harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{pmatrix}T_1 T_2 T_3 H_1 H_2 H_3 \end{pmatrix}^t J \begin{pmatrix}T_1 T_2 T_3 H_1 H_2 H_3 \end{pmatrix}= 2\pi \imath \begin{pmatrix}0 0 0 0 0 1 0 -1 0 \end{pmatrix}.
\begin{bmatrix}g_{11}g_{12}g_{21}g_{22}\end{bmatrix}\left(\begin{matrix}a^1 a^2\end{matrix}\right) =\left(\begin{matrix}\partial_1 \eta \cdot \overline{\partial}_A \hat{n}\partial_2 \eta \cdot \overline{\partial}_A \hat{n}\end{matrix}\right).
\begin{array}{rcl}- \delta_f (\tilde{P}(x)) +\begin{pmatrix}A 0 2 x^\top S 0 \end{pmatrix}\tilde{P}(x)+ \tilde{P}(x) \begin{pmatrix}A^\top 2Sx 0 0 \end{pmatrix}begin{align*}2ex]+ \begin{pmatrix}BB^\top 2BB^\top Mx 2 x^\top M BB^\top 4 x^\top M B B^\top M x \end{pmatrix}= 0 \end{array}
(q^{\mu-2}T_{N-2}T_{N-3}\dots T_{N-\mu +1}) (q^{\mu -1}T_{N-1}T_{N-2}\dots T_{N-\mu+1}) &= (q^{\mu -1}T_{N-1}T_{N-2}\dots T_{N-\mu+1}) (q^{\mu-2}T_{N-1}T_{N-2}\dots T_{N-\mu +2}).
\gamma_{2 a}\equiv - \left( \left( \mathrm{D}_{i}\right)_{a}^{b}\pi_{b}^{i}- \mathrm{i g}\bar{\psi}_{\alpha}^{B}\left( \gamma^{0}\right)_{\beta}^{\alpha}T_{a B}^{A}\psi_{A}^{\beta}\right) \approx 0 ,
\frac{\phi}{\mathsf{M}}\eta(\mathsf{A}_1,\mathsf{A}_{\mathsf{M}+1}) \geq \frac{2^{-6}\,\mathsf{q}^{-1}\,\pmb{\eta}}{\lfloor 2^{-10}\,\pmb{\eta}\,\mathsf{q}^{-1}\lambda^{-1}\rfloor}\geq \frac{2^{-6}\,\mathsf{q}^{-1}\,\pmb{\eta}}{2^{-10}\,\pmb{\eta}\,\mathsf{q}^{-1}\lambda^{-1}}= 2^4\lambda\,.
\begin{cases}\frac{C_2(\lambda_2\rho_0)}{2}+\frac{C_3(\lambda_2\lambda_{2,3}\rho_0)}{2}=C_1(\lambda_1\rho_0) \frac{C_2(\lambda_2\rho_0)}{2}+\frac{C_4(\lambda_2\lambda_{2,4}\rho_0)}{2}=C_1(\lambda_1\rho_0) \lambda_1+\lambda_2=\lambda_{2,3}+\lambda_{2,4}=1\end{cases}.
r_\chi c = \sum_{i \in T^L}r_\chi \langle \lambda, \beta_i \rangle > \sum_{i \in T^L}(1-r_\chi) \langle \lambda, \beta_i \rangle = -\sum_{i \in U^L}\langle \lambda, \beta_i \rangle - \sum_{j \in T^L}r_\chi \langle \lambda, \beta_j \rangle \ge \alpha.
J_n^+=\begin{pmatrix}d_n(1) a_n(1) 0 0 \dots a_n(1) d_n(2) a_n(2) 0 \dots 0 a_n(2) d_n(3) a_n(3) \dots 0 0 a_n(3) d_n(4) \dots \vdots \vdots \vdots \vdots \ddots \end{pmatrix}
P_{\dagger}^* \mathcal Q(\ell, \omega) P_{\dagger}= \frac1\ell P_{\dagger}^* \mathcal Q_0 P_{\dagger}- \frac{4 | \Gamma| \cdot |\partial_2 \psi_{\varkappa, 2}(0)|^2}{\sqrt{\Lambda -\omega}\cdot \sqrt{2 \zeta_1 ''(\varkappa)}}\; P_{\dagger}^* T_\ell^* \bigr( P_+ + P_- \bigr) T_\ell P_{\dagger}+ P_{\dagger}^* R(\ell,\omega) P_\dagger
f \in \left( \prod^{r-1}_{\ell=t}I_\ell(X)\right)^2 I_r(X)^{s-r+1}&\subseteq \left( \prod^{r-1}_{\ell=t}\I_t(X)^{(\ell-t+1)}\right)^2\left( I_t(X)^{(r-t+1)}\right)^{s-r+1}\hbox{\cite[Proposition 10.2]{BookDet}}&\subseteq I_t(X)^{((r-t)(r-t+1))}I_t(X)^{((r-t+1)(s-r+1))}&\subseteq I_t(X)^{(s-t+1)(r-t+1)}= I_t(X)^{(h)}.
\int{d\rho_0 d^4 x_{0}\over \rho_0^5}\, \prod_{p=1}^4{\left[{\rho_0 \over{\rho_0^2 + (x_p - x_{0})^2}}\right]}^4 = \frac{5\pi^{2}}{108}\prod_{p<q}{\partial \over \partial x_{pq}^2}\int \prod_{p}d\alpha_p{\delta \left( 1-\sum_q \alpha_q \right) \over (\sum_{p,q}\alpha_p \alpha_q x_{pq}^2 )^{2}}\; ,
\left|\int_0^t C_{42}\; d\tau\right|\leq \sum_{k=1}^{\lfloor (l-1)/2\rfloor-2}\int_0^T\int |u\partial_x^{(l-1)-k}u\partial_x^{3+k}u\partial_x^{l-1}u|\chi \; dxd\tau\leq c(\nu;8(l-6))
\hat{\Sigma}( v - u ) = \frac{1}{8 \pi^{2}}\left[ f^{2}+ 2 g^{2}( \alpha^{2}R + \beta^{2}L ) ( 1 - \Gamma ) \right] \frac{{\textstyle / \hspace{- 2 m m}v - / \hspace{- 2 m m}u}}{{\textstyle ( v - u )^{6}}}+ ( a L + b R ) \, / \hspace{- 2 m m}\partial_{v}\, \delta^{4}( v - u )
\Delta \underline{u}(x) = z''(\delta(x))+ z'(\delta(x))\Delta\delta(x) \ge z''(\delta(x)) + \frac{(N-1)}{\rho_0 +\delta(x)}z'(\delta(x)) = - \frac{\mu}{\delta(x)^2}z + z^p = - \frac{\mu}{\delta(x)^2}\underline{u}+ \underline{u}^p \,.
\mbox{tr}[a_1^{\dagger}(\frac{P^+}{3})a_2^{\dagger}(\frac{2P^+}{3})]|0\rangle +\mbox{tr}[a_2^{\dagger}(\frac{P^+}{3})a_1^{\dagger}(\frac{2P^+}{3})]|0\rangle,
(t+r+1)\cdots (t+n)\left[\underset{i=1}{\overset{s}{\sum}}\binom{m_i+r-1}{r}+\dim_K(I_{\alpha})_t \right] &=(r+1)\cdots (n-1)n\left[\underset{i=1}{\overset{s}{\sum}}\binom{m_i+n-1}{n}+ \dim_K I_t \right].
\Big| \big[V(\tau) - \beta_\varepsilon\big(V(\tau)\big)\big]\cdot \nabla_{\!x}\eta_\varepsilon\big(X(\tau)\big) \Big| = \big[v\!-\!\beta_\varepsilon(v)\big]_{\perp}\big(V(\tau)\big) \left|\frac{\partial\eta_\varepsilon}{\partial x_{\!\perp}}\big(X(\tau)\big) \right| begin{align*}3pt] = O(\varepsilon^4) O(1/\varepsilon) = O(\varepsilon^3) .
= x \sum_{j=1}^N \frac{A_j}{(\log x)^{j-1/2}}\sum_{d\le x/2}\frac{h_{1/\tau}(d)}{d (1-\frac{\log d}{\log x})^{j-1/2}}+ O\left( \frac{x}{(\log x)^{N+1/2}}\sum_{d\le x/2}\frac{|h_{1/\tau}(d)|}{d (1-\frac{\log d}{\log x})^{N+1/2}}\right) + \sum_{x/2<d\le x}h_{1/\tau}(d).
h_{j,i}^{(n)}:=\sum_{k=0,1}[(x_{j,i,k}^{(n)})^*,x_{j,i,k}^{(n)}]+z_{j,i}^{(n)}, h_{1,i}^{(n)}:=[(y_{i,0}^{(n)})^*,y_{i,0}^{(n)}]+[(y_{i,1}^{(n)})^*,y_{i,1}^{(n)}]-\sum_{j=2}^k z_{j,i}^{(n)}.
S_iM_1+\xi_iN_{i^\prime}={}S_iM_1-S_iN_i+S_iN_i+\xi_iN_{i^\prime}\geq{}S_iM_1-S_iN_i+\xi_i(N_i+N_{i^\prime}) >{}S_iM_1-S_iN_i+\xi_iM_1 \geq{}S_iM_1-S_iN_i+\xi_iM_2 ={}\Lambda_i-S_iN_i.
E\left[P[E_v\in C\cap f^{-1}(D)\mid W_v, f(E_v)]; F_v\in B, W_v\in A\right] = E\left[E\left[P[E_v\in C\cap f^{-1}(D)\mid W_v, f(E_v)]; F_v\in B, W_v\in A\mid W_v, f(E_v)\right]\right] = E\left[P[E_v\in C\cap f^{-1}(D)\mid W_v, f(E_v)]\,P\left[F_v\in B\mid W_v, f(E_v)\right]; W_v\in A\right].
\|f_1-f_0\|_t = \gamma2^{j(1/2-1/t)}\|\psi\|_t \asymp \varepsilon^{\beta/(\beta+1/2)}\varepsilon^{(1/t-1/2)/(\beta+1/2)}= \varepsilon^{(\beta+1/t-1/2)/(\beta+1/2)},
\sum_{k\not=0}\exp(-2Td^3(k,0))d^{-1}(k,0)\lesssim(T^\frac{1}{3})^{-\frac{3}{2}}, \sum_{k\not=0}\exp(-2Td^3(k,0))d(k,0)\lesssim(T^\frac{1}{3})^{-\frac{7}{2}},
m_{\infty}= \inf\left\{E(v):v\in\mathcal{N}(E)\right\}=E(U)U(\ref{eq:U}) E(v)= \int_{\mathbb{R}^{n}}\frac{1}{2}|\nabla v|^{2}+\frac{(a-\omega^{2})}{2}|v|^{2}-\frac{1}{p}|v^{+}|^{p}dx; \mathcal{N}(E)= \left\{v\in H^{1}(\mathbb{R}^{n})\smallsetminus\{0\}\ :\E(v)v=0\right\}.
S_2 \le \Gamma(2m)\frac{\Gamma(m+3/2)\Gamma(m-3/2)}{\Gamma(2m)}\int_{0}^1y^{m+1/2}(1-y)^{m-3/2}\,dy &= \frac{\Gamma(m+3/2)\Gamma(m-3/2)\Gamma(3)\Gamma(m-1/2)}{\Gamma(m+5/2)},
I_\lambda(\omega): =\frac{1}{2}\|\omega\|_X^2-\frac{\lambda}{2}\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2\ln|\omega(x,0)|dx &~~~~+\frac{\lambda}{4}\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2dx-\frac{1}{2_\alpha^*}\displaystyle\int_{\mathbb{R}^N}b(x)|\omega(x,0)|^{2_\alpha^*}dx,
\widehat{D}_{0}\pi_{a}+ i \varsigma{\frac{\partial E_{a}}{\partial \phi_{i}}}\tau_{i}- i \overline{{{p}}}^{\prime}{\frac{\partial \overline{{{J}}}^{a}}{\partial \overline{{{\phi}}}_{i}}}\overline{{{\tau}}}_{i}+ i E_{a}\beta - i \overline{{{J}}}^{a}\overline{{{\gamma}}}= 0
\begin{aligned}z_m( \phi_{\mathrm{t},m}, \phi_{\mathrm{r},m}) =\sum_{i\in\{\mathrm{t,r}\}}\Bigg([\mathbf{V}_i]_{m,m}|\phi_{i,m}|^2 + 2\Re\Big\{\underbrace{\Big(\sum_{n\ne m}[\mathbf{V}_i]_{m,n}\phi_{i,n}- [\tilde{\mathbf{v}}_i]_m\Big)}_{= \chi_{i,m}}\phi_{i,m}^*\Big\}\Bigg). \end{aligned}
\frac{d^2x}{d\tau^2}= \frac{\left(m \tau^2(x^2-1)\! -\!\omega^2\tau^2\right)\! \frac{dx}{d\tau}-\frac{m \left(k-m \right)}{3\omega^2}x^3\tau + \frac{k\,\left( k-m \right)}{\omega^2}x\tau+f(\tau^2+1)}{\omega^2\,\tau^3}.
\mathrm{P}\left( \bar{E}_{m,1}\right) = \frac{\gamma(M-m+1,\xi_m)}{(M-m)!}\times \prod^{m-1}_{n=1}\left[1-\frac{\gamma\left(M-n+1,\max\left\{\xi_n, \frac{\epsilon_{2,n}}{\rho \beta_n^2}\right\}\right)}{(M-n)!}\right] ,
E[Z(t)^2] = \frac{1}{C(H)^2}\int_{\mathbb{R}}\Bigg[(t - s)^{H - \frac{1}{2}}_+ - (-s)^{H- \frac{1}{2}}_+ \Bigg]^2 ds & = \frac{1}{C(H)^2}t^{2H}\int_{\mathbb{R}}\Bigg[(1 - u)^{H - \frac{1}{2}}_+ - (-u)^{H- \frac{1}{2}}_+ \Bigg]^2 du &= t^{2H}
h\,\left(\,t;{\bf x},{\bf x}\,|\,M\,\right) =\frac{1}{\left( 4\,\pi\,t \right)^{d/m}}\:\left[\,a_0\,(\,x\,|\,M\,) +a_1\,(\,x\,|\,M\,)\:t + \dots \,\right]\ .
C_{01\ell}(N) = \frac{(-1)^N}{N!}\left[ z^{N-\ell}\right]e^z \left(\frac{e^z - 1}{z}\right)^{\ell-1}\left(\frac{z}{e^z - 1}\right)\left(\frac{2z}{e^{2z}- 1}\right) \cdots \left(\frac{N z}{e^{Nz}- 1}\right).
A' \ = \ \sum_{\alpha = 0}^{a-1}\sum_{\delta = 0}^{a - \alpha - 1}\sum_{j=0}^\delta \binom{n}{\alpha + \delta}\binom{\alpha + \delta}{\alpha}\sum_{i=0}^{a - \alpha - \delta - 1}(-1)^{\alpha +j+i}\binom{n - \alpha - \delta}{i}2^{n-1-\alpha-j}\binom{\delta}{j}I(\alpha + j, 0).
\sigma A^{2}r_{p}^{2}=\sum\limits_{j=1,\textrm{}j\neq i}^{n}\frac{m_{j}\left(1-\sigma(q_{ip}\odot_{k}q_{jp})\right)}{(\sigma-\sigma(q_{ip}\odot_{k}q_{jp})^{2})^{\frac{3}{2}}},\textrm{}i\in\{1,...,n\}.
-(t*x)yh_{gt}-gt\beta_{(t*x)y}+ x(g*y)h_{gt}+gt\beta_{x(g*y)}= x[h_g, y]h_t+y[x, h_t]h_g+tx[h_g, \beta_y]+yg[\beta_x, h_t]+t[g, \beta_y]\beta_x+g[\beta_x, t]\beta_y.
\sum_{j=1}^{k}\prod_{i=k}^{k+j-2}\left(1-\frac{1}{i^\beta}\right)\le e^{ck^{1-\beta}}\sum_{j=k+1}^{2k-1}e^{-cj^{1-\beta}}\le e^{ck^{1-\beta}}\int_{k}^{\infty}e^{-cx^{1-\beta}}dx.
Z = \int{\cal D}h_{\mu \nu}{\cal D}\theta_{\mu}{\cal D}\varphi{\cal D}b_{\mu}{\cal D}b{\cal D}c_{\mu}{\cal D}\bar{c}_{\mu}{\cal D}c{\cal D}\bar{c}\exp i \int d^{4}x L_{\mathrm{T}, a = 1}^{\prime},
D(e_{ii}) =\sum_{x<i}C_{xi}^{ii}e_{xi}+\sum_{y>i}C_{iy}^{ii}e_{iy}, D(e_{ij}) =\sum_{x<i}C_{xi}^{ii}e_{xj}+C_{ij}^{ij}e_{ij}+\sum_{y>j}C_{jy}^{jj}e_{iy},\mbox{if}i<j.
\frac{1}{hn}\sum_{g \in G}g \Bigg( \sum_{i\in S}\sigma_iH\sigma_i^{-1}\Bigg) g^{-1}= \frac{1}{hn}\sum_{i \in S}\Bigg( \sum_{g \in G}g \sigma_i H \sigma_i^{-1}g^{-1}\Bigg) &= \frac{1}{hn}\sum_{i\in S}\Bigg(\sum_{g \in G}gHg^{-1}\Bigg) &= \frac{\epsilon}{hn}\sum_{g \in G}gHg^{-1}.
\int_{\Lambda^\infty}S_\lambda f^{m,v}\overline{f^{m',v'}}\, dM = \sum_{\mu \in D_v}c^{m,v}_\mu \overline{c^{m',v'}_\lambda}\rho(\Lambda)^{d(\lambda)/2}M(Z(\lambda\mu)) &= \overline{c^{m',v'}_\lambda}\rho(\Lambda)^{-d(\lambda)/2}\sum_{\mu \in D_v}c^{m,v}_\mu \rho(\Lambda)^{-d(\mu)}x^\Lambda_{s(\mu)}&= 0.
R_M(u, x, e^{z}x) = \int_0^u \int_{X}e^{-(u -s)\Delta^X}( x, y) A_M(s,y , e^{z}x) dv (y) ds + \int_0^u \int_X e^{-(u - s) \Delta^X}( e^z x, y) B_M(s, y, x) dv (y) ds\,.
K(t,x)=\pi^{-\frac{d}{2}}|x|^{-d}t^{-1}\textnormal{H}_{12}^{20}\left[\frac{1}{4}t^{-\alpha}|x|^{2}\ \begin{array}{cc}(0,\alpha) (\frac{d}{2},1) (1,1)\end{array}\right].
t u = - 1 / 2 ( \eta^{\mu \alpha}\eta^{\nu \beta}+ \eta^{\nu \alpha}\eta^{\mu \beta}- \eta^{\mu \nu}\eta^{\alpha \beta}) k_{\mu}^{1}k_{\nu}^{2}k_{\alpha}^{3}k_{\beta}^{4}\equiv - 1 / 2 K^{\mu \nu ; \alpha \beta}k_{\mu}^{1}k_{\nu}^{2}k_{\alpha}^{3}k_{\beta}^{4}
\frac{F}{V}= ( - 1 )^{f}\frac{1}{\beta}\int_{0}^{\infty}\frac{d p^{-}}{\sqrt{2}\pi}\int \frac{d^{D - 2}p_{T}}{( 2 \pi )^{D - 2}}\ln \left( 1 - ( - 1 )^{f}e^{- \frac{\beta}{\sqrt{2}}\left( \frac{{\vec{p}_{T}}^{2}+ M^{2}}{2 p^{-}}+ p^{-}\right)}\right)
\phi_{k+1}(x) := f(x_{k+1}) + A_{k+1}^{-1}D_h(x, z_{k+1}) & \overset{\eqref{Eq:Est}}{=}(1 - \tau_k) \phi_k(x) + \tau_k f_k(x) &\overset{~\ref{table:Table}}{=}\Big(1 - A_{k+1}^{-1}\alpha_k \Big)\Big(f(x_k) +A_k^{-1}D_h(x, z_k)\Big) +A_{k+1}^{-1}\alpha_k f_k(x).
X\circ Y = \frac 12 \begin{pmatrix}2s_1t_1 (t_1+t_2)x_3 (t_1+t_3)\overline{x_2}(t_1+t_2) \overline{x_3}2s_2t_2 (t_2+t_3)x_1 (t_1+t_3)x_2 (t_2+t_3)\overline{x_1}2s_3t_3\end{pmatrix}.
\mathrm{e}^{2\pi\mathrm{i}j/m}(z_1,z_2,y) :=(\mathrm{e}^{2\pi\mathrm{i}j/m}z_1,\mathrm{e}^{2\pi\mathrm{i}j/m}z_2,y), \qquad\qquad\tau(z_{1},z_{2},y):=(z_{2},z_{1},y), \alpha(z_1,z_2,y) :=(z_1,z_2,\alpha y)\quad\alpha\in O(N-4).
\hat{V(\overline{y}_{w})}=\frac{1}{\hat{N}^{2}}\sum_{i=1}^{n}((1-\pi_{i})\frac{y_{i}-\overline{y}_{w}}{\pi_{i}})^{2}=\frac{1}{(\sum_{i=1}^{n}w_{i})^{2}}\sum_{i=1}^{n}w_{i}^{2}(y_{i}-\overline{y}_{w})^{2}.
\chi_\ell(x)^g= \det \left( \rho_\ell(x):A[\ell] \to A[\ell] \right) = \prod_{W_i}\det \left( \rho_\ell(x) :W_i \to W_i \right) = \chi_\ell(x)^{m_1}\chi_\ell(x)^{m_2}\forall x \in I_{\mathfrak{l}}^t,
\mathcal{N}_{\ast}(s,k)=\frac{1}{(2\pi)^{2}}\iiint e^{is(-k^2+\ell^2-m^2+n^2)}f^{\#}(s,\ell)\overline{f^{\#}(s,m)}f^{\#}(s,n) \times \mu^{\#}_{\ast}(k,\ell,m,n)\,dndmd\ell, \ast\in\{0; L; R,\!1; R,\!2\}
\mathcal{T}:= \lbrace t\in [0,\overline{t}] :\ \exists k_t \geq 0 \mbox{s.t.}\Pi_s \mbox{is a graph of a}k_t \mbox{-Lipschitz function}&\Phi_s:\overline{\mathcal{D}( V_{\xi_0}\times I_s)}\rightarrow \mathbb{R}^n \mbox{for all}s\in [0,t] \rbrace.
\sum_{n \in \mathfrak{S}_N}c_n^c(u_0,f) \leqslant \frac{\beta_- - 1}{\pi c_-}\Big( 3 \times 2^{-\mu+\alpha+\beta_-}\, M \, + 2^{-\mu + \beta_-}(2^{\alpha}M + M') \Big) \, \frac{\alpha + 2 - \mu - \beta_-}{\alpha + 1 - \mu - \beta_-}& =: c_2^c(u_0,f) \; .
z=\frac{1}{a}(w-w_{0})-\frac{b}{a^{3}}(w-w_{0})^{2}-\frac{1}{a^{5}}(ac-2b^{2})(w-w_{0})^{3}-\frac{1}{a^{7}}(a^{2}d-5abc+5b^{3})(w-w_{0})^{4}+\ldots
\lim_{v \to x_{i}}\left[ (\partial_{x}+ \partial_{u})^{2}f(x,x_{i},u,v) \right] = - (2\pi)^{2}\frac{(x-u)^{2}}{(x-x_{i})^{2}(u-x_{i})^{2}}\: g(x,x_{i},u) \; .
\frac{1}{2}\frac{d}{dt}( |\theta|_{L^2(\Omega)}^2 + |u|_{L^2(\Omega)}^2) + \kappa |\nabla \theta|_{L^2(\Omega)}^2 + \sum_{i=1}^N \kappa_i |\nabla u_i|_{L^2(\Omega)}^2 \leq \hat{C}(|\theta|_{L^2(\Omega)}^2 + |u|_{L^2(\Omega)}^2) \mbox{a.e. on}[0,T],
\displaystyle \frac{\vartheta_{\sigma}^{( n )}( w^{\prime}, x^{- 1}z_{2 n - 1}| \zeta^{\prime \prime}, \zeta_{2 n - 1}, \epsilon x^{- 1}\zeta_{2 n - 1})}{\vartheta_{\epsilon \sigma}^{( n - 1 )}( w^{\prime}| \zeta^{\prime \prime})}= \displaystyle x^{2 n}z_{2 n - 1}^{- 2 n}\prod_{j = 1}^{2 n - 2}z_{j}^{- 1}\prod_{a \in A \atop a \neq a_{2 n}}w_{a}^{- 1}\Theta_{x^{2}}( x w_{a}/ z_{2 n - 1}) .
a_+(v) ( \psi^{(n)}) = ( a_{n}(v)\psi^{(n+1)}) a^{\dagger}_+(v) ( \psi^{(n)}) = (0, a^\dagger_0(v)\psi^{(0)},a_1^\dagger(v)\psi^{(1)},\dots ) \varphi_+(v) = a_+(v)+a_+^\dagger(v)
\mathbf{i}_{\alpha}\widetilde \alpha \equiv ( i_{W \alpha}\otimes 1 + 1 \otimes i_{\alpha}) \widetilde \alpha = 0 \quad , \quad \mathbf{L}_{\alpha}\widetilde \alpha \equiv ( L_{W \alpha}\otimes 1 + 1 \otimes L_{\alpha}) \widetilde \alpha = 0 \quad ,
\int_{0}^{1}\left(2A(0) - A(u) - A(-u)\right)\hat{\psi}(u) du +\kappa \int_{0}^{1}\left(A(u)- A(-u)\right)u\hat{\psi}(u) du -\int_{0}^{1}\left(h(-\kappa u)A(u) + h(\kappa u)A(-u)\right)\hat{\psi}(u) du .
\lim_{N \to +\infty}\int_0^{\hat t}\int_{\mathbb R^{d}}\nabla \phi(x) \cdot [K\star \mu_{N}(t)+ g(y_{N}(t)](x) d \mu_{N}(t)(x) dt &= \int_0^{\hat t}\int_{\mathbb R^{d}}\nabla \phi(x) \cdot [K\star \mu(t)+g(y(t))](x) d \mu(t)(x) dt,
(\beta\lambda u_1h(a_1)f^{p^{e'}}(a_1),\dots,\beta\lambda u_sh(a_s)f^{p^{e'}}(a_s), \lambda u_{s+1}h(a_{s+1})f^{p^{e'}}(a_{s+1}),\dots,\lambda u_nh(a_n)f^{p^{e'}}(a_n), & f_{k-1}^{p^{e'}}) = (u_1g(a_1),\dots,u_sg(a_s),u_{s+1}g(a_{s+1}),\dots,u_ng(a_n),-g_{n-k}).
\sum_{j=0}^{k_1(x)-1}\Psi_g(f)(\sigma_B^j(h(\sigma_A(x)))) = \sum_{i'=0}^{l_2^{k_1(x)}(h(\sigma_A(x)))-1}f(\sigma_C^{i'}(gh(\sigma_A(x))))-\sum_{j'=0}^{k_2^{k_1(x)}(h(\sigma_A(x)))-1}f(\sigma_C^{j'}(g(\sigma_B^{k_1(x)}(h(\sigma_A(x)))))).
v_{\theta}(r)=ar+\frac{b}{r},a=\frac{\Omega_{2}R_{2}^{2}-\Omega_{1}R_{1}^{2}}{R_{2}^{2}-R_{1}^{2}},b=\frac{(\Omega_{1}-\Omega_{2})R_{1}^{2}R_{2}^{2}}{R_{2}^{2}-R_{1}^{2}}.
\sum_{1\leq i_1,\ldots,i_m\leq k}\sum_{1\leq j_1<\ldots<j_p\leq m}f_p(X_{i_{j_1}},\ldots,X_{i_{j_p}}) ={m\choose p}\sum_{1\leq i_1,\ldots,i_m\leq k}f_p(X_{i_1},\ldots,X_{i_p}) & ={m\choose p}k^{m-p}\sum_{1\leq i_1,\ldots,i_p\leq k}f_p(X_{i_1},\ldots,X_{i_p}).
\|H(t)\|^2_{H^2\cap W^{2,q}}+ \|H_t(t)\|^2_{1}\leq Cc^4_1, \int_{0}^{t}|H_{tt}|^2_2s\leq Cc^3_1,t|H_t(t)|^2_{D^{1,q}}\leq Cc^3_2
n_{{\bf{k}}}= n^{\beta}({\bf{k}}) \frac{N}{\langle N \rangle}+ \sum_{{\bf{q}}}a_{{\bf{k}}-{\bf{q}}/ 2}^{\dagger}({\bf{q}}) a_{{\bf{k}}-{\bf{q}}/ 2}({\bf{q}}) - \sum_{{\bf{q}}}a_{{\bf{k}}+{\bf{q}}/ 2}^{\dagger}({\bf{q}}) a_{{\bf{k}}+{\bf{q}}/ 2}({\bf{q}})
E(x) = \left( \begin{array}{cc}I_R 2B(x) 0 I_L \end{array}\right) \;\;\;\;\; F_R(x) = \left( \begin{array}{c}G_R(x) 0 \end{array}\right) \;\;\;\;\; F_L(x) = \left( \begin{array}{cc}0 G_L(x) \end{array}\right).
\bigl\langle(f(A_0)-f(A_1))(\xi_k^{(1)}),\xi_i^{(0)}\bigr\rangle=f^{[1]}(\lambda_i^{(0)},\lambda_k^{(1)})\bigl\langle (A_0-A_1)(\xi_k^{(1)}),\xi_i^{(0)}\bigr\rangle.
\tilde{Q}^{n}_{\omega}(B\times R\times D) \doteq \frac{1}{n}\sum_{i=1}^{n}\delta_{\tilde{X}^{n}_{i}(\cdot,\omega)}(B)\cdot \delta_{\tilde{\rho}^{n,i}_{\omega}}(R)\cdot \delta_{W^{n}_{i}(\cdot,\omega)}(D),\omega\in \Omega_{n},
\frac{U(o,o\vert z)}{r}=\frac{-\left(\left(m+\lambda-1\right)-mU(o,o|z)-\left(d-1\right)z\right)+\sqrt{\left(\left(m+\lambda-1\right)-mU(o,o|z)-\left(d-1\right)z\right)^2+4\lambda dz^2}}{2m};
Lb_N(z,\xi,w) :=-Nb_Nz^{N-1}[a^-]_s(z,\xi,w)-[a^-]_s(z,\xi,w)\bar b_N\sum_i\bar z^i(-\bar z-\xi)^{N-1-i}&+\nabla [a^-]_s\cdot \Bigl(b_Nz^N, -\bar b_N(\bar z^N+(-\bar z-\xi)^N),\bar b_Nw(\bar z^{N-1}+(-\bar z-\xi)^{N-1})\Bigr).
I_{X,1}\ :=\ \langle\,_rT_{ij}\ -\_rf_{ij};~r=1,\ldots,\mathbf r,~0\le i<j\le n_r+2,\i\le n_r\,\rangle, I_{Y,1}\ :=\ \langle\,S_{ij}\ -\g_{ij};~1\le i<j\le n+2,\i\le n\,\rangle.
h_i = \frac{\tilde{\theta}_i^{b_i / d_i}}{\theta_i^{b_i p / d_i}}\cdot \frac{\theta_i^{p (s^\mathsf{T}A)_i}}{(\theta_i;\zeta_i^q)_{ps_i}}\cdot \prod_{t=1}^{p-1}\frac{1}{(\theta_i; \zeta_i)_{qt}}\in H^\times.
r_2=\dfrac{q_2^2-1}{4}\big(e_1^2-e_2\big)-\frac{1}{2}e_1q_1q_2+\frac{1}{2}q_2q_0+\frac{1}{4}q_1^2, r_1=\dfrac{q_2^2-1}{4}(e_3-e_2e_1)+\frac{1}{2}e_2q_1q_2-\frac{1}{2}q_1q_0,
P \ = \ \sum_{i , j = 1}^{r}| z^{i}\rangle \, \frac{1}{\langle z^{i}| z^{j}\rangle}\, \langle z^{j}| \ = \{\cal U}\, \sum_{n = 0}^{r - 1}| n \rangle \langle n | \;{\cal U}^{\dagger}\quad ,
\langle \mathcal{F}^{-1}(AH_{0}f), \phi \rangle = \langle ( \alpha \cdot x ) \mathcal{F}^{-1}f,\frac{\alpha \cdot x}{|x|^{2}}\phi \rangle &= \langle \mathcal{F}^{-1}f, \frac{(\alpha \cdot x)^{2}}{|x|^{2}}\phi \rangle= \langle \mathcal{F}^{-1}f, \phi \rangle.
\Gamma = \int d t \Bigl [ \sum_{a}{\frac{N_{a}}{2 g_{s}}}v_{a}^{2}+{\frac{1}{2}}\sum_{a , b}{\frac{1 5 N_{a}N_{b}}{1 6}}{\frac{v_{a b}^{4}}{r_{a b}^{7}}}-{\frac{1}{2}}\sum_{a , b}{\frac{1 5 N_{a}N_{b}}{8}}{\frac{\dot{g}_{s}}{g_{s}}}{\frac{v_{a b}^{2}v_{a b}\cdot x_{a b}}{r_{a b}^{7}}}\Bigr ] .
{\omega}_{\gamma_i}\sim{\rho^3R^{d_i}(\rho d^{4-d_i}x_0+x_0 d\rho\,d^{3-d_i}x_0)\over (\rho^2+x_0^2)^4},\quad{\Omega}_{\gamma_i}\sim{R^{d_i}(3\rho^2+x_0^2)x_0d^{3-d_i}x_0\over(\rho^2+x_0^2)^3},\quad{\rm for}\,\, x_0\rightarrow \infty,
G^{\prime}= \vec{e}_{R}\cdot \vec{g}= \left( \begin{array}{c c c c}{{0}}&{{- g_{3}}}&{{g_{2}}}&{{- g_{1}}}\\{{g_{3}}}&{{0}}&{{- g_{1}}}&{{- g_{2}}}\\{{- g_{2}}}&{{g_{1}}}&{{0}}&{{- g_{3}}}\\{{g_{1}}}&{{g_{2}}}&{{g_{3}}}&{{0}}\\ \end{array}\right) = \vec{e}^{~^{\prime}}\cdot \vec{g}
\begin{array}{l}{\cal D}_-{\cal X}^--{\cal X}^-{\cal D}_-=\mu_-^{-1}~,~~~~~~\mu_-{\cal X}^-=q^2{\cal X}^-\mu_-~, \mu_-{\cal D}_-=q^{-2}{\cal D}_-\mu_-~,~~~~~~\mu_-^{-1}\equiv1+(q^{-2}-1){\cal X}^-{\cal D}_-~,\end{array}
\sum_{i=1}^{N}\tilde g(|x|^2 + \tau^2 + 2\tau |x| |\langle \hat x, \xi_i \rangle|) + \tilde g(|x|^2 + \tau^2 - 2\tau |x| |\langle \hat x, \xi_i \rangle|) \geq N \Big{(}\tilde g(|x|^2 + \tau^2 + 2\tau |x| \frac{1}{\sqrt{N}}) + \tilde g(|x|^2 + \tau^2 - 2\tau |x| \frac{1}{\sqrt{N}}) \Big{)}.
\Gamma_{ij}(\omega)= \omega^2\delta_{ij}+ \omega^4\sum_{k=1}^\infty \frac{\left( \int_{Q_0}\phi^k_i \right) \left( \int_{Q_0}\phi^k_j \right)}{\alpha_k - \omega^2},\ \ \ \i,j=1,2,3,\ \ \ \ \ \ \ \ \ \omega^2\notin\{0\}\cup\{\alpha_k\}_{k=1}^\infty.
\mathrm{Cov}\left(\int f(t) G(t) dt, \int g(t)G(t) dt \right)= \sum_{k \in{\mathbb Z}}{\mathbb{E}}\left( \iint f(t)g(s) ({\mathbf{1}}_{X_0 \leq t}-F(t)) ({\mathbf{1}}_{X_k \leq s}-F(s)) \dt ds \right)\, .
V^{q}( \psi \otimes \bar{w}, z ) \; ( \chi \otimes \bar{v}) : = \sum_{\bar{w}^{\prime}, \bar{v}^{\prime}}\left( \widetilde{F}^{- 1}\right)_{\bar{w}^{\prime}, \bar{v}^{\prime}}^{\bar{w}, \bar{v}}V ( \psi \otimes \bar{w}^{\prime}, z ) \; ( \chi \otimes \bar{v}^{\prime}) ,
d_{i}^{+}(x) :=\#\left\{p \, | \, i<p \le k, \, \epsilon_{i}(x)=\epsilon_{p}(x) \right\}, d_{i}^{-}(x) :=\#\left\{p \, | \, 1\le i<p, \, \epsilon_{i}(x)=\epsilon_{p}(x) \right\}.
k(y)=\frac{(y_d \wedge 1)^{\beta_1}(y_d \vee1)^{\beta_2}}{|y|^{d+\alpha+\beta_1+\beta_2}}(1+|\log(y_d)|)^{\beta_3}\left(\log\left(1+\frac{|y|}{y_d \vee 1}\right)\right)^{\beta_4}.
F ( T ) \simeq - 0 . 2 2 9 2 4 \frac{T}{R}- \frac{3 T}{6 4 R}\ln \frac{R T}{2 \hbar c}- \frac{5 1}{6 5 5 3 6}\frac{\hbar^{2}c^{2}}{R^{3}T}+{\cal O}( T^{- 3}) .
\rho_{Q E D}= \frac{\exp [ - \beta \int d^{3}x ( T_{Q E D}^{0 0}( x ) - \mu e \overline{{{\Psi}}}\gamma^{0}\Psi ) ]}{{\mathrm{t r}}\exp [ - \beta \int d^{3}x ( T_{Q E D}^{0 0}( x ) - \mu e \overline{{{\Psi}}}\gamma^{0}\Psi ) ]}\ ,
X_*^*\Bigl(\eta_n^2\frac{\omega_n\varphi_n}{\psi_{nt}}\varphi F_{\tau,\mathbb T\setminus\sigma}\oplus \eta_n^2\frac{\omega_n\varphi_n}{\psi_{nt}}\varphi F_{\mathbb T,\mathbb T\setminus\sigma}\Bigr)=(\varphi\eta_n)(T)y_{nt}.
\hat{\Phi}_{\rm{st},\beta}(t) = \Phi^*_{\rm{st},\beta}(t) + \frac{1}{2}\frac{\partial^2 \Phi_{\rm{st},\beta}(t)}{\partial^2 R_{\beta}}\Big|_{{R}^{*}_{\beta}}\left[ \hat{R}^{*}_{\beta}-{R}^{*}_{\beta}\right]^2 + O\left[ \left( \hat{R}^{*}_{\beta}-{R}^{*}_{\beta}\right)^3 \right].
|I_{i}|=|I|:=|\ln\gamma|^{-b}, i\in \mathcal I :=\left\{-\left\lfloor\frac{\epsilon^{-1}L}{|I|}\right\rfloor,\ldots,\left\lfloor\frac{\epsilon^{-1}L}{|I|}\right\rfloor-1\right\}
\mathcal{C}_{-1}(\alpha, \nu, 1, \varnothing, \lbrace 2 \rbrace) = \mathcal{C}_{-1}([2,1], [\nu_1, \nu_2], 1, \varnothing, \lbrace 2 \rbrace) = \left \lceil \frac{\nu_1 + 1}{2}\right \rceil; \mathcal{C}_{-1}(\alpha, \nu, 2, \varnothing, \lbrace 1 \rbrace) = \mathcal{C}_{-1}([2,1], [\nu_1, \nu_2], 2, \varnothing, \lbrace 1 \rbrace) = \nu_2 + 1.
C =\sum_{k=1}^{\infty}p^2(1-p)^{k-1}\max_{\substack{p(x^k): \sum_{i=1}^{k}X_i\leq1}}H(Y^k)-h_2(\alpha), C_{\mathrm{fb}}=\sum_{k=1}^{\infty}p^2(1-p)^{k-1}\max_{\substack{p(x^k\|y^{k-1}): \sum_{i=1}^{k}X_i\leq1}}H(Y^k)-h_2(\alpha),
X_{jk}z_\ell =0 T_{jk}w_\ell =0 X_{jk}w_\ell = \begin{cases}z_k \ell=j -z_j \ell=k 0 otherwise\end{cases}T_{jk}z_\ell = \begin{cases}w_k \ell=j -w_j \ell=k 0 otherwise. \end{cases}
m_n\dotsm c_{i_j}\dotsm m_n =q^{-1}m_n\dotsm c_{i_j}\dotsm T_{i_j-1}m_n &=q^{-1}m_n T_{i_j-1}\dotsm c_{i_j-1}\dotsm m_n &=m_n\dotsm c_{i_j-1}\dotsm m_n.
\left\langle A^m_\nu F^{n\mu\nu}\right\rangle = \left\langle A^m_\nu \partial^\mu A^{n\nu}\right\rangle - \left\langle A^m_\nu \partial^\nu A^{n\mu}\right\rangle + \epsilon^{npq}\left\langle A^m_\nu A^{p\mu}A^{q\nu}\right\rangle .
\tilde m_1h_1 = \Big(\tilde m_1-\widetilde{\left(\frac{m_1}{q}\right)}\tilde q\Big)h_1+\widetilde{\left(\frac{m_1}{q}\right)}(\tilde qh_1) &= \Big(\tilde m_1-\widetilde{\left(\frac{m_1}{q}\right)}\tilde q\Big)h_1+\sum_{j=1}^s c_j^\prime\widetilde{\left(\frac{m_1}{q}\right)}\tilde m_j^\prime h_j^\prime.
H_{C}= \int{d^{2}}x \left\{\begin{array}{r}{{- \frac{1}{2}F_{i 0}\left({1 + e^{2}a \left[{1 - 1 2 \alpha^{2}g^{2}\partial^{2}}\right]}\right) F^{i 0}+ \frac{1}{4}F_{i j}\left({1 + e^{2}a \left[{1 - 1 2 \alpha^{2}g^{2}\partial^{2}}\right]}\right) F^{i j}}}\\{{+ \Pi^{i}\partial_{i}A_{0}- \frac{\chi}{2}\varepsilon^{i j}A_{0}\partial_{i}A_{j}+ A_{0}J^{0}}}\\ \end{array}\right\}.
W^{\Omega_{z_{1}}^{*}, \Omega_{z_{2}}}\left( \begin{array}{c c}{c}&{d}\\{b}&{a}\\ \end{array}\right) = ( - 1 )^{n - 1 + \mu + \kappa}W^{( n - 1 , 1 )}\left( \left. \begin{array}{c c}{c}&{d}\\{b}&{a}\\ \end{array}\right| z_{1}- z_{2}\right) ,
\frac 1{24}\times \frac 1{24}=\frac{25}{24}\times \frac{25}{24}=0+1+\frac 16\quad \quad \frac 1{24}\times \frac{25}{24}=\frac{25}{24}\times \frac 1{24}=\frac 23+2\left( \frac 32\right)
\Delta_{i,1}f(x) = \frac{1}{g(x)}\frac{\partial}{\partial x_i}\left[g(x)\left(f(F(x))-f(x)\right)\right]- \frac{\partial (f(y)-f(x))}{\partial x_i}(x,F(x)) &= \nabla f (F(x))\cdot\frac{\partial F}{\partial x_i}(x) - \frac{\partial f}{\partial x_i}(x) +\frac{\partial \log g(x)}{\partial x_i}(f(F(x))-f(x)) +\frac{\partial f}{\partial x_i}(x),
\partial_{t}( v - \langle v \rangle ) + v \partial_{x}v - \frac{{\Delta u}^{2}+{\Delta v}^{2}}{{\Delta q}^{2}}( x - \langle q \rangle ) + \frac{1}{2}\partial_{x}\bigg ( u^{2}+ \frac{\hbar}{m}\partial_{x}u \bigg ) = 0 \: .
h^T_\tau({\bf X}_i,{\bf X}_j,{\bf X}_k,{\bf X}_l ) := \sum_{1 \leq p < q \leq m}h_\tau^W ({\bf X}_i^{(pq)},{\bf X}_j^{(pq)},{\bf X}_k^{(pq)},{\bf X}_l^{(pq)}),
\frac{1}{\alpha k_{0}^{2}- | \mathbf{k}|^{2}- m^{2}+ i \varepsilon}+ \frac{1}{\alpha k_{0}^{2}- | \mathbf{k}|^{2}- m^{2}+ i \varepsilon}\bullet \frac{- \delta \alpha k_{0}^{2}}{\alpha k_{0}^{2}- | \mathbf{k}|^{2}- m^{2}+ i \varepsilon}+ . . . . . .
I_{mn}(x,y,z)=I_{0}(\frac{w_{0}}{w})^{2}[H_{m}(\frac{\sqrt{2}x}{w})exp(\frac{-x^{2}}{w^{2}})]^{2}[H_{n}(\frac{\sqrt{2}y}{w})exp(\frac{-y^{2}}{w^{2}})]^{2}