harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\|G(\overline{a})\|_{\mu}^{\mu[\mathcal{N}(\mathbf{G})]}(D) = \ominus_D \|G(\overline{a})\|^{\mathcal{N}(\mathbf{G})}= \ominus_D \bigcap_{E\in P(\mathcal{A})}\|\varphi(\overline{a})\|_{\mu}^{\mu[\mathcal{N}]}(E) = \|\Box \varphi(\overline{a})\|_{\mu}^{\mu[\mathcal{N}]}
\pi^{i j}= - 2 \gamma^{1 / 2}e^{- \frac{3}{2}( \phi - \tilde{\phi})}\left[ \frac{1}{2}\gamma^{i j}H + \gamma^{- 1 / 3}\left( \frac{\partial H}{\partial h_{i j}}- \frac{1}{3}\frac{\partial H}{\partial h_{k l}}h_{k l}h_{i j}\right) \right] ,
E[|Z(t) - Z(s)|^2] = \frac{1}{C(H)^2}\int_{\mathbb{R}}\Bigg[(t - u)^{H - \frac{1}{2}}_+ - (s-u)^{H- \frac{1}{2}}_+ \Bigg]^2 ds & = \frac{1}{C(H)^2}t^{2H}\int_{\mathbb{R}}\Bigg[(t-s-u)^{H - \frac{1}{2}}_+ - (-u)^{H- \frac{1}{2}}_+ \Bigg]^2 du &= |t-s|^{2H}
J=\left({\begin{array}{*{20}c}{b_1}{a_1}0 0 0{...}{a_1}{b_2}{a_2}0 0{...}0{a_2}{b_3}{a_3}0{...}0 0{a_3}{b_4}{a_4}{...}{...}{...}{...}{...}{...}{...}\end{array}}\right),
G_{\Psi}^{a}= \sum_{k = 1}^{n}( \Psi_{k})_{i}^{\dagger}T_{i j}^{a}( \Psi_{k})_{j}- \sum_{k = n + 1}^{N}( \Phi_{k})_{i}^{\dagger}T_{j i}^{a}( \Phi_{k})_{j}~ .
\lim_{n \to \infty}n^{d(r,s)-3}E(\zeta_{n}(r,t)\overline{\zeta_{n}(s,u)})=\lim_{n \to \infty}n^{d(r,s)-3}E(S_{\lfloor nt \rfloor}(r)\overline{S_{\lfloor nu \rfloor}(s)}).
M(1,\xi_n)=\frac{1}{1-\alpha_N^0(\xi_n)\beta_N^0(\xi_n)}\begin{pmatrix}\grave{m}_{N1}(\xi_n) \grave{m}_{N2}(\xi_n) \grave{m}_{N3}(\xi_n) \grave{m}_{N4}(\xi_n)\end{pmatrix}
\textsf{E}\Vert BA\Vert^{p}=\textsf{E}\Vert B(A-\textsf{E}\tilde{A})\Vert^{p}=\textsf{E}\Vert \textsf{E}(B(A-\tilde{A})|A)\Vert^{p}\le \textsf{E}\Vert B(A-\tilde{A})\Vert^{p}&=\textsf{E}\Vert B(\varepsilon_{ij}(a_{ij}-\tilde{a}_{ij}))_{N\times n}\Vert^{p}\le 2^{p}\textsf{E}\Vert B(\varepsilon_{ij}a_{ij})_{N\times n}\Vert^{p}.
\sum_{j<1-h}(j-1+h)k_j + \sum_{j>1-h}(j-1+h)k_j = w +m 0 \leq \sum_{j>1-h}(j-1+h)k_j = w + m - \sum_{j<1-h}(j-1+h)k_j -m \leq w + \sum_{j<1-h}(1-h-j) k_{j}=\begin{cases}w h\geq 1 begin{align*}-2mm] w+ k_{0}\leq w+1 h =0\end{cases}
r_n=\begin{cases}h_n x+z_n,\ \mathrm{if}\ \delta_n=1, \mathrm{sgn}(h_n x+z_n),\ \mathrm{if}\ \delta_n=0,\ |h_n|^2\mathcal{E}_\mathrm{s}\leq\mathcal{T}, \mathrm{sgn}(h_n x+z_n+z_n^{\mathrm{d}}),\ \mathrm{if}\ \delta_n=0,\ |h_n|^2\mathcal{E}_\mathrm{s}>\mathcal{T},\end{cases}
s_{j_1}\cdots s_{j_{{k_l}-1}}(\varpi_{j_{k_l}}) = s_{j_1}\cdots s_{j_{{k_l}-1}}(\varpi_n)={\rm wt}(\underbrace{-, \cdots, -}_{2l-2}, \underbrace{+, \cdots, +}_{n-2l+2}), & s_{j_1}\cdots s_{j_{k_l}}(\varpi_{j_{k_l}}) ={\rm wt}(\underbrace{-, \cdots, -}_{2l}, \underbrace{+, \cdots, +}_{n-2l}).
\left(\frac{1}{n}\sum_{0\le t<n}\exp\left\{i\,\frac{2\pi}{n}\,u\,t-i\,\frac{2\pi}{n}\,t\,v\right\}\right)=\begin{cases}\begin{array}{cc}1 \mbox{if}\:0\le u=v<n 0 \mbox{otherwise}\end{array}.\end{cases}
\left\{\mathrm{\mathrm{ML}}(R_{n,e,P})=k[X],\,\mathrm{\mathrm{AL}}(R_{n,e,P})=k[X,S],\, B_{n_{0},P}\,,\, R_{n,e_{0},P};\,\, n_{0}\in\{1,\ldots,n\},e_{0}\in\{1,\ldots,e\}\right\}
{_{mi}\mathcal{R}}(A\ast B)\simeq\bigoplus_{t=a}^{b}\mathbf{1}_{pt}^{f_{m,t}(v)}\boxtimes({_{ti}\mathcal{R}}(A)\ast{_{(m-t)i}\mathcal{R}}(B))[-P_t](-\frac{P_t}{2}),
\partial_\lambda \Delta(\lambda)|_{\Lambda_1}= \frac{1}{5}(4\gamma_4^3 + 27 \gamma_6^2)\, \big(\wp'(\alpha)\big)^6 \Big(\big(\tfrac{3}{5}\wp(\alpha)\big)^3, \big(\tfrac{3}{5}\wp(\alpha)\big)^2, \tfrac{3}{5}\wp(\alpha), 1 \Big),
[y_1, y_2]=-y_3+\frac{\alpha_4}{\beta_3}y_4+\theta_1y_5, [y_2, y_1]=y_3, [y_2, y_2]=y_4, [y_2, y_3]=-\theta_2y_5, [y_3, y_2]=(\theta_2-1)y_5, [y_1, y_4]=y_5.
{\cal J}\overline{{{A}}}_{n}^{i}{\cal J}^{- 1}= a_{n}^{i};{\cal J}\overline{{{\widetilde A}}}_{n}^{\scriptscriptstyle L}{\cal J}^{- 1}= L_{n}^{\scriptscriptstyle T};{\cal J}\overline{{{C}}}_{n}{\cal J}^{- 1}= c_{n}.
\left\{\begin{pmatrix}u f\end{pmatrix}\in L^2(\Omega)\times L^2(\partial\Omega),\ \Delta u\in L^2(\Omega),\ \frac{\partial u}{\partial \nu}\in L^2(\partial\Omega),\ \eta_1\Delta_{\partial \Omega}f\in L^2(\partial\Omega) \right\},
\bar{K}_G(v) = e^{-\frac{v^2}{2\theta^2}}, \bar{K}_e(v) = e^{-\frac{v}{2\theta^2}}, \bar{K}_C(v) = \frac{1}{1+\frac{v^2}{\theta^2}}, \bar{K}_t(v) = \frac{1}{1+v^{\theta}}, \bar{K}_i(v) = \frac{1}{\sqrt{v^2+\theta^2}}.
b_n(\mathrm{d}x)= \frac{v_{nF}(\mathrm{d}x)}{\sqrt{f(x)}}- \int_A \frac{v_{nF}(\mathrm{d}y)}{\sqrt{f(y)}}\frac{1}{\sqrt{\Delta}- \int_A \sqrt{f(y)}\,\mathrm{d}y}\bigl(\eta_A(x) -\sqrt{f(x)}\bigr) \,\mathrm{d}x
\int_{-1}^{0}\int_{x^{2}}^{1}\sqrt{y}dydx=\int_{-1}^{0}[\frac{y^{3/2}}{3/2}|_{x^{2}}^{1}]dx=\frac{2}{3}\int_{-1}^{0}(1-x^{3})dx=\frac{2}{3}(x-x^{4}/4)|_{-1}^{0}=\frac{5}{6}
q_{b}( a , r ) \approx \frac{( 1 - 2 \delta_{B 0}) \tilde{\nu}^{3 / 2}\exp \left[ - 2 \tilde{\nu}\ln ( a / r ) \right]}{( 8 \pi )^{1 / 2}S_{D}\sigma^{D - 1}r^{D}\sqrt{a^{2}- r^{2}}}f_{1}^{( q )},
\begin{dcases*}\nu=\cos\theta\,\overline\nu+\sin\theta\,\overline\eta \eta=-\sin\theta\,\overline\nu+\cos\theta\,\overline\eta\end{dcases*}, \begin{dcases*}\overline\nu=\cos\theta\,\nu-\sin\theta\,\eta \overline\eta=\sin\theta\,\nu+\cos\theta\,\eta\end{dcases*}.
\begin{aligned}\ S=((b-1)\deg x_{i+1}+(b-2)\deg x_{i+2}+\dots+\deg x_{i+b-1})+ &((a-1)\deg x_1+\dots+(a-i)\deg x_i+ &(a-i-1)\deg m_b(x_{i+1},\dots,x_{i+b})+ &(a-i-2)\deg x_{i+b+1}+\dots+\deg x_{a+b-2})\end{aligned}
\frac{\varepsilon_B}{2}= \frac{1}{2}\left| \mathbb{P}(B=1) - \mathbb{P}(B=0) \right| & = \frac{1}{2}\left| 2 \mathbb{P}(B=1) - 1 \right| & = \frac{1}{2}\left| 2 \mathbb{E}[B] - 1 \right|,
\biggl\{- 4z(1-z) \frac{d^2}{dz^2}+ 2 [- 4b_1z^2 + (4a+4b_1+3)z - 4a + 2l + d -3] \frac{d}{dz}+ [B_1 - 2b_1(4a+2b_1+3)]z - A + 2a(2a+1) + 2b_1(4a-2l-d+3)\biggr\}\phi(z) = 0,
4\,\left(N(X,T)\right)^\flat\wedge\left(JN(X,T)\right)^\flat =\frac{\left(X(\textbf{H})\right)^2+\left(JX(\textbf{H})\right)^2}{4}T^\flat\wedge(JT)^\flat, =\frac{\left(X(\textbf{H})\right)^2+\left(JX(\textbf{H})\right)^2}{4}d\theta\wedge d\varphi.
X_i^{\rm C}(R^\mu) = X_i^{\rm C}(L) + \mu_a B^a_{ij}v^j , \qquad X_i^{\rm V}(R^\mu) = X_i^{\rm V}(L),{\tilde E}_a^{\rm C}(R^\mu) = - \mu_c C^c_{ab}{\tilde{\rm v}}^b, \qquad\qquad{\tilde E}_a^{\rm V}(R^\mu) ={\tilde E}_a^{\rm V}(L) - \mu_a.
\mathcal{F}(\mathcal{S};\mu_1, \mu_2) = \left\{\pi \in \mathcal{P}(\mathcal{X}_1 \times \mathcal{X}_2): \pi \circ \operatorname{proj}_{\{j\}}^{-1}= \mu_j, \ \forall j = 1, 2 \right\}.
\mathcal{T}\left(J_{i}^{n}\left(\omega\right),a_{n}W_{i}\left(\omega\right)\right) =x_{t}\mathcal{T}\left(J_{i}^{n}\left(\omega\right),a_{n}U_{i}\left(\omega\right)\right) =y_{t}\left|B_{n}^{X}[0,T-\frac{\epsilon}{2}]\right| =M.
\tilde{m}_{b}={\frac{\beta^{2}}{8 \pi}}( 1 -{\frac{\tilde{m}_{0}^{2}}{\tilde{\mu}^{2}( \varphi_{0})}}) [{\frac{1}{\sqrt{1 - \tilde{m}_{b}^{2}}}}t a n^{- 1}\sqrt{{\frac{1 + \tilde{m}_{b}^{2}}{1 - \tilde{m}_{b}^{2}}}}-{\frac{\pi}{4}}]
\sum_{m=0}^{6}\sum_{k_0+\ldots+k_m+m=\ell_n^2}\left(\tilde{\chi}_{n,x}\overline{R}_n\right)^{k_0}\tilde{\chi}_{n,x}S_n\left(\tilde{\chi}_{n,x}\overline{R}_n\right)^{k_1}\ldots \tilde{\chi}_{n,x}S_n\left(\tilde{\chi}_{n,x}\overline{R}_n\right)^{k_m}f(x).
\tilde{c}( r , s ) = \left( \begin{matrix}{n}\\{r}\\ \end{matrix}\right) \left( \begin{matrix}{n}\\{s}\\ \end{matrix}\right) \Bigg [ c ( r , s ) + \frac{1}{1 + n}\Bigg ] - \frac{1}{2 ( 1 + n )}\left( \begin{matrix}{n + 1}\\{r}\\ \end{matrix}\right) \left( \begin{matrix}{n + 1}\\{s}\\ \end{matrix}\right) .
\left\Vert h_{D_{1}+D_{2}}\left(x\right)-h_{E_{1}+E_{2}}\left(x\right)\right\Vert =\left\Vert h_{D_{1}}\left(x\right)+h_{D_{2}}\left(x\right)-h_{E_{1}}\left(x\right)-h_{E_{2}}\left(x\right)\right\Vert .
e^{ic\langle \xi,-v\alpha_0\rangle}=\prod_{\alpha\in R^+_0}\left( \frac{1-t_{\alpha}e^{i\langle \xi,\alpha \rangle}}{e^{i\langle \xi,\alpha \rangle}-t_{\alpha}}\right)^{{\langle -v\alpha_0, \hat\alpha\rangle}},\forall v\in{W_0},
\mathcal{J}_S[(w + \overline{w})u](t) = \mathcal{J}_S[(W + \overline{W})U] (t), \mathcal{J}_{W_{+}}\Bigl[\frac{\Delta}{\langle \nabla \rangle}(u \overline{u}) \Bigr](t) = \mathcal{J}_{W_{+}}\Bigl[\frac{\Delta}{\langle \nabla \rangle}(U \overline{U}) \Bigr](t),
\begin{array}{l l}{{{\cal L}_{K}=}}&{{\frac{1}{4}F_{\mu \nu}F^{\mu \nu}- ( 1 - g^{2}\zeta \eta^{2}) \tilde{G}_{\mu}\tilde{G}^{\mu}+ 2 ( m + g h \zeta \eta^{2}) A_{\mu}\tilde{G}^{\mu}+}}\\{{}}&{{- 2 h \zeta \eta^{2}\Sigma \partial_{\mu}A^{\mu}+ h^{2}\zeta \eta^{2}A_{\mu}A^{\mu}+ \zeta \eta^{2}\partial_{\mu}\Sigma \partial^{\mu}\Sigma ,}}\\ \end{array}
\frac{n!}{n^n}= \frac{\binom{n}{t_1}t_1! \cdot t_2!}{n^n}\leq \frac{\binom{n}{t_1}t_1! \cdot t_2!}{\binom{n}{t_1}t_1^{t_1}\cdot t_2^{t_2}}= \prod_{\sigma = 1}^2 \frac{t_\sigma!}{t_\sigma^{t_\sigma}},
\phi(3u^4 - 3 y_1^2 u^2 + y_1^4) = 3u^4 - 3(u^2 - y_1^2)u^2 + (u^2 - y_1^2)^2 = 3u^4 -3u^4 + 3y_1^2 u^2 + u^4 - 2u^2 y_1^2 + y_1^4 = y_1^4 + y_1^2u^2 + u^4 \in J_2,
( \partial_{\bar{l}}A_{0}^{z}) ( \partial_{l}A_{0}^{\bar{z}}) - ( \partial_{\bar{l}}A_{0}^{\bar{z}}) ( \partial_{l}A_{0}^{z}) = \frac{{\cal L}_{0}}{2}= \frac{4}{\cosh ( l )^{2}}- 1 .
[ x_{r}, p_{r}] = i \hbar + i \hbar ( q^{2}- 1 ) \sum_{s \le r}\left( \frac{q^{2}+ 1}{2}\right)^{s - 1}\left( \frac{x_{s}^{2}}{4 L_{s}^{2}}+ \frac{p_{s}^{2}}{4 K_{s}^{2}}\right)
u (t)=-\rho_{r}(\check{\xi}_{r}(t_{k}))\check{\xi}_{r}(t_{k})+\Psi \eta (t_{k}) \dot{\hat{\xi}}(t)=A_{o}\hat{\xi}(t)+\lambda e(t_{k})+B(u(t)\!-\!\Psi \eta (t_{k})) \dot{\eta}(t)=M \eta (t)+N u (t),~\forall t\in[t_{k},t_{k+1}),~k\in\mathbb{S}
\mathbb A\bigl((\pi_t)_{t\in[0,T]}\bigr) =\frac 14\int_0^T \bigl\|\dot\rho_t -\Delta\phi(\rho_t) -\nabla\cdot(\chi(\rho_t)\nabla V)\bigr\|_{-1,\chi(\rho_t)}^2\;\!\mathrm dt.
\begin{array}{r l r l}{{1^{*}:}}&{{a \mapsto \epsilon ( a )}}&{{\mu^{*}( f_{1}, f_{2}) :}}&{{a \mapsto f_{1}\otimes f_{2}\circ \Delta ( a )}}\\{{\epsilon^{*}:}}&{{f \mapsto f ( 1 )}}&{{\Delta^{*}( f ) :}}&{{( a_{1}, a_{2}) \mapsto f ( \mu ( a_{1}, a_{2}) )}}\\{{S^{*}( f ) :}}&{{a \mapsto f ( S ( a ) )}}&{{~}}&{{~}}\\ \end{array}
(\mathcal{H}\mathbf{Q})(\mathbf{S}^{'}, \mathbf{A}^{'}, \overline{\pmb{\omega}}) = \max_{A^{'}\subseteq \overline{\mathbf{A}}, \pmb{\Omega}^{'}\subseteq \overline{\pmb{\Omega}}}\pmb{\Omega}^T\mathbf{Q}_{\pi_{\overline{\theta}^{*}}}(\mathbf{S}^{'}, \mathbf{A}^{'}, \overline{\pmb{\omega}}^{'}).
||x||_1\, U(T,x)= \ \mathbb{E}\big[ \Xi(0)\big] \ge \mathbb{E}\big[ \Xi(T)\big] =\mathbb{E}\big[X(T)L(T)U(0,\mathfrak{X}(T))\big] \ge \ \mathbb{E}\big[X(T)L(T)\big] = ||x||_1\, \mathfrak{u}_{\mathcal{M}}(T,x)\,,\mathrm{by\the\definition}\ \eqref{eq:um}\,.
\begin{aligned}\langle \sigma \partial_t^{1/2}y,\partial_t^{1/2}v \rangle = \langle \sigma \partial_t y,v^{\perp}\rangle \mbox{and}\langle \sigma \partial_t^{1/2}y,\partial_t^{1/2}v^{\perp}\rangle = \langle \sigma \partial_t y,v \rangle \end{aligned}
I_{1,1}= \Bigl|\int_0^\infty\langle \partial_t(u\theta \phi \chi),u\theta \phi \chi \rangle\,{\rm d}t\Bigr|, I_{1,2}= \Bigl| \int_0^\infty\langle u \theta \phi \chi\cdot\theta \chi \partial_t\phi,u \rangle\,{\rm d}t\Bigr|, I_{1,3}= \Bigl| \int_0^\infty\langle u\theta \phi \chi\cdot\theta \phi \chi',u\rangle dt\Bigr|.
\begin{aligned}\mathbb{E}\left\{\Big[ \frac{\hat F_{\lambda^*}}{F_\varpi(P \parallel Q)}-1 \Big]^2\right\}= \frac{{\rm MSE}(\hat F_{\lambda^*})}{F_\varpi^2(P \parallel Q)}=O(\varGamma^{-1}),\end{aligned}
g_2(x,w) = u_{5,3}x^5 w^3 + u_{4,4}x^4 w^4 + u_{3,5}x^3 w^5, g_3(x,w) = u_{7,5}x^7 w^5 + u_{6,6}x^6 w^6 + u_{5,7}x^5 w^7.
-\real\frac{L'_\chi}{L_\chi}(s) = \frac{1}{2}\log(\Delta N(\mathfrak f_{\chi})) + \real \left(\delta(\chi)\left(\frac{1}{s}+\frac{1}{s-1}\right) -\sum_{\rho \in R_{\chi}}\frac{1}{s - \rho}+ \psi_\chi(s)\right).
a_{0,3}(\ell,k)=\frac{1(\ell\le n_1)}{(2\pi i)^2}\int_{\Gamma_{D_1}}dz\int_{\gamma_{\tau_1}}d\zeta\frac{G_{\ell-1,\mu_1,\xi_1}(z)}{G_{k,\mu_1,\xi_1}(\zeta)(z-\zeta)}=1(\ell\le n_1)c_3(\ell,k).
X_{0,0}(x,x_{1}) |I_1\rangle\simeq \sum_{n=0}^{|I_1|}\sum_{\substack{J' \subseteq I_1 |{J'}|=n}}\mathcal{G}^{n}(0,I_1,{J'})W_{n+1,n+1}(x)\left|I_1\setminus{J'}\right\rangle \,.
\left(u(t)-u_0-t\partial_tv_01_{\alpha >1},\phi\right) =I_{t}^{\alpha}\left(f(t),\phi\right)+\sum_{k=1}^{\infty}I_{t}^{\alpha-\beta}\int_{0}^{t}\left(g^{k}(s),\phi\right)dw_{s}^{k}
\delta^{-n}\sigma_4^{-1}s_2 \underbrace{\sigma_4 \delta^n r_1 \delta^{-n}\sigma_4^{-1}s_2 \sigma_4 \delta^n \dots}_{= (\sigma_4 \delta^n \sigma_4^{-1}r_1 \sigma_4 \delta^{-n}\sigma_4^{-1}s_2)^{n-1}}\sigma_4 \delta^n \sigma_4^{-1}r_1 \sigma_4 \delta^{-n}\sigma_4^{-1}s_2 \sigma_4 \delta^{n}r_1
L_{2 k}= - \frac{2 y_{1}y_{2}}{\zeta}\cdot [ R_{k}+ \frac{y_{1}^{2}+ y_{2}^{2}}{\zeta y_{2}}P_{k}- \frac{( y_{1}^{2}- y_{2}^{2})}{2 y_{1}y_{2}}\cdot \frac{\partial_{2}P_{k}}{2 k}- \frac{\partial_{1}P_{k}}{2 k}+ \frac{y_{1}^{2}+ y_{2}^{2}}{y_{1}y_{2}^{2}}\cdot \frac{P_{k}}{4 k}] ,
\begin{aligned}b_0 = \tilde{b}_0 = 1, b_1 = \tilde{b}_1, b_2 = \tilde{b}_2, b_3 =\tilde{b}_3, b_4 = \tilde{b}_4, b_5 = b_6=0, b_7 = \tilde{b}_0 = 1, b_8 = \tilde{b}_1, b_9 = \tilde{b}_2, b_{10}= \tilde{b}_3 , b_{11}= \tilde{b}_4 = 1.\end{aligned}
P(N,n,l,\omega x^2) = L_N^{l + \frac{1}{2}}(-\omega x^2) L_n^{l + \frac{1}{2}}(\omega x^2) - L_{N-1}^{l + \frac{1}{2}}(-\omega x^2) L_{n-1}^{l + \frac{1}{2}}(\omega x^2),
\Phi_\delta(t)=\int_{0}^{t}\Phi_\delta'(s)ds \leq \int_{0}^{t}\int_{B_R}\frac{|\mathbf{B}_s(X_1(s,x))-\mathbf{B}_s(X_2(s,x))|}{\delta+|X_1(s,x)-X_2(s,x)|}dx ds & \leq \int_{0}^{t}\int_{B_R}\min\left\{\frac{2||\mathbf{B}_s||_{L^\infty}}{\delta},\frac{|\mathbf{B}_s(X_1(s,x))-\mathbf{B}_s(X_2(s,x))|}{|X_1(s,x)-X_2(s,x)|}\right\}dx.
\|f(t)\|_{L^2}^2 = \|1_{\Omega_{100}}h_{100}+ \sum_j f_j(t)\|_{L^2}^2 \leq 2\|1_{\Omega_{100}}h_{100}\|_{L^2}^2 + \sum_{j}j^2\|f_j(t)\|_{L^2}^2
S_\lambda (f) = W^* t_\lambda W ( f )= W^* t_\lambda \pi(f)\xi &= W^* \pi(f\circ \sigma^n) t_\lambda \xi = W^* \pi(f\circ \sigma^n)W W^* t_\lambda \xi &=(f\circ \sigma^n) \cdot f_\lambda.
A_{j}(\textbf{u}) = \left( \begin{array}{ccccc}v_{j}0 0 \cdots 0 \delta_{j1}f(\textbf{u}) v_{j}0 \cdots 0 \delta_{j2}f(\textbf{u}) 0 v_{j}\cdots 0 \cdots \cdots \cdots v_{j}\cdots \delta_{jd}f(\textbf{u}) 0 0 \cdots v_{j}\end{array}\right).
X=A_{1}^{\dagger}C_{1}+L_{A_{1}}D_{1}B_{1}^{\dagger}+L_{A_{1}}U_{1}R_{B_{1}},\Y=A_{2}^{\dagger}C_{2}+L_{A_{2}}D_{2}B_{2}^{\dagger}+L_{A_{2}}U_{2}R_{B_{2}}, Z=A_{3}^{\dagger}C_{3}+L_{A_{3}}D_{3}B_{3}^{\dagger}+L_{A_{3}}U_{3}R_{B_{3}},
\sum_{\phi \in \mathcal{B}(\Pi)^{{K}^S}}\mathcal{P}_{G^{\sigma}}(\Pi'(f^1)\phi) \overline{\mathcal{P}_{G^{\theta}}(\phi)}=\mathrm{tr}(\Pi(f^S)) \sum_{a_i}\mathcal{P}_{G^{\sigma}}(\phi^{S_0(ME)}_0 \otimes \Pi'(f_{S_0(ME)}^1)a_{i})\mathcal{P}_{G^{\theta}}(\overline{\phi^{S_0(ME)}_0 \otimes a_{i}})
\prod_{m=1}^{\infty}\left( \prod_{n=1}^{\infty}\frac{1}{1-(t^m)^n}\right)^{\mu(m)}=\prod_{k=1}^{\infty}\left(\prod_{m \vert k}\frac{1}{1-t^k}\right)^{\mu(m)}=\prod_{k=1}^{\infty}\left(\frac{1}{1-t^k}\right)^{\sum_{m \vert k}\mu(m)}=\prod_{k=1}^{\infty}\left(\frac{1}{1-t^k}\right)^{\sum_{m \vert k}\mu(m)}=\frac{1}{1-t},
\partial_{z_{+}}^{3}v^{+}( z_{+}) = z_{-}^{2}\partial_{z_{-}}\bigg ( z_{-}^{2}\partial_{z_{-}}\bigg ( z_{-}^{2}\partial_{z_{-}}\bigg ( z_{-}^{- 2}v^{-}( z_{-}) \bigg ) \bigg ) \bigg ) = 0 ,
\varpi_{1*}\circ\cdots\circ \varpi_{r*}(1_{Z_{\lambda}}) = \phi_1\left(t_1^{\lambda_1}\cdots t_r^{\lambda_r}\prod_{1\leq i\leq r \atop{\chi_i\geq 0}}\frac{1}{2+\beta t_i}\frac{\prod_{(i,j)\in \Delta_{r}}(1- \bar t_i/\bar t_j)}{\prod_{(i,j) \in C(\lambda)}(1- t_i/\bar t_j)}\right).
=\frac{e^{4}}{(k-k^{\prime})^{4}}((\overline{v}_{k^{\prime}}\gamma^{\mu}v_{k})(\overline{u}_{p}\gamma_{\mu}u_{p^{\prime}}))((\overline{v}_{k}\gamma^{\nu}v_{k^{\prime}})(\overline{u}_{p^{\prime}}\gamma_{\nu}u_{p}))
\partial_tM'(t)=\sum_{\substack{|k-k_0|\leq D\sigma k_0 |l-k_0|\leq D\sigma k_0}}\int_{\mathbb{R}^2}\zeta(k_0\sqrt{\sigma}v)\mu(t,k,l,v,w)\overline{\beta_k(t,v)}\beta_l(t,w)\, \mathrm{d}v \mathrm{d}w,
\left( \prod_{j=1}^d \frac1 r_j \right) \frac{\left( \log T \right)^d}{d!}\ \le \|E_d(T)|\ \le \ \left( \prod_{j=1}^d \frac1 r_j \right) \frac{\left( \log T +\sum_{j=1}^d r_j \right)^d}{d!}.
\mathcal{E}^R_2(\theta)= -\chi \Phi(g^R, \pi^R) - (1-\chi) \Phi((g^\theta)^R, (\pi^\theta)^R)- (2\psi_0 R^{-2}, \tfrac{1}{2}h^R\cdot_{\gamma^R}(\textup{div}_{\gamma^R}\tau^R +V^R)).
\begin{array}{lll}{\displaystyle \nabla \times \mathbf{E}+\frac{\partial \mathbf{B}}{\partial t}=0, \;\nabla\cdot\mathbf{B}= 0 ,}begin{align*}2mm]{\displaystyle \frac{1}{{\mu}}\nabla \times \mathbf{B}- \epsilon\frac{\partial \mathbf{E}}{\partial t}=\mathbf{J}, \nabla\cdot(\epsilon \mathbf{E}) = \rho.}\end{array}
2s_{n}=\frac{2}{2}+\frac{2}{4}+\frac{2}{8}+\frac{2}{16}+\cdot\cdot\cdot+\frac{2}{2^{n}}=1+[\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdot\cdot\cdot+\frac{1}{2^{n-1}}]=1+[s_{n}-\frac{1}{2^{n}}].
i \gamma^{\mu}\partial_{\mu}\left( \begin{array}{c}{{\psi_{L}}}\\{{\psi_{R}}}\\{{\psi_{\nu}}}\\ \end{array}\right) = \left( \begin{array}{c c c}{{0}}&{{\phi^{2}}}&{{0}}\\{{\overline{{{\phi^{2}}}}}}&{{0}}&{{\overline{{{\phi^{1}}}}}}\\{{0}}&{{\phi^{1}}}&{{0}}\\ \end{array}\right) \left( \begin{array}{c}{{\psi_{L}}}\\{{\psi_{R}}}\\{{\psi_{\nu}}}\\ \end{array}\right) .
\Psi_{\nu}^{( \nu^{2})}(x;\Lambda_{2})=\frac{\lambda_{2}+\int_{x_{0}}^{x}dt\big( \Psi_{\nu}^{(\nu)}(t;\lambda_{1})\big)^{2}}{\Psi_{\nu}^{(\nu)}(x;\lambda_{1})}, \lambda_{1},\lambda_{2}\in \mathbb{R},
\begin{aligned}\Delta r_{O2}\leq \ln \Big(1+ \frac{(P-P_{O2}^*)|\hat{h_1}|^2}{N_0}\Big) = \ln \Big(1+ \frac{(P-P_{O2}^*)|{h_1}|^2}{e^{\frac{r^*}{\alpha_i}}N_0}\Big) \leq \ln \Big(1+ \frac{(P-P_{O2}^*)|{h_1}|^2}{e^{r^*}N_0}\Big).\end{aligned}
\begin{aligned}-\iint g_s \partial_y v^\epsilon \epsilon g^\epsilon \leq \epsilon \Vert g_{s}\Vert_{L^\infty}\Vert v^\epsilon_y \Vert_{L^2}\Vert g^\epsilon\Vert_{L^2}\leq \epsilon L \Vert g_{s}\Vert_{L^\infty}\Vert v^\epsilon_y \Vert_{L^2}\Vert g^\epsilon_x\Vert_{L^2},\end{aligned}
q_{sqL}=\frac{1}{k}\sum_{i=1}^n\sqrt{s_i^*}= \frac{1}{k}\sum_{i=1}^n\frac{s_i^*}{\sqrt{s_i^*}}< \max_i \frac{1}{\sqrt{s_i^*}}\frac{\sum_{i=1}^ns_i^*}{k}&=\max_i \frac{1}{\sqrt{s_i^*}}= q_{L}
\phi_{ij}-\phi_j(\frac{i}{n})= \mathbf{e}_j^* \Omega_i^b\left(\gamma_i^b-\widetilde{\gamma}^b_i\right)+\mathbf{e}_j^*\Omega_i^b\left( \widetilde{\Gamma}^b_i-\Gamma^b_i \right) \widetilde{\Omega}_i^b\widetilde{\gamma}_i^b,
{\cal L}^{( 1 )}={\frac{1}{2}}{\epsilon^{j k i}\partial_{j}{A_{k}}^{a}}{\epsilon_{a b}\partial_{0}{A_{i}^{b}}}-{\frac{1}{4}}{F^{a , j k}}{{F^{a}}_{j k}}+{\dot{\lambda}^{( a )}}{A_{3}^{a}}
C ( \beta ) = \rho_{0}\frac{\beta^{- p q + 1}\Gamma ( q / p )^{p}}{2 p ! q ! \Gamma ( p / q )^{q}}\prod_{i = 1}^{p}\Gamma ( q i / p )^{- 2}\prod_{i = 1}^{q - 1}\Gamma ( - p i / q )^{- 2}.
\sum_{h=1}^{+\infty}\frac{1}{h^\alpha(k+h)^\beta}\leq \sum_{h=1}^{+\infty}\frac{1}{h^{\alpha+\beta-\gamma}(k+h)^\gamma}&= \frac{1}{k^\gamma}\sum_{h=1}^{+\infty}\frac{1}{h^{\alpha+\beta-\gamma}(1+h/k)^\gamma}&\leq \frac{1}{k^\gamma}\sum_{h=1}^{+\infty}\frac{1}{h^{\alpha+\beta-\gamma}}
\sum_{k =0}^{\infty}\sum_{a=\pm 1}\bigg|P\Big(a\big|\sigma_{-\infty}^{-(k+1)}\omega_{-k}^{-1}\Big)-P\Big(a\big|\tau_{-\infty}^{-(k+1)}\omega_{-k}^{-1}\Big)\bigg|^2< \infty
T_nT_mf(x)=\sum_{j=0}^{n-1}[T_mf(\frac{x+j}{n})-T_mf(\frac{j}{n})] &=\sum_{j=0}^{n-1}\sum_{k=0}^{m-1}\{[f(\frac{\frac{x+j}{n}+k}{m})-f(\frac{k}{m})]-[f(\frac{\frac{j}{n}+k}{m})-f(\frac{k}{m})]\}&=\sum_{j=0}^{n-1}\sum_{k=0}^{m-1}[f(\frac{x+j+nk}{mn})-f(\frac{j+nk}{mn})] &=\sum_{j=0}^{mn-1}[f(\frac{x+j}{mn})-f(\frac{j}{mn})] =T_{nm}f(x).
= 2 \pi \hbar \sum_j < \psi_j | \bigg\{\frac{-1}{3 i \hbar}\hat \theta [\partial_x \hat\theta , \partial_y \hat \theta ] + \frac{1}{2}\big( (\partial_x \hat \theta)(\partial_{\tilde x}\hat\theta) + (\partial_y \hat\theta)(\partial_{\tilde y}\hat\theta) \big) \bigg\}| \psi_j >,
\tfrac1N I(U_3^N;Y^N,U_1^{i_1},U_2^N) < I_3, \tfrac1N I(U_{1,i_1+1}^N;Y^N,U_1^{i_1},U_2^N) \le I_1', \tfrac1N I(U_{1,i_1+1}^N, U_3^N;Y^N,U_1^{i_1},U_2^N) = I_1' + I_3,
\rho^R_{0}\left( \rho_{t}\right) - \rho^R_{t -s}\left( \rho_{s}\right)=\int_{s}^{t}\rho^R_{t -u}\left( \mathcal{L}_{\star}^{full}\left( u \right) \, \rho_{u}- \mathcal{L}_{\star}^R \, \rho_{u}\right) du ,
h = \int d^{2}x \; \left[ \frac{1}{2 L}p_{-}^{3 \, \dagger}({x}_{\bot}) p_{-}^{3}({x}_{\bot}) + \frac{L}{2 \eta^{2}}( \nabla_{\bot}a_{-}^{3}({x}_{\bot}) )^{2}\right] .
I_{\tilde{a}}[u] =\frac{1}{p}[u]_{s,p}^p+\frac{1}{p}\int_{\mathbb{R}^N}\tilde{a}(x)|u|^p dx-\frac{b}{2}\int_{\mathbb{R}^N}(K\ast F(u))F(u)dx-\frac{\varepsilon_g}{p_g}\int_{\mathbb{R}^N}|u|^{p_g}dx.
M_{t \wedge T_n}= f(V_{t \wedge T_n}) - \int_0^{t \wedge T_n}\mathcal{A}f(V_s) \d s & = f(V_t^n) - \int_0^{t \wedge T_n}\mathcal{A}_n f(V_s) \d s & = f(V_t^n) - \int_0^{t}\mathcal{A}_n f(V_s^n) \d s,
I_n(\mu) =\left \{\int_0^{\frac{\pi}{6n+4}}+\int_{\frac{\pi}{6n+4}}^{\frac{\pi}{2}}\right \}\theta\sin\left(\mu\theta\right)\prod_{k=0}^{n}\cos((3k+1)\theta)\cos((3k+2)\theta)\mathrm{d}\theta begin{align*}5pt] =I^{(1)}_n(\mu)+I^{(2)}_n(\mu).
\frac{|G(x+h)-G(x)|}{|h|^{\rho_n-\lfloor\rho_n\rfloor}\psi(|h|)}= \frac{|G(x+h)-G(x)|}{|h|}\frac{|h|^{1-(\rho_n-\lfloor\rho_n\rfloor)}}{\psi(|h|)}&\quad\le{N^{1-(\rho_n-\lfloor\rho_n\rfloor)-\epsilon}}C_{\psi,\epsilon}{|h|^{1-(\rho_n-\lfloor\rho_n\rfloor)-\epsilon/2}}\le C_{\psi,\epsilon}N^{-\epsilon/2}\le \eta/2
i G_{F \bar{0}{0}}( x , y ) = \theta ( x^{0}- y^{0}) \sum_{k}\bar{u}_{k}( x ) u_{k}^{*}( y ) \bigl ( \alpha_{k}^{*}- \frac{| \beta_{k}|^{2}}{\alpha_{k}}\bigr ) + x \leftrightarrow y .
\begin{array}{l}R_{ijk}{}^l=\partial_{x_i}\Gamma_{jk}{}^l-\partial_{x^j}\Gamma_{ik}{}^l+\Gamma_{in}{}^l\Gamma_{jk}{}^n-\Gamma_{jn}{}^l\Gamma_{ik}{}^n, begin{align*}0.05in]\rho_{jk}=\partial_{x_i}\Gamma_{jk}{}^i-\partial_{x^j}\Gamma_{ik}{}^i+\Gamma_{in}{}^i\Gamma_{jk}{}^n-\Gamma_{jn}{}^i\Gamma_{ik}{}^n\end{array}
\begin{aligned}J_2(\beta,N,T) \leq CN^2\int_{0}^{\pi}\iint_{0\le s\le t\le T}\frac{1}{(t+s)^{3/2}}& \times\int_{\mathbf{B}_2(0)}\exp\left(-\frac{\big|z-D_t^{(k)}+D_s^{(\ell)}\big|^2}{2(t+s)}\right) dz dsdt d\theta. \end{aligned}
(T_0-T_h(\lambda))(u_0)(x) =\frac{\eta_0^1}{4\pi}\int_{B_1}\big(1-\exp({i\sqrt{\lambda}h|x-y|)}\big) \frac{u_0(y)}{|x-y|}dy &+\frac{\eta_0^2}{4\pi}\int_{B_2}\big(1-\exp({i\sqrt{\lambda}h|x-y|)}\big) \frac{u_0(y)}{|x-y|}dy
d s^{2}= g_{y y}d y^{2}+ g_{i j}^{( d )}d x^{i}d x^{j}={\frac{l^{2}d y^{2}}{4 y^{2}}}+{\frac{1}{y}}\tilde{g}_{i j}( x , y ) d x^{i}d x^{j}\,
\vec{\psi}_{i+1}(n)=\begin{pmatrix}cn 1\end{pmatrix}+\sum_{s=n_0+1}^n\Phi_1(s)R(s-1)\vec{\psi}_{i}(s-1)-\sum_{s=n+1}^\infty\Phi_2(n)R(s-1)\vec{\psi}_{i}(s-1).
\frac{\omega_1}{2\pi}\int_{t-\pi/\omega_1}^{t+\pi/\omega_1}v_0J_0^2(s) ds= v_0\left(\frac{A(x)^2}{2}+ \frac{B(x)^2}{2}+ A(x)B(x) \cos(\eta t)\right) + O\left(\frac{\left(|A(x)|^2 +|B(x)|^2\right) \eta}{\omega_1}\right).
\sup_{\pi \in \Gamma(\mathcal{G}, \varphi )}\int_{\mathcal{T}\times \mathcal{X}}\varphi(t,x) d\pi(t,x) = \sup_{\mu \in \mathcal{G}}\left\{\int_{\mathcal{T}}\left[ \sup_{x \in \mathcal{X}}\varphi(t,x) \right] d \mu(t) \right\}.
\phi_b(b; b-ie) =\begin{cases}(c_{w - i}; b-ie), \max\{0,w -r\}\leq i < w, (b-1; b-1-ie), \;\, 0 \leq i < w-r;\end{cases}\phi_b(c'_j; b-1 + (j-\tilde{w})e) =\begin{cases}(b-1; b-1 + (j-\tilde{w})e), 1\leq j \leq \min\{r-l,\tilde{w}\}, (c_{j+l}; b+(j-\tilde{w})e), \tilde{w}< j \leq r-l;\end{cases}
\Biggl[- \frac{1}{2}\frac{d^2}{dx_{\parallel}^2}-\frac{1}{2}\Biggl( \omega + q E_0 x_{\parallel}\Biggr)^2 \Biggr]\phi_{\omega,{\bf k}_{\perp}}(x_{\parallel}) = - \frac{1}{2}(m^2+{\bf k}_{\perp}^2) \phi_{\omega,{\bf k}_{\perp}}(x_{\parallel}).
\phi_{WW}(f_i)=\{\pi_{W_i}S_{W}^{-1}f_i\}_{i\in I}=\{\pi_{W_i}S_{W}^{-1}\pi_{W_i}f_i\}_{i\in I}=\{\frac{1}{\omega_i^2}f_i\}_{i\in I}.
\sum_{k=1}^{N'}\P\left( \mathfrak{E}_{B,N,N',y},\, d W_k\leq -(k^{\frac{9}{10}}+B^{9/5})/10 \right) + \sum_{k= N' + 1}^{N}\P\left( \mathfrak{E}_{B,N,N',y},\, d W_k +y\leq -((N-k)^{\frac{9}{10}}-B^{9/5})/10 \right) \leq \frac{\epsilon}{N^{3/2}}.
\hat{\mathrm{P}}_{\mathrm{R}}^{s}= \frac{1}{2}\hbar \int d x ({\psi}_{\mathrm{R}}^{\dagger s}( - i \partial_{x}){\psi}_{\mathrm{R}}^{s}-{\psi}_{\mathrm{R}}^{s}( i \partial_{x}){\psi}_{\mathrm{R}}^{\dagger s}) - \int d x \hat{\mathrm{E}}{\partial_{1}}A_{1}.
\begin{aligned}d\sum\limits_{i=0}^{q-2}\binom{q}{i}(2x+1)^{q-i}B_{i}=d(B_{0}.(2x+1)^{q}+qB_{1}.(2x+1)^{q-1}+\sum\limits_{i=1}^{\frac{q-2}{2}}\binom{q}{2i}(2x+1)^{q-2i}B_{2i}.\end{aligned}
M^j_k= \left[ \begin{array}{ccc}p_j(1,1)f_j(Y_k|X_{k}=1,Y_{k-1}) \cdots p_j(d,1)f_j(Y_k|X_{k}=1,Y_{k-1}) \vdots \ddots \vdots p_j(1,d)f_j(Y_k|X_{k}=d,Y_{k-1}) \cdots p_j(d,d)f_j(Y_k|X_{k}=d,Y_{k-1}) \end{array}\right]