harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}\sigma^{\kappa}(x,y \mid t) \doteq \frac{\kappa(1-\kappa) \eta^0_{(1)}(x\mid s)G(\pi^*)_{x,y}+\kappa(1-\kappa) \pi^*_xG(M^{0}(s))_{x,y}+ \kappa^2 \pi^{*}_x G(\pi^*)_{x,y}}{2\kappa(1-\kappa)+\kappa^2}.\end{aligned}
A x^2 y^2 + B x y^2 + C x^2 y + D x y \equiv (-P_1 y -P_0)(A x^2 + Bx) + C x^2 y + D x y &\equiv (-P_1 A y - P_0 A + C y)(-P_1 x - P_0) + (D - P_1 B) x y - P_0 B x &= (P_1 A P_1 + D - P_1 B - C P_1) x y + (P_1 A P_0 - C P_0) y + (P_0 A P_1 - P_0 B) x + P_0 A P_0 & = \phi(Y) ,
\begin{bmatrix}0_{n-n_D}0_{(n-n_D) \times n_D}0_{n_D\times (n-n_D)}I_{n_D}\end{bmatrix}\begin{bmatrix}I_{n-n_D}+X_1 X_2 X_3 I_{n_D}+X_4\end{bmatrix}=0,
S(\theta,\lambda,t)=\frac{\rho_{i}}{\gamma}G_{s}\otimes_{i}I+\frac{\rho_{w}}{\gamma}G_{s}\otimes_{o}S+S^{E}-\frac{\rho_{i}}{\gamma}\overline{G_{s}\otimes_{i}I}-\frac{\rho_{w}}{\gamma}\overline{G_{o}\otimes_{o}S},
\zeta_{p,0}(s,c)=\frac{\pi}{\sqrt{c}}\,\frac{1}{s-1}+\frac{\pi}{\sqrt{c}}\left(2\gamma-\log\left(\frac{c}{4}\right)+8\,\sum_{n=1}^{\infty}\frac{1}{2n-1}\cdot\frac{1}{\sigma\left(\sqrt{c}\lambda_{n}^{\prime}\right)e^{(2n-1)\pi\sqrt{c}}-1}\right)+O(s-1).
f\left(x_i^{c_{+}}\right) =-\frac{1}{3}\left[\alpha_i(\mu_i+K_i)-3\beta_i\right] -\frac{1}{27}\left[(\mu_i+K_i)+3\alpha_i\right]^2\left[2(\mu_i+K_i)-\sqrt{\Delta_i}\right]-\frac{1}{3}\alpha_i\sqrt{\Delta_i}< 0.
2\partial_{\bar z}\mbox{\bf Q}+A_1\mbox{\bf Q}-\mbox{\bf Q}A_2=0\quad\mbox{in}\,\,\Omega\setminus \mathcal X, \mbox{\bf Q}\vert_{\tilde\Gamma}=I,\partial_{\vec\nu}\mbox{\bf Q}\vert_{\tilde\Gamma}=0,
=\frac{\partial}{\partial t}\frac{f\left( \frac{r}{a(t)}\right)}{a(t)^{2}}+\left( \frac{\partial}{\partial r}\frac{f\left( \frac{r}{a(t)}\right)}{a(t)^{2}}\right) \frac{\dot{a}(t)r}{a(t)}+\frac{f\left( \frac{r}{a(t)}\right)}{a(t)^{2}}\frac{\dot{a}(t)}{a(t)}+\frac{f\left( \frac{r}{a(t)}\right)}{a(t)^{2}}\frac{\dot{a}(t)}{a(t)}
\begin{array}{lllll}\hat{f}(\xi) = \displaystyle \hat{u_*}+i\int_1^t\int e^{i(\eta^2-2\xi\eta)s}e^{-s}{\hat{f}(\xi-\eta,s)\hat{v_*}(\eta,s)}d\eta ds & \displaystyle +i\int_1^t\int_1^s\iint e^{2i\eta(\sigma-\xi)s}F(s,\xi,\sigma,\eta)dsd\eta d\sigma,\end{array}
V_{a_{1},a_{2},...a_{m}}^{c}=\bigoplus_{\{b_{j}\}}(V_{a_{1},a_{2}}^{b_{1}}\otimes V_{b_{1},a_{3}}^{b_{2}}\otimes V_{b_{2},a_{4}}^{b_{3}}...V_{b_{m-3},a_{m-1}}^{b_{m-2}}\otimes V_{b_{m-2},a_{m}}^{c})
F_{\nu}= \{- \frac{1}{2}\varepsilon_{\nu \gamma \rho \sigma}G^{\rho \sigma}C^{\gamma a}+ ({\cal D}_{\nu}C_{\gamma}^{a}-{\cal D}_{\gamma}C_{\nu}^{a}) B^{\gamma}+ B_{\nu}\partial^{\gamma}C_{\gamma}^{a}\}\mu_{a}.
\| h \|_{H^1_A}= \sup_{\substack{f \in (H^1_A)^* \| f \|_{(H^1_A)^*}\leq 1}}|f(h)| \asymp \sup_{\substack{f \in (H^1_B)^* \| f \|_{(H^1_B)^*}\leq 1}}|f(h)| = \| h \|_{H^1_B}
\begin{array}{rcl}U^{s,x}= u(s,x) + \int_s^{\cdot}\mathfrak{a}(u)(r,X_r)dV_r + M[u]^{s,x}&= u(s,x) - \int_s^{\cdot}f\left(r,X_r,u(r,X_r),\sqrt{\mathfrak{G}(u,u)}(r,X_r)\right)dV_r + M[u]^{s,x}&= u(s,x) - \int_s^{\cdot}f\left(r,X_r,U^{s,x},\sqrt{\mathfrak{G}(u,u)}(r,X_r)\right)dV_r + M[u]^{s,x}, \end{array}
\lim_{m_{\pi}^2\rightarrow 0}m_{\pi}^2\int d^3 \vec{x}\,\omega_{\pi}^{(1)}=-\lambda\int d^3 \vec{x}\, (\varphi_{\pi}^3+\varphi_{\pi}\varphi_{\sigma}^2+2v\varphi_{\pi}\varphi_{\sigma})\ne 0\quad,
I_{k}\left( \mu , d , \lambda \right) : = - \frac{\left( 2 \mu \right)^{\frac{d}{2}- 1}}{\left( 2 k \lambda \right)^{\frac{d}{2}}}\Gamma \left( \frac{d}{2}+ k \right) W_{- k , \frac{d - 1}{2}}\left( 4 \mu k \lambda \right) .
J_{\lambda,\mu}(u,0)=\frac{1}{p}\int_Q\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}\,dx\,dy -\frac{\lambda}{q}\int_\Omega |u|^{q}dx=-\frac{p-q}{pq}\|u\|_{X_0}^p<0.
F(D^{2}u,Du,u) =\int_\Omega \tilde{g}\left( \det(u_{ij})\right) dx+\int_\Omega \Tilde{G}(x)\det(u_{ij})dx & +\int_\Omega\int_\Omega W(x,y)\det(u_{ij}(y))\det(u_{ij}(x))dydx,
\displaystyle{\sum_{k}}\frac{(-1)^{k-l_{3}}}{(k-l_{1})!(k-l_{2})!(k-l_{3})!(l_{1}+l_{2}-k)!(l_{2}+l_{3}-k)!(l_{3}+l_{1}-k)!}& =(-1)^{l_{3}}{\sum_{k}}\frac{(-1)^{k}}{R(l_{1},l_{2},l_{3};k)}=(-1)^{l_{3}}\Sigma_{0}(l_{1},l_{2},l_{3})
P_{{\cal S}}( A ) = \sum_{j = 1 , 2}P_{{\cal S}_{j}/{\cal S}}P_{{\cal S}_{j}}( A ) + 2 \sqrt{P_{{\cal S}_{1}/{\cal S}}P_{{\cal S}_{1}}( A ) P_{{\cal S}_{2}/{\cal S}}P_{{\cal S}_{2}}( A )}\lambda ( A ;{\cal S},{\cal S}_{j}) .
\sum_{n=1}^{u}\frac{f_{n}}{(\alpha n+\beta)^{k}}z^{n}=[w^{u}](\sum_{j=1}^{u+u_{0}}\{\begin{matrix}k+2\\j\end{matrix}\}_{(\alpha,\beta)^{*}}\frac{(wz)^{j}F^{(j)}(wz)}{1-w}).
- [ Z_{\mu}, Z_{\nu}]^{2}={\cal S}\equiv 2 b^{2}r^{2}+ 2 z^{2}\, ( a^{2}+ b^{2}+ r^{2}) + ( a^{2}+ r^{2}) \, y^{2}+ b^{2}x^{2}- 4 a b x y .
\zeta + a_{0}\frac{1}{\zeta}x_{1}^{4 - n}z^{8 - n}\to \zeta + a_{0}\frac{1}{\zeta}\prod_{i = 1}^{N_{s}}( x_{1}- m_{i}^{4}) \prod_{j = 1}^{N_{f}}( z - m_{j}^{2}) .
\mathrm{R e s}_{z^{\prime}= z}^{\Psi}\partial^{\Psi}\! : \! \psi ( z^{\prime}) \! : \, \bar{\partial}^{\Psi}\! : \! \phi ( z ) \! : = - : \! ( \phi_{; \mu \eta}- i C_{\eta \nu \mu}\phi_{; \nu}) \psi^{; \eta}\bar{\partial}^{\Psi}\! X^{\mu}\! : + O ( \epsilon^{5})
\langle(\nabla_U S)A,U\rangle{}=\langle\nabla_U SA-S\nabla_UA,U\rangle=\langle\nabla_U(\gamma A),U\rangle-\langle \nabla_UA,SU\rangle &{}=(U\gamma)\langle A,U\rangle+\gamma\langle\nabla_UA,U\rangle-\alpha\langle\nabla_UA,U\rangle=(\alpha-\gamma)\langle\nabla_UU,A\rangle.
I=\int_{R^2}d^2z\frac{(y_2\bar y_2)^{\frac{1}{4}}}{\bar y_2}(y_1\bar y_1)^{-\frac{1}{4}}e^{-|x_\perp|(\sqrt{y_1\bar y_1}+\sqrt{y_2\bar y_2})}
\mathcal{T}= \left \{(m',n') \in \mathbb{Z}^2_{0,n, >}: |\eta| \le \frac{3}{2}~, ~\Big | (2\zeta-1) \eta-(2\zeta-1) g(\zeta) \Big | \le k (n) ~,~ |\eta-\zeta| \le k (n) \right \}\ ,
M_n =\sum_{j=0}^n (n-j+4) \frac{4}{\pi}2^{2j}\frac{1}{(2n-2j)!}\left(\frac{\pi}{2}\right)^{2n-2j}< &<2^{2n}\Bigl(\frac{16}{\pi}\cosh(\pi/4)+\frac12\sinh(\pi/4)\Bigr)<7.2 \cdot 2^{2n}, N_n =\sum_{j=0}^n \frac{4}{\pi}2^{2j}\frac{1}{(2n-2j)!}\left(\frac{\pi}{2}\right)^{2n-2j}< 2^{2n}\frac{4}{\pi}\cosh(\pi/4)<1.7\cdot 2^{2n}.
\begin{cases}\|a\|_{C_{T}L_{x}^{2}}+\|\nabla a\|_{L_{T}^{2}L_{x}^{2}}\leq C(c,\gamma)\|w_{0}\|_{L^{2}}, \|a\|_{C_{T}L_{x}^{p}\cap L_{T}^{\frac{4p}{3}}L_{x}^{2p}}\leq C(p,c,\gamma)\|w_{0}\|_{L^{p}}.\end{cases}
dX^{(n)}_i(t)=\sqrt{2((X_i^{(n)})^2(t)+1)}d\beta^{(n)}_i(t)+\left[\left(2-2n-2\Re(s)\right)X_i^{(n)}(t)+2\Im(s)\right]dt+\frac{1}{2}dK_i^{(n),-}(t)-\frac{1}{2}dK_i^{(n),+}(t),
\{1\},\{y^{2s+1}\}, \quad\{y^n\},\{y^{6s+3}\}, \{y^{2j},y^{-2j}\}1\leq j<2s+1=n/2, \{y^{2j+1},y^{n-2j-1}\},\enspace \{y^{n+2j+1},y^{2n-2j-1}\}1\leq 2j+1< l=n/2, \{x,xy^4,\dots ,xy^{2n-4}\}, \{xy,xy^5,\dots ,xy^{2n-3}\}, \{xy^2,xy^6,\dots ,xy^{2n-2}\}, \{x^3,xy^7,\dots ,xy^{2n-1}\}
X_1(t) = X_1(0) + \sqrt{2}W(t) - \beta t + \int_0^t\left(-X_1(s) + X_2(s)\right)ds - U_1(t), X_2(t) = X_2(0) + U_1(t) + \int_0^t(-X_2(s) + X_3(s))ds, X_i(t) = X_i(0) + \int_0^t(-X_i(s) + X_{i+1}(s))ds, 3 \le i \le k - 1, X_k(t) = X_k(0) + \int_0^t-X_k(s)ds, X_i(t) = 0, i \ge k + 1,
(T_{\alpha_{\flat}(u)}{\rm grad}_E\, V_{\alpha})f =g(T_{\alpha_{\flat}(u)}{\rm grad}_E\,V_{\alpha},{\rm grad}\,f) &=g({\rm grad}_E\,V_{\alpha},T_{\alpha_{\flat}(u)}{\rm grad}\,f) &=(T_{\alpha_{\flat}(u)}{\rm grad}\,f)^hV_{\alpha}.
\mathbb{P}\left( \mathbf{U}^t=\mathbf{m}\right) = \sum_{\mathbf{n}\in \mathbb{N}_0^S}\mathbb{P}\left( \mathbf{U}^t = \mathbf{m}| \mathbf{N}^t = \mathbf{n}\right) \mathbb{P}\left( \mathbf{N}^t = \mathbf{n}\right).
\sum\limits_{m=1}^{n}{\sum\limits_{k=1}^{m}{Z_{k}^{2}}}=\sum\limits_{m=1}^{n}{Z_{m+2}^{2}}-\sum\limits_{m=1}^{n}{Z_{m-1}^{2}}-\sum\limits_{m=1}^{n}{Z_{m-3}^{2}}+n\left( T-{{a}^{2}}\right)
\mathcal{M}(z_-,z_+,[0,1],\rho,H^{(k)},J) =\Bigg\{(\tau,u)\in [0,1]\times C^{\infty}(\mathbb{R}\times S^1\rightarrow M) \ \Bigg| \ \begin{matrix}u \textrm{\satisfies \}(\textrm{C}) \lim_{s\rightarrow \pm\infty}u(s,t)=x_{\pm}(t) [u_-\sharp u,x_-]=R^{-\frac{1}{k}\tau}(z_+) \end{matrix}\Bigg\}
\mu_1(A) = \frac{\sqrt{\lambda_1^A\lambda_N^A}}{\frac{1}{2}\left(\lambda_1^A+\lambda_N^A\right)}= \frac{2\sqrt{\kappa_A}}{\kappa_A+1}= \frac{\mathrm{GeometricMean(\lambda_1^A,\lambda_N^A)}}{\mathrm{ArithmeticMean(\lambda_1^A,\lambda_N^A)}},
\varphi_0 u =-\varphi_0 n\times(n\times u)+\varphi_0 n(n\cdot u) &= (n_2u_3-n_3u_2) Z_1 +(n_3u_1-n_1u_3)Z_2+(n_1u_2-n_2u_1) Z_3+\frac{u\cdot n}{d}Z_4.
\widehat{F}_{\mu \nu}= F_{\mu \nu}+ \frac{1}{4}\theta^{\rho \sigma}\bigg ( 2 \left\{F_{\mu \rho}, F_{\nu \sigma}\right\}- \left\{A_{\rho}, D_{\sigma}F_{\mu \nu}+ \partial_{\sigma}F_{\mu \nu}\right\}\bigg ) +{\cal O}\left( \theta^{2}\right) .
K_{\rm np}= d \left(\frac{S_R}{4\pi}\right)^{p/2}e^{-b\sqrt{S_R}}\left[1+{\cal O}\left(\frac{1}{\pi S_R},\frac{1}{\pi T_R}\right)\right]+{\cal O}(e^{-2\pi S},e^{-2\pi T}),
\!\!\!\!\!\!\!\!\!\!\!\!\!U:=\bigoplus_{k\in\mathbb{N}}U_k:\bigoplus_{k\in\mathbb{N}}\left(({\mathbb C}^d)^{\hat{\otimes}k}, \ \langle \cdot , \cdot\rangle_k\right)\rightarrow\bigoplus_{k\in\mathbb{N}}\left(\mathcal{P}_k, \ \langle \cdot , \cdot\rangle_{\mathcal{P}_k}\right)=\left(\mathcal{P},\ \langle \cdot ,\cdot \rangle\right)
R_{M N , M^{\prime}N^{\prime}}= ( \gamma_{+})_{M M^{\prime}}\delta_{N N^{\prime}}+ ( \gamma_{-})_{M M^{\prime}}\gamma_{N N^{\prime}}= \delta_{M M^{\prime}}( \gamma_{+})_{N N^{\prime}}+ ( \gamma )_{M M^{\prime}}( \gamma_{-})_{N N^{\prime}}
\mathrm{( i )}\phantom{m m}c^{e}\; + \; c^{i}\; = \; 2 6 \ , \quad \ \bar{c}^{e}\; + \; \bar{c}^{i}\; = \; 2 6 \ , \phantom{Z e i c h n u n g Z e i c h n u n g Z e i c h n}
{\cal L}^{\prime}( x ) = \overline{{{\psi}}}^{1}( x ) ( i \partial \! \! \! / - m ) \psi^{1}( x ) + \overline{{{\psi}}}^{2}( x ) ( i \partial \! \! \! / ) \psi^{2}( x ) - \overline{{{\psi}}}^{3}( x ) ( i \partial \! \! \! / ) \psi^{3}( x ) .
C_n^{\uparrow}= \frac{1}{n}\log_2\sum_{j=1}^{2^n}\exp_2\left(-\sum_{i=1}^{2^n}\begin{bmatrix}P_{n|s_0}^{-1}\end{bmatrix}_{j,i}H(Y^n|X^n = x_i^n)\right),
\mathcal{E}(t) := \frac{1}{2}\sum_{n \in \mathbb{Z}}\left[\mathcal{Q}_n^2(t) + \mathcal{U}_n^2(t) + \epsilon^2 p W^{p-1}(\epsilon(n-t),\epsilon^3t)) \mathcal{U}^2_{n}(t)\right].
h^D(p_C,p_W,\hat{\delta},\hat{f}) - l = 0 g_1^D(p_C,\hat{r}) \leq 0 p_W \leq \overline{p}_W g_2^D(\hat{\delta}, \hat{f}) \leq \overline{f}g_3^D(\hat{r}) \leq \overline{R}
\mathcal{G}^k_f(\vec{a};\vec{b})=f(\langle \vec{a},\vec{b}\rangle)\int \limits_{|a_n|}^{|b_n|}\int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}f\circ \mathbf{e}\bigg((i)^k \frac{\sqrt[k]{\sum \limits_{j=1}^{n}x^k_j}}{||\vec{a}||^{k+1}+||\vec{b}||^{k+1}}\bigg)dx_1dx_2\cdots dx_n
\left(\begin{array}{r|c|c}A B \vartheta^T \hline -B A \vartheta^T \hline \vartheta \vartheta 1 \end{array}\right),\begin{array}{l}A^TA-B^TB=I_n, B^TA+A^TB=O_n,\end{array}A, B\in GL(n;\mathbb{R}),
\frac{1}{\pi}=\frac{\sqrt{7}}{22^{3}}\sum_{k=0}^{\infty}s_{7A}(k)\frac{11895k+1286}{(-22^{3})^{k}},j_{7A}(\frac{7+\sqrt{-427}}{14})=-22^{3}+1=-(39\sqrt{7})^{2}=-10647
\Phi(x) := \mathrm{e}^{x}+\mathrm{e}^{-x}\ \mbox{if}\n=1, \Phi(x) := \int_{\mathbb{S}^{n-1}}\mathrm{e}^{x\cdot \omega}\, \mathrm{d}\sigma_\omega \,\mbox{if}\n\geqslant 2,
x_1 x_2+ \frac12 ({(1-x_1)^2+(1-x_2)^2}) =\frac 12+ \frac 12\left( 1-2x_1-2 x_2+2x_1 x_2+x_1^2+x_2^2 \right) &=\frac 12 + \frac{1}{2}(1-x_1-x_2)^2 & \geq \frac 12=2^{1-k}.
\binom{an+b}{r}^{p}\prod_{s=2}^{L}\binom{\alpha_{s}n+\beta_{s}}{r_{s}}^{p_{s}}=\sum_{i=0}^{rp+\sigma}A_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,i\right) \binom{n+rp+\sigma -i}{rp+\sigma}.
A=(\begin{matrix}w^{2}&w^{2}&0&0&0&0\\ 1&w^{2}&0&0&0&0\\ 1&1&w^{2}&w^{2}&0&0\\w&1&1&w^{2}&0&0\\ 0&w^{2}&w^{2}&w^{2}&0&w\\w^{2}&1&w^{2}&0&w^{2}&0\end{matrix})
\left( \begin{matrix}{{X^{I}}}\\{{F_{I}}}\\ \end{matrix}\right) \rightarrow \left( \begin{matrix}{{{\bf a}}}&{{0}}\\{{0}}&{{{{\bf a}^{t}}^{- 1}}}\\ \end{matrix}\right) \left( \begin{matrix}{{X^{I}}}\\{{F_{I}}}\\ \end{matrix}\right) \ ,
\begin{aligned}h_\lambda (s,t):= \exp\bigg\{-\lambda\Big( \int_s^t \| v(\tau)\|_{L^{\frac{dr}{r-1}}_x}^{2r}+ \int_s^t \| \nabla v(\tau)\|_{L^{\frac{dr}{2r-1}}_x}^{2r}+ \int_s^t \| \nabla v(\tau)\|_{L^{q}_x}^{2r}\Big)\Big\}\leq 1.\end{aligned}
\check{x}_{3 1 ; 2}= \tilde{x}_{3 1}+{\frac{4 i}{( 1 2 )}}\theta_{3 1}( \bar{\theta}_{2}^{+}- ( 2 1^{-}) \bar{\theta}_{1}^{+}) \quad \Rightarrow \quad \delta \check{x}_{3 1 ; 2}= 0 \; .
\Phi_{,1}^\tau({u^\ell}) = e^{\tau \mathcal{L}}\Big( u^\ell + \tau \mathcal{B}\left( F(u^\ell) \cdot \varphi_1\big(\tau \mathcal{A}\big)G(\overline{u^\ell})\right)\Big)= e^{\tau \mathcal{L}}\left( u^\ell + \tau \Psi^\tau_{, 1}(u^l)\right).
\pi^2/6 =[[0,2n^2-2n+1],[1,-n^4]]=[[2,2n^2-2n+3],[-1,-n^4]] =[[3/2,2n^2-2n+7],[1,-n^4]]=[[31/18,2n^2-2n+13],[-1,-n^4]] =[[115/72,2n^2-2n+21],[1,-n^4]]=[[3019/1800,2n^2-2n+31],[-1,-n^4]] =[[973/600,2n^2-2n+43],[1,-n^4]]\;,
\begin{aligned}\psi(Y^{\epsilon}(t)) =\psi(y^{\epsilon}_{0})+\int_0^t\mathcal{B}^{\epsilon}(s)\tilde{N}^{\epsilon^{-2}}(ds) &\quad+\int_0^t\langle\nabla\psi(Y^{\epsilon}(s-)),h^{\epsilon}(s)\rangle ds+\int_0^t\mathcal{A}^{\epsilon}(s)ds,\end{aligned}
Q_{TE_{mnp}}=\frac{Z_{0}lwh}{4R_{s}}\frac{k_{xy}^{2}k_{r}^{3}}{\zeta lh(k_{xy}^{4}+k_{x}^{2}k_{z}^{2})+\xi wh(k_{xy}^{4}+k_{y}^{2}k_{z}^{2})+lwk_{xy}^{2}k_{z}^{2}}
\frac{1-\sqrt{1-4\delta^2}}{2\delta}<\frac{1-\sqrt{1-4|\beta|^2}}{2|\beta|}\leq\frac{1-\sqrt{1-4|\beta|^2}}{2|\beta|\cos\theta}<|p|<\frac{1+\sqrt{1-4|\beta|^2}}{2|\beta|\cos\theta}<\frac{1+\sqrt{1-4\delta^2}}{2\delta}.
E [ \sigma ] = \int_{m}^{\infty}\frac{d t}{2 \pi}\frac{t}{\sqrt{t^{2}- m^{2}}}\left[ \ln g ( i t , 0 ) + \ln \left( g ( i t , 0 ) - \frac{1}{t}g^{\prime}( i t , 0 ) \right) \right]_{1}+ E_{\mathrm{F D}}^{( 1 )}+ E_{\mathrm{C T}}\, ,
f_n(w) := \frac1n \sum_{i=1}^n \log \bigg( w - \frac{x_i^{(n)}}n \bigg) -\frac1n \sum_{j=v_n+s_n-n}^{v_n}\log \left( w - \frac{j}n \right), \tilde{f}_n(z) := \frac1n \sum_{i=1}^n \log \bigg( z - \frac{x_i^{(n)}}n \bigg) -\frac1n \sum_{j=u_n+r_n-n+1}^{u_n-1}\log \left( z - \frac{j}n \right).
(N^{[0,k]}_{i})^{-1}(\vec a)=(a_{1},a_{1}^{-1}a_{2},\dotsc,a_{i-1}^{-1}a_{i},a_{i+1}^{-1}a_{i+2},\dotsc,a_{k-1}^{-1}a_{k},a_{k}^{-1}),
\footnotesize\left(\tilde{\alpha}_* \circ \chi_0 \right) \left(\bar{\varphi}'' \right)=\tilde{\alpha}_* \left( \chi_0 \left(\bar{\varphi}'' \right) \right)=\tilde{\alpha}_* \left([\varphi] \right)=\left(\alpha \circ \varphi, 0 \right)+\bar{B}_n.
\int_{G}d g \, \overline{{{\langle l_{1}, m_{1}| T_{g}| l_{1}, m_{1}^{\prime}\rangle}}}\langle l_{2}, m_{2}| T_{g}| l_{2}, m_{2}^{\prime}\rangle ={\frac{\delta^{l_{1}l_{2}}\delta_{m_{1}m_{2}}\delta_{m_{1}^{\prime}m_{2}^{\prime}}}{\mathrm{d i m}_{l_{1}}}},
Solv_{r+1}\, \equiv \,Solv \left( E_{r+1(r+1)}/{\cal H}_{r+1}\right )= Solv \left( SL(2,R)/ O(2) \right )\, \oplus \, Solv \left( \frac{GL(r)}{SO(r)}\right) \,\oplus \,{\bar{\cal W}}_{r+1}
\frac{(\sqrt{6}\pi\frac{1}{2})_{r-1}}{(n+1)^{r-\frac{1}{2}}\left(1-\frac{25}{24(n+1)}\right)^{r-\frac{1}{2}}}-\frac{(r-1)!}{(n+1)^{r}\left(1-\frac{24r-25}{24(n+1)}\right)^{r}}begin{align*}6pt] +\sum_{k=1}^{\infty}\left(\frac{k}{2}\right)_{r}\frac{6^{r}}{k\pi^{k}24^{\frac{k}{2}}\left(n+1\right)^{\frac{k}{2}+r}\left(1-\frac{25}{24(n+1)}\right)^{\frac{k}{2}+r}}.
u_\sigma(t)=\frac{\sigma}{(1-2\sigma^2)^{\frac{1}{2}}}\left(\cos\left((1-2\sigma^2)^\frac{1}{2}\frac{t}{\sigma}\right),\sin\left((1-2\sigma^2)^\frac{1}{2}\frac{t}{\sigma}\right),\frac{(1-2\sigma^2)^{\frac{1}{2}}}{\sigma}\sqrt{1-\frac{\sigma^2}{1-2\sigma^2}}\right)
\begin{array}{c}t_ie_jt_i^{-1}=q^{a_{ij}}e_j t_if_jt_i^{-1}=q^{-a_{ij}}f_j{}[e_i,f_j]=\delta_{ij}\frac{t_i-t_i^{-1}}{q-q^{-1}}\end{array}
\sum_{j = -N}^{N}\log \left| \tanh \left( \frac{\pi}{4d}(x - a(j)) \right) \right| & \leq \int_{a(-N-1)}^{a(N+1)}\log \left| \tanh \left( \frac{\pi}{4d}(x - z) \right) \right| b'(z) \, \d z + c' \log N + c'',
\int_{\varepsilon_0 < |x-y| < \varepsilon}\frac{d|\nu_1|(y)}{|x-y|^{m}}\le \sum_{j=0}^{k-1}\int_{5^j\varepsilon_0 \le |x-y| < 5^{j+1}\varepsilon}\frac{d|\nu_1|(y)}{|x-y|^{m}}&\le \sum_{j=0}^{k-1}(5^j\varepsilon_0)^{-m}|\nu_1|(B(x,5^{j+1}\varepsilon_0)) &\le M_{\mu}\nu_1(x) \sum_{j=0}^{k-1}(5^j\varepsilon_0)^{-m}\mu(B(x,5^{j+1}\varepsilon_0)).
S(\vec{\alpha},\vec{\nu}) =\sum_{\vec{z}(q)}\sum_{\vec{z}+q\vec{y}\in P\mathcal{B}}e(\vec{\alpha}\cdot(\vec{f}(\vec{z}+q\vec{y})-\vec{\nu})) &= \sum_{\vec{z}(q)}e_q(\vec{a}\cdot(\vec{f}(\vec{z})-\vec{\nu})) \sum_{\vec{z}+q\vec{y}\in P\mathcal{B}}e(\vec{\beta}\cdot(\vec{f}(\vec{z}+q\vec{y})-\vec{\nu})).
\mu^c(t) =\frac{1}{\pi 2^3}[51 \,{p}_{0}(t) - 2\cdot 11\;{p}_{2}(t)+{p}_{4}(t) ] =\frac{1}{\pi 2^3}\sqrt{\frac{2}{\pi}}[(-2 -36 t^2+38 t^4) e^{-t^2}+1+17 t^2-11 t^4+t^6]e^{-\frac{t^2}{2}}.
\int_{0}^{1}d s \int_{0}^{1}d t \int d k \frac{\exp \{i k ( s - t ) \}P ( \lambda k , p_{1}\ldots p_{n})}{( i \lambda k + p_{1}^{2}+ m^{2}) \ldots ( i \lambda k + p_{n}^{2}+ m^{2})}
\sum_{k=1}^{n-1}\frac{1}{\sqrt{kn-k^2}}\leq 2 \arctan\left( \frac{\sqrt{n-1}}{1}\right)+\frac{1}{\sqrt{n-1}}+ \frac{2}{\sqrt{n}}\leq \pi + \frac{1}{\sqrt{n-1}}+ \frac{2}{\sqrt{n}}.
\mathcal{L}_{F}^{\alpha,-1}\left[\frac{S_{F}^{\alpha}(s)^{\zeta-\mu}}{S_{F}^{\alpha}(s)^{\zeta}+S_{F}^{\alpha}(a)}\right] &= S_{F}^{\alpha}(x)^{\mu-1}E_{F,\zeta,\mu}^{\alpha}(-S_{F}^{\alpha}(a)S_{F}^{\alpha}(x)^{\zeta}).
\langle G \vert J_{GB}^i \vert G \rangle = - \epsilon^{ilm}\int d^3 x\langle G \vert x^l (i \partial_{(x)}^m [\delta^3 ({\bf x-y})]) \vert G \rangle = - \epsilon^{ilm}\int d^3 x \left(i \partial_{(x)}^m [\delta^3 ({\bf x-y})] \right) x^l
\mathbb{P}\bigl(\bigl\lbrace n'=k\bigr\rbrace\bigr) =\mathbb{P}\Biggl(\bigcup_{i=0}^{N-1}\bigl\lbrace\tilde{n}=k-i[N]\bigr\rbrace\cap\lbrace n=i\rbrace\Biggr) &=\sum_{i=0}^{N-1}\mathbb{P}\bigl(\bigl\lbrace\tilde{n}=k-i[N]\bigr\rbrace\bigr)\mathbb{P}\bigl(\lbrace n=i\rbrace\bigr) &=\frac{1}{N}.
\frac{\mathbf P_{\delta_x}[X_T(f)e^{-\theta X_T(f)}]}{\mathbf P_{\delta_x}[X_T(f)]}= \mathbf P_{\delta_x}^{X_T(f)}[e^{-\theta X_T(f)}] = \mathbf P_{\delta_x}[e^{-\theta X_T(f)}] \dot{\mathbf P}_x^{(T,f)}[e^{-\theta Y_T(f)}],
\int_{B_t}|Df_{j,t}|\, dx = \frac{t}{n-1}\int_{\partial B_t}|D f_j| \left\{1-\frac{\langle x, D f_j \rangle^2}{|x|^2|D f_j|^2}\right\}^{1/2}\, d\mathcal{H}^{n-1}\mbox{a.e in $t$}.
\int_{M}\frac{n-4}{2}\Phi^{\frac{n+4}{n-4}}Q_{g}+ \frac{(n-4)}{(n-2)^2}|\mathrm{Ric}_{\tilde{g}}|^2_{\tilde{g}}\Phi \, dv_{\tilde{g}}=\frac{n-4}{4(n-1)}\mathcal{E}(g).
\frac{\partial \, L_{n}}{\partial \, t}= \frac{1}{2}\left( p \, \frac{\partial \, L_{n}}{\partial \, p}- n \, \left( L_{n}- u \right) \right) \, u_{x}- \frac{\partial \, L_{n}}{\partial \, x}\, u - v^{n - 1}\, v_{x}
\overline\chi'(\overline p_0) ={\tiny\begin{pmatrix}0 0 1 -d_0 d_0 1 z_0 -\Re(z_0d_0) -1 0 0 -z_0 0 0 0 1\end{pmatrix}}, \overline\chi'(\overline p_0)^{-1}={\tiny\begin{pmatrix}0 0 -1 -z_0 -z_0 1 d_0 \Re(z_0d_0) 1 0 0 d_0 0 0 0 1\end{pmatrix}},
\left\langle \begin{array}{c c}{r}{t}\end{array}\right\rangle_{n - 1}\left\langle \begin{array}{c c}{r}{q}\end{array}\right\rangle_{n - 1}= \left\langle \begin{array}{c c}{r}{q}\end{array}\right\rangle_{n - 1}\left\langle \begin{array}{c c}{r}{t}\end{array}\right\rangle_{n - 1}
\mathrm{Re}\left[B(\overline{\mathbf{A}}_{h}^{k};\Psi_{h}^{k-1},\Psi_{h}^{k})-B(\overline{\mathbf{A}}_{h}^{k};\Psi_{h}^{k},\Psi_{h}^{k-1})\right] = 0.
\oint \frac{z^{m-j}dz}{(1-e^{-z})^{m+1-i}}= \oint \frac{z^{m-j}(e^z)^{m-i}de^z}{(e^z-1)^{m+1-i}}= \oint \frac{(\ln(1+x))^{m-j}(x+1)^{m-i}dx}{x^{m+1-i}}
\mathcal{B}^{(i_1,i_2)}(s_1,s_2) = \sum_{a_1+a_2=i_1}\binom{i_1}{a_1}\sum_{a_3+a_4=i_2}\binom{i_2}{a_3}\Big( \frac{\partial^{a_1}}{\partial s_{1}^{a_1}}\frac{\partial^{a_3}}{\partial s_{2}^{a_3}}\Theta(s_1+s_1+1,h) \Big) \tilde{\mathcal{B}}^{(a_2,a_4)}(s_1,s_2).
\forall v,w\in H^1_{\rm per}, a_q^V(v,w):= \int_\Gamma\left[ \overline{\left( - i \frac{d}{dx}+ q\right)v}\left(- i \frac{d}{dx}+ q\right) w \right] + \langle V , \overline{v}w \rangle_{H^{-1}_{\rm per}, H^1_{\rm per}}.
\Big\langle \nabla\mathrm{H};\mathrm{e}^{(1)}_{\mathrm{n}}\Big\rangle_{\mathbb{L}^{2}(\Omega)}= \Big\langle \mathbb{T}\Big[\nabla\mathrm{H}\Big];\ \mathrm{e}^{(1)}_\mathrm{n}\Big\rangle_{\mathbb{L}^2(\Omega)}.
J = 4 \Omega ({\cal V}_{1}^{( 1 )}\times{\cal V}_{2}^{( 1 )}) ({\cal V}_{1}^{( 2 )}\times{\cal V}_{2}^{( 2 )}) | |{\cal V}_{1}^{( 1 )}| |^{- 4}| |{\cal V}_{1}^{( 2 )}| |^{- 4},
q_{m+1}^{\frac{1}{1+q_m}}=\left(bq_m\left\lceil q_{m}^{-1}(2Kq_{m})^{\frac{1+q_{m}}{1+q_{m-1}}}\right\rceil + q_{m-1}\right)^{\frac{1}{1+q_m}}&>b^{\frac{1}{1+q_m}}(2Kq_m)^{\frac{1}{1+q_{m-1}}}&>(2Kq_m)^{\frac{1}{1+q_{m-1}}}
\langle{T_{\k}^{j}}\rangle - \langle T_{_{\ominus}k}^{\, j}\rangle \sim{\frac{\pi M^{2}}{2}}\Phi^{2}\Big ( ( 2 c^{- 2}- c_{_\mathrm{S}}^{- 2}) u^{j}u_{k}- \gamma_{_k}^{j}\Big ) \, \ln \Big \{{\frac{w_{\!_{\odot}}}{w}}\Big \}\ .
2m^{2}ds^{2}= (1-\frac{2mr}{\rho^{2}})dt^{2}-\frac{4amr}{\rho^{2}}\sin^{2}\theta dt d\phi -(r^{2}+ a^{2}-\frac{2a^{2}mr}{\rho^{2}})\sin^{2}\theta d\phi^{2}-\rho^{2}(\frac{1}{\Delta}dr^{2}+ d \theta^{2}),
M_{(\tau=1, \lambda_1, \lambda_2)}= \beta^{-1}_{22}(a=(1,1), b_0=1,\,b_1=1+ \lambda_1, \,b_2=1+\lambda_2) \times \times\beta_{1,0}^{-1}(a=1, b_0=\lambda_1+\lambda_2+2),
\mathcal R_1 (x)= \mathcal R_1 (x_1,x_2) ={M}_k \begin{pmatrix}x_1 x_2 \end{pmatrix},{M}_k= \begin{pmatrix}\cos\frac{2\pi}{k}-\sin\frac{2\pi}{k}begin{align*}3pt] \sin\frac{2\pi}{k}\cos\frac{2\pi}{k}\end{pmatrix}
\Leftrightarrow \qquad \delta_{q p}u_{\underline{{{m}}}}^{~ + +}= v_{q}^{+}\Gamma_{\underline{{{m}}}}v_{p}^{+}, \qquad \delta_{\dot{q}\dot{p}}u_{\underline{{{m}}}}^{--}= v_{\dot{q}}^{-}\Gamma_{\underline{{{m}}}}v_{\dot{p}}^{-}, \qquad \gamma_{q \dot{p}}^{i}u_{\underline{{{m}}}}^{i}= v_{q}^{+}\Gamma_{\underline{{{m}}}}v_{\dot{p}}^{-},
\dim_{L}C_{a}=\sup :\Bigl\{\beta :\exists \k_{\beta},\n_{\beta}\ 2^{n}\geq \left( \frac{s_{k}}{s_{k+n}}\right)^{\beta}\ \forall k\geq k_{\beta},n\geq n_{\beta}\Bigr\}& =\liminf_{n}\left( \inf_{k}\frac{n\log 2}{\log (s_{k}/s_{k+n})}\right).
\frac{\mbox{d}\mu_{\beta,\gamma}}{\mbox{d}x}=\frac{\gamma\,(1-a)^{2}}{\pi\!\left(\big((\beta+1)a\psi_{a}(x)-(\beta+a)\varphi_{a}(x)\big)^{2}+\gamma^{2}\big(a\psi_{a}(x)-\varphi_{a}(x)\big)^{2}\right)}\,,
q^{\tilde{\eta}}e_{r-1,s-1}^{\nu-1}=\lim_{\begin{tabular}{c}{\scriptsize $L\rightarrow \infty$}{\scriptsize $ LN \equiv 0 \,{\rm mod}\, (2\nu-2)$}\end{tabular}}q^{f^{\rm down}(L)}X_{L}(s|r,r+N)_{q^{-1}}
\mathsf{P}^i_j \triangleleft K_\lambda = \sum_{m, n}c^m_n (\mathsf{M}^n_m)^i_j \triangleleft K_\lambda = \sum_{m, n}q^{-(\lambda, \lambda_m - \lambda_n)}c^m_n (\mathsf{M}^n_m)^i_j.
\int_{0}^{t}|u|^2_{D^2}s\leq C \int_{0}^{t}\Big(|\rho u_t+\rho v\cdot \nabla v|^2_2+|\nabla P|^2_2+|H\times H|^2_2+|u|^2_{D^1_0}\Big)s\leq Cc^{10}_1.
\|{v}^n_{\Phi}\|_{\widetilde{L}_t^\infty(\dot{B}^{\frac{n-1}{p}-s,\frac{1}{2}}_{p,1})}+\|{v}^n_{\Phi}\|_{L_t^1(\dot{B}^{\frac{n-1}{p},\frac{1}{2}}_{p,1})}\lesssim Y_0 +(X(t)+Y(t))\Psi(t).
V(t,e(t))=\frac{1}{2}e'(t)P_R(t)e(t)\leq \frac{1}{2}e'(\bar{t}_0)P_{R}(\bar{t}_0)e(\bar{t}_0)\exp\Bigl(-\int_{\bar{t}_0}^{t}2\bar{d}_R(s)ds\Bigr), \forall t\in[\bar{t}_0,\hat{t}].
\frac{1}{\sqrt{2}k}\left[ 1 - \kappa \left( \frac{d}{d r}+ \frac{\kappa + 1}{r}\right) \right] R_{E \ell ( \kappa )}( r ) = \left( E + \frac{1}{2 k^{2}}\right)^{\mathrm{~ \frac{1}{2}~}}R_{E \ell ( - \kappa )}( r ) .
S_{\omega,\mathbf{c}}(V^{\lambda}) =L(V^{\lambda})+N(V^{\lambda})+\omega M(V^{\lambda})+\mathbf{c}\cdot \mathbf{P}(V^{\lambda}) &=\lambda^2L(V)+\lambda^{\frac{d}{2}+1}N(V)+\omega M(V)+\lambda \mathbf{c}\cdot \mathbf{P}(V),
\omega_m^{-1}r^{-m}\left|B_{g(b)}\left(x_0, r \right)\right|_{g(b)}\geq e^{-\frac{\delta_0}{100}}\cdot \left( 1-\frac{\delta_0}{100}\right) \cdot e^{-\frac{m\delta_0}{100m}}=e^{-\frac{\delta_0}{50}}\cdot \left( 1-\frac{\delta_0}{100}\right)> 1-\frac{\delta_0}{10}.
\prod_{x\in\mathcal{S}^1(\boldsymbol{\pi})}\Big(\frac{1}{\varepsilon^\alpha h}+(\lambda-\tilde{a}^\varepsilon_{x,x})\Big) \prod_{x\in\mathcal{S}^2(\boldsymbol{\pi})}\Big(\frac{1}{\varepsilon^\alpha h}-\tilde{a}^\varepsilon_{x,x}\Big) =\sum_{l=0}^{N-n-p}\beta^\varepsilon_{l,p,\boldsymbol{\pi}}(\lambda)\Big(\frac{1}{\varepsilon^{\alpha}h}\Big)^l.
A_2 \lesssim C_1 + \sup_{k\leq M-1}\sup_{x \in (x_k, x_{k+1})}\bigg(\int_{x}^{\infty}u\bigg)^{\frac{1}{q}}\bigg( \int_{0}^{x}\bigg(\int_0^t w \bigg)^{-\frac{p}{p-r}}w(t) V_r(0, t)^{\frac{pr}{p-r}}dt \bigg)^{\frac{p-r}{pr}}& = C_1 + C_3,
\beta_{n,n-1}=-\langle P_n(x;t),x^2P_{n-1}(x;t)\rangle H_{n-1}^{-1}=-\langle x^2P_{n}(x;t),P_{n-1}(x;t)\rangle H_{n-1}^{-1}&=-\langle x(a_nP_n(x;t)+b_nP_{n-1}(x;t)), P_{n-1}(x;t)\rangle H_{n-1}^{-1}&=-a_n\langle xP_n(x;t),P_{n-1}(x;t)\rangle H_{n-1}^{-1}-b_n\langle xP_{n-1}(x;t),P_{n-1}(x;t)\rangle H_{n-1}^{-1}