harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\theta(p+z|\tau)\theta(p+p'+z'|\tau)\theta(p'|\tau)\theta(z+z'|\tau)-\theta(p'+z'|\tau)\theta(p+z+z'|\tau)\theta(z|\tau)\theta(p+p'|\tau) +\theta(p+p'+z+z'|\tau)\theta(p'-z|\tau)\theta(p|\tau)\theta(z'|\tau)=0
[\phi(d^{n_1}e^{n_2}), \phi(a^{n_1}b^{-n_2}c^{n_3})]= \phi([d^{n_1}e^{n_2}, a^{n_1}b^{-n_2}c^{n_3}])= \phi(c^{n_1^2+n_2^2}).
\begin{aligned}1-\frac{2\sqrt{2}r}{\sqrt{y}}\leq Q(\Lambda_{(z,\theta)}, r) \leq 1-\frac{\sqrt{2}r}{\sqrt{y}}\sqrt{2}r\leq \sqrt{y}; Q(\Lambda_{(z,\theta)},r)=0 \sqrt{2}r\geq \sqrt{y}.\end{aligned}
\phi(X)=\left[\begin{array}{cc}vec(X(1)) \cdots vec(X(N))\end{array}\right],\psi(X)=\left[\begin{array}{cc}svec(X(1)) \cdots svec(X(N))\end{array}\right],\forall X=(X(1),\cdots,X(N))\in S_n^N,
\int_0^\infty K_\nu(x)x^{s-1}dx = 2^{s-2}\Gamma\left(\frac{s+\nu}{2}\right)\Gamma\left(\frac{s-\nu}{2}\right), \Re s > |\Re \nu|, \int_0^\infty \cos(x)x^{s-1}dx = \Gamma(s)\cos\left(\frac{\pi s}{2}\right), 0 <\Re s < 1.
\frac{\langle h_{1},...,h_{a}+1,...,h_{\mathsf{N}}|h_{1},...,h_{a}+1,...,h_{\mathsf{N}}\rangle}{\langle h_{1},...,h_{a},...,h_{\mathsf{N}}|h_{1},...,h_{a},...,h_{\mathsf{N}}\rangle}=\prod_{\substack{b=1 b\neq a}}^{\mathsf{N}}\frac{X_{a}^{(h_{a})}-X_{b}^{(h_{b})}}{X_{a}^{(h_{a}+1)}-X_{b}^{(h_{b})}},
d^2(\alpha_i) = d \left( -\sum_{i< j}\alpha_ix_{i,j}\right) = -\sum_{i< j}d(\alpha_i)x_{i,j}- \sum_{i< j}\alpha_id(x_{i,j}) = \sum_{i< j}\sum_{i< r}\alpha_ix_{i,r}x_{i,j}- \sum_{i<r<j}\alpha_ix_{i,r}x_{r,j}= \sum_{i<r<j}\alpha_ix_{i,r}x_{r,j}- \sum_{i<r<j}\alpha_ix_{i,r}x_{r,j}= 0;
\sum_{k\in\mathbb{T}_{j}}\int\limits_{\gamma(t_{j},\varepsilon)}a_{k}(t)\Psi_{k,t}(x)dt=\int\limits_{(t_{j}-\varepsilon,t_{j}+\varepsilon)}\sum_{k\in\mathbb{T}_{j}}a_{k}(t)\Psi_{k,t}(x)dt.
\left(J_b^\alpha f_n\right)(x)= & \sum_{k=0}^n \alpha^k C_n^k x^{n-k}\left[\frac{1}{k+1}\left(\frac{b-x}{\alpha}\right)^{k+1}E_1\left(\frac{b-x}{\alpha}\right)-\frac{k!}{k+1}e_k\left(\frac{b-x}{\alpha}\right)e^{\frac{-(b-x)}{\alpha}}+1\right],
\begin{array}{l}{{F = u_{0}+ v = \alpha + \bar{\beta},}}\\{{\quad q u a d \leftrightarrow \quad v = W = \alpha + \bar{\beta}- u_{0},}}\\{{\bar{F}= u + v_{0}= \bar{\alpha}+ \beta ,}}\\{{\quad \quad \leftrightarrow \quad u = \bar{W}= \bar{\alpha}+ \beta - v_{0}.}}\\ \end{array}
\frac{1}{2\pi i}\,\intop_{\frac{1}{2}-\mu-i\infty}^{\frac{1}{2}-\mu+i\infty}\Gamma(z)\,\Gamma(s-z)\,\zeta_{p}(2z)\,x^{2z}dz=\sqrt{\pi}\,x\sum_{m=1}^{\infty}\,\frac{1}{2\pi i}\,\intop_{\mu-i\infty}^{\mu+i\infty}\,\Gamma\left(z\right)\,\Gamma\left(s+z-\frac{1}{2}\right)\,\left(2z,2\pi pm\right)_{m}\,(\pi xm)^{-2z}\,dz,
lim_{n\rightarrow\infty}n(1-\sqrt[n]{x})=lim_{n\rightarrow\infty}\frac{1-\sqrt[n]{x}}{1/n}=lim_{n\rightarrow\infty}\frac{-ln(x)\sqrt[n]{x}\frac{-1}{n^{2}}}{\frac{-1}{n^{2}}}=-ln(x)
c^{2}-b^{2}=\left(\frac{-x^{2}-2dx+d^2}{2y}\right)^{2}-\left(\frac{x^{2}+d^2}{2y}\right)^{2}& = \left(\frac{2d^{2}-2dx}{2y}\right)\left(\frac{-2x^2-2dx}{2y}\right) & = \left(\frac{d^{2}-dx}{y}\right)\left(\frac{-x^2-dx}{y}\right) & = \frac{dx^3-d^{3}x}{y^2}& = \frac{d(x^3-d^{2}x)}{x^3-d^{2}x}& = d.
\left(\alpha(q)\alpha(q^5)\right)^{1/2}+\left(\left(1-\alpha(q)\right)\left(1-\alpha(q^5)\right)\right)^{1/2}&+2\left(16\alpha(q)\alpha(q^5)\left(1-\alpha(q)\right)\left(1-\alpha(q^5)\right)\right)^{1/6}= 1.
\rho \ddot{u}_i = \big( A_{iJKj}u_{j,K}- \beta_{Ji}\dot{\tau}-(C_{iJKLIj}u_{j,IL}+ M_{iJKL}\tau_{,L})_{,K}\big)_{,J}-E(|\dot{u}|\nu_m) \dot{u}_i a \ddot{\tau}=-\beta_{Ki}\dot{u}_{i,K}+ m_{IJ}q_{I,J}+ M_{jLKI}u_{j,LKI}+K_{IJ}\tau_{,IJ}, \kappa \dot{q}_i = \dot{\tau}_{,i}- q_{i}
V = \left(\frac{b}{6}\right)^2 \frac{|z|^4}{(9/\gamma)^2 (1-|z|^2)^2}\left\{3\left| 1+\ln(c\omega(S)h(T) e^{-\tilde{K}/2}z^3)\right|^2 + C_2 |z|^2 \right\}
\; = \frac{1}{Z}\sum_{n=0}^\infty\frac{1}{n!}\Bigg\langle\left( \sum_{b=1}^N m^{(b)}\int_\Lambda d^2 x \; t(x) \Big[\overline{\psi}^{(b)}(x) P_+ \psi^{(b)}(x) +\overline{\psi}^{(b)}(x) P_- \psi^{(b)}(x) \Big] \right)^n
S_{0}^{+ Q}L = \{\left( \begin{matrix}{0}&{- i}\\{- i}&{0}\\ \end{matrix}\right) , \left( \begin{matrix}{i}&{0}\\{0}&{i}\\ \end{matrix}\right) , \left( \begin{matrix}{- 1}&{0}\\{0}&{1}\\ \end{matrix}\right) , \left( \begin{matrix}{0}&{- i}\\{i}&{0}\\ \end{matrix}\right) \}
{\bf S}_2 \,{\bf \Xi}^{-1}\,{\bf D}_\xi^{(k)}(x|\lambda)={\bf S}_2 \,{\bf S}_1 \,{\bf \Lambda B}^{(k)}\left(\frac{x}{\lambda}\right)={\bf \Lambda B}^{(k)}\left(\frac{x}{\lambda}\right).
(f.\phi)(\gamma): =f(\gamma)\alpha_{\gamma}(\phi(s(\gamma))) \textrm{for $\gamma \in \mathcal{G}$, and}\langle f,g \rangle(x): = (f^{*}*g)(x) = \int \alpha_{\gamma}(f(\gamma^{-1})^{*})\alpha_{\gamma}(g(\gamma^{-1}))d\lambda^{(x)}(\gamma) \textrm{for $x \in \mathcal{G}^{(0)}$}.
R_{\kappa \lambda \mu \nu}\,=\,R_{\kappa \mu}g_{\lambda \nu}-R_{\lambda \mu}g_{\kappa \nu}+R_{\lambda \nu}g_{\kappa \mu}-R_{\kappa \nu}g_{\lambda \mu}-\frac{R}{2}\left(g_{\kappa \mu}g_{\lambda \nu}-g_{\kappa \nu}g_{\lambda \mu}\right).
C_1[f,g,h] = \sum\limits_{N_1\sim N_2}\partial_x^{-1}P_{N_1}(\partial_xf_{N_2}g_{\leq N_2}h_{\leq N_2}),C_2[f,g,h] = \sum\limits_{N_1\sim N_2}\partial_x^{-1}P_{N_1}(f_{N_2}g_{\leq N_2}\partial_xh_{\leq N_2}),
\Xi = \left\{\xi\in U \,\left|\, \exists \Omega\subset A\times A \ \mbox{with}\begin{array}{l}N(\Omega,\delta)\geq\delta^{\epsilon}N(A,\delta)^2 \mbox{and}\N(\pi_\xi(\Omega),\delta)\leq\delta^{-\epsilon}N(A,\delta)\end{array}\right.\right\},
{\cal Z}_{o p e n , B P S}^{p - p}\int_{0}^{\infty}\frac{d s}{s}\left[ \mathrm{T r}_{\mathrm{N S}}\left( \frac{1 + ( - 1 )^{F}}{2}\right) q^{2 L_{0}- 1}- \mathrm{T r}_{\mathrm{R}}\left( \frac{1 + ( - 1 )^{F}}{2}\right) q^{2 L_{0}}\right] ~ ~ ,
\beta' := \frac{d \beta}{p(d-1)}\, , \beta_0 := \max\{\beta,\beta'\}, \mathcal{E}_{r,\beta,d,B}^p := \{\chi_K \cdot g \,:\, g \in \mathcal{F}_{\beta',d,B}K \in \mathcal{K}_{r,\beta,d,B}\}\, .
\frac{\varphi_{n + 1}- 2 \varphi_{n}+ \varphi_{n - 1}}{T^{2}}+ \left( \mu^{2}- \nabla^{2}\right) \frac{\left( \varphi_{n + 1}+ 4 \varphi_{n}+ \varphi_{n - 1}\right)}6 ={}_{{}_{\! \! \! \! \! \! \! c}}j_{n}.
C_{k \pm \mathbf e_1}= C_{k-[\frac{k-1}{L}] L \pm 1,[\frac{k-1}{L}] +1}, k\neq 0,1\, \mod L,(\pm), C_{k \pm \mathbf e_2}= C_{k-[\frac{k-1}{L}] L ,[\frac{k-1}{L}] +1 \pm 1}, k\leq L(M-1),\k\geq L+1, (\pm).
d_{\Omega_M}(\omega,\tilde{\omega}) = \sum_{i=1}^{\infty}\frac{1}{2^i}\frac{\sup_{0\leq t\leq i}d_M(\omega_t,\tilde{\omega}_t)}{1+ \sup_{0\leq t\leq i}d_M(\omega_t,\tilde{\omega}_t)}.
\frac{\partial \Phi}{\partial \nu}(\tfrac{1}{2},\kappa,\nu) = (1-\kappa) \ln(\tfrac{1}{2}) (\tfrac{1}{2})^{1+\nu}- \ln (\tfrac{1+\kappa}{2}) (\tfrac{1+\kappa}{2})^{1+\nu}& \geq (\tfrac{1}{2})^2 ((1-\kappa) \ln(\tfrac{1}{2}) - \ln (\tfrac{1+\kappa}{2}) (1+\kappa)^{1+\nu}),
\frac{1}{2}\sum_{i=1}^{m}\sum_{j=1}^{m}\sum_{\alpha=1}^{n}\sum_{\beta=1}^{n}g^{ij}\frac{\partial f^{\alpha}}{\partial x^{i}}\frac{\partial f^{\beta}}{\partial x^{j}}(h_{\alpha\beta}\circ f).
\lim_{N \to \infty}\frac{1}{N^{1+m}}\sum^N_{i_1, \ldots, i_{2m}=1}\prod_{\{x,y\}\in s_\chi^{-1}\cdot \pi}\delta_{i_{x}, i_{y + 1}}\delta_{i_{y}, i_{x + 1}}c_{\chi(s_\chi(x)), \chi(s_\chi(y))}= 0
o_\lambda(\mathbf{x}^{\pm};1)=\frac{\det(x^{\lambda_j+n-j+\frac{1}{2}}_i-x^{-(\lambda_j+n-j+\frac{1}{2})}_i)^{n}_{i,j=1}}{\det(x^{n-j+\frac{1}{2}}_i-x^{-(n-j+\frac{1}{2})}_i)^{n}_{i,j=1}}
\lefteqn{|X^N(t)-X^N(s)|^2}&=2\int_s^t(X^N(u)-X^N(s),\sigma(X^N(u))dB^N(u))+2\int_s^t(X^N(u)-X^N(s),b(X^N(u)))du &+2\int_s^t(X^N(u)-X^N(s), d\Phi^N(u)) &\le2\int_s^t(X^N(u)-X^N(s),\sigma(X^N(u))dB^N(u))+2\int_s^t(X^N(u)-X^N(s),b(X^N(u)))du & =:I_1+I_2,
|\partial_{t}^{2}v_{4}^{0}(t,r)|\leq \begin{cases}\frac{C}{t^{3}\log^{2N+b-2}(t)}, r\leq \frac{t}{2}\frac{C}{t^{2}\log^{3N+b}(t)}, r \geq \frac{t}{2}\end{cases}
\int_{\mathbb{R}^{2N}}\frac{|u_n(x) - u_n(y)|^{p-2}\, (u_n(x) - u_n(y))\, (\varphi(x) - \varphi(y))}{|x - y|^{N+s\,p}}\, dx\, dy=\int_{\Omega}\frac{f_n(x)}{(u_n+1/n)^\gamma}\,\varphi\, dx,
a_1^4(1+a_1^2+b+b^2+b^3)^2k+a_1^2(1+a_1^4+a_1^4b^2+a_1^2b^3+a_1^2b^7+b^8) =\, (a_1^2k^2+a_1^2bk+b+b^3)h_1+h_2=0,
\varphi_{k}:= -1 + \sum_{i=1}^{k}\alpha_{i}^{k}w_{i}, \mu_{k}:= \mu_{\infty}+ \sum_{i=1}^{k}\beta_{i}^{k}w_{i}, \sigma_{k}:= \sigma_{\infty}+ \sum_{i=1}^{k}\tau_{i}^{k}w_{i},
C_{i_{1}i_{2}... i_{n}}= \sqrt{\frac{[r_1]_{q^2}!}{\left([r_1]_{q}!\right)^3}\cdot \frac{[r_2]_{q^2}!}{\left([r_2]_{q}!\right)^3}\cdot ... \cdot\frac{[r_m]_{q^2}!}{\left([r_m]_{q}!\right)^3}}\mbox{\quad with \quad}m\le n
\frac{d}{dt}\mathcal{H}(\rho|\rho^\infty)= -\beta\frac{d}{dt}\mathcal{\bar F}(\rho(t))=\beta(\nabla\bar F, \nabla \bar F)_\rho = \beta\sum_{(i,j)\in E}[(\bar F_j(\rho)-\bar F_i(\rho))_+]^2\rho_i = \beta\sum_{(i,j)\in E}[(\log\frac{\rho_i}{e^{F_i(\rho)/\beta}}-\log \frac{\rho_j}{e^{F_j(\rho)/\beta}})_+]^2\rho_i\ .
\mathfrak{S}_{\delta, \flat}^{\tau_k}(H', M, N) := \frac{1}{H'}\sum_{h\sim H'}a_h\mathop{\sum_{m\sim M}\sum_{n\sim N}}_{D<mn\le 2D}\tau_{k-1}(m) \Big(\frac{hx}{mn+\delta}\Big), \mathfrak{S}_{\delta, \dagger}^{\tau_k}(M, N) := \mathop{\sum_{m\sim M}\sum_{n\sim N}}_{D<mn\le 2D}\tau_{k-1}(m) R_H\Big(\frac{x}{mn+\delta}\Big).
\int_{0}^{\infty}{\frac{z^{\nu - 2 m}}{\sinh{( \pi z )}}F_{1}( z ) \bar{J}_{\nu}( z ) d z}= A_{\nu m}^{( 0 )}\left[ \frac{z F_{1}( z )}{\sinh{( \pi z )}}\right] + \frac{2}{\pi}\sum_{k = 1}^{\infty}( - 1 )^{m + k}k^{\nu - 2 m}\bar{K}_{\nu}( k ) F_{1}( i k ) ,
=\int_{r>0}^{R}\frac{2r}{R^2}\mathcal{L}_{I_{\phi\setminus M_0}}(T r^\alpha \eta \frac{P_m}{P_p})\mathcal{L}_{I_{\psi\setminus P_0}}(Tr^\alpha) \exp\{-Tr^\alpha \frac{P_m}{P_p}{q}^{-\alpha}{h_0}\}r.
\mathcal{L}_{\boldsymbol{\lambda}}(f\ast_{\boldsymbol{\lambda}}g)=\mathcal{L}_{\boldsymbol{\lambda}}(f)\mathcal{L}_{\boldsymbol{\lambda}}(g), \mathcal{B}_{\boldsymbol{\lambda}}(fg)=\mathcal{B}_{\boldsymbol{\lambda}}(f)\ast_{\boldsymbol{\lambda}}\mathcal{B}_{\boldsymbol{\lambda}}(g),
\frac{1}{2}\int d^{4}x k_{a b}{\bf B}^{a}({\bf \dot{A}}^{b}+ \mathrm{\boldmath ~ \a l p h a^{b}~}) - \frac{1}{2}\sum_{n}k_{a b}q_{n}^{a}\int{\bf A}^{b}( z_{n}) \cdot d{\bf z}_{n}= \int d^{4}x \left( \partial_{\mu}V^{\mu}+ \frac{1}{2}\, k_{a b}\mathrm{\boldmath ~ \a l p h a^{a}\b e t a^{b}~}\right) \ .
N_{(a,b)}(n) = ~ 8\sigma(\frac{n}{a}) - 32\sigma(\frac{n}{4a}) + 8\sigma(\frac{n}{b}) - 32\sigma(\frac{n}{4b}) + 64\, W_{(a,b)}(n) + 1024\, W_{(a,b)}(\frac{n}{4}) - 256\, \biggl( W_{(4a,b)}(n) + W_{(a,4b)}(n) \biggr).
\sum_{j=1}^{n+1}\| \varphi_j(x) \widetilde{U^j}\|_{C^{t_j, \alpha}(B_{\frac{1}{2}}(0))}&\le C\left( \sum_{j=1}^{n+1}\| (\phi(x))^{t_j+s_j}\varphi_j(x) \widetilde{(LU)_j}\|_{C^{-s_j, \alpha}(B_1(0))}+ \sum_{j=1}^{n+1}\| \varphi_j(x) \widetilde{U^j}\|_{L^2(B_1(0))}\right).
\mathcal{B}_\pm(U, -\varGamma) = \ker (- \varGamma \pm 1) \cap \ker (-C + 1) & = \ker (\varGamma \mp 1) \cap \ker (C - 1)= \mathcal{T}_\pm(U,\varGamma), \mathcal{T}_\pm(U, -\varGamma) = \ker (- \varGamma \mp 1) \cap \ker (-C - 1) & = \ker (\varGamma \pm 1) \cap \ker (C + 1) = \mathcal{B}_\pm(U,\varGamma),
[\begin{matrix}Y_{1}(s)\\Y_{2}(s)\end{matrix}]=[\begin{matrix}H_{11}(s)&H_{12}(s)&H_{13}(s)\\H_{21}(s)&H_{22}(s)&H_{23}(s)\end{matrix}][\begin{matrix}U_{1}(s)\\U_{2}(s)\\U_{3}(s)\end{matrix}].
\frac{1}{|B_{r}|}\int_{B_{r}}|X_{i}X_{j}h(x)|dx =\frac{1}{r^{Q}|B_{1}|}\int_{B_{1}}|\left( X_{i}X_{j}h\right) (D_{r}(y))|r^{Q}dy\ & =\frac{1}{|B_{1}|}r^{-2}\int_{B_{1}}|X_{i}X_{j}\widetilde{h}(y)|dy
\zeta_{R,p}(s) = \frac{1+px^2+p^2x^3-p^4x^5-2p^5x^6-2p^6x^7-p^7x^8+p^9x^{10}+ p^{10}x^{11}+p^{11}x^{13}}{(1-p^5x^6)(1-p^4x^4)(1-p^3x^3)(1-p^2x^2)(1-x)}
{F_{T_w}}\left( t \right) = \Pr \left({{T_w}< t}\right) = \Pr \left({{\kappa^2}+{C_{2w}}Z_w < t}\right) &= \Pr \left({Z_w < \frac{{t -{\kappa^2}}}{{{C_{2w}}}}}\right) ={F_{Z_w}}\left({\frac{{t -{\kappa^2}}}{{{C_{2w}}}}}\right),
{\cal H}= \frac{1}{2}\pi_{\varphi}^{2}+ \frac{1}{2}\left( \partial_{1}\varphi \right)^{2}+ \frac{1}{2}\mu^{2}\varphi^{2}+ \frac{1}{2}\left( \frac{\phi}{\mu}\right)^{2}+ \left( \frac{\phi}{\mu}\right) ( \partial_{1}\varphi ) \ ,
\tilde{P}_{n,\phi}^{i+1}:=\min\left\{\frac{\bar{S}_n}{\sqrt{(\breve{P}_{n,\phi}^i)^2+(\breve{Q}_{n,\phi}^i)^2}},1\right\}\breve{P}_{n,\phi}^i \tilde{Q}_{n,\phi}^{i+1}:=\min\left\{\frac{\bar{S}_n}{\sqrt{(\breve{P}_{n,\phi}^i)^2+(\breve{Q}_{n,\phi}^i)^2}},1\right\}\breve{Q}_{n,\phi}^i.
A_{r}^{a}= \frac{1}{L_{\perp}\sqrt{4 \pi}}\sum_{\lambda = \pm 1}\sum_{k = 1}^{\Lambda}\frac{1}{\sqrt{k}}\left( a_{k , \lambda}^{a}\epsilon_{r}^{- \lambda}e^{- 2 \pi i k x^{-}/ L}+{a_{k , \lambda}^{a}}^{\dag}\epsilon_{r}^{\lambda}e^{2 \pi i k x^{-}/ L}\right) .
H^1_D(\Omega^{(4)}\setminus \Gamma^{(4)}(t_0)) :=\{v \in H^1(\Omega^{(4)}\setminus \Gamma^{(4)}(t_0)) \ :\v = 0 \ \hbox{on}\partial_D \Omega^{(4)}\}\,, &{\mathcal H}:= \{v \in H^2(\Omega^{(4)}\setminus \Gamma^{(4)}(t_0)) \ : \eqref{bc-v}\ \hbox{hold true}\}\oplus \{k \zeta S\ :\k\in \mathbb R\}\,,
P_i^{(m)}Q_i^{(n)}\oplus \bigoplus_{k \geq 1}[ \bigoplus_{l=0}^k (P_i^{(m-k)}Q_i^{(n-k)}\langle k-2l \rangle \oplus P_i^{(m-k)}Q_i^{(n-k)}\langle k-2l \rangle \lbrace 1 \rbrace) \oplus &\bigoplus_{l=0}^{k-2}(P_i^{(m-k)}Q_i^{(n-k)}\langle k-2l-2 \rangle \oplus P_i^{(m-k)}Q_i^{(n-k)}\langle k-2l-2 \rangle \lbrace 1 \rbrace)]
d s_{1 , 3}^{2}= - d \tau^{2}+ d r^{2}+{\frac{x_{1}x_{2}}{y_{3}}}d \psi^{2}+ \Big ({\frac{x_{2}x_{3}}{y_{1}}}\cos^{2}\psi +{\frac{x_{1}x_{3}}{y_{2}}}\sin^{2}\psi \Big ) d \theta^{2}- \sum_{i , j = 1}^{4}A_{i}h_{i j}^{- 1}A_{j}.
\Phi(x) - \frac{1}{2}= \frac{1}{\sqrt{2 \pi}}\int_0^x e^{-\frac{1}{2}z^2}dz = \frac{1}{\sqrt{2 \pi}}\int_0^x \underset{i=0}{\overset{\infty}{\sum}}\frac{(-1)^i z^{2i}}{2^i i!}dz = \frac{x}{\sqrt{2 \pi}}\Big(\underset{i=0}{\overset{\infty}{\sum}}\frac{(-1)^i ( x/\sqrt{2})^{2i}}{(2i+1) i!}\Big),
\frac{1}{N}\sum_{n=1}^N F_1(U^{an}(x, y, z)) F_2(U^{bn}(x, y, z))\exp \left[ -12\pi i (b-a)(bq_2y\alpha - 3bq_2 + c_2) \alpha^2n^2 + c_1\alpha^2n + \alpha^2c_0 \right] &= \frac{C_{y, z}}{N}\sum_{n=1}^N f_1(T^nx) f_2(T^{2n}x) \exp(2\pi i [12bq_2(b^2-a^2) n^3 + c_2n^2 + c_1n + c_0]\alpha^2)
r_n^{[3]}(h):=\frac{2^{3p+3}\cdot y_{8n}\left(\frac{h}{8}\right)-7\cdot 2^{2p+1}\cdot y_{4n}\left(\frac{h}{4}\right)+7\cdot 2^{p}\cdot y_{2n}\left(\frac{h}{2}\right)- y_n(h)}{\left(2^p-1\right) \left(2^{p+1}-1\right)\left(2^{p+2}-1\right)}.
7 + 7 = 1 4 1 4 + 7 = 2 1 2 1 + 7 = 2 8 2 8 + 7 = 3 5 3 5 + 7 = 4 2 4 2 + 7 = 4 9 4 9 + 7 = 5 6 5 6 + 7 = 6
\phi ( x^{0}\, , \, \vec{x}\, ) \, = \, c \, \int d^{n}x^{\prime}{\frac{( x^{0})^{n}}{\left( ( x^{0})^{2}\, + \, ( \vec{x}\, - \,{\vec{x}}^{\prime}\, )^{2}\, \right)^{n}}}\phi_{0}({\vec{x}}^{\prime}) \, \, .
\left\lfloor (F_n + 1) \phi \right\rfloor = F_n + 1 + \left\lfloor (F_n + 1) \tau \right\rfloor &= F_n + 1 + \left\lfloor F_{n-1}- (-\tau)^n + \tau \right\rfloor &= F_n + 1 + F_{n-1}+ \left\lfloor \tau - (-\tau)^n \right\rfloor &= F_{n+1}+ 1
\widetilde{\tau}_B \left( \widetilde{\theta}_B (y,t) \right) = \widetilde{\tau}_B \left( \theta_{B,1}(y), \theta_{B,2}(y), \theta_{B,2}(y) - t \right) &= \left( \tau \left( \theta (y) \right) , \theta_{B,2}(y) - \left( \theta_{B,2}(y) - t \right) \right) &= (y,t) \; .
\left( \begin{matrix}{{\cos \alpha}}&{{- \sin \alpha}}&{{0}}\\{{\sin \alpha}}&{{\cos \alpha}}&{{0}}\\{{0}}&{{0}}&{{1}}\\ \end{matrix}\right) \left( \begin{matrix}{{A_{1}}}\\{{A_{2}}}\\{{A_{u}}}\\ \end{matrix}\right) = \left( \begin{matrix}{{A_{1}\cos \alpha - A_{2}\sin \alpha}}\\{{A_{1}\sin \alpha + A_{2}\cos \alpha}}\\{{A_{u}}}\\ \end{matrix}\right) ,
\sup_{x\in B_{\sigma}(x_{0})}|F_{A}| \leq\sup_{x\in B_{r_{0}+\sigma}(p)}|F_{A}|(x) &\leq\frac{(1-2r_{0})^{2}}{(1-2r_{0}-2\sigma)^{2}}\sup_{x\in B_{r_{0}}(p)}|F_{A}|(x)=4b.
\begin{aligned}\|v_i \|_{C^{(1+\alpha)/2,1+\alpha}}\leq C r^{-1-\alpha}\|\nabla v\|_{L^p(D_T)}&+ C\big(r^{-1-2\alpha}+r^{-2}\big)\big(\|v_{0}\|_{C^{1+\alpha}}+\|f\|_{L^p(D_T)}+\|g \|_{C^{\alpha/2,\alpha}}\big).\end{aligned}
I = \int_{\Omega}\left( \frac{1}{2}\epsilon_{\mu \nu \lambda}B_{\mu}^{*}( \partial_{\nu}- i g a_{\nu}) B_{\lambda}+ M B_{\mu}^{*}B^{\mu}+ \frac{1}{2}\epsilon_{\mu \nu \lambda}a_{\mu}\partial_{\nu}a_{\lambda}\right) \; .
[r_{21},r_{32}] =[r^{ij}X_{j}\otimes X_{i}\otimes 1,r^{lk}1\otimes X_{k}\otimes X_{l}]= -r^{ij}r^{lk}X_{j}\otimes \mathrm{ad}^{(r)}_{X_{i}}X_{k}\otimes X_{l}&=-r^{ij}r^{lk}X_{j}\otimes [X_{k},X_{i}]\otimes X_{l}=-r^{ij}r^{lk}f_{ki}\,^{s}X_{j}\otimes X_{s}\otimes X_{l},
A_{1,\phi_{1}}(t,k) = (\sqrt{2\pi})^3\int_{0}^{t}\phi_{1}(k) \, ik \, e^{-isk^{2}}\Big(\int \overline{\mathcal{K}}^{\#}_{R}(x,k) \, s \, u_{M_1}(s,x) u_{M_2}(s,x) u_{M_3}(s,x) \, dx \Big) ds.
\delta R_{\theta \theta}= \frac{1}{1 2}g_{\theta \theta}\delta [ e^{\frac{8}{3}\lambda}F^{( K ) 2}+ e^{- \frac{4}{3}\lambda + 4 \nu}F^{2}+ e^{- \frac{4}{3}\lambda - 4 \nu}H^{2}] .
(\overline\alpha - \underline\alpha) \int_D (\hat u_\epsilon - m)\,dx = \int_\Omega h_\epsilon(x)\,d\alpha dx = \int_\Omega \frac{\alpha |\nabla_x u_\epsilon|^2 + \epsilon^2 |(u_\epsilon)_\alpha|^2}{u_\epsilon^2}>0.
y^{(m)}_{r_1,\ldots,r_k}=\Bigl(\prod_{i=1}^k\mu(r_i)g(r_i)\Bigr)\sum_{\substack{d_1,\ldots,d_k r_i\mid d_i\forall i d_m=1}}\frac{\lambda_{d_1,\ldots,d_k}}{\prod_i\varphi(d_i)},
= \, \frac{1}{2}\int_{0}^{l}d x \left( \dot{y}^{2}\, + \, 2 a_{3}\dot{y}\stackrel{\ldots}{y}\, - \, a_{2}\dot{y}\dot{y}^{\prime \prime}\, - \, a_{3}\ddot{y}^{2}\, + \, a_{1}y^{2}\right) .
V ={x^{p-2}}\,{y^{p-2}}-{x^{p-4}}\,{y^{p-4}}\,\left({x^3}+{y^3}\right) +{x^{p-6}}\,{y^{p-6}}\,\left({x^6}+{y^6}\right) + \cdots +x\,y\,\left({x^{3\,{p-3\over2}}}+{y^{3\,{p-3\over2}}}\right)
\frac{(\log m)^2}{m}\max_{(k'-1)m < \ell \le k' m}\{\overline{R}_{\ell}\}\le \sum_{j=1}^{k'}D_j^{(m)}+ \frac{(\log m)^2}{m}\Big( k' \varphi_m - \varphi_{k' m}\Big)+ c' \,,
L(x,y) \,=\, L^0(x,y) + L^{111}(x,y) + L^{112}(x,y) + L^{12}(x,y) \,=\, \frac{x \otimes y + x^\bot \otimes y^\bot}{4\pi |x|^{2}}+ \frac{-3(x \otimes y) + x^\bot \otimes y^\bot}{8\pi |x|^{2}}+ \frac{x \otimes y}{4\pi |x|^{2}}- \frac{x \otimes y + x^\bot \otimes y^\bot}{8\pi |x|^{2}}\,.
- \frac{e^{i \frac{3}{4}\pi \mathrm{s i g n}\left( \frac{H}{P}\right)}}{2 \sqrt{| P |}}\mathrm{s i g n}( P ) e^{- \frac{i \pi}{2 K}\phi}\int_{- \infty}^{+ \infty}d \beta \, e^{- \frac{i \pi}{2 K}\frac{H}{P}\beta^{2}}\frac{\prod_{i = 1}^{3}2 i \sin \frac{\pi \beta}{p_{i}}}{2 i \sin \frac{\pi \beta}{K}}.
\left( N^{q}\sin^{q}\left( at\right) \right) \left( u;v\right) =\frac{1}{uv}\sum\limits_{0}^{\infty}\left( -1\right)^{n}a^{2n+1}\frac{u^{2n+1}}{v^{2n+1}}q^{-2n\frac{\left( 2n-1\right)}{2}}.
\bar{J}=\xi_1\|u_1\|_{L^2}^2+\xi_2\|u_2\|_{L^2}^2, \ell_1= \|u_1\|_{L^2}^2(2Q+\sigma_1^2-|\xi_1|^2), \ell_2= \|u_2\|_{L^2}^2(2Q+\sigma_2^2-|\xi_2|^2).
B_1(m) = - p^{1-3m/2}(-1)^m \chi_{\psi}(p^m) \int_{\mathcal B_1(m)}(c,p^m)_p\chi_{\psi}(c) e(h) \chi_{\delta}(c)|c|^{-3/2}_p\mathbf h_p(h) dh,
\sum_{m=\ell_2}^{\infty}\sum_{k=\max\{0,\ell_1+\ell_2-m\}}^{\ell_1}A_{m,k}=\sum_{k=0}^{\ell_1}\sum_{m=\ell_1+\ell_2-k}^{\ell_1+\ell_2}A_{m,k}+\sum_{k=0}^{\ell_1}\sum_{m=\ell_1+\ell_2+1}^{\infty}A_{m,k},
nR \stackrel{(a)}{\leq}nC_1+nC_2 -\sum_i \left[I(X_{1i}X_{2i};U_i|U^{i-1})+I(X_{2i};U_i|U^{i-1}X_{1i})+I(X_{1i};U_i|U^{i-1}X_{2i})\right] &= nC_1+nC_2 -n I(X_{1I}X_{2I};U_I|U^{I-1}I)+nI(X_{2I};U_I|U^{I-1}X_{1I}I)+nI(X_{1I};U_I|U^{I-1}X_{2I}I) &\stackrel{(b)}{=}nC_1+nC_2 -n I(X_{1I}X_{2I};U_I|Q)+nI(X_{2I};U_I|X_{1I}Q)+nI(X_{1I};U_I|X_{2I}Q).
|I_{R_i}|=|V(R_i)\setminus (Q_{R_i}\cup S_{R_i})| =|V(R_i)|+|S_{R_i}|-(|Q_{R_i}|+2|S_{R_i}|) &>|V(R_i)|+|S_{R_i}|-\left(\frac{1}{3}- 3\sqrt \delta \right)v(F).
\frac{\partial}{\partial y}H(t,y)=\displaystyle\int_{\left\{x:\,w(x,y)>t\right\}}\frac{\partial}{\partial y}f(x,y)\,d\gamma(x)+\displaystyle\int_{\left\{x:\,w(x,y)=t\right\}}\frac{\partial}{\partial y}w(x,y)\frac{f(x,y)}{|\nabla_{x}w|}\varphi(x)\,d\mathcal{H}^{n-1}(x).
f_4 =1728^{-1/6}\eta^{8}K^{1/6}\ \_{3}F_2(1/6, 1/2, 5/6; 5/4, 3/4; K) = f_2^2 - 16f_3^2, f_5 =1728^{1/12}\eta^{8}K^{-1/12}\_{3}F_2(-1/12, 1/4, 7/12; 3/4, 1/2; K)= f_1f_2, f_6 =1728^{-5/12}\eta^{8}K^{5/12}\ \_{3}F_2(5/12, 3/4, 13/12; 5/4, 3/2; K) = f_1f_3.
\bar{\cal K}[A]={\cal K}[A]-\frac{1}{2i}{\rm Tr}\ln A^{-1}=\int{\cal D}\varphi \left.e^{-\frac{1}{2}\varphi A^{-1}\varphi-\frac{i\lambda}{4!}\varphi^4}\right|_{\rm excl. \; 1VR}
\frac{d}{dt}F_r(\phi,\psi)\leq C\cdot s^{-1}\sum_{l=0}^{2r+4}F_r(\phi,\psi)^{l/2+1}+C(n-2) \dot{s}s^{1-n}\sum_{l=0}^{r}F_r(\phi,\psi)^{l/2+2},
g(p) \triangleq \frac{1-p}{2p}\log\left(\frac{1}{1-p}\right) + H(p) &= \frac{1-p}{2p}\log\left(\frac{1}{1-p}\right) + p \log\left( \frac{1}{p}\right) + (1-p) \log\left( \frac{1}{1-p}\right)
\left[ \frac{\partial^{2}}{\partial \xi^{2}}+ 2 \, \kappa \, \delta \, \frac{s}{c}\frac{\partial}{\partial \xi}- \left\{\frac{1}{\kappa^{2}}- \kappa^{2}\delta ( 1 + \delta ) \right\}\frac{s^{2}}{c^{2}}+ ( \varepsilon + \kappa^{2}\delta ) \right] \Psi ( \xi ) = 0 \; ,
\Theta_{\tilde{\rho},\rho}(F) + \widetilde{\Theta}_{\tilde{\rho},\rho}(F) = \Xi_{\tilde{\rho},\rho, 2^{k_0}}(F) - \Xi_{\tilde{\rho},\rho, 2^{k_m}}(F), \Xi_{\tilde{\rho},\rho,t}(F) \leq \|\tilde{\rho}\|_{\textup{L}^1(\mathbb{R})}\|\rho\|_{\textup{L}^1(\mathbb{R})}.
\overset{0}{R}(X,Y,AZ,AW) = \overset{0}{R}(AX,AY,Z,W), \overset{0}{R}(X,AY,AZ,W) = \overset{0}{R}(AX,Y,Z,AW), \overset{0}{R}(AX,AY,AZ,AW) = \overset{0}{R}(X,Y,Z,W).
x_1=x_0+F(x_0)+l\chi_1\ge x_0+l- \frac{l\varepsilon}{2C}+l\left(-1+\frac{\varepsilon}{C}\right)=x_0+ \frac{l\varepsilon}{2C}>u_l-\delta+ \frac{l\varepsilon}{2C}_l \geq u_l.
Q(X,L,\tau)=\left\{\left(\prod_{j=1}^{2k+1}\frac{x_j\theta'(0,\tau)}{\theta(x_j,\tau)}\right)\frac{\sqrt{-1}\theta(u,\tau)}{\theta_1(0,\tau)\theta_2(0,\tau)\theta_3(0,\tau)}\right\}^{(4k+2)}.
{\cal L}^{( 0 )}= \frac{\kappa}{2 \pi}\epsilon^{m n}A_{n}\dot{A}_{m}+ \pi_{a}\dot{z}_{a}+ \pi_{a}^{*}\dot{z}_{a}^{*}+ \Omega_{1}^{( 0 )}\dot{\alpha}-{\cal H}^{( 0 )}, m , n = 1 , 2 ,
R_{\ell}( r ) = C_{\ell}\; \left( 1 -{\frac{r_{0}}{r}}\right)^{i k r_{0}}j_{\ell}( k r ){}_{2}F_{1}( - \ell + 2 i k r_{0}\; , \; - \ell \; ; \; - 2 \ell \; ; \; r_{0}/ r \, )
\mathcal{B}_{n-1}^{\ell+1}-\mathcal{A}_n^\ell=\mathcal{B}_n^{\ell-1}-\mathcal{A}_{n+1}^{\ell-1},(\mathcal{B}_n^{\ell+1}-\mathcal{A}_{n+1}^\ell)\mathcal{B}_n^\ell=\mathcal{B}_{n+1}^{\ell-1}(\mathcal{B}_n^\ell-\mathcal{A}_{n+1}^{\ell-1}).
\Theta(D_e;D_{\bar e})= -\int_0^{\infty}dl' \frac{df(D_e^{\kappa}l')}{dl'}f(D_{\bar e}^{\kappa}l') =-\int_0^{\infty}dz \frac{df(z)}{dz}f\left(\left(\frac{D_{\bar e}}{D_e}\right)^{\kappa}z\right).
\begin{array}{c}\left[ A^{-},A^{+}\right]_{q\left( x\right)}=-2\alpha \left( x\right)\sinh \frac{i}{2}\frac{d}{dx}\left[ \alpha \left( x\right) \sinh 2\rho_{\frac{s}{2}}\left( x\right) \right] \rightarrow \rightarrow \frac{d^{2}\rho \left( x\right)}{dx^{2}}\end{array}
1 \leq |\cos(\omega_{N-1}-\omega_{N-1}'+\xi)|\sin^2(\psi_{N-1})+|\cos((N-1)(\omega_{N-1}-\omega_{N-1}'))|\cos^2(\psi_{N-1}) &\leq \sin^2(\psi_{N-1})+\cos^2(\psi_{N-1})=1.
\nabla\times\mathbf{A}=\frac{1}{r^{2}\sin\theta}\begin{vmatrix}\begin{array}{ccc}\mathbf{e}_{r}r\mathbf{e}_{\theta}r\sin\theta\mathbf{e}_{\phi}\frac{\partial}{\partial r}\frac{\partial}{\partial\theta}\frac{\partial}{\partial\phi}A_{r}rA_{\theta}r\sin\theta A_{\phi}\end{array}\end{vmatrix}
\sum_{j=1}^\infty |c_j|^q\frac{\mu(D(z_j,r))}{(1+|z_j|)^{mq}}\lesssim\|\mu\|^q\|(c_j)\|_{\ell^p}^q= \|\mu\|^q \Big(\sum_{j=1}^\infty |c_j|^p\Big)^{q/p}.
\tilde{\mathfrak{gl}}^*_kF_{\overrightarrow{\beta}}\xrightarrow{\varphi}Z\times_{(I_{\mu}(X^{ss}))^k}\prod_{i=1}^kF_{*,d_i}\xrightarrow{pr_Z}Z\rightarrow\tilde{Q}^{\epsilon_{+}}_{n+k}([E^{ss}/G],\overline{\beta}).
\frac{1}{\lambda^2}\log \left( \int e^{-t \lambda Lf}d\mu \right) = \frac{1}{\lambda^2}\log \left( 1 + \sum_{k=2}^\infty \frac{(-t \lambda)^k}{k!}\int (Lf)^k d\mu \right) (\mbox{since}\int Lfd\mu=0) & \leq \sum_{k=2}^\infty \frac{t^k \lambda^{k-2}}{k!}\int |Lf|^k d\mu (\mbox{since}\log(1+x) \leq x) & \leq \int e^{t |Lf|}d\mu .
V_{x}= 3 \,{x^{8}}+ 8 \,{x^{7}}\,{y^{2}}- 7 \,{x^{6}}\,{y^{4}}+ 6 \,{x^{5}}\,{y^{6}}- 4 \,{x^{3}}\,{y^{7}}+ 2 \, x \,{y^{8}}= 0
\begin{array}{rcl}{{J}_{v}}({{x}_{0}},u,[{{t}_{0}},{{t}_{f}}]) = g({{t}_{0}},{{x}_{0}},{{t}_{f}},{{x}_{f}}) & +\int_{{{t}_{0}}}^{{{t}_{f}}}{{{L}_{v}}(t,x,u)dt}\end{array}
f(10) = f(2)\,f(5) * = f(1^2+1^2+2^2+2^2) = 2f(1)+2f(4) = 10 f(12) = f(4)\,f(3) = 4f(3) * = f(1^2+1^2+1^2+3^2) = 3+f(9) f(15) = f(3)\,f(5) * = f(1^2+1^2+2^2+3^2) = 6+f(9) f(18) = f(2)\,f(9) * = f(1^2+2^2+2^2+3^2) = 9+f(9) f(20) = f(4)\,f(5) = 4f(5) * = f(1^2+1^2+3^2+3^2) = 2+2f(9),
f^{(j)}(x) = \sum_{k=0}^j \binom{j}{k}(p)_k (x - \lambda)^{p-k}g^{(j-k)}(x) & = (x - \lambda) \sum_{k=0}^j \binom{j}{k}(p)_k (x - \lambda)^{p-1-k}g^{(j-k)}(x)