harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{matrix}\mu_{\alpha}(x)&=&-\frac{1}{2}\frac{W_{-\alpha,-1}(-x^{\alpha})}{\Gamma(\frac{1}{\alpha}+1)N_{\alpha}(x)}\\ &=&-\frac{1}{2}\frac{W_{-\alpha,-1}(-x^{\alpha})}{W_{-\alpha,0}(-x^{\alpha})}\end{matrix}
P_{\Lambda_L,\eta, h}^{a}(\sigma)=\frac{1}{Z^a_{\Lambda_L,\eta,h}}e^{\beta_c\sum_{\{u,v\}}\sigma_u\sigma_v+\beta_c\sum_{\{u,v\}:u\in\Lambda_L^a, v\in\partial_{ex}\Lambda_L^a}\sigma_u\eta_v+a^{15/8}h\sum_{u\in\Lambda^a_{L}}\sigma_u},
e^{\psi}= \left| \frac{Q_2 \sqrt{c_1}}{Q_1 \sqrt{c_2}}\frac{\sinh \left[ \sqrt{c_2}(I(A) +\tilde{c}_2 ) \right]}{\sinh \left [\sqrt{c_1}(I(A) + \tilde{c}_1 ) \right ]}\right|^{1/2},
1 = \sum_{i=1}^r\frac{-((\lambda-1) - (m_i-1)R_\mathbb{G}) +[ ( (\lambda-1) - (m_i-1) R_\mathbb{G})^2 + 4\lambda m_iR_\mathbb{G}^2]^{1/2}}{2m}&=\sum_{i=1}^r\frac{(1-\lambda)+ (m_i-1) R_\mathbb{G}+[((1-\lambda) + (m_i-1) R_\mathbb{G})^2 + 4\lambda m_iR_\mathbb{G}^2]^{1/2}}{2m}.
S_f= - \frac{1}{2 g^2}Tr \left(\sum_{i=0}^{3}\overline{\psi_i}\Gamma^{\mu}[A_{\mu},\psi_i]+ 2 \sum_{i=1}^{3}\overline{(\psi_i)^c}\bar{\Gamma}^{(i)}[B^{\dagger}_i,\psi_0]+ \sum_{i,j,k=1}^3 |\epsilon_{ijk}| \overline{(\psi_i)^c}\Gamma^{(j)}[B_j, \psi_k] + h.c.\right)
\mathcal{D}_{p}^{\mathcal{T}}(F,f,\kappa)=\inf{\huge \{}\sup_{\mu(E)=\kappa}\int_{E}(M_{\mathcal{T}}\phi)^{p}d\mu:\phi\geq0\text{measurablewith}\int_{X}\phi d\mu=f\int_{X}\phi^{p}d\mu=F{\huge \}}
\{{\cal Q},\omega_d\}=(-1)^gi_{v_1}\cdots i_{v_g}d\Omega_{d+g}+\sum_{k=1}^g(-1)^{k+1}{\cal L}_{v_k}i_{v_1}\cdots i_{v_{k-1}}i_{v_{k+1}}\cdots i_{v_g}\Omega_{d+g}=d\omega_d.
\nabla^{2}\phi=\frac{1}{h_{1}h_{2}h_{3}}[\frac{\partial}{\partial q^{1}}(\frac{h_{2}h_{3}}{h_{1}}\frac{\partial\phi}{\partial q^{1}})+\frac{\partial}{\partial q^{2}}(\frac{h_{3}h_{1}}{h_{2}}\frac{\partial\phi}{\partial q^{2}})+\frac{\partial}{\partial q^{3}}(\frac{h_{1}h_{2}}{h_{3}}\frac{\partial\phi}{\partial q^{3}})]
f'(x)(v) =p\left \| x \right \|_H^{p-2}\langle x, v \rangle_H, f''(x)(v,w) = \begin{cases}p\left \| x \right \|_H^{p-2}\langle v, w \rangle_H+p(p-2)\left\| x \right\|_H^{p-4}\langle x, v \rangle_H\langle x, w \rangle_H, x\neq 0 p\left \| x \right \|_H^{p-2}\langle v, w \rangle_H, x = 0.\end{cases}
Q_6 = 4 q_1^3 q_3 - 2 q_1^2 q_2^2 - 10 q_1^2 q_4 + 12 q_1 q_2 q_3 - 5 q_2^3 - 15 q_2 q_4 + 5 q_3^2 + 35 q_6.
r^{2}G ( r ) \rightarrow 1 + \frac a 4 \ln ( \mu r ) + \frac{a^{2}}{6 4}\ln ( \mu r ) \left( 1 + 2 \ln ( \mu r ) \right) + \frac{a^{3}}{1 5 3 6}\ln ( \mu r ) \left( 9 + 6 \ln ( \mu r ) + 4 \ln^{2}( \mu r ) \right) + O ( a^{4}) .
2 L_{z}= \left( \begin{matrix}{\ell}&{0}\\{\noalign{\smallskip}0}&{- \ell}\\ \end{matrix}\right) \qquad 2 L_{y}= \left( \begin{matrix}{0}&{1}\\{\noalign{\smallskip}- 1}&{0}\\ \end{matrix}\right) \qquad 2 L_{x}= \left( \begin{matrix}{0}&{\ell}\\{\noalign{\smallskip}\ell}&{0}\\ \end{matrix}\right)
F^{\circ k}\left( v_{1}\otimes \cdots \otimes v_{k}\right) :=\left( F^{\circ1}\left( v_{1}\right) \right) \circ \left( F^{\circ 1}\left( v_{2}\right)\right) \circ \cdots \circ \left( F^{\circ 1}\left( v_{k}\right) \right) .
a_{11}=\frac{\sigma r_{1}^{2}+\overline{\sigma}r_{2}^{2}+ \sigma r_{3}^{2}}{R^{2}}, \:a_{12}=a_{21}=-\frac{(\sigma-\overline{\sigma})r_{3}r_{2}^{2}}{r R^{2}}, \:a_{22}=\frac{r_{3}(\sigma r_{1}^{2}+\overline{\sigma}r_{2}^{2}+ \sigma r^{2})}{r R^{2}}
\widehat{L}_{\beta}( \lambda ) = \frac{1}{\sin \gamma}\left( \begin{array}{c c}{{\sinh [ \gamma ( \lambda + \mathrm{i}S^{3}) ]}}&{{\mathrm{i}\, \sin \gamma \, e^{( \gamma - \beta ) \lambda}\, S^{-}}}\\{{\mathrm{i}\, \sin \gamma \, e^{( \beta - \gamma ) \lambda}\, S^{+}}}&{{\sinh [ \gamma ( \lambda - \mathrm{i}S^{3}) ]}}\\ \end{array}\right)
\MoveEqLeft A(t)u(t) = \int_0^t A(t)^2 e^{-(t-s) A(t)}(A(s)^{-1}- A(t)^{-1}) A(s) u(s) \d s & + \int_0^t A(t) e^{-(t-s) A(t)}f(s) \d s + A(t) e^{-t A(t)}u_0.
S_{R^{4}}^{A}= \int d^{1 0}x \sqrt{- g}~ 2 \zeta ( 3 ) e^{- 2 \phi}( t_{8}t_{8}+ \frac{1}{8}\epsilon_{1 0}\epsilon_{1 0}) R^{4}+ \frac{2 \pi^{2}}{3}( t_{8}t_{8}- \frac{1}{8}\epsilon_{1 0}\epsilon_{1 0}) R^{4}
\Sigma_{1}^{i}( z - v ) = \frac{- 3 f_{l}^{2}}{1 6 ( 4 \pi^{2})}\left( 2 \frac{Y_{L}^{( e )}}{Y_{R}^{( e )}}L \delta_{i ( e )}+ \frac{Y_{R}^{( e )}}{Y_{L}^{( e )}}R \right) / \hspace{- 2 m m}\partial \delta^{4}( z - v ) \, .
\sum_{|\lambda/\mu|=n+1}f_{\lambda/\mu}g(\lambda) =\sum_{|\nu/\mu|=n}f_{\nu/\mu}g(\nu)+\sum_{|\nu/\mu|=n}f_{\nu/\mu}Dg(\nu) &=\sum_{k=0}^n\binom{n}{k}D^kg(\mu)+\sum_{k=0}^n\binom{n}{k}D^{k+1}g(\mu) &=\sum_{k=0}^{n+1}\binom{n+1}{k}D^kg(\mu).
\begin{aligned}\pi^{Sp(n)\times Sp(1)}_{d,e}= \pi^{Sp(n)}_{d,e}\otimes \pi^{Sp(1)}_{d-e}\dim \pi^{Sp(n)\times Sp(1)}_{d,e}= \frac{(d+e+2n-1)(d-e+1)^2\prod_{j=1}^{2n-3}(d+j+1)(e+j)}{(2n -1)(2n-2)\cdot [(2n-3)!]^2}. \end{aligned}
( \Delta^{a}+ \mathrm{\Large \frac{i}{\hbar}}V^{a}) \exp \big \{\mathrm{\Large \frac{i}{\hbar}}W \big \}= 0 , \qquad ( \Delta^{a}- \mathrm{\Large \frac{i}{\hbar}}V^{a}+ \ldots ) \exp \big \{\mathrm{\Large \frac{i}{\hbar}}X \big \}= 0 .
\frac{1}{V}< H_{A}^{\theta}> = \frac{1}{4}\int \frac{d^{2}k}{( 2 \pi )^{2}}[ G^{- 1}( k ) + k^{2}G ( k ) - \frac{4 \pi^{2}}{g^{2}}z k^{- 2}G^{- 2}( k ) ]
\begin{aligned}W(F) = \Phi(v_1,v_2,v_3) &= \phi(v_1) + \phi(v_2) + \phi(v_3) +g(v_2v_3) + g(v_1 v_3) + g (v_1 v_2) + h(v_1 v_2 v_3) \, ,\end{aligned}
\tilde{L}^\alpha_T \tilde{L}^\beta_v (B^s_{pr}):= \left\{f(\cdot,v,t) \in \mathcal{S}' \ \vert\ \Vert f \Vert_{\tilde{L}^\alpha_T \tilde{L}^\beta_v (B^s_{pr})}< \infty \right\},
\begin{aligned}\Phi_\sqcup^\alpha:a\sqcup (b\sqcup c) \rightarrow (a\sqcup b)\sqcup c \Phi_\sqcup^\alpha((0,x)) = (0,(0,x))\;, \Phi_\sqcup^\alpha(1,(0,y)) = (0,(1,y))\;, \Phi_\sqcup^\alpha(1,(1,z)) = (1,z)\;,\end{aligned}
D^+_kf(x_0) = \sum_{j=1}^n D^+_k a_j(x_0)\prod_{i\neq j}a_i(x_0) + \sum_{j=2}^N h^{j-1}\sum_{|I|=j}\prod_{i \in I}D_k^+ a_i(x_0) \prod_{i\not\in I}a_i(x_0) & \geq \phi(x_0) \sum_{j=1}^n \prod_{i\neq j}a_i(x_0).
q(G)(z) = k^{-n}\sum_{\phi:V\to[k]}\prod_{e=\{u,v\}\in E}(J+(z(A^{e}-J)))_{\phi(u),\phi(v)}&=k^{-n}\sum_{i=0}^{|E|}z^i\bigg( \sum_{\substack{F\subseteq E |F|=i}}\sum_{\phi:V\to[k]}\prod_{e=\{u,v\}\in F}(A^{e}-J)_{\phi(u),\phi(v)}\bigg).
\begin{aligned}\textbf{I}_3 =\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n}\left(-r!\frac1{n^{r+1}}\right)\sin\left(\frac{r\pi}{2}\right) &=-r!\sin\left(\frac{r\pi}{2}\right)\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^{r+2}}&=-r!\sin\left(\frac{r\pi}{2}\right)\left(1-\frac1{2^{r+1}}\right)\zeta(r+2).\end{aligned}
\lim_{{\vec k}\rightarrow 0}k^{\mu}{\cal F}_{\mu}(k)~\equiv~\lim_{{\vec k}\rightarrow 0}{\bar \epsilon}\int d^4 x \sqrt{-g}~k^{\mu}p_k(x) <0_-|{\cal T}{\cal S}_{\mu}(x){\cal O}_(y) |0_->~,
\begin{array}{llll}(i) \Sigma_{n}= \{t_{i_{1}}^{k_{1}}\ldots t_{i_{r}}^{k_{r}}\cdot \sigma \},\{\rm where}\ 0\le i_{1}<\ldots <i_{r}\le n-1, (ii) \Sigma^{\prime}_{n}= \{{t^{\prime}_{i_1}}^{k_{1}}\ldots{t^{\prime}_{i_r}}^{k_{r}}\cdot \sigma \},\{\rm where}\ 0\le i_{1}< \ldots <i_{r}\le n-1, \end{array}
\Delta = \frac{1}{2}\Delta_{h}^{\prime \prime}( r - r_{h})^{2}+ \frac{1}{6}\Delta_{h}^{\prime \prime \prime}( r - r_{h})^{3}+ \frac{1}{2 4}\Delta_{h}^{( i v )}( r - r_{h})^{4}+ \frac{1}{1 2 0}\Delta_{h}^{( v )}( r - r_{h})^{5}+ O ( r - r_{h})^{6}.
\left\|\frac{1}{m+1}\sum_{i=0}^mw_{n+i}-w\right\| <C\left[\left\|\frac{1}{m+1}\sum_{i=0}^mx_{n+i}-x\right\|+\left\|\frac{1}{m+1}\sum_{i=0}^my_{n+i}-y\right\|\right. &+\left.\left\|\frac{1}{m+1}\sum_{i=0}^mz_{n+i}-z\right\|\right].
0 =\partial_{x_1}\tilde u_{k_m}(tz_{l_m,k_m}+(1-t)z_{k_m(m)})|_{t=1}-\partial_{x_1}\tilde u_{k_m}(tz_{l_m,k_m}+(1-t)z_{k_m(m)})|_{t=0}&=\int_0^1 \partial_{e^m x_1}\tilde u_{k_m}(sz_{l_m,k_m}+(1-s)z_{k_m(m)}) |z_{l_m,k_m}-z_{k_m(m)}| ds.
V = G_{N}{\frac{m_{1}m_{2}}{r}}+ \int_{0}^{\infty}{\frac{d m}{k}}G_{N}{\frac{m}{k}}{\frac{m_{1}m_{2}e^{- m r}}{r}}= G_{N}{\frac{m_{1}m_{2}}{r}}\left( 1 +{\frac{1}{k^{2}r^{2}}}\right) .
N_{\ell}\; : = \int d \underline{{{x}}}\; \widehat{{\mathcal J}}_{\ell}^{0}\left( t , \underline{{{x}}}\right) ~ , ~ ~ N_{r}\; : = \int d \underline{{{x}}}\; \widehat{{\mathcal J}}_{r}^{0}\left( t , \underline{{{x}}}\right) ~ ,
\begin{aligned}\Big\| \Lambda^\alpha(uv)- \sum_{|k|< \alpha_1}\frac{1}{k!}\partial^k u \,\Lambda^{\alpha,k}v - \sum_{|m| \le \alpha_2}\frac{1}{m!}\partial^{m}v\, \Lambda^{\alpha,m}u \Big\|_{L^p}\lesssim_{\alpha,\alpha_1,\alpha_2,p,N}\| \Lambda^{\alpha_1}u \|_{L^p}\| \Lambda^{\alpha_2}v \|_{\operatorname{BMO}},\end{aligned}
I(q)=\Psi_{s}(q)\times\Psi_{s}^{*}(q)=\sum_{j=1}^{N}f_{j}e^{-iq\cdot R_{j}}\times\sum_{k=1}^{N}f_{k}e^{iq\cdot R_{k}}=\sum_{j=1}^{N}\sum_{k=1}^{N}f_{j}f_{k}e^{-iq\cdot(R_{j}-R_{k})}
K_{D}({\bf r^{\prime \prime}},{\bf r^{\prime}}; t^{'^{\prime}}, t^{'}) = \left\langle{\bf r^{\prime \prime}}\left| \hat{T}\exp \left[ - \frac{i}{\hbar}\int_{t^{\prime}}^{t^{\prime \prime}}\hat{H}d t \right] \right|{\bf r^{\prime}}\right\rangle ,
\frac{t^{b}}{\prod_i(1-t^{a_i})}=-\frac{t^{b-a_{i_0}}}{\prod_{i\neq i_0}(1-t^{a_i})}+ \frac{t^{b-a_{i_0}}}{\prod_{i}(1-t^{a_i})}=\sum_{\substack{x_i\geq 1 \sum_i x_ia_i\leq b}}p_{(x_i)}\cdot t^{b-\sum_i x_ia_i}+ \substack{\textnormal{negative degree}\textnormal{rational function}},
\lim_{\lambda \to \infty}\frac{p_{\lambda p(p+1)+\rho+1,k}(z)}{p_{\lambda p(p+1)+\rho,k}(z)}= \left(w_k^{(\rho)}\right)^{1/2}\widetilde{F}_{k}^{(\rho)}(z),z\in\mathbb{C}\setminus[a_{k},b_{k}], k=0,\ldots,p-1.
R^\nabla_{X,Y}Z=\left(R^W_{X,Y}\right)^{J,+}Z -\frac{1}{2}(dJ\theta)(X,Y)JZ-\frac{1}{2}(d\theta)(X,Y)Z +\frac{1}{4}\left(\left(D^W_XJ\right)\left(D^W_YJ\right) -\left(D^W_YJ\right)\left(D^W_XJ\right) \right)Z,
J_R(z)=\begin{cases}P^{(0)}(z)P^{(\infty)}(z)^{-1}, \mbox{for $z\in\partial U^{(0)}$}, P^{(b)}(z)P^{(\infty)}(z)^{-1}, \mbox{for $z\in\partial U^{(b)}$}, P^{(\infty)}(z)J_S(z)P^{(\infty)}(z)^{-1}, \mbox{for $z\in\Sigma_R\setminus\left(\partial U^{(0)}\cup \partial U^{(b)}\right)$,}\end{cases}
I^{\pm}(z)=\sum_{i_1+j_1=q_1}\frac{1}{i_1!j_1!k_1!}\int_{0<t_1<z}L_1(t_1,z)^{i_1}L_{-1}(t_1,z)^{j_1}L_0(t_1,z)^{k_1}\left(\frac{dt_1}{1-t_1}+\frac{-dt_1}{1+t_1}\right).
C_1 \Big(\displaystyle{\sum_{i=1}^n e_{i-n}(U_i)}+ \sum_{j=1}^p e_0(U_{n+j}) \Big) \le \Big(\displaystyle{\sum_{i=1}^n e_{1+i-n}(W_i)}+ \sum_{j=1}^p e_1(W_{n+j}) \Big)\le C_2 \Big(\displaystyle{\sum_{i=1}^n e_{i-n}(U_i)}+ \sum_{j=1}^p e_0(U_{n+j}) \Big)
\sum_{(A, B, C) \,\in\, \mathcal P_t}\frac1{|A|^2}\: \:\ll \:\: \sum_{A=1}^{\left\lfloor 2^{(1/2+\epsilon_2)t}\right\rfloor}\frac1{A}\: \: \:\ll \: \left(\frac 1 2 + \epsilon_2\right)\cdot t \:\: \ll \:\: t.
M_k(x; \boldsymbol{a})=\frac{1}{\phi^k(q)}\sum_{\substack{p_1, \cdots, p_k ~ p_1\cdots p_k\leq x}}(\chi_0(p_1)+\chi_a(p_1))\cdots (\chi_0(p_k)+\chi_a(p_k)).
y^{(m)}_{r_1,\ldots,r_k}=\Bigl(\prod_{i=1}^k\mu(r_i)g(r_i)\Bigr)\sum_{\substack{a_1,\ldots,a_k r_i\mid a_i\forall i}}\frac{y_{a_1,\ldots,a_k}}{\prod_{i=1}^k\varphi(a_i)}\prod_{i\neq m}\frac{\mu(a_i)r_i}{\varphi(a_i)}.
S = S' + \int d\tau \left( \sum_a q_s^* \delta_a q^s \eta_a - \sum_{ab}\alpha^*_b \left(\zeta^{-1}\right)_{ab}\eta_a + \sum_{abc}\frac{1}{2}C_{ab}{}^c \eta^*_c \eta_a\eta_b + \sum_a \bar{\eta}^*_a b_a \right) .
e^{- \mu}( \partial_{1}+ i A_{1}+ i \partial_{2}- A_{2}+{\frac{1}{2}}\partial_{1}\mu +{\frac{i}{2}}\partial_{2}\mu ) \psi_{1}+ e^{- \nu}( - \partial_{3}- i A_{3}+ i \partial_{4}- A_{4}-{\frac{1}{2}}\partial_{3}\nu +{\frac{i}{2}}\partial_{4}\nu ) \psi_{2}= 0 .
\left({{z_{1}}, \{z_{2}}}\right) = \zeta \left({{\frac{\xi}{\sqrt{1 - \left| \xi \right|^{2}}}}, \ 1}\right) , \quad \left({d z^{*}d z}\right)^{2}= | \zeta |^{2}d \zeta^{*}d \zeta{\frac{d \xi^{*}d \xi}{\left({1 - \left| \xi \right|^{2}}\right)^{2}}}\quad ,
g^* f_* \mathcal M = \varprojlim_k g^* f_* \tau_{\le k}\mathcal M = \varprojlim_k f'_* g'^* \tau_{\le k}\mathcal M = f'_* \varprojlim_k g'^* \tau_{\le k}\mathcal M = f'_* g'^* \mathcal M,
\langle \Phi_{H.W.}^{(\lambda_{1}, \mu_{1})}(z_{1},\bar{z_{1}})\Phi_{H.W.}^{(\lambda_{2}, \mu_{2})}(z_{2},\bar{z_{2}})\Phi_{T_{z}^{3}, Y^{3}}^{(\lambda_{3}, \mu_{3})}(z_{3},\bar{z_{3}})\tilde\Phi_{L.W.}^{(\lambda_{4}, \mu_{4})}(z_{4}, \bar{z}_{4})\rangle =
[\begin{matrix}N_{11}\\N_{22}\\N_{12}\end{matrix}]=\{\int_{-h}^{h}[\begin{matrix}C_{11}&C_{12}&0\\C_{12}&C_{22}&0\\ 0&0&C_{66}\end{matrix}]dx_{3}\}[\begin{matrix}u_{1,1}^{0}\\u_{2,2}^{0}\\ \frac{1}{2}(u_{1,2}^{0}+u_{2,1}^{0})\end{matrix}]
\Vert \L[\nabla\phi, \L^{-\alpha+\delta}]\L^\alpha \psi \Vert_{L^2}=\Vert [\nabla\phi, \L^{-\alpha+\delta}]\L^\alpha \psi \Vert_{H^1}\le C\Vert \nabla\phi\Vert_{W^{1, \infty}}\Vert \psi\Vert_{D(\L^\alpha)}
\mathbf{\dot{H}}_k = \sqrt{\frac{N_{BS}N_{MS}}{N_cN_p}}\sum_{i = 1}^{N_c}\sum_{l=1}^{N_p}\alpha_{il}^k \mathbf{a}_{MS}^k(\theta_{il}^k) \mathbf{a}_{BS}^k(\phi_{il}^k)^H,
1^{2}= \frac{m}{V}v = \frac{m}{p}= \frac{6 7 5 \times 1 0^{- 3}m}{0 . 9 \times 1 0^{- 3}m^{3}g / m^{3}}= 0 . 7 5 \times 1 0^{- 3}m^{3}
{\frac{\kappa}{4 \pi}}\partial_{z}\Bigl ( \{\xi_{\alpha A}( x ) , \xi_{\beta L}( z ) \}\; L_{\beta \rho}( z ) \Bigr ) = L_{\rho \gamma}^{- 1}( z ) \; P_{( \alpha A ) ( \gamma L )}( x , z ) \quad ,
\begin{array}{ll}\displaystyle C_{n-1}^{k-1}(\frac{U_r}{r})^{k-1}U_{rr}+C_{n-1}^k(\frac{U_r}{r})^k begin{align*}3mm] =\displaystyle\left[\frac{2\theta}{(1+r^2)^{\theta+1}}\right]^k\left[\frac{C_{n-1}^{k-1}+C_{n-1}^k+[C_{n-1}^k-C_{n-1}^{k-1}(2\theta+1)]r^2}{1+r^2}\right].\end{array}
\det(w_i'(I_d-P_{B,N_{(i-1)}})w_i) \quad= \frac{\det([B,N_{(i)}]'[B,N_{(i)}])}{\det([B,N_{(i-1)}]'[B,N_{(i-1)}])}\det(w_i'(I_d-P_{N_{(i-1)}})w_i) \quad= \frac{\det(N_{(i)}'N_{(i)})}{\det(N_{(i-1)}'N_{(i-1)})},
P_{N}^{( N_{f})}( W , M ) \ = \ \frac{1}{\widetilde{Z}_{N}^{( N_{f})}( M )}\prod_{f = 1}^{N_{f}}\det ( W W^{\dagger}+ M_{f}^{2}) \exp ( - N \Sigma^{2}{\mathrm{t r}}W W^{\dagger})
T^{+q}_{+p\pm2}=\frac12\gamma^i_{q\dot q}\Omega^{+2i}_{+p}\psi^{1-}_{\pm2\dot q},\quad T^{+q}_{-p\pm2}=\frac12\gamma^i_{q\dot q}\Omega^{+2i}_{-p}\psi^{1-}_{\pm2\dot q};
l_n^\alpha\big(x_1 \otimes \dots \otimes x_n\big)=\sum_{i=0}^{\infty}\frac{1}{i !}l_{n+i}\big(\alpha^{\otimes i}\otimes x_1 \otimes \dots \otimes x_n\big), \forall x_1, \ldots, x_n \in \mathfrak{L},
R_{\kappa \lambda \mu \nu}= R_{\kappa \mu}g_{\lambda \nu}- R_{\lambda \mu}g_{\kappa \nu}+ R_{\lambda \nu}g_{\kappa \mu}- R_{\kappa \nu}g_{\lambda \mu}- \frac{R}{2}\left( g_{\kappa \mu}g_{\lambda \nu}- g_{\kappa \nu}g_{\lambda \mu}\right) .
\begin{aligned}p_k'(\xi) =2\sigma |\xi+k|^{2\sigma-2}(\xi+k)-2\sigma|k|^{2\sigma-2}k \Rightarrow p_k'(0)=0, p_k''(\xi) =2\sigma(2\sigma-1)|\xi+k|^{2\sigma-2}\Rightarrow p_k''(\xi)\geq0.\end{aligned}
n ( x^{\prime}) = \delta^{( 2 )}( x^{\prime}- x ) - \delta^{( 2 )}( x^{\prime}- y ) \makebox [ . 5 i n ]{,}j_{\mu}( x^{\prime}) = \int_{\gamma}d x_{\mu}^{\prime \prime}\delta^{( 2 )}( x^{\prime}- x^{\prime \prime})
\begin{aligned}y\left[ n \right] = \underbrace{{\bf{h}}_1^H{\bf{f}}s\left[{n -{n_1}}\right]}_{{\rm{desired}}\;{\rm{signal}}}+ \underbrace{\sum\nolimits_{l \ne 1}^L{{\bf{h}}_l^H{\bf{f}}s\left[{n -{n_l}}\right]}}_{{\rm{ISI}}}+ z\left[ n \right], \end{aligned}
\int_{\mathbb{R}^{2N}}\frac{|\varphi_{h,t}(x) - \varphi_{h,t}(y)|^{p-2}\, (\varphi_{h,t}(x) - \varphi_{h,t}(y))\, (\varphi_h(x)-\varphi_h(y))}{|x - y|^{N+s\,p}}\, dx\, dy-\int_\Omega\,f(x)\cdot \Phi_{k}'(\varphi_{h,t})\varphi_h &\geq -\int_\Omega f(x)\cdot|\Phi_{k}'(\varphi_{h,t})-\Phi_{k}'(w)||\varphi_{h}|\,dx\,.
f ({\vec{\theta}}^{1},{\vec{\theta}}^{2}) = \theta_{a}^{1}\theta_{b}^{1}\theta_{c}^{1}\theta_{d}^{2}\theta_{e}^{2}\theta_{f}^{2}+ C_{a}\theta_{a}^{1}\theta_{b}^{2}\theta_{c}^{2}\theta_{d}^{2}+ D_{a}\theta_{a}^{2}\theta_{b}^{1}\theta_{c}^{1}\theta_{d}^{1}\ .
\sum_{1\leq j,k \leq n}f_j(\tau_k)^mf_k(\tau_j)^m =\sum_{j=1}^n\sum_{k=1}^nf_j(\tau_k)^mf_k(\tau_j)^m \geq \frac{1}{{d+m-1 \choose m}}\left(\sum_{j=1}^n f_j(\tau_j)^m\right)^2 =\frac{1}{(^m(\mathcal{X}))}\left(\sum_{j=1}^n f_j(\tau_j)^m\right)^2.
\Delta L(t)=-\frac{c_{1}e^{-a_{1}t}+c_{2}e^{-a_{2}t}}{3k_{v}\rho^{\frac{1}{3}}(c_{4}+c_{1}e^{-a_{1}t}+c_{2}e^{-a_{2}t})^{\frac{2}{3}}}
\begin{multlined}x^{k+3}\cdot x^{k+1}\cdot 17 x^4+x^{k+3}\cdot (2^k+k)x^k \cdot x^5+(2^{k+2}+k+2)x^{k+2}\cdot x^{k+1}\cdot x^5 = (2 k + 2^k + 2^{k + 2}+ 19) x^{2 k + 8},\end{multlined}
= tr_{12}(K_1^{t_1}K_2\tilde B_{12}^{t_2}B_{12}^{t_1}M_1^{t_1}M_2) =tr_{12}[{(K_1^{t_1}\tilde B_{12}K_2^{t_2})}^{t_2}{(M_1B_{12}M_2)}^{t_1}]=
\Vert v_{i}(C^{*}\pi_{W_{i}}C^{\prime})^{\frac{1}{2}}f \Vert^{2}=\sum_{i \in I}v_{i}^{2}\langle \pi_{W_{i}}C^{\prime}f, \pi_{W_{i}}C f \rangle\geq A\Vert f \Vert^{2}.
C_{n-1}(0) =\mu_0(n-1)(\alpha_{n-1}-1)+C_n(0)+\log(1+2^{\mu_0(n-1)+\Delta{C}_n})-H(\alpha_{n-1}), ~C_{n-1}(1) =\mu_1(n-1)(\beta_{n-1}-1)+{C}_n(0)+\log(1+2^{\mu_1(n-1)+\Delta{C}_n})-H(\beta_{n-1}).
|\Omega| =\int_\Omega u^{\frac{p}{p+1}}(t,x;s,u_0) u^{-\frac{p}{p+1}}(t,x;s,u_0)dx &\le \big(\int_\Omega u(t,x;s,u_0)dx\big)^{\frac{p}{p+1}}\big(\int_\Omega u^{-p}(t,x;s,u_0)dx \big)^{\frac{1}{p+1}}\quad\forall\, t> s.
1 - \bar{S}_{F}\ast{\cal V}= \left[ 1 - S_{F}^{c o v}\left( \gamma^{-}{A}_{-}+ \gamma^{j}{A}_{j}\right) \right] \ast \left[ 1 + \frac{e}{2}\gamma^{+}\frac{1}{i \partial_{-}}\ast{\cal W}_{- 1}[ \widehat{a}] \left( \gamma^{-}{A}_{-}+ \gamma^{k}{A}_{k}\right) \right] ,
\sum_{i_{1}+i_{2}+\cdots+i_{r}=N}\frac{a_{1}^{i_{1}}a_{2}^{i_{2}}\cdots a_{r}^{i_{r}}q^{\binom{i_1}{2}+\binom{i_2}{2}+\cdots+\binom{i_r}{2}}}{(q;q)_{i_{1}}(q;q)_{i_{2}}\cdots(q;q)_{i_{r}}}.
\nu_{LEQ}(\overrightarrow{\eta}) = \nu_{EQ}(\overrightarrow{\eta})\left(1+\varepsilon \sum_{i=1}^N \frac{\theta_i^{(1)}}{\theta_i^{(0)}}\eta_i \right) + O(\varepsilon^2) = \nu_{EQ}(\overrightarrow{\eta})\left(1+\varepsilon \sum_{i=1}^N \frac{m \rho_i^{(1)}}{\rho_i^{(0)}(m+\rho_i^{(0)})}\eta_i \right) + O(\varepsilon^2),
\prod_{j=0}^{k-1}\left(\frac{kp-j+i}{kp+j+1+i}\right)^{\frac{1}{2}}+ \prod_{j=0}^{k-1}\frac{1}{\sqrt{kp+j+1+i}}- 1 < \left(\frac{kp+i}{kp+1+i}\right)^{k/2}+ \left(\frac{1}{kp+1+i}\right)^{k/2}- 1 & < 1 - \frac{k}{2(kp+1+i)}+ \frac{1}{kp+1+i}- 1 < - \frac{k-2}{2(kp+1+i)}
\sum_{\mu: \ \mu_1=a+d+1}m_{\mathcal{M}(\alpha)}(\mu) \ - \sum_{\mu: \ \mu_1=a+d+1}m_{\mathcal{M}(\beta)}(\mu) = \sum_{\substack{(i,j)\in S_\alpha: i\geq 2j\geq 2}}2^{\max\{i-1,0\}+\max\{j-1,0\}}-\sum_{\substack{(i,j)\in S_\beta: i\geq 2j\geq 2}}2^{\max\{i-1,0\}+\max\{j-1,0\}},
{\cal S}={\frac{1}{2}}\int d x d t \left[ \frac{\dot{\phi}^{a}\dot{\phi}^{a}+ \phi^{a}\partial_{i}\partial_{i}\phi^{a}}{\phi^{a}\phi^{a}}- \frac{( \dot{\phi}^{a}\dot{\phi}^{a})^{2}}{( \phi^{a}\phi^{a})^{2}}\right] ,
<p_{1},...,p_{n}|S|q_{1},....,q_{m}>= lim_{T\rightarrow \infty}\frac{1}{T}\int_{0}^{T}d\tau E\Big[\prod_{j}\prod_{k}( \overline{f_{p_{j}}},\partial_{\tau}\phi_{\tau})( f_{q_{k}},\partial_{\tau}\phi_{\tau})\Big]
\sup_{0\le j\le N}\ \Vert P_{n}T^{\,j}P_{l}\,x\Vert \le \dfrac{1}{4}\,\beta_{l}\,\Bigl(\ \sup_{b_{l+1}-N\le k<b_{l+1}}\prod_{s=k+1}^{b_{l+1}-1}|w_s|\Bigr)\, \Vert P_{l}\,x\Vert
e_b(x,w) = \exp\left(x\sum_{i=0}^{\infty}w^{b^i}\right) \prod_{i=0}^{\infty}\left(\frac{\Gamma(b,xw^{b^i})}{(b-1)!}\right) \simeq \exp\left(x\sum_{i=0}^{\infty}w^{b^i}\right) \simeq e^{xw+xw^{b}}
\int \phi \mathcal D_2 =\frac 1 2 \int_{\Gamma^0}d\mathcal{H}^{n-1}(\dot \eta^0\circ \pi_1)^2 \, \frac{N\cdot\nu^0}{J}\partial_N\, \big(J \Delta h^0\, \phi \big) -\frac 1 2 \int_{\Gamma^0}d\mathcal{H}^{n-1}(N\cdot \nu^0)\,(\ddot\eta^0\circ\pi_1) \, \Delta h^0\phi.
\lambda (I_{i,n})\leq \beta_{n}^{-1}\,p_{i,n}\leq 4\,\beta_{n}^{-1}\,\sigma^{-2}\sigma_{n}^{2}(I_{i,n})=4\,\beta_{n}^{-1}\,\sigma^{-2}\sigma_{n}^{2}(C_{n})\,\mbox{{\rm Var}}(\delta_{i,n}).
\iint_Q{g_\epsilon \varphi}dx dt =\iint{(u^\epsilon)_+^\frac{n}{2}\partial_x I(u^\epsilon) \varphi}dx dt & = - \iint{\frac{n}{2}(u^\epsilon)_+^{\frac{n}{2}-1}\partial_x u^\epsilon I(u^\epsilon) \varphi}dx dt - \iint{(u^\epsilon)_+^\frac{n}{2}I(u^\epsilon) \partial_x \varphi}dx dt.
\begin{aligned}S_1 = \{n \in \mathbb Z \mid |p_n| > R |n| < \lfloor \alpha R \rfloor \}, S_2 = \{n \in \mathbb N \mid n \geq \lfloor \alpha R \rfloor p_n \geq -p_{-n}\}, S_3 = \{n \in \mathbb N \mid n \geq \lfloor \alpha R \rfloor p_n < -p_{-n}\}. \end{aligned}
(e_A \wedge e_B)g_v = e'_{(e_A \wedge e_B)f}&= e'_{(A \wedge B)f}&= e'_{Af \wedge' Bf}&= Af \wedge' Bf &= e_{Af}\wedge' e_{Bf}&= (e_A )g_v \wedge' (e_B )g_v .
{{\mathop{\rm P}\nolimits}_{out}}\left({{\gamma_{th}}}\right) ={I_R}+{I_L}- \int\limits_{{\Phi_R}\cap{\Phi_R}}{{f_{\hat{\textbf{p}}_u}}\left( \hat{\textbf{p}}_u \right)d}\hat{\textbf{p}}_u
J_\mu =i\overline{\Psi}(x)\Gamma_\mu \Psi (x) ,\hspace{0.3in}K_\mu = \overline{\Psi}(x)\Gamma_\mu\overline{\Gamma}_5\Psi (x) ,\hspace{0.3in}R_{\mu \alpha}=\overline{\Psi}(x)\Gamma_\mu \overline{\Gamma}_5\overline{\Gamma}_\alpha \Psi (x) ,
\frac{\lambda}{1 2}\sum_{\{x_{i}\in G \}}\sum_{k_{i}= 1}^{\mu}\; \phi_{x_{1}x_{2}x_{3}}^{k_{1}k_{2}k_{3}}\phi_{x_{1}x_{4}x_{5}}^{k_{1}k_{4}k_{5}}\phi_{x_{2}x_{5}x_{6}}^{k_{2}k_{5}k_{6}}\phi_{x_{3}x_{6}x_{4}}^{k_{3}k_{6}k_{4}}=
K_-(z) w_j = \theta(z+(j+1)\hbar) w_j, E(z) w_j = -\frac{\theta(z-x+(j-1)\hbar)\theta(j\hbar)}{\theta(z+j\hbar)\theta(x+\hbar)}w_{j-1}, F(z) w_j = \frac{\theta(z+x+(\ell-j)\hbar)\theta((\ell-j)\hbar)}{\theta(z+(j+1)\hbar)\theta(x+(\ell-2j-1)\hbar)}w_{j+1}.
\omega^{\Lambda}= ( \mathrm{d}\varphi_{I}+ \vec{W}_{I J}\cdot \mathrm{d}\vec{x}^{J}) \wedge \mathrm{d}x^{\Lambda I}+ U_{I J}\varepsilon^{\Lambda \Sigma \Pi}\mathrm{d}x^{\Sigma I}\wedge \mathrm{d}x^{\Pi J}\ .
[P_{2a+b}, P_{a+b+c}, P_{a+2b}, P_{a+b+d}][P_{a+c+d}, P_{2c+d}, P_{b+c+d}, P_{c+2d}] &= \frac{(1-[P_{2a+c}, P_{a+2c}, P_{a+b+c}, P_{a+c+d}])(1-[P_{a+b+d}, P_{b+c+d}, P_{2b+d}, P_{b+2d}])}{(1-[P_{2a+d}, P_{a+c+d}, P_{a+b+d}, P_{a+2d}]^{-1})(1-[P_{a+b+c}, P_{b+2c}, P_{2b+c}, P_{b+c+d}]^{-1})}
S_{\psi}^{( 2 )}= \sum_{n \leq p + 1}\int_{n \mathrm{\tiny - b r a n e}~ \subseteq ~ \mathrm{\tiny ~ D - b r a n e}}d y^{m_{1}}\cdots d y^{m_{n}}~ A_{m_{1}, \cdots , m_{n}}( y_{1}, \cdots , y_{p}) ,
f ( x , y ) = \left( \begin{array}{c c}{{- 2 \partial_{x^{-}}}}&{{\alpha \partial_{x^{-}}^{2}- 2 \partial_{x^{-}}^{2}\phi - 2 \partial_{x^{-}}\phi \partial_{x^{-}}}}\\{{- \alpha \partial_{x^{-}}^{2}- 2 \partial_{x^{-}}\phi \partial_{x^{-}}}}&{{0}}\\ \end{array}\right) \delta ( x - y )
b\left(U,T_{1},T_{2}\right)=\begin{cases}\mathcal A_2b_{2}\left(U,T_{1},T_{2}\right), U<N_{1}-N_{2}\mathcal A_1 b_{1}\left(U,T_{1},T_{2}\right)+ \mathcal A_2 b_{2}\left(U,T_{1},T_{2}\right), U=N_{1}-N_{2}\mathcal A_1 b_{1}\left(U,T_{1},T_{2}\right), U>N_{1}-N_{2}\end{cases}.
\|\theta(t)\|_{B^0_{\ell,\ell}}\lesssim \|\theta(t)\|_\infty^{1-\frac{2}{\ell}}\|\theta(t)\|_2^{\frac{2}{\ell}}&\lesssim R_{\infty}^{1-\frac{2}{\ell}}R_{2}^{\frac{2}{\ell}},
\frac{\partial}{\partial t}\left(I^{\alpha}\sum_{k=1}^{\infty}\int_{0}^{\cdot}h^{k}(s)dw_{s}^{k}\right)(t) =\frac{1}{\Gamma(\alpha)}\sum_{k=1}^{\infty}\int_{0}^{t}(t-s)^{\alpha-1}h^{k}(s)dw_{s}^{k}
\delta ( \tilde{g}( A_{\nu}^{\omega}) ) \equiv \frac{\operatorname*{l i m}_{\alpha \rightarrow 0}e^{- \frac{1}{2 \alpha}\int g ( A_{\nu}^{\omega}) d x}}{\operatorname*{l i m}_{\alpha \rightarrow 0}\int{\cal D}\Lambda e^{- \frac{1}{2 \alpha}\int d x \Lambda^{2}}}
(P_N\partial_t \widehat{u}(t),v) =(P_Nv_0,v)+\int_0^t (P_N\widehat{u}(s), \Delta v)ds &\quad+\int_0^t (f(\widehat{u}(s)),v)ds+\int_0^t (\widehat{\xi}(s), v)ds.
\int_{\big\{x \,:\, x^{\eta - 1}\| f \|_{L^1 (x^{-\eta}dx)}> \lambda \big\}}x^{-\eta}\, dx=\int_{\big\{x \,:\, x< \big(\lambda^{-1}\| f \|_{L^1 (x^{-\eta}dx)}\big)^{1/(1-\eta)}\big\}}x^{-\eta}\, dx\simeq\frac{\| f \|_{L^1 (x^{-\eta}dx)}}{\lambda},
M\models \forall \; \overline{z}\; \bigvee_{i=1}^{K}\exists \; \overline{b}\; \exists \; c \; (\theta^{\prime}_{i}(c, \overline{b}) \; \& \; \xi_{0}(\overline{z}, \theta_{i}(\cdot, \overline{b}), c, \rho(\cdot, \overline{a})))
\rho\left(B_{m}^{L}(2)\right) -\rho\left(B_{m}^{P}\right) \geqslant x^{T}\left(\mathcal{A}(B_{m}^{L}(2))x\right)-x^{T}\left( \mathcal{A}(B_{m}^{P})x\right) & =x_{v}x_{a}x_{b}(x_{a_{1}})^{k-3}+x_{v}x_{a_{1}}(x_{b_{1}})^{k-2}-x_{v}(x_{a_{1}})^{k-2}x_{a}-x_{v}(x_{b_{1}})^{k-2}x_{b}& =x_{v}(x_{a_{1}})^{k-3}\left(x_a-x_{a_1}\right)^{2}& >0.
\frac{\partial F_j}{\partial \nu_i}(e_d/2) =\int_{\partial B_1^+}\!\!\!x_i x_d x_{j-1}\phi(x_d)\,dx=\delta_{i(j-1)}\int_{\partial B_1^+}\!\!\!x_i^2 x_d \phi(x_d)\,dx=\frac{\delta_{i(j-1)}}{d-1}\int_{\partial B_1^+}\!\!\!(1-x_d^2) x_d\phi(x_d)\,dx\,,
V = - \frac{\Gamma ( - 1 / 2 )}{\sqrt{8 \pi^{2}\alpha^{\prime}}}[ ( \frac{r^{2}}{2 \pi \alpha^{\prime}}- \pi \epsilon )^{1 / 2}+ ( \frac{r^{2}}{2 \pi \alpha^{\prime}}+ \pi \epsilon )^{1 / 2}- 2 ( \frac{r^{2}}{2 \pi \alpha^{\prime}})^{1 / 2}]