harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left. \begin{array}{l l}\frac{\partial V(\textbf{x})}{\partial \textbf{x}_i}\!\!\!\! = - \frac{\nu_i}{\Lambda}[x_i-x^*(\textbf{x}), \y_i - y^*(\textbf{x}), \z_i- z^*(\textbf{x}) ] \end{array}\right.
\begin{aligned}\|f_N\|_{L^1(\nu)}=\frac{1}{N+1}\sum_{n=0}^N \int_{\mathbb T}|W^n\gamma|^2\nu &=\frac{1}{N+1}\sum_{n=0}^N \|W^n\gamma\|_{L^2(\nu)}^2=\frac{1}{N+1}\sum_{n=0}^N \|\gamma\|_{L^2(\nu)}^2=\|\gamma\|_{L^2(\nu)}^2 \end{aligned}
\frac{da}{dt}=\frac{\mathcal{P}a}{t}\left(b-\frac{1}{\mathcal{P}}\right), \frac{db}{dt}=\frac{\mathcal{P}}{2t}\left(1-b^2\right)-\frac{\mathcal{P}\mathcal{Q}}{2}\left(1-a^2\right)-\mathcal{Q}b.
\Big\langle a_1,b_1,a_2,b_2 \;\Big|\, a_1 a_2 = a_2 a_1,\, a_1 b_2 = a_2 b_1,\, b_1 a_2 = b_2 a_1,\, b_1 b_2 = b_2 b_1 \Big\rangle^+
u ( \vec{p}) \, \, = \, \, \left( \frac{w_{p}+ m}{2 \, w_{p}}\right)^{\frac{1}{2}}\, \left[ \begin{array}{c}{{1}}\\{{\frac{p^{2}\, - \, i p^{1}}{w_{p}\, + \, m}}}\\ \end{array}\right] \, \, \quad v ( \vec{p}) \, \, = \, \, \left( \frac{w_{p}+ m}{2 \, w_{p}}\right)^{\frac{1}{2}}\, \left[ \begin{array}{c}{{\frac{p^{2}+ i p^{1}}{w_{p}\, + \, m}}}\\{{1}}\\ \end{array}\right] \, \, .
\psi_q(a)^* (h_p^- - h_q^-) = ((h_p^- - h_q^-) \psi_q(a))^* = ((h_p^- - h_q^-) \psi_q(a)h_q^+)^* &= ((h_p^- - h_q^-) \psi_q(a)(h_q^+ - k_q))^*,
z_{a b}^{V}= \left( \begin{array}{c c c c}{z_{W}}&{0}&{0}\\{0}&{z_{W}}&{0}&{0}\\{0}&{0}&{z_{Z}\cos \theta_{Z}}&{- z_{A}\sin \theta_{A}}\\{0}&{0}&{z_{Z}\sin \theta_{Z}}&{z_{A}\cos \theta_{A}}\\ \end{array}\right)
A'(\overline{w})=\cdot \cap [X'^{(3)}\cup_w D^4] =\cdot \cap [X'^{(3)}\cup_{\overline{h}\varphi}D^4] - \cdot \cap [X'^{(3)}] & = \overline{h}_* \circ (\cap [X]) \circ \overline{h}^* - \cdot \cap [X'].
J_9= 2\int \nabla (\phi^k)^2\cdot (Q(u^{k})-Q(u^{k-1})) \cdot \overline{u}^{k+1}\leq C |\nabla \phi^k|_\infty|\phi^k \nabla \overline{u}^k|_2| \overline{u}^{k+1}|_2\leq \frac{C}{\nu}|\nabla \phi^k|^2_\infty | \overline{u}^{k+1}|^2_2+\nu |\phi^k \nabla \overline{u}^k|^2_2,
[0,1] \times [0,1] = \bigg[ \bigcup_{4 \leq 2^n \leq \lambda^{\frac{1}{2}}}\Big[ \bigcup_{\substack{|\mathcal I|=|\mathcal J|=2^{-n}\mathcal I \sim \mathcal J}}(\mathcal I \times \mathcal J) \Big] \bigg] \bigcup D
|I_i|\lesssim h^{-1}\|\sigma_{\rm HHJ}-\Pi_{\rm HHJ}\sigma\|\left(\sum_{N_{f_i}}\bigg|\int_{N_{f_i}}(\sigma-\Pi_{\rm HHJ}\sigma)dx\bigg|^2\right)^{\frac{1}{2}}+\|\sigma_{\rm HHJ}-\Pi_{\rm HHJ}\sigma\|_{0,\partial\Omega_{f_i}}\|\sigma-\Pi_{\rm HHJ}\sigma\|_{0,\partial\Omega_{f_i}}.
\begin{aligned}\hat J (\beta,N,T,\hat{\mathbf{R}}_1) \le N \iint_{\hat{\mathbf{R}}_1}\left( \left( \frac{C}{\sqrt{2\pi(t-s)}}\exp\left(-\frac{1}{4}T^{-\frac{1}{2}}\zeta^2(t-s)\right) \right) \wedge 1 \right) ds dt . \end{aligned}
p_{N{\mathrm{( i n t )}}}^{( 2 )}( \xi_{1}, \xi_{2}) = \frac{A_{d}}{2 \xi_{2}^{2}}\int_{0}^{\infty}d k k^{d - 2}\int_{0}^{\infty}d \omega \frac{I_{\omega}^{\prime}( k \xi_{1}) ( 1 + \omega^{2}/ k^{2}\xi_{2}^{2})}{I_{\omega}^{\prime}( k \xi_{2}) N_{\omega}^{\prime}( k \xi_{1}, k \xi_{2})}.
\gamma_{\textrm{p},k}=\frac{{\lvert \mathbf{h}_{k}^{H}\mathbf{p}_{\textrm{p},k}\rvert}^{2}}{\sum_{i=1, i\neq k}^{K}{\lvert \mathbf{h}_{k}^{H}\mathbf{p}_{\textrm{p},i}\rvert}^{2}+\sigma_{z}^{2}}.
Z_{T M}^{r e d}= \int{\cal D}p{\cal D}q d e t{\bigl \{\Theta_{a}, \Upsilon_{b}\bigr \}}\delta ( \theta ) \delta ({\theta}_{i}^{T}) \delta ( \Upsilon ) \delta ({\Upsilon}_{i}^{T}) e^{i \int ( p \dot{q}-{\cal H}_{T M})},
\frac{\partial S}{\partial V}=\frac{\partial}{\partial V}\{-\sum_{i}k_{B}ln[1-exp(-\frac{\hbar\omega_{i}(V)}{k_{B}T})]+\sum_{i}\frac{1}{T}\frac{\hbar\omega_{i}(V)}{exp(\frac{\hbar\omega_{i}(V)}{k_{B}T})-1}\}
\int\limits_{A}^{0}\int\limits_{\mathbb{R}^{3}_{+}}\partial_{y_{q}}\partial_{\tau}^{l}\varphi(y,\tau)U_{i(k)}(y,\tau) dy d\tau= (-1)^{l+1}\int\limits_{A}^{0}\int\limits_{\mathbb{R}^{3}_{+}}\varphi(y,\tau)\partial_{y_{q}}\partial_{\tau}^{l}U_{i(k)}(y,\tau) dy d\tau.
\bigg[\partial_{n+1}x_{n+1}^{1-2s}\partial_{n+1}+x_{n+1}^{1-2s}\sum_{j,k=1}^{n}\partial_{j}a_{jk}\partial_{k}\bigg]u_{2}=0\quad\mathbb{R}_{+}^{n+1}, u_{2}=u-\zeta u\quad\mathbb{R}^{n}\times\{0\}.
\lambda_1|\boldsymbol{\xi}|^2=\left(\min_{1\leq j\leq N}\lambda_j\right) | \boldsymbol{\xi}|^2 \leq \langle (D^2 w_2)(x^*)\,\boldsymbol{\xi},\boldsymbol{\xi}\rangle \leq \left(\max_{1\leq j\leq N}\lambda_j\right) | \boldsymbol{\xi}|^2 = \lambda_N |\boldsymbol{\xi}|^2
0 =\int_N f(xy^{-1}) \Delta_N (y^{-1}) 1_{q^{-1}(A)\cap N}(x) d\omega(x) &=\int_N f(x) 1_{q^{-1}(A)\cap N}(xy) d\omega(x) &=\int_N f(x) 1_{A \cap q(N)}\big(x \cdot q(y)\big)d\omega(x).
\sum_{i,j=1}^n(\nu_iW_{ij})(y,t)\underline{D}_j\eta(y,t,r) = (W^T\nu)(y,t)\cdot\nabla_\Gamma\eta(y,t,r) = 0, \sum_{i,j=1}^n(W_{ij}W_{ji})(y,t) = \sum_{i,j=1}^nW_{ij}(y,t)^2 = |W(y,t)|^2
[\tilde{f}_{i1}(a),\tilde{g}_{1j}(b)] =[[\tilde{t}_{ii'}(a),\tilde{f}_{i'1}(1)],\tilde{g}_{1j}(b)] &=[\tilde{t}_{ii'}(a),[\tilde{f}_{i'1}(1),\tilde{g}_{1j}(b)]]-(-1)^{|a|}[\tilde{f}_{i'1}(1),[\tilde{t}_{ii'}(a),\tilde{g}_{1j}(b)]] &=[\hat{t}_{ii'}(a),\tilde{t}_{i'j}(b)]] &=\tilde{t}_{ij}(ab),
H(p,q) = h_1(z) \left( \frac{1}{z-w}+ \widetilde{C}(z,w) \right) + h_1(w) \left(\frac{1}{w-z}+ \widetilde{C}(w,z) \right) = \frac{h_1(z)- h_1(w)}{z-w}+ h_1(z) \widetilde{C}(z,w) + h_1(w) \widetilde{C}(w,z)
\partial_{\underline y}\left(\underline t\,\underline\nu NP_{\ell}\right) =-\underline t\,\partial_{\underline y}\left(\underline\nu NP_{\ell}\right) &=\underline t\left(\partial_{\rho}N+\frac{2\ell+q-1}{\rho}N\right)P_{\ell}.
[1,g_2]\cdot [g_1,1] \cdot [1,g_2]^{-1}= Z^{d(g_2)d(g_1)}[g_1,g_2] \cdot [1,g_2^{-1}] =[Z_1^{d(g_2)d(g_1)}g_1,1] = Z^{d(g_2)d(g_1)}[g_1,1] .
D_{\alpha\beta}(p\|q)=T\left[\sum_{i}p^{\alpha+\beta-1}_{i}+\frac{\beta-1}{\alpha}\sum_{i}q^{\alpha+\beta-1}_{i}-\frac{\alpha+\beta-1}{\alpha}\sum_{i}p^{\alpha}_{i}q^{\beta-1}_{i}\right]
\left\langle{{\xi}\left\vert{\xi^{\prime}}\right\rangle}\right.={{\left({1-{\vert\xi\vert}^{2}}\right)^{K}\left({1-{\vert\xi^{\prime}\vert}^{2}}\right)^{K}}\over{\left({1-\xi^{*}\xi^{\prime}}\right)^{2K}}}\ ,
\frac{1}{2}\,(z_0 G\,(z_0)z_1^2G'\,(z_1)z_2^3z_3^3+11 \,\mbox{moreterms})-2\,(z_0G\,(z_0)z_1G\,(z_1)z_2^3z_3^3+5 \,\mbox{more})\,\equiv\, 0
H_{\gamma, \phi}(\sigma) := -\frac{1}{2}\sum_{\substack{x,y\in \Lambda, x\sim y}}\sigma_x\sigma_y - \sum_{\substack{x\in \Lambda, \, y\in \partial \Lambda, x\sim y}}\sigma_x\gamma_y- \sum_{x\in \Lambda}\phi_x \sigma_x,
\mathbb{P}(X_k = 1) = \frac{M^n - (M-2)^n}{M^n}, \mathbb{P}(X_k = M-2n+1) = \frac{(M-2)^n - (M-2n)^n}{M^n}, \mathbb{P}(X_k = 1/L) = \frac{(M-2n)^n}{M^n}.
g_1(t,s) =\frac{(t-a)^{\alpha -1}(b-s)^{\alpha-1-\beta}}{(b-a)^{\alpha-1-\beta}}-(t-s)^{\alpha-1}&=(t-a)^{\alpha-1}\left(\frac{b-s}{b-a}\right)^{\alpha-1-\beta}-(t-a)^{\alpha-1}\left(\frac{t-s}{t-a}\right)^{\alpha-1}&=(t-a)^{\alpha-1}\left[\left(\frac{b-s}{b-a}\right)^{\alpha-1-\beta}-\left(\frac{t-s}{t-a}\right)^{\alpha-1}\right].
\Xi_{n , m}( - \frac{1}{\tau}) = \widetilde{\Xi}_{n , m}( \tau ) = \sqrt{\frac{- i \tau}{2 m}}\sum_{n^{\prime}\, \mathrm{m o d}\, 2 m}\sin ( - 2 \pi \frac{n n^{\prime}}{2 m}) \Xi_{n^{\prime}, m}( \tau ) \, ,
m_{b_2}= m_{b_1}\cos\theta^{b_2}_{(b_1 b_1)}+ m_{b_1}\cos\theta^{b_2}_{(b_1 b_1)}, m_{b_1}= m_{b_1}\cos\theta^{b_1}_{(b_1 b_2)}+ m_{b_2}\cos\theta^{b_1}_{(b_2 b_1)}.
S^{WZ}(\gamma)=-\frac{1}{48\pi^{2}}\int_{B^{3}}d^{3}y\epsilon^{ijk}K(\gamma^{-1}\frac{\partial\gamma}{\partial y^{i}},[\gamma^{-1}\frac{\partial\gamma}{\partial y^{j}},\gamma^{-1}\frac{\partial\gamma}{\partial y^{k}}])
H_{n+p+m}=\left[\begin{smallmatrix}H^{x,x}_{n\times n}H^{x,u}_{n\times p}H^{x,y}_{n\times m}H^{u,x}_{p\times n}H^{u,u}_{p\times p}H^{u,y}_{p\times m}H^{y,x}_{m\times n}H^{y,u}_{m\times p}H^{y,y}_{m\times m}\end{smallmatrix}\right],
v(z)|_{W^i}=\begin{pmatrix}p^i(z)^{-1}0 0 p^i(z)\prod_{j\neq i, j\in I_w\cup I_g}p^j(z)^{a_{ji}}\prod_{j\neq i, j\in I_b}p^j(z)^{a_{ji}/2}\end{pmatrix}\begin{pmatrix}1 - \frac{q^i_{-}(z)}{q^i_{+}(z)}0 1 \end{pmatrix}
\widetilde{\mathcal{M}}_1 :=\sup_{0<|h|<h_0}\left[\frac{\delta_h (u_R\,\widetilde\eta)}{|h|^\frac{1+t}{p}}\right]^p_{W^{s,p}(B_{3/8})}&\le \frac{C}{(1-s)\,s}\,i_0^{2\,p}\,\sup_{0<|h|<h_0}\left\|\frac{\delta_h u_R}{|h|}\right\|^p_{L^p(B_{3/8})}+C\,i_0^{4\,(N+p)}\,\mathcal{A}_1.
\sum_{k=1}^\kappa (\kappa-k+1) q(k) \ge \frac{\kappa^2-1}{4}\left(1 - \kappa\sqrt{\frac{2}{\pi n/4}}\right) \sum_{m=n/4}^n \frac{\min\left\{{n-\kappa \choose m-\kappa},{n-\kappa \choose m}\right\}}{2^{n-\kappa}}.
J_1(v)(t,x) :=\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\big((1+\gamma)v^{\delta+1}-\gamma v-v^{2\delta+1}\big)(s,y)\d y\d s &=:((1+\gamma)J_{11}-\gamma J_{12}-J_{13})v(t,x), J_2(v)(t,x) :=\int_{0}^{t}\int_{0}^{1}\frac{\partial G}{\partial y}(t-s,x,y)v^{\delta+1}(s,y)\d y \d s,
\langle\mathcal{N}_\perp\rangle_{\psi_\mathrm{gs}}\le\; C \langle \mathrm{d}\Gamma_\perp(h_\mathrm{MF}-\mu_+)\rangle_{\psi_\mathrm{gs}}\le\; C \langle \mathcal{N}_-\rangle_{\psi_\mathrm{gs}}\le\; C_\varepsilon\min\{N,T^{-1-\varepsilon}\}.
u \in \mathrm{U S p}( 2 N ) : \qquad u_{i}^{I}u_{J}^{i}= \delta_{J}^{I}, u_{i}^{I}\Omega^{i j}u_{j}^{J}= \Omega^{I J}, u_{i}^{I}= ( u_{I}^{i})^{*}.
F(x)=\int_{-\infty}^{x}f(u)du=\Phi(x)-\phi(x)\sum_{n=3}^{\infty}\frac{1}{n!\sigma^{n-1}}B_{n}(0,0,\kappa_{3},...,\kappa_{n})He_{n-1}(\frac{x-\mu}{\sigma}),
\Vert P_{t-s}^\# \Pi_0 \Phi_s \sqrt{Q}\Vert_{L_2(L^2(1+\rho), L^2(\rho))}^2 = \sum_k \Vert P_{t-s}^\# \Pi_0 \Phi_s \sqrt{Q}e_k \Vert_{L^2(\rho)}^2 \leq e^{-2 \kappa (t-s)}\sum_k \Vert \sqrt{Q}e_k \Vert_{\rho_s}^2 \leq e^{-2 \kappa (t-s)}\Vert \sqrt{Q}\Vert_{L_2(L^2(1+\rho), L^2(\rho))}^2.
\delta U=\int_{\Omega^{0}}[-N_{\alpha\beta,\alpha}\delta u_{\beta}^{0}-M_{\alpha\beta,\beta\alpha}\delta w^{0}]d\Omega+\int_{\Gamma^{0}}[n_{\alpha}N_{\alpha\beta}\delta u_{\beta}^{0}+n_{\alpha}M_{\alpha\beta,\beta}\delta w^{0}-n_{\beta}M_{\alpha\beta}\delta w_{,\alpha}^{0}]d\Gamma
f^\beta_\alpha = \frac{1}{6}x^\gamma x^\delta R_{\alpha\gamma\delta}^{\beta}(q), h^i_\alpha = \frac{1}{8}R_{\alpha\beta j}^{i}(q) z^j x^\beta + \frac{1}{24}J_{\beta\gamma}^i(q) R_{\alpha \beta^\prime \gamma^\prime}^\gamma (q) x^\beta x^{\beta^\prime}x^{\gamma^\prime}+ \frac{1}{4}x^\beta J_{\alpha\beta}^{i(2)}.
\bigoplus_{\prod_j\epsilon_j=1}\bigotimes_jU^{\epsilon_j}=\bigoplus_{\prod_l\epsilon_l=1}\bigotimes_lV_l^{\epsilon_l}\bigoplus_{\prod_j\epsilon_j=-1}\bigotimes_jU^{\epsilon_j}=\bigoplus_{\prod_l\epsilon_l=-1}\bigotimes_lV_l^{\epsilon_l}.
D(z,p,q,r,s,t) =dx\otimes(\partial_x(z)-p,\partial_x(p)-r,\partial_x(q)-s) +du\otimes(\partial_u(z)-q,\partial_u(p)-s,\partial_u(q)-t) D(\omega\otimes(z,p,q)) =d\omega\otimes\pi_0(z,p,q)-\omega\wedge D(z,p,q) =d\omega\otimes (z)-\omega\wedge(dx\otimes(\partial_x(z)-p)+du\otimes(\partial_u(z)-q)).
D A_{( p )}^{\mu_{1}\ldots \mu_{p - 2 k}}= \partial^{\left[ \mu_{1}\right.}\eta_{( p )}^{\left. \mu_{2}\ldots \mu_{p - 2 k}\right]}+ \left( p - 2 k + 1 \right) \partial_{\nu}\eta_{( p )}^{\nu \mu_{1}\ldots \mu_{p - 2 k}}, \; k = 1 , \ldots , b_{p},
\mathcal{C}_{J,K}^{r,alt}:=\left\{\eta \in \mathcal{C}_{J,K}:\:\max_{i\in\{0,1\}}\liminf_{n \to -\infty}\mathbf{1}_{\{\eta_{2n+i}\ge K-r,\:\eta_{2n+1+i}\le J-K+r \}}=1\right\}
\mu\left(t,\left(\sum_{i=1}^n |x_i|^p\right)^{\frac1p}\right)=\left( \mu\left(t,\sum_{i=1}^n |x_i|^p\right)\right)^{\frac1p},\ \ \mu\left(t,\sum_{i=1}^n |x_i|^p\right)\leq \sum_{i=1}^n \mu\left(\frac{t}{n},\sum_{i=1}^n |x_i|^p\right),
\cos 2 x 0 = \cos [ ( 2 \times 0 + \frac{\pi}{6}) - \frac{\pi}{6}] = \cos ( 2 x 0 + \frac{\pi}{6}) \cdot \frac{\sqrt{3}}{2}+ \sin ( 2 x + \frac{\pi}{6}) \times \frac{1}{2}
{\cal C}={\cal C}^{- 1}= \left( \begin{matrix}{- 1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\ \end{matrix}\right) \quad , \quad{\cal C}\alpha^{\mu}{\cal C}^{- 1}= \overline{{\alpha}}^{\mu \ast}\quad .
\begin{aligned}\frac{2\lambda^k}{m}\sum_{i=1}^m\left\langle\nabla f_i(\bar{x}^k), \bar x^k -\theta^* \right\rangle = 2\lambda^k \left\langle\nabla F(\bar{x}^k), \bar x^k -\theta^* \right\rangle \geq 2\lambda^k (F(\bar{x}^k)-F(\theta^*)) \end{aligned}
\sigma_2(\tilde{\textbf{r}})=\left(\overline{r_1},\ldots,\overline{r_s},\widehat{r_1''},{r_1'},\ldots,\widehat{r_t''},{r_t'}\right) \left(\prod_{i=1}^sG_i\right)\times\left(\prod_{j=1}^t\left(H'_j\times H'_j\right)\right),
\mathrm{supp}((\mu^{\boxtimes s})^{\mathrm{ac}})\cap [\epsilon, \infty)\subset \overline{\{1/x: x=h_s(r), r\in V_s^+\cap [a, b] \}}\subset B_\epsilon(\mathrm{supp}(\mu^{\boxtimes t_0})^{\mathrm{ac}}),
l_1' (x) = l_2 (\alpha, x) - \frac{1}{2}~ l_3 (\alpha, \alpha, x) ,~l_2' (x,y) = l_2 (x, y) - l_3 (\alpha, x, y) ~~ ~~l_3' (x, y, z) = l_3 (x, y, z).
P_{n}^{(\alpha,\beta)}(z)=\frac{\Gamma(\alpha+n+1)}{n!\Gamma(\alpha+\beta+n+1)}\sum_{m=0}^{n}(\begin{matrix}n\\m\end{matrix})\frac{\Gamma(\alpha+\beta+n+m+1)}{\Gamma(\alpha+m+1)}(\frac{z-1}{2})^{m}.
S_{n,p}\left(\frac{n-p}{p-1}\right)^{\frac1n-1}\left( \int_{B_{R}}\frac{|u(x)|^{p^{*}}}{\left[ \log_{{\frac{n-1}{p-1}}}\frac{R}{|x|}\right]^{\frac{p(n-1)}{n-p}}}dx \right)^{\frac{1}{p^{*}}}\le\|\nabla u\|_{L^p(B_R)}.
\begin{aligned}\nu_c^{\pm}(Y_1,Y_2,\dots,Y_m) =\nu_c^{\pm}(Y_1,Y_2,\dots,Y_{l})+\nu_c^{\pm}(Y_1,Y_l,\dots,Y_{m})=\nu_c^{\pm}(Y_1,Y_2,\dots,Y_{l},Y_m)+\nu_c^{\pm}(Y_l,\dots,Y_{m}),\end{aligned}
{\tilde{\cal{E}}}_{m > 0}={\cal{E}}_{m > 0}( B ) -{\cal{E}}( 0 ) = \frac{B^{2}}{2}+ \frac{( e B )^{\frac{3}{2}}}{2 \pi}g \left( \frac{e B}{m^{2}}\right) + \frac{e B}{2 \pi}| m | .
\left\{\mathrm{e}^{- \frac{A}{2}}\not{\partial}_{x}+ \Gamma^{r}\left( \partial_{r}+{\frac{D_{1}}{4}}A^{\prime}+{\frac{D_{2}}{4}}B^{\prime}\right) + \mathrm{e}^{-{\frac{B}{2}}}\not{\Delta}_{y}\right\}\Psi = 0 ,
\left(1+\frac{e_1^*}{\sigma_1^2}+\sum\limits_{j=2}^K\frac{e_j^*}{\sigma_j^2}\right) \left(1+\frac{(1-e_1^*)}{\sigma_1^2}+\sum\limits_{j=2}^K\frac{(1-e_j^*)}{\sigma_j^2}\right) &\geq \left(1+\frac{e_1}{\sigma_1^2}+\sum\limits_{j=2}^K\frac{e_j^*}{\sigma_j^2}\right) \left(1+\frac{(1-e_1)}{\sigma_1^2}+\sum\limits_{j=2}^K\frac{(1-e_j^*)}{\sigma_j^2}\right),
(\lambda_{1},\ldots,\lambda_{d})\cdot z_{i}=\lambda_{1}^{a_{i1}}\cdots\lambda_{d}^{a_{id}}z_{i}, (\lambda_{1},\ldots,\lambda_{d})\cdot w_{i}=\lambda_{1}^{-a_{i1}}\cdots\lambda_{d}^{-a_{id}}w_{i}.
0= \langle \tau_{H}, \alpha_{1}\rangle\{(m_{1}+2m_{2}+m_{3}) (1-(\cot \langle \alpha_{1}, H\rangle )^{2}) +4m_{3}&+ (n_{1}+m_{3}) (1-(\tan \langle \alpha_{1}, H\rangle )^{2})\}e_{1}.
\lim_{n\to\infty}\mu_n (k) \ = \ \frac{\pi^Q (H_1) \nu(k-1)}{M_{\pi^Q_1}}. \ \ \ \ \ \ +\ \frac{\sum_{z \in H_2}\pi^Q ( z ) \left ( h (z(2)-1,k) -\rho(z(2))\nu (k-1)\right)}{M_{\pi^Q_1}}.
K ( t ; 2 ) = V ( t ) + 2 N | f_{0}|^{2}\int_{| U |^{\frac{k + 1}{2}}< R^{\prime}}d^{2}U \int_{| t - U^{k + 1}| / R < | Y | < R}{\frac{d^{2}Y}{| Y |^{2}}}~ .
\langle\nabla^{2}(\frac{-e^{2}}{4\pi\epsilon_{0}r})\rangle_{at}=\frac{-e^{2}}{4\pi\epsilon_{0}}\int d\vec{r}\psi^{*}(\vec{r})\nabla^{2}(\frac{1}{r})\psi(\vec{r})=\frac{e^{2}}{\epsilon_{0}}|\psi(0)|^{2},
\begin{aligned}\mu \bar h_{222}^{2^*}= \bigl\{12\bar\lambda^2+(\bar \lambda_1-3\bar \lambda_2)^2\bigl\}A , \bar h_{222}^{1^*}= -\bar h_{211}^{1^*}, \bar h_{221}^{1^*}= -\bar h_{111}^{1^*}, \mu\bar h_{211}^{1^*}= \bar \lambda(\bar \lambda_1-3\bar \lambda_2)A, \mu\bar h_{111}^{1^*}= -4\bar \lambda^2A \end{aligned}
\| W_{\psi_{i,1,*},\varphi_{i,1,*}}- T^{-1}_{(V^*b^0)_{[j]}}C_{U^*}W_{\chi,\phi}C_{V^*}\| = \; \|T^{-1}_{(V^*b^0)_{[j]}}C_{U^*}W_{\psi_{i},\varphi_{i}}C_{V^*}- T^{-1}_{(V^*b^0)_{[j]}}C_{U^*}W_{\chi,\phi}C_{V^*}\| \leq \; \|T^{-1}_{(V^*b^0)_{[j]}}\| \|W_{\psi_i,\varphi_i}- W_{\chi,\phi}\| \to 0, i \to \infty.
=2\pi\sum\limits_{k=1}^{\infty}\int\limits_{\frac{1}{k+1}}^{\frac{1}{k}-2^{-4k-1}}\,r\,dr+2\pi\sum\limits_{k=1}^{\infty}\int\limits_{\frac{1}{k}-2^{-4k-1}}^{\frac{1}{k}}r\cdot 2^{k-1}\,dr\leqslant
J_{k2}\le (2k+1)^{p-1}(t_{k+1}-t_k) \left[\sum_{j=1}^{k}I_j+\sum_{j=1}^{k}\left|e^{\lambda \tau_j}\Delta w_j-\Delta a_j\right|^p+\sup_{\tau_k\le t<\tau_{k+1}}\left|\int_{\tau_{k}}^{t}e^{\lambda s}dw_{l_s}\right|^p\right] &=(2k+1)^{p-1}(t_{k+1}-t_k) \left[\sum_{j=1}^{k+1}I_j+\sum_{j=1}^{k}\left|e^{\lambda \tau_j}\Delta w_j-\Delta a_j\right|^p\right].
\begin{array}{lll}\mbox{\rm minimize}C \cdot U \hat{X}U^T \mbox{\rm subject to}A_i \cdot U \hat{X}U^T = b_i \;\; \forall i \in \{1,\ldots,m\}& \hat{X}\in \mathbb{S}^d_{+}\end{array}\begin{array}{lll}\mbox{\rm maximize}b^Ty \mbox{\rm subject to}U^T (C - \sum^{m}_{i=1}y_i A_i) U \in \mathbb{S}^d_{+}, \end{array}
\delta \Gamma_{eff}=tr\int d^{d}x\, (D_{\mu}\epsilon^{a}) \frac{\delta\Gamma_{eff}}{\delta A_{\mu}^{a}}=-tr\int d^{d}x\, \epsilon^{a}D_{\mu}\left(\frac{\delta \Gamma_{eff}}{\delta A_{\mu}^{a}}\right)\;.
A_{2}=\begin{bmatrix}0 -4 -3 \varepsilon \varepsilon -4 \varepsilon \varepsilon \varepsilon -3 \varepsilon \varepsilon \varepsilon -2 \varepsilon \varepsilon -1 \varepsilon \varepsilon \varepsilon -1 \varepsilon \varepsilon 1 \varepsilon\end{bmatrix},
\delta R_{M N}( h ) = -{\frac{1}{2}}( \bar{\nabla}_{M}\bar{\nabla}_{N}h_{~ ~ P}^{P}+ \bar{\nabla}^{P}\bar{\nabla}_{P}h_{M N}- \bar{\nabla}^{P}\bar{\nabla}_{N}h_{M P}- \bar{\nabla}^{P}\bar{\nabla}_{M}h_{N P}) ,
T_{\mu\nu}=\phi_{,\mu}\phi_{,\nu}-\frac 12 g_{\mu\nu}\left(\phi_{,\rho}\phi^{,\rho}+m^2\phi^2\right)+\xi\left[(R_{\mu\nu}-\frac 12 g_{\mu\nu}R)\phi^2+g_{\mu\nu}(\phi^2)^{,\rho}_{~;\rho}-(\phi^2)_{;\mu\nu}\right] \, .
\pi P_vf(r_{k_n}) =-\arctan\bigg(\frac{r_{k_n}-x}{v}\bigg)+\arctan\bigg(\frac{r_{k_n}-x+\delta}{v}\bigg)+\pi P_v\varphi(r_{k_n}) &=\arctan\bigg(\frac{v}{r_{k_n}-x}\bigg)-\arctan\bigg(\frac{v}{r_{k_n}-x+\delta}\bigg)+\pi P_v\varphi(r_{k_n})
B_\lambda(s) =\alpha_ni^{nk}\,2^{-n(2n+1)(s+\frac12)+n}\int\limits_T\bigg(\prod_{1\leq i<j\leq n}(t_i^2-t_j^2)\bigg)\Big(\prod_{j=1}^nt_j\Big)^{-(2n+1)s+\frac12-n-k}\,d\mathbf{t}.
K(x) = R^X(E_1,E_2,E_2,E_1) + \langle (D_{E_1}E_1)^\perp, (D_{E_2}E_2)^\perp \rangle -\langle (D_{E_1}E_2)^\perp, (D_{E_1}E_2)^\perp \rangle &= R^X(E_1,E_2,E_2,E_1) + \langle A_{11}, A_{22}\rangle - \langle A_{12}, A_{12}\rangle.
\begin{aligned}\lVert u \rVert_E^{\sup}:=\lVert u \rVert_{E, \Omega_0}^{\sup}:=\sup_{\omega\in\Omega_0}\lVert u(\omega) \rVert_E, &\lVert u \rVert_{E}^{lip}:=\lVert u \rVert_{E, \Omega_0}^{lip}:=\sup_{\omega_1\neq \omega_2}\frac{\lVert u(\omega_1)-u(\omega_2)\rVert_E}{\lvert \omega_1-\omega_2\rvert},\end{aligned}
\Delta_E^M (k)=3D\frac{k_\mu= (\delta_{\mu\nu}+{\bar{\sigma}}_{\mu\nu})k_\nu}{k^4}\hspace{.5cm},\hspace{.5cm}\begin{array}{lcl}{\bar{\sigma}}_{ij}=3D i\varepsilon_{ijk}\sigma_k={\bar{\sigma}}_{i4}=3D i\sigma_i=3D-{\bar{\sigma}}_{4=i}\end{array}.
\mathbf{S}_t = \begin{bmatrix}\beta_{\mathrm{Re},n_s}(t), \beta_{\mathrm{Tr},n_s}(t), \mathbf{\Phi}_{\mathrm{Re},n_s}(t), \mathbf{\Phi}_{\mathrm{Tr},n_s}(t), P_t(t) \end{bmatrix}.
\frac{dx_i}{dt}=r_ix_i\left(1-\frac{x_i}{K_i}\right)-\frac{a_i x_iy_i}{(1+x_i)}=\frac{a_i x_i}{1+x_i}\left[\frac{r_i(K_i-x_i)(1+x_i)}{a_iK_i}-y_i\right]=p_i(x_i)\left[q_i(x_i)-y_i\right]. \rho_j\frac{dy_i}{dt}+ \rho_i\frac{dy_j}{dt}= \rho_jy_i[\frac{a_ix_i}{1+x_i}-d_i]+ \rho_iy_j[\frac{a_jx_j}{1+x_j}-d_j] = \rho_jy_i[p_i(x_i)-d_i]+ \rho_iy_j[p_j(x_j)-d_j]
\int_{p_1}^{p_2}U(p) \, e^{i \omega \psi(p)}\, dp = \int_{p_1}^{p_1 + \delta}\dots + \int_{p_1 + \delta}^{p_0-\delta}\dots + \int_{p_0-\delta}^{p_0 + \delta}\dots + \int_{p_0 + \delta}^{p_2}\dots & =: I^{(1)}(\omega) + I^{(2)}(\omega) + I^{(3)}(\omega) + I^{(4)}(\omega) \; .
\int_{R^{8+1}}{\hat C}_{{\hat \mu}_1{\hat \mu}_2{\hat \mu}_3}D{\hat X}^{{\hat \mu}_1}D{\hat X}^{{\hat \mu}_2}D{\hat X}^{{\hat \mu}_3}\wedge{\rm Tr}({\hat F}\wedge{\hat F}\wedge{\hat F})\, .
\psi_{Q}= \left( \begin{array}{c}{{\frac{\psi_{( 0 )}}{\sqrt{N_{f}}}I_{N_{f}}+ \psi_{( 1 )}}}\\{{\psi_{( 2 )}}}\\ \end{array}\right) \; , \qquad \quad \psi_{\bar{Q}}= \left( \begin{array}{c c}{{\frac{\psi_{( \bar{0})}}{\sqrt{N_{f}}}I_{N_{f}}+ \psi_{( \bar{1})}}}&{{\psi_{( \bar{2})}}}\\ \end{array}\right)
\frac{\partial^2 l}{\partial \theta_i^2}= \frac{2}{\theta_i^2}- \frac{6\phi}{\theta_i^4};\frac{\partial^2 l}{\partial \theta_i \partial \phi}= \frac{2}{\theta_i^3};\frac{\partial^2 l}{\partial \phi^2}= -\frac{1}{2\phi^2};\frac{\partial^3 l}{\partial \theta_i^3}= -\frac{4}{\theta_i^3}+ \frac{24\phi}{\theta_i^5}.
U(x)=\frac{1}{2\pi i}\int_{(2)}\frac{\Gamma(\frac{s+\iota}{2}+\frac{1}{4})}{\Gamma(\frac{\iota}{2}+\frac{1}{4})}(\pi^{\frac{1}{2}}x)^{-s}\frac{ds}{s}, V(x)=\frac{1}{2\pi i}\int_{(2)}\frac{\Gamma(\frac{s+\iota}{2}+\frac{1}{4})^2}{\Gamma(\frac{\iota}{2}+\frac{1}{4})^2}(\pi x)^{-s}\frac{ds}{s}.
\left(n^{A}({\cal R}_{AB}-\frac{1}{2}{\cal R}G_{AB}-8\pi{\bf G}{\cal T}_{AB})y^{B}_{\,,\mu}\right)^{(1)}+\left(n^{A}({\cal R}_{AB}-\frac{1}{2}{\cal R}G_{AB}-8\pi{\bf G}{\cal T}_{AB})y^{B}_{\,,\mu}\right)^{(2)}=0~.
\chi_\lambda(gx)=\begin{cases}\chi_{\lambda_0}(gx_0s_0)\overline{\chi}_{\lambda\backslash\{0\}}^+(x's')+\chi_{\lambda_0}(gx_0s_0)\overline{\chi}_{\lambda\backslash\{0\}}^-(x's') x's'\in\tilde{A}_{p(t-t_0)}, 0 \end{cases}
D_x \L_{(a,b)}(0,0) = 0 D_x^2 \L_{(a,b)}(0,0) = \frac{2 a}{\pi}\tan^{-1}\left( \sqrt{\frac{b}{a}}\right) D_y \L_{(a,b)}(0,0) = 0 D_y^2 \L_{(a,b)}(0,0) = \frac{2 b}{\pi}\tan^{-1}\left( \sqrt{\frac{a}{b}}\right)
\left(V_0+ V_1 E_{\gamma}\right)\cdot \alpha+\left(V_0E_{\gamma}+V_1\right)\cdot \beta =z_C, \left(-\gamma W_0+\gamma W_1E_{\gamma}\right)\cdot \alpha+\left(\gamma W_0E_{\gamma}-\gamma W_1\right)\cdot \beta =z_K
J =\frac{(1-r^2)^{L-\frac n2+\frac j2}}{2^{L-\frac n2-\frac j2}}\frac{2^{\frac n2-L+\frac j2}(\cos\theta)^{n-2L+j}}{(n-2L+j) (1-r^2)^{\frac n2-L+\frac j2}}+\cdots =\frac{2^{n-2L+j}}{n-2L+j}(\cos\theta)^{n-2L+j}(1-r^2)^{2L-n}+\cdots
\begin{array}{lll}G_W(0) = g(X_{\frak{m}}+Y_{\frak{m}}+Z_{\frak{m}},[W, X+Y+Z]_{\frak{m}})_o\ &+ g(W,[X, Y+Z]_{\frak{m}}+[Y, Z]_{\frak{m}})_o=0.\end{array}
x(t) = \begin{cases}e^{i t A_1}\left( h - i \int_0^{t}e^{-i s A_1}\Phi^* u(s) ds\right), t > 0; e^{i t A_1^*}\left( h + i \int_t^{0}e^{-i s A_1^*}\Phi^* y(s) ds\right), t < 0; \end{cases}
d_{G_i}(\tilde x, \tilde y) \leq d_X(\tilde x_1, \tilde y_1) + |\tilde x - \tilde x_1| + |\tilde y - \tilde y_1| \leq L d_i(\tilde x_1, \tilde y_1) + L|\tilde x - \tilde x_1| + L|\tilde y - \tilde y_1| &= L\tilde d_{G_i}(\tilde x, \tilde y).
\frac{\zeta'(s)}{\zeta(s)}=B-\frac{1}{s-1}+\frac{1}{2}\log \pi -\frac{\Gamma'(s/2+1)}{2\ \Gamma(s/2+1)}+\sum_{\rho}\left(\frac{1}{s-\rho}+\frac{1}{\rho}\right)
\Omega_4= G_m \sum_{z\in \mathbb{F}_p^*}\eta(z)\sum_{y\in \mathbb{F}_p^*}\zeta_p^{-\frac{m_p}{4z}y^2-y}\sum_{\delta \in \mathbb{F}_p^*}\zeta_p^{- \frac{A}{4z}\delta^2 -\big( \frac{By}{2z}+\rho \big)\delta}.
B_j(F,G)= \sum_{\ell \in \mathbb{Z}}\int_{2^j \ell}^{2^j (\ell+1)}\int_{t-I_j}\Big\langle |\nabla|^{2\sigma}e^{i(t-s) \sqrt{1-\Delta}}P_k^2 F(\cdot, s), G(x,t) \Big\rangle_{L_x^2}dsdt.
+\left[ \partial_{\mu}C^{a}{}_{c}{}^{\nu}-C^{a}{}_{d\mu}C^{d}{}_{c}{}^{\nu}\right] A^{\prime c \mu}+ A^{\prime b}{}_{\mu}R^{\prime a}{}_{b}{}^{\mu \nu}+\left[ C^{a}{}_{c\mu}+f^{a}{}_{cb}A^{\prime b}{}_{\mu}\right] \partial^{\nu}A^{\prime c\mu}
\mu_* (\cup_{k\in T}W_k (~\mu_*)) =\sum_{k\in T}p_k \mu_* \circ W_k^{-1}(\cup_{k\in T}W_k (~\mu_*))) &=\sum_{k\in T}p_k \mu_* (W_k^{-1}(\cup_{k\in T}W_k (~\mu_*))) &\ge \sum_{k\in T}p_k \mu_* (W_k^{-1}( W_k (~ \mu_*))) &\ge \sum_{k\in T}p_k \mu_* (~ \mu_*) &=1.
v_{i,j}v_{k,l}(w_1,\dots,w_{\chi(1)}) =v_{i,j}(w_1,\dots,w_{\chi(1)})E_{l,k}&=(w_1,\dots,w_{\chi(1)})E_{j,i}E_{l,k}&=\delta_{i,l}(w_1,\dots,w_{\chi(1)})E_{j,k}&=\delta_{i,l}v_{k,j}(w_1,\dots,w_{\chi(1)}).
\frac{D}{2\rho hW}[\frac{\partial^{4}W}{\partial x_{1}^{4}}+2\frac{\partial^{4}W}{\partial x_{1}^{2}\partial x_{2}^{2}}+\frac{\partial^{4}W}{\partial x_{2}^{4}}]=-\frac{1}{F}\frac{d^{2}F}{dt^{2}}=\omega^{2}
y_1 = X_1+X_2+X_3, y_2 = \frac{X_1X_2^2+X_2X_3^2+X_3X_1^2-3X_1X_2X_3}{X_1^2+X_2^2+X_3^2-X_1X_2-X_2X_3-X_3X_1}, y_3 = \frac{X_1^2X_2+X_2^2X_3+X_3^2X_1-3X_1X_2X_3}{X_1^2+X_2^2+X_3^2-X_1X_2-X_2X_3-X_3X_1}.
\tilde{f}_S(x,a,b) = \frac{I_a^{(q,r)}(x)}{I_a^{(q,r)}(b)}\Big[ \frac{W^{(q+r)}(b)}{W^{(q+r)\prime}(b+)}+ \tilde{f}_S(0,a,b) \Big( I_a^{(q,r)}(b) - \frac{W^{(q+r)}(b)}{W^{(q+r) \prime}(b+)}(I_a^{(q,r)})'(b+) \Big) \Big].
b_{1}={\frac{a_{1}}{2}}[ 1 +{\frac{({\bar{u}}{\bar{z}}^{\prime}{\bar{u}})}{( u x^{\prime}{\bar{u}})}}] ,{\bar{b}_{1}}={\frac{a_{1}}{2}}[ 1 +{\frac{( u{z}^{\prime}u )}{( u x^{\prime}\bar{u})}}] .
F\Gamma = \langle r^{(d,e)}\rangle \cup \langle r^{(d,e)}\rangle s \cup \langle r^{(d,e)}\rangle r^ks\cup \langle r^{(d,e)}\rangle r^{k}&= \langle r\rangle \cup \langle r\rangle s \cup \langle r\rangle r^ks \cup \langle r\rangle r^{k}= D_n.