harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\kappa_{1_\chi}\left(T_1 \cdots T_{k(1)}, T_{k(1)+1}\cdots T_{k(2)}, \ldots, T_{k(m-1)+1}\cdots T_{k(m)}\right) = \sum_{\substack{\sigma \in BNC(\widehat{\chi}) \sigma \vee \widehat{0_\chi}= 1_{\widehat{\chi}}}}\kappa_\sigma(T_1, \ldots, T_n).
\widetilde{\eta}^i \equiv U^i{}_j \eta^j\,,\qquad\widetilde{\eta}_i \equiv \eta_j (U^{-1})^j{}_i \ ,\qquad\widetilde{\theta}^i \equiv \tilde{U}^i{}_j \theta^j\,,\qquad\widetilde{\theta}_i \equiv \theta_j (\tilde{U}^{-1})^j{}_i \ ,
\overline{\alpha}(z) =\left(\prod_{i=1}^{s/2-1}\frac{1}{a_i(n-a_i)}\right)\frac{1}{\frac{n}{2}}\left(\prod_{i=1}^{s/2-1}\left(-\Big(a_i-\frac{n}{2}\Big)^2+\frac{x^2}{4}\right)\right)\frac{x}{2},
\|N_tu_0\|_{H^{1+\varepsilon}(\mathbb{T})}= \left\|\left(L_{t}[S_tu_0]S_t - W_t^\gamma\right) u_0\right\|_{H_x^{1+\varepsilon}(\mathbb{T})}\leq \tilde{C}(\|f\|_{H^1_x(\mathbb{T})}, \gamma, \varepsilon),
\begin{array}{l}{\Psi ( z , \bar{z}) = S_{- 1 / 2}\phi ( z , \bar{z}) ;}\\{\bar{\Psi}( z , \bar{z}) = \bar{S}_{- 1 / 2}\phi ( z , \bar{z}) ;}\\{F ( z , \bar{z}) = S_{- 1 / 2}\bar{S}_{- 1 / 2}\phi ( z , \bar{z})}\\ \end{array}
\big\|\chi_{S_{-\sigma,0}(\nu)}e^{\beta\langle(x,t),\nu\rangle}(i\partial_t+\Delta)u\big\|_{L_{t,x}^2(|V|^{-1})}\leq\big\|\chi_{S_{-\sigma,0}(\nu)}e^{\beta\langle(x,t),\nu\rangle}Vu\big\|_{L_{t,x}^2(|V|^{-1})}&=\big\|\chi_{S_{-\sigma,0}(\nu)}e^{\beta\langle(x,t),\nu\rangle}u\big\|_{L_{t,x}^2(|V|)}.
E[Y^o M] =E[\int_0^t(\nabla_W^{\mathbb{P}}Y(s) - \psi(s))\xi(s) \sigma^2 ds &+\int_0^T\int_{\mathbb{R}_0}(\nabla_J^{\mathbb{P}}Y(s,z) - z \psi(s)).z \xi(s)\mu(ds\dz)] & = E\left[\int_0^t \xi(s) \left[ (\nabla_W^{\mathbb{P}}Y(s)-\psi(s)) \sigma^2 +\int_{\mathbb{R}_0}z (\nabla_J^{\mathbb{P}}Y - z \psi(s)) \right] ds\right],
\lbrack{\hat{x}}_{C}^{I I},{\hat{x}}_{P}^{I I}] = - 3 2 u \partial_{t u u}^{3}- 1 6 ( 1 + 2 p_{2}- 2 u ) \partial_{t u}^{2}+ 3 2 t u \partial_{u u}^{2}+
P_{\mathbf{m}}\left(\mathbf{1};\frac{d}{2}\right) = \prod_{(i,j) \in \mathbf{m}}\frac{j-1+\frac{d}{2}(r-i+1)}{m_{i}-j+\frac{d}{2}(m_{j}^{\prime}-i+1)}= \prod_{1\leq i<j\leq r}\frac{\left(\frac{d}{2}(j-i+1)\right)_{m_{i}-m_{j}}}{\left(\frac{d}{2}(j-i)\right)_{m_{i}-m_{j}}}.
\text{L.H.Sof}~\eqref{3.9}&\lesssim\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\int_{0}^\infty\int_{0}^\infty\big|\rho^{2\alpha+\frac12}(r\rho)^{-\frac{n-2}2}J_{\nu(k)}(r\rho)b^0_{k,\ell}(\rho)\rho^{n-2}\big|^2\frac{\mathrm{d}\rho ~r^{n-1}\mathrm{d}r}{(1+r)^\beta}.
\frac{\int_0^1 P_{4n,+}^p}{\int_0^1 P_{4n,-}^p}\geq \frac{1}{p+1}\frac{1}{6\pi^2}\frac{(4n+ \frac{1}{2})^2}{n(4n+1)}\Big(\sum_{i=1}^{n}iy_{2i-1,4n}^p\Big)^{-1}= \frac{1}{p+1}\Big(\displaystyle\sum_{i=1}^{n}iy_{2i-1,4n}^p\Big)^{-1}\Big( \frac{2}{3\pi^2}+\mathcal{O}(n^{-1})\Big).
T(h_{1},h_{2})= \int L_{g_{0}}h_{1}h_{2}& -c_{n}n(n+2)\int \left[\frac{K\omega^{\frac{4}{n-2}}}{4n(n-1)}+\sum_{i}\varphi_{i}^{\frac{4}{n-2}}ab\right]h_{1}h_{2}.
\frac{d}{d t}\left\langle D \right\rangle_{\scriptstyle \! \Psi_{\mathrm{\!{\scriptscriptstyle ( g s )}}}}\stackrel{( a \rightarrow 0 )}{=}E_{\mathrm{( g s )}}\left\{1 + O \left( \left[ \kappa a \right]^{2}\ln [ \kappa a ] \right) \right\}.
- i \Gamma_{5 m e a s}^{\mu \alpha \beta}= - \frac{1}{2 \pi^{2}}\epsilon^{\mu \nu \alpha \beta}[ q_{2 \nu}( M_{a}( p ; q_{1}, q_{2}) + M_{a}( q_{2}; p , q_{1}) ) - q_{1 \nu}( M_{a}( p ; q_{1}, q_{2}) + M_{a}( q_{1}; p , q_{2}) ) ] .
\Psi_{E}= \left( \begin{array}{c}{\varphi_{E}\left( r , \theta \right)}\\{\chi_{E}\left( r , \theta \right)}\\ \end{array}\right) \qquad , \quad \mathrm{s a t i s f y :}\quad \begin{array}{c}{L \varphi_{E}= E \chi_{E}}\\{L^{\dag}\chi_{E}= E \varphi_{E}}\\ \end{array}
- \partial^{\alpha}\big((u^s + \tilde{u}_\epsilon)\partial_x \tilde{w}_\epsilon \big) = - (u^s + \tilde{u}_\epsilon)\partial_x \partial^{\alpha}\tilde{w}_\epsilon + [ (u^s + \tilde{u}_\epsilon), \partial^{\alpha}]\partial_x \tilde{w}_\epsilon.
\Phi_{\mathrm{IPLA}}(t;i,R_{\mathrm{IPLA},u}^*) = \frac{e^{-jt(2^{R_{\mathrm{IPLA},u}^*}-1)}}{t}\cdot \left[ \frac{2\left( - \frac{jst}{\bar{L}}\right)^{\frac{K}{2}}}{(K-1)!}\cdot K_{K}\left( \sqrt{- \frac{4jst}{\bar{L}}}\right) \right]^i.
\lim_{x\rightarrow \infty}\frac{1}{\pi(x)}\sum_{p\le x}N_{p}^k(E[\ell])=\frac{\ell^4-2\ell^3-\ell^2+3\ell}{(\ell^2-\ell)(\ell^2-1)}+\ell^k \frac{\ell^3-2\ell-1}{(\ell^2-\ell)(\ell^2-1)}+\ell^{2k}\frac{1}{(\ell^2-\ell)(\ell^2-1)}.
( a \nabla_{\nu}b)^{2}\leqslant K(h^{\frac{1}{2^{n-1}}},2)^{-r_{n}}( a \sharp_{\nu}b )^{2}+ R_{0}^{2}(a-b)^{2}- \sum_{k=1}^{n-1}r_{k}\big[ a^{1-\frac{m_k}{2^k}}b^{\frac{m_k}{2^k}}- a^{1-\frac{m_k+1}{2^k}}b^{\frac{m_k+1}{2^k}}\big]^{2},
\begin{aligned}\Delta r_{O1}\leq \ln \Big(1+ \frac{(P-P_O^*)|\bar{h_1}|^2}{N_0}\Big) \leq \ln \Big(1+ \frac{(P-P_N^*)|\bar{h_1}|^2}{N_0}\Big) &= \ln \Big(1+ \frac{(P-P_N^*)|{h_1}|^2}{N_0e^{Kr^*}}\Big) & = \Delta r_N.\end{aligned}
\varpi_{n,1}\left({\nu ,p}\right) =\sum\limits_{s=0}^{n-2}{\frac{4}{\nu^{s}}\int_{1}^{p}{\left\vert{\frac{\tilde{{F}}_{s+1}\left( q\right) dq}{q^{2}\left({1-q^{2}}\right)}}\right\vert .}}
T_z(0) := B^{z-1}[BA^{\frac{1}{2}},A^{z-\frac{1}{2}}]+[BA^{\frac{1}{2}},A^{\frac{1}{2}}]Y^{z-1}, T_z(s) := B^{z-1+is}[BA^{\frac{1}{2}},A^{z-\frac{1}{2}+is}]Y^{-is}+B^{is}[BA^{\frac{1}{2}},A^{\frac{1}{2}+is}]Y^{z-1-is},s \neq 0.
(u[0],p[0])\buildrel \mu^{B[0]}_{k_0,\varepsilon_0}\over \mapsto(u[1],p[1])\buildrel \mu^{B[1]}_{k_1,\varepsilon_1}\over \mapsto\cdots\buildrel \mu^{B[T-1]}_{k_{T-1},\varepsilon_{T-1}}\over \mapsto(u[T],p[T]),
H_{R}= m_{1}+ m_{2}+ 2 \Delta + \frac{p^{2}}{2 m_{1}}+ \frac{p^{2}}{2 m_{2}}- \frac{\Delta}{2}\left( \frac{p^{2}}{m_{1}^{2}}+ \frac{p^{2}}{m_{2}^{2}}\right) + V_{0}+ V_{SD}+ V_{v^{2}}.
- \ \frac{1}{4 \, s^{2}}\: \left( \partial^{\mu}s \, \partial_{\mu}s \, + \ \partial^{\mu}a \, \partial_{\mu}a \right) \ + \ \frac{\delta}{4 \, s^{2}}\: \partial_{\mu}a \, A_{X}^{\mu}\ - \ \frac{\delta^{2}}{1 6 \, s^{2}}\: A_{X}^{\mu}A_{X \mu}\ + \ \frac{\delta}{4 \, s}\: D_{X}
\delta(C\wedge G\wedge G) =\delta C\wedge G\wedge G+C\wedge \delta dC\wedge G+C\wedge G\wedge\delta dC &=\delta C\wedge G\wedge G+2C\wedge d\delta C\wedge G &=\delta C\wedge G\wedge G-2d(C\wedge \delta C\wedge G)+2 dC\wedge \delta C\wedge G &=d(\ldots)+3\delta C\wedge G\wedge G.
E := E_1 + E_2 = u_{q+1}^\sharp e - (w^\sharp e + \delta_{q+1}\rho^2 (\nabla \Phi_1 \otimes \nabla \Phi_1 + \nabla \Phi_2 \otimes \nabla \Phi_2)) = \, u_{q+1}^\sharp e - w^\sharp e - \delta_{q+1}h_q = u_{q+1}^\sharp e + \delta_{q+2}e - g*\varphi_\ell = \, u_{q+1}^\sharp e - g_{q+1}+(g-g*\varphi_\ell)\, .
\left\|x^v\!\!\int_x^\infty [K_x(x,s)-K_s(x,s)]F(s+x)ds\right\| \le cx^v\!\!\int_x^\infty \!\!\sigma (x)\sigma\left(\!\frac{x+s}{2}\!\right)\|F(s+x)\|ds &\le c\int_x^\infty\sigma (y) \|F(2y)\|dy &\le c\sigma(x)\in L^1(\mathbb{R}^+),\;\;v=0,1.
H^{2}= - \frac{k}{a^{2}}+ \frac{\omega_{4}\, M}{a^{4}}- \frac{3 \, \omega_{4}^{2}\, Q^{2}}{1 6 \, a^{6}}- \frac{1}{L^{2}}+ \frac{4 \, \pi}{3 \, M_{p}^{2}\, \rho_{0}}\left( \rho_{0}+ \rho_{\mathrm{b r}}\right)^{2}\, \, ,
||\tilde{\mathcal{A}}\mathcal{F}||^2_{s-1,\ast,t}\leq C(K)\Big(||\mathbf{F}||^2_{s-1,\ast,t}+||\mathbf{F}||^2_{W^{1,\infty}_{\ast}(\Omega_t)}||\hat{W}||^2_{s-1,\ast,t}\Big).
\gamma_{\xi'+\frac{2k\pi}{\ell}}(x,y)=e^{\frac{2ik\pi}{\ell}\cdot(x-y)}\,\gamma_{\xi'}(x,y)\xi'\in Q_\ell^*, \, k\in \mathbb{Z}^3 x,\,y\in \mathbb{R}^3.
T_{d}(x) = \tanh \left( \frac{\pi}{4d}x \right), B_{N}(x; \{s_{\ell}\}, \mathcal{D}_{d}) = \prod_{k=-N}^{N}\tanh \left( \frac{\pi}{4d}(x - s_{k}) \right), B_{N;k}(x; \{s_{\ell}\}, \mathcal{D}_{d}) = \prod_{\begin{subarray}{c}-N\leq m \leq N, m \neq k \end{subarray}}\tanh \left( \frac{\pi}{4d}(x - s_{m}) \right),
ln\frac{P_{t}}{P_{t-1}}=\frac{1}{2}\sum_{i=1}^{n}(\frac{p_{i,t-1}q_{i,t-1}}{p_{t-1}q_{t-1}}+\frac{p_{i,t}q_{i,t}}{p_{t}q_{t}})ln(\frac{p_{i,t}}{p_{i,t-1}})
\Upsilon = g(R(X,Z)(Y), W) - g(R(Y,Z)(X), W) - g(R(Z,W)(X), Y) + g(\nabla_Z \nabla_X Y - \nabla_Z \nabla_Y X, W) = \underbrace{g(R(X,Z)(Y), W) + g(R(Z,Y)(X), W) + g(R(Y,X)(Z), W)}_{=0 \;}+ g(\nabla_Z [X,Y], W),
\begin{array}{l}{{\dot{L}_{j k}= \frac{1}{2}\coth ( ( q_{j}- q_{k}) / 2 ) ( \dot{q}_{k}- \dot{q}_{j}) L_{j k}}}\\{{\quad \quad \quad + \sum_{l \neq j , k}\frac{1}{2}( \coth ( ( q_{j}- q_{l}) / 2 ) - \coth ( ( q_{l}- q_{k}) / 2 ) ) L_{j l}L_{l k}}}\\ \end{array}
f_{1d,}= p_{2,\mu}q_{0,d-\mu}- p_{0,\mu}q_{2,d-\mu}= \alpha_{d-\mu_1}'A_{1,\mu_1}+ \beta_{d-\mu_2}'B_{1,\mu_2}, f_{2,d}= p_{0,\mu}q_{1,d-\mu}- p_{1,\mu}q_{0,d-\mu}= \alpha_{d-\mu_1}'A_{2,\mu_1}+ \beta_{d-\mu_2}'B_{2,\mu_2}.
\int_{\gamma_{n}}^{\gamma_{m}}Du(x)v(x)\dx=-\int_{\gamma_{n}}^{\gamma_{m}}u(x)Dv(x)\dx+\sum_{i=n}^{m-1}\left[ u^{-}(\gamma_{i+1})v^{-}(\gamma_{i+1})-u^{+}(\gamma_{i})v^{+}(\gamma_{i})\right] ?
\sum_{k=0}^{\frac{p^r-1}{2}}\frac{(6k+1)(\frac{1}{2})_k^3}{k!^3 4^k}\equiv (-1)^\frac{p-1}{2}p \sum_{k=0}^{\frac{p^{r-1}-1}{2}}\frac{(6k+1)(\frac{1}{2})_k^3}{k!^3 4^k}\pmod{p^{4r}}\quad.
\psi_R^a(0,x^-) = \frac{1}{\sqrt{2L}}~ \sum_{n= \frac{1}{2}, \frac{3}{2}, \dots}^{\infty}\left( b_n^{\, a}~ e^{- i k_n^+ x^- /2}+ d_n^{\, a \dag}~ e^{\, i k_n^+ x^- /2}\right) ~,
r(s)=r(0)+(s-\frac{s^{3}\kappa^{2}(0)}{6})T(0)+(\frac{s^{2}\kappa(0)}{2}+\frac{s^{3}\kappa^{\prime}(0)}{6})N(0)+(\frac{s^{3}\kappa(0)\tau(0)}{6})B(0)+o(s^{4}).
C^{'}_{FTN}(\alpha,W) = (1+\alpha)W \log_{2}\left( 1 + \frac{P}{WN_{0}}\right) \quad\;- 2\alpha W \log_{2}\left( \frac{1 + \frac{P}{2WN_{0}}+ \sqrt{1 + \frac{P}{WN_{0}}}}{2}\right).
\check{\mu}_{\delta}*\mu_{\delta}(\tilde H^2\cap U)\geqslant\check{\mu}_{\delta}((v\tilde H\cap A_i')^{-1})\mu_{\delta}(v\tilde H\cap A_i')=\mu_{\delta}(v\tilde H\cap A_i')^2\geqslant\delta^{O(\varepsilon)}.
P(0) = \mathrm{Tr}\left( \begin{bmatrix}1 \omega^{n-1}0 0 \end{bmatrix}\begin{bmatrix}1 \omega^{n-2}0 0 \end{bmatrix}\cdots \begin{bmatrix}1 1 0 0 \end{bmatrix}\right) = \mathrm{Tr}\left( \begin{bmatrix}1 1 0 0 \end{bmatrix}\right)= 1.
\frac{p_1(x+y)}{p_1(x)}= p_1\left(\frac{x}{p_1(x)}+\frac{y}{p_1(x)}\right) & = p_1\left(u+ \tau v\right) =p_1\left((1 - \tau)u+ \tau (u + v)\right) \leq (1 - \tau) + \tau p_1(u + v) & \leq (1 - \tau) + 2 \tau (1 - \delta_1(p_1(u - v))) = 1 + \tau -2 \tau \delta_1(p_1(u - v)),
\mathcal{N}_{n}^{\prime}\left( e_{n}+\Delta y_{n-1}\right) \leq\gamma\left(B^{\left( 2\right)}+2\right)^{-1}N_{n+1}^{-\mu}R_{n}=\gamma c\left(B^{\left( 2\right)}+2\right)^{-1}N_{n+1}^{-\mu}N_{n}^{\kappa\beta-\sigma}\Vert\bar{y}\Vert_{\delta}
\phi_2(t,q) := (1+t^{a(1)}q+t^{a(1)}q^2+\ldots)(1+t^{a(2)}q^2+t^{a(2)}q^4+\ldots)\times\cdots &\times(1+t^{a(n)}q^n+t^{a(n)}q^{2n}+\ldots)\times\cdots.
\frac{d||\mathbf{\ddot{p}}_d||}{dt}= 0 \frac{d}{dt}\sqrt{v^2 \dot{\phi}^2 + \dot{v}^2}= 0 v^2 \dot{\phi}\ddot{\phi}+ \dot{\phi}^2 v \dot{v}+ \dot{v}\ddot{v}= 0
\pi_{\lambda}\left( \exp\left( t_{1}Z_{1}+t_{2}Z_{2}+t_{3}Z_{3}\right)\right) f\left( x_{1},x_{2},x_{3}\right) =e^{\left( 2\pi i\left\langle\left[\begin{array}[c]{c}\lambda_{1}\lambda_{2}\lambda_{3}\end{array}\right] ,\left[\begin{array}[c]{c}t_{1}t_{2}t_{3}\end{array}\right] \right\rangle \right)}f\left( x_{1},x_{2},x_{3}\right).
\Psi(r,\theta,t)=\sqrt{{i \over r}}\,\,{1\over\sqrt{2\pi k_0}}\,\,{\sin{\pi\over\alpha}\over\cos{\pi\over\alpha}+\cos\theta}\,\,e^{-{b^2\over4\xi^2}}\,\,\Psi_{{\rm{free}}}(r,\alpha^2 t),
\frac{dN(\omega)}{d\sigma}=\frac{1}{2\pi}\frac{e^2}{\hbar c^3}(2a_0\omega_0)^3\langle x^2\rangle\int_{-\infty}^{+\infty}ds\, e^{-is\omega/a_0\omega_0}{\rm csch}^4 (s-i\epsilon)~,
I_{ik}=-\sum_{1\le m,n\le d}\Big\{\bigl((\sigma_{ik})_{mn}\bigr)_{x_m}\rho_{x_n}w_{ik}+(\sigma_{ik})_{mn}\rho_{x_mx_n}w_{ik}+(\sigma_{ik})_{mn}\rho_{x_n}(w_{ik})_{x_m}+(\sigma_{ik})_{mn}\rho_{x_m}(w_{ik})_{x_n}\Big\},
c(\mu \otimes A)=(A\otimes \mu )(c\otimes A)(A\otimes c), c(A\otimes \mu )=(\mu \otimes A)(A\otimes c) (c\otimes A), c(u\otimes A)l_{A}^{-1}= ( A\otimes u ) r_{A}^{-1},c(A\otimes u)r_{A}^{-1}= ( u\otimes A ) l_{A}^{-1}.
m \equiv \left( \partial_{\underline{{{y}}}}\Lambda \right) \Lambda^{- 1}={\textstyle \frac{1}{2}}m^{i}T_{i}={\textstyle \frac{1}{2}}\left( \begin{array}{c c}{{m^{1}}}&{{m^{2}+ m^{3}}}\\{{}}&{{}}\\{{m^{2}- m^{3}}}&{{- m_{1}}}\\ \end{array}\right) \, .
d s^{2}= -{\frac{1}{\sqrt \Delta}}\left( 1 -{\frac{r_{0}}{r}}\right) d t^{2}+ \sqrt \Delta \left({\frac{d r^{2}}{( 1 - r_{0}/ r )}}+ r^{2}d \theta^{2}+ r^{2}\sin^{2}\theta d \phi^{2}\right)
e T_{\mu \nu}= e \Theta_{\mu \nu}+ \frac{1}{2}g_{\mu \nu}\left({\tilde{\overline{\Psi}}}_{L}\frac{\delta{S}^{-}}{\delta{\tilde{\overline{\Psi}}}_{L}}+{\tilde{\Psi}}_{L}\frac{\delta{S}^{-}}{\delta{\tilde{\Psi}}_{L_{0}}}\right) .
{\cal G}= \left({\cal G}_X\mid{\cal G}_Y\right)= \left ( \begin{array}{cc|ccc}I_{\kappa}T_b 2T_2{\mathbf{0}}{\mathbf{0}}{\mathbf{0}}{\mathbf{0}}2T_1 2I_{\gamma-\kappa}{\mathbf{0}}\hline{\mathbf{0}}S_b S_q R I_{\delta}\end{array}\right ),
\left( \begin{array}{c}{a_{D}}\\{a}\\ \end{array}\right) \rightarrow \left( \begin{array}{c c}{0}&{- 1}\\{1}&{0}\\ \end{array}\right) \left( \begin{array}{c}{a_{D}}\\{a}\\ \end{array}\right) = \left( \begin{array}{c}{- a}\\{a_{D}}\\ \end{array}\right) ,
\begin{array}{l c l}{{{\mathcal L}_{2}^{( 0 , 2 )}}}&{{=}}&{{{u_{0}\star p}^{2}+ u_{1}\star p + u_{2}}}\\{{}}&{{=}}&{{u_{0}{p}^{2}+ ( u_{1}- 2 \theta u_{0}^{\prime}) p + ( u_{2}- \theta u_{1}^{\prime}+ \theta^{2}u_{0}^{\prime \prime})}}\\ \end{array}
(p_{1}+\cdot\cdot\cdot+p_{n})^{c}=\sum_{k_{1},...,k_{n}\in\mathbb{N}:k_{1}+\cdot\cdot\cdot+k_{n}=c}(\begin{matrix}c\\k_{1},...,k_{n}\end{matrix})p_{1}^{k_{1}}\cdot\cdot\cdot p_{n}^{k_{n}}
B_i=\bigcup_{u_{i+1}= 0}^{s_{i+1}-1}\bigcup_{u_{i+2}= 0}^{s_{i+2}-1}\dots\bigcup_{u_{h+1}= 0}^{s_{h+1}-1}\bigcup_{q_{i+1}= 0}^{p_{i+1}-1}\bigcup_{q_{i+2}= 0}^{p_{i+2}-1}\dots\bigcup_{q_h = 0}^{p_h-1}\Big( G_i \beta^{\sum_{j=i+1}^{h+1}u_j t_j}\prod_{i< j \le h}\alpha_j^{q_j}\Big).
f^{( 2 )}= \left( \begin{array}{c c c c}{0}&{- \delta_{i j}}&{2 a_{i}}&{\pi_{i}}\\{\delta_{i j}}&{0}&{0}&{a_{i}}\\{- 2 a_{i}}&{0}&{0}&{0}\\{- \pi_{i}}&{- a_{i}}&{0}&{0}\\ \end{array}\right) ,
H=-J\sum_{r}M_{r}\cdot(M_{r+e_{x}}+M_{r+e_{y}})-D\sum_{r}(M_{r}\times M_{r+e_{x}}\cdot e_{x}+M_{r}\times M_{r+e_{y}}\cdot e_{y})-B\cdot\sum_{r}M_{r}-A\sum_{rI}M_{zr}^{2}
i\hbar\frac{\partial}{\partial t}\Delta\langle\hat{N}\rangle=T[\Delta\langle\hat{N}\rangle]+NL[\langle\hat{1}\rangle,\Delta\langle\hat{2}\rangle,\cdot\cdot\cdot,\Delta\langle\hat{N}\rangle]+Hi[\Delta\langle\hat{N}+1\rangle],
\sum_a - \Omega_{kj}(e_a)\wedge (\omega_{ki}\circ J_{ij})(e_a) =\omega_{ki}(J_{ik}\cdot,\cdot) + \omega_{ki}(\cdot, J_{ik}\cdot) \sum_a (\Omega_{kj}\circ J_{ij})(e_a) \wedge \omega_{ki}(e_a) =\omega_{ki}(J_{ik}\cdot,\cdot) + \omega_{ki}(\cdot, J_{ik}\cdot)
\sup_{(\delta, \alpha) \in W_n^{(\eta)}}||\int_0^1 \nabla_{(\delta, \alpha)}\overline{G}_n(a_0, b_0, \delta_0 + t (\delta - \delta_0), \alpha_0 + t(\alpha - \alpha_0)) dt \begin{pmatrix}\delta - \delta_0 \alpha - \alpha_0\end{pmatrix}|| \rightarrow 0.
f_0 = x_1^{a_1}f_0 = x_1^{a_1}x_2 + \dots + x_K^{a_K}x_1, & f_1 = x_1 x_{K+1}^{b_1}+ x_{K+1}x_{K+2}^{b_2}+ \dots + x_{K+L-1}x_{K+L}^{b_L}
\{\hat{X}^{\mu}, \hat{S}^{\nu \lambda}\}^{\prime}= 2 \frac{\bar{P}^{\nu}\hat{S}^{\lambda \mu}- \bar{P}^{\lambda}\hat{S}^{\nu \mu}}{( P + \bar{P})^{2}}- \frac{4 P^{\mu}}{( P + \bar{P})^{2}}( \bar{P}^{\lambda}D^{\nu \rho}A_{\rho}- \bar{P}^{\nu}D^{\lambda \rho}A_{\rho}) . \nonumber \,
S ={\textstyle{g^2\over2}}\int dt \int d^3{x}\int d^3{y}\left\{\, -\dot{X}^{ax}(\gamma) \frac{1}{4\pi \mid x-y \mid}\dot{X}^{ay}(\gamma) - X^{ax}(\gamma) \, \delta(x-y) \, X^{ay}(\gamma) \, \right\}
f=\sum_{i=1}^{2}\big( \sum_{k=1}^{J_{0}}\|f_{i}^{k}\|_{L^{\infty}}|B(y_{i},2^{k}r)|^{1/p}a_{i}^{k}\big) + \sum_{i=1}^{2}\alpha_{i}^{J_{0}}\chi_{B(y_{i},2^{J_{0}}r)}.
\sigma_{\mathfrak{X}(X)}(D) = \inf{\{\sigma \in \mathbb{R}\colon \mbox{$\sup_{N}\big\| \sum_{n=1}^{N}{\frac{a_{n}}{n^{\sigma}}n^{-s}}\big\|_{\mathfrak{X}(X)}< \infty$}\}}begin{align*}2mm] \sigma_{\mathfrak{X}(X)}(D) \leq \limsup_{N}{\frac{\log{\left\| \sum_{n=1}^{N}{a_{n}n^{-s}}\right\|_{\mathfrak{X}(X)}}}{\log{N}}}
B_{1}=\int_{\mathbb{T}}\left(\sqrt{\rho_{\epsilon}(x)}-\left(\sqrt{\rho(x)}\right)_{\epsilon}\right)^{2}\frac{\left(\nabla\phi_{\epsilon}(y)\right)^{2}}{\phi_{\epsilon}(y)}dy B_{2}=\int_{\mathbb{T}}\left(\left(\sqrt{\rho(x)}\right)_{\epsilon}-\sqrt{\rho(x+y)}\right)^{2}\frac{\left(\nabla\phi_{\epsilon}(y)\right)^{2}}{\phi_{\epsilon}(y)}dy.
x_0=\frac{1-x_0}{1+\sum_{l \in \mathbb Z_0}{\lambda_l\widetilde{z}_l}}+x_0\cdot c_{00}, \ \x_j=\frac{(1-x_0)\lambda_{j}\widetilde{z}_{j}}{1+\sum_{l \in \mathbb Z_0}{\lambda_l\widetilde{z}_l}}+x_0\cdot c_{0j}, \ \j\in \mathbb Z_0.
p_{\boldsymbol{\bar{r}}[i]}= \frac{1}{N_b\sqrt{2\pi\sigma^2_{n}{\boldsymbol w}^\emph{H}_j[i]{\boldsymbol w}_j[i]}}\sum_{l=1}^{N_b}\exp\big(-\frac{(\bar{s}_{l_j}[i]-\bar{s}'_{l_j}[i])^2}{2\sigma^2_{n}{\boldsymbol w}^\emph{H}_j[i]{\boldsymbol w}_j[i]}\big),
Z ( t ) = \sum_{j \geq 0}( j + 1 )^{2}e^{- t ( j + 1 )^{2}+ t}= \frac{e^{t}}{2}\sum_{j \in Z}( j + 1 )^{2}e^{- t ( j + 1 )^{2}}= - \frac{e^{t}}{2}\frac{d}{d t}\sum_{j \in Z}e^{- t j^{2}}
\bigl(1+(2d-1)u^2\bigr)^{-(h+1+2|n|)}=\sum^{\infty}_{l=0}\binom{h+l+2|n|}{l}\bigl(-(2d-1)u^2\bigr)^{l}&=\sum^{\infty}_{l=0}\frac{(h+1)_{l+|n|}(l+|n|+h+1)_{|n|}}{l!2^{2|n|}\bigl(\frac{h}{2}+\frac{1}{2}\bigr)_{|n|}\bigl(\frac{h}{2}+1\bigr)_{|n|}}\bigl(-(2d-1)u^2\bigr)^{l}
\mathcal{R}_\varepsilon c(x) = \int_{\mathbb{R}^n_-}J_\varepsilon(|x-y|)(c(x) - \tilde{c}(y))y = \int_{\mathbb{R}^n_+}J_\varepsilon(|x-\hat{y}|)(c(x)- c(\hat{y}))y
\left\Vert V\right\Vert^{2}=BK\bar{B}^{t}=\left( VK^{-1}\right)K\overline{\left( VK^{-1}\right)}^{t}& =V\overline{K^{-1}}^{t}\overline{\,V^{t}}=VK^{-1}V^{t}=\left( \gamma_{11}\right) ,
f_{n+1,k}(z_1,...,z_{n+1})=-\sum_{i=1}^{n+1}\frac{g(z_1^{-1}...z_i^k...z_{n+1}^{-1})}{g(z_1^{-1}...z_{n+1}^{-1})}f_{n,k}(z_1,...,\widehat{z_i},...,z_{n+1}).
\lbrace A,B\rbrace{}^{*}=\lbrace A,B\rbrace -\int d^3\sigma [\lbrace A,\zeta^{\mu}(\tau ,\vec \sigma )\rbrace \lbrace{\cal H}_{\mu}(\tau ,\vec \sigma ),B\rbrace -\lbrace A,{\cal H}_{\mu}(\tau ,\vec \sigma )\rbrace \lbrace \zeta^{\mu}(\tau ,\vec \sigma ),B\rbrace ],
({\cal E}\xi^{\mu})_{\nu}\equiv [ \bot_{\mu \nu}D^{a}D_{a}+ ( n_{\nu i}\widetilde{\nabla}^{a}n_{\mu}^{i}) D_{a}+ h^{\alpha \beta}\bot^{\sigma}{_{\nu}}\bot^{\lambda}{_{\mu}}R_{\sigma \alpha \lambda \beta}- 2 \omega^{a}{_{i j}}n_{\nu}{^{i}}\widetilde{\nabla}_{a}n_{\mu}^{j}] \xi^{\mu}= 0 ;
\epsilon_r := \Bar{c}_0 \times \frac{1}{\sqrt{\sum_{i \in \widehat{\mathcal{C}}^{(r)}_j}N_i}}+\bar{c}_1\Delta_{\max}\sum_{i \in [M]}\sum_{t=0}^{T-1}\exp\left(-\bar{c}_2 N_in_x \left(\frac{\alpha^{(r)}\rho^{(i)}\|\Sigma^{(i)}_t\|}{\rho^{(i)}\|\Sigma^{(i)}_t\| + \sqrt{n_x}}\right)^2\right),
{}\displaystyle\sum_{n=1}^{\infty}| A_n-A_{n-1}|^{2}\geq\frac{1}{4}\displaystyle\sum_{n=1}^{\infty}\frac{| A_n|^{2}}{n^{2}}+\displaystyle\sum_{k=2}^{\infty}\binom{4k}{2k}\frac{1}{(4k-1)2^{4k-1}}\displaystyle\sum_{n=2}^{\infty}\frac{|A_n|^{2}}{n^{2k}}>\frac{1}{4}\displaystyle\sum_{n=1}^{\infty}\frac{|A_n|^{2}}{n^{2}},
N(\star_{i=1}^{m+k}G,\alpha^{m+k+1}) = N(\star_{i=1}^{m+k}G, (\star_{i=1}^{m}\alpha^{2}) \star (\star_{i=m+1}^{m+k}\alpha^{2}) ) = & N( \star_{i=1}^{m+k}G,(\star_{i=1}^{m}\alpha^{2}) \star (\star_{i=1}^{k}\alpha^{2}) ) \le N(\star_{i=1}^{m}G,\star_{i=1}^{m}\alpha^{2}) N( \star_{i=1}^{k}G,\star_{i=1}^{k}\alpha^{2}) ,
\langle \Phi(x) \Phi(0) \rangle \sim A_1 \frac{e^{-m_1 r}}{\sqrt{m_1 r}}+ A_2 \frac{e^{-m_2 r}}{\sqrt{m_2 r}}+ A_3 \frac{e^{-m_3 r}}{\sqrt{m_3 r}}\cdots
\begin{array}{c c c}{{\cal L}_{( 1 )}^{'}}&{}&{{\cal L}_{( 2 )}^{'}}\\{\downarrow}&{}&{\downarrow}\\{\overline{{{\cal M}}}_{g , 1}^{( 1 )}}&{}&{\overline{{{\cal M}}}_{g , 1}^{( 2 )},}\\ \end{array}
\Xi^{\omega}(\varphi):= \gamma^{\partial B_0}(\nu^{\partial B_1})Q^{\partial B_0}\zeta^{\partial B_1}\gamma^{X}(\nu^{\partial B_0})Q^{X}\zeta^{\partial B_0}\omega-\mu_{\partial B_1}=F^{\omega}(\varphi,T\varphi),
{\cal S}= \int_{M_{4}}d^{4}x \sqrt{g_{4}}\left\{- \frac{R_{4}}{2 \kappa_{4}}+ \frac{1}{2}( \partial \phi )^{2}+ \Lambda_{4}\exp \left[ \sqrt{\frac{2 \kappa_{4}}{3}}\phi \right] \right\}
\begin{array}{rcl}U_g/m^4 = \displaystyle +6D_m(k_1)D(k_1+p)D_m(k_2)D(k_2+p)D_m(k_1-k_2)(1+K_{12}) & \displaystyle +4D_m(k_1)D(k_1+p)D(k_2+p)D_m(k_2)D(k_1+k_2+p)K_{12}, \end{array}
\sum_{w\in S_{n+1}/S_n}\!\!\!w\cdot \frac{\prod_{1\le i\le n+2}\vartheta(z_{n+1}-y_i) \prod_{1\le i\le n}\vartheta(Y-z_i)}{\prod_{1\le i\le n}\vartheta(z_{n+1}-z_i)}= \prod_{1\le i\le n+2}\vartheta(Y-y_i),
\lVert T_{\ast , \mathcal I}\,:\, L^q \to L^q \rVert \lesssim \sum_{k=k_d}^{\infty}\lVert T_{\ast , \mathcal I (k)}\,:\, L^q \mapsto L^q \rVert\lesssim \sum_{k=k_d}^{\infty}2^{-k/2q}\lesssim q.
Vh= \left(nv_n-\sum_{k=1}^{n-1}v_k\frac{\partial s_1}{\partial x_k}\right)x_n^{n-1}+\left(-v_n(n-1)s_1 +\sum_{k=1}^{n-1}v_k\frac{\partial s_2}{\partial x_k}\right)x_n^{n-2}\hphantom{Vh=}{}+\cdots +(-1)^n\left(-v_ns_{n-1}+\sum_{k=1}^{n-1}v_k\frac{\partial s_n}{\partial x_k}\right).
\lbrack K,L]_{\operatorname*{FN}}(X,Y) =[KX,LY]-[KY,LX]-L\left([KX,Y]-[KY,X]\right) \phantom{\lbrack K,L]_{\operatorname*{FN}}(X,Y)=}{}-K\left([LX,Y]-[LY,X]\right) +(LK+KL)[X,Y],
\sigma\left(y_{1},y_{2}\right)=\begin{cases}\left(\frac{y_{1}^{-2}+y_{2}^{-2}}{2}\right)^{-\frac{1}{2}}, y_{1},y_{2}<\infty \sqrt{2}\, y_{1}, y_{1}<\infty,\: y_{2}=\infty \sqrt{2}\, y_{2}, y_{1}=\infty,\: y_{2}<\infty\end{cases}.
\delta S = - \frac{i}{2 \pi \alpha^{\prime}}\int_{z^{2}= 0}d z^{1}\frac{1}{2}\partial_{\nu}F_{\mu \lambda}\partial_{1}\bar{X}^{\mu}\Delta^{\nu \lambda}\left( z^{1}, z^{\prime 1}\right)_{| z^{\prime}\to z}.
\begin{array}{l l}{{\mathrm{S e c t o r}\ \}}&{{\mathrm{F i e l d ~ c o n t e n t}}}\\{{\mathrm{e v e n}}}&{{[ 1 , 1 ] , \, [ 3 , 3 ] , \, [ 5 , 5 ] , \, [ 7 , 7 ] , \, [ 9 , 9 ] , \,}}\\{{\mathrm{o d d}}}&{{[ 2 , 2 ] , \, [ 4 , 4 ] , \, [ 6 , 6 ] , \, [ 8 , 8 ] , \,}}\\{{\mathrm{t w i s t e d}}}&{{[ 1 , 9 ] , \, [ 3 , 7 ] , \, [ 5 , 5 ] , \, [ 7 , 3 ] , \, [ 9 , 1 ] .}}\\ \end{array}
\begin{aligned}F(\pi-\Theta(0,\phi,\psi),\phi,\psi) =-F(0,\phi,\psi), G(\pi-\Theta(0,\phi,\psi),\phi,\psi) =-H(0,\phi,\psi), H(\pi-\Theta(0,\phi,\psi),\phi,\psi) =G(0,\phi,\psi).\end{aligned}
Im \Gamma = i \sum_{i, \, j = 1}^N \int d^{\, 4}x \, d^{\, 4}y \, (\varphi_1 (x) - \varphi_2 (x))_i \, \gamma_{i j}(x, y) (\varphi_1 (y) - \varphi_2 (y))_j ,
\tilde{\cal L}^{(0)}= \pi_a\dot{\sigma}^a + \Psi\dot{\theta}- \frac{1}{2}\,{\sigma^\prime}^2_a -\frac{1}{2}\,\lambda\,\bigl(\sigma^2_a - 1\bigr) - \frac{1}{2}\,\pi^2_a + \sigma_a\pi^a\theta - \frac{1}{2}\sigma_a^2 \theta^2.
\begin{matrix}\begin{aligned}z_{1}z_{2}z_{3}z_{4}\times\prod_{i=1}^{3}(1+z_{i})(1+z_{i}z_{4})\times\prod_{1\leq i<j\leq3}(1-z_{i}z_{j}z_{4}) \;\;\times\left(1-(z_{1}z_{2}+z_{1}z_{3}+z_{2}z_{3}+z_{1}z_{2}z_{3})z_{4}-z_{1}z_{2}z_{3}z_{4}^{2}\right).\end{aligned}\end{matrix}
M' (z) = \mathfrak{X}_{[0, R]}M' (z) \mathfrak{X}_{[0, R]}+\mathfrak{X}_{(R, \infty)}M' (z) \mathfrak{X}_{[0, R]}+ \mathfrak{X}_{[0, R]}M' (z) \mathfrak{X}_{(R, \infty)}+ \mathfrak{X}_{(R,\infty)}M' (z) \mathfrak{X}_{(R, \infty)}.
\Vert \mathfrak{e}\Vert_{C^{m}}\ \doteq \ \max\limits_{\underline{n}\in{\mathbb{N}}_{0}^{d},\;|\underline{n}|=m}\;\max_{p\in \Gamma^{\ast}}\big|\partial_{p}^{\underline{n}}\mathfrak{e}(p)\big|.
\int_{\Omega}\rho|\psi_t|^2dx+\frac{\nu}{2}\frac{d}{dt}\|\nabla\times\psi\|^2 = \int_{\Omega}-[-\Delta,\rho^{(m)}]u_t\psi_tdx+\int_{\Omega}\Delta(\rho u\cdot\nabla u)\psi_tdx = J_1+J_2,
\underset{\left\{\pmb{F}_k\right\}_{k=1}^K,\left\{\pmb{W}_k\right\}_{k=1}^{K}}{}\sum\limits_{k=1}^K\mathrm{rank}\left(\pmb{J}_k\right) + \mathrm{rank}\left(\pmb{S}_e\right) \,\,\,\,\,\qquad \mathrm{rank}\left(\pmb{S}_k\right) = d, \,\,\, \forall \ell \in \mathcal{K}.
\mathbb{P}\left( M_\ell \notin \mathcal{M}_\ell \right) \leq \sum_{j=1}^m \mathbb{P}\left( S_{j, \ell}\ \hbox{is such that}\ |s_{j,\ell,i}| \geq \sqrt{\ell}\epsilon_\ell \ \hbox{for some}\i \right) &\leq \sum_{j,i}\mathbb{P}\left( S_{j, \ell}\ \hbox{is such that}\ |s_{j,\ell,i}| \geq \sqrt{\ell}\epsilon_\ell \right) \to 0.
\min_{0 \leq i \leq N}\{\|\nabla f(x_i)\|^2\}{}\leq{}\frac{q}{S_*}\|\nabla f(x_N)\|^2 + \sum_{i=0}^{N-1}\frac{p_i}{S_*}\min\{\|\nabla f(x_i)\|^2 \,,\, \|\nabla f(x_{i+1})\|^2\}.
z^{\epsilon_0}xz^{\epsilon_1}x\cdots z^{\epsilon_{p-1}}xz^{\epsilon_q}=z^{\epsilon_0}\sigma(z^{\epsilon_1})\sigma^2(z^{\epsilon_2})\cdots \sigma^q(z^{\epsilon_q}) x^q =z^{\epsilon_0}(\sigma(z))^{\epsilon_1}(\sigma^2(z))^{\epsilon_2}\cdots (\sigma^q(z))^{\epsilon_q}x^q
\langle\partial_t u,v\rangle +\int_{Q_T}\mathsf{A}\nabla u\cdot \nabla v\mathrm{dx}\mathrm{dt}+\int_{\Sigma_T}b(u)u v\mathrm{ds}\mathrm{dt}= =\int_{Q_T}u{\bf E}\cdot \nabla v\mathrm{dx}\mathrm{dt}+\int_{Q_T}fv\mathrm{dx}\mathrm{dt}+\int_{\Sigma_T}h v\mathrm{ds}\mathrm{dt}