harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\mathrm{Tr\,}}\left\{g(X^*AX+Y^*BY)+g(Y^*AY+X^*BX)\right\}&\le{\mathrm{Tr\,}}\left\{X^*g(A)X+Y^*g(B)Y)+Y^*g(A)Y+X^*g(B)X\right\}&={\mathrm{Tr\,}}\left\{(g(A) + g(B))(XX^* +YY^*)\right\}={\mathrm{Tr\,}}\left\{g(A)+g(B)\right\}
|u(t,x_1;y_1)-u(t,x_2;y_2)| \leq MC\sum_{j=0}^{N-1}\left(\frac{\rho_i((x^{(j)};y^{(j)}),(x^{(j+1)};y^{(j+1)}))}{\sqrt{t}}\right)^{\gamma}&\leq MCN^{1-\gamma}\left(\sum_{j=0}^{N-1}\frac{\rho_i((x^{(j)};y^{(j)}),(x^{(j+1)};y^{(j+1)}))}{\sqrt{t}}\right)^{\gamma}.
{\rm Coeff}_{m_{1},\cdots m_{n-1}}(\sum_{(\lambda_{1}\cdots \lambda_{n-1})\in\mathbb{N}^{n-1}}M_{1}x_1^{\lambda_1}\cdots x_{n-1}^{\lambda_{n-1}}) &={\rm Coeff}_{m_{1},\cdots m_{n-1}}(\sum_{(k_{1}\cdots k_{n-1})\in\mathbb{N}^{n-1}}N_{1}x_1^{k_1}\cdots x_{n-1}^{k_{n-1}})
\mathfrak{d}_2^{m+1}\dot{\phi}(n) = \mathfrak{d}_2^{m}\mathfrak{d}_2\dot{\phi}(n) &= (-1)^m\int_0^2\cdots\int_0^2 g^{(m)}(n+t_1+\cdots +t_m)\,dt_1\cdots dt_m,
\sum_{k=0}^{\frac{p-1}{2}}(-1)^k q^{k^2}[4k+1]\frac{(q;q^2)_k^3}{(q^2;q^2)_k^3}\equiv [p]q^{\frac{(p-1)^2}{4}}(-1)^{\frac{p-1}{2}}\pmod{[p]^3}\quad,
\mathbf{A}(t)=\begin{pmatrix}-c_N^{-1}c_{N-1}\chi(t+(N-1)\varepsilon) \ldots -c_N^{-1}(\alpha+c_0\chi(t)) \ldots -c_N^{-1}c_{-N}\chi(t-N\varepsilon) I_d 0 \ldots \ldots 0 0 \ddots \ddots \vdots \vdots \ddots \ddots \ddots \vdots 0 \ldots 0 I_d 0\end{pmatrix}.
{\vec{A}}_{3}= \alpha{\vec{B}}_{3}- \frac{1}{|{\vec{B}}_{3}|^{2}}{\vec{B}}_{3}\times \left[ ({\vec{A}}_{2}\times{\vec{B}}_{2}) + ({\vec{A}}_{1}\times{\vec{B}}_{1}) + ( \partial_{i}{\vec{B}}_{i}) \right] .
\frac{1}{n}\sum_{i=1}^n(Y_i-\hat{f}(X_i))^2 + \frac{_n(\hat{f})}{n}\leq \inf_{f\in\mathcal{F}}\left\{\frac{1}{n}\sum_{i=1}^n(Y_i-f(X_i))^2 + \frac{_n(f)}{n}+ A_f \right\},
\overline{w}_{1L}(s,2n+\gamma_{1L};L)=\delta_{n0}\overline{\Gamma}_{1L}(s)+\sum_{c=1}^{[L/2]}(-1)^{c+1}\overline{w}_{1L}(s,2(n-c)+\gamma_{1L};L)\overline{h}(s,c;L).
\mathrm{\boldmath \Psi \unboldmath}= \left( \begin{array}{c}{\Psi_{+ 1}( x )}\\{\Psi_{- 1}( x )}\\ \end{array}\right) , \Psi_{\zeta}( x ) = \left( \begin{array}{c}{\chi_{\zeta}( x )}\\{\varphi_{\zeta}( x )}\\ \end{array}\right) ,
S=\frac{1}{2}\int ee_{I}^{\alpha}e_{J}^{\beta}(F_{\alpha\beta}^{IJ}-\alpha*F_{\alpha\beta}^{IJ})\equiv\frac{1}{2}\int ee_{I}^{\alpha}e_{J}^{\beta}(F_{\alpha\beta}^{IJ}-\frac{\alpha}{2}\epsilon_{KL}^{IJ}F_{\alpha\beta}^{KL})
\mathcal{A}^\mathrm{uni}(X) := \left\{A \in \mathcal{L}(X) \;:\; \limsup_{D \rightarrow \infty}\sup_{x>1}\bar{\xi}_A(D,x) = \liminf_{D \rightarrow \infty}\inf_{x>1}\bar{\xi}_A(D,x) \right\}
\int_{\epsilon}^{\sqrt{\lambda^{2}- 4}}{d x{\frac{2}{\left( \lambda^{2}- 1 - x^{2}\right)}}\left[{\frac{1}{x^{2}}}-{\frac{2}{\left( \lambda^{2}- 1 - x^{2}\right)}}\right]}+ O ( \delta^{2}) .
\frac{d}{dt}\left (e^{-\theta_{2}t}G(t)\right ) =-\theta_{2}e^{-\theta_{2}t}G(t)+e^{-\theta_{2}t}g(t) & \leq -\theta_{2}e^{-\theta_{2}t}G(t)+e^{-\theta_{2}t}\cdot\left (\theta_{1}+\theta_{2}G(t)\right )=\theta_{1}e^{-\theta_{2}t}\mbox{a.e. on $[0,T]$}.
j_k' = \begin{cases}j_k, 1\leq k \leq n-\nu(m)-1; j_{n-\nu(m)}+p, k=n-\nu(m); 0, n-\nu(m) < k < n; \sum_{k=n-\nu(m)+1}^n (2^{n-k+1}-1)j_k - (2^{\nu(m)+1}- 1)p, k=n, \end{cases}
\sum_{i=0}^{p^k-1}(1+t)^{i p^m}= \sum_{i=0}^{p^k-1}(1+t^{p^m})^i = \frac{(1+t^{p^m})^{p^k}-1}{(1+t^{p^m}) -1}= t^{(p^k-1)p^m};
\frac{d}{dt}( p_i^T \tilde x_i) = \tilde x_i^T \dot{p_i}+ p_i^T\frac{d\tilde x_i}{dt}=- [\mathbb{E}x_i- (\Gamma m_i+\eta)]^T Q (I-\Gamma)\tilde x_i + p_i^T \tilde f .
\frac{X}{V}\sum_{u_1,\ldots,u_k}\left(\prod_{i=1}^k\varphi(u_i)\right)\sideset{}{^\prime}\sum_{\substack{d_1,\ldots,d_k e_1,\ldots,e_k u_i\mid d_i,e_i\forall i}}\frac{\lambda_{d_1,\ldots,d_k}\lambda_{e_1,\ldots,e_k}}{\prod_{i=1}^kd_ie_i}.
e(S_{ij},T_{ij}) \geq|S_{ij}|\cdot(\delta(G)-2)-\left(\sum\limits_{q=1}^{2m+1}e(S_{ij},S_q-S_{ij})+2e(S_{ij})\right) &\geq|S_{ij}|\cdot(\delta(G)-2)-(2k+1)kn &\geq \epsilon n\cdot (\delta(G)-2)-(2k+1)kn &>\frac{\epsilon n}{2}\delta(G)
|\textbf{F}_u(i+1)| \leqslant 2\cdot \Bigl[\epsilon cn^{1-s}\cdot \exp[{-i}]+c^{\frac 23}n^{\frac{2}{3}(1-s)}\Bigr]\cdot3cn^{1-s}&=\bigl(6\epsilon c^2n^{1-2s}\cdot\exp[{-i}]+ 6c^{\frac 53}n^{\frac{2}{3}-\frac{5}{3}s}\bigr)n &<\epsilon n \cdot \exp[{-i-1}],
\kappa \kappa^{-1}=\kappa^{-1}\kappa=1,\kappa J_{+}\kappa^{-1}= q J_{+},\kappa J_{-}\kappa^{-1}=q^{-1}J_{-},[J_{+},J_{-}]=\frac{\kappa-\kappa^{-1}}{q^{1/2}-q^{-1/2}}.
\nu_{T(x)}(Z(\lambda) )^2 \leq \Vert(S^{univ}_\lambda (S^{univ}_\lambda)^*)x\Vert^2 \ \Vert T^* T x \Vert^2 = \nu_{x}(Z(\lambda) )^2 \ \Vert T^* T x \Vert^2 ,
\mathcal{M}_0=\left\{w_1=\frac{1}{8}{\varphi_2^2 + \frac{1}{4}\sin^2\varphi_1-\frac{c_0}{2}\sin\varphi_1},\w_2=0,\ (\varphi_1,\varphi_2)\in \mathbb{R}^2 \right\}.
\mathbb{E}\{\mathbf{z}^{H}_{}\mathbf{z}_{}\}= \mathbf{q}^{H}\left(\tilde{\mathbf{L}}\tilde{\boldsymbol\Lambda}\tilde{\mathbf{L}}^{H}+\tilde{\mathbf{L}}_{}\tilde{\boldsymbol\Lambda}_{}\tilde{\mathbf{L}}_{}^{H}+\tilde{\mathbf{W}}\right)\mathbf{q},
c+o(1) =I[u_n]=\frac{1}{p}([u_n]_{s,p}^p+a\|u_n\|_p^p)-\frac{b}{2(p^\flat)^2}\int_{\mathbb{R}^N}(K\ast |u_n|^{p^\flat}) |u_n|^{p^\flat}dx+o(1), o(1) =I'[u_n]u_n= [u_n]_{s,p}^p+a\|u_n\|_p^p -\frac{b}{p^\flat}\int_{\mathbb{R}^N}(K\ast |u_n|^{p^\flat}) |u_n|^{p^\flat}dx+o(1).
u_i := 1\otimes\left[e_{i(k+1),\ell}\otimes e_{1,1}\right. + \left(e_{i(k+1)+1,\ell+1}+ \cdots + e_{(i+1)(k+1)-1,\ell+k}\right)\otimes 1 \left. +e_{(i+1)(k+1),\ell+k+1}\otimes e_{0,0}\right]
d s^{2}= 2 d x^{+}d x^{-}+ \left( m_{x}^{2}( x^{+}) x_{a}^{2}+ m_{y}^{2}( x^{+}) y_{l}^{2}+ m_{z}^{2}( x^{+}) z^{2}\right) ( d x^{+})^{2}+ d x_{a}^{2}+ d y_{l}^{2}+ d z^{2}.
n! \big|(-A) R(\lambda,A)^{n+1}x\big|_{L^p}=\big|\int_{0}^{\infty}t^n e^{-\lambda t}(-A) e^{tA}x dt\big|_{L^p}&\le C(p,\beta)\int_{0}^{\infty}e^{-\lambda t}t^{n-1}dt |x|_{L^p}&\le C(p,\beta) (n-1)! \lambda^{-n}|x|_{L^p}.
\omega_N \big(\tau_{\lambda,N}^t (A)\big) = \omega_n (A(t)) + \sum_{\nu=0}^{\infty}N^{-\nu}\big\{X_\nu(A(t)) + o_N\big\}+ N^{-\nu-1/2}\big\{Y_\nu(A(t)) +o_N\big\},
\mathcal{B}= * B^{A}\left[ \begin{array}{c}{v_{A}}\\{\overline{{v}}_{A}}\\ \end{array}\right] - i B^{A}\left[ \begin{array}{c}{v_{A}}\\{- \overline{{v}}_{A}}\\ \end{array}\right] + b^{\alpha}\left[ \begin{array}{c}{e_{\alpha}}\\{\overline{{e}}_{\alpha}}\\ \end{array}\right] + i * b^{\alpha}\left[ \begin{array}{c}{e_{\alpha}}\\{- \overline{{e}}_{\alpha}}\\ \end{array}\right]
\langle{\cal O}_{\mu_{1}\cdots \mu_{s}}( x ) ~{\cal O}_{\nu_{1}\cdots \nu_{s}}( 0 ) \rangle = c_{s}\frac{1}{( | x | \mu )^{2 h_{s}}}{\prod}_{\mu_{1}\cdots \mu_{s}, \nu_{1}\cdots \nu_{s}}^{( s )}\left( \frac{1}{| x |^{4}}\right) ,
| \i< \eta, \sum_i \Omega_\xi (a_i) x_i \zeta >| = | \i< \xi \otimes \eta, \theta_X^H (\sum_i a_i \otimes x_i ) \xi \otimes \zeta >|\leq \| \theta_Y^H (\sum_i a_i \otimes x_i ) \| | \xi \otimes \eta \| \| \xi \otimes \zeta \| \leq \| \theta_Y^H (\sum_i a_i \otimes x_i ) \| \| \xi \|^2 \| \eta \| \| \zeta \| ,
[ A ( s - t ) t r ( \lambda^{a_{2}}\lambda^{a_{1}}\lambda^{a_{3}}\lambda^{a_{4}}) + A ( u - t ) t r ( \lambda^{a_{2}}\lambda^{a_{3}}\lambda^{a_{1}}\lambda^{a_{4}}) + A ( s - u ) t r ( \lambda^{a_{1}}\lambda^{a_{2}}\lambda^{a_{3}}\lambda^{a_{4}}) + 3 \leftrightarrow 4 ]
\begin{aligned}2 i k z b(k)= \Psi^+_{11}(0 ; z)\left(z \Psi^-_{21}(0;z)-\widehat{\Psi}^-_{21}(0)\right)-\Psi^-_{11}(0;z)\left(z \Psi^+_{21}(0; z)-\widehat{\Psi}^+_{21}(0)\right) &+\widehat{\Psi}^-_{21}(0)\left(\Psi^+_{11}(0;z)-e^{ic_+(0)}\right)-\widehat{\Psi}^+_{21}(0)\left(\Psi^-_{11}(0;z)-e^{ic_-(0)}\right)\end{aligned}
\begin{aligned}0=S(1-\dfrac32S)+2\bar H^2S-\dfrac12\bar H^4-\bar H^2(\bar \lambda_1^2+\bar\lambda_2^2) &=-(S-\bar H^2)^2-\dfrac12\bar H^2(\bar \lambda_1-\bar\lambda_2)^2.\end{aligned}
\begin{aligned}\| (\bold{I}_{j-1}-\bold{\Pi}_{j-1}-\bold{\Xi}_{j-1}(\bold{I}_{j-1}-\bold{\Pi}_{j-1}^{T})^{-1}\bold{\Gamma}_{j-1})^{-1}_{(m)}\|_{1}& \| [\bold{S}_{j}^{-1}]^{(1,1)}_{(m)}\|_{1}\le \| [\bold{S}_{j}^{-1}]^{(m)}\|_{1}< \infty. \end{aligned}
p(x_m,x_n) \leq p(x_m,x_{m-1})+p(x_{m-1},x_{m-2})+\ldots+p(x_{n+1},x_n)-\sum_{i=1}^{m-n-1}p(x_{m-i},x_{m-i}) &\leq p(x_m,x_{m-1})+p(x_{m-1},x_{m-2})+\ldots+p(x_{n+1},x_n) &\leq(k^{m-1}+k^{m-2}+\ldots k^n)p(x_1,x_0) &=\frac{k^n}{1-k}p(x_1,x_0)
B^+_{\nu}= (\alpha+\beta+2+2\nu) \sin2\phi \partial_{\phi}+ (\alpha+\beta+1+2\nu) (\alpha+\beta+2 +2\nu) \cos2\phi - \alpha^2 + \beta^2, B^-_{\nu}= - (\alpha+\beta+2\nu) \sin2\phi \partial_{\phi}+ (\alpha+\beta+1+2\nu) (\alpha+\beta+2\nu) \cos2\phi - \alpha^2 + \beta^2,
X_{\pm}v_{\pm}= \sum_{i=0}^n \left(\kappa\sqrt{\pm 1}\, (q^{n/2-i}\mp q^{-n}q^{i-n/2}) \lambda_i v_i \, - \, s^{-1}(\pm 1) [i] \l_i v_{i-1}\right) = \sum_{i=0}^n \left( - q^{-n/2}\sqrt{\pm 1}\, [n-i] \lambda_i \, \mp \, s^{-1}[i+1] \l_{i+1}\right) v_i = 0
M_{U \ \scriptstyle \mathrm{m i n}}^{E_{8}- E_{6}}\approx 5 . 4 9 \times 1 0^{1 7}\times g_{U}^{E_{8}- E_{6}}\times \frac{1}{\sqrt{1 + 0 . 0 8 \times \left( g_{U}^{E_{8}- E_{6}}\right)^{2}}}\, \mathrm{G e V}\, .
\partial_tv^n(t,x)=-\frac{n\alpha}{nt+1}v(t,x)+\frac{n^2(x-a)^2}{2(nt+1)^2}v(t,x), partial_xv^n(t,x)=-\frac{n(x-a)}{nt+1}v(t,x), &\partial_{xx}^2v^n(t,x)=-\frac{n}{nt+1}v(t,x)+\frac{n^2(x-a)^2}{(nt+1)^2}v(t,x).
(S,\underline{S})^+ := \overline{((a_1,a_2,\ldots,a_k),(b_1,b_2,\ldots,b_{n-k}))}, (S,\underline{S})^- := \overline{((a_2,a_1,\ldots,a_k),(b_1,b_2,\ldots,b_{n-k}))}.
u_{\Omega,Z,A}^{(\sigma)}=\sum_{\{T,\lambda\}}\int_{-\infty}^\infty\! \frac{dk}{\sqrt{2\pi\omega_k}}\left(\frac{\omega_k+k}{\omega_k-k}\right)^{\!-\left(i\sigma\Omega+\sum_n\lambda_n\right)/2}\kern-4pt\left(\alpha_{\Omega,Z,\{T,\lambda\}}^{(\sigma)}U_k+ \beta_{\Omega,Z,\{T,\lambda\}}^{(\sigma)}U_k^*\right)\Phi_{\{T,\lambda\}}\cdot\phi_A ,
{}r \neq \frac{n^2}{4}: d_{N}(n, r) = r^{\frac{N}{2}}\, \frac{\xi^{N+1}- \xi^{-N-1}}{\xi-\xi^{-1}}\,, \xi+\xi^{-1}= \frac{n}{\sqrt{r}}\,,{}r = \frac{n^2}{4}: d_{N}(n, r) = r^{\frac{N}{2}}\, (N+1) \,.
(0,1;{\mathbf 0}) + (1,0;{\mathbf 0}) = \deg^\vee ~(= 0 \in \overline{N}), &(1,0;e_1) + (1,0;e_2) + (0,1;e_3) = \deg^\vee + (0,1;e_0) ~(= 0 \in \overline{N}).
w(m,m')= \frac{1}{\sqrt{2\pi N |\Phi_{m}^{(2)}(t_{+})|}}\frac{1}{\sqrt{2\pi N |\Phi_{m'}^{(2)}(s_{+})|}}{\rm e}^{-N(\Phi_{m}(t_{+})+\Phi_{m'}(s_{+}))}{\rm det}M(t_{+},s_{+}) ,
\sum_{i=1}^m \omega_i \langle x^k - P_{C_i}(x^k - \alpha_k g^k), y - P_{C_i}(x^k -\alpha_k g^k)\rangle \leq \sum_{i=1}^m \omega_i \langle \alpha_k g^k, y - P_{C_i}(x^k -\alpha_k g^k)\rangle.
g_i(t) = g_i(t, r) := \frac{\partial}{\partial t_i}g(t, r) = \log \Big| \frac{(t_i + (-1)^i r_1)(t_i + (-1)^i r_2)(t_i + (-1)^i r_3)}{\Upsilon_i t_i^2(t_1+t_2)}\Big|.
\mathcal{T}=\left\{t_{a,b}:=\left(\begin{array}{ccccc}a 0 0 \dots 0 0 b 0 \dots 0 0 0 ab \dots 0 \vdots \vdots \vdots \ddots \vdots 0 0 0 \dots a^{n-2}b\end{array}\right):a,b\in \mathbb{C}^{*}\right\}\cong \mathbb{C}^{*}\times \mathbb{C}^{*}.
f^{\prime}(x_{0})=\frac{f(x_{0}+h)-f(x_{0})}{h}-\frac{f^{(2)}(x_{0})}{2!}h-\frac{f^{(3)}(x_{0})}{3!}h^{2}-\frac{f^{(4)}(x_{0})}{4!}h^{3}+\cdot\cdot\cdot
B_e\, \left(A_dY^d+\sum_{j=0}^{d-1}A_jY^j\right) = \left(B_eY^e+\sum_{j=0}^{e-1}B_jY^j\right) \left(C_\ell Y^\ell+\sum_{j=0}^{\ell -1}C_jY^j\right).
f(x_{1},...,x_{k})=a(a+1)\cdot\cdot\cdot(a+k-1)(\prod_{i=1}^{k}\theta_{i})^{-1}(\sum_{i=1}^{k}\frac{x_{i}}{\theta_{i}}-k+1)^{-(a+k)},x_{i}>\theta_{i}>0,a>0,(1)
\begin{cases}\underline u(|x|, T+t)- q_{i_k}\geq -\sigma M_0 e^{-T/2}-\sigma e^{-\delta^* t}, \overline u(|x|, T+t)- q_{i_k}\leq \sigma M_0 e^{-T/2}+\sigma e^{-\delta^* t}\end{cases}
\{\hat\theta_N,\hat\sigma_N,\hat\mu_N\}=\textrm{argmin}\, f(\hat\theta,\hat\sigma,\hat\mu)= \sum_{s=1}^{q_N}\big[g_{s}(\hat\theta,\hat\sigma,\hat\mu)\big]^2 \textrm{where}g_{s}(\hat\theta,\hat\sigma,\hat\mu) = \hat\mu^2+ \frac{\hat\sigma^2}{s}\Big[1 + 2 \, \sum \limits_{k=1}^{s-1}\big(1 - \frac{k}{s}\big) \, \hat\rho(k;\hat\theta)\Big]- \bar{m}_s^2,
T_{\ \mu \rho \sigma}^{\mu}= - p^{2}T_{1}\Biggl [ \left( p_{\rho}p_{\sigma}- p^{2}g_{\rho \sigma}\right) \mp \frac{1}{4}\left( \varepsilon_{\rho \lambda}p^{\lambda}p_{\sigma}+ \varepsilon_{\sigma \lambda}p^{\lambda}p_{\rho}\right) \Biggl ] \, .
g_+(x) + g_-(x) - x - \ell = 0, x \in \mathcal{S}, 2g(x) - x - \ell < 0, x \in (4,\infty), g_+(x) - g_-(x) = 2\pi i \int^{4}_x \rho(s) ds, x \in \mathcal{S}, g_+(x) - g_-(x) = 2\pi i , x \in (-\infty,0).
G_{D}({\bf r''},{\bf r'}; E) =\left( r'' r' \right)^{-(D-1)/2}\,\sum_{l= 0}^{\infty}\sum_{m=1}^{d_{l}}Y_{l m}({\bf \Omega''})Y_{l m}^{*}({\bf \Omega'})G_{l +\nu}(r'',r';E)\;
\limsup_{t\to0}\int_{B_r(x_0)}|u(x,t)-u_0(x)|dx \leq \limsup_{t\to0}\left\{\int_{B_r(x_0)}|u-u_{R_i}|(x,t)dx+ \int_{B_r(x_0)}|u_{R_i}(x,t)-u_0(x)|dx \right\}\leq \limsup_{t\to0}Cr^{n-\frac{2}{1-m}}\,t^{\frac{1}{1-m}}=0
\|\sum_{T_1,T_2}\phi_{T_1}\phi_{T_2}\|_{1}\leq \|\sum_{T_1}\phi_{T_1}\|_2\|\sum_{T_2}\phi_{T_2}\|_{2}\leq \Big(\frac{R'^2}{\delta^2}\#T_1\Big)^{1/2}\Big(\frac{R'^2}{\delta}\# T_2\Big)^{1/2}= \delta^{-3/2}R'^2(\# T_1\, \# T_2)^{1/2}.
\tilde c_{jk}= \begin{pmatrix}\frac{dz_j/dz_k}{|dz_j/dz_k|}0 0 \frac{(dz_j/dz_k)^2}{|dz_j/dz_k|^2}\end{pmatrix}\begin{pmatrix}w_j^{p_j}w_k^{-p_k}0 0 w_j^{q_j}w_k^{-q_k}\end{pmatrix}.
B_{D}(\mathbf{n})=h(\tilde{Z},D\tilde{Z})=\sum_{i=0}^{n}d_{i}b_{i}^{2}z_{1}^{n-i}z_{2}^{i}\overline{z_{1}}^{n-i}\overline{z_{2}}=\sum_{i=0}^{n}d_{i}b_{i}^{2}|z_{1}|^{2(n-i)}|z_{2}|^{2i}=0
\rho_{i+\frac{1}{2}}^{*}=\rho_{i+\frac{1}{2}}^{*}(\rho_{i+\frac{1}{2}}^{L},\rho_{i+\frac{1}{2}}^{R}),\rho_{i-\frac{1}{2}}^{*}=\rho_{i-\frac{1}{2}}^{*}(\rho_{i-\frac{1}{2}}^{L},\rho_{i-\frac{1}{2}}^{R}),
\mathcal{F}(e^{i t \Lambda}C_{\iota_1, \iota_2, \iota_3}(V^{\iota_{1}}, V^{\iota_{2}}, V^{\iota_{3}})) (\xi):=iI^{\iota_1,\iota_2,\iota_3}( t,\xi) = i\sum_{k_{1}, k_{2}, k_{3}\in \mathbb{Z}}I^{\iota_1,\iota_2,\iota_3}_{k_{1}, k_{2}, k_{3}}( t,\xi) ,
\bar{\mu}^{\dot{A}}= \left( x^{B \dot{A}}- i \Theta^{B}\bar{\Theta}^{\dot{A}}\right) \lambda_{B}+ 2 \bar{z}^{\dot{A}\dot{B}}\bar{\lambda}_{\dot{B}}- i \bar{\Theta}^{\dot{A}}\bar{\Theta}^{\dot{B}}\bar{\lambda}_{\dot{B}}
\sum_{\substack{f_v(x, f) = x_{v}v \in C_1}}\delta_{1}(f_{C_1}) = c_1 \cdot \lambda_{1}(x_{A_1}) \sum_{\substack{f_v(x, f) = x_{v}v \in C_2}}\delta_{2}(f_{C_2}) = c_2 \cdot \lambda_{2}(x_{A_2}).
\begin{aligned}\|P_{\mu}(u_{\mu_1}u_{\mu_2}g(u_{<\mu_2})\|_{L^2}\lesssim \ \|u_{\mu_1}\|_{L^6}\|u_{\mu_2}\|_{L^6}\|P_{\approx \mu}g''(u_{<\mu_2})\|_{L^6}\lesssim \C^3 \epsilon^3 c_{\mu_1}c_{\mu_2}^2 \mu_1^{-s}\mu_2^{-s}\left(\frac{\mu_2}{\mu}\right)^N .\end{aligned}
\eta^{[i,j]}_{k}= \left\|\textrm{Proj}_{\bot \mathbf{Q}_k}\left( \mathbf{H}_{k}^{[i,j]}\mathbf{w}^{[i,j]}\right)\right\|^2 &= \left\|\mathbf{U}_k^{H}\mathbf{H}_{k}^{[i,j]}\mathbf{w}^{[i,j]}\right\|^2,
\lambda_{4}= \lambda_{v}( u ) \lambda_{w}( v ) \lambda_{v}( v ) \lambda_{u}( u ) , \quad \lambda_{5}= \lambda_{v}( v ) \lambda_{w}( w ) \lambda_{v}( w ) \lambda_{u}( v ) , \quad \lambda_{6}= \lambda_{v}( w ) \lambda_{w}( u ) \lambda_{v}( u ) \lambda_{u}( w )
\int_{\sigma-iT}^{\sigma+iT}\left(- s \int_{T'}^{\infty}\frac{\{v\}}{v^{s+1}}dv\right) \frac{x^s}{s}ds = -\int_{T'}^{\infty}\frac{\{v\}}{v}\left( \int_{\sigma-iT}^{\sigma+iT}\left(\frac{x}{v}\right)^s ds\right) dv.
S [ \phi , g ] = S_{c}[ \phi_{c}, g ] + \left. \frac{\delta^{2}S [ \phi , g ]}{\delta \phi^{2}}\right|_{\phi = \phi_{c}}\frac{\tilde{\phi}^{2}}{2}+ \mathrm{h i g h e r o r d e r t e r m s i n}\tilde{\phi}, \nonumber
\big\|{\widetilde{V}}^{(<)}\big\|_{\infty}\ =\ (1-{\varepsilon})\eta (\mathfrak{e})\ <\ \eta (\mathfrak{e})-\frac{1}{4}\,C_{\ref{lem-4.3}}(d,\mathfrak{e})\,\eta (\mathfrak{e})^{2}r_{0}^{-1/2},
\mathbb{E}[\eta_{m}\eta^{\mathcal{H}}_{m}]=p(\mathbf{g}^{(m)})^{\mathcal{H}}&\times \bigg(\mathbf{h}\mathbf{h}^{\mathcal{H}}\left(\kappa^{2}_{T}(\omega+1)+\omega\right)+\left(\kappa^{2}_{R}m+\frac{N_{0}}{p}\right)\mathbf{I}_{n}\bigg)\mathbf{g}^{(m)},
\langle\langle\Delta''u,\,u\rangle\rangle = \langle\langle\Delta''u_{h^\perp},\,u_h+u_{h^\perp}\rangle\rangle = \langle\langle\Delta''u_{h^\perp},\,u_{h^\perp}\rangle\rangle\geq\delta''_k\,||u_{h^\perp}||^2
\begin{aligned}f-Qf =\lim_{\mbox{\scriptsize supp}\psi_a\downarrow\mathcal{T}_a}\lim_{t\to\infty}e^{itH}(\psi_a(H^a)-P^a)P_a^{{\varepsilon}_a}(t)e^{-itH}f. &=\lim_{\mbox{\scriptsize supp}\psi_a\downarrow\mathcal{T}_a}\lim_{t\to\infty}e^{itH}\varphi(H^a)(\psi_a(H^a)-P^a)P_a^{{\varepsilon}_a}(t)e^{-itH}f &=\sum_{j=1}^M\Omega_jf.\end{aligned}
\vec{\psi}(n)=\begin{pmatrix}cn 1\end{pmatrix}+\sum_{s=n_0+1}^n\Phi_1(s)R(s-1)\vec{\psi}(s-1)-\sum_{s=n+1}^\infty\Phi_2(n)R(s-1)\vec{\psi}(s-1).
\varphi_{n}(\lambda)-\varphi_{n}(\lambda b) \underset{n\rightarrow\infty}{=}\lambda^{2}(b^{2}-1)\frac{(n+1)^{3}-n^{3}}{4n^{3}(n+1)^{3}}+o_{\lambda,b}\left(\frac{1}{n^{4}}\right) &\underset{n\rightarrow\infty}{=}\frac{3\lambda^{2}(b^{2}-1)}{4n^{4}}+o_{\lambda,b}\left(\frac{1}{n^{4}}\right).
(\rho_0c_0+\alpha_1) u_0^{\mathcal{R}}+i\omega\sqrt{d_1}u_1^{\mathcal{R}}=x(\rho_0u_0^{\mathcal{R}}+\sqrt{d_1}u_1^{\mathcal{R}}), &-i\omega\sqrt{d_{k}}u_{k-1}^{\mathcal{R}}-f_{k}^{-1}s_{k}u_{k}^{\mathcal{R}}+i\omega\sqrt{d_{k+1}}u_{k+1}^{\mathcal{R}}=x(\sqrt{d_{k}}u_{k-1}^{\mathcal{R}}+\rho_ku_{k}^{\mathcal{R}}+\sqrt{d_{k+1}}u_{k+1}^{\mathcal{R}}),
\|x\|_{M \otimes_{R}N}' = \inf \{\sum \|a_i\|'_M\|b_i\|'_N \ \ | \ \ \sum a_i \otimes b_i = x \}\le \inf \{\sum C_M \|a_i\|_M C_N\|b_i\|_N \ \ | \ \ \sum a_i \otimes b_i = x \}= C_M C_N \inf \{\sum \|a_i\|_M \|b_i\|_N \ \ | \ \ \sum a_i \otimes b_i = x \}= C_N C_M \|x\|_{M \otimes N}
\sum_{\rho}\frac{x^{\rho/2}\Gamma(1-\frac{\rho}{2})}{\zeta'(\rho)}-\frac{x}{(2\pi)^2}\int_{0}^{\infty}e^{-xt/(2\pi)^2}\left(\sum_{n\ge1}\frac{\mu(n)}{n}\cos(\frac{\sqrt{t}}{n})\right)dt=O(x^{\frac{1}{4}+\epsilon}).
{\cal H}= - K \cdot \left({\cal K}\frac{\delta}{\delta K}\right) - K \cdot \left( \frac{\delta}{\delta K}\vee \frac{\delta}{\delta K}\right) - \frac{1}{N^{2}}K \cdot \left( K \cdot \left( \wedge \frac{\delta}{\delta K}\right) \right) - K \cdot T .
D_{e_i}\hat{\psi}(\mathbf{x}) = e_i(\hat{\psi})(\mathbf{x}) - ((e_i(\hat{\psi}) \cdot \mathbf{x})\mathbf{x}= e_i(\hat{\psi})(\mathbf{x})+(\hat{\psi}(\mathbf{x})\cdot e_i(\mathbf{x})) \mathbf{x}
P^{\vec{h}}_{\Lambda,\eta,\beta,H}(\sigma^{\eta}_x=sgn(h_x))\geq a(\beta,H,|h_x|):=\frac{e^{-2d\beta+H|h_x|}}{e^{-2d\beta+H|h_x|}+e^{2d\beta-H|h_x|}},
{\rm (T3a)}\hskip 10pt{\lambda_r+i\choose i}y(s,t)_j\cr = \sum_{0\leq h< i}{\lambda_s-i+j\choose j-h}{\lambda_r+i\choose h}y(r,s)_{i-h}+{\lambda_s-i+j\choose j-i}y(r,t)_i,\cr \phantom{}\hskip 200pt \hbox{for}1\leq i\leq j\leq \lambda_t.
\begin{aligned}\int_{Q_T}\!\!\!\left(s\Theta (v_x)^2 + s^3 \Theta^3\left(\frac{x-x_0}{a}\right)^2v^2\right)e^{2s\varphi}dxdt\le C\left(\int_{Q_T}h^{2}\frac{e^{2s\varphi}}{a}dxdt +\int_0^T \int_\omega v^2e^{2s\varphi}dxdt\right).\end{aligned}
\int_{0}^{\infty}\frac{1}{4}\left( C - e^{-t/2}\Phi(e^{-2t},2,1/4) \right) e^{izt}\, dt =- \frac{1}{2z^2}\left(\frac{\Gamma'}{\Gamma}\left(\frac{s}{2}\right)-\frac{\Gamma'}{\Gamma}\left(\frac{1}{4}\right)\right)
(1-d q^{n-1}) v_n+ (1-q^n) v_{n+1}- \frac{\delta}{\beta}(1-e q^{n-1}) v_n -\frac{\delta}{\beta}(1-edq^{n-2})v_{n-1}= \left( \frac{1}{1-\rho_b}-\frac{\delta}{\beta\rho_b}\right) v_n,
\widetilde N(\widetilde Y,c) = \begin{pmatrix}\widetilde \beta d_1(c) c \widetilde \alpha + d_2(c) \widetilde \alpha^2 +d_3(c) \left( \widetilde A^2 + \widetilde B^2\right) -\widetilde B\left( 1 + c\omega_1(c) + m(c)\widetilde \alpha\right) \widetilde A\left( 1+ c\omega_1(c) + m(c)\widetilde \alpha\right)\end{pmatrix}.
\mathcal{T}_4(\varepsilon ;p;{\mathbf F})^p \leqslant(2C_{13})^p \varepsilon^{p/2}\left( (c_*/2)^{-1/4}\Gamma (1/4)\right)^{1+p/p'}\Vert \mathbf{F}\Vert^p_{\mathfrak{H}_p(T)}.
\tilde{F}( P_{-}, x^{+}) ={\frac{1}{2 L}}\langle{\cal O}( P_{-}, x^{+}){\cal O}( - P_{-}, 0 ) \rangle \ = \sum_{i}{\frac{1}{2 L}}\langle 0 |{\cal O}( P_{-}) | i \rangle e^{- i P_{+}^{i}x^{+}}\langle i |{\cal O}( - P_{-}, 0 ) | 0 \rangle \ .
\mathbb{E}\left[\varepsilon\right]={\mathbb{E}\left[\mathbf{s}^{\left(\right)^H}\mathbf{s}^{\left(\right)}\right]}-{\mathbb{E}\left[\mathbf{s}^{\left(\right)^H}\mathbf{y}'\right]}-{\mathbb{E}\left[\mathbf{y'}^H\mathbf{s}^{\left(\right)}\right]}+{\mathbb{E}\left[\mathbf{y'}^H\mathbf{y}'\right]}.
\pi_1(e_i) = \ \frac{a_i}{\sqrt{1+{\lambda_i}^2}}\; , \pi_2(e_i)= \ \frac{\lambda_ia_{n+i}}{\sqrt{1+{\lambda_i}^2}}\,\, ,\, 1 \le i \le r \,\, , \ \pi_1(e_i) = \a_i \;, \pi_2(e_i)= \ 0\,\, \,\, r+1 \le i \le n.
\vec{\phi}=(\underbrace{\frac{W+\bar{W}}{|W|^{2}+1}}_{\phi_1},\underbrace{i\frac{-W+\bar{W}}{|W|^{2}+1}}_{\phi_2},\underbrace{\frac{|W|^{2}-1}{|W|^{2}+1})}_{\phi_3},
\mathrm{d e t}( \delta_{\mu \nu}+ 2 \pi \alpha^{\prime}\tilde{F}_{\mu \nu}) = 1 + \frac{1}{2}( 2 \pi \alpha^{\prime})^{2}\tilde{F}_{\mu \nu}^{2}, \hspace{1 c m}\mathrm{f o r}\hspace{0 . 3 c m}d = 2 , 3
W^C_{82}= 1+( 3280 + 2 c )y^{14}+( 36244 + 128 b - 2 c )y^{16}&+( 514345 - 8192 a - 896 b - 26 c )y^{18}+ \cdots, W^S_{82}= a y^5+(- 18 a + b) y^9+( 153 a - 16 b - c) y^{13}&+( 304384 - 816 a + 120 b + 14 c) y^{17}+ \cdots,
e_N(x) :=u(x)-\pi_N^{\lambda}u(x)=\sum_{n=N+1}^\infty \hat u_n R_n^\lambda(x)=S(t)\sum_{n=N+1}^\infty\widehat{\breve U}_n C_n^\lambda(t) & =S(t)\big(\breve U(t)-\varPi_N^\lambda \breve U(t)\big):=S(t) \breve e_N(t).
\aligned S^3_{\alpha p}=a_3S^2_{\alpha p}-b_3T^2_{\alpha p},T^3_{\alpha p}=b_3S^2_{\alpha p}+a_3T^2_{\alpha p}, &S^4_{\alpha p}=a_4S^2_{\alpha p}-b_4T^2_{\alpha p},T^4_{\alpha p}=b_4S^2_{\alpha p}+a_4T^2_{\alpha p}, &S^5_{\alpha p}=a_5S^2_{\alpha p}-b_5T^2_{\alpha p},T^5_{\alpha p}=b_5S^2_{\alpha p}+a_5T^2_{\alpha p}.\endaligned
{R_{\mu \nu \rho}}^{\sigma}= \partial_{\nu}{\Gamma^{\sigma}}_{\mu \rho}- \partial_{\mu}{\Gamma^{\sigma}}_{\nu \rho}+{\Gamma^{\alpha}}_{\mu \rho}{\Gamma^{\sigma}}_{\alpha \nu}-{\Gamma^{\alpha}}_{\nu \rho}{\Gamma^{\sigma}}_{\alpha \mu},
r^2 q^S = \ \frac{1}{2i}\{r^2 p_S,X\}+( q - r^{-1}f(r)(r-r_{ps})) (r^2 p_S) = \ \alpha_S^2(r) \tau^2 + \beta_S^2(r) \xi^2 + \tilde q(r)(r^2 p_S)
QX=\frac{S(\xi ,\xi )}{2}\phi^{2}X-2\alpha \phi hX-\phi (\nabla_{\xi}h)X-S(\xi ,\xi )X+S(\phi^{2}X,\xi )\xi +\eta (X)S(\xi ,\xi )\xi +\eta(X)Q\xi +\frac{r}{2}\phi^{2}X.
\mathbb{E}\big[ |A_0(n)|^2\big] = \sum_{\substack{|\lambda|=n \lambda_0 \leqslant y_0}}\prod_k \frac{1}{k^{m_k}m_k!}\leqslant \frac{1}{r^n}\exp\bigg(\sum_{k\leqslant y_0}\frac{r^{k}}{k}\bigg).
A_{n,j,k}(t,z) \coloneqq(1-z)^{n}P_{n,j,k}\left(\frac{1+z}{1-z},t\right) =(1-z)^{n}t^{j+1}\left(\frac{1+z}{1-z}+t\right)^{k-j}\left(1+\frac{1+z}{1-z}t\right)^{n-j-k-1}\left(1+\frac{1+z}{1-z}\right)^{2j+1}=2^{2j+1}t^{j+1}(1+t+z(1-t))^{k-j}(1+t-z(1-t))^{n-j-k-1},
\mathcal{S'}_{\chi}(m):=\sum_{\substack{2j+3l+6k+3i=m 0\leq j,l \leq n-h-2 0\leq k,i}}(-1)^{i}q^{\frac{j+k+l}{2}}\Lambda_{j}(\chi^{2})\Lambda_{l}(\chi^{3})Sym^{k}(\chi^{6})
R_0 =\left\{(s,t;m(e))=(t,s;m(e))\mid e=\{s,t\}\in E\right\}&\mathrel{\hphantom{=}}\cup\{\alpha v = (\alpha(v))\alpha\mid v\in V\}\cup\{\varepsilon_i v = (\varepsilon_i(v))\varepsilon_i\mid v\in V, i=1,2\}\cup\{\iota v = v^{-1}\iota\mid v\in V\}
E_{k,\chi_{-4}}^{\infty}(\tau) =1-\frac{2k}{B_{k,4}}\sum_{n=1}^{\infty}\sigma_{k-1,\chi_{-4}}^{\infty}(n)q^{n}\intertext{and}E_{k,\chi_{-4}}^{0}(\tau) =\delta_{k,1}-\frac{2k}{B_{k,4}}\sum_{n=1}^{\infty}\sigma_{k-1,\chi_{-4}}^{0}(n)q^{n},
s_{l}\ \mapsto \ \epsilon^{H U_{l}}( h ) \cdot s_{l}\quad \quad \Rightarrow \quad \quad c_{l}\ \mapsto \ \epsilon_{L}^{U_{l}G}\circ \epsilon^{H U_{l}}\bigl ( h \bigr ) \cdot c_{l}\cdot \epsilon_{R}^{U_{l}G}\circ \epsilon^{H U_{l}}\bigl ( h^{- 1}\bigr ) \ \ .
I_n^\flat (0,t;v) = \int_{0}^t v(t_n) \otimes I_{n-1}^\flat (0,t_n;v) d t_n =- \int_{t}^1 v(t_n) \otimes I_{n-1}^\flat (0,t_n;v) d t_n =(-1)^{n}\int_{t}^1 v(t_n) \otimes I_{n-1}(t_n,1 -t_n;v) d t_n =(-1)^{n}I_n (t,1-t;v).
T_{0}( z^{\prime}) t ( z ) = \frac{\frac{c_{1}}{2}\omega ( z ) + \frac{c_{2}}{2}\Omega ( z )}{( z^{\prime}- z )^{4}}+ \frac{T_{0}( z ) + 2 t ( z )}{( z^{\prime}- z )^{2}}+ \frac{\partial_{z}t ( z )}{z^{\prime}- z}+ \cdots \: \: \: .
\langle{}^{f\!}\chi(bb'),c\rangle = (\chi f)(bb'\otimes c) = \chi\sum f(b,c_1)\,f(b',c_2) &= \sum \chi f(b,c_1)\, \chi f(b',c_2) = \langle{}^{f\!}\chi(b),c_1\rangle \langle{}^{f\!}\chi(b'),c_2\rangle &= \langle{}^{f\!}\chi(b) \cdot{}^{f\!}\chi (b'), c\rangle .
\tilde{Z}^{(q,r)}(x,\theta ) := \frac{rZ^{(q)}(x,\theta )+(q-\psi(\theta ))Z^{(q)}(x,\Phi(q+r))}{\Phi(q+r) - \theta}, \theta \geq 0, \tilde{Z}^{(q,r)}(x) :=\tilde{Z}^{(q,r)}(x, 0) = \frac{rZ^{(q)}(x)+qZ^{(q)}(x,\Phi(q+r))}{\Phi(q+r)},