harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{c c}{{D_{a \alpha}h^{\mu}= - 2 i ( \sigma^{\mu}\bar{\lambda}_{a})_{\alpha}\, , ~}}&{{~ \bar{D}{}_{\dot{\alpha}}^{a}h^{\mu}= 2 i ( \lambda^{a}\sigma^{\mu})_{\dot{\alpha}}\, ,}}\\ \end{array}
\mu_T(x,\mu_T(y,z))= \mu_T(x,\sum_{i=1}^k (y|v_i)(z|v_i)v_i)= \sum_{j=1}^k \sum_{i=1}^k (y|v_i)(z|v_i)(x|v_j)(v_i|v_j)v_j = \sum_{i=1}^k (y|v_i)(z|v_i)(x|v_i)(v_i|v_i)v_i
z_{1}^{2}z_{2}^{2}\cdot\cdot\cdot z_{k}^{2}\equiv\prod_{p_{i}\in P}p_{i}^{a_{i,1}+a_{i,2}+\cdot\cdot\cdot+a_{i,k}}(modN)(wherea_{i,1}+a_{i,2}+\cdot\cdot\cdot+a_{i,k}\equiv0(mod2))
\mathbf{E}\left(F\left((\eta^{N,b}_n)_{n=1}^{N}\right)\right)=\mathbf{E}\left(F\left((\eta_n)_{n=1}^{N}\right)\:\vline\:\hat{W}_N=\hat{W}_0,\:\max_{1\leq n\leq N}\hat{W}_n\leq K,\:S_N>0\right).
\mathbb{E}_{{\mu}_{\lambda_0}}\left[ \left( \sum_k n^{- 1}| \Delta_n \eta (n^{- 1}k + c_n r) |^2 | V' (u (k)) - \varphi_{V'}(\rho' (\lambda_0)) |^2 \right)^p \right] \leqslant \left( \sum_k n^{- 1}| \Delta_n \eta (n^{- 1}k + c_n r) |^2 \right)^p E_{\lambda_0}[(V' (u (0)) - \varphi_{V'}(\rho' (\lambda_0)))^{2 p}],
\varphi(w) =-2\log |w-w_i|-2\log \left|\log \left|\frac{w-w_i}{a_i(1)}\right|\right|+O(|w-w_i|),\ \i\neq n, \varphi(w) =-2\log\left|w\right|-2\log\log \left|\frac{w}{a_n(-1)}\right|+O(|w|^{-1}),\ \i=n.
\sum_{j=1}^{n+1}\omega_{j}\Omega_{j}=0\quad\quad\sum_{j=1}^{n+1}\Omega_{j}\omega_{j}=n\quad\mathcal{S}_{+}^{n}, \sum_{j=1}^{n}\omega_{j}\Omega_{j}=0\quad\quad\sum_{j=1}^{n}\Omega_{j}\omega_{j}=n\quad\partial\mathcal{S}_{+}^{n}.
\partial_{\mu}(\sqrt{g}R^{\mu}) = - \frac{1}{3g^3}\widetilde{\beta}(g) (F \widetilde{F}) - \frac{\tilde{b}(g)}{48\pi^2}(B \widetilde{B}) + \frac{\tilde{c}(g) - a(g)}{24 \pi^2}R \widetilde{R}+ \frac{5a(g) - 3 \tilde{c}(g)}{9 \pi^2}(V \widetilde{V})
\ddot Z={\dot Z^2\over Z}-{1\over \dot \eta}\bigl({\ddot \eta\over \dot \eta}-{\dot g\over g}\bigr)\dot Z+{1\over \dot \eta^2\mu}\bigl(\ddot \mu-{\dot \mu^2\over \mu}-{\dot g\over g}\dot \mu \bigr)Z+{h^2-l\over 4\dot\eta^2\mu^2}{Z^4-\mu^4\over Z}.
\left(\frac{\left|\sum_{i=1}^n[0,z_i]\right|}{\left|\sum_{i=1}^{n-1}[0,z_i]\right|}\right)^2\ge\left(\frac{\left|\sum_{i=1}^n[0,P_Ez_i]\right|}{\left|\sum_{i=1}^{n-1}[0,P_Ez_i]\right|}\right)^2+\left(\frac{\left|\sum_{i=1}^n[0,P_{E^\bot}z_i]\right|}{\left|\sum_{i=1}^{n-1}[0,P_{E^\bot}z_i]\right|}\right)^2.
M'(t) = 2{\rm Re}(u,u_t) +b||u(t)||^2_{L^2(\mathbb{H}^n)}-b||u_0||^2_{L^2(\mathbb{H}^n)}&=2{\rm Re}(u,u_t) +2b \int_{0}^t{\rm Re}(u(s),u_s(s))ds,
H_{2\alpha}(r, \phi, r', \phi', t) = \frac{e^{-(r^2 + r'^2)/4t}}{8 \pi \alpha t}\int_{A_\phi}e^{r r' \cos(z-\phi)/2t}\frac{e^{i \pi z/\alpha}}{e^{i \pi z/\alpha}- e^{i \pi \phi'/\alpha}}dz.
\Omega(\mathbf{x}, \mathbf{u})= H(x_1) H(x_2) \cdots=\prod_{j=1}^\infty\prod_{i=1}^{N}(1-x_ju_i)^{-1}=\prod_{j=1}^\infty \sum_{\ell = 0}^\infty x_j^\ell h_\ell(\mathbf{u})
(\frac{e^{n+1}_{u}-e^{n}_{u}}{\Delta t},v_{h}) + \nu(\nabla e^{n+1}_{u},\nabla v_{h}) + b(u^{n+1}-u^{n},u^{n+1},v_{h}) + b(e^{n}_{u},u^{n+1},v_{h}) + b(u^{n}_{h},e^{n+1}_{u},v_{h}) - (e^{n+1}_{p},\nabla \cdot v_{h}) = (\frac{u^{n+1}-u^{n}}{\Delta t}-u^{n+1}_{t},v_{h}) \; \; \forall v_{h}\in X_{h}.
\frac{1}{z^{4/3}}\begin{pmatrix}-t^{1/3}0 0 0 -\omega t^{1/3}0 0 0 -\omega^2 t^{1/3}\end{pmatrix}+\frac{1}{z}\begin{pmatrix}\theta^\infty_1/3 0 0 0 \theta^\infty_1/3 0 0 0 \theta^\infty_1/3 \end{pmatrix}.
[x_{(i\,j)}, x_{(i\,k)}]_c = 0, [x_{(i\,k)}, x_{(j\,k)}]_c =0, [x_{(i\,l)}, x_{(j\,k)}]_c =0, [x_{(i\,k)}, x_{(j\,l)}]_c = 2\chi_{(j\,k)}(g_{(i\,k)}) x_{(j\,k)}x_{(i\,l)}.
\frac{u e^{-2\pi i kn\theta}}{2k\pi}\exp\left\{-\frac{n\theta(1-\theta)}{2}(u-2k\pi)^2\right\}\cdot \left[1 + \frac{1-2\theta}{6}n\theta(1-\theta)(iu - 2\pi k i)^3 \right] &=: \hat{\delta}_{n,k}(u;\theta),
F_{A_{/ 1}}= 0 \Longleftrightarrow q^{a_{i / 1}\left[ i \right]}= \phi^{a_{i / 1}}\left( q^{b_{i / 1}\left[ i - 1 \right]}, q^{b_{i / 2}\left[ i - 1 \right]}, q^{b_{i / 2}\left[ i \right]}\right) \, .
x_1(t) = b_1 + y_1(t) + \int_0^t(-x_1(s) + x_2(s))ds - u_1(t), x_2(t) = b_2 + y_2(t) + \int_0^t(-x_2(s) + x_3(s))ds + u_1(t) - u_2(t), x_i(t) = b_i + y_i(t) + \int_0^t(-x_i(s) + x_{i+1}(s))ds,3 \le i \le k-1, x_k(t) = b_k + y_k(t) + \int_0^t-x_k(s)ds, x_1(t) \le 0, 0 \le x_2(t) \le B, x_i(t) \ge 0, t \ge 0,
\sup_{R\in\mathcal{D}_{\mu}}{\Big( \frac{1}{|R|}\int_R{\sum_{k=0}^{\mu-4}2^{(k-\mu)(N-\epsilon-s)q}2^{k(m+s)q}\big( \mathfrak{M}_{\sigma,2^k}\Pi^*_kf(x)\big)^q}dx\Big)^{1/q}}&\lesssim \sup_{0\leq k\leq \mu-3}\big\Vert 2^{k(m+s)}\Pi_kf\big\Vert_{L^{\infty}}\leq \sup_{0\leq k\leq \mu-1}\big\Vert 2^{k(m+s)}\Pi_kf\big\Vert_{L^{\infty}}
W_n(\lambda)=\frac{1}{n d_n}\sum\limits_{i=1}^{\lfloor n\lambda\rfloor}\sum\limits_{j=\lfloor n\lambda\rfloor+1}^n\left(1_{\left\{X_i\leq X_j\right\}}-\int_{\mathbb{R}}F(x)dF(x)\right), \ 0\leq \lambda\leq 1,
\tau_n = \inf\left\{t>0 ; \sup_{\{\phi\}, |\phi_s|<1}\left|\int_0^t\frac{\partial}{\partial x}M(ds,\theta^H_s + \phi_s)- \int_0^t \frac{\partial}{\partial x}M(ds,\theta^H_s) \right|\geq n \right\},
\|f^{=d}\|_q \leq (1/\rho)^d \gamma' = \gamma \left( 99\cdot2 \, r \, r' q \, (r')^{-2/q}\sqrt{\frac{\log(1 / \gamma)}{d}}\right)^d = \gamma \left( 99\cdot2\cdot\sqrt{2}\, r \, q \, (r')^{-2/q}\right)^d
_{3}\digamma_{2}\left(\begin{array}{ccc}a b c . d e\end{array}\mid 1\right) =\frac{\Gamma \left( e\right) \Gamma \left( d+e-a-b-c\right)}{\Gamma \left( e-a\right) \Gamma \left( d+e-b-c\right)}_{3}\digamma_{2}\left(\begin{array}{ccc}a d-b d-c . d d+e-b-c\end{array}\mid 1\right)
y\left( x\right) =\frac{\left( \psi \left( x\right) -\psi \left( a\right) \right)^{\gamma -1}}{\Gamma \left( \gamma \right)}I_{a+}^{\left( 1-\beta \right) \left( 1-\alpha \right) ;\psi}f\left( a\right) +I_{a+}^{\alpha ;\psi}f\left( x,y\left( x\right) \right).
s_3=\frac{(\lambda+\mu)(\lambda+2\mu)}{\lambda+3\mu}, s_4=\frac{2\mu(\lambda+2\mu)}{\lambda+3\mu}, t_1= \frac{\mu}{2|\xi|}\biggl(\frac{1-2s_1}{\tau-2\mu s_1|\xi|}-\frac{1}{\tau-2\mu|\xi|}\biggr).
\begin{array}{l}\displaystyle h(f_{\beta}^{(n)})= -\left({\rm log}\left[(n+1){n\choose x}\right] + (n+1){n\choose x}x I_1 + (n+1){n\choose x}(n-x) I_2 \right)= \displaystyle = -(U_0+U_1 + U_2) \end{array}
\frac{d}{dt}\bigg|_{t=0}V_k(g_{t}) = - K J \cdot \frac{d}{dt}\bigg|_{t=0}\int_\mathbb T \sigma g_{t}~ d\theta =-K\frac{d}{dt}\bigg|_{t=0}\int_\mathbb T (\sigma\cdot J)g_{t}~ d\theta \cr = - \int_\mathbb T \langle \nabla\varphi, K\nabla(\sigma\cdot J)\rangle \rho ~ d\theta.
\mathcal Q_3^{(3)}=-\frac{1}{\vert \Lambda \vert}\sum_{k \in \mathcal{P}_H, p \in \mathcal P_L}\frac{\widehat g(k) \alpha_{p-k}}{\sqrt{1-\alpha_k^2 \vphantom{\alpha_{p-k}^2}}\sqrt{1-\alpha_{p-k}^2}}\big( \bar z b^\dagger_{k-p}a^\dagger_p b_{k}+ z b_{k}^\dagger a_p b_{k-p}\big),
c_{1}=c_{2}=\cdots=c_{k_{i}}=i, \c_{k_{i}+1}=c_{k_{i}+2}=\cdots=c_{k_{i}+k_{j}}=j, \ \ldots, &\ldots, \c_{k_{i}+k_{j}+\cdots+k_{n-1}+1}=c_{k_{i}+k_{j}+\cdots+k_{n-1}+2}=\cdots=c_{k_{i}+k_{j}+\cdots+k_{n}}=n.
I_{p_{X_{\mathcal{I}}}^*q_{Y_{T^c}|X_\mathcal{I}}}(X_T; Y_{T^c}|X_{T^c})= \max_{p_{X_\mathcal{I}}}\{I_{p_{X_{\mathcal{I}}}q_{Y_{T^c}|X_\mathcal{I}}}(X_T; Y_{T^c}|X_{T^c})\}.
\int_0^{\infty}\langle \sigma(t) u_t(w), u_t(w) \rangle dw = \int_0^{\infty}\langle u(w), u(w) \rangle dw, \int_{-\infty}^0 \langle \sigma(t) y_t(w), y_t(w) \rangle dw = \int_{-\infty}^0 \langle y(w), y(w) \rangle dw.
\delta(a\star g) = \delta\circ\left(\sum_{i=0}^{f-1}\sigma^{i}(a) g_i\right)\circ \delta^{-1}&=\sum_{i=0}^{f-1}\sigma^{i}(\delta(a)) \delta\circ g_i \circ \delta^{-1}=\delta(a)\star\delta(g)
L(\boldsymbol{\rho};\boldsymbol{\lambda})=\left\|\boldsymbol{\rho}-\boldsymbol{\rho}\left(i+\frac{1}{2}\right)\right\|^{2}+\sum_{m=1}^{M}\lambda_m\left(\boldsymbol{e}_m^{\mathrm{T}}\boldsymbol{\rho}-1\right),
\begin{aligned}\|v\|_{L^2(\partial\Omega)}\leq C\Big\{\|v\|_{L^2(\Omega)}+\|v\|_{L^{q^\prime}(\Omega)}^{\frac{1}{2}}\|\nabla v\|_{L^q(\Omega)}^{\frac{1}{2}}\Big\}\leq C\|\nabla v\|_{L^q(\mathbb{R}^d)}\leq C\Big\{\|F\|_{L^{q}(\Omega)}+\|h\|_{L^{q}(\Omega)}\Big\},\end{aligned}
(1-\delta)\upsilon_s r_n^s \leqslant \mathcal{H}_s(B_d(x,r_n)\cap K) \leqslant (1+\delta)\upsilon_sr_n^s, &(1-\delta)\upsilon_sq_n^s \leqslant \mathcal{H}_s(B_d(x,q_n)\cap K) \leqslant (1+\delta)\upsilon_s q_n^s = (1-\delta)\upsilon_s r_n^s.
\| \Re W W_\alpha\|_{L^2}+ \|\partial^5 (\Re W W_\alpha)\|_{L^2}\lesssim \ \|W\|_{L^\infty}(\| W_\alpha\|_{L^2}+ \| \partial^5 W_\alpha\|_{L^2})+ \| W_\alpha\|_{L^\infty}\| \partial^5 W\|_{L^2}\lesssim \ \epsilon (\| W_\alpha\|_{L^2}+ \|\partial^6 W\|_{L^2}),
\begin{aligned}\Pi_p \coloneqq \{\boldsymbol{\pi}\in S_N\colon px_1,\ldots,x_p\in\mathcal{S}\setminus\mathcal{M}& \pi_{x_q}\notin\left\{x_q,y^*(x_q)\right\}1\leq q\leq p\}.\end{aligned}
\big| T_{[d_k]}f(x)-T_{[d_k]}f(y) \big| =\Big| \int_{\mathbb{R}^d}{\big( U_k(x,z)-U_k(y,z) \big)\Pi^*_kf(z)}dz \Big| \leq \big\Vert \Pi^*_kf \big\Vert_{L^{\infty}}\int_{\mathbb{R}^d}{\big| U_k(x,z)-U_k(y,z) \big|}dz.
n^{\alpha/2}\Big(\sum_{i = 1}^{n}||\mathbf{d_i}||^{6 +2\alpha}\Big)^{1/2}\leq n^{\alpha/2}||\mathbf{D^{-1}}||^{3+\alpha}\Big{(}\sum_{i=1}^{n}||\mathbf{x_{i}}||^{6+2\alpha}\Big{)}^{1/2}& =O(n^{-1})\;\;\; w.p. 1
v^\mu(x', x_d) - u^\mu(x', x_d) =v^\mu(x', H_u^\mu(x', y_d)) - y_d &=v^\mu(x', H_u^\mu(x', y_d)) - v^\mu(x', H_v^\mu(x', y_d)) &= \int_{H_v^\mu(x', y_d)}^{H_u^\mu(x', y_d)}D_d v^\mu(x', t)\, dt \,.
O_i(Q):=\left\langle\prod_{\ell\in I_i}\tau^{(a_\ell,b_\ell)}_{d_\ell}\sigma_1^{k_1(i)}\sigma_2^{k_2(i)}\sigma_{12}\right\rangle^{\mathbf{s}^{\Gamma_{0,k_1(i),k_2(i),1,I_i}},o}
(x\ast y)\ast z-(y\ast x)\ast z =(x\circ(uy))\circ(uz)-(y\circ(ux))\circ(uz) =((x\circ y)u)\circ (uz)-((y\circ x)u)\circ(uz) =x\circ(y(u\circ(uz)))-y\circ(x(u\circ(uz))) =x\circ((uy)\circ(uz)))-y\circ((ux)\circ(uz))) =x\circ(u(y\circ(uz)))-y\circ(u(x\circ(uz))) =x\ast(y\ast z)-y\ast(x\ast z),
\delta([\nu\rho,\nu^b\rho])\rtimes\sigma =L_{proper}(\delta([\nu\rho,\nu^b\rho]);\sigma) +L_{proper}(\delta([\nu\rho,\nu^{-a}\rho]);\sigma') \intertext{and}\delta([\rho,\nu^b\rho])\rtimes\sigma = L_{proper}(\delta([\rho,\nu^b\rho]);\sigma)+ L_{proper}(\delta([\rho,\nu^{-a}\rho]);\sigma')
S_s(x_1,y_1, \dots,x_s,y_s;m):= &\sum_{\tiny \begin{array}{c}\epsilon_j=0,1, i_j\geq0\end{array}}(-1)^{\epsilon_1+i_1+\cdots+\epsilon_s+i_s}u_s^{\epsilon_s}\cdots u_1^{\epsilon_1}v_1^{-(p-1)i_1-\epsilon_1}\cdots v_s^{-(p-1)i_s-\epsilon_s}&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\otimes (\beta^{\epsilon_1}P^{i_1}\cdots\beta^{\epsilon_s}P^{i_s})(m).
\int_\Omega p_h \cdot \tau_h dx = - \int_\Omega u_h\,\nabla_h\cdot\tau_h dx + \sum_{K\in\mathcal{T}_h}\int_{\partial K}\hat{u_h}\,n_K\cdot\tau_h ds \forall \,\tau_h \in W_h, &\int_\Omega (p_h \cdot \nabla_h v_h + u_h v_h)\, dx = \int_\Omega f\,v_h dx + \sum_{K\in \mathcal{T}_h}\int_{\partial K}\hat{p_h}\cdot n_K v_h ds \forall\,v_h\in V_h,
-\frac{d}{dt}\delta (t-t')g_{\sigma 0}+\varepsilon \int d^{3}\mathbf{x}a_{\nu 0}\frac{\partial}{\partial t}\frac{\delta^{2}W}{\delta J_{\nu}^{\alpha}(\mathbf{x},t)\delta J_{\sigma}^{\alpha}(\mathbf{y},t')}\mid_{_{_{0=J=...}}}=0
S = \int d \tau \{p_{m}( \dot{x}^{m}+ i \psi \sigma^{m}\bar{\rho}- i \rho \sigma^{m}\bar{\psi}) - \frac{e p^{2}}{2}-{\rho^{\alpha}}{\dot{\theta}}_{\alpha}-{\bar{\rho}}_{\dot{\alpha}}{\dot{\bar{\theta}}}^{\dot{\alpha}}\},
A=A^{}_{\pi_{1}}+a(r)\left(\theta^{2}\otimes e_{3}-\theta^{3}\otimes e_{2}+\theta^{4}\otimes e_{1}\right)+\left(b(r)-1\right)\left(\theta^{5}\otimes e_{3}-\theta^{6}\otimes e_{2}+\theta^{7}\otimes e_{1}\right),
\Gamma_{\mu\nu\lambda}^{\rm static}= -8ie^{3}T \int\frac{d^{3}k}{(2\pi)^{3}}\left( \sin \tilde{p}_{1}\cdot k + \sin\tilde{p}_{2}\cdot k + \sin \tilde{p}_{3}\cdot k \right) \sum_{n}\left[\frac{4k_{\mu}k_{\nu}k_{\lambda}}{((2\pi nT)^{2}+k^{2})^{3}}- \left(\frac{\delta_{\mu\nu}k_{\lambda}}{((2\pi nT)^{2}+k^{2})^{2}}+{\rm cyclic}\right)\right]
\left| \Phi(X(t),A,p(t))-\Phi(X(0),A,q)-t \left.\frac{\partial \Phi(X(u),A,p(u))}{\partial u}\right|_{u=0}\right| &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\left| \int_0^t (t-v) \frac{\partial^2\Phi(X(v),A,p(v))}{\partial v^2}dv\right|\le Kt^2\,.
b = r^{q+1}+ r = \left(\frac{h}{h+h^q}\right)^{q+1}+ \frac{h}{h+h^q}= \frac{h^{q+1}+h(h^q+h)^q}{(h^q+h)^{q+1}}= \frac{h^{q^2+1}}{(h^q+h)^{q+1}}.
{}_{4}\phi_3^{(m)}\left(\begin{matrix}q^{1-m-n},a,b,c dq^{m-1},e,f\end{matrix};q,q\right) =\left(\frac{bc}d\right)^{mn}\prod_{j=1}^m\frac{(b,c;a)_{j-1}(de/bc,df/bc;q)_{n+j-1}}{(d/b,d/c;q)_{j-1}(e,f;q)_{n+j-1}}&\quad\times\,{}_{4}\phi_3^{(m)}\left(\begin{matrix}q^{1-m-n},a,d/b,d/c dq^{m-1},de/bc,df/bc\end{matrix};q,q\right).
\frac{p_{\psi}\dot{\psi}}{2}=\frac{p_{\psi}^{2}}{2I_{3}}-\frac{p_{\psi}p_{\phi}cos\theta}{2I_{1}sin^{2}\theta}+\frac{p_{\psi}^{2}cos^{2}\theta}{2I_{1}sin^{2}\theta}
\big( \partial_{\tau}\nabla{\theta}^{k}_{\phi}, \,\nabla u\big) = \big( \partial_{\tau}\nabla e_{\phi}^{k}, \,\nabla u\big) + \frac{1}{\tau}\big(\vert\Psi^{k}_{h}\vert^{2}-\vert\Psi^{k}\vert^{2}- \vert\Psi^{k-1}_{h}\vert^{2}+\vert\Psi^{k-1}\vert^{2},\, u \big), \; \forall u \in X_{h}^{r}.
u_1(t) = M(t)D(t) \mathcal{F}M(t) \mathcal{F}^{-1}v_1(t) & = M(t)D(t) v_1 (t) + O(t^{-3/4 + C_1 \varepsilon^2_1}) & = M(t) D(t) e^{-3i\lambda_1 \Phi_1 (t)}\alpha + O(t^{-3/4 + 2C_1 \varepsilon^2_1})
Q_{[H,iA]}(f,g) := Q_H (f,iAg) - Q_H (Af, ig) = Q_H (f,g) - \langle (2V + x \cdot \nabla V) f , g \rangle , Q_{[[H,iA],iA]}(f,g) := Q_H (f,iAg) - Q_H (Af, ig) = 2 Q_{[H,iA]}(f,g) + \langle (2 x \cdot \nabla V + (x \cdot \nabla V )^2 V) f , g \rangle .
g_{N}(\mathsf{x},t,\mathsf{y})=\sum_{\mathsf{n}:0\leq n_{j}\leq N-1}\exp \left[ -tE_{\mathsf{n}}\right] \mathsf{h}_{\mathsf{n}}\left( \mathsf{x}\right) \mathsf{h}_{\mathsf{n}}\left( \mathsf{y}\right)
\varphi'(\mathbf{t}_{ij}(a)) =\tilde{t}_{ij}(a), \varphi'(\mathbf{u}_{kl}(a)) =\tilde{u}_{kl}(a), \varphi'(\mathbf{v}_{kl}(a)) =\tilde{v}_{kl}(a), \varphi'(\mathbf{w}_{kl}(a)) =\tilde{w}_{kl}(a), \varphi'(\mathbf{f}_{ik}(a)) =\tilde{f}_{ik}(a), \varphi'(\mathbf{g}_{ki}(a)) =\tilde{g}_{ki}(a).
\int | \Phi_{0}( w )^{2}| d w = \mathrm{K}^{2}\int \frac{d w}{z^{2}}\equiv \mathrm{K}^{2}\int_{- \infty}^{+ \infty}\frac{{\cal H}^{2}}{a^{3}{\varphi^{\prime}}^{2}}= 1 ,
\omega^{(2,2)}(\xi)= I_{0}(1-\xi)+I_{0,1}^{2}(1-\xi)^{2}+(1-\xi)\sum_{n=0}^{\infty}I_{n+1}\xi^{n}+(1-\xi)^{2}\sum_{n=0}^{\infty}I_{n+1,2}^{2}\xi^{n}= I_{0,1}^{2}(1-\xi)(3-\xi)+(1-\xi^{2})\sum_{n=0}^{\infty}I_{n+1,1}^{2}\xi^{n}+(1-\xi)\left(I(\xi)-2I_{1}^{2}(\xi)\right),
\begin{gathered}b_{-1}(x)\stackrel{}{=}\frac{c_0(x)\hat c_0(x)}{W(c_0,\hat c_0)}, b_0(x)\stackrel{}{=}\, \frac{-(c_0(x)\hat c_1(x)+c_1(x) \hat c_0(x))}{W(c_0, \hat c_0)}, \end{gathered}
\frac{|\partial_{R}(v_{corr}(x,R\lambda(x)))|}{(1+R^{2})\lambda(x)^{2}}\leq \begin{cases}\frac{C \log(\log(x))}{x^{2}\log(x) (1+R^{2})}, R \lambda(x) \leq \log^{N}(x) \frac{1}{x^{2}(1+R^{2})}, \log^{N}(x) \leq R\lambda(x) \leq \frac{x}{2}\frac{\log^{b}(x)}{\sqrt{x}}\cdot \frac{1}{1+R^{2}}, R\lambda(x) > \frac{x}{2}\end{cases}
\sum\limits_{i=1}^{j-1}\Big|\Big\langle{\frac{T^\ast(\tilde h_i)}{\|h_i\|_{F^\ast}}},{\frac{1_{\Delta_j^\ast}r^{\gamma_{j, m_j}}_{m_j}}{\|h_j\|_F}}\Big\rangle\Big| <{\frac{\eta_j}{2}}.
x_2 =x/y, y_3 =y/x, x_{11}=x/y, x_{12}=x, z_2 =1/y, z_3 =1/x, y_{11}=y, y_{12}=y/x, u' =u^{-1}, x_2' =u^{d/r}x/y, y_3' =u^{-d/r}y/x, x_{11}' =u^{d/r}x/y, x_{12}' =u^{-d}x, z_2' =u^{d+d/r}/y, z_3' =u^d/x, y_{11}' =u^{-d-d/r}y, y_{12}' =u^{-d/r}y/x.
ds^{2}=-\left( \gamma +\frac{r^{2}}{l^{2}}-\frac{\mu}{r^{d-3}}\right) dt^{2}+\frac{dr^{2}}{\left( \gamma +\frac{r^{2}}{l^{2}}-\frac{\mu}{r^{d-3}}\right)}+r^{2}d\sigma_{\gamma}^{2},
a_s(z) = A_s(\mu,z)+2\mu\sum_{r=0}^{s-1}A_r(\mu,z)B_{s-1-r}'(-\mu,0), b_s(z) = B_s(\mu,z)+2\mu\sum_{r=0}^{s-1}B_r(\mu,z)B_{s-1-r}'(-\mu,0).
\det\left(\left(\frac{z_j}{1-z_j}\right)^{i}\right)_{1\le i,j\le n_2}=\left(\prod\limits_{j=1}^{n_2}\frac{z_j}{(1-z_j)^{n_2}}\right)\det\left(z_j^{i-1}\right)_{1\le i,j\le n_2}.
\sum\limits_{r=0}^{m-1}\zeta_m^{sr}\hat{g}(r)\sum\limits_{t=0}^{m-1}\zeta_m^{-st}\hat{g}(t)=\hat{\hat{g}}(-s)\hat{\hat{g}}(s)=m^2g(s)g(-s)=m^2,s=1,\dots,m-1
Y_{t}^{u}=\Phi^{0}\left( \mathbb{P}_{X_{T}^{u}}^{W}\right)+\int_{t}^{T}f^{0}\left( \mathbb{P}_{X_{s}^{u}}^{W},Y_{s}^{u},Z_{s}^{u},\mathbb{P}_{\left( Y_{s}^{u},Z_{s}^{u}\right)}^{W},u_{s}\right) \mathrm{d}s-\int_{t}^{T}Z_{s}^{u}\mathrm{d}W_{s}.
\sum_{j=1}^{J}\mathcal{V}_r(\sigma_j^\ell)\lesssim_r \|F\|_{\ell^r}^{r-1}\Big(\sum_{j=1}^J \mathcal{V}_r^r(\sigma_j^\ell)\Big)^{\frac 1r}={\|F\|_{\ell^r}^{r-1}}\Big(\sum_{n\in\mathbb{J}}|F_n|^r\Big)^{\frac 1r}\leq \|F\|_{\ell^r}^r\leq \mathcal{V}_r^r(\gamma).
\int_{-L}^L \Vert (t,1,0) - (s,0,1) \Vert^p \,ds = \int_{-L}^{t-1}\! \left( t-s \right)^{p}{ds}+2\cdot 1 +\int_{t+1}^{L}\! \left( s-t \right)^{p}{ds}= \frac{\left( t+L \right)^{p+1}-1}{p+1}+2 + \frac{\left( L-t \right)^{p+1}-1}{p+1},
\pi_{posterior}\left(x\left|y\right.\right)\propto exp\left\{-\sum_{i=1}^{N}\frac{1}{2\sigma^{2}}\left(x_{i}-y_{i}\right)^{2}-\sum_{\left\langle i,j\right\rangle}\frac{1}{2}\left[\gamma\left(x_{i}-x_{j}\right)\right]^{2}\right\}
x^{\dagger}Gx=\sum_{i,j}x_{i}^{*}x_{j}\langle v_{i},v_{j}\rangle=\sum_{i,j}\langle x_{i}v_{i},x_{j}v_{j}\rangle=\langle\sum_{i}x_{i}v_{i},\sum_{j}x_{j}v_{j}\rangle=\|\sum_{i}x_{i}v_{i}\|^{2}\ge0.
\delta I [ g , \phi ] = \int_{\cal M}\sqrt{- g}d^{4}x \left[ E^{\mu \nu}\delta g_{\mu \nu}+ E_{\phi}\delta \phi + \nabla_{\mu}\theta^{\mu}( \delta g , \delta \phi ) \right] - \delta \left[ \int_{\partial{\cal M}}\sqrt{- h}d^{3}x B [ g , \phi ] \right] .
s_n = L(x^n) = L(\lim\limits_{k \to +\infty}g_{n,k}(x)) = \lim\limits_{k \to +\infty}L(g_{n,k}(x)) = \lim\limits_{k \to +\infty}\int_\mathbb{R}g_{n,k}(x) d\mu(x) = \int_\mathbb{R}\lim\limits_{k \to +\infty}g_{n,k}(x) d\mu(x) = \int_\mathbb{R}x^n d\mu(x).
f_t \circ ( \phi_1^* * \phi_2^* ) (x) = \int_{0}^{\infty}\int_{0}^{\infty}f_{T(t_1,t_2)}\circ ( 1_{(-J_1(t_1),J_1(t_1))}* 1_{(-J_2(t_2),J_2(t_2))}) (x) dt_1 dt_2
\frac{( \theta u )^{i}}{2 \pi}i ( k \theta p \mp 2 \pi ) e^{i ( k \theta p \mp 2 \pi ) \tau}= \frac{( \theta u )^{i}}{2 \pi}\frac{\partial}{\partial \tau}e^{i ( k \theta p \mp 2 \pi ) \tau}.
x_{\ell}(t)(1 - e^{-st})^{-j}= (e^{st}- 1)^{\ell}(e^{st^*}- 1)^{-\ell}(1 - e^{-st})^{-j}x_{\ell}(t^*) &= (e^{st}- 1)^{\ell-j}(e^{st^*}- 1)^{-\ell}e^{stj}x_{\ell}(t^*) &\leq (e^{st^*}- 1)^{-j}e^{st^*j}x_{\ell}(t^*) &= (1 - e^{-st^*})^{-j}x_{\ell}(t^*),
\overline a_g\overline a_h=(u_g\otimes 1)\overline s_e(u_h\otimes 1)\overline s_e=(u_g\otimes 1)\overline s_e \overline s_h(u_h\otimes 1)=0, & \overline a_g^*\overline a_h=\overline s_e(u_g\otimes 1)^*(u_h\otimes 1)\overline s_e=(u_g\otimes 1)^*\overline s_g\overline s_h(u_h\otimes 1)=\delta_{g,h}\overline s_e^2=\delta_{g,h}\overline a_{g_0}^*\overline a_{g_0},
{\cal P}_{2 m}^{( p u r e )}= \left( \frac 2{1 + T}\right)^{\frac 1 2}\frac{( 2 m ) !}{\left( 2^{m}m ! \right)^{2}}\left( \frac{T - 1}{T + 1}\right)^{m}, \qquad{\cal P}_{2 m + 1}^{( p u r e )}= 0 ,
W_{( 0 )}( x_{\mu}) - W_{( 0 )}( x_{\mu}^{'}) = \int_{x_{\mu}}^{x_{\mu}^{'}}\partial_{\mu}W_{( 0 )}d x^{\mu}= \left\{Q_{W}, \int_{x_{\mu}}^{x_{\mu}^{'}}W_{( 1 )}\right\},
\Pr_x[T_{V(H)}> t/2, T_{R_i \cup R_{i-1}}>t/2 ] \|\Pr_x^{t/2}[\cdot \mid T_{V(H)}> t/2, T_{R_i \cup R_{i-1}}>t/2 ] - \pi(\cdot) \|_{\ell,\pi}=o(1)
A(\theta, \mu) =2 u(\theta, \mu) -\int_{-\infty}^{\infty}\frac{d\theta'}{2\pi}\frac{1}{\cosh(\theta-\theta')}\ln(1+ B(\theta', \mu)^2), B(\theta, \mu) = -\int_{-\infty}^{\infty}\frac{d\theta'}{2\pi}\frac{1}{\cosh(\theta-\theta')}{\rm e}^{-A(\theta', \mu)}.
\frac{\tau^2_h(\varphi v)(x)}{|h|^\alpha}= \frac{\tau_h(\varphi(x+h) \tau_hv(x)+v(x) \tau_h\varphi(x))}{|h|^\alpha}=\varphi(x+2h)\frac{\tau^2_hv(x)}{|h|^\alpha}+2 \frac{\tau_hv(x)\tau_h\varphi(x+h)}{|h|^\alpha}+v(x) \frac{\tau^2_{h}\varphi(x)}{|h|^\alpha}
W [ A_{+}, A_{-}] = -{\frac{N}{2}}\int d^{2}k \, \left( \mid \psi_{+}( k ) \mid^{2}+ \mid \psi_{-}( k ) \mid^{2}+ 2 \cos \theta \psi_{+}^{*}( k ) \psi_{-}( k ) \right)
\begin{bmatrix}V_{\ell, YY}V_{\ell, YX}V_{\ell, XY}V_{\ell, XX}\end{bmatrix}= \begin{bmatrix}(Q_{\ell}/ Q_{\ell, XX})^{-1}- Q_{\ell, YY}^{-1}Q_{\ell, YX}(Q_{\ell}/ Q_{\ell, YY})^{-1}- (Q_{\ell}/ Q_{\ell, YY})^{-1}Q_{\ell, XY}Q_{\ell, YY}^{-1}(Q_{\ell}/ Q_{\ell, YY})^{-1}\end{bmatrix},
\left\Vert \Box_{k}^{\alpha_{1}}|~M_{1}\rightarrow M_{2}\right\Vert\gtrsim \frac{\Vert \Box_{k}^{\alpha_{1}}f_{k}^{\alpha_{1}}\Vert_{M_{2}}}{\Vert f_{k}^{\alpha_{1}}\Vert_{M_{1}}}\gtrsim2^{j\widetilde{A_{2}}(\mathbf{p},q,\alpha_{1},\alpha_{2})}.
\ln\left(\int_{\mathbb{S}^n}\tilde{K}e^{n(\psi-w_0\circ \pi)}dg_0\right) \geq \frac{1}{|\mathbb{S}^n|}\int_{\mathbb{S}^n}\ln \left(\tilde{K}e^{n(\psi-w_0\circ \pi)}\right)dg_0+\ln |\mathbb{S}^n| &\geq \frac{1}{|\mathbb{S}^n|}\int_{\mathbb{S}^n}\ln \left(\tilde{K}e^{-nw_0\circ \pi}\right)dg_0+n\bar{\psi}+\ln |\mathbb{S}^n|,
\sum_{i=1}^n \alpha_n^i\partial_t^{i+n+2}u_{n+1}+\partial_t^{n+2}u_{n+1}-\sum_{i=1}^{2n}\beta_n^i\Delta \partial_t^{i+1}u_{n+1}-\omega_{n+1}\kappa_n \Delta \partial_t u_{n+1}&\quad-(1-\omega_{n+1})\kappa_n \int_0^\infty g_{n+1}(s)\Delta \partial_t u_{n+1}(t-s)ds-\kappa_{n+1}\Delta u_{n+1}=0.
S_{-}( \eta ) = \left( \begin{matrix}{\cosh \eta}&{\sinh \eta}&{0}&{0}\\{\sinh \eta}&{\cosh \eta}&{0}&{0}\\{0}&{0}&{\cosh \eta}&{- \sinh \eta}\\{0}&{0}&{- \sinh \eta}&{\cosh \eta}\\ \end{matrix}\right) ,
\begin{aligned}e_u := u - u_h = (u - Q^x_\lambda u) + (Q^x_\lambda u - u_h) = \theta_u - \eta_u, e_w := w - w_h = (w - Q^x_{1 - \lambda}w) + (Q^x_{1 - \lambda}w - w_h) = \theta_w - \eta_w.\end{aligned}
R_3^{\frac12}= \mathcal{D}_t(u(t_{\frac12})-u^{\frac12})+\gamma\bigtriangleup (\sigma(t_{\frac12})-\sigma^{\frac12})-\bigtriangleup (u(t_{n-\theta})-u^{n-\theta})+(f(u(t_{\frac12}))-f^{\frac12}(u)) = O(\Delta t^2),
2\kappa^2 S_{\Phi}={{\kappa^2 N}\over{2\pi g_s}}\,{1\over{\sqrt{G_{\Omega\Omega}}}}\left({3 \over 4}{e^{3\Phi(r)/4}}\mu_6 +{1 \over 4}{e^{-\Phi(r)/4}\over V_{\rm E}(r)}\mu_2 \right)\,.
= \begin{pmatrix}-r_3 r^T 0 n N_0 r 0 n^T r_3\end{pmatrix}_x + \begin{pmatrix}k_0n_1-k_1r_1-k_2r_2 k_0r_3 k_0 n_0 0 -k_1r_3-k_2n_0 0 k_1r_2-k_2r_1+k_0n_2 \ast -k_2r_3+k_1n_0 \ast 0 \ast 0 \ast \ast \ast\end{pmatrix}
(\underbrace{(2+(2+ \dots + (2}_{k}+ 1/z^{2^{k+1}})^{\frac{1}{2}}\ldots )^{\frac{1}{2}})^{\frac{1}{2}})^{-1}= z^2 ((2 z^{2^2}+ (2 z^{2^3}+ \ldots (2 z^{2^{k +1}}+1)^{\frac{1}{2}}\ldots )^{\frac{1}{2}})^{\frac{1}{2}})^{-1}.
F_t(f)(v)=\sum_{\omega\in J(H_{\Delta_P,0}/tH_{\Delta_P,0}t^{-1})}\sum_{u\in J}\sum_{u'\in J(N_{L_P,0}/tN_{L_P,0}t^{-1}H_{\Delta_P,0})}u'\omega t\cdot f(t^{-1}\omega^{-1}u'^{-1}vut)= =\sum_{u\in J}\sum_{u'\in J(N_{L_P,0}/tN_{\Delta_P,0}t^{-1}H_{\Delta_P,0})}u'F_t(f(t^{-1}u'^{-1}vut))
|X_1^{-1}(t,x) - X_2^{-1}(t,x) | = |X_1^{-1}(t,x) -X_1^{-1}(t,y) | & \leq C| x-y |^{\theta(t)}= C| X_1(t,a) - X_2(t,a) |^{\theta(t)}.
\left[(t-\eta_n(s))^{\alpha}-(t-s)^{\alpha}\right]^2 \leq C\left[(t-\eta_n(s))^{2\alpha}+(t-s)^{2\alpha}\right] & \leq \begin{cases}C n^{-2\alpha}\alpha \geq 0 C(t-s)^{2\alpha}\alpha < 0 \end{cases}.
y_1=x_1, \, y_2=\frac{1}{\sqrt{\tau-1}}(x_1-x_2), \, y_3=\frac{1}{\sqrt{-\nu}}x_3; &[y_1,y_2]=0, \,[y_2,y_3]=\frac{-2w}{\sqrt{\nu(1-\tau)}}y_1 + \frac{w-1}{\sqrt{-\nu}}y_2, \,[y_3,y_1]=\frac{1+w}{\sqrt{-\nu}}y_1
r(u,1,x)_{2n}=e^{-\frac{h_+(0,\sqrt{u}x)}{u}}\chi(\sqrt{u}x)\Bigr(u^{-n}[b_n(0,\sqrt{u}x)]_{2n}+u^{-n+1}[b_{n-1}(0,\sqrt{u}x)]_{2n}+\cdots\Bigr).
det(A)=a_{i1}C_{i1}+a_{i2}C_{i2}+a_{i3}C_{i3}+\cdot\cdot\cdot+a_{in}C_{in}=\sum_{j=1}^{n}a_{ij}C_{ij}=\sum_{j=1}^{n}a_{ij}(-1)^{i+j}M_{ij}
\mathrm{CM}(X^{n})= (-1)^{n}\mu^{n}+(-1)^{n-1}n\mu^{n}+\sum_{k=2}^{n}\binom{n}{k}(-\mu)^{n-k}\mu^{k}&+\sum_{k=2}^{n}\sum_{i=2}^{k}\binom{n}{k}\binom{k}{i}(-1)^{n-k}\mu^{n-i}\sigma^{i}(i-1)\frac{c_{1}}{c_{(1)}^{\ast}}L_{2},~n\geq2,
\frac{\partial C}{\partial z^j_{\eta}}\frac{\partial^2\varphi^j}{\partial x^{\epsilon}\partial x^{\eta}}=\frac{f'(x^1)}{\sqrt{1+f'^2(x^1)}}\delta_{\epsilon 1}\left[ f(x^1)f''(x^1)+1+f'^2(x^1)\right],
w_2(\pi) =\left(\sum_{1\leq i<j\leq n}N_iN_jc_ic_j+\sum_{i=1}^{n}N_i\binom{c_i}{2}+\sum_{i=1}^{n}c_i^2\binom{N_i}{2}\right)\left(\sum_{t=1}^{n}v_t^2\right) &+\left(\sum_{i=1}^{n}c_i N_i''+2\sum\limits_{1 \leq i < j \leq n}N_iN_jc_ic_j\right)\cdot\left(\sum_{k<l}v_kv_l\right).
y^{k-1}=\sum_{i=1}^{k}\frac{\alpha_i^{k-1}}{q'(\alpha_i)}\prod_{j\neq i}(y-\alpha_j) =\sum_{i=1}^{k}\frac{\alpha_i^{k-1}}{q'(\alpha_i)}\frac{q(y)}{(y-\alpha_i)}
Z [ J ] = \int{\cal D}\phi \exp \{- \int d^{\nu}r [{\frac{1}{2}}\partial_{\tau}\phi_{r}\partial_{\tau}\phi_{r}+{\frac{1}{2}}\partial_{\vec{x}}\phi_{r}\partial_{\vec{x}}\phi_{r}+ V ( \phi_{r}) - J_{r}\phi_{r}] \}