harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
U_{\theta}^{c,\gamma}(-1)= \begin{cases}2+2\sqrt{1+c_{1}}, \mathrm{if}\;\gamma=\gamma^{+}(c), 2-2\sqrt{1+c_{1}}, \mathrm{otherwise},\end{cases}U_{\theta}^{c,\gamma}(1)= \begin{cases}-2-2\sqrt{1+c_{2}}, \mathrm{if}\;\gamma=\gamma^{-}(c), -2+2\sqrt{1+c_{2}}, \mathrm{otherwise}.\end{cases}
\alpha = \alpha(P_{i+1}^{(1)},P_{i+1}^{(2)}) + \alpha(P_i^{(2)},P_{i+1}) = (1-\chi(c - x') - \chi(x'-1-d) + \chi(c -d - 1)) + (-\chi(c-b-1)+\chi(c-x)+\chi(d-x+1)) = 1+\chi(c-d-1)-\chi(c - b - 1).
S_{+}= \int d^{4}x\sqrt{-g^{0}}\frac{1}{2}V_{0}^{+}\Phi\Bigr|_{y=0}+\int d^{4}x\sqrt{-g^{\pi r_{c}}}\frac{1}{2}V_{\pi}^{+}\Phi\Bigr|_{y=\pi r_{c}}\ ,
\kappa_0^{(2r+1)}(x)=\frac{t^{2 r}\left(t-x^2\right) h_0\left(a,c,x/t^{r}\right)}{\left(t^{2 r+1}-x^2\right) h_0(a,c,x)}, \kappa_n^{(2r+1)}(x)=\frac{\left(t-x^2\right) h_n\left(b,d,t^r/x\right)}{\left(t^{2 r+1}-x^2\right) h_n\left(b,d,1/x\right)}.
\begin{cases}x^0,y^0\in C, x^{n+1}=\underset{y\in C}{\textup{argmin}}\bigg\{\lambda_n f(y^n,y)+\dfrac{1}{2}\|y-x^n\|^2\bigg\}, y^{n+1}=\underset{y\in C}{\textup{argmin}}\bigg\{\lambda_n f(y^n,y)+\dfrac{1}{2}\|y-x^{n+1}\|^2\bigg\},\end{cases}
M_{n}= \left({\frac{\int d^{2}p ( F \Phi_{n})^{2}}{\int d^{2}p G ( p ) F \Phi_{n}F ( - 2 C \Phi_{n}^{3}+ j \Phi_{n}( u^{2}+ v^{2}- w )_{n})}}\right)^{\alpha}\, .
\sum_{k=K+1}^{m_{n_{{l_{q}}_{m}}}}\frac{2\mu_{k}\left(C_{n_{{l_{q}}_{m}},k}- (n_{{l_{q}}_{m}}-1)\mathbb{I}_{k=2}\right)- \mu_{k}^2}{4k}\left| \mathbb{P}_{n_{{l_{q}}_{m}}}\right. \stackrel{d}{\to}M_{1}-M_{2,K}.
x_{n}\sum_{a=1}^{\mathsf{N}}g_{a}(\zeta_{n}^{(1)})x_{a}+x_{n}f(\zeta_{n}^{(1)})=q_{n},q_{n}=\frac{\mathrm{det}_{q}K_{+}(\xi_{n})\mathrm{det}_{q}\,\mathcal{U}_{-}(\xi_{n})}{\eta -4\xi_{n}^{2}},\forall n\in \{1,...,\mathsf{N}\},
d\Big(F_\beta\big(P^{\mathcal{P}(\textbf{x}, h),\mathcal{P}(\textbf{x}, h)}(\textbf{y}'_{\mathcal{P}(\textbf{x}, h)})\big),\,e\Big) =d\big(\prod_{i=m}^2 p_{\beta,a_i}(f_i',f'_{i-1}),e\big)\leq\sum_{i=2}^md(p_{\beta,a_i}(f_i',f'_{i-1}),e),
\langle \Phi(z_1,\dots,z_N,\{\overline{\alpha}\})| \overline{x_1}\cdots \overline{x_N}\rangle = t^{N(M-N)}\prod_{j=1}^N z_j^{j-1-N}(1+tz_j^2)\prod_{1 \le j<k \le N}(1+tz_j z_k)(1+tz_j z_k^{-1})sp_{\overline{\lambda}}( \{tz \}_N |\{-\overline{\alpha}\}).
\frac{1}{n!}\int_a^{b}\cdots\int_a^b \det \left[ f_j(x_k) \right]_{1\leq j,k\leq n}\det \left[ g_j(x_k)\right]_{1\leq j,k\leq n}\prod_{1\leq j\leq n}(x_j)\, x_1\cdots x_n =\det \left[ \int_a^b f_j(x)g_k(x) (x) x \right]_{1\leq j,k\leq n}.
h=\left(\begin{array}{cc}1 0 0 -1 \end{array}\right)\,,\ \ \x=\left(\begin{array}{cc}0 1 0 0 \end{array}\right)\,,\ \ \y=\left(\begin{array}{cc}0 0 1 0 \end{array}\right)\ \ \ \ \ \ \ \ [h,x]=2x\,,\ \ [h,y]=-2y\,,\ \ [x,y]=h\ .
D(-2,\varepsilon) =\sigma_{\varepsilon}(-2I)=\sigma_{\varepsilon}([I\bullet iI,\frac{iI}{2}]_{\ast}) &=\sigma_{\varepsilon}([\psi(I)\bullet \psi(iI),\psi(\frac{iI}{2}))]_{\ast})=\sigma_{\varepsilon}(4i\psi(\frac{iI}{2})).
f\left( t,x,1\right) =-\frac{1}{2}x-\left \vert \sin t\right \vert\sqrt[4]{\left \vert x\right \vert}\sqrt[3]{x},\;f\left( t,x,2\right)=-\frac{1}{2}x, g\left( t,y,1\right) =-\sqrt{\left \vert b\left( t\right) \right \vert}y\cos t,\;g\left( t,y,2\right) =\sqrt{\left \vert b\left( t\right)\right \vert}y\sin t,
H_{D}^{f e r}= \frac{1}{\sqrt{2}}\int d^{3}x \xi^{\dagger}( \vec{x}) \left( \frac{1}{i \partial_{-}- e V_{-}}\ast \xi \right) ( \vec{x}) + \frac 1 2 \frac{e^{2}}{2 L}\int d^{2}x_{\perp}\Gamma^{i}( x_{\perp}) \left({\cal G}_{( \perp )}[{\cal M}^{2}] \ast \Gamma^{i}\right) ( x_{\perp})
I_{\alpha}(n) = \frac{\sqrt{2\pi}u}{\sqrt{(1+u)n}}\left(\frac{u}{e^{1/u}}\right)^n \left( 1+ \sum_{r=1}^{R-1}\frac{a_r(u)}{n^r}+ O\left( \frac{(\log n)^R}{n^R}\right) \right)
\nu_k(t) =\left|\{({\bf x}^1,{\bf x}^2,\ldots,{\bf x}^k)\in E_1\times \ldots \times E_k: \|{\bf x}^1+\cdots +{\bf x}^k\|=t\}\right| &=\sum_{({\bf x}^1,{\bf x}^2,\ldots,{\bf x}^k)\in E_1\times E_2 \times \cdots \times E_k}S_t({\bf x}^1+{\bf x}^2+ \cdots+{\bf x}^k).
\boldsymbol{R_v}\boldsymbol{R}+\boldsymbol{R}\boldsymbol{R_v}- 2 \mu \boldsymbol{R}\boldsymbol{R_v}\boldsymbol{R}& \;\;\; = \mu \boldsymbol{R}\left( \sigma^2_{\rm n}+ \left\{\boldsymbol{R}\boldsymbol{R_v}\right\}\right) +\frac{1}{\mu}\lim_{k\to\infty}\left( \boldsymbol{B}_k -\boldsymbol{A}_k \right).
U_{\alpha}(\omega_{j,n}) = \frac{P'_{\alpha}(\omega_{j,n})}{f'_{\beta}(\omega_{j,n})}\neq 0 \, ,U_{\alpha}(\overline{\omega_{j,n}}) = \frac{P'_{\alpha}(\overline{\omega_{j,n}})}{f'_{\beta}(\overline{\omega_{j,n}})}\neq 0 \,\mbox{and}~~U_{\alpha}(\frac{1}{\beta}) = \frac{P'_{\alpha}(\frac{1}{\beta})}{f'_{\beta}(\frac{1}{\beta})}\neq 0.
|K_{U}^{(D)}(x,y;t)-K_{0}(x,y;t)|\leq (4\pi t)^{-d/2}\exp({-\frac{|x-y|^2+4\delta^2}{4t}})\sum_{j=1}^{d}\frac{2^j\delta^{2j-2}}{(j-1)!t^{j-1}}.
\sqrt{a_{11}^{2}+a_{21}^{2}+2a_{11}a_{21}\frac{a_{12}^{2}+a_{22}^{2}-(a_{11}^{2}+a_{21}^{2})}{4a_{11}a_{21}}}=\sqrt{a_{11}^{2}+a_{21}^{2}+\frac{a_{12}^{2}+a_{22}^{2}-(a_{11}^{2}+a_{21}^{2})}{2}}& =\sqrt{\frac{a_{12}^{2}+a_{22}^{2}+a_{11}^{2}+a_{21}^{2}}{2}}& =2^{-\frac{1}{2}}\sqrt{a_{12}^{2}+a_{22}^{2}+a_{11}^{2}+a_{21}^{2}}.
\det\mathcal{T}_{\nu,\theta}=\det\Big\{P^{\top}\mathcal{T}_{\nu,\theta}P\Big\}\cr = \det\Big\{(1-\theta)\big\{(1-\nu)T_{tr}Id+\nu P^{\top}\Theta P\big\}+\theta T_{\delta}Id\Big\}\cr = \prod_{1\leq i\leq 3}\big\{(1-\theta)\big((1-\nu)T_{tr}+\nu\Theta_i\big)+\theta T_{\delta}\big\}.
\|Lu\|_{Z}=\|Lu-\psi+\psi\|_{Z}\leq \|Lu-\psi\|_{Z}+\|\psi\|_{Z}& \leq C_{1}M^{p}+C_{2}\varepsilon & \leq cM+C_{2}\varepsilon_{0}& =cM+C_{2}\frac{M(1-c)}{C_{2}}& =M,
f_{L}=\frac{K_{1}+\frac{4K_{2}}{(K_{1}+\frac{K_{1}K_{2}}{K_{1}^{4}+3K_{2}})^{3}}}{1+\frac{3K_{2}}{(K_{1}+\frac{K_{1}K_{2}}{K_{1}^{4}+3K_{2}})^{4}}}
\nabla \tilde{D}(x)=\nabla \tilde{B}(x,x)+\nabla (\tilde{C}x)=\left(\begin{array}{ccc}0 \tilde{B}_1 x_3 \tilde{B}_1 x_2 \tilde{B}_2 x_3 0 \tilde{B}_2 x_1 \tilde{B}_3 x_2 \tilde{B}_3 x_1 0\end{array}\right)+\tilde{C},
\frac{3m^2}{4}\leq \sum_{i=1}^{\frac{m}{2}}|A+_iA| & = \sum_{x \in A+A}\min\left(\frac{m}{2}, r_{A+A}(x)\right) & \leq \sum_{x \in (A+A) \cap A}\frac{m}{2}+ \sum_{x \in (A+A) \setminus A}r_{A+A}(x)
a_{S}^{\dagger}( k ) \, a_{S}^{\dagger}( k^{\prime}) - q^{- 2}\sum_{l , l^{\prime}\in{\cal I}}{\hat{\cal R}}\left( k , k^{\prime}; l , l^{\prime}\right) a_{S}^{\dagger}( l ) \, a_{S}^{\dagger}( l^{\prime}) = 0 \; ,
\left(\bigcap_{i<k}X_i\right)^c \subseteq \left(\bigcap_{i < k}X_{i,\sigma(i)}\right)^c = \left(\bigcap_{i < k}X_{i,m_{i,n}^{\infty}}\right)^c = \bigcup_{i < k}[\mathbf{I_{\sigma \upharpoonright i,n}^{\infty}}]
G''(x) =p(p-1)x^{p-2}\left[\frac{(2-p)\gamma^{p-1}+\gamma^{p-2}}{p-1}-\left(\frac{1-x}{x}\right)^{p-2}\right] &\geq p(p-1)x^{p-2}\left[\frac{(2-p)\gamma^{p-1}+\gamma^{p-2}}{p-1}-\gamma^{p-2}\right] &=p(2-p)(\gamma x)^{p-2}(1+\gamma)\geq 0.
B_{n}^{+}:=\bigl\{L_{n}^{+}(T_{n,+})\geq2^{1-\bar{\eta}_{\mathrm{c}}n}\,n^{1/(1+\beta)-\varepsilon}\bigr\},B_{n}^{-}:=\bigl\{L_{n}^{-}(T_{n,-})\leq2^{-\bar{\eta}_{\mathrm{c}}n}\,n^{1/(1+\beta)-\varepsilon}\bigr\}
\delta_{t o t}| \phi \rangle = \Bigl ( \xi \partial + \frac{d - 2}{2 z}\xi^{z}+ \frac{1}{2}\partial^{I}\xi^{J}M^{I J}+ M^{I J}\partial^{+}\xi^{I}\frac{\partial^{J}}{\partial^{+}}- \frac{\partial^{+}\xi^{I}}{2 z \partial^{+}}\{M^{z j}, M^{j I}\}\Bigr ) | \phi \rangle \, .
\left| \int\limits_0^1 x \varphi_{yyy}(x,y) J_0(\gamma_m x) dx \right| = \left| \int\limits_0^1 \sqrt{x}(\sqrt x \, \varphi_{yyy}(x,y)) J_0(\gamma_m x) dx \right| \leq \frac{C_8}{(\gamma_m)^{5/2}}\max_{x, y \in [0,1]}\left(\sqrt x\,\varphi_{yyyxx}(x,y)\right) \leq \frac{C_9}{m^{5/2}}
\left\Vert f\right\Vert_{\boldsymbol{B}_{p(\cdot ),q(\cdot )}^{\alpha(\cdot )}}:=\left\Vert \Phi \ast f\right\Vert_{p(\cdot )}+\left\Vert\left\Vert t^{-\alpha (\cdot )}(\varphi_{t}\ast f)\right\Vert_{p(\cdot)}\right\Vert_{L^{q(\cdot )}((0,1],\frac{dt}{t})}<\infty .
L_{1}^{\ast\ast}=\frac{\Gamma\left((m+1)/2\right)\sqrt{m}\left[\xi_{p}^{3}\left(1+\frac{\xi_{p}^{2}}{m}\right)^{-(m-1)/2}-\xi_{q}^{3}\left(1+\frac{\xi_{q}^{2}}{m}\right)^{-(m-1)/2}\right]}{\Gamma(m/2)(m-1)\sqrt{\pi}F_{Y}(\xi_{p},\xi_{q})},
\mathbb E[(\xi_1+ \ldots+\xi_{t_1})^p \mid X_0^{(0)}=x<0] \leq \mathbb E[({\xi_1^+}+ \ldots+{\xi_{t_1}^+})^p\mid X_0^{(0)}=x<0] \leq \mathbb E[(\xi_1^+)^p+\ldots +(\xi_{t_1}^+)^p\mid X_0^{(0)}=x<0] = \mathbb E[(\xi_1^+)^p] \, \mathbb E[t_1\mid X_0^{(0)}=x<0]<+\infty
\gamma A_{0}^{a}= \eta_{1}^{a}, \gamma A_{i}^{a}= \partial_{i}\eta_{2}^{a}+ f_{b c}^{a}\eta_{2}^{b}A_{i}^{c}, \gamma B^{0 i}= \eta_{1}^{i}, \gamma B^{i j}= \partial^{\left[ i \right.}\eta_{2}^{\left. j \right]},
\left(v_{n}^{\varepsilon}\right)_{t}+A\Delta v_{n}^{\varepsilon}-\rho_{\varepsilon}v_{n}^{\varepsilon}=e^{\rho_{\varepsilon}\left(t-T\right)}F\left(x,t;e^{\rho_{\varepsilon}\left(T-t\right)}v_{n}^{\varepsilon}\right)+\mathbf{P}_{\varepsilon}^{\beta}v_{n}^{\varepsilon}\in\left(H^{1}\left(\Omega\right)\right)',
y_i =\left(1-\frac{1}{\theta_{i,N}}\right)x_{i-1}+\frac{1}{\theta_{i,N}}x_0 d_i =\left(1-\frac{1}{\theta_{i,N}}\right)f'(x_{i-1})+\frac{2}{\theta_{i,N}}\sum_{j=0}^{i-1}\theta_{j, N}f'(x_j) x_{i}=y_i- \frac{1}{L}d_i
M(n)=\sum_{k=1}^{2^{2^{n}}}\prod_{j=1}^{2^{n}-1}\prod_{i=0}^{j-1}(1-b_{i}^{k}b_{j}^{k}\prod_{m=0}^{log_{2}i}(1-b_{m}^{i}+b_{m}^{i}b_{m}^{j})),
\frac{1}{{}_2 F_1(1,N;N+1;-x)}=\frac{(-1)^{N-1}x^N/N}{\log(1+t)-\sum_{n=1}^{N-1}(-1)^{N-1}x^n/n}=\sum_{n=0}^\infty c_{N,n}\frac{x^n}{n!}\,,
S_0 = \left(\begin{array}{ccccc}1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \end{array}\right) \;\;\;\; B = \left(\begin{array}{ccccc}0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 \end{array}\right) .
\gamma_m = \gamma_m(\chi, \sigma, T,r,\kappa) := \gamma_{\ref{TheoremExistenceAndUniquenessOnDomain},m}(\eta(\chi),R_1(\chi,\sigma,m,R(\chi,\sigma,m,r,\kappa),\kappa),T), C_m = C_m(\chi,\sigma,T,r) := C_{\ref{TheoremExistenceAndUniquenessOnDomain},m}(\eta(\chi), R_1(\chi,\sigma,m,R(\chi,\sigma,m,r,\kappa),\kappa),T),
N = \int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}d t H \simeq \frac{- 1}{M_{P l}^{2}\sqrt{1 + 4 \alpha \Lambda_{b}/ 3 M^{5}}}\int_{\phi_{\mathrm{i}}}^{\phi_{\mathrm{f}}}d \phi{\frac{V}{V^{\prime}}}\left[ 1 +{\frac{V}{2 \Lambda_{2}^{+}}}\right] .
S ( d = 1 1 ) = - T \int d^{p + 1}\hat{x}| \hat{k}|^{\beta}| \hat{k}^{\prime}|^{\gamma}\sqrt{\mid \partial_{i}\hat{X}^{\hat{\mu}}\partial_{j}\hat{X}^{\hat{\nu}}\hat{\Pi}_{\hat{\mu}\hat{\nu}}\mid}.
I_{1}= \int_{T_{\epsilon}^{2}}d^{2}z \partial_{\bar{z}}\overline{{\rho}}\partial_{z}\rho = \frac{1}{2 i}\int_{T_{\epsilon}^{2}}\bar{\partial}\overline{{\rho}}\wedge \partial \rho = \frac{1}{2 i}\int_{| z | = \epsilon}d z \overline{{\rho}}\partial_{z}\rho ,
\Pi_{\mu\nu}^{(I)}={e^{2}C^{(I)}\over 2}\, \int{d^{n}k\over(2\pi)^{n}}\,{(1-\cos p\times k)\over k^{2}(k+p)^{2}}\,N_{\mu\nu}^{(I)};\;\;\;\;\;\;\;\;\;\;\; I =a,b,c
\frac{1}{m}\sum_{j=0}^{m-1}\frac{e^{\frac{2\pi i r}{m}j}}{z+\sin\left(\frac{2\pi j}{m}\right)}= e^{-i\pi \ell/2}\cdot\frac{U_{m-\ell-1}(z) + e^{i\pi m/2}U_{\ell-1}(z)}{T_m(z) - \cos(m\pi/2)}.
R_\phi[\pi_\phi(u)\pi_\phi(b_1) \pi_\phi(u) \pi_\phi(b_2) \cdots \pi_\phi(u) \pi_\phi(b_n)] = R_\phi[\pi_\phi(ub_1ub_2\cdots ub_n)] = \phi(u b_1 u b_2 \cdots u b_n) = R[\beta(u b_1 u b_2 \cdots u b_n)] = R[ U \alpha(b_1) U \alpha(b_2) \cdots U \alpha(b_n)],
\log F ( A , \! B ) \! = \! - \! T_{2}\! \left( \alpha l \! + \! A_{0}\right) / \alpha \! - \! T_{2}\! \left( \beta l \! + \! B_{0}\right) / \beta \! + \! T_{2}\! \left( ( \alpha \! + \! \beta ) l \! + \! D_{0}\right) / ( \alpha \! + \! \beta ) ,
\Biggl.-{\bf 1}_{\{i_6=i_5\ne 0\}}{\bf 1}_{\{j_6=j_5\}}{\bf 1}_{\{i_1=i_3\ne 0\}}{\bf 1}_{\{j_1=j_3\}}{\bf 1}_{\{i_2=i_4\ne 0\}}{\bf 1}_{\{j_2=j_4\}}\Biggr),
\hat \theta_{{\rm{MLE}}}^{(u)}= \tau -{F^{- 1}}\left[{1 - \frac{1}{{\left({M + K}\right)N}}\sum\limits_{j \in{\cal N}}{\left({\sum\limits_{m = 1}^M{{\check{u}}_j^{(m)}}+ \sum\limits_{i = 1}^K{u_j^{(i)}}}\right)}}\right]
k=\frac{(a-1) \ell (a+\ell)}{2 a(\ell-a^2)},\lambda=\frac{(a+\ell) (3 a^2+a\ell-a-3 \ell)}{4 a(\ell-a^2)},\mu=\frac{(a-1) \left(\ell^2-a^2\right)}{4 a(\ell-a^2)}.
+ \frac{1}{2}h^{- 5 / 3}v^{\hat{I}}\partial_{\hat{J}}h \bar{\theta}\Gamma_{\tilde{I}}^{0 \tilde{J}}\theta + h^{- 1 / 6}v^{\hat{I}}\bar{\theta}( \Gamma_{\tilde{I}\tilde{1}}\partial_{2}- \Gamma_{\tilde{I}\tilde{2}}\partial_{1}) \theta + \cdots \Big ] d^{3}\zeta
\textstyle\mathbf{S_{_\mathrm{SM}}}=\mathbf{V}\left(\frac{\mu_{_{\rm SM}}}{\ln2}\left(\nu_{_{\rm SM}}\,\mathbf{I}_r-\rho_{_{\rm SM}}\,\boldsymbol{\Lambda}^{\rm H}\boldsymbol{\Lambda}\right)^{-1}-\frac{\sigma^2}{1-\rho_{_{\rm SM}}}\,\mathbf{\Lambda}^{-2}\right)\mathbf{V}^{\rm H}.
\|g_0\| + \|g_1\|+\cdots + \|g_m\| & = \|b_1\|+ (\|a_1\|-\|a_{2}\|) + \cdots + (\|a_m\|-\|a_{m+1}\|) & = \|b_1\|+ \|a_1\|-\|a_{m+1}\| = \|g\|.
F(\tau(g,x)) =F(gx)=(f_{g_1}(gx),\ldots,f_{g_n}(gx))= (f(g_1(gx)),\ldots,f(g_n(gx))) &= (f((g \ldotp g_1)x),\ldots, f((g\ldotp g_n)x))= (f_{g\ldotp g_1}(x) ,\ldots, f_{g\ldotp g_n}(x)) & = \tau_2(g, (f_{g_1}(x),\ldots, f_{g_n}(x)))= \tau_2(g,F(x)).
N(q)=-\bigg(\cos\theta+\lambda(q_2-x_2)\bigg)X_1(q)-\bigg(\sin\theta-\lambda(q_1-x_1)\bigg)X_2(q), &T(q)=-\bigg(\sin\theta-\lambda(q_1-x_1)\bigg)X_1(q)+\bigg(\cos\theta+\lambda(q_2-x_2)\bigg)X_2(q), &Z(q)=f(q)X_3(q),
S \left[ \left( m + \frac{\delta}{2}\right) i w + q \right] = \frac{( 1 + \gamma_{3}) / 2}{[ - ( 1 / 2 ) \delta + i q_{3}+ \left|{\bf q}\right|^{2}/ 2 m ] + \Sigma_{+}( q_{3}, \left|{\bf q}\right| )}\, + \, O \left( \frac{\ln m}{m}\right)
d(F(x+ s_2e_k),F(x+s_1e_k)) - \varepsilon \leq d(F(x+ s_2e_k),F(x+s_1e_k)) - 2 d(F(x+ s_1e_k), p_i) &\leq d(F(x+ s_2e_k),p_i) - d(F(x+ s_1e_k), p_i) &=\phi_i(F(x+ s_2e_k)) - \phi_i(F(x+s_1e_k)) &= \int_{s_1}^{s_2}\frac{d}{d \tau}\, (\phi_i \circ F)(x+\tau e_k) \, d \tau &\leq \int_{s_1}^{s_2}g(x+\tau e_k) \, d\tau.
\mu\left\{x \in X \mid \left\lvert A^{}_{N}f^{N,\gamma}(x)\right\rvert \geq \frac{1}{N^\gamma}\right\}\leq C^{-2}\cdot N^{2\gamma}\cdot N^{-2\alpha''}\cdot 3\cdot \dim\left<K.f^{N,\gamma}\right>\cdot \|f^{N,\gamma}\|_{L^2(m)}^{2}\leq 3\cdot C^{-2}\cdot N^{3\gamma+2\epsilon-2\alpha''}\|f\|_{L^2(X,\mu)}^{2}.
p_{k+1}(y'',y_n,y_{n+1}) = p_k(y'',y_{n},y_{n+1}) + r_0^{(3+{2\alpha})k}q\left(\frac{y''}{r_0^{2k}}, \frac{y_n}{r_0^k}, \frac{y_{n+1}}{r_0^k}\right),
\mbox{Prod}\left(\mathbf{A}^{(1)}\otimes\mathbf{B}^{(1)},\cdots,\mathbf{A}^{(k)}\otimes\mathbf{B}^{(k)},\cdots,\mathbf{A}^{(m)}\otimes\mathbf{B}^{(m)}\right)=
\mathcal{A}_{\alpha\beta}^{ij}(p,\lambda):=2a_{\alpha\beta}^{ij}-\sigma_{\gamma\alpha}^{ik}\sigma_{\gamma\beta}^{jk}-(p-2)(\sigma_{\gamma\alpha}^{ik}-\lambda_{\gamma\alpha}^{ik})(\sigma_{\gamma\beta}^{jk}-\lambda_{\gamma\beta}^{jk})
\delta_{gt}|\Phi_{ph}\rangle=(\alpha^I\hat{\partial}^I+(s-1)\alpha^z)|\Lambda_0\rangle-\frac{\alpha_J^2}{2s+d-4}\Bigl((\bar{\alpha}^I\hat{\partial}^I+(s-1)\bar{\alpha}^z)|\Lambda_0\rangle+\hat{\partial}^+|\Lambda_1\rangle\Bigr)\,.
1 + 2 \sum_{m=1}^\infty \frac{B_{2m}}{(2m)!}\omega_m(n) x^{2m}= \exp \left( \sum_{m=2}^\infty \frac{B_m}{m!}\left(2 \sinh^{-1}\left(\frac{x}{2}\right)\right)^m \frac{J_m(n)}{m}\right) &= \prod_{d \mid n}\exp \left(\sum_{m=2}^\infty \frac{B_m}{m \cdot m!}\left(2d \sinh^{-1}\left(\frac{x}{2}\right) \right)^m \right)^{\mu(n/d)}.
\Phi^{\hbar}(z) =\Phi^{2\hbar}(z+i\pi \hbar)\, \Phi^{2\hbar}(z-i\pi \hbar) &=\Phi^{3\hbar}(z+2i\pi \hbar)\, \Phi^{3\hbar}(z)\, \Phi^{3\hbar}(z-2i\pi \hbar) &=\prod_{j=1}^k \Phi^{k\hbar}(z+(k-2j+1)i \pi \hbar) \;.
\mathbf{e}^{j}(l,m)=\frac{(-1)^{l}}{\mu_{l,m}^{n}}\{\displaystyle\sum\limits_{k=0}^{l-m}}(-1)^{k}\binom{l-m}{k}J_{-}^{l-m-k}J_{+}^{l}J_{-}^{k}\ ,
\sum_{n=0}^{\infty}p_{-6}\left(5^{j}n+\dfrac{3\times5^{j}+1}{4}\right)q^{n}=\sum_{l=1}^{\infty}b(j,l)q^{l-1}\dfrac{E_{5}^{6l}}{E_{1}^{6l+6}}.
P^{(i)}_{BPSK}=\frac{1}{\Gamma(p)}\int^{\infty}_{0}\lambda^{p-1}\mbox{exp}(-\lambda) &\times\left\{\frac{1}{\pi}\int^{\pi/2}_{0}\!\!{\left(1+\frac{a^2\alpha_j\bar{\gamma}_j\lambda}{Z_1(1+\frac{Z_2}{Z_1}\lambda)\mbox{sin}^2\theta}\right)}^{-1}{d\theta}\right\}d\lambda.
{D^{{\rm{opt}}}}= \frac{1}{{{{\left({\frac{{\omega \left({1 - \omega}\right)\left({{B_2}-{B_1}}\right) + \left({1 - \omega}\right)\left({{B_1}+{C_1}}\right)}}{{\omega \left({1 - \omega}\right)\left({{B_1}-{B_2}}\right) + \omega \left({{B_2}+{C_2}}\right)}}}\right)}^{\frac{1}{{v - 1}}}}+ 1}}
\sum_{n=1}^{N}a_{n}n^{-s}= \Big(\sum_{n=1}^{N}\frac{a_{n}}{n^{\sigma_{0}+s}}\Big) N^{\sigma_{0}}+ \sum_{n=1}^{N-1}\Big( \sum_{k=1}^{n}\frac{a_{k}}{k^{\sigma_{0}+s}}\Big) \big( n^{\sigma_{0}}- (n+1)^{\sigma_{0}}\big)
\mu^{(-,\lambda_0)}(\Lambda,W) =\sum_{0\leq\theta\leq\varepsilon}\dim\ker(d(\lambda_0)-e^{i(\pi+\theta)})-\sum_{0\leq\theta\leq\varepsilon}\dim\ker(d(0)-e^{i(\pi+\theta)}) &=\dim(\Lambda(\lambda_0)\cap W).
\begin{array}{ll}|\lambda \mathbf{I}_{5}- \mathbf{M}(\eta)| =\left|\begin{array}{ccc}\lambda ik_2\eta^{\top}0 ik_2\eta (\lambda+a)\mathbf{I}_3 0 0 0 \lambda \end{array}\right|= \lambda \left|\begin{array}{ccc}\lambda ik_2\eta^{\top}ik_2\eta (\lambda+a)\mathbf{I}_3 \end{array}\right| =0.\end{array}
\chi^{R}_{ll}\approx 0,\quad\chi^{R}_{lk}\approx \chi^{R}_{kl}, \quad \chi^{J}_{lk}\approx \chi^{J}_{kl}, \quad (l \neq k)\quad\sin\chi^{J}_{11}+\sin\chi^{J}_{22}+\sin\chi^{J}_{33}\approx 0.
T \left( \begin{smallmatrix}X \chi \end{smallmatrix}\right) = \left( \begin{matrix}\tilde{A}X + \tilde{B}\left[ \begin{smallmatrix}\chi(0) \chi(1) \end{smallmatrix}\right] c \chi_x \end{matrix}\right), T: \mathcal{D}(T) \to \mathcal{H},
\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right] \frac{(q)_{n-1}(q)_n(-q)_{N-n}}{(-q)_{n-1}(-q)_{N}}\frac{(-q)^n}{1-q^{2n}}=\sum_{n=1}^{\infty}\frac{n (-q)^n(1-q^{Nn})}{1- q^{n}}.
\binom{n-\ell}{r_k(\ell)-\ell}^{-1}\binom{n}{r_k(\ell)}=\frac{n\cdot \ldots\cdot(n-\ell+1)}{r_k(\ell)\cdot\ldots (r_k(\ell)-\ell+1)}\ge \left(\frac{n}{r_k(\ell)}\right)^\ell
\sum_{i=1}^\infty\sum_{x\in D_i}\eta_x f(x,0) \geq \sum_{m=1}^\infty \delta^{\alpha^{2i_m}}M_{i_m}\geq \sum_{m=1}^\infty \delta^{\alpha^{2i_m}}\exp\bigl(c\alpha^{2i_m}\bigr) &=\sum_{m=1}^\infty 1=\infty\,.
\sum_{P}( - 1 )^{P}{\cal J}^{- 2}( \theta ){\frac{\exp \{N \beta \sum_{k}\cos ( \theta_{k}- \theta_{P ( k )}+{\frac{2 \pi}{L}}) \}}{\prod_{i , j}\cos \bigl ({\frac{\theta_{i}- \theta_{P ( i )}+ \theta_{j}- \theta_{P ( j )}}{2}}\bigr )}},
\Gamma(\pi)\ge\frac{1}{N}\sum_{\ell=1}^N\left(\frac{\rho_\ell\beta_\ell^\pi}{2T_{\ell,\pi}^{av}}+\frac{\rho_\ell T_{\ell,\pi}^{av}}{2}+\frac{c_\ell}{T_{\ell,\pi}^{av}}\right),
^{\rho}I^{1-\gamma}u(0)=\frac{\Gamma(\gamma)}{\Gamma(\alpha)}\Omega\sum_{i=1}^{m}\omega_{i}\int_{0}^{\xi_{i}}\left(\frac{\xi_{i}^{\rho}-s^{\rho}}{\rho}\right)^{\alpha-1}s^{\rho-1}\left(f(s,u(s),^\rho D^{\alpha,\beta}u(s))-p(s)u(s)\right)ds.
\hat x_e= \sum_{x_e}x_e \mu_e(x_e|B,A) = \frac{\sum_{x_e}x_e \mu_e(x_e|B_{\backslash e},A) p_\theta(B_e|x_e)}{\sum_{x_e}\mu_e(x_e|B_{\backslash e},A) p_\theta(B_e|x_e)}
{\cal L}_{\mathrm{B I}}= - \sqrt{- \mathrm{d e t}\, \big ( \eta_{\mu \nu}+ F_{\mu \nu}+ \bar{\chi}\Gamma_{\mu}\partial_{\nu}\chi +{\textstyle{\frac{1}{4}}}\bar{\chi}\Gamma^{a}\partial_{\mu}\chi \bar{\chi}\Gamma_{a}\partial_{\nu}\chi \big )}\, \, .
\begin{array}{cc}Z_{-}(z)=C^{-1}\bar{Z}^{t}_{+}(z)\left(\begin{array}{cc}1 0 0 \bar{\cal E}\end{array}\right)~~ ~~\bar{Z}_{-}(z)=\left(\begin{array}{cc}1 0 0 -\bar{\cal E}\end{array}\right) Z_{+}(z){}^{t}C \end{array}
(L_0 G)(\eta) := - |\eta| G(\eta); (L_1 G)(\eta) := z \sum_{\xi\subset\eta}\int_{{\mathbb R}^d}e^{-E^\phi(x,\xi)}G(\xi\cup x)e_\lambda(e^{-\phi (x -\cdot)}-1,\eta\setminus\xi) dx.
\phi = \left( \begin{matrix}{\phi^{+}}\\{\phi^{0}}\\ \end{matrix}\right) = \frac{1}{\sqrt{2}}\left( \begin{matrix}{\chi^{2}( x ) + i \chi^{1}( x )}\\{v + \psi ( x ) - i \chi^{3}( x )}\\ \end{matrix}\right) ,
\big[R_{ijk}, Q^{(ij)}\big] = -2\big(Q^{(ij)}\big)^2-2\big\{Q^{(ij)}, Q^{(jk)}\big\}+2\big(Q^{(ijk)}+Q^{(i)}+Q^{(j)}+Q^{(k)}\big)Q^{(ij)}\hphantom{\big[R_{ijk}, Q^{(ij)}\big] =}{}-2\big(Q^{(i)}-Q^{(j)}\big)\big(Q^{(ijk)}-Q^{(k)}\big)
\begin{aligned}\beta_{ij}= p^2 \Big(\big|\mathcal{N}_i^+ \cap \mathcal{N}_j^+\big| - d_i^- \mathbb{I}_{\mathcal{N}_i^-}(v_j) - d_j^- \mathbb{I}_{\mathcal{N}_j^-}(v_i) \Big) - p(1-p) \Big(\mathbb{I}_{\mathcal{N}_i^-}(v_j) + \mathbb{I}_{\mathcal{N}_i^+}(v_j)\Big). \end{aligned}
\Delta_{\nu}(x)=\sum_{m\geq0}\frac{(2\nu+m+1)_m}{m!}\frac{\left(\frac{x}{2}\right)^{2\nu+2m}}{\Gamma(\nu+m+1)\Gamma(\nu+m+2)}=\frac{1}{\sqrt{\pi}}\sum_{m\geq0}\frac{\Gamma\left(\nu+m+\frac{1}{2}\right)x^{2\nu+2m}}{m!\Gamma(\nu+m+2)\Gamma(2\nu+m+1)}.
{\rm dim}_k \{\rm H}^0\left(\mathbb{P},\mathcal{O}_\mathbb{P}(\delta a)\right)={\rm dim}_k \S_{\delta a}= \sum_{l+ai=\delta a}l+1 = \sum_{i=0}^\delta \delta a -ia+1 = (\delta a +1)(\delta +1)-a\frac{\delta(\delta+1)}{2}= \frac{(\delta+1)(\delta a +2)}{2}.
\|U\|^2 =\int_{\mathbb{R}^{n+1}_{+}}c^2 |\nabla U (x,y)|^2 + m^2 c^4 U (x,y)^2\, dx dy - c (mc^2 - \mu) \int_{\mathbb{R}^n}U(x,0)^2\, dx &\qquad\geq (1-\alpha) \int_{\mathbb{R}^{n+1}_{+}}c^2 |\nabla U (x,y)|^2 + m^2 c^4 U (x,y)^2\, dx dy,
L(x^{k+1},\lambda^{k+1})- L(x,\lambda^{k+1}) = f(x^{k+1})-f(x) -(\lambda^{k+1})^TA(x^{k+1}-x) \leq (\nabla f(x^k)-A^T\lambda^{k+1})^T(x^{k+1}-x)+\frac{L}{2}\|x^{k}-x^{k+1}\|_2^2\ .
\ell(w_\Delta w^{-1}sw) = \ell(w_\Delta s_{w^{-1}(\alpha_l)}) & = \ell(w_\Delta) - \ell(s_{w^{-1}(\alpha_l)}) & < \ell(w_\Delta) - 1 & = (\ell(w_\Delta w^{-1}) - 1) + \ell(w) & = \ell(w_\Delta w^{-1}s) + \ell(w).
\frac{1}{2\pi}\,e^{-\frac{a_i^2}{2}}\, \int_0^{\rho}\frac{1}{\sqrt{1-x^2}}\, e^{-\frac{(b_i-a_ix)^2}{2 (1-x^2)}}dx\geq \frac{1}{2\pi}\,e^{-6}\, \int_0^{1/\sqrt 3}\frac{1}{\sqrt{1-x^2}}\, e^{-\frac{24}{1-x^2}}dx = 8\kappa.
\alpha_{( \kappa ) \hat{\imath}}= ( \underbrace{0 , \dots , 2 ( D - \kappa - 3 ) \beta_{D - \kappa - 1}, - 2 ( D - \kappa - 1 ) \beta_{D - \kappa}, 0 , \dots , 0}_{k = D - 3 \mathrm{~ t e r m s}}\, ; \underbrace{0 , \dots , 0}_{q \mathrm{~ t e r m s}}) \, ,
P_c (u \cdot \nabla) y = \frac{1}{|\Omega|}\int_{\Omega}(u \cdot \nabla) y \ \d x = \frac{1}{|\Omega|}\bigg( \int_{\Omega}u \cdot \nabla y_k \ \d x \bigg)_{1 \leq k \leq 3}= 0 \hbox{for}u\in L_{\sigma}^p(\Omega),\y\in W^{1,p}(\Omega)^3,
s_q =2+\sum_{k=1}^{r}\left(\sum_{r\ge i_1\ge i_2> i_3\ge i_4>i_5\ge \dots >i_{2k-1}\ge i_{2k}\ge 1}m_{i_1}n_{i_2}m_{i_3}n_{i_4}\cdots m_{i_{2k-1}}n_{i_{2k}}\right) &+\sum_{k=1}^{r-1}\left(\sum_{r\ge i_1> i_2\ge i_3>i_4\ge i_5 >\dots\ge i_{2k-1}> i_{2k}\ge 1}n_{i_1}m_{i_2}n_{i_3}m_{i_4}\cdots n_{i_{2k-1}}m_{i_{2k}}\right).
F(\mu)+\int f_1d\mu+\epsilon \geq \frac{L(f_1)}{L(1)}+\inf_{f\in\mathcal{C}+\frac{\epsilon}{2}}\int fd\mu+\frac{\epsilon}{2}& = \frac{L(f_1)}{L(1)}+\inf_{g\in \mathcal{C}+\frac{\epsilon}{2}}\frac{L(g)}{L(1)}+\frac{\epsilon}{2}&= \frac{1}{L(1)}(L(f_1)+\inf_{g\in \mathcal{C}+\frac{\epsilon}{2}}{L(g)}) +\frac{\epsilon}{2}>0.
\lim_{N\to\infty}Q_{N}\Big[\varepsilon_0 \frac{\gamma}{\beta+\gamma}N^{4/3}(\log N)^{-2/3}\leq R_{N,d,D}\leq K_{0}\Big(\frac{\beta+\gamma}{\beta}\Big)^{1/2}N^{4/3}(\log N)^{4/3}\Big] =1.
\frac{2}{3}q^- \geq \gamma=\frac{2}{3}\Psi(\tau)^{-1}\log\left(\frac{(\Psi(\tau)/2)^{p-1}}{\overline{C}}\right) \textrm{and}0<\log\left(\frac{(\Psi(\tau)/2)^{p-1}}{\overline{C}}\right) \leq q^- \Psi(\tau).
Q^{(n)}(t) =P^{(n)}(t)+4\bigl[ \cdot \arctan \cdot \bigr]^{(n-1)}(t) &=P^{(n)}(t)+4\Biggl[ t\frac{(-1)^{n-1}(n-1)!}{(1+t^{2})^{n/2}}\sin (n\pi/2-n \arctan t) &+ \frac{(-1)^{n}(n-1)!}{(1+t^{2})^{(n-1)/2}}\sin ( (n-1)\pi/2-(n-1) \arctan t)\Biggr].
\begin{aligned}\mathfrak{S}_{\delta, \flat}^{\tau_k}(H', M, N) \ll \big((x^kD^{4k-1})^{1/6k}+ D^{1-1/(2k)}+ H'^{-1/2}M^{1/2}N+ (x^{-1}D^3)^{1/2}\big)x^{\varepsilon}& \ll \big((x^kD^{4k-1})^{1/6k}+ H'^{-1/2}M^{1/2}N\big)x^{\varepsilon}\end{aligned}
\partial_i(x_{i+1}f) + f \overset{(\ref{eq:extleibniz})}{=}- f + x_{i+1}\partial_i(f) + (x_i - x_{i+1})\partial_i(f) &= x_i\partial_i(f), \partial_i(x_{i}f) \overset{(\ref{eq:extleibniz})}{=}f + x_i\partial_i(f) - (x_i - x_{i+1})\partial_i(f) &= x_{i+1}\partial_i(f) + f,
\mathfrak{d}^{\rm (out)}_{n,d}= \mathfrak{d}^{\rm (out)}_{n-2,d+2}+ \mathfrak{d}^{\rm (out)}_{n-1,d-1}+ \mathfrak{d}^{\rm (out)}_{n-2,d}+ \sum\limits_{i=0}^{n-2}\mathfrak{d}^{\rm (out)}_{i,1}\cdot \mathfrak{d}^{\rm (out)}_{n-i-2,d-1}+ \sum\limits_{i=0}^{n-2}\sum\limits_{j=0}^{d-2}\tilde{s}_{i,j}\cdot \mathfrak{d}^{\rm (out)}_{n-2-i,d-2-j},
\int\limits_0^T dt\frac{\delta}{{\delta s_{\mu \nu}(t)}}\exp ( - \frac{1}{4}\int\limits_0^T{dt\dot x^2 ) = \frac{1}{2}}\int\limits_0^T dt\,\omega_{\mu \nu}\left[ \dot{x}(t) \right]\exp ( - \frac{1}{4}\int\limits_0^T{dt\dot x^2}),