harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
d R_{2}= \delta^{4}( p_{1}+ p_{2}- p_{\mathrm{T}}- p_{\mathrm{B}}) \delta ( p_{1}^{2}+ m_{1}^{2}) \delta ( p_{2}^{2}+ m_{2}^{2}) \frac{d^{4}p_{1}}{( 2 \pi )^{4}}\frac{d^{4}p_{2}}{( 2 \pi )^{4}}
\overline{\bigcup_{i\geq n}\Phi_i^{-1}(0)}\subseteq \left( \bigcup_{j=1}^l \overline{B_{c m_n^{-1/2}}(x_j)}\right) \cup \left( \bigcup_{i \geq n}\bigcup_{j=l+1}^k \overline{B_{c m_n^{-1/2}}(m_i x_j)}\right),
\begin{cases}\partial_{t}\rho+u\cdot \nabla \rho + \rho\nabla\cdot u=0 begin{align*}10pt] \partial_{t}u + u\cdot \nabla u + \frac{1}{\rho}\frac{1}{\kappa^{2}}\nabla \overline{P}\left(\rho\right)=0 \,. \end{cases}
N_{dr}=\left(\sum_{i=1}^n2^ix_i\right)a+\sum_{i=1}^n(2^i-1)x_id &=\left(\sum_{i=1}^{n-1}2^ix_i\right)a+\sum_{i=1}^{n-1}(2^i-1)x_id+(2^na+(2^n-1)d)x_n.
U_{p,i}=\sum_{\begin{subarray}{c}j,k=1 j\neq i,k\neq i\end{subarray}}^N\gamma_p(j,k)W^{(i)}_jW^{(i)}_k,V_{p,i}=2\sum_{j=1}^N\gamma_p(i,j)W^{(i)}_{j}
\delta(q_{i,(i+k)\bmod p,\, k},a)=\begin{cases}q_{i,(i+k+1)\bmod p,\, k+1}a=t_{k}^{(i)}k\ne m_i+n_i-1 q_{i,(i+m_i)\bmod p,\, m_i}a=t_{k}^{(i)}k= m_i+n_i-1 q_{(i+k+1)\bmod p,(i+k+1)\bmod p,\, 0}a\in [\![0, t_k^{(i)}-1]\!].\end{cases}
\mathbb{E}_{\mathbf{H}}\{\Vert\hat{\mathbf{x}}_{blue}-\mathbf{x}\Vert^2\}=\mathbb{E}_{\mathbf{H}}\textsf{Tr}\left(\mathbf{\Sigma}_{e,blue}\right) &=\mathbb{E}_{\mathbf{H}}\textsf{Tr}\left(\left(\mathbf{H}^*\mathbf{\Sigma}_z^{-1}\mathbf{H}\right)^{-1}\right) &=m\mu_{\mathbf{\Lambda}}\left(-1\right),
I = \left|\frac{\partial\chi}{\partial c_m}\right|\int{\cal D}\{c_m\}{\cal D}\{c\}{\cal D}\{\bar{c}\}{\cal D}\{a\}\exp\left\{-\sum_{m\neq 0}c_m^2E_m - \pi c_0^2M -c\bar{c}M\right\}
\nabla_{\rho}\Lambda^{\rho \mu_{3}\ldots \mu_{s}}= \phi^{\frac{d + 2 ( s - 2 )}{2}}\partial_{\rho}\left( \phi^{-{\frac{d + 2 ( s - 2 )}{2}}}\Lambda^{\rho \mu_{3}\ldots \mu_{s}}\right)
0=\langle \oplus_{i\neq j=1}^df_j,D_{V_1^*}V_2^*\cdots V_{i-1}^*V_{i+1}^*\cdots V_d^*h \oplus D_{V_2^*}V_3^* \cdots V_{i-1}^*V_{i+1}^*\cdots V_d^*h &\oplus\cdots\oplus D_{V_{d-1}^*}V_{d}^*h\oplus D_{V_d^*}h\rangle.
\underset{n_0,n_d\ge0, d\not\in C_J ,\atop n_0+\sum_{d\not\in C_J}n_d =\#C_J-1}{\sum}\frac{(2\pi i)^{n_0}}{\underset{d\not\in C_J}{\prod}{(\alpha_d-a_J)^{n_d}}}\epsilon^{\#C_J-1}+O(\epsilon^{\#C_J}).
\limsup_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{E}_{N}\exp\left\{\int_{0}^{T}\sum_{i=1}^{N}\left[\nabla G(t,\,x_{i}^{N}(t))-2G^{2}(t,\,x_{i}^{N}(t))\right]dt\right\}\le0
r^{2}\frac{d}{d r}\left\langle f^{\prime}( r^{\prime}) f^{\prime}( r ) \right\rangle = \left\langle f^{\prime}( r^{\prime}) f^{3}( r ) \right\rangle - \left\langle f^{\prime}( r^{\prime}) f ( r ) \right\rangle \left( 1 + g^{2}( r ) \right)
\sum_{j=-\infty}^\infty (-1)^j d(k)_{m+j}d(k)_{m-j}- (e(k)_{m})^2 = (-1)^{m}\,c_{2m}( E_{p_i}^\perp/E_{p_i}+ (F_{q_i}^\perp/F_{q_i})^* ) - c_{m}( E/E_{p_i}+ F/F_{q_i})^2,
\lambda_{i_1 ,\dots ,i_{n-1}}=(-1)^{n+i_1+\dots +i_{n-1}}\frac{(\alpha \beta +(-1)^{i_1+\dots +i_{n-1}}\prod_{k=1}^{n-1}\gamma_k^{i_k})\prod_{k=1}^{n-1}\gamma_k^{1-i_k}}{\alpha \beta}.
W_{i,k}^{\textit{fluid}}(u_k) = \underbrace{\sum_{j=i-N}^{k-1}\hat\rho_jc\hat\gamma_{i,j}b_i+u_k\hat\rho_kc\hat\gamma_{i,k}b_i}_{\mathcal{P}_{i,k}(u_k)}+ \underbrace{(1-u_k)\hat\rho_k c+\sum_{j=k+1}^{i-1}\hat\rho_j c}_{\mathcal{R}_{i,k}(u_k)},
s_{m, k, n, mn}(r) = 2^{mn}\sum_{\substack{A \in B_r^{nk}rank(A) = n}}\gcd(A)^{-m}\leq 2^{mn}\sum_{\substack{A \in B_r^{nk}rank(A) = n}}1 \leq 2^{mn}\left| B_r^{nk}\right| = 2^{mn}(2r + 1)^{nk}.
r_p=d_p\frac{(w_{p-1}v_{\omega_p},nw^{-1}hww_p v_{\omega_p})}{(w_{p}v_{\omega_p},nw^{-1}hww_{p}v_{\omega_p})}=d_p\frac{(w_{p-1}v_{\omega_p},gn_{w^{-1}}^{-1}ww_p v_{\omega_p})}{(w_{p}v_{\omega_p},gn_{w^{-1}}^{-1}ww_{p}v_{\omega_p})},
\eta(\eta(\tau_i, \tau_j),\eta(\tau_i,\tau_\ell)) =\eta((\tau_n^q\tau_1)^{n-i}\tau_{n-i+j}, (\tau_n^q\tau_1)^{n-i}\tau_n^{q-1+D(n-i+\ell)}\tau_{n-i+\ell+1}) &=\eta(\tau_{n-i+j}, \tau_n^{q-1+D(n-i+\ell)}\tau_{n-i+\ell+1}).
U=[\begin{matrix}cos\theta&sin\theta&0\\ -\frac{1}{\sqrt{2}}sin\theta&\frac{1}{\sqrt{2}}cos\theta&\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}sin\theta&-\frac{1}{\sqrt{2}}cos\theta&\frac{1}{\sqrt{2}}\end{matrix}].
\frac{\partial^{2}}{\partial p^{j}\partial p^{k}}T_{a}( p ) \: \frac{q^{j}}{2}\frac{q^{k}}{2}\; = \; - \frac{\partial}{\partial p^{j}}\left( ( p q ) \: T_{a}^{( 1 )}( p ) \right) \frac{q^{j}}{2}\; = \; ( p q )^{2}\: T_{a}^{( 2 )}( p ) \: - \: \frac{q^{2}}{2}\: T_{a}^{( 1 )}( p ) .
\sum_{n=0}^\infty \overline{p}(6n)q^n \equiv \frac{f_{12}^{15}}{f_6^6 f_{24}^6}\left( \frac{f_6^4 f_9^6}{f_3^8f_{18}^3}+ 2q\frac{f_6^3f_9^3}{f_3^7}+4q^2\frac{f_6^2f_{18}^3}{f_3^6}\right)^4 + \cdots \pmod{16}
\frac{F-\mathbb{E}(F)}{\sqrt{\mathbb{V}(F)}}=\frac{{\sqrt{\mathbb{V}(F')}}}{{\sqrt{\mathbb{V}(F)}}}\frac{F'-\mathbb{E}(F')}{\sqrt{\mathbb{V}(F')}}+\frac{F-F'-\mathbb{E}(F-F')}{{\sqrt{\mathbb{V}(F)}}},
\frac{\partial}{\partial y}\lambda_E(x,y) =\frac{1}{\frac{\partial x}{\partial \lambda}}&=\frac{1}{\frac{\partial \phi_{+}(\alpha k-1,x,\lambda, E)}{\partial \lambda}}&=-\frac{R_{+}^{2}(\alpha k -1,x,\lambda,E)}{\sum_{i=0}^{\alpha-1}f_iu_{+}^2(\alpha k+i,x,\lambda,E)},
\Delta=(\partial_{\bar{z}}A_0^{\bar{z}})(\partial_zA_0^z)-(\partial_zA_0^{\bar{z}})(\partial_{\bar{z}}A_0^z)={\cal L}|_{\varphi=\varphi_0}={\cal L}_0=\partial_z\varphi_0\partial_{\bar{z}}\varphi_0-V(\varphi_0)
m_{\pm}^{2}= \frac{1}{2 Z^{2}}\left( m_{r}^{2}+ ( 1 + 6 \xi \gamma^{2}) m_{h}^{2}\pm ( ( m_{r}^{2}- m_{h}^{2}( 1 + 6 \xi \gamma^{2}) )^{2}+ 1 4 4 \gamma^{2}\xi^{2}m_{r}^{2}m_{h}^{2})^{1 / 2}\right)
\P\big(\sum_{j=1}^m \beta_j Z_j(p_j)>A \big)=\sum_{V\in \mathcal{V}}\prod_{j\in V}p_j\prod_{j\in V^c}(1-p_j)\geq \P\big(\sum_{j=1}^m \beta_j Z_j(q_j)>A \big) \inf_{V\in \mathcal{V}}\prod_{j\in V}\frac{p_j}{q_j}\prod_{j\in V^c}\frac{1-p_j}{1-q_j}.
\int_1^{\frac{n}{2}}\int_1^{k-1}\frac{dy_1dy_2}{\big[(k-y_1)^{\alpha_1}+(n-y_2)^{\alpha_2}\big](y_1^{\alpha_1}+y_2^{\alpha_2})}&=\int_{\frac{n}{2}}^{n-1}\int_1^{k-1}\frac{dtds}{(t^{\alpha_1}+s^{\alpha_2}) \big[(k-t)^{\alpha_1}+(n-s)^{\alpha_2}\big]}.
D_\lambda(S||f_{a,b}) = \frac{1}{2\sum_{j,k}(x_i - x_k)^2}\lambda^2 \sum_{j,k}[y_j - y_k - a (x_j - x_k)]^2 &+ \frac{1}{2\sum_{j,k}(x_i - x_k)^2}\sum_{j,k}[x_j y_k - x_k y_j - b (x_j - x_k)]^2,
s_2 = \frac{s_1 \cdot{r}^{\top}B_{11}{r}+ s_3 \cdot{r}^{\top}B_{12}{r}+m r_2 q_{12}}{{r}^{\top}Q_2{r}}+ O_g(\delta m^{1/2}), s_4 = \frac{s_1 \cdot{r}^{\top}B_{21}{r}+ s_3 \cdot{r}^{\top}B_{22}{r}+ m r_4 q_{12}}{{r}^{\top}Q_2{r}}+ O_g(\delta m^{1/2})
\beta_e = \frac{\rho}{w}\left[ \frac{\pi^{\frac{3}{2}}\lambda_e \exp \!\! \left(\! \frac{\theta^2}{4w}\! \right) \mathrm{Erfc}\! \left( \! \frac{\theta}{2 \sqrt{w}}\! \right )}{2 \ln \frac{1}{1 - \epsilon}}\right]^2,
\begin{matrix}Y_{6}=[1\,2\,6][3\,4\,5], Y_{7}=[1\,3\,6][2\,4\,5], Y_{8}=[1\,4\,6][2\,3\,5], Y_{9}=[1\,5\,6][2\,3\,4], Y_{10}=[1\,4\,5][2\,3\,6].\end{matrix}
\mathfrak{S}_B(\mathcal{L})=\prod_{p\nmid B}\Bigl(1-\frac{\omega(p)}{p}\Bigr)\Bigl(1-\frac{1}{p}\Bigr)^{-k}\ge \prod_{p\le k, p\nmid B}\frac{1}{p}\prod_{p> k,p\nmid B}\Bigl(1-\frac{k}{p}\Bigr)\Bigl(1-\frac{1}{p}\Bigr)^{-k}.
\mathcal{A}:=\{a=(i\to j)\in A\,\vert\,f(i)=f(j)\mbox{and}w_A(a)=s(A)\}, \mathcal{A}':=\{c=(i\to j)\in C\,\vert\,g(i)=g(j)\mbox{and}w_C(c)=s(A)\}.
\dot z^j_*(t) =g^{kj}(t,z_*(t)) &\cdot\Big(\lambda_k+\dot\gamma^i(t)\frac{\partial f_i}{\partial z^k}(t,z_*(t))+\int_t^\omega\dot\gamma^i(s)\frac{d}{ds}\Big(\frac{\partial f_i}{\partial z^k}(s,z_*(s))\Big)ds &-\int_t^{\omega}\frac{\partial L}{\partial z^k}(s,z_*(s),\dot z_*(s))ds -a_k(t,z_*(t))\Big).
W(x,x')=W(x)-\frac{1}{2}W_{;\alpha}(x) \sigma^{;\alpha}+\frac{1}{2}W_{\alpha \beta}(x)\sigma^{;\alpha}\sigma^{;\beta}+\frac{1}{4}\{\frac{1}{6}W_{;\alpha\beta\gamma}(x)-W_{\alpha\beta;\gamma}(x)\}\sigma^{;\alpha}\sigma^{;\beta}\sigma^{;\gamma}+0(\sigma^{2})
F ( \mu , y ) = 2 \mu \tau_{0}+ \frac{2 i}{\pi}\int_{0}^{\pi}\frac{d \theta}{\cos \theta}\, \ln \left[ \frac{\Gamma ( 1 - i \mu \cos \theta )}{\Gamma ( 1 - i \mu y \cos \theta ) \Gamma ( 1 - i \mu ( 1 - y ) \cos \theta )}\right] .
\exp \left\{-\frac{x^{2}}{2}+C\frac{x^{3}}{\sqrt{n}}\right\}=\exp \left\{-\frac{x^{2}}{2}\left(1-C\frac{x}{\sqrt{n}}\right)\right\}\leq\exp\left\{-\frac{x^{2}}{2(1+Cx/\sqrt{n})}\right\}.
\mathbf{\tilde{K}}_{l}:=2dn^{d}\left( 8+\left\Vert \mathbf{F}\right\Vert_{1,\mathfrak{L}}\right) \underset{z\in \mathfrak{L},|z|=1}{\sum}\frac{1}{\left\vert \Lambda_{l}\right\vert}\underset{x\in \mathfrak{L}}{\sum}\mathbf{1}\left[ \left( x,x+z\right) \in \partial (lb_{0})\right] \ .
1= \frac{F(DF^{-1}(\bar y))}{F(DF^{-1}(\bar x))}\le C\, \frac{|DF^{-1}(\bar y)|}{|DF^{-1}(\bar x)|}\, \eta_H\left(\frac{|DF^{-1}(\bar y)|}{|DF^{-1}(\bar x)|}\right)=C\, \eta_{1+H, 1+1/H}\left(\frac{|DF^{-1}(\bar y)|}{|DF^{-1}(\bar x)|}\right)
|Q'| =\frac{1}{\sqrt{|b|}}|\sqrt{a}+\sqrt{b}|\cdot|r+\sqrt{ab}|^n \geqslant \frac{|b-a|}{\sqrt{|b|}}\cdot\frac{1}{|\sqrt{b}-\sqrt{a}|}|\sqrt{ab}|^3 &\geqslant \frac{2}{\sqrt{|b|}}\cdot\frac{1}{|\sqrt{b}|+|\sqrt{a}|}|ab|^{3/2}\geqslant 12\frac{|b|}{|a|},
\Sigma_{\sigma,x+1}= \sum_{s = \pm 1}\left( \sum_{0 \leq n \leq x}\sum_{k=1}^{\left\lfloor \frac{\sqrt{24n+25}-s}{6}\right\rfloor}(-1)^{k+1}\frac{k(3k+s)}{2}\cdot p(x-n)\right).
F_2\left(a, b_1, b_2 ; c_1, c_2 ; x, y\right)=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{(a)_{m+n}\left(b_1\right)_m\left(b_2\right)_n}{\left(c_1\right)_m\left(c_2\right)_n}\frac{x^m y^n}{m ! n !}
\mathcal{H}_{n}= \{(\beta,\gamma) \mid \frac{s_1+s_2-s}{s_1}\gamma \geq \beta \geq \gamma \geq 0 \textrm{and}\sinh\beta\sinh \gamma \in [\sinh\beta^{(n)}\sinh\gamma^{(n)},\sinh\beta^{(n+1)}\sinh\gamma^{(n+1)}]\}.
E_{0}^{( p )}( \omega ) = \sum_{n = 0}^{p}{\frac{A_{n}}{4^{n}}}{\frac{\Gamma ( p + n / 2 + 1 / 2 )}{\Gamma ( 3 n / 2 + 1 / 2 ) \Gamma ( p - n + 1 )}}\omega ({\frac{\lambda}{\omega^{3}}})^{n},
\left(\Lambda^{-1}\vartheta_{j,k}\right)^\wedge(\xi) = - \chi_j(t) \frac{k \cdot \xi}{|\xi|}\hat K_{\approx 1}\left(\frac{\xi}{\lambda_{q+1}}- k\right) \left( a_{k,j}\psi_{q+1,j,k}\right)^\wedge (\xi - k\lambda_{q+1})
| \lambda_{j}> = \left( \begin{array}{c}{{H_{1 3}( H_{2 2}- \lambda_{j}) - H_{2 3}H_{1 2}}}\\{{H_{2 3}( H_{1 1}- \lambda_{j}) - H_{1 3}H_{2 1}}}\\{{| H_{1 2}|^{2}- ( H_{1 1}- \lambda_{j}) ( H_{2 2}- \lambda_{j})}}\\ \end{array}\right) ,
\begin{array}{@{}l@{}}{\displaystyle \sum_{k=1}^{M}{(a_{k}-a_{k-1})b_{k}}=a_{M}b_{M}-a_{0}b_{1}-\sum_{k=1}^{M-1}{a_{k}(b_{k+1}-b_{k})},}begin{align*}2mm]{\displaystyle \sum_{k=1}^{M}{(a_{k}-a_{k-1})b_{k}}=a_{M}b_{M}-a_{0}b_{0}-\sum_{k=1}^{M}{a_{k-1}(b_{k}-b_{k-1})}.}\end{array}
\frac{\vartheta(v+z_j+\lambda_j)\vartheta(u+ z_j+a) \vartheta(\lambda_j+2a)}{\vartheta(v+z_j) \vartheta(u+ z_j-a)}-\frac{\vartheta(u-v+ a)\vartheta(v+z_j+\lambda_j+2a) \vartheta(\lambda_j)}{\vartheta(u-v- a) \vartheta(v+z_j)}&-\frac{\vartheta(2a)\vartheta(u-v- a -\lambda_j) \vartheta(u+z_j+\lambda_j+a )}{\vartheta(u-v- a)\vartheta(u+z_j-a )}=0.
S [ \phi , h , B ] = \frac{1}{\pi}\int d^{2}x \{- \frac{1}{2}\bar{\partial}\phi^{i}\partial \phi^{i}+ \frac{1}{2}\partial \phi^{i}h \partial \phi^{i}+ \frac{1}{3}B d^{i j k}\partial \phi^{i}\partial \phi^{j}\partial \phi^{k}\}.
Y_t^{0,x;u_{\cdot}}= \xi + \int_t^T \hat{G}\bigl(s, X_s^{0,x;u_{\cdot}}, Y_s^{0,x;u_{\cdot}}, Z_s^{0,x;u_{\cdot}}\bigr) ds - \int_t^T Z_s^{0,x; u_{\cdot}}dB_s, \,\, t \in [0, T].
\beta^{(a)}(\tau)={2\pi i \large{\int_{-1}^1}dx\sqrt{{x\Theta_3^4+\Theta_2^4-\Theta_4^4\over x^2-1}}\over (\Theta_2^8-\Theta_2^4\Theta_3^4)\left[\large{\int_{-1}^1}dx \sqrt{{1\over (x^2-1)(x\Theta_3^4+\Theta_2^4-\Theta_4^4)}}\right]^3}.
\frac{R_{t t}}{A e^{2 \Omega}}+ A R_{r r}+ \frac{( d - 2 ) R_{\theta_{a}\theta_{a}}}{r^{2}\displaystyle{\prod_{b = 1}^{a - 1}\sin^{2}\theta_{b}}}= \frac{d - 2}{r^{d - 2}}\frac{d}{d r}\left[ r^{d - 3}( 1 - A ) \right] \; .
K(L,r)=\sum_{j=0}^{n-2}\ \binom{n-2}{j}(-1)^j \int_0^{\alpha_0(r)}(2\cos \theta)^{n-2-j}\left[\int_{A(\theta)}\frac 1s (2\cos\alpha(s,r))^j ds\right]\; d\theta+\cdots.
\frac{\sqrt[m]{K}n^2(m+1)^2}{4m\lambda}\left(\frac{m-1}{m+1}\right)^{\frac{m-1}{m}}\max\{1,\lambda^{n-1}\}\sum_{i=1}^{n}\frac{1}{\lambda^{n-i}}\prod_{j=i}^{n}(1+\frac{\sigma_j^2}{2j\lambda})<1
h_2(\beta) = \frac{\beta^2}{(1-\beta) \log (1-\beta) + \beta}= \frac{\beta^2}{-(1-\beta) \log (1 + \frac{\beta}{1-\beta}) + \beta}& \le \frac{\beta^2}{-(1-\beta)(\frac{\beta}{1-\beta}- \frac{1}{2}(\frac{\beta}{1-\beta})^2 + \frac{1}{3}(\frac{\beta}{1-\beta})^3) + \beta}= \frac{6(1-\beta)^2}{3 - 5\beta}\le 4.
\widetilde{W}( z ) / g_{M}= \sum_{j = 1}^{r}\log ( z - \cos 2 \bar{q}_{j}) + \frac{k_{1}+ k_{2}}{2}\log ( 1 - z ) + \frac{k_{2}}{2}\log ( 1 + z ) + ( \mathrm{~ z ~ - i n d e p .}) .
\mathbb{P}_{x}[\sigma_{n}\leq t]\leq\mathbb{P}_{x}\Biggl[\bigcup_{k=1}^{n-1}\{\sigma_{k}\leq t,\,\tau_{U}\circ\theta_{\sigma_{k}}\leq{\textstyle\frac{t}{n-1}}\}\Biggr]\leq\sum_{k=1}^{n-1}\mathbb{P}_{x}[\sigma_{k}\leq t,\,\tau_{U}\circ\theta_{\sigma_{k}}\leq{\textstyle\frac{t}{n-1}}].
L = \frac{1}{2 h^{2}}g^{\mu \nu}\partial_{\mu}\sigma \partial_{\nu}\sigma - \frac{N}{2 \lambda_{1}h^{2}}\sigma^{2}+ \frac{N}{2 \xi_{1}h^{2}}R \sigma^{2}- \frac{N}{4 ! \lambda_{2}h^{4}}\sigma^{4}+ \bar{\psi}( i \gamma^{\mu}( x ) \nabla_{\mu}- \sigma ) \psi ,
dq_t^m= \frac{1}{m}(p_t^m-\psi(t,q_t^m))dt, d(p_t^m)_i= \left(-\frac{1}{m}\gamma_{ij}(t,q_t^m)\delta^{jk}((p_t^m)_k-\psi_k(t,q_t^m))+\tilde F_i(t,x_t^m)-\partial_{q^i}V(t,q_t^m)\right. &\left.+\frac{1}{m}\partial_{q^i}\psi_k(t,q_t^m)\delta^{jk}((p_t^m)_j-\psi_j(t,q_t^m))\right)dt+\sigma_{i\rho}(t,q_t^m)dW^\rho_t.
\gamma_{\Psi}A^{* \mu}= 0 , \; \gamma_{\Psi}\phi^{*}= 0 , \; \gamma_{\Psi}C^{*}= - \partial_{\mu}A^{* \mu}, \; \gamma_{\Psi}b^{*}= - \overline{{{C}}}^{*}, \; \gamma_{\Psi}\overline{{{C}}}^{*}= 0 .
H_{D}= \int d^{3}\sigma [ - A_{\tau}( \tau , \vec{\sigma}) \Gamma ( \tau , \vec{\sigma}) + \lambda ( \tau , \vec{\sigma}) \pi^{\tau}( \tau , \vec{\sigma}) + \lambda^{\mu}( \tau , \vec{\sigma}){\cal H}_{\mu}( \tau , \vec{\sigma}) ] .
K_N(\lambda,\lambda')~=~ \sqrt{\lambda \lambda'}\prod_{f=1}^{\alpha}\sqrt{\lambda^2+m_f^2}\sqrt{{\lambda'}^2+m_f^2}e^{-\frac{N}{2}(V(\lambda^2)+V({\lambda'}^2))}\frac{1}{N}\sum_{j=0}^{N-1}\frac{Q_j(\lambda) Q_j(\lambda')}{r_j}~,
\varphi(f) = \frac{-u^{-1}u^{-6}v^{-4}(1-u^2)^2(u^2+uv^2-1)^2}{-u^{-8}v^{-4}(1+u)^2(1-u)^2(u^2v^2-u^2-2u-1)(u^2v^2+u^2-2u+1)}&= \frac{(u^2 + uv^2 - 1)^2u}{(u^2v^2 - u^2 - 2u -1)(u^2v^2 + u^2 - 2u+1)}&= f,
\|\Phi^{N,n,F}_{\varepsilon,\theta}(Z)-\Phi^{N,n,F}_{\varepsilon,\theta}(W)\| =\sqrt{N^F-1}\cdot\|Q^F(Z)-Q^F(W)\| &\le \frac{\sqrt{N^F-1}}{\min\{\lambda_n(Z), \lambda_n(W)\}}\|Z-W\|\leq \frac{1}{1-L(n,\varepsilon,\theta)}\|Z-W\|.
\Delta_{k}\theta(t_{1})-\Delta_{k}\theta(t_{2}) = \int_{t_{1}}^{t_{2}}{\frac{d}{d\tau}\Delta_{k}\theta(\tau)\,d\tau}= -\int_{t_{1}}^{t_{2}}{\Delta_{k}[(u\cdot\nabla)\theta+ \Lambda_{x_{1}}^{2\alpha}\theta +\Lambda_{x_{2}}^{2\beta}\theta](\tau)\,d\tau}.
M(d\mid x,k)\propto M(d)\prod_{i=1}^{d}\bigg{\{}1+\bigg{(}\frac{d}{i}\bigg{)}^{2k+1}\bigg{\}}^{-1/2}\exp\bigg{(}\sum_{i=1}^{d}\frac{x^{2}_{i}}{2\varepsilon^{2}}\frac{(d/i)^{2k+1}}{(d/i)^{2k+1}+1}\bigg{)},
H_n^{-}=\sum_{j=1}^n\left(\chi_j^{-}c_j^\dagger c_j+\chi_j^{+}c_{\kappa(j)}c_{\kappa(j)}^\dagger\right)=\sum_{j=1}^n\left(\chi_j^{-}c_j^\dagger c_j+\chi_{\kappa(j)}^{+}c_jc_j^\dagger\right)
\left\lbrace \begin{array}{c}\phi(+a/2)=e^{i\varphi}.\phi(-a/2) \partial_{x}\phi(+a/2)=e^{i\varphi}.\partial_{x}\phi(-a/2) \partial_{x}^{2}\phi(+a/2)=e^{i\varphi}.\partial_{x}^{2}\phi(-a/2) \partial_{x}^{3}\phi(+a/2)=e^{i\varphi}.\partial_{x}^{3}\phi(-a/2)\end{array}\right. .
\partial_{yy}w=-C_s\sum_{k=1}^{\infty}\big[(\lambda_k^{1/2}y)^{s-1}\mathcal{K}_{s-1}(\lambda_k^{1/2}y)-(\lambda_k^{1/2}y)^{s-1}\mathcal{K}_{s-2}(\lambda_k^{1/2}y)y\big]\frac{\langle f,\psi_{k}\rangle_{L^{2}(\Omega,\gamma)}}{\lambda_k^{s-1}}\psi_{k}(x).
D_{\mathcal{H}, \pm}\big(a._{\mathcal{H}_{-b}}\Psi\bigr)=a._{\mathcal{H}_{-b}}\bigl[D_{\mathcal{H}, \pm}\big(\Psi\bigr)\bigr]+ b\cdot\left( \mathsf{M}^{\mathbb{Z}}\mp \frac{1}{2\pi}\frac{\partial\;}{\partial \Theta}\right) (a)._{\mathcal{H}_{-b}}\Psi.
\lim_{T_0\to\infty}\frac{1}{T_0}\,\mathbb{E}\big[\widetilde{v}_{,,m}(f)\,\widetilde{v}^*_{,n}(f)\big] = 0\quad\forall n,m, \lim_{T_0\to\infty}\frac{1}{T_0}\,\mathbb{E}\big[\widetilde{v}_{,,m}(f)\,\widetilde{v}_{,m}^*(f)\big] = 0\quad\forall m.
D_{\circ}(ab)=\lim_{\alpha}D_{\alpha}(ab)= \lim_{\alpha}(\varphi\circ\psi(a)D_{\alpha}(b)+\varphi\circ\psi(b)D_{\alpha}(a)) = \lim_{\alpha}\varphi\circ\psi(a)D_{\alpha}(b)+\lim_{\alpha}\varphi\circ\psi(b)D_{\alpha}(a) = \varphi\circ\psi(a)D_{\circ}(b)+\varphi\circ\psi(b)D_{\circ}(a).
h_{\mu_{1}}^{\alpha_{1}\dot{\alpha}_{1}}...h_{\mu_{s-1}}^{\alpha_{s-1}\dot{\alpha}_{s-1}}\omega_{\mu_{s}|\alpha_{1}...\alpha_{s-1},\dot{\alpha}_{1}...\dot{\alpha}_{s-1}}\sim\Phi_{\mu_{1}...\mu_{s}}
{\bf X_L}:={\partial\over\partial t}+ \dot q^{s}{\partial\over\partial q^{s}}+a^{s}(q,\dot q){\partial\over\partial \dot q^{s}}+\eta^{\mu}{\bf\Gamma}_{\mu}=:{\bf X}_{0}+\eta^{\mu}{\bf\Gamma}_{\mu},
m_{q}(\|z\|_{X}) \sim_{q}\min\{\|z\|_{X}^{2},\|z\|_{X}^{q}\}\;\;\qquad, m_{q}(\|z\|_{X}) \sim_{q}\max\{\|z\|_{X}^{2},\|z\|_{X}^{q}\}\;\;\qquad\!
max_{a_{0}}\{F(x_{0},a_{0})+\beta[max_{\{a_{t}\}_{t=1}^{\infty}}\sum_{t=1}^{\infty}\beta^{t-1}F(x_{t},a_{t}):a_{t}\in\Gamma(x_{t}),x_{t+1}=T(x_{t},a_{t}),\forall t\ge1]\}
\begin{array}{l l}{\{\phi_{a}^{i}, \phi_{b}^{j}\}}&{= \Omega_{a b}^{i j}}\\{\{\phi_{a}^{i}, G_{b}\}}&{= C_{a b}^{c}\phi_{c}^{i}}\\{\{G_{a}, G_{b}\}}&{= C_{a b}^{c}G_{c}}\\ \end{array},
|\partial_{tr}N_{2}(f)|(t,r) \leq \begin{cases}\frac{C}{t^{5}\log^{3b}(t) (r^{2}+\lambda(t)^{2})}, r \leq \frac{t}{2}\frac{C \log^{3}(r)}{r^{2}|t-r|^{5}}+ \frac{C \log^{3}(r)}{r^{5/2}\sqrt{t}(t-r)^{4}}+ \frac{C \log^{2}(r)}{r^{2}t^{3/2}|t-r|^{3}\log^{3b-1+\frac{5N}{2}}(t)}, t>r>\frac{t}{2}\end{cases}
\int_\Sigma f^{p-1}\Delta f = -(p-1) \int_\Sigma f^{p-2}|\nabla f|^2 +\int_{\partial \Sigma}f^{p-1}N(f) \leq -(p-1) \int_\Sigma f^{p-2}|\nabla f|^2 + C(S)\int_{\partial \Sigma}f^{p-1}\tilde{H}^\sigma,
I({F};{Y})\leq \frac{4L^2\tilde{C}\|\psi\|_\infty^2\delta_n |K_{j_n}| n}{m}\sum_{i=1}^m\Big( (2^{12}\log(n)|K_{j_n}|^{-1}B^{(i)})\wedge 1\Big)+O(1).
\frac{12}{\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{5}+\frac{1}{7}+\frac{1}{10}+\frac{1}{14}+\frac{1}{20}+\frac{1}{28}+\frac{1}{35}+\frac{1}{70}+\frac{1}{140}}=5.
\#\tilde{E}_{q_1}(\mathbb{F}_{q_1}) = q_2\,, \ldots,\#\tilde{E}_{q_i}(\mathbb{F}_{q_{i}}) = q_{i+1}\,,\ldots,\#\tilde{E}_{q_m}(\mathbb{F}_{q_m}) = q_1\,.
U\left(\sum_{j\ge 1}d_{2\alpha}(j)\xi X^j\right) =\sum_{j\ge 1}d_{2\alpha}(j)U(\xi X^j)=\sum_{j\ge 1}d_{2\alpha}(j) \sum_{\ell\ge 1}b(j,\ell)X^\ell &=\sum_{\ell\ge 1}\left(\sum_{j\ge 1}b(j,\ell) d_{2\alpha}(j)\right)X^\ell=\sum_{\ell\ge 1}d_{2\alpha+1}(\ell)X^\ell.
\mathcal{R}_{1,1}(t) = \int_0^t \int_0^\xi e^{(t- s)\mathcal{L}}\mathcal{C}[f, \mathcal{L}] \left(e^{s\mathcal{L}}v, e^{s\mathcal{L}}e^{\xi \mathcal{A}}\overline v \right)d s d\xi
\int_0^1 f(x)\,dx =\int_0^1 [\sum_{j=0}^{n-1}[f(\frac{x+j}{n})-f(\frac{j}{n})]\,dx &=\sum_{j=0}^{n-1}n\int_{\frac{j}{n}}^{\frac{j+1}{n}}f(x)\,dx-\sum_{j=0}^{n-1}f(\frac{j}{n}) &=n\int_0^1 f(x)-\sum_{j=0}^{n-1}f(\frac{j}{n}).
\chi_{r , s ; \ell}^{( P , P^{\prime}; N )}( q ) = q^{- \frac{c}{2 4}+ \Delta_{r , s}+ \frac{\ell ( N - \ell )}{2 N ( N + 2 )}}\hat{\chi}_{r , s ; \ell}^{( P , P^{\prime}; N )}( q )
\widetilde{t}_{j,k}^{\left[\widetilde{p},q,L1a\right]}= \widetilde{t}_{j,k}^{\left[\widetilde{p},q\right]}+{q \over \widetilde{p}-1}\,\widetilde{t}_{j,k+1}^{\left[\widetilde{p}-1,q-1\right]}\,\left(t_{j,k}-t_{j,k+1}\right).
\chi_M =X^n+(\alpha_{n-1}+c_{n-1})X^{n-1}+(\alpha_{n-2}+\alpha_{n-1}c_{n-1}+c_{n-2})X^{n-2}+\ldots &+(\alpha_{n-i}+\alpha_{n-i+1}c_{n-1}+\ldots+\alpha_{n-1}c_{n-i+1}+c_{n-i})X^{n-i}+\ldots &+(\alpha_1+\alpha_2c_{n-1}+\ldots+\alpha_{n-1}c_2+c_1)X+(\alpha_0+\alpha_1c_{n-1}+\ldots+\alpha_{n-1}c_1+c_0),
d_{GH}(B(x^j, \frac{1}{\epsilon}r_{x^j}), B_{W^j}(w^j, \frac{1}{\epsilon}r_{x^j}))<\epsilon r_{x^j}; (W^j, w^j) = (\mathbb{R}^{2s}, 0)\times (H^j, (h^j)^*).
\Lambda^{\prime +}_{IR}(p,p)=-{{\lambda e^3}\over{2\pi^3}}\int{dk^+\over k^{+}}\int d^2k_\perp{\Theta_\Delta(k) \over{k_\perp^2 +{m^2k^{+2}\over p^{+2}}}}\biggl[ 1 -{{m^2k^{+2}\over 2p^+}\over{k_\perp^2 +{m^2k^{+2}\over p^{+2}}}}\biggr] \;.
\displaystyle \int_{H(F)}f(h)\xi(h)dh =meas(K_a)^{-1}\int_{K_a}\int_{H(F)}f(a(t)^{-1}ha(t))\xi(h)dhd^\times t =meas(K_a)^{-1}\int_{H(F)}f(h)\int_{K_a}\xi(a(t)ha(t)^{-1})d^\times tdh =meas(K_a)^{-1}\int_{H(F)}f(h)\int_{K_a}\psi\left(t\lambda(h)\right)\lvert t\rvert^{-1}dtdh
u_t - \sum_{j, \, l=1}^{n}(k_{j\,l}(t,x,y) u)_{x_j x_l}+ \sum_{j=1}^{n}(f_j(t,x,y) u)_{x_j}+ \sum_{j=1}^{n}x_j u_{y_j}= 0, (x,y,t) \in \mathbb{R}^{2n+1}.
\tilde Uf:=-\dfrac{1}{y}\sum_{\ell =-\infty}^\infty y^{-\ell}\sum_{r=0}^\infty\Bigl(\partial_\theta^r\theta^\ell\bigm|_{\theta=1}\Bigr)[w^r x^{\ell}](-x)^{r+1}e^{w/x}f(w,x).
\sum_{p \in \mathcal{S}(j)}(\log p) \left| L(\tfrac{1}{2},f \otimes \chi_{8p})\right|^2 \left|\mathcal{M}(p) \right |^2 \ll X \exp \Big(\frac{4}{\alpha_j}\Big) (\mathcal{I}-j)e^{-122/\alpha_{j+1}}\ll (\mathcal{I}-j)e^{-42/\alpha_{j+1}}X \ll e^{-1/(10\alpha_{j})}X.
\begin{array}{c}{{\beta_{1}= \frac 2 N \delta_{a b}\delta_{c d}+ d_{a b e}d_{c d e}}}\\{{\beta_{2}= \frac 2 N \delta_{a d}\delta_{b c}+ d_{a d e}d_{b c e}}}\\{{\beta_{3}= \frac 2 N \delta_{a c}\delta_{b d}+ d_{a c e}d_{b d e}}}\\ \end{array}
H(G) = F_1(G) \le \frac{1}{2}Per(A_G) & \le (1+o(1))\frac{1}{e}(\sqrt{2\pi})^{\frac{1}{p}-1}p^{\frac{1}{2p}}n^{\frac{1}{2}+\frac{1}{2p}}p^{n}\frac{(n-1)!}{2}& = (1+o(1))c_p n^{\frac{1}{2}+\frac{1}{2p}}E(n,p)
\Delta_T (x,\tau) = \Delta^> (x,-i\tau)= \int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\int_{-\infty}^{+\infty}\frac{d p}{2\pi}e^{-\omega \tau}e^{i px}\rho(\omega,p)(1+f(\omega)).
\left( \begin{array}{c}{{m^{\prime}}}\\{{n^{\prime}}}\\ \end{array}\right) = ( ( \Lambda_{0}^{5})^{- 1})^{T}\left( \begin{array}{c}{{0}}\\{{\bar{\Delta}_{( m^{\prime}, n^{\prime})}^{1 / 2}}}\\ \end{array}\right) .
{\vec{f}}_{i}( x ) = \sum_{j = 1}^{N}\, \sum_{k = 1}^{n}\, \left( 1 + x \Lambda + \frac{x^{2}}{2 !}\Lambda^{2}+ \cdots + \frac{x^{n - 1}}{( n - 1 ) !}\Lambda^{n - 1}\right)_{i k}C_{k j}{\vec{e}}_{j}.
\begin{array}{c c c}{K_{a b}^{- 1}=}{p \left( \begin{array}{c c}{\sqrt{g}}{{\frac{1}{\sqrt{g}}}}{{\frac{1}{\sqrt{g}}}}{\sqrt{g}}\end{array}\right)}{+ ( 1 - p ) \left( \begin{array}{c c}{g}{{\frac{1}{g}}}{{\frac{1}{g}}}{g}\end{array}\right)}\end{array}
0= \int_{Q}m(t,x) \left( \frac{\partial H}{\partial p}[m(t,\cdot)]( x, \nabla u(t,x)) - \frac{\partial H}{\partial p}[m(t,\cdot)]( x, \nabla \tilde u(t,x))\right)\cdot (\nabla u-\nabla \tilde u) (t,x).
= \sum_{0\le m_1\le m_2\le\ldots\le m_r}\frac{1}{m_1^{k_1}m_2^{k_2}\ldots m_r^{k_r}}\sum_{j=0}^{m_r}(-1)^j\binom{m_r}{j}\sum_{n\ge0}(rx-j)^n\frac{t^n}{n!}= \sum_{n\ge0}\left(\sum_{0\le m_1\le m_2\le\ldots\le m_r}\sum_{j=0}^{m_r}\frac{(-1)^{j}(rx-j)^n\binom{m_r}{j}}{m_1^{k_1}m_2^{k_2}\ldots m_r^{k_r}}\right)\frac{t^n}{n!}.
\begin{aligned}\frac{1}{\tau^k}[f_2(x_2)-f_2(\breve{x}_2^{k})]-\frac{1}{\tau^{k-1}}[f_2(x_2)-f_2(\breve{x}_2^{k-1})]+(x_2-\widetilde{x}_2^{k})^T[-A_2^T\widetilde{\lambda}^k &+\tau^k\beta^kD(\widetilde{x}_2^{k}-\bar{x}_2^k)+(1-\gamma)(1-\tau^k)\beta^kA_2^T(A\breve{x}^{k-1}-b)]\geq 0,~\forall x_2.\end{aligned}
b_{1}( x , y ) = \sum_{n = 1}^{\infty}\Bigl [ b_{1}^{( c , 2 n - 1 )}( x ) E c_{1}^{2 n - 1}( \omega y , k ) + b_{1}^{( s , 2 n - 1 )}( x ) E s_{1}^{2 n - 1}( \omega y , k ) \Bigr ] .
\widetilde G_{n0}\le \sqrt{\dfrac{2}{\pi}}\sum_{l=1}^\infty \left|{\langle u_T,{\phi_{2ln}}\rangle}\right|\le \sqrt{\frac{2}{\pi}}\sum\limits_{l = 1}^\infty \frac{\widetilde E_1}{2^{2\beta}l^{2\beta}n^{2\beta}}.
\varphi_t:T_t(B) \to \mathbb{H}^1(M_t,\Theta_{M_t}^\bullet) \frac{\partial}{\partial t}\mapsto \left[\rho_t\left(\frac{\partial}{\partial t}\right)\left(=\frac{\partial{M}_t}{\partial t}\right), \frac{\partial{\Lambda_t}}{\partial t}\right]=\frac{\partial (M_t,\Lambda_t)}{\partial t}
S_U =\bigcup_{i=1}^s\Omega_{\alpha_i}=\bigcup_{i=1}^s\big(G(\ell,m)\cap Z_{\overline{\mathbb F}_q}(x_{\beta}:\beta\in I(\ell,m), \; \beta\not\leq\alpha_i)\big) &=G(\ell,m)\cap \bigcup_{i=1}^sZ_{\overline{\mathbb F}_q}\big(x_{\beta}:\beta\in I(\ell,m), \; \beta\not\leq\alpha_i\big).
H = \frac{J}{2}\left[ \sum_{n = 1}^{N - 1}\left( \sigma_{n}^{x}\sigma_{n + 1}^{x}+ \sigma_{n}^{y}\sigma_{n + 1}^{y}+ c o s \gamma \; \sigma_{n}^{z}\sigma_{n + 1}^{z}\right) + \right.
\dot{\cal F}_{\epsilon}^{i}= - \epsilon \varepsilon^{0 i j}\, \Delta_{j k}\,{\cal F}_{\epsilon}^{k}, \qquad \Delta_{i j}= m \delta_{i j}- \frac{1}{m}\varepsilon_{0 i k}\partial^{k}\varepsilon_{0 j l}\partial^{l}.