harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Psi_{\kappa}^{(P)}=\frac{i}{2^{3/2}}\,\exp\left(-\frac{\pi\kappa}2-i\frac{\kappa}2\ln\left(\frac{-x_+}{-x_-}\right)\right)H_{i\kappa}^{(1)}\left(m\sqrt{(-x_-)(-x_+)}\right).
\Psi^{-1}(\mathbf p + t \mathbf e) =\mathbf o + \frac{2RD (\mathbf p - \mathbf o)}{R^2 - (t-D+R)^2 +(t -D)^2}=\mathbf o + \frac{D (\mathbf p - \mathbf o)}{D - t}.
B(0)=\bigg(\frac{|J_{\rho_0}|e^{-T_{0}}}{2|J_{\rho_0}|e^{-T_{0}}+2}\bigg({\rm \exp}\bigg[\bigg(2+\frac{2}{|J_{\rho_{0}}|e^{-T_{0}}}\bigg)T_{0}\bigg] -1\bigg) + C_*{\rm \exp}\bigg[\bigg(2+\frac{2}{|J_{\rho_{0}}|e^{-T_{0}}}\bigg)T_{0}\bigg] \bigg)^{-1}.
\frac{1}{4\pi}\int_{S^{2}}W_{P}dS =\int_{SU(2)}W_{P}(g\mathbf{n}_{0})dg=\int_{SU(2)}trace(PK^{g})dg = & =trace\left( P\int_{SU(2)}K^{g}dg\right) =trace\left[ P\left( \frac{1}{n+1}I+\sum_{l=1}^{n}\left( \int_{SU(2)}\tilde{\varphi}_{l}(g)dg\right)K_{l}\right) \right] ,
\overline{v}(k)a(k)(\overline{F}_{\eta}(v,\omega)-\lambda)^{-1}v = \lvert v(k)\lvert^2(F_{-\eta}(v,\omega)+\omega(k)+\lambda )^{-1}\Omega &-\lvert v(k)\lvert^2(F_{-\eta}(v,\omega)+\omega(k)-\lambda)^{-1}(\overline{F}_{\eta}(v,\omega)-\lambda)^{-1}v.
\omega^{3}_{4}= i \{\frac{1}{2}d\ln (\frac{d\overline{g}_{2}}{d\overline{z}_{2}}\frac{dz_{2}}{dg_{2}}) + \frac{(g_{2}d\overline{g}_{2}- \overline{g}_{2}dg_{2})}{(1+ g_{2}\overline{g}_{2})}\},
\cos \phi = \frac{m_{31}}{\sqrt{m_{21}^2 + m_{31}^2}}, \sin \phi = \frac{m_{21}}{\sqrt{m_{21}^2 + m_{31}^2}}, b_{22}= -m_{22}m_{31}+ m_{21}m_{32}, b_{32}= -m_{23}m_{31}+ m_{21}m_{33}.
\begin{cases}\begin{aligned}\bar h^{1^*}_{11k}+\bar h^{1^*}_{22k}=0, &\bar h^{2^*}_{11k}+\bar h^{2^*}_{22k}=0, &\bar \lambda_1\bar h^{1^{\ast}}_{11k}+3\bar \lambda \bar h^{1^{\ast}}_{12k}+3\bar \lambda_2 \bar h^{2^{\ast}}_{12k}-\bar \lambda\bar h^{2^{\ast}}_{22k}=0.\end{aligned}\end{cases}
q^{\frac{3}{2}(\Lambda,\Lambda+2\rho)}=\sum_{\nu\in M(\Lambda)}q^{\frac{1}{2}(\nu,\nu+2(\Lambda+\rho))}\frac{D_q[\Lambda+\nu]}{D_q[\Lambda]}\sum_{s=0}^\infty (m_{\nu-s\delta}^+-m^-_{\nu-s\delta})\, q^{-s(3g+2\kappa_\Lambda)}\;.
\mathfrak{sp}(2)\ni u=\begin{pmatrix}x y -\overline{y}z\end{pmatrix}=\begin{pmatrix}x 0{0}z\end{pmatrix}\oplus\begin{pmatrix}0 y -\overline{y}0\end{pmatrix}\,\in \,V^{G}_{Id}\oplus H^{G}_{Id}.
\begin{aligned}(|(-\Delta_x)^{1/3}u|^2)^{1/2}_{Q_{1,R}}\le N (d) \delta^{-\theta}\sum_{k=0}^\infty 2^{-2k}(f^2)^{1/2}_{Q_{1,2^{k}R/\delta^2}}.\end{aligned}
\mathbf{e}^{j}(l_{1},m_{1})\mathbf{e}^{j}(l_{2},m_{2})=\sum_{l=0}^{2j}(-1)^{2j+m}\left[\begin{array}[c]{ccc}l_{1}l_{2}l m_{1}m_{2}-m\end{array}\right] \!\!{[j]}\ \mathbf{e}^{j}(l,m)
||\tilde v - x_n Q_0||_{L^{\infty}(\tilde G \cap Q_{\rho}(0,0))}\leq ||\tilde v - v_0||_{L^{\infty}(\tilde G \cap Q_{\rho}(0,0))}+ ||v_0 - x_n Q_0||_{L^{\infty}(\tilde G\cap Q_{\rho}(0,0))}& \le \frac{1}{4}\rho^{2+ \alpha}+ \frac{1}{4}\rho^{2+ \alpha}= \frac{1}{2}\rho^{2+ \alpha}.
{\frac{1}{1 6 \pi G_{r e n}}}={\frac{1}{1 6 \pi G_{B}}}+{\frac{1}{1 2}}~{\frac{1}{( 4 \pi )^{3 / 2}}}~ \int_{\epsilon^{2}}^{\infty}{\frac{d s}{s^{3 / 2}}}e^{- \mu s / l^{2}}
S(n,j) = (1-w^2)^j\left\{(-1)^{n-j}P_{n-j}^{2j,0}(2w^2-1) - (-1)^{n-j-1}P_{n-1-j}^{2j,0}(2w^2-1)\right\}& = (1-w^2)^j\left\{P_{n-j}^{0,2j}(1-2w^2) - P_{n-1-j}^{0,2j}(1-2w^2)\right\}.
-\lim_{s\to 0}\left(\frac{d}{ds}\zeta(s)^{-1}Z_{r_2}(f,s)\right)+\epsilon(1)\lim_{s\to 0}\left(\frac{d}{ds}\zeta(1+s)^{-1}Z_{r_1}(d_2(f),s+1)\right) =\sum_{v}(\log q_v)\bigg((e(\psi_v)+1)\tilde{c}_{2,v}(I(f_v))-\tilde{a}_{2,v}(I(f_v))\bigg)\prod_{v'\neq v}\tilde{c}_{2,v'}(I(f_{v'})).
D_{\eta}^{i}\equiv D^{i}= \frac{\partial}{\partial \eta_{i}}- i \bar{\eta}^{i}\frac{\partial}{\partial t}\quad , \quad \bar{D}_{\eta \, i}\equiv \bar{D}_{i}= \frac{\partial}{\partial \bar{\eta}^{i}}- i \eta_{i}\frac{\partial}{\partial t}
= \;\left[\prod_{a<b}^N \frac{(2n_a+2n_b)!}{(2n_a)!(2n_b)!}\right]^{\frac{1}{q(N-1)}}\; = \; \left[\prod_{a<b}^N{2n_a+2n_b \choose 2n_a}\right]^{\frac{1}{q(N-1)}}
V_{Q \bar{Q}}( \ell ) \sim \min_{w}\left[ \pi w^{2}\ell \left\{\frac{1}{2}\left( \frac{Q}{\pi w^{2}}\right)^{2}+ \frac{c_{e}}{4 d}\right\}\right] \sim \frac{Q^{2}}{\pi w_{\mathrm{m i n}}^{2}}\times \ell .
=\sup_{(x,t)\in\overline{Q},\overline{h}\in\overline{H},|\overline{h}|\leq\varepsilon x_{N}}x_{N}^{\omega\gamma}\frac{|\Delta_{\overline{h},x}^{2s}\left( x_{N}^{n-j}D_{x}^{\alpha}u(x,t)\right) |}{|\overline{h}|^{\gamma}},\quad|\alpha|=m-j,j\leq n.
\sum\limits_{k=1}^{+\infty}\left( \mathcal{D}_{\eta^{k}}(\eta^{k+1},\eta^{k}; \lambda^{k}) + \beta\|r^{k+1}\|^2\right) \leq \mathcal{D}_{\eta^{1}}(\hat{\eta},\eta^{1}; \lambda^{1})<+\infty,
E ={\frac{1}{2 i}}\int_{- \infty}^{\infty}{\frac{d \omega}{2 \pi}}e^{- i \omega \tau}\sum_{l = 1}^{\infty}( 2 l + 1 ) \int_{0}^{\infty}r^{2}d r 2 k^{2}[ F_{l}( r , r ) + G_{l}( r , r ) ] .
\int d^{p + 1}y \; \Phi ( X , y ) \; \partial_{\alpha}X^{i}\partial^{\alpha}X^{i}= \sum_{l}{\frac{1}{l !}}\int d^{p + 1}y \; ( \partial_{i_{1}}\dots \partial_{i_{l}}\Phi ) X^{i_{1}}\dots X^{i_{l}}\; \partial_{\alpha}X^{i}\partial^{\alpha}X^{i}.
\left| \sum_{\alpha \beta \gamma \delta}\langle \Phi_{\alpha}\Phi_{\beta}\Phi_{\gamma}\Phi_{\delta}\rangle_c\left. \right/ \sum_{\alpha \beta \gamma \delta}\langle \Phi_{\alpha}\Phi_{\beta}\Phi_{\gamma}\Phi_{\delta}\rangle\right|=\left| \langle \Phi^4(x) \rangle_c \left. \right/\langle \Phi^4(x) \rangle \right|\; (n=4) \; ,
([\hat{D}]_{q})_{ij}:=[\ell_{i}]_{q}\delta_{ij},\quad \quad ([I^{\omega}]_{q})_{ij}:=\left[ \sum_{n=1}^{\ell_{i}}\hat{I}_{\omega^{n}(i)j}\right]_{q}\;,\quad i,j\in \hat{\Delta}^{\omega}
\frac{d^{n}}{dx^{n}}f(g(x))=\sum\frac{n!}{m_{1}!m_{2}!\cdot\cdot\cdot m_{n}!}\cdot f^{(m_{1}+\cdot\cdot\cdot+m_{n})}(g(x))\cdot\prod_{j=1}^{n}(\frac{g^{(j)}(x)}{j!})^{m_{j}}.
\frac{\partial\nu}{\partial z}=-\frac{\frac{\sqrt{2m_*}}{\hbar}\nu^2 K_1\left(\frac{\sqrt{2m_*}}{\hbar}\nu z\right)}{\left[1+\frac{\sqrt{2m_*}}{\hbar}\nu z K_1\left(\frac{\sqrt{2m_*}}{\hbar}\nu z \right)\right]}.
p_\theta ({\bf{x}}) = S(\theta)^{\frac{1}{\alpha -1}}N_{\theta,\alpha}\big[ 1 + b_\alpha S(\theta)^{-1}\lbrace \rm{vec}^\top (\boldsymbol{\Sigma}^{-1}) \rm{vec}({\bf{x}}{\bf{x}}^\top) - 2 (\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu})^\top{\bf{x}}\rbrace\big]^{\frac{1}{\alpha -1}}.
\widetilde{\mathbf{y}}_\ell = \left[ \boldsymbol{\lambda}_\ell - \mathbf{u}_\ell \right] \bmod \Lambda_{C,\ell}\stackrel{(a)}{=}\left[ \mathbf{y}_\ell + \mathbf{q}_\ell \right] \bmod \Lambda_{C,\ell}
\langle \Phi_{1}^{\ell}\Phi_{2}^{\ell^{\prime}}\rangle = \frac{\sqrt{( 2 \ell + 1 ) ( 2 \ell^{\prime}+ 1 )}}{4 \pi^{2}}\int_{| t | < 1}\frac{d t d \bar{t}}{( 1 - | t |^{2})^{2}}\int_{- 2 \pi}^{2 \pi}d \psi \ \Phi_{1}^{\ell}\bar{\Phi}_{2}^{\ell^{\prime}},
\begin{bmatrix}\mathbf{x}_{S}^{1}\mathbf{x}_{S^c}^{0}\end{bmatrix}\mathbf{B}^S = \mathbf{e}_i \Longrightarrow \begin{bmatrix}\mathbf{x}_{i S}^{1, S}\mathbf{x}_{i S^c}^{0, S}\end{bmatrix}= \mathbf{e}_i \mathbf{H}^S.
\mathcal{H}_N\Psi_N(\lambda_1,\ldots,\lambda_\ell)=\mathcal{E}_{\lambda_1,\ldots,\lambda_\ell}\Psi_N(\lambda_1,\ldots,\lambda_\ell),\mathcal{E}_{\lambda_1,\ldots,\lambda_\ell}:=-\frac{J}{2}\sum_{j=1}^\ell\frac{1}{\lambda_j^2+\frac{1}{4}}.
\left\vert \!\left\vert \!\left\vert t\right\vert \!\right\vert\!\right\vert :=\sum_{k=1}^{n}\left\Vert \pi_{k}\left( t\right) \right\Vert^{\frac{1}{k}}\forall t\in T^{n}\left( \mathcal{V}\right)
\operatorname{E}\left(\boldsymbol{Y}\right) = \mu_1 \cdot \boldsymbol{\pi}, \operatorname{Cov}\left(\boldsymbol{Y}\right) = \mu_1 \cdot \textrm{diag}(\boldsymbol{\pi}) + (\mu_2-\mu_1^2) \cdot \boldsymbol{\pi}\boldsymbol{\pi}^t,
\frac{d}{dt}R_ku_{N+1}(t) + AR_ku_{N+1}(t)=\frac{d}{dt}R_ku_{N+1}(t) + kR_ku_{N+1}(t) &=e^{-kt}\frac{d}{dt}\big(e^{kt}R_ku_{N+1}(t)\big)=e^{-kt}\frac{d}{dt}\big(e^{(k-N-1)t}p_{N+1,k}(t)\big) &=e^{-kt}\Big[ -e^{(k-N-1)t}\sum_{m+j=N+1}R_kB(q_m(t),q_j(t))\Big] =-\sum_{m+j=N+1}R_kB(u_m(t),u_j(t)) .
\tilde{B}_{f,\mu,\nu}(x,a)=(\mu_{a}\circ \tilde{B}_{f})(x,\nu(a))=a^{d}\left(\sum_{i}c_{i}x^{p^{i}}a^{p^{i}-d}\right)=\sum_{i}c_{i}(xa)^{p^{i}},
\sum_{j=2}^N\sum_{k=N+1}^{2N}\frac{4jk\delta_j\delta_k}{\big(k^2-j^2\big)^2}= \frac 1{L^2\pi^2}\sum_{j=2}^N\sum_{k=N+1}^{2N}\frac{4\frac j L\frac k L\delta\big(\frac{j\pi}L\big)\delta\left(\frac{k\pi}L\right)}{\big(\big(\frac k L\big)^2- \big(\frac j L\big)^2\big)^2}.
\int_0^1\int_\Omega \partial_t \phi\,d\mu_t\,dt+\int_0^1\int_\Omega \langle \nabla \phi,\mathbf{v}_t\rangle\,d\mu_t\,dt=\int_\Omega \phi(1,\cdot)\,d\rho_1-\int_\Omega \phi(0,\cdot)\,d\rho_0;
y_{(2k-1)m+i,j}+ y_{(2(r_0-k+1)-1)m +(m+1-i),n+1-j}= \tfrac{1}{2}mnr +\tfrac{3}{2}, \mbox{\and}y_{(2k)m+i,j}+ y_{2(r_0-k)m +(m+1-i),n+1-j}= 2MN -\tfrac{1}{2}mnr +\tfrac{3}{2}.
\|\underline{B}^{(k-1)}\tilde{H}B^{(\ell-k-1)}\|\leq \sum_{i=1}^3 \|\underline{B}^{(k-1)}M^{(i)}B^{(\ell-k-1)}\|\leq \sum_{i=1}^3 \|\underline{B}^{(k-1)}\| \|M^{(i)}B^{(\ell-k-1)}\|.
\begin{array}{rcl}L_{\epsilon}\phi = \partial_{t}\phi - \nabla \cdot (A(t,x) \nabla \phi) - (eA(t,x)e+\epsilon) \partial_{zz}\phi - \nabla \cdot (A(t,x)e \partial_{z}\phi) & -\partial_{z}(e A(t,x)\nabla \phi) +q(t,x)\cdot\nabla\phi+q(t,x)\cdot e\partial_{z}\phi+c\partial_{z}\phi. \end{array}
\mathcal{D}(S_{min})=\left\{f\in{W}_2^2(\mathbb{R}\backslash\{0\}) : \begin{array}{cc}f_r(0)=0 f_s(0)=(q_2, f) f_r'(0)=0 f_s'(0)=(q_1, f)\end{array}\right\}.
{J_{5}^{\mu}}= - \sum_{r = 0 , 2 , . . .}{\tilde{\overline{\Psi}}}_{L_{r}}\gamma^{\mu}\tilde{\Psi}_{L_{r}}+ \sum_{s = 1 , 3 , . . .}{\tilde{\overline{\Phi}}}_{L_{s}}\sigma^{3}\gamma^{\mu}\tilde{\Phi}_{L_{s}}.
g(A_V(X),A_V(X)) =g(tan(A_U(X)),tan(A_U(X)))-2\delta g(U,N)g(\nabla_XN,A_U(X)) &+g(U,N)^2g(\nabla_XN,\nabla_XN) &=g(A_U(X),A_U(X))-\delta g(A_U(X),N)^2-2\delta g(U,N)g(\nabla_XN,A_U(X)) &+g(U,N)^2g(\nabla_XN,\nabla_XN).
{\cal K}_{{\bf C}^n}(z, \bar z) = N \,{\rm exp}\left({\rm i}\sum_{j=1}^n\bar z^j z^j\right){\rm d}z^1\wedge\ldots\wedge{\rm d}z^n\wedge{\rm d}\bar z^1\wedge\ldots\wedge{\rm d}\bar z^n,
\sum_{k=0}^{n}(\begin{matrix}n\\k\end{matrix})=(\begin{matrix}n\\ 0\end{matrix})+(\begin{matrix}n\\ 1\end{matrix})+\cdot\cdot\cdot+(\begin{matrix}n\\n-1\end{matrix})+(\begin{matrix}n\\n\end{matrix})=2^{n}.
\left [\begin{array}{cc}D -C B -A\end{array}\right ] \left [\begin{array}{cc}I_n \Psi \Phi I_m\end{array}\right ]=\left [\begin{array}{cc}I_n \Psi \Phi I_m\end{array}\right ]\left [\begin{array}{cc}R 0 0 -S\end{array}\right ],
S_{f,i}= \lim_{\tau \rightarrow -\infty,\tau' \rightarrow +\infty}(-\sqrt{2}e^{\tau'}\stackrel{\leftrightarrow}{\frac{\partial}{\partial\tau'}})(\sqrt{2}e^{-\tau}\stackrel{\leftrightarrow}{\frac{\partial}{\partial\tau}})G(\tau',\tau)
\mathcal{L}_{t}\Phi(v,t) =\big(b^{2}q_{0}\varepsilon+c^{2}\varepsilon\int_{0}^{\infty}z^{2}\nu(dz)-2\big)\|v\|^{4}_{L^{4}}&\quad+\big(b^{2}q_{0}\varepsilon^{-\frac{1-m\alpha}{m\alpha}}+c^{2}\varepsilon^{-\frac{1-n\beta}{n\beta}}\int_{0}^{\infty}z^{2}\nu(dz)+2\big)\|v\|^{2}_{L^{2}}.
\hat{\Lambda}_{A}^{\C}\hat{\Lambda}_{B}^{\D}\hat{\eta}_{C D}= \hat{\eta}_{A B}\ \ \ \mathrm{a n d}\ \ \ \hat{\Lambda}_{A}^{\C}\hat{\Lambda}_{B}^{\D}\hat{\varepsilon}_{C D}= \hat{\varepsilon}_{A B}
P_{M_{n,p,b}}(x_{ij}=1,\, \forall\, i,j)(\beta=0)={\frac{1}{n p - \overline{b}+1}}{\binom{n(p+1) - \overline{b}}{n}}={\frac{1}{n}}{\binom{n(p+1)-\overline{b}}{n-1}},
v_{3}^{\lambda}(t,r) = -r \int_{t+6r}^{\infty}ds \lambda''(s) (s-t) \left(\frac{1}{1+(s-t)^{2}}-\frac{1}{\lambda(t)^{2-2\alpha}+(s-t)^{2}}\right) + E_{5}^{\lambda}(t,r)
\frac{\partial}{\partial \theta}\begin{array}{c}{l i m}\\{\varepsilon \rightarrow 0}\\ \end{array}I ( p , m , \theta , \varepsilon ) \mid_{_{\theta_{0}^{-}}}\neq \frac{\partial}{\partial \theta}\begin{array}{c}{l i m}\\{\varepsilon \rightarrow 0}\\ \end{array}I ( p , m , \theta , \varepsilon ) \mid_{_{_{\theta_{0}^{+}}}}
\operatorname{Cov}\bigl(Z_t^H(a,b), Z_s^H(a,b) \bigr) &= \frac{1}{2}(a+b)^2 \bigl( s^{2H}+ t^{2H}\bigr) - ab ( t+ s )^{2H}- \frac{a^2 + b^2}{2}| t- s|^{2H}.
\sum_{n \geq 0}\left(\sum_{k_{1},\ldots,k_{m}\geq 0}W_{\underline{r},r_{1}^{{k_{1}}}{r_{2}^{k_{2}}}\cdots r_{m}^{k_{m}}j,n}\right)q^n=\Phi_{\underline{r}}(q)\frac{q^j}{1-q^j}.
\gamma = \sigma_2 \sigma_1^2 \sigma_2 \sigma_3 \sigma_4 \sigma_5 w_1 \sigma_2 \sigma_3 \sigma_4 \sigma_3^{-1}\sigma_2^{-1}w_2 \sigma_2 \sigma_3 \sigma_2^{-1}w_3 \sigma_2 \sigma_5^{-1}\sigma_4^{-1}\sigma_3^{-1}\sigma_1^2.
\{x_2 = u_i(y) \colon i\in [l]\}\cup \{x_2 < u_i(y) \colon i \in [l] \}\cup \, \{x_2 > u_i(y) \colon i\in [l] \}\cup \{x_1=h_i(x_2,y)\colon i\in [k]\}\cup \, \{x_1<h_i(x_2,y)\colon i\in [k]\}\cup \{x_1>h_i(x_2,y)\colon i\in [k]\}.
L_1u(x) =b(x) \Bigl(P.V.\displaystyle\int_{\mathbb{R}^{N-1}}\frac{(u(x)-u(x+y))}{|y|^{N+\alpha}}A_K^e(x,0,y/|y|)\;dy\Bigr) &=\frac{b(x)}{2}\int_{\mathbb{R}^{N-1}}\frac{(2u(x)-u(x+y)-u(x-y))}{|y|^{N+\alpha}}A_K^e(x,0,y/|y|)\;dy
dd^c \omega_0^{n-k-1}= dJd\omega_0^{n-k-1}=(n-k-1) dJ(\theta \wedge \omega_0^{n-k-1}) & = (-1)^{n-k-1}(n-k-1) d(J\theta \wedge (dJ\theta)^{n-k-1}) & = -(n-k-1) (-dJ\theta)^{n-k}\leq 0 .
\psi(t)=\begin{cases}O\left(t^{\theta}\right) \hbox{if}\theta<-\frac{M}2+3 O\left(t^{-\frac{M}2+3}\right) \hbox{if}\theta>-\frac{M}2+3 O\left(t^{-\frac{M}2+3}|\log t|\right) \hbox{if}\theta=-\frac{M}2+3\end{cases}
P_{2m+1}(A^p,B^p) =A^pB^{-p}P_{2m-1}(A^p,B^p) B^{-p}A^p &\le A^pB^{-p}f(A)^{2(n-m)p}B^{-p}A^p &\le A^pB^{-p}f(X)^{2(n-m)p}B^{-p}A^p &=A^p \left({{f(X)^{n-m-1}}\over{X}}\right)^{2p}A^p &\le A^p \left({{f(A)^{n-m-1}}\over{A}}\right)^{2p}A^p =f(A)^{2(n-m-1)p}.
\int_X f_1(\pi_1(x_1,x_2)) f_2(\pi_2(y_1,y_2))\> g(y_1)g(y_2) \> d(\omega_{X_1}\times\omega_{X_2})(y_1,y_2)= &=\int_D\int_X g(y_1)g(y_2)\> K_{h_1,h_2}((x_1,x_2), d(y_1,y_2)) \> f_1(h_1)f_2(h_2) d\omega_D(h_1,h_2).
\tau_a(\alpha) = \frac{1}{q}\sum_{u \in \mathbb{F}_{q}}\psi_1\left(_{\mathbb{F}_{q^m}/\mathbb{F}_{q}}(u\alpha)-ua \right) = \frac{1}{q}\sum_{u \in \mathbb{F}_{q}}\hat{\psi_1}(u\alpha)\psi_1(-ua),
m_\nu(a,b)=\inf_{\mathcal{T}(a,b)}\max_{t\in\mathbb{R}}J_\nu(t\star(u,v)),m_{r,\nu}(a,b)=\inf_{\mathcal{T}_r(a,b)}\max_{t\in\mathbb{R}}J_\nu(t\star(u,v)).
R_j(\mu,c){:=}{b^\#_{j-l}}V_{q}[\omega, \mu](x) +{b^\#_j}c\forall j \in \mathbb{N}\setminus \{0\}\, ,R[k](\mu,c){:=}\sum_{j=1}^{+\infty}R_j(\mu,c)k^j\, ,
{\widetilde \xi}| \Psi \rangle = \frac{i}{c o s^{2}( \theta ) - s i n^{2}( \theta ) e^{2 i \phi}}( \sqrt{2}s i n ( \theta ) c o s ( \theta ) e^{i \phi}- ( c o s^{2}( \theta ) + s i n^{2}( \theta ) e^{2 i \phi}) \xi ) | \Psi \rangle ,
R_{P}^{D_0}= R^{D_0}\setminus R_{K}^{C}= \{\epsilon_3 - \epsilon_1, \epsilon_3 - \epsilon_2 \}R_{P}^{D_1}= R^{D_1}\setminus R_{K}^{C}= \{\epsilon_1 - \epsilon_3, \epsilon_3 - \epsilon_2 \}R_{P}^{D_2}= R^{D_2}\setminus R_{K}^{C}= \{\epsilon_1 - \epsilon_3, \epsilon_2 - \epsilon_3 \}.
\Gamma_2 =-2\sum_i\Theta ||d^\top P\hat x_{vi}||_1-2\bar x^\top (H\otimes Pd) v-2\bar x^\top [H(E-\Theta I_N)\otimes Pd]\mbox{sgn}[(H\otimes d^\top P)\bar x] &\leq -2\sum_i(\Theta-l) ||d^\top P\hat x_{vi}||_1-2\sum_i(\theta_i-\Theta) ||d^\top P\hat x_{vi}||_1
\sum_{q=1}^{N_1}|Z^{\beta}_{t_{i\left(n,q\right)-1,n}}|{\bf 1}_{\{T_{q+1}-T_q < t_{i\left(n,q\right),n}- T_q + \Delta_n\}}\lesssim \frac{\lambda_n^3 \Delta_n}{\beta_n}
\mathcal{P}_1\left({x}\right)=\exp \left\{- 2\pi{\lambda_e}\int_0^\infty{{f_{\Pr}}({r_e})}{r_e}\sum\limits_{\ell,n \in \left\{{{},{}}\right\}}{\rm{\mathbf{1}}}\left({\max \{{r_e},d\}}< \big(\frac{{P_t}{G_\ell}{G_n^e}\beta}{x \sigma_e^2}\big)^{\frac{1}{{\alpha_{\mathrm{LoS}}}}}\right){{{\Pr}_{\ell n}}}d{r_e}\right\}
\sigma_3 \left(2^{4g(0)}\sigma_3 + 2^{3g(0)+2}+ 2^{2g(0)+1}+ 2^{g(0)+1}- 1\right) = 2^{p-4g(0)}- 2^{2g(0)+2}- 2^{g(0)+2}- 2^2 - 1.
\alpha_k (t)=\beta_k \int^t_0 e^{-i (k\pi )^2 (t-\tau)}h'_1(\tau ) d\tau = \beta_k h_1 (t) - i\beta_k (\pi k)^2 \int^t_0e^{-i (k\pi )^2 (t-\tau)}h_1 (\tau ) d\tau \, .
\rho^{\mathrm{t o t}}= \rho +{\frac{\rho^{2}}{2 \lambda}}+{\frac{\widetilde{\kappa}^{4}{\cal U}}{\kappa^{6}}}+{\frac{\Lambda}{\kappa^{2}}}, p^{\mathrm{t o t}}= p +{\frac{\rho^{2}}{2 \lambda}}+{\frac{\rho p}{\lambda}}-{\frac{\widetilde{\kappa}^{4}{\cal U}}{3 \kappa^{6}}}-{\frac{\Lambda}{\kappa^{2}}}.
\left(\ell\right) =\frac{\left(\frac{1}{4}\left(1-(1-2q)^{2\ell}\right)\right)^2}{\frac{1}{4}\left(1-(1-2q)^{2\ell}\right)\left(1+(1-2q)^{2\ell}\right)}&=\frac{1}{4}\cdot\frac{1-(1-2q)^{2\ell}}{1+(1-2q)^{2\ell}}.
\frac{d}{ds}b^{(1)}(s,\tau',\alpha^{(1)}(s,\tau', y'))\Big|_{y'=\beta^{(1)}}=\frac{d}{ds}b^{(2)}(s,\tau',\alpha^{(2)}(s,\tau', y'))\Big|_{y'=\beta^{(2)}}\ \ \mbox{on}\ \ \tilde R_{2T_1}^{(1)}(\tau').
F_1\left(\frac 1{w_1},\ldots, \frac 1{w_n}, t\right)=\left(\frac 1{w_1}\right)^{\deg_{w_1}Q_1}\cdot \ldots \cdot\left(\frac 1{w_n}\right)^{m_{1n}}\cdot \left(\widetilde q_1(w)+t \cdot \widetilde Q_1(w)\right),
\mathfrak{J}_{i}(n)=\sum\limits_{m_{1}+\cdots+m_{5}=n\atop U_{i}^{3}<m_{2},m_{3}\leq 8U_{i}^{3},V_{i}^{3}<m_{4},m_{5}\leq 8V_{i}^{3}}(m_{2}\ldots m_{5})^{-2/3}
\gamma = \frac{2}{3}{\alpha}+\frac{1}{2}= \frac{2}{3}\beta + \frac{1}{3}, M_1 = \frac{1}{3}\alpha^2 + \frac{1}{6}\alpha - \frac{1}{36}, M_2 = \frac{1}{18}\alpha^4 + \frac{1}{54}\alpha^3 - \frac{17}{216}\alpha^2 - \frac{1}{54}\alpha + \frac{25}{2592},
\Bigg|\Big\langle \widehat{\nabla \mathrm{H}}^{\textbf{in}};\mathrm{e}^{(2)}_{\mathrm{n}_{0}}\Big\rangle_{\mathbb{L}^{2}(\mathrm{B})}\Bigg|^2 = \mathcal{O}\Big(\delta^2\Big).
\sigma=\frac{1}{\lambda_{1}\lambda_{2}\lambda_{3}}[\lambda_{1}\frac{\partial W}{\partial\lambda_{1}}n_{1}\otimes n_{1}+\lambda_{2}\frac{\partial W}{\partial\lambda_{2}}n_{2}\otimes n_{2}+\lambda_{3}\frac{\partial W}{\partial\lambda_{3}}n_{3}\otimes n_{3}]
\sum_{\rho\in\Delta_+}{g_{|\rho|}}|\rho\cdot\mu| =\sum_{\rho\in\Delta_+}{g_{|\rho|}}|g(\rho)\cdot\lambda|=(\sum_{\rho\in\Delta_+}{g_{|\rho|}}\rho)\cdot\lambda=2\varrho\cdot\lambda,
u_{g_i}vu_{g_j}^*b(u_{g_j}v^*u_{g_i}^*v)v^*= u_{g_i}vu_{g_j}^*(u_{g_j}v^*u_{g_i}^*v)bv^*= (u_{g_i}e_iu_{g_i}^*)vbv^*
S_n(x,y) = S_n(y) - S_n(x), \ \ \ \F(x,y) = F(y) -F(x) & \tilde F_{n,l}(x,y) = \tilde F_{n,l}(y) - \tilde F_{n,l}(x), \ \ \ \J_m(x,y) = J_m(y) -J_m(x).
K_w^{\Theta}(z) = \sqrt{2}(I-\Theta (z) )^{-1}\left( \frac{I - \Theta (z) \Theta (w)^*}{1 - z \overline{w}}\right) \sqrt{2}(I - \Theta (w)^* )^{-1}& = V(z) k_w^\Theta (z) V(w)^*,
\frac{\mu(D(z,r))}{(1+|z|)^s}= \frac{1}{(1+|z|)^s}\int_{D(z,r)}d\mu(w)\leq\frac{e^{\frac{\alpha t r^2}{2}}}{(1+|z|)^s}\int_{D(z,r)}e^{-\frac{\alpha t}{2}|z-w|^2}d\mu(w) \lesssim e^{\frac{\alpha t r^2}{2}}\int_{D(z,r)}\frac{e^{-\frac{\alpha t}{2}|z-w|^2}}{(1+|w|)^s}d\mu(w)\lesssim \widetilde{\mu}_{(t,s))}(z)
\pi_{A}( x ) = \frac{\partial{\cal L}( x )}{\partial \left( \partial_{0}\Psi_{A}( x ) \right)}= i \left( \overline{{{\Psi}}}( x ) \Gamma_{4}\right)_{A}= i \left( \Psi^{+}( x ) \overline{{{\Gamma}}}_{4}\right)_{A}.
f_{++}(x)=\lim_{t_1,t_2\rightarrow0^+}F(x_1+it_1,x_2+it_2), f_{+-}(x)=\lim_{t_1,t_2\rightarrow0^+}F(x_1+it_1,x_2-it_2), &f_{-+}(x)=\lim_{t_1,t_2\rightarrow0^+}F(x_1-it_1,x_2+it_2), f_{--}(x)=\lim_{t_1,t_2\rightarrow0^+}F(x_1-it_1,x_2-it_2).
2m_0 \leq J(t_{\widehat{w}^+}\widehat{w}^+)+J(s_{\widehat{w}^-}\widehat{w}^-) & <J(t_{\widehat{w}^+}\widehat{w}^++s_{\widehat{w}^-}\widehat{w}^-) &\leq J(\widehat{w}) &=\inf\limits_{\mathcal{M}}J=m_1.
\Phi=\left[\begin{array}{cccc}0 0 0 -1_{2m}0 0 -1_r 0 0 1_r 0 0 1_{2m}0 0 0\end{array}\right],\Psi=\left[\begin{array}{cccc}0 0 0 J_m'I_m' 0 -\alpha^{-1}0 0 0 0 1_r 0 -J_m'I_m' 0 0 0\end{array}\right].
{\cal D}X^{\mu}= D X^{\mu}- \frac{l^{2}( k_{\rho}D X^{\rho}) - ( k \cdot l ) ( l_{\rho}D X^{\rho})}{k^{2}l^{2}- ( k \cdot l )^{2}}\, k^{\mu}\ - \frac{k^{2}( l_{\rho}D X^{\rho}) - ( k \cdot l ) ( k_{\rho}D X^{\rho})}{k^{2}l^{2}- ( k \cdot l )^{2}}\, l^{\mu},
\operatorname{det}\left(\begin{array}{cccc}f f_x f_{xx}f_{xxx}(z) (z)_x (z)_{xx}(z)_{xxx}(z\log q) (z\log q)_x (z\log q)_{xx}(z\log q)_{xxx}(z\log^2 q) (z\log^2 q)_x (z\log^2 q)_{xx}(z\log^2 q)_{xxx}\end{array}\right) =0
\bigg| \frac{\partial^{p+2}f}{\prod_{j=1}^{p+2}\partial x_{i_j}}(x) - \frac{\partial^{p+2}f}{\prod_{j=1}^{p+2}\partial x_{i_j}}(y) \bigg| \leq - [h]_{\alpha,p}\log \eta + [h]_{\alpha,p}2^{\alpha/2+1}\frac{\alpha+d+1}{\alpha}\frac{\Gamma(\frac{\alpha+d}{2})}{\Gamma(d/2)}\eta^{\alpha/2}.
-\frac{1}{2\pi}\int_{-\infty}^\infty \hat{g}(-\frac12+i(r-b))\cdot pv\int_0^\infty \frac{f_0(\nu)^{2a}}{f_1(\nu)}\nu^{ir}d^\times\nu\dr = -pv\int_0^\infty \frac{f_0(\nu)^{2a}}{f_1(\nu)}\cdot \frac{1}{2\pi}\int_{-\infty}^\infty \hat{g}(-\frac12+i(r-b))\nu^{ir}dr \d^\times\nu.
p_2-p_1 =\frac{a(M+1)(M\mu-1)}{2M\mu(M\mu+a)}=\frac{a(M+1)(a+(M-2)f)}{2M\mu(M\mu+a)}>0, p_2-p_3 =\frac{(1-a)^2 (M+1)}{2(M\mu+1)(M\mu+a)}>0,
\begin{cases}\left(\gamma_{1}y-1\right)x_{1}+\left(\delta_{1}y\right)x_{2}=\;0 \left(\gamma_{2}y\right)x_{1}+\left(\delta_{2}y-1\right)x_{2}=\;0 \gamma x_{1}+\delta x_{2}=\;1\end{cases}
v^{LR}_{b}(x) =- C^1_b\left( Z_b^{(\theta,r)}(x) - r Z^{(\theta)}(b) \overline{W}^{(r+\theta)}(x-b) \right) - r \overline{\overline{W}}^{(r+\theta)}(x-b) + \beta \left( \overline{Z}_b^{( \theta ,r)}(x) + \dfrac{\psi_X'(0+)}{\theta}- r \overline{Z}^{(\theta)}(b) \overline{W}^{(\theta+r )}(x-b) \right),
A=[Z';H_{\gamma}'\otimes K_{\gamma}'; \psi_{\gamma,\delta}^{H'}\otimes \psi_{\gamma,\delta}^{K'}] & B= [Z'';H_{\tau}''\otimes K_{\tau}''; \psi_{\tau,\sigma}^{H''}\otimes \psi_{\tau,\sigma}^{K''}]
\Bigl\{\|(R_{n}+a(R_{n})\rho)\theta + y_i\| \geq R_n\,, \i=1,\dots,2k-\ell-1 \Bigr\}&=\bigl\{\rho + \xi_n(\rho, \theta, y_i) \geq 0, \i=1,\dots,2k-\ell-1 \bigr\}\,,
\sum_{g\in G}|\xi'_x(g)-\xi'_{x'}(g)| =\sum_{g\in K}|\xi'_x(g)-\xi'_{x'}(g)|+\sum_{g\not\in K}|\xi'_x(g)-\xi'_{x'}(g)| &\leq \sum_{h\in F}|\xi_{\phi(x)}(h)-\xi_{\phi(x')}(h)|+\sum_{h\not\in F}|\xi_{\phi(x)}(h)|+\sum_{h\not\in F}|\xi_{\phi(x')}(h)| &\leq \epsilon/4+\epsilon/4+\epsilon/2=\epsilon.
\phi'(x) = \log \prod_{j \in J}m_j(x)^{\tau_j}\prod_{i \in I}l_i(x)^{-\sigma_i}, \phi''(x) = \sum_{j \in J}\dfrac{\tau_j^2}{m_j(x)}- \sum_{i \in I}\dfrac{\sigma_i^2}{l_i(x)}.
\begin{array}{ll}C_{34}:=\min \left[\{2^{r-1}3rBE\|u_x^0\|_\infty(5N)^{r-1}N^{-1}\}^{-1/(r-1)},\right. &\quad\left.\{2^{r-1}3rBE\|u^0\|_\infty^{r-1}5\}^{-1}\right].\end{array}
f_{TTT}= -{2i\over\pi}\frac{j_T(T)}{j(T)-j(U)}\left\{\frac{j(U)}{j(T)}\right\}\left\{\frac{j_T(T)}{j_U(U)}\right\}\left\{\frac{j(U)-j(i)}{j(T)-j(i)}\right\}.
\widetilde{\Gamma}_{i}^{( 9 )}= \left( \begin{matrix}{{0}}&{{t_{i}}}\\{{- t_{i}}}&{{0}}\\ \end{matrix}\right) \, , \qquad \widetilde{\Gamma}_{i}^{( 0 )}= \left( \begin{matrix}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{matrix}\right) \, , \qquad \widetilde{\Gamma}_{i}^{( 8 )}= \left( \begin{matrix}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{matrix}\right) \, ,
\int_{x_2=0^+}\left[\mathbf{R}\mathbf{U}(x_1'-x_1,0^+)\cdot \mathbf{t}(x_1,0^+) - \mathbf{R}\mathbf{\Sigma}(x_1'-x_1,0^+)\cdot \mathbf{u}(x_1,0^+)\right]\mathrm{d}x_1=0,
\widetilde{F_\alpha(f)}(\overline{v_1})=\sum_{\substack{v_2\in N_0(t,v)\backslash N_0/H_{\Delta,0}v_1\in N_0(t,v)t_\alpha v_2t_\alpha^{-1}H_{\Delta,0}}}\chi^w(t_\alpha)f(vv_2)=\sum_{u\in J((N_{\Delta,0}\cap t_\alpha^{-1}N_{\Delta_1,\ast}t_\alpha)/ N_{\Delta_1,\ast})}\chi(t_\alpha)\widetilde{f}(t_\alpha^{-1}\overline{v_1}t_\alpha u)
g_{j}(\zeta)=\sum_{n=1}^{\infty}\overline{g}_n\,\overline{\lambda}_n^2\left[\frac{d^2}{dy^2}\int_{\mbox{arccosh}(\large{y})}^{\infty}\frac{udu}{\sqrt{\cosh u-y}}\frac{1}{\sinh^j(u/2)}\right]_{y=1+n^2\zeta^2/2},
\imath_K\omega(Z_1,\cdots ,Z_{k+p}) & =\sum_{\sigma\in S_{k+1,p-1}}\mathrm{sgn}(\sigma)\omega(K(Z_{\sigma(1)},\cdots,Z_{\sigma(k+1)}),Z_{\sigma(k+2)},\cdots,Z_{\sigma(k+p)}),
\Phi(k,p)=\frac{1}{\sqrt{4\pi}}\int_{-1}^{1}dz^{\prime}\int_{0}^{\infty}d\gamma^{\prime}\frac{g(\gamma^{\prime},z^{\prime})}{[\gamma^{\prime}+m^{2}-\frac{1}{4}M^{2}-k^{2}-p\cdot kz^{\prime}-i\epsilon]^{3}},
\widetilde{e}\left(\frac{kb}{\varpi^l}\right)\sum_{c \bmod{\varpi^{l-1}}}\widetilde{e}\left(\frac{k\varpi^{-h}c}{\varpi^{l-h-1}}\right)=\widetilde{e}\left(\frac{kb}{\varpi^l}\right)\sum_{c \bmod{\varpi^{l-1}}}\widetilde{e}\left(\frac{c}{\varpi^{l-h-1}}\right).
X = \begin{pmatrix}\psi \phi \dot\phi \end{pmatrix}, A = \begin{pmatrix}-\alpha \partial_x - iM\beta 0 0 0 0 1 0 \partial_x^2-m^2 0 \end{pmatrix}, \mathcal M(X) = \begin{pmatrix}i\beta\psi \mathfrak K_1 0 \phi \mathfrak K_2 \end{pmatrix}.
\delta(s'(e)) = \tilde{\mathbf{Q}}\left( \delta\big( e^{-\pi i( \frac{k_R}{3}+ \frac{1}{2})}\tilde{\alpha}_0^{-1}\big) g_q\!\left( \tilde{\chi}_{\sigma_R}^{-1}\circ \tilde{\chi}_{\sigma_L}\right) \delta\big( e^{\pi i \frac{k_L}{3}}\tilde{\alpha}_0 \big) \right) \ ,