harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathbb{P}(X \ge (1 -\epsilon)\ell\alpha\ \mathrm{and}\Y \ge (1 -\epsilon)\ell\alpha) = 1 - \mathbb{P}(X < (1 -\epsilon)\ell\alpha\ \mathrm{or}\Y < (1 -\epsilon)\ell\alpha) &\ge 1 - \mathbb{P}(X < (1 -\epsilon)\ell\alpha) - \mathbb{P}(X < (1 -\epsilon)\ell\alpha) &\ge 1 - 4 e^{-\frac{\epsilon^2 \ell \alpha}3}
V(s) \colon H^{-1/2}(\Gamma) \rightarrow H^{1/2}(\Gamma) K(s) \colon H^{1/2}(\Gamma) \rightarrow H^{1/2}(\Gamma) K^t(s) \colon H^{-1/2}(\Gamma) \rightarrow H^{-1/2}(\Gamma) W(s) \colon H^{1/2}(\Gamma) \rightarrow H^{-1/2}(\Gamma)
\int_{0}^{+\infty}h(t,s)\d s = \int_{0}^{+\infty}[p(N_*,s) - p(N(t),s)]n(t,s) \d s &+ \int_{0}^{+\infty}\delta_{0}(x) \int_{0}^{+\infty}[p(N(t),s)-p(N_*,s)] n(t,s)\d s \d x = \int_{0}^{+\infty}[p(N_*,s) - p(N(t),s)]n(t,s) \d s + \int_{0}^{+\infty}[p(N(t),s)-p(N_*,s)] n(t,s)\d s &= 0,
\sup_{v \in \mathcal{S}}| P_s^n \varphi(v)| = \sup_{v \in \mathcal{S}}\left| \int \frac{1}{\| g_n\ldots g_1 v \|^{-s}}\varphi[(g_n\ldots g_1)\!\cdot\! v] \mu(dg_1)\ldots \mu(dg_n) \right| & \leq c_s \|\varphi\|_\infty \mathbb{E}\left(\|g_n \ldots g_1\|^s\right).
J^{2}=P^{2}=-\frac{1}{sin^{2}\beta}(\frac{\partial^{2}}{\partial\alpha^{2}}+\frac{\partial^{2}}{\partial\gamma^{2}}-2cos\beta\frac{\partial^{2}}{\partial\alpha\partial\gamma})-\frac{\partial^{2}}{\partial\beta^{2}}-cot\beta\frac{\partial}{\partial\beta}.
\mu_1 + \cdots + \mu_n = \sum_{j = 1}^n j r_j + \sum_{i = 1}^{r - 1}M_i (s_{L_i + 1}+ \cdots + s_{L_{i+1}}), \lambda_1 + \cdots + \lambda_n = \sum_{j = 1}^n j s_j + \sum_{i = 1}^r L_i (r_{M_{i-1}+ 1}+ \cdots + r_{M_i}).
\det\left(\dfrac{\partial\varphi}{\partial t_1}, \dfrac{\partial\varphi}{\partial t_2}\right)\Big|_{(0, 0)}= &\det\begin{pmatrix}-a(+)\bar s_0\bar i_0 +c(+)(N-\bar s_0-\bar i_0) -a(-)\bar s_0\bar i_0 +c(-)(N-\bar s_0-\bar i_0) a(+)\bar s_0\bar i_0 - b(+)\bar i_0 a(-)\bar s_0\bar i_0 - b(-) \bar i_0\end{pmatrix}\ne 0.
f(x,y)+f(y,z)-f(x,z) =\langle Ax,y-x\rangle+\langle Ay,z-y\rangle-\langle Ax,z-x\rangle &=-\langle Ay-Ax,y-z\rangle &\geq -\|Ax-Ay\|\big\|y-z\big\| &\geq -L\|x-y\|\big\|y-z\big\| &\geq -\dfrac{L}{2}\|x-y\|^2-\dfrac{L}{2}\|y-z\|^2 &=-c_1\|x-y\|^2-c_2\|y-z\|^2.
\omega Z_{i ( 2 )}\rightarrow Z_{i ( 2 )}, \o m e g a \tilde{B}_{i ( 1 2 )}^{1}\rightarrow \tilde{B}_{i ( 1 2 )}^{1}, \omega \tilde{\psi}_{i ( 1 2 )}^{1}\rightarrow \tilde{\psi}_{i ( 1 2 )}^{1}( i = 1 , 2 , 3 ) ,
W_{eff}~=~ m_{ij}\Phi_i \Phi_j \hspace{.3cm}f\left( \frac{\lambda_{a}\Phi_a}{m_{ij}\Phi_i \Phi_j}; \frac{g_{abc}\Phi_a \Phi_b\Phi_c}{m_{ij}\Phi_i \Phi_j}\right) .
\mathcal{E}(\theta_2\wedge \theta_3\wedge \theta_4) (\psi_0,\dots,\psi_{3})=\frac{1}{2}\sum_{\sigma \in S_3}(-1)^{\sigma}(x_{\sigma(1)}-x_0) (y_{\sigma(2)}-y_0) (z_{\sigma(3)}-z_0)
S_n(a+1)+S_n(a-1)-1 =\sum_{j=1}^{a}\left((1+j)^n+(1-j)^n\right)= \sum_{j=1}^{a}\sum_{k=0}^{n}\binom{n}{k}j^k(1+(-1)^k) &= 2 \sum_{j=1}^{a}\sum_{k=0}^{n/2}\binom{n}{2k}j^{2k}= 2 \sum_{k=0}^{n/2}\binom{n}{2k}\sum_{j=1}^{a}j^{2k}.
H(U_n) =\{abcd\in W_2(U_n): (a,b^{-1}), (c,d^{-1})\in U_n\}, I(U_n) =\{abcd\in W_2(U_n): (a,d^{-1}), (b,c^{-1})\in U_n, a\neq d^{-1}\}J(U_n) =\{abcd\in W_2(U_n): (a, d^{-1}), (b,c^{-1})\in U_n, a=d^{-1}\}&=\{abca^{-1}\in W_2(U_n): (b, c^{-1})\in U_n\}.
\frac{d^{2}\tilde{\phi}}{d\lambda^2}=\left[\tilde{A}^{2}+\tilde{B}\tilde{C}-\tilde{A}'+\tilde{A}\tilde{C}^{-1}\tilde{C}'+\frac{3}{4}(\tilde{C}^{-1}\tilde{C}')^{2}-\frac{1}{2}\tilde{C}^{-1}\tilde{C}''\right]\tilde{\phi},
J(u)_i =\begin{cases}0, u_i\in A 1, u_i\in B,\end{cases}\forall i\in\{1,...,N\}. \psi(\omega) =\begin{cases}\Psi^{-1}(\omega), \omega\in A \Psi^{-1}\circ f^{-1}(\omega), \omega\in B.\end{cases}
S_3 =\sum_{y\in \mathbb{F}_p^*}\zeta_p^{-By}\sum_{z\in \mathbb{F}_p^*}\zeta_p^{-Az}\sum_{x \in \mathbb{F}_r}\zeta_p^{\mathrm{Tr}(zx^2+yx )}& = G_m \sum_{z\in \mathbb{F}_p^*}\eta_m(z)\zeta_p^{-Az}\sum_{y\in \mathbb{F}_p^*}\zeta_p^{-\frac{m_p}{4z}y^2-By}.
\Delta_b=d_bd_b^\ast+d_b^\ast d_b\,,\,\Delta_{\partial_b}=\partial_b\partial_b^*+\partial_b^*\partial_b\,\,,\,\,\Delta_{\overline{\partial}_b}=\overline{\partial}_b\overline{\partial}_b^*+\overline{\partial}_b^*\overline{\partial}_b,
a_n = \sum_{j=0}^n h_1(j) \prod_{i=j+1}^n f(i) \equiv \sum_{j=n-kp+1}^n h_1(j) \prod_{i=j+1}^n f(i) = \sum_{j=0}^{kp-1}h_1(n-j) \prod_{i=0}^{j-1}f(n-i)\equiv f_{p,k}(n) \pmod{p^k},
\begin{aligned}\zeta(n_1,\ldots,n_r) =\mathop{\int \cdots \int}_{0<t_1<t_2<\cdots<t_N<1}\omega_{a_1}(t_1)\wedge \omega_{a_2}(t_2)\wedge \cdots \wedge \omega_{a_N}(t_{N}),\end{aligned}
L_{0}= \mathrm{D i a g o n a l}( 0 , 2 , 3 , 4 , 4 , 5 , 5 , 6 , 6 , 6 , 6 , 7 , 7 , 7 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 9 , 9 , 9 , 9 , 9 , 9 , 9 , 9 , \ldots )
J^2\Big(\int_0^{-x_1-x_3}x_1^{r_1}x_2^{r_2}x_3^{r_3}(x_1+x_2+x_3)^{-s}\,dx_2\Big) = \sum_{i+j=r_2+1-s}\binom{r_2+1-s}{i,j}(-1)^{r_1+r_3-s}\beta_{r_1+i+1}\beta_{r_3+j+1}\tau(r_2,-s),
K'\setminus\mathring K=\bigcup_{j\in I_X}\overline\Omega_j=\bigcup_{j\in I_X}\Upsilon_j\cup\overline{\Omega_j\setminus\Upsilon_j},bK'=\bigcup_{j\in I_X}\beta_j\subset \bigcup_{j\in I_X}\Upsilon_j\cup \overline{\beta_j\setminus\beta_j'}.
v_1= \begin{pmatrix}1-g_2 y_2 y_2^* 1-h_2 \end{pmatrix},v_2 = \begin{pmatrix}1-f_2 x_2^* x_2 1-e_2 \end{pmatrix}, \, \, \,\, V= \begin{pmatrix}v_1 0 0 v_2 \end{pmatrix}.
-\varepsilon u_{\varepsilon}^{\prime\prime}+u_{\varepsilon}^{\prime}+\xi_{\varepsilon}-f(u_{\varepsilon}) =0\text{\ \ \a.e. in}(0,T), \xi_{\varepsilon}\in\partial\phi(u_{\varepsilon})\text{\ \ \a.e. in}(0,T), u_{\varepsilon}^{\prime}(T) =0, u_{\varepsilon}(0) =u_{0},
\left\lbrace\begin{array}{ll}\psi''-\Delta\psi +b(\psi)=-e^{\lambda h}(e, AU) \hbox{in}(0,T) \times\Omega, \partial_\nu \psi =0 \hbox{on}(0,T) \times\Gamma, t=0: \psi=0, \psi'=0 \hbox{in}~ \Omega \end{array}\right.
\begin{array}{@{}l@{}}{\displaystyle \sum_{j=1}^{4}|I_3^{k,j}(\overline{\partial \theta}_{\mathbf{A}}^{k})| \leq C \left\{(\Delta t)^{4}+ h^{2r+2}+D(\widetilde{\theta}_{\mathbf{A}}^{k}, \widetilde{\theta}_{\mathbf{A}}^{k})+\|\overline{\partial \theta}_{\mathbf{A}}^{k}\|_{\mathbf{L}^2}^{2}\right\}.}\end{array}
\begin{matrix}[x,y,z]=[y,x,z] [x, y, z] + [z, x, y] + [y, z, x] = 0, [u, v, [x, y, z]] = [[u, v, x], y, z] + [x, [u, v, y], z] + [x, y, [u, v, z]]. \end{matrix}
\sigma_2 \int_0^1 \partial_{\sigma_2}\tilde{P}(0,t\sigma_2, x, z')\,dt =\int_0^1 \frac{(1-a_0^2)t\sigma_2^2}{(1+t^2\sigma_2^2)^{3/2}\sqrt{1-\sin^2 \theta}\sqrt{t^2\sigma_2^2-s(\theta)}}\,dt &=\frac{1-a_0^2}{2\sqrt{1-\sin^2 \theta}}\int_0^{\sigma_2^2}\frac{d\tau}{(1+\tau)^{3/2}(\tau-s(\theta))^{1/2}}.
H_{\perp s c a l a r \vec{0}}= \frac{p_{0}^{2}}{2}+ \frac{1}{2}\sum_{\vec{k}> 0}\omega_{\vec{k}}( a_{\vec{k}}^{+}a_{\vec{k}}+ \bar{a}_{\vec{k}}^{+}\bar{a}_{\vec{k}}) .
q'(A)= -\frac{3-A}{3-A}-\ln(3-A)-\left[\frac{A}{A}+\ln(A)\right]-\left[-\frac{2(1-A)}{1-A}-2\ln(1-A)\right] &-\left[\frac{1+A}{1+A}+\ln(1+A)\right]-\left[-\frac{2-A}{2-A}-\ln(2-A)\right] = -\ln(3-A)-\ln(A)+2 \cdot \ln(1-A)-\ln(1+A)+\ln(2-A) = \ln\left(\frac{(1-A)^{2}\cdot (2-A)}{(3-A) \cdot A \cdot (1+A)}\right).
\begin{array}{r c l}{{Q}}&{{=}}&{{\sqrt{\frac{2 5 - D}{6}}}}\\{{\alpha_{\pm}}}&{{=}}&{{\frac{Q}{2}\pm \sqrt{\frac{Q^{2}}{4}- 1}= \frac{1}{\sqrt{2 4}}\left( \sqrt{2 5 - D}\pm \sqrt{1 - D}\right) \ .}}\\ \end{array}
|\phi_n|_{\alpha,\sigma}\le |A \zeta_n|_{\alpha,\sigma}+(n-1)|\zeta_{n-1}|_{\alpha,\sigma}+ \sum_{k=1}^{n-1}|B(\zeta_k,\zeta_{n-k})|_{\alpha,\sigma}\le | \zeta_n|_{\alpha+1,\sigma}+(n-1)|\zeta_{n-1}|_{\alpha+1,\sigma}+ K^\alpha \sum_{k=1}^{n-1}|\zeta_k|_{\alpha+1/2,\sigma}|\zeta_{n-k}|_{\alpha+1/2,\sigma}.
\sum_{x}\frac{1}{2}\sum_{C}\oint_{C}dx^{t}\frac{\delta^2 \chi [C]}{\delta\sigma^{\nu t}(x)\delta\sigma^{\mu t}(x)}\epsilon^{1, \cdots, 7\nu\mu t}\times \frac{1}{7!}W_{1, \cdots, 7}(x),
|b| > E_{|A|,\alpha,\delta,|u|,|v|}\geq C_{\delta,|v|}+ W_{A,2\delta}C_{\delta,2\delta}+ D_{\alpha,\delta,|u|,2\delta}&\geq C_{\delta,|v|}+ \sum_{i=1}^{k}C_{\delta,|t_i|}+ D_{\alpha,\delta,|u|,|t_{k+1}|},
(a^2+ b^2 +c^2)(A^2+B^2+C^2) - (aA+bB+cC)^2 & - 2(a^2+b^2+c^2)(a^2A+b^2B+c^2C) + 2(a^3+b^3+c^3)(aA+bB+cC) & = 3 (bA-aB)^2 - 2 (a-b)(2a+b)(a+2b)(bA-aB) & = 3 \left\{(bA-aB) - \frac{1}{3}(a-b)(2a+b)(a+2b)\right\}^2 - \frac{1}{3}(a-b)^2(2a+b)^2(a+2b)^2
\frac{\partial}{\partial t}\left(\nabla_i\nabla_{\bar{j}}A\right)= \Delta\left(\nabla_i\nabla_{\bar{j}}A\right)+R_{i\bar{j}l\bar{k}}\nabla_k\nabla_{\bar{l}}A &-\frac{1}{2}\left(R_{l\bar{j}}\nabla_i\nabla_{\bar{l}}A+R_{i\bar{l}}\nabla_l\nabla_{\bar{j}}A\right)+\nabla_i\nabla_{\bar{j}}B.
\tilde{u}_{xxx}(t,x) =\tilde{w}_{xxx}(t,x)+\frac{d^2}{dx^2}p(x,x)\tilde{w}(t,x) &+2\frac{d}{dx}p(x,x)\tilde{w}_x(t,x) &+p(x,x)\tilde{w}_{xx}(t,x)+\frac{d}{dx}p_x(x,x)\tilde{w}(t,x) &+p_x(x,x)\tilde{w}_x(t,x)+p_{xx}(x,x)\tilde{w}(t,x) &-\int_x^L p_{xxx}(x,y)\tilde{w}(t,x)dy
S_{f}(x';Q,A)= S_{f}(x;Q,A) + O\Bigg(\varphi_C(x) \frac{Q}{\phi(Q)}\frac{1}{\log x}\prod_{\substack{p \leq x p\nmid Q}}\left(1 + \frac{|f(p)|}{p}\right) \Bigg)
\phi_b'(b_{1,i}^0,...,b_{P,i}^0,...,b_{1,i}^{t-1},...,b_{P,i}^{t-1},w_i):=\mathbb{E}_Z[\tilde{\phi}_{b,i}(Z)] &=\mathbb{E}_Z\left[ \phi_b(b_{1,i}^0,...,b_{P,i}^0,...,b_{1,i}^{t-1},...,b_{P,i}^{t-1},\sum_{r=0}^{t-1}\hat{\gamma}^t_{1,r}b_{1,i}^r+(\sigma_1^t)_{\perp}Z_1,...,\sum_{r=0}^{t-1}\hat{\gamma}^t_{P,r}b_{P,i}^r+(\sigma_P^t)_{\perp}Z_P,w_i) \right]
[\widetilde{\varphi}_{(123)}]=\begin{pmatrix}0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 -1 -1 -1 -1 -1 0 0 0 1 0 \end{pmatrix}, [\widetilde{\varphi}_{(132)}]=\begin{pmatrix}0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 -1 -1 -1 -1 -1 \end{pmatrix}.
\int_{\mathbb{R}_{\geqslant0}}y^{\kappa}m_{\alpha(z,s),\beta(z,s)}(\mathrm{d}y)\geqslant c_{3}\left(\frac{\alpha(z,s)}{\beta(z,s)}\right)^{\kappa}=c_{3}z^{\kappa}e^{-\kappa bs},s\in[0,t],\z>1.
\Delta_h e^u = (1-|x|^2)^2 |\nabla u|^2 e^u &+ e^u((1-|x|^2)^2\Delta u(x)+2(n-2)(1-|x|^2)\sum_{i=1}^{n}x_{i}\frac{\partial u}{\partial x_{i}}(x)) &=(1-|x|^2)^2 |\nabla u|^2 e^u +e^u \Delta_h u\ge 0.
\frac{1}{2}\frac{d}{dt}\int_{\Omega}\Big(\rho |u|^2+H^2\Big) x+\int_{\Omega}\Big(\frac{1}{\sigma}|\nabla H|^2+\mu |\nabla u|^2+(\lambda+\mu)(u)^2\Big)x=\int_{\Omega}P ux,
\frac{1}{2q}\frac{1}{dt}\int (u\varphi_N)^{2q}dx=-\frac{1}{2}\int (uv-u_xv_x)u_x(u\varphi_N)^{2q-1}\varphi_Ndx &\phantom{\frac{1}{2q}}+\int \partial_x(G\ast F_2)(u\varphi_N)^{2q-1}\varphi_N dx+\int (G\ast F_1)(u\varphi_N)^{2q-1}\varphi_N dx.
\left. \begin{array}{l}{{\left[ q_{\mu}, q_{\nu}\right] = i Q_{\mu \nu}( g )}}\\{{{}}}\\{{R_{\mu \nu}- ( 1 / 2 ) R g_{\mu \nu}= 8 \pi T_{\mu \nu}( \psi )}}\\{{{}}}\\{{F ( \psi ) = 0 ,}}\\ \end{array}\right.
\Phi_{R}\left( p^{\mu}\right) = \Lambda_{R}\left( \dot{p}^{\mu}\to p^{\mu}\right) \Phi_{R}\left( \dot{p}^{\mu}\right) = e^{i \vec{J}\cdot ( \vec{\vartheta}- i \vec{\varphi})}\Phi_{R}\left( \dot{p}^{\mu}\right) ,
_{s+1}F_{s}(a_1,\ldots,a_{s+1};b_1,\ldots,b_s;x):=\sum\limits_{i=0}^\infty\frac{(a_1)_i\cdots(a_{s+1})_i}{(b_1)_i\cdots(b_s)_i}\frac{x^i}{i!},
\psi(x)=\phi\bigl(\rho(x)\bigr)\ \Rightarrow\frac{d^2\phi(\rho)}{d\rho^2}+\frac1{\rho}\frac{d\phi(\rho)}{d\rho}-\left(1-\frac{\nu^2}{\rho^2}\right)\phi(\rho)=0.
\leq C\left( \left( x_{N}^{0}\right)^{m+\gamma}{\displaystyle\sum\limits_{i=1}^{N-1}}\left\langle D_{x_{i}}^{m}v\right\rangle_{x_{i},\overline{Q}_{(3/4)x_{N}^{0}}}^{(\gamma)}+\right.
\left[ \left(f_1f_5\right)^{1/2}\Gamma^0\partial_0 + \Gamma^1\left(\partial_r +\frac{3}{2r}+ \frac{1}{8}d_r (\ln (f_1f_5))\right) +\left(f_1f_5\right)^{1/2}\Gamma^{5}\partial_{x_5}+ \Gamma^{b}D_b\right]\chi + g(r)\chi=0
\big\{a_{11}(x/q)a_{22}(x/q) - a_{12}(x/q)a_{21}(x/q)\big\}\frac{a_{12}(x)}{a_{12}(x/q)}g_1(x/q) + g_1(qx) \qquad{}-\bigg\{a_{11}(x)+\frac{a_{12}(x)}{a_{12}(x/q)}a_{22}(x/q)\bigg\}g_1(x) - b_1 (x) + \frac{a_{12}(x)}{a_{12}(x/q)}a_{22}(x/q) b_1 (x/q){}- a_{12}(x) b_2 (x/q) = 0 .
\frac{1}{\, n\,}\sum_{j=1}^{n}\sum_{k=1}^{n}\mathbb E [ \overline{b}(s, \overline{X}_{s,i}, \overline{X}_{s,j}) \overline{b}(s, \overline{X}_{s,i}, \overline{X}_{s, k}) ] \le \frac{2 C}{n}\sum_{j=1}^{n}\sum_{k=j}^{n}\Big[ \frac{\,c^{k-j}\,}{\,(k-j)!\,}\Big]^{1/2}\le 2C \sum_{k=0}^{\infty}\Big[ \frac{\,c^{k}\,}{\,k!\,}\Big]^{1/2}< + \infty \, .
\!\!\!\!\!\!\!\!\!\! y_{\chi}(t_o) = \left. \left(A \sqrt{P_{r_o}h_o}s_{\chi}(t) + \sum_{k \in \tilde{\Psi}_\chi}i_{k}^{(\chi)}(t)+ \sum_{j \in \tilde{\Psi}_{\bar{\chi}}}i_{j}^{(\bar{\chi})}(t) + i_{\rm s}(t) + n_{o}(t)\right).*h_\chi(t-t_o) \right|_{t=t_o}.
|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>\,{,}
j^{\mu \nu}= \frac 1 2 \frac 1{\sqrt{g_{x}}}\epsilon^{\mu \nu \lambda \rho}T_{\lambda \rho}= \frac 1 2 \frac 1{\sqrt{g_{x}}}\epsilon^{\mu \nu \lambda \rho}( \partial_{\lambda}A_{\rho}- \partial_{\rho}A_{\lambda}) .
\frac{3(72\kappa^{10}-360\kappa^{8}-44\kappa^{6}-32\kappa^{4}+7\kappa^{2}-3)}{(4\kappa^{2}-1)^{-1}(2\kappa^{4}+1)^{2}(144\kappa^{4}-25\kappa^{2}+1)}
W(G_{1}(n,d,x,1)) =W(G_{1}(n,d,x,0))+1, W(G_{1}(n,d,x,2)) =W(G_{1}(n,d,x,0))+6, W(G_{1}(n,d,x,-1)) =W(G_{1}(n,d,x,0))+3.
(W(\rho^0,\rho^1))^2=\inf_{v_t,\rho_t}~~\big\{{\int_0^1\int_{M}v_t^2\rho_t dx dt}\colon \frac{\partial \rho_t}{\partial t}+\nabla\cdot (\rho_tv_t)=0,~\rho_0=\rho^0,~ \rho_1=\rho^1\big\},
\frac{1}{8 \pi}\int_{\partial{\cal M}}T r{\cal K}= \frac{1}{8 \pi}\frac{\partial}{\partial{\bf n}}( \mathrm{V o l}\partial{\cal M}) = \frac{1}{8 \pi}\frac{\partial}{\partial t}\left[ \left( \frac{3}{2}t \right)^{1 / 3}{\cal V}\right] \ .
2\lambda K+\frac{4(\lambda-1)}{n}\left(2\left\vert b\right\vert-b\right)+(2(2-\lambda)\left\vert b \right\vert-\lambda b)p\omega^{-1}-2\lambda\left\vert b\right\vert p(p-1)\omega^{-2}\leq 2\lambda K+\frac{12(\lambda-1)}{n}\left\vert b\right\vert+(4-\lambda)p\left\vert b\right\vert \left\vert J\right\vert.
J(\Sigma_{H0}^*) = \left[ \begin{array}{ccc}-\mu - \beta_2 -\beta_2 \eta \noalign{\medskip}0 \beta_2-\rho-\mu \beta_2 \eta + \alpha \noalign{\medskip}0 \rho -\alpha- d_A - \mu\end{array}\right] \, .
\pi_1 = \min\{c_{\alpha_1}+ c_{\alpha_1+\alpha_2}, c_{\alpha_1}+ c_{\alpha_2}, c_{\alpha_1+2\alpha_2}+ c_{\alpha_2}\}, \pi_2 = \min\{2 c_{\alpha_1}+ c_{\alpha_1+\alpha_2}, 2 c_{\alpha_1}+ c_{\alpha_2}, 2 c_{\alpha_1+2\alpha_2}+ c_{\alpha_2}\}.
H(G^{\{v\}}_{-1/2}v_{\nu,\ell_0},G^{\{v\}}_{-1/2}v_{\nu,\ell_0}) =H(G^{\{\phi(v)\}}_{1/2}G^{\{v\}}_{-1/2}v_{\nu,\ell_0},v_{\nu,\ell_0}) &=H([G^{\{\phi(v)\}}_{1/2},G^{\{v\}}_{-1/2}]v_{\nu,\ell_0},v_{\nu,\ell_0}).
\lim_{\stackrel{z \to \infty}{|\arg z| \le \theta < \pi/2}}\frac{z^{r-\alpha}\, \Gamma(r+1)}{\Gamma(r-\alpha)}\left[ \Lambda_r W(z) - \sum_{j=1}^k \frac{A_j}{z^{r+j}}\right] \, = \, \gamma.
\mu_{0}^{t+1}=\frac{\sum_{i}\pi_{i}^{t}m_{i}^{t}}{\sum_{i}\pi_{i}^{t}}, \tau_{0}^{t+1}=\frac{1}{\sum_{i}\pi_{i}^{t}}\underset{i}{\sum}\pi_{i}^{t}\bigl[\bigl(\mu_{0}^{t}-m_{i}^{t}\bigr)^{2}+V_{i}\bigr].
\sum_{k=1}^{n}\dfrac{1}{\binom{l}{p_{k}}}\sum_{\mathrm{card}(I)=p_{k}}\left(\prod_{j\in \left\{1, \ldots, l\right\}\setminus I}x_{j}\right) \cdot f_{k}\left(\prod_{i\in I}x_{i}\right)=0.
\ell_{R}\left(R/{I_{q}}\right)=\sum_{h=1}^{[\mathbb{K}_{\vartheta}: \mathbb{K}]}\# (\vartheta^{h}(R) \cap qH) - \sum_{h=1}^{[\mathbb{K}_{\vartheta}: \mathbb{K}]}\# (\vartheta^{h}(I_{q}) \cap qH)
\pi_*(\theta^N)= \pi_* q_*( \xi_0^{d-1}\xi_1^{d-2}\cdots \xi_{d-2}q^*\theta^N) = \pi_*q_*\Big(\prod_{i=0}^{d-1}\xi_i^{d-1-i}\Big(\sum_{i=0}^{d-1}\xi_i \Big)^N \Big)= p_*\Big(\prod_{i=0}^{d-1}\xi_i^{d-1-i}\Big(\sum_{i=0}^{d-1}\xi_i \Big)^N \Big),
\tau_{e m}= \frac{1}{1 + b \chi + \chi^{2}}\left( \begin{array}{c c c}{\chi^{2}}&{- 2}&{- \chi}\\{- \frac{1}{2}}&{( b + \chi )^{2}}&{- \frac{b + \chi}{2}}\\{2 \chi}&{4 ( b + \chi )}&{- 1 + b \chi + \chi^{2}}\\ \end{array}\right)
0 =\pi_1^*\overline{L}_1\cdots \pi_{n-1}^*\overline{L}_{n-1}\cdot\pi_1^*\overline{L}_1 &= (\pi_1^*\overline{L}_1\cdots \pi_{n-1}^*\overline{L}_{n-1}|\mathrm{div}(\pi_1^*1_a))- \sum_{v \in M(K)}\int_{H^{\mathrm{an}}_{\overline{K}_v}}\log ||\pi_1^*1_a||_{\pi_1^{*}\overline{L}_{1,v}}\prod_{j = 1}^{n-1}c_1(\pi_j^*\overline{L}_{j,v}).
z_{\alpha \beta}= m \left( \begin{matrix}{{\gamma_{a b}^{0}\delta_{q p}+{\cal P}_{a q ~ b p}^{( - )}}}&{{0}}\\{{0}}&{{{\cal P}^{( - )}{}_{\dot{q}\dot{p}}^{a b}}}\\ \end{matrix}\right) = \Sigma_{i = 1}^{2 4}\lambda_{\alpha}^{i}\lambda_{\beta}^{i}\; ,
G_{r}^{( 2 )}( x ) = \sum_{n_{0}, n_{1}, \ldots , n_{\nu - 2}= 0}^{\infty}x^{\frac{r - 1}{2}+ \sum_{i = 0}^{\nu - 2}n_{i}( \nu - 1 - i )}\times \prod_{i = 1}^{\nu - 2}\left[ \begin{array}{c}{{n_{i}+ \tilde{n}_{i}+ \tilde{V}_{i , r}}}\\{{n_{i}}}\\ \end{array}\right] .
\prod_{j=1}^t\P\Big(Z_{(k+g)(j-1)+i}^{(k+g)(j-1)+i+k-1}=s_j\Big) = 2^{-t(\hat{H}_1(s^t)+D_{\rm KL}(\hat{p}^{(1)}(\cdot|s^t),\mu_k^{(b)}))}.
{\widetilde{\gamma}_{{e^*}}}= \mathop{\max}\limits_{e \in{\Phi_e}}\left\{{\frac{{{P_S}G_e^{S}L\left(\left|X_e\right|\right)}}{{{{{P_A}G_e^{A}L\left(\left|X_e\right|\right)}+ \sigma_e^2}}}}\right\}.
\boldsymbol{\psi}_\alpha(z)\boldsymbol{\psi}_\beta(w)=(z-w)^{\Delta_{\alpha+\beta}-\Delta_\alpha-\Delta_\beta}[\delta_{\alpha+\beta,0}I_{\boldsymbol{\psi}_{\alpha}(z)\boldsymbol{\psi}_{\beta}(w)}+\varepsilon_{\alpha,\beta}\boldsymbol{\psi}_{\alpha+\beta}(w)+\ldots].
\{(q_{T},\alpha_{T})\in Z^{A}(T),\varphi_{T}:\Lambda^{2}\tilde{\mathcal{E}}\rightarrow\mathcal{O}_{X_{T}}\mid\varphi_{T}\mid_{D\times T}=\Omega\circ\Lambda^{2}\alpha_{T}\mid_{D\times T}\}
d\varphi=-(\frac{1}{2}(\hat{x}-\langle\hat{x}\rangle)^{2}+\frac{i}{\hbar}\frac{\hat{p}^{2}}{2m})\varphi dt+(\hat{x}-\langle\hat{x}\rangle)\varphi[dy-2\langle\hat{x}\rangle dt]
\begin{aligned}E(\|g_{k-1}\|^2) \leq \beta^{k-1}E(\|g_0\|^2) &\leq \beta^{k-1}\|E(g_0)\|^2 &\leq \beta^{k-1}\|\nabla f(\omega_0)\|^2 &\leq 2 \beta^{k-1}L(f(\omega_0)-f(\omega^*))\end{aligned}
\Upsilon(x,t)= \sum_{j=0}^{(k_2-1)/2}c_{k_1,k_2,j}\frac{\omega^{k_2-2j}}{|x|^{2j}}+ O\left(\sum_{j=0}^{(k_2+1)/2}\frac{\omega^{k_2-2j+1}}{|x|^{2j}}\right);
\| P_{> 0}\sum_{k > 0}(2^{-k}|D|)^\sigma( \chi_Q u_k v_k)\|_{L^2}^2\lesssim sum_{k > 0}\| \chi_Q u_k \|_{L^2}^2 \|v_k\|_{L^\infty}^2\lesssim \| \chi_Q S_{>0}(u) \|_{L^2}^2 \|v\|_{\dot B_{\infty,\infty}^0}\lesssim \ \| u\|_{BMO}^2 \|v\|_{\dot B_{\infty,\infty}^0}.
{\ddot{a}_0\over a_0}=-{{16\pi^2G^2_5}\over{9c^6}}(2\varrho^2c^4+\sigma\varrho c^2+3\sigma\wp+3\wp\varrho c^2)-{{{\cal C}c^2}\over a^4_0}+{{4\pi G_5}\over{3c^2}}\left(\Lambda+{{4\pi G_5}\over{3c^4}}\sigma^2\right),
\delta = \begin{cases}\lfloor \frac{r}{2}\rfloor v(\Delta) \geq r, \delta_0 v(\Delta) = 2\delta_0 <r, \end{cases}Y = \begin{cases}0 v(\Delta) \geq r, Y_0 (\Delta)_0 = Y_0^2 v(\Delta)<r.\end{cases}
\sum_{m=1}^{\infty}\frac{\tau_A(p^{m}) \tau_B(p^{m}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1}\right)}{p^{m(1+s_1+s_2 )}}\ll \sum_{m=1}^{\infty}\frac{1}{p^{m(1+\varepsilon)}}\ll \frac{1}{p^{1+\varepsilon}}.
x = \left( \begin{array}{c c}{\bar{u}_{2}}&{u_{1}}\\{- \bar{u}_{1}}&{u_{2}}\\ \end{array}\right) , \bar{x}= \left( \begin{array}{c c}{u_{2}}&{- u_{1}}\\{\bar{u}_{1}}&{\bar{u}_{2}}\\ \end{array}\right)
= \frac{p^4x^5\big((1-p^4x^4)(1-p^5x^6) - p^2x^2(1-p^3x^3)(1-p^3x^4)\big)}{(1-px)(1-p^3x^3)(1-p^3x^4)(1-p^4x^4)(1-p^5x^6)}.
\Delta ( J^{\pm}) \langle v | q^{- J^{0}}| v \rangle = ( J^{\pm}\langle v | ) | v \rangle + ( q^{- J^{0}}\langle v | ) J^{\pm}q^{- J^{0}}| v \rangle = - \langle v | J^{\pm}| v \rangle + \langle v | J^{\pm}| v \rangle = 0 ,
{\bf E}\Bigl[Z_{\varepsilon}(q\beta) Z^{-n}_{\varepsilon}(\beta)\Bigr] \approx N^{1+q^2\beta^2-(1+\beta^2)n}\prod\limits_{j=0}^{n-1}\frac{\Gamma(1-q\beta^2+(j+1)\beta^2)^2 \,\Gamma(1-\beta^2)}{\Gamma(1-2q\beta^2+(j+1)\beta^2)\,\Gamma(1+j\beta^2)}.
3\sum_{\substack{(a,b,c)\equiv(1,1,0)\bmod{2}a+b+c=w+2,a,b>1}}r{}_{a,b,c}I_{{\rm bl}}^{\mathfrak{m}}(a,b,c)-\sum_{\substack{(a,b,c)\equiv(0,0,0)\bmod{2}a+b+c=w+2}}r_{a,b,c}'I_{{\rm bl}}^{\mathfrak{m}}(a,b,c) \overset{?}{=}\frac{-220w^{6}+2751w^{5}-10375w^{4}+16620w^{3}-35620w^{2}-7536w+285120}{1440}I_{{\rm bl}}^{\mathfrak{m}}(w+2)
\frac{1}{2}\int_\mathbb{R}\big(|W_x|^2 + 2W_{x}W_{xt}\big)(x,t) dx+ a^2\int_0^t \|W_{xx}\|^2d\tau - \int_0^t \|W_{x\tau}\|^2d\tau \leqslant O(1) \delta^{-1/4}( \|W(0)\|_1^2 + \|W_t(0)\|_1^2)+ O(1) \delta_0^{{1}/{4}}+ O(1)\delta^{1/4}\int_0^t \|W_{xx}\|^2d\tau,
(-\Delta_p)^s w(x) \leq \left(C+\frac{C(\varepsilon_0)}{(1-\theta)^{N+s\,p}}\right)\,\frac{1}{|x|^{N+s\,p}}&-\left[c\,\Big(1-\theta^\frac{s\,p}{2}\Big)\,(M-m)^{p-1}-\frac{C\,(\varepsilon+\varepsilon_0)}{(1-\theta)^{N+s\,p}}\right]\,\frac{R^{s\,p}\,\theta^{s\,p}\,}{|x|^{N+s\,p}}.
\begin{array}{rcl}{\cal L}_{GG\phi}= \displaystyle+\frac{g^2v}{2\sqrt{2}}\left( A^{\mu}(\pi_+W^-_{\mu}+W^+_{\mu}\pi_-)+Z^{\mu}(W^-_{\mu}\pi_+ +\pi_-W^+_{\mu})\right) & \displaystyle+\frac{g^2v}{2}(W^-_{\mu}W^{+\mu}+Z_{\mu}Z^{\mu})\sigma,\end{array}
\mathrm{b o s o n s :}\lambda = \left( \begin{array}{c c}{0}&{B}\\{B^{T}}&{0}\\ \end{array}\right) , \mathrm{f e r m i o n s :}\lambda = \left( \begin{array}{c c}{A_{1}}&{0}\\{0}&{A_{2}}\\ \end{array}\right) ,
K ~=~ \frac{8\pi^{2\mu}(2\mu-1)^2(\mu-2)^2}{(2\mu-3)^2 \Gamma(\mu)\Gamma(\mu+1)}~~,~~ P ~=~ \frac{16\pi^{2\mu}(2\mu-1)(2-\mu)}{(2\mu-3)\Gamma(\mu) \Gamma(\mu+1)}
\sum_{i=1}^{t-1}\sum_{j = r}^{v_i}\binom{n}{j-2}2^{\binom{j}{2}}p^{\lambda(j-2)+1}\leq \sum_{i=1}^{t-1}\sum_{j=r}^{v_i}n^{j-2}2^{j^2}n^{-(\lambda (j-2) + 1)\frac{v_t-2}{e_t}}& = \sum_{i=1}^{t-1}\sum_{j=r-2}^{v_i-2}n^j 2^{(j+2)^2}n^{-(\lambda j + 1)\frac{v_t-2}{e_t}}.
\Delta_{kh}^{n}(f,x)=\sum_{i_{1}=0}^{k-1}\sum_{i_{2}=0}^{k-1}\cdot\cdot\cdot\sum_{i_{n}=0}^{k-1}\Delta_{h}^{n}(f,x+i_{1}h+i_{2}h+\cdot\cdot\cdot+i_{n}h).
H ({\lambda}) = - \frac{1}{s i n \gamma d_{0}^{2}d_{1}^{2}}\sum_{k = 1}^{L}\frac{{\lambda}q +{\lambda}^{- 1}q^{- 1}}{2}( S_{k}^{X}S_{k + 1}^{X}+ S_{k}^{Y}S_{k + 1}^{Y}) - \frac{1}{2}S_{k}^{Z}S_{k + 1}^{Z}
\partial_t u - \alpha \Delta u = \gamma u (1 - |u|^2), \, \alpha, \, \gamma \in \mathbb{C}, \mbox{Re}\alpha \geq 0, \mathcal{L}=\alpha\Delta, \, \, \mathcal{A}= - 2 i \mbox{Im}\alpha \Delta, \, f(u, \overline{u})= \gamma u (1 - u \overline{u})
\big\| P_s\big[ \big( \frac{\gamma_\beta}{\gamma}\big)^\frac1p \big] \big\|_{L^q(\gamma)}=\beta^{\frac{n}{2p'}}\beta_{s,p}^{-\frac{n}{2q'}},\;\;\; \beta_{s,p}\coloneqq 1 + (\beta-1) \frac{q}{p}e^{-2s}
[1a]_n(1,t) = t^3 (t+1)^{n-3}- t^n + \sum_{j=1}^{n-2}\left( t^3 (t+1)^{n-j-2}- t^{n-j+1}+ t^{n-j}\right) &= t^3 \left( \frac{(t+1)^{n-2}-1}{(t+1)-1}\right) - t^n + t^2 &= t^2(t+1)^{n-2}- t^n.
\sum_{k=0}^\infty\frac{q^{k^2}}{(1-q)(1-q^2)\dotsm(1-q^k)}=\prod_{j=0}^\infty\frac 1{(1-q^{5j+1})(1-q^{5j+4})}, \sum_{k=0}^\infty\frac{q^{k(k+1)}}{(1-q)(1-q^2)\dotsm(1-q^k)}=\prod_{j=0}^\infty\frac 1{(1-q^{5j+2})(1-q^{5j+3})}
\int_{\Xi}x^{nr+k}e^{x^r/\hbar}dx ={}(-1)^n\hbar^n \left(\prod_{i=1}^n (i -1+ \frac{k+1}{r})\right) \int_{\Xi}x^{k}e^{x^r/\hbar}dx ={}(-1)^n\hbar^n \frac{\Gamma\left(n+\frac{k+1}{r}\right)}{\Gamma\left(\frac{k+1}{r}\right)}\int_{\Xi}x^{k}e^{x^r/\hbar}dx.
g^{2}R_{2}^{( \pm )}( \nu ; 0 ) = g^{2}\int_{1}^{\infty}d \alpha ( \sum_{\ell = 1}^{7}\frac{\Phi_{\ell}^{( 3 )}( \alpha )}{\alpha^{\ell}}) [ \alpha^{\frac{\nu}{2}}+ \alpha^{\frac{- \nu}{2}}] = 2 \xi ( \frac{1}{2}+ \nu ) .
\mathcal{X}_{k, \rho}^{\theta}( s,x, t ) \approx X^{s,x}_t, \mathcal{I}_{k, \rho}^{\theta}( s,x, t ) \approx \big( 1 , \tfrac{[\sigma(s,x)]^{*}}{t - s}\smallint\nolimits_s^t \big[ \sigma( r, X_r^{s, x})^{- 1}D_r^{s, x}\big]^{*}\, d W_r \big).
S_n := \{u\in V(G_n): |f_n(u) - \alpha_1| < \varepsilon_0,~\, |\overline{f}_n(u) - \alpha_2| < \varepsilon_0\}, L_n := \{u\in V(G_n): |f_n(u) - \beta_1| < \varepsilon_0,~\, |\overline{f}_n(u) - \beta_2| < \varepsilon_0\}, T_n := V(G_n)\setminus (S_n \cup L_n).
| \Phi |^{2}( t , \vec{x}) \sim f_{n}^{2}( x_{1}^{2}+ x_{2}^{2})^{n}+ 2 n t f_{n}^{2}k_{- n}\sum_{p = 0}^{[ n / 2 ]}( - 1 )^{n + p}\left( \begin{array}{c}{{n}}\\{{2 p}}\\ \end{array}\right) x_{1}^{n - 2 p}x_{2}^{2 p}+ K t^{2},
D_{F}^{N}( x ) = \frac{- i}{( 2 \pi )^{4}}\int \frac{S_{\vec{k}}( N + 1 )^{2}e^{i \vec{k}. \vec{r}- i k_{0}h ( N_{\vec{k}}) t}d^{4}k}{k^{2}+ m^{2}}- ( N \rightarrow N - 1 ) ,
\lVert \mathfrak{I}_0 \rVert_{s_0+\mu}^{Lip(\gamma)}\le \varepsilon^{6-2b}\gamma^{-1}, \lVert Z \rVert_{s_0+\mu}^{Lip(\gamma)}\le \varepsilon^{6-2 b}, \gamma=\varepsilon^{2+a}, a\in (0, 1/6),
\eta(\sigma_{*}^{-1}V,\sigma_{*}^{-1}V)=\sum_{j=1}^{n}(Vx^{j})^{2}-(V\tau)^{2}=\frac{4R^{4}|V|^{2}}{(R^{2}-|u|^{2})^{2}}=h_{R}^{2(n)}(V,z,V),
\partial_{t}\omega_{s}^{\nu}=\nu\partial_{yy}\omega_{s}^{\nu}+P_{0}\left(U_{n}^{\nu}\cdot\nabla\omega_{n}^{\nu}\right) =\nu\partial_{yy}\omega_{s}^{\nu}+\partial_{y}P_{0}\left( v_{n}^{\nu}\omega_{n}^{\nu}\right) ,