harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Gamma^{(3)}=\setlength{\unitlength}{1pt}\begin{picture}(40,24)(-18,-3)\thicklines\put(0,0){\circle{24}}\put(12,0){\line(-5,3){18.174}}\put(12,0){\line(-5,-3){18.174}}\put(-6.174,10.290){\line(0,-1){20.58}}\end{picture},
\partial_\alpha h^\alpha{}_\mu-\frac{1}{2}\partial_\mu h^\alpha{}_\alpha=\frac{1}{2}\exp\!\left(-\frac{\lambda_3}{3m_4^3}|x^\perp|\right)\partial_\mu h_{\perp\perp}-\partial_\perp\left[\exp\!\left(-\frac{\lambda_3}{3m_4^3}|x^\perp|\right) h_{\perp\mu}\right]
\sum_{n=1}^{\infty}\frac{\Lambda_{\pi}(n)}{n^s}=-\frac{L'}{L}(s,\pi),\sum_{n=1}^{\infty}\frac{\Lambda_{\pi\otimes\chi}(n)}{n^s}=-\frac{L'}{L}(s,\pi\otimes\chi),\mathrm{Re}(s)>1.
\lim_{l\rightarrow \infty}\, \lim_{t\rightarrow \infty}\ \ \sup_{\xi\geq 0 , u\in K}\Biggl[ \frac{1}{l}\int_{\xi}^{\xi+l}\bigl \| q(t+s,u)\bigr\|^{p}\, ds\Biggr]^{1/p}=0.
\mathbb{P}\left(\displaystyle\sum_{j=1}^{(N-1)\wedge k}\mid M_j^{(N)}-M_j^{(\infty)}\mid <\varepsilon\right)>1-\delta, \mathbb{P}\left(\displaystyle\sum_{j=1}^{(N-1)\wedge k}\mid Y_j^{(N)}-Y_j^{(\infty)}\mid <\varepsilon\right)>1-\delta,
G^{(4)}_{\mathcal R}(x,y)= &-\frac{i}{2\pi}\left[\int_{\theta_c}^{\theta_{\hat x}}+ \int_{\mathcal D_{\theta_{\hat x}, \frac\pi2+\theta_{\hat x}-i\infty}}\right] \frac{2i\sin\zeta\widetilde{\mathcal S}(\cos\zeta,n)}{n^2-1}{e^{ik_{+}\left(-\vert y \vert\cos(\zeta+\theta_{\hat y})+ \vert x \vert\cos(\zeta-\theta_{\hat x})\right)}}d\zeta,
_{m,k}= \log\left(1+ \frac{\frac{\alpha_0^2|h_{mk}|^2}{L\left(||y_{m,k}+x_m||\right)}}{\sum^{M_s}_{l=1}\frac{\alpha_l^2|h_{mk}|^2}{L\left(||y_{m,k}+x_m||\right)}+\frac{1}{\rho}}\right),
7{\frac{R^{6}}{N}}( \partial_{\rho}R )^{2}- 2 \partial_{\rho}({\frac{R^{7}\partial_{\rho}R}{N}}) + 7 N R^{6}-{\frac{9}{1 4 4}}{\frac{R^{8}}{N}}( \partial_{\rho}\phi )^{2}+ 9 ({\frac{Q}{\pi^{5}}})^{2}{\frac{N e^{- 2 \phi}}{R^{1 0}}}= 0 .
\sum_{i=0}^{d-1}I_{\mathrm{bl}}^{\mathfrak{m}}(l_{i+1},\dots,l_{d},l_{1},\dots,l_{i})=\begin{cases}I_{\mathrm{bl}}^{\mathfrak{m}}(l_{1}+\cdots+l_{d}) d:\,{\rm odd}0 d:\,{\rm even}.\end{cases}
\lim_{k\to\infty}\nabla g(\overline{x})\nabla\Xi(\overline{y})\frac{\mu^k-\overline{\mu}}{t_k}=w-\nabla g(\overline{x})\nabla\big(\nabla\Xi(\cdot)\overline{\mu}\big)(\overline{y})(g'(\overline{x})\xi).
b=\frac{1}{2}(r-\frac{1}{2}\sigma_{G}^{2}),\sigma_{G}=\frac{\sigma}{\sqrt{3}},d_{1}=\frac{log\frac{S_{0}}{K}+(b+\frac{1}{2}\sigma_{G}^{2})T}{\sigma_{G}\sqrt{T}},d_{2}=d_{1}-\sigma_{G}\sqrt{T}
\psi^*\left(\begin{array}{c c}a b c d \end{array}\right) = \left(\begin{array}{c c c}a^{2 \, p^{e_1}}a^{p^{e_1}}\, b^{p^{e_1}}b^{2 \, p^{e_1}}0 1 0 c^{2 \, p^{e_1}}c^{p^{e_1}}\, d^{p^{e_1}}d^{2 \, p^{e_1}}\end{array}\right) .
\begin{array}{ll}p(t,x;s,y)=p^*(s,y;t,x),~p_{,i}(t,x;s,y)=p^*_{,i}(s,y;t,x), ~p_{,i,j}(t,x;s,y)=p^*_{,i,j}(s,y;t,x).\end{array}
i(v) \psi =i (\phi \iota_{*}Y) i \left(r \frac{\partial}{\partial r}\right) \Omega |_{r=1}&=i (J \iota_{*}Y) i \left(r \frac{\partial}{\partial r}\right) \Omega |_{r=1}&=i \cdot i (\iota_{*}Y) i \left(r \frac{\partial}{\partial r}\right) \Omega |_{r=1}=i \cdot i (\iota_{*}Y) \psi.
\left({\cal X}{\bar H}-E_i\right)|i\,\rangle=\left[\Xi^{\dagger}{\cal A}^{\dagger}H_{\mbox{\scriptsize I}}+\left({\cal X}{\bar H}-E_i\right)\left(1-\Xi^{\dagger}{\cal A}^{\dagger}\right)\right]|i\,\rangle\,.
D_{\mathrm{G}}^{2}[ U , U^{\prime}] = \sum_{x , i}\sum_{\alpha = 0}^{3}\left[ ( ( u_{\alpha})_{x , i}- ( u_{\alpha}^{\prime})_{x , i})^{2}+ ( ( \dot{u}_{\alpha})_{x , i}- ( \dot{u}_{\alpha}^{\prime})_{x , i})^{2}\right] ,
\begin{array}{lccc}\overline{P_0}^G = 0 \overline{Q_0}^G = s_2^2 \overline{P_1}^G = -s_1 + s_2^2 \overline{Q_1}^G = s_2^2 \overline{P_{-1}}^G = s_1 \overline{Q_{-1}}^G = s_2^2 \end{array}
\gamma_{r}^{\pm}= \oint \frac{d z}{2 \pi i}z^{r - 3 / 2}\gamma^{\pm}( z ) , \hspace{1 c m}d_{r}^{\pm \mu}= \oint \frac{d z}{2 \pi i}z^{r - 1 / 2}i \psi^{\pm \mu}( z )
\alpha \int_{\mathbb{R}^3}\mathrm{d}k \; f_y(k) \, \bigg\langle x u_{\alpha}\,\bigg|\, \frac{(h_{\alpha}-e_{\alpha})^2}{(h_{\alpha}-e_{\alpha}+|k|)}\,\bigg|\, x u_{\alpha}\bigg\rangle = O\Big( \frac{\alpha^4}{L}\Big).
\begin{array}{r r}{( 8 , 1 , 1 ; 1 , 1 , 8 ) ,}&{({\bar{8}}, 1 , 1 ; 1 , 8 , 1 )}\\{( 1 , 8 , 1 ;{\bar{8}}, 1 , 1 ) ,}&{( 1 , 1 , 8 ; 8 , 1 , 1 )}\\ \end{array}
\gamma =\gamma_1^s\oplus\gamma_1^\ell\oplus\gamma_2^s\oplus\gamma_2^\ell\oplus\ldots\oplus\gamma_{J}^s\oplus\gamma_{J}^\ell\oplus\gamma_{J+1}^s, \sigma =\sigma_1^s\oplus\sigma_1^\ell\oplus\sigma_2^s\oplus\sigma_2^\ell\oplus\ldots\oplus\sigma_{J}^s\oplus\sigma_{J}^\ell\oplus\sigma_{J+1}^s.
\frac{d}{dt}\bigg|_{t=0}\int_M \sqrt{1+|\alpha(t)|^2}dV_g = \int_M \frac{\frac{1}{2}\frac{d}{dt}|\alpha(t)|^2}{\sqrt{1+|\alpha_0|^2}}dV_g & = \left< \frac{\alpha_0}{\sqrt{1+|\alpha_0|^2}}, d\psi \right>.
{\bf B}^{\mathrm{l i n .}}= U{\bf B}^{\mathrm{c i r c .}}= B^{( 0 )}\left[ \left( \begin{matrix}{{1}}\\{{0}}\\{{0}}\\ \end{matrix}\right) e^{+ i \phi}+ \left( \begin{matrix}{{0}}\\{{1}}\\{{0}}\\ \end{matrix}\right) e^{- i \phi}\right] \ .
V(x,y,t)\left[\left(\frac{v(x,t)}{v(y,t)}\right)^{p-1}\phi(x,t)-\left(\frac{v(y,t)}{v(x,t)}\right)^{p-1}\phi(y,t)\right] begin{align*}3mm] \geq \left|v(x,t)-v(y,t)\right|^{p-2}\left(v(y,t)-v(x,t)\right)\left(\phi(y,t)-\phi(x,t)\right) begin{align*}3mm] =V(x,y,t)\left(\phi(x,t)-\phi(y,t)\right)
\mathrm{T r ~}[ ( D^{2}+ B + m^{2})^{- 1}- ( P^{2}+ m^{2})^{- 1}] = \int_{0}^{\infty}\, d t e^{- t m^{2}}\int \, d^{2}r < \!{\bf r}\, | \, e^{- ( D^{2}+ B ) t}- e^{- P^{2}t}\, | \,{\bf r}\! > \, .
xK_{n}^{p,N}\left( x\right) =K_{n+1}^{p,N}\left( x\right) +\alpha_{n}^{p,N}K_{n}^{p,N}\left( x\right) +\beta_{n}^{p,N}K_{n-1}^{p,N}\left(x\right) ,n\geq 0,
B(k) := \textup{\textbf{w}}\big( p \otimes s(k) \big) \textup{\textbf{w}}^{\ast}+M \big( z(k) \big) \textup{\textbf{w}}p \otimes \mathcal{R}(k^{2}) \textup{\textbf{w}}^{\ast}& + \Big( 1 + M \big( z(k) \big) \Big) \textup{\textbf{w}}R_{1}\big( z(k) \big) Q \textup{\textbf{w}}^{\ast},
\begin{array}{lcr}\{X^1,X^3\}+\{X^4,X^2\}+\{X^5,X^7\}+\{X^8,X^6\}=0 \{X^1,X^4\}+\{X^2,X^3\}+\{X^8,X^5\}+\{X^7,X^6\}=0 \{X^1,X^5\}+\{X^4,X^8\}+\{X^7,X^3\}+\{X^6,X^2\}=0 \{X^1,X^6\}+\{X^2,X^5\}+\{X^3,X^8\}+\{X^4,X^7\}=0 \{X^1,X^7\}+\{X^3,X^5\}+\{X^8,X^2\}+\{X^6,X^4\}=0 \{X^1,X^8\}+\{X^5,X^4\}+\{X^2,X^7\}+\{X^6,X^3\}=0 .\end{array}
\begin{aligned}\|O^b_2\|_{L^1_{t,x}}\lesssim \|(w^{(c)}_{q+1}, w^{(t)}_{q+1}, d^{(c)}_{q+1}, d^{(t)}_{q+1})\|_{L^2_{t,x}}\|(d^{(p)}_{q+1}, d_{q+1}, w^{(p)}_{q+1}, w_{q+1})\|_{L^2_{t,x}}\lesssim \lambda^{C_0}_q\lambda^{\gamma-\frac{1}{2}}_{q+1}+\lambda^{C_0}_q\lambda^{-\iota}_{q+1}.\end{aligned}
J^{\pm}_{it}(x/2) = \Re \left( e^{i \tilde{\omega}(x, t)}\tilde{f}_M^{\pm}(x, t)\right) + O(|t|^{-M}), \tilde{\omega}(x, t) = |t| \cdot \frac{|t|}{x}- \sqrt{t^2 + x^2},
M(a,k,c):= \int_0^{2\pi/k}u(z;a,k,c)~dz=2\pi-\frac{\pi}{k}(\sqrt{c^2-4a}+c)\intertext{and}P(a,k,c):= \frac{1}{2}\int_0^{2\pi/k}u^2(z;a,k,c)~dz=-c\pi+\frac{\pi}{4k}(\sqrt{c^2-4a}+c)^2.
V'_{m}(\phi) = \sum_{n=1}^{\infty}v^{(n)}_{m m_{2}\ldots m_{n}}\phi_{m_{2}}\ldots \phi_{m_{n}}, \;\;\;\;\;\; \tilde{V}'_{m}(\tilde{\phi}) = \sum_{n=1}^{\infty}v^{(n)}_{m m_{2}\ldots m_{n}}\tilde{\phi}_{m_{n}}\ldots \tilde{\phi}_{m_{2}}
r_1 = \sup_{(0,T] \times D_R}|u(t,x)|, r_2 = \sup_{(0,T] \times D_R}|q(t,x)|, C= \sup_{(0,T] \times D_R}|c(t,x)|, L= \sup_{(0,T] \times D_R}|\gamma(t,x)|.
\|K'(u_n)-K'(u)\|_{W_0^{s,p}(M)^{*}}= \sup_{v\in W_0^{s,p}(\Omega),\, \|v\|_{W_0^{s,p}(\Omega)}\leq1}\|\langle K'(u_n)-K'(u),v\rangle\| & \leq \|f(x,u_n)-f(x,u)\|_{L^{q'}(\Omega)}\|v\|_{L^q(\Omega)}\leq \|f(x,u_n)-f(x,u)\|_{L^{q'}(\Omega)}\to 0,
D_B\big(F_{i,j}\big):=\det\left(\begin{array}{ccc}F^{(0)}_{i-\mu,j}F^{(1)}_{i-\mu,j}F^{(2)}_{i-\mu,j}Z_0 Z_1 Z_2 A_{0,\mu_1}A_{1,\mu_1}A_{2,\mu_1}\end{array}\right).
\frac{1}{2\pi}\int_{-\pi}^{\pi}P_n(e^{\imath \theta};\alpha, \beta)Q_m(e^{-\imath \theta};\alpha, \beta)\omega(\theta)d\theta= \frac{\Gamma(2\alpha+1)}{\Gamma(\alpha+\beta+1)\Gamma(\alpha-\beta+1)}\frac{n!}{(2\alpha+1)_n}\delta_{n,m},
(\lambda_j-\lambda_k)\Gamma_{ij}^k +(\lambda_k-\lambda_i)\Gamma_{jk}^i+(\lambda_i-\lambda_j)\Gamma_{ki}^j =-(\nabla_{e_i}\lambda_j)\delta_{jk}-(\nabla_{e_j}\lambda_k)\delta_{ki}-(\nabla_{e_k}\lambda_i)\delta_{ij}.
- i g \frac{\tau^{\gamma}}{2}A_{{\sf G I}i}^{\gamma}({\bf{r}}) = - i g V_{C}({\bf{r}}) \frac{\tau^{\gamma}}{2}A_{i}^{\gamma}({\bf{r}}) V_{C}^{- 1}({\bf{r}}) + V_{C}({\bf{r}}){\partial}_{i}V_{C}^{- 1}({\bf{r}}) ,
{\mathbb P}_{0}\Big( \int^{t}_{0}\big \lvert{\mathbb E}_{0}\big[ \lvert b ( s, X_{s}, F_{s})\rvert \, \big \vert \, \mathcal F_{T}^{X}\big] \big \rvert^{2}{\mathrm d}s < \infty \Big) \, =\, 1 \, ; 0 \le t \le T \, .
\Omega(s) =\int_{0}^{\infty}\Big(\int_{\Gamma_{0}(N) \backslash H}\overline{f(x+iy)}\Theta(x+iy,g_t)\frac{dx dy}{y^{2}}\Big)t^{s}dt/t &=\int_{\Gamma_{0}(4k) \backslash H}\overline{f(x+iy)}\Big( \int_{0}^{\infty}\Theta(x+iy,g_t) t^{s}dt/t \Big) \frac{dx dy}{y^{2}}.
(a,b,c,\lambda_1(a,b,c),\lambda_2(a,b,c),\lambda_3(a,b,c))^h = (a^3,b^3,c^3,\lambda_1(a^3,b^3,c^3),\lambda_2(a^3,b^3,c^3),\lambda_3(a^3,b^3,c^3))
q^{\frac{1}{2}}(q^{\frac{1}{2}}X_2+h)(q^{\frac{1}{2}}X_3+h)= q^2X(-1,3)+(q+q^2)hX(-1,2) +qhX(0,1) &+(qh^2+q^2)X(-1,1)+qhX(-1,0)+q^{\frac{1}{2}}h^2.
\begin{array}{l l}{\begin{array}{l l}{\frac{d}{d r}\sigma ( r ) = 0 ,}&{r = 0 ,}\\{\sigma ( r ) = 0 ,}&{r \rightarrow \infty ,}\\ \end{array}}&{\begin{array}{l l}{A ( r ) \neq 0 ,}&{r \rightarrow \infty ,}\\{A ( r ) = 1 ,}&{r = 0 .}\\ \end{array}}\\ \end{array}
(11)(g_{j}H,g_{\zeta_{j}(j)}H,g_{\zeta_{j}^{2}(j)}H,...,g_{\zeta_{j}^{f_{j}-1}(j)}H)=(g_{j}H,xg_{j}H,x^{2}g_{j}H,...,x^{f_{j}-1}g_{j}H),
x_{+}(t).e_{-2r}= \left(\sum_{j = 0}^{r-1}\binom{r-1}{j}t^{r-1-j}v_{1}^{p-j-2}v_{2}^{j}\right) \otimes \left(f_{-1}+ t^{p}f_{1}\right) &= \left( \sum_{s = 1}^{r}\binom{r-1}{s-1}t^{r-s}e_{-2s}\right)+ \left( \sum_{s = p-r}^{p-1}\binom{r-1}{s+r-p}t^{r+s}e_{2s}\right)
\langle u(t),\varphi \rangle - \langle f,\varphi\rangle & + \mu\langle \int_0^t u ds,\varphi \rangle + \bigl\langle b \cdot \nabla \int_0^t uds,\varphi \bigr\rangle - \sigma \bigl\langle \int_0^t u dB_s,\nabla \varphi \bigr\rangle + \frac{\sigma^2}{2}\bigl\langle \nabla \int_0^t u ds, \nabla \varphi\bigr\rangle=0.
\rho=\frac{i\hbar}{2mc^{2}}(\psi^{*}\frac{\partial\psi}{\partial t}-\psi\frac{\partial\psi^{*}}{\partial t}),j=-\frac{i\hbar}{2m}(\psi^{*}\nabla\psi-\psi\nabla\psi^{*})\rightleftharpoons J^{\mu}=\frac{i\hbar}{2m}(\psi^{*}\partial^{\mu}\psi-\psi\partial^{\mu}\psi^{*})
\int_{a}^{b}\omega(x)r(x)dx=\int_{a}^{b}\omega(x)\sum_{i=1}^{n}l_{i}(x)r(x_{i})dx=\sum_{i=1}^{n}r(x_{i})\int_{a}^{b}\omega(x)l_{i}(x)dx=\sum_{i=1}^{n}r(x_{i})w_{i},
\tilde{B}_{d-1,i}=\frac{G_{d-1,i}}{G_{d-1,d-1}},\;\;\; \tilde{B}_{ij}= B_{ij}+ \frac{G_{i,d-1}B_{d-1,j}+ B_{i,d-1}G_{d-1,j}}{G_{d-1,d-1}}.
B \, = \, m + \frac{e^{2}}{2 \pi}\frac{\lambda + 1}{\lambda}\int_{0}^{\infty}d k \frac{k B}{A^{2}k^{2}+ B^{2}}\frac{1}{\sqrt{( p^{2}+ k^{2}+ \mu^{2})^{2}- 4 p^{2}k^{2}}}
\tilde\phi(\lambda)\sim i\sqrt{\frac{4\lambda}{\pi}}e^{-2\lambda}K_{\nu}(-2\lambda)\sim\sum_{n\geq 0}(-1)^n\frac{\Gamma(\nu+n+1/2)}{4^nn!\Gamma(\nu-n+1/2)}\lambda^{-n},\ \ \lambda\rightarrow\infty,
\prod_{b\in \lambda}\dfrac{1-q^{coarm_\lambda(b)}t^{n-coleg_\lambda(b)}}{1-q^{arm_\lambda(b)}t^{leg_\lambda(b)+1}}=\prod_{1\leq i < j \leq n}\prod_{r=0}^{\lambda_i-\lambda_j-1}\dfrac{1-q^rt^{j-i+1}}{1-q^rt^{j-i}}.
e(\bar R_{31}) =\left(\frac{(q^2-1)^r}2+\frac16 (q-1)^{2r}+ \frac13 (q^2+q+1)^r - (q-1)^r \right) q^6 + \left(- \frac{(q^2-1)^r}2+\frac16 (q-1)^{2r}+ \frac13 (q^2+q+1)^r\right) q^3 + \left(\frac13 (q-1)^{2r}-\frac13 (q^2+q+1)^r -(q-1)^r+1\right) (q^5+q^4).
\Gamma_{c . t .}^{\prime}= -{\frac{i}{{( 2 \pi )}^{3}}}\int B_{0}^{( 9 )}X_{6}^{( 9 )}+{\frac{i}{{( 2 \pi )}^{3}}}\int B_{0}^{( 5 )}X_{6}^{( 5 )},
\; + \; \frac{1}{2\pi}m c\int_{\Lambda}d^2 x \;\left[ e^{+ 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi+g}}\big( \delta(x), C_{m_d}\alpha \big) + i\theta}\; + \;e^{- 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi+g}}\big( \delta(x), C_{m_d}\alpha \big) - i\theta}\right]
\mathcal{H}_{[\cdot]}(\varepsilon,\mathcal{S}^{\mathcal{B}}_{(K,J,\mathbf{R})}, d_H^{\otimes_n}) \leq \mathcal{H}_{[\cdot]}(\varepsilon,\Pi_K, d_H^{\otimes_n}) + \sum_{k=1}^K \mathcal{H}_{[\cdot]}(\varepsilon,\mathcal{F}_{(J,\mathbf{R}_k)}, d_H^{\otimes_n}),
\Delta(X,K_X+\Lambda) \ge \Delta(X_1,(K_X+\Lambda)|_{X_1})+\Delta(X_2,(K_X+\Lambda)|_{X_2}) &+\max\{d_{X_1\to C}((K_X+\Lambda)|_{X_1}),d_{X_2\to C}((K_X+\Lambda)|_{X_2})\}-2.
\widetilde \Phi(z;r) = \widetilde E(r) \Phi(rz;r\vec{x},\vec{s}), \widetilde E(r) = \begin{pmatrix}1 0 \frac{i}{\sqrt{r}}\Phi_{1,12}(r\vec{x},\vec{s}) 1\end{pmatrix}e^{\frac{\pi i}{4}\sigma_{3}}r^{\frac{\sigma_{3}}{4}},
[w(\tau_0)]^{-1}\cdot \left|\frac{\partial^{n-m}}{\partial \upsilon_j^{n-m}}w(\upsilon+\tau_0)\right| \leq D_{n-m}\cdot [v_0(\upsilon)]^{d}\leq \left(\max_{0 \leq m \leq n}D_{n-m}\right) \cdot [v_0 (\upsilon)]^d =: \widetilde{D}_{n}\cdot [v_0 (\upsilon)]^d ,
= t r_{1 2}( K_{1}^{t_{1}}K_{2}\tilde{B}_{1 2}^{t_{2}}B_{1 2}^{t_{1}}M_{1}^{t_{1}}M_{2}) = t r_{1 2}[{( K_{1}^{t_{1}}\tilde{B}_{1 2}K_{2}^{t_{2}})}^{t_{2}}{( M_{1}B_{1 2}M_{2})}^{t_{1}}] =
\left[ \frac{1}{\pi}\begin{pmatrix}1 0 0 \pi \end{pmatrix}, \begin{pmatrix}1 0 0 \pi \end{pmatrix}, \begin{pmatrix}1 0 0 1 \end{pmatrix}, \begin{pmatrix}1 0 0 1 \end{pmatrix}\right]
S = \int d^{4}x \sqrt{- \hat{g}}e^{\psi}\left( \hat{R}- \omega ( \hat{\nabla}\phi )^{2}\right) + \sum_{i}m_{i}\int \left( - \hat{g}_{\mu \nu}d x_{i}^{\mu}d x_{i}^{\nu}\right)^{\frac{1}{2}},
P^{n+i}_{\mathbb{I}}=\sum_{k=1}^n\left(\frac{\partial P_\mathbb{I}^{n+ik}}{\partial x^k}+\frac{\partial P_\mathbb{I}^{n+in+k}}{\partial x^{n+k}}\right)=-\frac{1}{4}\sum_{k=1}^n\left(\frac{\partial P^{ik}}{\partial z^k}+\frac{\partial \overline{P^{ik}}}{\partial \overline{z}^k}\right).
{\Omega}^{u}_{ijk}=\frac{{\partial}{\omega}^{u}_{jk}}{{\partial}{\xi}^{i}}+ \frac{{\partial}{\omega}^{u}_{ki}}{{\partial}{\xi}^{j}}+ \frac{{\partial}{\omega}^{u}_{ij}}{{\partial}{\xi}^{k}}.
\|f\|_{L^2}^2=\|D_E\mathfrak{w}\|_{L^2}^2=\Bigg\| D_E \Bigg[\left(\begin{array}{c}w^+(t)\sqrt{z}w^-(t)\sqrt{\bar{z}}\end{array}\right)\chi+\mathfrak{w}_{\mathfrak{R}}\Bigg] \Bigg\|_{L^2}^2.
K(h^{\frac{1}{2^{n}}},2)^{r_{n}}H_{\nu}(A, B) \leqslant A \nabla B-\sum_{k=0}^{n-1}r_{k}[ H_{\frac{m_k}{2^k}}(A,B) -2H_{\frac{2m_k+1}{2^{k+1}}}(A,B) + H_{{\frac{m_k+1}{2^k}}}(A,B)] & \leqslant K(h^{\frac{1}{2^{n}}},2)^{R_{n}}H_{\nu}(A, B)
(\cdots)_{t=T}'\ll(n_1+n_2)\frac{(\log T)^\alpha}{T^{\beta+2}}+\frac{(\log T)^{\alpha-1}}{T^{\beta+1}}+\frac{(\log T)^\alpha}{T^{\beta+1}}\ll\frac{(\log T)^\alpha}{T^{\beta+1}}, &(\cdots)''\ll(n_1+n_2)\frac{(\log t)^\alpha}{t^{\beta+3}}+(n_1^2+n_2^2)\frac{(\log t)^\alpha}{t^{\beta+4}}+n_1n_2\frac{(\log t)^\alpha}{t^{\beta+4}}\ll\frac{(\log t)^\alpha}{t^{\beta+2}}.
\mathbf{\theta}[\diamond_i, q]= \mathbf{A}[\diamond_i, q] + \sum_{*}{\alpha_{* \diamond_i}\mathbf{A}[*, q]}\mathbf{\theta}[\diamond_j, q]= \mathbf{A}[\diamond_j, q] + \sum_{*}{\alpha_{* \diamond_j}\mathbf{A}[*, q]}.
L_{k-1, l}-L_{k, l}=\int_X -\chi(\varphi)dd^c(\varphi)\wedge dd^c(u_3-u_1) \wedge S_{k-1,l}&=\int_X -\chi(\varphi) dd^c(u_3-u_1) \wedge T_{k,l}-\int_X -\chi(\varphi) dd^c(u_3-u_1) \wedge T_{k-1,l+1}&=\int_X\chi'(\varphi) d\varphi\wedge d^c(u_3-u_1) \wedge T_{k,l}-\int_X\chi'(\varphi) d\varphi\wedge d^c(u_3-u_1) \wedge T_{k-1,l+1}
M^{6}r_{c}^{4}\sim \frac{M_{s}^{2}}{T_{2}}E_{2}( U , \bar{U}) = \frac{M_{s}^{2}}{T_{2}}\sum_{( m , n ) \in ( Z , Z ) - ( 0 , 0 )}\frac{U_{2}^{2}}{| m + n U |^{4}}\, .
\mathcal{V}=\frac{\log|\zeta(\tfrac{1}{2}+i(\tau+h))|}{\mathfrak{s}}\mathcal{W}=\frac{\log|\zeta(s_0)|}{\mathfrak{s}}\mathcal{V}' =\frac{\log|\zeta(\tfrac{1}{2}+i(\tau+h'))|}{\mathfrak{s}}\mathcal{W}' =\frac{\log|\zeta(s_0')|}{\mathfrak{s}}.
\sigma_t^1(m)=\sigma_x^{-1}[\dot s_t(m)],\sigma_t^2(m)=\left[s_t(m),\Gamma_0\!\cdot\!j^1s_t(m)\right]_G,U_t=\dot\varphi_t\circ\varphi_t^{-1},\quad\quad\Gamma_t=(\varphi_t)_*\Gamma_0.
\|\nabla^k g\|_{L^\infty(\Omega\cap A_j)}= \sup_{x\in \Omega \cap A_j}\left| \int_{\operatorname{supp}\eta}\partial_y^m \nabla_x^kG(x, y) \eta(y) \, dy \right| \le \sup_{|x - y| \ge \frac34 d_{j-1}}|\partial_y^m\nabla_x^kG(x, y)| &\le C(1 + d_j^{2-m-k-N}),
\Psi(x) \,= \,\sum_{\sigma=0,\pm 1}\int{d^3p\over (2\pi)^{3}}{1\over 2\,\omega_{\vec p}}\Big[ u_\sigma(\vec p\,)\, a_\sigma(\vec p\, )\, e^{-i p\cdot x}+ v_\sigma(\vec p\,) \,b^\dagger_\sigma(\vec p\,) \,e^{i p\cdot x}\Bigr ]\quad,
\epsilon_{\nu}^{1-2\beta'}\frac{2^{\nu}B_{\nu}}{\alpha_{0}}\leq(\epsilon_{0}\prod_{\mu=0}^{\infty}(\frac{2^{\mu}B_{\mu}}{\alpha_{0}})^{\frac{1}{3\kappa^{\mu+1}}})^{\kappa^{\nu}(1-2\beta')}\leq(\frac{\gamma_0}{80})^{\kappa^{\nu}},
\mu_{\partial B_1}=\det h|_{\partial B_1}\left(\left\{\frac{\partial}{\partial z_1},\dots,\frac{\partial}{\partial z_{n-1}},\frac{\partial}{\partial \bar{z}_1},\dots,\frac{\partial}{\partial \bar{z}_{n-1}}\right\}\right)^{\frac{1}{2}}dz_1\wedge\dots\wedge dz_{n-1}\wedge d\bar{z}_1\wedge\dots\wedge d\bar{z}_{n-1}
\varepsilon_{k+1}= \frac{p-1}{p}\sigma^{-\frac{1}{p-1}}\frac{(A_{k+1}- A_k)^{\frac{p}{p-1}}}{\delta}\|\nabla f(y_{k+1})\|^{\frac{p}{p-1}}+ \frac{A_{k+1}}{\delta}\langle \nabla f(y_{k+1}), y_{k+1}- x_{k+1}\rangle.
\hphantom{=}\,\lambda_1(\Omega)\int_\Omega \phi_1(x)^2\,dx-\int_\Omega\sigma(x)\phi_1(x)^2\,dx = \,\int_0^\pi \sin^2 x\,dx -\varepsilon^{-29/10}\int_0^\varepsilon \sin^2 x\,dx = \,\frac{\pi}{2}-\frac{\varepsilon^{-29/10}}{2}\big( \varepsilon-\sin\varepsilon\cos\varepsilon\big)\,.
B^{(\theta)}f = \sum_{I\in \mathcal{D}}\frac{\langle f, h_I\rangle}{\|h_I\|_2^2}b_I^{(\theta)}\qquad A^{(\theta)}f = \sum_{I\in \mathcal{D}}\frac{\langle f, b_I^{(\theta)}\rangle}{\|b_I^{(\theta)}\|_2^2}h_I
\sum_{p=1}^{\infty}(-1)^{n}Q_{n,j,k}(p,z)t^{p}=\frac{(-1)^{n}A(t,z)}{(1-t)^{n+1}}=\frac{(-1)^{n}t^{n+1}A(1/t,-z)}{(1-t)^{n+1}}=\frac{\bar{A}(t,-z)}{(1-t)^{n+1}}=\sum_{p=1}^{\infty}Q_{n,j,k}(-p,-z)t^{p}.
\mathsf{E}\left[\overline{X}{\bf 1}_{[\overline{Y}\geq a]}\right] = \mathcal{K}_\gamma(0) \cdot \mathsf{P}[\overline{Y}\geq a] + \frac{1}{2\pi i}\int_{\tau-i\infty}^{\tau+i\infty}\frac{\mathcal{K}_\gamma(\eta) - \mathcal{K}_\gamma(0)}{\ \eta}\exp \left[ n \left( \mathcal{K}_{Y}(\eta) - a \eta \right)\right] d \eta.
B_{0}(\rho_0,\rho_1) =\ \tau_1(1+\alpha\frac{\rho_0}{2})\rho_1\lambda_1^{-1}\langle\sin^2\theta h_0^-\rangle_{M_{\lambda_1}}- \tau_1\alpha \frac{c_0\rho_0}{2}\rho_1\lambda_1^{-1}\langle\sin^2\theta\cos\theta h_0^-\rangle_{M_{\lambda_1}}- \tau_0\alpha \frac{c_1\rho_1}{2}\rho_0\langle\sin^2\theta h_0\rangle_{M_{\lambda_0}}.
\exp\left(-\sum^\infty_{n=1}\frac{a_n}{n}z^{-n}\right)|\lambda^{sp}\rangle=(Y_{-\lambda_1}-z^{-1}Y_{-\lambda_1+1})\cdots (Y_{-\lambda_{n+1}}-z^{-1}Y_{-\lambda_{n+1}+1})|0\rangle.
J_1 = D_{u}v(x,G_n(hx,0))- D_{u}v(x,G_n(hx,z)), J_2 = D_{s}v(x,G_n(hx,0)) \D_uG_n(hx,0) \Dh(x) - D_{s}v(x,G_n(hx,z)) \D_uG_n(hx, z) \Dh(x), J_3 = D_{s}v(x,G_n(hx,z)) \D_sG_n(hx, z) \D_{u}G_m(\ell x,0) \D\ell(x).
v_2\left( n!^{2+j}\prod_{k=1}^{n}\left((k-1)!(n-k)!\right)^{i_k}\right) \geqslant (2+j)(n-m) + (i_1+\cdots+i_n)(n-2m+2) - i_{*}= (2+j)(m-2) + (i_1+\cdots+i_n+j+2)(n-2m+2)-i_{*}\geqslant 2(m-2) + (s+2)(n-2m+2) - i_{*}.
J_{q}( L , r ) = \sum q^{X_{r}( L )}\prod_{i = 1}^{\nu - 2}\left[ \begin{array}{c}{{n_{i}+ m_{i}}}\\{{n_{i}}}\\ \end{array}\right]_{q}= \sum q^{X_{r}( L )}\prod_{i = 1}^{\nu - 2}\left[ \begin{array}{c}{{n_{i}+ \tilde{n}_{i}- V_{i , r}}}\\{{n_{i}}}\\ \end{array}\right]_{q}; \; \; L + r \equiv e v e n \; \;
\int_0^\infty \Psi^{(5)}(Z) (\Psi w)'(Z)\:dZ=\int_0^\infty (\Psi^{(3)}(Z)^2 w(Z) + \sum_{k=1}^3 \begin{pmatrix}3 k\end{pmatrix}\int_0^\infty \Psi^{(3)}(Z) \Psi^{(3-k)}(Z) w^{(k)}(Z)\:dZ.
\limsup_{M \rightarrow \infty}\limsup_{n \rightarrow \infty}\frac{1}{n}\sum_{|x| < M n}\sup_{s \in [0,T]}| [ -(-\Delta)^{\gamma/2}G_s](\tfrac{x-1}{n}) - [ -(-\Delta)^{\gamma/2}G_s](\tfrac{x}{n}) | \leq \limsup_{M \rightarrow \infty}\limsup_{n \rightarrow \infty}\frac{2MK}{n}=0.
\|tL_{\mu_2}(I+tL_{\mu_2})^{-1}f\|^2_{L^2(\mathbb{R}^n, \mu_2)}\leq \sum_{j\in\mathbb{N}}\|tL_{\mu_2}(I+tL_{\mu_2})^{-1}f\|^2_{L^2(B^g(x^t_j,2 \sqrt{t}), \mu_2)}\leq \sum_{j\in\mathbb{N}}\|tL_{\mu_2}(I+tL_{\mu_2})^{-1}g^{j,t}\|^2_{L^2(B^g(x^t_j,2 \sqrt{t}), \mu_2)}.
e_{\mathbb{C}^*}(\pi_*(\mathcal{K}^\vee_i(a_j-a_{i}))) = (h_i+w_i\epsilon-w_j\epsilon)^{d_i+a_j-a_i+1}e_{\mathbb{C}^*}(\pi_*(\mathcal{K}^{\vee}_i \otimes\mathcal{K}_j(a_{j}-a_{i}))) = (h_i+w_i\epsilon-h_j-w_j\epsilon)^{d_i-d_j+a_j-a_i+1}.
[(u\cdot\nabla)A]_{j}=\sum_{i}\frac{u_{i}}{h_{i}}\frac{\partial A_{j}}{\partial q^{i}}+\frac{A_{i}}{h_{i}h_{j}}(u_{j}\frac{\partial h_{j}}{\partial q^{i}}-u_{i}\frac{\partial h_{i}}{\partial q^{j}}),
\Delta e^{-f}+ D_X e^{-f}+ 2P e^{-f}=-\Delta f e^{-f}+|\nabla f|^2 e^{-f}-|\nabla f|^2 e^{-f}+2P e^{-f}=-(R-2P) e^{-f}\leq -\frac{1}{m+1}Re^{-f}.
\theta^{-1}(H_{i,1})=-(-d)^{-i}[[\cdots[[\cdots [F_{0,0},F_{n-1,0}]_{q},\cdots,F_{i+1,0}]_{q},F_{1,0}]_{q},\cdots,F_{i-1,0}]_{q},F_{i,0}]_{q^{2}}\,, &\theta^{-1}(H_{i,-1})=-(-d)^{i}[E_{i,0},[E_{i-1,0},\cdots,[E_{1,0},[E_{i+1,0},\cdots,[E_{n-1,0},E_{0,0}]_{q^{-1}},\cdots]_{q^{-1}}]_{q^{-1}},\cdots]_{q^{-1}}]_{q^{-2}}\,
|k'(x(r))| =\left|\frac{(1/2\pi\beta)f'(r)}{(x'(r)/x(r))x(r)}\right| \le\frac{1}{\pi f(r)^{2}x(r)}\frac{r|f'(r)|}{\sqrt{1+r^{2}f'(r)^{2}}}\le\frac{1}{\pi f(r)^{2}x(r)}.
I(u, v)\leq 2^{n-1}\|d(u-v)\|_{\phi}^2\leq C (\|d(u-w)\|_\phi^2+\|d(v-w)\|_\phi^2) \leq CI(u, w)^{1/2^{n-1}}\left(I(u, \phi)^{1-1/2^{n-1}}+I(w, \phi)^{1-1/2^{n-1}}\right) &+CI(v, w)^{1/2^{n-1}}\left(I(v, \phi)^{1-1/2^{n-1}}+I(w, \phi)^{1-1/2^{n-1}}\right)
-\int_0^T(w,\phi_t)\, dt+\kappa\int_0^T(\Lambda^{\frac{1}{2}}w, \Lambda^{\frac{1}{2}}\phi)\, dt -\int_0^T(R^\perp\theta w,\nabla\phi)\, dt+\int_0^T(R^\perp w\cdot\nabla\Theta,\phi)\, dt &=(w_0,\phi(x,0))+\int_0^T(f,\phi)\, dt.
I_Q(V;Y,S|U)-I_Q(V;Z|U) =I_P(S,T;Y,S)-I_P(S,T;Z) =I_P(T;Y|S)-I_P(T;Z|S)+H_P(S|Z), I_Q(U,V;Y,S)-I_Q(U,V;S) =I_P(S,T;Y,S)-I_P(S,T;S)=I_P(T;Y|S),
\frac{3}{2}>\; b >\frac{2}{(2-\alpha)(1-2\alpha)}&c > \frac{2(2-\alpha) b^2 - (3-2\alpha) b -1}{b((2-\alpha) (1-2\alpha) b -2)}= \frac{((4-2\alpha) b +1) (b-1)}{b ((2-5\alpha + 2\alpha^2) b -2)}\, .
\begin{array}{l l}{n \left( \frac{1 - k}{1 + k}\right) = 2 G \left( U - T - J^{2}- \nu S^{2}\right)}\\{B^{2}= 1 - 8 G \left( U + \frac{( J^{2}+ \rho S^{2})}{2}\right)}\\{m^{2}= 4 G ( J^{2}+ \nu S^{2}) .}\\ \end{array}
V_{\mathrm{W_{B}}}( \theta ) = \frac{N T_{0}}{L \cos{\theta}}\Bigl ( 1 - \frac{L^{4}\sin^{4}{\theta}}{4 \sqrt{N^{2}- 1}}+ \frac{L^{8}\sin^{6}{\theta}\cos^{2}{\theta}}{3 2 ( N^{2}- 1 )}\Bigr ) \, ,
\mbox{var}(Y_i) = \sum_{i=1}^{\infty}\mbox{var}(E_i) +2 \sum_{i<j}^{\infty}\mbox{cov}(E_i,E_j) = \frac{mp}{1-p^2}+2 \sum_{i<j}\mbox{cov}(E_i,E_j)
\mathcal{M}_{\varepsilon}\Big(\mathcal{G}_{h,N_{1}}~\big|~{\bf{L^1}}([0,L],E) \Big)~\geq~{\mathrm{Card}\left(\mathcal{G}_{h,N_{1}}\right)\over \mathrm{Card}\left(\mathcal{I}_{\tilde{\delta}}(2\varepsilon)\right)}~\geq~{2^{N_1({\bf \tilde{p}}+2)}\over 2^{N_1(2+{\bf \tilde{p}}/2)}}~=~2^{N_1{\bf \tilde{p}}/2}~.
\Phi_n(x)=\mathrm{e}^{-\lambda/2^n}\int_{(a,b)}p(1/2^n;x-y)\,\Phi_n(y)\,\mathrm{d}y +\mathrm{e}^{-\lambda/2^n}\int_{\mathbb{R}\backslash(a,b)}p(1/2^n;x-y)\,\varphi(y)\,\mathrm{d}y.
\nabla_X \varphi Y = \frac{1}{2}\nabla_X Y + \frac{\sqrt{5}}{2}\nabla_X J_{\varphi}Y = \frac{1}{2}\nabla_X Y + \frac{\sqrt{5}}{2}J_{\varphi}\nabla_X Y = \varphi(\nabla_X Y), \forall X, Y \in{\mathfrak X}(M),
[ \Sigma_{\lambda \alpha}]_{a b}= i \int \frac{d^{4}S}{( 2 \pi )^{4}}( - i e \gamma_{\beta \alpha}^{\mu}) ( - i g_{\mu \nu}[ \tau_{3}D ( S ) \tau_{3}]_{b a}) ( - i e \gamma_{\lambda \gamma}^{\nu}) i ( P + S )_{\tau}\gamma_{\gamma \beta}^{\tau}[ D ( P + S ) ]_{a b}.
L_{\phi}L_{\phi}^{*}f_{j}^{(1)}= L_{\phi}a_h d_{\phi , h}^{*}Q^{*}( Q^{*})^{-1}f_{j}^{W , (1)}&= L_{\phi}a_{h}\big( d_{\phi , h}^{*}f_{j}^{W , (1)}\big) .
X(z) = \mathcal{O}\begin{pmatrix}1 z^{-\frac{1}{4}}h_{\alpha+\frac{1}{2}}(z) z^{-\frac{1}{4}}h_{\alpha+\frac{1}{2}}(z) 1 z^{-\frac{1}{4}}h_{\alpha+\frac{1}{2}}(z) z^{-\frac{1}{4}}h_{\alpha+\frac{1}{2}}(z) 1 z^{-\frac{1}{4}}h_{\alpha+\frac{1}{2}}(z) z^{-\frac{1}{4}}h_{\alpha+\frac{1}{2}}(z)\end{pmatrix}.
\|v_{N}\|_{L^2(m)}^2= \sum_{k=0}^{2^N}(\sum_{n\ge 2^N+1}(n+1)^{-\alpha}c_{k+n})^2\le C2^{N}(\sum_{n\ge 2^N+1}(n+1)^{-\alpha}c_{n})^2 \le \frac{C'}{N(\log (N+1))^3}\, .
\lim_{\vec{p}\rightarrow 0}\frac{\left(C_1\left(\left|\vec{p}\right| \right) +C_1^{*}\left(\left| \vec{p}\right| \right)+2\left| C_1\left(\left| \vec{p}\right| \right) \right|^2\right)\left| \vec{p}\right|^3\frac 1{\left(1-\left(1-\kappa \left|\vec{p}\right|^2\right)^2\right)}}{4p_0}=\frac C{2\left(2\pi\right)^4}