harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R := k[x_F \mid F ]/(I_1 + I_2), I_1 := \left(\sum_{i_1 \in F}x_F - \sum_{i_2 \in F}x_F \; \Big| \; i_1, i_2 \in E \right)\!, \; I_2 := \left(x_Fx_{F'}\mid F, F' \right)
\Gamma_t[\phi] = \int\!\! d^Dx\, \left \{V(\phi^2,t) + \frac{1}{2}K(\phi^2,t)\left (\partial_{\mu}\phi^a \right)^2 + \frac{1}{2}Z(\phi^2,t) \left ( \phi^a \partial_{\mu}\phi^a \right)^2 + \cdots \right \}
\lim_{N\rightarrow\infty}\frac{1}{N}{\displaystyle\sum\limits_{n=1}^{N}}\iota(n)\iota(n+1) ={\displaystyle\prod\limits_{p}}\left( 1-\frac{2}{p^{2}}\right) =0.3226340989...=-1+2(0.6613170494...) \ =\frac{6}{\pi^{2}}{\displaystyle\prod\limits_{p}}\left( 1-\frac{1}{p^{2}-1}\right) =\frac{6}{\pi^{2}}(0.5307118205...),
\frac{(q)_{N}}{(zq)_{N}(z^{-1}q)_{N}}=\frac{1}{(q)_N}+(1-z)\sum_{n=1}^{N}\left[\begin{matrix}N n \end{matrix}\right]\frac{(-1)^n(q)_nq^{n(n+1)/2}}{(q)_{n+N}}\left(\frac{1}{1-zq^n}-\frac{1}{z-q^n}\right).
F(G(y))G'(y) =\frac{G'(y)}{G(y)}=\frac{\sum_i i r_i y^{i-1}}{\sum_i r_i y^i}&=\frac{\sum_{i\ge 1}i r_i y^{i-1}}{y \sum_{i\ge 1}r_i y^{i-1}}=\frac{H(y)}{y},
\max\limits_{m\in\mathcal{M}_n}\Big|\Big|Q^{(\mathcal{B}_n)}_{I,\mathbf{U}}Q^{(\mathcal{B}_n)}_{\mathbf{Z}|M=m,I,\mathbf{U}}-Q^{(\mathcal{B}_n)}_{I,\mathbf{U}}Q^n_{Z|U}\Big|\Big|_{\mathsf{TV}}\leq e^{-n\gamma_1},
((\phi_{\alpha})_\ast \pi_{\alpha\beta})^o\circ\Pi_{\alpha}=((\tilde\phi_{\alpha\beta})_\ast\Pi_{\beta})^o\circ \widetilde{\pi_{\alpha\beta}}^o, ((\tilde \pi_{\alpha\beta})^{-1}(\widetilde{\hat\phi_{\alpha\beta}}(\tilde F_{\beta}))) =((\tilde \Pi_{\alpha})^{-1}(\hat\phi_{\alpha}(\tilde E_{\alpha\beta}))),
F(k)=(2\pi)^{d/2}k^{1-d/2}\sum_{l=0}^{+\infty}(-i)^{l}\sum_{m}Y_{l,m}(\Omega_{k})\int_{0}^{+\infty}J_{d/2-1+l}(kr)r^{d/2}dr\int f(r)Y_{l,m}^{*}(\Omega_{r})d\Omega_{r}.
K^x_{uv}\, k^u_\mathbf{I}\, k^v_\mathbf{J}\, - \,{\frac{\lambda}{2}}\, \varepsilon^{xyz}\,{\cal P}_\mathbf{I}^y \,{\cal P}_\mathbf{J}^z\,= \,{\frac{1}{2}}\,f^{\mathbf{K}}_{\phantom{\mathbf{K}}\mathbf{I}\mathbf{J}}\,{\cal P}_\mathbf{K}^{x}
\left(\begin{matrix}z^{-n}0 0 z^n\end{matrix}\right)\left(\begin{matrix}1 0 \sum_{k=-2n+1}^{0}x_kz^k 1\end{matrix}\right)=\left(\begin{matrix}z^{-n}0 \sum_{-n+1}^{n}c_{k}z^k z^n\end{matrix}\right).
\begin{aligned}|\sigma_n^2(z)-\sigma_n^2(y)| & \le (|\sigma_n(z)|+|\sigma_n(y)|)\cdot |\sigma_n(z)-\sigma_n(y)|\le (|\sigma(z)|+|\sigma(y)|)\cdot |\sigma_n(z)-\sigma_n(y)| & \le c\cdot \bigl((1 + |z|^{2\ell_\sigma+1}+ |y|^{2\ell_\sigma+1}) \cdot \bigl(|z-y|+n^{-1/2}\bigr).\end{aligned}
\rho_i (z) :=2h_{E_{3-i},\mathfrak{p}}(f_{3-i}(z )) -h_{E_i ,\mathfrak{p}}(f_i (z )+Q_i )-h_{E_i ,\mathfrak{p}}(f_i (z )-Q_i ) &-2\alpha_{3-i}\log_{E_{3-i}}(f_{3-i}(z))^2+2\alpha_i (\log_{E_i}(f_i (z))^2 +\log_{E_i}(Q_i )^2 ) \in \Omega_i.
\mathrm{\boldmath E}= q [ \frac{1 - v^{2}}{( R - \mathrm{\boldmath R \cdot v})^{3}}( \mathrm{\boldmath R}- R \mathrm{\boldmath v}) + \frac{1}{( R - \mathrm{\boldmath R \cdot v})^{3}}\mathrm{\boldmath R \times}\{( \mathrm{\boldmath R}- R \mathrm{\boldmath v}) \mathrm{\boldmath \times a}\}]
\max_{q\in S^{2}:\,q\cdot v'=0}(q\cdot x)=\max_{q\in S^{2}:\,q\cdot(Ov)=0}(q\cdot x)=\max_{q\in S^{2}:\,(O^\intercal q)\cdot v=0}((O^\intercal q)\cdot(O^\intercal x))=\max_{q\in S^{2}:\,q\cdot v=0}(q\cdot(O^\intercal x)).
Q_{a,b}(\mu , \nu ; N)=p_{a,b}(\mu , \nu ; 0 ; N) +\sum_{\lambda =1}^{\infty}(-1)^{\lambda}p_{a,b}(\mu , \nu ; \lambda ; N) +\sum_{\lambda =1}^{\infty}(-1)^{\lambda}m_{a,b}(\mu , \nu ; \lambda ; N).
\begin{array}{rl}f(t) =\displaystyle f(1)+\beta(t-1)+\int_0^\infty (1+\lambda)\int_1^t\ \frac{1}{s+\lambda}\,ds\,d\mu(\lambda) begin{align*}2.5ex] =\displaystyle f(1)+\beta(t-1)+\int_0^\infty (1+\lambda)\log\frac{t+\lambda}{1+\lambda}\,d\mu(\lambda).\end{array}
\Phi_0=a_0^{-1}\sum_{M\geq0}\frac{x^M}{\Gamma^2(M+1)}\; \frac1{2\pi i}\int_C\frac{\Gamma(4s+1)\Gamma(2M-s)}{\Gamma^3(s+1)}(e^{-\pi i}y)^s\,ds,
\mathcal{N}(u;f,\theta_f) - \mathcal{N}^h(u_h;f_h,\theta_f) = \sum_{i,j}\int_{I_i}\left(\frac{2\lambda_2 - 1}{2}\right)\left\vert(u_h - v)\right\vert[\![\theta_f]\!]^2_{x,j-1/2}\,{\rm d}x & + \sum_{i,j}\int_{T_{ij}}\left((u-u_h)\,\partial_v f\,\theta_f - \frac{1}{2}\theta^2_f\right)\,{\rm d}v\,{\rm d}x + \mathcal{K}^2(u_h - v,f,\theta_f),
D(0, \varepsilon) = \sigma_{\varepsilon}([\varphi(\frac{I}{2})\bullet \varphi(T),\varphi(iI)]_{\ast})=\sigma_{\varepsilon}([\frac{I}{2}\bullet T,iI]_{\ast}) &=\sigma_{\varepsilon}(i(T-T^{\ast})),
\frac{1+\sum_{i=0}^hk_1(i)}{r+k_1(0)}a^{\widehat{P}^{Cont,+r}}+\frac{1+\sum_{i=0}^hk_2(i)}{s+k_2(0)}b^{\widehat{P}^{Cont,+s}}&=1+ \sum_{\substack{j=1 2 \notin I_j}}^h\left(\frac{k_1(j)}{k_1(0)+k_1(j)}a^{\widehat{P}^{Cont}_j}+ \frac{k_2(j)}{k_2(0)+k_2(j)}b^{\widehat{P}^{Cont}_j}\right).
\left(D_{0+}^{\alpha,\alpha',\beta,\beta',\gamma}t^{\rho-1}\right)(x)=\frac{\Gamma(\rho)\Gamma(-\beta+\alpha+\rho)\Gamma(\alpha+\alpha'+\beta'-\gamma+\rho)}{\Gamma(-\beta+\rho)\Gamma(\alpha+\alpha'-\gamma+\rho)\Gamma(\alpha+\beta'-\gamma+\rho)}x^{\alpha+\alpha'-\gamma+\rho-1}.
\alpha\circ \beta (x_1, \ldots , x_{k+l-1}) & = \sum_{i=1}^k(-1)^{(i-1)(l-1)}\alpha(x_1, \ldots, x_{i-1}, \beta(x_i, \ldots , x_{i+l-1}), x_{i+1}, \ldots , x_{k+l-1})
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 \; .
\int_{\overline{\mathcal{M}}_{0,n+1}}{\xi_{T}}_* \left(\prod_{v\in V(T)}\frac{1}{\psi_{h(v)}-1}\right) ={}{\xi_{T}}_* \left(\prod_{v\in V(T)}\int_{\overline{\mathcal{M}}_{0,{\rm val}(v)}}\frac{1}{\psi_{h(v)}-1}\right) ={}(-1)^{|V(T)|}= (-1)^{|E(T)|+1}
\begin{aligned}\overline H(f) =\sup_{\phi}\, \sup_{\tilde\pi\in\widetilde\Pi}\,\inf_{g}\bigg[\int \d\P_0(\omega)\,\phi(\omega) \bigg\{\sum_{e\in \mathcal U_d}\tilde\pi(\omega,e) \bigg(f(\omega,e) -\log\frac{\widetilde\pi_\omega(0,e)}{\pi_\omega(0,e)}+\big(g(\omega)-g(\tau_e\omega)\big)\bigg)\bigg\}\bigg] \end{aligned}
\Vert\Phi(a)\Vert\le\sum_{j=1}^m \left\Vert P(a_j)\right\Vert \le\sum_{j=1}^m \left\Vert P \right\Vert \left\Vert a_j\right\Vert^n=\Vert P\Vert\sum_{j=1}^m \left\Vert a_j \right\Vert^n.
\mathcal{L}_{m}( g ) = \frac{( - 1 )^{m}}{2^{m}}\delta_{\mu_{1}\nu_{1}\ldots \mu_{m}\nu_{m}}^{\rho_{1}\sigma_{1}\ldots \rho_{m}\sigma_{m}}{R^{\mu_{1}\nu_{1}}}_{\rho_{1}\sigma_{1}}\cdots{R^{\mu_{m}\nu_{m}}}_{\rho_{m}\sigma_{m}},
([K\delta K] (\chi \cdot \phi))(\sigma,g) = \sum_i \delta_i \cdot (\chi(\phi))(\sigma,g \delta_i) &= \sum_i \chi(\det(g \delta_i)) \delta_i \cdot \phi(\sigma, g \delta_i) &= \chi(\det \delta) \chi(\det g) \sum_i \delta_i \cdot \phi(\sigma, g \delta_i).
N^{-s-\varepsilon-}\|P_{N}\eta\|_{X^{0, 1/2-}}N^{s-}_{1}\|P_{N_{1}}v\|_{X^{0, 1/2-}_T}\prod_{j=2}^3N_{j}^{\frac{p-5}{2(p-1)}+\varepsilon}\|P_{N_{j}}v\|_{X^{0, 1/2+}_T}\prod_{j\geq 4}^p N^{s-}_{j}\|P_{N_{j}}v\|_{X^{0, 1/2+}_T}.
X= L^{2},~U=u,~U_{0}=u_{0}, A(t)= (A^{0})^{-1}\left\lbrace B^{ij}(t)\partial_{i}\partial_{j}\cdot-A^{i}(t)\partial_{i}\cdot-D(t)\cdot\right\rbrace, D(A(t))= Y=H^{2},S(t)=\lambda_{0}I-A(t),~B(t)=0,
G ( x , x^{\prime}) \sim \frac{\sqrt{\Delta_{V V M}( x , x^{\prime})}}{8 \pi^{2}}\left( \frac{4}{\Delta ( x , x^{\prime})}+ v ( x , x^{\prime}) \ln \frac{1}{2}\Delta ( x , x^{\prime}) + w ( x , x^{\prime}) \right)
\lim_{N\rightarrow\infty}\lim_{t \rightarrow \infty}\frac{\left|z_N(-\sigma+ j\omega,t)\right|}{\left|z_N(\sigma-j\omega,t)\right|}= \frac{\left|\left(\frac{1}{2}-\sigma\right) + j\omega\right|}{\left|\left(\frac{1}{2}+\sigma\right) - j\omega\right|}
f_{n}^{2}= \frac{\sqrt{2 \alpha^{\prime}}}{R_{0}\cos^{2}\tau}\sum_{m = - \infty}^{\infty}\left\{C_{m}^{1}\left( \ddot{C}_{n - m}^{2}- n ( n - m ) C_{n - m}^{2}+ \frac{2 \sin \tau}{\cos \tau}\dot{C}_{n - m}^{2}\right) + \dot{C}_{m}^{1}\dot{C}_{n - m}^{2}\right\}
+{\textstyle (-1)^{k}i_{x_{\{v_{1},\ldots,v_{k}\}}}}(v_{k+1}\wedge v_{k+2}) -{\textstyle (-1)^{k}\{v_{1},\ldots,v_{k},v_{k+1}\}\wedge v_{k+2}}-{\textstyle (-1)^{k+(\sum_{i=1}^{k}|x_{i}|-1)|v_{k+1}|}}v_{k+1}\wedge(\{v_{1},...,v_{k},v_{k+2}\}
p_{1\rightarrow0}=\mathbb{E}_{R_0}\left\{\mathbb{E}_{\Theta|\Theta<T}\left\{p_{1\rightarrow0}(\mathbf{x}_{a,0},\mathbf{x}_{a,H}|r_0,\theta)\right\}\right\}, p_{0\rightarrow1}=\mathbb{E}_{R_0}\left\{\mathbb{E}_{\Theta|\Theta<T}\left\{p_{0\rightarrow1}(\mathbf{x}_{a,0},\mathbf{x}_{a,L}|r_0,\theta)\right\}\right\},
T(\chi_{V\times U})(\xi,z) =\lim_{i\to \infty}T(\chi_{V\times U})(\xi,x_{\lambda_{i}})\leq\frac{2p+q}{3}&<\frac{2q+p}{3}\leq \lim_{i\to \infty}T(\chi_{V\times U})(\xi,y_{\lambda_{i}})=T(\chi_{V\times U})(\xi,z),
+ \frac{1}{2}F_{a b}\xi^{a}\partial_{\tau}\xi^{b}- \frac{1}{4}D_{a}D_{b}F_{c d}\xi^{a}\xi^{b}\psi^{c}\psi^{d}+ \sum_{n = 3}^{\infty}\left( \frac{1}{n !}D_{a_{1}}\ldots D_{a_{n - 1}}F_{{a_{n}}b}\xi^{a_{1}}\ldots \xi^{a_{n}}\partial_{\tau}\overline{{{X}}}^{b}\right.
\begin{array}{c}2\pi{\cal E}=\int_{M_3}d({\rm tr}(\Phi_AH^A))+\int_{M_3}d({\rm tr}(\Phi_BK^B)) =\int_{S^2}{\rm tr}(\Phi_AH^A)+\int_{S^2}{\rm tr}(\Phi_BK^B) \end{array}
B_{n+1}(z)+B_n(z) =(z-\alpha_n)A_n(z)-v'(z), 1+(z-\alpha_n)\left(B_{n+1}(z)-B_n(z)\right) =\beta_{n+1}A_{n+1}(z)-\beta_n A_{n-1}(z), B_n^2(z)+v'(z)B_n(z)+\sum\limits^{n-1}_{j=0}A_j(z) =\beta_nA_n(z)A_{n-1}(z).
\tilde{Z}_{x_0}(t) =e^{\Delta t}Ax_0+\tilde{Z}(t) \tilde{Y}(t) =\tilde{X}(t)-\tilde{Z}_{x_0}(t)=\int_0^te^{(t-s)A}\frac{F(X_{\ell(s)})}{1+\Delta tM_{\ell(s)}}ds R(t) =\int_{0}^{t}e^{(t-s)A}F\bigl(\tilde{Y}(s)+\tilde{Z}_{x_0}(t_{\ell(s)})\bigr)ds r(t) =\tilde{Y}(t)-R(t).
T\beta T^\star = \left(\begin{array}{cc}0 -i1_2 i1_2 0\end{array}\right)\,,T\alpha_kT^\star = \alpha_k\,,T(i\beta\alpha_{3})T^\star=\left(\begin{array}{cc}\sigma_3 0 0 -\sigma_3\end{array}\right) =: \mathcal{B}^0\,.
[v,a]\gamma_{z,s}= \gamma_{z,s}(a, v, 1)= \bigl( a,\, v+za,\, 1 - h(z,v) - sa \bigr) &= \bigl[ (v + za)\bigl( 1 - h(z,v) - sa \bigr)^{-1},\, a\bigl( 1 - h(z,v) - sa \bigr)^{-1}\bigr] .
a^{\textnormal{hom}}_{ijrs}:= \sum_{k=1}^n \sum_{l=1}^d\int_Y a_{ijkl}\big( \partial_{l}N^{(rs)}_{k}(y) +\delta_{kr}\delta_{ls}\big)\,{\textnormal{d}}y ({i,r \in \{1,\ldots,n\}, j,s \in \{1,\ldots,d\}}),
\left(A_1',D_1'\right)=\left(\Lambda\left(p_1,\cdots, p_{\lfloor\frac{m-1}{2}\rfloor},e_m,b_1,\cdots, b_{\lfloor\frac{k-1}{2}\rfloor},e_k\right)\otimes \Lambda\left(x_1,\cdots, x_{\lfloor\frac{m+k-1}{2}\rfloor}, \bar{e}_{m+k},a_1,\cdots, a_{\lfloor\frac{m+k-1}{2}\rfloor},{e}_{m+k}\right),D_1'\right)
\Psi = \dfrac{1}{p!q!}\psi_{A_p\bar{B}_q}\mathcal{Z}^{A_p}\wedge\mathcal{Z}^{\bar{B}_q}, \Phi = \dfrac{1}{p!q!}\phi_{A_p\bar{B}_q}\mathcal{Z}^{A_p}\wedge\mathcal{Z}^{\bar{B}_q},
\Delta_{\varepsilon , \overline{{{\varepsilon}}}}^{( 1 )}= \frac{\sigma^{2}}2 \cdot \frac{m - i \varepsilon \widehat{p}}{2 m}\cdot \frac{m - i \overline{{{\varepsilon}}}\overline{{{p}}}}{2 m}\cdot \left( 1 - \sigma_{p}^{2}\right) = \Psi_{\varepsilon , \overline{{{\varepsilon}}}}\cdot \overline{{{\Psi}}}_{\varepsilon , \overline{{{\varepsilon}}}}
P_{\mathcal{R}}\doteq \frac{\left(1-\frac1{d_c}\right)^{d_v}}{\sum\limits_{\ell=0}^{d_v}\binom{d_v}{\ell}\binom{wM\frac{d_v}{d_c}-d_v}{d_v-\ell}\left(1-\frac1{d_c}\right)^\ell}.
\Vert Dw\Vert(\Omega^*)= \Vert Du\Vert(\Omega)+\Vert Dv\Vert(\Omega^*\setminus\overline{\Omega})+\int_{\partial \Omega}|T_{\Omega}u-T_{\Omega^*\setminus \overline{\Omega}}\,v|\theta_{\Omega}\,d\mathcal H.
\implies \begin{cases}a\left(r^k+\dfrac{1}{r^k}\right)=a\left(r^k+\dfrac{-1_(-1)^{k+1}}{r^k}\right)=F_k, \left(r^k-\dfrac{1}{r^k}\right)=\left(r^k+\dfrac{-1(-1)^{k+1}}{r^k}\right)=L_k\end{cases}
\sum_{k+3\Delta^{(k)}\geq r; k\leq N/2}2^{r+3\Delta^{(k)}}e^{- (k + 3 \Delta^{(k)}-r)^{\alpha_-}}\leq \sum_{k + 3\Delta^{(k)}\geq r}2^{k + 6\Delta^{(k)}}e^{- (k + 3 \Delta^{(k)}-r)^{\alpha_-}}\leq 2^{-k'}\sum_{s = r}^{\infty}2^{2s}e^{-(s-r)^{1/10}}\leq 2^{3r/2}\sum_{t = 0}^{\infty}2^{2t}e^{-t^{1/10}}\ll 2^{3r/2}.
g_k(q) = (q;q)_\infty (-q;q^2)_\infty \sum_{n=0}^\infty a_k(n)q^n \qquad\ =(-q;q^2)_\infty (q^{k};q^{k})_\infty\sum_{n=1}^{\infty}\frac{q^{kn}+ 2q^{2kn}+\dots+ (k-1)q^{(k-1)kn}}{1 + q^{kn}+ q^{2kn}+\dots+ q^{(k-1)kn}}.
\delta \chi^j= i/\kern-.5em J^j \epsilon -\frac 12 \varepsilon^{jkl}(\gamma_p \epsilon ~\bar \chi^k \gamma_p \chi^l + \gamma_\mu \epsilon ~ \bar \chi^k \gamma^\mu \chi^l ) ,\qquad \delta J_\mu^j=-\bar\epsilon(\partial_\mu \chi^j + \varepsilon^{jkl}J_\mu^k\chi^l)~.
\left.C(p_{Y|X,S},p_S,\mbox{BEC}(\epsilon))\right|_{\epsilon=1}= \left.C'(p_{Y|X,S},p_S,\mbox{BEC}(\epsilon))\right|_{\epsilon=1}&=\underline{C}(p_{Y|X,S},p_S), \left.C(p_{Y|X,S},p_S,\mbox{BEC}(\epsilon))\right|_{\epsilon=0}= \left.C'(p_{Y|X,S},p_S,\mbox{BEC}(\epsilon))\right|_{\epsilon=0}&=\overline{C}(p_{Y|X,S},p_S).
F^{k}\!_{i}F^{l}\!_{j}\left( g^{i j}\left\{\begin{array}{c}{{a}}\\{{a k}}\\ \end{array}\right\}- g^{j b}\left\{\begin{array}{c}{{i}}\\{{k b}}\\ \end{array}\right\}\right) \partial_{l}+ F^{l}\!_{i}g^{i j}( \partial_{k}F^{k}\!_{j}) \partial_{l}= 0 \; ,
\left( \begin{matrix}{m^{i}}\\{n^{i}}\\{p^{i}}\\ \end{matrix}\right) = \left( \begin{matrix}{m}&{j}&{k}&{m^{4}}&{m^{5}}&{\cdots}\\{0}&{n}&{l}&{n^{4}}&{n^{5}}&{\cdots}\\{0}&{0}&{p}&{p^{4}}&{p^{5}}&{\cdots}\\ \end{matrix}\right)
\det_{1\leq j,k\leq m}\left(w\big((x_1+j-1,y_1+m-j);(x_2+l_k,y_2-l_k)\big)\right) &\quad={(-1)^{\binom m2}2^{\binom m2-m(x_2-x_1)-|l|}}q^{X}\Delta(q^l) &\quad\quad\times\prod_{j=1}^m\frac{(q;q)_{x_2+y_2-x_1-y_1-m+j}(-q^{Z-x_2-y_1-y_2-m+j};q)_{x_2-x_1-j+1+l_j}}{(q;q)_{x_2-x_1+l_j}(q;q)_{y_2-y_1-l_j}},
\hat C_{w_k}=\sum_{\substack{m=1 m\neq k}}^{K}({\bf h}^{H}_{kB}{\bf\hat{t}}_{B_m})^{2}S_{B_m}+\sum_{j=1}^{2}({\bf h}^{H}_{kj}{\bf\hat{t}}_j)^{2}S_j,\ \forall k\in\mathcal{C}, \hat C_{q_j}=\sum_{m=1}^{K}({\bf g}^{H}_{jB}{\bf\hat{t}}_{B_m})^{2}S_{B_m}+\kappa({\bf g}^{H}_{jj}{\bf\hat{t}}_j)^{2}S_j, \ \forall j\in\mathcal{D}.
= ( - 1 )^{l_{1}- 1 / 2}c_{0}l_{1}\sqrt{\frac{( k_{1}- l_{0}- 1 ) ( k_{1}+ l_{0})}{( l_{1}- l_{0}) ( l_{1}- l_{0}- 1 ) ( l_{1}+ l_{0}) ( l_{1}+ l_{0}+ 1 )}},
\sum_{k = 0}^{s}\big\|\partial_{t}^{k}z(t, \cdot)\big\|_{H^{s - k}(\Omega)}^{2}+\sum_{k = 0}^{s-2}\big\|\partial_{t}^{k}\theta(t, \cdot)\big\|_{H^{s + 1 - k}(\Omega)}^{2}+\big\|\partial_{t}^{s-1}\theta(t, \cdot)\big\|_{H^{1}(\Omega)}^{2}\to \infty t \nearrow T^{\ast}_{J}.
\sum_{j=0}^{k-1}\Biggl[ \binom{2k-1}{j}-\binom{2k-1}{j-1}\Biggr]\frac{c_1^{j}\,c_2^{2k-1-j}}{(c_1+c_2)^{2k-1}}=\sum_{j=0}^{k}\Biggl[ \binom{2k}{j}-\binom{2k}{j-1}\Biggr]\frac{c_1^{j}\,c_2^{2k-j}}{(c_1+c_2)^{2k}}
II =\int_{2x_1}^{\frac{1}{2}x_2}\left[K\left(\frac{x_1}{y}\right)-K\left(\frac{x_2}{y}\right)\right]\frac{\omega(y)}{y}~dy \geq\int_{2x_1}^{\frac{1}{2}x_2}(2-\frac{1}{2}\log(3))\frac{1}{y}~dy &= (2-\frac{1}{2}\log(3))\log\left(\frac{x_2}{x_1}\right)-C.
\nabla_{\mu}\zeta^{A}=\widetilde{\nabla}_{\mu}\zeta^{A}+\vartheta_{\mu B}^{{\footnotesize (T)}}{}^{A}\zeta^{B},\nabla_{\mu}\xi_{A}=\widetilde{\nabla}_{\mu}\xi_{A}-\vartheta_{\mu A}^{{\footnotesize (T)}}{}^{B}\xi_{B}.
\frac{d\alpha}{ds}(s) = \int_{\mathbb{R}^n}w(s,y)dy, \frac{e^{-s}}{b(t(s))^2}\frac{d^2\alpha}{ds^2}(s) = \frac{e^{-s}}{b(t(s))^2}\frac{d\alpha}{ds}(s)- \frac{d\alpha}{ds}(s) + \int_{\mathbb{R}^n}r(s,y)dy,
g_1 = 1+2\sum_{n=1}^N\epsilon^{n(n+1)}g^{-n}d_{n+1}e^{nx}D_{N-n}^0+o(\epsilon^{N(N+1)}), g_2= 1+\sum_{n=1}^N\epsilon^{n(n-1)}g^{-n}d_ne^{nx}D_{N-n}^2+o(\epsilon^{N(N-1)}),
\Delta < \left( \frac{7c}3 + 7 \right)\rho + \frac 1c - \left(\frac{4c}3+5\right)\nu = -\frac{(c-3) (c^3 + 21c^2 + 22c + 24)}{12 c (c+1) (c+2) (2c^2 + 3c + 3)}< 0.
S_4=\left(\frac{\phi(\mathfrak{m})}{|\mathfrak{m}|}\right)^{k-1}c_K^{k-1}(\log R)^{k-1}I_{3k}^{(m)}(F(w_1)) +O\left(F_{\max}^2 \left(\frac{\phi(\mathfrak{m})}{|\mathfrak{m}|}\right)^{k-1}\frac{1}{D_{0}}(\log R)^{k-1}\right)
\begin{aligned}\|u-P_m\|^*_{L^2(B_{\eta^m})}\leq \eta^{2m}, \Delta P_m=f_{B_{\eta^m}}, |(P_m-P_{m-1})(0)|+\eta^{m-1}|D(P_m-P_{m-1})(0)|+\eta^{2(m-1)}|D^2(P_m-P_{m-1})(0)|\leq \bar{C}\eta^{2(m-1)}, \end{aligned}
\eta_{\rm res}^2 = \sum_{E \in{\cal T}_h}\eta_{\rm res}^2(E) \,, \eta_{\rm coef}^2 = \sum_{E \in{\cal T}_h}\sum_{k=1}^6\eta_{{\rm coef},k}^2(E) \,, \eta_{\rm rhs}^2 = \sum_{E \in{\cal T}_h}\sum_{k=1}^2\eta_{{\rm rhs},k}^2(E)\,.
|x_{N}^{n-(1-\omega)\gamma}D_{x_{N}}^{m}u(x,t)|\leq C||v||_{C_{x}^{m-n+(1-\omega)\gamma}(R_{T}^{N-1})}x_{N}^{n-(1-\omega)\gamma}x_{N}^{-m+(m-n+(1-\omega)\gamma)}=
\left| \sum \limits_{k=0}^{N-1}\partial_\theta c(\theta_k) \cdot p(x_k) \cdot \prod \limits_{j=k+1}^{N}c(\theta_j) \cdot p'(x_j) \right| \leq\frac1{25}\cdot C_{N-k-1}\left| \sum \limits_{k=0}^{N-1}\partial_\theta c(\theta_k) \cdot (1-x_k) \right|.
\|u_{\infty}\|_{L_{\infty}(-\infty,0; L^{3,q}(\mathbb{R}^{3}_{+}))}\leqslant \sup_{k}\|u_{k}\|_{L_{\infty}(-\infty,0; L^{3,q}(\mathbb{R}^{3}_{+}))}=L<\infty.
(\ell,m | n,k) = \langle a,b \mid a^\ell, b^m, (ab)^n, (a^{-1}b)^k \rangle, (\ell,m,n;q) = \langle a,b \mid a^\ell, b^m, (ab)^n, [a,b]^q \rangle, G^{m,n,p}= \langle a,b \mid a^m, b^n, c^p, (ab)^2, (ac)^2, (bc)^2, (abc)^2 \rangle.
\frac{-i\langle T a^{\dagger t}_{{\bf{k}}}(-{\bf{q}})a^{\dagger}_{{\bf{k}}^{'}}({\bf{q}}^{'}) \rangle}{\langle T 1 \rangle}= \sum_{n}\mbox{}exp(\omega_{n}t)\mbox{}\frac{-i\langle T a^{\dagger n}_{{\bf{k}}}(-{\bf{q}})a^{\dagger}_{{\bf{k}}^{'}}({\bf{q}}^{'}) \rangle}{\langle T 1 \rangle}
e^{-\int_{\xi_j}^u G_j(v) \: dv}\mu X_{j-1}(u) \leq (1 + 3 \delta) s e^{-\int_{\xi_j}^u G_j(v) \: dv}e^{\int_{\tau_j}^u G_{j-1}(v) \: dv}&= (1 + 3 \delta) s e^{\int_{\tau_j}^{\xi_j}G_j(v) \: dv}e^{-s(u - \tau_j)}.
\begin{array}{l l l l l l}{{\displaystyle D}}&{{= 3}}&{{: \quad}}&{{S p ( 4 ; R )}}&{{\to}}&{{O S p ( N ; 1 | R )}}\\{{\displaystyle D}}&{{= 4}}&{{: \quad}}&{{S U ( 2 , 2 )}}&{{\to}}&{{S U ( 2 , 2 | N )}}\\{{\displaystyle D}}&{{= 6}}&{{: \quad}}&{{U_{\alpha}( 4 ; H )}}&{{\to}}&{{U U_{\alpha}( N ; 4 | R )}}\\ \end{array}
L_{n_{1}}(z) + C\kappa >\frac{1}{n_{1}}\log \lVert M^{z}_{n_{1}}(x_{0}+j \omega)\rVert > 2L_{n_{1}}(z) + O \Big ( \frac{1}{n_{1}}\Big) - (L_{n_{1}}(z) + C\kappa) > L_{n_{1}}(z) - C\kappa.
\ell \neq 5 \quad S p D_{5 6}\left( E^{{\vec{\alpha}}_{\ell}}\right) = \left( \begin{matrix}{A [{{\vec{\alpha}}_{\ell}}]}&{{\bf 0}}\\{{\bf 0}}&{- A^{T}[{{\vec{\alpha}}_{\ell}}]}\\ \end{matrix}\right)
U ( L_{1}, L_{2}) = \frac{\pi}{4 8}\biggl ( \frac{1}{L_{1}}+ \frac{1}{L_{2}}\biggr ) - \frac{L_{1}L_{2}}{3 2 \pi}\sum_{p , q = - \infty}^{\infty}^{\prime}( p^{2}L_{1}^{2}+ q^{2}L_{2}^{2})^{- \frac{3}{2}}.
G(\mu_1,\mu_2)=\left\{(x^1,x^2,x^3)=\left(\begin{array}{cccc}e^{\mu_1\>x^3}0 x^1 0 e^{\mu_2\>x^3}x^2 0 0 1 \end{array}\right)\right\}\subset \mathrm{GL}_{3}\mathbb{R}.
J_1 = \sup_{j\le r+1}\sup_\alpha \sup_{\beta \in I_j}|b_{\alpha ,\beta}e^{r (|\alpha |^{\frac 1{2s}}+|\beta |^{\frac 1{2s}})}|\intertext{and}J_2 = \sup_{j> r+1}\sup_\alpha \sup_{\beta \in I_j}|b_{\alpha ,\beta}e^{r (|\alpha |^{\frac 1{2s}}+|\beta |^{\frac 1{2s}})}|
\frac{y_{n+1}-y_{n}}{x_{n+1}-x_{n}}=\frac{\left (p_{n+1}^2-1 \right )}{\left ((p_{n+1}^2-1)-1 \right )}\frac{y_{n}}{x_{n}}=\frac{\left (p_{n+1}^2-1 \right )}{\left (p_{n+1}^2-2 \right )}\frac{y_{n}}{x_{n}},
{\bf{b}}\widetilde{\mathbb{S}}_{\lambda^i}{\bf{b}}\widetilde{\mathbb{S}}_{\lambda^j}=\sum_{\lambda^q\succeq \lambda^i,\lambda^j}\sum_{P : \delta P=\Delta_{\sigma_r\lambda^i~\sigma_r\lambda^j}^{\sigma_r\lambda^q}}\sum_{s=0}^{|P|}a^{(P)}_s({\bf{b}}\widetilde{\mathbb{S}}_{\lambda^1})^s{\bf{b}}\widetilde{\mathbb{S}}_{\lambda^q}
\begin{cases}\partial_t \psi - \Delta \psi + \psi \cdot \nabla \psi + \nabla \pi + \phi( \partial_t \eta - \Delta \eta) - 2 \nabla \phi \cdot \nabla \eta + (\psi \cdot \nabla \eta) \phi =0 \div \psi = - \nabla \eta \cdot \phi \, . \end{cases}
\frac{p_1}{a_1^{d_1}}\, \frac{1}{\Gamma \left(\frac{d_1}{p_1}\right)}\, (d_1 - d_2) \, \frac{a_1^{d_1}}{p_1^2}\, \Gamma \left(\frac{d_1}{p_1}\right) \, \psi\left( \frac{d_1}{p_1}\right) = \frac{1}{p_1}\, \psi\left( \frac{d_1}{p_1}\right) \, (d_1 - d_2).
\langle \mathbf{w}_1 ,\, \mathbf{w}_2 \rangle \langle \mathbf{w}_2 ,\, \mathbf{w}_3 \rangle \langle \mathbf{w}_3 ,\, \mathbf{w}_1 \rangle = \, \frac{1}{2}\left(-1 + \rho_{12}+ \rho_{13}+ \rho_{23}\right) + \frac{i}{4}\det(\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3).
\ll N \left( C p^{24}\log \log N \right)^{2 p}\sum_{\sigma \in S_{2 p}}\sum_{0 \le h_1, \dots, h_{2 p}\le \log K}\bigg( \prod_{\substack{i = 1 i \ne i_0}}^{2 p}\alpha(h_i)^2 e^{h_i}\bigg) \mathrm{vol}(I_{\sigma,h}).
\begin{array}{l}\delta\theta^\alpha =\epsilon^\alpha, \qquad\delta\bar\theta_\alpha =\bar\epsilon_\alpha, \delta x^{\bar\mu}= -i\bar\epsilon_\alpha\tilde\Gamma^{{\bar\mu}\alpha\beta}\bar\theta_\beta+i\epsilon^\alpha\Gamma^{\bar\mu}_{\alpha\beta}\theta^\beta, \qquad\delta x^{11}=0,\end{array}
\prod_{\ell=1}^{\infty}(1+\epsilon^{\ell}t_{d\ell})^{X_{\ell}}= \sum_{\mu}\binom{X}{\mu}\epsilon^{1\mu_1+2\mu_2+\ldots}t_d^{\mu_1}t_{2d}^{\mu_2}\ldots = \sum_{\mu}\binom{X}{\mu}\epsilon^{\|\mu\|}t_d^{\mu_1}t_{2d}^{\mu_2}\ldots.
\sum_{n = q_k}^{q_{k+1}-1}h(n) = \sum_{n=q_k}^{q_k+q_{k-1}-1}h(n) + \sum_{i=1}^{a_{k+1}-1}\left( \sum_{n = iq_k + q_{k-1}}^{(i+1)q_k+q_{k-1}-1}h(n) \right) &\le q_{k-1}h(q_k) + \sum_{i=1}^{a_{k+1}-1}q_k h(iq_k + q_{k-1}) &= q_{k-1}h(q_k) + \sum_{i=1}^{a_{k+1}}q_k h(iq_k + q_{k-1}) - q_k h (q_{k+1}) &= q_{k-1}h(q_k) + \mu(G_k) - q_k h(q_{k+1}),
(\nabla_{X}f)Y-A_{hY}X-t(\sigma(X,Y)) =0, (\overline{D}_{X}h)Y+\sigma(X,fY)-s(\sigma(X,Y)) =0, (\overline{\nabla}_{X}t)V-A_{sV}X+f(A_{V}X) =0, (D_{X}s)V+\sigma(X,tV)+h(A_{V}X) =0,
[\cdots [F_{0|1,0},\tilde F_{n-2,0}]_q\cdots\tilde F_{i+1,0}]_q \equiv-q^{-1}[\cdots [[F_{0,0},F_{1,0}]_q,F_{n-1,0}]_q\cdots F_{i+2,0}]_q &=-q^{-1}[\cdots [[F_{0,0},F_{n-1,0}]_q\cdots F_{i+2,0}]_q F_{1,0}]_q
\sum_{m=0}^N\alpha^mc_1^{\frac{m}{2}}\|\tilde{S}_m^0\|_{L^1}\le\frac{1}{2\beta h}\sum_{n=2}^N\frac{1}{(n-2)!}\left(b\beta L^d U_{max}\sum_{m\in\{2,4\}}(\alpha+1)^mc_1^{\frac{m}{2}}\right)^n &\le \frac{1}{2\beta h}\left(b\beta L^d U_{max}\sum_{m\in\{2,4\}}(\alpha+1)^mc_1^{\frac{m}{2}}\right)^2e^{b\beta L^d U_{max}\sum_{m\in\{2,4\}}(\alpha+1)^mc_1^{\frac{m}{2}}}.
\|\nabla g_u(g_u^{-1}(x))\|_{\mathrm{F}}^2 = \left( \nabla g_u(g_u^{-1}(x))^\top \nabla g_u(g_u^{-1}(x)) \right)\leq \Lambda^2 d, &\|\nabla g_v(g_v^{-1}(x))\|_{\mathrm{F}}^2 = \left( \nabla g_v(g_v^{-1}(x))^\top \nabla g_v(g_v^{-1}(x)) \right)\leq \Lambda^2 d,
y(t)=\begin{cases}w_1(t)-\frac{t}{s_1}w_1(s_1), t\in[0;s_1] w_2(t-s_1)-\frac{t-s_1}{s_2-s_1}w_2(s_2-s_1), t\in[s_1;s_2] \cdot \cdot \cdot w_{N+1}(t-s_N)-\frac{t-s_N}{1-s_N}w_{N+1}(1-s_N), t\in[s_N;1].\end{cases}
{dM^{(1)}\over dm}={d(\delta M)\over dm}+{1\over2}\sum_n{d\omega^B_n \over dm}-{1\over2}\sum_n{d\omega^F_n\over dm}- \left({1\over2}\sum_n{d\tilde\omega^B_n\over dm}-{1\over2}\sum_n{d\tilde\omega^F_n\over dm}\right)
\ell(xs_\alpha, s_\alpha v\beta) = \ell(x,v\beta) + \ell(s_\alpha,s_\alpha v\beta) =~\, \ell(x,v\beta)+\begin{cases}1, v\beta = -\alpha, -1, v\beta=\alpha, 0, v\beta\neq \pm \alpha.\end{cases}
\sigma_3 = x_3(0) = -3 \;l\;\sigma_1^2~2^{\frac{1}{p}-1}, \sigma_2 =x_2(0) = -l~ \sigma_1^2~2^{\frac{1}{p}-1}+ \frac{3}{p}~m \sigma_1^2~ 2^{\frac{1}{p}-2},
S_{l c}^{F}= \frac{i}{4 \pi l_{s}^{2}p^{+}}\int d \tau \int_{0}^{2 \pi l_{s}^{2}p^{+}}d \sigma \left[ S^{1}\left( \partial_{+}- \frac{\mu}{2}\gamma^{1 2}\right) S^{1}+ S^{2}\left( \partial_{-}- \frac{\mu}{2}\gamma^{1 2}\right) S^{2}+ 2 \mu S^{1}\gamma^{2 3 4}S^{2}\right] ,
Q_1\colon (\mu-\mu\overline{\mu}+\overline{\mu})y_1^2-2y_1y_2+y_2^2+(1-\overline{\mu}+\mu\overline{\mu}-\mu)y_3^2+y_4^2=0, Q_2\colon \mu\overline{\mu}y_1^2-2\mu\overline{\mu}y_1y_2+(\mu-1+\overline{\mu})y_2^2+\mu\overline{\mu}y_4^2+(1-\overline{\mu}+\mu\overline{\mu}-\mu)y_5^2=0.
\tau_{H}= -m_{2}\cot \langle \alpha_{1}, H \rangle \alpha_{1}-m_{1}\cot \langle \alpha_{1}+\alpha_{2}, H \rangle (\alpha_{1}+\alpha_{2}) &-m_{2}\cot \langle \alpha_{1}+2\alpha_{2}, H \rangle (\alpha_{1}+2\alpha_{2}) = -(2m_{2}+m_{1})\cot \langle \alpha_{1}, H \rangle (\alpha_{1}+\alpha_{2}).
f(x) \, = \, \underbrace{\displaystyle{\max_{1 \leq i \leq I_1}}\, \left( \, x^T a^i + \alpha_i \, \right)}_{\mbox{convex in $x$}}-\underbrace{\displaystyle{\max_{1 \leq i \leq I_2}}\, \left( \, x^T b^i + \beta_i \, \right)}_{\mbox{convex in $x$}}
\sigma ({\bf q})=\left\{N_{0}\left( 1-e^{-q^{2}\overline{u_{t}^{2}}}\right)+e^{-q^{2}\overline{u_{t}^{2}}}\left| \sum_{n}e^{i{\bf qr}_{n0}}\right|^{2}\right\}\sigma_{0}({\bf q}),
l_0:=\sup\left\{l>0:\frac{l}{l_{cr^*}}\frac{1}{2s_d m}g_d(u)>-\sqrt{\frac{1}{m}\left(\alpha-\frac{l}{l_{cr^*}}\right)}+\frac{1}{2m}\frac{l}{l_{cr^*}}u \forall u\in(0,\infty)\right\}.
\Phi_{w_4}(y) = \int_{\Re \mu = 0}h_T(\mu) K_{w_4}(y; \mu )\, (\mu) d \mu, & \Phi_{w_5}(y) = \int_{\Re \mu = 0}h_T(\mu) K_{w_4}(-y; -\mu )\, (\mu) d \mu , & \Phi_{w_6}(y_1, y_2) = \int_{\Re \mu = 0}h_T(\mu) K^{(y_1), (y_2)}_{w_6}((y_1, y_2) ; \mu )\, (\mu) d \mu.