harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac1{2\pi}\int_{-1}^1(1-t^2)^2f_0(\frac1x+it) e^{(\frac1x+it)x}dt+\frac1{2\pi}\int_{-1}^1\frac{(1-t^2)^2}{\frac1x+it}f(0)e^{(\frac1x+it)x}dt =b(x)+O(1),x\to\infty.
g^{- 1}= \left( \begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{1}}\\{{0}}&{{0}}&{{\left( 1 + e^{2}\right)^{- 1}}}&{{0}}\\{{0}}&{{- \left( 1 + e^{2}\right)^{- 1}}}&{{0}}&{{0}}\\{{- 1}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right) ,
\sum_{l=1}^{h(m,n)}C_{m}^{l}(G^{0})(\lambda)(C_{m}^{l}(G^{0})(\lambda))^{*}=d_{m}^{0}(\lambda )^{\top}\vec{\alpha_{m}}\cdot \vec{\alpha_{m}}^{*}\overline{d_{m}^{0}(\lambda )},
\psi_s\overline{k_{0}^{\alpha}(\eta_p)}+k_{0}^{\beta}(\zeta_s)\overline{\chi_p}=(1-\overline{\eta}_p\zeta_s)r_{s,p}\sqrt{|\alpha'(\eta_p)|}\sqrt{|\beta'(\zeta_s)|},\begin{array}{l}1\leq p\leq m \phantom{1}1\leq s\leq n\end{array}.
c_1(\mathcal E_m) =3mL+D,c_2(\mathcal E_m)=3m^2L^2+2mL\cdot D+c_2(X) + D^2, c_3(\mathcal E_m) =m^2L^2\cdot D+m L \cdot D^2+c_2(X)\cdot D-c_3(X) + D^3,
y_{\ell}^{\vphantom +}( r ) = \sqrt{r}\left( C_{\ell}^{-}\left( \frac{R}{r}\right)^{\left( ( \ell + 1 )^{2}- M^{2}N \right)^{1 / 2}}+ C_{\ell}^{+}\left( \frac{r}{R}\right)^{\left( ( \ell + 1 )^{2}- M^{2}N \right)^{1 / 2}}\right) \ , \ \M^{2}\leq \frac{( \ell + 1 )^{2}}{N}.
\begin{aligned}\tilde a =-2{\tilde f_1}'{\tilde f_1}^{-1}-\tilde f_1\varphi''(\varphi')^{-1}{\tilde f_1}^{-1}=-2(h'f_1+hf_1')f_1^{-1}h^{-1}-h(f_1\varphi''(\varphi')^{-1}f_1^{-1})h^{-1}=-2h'h^{-1}+h(-2f_1'f_1^{-1}-f_1\varphi''(\varphi')^{-1}f_1^{-1})h^{-1}.\end{aligned}
\sum_{h=1}^{d-2}\lfloor\langle\frac{hp^i}{d-1}\rangle-\frac{mp^i}{q-1}\rfloor+\sum_{h=1,~h\neq \frac{d}{2}}^{d-1}\lfloor\langle \frac{-hp^i}{d}\rangle+\frac{mp^i}{q-1}\rfloor=0
\varphi_k(x)\varphi_k(y) =\lim_n\varphi_n\big(\varphi_{n,k}(x)\varphi_{n,k}(y)\big)\overset{\eqref{eq: jump}}{=}\lim_n\varphi_n\big(\psi_n(\varphi_k(x))\psi_n(\varphi_k(y))\big),
\int_{B(0, R)}e^{-\frac{|z|^{2}}{4t}}\partial_{x_{i}}u_i(x-z)dz = \frac{1}{2t}\int_{B(0, R)}z_{i}e^{-\frac{|z|^{2}}{4t}}u_i(x-z)dz-\int_{\partial B(0, R)}e^{-\frac{|z|^{2}}{4t}}u_i(x-z)\nu_{i}(z)ds(z) & = \frac{1}{2t}\int_{B(0, R)}z_{i}e^{-\frac{|z|^{2}}{4t}}u_i(x-z)dz-e^{-\frac{R^{2}}{4t}}\int_{\partial B(0, R)}u_i(x-z)\frac{z_{i}}{R}ds(z).
T_{\textrm{a}}^*= T_{\textrm{c}}^* \triangleq \min_{T_{\textrm{c}}\in \mathbb{N}, \, \mathbf{d}_{t}\in \mathcal{D}}\; T_{\textrm{c}}+ \mathbb{I}_{\mathcal{A}'}({T_{\textrm{c}}}).
d s^{2}= \left( 1 -{\frac{r_{-}}{r}}\right)^{- 1}( d x^{5}+{\frac{2 Q}{r}}d t )^{2}- \left( 1 -{\frac{r_{+}}{r}}\right) d t^{2}+{\frac{1 - r_{-}/ r}{1 - r_{+}/ r}}d r^{2}+ r^{2}( 1 -{\frac{r_{-}}{r}})^{2}d \Omega_{2}^{2},
B_{3 / 2}= \frac{\alpha \, \sqrt \pi}{6 4 \, R}, \quad B_{2}= \frac{4 \, \alpha}{3 1 5 \, R^{2}}, \quad B_{5 / 2}= \frac{3 7 \, \alpha \, \sqrt \pi}{2^{1 3}}\,{.}
\mathbb{V}_{1}=Mod\left( Eq\left( \mathbb{AOL}\right) \cup\left\{\right\}\right) \mathbb{V}_{2}=Mod\left( Eq\left( \mathbb{AOL}\right) \cup\left\{\right\}\right) \mathbb{V}_{3}=Mod\left( Eq\left( \mathbb{AOL}\right) \cup\left\{\right\}\right)
\sum_{t=0}^{+\infty}\frac{t!}{\Gamma(k+t+1)}L_t^{k}(x) L_t^{k}(y) z^t = \frac{(xyz)^{-\frac{k}{2}}}{(1-z)}\exp\left(-\frac{(x+y)z}{1-z}\right) I_{k}\left(\frac{2\sqrt{xyz}}{1-z}\right)
K_{a,\psi}f(\xi)= \psi \circ (\Pi \tilde L_a-\tilde L_a) f(\xi) = \sum_{m\in\mathbb{Z}^d \backslash \{0\}}\psi(\xi) \int_{\mathbb{R}^d}\int_{\mathbb{T}^d}e^{i(\varphi_a(x,\xi+2\pi m)-\varphi_a(x,\eta))}f(\eta) d\eta dx = \int_{\mathbb{T}^d}k_{a,\psi}(\xi,\eta) f(\eta) d\eta ,
\hat{h}_{\mu \nu}( x , 0 ) = \frac{2 G_{5}k m_{0}}{r}\left[ \textrm{d i a g}( 1 , 1 , 1 , 1 ) + \frac{1}{3 k^{2}r^{2}}\textrm{d i a g}( 2 , 1 , 1 , 1 ) + \mathcal{O}( 1 / r^{3}) \right]
f_1 \textstyle = \sum_{r - 1 > i(p-1) + 1 > 1}\Upsilon_i^{(1)}x^{i(p-1)+1}y^{r-i(p-1)-1}, f_2 \textstyle = \sum_{r - 1 > i(p-1) + 1 > 1}\Upsilon_i^{(2)}x^{i(p-1)+1}y^{r-i(p-1)-1},
\begin{aligned}4 \pi -\frac{(1+k)^{2}}{(1-k)^{2}}(\eta(r_{0}))^{2}\int_{\Sigma}|\nabla u|^2 \le 8\pi a\left(\frac{\mathfrak{m}_{ADM}}{m}-1\right) (1-a\eta(r_{0})) .\end{aligned}
J_{1}:=\int_{-h}^{h}\rho dx_{3}=2\rho h;J_{2}:=\int_{-h}^{h}x_{3}\rho dx_{3}=0;J_{3}:=\int_{-h}^{h}x_{3}^{2}\rho dx_{3}=\frac{2}{3}\rho h^{3}
Z_2=\Biggl(\frac{a_1+Z_1}{{\bar a}_1 +Z_1^{-1}}\Biggr)=Z_0^{-2}\Biggl(\frac{a_0+Z_0}{{\bar a}_0+Z_0^{-1}}\Biggr)\Biggl(\frac{\mu_+ +Z_0}{{\bar \mu}_++Z_0^{-1}}\Biggr)\Biggl(\frac{\mu_-+Z_0}{{\bar \mu}_- +Z_0^{-1}}\Biggr)
\int_{d(R)}^{\rho_1}f(\rho,R) \, d\rho \leq \frac{1}{1 - \rho_1^2}\, \arcsin(\xi) \Big|_{\xi(\rho_1)}^{\xi(d(R))}= \frac{1}{1 - \rho_1^2}\, \Big( \frac{\pi}{3}\, - \arcsin(\xi(\rho_1)) \Big) < \frac{1}{1 - \rho_1^2}\, \frac{\pi}{3}.
\pi_{t,k}(\varphi) \, :=\, \mathbb E [ \varphi (t, \overline{X}_{t, 1}, \ldots , \overline{X}_{t,k}) \, \vert \, \mathcal F_{t}^{X}] \, ; k \, =\, 2, \ldots , n , \, \, 0 \le t \le T\,
\gamma_{\textrm{FDF}}^{\textrm{NOMA}}=\frac{\left(1-\beta_{\textrm{F}}\right)P_\textrm{F}}{\left(1-\beta_{\textrm{F}}\right)P_\textrm{N}+{d_{\textrm{BF}}^{\alpha}\sigma_{\textrm{F}}^2}/{|h_{\textrm{BF}}|^2}}.
d_{m}=b_{m}r_{m}=\frac{2(2m-1)}{z}\frac{z}{2(2m-1)}=1, c_{m+1}=r_{m+1}r_{m}a_{m+1}=\frac{m^{2}(m^{2}-r^{2})z^{2}}{4(2m+1)(2m-1)}.
\frac{1}{\log(\log(t))}\int_{t}^{\infty}\frac{ds}{s^{2}\log^{b+1}(s)(1+s-t)^{3}}|\frac{1}{\lambda_{1}(t)^{1-\alpha}+s-t}-\frac{1}{\lambda_{2}(t)^{1-\alpha}+s-t}| \leq \frac{C ||e_{1}-e_{2}||_{X}}{(\log(\log(t)))^{3/2}t^{2}\log^{b+1}(t)}
I_{AB}(\hat{{\cal X}}_{12})=\textstyle{\frac{1}{4}}\mbox{tr}(\gamma_{A}\hat{{\cal X}}_{12}\gamma_{B}\hat{{\cal X}}_{21})=\textstyle{\frac{1}{4}}\mbox{tr}(\tilde{\gamma}_{A}\hat{{\cal X}}_{21}^{-1}\tilde{\gamma}_{B}\hat{{\cal X}}_{12}^{-1})
\dfrac{\partial}{\partial x_i}(\gamma(\tilde{r})) = \, \gamma^{\prime}(\tilde{r}) \dfrac{\partial}{\partial x_i}\sqrt{(2\xi_1(x) - x_1 - a_1)^2 + \cdots + (2\xi_{n+1}(x) - x_{n+1}- a_{n+1})^2}= \, \dfrac{\gamma^{\prime}(\tilde{r})}{\tilde{r}}\left\{\left(\sum_{k}2\dfrac{\partial \xi_k(x)}{\partial x_i}(\tilde{x}-a)_k\right) - (\tilde{x}-a)_i\right\}.
W(x) = 1 - e^{-\lambda(x-\sigma +T)}+ \alpha_{\lambda}[e^{-\lambda(K-\sigma)}- e^{-\lambda(x-\sigma+T)}] & + \int_{0^+}^{K-0}1 - e^{-\lambda (x - w - \sigma + T)}dW(w) + \alpha_{\lambda}\int_{0^+}^{K-0}e^{-\lambda(K - w - \sigma)}- e^{-\lambda (x - w - \sigma + T)}dW(w),
\left\|u-u_h\right\|_{dG}^2\leq \sum_{k=1}^N C^{(k)}\left(\left({h^{(k)}}\right)^{2r}+\sum_{j \in{\mathcal{I}}_{\mathcal{F}}^{(k)}}\alpha^{(k)}\frac{h^{(k)}}{h^{(j)}}\left({h^{(k)}}\right)^{2r}\right),
\begin{gathered}\left\|\phi_\epsilon \circ \big(\tilde{y}_{T_m}^K\big)^{-1}- f_0 \circ \big(\tilde{y}_{T_m}^K\big)^{-1}\right\|_{L^1(\mathbb{T}^2)}\le\epsilon, \left\|\phi_\epsilon \circ z_m - f_0 \circ z_m\right\|_{L^1(\mathbb{T}^2)}\le\epsilon.\end{gathered}
\sup \left|\frac{\partial\varphi_w}{\partial w}(z)\right| |w_{m+1}- g(w_m)| = \sup \left|\frac{\partial\chi_w}{\partial w}(\psi_w^{-1}(z)) + \frac{\partial\chi_w}{\partial \mathcal{Z}}(\psi_w^{-1}(z)) \frac{\partial\psi_w^{-1}}{\partial w}(z)\right| |w_{m+1}- g(w_m)| = o(w_m)
J_\circ (\ell) =\begin{cases}\, J(\ell), \textrm{for}\, \ell = -1,0,\cdots, d-3-[d-\alpha], \, [\,2 \beta_{d-\ell-1}(\alpha -\ell-1),\, 2 \beta_{d-\ell}(\alpha -\ell)\,], \textrm{for}\ell = d-2-[d-\alpha], \, [\, 1,\, 2 \beta_{[d-\alpha]+1}(1-\langle d-\alpha\rangle) \,], \textrm{for}\ell = d-1-[d-\alpha].\end{cases}
\mu_P(U_P) = \left(1-\mu_P(P\widehat{\mathcal{O}}_P)\right) \mu_P(P\widehat{\mathcal{O}}_P)^{d-1}\left(\mu_P(P\widehat{\mathcal{O}}_P)-\mu_P(P \widehat{\mathcal{O}}_P)^2 \right) = \left(1-q^{-\deg(P)}\right) q^{-(d-1) \deg(P)}\left( q^{-\deg(P)}- q^{-2\deg(P)}\right) = \frac{(1-q^{-\deg(P)})^2}{q^{d \deg(P)}}.
E\left[\widetilde{Z}_n^*(t)^2|\mathcal{F}^X\right] =\frac{2}{3\mathfrak{s}^2_n(t)}\sum_{i=1}^nK_h(t_{i-1}-t)^2(X_{t_{i}}-X_{t_{i-1}})^4\leq c_0r_n^2nh\cdot\frac{nh}{h^2}=c_0n^2r_n^2
X\left( t\right) \in \overline{\Omega}_{t},\left\vert \Lambda\right\vert \left( t\right) =\int_{\left( 0,t\right]}I_{\left\{X\left(s\right) \in \partial \Omega_{s}\right\}}d\left\vert \Lambda \right\vert\left( s\right) <\infty ,
\big| h^{(M)}(f(x)) - h^{(M)}(f(y)) \big|\leq CR^{1-\frac{k}{\beta}}\sum \sum_{r=0}^p\frac{f(y)^{\frac{k}{\beta}-1}+ (1-f(y))^{\frac k{\beta}-1}}{f(y)^{r+q+\frac 12}(1-f(y))^{p+q-r+\frac 12}}|x-y|^{\beta-k}.
\widehat{b}= \frac{\sum_{i = 1}^{5}x_{i}y_{i}- 5 \overline{x}y}{\sum_{i = 1}^{5}x_{i}^{2}- 5 \overline{x}^{2}}= \frac{1 1 2 - 1 0 2}{2 0 0 - 1 8 0}= 0 . 5
f\left(e_{(\mu, i) (\nu, j)}\right) =f\left(\sigma \left(r_{\mu}(c_{ij})a_{\nu}\right)\right)=\sigma'\left(f\left(r_{\mu}(c_{ij})a_{\nu}\right)\right) &=\sigma'\left(f\left(r_{\mu}(c_{ij})\right)f\left(a_{\nu}\right)\right)=\sigma'\left(r'_{\mu}(c_{ij})a'_{\nu}\right).
g= \sum_{n}c_{n}[2\theta(y-y_{n})-1]\, , \hspace{2cm}c_{n}= \left.-\frac{\hat{\chi}T_{n}}{2(\hat{d}-2)}\frac{1}{\sum_{m}c_{m}|y-y_{m}|+C}\right|_{y=y_{n}}\, .
W^{1,p}(\Omega,w)\hookrightarrow W^{1,p_s}(\Omega)\hookrightarrow \begin{cases}L^q(\Omega), p_s\leq q\leq p_s^{*}, 1\leq p_s<N, L^q(\Omega), 1\leq q< \infty, p_s=N, C(\overline{\Omega}), p_s>N,\end{cases}
d s^{2}= - e^{- 2 \phi ( r )}\left( 1 - b ( r ) / r \right) d t^{2}+{\frac{d r^{2}}{\left( 1 - b ( r ) / r \right)}}+ r^{2}( d \theta^{2}+ \sin^{2}\theta \; d \varphi^{2}) .
M_{0}( k_{1}, k_{2}, \dots , k_{r}) = \frac 1{K}\sum_{D | k_{\nu}, \, \nu = 1 , 2 , \dots , r}\frac{\mu ( D ) ( K / D ) !}{\prod_{\nu = 1}^{r}( k_{\nu}/ D ) !}, \qquad \qquad K : = \sum_{\nu = 1}^{r}k_{\nu}.
P((x_1,x_2),\mu)= \begin{pmatrix}y_1 y_1 y_2 -y_2 y_3 y_4 \end{pmatrix}\begin{pmatrix}c_E 0 0 \overline{c_E}\end{pmatrix},y_i=\int_{\beta_i}\psi_1\quad(i=1,2,3,4).
\begin{aligned}\mathcal{J}(y(u),u) \leq \,\, \mathcal{J}^\oplus(\alpha,\beta;\eta,\boldsymbol{\tau},v) \forall \, \eta \in H^{1,1}_{0,per}(Q_T), \, \boldsymbol{\tau}\in H(\emph{div},Q_T), \, v \in L^2(Q_T), \, \alpha, \beta > 0. \end{aligned}
=(-1)^{n-1}n^{n-2}(n+1)^{n-1}\times\left|\begin{array}{cccccc}-1 -2 -3 \cdots -(n-2) n^{2}-n+1 -1 1 0 \cdots 0 0 0 -1 1 \cdots 0 0 0 0 -1 \cdots 0 0 \cdots \cdots \cdots \cdots \cdots \cdots 0 0 0 \cdots -1 1 \end{array}\right|
||\partial_{r}F_{6}(t,r)||_{L^{2}(r dr)}\leq \frac{C}{t^{9/2}\log^{5b-1+\frac{5N}{2}}(t)}+ ||\partial_{r}v_{5}||_{L^{2}(r dr)}\cdot ||\frac{\chi_{\geq 1}(\frac{4r}{t}) \lambda(t)^{2}}{(r^{2}+\lambda(t)^{2})^{2}}||_{L^{\infty}_{r}}&\leq \frac{C}{t^{9/2}\log^{5b-1+\frac{5N}{2}}(t)}
\Psi_i(p,p)-\Psi_i(0,0)\equiv p\cdot\frac{\partial\Psi_i(x,0)}{\partial x}\bigg|_{x=0}+p\cdot\frac{\partial\Psi_i(0,y)}{\partial y}\bigg|_{y=0}\pmod{p^2},i=1,2.
R(z) = \left( \begin{array}{cc}0 0 0 R_{\Omega}(z)\end{array}\right) +\left(\begin{array}{c}1 R_{\Omega}(z)|1 \rangle\end{array}\right)\frac{1}{C(z)}\left( 1, \langle 1 | R_{\Omega}(z) \right )
d\left( \chi_{\sigma_i+1}^{-1}\circ \chi_{\sigma_i}\right)_* \vert_{\zeta_i(1)}= e^{2\pi i\left( \frac{k_{i+1}}{3}- \frac{k_i+\eta_i}{3}+ \frac{1}{2}\right)}\alpha t_i\alpha^{-1},
\min \sigma\big(\mathbf{H}(t, \cdot)\big) \geq \alpha \exp\big(-t/\tau\big) +\omega \int_{0}^{t}\exp\big(-(t - s)/\tau\big) \mathrm{d}s &\geq \alpha \exp\big(-t/\tau\big) +\frac{\omega}{\tau}\big(1 - \exp\big(-t/\tau\big)\big) &\geq \min\big\{\alpha \exp(-1), \omega/(2 \tau)\big\}=: \kappa > 0
R^{\omega,\omega_\alpha}h(z) =4\int_{\mathbb{D}}h'(\xi)\overline{(B_z^{\omega_\alpha})'(\xi)}\omega^*(\xi)dA(\xi) &=4\int_{\mathbb{D}}(1-|\xi|)h'(\xi)\overline{(B_z^{\omega_\alpha})'(\xi)}\frac{\omega^*(\xi)}{1-|\xi|}dA(\xi).
\begin{cases}\underline h(t):=c_0t+\delta(t), \ \ \t\geq 0, &\underline U(t,x):= (1-\epsilon(t))[\Phi(x-\underline h(t))+\Phi(-x-\underline h(t))-\mathbf{u}^*], \ \ \t\geq 0, x\in [-\underline h(t),\underline h(t)],\end{cases}
\bar{\Psi}^{\downarrow}= \psi : e^{\frac{i}{\sqrt{2}}( \varphi_{c}+ \varphi_{s})}: , \qquad \Delta_{s}^{\uparrow \uparrow}= : e^{i \sqrt{2}\varphi_{s}}: , \qquad \Delta_{c}= : e^{i \sqrt{2}\varphi_{c}}: \ ,
E_{p,q}^{1}=Z_{p,q}^{1}=Z_{p,q}^{r}\oplus M_{p,q}^{r-1}\oplus\cdots\oplus M_{p,q}^{1}= H_{p,q}^{r}\oplus B_{p,q}^{r}\oplus M_{p,q}^{r-1}\oplus\cdots\oplus M_{p,q}^{1}.
E_{TM}(a) = \frac{\hbar c}{a}\, \left\{-\frac{\,{n_{0}}^3\,a^4\,{\Omega}^4}{24\,\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\}.
\sum_{\substack{B\subset \{1,2,\cdots , n\}|B| = k}}\prod_{i\in B}b(i;n) \leq \frac{\lambda_n}{k}\, \sum_{\substack{A\subset \{1,2,\cdots , n\}|A| = k-1}}\prod_{i\in A}b(i;n),
\bigl\|\sum_j\bigl\langle G_R(\Lambda_{\psi}(q_j)\otimes p_j^*\zeta_2),\pi_R(\rho_2)^*\zeta_1\otimes\rho_1\bigr\rangle\bigr\| &=\bigl\|\sum_j\langle\rho_1\otimes\pi_R(\rho_2)^*\zeta_1,G_{\rho}(p_j^*\zeta_2\otimes\Lambda_{\psi}(q_j))\rangle\bigr\|.
f_\nu = C_2 J_{\nu}(\sqrt{-C_1}x) + C_3 Y_{\nu}(\sqrt{-C_1}x) C_1<0, &f_\nu = C_2 I_{\nu}(\sqrt{C_1}x) + C_3 e^{\nu\pi\i}K_{\nu}(\sqrt{C_1}x) C_1>0.
\left( -\frac{d^{2}}{d\beta^{2}}+\frac{(b_{n_1}^{(1)})^2-1/4}{\sin^{2}\beta}+ \frac{(b_{n_2}^{(2)})^2-1/4}{\cos^2 \beta}\right) H_{m,n_1,n_2}(\beta )= C_{m,n_1,n_2}H_{m,n_1,n_2}(\beta ),
(d_A \alpha)(X_0, X_1, \ldots, X_n) = \sum_{i=0}^n (-1)^{i}\rho (X_i) \alpha(X_0, \ldots , \hat{X_i}, \ldots , X_n) & + \sum_{i < j}(-1)^{i+j}\alpha ([X_i,X_j], X_0, \ldots , \hat{X_i},\ldots , \hat{X_j}, \ldots , X_n),
\left\vert\xi_{h,i}-1\right\vert \le \max\left\{\left\vert 1-\|a-(h/2)e_i\|_{l_2^d}^2\right\vert~,~\left\vert1-\|a+(h/2)e_i\|_{l_2^d}^2\right\vert\right\}&=\max\left\{\left\vert -ha_i-h^2/4\right\vert~,~\left\vert ha_i-h^2/4\right\vert\right\}&\leq h+h^2/4.
\Delta_{T}( x , \tau ) = \Delta^{>}( x , - i \tau ) = \int_{- \infty}^{+ \infty}\frac{d \omega}{2 \pi}\int_{- \infty}^{+ \infty}\frac{d p}{2 \pi}e^{- \omega \tau}e^{i p x}\rho ( \omega , p ) ( 1 + f ( \omega ) ) .
D_{{ia}}(T) =\zeta(1+{ia})T+\zeta(1-{ia})T^{1-{ia}}\frac{1}{1-{ia}}+\mathcal{O}(T^{1/2+\varepsilon}) -\;\frac{1}{2\pi i}\bigg[\int_{1+\varepsilon_T+iT^{1/2}}^{\frac{1}{2}+iT^{1/2}}+\int_{\frac{1}{2}+iT^{1/2}}^{\frac{1}{2}-iT^{1/2}}+\int_{\frac{1}{2}-iT^{1/2}}^{1+\varepsilon_T-iT^{1/2}}\bigg] \zeta(s)\zeta(s+{ia})T^s\frac{ds}{s},
\Rightarrow Q_{2}O^{\prime}=\frac{R\alpha_{2}}{\alpha_{2}-\frac{\alpha_{2}+\beta_{2}}{n}}=\frac{R\frac{P_{2}Q_{2}}{PQ}\alpha}{\frac{P_{2}Q_{2}}{PQ}(\alpha-\frac{\alpha+\beta}{n})}=\frac{R\alpha}{\alpha-\frac{\alpha+\beta}{n}}=QO^{\prime}
\mathbb{E}(\overline{\Vert X(t)\Vert}^2) =\mathbb{E}\left(\left[\frac{1}{t}\int_0^t \Vert X(s)\Vert\right]^2\right) &=\frac{1}{t^2}\mathbb{E}\left(\left[\int_0^t \Vert X(s)\Vert\right]^2\right) &\leq \frac{1}{t}\mathbb{E}\left(\int_0^t \Vert X(s)\Vert^2\right)=\mathbb{E}( \overline{\Vert X(t)\Vert^2}).
M^N_\tau=\frac{B^N_{\tau_t}-B^N_{\tau_t+\tau(t-\tau_t)}}{\sqrt{t-\tau_t}},M^T_\tau=\frac{B^T_{\tau_t+\tau(t-\tau_t)}-B^T_{\tau_t}}{\sqrt{t-\tau_t}}.
\widetilde{\chi}_l = \begin{cases}\widetilde{\chi}_{\nearrow}= \mbox{top-right shift of $\widetilde{\chi}$,}\mbox{if $l$ is a multiple of $\sqrt{K}$}, \widetilde{\chi}_{\rightarrow}= \mbox{right shift of $\widetilde{\chi}$,}\mbox{otherwise.}\end{cases}
\begin{aligned}s_1=\iint\frac{h^0_e}{u^0_e}\nabla w_2\cdot\nabla w_1\lesssim |\frac{h^0_e}{u^0_e}|\Vert\nabla w_1\Vert_{L^2}\Vert\nabla w_2\Vert_{L^2},\end{aligned}
S_{3 / 2}= \int_{M}d^{4}x \sqrt{g}\biggr [{\frac{1}{2}}\tilde{\kappa}_{\mu}{\widehat \nabla}\kappa^{\mu}+ \tilde{\epsilon}{\widehat \nabla}\epsilon +{\frac{1}{2}}\tilde{\eta}{\widehat \nabla}\eta \biggr ] ,
0 = \int_{-{\infty}}^{\infty}d x^{-}\left[{\theta}_{-}^{\; \; +}\right]_{x^{+}= -{\infty}}^{x^{+}={\infty}}+ \frac{1}{2}{\int}_{-{\infty}}^{\infty}d x^{+}\left[ B ( x ) \frac{n_{-}}{m^{2}- n_{-}{\partial}_{+}^{\; \; 2}}B ( x ) \right]_{x^{-}= -{\infty}}^{x^{-}={\infty}},
\int_{- 1}^{x_{n,j}^{(\alpha )}}{{P_{B,n}}f(x)\,dx}= \frac{{x_{n,j}^{(\alpha )}+ 1}}{2}\sum\limits_{i = 0}^n{f_{n,i}^{(\alpha )}{\mkern 1mu}\int_{- 1}^1{{\cal L}_{B,n,i}^{(\alpha )}\left({t; - 1,x_{n,j}^{(\alpha )}}\right)\,dt}},j = 0, \ldots ,n.
x_{k}u_{k}\, a_{ij}(x) u_{ij}= \partial_i\big(x_{k}u_{k}\, a_{ij}(x) u_{j}\big) - x_{k}u_{k}\partial_i a_{ij}(x) u_{j}- a_{ij}(x)u_{i}u_{j}- x_{k}a_{ij}(x)u_{ik}u_{j}.
F_{z^{eq}_{n}}(z^{eq}_{n}|\tilde{a}) = 1 - \frac{1/v}{z^{eq}_{n}\frac{\overline{\gamma}_{sp}}{\overline{\gamma}_{s}}+\frac{1}{v}}\, \exp\left(-\frac{\tilde{a}}{v}+\frac{\tilde{a}}{\frac{\overline{\gamma}_{sp}}{\overline{\gamma}_{s}}z^{eq}_{n}v^{2}+v}\right).
\limsup_\lambda\| e_\lambda^*(b-\omega(b)I)e_\lambda\| = \limsup_\lambda \|e_\lambda^* (s^*+t) e_\lambda\| &\leq \limsup_\lambda (\|e_\lambda s^*\|+\|t e_\lambda\|) &= \limsup_\lambda (\|se_\lambda\|+\|t e_\lambda\|) &=0.
d s^{2}= \sinh 2 t \left( - d t^{2}+ d x^{2}+ e^{- 2 x}\left[ \left( \mathrm{t a n h}t \right)^{\sqrt{3}}d y^{2}+ \left( \mathrm{t a n h}t \right)^{- \sqrt{3}}d z^{2}\right] \right)
\| D^s h \|_p \lesssim \| (2^{ks}P_k h)_{l_k^2}\|_p & = \| ( 2^{ks}P_k( \sum_{j\ge k-10}f_j \tilde g_j ) )_{l_k^2}\|_p & \lesssim \sum_{l\ge -10}\| ( 2^{ks}f_{k+l}\tilde g_{k+l})_{l_k^2}\|_p & \lesssim \sum_{l \ge -10}2^{-ls}\| (2^{ks}f_k \tilde g_k )_{l_k^2}\|_p\lesssim \| (2^{ks}f_k \tilde g_k )_{l_k^2}\|_p.
(2^*-1) \alpha_n\sum_{i=1}^k\mu_i^{\frac{n-2}{2}}\int_\Omega U_{\mu_i,\xi_i}^{2^*-2}(x)H(x,\xi_i) \psi^h_{\mu_j,\xi_j}dx = o\Big(\frac{\epsilon}{|\ln\epsilon|^2}\Big).
D_{\Phi}^{(g)}(f,y) = g^{-1}D_{\Phi}(g(f,y)) = g^{-1}D_{\Phi}(gf\alpha(g,f),\alpha(g,f)^{-1}y) &= g^{-1}(\delta_{gf\alpha(g,f)}\times D_{\pi}(\alpha(g,f)^{-1}y)) &= \delta_{f}\times \alpha(g^{-1},gf\alpha(g,f))^{-1}D_{\pi}(\alpha(g,f)^{-1}y) &= \delta_{f}\times D_{\pi}^{(\alpha(g,f)^{-1})}(y).
E_1^{-k,i+k}=\underset{j\ge0\atop j\ge -k}\bigoplus\underset{N\in \Theta_{2j+k+1}}\bigoplus H_{conv}^{i-2j-k}(M\cap N/K_0)\Longrightarrow H_{conv}^i(M/{\mathfrak S}^{0})
u(t+\Delta t)= \,u(t+\tfrac{\Delta t}2)+\tfrac{\Delta t}2u'(t+\tfrac{\Delta t}2)+\mathcal{O}(\Delta t^2), u(t)= \,u(t+\tfrac{\Delta t}2)-\tfrac{\Delta t}2u'(t+\tfrac{\Delta t}2)+\mathcal{O}(\Delta t^2),
\begin{aligned}\kappa_3(t) =\frac{\left(\alpha_n'(t)\right)^2}{4\alpha_n(t)}-\frac{\alpha_n'(t)}{2\alpha_n(t)}-\alpha_n^3(t)+2t\alpha_n^2(t)+(-t^2+2n+1)\alpha_n(t),\kappa_4(t)=\frac{\alpha_n'(t)}{2\alpha_n(t)},\end{aligned}
[K^{cyc}(a):K^{cyc}(a)]=\frac{[G_{K^{cyc}}:G_{K^{cyc}(a)}]}{[K^{cyc}(a):K^{cyc}]}=\frac{\#(2,\mathbb{Z}/n_{i}\mathbb{Z})}{D(n_{i})}=n_{i},
Q_{0}( \zeta ) \; = \;{\frac{\zeta}{2 \tilde{z}_{0}}}\left[ \left( 1 -{\frac{3}{4 \tilde{z}_{0}}}\right) +{\frac{1}{4 \tilde{z}_{0}}}{\frac{\zeta}{2 \tilde{z}_{0}+ \zeta}}\right] \; ,
\phi b^{n_0}xb^{m_k}\sim \phi b^{n_0+1}xb^{m_k}-\sum_{s\in S_b}\phi sb^{n_0+1}xb^{m_k}&\sim ... &\sim \phi b^{n_0+(m_0-n_0)}xb^{m_k}-\sum_{i=1}^{m_0-n_0}\sum_{s\in S_b}\phi sb^{n_0+i}xb^{m_k}&=\phi b^{m_0}xb^{m_k}-\sum_{i=n_0+1}^{m_0}\sum_{s\in S_b}\phi sb^ixb^{m_k}.
p_{s}=p^{*}\frac{v^{*2}[v_{f}v_{g}-v^{*}(v_{f}+v_{g})]}{v_{f}^{2}v_{g}^{2}}T_{s}=T^{*}\frac{v^{*}(v_{f}+v_{g})(v_{f}-v^{*})(v_{g}-v^{*})}{v_{f}^{2}v_{g}^{2}}
\sigma_{1 , +}= \sigma_{1 , 1}= \frac{1}{2}\left( u_{1 , 1}+ u_{1 , 2}\right) + i \sqrt{u_{1 , 3}- \left[ \frac{1}{2}( u_{1 , 1}- u_{1 , 2}) \right]^{2}}= \sigma_{1 , 3}^{*}= \sigma_{1 , -}^{*}
f'(x_1^*,\ldots,x_L^*;0,\ldots,d_l,\ldots,0) =\nabla_{x_l}f(x_1^*,\ldots,x_L^*)^\top d_l &\le\|\nabla_{x_l}f(x_1^*,\ldots,x_L^*)\|_2 &\le\|\nabla_{x_l}f(0,\ldots,0)\|_2+\|\nabla_{x_l}f(x_1^*,\ldots,x_L^*)-\nabla_{x_l}f(0,\ldots,0)\|_2 &\le\|\nabla_{x_l}f(0,\ldots,0)\|_2+M\|(x_1^*,\ldots,x_L^*)\|_2 &\le\|\nabla_{x_l}f(0,\ldots,0)\|_2+M\sqrt{\sum_{l\in[L]}C_l^2},
\textsf{Var}(|F_1|^2) &= \mathbb{E}[|F_1|^4] - (\mathbb{E}[|F_1|^2])^2 &= \frac{711 - 640 a - 72 a^2 + a^4 + 12 (33 + 32 a) \log{a}+ 72 (\log{a})^2}{18 (\log{a})^4}
\frac{1}{b_0}= \frac{a_0}{\pi^2 \big( \prod_{k=1}^\infty \frac{\mu_k - \mu_0}{k^2}\big)^2}, \frac{1}{b_n}= \frac{a_n n^4}{\pi^2 [\mu_0-\mu_n]^2 \big( \prod_{k=1, k \neq n}^\infty \frac{\mu_k - \mu_n}{k^2}\big)^2}, n = 1, 2, \ldots .
Z_{bif}(\alpha | P', \vec{Y}'; P, \vec{Y}) = \prod_{i,j = 1}^2 \Bigg( \prod_{s \in Y_i}( P_j' - P_i + b (l_{Y'_j}(s) +1 ) - b^{-1}a_{Y_i}(s) -\alpha) \prod_{t \in Y'_j}( P_j' - P_i - b l_{Y_i}(t) + b^{-1}(a_{Y'_j}(t) + 1) -\alpha) \Bigg),
\int\displaylimits_{-\infty}^{+\infty}e^{\pm i\frac{x_0y_0}{2c}}e^{-\frac{x_0^2}{4c}}\mathrm{d}x_0 = 2\sqrt{\pi c}\e^{-\frac{y_0^2}{4c}}\int\displaylimits_{0}^{+\infty}\cos\left( \frac{rg}{2c}\right) e^{-\frac{r^2}{4c}}\mathrm{d}r = \sqrt{\pi c}\e^{-\frac{g^2}{4c}},
u_{r}={u^{cl}}_{r}\pm 2 \Lambda^{N_{c}-\frac{1}{2}}\Big[\delta_{N_{c},r}(\sqrt{\lambda M + m_Q}-\frac{\lambda M}{2 \sqrt{\lambda M + m_Q}})+\delta_{N_{c}-1,r}\frac{\lambda}{2\sqrt{\lambda M+m_Q}}\Big].
\Upsilon_{t}^{s}=\exp\left\{\int_{t}^{s}\left(g_{1}(r)-\frac{1}{2}f_{2}^{2}(r)\right)dr +\int_{t}^{s}f_{2}(r)dw_{2}(r)\right\}, & g_{1}=a+b_{1}^{2}m_{1}^{-1}\gamma_{1}, & g_{2}=(b_{1}^{2}m_{1}^{-1}\gamma_{2}+b_{2}^{2}m_{2}^{-1}\alpha_{2})\tilde{y}+b_{1}^{2}m_{1}^{-1}\gamma_{3}+b_{2}^{2}m_{2}^{-1}\beta_{2}+b_{1}n_{1}+b_{2}n_{2}+c.
(P_{1,1}(t),P_{1,2}(t),\,P_{2,1}(t),P_{2,2}(t),\,P_{3,1}(t),P_{3,2}(t),\, Q(t)) = (0,\infty,\,1,-1,\,i + \varepsilon e^{2 \pi i t},-i,\,0),
\hat f_m(x):= \begin{cases}\frac{\nu(J_1)}{\mu_n}\textnormal{if}x\in I\cap (-\infty, x_1^*], \frac{1}{x_{j+1}^*-x_j^*}\bigg[\frac{\nu(J_{j+1})}{\mu_n}(x-x_j^*)+\frac{\nu(J_{j})}{\mu_n}(x_{j+1}^*-x)\bigg] \textnormal{if}x\in (x_j^*,x_{j+1}^*] j = 1,\dots,m-1, \frac{\nu(J_m)}{\mu_n}\textnormal{if}x\in I\cap (x_m^*,\infty). \end{cases}
\epsilon-\delta <\sum_{i=1}^{\infty}|x_{N_{k}}(i)|\chi_{F_{k}}(i) &=\sum_{i=1}^{n}|x_{N_{k}}(i)|\chi_{F_{k}}(i)+\sum_{i=n+1}^{\infty}|x_{N_{k}}(i)|\chi_{F_{k}}(i) &\leq \sum_{i=1}^{n}\chi_{F_{k}}(i)+\sup_{x\in A}\sum_{i=n+1}^{\infty}|x(i)|.
K_{1 2}\, \approx \, \, < K_{T 1 2}( P_{0}- h_{3}) > \, \, = \, \,{\frac{- 1}{2 i \pi}}\, \beta_{1}\beta_{2}\, \Lambda_{1 2}^{+ +}\, V_{1 2}( P_{0}\! - \! h_{3}) \, \Lambda_{1 2}^{+ +}\, , \cdots .
\|Tf\|_{L^q(X)}= \Big(\int_X\Big|\int_X K(z,z')f(z')\;d\mu(z')\Big|^q\;d\mu(z)\Big)^{1/q}\lesssim \Big(\int_0^\infty\Big|\int_{r<r'}\tilde{K}(r,r')\tilde{f}(r') r'^{-1}\;dr'\Big|^q\; r^{-1}dr\Big)^{1/q},
\begin{array}{r c l}{{{\cal Q}_{e}^{I \ \prime}}}&{{=}}&{{( c \hat{a}_{0}+ d ){\cal Q}_{e}^{I}+ c e^{- \hat{\phi}_{0}}{\cal Q}_{m}^{I}\, ,}}\\{{}}&{{}}&{{}}\\{{{\cal Q}_{m}^{I \ \prime}}}&{{=}}&{{- c e^{- \hat{\phi}_{0}}{\cal Q}_{e}^{I}+ ( c \hat{a}_{0}+ d ){\cal Q}_{m}^{I}\, .}}\\ \end{array}
M_{s}^k=\frac{\left(M_{s}^{k-1}\backslash \phi_k( (0,s^{1/4}]\times X\times[-\eta/2,\eta/2])\right) \bigsqcup \left(\{\mathbf{r}_s\leq 1\}\times I_{\eta}^{k}\right)}{\thicksim_k},
i(t)=i_{0}e^{-\frac{R}{L}t}+L^{-1}[\frac{E\omega}{(s^{2}+\omega^{2})(sL+R)}]=i_{0}e^{-\frac{R}{L}t}+L^{-1}[\frac{E\omega}{2j\omega}(\frac{1}{s-j\omega}-\frac{1}{s+j\omega})\frac{1}{(sL+R)}]
\Gamma^{(2)}[\varphi_{1}]=\frac{\sigma^{2}}{4(2\pi)^{d}}\int d^{d}x \int d^{d}y\,\varphi_{1}^{2}(x) \varphi_{1}^{2}(y)\int d^{d}p\, e^{-ip(y-x)}m_{2}^{d-4}I(p^{2},m_{2}^{2}).
\frac{\partial h(\gamma, K)}{\partial K}= -(1-\gamma)^{K+1}\log(1-\gamma)/\gamma - (1-\gamma)^{K+1}- K(1-\gamma)^{K+1}\log(1-\gamma) = -(1-\gamma)^{K+1}\left(\log(1-\gamma)/\gamma +1+K\log(1-\gamma)\right)\geq 0
F_{a s y m}^{q}( \nu , \tilde{w}) = \int_{0}^{1}\sqrt{\frac{1 + \frac{1}{\nu}\log \left( \frac{2 ( \tilde{y}^{\prime}\tilde{y}_{c}+ 1 ) ( \tilde{w}- \tilde{y}^{\prime}\tilde{y}_{c}- 2 )}{( \tilde{y}^{\prime}\tilde{y}_{c}+ 2 ) ( \tilde{w}- 2 )}\right)}{1 + \frac{1}{\nu}\log \left( \frac{2 \left( 1 - \sqrt{\tilde{w}}\right)^{2}}{\tilde{w}- 2}\right)}}~ d \tilde{y}^{\prime}.
{\tilde{M}}: = M \sqcup_{{p_{1}, \varepsilon}}X_{\Gamma_1}\sqcup_{{p_{2},\varepsilon}}\dots\sqcup_{{p_{N}, \varepsilon}}X_{\Gamma_{N}}\sqcup_{{q_{1}, \varepsilon}}X_{\Gamma_{N+1}}\sqcup_{{q_{2},\varepsilon}}\dots \sqcup_{{q_{N+K}, \varepsilon}}X_{\Gamma_{N+K}},
\tilde{m}_{sb}={\frac{{\tilde{m}_f}^2}{2\tilde{m}_s}}[{\frac{1}{\sqrt{1-\tilde{m}_{sb}^2}}}tan^{-1}\sqrt{{\frac{1+\tilde{m}_{sb}^2}{1-\tilde{m}_{sb}^2}}}-{\frac{\pi}{4}}]
\lim_{L\rightarrow 0}k_{\lambda e}(A_L,q_L,\mu_L) \geq \max_\beta\big( -\frac{\lambda^2}{|C|}\int_C (\nabla\beta+e) A (\nabla \beta+e) -\overline{\mu}\big) = -\lambda^2 D_e(A) -\overline{\mu}.
\sum_{p=1}^{r_{i}^{(1)}}(\frac{2}{3}\sum_{p>p'>0}\mathrm{min}\{n_{p,i},n_{p',i}\}+\frac{1}{3}n_{p,i}) =\frac{1}{3}\sum_{s=1}^{k}r^{(s)^{2}}_{i}.
\tilde u_i^{(k)}=(y^{(k-1)},z^{(k-1)})u_i^{(k)}(y^{(k-1)},z^{(k-1)})^{-1}, \tilde v_i^{(k)}=(y^{(k-1)},z^{(k-1)})v_i^{(k)}(y^{(k-1)},z^{(k-1)})^{-1}, \tilde w^{(k)}_j =(y^{(k-1)},z^{(k-1)})w_j^{(k)}(y^{(k-1)},z^{(k-1)})^{-1}, \tilde x^{(k)}_j =(y^{(k-1)},z^{(k-1)})x_j^{(k)}(y^{(k-1)},z^{(k-1)})^{-1},