harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\max_{P(t), \t \in [0,D]}\int_0^D \frac{1}{2}\log\left(1 + P(\tau)\right)d\tau \mbox{s.t.}\frac{d}{dt}T(t) = f_1(T,B,P), \ \frac{d}{dt}B(t)= f_2(T,B,P) & g_1(T,B,t) \leq 0, \g_2(T,B,t) \leq 0
\widetilde h(x,t) = e^{2\pi i (t+t_0)\nu_0}\sum_k \sum_{(q,m)\in\Gamma}\big[b_{(q,m),k}& \hskip-.25in\widetilde H \widetilde g (t - (q-k)/\sqrt{L}) e^{-2\pi i m(q-k)/L}& \widetilde \Phi_{(q,m)}(t{+}t_0,x-(t{+}t_0)+(q-k)/\sqrt{L})\big],
R(X ,Y)Z= R(\bar X,\bar Y)\bar Z+\eta(X)R(\xi,\bar Y)\bar Z-\eta(Y)R(\xi,\bar X)\bar Z +\eta(Z)R(\bar X,\bar Y)\xi -\eta(Z)\eta(X)R(\bar Y,\xi)\xi+\eta(Z)\eta(Y)R(\bar X,\xi)\xi.
\sum_{x=4}^{a}\sum_{y=b+1}^{\infty}u_{x,y,a,b}~=~ \sum_{x=4}^{a}\sum_{y=b+1}^{\infty}\Bigl( x \sqrt{(x \wedge a)(x \wedge b)}- y \sqrt{(y \wedge a)(y \wedge b)}\Bigr)^2 p(x) p(y) begin{align*}.5em] = ~ \sum_{x=4}^{a}\sum_{y=b+1}^{\infty}\bigl( x^2 - y \sqrt{ab}\bigr)^2 p(x) p(y).
\int \, \prod_{A = 1 , 2 , 3 , 4}d \nu^{A}\, \exp \big [ \sqrt 8 \pi i g^{- 1}\mathrm{t r}_{k^{\prime}}\, \chi_{A B}( \nu^{A})^{t}\nu^{B}\big ] = 2^{6 k N - 6{k^{\prime}}^{2}}( \pi / g )^{4 k N - 4{k^{\prime}}^{2}}\left( \mathrm{d e t}_{4 k^{\prime}}\chi \right)^{N^{\prime}/ 2 - k^{\prime}}\ .
[C_{IJ},C_{JK}]=[C_{JK},C_{IK}]=[C_{IK},C_{IJ}]\,, &\big[C_{JK},[C_{IJ},C_{JK}]\big]=2 C_{IK}C_{JK}-2C_{JK}C_{IJ}+ 2(C_J-C_K)(C_{IJK}-C_I)\,,
\mp\left[-e^{3A}\dot{\Psi}^{(0)}F_{ww}+2e^A\dot{\varphi}+2e^{2A}(e^A\dot{\Psi}^{(0)})^{\cdot}\phi_{\pm w}\right] ={V''_{\pm}}^{(0)}(\varphi+e^{2A}\dot{\Psi}^{(0)}\phi_{\pm w}) + 2{\alpha'_{\pm}}^{(0)}e^{2A}\bar{\tau}_{\pm(Y)},
\left\langle \psi_{0,k_1}, \psi_{0,k_2}\right\rangle = 2\overline{\mathbf{a}}_\mathbf{1}(2c)^{k_1+1}\pi \frac{2\pi^{m/2}}{\Gamma(m/2)}\delta_{k_1k_2}\int_0^{+\infty}R^{2k_1+1}e^{\left(-R^2\right)}\mathrm{d}R\ \mathbf{a_2}&= 2\overline{\mathbf{a}}_\mathbf{1}(2c)^{k_1+1}\frac{2\pi^{m/2+1}}{\Gamma(m/2)}\delta_{k_1k_2}\frac{\Gamma(k_1+1)}{2}\mathbf{a_2},
\widetilde{\alpha}_{n,q}(l)= \sum_{k=q}^{2n-2}c_{n,q,k}\sum_{i=0}^{2n-2}b_{2n-1,i}\sum_{p=0}^i(q-n-1-k)^{i-p}\binom{i}{i-p}l^p \forall l\geq 2n-1.
\sum_{n \leq N}n \cdot \pi_{X}(n) R_{X}^{n}= N- a(N)+ A_{X}+ K_{X}+ O \left( (\rho_{X}R_{X})^{N}\right) & \left( = \left[ \frac{N}{\Delta_{X}}\right] \Delta_{X}+ A_{X}+ K_{X}+ O \left( (\rho_{X}R_{X})^{N}\right) \right),
< \Psi_{1}| \Psi_{2}> = N \int_{- \infty}^{\infty}\frac{d^{2}\vec{p}}{p^{0}}\int_{0}^{2 \pi}\frac{d \varphi}{( p , n )}\overline{{\Psi_{1}( p , \varphi )}}\Psi_{2}( p , \varphi ) ,
\int{\cal D}A_{\mu}^{a}{\cal D}B^{a}{\cal D}\bar{c}^{a}{\cal D}c^{a}\exp\{-S_{YM}(A_{\mu}^{a})+\int dx[-iB^{a}(\partial^{\mu}A_{\mu}^{a})+\bar{c}^{a}(-\partial_{\mu}(D^{\mu}c)^{a}]\}
\left(e^{-2\pi h};e^{-2\pi h}\right){}_{\!\infty}=\frac{1}{\sqrt{h}}\,\exp\!\left(\frac{\pi}{12}\left(h-\frac{1}{h}\right)\right) \left(e^{-2\pi/h};e^{-2\pi/h}\right){}_{\!\infty},\Re h>0.
X((\alpha_1 + \beta_1f_1) + \gamma(\alpha_2 + \beta_2f_2)) = X(\alpha_1 + \gamma \alpha_2 + \beta_1f_1 + \gamma \beta_2f_2) & = \alpha_1 + \gamma \alpha_2 + T^{-1}(\beta_1f_1 + \gamma \beta_2f_2) & = \alpha_1 + T^{-1}(\beta_1f_1)+ \gamma (\alpha_2 + T^{-1}(\beta_2f_2)) &=X(\alpha_1 + \beta_1f_1) + \gamma X(\alpha_2 + \beta_2f_2).
c^\tau(x*y) =\tau(g_1,g_2)c^\tau(xy) =\tau(g_1,g_2)\tau(h,g_1g_2)c(xy) =\tau(h,g_1)\tau(hg_1,g_2)c(x)y =\tau(hg_1,g_2)c^\tau(x)y=c^\tau(x)*y
\pi^{a\,\prime}_A\,=\,\phi^{\ast\,a}_A\,\,,\,\,\,\,\phi^\prime_A\,=\,{\bar \phi}_A\,\,\,\,,\,\,\,\,\phi^{\ast\,a\,\prime}_A\,\,=\,\,\,\,\pi^{\,a}_A\,\,\,,\,\,\,\,{\bar \phi}^{A\,\prime}\,=\,\phi^A \,\,,
\begin{aligned}\frac{d}{ds}\Big|_{s=0}H(\gamma(s) | e^{-\Psi}) = \int_{\mathbb{S}^{1}}f e^{-\Psi}d\omega = \int_{\mathbb{S}^{1}}(\rho - e^{-\Psi}) d\omega=0, &\frac{d}{ds}\Big|_{s=0}I(\gamma(s) | e^{-\Psi}) = 0,\end{aligned}
V=\sum_{i=1}^{n}PV_{i}=\sum_{i=1}^{n}\frac{CF_{i}}{(1+y/k)^{k\cdot t_{i}}}=\sum_{i=1}^{4}\frac{10}{(1+.04/2)^{i}}+\frac{100}{(1+.04/2)^{4}}
D_{\mu \nu}^{a b}( p , p_{3}) = \frac{\delta^{a b}\delta^{a 3}}{p^{2}- p_{3}^{2}+ i \epsilon}\left[ - g_{\mu \nu}+ p_{\mu}p_{\nu}\left( ( 1 - \delta_{p_{3}, 0}) \frac{1}{p_{3}^{2}}+ \delta_{p_{3}, 0}( 1 - \xi ) \frac{1}{p^{2}+ i \epsilon}\right) \right]
\frac{d \rho}{d x_{1}}= \sin h ( \gamma^{K}, \rho ) - \rho^{n - 1}\sin [ n h ( \gamma^{K}, \rho ) ] \hspace{0 . 4 c m}, \hspace{0 . 4 c m}\frac{d \rho}{d x_{1}}= \cosh_{\perp}( \gamma_{\perp}^{K}, \rho ) - \rho^{n - 1}\cos [ n h_{\perp}( \gamma_{\perp}^{K}, \rho ) ]
|\sum_{i=1}^{n}a_{i}b_{i}|^{2}+\sum_{i<j}^{n}|a_{i}\overline{b}_{j}-a_{j}\overline{b}_{i}|^{2}=(\sum_{i=1}^{n}|a_{i}|^{2})(\sum_{i=1}^{n}|b_{i}|^{2}).
\int_0^{1/2}g^*(2x) - g_i(2x) dx = \frac{1}{2}\int_0^1 g^*(u) - g_i(u) du &\int_{1/2}^{1}g^*(2x-1) + g_i(2x-1) dx = \frac{1}{2}\int_{0}^1 g^*(u)+g_i(u)du,
\delta_\square(G,G')=\min_{X\in \chi(G,G')}\max_{S, T\subset V\times V'}\left\vert \sum_{\substack{(i,u)\in S (j,v)\in T}}X_{iu}X_{jv}\left(\beta_{ij}(G) - \beta_{uv}(G')\right) \right\vert.
H_{min}\left( (E_1[t]\times E_1[t]^{\vee}) \otimes F\right) =H_{min}(E_1[t]\times E_1[t]^{\vee})H_{min}(F) &=H_{min}(E_1[t])H_{min}(F) &=t H_{min}(E_1)H_{min}(F).
\begin{aligned}j-invariant = const\frac{(a^2-3b)^3}{a^2b^2-4b^3-4a^3c+18abc-27c^2}= const\frac{(r_1^2+r_2^2+r_3^2-r_1r_2-r_2r_3-r_3r_1)^3}{(r_1-r_2)^2(r_2-r_3)^2(r_3-r_1)^2}\end{aligned}
K^p_4[H_0] =\mathring{K}^p_4[H_0]+Err^p_4[H_0], Err^p_4[H_0] =\eta\left(\frac{r^p}{r(1-\frac{2M}{r})}\nabla_L \Big(rH_0\Big)+r^{p-2}\nabla_T H_0 \right)\cdot\left(-\frac{2M}{r^3}P\right).
g(x,y,n,z) :=\frac{n^2}{\pi^2}\sin^2\Big(\frac{\pi x}{n}\Big)+\frac{n^2}{\pi^2}\sin^2\Big(\frac{\pi y}{n}\Big) -\frac{2 n^2}{3\pi^2}\sin^2\Big(\frac{\pi x}{n}\Big)\sin^2\Big(\frac{\pi y}{n}\Big) + z^2.
\begin{array}{l}a^{\prime}_{k}= a_{k}+ \alpha_{kl\,}b_{l}, begin{align*}1ex]b^{\prime}_{k}= b_{k}+ \beta_{kl\,}a_{l},\end{array}\begin{array}{l}a^{\prime\,\dagger}_{k}= a^{\dagger}_{k}- \alpha_{lk\,}b^{\dagger}_{l}, begin{align*}1ex]b^{\prime\,\dagger}_{k}= b^{\dagger}_{k}- \beta_{lk\,}a^{\dagger}_{l}.\end{array}
i_{221}=-D_{1,1}D_{2,1}x_{3,1}, i_{212}=-D_{1,1}x_{2,1}D_{3,1}, i_{122}=-x_{1,1}D_{2,1}D_{3,1}, i_{211}=D_{1,1}x_{2,1}x_{3,1}, i_{121}=x_{1,1}D_{2,1}x_{3,1}, i_{112}=x_{1,1}x_{2,1}D_{3,1}, i_{111}=-x_{1,1}x_{2,1}x_{3,1},
|\cos u-\cos(u+\delta)| =|(1-\cos\delta)\cos u+\sin\delta\sin u|\leq (1-\cos\delta)+|\sin\delta|\cdot|\sin u|\leq &\leq \frac{\delta^2}{2}+|\delta|\cdot|\sin u|\leq\frac{1}{n}\left(\frac{1}{n}+|\sin u|\right)=\Delta_n(\cos u).
y''(z)+\bigg(2\frac{\phi'_{e}(z)}{\phi_{e}(z)}+\frac{\widetilde{\tau_{e}}(z)}{\sigma_{e}(z)}\bigg)y'(z)+\bigg(\frac{\phi''_{e}(z)}{\phi_{e}(z)}+\frac{\phi'_{e}(z)}{\phi_{e}(z)}\frac{\widetilde{\tau_{e}}(z)}{\sigma_{e}(z)}+\frac{\widetilde{\sigma_{e}}(z)}{\sigma_{e}^{2}(z)}\bigg)y(z)=0.
{3\over b^2}{a^{\prime}\over a}\left({a^{\prime}\over a}+{n^{\prime}\over n}\right)-{3\over{c^2n^2}}\left[{\dot{a}\over a}\left({\dot{a}\over a}-{\dot{n}\over n}\right)+{\ddot{a}\over a}\right]-{{3k}\over a^2}=-{{8\pi G_5}\over c^4}\Lambda,
\Omega_{0, F}= p\circ(K^x + \Delta^x) - p\circ K^x + K^y \cdot (q\circ(K^x + \Delta^x) - q\circ K^x) & \quad+ \, \Delta^y \cdot q\circ (K^x + \Delta^x) + u \circ (K + \Delta) - u \circ K+ g\circ (K + \Delta)
\mathbb E\left[x^T(t) P^{-1}x(t)\right] \leq\mathbb E\int_0^t x^T(s) (A^T P^{-1}+P^{-1}A+ \sum_{k=1}^m N_k^TP^{-1}N_k +\sum_{i, j=1}^v H_i^T P^{-1}H_j k_{ij}) x(s)ds \quad+\mathbb E\int_0^t 2 x^T(s) P^{-1}B u(s)ds +\int_0^t \mathbb E\left[x^T(s) P^{-1}x(s)\right] \left\|u(s)\right\|_2^2 ds.
F^{\mathrm{out}}(y) = (1\ 5)(2\ 3\ 4) \cdot y \cdot (2\ 3\ 4) \cdot y \cdot (3\ 4) \cdot y^2 \cdot (2\ 3)(4\ 5) \cdot y \cdot (2\ 3\ 4) \cdot y \cdot (3\ 4) \cdot y^2 \cdot (1\ 4\ 2\ 5) \enspace.
x =\left(a_{s-\ell}k^s+\sum_{r=\ell}^sa_{r-\ell}k^r\right)t+ik^\ell+\sum_{r=0}^{\ell-1}a_{r+s+1-\ell}k^r &=a_{s-\ell}n+\left(\sum_{r=\ell}^{s-1}a_{r-\ell}k^r\right)t+ik^\ell+\sum_{r=0}^{\ell-1}a_{r+s+1-\ell}k^r
{\cal L}= - \frac{1}{4}F_{\mu \nu}F^{\mu \nu}+ \frac{1}{2}m^{2}( A_{\mu}+ \partial_{\mu}\theta )^{2}+ A^{\mu}\partial_{\mu}B + d_{2}\partial_{\mu}\bar{\cal C}\partial^{\mu}{\cal C}- \frac{1}{2}\alpha B^{2}.
g = \int_{S^{2}_{\infty}}d\vec{\sigma}\cdot \frac{1}{v}\Phi^{a}\vec{B}_{a}= \frac{1}{v^{2}}\int_{S^{2}_{\infty}}d\vec{\sigma}\cdot \Phi_{a}\Phi^{a}\vec{B}= \int_{S^{2}_{\infty}}d\vec{\sigma}\cdot \vec{B}= \frac{4\pi n}{e}\, .
k^{P;P^k_j}(t,p,w) := \sum_{i=1}^{d_{N-1}}\mathbf{1}_{\{P^{N-1}_i\cap P^k_j\neq\emptyset\}}k^{P;P^{N-1}_i;P^k_j}(t,p,w), k^{P;P^0_j}(t,p,w) := \sum_{i=1}^{d_{N-1}}\mathbf{1}_{\{P^{N-1}_i\cap P^0_j\neq\emptyset\}}k^{P;P^{N-1}_i;P^0_j}(t,p,w),
\widehat{h}_{5}= h_{5}^{[ K P , S G ]}\left( r , \theta , \chi \right) = \eta_{5}^{[ K P , S G ]}\left( r , \theta , \chi \right) \eta_{5}\left( r , \theta \right) h_{5}( r ) , \qquad h_{5}( r ) = - r_{0}^{2}e^{2 \psi ( r )},
\mathrm{R e}\chi_{T}^{\mathrm{D}}( \omega ) = - \frac{\hbar A}{\pi^{2}}\int \int_{R_{4}}d \omega_{\mathrm{i n}}d k_{\|}k_{\|}\sqrt{k_{\|}^{2}- ( \omega + \omega_{\mathrm{i n}})^{2}}\sqrt{\omega_{\mathrm{i n}}^{2}- k_{\|}^{2}}{\overline{n}}( - \omega_{\mathrm{i n}})
\bar{C}^{r}_{x}=\{C^{r}_{x},\pm^{r}_{x},\times^{r}_{x},\div^{r}_{x},0^{r}_{x}, 1^{r}_{x}\}=\{rC_{x},\pm_{x}\frac{\times_{x}}{r},r\div_{x},0_{x},r_{x}\}
T^{\ast}|T|^{2k}T =M_{(E(|u|^{2}))^{k-1}\chi_{S}(E(|w|^{2}))^{k}|E(uw)|^2}M_{\bar{u}}EM_{u}, T^{\ast}T =M_{E(|w|^{2})}M_{\bar{u}}EM_{u},
{\cal O}_k=\delta_{k,0}+\sum_{j=0}^{\infty}\frac{(L_{-1})^j L_{k+j}}{(k+j+1)!},\hspace{1em}\bar{{\cal O}_k}=\delta_{k,0}+\sum_{j=0}^{\infty}\frac{(\bar{L}_{-1})^j\bar{L}_{k+j}}{(k+j+1)!}.
F_{sce}:= F^0 \setminus \overline{r(E^1)}, F^0_{fin}:=\{v \in F^0: \exists V~~v ~~ r^{-1}(V)~ \}, F^0_{rg}:=F^0_{fin}\setminus \overline{F^0_{sce}}, F^0_{sg}:=F^0 \setminus F^0_{rg}.
\begin{aligned}\widetilde{f}(\nabla^{E}, h^{E}_{0}, h^{E}_{1}):=\sum_{i=0}^{k}(-1)^{i}\widetilde{f}(\nabla^{E^{i}}, h^{E^{i}}_{0}, h^{E^{i}}_{1}), &\widetilde{f}(\nabla^{H(E)}, h^{H(E)}_{0}, h^{H(E)}_{1}):=\sum_{i=0}^{k}(-1)^{i}\widetilde{f}(\nabla^{H^{i}(E)}, h^{H^{i}(E)}_{0}, h^{H^{i}(E)}_{1}).\end{aligned}
p_{k+1}\leq s p_k + \frac{(s p_k- q_{k+1})q_k}{1 - q_k - q_{k+1}}= &= s p_k \frac{1 - q_k}{1 - q_k - q_{k+1}}- \frac{q_{k+1}q_k}{1 - q_k - q_{k+1}}\leq s \frac{p_k}{1 - \frac{q_{k+1}}{1 - q_k}}.
\mathbb{E}_0\left[\bar{U}^{(pq)}_h\middle| \mathcal{F}_{n, q-1}\right] = \mathbb{E}_0\left[W_h^{(pq)}\middle| \mathcal{F}_{n, q-1}\right] = \mathbb{E}_0\left[U_h^{(pq)}\middle| \mathcal{F}_{n, q-1}\right] = 0
w (x,y)= &=\frac{1}{x_1y_1+x_1y_1\left|x-y\right|+\left|x-y\right|^2}\frac{1}{(1-x_1)(1-y_1)+(1-x_1)(1-y_1)\left|x-y\right|+\left|x-y\right|^2}.
{\hat v}(\eta ,{\bf x}) \, = \, \frac{1}{(2\pi )^{3/2}}\int{\rm d}^3{\bf k}[{\hat c}_{\bf k}(\eta )e^{i{\bf k}\cdot{\bf x}}+{\hat c}^{\dagger}_{\bf k}(\eta )e^{-i{\bf k}\cdot{\bf x}}] \, ,
{\bf H}_1 =\begin{bmatrix}2.03e^{-i0.68}2.1e^{i2.64}3.2e^{i1.48}4.7e^{i1.97}4.5e^{-i0.66}2.85e^{i2.41}\end{bmatrix},{\bf H}_2 =\begin{bmatrix}3.2e^{-i0.72}2.3e^{i2.52}1.9e^{i1.35}2.8e^{i1.68}2.5e^{-i0.76}3.4e^{i2.23}\end{bmatrix}.
3 ! F_{3}^{( 4 )}= [ L_{3}^{2}L_{1}]^{( 4 )}+ [ L_{2}^{2}L_{1}]^{( 4 )}+ [ L_{3}^{2}L_{2}]^{( 4 )}+ 2 [ L_{3}L_{2}L_{1}]^{( 4 )}- [ \Sigma F_{2}]^{( 4 )}
P^{0,1}\mu_1\partial\Delta^{-1}_0\star{\rm ad}\bar\nu_i^T\star P^{1,0}\bar\mu_2\bar\partial\Delta^{-1}_0\star{\rm ad}\nu_j\star P^{0,1}\nu_i =i P^{0,1}\mu_1\partial\Delta^{-1}_0\star{\rm ad}\bar\nu_i^T\star P^{1,0}\bar\mu_2\bar\partial\Delta^{-1}_0{\rm ad}\nu_i\star \nu_j
\nu_{(f,H),\leqslant k}(z) =\begin{cases}0 \nu_{(f,H)}(z)>k, \nu_{(f,H)}(z) \nu_{(f,H)}(z)\leqslant k,\end{cases}&\nu_{(f,H),\geqslant k}(z) =\begin{cases}0 \nu_{(f,H)}(z)<k, \nu_{(f,H)}(z) \nu_{(f,H)}(z)\geqslant k,\end{cases}
\begin{cases}\forall \gamma^1,p^X,x^X\left(\gamma>_\mathbb{R}0\land \gamma^{-1}(x-_Xp)\in Ap\rightarrow p=_X J^A_{\gamma}x\right), \forall \gamma^1,p^X,x^X\left(\gamma>_\mathbb{R}0\land p=_X J^A_{\gamma}x\rightarrow\gamma^{-1}(x-_Xp)\in Ap)\right),\end{cases}
|\Psi_{i}\rangle=\exp\left( \sum_{n=1}^{\infty}\mu_{n}^{(i)}\left(a_{n}^{(i)}\right)^{\dagger}\right) |p^{(i)}\rangle=\exp\left(ip^{(i)}x_{0}+\sum_{n=1}^{\infty}\kappa_{n}^{-1/2}\mu_{n}^{(i)}\alpha_{-n}^{(i)}\right) |0\rangle.
\Lambda(x) = \frac{q-1}{\sqrt{\sum_{i=1}^n x^{2(q-1)}_i}}{\rm{diag}}(x^{q-2}_i) ~,~ U = \langle \Lambda \nu, \nu \rangle \langle \Lambda^{-1}\nu, \nu \rangle \hat{e}_2 \otimes \hat{e}_2 + \sum_{k >2}^n \hat{e}_k \otimes \hat{e}_k ~ ,
\sum_{i=1}^{k_n}\frac{\kappa_i^2f_{0,i}^2}{(1+n\lambda_i\kappa_i^2)^2}\leq n^{-2}\sum_{i=1}^{k_n}\lambda_i^{-2}\kappa_i^{-2}i^{-2\beta}i^{2\beta}f_{0,i}^2 &= n^{-2}\sum_{i=1}^{k_n}i^{2(\alpha-\beta)}\exp(2(\xi+\gamma)i^p)i^{2\beta}f_{0,i}^2,
p_{r-1}^{(d-1)}(s) \equiv\sum_{h=0}^{d-1}(-3)^h2^{r-h-1}\binom{d-h-1}{r-h-1}\binom{s}{h}\pmod{3^{k-1}}&\equiv \sum_{h=0}^{k-2}(-3)^h2^{r-h-1}\binom{d-h-1}{r-h-1}\binom{s}{h}\pmod{3^{k-1}}.
\int \partial_{\alpha}\left(P_{\alpha}^k(\hat \Psi_{\alpha}h_{\alpha}) \hat \Psi_{\alpha}\: dm \right) = \int P_{\alpha}^k(\hat \Psi_{\alpha}h_{\alpha}) \partial_{\alpha}\hat \Psi_{\alpha}\: dm + \int \partial_{\alpha}[P_{\alpha}^k(\hat \Psi_{\alpha}h_{\alpha})] \hat \Psi_{\alpha}\: dm &=(I) + (II)
( X^{(k)}\cdots X^{(1)})e_i = \sum_{j_{1}, \ldots, j_{2k}=-(p-1)}^{n-1}\prod_{r=1}^{k}a^{(r)}_{j_{2r-1}}a^{(r)}_{j_{2r}}\prod_{s=1}^{k}\chi_{[1,n]}(i+\sum_{\ell=1}^{2s-1}(-1)^\ell j_\ell) \times \chi_{[1,p]}(i+\sum_{\ell=1}^{2s}(-1)^\ell j_\ell ) e_{i-\sum_{q=1}^{2k}(-1)^{q}j_{q}}.
\frac{2e-(pd_p+qd'_q-d_pd'_q)+ \sqrt{(pd_p+qd'_q-d_pd'_q)^2-4d_pd'_q(pq-e)}}{\partial d_{p}}= -p+d'_{q}+\frac{(pd_{p}+qd'_{q}-d_{p}d'_{q})(p-d'_{q})-2d'_{q}(pq-e)}{\sqrt{(pd_{p}+qd'_{q}-d_{p}d'_{q})^2-4d_{p}d'_{q}(pq-e)}}
\lambda : \mathbb{R}\times \Sigma \rightarrow \mathbb{R}^+, \stackrel{}{\longrightarrow}g_{\lambda}=-\lambda^2 dt^2 + g_t\nabla \phi = 0, h , \stackrel{}{\longrightarrow}g_{h}\nabla \phi = 0, h-E^{g,h}=0.
{\cal L}^{( 0 )}= - \frac{e^{2}B^{2}}{2 4 \pi^{2}}\ln \frac{\Lambda^{2}}{m^{2}}- \frac{e^{2}B^{2}}{8 \pi^{2}}\int_{0}^{\infty}\frac{d s}{s^{2}}e^{- m^{2}s / ( e B )}( \coth s - \frac{1}{s}- \frac{s}{3})
L_w^{2^a}= \bigg(\frac{R_w}{h_1(X)^{2^a d}}\bigg)^{2^{8}}\begin{array}{l}L_w = \tilde{w}_1(X)X^{2^{8-a}}+ \tilde{w}_0(X) \:, R_w = h_1(X)^{2^a d}\big(w_1 \big( \big(\frac{h_0(X)}{h_1(X)}\big)^{2^a}\big)X+ w_0 \big( \big( \frac{h_0(X)}{h_1(X)}\big)^{2^a}\big) \big)\end{array}
N\sum_{ijk}\int_x \Biggl[\tilde{A}^\mu_{ij}(x)\tilde{A}_{\mu kl}(x)\tilde{A}^\nu_{jk}(x)\tilde{A}_{\nu li}(x) W^{ijkl}+\tilde{A}^\mu_{ij}(x)\tilde{A}_{\mu jk}(x)\tilde{A}^\nu_{kl}(x)\tilde{A}_{\nu li}(x) L^{ijkl}\Biggr].
\{R_{1 1 1}^{1 1 1}, R_{1 1 2}^{1 1 2}, R_{1 2 1}^{1 2 1}, R_{1 2 2}^{1 2 2}, R_{2 1 1}^{2 1 1}, R_{2 1 2}^{2 1 2}, R_{2 2 1}^{2 2 1}, R_{2 2 2}^{2 2 2}\}
B(1, (\gamma(m)^{-1}, \gamma(m))) = (1, (1, \gamma(m)^{-1}, \gamma(m))) - (1, (1, \gamma(m), \gamma(m)^{-1})) + (1, (\gamma(m)^{-1}, 1, \gamma(m))) - (1, (\gamma(m) , 1, \gamma(m)^{-1})).
l_{c}= \int d \varphi \sqrt{g_{\varphi \varphi}}= 2 \pi \rho_{c}\left( 1 -{\frac{\sum_{i}{\hat{q}}_{i}^{2}}{2 \rho_{c}^{2}}}-{\frac{{\cal F}( 1 )^{2}}{2 A^{2}\rho_{c}^{2}}}\right) ,
\partial_{\underline{{i}}}\epsilon_{I}+{\textstyle \frac{1}{4}}\biggl ({\frac{\partial_{\underline{{i}}}{\overline{{{\cal H}}}_{2}}}{\overline{{{\cal H}}}_{2}}}-{\frac{\partial_{\underline{{i}}}{\cal H}_{2}}{{\cal H}_{2}}}\biggr ) \epsilon_{I}-{\textstyle \frac{1}{2}}\left( \partial_{\underline{{i}}}U \right) \epsilon_{I}= 0 \, .
\begin{matrix}1&\frac{1}{2}&\frac{1}{4}&\frac{1}{8}&\frac{1}{16}&\cdot\cdot\cdot\\ &\frac{1}{2}&\frac{1}{4}&\frac{1}{8}&\frac{1}{16}&\cdot\cdot\cdot\\ &&\frac{1}{4}&\frac{1}{8}&\frac{1}{16}&\cdot\cdot\cdot\\ &&&\frac{1}{8}&\frac{1}{16}&\cdot\cdot\cdot\\ &&&&\cdot\cdot\cdot&\cdot\cdot\cdot\end{matrix}
\frac{a_2(a_1+a_4)}{1-a_3}+ a_3(1-a_3) \frac{a_2^2}{(1-a_3)^2}= a_2\frac{(1-a_2-a_3)+a_2a_3}{1-a_3}&= a_2\frac{(1-a_3) - a_2(1-a_3)}{1-a_3}= a_2(1-a_2).
\|(-A_n)^{\frac{1}{2}}\exp(\!-A_n^2(t\!-\!s))\Sigma_n(U(s))\|^2_{\mathrm F}\!=\!\sum_{k,l=1}^{n-1}\langle(-A_n)^{\frac{1}{2}}\exp(-A_n^2(t\!-\!s))\Sigma_n(U(s))e_k,e_l\rangle^2 &=\sum_{l=1}^{n-1}(-\lambda_{l,n})\exp(-2\lambda^2_{l,n}(t-s))\|\Sigma_n(U(s))e_l\|^2\le C_\epsilon\sum_{l=1}^{n-1}(-\lambda_{l,n})^{-\frac{1}{2}-2\epsilon}(t-s)^{-\frac{3}{4}-\epsilon},
T_0 = T_\pi T_{n-1}T^{-1}_\pi = T_\pi (t^{-\frac12}x_n\partial_{n-1}-t^{\frac12}\partial_{n-1}x_n)T^{-1}_\pi= t^{-\frac12}q^{-1}x_1\partial_0-t^{\frac12}\partial_0 q^{-1}x_n,
[ \frac{\alpha^{2}}{4}+ ( q^{\alpha )}+ r^{\alpha )}) t^{( \alpha )}] h_{i}^{( \alpha ) 2}C_{i}+ [ \frac{\alpha^{2}}{4}+ q^{\alpha )}t^{\alpha )}- r^{( \alpha )}u^{( \alpha )}] \sum_{j \neq i}h_{j}^{( \alpha ) 2}C_{j}= C_{i}, i = 1 , . . . n
\sum_{n = 1}^{\infty}{\frac{1}{n^{2}+ \left({\frac{E}{\kappa}}\right)^{2}}}={\frac{\pi \kappa}{2 E}}\left[{\frac{2}{e^{2 \pi E / \kappa}- 1}}+ 1 -{\frac{\kappa}{\pi E}}\right] ,
N_{m}=N_{m-1}+N_{m-3}&=(6N_{m-6}+5N_{m-11}+N_{m-16})+(6N_{m-8}+5N_{m-13}+N_{m-18}) &=6(N_{m-6}+N_{m-8})+5(N_{m-11}+N_{m-13})+(N_{m-16}+N_{m-18}) &=6N_{m-5}+5N_{m-10}+N_{m-15},
\varepsilon_{a b}= \left( \begin{array}{c c}{0}&{1}\\{- 1}&{0}\\ \end{array}\right) = \varepsilon_{\dot{a}\dot{b}}, \quad \varepsilon^{a b}= - \varepsilon_{a b}, \quad \varepsilon^{\dot{a}\dot{b}}= - \varepsilon_{\dot{a}\dot{b}}.
\int_{-\infty}^{\infty}xyf(x)dx=-2(\int_{0}^{c}\frac{e^{-t^{2}/2}}{\sqrt{2\pi}}t^{2}dt-\int_{c}^{\infty}\frac{e^{-t^{2}/2}}{\sqrt{2\pi}}t^{2}dt)=1-4\int_{0}^{c}\frac{e^{-t^{2}/2}}{\sqrt{2\pi}}t^{2}dt
J_{g f}= \left( \begin{array}{c c}{{0}}&{{J_{+}}}\\{{1}}&{{0}}\\ \end{array}\right) \quad ; \quad \tilde{J}_{g f}= \left( \begin{array}{c c}{{0}}&{{1}}\\{{J_{+}}}&{{0}}\\ \end{array}\right) \quad ; \quad g = \left( \begin{array}{c c}{{- J_{+}g_{2 2}}}&{{- g_{2 1}}}\\{{g_{2 1}}}&{{g_{2 2}}}\\ \end{array}\right) .
\int_{\mathcal{V}}\psi(v) \mathcal{L}\phi(v)\,{\rm d}\mu(v)= \int_{\mathcal{V}}\phi(v) \mathcal{L}\psi(v)\,{\rm d}\mu(v)\mbox{for all}\phi,\psi\in L^2(\mathcal{V};{\rm d}\mu).
\bar{M}= \sum_{l}( l + 1 ) \partial^{( l )}\Phi^{A}\frac{\partial}{\partial ( \partial^{( l )}\Phi^{A})}+ \sum_{l}( l + 2 ) \partial^{( l )}\Phi_{A}^{*}\frac{\partial}{\partial ( \partial^{( l )}\Phi_{A}^{*})}.
\sum_{k=1}^{N}\big|a_{k}^2\big|\le C\Delta^{\frac12-2\kappa}(1+|x|_{L^{\max(p,2q)}})^{K+1}\int_{0}^{T}\frac{1}{(T-t)^{1-\kappa}}\bigl(1+\frac{1}{t^{1-\kappa}}\bigr)dt.
T_{A B}= -{\cal L}_{\mathrm{b r a n e}}G_{A B}\, + \, \frac{1}{2}\left[ ({\cal D}_{A}\phi )^{*}{\cal D}_{B}\phi + ({\cal D}_{B}\phi )^{*}{\cal D}_{A}\phi \right] - \, F_{A C}{F_{B}}^{C}.
T(p;n_1,n_2,n_3,m)= S(p,1)^3\sum_{\substack{a=1 (a,p)=1}}^pe\Big(\frac{-\overline{4a}\,(n_1^2+n_2^2+n_3^2-4m)}{p}\Big)\Big(\frac{a}{p}\Big) = S(p,1)^3\Big(\frac{-1}{p}\Big)\Big(\frac{n_1^2+n_2^2+n_3^2-4m}{p}\Big)\sum_{\substack{b=1 (b,p)=1}}^pe\Big(\frac{b}{p}\Big)\Big(\frac{b}{p}\Big).
S(\nabla) \, = \, \omega\circ \nabla \, : \, \mathrm{T}_X(-D_0) \, \longrightarrow \, \mathrm{At}_{D_0}( D-D_0) \, \longrightarrow \, E_P({\mathfrak g}/{\mathfrak p})(D-D_0) \, .
\lim_{n\to\infty}n\, x\, \{g_{\mu/n,\lambda_1}\star g_{\mu/n,\lambda_2}\}(x) = \mu\, e^{-(\lambda_1+\lambda_2)x/2}\, \sqrt{\pi(\lambda_2-\lambda_1)x}\;I_{-\tfrac{1}{2}}\left(\frac{\lambda_2-\lambda_1}{2}\, x\right) = \mu\,e^{-\lambda_1 x}+\mu\, e^{-\lambda_2 x}
r_{\ell}\leq e^{40\varepsilon q_n}\frac{e^{-q_nL}}{\max(|\ell|,1)}\max(r_{\ell-1}, r_{\ell+1})\times\begin{cases}\max(|\ell|, e^{\delta_n q_n}), \beta_n\geq \delta_n+200\varepsilon e^{\beta_n q_n}, \beta_n<\delta_n+200\varepsilon\end{cases}.
\mathcal{M}(t) =||({\mathbf V},\mathbf{F})||^2_{s,\ast,t}+(||\dot{{\mathbf U}}||^2_{W^{2,\infty}_{\ast}(\Omega_t)}+||\mathbf{F}||^2_{W^{1,\infty}_{\ast}(\Omega_t)})||\hat{W}||^2_{s+2,\ast,t}
\begin{aligned}\hat{Q}_{1}=-r,\hat{Q}_{2}=i\vartheta\frac{\partial}{\partial r}, &\hat{P}_{1}=\hbar\kappa+i\hbar\frac{\partial}{\partial r},\hat{P}_{2}=\hbar\delta+\frac{\hbar r}{\vartheta},\end{aligned}
\left\lbrace \begin{array}{cc}\Theta_1^+(x) =-\Theta_2^-(x) = \dfrac{\delta}{\left(2\sqrt{x^2+\delta^2}(\sqrt{x^2+\delta^2}-x)\right)^{1/2}}, \Theta_2^+(x) =\Theta_1^-(x) = \dfrac{\sqrt{x^2+\delta^2}-x}{\left(2\sqrt{x^2+\delta^2}(\sqrt{x^2+\delta^2}-x)\right)^{1/2}}.\end{array}\right.
|S| >(2m+1)\cdot (\frac{2n}{2k+3}-2)-\binom{2m+1}{2}\cdot \epsilon n &=\frac{2n}{2k+3}\cdot (2m+1)-2(2m+1)-(2m+1)\cdot m\cdot \epsilon n &>\frac{6k}{(2k+3)\cdot 3k+1}\cdot (2m+1)n
g_n(Ux) =g_n(\theta_\omega \theta_fx)=g_n\left(\theta_\omega\left(\sum_{m=1}^{\infty}f_m(x)e_m\right)\right) &=g_n\left(\sum_{m=1}^{\infty}f_m(x)\theta_\omega e_m\right)=g_n\left(\sum_{m=1}^{\infty}f_m(x) \omega_m\right) &=\sum_{m=1}^{\infty}f_m(x) g_n(\omega_m)=f_n(x), \forall x \in \mathcal{X},
(u_{\xi},v_{\eta})_{L^2}=0 \,\,\,\, \hbox{for}\,\,\,\, \xi\neq\eta, \,\,\,\, \hbox{and}\,\,\,\,(u_{\xi},v_{\eta})_{L^2}=1 \,\,\,\, \hbox{for}\,\,\,\, \xi=\eta,
\hat{H}=\frac{1}{2}\hat{p}^2_z+\hbar\omega\left[a_+^\dagger a_+\,+\,a_-^\dagger a_-\right]\ \ \ ,\ \ \ \hat{\phi}=\hat{p}_z+\hbar g\left[a_+^\dagger a_+\,-\,a_-^\dagger a_-\right]\ \ \ .
\lim_{n\to\infty}\int_{\mathbb{R}^N}(K\ast |u_n|^{p^\sharp}) |u_n|^{p^\sharp}dx= \lim_{n\to\infty}\int_{\mathbb{R}^N}(K\ast |u_n|^{p^\flat}) |u_n|^{p^\sharp}dx=0.
{\cal A}_{4}={}_{41i}\langle V_3| ~_{j23}\langle V_3| Y_{-2}^{(i)}Y_{-2}^{(j)}\frac{b^{(i)}_o}{L^{(i)}}WQ \frac{b^{(i)}_o}{L^{(i)}}|V_2\rangle_{ij}|\tilde{A}_{4}\rangle_{4}|\tilde{A}_{1}\rangle_{1}|\tilde{A}_{2}\rangle_{2}|\tilde{A}_{3}\rangle_{3}.
r_{0}^{6}={\frac{1}{6^{2}}}\{( t^{I}q_{I})_{\mathrm{h o r}}\}^{3}\qquad \Longrightarrow \qquad{\frac{1}{2^{1 1 / 2}}}( - q_{0})^{2}q_{1}={\frac{8 ( Q_{H}Q_{F})^{2}}{\pi^{2}}}\ ,
( \gamma_{+}t^{2}- 2 h t + \gamma_{-})^{1 / 2}( \gamma_{-}t^{2}- 2 h t + \gamma_{+})^{1 / 2}\frac{d \alpha_{l}^{\pm}}{d t}= \pm \mathrm{i}\tilde{\lambda}_{l}\alpha_{l}^{\mp}( t )
L_{11}=(\Delta')^{-1}([d,a]a^{-1}(ac^{-1}d-b)-([d,b]-[d,a]a^{-1}b))(ac^{-1}d-b)^{-1}=(\Delta')^{-1}([d,a]c^{-1}d-[d,b])(ac^{-1}d-b)^{-1}=(\Delta')^{-1}(d(a-bd^{-1}c)-\Delta' )(a-bd^{-1}c)^{-1}=((\Delta')^{-1}d-(a-bd^{-1}c)^{-1}).
\int_{g}{}_{\nu \bar{\nu}}\langle O_{2}| \biggl ( \Sigma_{i , j = 3}^{k}( g^{\dagger}\lambda^{[ 1 2 ]}g )_{j}^{i}( \nu_{i}^{3}{\bar{\nu}}^{4 j}- \nu_{i}^{4}{\bar{\nu}}^{3 j}) \biggr )^{k - 2}{| O_{k}\rangle}_{\nu \bar{\nu}}
\gamma= \frac{\mathbf{h}_\ast^\top\boldsymbol{\Sigma}^{-1}\mathbf{h}_\ast}{2\sqrt{\mathbf{h}_\ast^\top\boldsymbol{\Sigma}^{-1}\mathbf{h}_\ast+\mathbf{h}_\ast^\top\boldsymbol{\Sigma}^{-1}\boldsymbol{\Sigma}_c\boldsymbol{\Sigma}^{-1}\mathbf{h}_\ast}}
f(\alpha \mathbf{z}) = \lvert \langle\alpha \mathbf{z},\overline{\alpha}\mathbf{\overline{z}}\rangle \rvert^2 - \langle \alpha \mathbf{z},\alpha \mathbf{z}\rangle^2 & = \lvert \alpha \rvert^4 (\lvert \langle\mathbf{z},\mathbf{\overline{z}}\rangle \rvert^2 - \langle\mathbf{z},\mathbf{z}\rangle^2) & = \lvert \alpha \rvert^4 f(\mathbf{z}).
\left\{f_k(u)\left(\prod_{\scriptsize{j\ne k}\atop\scriptsize{1\le j\le \lambda}}q_j\right)(u-1)^2+(-1)^{\lambda}\left(\prod_{\scriptsize{j\ne k}\atop\scriptsize{1\le j\le \lambda}}p_j\right)u\right\}\cdot\frac{u^a-1}{u-1}\cdot\frac{u^b-1}{u-1}=\pm u^l \cdot \frac{u^c-1}{u-1}.
\Phi_n(x) = d^{2n}\sum_{b}b e_1 e_2\cdots e_{n+1}x e_n \cdots e_2 e_1 b^* &= d^{2n}\sum_{b}b e_1 e_2\cdots e_ke_{k+1}\cdots e_ne_{n+1}e_n\cdots e_{k+1}x e_k \cdots e_2 e_1 b^* &= d^{2k}\sum_{b}b e_1 e_2\cdots e_{k+1}x e_k \cdots e_2 e_1 b^* &= \Phi_k(x).
\|b_\varepsilon \varphi\|_2^2 \leq \langle E_\varepsilon |b|^2,\varphi^2\rangle=\|b\sqrt{E_\varepsilon \varphi^2}\|^2_2 &\leq \delta\|\nabla\sqrt{E_\varepsilon \varphi^2}\|_2^2+c_\delta\|\varphi\|_2^2, \varphi \in W^{1,2},
|\mathcal{T}(T)\rangle_{\alpha_{-1}\cdot \tilde{\alpha}_{-1}\tilde{c}_1 c_1 |0\rangle}\sim\frac{1}{T^2}\,\, U_{f(v)}^{\dagger}c(i)\partial X^{\mu}(i)\,\tilde{c}(i) \bar{\partial}X_{\mu}(i) |0\rangle,
\kappa_2=\frac{25}{64},\lambda_2=\frac 14;\kappa_3=\frac{49}{64},\lambda_3=\frac 14;\kappa_4=\frac{81}{64},\lambda_4=\frac 14; \kappa_5=\frac{121}{64},\lambda_5=\frac 14;\kappa_6=\frac{169}{64},\lambda_6=\frac 14;\quad\kappa_7=\frac{225}{64},\lambda_7=\frac 14.
\|g_k^{j,t}\|^2_{L^2(C^{j,t}_k, d\mu_2)}\leq \frac{C}{\omega(B^g(2^k\sqrt{t}))}\int_{B^g(x^t_j, 2^{k+2}\sqrt{t})}\int_{B^g(x^t_j, 2^{k+2}\sqrt{t})}|f(x)-f(y) |^2 d\mu_2(x)d\mu_2(y).
S= -i \left[ \int d^2 \sigma d\theta_0^+ d\theta_0^- g_{ij}D_{0+}\varphi^i D_{0-}\varphi^j + \int d^2 \sigma dt d\theta_0^+ d\theta_0^-H_{ijk}\partial_t D_{0+}\varphi^j D_{0-}\varphi^k \right] ,