harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{DTS}_{(p,q)}(X) =\mathrm{DTV}_{(p,q)}^{-3/2}(X)\bigg\{\sum_{k=2}^{3}\binom{3}{k}[-\mathrm{DTE}_{(p,q)}(X)]^{3-k}[\mu^{k}+k\mu^{k-1}\sigma\mathrm{DTE}_{(p,q)}(Y)] ~~+2\mathrm{DTE}_{(p,q)}^{3}(X)+3[\mu-\mathrm{DTE}_{(p,q)}(X)]\sigma^{2}\left(L_{1}+1\right)+\sigma^{3}\left(L_{1}^{\ast}+2L_{2}^{\ast}\right)\bigg\},
S_{3}(\lambda) = \frac{2(-1)^\lambda}{(1+2\lambda)^2}\frac{1}{3+6\lambda}\sum\limits_{I_{13}=-\lambda}^\lambda (-1)^{I_{13}}\binom{2\lambda}{\lambda+I_{13}}\frac{1}{\binom{1+3\lambda+I_{13}}{1+2\lambda}}\Biggl[\frac{1}{\binom{1+3\lambda-I_{13}}{1+2\lambda}}+ \frac{1}{\binom{2+5\lambda+I_{13}}{1+2\lambda}}\Biggr].
\int_{-\partial D_{\lambda_1}\cup\partial D_{\lambda_2}}\psi d^cV\wedge(dd^cV)^{n-1}\ = \ \int_{D_{\lambda_2}\setminus\bar D_{\lambda_1}}\psi(dd^cV)^n \ \hskip2cm + \ \int_{D_{\lambda_2}\setminus\bar D_{\lambda_1}}d\psi\wedge d^cV\wedge (dd^cV)^{n-1}& := \I + II.
\alpha_1= -\frac{1}{2}\gamma^{-1}g_1, \alpha_2= -\frac{1}{2}(\gamma+ \frac{1}{2}\tilde\gamma^T)^{-1}g_2, \alpha_3= (\tilde\gamma^T)^{-1}\left((2n+4)\gamma \beta^{-1}(t,q) \alpha_2-g_3\right).
\sigma_{0}=2(\mathit{e}-1)\big((2-2\alpha+\beta)(\mathit{e}+1) &+\sqrt{((\mathit{e}+1)(2-2\alpha+\beta))^2-4(\mathit{e}-1)(3+\mathit{e}-2\alpha-2\alpha\mathit{e}-\beta-\beta\mathit{e})}\big)^{-1}.
\| \left( |\cdot|^{-(N-\gamma)}* (|u_1|^{p-2}u_2 u_3) \right) |u_4|^{p-2}|u_5| \|_{S'(L^2)}&\lesssim \min_{j=2,3,5}\left\lbrace \| u_1 \|_{S(\dot{H}^{s_c})}^{p-2}\| u_4 \|_{S(\dot{H}^{s_c})}^{p-2}\| u_j \|_{S(L^2)}\prod_{i ,k \not \in \{1,4,j \}}\| u_i \|_{S(\dot{H}^{s_c})}\| u_k\|_{S(\dot{H}^{s_c})}\right\rbrace
\mathrm{t r}\exp \left[ \frac{e_{0}\tau}2 \Sigma_{\mu \nu}\left( g F_{\mu \nu}- \sigma \widetilde{F}_{\mu \nu}\right) \right] = 2 \mathrm{R e}\frac{\sinh \left[ ( s + 1 / 2 ) e_{0}\tau \left( g X_{0}- \sigma \widetilde{X}_{0}\right) \right]}{\sinh \left[ \left( e_{0}\tau / 2 \right) \left( g X_{0}- \sigma \widetilde{X}_{0}\right) \right]}.
\tilde C = C \int_0^{t_{j+1}}e^{As}\, ds -C\frac{t_b-t_{j+1}}{t_b-t_a}\int_0^{t_a}e^{As}\, ds -C\frac{t_{j+1}-t_a}{t_b-t_a}\int_0^{t_b}e^{As}\, ds &=C \left( \frac{t_b-t_{j+1}}{t_b-t_a}\int_{t_a}^{t_{j+1}}e^{As}\, ds - \frac{t_{j+1}-t_a}{t_b-t_a}\int_{t_{j+1}}^{t_b}e^{As}\, ds \right)
-\iint_{Q_t}\mathbf{w}_t\cdot \mathbf{w}_{xx}\omega^2 dxdt+\mu\iint_{Q_t}|\mathbf{w}_{xx}|^2\frac{\omega^2}{\rho}dxds =\iint_{Q_t}u\mathbf{w}_x\cdot \mathbf{w}_{xx}\omega^2 dxds-\iint_{Q_t}\mathbf{b}_x\cdot \mathbf{w}_{xx}\frac{\omega^2}{\rho}dxds.
\langle\Psi_{1}^{\prime I_{1}}(z_{1})\Psi_{2}^{\prime I_{2}}(z_{2}) \cdots\Psi_{n}^{\prime I_{n}}(z_{n})\rangle=\langle \Psi_{1}^{I_{1}}(z_{1})\Psi_{2}^{I_{2}}(z_{2})\cdots \Psi_{n}^{I_{n}}(z_{n})\rangle
\Big(A^{\frac{1}{2}}\big((1-\lambda)A^{-1}+\lambda B^{-1}\big)A^{\frac{1}{2}}\Big)^{\frac{1}{2}}A^{-\frac{1}{2}}\big((1-\lambda)A+\lambda B\big)A^{-\frac{1}{2}}\Big(A^{\frac{1}{2}}\big((1-\lambda)A^{-1}+\lambda B^{-1}\big)A^{\frac{1}{2}}\Big)^{\frac{1}{2}}&-I=\lambda(1-\lambda)A^{-\frac{1}{2}}(A-B)B^{-1}(A-B)A^{-\frac{1}{2}}.
{}I_1= \frac{1}{N-3}\,{d^{3-N}}{Y^{-1}}\left(\cos^2\frac{\phi_x}{2}\right )^{\frac{3-N}{2}}{}_2 F_1 \left( \frac{N-3}{2}, \frac{5-N}{2}; \frac{N-1}{2}; \sin^2\frac{\phi_x}{2}\right).
\mathrm{\mathrm{}~^{*}\! ~}F_{\mu \nu}( x ) = 4 \pi \{\mathrm{\small ~ \frac{1}{2}~}\epsilon_{\mu \nu \rho \sigma}[ \partial^{\sigma}{\tilde{\lambda}}^{\rho}( x ) - \partial^{\rho}{\tilde{\lambda}}^{\sigma}( x ) ] \},
\rho_{D}^{-2}(z) |Sf(z)| = (2{\rm Re}z)^2 |Sf(z)| \left(\frac{\rho_D^{-1}(z)}{2{\rm Re}z}\right)^2\leq 6 \left(\frac{\rho_D^{-1}(z)}{2{\rm Re}z}\right)^2 \leq \sigma(t_0)
\sum_{n=0}^{\infty}(\frac{(-1)^{n}}{2n+1})^{3}=\frac{1}{1^{3}}-\frac{1}{3^{3}}+\frac{1}{5^{3}}-\frac{1}{7^{3}}+\cdot\cdot\cdot=\frac{\pi^{3}}{32}
w'_{l'}={z'}_{l'}\overline{{z'}}_{\tau'\left(l'\right)}+{\lambda'}_{l'}\left({z'}_{l'}{z'}_{\sigma'\left(l'\right)}+\overline{{z'}}_{l'}\overline{z}_{\sigma'\left(l'\right)}\right) +\displaystyle\sum_{k\geq3}{\varphi'}_{k}^{\left(l'\right)}\left(z',\overline{z'}\right),\quad\mbox{for all $l'=1,\dots,N'$,}
{\rm Coeff}_p[E_{\mu},m]\propto\left[\prod_{\substack{\kappa \prec \nu[1]}}\frac{Y(w)-y_{\kappa}(w)}{y_{\mu}(w)-y_{\kappa}(w)}\cdot\left( z^{\nu[1]}+ \sum_{\nu \prec \nu[1]}h_{\nu}(t;w) z^{\nu}\right)\right]_{q=t^{-m}}
X:=ax+a\sum_{m\ge1}\binom{m+1}{1}(2b)^mt_m, &T_1:=a^2t_1+a^2\sum_{m\ge2}\binom{m+1}{2}(2b)^{m-1}t_m, &T_2:=a^3t_2+a^3\sum_{m\ge3}\binom{m+1}{3}(2b)^{m-2}t_m, &\quad\ldots
\|\Gamma_{m, bc}\|_2\leq \frac{\omega^{2k}}{\pi^{2k}(2m+1)}\cdot\begin{cases}1/4(2m+\theta)^{2k-2}, \, m\in\mathbb{Z}_+, \, bc\in\{per, ap\}, 1/m^{2k-2}, m\in\mathbb{N}, bc=dir.\end{cases}
P_n=\left(\begin{array}{ccccc}1 0 \dots \dots 0 0 \xi 0 \dots 0 \vdots \ddots \ddots \ddots \vdots 0 \ldots 0 \xi^{n-2}0 0 \ldots \ldots 0 \xi^{n-1}\end{array}\right),Q_n=\left(\begin{array}{ccccc}0 1 0 \dots 0 0 0 1 \dots 0{\vdots}\ddots \ddots \ddots \vdots 0 \ldots 0 0 1 1 0 \ldots \ldots 0\end{array}\right),
\begin{array}{c c c c c}{{}}&{{}}&{{1}}&{{}}&{{}}\\{{}}&{{0 \!}}&{{}}&{{\! 0}}&{{}}\\{{1 \!}}&{{\!}}&{{2 0}}&{{\!}}&{{\! 1}}\\{{}}&{{0 \!}}&{{}}&{{\! 0}}&{{}}\\{{}}&{{}}&{{1}}&{{}}&{{}}\\ \end{array}.
Z^{( 1 )}[ \eta ] = \mathrm{S d e t}^{- \frac 1 2}{\sf Y}[ \eta ] = \mathrm{S d e t}^{- \frac 1 2}\left( \begin{array}{c c}{p^{2}}{\lambda \bar{\eta}}{\lambda \eta}{i p \llap /}\end{array}\right) .
\int_{x}^{0}m_{-2,\beta}[f](t)\overline{g}(t)(1-t)^{-2}(1+t)^{\beta}dt-\int_{x}^{0}f(t)\overline{g}(t)(1-t)^{-2}(1+t)^{\beta}dt+f^{\prime}(0)\overline{g}(0) & =(1-x)^{-1}(1+x)^{\beta+1}f^{\prime}(x)\overline{g}(x)+\int_{x}^{0}f^{\prime}(t)\overline{g}^{\prime}(t)(1-t)^{-1}(1+t)^{\beta+1}dt,
\limsup\limits_{n \rightarrow \infty}\tilde{P}(\hat{Y}^n)=\lim_{m \rightarrow \infty}\limsup\limits_{n \rightarrow \infty}\tilde{P}(\hat{Y}^n)\leq \lim_{m \rightarrow \infty}\tilde{P}(Y^m)=\tilde{P}\left(\cap_m Y^m\right)=0.
\alpha=\alpha(\lambda_1/\lambda_2,d,N):=\left(1- \frac{\rho(d)-\rho(d-1)}{\sqrt{2\omega^2\frac{(\rho(d-1)+\omega)^2+\omega^2}{(\rho(d-1)+2\omega)^2}}+ \rho(d)}\right)^{-\frac{2}{4-N}},
\epsilon(s,\pi,\wedge^2,\theta)\frac{J_\theta(s,W,\phi)}{L(s,\pi,\wedge^2)}=\frac{J_{\theta^{-1}}(1-s,\rho(\tau) \widetilde{W},\widehat{\phi})}{L(1-s,\pi^\vee,\wedge^2)}.
h_{1}=-f_{\;bc}^{a}\left( \frac{1}{2}A_{j}^{b}A_{k}^{c}B_{a}^{jk}+\left(\partial^{l}B_{0l}^{b}\right) \left( B_{a}^{0i}A_{i}^{c}+\eta_{a}^{\left(2\right) i}{\cal P}_{i}^{\left( 2\right) c}+\eta_{a}{\cal P}^{c}\right)\right) ,
T^{s}_{(a', [g, x']), (a, [g, x])}=T^{e}_{(h^{-1}g^{-1}a', [e, h^{-1}x']),(h^{-1}g^{-1}a, [e, h^{-1}x])}&=T_{(h^{-1}g^{-1}a', h^{-1}x'),(h^{-1}g^{-1}a, h^{-1}x)}&=T_{(g^{-1}a', x'),(g^{-1}a, x)}&=T^{g}_{(a', [g, x']), (a, [g, x])}
\leq \left(1+O\left(\frac{1}{k^3\left(\log n\right)^4}\right)\right)e^{-2i}\left(1+e^{-n^{1/100}}\right)\frac{(n-1)!}{2}=\left(1+O\left(\frac{1}{k^3\left(\log n\right)^4}\right)\right)e^{-2i}\frac{(n-1)!}{2}=\left(1+O\left(\frac{1}{k^2\log^2 n}\right)\right)e^{-2i}\frac{(n-1)!}{2}.
\min_{i=1,\ldots,N}\|\mathcal{G}_{1/\tilde\mu}(y_j)\|^2\leq \frac{8\cdot 24 L\tilde{\mu}\sum_{j=1}^N \frac{\varepsilon_j}{a_j^2}}{N(N+1)(2N+1)}&+\frac{48\tilde\mu^2}{\tilde{\mu}-\mu}\left( \frac{\|x^*-v_0\|^2}{N(N+1)(2N+1)}+\frac{M^2(r+\frac{\rho}{2}(N+3))}{(N+1)(2N+1)}+ \frac{4L\sum_{j=1}^N\frac{\varepsilon_j+\delta_j}{a_j^2}}{N(N+1)(2N+1)}\right)
\begin{aligned}\int_{T_1}^t \left<\Phi (\tau),C_2 f_N^{(2)}(\tau) \right> d\tau = & \int_{T_1}^t\int d\tau d\omega dx_1 dv_1 dv_2 \; \omega \cdot (v_2 - v_1)\Phi (\tau,x_1,v_1)f_N^{(2)}(\tau,x_1,v_1,x_1+\varepsilon \omega,v_2)\end{aligned}
k_n(z)= -(2n+2)z^{2n+1}+2z\frac{1-z^{2n+2}}{1-z^2}&+\sum_{i=0}^{n-1}\left\{(1-4a_{i+1}^2)z^2(\frac{1-z^{2n-2i-2}}{1-z^2})-2b_i z(\frac{1-z^{2n-2i}}{1-z^2})\right\}'(1-z^2)\frac{z^i p_i(x)}{\beta_i}&+\sum_{i=0}^{n-1}\left\{(1-4a_{i+1}^2)z^2(\frac{1-z^{2n-2i-2}}{1-z^2})-2b_i z(\frac{1-z^{2n-2i}}{1-z^2})\right\}k_i(z).
\delta(c\frac{d\tau}{dq})^{2}=2c^{2}\frac{d\tau}{dq}\delta\frac{d\tau}{dq}=\delta[(1-\frac{r_{s}}{r})c^{2}(\frac{dt}{dq})^{2}-\frac{1}{1-\frac{r_{s}}{r}}(\frac{dr}{dq})^{2}-r^{2}(\frac{d\varphi}{dq})^{2}].
\mathsf{d}(x,y) = \inf_{s \in G , w \in{\rm Pic}^0(X)}\mathsf{d}_{t-s}(t+v_1 - s-w,t+v_2- s-w) &= \inf_{w \in{\rm Pic}^0(X)}\mathsf{d}_{0}(v_1 -w,v_2-w) &= \mathsf{d}_{0}(v_1,v_2)
F ( T ) = - \frac{\xi_{0}^{2}x_{0}^{4}t}{8 \pi a}\sum_{l = 1}^{\infty}\left[ \frac 1{\nu ( x_{0}^{2}+ l^{2})^{2}}+ \sum_{n = 1}^{\infty}\frac{2 \nu^{5}}{( x_{0}^{2}+ l^{2}+ n^{2}t^{2})^{2}( \nu^{2}+ n^{2}t^{2})^{3}}\right] ,
T_{K}^{+}= K_{q}^{+}\frac{( K_{q}^{3}- k + 1 )}{[ K_{q}^{3}- k + 1 ]}\mathrm{a n d}T_{K}^{-}= \frac{( K_{q}^{3}- k + 1 )}{[ K_{q}^{3}- k + 1 ]}K_{q}^{-}\ ,
\left({H}_P(x,V)\right)_{i,j}:=\frac{1}{2}{F_P^i\bullet\left(\mathcal{L}_{F_P(x)}^{-1}\mathcal{L}_{V_P}+\mathcal{L}_{V_P}\mathcal{L}_{F_P(x)}^{-1}\right)F_P^j}
(e_B|xe_B y)_{L^2({\rm Tr})}={\rm Tr}(e_Bxe_B y) &={\rm Tr}(E_B(x)e_By) &=\tau(\Phi (E_B(x)e_By)) &=\tau(E_B(x)y) &=\Phi_B(x\otimes y^\circ)\, .
\mathcal{D}(M^T_\lambda\mathcal{Q}_\lambda M_\lambda) =M^{-1}_\lambda\mathcal{D}(\mathcal{Q}_\lambda) &=\{u\in H^1([-t_0,t_0],E):\, u(-t_0)\in\Psi_\lambda(-t_0)E^u_\lambda(0),\, u(t_0)\in\Psi_\lambda(t_0)E^s_\lambda(0)\}&=\{u\in H^1([-t_0,t_0],E):\, u(-t_0)\in E^u_\lambda(-t_0),\, u(t_0)\in E^s_\lambda(t_0)\}=\mathcal{D}(\mathcal{A}^0_\lambda),
\frac{\sqrt{\widehat{w}_s^{(n)}(x_1^{(n)}) \widehat{w}_s^{(n)}(x_2^{(n)})}}{2n^2}\cdot \frac{p_0^{(n)}(x_1^{(n)}) p_0^{(n)}(x_2^{(n)})}{\widehat{h}_0^{(n)}}= \frac{(z_1z_2)^{\frac{s}{2}}}{(z_1 + u)(z_2+ u)}\cdot \frac{(2n^2)^{1-s}}{\int_{-1}^1 \frac{(1-t)^s}{( 1 + \frac{u}{2n^2}- t)^2}dt}.
D(\Lambda)\simeq \prod_{i=1}^r\exp(_{\mathbb{R}}\{x_{{\varphi_i}}, y_{{\varphi_i}}, \sqrt{-1}h_{{\varphi_i}}\})/\exp(_{\mathbb{R}}\{\sqrt{-1}h_{{\varphi_i}}\}, &S(\Lambda)\cap (N_+B/B)\simeq\prod_{i=1}^r\exp(_{\mathbb{C}}\{e_{{\varphi_i}}\}).
\langle e^A \rangle_{\beta}= \exp \left[ \frac{1}{2}\, \langle A^2\rangle_{\beta}\right] \ ; \ \langle \, : e^A : \, \rangle_{\beta}=\exp \left[ \frac{1}{2}\, \langle \, : A^2 : \, \rangle_{\beta}\right] \, ,
\phi_1(K_1) := A_1^T K_1 + K_1 A_1 + Q_1 - K_1 B_1 B_1^{\top}K_1 = 0, A_1(K_1)^{\top}K_{12}+ K_{12}A_{2}= -(Q_{12}+ K_1A_{12}), A_2^{\top}K_2 + K_2A_2 = K_{12}^{\top}B_1B_1^{\top}K_{12}- K_{12}^{\top}A_{12}- A_{12}^{\top}K_{12}- Q_2 \,.
X^k=x^k+\frac{i\pi_r}{m(\omega+m)}\sum_{a=1}^{N}[\psi^k_a,\psi^r_a]\;,\;\;\theta^k_a=\psi_a^k+\frac{\pi_k(\omega-m)}{{m \vec\pi}^2}\pi_r\psi^r_a \;.
\left(\begin{matrix}x=0\,\left( \frac12 \right) x=\infty \overbrace{\begin{matrix}\sqrt{t_1}0 -\sqrt{t_1}0\end{matrix}}\overbrace{\begin{matrix}0 0 \theta^\infty_1 1 -t_2 \theta^\infty_2\end{matrix}}\end{matrix}\right) ,
P(\max_{1 \leq j \leq n}V_j \geq 2e) \leq \sum_{j=1}^n P(V_j \geq 2e) \leq \sum_{j=1}^n \left(\frac{\|V_j\|_{L_p}}{2e}\right)^p \leq n\left(\frac{1}{e}\right)^p=\frac{1}{w^{\kappa_2}n^{\kappa_2-1}}.
i \overline{{{\psi}}}_{R}\gamma_{\mu}\psi_{R}= i \overline{{{\psi}}}\gamma_{\mu}\frac{1 - \gamma_{3}}{2}\psi \longleftrightarrow \frac{1}{\sqrt{2 \pi}}( \partial_{\mu}\Lambda + \epsilon_{\mu \nu}\partial^{\nu}\Lambda ) .
D_{a,b}= \sum_{w=0}^{p}(-1)^w \tbinom pw \tbinom{p-w}{(p-w)-b}\tbinom w{a-(p-w)}\sum_{i=-1}^{r}\tbinom{(p-w)-b}{2(p-w)-2b+i-r}\tbinom{a-(p-w)}{2a-2(p-w)-i-1}.
f^{[ab]}_{[cd][ef]}={1 \over 2}\bigg( \eta_{ce}\delta^a_d\delta^b_f - \eta_{cf}\delta^a_d\delta^b_e + \eta_{df}\delta^a_c\delta^b_e - \eta_{de}\delta^a_c\delta^b_f - (a \leftrightarrow b) \bigg).
\int_{\alpha t}^{\beta t}\frac{e^{ix^2/(4t)}}{\sqrt t}\widehat h(x/(2t))e^{ixk}dx=\frac{e^{-itk^2}}{\sqrt t}\int_{\alpha t}^{\beta t}\exp\left(i\left( \frac{x}{2\sqrt t}+k\sqrt t \right)^2 \right)\widehat h(x/(2t))dx
\begin{array}{l}{{( 1 ) \quad{\displaystyle{\frac{\delta}{\delta b_{a}}}}\Delta = 0 \, , \qquad ( 2 ) \quad \displaystyle{\int}d^{3}x \,{\displaystyle{\frac{\delta}{\delta \, c^{a}}}}\Delta = 0 \, , \qquad ( 3 ) \quad{\cal G}^{a}\Delta = 0 \, ,}}\\{{( 4 ) \quad{\cal W}_{X}\Delta = 0 \, , \qquad X = \mathrm{d i f f . , \, L o r e n t z \, , \, r i g i d}\, .}}\\ \end{array}
C_{1}(\sum_{k=1}^{n}|\alpha_{k}|^{q^{*}})^{\frac{1}{q^{*}}}\leq \|\sum_{k=1}^{n}\alpha_{k}z_{k}\|\leq C_{2}(\sum_{k=1}^{n}|\alpha_{k}|^{q^{*}})^{\frac{1}{q^{*}}}.
A\left( \gamma^{1/2}|a| \right)\,{\bf I}+ B\left( \gamma^{1/2}|a| \right) \sum_{i,j=1}^3 \frac{y_i y_j}{|y|^2}\, \frac{(v_i - u_{0i}) (v_j - u_{0j})}{|v-u_0|^2}\geq \alpha\, 8 \sqrt{\pi\, \gamma}
F_X(t^{c_0}u_{00}, \ldots,t^{c_N}u_{0N};\ldots ; t^{c_0}u_{n0},\ldots,t^{c_N}u_{nN}) = t^{e_0}G_0(\textbf{u}_0,\ldots,\textbf{u}_N)+\cdots +t^{e_r}G_r(\textbf{u}_0,\ldots, \textbf{u}_N),
{\frac{1}{2}}( \Psi_{n^{\prime}}^{*}( \varphi + \lambda ) + \Psi_{n^{\prime}}^{*}( \varphi - \lambda ) ) \ + \{\frac{1}{2}}( \Psi_{n^{\prime}+ 1}( \varphi ) + \Psi_{n^{\prime}- 1}( \varphi ) )
\alpha_{ij}=-2\sum_{k^{\prime}\ne k}\frac{\langle\psi_{k}^{0}|\mu_{i}|\psi_{k^{\prime}}^{0}\rangle\langle\psi_{k^{\prime}}^{0}|\mu_{j}|\psi_{k}^{0}\rangle}{E_{k}^{(0)}-E_{k^{\prime}}^{(0)}}.
b_1 =\dim(W_0\cap V),b_2=a_0-b_1, b_3 =\dim(W_+\cap V)-b_1,b_4=\dim(W_-\cap V)-b_1, b_5 =\dim(U_+\cap V)-b_1-b_3,b_6=\dim(U_-\cap V)-b_1-b_4, b_7 =\dim(W\cap V)-b_1-b_3-b_4, b_8 =\dim((W_++U_-)\cap V)-\dim(W\cap V)-b_6, b_9 =\dim((U_++W_-)\cap V)-\dim(W\cap V)-b_5, b_{10}=a_+-b_3-b_7-b_8,b_{11}=a_--b_4-b_7-b_9.
\kappa_{rF}(A,\lambda b)=\lim_{\delta\rightarrow 0}\sup_{\left\|[\Delta A \; \lambda \Delta b]\right\|_F<\delta}\frac{\frac{\left\|F\left(A+\Delta A,\lambda (b+\Delta b)\right)-F(A, \lambda b)\right\|_2}{\|F(A,\lambda b)\|_2}}{\frac{\left\|[\Delta A \; \lambda \Delta b]\right\|_F}{\left\|[A \; \lambda b]\right\|_F}}.
\frac{s_3}{2\pi i (s_1+s_3-1) \Gamma(s_3+1)}\sum_{n>at_3}\int_n^\infty \int_{(c)}\frac{\Gamma(s_3+1+z)\Gamma(-z)}{u^{s_1+s_3+1+z}n^{s_2-1-z}}dz du. &= \frac{s_3}{2\pi i (s_1+s_3-1) \Gamma(s_3+1)}\sum_{n>at_3}\frac{1}{n^{s_1+s_2+s_3-1}}\int_{(c)}\frac{\Gamma(s_3+1+z)\Gamma(-z)}{s_1+s_3+z}dz .
{\rm V}({\rho}_{\rm N};{\rho}_{\rm N}) = \frac{1}{4}(1+{\rm N})^2{\rm V}({\rho};{\rho}) + \frac{1}{4}(1-{\rm N})^2{\rm V}({\sigma};{\sigma}) + \frac{1}{2}(1-{\rm N}^2){\rm V}({\rho};{\sigma}).
R^+[u](x_0, \theta) = \inf_{\theta' \in (\theta - \frac{\pi}{2},\theta + \frac{\pi}{2})}\frac{\big( \partial_{\theta'}u(x_0) \big)^+}{\cos(\theta - \theta')}R^-[u](x_0, \theta) = \sup_{\theta' \in (\theta - \frac{\pi}{2},\theta + \frac{\pi}{2})}\frac{\big( - \partial_{\theta' + \pi}u(x_0) \big)^+}{\cos(\theta - \theta')}.
\int_{\Phi_\rho( B_{\rho})}\int_{\Phi_\rho( B_{\rho})}\frac{(d^s(x)-d^s(y))^2}{|x-y|^{N+2s}}\, dxdy =\int_{B_{\rho}}\int_{B_{\rho}}\frac{(d_+^s(x)-d_+^s(y))^2}{|\Phi_\rho( x)-\Phi_\rho( y)|^{N+2s}}\, dxdy & \leq C \int_{B_{\rho}}\int_{B_{\rho}}\frac{(d_+^s(x)-d_+^s(y))^2}{|x-y|^{N+2s}}\, dxdy<\infty ,
u(r,z,t) \,=\, \frac{\Gamma}{\sqrt{\nu t}}\,U^\epsilon\Bigl(\frac{r-\bar r}{\sqrt{\nu t}}\,,\,\frac{z-\bar z}{\sqrt{\nu t}}\,,\,t\Bigr)\,, (r,z) \in \Omega\,,t \in (0,T)\,,
\max_{x\in L_0}h_i(x)-\min_{y\in L_0}h_i(y) = \left|\int_0^{d_{L_0}(x_i,y_i)}(dh_i)_{\gamma_i(t)}(\dot{\gamma}_i(t))dt\right| &\leq ||dh_i|| \int_0^{d_{L_0}(x_i,y_i)}|\dot{\gamma}_i(t)|dt &\leq \mathrm{Diam}(L_0)||dh_i||.
\mathop{l.i.m}\limits_{\|\bigtriangleup_{n}\| \longrightarrow 0}\sum_{\substack{j\in J\setminus J_{K(n)}}}\Phi(\frac{B_{t_{j}}+B_{t_{j+1}}}{2},t_{j})[B_{t_{j+1}}-B_{t_{j}}]=\int_{0}^{T}\Phi(B_{t},t) \circ dB_{t}
||A_0||_{eu}^2=\frac{1}{3g-3}<\sum_{i=1}^{3g-3}\frac{\partial}{\partial x_{i}}(X)\wedge \frac{\partial}{\partial y_{i}}(X),\sum_{i=1}^{3g-3}\frac{\partial}{\partial x_{i}}(X)\wedge \frac{\partial}{\partial y_{i}}(X)>_{eu}=1.
\sup_{t\in[0,T]}\| \widehat{f}(t)\|_{Z}=\sup_{t\in[0,T]}\| g (t)\|_{Z}\lesssim \epsilon_0, \sup_{t\in[0,T]}(1+t)^{1/2}\| V(t)\|_{W^{N_2}}\lesssim \epsilon_0.
\left(\int_G g(\theta_{e_1,e_2})\,dg\right)(e)(x) :=\int_M\left(\int_G\theta_{g(e_1)(x),g(e_2)(y)}\,dg\right)e(y)\,d\mu &=\int_M\int_G g(e_1)(x)\langle gB^s e_2(y),B^s e(y)\rangle_{E_y}\,dg\,d\mu.
i_*J(\tau,z)=5Hz+5H\tau(-z)+\sum_{n,d \atop 0\leq i\leq 4}\frac{q^d}{n!}\left\langle \tau(\psi)^n\frac{H^i}{z-\psi}\right\rangle_{0,n+1,d}^\mathrm{GW}H^{4-i},
F_{\tau}(\chi) =\sum_{x \in X_{\tau}}\prod_{i=1}^{k}\chi(x_i) &=\sum_{x \in X_{\tau}}\prod_{i=1}^{c_1}\chi(x_i)\prod_{i=1}^{c_2}\chi^2(x_{c_1+2i})\cdots\prod_{i=1}^{c_k}\chi^k(x_{c_1+c_2+\cdots+k i}) =\prod_{i=1}^{k}(\sum_{a\in D}\chi^i(a))^{c_i}= n^{\sum c_i m_i(\chi)}s_{\chi}(D)^{{\sum c_i (1-m_i(\chi))}},
f_{\varepsilon}^{\prime\prime}=\left( 1+2a_{\varepsilon}\right) \frac{1}{\mu_{\varepsilon}}f_{\varepsilon}^{\prime}(\frac{1}{2})=f^{\prime}(\frac{1}{2}),f_{\varepsilon}(\frac{1}{2})=f(\frac{1}{2})
[E,F]v_i \; =\; \pm \big( [n-i] Ev_{i+1}-[i] Fv_{i-1}\big) & \;=\; \pm \big( [n-i] [i+1] [i] [n-i+1\big) v_{i}& \;=\; \pm [n-2i] v_{i}&\; = \; \frac{K^2-K'^2}{q-q^{-1}}\, v_i ,
\int_{- 1 / K_{p}}^{1 / K_{p}}d y \, u_{0}^{2}( y ) w ( y ) \sim \left. ( 1 - K_{p}| y | )^{\frac{8 ( q + 1 ) ( n - 1 ) + 4 ( p + 1 ) ( n + 1 ) + ( p + n )^{2}a_{p}^{2}}{( p + n )^{2}a_{p}^{2}+ 4 ( p + 1 ) n}}\right|_{- 1 / K_{p}}^{1 / K_{p}}
\sum_{X_0 < p \leq X}e(qf(p)) = \sum_{X_0 < n \leq X}\frac{1}{\log n}\Lambda(n) \, e(qf(n)) + O (\sqrt{X}) \ll \frac{1}{\log X}\max_{X_0 < X_1 \leq X}\left|\sum_{X_0 \leq n \leq X_1}\Lambda(n) \, e(qf(n)) \right| + \sqrt{X}.
\bar{g}_{a b}= \left( \begin{array}{c c}{{\displaystyle{\cosh \chi + \cos \phi \sinh \chi}}}&{{\displaystyle{\sin \phi \sinh \chi}}}\\{{\displaystyle{\sin \phi \sinh \chi}}}&{{\displaystyle{\cosh \chi - \cos \phi \sinh \chi}}}\\ \end{array}\right) \qquad .
( u^{- 1})_{\underline{{b}}}^{~ \underline{{a}}}= \eta^{\underline{{{a c}}}}u_{\underline{{c}}}^{~ \underline{{d}}}\eta_{\underline{{{d b}}}}\equiv ( u_{a}^{~ \underline{{a}}}, u_{i}^{~ \underline{{a}}}) .
H=\frac{1}{2}p_i^A p_i^A+\frac{1}{4}G_{ij}^A G^{ij\, A}-A^{0\, A}\nabla^i p_i^A -\frac{m^2}{2}A^{0\, A}A^{0\, A}+\frac{m^2}{2}A^{i\, A}A^{i\, A}.
(I^a f)(t) := \frac1{\Gamma(a)}\int_0^t \frac{f(s)}{(t-s)^{{1-a}}}\, ds t > 0, (D^a f)(t) := \frac1{\Gamma(1-a)}\frac d{dt}\int_0^t \frac{f(s)}{(t-s)^a}\, ds = \frac d{dt}(I^{1-a}f)(t) t > 0,
\begin{aligned}T_2 =-\int h \left(\nabla_{\xi}\psi,\nabla_{x}h\right)\varrho\,m_0 = -\frac{1}{2}\int \left(\nabla_{\xi}\psi,\nabla_{x}h^2\right)\varrho\,m_0 &= \frac{1}{2}\int h^2 \left[\frac{(\nabla_{\xi}\psi,\nabla_{x}\varrho)}{\varrho}\right]\varrho\,m_0,\end{aligned}
\mathcal L_{\mathrm{m a s s}}= \frac{1}{2}\left( \psi_{\mu L}^{1}\ \psi_{\mu L}^{2}\right) \sigma^{\mu \nu}\left( \begin{array}{c c}{\omega / R}&{i p_{5}}\\{- i p_{5}}&{\omega / R}\\ \end{array}\right) \left( \begin{array}{c}{\psi_{\nu L}^{1}}\\{\psi_{\nu L}^{2}}\\ \end{array}\right) + h . c .
G \left({}_{\zeta}|{}^{\zeta^{\prime}}\right)_{l l^{\prime}}= \delta_{l , l^{\prime}}\; g \left({}_{\zeta}|{}^{\zeta^{\prime}}\right) \; , \; \; l = ( p_{D}, n ) \; , l^{\prime}= ( p_{D}^{\prime}, n^{\prime}) \; ,
\frac{\#\{n,\,\,\lambda_n\,({\rm mod}\,a^{-1})\in I,\,\,M\le n\le M+N\}}{N}&=\frac{\#\{\lambda\in \Lambda,\,\,\lambda_M\le \lambda\le\lambda_{M+N-1},\,\,\lambda\,({\rm mod}\,a^{-1})\in I \}}{\lambda_{M+N-1}-\lambda_M}\,\left(\frac{\lambda_{M+N-1}-\lambda_M}{\#A_{M,N}}\right)
S_\xi{}=\begin{pmatrix}\frac{\sqrt{-3c}}{2}0 0 \frac{\sqrt{-3c}}{6}\end{pmatrix}, S_\eta{}=\begin{pmatrix}0 \frac{\sqrt{-3c}}{6}\frac{\sqrt{-3c}}{6}0\end{pmatrix}, a{}=\frac{1}{3}, b_1{}=\frac{2\sqrt{2}}{3}, b_2{}=0.
\frac{1}{4 !}\int_{{\Lambda}}\frac{d^{4}\vec{p}_{1}}{( 2{\pi})^{4}}\frac{d^{4}\vec{p}_{2}}{( 2{\pi})^{4}}\frac{d^{4}\vec{p}_{3}}{( 2{\pi})^{4}}\frac{d^{4}\vec{p}_{5}}{( 2{\pi})^{4}}{\delta}{\lambda}_{4}( 1 2 3 5 ){\delta}^{4}( \vec{p}_{1}+ \vec{p}_{2}+ \vec{p}_{3}+ \vec{p}_{5}) \hat{\phi}_{2}( \vec{p}_{1}) \hat{\phi}_{2}( \vec{p}_{2}) \hat{\phi}_{2}( \vec{p}_{3}) \hat{\phi}_{2}( \vec{p}_{5}) .
\begin{matrix}\Omega_{xx}=ax_{1}^{2}+2bx_{1}x_{2}+cx_{2}^{2}\\ \Omega_{yy}=ay_{1}^{2}+2by_{1}y_{2}+cy_{2}^{2}\\ \Omega_{xy}=ax_{1}y_{1}+b(x_{1}y_{2}+x_{2}y_{1})+cx_{2}y_{2}\end{matrix}
\sigma_{j}^{B W}( E ) \sim \left| a_{j}^{B W}( E ) \right|^{2}= \left| \frac{r_{\eta}}{E - ( E_{R}- i \frac{\Gamma}{2})}\right|^{2}\sim \frac{1}{( E - E_{R})^{2}+ ( \frac{\Gamma}{2})^{2}},
\Gamma_{I,J,\mu}:=\bigl\{(x,x_1,\dots,x_\ell,y_1,\dots,y_{n+\ell})\mid x=x_a=y_b\,\forall\, a\in I\,,\, b\in J x_c=y_{\mu(c)}\,\forall\,c\in \bar I\bigr\}\,.
a_{n}(T) =\sup_{\dim E=n}\Big[\inf_{f\in S_E}\Vert Tf\Vert\Big]\,, a_{n}(T) =\inf_{{\rm codim}E <n}\Vert T_{|E}\Vert=\inf_{{\rm codim}E <n}\Big[\sup_{f\in S_E}\Vert Tf\Vert\Big]\,.
\begin{array}{cc}\mathring{K}_2[H_0] \gtrsim_s e[H_0]\ \textup{in}\ [2M,r_0], \Big|\mathring{K}_2[H_0]\Big| \lesssim e[H_0]\ \textup{in}\ [r_0,r_1], \mathring{K}_2[H_0] = 0\ \textup{in}\ [r_1,\infty ).\end{array}
[\varphi^* (\tau \cdot \psi)] (m) = (\tau \cdot \psi)( \varphi (m)) = \psi (\tau^{-1}\cdot \varphi (m)) = \psi (\varphi (\tau^{-1}\cdot m)) = [\varphi^* (\psi)] (\tau^{-1}\cdot m) = [\tau \cdot \varphi^* (\psi)] (m)
\begin{aligned}A_7\lesssim \varepsilon d_k^22^{-k}\left(||\varepsilon\partial_x(Q_{11}, \varepsilon Q_{12})_\psi||_{\tilde L^2_{t,\eta'(t)}(B^\frac12)}+||\partial_y(Q_{11}, \varepsilon Q_{12})_\psi||_{\tilde L^2_{t,\eta'(t)}(B^\frac12)}\right)||(\varepsilon\partial_xu, \varepsilon^2\partial_x v)_\psi||_{\tilde L^2_{t,\eta'(t)}(B^\frac12)}\end{aligned}
\mu_{\hat{\rho}\tau}:= \mathbb{E}_0\left[ \hat{\rho}^{(pq)}\tau^{(pq)}\right] \;=\; \frac{(n+1)^2}{6(n-2)}\left[\mu_{\rho^2}- \left( \frac{n -2}{n+1}\right)^2 \mu_{\hat{\rho}^2}-\frac{9}{(n+1)^2}\mu_{\tau^2}\right],
\bigg\| \frac{\partial_rf}{|x|}\bigg\|_{2}^2 = (\mathbf{H}_{3}- 1)^2 \bigg\| \frac{f}{|x|^2}\bigg\|_{2}^2 + \bigg\|\partial_r \frac{f}{|x|}+ \mathbf{H}_{3}\frac{f}{|x|^2}\bigg\|_{2}^2.
\mathrm{\large \frac{1}{2}}( W_{m}^{( 0 )}, W_{m}^{( 0 )})^{a}- V_{m}^{a}W_{m}^{( 0 )}= 0 , \qquad \mathrm{\large \frac{1}{2}}\{W_{m}^{( 0 )}, W_{m}^{( 0 )}\}_{A}- V_{A}W_{m}^{( 0 )}= 0 ,
P_{F_{4}}^{\Delta_{S}}( 2 | x ) = P_{E_{6}}^{\bf 2 7}( x ) / x^{3}, \quad P_{F_{4}}^{\Delta_{L}}( 2 | x ) \left( P_{F_{4}}^{\Delta_{S}}( 2 | x ) \right)^{2}= P_{E_{6}}^{\Delta}( x ) .
c_0^k \cdot \|A\| \leq \frac{1}{k+2}\cdot\sum_{i=0}^k (\frac{\beta}{{k+2 \choose i+1}}\cdot \|L_\eta(A,i)\| - (k+2)\cdot \|L_\eta(A,i-1)\|)\leq \|\delta(A)\| + \|\Upsilon_\eta(A)\|.
\tilde \phi_{k_1,k_2}(R,\xi )\coloneqq \begin{cases}e^{R \, r(-\xi )}\frac{(\pi \xi )^{n+k_1}\cos(\pi \xi )^{k_1}(1-\xi )^{n+k_1+k_2}}{\sin(\pi \xi )^{n+k_1}}, 1, \end{cases}
\frac{{\partial H(\tilde p,\tilde q)}}{{\partial{\psi_j}}}= \frac{{\partial \tilde p}}{{\partial{\psi_j}}}\ln \frac{{\tilde p\left({1 - \tilde q}\right)}}{{\tilde q\left({1 - \tilde p}\right)}}- \frac{{\tilde p - \tilde q}}{{\tilde q\left({1 - \tilde q}\right)}}\frac{{\partial \tilde q}}{{\partial{\psi_j}}}.
\left( \prod_{1 \le j \le m}a_j^{\binom{\nu_j}{2}}\right)\prod_{1 \le i < j \le m}(a_i - a_j)^{\min \left\{\nu_i^2,\, \nu_j^2 \right\}}\underset{\mathbb{Z}}{\bigg|}D(\overline{a}),
\left(1+\frac{q^{m-1}-q}{q-1}\right)Q^{(m)}_\nu(k) = \left(\frac{q^{k+1}-1}{q-1}+\frac{q^k(q^{m-1}-q)}{q-1}\right)Q^{(m-1)}_\nu(k) &\Rightarrow \frac{q^{m-1}-1}{q-1}Q^{(m)}_\nu(k) = \frac{q^{m+k-1}-1}{q-1}Q^{(m-1)}_\nu(k) &\Rightarrow Q^{(m)}_\nu(k) = \frac{q^{m+k-1}-1}{q^{m-1}-1}Q^{(m-1)}_\nu(k).
H_{\mu \nu}=\mu \delta_{\mu, \nu},\quad D_{\mu \nu}= \delta_{\mu, \nu}D_{\mu},\quad D_{\mu}=-i\left(g_s\,z(\mu\cdot q) +g\sum_{\gamma\in\Delta,\ \gamma\cdot\mu=1}z(\gamma\cdot q)\right).
{{d^2x^{\rho}}\over{d\lambda^2}}+\Gamma^{\rho}_{\mu\nu}{{dx^{\mu}}\over{d\lambda}}{{dx^{\nu}}\over{d\lambda}}+{\cal W}^{-1}g^{\rho\sigma}\partial_y({\cal W}g_{\sigma\mu}){{dx^{\mu}}\over{d\lambda}}{{dy}\over{d\lambda}}=0,
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{(\tau,v_{s+k+1},\omega_{k+1}) \in \mathcal{B}_{III}^+}d\omega_{k+1}dv_{s+k+1}d\tau \leq C_{d,s,k}C_{d,\alpha}T R \eta^{d-1}\end{aligned}
c(x,\sigma)=\frac{1}{Z_{\gamma}(\sigma_{x^c})}e^{-\frac{\beta}{2}\Delta_x H_{\gamma}(\sigma)},\,\,\,\,\,\,\,\Delta_x H_{\gamma}(\sigma)=H_{\gamma}(\sigma^x)-H_{\gamma}(\sigma)=2\sigma(x)\sum_{y\neq x}J_{\gamma}(x,y)\sigma(y),
\mathcal{E}_p(r_1, \ldots, r_{n+1}) \ge \mathcal{E}_p(r_1, \ldots, r_n, u_{n+1}q') = \mathcal{E}_p(u_1+ \epsilon_1, \ldots, u_n + \epsilon_n, u_{n+1}) q' + \sum_{i\le n, \epsilon_i=0}v_i
v^{-1}\pi P_v f(u) =2\int_{-\infty}^{+\infty}\frac{(t-u)}{((t-u)^2+v^2))^2}\int_{u}^{t}f(t')dt'dt &\geq \int_{u}^{+\infty}\frac{(t-u)^2}{((t-u)^2+v^2))^2}(M_Rf(u,t-u)\chi_{t>u})dt.
\phi(z_3)=1-\sigma(z_3) \leqslant 1-\left( \lambda \sigma(z_1)+(1-\lambda)\sigma(z_2) \right) &=\lambda(1-\sigma(z_1))+(1-\lambda)(1-\sigma(z_2)) &=\lambda \phi(z_1)+(1-\lambda)\phi(z_2).