harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\kappa_\lambda = \det \left(\begin{array}{cccc}\kappa_{(\lambda_1)}\kappa_{(\lambda_1+1)}\cdots \kappa_{(\lambda_1+k-1)}\kappa_{(\lambda_2-1)}\kappa_{(\lambda_2)}\cdots \kappa_{(\lambda_2+k-2)}\multicolumn{2}{c}{\dotfill}\kappa_{(\lambda_k-k+1)}\kappa_{(\lambda_k-k+2)}\cdots \kappa_{(\lambda_k)}\end{array}\right).
\tilde\Omega_\psi(m)(x^-_{n_1}\cdots x^-_{n_k}v_{\lambda_i}) :=\Omega_\psi(m)(x^-_{n_1}\cdots x^-_{n_k})v_{\lambda_i}\tilde x_m^-(x^-_{n_1}\cdots x^-_{n_k}v_{\lambda_i}) := x_m^-x^-_{n_1}\cdots x^-_{n_k}v_{\lambda_i}.
\begin{array}{l}\lambda\left(\kappa^S_{ad}\epsilon_{dbc}\alpha_c+\kappa^S_{bd}\epsilon_{dac}\alpha_c\right) + R\left(\beta_ak_b+\beta_bk_a\right)=0 begin{align*}2mm]\ \ \lambda R\left(\alpha_ak_b-\alpha_bk_a\right)+\kappa^S_{ad}\epsilon_{dbc}\beta_c -\kappa^S_{bd}\epsilon_{dac}\beta_c =0. \end{array}
\begin{cases}\frac{du_n(t)}{dt}= [\mathcal{D}u]_n(x)+u_n(t)g(u_n(t),u_n(t-\tau)),n\in \mathbb{Z}, t>0, u_n(s)=\varphi_n(s), n\in \mathbb{Z}, s\in [-\tau, 0],\end{cases}
\left[\delta\left\{\left(U_j, \sum_{\lambda=1}^r\sum_{|\beta|=n+1}a_{j, \beta}^\lambda\cdot e_{j, \beta}^*\otimes e_j^\beta\right)\right\}\right]=u_n(Y, X; \{w_j\})-u_n(Y, X; \{\widehat{w}_j\})
\begin{cases}v_i(x,t)=S_{d,i}(t,t_{i,ex}(x,t))S_{c,i}(t_{i,ex}(x,t),t_{i,en}(x,t))S_{d,i}(t_{i,en}(x,t),0)v_{i,0}(x) \forall\:i\in[1,p], v_i(x,t)=S_{d,i}(t,0)v_{i,0}(x) \forall\:i\in[p+1,n]; \end{cases}
\begin{aligned}|W_\epsilon(x_j,x_k)|\leq \frac{1}{|x_j-x_k|}, &|\nabla_jW(x_j,x_k)|\lesssim \frac{1}{|x_j-x_k|^2}, &|\nabla_j\nabla_kW(x_j,x_k)|\lesssim \frac{1}{|x_j-x_k|^3},\end{aligned}
(\lambda_k-\theta)^T (g(x_k)+\delta) & =(\lambda_k-\theta)^T (g(x_k)+\delta) + f(x_k) - f(x_k) & = L (x_k,\lambda_k,\delta) - L(x_k,\theta,\delta) & \le L (x_k,\lambda_k,\delta) - h(\theta,\delta),
\wp^{- 1}= \sigma{_{2}F_{1}}\left({\frac{1}{2}},{\frac{1}{3}};{\frac{4}{3}}; - 4 \sigma^{3}\right) -{\frac{4^{1 / 3}}{3}}B \left({\frac{1}{3}},{\frac{1}{6}}\right)
[\tilde\Omega] = \left( \prod_{i=1}^s (1-\beta T_i)^{-q_i}\right) \left( \prod_{1\leq i<j\leq s}(1-R_{ij})\right) \cdot \tilde{c}_{q_1}(1) \cdots \tilde{c}_{q_s}(s)
d^{m,l}= \sum_{i=0}^{l}d_{1,i}^{m,l}= \sum_{i=0}^{l}\binom{l}{i}((\theta+1)_{m-1\uparrow})^{i}(\sum_{j=1}^{m-1}(\alpha + \theta )_{m-j\uparrow}(\theta+m-1)_{j-1\downarrow})^{l-i}=( (\theta +1)_{m-1\uparrow}+ \sum_{j=1}^{m-1}(\alpha + \theta)_{m-j\uparrow}(\theta+1+m-j)_{j-1\uparrow})^{l}.
_{v_{4,2}^{1}}= \frac{-\partial_{12}v_{4,c}^{1}(s,|x+y|)}{|x+y|^{2}(s-t)}((x+y)\cdot y)(\widehat{x}\cdot(x+y)) &+\frac{\partial_{1}v_{4,c}^{1}(s,|x+y|)}{(s-t)|x+y|}\left(-y\cdot \widehat{x}+ \frac{(\widehat{x}\cdot(x+y))(y\cdot(y+x))}{|x+y|^{2}}\right)
dJ_t = \sum_{i=1}^r \nabla V_i ( \phi_t) J_t dh_t^i \mbox{with $J_0 ={\rm Id}_d$,}dK_t = - \sum_{i=1}^r K_t \nabla V_i ( \phi_t) dh_t^i \mbox{with $K_0 ={\rm Id}_d$.}
\widehat{\Gamma}_{\mathrm{s}}(X)=\left\{\widehat{\gamma}\in \widehat{\Gamma}(X):\ \sup_{k\in \mathbb{Z}^{d}}\ \left[ N(\gamma_{k})^{P}+ \sum_{x\in \gamma_{k}}\left\vert \sigma_{x}\right\vert^{q}\right] \cdot \left[ \log \left( 1+|k|\right) \right]^{-1}<\infty\right\},
\frac{\mathbb{P}[^{+}_{\omega \ge \omega_1}X^{h}(\omega) \in d\omega_2, \max\limits_{\omega \ge \omega_1}X^{h}(\omega) \in dx^{*}]}{d\omega_2 dx^{*}}= 2\int_{-\infty}^{x_{*}}\frac{j(\omega_2-h)}{j(\omega_1)}\Phi(\omega^*-h,y-x^{*},\omega_2-h)\tilde{\Phi}(-\omega^{*},y-x^{*},-\omega_1)dy
\delta L^{+ 4}= ( \epsilon_{b}^{\beta}\theta^{+ b}\partial_{\beta}^{-}- \epsilon^{+ b}\partial_{b}^{-}) L^{+ 4}+{\frac{1}{2}}( \theta^{+ b}\epsilon_{b}^{\beta}) \partial^{+ \alpha}\mathrm{T r}( A_{\alpha}^{+}A_{\beta}^{+}) \ ,
\iota_{KZB}\circ j^{dR}(X,b)(x)=\int_{b}^{x}\alpha A+\sum_{r,s\geq 0}g_{r,s}\sigma_{r,s}-\int_{b}^{x}\beta S\left(\int_{b}^{x}\alpha\mathrm{ad}_{A}, \int_{b}^{x}\beta\mathrm{ad}_{B}\right)\left(-\int_{b}^{x}\alpha \sigma_{0,0}+\sum_{r,s\geq 0}g_{r,s}\sigma_{r+1,s}\right)
q_1=u^3_{10}u^2_{01}-u^3_{01}u^2_{10}-u^4_{10}u^1_{01}+u^4_{01}u^1_{10}\quad{\rm and}q_2=u^4_{10}u^2_{01}-u^4_{01}u^2_{10}+u^3_{10}u^1_{01}-u^3_{01}u^1_{10}\,.
\|T_\lambda f\|_{L^2}=\bigg\|\frac1{(|\xi|^2-z)^\lambda}\widehat{f}(\xi)\bigg\|_{L^2}&\leq\sup_{\xi\in\mathbb{R}^n}\bigg|\frac1{(|\xi|^2-z)^\lambda}\bigg|\|f\|_{L^2}&\leq\sup_{\xi\in\mathbb{R}^n}e^{Im(\lambda)\arg(|\xi|^2-z)}\|f\|_{L^2}&\leq e^{\pi|Im(\lambda)|}\|f\|_{L^2}.
\mathbb{E}y(t)=\bar{\Gamma}_{t}^{T}\mathbb{E}\xi+\int_{t}^{T}\bar{\Gamma}_{t}^{s}\Big[\big(b_{1}^{2}(s)m_{1}^{-1}(s)\beta(s)+b_{1}n_{1}+b_{2}n_{2}+c(s)\big)\Big]ds
\frac{\partial P_n}{\partial y}(-\bar\beta,\bar\alpha)=\left((\lambda-n)(\alpha y-\beta x)\bar\beta-(n+\lambda)(\bar\beta y+\bar \alpha x)\alpha\right)P_{n-1}(-\bar\beta,\bar\alpha)=\left(2n(\beta x-\alpha y)\bar\beta-(n+\lambda)x\right)P_{n-1}(-\bar\beta,\bar\alpha).
\beta\Big(\big[[x,y],t\big] \Big) =-\Big[x, [\alpha(y),t] \Big] -\Big[y, [\alpha(x),t] \Big] \beta\Big(\rho(t)\theta(x,y) \Big) =-\rho\left( x\right) \theta (\alpha(y),t)-\rho\left( y\right) \theta (\alpha(x),t).
G({\vec r},{\vec y};{\vec r}_0,{\vec y}_0) ={1\over V}\sum_{\vec{q},j,m,n}G_{j \vec q}(r;r_0)D^{*j}_{mn}(\psi_0,\theta_0,\phi_0)D^j_{mn}(\psi,\theta,\phi) e^{i{\vec q}.({\vec y}-{\vec y}_0)},
E_1^{-k,i+k}=\underset{j\ge0\atop j\ge -k}\bigoplus\underset{N\in\Theta_{2j+k+1}}\bigoplus E(N)\otimes_{K_0}H_{conv}^{i-2j-k}(N/K_0)\Longrightarrow H_{conv}^i(Y/{\mathfrak S}^{0},E).
(f \star g)(\tilde{\bf r}) = \frac{1}{(2\pi )^2}\sum_{P', P''}\int_{-\pi}^{+\pi}d\theta 'd \theta '' e^{[-2iA(\tilde{\bf r}, \tilde{\bf r}', \tilde{\bf r}'')]}f(\tilde{\bf r}') g (\tilde{\bf r''})
\frac{\psi^{+0}_{n,p-1}}{\psi^{+0}_{n,p}}\ \frac{\rho^{0+}_{n,p}}{\rho^{0+}_{n-1,p}}=\frac{(n-p+a_{12}-a_{123}+j^{(3)}+1)(n-p+a_{12}-a_{123}-j^{(3)})}{(n-p+a_{12}-a_{123}+j^{(3)})(n-p+a_{12}-a_{123}-j^{(3)}-1)}\ .
g_1 < g_2 \Leftrightarrow \begin{cases}g_1^{-1}g_2\notin H \ p(g_1)\prec p(g_2), g_1^{-1}g_2\in H s^{-1}g_1<_H s^{-1}g_2 \textit{where $s\in S$ is such that $sH=g_1H=g_2H$}.\end{cases}
\sum_{n=-\infty}^\infty \left[{a\atop b+\alpha n}\right]_{p}\frac{1}{(-zq^n,-q^{1-n}/z;q)_\infty}=\int_{-\infty}^\infty\left[{a\atop b+\alpha x}\right]_{p}\frac{1}{\left(-zq^x,-q^{1-x}/z;q\right)_\infty}\,dx,
\sum_{k=0}^{K}\alpha_k^i \widehat{P_{ij}(k)}= \sum_{k=0}^{K}\alpha_k^i \frac{N_{ij}(k)}{N_i(k)}= \sum_{k=0}^{K}\frac{N_i(k)}{N_i}\frac{N_{ij}(k)}{N_i(k)}=\frac{1}{N_i}\sum_{k=0}^{K}N_{ij}(k) = \frac{N_{ij}}{N_i}=\left(\widehat{\mathbf{P}_M^S}\right)_{ij}
V_{n}(\mathbf{u}_1,\mathbf{u}_2)-V_{n}(\mathbf{u}_1,\mathbf{0}_{p-s}) > -M_1 \Vert \mathbf{u}_2 \Vert + M_2 \iota_n n^{(\varsigma -t)/2}\Vert \mathbf{u}_2 \Vert^{t}= \Vert \mathbf{u}_2 \Vert^{t}(-M_1 \Vert \mathbf{u}_2 \Vert^{1-t}+ M_2 \iota_n n^{(\varsigma -t)/2}).
a_l^0(\Upsilon_{\!\rho}) =\sum_{k\in\mathcal M_{n-1}(l)}a_l^k(\Psi_\rho^L)\int_{SO(n)}\left<Y_l^0,\sum_{m\in\mathcal M_{n-1}(l)}T_l^{k m}(\tilde g)Y_l^m\right>d\nu(\tilde g) &=\sum_{k\in\mathcal M_{n-1}(l)}a_l^k(\Psi_\rho^L)\int_{SO(n)}T_l^{k0}(\tilde g)\,d\nu(\tilde g),
I_{1}+\frac{I_{2}}{e^{\zeta\omega_{n}t_{2}}}+\frac{I_{3}}{e^{\zeta\omega_{n}t_{3}}}+\frac{I_{4}}{e^{\zeta\omega_{n}t_{4}}}=I+\frac{3I}{K}+\frac{3I}{K^{2}}+\frac{I}{K^{3}}=1,K=e^{\zeta\pi/\sqrt{1-\zeta^{2}}}
\mathbb{E}_N^m[(\mu_t^{1,N},\mu_t)] \leq \mathbb{P}_N^m((\mu_t^{1,N},\mu_t)>\epsilon) + \epsilon\mathbb{P}_N^m((\mu_t^{1,N},\mu_t)\leq\epsilon) &\leq \mathbb{P}_N^m((\mu_t^{1,N},\mu_t)>\epsilon) + \epsilon \overset{N \to \infty}{\longrightarrow}\epsilon,
\frac{\phi''_{e}(z)}{\phi_{e}(z)}+\frac{\phi'_{e}(z)}{\phi_{e}(z)}\frac{\widetilde{\tau_{e}}(z)}{\sigma_{e}(z)}+\frac{\widetilde{\sigma_{e}}(z)}{\sigma_{e}^{2}(z)}=\frac{\bar{\sigma_{e}}(z)}{\sigma_{e}^{2}(z)}.
\int_{\Gamma_{\varepsilon,a}}h_x\left(F^{\parallel}(x), \frac{\exp_{x; h}(y)}{|\exp_{x; h}(y)|_h}\right) v(y)d\mu_h(y) = \varepsilon^2 R_{F,a}\tilde v(t') + \varepsilon^3\mathcal{A}_\varepsilon\tilde v(t'),
=\log(\frac{1}{a})\log(1+\frac{1}{a})^{\lambda'-1}+(1-\lambda') h(a)+(1-\lambda')\int_1^a \frac{\log(1+\frac{1}{b})^{\lambda'-1}}{b}(\frac{\log(b)}{(1+b)\log(1+\frac{1}{b})}+1)
\begin{cases}-\dfrac{\partial y}{\partial t}(x,t)+\triangle y(x,t)\, =\,u(x,t)\,\,\, in\,\, Q:=\Omega\times (0,T) y(x,t)\, =\,h(x,t)\,\,\, on\,\, \Sigma :=\Gamma\times (0,T) y(x,0)\, =\,y_0(x)\,\,\, in\,\, \Omega\end{cases}
\log \left( \# C_{\sigma}\right) = \log \left( \sum_{M \in C_{\sigma}}\frac{1}{\# \mathrm{Aut}(M)}\right) + O \left( \log \left( \max_{M \in C_{\sigma}}\# \mathrm{Aut}(M) \right) \right) &= [K : \mathbb{Q}] n^2 \log n + O([K : \mathbb{Q}]^2 n^2).
t^{2\beta - 1}|y|^{-d-\alpha}\int_{|y|^{-\alpha/\beta}t}^{1}z^{-2\beta}w(z, \beta, 1)dz \le Ct^{2\beta - 1}|y|^{-d-\alpha}\int_{|y|^{-\alpha/\beta}}^{1}z^{-2\beta + q - 3}dz = C t^{2\beta - 1}|y|^{-d-\alpha}\left(1 - (|y|^{-\alpha/\beta}t)^{-2\beta + q - 2}\right).
{\rm det}\begin{bmatrix}c_5 c_4 c_3 c_4 c_3 c_2 c_3 c_2 c_1 \end{bmatrix}\,\, = \,\,c_1 c_3 c_5-c_1c_4^2-c_2^2 c_5+2c_2c_3 c_4-c_3^3 = 0.
L_{1}^{\left( 0 \right)}= - \frac{a}{2}A_{\mu}^{\left( 0 \right)}A^{\mu \left( 0 \right)}+ \frac{b}{2}\left( F_{\mu \nu}^{\left( 0 \right)}\right)^{2}+ \theta \epsilon^{\mu \nu \rho}A_{\mu}^{\left( 0 \right)}\partial_{\nu}A_{\rho}^{\left( 0 \right)}.
\|q\|_{H^s}\leq \Big(\sum_{n=0}^s \|\partial_{\alpha}^n q\|_{L^2}^2\Big)^{1/2}\leq \Big(\sum_{n=0}^s (400|\lambda x(0)| n! (n+1)! d_I(t)^{-3/2})^2\Big)^{1/2}\leq 400((s+2)!)^2 |\lambda x(0)| d_I(t)^{-3/2}\leq K_s^{-1}\epsilon d_I(t)^{-3/2}.
w =w(1)\cdots w(r) \cdots w(h(r)) \w(h'(r)) \w(h''(r)) \cdots w(h(r+1)) \cdots w(n), w' =w(1)\cdots w(r) \cdots w(h(r)) \w(h''(r)) \w(h'(r)) \cdots w(h(r+1)) \cdots w(n).
V_{k+1}= \{m \in \bar{I}_{k-1}\mid j_{2p+2}<m<j_{2p+1}\mbox{for some}p\geq 0\}begin{align*}3pt] +\{m \in \bar{I}_{k}\mid m>i_k,\,j_{2p+1}<m<j_{2p}\mbox{for some}p\leq q\}.
\zeta_{2,\varepsilon}\mathcal{T}_{a^+,a^-}(c^k) =\mathcal{T}_{a^+,a^-}(\zeta_{2,\varepsilon}c^k)=\zeta_{2,\varepsilon}f^k &= \bar{a}^-(\zeta_{2,\varepsilon}c^k)-a^+\overline{(\zeta_{2,\varepsilon}c^k)^{aps}}=\zeta_{2,\varepsilon}f^k
{\cal E}^{\mu}(f,f)=\frac{1}{2}\sum_{x,y\in\Gamma}\mu(x^{-1}y)(f(x)-f(y))^2\leq\sum_{x,y\in\Gamma}\mu(x^{-1}y)(f(x)^2+f(y)^2)=2\|f\|_{\ell_2(\Gamma)}^2.
S_{n,l}^{\ast}(x) = \frac 1{d_l p^{1/2}}\sum_{i=1}^{pl}\left (1_{\{Y^{\ast}_i \leq x \}}- \tilde F_{n,l}(x) - J_m(x)/m! \left ( H_m(X_i^{\ast}) - \tilde \mu_{n,l}(H_m) \right ) \right ),
\int_0^{2\pi}| T(f)(re^{i\theta})|^p \,d\theta &= \int_{D_{r,*}}| T(f)(re^{i\theta})|^p \,d\theta + \int_{D^{r,*}}| T(f)(re^{i\theta})|^p \,d\theta &\leq 2\pi + \int_0^{2\pi}| T(f)(re^{i\theta})|^\gamma \,d\theta &\leq 2\pi + C||f||_{L^q(D)}< \infty.
u_1(y) =u_1(y,t)=\tilde u_1(x_M+n_1(x_M)y_1+n_2(x_M)y_2+\tau(x_M)y_3,t), u_2(y) =u_2(y,t)=\tilde u_2(x_M+n_1(x_M)y_1+n_2(x_M)y_2+\tau(x_M)y_3,t), u_3(y) =u_3(y,t)=\tilde u_3(x_M+n_1(x_M)y_1+n_2(x_M)y_2+\tau(x_M)y_3,t).
\pi_{i}^{( 1 ) a}= - \, \frac{1}{m^{2}}\, \eta^{1 b}\, G_{1 i}^{( 0 ) b a}+ \frac{g}{2 m^{2}}\, f^{b c d}\, \eta^{2 b}\, ( D_{i}\pi^{i})^{c}\, G_{1 i}^{( 0 ) d a}- \eta^{2 b}\, G_{2 i}^{( 0 ) b a}\, ,
\frac{\sqrt{n}g_{k}}{4\sqrt{x_k}}\left( \int_{x_{k-1}}^{x_{k+1}}\frac{L_Bs(x)}{e^{I(x)}}\, dx -\frac{2L_B s_k}{ne^{I_k}}\right) &=\frac{\sqrt{n}g_{k}}{4\sqrt{x_k}}\int_{x_{k-1}}^{x_{k+1}}e^{-I(x)}( L_Bs(x) -L_B s_k )\,dx +\frac{\sqrt{n}g_{k}}{4\sqrt{x_k}}L_Bs_k\int_{x_{k-1}}^{x_{k+1}}\left( e^{- I(x)}- e^{- I_k}\right)\, dx &=A'_k +B'_k.
m^2_I = \Bigl[{2\pi^3\over 3}-{\pi\over 3}\Big({\rm ln}\Big({2\pi^2\over 3}\Big)\Big)^2\Big] T^2{\hat g}^4+{4\pi\over 3}T^2{\rm ln}\Big({1\over{\hat g}}\Big) -{4\pi\over 3}T^2\Big({\rm ln}\Big({1\over{\hat g}}\Big)\Big)^2+ 0({\hat g}^5)
f(g_s,s\xi,\hat{\psi})=\big(\mathcal{T}_{a^+,a^-}\circ B(s\mathfrak{k}_s),P_{\ker(D|_{L^2_1})}(D_{pert,\eta_s}(\psi_s))\big)\in \mathbb{H}_1\times\ker(D|_{L^2_1})\cong\mathbb{K}_1
Q^{(m)}_{2r}(1) =\mu^{(m)}_{2r,1}+\mu^{(m)}_{2r,-1}-q^m\mu^{(m-1)}_{2r-1}, Q^{(m)}_{2r+1}(1) =\mu^{(m)}_{2r+1}-q^m\big[\mu^{(m-1)}_{2r,1}+\mu^{(m-1)}_{2r,-1}\big].
P_R(\tilde{r};0) = P_{\rm affine}\left( \tilde{r}; \lambda_R, \omega_R, -\tfrac{\beta_R}{a_{2R}}\right), T_R(\tilde{r};0) = T_{\rm affine}\left( \tilde{r}; \lambda_R, \omega_R, -\tfrac{\beta_R}{a_{2R}}\right),
Q_1^k = E\bigg[e^{-\beta n}\Big(M^{\phi}(n)-\max_{0\le i\le n}D_i^{\phi}(n-\tau_i)\Big)\bigg]^{\!k}&\le E\bigg[\sum_{i>n}e^{-k\beta n}\big(D_i^{\phi}(t-\tau_n)\big)^k\bigg] &=E\bigg[\sum_{i>n}e^{-k\beta \tau_i}\zeta_i^k(n-\tau_i)\bigg]
\mathbf{E}_{t}^{\mathbf{A}}\left( \mathbf{x}\right) :=\int\nolimits_{0}^{1}\left[ E_{\mathbf{A}}(t,\alpha x^{(2)}+(1-\alpha )x^{(1)})\right](x^{(2)}-x^{(1)})\mathrm{d}\alpha \ .
\begin{aligned}\left(\chi + t \chi + \tilde \chi + \sqrt{-1}\partial\bar\partial\varphi_t\right)^n = c \left(\chi + t\chi+ \tilde \chi + \sqrt{-1}\partial\bar\partial \varphi_t\right)^m \wedge \omega^{n - m}+ b_t f \omega^n , \end{aligned}
P_{\textbf{n}_2+\textbf{e}_i,\textbf{n}_1-\textbf{e}_j}(x):=\sum_{k=1}^q B_{\textbf{n}_2+\textbf{e}_i,\textbf{n}_1-\textbf{e}_j,k}(x)w_{2,k}(x),
\tilde \varphi^1_{z\bar z}- \frac{1}{2}(\tilde \varphi^1_{z}\tilde \varphi^3_{\bar z}+\tilde \varphi^1_{\bar z}\tilde \varphi^3_z)=0,\; \; \tilde \varphi^2_{z\bar z}+ \frac{1}{2}(\tilde \varphi^2_{z}\tilde \varphi^3_{\bar z}+\tilde \varphi^2_{\bar z}\tilde \varphi^3_z)=0,
\int_{0}^{\infty}R dR \left(\frac{\cos(2Q_{1}(R))-1}{R^{2}\lambda(t)^{2}}\right)\phi_{0}(R) v_{2}(t,R\lambda(t)) = \frac{4b}{\lambda(t) t^{2}\log^{b}(t)}+E_{v_{2},ip}(t,\lambda(t))
a_j\left[t+1\right]= a_j\left[t\right]-4\mu a_j\left[t\right]\sum_{l=1}^{M}\lvert\phi^{\left(l,j\right)}\rvert^2+2\mu\Re\left\{\phi^{\left(j,j\right)}\right\}-4\mu a_j\left[t\right]\sum_{q=1}^{M-1}\sum_{r=q+1}^{M}\Re\left\{\phi^{\left(q,j\right)^*}\phi^{\left(r,j\right)}\right\}.
i_{p^n_{U,W}}(\mathbf{u},\mathbf{w}) =\sum_{t=1}^ni_{p_{U,W}}(u_t,w_t) i_{p^n_{U,V,W}}(\mathbf{u},\mathbf{v},\mathbf{w}) =\sum_{t=1}^ni_{p_{U,V,W}}(u_t,v_t,w_t),
L^{(c)}(z)\begin{pmatrix}F_+ F_-\end{pmatrix}=\begin{pmatrix}L^{(c)}_{11}(z) L^{(c)}_{12}(z) L^{(c)}_{21}(z) L^{(c)}_{22}(z)\end{pmatrix}\begin{pmatrix}F_+ F_-\end{pmatrix}=\lambda \begin{pmatrix}F_+ F_-\end{pmatrix},
E_{\mu_L^{\delta_x}}\big[e^{\frac14(t_y-t_x)}\big] = E_{\mu_L^{\delta_x}}\big[e^{\frac14(t_y-t_z)}e^{\frac14(t_z-t_x)}\big]\le E_{\mu_L^{\delta_x}}\big[e^{\frac12(t_y-t_z)}\big]^{\frac12}E_{\mu_L^{\delta_x}}\big[e^{\frac12(t_z-t_x)}\big]^{\frac12}.
\mathcal{P}= \sum_{k=0}\frac{1}{k!}\: t^{1-k}\: \mathcal{P}^{a_{1}\cdots a_{k}}(x) x^{*}_{a_{k}}\cdots x^{*}_{a_{1}}+ \sum_{k=0}\frac{1}{k!}\: t^{1-k}\: \bar{\mathcal{P}}^{a_{1}\cdots a_{k}}(x)x^{*}_{a_{k}}\cdots x^{*}_{a_{1}}t^{*},
E_\lambda\big(\zeta_t(u)\zeta_t(v)\big)=F_t(u-v,1)=F_t(v-u,1), &E_\lambda\big(\zeta_t(u)g_t(v)\big)=E_\lambda\big(\zeta_t(v)g_t(u)\big)=F_t(u-v,2)=F_t(v-u,2), &E_\lambda\big(g_t(u)g_t(v)\big)=F_t(u-v,3)=F_t(v-u,3), &F_t(e_1,i)=F_t(y,i)
\left[ \psi_{a , b}^{\left[ A , B \right]}( x ) , \psi_{a^{^{\prime}}, b^{^{\prime}}}^{\left[ A , B \right]}( y ) \right]_{\pm}= 0 \, \, \, \, \, \, \, \, f o r \, \, \, \, \, ( \, x - y )^{2}< 0
p(0,0)=p(1,1)=(0,0), p(2,2)=(\frac{1}{4},0), &p(0,1)=p(1,0)=(\frac{1}{6},0), &p(0,2)=p(2,0)=(\frac{7}{10},0), &p(1,2)=p(2,1)=(\frac{1}{2},0).
\mathbb E_x\left[e^{-p\int_{0}^{t}\rm{\bf{1}}_{\{X_s > b\}}ds}\rm{\bf{1}}_{\{X_{t}>y\}}\right]=e^{-pt}\mathbb E_x\left[e^{-(-p)\int_{0}^{t}\rm{\bf{1}}_{\{X_s \leq b\}}ds}\rm{\bf{1}}_{\{X_{t}>y\}}\right].
{\rm Z}\approx{\bf N_{2}}\int d\Omega e^{\gamma \Omega^{00}(1-\frac{1}{2F^{2}}\frac{1}{V}\sum_{n}' [1/p_{n}^{2}])}={\bf N_{3}}Y_{N}\left( \gamma \Big[ 1- \frac{N-1}{2F^{2}}G_{1}\Big]\right)
(1+|x|^{\alpha}) \Delta f_{\alpha,\beta,\lambda}(x)-|x|^{\beta}f_{\alpha,\beta,\lambda}(x) =o(1)f_{\alpha,\beta,\lambda}(x)+\lambda \frac{1+|x|^{\alpha}}{|x|^2}f_{\alpha,\beta,\lambda}(x).
\int_{\mathbb{R}_{+}^{n}}\nabla U\nabla v_{q}dy =-\int_{\mathbb{R}_{+}^{n}}U\Delta vdy+\int_{\partial\mathbb{R}_{+}^{n}}U(0,z)\frac{\partial v_{q}}{\partial t}dz =2h_{ij}\int_{\mathbb{R}_{+}^{n}}Ut\frac{\partial^{2}U}{\partial z_{i}\partial z_{j}}-n\int_{\partial\mathbb{R}_{+}^{n}}U(0,z)\left(U(0,z)^{\frac{2}{n-2}}v_{q}\right)dz=0
\begin{aligned}\langle \nabla_i\nabla_j \Psi_t-\nabla_l\Psi_t\Gamma_{ij}^l, \nabla_m\nabla_m \Psi_t-\nabla_l\Psi_t\Gamma_{mm}^l-\frac{\sum_k(\nabla_k\nabla_k \Psi_t-\nabla_l\Psi_t\Gamma_{kk}^l)}{n}\rangle =\frac{1}{2t}O(t)+\sum_l\Gamma_{ij}^l(\Gamma_{mm}^l-\frac{\sum_k\Gamma^l_{kk}}{n}). \end{aligned}
P(\delta,c,1)= \mathcal I\Bigl(-\mu\delta/\sigma,\delta,|c|,1\Bigr) =\begin{cases}-\dfrac{\mu}{\sigma}\delta e^{\mu r}+ \delta \dfrac{\sigma}{\mu}(1-e^{\mu r}), \mu\ne0, -\delta\sigma r, \mu=0.\end{cases}
\log L-\dfrac{1}{n}\underset{m=0}{\overset{n-1}{\sum}}\log \left| \det B\left(L^{m}k\right)\right|\underset{a.e.k}{\xrightarrow{n\rightarrow\infty}}\log L-\underbrace{\int_{0}^{1}\log\left|p^{}_{L}\right|dk}_{=0}-\underbrace{\int_{0}^{1}\log\left|Q-R\right|dk}_{=m\left(Q-R\right)},
F^{-j}\left(V^{\otimes n}_{dR}\otimes \mathcal{O}_{C}\right)=\begin{cases}\mathrm{Span}_{\mathcal{O}_{C}}\left\{w A_iB_j\sum_{i=1}^{g}\mathrm{deg}_{A_i}w\leq |j|\right\}j\leq 0 0 j>0. \end{cases}
(\|\mathfrak{F}_2\|_{\nu}^{(2)})^2 =\sum_{j+k\leq\nu}\int_0^T\|(-\partial_t^2)^j\mathfrak{F}_2(t,\cdot)\|_k^*)^2dt &=\sum_{j+k\leq\nu}\int_0^T(\|(-\partial_t^2)^j\mathfrak{F}_2^{[0]}(t,\cdot)\|_{[0]k}^*)^2dt + &+\sum_{j+k\leq\nu}\int_0^T(\|(-\partial_t^2)^j\mathfrak{F}_2^{[1]}(t,\cdot)\|_{[1]k}^*)^2dt.
\|K(x-y)-K(x)\|_{\ell^s(\mathbb{Z}^+)}=\|\Phi_k(x,y)-\Phi_{k-1}(x,y) \|_{\ell^s(\mathbb{Z}^+)}&\leq \|\Phi_k(x,y)\|_{\ell^s(\mathbb{Z}^+)}+\|\Phi_{k-1}(x,y) \|_{\ell^s(\mathbb{Z}^+)}&\leq 2\|\Phi_{k-1}(x,y)\|_{\ell^s(\mathbb{Z}^+)},
f_{\alpha, \beta}(X+Y) = \alpha(b_{d}(X+Y)^d+\cdots+b_{d-k}(X+Y)^{d-k})+\beta =\alpha(b_dX^d+\cdots+b_{d-k}X^{d-k})+\beta+\alpha \sum_{j=d-k}^d b_j\sum_{i=1}^j{j \choose i}X^{j-i}Y^i =f_{\alpha,\beta}(X)+h_1(X,Y).
\left\| \sum_{\{u\in B_{\ell p}: u_i = 0, u_j = \ell p \}}(\alpha_\omega)^u(w) \right\| \leq \left\| \sum_{\{u\in B_{\ell p}: u_i=0, u_j = \ell p \}}(\alpha_\omega)^u(w^*w) \right\|^{1/2}\leq \sqrt{(\ell p)^{m-2}}\cdot \|w^*w\|^{1/2}\leq \sqrt{(\ell p)^{m-2}}
\left[ \bar \gamma^2\bar \gamma^0\bar \gamma^1\partial_\vartheta +\frac{\bar \gamma^3\bar \gamma^0\bar \gamma^1}{\sin \vartheta}\partial_\varphi +\bar \gamma^2\bar \gamma^0\bar \gamma^1S\partial_\vartheta S^{-1}+\frac{\bar \gamma^3\bar \gamma^0\bar \gamma^1}{\sin\vartheta}S\partial_\varphi S^{-1}\right] \tilde \Phi =-i\kappa \tilde\Phi
\theta(-\bar R_1,t) \,, \kern 0.25in\theta(\bar R_2,t) \,, \kern 0.25in\phi(-\bar R_1,t) \,, \kern 0.25in\phi(\bar R_2,t) \,, \kern 0.25inr_1(t) \,, \kern 0.25inr_2(t) \,.
\begin{cases}\dot x_{N_n}^n(t) = \bar{w}_{N_n-1}^n , begin{align*}7pt] \dot x_i^n(t) = v_i^n \left(\dfrac{\kappa_n}{x_{i+1}^n(t) - x_i^n(t)}\right) , i=0, \ldots, N_n-1 , begin{align*}7pt] x_i^n(0) = \bar{x}_i^n , i=0, \ldots, N_n ,\end{cases}
\begin{aligned}J(p):=\sup\limits_{\tau_I\leq\tau_O}\mathbb{E}^p\left[\int_{\tau_I+\delta}^{\tau_O+\delta}\exp(-r t)(P(t)-C)\mathrm{d}t \right. -\exp(-r (\tau_I+\delta))K_I &\left.-\exp(-r(\tau_O+\delta))K_O\right],\end{aligned}
j \left(E-1\right) = D j \left(\frac{E-1}{D}\right) \ge D G \ge D E - D (D-j)T' &= D (E-1) + D - D (D-j)T' &= j (E-1) + D + (D-j) (E-1- D T') &\ge j (E-1) + D + (D-j)(0) &= j(E-1) + D
\sum_{k=1}^{\infty}\frac{1}{3^{k}k}+\sum_{k=1}^{\infty}\frac{1}{4^{k}k}=(\frac{1}{3}+\frac{1}{4})+(\frac{1}{18}+\frac{1}{32})+(\frac{1}{81}+\frac{1}{192})+(\frac{1}{324}+\frac{1}{1024})+\cdot\cdot\cdot=ln2
I(t_1u_1+t_2u_2+t_3u_3) <\sum_{i=1}^3\left(I(u_i)-\frac{1}{4}I'(u_i)[u_i]\right) =\sum_{i=1}^3\left(I(u_i)-\frac{1}{4}I'(u)[u_i]\right)+\frac{\lambda}{2}\sum_{i<j}\int_{\mathbb R^3}\phi_ju_i^2dx =\sum_{i=1}^3I(u_i)+\frac{\lambda}{2}\sum_{i<j}\int_{\mathbb R^3}\phi_ju_i^2dx =I(u_1+u_2+u_3) =I(u),
\begin{array}{ll}f_1:= v+ t^{5}~ v^2 + t^{3}~uv^2 + 3u^2v -4 u^4v^2, begin{align*}4pt] f_2:= -v + 2t^{2}~u^2v -3 t^{2}~u^3v^2 +5t^4~u^4v^2.\end{array}
p_{l}^{\eta,\psi}(t,x,y) =\int_{M_{x,y}}\eta(u)\psi\left(\frac{\Lambda\|u\|_{\textsc{CC}}}{t^{H}}\right)K(u,x)\rho_{t}(u)m_{x,y}(du) =\int_{M_{x,y}\cap W}\eta(u)\psi\left(\frac{\Lambda\|u\|_{\textsc{CC}}}{t^{H}}\right)K(u,x)\rho_{t}(u)m_{x,y}(du),
\sigma ( \rho ) = \frac{\cosh^{4 / 5}\left[{\frac{5}{4}}k ( \rho - \rho_{0}) \right]}{\cosh^{4 / 5}\left[{\frac{5}{4}}k \rho_{0}\right]}, \gamma ( \rho ) = A \frac{\sinh^{\alpha}\left[{\frac{5}{4}}k ( \rho - \rho_{0}) \right]}{\cosh^{6 / 5}\left[{\frac{5}{4}}k ( \rho - \rho_{0}) \right]}
P_{ed}=\frac{2\pi\lambda_m\int\limits_{0}^{\infty}re^{-\pi\lambda_m r^2}e^{-\pi\lambda_m r^2\rho(T,\delta)}(1-e^{-\pi\lambda_m r^2\rho( S_{tm},1)})r}{2\pi\lambda_m\int\limits_{0}^{\infty}re^{-\pi\lambda_m r^2}(1-e^{-\pi\lambda_m r^2\rho( S_{tm},1)})r}
\left|F^{(k)}\right| \leq C \varepsilon |\nabla^{[2,k+1]}\Psi|, (P=1), \left|F^{(k)}\right| \leq C \varepsilon (1 + \Psi) |\nabla^{k+1}\Psi|+\varepsilon |\nabla^{[2,k]}\Psi|, (P=2).
\det(X_{N-1})= \cos(\psi_{N-2})\sin(\psi_{N-2})\cos^{2(N-2)-2}(\psi_{N-2})\left[\det\begin{pmatrix}\cos(\phi_{N-1}) \sin(\phi_{N-1}) -\sin(\phi_{N-1}) \cos(\phi_{N-1})\end{pmatrix}\right]^{N-3}\det\left(X_{N-2}\right) = \cos^{2(N-2)-1}(\psi_{N-2})\sin(\psi_{N-2})\det\left(X_{N-2}\right).
\lim_{\beta\rightarrow\infty}w_{j}\left( \beta\right) =\mathring{w}_{j},\lim_{\beta\rightarrow\infty}\operatorname{Re}\zeta_{j}\left( \beta\right) =\rho_{j},\lim_{\beta\rightarrow\infty}\operatorname{Im}\zeta_{j}\left( \beta\right) =0
\overline{\Sigma}^n[s,x,\mu] = \phi\left(\int \sigma^n(s,x,y)\,\mu(dy)\right)= \phi(\Sigma^n[s,x,\mu]), \overline{B}^n[s,x,\mu] = \psi\left(\int b^n(s,x,y)\,\mu(dy)\right)= \phi(B^n[s,x,\mu]),
\|a(x,hD)q(x,hD)u\|_{L^2}^2 =\int |a|^2|q|^2d\mu +o(1), \|a(x,hD)Pq(x,hD)u\|_{L^2}^2 =h^2\int |a|^2|H_pq|^2d\mu +o(h^2).
\int_0^1|\psi_{j_l,k_l}(x_l)|dx_l=\int_{2^{-j_l}(-N_l+1+k_l)}^{2^{-j_l}(N_l+k_l)}2^{j_l/2}|\psi_{k_l}(2^{j_l}x_l)|dx_l\lesssim2^{-j_l/2},
\epsilon_{D}( a ) = - \frac{\pi^{\frac{D - 1}{2}}}{2^{D}}\frac{1}{a^{D - 1}}\Gamma \left( \frac{1 - D}{2}\right) \sum_{n = 1}^{\infty}( n^{2}+ 4 n \beta - 2 \beta )^{\frac{D - 1}{2}}.
\bar{p}_{I}\to{\frac{3 \bar{p}_{I}+ \bar{s}}{3 + \bar{s}}}\ \ \forall \I < 8 \quad , \quad \bar{p}_{I}\to{\frac{3 \bar{p}_{I}- 2 \bar{s}}{3 + \bar{s}}}\ \ \forall \I \geq 8
\Gamma_{NP}^{(1)}\,=\,\frac{g^2}{4}\,\frac{1}{16 \pi^2}\int_0^1 dx \int\frac{d\alpha}{\alpha}e^{i\alpha[x(1-x)p^2 - m^2] + i\frac{p\circ p}{4\alpha}- \alpha \epsilon}\Phi(-p,\theta)\bar{\Phi}(p,\theta)
J_1 =p_x\ , J_2 =p_y\ , J_3 =q_x p_x\ , begin{align*}3pt]J_4 =q_y p_x\ , J_5 = q_x p_y\ , J_6 =q_y p_y\ , begin{align*}3pt]J_7 =q_x \left(q_x p_x + q_y p_y + 3\nu\right)\ , J_8 =q_y \left(q_x p_x + q_y p_y + 3\nu\right)\ ,
\psi(tv) =\frac{t^{p}}{p}\|u\|^p_{W_{0}^{s,p}(M)}+ \frac{t^{p}}{p}\|u\|^{p}_{L^{p}(M)}- \int_{\Omega}F(x,tv)d\mu_g(x) \leq \frac{t^{p}}{p}(\|u\|^p_{W_0^{s,p}(M)}+C \|u\|^p_{W_0^{s,p}(M)}-\int_{\Omega}F(x,tv)d\mu_g(x) \leq C' \frac{t^{p}}{p}\|u\|^p_{W_0^{s,p}(M)}-t^{\mu}\int_{\Omega}f(x,v)d\mu_g(x).
\mathcal I_2(n) = \int_{\mathcal A_2^+(n)}\chi_{\psi}(c) \chi_{\delta}(c)|c|^{-3/2}_p\mathbf h_p(h) dh + \int_{\mathcal A_2^-(n)}(c,d)_p\chi_{\psi}(c) \chi_{\delta}(c)|c|^{-3/2}_p\mathbf h_p(h) dh.
{\tilde{X}}_{{\bf{q}}}= ( - \frac{1}{2 \mathrm{~}i \mathrm{~}N \mathrm{~}\epsilon_{{\bf{q}}}}) \sum_{{\bf{k}}}[ \Lambda_{{\bf{k}}}( -{\bf{q}}) \omega_{{\bf{k}}}({\bf{q}}) a_{{\bf{k}}}({\bf{q}}) - \Lambda_{{\bf{k}}}({\bf{q}}) \omega_{{\bf{k}}}( -{\bf{q}}) a_{{\bf{k}}}^{\dagger}( -{\bf{q}}) ]
{\cal Z}= \int D S_{\mu \nu}\exp \left\{- \left[ ( \pi \eta )^{2}\int d^{4}x \int d^{4}y S_{\mu \nu}( x ) D_{m}^{( 4 )}( x - y ) S_{\mu \nu}( y ) + V \left[ \pi \Lambda S_{\mu \nu}\right] \right] \right\}
\cosh(b) = \cosh(a) \cosh(r) - \sinh(a) \sinh(r) \cos(\theta) &= \cosh(a) \cosh(r) - \sinh(a) \cosh(r) \tanh(a) = \frac{\cosh(r)}{\cosh(a)}[\cosh(a)^2 - \sinh^2(a)] = \frac{\cosh(r)}{\cosh(a)}.
| D^{i}W | = | h | \Big ( A_{i}^{2}e^{- 2 J \beta x}+ B_{i}^{2}e^{- 2 J \beta ( 1 - x )}+ 2 A_{i}B_{i}e^{- \beta J}\cos ( 2 \alpha - \beta \chi ) \Big )^{\frac{1}{2}}.
\|P(x)\|^2 = \sum_{k\geq 0}\|P_k(x)\|^2\|\tilde{B}e_k\|^2 &= \sum_{k\geq 0}|\mathcal{A}(x,k)|\|BDe_k\|^2 &\leq M\sum_{k\geq 0}\tbinom{N-1+k}{N-1}\tbinom{N-1+k}{N-1}^{-1}\|Be_k\|^2 &= M\|B\|_{S_2}^2.
c_++(-1)^{|m|+|n|}c_- = \lim_{k\to\infty}\tilde{\phi}(m+n+2k\chi^J) & =\lim_{k\to\infty}\sum_{\substack{I\subset[N] |I|\geq l}}f_\phi^I(U(m+k\chi^J,n+k\chi^J)) &= f_\phi^\varnothing(U(m,n)) + f_\phi^{\tilde{I}}(U(m,n)),