harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L_{g_{0}}\varphi_{a, \lambda}= u_{a}^{\frac{n+2}{n-2}}L_{g_{a}}\frac{\varphi_{a, \lambda}}{u_{a}}= 4n(n-1)\varphi_{a, \lambda}^{\frac{n+2}{n-2}}\gamma_{n}\vert \nabla G_{a}^{\frac{1}{2-n}}\vert^{2}_{g_{a}}+\frac{u_{a}^{\frac{2}{n-2}}R_{g_{a}}}{\lambda}\varphi_{a, \lambda}^{\frac{n}{n-2}},
\frac{d}{dt}\varphi_{t}^{\epsilon}=\frac{d}{dt}\varphi_{t}+\epsilon\frac{d}{dt}(w_{t}\circ\varphi_{t})=v_{t}\circ\varphi_{t}+(Dw_{t})\circ\varphi_{t}v_{t}\circ\varphi_{t}+o(\epsilon).
F_{2}(\beta; \xi, \tau) = \frac{e^{-\tau\sqrt{1-\partial_{\xi}^{2}}}}{2K_{1}(\beta)}\int_{-\infty}^{\infty}e^{-\beta\sqrt{1+k^{2}}+ ik\xi}dk & = \frac{1}{2K_{1}(\beta)}\int_{-\infty}^{\infty}e^{- (\beta+\tau) \sqrt{1+k^{2}}+ ik\xi}dk.
\begin{cases}u_t -d_u\Delta u = f(x,y,t), (x,y) \in \Omega^S, t\in [0,1]; &u(x,y,t) = 0, (x,y) \in \partial \Omega^S, t\in [0,1]; &u(x,y,0) = y(y - 1)\left(- \frac{y^2}{2}+ x + 1\right)\left(\frac{y^2}{2}+ x - 1\right), (x,y) \in \Omega^S,\end{cases}
B(E, L^{\infty}_{\sigma}(\Omega; F^*)) = \bigl(E \overset{\wedge}{\otimes}L^1(\Omega;F) \bigr)^* & = \bigl( E \overset{\wedge}{\otimes}L^1(\Omega) \overset{\wedge}{\otimes}F \bigr)^* & = L^1(\Omega; E \overset{\wedge}{\otimes}F)^* & = L^{\infty}_{\sigma}(\Omega; B(E, F^*)).
[n]_q! S_n(A) = \sum_I\tilde c_I(q)\tilde \Theta^I(q)= \sum_I \frac{[n]_q!}{[i_1]_q[i_1+i_2]_q\cdots[i_1+i_2+\cdots i_r]_q}\tilde \Theta^I(q)
u(\varphi,\theta)=a v_z^\perp+b\xi+\alpha v_x^\perp+\beta v_y^\perp+\gamma v_{(x,z)}^\perp+\delta v_{(y,z)}^\perp+\sum_{n= 2}^\infty(a_n(\varphi)\cos(n\theta)+b_n(\varphi)\sin(n\theta)),
\sum_{z\in\Lambda^e_n}\sum_{w\sim z}{\bf 1}_{\{w,z\notin x_1^e\}}+ \Big(\sum_{z\in x_1^e}\sum_{w\sim z}+ \sum_{z\in x_2^e}\sum_{w \in x_1^e}\Big) \;:=\;\eqref{term1}(i)\,+\,\eqref{term1}(ii),
A^{S_1G_1S_2G_2P}(z)=\frac{1}{z^{4/3}}\begin{pmatrix}-t^{1/3}0 0 0 -\omega t^{1/3}0 0 0 -\omega^2 t^{1/3}\end{pmatrix}\hphantom{A^{S_1G_1S_2G_2P}(z)=}{}+\frac{1}{z}\begin{pmatrix}\theta^\infty_1/3-2/3 0 0 0 \theta^\infty_1/3-2/3 0 0 0 \theta^\infty_1/3-2/3 \end{pmatrix}+\cdots.
y_{(2k-1)m+2i-1,j}+ y_{(2k-1)m+2i-1,j+1}= 2MN+3 -( x_{2i-1,j}+ x_{2i-1,j+1}) \mbox{and}y_{(2k-1)m+2i,j}+ y_{(2k-1)m+2i,j+1}= 2MN+3 -( x_{2i,j}+ x_{2i,j+1}).
\mathcal{A}^h(z) = a_0+2\sum_{j=1}^N a_j T_{2j}(\rho \cos \tfrac\theta2), \mathcal{B}^h(z) = 2 (\sin\tfrac\theta2 + |\alpha| \cos\tfrac\theta2) \sum_{j=1}^N b_j U_{2j-1}(\rho \cos \tfrac\theta2).
(\partial_{t}-\underline{\Delta})\left\|\overline{\nabla}{f}\right\|^{2}&=-2f^{2}_{ij}+2g'(\overline{\nabla}{f},\overline{\nabla}\left\|\overline{\nabla}{f}\right\|^{2})-2g'(\overline{\nabla}{f},\overline{\nabla}\,Scal')-4Ric'_{ij}\overline{\nabla}_{i}f\overline{\nabla}_{j}f.
Z_{b^{-1}}\phi(x, \omega) = \sqrt{\tfrac{\pi}{2}}\, b \, \vartheta_1'(0, i b) \, e^{-\pi b x^2}\frac{\vartheta_3 (\omega - i b x, i b)}{\vartheta_4 (\omega, i b) \vartheta_4 (x, i/b)}& = \sqrt{\tfrac{\pi}{2^{3/2}}}\, b^{3/4}\, \vartheta_1'(0, i b) \,\frac{Z_{b^{-1/2}}\, \varphi (x, \omega)}{\vartheta_4 (\omega, i b) \vartheta_4 (x, i/b)},
U_1(1)V_1(1)\bar{\Lambda}'(1)V_1(1)^*U_1(1)^*U_1(0)e_i &=U_1(1)V_1(1)\bar{\Lambda}'(1)V_1(1)^*e_{\sigma(i)}&=U_1(1)V_1(1)\bar{\Lambda}'(1)e_{\sigma'(i)}&=U_1(1)V_1(1)\lambda'_{\sigma'(i)}(1)e_{\sigma'(i)}&=U_1(1)\lambda'_{\sigma'(i)}(1)e_{\sigma(i)}&=\lambda'_{i}(0)U_1(0)e_i,~~~\forall i\in S_n
\lVert f\circ X_w^{-1}-f \rVert_\infty = \sup_{v\in F_n}\lvert f(X_w^{-1}v)-f(v)\rvert = \sup_{v\in F_n}\lvert f(wvw^{-1})-f(v)\rvert \leq \sup_{v\in F_n}\lvert f(w)+f(vw^{-1})-f(v)\rvert+D(f) \leq \sup_{v\in F_n}\lvert f(w)+f(w^{-1})\rvert+2D(f) < \infty
\Pr\left({\boldsymbol G}_u \to{\boldsymbol G}_t\right) = \frac{1}{2}- \frac{\mu_{u, t}}{2}\sum_{r = 0}^{K - 1}\binom{2 r}{\!r}\left(\frac{1 - \mu_{u, t}^2}{4}\right)^{\!r}
\|F_A\|_{L^{7/2}(B_r(x))}= \|F_{A'}\|_{L^{7/2}(\mathbb{B})}\hat{\mathcal{E}}(A, B_r(x)) = \hat{\mathcal{E}}(A', \mathbb{B}) = \|F_{A'}\|^2_{L^2(\mathbb{B})}.
\alpha^{(N)}_n=(1-aq^{2n-N})(aq^{1-N})_N\sum_{j\in \mathbb{Z}}(-1)^j\frac{q^{N(n-j)+j(j-1)/2}}{(aq^{2n-N-j})_{N+1}}\left[{N\atop j}\right]\alpha_{n-j},
\frac{d}{d\theta}\int T(X_{1},...,X_{n})\prod_{i=1}^{n}f(x_{i}|\theta)dx_{1}...dx_{n}=\int T(X_{1},...,X_{n})[\frac{\partial}{\partial\theta}\prod_{i=1}^{n}f(x_{i}|\theta)]dx_{1}...dx_{n}
|T_{b}^{m}f(x)|\leq c_{n,T}\sum_{j=1}^{3^{n}}\sum_{k=0}^{m}\binom{m}{k}\sum_{Q\in\mathcal{S}_{j}}|b(x)-b_{Q}|^{m-k}\left(\frac{1}{|Q|}\int_{Q}|b-b_{Q}|^{k}|f|\right)\chi_{Q}(x).
k_{x}=\limsup_{z\rightarrow 0}\frac{\ln\|(z-T)^{-1}x\|_p}{\ln\|(z-T)^{-1}\|}\geq\limsup_{z\rightarrow 0}\frac{\ln\big|f_0\big((z-T)^{-1}x\big)\big|}{\ln\|(z-T)^{-1}\|}.
\tilde{\alpha}(d,d,a_3, \dots, a_k) = \alpha'(d,d,a_3, \dots, a_k) + \tilde{\sigma}(d,d,a_3, \dots, a_k) = \sigma'(d,a_3, \dots, a_k) + \delta'(d,a_3, \dots, a_k) + \sigma'(d,a_3, \dots, a_k) = \delta'(d,a_3, \dots, a_k)
( q^{2}+ y^{2}M_{\ast}^{2})^{k}\, C_{2 k}^{\frac{1}{2}d - 1}\left( \frac{y \left( q^{2}+ M_{\ast}^{2}\right)}{\sqrt{q^{2}+ y^{2}M_{\ast}^{2}}}\right) = \sum_{l = 0}^{2 k}B_{l}\, ( q^{2}, M_{\ast}^{2}) \, C_{l}^{\frac{1}{2}d - 1}( y )
\vert F ( z ) \vert < \vert z \vert \varepsilon e^{( c - 2 t ) \vert y \vert}\quad \textrm{o r}\quad \vert f ( z ) \vert < \frac{M e^{2 ( 1 - t ) \vert y \vert}}{\vert z \vert^{\alpha - 1}}, \quad z = x + i y , \quad \vert z \vert \to \infty
Q(t^{\frac{a+\epsilon}{2}},t^{\frac{1-a-2\epsilon}{2}}) =P(|B(H(T_{t^{\frac{a+\epsilon}{2}}}))|>|\tilde{B}(\tilde{H}(\tilde{T}_{t^{\frac{1-a-2\epsilon}{2}}}))|) &=P(|\theta(T_{t^{\frac{a+\epsilon}{2}}})|>|\tilde{\theta}(\tilde{T}_{t^{\frac{1-a-2\epsilon}{2}}})|).
\hat{Z}_{8 8 8}^{1 0 , 1 0}( \Omega ) = \frac{( d e t \I m \ \Omega )^{2}}{2^{3}}\left| \Theta \left[ \begin{array}{l l l}{{1 / 2}}&{{1 / 2}}&{{1 / 2}}\\{{1 / 2}}&{{0}}&{{1 / 2}}\\ \end{array}\right] ( 0 \vert \Omega ) \right|^{8}
\partial_i (b_3-m_3)[\hat{\imath}] =\Upsilon'(0)\{M_x[\partial_i (g(a_1-1)-g(0))[\hat{\imath}]]-M_{\varphi, x}[\partial_i (g(a_1-1)-g(0))[\hat{\imath}]] \}&+\partial_i \{\Upsilon_{\geq 2}[M_x[g(a_1-1)-g(0)]]-M_{\varphi}[\Upsilon_{\geq 2}[M_x[g(a_1-1)-g(0)]]]\}[\hat{\imath}]
F_5 := E\int_{t_0}^{u \wedge \tau_{nR}}|b_s(x^n_{\kappa(n,s)})-b^n_s(x^n_{\kappa(n,s)})|^2 I_{B(R)}ds &\leq E\int_{t_0}^{t_1}I_{\{t_0 \leq s <\tau_{nR}\}}I_{B(R)}|b_s(x^n_{\kappa(n,s)})-b^n_s(x^n_{\kappa(n,s)})|^2 ds.
\Theta(p,q,m)=\left\{H\in C^{m\times n}\ \forall n|\mbox{Elements of}H\sim p,\right. &\arg\max_{\mathbf{\Sigma}_{X}:\mbox{tr}(\mathbf{\Sigma}_{X})\leq P}E\left[\log \det(\textbf{I}_{n}+H^H\mathbf{\Sigma}_XH)\right]=\frac{P}{m}\textbf{I}_m , &\mbox{and}\lambda_i(H^HH)\sim q, \forall i=1,\ldots,\mbox{rank}(H)\}.
J_{\mathrm{a s y m}}= - \int_{R_{*}^{2}}d^{2}x \; \epsilon_{i j}x^{i}\, \biggl \{\dot{f}\partial_{j}f - \kappa \bar{A}_{j}\epsilon_{k l}\partial_{k}\bar{A}_{l}\biggr \}- n \pi \rho_{e}| \vec{q}|^{2}.
\begin{aligned}A\geq \frac{2}{n}\phi G-C_{4}\phi^{1/2}G^{1/2}-\frac{C_{1}^{2}}{C_{4}R^{2}}\phi^{1/2}G^{1/2}-2K\phi-2\lambda_{1}\phi \geq \frac{1}{n}\phi G-\frac{n}{4}(C_{4}+\frac{C_{1}^{2}}{C_{4}R^{2}})^{2}-2K\phi-2\lambda_{1}\phi.\end{aligned}
(1-1.5)\ge Q^{tide}\equiv\frac{GM_{\odot}/R_{\odot}^{2}}{[GM_{\bullet}/s_{Hill}^{2}-GM_{\bullet}/(s_{Hill}+R_{\odot})^{2}]},s_{Hill}\rightarrow R_{\odot}(\frac{(2-3)GM_{\bullet}}{GM_{\odot}})^{\frac{1}{3}},
E_{T M}( a ) = \frac{\hbar c}{a}\left\{- \frac{{n_{0}}^{3}a^{4}{\Omega}^{4}}{2 4 \pi c^{4}\left( 1 + s \right)}+ \frac{n_{0}a^{2}\Omega^{2}}{4 \pi c^{2}( 1 + s )}+ \mathcal{O}( s + 1 )^{0}\right\}.
\lambda_{c_n\cdots c_1}(\theta) =\frac{d(c_n\cdots c_1\underline a, c_n\cdots c_1\underline b)}{d(c_m\cdots c_1\underline a,c_m\cdots c_1\underline b)}\frac{d(c_m\cdots c_1\underline a,c_m\cdots c_1\underline b)}{d(\underline a,\underline b)}&= \lambda_{c_n\cdots c_{m+1}}(\phi_{c_1\cdots c_m}^{-1}(\theta))\lambda_{c_m\cdots c_1}(\theta).
\begin{aligned}\sum_{k=n+2}^{\infty}\frac{1}{k^{\omega}(k^\gamma-n^\gamma)}= \left( \sum_{k=n+2}^{2n}+ \sum_{k=2n+1}^{\infty}\right) \frac{1}{k^{\omega}(k^\gamma-n^\gamma)}& \leq C \left(\frac{1}{n^\omega}\sum_{k=n+2}^{2n}\frac{1}{k^\gamma-n^\gamma}+ \sum_{k=2n+1}^{\infty}\frac{1}{k^{\omega + \gamma}}\right).\end{aligned}
I_{h,\gamma}=\inf \left\{E[b] \ \Bigg|\b \in L^{2}_{\sigma,\textrm{axi}}(\mathbb{R}^{3}),\H[b]=h \right\}, S_{h,\gamma}=\left\{b\in L^{2}_{\sigma,\textrm{axi}}(\mathbb{R}^{3})\ \middle|\E[b]=I_{h,\gamma},\H[b]=h \right\}.
G_{\delta}= \left\{x\in\partial\Omega\,:\, \exists r^x_i\searrow 0\;\;\; \dfrac{|A\cap B(x,r^x_i)|}{|B(x,r^x_i)\cap \Omega|}<1-\delta \right. &\left. \dfrac{|A'\cap B(x,r)|}{|B(x,r)\cap \Omega|}>\frac12 0 < r < \delta\right\}.
Z_{gYM_2}(M, G)= \sum_{R}\,(\dim R)^{2-2p}\, e^{-\sum_{r=1}^n \, g_r A C_r(R)\, +\, \sum_{\{k_i \}}\, a(k_1,k_2, \ldots) \prod_s \, (C_s(R))^{k_s}}
\int_{- \infty}^{\infty}\frac{d x}{\sqrt{\pi}}\, e^{i \lambda x^{2}}= \lim_{\epsilon \rightarrow 0}\int_{- \infty}^{\infty}\frac{d x}{\sqrt{\pi}}\, e^{i \lambda x^{2}}\, e^{- \epsilon x^{2}}= \frac{1}{| \sqrt{\lambda}|}\exp \left( \frac{i \pi}{4}\mathrm{s g n}\, \lambda \right) ,
\gamma^{0}= \left( \begin{array}{r r}{{0}}&{{- i}}\\{{i}}&{{0}}\\ \end{array}\right) \; \; \gamma^{1}= \left( \begin{array}{r r}{{0}}&{{i}}\\{{i}}&{{0}}\\ \end{array}\right) \; \; \gamma^{2}= \left( \begin{array}{r r}{{i}}&{{0}}\\{{0}}&{{- i}}\\ \end{array}\right)
\begin{aligned}R(t)S(t) = KR^2(t)\iint_{\mathbb T \times \mathbb R}\sin^{2}(\theta-\phi)f(\theta,\omega,t) ~ d\theta d\omega = K\iint_{\mathbb T \times \mathbb R}(-R\sin(\theta-\phi))^2 f(\theta,\omega,t) ~ d\theta d\omega = K\iint_{\mathbb T \times \mathbb R}|\partial_{\theta}(\sigma\cdot J)|^{2}f ~ d\theta d\omega. \end{aligned}
\bar \omega(y,s) \,=\, \frac{\Gamma}{\nu s}h_s\Bigl(\frac{y_1}{\sqrt{\nu s}}\,,\frac{y_3}{\sqrt{\nu s}}\Bigr)\,e_2 \,=:\, \bigl(0,\bar \omega_2(y_1,y_3,s), 0\bigr)\,.
G(Z) =\eta_{\ell+2,m-2}(D_Z)\frac{\partial^2}{\partial S^2}(P_{0+}\vec\Phi)(b_Z) \qquad-\eta_{\ell+2,m-2}(D_Z)\frac{\partial^2}{\partial S\partial T}(P_{1+}\vec\Phi)(b_Z) \qquad+\eta_{\ell+2,m-2}(D_Z)\frac{\partial^2}{\partial T^2}(X_+\vec\Phi)(b_Z).
\tilde{E}^{(2)}_{a,b}(\tau) = a_0(\tilde{E}^{(2)}_{a,b}) + \sum_{\substack{m,n\geq 1 m \equiv a (N)}}\zeta_N^{bn}n q^{mn/N}+ \sum_{\substack{m,n\geq 1 m \equiv -a (N)}}\zeta_N^{-bn}n q^{mn/N}- 2 \sum_{m,n\geq 1}n q^{mn}.
H_n[f]=\sum_{k_1=1}^\infty\dotsi\sum_{k_n=1}^\infty\frac{a_{k_1}\dotsb a_{k_n}}{k_1^2k_2^3\cdots k_n^{n+1}}\prod_{1\le i<j\le n}\big(k_i^{-1}-k_j^{-1}\big).
\varphi_{s'}^*(\varphi_s^*)^{-1}(\omega_{T|_U}+\pi_{T|_U}^*\Theta) = (\varphi(-t_{ss'}))^*(\omega_{T|_U}+\pi_{T|_U}^*\Theta) = \omega_{T|_U}-\pi_{T|_U}^*dt_{ss'}+\pi_{T|_U}^*\Theta = \omega_{T|_U}+\pi_{T|_U}^*s'^*\omega = \varphi_{s'}^*\omega
\begin{aligned}u_1(t)=\lambda_1\int_{0}^{1}k_1(t,s) f_1(u_1(s),u_2(s))\,ds, u_2(t)= \lambda_2\int_{0}^{1}k_2(t,s) f_2(u_1(s),u_2(s))\,ds,\end{aligned}
\varphi(K,\phi) = \int^{\infty}_{0}\log \left(1+x\phi \right) f_{x_K}(x)dx = K \log(e) \int^{\infty}_{0}\ln \left(1+x\phi \right) e^{-Kx}dx = - \log(e) e^{\frac{K}{\phi}}\mathbf{E_i}\left(-\frac{K}{\phi}\right).
\left\{[0,1]^2;\varphi_1(x,y) = \left(\frac{x+1}{2},\frac{y}{2}\right),\varphi_2(x,y) = \left(\frac{x}{2},\frac{y+1}{2}\right),\varphi_3(x,y) = \left(\frac{x}{2},\frac{y}{2}\right)\right\}.
\lbrack A, \lbrack A, \lbrack A, A^*\rbrack_q \rbrack_{q^{-1}}\rbrack = (q^2-q^{-2})^2 \lbrack A^*, A\rbrack, & \lbrack A^*, \lbrack A^*, \lbrack A^*, A \rbrack_q \rbrack_{q^{-1}}\rbrack = (q^2-q^{-2})^2 \lbrack A, A^*\rbrack.
\int_{\Psi_{j+1}- \Psi_j}x^k e^{x^r / \hbar}dx = \frac{1}{r}e^{\pi i\frac{k+1}{r}+ 2\pi i j \frac{k+1}{r}}\hbar^{(k+1)/r}\Gamma\left(\frac{k+1}{r}\right) \left(e^{2\pi i(k+1) / r}- 1\right)= \frac{1}{r}C_k \zeta^{j(k+1)}
\sum_{j\ne i}\frac{2}{z_i-z_j}{}+ \frac{(4a+3)z_i^4 - (4a-4b_1-2l-d+3)z_i^3 - 4(b_1-2b_2)z_i^2 - 4(2b_2-3b_3)z_i - 12b_3}{2z_i^4(z_i-1)}= 0, i=1, 2, \ldots, n,
\begin{aligned}\left|\int_0^\infty\rho^{\gamma-3}\rho_t^2\frac{u}{r}dx\right|\leq\epsilon\int_0^\infty\rho^\gamma r^4(\log\rho)_{xt}^2dx+C_\epsilon\int_0^\infty\rho^{-1}\frac{u^2}{r^2}dx\int_0^\infty(\rho^\frac{\gamma-1}{2})_t^2dx.\end{aligned}
\left|\int_{\mathcal{I}_R}e^{zw}\psi_{\varepsilon}(z)u(z) dz \right| < 2\pi R \sup_{z\in \mathcal{I}_R}|e^{zw}u(z)| = 2\pi R \sup_{z\in \mathcal{I}_R}|e^{z(w-\varepsilon' \xi_0)}|\sup_{z\in \mathcal{I}_R}|e^{z(\varepsilon'\xi_0)}u(z)|.
p(\gamma) =s(\gamma)\prod_{i=1}^{k}(\sum_{l_i\geq0}P(\mu\geq l_i+1)(q(z_{i-1}-a))^{l_i}) &\leq s(\gamma)\prod_{i=1}^{k}(\sum_{l_i\geq0}P(\mu\geq l_i+1)) &=s(\gamma)\prod_{i=1}^{k}(E\mu) &=s(\gamma).
\phi_{h}= \sum_{r = 1}^{h}{\frac{( - 1 )^{r}}{r}}\left({2 h - 1 \atop r - 1}\right) P_{r , 2 h - r}-{\frac{1}{2}}{\frac{( - 1 )^{h}}{h}}\left({2 h - 1 \atop h - 1}\right) P_{h , h}\, ,
Z_{0}:=\{0\},Z_{m}:=\{e_{1},...,e_{m}\}, Y_{m}:=\{w\in H_{0}^{1}(\Theta):\left\langle w,z\right\rangle_{a}=0z\in Z_{m}\}, T_{0}:=(-\infty,\lambda_{1}^{a,b}), \quad T_{m}:=[\lambda_{m}^{a,b},\lambda_{m+1}^{a,b}) m\in\mathbb{N}.
(y_i-y_{i+1}) e_k(y_1,\ldots,y_i)+ \sum_{\substack{0\le p\le i,\ 1\le q\le n-i p+q=k}}(y_i-y_{i+1}) e_{p}(y_1,\ldots,y_i) e_{q}(y_{i+1},\ldots,y_n) =0.
h_{\Pi_{\mu}K}(\theta) =\frac{1}{2}\int_{\partial K}|\langle\theta,n_K(y) \rangle| \phi(y)\, dy-\frac{1}{2}\int_{\partial K}\langle n_K(y),\theta\rangle \phi(y)\, d y &= \int_{\partial K}\langle n_K(y),\theta\rangle_{-}\phi(y)\, d y,
{\cal L}_{\mathrm{h y p . m a s s}}= - c \sigma^{\prime}\bar{\Psi}\Psi + c \sigma^{\prime}( F_{i}^{*}\phi^{i}+ \phi_{i}^{*}F^{i}) - c k^{2}( \sigma_{3})_{j}^{i}\phi_{i}^{*}\phi^{j},
\lambda_{i}=\lambda_{i}(x, y, 1, w_{1}, w_{2}, \ldots,w_n, w_{n+1},w_{n+2},w_{n+1},w_{n+2},\ldots), \rho_{i}=\rho_{i}(x, y, 1, w_{1}, w_{2}, \ldots,w_n, w_{n+1},w_{n+2},w_{n+1},w_{n+2},\ldots).
\mathsf{Z}_{\alpha}(u, \mathbf{z}_2, q_2)=\Big[ \mathsf{Z}(t, u, \mathbf{z}_1, \mathbf{z}_2, q_1, q_2) \Big]_{t^k q^{W_1 \cdot \alpha}e(\mathbf{z}_1 \cdot \alpha)}.
I_\mu(u)= \frac{1}{2}\int_{\mathbb{R}^N}\left(|\nabla u|^2+(\mu+1)u^2\right) d x+\int_{\mathbb{R}^N}F_1(u)d x-\int_{\mathbb{R}^N}F_2(u) d x \geq \frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^2 d x+\int_{\mathbb{R}^N}F_1(u)d x-C C_1 a^{\left(1-\beta_p\right) p}\left(\int_{\mathbb{R}^N}|\nabla u|^2 d x\right)^{\frac{p\beta_p}{2}}.
\phi_1 := \phi_{6A}(\tau, z) = 2 \zeta^{\pm 1}- 2 + (-2 \zeta^{\pm 2}+ 6 \zeta^{\pm 1}- 8) q + (2 \zeta^{\pm 3}- 8 \zeta^{\pm 2}+ 16 \zeta^{\pm 1}- 20) q^2 + O(q^3)
\left|\sum_{j=0}^{M-1}\log J^u(f^jx)-M\beta\right| \leq\sum_{n=1}^{N-1}\frac{2q_{n}\kappa_{n}}{n}+\frac{2q_{N}s}{N}+(M-(\kappa_1q_1+\cdots+\kappa_{N-1}q_{N-1}+sq_{N}))(\log 5-\beta) &\leq \frac{3q_{N-1}\kappa_{N-1}}{N}+\frac{2q_{N}s}{N}+q_N(\log 5-\beta)\leq\frac{4M}{N},
s_\Sigma(\bar x) \le s_\Sigma (x) + \frac{1}{\lambda}d(x,\bar x) \le r_m + \frac{\bar K}\lambda r_{m'}< \left(\frac 1{\bar K^2}+ \frac 1{\lambda\bar K}\right) r_{m'-2}\le \frac{r_{m'-2}}{\bar K}\le \frac{r_i}{\bar K}.
\frac{1}{2\pi}\langle e^{-\kappa^2|x|^2/8}\chi_{\kappa x/2}, e^{-\kappa^2|y|^2/8}\chi_{\kappa y/2}\rangle = \frac{1}{2\pi}e^{-\kappa^2|x|^2/8}e^{-\kappa^2|y|^2/8}e^{\kappa^2 x\cdot y/4}= \frac{1}{2\pi}e^{-\kappa^2|x-y|^2/8},
S_{\mbox{\footnotesize gf}}= - \frac{1}{64 e^2}\int d^4x\,d^4\theta\,\Bigg(V D^2 \bar D^2\Big(1 + \frac{\partial^{2n}}{\Lambda^{2n}}\Big) V+ V \bar D^2 D^2\Big(1+ \frac{\partial^{2n}}{\Lambda^{2n}}\Big) V\Bigg),
\delta{\widetilde \varphi}= - \frac{\mathrm{e}^{\varphi_{0}}\, ( m - \chi_{0}\, n )^{2}- \mathrm{e}^{- \varphi_{0}}n^{2}}{2 \,{\widetilde \Delta}_{m , n}}\, \frac{{\widetilde Q}_{1}}{r^{6}}, \delta{\widetilde \chi}= \frac{\mathrm{e}^{- \varphi_{0}}\, ( m - \chi_{0}\, n ) \, n}{{\widetilde \Delta}_{m , n}}\, \frac{{\widetilde Q}_{1}}{r^{6}}~ ~ ,
- \ln ({\det}^{\prime}( L^{\dag}L ) ) = Z^{\prime}( s ) |_{s = 0}= \gamma_{E}Z ( 0 ) +{\mathrm{F i n i t e}}_{\epsilon \rightarrow 0}\int_{\epsilon}^{\infty}\frac{d t}{t}\mathrm{T r}^{\prime}( e^{- t L^{\dag}L})
\pi_\ell^{O(4p,4q)}|_{[Sp(p)\times Sp(1)]\times [Sp(q)\times Sp(1)]}=\sum_{\substack{m=0 begin{align*}.1ex] 0 \le k \le m/2}}^\infty sum_{\substack{0\le e \le d begin{align*}.1ex] 0\le e'\le d' begin{align*}.2ex] d+e = m+\ell+4q begin{align*}.1ex] d'+e' = m-2k}}\pi^{Sp(p)\times Sp(1)}_{d,e}\otimes \pi^{Sp(q)\times Sp(1)}_{d',e'}.
\begin{aligned}Z_{s,s+k+1}\left[ Z_s,t+\tau; t_1+\tau,\dots,t_k+\tau,0;v_{s+1},\dots,v_{s+k},v_{s+k+1}; \right. & \left.\omega_1,\dots,\omega_k,\omega_{k+1}; i_1,\dots,i_k,i_{k+1}\right] & \in \mathcal{K}_{s+k+1}\cap \mathcal{U}_{s+k+1}^\eta\end{aligned}
K=\sqrt{\frac{(ab^{2}-a^{2}b-ad^{2}+bc^{2})(ab^{2}-a^{2}b-ac^{2}+bd^{2})}{4(b-a)^{2}}-(\frac{c^{2}+d^{2}-a^{2}-b^{2}}{4})^{2}}.
\begin{array}{ccc}K^{-1}_{ab}= p \left(\begin{array}{cc}\sqrt{g}{1 \over \sqrt{g}}{1 \over \sqrt{g}}\sqrt{g}\end{array}\right) + (1-p)\left(\begin{array}{cc}g{1 \over g}{1 \over g}g\end{array}\right)\end{array}
\begin{aligned}|- b(x) \cdot D\varphi (x) + f (x)| = |\bar f (x)| \leq \left( \log \frac{\kappa}{r}- \frac{\log 2}{2}\right)^{-1}\int_{\tau_-^\delta (y)}^{\tau_+^\delta (y)}|f (X(t, y))| \, dt,\end{aligned}
[\begin{matrix}1&-S_{1}&-S_{2}&0&0&0\\ 0&1&1&1&0&0\\ 0&F_{1}&F_{2}&0&1&0\\ 0&P_{1}&P_{2}&0&0&1\\ \end{matrix}][\begin{matrix}z\\x_{1}\\x_{2}\\x_{3}\\x_{4}\\x_{5}\end{matrix}]=[\begin{matrix}0\\L\\F\\P\end{matrix}],[\begin{matrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\\x_{5}\end{matrix}]\ge0.
{\mathcal{J}}_2(x,t)\leq C \left(\int_{|x-y|\leq 3R^N(t)}\ \rho^N(y,t)^{\frac53}\,dy\right)^{\frac35}\left(\int_{a< |x-y|\leq 3R^N(t)}\frac{1}{|x-y|^5}\,dy\right)^{\frac25}&\leq CQ^N(3R^N(t),t)^{\frac35}\left[ a^{-\frac45}+R^N(t)^{-\frac45}\right].
q^{(i)}_{n,d}=q^{(i)}_{n-1,d+r-2}+i\cdot s_{n-1,d+i-2}+2\sum\limits_{j=0}^{n-1}\sum\limits_{k=0}^{d-2}q^{(i)}_{j,k}\cdot s_{n-j-1,d-k-2}.
-{\textstyle\frac{1}{2\sqrt{2}}}e^{-\phi}\sigma^{ab}F^{+}_{ab}\alpha_{IJ}\gamma_{\hat{\imath}}\epsilon_{J}=\gamma^{j}[-{\textstyle\frac{1}{\sqrt{2}}}\omega_{\hat{\imath}}F^{+}_{0j}+{\textstyle\frac{1}{\sqrt{2}}}e^{-2\phi}F^{+}_{0j}\gamma^{i}\gamma^{0}] \alpha_{IJ}\epsilon_{J}\, .
0 \geq Q'' = \frac{v_\epsilon^{(5)}}{v_\epsilon''}- 3 \frac{v_\epsilon''' v_\epsilon^{(4)}}{(v_\epsilon'')^2}+ 2 \frac{(v_\epsilon''')^2}{(v_\epsilon'')^2}+ (1-\alpha)(\log \theta)'''.
\xi=\frac{\sqrt{\ell_r}}{(2+o(1))\ell_{r-1}\left(\log\tfrac{\ell_r}{\ell_{r-1}}\right)^2}\ll \zeta^{-(r-2)/2}\ll(\log k)^{-1}=O\bigl((\log \ell_r)^{-1}\bigr),
\left( \begin{array}{cc}T_{n1}T_{n2}\end{array}\right) = \left( \begin{array}{cc}\overline{X}_1^* \overline{X}_2^* \end{array}\right) - \left( \begin{array}{cc}\sqrt{n_1}G_m^{-1}(p) \sqrt{n_2}G_m^{-1}(p) \end{array}\right) + o_P(1),
\left| \mathcal{B}_{p h y s}\right\rangle = \int_{- \infty}^{\infty}d \rho \sum_{m \in Z}\ \sum_{r \in Z_{+}}\, e^{- i \rho \Theta}\, \left| B_{p^{+}= m + \alpha , \, p^{-}= - \frac{r}{m + \alpha}, \, p_{2}= \rho}\right\rangle .
\sum_{1\leqslant k_1\leqslant k_2\leqslant k_3\leqslant n}k_1 k_2 k_3(n+3)^{k_1-1}(n+2)^{k_2-k_1}(n+1)^{k_3-k_2}n^{n-k_3}={n+2\choose 3}n^{n+2}.
\binom{a}{t}\,\geq\, \left(\frac{a}{t}\right)^t\,\geq\,\left(\frac{\sqrt{d}+\alpha}{2\left(\tau + 1 /\sqrt{d}\right)}\right)^{\tau \sqrt{d}}\,=:\, \gamma_{\alpha\tau}(d) \,\geq\, 1 \,.
\pi^2_{(\underline\alpha)}=\left(\pi^{2-}_q\atop\pi^{2+}_{\dot q}\right)=\left(\nabla\theta^{2-}_q+\frac12\gamma^i_{q\dot r}\theta^{2+}_{\dot r}\Omega^{-2i}\atop \nabla\theta^{2+}_{\dot q}+\frac12\tilde\gamma^i_{\dot qr}\theta^{2-}_r\Omega^{+2i}\right).
i ( 2 \pi )^{4}G ( \xi \eta ;{\xi}^{\prime}{\eta}^{\prime}) = \sum_{1 2 3}{\Delta_{1}}^{- 1}{\Delta_{2}}^{- 1}\int d{{\hat{q}}_{1 2}}^{\prime \prime}M d{\sigma_{1 2}}^{\prime \prime}K ({\hat{q}}_{1 2},{{\hat{q}}_{1 2}}^{\prime \prime}) G ({\xi_{3}}^{\prime \prime}{\eta_{3}}^{\prime \prime};{\xi_{3}}^{\prime}{\eta_{3}}^{\prime})
D_{\widetilde{x}}\widetilde{u^{\alpha}_{J_1;J_2}}=\left(D_x\widetilde{u^{\alpha}_{J_1;J_2}}\right) (D_x\widetilde{x})^{-1}\frac{\operatorname{d}}{\operatorname{d}\!\varepsilon}(M(\varepsilon)^{-1})=-M(\varepsilon)^{-1}\frac{\operatorname{d}\!M(\varepsilon)}{\operatorname{d}\!\varepsilon}M(\varepsilon)^{-1}.
\begin{array}{c}eC \supset \cup_{\alpha \in J}h (\psi_{\alpha}d_{\alpha}\psi_{\alpha}^{-1}) h^{-1}C =h ( \cup_{\alpha \in J}\psi_{\alpha}d_{\alpha}\psi_{\alpha}^{-1}) h^{-1}C \,,\end{array}
\tilde{\phi}_n^k (t,y) = \begin{cases}0 t= x_n, 5 \psi_n^k (t)\theta_n |g_n(t)| |y| \geq 5 \theta_n |g_n(t)|, \psi_n^k(t) |y| |y| \leq 5\theta_n |g_n(t)|;\end{cases}
\nabla_{KZB}=d-\beta B-\alpha A-\alpha\sum_{k=2}^{\infty}p_{k}\mathrm{ad}_{B}^{k}(A), \nabla_{KZB}^{\prime}=d-\left(\beta-df \right)B-\alpha A-\alpha\sum_{k=1}^{\infty}q_{k}\mathrm{ad}_{B}^{k}(A),
\Theta_\mu(100B(Q)) \leq A_0^{-10n(J(Q)-J(R)-1)}\,\frac{\mu(100B(R))}{r(100B(Q))^n}& = A_0^{-10n(J(Q)-J(R)-1)}\Theta_\mu(100B(R))\,\frac{r(B(R))^n}{r(B(Q))^n}.
\tau_2=\begin{cases}\lbrack (m+1,1),(m,2),(m+1,1),(2,1)^4\rbrack,\left(\frac{p_2}{q}\right)=-1, \lbrack (m+1,n),(m,n+1),(\max(m+1,n+1),\min(m,n)),(1^3)^4\rbrack,.\end{cases}
\left[\begin{array}{cc}a_0 1 1 0 \end{array}\right]\left[\begin{array}{cc}a_1 1 1 0 \end{array}\right]\cdots\left[\begin{array}{cc}a_i 1 1 0 \end{array}\right]=\left[\begin{array}{cc}p_i p_{i-1}q_i q_{i-1}\end{array}\right].
F'^{\dag}\left( \frac{\partial{g'}_{\mu\nu}(x,l)}{\partial l_{i}}\right) \equiv - 4{\nabla'}^{\mu}\frac{\partial{g'}_{\mu\nu}(x,l)}{\partial l_{i}}- 2 \left( C - \frac{2}{D}\right) \partial_{\nu}\left[{g}'^{\alpha\beta}(x,l) \frac{\partial{g'}_{\alpha\beta}(x,l)}{\partial l_{i}}\right]=0 \; .
\kappa<\frac{4(n-1)(n-2)}{5n-4+\sqrt{9n^2+8n-16}}=\frac{4(n-1)(n-2)\big\{5n-4-\sqrt{9n^2+8n-16}\big\}}{(5n-4)^2-9n^2-8n+16}&=\frac{5n-4-\sqrt{9n^2+8n-16}}{4}.
\pi=\sqrt{12}\sum_{k=0}^{\infty}\frac{(-3)^{-k}}{2k+1}=\sqrt{12}\sum_{k=0}^{\infty}\frac{(-\frac{1}{3})^{k}}{2k+1}=\sqrt{12}(1-\frac{1}{3\cdot3}+\frac{1}{5\cdot3^{2}}-\frac{1}{7\cdot3^{3}}+\cdot\cdot\cdot)
d_T(t,t') \ge \frac{1}{j}\sum_{t_0=t,t_{1},\dots,t_{j-1}, t_j=t', (t_{i},t_{i+1}) \in F}d_T(t_i,t_{i+1}) = \frac{1}{j}\sum_{t_0=t,t_{1},\dots,t_{j-1}, t_j=t', (t_{i},t_{i+1}) \in F}d(x_{t_i},x_{t_{i+1}}) \ge \frac{1}{|V(T)|}d(x_t,x_{t'}).
\lambda_{\alpha}=\displaystyle\frac{\displaystyle\int_{\Omega}\displaystyle\frac{|\nabla u_{\alpha}|^{2}}{c+\alpha}dx}{\displaystyle\int_{\Omega}u_{\alpha}^{2}\left\{-div\left[\displaystyle\frac{\nabla c}{(c+\alpha)^{2}}\right]\right\}dx}.
\int_{G_n}|\phi(t, u - w) - \phi(t, u - w_n)| \leq 2 \int_{G_n}\|\phi\|_{\infty}\leq 2C \lambda(G_n) \leq 2C \frac{T_{n+1}^2}{16C}= \frac{T_{n+1}^2}{8}.
1+(1-z)(1-z^{-1})\sum_{n=1}^{N}\frac{(-1)^n(1+q^n)q^{\frac{n(3n+1)}{2}}(q)_{N}^2}{(1-zq^n)(1-z^{-1}q^n)(q)_{N-n}(q)_{N+n}}&=\frac{(q)_N^{2}}{(zq)_N(z^{-1}q)_N}\sum_{n=0}^{N}\frac{(z)_{n}(z^{-1})_{n}q^n}{(q)_n}.
\phi_{\chi}^o(s) = \frac{\sqrt{\pi}}{4}\left[P^{-\frac{s}{2}}\frac{z(s)}{1+s}-j \frac{P^{-\frac{1-s}{2}}}{G\left(\overline{\chi},P\right)}\frac{z(1-s)}{2-s}\right]; \mathfrak{Re}(s) \in (0,1)
\varepsilon_{0}^{\beta}= \frac{| e | H}{( 2 \pi )^{2}}\sum_{n = 0}^{\infty}\int_{- \infty}^{\infty}d p \epsilon_{n}\left\{\frac{- 2 + \mathrm{e}^{\beta \epsilon_{n}}}{1 - \mathrm{e}^{\beta \epsilon_{n}}}\right\}.
\|I_1\|_{X^{\beta}}=\|(e^{-Ah}-I)A^{-\sigma}(A^{\sigma+\beta}e^{-A(t-t_0)}u_0)\|_{L^p(\Omega)}\leq Ch^{\sigma}\|A^{\sigma+\beta}e^{-A(t-t_0)}u_0\|_{L^p(\Omega)}&\leq Ch^{\sigma}(t-t_0)^{-\sigma-\beta}e^{-(t-t_0)}\|u_0\|_{C^0(\bar\Omega)}.
+e^2A_\mu{{\{-4\alpha+8g(1-\alpha )\partial^2(2\alpha^2g-a+a^2g\partial^2)\}}\over{(ag\partial^2-1)^2+4\alpha^2g^2\partial^2}}\epsilon^{\mu\nu\lambda}\partial_\nu A_\lambda -4J_\mu A^\mu].
\begin{array}{rl}{\rm E}[z\mid E_B] ={\rm E}[c_B^t(A_B)^{-1}b \mid E_B] & ={\rm E}[c_B^t(A_B)^{-1}(A_Bx_B+A_{B^c}x_{B^c}) \mid E_B] & ={\rm E}[c_B^tx_B+c_B^t (A_B)^{-1}A_{B^c}x_{B^c}\mid E_B]. \end{array}
m [ ( \phi_{a^{\prime}}\otimes \phi_{R_{a^{\prime}}}) ( i d \otimes S ) \Delta ( t ) ] = \phi_{a^{\prime}}( m [ ( i d \otimes S ) \Delta ( t ) ] ) = \phi_{a^{\prime}}( \bar{e}( t ) ) = \phi_{a^{\prime}}( 0 ) = 0 ,
\Box_{g_O}\psi_O = \Omega^{-3}\Box_{g_I}\psi_I \Rightarrow \Box_{g_O}\psi_O = \Omega \cdot g^{\mu \nu}_O \cdot \partial_{\mu}\psi_O \cdot \partial_{\nu}\psi_O - \frac{2}{M}\partial_u \psi_O \cdot \psi_O - \frac{2D}{M}\partial_{r'}\psi_O \cdot \psi_O - \frac{r' - M}{(r')^2}\psi_O^2 .
\int_\Omega \Big(\big\langle \frac{\partial d_\epsilon(t)}{\partial x_i},\frac{\partial d_\epsilon(t)}{\partial x_j}\big\rangle \frac{\partial Y^i}{\partial x_j}-\big(\frac12|\nabla d_\epsilon(t)|^2+\frac{1}{4\epsilon^2}(1-|d_\epsilon(t)|^2)^2\big){\rm{div}}Y+\big\langle \tau_\epsilon(t), Y\cdot\nabla d_\epsilon(t)\big\rangle\Big)=0.