harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E_2 \ll \widehat{P}^{n-5}\widehat{Y}^{4-n/2 -2/n+\varepsilon}\widehat{Z}^{3-n/2}\widehat{V}^{n-2}= \widehat{P}^{-3}\widehat{Z}^{1+n/2}\widehat{Y}^{2+n/2-2/n +\varepsilon}\ll \widehat{P}^{5n/6-3 +5/3}\widehat{Y}^{1-2/n+\varepsilon}\ll \widehat{P}^{5n/6 -3 +10/3 + \varepsilon},
\mathcal{F}_S\left(\{M_j\},\tilde{\mathcal{B}}_T\times(-T,T)\right) \geq \sum_{j=0}^2 \alpha_j \int^T_{-T}\, dt \int_{\tilde{\mathcal{B}}_T}|D\chi_{A_j}| = \mathcal{F}_S\left(\{Q_j\},\tilde{\mathcal{B}}_T\times(-T,T)\right)
|\tilde A_n\cap B| \leqslant \sum_{i = 1}^{p_n}\vert A_ng_i^{-1}\cap B \vert \leqslant \sum_{i = 1}^{p_n}\vert A_n \cap Bg_i \vert \tau \leqslant \tau p_n\vert A_n \cap D \vert \leqslant \frac{\vert B \vert}{2}, \;\;\;.
(\#S_1) \mu_{S_1}= \sum_{i \in S_1}(|x_i|^2 - |y_i|^2 - \alpha_i) = \frac{\gamma}{2}- \delta - \alpha_{S_1}, (\#S_2) \mu_{S_2}= \sum_{i \in S_1}(|x_i|^2 - \alpha_i) = \frac{\gamma}{2}- \alpha_{S_2},
\pi_\beta d_0u_j =\left(\sum_{k=0}^{l+1}(-1)^{l}R_{l+1-k}^{g\upsilon_j\beta'\tau_{l+1-k}}\pi_{\beta'\sigma_k}-\sum_{i=1}^{l}(-1)^i\pi_{\beta'\delta_i}\right)u_j &= \sum_{k=0}^{l+1}(-1)^{l}R_{l+1-k}^{g\upsilon_j\beta'\tau_{l+1-k}}\pi_{\upsilon_j\beta'\sigma_k}-\sum_{i=1}^l (-1)^i\pi_{\upsilon_j\beta'\delta_i}.
\gamma_{1}:=\tilde{\mu}_{3}=E[(\frac{X-\mu}{\sigma})^{3}]=\frac{\mu_{3}}{\sigma^{3}}=\frac{E[(X-\mu)^{3}]}{(E[(X-\mu)^{2}])^{3/2}}=\frac{\kappa_{3}}{\kappa_{2}^{3/2}}
\lim_{n\rightarrow \infty}n^{\alpha+\frac 12}\int_0^{\tau}(t-s)^\alpha\sigma'(X_s) \left( \int^s_{\eta_n(s)}\left(s-\eta_n(u)\right)^{\alpha}\sigma(X_{\eta_n(u)})\, dW_u \right) ds=0
( Z_{1}\widehat{\phi})_{j}( x ) = \widehat{\phi}_{j - a}( x - \frac{1}{m}) , ( Z_{2}\widehat{\phi})_{j}( x ) = e^{- 2 \pi i j / m}e^{2 \pi i x / ( n - m \theta )}\widehat{\phi}_{j}( x ) .
{\cal V}( \lambda ) = \int_{0}^{\infty}d \rho \, \rho^{D - 2}\ln \left( 1 -{\frac{I_{\eta - 1}( \lambda \rho )}{K_{\eta - 1}( \lambda \rho )}}{\frac{K_{\eta - 1}( \rho )}{I_{\eta - 1}( \rho )}}\right) ,
P_{s_{13}}(x_{ij}=1;\beta)=(3,2)_{\beta},P_{s_{14}}(x_{ij}=1;\beta)=(26,42,19,2)_{\beta}, P_{s_{15}}(x_{ij}=1;\beta)=(381,988,917,362,55,2)_{\beta},P_{s_{15}}(x_{ij}=1;1)= 2705.
\sum_{i=0}^{k(i)}\exp\left[\gamma_\mathrm{int}(p)r_i\right] \leq \exp\left[\gamma_\mathrm{int}(p)r\right] \sum_{i=0}^{k(i)}e^{-ci}\preceq \exp\left[\gamma_\mathrm{int}(p)r\right].
{\bf m}= \{m^{(-2k_{i}-i+3)}_{0,\ldots, 0,\underset{i}{k_{i}},0, \ldots, 0}\mid i , \k_{i}\in \mathbb{Z}_{\geq0}\}\cup \{m^{(-2k_{i}-i+2)}_{0,\ldots, 0,\underset{i}{k_{i}},0, \ldots, 0}\mid i , \k_{i}\in \mathbb{Z}_{\geq0}\}.
\psi ( p ) = \frac{1}{\sqrt{8 \pi \sqrt{\beta}}\hbar}\int_{- \infty}^{+ \infty}d{\xi}{\mathrm{}}( 1 + \beta p^{2})^{1 / 2}{\mathrm{}}e^{- i{\xi}\frac{\tan^{- 1}( \sqrt{\beta}p )}{\hbar \sqrt{\beta}}}{\mathrm{}}\psi ({\xi})
\tau^{(1)}_1(E)=\xi_0 \frac{D^{[0]}(E)}{D^{[0]}(E \Omega^3)}+\xi_1 \frac{D^{[0]}(E\Omega^6)}{D^{[0]}(E \Omega^3)}\frac{D^{(2)}(E)}{D^{(2)}(E \Omega^3)}+\xi_2 \frac{D^{(2)}(E\Omega^6)}{D^{(2)}(E \Omega^3)},
D = v^*v (\theta(eAe)' \cap fMf) v^*v =(\theta(eAe) v^* v)' \cap v^* v M v^*v &= (v^* A v)' \cap v^* v M v^*v &= (v^* \, A vv^* \, v)' \cap v^* \, vv^* M vv^* \, v &= v^* ((A vv^*)' \cap vv^* M vv^*) v &= v^* \, vv^*(A'\cap 1_A M 1_A)vv^* \, v &= v^*(A' \cap 1_A M1_A) v.
\alpha_{\widetilde{\tau}}= \left(\begin{array}{ccc}-6 2 \theta^2 + 2 -\theta^2 + 11 -2 \theta^2 + 2 -10 -3 \theta^2 + 1 \theta^2 + 11 3 \theta^2 + 1 2 \end{array}\right) .
|\mathbb P(\tau_k^*\in I_1)-\mathbb P(\widetilde{\tau}_k\in I_1)|\leq \mathbb P^{GUE(n)}(\xi^{(n)}(L_k^{(1)})=0)+\mathbb P^{GUE(n)}( \xi^{(n)}(L_k^{(0)})=0) \leq n^{-1/4+o(1)}+n^{-1/4+o(1)}=n^{-1/4+o(1)},
\phi_{k+1}\leq \Big(1 -\frac{A_0}{A_{k+1}}\Big) f(x) + \frac{A_0}{A_{k+1}}\phi_0(x)= \Big(1 -\frac{1}{A_{k+1}}\Big) f(x) + \frac{1}{A_{k+1}}\phi_0(x).
\overline{\psi}^{T}=(\psi^{\dagger}\gamma^{0})^{T}=(\begin{matrix}0&I\\I&0\end{matrix})(\begin{matrix}\psi_{L}^{*}\\ \psi_{R}^{*}\end{matrix})=(\begin{matrix}\psi_{R}^{*}\\ \psi_{L}^{*}\end{matrix})
s = \bigg ( \begin{array}{c}{{r}}\\{{0}}\\ \end{array}\bigg ) ~ \textrm{w i t h s t a b i l i z e r}G = \bigg \{\bigg ( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{q}}\\ \end{array}\bigg ) \bigg | ~ q \in S p ( 1 ) \bigg \}\simeq S U ( 2 )
L_{1}(x(u,z))^{\ast}=\sum_{m=0}^{\infty}\alpha_{m}(u)^{\ast}\overline{z}^{m}\rho \big( \overline{z}^{-\kappa},\overline{z}^{\kappa}\big) C^{^{\prime}\ast}(\overline{x_{N}x_{N-1}})^{-\gamma\kappa};
H^2(\tilde Y, \mathbb Q) =(\pi_{4*}H^2(X, \mathbb Z)\oplus M)\otimes\mathbb Q=(\pi_{4*}H^2(X, \mathbb Z)\oplus\widehat{\pi_{2}}_*N\oplus N_Y)\otimes\mathbb Q= =(\widehat{\pi_{2}}_*(\pi_{2*}H^2(X, \mathbb Z)\oplus N)\oplus N_Y)\otimes\mathbb Q=(\widehat{\pi_{2}}_*H^2(\tilde Z, \mathbb Z)\oplus N_Y)\otimes\mathbb Q.
d \theta^r = 0, d \theta^x = 0, d\theta^q = \theta^x \wedge \theta^r, d\theta^p = \theta^x \wedge \theta^q, d \theta^y = \theta^x \wedge \theta^p, d \theta^z = \theta^r \wedge \theta^y + \theta^p \wedge \theta^q
\lim_{n\to\infty}\frac{1}{-\log b_c^{(n)}}\log P(B_1^{(n)}) =\lim_{n\to\infty}\frac{a_c^{(n)}}{-\log b_c^{(n)}}\lim_{n\to\infty}\frac{1}{a_c^{(n)}}\log P(B_1^{(n)}) &=-J(x_0)\lim_{n\to\infty}\frac{a_c^{(n)}}{-\log b_c^{(n)}}=-\infty.
H(e_i)=(\alpha-2[i])e_i ~ ; &E(e_0)=0 ~ ; E(e_i)= [i](\alpha-([i]-1))e_{i-1}, 1 \leq i \leq m ~ ; &F(e_m)=\beta e_0 ~ ; F(e_i)= e_{i+1}, 0 \leq i \leq m-1.
\frac{z_{1}}{z_{2}}= \frac{x_{1}- y_{1}c}{x_{2}- y_{2}i}= \frac{[ x_{1}- y_{1}i ) ( x_{2}+ y_{2}i )}{( x_{2}- y_{2}i ) ( x_{2}+ y_{1}i )}= \frac{( x_{1}x_{2}+ y_{1}y_{2}) - ( x_{2}y_{1}^{\prime}- x_{1}y_{2}) i}{x_{2}^{2}+ y_{2}^{2}}
S(yz) =S\bigl((\psi\otimes\operatorname{id})((a^*\otimes 1)(\Delta(yb)))\bigr)=(\psi\otimes\operatorname{id})\bigl(\Delta(a^*)(yb\otimes 1)\bigr) &=(\psi\otimes\operatorname{id})\bigl(\Delta(a^*)(b\otimes\gamma_B(y))\bigr) &=(\psi\otimes\operatorname{id})\bigl(\Delta(a^*)(b\otimes1)\bigr)\gamma_B(y)=S(z)\gamma_B(y).
E[p] = \frac{Lg^2}{2N}s_\theta[n] = \frac{Lg^2}{2N}\left[ \sum_{j=1}^N \left( p_j - \frac{\theta}{2 \pi}\right)^2 - \sum_{j=1}^N \left( \frac{N+1}{2}- j - \frac{\theta}{2 \pi}\right)^2 \right]
\Vert Y_1 - X_* \Vert_F^2 \leq \delta^2 \Vert Y_0 - X_*\Vert_F^2 + + \Big(\delta^2 \Vert Y_0 - X_*\Vert_F^2 - \sum_{k=1}^r\limits s_k^2 \sin^2 \phi_{R1, k}\Big) \frac{\sum_{k=1}^r\limits s_k^2\sin^2 \phi_{R0, k}}{s_r - \sum_{k=1}^r\limits s_k^2\sin^2\phi_{R0, k}- \sum_{k=1}^r\limits s_k^2 \sin^2 \phi_{R1, k}}.
\begin{aligned}-16L\mathcal Q(t) = ~{}2(9+\tilde \sigma) \int g^2 + (-3+ 5\tilde\sigma) \int g_x^2 & ~{}+4 (-1 + \tilde\sigma) \int g_{xx}^2 + (1+\tilde \sigma)\int g_{xxx}^2 +O\left(\frac1L \int (g^2 +g_x^2 + g_{xx}^2)\right).\end{aligned}
G^+=\{x\in\mathbb{R}^2\, | \ \begin{bmatrix}0 0 \beta_2v_{12}\beta_2v_{22}\beta_1v_{11}\beta_1v_{21}\end{bmatrix}\begin{bmatrix}x_1 x_2\end{bmatrix}\succeq_{\mathcal{L}^3}\begin{bmatrix}-\eta \beta_2\alpha_2 \beta_1\alpha_1\end{bmatrix}\},
k+\Theta_k-1= N(\mu_k^*;b_k^*) =\frac14N_0(\mu_k^*;b_k^*)+\frac12\bigg\lfloor \frac{{b_k^*}(\mu_k^*)^{1/2}}{\pi}\bigg\rfloor+\frac12\bigg\lfloor \frac{(\mu_k^*)^{1/2}}{b_k^*\pi}\bigg\rfloor-\frac14 \ge\frac14N_0(\mu_k^*;b_k^*)+ \frac{b_k^*(\mu_k^*)^{1/2}}{2\pi}+ \frac{(\mu_k^*)^{1/2}}{2\pi b_k^*}-\frac54.
\max \left\{\|x^A-x_0\|, \|x^B-x_0\| \right\}+ \frac{M}{\eta}\|x^A-x^B\| &\le \zeta + \frac{M}{\eta}\left( \|x^A-x_0\| + \|x_0-x^B\| \right)\le \zeta + \frac{M}{\eta}2\zeta = \delta \, .
\textup{div}_g \pi = r^{-1-p}\left[ (2-p)\beta + \tilde{\nabla}^a \alpha_a - h\right] dr &+ r^{-p}\left[ (3-p) \alpha_a + \tilde{\nabla}_b \hat{h}^b_a + \frac{1}{2}\frac{\partial}{\partial u^a}h\right] du^a + O(r^{-1- p-q}).
|\mathring{Ric}|^2 = \sum_i (\frac R4 - \lambda_i)^2 =\lambda_1^2 + \lambda_2^2 + \lambda_3^2 + \lambda_4^2- \frac 14 R^2 = (\frac{1-\eta}2 )^2 R^2 - 2 \lambda_1 \lambda_2 + 2 (\frac{1+\eta}4 )^2 R^2 + 2 y^2 - \frac 14 R^2 = \frac 18 \cdot ( 3\eta^2 -2\eta +1) R^2 + 2 y^2 - 2 \lambda_1 \lambda_2 .
\lim_{j \to \infty}\int_{\Omega}\langle A^{-1}_{\varepsilon, \delta+\delta'}\beta_{j}, \beta_{j}\rangle_{\omega_{\varepsilon}}e^{\psi - (\delta-\delta') \phi}dV_{\omega_{\varepsilon}}\leq \lim_{j \to \infty}\frac{1}{\delta(n-1)}\int_{\Omega}|\beta_{j}|^{2}_{\omega_{\varepsilon}}e^{\psi - (\delta-\delta') \phi}dV_{\omega_{\varepsilon}}= 0.
t_{\alpha,\varepsilon}= C_{\alpha,1}\, \varepsilon^2\sqrt{\sum_{k=1}^{D_\varepsilon}b_k^{-4}}\mathrm{and}s_{\alpha,\varepsilon}= C_{\alpha,2}\, \varepsilon^2 \sqrt{D_\varepsilon}\forall \varepsilon \in (0,1),
{\cal L}_{B I}= 4 b e^{2 \alpha \varphi}\left\{1 - \left[ 1 + \frac{e^{- 4 \alpha \varphi}}{2 b}F^{2}- \frac{e^{- 8 \alpha \varphi}}{1 6 b^{2}}( F \star F )^{2}\right]^{\frac{1}{2}}\right\}.
|D|^m[f(s|D|),x] = s\int_{\mathbb{R}}\int_0^1 \widehat{h'}(u)e^{ius(1-v)|D|}|D|^mh(s|D|)\delta(x)e^{iusv|D|}\,dvdu \quad+s\int_{\mathbb{R}}\int_0^1 \widehat{h'}(u)e^{ius(1-v)|D|}|D|^m\delta(x)h(s|D|)e^{iusv|D|}\,dvdu.
\sum_{n=0}^{M}f(b+n c)= \frac{1}{c}\int_{b}^{b+M c}f(\theta) d\theta +\frac{1}{2}[f(b)+f(b+M c)] +\frac{1}{12}c^2 \sum_{n=0}^{M-1}f^{(2)}(b+c(n+\phi)),
\frac{d^{2}N}{d\eta dt}=\frac{1}{\zeta\hbar\chi}\frac{dP}{d\omega}=\frac{\sqrt{3}}{2\pi}\frac{e^{2}mc}{\hbar^{2}}\frac{\chi}{\gamma\eta}F(y)whereF(y)=y\int_{y}^{\infty}K_{\frac{5}{3}}(x)dxandy=\frac{2\eta}{3\chi^{2}}
y_1 = \frac{-x_1+x_2+x_3-x_1x_2x_3}{1-x_1x_2+x_2x_3-x_3x_1},\y_2 = \frac{x_1-x_2+x_3-x_1x_2x_3}{1-x_1x_2-x_2x_3+x_3x_1},\y_3 = \frac{x_1+x_2-x_3-x_1x_2x_3}{1+x_1x_2-x_2x_3-x_3x_1}.
- 2 i \int \frac{d^{4}P}{( 2 \pi )^{4}}e^{- i P \cdot ( x - y )}\sum_{\tau = \pm}\theta ( \tau p_{0}) \left[ P \cdot \partial_{X}f^{( \tau )}( X ;{\bf p}) \right] \hat{\Delta}^{( \tau )}( X ; P ) = 0 ,
{}~ \times \Bigg \{\sum_{\sigma \in Z_{N}}\frac{w ( x_{3}, x_{1 3}, x_{1}| d , h + \sigma ) w ( x_{4}, x_{2 4}, x_{2}| a , g + \sigma )}{w ( x_{8}, x_{5 8}, x_{5}| e , c + \sigma ) w ( x_{7}/ \omega , x_{6 7}, x_{6}| f , b + \sigma )}\Bigg \}_{0}
\frac{1}{2}\int \frac{d^{4}k}{( 2 \pi )^{2}}\Bigg ( \frac{1}{k^{2}( k + p )^{2}}- \sum_{i}c_{i}\frac{1}{\Big ( k^{2}+ M_{i}^{2}\Big ) \Big ( ( k + p )^{2}+ M_{i}^{2}\Big )}\Bigg ) .
\begin{array}{r c l}{{i T^{\mathrm{1 P I}}}}&{{=}}&{{\displaystyle i \lambda^{2}v \int \frac{d^{4}k_{1}}{( 2 \pi )^{4}}\int \frac{d^{4}k_{2}}{( 2 \pi )^{4}}T ( k_{i}) ,}}\\{{T ( k_{i})}}&{{=}}&{{\displaystyle T_{a}+ T_{b}+ T_{c},}}\\ \end{array}
B\, dW(t)=\begin{pmatrix}0 0 0 \dots \frac{\sqrt{2\gamma}}{m}0 0 \cdots 0 \sqrt{2\lambda_1}0 \cdots \vdots \vdots \vdots \ddots \end{pmatrix}\begin{pmatrix}dW_0 dW_1 dW_2 \vdots \end{pmatrix}=\begin{pmatrix}0 \frac{\sqrt{2\gamma}}{m}dW_0(t) \sqrt{2\lambda_1}dW_1(t) \sqrt{2\lambda_2}dW_2(t) \vdots\end{pmatrix}.
H_p(T(4),T(4)) =H_p(\{0,1\},\{0,1\}) &=\max\left\{p(x,\{0,1\})x\in\{0,1\}\right\}&=\max\left\{p(0,0),p(1,1)\right\}&=(0,\frac{1}{2}) &\leq\frac{1}{3}(\frac{3}{2},4) &=\lambda\left[p(T(4),4)+p(T(4),4)\right].
df \big( \sum_{\sigma \in S_n}sgn(\sigma)^{m - 1}[x_{\sigma(1)}| \dots | x_{\sigma(n)}] \big) & = 2 \sum_{\sigma \in S_n}sgn(\sigma)^{m - 1}f( [x_{\sigma(1)}| \dots | x_{\sigma(n-1)}] ) x_{\sigma(n)}.
\begin{array}{c}{{R ( u / v ) ( K_{l}( u ) \otimes{\bf 1}) R ( u v q^{- 1}) ({\bf 1}\otimes K_{l}( v ) =}}\\{{= ({\bf 1}\otimes K_{l}( v ) ) R ( u v q^{- 1}) ( K_{l}( u ) \otimes{\bf 1}) R ( u / v ) \, ,}}\\ \end{array}
I_{\mathrm{D}p}\; = \; \int d^{p + 1}\zeta \; \left( T_{( p )}\; e^{( p - 3 ) \Phi / 4}\; \sqrt{- \mathrm{d e t}\;{\widehat g}_{\alpha \beta}}\; + \; \rho_{( p )}\; \widehat C^{( p + 1 )}\right) .
0 =-2\,\langle w_0\cdot\delta_G,\theta'\rangle+2\,\|\theta_i\|^2 &=-2\,\langle w_0\cdot\delta_G,\theta'\rangle+\|\theta'\|^2 &=-2\,\langle w_0\cdot\delta_G,\theta'\rangle+2\, \langle \theta',w_0\cdot\theta_i\rangle\,,
{\bf y}_0 = \left[\begin{matrix}-a^3(\overline{a}^2\overline{\tau}+a)^2 (a^2\overline{\tau}+\overline{a}\tau) (a^2\overline{\tau}-\overline{a}) (a^2\overline{\tau}+\overline{a}\tau) (\overline{a}^2\tau-a) \end{matrix}\right],
\Theta =a_{n}{\rm G}_{n}=a_{n}(-1)^{\frac{n}{2}}\varepsilon_{\mu_{1}\nu_{1}\cdots \mu_{\frac{n}{2}}\nu_{\frac{n}{2}}}\varepsilon^{\alpha_{1}\beta_{1}\cdots \alpha_{\frac{n}{2}}\beta_{\frac{n}{2}}}\prod_{i=1}^{\frac{n}{2}}R_{\alpha_{i}\beta_{i}}^{\mu_{i}\nu_{i}}
\forall z\in B_{\epsilon}(0):f(z)=\frac{1}{2\pi i}\int_{\partial B_{\epsilon}(0)}\frac{f(\xi)}{\xi-z}d\xi+\frac{1}{2\pi i}\iint_{B_{\epsilon}(0)}\frac{\partial f}{\partial\overline{\xi}}\frac{d\xi\wedge d\overline{\xi}}{\xi-z}.
\delta_{[ \mu_{1}. . . \mu_{n}]}^{[ \nu_{1}. . . \nu_{n}]}= \sum_{k = 1}^{n}( - 1 )^{k + 1}\delta_{\mu_{1}}^{\nu_{k}}\delta_{[ \mu_{2}. . \mu_{k}. . \mu_{n}]}^{[ \nu_{2}. . \nu_{1}. . \nu_{n}]}.
\int_\Omega \Big| \chi\Big( \frac{\nu(x)}r \Big)\Big|^2 \, e^{-\nu(x)\, \mbox{\scriptsize Re}\, \sqrt{s}}\, d\omega = \int_\Omega \big| \chi\big( \nu(\omega) \big)\big|^2 e^{-r\, \nu(\omega)\, \mbox{\scriptsize Re}\, \sqrt{s}}\, d\omega \le c\, |s|^{-1/2}\, r^{-1},
p_{xi}=\frac1n\times \frac{\binom{i-1}{2}}{\binom{n-1}{2}}\times \frac{\binom{i-2}{1}}{\binom{i-1}{2}}\cdot \frac{n-i}{n-3}&=\frac{2(i-2)(n-i)}{n(n-1)(n-2)(n-3)}.
g_1=e_1+a_1e, g_2=e_2+e_3+a_2e,g_3=e_2+ a_3e, g_4=e_1+e_3+ a_4e , g_5=e_3+ a_5e , g_6=e_1+e_2+a_6e , g_7=e_1+e_2+e_3+ a_7e,\a_1, \ldots , a_7\in [0,n-1].
\psi^{(e)}(x)=A\, I^{(+)}_{i\nu}\bigl(\rho(x)\bigr)+B\, K_{i\nu}\bigl(\rho(x)\bigr),A,B\in\mathbb{R}, \psi^{(o)}(x)=C\, I^{(+)}_{i\nu}\bigl(\rho(x)\bigr)+D\, K_{i\nu}\bigl(\rho(x)\bigr),C,D\in\mathbb{R},
H_T = \int dx^1 \mbox{Tr}\left\{- \omega_0 ( D_1 \pi_{\omega_1}- [ \pi_{\phi}, \phi ] ) - ( \phi^2 + \{\pi_{\omega_1}, \phi \}) B + \lambda_{\omega_0}\pi_{\omega_0}+ \lambda_B \pi_B \right\}.
L = - m \sqrt{- \dot{x}^{2}\left( 1 - \frac{2 s}{m}\frac{\dot{\varphi}}{( \dot{x}, n )}- \frac{\varrho^{2}}{m^{2}}\frac{\dot{\varphi}^{2}}{( \dot{x}, n )^{2}}\right)}+ \varrho \frac{( \dot{x}, \partial_{\varphi}n )}{( \dot{x}, n )}\dot{\varphi}\ .
\begin{array}{rl}\rho_{\textnormal{pop}}= \displaystyle \sup_{\lambda, \{\sigma_i\}}\lambda \textnormal{subject to}\displaystyle f - \lambda= \sigma_0 + \sum_{i=1}^m \sigma_ig_i \geq 0, & \displaystyle \sigma_i \textnormal{is a sum of squares for}i = 0,\dots,m.\end{array}
\tilde{W}_{1}^{B W}( \beta , \alpha , y , \epsilon ) ={\frac{1}{1 2}}\left( \alpha +{\frac{1}{\alpha}}\right) \ln{\frac{2 \pi \alpha \epsilon}{\beta}}- \frac 1 2 \ln{\frac{\pi \alpha}{\ln{( \beta / 2 \pi \alpha \epsilon )}}}
\mathbb{P}\left(\sum_{j=0}^{r}L(i,j) \geq c\right) \leq e^{-sc}\prod_{j=0}^{r}\mathbb{E}e^{sL(i,j)}= e^{-sc}\left(e^{s}p_1 + e^{-s}(1-p_1-p_\alpha)\right)^{r+1}.
\epsilon (\partial_{t}^+e_h^{u^n},e_h^{u^n})+\|e_h^{\phi^n}\|^2_{\mathcal{T}_h}=-(\phi^n - \phi_{Ih}^n,e_h^{\phi^n})_{\mathcal{T}_h}&-\epsilon^{-1}(f^n(u_h^n)-f(u^n),e_h^{\phi^n})_{\mathcal{T}_h}+\epsilon (\partial_{t}^+u_{Ih}^n-\partial_tu^n,e_h^{u^n})_{\mathcal{T}_h}.
S ( \phi , J ) = \sum_{n}a^{D}\left\{\frac{1}{a^{2}}\sum_{\mu}\frac{1}{2}\left( \phi_{n + \mu}- \phi_{n}\right)^{2}+ m^{2}\phi_{n}^{2}+ \lambda \phi_{n}^{4}+ J_{n}\phi_{n}\right\}
\mathbf{D}\mathbf{C}_M(\tau-N\varepsilon)\mathbf{U}(\tau-N\varepsilon)=\begin{pmatrix}\mathbf{f}(\tau+(M-N)\varepsilon) \vdots \mathbf{f}(\tau+(M-1)\varepsilon)\end{pmatrix}-\mathbf{D}\mathbf{F}_M(\tau-N\varepsilon).
W_{\mathrm{T}}^{( \xi )}= \frac{1}{4 \xi}\int d^{4}x \mathrm{t r}F_{\mu \nu}\tilde{F}^{\mu \nu}- \frac{1}{2 4 \xi}\epsilon^{\mu \nu \rho \sigma}\bar{Q}_{\mu}^{( \xi )}\bar{Q}_{\nu}^{( \xi )}\bar{Q}_{\rho}^{( \xi )}\bar{Q}_{\sigma}^{( \xi )}\int d^{4}x \mathrm{t r}\phi^{2},
P(\omega_N)^2 = \prod_{i=1}^N \prod_{j\not= i}d_{i,j}\leq \prod_{i=1}^N \bigg( \Big(\sum_{j\not= i}d_{i,j}\Big)/(N-1)\bigg)^{N-1}=\bigg( \frac{2N}{N-1}\bigg)^{N(N-1)},
{\mathcal Z}_{A}^{r e g}[ J ] \, = \, \int{\mathcal D}{\mathcal A}_{\mu}\, \, \exp \left\{i \int d^{3}x [ - \frac{\xi^{2}}{4}{\mathcal F}_{\mu \nu}{\mathcal F}^{\mu \nu}\, + \, \frac{\kappa}{2}\epsilon^{\mu \nu \lambda}{\mathcal A}_{\mu}\partial_{\nu}{\mathcal A}_{\lambda}+ J^{\mu}{\mathcal A}_{\mu}] \right\}\, ,
X^{2^m}-1=(X-1)(X+1)\prod\limits_{h\in H_{i}, ~\atop{1\leq i\leq(a-1)}}(X^2-2hX+1)\prod\limits_{h\in H_{a},~\atop{0\leq k\leq(m-a-1)}}(X^{2^{k+1}}-2hX^{2^{k}}-1).
\vert\{\lambda\},\{\rho\},\chi\rangle{}~\equiv{}~ (\alpha^0_{-{n_1}})^{\lambda_1}...(\alpha^0_{-{n_p}})^{\lambda_p}(\epsilon^{(\pm)0}_{-r_1})^{\rho_1}...(\epsilon^{(\pm)0}_{-r_s})^{\rho_s}\vert\chi;q\rangle{}~,
D^\sigma_{\!x}\sum_{k=0}^{\infty}a_k x^k = \!\!\sum_{k\,=\left\lceil\sigma\right\rceil}^{\infty}\!a_k\,\frac{\Gamma(1\!+\!k)}{\Gamma(1\!+\!k\!-\!\sigma)}\,x^{k-\sigma}\quad (\sigma>0)
0=\frac{d}{dc}\left((\tilde{L}_c - \tilde{\lambda}_c)\phi_c\right)\big|_{c=1}= \left(\frac{d\tilde{L}_c}{dc}-\frac{d\tilde{\lambda}_c}{dc}\right)\phi_c\big|_{c=1}+ \left(\tilde{L}_c - \tilde{\lambda}_c\right)\frac{d\phi_c}{dc}\Big|_{c=1}.
\bar{w}_k(z):=\tfrac{\varepsilon_k^{N-1}}{|Du|(Q_k)}\left(\tilde{u}_k(x_0+\varepsilon_k z)-a_k\right), \;w_k(z):=\tfrac{\varepsilon_k^{N-1}}{|Du|(Q_k)}\left(\bar{u}_k(x_0+\varepsilon_k z)-a_k\right).
K(x,\lambda(t)) = \int_{0}^{\infty}dR \frac{R}{(1+R^{2})^{3}}\int_{0}^{x}\rho d\rho\left(\frac{1}{\sqrt{x^{2}-\rho^{2}}}-\frac{1}{x}\right) \left(1+\frac{R^{2}\lambda(t)^{2}-1-\rho^{2}}{\sqrt{(R^{2}\lambda(t)^{2}-1-\rho^{2})^{2}+4R^{2}\lambda(t)^{2}}}\right) \geq 0
(z,w) \star \begin{pmatrix}a -\bar{b}b \bar{a}\end{pmatrix}=\begin{pmatrix}z^p & z^q\end{pmatrix}\begin{pmatrix}a -\bar{b}b \bar{a}\end{pmatrix}\begin{pmatrix}w & \bar{w}\end{pmatrix}^{-1}
Q/Q(2) \downarrow X = 3^{[r]}\otimes 2^{[s]}\otimes 1^{[r+1]}, Q(2)/Q(3) \downarrow X = 4^{[r]}\otimes 2^{[s]}+ 2^{[s]}, Q(3)/Q(4) \downarrow X = 3^{[r]}\otimes 1^{[r+1]}, Q(4)/Q(5) \downarrow X = 2^{[s]}.
e_{s}(z_{b_{1}}, qz_{b_{2}}, \ldots , q^{r+s-1}z_{b_{r+s}}) =e_{s}(z_{b_{1}}, qz_{b_{2}}, \ldots , q^{r+s-2}z_{b_{r+s-1}}) +q^{r+s-1}z_{b_{r+s}}e_{s-1}(z_{b_{1}}, qz_{b_{2}}, \ldots , q^{r+s-2}z_{b_{r+s-1}}),
\mid 2^f h^{i'}\overline{h}^{j'}\sum\limits_{a_m\neq a_i}\chi_4(g(a_m))\mid &=\mid 2^fh^{i'}\overline{h}^{j'}\left\lbrace \sum\limits_{a_m}\chi_4(g(a_m))-\chi_4(g(a_1))-\cdots-\chi_4(g(a_{m-1})) \right\rbrace \mid &\leq 2^{3m}\cdot 3m\sqrt{q}+2^{2m-3}\leq 2^{2m}(1+2^m\cdot 3m\sqrt{q}),
d s^{2}= d r^{2}+ f^{2}( r ) \lambda^{2}+ a^{2}( r ) ( \mu_{1}^{2}+ \mu_{2}^{2}) + b^{2}( r ) ( \Sigma_{1}^{2}+ \Sigma_{2}^{2}) + c^{2}( r ) ( \nu_{1}^{2}+ \nu_{2}^{2}) \, .
F_4= a^2\Delta\delta u-u^2\cdot\nabla \delta u-\delta u\cdot\nabla u^1+\delta a(\Delta u^1-\nabla\Pi^1) &+(1+a^2)(\delta B\cdot\nabla B^2+ B^1\cdot\nabla \delta B)+\delta a( B^1\cdot\nabla B^1).
\frac{\gamma}{\lambda_1}\int_{D}|\nabla u^-_\infty|^2(x) d\mu \geq \gamma \int_{D}(u^-_\infty(x))^2 d\mu & = \int_{D}|\nabla u^-_\infty|^2(x) d\mu - \int_{D}u^-_\infty(x) \Delta_\mu u^+_\infty(x) d\mu.
P\left(\frac{n-A_n^*}{f_3(n)}\in O\right) \geq\mathrm{e}^{(\eta+x-\delta)f_3(n)\log(b_c^{(n)}/f_3(n))}-\mathrm{e}^{(x+\delta-\eta)f_3(n)\log(b_c^{(n)}/f_3(n))}&=\mathrm{e}^{(x-\delta+\eta)f_3(n)\log(b_c^{(n)}/f_3(n))}\left(1-\mathrm{e}^{2(\delta-\eta)f_3(n)\log(b_c^{(n)}/f_3(n))}\right),\quad
0 \ \square \f_i(x,y) \Leftrightarrow 0 \ \square \a_{i\mu_i}(x) F_i(x,y), v(f_i(x,y)) = 1 \Leftrightarrow v(a_{i\mu_i}(x) F_i(x,y)) = 1, v(f_i(x,y)) < v(f_j(x,y)) \Leftrightarrow v(a_{i\mu_i}(x) F_i(x,y)) < v(a_{j\mu}(x) F_j(x,y)),
\partial F =\partial(e^{2m_{1}(\varphi-\eta)}a_{1}+e^{2m_{2}(\varphi-\eta)}a_{2}) &=e^{2m_{1}(\varphi-\eta)}\partial a_{1}+e^{2m_{2}(\varphi-\eta)}\partial a_{2}&+2m_{1}a_{1}e^{2m_{1}(\varphi-\eta)}\partial (\varphi-\eta)+2m_{2}a_{2}e^{2m_{2}(\varphi-\eta)}\partial (\varphi-\eta),
\lim_{N\rightarrow+\infty}\frac{1}{N}\log\frac{dP}{dP^N_{f_1, f_2, f_3}}=\int_{\mathbb{T}}\Bigg(\sum_{k=1}^3f_k(u)\log\left(\frac{\rho_{k-1}(u)}{f_k(u)}\right) &+\left(1-\sum_{k=1}^3f_k(u)\right)\log\left(\frac{1-\sum_{k=1}^3\rho_{k-1}(u)}{1-\sum_{k=1}^3f_k(u)}\right)\Bigg)du=-I_{ini}(\hat{W}_0)
(G^{-1}_{OM})_{ab}={\left( (2K^2 -1) - 2K^2 \sqrt{1-K^{-2}}\right)^{1\over6}\over K}\, \left[ \hat{\eta}_{\hat{a}\hat{b}}+ \,{1\over 4}(\hat{{\cal H}}^2)_{\hat{a}\hat{b}}\right]
\frac12\,\Psi_{\varepsilon}^{(k)}(a)=(-1)^{k-1}\sum_{\ell=0}^{k-1}(-1)^{\ell}p^{(k-\ell-1)}(0)\Psi_{\varepsilon}^{(\ell)}(a)\,\varepsilon^{(\ell-k)/(2N)}+(1-\mathrm{e}^{\lambda \varepsilon})\Phi_{\varepsilon}^{(k)}(a)+R_{k,\varepsilon}
Y^{( 1 )}= - \frac{2}{\sqrt{- g}}( \kappa v^{i}\bar{\tilde{\cal H}}^{j}\gamma_{i j k}\gamma^{( 4 )}E^{k}+ \bar{\kappa}v^{i}\tilde{\cal H}^{j}\gamma_{i j k}\gamma^{( 4 )}\bar{E}^{k}) ,
G^{+}\left( x,x^{\prime}\right) =\left| A\right|^{2}G_{E}^{+}\left(x,x^{\prime}\right) +\left| B\right|^{2}G_{E}^{+}\left( x^{\prime},x\right) +AB^{\ast}G_{E}^{+}\left( x,x_{A}^{\prime}\right) +BA^{\ast}G_{E}^{+}\left( x_{A},x^{\prime}\right) .
\mathfrak{W}(c) \equiv \big\{\lambda\in \mathfrak B^d_\rho: p^\prime \lambda = 0 \cap \mathfrak w_{j}(\lambda)\le c, \: \forall j=1,\dots,J\big\}, c_{\pi^*}\equiv \inf\{c\in\mathbb R_+:\mathrm{Pr}(\mathfrak{W}(c)\ne\emptyset)\ge 1-\alpha\}.
\beta F = - \frac{N}{\beta^{3}}\int d \theta d \phi \int_{r_{+}+ h}^{L}d r \frac{\sqrt{g_{4}}}{( - g_{t t}^{'})^{2}}= - N \int_{0}^{\beta}d \tau \int d \theta d \phi \int_{r_{+}+ h}^{L}d r \sqrt{g_{4}}\frac{1}{\beta_{l o c a l}^{4}},
\Big(\sum_{\substack{m_1>0 m_2\geq 0 m_1+m_2=m}}\sum_{\substack{\mu_1\in L_0^\vee/L_0 \mu_2\in \Lambda^\vee/\Lambda (\mu_1,\mu_2) \in (\mu + L)/(L_0 \oplus \Lambda)}}R_\Lambda(m_2,\mu_2) \cdot \mathcal{Z}_0(m_1 , \mu_1 ) \ , \i^* \Phi_{m,\mu}\Big) \in \widehat{\mathrm{Div}}(\mathcal{Y})
A_7(a,b) = \bigl( \frac{-14}{3}a+\frac{14}{3}b \bigr)Jq^7+\bigl( \frac{29}{3}a-\frac{14}{3}b \bigr)Jq^6Jq^1 +\bigl( \frac{-28}{3}a+\frac{7}{3}b \bigr)Jq^5Jq^2+\bigl( \frac{28}{15}a-\frac{7}{15}b \bigr)Jq^4Jq^3 \quad+\bigl( \frac{4}{3}a-\frac{1}{3}b \bigr)Jq^3Jq^4+aJq^2Jq^5+bJq^1Jq^6=0.
X_+X_+'= \sum_{k+\ell\leq s}\sum_{i:\lambda_i>0}\left(\lambda_i |\xi^{(k,\ell)}_i(t)|^2+\xi^{(k,\ell)}_i(t)\mathcal{E}^{(k,\ell)}_i(t)\right) = \sum_{k+\ell\leq s}\sum_{i:\lambda_i>0}\left(\lambda_i |\xi^{(k,\ell)}_i(t)|^2 \right)+X_+Y_+,
E_{a,b}^{(M)}E_{c,d}^{(N)}=\begin{cases}E_{a,d}+q_c^{-1}E_{c,d}E_{a,b}, M=N=1; \sum_{t=0}^{\min(M,N)}q_{b}^{-(N-t)(M-t)}E_{c,d}^{(N-t)}E_{a,d}^{(t)}E_{a,b}^{(M-t)}, .\end{cases}
(X_1\otimes_{eh}X_2)\otimes_{eh}Y =\Gamma_R((X_1\otimes_{eh}X_2)^*,Y)\cap CB^{\sigma}((X_1\otimes_{eh}X_2)^*,Y^{**}) &=\Gamma_R((X_1\otimes_{eh}X_2)^*,Y)\cap CB_m^{\sigma}(X_1^*\times X_2^*, Y^{**})\to X_1\otimes_{eh}X_2\otimes_{eh}Y.
2\big(N+\sqrt{N[1+(N-1)c]}\big) =2\Big(N+N\sqrt{\frac{1}{N}+\frac{N-1}{N}c}\Big) begin{align*}2 pt] <2\Big(N+N\sqrt{\frac{1}{N}+c}\Big) begin{align*}2 pt] <2(N+N\sqrt{2c}) begin{align*}2 pt] =2N(1+\sqrt{2c}).
\exp(\alpha+\beta) =\sum\frac{(\alpha+\beta)^n}{n!}=\sum_{n=0}^\infty\sum_{k=0}^n\binom{n}{k}\frac{\alpha^k\beta^{n-k}}{n!}&=\sum_{n=0}^\infty\sum_{k=0}^n\frac{\alpha^k\beta^{n-k}}{k!(n-k)!}=\sum\frac{\alpha^n}{n!}\cdot\sum\frac{\beta^n}{n!}=\exp(\alpha)\exp(\beta).
e^{- 2 \phi}= \left( ( e^{- 2 \hat{\phi}_{0}}+{\frac{2 \tilde{m}_{k}}{r}}) ( ( A_{u})_{0}+{\frac{2 \hat{m}_{k}}{r}}) - \left[ ( A_{i})_{0}+{\frac{2 ( q_{k})_{i}}{r}}\right]^{2}\right)^{\frac{1}{2}}\ .
\frac{\partial^\gamma u}{\partial t^\gamma}+ u \frac{\partial u}{\partial x}-\nu \frac{\partial^2 u}{\partial x^2}=0,\,\,\,\,\,\,-3\leq x \leq 3,\,\,\,\,\,0 < \gamma < 1,\,\,\,\,t>0.
{\cal V}(\phi_0,\mu) = V(\phi_0,\mu_0) +{\hbar \over{4\pi}}\left\{V''(\phi_0,\mu) - m^2(\mu) \right\}\ln{\mu\over m(\mu)}+\hbar m^2(\mu) \; X[V(\phi_0,\mu)/m^2(\mu),\mu]+O(\hbar^2).
\begin{array}{rcl}\lambda_n(\nabla_{xx}\mu_f(x,t)) \ge 12\displaystyle \min_{i=1,\ldots,n}a_i x_i^2 +2 \lambda_n(B) +4 t^2 a_k & \ge 2 \lambda_n(B) + 4 t^2 a_k\,, \end{array}
I \equiv \int_{- \infty}^{\infty}d x \, \partial_{x}\theta ( x ) \exp \left[{\frac{i}{\hbar}}\int_{a_{1}}^{x}d x \, p_{1}( x ) -{\frac{i}{\hbar}}\int_{a_{2}}^{x}d x \, p_{2}( x ) \right] .
G^{W}_{\alpha,\beta}(\tau)={1\over 4\pi^{2}}~\left({\cosh^{2}\alpha\over (2\sinh{\tau-i\epsilon\over 2})^{2}}+{\sinh^{2}\alpha\over (2\sinh{\tau+i\epsilon\over 2})^{2}}+{\sinh 2\alpha \cos\beta\over 2\cosh^{2}{\tau\over 2}}\right)~,
\lim_{|b|\to\infty}\frac{\mathcal{H}(b,u,w)}{|b|\ln(|b|+1)}=\frac{1}{2}\qquad\mathrm{and}\lim_{|b|\to 0}\frac{\mathcal{H}(b,u,w)}{b^2}=\frac{1}{4(1+uw)}.
V ( | z | ) = \left( \frac{b}{6}\right)^{2}\frac{| z |^{4}}{( 1 - | z |^{2})^{2}}\left\{3 \left( \log \left| \frac{z}{z_{t r}}\right|^{3}\right)^{2}+ C_{2}| z |^{2}\right\}.