harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left(B_t:0\leq t<T;\,\mathbb{P}_{0\to \mu}^{\delta,T}\right) \stackrel{\mathcal{L}}{=}\left(\frac{T-t}{T}X_\frac{Tt}{T-t}:0\leq t<T;\,\mathbb{P}_0^{\delta,\frac{\mu}{T}}\right) &\stackrel{\mathcal{L}}{=}\left((T-t)X_\frac{t}{T(T-t)}:0\leq t<T;\,\mathbb{P}_0^{\delta,\mu}\right).
\begin{array}{ccc}\hbox{d}^\mathbf{A}p_i = (\partial^{\gamma,\mathbf{A}}_bp{_i}^{ab}+ \partial_kp{_i}^{ak})\e^{(3)}_a\wedge \bar{e}^{(r)}+\ (\partial^{\gamma,\mathbf{A}}_bp{_i}^{jb}+ \partial_kp{_i}^{jk}+ \frac{1}{2}\mathbf{F}{^j}_{ab}p{_i}^{ab}+ \frac{1}{2}c^j_{k\ell}p{_i}^{k\ell})\e^{(4)}\wedge \bar{e}^{(r-1)}_j\end{array}
\Sigma^{ab}= \frac{1}{4}(\gamma^a \gamma^b - \gamma^b \gamma^a)=\frac{1}{4}\left( \begin{array}{cc}\sigma^a \bar\sigma^b - \sigma^b \bar\sigma^a 0 0 \bar\sigma^a \sigma^b - \bar\sigma^b \sigma^a\end{array}\right).
\begin{array}{rl}\underset{\boldsymbol{\phi}}{}\boldsymbol{\phi}^{H}\mathbf{M}\boldsymbol{\phi}\boldsymbol{\Phi}_{\boldsymbol{\phi}}^H\boldsymbol{\Phi}_{\boldsymbol{\phi}}=N_c\mathbf{I}_{N_c}, \boldsymbol{\Phi}_{\boldsymbol{\phi}}=\mathrm{circ}\left(\boldsymbol{\phi}\right).\end{array}
\mathbf{y}_t = \sqrt{P}\widehat{\mathbf{H}}_{t}\mathbf{x}_t+\sqrt{P}\widetilde{\mathbf{H}}_{t}\mathbf{x}_t+\mathbf{w}_t = \sqrt{P}\widehat{\mathbf{H}}_{t}\mathbf{x}_t + \widetilde{\mathbf{w}}_{t}.
\frac{d}{dz}\left[{}^{M\!L\!}\textrm{F}_{p}(a,b;c;z) \right] = \frac{ba}{c}\left[\sum_{n=0}^{\infty}(a+1)_n \frac{{}^{M\!L\!\!}B_{p}(b+n+1,\;c-b)}{B(b+1,\;c-b)}\frac{z^n}{n!}\right] = \frac{ba}{c}{}^{M\!L\!}\textrm{F}_{p}(a+1,b+1;c+1;z).
j_1(\tau) = 2\cdot 3^5\cdot\chi_{12}^5\chi_{10}^{-6}\ , j_2(\tau) = 2^{-3}\cdot3^3\cdot\psi_4\chi_{12}^3\chi_{10}^{-4}\ , j_3(\tau) = 2^{-5}\cdot 3\cdot \left( \psi_6\chi_{12}^2\chi_{10}^{-3}+ 2^2\cdot3\cdot\psi_4\chi_{12}^3\chi_{10}^{-4}\right) \ .
D^{\psi}_f(x): =D^{\phi}_{f}(t^df(z_1)x) &=1+c(f)p_1t^df(z_1) x+ c(f)p_2t^{2d}f(z_1)^2fx^2 + \dots + c(f)t^{dr_0}f(z_1)^{dr_0}f^{r_0-1}x^{r_0},
0=\sum_{d_1,d_2<D}\frac{1}{\sqrt{d_1d_2}}\sum_{Id_1, Jd_2<D}\alpha_{Id_1}\bar{\alpha}_{Jd_2}\sum_{L, M\geq 1}\frac{G_k(LMd_1d_2\slash p)}{\sqrt{LM}}\sum_{f\in \mathcal{B}_{2k}(p)}\omega_f\lambda_f(IL)\lambda_f(JM).
E^{\mathrm{( s u r f )}}( a ) = \frac{4 \xi - 1}{2 \pi a}\sum_{l = 0}^{\infty}D_{l}\int_{m a}^{\infty}\frac{z d z}{\sqrt{z^{2}- m^{2}a^{2}}}\left[ 1 - \left( \beta + \frac{n}{2}\right) \frac{\left( \tilde{I}_{\nu}( z ) \tilde{K}_{\nu}( z ) \right)^{\prime}}{z \tilde{I}_{\nu}^{\prime}( z ) \tilde{K}_{\nu}^{\prime}( z )}\right] \;
\mathrm{ch}(c)\cdot D^2e^{-s^2D^2}= \Gamma\left(\prod_{k=0}^{p-2}[F,a_k]\right) |D|^{p-1}\cdot |D|^{1-p}L(a_{p-1})L(a_p)e^{-s^2D^2}\quad-\Gamma\left(\prod_{k=0}^{p-1}[F,a_k]\right)|D|^{p-1}\cdot|D|^{1-p}L(\delta(a_p))e^{-s^2D^2}.
\left[\begin{array}{cc}a_{11}(A) a_{12}(A) a_{21}(A) a_{22}(A) \end{array}\right]=\left[\begin{array}{cc}a_{11}(A) a_{12}(A) 0 0 \end{array}\right]=\left[\begin{array}{cc}a_{11}(A) 0 a_{21}(A) 0 \end{array}\right].
\xi(\xi+1) = L(L+1), - \kappa \xi^2 + 2\xi\zeta + \eta^2 - \frac{\zeta^2}{\kappa}= - E_0, 2\xi\eta = -Q, 2\eta\zeta = \kappa B_1, \frac{\zeta}{\kappa}\left(\frac{\zeta}{\kappa}-1\right) = B_2.
||\mathtt{g}\cdot\mathtt{h}||_\nu \leq ||\mathtt{g}||_\nu+||\mathtt{h}||_\nu, ||\mathtt{h}\cdot\mathtt{g}\cdot\bar{\mathtt{h}}||_\nu =||\mathtt{g}||_\nu, ||\bar{\mathtt{g}}||_\nu =||\mathtt{g}||_\nu.
\dot{\tilde{\rho}}=-\sum_{i}\int_{0}^{t}dt^{\prime}tr_{R}\{[\tilde{\alpha}_{i}(t)\tilde{\Gamma}_{i}(t),[\tilde{\alpha}_{j}(t^{\prime})\tilde{\Gamma}_{j}(t^{\prime}),\tilde{\rho}(t)R_{0}]]\}
\sum_{n=0}^{\infty}\frac{y^nz^nq^{n^2+n}}{(yq;q^2)_{n+1}}=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\sum_{\ell=0}^{\infty}b_\nu(\ell,m,n)z^\ell y^m q^n =\sum_{n=0}^{\infty}(-zq;q^2)_n(yq)^n.
\sum_{y \notin B_n(\hat x)}|\phi(y)|^2 = \sum_{m > R_n}\sum_{d(y, \hat x) = m}| \phi(y)|^2 \leq \frac 32 \sum_{m > R_n}K^{m}C_n^2 e^{-2\mu m}= \frac 32 C_n^2 \frac{e^{-2 \mu_K R_n}}{e^{2\mu_k}-1}\, .
\chi_k^n(\tau) ={1 \over{(\eta(\tau))^3}}\cdot 2(k+2) \sum_{m=-\infty}^\infty \left( m +{{n+1}\over{2(k+2)}}\right) e^{2 \pi i (k+2) \tau \left( m +{{n+1}\over{2(k+2)}}\right)^2}.
(D_L)_{\beta \gamma}= \delta_{\beta, \gamma}(D_L)_{\beta},\quad (D_L)_{\beta}=-i\sum_{\Upsilon\in\Delta_l,\ \beta\cdot\Upsilon=2}\left[g_{l_1}z(\Upsilon\cdot q) +g_{l_2}z^{(1/2)}(\Upsilon\cdot q)\right].
Z_{H_G,k}(t)=\int_{\mathbb{R}^k}' \left[ \int_0^t \prod_{j=1}^k (s-x_j)_+^{\gamma}ds\right] B(dx_1)\ldots B(dx_k),\gamma=H-\frac{3}{2}=\frac{H_G-1}{k}-\frac{1}{2},
\frac{\lambda_i}{x - a_i}= \frac{\lambda_i}{\gamma^{-1}(\gamma(x)) - a_i}= \frac{\lambda_i d}{b - a_i d}+ \frac{\lambda_i (be-cd)(b-a_id)^{-2}}{\gamma(x) +(c-a_ie)(b-a_id)^{-1}}.
\omega \lambda(t)^{2}T(y)(t,\omega) &= - \int_{t}^{\infty}\sin((t-x)\sqrt{\omega}) \frac{\lambda(t)^{2}}{\lambda(x)}\lambda(x) \sqrt{\omega}\left(F_{2}(y)(x,\omega) - \mathcal{F}(\sqrt{\cdot}F(x,\cdot \lambda(x)))(\omega \lambda(x)^{2})-\mathcal{F}(\sqrt{\cdot}F_{3}(x,\cdot\lambda(x)))(\omega \lambda(x)^{2})\right) dx
\begin{array}{rl}b_k'(\rho_s)-\mu (\bar\sigma-\tilde\sigma)c_k [0.3 cm] = \, \displaystyle -\mu a_k'(\rho_s)+(\gamma k^2-\mu \sqrt{\bar\sigma^2-\hat\sigma^2})c_k k\tanh k\rho_s -{k\, e_k\over\sinh k(\rho_s-\eta_s)}-\mu (\bar\sigma-\tilde\sigma)c_k [0.3 cm] = \, \lambda_k(\gamma) c_k, \end{array}
p(y_{ij}) H\left(\frac{p(y_i)P_{X|Y=y_j}+p(y_j)P_{X|Y=y_j}}{p(y_{ij})}\right) & - \left( p(y_i) H(P_{X|Y=y_i})+ p(y_j) H(P_{X|Y=y_j}) \right).
2^{-j-1}\Bigg\Vert \prod_{i=0}^{j}\big(\rho(g_{k_i})-\mathrm{Id}\big)\rho(g_p)f\Bigg\Vert \leqslant 2\times 2^{-j-2}\Bigg\Vert \prod_{i=0}^{j}\big( \rho(g_{k_i})-\mathrm{Id}\big)(\rho(g_p)-\mathrm{Id})f\Bigg\Vert &+2^{-j-1}\Bigg\Vert \prod_{i=0}^{j}\big(\rho(g_{k_i})-\mathrm{Id}\big)f\Bigg\Vert
\stackrel{(3)}{\varphi}_\mu \,=\,p_{1\mu}\,-\,m\,\frac{q_{2\mu}}{\sqrt{q_3^2}}\,-\,\beta \,\frac{\varepsilon_{\mu \nu\lambda}\,q_2^\nu\,q_3^\lambda}{q_2^2\,\sqrt{q_2^2}}\,=\,0, \quad \mu\,=\,0,\,1,\,2\,{.}
\Pi_T \Big( b: \sum_{j=1}^d \|b_j-b_{0,j}\|_{\infty}\geq (\log T)^\delta T^{-\frac{s\wedge[a-(d/2-2)_+]}{2a+d}}\big| X^T \Big) \rightarrow^{P_{b_0}}0~ T \to \infty.
\left( \begin{array}{c}{u_{n}^{1}( y )}\\{u_{n}^{2}( y )}\\ \end{array}\right) \propto \left( \begin{array}{c}{\cos \left( \mu_{n}y \right) + \frac{m_{-}}{\mu_{n}}\sin \left( \mu_{n}y \right)}\\{\frac{M_{n}- m_{+}}{\mu_{n}}\sin \left( \mu_{n}y \right)}\\ \end{array}\right)
\alpha \Vert y \Vert_U = \Vert x - z(\alpha) \Vert_U \leq K_\rho \Vert F(x) - F(z(\alpha)) \Vert_V \leq K_\rho ( \alpha \Vert DF(x)y \Vert_V + \Vert F(x+\alpha y) - F(x) - DF(x) \alpha y \Vert_V ).
n!{\|h_n(.,t,x)\|}_{\mathcal{H}^{\otimes n}}^2 \leq\frac{1}{n!}C_1 e^{-(2-\delta)\mu_1 t}&\times\int_{[0, t]^{2n}}\prod_{i=1}^n \gamma(t_i-s_i)\prod_{i=1}^n\Big(t_{\tau(i+1)}+s_{\iota(i+1)}-(t_{\tau(i)}+s_{\iota(i)})\Big)^{-\beta/\alpha}d{\bf t}d{\bf s}.
\tfrac12 \alpha_1^2+\tfrac12 \alpha_2^2 +\tfrac 12(\alpha_1+\alpha_2)^2+\sum_{i=3}^{r}\alpha_i^2\le \tfrac{3r-5}{3r-2}-\Bigl(2\sum_{i<j}a_ia_j-a_1a_2\Bigr)\,,
\Vert G'_{4,\gamma}(t)\Vert_{L^2(\mathbb T)}=\Vert 2\left[\mathbb H\,;\,\phi_{0,x}\right]\partial^m_x\phi_{x,\gamma}\Vert_{L^2(\mathbb T)}\le C\Vert\phi_{0,x}\Vert_{H^1(\mathbb T)}\Vert\partial^m_x\phi_{x,\gamma}\Vert_{L^{2}(\mathbb T)}&\le C\Vert\varphi_{0}\Vert_{H^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^{m}(\mathbb T)}\,.
\Pi_{\Psi_1}(\ell_F) = \left\|\ell_F\right\|_{(\Psi_1)}\max\{1, \varrho\left(\ell_F; \Psi_1\right)\}\le \max\{1, \varrho\left(\ell_F; \Psi_1\right)\}^2 \le\left\|Q_F\right\|_{L_{p_1}}^{2p_1}.
P(x^mP(x^{m+k\theta(m)}))+P(P(x^m)x^{m+k\theta(m)}) = \beta(m+k\theta(m))\beta(m+(k+1)\theta(m))x^{\theta(m+(k+1)\theta(m))}+\beta(m)\beta(m+(k+1)\theta(m))x^{\theta(m+(k+1)\theta(m))}= \frac{k+2}{k+1}\beta(m)\beta(m+(k+1)\theta(m))x^{\theta(m+(k+1)\theta(m))}.
G \vcentcolon = F_C^{(3)}\left(1,1-\nu,1+\nu,1-\nu,1+\nu,z_1,z_2,z_3\right) &=\sum_{n_1,n_2,n_3 =0}^\infty \frac{(1)_{n_1+n_2+n_3}(e)_{n_1+n_2+n_3}}{(2-e)_{n_1}(2-e)_{n_3}(e)_{n_2}}\frac{z_1^{n_1}z_2^{n_2}z_3^{n_3}}{n_1! n_2! n_3!}
c_r(p,q,2N_1) = -\frac{1}{2}c_r(p,q,N_1) + \frac{1}{2}c_r(p,q^2,N_1) = -\frac{1}{4}a_r^+(p,q,N_1)-\frac{1}{4}a_r^-(p,q,N_1) + \frac{1}{4}a_r^+(p,q^2,N_1)+\frac{1}{4}a_r^-(p,q^2,N_1)
U_0(x',x_m,y)=\begin{cases}U_0(x',x_m,y)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \x_m\geq 0, \sum^3_{i=1}c_i U_0(x',-x_m/i,y)\ \ \ \ \ \ \ \ \ \ \x_m< 0,\end{cases}
\phi_{1 \mu}\equiv p_{\mu}-{\frac{m \dot{n}_{\mu}}{\sqrt{\dot{n}^{2}}}}\approx 0 ; \, \nonumber \phi_{2 \mu}\equiv p_{\mu}^{( n )}-{\frac{m}{\sqrt{\dot{n}^{2}}}}({\dot{x}}_{\mu}-{\frac{( \dot{x}\dot{n}){\dot{n}}_{\mu}}{{\dot{n}}^{2}}}) \approx 0 ; \phi_{3}\equiv p^{( \lambda )}\approx 0 .
\#R^i_{n_0;l_1,\cdots,l_{n_0}}=\underset{\tilde{l}=(\tilde{l}_{i},\tilde{l}_{i+1},\cdots,\tilde{l}_{n_0})\in L}{\sum}c_{\tilde{l}}\#P_{l_1,l_2,\cdots,l_{i-1},\tilde{l}_i,\tilde{l}_{i+1},\cdots,\tilde{l}_{n_0}}
A_{\mathrm{n o n d e g}}= \sum_{m^{i j k}}\mu^{\prime}\left( m^{i j k}\right) \frac{e^{- 2 \pi \sqrt{( m^{i j k})^{2}}+ 2 \pi i C_{i j k}m^{i j k}}}{\sqrt{( m^{i j k})^{2}}}.
\begin{array}{r c l}{\{\Gamma_{0}^{A}, \Gamma_{0}^{B}\}}&{=}&{0}\\{\{\Gamma_{0}^{A}, \Gamma_{1}^{B}\}}&{=}&{E^{A B}( \Gamma_{0})}\\{\{\Gamma_{1}^{A}, \Gamma_{1}^{B}\}}&{=}&{\Gamma_{1}^{C}\partial_{C}E^{A B}( \Gamma_{0}) .}\\ \end{array}
\bigoplus_{s,t=1}^n\bigoplus_{l=1}^{k_{st}}Y_{jj}\otimes_B Y_{st}^{(l)}\oplus \bigoplus_{l=1}^{k}Y_{jj}^{(l)}\overset{Y_{jj}\otimes_B \Phi}{\longrightarrow}Y_{jj}.
f \left( x \right) \rightarrow \widetilde{f}\left( x \right) \equiv - \left( \frac{r}{\beta}\right)^{3}x^{2}\log \left[ 1 - e^{- \left( x^{2}- \left| m_{e f f}^{2}\right| \beta^{2}\right)^{1 / 2}}\right] ,
E[G(X_{\rho_3,m})] = E\left[\left. G(X_{\rho_3,m})\right|\zeta=1 \right] P(\zeta=1) + E\left[\left. G(X_{\rho_3,m})\right| \zeta=0\right]P(\zeta=0) &= \theta E[G(X_{\rho_1,m})] + (1-\theta) E[G(X_{\rho_2,m})]
\Vert p_{1}^{\varphi_{t}}f(x_1) \Vert =\Vert p_{1}^{\varphi_{t}}|f(x_1)|^2 p_{1}^{\varphi_{t}}\Vert^{1/2}\leq C(t)\Lambda^{-1/2}\Vert f\Vert_2\;.
\psi_{1,B}(x,t) = \mathrm{e}^{\,\mathrm{i}\,(t-t_0)\,\left(|\psi_{1,B}(x,t_0)|^2 + e\,|\psi_{2,B}(x,t_0)|^2\right)}\psi_{1,B}(x,t_0), \psi_{2,B}(x,t) = \mathrm{e}^{\,\mathrm{i}\,(t-t_0)\,\left(e\,|\psi_{1,B}(x,t_0)|^2 + |\psi_{2,B}(x,t_0)|^2\right)}\psi_{2,B}(x,t_0).
\Theta =\begin{pmatrix}0 \theta_0 \theta_1 \theta_2 \cdots \theta_{d-3}\theta_{d-2}begin{align*}4pt] 0 \theta_0 \theta_1 \theta_2 \cdots \theta_{d-3}& 0 \theta_0 \theta_1 \ddots \vdots 0 \theta_0 \ddots \theta_2 & 0 \ddots \theta_1 & \ddots \theta_0 begin{align*}4pt] 0\end{pmatrix}
K_{\hbar}^{top}=\left( -\hbar\right)^{M}\partial^{2M}\preceq_{\mathcal{S}}\frac{1}{c_{\mathcal{B}_{M}}}\mathcal{L}_{\hbar}^{top}\preceq_{\mathcal{S}}\frac{1}{c_{\mathcal{B}_{M}}}\left( \mathcal{L}_{\hbar}+EK_{\hbar}^{<}\right) .
\eta_1t+\frac{(\sum_{j\neq 1}\eta_j)(1-t)}{\pi_2}= \left(\eta_1-\left(\frac{\sum_{j\neq 1}\eta_j}{\pi_2}\right)\right)t+\left(\frac{\sum_{j\neq 1}\eta_j}{\pi_2}\right) &=\begin{cases}\eta_1 t=1 \frac{\sum_{j}\eta_j}{1+\pi_2}t=\frac{1}{1+\pi_2}, \end{cases}
P_{s_{14}}(t,\alpha,\beta)= t^{5}+3(1+\beta) t^{4}+((3,5,2)_{\beta}+3 \alpha) t^{3}+ (2(1+\beta)^2+\alpha(5+4 \beta)) t^{2}\hphantom{P_{s_{14}}(t,\alpha,\beta)=}{}+((1+\beta((1+3 \alpha)+2 \alpha^{2}) t +\alpha + \alpha^{2}.
0=\left[ \Gamma_{0}f,\Gamma_{0}h \right]=\left[ \Gamma_{0}f,\left( A-z \right)^{-1}\Gamma_{0}h \right]=\left( f,Q\left( z \right)h \right),\thinspace \forall z\in D\left( Q \right),\thinspace \forall h\in H.
\pi_{1}(\phi_{\theta})S_{1}e_{n}^{2}= e^{-i \left(n + 1 + m + \frac{\lambda}{2}\right) \theta}S_{1}e_{n}^{2},n \geq 0;\,\,\,\pi_{1}(\phi_{\theta})S_{1}e_{-n}^{2}= e^{i \left(n - 1 + p - \frac{\lambda}{2}\right) \theta}S_{1}e_{-n}^{2}, n \geq 1
\mathbb{P}\left\{\mathcal{M}_i^{1, j}\right\}= \mathbb{P}\left\{\sum_{t=0}^{T-1}\sum_{l=1}^{{N_i}}v^{(i),\top}_{l,t}\bar{\Sigma}^{(i)}_tv^{(i)}_{l,t}\geq \sum_{t=0}^{T-1}\sum_{l=1}^{{N_i}}u^{(i),\top}_{l,t}\Tilde{\Sigma}^{(i)}_t u^{(i)}_{l,t}\right\},
{\cal V}^{\prime \prime}+ \left[ \frac{\eta_{c}q^{2}}{( \eta_{c}- \eta )}+ \frac{1}{4 ( \eta_{c}- \eta )^{2}}\right]{\cal V}= 0 ,{\cal W}^{\prime \prime}+ \left[ \frac{\eta_{c}q^{2}}{( \eta_{c}- \eta )}+ \frac{1}{4 ( \eta_{c}- \eta )^{2}}\right]{\cal W}= 0
-\frac{1}{2\pi i}\int^{i\infty}_{-i\infty}\frac{\zeta'}{\zeta}(s)\hat{g}(s) ds=\sum_{\rho}\hat{g}(\rho)-\hat{g}(1)-\frac{1}{2\pi i}\int_{(c)}\frac{\zeta'}{\zeta}(s)\hat{g}(s)ds,
G_{\ell}(y) := \frac{1}{2\pi i}\int_{(\sigma)}(\pi^3 y)^{-s}\, \prod_{i=1}^{3}\frac{\Gamma\left(\frac{1+s+{\bf \alpha}_{i}+ \ell}{2}\right)}{\Gamma\left(\frac{-s-{\bf \alpha}_{i}+ \ell}{2}\right)}\tilde \psi(-s)ds.
Q_{\alpha}^{( 1 )}= - \frac{i}{2}[ A_{\mu}, A_{\nu}]^{i j}\Gamma_{\alpha \beta}^{\mu \nu}\frac{\partial}{\partial \bar{\psi}_{\beta}^{i j}}+ i \Gamma_{\mu \alpha \beta}\psi_{\beta}^{i j}\frac{\partial}{\partial A_{\mu}^{i j}}
J_{1,n}:= \left( \sum_{s=1}^n \Vert \tilde{A}_s \Vert\right) \sum_{t=n+1}^{\infty}\Vert \tilde{a}_{t-1}\Vert \cdot \Vert y_t \Vert, J_{2,n}:= \sum_{s=1}^n \sum_{t=n+1}^{\infty}\sum_{u=0}^{s-2}\Vert \tilde{A}_{u+1}\Vert \cdot \Vert \tilde{a}_{t-s+u+1}- \tilde{a}_{t-s+u}\Vert \cdot \Vert y_t \Vert.
\begin{cases}\overline{V}_{\cdot}^{(m),i,n}:=\overline{\Psi}_{\cdot}^{i}(\overline{\textbf{V}}^{(m-1),n}), \,\, \overline{V}_{\cdot}^{(0),i,n}=V_{\cdot}^{(0),i,n}, \underline{V}_{\cdot}^{(m),i,n}:=\underline{\Psi}_{\cdot}(\underline{\textbf{V}}^{(m-1),n}), \,\, \underline{V}_{\cdot}^{(0),i,n}=V_{\cdot}^{(0),i,n}. \end{cases}
\int\limits_{\mathbb{R}^d}a_{\rm sym}(\xi-q) \mu(\xi,q) \big( \tilde\varphi^i_{0}(q) - \tilde\varphi^i_{0}(\xi) \big) \, dq \ = \ 2 \, \int\limits_{\mathbb{R}^d}c^i(\xi-q) \mu(\xi,q) \, dq.
\frac{d\mu\circ f}{d\mu}(x) =\frac{\mu_{k}(x_{k}+1)}{\mu_{k}(x_{k})}\prod_{i=1}^{k -1}\frac{\mu_i(0)}{\mu_i(\max{A_i})}=\frac{\mu_{k}(x_{k}+1)}{\mu_{k}(x_{k})}\prod_{\substack{i=1 \textrm{$i$ odd}}}^{k -1}\dfrac{1-2^{-i}}{2^{-i}}.
\phi_h^\prime(x) = \frac{I - \frac{h}{2}f^\prime(x) + \frac{h}{2}f^\prime\left(\frac{x+\phi_h(x)}{2}\right)}{I + \frac{h}{2}f^\prime(\phi_h(x)) - \frac{h}{2}f^\prime\left(\frac{x+\phi_h(x)}{2}\right)}.
(\omega_{P}^{p})_{A}(a,b) =r(r-1){\textstyle\int\nolimits_{M}}p(a,b,F_{A},\overset{(r-2)}{\ldots},F_{A}), (\mu_{P}^{p})_{A}(X) =-r{\textstyle\int\nolimits_{M}}p(v_{A}(X),F_{A},\overset{(r-1)}{\ldots},F_{A})
\mathbf{u}(t)=c_1^{-1}\mathbf{f}(t-\varepsilon)-c_1^{-1}(\alpha+c_0)c_1^{-1}\mathbf{f}(t-2\varepsilon)+c_1^{-1}(\alpha+c_0)c_1^{-1}\alpha\varphi(t-2\varepsilon).
E\left[\exp\left\{\frac{c_q}{2}\left(\frac{|\Delta_{i,j}|}{\sigma_{i,j}}\right)^{2/q}\right\}-1\right]=\int_0^\infty P\left(\exp\left\{\frac{c_q}{2}\left(\frac{|\Delta_{i,j}|}{\sigma_{i,j}}\right)^{2/q}\right\}-1>u\right)du,
\Gamma\left(s-r+j\right) \left(\mathbf{b}_{s,i}^t-\mathbf{b}_{i}^t\right)\mathbf{\Psi}_s^{-1}\mathbf{a}_{s,j}= \frac{\left(-1\right)^{r-j}\log\left(\theta_{n-m+i}\right)}{\left(r-j\right)!}\mathbf{b}_i^t\mathbf{\Psi}^{-1}\mathbf{a}_{j}+o(s),
d_{m^{\prime}m}^{j}(\beta)=(-1)^{\lambda}(\begin{matrix}2j-k\\k+a\end{matrix})^{\frac{1}{2}}(\begin{matrix}k+b\\b\end{matrix})^{-\frac{1}{2}}(sin\frac{\beta}{2})^{a}(cos\frac{\beta}{2})^{b}P_{k}^{(a,b)}(cos\beta),
\begin{array}{l}{{T ( z ) = \hat{T}( z ) + T_{Q}( z )}}\\{{\}}\\{{G^{+}( z ) = \hat{G}^{+}( z ) + G_{Q}^{+}( z )}}\\{{\}}\\{{G^{-}( z ) = \hat{G}^{-}( z ) + G_{Q}^{-}( z )}}\\{{\}}\\{{J ( z ) = \hat{J}( z ) + J_{Q}( z )}}\\ \end{array}
\begin{array}{c}h(\tilde{y}^{(k-1)})-h(\tilde{y}^{*})\geq h(\tilde{y}^{(k)})-h(\tilde{y}^{*})+\underset{i=1}{\overset{m}{\sum}}\frac{\mu}{2}\|y_{i}^{(k)}-y_{i}^{(k-1)}\|^{2}.\end{array}
[\begin{matrix}3&1&0&0&0&0&0\\ 1&2&1&0&0&0&0\\ 0&1&1&1&0&0&0\\ 0&0&1&0&1&0&0\\ 0&0&0&1&1&1&0\\ 0&0&0&0&1&2&1\\ 0&0&0&0&0&1&3\\ \end{matrix}].
F_p(p,a,2) =F_p(p,a,3)-\sum_{i=3}^{a-1}F_p(p,i,2)\cr =F_p(p,a,3)-\sum_{i=3}^{a-1}\dim(i,2)\ \ \ \ \ \ \ \ \ \ \ (\dagger)\cr =F_p(p,a,3)-\dim(a-1,3).
R_s = \sum_{w' \in W(D_{p+q+1}\times C_{p-1})}(-1)^{l(w')}X\left(\begin{array}{cccccccccc}p+q, \dots, 1, 0; p-1, \dots, 2,1 w'( p+q, \dots, 1, 0; p-1, \dots, 2,1)\end{array}\right).
M_t= \exp^{\triangleleft}(th)=1+ \sum_{\tau\in\mathcal T}\operatorname{cm}(\tau)P_h(\tau)\frac{t^{|\tau|}}{|\tau|!}=1+ \sum_{\tau\in\mathcal T}\frac{1}{\tau!\sigma(\tau)}P_h(\tau)t^{|\tau|},
G(c_1,c_2,\dots,c_n,m) &=G(H_1(c_1,\dots,c_n,m),\dots,H_n(c_1\dots,c_n,m)) &=(p_1(c_1,c_2,\dots, c_n),\dots,p_n(c_1,...,c_n)) &=(c_1,c_2,\dots,c_n).
\underset{n\le X^{1+\epsilon}, \, (n,M)=1}{\sum}|a'_f(n)|^{2}e^{-n/X}+ \underset{n > X^{1+\epsilon}, \, (n,M)=1}{\sum}|a'_f(n)|^{2}e^{-n/X}\gg_f X.
\varepsilon_{k}A_{k}^{-1}\mathbf{x}_{k}=\mathbf{x}\mathbf{x}^{T}\mathbf{x}_{k}-\mathbf{x}\mathbf{c}^{T}L_{k}^{-1}\mathbf{V}^{T}\mathbf{x}_{k}+\varepsilon_{k}VL_{k}^{-1}V^{T}\mathbf{x}_{k}.
\!\begin{multlined}[t]x^{k+2}\cdot x^{k+1}\cdot 11x^3 + x^{k+2}\cdot (2^k+k)x^k \cdot x^4 + (2^{k+1}+k+1)x^{k+1}\cdot x^{k+1}\cdot x^4 = (3\cdot2^k + 2k + 12) x^{2k + 6},\end{multlined}
r_{t}(\vec{\zeta},\vec{\xi})=\bigg(\prod_{n=1}^Ne^{\mu_n(\xi_n-\zeta_n)-{t}\mu_n^2/2}\bigg)\det_{1\leq k,l\leq N}[F_{k,l}(\xi_{N+1-l}-\zeta_{N+1-k},{t})],
\overline{\chi}\left( -2\right) bc^{p}S_{p}\left( b,c:\chi\right)+\chi\left( -2\right) cb^{p}S_{p}\left( c,b:\overline{\chi}\right) & =\frac{p}{2^{p+1}}\sum\limits_{j=1}^{p}\left( -1\right)^{j}\binom{p-1}{j-1}c^{j}b^{p+1-j}\overline{E}_{j-1,\chi}\left( 0\right) \overline{E}_{p-j,\overline{\chi}}\left( 0\right) .
(x|a)_{n}=\sum_{k=0}^{n}(-1)^{n-k}e_{n-k}(a_1,\dots, a_{n}) \,x^k, \frac{1}{(x|a)_{n}}=\sum_{k=n}^{\infty}h_{k-n}(a_1,\dots, a_{n}) \,x^{-k}, x^n =\sum_{k=0}^{n}h_{n-k}(a_1,\dots, a_{k+1}) \,(x|a)_k, x^{-n}=\sum_{k=n}^{\infty}(-1)^{n-k}e_{k-n}(a_1,\dots, a_{k-1}) \frac{1}{(x|a)_{k}}.
\Omega g(i) =b_i(g_{i+1}-g_i)+a_i(g_{i-1}-g_i)+\mu_ig_i =b(i+1)-a(i+1)+\gamma (i+1) \leq -\lambda(i+1)=-\lambda g_i, \i\geq 1, \Omega g(0) =b_0(g_1-g_0)+\mu_0 g_0=(b-c)g_0\leq -\lambda g_0.
\left|\left(I-\operatorname{Tayl}_x^{|\alpha|}\right)a(y)\right| \leq|a(y)|+\sum_{\beta\in\mathbb{Z}_+^n,\,|\beta|\leq|\alpha|}\frac{|(y-x)^\beta|}{\beta!}|\partial^\beta a(x)| &\lesssim1+\sum_{k=0}^{|\alpha|}|y-x|^k\sim1+|y-x|^{|\alpha|}.
\|\rho(t)-\overline{\rho}\|_{W^{2,r}}\leq \Big(\|\rho_0-\overline{\rho}\|_{W^{2,r}}+\overline{\rho}\int_0^t \|\nabla v(s)\|_{W^{2,r}}s\Big) \exp\Big(C\int_0^t \|\nabla v(s)\|_{H^2\cap W^{2,q}}s\Big).
\kappa = \begin{cases}\beta^{\frac{1}{3}}N^{\frac{1}{3}}, d=1, \beta^{\frac{1}{4}}N^{\frac{1}{4}}, d=2,3, \end{cases}\alpha = \begin{cases}0, \d=1, - \frac{1}{4}, \d=2,3. \end{cases}
\!\!{\cal F}(\psi^1) *{\cal F}(\psi^2) = \int \phi_2 \phi_3 \phi_4(\bar{\psi}_1 - \phi_1)(\bar{\psi}_3 - \phi_3)(\bar{\psi}_4 -\phi_4)d\phi = \bar{\psi}_3 \bar{\psi}_4.
a \left( \eta \right) = \sqrt{\frac{\omega_{4}M}{2}}\Big [ 1 - c_{1}\cos \left( 2 \eta \right) \Big ]^{\frac{1}{2}}, \mathrm{w i t h}c_{1}= \sqrt{1 -{\frac{3 Q^{2}}{4 M^{2}}}}.
H_{2,1}^{1,1}=\int_{-\frac{\pi}{2}}^{0}\!\int_\gamma^{\frac{3}{2}}\frac{dr d\theta}{(r^2-\gamma^2)^\frac{\tau}{2}|\theta-\pi-\theta_1(r)+\varphi|^\tau |re^{i\theta}+1|^q}.
\P(S(\zeta(t+1))=S(\zeta(t))+1\|\zeta(t)=\zeta) =1-\frac{\sum_{x\in \L}1_{\{\zeta_x>0\}}}{\sum_{x\in \L}e^{U(x,\zeta)}+\sum_{x\in \L}1_{\{\zeta_x>0\}}}&\geq1-\frac{|\L|}{{\displaystyle\max_{x\in \L}}\e^{U(x,\zeta)}}\geq 1-|\L|e^{-C_1 S(\zeta)}.
f^0(m)(w_1s_1+w_2s_2+w_{1,2}\gamma) = f^0(m(w_1s_1+w_2s_2+w_{1,2}\gamma)) \ \ + f^1(d(w_1s_1+w_2s_2+w_{1,2}\gamma))(m) f^0(m)w_1s_1+f^0(m)w_2s_2+f^0(m)w_{1,2}\gamma = f^0(m)w_1s_1+f^0(m)w_2s_2 + f^0(mw_{1,2}\gamma) \ \ + f^1(d(w_{1,2}\gamma))(m),
\left|N_{\alpha, \lambda_j,\gamma}(n_{r,j}, R)-\gamma R \right|\leq &30^2\left( \frac{\tau_{r,j}}{A_{r,j}}+ D_{r,j}(a_r) + \sum_{m_1,m_2=-A_{r,j}}^{A_{r,j}}\!\!\!\!\!\!\!\!^{\Large '}\ \ \ \ \ \ \ \frac{1}{\overline{m_1}\overline{m_2}}\left| U(m_1,m_2,a_r)\right|\right)
\Delta_{\varepsilon , \overline{{{\varepsilon}}}, s_{p}, \overline{{{s}}}_{p}}= \frac{m - i \varepsilon \widehat{p}}{2 m}\cdot \frac{m - i \overline{{{\varepsilon}}}\overline{{{p}}}}{2 m}\cdot \left( \frac 1 2 + 2 s_{p}\sigma_{p}^{( 1 / 2 )}\right) \left( \frac 1 2 + 2 \overline{{{s}}}_{p}\overline{{{\sigma}}}_{p}^{( 1 / 2 )}\right)
P(N_{t-d+h}-N_{t-d}=1,Z_t = j, d-h<U_t|Z_{t-d}= k, U_{t-d}= v) =\sum_{n=0}^\infty P(T_{n+1}\in A,T_{n+2}>t,Z_t = j|Z_{t-d}= k, U_{t-d}= v,N_{t-d}= n)g(n).
Q = +{\frac{\pi}{2}}- \rho \int_{0}^{n^{2}- 1}{\frac{d u^{\prime \prime}}{u^{\prime \prime}}}{\frac{\sqrt{n^{2}- 1 - u^{\prime \prime}}\sqrt{u^{\prime \prime}}}{( n^{2}- 1 - u^{\prime \prime}) + \rho^{2}u^{\prime \prime}}}.
0\leq|\tilde{\nabla}\psi|^2=\langle\psi,D^2\psi\rangle-\frac{R}{4}|\psi|^2+\frac{1}{2}\Delta|\psi|^2-\frac{|d|\psi|^2|^2}{2|\psi|^2}-\frac{|T|^2}{4|\psi|^2}+2\langle D\psi,\alpha\cdot\psi\rangle
l_0 := \sup\big\{l\in \mathbb{N}^+: \|u\|_{L^2(A^g(\mathbf{0}, \tau/K,\tau); \|\Sigma^0\|_g)}\leq \delta_1' \cdot \tau^{1+ n/2}, \ \forall \tau = K^{-m}\in [K^{-l+1}, 1],\m \big\}.
d e t [ \delta_{b}^{a}\partial_{t}- \omega^{a c}\partial_{c}\partial_{b}H ] = \int{\cal D}c^{a}{\cal D}{\bar{c}}_{a}\exp \left\{- \int{\bar{c}}_{a}[ \delta_{b}^{a}\partial_{t}- \omega^{a c}\partial_{c}\partial_{b}H ] c^{b}d t \right\}.
w\left( \left[\begin{array}{cc}0 AXB^{\ast}BYA^{\ast}0\end{array}\right] \right) =w(CSC^{\ast}) & \leq \Vert C\Vert^{2}w(S) & =\max \{||A||^{2},||B||^{2}\}w\left( \left[\begin{array}{cc}0 X Y 0\end{array}\right] \right)
a_k(x) =x\sum_{j\geq k-1}a_j(x)\,, b_k(x) =x\big(a_{k-1}(x)+b_{k-1}(x)+c_{k-1}(x)\big)\,, c_k(x) =x\sum_{j\geq k}\big(b_j(x)+c_j(x)\big)\,,
\gamma(\eta) =a_\gamma\eta+c_\gamma\xi; \gamma\mapsto\smash{\begin{pmatrix}a_\gamma b_\gamma 0 c_\gamma d_\gamma 0 0 0 e_\gamma \end{pmatrix}}, \qquad\, \gamma(\xi) =b_\gamma\eta+d_\gamma\xi; \gamma(\zeta) =e_\gamma\zeta.
{\cal Z}_{l o c a l}= 2 \sqrt{1 +{\frac{g^{2}}{2 \pi}}}\exp \left[ - \left( 1 -{1 2}{\frac{A^{2}}{1 +{\frac{g^{2}}{2 \pi}}}}\right) S_{L i o u .}( \sigma ) \right] ,
_{\mathcal U}(x,y):=_{\Psi,\mathcal U}(x,y):= A^{-1}\sup_{z\in Q_y}|\langle \psi_x,\psi_y-\psi_z \rangle| = A^{-1}\sup_{z\in Q_y}|K_\Psi(x,y)-K_\Psi(x,z)|,
\begin{aligned}c_0 =\min\{c_{0,\alpha}^*, c_{\alpha,\gamma}^*\}&\ge \min\big\{\inf_{[0,\alpha)}\delta(f,\alpha)-2\sup_{[0,\alpha)}\sqrt{\Delta(Dg, \alpha)}, \inf_{(\alpha,\gamma]}\delta(f,\alpha)-2\sup_{(\alpha,\gamma]}\sqrt{\Delta(Dg, \alpha)}\big\}&\ge \inf_{[0,\gamma]}\delta(f,\alpha)-2\sup_{[0,\gamma]}\sqrt{\Delta(Dg, \alpha)}=s_{0,\gamma}.\end{aligned}
\boldsymbol{V}_{B\rightarrow A}^{ext}=\left((\boldsymbol{V}_{B}^{post})^{-1}-(\boldsymbol{V}_{B}^{pri})^{-1}\right)^{-1}, \boldsymbol{x}_{B\rightarrow A}^{ext}=\boldsymbol{V}_{B\rightarrow A}^{ext}\left((\boldsymbol{V}_{B}^{post})^{-1}\boldsymbol{x}_{B}^{post}-(\boldsymbol{V}_{B}^{pri})^{-1}\boldsymbol{x}_{B}^{pri}\right),
\phi_{1}= \frac{m}{\sqrt \lambda}\left\{- \frac{3}{4}m^{2}t^{2}{\eta}_{0}- \frac{1}{4}{\eta}_{k{_0}}+ a_{1}e^{i \omega_{1}t}\frac{\mathrm{{s h}}z}{\mathrm{{c h}}^{2}z}+ \sum_{k = 0}^{\infty}a_{k}{\widetilde C}_{k}( t ) \eta_{k}( x ) \right\}
\mathrm{w i t h ~}\omega_{l , m , n}\equiv \sqrt{\Biggl ( \frac{l \pi}{2 L_{x}}\Biggr )^{2}+ \Biggl ( \frac{m \pi}{2 L_{y}}\Biggr )^{2}+ \Biggl ( \frac{n \pi}{2 L_{z}}\Biggr )^{2}}\; \; ,
\Delta_{u}\mathcal{H}(t)=\mathcal{H}(t,X^{u}(t),p^{u}(t),q^{u}(t),P^{u}(t),v(t),u(t))-\mathcal{H}(t,X^{u}(t),p^{u}(t),q^{u}(t),P^{u}(t),u(t),u(t)),
X_4(y_1, y_2, y_3, y_4)= \frac{1}{2N}\int dx_1dx_2 \Big\{&\quad\u(y_1, x_1)u(x_2, y_2)v_N(x_1-x_2) \bar c(y_3, x_1)c(x_2, y_4) \Big\}.
E_{Q}- E ={\frac{1}{2}}\int d x^{1}\left[{\frac{e^{2}}{\pi + a e^{2}}}\chi^{2}+ ({\frac{e^{2}}{\pi + a e^{2}}})^{2}\chi ( \partial_{1}^{2}-{\frac{e^{2}}{\pi + a e^{2}}})^{- 1}\chi \right] .
(1-\kappa)^{-1}\lambda_k(\xi)\geq \inf_{\substack{V\,\textrm{subspace of}\, \Lambda_\xi^+ H_\xi^{1/2}\dim V=k}}\sup_{u_\xi\in V\setminus \{0\}}\frac{\left(|D_{\xi}|u_\xi,u_\xi\right)}{\|u_\xi\|_{L^2_\xi}^2}.
\frac{\partial}{\partial y}y(3-2\ln y)+(2y^2-(1-2\kappa)y+\kappa)\ln k =1-2\ln y+4y\ln k-(1-2\kappa)\ln k, \frac{\partial^2}{\partial y^2}y(3-2\ln y)+(2y^2-(1-2\kappa)y+\kappa)\ln k =-\frac2y+4\ln k,\frac{\partial^3}{\partial y^3}y(3-2\ln y)+(2y^2-(1-2\kappa)y+\kappa)\ln k=2.
\begin{array}{lcll}\mbox{floor}\mbox{SU(2/1)}\mbox{field}\mbox{helicity}\mid \mbox{gnd}> (0~+\frac{5}{2}) A_\mu -1 \mid \mbox{2nd}> (1~+\frac{5}{2}) 2 \Psi_\mu -\frac{3}{2}\mid \mbox{3rd}> (0~+\frac{7}{2}) e_\mu^a -2. \end{array}
[ \delta_{1}, \delta_{3}] = \delta_{3}( L^{\nu \rho}= [ L , L^{\nu \rho}] + \delta_{2}( L^{\rho}= [ \partial_{\nu}L , L^{\rho \nu}] ) + \delta_{1}( L^{\prime}= [ \partial_{\nu}L , [ A_{\rho}, 2 L^{[ \nu \rho ]}] ] ) .