harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_s^t \delta \tilde{B}_{sr}^i dZ_r^j - \int_s^t \delta B_{sr}^i dZ_r^j = (\delta \tilde{B}_{st}^i - \delta \tilde{B}_{st}^i) \delta Z_{st}^j + \int_s^t \delta Z^j_{sr}dB^i_r - \int_s^t \delta Z^j_{sr}d\tilde{B}^i_r & = - \int_s^t u(r,X_r)^i dr \delta Z_{st}^j + \int_s^t \delta Z_{sr}^i u(r,X_s)^j dr
\hat{H}_0=\hbar g\omega\left[{\alpha^a_i}^\dagger\alpha^a_i+\frac{1}{2}2d\right]=\hbar g\omega\left[\hat{N}+d\right]\ \ ,\ \ \hat{\phi}=i\hbar\epsilon^{ab}{\alpha^a_i}^\dagger\alpha^b_i\ \ .
F_{max}(r_G(t))=\frac{S_{max}}{\sqrt{2\pi}}\int_0^{r_G(t)}\frac{e^{-\frac{S^2_{max}}{2s}}}{s^{\frac{3}{2}}}ds F_{min}(r_G(t))=\frac{S_{min}}{\sqrt{2\pi}}\int_0^{r_G(t)}\frac{e^{-\frac{S^2_{min}}{2s}}}{s^{\frac{3}{2}}}ds.
\alpha = \sum_{i<j}\epsilon_i \epsilon_j \alpha_{ij}e^{\flat}_i \wedge e^{\flat}_j \Rightarrow \alpha_E = \sum_{i<j}\epsilon_i \epsilon_j \alpha_{ij}E_{ij}\beta = \sum_{k<l}\epsilon_k \epsilon_l \beta_{kl}e^{\flat}_k \wedge e^{\flat}_l \Rightarrow \beta_E = \sum_{k<l}\epsilon_k \epsilon_l \beta_{kl}E_{kl}.
g^{\mu \nu}( t , \vec{x}) \equiv{\frac{1}{c^{2}}}\left[ \begin{matrix}{- 1}&{\vdots}&{- v \hat{r}^{j}}\\{\cdots \cdots}&{\cdot}&{\cdots \cdots \cdots \cdots}\\{- v \hat{r}^{i}}&{\vdots}&{( c^{2}\delta^{i j}- v^{2}\hat{r}^{i}\hat{r}^{j})}\\ \end{matrix}\right] .
P(P^{-1}([P^{-1}(x),y])) = [P^{-1}(x),y]= [P^{-1}(x),P(P^{-1}(y)] = [P(x'),P(P^{-1}(y)]= P([P(x'),P^{-1}(y)]) = P([P^{-1}(x),P^{-1}(y)]).
M = \frac{i}{\sqrt{{\mathrm{\boldmath ~ \beta ~}}^{2}}}\left( \begin{array}{l l}{E_{\mathrm{\boldmath ~ \beta ~}}}{- E_{-{\mathrm{\boldmath ~ \mu ~}}}}{E_{\mathrm{\boldmath ~ \mu ~}}}{\, \, \, E_{-{\mathrm{\boldmath ~ \beta ~}}}}\end{array}\right) \, .
\phi^{(+)}_0(x)^2 =\phi^{(+)}_0(0)^2\frac{q^x(bq;q)_x}{(q,-c^{-1}q;q)_x}, \phi^{(-)}_0(x)^2=\phi^{(-)}_0(0)^2\frac{q^x(-bcq;q)_x}{(q,-cq;q)_x}.
K_n(x,y) = \frac{\theta}{(2\pi i)^2}\int_{c+i\mathbb R}ds \int_{\Sigma_n}dt \frac{x^{-\theta s-1}y^{\theta t}}{s-t}n^{-\theta s+\theta t}\frac{\Gamma(s+1)\Gamma(\alpha+1+\theta s)\Gamma(t-n+1)}{\Gamma(t+1)\Gamma(\alpha+1+\theta t)\Gamma(s-n+1)},
\mathcal{N}_{n}(u) =\Vert u\Vert_{0}+N_{n}^{-\sigma}\Vert u\Vert_{\sigma}\,E_{N_{n+1}}\mathcal{N}_{n}^{\prime}(v) =\Vert v\Vert_{0}^{\prime}+N_{n}^{-\sigma}\Vert v\Vert_{\sigma}^{\prime}\,E_{N_{n+1}}^{\prime}
\alpha_{\tau}\beta_{\tau}+\alpha_{\tau}\gamma_{\tau}+\beta_{\tau}\gamma_{\tau}=\frac{1}{4}(\lambda^{2}a^{4}+\mu^{2}b^{4}+\nu^{2}c^{4}-2\lambda\mu a^{2}b^{2}-2\lambda\nu a^{2}c^{2}-2\mu\nu b^{2}c^{2}).
=\sum_{i=1}^n(-1)^{i-1}e_{x_1}\cdots e_{x_{i-1}}e_{x_{i+1}}\cdots e_{x_n}-\sum_{i=1}^n(-1)^{i-1}e_{x_1}\cdots e_{x_{i-1}}x_ie_{x_{i+1}}\cdots e_{x_n}.
\lbrace{\cal F}_{i},{\overline{{Q}}}_{a}\rbrace^{\ast}= \frac{i}{2}{( \sigma_{i})}_{a}^{c}{\overline{{Q}}}_{c}, \quad \lbrace{\cal F}_{i}, Q^{a}\rbrace^{\ast}= - \frac{i}{2}{( \sigma_{i})}_{c}^{a}Q^{c}.
\psi\bigl(s\sigma^{\nu}_{-i}(y)\bigr) =(\psi\otimes\nu)\bigl((\Delta s)(1\otimes\sigma^{\nu}_{-i}(y))\bigr)=(\psi\otimes\nu)\bigl((1\otimes y)(\Delta s)\bigr) &=(\psi\otimes\nu)\bigl(\Delta(ys)\bigr)=\psi(ys).
I{}=\int\limits_0^{T}(T-u) \bigl(e^{-2au}+e^{-a| u+\tau|}e^{-a| u-\tau|}\bigr)du{}=\int\limits_0^{T}Te^{-2au}du+T\int\limits_0^{T}e^{-a| u+\tau|}e^{-a| u-\tau|}du -\int\limits_0^{T}ue^{-2au}du \quad{}-\int\limits_0^{T}ue^{-a| u+\tau|}e^{-a| u-\tau|}du{}=I_1+I_2+I_3+I_4.
u_{0}^{\prime}=\epsilon,u_{1}^{\prime}=\epsilon(2c-1),\u_{n+2}^{\prime}=(2c-2)u_{n+1}^{\prime}-u_{n}^{\prime},\n\geq0, & z_{0}=\epsilon,z_{1}=\epsilon(2c-3),\z_{n+2}=(2c-2)z_{n+1}-u_{n},\n\geq0.
\sum_{m=-\infty}^{\infty}\sum_{n=1}^{\infty}\,\ln(\bar\omega_m^2\,+\,k_n^2)\,=\,\frac{2}{T}\,\sum_{n=1}^{\infty}\left[\frac{k_n}{2}\,+\,T\,\ln(1\,-\,e^{-k_n/T})\right]\,{,}
\frac{1}{g_{Y M}^{\ 2}}= \frac{\left( \alpha^{\prime}\right)^{\frac{3 - p^{\prime}}{2}}}{\left( 2 \pi \right)^{p^{\prime}- 2}G_{s}}= \frac{\left( \alpha^{\prime}\right)^{\frac{3 - p^{\prime}}{2}}}{\left( 2 \pi \right)^{p^{\prime}- 2}g_{s}}\left( \frac{\det ( g + 2 \pi \alpha^{\prime}B )}{\det G}\right)^{\frac{1}{2}}~ .
\det(M) = \det(P)^{-1}\cdot \det(\int_{T_{n_a}c}2i\pi F_b(z)dz)_{1 \leq a,b \leq g}=\det(P)^{-1}\cdot \det(\int_{c}2i\pi a_{n_a}(F_b)\cdot F_b(z)dz)_{1 \leq a,b \leq g}& \sim \det(a_{n_a}(F_b))_{1 \leq a,b \leq g}^{-1}\cdot \det(\int_{c}2i\pi a_{n_a}(F_b)\cdot F_b(z)dz)_{1 \leq a,b \leq g}
f(u,v)+g(u,v)=-\frac{E^\prime}{2cE}\left(cu-c(2\alpha+1)v+(u^2-v^2)(u+v)\right), f(u,v)-g(u,v)=-\frac{E^\prime}{2cE}\left(cu+c(2\alpha+1)v+(u^2-v^2)(u-v)\right).
\beta(n,s)=\frac{-i}{2\pi}\int\! d\alpha\frac{e^{-in\alpha}}{\sqrt{4\omega_n\omega_s}}\left\{i\omega_n f_s^{+\ast}(t'(0,\alpha))e^{-is\alpha'(0,\alpha)}-\partial_t\left\{f_s^{+ \ast}(t'(t,\alpha))e^{-is\alpha'(t,\alpha)}\right\}\left|_{t=0}\right.\right\}
(-a)*a_{m-k}=-a_{m-k+1}-(-a)*a_{m-k+1}=-a_{m-k+1}+\sum_{j=1}^{k-1}(-1)^{j+1}a_{m-k+1+j}=-a_{m-k+1}+\sum_{j=2}^k(-1)^ja_{m-k+j}=\sum_{j=1}^k(-1)^ja_{m-k+j}.
\begin{aligned}\|z\|_{C^{\frac{1+\beta_0}{2},1+\beta_0}(D)}\leq C_2(\|F_2\|_{L^p(D)}+ \|w\|_{L^p(D)}+ \|z_0\|_{W^{2,0}_{p}([0,1])}) & \leq C_2(\|F_2\|_{C^0(D)}+ \|w\|_{C^0(D)}+ \|z_0\|_{C^{2}([0,1])}),\end{aligned}
(1, \mu)=1, (\vert v_i \vert^2, \mu)=1, (\vert v \vert^2, \mu)=3, &(\vert v_i \vert^2 \vert v_j \vert^2, \mu)=1\ (i \neq j), (\vert v_i \vert^4, \mu ) =3, &(\vert v \vert^2 \vert v_i \vert^2, \mu) = 5, (\vert v \vert^4, \mu) = 15, ( \vert v\vert^4 \vert v_i \vert^2 \mu )=35.
\begin{array}{c c}{{\displaystyle{\langle \phi^{\alpha}( z_{1}) \phi^{\beta}( z_{2}) \rangle = C_{\phi}\frac{I^{\alpha \beta}( \hat{{\cal X}}_{1 2})}{( \det{\cal X}_{1 2})^{\frac{1}{2}\eta}}}~}}&{{~ I^{\alpha \beta}( \hat{{\cal X}}_{1 2}) = \hat{{\cal X}}_{1 2}^{\alpha \beta}}}\\ \end{array}
g_{2a+1}^j =\frac{(-1)^{\frac{j-1}{2}+a+\frac{k}{2}}j!}{(k+2a+1)\prod\limits_{d=0,\;d\neq a}^{\lfloor l/2\rfloor-1}((k+2a+1)^2-(k+2d+1)^2)}\sum_{b=0}^{(j-1)/2}\frac{\prod\limits_{d=0,\;d\neq a}^{\lfloor l/2\rfloor-1}((2b+1)^2-(k+2d+1)^2)}{\prod\limits_{d=0,\;d\neq b}^{(j-1)/2}((2b+1)^2-(2d+1)^2)},
\left\{\begin{array}{c}{{( m - 1 ) \partial_{A}\partial_{B}x^{j}+ L^{i}{}_{j k}( x , y ) \partial_{A}x^{i}\partial_{B}x^{j}= 0}}\\{{( n - 1 ) \partial_{A}\partial_{B}y^{b}+ C^{a}{}_{b c}( x , y ) \partial_{A}y^{a}\partial_{B}y^{b}= 0}}\\ \end{array}\right\}
\gamma_j(r\xi)=\frac i{2\pi}r d\bar\xi_j\wedge \beta_{n-1}(r\xi)= \frac i{2\pi}\left(r\sum_{m=1}^n \bar v^m_j d\bar\alpha_j\right)\wedge \beta_{n-1}(r\alpha)=\sum_{m=1}^n \bar v^m_j \gamma_m(r\alpha)\ .
\mathrm{t r}\left[ Q \, \Gamma^{, \rho}\Gamma^{, \sigma}\right] = - 4 k^{\rho}k^{\sigma}t^{2}\xi^{- 2}- \frac{2}{k^{2}}P^{\rho \sigma}\left[ \cosh \left( k^{2}( - 1 + \xi^{- 1}) t \right) - 1 \right]
\hat{F}_{\infty}(z) - \frac{\hat{A}_{n}(z)}{R_{n}(z)}= \frac{(1+c^2)\gamma_{n}}{4d_1 r_{n,n}}\, \frac{1}{z^{n+1}}+ O\big((1/z)^{n+2}\big), n \geq 1.
{u_2}_x=- \frac{1}{W^{(2)}_{u_1 u_1 u_1}}\, ,{u_1}_x =-\frac{W^{(2)}_{u_1 u_1 u_1 u_1}}{(W^{(2)}_{u_1 u_1 u_1})^2}\, , &{u_2}_t=-\frac{u_1}{W^{(2)}_{u_1 u_1 u_1}}\, ,{u_1}_t=-\frac{u_1 W^{(2)}_{u_1 u_1 u_1 u_1}}{(W^{(2)}_{u_1 u_1 u_1})^2}-\frac{1}{W^{(2)}_{u_1 u_1 u_1}}\, .
\tilde{C}^{(q,r)}(b;h)=\tilde{g}^{(q)}(b;h)=\dfrac{\Xi^{(q,r)}(b;h)-\dfrac{\rho^{(q)}_{b}(b;h)}{W^{(q)}(b)}Z^{(q)}(b;\Phi(q+r))}{\displaystyle\dfrac{Z^{(q)}(b;\Phi(q+r))}{W^{(q)}(b)}-\dfrac{r}{\Phi(q+r)}},
\varphi_{k,k-1}(\gamma_1,\beta_1,\omega_k,\alpha_2,\alpha_3,\alpha_5) := \Big(\Phi_{1}^{(1)}(\gamma_1;V_a),\Phi_{1}^{(2)}(\gamma_1;V_a)\Big)\cdot \textbf{n}_{k}-\Big(\Phi_{1}^{(1)}(\beta_1;\tilde{\Phi}(\alpha_5,\alpha_3,\alpha_2;V_a)),\Phi_{1}^{(2)}(\beta_1;\tilde{\Phi}(\alpha_5,\alpha_3,\alpha_2;V_a))\Big)\cdot\textbf{n}_{k-1}.
=\sum_{l=0}^{\min \left( k,m\right)}c_{l}\left( \left( \frac{k}{n}\right)^{l}- \frac{k!\left( n-l\right) !}{n! \left( k-l\right) !}\right) \left(b-a\right)^{l}+\sum_{l=k + 1}^{m}c_{l}\left(\frac{k}{n}\right)^{l}\left(b-a\right)^{l}.
\lambda_{1}\left( B\right) =\cdots=\lambda_{k}\left( B\right)=\frac{n}{\sqrt{2k}}, \lambda_{n-k+1}\left( B\right) =\cdots=\lambda_{n}\left( B\right)=-\frac{n}{\sqrt{2k}}, \lambda_{i}^{\ast}\left( B\right) =0k<i\leq n-k
\Delta_h [S,a] =\Delta_h Sa-\Delta_h aS=S\Delta_h a-\Delta_h aS &=S([\Delta_h,a]+a\Delta_h)-([\Delta_h,a]+a\Delta_h)S &=S((\Delta_h a)\delta_h+a\Delta_h)-((\Delta_h a)\delta_h+a\Delta_h)S &=[S(\Delta_ha)\delta_h-(\Delta_ha)S\delta_h]+[Sa\Delta_h-aS\Delta_h]=[S,\Delta_h a]\,\delta_h +[S,a]\,\Delta_h.\blacksquare
r_{+}(k)=-r^*_{-}(k), k\in I_{R}, r_{+}(k)=\frac{1}{r_{-}^{*}(k)}, \vert r_{\pm}(k) \vert=1, k\in I_{L}\backslash I_{R}, r_{+}(k)=r_{-}(k), k\in \mathbb{R}\backslash(I_L\cup I_R).
X = \sum_{i \in \mathcal{I}}X_i, \mu = \mathbb{E}[X] = \sum_{i \in \mathcal{I}}p^{e(H_i)}, \delta = \sum_{\substack{(i, j) \in \mathcal{I}\times \mathcal{I}H_i \sim H_j}}\mathbb{E}[X_i X_j] = \sum_{\substack{(i, j) \in \mathcal{I}\times \mathcal{I}H_i \sim H_j}}p^{e(H_i) + e(H_j) - e(H_i \cap H_j)}
\frac{1}{n}(q^{0})^* q^{t+1}\overset{\mathbf{. .}}{=}\tilde{E}_{0,t+1}, \frac{1}{n}(q^{r+1})^*q^{t+1}\overset{\mathbf{. .}}{=}\tilde{E}_{r+1,t+1}, \frac{1}{n}(m^r)^* m^t \doteq \breve{E}_{r,t}.
-\left[J^1(x)\right]_{-L/2}^{L/2}= \left[(/\hspace{-0.5em}\partial\phi + U)\gamma^1\psi - (-/\hspace{-0.5em}\partial\phi + U)\gamma^1\gamma^3\psi\right]_{L/2}= \left.(1 + \gamma_3)(/\hspace{-0.5em}\partial\phi + U)\gamma^1\psi \right|_{L/2}.
\| S(t)g \|_{L^{q}_{t}L^{r}_{x}}\lesssim \|g \|_{L^{2}}, \frac{2}{q}+\frac{1}{r}=\frac{1}{2}, 2\leq q,r \leq \infty , \| S(t)g \|_{L^{4}_{x}L^{\infty}_{t}}\lesssim \|g \|_{H^{\frac14}}.
G_{0}^{( 2 )}( \vec{\rho}, u ) = \frac{g}{L}\frac{1}{( 2 \pi )^{d - 1}}\sum_{n = 1}^{\infty}\cos ( \frac{n \pi u}{L}) \int d^{d - 1}p \frac{e^{i \vec{p}. \vec{\rho}}}{( \vec{p}^{\, 2}+ ( \frac{n \pi}{L})^{2}+ m^{2})},
c_{g}( 0 , 0 ) = m_{g}\, c_{0}( 0 , 0 ) \, , \quad ( g \neq 1 ) \, , \qquad c_{1}( 0 , 0 ) = m_{1}\, c_{0}( 0 , 0 ) - 2 c_{0}( - 1 , 0 ) \, ,
P(B_2^{(n)}) \leq\sum_{j=\lceil K\rceil}^{J_n-1}P\left(\bigcup_{t\in [\theta_j^{(n)},\theta_{j+1}^{(n)}]\cap\mathbb N}\{S_n'(t)\leq t\}\right)\leq\sum_{j=\lceil K\rceil}^{J_n-1}P\left(S_n'(\theta_j^{(n)})\leq\theta_{j+1}^{(n)}\right),
c_2(f) =\lim_{s\to 0}\left(\frac{d}{ds}s\zeta(s)\zeta(s)^{-1}Z_{r_2}(f,s)\right) =\big(\mathrm{Res}_{s=0}\zeta(s)\big)\lim_{s\to 0}\left(\frac{d}{ds}\zeta(s)^{-1}Z_{r_2}(f,s)\right)+\kappa_1\left(\zeta(s)^{-1}Z_{r_2}(f,s)\right)\big|_{s=0}=-\epsilon(0)\kappa_0\lim_{s\to 0}\left(\frac{d}{ds}\zeta(s)^{-1}Z_{r_2}(f,s)\right)+\kappa_1\tilde{c}_2(I(f)).
\frac{\partial^2}{\partial t^2}\phi_{\alpha j}(t)+ 3 \frac{\dot{R}(t)}{R(t)}\frac{\partial}{\partial t}\phi_{\alpha j}(t) +\left(m^2 + \xi{\bf R}+\frac{{\bf k}^2}{R^2 (t)}\right)\phi_{\alpha j}(t) = 0,
\sigma_{k}(r):=\int_{0}^{r}\int_{0}^{\tau}\frac{s_{k}(t)^{2}}{s_{k}(\tau)^{2}}d t d \tau=\begin{cases}\frac{1}{2 k}(1-\sqrt{k}r \cot (\sqrt{k}r)) \textrm{if}k>0 \frac{r^2}{6}\textrm{if}k=0 \frac{1}{2k}\left(1 -\sqrt{-k}r \coth \left(\sqrt{-k}r \right)\right) \textrm{if}k<0.\end{cases}
g ( z ) = \sum_{i j}\frac{\sigma ( z + s_{i j})}{\sigma ( z ) \sigma ( s_{i j})}e^{- \frac{\zeta ( \pi )}{\pi}z s_{i j}}e^{i s_{i j}\frac{z - \bar{z}}{\Delta - \overline{{\Delta}}}}Y_{i j}E_{i j}= \sum_{i j}W ( z , s_{i j}) Y_{i j}E_{i j}.
(1-\epsilon)\int_{\psi}(|\psi\rangle\langle\psi|)^{\otimes t}d\psi\le\sum_{i}p_{i}(|\phi_{i}\rangle\langle\phi_{i}|)^{\otimes t}\le(1+\epsilon)\int_{\psi}(|\psi\rangle\langle\psi|)^{\otimes t}d\psi
\overline{[A_{a,0}]}= [A_{a,0}], \overline{[A_{a,1}]}= [A_{a,1}] +(v^{-a-1}-v^{a+1}) [A_{a+1,0}], \overline{[A_{a,2}]}= [A_{a,2}] +(v^{-a-1}-v^{a+1}) [A_{a+1,1}]+ \frac{v^{-2a-3}(1 -v^{2(a+1)})( 1- v^{2(a+2)})}{1- v^4}[A_{a+2,0}].
- c_\alpha^\top b_{1:k}+ c_\alpha^\top \alpha - c_\beta^\top \beta \to \min &\begin{cases}B \beta - A \alpha + y = \hat b \bar B \beta - \bar A \alpha \leq \hat b_d \alpha \in \mathbb{Z}_+^k,\, y \in \mathbb{Z}_+^s,\, \beta \in \mathbb{Z}^s, \end{cases}
\alpha = -6\left(\left(\sum_{i=1}^n(k_{i-1}(f)-f(i))\right)^2 + \sum_{i=1}^n(k_{i-1}(f)^2-f(i)^2)\right) = 12\left(\sum_{i=1}^n f(i)(k_{i-1}(f)-f(i)) - 2\sum_{1\leq j < i \leq n}(k_{i-1}(f) - f(i))(k_{j-1}(f) - f(j))\right) = 12\sum_{i=1}^n(k_{i-1}(f) - f(i))\left(\sum_{j=1}^{i}f(j) - \sum_{j=1}^{i-1}k_{j-1}(f)\right)
(a'(x)y)_i =\sum_{j=1}^da'_{ij}(x)y_j=\Bigg(\sum_{l;l\neq i}\phi'_{il}(x_i-x_l)\Bigg)y_i+\sum_{j;j\neq i}(-\phi'_{ij}(x_i-x_j))y_j &=\sum_{l;l\neq i}\phi'_{il}(x_i-x_l)(y_i-y_l).
\begin{array}{l l l}{\alpha_{i k}}&{=}&{1 \otimes \cdots \otimes 1 \otimes \alpha_{i}\otimes 1 \otimes \cdots \otimes 1 \in H^{1}( ( M )^{N},{\bf{Z}})}\\{\beta_{k}}&{=}&{1 \otimes \cdots \otimes 1 \otimes \beta \otimes 1 \otimes \cdots \otimes 1 \in H^{2}( ( M )^{N},{\bf{Z}}) ,}\\ \end{array}
V_n^{(0)}=V_n(X_1,X_2,\dots,X_n)=E[\max\{U_1(X_1),W_1(X_1)\}] &=\int_{-\infty}^{\infty}\max\{U_1(x),W_1(x)\}dF_1(x) &= \int_{-\infty}^{x_1^*}W_1(x) dF_1(x)+ \int_{x_1^*}^{\infty}U_1(x) dF_1(x) \geq \int_{-\infty}^{x_1^*}W_1(x_1^*) dF_1(x)+ \int_{x_1^*}^{\infty}U_1(x_1^*) dF_1(x) =U_1(x_1^*)=W_1(x_1^*) &=V_n(X_1^{'},X_2,\dots,X_n)=V_n^{(1)},
A_{\varpi}= \sum_{l_1 \not= l_2}\left(x(l_1l_2) \lambda(l_1l_2, F_{\varpi}) + \frac{x(l_1^2l_2^2)}{(l_1l_2)^{3/2}}\lambda(l_1^2l_2^2, F_{\varpi}) + \frac{x(l_1^4l_2^4)}{(l_1l_2)^{9/2}}\lambda(l_1^4l_2^4, F_{\varpi})\right) & + \sum_{0 \leq b \leq s \leq 4}\sum_{l}\xi_{b, s}(l) \lambda_{0, b}^{( 2s)}(l, F_{\varpi})
\pi_{\nu}(f) = f(\pi_{\nu}(S)) =(I \otimes c^*)\left(I - \sum_{j=1}^d \pi_{\nu}(S_j) \otimes T_{0,j}^*\right)^{-1}\left(\sum_{j=1}^d \pi_{\nu}(S_j) \otimes T_j^* x\right).
\Gamma^{a_1\cdots a_N}[k_1,\varepsilon_1;\ldots;k_N,\varepsilon_N] = (-ig)^N{\rm tr}(T^{a_1}\cdots T^{a_N}){{{(4\pi\mu^2)}^{-{\epsilon\over 2}}}\over{32{\pi}^2}}\sum_{{\rm diagrams}}D_i
G_R(x,x')=-\int{d^4 k \over (2\pi)^4}e^{ik_{\mu}(x^{\mu}-x'{}^{\mu})}\Biggl[{a(y)^2 a(y')^2 \ell^{-1}\over{\bf k}^2-(\omega+i\epsilon)^2}+\int_0^{\infty}dm\,{u_m(y) u_m(y')\over m^2+{\bf k}^2-(\omega+i\epsilon)^2}\Biggr],
B_1 = \Big\{\Big| \sum_{j=1}^{N_n^+}\Delta_{\tau,j}- \mu_s N_n^+ \Big| \leq (N_n^+)^{\frac{1}{2}+ \epsilon_2}\Big\}~~\mbox{and}~~B_2 = \Big\{\Big| \sum_{j=1}^{N_n^-}\Delta_{\tau,j}- \mu_s N_n^- \Big| \leq (N_n^-)^{\frac{1}{2}+ \epsilon_2}\Big\}.
\prod_{v \in S \cup \{v_0 \}}\mu(\Pi_v \otimes \mathcal{X}|-|_v^s , \Psi_v) = \prod_{v \in S \cup \{v_0 \}}\prod_i \gamma(r_i \circ (\rho_v \otimes \mathcal{X}_v |-|_v^s), \Psi_v) \cdot \gamma(r_i^{\vee}\circ (\rho_v \otimes \mathcal{X}_v |-|_v^s), \overline{\Psi}_v).
\begin{array}{rcl}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{i},T_{j}\right)\left|\mathcal{F}_{0}\right.;\tilde{A_{i}}\left(T_{j}\right)\right] = \mathbb{E}^{\mathbb{P}}\left[\tilde{A}\left(t_{i},T_{j}\right)\left|\mathcal{F}_{0}\right.\right] = \tilde{A}\left(t_{i},T_{j}\right)\end{array}
\rho_{f,\epsilon,\phi}^T:=\frac{1}{T^{k/2}}\sum_{j_1 \neq \ldots \neq j_k}\sigma_{j_1}\ldots \sigma_{j_k}\langle \Lambda_\epsilon^f(x^{j_1}+\xi^{j_1},\ldots,x^{j_k}+\xi^{j_k};T),\phi \rangle,
\frac{\delta \rho}{\rho}\simeq A_{e i}\left( T_{0}\lambda_{0}\sqrt{2 \omega + 1}\right)^{\frac{4}{2 \omega - 1}}( 2 \omega + 1 )^{\frac{3}{2}}t_{e}^{- \frac{2 \omega + 1}{2 \omega - 1}}
B_{\theta}(\theta)=\vec{w}^{H}\vec{v}_{\theta}(\theta)=\sum_{n=0}^{N-1}w_{n}^{*}[\vec{v}_{\theta}(\theta)]_{n}=\sum_{n=0}^{N-1}w_{n}^{*}e^{j(n-\frac{N-1}{2})\frac{2\pi}{\lambda}dcos\theta}
Z=\sum_{\{{\sigma}_0 \}}exp\left\{J_3\sum_\triangle{\sigma}_0^{(1)}{\sigma}_0^{(2)}{\sigma}_0^{(3)}+h\sum_j\sigma_0^{(j)}\right \}{[g_n({\sigma}_0^{(1)})]}^{\gamma -1}{[g_n({\sigma}_0^{(2)})]}^{\gamma -1}{[g_n({\sigma}_0^{(3)})]}^{\gamma -1},
x_{m+1}\sqrt{c}+z_{m+1}\sqrt{a}= (x\sqrt{c}+z\sqrt{a})(s+\sqrt{ac})^{m+1}&= (x\sqrt{c}+z\sqrt{a})(s+\sqrt{ac})^{m}(s+\sqrt{ac}) &= (x_{m}\sqrt{c}+z_{m}\sqrt{a})(s+\sqrt{ac}) &= (sx_m+az_m)\sqrt{c}+(sz_m+cx_m)\sqrt{a},
\left| \Omega \right\rangle \equiv \left|\overline \Psi \left({\bf y}\right)\Psi \left({\bf 0}\right) \right\rangle =\overline \psi \left({\bf y}\right)\exp \left({ig\int\limits_{\bf 0}^{\bf y}{dz^i A_i \left( z \right)}}\right)\psi \left({\bf 0}\right)\left| 0 \right\rangle ,
T = \left( \begin{array}{c c c c c c}{{e^{i \theta_{1}}}}&{{}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{\ddots}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{}}&{{e^{i \theta_{N}}}}&{{}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{e^{- i \theta_{1}}}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{}}&{{\ddots}}&{{}}\\{{}}&{{}}&{{}}&{{}}&{{}}&{{e^{- i \theta_{N}}}}\\ \end{array}\right)
\nabla_{\mathsf{R}_{\tau}}\left( \begin{matrix}\alpha_1 \alpha_2 \alpha_3\end{matrix}\right) = \left( \begin{matrix}0 0 0 0 0 -1 0 0 0\end{matrix}\right)\left( \begin{matrix}\alpha_1 \alpha_2 \alpha_3\end{matrix}\right)
[(a_0\xi+A(u)\xi^2)\frac{\partial}{\partial \xi}\wedge \frac{\partial}{\partial u}, (b_0\xi+B(u)\xi^2)\frac{\partial}{\partial \xi}]=(a_0B(u)-b_0A(u)) \xi^2\frac{\partial}{\partial \xi}\wedge \frac{\partial}{\partial u},\,\,\,\,\,b_0,t_0,t_1,...,t_{n-1}\in \mathbb{C}.
\rho[\frac{\partial\overline{u_{i}}}{\partial t}+\frac{\partial\overline{u_{i}}\overline{u_{j}}}{\partial x_{j}}+\frac{\partial\overline{u_{i}^{\prime}u_{j}^{\prime}}}{\partial x_{j}}]=-\frac{\partial\overline{p}}{\partial x_{i}}+\mu\frac{\partial^{2}\overline{u_{i}}}{\partial x_{j}\partial x_{j}}.
[Z_i] = -[X_{i-1}] + [X_{i+1}] + [Z_{i-1}] \; i=3,\ldots,n-1 [Y_i] =-[X_{i+1}]+[X_i] \; i=2,\ldots,n-1 [Z_2] =[Y_1]+[X_3] [Y_n] =[Z_{n-1}]-[X_{n-1}]
\zeta \left( \begin{array}{r c r}{s_{1}}{\ldots}{s_{k}}{\sigma_{1}}{\ldots}{\sigma_{k}}\end{array}\right) : = \sum_{n_{j}> n_{j + 1}> 0}\quad \prod_{j = 1}^{k}\frac{\sigma_{j}^{n_{j}}}{n_{j}^{s_{j}}}
\mathbb{P}_{a,b,r,p}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( z\right) =\frac{1}{\left( r!\right)^{p}\prod_{s=2}^{L}\left( r_{s}!\right)^{p_{s}}}\sum_{k=0}^{rp+\sigma}k!S_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,k\right) \left( z-1\right)^{rp+\sigma -k}.
\varphi(\tau)^8{\rm ch}(\mathcal{V}_i)=\frac{1}{2}\left(\prod_{l=1}^8\theta_1(y_l^i,\tau)+\prod_{l=1}^8\theta_2(y_l^i,\tau)+\prod_{l=1}^8\theta_3(y_l^i,\tau)\right),
\tilde{G}|_E = (x_{i_0}+ \zeta^2 = x_{i_1}+ \zeta c = a c^2 + 2 x_{i_1}d = 0) &= (x_{i_0}+ \zeta^2 = x_{i_1}+ \zeta c = c (a c - 2 \zeta d) = 0) &= (c (a c - 2 \zeta d) = 0)|_E,
_{1 2}\langle R | =_{1}\langle{\cal O}| \otimes_{2}\langle{\cal O}| \exp [ - \sum_{r = 1}^{2}\int_{0}^{\pi / 2}d \sigma \, b_{r}( \sigma ) b_{r - 1}( \pi - \sigma ) ] , ~ ~ r = 1 , 2 ,
\lim_{\beta\downarrow0}\frac{1 + \frac{1-\beta}{p\beta}( 1+ (q-1)e^{-2s})}{1+ \frac{1-\beta}{p\beta}(1-e^{-2s})}&=\frac{\frac{1 - e^{-2s}+ qe^{-2s}}{p}}{\frac{1 - e^{-2s}}{p}}=\frac{1 - e^{-2s}+ qe^{-2s}}{1 - e^{-2s}}=: C_{q,s}>0.
{\cal C}_{\theta}(Z)=\left\{H \in{\cal S}^q:\begin{array}{l}Q^{T}_{b_S}H[Q_{b}]=0, Q^{T}_{b_U}H[Q_{b_L}]=0 begin{align*}4pt] Q^{T}_{b_U}HQ_{b_U}\in{\cal S}^{|b_U|}_+, Q^{T}_{b_L}HQ_{b_L}\in{\cal S}^{|b_L|}_- \end{array}\right \}.
\begin{aligned}\bigl(1+2\alpha\beta-\beta-\alpha\bigl)|\nabla B|^2-2(1-\beta) (B+\varepsilon)\sum_{i,j,k=1}^nh_{ijk}^2 &\leq \dfrac{2}{n+2}B\biggl\{(n-2)(1-\beta)-4n\alpha(1-\beta)+2n\alpha \biggl\}\bigl(\sum_{k=1}^nh_{kkk}^2+3\sum_{i\neq k}h_{iik}^2\bigl)=0.\end{aligned}
H_{\alpha}^{\mathrm{XY}}= -\frac\alpha2 \sum_{j=0}^{M-1}\left( (1+\gamma)\sigma^x_j\sigma^x_{j+1}+ (1-\gamma)\sigma^y_j\sigma^y_{j+1}\right) - \sum_{j=0}^{M-1}\sigma^z_j,
2C_{1}[\frac{\lambda^{2}}{J^{5/3}}-\frac{1}{3J}(2\lambda^{2}+\frac{J^{2}}{\lambda^{4}})]+2D_{1}(J-1)=\frac{2C_{1}}{J^{5/3}}(\lambda^{2}-\frac{J^{2}}{\lambda^{4}})
\sigma(H^{\epsilon,\kappa})\cap \left [a_0,b_N\right ]\subset \bigcup_{k=0}^N [a_k,b_k]\,,\quad{\rm dim}\big({\rm Ran}E_{[a_k,b_k]}(H^{\epsilon,\kappa}) \big )= + \infty\,, & b_k-a_k\leq C_0 \,\kappa\epsilon + C_1\, \epsilon^{4/3}\, , \; 0\leq k\leq N\,,{\rm and}a_{k+1}-b_k\geq\frac{1}{C_2}\epsilon \, ,\; 0\leq k\leq N-1\,.
\Pi ( k_{0}) = e^{2}\,{\frac{\mathrm{s e c h}^{2}{\frac{\beta m}{2}}}{1 + \operatorname{t a n h}^{2}{\frac{\beta m}{2}}\operatorname{t a n}^{2}{\frac{\beta k_{0}}{2}}}}\,{\frac{\operatorname{t a n}{\frac{\beta k_{0}}{2}}}{k_{0}}}
G^{1,1}_{1}={\rm span}\left\{\right. (E_{1,2})_{(k_{1})}(E_{1,3})^{l}_{(k_{2})},\,(E_{1,4})_{(k_{1})}(E_{1,3})^{l}_{(k_{2})},\,(E_{2,3})_{(k_{1})}(E_{1,3})^{l}_{(k_{2})},\, (E_{2,4})_{(k_{1})}(E_{1,3})^{l}_{(k_{2})},\,(E_{3,4})_{(k_{1})}(E_{1,3})^{l}_{(k_{2})},\,|l\in \mathbb{N},k_{1},k_{2}\in \mathbb{Z}_{<0}\left. \right\}+G^{1,1}_{0},
i \partial_t U_k^A + \Delta U_k^A + \left( |\cdot|^{-(N-\gamma)}* \big|U_k^A \big|^p \right) \big|U_k^A \big|^{p-2}U_k^A = O(e^{-(k+1) e_0 t}) \mathcal{S}(\mathbb{R}^N).
2^{2l-1}\binom{\frac{q-1}2}l=2^{2l-1}\binom{-\frac 12}l=2^{2l-1}\frac{(-\frac 12)(-\frac 32)\cdots(-\frac{2l-1}2)}{l!}\cr =\frac{2^{l-1}(-1)(-3)\cdots(-2l+1)}{l!}=\frac{(-l)(-l-1)\cdots(-2l+1)}{l!}=\binom{-l}l.
T_F^{-1}=\frac{i}{2\sqrt{2}}\sum_k e^{-iH_k}\biggl[ (1-i)e^{iH_{k'}}e^{iH_{\bar k'}}+(1+i)e^{iH_{k'}}e^{-iH_{\bar k'}}+(1+i)e^{-iH_{k'}}e^{iH_{\bar k'}}+(1-i)e^{-iH_{k'}}e^{-iH_{\bar k'}}\biggr]
\left(\begin{array}{c}h_\alpha (x) h_\beta (x)\end{array}\right )={D^{\alpha,\beta}_{\sin \beta, \sin \alpha}}^{-1}x= \frac{1}{C_{\alpha,\beta}}\left ( \begin{array}{cc}\sin \alpha \langle x, e_{\beta-\frac{\pi}{2}}\rangle \sin \beta \langle x, e_{\alpha + \frac{\pi}{2}}\rangle\end{array}\right )
\left( \prod_{1\leq i<j\leq n-r}\frac{1-\delta_i\delta_j R_{ij}}{1+\delta_i \delta_j (R_{ij}-\beta T_i)}\right) \left(\prod_{i=1}^{n-r}\frac{1-\beta\tilde T_i}{2-\beta\tilde T_i}d_{\lambda_i}(i) \right),
|\xi g|^{2}_{L^{2}_{\xi}}\leq C \big<\Lambda g,\Lambda g\big>_{\xi}=\big<L g-Kg,Lg-Kg\big>_{\xi}&\leq C_{1}|Lg|^{2}_{L^{2}_{\xi}}+C_{2}|g|^{2}_{L^{2}_{\xi}}\,.
( U ( g ) e_{r}) ( z ) = \pm ( \alpha + \overline{{\beta}}\overline{{z}})^{- \frac{1}{2}}( \overline{{\alpha}}+ \beta z )^{- \frac{1}{2}}\left( \frac{\alpha z + \overline{{\beta}}}{\beta z + \overline{{\alpha}}}\right)^{r}
\tilde{g}(\eta,t)=\mathcal{O}\left(\frac{1}{\eta}\right), \eta\rightarrow 0;~~~~\tilde{g}(\eta,t)=\mathcal{O}\left(\frac{1}{(\eta-\frac{1}{2})^{2}}\right), \eta\rightarrow \frac{1}{2}; &\tilde{g}(\eta,t)=\mathcal{O}\left(\frac{1}{\eta^{2}}\right), \eta\rightarrow \infty.
\sum_{k + l \geq 0}a_{kl}^{2}(\varphi(x)) (\mu^{p}z_{1}+ z_{2}^{p})^{k}(\mu z_{2})^{l}= \sum_{k + l \geq 0}\mu a_{kl}^{2}(x) z_{1}^{k}z_{2}^{l}.
f^A_{D0}= 2\pi V_{11}\int_{0}^{\infty}\frac{dt}{t^{5/2}}\sum_{n^I\ne 0}e^{-\frac{\pi}{t}{\cal M}^2}= V_{11}\sum_{n^I\ne 0}\left({\cal M}^2\right)^{-3/2}, \quad{\cal M}^2 = n^I g_{IJ}n^J,
\hat{R}_{1 2}( \sigma^{1}K^{\prime}\sigma^{1})_{1}\hat{R}_{1 2}^{- 1}( \sigma^{1}K^{\prime}\sigma^{1})_{1}= ( \sigma^{1}K^{\prime}\sigma^{1})_{1}\hat{R}_{1 2}^{- 1}( \sigma^{1}K^{\prime}\sigma^{1})_{1}\hat{R}_{1 2}\; .
[w_\rho \eta_\delta]_{s,p}^p \le \int_{A_1}\frac{|w_\rho(x) - w_\rho(y)|^p}{|x - y|^{N+sp}}\, dx\, dy + \int_{A_2}\frac{|w_\rho(x)\, \eta_\delta(x) - w_\rho(y)\, \eta_\delta(y)|^p}{|x - y|^{N+sp}}\, dx\, dy & + 2 \int_{A_3}\frac{|w_\rho(x)\, \eta_\delta(x) - w_\rho(y)|^p}{|x - y|^{N+sp}}\, dx\, dy =: I_1 + I_2 + 2 I_3,
T=\frac{dt}{\sqrt{(1-\frac{2GM}{rc^{2}})dt^{2}-\frac{1}{c^{2}}(1-\frac{2GM}{rc^{2}})^{-1}dr^{2}-\frac{r^{2}}{c^{2}}d\theta^{2}-\frac{r^{2}}{c^{2}}sin^{2}\theta d\phi^{2}}}
\delta_{l}^{( 2 )}\approx \eta \left[ \ln \left( \frac{z_{l}}\alpha \right) + \frac 1{2 4 z_{l}^{2}}- \frac{q^{2}}{2 z_{l}^{2}}+ O \left( \frac 1{z_{l}^{4}}\right) \right] + O ( \eta^{3}) \ .
- \int_{\Omega_i}\nabla \cdot \Big( \varepsilon \nabla u^* \Big) \varphi\, dx=\int_{\Omega_i}\varepsilon \nabla u^* \cdot \nabla \varphi\, dx-\int_{\partial \Omega_i}\varepsilon \nabla u^* \cdot \nu \varphi\, dx.
\left( \begin{matrix}{\Gamma_{S}^{1 1}( p )}&{\Gamma_{S}^{1 2}( p )}\\{\Gamma_{S}^{2 1}( p )}&{\Gamma_{S}^{2 2}( p )}\\ \end{matrix}\right) ={\bf \sf M}^{- 1}\left( \begin{matrix}{\Gamma_{R F}^{\phi_{S}}( p )}&{0}\\{0}&{{\Gamma_{R F}^{\phi_{S}}}^{*}( p )}\\ \end{matrix}\right){\bf \sf M}^{- 1},
\mathbf{Q}_\mathrm{z}=\mathbf{V}_\mathrm{z}\left[\!\!{\begin{array}{*{20}c}{\mathbf{Q}_\mathrm{z}^{\prime}}{\mathbf{0}}{\mathbf{0}}{\mathbf{0}}\end{array}}\!\!\right]\mathbf{V}_\mathrm{z}^\mathrm{H}.
\Sigma^{+}= \{\alpha_{1}=e_{1}-e_{2}, \alpha_{2}=e_{2}, \alpha_{1}+\alpha_{2}, \alpha_{1}+2\alpha_{2}, 2\alpha_{2}, 2\alpha_{1}+2\alpha_{2}\}, W^{+}= \emptyset, delta =2\alpha_{1}+2\alpha_{2},
G^{(k)}(x,it)G^{(k)}(x,-it) = \frac{1}{4}\left((x+\sqrt{x^2-1})^{2k}+ (x-\sqrt{x^2-1})^{2k}\right) +\frac{1}{4}\left(e^{i2t\sqrt{x^2-1}}+e^{-i2t\sqrt{x^2-1}}\right) &= \frac{1}{2}T_{2k}(x) + \frac{1}{2}\cos\left(2t\sqrt{x^2-1}\right) .
\int \limits_{{\cal L}^{d}_{d-1}}f(H)d\nu_{d-1}(H) = c_{3}\int \limits_{{\cal L}^{A}_{2}}\int \limits_{{\cal L}^{L}_{d-1}}f(H)[H,A]^{2}d\nu_{d-1}^{L}(H)d\nu_{2}^{A}(L).