harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Z_N= \prod\limits_{j=1}^N \int\limits_{\rm PBC}{d\mu \left({\xi_j^*,\xi_j}\right)}\exp \left[ iJ \left\{2i\ln \left({{1+\xi_j^*\xi_j}\over{1+\xi_j^*\xi_{j-1}}}\right) + \epsilon h{{1 - \xi_j^*\xi_{j-1}}\over{1+\xi_j^*\xi_{j-1}}}\right\}\right].
\left( \cdots \left( \varphi_{k_1l_1}^{r_2'\cdots r_s'}\right) \cdots \right)_{k_jl_j}^{r_{j+1}'\cdots r_s'}(X_1,\cdots, X_j+2\pi ,\cdots, X_n, Y_1,\cdots, Y_n) &=\left( \cdots \left( \varphi_{k_1l_1}^{r_2'\cdots r_s'}\right) \cdots \right)_{(k_j+1)(l_j+1)}^{r_{j+1}'\cdots r_s'}(X_1,\cdots, X_n, Y_1,\cdots, Y_n),
df^!{\rm pr}^!_{i_v}(Z) \overset{(a)}{=}{\rm pr}_{2\ast}df^!{\rm pr}^!_{i_v}(Z)\overset{(b)}{=}0_{X_v}^!di_{v\ast}{\rm pr}_{i_v}^!(Z)\overset{\eqref{eq:pullbackCCdef}}{=}0_{X_v}^!(i_v^\ast Z),
\mathcal I_j(x,y)= \iint a_{1,j}(\tau,s,\xi)e^{2\pi i\phi_1(x,y,t;,\xi,\tau,s,\theta)}d\theta ds d\tau d\xi, \mathcal J_j(x,y)= \iint a_{2,j}(\tau,s,\xi)e^{2\pi i\phi_2(x,y,t;,\xi,\tau,s,r)}dr ds d\tau d\xi,
R'_V =\{g\in R_V\mid g(S_3\oplus U(s_7,s_4,s_6)\oplus U_{(8)}^{+2,s_2}\oplus U_{(6)}^{+,s_1}) & =S_3\oplus U(s_7,s_4,s_6)\oplus U_{(8)}^{+2,s_2}\oplus U_{(6)}^{+,s_1}\}
\phi_{(p_{2},q_{2};r_{2},s_{2})}\otimes\phi_{(p_{1},q_{1};r_{1},s_{1})}= \sum_{(p,q)\in\Lambda(p_{1},q_{1},p_{2},q_{2})}\; \sum_{(r,s)\in\Lambda(r_{1},s_{1},r_{2},s_{2})}\left[ \phi_{(p,q;r,s)}\right] ,
\sum_{j=-\infty}^{nt}\Bigl[\sum_{k=nu+1}^{nt}v_{k-j}(s)\Bigr]^p=\sum_{j=-nu+1}^\infty\Bigl[\sum_{k=nu+1}^{nt}(k+j)^{-d(s)}\Bigr]^p+\sum_{j=nu+1}^{nt}\Bigl[\sum_{k=1}^{nt-j+1}k^{-d(s)}\Bigr]^p
\delta_{t}u^{n}-\frac{\mu}{\tau}\int\limits^{t_{n}}_{t_{n-1}}\Delta u(\cdot,t)dt -\frac{1}{\tau}\int\limits^{t_{n}}_{t_{n-1}}I^{(\alpha)}\Delta u(\cdot,t)dt =\frac{1}{\tau}\int\limits^{t_{n}}_{t_{n-1}}f(\cdot,t)dt+ \frac{1}{\tau}\int\limits^{t_{n}}_{t_{n-1}}g(u(\cdot,t))dt.
\delta_{\vec{u}}( t ) < W ( \gamma ) > = -{\frac{4 \pi i}{k}}\int_{0}^{1}d s u^{a}( s ) ({\eta^{\prime}}^{b}( s ) + t{u^{\prime}}^{b}( s ) ) \epsilon_{a b c}\oint_{\gamma}d y^{c}\delta^{3}( \eta ( s , t ) - y ) \left< \mathrm{T r}\left( \tau^{l}H ( \eta ( t )_{x}^{o}) H ( \gamma_{o}^{y}) \tau^{l}H ( \gamma_{y}^{o}) H ( \eta ( t )_{o}^{x}) \right) \right> .
\delta_{\gamma}^{\alpha}\delta_{\delta}^{\beta}= - \frac 1 4 \hat{C}_{\gamma \delta}( \hat{C}^{- 1})^{\alpha \beta}- \frac 1 4 ( \hat{C}\hat{\gamma}_{A})_{\gamma \delta}( \hat{\gamma}^{A}\hat{C}^{- 1})^{\alpha \beta}- \frac 1 2 ( \hat{C}\hat{S}_{A B})_{\gamma \delta}( \hat{S}^{A B}\hat{C}^{- 1})^{\alpha \beta},
d_i(x,z) = J_{2k+1}(x,d_{i-1}(x,z),d_{i-1}(x,z)) 1\leq i< 2^k, d_i(x,z) = J_{2k+1}(b(x,z),d_{i-1}(x,z),d_{i-1}(x,z)) 2^k\leq i<2^{k+1},
\int \prod_{a=1}^N \prod_{j=1}^{n_a}d^Dz_j^{(a)}\bigg| \prod_{b<c}^N \; f^{(b,c)}\bigg| \; \leq \;\prod_{a<b}^N \Bigg[ \int \;\prod_{i=1}^{n_a}d^Dz_i^{(a)}\prod_{j=1}^{n_b}d^Dz_j^{(b)}\;\bigg| f^{(a,b)}\bigg|^{N-1}\Bigg]^{\frac{1}{N-1}}\; .
T_{\it 3}= m_{1}d z^{1}\wedge d \bar{z}^{2}\wedge d \bar{z}^{3}+ m_{2}d \bar{z}^{1}\wedge d z^{2}\wedge d \bar{z}^{3}+ m_{3}d \bar{z}^{1}\wedge d \bar{z}^{2}\wedge d z^{3}+ m_{4}d z^{1}\wedge d z^{2}\wedge d z^{3}\ .
\alpha_{(X,a_{1})}\alpha_{(Y,b_{1})}^{-1}= \alpha_{(X,a_{1})}\alpha_{(Xa_{1},c)}\alpha_{(Xa_{1},c)}^{-1}\alpha_{(Y,b_{1})}^{-1}= \alpha_{(X,a_{1})}\alpha_{(Xa_{1},c)}(\alpha_{(Y,b_{1})}\alpha_{(Yb_{1},c)})^{-1}=\alpha_{(X,a_{1}c)}\alpha_{(Y,b_{1}c)}^{-1}
\frac{\partial}{\partial t}C_{ij}=\Delta_g C_{ij}-5g^{pq}R_{ip}C_{qj}-5g^{pq}C_{iq}R_{pj}+2\langle C_2,\mathrm{Ric}\rangle_gg_{ij}+4RC_{ij}&+\frac{1}{2}g_{ik}g_{jm}\varepsilon^{klm}\nabla_l|\mathrm{Ric}|_g^2+\frac{R}{2}g_{ik}g_{jl}\varepsilon^{klm}\nabla_mR+2g_{ik}g^{pq}\varepsilon^{klm}R_{pl}\nabla_mR_{qj}&+g_{ik}\varepsilon^{klm}R_{jm}\nabla_lR.
\varphi_j^+(\theta)=e^{-i\bar A\theta}\Big(e^{i(\bar A+j)\theta}+O\big(\tfrac1{|j|^3}\big)\Big),j\to+\infty, &\varphi_j^-(\theta)=e^{-i\bar A\theta}\Big(e^{i(\bar A-j)\theta}+O\big(\tfrac1{|j|^3}\big)\Big),j\to+\infty.
a = \left( \begin{array}{l l l l}{{a_{1 1}}}&{{a_{1 2}}}&{{0}}&{{0}}\\{{a_{2 1}}}&{{a_{2 2}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{a_{1 1}}}&{{a_{1 2}}}\\{{0}}&{{0}}&{{a_{2 1}}}&{{a_{2 2}}}\\ \end{array}\right) \simeq \left( \begin{array}{l l}{{a_{1 1}}}&{{a_{1 2}}}\\{{a_{2 1}}}&{{a_{2 2}}}\\ \end{array}\right) \otimes \underline{{{1}}}
\begin{aligned}[\bold{S}_{j-1}^{-1}]^{(1,2)}=[\bold{I}_{j-1}-\bold{\Pi}_{j-1}-\bold{\Gamma}_{j-1}\times &(\bold{I}_{j-1}-\bold{\Pi}_{j-1}^{T})^{-1}\bold{\Xi}_{j-1}]^{-1}\bold{\Gamma}_{j-1}(\bold{I}_{j-1}-\bold{\Pi}_{j-1}^{T})^{-1},\end{aligned}
{\mathcal J}^{\mu}= \sum_{i = 1 \atop s = \uparrow , \downarrow}^{N}{\mathcal J}^{( i , s ) \mu}, \widehat{{\mathcal J}}_{\ell / r}^{\mu}= \sum_{i = 1 \atop s = \uparrow , \downarrow}^{N}\widehat{{\mathcal J}}_{\ell , r}^{( i , s ) \mu}.
\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}_{II}}d\omega_{k+1}dv_{s+k+1}d\tau \leq C_d T R^d \alpha\end{aligned}
|B^R_{(g^\theta, \pi^\theta)}(x^k, 0) - 16 \pi \mathcal C^\theta_k | \le \kappa |\theta|^2R^{-1}|B^R_{(g^\theta, \pi^\theta)}(0, x\times \frac{\partial}{\partial x^k}) - 8 \pi \mathcal J^\theta_k | \le \kappa |\theta|^{2}R^{-1}.
\hat{U}_{\mathrm{B R S T}}( t_{f}, t_{i}) = \left[ e^{- \frac{i}{\hbar}\epsilon \hat{H}_{\mathrm{e f f}}}\right]^{N}\ \ \ , \ \ \, e p s i l o n = \frac{\Delta t}{N}= \frac{t_{f}- t_{i}}{N}\ \ \ .
\Lambda^{(3)}(g_2, g_3)^{\alpha}\: + \:\Lambda^{(3)}(g_1, g_2 g_3)^{\alpha}\: = \:g_3^* \Lambda^{(3)}(g_1, g_2)^{\alpha}\: + \:\Lambda^{(3)}(g_1 g_2, g_3)^{\alpha}\: + \: d \log \gamma_{\alpha}^{g_1, g_2, g_3}
\delta_{\hat{\chi}}\hat{L}_{\hat{\mu}_{1}\ldots\hat{\mu}_{r}}= m\hat{\lambda}_{\hat{\mu}_{1}}\hat{k}^{\hat{\nu}}\hat{L}_{\hat{\nu}\hat{\mu}_{2}\ldots\hat{\mu}_{r}}+\ldots +m\hat{\lambda}_{\hat{\mu}_{r}}\hat{k}^{\hat{\nu}}\hat{L}_{\hat{\mu}_{1}\ldots\hat{\mu}_{r-1}\hat{\nu}}\, .
\parallel\int\limits_{E}\sum_{k>n}a_{k}(t)\Psi_{k,t}dt\parallel_{(-m,m)}^{2}\leq2\pi\int\limits_{(-m,m)}\int\limits_{E}\mid\sum_{k>n}a_{k}(t)\Psi_{k,t}(x)\mid^{2}dtdx=I_{n}\rightarrow0
| \Sigma_{R}, \xi \! >_{\epsilon}\: \equiv \: | \Sigma_{R}, \xi \! >_{0}+ \: \overbrace{\frac{\epsilon}{2 \pi i}\int_{P \in \Sigma_{R}/ D_{\xi}}< \! \phi | \Sigma_{R}, \xi , u \! >_{0}d u |_{P}\wedge d \bar{u}|_{P}}^{\Delta | p \! >},
(3,2) = \frac{3 s^2 - 24 s + 60}{s^3 + 9 s^2 + 36s + 60}, G_{23}^{\gamma}(s) = \frac{5 (\gamma -1)^2 s^2 + 15 (\gamma -1) s + 15}{\gamma^3 s^3 +6 \gamma^2 s^2 + 15 \gamma s + 15}.
B=\left(\begin{array}{@{}c|c@{}}B_j 0 \hline 0 \begin{matrix}2^{k_j}u_1 0 0 0 2^{k_j+1}u_2 0 0 0 \ddots \end{matrix}\end{array}\right)\sim\left(\begin{array}{@{}c|c@{}}B_j 0 \hline 0 \begin{matrix}2^{k_j}u_1' 0 0 0 2^{k_j+1}u_2' 0 0 0 \ddots \end{matrix}\end{array}\right)=B'.
\| f\|_{L^r(S_t, d\sigma)}^r=\frac{1}{|S_t|}\sum_{\textbf{x}\in S_t}|f(\textbf{x})|^r\mbox{for}~~ 1\le r<\infty, & \| f\|_{L^\infty(S_t, d\sigma)}= \max_{\textbf{x}\in S_t}|f(\textbf{x})|.
\big( f' , f' \big) = \int \, (\, \mathfrak{J}_L ((\mathfrak{J}^L f')_x ), f'_x \,) \, d\mu_{\mathfrak{G}/H}= \int \, (\, \mathfrak{J}_L L_{h(x)}\mathfrak{J}_L L_{h(x)^{-1}}\sqrt{\lambda ([x], g_0)}\, f_{xg_0}, \sqrt{\lambda ([x], g_0)}\, f_{xg_0}\,) \, d\mu_{\mathfrak{G}/H}(x) < \infty,
(-1)^{j+r-1}\binom{j+r-1}{j}\frac{1}{2}S(n,r)\binom{2n-j}{n}\sum_{l=0}^{r-1}(-1)^l\binom{2n-j+l}{l}\frac{\binom{2(j+r-1-l)}{j+r-1-l}\binom{2(n-j+l+1)}{n-j+l+1}\binom{n-j}{j+r-l-1}}{2\binom{2n-j+l+1}{n}}
\delta \Psi \bigg ( \phi^{A}, \phi_{A}^{*}; \frac{\delta_{l}}{\delta \phi^{A}}, \frac{\delta}{\delta \phi_{A}^{*}}\bigg ) = \delta \Psi \bigg ( \varphi^{i}, \psi^{\alpha}, \phi_{A}^{*}; \frac{\delta_{l}}{\delta \varphi^{i}}, \frac{\delta_{l}}{\delta \psi^{\alpha}}, \frac{\delta}{\delta \phi_{A}^{*}}\bigg )
R(z) = \left( \begin{array}{cc}0 0 0 R_{\Omega}(z)\end{array}\right) +\left( \begin{array}{c}1 R_{\Omega}(z)|1 \rangle\end{array}\right)\frac{1}{z+\i \sigma(z)}\left( 1, \langle 1 | R_{\Omega}(z) \right ) .
\frac{(1-v+xv^2)^2}{(1-xv)(1-v)^2}C(x,t) &\qquad+\frac{xv^2(1-v+xv^2)(2v^2x^2-2x^2v+2x-1)}{(1-v)^2(1-2x)(1-xv)}C(x,1) &\qquad-\frac{(2v^2x^2-2xv+1)x^2v^3}{(1-xv)(1-2x)(1-v)}A(x,1)-\frac{(2v^2x^2-v+1)v^3x^3}{(1-xv)(1-2x)(1-v)}=0
q^Ha^{\varepsilon}_1=a^{\varepsilon}_1q^{H+1}\,,\,\,q^Ha^{\varepsilon}_2=a^{\varepsilon}_2q^{H-1}\,,\,\,q^{\bar H}{\bar a}_{\varepsilon}^1={\bar a}_{\varepsilon}^1q^{{\bar H}-1}\,,\,\,q^{\bar H}{\bar a}_{\varepsilon}^2={\bar a}_{\varepsilon}^2q^{{\bar H}+1}\,,
i e \left( 2 \pi \right)^{4}\delta ( p^{\prime}- p - q ) \Gamma_{a b c}^{\mu}( q , p ) = \int d x d y d z e^{i \, ( p^{\prime}z - p y - q x )}{\frac{\delta^{3}\Gamma}{\delta \bar{\psi}_{c}( z ) \delta \psi_{b}( y ) \delta A_{a}^{\mu}( x )}}\, .
\langle g_{x y}\rangle = \left( \begin{array}{c c c c}{b}&{0}&{0}&{0}\\{0}&{b}&{0}&{0}\\{0}&{0}&{\langle g_{3 3}\rangle}&{\langle g_{3 4}\rangle}\\{0}&{0}&{\langle g_{4 3}\rangle}&{\langle g_{4 4}\rangle}\\ \end{array}\right)
\tabcolsep 1 p t \begin{array}{c c c c c}{d s^{2}=}&{d X^{\mu}\eta_{\mu \nu}d X^{\nu}}&{-}&{d X^{+}d X^{-}}&{}\\{}&{( d - 2 , 1 )}&{+}&{( 1 , 1 )}&{\Rightarrow ( d - 1 , 2 )}\\ \end{array}\tabcolsep 6 p t
\sum_{i = 1}^{n}q_{a}^{i}[ \varphi_{i}^{\alpha}\overline{{\varphi}}_{i \beta}+ \varphi_{i \beta}\overline{{\varphi}}_{i}^{\alpha}] = \vec{\xi_{a}}\vec{\sigma}_{\beta}^{\alpha}\qquad ; a = 1 , . . . , r .
\partial_{t}u - \mathrm{div}\, \left(g\Big(\int_{0}^{t}\theta(t - s) \big|\nabla u(s, \cdot)\big|^{2}\mathrm{d}s\Big) \nabla u\right) = 0 (0, \infty) \times G, \frac{\partial u}{\partial \mathbf{n}}= 0 (0, \infty) \times \Gamma, u(0, \cdot) = u^{0}\Omega,
\left|D_t^\sigma(-\Delta)^{\gamma/2}q_{\alpha,\beta}(1,x)\right|+\left|D_t^\sigma(-\Delta)^{\gamma/2}\partial_t q_{\alpha,\beta}(1,x)\right| & \leq N\left(\{|x|^{1-\gamma}(1+\ln|x|1_{\gamma=1})\}\wedge|x|^{-1-\gamma}\right)
A_{10}A_{ij}= (k-i-j+1)(r-2) A_{i-1,j}+ \left( i(r-3)+j(r-2) \right) A_{ij}+(i+1)A_{i+1,j}, A_{01}A_{ij}= (k-i-j+1)(r-2)j A_{i-1,j}+ (k-i-j+1)(n-k-j+1)(r-1)A_{i,j-1}+ (i+1)j A_{i+1,j}+ (j+1)^2 A_{i,j+1}+ (i+1)(n-k-j+1)(r-1) A_{i+1,j-1}&+ (j+1)^2(r-2) A_{i-1,j+1}+j\left( k-i-j+(r-2)i+(n-k-j)(r-1)\right)A_{i,j}.
H^2=\left(\frac{\dot{R}}{R}\right)^2=\frac{8\pi G_4}{3}\,\Lambda_4-\frac{k}{R^2}+ \left(\mu+\frac{8\pi G_4}{3}\,\hat{\rho}\right)\frac{1}{R^4}-\frac{q^2}{R^6}+\frac{4\pi G_4}{3\sigma}\,\frac{\hat{\rho}^2}{R^8}\,.
\zeta(s)\int_{0}^\infty\frac{\d{t}}{t}\;t^{\frac{s}{\lambda}}f(t)=\frac{-\lambda}{2}\left[\frac{{F}_\lambda(0)}{s}+\frac{\widehat{F}_\lambda(0)}{1-s}\right]+\int_{1}^\infty\frac{\d{t}}{t}\sum_{n=1}^\infty\left[t^{\frac{s}{\lambda}}{F}_\lambda\bigr(n{t}^{\frac{1}{\lambda}}\bigr)+t^{\frac{1-s}{\lambda}}\widehat{F_\lambda}\bigr(n{t}^{\frac{1}{\lambda}}\bigr)\right]
\phi_\gamma=\phi_{\alpha}\phi_{\beta}-\eta_{\beta}\otimes\xi_{\alpha}=-\phi_{\beta}\phi_{\alpha}+\eta_{\alpha}\otimes\xi_{\beta}, \xi_{\gamma}=\phi_{\alpha}\xi_{\beta}=-\phi_{\beta}\xi_{\alpha}, \ eta_{\gamma}=\eta_{\alpha}\circ\phi_{\beta}=-\eta_{\beta}\circ\phi_{\alpha}
f_{R}(R|\alpha,p,\nu)=\frac{2}{\alpha^{1/2}\nu^{1/2}B(\frac{1}{2}p,\frac{1}{2}\nu)}(\frac{R^{2}}{\alpha\nu})^{(p-1)/2}(1+\frac{R^{2}}{\alpha\nu})^{-(\nu+p)/2}
\mu^{l}(B_t(w_0))= \sum_{z\in Z}\mu^{l}(W^z) +\mu^{l}\big(W^{(2)}\setminus E\big)=\sum_{z\in Z}\mu^{l}(W^z) +\mu^{l}_1\big(W^{(2)}\setminus E\big).
c_{15}\delta^{-1}L N_*^{-\beta+1}[B_{N_*}(f_0)]^{-1}\leq c_{15}c_5 \delta^{-1}LN_*^{-\beta-1}=c_{15}c_5 c_{21}^{-\beta-1}\delta^{-1}T^{-1/2}\leq c_{15}c_5 c_{21}^{-\beta-1}C_1^{-1},
E = - \sum_{l = 1}^{\infty}( 2 l + 1 ) \int_{- \infty}^{\infty}{\frac{d \omega}{2 \pi}}e^{- i \omega \tau}\left\{\int_{0}^{a}r^{2}d r \, k^{\prime 3}( A_{F}+ A_{G}) j_{l}^{2}( k^{\prime}r ) + \int_{a}^{\infty}r^{2}d r \, k^{3}( B_{F}+ B_{G}) h_{l}^{2}( k r ) \right\}.
f^{\#}(t,k) = f^{\#}(0,k) \pm i \int_{0}^{t}\mathcal{N}_{\mu^\sharp}[f,f,f](s,k) \, ds, \mathcal{N}_{\mu^\sharp}[f,f,f](s,k) := \!\! \iiint e^{is(-k^2+\ell^2-m^2+n^2)}f^{\#}(s,\ell)\overline{f^{\#}(s,m)}f^{\#}(s,n) \, \mu^\sharp(k,\ell,m,n) \,dndmd\ell,
H_2(u) = \frac{1}{2}\sum_{j \neq 0}j^2 |u_j|^2, H_4(u) = - \frac{\varsigma}{4}\, \sum_{j_1 + j_2 + j_3 + j_4 = 0}u_{j_1}u_{j_2}u_{j_3}u_{j_4}\,.
\sum_{k=0}^{\tilde{K}}r^k |\nabla_{g_0}^k([H_{1,i}^{-1}\circ \Phi_{\overline{V}_i}\circ H_{0,i}]^*g_i - g_0)|_{g_0}= O_{\tilde{K},\tilde{\epsilon},i}(r^{\max\{\lambda,-2\}+2\tilde{\epsilon}})\;\,\;\,r\to\infty,
0 \leq E\left[\psi(X'(T),m(T))-\psi(X(T),m(T)) + \psi(X(T),m'(T))-\psi(X'(T),m'(T))\right] &+E\left[\int_0^T [c_1(X'(t),m(t)) -c_1(X(t),m(t)) \right. &\left. +c_1(X(t),m'(t))-c_1(X'(t),m'(t))]dt\right] &= \sum_{x\in\Sigma}(\psi(x,m(T))-\psi(x,m'(T)))(m'_x(T)-m_x(T)) &+\int_0^T \left[\sum_{x\in\Sigma}(c_1(x,m(t))-c_1(x,m'(t)))(m'_x(t)-m_x(t))\right]dt.
\begin{array}{c}|\xi\wedge\eta|^2=2|P_1(\xi\wedge\eta)|^2=-{1\over 4}\left(\xi\wedge\eta, \sum\limits_{a=1}^3 I_a^2 (\xi\wedge\eta)\right)= ={3\over 2}|\xi\wedge\eta|^2-{1\over 2}\left(\xi\wedge\eta,\sum\limits_{a=1}^3 I_a\xi\wedge I_a\eta\right)= ={3\over 2}|\xi\wedge\eta|^2+{1\over 2}(\xi,I_a\eta)(\eta,I_a\xi)\end{array}
F_X(0) =\frac{(1-\beta^2/\alpha^2)^{\nu+1/2}}{2\sqrt{\pi}\Gamma(\nu+1/2)}\sum_{k=0}^\infty\frac{(-1)^k}{k!}\bigg(\frac{2\beta}{\alpha}\bigg)^k\Gamma\bigg(\frac{k+1}{2}\bigg)\Gamma\bigg(\nu+\frac{k+1}{2}\bigg) &=\frac{(1-\beta^2/\alpha^2)^{\nu+1/2}}{2\sqrt{\pi}\Gamma(\nu+1/2)}\big(S_1+S_2),
\sum_{i=1}^{n}d_{i}^{2m}=n((R^{2}+L^{2})^{m}+\sum_{k=1}^{\lfloor\frac{m}{2}\rfloor}(\begin{matrix}m\\ 2k\end{matrix})(\begin{matrix}2k\\k\end{matrix})R^{2k}L^{2k}(R^{2}+L^{2})^{m-2k})
b^{( 3 )}= \frac{2}{3}\Pi_{\tilde{\mu}\tilde{\nu}}\Pi^{\tilde{\mu}}\Pi^{\tilde{\nu}}- \frac{2}{1 5}\Pi_{\alpha \beta}\Pi^{\alpha}\Pi^{\beta}- \frac{3}{5}\Pi_{\tilde{\mu}\alpha}\Pi^{\tilde{\mu}}\Pi^{\alpha}\quad .
< \psi \left( R \right) \psi \left( 0 \right) \omega \left( f_{1}\right) \ldots > ~ = ~ \sum_{N}{< 0 | \psi \left( R \right) | \Delta + N > < \Delta + N | \psi \left( 0 \right) \omega \left( f_{1}\right) \ldots | 0 >}
\begin{pmatrix}B^0_{00}=0 B^0_{10}=0 B^0_{01}=\frac{\beta_0}{\eta_0\eta_1}B^0_{11}=0 B^1_{00}=0 B^1_{10}=\frac{\beta_1}{\eta_0\eta_1}B^1_{01}=0 B^1_{11}=0 B^2_{00}=\frac{\beta_2}{{\eta_0}^2}B^2_{10}=0 B^2_{01}=0 B^2_{11}=0 B^3_{00}=\frac{\beta_3}{{\eta_0}^2}B^3_{10}=0 B^3_{01}=0 B^3_{11}=0\end{pmatrix}
\frac{2}{\pi}\int_{-(\log{t})^{2}}^{(\log{t})^{2}}\log\zeta(\sigma+i(t+u))\left( \frac{\sin{\psi u}}{u}\right)^2 e^{iHu}du = \sum_{n=1}^{\infty}\frac{\Lambda(n)\max(0,\psi-|H-\log{n}|)}{n^{\sigma+it}\log{n}}+ O\left( \frac{e^{|H|+2\psi}}{(\log{t})^{2}}\right).
\Vert \mathcal{Y}_2\mathbf{u}\Vert^2_{L_2(\mathbb{R}^d)}\leqslant \nu \mathfrak{a}[\mathbf{u},\mathbf{u}]+C(\nu)\Vert \mathbf{u}\Vert^2_{L_2(\mathbb{R}^d)},\mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^n).
\|U(t)\|+\gamma\int_0^te^{\gamma(t-s)}\,\|U(s)\|ds &\leq \frac{1}{\sqrt{1-\mu^2}}\Big(e^{\gamma t}\,\|U(0)\|+\int_0^te^{\gamma(t-s)}\,\|{\tilde L}_aU(s)\|ds\Big).
\begin{aligned}\int |\Delta d|^2dx+\int_s^t\int |\nabla d_t|^2dxd\tau &\le 2\int |\Delta d|^2(s)dx +C\int_s^t\int (|\nabla d|^2 |\nabla^2 d|^2+|\nabla u|^2 |\nabla d|^2+|u|^2 |\nabla^2 d|^2+|\nabla d|^6)dxd\tau.\end{aligned}
A_{\mathcal{C},\mathrm{B},\beta}^{\alpha}=\int_{\mathcal{C}}\, (gId-A^{-1})^{-\beta}\,g^{\beta-\alpha}\;d\nu(g_{\mathcal{C}_{\mathrm{B}}})=A^\alpha\;,\;\;\;\;\;\alpha\in\langle0,\Re(\beta)\rangle\;.
\zeta_{3,m}^{\nu}(r,\theta,\phi) := \xi(r)r^{\sqrt{1-\nu^2}-1}\begin{pmatrix}\nu \Omega_{\frac{1}{2}+m_2,m_1,-m_2}(\theta,\phi) -\mathrm{i}\big(\sqrt{1-\nu^2}+(-1)^{\frac{1}{2}-m_2}\big) \Omega_{\frac{1}{2}-m_2,m_1,m_2}(\theta,\phi) \end{pmatrix};
\|\phi\|_{\ast\ast}:=\sum_{i=1}^{k}\|\phi\|_{C^{2m,\alpha}(r_{i}<2)}+\sum_{i=1}^{k}\sum_{|\alpha|\leq 2m-1}\|r_{i}^{|\alpha|}D^{\alpha}\phi\|_{L^{\infty}(r_{i}\geq 2)}.
c_1v_1^2+c_2\left(v_1-\tilde{d}_1\right)^2=\left(c_1+c_2\right) \left(v_1-\frac{\tilde{d}_1c_2}{c_1+c_2}\right)^2+\frac{\tilde{d}_1^2c_1c_2}{c_1+c_2}
D_t (P\widetilde F_i-F_i) =(D_tP)\widetilde F_i +PD_t\widetilde F_i -D_t F_i &=(D_tP)\widetilde F_i +P \widetilde \nabla_i \widetilde H-\nabla_i H &=(D_t P)\widetilde F_i+(\tilde g^{kl}-g^{kl})(P\widetilde\nabla_i \widetilde A_{kl})+g^{kl}(P\widetilde\nabla_i \widetilde A_{kl}-\nabla_i A_{kl}).
\det A(a_1,\dots,a_n) =\det(a_{j+1}^{i+1}-a_1^{i+1})_{i,j=1}^{n-1}=\det\left(\left(a_{j+1}-a_1\right) \left(\sum_{k=0}^{i}a_{j+1}^{k}a_1^{i-k}\right)\right)_{i,j=1}^{n-1}= \prod_{j=2}^{n}(a_{j}-a_1) \det\left(\sum_{k=0}^{i}a_{j+1}^{k}a_1^{i-k}\right)_{i,j=1}^{n-1}.
{\hat A}_{T,t}^{(i_1 i_2)}=\frac{T-t}{2\pi}\sum_{r=1}^{\infty}\frac{1}{r}\left(\zeta_{2r}^{(i_1)}\zeta_{2r-1}^{(i_2)}-\zeta_{2r-1}^{(i_1)}\zeta_{2r}^{(i_2)}+\sqrt{2}\left(\zeta_{2r-1}^{(i_1)}\zeta_{0}^{(i_2)}-\zeta_{0}^{(i_1)}\zeta_{2r-1}^{(i_2)}\right)\right).
r\partial_r (r^{-1}n_{ab}) = R_{Lab\underline L}- (l_b^c+r^{-1}\delta_b^c) n_{ac}+ \nabla_a \eta_b - \eta_a\eta_b = W_{Lab\underline L}+ \sigma_{ab}- (l_b^c+r^{-1}\delta_b^c) n_{ac}+ \nabla_a \eta_b - \eta_a\eta_b.
\rho=\frac{V}{\mathcal{E}}=\frac{2\mu V}{\hbar^{2}k_{\mathrm{o}}^{2}}\approx\frac{3\sqrt{\Sigma}}{2\sqrt{\pi}k_{\mathrm{o}}^{2}L^{3}}\approx\frac{3a_{\mathrm{s}}}{k_{\mathrm{o}}^{2}L^{3}}
(1-t^{2}\mathbf{h}(x)\mathbf{e}(xy))^{-1}(1+t\mathbf{h}(x))=\frac{1}{1-t}+\sum_{n=1}^{\infty}\sum_{L\vDash n}\frac{N_L}{(1-t)(1-t^{2})^{n}}x^{n}\mathbf{r}_{L}
\left| \langle\partial^{m}\partial_l^j \mathcal F(h,h), \, f\rangle_{L^2_{x,v}}\right| \leq\begin{cases}\displaystyle\mathcal G_{x,v,z}^{s,m}(h,h)\, ||f||_{\Lambda}\,, j \neq 0, begin{align*}2pt] \displaystyle \mathcal G_{x,z}^{s,m}(h,h)\, ||f||_{\Lambda}\,, \qquad j=0.\end{cases}
Z_{Q,(\epsilon_{\tau},\epsilon_{\alpha},\epsilon_{\beta})}(\lambda )=\mathbf{A}_{(\epsilon_{\tau},\epsilon_{\alpha},\epsilon_{\beta})}(\lambda )(\lambda )Q(\lambda -\eta )+\mathbf{A}_{(\epsilon_{\tau},\epsilon_{\alpha},\epsilon_{\beta})}(-\lambda )Q(\lambda +\eta ).
0\neq \frac{24}{7}F\left(\frac{11}{3},\frac{1}{3},\frac{13}{3};\frac{1}{\lambda}\right)(\lambda^2-\lambda+1)-\frac{10}{21}F\left(\frac{8}{3},\frac{1}{3},\frac{10}{3};\frac{1}{\lambda}\right)\left(\lambda+1 \right)(7\lambda^2-10\lambda+7)+ &2F\left(\frac{5}{3},\frac{1}{3},\frac{7}{3};\frac{1}{\lambda}\right)\lambda(1-\lambda)^2,
\lbrack{H_{0}^{-}, G_{r i}^{-}}\rbrack = 0 , \quad \lbrack{{\bf P}_{0}^{-}, Q_{r i}^{-}}\rbrack = 0 , \quad \lbrack{{\bf P}_{0}^{-}, G_{r i}^{-}}\rbrack = 0 .
\Big(1+ 2^{p-1}\displaystyle\sum_{n= 3}^{\infty}\frac{1}{n^p}\Big)\langle f(A)\hat{k}_\lambda, \hat{k}_\lambda \rangle^p< \Big(\frac{p}{p-1}\Big)^{p}\sup_{\lambda \in \Omega}|\langle f(A)^p\hat{k}_\lambda, \hat{k}_\lambda \rangle|.
\Gamma_{\rm{b4}}\left(a\right)=\left({\cal C}_{66}a+{\cal C}_{44}\right)\left[\left({\cal C}_{13}+{\cal C}_{44}\right)\left(e_{15}a+e_{33}\right)-\left({\cal C}_{44}a+{\cal C}_{33}\right)\left(e_{31}+e_{15}\right)\right]
T\left(\mathcal{F}^{\sharp}_{[N_1,h_2]}\right)~=~\left\{\varphi_{f^{\sharp}}~\Big|~f^{\sharp}\in \mathcal{F}^{\sharp}_{[N_1,h_2]}\right\}~\subseteq~\mathcal{I}_{[N_1,h_2]}~.
\begin{cases}\|e^\pm_{k}\|_{s,X}\le C\delta^2\theta_k^{L_4(s)}\Delta_k, s_0\le s\le s_1-2\cr\|\overline{e}^{\pm}_k\|_{s,X}\le C\delta^2\theta_k^{L_6(s)}\Delta_k, s_0+\frac 32\le s\le s_1-\frac 72\cr\|\tilde{e}_{k}\|_{H^s(\omega_X)}\le C\delta^2\theta_k^{L_5(s)}\Delta_k, s_0+1 \le s\le s_1-4\end{cases}
\eta(\eta(\tau_j, \tau_i),\eta(\tau_j,\tau_\ell)) = \eta((\tau_n^q\tau_1)^{n-j}\tau_{{n-j+i}},(\tau_n^q\tau_1)^{n-j}\tau_n^{q-1+D(n-j+\ell)}\tau_{{n-j+\ell+1}}) &=\eta(\tau_{{n-j+i}},\tau_n^{q-1+D(n-j+\ell)}\tau_{{n-j+\ell+1}}).
\Gamma_{\gamma_0, \omega}^+(t):=\int_{L^+_{\gamma_0}(t)}|f(\theta, \omega, t)|^2 ~ d\theta = \int_{\phi(t) - \frac{\pi}{2}+ \gamma_0}^{\phi(t) + \frac{\pi}{2}- \gamma_0}|f(\theta,\omega, t)|^2 ~ d\theta.
\partial_{\tau}\langle\Psi_{j}^{\tau},\Phi_{\ell}^{\tau}\rangle\big|_{\tau = 0}= 0 \implies \left\langle\partial_{\tau}\Psi_{j}^{\tau}\big|_{\tau = 0}, \Phi_{\ell}^{0}\right\rangle = - \left \langle\Psi_{j}^{0},\partial_{\tau}\Phi_{\ell}^{\tau}\big|_{\tau = 0}\right \rangle.
_0 \left< \right| T \left( \Psi ( x_{2}) \bar{\Psi}( x_{1}) \right) \left| \right>_{0}= \frac{1}{( 2 \pi )^{3}}\int{\mathrm d}^{3}p \frac{- i \hat{p}+ m}{2 E}e^{i p ( x_{2}- x_{1})}
\lvert A\psi_{u}(s)(y)\rvert =\left\lvert\partial_{yy}\psi(s)(y)+\left(\frac{d-1}{y}-\frac{y}{2}\right)\partial_{y}\psi_{u}(s)(y)+\frac{d-1}{y^{2}}\psi_{u}(s)(y)\right\rvert \lesssim y^{-2}+y^{2}.
|w_{n,R}(x)| \leq 2^{-d}|R|^{1/2}\prod_{i=1}^d \bigg(|g_{I_i}\bigl( |I_i|(x_i +e_{n_i,I_i}) \bigr)|+ |g_{I_i}\bigl( |I_i|(x_i -e_{n_i,I_{i}})|\bigr) \bigg) \leq 2^{-d}|R|^{1/2}\sum_{m=1}^{2^d}|G_R(\Delta(x+U_m e_{n,R}))|,
\widehat{\deg}_+(\overline V):=\sup_{W\subset V}\widehat{\deg}(\overline W)=\int_0^{+\infty}\dim_{k}(\mathcal F^t(\overline V))\,\mathrm{d}t, \widehat{\deg}(\overline V)=-\int_{\mathbb R}\,t\,\mathrm{d}\dim_k(\mathcal F^t(\overline V))\,\mathrm{d}t.
[L^p(\tilde{\Omega}), H_{b.c.}^{2,p}(\tilde{\Omega})]_{\theta}= \begin{cases}H^{2\theta,p}(\tilde{\Omega}) \hbox{all b.c.}, 1+1/p <\theta \leq 2, H^{2\theta,p}(\tilde{\Omega}) \hbox{only Dirichlet part}, 1/p <\theta \leq 2, H^{2\theta,p}(\tilde{\Omega}) \hbox{without b.c.}, 0 \leq \theta \leq 1/p.\end{cases}
\rho (r)\rho (m) = \alpha (r)\alpha (m)\sum_{e}\bigg(\chi (\xi (\textnormal{Gr}_{e-[\textnormal{Ker}\,G\mu ]}(Ga)) + \chi (\nu (\textnormal{Gr}_{e}(Gb)))\bigg)\beta (e) & = \alpha (r)\alpha (m)\sum_{e}\bigg(\chi (\textnormal{Gr}_{e-[\textnormal{Ker}\,G\mu ]}(Ga)) + \chi (\textnormal{Gr}_{e}(Gb))\bigg)\beta (e).
R \ll \sum_{\substack{x/2 < \l_n < 2x n \ne x}}|a_n| \min \bigg( 1, \frac{x}{T|x-\l_n|}\bigg) + \frac{4^{\sigma_0}+x^{\sigma_0}}{T}\sum_{n=1}^{\infty}\frac{|a_n|}{\l_n^{\sigma_0}}.
\alpha(x) = \frac 1{\mathsf{d}(x)}\alpha(\mathsf{n}(x)) = \frac 1{\mathsf{d}(x)}\sum_{i=1}^k n_i \alpha (s_i) = \frac 1{\mathsf{d}(x)}\sum_{i=1}^k n_i s_i \frac{\alpha (s_j)}{s_j}= xa.
S \, \simeq \, \frac{M_{0}^{2}}{2}\, \int_{t_{1}}^{t_{2}}\! d t \int_{0}^{R}\! d r \, \left[ \dot{\bf{u}}^{2}( t , r ) \, - \,{\bf{u}}^{' 2}( t , r ) \right] \, + \, \sum_{a = 1}^{2}\, \frac{m_{a}}{2}\, \int_{t_{1}}^{t_{2}}\! d t \, \dot{\bf{u}}^{2}( t , r_{a}) \,{,}
L_n \le \big(d^{\alpha_0}_n + |T-\bar t_n|^{\frac{\alpha_0}\gamma}\big) \frac{M_n^{\frac2\gamma+\alpha_0\frac{\gamma-1}\gamma}}{r_n^{\alpha_0}}= \big(d^{\alpha_0}_n + |T-\bar t_n|^{\frac{\alpha_0}\gamma}\big) \frac{M_n}{r_n^{\alpha_0}}\le 2L_n.
\bar{\eta}{}^{A}= \frac{i}{\sqrt{2}}\left[ \begin{array}{l}{\operatorname{c o s}\left( \frac{\tau}{2}- \frac{\pi}{4}\right) \mathrm{e}^{\hspace{2 m m}i \sigma / 2}}{\, \operatorname{s i n}\left( \frac{\tau}{2}- \frac{\pi}{4}\right) \mathrm{e}^{- i \sigma / 2}}\end{array}\right] .
R ( \phi ) D ( u , v ) = \left( \begin{matrix}{{\cos \phi}}&{{- \sin \phi}}&{{0}}&{{0}}\\{{\sin \phi}}&{{\cos \phi}}&{{0}}&{{0}}\\{{u}}&{{v}}&{{1}}&{{0}}\\{{u}}&{{v}}&{{0}}&{{1}}\\ \end{matrix}\right) .
\delta\left( \alpha\right) \left[ \psi\left( \mathrm{i}\alpha\right)-\int K\left( \alpha,s\right) \mathrm{d}\mu\left( s\right) \right]+\mu^{\prime}\left( \alpha\right) & =\widetilde{\delta}\left( \alpha\right) \left\{\psi\left(\mathrm{i}\alpha\right) -\int K\left( \alpha,s\right) \mathrm{d}\mu\left(s\right) \right\},
\begin{aligned}P(t,T) = \exp \left\{- \frac{a}{b}\left( T-t-C(t,T)\right) - C(t,T)r(t) \right\}&\qquad\qquad\qquad\times E^Q\left.\left[\exp\left\{\sigma\int_{t}^{\infty}f(s)dZ(s)\right\}\right|\mathcal{F}_t\right].\end{aligned}
C_{t+1}= C_t \cup \{e_t, e_{t, 1}, \dots, e_{t, k_t - 1}\}, A_{t+1}= A_t \setminus \{e_t, e_{t, 1}, \dots, e_{t, k_t - 1}\}\bigcup_{j=1}^{k_t -1}\left ( \Delta^{(n)}_{u_{t,j}}\cap U_t \right ), U_{t+1}= U_t \setminus \bigcup_{j=1}^{k_t - 1}\Delta^{(n)}_{u_{t, j}}, N_{t+1}= N_t \cup \{u_{t, 1}, \dots, u_{t, k_t - 1}\}.
\underset{k=i\kappa}{\rm Res}\tilde{M}^{(1)}(x,t,k)=\frac{\gamma_0}{a_{1}'(i\kappa)\delta^2(i\kappa,\xi)}e^{-2\kappa x+8\kappa^3t}\tilde{M}^{(2)}(x,t,i\kappa), \gamma_0^2=1,
\left( Q_{2}^\dagger \right)^{31}u_B^- ({\bf r}) \propto \sum_{n=0}^\infty{(-1)^n \over (2 n)!! (2n + 2 \left( 2 \ell +1 - \lambda \right))!!}\left( \sqrt{E}r\right)^{2n + \left( 2 \ell +1 - \lambda \right)}e^{-i \left( 2 \ell +1 \right) \theta}
n-1 =H(V_{1}^{n}) =H(V_{m_{n}+1}^{n-m_{n}})+H(V_{1}^{m_{n}}|V_{m_{n}}^{n-m_{n}})+H(V_{n-m_{n}+1}^{n}|V_{1}^{n-m_{n}}) \geq H(V_{m_{n}+1}^{n-m_{n}})+H(V_{1}^{m_{n}}|V_{m_{n}}^{n})+H(V_{n-m_{n}+1}^{n}|V_{1}^{n-m_{n}}).
\left(\kern-.1em\left(\kern-.1em\mathbf{u},\mathbf{v}\kern-.1em\right)\kern-.1em\right) :=k^{2}\left( \mathbf{u},\mathbf{v}\right) +\operatorname*{i}kb_{k}\left( \mathbf{u}^{\nabla},\mathbf{v}^{\nabla}\right) .