harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
j_i=-s\sum\limits_{n=0}^{N}(\partial_{i}u_{n}-\imath A_n^iu_n,-\imath u_n)\chi_{\Omega}(x_1,x_2)dx_{1}dx_{2}\delta_{ns}(x_3) \quad &j_3=s\sum\limits_{n=0}^{N-1}\dfrac{1}{\lambda^{2}s^2}(u_{n+1}-u_{n}\Upsilon_{n}^{n+1},\imath u_{n}\Upsilon_{n}^{n+1})\chi_{\Omega}(x_1,x_2)\chi_{[ns,(n+1)s]}(x_3).
\tilde{C}_{N}(z_{1},z_{2}) := \frac{1}{d_{N}^{2}}\frac{1}{(2\pi i)^{2}}\oint_{\mathcal{S}_{z_{1}}}d\omega_1 \oint_{\mathcal{S}_{z_{2}}}d\omega_2 \frac{s(\omega_1)s(\omega_2)}{(z_{1}-\omega_1)^{2}(z_{2}-\omega_2)^{2}}\,K_{N}(\omega_1,\omega_2)
\sigma^k(x) := \begin{cases}\sigma(x) x \in D_k \sigma(l_k) x \leq l_k \sigma(u_k) x \geq u_k \end{cases}b^k(x) := \begin{cases}b(x) x \in D_k b(l_k) x \leq l_k b(u_k) x \geq u_k. \end{cases}
\sum_{\gamma\in\mathcal{C}_{[n,r,s]}}w(\gamma)=a_{-s}^{(r+s)}\sum_{k=r}^{n-s-1}\sum_{\ell=k+s}^{n-1}A_{[k-1,r-1]}^{(1)}B_{[\ell-k-1,s-1]}^{(1)}W_{[n-\ell-1,0]}.
\psi'(x) =(\log(e^2/x))^{-1/2}+ x \cdot (-1/2) \cdot(\log(e^2/x))^{-3/2}\cdot (x/e^2) \cdot (-e^2/x^2) &= (\log(e^2/x))^{-1/2}(1+ \frac12(\log(e^2/x))^{-1}) =h(x)(1+h(x)^2/2),
P_{22,1}= \partial_1 P_{22}- 2\Gamma_{12}^2 P_{22}&= \left( - \frac{1}{n-1}f f'' \right)' - 2 \left(- \frac{1}{n-1}f f'' \right)\left( \frac{f'}{f}\right) &= \frac{1}{n-1}\left( f' f'' - f f''' \right).
A(q^{n-1},t)A(q^{n-2},t)\cdots A(1,t)\Big|_{q=\omega}= \left(A(\omega^{\frac{n}{(g',n)}-1},t)A(\omega^{\frac{n}{(g',n)}-2},t) \cdots A(1,t)\right)^{(g',n)}.
\begin{aligned}\tilde{Q}^y_t = \frac{\widehat{Q}^y_t(\dot{\varphi}_y)}{1+ \varphi_\sigma(y) \big( A_t - q_{\xi, t}(y) - \varphi_\xi(y) \big)}+ O \left( (\ln_2 t)^{-2|y| -1}\right)\end{aligned}
\smashoperator{\int_{\mathbb{S}^{n-1}}}|x_1|^{2\mu_1}\mathrm{d}x_1\cdots |x_n|^{2\mu_n}\mathrm{d}x_{n}\;\big[\Psi_{\mathbf{\mathbf{j}'}}^{s'}(x_1,\ldots, x_{n})\big]^{*}\cdot \Psi_{\mathbf{j}}^{s}(x_1,\ldots, x_{n})=\delta_{\mathbf{j}\mathbf{j}'}\delta_{ss'},
H_{0}= 2 \pi \left\{\frac{1}{4 r}( f_{A}^{\prime 2}+ f_{B}^{\prime 2}) + \frac{\eta^{2}}{2}\left[ r ( H^{\prime 2}+ K^{\prime 2}) + \frac{1}{r}( K f_{A}- H f_{B})^{2}\right] + \frac{\lambda \eta^{4}}{4}r ( 1 - H^{2}- K^{2})^{2}\right\}
F_l=\begin{pmatrix}\alpha_{-1}^{l-1}\alpha_{0}^{l-1}\dots \alpha_{l-2}^{l-1}\alpha_{-1}^{l-2}\alpha_{0}^{l-2}\dots \alpha_{l-2}^{l-2}\vdots \vdots \ddots \vdots \alpha_{-1}^{0}\alpha_{0}^{0}\dots \alpha_{l-2}^{0}\end{pmatrix},
\langle \bar{\psi}\psi \rangle_{T \mu}~ \simeq ~ - ~ \frac{2 \vert e H \vert}{\pi^{2}}T \sum_{l}\int_{0}^{\sqrt{\vert e H \vert}}d p_{3}\frac{m ( T , \mu )}{[ ( 2 l + 1 ) \pi T + i \mu ]^{2}+ p_{3}^{2}+ m^{2}( T , \mu )}_{,}
\sigma_3(s) = \bigg(\sum_{j=1}^{2}R_j +(1-e^{-is})R_3, 2\big(\frac{\pi}{2}-1\big)\big(\sum_{j=1}^{2}\|R_j\|^2\big)+2(s-\sin s) \|R_3\|^2 &-2 \langle R_1, R_2 \rangle -2 \sin s \langle R_1+R_{2}, R_3 \rangle\bigg)
\Phi_{A^*}\Big\vert_{\partial A^*}\!\! - \sup\,\Phi_{A}\Big\vert_{\partial B_r}\ge\omega_n \left( R_A^2-r^2 + \frac{r^n-R_A^n}{(\sqrt2 r)^{n-2}}\right) &=\omega_n R_A^2 \bigl(-2+n2^{1-\frac n2}\bigr)\bigl(\tfrac{r}{R_A}-1\bigr)+ O\bigl(\tfrac{r}{R_A}-1\bigr)^2
\lim_{\mathsf{SNR}\rightarrow \infty}\left(N \cdot\alpha^{2(S-1)}\right) \cdot \left( c_0 (\epsilon_I)^{\tau}{\mathsf{SNR}}^{-\tau}+ \Omega\left({\mathsf{SNR}}^{-(\tau-1)}\right)\right) \cdot\frac{N^{-(\epsilon_D/2)}}{(S-s)!}= \infty.
\begin{aligned}\psi''(\alpha) = -{h^{-1}}'(\alpha) - \frac{1}{\widehat{E}}&=\frac{1}{\bigl(E-\sigma'(h^{-1}(\alpha))\bigr)}-\frac{1}{\bigl(E-\frac{\gamma}{2}\bigr)}\geqslant \frac{\gamma}{2(E-\gamma)(E-\frac{\gamma}{2})}> 0.\end{aligned}
\left|\underset{\Gamma_{2p-1}}{\int\sum}\overline{\widehat{\eta}}_k\prod_{\substack{j=1 odd}}^{2p-1}\widehat{v}_{k_j}\prod_{\substack{j=2 even}}^{2p-1}\overline{\widehat{v}}_{k_j}\,d\Gamma\right|\lesssim \|\eta\|_{X^{-s,1/2-}}\|v\|_{X^{s,1/2-}_T}\|v\|^{2p-2}_{X^{s,1/2+}_T}.
|u^\infty(t,x)-u^\infty(t,x')| \le C_\alpha' \sum_{n= 1}^{N-1}a_n^3 \lambda_\beta(2^{-4n-3})^{-2-2\alpha}|x-x'| + C_\alpha' \sum_{n= N}^{\infty}a_n^{2-2\alpha}
v_{0}:= \phi(\beta_{0}) = c_{5}= a_{2}v_{1}:= \phi(\beta_{1}) = b_{1}b_{2}a_{2}b_{2}v_{2}:= \phi(\beta_{2}) = b_{1}b_{2}a_{2}a_{1}^{-1}a_{2}b_{2}a_{1}v_{c}:= \phi(c) = b_{1}b_{2}a_{2}a_{1}^{-1}b_{2}^{-1}a_{2}a_{1}^{-1}b_{1}^{-1}
\begin{cases}\partial_t \omega + u \cdot \nabla \omega = \frac 1 r (u\cdot e_r) \omega, \partial_t \tilde{\omega}+ u^{(l)}\cdot \nabla \tilde \omega=\frac 1 r (u^{(l)}\cdot e_r) \tilde \omega, \omega \Bigr|_{t=0}= \tilde \omega \Bigr|_{t=0}= \omega_0^{(l)}+ \omega_0^{(h)}.\end{cases}
D_1(\iota_{y*}1) =\sum\limits_{\bar{w}}\frac{(D_1(\iota_{y*}1), \iota_{\bar{w}*}1)}{e(T_{\bar{w}}T^*\mathcal{P})}\iota_{\bar{w}*}1 &=\prod\limits_{\alpha\in R^+_P}(y\alpha-\hbar)\iota_{\bar{y}*}1,
\sum_{\substack{p\leq n p\notin\mathcal{S}}}\dfrac{\log p}{p-1}\leq \dfrac{\log(2n^2+l)}{n-1}\left(2\log 4\dfrac{n}{\log n}+\sqrt{n}\right)+\sum_{i=1}^{r}\dfrac{e_i\log p_i}{\lambda (n-1)}+\dfrac{n\log 4}{n-1}.
\psi_{1}^{' \pm}= \gamma^{\mp}\left[ \begin{array}{c}{\mp \sin \theta ( 1 + \xi^{2})}\\{0}\\{i ( 1 - \xi^{2}+ 2 i \xi_{3}) ( 1 \mp \cos \theta )}\\{- 2 ( 1 \mp \cos \theta ) ( \xi_{1}- i \xi_{2})}\\ \end{array}\right]
\log F(z+\omega_2\,|\,\overline\omega_1,\omega_2) - \log F(z\,|\,\overline\omega_1,\omega_2) = f_{-1}^c(z,\overline\omega_1), & \log G(z+\omega_2\,|\,\omega_1,\widetilde\omega_1,\omega_2)- \log G(z+\omega_2\,|\,\omega_1,\widetilde\omega_1,\omega_2) = g_{-1}^c(z,\omega_1,\omega_2),
\Omega(\l) \; :=\; EF \; + \; \frac{q^{-1}K^2+qK^{'2}}{(q-q^{-1})^2}\;-\; \frac{q\l+q^{-1}\l^{-1}}{(q-q^{-1})^2}\,KK' \; =\; EF \; + \; \kappa^2 \big( q^{-1}K-q\l K')(K-\l^{-1}K') \; =\; FE \; + \; \kappa^2 \big( qK-q^{-1}\l^{-1}K')(K-\l K').
\psi_n(x) = \kappa_n \sum_{j=0}^n \frac{\prod_{k=0}^{n-1}(\theta j + k +a )}{\theta^n \prod_{k \neq j}(j-k)}x^{\theta j}& = \frac{\kappa_n}{2\pi i}\oint_{\Sigma_n'}\frac{\prod_{j=0}^{n-1}(\theta w + j +a )}{\theta^n \prod_{j=0}^{n}(w-j)}x^{\theta w}dw .
s_k =\binom{p-1}{k}f(x,v)^{k-1}f(y,v)^{p-k-1}(f(x,v)y-f(y,v)x) = \binom{p-1}{k}f(x,v)^{k}f(y,v)^{p-k-1}y - \binom{p-1}{k}f(x,v)^{k-1}f(y,v)^{p-k}x.
2^{jn\alpha_2/2}\left(\sum_{l\in \Gamma_k^{\alpha_2,\alpha_1}}\|\Box_l^{\alpha_2}\Box_k^{\alpha_1}f\|^q_{L^{2}}\right)^{1/q}\lesssim 2^{jn\alpha_2/2}2^{jn(\alpha_1-\alpha_2)(1/q-1/2)}\|f\|_{M_1}.
\frac{d}{ds}aslh(s)=[\frac{d}{ds}arctan(s)]D(s)^{2}=\frac{1}{s^{2}+1}D(s)^{2}=\frac{1}{s^{2}+1}(\frac{\sqrt{s^{2}+1}}{\sqrt[4]{s^{4}+1}})^{2}=\frac{1}{\sqrt{s^{4}+1}}
\textbf{err}^{(4)}:= \mathcal{O}\Bigg(\alpha_\mathrm{p}^{\frac{1}{4}}\delta^{\frac{7}{4}}\Big\Vert \gamma^{\textbf{int}}_{1}\mathrm{U}_{\mathrm{i}}\Big\Vert_{\mathrm{H}^{-\frac{1}{2},-\frac{1}{4}}\Big(\partial \Omega\times \mathbb{R}\Big)}\sqrt[4]{\mathbb{V}_1 \cdot \mathbb{V}_2}\Bigg).
\varepsilon_{10}^{\mu\kappa r_1\cdots r_8}\varepsilon_{10}^{\nu\lambda s_1\cdots s_8}\times\eta_{\mu\nu}\times \zeta^{(5)}_{\kappa\lambda}\times\prod_{i=1}^4 \zeta_{r_{2i-1}s_{2i-1}}^{(i)}k^{(i)}_{r_{2i}}k^{(i)}_{s_{2i}}\ .
V ( z ) ={\frac{D - 2}{1 6}}\left[ ( D - 6 ) \left({\frac{{\cal C}^{\prime}}{\cal C}}\right)^{2}+ 4{\frac{{\cal C}^{\prime \prime}}{\cal C}}\right] = -{\frac{( 1 + \Delta ) \Lambda}{2 \Delta ( 1 + \bar{K}| z | )^{2}}}+{\frac{2 \bar{K}}{\Delta + 2}}\delta ( z ) .
\sum_{t=T_{1}+1}^{T}\mathbb{E}[f(x_{t})-f^*]\leq \frac{\mathcal{B}_{3}}{\alpha}r_{1}+\frac{\mathcal{B}_{3}\mathcal{B}_{4}}{18\bar{L}^2\kappa}\beta\frac{d}{\alpha}T_{1}^{\frac{1}{\beta}}+\mathcal{B}_{4}\frac{d}{\alpha}T^{\frac{1}{\beta}}.
\mathbb{P}\left( \frac{1}{n}M_{n}\left( t \right) \in G \right) \geq \sup_{a \in G}\sup_{\lambda \in \mathcal{R}\left( t \right)}\left[ - \left[ \ell \left( \lambda ; x \right) + \psi \left( \lambda \right) \right] \right] &= - \inf_{a \in G}I \left( a \right),
N_{\varphi}= \left( \begin{array}{lr}-\frac{2 q \sin^2 \frac{\varphi x}{2}}{(p + q)}- \frac{i q (p - q)^{-}(p + q)^{+}\sin \varphi x}{(p^2 - q^2)}\frac{i q (p - q)^{+}(p + q)^{-}\sin \varphi x}{(p^2 - q^2)}\frac{2 q \sin^2 \frac{\varphi x}{2}}{(p - q)}\end{array}\right).
\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{A(x,y)}{|x-y|^{n+2s}}(u(x)-u(y))(\varphi(x)-\varphi(y))dydx = \int_{\Omega}\varphi d\mu \forall \varphi \in C_0^\infty(\Omega).
V(m) = \frac{N}{4\pi}\int (1 + \frac{g}{4\pi}\mbox{ln}(\frac{\Lambda^2}{m^2}+ 1))(\mbox{ln}(\frac{\Lambda^2}{m^2}+ 1) - \frac{2}{1 + \frac{m^2}{\Lambda^2}})dm^2.
f''(\eta) +\left[\frac{1+4s}{2(\eta-1)}+\frac{1+\beta}{\eta}-\alpha\right]f'(\eta)+\left[\frac{-\alpha+\beta+\beta^2+4s-4\alpha s+4\beta s}{4 (\eta-1)}+\frac{-2\alpha-\beta-2\alpha\beta-\beta^2-4s-4\beta s+\alpha^2}{4 \eta}\right]f(\eta)=0
A_\varnothing (\xi) : = \begin{pmatrix}2 \xi_1^2 + \xi_2^2 - \partial_3^2 \xi_1 \xi_2 - \mathrm{i}\xi_1 \partial_3 begin{align*}4pt] \xi_1 \xi_2 \xi_1^2 + 2 \xi_2^2 - \partial_3^2 - \mathrm{i}\xi_2 \partial_3 begin{align*}4pt]- \mathrm{i}\xi_1 \partial_3 - \mathrm{i}\xi_2 \partial_3 \xi_1^2 + \xi_2^2 - 2 \partial_3^2 . \end{pmatrix}.
\langle T_\mu^\nu(x)\rangle_{Ren.}(\mu)-\langle T_\mu^\nu(x)\rangle_{Ren.}(\mu')=\frac1{(4\pi)^{n/2}}\frac1{\sqrt{g}}\frac\delta{\delta g^{\mu\nu}}\int d^nx\sqrt{g}a_{n/2}(x)\ln(\mu/\mu') \ .
\hat{\delta}_{V}( z - z^{\prime}) = \int d k \Theta ( \Lambda - | k | ) \xi_{V}^{*}( k , z^{\prime}) \xi_{V}( k , z ) = \frac{1}{2 \pi}\int_{- \Lambda}^{\Lambda}d k e^{i k ( z - z^{\prime})}= \frac{1}{\pi}\frac{\sin [ \Lambda ( z - z^{\prime}) ]}{( z - z^{\prime})}
H(x_1^{k-l}h(x'))=h(x')Hx_1^{k-l}=\frac{(k-l)!}{2^{k-l}}\|x\|^{k-l}C_{k-l}^{\frac{n-2}{2}+l}\left(\frac{x_1}{\|x\|}\right)h(x')
\left\langle A_{\nu}^{m}F^{n \mu \nu}\right\rangle = \left\langle A_{\nu}^{m}\partial^{\mu}A^{n \nu}\right\rangle - \left\langle A_{\nu}^{m}\partial^{\nu}A^{n \mu}\right\rangle + \epsilon^{n p q}\left\langle A_{\nu}^{m}A^{p \mu}A^{q \nu}\right\rangle .
|P(t)\rangle=\frac{1}{2p}(|P_{L}\rangle e^{-\frac{i}{\hbar}(m_{L}-\frac{i}{2}\gamma_{L})t}+|P_{H}\rangle e^{-\frac{i}{\hbar}(m_{H}-\frac{i}{2}\gamma_{H})t})=g_{+}(t)|P\rangle-\frac{q}{p}g_{-}(t)|\overline{P}\rangle
\nu^{1}_\epsilon \star \nu^{1}_\iota = \begin{cases}1,0,\lambda-\frac{1}{2}, \nu^{0}_+, \nu^{0}_- \mbox{if}\epsilon = \iota \mbox{and}n \geq 5 1,0,\lambda-\frac{1}{2}\mbox{if}\epsilon = \iota \mbox{and}n=m =3,4 \lambda \mbox{if}\epsilon = -\iota\end{cases}
- \left. \frac{i}{4}\mathrm{T r}\, \log \widehat{\Omega}^{2}\right|_{d i v}= \left. \frac{i}{4}\mathrm{T r}\, \log \left( - \Delta +{\frac{R}{4}}\hat{1}\right) \right|_{d i v}= -{\frac{1}{2 \epsilon}}\int d^{2}x \, \sqrt{g}\,{\frac{n}{1 2}}R \ .
\sum_{n=-\infty}^\infty \frac{(-q;q^2)_n q^n}{(q^3;q^2)_n}= \sum_{n\ge 0}\frac{(-q;q^2)_n q^n}{(q^3;q^2)_n}+\sum_{n\ge 1}\frac{(-q;q^2)_{-n}q^{-n}}{(q^3;q^2)_{-n}}&=(1-q)\left(\sum_{n\ge 0}\frac{(-q;q^2)_nq^n}{(q;q^2)_{n+1}}+\sum_{n\ge 0}\frac{(q;q^2)_n(-q)^n}{(-q;q^2)_{n+1}}\right).
I_2 = \sum_{j=1}^{s-1}\min\{I(W_{j},Y_1|X^{j-1},W^{j-1}),\,I(W_{j};Y_2|X,W^{j-1})\}+ I(W_{s};Y_2|X,W^{s-1}) &\le \min\{I(W_{1},Y_1|X_{1}),\,I(W_{1};Y_2|X)\}+ \sum_{j=2}^{s}I(W_{j};Y_2|X,W^{j-1}) &= \min\{I(W_{1},Y_1|X_{1}),\,I(W_{1};Y_2|X)\}+ I(W;Y_2|X,W_{1}).
\frac{1}{T}\sum_{t=2}^{T+1}\Delta_{t}\le\frac{8}{Tc_{1}}\left(R_{1}+\frac{c_{1}}{3}\left(\gamma+\frac{2\sigma^{p}}{c_{2}}\right)\right)^{2}\max\left\{\left(52T(1+\log T)^{2}c_{2}\right)^{1/p};4R_{1}L+\frac{2c_{1}}{3}L\left(\gamma+\frac{2\sigma^{p}}{c_{2}}\right)+2\nabla_{1};\frac{Lc_{1}}{6}\right\}=\widetilde{O}\left(T^{\frac{1-p}{p}}\right).
\big\langle \widetilde \Lambda_{v_1}\widetilde P_{v_1, \rm R}\widetilde F (v_1), \widetilde P_{v_1, \rm R}\widetilde F (v_1) \big\rangle = \alpha_{v_1}|\widetilde f (v_1)|^2 = \alpha_{v_1}|f (v_1)|^2 & = \big\langle \Lambda_{v_1}P_{v_1, \rm R}F (v_1), P_{v_1, \rm R}F (v_1) \big\rangle
\frac{1}{n}H(A_R;n)= \frac{1}{n}\sum_{k=0}^{n-1}H(A_R;k+1|k) \le \frac{1}{n+1}\sum_{k=0}^{n}H(A_R;k+1| k)= \frac{1}{n+1}H(A_R;n+1).
\Delta^mu=\sum_{\substack{i,j,q\geq0 i+j+q=m-1 (i,j,q)\neq(0,m-1,0)}}c_{ijq}^{m-1}\Delta^i\nabla^q(P(u))\Delta^{j+1}\nabla^qu -\Delta^{m-1}(A(u)(\nabla u,\nabla u)),
I'_{\mathrm{b2u}}\left( k \right) = \frac{P_{\mathrm{bs}}a_{\mathrm{b2u},k}}{N_{\mathrm{bs}}}\left(\max \{\|\mathcal{H}^{\dagger}_{\mathrm{b2s},i}\| \}_{i=1}^{N_{\mathrm{bs}}}\right)^{-2}\| \mathbf{H}_{\mathrm{b2u},k}\mathbf{G}\|_\mathrm{F}^2.
(x^{-1}I+F(x)^{-1/\gamma})^{-1}&\quad=x\bigl(I+\{\Phi((A+x^{-1}H)^p)^{1/2}\Psi((B+x^{-1}K)^q)\Phi((A+x^{-1}H)^p)^{1/2}\}^{-1/\gamma}\bigr)^{-1}.
\int_X f_1 \cdot \tilde T_{h}f_2 \> d\tilde\omega_X = \int_X\int_X f_1(x) f_2(y) \tilde K_{h}(x,dy) \> d\tilde\omega_X(x) &=\int_X\int_X \frac{\phi_0(y)\> f_1(x) \> f_2(y)}{\alpha_0(h) \>\phi_0(x)}K_{h}(x,dy) \>\phi_0(x)^2\> d\omega_X(x) &= \frac{1}{\alpha_0(h)}\int_X \phi_0f_1 \cdot T_{h}(\phi_0f_2) \> d\omega_X.
(4.13)=\frac{2C_1C_2}{\pi}\int\limits_0^{\infty}dw_1\;w_1^{-\frac{1}{2}}\left(w_1+{\sigma}_1^2\right)^{\frac{\lambda+m}{2}}\int\limits_0^{\infty}dw_2\;w_2^{-\frac{1}{2}}\left(w_2+{\sigma}_2^2\right)^{\frac{\lambda+m}{2}}=
\Big\Vert \partial_{t}^{\frac{1}{2}}\mathbb{I}_{(\Omega)}\Big[\psi\Big] \Big\Vert_{\mathrm{L}^2(\Omega\times\mathbb{R_+})}^2 = \alpha\delta^2 \Big\Vert\partial_{\Tilde{t}}^{\frac{1}{2}}\mathbb{I}_{(B)}\Big[\hat{\psi}\Big] \Big\Vert_{\mathrm{L}^2(B\times\mathbb{R_+})}^2.
\left|\Phi_{i}(x,R)\right|\leq C(p,n,\sigma,\alpha,\mathcal{M})\begin{cases}\varepsilon^{\min\{(1-\sigma)p,(\sigma p+\alpha)(p-2)+\alpha\}}, , \varepsilon^{\min\{(1-\sigma)p,p+\alpha-2\}}, ,\end{cases}
Z^t = t (t A^T A + (1-t) I)^{-1}A^T Y = \sum_{i=1}^d \frac{t \sigma_i}{t \sigma_i^2 + (1-t)}\langle Y, u_i \rangle v_i = \sum_{i=1}^d \left( \frac{t \sigma_i^2}{t \sigma_i^2 + (1-t)}\langle X, v_i \rangle + \frac{t \sigma_i}{t \sigma_i^2 + (1-t)}\langle \eta, u_i \rangle \right) v_i,
480\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})\right\}^{(16)}=\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})\right. &\left.\cdot{\rm ch}(\widetilde{TX}+2\wedge^2\widetilde{L_R\otimes C}-\widetilde{L_R\otimes C}\otimes \widetilde{L_R\otimes C}+\widetilde{L_R\otimes C})\right\}^{(16)}.
x_{1}^{k+1}= \phi(W_1 x_{1}^{k}+V_n x_{n}^{k}+b_1), x_{2}^{k+1}= \phi(W_2 x_{2}^{k}+V_{1}x_{1}^{k+1}+b_2), \cdots x_{n}^{k+1}= \phi(W_n x_{n}^{k}+V_{n-1}x_{n-1}^{k+1}+b_n),
L_3^{-1}=(1-T) (3^2T^4 + 3T^3 + T + 1), f_3=1, L_{13}^{-1}=(1+T^2) (13T^2 + 5T + 1),f_{13}=2, L_{97}^{-1}=(1+T) (97^2T^4 + 582T^3 + 78T^2 + 6T + 1), f_{97}=1.
\mathcal{K}_Y(x,y)=\frac{c_{N, s}}{2}\left[\operatorname{div}Y(x)+\operatorname{div}Y(y)-(N+2 s) \frac{(Y(x)-Y(y)) \cdot(x-y)}{|x-y|^{2}}\right]|x-y|^{-N-2 s},
\psi (\vec{p}\,;\rho,\varphi)\sim e^{i\varepsilon N(\varphi-\varphi_p-\pi)}\left[e^{i\varepsilon \delta (\varphi-\varphi_p-\pi)}\; e^{ip_\perp\rho\cos(\varphi-\varphi_p)}+ f_{AB}(\varphi -\varphi_p)\;{e^{ip_\perp\rho}\over \sqrt{\rho}}\right]
(e_{s-1})_A\overset{\mathcal{X}^+(2)}{W}\!\!\!\!_{vB^{-1}A}\triangleright \big(\chi^+_s + \chi^-_{p-s}\big) = \chi^+_{s-1}+ \chi^-_{p-s+1}, (e_{s+1})_A\overset{\mathcal{X}^+(2)}{W}\!\!\!\!_{vB^{-1}A}\triangleright \big(\chi^+_s + \chi^-_{p-s}\big) = \chi^+_{s+1}+ \chi^-_{p-s-1}.
\widehat{\mathfrak{m}}^b_0 = q\mathbf{m}_0(L,\xi_b) = qW(\mathbf{m}_b). \widehat{\mathfrak{m}}^b_1(x) = d_b(\widehat{\mathfrak{m}}^b_0)(x),\ \ \x\in H^1(L,\mathbb{C}).
X^{\prime}=[\begin{matrix}ct^{\prime}\\x^{\prime}\\y^{\prime}\\z^{\prime}\end{matrix}],\eta=[\begin{matrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{matrix}],X=[\begin{matrix}ct\\x\\y\\z\end{matrix}]
\textrm{P}_\textrm{so}^{u}(\beta_e) = 1 - \exp \! \left[ - 2 \pi \lambda_e \!\! \int_{0}^{\infty}\!\! e^{- \left( \frac{\beta_e}{\rho}\right )^{\frac{2}{\alpha}}\theta r^2}\mathcal{L}_{W}\!\! \left(\frac{\beta_e}{\rho}r^\alpha \! \right) r \mathrm{d}{r}\right],
\frac{\Gamma ( \tilde{\sigma}_{i}\tilde{\sigma}\lambda )}{\Gamma ( \tilde{\sigma}\lambda )}= \frac{[ i ( \tilde{\sigma}\lambda )_{i + 1}- ( \tilde{\sigma}\lambda )_{i}- c ]}{[ i ( \tilde{\sigma}\lambda )_{i + 1}- ( \tilde{\sigma}\lambda )_{i}+ c ]}, \frac{\Gamma ( \bar{\sigma}_{N}\tilde{\sigma}\lambda )}{\Gamma ( \tilde{\sigma}\lambda )}= - 1 .
{J_n(k)}=C(k)\sum_{M_n(2k)}\prod_{\substack{1\leq j \leq 2k -1 j\neq 1, k+1}}\!\!\!\! p(m_j)\times \Big\{p(m_1+m_{k +1}) - p(m_1 + \cdots +m_{k+1})\Big\}.
\sum_{n=0}^{\infty}\textup{PDO}_\textup{t}(n)q^n =q\dfrac{f_2f_{12}^2}{f_6}\cdot\dfrac{f_3^2}{f_1^2}&=q\dfrac{f_2f_{12}^2}{f_6}\left(\dfrac{f_4^4f_6f_{12}^2}{f_2^5f_8f_{24}}+2q\dfrac{f_4f_6^2f_8f_{24}}{f_2^4f_{12}}\right),
\frac{n!}{n^n}\leq \frac{t_1!}{t_1^{t_1}}\cdot \frac{(n-t_1)!}{(n-t_1)^{n-t_1}}\leq \frac{t_1!}{t_1^{t_1}}\cdot \prod_{\sigma = 2}^{u+1}\frac{t_\sigma!}{t_\sigma^{t_\sigma}}= \prod_{\sigma = 1}^{u+1}\frac{t_\sigma!}{t_\sigma^{t_\sigma}},
R_1,~R_2,~R_3,~R_4,~R_7,~R_8, A=\frac{1}{2}(R_2+R_3+R_4+R_5),~B=\frac{1}{2}(R_1+R_2+R_3+R_6),~C=\frac{1}{2}(R_7+R_8+R_9+R_{12}), \frac{1}{2}(R_1+R_2+R_5+R_8+R_{10})+\frac{1}{4}(R_2+R_3+R_4+R_5).
\frac{1}{r_n^N}\sum_{i=1}^{d}\int_{B_{r_n}(x_n)}|\nabla u_i|^2 \leq \frac{1}{(C \rho_n)^N}\sum_{i=1}^{d}\int_{B_{4\rho_n}(\bar x_n)}|\nabla u_i|^2 \leq \frac{C}{\rho_n^N}\sum_{i=1}^{d}\int_{B_{2\rho_n}(\bar x_n)}|\nabla u_i|^2.
\sum_{j=1}^{N-1}\sum_{k=0}^n (-1)^k \binom{n}{j, \k}B_kB_{N,n-k-j}= (N+1)\delta_{n-N-1}+(-1)^{n-N-1}\binom{n}{N}B_{n-N}+ \sum_{j=0}^{N-1}\binom{n}{1, \j}B_{N,n-1-j}
U_{\underline{{{m}}}}^{+ +}\sigma_{\underline{{{\mu}}}\underline{{{\nu}}}}^{\underline{{{m}}}}= 2 V_{\underline{{{\mu}}}q}^{~ +}V_{\underline{{{\nu}}}q}^{~ +}, \qquad U_{\underline{{{m}}}}^{--}\tilde{\sigma}^{\underline{{{m}}}\underline{{{\mu}}}\underline{{{\nu}}}}= 2 V_{q}^{- \underline{{{\mu}}}}V_{q}^{- \underline{{{\nu}}}}, \qquad
c^k_{\mathrm{s2b},1}\left( N_{\mathrm{sc}}\right) = \mathsf{E}\left\lbrace B \sum_{i=1}^{N_{\mathrm{sc}}}\log_2\left[ \frac{P_{\mathrm{sc}}}{BN_{\mathrm{sc}}\sigma^2_{\mathrm{bs}}}\|(\mathcal{H}^{\dagger}_{b2s})^\mathrm{T}_j \|^{-2}+1 \right] \right\rbrace,
\frac{z}{1-q^{1/2}}_{2}\phi_{1}[\begin{matrix}qq^{1/2}\\q^{3/2}\end{matrix};q,z]=\frac{z}{1-q^{1/2}}+\frac{z^{2}}{1-q^{3/2}}+\frac{z^{3}}{1-q^{5/2}}+...
\begin{array}{rcl}Z(t,\tau) = E(t,\zeta_{j})E(t_{j},\zeta_{j})^{-1}\prod\limits_{k=i+2}^{\leftarrow j}E(t_{k},\gamma(t_{k-1}))E(t_{k-1},\gamma(t_{k-1}))^{-1}& E(t_{i+1},\gamma(\tau))E(\tau,\gamma(\tau))^{-1},\end{array}
\lim_{a\to\infty}\nabla (Z^{a,n})(t,t,z) = \int_{\overline{X^n}-X^n(t) - z}^{\overline{X^n}-X^n(t)}F_{T-t}(u) du =: \nabla P^n(t,z) \lim_{a\to\infty}\nabla_X (Z^{a,n})(t,t) = \sigma F_{T-t}(\overline{X^n}(t)-X^n(t))
\frac{\tau^{-\alpha}}{\Gamma(2-\alpha)}b_0 \left (U_h^n, v_h \right )+ \left(\nabla U_h^{n},\nabla v_h \right)= \left(f(U_h^{n-1}),v_h \right)- \frac{\tau^{-\alpha}}{\Gamma(2-\alpha)}\sum_{j=0}^{n-1}b_{n-j}\left (U_h^j, v_h \right ).
C_h = \lim_{q\to1}\left(g^{st}C_qg\right) = \begin{pmatrix}0 0 -1 0 1 0 1 0 h \end{pmatrix}, \hat{B_h}= \lim_{q\to1}\left[(g\otimes g)^{-1}\hat{B_q}(g\otimes g)\right],
Y_\alpha[s][x_\alpha^+\otimes t^{s},Y_\alpha[s]^{(r)}] =Y_\alpha[s]\partial_{u_s}(Y_\alpha[s]^{(r-1)})\mod{\mathcal I}&=Y_\alpha[s]Y_\alpha[s]^{(r-2)}\partial_{u_s}(Y_\alpha[s])\mod{\mathcal I}&=(r-1)Y_\alpha[s]^{(r-1)}\partial_{u_s}Y_\alpha[s]\mod{\mathcal{I}}.
U=\left(\begin{array}{ll}\frac{1}{p^T Hp}Hp \frac{1}{p^T Bp}Bp\end{array}\right) \left(\begin{array}{cc}p^THp+\varphi -\varphi -\varphi -p^TBp+\varphi\end{array}\right) \left(\begin{array}{c}\frac{1}{p^T Hp}p^TH \frac{1}{p^T Bp}p^TB \end{array}\right),
a_1 \, \gamma \, - \, a_2 \alpha \, h^{-1}\, \left( h^{-1}+ \mu \mathcal{F}\right)^{-1}_S \, h^{-1}\, \left[ \mbox{det}\, \left( h^{-1}+ \mu \mathcal{F}\right) \right]^\alpha = 0
\mathrm{I = \int \frac{d q_{0}}{2 \p i}\int \frac{d k_{0}}{2 \p i}\frac{k_{0}}{( k_{0}^{2}- \vec{k}^{2}+ i \e p s i l o n )}\frac{q_{0}}{( q_{0}^{2}- \vec{q}^{2}+ i \e p s i l o n )}, \qquad \e p s i l o n > 0 ,}
\mathrm{Ad}(u_{\beta}(x))(A)+ l\delta (u_{\beta}(x)) = A_{\Delta}^- + (c_{\beta, -\alpha_s,1}x + c_{\bar{\alpha}})X_{\bar{\alpha}}+ \sum_{\gamma_i \in \Gamma_k}Z_{\gamma_i}+ \sum_{\alpha \in \Phi_{k,j+1}}\tilde{Z}_{\alpha}+ \sum_{\alpha \in \Phi_{k-1}^+}\tilde{Z}_{\alpha}.
\widetilde{V}^{( \nu^{m})}(x;\Lambda_{m-1}) =V(x)-2\left( \log \prod\limits_{j=0}^{m-1}\Psi_{\nu}^{( \nu^{j})}(x;\Lambda_{j})\right)^{\prime \prime}=V(x)-2\sum_{j=0}^{m-1}\big( \log \Psi_{\nu}^{( \nu^{j})}(x;\Lambda_{j})\big)^{\prime \prime},
( * ) \; - \ln [ Q ( \rho_{1}) / Q ( \rho ) ] = \ln \left| \frac{a ( \rho_{1}+ 1 - \cos \chi ( \rho_{1})}{a + 1 - \cos \chi}\right| + \int_{\sigma = \rho}^{\rho_{1}}\frac{f ( \sigma ) \, d \sigma}{\sigma^{2}[ a ( \sigma ) + 1 - \cos \chi ( \sigma ) ]}
\partial_\lambda \Big( \lambda^{\nu}\, J_{-\nu}(\sqrt{\lambda}\, r) \, J_{-\nu}(\sqrt{\lambda}\, r') \Big ) = -\frac{\lambda^{\nu-\frac 12}}{2}\left( r\, J_{1-\nu}(\sqrt{\lambda}\, r) \J_{-\nu}(\sqrt{\lambda}\, r') + r'\, J_{1-\nu}(\sqrt{\lambda}\, r') \J_{-\nu}(\sqrt{\lambda}\, r) \right).
\begin{aligned}\frac{1}{2}\frac{{\rm d}}{{\rm d}t}\|\theta_u\|^2_{0,I_h}+ \frac{1}{2}\|\theta_w\|^2_{0,I_h}\leq Ch^{2k+2}+ C\|f - f_h\|^2_{0,\mathcal{T}_h}+ \|\theta_u\|^2_{0,I_h}+ h^{-1}\|\theta_u\|^4_{0,I_h}& + \kappa_1 + \kappa_2 + \kappa_3 + \kappa_4 + \kappa_5.\end{aligned}
\star(\theta^1\wedge\theta^2\pm\theta^3\wedge\theta^4) =\theta^3\wedge\theta^4\pm\theta^1\wedge\theta^2 , \star(\theta^1\wedge\theta^3\pm\theta^4\wedge\theta^2) =\theta^4\wedge\theta^2\pm\theta^1\wedge\theta^3, \star(\theta^1\wedge\theta^4\pm\theta^2\wedge\theta^3) =\theta^2\wedge\theta^3\pm\theta^1\wedge\theta^4 .
E_{T M}( a ) = \frac{\hbar c}{a}\, \left\{- \frac{\,{n_{0}}^{3}\, a^{4}\,{\Omega}^{4}}{2 4 \, \pi \, c^{4}\, \left( 1 + s \right)}+ \frac{n_{0}a^{2}\Omega^{2}}{4 \pi c^{2}( 1 + s )}+ \mathcal{O}( s + 1 )^{0}\right\}.
\left(1-c+(a^{p^m}-a)^p\left(1-\dfrac{1-c}{(1-c)^{p^m}}\right)\right)^p=-\left(1-\dfrac{1-c}{(1-c)^{p^m}}\right)^{p-1}(a^{p^m}-a)\left(1-\dfrac{(1-c)}{(1-c)^{p^m}}\right).
\sum_a p_a \dot{\mu}_a + \sum_{(a,b) \in \Gamma}p_{(a,b)}\dot{w}_{(a,b)}& = \sum_{a}p_a \left(\dot{\mu}_a - \sum_{b: (a,b) \in \Gamma}\left(\dot{w}_{(b,a)}- \dot{w}_{(a,b)}\right)\right) + \sum_{(a,b) \in \Gamma}\dot{w}_{(a,b)}\left(p_{(a,b)}- p_a + p_b\right).
u\left( x,\varepsilon \right) =D_{\lambda}^{-1}\left( \varepsilon \right) \left[ U_{1,\lambda}\left( x,\varepsilon \right) D_{1,\lambda}\left(\varepsilon \right) +U_{2,\lambda}\left( x,\varepsilon \right) D_{2,\lambda}\left( \varepsilon \right) \right] =
L'_0:\ \mbox{Eq.}(\ref{1}) \mbox{with}h_{1,1}y_1(0)+h_{1,2}y_2(0)=h_{0,1}y_1(a)+h_{0,2}y_2(a)=0, &L'_1:\ \mbox{Eq.}(\ref{1}) \mbox{with}h_{1,1}y_1(0)+h_{1,2}y_2(0)=y_1(T)=0, &L'_2:\ \mbox{Eq.}(\ref{1}) \mbox{with}h_{0,1}y_1(a)+h_{0,2}y_2(a)=y_1(T)=0.
{\check c}_{\gamma, \sigma \dot \beta}^{\enspace \enspace \enspace \,\sigma \dot \sigma}=\bar{c_{a}}^{\sigma \dot \sigma}E_{\gamma}{c_{\sigma \dot \beta}}^{a}+{c_{\sigma \dot \beta}}^{a}\bar{c_{b}}^{\sigma \dot \sigma}{c_{\gamma a}}^{b}-{c_{\gamma \dot \beta}}^{\dot \sigma}
\beta_{k0}(t\to\infty)=\sqrt{\frac{\omega_k}{\tilde{\omega}}}\frac{(\tilde{\omega}-\omega_k)(2\omega_k^2-2\tilde{\omega}^2+i\pi g\omega_k)}{\left[4\left(\omega_k^2-\tilde{\omega}^2\right)^2+\pi^2g^2\omega_k^2\right]}\sqrt{g\Delta \omega}{\rm e}^{i\omega_k t}\;.
d s^{2}= H^{\frac{p - 3}{4}}\Big ( - f d t^{2}+ \sum_{i = 1}^{p}( d x^{i})^{2}\Big ) + H^{\frac{p + 1}{4}}\left( f^{- 1}d r^{2}+ r^{2}d \Omega_{4 - p}^{2}\right) \, .
Q(t,t) \leq Q(r,r)+\sum_{p=1}^{n}Q(pc_*H^{-\frac{1}{7}}+r,c_*H^{-\frac{1}{7}}) & \lesssim (rH)^\frac{1}{2}+n H^\frac{3}{7}& \lesssim c_*(rH)^\frac{1}{2}+nc_*H^{-\frac{1}{7}}H^\frac{4}{7}& \lesssim c_* t^\frac{1}{2}H^\frac{1}{2}+t H^\frac{4}{7}
| (-\Delta)^{\sigma}f (x)| \leq C{|x|^{-n-2 \sigma}}\int_{|y| < 3 |x|}|f(y)| dy+ |f(x)| |x|^{-2 \sigma}&+ C \frac{2^{3-2 \sigma}}{2-2 \sigma}|x|^{2-2 \sigma}\sum_{|\alpha|=2}\frac{|\alpha|}{\alpha!}|\partial^\alpha f ( \frac x 2 ) |,
\begin{aligned}F(t;q) = \frac{1}{2\pi\mathrm{i}}\int_{\Gamma_{\theta,\kappa}}e^{zt}z^{\alpha-1}(z^\alpha + A(q))^{-1}\,\d z~~~~E(t;q) = \frac{1}{2\pi\mathrm{i}}\int_{\Gamma_{\theta,\kappa}}e^{zt}(z^\alpha +A(q))^{-1}\,\d z, \end{aligned}
S_{1D}= - \int d^5 \sigma \left(\sqrt{-G}\sqrt{1+z_1+\frac{z_1^2}{2}-z_2}+\frac{\epsilon_{\mu\nu\lambda\sigma\tau}C^{\mu}\tilde{{\cal H}}^{\nu\lambda}\tilde{{\cal H}}^{\sigma\tau}}{8(1+C_1^2)}\right).
\max_{0\le j<J}\frac{|f^{(j)}(q)|}{j!}\le \max_{0\le j < J}\sum_{k+ l=j}\frac{|\bar{f}^{(k)}(q)|}{k!}\frac{|(u^d)^{(l)}(q)|}{l!}\le \left(J \max_{0\le j < J}\frac{|(u^d)^{(j)}(q)|}{j!}\right) \max_{0\le j < J}\frac{|\bar{f}^{(j)}(q)|}{j!}
F_2(z) = B^zA^z-B^{z-1}A^{z-1}BA = B^{z-1}A^{z-1}\cdot [A,B]+[B,B^{z-1}A^z] = B^{z-1}A^{z-1}\cdot[A,B]+B^{z-1}\cdot[B,A]\cdot A^{z-1}+B^{z-1}A\cdot[B,A^{z-1}].