harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
V \left( t , \bar{t}, ( E , M ) \right) \sim | Z^{\mathrm{l o c a l}}( t , \bar{t}, ( q , p ) ) |^{2}= \left| e^{\frac{K ( t , \bar{t})}{2}}[ X^{\Lambda}( t ) q_{\Lambda}- F_{\Lambda}( t ) p^{\Lambda}] \right|^{2}\ ,
{\cal L}^\psi_{mass}= - 2{\beta \over{\pi^2}}\left({\alpha \over 2}\right)^{12}{1 \over{m^3}}\left\{F^{ij}F^{kl}\psi^{\dag}_{ij}\psi^{\dag}_{kl}+ 4 F^{ij}F^{kl}\psi^{\dag}_{li}\psi^{\dag}_{jk}\right\},
f^{[n]}(\lambda_0,\ldots,\lambda_n)= (fu)^{[n]}(\lambda_0,\ldots,\lambda_n)u^{-1}(\lambda_j) &-f^{[n-1]}(\lambda_0,\ldots,\lambda_{j-1},\lambda_{j+1},\ldots,\lambda_n)u^{-1}(\lambda_j).
|H'(x)-H'(y)| =\Bigl|\frac{1}{G'(G^{-1}(x))}-\frac{1}{G'(G^{-1}(y))}\Bigr|\leq c_1\cdot |G'(G^{-1}(y))-G'(G^{-1}(x))| &\leq c_2\cdot |G^{-1}(y)-G^{-1}(x)|\leq c_3\cdot |x-y|.
\Delta y \Delta p \ge \frac{\hbar_R}{2}\left| <\frac{1}{1 -{\epsilon \over g_s^R}\delta{\hat y}-2 g_s^R \delta{\hat p}+ \dots}>\right|\sim \frac{\hbar_R}{2}(1 +{\epsilon^2\over (g_s^R)^2}\Delta y^2 + O(\alpha_s^{\prime})\Delta p^2 + \dots )
Z_{left}(\tau) = \prod_{I=1}^{16}Z^{0,2mRA_y^I}_{0,2nRA_y^I}(\tau) +\prod_{I=1}^{16}Z^{0,2mRA_y^I}_{1,2nRA_y^I}(\tau) +\prod_{I=1}^{16}Z^{1,2mRA_y^I}_{0,2nRA_y^I}(\tau) +\prod_{I=1}^{16}Z^{1,2mRA_y^I}_{1,2nRA_y^I}(\tau)
\lim_{x\rightarrow 0^{+}}H_{e^{xa}F,e^{xa}G}\left( x\right)=\lim_{x\rightarrow 0^{+}}\left( \frac{\left( e^{xa}F\left( x\right) \right)^{\prime}}{\left( e^{xa}G\left( x\right) \right)^{\prime}}e^{xa}G\left(x\right) -e^{xa}F\left( x\right) \right) \geq \left( \leq \right) 0.
\|x_\alpha^A -x_\alpha^B \| = \|x_\beta^A-x_\beta^B + h_\beta(w^A_\beta-w^B_\beta)\| \le \|x_\beta^A-x_\beta^B\| - h_\beta \eta &\le \|x_1^A-x_1^B\| -(t_\beta+ h_\beta) \eta \, .
z^{t+(3n^{d-2}+n^{d-3}+\cdots+n^2)}&= z^{(3n^{d-2}+n^{d-3}+\cdots+n^2)}(z^t-z^nx^n)+(x^{n-1})(xz^{3n^{d-2}+n^{d-3}+\cdots+n^2+n}) \in I_n.
\mathcal{E}^q_{OP}(u):=\int_\Omega \left[|\nabla u|^2+q(x)\,\chi_{\{u>0\}}\right]\,dx , \mathcal{E}^q_{DP}(u):=\int_\Omega \left[ |\nabla u|^2+q_1(x)\,\chi_{\{u>0\}}+q_2(x)\,\chi_{\{u<0\}}\right]\,dx , \mathcal{E}^q_{V}(u):=\int_\Omega \left[|\nabla u|^2+q(x)\,\chi_{\{|u|>0\}}\right]\,dx ,
U_{\varepsilon_k}=\{E+\xi(2E(k,0)-E(0,-k))-\xi^2E(k,-k)\mid\xi\in\Bbbk\}, U_{-\varepsilon_k}=\{E+\xi(E(0,k)-2E(-k,0))-\xi^2E(-k,k)\mid\xi\in\Bbbk\}.
\hat{D}_{\mu \nu \rho \underline{{x}}}={\textstyle \frac{3}{8}}\left( C_{\mu \nu \rho}- A_{[ \mu}^{( i )}B_{\nu \rho ]}^{( i )}- \epsilon^{i j}A_{[ \mu}^{( i )}A_{\nu}^{( j )}B_{\rho ]}\right) , \hat{\rho}_{\mu \nu \underline{{x}}}={\textstyle \frac{1}{2}}\chi_{\mu \nu}.
\underline{\sigma}(\alpha,\zeta)= \liminf_{n\to\infty}-\frac{\log \Vert \alpha \zeta^{n}\Vert}{n\cdot \log \zeta},\overline{\sigma}(\alpha,\zeta):= \limsup_{n\to\infty}-\frac{\log \Vert \alpha \zeta^{n}\Vert}{n\cdot \log \zeta}.
\partial_{0}J_{1}- \partial_{x}J_{0}= \frac{1}{2}(\partial_{0}L_{1}- \partial_{x}L_{0}) + \frac{1}{2}(\partial_{0}L_{0}-\partial_{x}L_{1})g + \frac{1}{2}(L_{0}^{2}- L_{1}^{2})g \;\;\; ,
\mathrm{( f )}+ \mathrm{( g )}= - \frac{d_{\Gamma}g^{2}}{2 x_{i j}^{4}}\sum_{k}\frac{x_{j k}^{\mu}x_{i k}^{\nu}+ x_{j k}^{\nu}x_{i k}^{\mu}- \delta^{\mu \nu}x_{j k}\cdot x_{i k}}{x_{j k}^{2}x_{i k}^{2}},
\sum_{M_{1}+ M_{2}= M}\bigl ( J_{1}, M_{1}; J_{2}, M_{2}\vert J_{1}, J_{2}; J , M \bigr ) \bigl ( J_{1}, M_{1}; J_{2}, M_{2}\vert J_{1}, J_{2};{\overline{{J}}}, M ) = \delta_{J , \,{\overline{{J}}}}
\mathfrak{M}= [\mathfrak{A}_{i}, \mathfrak{M}] = [\mathfrak{A}_{i}, \mathfrak{D}_{1}, \mathfrak{C}_{1}] = [[\mathfrak{A}_{i}\mathfrak{D}_{1}], [\mathfrak{A}_{i}\mathfrak{C}_{1}]].
\frac{d}{dt}(\begin{matrix}M_{x}^{\prime}\\M_{y}^{\prime}\\M_{z}^{\prime}\end{matrix})=(\begin{matrix}-\frac{1}{T_{2}}&\Delta&-\epsilon\\ -\Delta&-\frac{1}{T_{2}}&\epsilon\\ -\epsilon&-\epsilon&-\frac{1}{T_{1}}\end{matrix})(\begin{matrix}M_{x}^{\prime}\\M_{y}^{\prime}\\M_{z}^{\prime}\end{matrix})+(\begin{matrix}0\\ 0\\ \frac{M_{0}}{T_{1}}\end{matrix})
\mathcal{S}:=\prod_{d\geqslant 0}\bigg(\bigoplus_{\alpha\in X_\xi}\mathcal{O}_\alpha\otimes \mathcal{M}_\eta^d\theta^d\bigg)\otimes \mathcal{O}_{A_\eta}=\bigoplus_{(d,\alpha)\in S(X_\xi)}\mathcal{S}_{d,\alpha}\theta^d.
u_s\in L^q(\Omega)\cap C^\infty(\Omega), \|u_s\|_q\leq C|u_\infty|^{1/2},\forall q\in (2,\infty], &\nabla u_s\in L^r(\Omega), \|\nabla u_s\|_r\leq C^\prime |u_\infty|^{1/2},\forall r\in (4/3,\infty], &\mbox{provided $0<|u_\infty|\leq\delta_0$}.
\limsup_{n\to \infty}\frac{\sum_{i=0}^n \sigma_i\xi_{i+1}}{\sqrt{\sum_{k=0}^{n}\sigma^2_k}}=\infty, \liminf_{n\to \infty}\frac{\sum_{i=0}^n \sigma_i\xi_{i+1}}{\sqrt{\sum_{k=0}^{n}\sigma^2_k}}=-\infty, \quad,
=\int \left( \sum_{n \in I}\frac{w_n \mathbf{1}_{U(x_{n}, r_{n})}}{\mu U(x_{n}, r_{n})}\right) M^u h \d\mu\le\left\| \sum_{n \in I}\frac{w_n \mathbf{1}_{U(x_{n}, r_{n})}}{\mu U(x_{n}, r_{n})}\right\|_p\left\|M^u_{\mu}h \right\|_q.
\forall w \in C^2_e(S^{n-1}) \;\;\; \int w dS_K = 0 \;\; \Rightarrow \int_{S^{n-1}}(-H_K w) w d\theta \geq \frac{1-p}{n-1}\int_{S^{n-1}}\frac{w^2}{h_K}dS_K ,
\frac{Z(\boldsymbol{\delta})}{{\sigma_n}}=\frac{\boldsymbol{\delta}^\top\mathbf{H}_\ast^\top\boldsymbol{\Sigma}^{-1}\mathbf{H}_\ast\boldsymbol{\delta}}{2\sqrt{\boldsymbol{\delta}^\top\mathbf{H}_\ast^\top\boldsymbol{\Sigma}^{-1}\boldsymbol{\Sigma}_\ast\boldsymbol{\Sigma}^{-1}\mathbf{H}_\ast\boldsymbol{\delta}}}.
Q(i)-P(Q(i)) = |\{j\in J/\, Q(j)<Q(i), \phi(Q(j),Q(i))<0\}| &= |\{j\in J/\, Q(j)<Q(i), j>i\}| &= |\{j\in J/\, j>i,\psi(j,i)>0\}| &= Q(i)-i\,\,,
A_0^{(1)}=\begin{pmatrix}0 0 q_2 -t_2\end{pmatrix},A_0^{(0)}=\begin{pmatrix}-p_2q_2 t_2p_2 1-p_1 p_2q_2+\theta^0\end{pmatrix},N= \begin{pmatrix}0 1 0 0 \end{pmatrix}, A_{t_1}=\begin{pmatrix}q_1 1\end{pmatrix}\begin{pmatrix}p_1 \theta^{t_1}-p_1q_1\end{pmatrix},N_1=\frac{q_1(p_1q_1-\theta^{t_1})}{t_1}N,N_2=-p_2N.
\tilde F_\lambda (x_1,\dots, x_n;t)= \frac{(1-t)^n}{\prod_{i=1}^{n-l}(1-t^i)}\sum_{\sigma \in S_n}\sigma\left((1-x_{1}^{\lambda_1})\dots (1-x_{l}^{\lambda_n})\prod_{i=1}^{n}\frac{x_i}{1-x_i}\prod_{i < j}\frac{x_{i}-tx_{j}}{x_{i}-x_{j}}\right),
\xi(x)=\begin{cases}0 x\in [-1,-\bar\beta_1]\cup[\bar \beta_1, \bar\lambda_2] \cup[2-\bar \lambda_2, 2], 1 x\in[-\tilde \lambda_1,\tilde \lambda_1] \cup [\tilde\lambda_2, 2-\tilde\lambda_2],\end{cases}
\tau_{i}^{n+1}=\tau_{i}^{n+\frac{1}{2}}, & \frac{u_{i}^{n+1}-u_{i}^{n+\frac{1}{2}}}{\Delta t}=-\frac{1}{\varepsilon^{2}}\left(\sigma\,u_{i}^{n+1}+(1-\varepsilon^{2})\frac{p_{i+\frac{1}{2}}^{n+1}-p_{i-\frac{1}{2}}^{n+1}}{\Delta x}\right).
\begin{cases}\frac{\partial x_{1}}{\partial y}=\left(\frac{\partial y}{\partial x_{1}}\right)^{-1}, \frac{\partial x_{1}}{\partial z}=-\left(\frac{\partial y}{\partial x_{1}}\right)^{-1}\cdot\frac{\partial y}{\partial x_{2}}, \frac{\partial x_{2}}{\partial y}=0, \frac{\partial x_{2}}{\partial z}=\mathrm{I}_{m-n}.\end{cases}
\Bigl( \frac{M_n(i,j)}{n^{\delta_{ij}+1/2}}; 0\leq i\leq I,0\leq j\leq O\Bigr) =: K_{IO}\left(\frac{N_n(k,l)-\nu_n(k,l)}{\sqrt{n}}:\, 0\le k\le I,\, 0\le l\le O\right)^T,
\Gamma_{00}^{\;\; 0}= \frac{V'}{V}\; , \; \Gamma_{ii}^{\;\; 0}= -\frac{f_i'}{2V^2}\; , \;\; \Gamma_{i0}^{\;\; i}= \frac{f_i'}{2 f_i}\;\;\; ; \; \; i=1, 2, 3
\delta{\tilde e}_\mu^{\ \underline \nu}= -\bar{\tilde\epsilon}_{+i}\tilde \Gamma^{\underline \nu}\tilde\psi^i_{\mu +}-2 \kappa^{D-2}W\bar{\tilde\epsilon}_{+i}\tilde \Gamma^{\underline{\nu\lambda}}\tilde\eta^i_-\tilde e_{\mu\underline\lambda}+ c.c.\ .
E_{0}= \left(\begin{array}{cccc}0 0 0 1 0 \mu_1 0 0 0 0 \mu_2 0 0 0 0 0\end{array}\right), E_{1}= \left(\begin{array}{cccc}0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0\end{array}\right),E_{2}=\left(\begin{array}{cccc}0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0\end{array}\right).
\int_{\mathbb{R}^{n-1}}\int_{-\infty}^{g(x^{\prime})}u_{x_{i}}(x^{\prime},x_{n})dx_{n}dx^{\prime}=\int_{\mathbb{R}^{n-1}}\int_{-\infty}^{0}u_{x_{i}}(x^{\prime},g(x^{\prime})+s)dsdx^{\prime}
F(z_1,z_2)=e^{z_1+z_2}\sum_{\rho,\sigma,\mu,\nu}\binom{i}{\rho}\binom{i-\rho}{\sigma}\binom{k}{\mu}\binom{k-\mu}{\nu}u_{11}^{\rho}u_{12}^{\sigma}u_{21}^{\mu}u_{22}^{\nu}\;z_1^{\rho+\mu}z_2^{\sigma+\nu},
\mathcal{T}(V_{\varphi_i})= \left\{F\in L^2\left(\mathcal O, l^2(\mathcal Z)\right): F(\xi)=r(\xi)\mathcal{T}(\varphi_i)(\xi), r \in L^2\left(\mathcal O, \Omega_i\right)\right\}= \left\{F\in L^2\left(\mathcal O, l^2(\mathcal Z)\right): F(\xi)\in J_i(\xi)\right\},
0 =dy'(z_2)-cy(z_2)+sv(z_2)\left(1-\frac{v(z_2)}{q(u(z_2))}\right) begin{align*}0.2cm] \geq sv(z_2)\left(\frac{q(1)}{q(0)}-\frac{v(z_2)}{q(u(z_2))}+1\right)>0,
( - 1 )^{F}= \Gamma_{1 1}( - 1 )^{\sum_{m = 1}^{\infty}\psi_{- m}\cdot \psi_{m}},{G}= - \gamma_{0}\beta_{0}- \sum_{m = 1}^{\infty}\left[ \gamma_{- m}\beta_{m}+ \beta_{- m}\gamma_{m}\right] .
( \eta_{0 , 1}, \; \eta_{2 , 1}, \; \eta_{2 , 2}, \; \eta_{2 , 3}) = ( 0 . 3 9 7 6 5 , \; - 0 . 1 3 8 9 7 , \; 0 , \; 0 . 0 4 0 8 9 3 ) .
G_r^{(f)}(\omega)=\frac{1}{\sqrt{7}}\{(\tilde{\chi}\Gamma_i\chi\sigma_i)_r + (\tilde{\chi}[\Gamma_j,\Gamma_k]\chi\sigma_{jk})_r + (\sigma_i\sigma_j\sigma_{ij})_r + (\sigma_{ij}\sigma_{jk}\sigma_{ki})_r\},
\begin{array}{lll}W^{+}_{i}= (R^1_{\pi_{S^{i}X}})_{*}\xi^{-1}(2\Delta_{i}) &= (R^1_{\pi_{S^{i}X}})_{*}\{\pi_{X}^{*}(\xi^{-1})\otimes\mathcal{O}_{X\times S^{i}X}(2\Delta_{i})\}&= (R^1_{\pi_{S^{i}X}})_{*}\{(\xi^{-1}\boxtimes\mathcal{O}_{S^{i}X})(2\Delta_{i})\}.\end{array}
T_{M_{2}}l \int d^{2}\xi ( 2 \pi \alpha^{\prime}) \epsilon^{i j}\partial_{i}\varrho \ \left({\tilde{\cal G}}_{j}^{( 1 )}-{\textstyle \frac{1}{2 \pi \alpha^{\prime}}}{\tilde{A}}^{( 1 )}{}_{j}+{\textstyle \frac{m}{2}}b_{j}\right) \, .
=(\mathrm{Ind}^G_{P_{S'}}\circ [\mu_{S'}] )(\bigoplus_{w \in W_{\rho}}\delta^{\frac{1}{2}}_{P_{S'}}\otimes I^{M_{S'}}_{w(M_S)}(w(\rho)))\otimes [1][|\cdot|^{\langle \rho_G, \mu_{S'}-\mu \rangle}],
\begin{array}{r c l}{{\eta_{1 2}}}&{{=}}&{{- \left( P_{1 2}{}^{t_{2}}\left( ( r_{1 2}{}^{t_{2}})^{- 1}\right) \right)^{t_{2}}}}\\{{}}&{{=}}&{{- \mathrm{t r}_{3}\left( P_{2 3}( r_{2 3}{}^{t_{3}})^{- 1}\right) P_{1 2}}}\\{{}}&{{=}}&{{- D_{2}P_{1 2}.}}\\ \end{array}
\frac{\partial^k}{\partial t^k}H(x(t))=\sum_{i=1}^k\sum_{\substack{k_1+\dots+k_i=k k_j\geq 1}}D^iH(x(t)) \left[ \frac{d^{k_1}x (t)}{dt^{k_1}}, \dots ,\frac{d^{k_i}x (t)}{dt^{k_i}}\right].
\widehat{u}(x,y,z) =0Y_{s}\times Z_{s},\Omega, u_{\varepsilon\delta}\rightarrow\frac{1}{\left\vert Y\right\vert\left\vert Z\right\vert}\int_{Y\times Z}\widehat{u}\left( x,y,z\right)dydz(L^{2}\left( \Omega)\right)^{n},
\frac{\partial}{\partial \eta^\sigma_k}= \frac{\partial}{\partial \xi^\sigma}+ \sum_l \bigg(\frac{\partial h^k_l}{\partial \eta^\sigma_k}\bigg) \frac{\partial}{\partial x^l}+ \sum_\tau \bigg(\frac{\partial \omega^k_l}{\partial \eta^\sigma_k}\bigg) \frac{\partial}{\partial \xi^\tau},
T =\Big\{y\in\mathbb{R}^{k}:\;y_1>\delta_1,\dots,y_{k}>\delta_{k},\;y_1\cdots y_{k}<\frac{h}{2^k}\,\delta_1\cdots\delta_{k}\Big\}, E_y =\Big\{x\in E:\;(x_{1},\dots,x_k)=y\Big\}\;\; (y\in T).
f_k =[ u^k] F(u)=\frac{N_1^{t-k}}{D_t}(-1)^{k}z^k, g_k =[ u^k] G(u)=\frac{N_2^{t-k}}{D_t}(-1)^{k}z^k, \mbox{and}h_k =[ u^k] H(u)=\frac{N_3^{t-k}}{D_t}(-1)^{k}z^k.
\varrho_{\Phi}=\left({1\over 2}{\dot{\Phi}^2\over c^2}+V\right){1\over c^2},\ \ \ \ \ \wp_{\Phi}={1\over 2}{\dot{\Phi}^2\over c^2}-V,\ \ \ \ \I\equiv 1+B{\dot{\Phi}^2\over c^2}.
\Theta_{\lambda, T}(\tau, z, t) = e^{2 \pi i k t}\ \sum_{\gamma \in L}\ \frac{q^{\frac{|\bar{\lambda}+ k \gamma |^2}{2k}}e^{2 \pi i (\bar{\lambda}+ k \gamma ) (z)}}{\prod_{\beta \in T}(1-q^{-(\gamma | \beta)}e^{-2 \pi i \beta (z)})}\,,
Z_{N,\beta}=\int_{\mathbb{T}^N}\prod_{1\leq j< k\leq N}\left|e^{i\theta_j}-e^{i\theta_k}\right|^\beta \*d\theta= (2\pi)^N \* \frac{\Gamma\left(1+\frac{\beta N}{2}\right)}{\Gamma\left(1+ \frac{\beta}{2}\right)^N}.
F(u)\,=\;\frac 12(u^2-1)^2, & G(u)\,=\,\frac 13(u-2)(u+1)^2\log\vert 1+u\vert\,-\,\frac 13(u+2)(u-1)^2\log\vert 1-u\vert\,+\,\frac{u^2}3\,+\,\frac 43\log 2\,-\,\frac 13, W_\epsilon(u)\,=\,\frac{F(u)}{2\epsilon^{1-\kappa}}\,-\,\frac{\dot\kappa}{K(\epsilon)}G(u).
\eta ( \alpha ) u^{\beta ( \alpha )}( - \vec{p}) = - ( - 1 )^{\alpha}\gamma^{2}\gamma^{4}\gamma^{5}u^{\alpha \, *}( \vec{p}) \, , \qquad e_{\mu}^{- \ell *}( - \vec{q}) = - R_{\mu \nu}e_{\nu}^{\ell}( \vec{q}) \, ,
E_{N}( x ) = -{\frac{R}{2 \pi}}\ln \left|{\frac{g_{N}^{x}( i R )}{g_{N}^{x = 1}( i R )}}\right| +{\frac{1}{2 \pi}}\int_{0}^{R}\ln \left|{\frac{g_{N}^{x}( i \xi )}{g_{N}^{x = 1}( i \xi )}}\right| d \xi .
\partial_\eta F(\phi,\eta)(\rho) =\frac{d}{d\epsilon}\Big|_{\epsilon=0}F(\phi,\eta+\epsilon \rho) &=\frac{1}{2}\int_{-\infty}^xe^{-\eta(x)+\eta(y)}\phi(y)\left(\rho(x)\eta'(y)-\rho(y)\eta'(y)-\rho'(y)\right)\;dy &\qquad+\frac{1}{2}\int_{x}^{\infty}e^{\eta(x)-\eta(y)}\phi(y)\left(\rho(x)\eta'(y)-\rho(y)\eta'(y)+\rho'(y)\right)\;dy.
x - \frac{1}{V'(x)}- \sqrt{2W(x)}=- \frac{1}{V'(x)}- \frac{2 \log H(x)}{x + \sqrt{2W(x)}}& \leq - \frac{1}{x}- \frac{2 \log H(x)}{x}& = - \frac{2}{x}\log (\sqrt{e}H(x)) .
(\psi^-,\phi^+)=\sum_j\int_{m_0^2}^{\infty}d{\mathsf s}\, \sum_{j_3}\int \frac{d^3\hat{\bf p}}{2\hat{E}}\langle \psi^-|[j,{\mathsf s}],j_3, \hat{\bf p}^-\rangle S_j({\mathsf s}) \langle^+j_3,\hat{\bf p},[j,{\mathsf s}]|\phi^+\rangle \, .
\begin{array}{lll}z^1z^{2}=p^{2}z^{2}z^1 \;,\qquad z^{2}z^{3}=p^{2}z^{3}z^{2}\;,\qquad z^{3}z^{4}=p^{2}z^{4}z^{3}\;, z^{4}z^1=p^{2}z^1z^{4}\;,\qquad z^{4}z^{2}=z^{2}z^{4}\;,\qquad z^1z^{3}=z^{3}z^1 \;.\end{array}
\left [ ( h r^3 \partial_r)^2 + \omega^2 r^6 f -{{8 h r^4 r_{\pm}^4}\over (r^2 + r_{\pm}^2)^2}\left ( 1 +{r_0^2 \over r_{\pm}^2}\right ) \right ] \tilde \phi_{\pm}=0,
c_1^{22}(A) = \bar{a}(1-\bar{b}) \int_Y r_B b_{11}\partial_1 w_A = -2\bar{a}(1-\bar{b}) \int_Y (\partial_1 w_A)(\partial_{22}^2 w_B), c_2^{22}(A) = \bar{a}(1-\bar{b}) \int_Y r_B b_{22}\partial_2 w_A = 2\bar{a}(1-\bar{b}) \int_Y (\partial_2 w_A)(\partial_{11}^2 w_B).
{\cal I}_{\mathrm{d e g}}^{\mathrm{h e t}}= \frac{V^{( 8 )}T_{2}}{2^{8}\pi^{4}}t_{8}F_{+}^{4 - l}F_{-}^{l}\Delta_{F_{+}^{4 - l}F_{-}^{l}}^{\mathrm{d e g}}( T_{2}, U ) \leftrightarrow{\cal I}_{\mathrm{1 - l o o p}}^{\mathrm{I}}.
\begin{array}{c}\dfrac{\partial U}{\partial t}-\dfrac{\partial^{2}U}{\partial x^{2}}+k_{1}U\dfrac{\partial U}{\partial x}+k_{2}(UV)_{x}=0 \dfrac{\partial V}{\partial t}-\dfrac{\partial^{2}V}{\partial x^{2}}+k_{1}V\dfrac{\partial V}{\partial x}+k_{3}(UV)_{x}=0\end{array}
W^s_\varepsilon(x) :=\left\{y\in M:d\big(f^n(x),f^n(y)\big) \leq\varepsilon,\ \forall n\geq 0\right\}; W^u_\varepsilon(x) :=\left\{y\in M:d\big(f^{-n}(x),f^{-n}(y)\big) \leq\varepsilon,\ \forall n\geq 0\right\},
Z ={\cal N}\, \int \prod_x \{{\rm d}v_{\mu}\,{\rm d}j^{\mu}\,{\rm d}j^{\mu}_5 \,{\rm d}|h|^2 \,{\rm d}\alpha \}\, e^{{\rm i}\int{\rm d}^4 x \,{\cal L}[v_{\mu},j^{\mu},j^{\mu}_5,|h|,\alpha]},
\begin{array}{@{}l@{}}{\displaystyle \tau \sum_{k=1}^{m}|U_1^{k}(\overline{\partial_{\tau}\theta}_{\mathbf{A}}^{k})| \leq C\left(h^{2r}+\tau^{4}\right)+ C\tau\sum_{k=1}^{m}\|\overline{\partial_{\tau}\theta}_{\mathbf{A}}^{k}\|_{\mathbf{L}^2}^{2}.}\end{array}
\bar{\mathbf{y}}(v) =\chi_{\{v\geq 0\}}\int_0^{v}(\frac12{e^{\frac{-y^2}{2}}}-e^{\frac{y^2}{2}})dy-\chi_{\{v<0\}}\int_0^{v}\frac12{e^{-\frac{y^2}{2}}}dy; \bar{\mathbf{z}}(v) =\chi_{\{v\geq 0\}}(\frac12{e^{\frac{-v^2}{2}}}-e^{\frac{v^2}{2}})-\chi_{\{v<0\}}\frac12{e^{-\frac{v^2}{2}}},
\Pi_{\mu \nu}\equiv -{\frac{1}{4}}T_{\mu \lambda}T_{\nu}^{\lambda}+{\frac{1}{1 2}}T T_{\mu \nu}+{\frac{1}{8}}g_{\mu \nu}\left( T_{\lambda \kappa}T^{\lambda \kappa}-{\frac{1}{3}}T^{2}\right) \, .
\int_{\mathfrak{m}}\big|S(\alpha)\big|d\alpha \le\mathcal{I}_1^{1/2}\mathcal{I}_2^{1/2}\sup_{\substack{\alpha\in\mathfrak{m}\boldsymbol{\beta}\in [0,1]^3}}\Big|\prod_{j=3}^{n-6}f_j(\alpha,\boldsymbol{\beta})\Big|,
\widetilde{J}(z, u) = \sum_{j=0}^{K-1}\Big[r\big(\tilde{z}^u(t_j), \nu_j\big) + \int_{t_j}^{t_{j+1}}\phi\big(\tilde{z}^u(s), \nu_j\big)\, ds \Big] + \psi\big(\tilde{z}^u(T)\big)\,,
\Gamma_{I}^{\phi^{-}}( - i \Omega_{m}, \stackrel{\rightharpoonup}{p}) = \frac{G}{2}+ G L ( - i \Omega_{m}, \stackrel{\rightharpoonup}{p}) \Gamma_{I}^{\phi^{-}}( - i \Omega_{m}, \stackrel{\rightharpoonup}{p}) ,
\begin{array}{ll}U_+=\left\{S_1-S_2>0\right\}\times \left(-\dfrac{\pi}{4},\dfrac{\pi}{4}\right), V_+=\left\{x-y>0\right\}\times \mathbb{R}, U_-=\left\{S_1-S_2<0\right\}\times \left(-\dfrac{\pi}{4},\dfrac{\pi}{4}\right), V_-=\left\{x-y<0\right\}\times \mathbb{R}.\end{array}
\int_{\mathbb{S}^{d-1}}\frac{\sin^2{(\pi T\|k \| \xi_1)}}{\pi^2 || k||^2 \xi_1^2}d\xi = C \int_{0}^\pi \frac{\sin^2{(\pi T \|k \| \cos \phi)}}{\pi^2 || k||^2 \cos^2 \phi}(\sin \phi)^{d-2}\, d\phi,
\sum_{i=0}^{n-1}|R_{4,i}|\lesssim \sum_{i=0}^{n-1}(t_{i+1}-t_i)^{H_0+1/2}\leq \max_{0\le i\le n-1}(t_{i+1}-t_i)^{H_0-1/2}\sum_{i=0}^{n-1}(t_{i+1}-t_i) \to 0,\,\n\to \infty.
L_{+}={\bar{M}}_{+}^{- 1}M_{+}\, , \quad L_{-}={\bar{M}}_{-}^{- 1}M_{-}\quad ( \mathrm{s a t i s f y i n g}\quad [ \, L_{\pm}\, , \, u ( x ){\bar{u}}( \bar{x}) \, ] = 0 \, )
h^{(n)}(x) = \sum_{i=0}^{n-1}(1-g(x))^{-(i+2)}&\cdot \left[\sum_{j_1+j_2 \cdots +j_{i+1}=n}\left(\begin{array}{c}n j_1,j_2,\ldots ,j_{i+1}\end{array}\right)g^{(j_1)}(x)g^{(j_2)}(x) \cdots g^{(j_{i+1})}(x)\right],
V_f = \frac{\mu}{2}\sum_{p=0}^{N-1}u_p^*f(2{\hat 1}-D-D^{\dagger})u_p =: -\frac{1}{2}\sum_{p=0}^{N-1}\sum_{q=0}^{N-1}u_q^*\Delta_f(|p-q|)u_p
l(\langle\rangle) =\langle\forall_{\vec v}\forall_x\exists_y(\forall_{z\in y}(z\in x\land\theta(x,z,\vec v))\land\forall_{z\in x}(\theta(x,z,\vec v)\rightarrow z\in y))\rangle, o(\langle\rangle) =\Omega+\omega\cdot(k+1),
\mathbf{\Sigma}_{\mathbf{\hat{z}}}=\begin{bmatrix}\alpha_1 \alpha_3 0 \cdots 0 \alpha_3 \alpha_2 \alpha_3 \cdots 0 0 \alpha_3 \ddots \ddots \vdots \vdots \cdots \ddots \ddots \alpha_3 0 \cdots \cdots \alpha_3 \alpha_2\end{bmatrix},
\sum_{|\vec{n}|=n}b^{\vec{n}}\left(\sum_{m=0}\alpha_{m}^{\vec{n}^{'},\vec{n}}(c)(\Delta+\theta)^{m}\right)\langle\Delta^{'}+\theta|\Delta+\theta\rangle=0,\hspace{1cm}\forall\:\:\:\vec{n}^{'}:|\vec{n}^{'}|=n,
= i\int \frac{d^4 x_{1}d^4 p^{\prime}}{(2 \pi)^4}e^{-ip_{\mu}^{\prime}x_{2}^{\mu}}\frac{p_{\nu}^{\prime}\gamma^{\nu}+ m}{p_{\tau}^{\prime}p^{\tau \, \prime}- m+i\epsilon}e^{i p_{\eta}^{\prime}x_{1}^{\eta}}N_{\sigma}\gamma^{\sigma}\psi(1)
\mu(E_{\mu}^{[\alpha,\gamma],\varepsilon}) =\mu(\left\{x:\underline{\dim}(\mu,x)\in [\alpha,\gamma]\right\}) \mu(F_{\mu}^{[\alpha,\gamma],\varepsilon}) =\mu(\left\{x:\overline{\dim}(\mu,x)\in [\alpha,\gamma]\right\})
\frac{\partial V_{2}(\zeta)}{\partial \zeta}\varphi(\zeta,\psi,\nu,\mu) \leq -\|\zeta_{1}\|^{2}-|\zeta_{2}|^{2}+\nu^{2}+(\bar{\pi}_{1}(\psi)+|\omega_{2}(\psi)|^{2})\|\psi\|^{2}\leq -\gamma_{2}(\|\zeta\|)+\nu^{2}+\pi_{2}(\psi).
\sum_{1<n\le N}\pi(n)R_X^n =A(N)\cdot\dfrac{1}{N}-A(1)\cdot\dfrac{1}{1}-\int_2^N A(t)\cdot \left\{\dfrac{1}{t}\right\}'\,dt &=\dfrac{A(N)}{N}-A(1)+\int_2^N \dfrac{A(t)}{t^2}\,dt.
T^*T_\Lambda\{g_i\}_{i\in\mathbb{Z}}=\sum_{i\in\mathbb{Z}}T^*\Lambda_i^*g_i =\sum_{i\in\mathbb{Z}}(\Lambda_i T)^*g_i &=\sum_{i\in\mathbb{Z}}\Lambda_{i+1}^*g_i=T_\Lambda\mathcal{T}\{g_i\}_{i\in\mathbb{Z}},
\delta I = [ \mathrm{t e r m s ~ t h a t ~ v a n i s h ~ w h e n ~ t h e ~ e q u a t i o n s ~ o f ~ m o t i o n ~ h o l d}]^{\mu \nu}\delta g_{\mu \nu}+ \int_{\partial{\cal M}}d^{3}x ( P^{i j}+ Q^{i j}) ]
[x_1, (y_1 \alpha^j)\varphi] = [x_1, y_1 \alpha^j]\varphi = (z_1 \alpha^j) \varphi = z_1 \alpha^j [y_1,(y_1 \alpha^j)\varphi] = [y_2, (y_1 \alpha^j) \varphi] = 0
= 2^{3m-6s}\pi^{m}M \cdot \max\left\{\frac{(n-\alpha)^{2s}}{(n-\beta)^{2s}}, \frac{(n-\beta)^{2s}}{(n-\alpha)^{2s}}\right\}\le 2^{3m-6s}\pi^{m}MK,
\beta ( B_{1 2}+ F_{1 2}) m_{( r ; s )}= \beta ( B + F ) m_{( r ; s )}= \beta ( r - s ) m_{( r ; s )}+ \sum_{\ell = 1}^{r - s - 1}2 \beta ( r - s - \ell ) m_{( r - \ell ; s + \ell )}\, .
\theta \in L^{\infty}\left(0,\infty; H^{1/2}(\mathbb{T})\right) \cap L^{2}\left(0,\infty; \dot{H}^{1}(\mathbb{T})\right), \Lambda^{\gamma/2}\theta \in L^{2}\left(0,\infty; H^{1/2}(\mathbb{T})\right).
\sigma_{1,p}= \sup \{\Vert D_{(v,w)}(\tilde{a}_{p},\tilde{b}_{p})\Vert: (v,w)\in B^{k}(0,\beta_{i})\times B^{d-k}(0,\beta_{i})\}\sigma_{2,p}= \sup \{\Vert D_{(v,w)}(\tilde{c}_{p},\tilde{d}_{p})\Vert: (v,w)\in B^{k}(0,\beta_{i+1})\times B^{d-k}(0,\beta_{i+1})\}.
Q_u=\begin{pmatrix}1 . -1 -1 -1 . . -1 -1 -1 . 1 1+\rho 1+\rho -\rho -\rho 1 1+\rho 1 -\rho -1-\rho -1-\rho \rho \rho -1-\rho -1 \rho -1\end{pmatrix}, Q_v=\begin{pmatrix}1 . -1 -1 . -1 . -1 -1 -1 1 . -1 -1-\rho -1 \rho \rho \rho -1-\rho -1-\rho 1+\rho 1 -\rho 1 -\rho -\rho 1+\rho 1+\rho\end{pmatrix}.
-\log\vert R_{\ell,i,s}( \beta)\vert_{v_0}\leq n\log\vert \beta \vert_{v_0}-(rm+1)n \log\| \boldsymbol{\alpha}\|_{v_0}-nc(x,v_0)+(1-\varepsilon_v)\limsup_{n\to \infty}\dfrac{1}{n}\log|D_{\boldsymbol{c},rmn}\cdot D'_{\boldsymbol{c},rmn}|^{-1}_v+o(n) &=\mathbb{A}_{v_0}(\boldsymbol{\eta},\boldsymbol{\zeta},\boldsymbol{\alpha},\beta)n+o(n)\enspace.
\left[ \left( \frac{H_{0}}{f}\right)^{2}- \left( \frac{f^{\prime}}{f}\right)^{2}\frac{1}{b_{0}^{2}}\right] + \frac{2 \alpha}{M^{2}}\left[ \left( \frac{H_{0}}{f}\right)^{2}- \left( \frac{f^{\prime}}{f}\right)^{2}\frac{1}{b_{0}^{2}}\right]^{2}= \frac{\Lambda_{b}}{6 M^{3}}.
O=\sum_{i=1}^{n}L(y_{i},F_{m-1}(x_{i})+h_{m}(x_{i}))\approx\sum_{i=1}^{n}L(y_{i},F_{m-1}(x_{i}))+h_{m}(x_{i})\nabla_{F_{m-1}}L(y_{i},F_{m-1}(x_{i}))+...
\mu_Y(B_i) = G_*\mu_Y( x_i < \varepsilon,\x_j > r-\varepsilon \ ) = G_*\mu_Y(U_\varepsilon(x_i \le 0,\x_j \ge r\ )) \ge F_*\mu_X(x_i \le 0,\x_j \ge r\ ) - \varepsilon = \mu_X(A_i)-\varepsilon \ge \kappa_i-\varepsilon-\delta.
\Delta_s\frac{\lambda(s)}{B_{k_1k_2}(s^2)}=\frac{1}{B_{k_1k_2}(s^2)}\sum_{j=1}^2 \sum_{p=1}^{k_j}\left( \frac{\partial^2}{\partial s_{pj}^2}+ \frac{1}{s_{pj}}\frac{\partial}{\partial s_{pj}}-4D_{pj}\right)\lambda(s) \ ,
b_{m-l+1}=C\lceil a_{m-1}^2\rceil+l &\leq C\lceil (a_{m}-1)^2\rceil+l &=C\lceil a_m^2-2a_m+1\rceil+l &\leq C(a_m^2-2a_m+2)+l &\leq Ca_m^2-2Ca_m+2C+Cl &=Ca_{m+l-1}^2-C(2a_{m+l-1}-2-l) &\leq Ca_{m+l-1}^2-C(4(m+l-1)-2-l) &=Ca_{m+l-1}^2-C(4m+3l-6)\leq Ca_{m+l-1}^2.
\bigl\|x\pm\frac{y}{\|y\|}\bigr\| \leq \bigl\|x-\sum_{j=1}^m\alpha_j\, m_{a_j,b_j}\bigr\|+\bigl\|\sum_{j=1}^m\alpha_j\, m_{a_j,b_j}\pm y\bigr\|+\bigl|\|y\|-1 \bigr| &< \frac{\varepsilon}{5}+\sum_{j=1}^m\alpha_j \|m_{a_j,b_j}\pm y_j\|+\frac{2\varepsilon}{5}\leq 1+\varepsilon.
\left<\displaystyle\sum_{k=1}^{q}R_{k}\left(Z_{i}\right),\displaystyle\sum_{l=1}^{q}R_{l}\left(Z_{j}\right)\right>=\displaystyle\sum_{k,l=1}^{q}\left<R_{k}\left(Z_{j}\right),\displaystyle R_{l}\left(Z_{j}\right)\right>=\left<Z_{i},Z_{j}\right>,\quad\mbox{for all $i,j=1,\dots,q$,}
\|x-z\|\ge \max(\|x-\sigma(x)\|/2,{\mathfrak R}^k)\quad\mbox{for}z\in \sigma(\Gamma), &5r\le \|\sigma(x)-z\|=d(x,z)\le \|x-z\|\quad\mbox{for}z\in \sigma(\Gamma), z\notin{\mathcal O}(5B).
\big|\overline{\rho}(\alpha_t(f))(z)-\beta_t(\overline{\rho}(f))(z)\big| =\big|\alpha_t(f)(\rho(z))-\overline{\rho}(f)(\eta_{t^{-1}}(z))\big| &=\big|f(\theta_{t^{-1}}(\rho(z)))-f(\rho(\eta_{t^{-1}}(z)))\big|<\varepsilon.
\mathbb{P}(w_{n+1}\leq x , w_{n-k}< K, w_{n-j}= K, j = 0, \ldots k-1) &=\int_{0-}^{K-0}[G(a_k^x(w)) - G(b_k(w))]dW_{n-k}(w), k = 0,\ldots, n,
S[\psi^*,\psi]=\int [Dx^\mu(s)]\,\psi^*[x(s)]\left[-\hbar^2\left(\int_0^1 ds\sqrt{x^{\,\prime\, 2}}\right)^{-1}\int_0^1{ds\over\sqrt{x^{\,\prime\, 2}}}{\delta^2\over \delta x^\mu(s)\ \delta x_\mu(s)}+m^4\right]\psi[x(s)]
\mathrm{P}_{m,i}= \mathrm{P}\left(z_t<\frac{\epsilon_1}{\rho}\right) + \underset{Q_1}{\underbrace{\mathrm{P}\left( z_t>\frac{\epsilon_1}{\rho}, z_m<\max\left\{\frac{\epsilon_2}{\rho\xi_2},\cdots, \frac{\epsilon_i}{\rho\xi_{i}}\right\}\right)}}.
r^{S_0 \oplus S_1}_{(1, i)}= R_i + \beta \sum_{j \in N}p_{ij}f^{S_0 \oplus S_1}_{(1, j)}- \beta f^{S_0 \oplus S_1}_{(0, i)}= R_i + \beta \sum_{j \in S_1}p_{ij}f^{S_1}_j - \beta (f^{S_1}_i - c_i) = r^{S_1}_i + \beta c_i,
\mathcal{R}(A\circ B)^{-1}\leq\mathcal{R}^{-1}(A\circ B) &\leq (\mathcal{R}A\circ\mathcal{R}B)^{-1}&\leq \mathcal{R}^{-1}A\circ\mathcal{R}^{-1}B &\leq \sec^2\theta\mathcal{R}A^{-1}\circ \sec^2\theta\mathcal{R}B^{-1}&=\sec^4\theta(\mathcal{R}A^{-1}\circ \mathcal{R}B^{-1}).
F_0(z) = B^{z-1}[B,A^{z-1}A^{1/2}]A^{1/2}+[B,A^{1/2}]A^{1/2}Y^{z-1}= F_2(z)+B^{z-1}A^{z-1}[B,A^{1/2}]A^{1/2}+[B,A^{1/2}]A^{1/2}F_3(z) = F_2(z)+F_1(z)[B,A^{1/2}]A^{1/2}+[B,A^{1/2}]F_3(z).
c_{\ell,n+1}=\sum_{i=0}^{\ell-1}c_{in}+ c_{\ell,n}= c_{\ell-1,n+1}+c_{\ell,n}=\binom{n+\ell-1}{\ell-1}+\binom{n+\ell-1}{\ell}=\binom{n+\ell}{\ell},
H(t;x,y,1,u,1,v) =\frac{xvt^2(1-yr)}{(1-ytu)(1-tuv(y-yr+1))(tux+y^{-1}-tu)}&\quad+\frac{yu^2vt^2(1-v)(1-yr)}{(1-ytu)(1-tuv(y-yr+1))}F(t;x,y,1,u,1,v).