harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
WN(\theta;\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\sum_{k=-\infty}^{\infty}exp[\frac{-(\theta-\mu-2\pi k)^{2}}{2\sigma^{2}}]=\frac{1}{2\pi}\vartheta(\frac{\theta-\mu}{2\pi},\frac{i\sigma^{2}}{2\pi})
d_n=d_{n-1}+b_{n-1}-b_{n-2}+\binom{n-3}{2}-\binom{n-3}{5}+\sum_{a=3}^{n-3}\sum_{\ell=0}^{n-3-a}\sum_{m=1}^{n-2-a-\ell}\binom{n-5-\ell-m}{a-3}m,
f_{0}( N ; x , p ) ={\frac{1}{\pi}}\! \int_{0}^{\infty}\! \! \! \! d y ~{\frac{\cos ( y p )}{( \cosh ( 2 x / \hbar ) + \cosh ( y ) )^{N}}}={\frac{1}{( N - 1 ) !}}~ \left( \frac{- \hbar}{2 \sinh ( 2 x / \hbar )}\partial_{x}\right)^{N - 1}f_{0}( 1 ; x , p ) ,
\begin{array}{c}\chi (a,b)=\chi (a)+\chi (b)+n_{2}(a)n_{1}(b) \chi (c,d)=\chi (c)+\chi (d)+n_{2}(c)n_{1}(d) \chi (a,d)=\chi (a)+\chi (d)+n_{2}(a)n_{1}(d) \chi(b)=\chi(c) (\epsilon_{(b,c)})_{n}=(-)^{n_{2}(b)}q^{\frac{1}{2}(n_{1}(b)-n_{2}(b))}\end{array}
M(\theta_{1},\theta_{2})=[\begin{matrix}(m(\theta_{1})+m(\theta_{2}))/2\\ (m(\theta_{1})-m(\theta_{2}))/(\theta_{1}-\theta_{2})\end{matrix}]=C_{0}[\begin{matrix}\mu\\ 1\end{matrix}]+\sum_{k=1}^{n}C_{k}F_{k}(\theta_{1},\theta_{2}),
q_{k+1}(\lambda) = \lambda * q_{k}(\lambda) - q_{k}(\lambda)\times \boldsymbol{\alpha}_{k}^H - q_{k-1}(\lambda)\times \boldsymbol{\beta}_{k}^H p_{k+1}(\lambda)\times\boldsymbol{\beta}_{k+1}= \lambda * p_{k}(\lambda) - p_{k}(\lambda)\times\boldsymbol{\alpha}_{k}- p_{k-1}(\lambda),
A-B = \sum_{j=2}^{i}x_{j-1}n_{j-1}\Delta x_j \int_{x_{i+1/2}-x_j}^{x_{\alpha_{i,j}-1/2}}(v-x_{\alpha_{i,j}-1})f'(x_{j-1},x_{\alpha_{i-1,j-1}-1}) dv &-\sum_{j=2}^{i}x_{j-1}n_{j-1}\Delta x_{j-1}\int_{x_{i-1/2}-x_{j-1}}^{x_{\alpha_{i-1,j-1}-1/2}}(v-x_{\alpha_{i-1,j-1}-1}) f'(x_{j-1},x_{\alpha_{i-1,j-1}-1}) dv +{\cal O}(\Delta x^3).
K_{\hat{w}_\lambda}\subseteq[\hat{v}_\lambda^*,\hat{u}_\lambda^*]\cap C^1(\overline{\Omega}),\; K_{\hat{w}_\lambda^+}=\{0,\hat{u}_\lambda^*\},\;K_{\hat{w}_\lambda^-}=\{0,\hat{v}_\lambda^*\}.
\begin{aligned}ric_g =\kappa(n-1)g_\kappa+ +h^{-2}\Big\{(n-2)h(\nabla^2h)_{g_\kappa}+\Big[h(\Delta h)_{g_\kappa}-(n-1)\|(\nabla h)_{g_\kappa}\|^2\Big]g_\kappa\Big\}, \end{aligned}
\|u\|_k=\Big( (\|u^{[0]}\|_{[0]k})^2+(\|u^{[1]}\|_{[1]k})^2\Big)^{1/2}, &\|\vec{u}\|_k=\Big( (\|y\|_{k+1})^2+(\|v\|_k)^2\Big)^{1/2}
7{R^6 \over N}(\partial_\rho R)^2 - 2 \partial_\rho({R^7 \partial_\rho R \over N}) + 7 N R^6-{9 \over 144}{R^8 \over N}(\partial_\rho \phi)^2 + 9 ({Q \over \pi^5})^2{N e^{-2 \phi}\over R^{10}}=0.
\frac{1}{2^{i}}K^{n_{i-1}}\left( E^{0}(S_{2^{i}}^{2})\right) =\frac{1}{2^{i}}\sum\nolimits_{j=1}^{2^{i}}K^{j}\left( K^{n_{i-1}}(V_{0}^{2}\right) ) & +2\sum\nolimits_{k=0}^{i-1}\frac{1}{2^{i-k}}\left( \sum\nolimits_{j=1}^{2^{i-k}-1}K^{(j-1)2^{k+1}}\right) \left( \frac{1}{2^{k}}K^{n_{i-1}}(g_{k})\right) .
=B^\alpha{\cal T}_\alpha+\frac{1}{2}B^\beta B^\alpha U^\gamma_{\alpha\beta}\Pi_\gamma(-1)^{\varepsilon_\beta+\varepsilon_\gamma}+\Pi^\alpha_*\bar{\cal P}_\alpha(-1)^{\varepsilon_\alpha}-\Pi_*^\beta B^\alpha U^\gamma_{\alpha\beta}B^*_\gamma(-1)^{\varepsilon_\beta},
\int_{E_{N}}\rho^{n-1}\,d\rho=\sum_{k=1}^{N}\int_{k}^{k+k^{-n-1}}\!\!\!\rho^{n-1}\,d\rho\le\sum_{k=1}^{\infty}(k+1)^{n-1}k^{-n-1}<C.
\sum\limits_{k=0}^{l}{l\choose k}\left(-\frac{\sigma(T^{2H}-r^{2H})}{(T-r)^{H}}\right)^kh_{l-k}\left( \frac{B_{T}^{H}-B_{r}^{H}}{(T-r)^{H}}\right)=h_l\left(\frac{-\sigma(T^{2H}-r^{2H})+B_T^H-B_r^H}{(T-r)^H}\right).
S^*(X,\mathbf{d},\mathbf{d}) =C^*(\mathbf{d})\operatorname{vol}(\mathcal{R})X^2(\log X)^3+ O_\varepsilon((d_1d_2d_3)^{\varepsilon-1}X^2 (\log X)^2)+O_\varepsilon((d_1d_2d_3)^\varepsilon X^{\frac{23}{12}+\varepsilon}).
F_{2}( q^{2}) ={\frac{M e^{2}}{4 \pi^{2}}}\int_{0}^{1}d \alpha \, \int_{0}^{1}d x \, \,{\frac{m - x M}{\alpha ( 1 - \alpha ) \,{\frac{1 - x}{x}}\,{\vec{q}}_{\perp}^{2}- M^{2}+{\frac{m^{2}}{x}}+{\frac{\lambda^{2}}{1 - x}}}}\ .
\mathcal{J}(u)=\frac{1}{2}\, \int_{\mathbb{R}^N}|\nabla u|^2\, dx+\frac{c_2^{2-q}}{q}\, \int_{\mathbb{R}^N}|u|^{q}\, dx-\int_{\mathbb{R}^N}\frac{u}{(1+|x|^2)^{\alpha/2}}\, dx,
\left( \xi_{,t}+\dot{x}^{r}\xi_{,r}\right) \left( \frac{1}{2}g_{ij}\dot{x}^{i}\dot{x}^{j}-V\right) =\frac{1}{2}\left( \xi_{,t}g_{ij}\dot{x}^{i}\dot{x}^{j}-2V\xi_{,t}+g_{ij}\xi_{,k}\dot{x}^{i}\dot{x}^{j}\dot{x}^{k}-2\dot{x}^{k}\xi_{,k}V\right)
\lambda_n(c) \sim \widetilde{\lambda_n}(c) = \frac{1}{2}\exp\left(-\frac{\pi^2(n+\frac 12)}{2}\int_{\Phi\left(\frac{2c}{\pi (n+\frac 12)}\right)}^1 \frac{1}{t(\mathbf E(t))^2}\,\mathrm{d}t \right).
\pi_a^j\circ \iota_a^i = (\pi_1^{\lfloor \frac{j}{n_2}\rfloor}\otimes \pi_2^{j\;\mathrm{mod}\;n_2})\circ (\iota_1^{\lfloor \frac{i}{n_2}\rfloor}\otimes \iota_2^{i\;\mathrm{mod}\;n_2}) &=(\pi_1^{\lfloor \frac{j}{n_2}\rfloor}\circ \iota_1^{\lfloor \frac{i}{n_2}\rfloor})\otimes (\pi_2^{j\;\mathrm{mod}\;n_2}\circ \iota_2^{i\;\mathrm{mod}\;n_2}),
B_{j,m}(f,g)(x)=2^{-\frac{(b-a)j}{2}}\iint \hat{f}(\xi)\hat{g}(\eta)e^{2\pi i\left(\frac{\xi}{2^{(b-a)j}}+\eta\right)x}I_{\rho,m}\hat{\Phi}(\xi)\hat{\Phi}(\eta)\,d\xi\,d\eta,
L(x^a,D^a_{\;\;b}x^b) \rightarrow L(x'^a,D'^a_{\;\;b}x'^b) = L(G^a_{\;\;b}x^b,G^a_{\;\;b}D^b_{\;\;c}x^c) &= L(x^a,D^a_{\;\;b}x^b),
A' \ = \ 2^{n-1}\sum_{\omega = 0}^{a-1}2^{-\omega}(-1)^{\omega}I(\omega, 0) \sum_{\alpha = 0}^\omega \binom{\omega}{\alpha}\sum_{m = 0}^{a - 1 - \omega}\binom{n}{m +\omega}\binom{m+\omega}{m}\sum_{i=0}^m (-1)^i \binom{m}{i}.
B^{k}\triangleq \left\lbrace w \in X^k: \| w \|_{X^k}\le 1 \right\rbrace, & X^k \triangleq \Big\{w: \left\langle \nabla \right\rangle w \in S([t_k, +\infty), L^2 ): \| w \|_{X^k}= \sup_{t \ge t_k}\| \left\langle \nabla \right\rangle w \|_{S( [t, + \infty), L^2)}e^{(k+\frac{1}{2}) e_0 t}< \infty \Big\},
\frac{d}{dy}2E(\frac{1}{2}\sqrt{2})-K(\frac{1}{2}\sqrt{2})-2E[arccos(xy);\frac{1}{2}\sqrt{2}]+F[arccos(xy);\frac{1}{2}\sqrt{2}]=\frac{\sqrt{2}x^{3}y^{2}}{\sqrt{1-x^{4}y^{4}}}
VII_{A}=\varepsilon\int_{t}^{T}\int_{\mathbb{T}^{d}}(\partial^{\alpha}\partial_{x_{j}}w^{n+1})\sum_{i=1}^{d}\left[\left(\Theta_{p_{i}}(\tau,x,\mu^{n},Dw^{n})\right)\left(\partial^{\alpha}\partial_{x_{i}x_{j}}^{2}w^{n}\right)\right]\dxd\tau,
W = \lambda [ 1 / 6 ( A A_{s y m})^{3}( Q{\bar{Q}})^{2}+ 4 ( A A_{s y m}) ( A A Q{\bar{Q}}_{a n t i})^{2}+ 6 4 ( A A A{\bar{Q}}{\bar{Q}}) ( A A A Q Q ) - \Lambda^{8}( Q{\bar{Q}}) ] ,
\|g(|\lambda_k \cdot|)\|_{L^q(B_{2\rho}(0))}=(4\pi\rho^2)^{\frac1q}\bigg(\dfrac1{|B_{2\rho\lambda_k}(0)|}\int_{B_{2\rho\lambda_k}(0)}|g(y)|^qdy\bigg)^{\frac1q}\leq(4\pi\rho^2)^{\frac1q}g(0)+\varepsilon
\aligned \widetilde{\mathcal W}{}^i_{jmn}=R^i_{jmn}+\frac1{N+1}\delta^i_jR_{[mn]}-\frac12\big(F^i_j\sigma_m+F^i_m\sigma_j\big)_{|n}+ \frac12\big(F^i_j\sigma_n+F^i_n\sigma_j\big)_{|m}-\delta^i_{[m}\zeta_{jn]}, \endaligned
L^{\psi}V ={\binom{L_{0}\L_{1}}{L_{1}\L_{0}}}^{\psi}{\binom{V_{0}}{V_{1}}}={\binom{\psi_{0 0}L_{0}\ \psi_{1 1}L_{1}}{\psi_{1 0}L_{1}\ \psi_{0 1}L_{0}}}{\binom{V_{0}}{V_{1}}}.
(cf_2^v)^4(a_j^h)^2\left(\sum_{i=j+1}^{m_1}\frac{(a_i^h)^2}{(1+(cf_2^v)^2B_i^h)(1+(cf_2^v)^2B_{i+1}^h)}\right)+\frac{1+(cf_2^v)^2B_j^h}{1+(cf_2^v)^2B_{j+1}^h}=1-\frac{(cf_2^va_j^h)^2}{1+(cf_2^v)^2b_1^h},
\partial_{t}\varphi + \frac{1}{2m}({\mbox{\boldmath $\nabla$}}\varphi)^2 + \frac{\hbar^2}{4m}\left[\frac{1}{2}\left(\frac{{\mbox{\boldmath $\nabla$}}\rho}{\rho}\right)^2 - \frac{\triangle \rho}{\rho}\right] + U = 0
f_{10}f_{15}t = f_{19}\overline{c}= f_{22}\overline{b}\mbox{where}f_{20}\overline{c}\in f_{15}\overline{A}^\ast \mbox{and}f_{23}\overline{b}\in f_{26}(\widehat{C}\cap f_{25}^{-1}f_{15}\overline{A}^\ast).
-{\cal A}_{\alpha}i \Omega ={\cal D}_{\alpha}{\cal V}\Omega{\cal V}^{T}= \left( \begin{matrix}{{\langle{\cal D}_{\alpha}U_{\beta}, U_{\gamma}\rangle}}&{{\langle{\cal D}_{\alpha}U_{\beta}, \bar{U}^{\gamma}\rangle}}\\{{\langle{\cal D}_{\alpha}\bar{U}^{\beta}, U_{\gamma}\rangle}}&{{\langle{\cal D}_{\alpha}\bar{U}^{\beta}, \bar{U}^{\gamma}\rangle}}\\ \end{matrix}\right)
- \frac{1}{x_{1 2}^{4}x_{3 4}^{4}}( 2 0 - 2 0 Y - \frac{1 6}{3}v + \frac{1 0}{3}Y^{2}+ \frac{8}{3}v Y + \frac{1}{3}v^{2}) \Phi ( v , Y )
M_{S}(Z_0,J,\rho_1 \rho_2)=\|f_0\|^2_{S,\rho_1 \rho_2}=\|F_2\|^2_{S,\rho_1 \rho_2}+\|f_1f_2\|^2_{S,\rho_1 \rho_2}\ge M_{\partial D_1}\times M_{S_1}.
w\left( re^{i\varphi}\right) =w\left( r\cos\varphi,r\sin\varphi\right) :={\displaystyle\sum_{k=0}^{\infty}}{\displaystyle\sum_{\ell=1}^{a_{k}}}w_{\left( k,\ell\right)}\left( r\right) Y_{\left( k,\ell\right)}\left( \varphi\right) .
W=\int_{t_{1}}^{t_{2}}F\cdot vdt=\int_{t_{1}}^{t_{2}}Fvdt=\int_{t_{1}}^{t_{2}}mavdt=m\int_{t_{1}}^{t_{2}}v\frac{dv}{dt}dt=m\int_{v_{1}}^{v_{2}}vdv=\frac{1}{2}m(v_{2}^{2}-v_{1}^{2}).
\exp_{\*}\left( it\left( x+p^{2}\right)\right) =2\pi \int_{-\infty}^{\infty}dE\,e^{iEt}\,\frac{2^{2/3}}{2\pi}\,\mathrm{Ai}\left( 2^{2/3}\left( x+p^{2}-\frac{E_{1}+E_{2}}{2}\right)\right) =\exp \left( it\left( x+p^{2}+t^{2}/12\right) \right) .
\gamma^{0}= \sigma_{2}= \left( \begin{array}{c c}{{0}}&{{- i}}\\{{i}}&{{0}}\\ \end{array}\right) \hspace{1 c m}\gamma^{1}= i \sigma_{1}= \left( \begin{array}{c c}{{0}}&{{i}}\\{{i}}&{{0}}\\ \end{array}\right)
\frac{1}{8 \pi}\int_{\Sigma_r}(h_0^{(1)}-h^{(1)}) dS^2 = \frac{4\pi}{3}T(e_0, e_0) \frac{1}{8 \pi}\int_{\Sigma_r}\tilde X^i \tilde \nabla^a (\alpha_H^{(2)}-\alpha_{H_0}^{(2)})_a dS^2 = \frac{4\pi}{3}T(e_0, e_i)
\Gamma_2(x_1+x_2-2q\,|\,a) = 2^{-B_{2, 2}(x_1+x_2-2q\,|\,a)/2}\Gamma_2((x_1+x_2)/2-q\,|\,a)\times \times\Gamma_2((x_1+x_2+a_1)/2-q\,|\,a)\times\Gamma_2((x_1+x_2+a_2)/2-q\,|\,a)\times \times\Gamma_2((x_1+x_2+a_1+a_2)/2-q\,|\,a).
\zeta^{\,-\ln N}\phi_N(\xi) = 1 + \left[i \xi + O\left(\frac{1}{N}\right)\right]\int_0^{\infty}e^{[i \xi \, + \, O(N^{-1})] \, s}\left[1 - \left(1 - e^{- s}\right)^N\right] ds &= 1 + i \xi \int_0^{\infty}e^{i \xi s}\left[1 - \left(1 - e^{- s}\right)^N\right] ds + O\left(\frac{1}{N}\right)
\m^{2}( \beta ) = m_{0}^{2}+ \frac{\lambda_{0}}{( 2 \pi )^{D / 2}}\sum_{n = 1}^{\infty}\biggl ( \frac{m ( \beta )}{\beta n}\biggr )^{\frac{D}{2}- 1}K_{\frac{D}{2}- 1}\left( m ( \beta ) n \beta \right) .
\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{F}(t) = 2\,\mathrm{Re}\langle\dot{\phi},\ddot{\phi}\rangle + 2\,\mathrm{Re}\left\langle -\frac12\Delta\phi+V\phi+\beta|\phi|^2\phi-\Omega L_z\phi,\, \dot{\phi}\right\rangle &= 2\,\mathrm{Re}\left\langle\dot{\phi},-\eta(t)\dot{\phi}+\lambda_{\phi}(t)\phi\right\rangle &= - 2\,\eta(t)\|\dot{\phi}\|^2.
d s^{2}= 2 e^{2 \phi}d \tilde{u}( d \hat{v}+ \omega_{i}d \tilde{x}^{i}) + \xi^{2}d \tilde{u}^{2}- \sum_{i = 1}^{i = 3}d \tilde{x}^{i}d \tilde{x}^{i}- ( d \hat{x})^{2}- \sum_{i = 5}^{i = 8}d \tilde{x}^{i}d \tilde{x}^{i}\ ,
\frac{\partial\,L_{n}}{\partial\,t}=\frac{1}{2}\left(p\,\frac{\partial\,L_{n}}{\partial\,p}-n\,\left(L_{n}-u\right)\right)\,u_{x}-\frac{\partial\,L_{n}}{\partial\,x}\,u-v^{n-1}\,v_{x}
\lim\limits_{N \rightarrow \infty}\mathbb{P}(R_{N}\leq N^{1+\delta}) = \lim\limits_{N \rightarrow \infty}\mathbb{P}( \sigma^{+}(N^{\gamma}+ 1) = \sigma^{+}(N^{\gamma})) \leq \lim\limits_{N \rightarrow \infty}\frac{N^{\gamma}}{N}= 0
\underline{f}(t,x,\cdot,\cdot,\cdot)=f_1(t,x,\cdot+u_2,\cdot+\nabla u_2,\cdot+v_2)-f_2(t,x,u_2,\nabla u_2,v_2) &\underline{g}(t,x,\cdot,\cdot,\cdot)=g(t,x,\cdot+u_2,\cdot+\nabla u_2,\cdot+v_2)-g(t,x,u_2,\nabla u_2,v_2) &\underline{G}=G_1-G_2.
\nabla^2 \Psi_r (a,b) = \frac{2r(2r-1)|b|^{2r-2}}{a^{2r+1}}\begin{pmatrix}|b|^2 -ab^* - ab \frac{a^2}{2r-1}\big(I_3+(2r-2)\frac{bb^*}{|b|^2}\big)\end{pmatrix}\geq \frac{2r(2r-1)|b|^{2r-2}}{a^{2r+1}}\begin{pmatrix}|b|^2 -ab^* - ab a^2 \frac{bb^*}{|b|^2}\end{pmatrix}
E \cdot \nabla_v \left( \partial_{v_i}F \right) + \nu \left( \partial_{v_i}F \right) = \int \sigma ( v, v') F ( v') \d v' \, \partial_{v_i}M ( v) + \int \partial_{v_i}\sigma ( v, v') F ( v') \d v' M( v)- \left( \partial_{v_i}\nu \right) F .
\tilde{\mathbf{y}}_{k}=\sqrt{P_{T}/2N}G_{R}\mathbf{H}_{k}^{H}\mathbf{\breve{E}}_{k}\mathbf{\breve{d}}_{+,k}+G_{R}\mathbf{H}_{k}^{H}\mathbf{n}_{R}+\mathbf{n}_{k},
A(y+1) ={\rm argmin}\Bigl\{Tz + W(z) +{1\over 2}\gamma (z-y-1)^2\Bigr\}=1+{\rm argmin}\Bigl\{T(z-1) + W(z-1) +{1\over 2}\gamma (z-y)^2\Bigr\}=1+{\rm argmin}\Bigl\{Tz + W(z) +{1\over 2}\gamma (z-y)^2\Bigr\}=1+ A(y).
\delta =\epsilon^0\,\underline \epsilon_0+\epsilon^+\,\underline\epsilon_+ +\epsilon^-\,\underline \epsilon_- +\alpha^I\,\underline \alpha_I+\beta_I\,\underline \beta^I+\hat\alpha{}^I\,\underline{\hat \alpha}_I+\hat\beta_I\,\underline{\hat\beta}{}^I+\omega^i\,\underline \omega_i\ .
\int_{a}^x \frac{dt}{\sqrt t \log(t)^2}= \int_{\log a}^{\log x}\frac{e^{u/2}}{u^2}\, du &=\int_{\log a}^{\log a + i\infty}\frac{e^{u/2}}{u^2}\, du - \int_{\log x}^{\log x + i\infty}\frac{e^{u/2}}{u^2}\, du.
\Lambda(\mu_{1},\dots,\mu_{n}) :=(\Lambda_{n})_{*}(\mu_{1}\times\cdots\times\mu_{n}),{\mathcal A}(\mu_{1},\dots,\mu_{n}) :=({\mathcal A}_{n})_{*}(\mu_{1}\times\dots\times\mu_{n}),{\mathcal H}(\mu_{1},\dots,\mu_{n}) :=({\mathcal H}_{n})_{*}(\mu_{1}\times\dots\times\mu_{n}).
P_+ = \frac{f(X_-^*(y^*),y^*)}{f(X_-^*(y^*),y^*) + f(X_+^*(y^*),y^*)}, P_- = \frac{f(X_+^*(y^*),y^*)}{f(X_-^*(y^*),y^*) + f(X_+^*(y^*),y^*)}.
P_{1}^{\mathrm{PNC}}(g_{1r},g_{2r}) =\left[\beta_1^* \log_2 e - \frac{1}{2}\left( \frac{1}{g_{1r}}+\frac{1}{g_{2r}}\right)\right]^+ , P_{1i}^{\mathrm{PNC}}(g_{1r},g_{2r}) =\left[\beta_1^* \log_2 e \frac{g_{3-i,r}}{g_{1r}+g_{2r}}-\frac{1}{2g_{ir}}\right]^+, P_2^{\mathrm{PNC}}(g_{r1},g_{r2}) =\left[\beta_2^*\log_2 e - \frac{1}{g_{2r}}\right] ,
S^{c}={\cal C}{\cal K}= e^{i \vartheta_{[ 1 ]}^{c}}\left( \begin{matrix}{0}&{\Theta_{[ 1 ]}}\\{- \Theta_{[ 1 ]}}&{0}\\ \end{matrix}\right){\cal K}, \quad \Theta_{[ 1 ]}= \left( \begin{matrix}{0}&{0}&{1}\\{0}&{- 1}&{0}\\{1}&{0}&{0}\\ \end{matrix}\right)
\mathbb{P}\left\lbrace X_s^{\omega^\prime}=Y_s^{\omega^\prime}\right\rbrace = \lim_{R\to\infty}\left[\mathbb{P}\left\lbrace X_s^{\omega^\prime}=Y_s^{\omega^\prime}, s\le \tau(R)\right\rbrace+\mathbb{P}\left\lbrace s>\tau(R)\right\rbrace\right]=1
U^-_1 = \Big( U \setminus \{u_2\}\Big) \cup \{q_i^{-1}u_2, q_{s}u_2, q_s^{-1}u_2, q_i u_2 \}W^-_1 = W \cup \{u_2, q_i^{-1}q_{s}^{-1}u_2, q_i q_s u_2 \},
T^k P_{\mathcal{K}_i}x_n\xi_\phi = u^k \cdot (P_{\mathcal{K}_i}x_n\xi_\phi)\cdot u^{-k}= P_{u^k\cdot\mathcal{K}_i\cdot u^{-k}}u^k x_n u^{-k}\xi_\phi = P_{T^k\mathcal{K}_i}u^k x_n u^{-k}\xi_\phi.
L^{s,(N)}_{\theta_i}=4\cos^2\left(\frac{\theta_i}{2}\right)\frac{d^2}{d\theta_i^2}+\left[\left(-4N-4\Re(s)\right)\sin\left(\frac{\theta_i}{2}\right)\cos\left(\frac{\theta_i}{2}\right)+4\Im(s)\cos^2\left(\frac{\theta_i}{2}\right)\right]\frac{d}{d\theta_i},
\begin{aligned}c'_{\alpha_2}+ 2 c'_{\alpha_1+2\alpha_2}+ c'_{\alpha_1+\alpha_2}= \# )\ +\ 2 \# \Big ) = y, c'_{\alpha_1}+ c'_{\alpha_1+\alpha_2}+ c'_{\alpha_1+2\alpha_2}= \# (\ +\ \# \Big (= x.\end{aligned}
\Omega_{2 g + k - l}={\frac{( - 1 )^{l + 1}}{2 g + 1 + k - l}}\sum_{j = 0}^{2 g + 1}( - 1 )^{j}\, ( 2 g + 1 - j ) \, s_{j}\,{\frac{\partial}{\partial s_{l}}}\Omega_{2 g + k - j}.
W_{k = 0}^{\beta}= \mathrm{}- \frac 1 2 \int_{0}^{\infty}\! \frac{{\mathrm{d}}s}{s}{\mathrm e}^{- s m^{2}}\Big ({\mathrm T r}{K}^{\beta}( s ) |_{k = 0}-{\mathrm T r}K ( s ) |_{\phi = 0}\Big ) \Big |_{m^{2}= 0}.
\mbox{Agent}~1:\,\begin{cases}\dot{x}_{11}=x_{12}\dot{x}_{12}=\Delta_1+u_1 y_1=x_{11}\end{cases}\mbox{Agent}~2:\,\begin{cases}\dot{z}_{21}=-c_2z_{21}+b_2x_{21}\dot{x}_{21}=\Delta_2+u_2 y_{2}=x_{21}, \, c_2>0\end{cases}\mbox{Agent}~3:\,\begin{cases}\dot{x}_{31}=x_{32}\dot{x}_{32}=\Delta_3+u_3 y_3=x_{31}.\end{cases}
\textrm{E}[ \| I_2^n(\theta',\theta) \|^2] &\leq \widetilde{C}(\|b_n\|_{L_{\infty}^{\infty}}) E\left[ \left\|\sum_{m=1}^{\infty}\int_{\Delta_{\theta,t}^m}\prod_{j=1}^m b_n'(s_j,x+B_{s_j}^H) \left( K_H(s_m,\theta')I_d - K_H (s_m,\theta)I_d \right) ds_m \cdots ds_1\right\|^4 \right]^{1/2},
\rho^\ast \mathrm{d}L^\textup{hor}(a) + \overline{\nabla}_a^\ast \mathrm{d}L^\textup{ver}(a) = 0 , \rho^{\prime \ast}\mathrm{d}L^{\prime \textup{hor}}(a^\prime ) + \overline{\nabla}_{a^\prime}^{^\prime \ast}\mathrm{d}L^{\prime \textup{ver}}(a^\prime ) = 0 .
\Psi(t) = -iM S_{\pm\xi}(-t) P_\mu^2 \psi_{\mp}^\mu(t) + i S_{\pm\xi}(-t) P_\mu ( \Theta P_\mu \phi^\mu \cdot \Theta P_\mu \psi_{\mp}^\mu)(t) - M_{\mathfrak K_1}S_{\pm\xi}(-t) P_\mu^2 \psi_\pm^\mu(t)
4\sum_{j,k=1}^\infty \int_0^\infty m^s \int_{|x|>R}\partial^2_{jk}\varphi_R \partial_j \overline{u}_m(t) \partial_k u_m(t) dx dm & = 8s\|u(t)\|^2_{\dot{H}^s(|x|>R)}-4 \int_0^\infty m^s \int_{|x|>R}(2-\varphi''_R) |\nabla u_m(t)|^2 dx dm &\leq 8s \|u(t)\|^2_{\dot{H}^s(|x|>R)}.
\nabla_H \varphi_n (p_n) = \Bigg( \frac{x_\xi}{n}+2n y_\xi (x_{p_n}y_\xi -y_{p_n}x_\xi)- n y_{p_n}z_{p_n}, \frac{y_\xi}{n}-2nx_\xi (x_{p_n}y_\xi -y_{p_n}x_\xi)+ n x_{p_n}z_{p_n}\Bigg).
h_{n,k}(s) = 2^{\frac{k+2s}{2}[\frac{n}{2}]}\prod_{i=0}^{n-1}\xi (2s+k-2i) \cdot \prod_{i=0}^{n-1}\prod_{j=1}^{\frac{k}{2}+n-1-i}(s+k-2i-j) \cdot \prod_{i=0,\,}^{n-1}{(1-2^{2i-k-2s})},
c_{2i+3}& =\frac{1}{2}\frac{\left( N\kappa-\kappa+m+2+3i\right)}{N\kappa+m+2+i}c_{2i+1}-\frac{1}{4}\frac{\left( 2i+1\right) i}{\left(N\kappa+m+2+i\right) \left( N\kappa+m+1+i\right)}c_{2i-1}.
{\frac{1}{2 \pi}}\sum_{l = 1 / 2}^{\infty}2 l e^{-{\frac{l^{2}}{\rho^{2}}}t}={\frac{\rho^{2}}{4 \pi t}}+{\frac{\rho^{2}}{( 4 \pi t )^{\frac{3}{2}}}}\mathrm{P}\int_{- \infty}^{\infty}d x ({\frac{x}{2 \rho}}\mathrm{c o s e c}{\frac{x}{2 \rho}}- 1 ) e^{-{\frac{x^{2}}{4 t}}},
\mathcal{S}_k \coloneqq \left\{\mathbf{P}_k \in \mathcal{W}_{k}(0): \mathbf{p}_{k-1,1}=\mathbf{p}_{k-1,2},~\mathbf{p}_{k-1,3}=\mathbf{p}_{k-1,4}\right\}, &= \big\{\mathbf{P}_k \in \mathcal{W}_{k}(0): (\mathbf p_{k-1,1}\cup \mathbf p_{k-1,2}) \cap (\mathbf p_{k-1,3}\cup \mathbf p_{k-1,4})=\varnothing \big\}.
\varphi^{(p,q)}(z)=\frac{(-1)^{q+\nu+1}}{q}(1-z\bar{z})z^{m+\nu-p+1}\bar{z}^{m+\nu-q+1}\sum_{j=0}^{\nu}(-1)^j\frac{(j+\nu+m+1)!}{j!(j+m)!(\nu-j)!}(z\bar{z})^j,
\begin{cases}\partial_t \psi(x,t) + \Delta \psi(x,t)= -\theta(x,t), (x,t)\in\Omega_{\tau,T}, \nabla\psi(x,t) \cdot \nu(x) = 0, (x,t)\in \partial\Omega\times(\tau,T), \psi(x,T) = 0, x\in\Omega.\end{cases}
\frac{f^{\prime}\left( f^{\prime \prime}\right)^{2}}{f^{2}f^{\prime\prime \prime}}=\frac{\left( g^{\prime \prime}\right)^{2}}{g^{\prime\prime \prime}}\left( \frac{g^{\prime \prime \prime}}{g^{\prime}g^{\prime\prime}}\right)^{\prime}.
\bar r_1\big(a_{1,\inf}-k\frac{\chi_1}{d_3}\big) = a_{0,\sup}-a_{2,\inf}\big\{\frac{b_{0,\inf}-b_{1,\sup}\bar r_1-l\frac{\chi_2}{d_3}\bar r_2}{b_{2,\sup}-l\frac{\chi_2}{d_3}}\big\}\,\,\,\, -k\frac{\chi_1}{d_3}\big\{\frac{a_{0,\inf}-a_{2,\sup}\bar r_2-k\frac{\chi_1}{d_3}\bar r_1}{a_{1,\sup}-k\frac{\chi_1}{d_3}}\big\}.
\begin{aligned}I_\nu(z) = \frac{(z/2)^{\nu}}{\Gamma(1+\nu)}(1+\mathcal{O}(z^2)), \nu \geq 0, K_\nu(z) = \frac{\Gamma(\nu)\,(z/2)^{-\nu}}{2}\, (1+\mathcal{O}(z^2)) - \frac{\pi\, (z/2)^{\nu}}{2\sin(\nu\pi)\, \Gamma(\nu+1)}\, (1+\mathcal{O}(z^2)), \nu>0, \end{aligned}
\int_{a}^{b}\psi^{*}\frac{d^{2}\psi}{dx^{2}}dx=[\psi^{*}\frac{d\psi}{dx}]_{a}^{b}-\int_{a}^{b}\frac{d\psi^{*}}{dx}\frac{d\psi}{dx}dx=[\psi^{*}\frac{d\psi}{dx}]_{a}^{b}-\int_{a}^{b}|\frac{d\psi}{dx}|^{2}dx
\tilde{\Psi}(k, \tau) = e^{\tau\mu_{0}\partial_{k}}\tilde{\Phi}(k, \tau) = \tilde{\Phi}(k + \mu_{0}\tau, \tau) & = e^{-i\int_{0}^{\tau}\!\! \sqrt{1 + (k + \mu_{0}\tau')^{2}}d\tau'}\tilde{\Psi}_{0}(k + \mu_{0}\tau).
\frac{\Gamma(k-1)}{(4\pi\sqrt{mn})^{k-1}}\sum_{f\in F}\overline{\hat{f}}(m)\hat{f}(n)=\delta_{mn}+2\pi i^{-k}\sum_{c>0}\frac{S(m,n;c)}{c}J_{k-1}(\frac{4\pi\sqrt{mn}}{c}),
\lim_{n\to +\infty}{\rm Prob}\left(c_n^{(j)}x^*>s\right)=\lim_{n\to +\infty}\det\left(1-\left.K_n^{(j)}\right|_{[0,s/c_n^{(j)}]}\right) = \det\left(1-\left.\mathbb K^{(j)}\right|_{[0,s]}\right).
\sum_{i=0}^{n-1}\phi(x_i,i+k) = \sum_{i=0}^{n-1}\phi(z_{i+k},i+k) + \sum_{i=0}^{n-1}\left(\phi(x_i,i+k)-\phi(z_{i+k},i+k)\right) &= n +\sum_{i=0}^{n-1}\chi_{[1]}(\sigma^{i+k}z) - \sum_{i=0}^{n-1}\mathfrak{d}(\sigma^ix,\sigma^{i+k}z),
K^{\pm 1}m^k = (\lambda q^{-2k})^{\pm 1}m^k , begin{align*}1ex]Fm^k = \frac{\lambda q^{-k}- \lambda q^{k}}{q-q^{-1}}\lambda q^{-2k-1}m^{k+1}, begin{align*}1ex] Em^{k}= [k] \lambda^{-1}q^{2k-1}m^{k-1}.
(T_\lambda g)(x) = \frac{1}{2\alpha\sqrt{\lambda}}\left(u(x)\int_{-\infty}^{x}\!v(y)\operatorname{sgn}(y)g(y)\,\mathrm dy+ v(x)\int_{x}^{\infty}\!u(y)\operatorname{sgn}(y)g(y)\,\mathrm dy\right).
\mathbb{E}(Z_t) = \mathbb{E}(Y_t)+\sum_{u=1}^{a}\sum_{M\in\mathcal{M}_{t,u}}\mathbb{E}(X_M) &= 2^{-t}\sum_{u=1}^{a}\binom{n}{u}\sum_{i=0}^{b}i\binom{t}{b-i}&+2^{-t(n-k)}\sum_{r=0}^{n-k}(n-k-r){n-k \brack r}_2\prod_{i=0}^{r-1}(2^t-2^i).
\frac{\partial}{\partial t}\nabla_i\nabla_jw= \nabla_i\nabla_j\frac{\partial w}{\partial t}-\big(\frac{\partial}{\partial t}\Gamma^k_{ij}\big)\nabla_kw = \nabla_i\nabla_j\big((1+\frac{\alpha'}{4}R+\frac{\alpha'}{4}a)\Delta w-af\big) &+\frac{1}{2}\big(1+\frac{\alpha'}{2}R\big)\big(\nabla_iR\nabla_jw+\nabla_jR\nabla_iw-\langle\nabla R,\nabla w\rangle g_{ij}\big),
F_Y\Bigl(\sum_{i=1}^{k}\frac{t_i}{10^i}+\eta\Bigr) \le\prod_{i=1}^{k-4}\Bigl( G(t_i,\dots,t_{i+4})+O(10^{i-1}\eta) \Bigr) &=(1+O_J(Y\eta))\prod_{i=1}^{k-4}G(t_i,\dots,t_{i+4}).
\left\|N(h,\mu){\rm Op}_h(\chi)U^{-1}{\rm Op}_h(\phi\varrho^k)U{\rm Op}_h(\psi)-\sum_{j=0}^{s}h^j{\cal T}_j^{(k)}\right\|\lesssim h^{s+1}|{\rm Im}\,z|^{-(3s+2-k)/2}.
\Psi_{\kappa}^{( P )}= \frac{i}{2^{3 / 2}}\exp \left( - \frac{\pi \kappa}2 - i \frac{\kappa}2 \ln \left( \frac{- x_{+}}{- x_{-}}\right) \right) H_{i \kappa}^{( 1 )}\left( m \sqrt{( - x_{-}) ( - x_{+})}\right) .
S_{(\lambda,\mu),(\lambda',\mu')}= \frac{\sum_{w\in W}(-1)^{l(w)}\xi^{\langle 2\rho+\lambda,(w\bullet(\lambda',\mu'))_2\rangle + \langle\mu,(w\bullet(\lambda',\mu'))_1+2\rho\rangle}}{\sum_{w\in W}(-1)^{l(w)}\xi^{\langle 2\rho,w\bullet 0\rangle}}.
y_{2i-1,4i}\leq \frac{1}{\sqrt{2\pi i}}(1-z_{2i-1,4i}^2)^{-1/4}\leq \frac{1}{\sqrt{2\pi i}}\Big(1-\cos^2\Big(\frac{2i-1}{4i+\frac{1}{2}}\pi\Big)\Big)^{-1/4}&=\Big(2\pi i\sin\Big(\frac{2i-1}{4i+\frac{1}{2}}\pi\Big)\Big)^{-1/2}.
\|T_g(s_n)\|_{\infty}\geq \limsup_{t \to 1^{-}}\textnormal{Re}\left(\int_{0}^{t}{f_n(re^{i\theta_n})g^{\prime}(re^{i\theta_n})r^{k_n}dr}\right) &= \textnormal{Re}\left(\int_{0}^{1}{f_n(re^{i\theta_n})g^{\prime}(re^{i\theta_n})r^{k_n}dr}\right),
\begin{array}{cccc}({\sf JB'})^\circ={\sf KB}, ~~ ({\sf JTB'})^\circ={\sf TB}, ({\sf JDB'})^\circ={\sf DB}, ({\sf JB4'})^\circ=({\sf JB5'})^\circ=({\sf JB45'})^\circ={\sf KB5}, \multicolumn{4}{l}{({\sf JTB5'})^\circ=({\sf JDB5'})^\circ=({\sf JTB45'})^\circ=({\sf JDB45'})^\circ=({\sf JTB4'})^\circ=({\sf JDB4'})^\circ={\sf S5}.}\end{array}
\mathrm{Res}_{s=1/2}\mathrm{I}_k= \mathrm{Res}_{s=1/2}\int_{K}Z_{\leq d}(f_{k\rho},k\varphi^{U_k,\psi_k},k{\varphi'}^{U_k,\psi_k^{-1}},s)\, dk.
\sigma^{0 k}\equiv i \alpha^{k}= i \left( \begin{array}{c c}{0}&{\sigma^{k}}\\{\sigma^{k}}&{0}\\ \end{array}\right) ; \sigma^{i j}= \epsilon_{i j k}\left( \begin{array}{c c}{\sigma^{k}}&{0}\\{0}&{\sigma^{k}}\\ \end{array}\right) ,
\mathcal{R}_1^{\mu}[\beta b, h, \overline{V}] = - \frac{h^2}{2}(\nabla_X \cdot \overline{V})^2 - \frac{1}{3h}\big{(}\nabla_X(h^3 \nabla_X \cdot \overline{V})\big{)}\cdot \overline{V}- \frac{1}{2}h^3\Delta_X(|\overline{V}|^2) + \frac{1}{6h}h^3 \Delta_X (|\overline{V}|^2).
\mathcal{E}_{k}^{(m)}(Z,s)=\sum_{M=\binom{*\;*}{C\,D}\in\Gamma_{\boldsymbol{K},\infty}^{(m)}\backslash\Gamma_{\boldsymbol{K}}^{(m)}}(CZ+D)^{-k}|(CZ+D)|^{-s}, & (Z,s)\in \mathcal{H}_m\times\mathbb{C}.
\int_{0}^{\infty}{\frac{x^\nu J_\nu (x)}{e^{2\pi x/b}-1}F_1(x)dx}=\frac{2}{\pi}\sum_{k=0}^{\infty}{'}(bk)^\nu K_\nu (bk)F_1(ibk)-\frac{1}{\pi}\int_{0}^{\infty}{x^\nu K_\nu (x)F_1(ix)dx},
q\geq q_0:=4+\frac{m}{\log{\left(\frac{1}{\l}\right)}},\quad\mbox{where}m=\max{\left\{2,\frac{Q}{\log{6}},\frac{2\sigma_C^2}{\log{6}},\frac{Q\log{2}}{\log{8}},\frac{2Q\log{(\frac{n+1}{\sigma})}}{\log{8}}\right\}},
\begin{aligned}S(U,f^n) \leq S\!\left(\sqrt{R},f^n\right)\leq \frac{2T_0(R,f^n)}{\log R}\leq \frac12 \left(T(R,f^n)+C\right) \leq \frac12 \left(\log M(R,f^n)+C\right)\leq \log M^n(R,f)\end{aligned}
Z_{N}=\sum_{\sigma,\sigma^{\prime},\sigma^{\prime\prime},\sigma^{\prime\prime\prime}}A_{\sigma|\sigma^{\prime}}A_{\sigma^{\prime}|\sigma^{\prime\prime}}A_{\sigma^{\prime\prime}|\sigma^{\prime\prime\prime}}A_{\sigma^{\prime\prime\prime}|\sigma}=trA^{4}.
h_{\i}^{\underline{{a}}}\dot{\xi}_{\alpha}^{\underline{{a}}}- \varepsilon_{i j}B_{\alpha}^{\underline{{a}}}\xi_{\alpha}^{\underline{{a}}}\dot{x}^{j}-{\cal E}_{j ; \alpha}^{\underline{{a}}}\xi_{\alpha}^{\underline{{a}}}= 0 .
\begin{aligned}b_{\gamma_+}(0_n,\theta) =\lim\limits_{t\rightarrow\infty}[s_E(y)-s_{p_0}(y)] &=\lim\limits_{t\rightarrow\infty}[t-\cosh^{-1}(\cosh t\cdot\cosh(\lambda|\theta-\theta_0|))] = -\ln\cosh (\lambda|\theta-\theta_0|).\end{aligned}