harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f\big( (k-1)^2+(k-1)^2+(k-1)^2 + 3^2(k-6) + 4^2+4^2+4^2 \big) &= 3f\big( (k-1)^2 \big) + (k-6)f(9) + 3f(16) = 3(k^2 + k - 1) &= f(3k^2+3k-3) = f(3)\,f(k^2+k-1) = f(3)\,(k^2+k-1),
\|\Delta{\bf{J}}^{}({\bf{n}})-\Delta{\bf{J}}^{}({\bf{\hat{n}}})\| \leq{Q}_1 \sum_{i=1}^I \Delta x_i \sum_{k=1}^{I}\Delta x_k \left|n_k-\hat{n}_k\right| +{Q}_1 \sum_{i=1}^I \left|n_i-\hat{n}_i\right| \Delta x_i \sum_{j=1}^{I}\Delta x_j.
B(\Phi(x),\Phi(v))=B(x-\tau(x)-\frac{1}{2}B_{\mathfrak a}\big(\tau(x),\tau(\cdot)\big),v+B_{\mathfrak a}(v,\tau(\cdot))) =-B_{\mathfrak a}(\tau(x),v)+B_{\mathfrak a}(v,\tau(x))=0
\operatorname{Re}\psi_3(i\rho,i\tau,\xi+i\eta)=-(1-\alpha)\tau^2-\beta\rho\eta\leqslant 0,\shortintertext{and}\operatorname{Re}\psi_4(i\rho,i\tau, \xi+i\eta)=-(1-\alpha)\rho^2-\beta\rho\eta \leqslant 0,
| 0_{M}> = Z \exp \left[{\int_{0}^{\infty}d \tilde{\omega}\int d^{n - 2}k \sum_{\sigma}e^{- \pi \tilde{\omega}/ a}b_{k}^{( \sigma ) \dagger}\bar{b}_{\tilde{k}}^{( - \sigma ) \dagger}}\right] | 0_{R}>{,}
d_X(F(x,y),x) \leq d_X(F(x,y),F^{n+1}(x_0,y_0))+ d_X(F^{n+1}(x_0,y_0),x) &= d_X(F(x,y),F(F^n(x_0,y_0),G^n(y_0,x_0)))+ d_X(F^{n+1}(x_0,y_0),x) &\leq k\d_X(x,F((F^n(x_0,y_0),G^n(y_0,x_0)))+ l\d_X(F^n(x_0,y_0),F(x,y)) &\ \ \ + d_X(F^{n+1}(x_0,y_0),x) &= k\d_X(x,F^{n+1}(x_0,y_0))+ l\d_X(F^n(x_0,y_0),F(x,y)) &\ \ \ + d_X(F^{n+1}(x_0,y_0),x)
S_{\nu}(N;y) = \sum_{d\in \Phi(N,y)}\sum_{m\in \Phi(N/d,y)}\tau_{\nu-1}(m)= \sum_{d\in \Phi(N,y)}S_{\nu-1}(N/d;y) & \le \sum_{d\in \Phi(N/y,y)}S_{\nu-1}(N/d;y) + S_{\nu-1}(N;y).
\mathbf{T}_2(E) = \xi^{-3}D^+(E \Omega^3,\ell) D^-(E\Omega^{-3},\ell) -\xi^{3}D^-(E\Omega^3,\ell) D^+(E\Omega^{-3},\ell) &= -2ih Q^+(-E) Q^-(-E) +O(h^2).
\tilde{\varphi}_{p}(x):=\sum_{n=1}^{\infty}\frac{p^{2}+\lambda_{n}^{2}}{p\left(p+\frac{1}{\pi}\right)+\lambda_{n}^{2}}\,\lambda_{n}^{s-\frac{1}{2}}\,K_{s-\frac{1}{2}}(2\pi\lambda_{n}x),\,\,\,\,x>0.
\sum_{\substack{\gamma\in\Gamma^S(\chi) \gamma\le T}}\frac2{\sqrt{\frac14+\gamma^2}}\ge \sum_{\substack{\gamma\in\Gamma^S(\chi) \gamma\le T}}\frac2\gamma - \frac14 \sum_{\substack{\gamma\in\Gamma^S(\chi) \gamma\le T}}\frac{1}{\gamma^3}> W(1) - \frac14 \sum_{\gamma\in\Gamma^S(\chi)}\frac{1}{\gamma^3}= \max\{8,|2-n|\}.
\int{\cal D}x{\cal D}y\exp\left(\frac{i}{\hbar}\int\,dt\left(\frac{1}{2}(\dot{x}^{2}+\dot{y}^{2})\right)\right) =\int{\cal D}r{\cal D}\theta J[r] \exp\left(\frac{i}{\hbar}\int\,dt\left(\frac{1}{2}(\dot{r}^{2}+r^{2}\dot{\theta}^{2})\right)\right),
\langle \omega_i,\omega_j \rangle =0,{\rm \if}\i+j\leq n+1\, , \langle \omega_1,\omega_{n+1}\rangle =(-(n+2))^n\frac{c_n}{t_1^{n+2}-t_{n+2}},{\rm \where}\c_n \{\rm is \a \constant}\, , \langle \omega_j,\omega_{n+2-j}\rangle =(-1)^{j-1}\langle \omega_1,\omega_{n+1}\rangle,{\rm \for}\j=1,2,\ldots,n+1\, ,
\mathrm{m}_q = \frac{1}{2}c_q(n) \int_M \left(L_{(q)}+ (2q-1)! \left\langle \left[ T_{(2q-1)\alpha}, A_{\beta}\right] \cdot e_{\alpha}^{\top}, e_{\beta}^{\top}\right\rangle \right) \frac{1}{\sqrt{G}}dM & + \frac{1}{2}(2q-1)! c_q(n) \int_{\Sigma}\left( \frac{|Df|^2}{1 + |Df|^2}\right)^q H_{(2q-1)}d\Sigma,
F_{\mu \nu}= \left( \begin{array}{c c c c c}{0}&{- \omega_{1}}&{}&{}&{}\\{\omega_{1}}&{0}&{}&{}&{}\\{}&{}&{\ddots}&{}&{}\\{}&{}&{}&{0}&{- \omega_{5}}\\{}&{}&{}&{\omega_{5}}&{0}\\ \end{array}\right) ,
\left( \begin{array}{c}{\partial_{\mu}\theta}\\{\sin \theta \partial_{\mu}\varphi}\\ \end{array}\right) = L \left( \begin{array}{c}{G_{\mu}}\\{H_{\mu}}\\ \end{array}\right) , L = \left( \begin{array}{c c}{\beta}&{\gamma}\\{- \gamma}&{\beta}\\ \end{array}\right) ,
\widetilde{s_P}= \i\hat{c}(\tau+|\xi|^2+|\eta|^2)\widetilde{cF}(\tau, \xi, \eta) = \ \chi_{\{|\tau+|\xi|^2+|\eta|^2|> 1\}}\widetilde{s_P}+\chi_{\{|\tau+|\xi|^2+|\eta|^2|\le 1\}}\widetilde{s_P}=: \widetilde{s_1}+\widetilde{s_2}
\begin{aligned}\mathbb{E}\left[\|\bar x^{k+1}-\theta^*\|^2|\mathcal{F}^k\right]\le (1+a^k)\|\bar x^k -\theta^*\|^2 \cr +\left(\frac{\gamma^k}{m}+a^k\right) \sum_{i=1}^m \|x_i^k - \bar x^k\|^2 + b^k -c^k (F(\bar x^k)- F(\theta^*))\end{aligned}
\begin{bmatrix}H_1(x_{[0,N-L]}) H_L(u)\end{bmatrix}= \begin{bmatrix}x_0 x_1 \dots x_{N-L}u_0 u_1 \dots u_{N-L}u_1 u_2 \dots u_{N-L+1}\vdots \vdots \ddots \vdots u_{L-1}u_L \dots u_{N-1}\end{bmatrix}.
\sum_{m=0}^{\infty}\frac{q^{m^{2}+m\nu}}{\left(q;q\right)_{m}z^{m}}I_{\nu+m}^{(3)}(2zq^{m/2};q)=\frac{z^{\nu}}{(q;q)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{\binom{n}{2}}(qz^{2})^{n}}{\left(q;q\right)_{n}}=\frac{\left(-qz^{2};q\right)_{\infty}}{(q;q)_{\infty}}z^{\nu},
(dJdu)^{(1,1)}={}\frac{1}{2}\left(- J^k_{\ \ell}\partial_k \partial_{i}u - J^a_{\i}J^b_{\ \ell}J^k_{\b}\partial_k \partial_a u \right. \left. - (\partial_{i}J^k_{\ \ell}) \partial_k u - J^a_{\i}J^b_{\ \ell}(\partial_a J^k_{\b}) \partial_k u \right) dx^i \wedge dx^{\ell}.
|\alpha(n)-\alpha(m)| =\int_{m\leq d(x,x_{0})\leq n}|f(x)|d{\rm vol}(x) &\leq C \int_{m\leq d(x,x_{0})\leq n}e^{-qd(x,x_{0})}d{\rm vol}(x)\quad\mbox{for all}~q\in\mathbb{N}, &\leq C' e^{-qm+np},
L_{1}u=\sum\limits_{i=0}^{m_{1}}\varepsilon^{\frac{i}{2}}\alpha_{i}u^{\left( i\right)}\left( 0,\varepsilon \right) =f_{1}\left(\varepsilon \right) ,L_{2}u=\sum\limits_{i=0}^{m_{2}}\varepsilon^{\frac{i}{2}}\beta_{i}u^{\left( i\right)}\left( T,\varepsilon \right)=f_{2}\left( \varepsilon \right) ,
\int_0^{\eta}\frac{{\rm d}\tau}{a^2({\tau})}= \frac{\eta_0}{2\ell_0}\frac{1}{a(\eta )}\biggl[\frac{\eta}{\eta_0}+\frac{\sqrt{2}}{\ell_0}a(\eta ) \tan^{-1}\biggl(\frac{\eta}{\sqrt{2}\eta_0}\biggr)\biggr].
\Pi_{1}\overline{N}(^{\ast}b) = -\pi_{1}p\xi_{be_{k}}\Pi_{2}\overline{N}(^{\ast}b) = - \gamma' \xi_{be_{k}}\Pi_{3}\overline{N}(^{\ast}b) = 0 \Pi_{4}\overline{N}(^{\ast}b) =0
\left( \delta F_{p}\right) \left( \phi_{0}, \cdots , \phi_{p + 1}\right) = \sum_{i = 0}^{p + 1}( - 1 )^{i}\, F_{p}\left( \phi_{0}, \cdots , \phi_{i - 1}, \widehat{\phi}_{i}, \phi_{i + 1}, \cdots , \phi_{p + 1}\right)
\psi_{\Theta,\ell, \sigma}(\xi)= \psi_\ell(\xi) \varphi_\sigma\left (\frac{\xi}{\|\xi\|}\right )\, ,\, \, \tilde \psi_{\Theta,\ell, \sigma}(\xi)=\tilde \psi_\ell(\xi) \varphi_\sigma\left (\frac{\xi}{\|\xi\|}\right )\, .
p^*_{\{x,y\}^*}= \frac{2-2p_{\{x,y\}}}{2-p_{\{x,y\}}}=\frac{2\exp(-2|\varphi_x \varphi_y|)}{1+\exp(-2|\varphi_x \varphi_y|)}=1- \tanh |\varphi_x \varphi_y|
\psi^{( 5 )}( 1 ) + \frac{4 5}{4}\psi^{( 4 )}( 1 ) + \frac{2 3 5}{4}\psi^{( 3 )}( 1 ) + \frac{9 7 5}{8}\psi^{( 2 )}( 1 ) + \frac{1 6 3 5}{3 2}\psi^{( 1 )}( 1 ) = 0 .
\|ab\|_{B^{s-i}_{p,r}}\lesssim \|a\|_{B^{s-i}_{p,r}}\|b\|_{B^{s-i}_{p,r}},\|a^2\|_{B^s_{p,r}}\lesssim \|a\|_{L^\infty}\|a\|_{B^s_{p,r}},\|a\|_{L^\infty},\|\nabla a\|_{L^\infty}\lesssim \|a\|_{B^s_{p,r}},i=0,1,
{\cal L}^{\prime}=\bar{\psi}\gamma^{\mu}\left[ i\partial_\mu +e{\cal A}_\mu \right]\psi -\frac{1}{2G}{\cal A}_\mu{\cal A}^\mu -\frac{\kappa}{2}\epsilon_{\mu \nu \rho}{\cal A}^{\mu}\partial^{\nu}{\cal A}^{\rho},
E\left({a;b}\right) =\int_{0}^{a}{\left({\frac{1-b^{2}t^{2}}{1-t^{2}}}\right)^{1/2}dt}=b\int_{0}^{a}{\left({\frac{b^{-2}-t^{2}}{1-t^{2}}}\right)^{1/2}dt}.
0>-\frac{\partial G}{\partial \nu}(x_0)= \frac{\partial G}{\partial x_n}(x_0) = \frac{2w_kw_{kn}}{|Dw|^2}+ah_n=\frac{2w_{1n}}{w_1}- a = \frac{2u_{1n}}{w_1}- \frac{2\varphi_{1}h_n+2\varphi_{n}h_1+2\varphi h_{1n}}{w_1}- a = \frac{2u_{1n}}{w_1}-\frac{-2\varphi_{1}+2\varphi h_{1n}}{w_1}- a = 2{\nu^1}_1 - a \geq 2 \kappa_2 - a \geq 0,
\phi_1 := \phi_{14AB}(\tau, z) = 2 \zeta^{\pm 1}- 3 + (-3\zeta^{\pm 2}+ 7 \zeta^{\pm 1}- 8) q + (2 \zeta^{\pm 3}- 8 \zeta^{\pm 2}+ 19 \zeta^{\pm 1}- 26)q^2 +O(q^3)
\frac{\alpha\rho}{4}C_{i+1}\overset{\eqref{eq:dgdr}}{\geq}\frac{a_i^{36k^2+4k}}{2^{109k^4+13k^3+2}}\cdot C_{i+1}\overset{\eqref{rec}}{=}\frac{a_i^{37k^2}}{2^{110k^4}}\cdot \frac{2^{122k^4}C_i}{b_i^4a_i^{37k^2}}= \frac{2^{12k^2}C_i}{b_i^4}\ge C_i\,.
H_1(t,x,y)=2^{\sigma_1 l+k_1}\int_{n_1 2^{-(\sigma_1 l+k_1)}+P_1(2^{-l}t)}^{(n_1+1)2^{-(\sigma_1 l+k_1)}+P_1(2^{-l}t)}|f_1|\left(x+u,y\right)\,\mathrm{d}u
G = K_{i-1}(x,k-y,r-1)\frac{\binom{n}{i-1}}{\binom{k}{i-1}}\Big(\frac{(n-j-k)(j+1)}{n-i-2j}H_{j+1}(y,n-(i-1),k-(i-1)) & \quad+\frac{(k-i-j+1)(n-i-j+2)}{n-i-2j+2}H_{j}(y,n-(i-1),k-(i-1))\Big).
V(t,z) = - \frac 3 8 \left(1+u_0(t)\right)^3 + \sum_{\substack{i \in K_{N_0}1 \le i < N_0}}\tilde V_i(t) \tilde x^i + O\left(\tilde x^{N_0}\right) \mbox{as}\, \tilde x \searrow 0 \, \mbox{and}\, t > 0,
(g\cdot h)^\vee_M(m) =\big((g\cdot h)(m)\big)(1)=(h(m)\cdot g^{-1})(1)=h(m)(g) &=\big(g^{-1}\cdot h(m)\big)(1)=h(g^{-1}\cdot m)(1)=h^\vee_M(g^{-1}\cdot m),
\left\lbrace\begin{matrix}A_k(\theta, E)=A(\theta+(k-1)\alpha, E)\cdots A(\theta+\alpha, E)A(\theta, E)\ \mathrm{for}\k\geq 1, A_0(\theta, E)=\mathrm{Id}, A_k(\theta, E)=(A_{-k}(\theta+k\alpha, E))^{-1}\ \mathrm{for}\k\leq -1.\end{matrix}\right.
\frac{\partial g_{jk}}{\partial x^{i}}+\frac{\partial g_{ki}}{\partial x^{j}}-\frac{\partial g_{ij}}{\partial x^{k}}=2\langle\frac{\partial\vec{\Psi}}{\partial x^{k}},\frac{\partial^{2}\vec{\Psi}}{\partial x^{i}\partial x^{j}}\rangle
\begin{array}{c}{d s^{2}= - \lambda d t^{2}+ \lambda^{- 1}d r^{2}+ R^{2}d \Omega ,}\\{e^{2 \phi}= e^{2 \phi_{0}}\frac{r + \Sigma}{r - \Sigma},}\\{F = \frac{Q e^{\phi_{0}}}{( r - \Sigma )^{2}}d t \wedge d r , G = P e^{\phi_{0}}\sin \theta d \theta \wedge d \varphi ,}\\ \end{array}
{\displaystyle V_{\alpha}^{( 1 ) i}\frac{i \tilde{\cal P}^{\dot{\beta}\alpha}}{( m + i e \bar{W}/ 2 )}\sqrt{\frac{m^{* 2}}{-{\cal P}^{2}}}= \bar{V}^{( 1 ) \dot{\beta}i}\mathrm{, ~}V_{\alpha}^{( 2 ) i}\frac{- i \tilde{\cal P}^{\dot{\beta}\alpha}}{( m + i e \bar{W}/ 2 )}\sqrt{\frac{m^{* 2}}{-{\cal P}^{2}}}= \bar{V}^{( 2 ) \dot{\beta}i}}.
T_{\mu \nu \rho \sigma}^{d v , n i}( p ) = - i \frac{2^{\omega}}{3 2}m^{2}g_{\mu \rho}\int_{0}^{1}\! d x \int \! \! \frac{d^{2 \omega}l}{( 2 \pi )^{2 \omega}}\frac{( 2 l - p ( 1 - 2 x ) )_{\nu}( 2 l - p ( 1 - 2 x ) )_{\sigma}}{[ l^{2}- \Delta ]^{2}}\, \, ,
\begin{aligned}\lambda^{J}_1\leq -n(1+H^2) &-2\alpha(1-\beta)n\dfrac{4(n-1)\big(1-2\alpha(1-\beta)\big)(1+H^2)-2\alpha(1-\beta)(n-2)^2H^2}{4(n-1)\big(1-2\alpha(1-\beta)\big)}.\end{aligned}
\Gamma_{0}^{M}( k ) = i \int{\frac{d^{4}u}{( 2 \pi )^{4}}}\hat{I}_{a b}\big ( k - u ,{\frac{k + u}{2}},{\frac{k + u}{2}}\big ) \sigma^{a}\hat{H}( u ) \Gamma_{0}^{M}( u ) \hat{H}( u ) \sigma^{b}\ .
V_{(a,b)}(n+p^{k-a+1}) = (n+p^{k-a+1}-2)\rho_G - 1 -\bigoplus_{i=1}^{\ell'}\lambda(i) = (n-1)\rho_G -1+(p^{k-a+1}-1)\rho_G - \bigoplus_{i=1}^{\ell'}\lambda(i)
ie_0^2\int\frac{d^D\!K}{(4\pi)^Dk^2}\gamma_\nu\frac{1}{\not{p}+\not{k}}\left[ f_n\gamma_\mu - g_n\frac{(\not{p}+\not{k})\gamma_\mu(\not{p}+\not{k})}{(p+k)^2}\right]\left(\frac{\mu^2}{(p+k)^2}\right)^{nx}\frac{1}{\not{p}+\not{k}}\gamma^\nu
S^{ab}(x) S^{Tbc}(x) = S^{ab}(x) S^{cb}(x) = \frac{1}{4}\tau_{ij}^b\tau_{kl}^b (U(x) \tau^a U^{\dagger}(x))_{ij}(U(x) \tau^cU^{\dagger}(x))_{kl}= \frac{1}{2}tr[\tau^a \tau^c] = \delta^{ac}
\textstyle \Xi_{ij}\textstyle = \delta_{i+p-j-1-s \equiv_{p-1}0}\delta_{i+p-j-1-s \ne 0}\delta_{r-i+j\equiv_{p-1}0}\delta_{r-i+j \ne 0}\textstyle= \delta_{j \equiv_{p-1}i - s}\delta_{(i,j) \not\in \{(0,p-1-s), (r,0)\}}\textstyle= \delta_{j \equiv_{p-1}i - s}\delta_{i \not\in \{0,r\}}.
\int_{\Omega}n\min\limits_{1\le k \le n}|x - \alpha(k,\omega)|\, dP(\omega) & =\int_0^\infty P(\{\omega: n\min\limits_{1\le k \le n}|x - \alpha(k,\omega)| \ge t\}) \, dt= \int\limits_0^{n/2}(1 - 2t/n)^n \, dt= \frac n{2(n+1)}.
\sum_{l=-r}^{r}\dfrac{1}{\omega_{l}}\, \biggl(u_{(2r+1)+l}+\sum_{p\ge 2}\dfrac{\overline{\lambda}^{p-1}}{\omega_{(p-1)(2r+1)+l}\dots\omega_{(2r+1)+l}}u_{p(2r+1)+l}&+\sum_{p\ge 1}\dfrac{\omega_{l-p(2r+1)}\dots\omega_{l-(2r+1)}}{\overline{\lambda}^{p}}u_{-p(2r+1)+l}\biggr)=0
\log\Gamma\bigl(1+\frac{z-q}{\tau}\bigr) - \log\Gamma\bigl(1+\frac{z}{\tau}\bigr) = \int\limits_{0}^\infty \frac{dt}{t}\Bigl[ e^{-tz}\bigl(\frac{e^{tq}-1}{e^{t\tau}-1}\bigr) - \frac{q}{\tau}e^{-t\tau}\Bigr].
\tilde{\psi}((x_{ij})) =\frac{1}{2}\left(\psi((x_{ij})) +\overline{\psi((x_{ij}))}\right)=\frac{1}{2}\left(\sum_{i,j=1}^{n}x_{ij}\psi(E_{ij}) +\sum_{i,j=1}^{n}x_{ij}\overline{\psi(E_{ij})}\right) &=\frac{1}{2}\left(\sum_{i=1}^{n}x_{ii}(\psi(E_{ii})+\psi(E_{ii})^{t})+\sum_{1\leq i\neq j \leq n}x_{ij}(\psi(E_{ij}) + \psi(E_{ji}))\right)
\mu_\psi(\phi) = \lim_{n\to\infty}\frac{\sum_{\gamma \in P(n)}\exp(\ell_{\psi}(\gamma))\delta_\gamma(\phi)}{\sum_{\gamma \in P(n)}\exp(\ell_{\psi}(\gamma))}\leq \lim_{n\to\infty}|\phi|_\infty\frac{\sum_{\gamma\in Q'(n)}\exp(\ell_{\psi}(\gamma))}{\sum_{\gamma \in P(n)}\exp(\ell_{\psi}(\gamma))}.
\det(\Theta_1)=\frac{[H^1(D_{\mathcal{F}})_{tor}][H^0_{et}(U,\mathcal{F})_{tor}][Q_1]\prod_{v\in S_{\infty}}h(G_v,\mathcal{F}_v)^{-1}}{[H^0(D_{\mathcal{F}})_{tor}]\prod_{v\in S}[H^0(K_v,\mathcal{F}_v)_{tor}][\mathrm{cok}(\delta(\mathcal{F})_{tor})][\ker(\Psi^1(\mathcal{F}))]}.
\Theta_{\Lambda_{24}}(z\alpha^{I}|\tau)=C_{1}(\tau)\left[\frac{\theta_{1}(z|\tau)}{\theta_{1}^{'}(0|\tau)}\right]^{2}+\sum_{i=2}^{4}C_{i}(\tau)\left[\frac{\theta_{i}(z|\tau)}{\theta_{i}(0|\tau)}\right]^{2}.
\|\sum^{m_2}_{m=m_1+1}p_m(v_mx_{n(m,k)}v_m^*+v_m^*x_{n(m,k)}v_m)p_mq_{m_1,m_2}\|_{\mathcal M}\leq \sum^{m_2}_{m=m_1+1}\|p_m(v_mx_{n(m,k)}v_m^*+v_m^*x_{n(m,k)}v_m)p_mq_{m}\|_{\mathcal M}\leq 2^{-m_1}(k+1)^{-1}.
\rho \mathbf{U}_{tt}+ \operatorname{div}\boldsymbol{\sigma}^{\mathrm{elastic}}+ \gamma \nabla \tau = 0 (0, \infty) \times \mathcal{B}_{h}, \rho c \tau_{t}- \lambda_{0}\triangle \tau + \gamma T_{0}\operatorname{tr}\big(\boldsymbol{\varepsilon}^{\mathrm{elast}}_{t}\big) = 0 (0, \infty) \times \mathcal{B}_{h}.
X'' + \frac{1}{r}X' = \Big(b^2\,s_+ + O(|\hat u| + |\hat v| + r^{-2}) \Big) X+ \frac{k^2s_+}{\sqrt{2}r^2}, Y'' + \frac{1}{r}Y' = \frac{1}{3}\Big(4c^2\,s_+^2 - b^2\,s_+ + O(|\hat u| + |\hat v|+ r^{-2}) \Big)\,Y + \frac{\sqrt{3}k^2\,s_+}{\sqrt{2}r^2}.
\theta_1(\tau, z+1) = - \theta_1(\tau, z), \theta_1(\tau, z+\tau) = - e^{-2\pi i z - \pi i \tau}\theta_1(\tau, z), \theta_1(\tau+1, z) = e^{\pi i/4}\theta_1(\tau, z), \theta_1(- \frac{1}{\tau}, \frac{z}{\tau}) = - i \sqrt{\frac{\tau}{i}}e^{\frac{\pi i z^2}{\tau}}\theta_1(\tau, z).
{[\ell_a,\ell_b,\ell_a]}={\textstyle \big(\frac{\ell_b}{\ell_a}\big)_4}\hbox{if $\ell_a \not\equiv 3 \pmod{4}$ and $\ell_b \not\equiv 3 \pmod{4}$,}&{[\ell_a,\ell_b,\ell_a]}={[\ell_a,\ell_b,\ell_b]}={\textstyle \big(\frac{-\ell_b}{\ell_a}\big)_4}\hbox{if $\ell_a \not\equiv 3 \pmod{4}$ and $\ell_b \equiv 3 \pmod{4}$.}
Y_s^{t,x}\ = \Y_T^{t,x}+ \int_s^T f(r,X_r^{t,x},Y_r^{t,x},Z_r^{t,x}) dr - (K_T^{t,x}- K_s^{t,x}) - \int_s^T \langle Z_r^{t,x},dW_r\rangle, t \leq s \leq T.
\inf_{L(\gamma)=r}\int_0^re^{-2\frac{\phi(\gamma_t)}{n-m}}\d t\geq \int_{r_0}^re^{-\frac{2C\log (1+t)}{n-m}}\d t=\int_{r_0}^{r}\frac{\d t}{(1+t)^{2C/(n-m)}}\to +\infty r\to\infty.
2\cosh(4t)= (1-z+\langle x,y\rangle)^{2}(1+z)^{2}+(1-z+\langle x,y\rangle)^{2}\|y\|^{2}+(\langle x,y\rangle-z)^{2}&+(1+z)^{2}\|x\|^{2}+2\langle x,y\rangle+\|x\|^{2}\|y\|^{2}+\|x\|^{2}+z^{2}+\|y\|^{2}+1.
L_{G}( u )_{j}^{i}= E_{0}+ \sum_{\alpha \neq ( 0 , 0 )}E_{\alpha}( I_{\alpha})_{j}^{i}= \sum_{k}\phi ( u , z )_{k}^{i}\phi^{- 1}( u , z )_{j}^{k}\partial_{u}- l \sum_{k}\partial_{u}\phi ( u , z )_{k}^{i}\phi^{- 1}( u , z )_{j}^{k},
a_2 = z + c_{n,m}\mathbb{E}[\delta_2(z)] a_1 = z^2 -c_{n,m}+1 +z c_{n,m}\mathbb{E}[\delta_2(z)]-z \mathbb{E}[\delta_1(z)]-c_{n,m}\mathbb{E}[\delta_1(z)]\mathbb{E}[\delta_2(z)] a_0 = z - z^2 \mathbb{E}[\delta_1(z)] + c_{n,m}\mathbb{E}[\delta_1(z)] - z c_{n,m}\mathbb{E}[\delta_1(z)]\mathbb{E}[\delta_2(z)].
h= \left[\begin{array}{cc}f_{1}-\frac{a_{12}+i_1(i_1-i_2)-\lambda(a_1-a_2)f_{1}}{\lambda(a_1-a_2)}-\frac{a_{12}}{(a_1-a_2)}f_{2}i_2-\lambda f_2 \end{array}\right], h'= \frac{a_{12}+i_1-\lambda(a_1-a_2)f_{1}}{\lambda(a_1-a_2)}.
[ c, D^n ] = \sum_{i=0}^{n-1}(-1)^{(i+1)|c| + 1}\sum_{i \le p \le n-1}\binom{p}{i}\sum_{\bold{q}= (q_1,\ldots,q_{i+1})}\tau_{\bold{q}}b_{\bold{q}}D^{n-i-1}\,.
L_{k}M := \Lambda \otimes_{\mathbb{Z}[\pi_{1}(X_{k})]}M L_{k}\alpha := id_{\Lambda}\otimes_{\mathbb{Z}[\pi_{1}(X_{k})]}\alpha \colon \Lambda\otimes_{\mathbb{Z}[\pi_{1}(X_{k})]}M \longrightarrow \Lambda \otimes_{\mathbb{Z}[\pi_{1}(X_{k})]}N
-2-n =w = \sum_{j}k_j(j-2+h) = \sum_{j}k_j(j-2) = -2k_{0}-k_{1}+k_{3}+ 2 k_{4}+ \cdots \;. 2k_{0}-2 +k_{1}-n = k_{3}+ 2 k_{4}+ \cdots \;.
M(\vec{\nu}, \vec\gamma) :=\bigl\{f \in R \; \big| \; T_{(\alpha, 2(\mu_\alpha-\nu_\alpha))}^-(f) = \gamma_\alpha \cdot T_{(\alpha, 2(\mu_\alpha-\nu_\alpha))}^+(f) \mbox{\rm for all}\alpha \in \Pi\bigr\}.
- \delta^{a b}C_{1}C_{2}{\frac{1}{1 6}}{\frac{g^{4}}{p^{2}}}\Lambda^{3 - \omega}\int{\frac{d^{\omega}k}{( 2 \pi )^{\omega}}}{\frac{p^{2}k^{2}- ( k \cdot p )^{2}}{k^{3}( k + p )^{2}}}.
\rho_{\alpha, \varepsilon}(x) := 2 \pi \sqrt{1-x^2}\psi_{\alpha, \varepsilon}(x) = \varepsilon_{\alpha}- \varepsilon x + \frac{2}{\alpha}\sqrt{1-x^2}\arctan \left( \frac{\sqrt{1-x^2}}{\sqrt{\alpha^2-1}}\right),x \in [-1,1]
S(\mathfrak{x}) := S(x_{n_{1}}\otimes \cdots \otimes x_{n_{r+1}}\otimes x_{n_{r}}\otimes \cdots \otimes x_{n_{i+1}}) + S(\mu^{i-1}\alpha_{T}(x_{n_{1}}\otimes \cdots\otimes x_{n_{r-1}})\otimes [x_{n_{r}},x_{n_{r+1}}]_{\mathfrak{g}}\otimes \alpha_{T}(x_{n_{r+2}}\otimes \cdots \otimes x_{n_{i+1}})).
\operatorname*{Var}\left[{\displaystyle\sum\limits_{x=1}^{2n-1}}\left( f(x,V)-1\right) \right] =7n^{2}-4(n+1)^{2}H_{n}^{(2)}-2(n+1)H_{n}+13n & =\left( 7-\frac{2\pi^{2}}{3}\right) n^{2}-2n\ln(n)+\left( 17-2\gamma-\frac{4\pi^{2}}{3}\right) n & -2\ln(n)+\left( 5-2\gamma-\frac{2\pi^{2}}{3}\right) +o(1)
{\cal D}\bar{\cal M}\equiv{\cal D}\bar{B}_{\mu \nu}{\cal D}\bar{B}_{\mu 5}{\cal D}\bar{B}_{5 5}{\cal D}\bar{\zeta}_{\mu}{\cal D}\bar{\zeta}_{4}{\cal D}\bar{\zeta}_{\mu , \theta}{\cal D}\bar{\zeta}_{4 , \theta}{\cal D}\bar{\eta}{\cal D}\bar{\eta}_{, \theta}\, ,
\lim_{\epsilon \searrow 0}\int_{-r}^0 DF(s,X_s,X(s))(\beta) \cdot \left (\frac{1}{\epsilon}\int_{s+\beta}^{s+\beta+\epsilon}\nabla_\theta X(r) dr - \nabla^+_\theta X(s+\beta)\right ) d\beta = 0.
f^{\Sigma}_{\Lambda}:= v^{\Sigma}_{\Lambda}+ Op_h^w(\Gamma) \left[h \frac{\tilde{\theta}_h(x)}{\varphi(x)}\left(k_1 \Phi^{\Sigma}_3(x) (v^{\Sigma}_{\Lambda})^3 + k_2 \Phi^{\Sigma}_{-1}(x)|v^{\Sigma}_{\Lambda}|^2\overline{v^{\Sigma}_{\Lambda}}+ k_3 \Phi^{\Sigma}_{-3}(x)(\overline{v^{\Sigma}_{\Lambda}})^3 \right)\right] \, ,
-\frac{\partial LD_{\alpha}I(p\|q)}{\partial q_{j}}=-\frac{1}{\alpha}\left[\left(\sum_{i}q_{i}\right)^{-1}-\left(\sum_{i}p^{\alpha}_{i}q^{1-\alpha}_{i}\right)^{-1}p^{\alpha}_{j}q^{-\alpha}_{j}\right]
\left(a_{h-2},a_{h-1},a_h\right)=\left\{\begin{aligned}\left(n-5,n-4,n-3\right), \left(n-5,n-4,n-2\right), \left(n-5,n-3,n-1\right), \left(n-5,n-2,n-1\right), \left(n-4,n-3,n\right), \left(n-4,n-2,n\right), \left(n-3,n-1,n\right), \left(n-2,n-1,n\right) \end{aligned}\right\}.
\mathrm{Var}\left[\frac{1}{n_1}\tilde{i}(x^{n_1}; Y_1^{n_1}, Y_2^{n_2})\right] = \frac{\mathsf{V}_{\rho,1}}{n_1}+ \frac{\mathsf{V}_{\rho,2}}{n_1^2}\cdot \sum_{i=n_2+1}^{n_1}x_i^2,
\begin{aligned}\Lambda(Z_{\alpha},Z_{\beta})(Pj+k) =\Lambda({C_{k_1},C_{k_2}})(j)\Lambda(\mathbf{b_p},\mathbf{b_p'})(k) +\Lambda(C_{k_1},C_{k_2})(j+1)\Lambda(\mathbf{b_p},\mathbf{b_p'})(-P+k), \end{aligned}
\left( \int_{0}^{t_{1}-t_{0}}\left( \int_{B_{1}}(\psi w)^{2}dx \right)^{p}ds \right)^{1/p}\leq C \left( (g_{1-\alpha}*W)(t_{*}) + \|F\|_{L^{1}([0,t_{2}-t_{0}])}\right),
\delta'(\boldsymbol{x}_1,\ldots,\boldsymbol{x}_n)^{q^{-m}}=(\xi^{q^{-m}})^{-(1+q+\cdots+q^{n-1})}\delta(\boldsymbol{\xi}\boldsymbol{x}_1,\ldots,\boldsymbol{\xi}\boldsymbol{x}_n)^{q^{-m}}
{\rm tr}\left[\mathbf{M}(\tau)\right] = \big[\nabla_{\!x}\cdot\mathbf{W_{\!\varepsilon}}- \nabla_{\!v}\cdot\mathbf{B}\big] \big(\tau,X(\tau),V(\tau)\big) begin{align*}2pt] = \big[V(\tau) - \beta_\varepsilon\big(V(\tau)\big)\big]\cdot \nabla_{\!x}\eta_\varepsilon\big(X(\tau)\big) \; -\big(\nabla_{\!v}\cdot\mathbf{B}\big) \big(\tau,X(\tau),V(\tau)\big)
\alpha'(i) C_{\alpha',\beta',a',b'}^{(k)}= (\alpha+\delta_i-\delta_{i+1})(i) C_{\alpha+\delta_i-\delta_{i+1},\beta,a,b}^{(k)}=\frac{\alpha(i)(i+2)k!}{\alpha\ddot!\beta\ddot!a\ddot!b\ddot!}.
\lim_{\lambda\to\infty}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_{M_r^\alpha(\mathbb{R}^n)}=\left[\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{M_r^\alpha(\mathbb{R}^n)};
F_{( e )}^{\mu \nu}= \left( \begin{array}{c c c c}{{0}}&{{- E_{x}}}&{{- E_{y}}}&{{- E_{z}}}\\{{E_{x}}}&{{0}}&{{- B_{z}}}&{{B_{y}}}\\{{E_{y}}}&{{B_{z}}}&{{0}}&{{- B_{x}}}\\{{E_{z}}}&{{- B_{y}}}&{{B_{x}}}&{{0}}\\ \end{array}\right) .
2\frac{F^{ii}h_{11i}^2}{\kappa_1 (\kappa_1 -\tilde{\kappa}_i )}-\frac{F^{ii}h_{11i}^2}{\kappa_1^2}=\frac{F^{ii}(\kappa_1 +\tilde{\kappa}_i )}{\kappa_1^2 (\kappa_1 -\tilde{\kappa}_i )}h_{11i}^2\geq c(n)\frac{F^{ii}h_{11i}^2}{\kappa_1^2},
Area_{g'}(graph_{\Sigma, g}(\phi)) = \int_{\Sigma}F^{g'}(x, \phi(x), \nabla_{\Sigma}\phi(x))\dvol_{g|_{\Sigma}}(x)\ \ \ \ \forall \phi\in C^1(\Sigma) \|\phi\|^*_{1; \Sigma}\leq \delta_1
\min_{0{}\leq{}i{}\leq{}N}\big\{\|\nabla f(x_i)\|^2\big\}\leq{}\frac{2L\,\big[f(x_0)-f_*\big]}{1 + \sum\limits_{i=0}^{N-1}\left(2 h_i - h_i^2 \right)}\, , \, \forall h_i \in (0,2).
\lim_{\beta \to 0}{{\cal W}^{(3)}\pi^{2\omega}e^{2i\pi \omega}\over C_FC_A (2T)^{4-4\omega}(LT)^2}=-{3\over 2(\omega-1)^2}+{3\gamma -11/2\over(\omega -1)}-{35\over 2}+11\gamma -3\gamma^2 +{\pi^2\over 12}+{\cal O}(\omega -1)\ .
\frac{d}{dt}\frac{W_2^2(\mu_t,\nu)}{2}= \lim_{s\downarrow 0}\frac{W_2^2(\mu_{t+s},\nu) - W_2^2 ( \mu_t , \nu )}{2s}&\ge\lim_{s \downarrow 0}\frac{1}{s}\left( \int_X \varphi_t \, d \mu_{t+s}- \int_X \varphi_t \, d \mu_t \right)= - \int_X \langle \nabla u , \nabla \varphi_t \rangle \,d\mu_t,
{\cal M}^{(n)}(E)=\begin{pmatrix}\tau^{(1)}_1(E) 1 0 \cdots 0 -\tau^{(2)}_1(E) 0 1 \cdots 0 \vdots 1 (-1)^{n-1}\tau^{(n)}_1(E) 0 0 \cdots 0 \end{pmatrix}.
\lim_{k\to\infty}\|\phi_k \nabla u_k\|_{L^2(\mathbb{R}^n)}= 2 \lim_{k\to\infty}\|\phi_k \nabla u_k\|_{L^2(\Omega)}, & \lim_{k\to\infty}\|\phi_k D(u_k)\|_{L^2(\mathbb{R}^n)}= 2 \lim_{k\to\infty}\|\phi_k D(u_k)\|_{L^2(\Omega)}.
\begin{array}{llllll}L_{3, a}^{1}= \langle x_1, x_2, (1 + \sqrt{1+4a}) h_1+ (1 - \sqrt{1+4a}) h_2 \rangle, L_{3, a}^{2}= \langle x_1, x_2, (1 - \sqrt{1+4a}) h_1+(1 + \sqrt{1+4a}) h_2 \rangle.\end{array}
V_m=\tilde{\Phi}(\alpha_5,\alpha_3,\alpha_2,\alpha_1;V_a))),Z_m=Z_a+\alpha_4, &V_b=\tilde{\Phi}_5(\beta_5;\mathcal{F}(\sigma_3,\sigma_2;\tilde{\Phi}_1(\beta_1;V_m))),Z_b=Z_m+\beta_4, &V_b=\tilde{\Phi}_5(\gamma_5;\mathcal{F}(\sigma'_3,\sigma'_2;\tilde{\Phi}_1(\gamma_1;V_a))),Z_b=Z_a+\gamma_4,
\|\nabla^{k_0+1}u\|^2_{L^2(B)}+\|\nabla^{k_0}u\|^2_{L^4(B)}+\sum_{1\le j\le k_0-1}\|\nabla^{j}u\|^2_{L^{\infty}(B)}\le C_{k_0}\|\nabla^{k_0+1}u_0\|^2_{L^2(B)}+C_{k_0}\le C_k<\infty
\mathrm{d}\varepsilon_{x,y}(\omega_1,\omega_2) = \frac{1}{128 \pi^4}J_{x,y}(\omega_1,\omega_2)^2 \, \mathrm{d}_{2(k+3)}\, \omega_1 \, \mathrm{d}_{2(k+3)}\, \omega_2,
\left\|\sum_{i=1}^{L}b_i\sum_{j=1}^N \varepsilon_{j,i}\langle x,f_j\rangle x_j \right\| \leq \sum_{i=1}^{L}b_i\left\|\sum_{j=1}^N \varepsilon_{j,i}\langle x,f_j\rangle x_j \right\| \leq \sum_{i=1}^L b_i C\|x\|=C\|x\|.
\Lambda =d\Lambda_0 +\sum_{i=1}^{r}\mu_i\epsilon_i -\epsilon\sum_{i=1}^{r}\mu_i, \rho =r\Lambda_0+\sum_{i=1}^r (r-i)\epsilon_i-\epsilon\sum_{i=1}^r (r-i)=r\Lambda_0-\sum_{i=1}^r i\epsilon_i+\epsilon\sum_{i=1}^r i, \Lambda+\rho =(r+d)\Lambda_0 +\sum_{i=1}^{r}(\mu_i-i)\epsilon_i -\epsilon\sum_{i=1}^{r}(\mu_i-i)
R [ f ( z ) , g ( z ) ] = \pi i \left[ \sum_{k}{\mathrm{R e s}}_{z = z_{g , k}}g ( z ) + \sum_{k}{\mathrm{R e s}}_{{\mathrm{I m}}z_{f , k}> 0}f ( z ) - \sum_{k}{\mathrm{R e s}}_{{\mathrm{I m}}z_{f , k}< 0}f ( z ) \right] .
J[u^*](\hat{t},\hat{x})+K_{(0,\rho)}[\varphi](\hat{t},\hat{x})+K_{(\rho,\hat{t})}[u^*](\hat{t},\hat{x}) &+F_*(\hat{t},\hat{x},u^*(\hat{t},\hat{x}),\nabla\varphi(\hat{t},\hat{x}),\nabla^2\varphi(\hat{t},\hat{x}))\le0.
\mu_0 = - \frac{\sqrt{2m}}{\pi}, ~~~~~ \mu_1 = -\frac{\sqrt{2m}}{4\pi}\left(m - \frac{1}{2}\right), ~~~~~ \mu_2 =-\frac{\sqrt{2m}}{16\pi}\left( m^2 - 5m + \frac{9}{4}\right).
\psi^{x_1}_{-\ell}(\lambda^c_xs)=\Phi^c_{x_{-\ell+1}}\left((\lambda^c_{x_{-\ell+1}}\cdots\lambda^c_{x_{-1}}\lambda^c_x)^{-1}\lambda^c_xs\right)=\psi^x_{-\ell+1}(s).
\int_{0}^{2\pi}cos^2(\theta)d\theta =\int_{0}^{2\pi}sin^2(\theta) d\theta= \pi &\int_{0}^{2\pi}cos^{4}(\theta) d\theta =\int_{0}^{2\pi}sin^4(\theta) d\theta= \frac{3\pi}{4}&\int_{0}^{2\pi}cos^2(\theta) sin^2(\theta) d\theta=\frac{\pi}{4},
(\widetilde{\chi}a)(T) =\widetilde{\chi}(V_{a}^{*}TV_{a}) =\chi(V_{a}V_{a}^{*}TV_{a}V_{a}^{*}) = \chi(E_{a})\chi(T)\chi(E_{a}) = 1_{A}(a)\chi(T)1_{A}(a) = \chi(T).