harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{cases}\partial_t u+u\cdot\nabla u+\nabla P=\mu \Delta u-\nabla\cdot(\nabla d\odot\nabla d), \nabla\cdot u = 0, \partial_t d+u\cdot\nabla d=\Delta d+\frac{1}{\epsilon^2}\left(1-|d|^2\right)d.\end{cases}
\sigma(P_{\rho}^*)(\xi) = \left[\sum_{|\ell|<N_1}\sum_m\frac{(-m)^{\ell}}{\ell!}(\partial^{\ell}\rho_m)(\xi)\prod_{j=1}^mU_j^{m_j}+T_{N_1}(\xi,y)\right]^* &= \sum_{|\ell|<N_1}\frac{\partial^{\ell}\delta^{\ell}[(\rho(\xi))^*]}{\ell!}+T_{N_1}(\xi,y)^*
\Biggl(\frac{1}{(3+x)^s(1-x)}\Biggr)_{k-s}&= \frac{1}{4^s}\Biggl(1 +\biggl(-\frac{1}{3}\biggr)^{k-1}\biggl(\frac{1}{3}+\frac{4}{3^2}\binom{k-s+1}{1}+\frac{4^2}{3^3}\binom{k-s+2}{2}+\cdots+\frac{4^{s-1}}{3^s}\binom{k-1}{s-1}\biggr)\Biggr) &=\frac{1}{4^s3^k}\Biggl( 3^k+(-1)^{k-s}\sum_{i=0}^{s-1}\binom{k-s+i}{i}3^{s-1-i}4^i \Biggr).
\mathcal{S}=\sum\limits_{p=0}^{n-1}\gamma_{p}^{(\nu ,n)}\left[ \left( -\frac{d}{dt}\right)^{p}\vartheta_{2}(t)+\sum\limits_{\ell =0}^{+\infty}\frac{(-1)^{\ell}}{(p+\ell +1)\ell !}\left[ B_{2(p+\ell +1)}\left(\frac{1}{2}\right)-B_{2(p+\ell +1)}\left(\nu+\frac{1}{2}\right)\right] \t^{\ell}\right]
\omega_{1,1}= (jkx_1)(\overline{jk}x_1)+(jx_2)(\bar{j}x_2)+(kx_3)(\bar{k}x_3)+(x_3)(x_3) = jk\overline{jk}x_1x_1+j\bar{j}x_2x_2+k\bar{k}x_3x_3+x_3x_3 & = x_1^2+x_2^2+2x_3^2
\phi_{1}^{[ k ]}( z_{1}) = 3 D \frac{2 f^{\prime}= ( z_{1}) V^{2}( z_{1})}{| V ( z_{1}) |^{2}+ | f ( z_{1}) V ( z_{1}) |^{2}}\hspace{1 c m}z_{1}= 3 D x_{1}+ i x_{2}\hspace{2 . 3 c m}( 8 )
\zeta(\tilde{\sigma_{0}}) = \dfrac{1}{4(-1-\mathit{e}+2\alpha+\beta)^2}\big (2(2+\beta-2\alpha)(1+\mathit{e}^2-2\mathit{e}(2\alpha+\beta -1)) &\quad \times((-2\alpha+2+\beta) -\sqrt{(-2\alpha+2+\beta)^2-4(\mathit{e}-1)(2\alpha-1+\beta-\mathit{e})}) &\quad{}+ 4(2\alpha+\beta-1-\mathit{e})(\mathit{e}^2-1)(2\alpha+\beta-2)\big),
\begin{aligned}\||u|+|D_v u|+|D_v^2 u|\|_{L_2((-1,0)\times B_{R^3}\times B_R)}&\le N (d) \delta^{-1}\sum_{k=0}^\infty 2^{-k(k-1)/4}R^{-k}\|f\|_{L_2(Q_{1,2^{k+1}R/\delta^2})},\end{aligned}
\begin{array}{l}{{\partial X ( z ) \sigma_{\vartheta}( w , \bar{w}) \sim ( z - w )^{- ( 1 - \vartheta )}\tau_{\vartheta}( w , \bar{w}) ,}}\\{{\partial \bar{X}( z ) \sigma_{\vartheta}( w , \bar{w}) \sim ( z - w )^{- \vartheta}\tau_{\vartheta}^{\prime}( w , \bar{w}) ,}}\\ \end{array}
v \in W_\ell \Longleftrightarrow h(z) \, v = z^{\ell}\, v \Longleftrightarrow P h'(z) P^{-1}\, v = z^{\ell}\, v \Longleftrightarrow h'(z) P^{-1}\, v = z^{\ell}\, P^{-1}\, v \Longleftrightarrow P^{-1}\, v \in{W'}_\ell .
\frac{1}{k}D_z p_T (\mathbf{y}) = \sum_{a+b+c=k-1}x_u^a x_v^b x_w^c \binom{k-1}{a,b,c}d_T(u,v,w) + \sum_{a+c=k-1}x_u^a x_w^c \binom{k-1}{a,c}(d_T(u,w)-d_T(u,v,w)) + x_u^{k-1}(d_T(u)-d_T(u,w)) = 2 (x_u+x_v+x_w)^{k-1}- (x_u+x_w)^{k-1}- x_u^{k-1}= 0 - (-\zeta)^{k-1}- 1^{k-1}= 0.
P_{Y^{j-1}, \underline S}(0^{j-1}, \underline S) H(P_{Y_j| Y^{j-1}\underline S}(1 | 0^{j-1}, \underline S)) & = (1- \frac{\sum_{k=1}^{j-1}b_k^0}{M}) H(\frac{b_j^0}{M- \sum_{k=1}^{j-1}b_k^0}).
\begin{array}{rcl}-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]-\lambda_{p}\hat{Q}v_{p}'-\hat{A}_{12}^{\top}\mu'_{p}= 0 -\hat{A}_{p}^{\top}\mu'_{p}-B_{p}^{\top}\eta'_{p}= 0 \hat{A}_{12}v_{p}'+\hat{A}_{p}v''_{p}= 0 B_{p}v''_{p}= b\end{array}
\begin{aligned}u_t+Bu= 0\ \ \ \ \ \{\rm{in}}\ \ \Omega \times (0,T] \begin{rcases}u\geq \psi, \ \ \alpha u_t+u_\nu\geq 0\ \ \ &(\alpha u_t+u_\nu)(u-\psi)=0\ \ \end{rcases}\ \ \{\rm{on}}\ \ \Gamma \times (0,T] u=\phi\ \ \ \ \ \{\rm{on}}\ \ \partial_p(\Omega \times (0,T])\setminus (\Gamma\times (0,T]) \end{aligned}
\|\operatorname{div}\vec{v}\| =\sup_{q\in Q,~\|q\|=1}({\rm div}\vec{v},q)=\sup_{q\in Q,~\|q\|=1}({\rm div}(\vec{v}-P\vec{v}),q) \leq \left\|{\rm tr}[\varepsilon(\vec{v}-P\vec{v})]\right\|\leq\sqrt{d}\left\|\varepsilon(\vec{v}-P\vec{v})\right\|.
(x-c)Q_{n-1}(x)=P_{n}(x)+(\lambda_{n}+\xi_{n-1}-c)P_{n-1}(x)-\sum_{i=2}^{d}\sum_{j=i}^{d}\frac{\eta_{n-j}^{d-j}}{\lambda_{n-i}...\lambda_{n-j}}P_{n-i}(x).
g(t)\leq g\left(\frac{m}{\lambda^{n_t-1}}\right) \leq c_3^{-n_t}g\left(\lambda^{n_t}\frac{m}{\lambda^{n_t-1}}\right) \leq c_3^{-\log_{\lambda}\frac{m}{t}}g(\lambda m) = \frac{g(\lambda m)}{m^{\log_{\lambda}c_3}}\, t^{\log_{\lambda}m}
U_k(s,x)=U(0,x)-\alpha\int_0^s \int_0^1 A(x,y)U_{k-1}(u,y)\mathrm{d}y\mathrm{d}u +\alpha\beta\int_0^s \mathrm{d}\xi_2(u,x)+\alpha\zeta\int_0^s \mathrm{d}\xi_3(u,x).
F_6(a,b,c,d,e,f) = \int\limits_0^{\infty}\!\! dx x J_0(ax) J_0(bx)J_0(cx) J_0(dx) J_0(ex) J_0(fx) &= \int\limits_0^{\infty}\!dt \,t \,F_3(a,b,t)\!\int\limits_0^{\infty}\!dq \,q \,F_3(c,d,q)\!\int\limits_0^{\infty}\!dp \,p \,F_3(e,f,p) \,F_3(t,q,p) &= \int\limits_0^{\infty}\!dt \,t \,F_4(a,b,c,t) \,F_4(d,e,f,t) \;.
\{\left(\chi_{-j_k}\right)^{m_k}\dots \left(\chi_{-j_2}\right)^{m_2}\left(\chi_{-j_1}\right)^{m_1}|0\rangle \ \arrowvert \ \j_k>\dots > j_2 > j_1 > 0, \j_i\in \mathbb{Z}+\frac{1}{2}, \m_i > 0, m_i\in \mathbb{Z}, \i=1, 2, \dots, k\}.
\begin{array}{c}\partial_{zz}^{2}w\left(z,\,\tau\right)\geq0 \partial_{z}w\left(z,\,\tau\right)\,\geq\,\partial_{z}w\left(\frac{\hat{w}\left(0,\,\tau\right)}{\sqrt{2}},\,\tau\right)=-1 \partial_{z}w\left(z,\,\tau\right)\,\leq\,\partial_{z}w\left(3\beta,\,\tau\right)\,\leq\,C\left(n,\,\Lambda\right)\beta^{\alpha-1}\leq\frac{1}{3}\end{array}
\left ( \begin{array}{l}\eta_0^*\nabla f(x^*)+\displaystyle \sum_{j=1}^p[y^I]^*_j\nabla^2c_j(x^*)\eta_1^*- A^TA\eta_1^*+{\cal J}c(x^*)^T{\rm Diag}(v_b)\xi^* begin{align*}6pt]{\cal J}c(x^*)\eta_1^*+{\rm Diag}(v_b)\xi^*+{\rm Diag}(v_a)\xi^* \end{array}\right )=0.
\begin{aligned}\partial f_{[{\scriptscriptstyle\frac12},{\scriptscriptstyle\frac12},0]}= e^{-1}_{[0,{\scriptscriptstyle\frac12},0]}\, e^{-1}_{[{\scriptscriptstyle\frac12},1,0]}\, e_{[1,{\scriptscriptstyle\frac12},0]}\, e_{[{\scriptscriptstyle\frac12},0,0]},\end{aligned}
\alpha (x + i) = e^{- n}\frac{e^{i n x}(x + i)^s}{(x + 2 i)^{r + 1}}\hphantom{\alpha (x + i)}{}= e^{- n}\sum_{k \geq 0}( - 1)^k \binom{r + k}{k}x^k (2 i)^{- r - k - 1}\sum_{j \geq 0}\binom{s}{j}x^j i^{s - j}\sum_{l \geq 0}\frac{(i n x)^l}{l!}.
\bar{\varepsilon}_{i}\gamma_{\mu_{1}}\gamma_{\mu_{2}}\cdots\gamma_{\mu_{n}}\psi^{i}_{m}=-\bar{\psi}_{mi}\gamma_{\mu_{n}}\cdots\gamma_{\mu_{2}}\gamma_{\mu_{1}}\varepsilon^{i}=-(\bar{\varepsilon}_{i}\gamma_{\mu_{1}}\gamma_{\mu_{2}}\cdots\gamma_{\mu_{n}}\psi^{i}_{-m})^{*}\,.
\mathcal{H}_{\varphi_0}: \{\psi\in\mathfrak{B}_{b}^{L}(d_0,M_0): \psi(s,0)=\varphi_0\}\to\{\psi\in\mathfrak{B}_{b}^{L}(d_0,M_0): \psi(s,0)=\varphi_0\}
\begin{cases}\partial_t \left( \frac{\omega}r \right) + (u \cdot \nabla ) \left( \frac{\omega}r \right) =0, 0<t\le 1, u=-\Delta^{-1}\nabla \times \omega, \omega \Bigr|_{t=0}=(\omega_{-1}^{\theta}+\omega_0^{\theta}+\sum_{j=1}^N \omega_j^{\theta}) e_{\theta}, \end{cases}
X= \left[ \begin{array}{rrrr}jkx_1 jx_2 kx_3 x_3 jx_2 jkx_1 x_3 kx_3 kx_3 x_3 jkx_1 jx_2 x_3 kx_3 jx_2 jkx_1\end{array}\right]\!,
\int e^{ax}((a)^{n}f(x)-(-1)^{n}\frac{d^{n}f(x)}{dx^{n}})dx=e^{ax}\sum_{k=1}^{n}(a)^{n-k}(-1)^{k-1}\frac{d^{k-1}f(x)}{dx^{k-1}}+C
A_{\Phi_S}^0 = \sum_{\{i,j\}\subseteq S}A_{\Phi_{\{i,j\}}}^0 - (\epsilon-2)\sum_{i \in S}A_{\Phi_{\{i\}}}^0 + \frac{(\epsilon-1)(\epsilon-2)}{2}A_{\Phi_{\{\varnothing\}}}^0.
\lim\limits_{n\to\infty}\frac{\rho^{\Pi_n}(p)}{\rho^{\Pi_n}(q)}=\lim\limits_{n\to\infty}\frac{(1-(1-p/2n^2))^s}{(1-(1-q/2n^2))^s}=\frac{p^s}{q^s}=\frac{\rho^{J_s}(p)}{\rho^{J_s}(q)}.
\left\langle u_{1}, u_{2}\right\rangle_{H_{\lambda}}=\int_{\mathbb{B}^{N}}u_{1}^{p}u_{2}\mathrm{~d}V_{\mathbb{B}^{N}}=\int_{\mathbb{B}^{N}}u_{2}^{p}u_{1}\mathrm{~d}V_{\mathbb{B}^{N}}, \;\left\|u_{i}\right\|_{H_{\lambda}}=\int_{\mathbb{B}^{N}}u_{i}^{p+1}\mathrm{~d}V_{\mathbb{B}^{N}}, \; i=1,2.
\Big{(}\dot{u}_t^*(-u_{t,j}^{*,i,j}+I_{H,i}(a))P_{DH}'\Big{)}_i = -\dot{u}^*_{t,i}u^{*,i,j}_{t,j}P_{DH}'+\dot{u}^*_{t,i}I_{H,i}(a)P_{DH}' -\dot{u}^*_{t}u^{*,i,j}_{t,i,j}P_{DH}'+\dot{u}^*_{t}u^*_{j,i}I_{H,i,j}(a)P_{DH}' -\dot{u}^*_{t}u^{*,i,j}_{t,j}P_{DH,i}'+\dot{u}^*_{t}I_{H,i}(a)P_{DH,i}'
I_{i}= \frac{h_{1}^{i}+ h_{2}^{i}}{\alpha^{\prime}\pi}\left[ \sum_{n = 1}^{\infty}\left( \overline{{{b}}}_{n}^{i}b_{n}^{i}- \overline{{{a}}}_{n}^{i}a_{n}^{i}\right) - \frac{\left| \epsilon^{i}\right|}{\epsilon^{i}}\overline{{{a}}}_{0}^{i}a_{0}^{i}\right] ,
\footnotesize \aligned dz' =\big(z_z'+i\,z_w'\,\varphi_z\big)\,dz +\big(i\,z_w'\,\varphi_{\overline{z}}\big)\,d\overline{z}+\big(i\,z_w'\,\varphi_u+z_w'\big)\,du & = \big(z_z'+i\,z_w'\,\varphi_z\big)\,dz + z_w' \big\{\underline{i\,\varphi_{\overline{z}}\,d\overline{z}+ (i\,\varphi_u+1)\,du}\big\}.\endaligned
(D \cdot E \cdot E_n)_X = (D\vert_E \cdot E_n \vert_E)_E (D \cdot E \cdot E_n)_X = \left( (\phi^\ast)^n(D) \cdot (\phi^\ast)^n(E) \cdot (\phi^\ast)^n(E_n)\right)_X = (\lambda^n D \cdot E_{-n}\cdot E)_X = \lambda^n(D \vert_E \cdot E_{-n}\vert_E)_E.
\widehat{R}(e_{i}\otimes e_{i}) = q e_{i}\otimes e_{i}, \widehat{R}(e_{i}\otimes e_{j}) = e_{j}\otimes e_{i}, i > j, \widehat{R}(e_{i}\otimes e_{j}) = e_{j}\otimes e_{i}+ (q - q^{-1}) e_{i}\otimes e_{j}, i < j.
\mathcal H_1 \to \mathcal H_{p,0}: w = (z_1,z_2,z_3) \mapsto w\otimes...\otimes w =\!\!\sum_{j+k+l=1}\sqrt{\dfrac{p!}{j!\,k!\,l!}}z_1^jz_2^kz_3^l \, e_{j,k,l}\ .
f^{\prime\prime}(x)\approx\frac{\Delta_{h}^{2}[f](x)}{h^{2}}=\frac{\frac{f(x+2h)-f(x+h)}{h}-\frac{f(x+h)-f(x)}{h}}{h}=\frac{f(x+2h)-2f(x+h)+f(x)}{h^{2}}.
\frac{X(s)^{\perp}}{2}=\frac{1}{2}(X(s)-g^{ij}\langle X(s),X_i(s)\rangle X_j(s)) &=\sqrt{2}\left(\frac{1}{2}+\frac{s}{2}f-s^2\left(\left(\frac{\partial f}{\partial\theta_1}\right)^2+\left(\frac{\partial f}{\partial\theta_2}\right)^2\right)\right)(e^{i\theta_1},e^{i\theta_2})+T_2(s),
( k + h )^{2}> > s^{2}- r^{2}> 0 , \ \ ( \tilde{k}+ \tilde{h})^{2}> > \tilde{s}^{2}- \tilde{r}^{2}> 0 , \ \ ( k + h ) ( \tilde{k}+ \tilde{h}) > > s \tilde{s}- r \tilde{r},
\lambda_\ell^1 = \inf\left\{\int_{\Omega_\ell}A\nabla u.\nabla u \,: \, u \in V(\Omega_\ell),\, \int_{\Omega_\ell}u^2 = 1\right\}=\inf_{u \in V(\Omega_\ell)\setminus \{0\}}\frac{\int_{\Omega_\ell}A(X_2)\nabla u.\nabla u}{\int_{\Omega_\ell}u^2}.
\delta \hat{\omega}_{m a b}= - i \kappa ( 2 \bar{\varepsilon}\gamma_{[ a}\hat{\cal R}_{b ] m}- \bar{\varepsilon}\gamma_{m}\hat{\cal R}_{a b}) + 2 i \kappa^{2}( 2 i \bar{\varepsilon}\vec{\tau}\gamma_{a b}\psi_{m}\vec{t}+ \frac{1}{\sqrt{3}}\bar{\varepsilon}\psi_{m}\hat{F}_{a b}+ \frac{1}{2}\bar{\varepsilon}\gamma_{a b c d}\psi_{m}v^{c d}) .
\dot{\eta}^{\prime}= \left\{\eta + \delta \eta , H \right\}^{( \eta , \Lambda )}+ \left\{\eta , \frac{\partial \delta W}{\partial t}\right\}^{( \eta , \Lambda )}= \left\{\eta + \delta \eta , H + \frac{\partial \delta W}{\partial t}\right\}^{( \eta , \Lambda )}\; .
da_1 = -1 + (1+b_1 b_2)b_7 + b_1(1+b_4b_3) (1+b_6 b_5) da_2 = 1- b_3(1+ b_2b_1) da_3 = 1+ b_3b_4 da_4 = 1+ b_5 b_4 da_5 = 1+ b_5 b_6 da_6 = 1+ b_7 b_6
{\sigma_{\tau}}_M(v, \xi) := \sum_{|\alpha| + |\beta| < M}\sum_{|\alpha|+|\beta|\le|\delta | \le N\cdot(|\alpha| + |\beta|)}k_\delta(\tau, \alpha, \beta,x-y) \partial_x^\alpha \partial_y^\beta \partial_\xi^\delta a(v, v, \xi).
\left\| \partial_{t}^{\ell}\nabla_{x}^{k}E_{3}(t)g\right\|_{2}\le Ct^{1- \ell}(1+t)^{-n(\frac{1}{r_{1}}-\frac{1}{2})-(k-\tilde{k}_{3})}\| \nabla^{\tilde{k}_{3}}_{x}g \|_{r_{1}}& + C e^{-ct}t^{(1-\ell)-n(\frac{1}{r_{2}}-\frac{1}{2})-(k-\tilde{k}_{4})}\| \nabla^{\tilde{k}_{4}}_{x}g\|_{r_{2}},
f_k(\lambda_1,\dots,\lambda_n)=\frac{\mathrm{i}g}{2}\frac{A''(\lambda_k)}{A'(\lambda_k)}=\mathrm{i}g\sum_{\substack{\ell=1 (\ell\neq k)}}^n\frac{1}{\lambda_k-\lambda_\ell},k=1,\dots,n,
f_{E_6}=\frac{e^{-S_1-i\frac{x_0 I_3}{y(x_0^2 + y^2)}}}{(y^2+x_0^2)S_1}\propto\frac{K_{1/2}(S_1)e^{-i\frac{x_0 I_3}{y(x_0^2 + y^2)}}}{(y^2+x_0^2)\,\sqrt{S_1}}\, .
V_{Q \bar{Q}}(\ell) \sim \min_{w}\left[\pi w^2 \ell \left\{\frac{1}{2}\left( \frac{Q}{\pi w^2}\right)^2 + \frac{c_{e}}{4 d}\right\}\right] \sim \frac{Q^2}{\pi w_{\rm min}^2}\times \ell.
\langle 1 , g \rangle = \langle 1, \theta_{k_1}\cdots \theta_{k_t}\overline{\theta}_{l_1}\cdots \overline{\theta}_{l_u}g^\prime \rangle &:= w (k_1, \dots, k_t) \delta_{t,u}\delta_{k_1, l_u}\cdots \delta_{k_t, l_1}\, \langle 1, g^\prime \rangle &= w(k) \delta_{t,u}\delta_{k, l^T}\, \langle 1, g^\prime \rangle.
V(\vec{x}) = -\dfrac{1}{2}[\boldsymbol{\lambda}^{*}(\vec{x})]^{\top}\vec{G}\vec{H}^{-1}\vec{G}^{\top}\boldsymbol{\lambda}^{*}(\vec{x})- [\boldsymbol{\lambda}^{*}(\vec{x})]^{\top}(\vec{W}+\vec{S}\vec{x}).
\mathbf{v}_{1,k}^H\mathbf{y}_{1,k}= \mathbf{v}_{1,k}^H\mathbf{H}_{1,k}\mathbf{p}_1 \left(\alpha_{1,1}s_{1,1}+\cdots+ \alpha_{1,K}s_{1,K}\right) +\sum_{m=2}^{M}\mathbf{v}_{1,k}^H\mathbf{H}_{1,k}\mathbf{p}_{m}\tilde{{s}}_{m}+\mathbf{v}_{1,k}^H\mathbf{n}_{1,k}.
- \tau_{5}\, \int d^{6}\xi \, \frac{z}{\lambda^{2}H}\, \sin{\widetilde{\theta}}\left[ \left( 1 + \frac{\lambda^{2}H \sigma^{2}}{z^{2}\tan^{2}{\widetilde{\theta}}}\right)^{1 / 2}- 1 \right] ~ ~ .
\partial_a(\epsilon_{i_1}\wedge \cdots \wedge \epsilon_{i_a}\otimes \delta_v)=\sum_{j = 1}^a (-1)^{j+1}\epsilon_{i_1}\wedge \cdots \wedge\widehat{\epsilon}_{i_j}\wedge\cdots\wedge \epsilon_{i_a}\otimes (1 - M_j^t)\delta_v,
q^{s(s-1)/2}\sum_{b=r+s}^{m}q^{b}\bigl\{e_{r-1}(q^{s+1}, \ldots , q^{b-1})e_{s}(qz_{1}, \ldots , q^{b-1}z_{b-1}) +z_{b}\,e_{r}(q^{s+1}, \ldots , q^{b})e_{s-1}(qz_{1}, \ldots , q^{b-1}z_{b-1})\bigr\}.
(1-H)(P_n x+w_k) = P_n (1-H)(P_nx+w_k) + Q_n (1-H)(P_nx+w_k) = P_n(1-H)x+ Q_n (1-H)(P_nx+w_k)\phantom{\mathop{\longrightarrow}^{k\to\infty}}= y + Q_n (1-H)(P_nx+w_k) \mathop{\longrightarrow}^{k\to\infty}y.
\exp(c/\bar \gamma)E_1\left(c/\bar \gamma \right)=\frac{c}{\bar\gamma},\textrm{as}\bar\gamma\to 0, &\exp(c/\bar \gamma)E_1\left(c /\bar \gamma\right)=-K_{EM}- \ln(c)+ \ln(\bar\gamma) ,\textrm{as}\bar\gamma\to \infty ,
\varpi_{1*}\circ\cdots\circ \varpi_{r*}(1_{Z_{\lambda}}) = \phi_1 \left( t_1^{\lambda_1}\cdots t_r^{\lambda_r}\frac{\prod_{(i,j)\in \Delta_r}(1- \bar t_i/\bar t_j)}{\prod_{(i,j) \in C(\lambda)}(1- t_i/\bar t_j)}\right).
W_{\delta,q}(\xi) =U_{\delta}\left(\left(\psi_{q}^{\partial}\right)^{-1}(\xi)\right)\chi\left(\left(\psi_{q}^{\partial}\right)^{-1}(\xi)\right)= =\frac{1}{\delta^{\frac{n-2}{2}}}U\left(\frac{y}{\delta}\right)\chi(y)=\frac{1}{\delta^{\frac{n-2}{2}}}U\left(x\right)\chi(\delta x)
(xq)^{2a-2}\sum_{h=1}^{k-a+1}(xq^2)^{2h}U_{2k,2k-2h}(xq^2;q)+(xq)^{2a-2}\sum_{h=0}^{k-a}(xq^2)^{2h}U_{2k,2k-2h}(xq^2;q).
\chi^{a b}= \sum_{n = 1}^{\infty}( - 1 )^{n}\left( \Lambda^{n}\right)^{a b}- \frac{g \bar{\theta}^{b a}}{2 !}+ \sum_{n = 1}^{\infty}d_{n}\left[ ( g \bar{\theta})^{2 n}\right]^{b a}.
{\cal Z}={\frac{y^{2}( q ) \, \prod_{i < j}A_{i j}B_{i j}}{c^{2}( \mathrm{d e t}K )^{5}\, \prod_{i < j}( a_{i}- a_{j})}}={\frac{y^{2}( q ) \, Q_{\delta}}{c^{2}( \mathrm{d e t}K )^{5}\, P ( a )}},
\det (v^t r+r^t v-\mu \mathbf I_n) =(-1)^n\mu^{n-2}\Bigl(\mu^2-\Bigl(2\sum_{i=1}^n \|\mathbf v_i\|^2\Bigr)\mu -\sum_{1\leq i<j\leq n}\left(\|\mathbf v_i\|^2-\|\mathbf v_j\|^2\right)^2\Bigr)=0.
\mu_k^{\mathrm{opt}}=\frac{\sum_{l\in\mathcal{M}_{k}}\overline{\mathbf{h}}_{kl}^{\mathbf{H}}\mathbf{z}_{kl}}{\sum_{i \in \mathcal{K}_{l}}\sum_{l\in\mathcal{M}_{i}}\overline{\mathbf{h}}_{kl}^{\mathbf{H}}\mathbf{z}_{il}\mathbf{z}_{il}^{\mathbf{H}}\overline{\mathbf{h}}_{kl}+\delta_{}^2}.
DF_K(\xi,\eta) = \left(\begin{array}{cc}\frac{\partial x}{\partial\xi}\frac{\partial x}{\partial\eta}\frac{\partial y}{\partial\xi}\frac{\partial y}{\partial\eta}\end{array}\right) = \left(\begin{array}{cc}a_1+a_{12}\eta a_2+a_{12}\xi b_1+b_{12}\eta b_2+b_{12}\xi\end{array}\right),
\bar{\Delta}_{\rm Reactive}=\frac{\mathbb{E}\sum_{t\in \mathcal{I}_j}\Delta{\left(t\right)}}{\mathbb{E}T^{\rm Reac}_j}=\frac{\mathbb{E}T^{\rm Proac}_j}{2\mathbb{E}\left(T^{\rm Reac}_j\right)^2}+\mathbb{E}\tau^{\rm Reac}_{V_j}-\tau_{\rm f}-\frac{1}{2}.
W_{F_{4}}^{\underline{{{2 7}}}}={\frac{a_{1}( x )}{2 4}}-{\frac{1}{2}}\left\{\left( - q + \sqrt{q^{2}+ 4 p^{3}}\right)^{1 / 3}+ \left( - q - \sqrt{q^{2}+ 4 p^{3}}\right)^{1 / 3}\right\},
\Phi_{pq}(X) = \sum_{i=0}^{[p^{q-2}]_q - 1}\sum_{j=0}^{[q^{p-2}]_p-1}X^{ip+jq}- \sum_{i=[p^{q-2}]_q}^{q-1}\sum_{j=[q^{p-2}]_p}^{p-1}X^{ip+jq-pq}
\begin{matrix}{\rm dim}\bigl(\mathcal{T}^n_{d,e,m}\bigr) \,\, \leq \,\,{\rm min}\bigl\{\, \binom{d+n}{n}+\binom{e+n}{n}-2\,,\,\binom{m+n}{n}-1 \, \bigr\}. \end{matrix}
P_n(x) = \log 2 \cdot \omega_0(n) + \frac14 \sum_{2\le q\le X}\omega_{q}(n)^2 \Lambda(q) D(x) = \log 2 \cdot \mu(\omega_0) + \frac{1}{4}\sum_{2\le q \leq X}\mu(\omega_q)^2 \Lambda(q),
= C(C_{Sob}, p^*_-, p^-) \left( \int |\nabla u|^{p(x)}+ |\nabla v|^{p(x)}+ 2 \right)^{\frac{p^*_-}{p_-}- 1}\left( \int |\nabla u|^{p(x)}+ |\nabla v|^{p(x)}+ 2 \right) &= C \int (|\nabla u|^{p(x)}+ \mathbb{B}(x)|u|^{r(x)}+ 2).
\widehat\alpha(mtu) = \alpha_Z(t)\in \alpha(t) \cdot E K_V= \alpha(m)^{-1}\cdot \alpha(m) \alpha(t) \alpha(u) \cdot \alpha(u^{-1}) E K_V &\subseteq K_M \cdot \alpha(mtu) \cdot K_U E K_V= \alpha(mtu) \cdot K_M K_U E K_V.
J[{\hat\xi}] ={1\over8\pi G}\int_{r=r_+}\!\! d^{n-2}x\, \Bigl\{n^a\nabla_a{\hat\xi}^t\sqrt{\sigma}+{\hat\xi}^a\pi_a{}^r + n_a{\hat\xi}^a K\sqrt{\sigma}\Bigr\}
\gamma_i \sum_{g\in G}c_g g = \Bigg(\prod_{j=1}^{i-1}\gamma_j^{a_j}\Bigg)\sum_{g\in G}c_g g\intertext{in $\ell_2 G$,using the fact that the $\gamma_j$'s pairwise commute. Reindexing,}\sum_{g\in G}c_g g = \sum_{g\in G}c_g\Bigg(\prod_{j=1}^{i-1}\gamma_j^{a_j}\gamma_i^{-1}\Bigg) g.
C^{(1,1,1,1)}_{abcd}=-4C^{(1,1,2)}_{ab\gamma_1}C^{(1,1,2)}_{cd \gamma_1}+C^{(1,1,2)}_{ab\gamma_2}C^{(1,1,2)}_{cd \gamma_1}+C^{(1,1,2)}_{ab\gamma_1}C^{(1,1,2)}_{cd \gamma_2},
\lambda^{(i)}_{(\vec{x}_i,\mu_i)}(\vec{y}_i,\mu'_i)= (f_{i}^{q_{i}(\vec{x}_i,\mu_i)}(\vec{y}_i), \mu'_i+b_{i}(\vec{x}_i,f_{i}^{q_i(\vec{x}_i,\mu_i)}(\vec{y}_i))),
\|\psi\|_{\mathcal S_N}:=\sup_{(x,y,z)\in\mathbb R^3}\Big( 1+ |(x,y,z)|^N \Big)\sum_{\alpha,\beta,\gamma=0}^N \Big| \partial_x^\alpha\partial_y^\beta\partial_z^\gamma\psi(x,y,z) \Big|.
\int_0^tII(s)ds\lesssim \int_0^t\sum_{j=0}^k\|\phi^{(\nu)}(s,\nu\cdot)\|_{\dot{H}^{j+2\sigma}}ds\sim (c |\log \nu|^c)^{1+2\sigma-\frac{d}{2}}\nu^{2\sigma-\frac{d}{2}}.
\max_{j\in B}\left\{\Pr_{v\sim\mu_t}\big[|vH|=j \big]\cdot \frac{2^N}{\binom{N}{j}}\right\}\leq \max_{j\in B}\left\{2^{-(1-h(j/N))(1-\eta) N}\cdot \frac{2^N}{\sqrt{\frac{8\pi}{e^4 N}}\cdot2^{h(j/N)N}}\right\}= \max_{j\in B}\left\{\sqrt{\frac{e^4 N}{8\pi}}\cdot 2^{(1-h(j/N))\eta N}\right\}.
\langle 0 | \varphi^{2}( x ) | 0 \rangle = \langle 0^{( \alpha )}| \varphi^{2}( x ) | 0^{( \alpha )}\rangle - \frac{\sigma^{1 - D}}{\pi n S_{D}r^{n}}\sum_{l = 0}^{\infty}D_{l}\int_{m}^{\infty}d z \frac{z \Omega_{\alpha \nu_{l}}( a z , b z )}{\sqrt{z^{2}- m^{2}}}G_{\nu_{l}}^{( \alpha ) 2}( \alpha z , r z ) ,
D^{1}( \psi ) = \left( \begin{matrix}{{\psi_{1}^{2}}}&{{- \sqrt{2}\psi_{1}\psi_{2}^{*}}}&{{\psi_{2}^{*}}}\\{{\sqrt{2}\psi_{1}\psi_{2}}}&{{\mid \psi_{1}\mid^{2}- \mid \psi_{2}\mid^{2}}}&{{- \sqrt{2}\psi_{1}^{*}\psi_{2}^{*}}}\\{{\psi_{2}^{2}}}&{{\sqrt{2}\psi_{2}\psi_{1}^{*}}}&{{\psi_{1}^{* 2}}}\\ \end{matrix}\right) = e^{i \Sigma_{3}\alpha}~ e^{i \Sigma_{2}\beta}~ e^{i \Sigma_{3}\gamma}
P_{k}(x) = \sum_{n=1}^{\infty}\frac{\mu(n)}{n^k}\exp\left({-\frac{x}{n^2}}\right) = \sum_{n=1}^{\infty}\frac{\mu(n)}{n^k}\sum_{m=0}^\infty \frac{(-1)^m x^m}{m! n^{2m}}= \sum_{m=0}^\infty \frac{(-1)^m x^m}{m! \zeta(k+2m)}.
|F|_{c_\lambda}= \sum\limits_{e\in F}c_\lambda(e),\F\subset E\left(\mathbb{Z}^d\right), |K|_\pi = \sum\limits_{x\in K}\pi(x), \K\subset\mathbb{Z}^d, \psi (\mathbb{Z}^d,\, t) = \inf\left\{|\partial_E K|_{c_\lambda}:\t\leq\vert K\vert_\pi<\infty\right\},\t>0;
E_{2}( r , a ) = \frac{( \varepsilon - 1 )^{2}}{1 6 \pi^{2}a^{4}}\frac{2 3}{6 0}\frac{( d^{4}- 1 0 d^{2}- 1 5 )}{( 1 + d )^{4}( 1 - d )^{4}}, d = \frac{r}{a}, r < a .
\eta_{\alpha}(\theta, x) \Bar{\psi}^{\hat v^*_{2, \alpha}}_{\alpha, 1}= \ \inf_{v_1 \in V_1}\Big[{\mathcal L}\Bar{\psi}^{\hat v^*_{2, \alpha}}_{\alpha, 1}(x, v_1, \hat v^*_{2, \alpha}(\theta,x))+ \theta r_1(x, v_1, \hat v^*_{2, \alpha}(x)) \Bar{\psi}^{\hat v^*_{2, \alpha}}_{\alpha, 1}\Big],
\varphi_{i}(y;s_{i},s_{i+1})=\sqrt{1-s_{i+1}}\frac{(1-s_{i})+\sqrt{1-s_{i+1}}\tanh (b_{i}c_{iN}y\sqrt{1-s_{i+1}})}{(1-s_{i})\tanh (b_{i}c_{iN}y\sqrt{1-s_{i+1}})+\sqrt{1-s_{i+1}}}
\begin{array}{ll}F^{-1}_{a,b}(z)=\rho(n,1)+\dfrac{b-1}{a+1}\rho(n,2) +\dfrac{(b-1)(a^2+3ab-a+5b-4)}{2(a+1)^2(a+2)}\rho(n,3)+O(z^{(4/a)})\end{array}
\sum_{j \geq 1}\frac{(4z)^j}{(1+R+z)^j}\left(1-2\frac{yw}{w-\kappa}\right)^{j-1}\left(1+\frac{1}{y}\right)^{j}e^{-2jty}(1-w^2)^j = \left(1+\frac{1}{y}\right)e^{-2ty}\frac{(4z(1-w^2))}{(1+R+z)}& \left(1-\frac{4z(1-w^2)}{1+R+z}\left(1-2\frac{yw}{w-\kappa}\right)\left(1+\frac{1}{y}\right)e^{-2ty}\right)^{-1}.
\hat{\mathbf{x}}_* = \mathbf{P}^{-\frac{1}{2}}\hat{\mathbf{A}}^\dag \hat{\mathbf{b}}+ \mathbf{P}^{-\frac{1}{2}}(\mathbf{I}- \hat{\mathbf{A}}^\dag \hat{\mathbf{A}}) \hat{\mathbf{x}}_0
\nabla\cdot\mathbf{A}=\frac{1}{h_{1}h_{2}h_{3}}\left[\frac{\partial}{\partial u_{1}}\left(h_{2}h_{3}A_{1}\right)+\frac{\partial}{\partial u_{2}}\left(h_{1}h_{3}A_{2}\right)+\frac{\partial}{\partial u_{3}}\left(h_{1}h_{2}A_{3}\right)\right]
x(t) \sim\exp\left( (\theta+1)^{\frac{1}{1+\theta}}\left[ (Mt)^{1/(1+\theta)}- \tilde{c}(t) (Mt)^{(1-\theta)/(1+\theta)}\right] \right) &= y(t)\exp\left( -(\theta+1)^{\frac{1}{1+\theta}}\tilde{c}(t) (Mt)^{(1-\theta)/(1+\theta)}\right), \mbox{as}t\to\infty,
-~ \frac{(\mu-1)(2\mu-3)}{2(\mu-2)}\left[ \hat{\Phi}(\mu) ~+~ \hat{\Psi}^2(\mu) ~-~ \frac{8(2\mu-3)}{(\mu-1)(\mu-2)\eta_1}~-~ \frac{2}{(\mu-2)^2}\right] \eta_1^2
\mu_{-}= \left( \begin{array}{c c}{I_{m_{1}}}&{0}\\{\mu_{- 2 1}}&{I_{m_{2}}}\\ \end{array}\right) , \qquad \mu_{+}= \left( \begin{array}{c c}{I_{m_{1}}}&{\mu_{+ 1 2}}\\{0}&{I_{m_{2}}}\\ \end{array}\right) ,
\{f,g\}(\psi)=\frac1{2\hbar}\operatorname{Im}\left\langle\frac{\delta f}{\delta \psi}\bigg|\frac{\delta g}{\delta \psi}\right\rangle = \left\langle \frac{\delta f}{\delta \psi},-\frac{i}{2\hbar}\frac{\delta g}{\delta \psi}\right\rangle .
|H_n(z_a)-H_n(z_a')| =\Bigl|\sum_{j=0}^{n-1}(H(G^jz_a)-H(G^jz_a'))\Bigr|\le C_2|h|_{C^\eta}\sum_{j=0}^{n-1}\gamma^{n-j}\varphi(G^ja)\gamma^{s(z,z')}.
P^-_{rh}={g^2 L \over 4\pi^2}\left[\sum_{N \neq 0}{C^3_N C^3_{-N}\over N^2}+\sum_N \left [{C^+_N C^-_{-N}\over (Z_R+N)^2}+{C^-_{N}C^+_{-N}\over (Z_R-N)^2}\right ]+\Pi^2_R \right]
\forall \psi\in{\cal P}^n_{u,v}([a,b]),\,\, \hbox{if}\,\, \psi^{(n-2)}(s_1)=\psi^{(n-2)}(s_2)=0, s_1,s_2\in[a,b],\,\, s_1\ne s_2 \hbox{then}\,\, \psi^{(n-2)}(s)=0, s\in[a,b];
\begin{pmatrix}\hat{u}\hat{v}\hat{w}\end{pmatrix}= \begin{pmatrix}0.1+0.2x+0.2x+0.1z+0.15x^2+0.2y^2+0.1z^2+0.15xy+0.1yz+0.1zx 0.15+0.1x+0.1y+0.2z+0.2x^2+0.15y^2+0.1z^2+0.2xy+0.1yz+0.2zx 0.15+0.15x+0.2y+0.1z+0.15x^2+0.1y^2+0.2z^2+0.1xy+0.2yz+0.15zx \end{pmatrix}
\begin{aligned}0 \leq \mathcal{I}(\eta_0, \delta) - \mathcal{I}(\eta, 0) = \mathcal{I}(\eta_0, \delta) - \mathcal{I}(\eta, \delta) + \mathcal{I}(\eta, \delta) - \mathcal{I}(\eta, 0) & \leq \Psi_1\left( \eta_0- \eta , M ( 1- \eta/ \eta_0) \right) + \Psi_2 \left(M\delta, \delta \right).\end{aligned}
\sin(\alpha_{1})\cos(\alpha_{2})=\frac{-1}{4}(\sin(-\alpha_{1}-\alpha_{2})\allowbreak+\allowbreak\sin(-\alpha_{1}+\alpha_{2})-\sin(\alpha_{1}-\alpha_{2})-\sin\left( \alpha_{1}+\alpha_{2})\right) =\frac{1}{2}(\sin(\alpha_{1}+\alpha_{2})+\sin(\alpha_{1}-\alpha_{2})).
[\begin{matrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{matrix}]=\frac{E}{(1+\nu)(1-2\nu)}[\begin{matrix}1-\nu&\nu&0\\ \nu&1-\nu&0\\ 0&0&\frac{1-2\nu}{2}\end{matrix}][\begin{matrix}\epsilon_{11}\\ \epsilon_{22}\\ 2\epsilon_{12}\end{matrix}]
g_{ij}=\frac{1}{2}(h_{ij}-ik_{ij})\,,\,g_{ij}=g\left(\frac{\partial}{\partial z^i},\frac{\partial}{\partial z^j}\right)\,,\,h_{ij}=h\left(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right)\,,\,k_{ij}=h_{in+j}=h\left(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^{n+j}}\right),
d(\alpha,\beta) \leq \left(\frac{1}{n^2}\sum_{i=1}^n\left(g_\alpha(X^i)+g_\beta(X^i)\right)^2\right)^{1/2}\max_{i=1,\dots,n}\left|g_\alpha(X^i)-g_\beta(X^i)\right| &\leq \frac{2}{\sqrt{n}}\sup_{\alpha\in T}\|g_\alpha\|_n\max_{i=1,\dots,n}\left|g_\alpha(X^i)-g_\beta(X^i)\right|.
V^\mu=mc\int_{\mathbb{R}^3}\int_0^\infty p^\mu F \phi(\mathcal{I}) \,d\mathcal{I}\,\frac{dp}{p^0},T^{\mu\nu}=\frac{1}{mc}\int_{\mathbb{R}^3}\int_0^\infty p^\mu p^\nu F\left( mc^2 + \mathcal{I}\right) \phi(\mathcal{I})\,d\mathcal{I}\,\frac{dp}{p^0}.
\int_{a}^{\infty}F_{0}\left( x\right) \mathrm{d}x =\int\left[ \int_{a}^{\infty}\mathrm{e}^{-2sx}\mathrm{d}x\right] \mathrm{d}\sigma\left(s\right) & =\int\mathrm{e}^{-2as}\frac{\mathrm{d}\sigma\left( s\right)}{2s}\leq\max_{s\in\operatorname*{Supp}\sigma}\mathrm{e}^{-2as}\cdot\int\frac{\mathrm{d}\sigma\left( s\right)}{2s}<\infty.