harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E(u_0)=E(u(t)) \geq \frac{1}{2}X(t)-\frac{C_{N,p,s_1}}{p}\|u_0\|_2^{p-\frac{N(p-2)}{2s_1}}\|(-\Delta)^{\frac{s_1}{2}}u(t)\|_{2}^{\frac{N(p-2)}{2s_1}}& \geq X(t)\left(\frac{1}{2}-\frac{C_{N,p,s_1}}{p}\|u_0\|_2^{p-\frac{N(p-2)}{2s_1}}X(t)^{\frac{N(p-2)-4s_1}{4s_1}}\right).
\sum_{n=1}^{N}\frac{(-1)^{n-1}q^{\frac{n(n+1)}{2}}}{(1-cq^n) (q)_n (q)_{N-n}}\left(n + \sum_{k=1}^n \frac{q^k}{1- q^k}\right) =\frac{-c}{(1-c)^2(q)_N}\left( 1- \frac{(q)_N}{(cq)_N}\right) + \frac{1}{(1-c)(cq)_N}\sum_{k=1}^N \frac{(cq)_k q^k}{(q)_k (1-q^k)}.
P(\Pi) = \prod_{i=1}^d (\Pi - 1^{\otimes (i-1)}\otimes \pi \otimes 1^{\otimes (d-i)}) &= \sum_{r=0}^d (-1)^r \Pi^{d-r}\sum_{\substack{(i_1,\dots,i_d)\in \{0,1 \}^d |\{s | i_s = 1 \}| = r}}\pi^{i_1}\otimes \dots \otimes \pi^{i_d}&= \sum_{r=0}^d (-1)^r e_{\alpha_r}\Pi^{d-r},
{\frac{d}{d t}}\left( \begin{array}{l}{p}{q}\end{array}\right) ={\frac{1}{\hbar}}\left( \begin{array}{l l}{0}{-{\cal H}}{{\cal H}}{0}\end{array}\right) \left( \begin{array}{l}{{\frac{\delta H_{0}}{\delta p}}}{{\frac{\delta H_{0}}{\delta q}}}\end{array}\right)
\theta\Big(\sum_{i=0}^{+\infty}x_it^{i},\sum_{j=0}^{+\infty}y_jt^{j}\Big)\Big(\sum_{k=0}^{+\infty}V_kt^{k}\Big)=\sum_{s=0}^{+\infty}\sum_{i+j+k=s}\theta(x_i,y_j)v_kt^{s},\;\forall x_i,y_j\in L,\;v_k\in V.
\begin{array}{c}m(x, \omega)= \sqrt{2+ \left(x_1+ \frac{Ex_1^2}{\lambda}\right)^2 + x_2^2 + 4 \pi^2 \omega_1^2 + 4 \pi^2 \omega_2^2}\geq \sqrt{2+ \left(x_1+ \frac{Ex_1^2}{\lambda}\right)^2 + x_2^2 + \omega_1^2 + \omega_2^2}\end{array}
h(0)e^{\pm \gamma_1}=\pm e^{\pm \gamma_1}, h(0)e^{\pm \gamma_2}=\pm e^{\pm \gamma_2}, x^{\pm}(0)e^{\mp\gamma_1}=\mp e^{\pm\gamma_2}, x^{\pm}(0)e^{\mp\gamma_2}=\pm e^{\pm\gamma_1}.
\begin{aligned}(v)\ h(x,y)+h_e \phi(y) \geq \frac{\vartheta_0}{2}>0, &u(x,y)+u_e\phi(y)+u_b(1-\phi(y))>h(x,y)+h_e\phi(y), &\Vert \langle y\rangle^{l+1}\partial_y (u,h)\Vert_{L^\infty}\leq \sigma_0, &\Vert \langle y\rangle^{l+1}\partial_y^2 (u,h)\Vert_{L^\infty}\leq \vartheta_0^{-1}.\end{aligned}
\begin{cases}\partial_{\mathcal{A}}\circ \partial_{\mathcal{A}}(x_i) = 0, \forall i\in\{1,2,\cdots,n\}\partial_{\mathcal{A}}(x_jx_k-x_kx_j) = 0, \forall 1\le j<k\le n.\end{cases}
S^{( 2 )}( p_{2}, p_{3}) S^{( 2 )}( p_{3}, p_{1}) S^{( 2 )}( p_{1}, p_{2}) = S^{( 2 )}( p_{1}, p_{2}) S^{( 2 )}( p_{1}, p_{3}) S^{( 2 )}( p_{2}, p_{3}) ,
2f_{\lambda}(\lambda)f_{\lambda\lambda}(\lambda)-\nabla f_{\lambda}\cdot \nabla f(\lambda) &=-f_{\lambda}(\lambda)\Delta f(\lambda)-f(\lambda)\Delta f_{\lambda}(\lambda)+2\sum_{\l-k>1}\dfrac{\partial}{\partial \lambda}\left(\det\left((\lambda I_N-H)_{k|k}\right)\det\left((\lambda I_N-H)_{\l|\l}\right)\right).
{\mathbb E}|\eta|\lesssim \biggl(\frac{\mu_r m_r^{5/2}}{\sqrt{n}}+ \frac{\mu_r m_r^3}{n}\biggr){\mathbb E}\|C_r X\|^2 = \biggl(\frac{\mu_r m_r^{5/2}}{\sqrt{n}}+ \frac{\mu_r m_r^3}{n}\biggr){\rm tr}(C_r \Sigma C_r) &\lesssim \frac{\|\Sigma\|_{\infty}^2}{\bar g_r^2}\biggl(\frac{m_r^{5/2}}{\sqrt{n}}\vee \frac{m_r^3}{n}\biggr){\bf r}(\Sigma).
\begin{aligned}i=i_1, j=i_2, t_1=T\gamma_g(2\tau+1), t_2=T\gamma_g(2\tau+1)+2\tau+1, & \alpha_d= \frac{h\psi(X[0] + \rho) (1-hn^{\infty}\kappa)^{3\tau+1}}{1+ n^{\infty}\kappa}\mbox{and}\underline{\psi}:=\psi(X[0] + \rho)\end{aligned}
\tilde{\omega}\big(\tilde{X},P^{\sharp}\mathrm{d}f\big) = \tau\big(P^{\sharp}\mathrm{d}Q(X),P^{\sharp}\mathrm{d}f\big)-\mathrm{d}Q\big(\tilde{X},P^{\sharp}\mathrm{d}f\big) =\{Q(X),f\}_{P}+{\mathcal L}_{P^{\sharp}\mathrm{d}f}Q(X)=0.
H_{1,1}=\int_0^{2\pi}\int_0^{\frac{1}{2}}\frac{r dr d\theta}{|\gamma+r\sin(\theta+\varphi)|^\tau r^q|re^{i\theta}-1|^q}, H_{1,2}=\int_0^{2\pi}\int_{\frac{1}{2}}^\gamma\frac{r dr d\theta}{|\gamma+r\sin(\theta+\varphi)|^\tau r^q|re^{i\theta}-1|^q}.
\prod\limits_{j=0}^{\ell-1}{(a\delta_jX-\gamma_jY)}= \prod\limits_{j=0}^{\ell-1}{(a\delta_jX-a\gamma_jY)}= a^{\ell}\prod\limits_{j=0}^{\ell-1}{(\delta_jX-\gamma_jY)}= \prod\limits_{j=0}^{\ell-1}{(\delta_jX-\gamma_jY)}\cdot
E_{P_{\alpha}}\left[ b_{a}^{2}\left( \mathbf{O};P_{\alpha}\right) \right]=E_{P_{\alpha}}\left[ \left\{O_{1}+O_{2}+\alpha O_{1}O_{2}\right\}^{2}\right] =E_{P_{\alpha}}\left[ O_{1}^{2}+O_{2}^{2}+\alpha^{2}(O_{1}O_{2})^{2}\right] =2+\alpha^{2}.
\int_{| k | < \frac{\Lambda}{s}}\frac{d^{D}k_{1}}{( 2 \pi )^{D}}\int_{| k | < \frac{\Lambda}{s}}\frac{d^{D}k_{2}}{( 2 \pi )^{D}}\int_{\frac{\Lambda}{s}< | k | < \Lambda}\frac{d^{D}k_{3}}{( 2 \pi )^{D}}\int_{\frac{\Lambda}{s}< | k | < \Lambda}\frac{d^{D}k_{4}}{( 2 \pi )^{D}}\phi_{4 f}( k_{4}) \phi_{3 f}( k_{3}) \phi_{2 s}( k_{2}) \phi_{1 s}( k_{1}) .
\iota : \Omega_{k}^\xi \hookrightarrow \Omega_{k(k+1)k}, \xi^i \mapsto \xi_{k+1}^i\sum_{\ell=0}^k (-1)^\ell x_{\ell,k}\otimes_{k+1}\xi_{k+1}^{k-\ell}, \eta : \Omega_{k}\hookrightarrow \Omega_{k(k+1)k}, 1 \mapsto \sum_{\ell=0}^k (-1)^\ell x_{\ell,k}\otimes_{k+1}\xi_{k+1}^{k-\ell},
\frac{1}{2}\vert \left({}_q\langle J_1,\gamma_1; J_2,\gamma_2\vert \left[\hat X_{1,q},\hat X_{2,q}\right] \vert J_1,\gamma_1; J_2,\gamma_2\rangle_q\right)\vert = \frac{\hbar^2}{4(\lambda_1 +\lambda_2)^2}\vert K_1\left(1-(1-q^2)J_1\right)- K_2\left(1 -(1-q^2)J_2\right)\vert;
P_{\hat p,p}(v) = v + \langle v,w \rangle \left( - \sqrt{1-\langle \hat p, p \rangle^2}\hat p + (\langle \hat p, p \rangle -1)w \right), w = \frac{p - \langle \hat p, p \rangle \hat p}{|p - \langle \hat p, p \rangle \hat p|}.
\Theta(x,v,m)=\begin{pmatrix}\frac{\partial^2}{\partial m^2}\left( m \tilde f\left(x,v,m\right) \right) m\frac{\partial \nabla_v^T \tilde f}{\partial m}\left(x,v,m\right) m\frac{\partial \nabla_v \tilde f}{\partial m}\left(x,v,m\right) m D^2_{vv}\tilde f\left(x,v,m\right) \end{pmatrix}.
m(t; \lambda) = \frac{4\pi}{3}\cdot \begin{cases*}(\frac{1+t}{1+\lambda})^3 if $0 < \lambda < \frac{1}{t}$, (\frac{1-t}{\lambda-1})^3 if $\frac{1}{t}\leq \lambda < 2$, 0 otherwise. \end{cases*}
\nabla^{k}I_{n,q}^{r}=\frac{(-\alpha)_{k+1}}{\Gamma(1-\alpha)}\int_{[0, 1]^{k+1}}(\sum_{i=1}^{k}\xi_{i}+n-k+1-s)^{-\alpha-k-1}\binom{s-q+r-1}{r}\mathrm{d}^{k}\mathbf{\xi}\mathrm{d}s
\int_{\{\Psi<-t_1\}}|\tilde F_j|^2 \le 2\int_{\{\Psi<-t_1\}}|(1-b_{t_{0},B_{j}}(\Psi))f_{t_{0}}|^{2}+2\int_{\{\Psi<-t_1\}}|\tilde{F}_{j}-(1-b_{t_{0},B_{j}}(\Psi))f_{t_{0}}|^{2}\le 2\int_{\{\Psi<-t_0\}}|f_{t_0}|^2+2(e^{t_0-t_1+B_j}-1)\frac{G(t_{0})-G(t_{0}+B_{j})}{B_{j}}.
{\overline{{{Q}}}}_{a}= \Big ( \frac{{\mathrm{\a e}}R^{- 1 / 2}P}{\sqrt{6}}+ i \frac{\sqrt{k}\sqrt{6 R}}{{\mathrm{\a e}}}\Big ){\overline{{{\lambda}}}_{a}}- 3 i{\mathrm{\a e}}( 6 R )^{- 3 / 2}({\overline{{{\lambda}}}}{\overline{{{\lambda}}}}) \lambda_{a},
(A^{(4)}(t,x) \xi) \cdot \xi \geq \frac12 \| DP^{T}(t,y)\xi \|^2 - \frac{m}{d^4}\xi_1^2 = \frac{1}{2 d^4}\left[ ( p^2 + q^2 - 2 m ) \xi_1^2 + 2 q d^2 \xi_1\xi_2 + d^4 \xi_2^2 \right] \geq \frac{1}{2}\left[ p^2 - \left ( \frac{1}{\varepsilon}-1 \right) q^2 - 2 |m| \right] \xi_1^2 + \frac{1}{2}(1-\varepsilon) \xi_2^2 \,,
A_{\mathbf{k}}(z)=2^{-n(z-1)}\bigg(\prod_{j=1}^n \prod_{i=1}^j\frac{1}{z+k-1-j+2i}\bigg)\bigg(\prod_{j=1}^n \prod_{i=0}^{\frac{k-k_j}2-1}\frac{z-(k-1-j-2i)}{z+(k-1-j-2i)}\bigg).
\lim\limits_{n\to \infty}\frac{n^{3}}{8}\P\left(V_n=2r_{1,n},\, \tau_n=2r_n, \, L_n=k_n,\, X^{\left(n\right)}_1=j_n|X_{0}^{\left(n\right)}=0\right) &=\frac{2p^2ql^2z}{[2\pi x(t-x)(T-t)]^{3/2}}e^{-\frac{z^2}{2(T-t)}-\frac{l^2}{2}\left(\frac{p^2}{x}+\frac{q^2}{t-x}\right)}.
\lim_{t\to\infty}\left|\frac{|\zeta_{n}^{t}(w_1)|_w}{|\zeta_{n}^{t}(w_1)|}-\frac{|\zeta^{t}(x_1)|_w}{|\zeta^{t}(x_1)|}\right|\le \lim_{t\to\infty}\left|\frac{n}{|\zeta^{t}(x_1)|}\right|= 0
\begin{aligned}T_a(\rho,\theta)=\left(\bigcap_{k=1}^{k_a}\{x\ |\ |z_{ak}|^2>\rho|x|^2\}\right)\cap\{x\ |\ |x_a|^2>(1-\theta)|x|^2\}, &{\tilde T}_a(\rho,\theta)=\left(\bigcap_{k=1}^{k_a}\{x\ |\ |z_{ak}|^2>\rho\}\right)\cap\{x\ |\ |x_a|^2>1-\theta\}. \end{aligned}
\widehat{a}_6 = \frac{2}{\sqrt{6}}\sqrt{\frac{\alpha_6}{1-\alpha_6}}\left(-\widehat{D}_6(e^{-\pi/\sqrt{6}})\right)+\frac{1}{2(1-\alpha_6)}&=-\left(\frac{1}{4}+\frac{1}{6}\sqrt{6}-\frac{1}{6}\sqrt{3}\right)+\frac{1}{2(1-\alpha_6)}&= \frac{2}{3}\sqrt{3}-\frac{5}{12}\sqrt{6}.
P(\max_{0\le u\le t^a}B(H(u))-B(H(t^a))>\max_{t^a\le u \le t}\tilde{B}(H(u))-\tilde{B}(H(t^a))) = P(\max_{0\le u\le t^a}B(H(u))>\max_{t^a\le u \le t}\tilde{B}(H(u))-\tilde{B}(H(t^a))).
H_{ij}(s,y) =\int_{s}^{T}\overline{\phi}_i(t) \left(\int_{0}^{\pi}G_{t-s}(x,y) \phi_j(x) \, \textrm{d}x \right) \,\textrm{d}t =\int_{s}^{T}\overline{\phi}_i(t) \phi_j(y) e^{-j^2(t-s)}\, \textrm{d}t.
\zeta(X)=\left|\begin{array}{l}\zeta_1=(x-u_1)^2+(y-v_1)^2+(z-w_1)^2 \zeta_2=(x-u_2)^2+(y-v_2)^2+(z-w_2)^2 \zeta_3=(x-u_3)^2+(y-v_3)^2+(z-w_3)^2 \zeta_4=(x-u_4)^2+(y-v_4)^2+(z-w_4)^2 \end{array}\right. \mbox{avec}X_1=x,X_2=y,X_3=z \mbox{les inconnues}
G^{1}(s):=r\left(s\right), G^{5}(s;x,y) :=\sum_i\sum_{j\leq i}d_{ij}\left(s \right)f_i(x)f_j(y), &G^{3}(s):=p\left(s\right), G^{6}(s;x) :=\sum_i\mu_i\left(s\right)f_i(x)=\mu\left(s ;x\right).
D_n^{(2)}=\begin{cases}8(n+3)(2n+1)(2n+5)(4n+3)(4n+5)f_0^3f_6 , 8(n+4)(2n+1)(4n+3)(4n+5)f_0^3 , 3(n+3)(3n+2)(3n+4)f_1^2 .\end{cases}
\begin{cases}e^{i \theta}y^{\prime}+ \mathcal{M}_{n+p}y=(0, \ldots, b_n v_n, b_{n+1}v_{n+1}, \ldots, b_{n+p}v_{n+p})^t \,, y(0)=(y_1^0, \ldots, y_{n+p}^0) \,,\end{cases}
\int_{0}^{\infty}d t_{1 1}\frac{1}{t_{1 1}^{~ ~ 2}}\exp \left( - \frac{A}{t_{1 1}}\right) = \frac{1}{A}, \int_{0}^{\infty}d t_{2 2}\exp \left( - B t_{2 2}\right) = \frac{1}{B}~ ,
So\frac{\partial f}{\partial v}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial v}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial v}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial v}+\frac{\partial f}{\partial y}\cdot0=\frac{\partial f}{\partial x}\frac{\partial x}{\partial v}.
\sigma_rwt(p_s).(p\widetilde{\mathbb{S}}_{\lambda^i}|_{z=0})={\bf{b}}wt(p_s).(p\widetilde{\mathbb{S}}_{\lambda^i}|_{z=0})+\frac{{\bf{b}}(p_s)}{b_{0}}(p\widetilde{\mathbb{S}}_{\lambda^1}|_{z=0})(p\widetilde{\mathbb{S}}_{\lambda^i}|_{z=0}).
W=\int_{\gamma}F(r)\cdot dr=\int_{\gamma}(kq\sum_{i=1}^{n}\frac{Q_{i}(r-p_{i})}{|r-p_{i}|^{3}})\cdot dr=kq\sum_{i=1}^{n}(Q_{i}\int_{\gamma}\frac{r-p_{i}}{|r-p_{i}|^{3}}\cdot dr)
I_1 = \sup_{(b,c,F) \in \Theta}\bigg\{\int_{|z|\leq 1}|\psi(z)-\psi(0)-D_x \psi(0) z| \, F(dz)\bigg\}, I_2 = \sup_{(b,c, F) \in \Theta}\bigg\{\int_{|z|>1}|\psi(z)-\psi(0)| \, F(dz)\bigg\}.
|u(t,x;0,u^*)-u^*(x)| \le |u(t,x;0,u^*)-u(t,x;0,u_{n_k})| +|u(t,x;0,u_{n_k})-u^*(x)|(\,\, k\gg 1) &= |u(t,x;0,u^*)-u(t,x;0,u_{n_k})| +|u(\tau_{n_k},x;0,u_{n_k})-u^*(x)| & \le |u(t,x;0,u^*)-u(t,x;0,u_{n_k})| +|u(\tau_{n_k},x;0,u_{n_k})- u(\tau_{n_k},x;0,u^*)| &+|u(\tau_{n_k},x;0,u^*)-u^*(x)| &\le 3\epsilon\forall\, x\in\bar\Omega.
\int_{-1}^1 e^{ru}u(1-u^2)^{\frac{p-3}{2}}du = 2^\frac{p-2}{2}\Gamma(\frac{p-1}{2})\sqrt{\pi}\frac{d}{dr}\Big(r^{-\frac{p-2}{2}}I_\frac{p-2}{2}(r)\Big) &=\Gamma(\frac{p-1}{2})\sqrt{\pi}\big(\frac{2}{r}\big)^\frac{p-2}{2}I_{\frac{p}{2}}(r).
[t(f_1', \ldots, f_k')](x_1, \ldots, x_m) = t(f_1'(x_1, \ldots, x_m),\ldots, f_k'(x_1, \ldots, x_m)) = t(f_1(x_1/\alpha, \ldots, x_m/\alpha), \ldots, f_k(x_1/\alpha, \ldots, x_m/\alpha)) = [t(f_1,\ldots,f_k)]'(x_1, \ldots, x_m) \in \varphi(C).
\int_{t}^{T}K_{ij}(s,t)\cdot\widehat{w}_{i}(s)ds-\sum_{v=u+1}^{N}\left (t_{v}-t_{v-1}\right )K_{ij}^{(v,u)}w_{i}^{(v)}& \quad\leq\bar{\epsilon}w_{i}^{(u)}\cdot\left [\eta +\sum_{v=u+1}^{N}\left (t_{v}-t_{v-1}\right )\right ]\leq\bar{\epsilon}w_{i}^{(u)}\cdot\left (\eta +p\cdot\parallel{\cal P}\parallel \right )
J (x^1 \frac{\partial}{\partial x^2}-x^2\frac{\partial}{\partial x^1}) &=-\frac{1}{24\pi}\int_{S^2}\tilde{x}^3* d\left( dA \left[A{\tilde{\Delta}}A-|{\tilde{\nabla}}A|^2\right]\right) \, d\mu_{S^2}= \frac{2}{3\cdot7\cdot 11}.
\mathrm{d e t}( K_{\Phi}) = - m^{3}( y_{1}y_{2}y_{3})^{3 / 2}\Big ( x_{1}x_{2}x_{3}+{\frac{1}{4}}m n^{2}\Big )^{3}= ( y_{1}y_{2}y_{3})^{9 / 2},
\frac{\partial^{4}w}{\partial x^{2}\partial y^{2}}=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}(\frac{m\pi}{a})^{2}(\frac{n\pi}{b})^{2}w_{mn}sin\frac{m\pi x}{a}sin\frac{n\pi y}{b}
\frac{1}{k^2}(|E(G)| - |E(G - A)|) \, \leq\, \frac{1}{6}\left((3\alpha-2)^2+4(3\alpha-2)-2\right) + \alpha (3-2\alpha) - \frac{1}{6}(\alpha-1)\, &=\,\frac{1}{6}(-3 \alpha^2+17 \alpha-5).
t_{\mu,\nu}(x) =\frac{x^{\mu+1}}{(\mu-\nu+1)(\mu+\nu+1)}{}_1F_2\bigg(1;\frac{\mu-\nu+3}{2},\frac{\mu+\nu+3}{2};\frac{x^2}{4}\bigg) &=2^{\mu-1}\Gamma\bigg(\frac{\mu-\nu+1}{2}\bigg)\Gamma\bigg(\frac{\mu+\nu+1}{2}\bigg)\sum_{k=0}^\infty\frac{(\frac{1}{2}x)^{\mu+2k+1}}{\Gamma\big(k+\frac{\mu-\nu+3}{2}\big)\Gamma\big(k+\frac{\mu+\nu+3}{2}\big)}.
\begin{cases}c_1=w_1+w_3\kappa, c_2=\big(w_1(p-2+s)+w_3(p-4+s)\kappa\big)\theta, c_3=w_2+w_4\kappa, c_4=\big(w_2(p-2+s-\gamma)+w_4(p-4+s-\gamma)\kappa\big)\theta.\end{cases}
\Big(\int_B|a(y)|dy\Big)^p \leq\Big(\int_B|a(y)|^{q_0}\omega(y)dy\Big)^{p/{q_0}}\Big(\int_B\omega(y)^{-{q_0}'/{q_0}}dy\Big)^{p/{q_0}'}&\lesssim\omega(B)^{p/{q_0}-1}\omega(B)^{-p/{q_0}}|B|^{p}=\omega(B)^{-1}|B|^{p}.
\mathbb{P}_{\mathbb{C}}^n\!\setminus\!\{[1\!:\! z_2\!:\!\cdots\!:\! z_{n+1}]: z_2,\!\cdots\!,\!z_{n+1}\!\in\!\mathbb{C}\}\!=\!\{[0\!:\! z_2\!:\!\cdots\!:\! z_{n+1}]: z_2,\!\cdots\!,\! z_{n+1}\!\in\!\mathbb{C}\}.
D^{\mu}\varphi_{a}= \partial^{\mu}\varphi_{a}+ g W_{a b}A^{b \mu}\varphi_{a}, \bar{F}_{\mu \nu}^{a}= \partial_{\left[ \mu \right.}A_{\left. \nu \right]}^{a}+ g \frac{\delta W_{b c}}{\delta \varphi_{a}}A_{\mu}^{b}A_{\nu}^{c}.
E_{q}^{h}+(E_{q}^{h}-\sum a^{hf}\pi_{P}^{f})\frac{dp}{dt}=\frac{dI^{h}}{dt}+\{\sum a^{hf}\pi_{z}^{f}\frac{dz^{f}}{dt}-E_{z}^{h}\frac{dz^{h}}{dt}\}
\mathcal{R}_{1,P}(R_3')= \left\{\begin{aligned}(R_1,R_2):R_1,R_2\geq 0,\R_1\leq I(U_1;Y_1) \triangleq r_i, R_1+R_2 \leq I(U_1;Y_1)+ I(X_1;Y_2|U_1,U_2)+ \min\{0,\I(U_2;Y_2|U_1)-I(U_2;Y_3)\}\triangleq r_j \end{aligned}\right\}.
(-1)^n\frac{c_{n,\rho,q}^{(k)}(z)}{n!}=\sum_{m=1}^n\binom{n-1}{m-1}\frac{\widehat c_{m,\rho,q}^{(k)}(z)}{m!}\,, (-1)^n\frac{\widehat c_{n,\rho,q}^{(k)}(z)}{n!}=\sum_{m=1}^n\binom{n-1}{m-1}\frac{c_{m,\rho,q}^{(k)}(z)}{m!}\,.
\lambda_{1,\epsilon}(0) =\frac{1}{3}a^2 \alpha^2 \epsilon^2+O(\epsilon^4), &\lambda_{2,\epsilon}(0) =\frac{1}{20}a^2 \alpha^4 \epsilon^4+O(\epsilon^6), &\lim_{\epsilon \rightarrow 0}\frac{\lambda_{2,\epsilon}(0)}{\lambda^2_{1,\epsilon}(0)}={\frac{9}{20 a^2}}.
\prod_{j=0}^{k-1}\left(1 + \frac{1}{x_{(j)}}\right) < \prod_{j=m_p}^{k-1}\left(1 + \frac{1}{x_{(j)}}\right) \prod_{h=1}^{p-2}\left(1 + \frac{1}{X_q(2^h)}\right)^{2^h-1}< \left(1 + \frac{1}{X_q(2^{p-1})}\right)^{k-m_p}\prod_{h=1}^{p-2}\left(1 + \frac{1}{X_q(2^h)}\right)^{2^h-1}.
\widehat{D}^{(\ell)}=\mathrm{Diag}\bigg(\underbrace{a_+\xi_i^{(\ell)}+ a_-(1-\xi_{i}^{(\ell)})}_{=:d_{i}^{(\ell)}},\, i=1,\ldots, n_\ell\bigg),\xi_i^{(\ell)}\sim \mathrm{Bernoulli}(1/2)\,\, iid.
e^{(\pm i\pi\sum_{k=1}^{j-1}f_{k}^{\dagger}f_{k})}=\prod_{k=1}^{j-1}e^{\pm i\pi f_{k}^{\dagger}f_{k}}=\prod_{k=1}^{j-1}e^{\pm i\pi\frac{\sigma_{k}^{z}+I}{2}}=\prod_{k=1}^{j-1}(-\sigma_{k}^{z}).
K^{\alpha_1, \alpha_2}_{w_6}(y; \mu) = \int_{-i\infty}^{i\infty}\int_{-i\infty}^{i\infty}|4\pi^2 y_1|^{-s_1}|4\pi^2 y_2|^{-s_2}G(s, \mu) S^{\alpha_1, \alpha_2}(s, \mu)\frac{ds_1\, ds_2}{(2\pi i)^2},
<\Delta R_{i}\cdot\Delta R_{j}>=\frac{3k_{B}T}{\gamma}[U\Lambda^{-1}U^{T}]_{ij}=\frac{3k_{B}T}{\gamma}\sum_{k=1}^{N-1}\lambda_{k}^{-1}[u_{k}u_{k}^{T}]_{ij}
H_{i}(u)=h_{i,0}u^{-1}+\big(h_{i,1}-\frac{1}{2}(h_{i,0})^2\big)u^{-2}+\big(h_{i,2}-h_{i,0}h_{i,1}+\frac{1}{3}(h_{i,0})^3\big)u^{-3}+\ldots.
\frac{\Gamma_q(\lambda_N - z)}{\Gamma_q(\lambda_1 + \theta N - z)}= (1 - q)^{\lambda_1 + \theta N - \lambda_N}\frac{\big(q^{\lambda_1 + \theta N - z}; q\big)_{\infty}}{\big(q^{\lambda_N - z}; q\big)_{\infty}}.
(1+2 \cos^2 \phi ) \, p_{0,\ell}( \phi)-5 \cos \phi \, p_{0,\ell+1}( \phi)+2 p_{0,\ell+2}( \phi) &= \sin \phi \, \sin \psi_{0,\ell-1}\sum_{k=0}^\infty (-1)^k h_0(2k) s_{0,2k}(\ell,\phi) +\sin \phi \, \cos \psi_{0,\ell-1}\sum_{k=0}^\infty (-1)^k h_0(2k+1) s_{0,2k+1}(\ell,\phi) &\;\;+ \phi^{-1/2}O(\ell^{-1-1/2}),
\int_{-\infty}^{\infty}\int_0^1 \left| \alpha \left(\frac{\partial u}{\partial s}, X^{\omega_s}_H \right)\right| dtds = \int_{-s_u}^{s_u}\int_0^1 \left|\alpha \left(\frac{\partial u}{\partial s}, X^{\omega_s}_H \right)\right|dtds.
\tilde{g}_x(x,y,z) = \frac{-g_{xz}(x,0,0)}{g_z(x,0,0)}\tilde{g}(x,y,z) \quad\quad+ \frac{g_z(0,0,0)}{g_z(x,0,0)}[g_x(x,y,g^*(0,y,h-z))-g_x(x,0,0)],
\lim_{a\to1}\bigg\{\frac{1}{(1-q)^2}-\frac{n(1-a)a^{(n-1)/2}}{(1-q/a)(1-aq)(1-a^n)}\bigg\}\frac{(1-aq^n)(a-q^n)}{(1-a)^2}begin{align*}5pt] =\lim_{a\to1}\frac{\{(1-q/a)(1-aq)(1-a^n)-n(1-a)a^{(n-1)/2}(1-q)^2\}(1-aq^n)(a-q^n)}{(1-q)^2(1-q/a)(1-aq)(1-a^n)(1-a)^2}begin{align*}5pt] =[n]^2\frac{n^2(1-q)^2-(1+22q+q^2)}{24(1-q)^2}.
\frac{\partial}{\partial t}\mathring{h}_i^j= -\nabla^j\nabla_iH^{-\alpha}-H^{-\alpha}\mathring{h}_i^k\mathring{h}_k^j-H^{1-\alpha}\mathring{h}_i^j+H^{-\alpha}(1-\frac{H^2}4)\delta_i^j-\frac 12\partial_tH\delta_i^j.
\left\langle{{E^{\prime}, l^{\prime}, m^{\prime}; i n}}\mathrel{\left|{\vphantom{{E^{\prime}, l^{\prime}, m^{\prime}; i n}{E , l , m}}}\right. \kern - \nulldelimiterspace}{{E , l , m ; i n}}\right\rangle ={\frac{V}{( 2 \pi )^{3}}}\delta_{l l^{\prime}}\delta_{m m^{\prime}}\delta ( E - E^{\prime})
H_{r}=A_{r}+qD_{r}=\sum_{|w|=r}^{\circlearrowleft}(\delta_{j_{1}}-\hat{t}_{j_{1}})\cdots (\delta_{j_{r}}-\hat{t}_{j_{r}})=\sum_{|w|=r}^{\circlearrowleft}t_{w}\rho_{t}(\pi_{w})~
S^{\mathrm{g r a v .}}= -{\frac{T_{K K 1 1}}{2}}\int d^{7}\hat{\xi}\sqrt{| \hat{\gamma}|}\ \left[ \hat{\hat{k}}^{4 / 7}\hat{\gamma}^{\hat{\imath}\hat{\jmath}}\hat{D}_{\hat{\imath}}\hat{\hat{X}}^{\hat{\hat{\mu}}}\hat{D}_{\hat{\jmath}}\hat{\hat{X}}^{\hat{\hat{\nu}}}\hat{\hat{g}}_{\hat{\hat{\mu}}\hat{\hat{\nu}}}- 5 \right] \, ,
C_0^{L^{\star}}=\begin{cases}\,\,\int_{-\infty}^{-\beta^{\star}\eta^{\star}}{p(\gamma)d\gamma}, \eta^{\star}> 0, \,\,\int_{-\beta^{\star}\eta^{\star}}^{\infty}{p(\gamma)d\gamma}, \eta^{\star}< 0;\end{cases}
\begin{pmatrix}(n-2)(k-1)+1 (2-2k) F_{1,1}^{(1)}\dots (2-2k) F_{1,n-1}^{(1)}\end{pmatrix}=\begin{pmatrix}(n-2)(k-1)+1 0 \dots 0 8(k-1) \end{pmatrix}
(\Box^h\Phi)_{A_p\overline{B}_q}= - h^{\bar{\varepsilon}\gamma}\nabla_{\mathcal{X}_\gamma}\circ\nabla_{\mathcal{X}_{\bar{\varepsilon}}}(\phi_{A_p\overline{B}_q}) + \sum_i h^{\bar{\varepsilon}\gamma}[\nabla_{\mathcal{X}_\gamma},\nabla_{\mathcal{X}_{\bar{\beta}_i}}]\phi_{A_p\bar{\varepsilon}\bar{\beta}_1\dots\hat{\bar{\beta}}_i\dots\bar{\beta}_q}.
f_{k,\ell}- f_{k-1,\ell+1}- \varrho ( f_{k,\ell-1}- f_{k-1,\ell}) = \beta \eta_{k,\ell}+ \frac{\beta^2}{2}\sum_{i=0}^{2\ell}\varrho^i - \beta \sum_{j=\ell}^{J}b_j \varrho^{j-\ell},
x_v =0 \forall v\in V,\,fs(v)\neq \{3,3,4,4\},\{3,4,4,4\}x_v+x_u \le 1 \forall v\in V,\, u\in fn(v) x_v \geq 0 \forall v\in V x_v \in\{0,1\}\forall v\in V
(\partial_{|x|}u)(t,r) = -\frac{\lambda''(s)}{|x|}\int_{0}^{t-s}\frac{\rho d\rho}{\sqrt{(t-s)^{2}-\rho^{2}}}\left(1+\frac{|x|^{2}-1-\rho^{2}}{\sqrt{(1+|x|^{2}+\rho^{2})^{2}-4 |x|^{2}\rho^{2}}}\right)
\partial_{t}\left(-16 \int_{t}^{\infty}\lambda''(s) \left(K_{3}(s-t,\lambda(t))-K_{3,0}(s-t,\lambda(t))\right)ds\right) &= -16 \int_{t}^{\infty}\lambda'''(s) \left(K_{3}(s-t,\lambda(t))-K_{3,0}(s-t,\lambda(t))\right) ds &-16 \int_{t}^{\infty}\lambda''(s) \left(\partial_{2}(K_{3}-K_{3,0})(s-t,\lambda(t))\right) \lambda'(t) ds
\frac{f_n(2k-2)}{(2k-2)_kn^{k-2}}=\frac1{(2k-2)_k}\left(\frac14\right)^{\binom{2k-1}2}d^{\binom{2k-2}2}n^k > \frac{d^{2k^2}}{2^{5k^2}}n^k
z_1 - \lfloor \sqrt{n}\rfloor < \sum_{j = 2}^N \frac{n^{\frac{1}{2^j}}}{2^{j-1}}+ \frac{1}{2^{N-1}}+ 2 < \frac{1}{2}n^\frac{1}{4}+ (N - 2)\cdot \frac{1}{4}n^\frac{1}{8}+ 2.25 < \frac{1}{2}n^\frac{1}{4}+ \log_2\log_5(n)\cdot \frac{1}{4}n^\frac{1}{8}+ 2.25 < n^\frac{1}{4}.
S_{TM}^4 = \int d^4x\Bigl[-{1 \over 4}F_{\mu\nu}F^{\mu\nu}-{1 \over 12\mu^2}H_{\mu\nu\lambda}H^{\mu\nu\lambda}-{1 \over 4}\varepsilon^{\mu\nu\lambda\rho}B_{\mu\nu}F_{\lambda\rho}\Bigl]\;,
\begin{array}{rcl}\frac{1}{2}\underset{j=1}{\overset{k}{\sum}}\underset{i=1}{\overset{m+1}{\sum}}\|y_{i}^{(j)}-y_{i}^{(j-1)}\|^{2}\leq h^{0}(\tilde{y}^{(0)})-h^{k}(\tilde{y}^{(k)}) \leq h^{0}(\tilde{y}^{(0)})-\bar{h}(\tilde{y}^{(k)}) \leq h^{0}(\tilde{y}^{(0)})-\min_{y}\bar{h}(y).\end{array}
r\sum_{\tau=t+h-1}^{n}Z_{N,J}^{\tau}+n Z_{I,J}^{t+h-2}+\sum_{\tau=t}^{t+h-2}n^{t+h-\tau-1}Z_{i_{\tau},J}^{\tau}\leq 2r+\sum_{\tau=1}^{h-2}n^{\tau}, \forall \t\in [n-h], \J\subseteq V, \l\in[n-1], \I\subseteq N, \r\in [n-1], \i_t,\dotsc,i_{t+h-2}\in N.
P(x) = \int_0^1 r^x G(1-r) \,dr = \int_0^1 \exp\Big( x \log r - m(1-r)\Big) dr \geq \int_0^1 \exp\Big( 2x(r-1) - m(1-r) \Big) dr = \int_{0}^1 \exp\Big(-2xt - m(t)\Big) dt.
a_{i}( x ) ={\frac{1}{\sqrt{2 \pi L_{1}}}}\sum_{n_{1}}\int d p_{2}\,{\frac{1}{p^{0}}}[ ({\frac{p_{i}}{\mu}}g ( \vec{p}) + i{\frac{\epsilon_{i j}p_{j}}{2}}h ( \vec{p}) ) e^{- i p \cdot x}+ c . c ]_{p^{0}= | \vec{p}|}\, ,
\left( \sum_{i_{1},...,i_{m}=1}^{N}\left\vert y_{i_{1}...i_{m}}\right\vert^{2}\right)^{\frac{1}{2}}\leq K_{m,r}\left({\displaystyle\int\limits_{I^{m}}}\left\vert \sum_{i_{1},...,i_{m}=1}^{N}r_{i_{1}}\left( t_{1}\right)...r_{i_{m}}\left( t_{m}\right) y_{i_{1}...i_{m}}\right\vert^{r}dt_{1}...dt_{m}\right)^{\frac{1}{r}}
0=\sum_{n \leq x}\frac{\Psi (n)\Lambda (n)}{n^s}=\sum_{n \leq x}\frac{\Lambda (n)}{n^s}\left(\frac{\varphi (p-1)}{p-1}\sum_{d \,| \,p-1}\frac{\mu (d)}{\varphi (d)}\sum_{(\chi ) = d}\chi (n)\right) ,
\partial_{t}\overline u - \Delta \overline u + \nabla\cdot(\overline u \nabla p_1) + \nabla\cdot(u_{2}\nabla\overline p) = -c\overline u + N \overline w\,, begin{align*}.3cm]\partial_{t}\overline w - \Delta \overline w + \nabla\cdot(\overline w \nabla v) = c\overline u - N\overline w\,, begin{align*}.3cm]-\Delta \overline p = \overline w - \delta\overline p\,.
\|g_{n,1}(z)\| =\|g_{n,p_j}\left(G_j(z)\right)\| \leq D^{n-p_j}D^{\frac{2^{j-1}}{k-x}}+ \frac{Z}{1-D}D^{-2^{j+1}\frac{k-2+\epsilon}{k-x}}\left(D^{\frac{2^{j-1}}{k-x}}\right)^2 & \leq D^{\frac{2^{j-1}}{k-x}}+ \frac{Z}{1-D}D^{2^{j+1}\frac{-k+5/2-\epsilon}{k-x}}.
\sum_{k=0}^{\infty}\frac{q^{2k^{2}-\left(2n+1\right)k}}{\left(q,q^{2};q^{2}\right)_{k}}=\frac{\left(-1\right)^{n}\left(-q;q\right)_{n}}{\left(q;q^{2}\right)_{\infty}q^{\binom{n+1}{2}}}.
(\tilde{u_p^n},\tilde{v_p^n},\tilde{h_p^n},\tilde{g_p^n})|_{y\rightarrow\infty}=(0,0,0,0), & (\tilde{u_p^n},\tilde{v_p^n},\tilde{h_p^n},\tilde{g_p^n})|_{y=0}=(-\overline u_e^n(x),0,-\overline h_e^n(x),0).
\mathbb{Q}(|\hat{\sigma}^2_n(Y_n) - (1 + \rho^*) | > \varepsilon ) = \mathbb{E}_{\mathbb{Q}}\left( Q_n(X_n^{(2)}, 0)(|\hat{\sigma}^2_n - (1 + \rho^*) | > \varepsilon )\right) \to 0,
\mathrm{P}^o_{i,k}= \sum^{k-1}_{p=0}{k-1 \choose p}\frac{(-1)^p K!\left[\frac{\gamma\left(N-M+1,\epsilon_{i,k}^*\right)}{(N-M)!}\right]^{K-k+p+1}}{(K-k)!(k-1)!(K-k+p+1)},
M_1(u,t) =\int_{[0,t-u)}v \mu(dv)+ \int_{[t-u,u)}(t-u) \mu(dv) + \int_{[u,t]}(t-v) \mu(dv) &\leq \int_{[0,t-u)}v \mu(dv)+ \int_{[t-u,u)}v \mu(dv) + \int_{[u,t]}(t - v)\mu(dv).
\begin{array}{llll}M_8 [z_1, z_2]=z_2, [z_3, z_4]=z_4 M_{12}[z_4, z_1]=z_1, [z_4, z_2]=2z_2, [z_4, z_3]=z_3, [z_3, z_1]=z_2 M_{13, a}[z_4, z_1]=z_1+az_3, [z_4, z_2]=z_2, [z_4, z_3]=z_1, [z_3, z_1]=z_2 M_{14}[z_4, z_1]=z_3, [z_4, z_3]=z_1, [z_3, z_1]=z_2 \end{array}
\begin{cases}a^{\varepsilon,\nu}~\doteq~ -\eta'(u^{\varepsilon,\nu})f_{x}^{\nu}(t,x,u^{\varepsilon,\nu})+ q_{x}^{\nu}(t,x,u^{\varepsilon,\nu}), b^{\varepsilon,\nu}~\doteq~-\varepsilon \eta''(u^{\varepsilon,\nu})\bigl(u^{\varepsilon,\nu}_{x}\bigr)^2, c^{\varepsilon,\nu}~\doteq~\varepsilon\eta(u^{\varepsilon,\nu})_{xx}. \end{cases}
f_{a}\ = \S \ + \ \sum_{k}\, s_{a k}\, M_{k}\ , \hspace{1 c m}K ( S , \bar{S}, M_{k}, \bar{M}_{k}) \ = \ - \, \ln ( S + \bar{S}) \ + \ \frac{1}{4}\: ( M_{k}+ \bar{M}_{k})^{2}\ .
\gamma_{o}(G) =\dfrac{n(T)-l(T)+s(T)+1}{3}+S_{c}& =\dfrac{n-(L_{c}+2S_{c})-\left( \ell-L_{c}+1\right) +1+s-S_{c}+3S_{c}}{3}& =\frac{n-\ell(G)+s(G)}{3}.
\|E^{(n)}(t)\|_{H^{N+1}}\leq\sum_{p=2}^{\infty}\sum_{\substack{n_1,\cdots,n_p\leq n; n_1+\cdots+n_p>n}}\sum_{i=1}^3\big\|\mathcal{N}_{p,i}(g^{(n_1)},\cdots,g^{(n_p)})\big\|_{H^{N+1}}&\leq \sum_{p=2}^{\infty}\sum_{\substack{n_1,\cdots,n_p\leq n; n_1+\cdots+n_p>n}}C^pt^{4p}\sum_{j=1}^p\|g^{(n_j)}(t)\|_{H^{N+2}}\prod_{i\neq j}\|g^{(n_i)}(t)\|_{H^{3}}.
\beta_{s, t}(R/I_{\Delta}) = \begin{cases}0 t \le s+1 \beta_{s, s+2}(F_0) + \beta_{s-1, s+2}(F_1) + \beta_{s-2, s+2}(F_2) t = s+2 begin{align*}.5ex]\beta_{2s - t, t}(F_{t-s}) = \sum_{k \ge t - s}\binom{k}{t-s}\binom{d-k}{2s - t}L_k t \ge s+3.\end{cases}
E[| Q^{n,4}_\tau|^2] \le C \sum_{j=0}^{\lfloor n\tau \rfloor}\left(\int_{\frac jn}^{\frac{j+1}n \wedge \tau}\gamma_s ds \right)^2 \le C \sup_{0\le j \le \lfloor n\tau \rfloor}\left(\int_{\frac jn}^{\frac{j+1}n \wedge \tau}\gamma_s ds \right) \int_0^{\tau}\gamma_s ds,