harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\varphi(b_0a_1b_1a_2b_2\cdots a_nb_n) =\frac{1}{\omega(p)}\omega(pb_0a_1b_1a_2b_2\cdots a_nb_n) &=\frac{1}{\omega(p)}\omega(pa_1\cdots a_n)\tau(b_0)\cdots \tau(b_n) &=\varphi(a_1\cdots a_n)\varphi(b_0)\cdots \varphi(b_n).
\boldsymbol{v}(\boldsymbol{x},t) = \boldsymbol{v}_0^c(\boldsymbol{x}) + \sum_{k=1}^{\infty}\left(\boldsymbol{v}_k^c(\boldsymbol{x}) \cos(k \omega t) + \boldsymbol{v}_k^s(\boldsymbol{x}) \sin(k \omega t)\right)
A_t= \left\{ct\leq |W^1_{\eta t}| \leq ct+o(t)\right\}, B_t= \left\{|W^1_s|\leq kt \ \ \forall \ 0\leq s \leq \eta t \right\}, C_t= \left\{|W^{d-1}_s|\leq r(t) \ \ \forall \ 0\leq s \leq \eta t \right\}.
S_{F}^{\alpha}(T)=-\frac{1}{\sqrt{2 g_{F}^{\alpha}}}\int_{S_{F}^{\alpha}(A)}^{S_{F}^{\alpha}(B)}\frac{1}{\sqrt{(S_{F}^{\alpha}(y)-S_{F}^{\alpha}(\eta))}}d_{F}^{\alpha}\mathfrak{s}_{F}^{\alpha}.
h_F(q_{13}) = (\frac{2 (a_1 - c_1 t)}{b_1 + d_1 t}, - \frac{2 (a_1 - c_1 t)}{b_1 + d_1 t}\cdot J_2)_F= (\frac{2 (a_1 - c_1 t)}{b_1 + d_1 t}, J_2)_F.
E_{n,k}(X_{n,k}^{\mathcal{T}}|W_{\mathcal{T}(n,k;R_n)})=E_{n,k}(Y^{(n,k)}_k|W_{\mathcal{T}(n,k;R_n)})\le\frac{E_{n,k}Y^{(n,k)}_k}{P_{n,k}(W_{\mathcal{T}(n,k;R_n)})},
\mathrm{GSp}_{2g}(R) =\left\{\alpha\in\mathrm{GL}_{2g}(R)~|~\alpha^TJ\alpha=\nu(\alpha)J~\textrm{with}~\nu(\alpha)\in R^\times\right\}, \mathrm{Sp}_{2g}(R) =\left\{\alpha\in\mathrm{GSp}_{2g}(R)~|~\nu(\alpha)=1\right\},
H_t \left[ \vec{v}_t - \vec{w}_t \right] = \lambda_t \vec{v}_t - \mu_t \vec{w}_t &= (\lambda_t - \mu_t) \vec{v}_t + \mu_t ( \vec{v}_t - \vec{w}_t) \Rightarrow H_t \left[ \frac{\vec{v}_t - \vec{w}_t}{\lambda_t - \mu_t}\right] = \left( \vec{v}_t + \mu_t \frac{\vec{v}_t - \vec{w}_t}{\lambda_t - \mu_t}\right)
\begin{array}{l c r}{\sum_{\nu = 1}^{8}X_{m_{1}n_{1}}^{\mu}( \sigma ) X_{m_{2}n_{2}}^{\nu}( \sigma ) X_{m_{3}n_{3}}^{\nu}( \sigma ) = - 8{\gamma}^{\mu}X_{m_{1}n_{1}}( \sigma ) X_{m_{2}n_{2}}( \sigma ) X_{m_{3}n_{3}}( \sigma ) .}\\ \end{array}
\frac{1}{L}F_{B}( \beta , L ) = \frac{\pi}{6 L^{2}}+ \frac{1}{\beta L}\ln{\frac{L}{\beta}}+ \frac{2}{\beta L}\ln{\eta \left( i \frac{L}{\beta}\right)}\longrightarrow - \frac{\pi}{6 \beta^{2}}\; .
\partial^{\nu}\Lambda_{\nu \mu}= \varepsilon_{\mu \nu}n^{\nu}\bar{c}^{a}\, \frac{\delta S}{\delta \phi^{a}}- A_{\mu}^{a}\, \frac{\delta S}{\delta c^{a}}+ \partial_{\mu}\bar{c}^{a}\, \frac{\delta S}{\delta b^{a}}+ \mathrm{t o t a l ~ d e r i v .}\quad .
V_{N B P S}( z ) = 2 m^{2}\left( \begin{array}{c c}{{\frac{2 \alpha}{\lambda}- ( 1 + \frac{2 \alpha}{\lambda}) s e c h^{2}( M z )}}&{{0}}\\{{0}}&{{\frac{\alpha}{2 \lambda}- \frac{3 \alpha}{\lambda}s e c h^{2}( M z )}}\\ \end{array}\right) .
\begin{aligned}A_{n,i}(x) =A_{n,i}^{\mu,\nu,a,b}(x),Q_n(x)=Q_n^{\mu,\nu,a,b}(x), B_{n,i}(x) =B_{n,i}^{\mu,\nu,a,b}(x), P_n(x)=P_n^{\mu,\nu,a,b}(x).\end{aligned}
E_{o_-\cdot w_{P'}}(w)T_n = E_{o_-\cdot w_{P'}}(w)E_{o_-\cdot n_{w_{P'}}w}(n) & = E_{o_-\cdot w_{P'}}(wn) & = E_{o_-\cdot w_{P'}}(n)E_{o_-\cdot n_{w_{P'}}n}(n^{-1}wn) & = T_nE_{o_-\cdot n_{w_{P'}}n}(n^{-1}wn).
\sum_{i}\gamma_{i}^{\left( D \right)}\left( a , b , c \right) \alpha_{i l}\left( a , b , c \right) \alpha_{i l^{\prime}}\left( a , c , b \right) = \left( - 1 \right)^{\left( m - 2 \right) \left( l - 1 \right)}\gamma_{l}^{\left( D \right)}\left( b , a , c \right) \delta_{l l^{\prime}}.
\Delta \lambda_n^{crit}= - 2 \theta a \left( \frac{\lambda_n^{crit}}{2(n+1)}\right)^2 \left( c_{n+1}+ \left( \frac{\lambda_{n+1}^{crit}}{2(n+2)}\right)^2 \left( c_{n+2}+ \cdots \right) \right).
\int d^3p\, \frac{Tr[\gamma_{\mu_1}(m+\slash\!\!\!p)\gamma_{\mu_2}(m+\slash\!\!\!p+\slash\!\!\!k_1)...\gamma_{\mu_n}(m+\slash\!\!\!p+...+\slash\!\!\!k_{n-1})]}{(m^2-p^2+i\epsilon)[m^2-(p+k_1)^2+i\epsilon]...[m^2-(p+...+k_{n-1})^2+i\epsilon]}
S_E^{(5)}= -\sum_{j=0}^{1}(-1)^j\left[\frac{z}{\tilde{\rho}_0\sinh^2\left(z/\tilde{\rho}_0\right)}-\coth\left(z/\tilde{\rho}_0\right)+\frac{4}{15}\cosh\left(z/\tilde{\rho}_0\right)\sinh\left(z/\tilde{\rho}_0\right)\right]S_E^{(5)j},
L(c_1,\dots,c_k,\lambda) = \sum\limits_{j=1}^k c_j^2\frac{2}{j+1}+ \lambda (\sum\limits_{j=1}^k c_j j -1) \textrm{with}\frac{\partial L}{\partial c_j}\stackrel{!}{=}0 \forall j \in \{1,\dots,k \}\textrm{and}\frac{\partial L}{\partial \lambda}\stackrel{!}{=}0.
f_0(t) = \phantom{a}(h_{p-1}t^{p-1}-g_{p-1}(at+b)^{p-1}) + \sum_{\ell = 0}^{p-2}\sum_{\theta=0}^{d}(h_{p\theta+\ell}t^{p\theta+\ell}- g_{p\theta+\ell}(at+b)^{p\theta+\ell}).
C \Big [ \begin{array}{c}{{\alpha + \beta}}\\{{\gamma}}\\ \end{array}\Big ] = ( - 1 )^{\gamma ( \psi^{\mu}, \tilde{\psi}^{\mu})}\, \exp \Big \{{\frac{i \pi}{2}}\, n ( \alpha \beta \gamma ) \Big \}\, C \Big [ \begin{array}{c}{{\alpha}}\\{{\gamma}}\\ \end{array}\Big ] C \Big [ \begin{array}{c}{{\beta}}\\{{\gamma}}\\ \end{array}\Big ] .
\sup_{\mathcal{T}>0}\ |\frac{1}{\mathcal{T}}\int_0^{\mathcal{T}}h(u(t),y_{\epsilon}(t),z_{\epsilon}(t))dt - \frac{1}{\mathcal{T}}\int_0^{\mathcal{T}}\tilde h(\mu(t),z(t))dt|\ \leq \alpha_h (\epsilon ), \ \ \ \ \{\rm where}\ \ \ \ \ \lim_{\epsilon\rightarrow 0}\alpha_h (\epsilon) = 0
LHS =\int_{0}^{t}\int_{\Gamma_1}\Phi(x-y,\tau)\,dS(y)\,d\tau &=\frac{1}{(4\pi)^{N/2}}\int_{0}^{t}\tau^{-N/2}\int_{\Gamma_1}e^{-|x-y|^{2}/(4\tau)}\,dS(y)\,d\tau.
m{\frac{d \Omega_{B H}}{d m}}\sim{\frac{m t_{e q}^{1 / 2}}{m_{p}^{2}t_{f}^{3 / 2}}}\sim 1 0^{2 7}\left({\frac{\lambda}{e}}\right)^{1 / 2}\left({\frac{M}{m_{p}}}\right)^{5 / 2}\left({\frac{M}{e m}}\right)^{\frac{n - 1}{2}}\; ,
\Big \langle : e^{i \varphi ( f )}:_{M}e^{i \varphi ( g )}:_{M}\Big \rangle_{C_{m = 0}}: = \lim_{m \rightarrow 0}\Big \langle : e^{i \varphi ( f )}:_{M}e^{i \varphi ( g )}:_{M}\Big \rangle_{C_{m}}.
\|\mathcal{A}(\mathcal{M}_d^{})^\nu\|_{\mathcal{L}\times \mathcal{L}}\le \eta(\nu) \, \| \mathcal{D}_d \|_{\mathcal{L}\times \mathcal{L}}\mathcal{D}_d = \begin{bmatrix}\hat{A}0 0 \hat{S}\end{bmatrix}.
(\nabla_{e_i}\Phi)(e_i,\phi e_k) =g(e_i,\nabla_{e_i}(\xi\times(\xi\times e_k))+g(\nabla_{e_i}(\xi\times e_k),\xi\times e_i) =g(e_i,\nabla_{e_i}(-e_k))+g(\nabla_{e_i}(\xi\times e_k),\xi\times e_i) =g(\nabla_{e_i}e_i,e_k)+g(\nabla_{e_i}(\xi\times e_k),\xi\times e_i)
\mathcal{A}(Y,\Delta\theta_j)= \frac{V_pL^{4-n}F(\Delta\theta_j)}{2^{p-2}(2\pi^2\alpha')^{(p+1-n)/2}}\sum_{\omega_k\in Z}\int_0^\infty t^{-\frac{p+n-5}{2}}\exp^{-\frac{t\sum_k(Y_k+2\pi \omega_k R)^2}{2\pi^2\alpha'}}dt\,,
P_\textup{ord}(y_1,\ldots,y_k;(ML)^{\chi(y)})\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\subset P(y_1,L^{\chi(y_1)})P^*(y_2,\ldots,y_k;(O_{k,s}(M)L)^{\chi(y_2)},\ldots,(O_{k,s}(M)L)^{\chi(y_k)})P(y_1^{L^{\chi(y_1)}};M^{|\chi(y_1)|})
-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}- \frac{1}{4}F'_{\mu\nu}F'^{\mu\nu}= \frac{-1}{8g^2}({\cal F}_{\ast}^{\mu\nu}{\cal F}_{\mu\nu}- i{\cal F}_{\ast}^{\mu\nu}{\cal F}_{\mu\nu}i) .
\left\Vert \mathbf{u}\right\Vert_{\operatorname*{curl},\Omega,k,\lambda}:=\left( k^{2}\left\Vert \mathbf{u}\right\Vert^{2}+\frac{1}{\lambda^{2}}\left\Vert \operatorname*{curl}\mathbf{u}\right\Vert^{2}\right)^{1/2}.
\hat{Z}_{8 8}^{1 0}= \hat{Z}_{8 8}^{\overline{{1 0}}}= \frac{( d e t \I m \Omega )^{2}}{4}\biggl \vert \Theta \left[ \begin{array}{l l}{1 / 2}&{1 / 2}\\{1 / 2}&{1 / 2}\\ \end{array}\right] ( 0 \vert \Omega ) \biggr \vert^{8}
\limsup_{n\to \infty}\big\| f_n \big\|_{\mu;p}= \Big ( \limsup_{n\to \infty}\int f^p_n d\mu \Big)^{\frac{1}{p}}\leq \Big ( \int \limsup_{n\to \infty}f^p_n d\mu \Big)^{\frac{1}{p}}= \big\| \limsup_{n\to \infty}f_n \big\|_{\mu;p}.
\tilde{Z}= \left( \begin{array}{c c c c}{\tilde{z}_{+ +}}&{0}&{0}&{0}\\{0}&{\tilde{z}_{--}}&{0}&{0}\\{0}&{0}&{\tilde{z}_{Z Z}}&{\tilde{z}_{Z A}}\\{0}&{0}&{\tilde{z}_{A Z}}&{\tilde{z}_{A A}}\\ \end{array}\right)
-[2m-1]_q P_q(2m-k-2,k-1) + [2m-2k]_q P_q(2m-k-1,k-1) \qquad\qquad= (-[2m-1]_q [2m-2k]_q + [2m-k-1]_q[2m-2k]_q)P_q(2m-k-2,k-2) \qquad\qquad= - q^{2m-k-1}[k]_q [2m-2k]_q P_q(2m-k-2,k-2).
\overline{E}^{D*}_{T}= \frac{\left(P_{CT_x}+ P_{CR_x}\right) T_{on}}{L}+ \frac{(1 + \alpha) N_{0}m_{S, D}T_{on}P_{L_{S,D}}}{L\, \Omega_{S,D}g_{QAM}}\left[\left(\frac{C}{p^{*}}\right)^{1/m}- 1\right] + \frac{2P_{LO}T_{tr}}{L}.
\int^{er}_r x^{-1}\, dx = 1 , \int^{er}_r x^{( q - 1 )}\, dx = \frac{r^q \, ( e^q - 1 )}{q}\geq r^q, \int^{er}_r x^{( - q - 1 )}\, dx = \frac{r^{- q}\, ( 1 - e^{-q})}{q}.
\mathbb{M}+B(j, k) =(24 c_1^2-12 c_4) D_S^6+(\frac{14}{3}c_2^2-4 c_6) D_S^4+(4 c_6-\frac{16}{3}c_2^2)D_S^3 U D_S (\mathrm{I}-\frac{1}{j^2+k^2+j k}D_S^2) &+12 (c_2 c_3-c_7) D_S^2-6 c_3^2 \mathrm{I}+(16 c_2 c_3-24 c_7) D_S U D_S (\mathrm{I}-\frac{1}{j^2+k^2+j k}D_S^2).
\operatorname{Disc}(\operatorname{Tr}_{\mathbb{F}_{q^a}/\mathbb{F}_{q}}( \langle u \rangle)) = \operatorname{Norm}(u) \cdot \operatorname{Disc}(\operatorname{Tr}_{\mathbb{F}_{q^a}/\mathbb{F}_{q}}( \langle 1 \rangle))
\sum_{k}f(x_k^{n,1})\,\psi(x_k^{n,1})\, \phi(\alpha + |I_k^n|) =\sum_{k}f(x_k^{n,1}) \psi(x_k^{n,1})\,\Big( \frac{\phi(\alpha +|I_k^n|)}{|I_k^n|}-\phi'_+(\alpha)\Big)\,|I_k^n| &+\phi'_+(\alpha)\,\sum_{k}f(x_k^{n,1})\psi(x_k^{n,1})\, |I_k^n| =C_n+D_n,
(f(c_{1,i}),f(c_{2,i})) =(i,i+n-1)=1.\intertext{Next, for $i=1$, we see that}(f(c_{1,1}),f(c_{2,1})) =(n-1,n)=1.\intertext{Finally, observe that}(f(c_{1,1}),f(c_{1,2})) =(n-1,2)=1, (f(c_{1,1}),f(c_{1,n})) =(n-1,2n)=1, (f(c_{1,1}),f(c_{1,2})) =(n-1,2)=1, (f(c_{1,n-2}),f(c_{2,n-2})) =(n-2,2n-3)=1
\frac{1}{2}\frac{d}{dt}\int_{\mathbb T^2}\varphi^2(\theta)dx+\int_{\mathbb T^2}\nabla\cdot\left(\frac{1}{2}\varphi^2(\theta)u\right)\,dx +\nu\int_{\mathbb T^2}\left|\Lambda^{\frac{\alpha}{2}}\varphi(\theta)\right|^2\,dx\leq\int_{\mathbb T^2}f\varphi(\theta)\,dx.
\int_{\Omega}{e^{u_0^{'}+v_1}}dx =\frac{|\Omega|}{2}-(k_{22}N_1-k_{12}N_2)\frac{\pi}{|K|}\equiv \eta_1>0 \int_{\Omega}{e^{u_0^{''}+v_2}}dx =\frac{|\Omega|}{2}-(k_{11}N_2-k_{21}N_1)\frac{\pi}{|K|}\equiv \eta_2>0.
e^{\tau L_{0}}= \left( \begin{matrix}{\strut 1}&{\tau}&{\tau}\\{\strut \tau}&{1 +{\frac{\tau^{2}}{2}}}&{{\frac{\tau^{2}}{2}}}\\{\strut - \tau}&{-{\frac{\tau^{2}}{2}}}&{1 -{\frac{\tau^{2}}{2}}}\\ \end{matrix}\right) \ .
2ni\cdot d_{0,s}(n,i) =n(i+s)d_{0,s}(0,i), 2ni\cdot d_{0,s}(0,i) =n(i+s)d_{0,s}(-n,i), 0 =n(i+s)d_{0,s}(0,i)+ni\cdot d_{0,s}(n,-i).
r_{i+1}= \left(\prod_{i=n}^1 p_i - \prod_{i=n}^1 q_i \right)(\kappa_{i+1}^{\epsilon})^{-1}= (\kappa_{i+1}^{\epsilon})^{-1}\left(\prod_{i=n}^1 p_i - \prod_{i=n}^1 q_i \right).
\hat{R}^{\dagger}\hat{R}-\bar{R}^{\dagger}\bar{R}=\bar{R}^{\dagger}(I_K+\gamma)^2\bar{R}-\bar{R}^{\dagger}\bar{R}\cr =2\bar{R}^{\dagger}\gamma\bar{R}+ \bar{R}^{\dagger}\gamma^2\bar{R}.
\| F(t) - P \| \le C_1 |t|; C_1 = \beta_1 \delta^{-1/2}\| X_1 \|, \| A(t)F(t) - t^2 SP \| \le C_2 |t|^3; C_2 = \beta_2 \delta^{-1/2}\| X_1 \|^3.
\frac{\sum_{i=0}^{d_x-1}w_i y^i}{r^{\ell}}\frac{dx}{r}= d\left(\frac{\sum_{i=0}^{d_x-1}v_i y^i}{r^{\ell}}\right)+\frac{\sum_{i=0}^{d_x-1}u_i y^i}{r^{\ell-1}}\frac{dx}{r}.
\varphi(x)\psi(\xi)-\psi(\xi)\varphi(x) =\sum_i \xi(a_i)\varphi(x)\varphi(b_i)- \sum_i \xi(a_i)\varphi(b_i)\varphi(x) &=\sum_i \xi(a_i)\varphi([x,b_i]) &=\psi(ad^\ast_x\xi)-\varphi(ad^\ast_\xi x),
H_{1 , 2}= - R / 2 \, ( 1 . 5 7 \, a_{1}^{2}- 2 . 0 9 \, a_{2}^{2}) + R^{2}/ 2 \, ( 1 . 2 3 \, a_{1}^{2}- 0 . 6 1 7 \, a_{2}^{2}) - 2 . 1 3 \, a_{1}^{2}\, a_{2}^{2}/ ( 2 \, R ) .
\left| \phi_{i, out, \beta}(y) - \sum_{j=0}^i c_{i,j}(\sqrt{b})^{2j-\gamma}T_{j}\left( \frac{y}{\sqrt{b}}\right) \right| \le C y^{2i +2 -\gamma}|\ln y| b^{1-\frac{\epsilon}{2}}.
\frac{1}{n}\sum_{i=1}^nK(p_{f_0,i},p_{f,i}) =\frac{1}{2}\log{\left(\frac{\sigma^2}{\sigma_0^2}\right)}-\frac{1}{2}\left(1-\frac{\sigma_0^2}{\sigma^2}\right)+\frac{\|f-f_0\|_n^2}{2\sigma^2}, \frac{1}{n}\sum_{i=1}^nV(p_{f_0,i},p_{f,i}) =\frac{1}{2}\left(1-\frac{\sigma_0^2}{\sigma^2}\right)^2+\frac{\sigma_0^2\|f-f_0\|_n^2}{\sigma^4}.
\Phi_{2}(x) =n^{2n}m^{2m}\prod_{p=1}^{m}\left[\left(x+\frac{\tilde{p}-p}{m}\right) \left(2\bar{p}+2 -x + \frac{2\alpha-2\tilde{r}+1}{n}+\frac{\tilde{p}+p-1}{m}\right)\right] \times \prod_{r=1}^{n}\left[\left(\bar{p}+1-x-\frac{\tilde{r}-r}{n}\right)\left(\bar{p}+1-x+ \frac{2\alpha-\tilde{r}+r}{n}\right)\right].
|f(z)| =|h(z)+\overline{g}(z)| \geq |h(z)|-|g(z)|=|h(z)|\left(1-\left|\frac{g(z)}{h(z)}\right|\right) &=|h(z)|\left(1-\left|\frac{G'(z)}{H'(z)}\right|\right)\geq \frac{|z|}{(1+|z|)^2}\left(1-\frac{\alpha+|z|}{1+\alpha|z|}\right), z \in \Delta.
S_{N,r+1}^{(4,b)}=\sum_{k=0}^{r+1}N_{4k+b}=S_{N,r}^{(4,b)}+N_{4(r+1)+b}&=(5S_{N,r-1}^{(4,b)}-2S_{N,r-2}^{(4,b)}+S_{N,r-3}^{(4,b)}+1)+(5N_{4r+b}-2N_{4(r-1)+b}+N_{4(r-2)+b}) &=5S_{N,r}^{(4,b)}-2S_{N,r-1}^{(4,b)}+S_{N,r-2}^{(4,b)}+1,
(q_{\lambda}f)\left(\mathbf{x}_0 \right)=\left\langle f(\mathbf{y}), q_{\lambda}(\mathbf{x}_{0}, \mathbf{y})\right\rangle=\left\langle f(\mathbf{x}_{0}+2^{-\lambda}\mathbf{y}),\varphi_{\lambda}(\mathbf{y})\right\rangle,
P_{N}(1^{n},0^{m}) := \mathbb{P}((X_{\infty})^{*}_{1}= 1,\dots,(X_{\infty})^{*}_{n}= 1,(X_{\infty})^{*}_{n+1}= 0, \dots ,(X_{\infty})^{*}_{n+m}= 0)\;,
\Phi(X(\cdot,T))-\Phi(X(\cdot,0)) =-(\sqrt{2\pi})^{n+1}\int_{0}^{T}\int_{\mathbb{S}^n}e^{\frac{r^2}{2}}Kh(fh^{p-1}-\frac{1}{(\sqrt{2\pi})^{n+1}}\frac{1}{K}e^{\frac{-r^2}{2}})^2dxdt & \leq -C_0\int_{0}^{T}\int_{\mathbb{S}^n}(fh^{p-1}-\frac{1}{(\sqrt{2\pi})^{n+1}}\frac{1}{K}e^{\frac{-r^2}{2}})^2dxdt.
J_1(z) = \frac{v(n^2 f(y);f(ry)/f(y))}{2 \sqrt{-f(z)}}P^{(\infty)}(z)\begin{pmatrix}-1 -ie^{-i \pi \alpha \theta(z)}-ie^{i \pi \alpha \theta(z)}1 \end{pmatrix}P^{(\infty)}(z)^{-1}.
\frac{\Gamma(t+1 + m_j-n)}{\Gamma(s+1+m_j-n)}= \begin{cases}\displaystyle \frac{\sin \pi s}{\sin \pi t}\, n^{t-s}\left(1 + O\left(\frac{1}{n}\right)\right), j = 0, \displaystyle(m_j-n)^{t-s}\left(1 + O\left(\frac{1}{m_j-n}\right)\right), j \in \{1, \ldots, r \}\setminus J\end{cases}
{\l}^{A}\hat{\bar{\mu}}^{\dot{B}}P_{A\dot{B}}+ \l^A \hat{\mu}^B Z_{AB}= 0, \qquad \hat{\mu}^{A}{\bar{\l}}^{\dot{B}}P_{A\dot{B}}+ \bar{\l}^{\dot{A}}\hat{\bar{\mu}}^{\dot{B}}\bar{Z}_{\dot{A}\dot{B}}= 0,
H_{i}(x,m^{\hat v}_{t},Du^{\hat v}_{i}(x,t)) =f_{i}(x,m^{\hat v}_{t},\hat{v}_{i}(x,m^{\hat v}_{t}))+Du^{\hat v}_{i}(x,t).g_{i}(x,m^{\hat v}_{t},\hat{v}_{i}(x,m^{\hat v}_{t})) \, ,
C^{\mathrm{dl},k}_{3}\left( N_{\mathrm{sct}}, N_{\mathrm{scr}}\right) = \left(\frac{T_{\mathrm{dl}}}{T}\right) \min\left\lbrace c^k_{\mathrm{b2s},3}\left(N_{\mathrm{scr}}\right), \right. \ \,\,\,\,\,\, \left. c^k_{\mathrm{s2u},1}\left( N_{\mathrm{sct}}\right) \right\rbrace.
\sum_{e,f}G^{ab}_{ef}(p_1,p_2; +)G^{ef}_{cd}(p_1,p_2; -)=\sum_{e,f}G^{ab}_{ef}(p_1,p_2; -)G^{ef}_{cd}(p_1,p_2; +)=\delta^a_c \delta^b_d .
\Psi_k^+ (u, \mathbf{v}_1 , \mathbf{v}_2 ,\vartheta, g\,T) := u\, \psi \left( \widetilde{\mu}_{g e^{-\imath \vartheta B/\sqrt{k}}g^{-1}}( x_{1k}),x_{2k}\right) -\frac{\vartheta}{\sqrt{k}}\, \nu.
(q+1 )\widetilde{m}^q (\nabla_y \widetilde{m})^T\nabla_y^2\widetilde{w}(\Lambda + \nabla_y \widetilde{w})+ (q+1 )\widetilde{m}^q (\nabla_y\widetilde{m})^T\nabla_yV= (q+1 )\widetilde{m}^{q-1}|\nabla_y\widetilde{m}|^2.
r_{n}(Sx) - r_{n}(x) = \begin{cases}-1 \textrm{if}x \not\in \xi^{n-1}(\Omega ({\xi})) \vert \xi^{n-1}(a) \vert -1 \textrm{if}x \in \xi^{n-1}([a]), a\in A. \end{cases}
R_{\mu\nu}-\frac{1}{2}\,G_{\mu\nu}\,{\cal R}-\frac{1}{2}\left(\partial_\mu\varphi\,\partial_\nu\varphi-\frac{1}{2}\,G_{\mu\nu}\,\partial_\rho\varphi\,\partial^\rho\varphi\right) =0
\langle \Phi(t^{-1}z_1,\dots,t^{-1}z_N)| \overline{x_1}\cdots \overline{x_N}\rangle = t^{MN}\prod_{j=1}^N z_j^{j-1-N}(1+t^{-1}z_j^2)\prod_{1 \le j<k \le N}(1+t^{-1}z_j z_k)(t^{-1}+z_j z_k^{-1})sp_{\overline{\lambda}}( \{z \}_N ).
\lambda_b=\frac{1}{2}\left[e^{ib\left(\frac{\alpha-\beta}{2}\right)}+e^{-ib\left(\frac{\alpha-\beta}{2}\right)}\right]\delta_0- \frac{1}{2}\left[\e^{ib\left(\frac{\alpha+\beta}{2}\right)}\,\delta_{b}+e^{-ib\left(\frac{\alpha+\beta}{2}\right)}\,\delta_{- b}\right].
f_{\varepsilon}(t_0)<\tilde{c}\quad\Longrightarrow\dot{p}_{\varepsilon}(t)=\begin{cases}\frac{d}{dt}u(f_\varepsilon)(t) \u(f_\varepsilon(t))\leq S_\varepsilon(t), 0 \u(f_\varepsilon(t))> S_\varepsilon(t),\end{cases}\quad [t_0,t_0+\delta].
\|(z-T)^{-1}e_0\|^{p}_{p}=\|\sum\limits_{n=0}^{\infty}\frac{T^n}{z^{n+1}}e_0\|^{p}_{p}=\|\frac{e_0}{z}+\sum\limits_{n=1}^{\infty}\frac{\prod\limits_{j=0}^{n-1}w_j}{z^{n+1}}e_n\|^{p}_{p}&=\frac{1}{|z|^p}+\frac{1}{|z|^p}\sum\limits_{n=1}^{\infty}\frac{\big(\prod\limits_{j=0}^{n-1}w_j\big)^p}{|z|^{pn}}.
\frac{h(t+\Delta t)-h(t)}{\Delta t}= -\int_x^\infty h(s)\frac{F(t+\Delta t+s)-F(t+s)}{\Delta t}ds. = -\int_x^\infty h(s)F'(t+\theta\Delta t+s)ds = -\int_{x+t+\theta\Delta t}^\infty h(s-t)F'(s)ds -\int_{x+t+\theta\Delta t}^\infty [h(s-t-\theta\Delta t)-h(s-t)]F'(s)ds
G(2z\,|\,\tau) = (2\pi)^{-z}2^{1+{}_2 S_0(2z)}\frac{G(z\,|\,\tau)\, G(z+1/2\,|\,\tau) \, G(z+\tau/2\,|\,\tau)\,G\bigl(z+(1+\tau)/2\,|\,\tau\bigr)}{G(1/2\,|\,\tau) \,G(\tau/2\,|\,\tau) \,G\bigl((1+\tau)/2\,|\,\tau\bigr)}.
\begin{aligned}f(t,t_1,t_2,t_3) =\kappa(Z(t_3),Z(t_1),Z(t),Z(t_2)) =(z(t_2) -z(t_1))^{-1}(z(t_1) -z(t))(z(t) -z(t_3))^{-1}(z(t_3) - z(t_2)),\end{aligned}
\mathbb{P}( (X_{C_1}(\bar t), \ldots, X_{C_r}(\bar t))= (Y_{C_1}(\bar t), \ldots, Y_{C_r}(\bar t)) ) \geq 1 - \sum_{h=1}^r \exp \left (-\hat{c}_N \left\lfloor\frac{{\bar \ell_h}^{N+1}}{N-1}\right \rfloor \right ).
\frac{1}{6}\frac{2M}{\eta}K_1 (2 \eta M)-2M \sum_{k=1}^{\infty}\frac{1}{(k\pi)^2}\frac{K_1 (2M\sqrt{\eta^2+k^2 \pi^2 S^2})}{\sqrt{\eta^2+k^2 \pi^2 S^2}}\hspace{0.5cm}.
P(x'):=\frac{2}{\omega_{n-1}\sqrt{{\rm det}\,(A_{\rm sym})}}\frac{1}{\big\langle(A_{\rm sym})^{-1}(x',1),(x',1)\big\rangle^{\frac{n}{2}}},\qquad\forall\,x'\in{\mathbb{R}}^{n-1},
\begin{aligned}f(D,R;\alpha)=\sum_{j=1}^J\frac{|b^j|^2}{8\pi}\frac{E}{1-\nu^2}\bigg( \,2+\frac{D^2}{R^2}\Big(\frac{D^2}{R^2}-2\Big)-2\log R &\,+\frac{1}{4(1-\nu)}\frac{D^2}{R^2}\Big(\frac{R^2}{D^2}-1\Big)\Big(\frac{D^2}{R^2}\Big(\frac{R^2}{D^2}+1\Big)-2\Big)\bigg)\,,\end{aligned}
\frac{1}{|G|}\sum_{r_{1}+r_{2}+...+r_{n}=k}(\begin{matrix}k\\r_{1},r_{2},...r_{n}\end{matrix})r_{1}![z^{r_{1}}]g(z)r_{2}![z^{r_{2}}]g(z)\cdot\cdot\cdot r_{n}![z^{r_{n}}]g(z)
T(m,\tau_1, \tau_2)=\sup\limits_{\lambda\in \left[\tau_1, \tau_2\right]}\frac{\left|W_m^*(\lambda)\right|}{\left\{\int_0^{\lambda}\left(W_m^*(r)-\frac{r}{\lambda}W_m^*(\lambda)\right)^2dr+\int_{\lambda}^1\left(W_m^*(r)-\frac{1-r}{1-\lambda}W_m^*(\lambda)\right)^2dr\right\}^{\frac{1}{2}}}.
\Delta(\lambda)= 2\cos \theta_N(\lambda)+ \frac{1}{1-\alpha_N^0\beta_N^0}\frac{1}{(2i\lambda)^N}\Big(\big(R_N(1,\lambda)-\alpha_N^0S_N(1,\lambda)\big)_1 & +\big (S_N(1,\lambda)-\beta_N^0 R_N(1,\lambda)\big)_2\Big)
\begin{aligned}\nu_c^-(Y_1,Y_2,\dots,Y_m):= \mu(Y_1,Y_2)+\mu(Y_2,Y_3)+\dots+\mu(Y_{m-1},Y_m)-\mu(Y_1,Y_m), \nu_c^+(Y_1,Y_2,\dots,Y_m):= \mu^*(Y_1,Y_2)+\mu^*(Y_2,Y_3)+\dots+\mu^*(Y_{m-1},Y_m)-\mu^*(Y_1,Y_m).\end{aligned}
\lambda_{1}(t) E_{0,1}(\lambda_{1}(t),\lambda_{1}'(t),\lambda_{1}''(t)) - \lambda_{2}(t) E_{0,1}(\lambda_{2}(t),\lambda_{2}'(t),\lambda_{2}''(t)) &=\int_{0}^{1}DF_{0,0,1}(\lambda_{\sigma}) \cdot (\lambda_{1}(t)-\lambda_{2}(t),\lambda_{1}'(t)-\lambda_{2}'(t),\lambda_{1}''(t)-\lambda_{2}''(t)) d\sigma
d s_{n + 3 + d}^{2}= e^{- 2 L n \rho}d x^{2}+ e^{4 L \rho}( 2 d x d t + d y^{i}d y^{i}) + d \rho^{2}+ d \psi^{2}+ ( L n )^{- 2}\sin^{2}\! ( L n \psi ) d \Omega_{d - 1}^{2}
( g^{ij}g^{kl}- 2 g^{jl}g^{ik})\partial_k \partial_l \partial^2_t q \partial^2_{ij}\eta^\mu &=( g^{ij}g^{kl}- g^{jl}g^{ik})\partial_k \partial_l \partial^2_t q \partial^2_{ij}\eta^\mu- g^{jl}g^{ik}\partial_k \partial_l \partial^2_t q \partial^2_{ij}\eta^\mu ,
e_{2}\cdot \left[ f_{\delta}(x_{0}+ t \theta_{x_{0}}) - f(x_{0}+ t \theta_{x_{0}}) \right] = e_{2}\cdot \left[ \frac{\delta}{(e_{2}\cdot \theta_{x_{0}}^{\perp})}(\theta_{C}^{\perp}) \right] = \delta.
\mathrm{c^{sm}}(\Omega_i)\mathrm{c^{sm}}(\Omega_j) =H^{m-i+1}(h+H)^{i-1}\cdot H^{m-j+1}(h+H)^{j-1}&=H^{2m-i-j+2}(h+H)^{i+j-2}&=H^{m-(i-1)+1+m-(j+1)+1}(h+H)^{(i-1)-1+(j+1)-1}&=H^{m-(i-1)+1}(h+H)^{(i-1)-1}\cdot H^{m-(j-1)+1}(h+H)^{(j-1)-1}&=\mathrm{c^{sm}}(\Omega_{i-1})\mathrm{c^{sm}}(\Omega_{j+1}).
\bigg( \sum_{\vert \alpha \vert = m}\Vert x_{\alpha}\Vert^{q}\bigg)^{1/q}\leq \big(C_q(X)\, K \big)^{m}\frac{m^m}{m!}(m!)^{1/q'}\bigg(\int_{\mathbb{T}^{\infty}}\Big\Vert \sum_{\vert \alpha \vert = m}x_{\alpha}z^{\alpha}\Big\Vert^2 dz\bigg)^{1/2}\,,
L_{n}^{( l l m )}( \xi ) = \left( \begin{array}{c}{{e^{p_{n}\Delta}~ f ( u_{n}) ~ \qquad{\Delta}{\xi}e^{\eta u_{n}}}}\\{{\frac{\Delta}{\xi}e^{\eta u_{n}}\qquad f ( u_{n}) ~ e^{- p_{n}\Delta}}}\\ \end{array}\right) \quad \mathrm{w h e r e}\ \f ( u_{n}) = [ 1 +{\Delta^{2}}e^{\eta ( 2 u_{n}+ i )}]^{{\frac{1}{2}}}. \ll{L l m}
-\mathcal{T}_0 = \frac{1}{4 \pi^2}\sum_{k\neq0}\int_t\int_\xi\int_\eta \left(A(k,\xi)-A(k,\xi-\eta)\right)\widehat{u_0^z}(\eta)i k\hat{f}(k,\xi-\eta) A(k,\xi)\bar{\hat{f}}(k,\xi)\,d\eta d\xi dt.
| \psi_{\mathrm{p h y s}}> = \sum_{\alpha_{n_{0}}}| E_{n_{0}}, \alpha_{n_{0}}> \, \psi_{\alpha_{n_{0}}}\ \ , \ \ \psi_{\alpha_{n_{0}}}= < E_{n_{0}}, \alpha_{n_{0}}| \psi_{\mathrm{p h y s}}> \ ,
\!\!\begin{array}{l}\mbox{\em The$\;\;$derivative$\;\;F^{(\mbox{\tiny $K$}\!+\!1)\!\,}(x)\;\;$is$\;\;$quotient$\;\;$of$\;\;$two$\;\;$mixed$\;\;$trigono-}\mbox{\em metric$\;\;$polynomial$\;\;$functions,$\;\,$where$\;\,\mbox{\small $K$}\!=\!\max\{k_{j}\,|\,j\!=\!1,\!...,m\}$.}\end{array}
Op_h^w\big(c(x,\xi)\widetilde{\Gamma}(\frac{x+p'(\xi)}{\sqrt{h}})(x+p'(\xi))\big)Op_h^w(c')v = &= h Op_h^w\big(c(x,\xi)\widetilde{\Gamma}(\frac{x+p'(\xi)}{\sqrt{h}})\big)\mathcal{L}Op_h^w(c')v & + h Op_h^w\big(\widetilde{\Gamma}(\frac{x+p'(\xi)}{\sqrt{h}})(\partial_{\xi}c-p''\partial_xc ) +r_1 \big)Op_h^w(c')v \, ,
=\sum_{l=1}^n \sum_{j<k}\phi_j \wedge \phi_k \wedge \bar \phi_l \left( \bar\partial_l T^{i}_{jk}+ \sum_{a=1}^n T^i_{aj}A_{ka, \bar l}- \sum_{b=1}^n T^i_{bk}A_{jb, \bar l}\, , \right)
{{\cal H}}_{\rm coll}=\frac{1}{2}{\cal I}^{\alpha \beta}I_{\alpha}I_{\beta}-{\cal I}^{\alpha \beta}I_{\alpha}\overline{J_{\beta}^{(2)}}+\frac{1}{2}{\cal I}^{\alpha \beta}\overline{J_{\alpha}^{(2)}J_{\beta}^{(2)}}.
\mathfrak{H}=\bigoplus_{M=-|\Lambda|/2}^{|\Lambda|/2}\mathfrak{H}_M,\ \ \ \mathfrak{H}_M=\Big(\ker[S^{(z)}-M]\cap \mathfrak{E}_{|\Lambda|}\Big)\otimes \mathfrak{P}, H =\bigoplus_{M=-|\Lambda|/2}^{|\Lambda|/2}H_M,\ \ \ \H_M=H\restriction \mathfrak{H}_M.
\bigl\{S , S , S-p , m+p-S | J , m \bigr\}_{q=1}= f_p(J,m) \, \left(\frac{(2J+1)(2S-p)!(2S-m-p)!(J+m)!}{p! (m+p)! (2S-J)! (2S+J+1)! (J-m)!}\right)^{\frac 12},
\int_{y}\Delta_{\varphi}( x_{1}, y ) \frac{\delta^{2}S_{B}}{\delta \varphi ( y ) \delta \varphi ( x_{2})}= \int_{y}\frac{\delta^{2}S_{B}}{\delta \varphi ( x_{1}) \delta \varphi ( y )}\Delta_{\varphi}( y , x_{2}) = \delta ( x_{1}- x_{2}) .
\rho= \, \, a_1 e^{123}+ a_2 e^{124}+ a_3 e^{135}+ a_4 (e^{234}-e^{125}) + a_5(e^{235}-e^{134}) + a_6 (e^{236}-e^{145}) begin{align*}2pt] +a_7 (e^{245}-2e^{126}) + a_8 (e^{146}+e^{256}) + a_9 (2e^{136}+ e^{345}) + a_{10}(e^{346}-e^{156}) + a_{11}e^{456},
v_t-\Delta v+v+(I-\Delta)^{-1}[\gamma(v)u]-\gamma'(v)|\nabla v|^2 &=\bigg(\partial_t(w+\Gamma(v))-\Delta (w+\Gamma(v))+\left(w+\Gamma(v)\right)\bigg)+\left(v\gamma(v)-\Gamma(v)\right).
\bigoplus_{i=1}^d D(\Gamma_m,A)e_i \to (M_{n_1}\oplus N_{n_1})\otimes_A D(\Gamma_m,A) \Phi : f(Z_m)e_i \to f(Z_m)\cdot(e_i\otimes 1) \Psi: f(Z_m)e_i \to e_i \otimes f(Z_m)
\frac{\partial^{2}\Phi_{2}}{\partial t^{2}}+g\frac{\partial\Phi_{2}}{\partial z}=-\eta_{1}\frac{\partial}{\partial z}(\frac{\partial^{2}\Phi_{1}}{\partial t^{2}}+g\frac{\partial\Phi_{1}}{\partial z})-\frac{\partial}{\partial t}(|u_{1}|^{2}),
{}_{L^2(a,b;\dot{H}^{-1})}\langle f,u\rangle_{L^2(a,b;\dot{H}^1)}= \int_a^b{}_{\dot{H}^{-1}}\langle f(t,.),u(t,.)\rangle_{\dot{H}^1}\,{\rm d}t & = -\int_a^b{}_{L^{2}}\langle \psi(t,.),\nabla u(t,.) \rangle_{L^{2}}\,{\rm d}t,
{\mathbb E}\biggl\|n^{-1}\sum_{j=1}^n \langle X_j,u\rangle X_j-{\mathbb E}\langle X,u\rangle X\biggr\|= &{\mathbb E}\biggl\|n^{-1}\sum_{j=1}^n (\langle X_j,u\rangle^2 -{\mathbb E}\langle X,u\rangle^2)\frac{\Sigma u}{\langle \Sigma u,u\rangle}+n^{-1}\sum_{j=1}^n \langle X_j,u\rangle X_j'\biggr\|,
\mathcal{\bar{Y}}_{i,j}= \{y:P_{X|\bar{Y}=y}(j)>P_{X|\bar{Y}=y}(j+1), &\frac{i-1}{k}\le H(P_{X|\bar{Y}=y})<\frac{i}{k}\},i=1,\dots,k, \mathcal{\bar{Y}}_{k+1}= \{y:P_{X|\bar{Y}=y}(0)=P_{X|\bar{Y}=y}(1)\}.
s_{l}= lq(u),0\le l\le L,L = [T/q(u)],q(u) = \theta \frac{\Delta(u)}{u}, \Delta(u)=\overleftarrow{\sigma}\left(\frac{\sqrt{2}\sigma^2(u\tau^*)}{u(1+c\tau^*)}\right) \tau_n = \tau(u)+ nq(u),0\le |n|\le N,N = [\tau^*(u)/q(u)], E_{l,n}(u)=[s_l,s_{l+1}]\times[\tau_n, \tau_{n+1}].
v_{\Lambda \Omega}= \frac{1}{2}\mathrm{{s g n}}( \sigma_{R_{i j}^{( \ell )}\Lambda}^{a}) v_{\Lambda_{i j}\left( R_{i j}^{( \ell )}\Lambda \right)_{i j}}n_{\omega^{s}}( \Lambda_{i}- \ell ) \bigl ( n_{\omega^{s}}( \Lambda_{i}- \ell ) - 1 \bigr ) ,