harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
+ e^{2}A_{\mu}{\frac{\{- 4 \alpha + 8 g ( 1 - \alpha ) \partial^{2}( 2 \alpha^{2}g - a + a^{2}g \partial^{2}) \}}{( a g \partial^{2}- 1 )^{2}+ 4 \alpha^{2}g^{2}\partial^{2}}}\epsilon^{\mu \nu \lambda}\partial_{\nu}A_{\lambda}- 4 J_{\mu}A^{\mu}] .
\mu_{\lambda \rho}(C_{\bf 0}\in\Lambda({\bf k})\mid ~ \Gamma_{{\bf 0}})= \frac{\lambda^{m-1}m!}{(m-1)!}\prod_{j=1}^d \frac{p_j^{k_j}}{k_j!}F_\lambda ({\bf k})=\lambda^{|{\bf k}|-1}|{\bf k}|\prod_{j=1}^d \frac{p_j^{k_j}}{k_j!}F_\lambda ({\bf k}).
\tilde{K}_{it}(x) = \int_{-\infty}^{\infty}\cos(2x \sinh v)\exp( it v) dv, J^{\pm}_{it}(x) = \int_{-\infty}^{\infty}\left\{\begin{array}{l}\sin \cos\end{array}\right\}(2x \cosh v)\exp( it v) dv
=\int\int \frac{1}{N}\sum\limits_{i=1}^N \left ( \frac{1}{2\pi\theta^2}\right ) \left ( \frac{1}{2\pi4\theta^2}\right ) exp \left ( - \frac{(u_1-b)^{2}}{2\theta^2}\right) exp \left( - \frac{(u_2-b')^{2}}{2\theta^{2}}\right) exp \left( - \frac{c}{4\theta^2}\right ) \mathrm{d}u_1\mathrm{d}u_2
\left\{\sup_{0\le t\le\eta}\left|\frac{1}{N}\sum_{j=1}^{N}\left[\phi_{\epsilon}\left(x_{j}^{N}(t)-x_{i}^{N}(0)\right)-\phi_{\epsilon}\left(x_{j}^{N}(0)-x_{i}^{N}(0)\right)\right]\right|\ge\gamma\epsilon\right\}.
U (\vec{a},\Lambda)\left(S, f_\alpha \otimes \rho(E^\alpha)\right) &:= \left(\Lambda S + \vec{a}, e^{-i[\vec{a}\cdot (\hat{H}(\rho_n)\hat{f}_0 + \hat{P}(\rho_n)\hat{f}_1 )]}f_{\alpha}(\Lambda^{-1}(\cdot-\vec{a}))\otimes \rho(E^\alpha)\right).
y^{2i_{1}-1}\eta(\S_{2i_{1},j_{1}})\eta(\S_{2i_{2},j_{2}})=\sum_{l=0}^{n^2-2i_{2}}y^{l}\eta(\S_{n^{2}+2i_{1}-2i_{2}-2l,\overline{j_{1}+j_{2}+l}}).
\zeta_{U}\left( s\right)=D^{s/2}\sum_{k=0}^{\infty}\binom{-s}{k}\left( -1\right)^{k}\frac{Q^{k}}{a^{s+2k}-Q^{k}}=D^{s/2}\sum_{k=0}^{\infty}\binom{-s}{k}\left( -1\right)^{k}\frac{a^{-s-k}b^k}{1-a^{-s-k}b^{k}},
\begin{array}{l c}{{{\mathrm{S e c t o r}}\quad \quad}}&{{S U ( 2 ) \otimes S U ( 2 ) ~{\mathrm{r e p .}}}}\\{{{\mathrm{N S}}}}&{{2 ({\bf 1},{\bf 1}) ~ ,}}\\{{{\mathrm{R}}}}&{{({\bf 1},{\bf 2}) ~ .}}\\ \end{array}
g_{\theta}\equiv \bar{\zeta}\tilde{p}d_{\theta}+ \bar{d}_{\theta}\tilde{p}\zeta = - i ( \bar{\zeta}\tilde{p}p_{\theta}+ \bar{p}_{\theta}\tilde{p}\zeta ) - 4 k^{2}m^{2}( \theta \zeta + \bar{\zeta}\bar{\theta}) \approx 0 ,
M_{ij}:= \begin{bmatrix}(A_1)_{ii}(B_1)_{jj}-(C_1)_{ii}(D_1)_{jj}\ddots \ddots (A_{r-1})_{ii}(B_{r-1})_{jj}-(C_{r-1})_{ii}(D_{r-1})_{jj}-(C_r)_{ii}(D_r)_{jj}(A_r)_{ii}(B_r)_{jj}\end{bmatrix},
\sigma_{\mathrm{i n}\to \mathrm{a l l}}( \theta ) ={\frac{1}{2 \pi p}}\left({\frac{1}{\sin^{2}{\theta}}}\right) 2 \left( 1 - \mathrm{R e}\langle \mathrm{i n}|{\cal R}^{2}| \mathrm{i n}\rangle \right) ,
\zeta(y,0)=y\mbox{in}\Omega_+,\zeta(\cdot,t):\Omega_+\mapsto \Omega_+\;\mbox{is one to one and belongs to}\; C^1(\Omega_+), &\det(\nabla \zeta(\cdot, t))\neq 0\mbox{on}\Omega_+ \;\mbox{for any given}\; t\in [0,T) ,
\left(T_u^{*}H_u T_u\right) (z_0) = \left(T_v^{*}H_v T_v\right) (z_0) + \mathcal{O}\left(|z_0|^{-2}\right) = -8v(z_0) + \mathcal{O}\left(|z_0|^{-2}\right) < 0
g(x_1,\cdots,x_{d-1},x_d)=\sum_{\alpha\in\{-m,\cdots,-1,0,1,\cdots,m\}^{d-1}}a_{\alpha}(x_d) e^{\alpha_1\frac{2\pi ix_1}{T_1}}\cdots e^{\alpha_{d-1}\frac{2\pi ix_{d-1}}{T_d}}
-\tilde{u}(x_{2}, t_{2})-\left(p_{2}-\frac{1}{\delta}(y_{2}-y_{1})\right)\cdot x_{2}&=h_{2}+\frac{1}{2\delta}(|y_{2}|^{2}-|y_{1}|^{2})+\frac{1}{2\delta}\left[(t_{2}-s_{2})^{2}-(t_{2}-s_{1})^{2}\right] &=h_{2}+\frac{1}{2\delta}\left[|y_{2}|^{2}-|y_{1}|^{2}+s_{2}^{2}-s_{1}^{2}-2t_{2}(s_{2}-s_{1})\right].
e^{-w(x,t)}=\Pi_{x}^{h}\left[e_{q+g}(t)e^{-l(\xi^{h}_{t})}\right]+\Pi^{h}_{x}\big[\int_{0}^{t}{\rm d}s\,q(\xi^{h}_{s})e_{q+g}(s) \quad\cdot\pi^{h}(\xi^{h}_{s},e^{-w_{t-s}})\int_{[0,+\infty)}e^{-\theta\pi(\xi^{h}_{s},V_{t-s}f)}\eta(\xi^{h}_{s},{\rm d}\theta)\big].
\sum_{n=0}^{\infty}p_{-(25r+k)}(5n+5-t)q^{n}=\sum_{5r+s+1}^{25r+5s+t}m(25r+5s+t,h)\dfrac{q^{h-5r-s-1}}{{E_{1}^{6h}}E_{5}^{25r+5s+t-6h}}.
y_- \le -y_1^{\textsc{d}}\mkern-.5mu-\mkern.5mu y_{1\mkern-1mu +}^{\textsc{g}}\mkern-.5mu+\mkern2.5mu[\mkern.3mu y_1^{\textsc{d}}\mkern-.5mu+\mkern.5mu y_{1\mkern-1mu +}^{\textsc{g}}]=-\lfloor y_1^{\textsc{d}}\mkern-.5mu+\mkern.5mu y_{1\mkern-1mu +}^{\textsc{g}}\rfloor.
\sum_{{\bf q}_{k}}\frac{1}{\sqrt{2 E_{q_{k}}V}}e^{- i \overline{{{Q}}}_{k}\cdot z_{2}}\sum_{p_{j}}\frac{1}{\sqrt{2 E_{p_{j}}V}}e^{i \overline{{{P}}}_{j}\cdot z_{1}}\langle b_{-{\bf p}_{j}}^{\dagger}b_{-{\bf q}_{k}}\rangle
C_{01\ell}(N) = \frac{(-1)^N}{N!}\left[ z^{N-\ell}\right]e^z \left(\frac{z}{e^z - 1}\right)^{1-\ell}\left(\frac{z}{e^z - 1}\right)\left(\frac{2z}{e^{2z}- 1}\right) \cdots \left(\frac{N z}{e^{Nz}- 1}\right).
C^{\diamond}\leq \max_{p(x_1,x_2)}\min_{p(u|x_1,x_2,y)=p(u|y)}\max_{p(q|x_1,x_2,y,u)=p(q|x_1,x_2)}\!\!\min\left\{\!\!\!\!\begin{array}{l}C_1+C_2, C_1+I(X_2;Y|X_1Q), C_2+I(X_1;Y|X_2Q), I(X_1X_2;Y|Q), C_1+C_2-I(X_1X_2;U|Q)+I(X_2;U|X_1Q)+I(X_1;U|X_2Q)\end{array}\!\!\!\!\right\}.
\begin{aligned}\left(n-\frac{1}{2}\right)\exp\left(-\frac{\log{R}}{2}\left(n-\frac{1}{2}\right)\right)\ge \sqrt{\frac{\mathcal{N}_F(T)}{K_{F,3}(T,\tau)}}\exp\left(\frac{\log{R}}{4}\right).\end{aligned}
\|g_n\|_{L^q(Q_{R,\tau})}= \frac{r_n^{2-\frac{N+2}{q}}}{M_n}\|f_n\|_{L^q( \widetilde{Q}_{n,R,\tau})}, \widetilde{Q}_{n,R,\tau}= B_{Rr_n}(\bar x_n) \times (\bar t_n, \bar t_n + r_n^2 \tau).
|\mathcal C_3(n,k)|\ =\ 3\ +\{n-2\choose k-2}-{n-k-2\choose k-2}+{n-3\choose k-2}-{n-k-2\choose k-2}+{n-4\choose k-2}-{n-k-3\choose k-2}+ \cdots{n-k-1\choose k-2}-{n-2k\choose k-2}.
\tau^2 - 2 \tau \xi \cdot \omega + |\xi|^2 = (\tau - \xi \cdot \omega)^2 + |\xi|^2 - (\xi \cdot \omega)^2 &= (\tau - \xi \cdot \omega)^2 + (\xi_1 \omega_2 - \xi_2 \omega_1)^2
({1_{-}2_{-}}) \equiv ( g_{1})_{-}{}^{i}\epsilon_{i j}( g_{2})_{-}{}^{j}= e^{- i{( \phi_{1}+ \phi_{2}) / 2}}\frac{\tau_{1}- \tau_{2}}{\sqrt{( 1 + | \tau_{1}|^{2}) ( 1 + | \tau_{2}|^{2})}}
g_{\mu \nu}\left( x + \pi ( \xi ) \right) = g_{\mu \nu}+ g_{\mu \nu ; \lambda}\xi^{\lambda}+{\frac{1}{2}}\left[ g_{\mu \nu ; \lambda \sigma}+{\frac{1}{3}}\left( R_{~ \lambda \mu \sigma}^{\kappa}g_{\kappa \nu}+ R_{~ \lambda \nu \sigma}^{\kappa}g_{\kappa \mu}\right) \right] \xi^{\lambda}\xi^{\sigma}+ . . .
\begin{array}{c | c c c c c c c c}{{p^{\prime}}}&{{5}}&{{7}}&{{9}}&{{1 1}}&{{1 3}}&{{1 5}}&{{\cdots}}\\ \hline{{\Delta}}&{{7}}&{{1 1}}&{{1 5}}&{{1 9}}&{{2 3}}&{{2 7}}&{{\cdots}}\\ \end{array}
Q^3 |\Phi ' \rangle =\int dy^{-}dy_{\bot}\left \{f^{3ab}A^a_{\bot}(y_{\bot}, y^-) \Pi^b_{\bot}(y_{\bot}, y^-)+ \rho^3_m(y_{\bot}, y^-) \right \}|\Phi ' \rangle =0 \,.
\inf_{\widehat{\theta}}\sum_{\theta_0\in \Theta}P_{\theta_0}^n \big(\rho(\widehat \theta, \theta_0) \geq A \big) \pi_{\pm}(\theta_0) = 1 - \sup_{\widehat{\theta}}\int \sum_{\theta_0 \in \Theta}\mathbf{1}(\rho(\widehat \theta, \theta_0) < A) p_{\theta_0}\pi_{\pm}(\theta_0) d\mu.
F_{10}: =-\frac{43867}{2^{10}\cdot 3^5\cdot 5^2\cdot 7\cdot 53}(E^{(3)}_{10}-E^{(3)}_4E^{(3)}_6), F_{12}: =\frac{131\cdot 593}{2^{11}\cdot 3^6\cdot 5^3\cdot 7^2\cdot 337}(3^2\cdot 7^2E_4^{(3)3}+2\cdot 5^3E_6^{(3)2}-691E^{(3)}_{12}).
\alpha^{di}_{max}:= \alpha_{n-i+1}+ 2\sum\limits_{j = n-i+2}^{n-2}\alpha_j + \alpha_{n-1}+ \alpha_n \; \; n \geq 4, \alpha^{d3}_{max}:= \alpha^{a3}_{max}:= \alpha_{n-2}+ \alpha_{n-1}+ \alpha_n.
\widetilde{\eta}^{i}\equiv U^{i}{}_{j}\eta^{j}\, , \qquad \widetilde{\eta}_{i}\equiv \eta_{j}( U^{- 1})^{j}{}_{i}\ , \qquad \widetilde{\theta}^{i}\equiv \tilde{U}^{i}{}_{j}\theta^{j}\, , \qquad \widetilde{\theta}_{i}\equiv \theta_{j}( \tilde{U}^{- 1})^{j}{}_{i}\ ,
\partial_{t}\langle\hat{x}^{2}\rangle=\frac{1}{2i\hbar m}\langle[\hat{x}^{2},\hat{p}^{2}]\rangle=-\frac{1}{2m}(\hat{x}\hat{p}+\hat{p}\hat{x})\approx2v_{0}\langle\hat{x}\rangle_{+}-2v_{0}\langle\hat{x}\rangle_{-},
F_{i}=\left(\begin{array}{ccc}1 0 n_{i,k_{i}}1 \end{array}\right) \left(\begin{array}{ccc}1 n_{i,k_{i}-1}0 1 \end{array}\right)\cdots \left(\begin{array}{ccc}1 0 n_{i,2}1 \end{array}\right)\left(\begin{array}{ccc}1 n_{i,1}0 1 \end{array}\right) ,
{\mathcal H}(\theta ,W)=\frac{1}{2d}\log\left(\frac{(1-\sin \theta )^{\beta^{3/2}}(1- \cos \theta)}{(1+\sin \theta)^{\beta^{3/2}}(1+\cos \theta )}\right)-\frac{\alpha \beta^{1/2}}{\big( \frac{d^{2}}{\beta}(\beta^{1/2}\sin \theta - \cos \theta)^{2}+W^{2}\big)^{1/2}}.
f_\varepsilon:= \restriction{(\kappa_{B,C}\circ f)}{(B \setminus C)^*}\colon (B\setminus C)^*\to (B\setminus C)^* \ \g_\varepsilon := \restriction{g}{(B \setminus C)^*}\colon (B\setminus C)^*\to A^*
t_{j}(x) = \dfrac{\tanh(\alpha x_{j})-\tanh(\alpha x)}{\tanh(\alpha x_{j})-\tanh(\alpha x_{j-1})}y_{j-1}+\dfrac{\tanh(\alpha x)-\tanh(\alpha x_{j-1})}{\tanh(\alpha x_{j})-\tanh(\alpha x_{j-1})}y_{j}.
f ( x ) = 1 - \frac{a}{x}+ \frac{2 x - 2 a}{x^{3}}= \frac{x^{3}- a x^{2}+ 2 x - 2 a}{x^{3}}= \frac{( x^{2}+ 2 ) ( x - a )}{x^{3}}( a > 0 ) ( x > 0 )
A_2= c^2(1-\theta_1)-2t^2\theta_2=\dfrac{9abc^3d-b^2c^4}{6 a c^3 +6b^3 d-4 b^2 c^2}-\dfrac{2b^2 (3bd-c^2)^2}{6 a c^3 + 6 b^3 d-4b^2 c^2}= \dfrac{3 b (4 b^2 c^2 d + 3 a c^3 d - 6 b^3 d^2-b c^4)}{6 a c^3 +6b^3 d-4 b^2 c^2}= \dfrac{3b^3c^2d}{2c^2(3 a c-b^2)+2b^2(3bd-c^2)}\cdot(4+\dfrac{3ac}{b^2}-\dfrac{6bd}{c^2}-\dfrac{c^2}{bd}).
[M_S, \mu_S]= (I^G_{M_S}\otimes \delta^{-\frac{1}{2}}_{P_S}\circ [\mu_S] \circ (\delta^{\frac{1}{2}}_{P_S}\otimes J^G_{P^{op}_S})) \otimes [1][|\cdot|^{\langle \rho_G, \mu_S - \mu \rangle}].
\mathrm{d}s^{2}= G_{\mu \nu}( \gamma , R ) \mathrm{d}X^{\mu}\mathrm{d}X^{\nu}= \mathrm{d}\rho^{2}+ \rho^{2}\left( \mathrm{d}\phi +{\frac{\gamma}{R}}\mathrm{d}y \right)^{2}+ \mathrm{d}y^{2}+ \mathrm{d}x_{i}\mathrm{d}x^{i},
\partial_{(1)}= \partial_\mu, \partial_{(a_1, ..., a_{k-1}, a_k)}= \begin{cases}\nabla_{v_{a_k}}\cdot \partial_{(a_1, ..., a_{k-1})}a_k \leq \max \{a_1, ..., a_{k-1}\}, \partial_\mu \cdot \partial_{(a_1, ..., a_{k-1})}a_k > \max \{a_1, ..., a_{k-1}\}. \end{cases}
Var(X)=\frac{\Psi(\frac{\alpha+2}{2},\frac{\gamma}{\sqrt{\beta}})}{\beta\Psi(\frac{\alpha}{2},\frac{\gamma}{\sqrt{\beta}})}-[\frac{\Psi(\frac{\alpha+1}{2},\frac{\gamma}{\sqrt{\beta}})}{\beta^{1/2}\Psi(\frac{\alpha}{2},\frac{\gamma}{\sqrt{\beta}})}]^{2}.
\sum_{t=1}^{T}\theta_t \left(\frac{1}{\gamma_t}V_{x_t}(x) - \frac{1}{\gamma_t}V_{x_{t+1}}(x) - \alpha V_{(x_t)}(x) + \frac{\gamma_t G^2}{2}\right) \leq \frac{2 G^2}{\alpha\,(T+1)},
p_{k}^{T}b=p_{k}^{T}Ax_{*}=\sum_{i=1}^{n}\alpha_{i}p_{k}^{T}Ap_{i}=\sum_{i=1}^{n}\alpha_{i}\langle p_{k},p_{i}\rangle_{A}=\alpha_{k}\langle p_{k},p_{k}\rangle_{A}
\mathrm{dim}~\mathrm{Hom}_{G_{n}}(\pi, \pi') \geq \mathrm{dim}~\mathrm{Hom}_{G_{n}}(\mathrm{ind}_{R_{n+1-i}}^{G_{n}}\pi^{[i]}\boxtimes \psi_{i-1}, \pi' ) = \mathrm{dim}~\mathrm{Hom}_{G_{n+1-i}}( \pi^{[i]},{}^{(i-1)}\pi') \neq 0 .
A\,\sum_{i=1}^{N}\,\int_{\mathbb{R}^d}\,|\hat{F_i}(\lambda)|^2\,d\lambda\leq \int_{\mathbb{R}^d}\,\epsilon^d\, \left|\sum_{i=1}^{N}\,\hat{F_i}(\epsilon\lambda)\,e^{-2\pi i x_i\cdot\lambda}\right|^2\,d\mu(\lambda)\leq B\, \sum_{i=1}^{N}\int_{\mathbb{R}^d}\,|\hat{F_i}(\lambda)|^2\,d\lambda.
q^{-1/4}\mathcal{S}_{q^{-1}}(q^{-1/4}z)=-q^{1/4}\mathcal{S}_{q}(q^{1/4}z) \mbox{and}\mathcal{C}_{q^{-1}}(q^{-1/4}z)=\mathcal{C}_{q}(q^{1/4}z).
\mathbf{y}_k=\sum_{j=1}^K\mathbf{H}_{kj}^H(\mathbf{{P}}_{\mathrm{c},j}\mathbf{s}_{\mathrm{c},j}+\mathbf{P}_{\mathrm{p},j}\mathbf{s}_{\mathrm{p},j})+\mathbf{n}_k,
\zeta(3)=1+\begin{array}{ccccccccccc}1 1^3 1^3 2^3 2^3 \cline{1-1}\cline{3-3}\cline{5-5}\cline{7-7}\cline{9-9}\cline{11-11}2^2\cdot 1 + 1 + 2^2\cdot 3 + 1 + 2^2\cdot 5 +\cdots\end{array}.
\textbf{I}=\int_{\mathbb{R}^n\times(\mathbb{R}^n\backslash B_\delta(0))}f_\lambda(x)f_\lambda(y)|x-y|^{\alpha-n}dxdy, \textbf{II}=\int_{B_\delta(0)\times(\mathbb{R}^n\backslash B_\delta(0))}f_\lambda(x)f_\lambda(y)|x-y|^{\alpha-n}dxdy.
ds_5^2=-\o^2dt^2 + (\delta+\o^2)\frac{r_-}{r_+}dy^2 +\frac{dw^2}{\delta+\o^2}\frac{4r_+^3}{r_-}\frac{1}{(1-\o^2)^4}+\frac{r_+^2}{(1-\o^2)^2}d\O_2^2,
\!{\cal K}_{p}( x^{\prime \prime}, y^{\prime \prime}, z^{\prime \prime}, N | x^{\prime}, y^{\prime}, z^{\prime}, 0 ) = \frac{\lambda_{p}( - 2 N )}{\left| 4^{\frac 1 3}N \right|_{p}^{\frac 3 2}}\chi_{p}\left( - \bar{S}( x^{\prime \prime}, y^{\prime \prime}, z^{\prime \prime}, N | x^{\prime}, y^{\prime}, z^{\prime}, 0 ) \! \right) .
\frac{\prod_{1\le i<j\le N}(q^{a_i}-q^{a_j})(q^{s_j}-q^{s_i})}{\prod_{i,j=1}^N(q^{s_i}-q^{a_j})}=\det\left(\frac{1}{q^{s_i}-q^{a_j}}\right)_{i,j=1}^N, &\frac{\prod_{1\le i<j\le N}\sin\pi (a_i-a_j)\sin\pi(s_j-a_i)}{\prod_{i,j=1}^N\sin\pi(s_i-a_j)}=\det\left(\frac{1}{\sin\pi(s_i-a_j)}\right)_{i,j=1}^N.
\int_{Q_T}T_\mathcal{M}( \phi_M )\nabla \theta_M \cdot \mathbf{v}\mathrm{dx}\mathrm{dt}=-\int_{Q_T[|\phi_M|<\mathcal{M}]}\theta_M \nabla \phi_M \cdot \mathbf{v}\mathrm{dx}\mathrm{dt},
T = i \tau_{i n t}\cdot \tau_{x}K , \quad \tau_{i n t}= \tilde{\gamma}^{1}\tilde{\gamma^{3}}, \quad \tau_{x}= d i a g ( - 1 , 1 , 1 , 1 ) , \quad \mathrm{a n d}\quad K i K^{- 1}= - i
\prod^{q_1}_{j=1}\bigg(\partial_x^{\alpha_j}\partial_y^{\beta_j + 1}\big( u^s +{u}^n \big) \bigg)\bigg|_{y=0}\times \prod^{q_2}_{l=1}\bigg(\partial_x^{\tilde \alpha_l}\partial_y^{\tilde\beta_l + 1}\big( u^s +{u}^{n-i}\big)\bigg)\bigg|_{y=0}\,\, ,
\overline{\delta}(k) = \sum_{\sigma\in\{+,- \}}\sum_{G=0}^\infty (2G+1)\left(\delta_{G,\sigma}(k) - \delta^{(1)}_{G,\sigma}(k) - \delta^{(2)}_{G,\sigma}(k) \right) + \delta_{\rm lim}(k) \, .
\tilde{B}({\bf x})=\frac{(x_2+x_1+\beta_2)(x_2-x_1+\beta_2-\beta_1)(N-x_2)(x_2+N+\beta_3)}{(2x_2+\beta_2+1)(2x_2+\beta_2)}, \tilde{E}({\bf x})=\frac{(x_2-x_1)(x_2+x_1+\beta_2)(N-x_2+\beta_3-\beta_2)(N+x_2+\beta_2)}{(2x_2+\beta_2-1)(2x_2+\beta_2)}.
P^2=1, PJ_0=J_0P, PJ_\pm=-J_\pm P, [J_0, J_\pm] = \pm J_\pm, [J_+,J_-]=2 J_0 + 2 (\gamma+\delta+1)J_0P - (2N+1)(\gamma-\delta)P + (\gamma-\delta)I.
\mathcal{C}= \sum \limits_j A_j(m_1,m_2) \overline{A_j(n_1,n_2)}\frac{\left|F^{\#}\left(\nu^{(j)}_1, \nu^{(j)}_2\right)\right|^2}{6 \mathcal{L}_j \prod \limits_{k=1}^3 \Gamma\left(\frac{1+3\nu_k^{(j)}}{2}\right)\Gamma\left(\frac{1-3\nu_k^{(j)}}{2}\right)},
\widetilde{V}^{(0^{2})}\big(x;\omega ,N,-\Vert \psi_{0}\Vert^{2}\big)=V(x;\omega,N) +\frac{4\omega}{N!}z^{1/2}\frac{d}{dz}\left[ z^{N+1/2}\Big/\left( \sum\limits_{l=0}^{N}z^{l}/l!\right)\right]
L\tau +\partial_{\vec v}L\cdot \bigl(\vec \Xi-\tau \vec v\bigr)= -\vec u\cdot\Bigl(\vec v\times \bigl(\vec r\times \vec v\bigr)- \frac{\mu}{\lVert \vec r\rVert}\,\vec r\Bigr)=-\vec u\cdot\vec A.
x_{n-2}x_{n-1}+ \sum_{j=0}^{n-3}x_j x_{j+2}+ \sum_{j=0}^{n-3}x_j & x_{n-2}x_{n-1}+ \sum_{j=0}^{n-3}x_j x_{j+2}+ \sum_{j=0}^{n-3}x_j + x_1 + 1
\phi(\exp^{\nabla}(t f^{\hat{p}}|_p)) |_{\hat p = p}= (\tilde \phi \circ e^{t H_{\bar{f}(\hat u)}})(P_{\hat p,p}\hat u) |_{\hat p = p}& = (\exp^{\cdot}(t H_{\bar{f}}) \rhd \tilde \phi)(P_{\hat p,p}\hat u) = (\exp^{\cdot}(t f) \rhd \phi)(p)
\begin{array}{rclrcl}B^{(1)\prime}= -B^{(2)}\, , B^{(2)\prime}= -B^{(1)}\, , & g_{\mu\nu}^{\prime}= \left( \ell^{2}+ke^{-2\phi}\right)^{\frac{1}{2}}g_{\mu\nu}\, ,\hspace{1cm}C^{\prime}= -C\, .\end{array}
R^{( 1 )}= R_{1 2}\, \mathrm{o r}\, R_{2 1}^{- 1}\, , \quad R^{( 2 )}= R_{2 1}\, , \quad R^{( 3 )}= R_{1 2}\, , \quad R^{( 4 )}= R_{1 2}^{- 1}\, \mathrm{o r}\, R_{2 1}\, .
Q^{1}= \left( \begin{array}{c c c}{{0}}&{{0}}&{{1}}\\{{1}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}\\ \end{array}\right) , Q^{2}= \left( \begin{array}{c c c}{{0}}&{{0}}&{{1}}\\{{j}}&{{0}}&{{0}}\\{{0}}&{{j^{2}}}&{{0}}\\ \end{array}\right) , Q^{3}= \left( \begin{array}{c c c}{{0}}&{{0}}&{{1}}\\{{j^{2}}}&{{0}}&{{0}}\\{{0}}&{{j}}&{{0}}\\ \end{array}\right)
\forall t\in [p,1]:\nabla_p s_{f}(t)=s_f(p)+\int_p^tf(s)-f(s-p) ds=\nabla_p s_{g}(t) \Longleftrightarrow &f(t)-f(t-p)=g(t)-g(t-p) ~\lambda-a.s. \mbox{for all}t\in [p,1],~~ s_f(p)=s_g(p).
\|z^{-2}k^{-1}b(k) \|_{L^2(-\delta,\delta)}= \| z^{-1}( z^{-1}k^{-1}b(k)) \|_{L^2(-\delta,\delta)}\leq \|z^{-1}\|_{L^{1/2}(-\delta,\delta)}^{1/5}\| z^{-1}k^{-1}b(k) \|_{L^8}^{4/5}(-\delta,\delta)\leq c.
\ll \frac{1}{\pi_N(x)^{2r}L^r}\sum_{a = 0}^{2r - 1}\pi_N(x)^a (\log \log x)^{2r - a}\pi_N(x)^{r-a}L^{2r}\sum_{a_2 = 0}^{[a/2]}\left(\frac{\pi_N(x)}{L}\right)^{a_2}&+ \frac{x^{E_3(r) \pi_N(x)}8^{\nu(N)}}{k N}&\ll \frac{L^{1/2}\log \log x}{\pi_N(x)^{1/2}}+ \frac{x^{E_3(r) \pi_N(x)}8^{\nu(N)}}{k N}.
\mathrm{H}(\mathrm{x}) -\alpha \int_\Omega\nabla\mathbb{G}^{(0)}(\mathrm{x},\mathrm{y})\cdot\nabla \mathrm{H}(\mathrm{y}) d\mathrm{y}= \mathrm{H}^{\textbf{in}}(\mathrm{x}) + \alpha \int_\Omega\nabla\Big(\mathbb{G}^{(\mathrm{k})}-\mathbb{G}^{(0)}\Big)(\mathrm{x},\mathrm{y})\cdot\nabla \mathrm{H}(\mathrm{y}) d\mathrm{y}.
\mathfrak{A}(n;q_1,q_2,q_2')=q_1\sideset{}{^\star}\sum_{b\bmod{q_1}}\;\sideset{}{^\dagger}\sum_{c,d\bmod{q_1}}e_{q_1}(-\ell \bar{b}cd\bar{q}_2^2q_2'^3\overline{(cdnq_2'^2-\ell q_2)}+b+c+d),
X(k_{1}+\frac{N}{2},k_{2})=S_{00}(k_{1},k_{2})+S_{01}(k_{1},k_{2})W_{N}^{k_{2}}-S_{10}(k_{1},k_{2})W_{N}^{k_{1}}-S_{11}(k_{1},k_{2})W_{N}^{k_{1}+k_{2}}
\Gamma_{\nu \lambda}^{\mu}( x ) \ = \ \frac 1 2 \g^{\mu \sigma}( x ) \left( \frac{\partial g_{\sigma \lambda}( x )}{\partial x^{\nu}}\ + \ \frac{\partial g_{\nu \sigma}( x )}{\partial x^{\lambda}}\ - \ \frac{\partial g_{\nu \lambda}( x )}{\partial x^{\sigma}}\right)
(a;q)_n :=\prod_{k=0}^{n-1}(1-aq^{k}), (a;q)_{\infty}:=\prod_{k= 0}^\infty (1-aq^{k}), (a_1,a_2,\cdots,a_m;q)_n :=(a_1;q)_n(a_2;q)_n\cdots(a_m;q)_n, (a_1,a_2,\cdots,a_m;q)_\infty :=(a_1;q)_\infty(a_2;q)_\infty\cdots(a_m;q)_\infty.
k_1 \left(I^{(\alpha_j)}B_jV^{\frac{1}{2}}, V^{1}\right) + \sum\limits_{n=2}^{N}k_n \left(I^{(\alpha_j)}B_jV^{n-\frac{1}{2}}, V^{n-\frac{1}{2}}\right) \geq 0.
<{\cal O}_{1}\cdots{\cal O}_{k}\, \mathrm{e x p}\left\{\sum_{k = 0}^{4}\sum_{p = 4 - k}^{4 n}\sum_{\beta_{k}= 1}^{b^{k}({\cal M})}\sum_{\alpha_{p}= 1}^{b^{p}({\cal N})}s_{\alpha_{p}, \beta_{k}}^{p , k}{\cal D}_{p , k}^{\alpha_{p}, \beta_{k}}\right\}> .
\frac{N!}{\prod_{h = 1}^{g}(N_h)!}\, Z_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\mathsf{A}}= N^{(\frac{\beta}{2})N + e}\exp\Big(\sum_{k =-2}^K N^{-k}\, F^{\{k\}}_{\frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}+ o(N^{-K})\Big)\,.
P_n(t)=\exp\left\{-\lambda\left(1-{\rm e}^{-\mu}\right)t\right\}\, \bigg[1+{\bf 1}_{\{n\geq 1\}}\sum_{k=0}^n \left(\lambda{\rm e}^{-\mu}t\right)^k \sum_{j=k}^n S_2(j,k)\,{\mu^{j}\over j!}\bigg], n=0,1,\ldots.
M_{\psi_\alpha-\xi}(s)+R_{\psi_\alpha}(s)=O(|t|^{1-\sigma-\alpha+\varepsilon})+O(|t|^{1-\sigma+(\alpha-\frac{1}{2})l+\varepsilon})=O(|t|^{\frac{1}{2}-\sigma+\varepsilon}),
h_{ij}=0 \i,j\in \{1,2,\cdots,n \}\ \i\neq j, &h_{nn}=\frac{n(n-1)}{r}h_{11}= \frac{n(n-1)}{r}h_{22}= \cdots =\frac{n(n-1)}{r}h_{n-1n-1}.
\sum_{m \in M}(-1)^{\sigma_{M \setminus \{m\}}(m)}\eta_{M \setminus \{m\}}= \sum_{m \in M}(-1)^{\sigma_{M \setminus \{m\}}(m)}\sum_{p}\eta_{M \setminus \{m\}, p}= \sum_{p}\sum_{m \in M}(-1)^{\sigma_{M \setminus \{m\}}(m)}\eta_{M \setminus \{m\}, p}= 0.
\begin{array}{c c c}{{\Psi_{I}}}&{{=}}&{{( w_{2})^{\frac{1}{2}}\Psi ~ ~ ;}}\\{{F_{I}^{\mu \nu}}}&{{=}}&{{( w_{3})^{\frac{1}{2}}F^{\mu \nu}~ ~ ;}}\\ \end{array}\quad \begin{array}{c c c}{{m_{I}}}&{{=}}&{{\frac{w_{0}}{w_{2}}m}}\\{{e_{I}}}&{{=}}&{{\frac{w_{1}}{w_{2}w_{3}^{\frac{1}{2}}}e}}\\ \end{array}
{\cal L}_{\psi \bar{\psi}H}= - \bar{\psi}[ i \gamma^{\mu}\sigma^{\nu \lambda}\left\{\frac{1}{M_{P}e^{k r_{c}\pi}}H_{\mu \nu \lambda}^{0}+ \frac{1}{\Lambda_{\pi}}\frac{2 J_{0}( x_{n})}{\pi x_{n}}\sum_{n = 1}^{\infty}H_{\mu \nu \lambda}^{n}\right\}] \psi
\alpha(D_{\Gamma})(e_\lambda) = D_{\Gamma}(\partial^N(e_\lambda)) - \partial^{N\otimes_B J}(D_{\Gamma}(e_\lambda)) = D_\Gamma\left(\sum_{\mu< \lambda}e_{\mu}b_{\mu\lambda}\right) - \partial^{N\otimes_B J}(\gamma_\lambda) & = \sum_{\mu<\lambda}\left(\gamma_{\mu}b_{\mu\lambda}+e_{\mu}\otimes \delta(b_{\mu\lambda})\right) - \partial^{N\otimes_B J}(\gamma_\lambda).
|\mathfrak{K}_2| \leq 2s^{2- n}\frac{\sup_{b= s}|\nabla u|^2}{D(s)}\int_{b(x)= s}\Big| |\nabla b|- |\nabla b|^{-1}\Big|dx &\leq 2s^{2- n}\cdot \frac{\Big(\sup_{b\leq 2^{4n}r}|\nabla u|^2\Big)}{D(s)}\Big| |\nabla b|- |\nabla b|^{-1}\Big|dx &\leq C(n, V_M, \gamma)s^{-n}\int_{b= s}\Big| |\nabla b|- |\nabla b|^{-1}\Big|
< q | n > = \left( \frac{m \omega}{\pi \hbar}\right)^{\frac{1}{4}}\, \frac{1}{\sqrt{2^{n}n !}}\, e^{- \frac{m \omega}{2 \hbar}q^{2}}\, H_{n}\left( q \sqrt{\frac{m \omega}{\hbar}}\right) \ \ \ , \ \ \n = 0 , 1 , 2 , \dots \ \ \ ,
(3I_{2}^{\prime})^{3}\frac{1+c_{3}cos3\theta+c_{6}cos^{2}3\theta}{1+c_{3}+c_{6}}=(\frac{\sigma_{eq}-\gamma_{1}I_{1}}{1-\gamma_{1}})^{6-l-m}(\frac{\sigma_{eq}-\gamma_{2}I_{1}}{1-\gamma_{2}})^{l}\sigma_{eq}^{m}
{\cal U}_{\Delta}^{\dagger}\vec{\nabla}\delta \vec{B}^{3}\,{\cal U}_{\Delta}= \frac{2 \pi}{3 g}\epsilon_{i j k}\epsilon_{a b c}c_{i}^{a}c_{j}^{b}c_{k}^{c}\delta^{( 3 )}( \vec{\phi}) \ .
\vert\nu_A\rangle={1\over\sqrt{1+\vert\nu_A\vert^2}}\left(\begin{array}{c}-\nu_A 1\end{array}\right)\quad,\quad\langle \nu_A\vert={1\over\sqrt{1+\vert\nu_A\vert^2}}\Bigl(-\nu_A^* \;\quad\; 1\Bigr)\;.
\begin{array}{l}A((1 + \sqrt{1+4a}) h_1+(1 - \sqrt{1+4a}) h_2)A^{-1}= (1 + \sqrt{1+4a}) h_1 +(1 -\sqrt{1+4a}) h_2 -\frac{2d\delta}{\gamma}x_1 -\frac{2e\zeta}{\epsilon}x_2. \end{array}
\alpha_{i,n}= 0\mbox{for}i<0,\n<1\mbox{or}i\geq n, \alpha_{0,1}= 1, \alpha_{i,n+1}= \frac1{{\Delta}n}\left(\left({\partial \over\partial x}Q\right){}\left({\partial \over\partial x}\alpha_{i,n}\right)+\left({\partial \over\partial x}Q\right)\alpha_{i-1,n}+4n\alpha_{i,n}\right){}\mbox{in the other cases}
T_{w_{y}}^{-1}= T_{w_{x}}^{-1}(T_{\sigma_{x}(i)}^{-1}\cdots T_{\sigma_{x}(i)+d_{i}^{+}(x)-1}^{-1})(T_{\sigma_{y}(i)-d_{i}^{-}(y)}\cdots T_{\sigma_{y}(i)-1}).
( \cos(\rho k) -1 +i \sin(\rho k) )( \cos(\sigma k) -1 -i \sin(\sigma k) ) &= ( \cos(\rho k) -1)( \cos(\sigma k) -1) + \sin(\rho k)\sin(\sigma k) &+ i (\cos(\sigma k) -1) \sin(\rho k)- i (\cos(\rho k) -1) \sin(\sigma k).
u\left(t,x\right)=\frac{e^{i\frac{x^{2}}{4t}}}{\sqrt{-2it}}\exp\left(-\frac{i}{2}\left|W_{+\infty}\left(-\frac{x}{2t}\right)\right|^{2}\log t\right) W_{+\infty}\left(-\frac{x}{2t}\right) + \mathcal{O}\left( \varepsilon t^{-\frac{1}{2}-\delta}\right), t\geq1.
a_{n,p,10}= \sum_{s=1}^n \binom{n-1}{s}r_{s,p,10}= \sum_{s=1}^n \binom{n-1}{s-1}\binom{p+s-2}{s-1}=\sum_{s=0}^{n-1}\binom{n-1}{s}\binom{p+s-1}{s}.
\begin{aligned}\|\rho^\varepsilon-\bar{\eta}\|_{C(\overline{Q_{\varepsilon,T}})}\leq c_T\Bigl(\|\rho_0^\varepsilon-\bar{\eta}_0\|_{C(\overline{\Omega_\varepsilon(0)})}+\|f^\varepsilon-\bar{f}\|_{C(Q_{\varepsilon,T})}\Bigr) +\varepsilon c_T\Bigl(\|\eta_0\|_{C(\Gamma(0))}+\|\eta\|_{C^{2,1}(S_T)}\Bigr) \end{aligned}
\begin{aligned}{q^{r(r+1)/2}\over (r+1,q - \varepsilon)}\prod_{i=1}^r(q^{i+1}-\varepsilon^{i+1}) \geq{q^{r(r+1)/2}\over q(q+1)}q^{(r+1)(r+2)/2}\prod_{i=1}^r(1-(\varepsilon q)^{-i-1}) & \geq{q^{(r+1)^2}\over q(q+1)}\prod_{i=1}^\infty \left(1-\frac{1}{q^{i+1}}\right),\end{aligned}
\frac{d}{dt}\left\langle D\right\rangle_{\scriptstyle \! \Psi_{\rm \!{\scriptscriptstyle (gs)}}}\stackrel{(a \rightarrow 0)}{=}E_{{\rm (gs)}}\,\left\{1 + O\left( \left[ \kappa a \right]^{2}\ln [ \kappa a ] \right) \right\}\; .
\int_\Omega\left({\mathcal D}_{\psi}f(w) + ( \frac{\partial p_0}{\partial x}+ \frac{\partial q_0}{\partial y})f + ( \frac{\partial p_1}{\partial x}+ \frac{\partial q_1}{\partial y}){i}f\right)dxdy =\int_{\partial \Omega}f d\sigma_{\psi}(w),
k\sum_{j=1}^k \frac{j}{k-j+1}=k\sum_{j=1}^k \frac{k-j+1}{j}&=k^2\sum_{j=1}^k \frac{1}{j}-k\sum_{j=1}^k 1 +k \sum_{j=1}^k \frac{1}{j}&\ge k^2 (\log (k)+\gamma+O(k^{-1})) -k^2 = k^2(\log(k) + \gamma -1 +O(\log k/k)).
[L_n+z,L_m+z']=(\{m\}-\{n\})L_{n+m}+c \frac{1}{q^{n-2}}\frac{1+q^{2}}{1+q^{n}}\frac{\{n+1\}\{n\}\{n-1\}}{\{12\}}\delta_{n+m,0}, \forall n>0 &[L_n+z,G_m+z']=(\{m+1\}-\{n\})G_{n+m}&[G_n+z,G_m+z']=0.
\mathrm{usp}_S(X,[af])=\limsup_n \frac{\lvert X^{n}[af]\rvert_S}{n}=\limsup_n \frac{\lvert a\rvert_0 \lvert X^{n}[f]\rvert_S}{n}=\lvert a\rvert_0\mathrm{usp}_S(X,[f]).
{1\over g_{1}^2}+{1\over g_{2}^2}={1\over 4\pi g_{s}};\qquad{1\over g_{1}^2}-{1\over g_{2}^2}={1\over 4\pi^2 g_{s}}\left(\frac{1}{2\pi\alpha'}\int_{S^2}B_{(2)}-\pi \right) .
\mathcal{C}\big ( \mathcal{U}_{(i,j)}\big ) \cong \begin{cases}C^\infty \big ( \mathcal{U}_{(0,0)}\big ) \mathcal{S}\big ( \mathcal{U}_{(i,j)}\big ) \mathcal{A}\big ( \mathcal{U}_{(i,j)}\big ) \end{cases}\, ,
P^m_l (x) = \frac{(l+m)!}{2^m\, l!}\, (1-x^2)^{\frac m2}\, P^{(m,\, m)}_{l-m}(x), \qquad \mathrm{and}\qquad P^m_l (x) = (1-x^2)^{\frac m2}\,\frac{d^m}{dx^m}P_l (x).
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}}\ ,