harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{cases}f \pm 1, 0, f'( \pm 1) =:-p < 0,\f'(0)=: \mu > 0, f(u)\leq C(1+|u|^q)C,q>0, f'(u)\leq cc>0, f f(u)\leq -p(u-1) \ (u\geq 1). \end{cases}
|B_\ell| \geq (1- \gamma)^\ell |B|-\ell &\geq (1-\gamma)^{-\ln(1-\beta')/\gamma}|B|-\ell &\geq e^{(\gamma+\gamma^2) \ln(1-\beta')/\gamma}|B|-\ell &=(1-\beta')^{1+\gamma}|B|-\ell,
\begin{aligned}\Re \delta W = \int_\Omega B_0 (v_0-v_1)\cdot(v_0-v_1) + \int_D (B_1-B_0)v_1\cdot v_1 &=- \int_\Omega B_1 (v_0-v_1)\cdot(v_0-v_1) + \int_D (B_1-B_0)v_0\cdot v_0, \end{aligned}
h ={\mathrm{e}}^{a_{0}L_{0}}{\mathrm{e}}^{a_{1}L_{1}}{\mathrm{e}}^{b_{- 1}J_{- 1}}{\mathrm{e}}^{b_{0}J_{0}}{\mathrm{e}}^{b_{1}J_{1}}{\mathrm{e}}^{b_{2}J_{2}}\tilde{h},
\|\, \mbox{\boldmath $u$}(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R}^{3})}}\;\!=\;\Bigl\{\,\sum_{i\,=\,1}^{3}\int_{\mathbb{R}^{3}}\!|\:u_{i}(x,t)\,|^{q}\,dx\,\Bigr\}^{\!\!\:\!1/q}
v_{4 , k}( - t ; 0 , 0 , 0 , 0 ; 0 ) = z_{\lambda}^{( k )}- \int_{0}^{t}d t^{\prime}\, \frac{1}{2}\int_{q}\frac{\Delta ( q )}{q^{2}}\, v_{6 , k - 1}( - t^{\prime}; q , - q , 0 , 0 , 0 , 0 ; 0 )
\left[{\hbar}^{2}\frac{{\partial}^{2}}{{\partial}x_{1}^{2}}\pm i \frac{e{\hbar}}{c}{\cal E}^{e x t}+ \left( \frac{e A_{0}^{e x t}- E}{c}\right)^{2}\right] u_{1 ( 2 )}= m^{2}c^{2}u_{1 ( 2 )},
u_*(\zeta)=u(z(\zeta))\,\sqrt{\frac{\partial z}{\partial \zeta}\,\frac{\partial\bar z}{\partial\bar\zeta}}= u(z(\zeta))\, \left|\frac{\partial z}{\partial \zeta}\right|, \psi_*(\zeta)=\psi(z(\zeta))\,\sqrt{\frac{\partial z}{\partial \zeta}},\ \ \psi^+_*(\zeta)=\psi^+(z(\zeta))\,\sqrt{\frac{\partial z}{\partial \zeta}},
\{f , g \}={\frac{\partial f}{\partial \Pi}}\cdot{\frac{\partial g}{\partial \nabla^{2}X}}+{\frac{\partial f}{\partial P}}\cdot{\frac{\partial g}{\partial \dot{X}}}+{\frac{\partial f}{\partial p}}\cdot{\frac{\partial g}{\partial X}}- ( f \leftrightarrow g ) .
V ( x ) = \sum_{n}\sum_{j_{1}, . . . , j_{n}}P_{j_{1}. . . j_{n}}( \partial^{x_{1}}, . . . , \partial^{x_{n}}) : A_{j_{1}}( x_{1}) . . . A_{j_{n}}( x_{n}) : |_{x_{1}= . . . = x_{n}= x},
R\left(GG\right)\coloneqq R\left(GG,\alpha,s,N\right) \coloneqq\frac{2}{N}\sum_{d\in \mathcal{D}_{N}\left(GG \right)}\mathrm{r}\left(d\right)I_{s,N}\left(d\alpha\right), I_{s,N}\left(x\right) \coloneqq\begin{cases}1 \left\Vert x\right\Vert \leq s/N 0 \mathrm{otherwise,}\end{cases}
{\bf H}_u =g(H,e_{\beta})e_{\beta},{\bf S}_u =\frac{1}{2}\left(g(H,e_{\beta})^2-|A_{\beta}|^2\right)H\circ\pi_P-g(H,e_{\beta})\sum_{\alpha}{\rm tr}(A_{\alpha}A_{\beta})e_{\alpha}+\sum_{\alpha}{\rm tr}(A_{\alpha}A_{\beta}^2)e_{\alpha}.
\frac{\mu^{s+1}}{2}\left(\frac{2\pi}{L}\right)^{-s}\frac{1}{\Gamma\left(\frac{s}{2}\right)}\int_{0}^{\infty}dt\,t^{(\frac{s}{2}-1)}e^{-(\frac{mL}{2\pi})^2t}\Theta(0,\frac{t}{\pi}\rfloor_{s=-1}\,,
\mathbb{E}(y_p^j)^2 = \xi'(q_{p+1})-\xi'(q_p), \ \ 0\leq p \leq k, \j=1,2, &\mathbb{E}y_p^1 y_p^2 = \eta t\xi'((q_{p+1})-\xi'(q_p)), \ \ 0\leq p <\tau, &y_p^1, y_p^2 \ \textrm{are independent}, \ \ \tau\leq p \leq k.
\sum_{i=1}^{n-1}{^\alpha}\gamma_{ji}\frac{\partial F_{\beta\alpha}^i}{\partial z^n_\beta}\biggr|_{z_\beta(p)}=\begin{cases}{^\beta}\gamma_{nn}\frac{\partial F^j_{\beta\alpha}}{\partial z^n_\beta}\Bigr|_{z_\beta(\gamma p)}\;\,1 \leq j \leq n-1, 0 \;\,j = n.\end{cases}
0=(\sum_{i=1}^6a_iX_i)^2 = \sum_{i=1}^6a_i^2\underbrace{X_i^2}_{=Y}+2\sum_{i=2}^6a_1a_iX_1X_i+\sum_{2\leq i<j\leq 6}a_ia_j\underbrace{X_iX_j}_{=-Y-X_1(X_i+X_j)}&=(\sum_{i=1}^6a_i^2-2\sum_{2\leq i<j\leq 6}a_ia_j)Y+2\sum_{i=2}^6(a_1a_i-\sum_{\substack{j=2 j\not=i}}^6a_ia_j)X_1X_i,
[\begin{matrix}f_{x1}\\f_{y1}\\f_{x2}\\f_{y2}\\ \end{matrix}]=[\begin{matrix}k_{11}&k_{12}&k_{13}&k_{14}\\k_{21}&k_{22}&k_{23}&k_{24}\\k_{31}&k_{32}&k_{33}&k_{34}\\k_{41}&k_{42}&k_{43}&k_{44}\\ \end{matrix}][\begin{matrix}u_{x1}\\u_{y1}\\u_{x2}\\u_{y2}\\ \end{matrix}]
{\cal W}= \epsilon_{\alpha \beta \gamma}\Phi_{\alpha}\chi_{\beta}\chi_{\gamma}+ \chi_{1}P^{1}{\widehat R}^{1}+{\widetilde \chi}_{1}P^{1}R^{1}+{\widehat \chi}_{1}{\widehat P}^{1}R^{1}+ \dots .
\left\langle \rho, u\alpha_{1}, v\alpha_{3}\right\rangle \cong C_{p}^{3}\A=\begin{pmatrix}u_{2}v_{2}u_{3}v_{3}\end{pmatrix}\in \mathrm{GL}_{2}(\mathbb{F}_{p}) \ \v_{2}=u_{3}+\mathrm{det}(A).
\dot x_\text s(t) =Ax_\text s(t)+Bu(t)+YC'(y(t)-Cx_\text s(t)), \dot x_\text a(t) =Ax_\text a(t)+Bu(t),x_\text a(t_i)=Z_\gamma x_\text s(t_i) u(t) =-B'Xx_\text a(t).
u_{\varepsilon_{n}}\rightarrow u_{*}=e^{i\phi}\prod_{j=1}^{N}( \dfrac{z-b_{j}}{|z-b_{j}|})^{d_j}\,\,\,\,in\, C^{1,\alpha}_{loc}\left(\overline{G}\setminus \left\lbrace b_{1},...,b_{N}\right\rbrace\right)
R ( \theta ) = \frac{\operatorname{s i n}\left( \frac{2 \pi}{5}- \frac{3 i \theta}{5}\right) \operatorname{s i n}\left( \frac{\pi}{5}- \frac{3 i \theta}{5}\right)}{\operatorname{s i n}\left( \frac{2 \pi}{5}+ \frac{3 i \theta}{5}\right) \operatorname{s i n}\left( \frac{\pi}{5}+ \frac{3 i \theta}{5}\right)}.
B = \bigg\{x\in X : d\bigg(x,X\setminus \bigsqcup_{i\in I}S_i'' V_i\bigg) > \eta \bigg\}, A = \bigg\{x\in X : d\bigg(x,X\setminus \bigsqcup_{i\in I}S_i V_i\bigg) \leq \eta \bigg\}
\frac{\partial U_{i-1}(X_{i-1})}{\partial X_{i-1}}F_{i-1}(X_{i-1},\check{\xi}_{i},\tilde{\chi},\mu)\leq-\alpha_{i-1}(\|X_{i-1}\|)+|\check{\xi}_{i}|^{2}+\pi_{i-1}(\tilde{\chi})
\lambda^{(\omega)}_{(a,b,0)}\lambda^{(\omega)}_{(c,d,1)}= \exp\left(-\frac{\sqrt{-1}\theta}{2}( a d - b c)\right) \lambda^{(\omega)}_{(a+c,b+d,1)}, \lambda^{(\omega)}_{(c,d,1)}\lambda^{(\omega)}_{(a,b,0)}= \exp\left(-\frac{\sqrt{-1}\theta}{2}(ca+ad+bd)\right) \lambda^{(\omega)}_{(c-a-b,d+a,1)}.
V_{0}= \frac 1 2 < \{( K^{\tilde{M}\tilde{A}}\vec{L}^{\tilde{A}})^{2}+ |{\cal Z}^{\tilde{M}}K^{\tilde{M}\tilde{A}}|^{2}+ 2 |{\cal Z}^{\tilde{M}}K^{\tilde{M}\tilde{A}}\vec{L}^{\tilde{A}}|^{2}\}> .
_{n}(\lambda q^{j_{n}-1})_{n}(\lambda q^{-j_{n}})=-q\frac{\beta_{n}a_{n}c_{n}}{\alpha_{n}}(\frac{1}{\lambda}-q^{2(j_{n}-1)}\frac{\alpha_{n}}{\beta_{n}}\lambda )(\frac{1}{\lambda}+q^{-1}\frac{d_{n}\alpha_{n}}{c_{n}\beta_{n}}\lambda ).
{\cal A}_{D_{2}}( k_{1}, \epsilon_{1}; k_{2}, \epsilon_{2}) = g_{c}^{2}e^{- \lambda}\int_{0}^{1}d y \langle c ( z ) V_{\epsilon_{1}}( z , \bar{z}) c ( w ) \tilde{c}( \bar{w}) V_{\epsilon_{2}}( w , \bar{w}) \rangle |_{z = i y , w = i}~ ,
\ln \mathrm{d e t}_{3}\lower 7 p t \mathrm{~ \scriptscriptstyle ~ m^{2}~ \to ~ \infty ~}\mkern - 2 5 m u = \ \frac{1}{9 0 \pi m^{6}}\int d^{2}x \, B^{4}+{\mathcal{R}}( m^{2}) ,
\partial_t u = \Delta (u\gamma(v)) \;\;\;\; (0,\infty)\times \Omega\,, \partial_t v = \Delta v - v + u \;\;\;\; (0,\infty)\times \Omega\,, \nabla (u\gamma(v))\cdot \mathbf{n}= \nabla v\cdot \mathbf{n}= 0 \;\;\;\; (0,\infty)\times \partial\Omega\,, (u,v)(0) = (u^{in},v^{in}) \;\;\;\; \Omega\,,
\nu^{3}\sum_{i \ne j \ne k}\sum_{l \ne m}\bigg [ \epsilon_{l}( y_{l m}K_{l m}^{\mathrm{t o t}}) , \xi_{i}( y_{i j}K_{i j}^{\mathrm{t o t}}) ( y_{j k}K_{j k}^{\mathrm{t o t}}) \bigg ] .
U_P = \left( P \widehat{\mathcal{O}}_P \setminus P^2 \widehat{\mathcal{O}}_P \right) \times \left(P \widehat{\mathcal{O}}_P \right)^{d-1}\times \left( \widehat{\mathcal{O}}_P \setminus P \widehat{\mathcal{O}}_P \right).
\left\{w_{A}, w_{B}\right\}= \frac{1}{2}\gamma w_{A}w_{B}\left[ ( 4 - w_{A}- w_{B}) ( \delta_{A + 1 , B}- \delta_{A - 1 , B}) + w_{A - 1}\delta_{A - 2 , B}- w_{A + 1}\delta_{A + 2 , B}\right]
S_{2}={\frac{1}{2}}\int d p_{1}d p_{2}d p_{3}d p_{4}\big [{\frac{- g^{2}}{4 ( 2 \pi )^{d}}}\delta ( p_{1}- p_{2}+ p_{3}- p_{4}) + i \delta ( p_{1}- p_{2}) \delta ( p_{3}- p_{4}){\sigma^{0}}^{- 1}( p_{1}){\sigma^{0}}^{- 1}( p_{3}) \big ] \eta ( p_{4}, p_{1}) \eta ( p_{2}, p_{3})
\|\alpha_*u_*+\lambda x_v+(1-\lambda)x_w\|^2= =\alpha_*^2\|u_*\|^2+\|\lambda x_v+(1-\lambda)x_w\|^2+2\alpha_*\langle u_*,\lambda x_v+(1-\lambda)x_w\rangle= =\lambda \|x_v\|^2 + (1-\lambda)\|x_w\|^2
\mathcal{S}_{\beta ,\varepsilon,k}(\zeta) :=\left[ \Re(\zeta) + \rho_{\beta , \varepsilon ,k}^{-}(\Re(\zeta)),\, \Re(\zeta)+\rho_{\beta ,\varepsilon,k}^{+}(\Re(\zeta))\right] & \times\left[\Im (\zeta )+\Im(\beta) -M_{\varepsilon,k}(\Re(\zeta)),\Im (\zeta )+\Im(\beta) +M_{\varepsilon,k}(\Re(\zeta))\right].
R_{hijk}\ =\g_{hs}R^{s}_{\, ijk}\ =\g_{hs}(\partial_k \Gamma^{s}_{ij}- \partial_j \Gamma^{s}_{ik}+ \Gamma^{r}_{ij}\Gamma^{s}_{rk}- \Gamma^{r}_{ik}\Gamma^{s}_{rj})
C_{i,j}\triangleq \max_{p_{X_{i,j}}}I_{p_{X_{i,j}}q_{Y_{i,j}|X_{i,j}}}(X_{i,j};Y_{i,j}) \ = I_{\bar p_{X_{i,j}}q_{Y_{i,j}|X_{i,j}}}(X_{i,j};Y_{i,j}).
\int_{B_{2R}}|R\nabla\eta|^2 u^2\lesssim \bigg(\int_{B_{2R}}\big(|R\nabla\eta|^\frac{2q-1}{q}|u|\big)^{2q}\bigg)^\frac{1}{2q-1}\bigg(\int_{B_{2R}}|u| \bigg)^\frac{2(q-1)}{2q-1}.
(\delta_\mathrm{mRBA})^2 (u) =~ \delta_\mathrm{mRBA}( \delta_\mathrm{Hoch}(u), -u ) =~ \big( (\delta_\mathrm{Hoch})^2(u), ~ \widetilde{\delta}_\mathrm{Hoch}(u) - \Psi^1 \circ \delta_\mathrm{Hoch}(u) \big) = 0
i \tilde{\Delta}(\rho, \sigma) \equiv \sum_{{\bf p}, \,{\bf p}'}\frac{1}{\sqrt{2 E_p V}\sqrt{2 E_{p'}V}}\langle a^\dagger_{{\bf p}'}\, a_{\bf p}\rangle \, e^{- i (P \cdot \rho - P' \cdot \sigma)}\, ,
a_{\mathrm{c o r r e l a t i o n}}( x , \tau ) = \frac{e c^{2}}{2 \pi^{2}}\frac{\partial}{\partial R^{2}}\ \int d u \ \frac{\theta \Bigl ( R^{2}- c^{2}u^{2}- c^{2}( \alpha - u )^{2}\Bigr )}{\sqrt{R^{2}- c^{2}u^{2}- c^{2}( \alpha - u )^{2}}}\ .
\left(\begin{array}{cc}0 A^{-}(\ell) A^{+}(\ell) 0\end{array}\right) \left(\begin{array}{c}\chi_{E\ell}(r) \chi_{E\ell+1}(r)\end{array}\right)=\sqrt{E+\frac{1}{2(\ell+1)^{2}}}\left(\begin{array}{c}\chi_{E\ell}(r) \chi_{E\ell+1}(r)\end{array}\right) ,
\frac{d^n}{dt^n}\left(f_1(t)f_2(t)\ldots f_m(t)\right)=\underset{k_j\in\mathbb{N}_0}{\sum_{\sum_{j=1}^{m}k_j=n}}n!\prod_{j=1}^{m}\frac{f_j^{(k_j)}(t)}{k_j!}.
\int_{D}\kappa(y,x) \nabla (u(y,x)-u_M^{N,h}(y,x))\cdot\nabla v(x)\mathrm{d}x &=\int_{D}(\kappa_M^{N,h}(y,x)-\kappa(y,x))\nabla u_M^{N,h}(y,x)\cdot\nabla v(x)\mathrm{d}x\forall v\in V.
\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\frac{1}{k^{\beta}+n^{\beta}+m^{\beta}}\iint_{D_{k,n,m}}\frac{1}{(|x_1|^{\alpha_1}+|x_2|^{\alpha_2}+|x_3|^{\alpha_3})}\times \frac{dxdy}{(|y_1|^{\alpha_1}+|y_2|^{\alpha_2}+|y_3|^{\alpha_3})}<\infty.
\left(\frac{\lambda_1+\frac{i}{2}}{\lambda_1-\frac{i}{2}}\right)^4=\frac{\lambda_1-\lambda_2+i}{\lambda_1-\lambda_2-i},\left(\frac{\lambda_2+\frac{i}{2}}{\lambda_2-\frac{i}{2}}\right)^4=\frac{\lambda_2-\lambda_1+i}{\lambda_2-\lambda_1-i}.
U = 1 \textnormal{in}A \ \Leftrightarrow \k^{-1}((\theta(U) q \Phi_0)^{-1}t (\theta(U) q \Phi_0))k = (\theta(U) q \Phi_0)^{-1}t (\theta(U) q \Phi_0) \textnormal{in}G
\left|I_5\right| =\delta \left|\left(\sigma(w_3),\varphi_2\right)\right|\leq D_{p,\alpha}\,\delta\left\|\nabla w_3\right\|^\alpha_p\left\|\nabla\varphi_2\right\|_{p}&\leq D_{p,\alpha}\,\delta\left\|\nabla w_3\right\|^\alpha_2\left\|\nabla\boldsymbol\varphi\right\|_{p}\leq F_5\,\delta\left\|D\boldsymbol\varphi\right\|_{p},
\left[ \pi_{l}^{\left( N \right)}\left( \bar{x}\right) , \pi_{k}^{\left( N \right)}( \bar{y}) \right]_{x^{0}= y^{0}}= - i \theta \left( - 1 \right)^{N - 1}\left( b + a \nabla^{2}\right) \epsilon_{i k}\partial_{l}\partial^{i}\delta^{2}\left( \bar{x}- \bar{y}\right) ,
\left\langle \alpha_{1}, \alpha_{2}, . . . , \alpha_{2 N}, t \right| U_{+}\left| \alpha_{1}^{\prime}, \alpha_{2}^{\prime}, . . . , \alpha_{2 N}^{\prime}, t \right\rangle = R_{\alpha_{1}\alpha_{2}}^{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}\: R_{\alpha_{3}\alpha_{4}}^{\alpha_{3}^{\prime}\alpha_{4}^{\prime}}\: . . . \: R_{\alpha_{2 N - 1}\alpha_{2 N}}^{\alpha_{2 N - 1}^{\prime}\alpha_{2 N}^{\prime}}
S_{m}^{( 1 )}= S_{m}^{( 1 )}( \Psi_{1}, A^{2}( Q , R ) g_{\mu \nu}) \hspace{0 . 5 c m}\mathrm{a n d}\hspace{0 . 5 c m}S_{m}^{( 2 )}= S_{m}^{( 2 )}( \Psi_{2}, B^{2}( Q , R ) g_{\mu \nu}) ,
g_{\mu \nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}= \lambda ( r ) \mathrm{d}t^{2}- \lambda^{- 1}( r ) \mathrm{d}r^{2}- R ( r ) ( \mathrm{d}\theta^{2}+ \mathrm{s i n}^{2}\theta \mathrm{d}\phi^{2}) ,
\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right] \frac{(-1)^{n-1}q^{\frac{n(n+1)}{2}}}{(1-cq^n)}= \frac{1}{c-1}\sum_{n=1}^{N}\frac{(1/c)_n (cq)^n}{(q)_n}\frac{(q^{N+1-n})_n}{(c q^{N+1-n})_n}& = \frac{1}{c-1}\sum_{n=1}^{N}\frac{(1/c)_n q^n}{(q)_n}\frac{(q^{-N})_n}{\left(\frac{q^{-N}}{c}\right)_n},
\L^{(de)}_{(X,\Delta)/Y}:={\iota_{Y^{de}}}_*\L^{(de)}_{(U,\Delta|_U)/Y},\quad\textup{and}\phi^{(de)}_{(X,\Delta)/Y}:={\iota_{Y^{de}}}_*(\phi^{(de)}_{(U,\Delta|_U)}/Y):{F^{(de)}_{X/Y}}_*\L^{(de)}_{(X,\Delta)/Y}\to\O_{X_{Y^{de}}}.
- i \frac{\delta}{\delta J_{+}( t_{1})}\langle t_{2}| t_{2}\rangle \Bigg |_{+ = -}= i \frac{\delta}{\delta J_{-}( t_{1})}\langle t_{2}| t_{2}\rangle \Bigg |_{+ = -}= \langle t_{2}| \Phi ( t_{1}) | t_{2}\rangle .
\sum_{i,j}|b_j^s - a_i^s| = \sum_{i,j}|b_j - a_i||b_j^{s-1}+ \dots + a_i^{s-1}| \geq \sum_{i,j}|b_j - a_i| b_j^{s-1}& \geq \sum_{i,j}(b_j - a_i) b_j^{s-1}= n \sum_j b_j^s - \sum_j b_j^{s-1}\cdot \sum_i a_i,
\left\{\begin{array}{c}{\partial_{\mu}( L A_{\nu}^{a}) - \partial_{\nu}( L A_{\mu}^{a}) + \epsilon^{a b c}A_{\mu}^{b}A_{\nu}^{c}L^{2}\left( \begin{array}{c c}{0}&{1}\\{- 1}&{0}\\ \end{array}\right)}\\ \end{array}\right\}\left( \begin{array}{c}{e^{a}}\\{f^{a}}\\ \end{array}\right) = 0 ,
(\frac{\partial^{2}}{\partial\theta_{i}^{\prime}\partial\theta_{j}^{\prime}}D(\theta,\theta^{\prime}))_{\theta^{\prime}=\theta}=-\int f(x;\theta)(\frac{\partial^{2}}{\partial\theta_{i}^{\prime}\partial\theta_{j}^{\prime}}log(f(x;\theta^{\prime})))_{\theta^{\prime}=\theta}dx=[I(\theta)]_{i,j}.
\begin{array}{llllllllll}J^1 = \langle x_1\rangle, J^2 = \langle x_2\rangle, J^3 = \langle h_1\rangle, J^{4}= \langle h_2\rangle, J^{5}= \langle x_1+x_2\rangle, J^{6}= \langle x_1+h_2\rangle, J^{7}= \langle h_1+x_2\rangle, J^{8, a}= \langle h_1+a h_2\rangle, for \,\, a \in \mathbb{C^{*}},\end{array}
\sum_{\mathfrak{p}\in \mathfrak{P}_{tdo}}q^{deg (v_{\mathfrak{p})}}= \Big(\sum_{m\in \mathbb{Z}_{\geq 0}}q^{T_m}\Big)\cdot \prod_{i\in \mathbb{Z}_{+}}(1+q^{\frac{2i-1}{2}}).
\mathbf{w}_{l}=\bigl[\underbrace{0,\cdots,0}_{k_{s,l}},\underbrace{\tilde{\mathbf{w}}_{l}^{T}}_{k_{c,l}-k_{s,l}},\underbrace{0,\cdots,0}_{k-k_{c,l}}\bigl]^{T}\in\mathbb{F}_{\gamma}^{k\times1}.
d V = \left[ \mathrm{d e t}( - g ) ]^{1 / 2}\right] \mathrm{T r}_{U ( N )}\Delta e_{a_{0}}^{0 \prime}\cdots \Delta e_{a_{d - 1}}^{( d - 1 ) \prime}d \xi^{a_{0}}\cdots d \xi^{a_{d - 1}}\quad ,
\delta(A^{[n]}b) = \delta(A)\sum_{j_2,\dots, j_\ell\geq 0}\prod_{r=2}^\ell \delta_{j_r}(A)+\sum_{j_1=1}^m\sum_{j_2\cdots j_\ell=0}\prod_{r=1}^\ell \delta_{j_r}(A).
\sum_{s=0}^{r-1}\Psi_{\alpha}\circ\theta_{X_s}(X^{i_0}_0\cdots X^{i_{r-1}}_{r-1})=\sum_{s=0}^{r-1}\frac{i_s\alpha^{\vert \underline i\vert}}{\prod_{s'=0}^{r-1}(i_{s'}+\gamma_{r-s'})\cdots (i_{s'}+\gamma_r)}=\vert \underline{i}\vert\frac{\alpha^{\vert\underline{i}\vert}}{\prod_{s'=0}^{r-1}(i_{s'}+\gamma_{r-s'})\cdots (i_{s'}+\gamma_r)}\enspace.
\langle wE(uf), f\rangle =\int_{X}wE(uf)\bar{f}d\mu &=\int_{X}ufE(w\bar{f})d\mu &=\int_{X}f\overline{\bar{u}E(\bar{w}f)}d\mu &=\langle f, \bar{u}E(\bar{w}f)\rangle.
K_\lambda(x_1,\ldots,x_n;q,t) = \frac{\prod_{b\in\mu}1-q^{l(b)+1}t^{a(b)+1}}{\prod_{b\in\delta}1-q^{l(b)}t^{a(b)+1}}E_{0^n\mu}(x_1,\ldots,x_n;q,t),
\mathcal{H}_1 = \ 2 \int dx\ \{(v_N\ast |\phi|^2)\phi(x)a^\ast_x+ (v_N\ast |\phi|^2)\bar\phi(x)a_x\}\mathcal{H}_2 = \ -\frac{1}{2}\int dxdy\ \{2g^T(x, y)a_x^\ast a_y + m(x, y)a_x^\ast a_y^\ast +\bar m(x, y) a_xa_y\}
\begin{array}{l l}{{z^{\prime}= z + \varepsilon \zeta \theta ;}}&{{\theta^{\prime}= \theta + \varepsilon \zeta}}\\{{\bar{z}^{\prime}= \bar{z}- \bar{\varepsilon}\bar{\zeta}\bar{\theta};}}&{{\bar{\theta}^{\prime}= \bar{\theta}+ \bar{\varepsilon}\bar{\zeta}.}}\\ \end{array}
\sigma^+=\begin{pmatrix}0 1 0 0\end{pmatrix},\sigma^-=\begin{pmatrix}0 0 1 0\end{pmatrix},\pi^+=\begin{pmatrix}1 0 0 0\end{pmatrix},\pi^-=\begin{pmatrix}0 0 0 1\end{pmatrix},
\mathcal{F}(u^2) =4\mathcal{F}((1+x^2)^{-2})= 8\sqrt{\pi}\left(\frac{|\xi|}{2}\right)^{3/2}K_{\frac{3}{2}}(|\xi|) &=4\sqrt{\pi}(|\xi|+1)\left(\frac{|\xi|}{2}\right)^{1/2}K_{\frac{1}{2}}(x) = 2(|\xi|+1) \mathcal{F}((1+x^2)^{-1}) = \mathcal{F}(|D|u+u).
\partial_t Q_0^1 - \nu \widetilde{\Delta}_t Q_0^1+ Q^2_0 = - ((\widetilde{U}_0 \cdot \nabla + U_{\neq}\cdot \nabla_t)Q^1)_0 - (Q^j \partial_j^t U^1)_0 - 2(\partial_i^t U^j \partial_j^t \partial_i^t U^1)_0.
\P \bigl( X_{t}\geq 1 - \alpha \eta' \bigr) \geq \P \bigl( 1>X_{t}\geq 1 - \alpha \eta', X_{t-}+\alpha \Delta e(t) =X_t \bigr) &= \P \bigl( 1-\alpha \eta \leq X_{t-}< 1-\alpha \Delta e(t), X_{t-}+\alpha \Delta e(t) =X_t \bigr) & = \P \bigl(1-\alpha \eta \leq X_{t-}< 1-\alpha \Delta e (t) \bigr) \geq \eta'.
\begin{array}{c}\displaystyle \left\{B_i^{H}\left(\frac{k_1}{r}\cos \theta_i+\frac{k_2}{r}\sin \theta_i\right);0\leqslant k_1,k_2\leqslant r\right\}\overset{\mathcal{L}}{=}\displaystyle \left(\frac{\cos \theta_i}{rq_i}\right)^{H}\{B_i^{H}(k_1q_i+k_2p_i);0\leqslant k_1,k_2\leqslant r\}\end{array},
\int_{\mathbb{R}^2}f^1_{\alpha a}V^{(\alpha,a)}dx\lesssim\sum_{\tiny\begin{matrix}\beta+\gamma=\alpha b + c = a \end{matrix}}\sum_{1\leq i,j \leq 2}\|\nabla_i V^{(\beta,b)}\nabla_j V^{(\gamma,c)}-\nabla_i H^{(\beta,b)}\cdot \nabla_j H^{(\gamma,c)}\|_{L^2}\|V^{(\alpha,a)}\|_{L^2}.
0= [(\varphi_{\lambda}(x, [{y_1}_\mu y_2])+ (-1)^{y_2(x+ y_1)}\varphi_{-\partial - \lambda}(y_2, [x_\mu y_1], )- (-1)^{xy_1}\varphi_\lambda(y_1, [ x_\mu y_2]))_{\lambda+ \mu}[w_\gamma v]].
S(x_0, x_1, x_2, \ldots ) = (0, x_0, x_1, x_2, \ldots) \,, \intertext{and by $S^*$ its adjoint (the \emph{backward shift}),}S^* (x_0, x_1, x_2, \ldots ) = (x_1, x_2, x_3, \ldots ),
\Delta{\cal L}_D \ =\m^\dagger m\G \left({A_D \over \Lambda},{A_D^\dagger \over \Lambda}, \tau_0, \tau_0^\dagger,{MM^\dagger \over \Lambda^2},{\tilde{M}M^\dagger \over \Lambda^2},{\tilde{M}\tilde{M}^\dagger \over \Lambda^2}\right)\Bigg|_D
\phi_{N,j}^2(x)=-\partial_x\left(\frac{\tilde{w}_{N,j}(x)}{u_N(x)}\cdot\frac{e^{-2k_jx}}{2k_j}\right), &\phi_{N,j}(x)\phi_{N,l}(x)=-\partial_x\left(\frac{\tilde{v}_{N;j,l}(x)}{u_N(x)}\cdot\frac{e^{-(k_j+k_l)x}}{k_j+k_l}\right),\tilde{v}_{N;j,j}(x)\equiv \tilde{w}_{N,j}(x),
L\left(\frac{1}{2}+\nu',\chi\right) = \sum_{n\leq x}\frac{\chi(n)}{n^{1/2+\nu'}}+ \epsilon(\chi)\left(\frac{q}{\pi}\right)^{-\nu'}\frac{\Gamma\left(\frac{1}{4}-\frac{\nu'}{2}\right)}{\Gamma\left(\frac{1}{4}+ \frac{\nu'}{2}\right)}\sum_{n\leq y}\frac{\overline{\chi}(n)}{n^{1/2-\nu'}}+ ,
\rho=\rho_{0}e^{(\frac{gM}{RL}-1)ln(1-\frac{Lh}{T_{0}})}\approx\rho_{0}e^{-(\frac{gM}{RL}-1)\frac{Lh}{T_{0}}}=\rho_{0}e^{-(\frac{gMh}{RT_{0}}-\frac{Lh}{T_{0}})}
\int_{\partial B(0,r)}g_{i}(x)g_{j}(x)dx=n\left(-\int_{\partial B(0,r)}x_{1}^{4}dx+\int_{\partial B(0,r)}x_{1}^{2}x_{2}^{2}dx\right)=-\frac{2}{n+2}r^{n+3}\mathrm{Vol}(S^{n-1}).
G_2(x,y,q): = - \frac{zJ_1^3J_2^3}{j(x;q)j(y;q)j(z;q)}\frac{j(xy;q^2)j(qxz;q^2)j(qyz;q^2)}{j(-xq;q^2)j(-yq;q^2)j(-z;q^2)}.
\int_{0}^{1}\overline{\xi(x)}\xi^{\prime}(x)dx+\int_{0}^{1}\overline{\eta(x)}\eta^{\prime}(x)dx+\int_{0}^{1}\overline{\xi(x)}\eta^{\prime}(x)dx+\int_{0}^{1}\overline{\eta(x)}\xi^\prime(x)dx \quad+\int_{0}^{1}\overline{\eta(x)}\eta^{\prime\prime}(x)dx=\lambda\int_{0}^{1}|\xi(x)|^2dx+\lambda\int_{0}^{1}|\eta(x)|^2dx.
L_{1}\, = \, - \frac{a}{4}\left( F_{\mu \nu}\right)^{2}+ \, b \, \partial_{\mu}F^{\mu \lambda}\, \partial^{\nu}F_{\nu \lambda}\, + \, \theta \, \epsilon^{\mu \nu \rho}\, \partial_{\sigma}\, A_{\mu}\, \partial_{\nu}\partial^{\sigma}\, A_{\rho},
\langle \mathfrak{x}_1,\mathfrak{y}_1,\ldots,\mathfrak{x}_b,\mathfrak{y}_b,\mathfrak{z}_1,\ldots,\mathfrak{z}_{s}\rangle =G, \mathfrak{z}_1+\cdots+\mathfrak{z}_{s}=0, b-1+\sum\limits_{i=1}^{s}\Big(1-\frac{1}{|\mathfrak{z}_{i}|}\Big) =\frac{g-1}{|G|}.
\frac{F}{V}=-\sum_{\tau\in{\cal F}}\frac{\nu^{-2}e^{-\frac{\beta^2|n\tau-m|^2}{4\pi\alpha'\tau_2}}}{n^2+\nu^{-2}r^2}\left(\frac{1}{4\pi\alpha'\tau_2}\right)^{13}\vert\eta(\tau)\vert^{-48}
\frac{2}{e^t+1}= \left( \frac{e^t-1}{2}+1 \right)^{-1}= \sum_{l=0}^\infty \left( \frac{e^t-1}{2}\right)^l (-1)^l &= \sum_{l=0}^\infty (-1)^l 2^{-l}l! \sum_{n=l}^\infty S_2(n,l) \frac{t^n}{n!}&= \sum_{n=0}^\infty \left( \sum_{l=0}^n S_2(n,l) (-1)^l 2^{-l}l! \right) \frac{t^n}{n!}.
\rule{.9cm}{0cm}\frac{\delta}{\delta\phi^*_A}+ \frac{i}{\hbar}\frac{\delta\Gamma}{\delta\phi^*_A}-(-1)^{\varepsilon_i(\varepsilon_A+1)}(\Gamma^{''-1})^{ij}\bigg(\frac{\delta_l}{\delta\varphi^j}\frac{\delta\Gamma}{\delta\phi^*_A}\bigg)\frac{\delta_l}{\delta\varphi^i}\bigg),
D_m\Big\{\big[v(t),w(t)\big];(v_0,w_0)\Big\}&=\sqrt{\left(\begin{array}{c}v(t)-v_0 w(t)-w_0\end{array}\right)^{\top}\Omega_{ij}^{-1}\left(\begin{array}{c}v(t)-v_0 w(t)-w_0\end{array}\right)},
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).
\mathbb{E}[\|\mathbf{H}_k\|^2] = \mathbb{E}[ \|\mathbf{H}_{k,}\|^2 ] + \mathbb{E}[ \|\mathbf{H}_{k,}\|^2 ] + \mathbb{E}[\mathbf{H}_{k,}\cdot \mathbf{H}^H_{k,}] + \mathbb{E}[\mathbf{H}_{k,}\cdot \mathbf{H}^H_{k,}].
(e^{\frac{1}{2}\ell_\nu}+1)^{-1}=(e^{\frac{1}{2}(\ell_\nu+2\ell_\mu)}+1)^{-1}+(e^{\frac{1}{2}(\ell_\nu+2\ell_{\mu'})}+1)^{-1}+(e^{\frac{1}{2}(\ell_\mu+\ell_{\mu'})}-1)^{-1}.
Z_{N}=\int_{-\infty}^{\infty}\cdot\cdot\cdot\int_{-\infty}^{\infty}d\sigma_{1}\cdot\cdot\cdot d\sigma_{N}exp[K\sum_{\langle jl\rangle}\sigma_{j}\sigma_{l}+h\sum_{j}\sigma_{j}]\delta[N-\sum_{j}\sigma_{j}^{2}]
A_6= \ell_1A_1,A_4=\ell_2A_2, A_5=1/(\ell_1\ell_2)A_3 \quad\mathrm{or}A_5= \ell_1A_1, A_6=\ell_2A_2, A_4=1/(\ell_1\ell_2)A_3\, .
\mathcal{L}:=\Lambda(0)=\mathfrak{p}_F^{a_{n}}w_n\oplus\mathfrak{p}_F^{a_{n-1}}w_{n-1}\oplus\cdots\oplus \mathfrak{p}_F^{a_{-n+1}}w_{-n+1}\oplus \mathfrak{p}_F^{a_{-n}}w_{-n}.
(i)~\Phi(\alpha f+\beta g)=\alpha\Phi(f)+\beta\Phi(g); &(ii)~\Phi(fg)=\Phi(f)\Phi(g)~~\Phi(f^*)=\Phi(f)^*; &(iii)~\|\Phi(f)\|=\|f\|:=\sup_{t\in Sp(A)}|f(t)|; &(iv)~\Phi(f_0)=1 ~~ \Phi(f_1)=A,~ f_0(t)=1~ f_1(t)=t, ~ &t\in Sp(A).
\sum_{j=1}^N \int_{I_j}\sum_{i=1}^m \partial_t c_{ih}p_{ih}dx = - \sum_{j=1}^N \int_{I_j}\sum_{i=1}^m c_{ih}|p_{ihx}|^2 dx - \sum_{j=1}^{N-1}\sum_{i=1}^m ( \{c_{ih}\}[p_{ih}] (\widehat{p_{ihx}}+\{p_{ihx}\}))_{j+1/2},
\begin{gathered}\tfrac12\hat b_0,\hat b_1\le\tfrac12\hat a_0,\hat a_1,\hat a_2,\hat a_3<\hat b_2,\hat b_3, \hat a_0+\hat a_1+\hat a_2+\hat a_3\le\hat b_0+\hat b_1+\hat b_2+\hat b_3-2.\end{gathered}
[ x^I, p^J ] = i \delta^{IJ}~,~~~[ \alpha^i_n, \alpha^j_m ] = \omega_n \delta^{ij}\delta_{n+m,0}~,~~~[ \alpha^{i'}_n, \alpha^{j'}_m ] = \omega'_n \delta^{i' j'}\delta_{n+m,0}~.
\big(\alpha_F(r),h_2\big(r\otimes\frac{rr_2'}{r_1'}\big)-nw'\big) =\big(\alpha_F(r),h_2\big(r\otimes\frac{rr_2s_2}{r_1s_1}\big)-nh_2\big(1\otimes\frac{s_2}{s_1}\big)-nw\big) &= \big(\alpha_F(r),h_2\big(r\otimes\frac{rr_2s_2}{r_1s_1}+r\otimes\frac{s_1}{s_2}\big)-nw\big) &= \big(\alpha_F(r),h_2\big(r\otimes\frac{rr_2}{r_1}\big)-nw\big) .
M=\begin{pmatrix}1 0 \frac{1}{2}\frac{1}{2}0 0 1 \frac{1}{2}\frac{1}{2}0 \frac{1}{2}\frac{1}{2}\frac{3}{2}\frac{1}{2}1 \frac{1}{2}\frac{1}{2}\frac{1}{2}\frac{3}{2}1 0 0 \frac{1}{2}\frac{1}{2}1\end{pmatrix}.
I_0(w'';w,\xi_x;w',\xi'_x;\tilde{w},\tilde{w}';\eta)= \frac{I_1(w'';w,w';\tilde{w},\tilde{w}';\eta)}{\lambda^{-1}(\xi_x-\xi'_x)}+I_2(w'';w,w';\tilde{w},\tilde{w}';\eta).
I_1 \leq \int e^{-r_2( |y-x|^{1/s}+ |y|^{1/s})}d y \leq e^{- r_2 c |x|^{1/s}}\int e^{- r_2 c |y|^{1/s}}d y = B e^{- r_2 c |x|^{1/s}},
Equ> p^s+1-(\tau+1)p-(\tau-1)p^{s-1}+\tau p^{s-2}+\frac{p(\tau-1)-\tau}{p-\tau+1}> p^s+1-p^2-(p-2)p^{s-1}-\tau = p^{s-1}-p^2+p^{s-1}-\tau+1 > 1,
\lim_{u\to g(0)}\frac{g_{\kappa,u}(g^{-1}(u)x)}{q(u)}=\lim_{u\to g(0)}\frac{\sum_{i=0}^\kappa(g(M^i g^{-1}(u)x)-u)}{q(u)}=-\sum_{i=0}^\kappa\log(x M^i) &=-\sum_{i=0}^\kappa(\log(x)+i\log M)=-(\kappa+1)\log(x)-\frac{\kappa(\kappa+1)}{2}\log M &=-(\kappa+1)(\log(x)+\kappa\log\sqrt{M})
\int_{\mathcal S}\zeta^3\, W(\zeta)\,d\zeta=a_{2,1}p_1^2,\int_{\mathcal S}\zeta^4\, W(\zeta)\, d\zeta=p_2^2+(a_{2,1}-na_{2,0}a_{2,2})p_1^2,\mbox{and so forth.}
P({\bf x}|F,{\bf y})= \frac{P_0({\bf x}) P({\bf y}|F,{\bf x})}{Z({\bf y},F)}, &= \frac{1}{Z({\bf y},F)}\prod_{i=1}^N P_0( x_i)\prod_{\mu=1}^M \frac{e^{-\frac{(y_\mu - \sum_{i=1}^N F_{\mu i}x_i)^2}{2\Delta}}}{\sqrt{2\pi \Delta}}.
\vartheta_{2^{k_j}}(x'-x-y)\psi_{2^l}(y'-x-y) & = \int_{\mathbb{R}^2}\widehat{{\vartheta}}(2^{k_j}\xi) e^{2\pi i x'\xi}\widehat{\psi}(2^l\eta) e^{2\pi i y'\eta}\widehat{\omega}(2^l(-\xi-\eta))e^{2\pi i x(-\xi-\eta)}\widehat{\omega}(2^l(-\xi-\eta)) e^{2\pi i y(-\xi-\eta)}d\xi d\eta.