harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f(t)=\sum_{k=1}^{+\infty}\bigg(1-\frac{1}{2k}\bigg)(\chi_{I_{2k}}(t)+\chi_{I_{2k}}(-t))+\sum_{k=1}^{+\infty}\frac{1}{2k+1}(\chi_{I_{2k+1}}(t)+\chi_{I_{2k+1}}(-t))+\chi_{(1/2,a]}(t),
\Big\| \sum_{i=1}^M \sum_{j=1}^N \Theta_{x_i \otimes y_j,x_i \otimes y_j}T - T \Big\| &\le \Big\| \sum_{i=1}^M \sum_{j=1}^N \Theta_{x_i \otimes y_j,x_i \otimes y_j}\Big\| \|T-S\|+ \Big\| \sum_{i=1}^M \sum_{j=1}^N \Theta_{x_i \otimes y_j,x_i \otimes y_j}S - S\Big\| + \|T - S\|<\varepsilon.
\begin{array}{c}A_{2\rightarrow 2+n}=s\frac{(s_1/m^2)^{\alpha(q_1^2)}}{(q_1^2-m^2)}\cdots \frac{(s_{n+1}/m^2)^{\alpha(q_{n+1}^2)}}{(q_{n+1}^2-m^2)}\Gamma_{AD_0}^{i_1}\gamma_{i_1i_2}^{D_1}\cdots\gamma_{i_ni_{n+1}}^{D_n}\Gamma_{BD_{n+1}}^{i_{n+1}}\end{array}
u_{\ell}\tilde r_{k+1}+ u_{\ell+1}= u_{\ell}( \tilde r_{k+1}+ 1) + u_{\ell-1}\ge u_\ell \sqrt{\frac{q_{k+1}}{q_k}}+ u_{\ell-1}\sqrt{\frac{q_{k}}{q_{k+1}}}&= \frac{u_\ell q_{k+1}+ u_{\ell-1}q_{k}}{\sqrt{q_kq_{k+1}}}= \frac{q_{k+\ell}}{\sqrt{q_kq_{k+1}}}.
\lambda_{\mathcal{H}}(\rho^{\infty})=\min_{\Phi\in \mathbb{R}^n}\{\frac{1}{\Delta x^4}\sum_{(i,j)\in E}\sum_{(k,l)\in E}h_{ij, kl}(\Phi_i-\Phi_j)_+(\Phi_k-\Phi_l)_+~:~ & \sum_{(i,j)\in E}(\frac{\Phi_i-\Phi_j}{\Delta x})^2_+\rho_i=1\},
\mathbf{E}[Z^*_{\nu_m};\nu_m=j,Z^*_{\nu_m}>3B_m] \le \mathbf{E}\left[\frac{3}{2}B_m+X_{j};T_g>\nu_m=j, X_{j}>\frac{3}{2}B_m\right] &\le \mathbf{E}[2 X_{j};T_g>\nu_m=j, X_{j}>B_m] &\le 2\mathbf{E}[ X_{j};T_g>j-1, X_{j}>B_m] &=2\mathbf{E}[ X_{j}; X_{j}>B_m]\mathbf{P}(T_g>j-1) &\le2\mathbf{E}[ X_{j}^2/B_m; X_{j}>B_m]\mathbf{P}(T_g>j-1).
|B_\delta|(1+\delta)\leq\int_{B_\delta}\frac{|\sigma|_r}{\alpha_0}dxdt\leq \liminf_{N\rightarrow\infty}\int_{B_\delta}\frac{|\sigma^N|_r}{\alpha_0}dxdt\leq\liminf_{N\rightarrow\infty}\Big(\frac{C}{\alpha_0}\Big)^{\frac{1}{N}}|B_\delta|^{\frac{N-1}{N}}=|B_\delta|.
||A||_{\rho-\delta/2}\le CE^4\delta^{-2\tau}||\tilde{e}||_{\rho}=C\delta^{-2\tau}||\tilde{e}||_{\rho}, ||B||_{\rho-\delta/2}\le CE^4\delta^{-2\tau}||\tilde{v}^0||_{\rho}=C\delta^{-2\tau}||\tilde{v}^0||_{\rho}, |G| \le CE^4||e||_{\rho}\le C\epsilon, |D| \le CE^4||v^0||_{\rho}\le C.
l_{X}(b,k):=\min\{l\in\mathbb{N}: (F\in\widehat{\mathcal{Q}}_{b},\,\mu_{X,b}(F)>l)\Rightarrow\nu_{X,b}(F)>k\}, l_{\varphi^{*}}(b,k):=\min\{l\in\mathbb{N}:[F\in\widehat{\mathcal{Q}}_{b},\forall_{a\in\varphi^{-1}(b)}\nu_{M,a}(\widehat{\varphi}^{*}_{a}(F))>l]\Rightarrow \nu_{X,b}(F)>k\}.
\P \left[ |X_{t}^{i}-v_{i}t| \geq \frac{\epsilon}{d}u, t \in [0,u]\right] & = \P\left[ |Y_{t}-\bar{v}_{i}t| \geq \frac{\epsilon}{d}u, t \in [0,(p(e_{i})+p(-e_{i}))u]\right].
\theta_{q}= i \hbar \_{q}< \zeta | d | \zeta >_{q}= \frac{i \hbar}{2}\ (_{q}< \zeta | T_{i}^{+}| \zeta >_{q}d \zeta - \_{q}< \zeta | T_{i}^{-}| \zeta >_{q}d \bar{\zeta}) .
|(V_\Lambda - V)(x)| = C_d\left|\int_{|k|>\Lambda}e^{ikx}\frac{1}{|k|^{d-1}}\hat{\rho}(k)\, dk \right| \leq C_d\left(\int_{|k|>\Lambda}\frac{1}{|k|^{2d}}dk\right)^{1/2}\left(\int |k|^2 |\hat\rho(k)|^2\, dk \right)^{1/2}\to 0(\Lambda\to\infty).
{\mathcal L}_V(D_V)\otimes\Omega^{s'}_V\cong\bigoplus_{s_1+s_2=s'}({\mathcal L}_{Y^1}(D_{Y^1})\otimes\Omega^{s_1}_{Y^1})\otimes_k({\mathcal L}_{Y^2}(D_{Y^2})\otimes\Omega^{s_2}_{Y^2}).
\left|\phi(\Lambda_{m,n}\eta,\eta)\right| =\Lambda_{m,n}\left|\sqrt{\eta^2+\frac{2p+2}{\Lambda_{m,n}}}-\sqrt{\eta^2+2m+2}\right| &\geq \frac{|a|}{\Lambda_{m,n}\sqrt{\eta^2+2m+2}}&\geq\frac{1}{\Lambda_{m,n}\sqrt{\eta^2+2m+2}}.
\hat M=\underset{A\in \mathbb R^{n_1\times n_2}}{\mathrm{argmin}}\left \{\left\Vert Y-A\right\Vert^{2}_{2}+\lambda\Vert A\Vert_0\log\left (\dfrac{e\,n_1\,n_2}{\Vert A\Vert_0\vee 1}\right )\right \}
F\left( \lambda \right) =\overset{\infty}{\underset{n=0}{\sum}}F\left(\lambda_{n}\right) \frac{h\left( \lambda \right) \widetilde{\omega}\left(\lambda \right)}{\left( \lambda -\lambda_{n}\right) h\left( \lambda_{n}\right) \widetilde{\omega}^{\prime}\left( \lambda_{n}\right)}.
\begin{aligned}T_k g_k^\pm = (\mu_{2k-1}+ d_k \pm d_k \tau_k) g_k^\pm, \tau_k = \sqrt{1-t_k^2}, g_k^\pm = \begin{pmatrix}1 G_k^\pm\end{pmatrix},G_k= \left(\frac{1+\tau_k}{1-\tau_k}\right)^\frac 12.\end{aligned}
= \frac{1}{4}{\eta}_{\mu \nu}{\eta}_{\rho \sigma}\sum_{n = 1}^{m - 1}\sum_{s = 1}^{m - 1}\langle 0 \mid \left({\alpha}_{m - n}^{\mu}\left[{\alpha}_{n}^{\nu},{\alpha}_{s - m}^{\rho}\right]{\alpha}_{- s}^{\sigma}+ \right.
\begin{aligned}\delta_1 = s \delta_2 = t \delta_3 = st + 1 \delta_4 = r^3 - 3rt + t^2 + t \delta_5 = r^3s - 3rst + st^2 + st + t \delta_6 = r^3s^2 - 3rs^2t - 3rs + s^2t^2 + s^2t + 2st + s + 1 \delta_7 = r^3s^2t + r^3s - 3rs^2t^2 - 3rst + s^2t^3 + s^2t^2 + 2st^2 + t. \end{aligned}
\begin{cases}-\nabla\cdot (\sigma\nabla u)=f=f^+-f^- \mbox{in}\Omega \sigma\nabla u\cdot n=0 \mbox{on}\partial\Omega |\nabla u|\leq 1 \mbox{in}\Omega, |\nabla u|= 1 \sigma-\mbox{a.e.}\end{cases}
\phi_{i j}= \frac{1}{\sqrt{4 \pi}}\sum_{n = 1}^{\infty}\frac{1}{\sqrt{n}}\bigg [ A_{i j}( n ) \mathrm{e}^{- i \pi n x^{-}/ L}+ A_{j i}^{\dag}( n ) \mathrm{e}^{i \pi n x^{-}/ L}\bigg ] ,
C^{(pq)}:= \mathbb{E}_0\left[ W^{(pl)}_hW^{(ql)}_h \middle|\ \mathcal{F}_{n, l-1}\right] C^{(pq)}:= \mathbb{E}_0\left[ U^{(pl)}_hU^{(ql)}_h \middle|\ \mathcal{F}_{n, l-1}\right],
H^{2}= \left( \frac{1}{S}\frac{d S}{d \tau}\right)^{2}= R^{' 2}\left[ \frac{d A}{d R}+ \frac{1}{4}\frac{d \phi}{d R}\right]^{2}\equiv \frac{8 \pi}{3}\rho_{\mathrm{e f f}}
\delta{I \! \! N}= ({I \! \! L}{I \! \! N}\dot{)}+ \frac{i}{2}D_{\bar{\eta}}{I \! \! L}D_{\eta}{I \! \! N}+ \frac{i}{2}D_{\eta}{I \! \! L}D_{\bar{\eta}}{I \! \! N}.
\Omega = 2 i J \left[ \frac{d{\bar{\xi}}\wedge d \xi}{1 + \mid \xi \mid^{2}}- \frac{( \xi d{\bar{\xi}}) \wedge ({\bar{\xi}}d \xi )}{( 1 + \mid \xi \mid^{2})^{2}}\right] , \ \ \ \mid \xi \mid^{2}= \sum_{p}\mid \xi_{p}\mid^{2}.
(\Delta - D_t)(uP)\ =\ m_n (m_{n}+1)x^{m-\overline{n}}t^\ell + \sum_{i \neq n}m_i (m_i - 1) x^{m - 2 \overline{i}+ \overline{n}}t^\ell - \ell x^{m+ \overline{n}}t^{\ell-1}+\ \big\{2 <DP_1,DP> + P_1(\Delta P -P_t)\big\}+ \big\{2 <Dz,DP> + z (\Delta P -P_t)\big\}.
\aligned\int_{\mathbb{R}^N}(I_\mu*|{u_\nu}|^{p})|{v_\nu}|^{q}= (1+o_\nu(1))[(\gamma_p+\gamma_q)\nu D^{\frac{2}{\gamma_p+\gamma_q}}_0]^{\frac{\gamma_p+\gamma_q}{2-\gamma_p-\gamma_q}}= (1+o_\nu(1))\nu^{\frac{\gamma_p+\gamma_q}{2-\gamma_p-\gamma_q}}\int_{\mathbb{R}^N}(I_\mu*|{u_0}|^{p})|{v_0}|^{q}.\endaligned
\tilde F_2(t)= \tfrac{1}{\nu}\left\|\tilde f(t)\right\|_2^2+c(1+\alpha)\left\|\tilde y\right\|_{L^\infty(I;H^3)}\left\|\psi_k\right\|^2_{L^\infty(I;L^2)}, &\tilde G_2=\tfrac{c(1+\alpha)}{\alpha}\left\|\tilde y\right\|_{L^\infty(I;H^3)}.
A(\Lambda_1 , K_1 ; \ldots ;\Lambda_N , K_N )=g^{(N-2)}\int\prod_{i=1}^N d^2 z_i\biggl\langle\prod_{j=1}^N e^{i/2\K_j\cdot X}W_\Lambda (z_j,{\bar z}_j)e^{i/2\K_j\cdot \bar X}\biggl\rangle \ .
\overline{\zeta}_q(s)= 2\xi_q(2s-1)+(n+1-q)(2s-1)\xi_q(2s)+\sum_{j=2}^{2n}(n+1-q)^j\frac{\Gamma(s+j-1)(2s-2+j)}{\Gamma(j+1)}\Bigl(\frac{\gamma_{n,q}(j-1)}{\Gamma(s+1)}&+s o(1)\Bigr)+\frac{1}{\Gamma(s)}\sum_{j=2n+1}^\infty (n+1-q)^j \frac{\Gamma(s+j-1)(2s-2+j)}{\Gamma(j+1)}\xi_q(2s+j-1).
\tilde{G}= \begin{bmatrix}-(Z-I) A^{-1}(Z-I)^{-1}G (Z-I) A^{-1}(Z-I)^{-1}e_1 e_1 \end{bmatrix}, \tilde{B}= \begin{bmatrix}(Z-I) A^{-*}(Z-I) B e_1 (Z-I) A^{-*}(Z-I) e_1 \end{bmatrix}.
\begin{aligned}\left(1+\frac{x}{i}+\frac{x^2}{4\pi^2}\right) \left(1+\frac{x}{i}+\frac{x^2}{16\pi^2}\right) &\left(1+\frac{x}{i}+\frac{x^2}{36\pi^2}\right)\left(1+\frac{x}{i}+\frac{x^2}{64\pi^2}\right)\cdots,\end{aligned}
\eta_{i,r}=\frac{\left(r-\frac{1}{2}\right)!\left(r-\frac{1}{2}\right)!}{(i-\frac{1}{2})!(r-i-\frac{1}{2})!r!}=\frac{(2r-1)!!(2r-1)!!}{2^{r}r!(2i-1)!!(2(r-i)-1)!!}.
{\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}.
S(\mathcal{A}'_d,X^\delta)=\Bigl(1-O\Bigl(\exp\Bigl(\frac{-\epsilon}{2\delta}\Bigr)\Bigr)\Bigr)\frac{\kappa\#\mathcal{A}{}}{d}\prod_{\substack{p\le X^\delta p\nmid 10}}\Bigl(1-\frac{1}{p}\Bigr)+O\Bigl( \sum_{\substack{e<X^{\epsilon/2}(e,10)=1 p|e\Rightarrow p\le X^\delta}}R_d(e)\Bigr).
E(w) :=E(\tilde{q}(w),t-t(k,x);f):=\det (DT_f^{t-t(k,x)}(\tilde{q}(w)))\ge 1\intertext{and}S(w) :=-F(\tilde{q}(w),t-t(k,x);f)/E(\tilde{q}(w),t-t(k,x);f)
-\int_0^{T}\int_\Omega \varphi \nu_t - \int_0^{T}\int_{\partial\Omega}d\ \nu\frac{\partial \varphi}{\partial\eta}+\int_0^{T}\int_\Omega d\ \nabla\nu.\nabla\varphi-\int_0^{T}\int_\Omega\theta\nu\ &= \int_{\Omega}\nu(x,0)\varphi(x,0)
\begin{array}{r c l}\mathrm{tr}_{(N)}\left(J_k\Delta(x)\right) = \frac{1}{N}e^{2\pi i k_ix^i/L}, \frac{1}{N}\sum_x\Delta(x)_{\mu\nu}\Delta(x)_{\lambda\rho}= \delta_{\mu\rho}\delta_{\nu\lambda}, \frac{1}{N}\mathrm{tr}_{(N)}\left(\Delta(x)\Delta(y)\right) = N^2\delta_{x,y(\mathrm{mod}\L)}.\end{array}
\sum_{j=1}^m\left|{\textbf E}(f_j\cdot \phi_l)\right|\le \sqrt m\left(\sum_{j=1}^m\left|{\textbf E}(f_j\cdot \phi_l)\right|^2\right)^{1/2}\le \sqrt{\frac{m\cdot \sum_{j=1}^m\|f_j\|_2^2}{n}}.
(-)^{a}\sum_{j=0}^{a}(-1)^{j}\left(\begin{array}{c}j a\end{array}\right)\mbox{exp}\left[2\pi i\frac{Q}{P}(h_{e_{1}+Pe,n_{1}+Pn-\frac{P}{a}j}-h_{e_{1},n_{1}})\right]|e_{1}+Pe,n_{1}+Pn-\frac{P}{a}j>
f_{n}=|\bigcup_{i<j\in\mathbb{N}_{n}}A_{ij}|=\sum_{k=1}^{\lfloor n/2\rfloor}[(-1)^{k+1}\sum_{\{(i_{1},j_{1}),...,(i_{k},j_{k})\}\in S_{k}}|\bigcap_{l=1}^{k}A_{i_{l}j_{l}}|]
{}-\sum_{j=1}^{q}\int_{0}^{t}K_{ij}(t,s)\widehat{z}_{j}(s)ds\leq-\sum_{j=1}^{q}\sum_{v=1}^{u-1}\left (t_{v}-t_{v-1}\right )K_{ij}^{(u,v)}z_{j}^{(v)}.
I'_s = \{i' = s-1-i \,:\, i \in \bar{I}_s \}, \bar{I}'_s = \{i' = s -1-i \,:\, i \in I_s \}, J'_s = \{j' = r-s-1-j \,:\, j \in \bar{J}_s \}(s \in \{p, \, q\}).
Q_aQ_bQ_a - Q_{Q_a b}= 4(Q_aQ_{ab}+Q_{ab}Q_a) + 2(Q_aQ_{a^2,b^2}+ Q_{a^2,b^2}Q_a) + Q_{a,b}Q_aQ_{a,b}&-( 2Q_{a^2b}+ 8 Q_{a(ab)}+ 2Q_{a^3,b^2a}+ 4Q_{a^3,b(ab)}).
\int_{\mathbb{R}^N}\frac{u^\theta}{|x|^\alpha}\, dx=\int_{\mathbb{R}^N}\frac{\displaystyle\int_0^\infty \chi_{\{t\,:\,u(x)^\theta>t\}}(s)\, ds}{|x|^\alpha}\, dx=\int_0^\infty \int_{\mathbb{R}^N}\frac{\chi_{\{t\,:\,u(x)>t^{1/\theta}\}}(s)}{|x|^\alpha}\, dx\, ds,
\delta_\alpha j_{\mu\beta}=\sigma_{\mu\nu\alpha\beta}\partial^\nu\left(\frac{8}{3}{\kern .35em\bar{\kern -.35em{\widetilde J}}}+\frac{1}{2}D^2(a_1 (X+{\bar X}) + a_2 (X-{\bar X}))\right)\displaystyle{\biggl\vert}\ .
K_{j}^{a}\partial_{a}( - \gamma^{c d}\partial_{c}X^{\mu}\partial_{d}X^{\nu}g_{\mu \nu}+ \frac{\varepsilon^{a_{1}a_{2}. . . a_{d + 1}}}{\sqrt{- \gamma}}\partial_{[ a_{1}}A_{a_{2}. . . a_{d + 1}]}) = 0
\varSigma^{(k,l)}:= \sum_{m=1}^M\beta_m^{k,l}\bar{g}_{k,l}^m - \sum_{r,s=1}^M \alpha_{r,s}^{k,l}\bar{g}_k^r \bigl(\bar{g}_l^s\bigr)^T \in \mathbb{R}^{d_k \times d_l}; \varSigma := \bigl(\varSigma^{(k,l)}\bigr)_{k,l=1,\dots,K}\in\mathbb{R}^{d \times d}.
\int \! \! \int d^{d}k_{1}~ d^{d}k_{2}~ \left( \frac{\partial}{\partial k_{1 \mu}}+ \frac{\partial}{\partial k_{2 \mu}}\right) ~ \left[ k_{1 \mu}~ P_{k_{1}, m_{1}}^{\nu_{1}}P_{k_{1}- k_{2}, m_{2}}^{\nu_{2}}P_{k_{2}, m_{3}}^{\nu_{3}}\right] \equiv 0 ,
\chi(A,B)=\sum\limits^{K_{1}}_{k=1}\frac{(-1)^{N+i_{k}}}{(i_{k}-M)!(N-i_{k})!}\sum\limits^{N-M}_{l=0}\sum\limits^{l}_{m=0}(-1)^{m}\sigma_{m}(M,N)i^{l-m}_{k}T_{N-M-l}
u_1 =\mu^{-1}p^{r-t}v_2^{-1}(p^{-a-b}x'y'-p^{-d}u'); u_2 =\lambda^{-1}p^{t-r}(v_1v_3)^{-1}(p^{s-c-d}u'z'-w'); u_3 = -v_3v'(v_4w')^{-1}p^{r-s-f}; u_4 =-\mu^{-1}v_3^{-1}x'p^{s-r-a}; u_5 =\lambda^{-1}(v_1v_4)^{-1}p^{t-s}(p^{-f}v'-p^{-b-c}y'z'); u_6 = \mu^{-1}v_4^{-1}p^{-s}.
\delta S^2 = E\int_0^T \int \varphi (t, g_t (x)) (\det \nabla g_t (x) -1)dtdx & + E\int_0^T \int (\nabla p (t,g_t (x)) .h(t,g_t (x)) (\det \nabla g_t (x) -1)dtdx &+ E\int_0^T \int p(t,g_t (x)) \left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}\det \nabla (g_t (x)+\epsilon h(t,g_t (x)) dt dx
a \left( x\right) \in \left[ 1,d_1 \right],~~ d_1 = const.>1,~ a(x) =1x\in \Omega_{FDM}, c\left( x\right) \in \left[ 1,d_2 \right],~~ d_2 = const.>1,~ c(x) =1x\in \Omega_{FDM}, a(x), c(x) \in C^{2}\left( \mathbb{R}^{3}\right) .
\widetilde{IV}_{A,ii}=\varepsilon\int_{0}^{t}\int_{\mathbb{T}^{d}}(\partial^{\alpha}\mu^{n+1})\sum_{i=1}^{d}\sum_{j=1}^{d}\left(\partial_{x_{i}}\left((\mu^{n}+\bar{m})\Theta_{p_{i}p_{j}}\right)\right)(\partial^{\alpha}\partial_{x_{j}}w^{n})\dxd\tau.
B =\bigcap_{j=0}^{q}\Bigl(y^{*}\bigl(\eta_{1},T_{j}(\eta_{2})\bigr)\le a_{j}\Bigr)\cap \Bigl(T_{q}(\eta_{2})<t\le T_{q+1}(\eta_{2})\Bigr) =\bigcap_{j=0}^{q}\Bigl(T_{j}(\eta_{2})\le y(\eta_{1},a_{j})\Bigr)\cap \Bigl(T_{q}(\eta_{2})<t\le T_{q+1}(\eta_{2})\Bigr)
e^{i \hat{\Gamma}[\vec{\varphi}_1, \vec{\varphi}_2]}= \langle T_C \mbox{exp}\left[ \frac{i}{\delta}\int d^{\, 4}x \sum_{j = 1}^2 \left\{(-)^{j - 1}{\cal L}' (\vec{\phi}_j (x) + \vec{\varphi}_j (x)) - \frac{\delta \hat{\Gamma}[\vec{\varphi}_1, \vec{\varphi}_2]}{\delta \vec{\varphi}_j (x)}\cdot \vec{\phi}_j (x) \right\}\right] \rangle ,
\sim\frac{3^{1/3}}{\Gamma(2/3)}\frac1{n^{1/3}}{\displaystyle\prod\limits_{1\leq m\equiv0\operatorname*{mod}3}}\frac{e^{1/m}+e^{\omega/m}+e^{\omega^{2}/m}}3 & =\frac{3^{5/6}\Gamma(1/3)}{2\pi\,n^{1/3}}{\displaystyle\prod\limits_{k=1}^{\infty}}\Psi\left( \frac1{3k}\right) =(1.0729979443...)n^{-1/3}.
C_{\alpha\beta}=[\begin{matrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{22}&C_{23}&0&0&0\\C_{13}&C_{23}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{55}&0\\ 0&0&0&0&0&C_{66}\end{matrix}].
\int_{(-\infty,-n^m)}\frac{ns^{2}\,d\sigma_{\mu}^{-}(s)}{(\widetilde{x}_n-s)^{2}}=n\sum_{k=n+1}^\infty\left(\frac{k^m}{\widetilde{x}_n+k^m}\right)^2\sigma_\mu^-(\{-k^m\}) &\leq16n^3\sigma_\mu^-((-\infty,-n^m))=\frac{16\widetilde{\ell}(n^m)}{n^{am-3}}
\pi(\mathsf{c}_s) = \frac{v^{2}\tilde{A}_{s}^{-2}- 1}{v - v^{3}}= \frac{v^{2}((v^{-1}-v)\tilde{A}_s + (v^{-2}-1+v^2)) - 1}{v - v^{3}}= \tilde{A}_s - v = c_s.
\frac{b_{n+1}-b_{n}}{v\left( v^{2}-1\right) \left( v^{2}-1/4\right)}=\left(2v-3\right) \left( v-\frac{1}{2}\right)^{2n-2}+\left( 2v+3\right) \left( v+\frac{1}{2}\right)^{2n-2}+4v^{2n-1}>0
X^{o}=(\begin{matrix}x_{11},x_{12},\cdot\cdot\cdot,x_{1N_{1}}\\x_{21},x_{22},\cdot\cdot\cdot,x_{2N_{2}}\\ \vdots\\x_{T1},x_{T2},\cdot\cdot\cdot,x_{TN_{T}}\end{matrix})
\Gamma_1(X) =X\sum\limits_{1\leq d\leq z}\frac{\mu(d)\lambda(d^2)}{d^2}+\mathcal{O}\big(zX^\varepsilon \big) &=\sigma X-X \sum\limits_{d>z}\frac{\mu(d)\lambda(d^2)}{d^2}+\mathcal{O}\big(zX^\varepsilon\big)\,,
\left[ x , p \right] = i , \quad \left[ a_{0}, a_{0}^{\dagger}\right] = 1 , \quad \left[ a_{n}^{I},{a_{m}^{I}}^{\dagger}\right] = \delta_{n m}, \quad \left[ a_{n},{a_{m}}^{\dagger}\right] = \delta_{n m}, \quad \left[ \tilde{a}_{n},{\tilde{a}_{m}}^{\dagger}\right] = \delta_{n m}
|\partial_{t}\langle \left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}})-1}{r^{2}}\right) v_{5}\left(1-\chi_{\geq 1}(\frac{4r}{t})\right)\vert_{r = R\lambda(t)},\phi_{0}\rangle| \leq \frac{C}{t^{9/2}\log^{2b-2+\frac{5N}{2}}(t)}+ \frac{C}{t^{7/2}\log^{b-3+\frac{5N}{2}}(t)}
x^{+}\simeq x^{+}+ \frac{2}{1 + \sqrt{1 + 4 R_{s}^{2}/ R^{2}}}\frac{R_{s}^{2}}{R}\ \ , \ \x^{-}\simeq x^{-}+ \frac{1 + \sqrt{1 + 4 R_{s}^{2}/ R^{2}}}{2}R
\varphi_{k , l}( r ) = j_{l}( k R ) \Theta ( R - r ) + \frac i 2 \left( f_{l}( k ) \hat{h}_{l}^{-}( k R ) - f_{l}^{\star}( k ) \hat{h}_{l}^{+}( k R ) \right) \Theta ( r - R )
\lim_{| y | \rightarrow \infty}\lim_{\kappa^{2}\rightarrow 0}\psi_{2}^{( \perp )}( y ; \kappa^{2}) = 1 \quad \Longrightarrow \quad \psi_{2}^{( \perp )}( \cdot ; \kappa^{2}= 0 ) \not \in L^{2}( I \! \! R ) .
\hat L_r(E'-E)-L_r(E'-E)= &\frac{1}{2\pi i}\oint_{\gamma_r}(R_{\hat \Sigma}(\eta)-R_{\Sigma}(\eta))(E'-E)R_{\hat \Sigma}(\eta)d\eta+\frac{1}{2\pi i}\oint_{\gamma_r}R_{\Sigma}(\eta)(E'-E)(R_{\hat \Sigma}(\eta)-R_{\Sigma}(\eta))d\eta,
W(f_{\ell+1}, z_4) = \mu( W[f_1 , \dots, f_{\ell +1}] ) = \mu(W^\tau) , X(f_{\ell+1}, z_4) = \mu( X[f_1 , \dots, f_{\ell +1}] ) = \mu(X^\tau) , Y(f_{\ell+1}, z_4) = \mu( Y[f_1 , \dots, f_{\ell +1}] ) = \mu(Y^\tau) .
{\xi_{k,{l},{j}}}={\rho_k}{\alpha_{{k,{\rm{R}}},{l}}}{\alpha_{{\rm{T}},k,{j}}}{\bf{a}}_{{\rm{S,}}k}^H\left({\Phi_{{k},{l}}^{\rm{D}}}\right){{\bf{a}}_{{\rm{S}},k}}\left({\Theta_{k,{j}}^{\rm{A}}}\right).
\sum_{j=0}^{n-1}\mu \prod_{i=1}^{j}P_i(\epsilon)r = \mu \sum_{j=0}^{n-1}\tilde{P}^j r + \epsilon\mu \sum_{i=1}^{n-2}i\tilde{P}^i \tilde{P}^{(1)}\sum_{j=i+1}^{n-1}\tilde{P}^{j-i-1}r + O(\epsilon^2n^4)
\delta H_{B}^{( m )}= \frac{m c_{m}\kappa}{2^{m - 1}}\; \oint_{B}d^{D - 2}\! x \left( \sqrt{\tilde{h}}\tilde{\delta}_{d_{1}\cdots c_{m}d_{m}}^{b_{1}\cdots a_{m}b_{m}}\; \delta \tilde{h}_{b_{1}}{}^{d_{1}}\cdots \tilde{R}_{a_{m}b_{m}}{}^{c_{m}d_{m}}\right) \ .
P_1(x) = 4.86x + 0.29x^2 - 0.96x^3 + 0.974x^4 - 0.17x^5, P_2(x) = -3.11x - 0.3x^2 + 0.87x^3 - 0.18x^4 - 0.53x^5, P_3(x) = 4.86x + 0.06x^2.
\sigma_{j}(\varepsilon_k) :=\varepsilon_k-(\varepsilon_k|\alpha_j)\alpha_j = \varepsilon_k-(\delta_{j,k}-\delta_{j+1,k})(\varepsilon_j-\varepsilon_{j+1}) = \begin{cases}\varepsilon_{j+1}j=k, \varepsilon_{j}j=k-1, \varepsilon_k .\end{cases}
\textstyle J(\exp Y)\, =\,c \prod\limits_{\alpha\in\Sigma^{+}}\, (\sinh\langle{\alpha,H}\rangle)^{m_{\alpha}}\, \asymp\, \prod\limits_{\alpha\in\Sigma^{+}}\Big\lbrace \frac{\langle\alpha, H\rangle}{1+\langle\alpha, H\rangle}\Big\rbrace^{m_{\alpha}}\, e^{2\langle\rho, H\rangle},
\chi(r)= \frac{\sqrt{2}}{2}\,\left(\begin{array}{c}1 e^{i\phi}\end{array}\right)\,\delta_{r(+)}\,+\, \frac{\sqrt{2}}{2}\,\left(\begin{array}{c}-e^{-i\phi}1 \end{array}\right)\,\delta_{r(-)},
E_{2}^{p , q}\: = \: H^{p}\left( T ,{\cal E}^{\vee}|_{T}\otimes{\cal F}\otimes \Lambda^{q}N_{S / X}|_{T}\right) \: \Longrightarrow \: \mathrm{E x t}_{X}^{p + q}\left( i_{*}{\cal E}, j_{*}{\cal F}\right) .
\prod_{j\geq 1}{E_j + N_j - 1\choose N_{j}}= \prod_{j\geq 1}{\sum_{i=0}^{m-1}N_{j-i}-1\choose N_{j}}= \prod_{j\geq 1}{s_{j}- s_{j+1}+ \chi (m-1 | j) \choose s_{j+1}- s_{j+m}}.
r=\begin{pmatrix}1 x_1 x_2 x_3 x_4 0 1 \alpha_1(X) \alpha_2(X) \alpha_3(X) 0 0 1 0 0 0 0 0 1 x_1 0 0 0 0 1 \end{pmatrix}, X=X(x_1,x_2,x_3,x_4).
\varphi(x) = \left| \frac{uJJ_1}{4s^2}\right|^{\frac{1}{2}}\varphi_1'(x_1') \varphi_3'(x_3') \int_F \int_F \varphi_2'(x_2') \varphi_4'(x_4') \psi(x_2'y_2' + x_4' y_4') \, dx_2' \, dx_4' = \left| \frac{uJJ_1}{4s^2}\right|^{\frac{1}{2}}\varphi_1'(x_1') \hat{\varphi}_2'(y_2') \varphi_3'(x_3') \hat{\varphi}_4'(y_4').
(\delta Z)_{s,t}=S_{t-s}Z_{s}+\int_{s}^{t}S_{t-\tau}F(Z_{\tau})\mathrm{d}\tau+\int_{s}^{t}S_{t-\tau}G(Z_{\tau})\circ\mathrm{d}\mathbf{X}_{\tau}, \ \Z_{0}=z_0, \s,t\in \mathbb{R},
\gamma^{0}= \left( \begin{array}{c c}{{\sigma_{3}}}&{{0}}\\{{0}}&{{- \sigma_{3}}}\\ \end{array}\right) , \ \gamma^{1}= \left( \begin{array}{c c}{{i \sigma_{1}}}&{{0}}\\{{0}}&{{- i \sigma_{1}}}\\ \end{array}\right) , \ \gamma^{2}= \left( \begin{array}{c c}{{i \sigma_{2}}}&{{0}}\\{{0}}&{{- i \sigma_{2}}}\\ \end{array}\right)
< \phi_{\eta}( x_{1}, \tau_{1}) \ldots \phi_{\eta}( x_{n}, \tau_{n}) >_{\eta}\equiv \int d \eta ( x , t ) e^{- \int \eta^{2}/ \Omega}\phi_{\eta}( x_{1}, \tau_{1}) \ldots \phi_{\eta}( x_{n}, \tau_{n}) .
\mu_{-}= \left( \begin{array}{c c}{I_{m}}&{0}\\{\mu_{- 2 1}}&{I_{m}}\\ \end{array}\right) , \qquad \mu_{+}= \left( \begin{array}{c c}{I_{m}}&{\mu_{+ 1 2}}\\{0}&{I_{m}}\\ \end{array}\right) ,
T^F(z)={2\sum_{i,j=1}^h\omega_i''A_{ij}\overline \omega_j-3\left(\sum_{i,j=1}^h\omega_i'A_{ij}\overline\omega_j\right)^2\over 2\left(\sum_{i,j=1}^h\omega_iA_{ij}\overline\omega_j\right)^2}.
D_{p p \Delta \Delta}|_{\log}= \frac{\pi^{\frac{d}{2}}}{2}( - 1 )^{\Delta - p - 1}\frac{\Gamma ( p + \Delta - 2 )}{\Gamma ( \Delta - p + 1 )}\frac{2}{( \Gamma ( p ) )^{2}}\frac{s^{\Delta}\ln s}{( x_{1 3}^{2})^{p}( x_{2 4}^{2})^{\Delta}}( \sqrt{\pi}\frac{\Gamma ( \Delta )}{\Gamma ( \Delta + \frac{1}{2})}) ,
\frac{5m_0 m_1 \log(d+1)}{n}+ \frac{2v v_f}{m_1}+ \frac{L_2v^2_fv^2_0}{m_0}+ \frac{L^2_2v^2_fv^4_0}{4m^2_0}+ \left(\frac{1}{n}\sum_{i=1}^n|Y_i|\right)\frac{v_fL_2 v^2_0}{m_0}+ \frac{T_n}{n}.
B=\begin{bmatrix}u_{11}J(T^{-i},T^{-r}) u_{12}J(T^{-i-2r},T^{-3r}) 0 0 u_{21}J(T^{-i},T^{-3r}) u_{22}J(T^{-i-2r},T^{-r}) 0 0 0 0 v_{11}J(T^{-i-3r},T^{-3r}) v_{12}J(T^{-i-3r},T^{-r}) 0 0 v_{21}J(T^{-i-r},T^{-r}) v_{22}J(T^{-i-r},T^{-3r}),\end{bmatrix}
\|s\mu_N(x)-\mu_N(sx)\| =\frac{1}{N}\left\|\sum_{j=1}^N\delta_{s(x_{[1,j]}0^\infty)}-\delta_{s(x)_{[1,j]}0^\infty}\right\| &\leq \frac{1}{N}\left\|\sum_{j=k(s)+1}^{N-k(s)}\delta_{s(x_{[1,j]}0^\infty)}-\sum_{l=k(s)+1+\alpha_i}^{N-k(s)+\alpha_i}\delta_{s(x)_{[1,l]}0^\infty}\right\| + \frac{4k(s)}{N}&=\frac{4k(s)}{N}.
Y(\lambda)\underset{\lambda\to\infty}=\left\{I+\sum_{k=1}^\infty\Phi_k(\infty)\left(\frac1{\lambda}\right)^k\right\}\begin{pmatrix}\lambda^{-\theta_\infty/2}0 0 \lambda^{\theta_\infty/2}\end{pmatrix},\quad\theta_\infty=-(s+q).
{\rm d}\mu_{\rm one-fl}= \frac{1}{2^{11}\pi^4 v^2}\, M_{\rm PV}^5 \left( \frac{8\pi^2}{g^2}\right)^2 \!\exp\left( -\frac{8\pi^2}{g^2}-4\pi^2|v|^2\rho^2_{\rm inv}\right) \frac{{\rm d}\rho^2}{\rho^2}\,{\rm d}^4 x_0\,{\rm d}^2\theta_0 \,{\rm d}^2\bar\beta_{\rm inv}\,{\rm d}^2\bar\theta_0 \, .
\mathrm{Int}_{E+(2L+1)\sqrt{k}}W_0(x,n) \subset \bigcup_{m\in \mathbb{G}_x}W(x,m), \bigcup_{m\in \mathbb{B}_x}W(x,m) \subset \bigcup_{n\in \mathbb{Z}^k}\partial_{E+ (2L+1)\sqrt{k}}W_0(x,n).
Q= \big(q^{-1}-q^3\big)KL[K,L]_q + q^2 ([K,L]_q )^2+B(KL+LK)+C_0q^2K^2+C_1q^{-2}L^2 \hphantom{Q=}{}+ D_0\big(1+q^2\big)K+D_1\big(1+q^{-2}\big)L.
\frac{r(q_k+q_{k-1}-2,{\mathbf x})}{q_k+q_{k-1}-2}\le \frac{|W_k| + q_k + 2q_{k-1}-2}{q_k+q_{k-1}-2}&= 1 + \frac{t_k +\eta_k}{1+\eta_k - \varepsilon_k}< 1 + \frac{t_k +\eta_k}{1+\eta_k}+ \varepsilon_k.
S(n,\alpha,\gamma,0) \ge \left(S(n,\alpha,0,0) - S(n,\alpha,\gamma,0)\right) \int_{\mathbb{R}^n}|(w_k - w) (x,0)|^{2_{\alpha}^*}dx+S(n,\alpha,\gamma,0) \int_{\mathbb{R}^n}|w_k (x,0)|^{2_{\alpha}^*}dx+o(1) =\left(S(n,\alpha,0,0) - S(n,\alpha,\gamma,0)\right) \int_{\mathbb{R}^n}|(w_k - w) (x,0)|^{2_{\alpha}^*}dx+S(n,\alpha,\gamma,0)+o(1) .
\frac{d}{dt}\|x_\nu^{(4)}(t)-x_\nu^{(N-1)}(t)\|^2 \leq -\alpha\nu\|x_\nu^{(4)}(t)-x_\nu^{(N-1)}(t)\|^2+\nu\|x_\nu^{(3)}(t)-x_\nu^{(N)}(t)\|^2 & +\nu\|x_\nu^{(5)}(t)-x_\nu^{(N-2)}(t)\|^2+\frac{1}{\nu}M_{T_1,T_2}^{4,N-1,2-\alpha}(\omega)
b_i m_j = \begin{cases}[2] m_i, \; j\equiv i \bmod d; z m_1, \; i-1\equiv 0\equiv j \bmod d; z^{-1}m_0, \; i\equiv 0\equiv j-1 \bmod d; m_j, \; i\equiv j\pm 1\bmod d,\;; 0, .\end{cases}
\langle \mathcal{L}_0,\dots,\mathcal{L}_g\rangle=\langle\mathcal{L}_0,\dots,\mathcal{L}_q,\langle \mathcal{L}_{q+1},\dots,\mathcal{L}_{g}\rangle(\mathcal{X}_g^{g+1}/\mathcal{X}_g^{q+2})\rangle (\mathcal{X}_g^{q+2}/\mathcal{X}_g).
|(\frac{1}{R\theta}\partial^{\alpha}\widetilde{E},\partial^{\alpha}(\bar{\rho}\bar{u}))| =|(\partial^{\alpha'}\widetilde{E},\partial_{x_i}[\frac{1}{R\theta}\partial^{\alpha}(\bar{\rho}\bar{u})])| &\leq C\|\partial^{\alpha'}\widetilde{E}\|\|\partial_{x_i}[\frac{1}{R\theta}\partial^{\alpha}(\bar{\rho}\bar{u})]\|\leq C\eta_{0}\varepsilon^{1-a}.
\int_0^1\xi^{\frac{\alpha}{2}}\frac{d^p}{d\xi^p}\left\{\xi(1-\xi)\right\}^{p-\frac{1}{2}}{\rm d}\xi = \left\{\xi^{\frac{\alpha}{2}}\frac{d^{p-1}}{d\xi^{p-1}}\left\{\xi(1-\xi)\right\}^{p-\frac{1}{2}}\right\}_0^1 -\frac{\alpha}{2}\int_0^1\xi^{\frac{\alpha}{2}-1}\frac{d^{p-1}}{d\xi^{p-1}}\left\{\xi(1-\xi)\right\}^{p-\frac{1}{2}}{\rm d}\xi
w_\ell := \left(\sum_{q \in \left[b\right]}p_{n}\left(q,\ell\right)\right) f_{\ell,1}:= \left(\sum_{j \in \left[b\right]}\frac{p_{n}\left(j,\ell\right)}{\sum_{q \in \left[b\right]}p_{n}\left(q,\ell\right)}h_{1,b,j}\right) f_{\ell,2}:= h_{1,b,\ell}f_{i,j}:= h_{1,b,1}, \forall j>2, \forall i.
\theta=\left(a_0(u)e^{-\frac{t_0}{u_1}}+\left(a_1(u)+\frac{t_0}{u_1^2}b(u)\right)\xi_1+a_2(u)e^{\frac{t_0}{u_1}}\xi_1^2\right)\frac{\partial}{\partial \xi_1}+b(u)\frac{\partial}{\partial u_1}.
\nu_p([n]_{p^s}|_{v=a/b})=\nu_p\left(\frac{a^{2np^s}-b^{2np^s}}{a^{2p^s}-b^{2p^s}}\right) =\nu_p(a^{2np^s}-b^{2np^s}) &=\nu_p(2np^s/t_0)+\gamma = \nu_p (n) + s+\gamma,
\|\psi \omega^{(2)}_{G}\|^2_{H^{-\frac{1}{2}-\epsilon}(M\times M)}&=\sum_{j=k}\int_{\mathbb{R}^2}(|\xi_0|^2+|\eta_0|^2+\lambda_j^2+\lambda_k^2)^{-\frac{1}{2}-\epsilon}\Big|{\mathcal F_{(t,s)\to(\xi_0,\eta_0)}}\Big(\frac{\psi(t,s)}{\lambda_j}e^{i{\lambda_j(t-s)}}\Big)(\xi_0, \eta_0)\Big|^2d\xi_0d\eta_0.
0 = 2(x_2-1)+\gamma_2y_2 \Longrightarrow \gamma_2y_2 \approx 2, 0 = 2(x_3-2)+\gamma_3y_3\Longrightarrow \gamma_3y_3 \approx 4, 0 = \nu+\gamma \circ x , \nu \geq 0, \gamma \circ x \geq 0
\begin{aligned}\lim_N \frac{1}{N+1}\sum_{n=0}^N (U_{(\theta)1}^{-n}\varphi(U_{(\theta)c})U_{(\theta)1}^nx,x)=(\varphi(U_{(\theta)1})x,x) &=(J_{\theta,1}\varphi(U_{(\theta)1})x,J_{\theta,1}x)= (\varphi(U_\mu)J_{\theta,1}x,J_{\theta,1}x)=\int\varphi|J_{\theta,1}x|^2\mu\end{aligned}