harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{cases}r=\frac{R_A^3}{2R_0^2}e^{2\rho}\cos \chi \sigma=\frac{R_A^3}{2R_0^2}\left[e^{2\rho}\left(e^{2\rho}-\frac{1}{4}\right) e^{5\lambda}\right]^{1/2}\sin \chi\end{cases}
n_{\mathrm{e f f}}^{2}( n , \eta ) \simeq n^{2}- n^{2}\biggl ( \frac{n}{n_{\mathrm{i n f}}}\biggr )^{2}( 1 - n_{\mathrm{b}}^{2}\eta^{2}) , \quad \frac{a^{\prime \prime}}{a}\simeq n_{\mathrm{b}}^{2}\biggl [ 1 - 6 \biggl ( \frac{\eta}{\eta_{0}}\biggr )^{2}\biggr ] .
K_v=\left\{\gamma=\left[\begin{array}{ccc}a b c g e f h l d\end{array}\right]\in G(\mathcal{O}_v):\gamma\equiv\left[\begin{array}{ccc}1_m \ast \ast 0 1_r \ast 0 0 1_m\end{array}\right]p_v^{n_v}\right\}.
e_1:=vw\frac{\partial}{\partial u}+u \frac{\partial}{\partial v}, e_2:=u\frac{\partial}{\partial u}+v \frac{\partial}{\partial v}, e_3:= v^2 \frac{\partial}{\partial u}+2u\frac{\partial}{\partial w}, e_4:= v\frac{\partial}{\partial v}-2w \frac{\partial}{\partial w}.
|C_\frak{s}\cap R_i(m) | = \begin{cases}O\left( \dfrac{m^2}{q_n(q_n+q_{n-1})\sqrt{\log m}}\right) i =1,2 O\left( \dfrac{m^2}{q_n(q_n+q_{n-1})(\log m)^{3/2}}\right) i=3.\end{cases}
\det(A_n) = \alpha_1^n \cdots \alpha_k^n c_{k_{1}}^n \cdots c_{k_p}^n \prod_{1 \le i < j \le k}(\alpha_j - \alpha_i) \prod_{1 \le i <j \le p}(c_{k_j}- c_{k_i}) \prod_{\substack{1 \le i \le k 1 \le j \le p}}(c_{k_j}- \alpha_i).
H_k^-(x) =\left(\frac{b-x}{b-a}\right)^{\!\!N}\frac{(x-a)^k}{k\,!}\sum_{\ell=0}^{N-k-1}\binom{\ell+N-1}{\ell}\!\left(\frac{x-a}{b-a}\right)^{\!\!\ell}, H_k^+(x) =\left(\frac{x-a}{b-a}\right)^{\!\!N}\frac{(x-b)^k}{k\,!}\sum_{\ell=0}^{N-k-1}\binom{\ell+N-1}{\ell}\!\left(\frac{b-x}{b-a}\right)^{\!\!\ell}.
\langle n_{m}, \ldots , n_{1}| n_{1}^{\prime}, \ldots , n_{m}^{\prime}\rangle = ( - 1 )^{\sum_{a}n_{a}( a - 1 )}\delta_{\bar{n}_{1}n_{1}^{\prime}}\cdots \delta_{\bar{n}_{m}n_{m}^{\prime}}.
\frac{d}{dt}\big(\|\partial^2_{\phi}u\varphi_2\|^2_{L^2(\partial B)}\big) +\frac12\|\nabla\partial^2_{\phi}u\varphi_2\|^2_{L^2(B)}\le C(1+\|\nabla\partial_{\phi}u\varphi_2\|^2_{L^2(B)}) \|\nabla\partial_{\phi}u\varphi_1\|^2_{L^2(B)}+C.
R_{ii^{\prime}}^{\prime \prime}M_{i}M_{i^{\prime}}^{\prime}=M_{i^{\prime}}^{\prime}M_{i}R_{ii^{\prime}}^{\prime \prime}R_{ii^{\prime}}^{\prime \prime \prime}M_{i}^{\prime}M_{i^{\prime}}=M_{i^{\prime}}M_{i}^{\prime}R_{ii^{\prime}}^{\prime \prime \prime}\;,
{c_A\over 24}\sum_{r}\left[H_1,A^c_i(r)\right]\left[H_1, A^c_i(-r)\right]\sum_k{1\over \omega^3_k}=-{c_A\over 24}{1\over 2\pi^2}log\left({\Lambda\over\mu}\right)\int d^3x\,E^2
\mathfrak{N}_{f,a}\geq \theta(d)^2\Theta(g)^2\Bigg[\lambda \left(\frac{2 \sum_{i=1}^{r}\theta(p_i)+2 \sum_{i=1}^{s}\Theta(g_i)+1}{\lambda}\right)\left(-(2n+1)W(d)^2W(g)^2q^{m/2}\right) + \left\{q^{m-1}-(n+1)-(2n+1)q^{m/2}\right\}\Bigg]
\sum_{n\le x}F(n-h)G(n) =M(x)\sum_{\underset{(d,d_1)\mid h}{d, d_1}}\frac{f(d)g(d_1)(d,d_1)}{dd_1}+O(|E_1(x)E_2(x)E_3(x)|+\frac{M(x)}{x}(|E_2(x)|+|E_3(x)|)).
\underbrace{g(x) y}_{u}\underbrace{g(g(x) y)}_{g(u)}\stackrel{(a)}{=}\underbrace{g(g(x) y)}_{g(u)}\underbrace{g(x) y}_u &= \underbrace{g(g(x) y)}_{g(u)}\underbrace{g(x)}_{g(x)}y &\stackrel{(b)}{=}\underbrace{g(x)}_{g(x)}\underbrace{g(g(x) y)}_{g(u)}y,
\Pi ( q^{2}) ={\frac{2}{q^{2}}}{\frac{1}{D - 1}}\int{\frac{d^{D}p}{( 2 \pi )^{D}}}\left[ D - 2 +{\frac{( q^{2}- 4 p^{2})}{q^{2}+ 2 q \cdotp}}\right] s ( p ) .
s^H_{\tilde{g}_t}= e^{-2f}\left(s^H_{g_t}+ m\Delta^{g_t}(f)\right)= u^{2/n}s^H_{g_t}- 2u^{2/n-1}\Delta^{g_t}(u) - 2u^{2/n-2}|du|^2_{g_t}.
\tilde c =c-\lambda (\mu_1-\mu_2 )\biggl(1-\frac{3 (\mu_1^3-\mu_1^2\mu_2+\mu_1\mu_2^2-\mu_2^3 )\, (\mathbb E [(\xi_i-\eta_i)^2 ] )^2}{2 (\mu_1^2-\mu_1\mu_2+\mu_2^2 )^2\,\mathbb E [(\xi_i-\eta_i)^3 ]}\biggr).
v_{p}((\begin{matrix}2N\\N\end{matrix}))=v_{p}((2N)!)-2v_{p}(N!)=\sum_{i=1}^{\infty}\lfloor\frac{2N}{p^{i}}\rfloor-2\lfloor\frac{N}{p^{i}}\rfloor\le log_{p}(N)+1
\begin{array}{rcl}{\sf s}f = Z_c\partial^{\mu}(\partial_{\mu}c+igZ_g[c,G_{\mu}]) & +Z_cZ_g(1+\delta v/v)g^2v^2\xi/2(\hat{\phi}_0^{\dagger}ct^A\hat{\phi}_0+\hat{\phi}_0^{\dagger}t^Ac\hat{\phi}_0)t^A & +Z_cZ_gg^2v\xi/\sqrt{2}(\phi^{\dagger}ct^A\hat{\phi}_0+\hat{\phi}_0^{\dagger}t^Ac\phi)t^A.\end{array}
D_{\mathrm{G}}^{2}[ U , U^{\prime}] = \sum_{x , i}\sum_{\alpha = 0}^{3}\left[ ( ( u_{\alpha})_{x , i}- ( u_{\alpha}^{\prime})_{x , i})^{2}+ ( ( \dot{u}_{\alpha})_{x , i}- ( \dot{u}_{\alpha}^{\prime})_{x , i})^{2}\right] \, ,
\mathbb{R}^{m+3}\supset\mathbb{S}^{m+2}\setminus\{p\}\ni x=(x',\vec{x}'')~\to~\frac{\vec{x}''}{1-x'}\in \mathbb{R}^{m+2},~~~~~x'\in \mathbb{R},~\vec{x}''\in \mathbb{R}^{m+2}.
\frac{d^2\gamma^1}{dt^2}-2\lambda_1 \frac{d\gamma^1}{dt}\frac{d\gamma^3}{dt}=0 \frac{d^2\gamma^3}{dt^2}+2\lambda_1 e^{-2\lambda_1 \gamma^3}\left(\frac{d\gamma^1}{dt}\right)^2 =0.
0 =\frac{1}{2}||f-p_1||_1+\frac{1}{2}||f-p_2||_1-||f-p||_1 &=\int\frac{1}{2}|f(x)-p_1(x)|+\frac{1}{2}|f(x)-p_2(x)|-|f(x)-p(x)|\, dx.
{\mathcal L}_H= \left\{(q, p, P_q, P_p)\;\left| \exists \; u\in U \; \mbox{s. t.}\begin{array}{l}P_q=\frac{\partial H}{\partial q}(q,p,u), P_p=\frac{\partial H}{\partial p}(q,p,u), \frac{\partial H}{\partial u}(q,p,u)=0\, \end{array}\right.\right\}\, .
[ L^{m n}, L^{p k}]_{Q ( m n , p k )}= g_{q}^{m k}Q_{m n}Q_{p k}L^{n p}+ g_{q}^{n p}L^{m k}- g_{q}^{m p}Q_{m n}L^{n k}- g_{q}^{n k}Q_{p k}L^{m p}\ ,
A_{2-n}A_{4-n}A_{6-n}\psi_{n-6}^{(0)}= A_{2-n}A_{4-n}A_{6-n}\alpha_{n-6}^{(0)}+(B_{2-n}A_{4-n}A_{6-n}+A_{2-n}B_{4-n}A_{6-n}+A_{2-n}A_{4-n}B_{6-n})\beta_{n-6}^{(0)}.
\lim_{c\to 0}\frac{t_-^{c,n}}{t_-^{c,n}}\left(\frac{1+t_{+}^{c, n}t}{t-t_{+}^{c, n}}\right)= \lim_{c\to 0}\frac{t_-^{c,n}-t}{t_-^{c,n}t+1}=\frac{1}{t}\,, \quad\lim_{c\to 0}\frac{t_-^{c,n}}{t_-^{c,n}}\left(\frac{1-t_{+}^{c, n}t}{t-t_{+}^{c, n}}\right)=\lim_{c\to 0}\frac{t_-^{c,n}+t}{t_-^{c,n}t+1}=\frac{1}{t}.
\begin{cases}{\cal R}_{1}=\{(\mu_{R},\mu_{T}): \mu_{R}+\mu_{T}\geq 1,\mu_{R}\leq 1,\mu_{T}\leq 1\}{\cal R}_{2}=\{(\mu_{R},\mu_{T}): \mu_{R}+\mu_{T}< 1,\mu_{R}\geq0, \mu_{R}+N_T\mu_{T}\geq 1\}\end{cases}.
1 \geq \int_{\Gamma_{\mu}}e^{- \frac{n-1}{2}\ell^2_{f,v}(g)}\mu(dg) = \int_{\Gamma_{\mu}}e^{- \frac{( \sigma n l + \lambda - \log \langle f, gv \rangle)^2}{2 \sigma^2 (n-1)}}\mu(dg) & \geq 1 - \int_{\Gamma_{\mu}}\frac{( \sigma n l + \lambda - \log \langle f, gv \rangle)^2}{2 \sigma^2 (n-1)}\mu(dg) = 1 - o(1).
V ( y_{i}, \theta ) = - V_{p + 1}( i )^{p}\int_{0}^{\infty}\frac{d t}{t}( 8 \pi^{2}\alpha^{\prime}\, t )^{- \frac{p + 1}{2}}\sum_{m_{i}}e^{-{\frac{t}{2 \pi \alpha^{\prime}}}\sum_{i}( y_{i}+ m_{i}L_{i})^{2}}\frac{\theta_{1 1}^{4}( i t / 2 , i t )}{e^{\pi t}\eta^{1 2}( i t )}\, .
\hat{R}_{ij}= \, 2 \hat \omega_{ik}\hat\omega_{j}^{\;k}+ \lambda \hat{g}_{ij}, \hat{\nabla}^i \hat \omega_{ij}= \, 0, \hat\omega_{ij}\hat\omega^{ij}= \, \frac{\alpha^2}{m}+ \lambda .
A := \prod_{i=1}^{s-1}\left(q- p_i-1 \right), B := \sum_{k=1}^{s-2}(q-p_1)\ldots(q-p_{k-1})(q-p_{k+1}-1)\ldots(q-p_{s-1}-1), C = \sum_{k=s-1}^{\ell}C_k, \mbox{with}C_k:= \prod_{n=1}^{s-2}\left(q- p_n+\tau_{n,k}\right),
s_{\mathrm{e f f}}[ \alpha , \varphi ] = \int \! d^{2}x \left({\frac{1}{2}}\partial_{\alpha}\varphi \partial_{\alpha}\varphi +{\frac{\alpha - v}{2}}\varphi^{2}-{\frac{\alpha}{2 N g^{2}}}\right) + s [ \alpha ] .
F(\beta_i, \tan^2{\phi_i})={\tan^2{\phi_i}}(1+{\beta^2_i}{\tan^2{\phi_i}})^2(1-\tilde{b}\beta_i)^2-(\beta_i-1)(1+\beta_i{\tan^2{\phi_i}})((\gamma+1-2\tilde{b})\beta_i &-(\gamma-1))((\gamma-1+2\tilde{b}{\beta_i}){\beta_i}{\tan^2{\phi_i}}+(\gamma+1)).
S = \frac{1}{2 \pi \alpha^{\prime}}\int d^{2}\sigma \left( \, - \frac{1}{2}\sqrt{- \gamma}\gamma^{m n}\Pi_{m}^{r}\Pi_{n}^{s}\eta_{r s}+ \frac{1}{2 !}\epsilon^{m n}\Pi_{m}^{A}\Pi_{n}^{B}B_{B A}\, \right) ~ ,
\bar{\Gamma}_{s}^{[k]}=\bar{G}_{\bar{k}(s)}^{[2]}(\bar{\lambda}_{0}^{[k]})\bar{Y}^{[k]}_{s}\left(\bar{Y}^{[k]H}_{s}\Theta^{-2}(\bar{k}(s))\bar{Y}^{[k]}_{s}\right)^{-1/2}
\P_{\pi}(X_{1}=1, X_{1+d}=1, T_Y>d) \le \P_{\pi}(X_{1}=1, X_{1+d}=1, T_Y>d, \mathcal{E}) + \P_{\pi}(X_1=1, \overline{\mathcal{E}}) \le \P_{\pi}(X_{1}=1, X_{1+d}=1, T_Y>d, \mathcal{E}) + e^{-c_1 d}\P_{\pi}(X_1=1).
S(T_0,T_1,\dots,T_r,T_{r+1}):=\sum_{(n,\alpha)\in \mathbb N^{r+1}}\sum_{\begin{smallmatrix}\beta\in\mathbb N^m \theta'(\beta,\alpha)\end{smallmatrix}}\sum_{\begin{smallmatrix}(k_j)_{j\in I}\in \mathbb N_{>0}^I (\ref{eq3.12})_{n,\beta}\end{smallmatrix}}T_0^nT_1^{\alpha_1}\cdots T_r^{\alpha_r}T_{r+1}^{\sum_{j\in I}k_j\nu_j}.
\frac{6 \ddot{a}}{a}+ \dot{\phi}^{2}= \frac{2 \ddot{a}}{a}+ \frac{4 ( \dot{a}^{2}+ k )}{a^{2}}= - \frac{1}{\alpha}\left( \ddot{\phi}+ \frac{3 \dot{a}\dot{\phi}}{a}\right) = \Lambda e^{\alpha \phi}.
C_{2n}^\lambda(z) = \frac{(\lambda)_n}{(\frac{1}{2})_n}P_n^{(\lambda-\frac{1}{2},-\frac{1}{2})}(2z^2-1) = (-1)^n\frac{(\lambda)_n}{(\frac{1}{2})_n}P_n^{(-\frac{1}{2},\lambda-\frac{1}{2})}(1-2z^2).
\beta ( g ) = - 2 ( g^{*}- g ) \left\{1 +{\frac{1}{N}}\left[{\frac{2}{\ln \Theta}}\left( 1 +{\frac{l ( \Theta )}{\ln \Theta}}\right) +{\frac{5}{2 \ln^{2}\Theta}}+ O \left({\frac{l ( \Theta )^{2}}{\ln^{3}\Theta}}\right) \right] \right\},
y_1=x_1, \, y_2=\frac{1}{\sqrt{2}}(x_2+x_3), \, y_3=\frac{1}{\sqrt{2}}(x_2-x_3); [y_1,y_2]=[y_3,y_1]=-\frac{1+w}{\sqrt{2}}y_1, \, [y_2,y_3]=\frac{1-w}{\sqrt{2}}(y_2+y_3).
\mathbf{H}(t, \cdot) \times \boldsymbol{\nu}+ \gamma_{1}\mathbf{g}\big(\mathbf{E}(t, \cdot) \times \boldsymbol{\nu}) \times \boldsymbol{\nu}+ \gamma_{2}\mathbf{g}\big(\mathbf{E}(t - \tau, \cdot) \times \boldsymbol{\nu}) \times \boldsymbol{\nu}= \mathbf{0}(0, \infty) \times \Gamma
A_{T,L}^{x}:=\sum_{\gamma\in S_{T,L}:a\rightarrow\alpha\backslash\{a\}}x^{l(\gamma)},B_{T,L}^{x}:=\sum_{\gamma\in S_{T,L}:a\rightarrow\beta}x^{l(\gamma)},E_{T,L}^{x}:=\sum_{\gamma\in S_{T,L}:a\rightarrow\epsilon\cup\overline{\epsilon}}x^{l(\gamma)}.
{\cal L}^{(1)}=\pi_{0}\dot{A}^{0}+\pi_i\dot{A}^i+\pi_\theta\dot\theta+B\dot{N}+\bar{\cal P}\dot{\cal C}+\bar{\cal C}\dot{\cal P}+\tilde{\Omega}_{2}\dot\rho-{\cal H}^{(1)}
\|\Gamma(y)-\Gamma(z)\|_{\mathcal{B}}^2\leq 2K^2\|y-z\|_{L^2(0,T;H^2(I))}^2\left(\| y\|_{L^2(0,T;H^2(I))}^2 +\|z\|_{L^2(0,T;H^2(I))}^2\right) \leq 2K^2\|y-z\|_{\mathcal{B}}^2\left(\| y\|_{\mathcal{B}}^2 +\|z\|_{\mathcal{B}}^2\right).
|\Omega(\zeta_1,\zeta_2)| = (|\xi_1+\xi_2|^\alpha(\xi_1+\xi_2) - |\xi_1|^\alpha\xi_1 - |\xi_2|^\alpha\xi_2) + (\xi_1+\xi_2)(\mu_1+\mu_2)^2 - \xi_1\mu_1^2 - \xi_2\mu_2^2 = (|\xi_1+\xi_2|^\alpha(\xi_1+\xi_2) - |\xi_1|^\alpha\xi_1 - |\xi_2|^\alpha\xi_2) + 2\mu_1\mu_2 (\xi_1+\xi_2) + \xi_1\mu_2^2+\xi_2\mu_1^2 \sim N_{max}^{\alpha+1}
W_i = - \frac{dz_i}{w_i}\exp\Big( \frac{(z_i-\frac{w_i\hbar}{2})^{r+1}-(z_i+\frac{w_i\hbar}{2})^{r+1}+(r+1)w_i\hbar z_i^r}{\hbar(r+1)}\Big); D_x = d \; \frac{1}{(rz^{r-1}-r\epsilon)dz}.
qp_{bl}(z,w) =qp_{bl}(Tz,Tw) \geq a_1[qp_{bl}(z,w)+qp_{bl}(w,z)]+a_2[qp_{bl}(z,Tz)+ & qp_{bl}(Tz,z)]+ a_3[qp_{bl}(w,Tw)+qp_{bl}(Tw,w)] & \quad+a_4[qp_{bl}(z,Tw)+qp_{bl}(Tw,z)] &=(a_1+a_4)[qp_{bl}(z,w)+qp_{bl}(w,z)]+2a_2 qp_{bl}(z,z)
\boldsymbol{F}(p,q;r,s;a,b)=\left( \frac{a^{pr}-b^{pr}}{a^{ps}-b^{ps}}{\frac{a^{qs}-b^{qs}}{a^{qr}-b^{qr}}}\right)^{\frac{1}{(p-q)(r-s)}}pqrs(p-q)(r-s)\neq 0;
\int_{\partial X}d^2 B^{f(\xi)}_{F_p(x)}(DF_p(v), DF_p(v)) d\mu^x_p(\xi) +p \int_{\partial X}< DF_p(v), \overrightarrow{F_p(x)f(\xi)}>^2 d\mu^x_p(\xi) = p \int_{\partial X}< DF_p(v), \overrightarrow{F_p(x)f(\xi)}> < v, \overrightarrow{x\xi}> d\mu^x_p(\xi)
V_{\parallel}= \frac{1 - 2 ( \dot{f}- H f )^{2}}{1 - ( \dot{f}- H f )^{2}}\dot{H}+ \frac{2}{f^{2}}\frac{( 1 + H f ( \dot{f}- H f ) )^{2}}{1 - ( \dot{f}- H f )^{2}}+ 2 H^{2},
v(t) = v(0) \big(1 - k_{\alpha}\big) t^{\alpha-1}E_{\alpha, \alpha}\big[(c + k_{\alpha}) t^{\alpha}\big] + \frac{cv(0)}{\Gamma(\alpha)}\int_{0}^{t}s^{\alpha-1}\big(t-s\big)^{\alpha-1}E_{\alpha, \alpha}\big[(c + k_{\alpha}) s^{\alpha}\big]ds,
Z\subseteq\bigcap_{n\geq 1}\bigcup_{j=1}^k \overline{B_{c m_n^{-1/2}}(x_j)}= \bigcup_{j=1}^k \bigcap_{n\geq 1}\overline{B_{c m_n^{-1/2}}(x_j)}= \bigcup_{j=1}^k \lbrace x_j \rbrace.
\begin{pmatrix}w 0 0 z \end{pmatrix}\tau \begin{pmatrix}w 0 0 z \end{pmatrix}^{-1}\tau^{-\frac{w}{z}}= I + \begin{pmatrix}* * c_i \left( \frac{z}{w}- \frac{w}{z}\right) * \end{pmatrix}p.
\widehat{A}=a\cdot (y^{-1}x^2y)+ b \cdot (y^{-1})+c \cdot (x^2 y)+d\cdot (1) &=a \cdot (x^2)+ b \cdot (y) +c \cdot (x^2 y)+d \cdot (1), \widehat{B}=e\cdot ((x^2y)^{-1}x^{-1}y)+ f \cdot ((x^2 y)^{-1})+ g\cdot (x^{-1}y)+ h\cdot (1) &=e \cdot (x^3) + f\cdot (x^2 y) + g\cdot (x^3y)+ h\cdot (1).
x_i\omega_j = \omega_jx_i, \partial_i\omega_j = \begin{cases}\omega_j\partial_i i \ne j , begin{align*}1.5ex] \omega_{i}\partial_i + \omega_{i+1}\bigl( \partial_i x_{i+1}- x_{i+1}\partial_i \bigr) i = j .\end{cases}
\begin{array}{l c l}{{X^{( I I )}( \sigma , \tau )}}&{{=}}&{{{\displaystyle X_{0}^{( I I )}( \tau ) +}}}\\{{}}&{{}}&{{{\displaystyle \frac{\sigma_{0}}{2 \pi}~ k ~ \sum_{n \neq 0}^{}\left\{\frac{1}{| n |}~ F_{\nu}\left( \frac{2 \pi}{\sigma_{0}}| n | \tau , \frac{2 \pi}{\sigma_{0}}| n | \tau_{0}\right) \phi_{n}( \sigma_{0}) ~ e^{i \frac{2 \pi}{\sigma_{0}}n \sigma}\right\}}}}\\ \end{array}
\Big|B\Big(g',{1\over 2}d_K(g')|b(g')|^{1\over Q}\Big)\cap S_\Gamma(g')\Big| &=\Big|B\Big(0,{1\over 2}d_K(g')|b(g')|^{1\over Q}\Big)\cap S_\Gamma(0)\Big| &=\Big|B\Big(0,{1\over 2}d_K(g')|b(g')|^{1\over Q}\Big)\Big|- \Big|B\Big(0,{1\over 2}d_K(g')|b(g')|^{1\over Q}\Big)\cap S_\Gamma(0)^{c}\Big| &>{1\over Q 2^Q}\big(\omega_Q-\varepsilon\big)d_K(g')^Q|b(g')|,
\left(\widehat{\Upsilon}_k^N(t)\right)^2= \sum\limits_{n=N+1}^{\infty}\left|\frac{e^{i t f(n)}-1}{t}\right|^2 \left|\Delta^{k}c_{n}\right|^2 \leq2 \sum\limits_{n=N+1}^{\infty}\left|f(n)\Delta^{k}c_{n}\right|^2.
F_{1}(N) \!= \! \Big\{\!(i_{1}, \! \! \dots, \! \! i_{N^{\gamma}t}) \! \in \! \{1, \! \dots, \! N\}^{N^{\gamma}t}; l = 1, \! \dots, \! N^{\gamma}t: \! \eta^{i_{1}, \! \dots, \! i_{l}}\! \notin \! \{\eta, \! \eta^{i_{1}}, \! \dots, \! \eta^{i_{1}, \! \dots, \! i_{l-1}}\}\! \Big\}
\Gamma_{22}^1= \frac{1}{2}g^{11}\left(\frac{\partial g_{21}}{\partial s}+\frac{\partial g_{21}}{\partial s}-\frac{\partial g_{22}}{\partial u}\right)+\frac{1}{2}g^{12}\left(\frac{\partial g_{22}}{\partial s}+\frac{\partial g_{22}}{\partial s}-\frac{\partial g_{22}}{\partial s}\right) = g^{11}\frac{\partial g_{12}}{\partial s}=\frac{\partial g_{12}}{\partial s},
{J}_{(\mu_{1}\dots \mu_{n};\mu_{n+1})}^{(n)}+\frac{\partial{J}_{(\mu_{1}\dots \mu_{n}}^{(n)}}{\partial \psi^{\sigma}}\Gamma_{\mu_{n+1})\lambda}^{\sigma}\psi^{\lambda}=\frac{i}{2}\psi^{\rho}\psi^{\sigma}R_{\rho \sigma \nu (\mu_{n+1}}{{J}^{(n+1)\nu}}_{\mu_{1}\dots\mu_{n})}
G_{H}( x , x^{\prime}) = \frac{\Delta^{1 / 2}( x , x^{\prime})}{2 ( 2 \pi )^{n / 2}}\frac 1{\sigma^{n / 2 - 1}( x , x^{\prime})}\sum_{k = 0}^{\frac{n - 3}2}a_{k}( x , x^{\prime}) \sigma^{k}( x , x^{\prime}) \frac{\Gamma ( n / 2 - k - 1 )}{2^{k}}\ ,
\Delta:=\{\{1,2,3,8,9,\ldots,k+4\},\{3,4,5,8,9,\ldots,k+4\},\{1,5,6,8,9,\ldots,k+4\}, &\{1,4,7,8,9,\ldots,k+4\},\{3,6,7,8,9,\ldots,k+4\},\{2,5,7,8,9,\ldots,k+4\}, &\{2,4,6,8,9,\ldots,k+4\}\}
|(x,f_{t}(x)) - (y,f_{t}(y))|^{2}\leq |x-y|^{2}+ |f_{t}(x)|^{2}+ |f_{r}(y)|^{2}\leq \frac{\epsilon^{2}}{4}+ \frac{\epsilon^{2}}{4}+ \frac{\epsilon^{2}}{2}= \epsilon^{2}.
\chi^{\alpha-\beta}\,[u]^q_{W^{\beta,q}(\mathbb{R}^N)}- \frac{C\,\chi^\alpha}{s}\,\|u\|_{L^p(\mathbb{R}^N)}^{q\,(1-\theta)}\,\|u\|_{L^r(\mathbb{R}^N)}^{q\,\theta}&\le \frac{C}{\alpha-\beta}\,\|u\|_{L^{r}(\mathbb{R}^N)}^{q\,\theta}\,\left((1-t)^\frac{1}{p}\, [u]_{W^{t,p}(\mathbb{R}^N)}\right)^{q\,(1-\theta)},
\psi\big(\Delta(s^{\Lambda^i})^{E\cap \Lambda^i}\big)=\sum_{\substack{\emptyset\neq G\subseteq E G\cap \Lambda^{e_i}\neq \emptyset \lambda\in \mathrm{MCE}(G)}}(-1)^{(|G|+1)}\Theta_{s_\lambda^\Lambda, s_\lambda^\Lambda}.
A_{0}:=y_{1}y_{2}-y_{3}^2,A_{1}:=y_{0}y_{2}-y_{4}^2,A_{2}:=y_{0}y_{1}-y_{5}^2, A_{3}:=y_{4}y_{5}-y_{0}y_{3},A_{4}:=y_{3}y_{5}-y_{1}y_{4},A_{5}:=y_{3}y_{4}-y_{2}y_{5},
\sum_{k=1}^{K+1}\mathbb{E}[X_k^4]\leq\sum_{k=1}^{K+1}C\log(d)^{2/3}|\Delta_k|^2\leq C\eta^{-2}\log(d)^{2/3}C_{d,G}^{2(1+\eta)}K^{1+2\eta}.
f ( x , t ) = 4 \operatorname{t a n}^{- 1}\left[ \frac{v \operatorname{s i n h}( \frac{\mu ( x - x_{0})}{\sqrt{1 - v^{2}}})}{\operatorname{c o s h}( \frac{( \mu v ( t - t_{0}) )}{\sqrt{1 - v^{2}}})}\right] ,
\sum\limits_{\ell=1}^{M}\varphi_\ell^1 \otimes \varphi_\ell^2 \mapsto \sum\limits_{\ell=1}^{M}G_1(\cdot) \varphi_\ell^1 \otimes G_2 \varphi_\ell^2, \sum\limits_{\ell=1}^{M}\varphi_\ell^1 \otimes \varphi_\ell^2 \mapsto \sum\limits_{\ell=1}^{M}G_2 \varphi_\ell^1 \otimes G_1(\cdot) \varphi_\ell^2
f^{i_{1}\ldots i_{m_{1}}j_{1}\ldots j_{m_{2}}\ldots k_{1}\ldots k_{m_{N - 1}}}\; u_{i_{1}}^{1}\ldots u_{i_{m_{1}}}^{1}u_{j_{1}}^{2}\ldots u_{j_{m_{2}}}^{2}\ldots u_{k_{1}}^{N - 1}\ldots u_{k_{m_{N - 1}}}^{N - 1}\; .
\rho_k(\zeta) = \alpha \beta^k \zeta^{k+1}\Big(\sum_{i\geq0}\zeta^i\Big)\Big(\sum_{j\geq 0}\tbinom{k+j}{k}(\eta\zeta)^j\Big) = \alpha \beta^k \zeta^{k+1}\sum_{n\geq 0}\Big(\sum_{i=0}^{n}\tbinom{k+i}{k}\eta^i\Big)\zeta^{n}= \sum_{n\geq 0}\Big(\alpha \beta^k\sum_{i=0}^{n}\tbinom{k+i}{k}\eta^i\Big)\zeta^{n+k+1}.
\left( \frac{\sinh ( \sqrt{- 3 \Lambda}\, t )}{\tanh \left( \frac{\sqrt{- 3 \Lambda}}{2}\, t \right)}\right)^{\frac{1}{3}}\, \left( \tanh \left( \frac{\sqrt{- 3 \Lambda}}{2}\, t \right) \right)^{\alpha_{k}}\ .
\begin{array}{c}k_1^\mu T_{\rho \sigma \mu \nu}^{(4)abcd}(p_1,p_2;k_1,k_2)=C_1\gamma_{\rho\sigma \nu}^{(1)}(k_1)+C_2\gamma_{\rho \sigma \nu}^{(2)}(k_1) +C_3\gamma_{\rho \sigma \nu}^{(3)}(k_1)\end{array}
L^1_{loc}\left({\mathbb{R}_+;W^{1,\infty}(\Omega \times\mathbb{R}_+;\mathbb{R}^3)}\right) = \bigcap\nolimits_{T > 0}{L^1 \big((0,T);W^{1,\infty}(\Omega \times\mathbb{R}_+;\mathbb{R}^3)\big)},
K_{\mu \nu}^{( 0 )}( y ) = \left. \frac{1}{2}e_{\mu}^{( 0 ) M}e_{\nu}^{( 0 ) N}{\cal L}_{n^{( 0 )}}g_{M N}^{( 0 )}\right|_{x = Z^{( 0 )}( y )},
F_{\mu \nu}= \partial_{\mu}( \hat{\phi}^{a}W_{\nu}^{a}) - \partial_{\nu}( \hat{\phi}^{a}W_{\mu}^{a}) -{\frac{1}{g}}\epsilon_{a b c}\hat{\phi}^{a}( \partial_{\mu}\hat{\phi}^{b}) ( \partial_{\nu}\hat{\phi}^{c})
b_{n+1}(x) =\Big(x-\omega(a)+\left(a-\beta \right)\left(a+p+\beta \right) - \alpha_{2}-\alpha_{1}\Big) b_n(x) -\Big(\alpha_{2}\alpha_{1}+\gamma \left(p+2\beta \right)\Big)b_{n-1}(x) - \gamma^2 b_{n-2}(x) \, ,\;
\|(rT)^{m}x\|_{rT}^{2}={}r^{2m}\sum_{k=1}^{\infty}k\|r^{k}T^{k+m}x-r^{k-1}T^{k-1+m}x\|^{2}\leq{}r^{2m}\sum_{k=1}^{\infty}kr^{2(k-1)}\left(2\|T^{k+m}x-T^{k-1+m}x\|^{2}+2\|(1-r)T^{k+m}x\|^{2}\right) \leq{}r^{2m}\sum_{k=1}^{\infty}kr^{2(k-1)}\left(\frac{2c_{1,T}^{2}}{(k+m)^{2}}+2(1-r)^{2}Pb(T)^{2}\right)\|x\|^{2},
S^{(b)}_{q,v(b+1)}(N) \geq \left(1+o(1)\right)\left(\frac{1}{q}\left(2\sin \frac{\pi l}{2l+1}\right)^{k_{1}+1}- \frac{2}{q}\left(2\sin \frac{\pi l}{2l+1}\right)^{k_{1}-2}- \frac{2}{q}\left(2\sin \frac{\pi l}{(2l+1)}\right)^{k_{1}-3}+\cdots\right) &=\left(1+o(1)\right)\left(\frac{\beta^{k_{1}+1}}{q}-\frac{2}{q}\frac{\beta^{k_1-1}}{\beta-1}\right).
\left.\sum_{j=1}^n\frac{\sf d}{{\sf d}{x^j}}\left(\varpi^{ij}\left( \frac{\sf d}{{\sf d}{x^i}}V^j -\frac{\sf d}{{\sf d}{x^j}}V^i\right)\right)\right\vert_{{\cal E}_L}=0,\forall i=1,\dots,n,\, V=\sum_{i=1}^n V^i\frac{\partial}{\partial A^i}.
\gamma_{n}=\frac{1}{2}\delta_{n,0}+\frac{1}{i}\int_{0}^{\infty}\frac{dx}{e^{2\pi x}-1}\{\frac{(ln(1-ix))^{n}}{1-ix}-\frac{(ln(1+ix))^{n}}{1+ix}\},n=0,1,2,...
PF(A) =\{N_{dr}-m(2^n-1) \mid r=m(2^n-1)-\{1,2,...,n-1\}\}&=\left\{m^22^{2n}-2^n(m^2-md+m)+m-md-d-\{0,d,2d,...,(n-2)d\}\right\}&=\{F(A),F(A)-d,...,F(A)-(n-2)d\}.
M(z)= \begin{pmatrix}1 -\frac{1}{i (z^2-1)^{1/2}}\frac{z- (z^2-1)^{1/2}}{2i}\frac{z+ (z^2-1)^{1/2}}{2 (z^2-1)^{1/2}}\end{pmatrix}, z \in \mathbb C \setminus [-1,1].
\gamma_{i}^{*}= \gamma_{1}^{*}+ ( \beta_{2 1}^{-})^{*}+ ( \beta_{3 2}^{-})^{*}+ \cdots + ( \beta_{i , i - 1}^{-})^{*}= \mu_{1}^{*}+ \mu_{2}^{*}+ \cdots + \mu_{i}^{*}.
\dot{b}\langle \eta_*, \psi_* \rangle_{L^2}=-12 |b|^2 b \langle \psi_*^2, \tilde{w}_2 \rangle_{L^2}+\frac{16}{3 \sqrt{c_*}}\left( \delta_0 + \frac{20}{3}|b|^2 \right) b +\mathcal{O}(\delta_0^2 |b| + |b|^5).
\alpha(x,s,y,z)=(\lambda x,\,\mu s+x^{n+e}a(x),\,\frac{\mu^{d}}{\lambda^{n}}y+W,\,\frac{\mu^{dm}}{\lambda^{nm+e}}z+\frac{(\frac{\mu^{d}}{\lambda^{n}}y+W)^{m}-\frac{\mu^{dm}}{\lambda^{nm}}y^{m}+x^{n+e}a(x)}{\lambda^{e}x^{e}})
\pi(u)= \cosh \left(\frac{\sigma}{2}\right)\left[(\sinh^2 \left(\frac{\sigma}{2}\right)\right]^{-\frac{1}{2}-k}(2)^{-2k}F\left(k,k+\frac{1}{2};2k;- cosech^{2}\left(\frac{\sigma}{2}\right)\right)
C_{\theta}( \tau ) = \langle{\cal A}( \tau ){\cal B}\rangle_{0}\; - \; \lim_{\tau^{\prime}\rightarrow \infty}\langle{\cal U}_{\tau^{\prime}}({\scriptstyle{\cal A}}){\cal A}\rangle_{0}\lim_{\tau^{\prime \prime}\rightarrow \infty}\langle{\cal U}_{\tau^{\prime \prime}}({\scriptstyle{\cal B}}){\cal B}\rangle_{0}\stackrel{\tau \rightarrow \infty}{\longrightarrow}
\Lambda_{n}(\theta):=\log E\left[\mathrm{e}^{\theta\mathrm{Bin}(n-a_n,\pi(n))}\right] =(n-a_n)\log E\left[\mathrm{e}^{\theta\mathrm{Be}(\pi(n))}\right] &=(n-a_n)\log(1+\pi(n)(\mathrm{e}^\theta-1)).
a(z,\tilde{z}) := \int_\Omega \nabla z \cdot \nabla \tilde{z}\,dx{\rm for~}z, \tilde{z}\in V, a_\Gamma (z_\Gamma,\tilde{z}_\Gamma) := \int_\Gamma \nabla_\Gamma z_\Gamma \cdot \nabla_\Gamma \tilde{z}_\Gamma \, d\Gamma{\rm for~}z_\Gamma, \tilde{z}_\Gamma \in V_\Gamma,
Q(t)= Q_{\chi,\tilde{\chi}}(t):=\begin{cases}1 \quad\mbox{if}t\ge 1, \left(Q_{0}(t)/Q_{0}(1)\right)^{1/2}\quad\mbox{if}0<t<1\quad\mbox{and}\quad\chi\in\widetilde{\mathcal{W}}^{-}, t^{1/2}\quad\mbox{if}0<t<1\quad\mbox{and}\quad\chi\in\mathcal{W}_M^{+}, \lim_{s\to 0^+}Q(s)\quad\mbox{if}t=0.\end{cases}
\mathrm{E}T_\phi u = \mathrm{E}T_\phi a + \mathrm{E}T_\phi b \in{\mathrm{L}}^{p}(\mathbb{R}^n) +{\mathrm{H}}^{1,p}(\mathbb{R}^n) \subset \mathrm{L}^p(\mathbb{R}^n) + \dot{\mathrm{H}}^{1,p}(\mathbb{R}^n)
\varepsilon_{0}^{\beta}= \frac{| e | H}{( 2 \pi )^{2}}\sum_{n = 0}^{\infty}\int_{- \infty}^{\infty}d p \; \epsilon_{n}\left\{\frac{- 2 + \mathrm{e}^{\beta \epsilon_{n}}}{1 - \mathrm{e}^{\beta \epsilon_{n}}}\right\}.
\begin{array}{rcl}{\hat A}^{(1)}_i = \left( 1 + e^{2 \phi}k^{-2}(i_k C^{(1)})^2 \right)^{-1}\left( A^{(1)}_i - e^{2 \phi}k^{-2}C^{(1)}_i (i_k C^{(1)}) \right)\, ,{\hat A}^{(1)}_6 = -\left( 1 + e^{2 \phi}k^{-2}(i_k C^{(1)})^2 \right)^{-1}e^{2 \phi}k^{-2}(i_k C^{(1)}) \, , \end{array}
\hat \eta_{K,M,j}^{(3)}\buildrel \Delta \over ={\bf{e}}_j^T\sum\limits_{l = 1}^{M+K}{{{{\bf{\bar W}}}^{Q\left({M+ K - l + 1}\right)}}{{\boldsymbol{\xi}}^{(l)}}}={\bf{e}}_j^T\sum\limits_{l = 1}^{K}{{{{\bf{\bar W}}}^{Q\left({K - l + 1}\right)}}{{\boldsymbol{\xi}}^{(l+M)}}}.
I^w_u=\sum\limits_{i=1}^{N}m_i\hat d_i^2=I_u\cdot \frac{1}{1-\langle\textbf{u}_0,\textbf{w}_0\rangle^2}=\frac{\langle I_O\textbf{u}_0,\textbf{u}_0\rangle}{1-\langle\textbf{u}_0,\textbf{w}_0\rangle^2}.
R_{\max}^{\rm{N}}= W{\log_2}\left({1 + \frac{{\left({P -{p_K}\left({\overline \varepsilon ,W}\right)}\right){{\left|{{{\widetilde h}_b}}\right|}^2}}}{{{p_K}\left({\overline \varepsilon ,W}\right){{\left|{{{\widetilde h}_b}}\right|}^2}+ W{N_0}}}}\right).
r_{corr}=r[1+\frac{1}{n}(\frac{1}{m_{x}}-\frac{s_{xy}}{m_{x}m_{y}})+\frac{1}{n^{2}}(\frac{2}{m_{x}^{2}}-\frac{s_{xy}}{m_{x}m_{y}}[2+\frac{3}{m_{x}}]+\frac{s_{x^{2}y}}{m_{x}^{2}m_{y}})]
W_+^{\lambda}W_-^{\mu}\phi \left(x\right) = \frac{\sin\left(\mu \pi\right)}{\sin\left( \left(\lambda +\mu\right) \pi \right)}W_-^{\lambda + \mu}\phi \left(x\right) + \frac{\sin\left(\lambda \pi \right)}{\sin\left( \left(\lambda +\mu\right) \pi \right)}W_+^{\lambda + \mu}\phi \left(x\right).
\begin{array}{@{}l@{}}{\displaystyle |\mathrm{Re}\big(\sum_{k=1}^{M}J_2^{(k)}\big)|\leq |\sum_{k=1}^{M}J_2^{(k),1}| +|\mathrm{Re}\big(\sum_{k=1}^{M}J_2^{(k),2}\big)|}begin{align*}2mm]{\displaystyle \leq C\big(h^{2r+2}+(\Delta t)^{4}\big)+C \|\theta_{\Psi}^{M}\|_{\mathcal{L}^2}^{2}+ C\Delta t \sum_{k=1}^{M-1}{\|\theta_{\Psi}^{k}\|_{\mathcal{L}^2}^{2}}.}\end{array}
E\left[\left(S^k_{j,+}(t)-\bar{S}^k_{j,+,\Delta}(t)\right)^2\right] &=\nu\Delta\sum_{\ell=0}^{\lfloor t/\Delta\rfloor}p^k_{j,+}(t-\ell\Delta)E[Z_0(\ell\Delta)] &+2\sum_{\ell_1<\ell_2}E\left[W^k_{\ell_1\Delta,t}(j)\left(W^k_{\ell_2\Delta,t}(j)-\nu\Delta Z_0(\Delta\ell_2)p^k_{j,+}(t-\Delta\ell_2)\right)\right].
m_{i,j}:= m_{j 2^{\ell-i}}m_{j 2^{\ell-i}+ 1}\cdots m_{(j+1)2^{\ell-i}-1}, A_{i,j}:= A_{j 2^{\ell-i}}A_{j 2^{\ell-i}+ 1}\cdots A_{(j+1)2^{\ell-i}-1}, C_{i,j}:= V A_{i,0}\cdots A_{i,j-1}\bmod m_{i,j}.
\sum_{i=1}^{M}\frac{U_i(-1)^{j}}{(\beta_{i,\xi}+\theta_k)^{j+1}}+\sum_{i=1}^{M}\frac{P_i}{(\gamma_{i,q}-\theta_k)^{j+1}}-\sum_{i=1}^{M}\frac{H_i(-1)^j}{(\beta_{i,q}+\theta_k)^{j+1}}e^{\beta_{i,q}(b-y)}=0.