harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f ( \hat{Z}_{-}) = \Bigl ( 1 -{\frac{Z_{-}^{2}}{2 N}}\Bigr ) \rho +{\cal O}( 1 / N^{2}) , \quad f ( \hat{Z}_{1}) = \Bigl ( 1 -{\frac{Z_{1}^{2}}{2 N}}\Bigr ) \rho +{\cal O}( 1 / N^{2}) .
I(Z^{j-1},X_j^{i-1};X_i|W,S,A^n,Y^n) &\;=I(Z^{j-1},X_{j+1}^{i-1};X_i|W,S,A^n,Y^n) &\;+I(X_j;X_i|W,S,A^n,Y^n,Z^{j-1},X_{j+1}^{i-1}) &\;\overset{(a)}{\geq}I(Z^{j-1},X_{j+1}^{i-1};X_i|W,S,A^n,Y^n) &\;+I(Z_j;X_i|W,S,A^n,Y^n,Z^{j-1},X_{j+1}^{i-1}) &\;=I(Z^{j},X_{j+1}^{i-1};X_i|W,S,A^n,Y^n)
V_{-}( x , a_{0}) = \frac{( \omega_{0}- \omega_{1}) ( 3 \omega_{0}+ \omega_{1})}{4{( \omega_{0}+ \omega_{1})}^{2}x^{2}}- \frac{\omega_{1}}{2}+ \frac{{( \omega_{0}+ \omega_{1})}^{2}x^{2}}{1 6}.
[{\cal F},{\cal H}_{A}]=\frac{\delta{\cal F}}{\delta g_{ij}}\frac{\delta{\cal H_{A}}}{\delta p^{ij(A)}}-\frac{\delta{\cal F}}{\delta p^{ij(A)}}\frac{\delta{\cal H_{A}}}{\delta g_{ij}}=\frac{\delta{\cal F}}{\delta s^{A}},
F_\varepsilon(u):=\int_0^{\infty}\int_{\mathbb{R}^n}e^{-t/\varepsilon}\,\left(|u''(t,x)|^2+\tfrac{1}{\varepsilon^2}|\nabla u(t,x)|^2+\tfrac{1}{\varepsilon^2}|u(t,x)|^{2k}\right)\,dx\,dt
Z(s)=e^{-\Gamma (s)}=\exp (-\eta \int d^3x\frac i2\widehat{s}\varepsilon\partial \widehat{s})=\int{\cal D}A_\mu \exp \int d^3x(-\frac i{2\eta}A\varepsilon \partial A-iA\varepsilon \partial \widehat{s})\;.
\sum_{b=a+2}^{n-2}\sum_{m=2}^{b-a}\binom{b-a-1}{m-2}\left[(b-m+1)(n-b-1)+\binom{n-b-2}{2}\right] &=2^{n-a-2}(n-4+a)-\frac{1}{6}(n^3-6n^2-(3a^2-9a-11)n+2a^3-3a^2-5a-18).
p_x (nxb) = \sum_{i \in I}n_i x e_{\mathcal{B}}x^\ast n_i^\ast nxb = \sum_{i \in I}n_i x \tau( x^\ast x) \tau(n_i^\ast n) b = \sum_{i \in I}n_i x \tau(n_i^\ast n) b = nxb.
k_{2}^{(2)}(\tau_{1},\tau_{2})=\frac{1}{2!A_{2}^{2}}\{E\{y^{(2)}(n)\prod_{i=1}^{2}x^{(2)}(n-\tau_{i})\}-A_{2}k_{0}^{(2)}\delta_{\tau_{1}\tau_{2}}\},
||h_0||^2 =\langle\gamma_1g+\gamma_2,\gamma_1g+\gamma_2\rangle &=|\gamma_1|^2+|\gamma_2|^2+2\Re\{\gamma_1\overline{\gamma_2}g(0)\}&=|c_0|^2(1-|a|^2)(1+|a|^2)^2.
\sigma_{\mathrm{s t r i n g}}^{( \mathrm{o p e n})}( D ) = A ( D ) \beta_{H}^{- \frac{( D + 1 )}{2}}\: r_{S}^{D - 2}\: \left( \frac{c^{2}\beta_{S}}{b}\right)^{- \frac{( D - 3 )}{2}}\: \cdot \: \lbrace \frac{\beta_{H}}{\beta_{H}- \beta_{S}}\: e^{- ( \beta_{H}- \beta_{S}) c^{2}m_{0}}\: - \: E_{i}\left( - ( \beta_{H}- \beta_{S}) c^{2}m_{0}\right) \rbrace
\mathbf{H}=\begin{pmatrix}H_{11}iH_{12}iH_{14}-iH_{12}H_{22}H_{24}-iH_{14}H_{24}H_{44}\end{pmatrix}, \mathbf{W}=\begin{pmatrix}W_{11}iW_{12}iW_{14}-iW_{12}W_{22}W_{24}-iW_{14}W_{24}W_{44}\end{pmatrix}.
\exp\left(N\left(\int_\mathbb{T}\log\left(1+\sum_{k=1}^3\rho_{k-1}(u)\left(\exp\left(f_k(u)\right)-1\right)\right)du+o(1)\right)\right) &\geq P(\mu^N\in C)\times &\exp\left(N\left(\inf_{W\in C}\left(\sum_{k=1}^3W_{0,k}(f_k)+\mathcal{I}_1(W,F,G,H)\right)\right)\right).
Q_g^6= \frac{n^4-20n^2+64}{32n^2(n-1)^3}R_g^3, -P_g^6= \Delta_g^3+\frac{-3n^2+6n+32}{4n(n-1)}R_g\Delta_g^2+\frac{3n^4-12n^3-52n^2+128n+192}{16n^2(n-1)^2}R_g^2\Delta_g-\frac{n-6}{2}Q_g^6.
\frac{1}{2t_2}\leq |z(t_1) - z(t_2)| = | \int\limits_{t_1}^{t_2}z'(t) dt | = | \int\limits_{t_1}^{t_2}\frac{f(t)}{\theta_z (t, z(t))}dt | \leq \int\limits_{t_1}^{t_2}\frac{|f(t) |}{|\theta_z (t, z(t)) |}dt \leq \int\limits_{t_1}^{t_2}4 y(t) |f(t)| dt \leq \frac{16}{t_2}\int\limits_{t_1}^{t_2}|f(t)| dt \leq \frac{16 \delta}{t_2}.
\frac{\sum_{y \in J_\alpha}\P(S^b_\delta (\tau_U)=y|S^b_\delta (\tau_D) \in A)\P ( S^y_\delta (\tau_D)=a_\delta | S^y_\delta (\tau_D) \in A)}{\sum_{y \not\in J_\alpha}\P(S^b_\delta (\tau_U)=y|S^b_\delta (\tau_D) \in A)\P ( S^y_\delta (\tau_D)=a_\delta | S^y_\delta (\tau_D) \in A)}.
m^k_t = \int_{[0, \max\{0,(\tau_k)^{-1}(t)\}]}\,\delta_{\Phi^{e_k}_t(y,\,0)}dm^k_0(x) + \int_{(\max\{0, (\varsigma_k)^{-1}(t)\},t]}\delta_{\Phi^{e_k}_t(0,\,s)}d m_{x=V}^k(s).
R_{s,2}^{g_1, g_2, g_3}(f)= \int_{\mathbb{T}^2}\lambda(u,v)\theta_s^S(u)\theta_s^I(v)f(u)(g_2(u)-g_1(u))dudv &-\int_{\mathbb{T}}\psi(u)\theta_s^E(u)f(u)(g_3(u)-g_2(u))du
\lim_{m^2 \to 0}m^2\int_0^{\infty}dt\,e^{-t m^2}\int d^2 r\int_0^M dk\,k \,e^{-k^2 t}\int_0^{2 \pi}d \Theta\,( | \psi^{(+)}({\bf k, r})|^2 - |\psi_0({\bf k, r})|^2 )= 0\,.
a^\prime = \frac{-a^2+b^2+c^2}{2bc}\,, b^\prime = \frac{a^2-b^2+c^2}{2ac}\,, c^\prime = \frac{a^2+b^2-c^2}{2ab}\,, &a(0)=a_0\,,b(0)=b_0\,,c(0)=c_0\,,
S_{0}^{( \Phi )}= - \sum_{t = 1}^{T}\ln \left\{\sum_{r = 0}^{\Omega}\frac{1}{r !}\left[ \Phi^{*}( t ) \right]^{r}\sum_{s = 0}^{r}{\binom{r}{s}}\frac{( \Omega - s ) ! \left[ \Omega - ( r - s ) \right] !}{( \Omega ! )^{2}}x^{2 ( r - s )}\Phi^{s}( t ) \Phi^{r - s}( t - 1 ) \right\}\ .
\begin{aligned}(\tilde{\Delta}_E\tilde{h},\tilde{h})_{L^2(\tilde{g})}=(n+1)\int_{0}^{\infty}(\varphi')^2\sinh^{n}dr+(n+1)\lambda_i\int_{0}^{\infty}\varphi^2\sinh^{n-2}dr &\qquad+2n(n+1)\int_0^{\infty}\varphi^2\sinh^ndr\geq2n\left\|\tilde{h}\right\|^2_{L^2(\tilde{g})}.\end{aligned}
\mbox{$N=4$ WZ gauge:}\qquad A^{++}= (\theta^+)^{2ab}\phi_{ab}(x) +(\theta^+)^3_au^{-\alpha'}\chi^{a}_{\alpha'}(x) + (\theta^+)^4 u^{-\alpha'}u^{-\beta'}G_{\alpha'\beta'}(x)\ ,
\varphi(m_{i}) \log m_{i}\ge \varphi(m_{i_0}) \log m_{i_0}+ \sum_{j=i_0}^{i-1}\min \left( \varphi(m_1) \log \log m_i , \ \log m_i \right) &\ge \varphi(m_{i_0}) \log m_{i_0}+ \varphi(m_1) \sum_{j=i_0}^{i-1}\log \log m_j,
2 \nabla_{[ \rho}\kappa_{\mu ] \nu}{}^{\rho}= i \nabla_{[ \rho}\Big ( ( \sigma_{\mu ]})^{\alpha \dot{\beta}}J_{\nu}{}^{\rho}{}_{\alpha \dot{\beta}}+ \{\mathrm{p e r m . a c c . t o ( \ref{e q : k a p p a_d e f})}\}\Big )
\partial_{\hat \xi}u(\hat \xi, 0) - \partial_{\hat \xi}w(\hat \xi, 0) \leq \partial_{\hat \xi}u(\hat \xi, 0) - \partial_{\hat \xi}w(0, 0) + [ \partial_{\hat \xi}\hat w]_{0,\beta}|\hat{\xi}|^\beta = \left[ U_\beta (1+\beta) \cos{((1+\beta) \frac{\pi}{2})}+ [ \partial_{\hat \xi}\hat w]_{0,\beta}\right ] |\hat \xi|^\beta,
\frac{(-1)^{|V(T)|}}{|{\rm Aut}(\Gamma)|}\sum_{j=0}^{n}{\xi_{\Gamma}}_*\left(\prod_{i=1}^{n}\left(1+3\omega_i\right) \prod_{v\in V(T)}\frac{1}{1+3\omega_{r'}}\prod_{(h,h')\in E_2(G)}\frac{1}{\psi_h-(1+3\omega_{h'})}\right)\cdot (-\lambda - \delta_1)^{j}.
( p^{+}- r^{+}) \int_{M}| D \mathrm{u}( z ) |^{p( z )}\,\, dv_{g}( z ) + ( q^{+}- r^{+}) \int_{M}\mu ( z ) | D \mathrm{u}( z ) |^{q( z )}\,\, dv_{g}( z ) & = ( r^{-}+ \gamma^{-}- 1) \int_{M}g( z ) | \mathrm{u}( z ) |^{1 - \gamma ( z )}\,\,dv_{g}( z ).
\|a\|_* \le \frac{\|g\|_*- \|g_1\|_*}2+2 \le \frac{r-2}2+2=\frac r2+1, \|w_3\|_* \le \frac{\|h\|_*- \|g_1^{w_2}\|_*}2+2\le \frac{r-2}2+2=\frac r2+1, \|w\|_* \le \|a\|_* + \|w_1\|_*+ \|w_2\|_* + \|w_3\|_* &\le \left(\frac r2+1\right) + 0 + \|w_2\|_* + \left(\frac r2+1\right)=\|w_2\|_*+r+2.
\frac{|\xi_{m-1}(x)^d-\xi_{m-1}(y)^d|}{|\xi_{m-1}(x)^d|}=\left|1-e^{d \log \frac{\xi_{m-1}(y)}{\xi_{m-1}(x)}}\right| & \le d \frac{|\xi_{m-1}(x)-\xi_{m-1}(y)|}{|\xi_{m-1}(x)|}\left(1+2\frac{|\xi_{m-1}(x)-\xi_{m-1}(y)|}{|\xi_{m-1}(x)|}\right) & \le d \frac{|\xi_{m-1}(x)-\xi_{m-1}(y)|}{|\xi_{m-1}(x)|}\left(1+\frac{2\mu}{m^2}\right),
\Omega^{( 2 )}= 3 2 \pi^{2}{\frac{a^{1 0}}{( \rho^{2}+ a^{2})^{5}}}\cos \beta \, d \alpha \, d \beta , \quad \quad{\cal A}= \int_{\cal M}\omega^{( 3 )}= 6 4 \pi^{3}.
{}_J\Delta_{\nu}(x)=\frac{2}{\pi}\int_0^{\frac{\pi}{2}}J_{n+2\nu+1}(2x\cos\theta)\left[\cos(k\theta)-\cos((k+2)\theta)\right]d\theta=\frac{1}{\pi x}\int_0^{2x}J_{n+2\nu+1}(u)\Phi(u)du,
\psi_p' \left(\begin{pmatrix}1 u_1 * \cdots * 1 u_2 \cdots * \cdots \cdots \cdots 1 u_{n-1}1\end{pmatrix}\right) = \xi( \nu_1' u_1+ \nu_2' u_2+ \nu_3' u_3+\cdots+ \nu_{n-1}' u_{n-1}),
\Phi^{\prime}= \frac{( 2 \varepsilon ( \hat{\bf p}) )^{- 1 / 2}}{( \varepsilon ( \hat{\bf p}) + m c^{2})^{\frac{1}{2}}}\left( ( \varepsilon ( \hat{\bf p}) + m c^{2}) \Phi ( \varepsilon ( \hat{\bf p}) + m c^{2}) + c^{2}\vec{\sigma}{\bf \hat{p}}\Phi \vec{\sigma}{\bf \hat{p}}\right) \frac{( 2 \varepsilon ( \hat{\bf p}) )^{- 1 / 2}}{( \varepsilon ( \hat{\bf p}) + m c^{2})^{\frac{1}{2}}}\, ;
c((\iota_{D}^{*}W^{+}_{i})^{\vee}(\nu^{j}))= (1-2x)^{(r''-g_{Y})}\cdot\prod_{i=1}^{g_{Y}}(1-4\sigma_{i}-2x), =\prod_{\begin{array}{c}\mbox{\scriptsize Chern roots $\alpha$ of $(\iota_{D}^{*}W^{+}_{i})^{\vee}(\nu^{j})$}\end{array}}(1+{\alpha}).
(1-A_{E(n)}^t) (\varphi_n(b)) = (1-A_{E(n)}^t) \Big(\sum_{i=0}^{n-1}(A_{E(n)}^i)^t (\iota_n(b)) \Big) = (1-A_{E(n)}^n)^t (\iota_n(b)) = \iota_n \big( (1 - A_{E(n)}^n)^t (b) \big)= 0.
{dn^{(\lambda)}(\omega) \over d\omega}=\int_{\Sigma_r}d\Sigma^\mu \sum_k\tilde{j}_\mu (\phi^{(\lambda)}_{\omega,k})=\int_{\Sigma_r}\sqrt{h}d^{D-1}x\sum_k2\omega |\phi^{(\lambda)}_{\omega,k}|^2~~~.
\lambda_{1,\Delta \left( 0\right)}\left( t\right) dt=\left[ z^{-1}\right]\psi_{t}\left( z\right) \cdot \lambda_{d}\left( t\right) \cdot \mathbf{E}_{\Delta \left( 0\right)}\left( N_{d}\left( t\right) +1\mid \Delta \left(t\right) =-1\right) dt
g_k= (y_\alpha-y_\beta)^{d-a_i}\sum_{j=0}^{a_i-k}\lambda_j(y_\alpha-y_\beta)^j(y_\alpha+y_\beta)^{a_i-k-j}= (\sum_{s=0}^{d-a_i}(-1)^s\binom{d-a_i}{s}y^s_\alpha y^{d-a_i-s}_\beta)( \sum_{t=0}^{a_i-k}\lambda_t(y_\alpha-y_\beta)^t(y_\alpha+y_\beta)^{a_i-k-t})
\bigl\langle G_R(\Lambda_{\psi}(p)\otimes\Lambda(b)),\Lambda_{\psi}(q)\otimes\Lambda(d)\bigr\rangle &=\eta\bigl(d^*\,(\psi\otimes\operatorname{id})[\Delta(q^*p)(1\otimes b)]\bigr)=(\psi\otimes\eta)\bigl((1\otimes d^*)\Delta(q^*)\Delta(p)(1\otimes b)\bigr).
\overline{JO_{n}^{(3)}}=J_{n}^{(3)}-\sum_{s=1}^{7}J_{n+s}^{(3)}e_{s},\ \overline{jO_{n}^{(3)}}=j_{n}^{(3)}-\sum_{s=1}^{7}j_{n+s}^{(3)}e_{s},
\sum_{k \geq 0}\left(32 k^5 + 240 k^4 + 680 k^3 + 900 k^2 + 548 k + 120 \right) c_k^{\ast}z^k = \phantom{=}120 C(z) + 2400 z C^{\prime}(z) + 5100 z^2 C^{\prime\prime}(z) + 2920 z^3 C^{(3)}(z) + 560 z^4 C^{(4)}(z) + 32 z^5 C^{(5)}(z),
\langle s,t\rangle_{\mathcal{E}^0}(g) =\mu(g)^{-1/2}\sum_{k}\langle\sqrt{\rho_k}s,\sqrt{\rho_k}gt\rangle_{L^2\left(E|_{U_k}\right)}&=\sum_{k}\langle\sqrt{\rho_k}s,\sqrt{\rho_k}t\rangle_{\mathcal{E}^0_{U_k}}(g),
x=\frac{1}{24}+\frac{6\left(2r-1\right)^{2}}{\pi^{2}}W\left(-\frac{\sqrt{46}\pi}{24(2r-1)}\left(\frac{\pi\left(\frac{1}{2}\right)_{r-1}}{20\sqrt{6}}\right)^{\frac{1}{2r-1}}\right)^{2}.
A =t^8+5t^5+7t^2 B =-t^9-7t^6-12t^3-9 C =t^{10}+6t^7+12t^4+12t D_f =t^{18}+10t^{15}+41t^{12}+90t^9+102t^6+40t^3-27 \Delta =3t^{18}+30t^{15}+123t^{12}+270t^9+306t^6+120t^3-81
\|f-{\mathsf{C}}_{d_x,d_y}[f]\|_1 \le \dfrac{4V_{k,l}}{kl\pi^2}\begin{cases}\Pi_{1}[s](d_x)\Pi_{1}[r](d_y),~~~~~~~~~\mbox{if}k=2s, l=2r, \Pi_{1}[s](d_x)\Pi_{0}[r](d_y),~~~~~~~~~\mbox{if}k=2s, l=2r+1, \Pi_{0}[s](d_x)\Pi_{1}[r](d_y),~~~~~~~~~\mbox{if}k=2s+1, l=2r, \Pi_{0}[s](d_x)\Pi_{0}[r](d_y),~~~~~~~~~\mbox{if}k=2s+1, l=2r+1,\end{cases}
\hat{\varphi}(\xi) = \frac{\left( (i\xi)^3+r(i\xi)+c\lambda \right)\left(\alpha+\beta e^{-iL\xi}\right)}{(1-a^2b)(i\xi)^6+r(i\xi)^4+(c+1)\lambda(i\xi)^3+r\lambda(i\xi)+c\lambda^2}.
{\mathbb E}\,f(W_\infty) =\frac{1}{{\mathbb E}(N+1)}{\mathbb E}\sum_{n=0}^N f(W_n)=p\,{\mathbb E}\sum_{n=0}^N f(W_n) &=p\,{\mathbb E}\sum_{n=0}^\infty f(W_n)1_{\{N\ge n\}}=\sum_{n=0}^\infty{\mathbb E}\,f(W_n)(1-p)^np &=\sum_{n=0}^\infty{\mathbb E}\,f(\bar{S}_n)(1-p)^np={\mathbb E}\,f(\bar{S}_N)\,,
L_{11}\leq C\Big(\int_{0}^{1}\Big(1+\frac{\sqrt{t}}{\rho(x)}\Big)^{-2(N-1)}\Big(\frac{\sqrt{t}}{\rho(x)}\Big)^{2(M-n-1)}\Big)^{1/2}M_{V,\eta}f(x) &\leq CM_{V,\eta}f(x).
\zeta^{\mathfrak{m}}(\{2\}^{k_{d}},3,\dots\{2\}^{k_{1}},3,\{2\}^{k_{0}}) =(-1)^{k_{0}+\cdots+k_{d}+d}I^{\mathfrak{m}}(\overbrace{0,1,\dots,0,1,0}^{2k_{d}+3},\overbrace{0,1,\dots,0,1,0}^{2k_{d-1}+3},\dots,\overbrace{0,1,\dots,0,1,0}^{2k_{1}+3},\overbrace{0,1,\dots,0,1}^{2k_{0}+2}) =(-1)^{k_{0}+\cdots+k_{d}}L_{B}(x_{2k_{0}+2}x_{2k_{1}+3}\cdots x_{2k_{d}+3}),
S ( T ) = - \frac{\partial F}{\partial T}= \Delta n^{2}\left( \frac{3 \zeta ( 3 )}{4 \pi^{2}}t^{2}- \frac{7}{1 3 5}t^{3}+ \frac{8 8}{4 7 2 5}t^{5}- \frac{7 3 6}{5 5 1 2 5}t^{7}+{\cal O}( t^{9}) \right){.}
\begin{array}{lcl}\nu(\sup\{a_1, a_2, \ldots, a_n \}) = \nu(\sup\{a_1, \sup\{a_2, \ldots, a_n \}\}) &= \sup\{\nu(a_1), \nu(\sup\{a_2, \ldots, a_n \})\}&= \sup\{\nu(a_1), \sup\{\nu(a_2), \ldots, \nu(a_n)\}\}&= \sup\{\nu(a_1), \nu(a_2), \ldots, \nu(a_n)\}. \end{array}
\left. C_{2}^{\left( N,M|R,S\right)}\right\vert_{N+M=R+S}=C_{2}^{SU\left(C\right)}+\frac{1}{2C}\left( \Delta+\left( N-R\right) C\right)^{2}+\frac{C}{2}\left( \Delta+\left( N-R\right) C\right) .
\begin{pmatrix}b_1 \vdots b_{j-t}\end{pmatrix}=\begin{pmatrix}c_{1,f_1}^{(t)}\hdots c_{1,f_{l-t}}^{(t)}\vdots \vdots c_{j-t,f_{1}}^{(t)}\hdots c_{j-t,f_{l-t}}^{(t)}\end{pmatrix}\begin{pmatrix}s_1 \vdots s_{l-t}\end{pmatrix}
(\partial + \Gamma)^{k+1}((\partial^{|\beta|}\eta)(\tilde{\mathbf{x}}_i) (\tilde{\mathbf{x}}-\tilde{\mathbf{x}}_i)^\beta) = (\partial^{|\beta|}\eta)(\tilde{\mathbf{x}}_i) \sum \partial^{a_1}\Gamma \cdots \partial^{a_\ell}\Gamma \cdot \partial^b (\tilde{\mathbf{x}}-\tilde{\mathbf{x}}_i)^\beta,
- \log \left\vert \det\left( 1-\mathcal{L}_{s,\lambda}^{N}\right) \right\vert = \mathrm{Re}\sum_{n=1}^{\infty}\frac{1}{n}\mathrm{Tr}\, \left(\mathcal{L}_{s,\lambda}^{nN}\right) \leq \sum_{n=1}^{\infty}\frac{1}{n}\left\vert \mathrm{Tr}\, \left(\mathcal{L}_{s,\lambda}^{nN}\right) \right\vert.
\tau^{(kerF_\ast)^\bot}(F)= - \frac{\lambda^2}{2}\sum_{j=1}^{n}\left\{2\lambda^2 X_j \left(\frac{1}{\lambda^2}\right)\tilde{X_j}- g(\lambda X_j, \lambda X_j) F_\ast\left(\nabla_{\mathcal{H}}\frac{1}{\lambda^2}\right) \right\}.
u_{\Omega , A}^{( \sigma )}= \int_{- \infty}^{\infty}\! d k \sum_{M , N}\left( \alpha_{\Omega}^{k \{M , N \}( \sigma )}U_{k}+ \beta_{\Omega}^{k \{M , N \}( \sigma )}U_{k}^{*}\right) \Psi_{\{M , N \}}\cdot \phi_{A}\ ,
\begin{aligned}0 = t_{n_0^0}^j(0) \leq t_{n_0^0+1}^j(0) = \dots = t_{n_0^0+q'}^j(0) &\leq t_{n_0^0+q'+1}^j(0) = \dots = t_{n_0^0+2q'}^j(0) &\leq \dots \leq t_{k-n_0^1+1}^j(0) = 1, \end{aligned}
\Big[\mu_{k_1}, \big[\mu_{k_2},\big[\ldots\big[\mu_{k_{r-1}},[\mu_{k_r}, w_{\alpha}(J_1, p)]\big]\ldots\Big] \Big[\mu_{l_1}, \big[\mu_{l_2},\big[\ldots\big[\mu_{l_{s-1}},[\mu_{l_s}, w_{\beta}(J_2, p)]\big]\ldots\Big] J\setminus J_2 = \{k_1,\ldots, k_r\}J\setminus J_1 = \{l_1,\ldots, l_s\}
\phi_{\Phi_k, \beta}(x, y) = \frac{1}{2\pi i}\left(\int_{b-i \tau}^{b+i \tau}+ \int_{b-i U}^{b-i \tau}+\int_{b+i \tau}^{b+i U}\right) \left(\sum_{n\le x}\left(\frac{\Phi_k(n)}{n^\beta}\right)^{-z}\right)\frac{y^z}{z}\,dz +O\left( \frac{y^{b}}{U}\sum_{n \leq x}\Phi_k(n)^{-b}n^{b\beta}\right).
\left| \sum_{h=1}^N \frac{\lambda_hk_h}{1+\sqrt{1+\frac{\lambda_j}{c^2}}}\right| \leq \frac{r \lambda_N}{1+\sqrt{1+\lambda_N/c^2}}\leq \frac{r\lambda_N}{1+\sqrt{1+\lambda_N/\lambda_l}}\leq \frac{r\lambda_N}{2},
\overline g_{M_n,n}(x,t)= \dfrac{1}{n}\sum_{k=1}^n \widetilde g(x_k,t)+\sum_{p=1}^{M_n}\Big[ \dfrac{\pi}{n}\sum_{k=1}^n \widetilde g(x_k,t) \phi_p(x_k) \Big] \phi_p(x),
\begin{array}{l}<((\phi_{\tau},f_{1})-E[(\phi_{\tau},f_{1})])((\phi_{\tau},f_{2})-E [(\phi_{\tau},f_{2})]) ....\cr((\phi_{\tau},f_{2n})-E[(\phi_{\tau},f_{2n})])>=\hbar^{2n}\sum_{pairs}\prod_{(j,k)}\frac{1}{2i}(f_{j},{\cal A}^{-1}f_{k})\end{array}
I_4= \int \nabla \phi^2\cdot \partial^\zeta_x Q(v) \cdot \partial^\zeta_x u \leq C|\phi|_\infty|\nabla \phi|_\infty \|\nabla v\|_2 | \partial^\zeta_x u|_2\leq Cc^3_3 |\partial^\zeta_x u |_2, |\zeta|\leq 2.
x_{n+1,i}= \begin{cases}T_{p_i}(x_{n,i}), \textnormal{if}i=1\textnormal{or}2^{12\cdot 2^i}<n-1, \langle 2^{\lceil\log_2(i)\rceil+1}x_{n,i}\rangle, \textnormal{if}i\geq 2\textnormal{and}2^{12\cdot 2^i}\geq n-1. \end{cases}
\lim_{t\rightarrow t'}{d\over dt'}tan^{-1}\biggl({x_2(t')-x_2(t)\over x_1(t')-x_1(t)}\biggr)={1\over 2}{d\over dt}tan^{-1}\biggl({{\dot x}_2(t)\over{\dot x}_1(t)}\biggr)\,.
{\pi}^-=\frac{{\delta}L}{{\delta}{\partial}_-A_-}=0,\;{\pi}^+=\frac{{\delta}L}{{\delta}{\partial}_-A_+}=F_{-+},\;{\pi}_B=\frac{{\delta}L}{{\delta}{\partial}_+B}=0,\;{\pi}_{\psi}=\frac{{\delta}L}{{\delta}{\partial}_-{\psi}}=i\bar{\psi}{\gamma}^-.
h_{a,b}(x):=\left\lbrace\begin{aligned}\mathbf{E}_x(T_a\wedge T_b),x\in [a,b]\cap E, &\frac{h_{a,b}(b_k)-h_{a,b}(a_k)}{b_k-a_k}\cdot (x-a_k)+h_{a,b}(a_k),x\in (a_k,b_k)\subset (a,b), k\geq 1. \end{aligned}\right.
\begin{aligned}\sup_{a\leq t\leq b}\|u(\cdot,t)\|_{L^2(e^{\beta x}dx)}&\leq K (\|u_a\|_{L^2(e^{\beta x}dx)}+ \|u_b\|_{L^2(e^{\beta x}dx)}+\|H\|_{L^1([a,b],L^2(e^{\beta x}dx))}).\end{aligned}
{\rm E}[|G_T^{d,p}|-r^dpe^{-T}]^2 ={\rm Var}(|G_T^{d,p}|) &={\rm Var}(\sum_{x\in T^d(r)}1_{\{\zeta_T^{d,p}(x)=1\}}) &=\sum_{x\in T^d(r)}{\rm Var}(1_{\{\zeta_T^{d,p}(x)=1\}}) &\leq r^d P\big(\zeta_t^{d,p}(x)=1\big)=r^dpe^{-T}.
\psi^{(0)}(u,T) =\psi^{(0)}(u,1)+\sum\limits_{k=0}^{u}\psi^{(1)}(u+1-k,T-1)a_k, \psi^{(1)}(u,T) =\psi^{(1)}(u,1)+\sum\limits_{k=0}^{u}\psi^{(2)}(u+1-k,T-1)b_k, \psi^{(2)}(u,T) =\psi^{(2)}(u,1)+\sum\limits_{k=0}^{u}\psi^{(0)}(u+1-k,T-1)c_k.
{\mathcal F}(F_6) =[0, 1, 1, 2, 3, 5, 0, 5, 5, 2, 7, 1],{\mathcal F}^2(F_6) =[0, 1, 1, 4, 1, 1],{\mathcal F}^3(F_6) =[0, 1, 1, 0, 3, 5, 0, 5, 5, 0, 7, 1],{\mathcal F}^4(F_6) =[0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1] =[0,1,1].
\mathcal{E}_s=\sum_{k=0}^s \Big\{\int |D_t\theta_k|^2+i\theta_k \overline{\partial_{\alpha}\theta_k}d\alpha+\int |D_t\sigma_k|^2+i\sigma_k \overline{\partial_{\alpha}\sigma_k}d\alpha\Big\}+O(\epsilon^3).
\sigma^0\left(m^2_2\right)=\frac{12\pi\tilde\Gamma_{e,2}\tilde\Gamma_{f,2}}{m^2_2\Gamma^2_2}\left(1-\frac{\Gamma^2_2}{m^2_2}\right)+\sigma_{\rm B}^0\left(m^2_2\right),
H_N^+(t,x) = C \sum_{f'( N) \leq m \leq f'(2 N )}(m-x)^{\frac{2 - \alpha}{2\alpha - 2}}e\left(t x_m^{\alpha}+ (x-m) x_m \right) + O( N^{1 - \frac\alpha2}) &= C \sum_{f'( N) \leq m \leq f'(2 N )}(m-x)^{\frac{r}2-1}e\left(c_{t,r}(m-x)^{r}\right) + O(N^{\frac12}),
ds^2= - \frac{r^2}{l^2}dt^2+ \left(\frac{r^2}{l^2}-r_0^2 \right) d \theta^2 + \left(\frac{r^2}{l^2}- r_0^2 \right)^{-1}dr^2 +dx_1^2 + \sinh^2 \left( \frac{\sqrt{2}x_1}{l}\right)dx_2^2
\|\frac{1}{\sqrt{p}}\nabla_p s_{f}\|_{p-Sl}= \frac{1}{\sqrt{p}}\inf_{g\in L^2[0,1]:\nabla_p s_{f}=\nabla_p s_{g}}\|s_g\|_{\mathcal{H}_{B_{[0,1]}}}= \frac{1}{\sqrt{p}}\inf_{g\in L^2[0,1]:\nabla_p s_{f}=\nabla_p s_{g}}\|g\|_{L^2[0,1]},
\frac{g \gamma_{d}^{\alpha}}{\sqrt{2}}\left( \begin{array}{c}{{0}}\\{{i W_{\alpha}^{2}\psi_{2 L}+ Z_{\alpha}\psi_{1 L}/ \cos \theta_{w}}}\\ \end{array}\right) + \frac{M_{i}+ \eta_{i}}{\sqrt{2}}\left( \begin{array}{c}{{\psi_{1 R}}}\\{{\psi_{1 R}}}\\ \end{array}\right) .
g =(v_1v_3v_5)(v_2v_4)=(v_1v_3v_5v_2)(v_4)=(v_1v_3v_5v_4)(v_2) &=(v_1v_3v_2)(v_5v_4)=(v_3v_5v_4)(v_1v_2)=(v_3v_5)(v_4v_1v_2).
\frac{\partial L}{\partial f_{i}}+\sum_{j=1}^{n}\sum_{\mu_{1}\le...\le\mu_{j}}(-1)^{j}\frac{\partial^{j}}{\partial x_{\mu_{1}}...\partial x_{\mu_{j}}}(\frac{\partial L}{\partial f_{i,\mu_{1}...\mu_{j}}})=0
\sum_{I \subseteq [5]}(-1)^{|I|}\Sigma_I = (y^5 - y^4 + y^3 + 2y^2 - y) -5(y(y-1)^3 + y^3) +y^3 +6(y^2 + (y-1)^3) &\qquad+3y(y^2 - y + 1) -3y^2 -6(y^2 - y + 1) -y^2 +5y -1 &= y^5 - y^4 -5y^2 + 13y - 7 - 5y(y^3 - 3y^2 + 3y - 1) + 6(y^3 - 3y^2 + 3y - 1) &= y^5 - 6y^4 + 21y^3 -38y^2 + 36y - 13.
e(\bar R_{33}) = \, \left(\frac{(q^2-1)^r}2+\frac12 (q-1)^{2r}- 2(q-1)^r+1 \right) \big( (q^{2r}-q^{r+1})(q^4+q^3+q^2) + (q^2-q^{r+1})(q^3+q^2+q)\big) + \left(- \frac{(q^2-1)^r}2+\frac12 (q-1)^{2r}- (q-1)^r+1 \right) \big((q^{2r}-q^{r+1}) (q^3+q^2+q)+ (q^2-q^{r+1})(q^4+q^3+q^2)\big) .
\left( \begin{array}{c c}{{A}}&{{( m_{i}+ \partial_{i}) I}}\\{{( m_{i}- \partial_{i}) I}}&{{- A}}\\ \end{array}\right)^{2}= ( A^{2}+ ( m_{i}+ \partial_{i}) ( m_{i}- \partial_{i}) I ) \left( \begin{array}{c c}{{I}}&{{0}}\\{{0}}&{{I}}\\ \end{array}\right)
X_{L}( a , b , c ; q ) = \displaystyle \sum_{\begin{array}{c}{\{\scriptstyle \mu |{\mu_{0}= a , \mu_{L}= b , \mu_{L + 1}= c}\}}\\ \end{array}}\! \! \! \! \! \! q^{\sum_{k = 0}^{L - 1}( k + 1 ) H ( \mu_{k}, \mu_{k + 1}, \mu_{k + 2})},
B ( j ) = -{\pi}^{2 j + 1}\left( \frac{\Gamma ( 1 - b^{2})}{\Gamma ( 1 + b^{2})}\right)^{2 j + 1}\frac{2 j + 1}{\pi}\frac{\Gamma \left( 1 + b^{2}( 2 j + 1 ) \right)}{\Gamma \left( 1 - b^{2}( 2 j + 1 ) \right)}; \ \b^{2}= \frac{1}{k - 2}.
S=\frac{1}{2}\sum_{j_1,j_2,j_3,m_1,m_2,m_3}\left|{A_{j_1j_2j_3}^{m_1m_2m_3}}\right|^2+\frac{\l}{4!}\sum_{j_1...j_6,m_1...m_6}(-1)^{\sum_i m_i}A_{j_1j_2j_3}^{-m_1-m_2-m_3}A_{j_3j_5j_4}^{m_3-m_5m_4}A_{j_4j_2j_6}^{-m_4m_2m_6}A_{j_6j_5j_1}^{-m_6m_5m_1}
\left|\frac{h}{N(\beta_1)}F_0^{l}(\beta_1)-\frac{h}{N(\beta_2)}F_0^{l}(\beta_2)\right| &+\sum_{n=2}^{\infty}\left|\frac{h}{N(\beta_1)}T_0^{l,(n)}(\beta_1)-\frac{h}{N(\beta_2)}T_0^{l,(n)}(\beta_2)\right| &\le c \beta_1^{-\frac{1}{2}}(M^{-1}\alpha^{-4}+(c_0+c_0')c_0'U_{max})\le \beta_1^{-\frac{1}{2}}\alpha^{-4}.
B(k) = \frac{e^{5.449}(6.274)^k}{(\frac{k}{4}-1)^{\frac{k-1}{2}}}+ \frac{e^{10.905}(4.793)^k}{\sqrt{(k-2)!}}+ \frac{e^{6.511}(10.096)^k}{\sqrt{(k-2)!}},
W^0_2(p,p_1)\ast W^0_{n+2}(p',p'_1,\dots,p'_{n+1})= W^0_{n+2}(p',p'_1,\dots,p'_{n+1})\ast W^0_2(p,p_1) = W^0_{n+2}(p,p'_1,\dots,p'_{n+1})
M = \left( \begin{matrix}{{m -{\frac{g^{2}}{4 \pi r}}}}&{{{\frac{g^{2}}{4 \pi r}}}}\\{{{\frac{g^{2}}{4 \pi r}}}}&{{m -{\frac{g^{2}}{4 \pi r}}}}\\ \end{matrix}\right) \, .
[\begin{matrix}C_{11}&C_{12}&C_{13}\\C_{12}&C_{22}&C_{23}\\C_{13}&C_{23}&C_{33}\end{matrix}]=\frac{1}{1-\nu_{12}\nu_{21}}[\begin{matrix}E_{1}&\nu_{12}E_{2}&0\\ \nu_{21}E_{1}&E_{2}&0\\ 0&0&2G_{12}(1-\nu_{12}\nu_{21})\end{matrix}].
\int_{\mu_{1}}^{\infty}d x \, \left( x^{2}- \mu_{1}^{2}\right)^{\nu - 1}e^{- \mu_{2}x}= \frac{1}{\sqrt{\pi}}\left( \frac{2 \mu_{1}}{\mu_{2}}\right)^{\nu - 1 / 2}\Gamma \left( \nu \right) K_{\nu - \frac{1}{2}}\left( \mu_{1}\mu_{2}\right) ,
\delta(s(e)) = \tilde{\mathbf Q}_\alpha \big( g_z( \tilde{\varphi}_i^{-1}\circ \tilde{\chi}_{\sigma_R}) \big) \cdot \begin{cases}: \delta\big(e^{-\pi i ( \frac{k_L}{3}-\frac{k_R}{3}+\frac{1}{2})}\big) : \delta \big( e^{-\pi i ( -\frac{k_R}{3}- \frac{1}{6}+ \frac{1}{2})}\big) \ .\end{cases}
\left(xq;q^{2}\right)_{\infty}\sum_{n=0}^{\infty}\frac{q^{n^{2}}z^{n}}{\left(q;q\right)_{n}\left(xq;q^{2}\right)_{n}}=\sum_{n=0}^{\infty}\frac{q^{n^{2}}\left(-x\right)^{n}}{\left(q^{2};q^{2}\right)_{n}}A_{q}\left(-q^{2n}z\right),
( A_\mu, \psi_\mu, \bar{\psi}_\mu, \phi, V_\mu, \bar{\phi}, \eta, \bar{\eta}, Y, \chi_{\mu\nu}, B_{\mu\nu})\rightarrow( A_\mu, \bar{\psi}_\mu, \psi_\mu, \phi, - V_\mu, \bar{\phi}, \bar{\eta}, \eta, - Y, \tilde{\chi}_{\mu\nu}, B_{\mu\nu}).
\mu^{\rm N}=\mu^*+\frac{\varkappa}{2},\varkappa^{\rm N}=\frac{\varkappa}{2},\qquad\,\lambda^{\rm N}=\lambda , \gamma^{\rm N}=\frac{\beta + \gamma}{2},\qquad\ \ \beta^{\rm N}=\frac{\gamma-\beta}{2}, \ \ \ \ \ \ \ \alpha^{\rm N}=\alpha \,.
n \: C \; \alpha_{1}^{1 - p - \gamma}\: \beta_{1}^{1 - q}\; \alpha_{i_{0}}^{\gamma}\left( \prod_{j = 2}^{p}\alpha_{1}^{a_{j}- \alpha_{j}}\: \alpha_{j}^{- a_{j}}\right) \left( \prod_{k = 2}^{q}\beta_{1}^{b_{k}- \beta_{k}}\: \beta_{k}^{- b_{k}}\right) \; ,
\tau(Q^n(K)) = \begin{cases}\tau(K) \tau(K) \le 0 \epsilon(K) \in \{0,1\}\tau(K) + 1 \tau(K) < 0 \epsilon(K) = -1 \tau(K) + n \tau(K) = 0 \epsilon(K) = -1 \tau(K) > 0.\end{cases}
[a_{(1\,p+1)},a_p]_c =[a_{(1\,p-2)},[a_{(p-1\,p+1)},a_p]_c]_c =-\lambda_{ppp+1}(1-\xi^2)[a_{(1\,p-2)},a_{p-1}g_{ppp+1}]_c &=-\lambda_{ppp+1}(1-\xi^2)\Big(a_{(1\,p-2)}a_{p-1}g_{ppp+1}&-\chi_{p-1}(g_{(1\,p-2)})a_{p-1}g_{ppp+1}a_{(1\,p-2)}\Big) &=-\lambda_{ppp+1}(1-\xi^2)a_{(1\,p-1)}g_{ppp+1},
\Delta\big(\big[x_i^+, x_{j,1}^+\big]\big) = \big[\square\big(x_i^+\big), \square\big(x_{j,1}^+\big) - \hbar \big[1 \otimes x_j^+,\Omega_+\big]\big] \hphantom{\Delta\big(\big[x_i^+, x_{j,1}^+\big]\big)}{}= \square\,T_i\big(x_{j,1}^-\big) - \hbar \big[\square\big(x_i^+\big),\big[1 \otimes x_j^+,\Omega_+\big]\big].
\gamma \cdot \left( \begin{array}{c c}{\Psi^{3}}{\Psi^{4}}\end{array}\right) = R_{D_{2}}^{r e g}( \gamma )^{- 1}\cdot \left( \begin{array}{c c}{\Psi^{3}}{\Psi^{4}}\end{array}\right) \cdot R_{D_{2}}^{r e g}( \gamma ) .
\mathrm{d}\mu = \frac{1}{2^{5}}\, \frac{\Lambda_{\mathrm{o n e - f l}}^{5}}{\Phi^{2}( x_{0}, \theta_{0})}\, \exp \left( - 4 \pi^{2}\, \bar{\Phi}\Phi \, \rho_{\mathrm{i n v}}^{2}\right) \frac{\mathrm{d}\rho^{2}}{\rho^{2}}\, \mathrm{d}^{4}x_{0}\, \mathrm{d}^{2}\theta_{0}\, \mathrm{d}^{2}\bar{\beta}\, \mathrm{d}^{2}\bar{\theta}_{0}\, .
\sum_{n=0}^{\infty}\frac{q^{n^{2}/2}\left(-e^{\xi}t\right)^{n}h_{n}\left(\sinh\left(\xi+\log q^{n/2}\right)\vert q\right)}{\left(q;q\right)_{n}}=\frac{A_{q}\left(te^{2\xi}\right)}{\left(t;q\right)_{\infty}}
\widetilde Q_j = -Q_j(e)P_eIR^{P_e}_{e-\i 0_+}-R^{P_e}_{e-\i 0_+}IP_eQ_j(e) +\d R^{P_e}_{e-\i 0_+}IP_eQ_j(e)P_eIR^{P_e}_{e-\i 0_+},j\ge 1,
{\cal G}_{a}^{( \pm )}\equiv \frac{L}{2}\eta_{a b}\epsilon^{i j}( \partial_{i}A_{j}^{( \pm ) b}- \partial_{j}A_{i}^{( \pm ) b}+ \epsilon_{\mathrm{~}c d}^{b}A_{i}^{( \pm ) c}A_{j}^{( \pm ) d}) \approx 0 ,