-1})\rightarrow (0,0)}{\lim}\left\{\left( \eta^{2}\left\vert \Lambda_{l}\right\vert \right)^{-1}\mathbf{S}^{(\omega ,\eta \mathbf{A}_{l})}\left( t\right) \right\}\ .
\Delta(D',D') = \Delta(G-\{D+\{0\}\},G-\{D+\{0\}\}) & = -2(k+1)G+nG+\Delta(D+\{0\},D+\{0\}) &= (n-2k-2)G+\Delta(D,D)+2D+\{0\}&=(n-2k-2)[\{0\}+D'+(G-D'-\{0\})]+k\{0\}+\lambda(G-D'-\{0\})+\mu D' &\ \ \ +2(G-D'-\{0\})+\{0\}&=(n-k-1)\{0\}+(n-2k+\mu-2)D'+(n-2k+\lambda)(G-D'-\{0\}
\begin{cases}J_h^1 = \{I = [t_k^1,t_{k+1}^1]: I\subseteq [s_h, s_{h+1}]\}, J_h^N = \{I = [t_k^N,t_{k+1}^N]: I\subseteq [s_h, s_{h+1}],\I\not\subseteq J\ \forall J\in\cup_{M < N}J_h^M\}, \end{cases}
\frac{1}{2 \pi}c^{( f )}: e^{\mp i 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi + g N}}U_{1 f}\Phi^{( 1 )}( x )}:_{M^{( 1 )}}\prod_{I = 2}^{N}: e^{\mp i 2 \sqrt{\pi}U_{I f}\Phi^{( I )}( x )}:_{M^{( I )}}e^{\pm i \frac{\theta}{N}}.
\int_u^v e^{f_2(t)-f_1(t)-2s(0,t)}dt = \int_u^v \frac{e^{g(t)}dt}{\Big(\int_0^t e^{g(s)}ds\Big)^2}=\frac{-1}{\int_0^t e^{g(s)}ds}\Big\vert_u^v = \frac{1}{\int_0^u e^{g(s)}ds}-\frac{1}{\int_0^v e^{g(s)}ds}.
I_2'(x) = \lambda P_n(0) \int_{z=x}^{\infty}B(z-x)\lambda e^{-\lambda(z-x)}dz + \lambda \int_{z=x}^{\infty}\int_{u=0}^{z-x}\bar{G}(u)B(z-x-u)f_n(u)\lambda e^{-\lambda(z-x)}du dz - P_n(0)\int_{z=x}^{\infty}f_{B}(z-x)\lambda e^{-\lambda(z-x)}dz - \int_{u=0}^{z-x}\bar{G}(u)f_{B}(z-x-u)f_n(u)\lambda e^{-\lambda(z-x)}du dz.
\nu(x_{2m+1,i}) =\nu{\left(\sum_{k=1}^\infty x_{2m,k}\beta_{k,i}\right)}\geq\min_{k\geq2}\{\nu(x_{2m,k})+\nu(\beta_{k,i})\}=\min_{k\geq4}{\left\{\big(\nu(x_{2m,2})+\nu(\beta_{2,i})\big),\big(\nu(x_{2m,3})+\nu(\beta_{3,i})\big),\big(\nu(x_{2m,k})+\nu(\beta_{k,i})\big)\right\}}.
\tilde{\rho}q (r\#\delta_h) =\rho(ju(r)\#\delta_h\Phi(rju(r)^{-1}))=\rho(1\#\Phi(rju(r)^{-1})) \rho(ju(r)\#\delta_h) &=\rho(rju(r)^{-1}\#1) \rho(ju(r)\#\delta_h) = \rho(r\#\delta_h)
Q^{( n )}( w | \zeta )^{\varepsilon}= \prod_{a \in A}\zeta_{a}\left( \prod_{j = 1}^{a - 1}( x z_{j}- w_{a}) \prod_{j = a + 1}^{2 n}( x w_{a}- z_{j}) \right) \left/ \prod_{a , b \in A \atop a < b}( x^{- 1}w_{b}- x w_{a}) . \right.
E_{u,v}(t)={}\frac{1}{2}\int_\Omega \rho(x)| u_t(x,t)|^2+\rho(x)|v_t(x,t)|^2 + \nabla u(x,t)^{\top}\cdot K(x) \cdot\nabla u(x,t) \,dx &{}+{}\frac{1}{2}\int_{\Omega}\nabla v(x,t)^{\top}\cdot K(x) \cdot\nabla v(x,t)+ (uv)^2(x,t) \,dx,
A = \left( \begin{array}{c c c}{2 \omega_{\eta}}&{- t}&{i v}\\{- t}&{2 \omega_{t}}&{- i r}\\{- i v}&{i r}&{2 \omega_{b}}\\ \end{array}\right) B = \left( \begin{array}{c c c}{0}&{- i t}&{u}\\{- i t}&{0}&{s}\\{u}&{s}&{0}\\ \end{array}\right) ,
\beta_N(\pi x^N,\pi B) &=\mathbb{P}_{N,m}\bigg(\Big(\Psi(0, x^N_{\pi(j)}, \frac{1}{N-1}\sum_{k\neq j}\delta_{x^N_{\pi(k)}}, \Phi^{N,m}_j(0,x^N_{\pi(j)}), \xi^{j,N,m}_1)\Big)_{j = 1}^N \in \pi B\bigg)=\star.
V_{\infty}= \begin{dcases}0 \infty \notin S \max\left\{\frac{7\zeta(3)}{4\pi^2}- \frac{0.95 d + 2}{2 d^2}- \frac{(d-2) \log{d}}{2 d (d-1)},0\right\}\infty \in S \end{dcases}
S_1 := C_R(a) = Z \cup (Z + a), S_2 := C_R(b) = C_R(c) = C_R(b + c) = Z \cup (Z + b) \cup (Z + c) \cup (Z + (b + c)), S_3 := C_R(a + b)= Z \cup (Z + (a + b)), S_4 := C_R(a +c) = Z \cup (Z + (a + c)) S_5 := C_R(a + b + c) = Z \cup (Z + (a + b + c)).
y_{k+1}- \widetilde{y_{k+1}}- (y_k - \widetilde{y_k}) +\frac{\rho_T}{T}\int_{s_k}^{s_{k+1}}\Big(M(y(s)) -M(\widetilde{y_k})\Big)\,ds=0 \,, \forall \k \in \mathbb{N}\,.
\Big(\frac{1}{|P|}\int_P{\sum_{k=\mu}^{\infty}{2^{skq}\Big|\Pi_k\Big(\sum_{n=3}^{\infty}{T_{[b_n]}f}\Big)(x) \Big|^q}}dx \Big)^{1/q}\lesssim \sup_{R\in\mathcal{D}_{\mu}}{\Big( \dfrac{1}{|R|}\int_R{\sum_{k=\max{(3,\mu-2)}}^{\infty}{2^{skq}\big|T_{[b_k]}f(x) \big|^q}}dx \Big)^{1/q}}.
\begin{aligned}x(t)-z_1(t) = \,x(0)-z_1(0) + \int_0^t\dot{x}(\tau)-\dot{x}_a(\tau) d\tau &- \gamma \int_0^t \dfrac{\partial \epsilon_1(\eta,t)}{\partial \eta}\Big|_{\eta = \eta_a(\tau)}\dot{\eta}_a(\tau) d\tau &- \gamma \int_0^t \dfrac{\partial}{\partial \tau}\epsilon_1(\eta_a(\tau),\tau) d\tau.\end{aligned}
T_{M_{2}}l \int d^{2}\xi ( 2 \pi \alpha^{\prime}) \epsilon^{i j}\partial_{i}\varrho \ \left({\tilde{\cal G}}_{j}^{( 1 )}-{\textstyle \frac{1}{2 \pi \alpha^{\prime}}}{\tilde{A}}^{( 1 )}{}_{j}+{\textstyle \frac{m}{2}}b_{j}\right) .
W_a(x):=\sup_{\zeta(\rho)=0, \rho=\beta+i\gamma \atop a<\Re \rho=\beta, |\gamma|<x}\frac{x^\beta}{|\rho|}, W(x):= \sup_{\zeta(\rho)=0, \rho=\beta+i\gamma \atop \theta<\Re \rho=\beta, |\gamma|<x}\frac{x^\beta}{|\rho|}.
{\cal C}_{a}( \nu_{I}) = \alpha^{\prime}\sum_{I , \overline{{J}}}2 \kappa_{a I}\overline{{\kappa}}_{a \overline{{J}}}G^{I \overline{{J}}}\left\{\gamma + \frac{1}{2}\Big ( \mathrm{\boldmath{\psi}}( \nu_{I}) + \mathrm{\boldmath{\psi}}( 1 - \nu_{I}) \Big ) \right\},
\epsilon^{\delta \alpha \beta \gamma}D_{\delta}^{i}( w_{\alpha \ldots \alpha_{m_{1}}}w_{\beta \ldots \beta_{m_{2}}}w_{\gamma \ldots \gamma_{m_{3}}}) = 0 \ \rightarrow \ \ell = 6 +{\frac{1}{2}}( J_{1}+ 2 J_{2}+ 3 J_{3}) \; .
e^{i \alpha Q_{5}}\, | 0 >_{\varphi}\otimes | \Omega_{\theta}> = | z_{0}>_{\varphi}\otimes | \Omega_{\theta}> \ \mathrm{w i t h}\z_{0}= - \frac{\alpha \lambda \sqrt{2 \mu}}{2 \sqrt{\pi}\sqrt{L}}\ ,
(i) :\left[ E(l_{1},m_{1}),E(l_{2},m_{2})\right] =\sum_{l\ \equiv\l_{1}+l_{2}+1}K_{m_{1},m_{2}}^{l_{1},l_{2},l}E(l,m_{1}+m_{2}), (ii) :\left[ \left[ E(l_{1},m_{1}),E(l_{2},m_{2})\right] \right]=\sum_{l\ \equiv\l_{1}+l_{2}}K_{m_{1},m_{2}}^{l_{1},l_{2},l}E(l,m_{1}+m_{2}),
A = a_{0}+ i \mathrm{\boldmath ~ a ~}\cdot \mathrm{\boldmath ~ \t a u ~}= \left( \begin{array}{c c}{{a_{0}+ i a_{3}}}&{{i a_{1}+ a_{2}}}\\{{i a_{1}- a_{2}}}&{{a_{0}- i a_{3}}}\\ \end{array}\right)
\sum_{\alpha\in \Phi^+(W)}\!\!T_\alpha -w\bigg(\sum_{\alpha\in \Phi^+(W)}\!\! T_\alpha\bigg) = \sum_{\alpha\in \Phi^+(W)}\!\! T_\alpha -\sum_{\alpha\in \Phi^+(W)}\!\! T_{w\alpha}= \sum_{\alpha\in \Phi^+(W)\cap w\Phi^-(W)}\!\! (T_\alpha-T_{-\alpha}),
{\cal R}_{1 2}^{s h}( x ){\cal R}_{1 3}^{s h}( x y ){\cal R}_{2 3}^{s h}( y ) ={\cal R}_{2 3}^{s h}( y ){\cal R}_{1 3}^{s h}( x y ){\cal R}_{1 2}^{s h}( x ) .
\Pi_T \left( \|b-b_0\|_{\mu_0}\leq 4\sqrt{d}\|\mu_0\|_\infty^{1/2}\varepsilon_T \right) \geq \Pi_T \left( \sup_{j=1,\dots,d}\|b_j-b_{0,j}\|_{L^2}\leq 4 \varepsilon_T \right) & \geq \prod_{j=1}^d \Pi_T \left( \|b_j-b_{0,j}\|_\infty \leq 4\varepsilon_T \right) \geq e^{-dT\varepsilon_T^2},
\Bigl[I_{n+1}\frac{\partial}{\partial x_n},Q\Bigr]f = I_{n+1}\frac{\partial}{\partial x_n}(Qf) - Q \Bigl(I_{n+1}\frac{\partial}{\partial x_n}\Bigr)f = \frac{\partial Q}{\partial x_n}f,f\in C^{\infty}(\Omega).
f(n, \phi, \tilde{p}, \tilde{q}) := \P_{(x_1,\dots,x_n) \sim \tilde{p}^{\otimes n}}( \phi(y_1,\dots,y_n) \ne p) + \P_{(x_1,\dots,x_n) \sim \tilde{q}^{\otimes n}}( \phi(y_1,\dots,y_n) \ne q).
E_{B}= \left( \begin{array}{c c}{E_{1}}&{E_{2}}\\ \end{array}\right){\cal E}_{B};{\cal E}_{B}= \left( \begin{array}{c}{1}\\{\nu_{B}^{c}}\\ \end{array}\right){\frac{1}{\sqrt{1 + \vert \nu_{B}\vert^{2}}}}.
E(x,t,k)=I-\frac{1}{2\pi i}\oint_{|s+k_0|=\epsilon}\frac{\mathfrak{B}^r(\xi,t)}{(s+k_0)(s-k)}ds+\frac{1}{2\pi i}\oint_{|s-k_0|=\epsilon}\frac{\overline{\mathfrak{B}^r(\xi,t)}}{(s-k_0)(s-k)}ds+R(\xi,t), |k\pm k_0|>\epsilon,
M(x,y,z) = \frac{z^{x - y}x^{y - z}}{y^{x - z}}= \left(\frac{z}{y}\right)^{x - y}\left(\frac{x}{y}\right)^{y - z}= \left[\left(\frac{z}{y}\right)^{\frac{x}{y}- 1}\left(\frac{x}{y}\right)^{1 - \frac{z}{y}}\right]^y.
|T_+-T|_{\mathcal{V}_+^*,\mu;s_+}=|T\circ T^+-T|_{\mathcal{V}_+^*,\mu;s_+}\leq (1+X_+)C_1 r_+\delta_{+\mu}<r_+\chi_+{\rm for}\,0\leq \mu \leq \alpha,
\lambda_{i_0 ,j_0 , k_0}x^{i_0}y^{j_0}z^{k_0}= \gamma \left( x^{|\alpha|}\right)^{i_1}\left( y^{|\alpha|}\right)^{j_1}\left( z^{|\alpha|}\right)^{k_1}\left( x^{|\alpha^3|}y^{|\alpha^3|}z^{|\alpha^3|}\right)^{i_2},
m_{x}^{\xi_1^\infty}([ x]_{\xi_0^{\infty}})= \lim_{N \to \infty}\frac{m([ x]_{\bigvee_{i=0}^{N-1}\sigma^{-i}\xi})}{m ([ x]_{\bigvee_{i=1}^{N-1}\sigma^{-i}\xi})}=\frac{u_{x_0}}{\lambda u_{x_1}}.