harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{\Omega}|\nabla_g X|^2 \rho_g \, d\mu_g &\le C_0\left( N^2\int_{\Omega}|\mathcal{D}_g X|^2 \rho_g \, d\mu_g + \int_{\Omega}|X|^2 \rho_g \, d\mu_g + \int_{\Omega}|X|^2 d_g^{-4}\rho_g \, d\mu_g \right),
\int\limits_{\mathbb{R}^d}a_{\rm sym}(\xi-q) \mu(\xi,q) \big( \tilde\varphi^i_{0}(q) - \tilde\varphi^i_{0}(\xi) \big) \, d q \ = \ 2 \, \int\limits_{\mathbb{R}^d}c^i(\xi-q) \mu(\xi,q) \, dq
\min_{y\in [-1,1]^2}\max \{|\Phi(y)|, |\nabla \Phi(y)|\}= 2\mu_{k_0}^{1/2}X\min_{t\in [0,2\pi]}\max \{|\cos t|, 2\pi R|\sin t|\}=2\mu_{k_0}^{1/2}X\left(1-\frac1{8\pi^2 R^2}+O\left(\frac1{R^4}\right)\right),
P_L(\tilde{r};0) = \tilde{\gamma}-P_{\rm affine}\left( \tilde{\gamma}- \tilde{r};\lambda_L, \omega_L, \frac{\beta_L}{a_{2L}}\right), T_L(\tilde{r};0) = T_{\rm affine}\left( \tilde{\gamma}- \tilde{r};\lambda_L, \omega_L, \frac{\beta_L}{a_{2L}}\right),
g(xt\otimes t^{-1}y) = (xt)*g^{-1}\otimes gt^{-1}y = (x*g^{-1})\cdot(gtg^{-1})\otimes gt^{-1}y &= x*g^{-1}\otimes (gtg^{-1})gt^{-1}y = x*g^{-1}\otimes gy = g(x\otimes y)
\boldsymbol{I}_{12}^{-}\boldsymbol{P}_{1}^{+},\;\;\boldsymbol{I}_{12}^{-}\boldsymbol{P}_{1}^{-}(=\boldsymbol{I}_{12}^{-}\boldsymbol{P}_{2}^{+}),\;\;-(\boldsymbol{I}_{12}^{-}\boldsymbol{P}_{3}^{+}\oplus \boldsymbol{I}_{12}^{-}\boldsymbol{P}_{3}^{-}).
f(T_2e^{i\theta}) =\frac{1}{q}\sum_{m=1}^q \sum_{n\ge 1}\langle\Phi_{0,m}^{(N)}(\phi),A(\phi)\Phi_n^{(N)}(\phi)\rangle\langle\Phi_n^{(N)}(\phi),B(\phi)\Phi_{0,m}^{(N)}(\phi)\rangle &\times \exp[i(E_n^{(N)}(\phi)-E_{0,m}^{(N)}(\phi))T_2e^{i\theta}],
E \Biggl[\zeta f_1\bigl(\check{\mathbf{L}}^n\bigr)f_2\bigl(\widetilde{\mathcal{W}}(\beta)\bigr)\prod_{j=1}^JY^n_{j-}Y^n_{j+}\Biggr] -E \Biggl[\check{\zeta}f_1\bigl(\check{\mathbf{L}}^n\bigr)\prod_{j=1}^{J-1}Y^n_{j-}Y^n_{j+}\Biggr]\to0.
\left(\int_{\overline\Omega}g_k^p\,d\mu\right)^{1/p}\le\sum_{j=k}^{N(k)}\lambda_{j,k}\left(\int_{\overline\Omega}g_{u_j}^p\,d\mu\right)^{1/p}=\sum_{j=k}^{N(k)}\lambda_{j,k}\left(I_{f_j}(u_j)+p\int_{\partial\Omega}u_jf_j\,d\nu\right)^{1/p}.
E(T_1)b = E_{D_1}(\mu_1(T_1), \mu_2(T_2),\ldots, \mu_n(T_n)) \mbox{and}(1-p_{\epsilon(1)})T_1 b = E_{D_2}(\mu_1(T_1), \mu_2(T_2),\ldots, \mu_n(T_n)).
\begin{aligned}H^s_{S^{\perp}}(\mathbb{T}^{\nu+1}):=\left\{u\in H^s(\mathbb{T}^{\nu+1}) : u(\varphi, \cdot)\in H_S^{\perp},\,\,\,\forall\varphi\in \mathbb{T}^{\nu}\right\}, & H^s_{S}(\mathbb{T}^{\nu+1}):=\left\{u\in H^s(\mathbb{T}^{\nu+1}) : u(\varphi, \cdot)\in H_S\,\,\,\forall\varphi\in \mathbb{T}^{\nu}\right\}.\end{aligned}
_2 F_{1}[ a , b ; c ; z ] = \sum_{l = 0}^{\infty}{\frac{( a )_{l}( b )_{l}}{( c )_{l}}}{\frac{z^{l}}{l !}}= 1 +{\frac{a b}{c}}{\frac{z}{1 !}}+{\frac{a ( a + 1 ) b ( b + 1 )}{c ( c + 1 )}}{\frac{z^{2}}{2 !}}+ \cdots .
E_{\pi|_{Y}}((T_1, \ldots, T_n)|_{Y})E_{\pi|_{V_{p}}}((T_1, \ldots, T_n)|_{V_{p}}) \cdots E_{\pi|_{V_m}}((T_1, \ldots, T_n)|_{V_m}) = E_{\pi|_{V}}((T_1,\ldots, T_n)|_{V})
\int_{\mathcal{H}}(\mathfrak{Re}\langle e_k, T^nx \rangle)^2\,(\mathfrak{Re}\langle e_\ell, x \rangle)^2\, dm(x)= \frac{2!}{2^2}(\mathfrak{Re}\langle RT^{*n}e_k,e_\ell \rangle)^2+\sigma_k^2\sigma_\ell^2.
\Delta v_{1\epsilon}+e^{v_{2\epsilon}+h_{2\epsilon}}(1-e^{v_{1\epsilon}+h_{1\epsilon}})=\frac{4(N_1+\beta_1)}{(1+|x|^2)^2}=g_{1}, \Delta v_{2\epsilon}+e^{v_{1\epsilon}+h_{1\epsilon}}(1-e^{v_{2\epsilon}+h_{2\epsilon}})=\frac{4(N_2+\beta_2)}{(1+|x|^2)^2}=g_{2}.
\mathcal{A}(\mathcal{M}_\ell)^\nu = \mathcal{P}_\ell (I - \mathcal{M}_\ell) (\mathcal{M}_\ell)^\nu = \mathcal{P}_\ell \mathcal{M}_2(\hat{S}) \mathcal{M}_1(\hat{H}^\top) \, (I - \mathcal{M}_s) (\mathcal{M}_s)^{\nu-1}\, \mathcal{M}_1(\hat{H}).
\gamma_a(2s) = v^{-2s^2+s}\prod_{k=1}^s \frac{1-v^{-2a-4k-2}}{1-v^{-4k}}, \gamma_a(2s+1) = v^{-a-2s^2-s-1}\prod_{k=1}^s \frac{1-v^{-2a-4k-2}}{1-v^{-4k}}.
\tilde t=\varepsilon_4t+\varepsilon_1 ,\tilde x=\varepsilon_5x+\varepsilon_2 ,\tilde u=\varepsilon_6u+\varepsilon_3 , \tilde A=\varepsilon_7 A ,\tilde G=\varepsilon_4^{-1}\varepsilon_5^2\varepsilon_7^{-1}G ,\tilde W=\varepsilon_4^{-1}\varepsilon_6W ,
V^{( 1 )}( \Phi^{( 1 )}) : = \frac{1}{2}\Big ( \Phi^{( 1 )}\Big )^{2}\; \frac{e^{2}N}{\pi + g N}\; - \; \frac{1}{\pi}m c N \cos \left( 2 \sqrt{\frac{\pi}{N}}\sqrt{\frac{\pi}{\pi \! + \! g N}}\Phi^{( 1 )}- \frac{\theta}{N}\right) \; .
e^{-|O(at)x-y|^2/4t}&=e^{-|x|^2/4t}+e^{-|x|^2/4t}\,\frac{\left(O(at)x\right)\cdot y}{2t}&+\frac{1}{2}\,e^{-|O(at)x-\theta y|^2/4t}\; y^T\,\frac{\left(O(at)x-\theta y\right)\otimes \left(O(at)x-\theta y\right)-2t\,\mathbb I}{4t^2}\, y.
\Delta \rho (k)=\frac{1}{\pi}\frac{1}{k^{2}+1}\int_{1}^{\infty}dx\left[ \Theta (\Lambda -\frac{1}{2x^{2}}\delta -|k|)-\Theta (\Lambda -|k|)\right] \left( \frac{\sqrt{x^{2}-1}}{x^{2}}+\frac{k^{2}}{\sqrt{x^{2}-1}}\right)
\Gamma(2+a+b+2\gamma s) = 2^{1+a+b+\gamma 2s}\Gamma(1+\gamma s + (a+b)/2)\Gamma(\gamma s+(a+b+3)/2)/\sqrt{\pi}&\Gamma(2+a+b+\gamma(2s+1)) &= 2^{1+a+b+\gamma(2s+1)}\Gamma(1+\gamma(s+1/2)+(a+b)/2)\Gamma(\gamma(s+1/2)+(a+b+3)/2)/\sqrt{\pi}
R^*\psi = \sum_{\beta \neq 0}(-1)^{|\alpha|+|\beta|}\, D^\alpha_x\Big(r_{\alpha,\beta}(x)\,D^\beta_{\xi}\psi(x,\xi)\Big) =:\sum_{\beta \neq 0}r^*_{\alpha,\beta}(x)\,D^\alpha_x D^\beta_{\xi}\psi(x,\xi).
0 < \delta < 1, 1< \delta^2 + \delta^6, d_* = \frac{1}{\sqrt{1-\delta^2}}, 0 < q_{max}, 0<s, \frac{1}{\delta^4 d_*^2}\Big( 1 +\frac{q_{max}}{\delta^2 d_*^2}\Big) \leq \delta^2 - \frac{s}{\delta^2 d_*^2}.
\chi_{1 1}^{R}= \chi_{R}( \theta_{+ 1}^{\prime}) - \chi_{R}( \theta_{+ 1}) , \quad \chi_{1 1}^{J}= \chi_{J}( \theta_{+ 1}^{\prime}) - \chi_{J}( \theta_{+ 1}) , \quad \chi_{1 2}^{J}= \chi_{J}( \theta_{+ 1}^{\prime}) - \chi_{J}( \theta_{+ 2}) .
s^2\mathcal{Q}_1 =\hat{\Theta}_s^0(s\mathfrak{e}_0); s^2\mathcal{Q}_2 =-\hat{\Theta}_s^0(s\mathfrak{c}^g_0-s\mathfrak{h}^g_0) s^2\mathcal{Q}_3 =\hat{\Theta}_s^0(s\mathfrak{e}'_0).
E_1^{-k,i+k}=\underset{j\ge0\atop j\ge -k}\bigoplus\underset{N\in \Theta_{2j+k+1}}\bigoplus H_{rig}^{i-2j-k}(M\cap N/K_0)\Longrightarrow H_{rig}^i(M/{\mathfrak S}^{0}).
\frac{x^\lambda}{(q;q)_\infty^{\mathrm{rk}(\mathfrak{g})}\prod_{\alpha\in\Delta_+}(1-x^\alpha)\prod_{\alpha\in\Delta}(qx^\alpha;q)_\infty}= \sum_{\mu\in P}m_{\lambda\mu}(q)\frac{E_\mu(x,q,0)}{(q)_\mu}.
|t|^{d/2}\|u(t,)\|_{L^\infty}\leq t^{d/2}\sum_{k\in\mathbb{Z}}\|u_k(t,)\|_{L^\infty}& \lesssim_{d}\sum_{k\in\mathbb{Z}}\left( \sum_{|\alpha| + |\beta| = d}\| ()^\alpha u_k(0,)\|_{L^2}\|()^\beta u_k(0,)\|_{L^2}\right)^{\frac12}
\begin{cases}z_{(0,1)}=0, z_{(0,0)}z_{(0,1)}=z_{(0,0)}z_{(1,0)}=z_{(0,0)}z_{(1,1)}=z_{(0,1)}z_{(1,0)}=z_{(0,1)}z_{(1,1)}=z_{(1,0)}z_{(1,1)}=0, z_{(0,0)}\vee z_{(0,1)}\vee z_{(1,0)}\vee z_{(1,1)}=1\end{cases}
(1-\frac{2}{2^{s}})\zeta(s)=\eta(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{s}}=\frac{1}{1^{s}}-\frac{1}{2^{s}}+\frac{1}{3^{s}}-\cdot\cdot\cdot,
\Upsilon_m(\lambda_1,\cdots, \lambda_m, x_1,\cdots x_m)= c_1\Bigg( \sum_{i=1}^m \tau(x_i) \lambda_i^{n-2}- \sum_{\substack{i, j=1 i \ne j}}^m G(x_i, x_j) (\lambda_i\lambda_j)^{n-2 \over 2}\Bigg) - c_2\log (\lambda_1 \cdots \lambda_m)
a_1 \frac{d b_1}{b_1}\wedge \frac{d b_2}{b_2}+a_2 \frac{d b_1}{b_1}\wedge \frac{d b_3}{b_3}+a_3 \frac{d b_2}{b_2}\wedge \frac{db_3}{b_3}= a_1' \frac{d b_1}{b_1}\wedge \frac{d b_2}{b_2}+a_2' \frac{d b_1}{b_1}\wedge \frac{d b_3}{b_3}+ \frac{d z_1}{z_1}\wedge \frac{dz_2}{z_2}
V ( r ) \sim - \frac{1}{\pi^{2}M_{5}^{3}\lambda r}\left[ \frac{\pi}{2}- \frac{\frac{2 R}{\lambda}}{\frac{2 R}{\lambda}- \frac{1 5}{8}}\left\{\frac{\pi}{2}- f \left( \frac{r}{R}\left( \frac{2 R}{\lambda}- \frac{1 5}{8}\right) \right) \right\}\right]
1\{\bar g(\theta_L)> c(\theta_L)\}\Big(1-\Phi\Big(\frac{t-R}{\varsigma}\Big)\Big) =1\{\bar g(\theta_L)> c(\theta_L)\}\Big(1-\Phi\Big(\frac{\bar g(\theta_L)- c(\theta_L)-s_L(\theta_L)R}{\varsigma s_L(\theta_L)}\Big)\Big).
A=\left[\begin{array}{cccc}C_{1}C_{2}\cdots C_{m}C_{2}^{t}C_{m+1}\cdots C_{2m-1}\vdots \vdots \ddots \vdots C_{m}^{t}C_{2m-1}^{t}\cdots C_{\binom{m+1}{2}}\end{array}\right],
f(t) := 16t^6 - 24t^4 - 8t^3 + 36t^2 + 6t + 1, g(t) := 64t^8 + 64t^7 + 64t^6 - 128t^5 - 56t^4 + 16t^3 + 64t^2 - 8t + 1.
Q = \frac{1}{2 \pi}\left\{\lim_{r{\rightarrow}{\infty}}\left[ \overline{{\cal{N}}}( r ) -{\sin}\left( \overline{{\cal{N}}}( r ) \right) \right] + 2 \pi \ell - \overline{{\cal{N}}}( 0 ) +{\sin}\left( \overline{{\cal{N}}}( 0 ) \right) \right\}.
\chi_{( \mu , 2 \kappa , - s )}( \sigma^{2}) = \chi_{( \lambda , \kappa , 0 )}( v ) \chi_{( \lambda , \kappa , 0 )}( v ) \chi_{( \mu , 2 \kappa , - s )}( v^{- 1}) = q^{( \mu , \mu + 2 \rho_{0}) - 2 ( \lambda , \lambda + 2 \rho_{0}) - 2 s ( 2 \kappa + g )}
\pi_\phi(m^*a_1m)\pi_\phi(m^*a_2m) =\pi_\phi(m^*a_1mm^*a_2m)=\pi_\phi(m^*mm^*a_1a_2m) &=\phi(m^*m)\pi_\phi(m^*a_1a_2m)=\pi_\phi(m^*a_1a_2m)
C^1(X) = \left\{f \in C^0(X) : \sum_{x \in X}f(x) = 0 \right\}, C^2(X) = \left\{f \in C^0(X) : \sum_{x \in X}f(x) = 0, \, \sum_{x \in X}f(x) x = 0 \right\}
\mathcal{U}= \left \{\alpha_{k_{0}}\times \alpha_{k_{1}}\times \ldots \times \alpha_{k_{M}}\times \alpha_{k_{M+1}}\times I^{\infty}: \{k_{j}\}_{j=0}^{M+1}\right. & \left. \{1, \ldots, n \}M+2 \right. \big \}.
\left( \int d^{d}k \ \frac{1}{k^{4}}\right)_{\infty}= \left( \lim_{L \to 0}\int d^{d}k \ \frac{1}{( k^{2}+ L )^{2}}\right)_{\infty}= - \frac{2 i \pi^{2}}{d - 4},
x_1(t) = b_1 + y_1(t) + \int_0^t(-x_1(s) + x_2(s))ds - u_1(t), x_2(t) = b_2 + y_2(t) + \int_0^t(-x_2(s))ds + u_1(t) - u_2(t), x_1(t) \le 0, 0 \le x_2(t) \le B, t \ge 0,
Q_{F_{1}, V_{8}}^{M a x w e l l}= \int_{V_{8}}*_{s}j_{D 2}^{M a x w e l l}= \frac{T_{F 1}}{2}\left( 1 - \cos \theta_{0}\ \{+ F_{2}\{\mathrm{f l u x \a b o v e \b r a n e}}\atop{- F_{2}\ \mathrm{f l u x \b e l o w \b r a n e}}}\right) ,
\mathbb{P}( Z^1_t - Z^2_t \in B(0, \delta), Z^1_t, Z^2_t \in B(0, R')) = \mathbb{P}( Z_t^1, Z_t^2 \in B(0,R')) - \mathbb{P}( Z^1_t - Z^2_t \notin B(0, \delta), Z^1_t, Z_t^1 \in B(0,R'))
\mathfrak{S}(f_k)=\mathfrak{S}(f_k|_{V\setminus\mathcal{F}})=\tilde{k}+\tilde{r}-1,\,\,\,\,\,\overline{\mathfrak{S}}(f_k)=\overline{\mathfrak{S}}(f_k|_{V\setminus\mathcal{F}})=n-|\mathcal{F}|-\tilde{k}+1.
[ \hat{C}^{i b}, \hat{Q}^{a}]_{+}= i \varepsilon^{a b}\hat{\pi}_{( B )}^{0 i}, \; \; [ \hat{\pi}_{( C ) i b}, \hat{Q}^{a}]_{+}= i \varepsilon_{0 i j k}\delta_{b}^{a}( \partial^{j}\hat{A}^{k}+ \frac{1}{2}\hat{A}^{j}\wedge \hat{A}^{k}) ,
\Gamma \left( - N - \frac 1 2 , w \right) = \frac{( - 1 )^{N}}{\Gamma \left( N + \frac 3 2 \right)}\left[ - \pi \, \mathrm{e r f c}\left( \sqrt{w}\, \right) + e^{- w}\sum_{l = 0}^{N}( - 1 )^{l}\Gamma \left( l + \frac 1 2 \right) w^{- l - \frac 1 2}\right] \, ,
T_{a}\Psi^{( 0 )}= \left( \begin{array}{c}{\phi^{( 0 )}( s^{i}+ a^{i})}\\{\phi^{\ast ( 0 )}( s^{i}+ a^{i})}\\{A_{k}^{( 0 )}( s^{i}+ a^{i})}\\ \end{array}\right) ,
\left( 1+\left\vert x\right\vert^{\beta}\right) \left\vert \partial_{x}^{\beta}\mathcal{B}_{\ell}\left( x\right) \right\vert =\left\vert\partial_{x}^{\beta}\mathcal{B}_{\ell}\left( x\right) \right\vert+\left\vert x^{\beta}\partial_{x}^{\beta}\mathcal{B}_{\ell}\left( x\right)\right\vert \leq C\mathcal{B}_{\ell}\left( x\right) .
\frac{\partial}{\partial t}\mu^p(x^{(p)},t)+\frac{\partial}{\partial x^{(p)}}\left(\frac{\mu^p(x^{(p)},t)}{\omega(x^{(p)},t)}\f\big(\omega(x^{(p)},t)\big)\right)=0, x^{(p)}\in P^p,\t>0,
I_{2 , 2}( \overline{{P_{1}}}, \overline{{P_{2}}}, Q_{1}, Q_{2}) = \sum_{N = 0}^{\infty}\int d x d y \overline{{P_{1}( x )}}\overline{{P_{2}( x )}}G_{N}( x , y ) Q_{1}( y ) Q_{2}( y ) .
\mod{\frac{I_2(w)}{G(w)}}=\mod{\frac{\cos(w)}{\sin(w)}}\frac{\mod{O(w^{-1})e^{-w^2+1+O(w^{-1})}+O(w^{-1})e^{O(w^{-1})}}}{\mod{e^{-w^2+1}-1}}\leq\frac{C}{\mod{w}}\mod{\frac{\cos(w)}{\sin(w)}}\frac{\mod{e^{-w^2+1}}+1}{\mod{e^{-w^2+1}-1}},
\mathcal{H}_\mathrm{approx}:= \mathrm{const}+ \int{dx \big\{\overline{B(x)}b_x + B(x) b_x^\ast\big\}}&+\int{dx dy\Big\{(b_x, b_x^\ast)\begin{pmatrix}-H^T(x,y) ( \upsilon_N \overline{m})(x,y) -( \upsilon_N m)(x,y) H(x,y)\end{pmatrix}\begin{pmatrix}-b_y^\ast b_y\end{pmatrix}\Big\}},
C_{\varphi}^{*,\beta}- M_{g}C_{\sigma}M_h^{*,\beta}= (C_{\varphi}^{*,\alpha}+K_1) - M_{g}C_{\sigma}(M_h^{*,\alpha}+K_2) & = C_{\varphi}^{*,\alpha}- M_{g}C_{\sigma}M_h^{*,\alpha}+ K_1 - M_{g}C_{\sigma}K_2 & = K + K_1 - M_gC_{\sigma}K_2,
\prod_{j=1}^{k-1}\Gamma(a+b+2-(k+j-3)\gamma) = \frac{\Gamma(2+a+b-(k-2)\gamma)}{\Gamma(2+a+b-(2k-2)\gamma)}\frac{\prod_{j=1}^{k}\Gamma(2+a+b-(k+j-2)\gamma)}{\Gamma(2+a+b-(2k-3)\gamma)}
(B){\cal T}_s:=\{t_k:\, a_k \int_{B_{\sigma}}e^{-t_k^2 x^2}dx \leq \delta\}\; \Leftrightarrow \; R_l=\min k : \, a_k \int_{B_{\sigma}}e^{-t_k^2 x^2}dx \leq \delta.
G_{\ell}{\underbar{c}^{\star}_{\ell}}^{(LI)}={\underbar{d}^{\star}_{\ell}}^{(LI)},\ \ \ell = 1,\dots, L m_k({\underbar{d}^{\star}_{k}}^{(CR)}) ={\underbar{c}^{\star}_{k}}^{(CR)},\ \k=1,\dots,K,
\mathbf{A}(\lambda )=(-1)^{\mathsf{N}}\frac{2\lambda +\eta}{2\lambda}(\lambda -\eta /2+p)(\sqrt{(1+\xi^{2})}(\lambda -\eta /2)+q)\prod_{b=1}^{\mathsf{N}}(\lambda -\zeta_{b}^{(0)})(\lambda +\zeta_{b}^{(1)}),
\langle \hat \theta_r-\theta_r,u\rangle=\frac{\langle (\hat P_r-P_r)\theta_r,u\rangle-\Bigl(\sqrt{1+\langle (\hat P_r-P_r)\theta_r,\theta_r\rangle}-1\Bigr)\langle \theta_r, u\rangle}{\sqrt{1+\langle (\hat P_r-P_r)\theta_r,\theta_r\rangle}}
K=\frac{1}{2}\sum_{i=1}^{N}m_{i}\dot{\textbf{x}}_{i}^{2}=\frac{1}{2}[\dot{\textbf{x}}_{1},...,\dot{\textbf{x}}_{N}]\textbf{M}\left[\begin{array}{c}\dot{\textbf{r}}_{1}... \dot{\textbf{x}}_{N}\end{array}\right]
[\vec w]_{A_{\vec p, \vec r}}\le [W]_{A_{\frac{p_m}{r_m},\frac{\delta_{m+1}}{r_m}}(\widehat{w})}^{\frac1{\delta_{m+1}}}[\widehat{w}]_{A_{\frac{1-r}{r}\varrho}}^{\frac1{\varrho}}\prod_{i=1}^{m-1}\Big[w_i^{\frac{\theta_i}{p_i}}\Big]_{A_{\frac{1-r}{r}\theta_i}}^{\frac1{\theta_i}}.
\Phi(x,y) =\#\{n\le x\colon P^-(n)>y\Omega(n)\le2\}&=\pi(x)-\pi(y)+1+ \sum_{y<p\leq \sqrt{x}}\sum_{p\le q\le x/p}1 &=\pi(x)-\pi(y)+1+ \sum_{y<p\leq \sqrt{x}}(\pi(x/p)-\pi(p)+1),
\|(I - PB_c (P^tAP)^{-1}P^tA)(I - MA)\|_A^2 \le \|(I - PB_c (P^tAP)^{-1}P^tA)\|_A^2 \|I - MA\|_A^2 & \le \|(I - PB_c (P^tAP)^{-1}P^tA)\|_A^2 & = \|T\|_A^2 + \|P(B_c - I_c) (P^tAP)^{-1}P^tA\|_A^2 & \le \|T\|_A^2 + \|B_c - I_c\|_{A_c}^2 \|(P^tAP)^{-1}P^tA\|_{A_c}^2 & \le \|T\|_A^2 + \|I - T\|_A^2 = 1.
S^{-1}(w_{ij}\Delta_{ij}^{c_i,c_j})=\underbrace{\cdots (s_i+c_iD_{s_i})(s_j+c_jD_{s_j})}_m-\underbrace{\cdots (s_j+s_jD_{s_j})(s_i+c_iD_{s_i})}_m
(A_2 w , w)_H = -\frac{3}{2}\int_0^L \frac{w_x^2}{(L-x)^2}dx + 3 \int_0^L \frac{w^2}{(L-x)^4}dx +\frac{1}{2}\int_0^L \frac{w^2}{(L-x)^2}dx -\frac{w_x^2(0)}{2L}
I_{n}^{p}\equiv \frac{\mu_{0}^{( p + 1 ) / 2}}{n}\int_{\Sigma}d^{p + 1}\sigma e ( \sigma ) \left[{\frac{( \mathrm{d e t}\gamma_{m n})^{n / 2}}{e ( \sigma )^{n}}}+ n - 1 \right]
{\frac{1}{G}}={\frac{1}{1 2 \pi}}q^{\prime}( 1 ) ={\frac{1}{1 2 \pi}}\left[ \sum_{s}( 1 - 6 \xi_{s}) m_{s}^{2}\ln m_{s}^{2}+ 2 \sum_{d}m_{d}^{2}\ln m_{d}^{2}\right] ~ .
{\rm Im}\,{\bf w}^j:={\bf \Phi}_j(z,\overline z,{\rm Re}\,{\bf w}^2, \ldots,{\rm Re}\,{\bf w}^{j-1})+{\rm O}(j) \ \ \ \ \ \{\scriptstyle (j\,=\,2\,,\,\ldots\,,\,\rho)},
\delta_i(\iota(x)) =\sum_{\sigma\in S_p}sgn(\sigma)P_{\sigma(1)}\otimes\ldots\otimes\widehat{P_{\sigma(i)}}\otimes\ldots\otimes P_{\sigma(p)}\otimes(P_{\sigma(i)}+Q) g(\delta(x)) =\sum_{j=1}^p\sum_{\tau\in S_{p-1}}(-1)^jsgn(\tau)P_{\tau'(1)}\otimes\ldots\otimes P_{\tau'(p-1)}\otimes(P_j+Q),
\operatorname{Ext}^1_{\mathcal{O}}(P,M) = H^1(\operatorname{Hom}(P,M)^{\mathcal{O}}) \rightarrow H^1(\operatorname{Hom}(\overline{P},\overline{M})^{\mathcal{O}}) = \operatorname{Ext}^1_{\mathbb{F}}(\overline{P},\overline{M})
\begin{aligned}\lim_{k \to \infty}D(c)^{n_k}= \lim_{k \to \infty}\left( \begin{matrix}(1+c)^{n_k}0 0 (1+c)^{-n_k}\end{matrix}\right) & = \lim_{k \to \infty}\left( \begin{matrix}\exp(n_k \log(1+c)) 0 0 \exp(-n_k \log(1+c)) \end{matrix}\right) & = \left( \begin{matrix}\exp(u \log(1+c)) 0 0 \exp(-u \log(1+c)) \end{matrix}\right) = D(d).\end{aligned}
\frac{\delta S}{\delta\phi}=\frac{\delta S}{\delta\hat{h}_{ab}}\frac{\delta\hat{h}_{ab}}{\delta\phi}=-\frac{1}{2}\sqrt{-h}T_{ab}e^{\phi}h^{ab}=-\frac{1}{2}\sqrt{-h}T_{a}^{a}e^{\phi}=0\Rightarrow T_{a}^{a}=0,
\Delta_{F}\left( x , x^{\prime}\right) = < I N \left| T ( \Phi ( x ) \Phi ( x^{\prime}) ) \right| I N > = ( - i ) \int \frac{d^{4}k}{\left( 2 \pi \right)^{4}}e^{i k ( x - x^{\prime})}\frac 1{k^{2}+ M^{2}- i \varepsilon}
\alpha_1:=\alpha_{\xi}=\left(\frac{2\cdot 2(1+\varepsilon)\cdot \sqrt{1-\varepsilon^2}}{2\cdot\frac{\sqrt{1-\varepsilon^2}}{1-\varepsilon}\cdot 1}\right)^{1/2}=\sqrt{2}\cdot\sqrt{1-\varepsilon^2}
\beta^{(2)}_0=\lim_m\frac{\sum_{n\in{\mathbb N}^m_0}\frac{1-c_{2n}}{1+c_{1n}}}{\sum_{n\in{\mathbb N}^m_0}\frac{1-c_{1n}}{1+c_{1n}}},\quad\beta^{(3)}_0=\lim_m\frac{\sum_{n\in{\mathbb N}^m_0}\frac{(1-c_{1n})(1-c_{2n})}{1+c_{1n}}}{\sum_{n\in{\mathbb N}^m_0}\frac{1-c_{1n}}{1+c_{1n}}}.
\forall\left( \mathbf{j},u\right) \in\left( \mathbf{H}\left(\mathbb{R}^{3},\operatorname*{div}\right) ,L^{2}\left( \mathbb{R}^{3}\right) \right) b\left( \left( \mathbf{j},u\right) ,\left(\mathbf{m},v\right) \right) =0.
{G}^m (t,u) =b\left[ f(u+\hat{v}(\cdot +\gamma^m(t))) - f(\hat{v}(\cdot +\gamma^m(t)))\right]{R}^m(t,u) ={G}^m (t,u) -b \,f'(\hat{v}(\cdot+\gamma^m(t)))\, u.
S_i=\sum_{k=1}^n E_{k,i},E_{k,i}=\frac{1}{n^2}(D_k(\tau_i) -D_{k+1}(\tau_i))(\sqrt{n}f_{1,k}+f_{3,k}) +\frac{1}{n^2}D_{k+1}(\tau_i)f_{2,k},
D_{2}^{\scriptscriptstyle S E}( x_{1}, x_{2}) = \bigl [ T_{1}^{h + u}( x_{1}) , T_{1}^{h + u}( x_{2}) \bigr ] \bigr |_{\scriptscriptstyle S E}= : h^{\alpha \beta}( x_{1}) h^{\mu \nu}( x_{2}) : d_{2}( x_{1}- x_{2})_{\alpha \beta \mu \nu},
\epsilon_{a,0}(\beta) = T \sum_{b=1}^{P}{\cal I}_{ab}^{(P)}\int_{-\infty}^\infty\frac{d\beta'}{2\pi}\frac{P-1}{\cosh[(P-1)(\beta-\beta')]}\ln\left(1+e^{-\epsilon_{b,0}(\beta')/T}\right)
\leq \; \left\langle \prod_{b = 1}^{N}2 \cosh \left( \; 2 \beta^{( b )}\int_{\Lambda}d^{2}x \; t ( x ) \; : \cos \left( 2 \sqrt{\pi}\varphi^{( b )}( x ) \right) :_{Q^{W}}\; \right) \; \right\rangle_{Q^{\mu}}
Q^{\mu\nu}=\left( \epsilon^{\mu\nu\sigma\tau}-\frac{i}{2}e^{\mu\nu\sigma\tau}\right) F_{\sigma\tau},P_{\mu\nu}=\left( \delta_{\mu}^{\lambda}\delta_{\nu}^{\rho}-\frac{i}{2}e_{\mu\nu\sigma\tau}\epsilon^{\sigma\tau\lambda\rho}\right) F_{\lambda\rho},
[\begin{matrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{matrix}]=\frac{E}{(1-\nu^{2})}[\begin{matrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{matrix}][\begin{matrix}E_{11}\\E_{22}\\E_{12}\end{matrix}]
\alpha\circ P_{d}=P_{d}\circ\alpha &\Longleftrightarrow\forall m\in\mathbb{Z},~q^{m}+q^{-m}=q^{m+d}+q^{-m-d}\Longleftrightarrow \forall m\in\mathbb{Z},~q^m(1-q^d)=q^{-m-d}(1-q^d) &\begin{array}[b]{l}\Longleftrightarrow \forall m\in\mathbb{Z},~q^m=q^{-m-d}\end{array}\Longleftrightarrow \forall m\in\mathbb{Z},~q^{2m+d}=1,
\{_{(p-2)}A\}_{d-n}= \phi_{d, d-n}^\jmath \circ \Phi_d^\jmath (\{_{p}A\}) &=\Phi_{d-n}^\jmath \circ \xi_{-2}^\jmath (\{_{p}A\}) =\{_{(p-2)}A\}_{d-n}+ \sum_{\stackrel{B \in \Xi_{d-n}}{B\sqsubset\ \!_{(p-2)}A}}f_B \{B\}_{d-n}.
\bar{\Phi}= 2 \sqrt 2 r , \bar{F}_{t r}= Q e^{- 2 \sqrt 2 r}, \bar{T}= 0 ,{}\bar{G}_{\mu \nu}= \left( \begin{array}{c c}{- f}&{0}\\{0}&{f^{- 1}}\\ \end{array}\right) ,
M: = \Bigl\{(\gamma_i, \delta_i) \Bigm| \gamma_{it}= \prod_{k \in \eta_i(t)}\delta_{ik}\prod_{k \notin \eta_i(t)}(1-\delta_{ik}) \; t \in \{1, \ldots, 2^{\lceil \log_2 l_i \rceil}\},\ \delta_i \in [0,1]^{\lceil \log_2 l_i \rceil}\Bigr\}.
\theta C+I_d=\begin{pmatrix}1 \theta 0 0 0 \hdots\cr \theta 1 \theta 0 0 \hdots\cr 0 \theta 1 \theta 0 \hdots\cr \ddots \ddots \ddots \ddots \cr0 \hdots 0 \theta 1 \theta\cr0 \hdots 0 0 \theta 1\end{pmatrix}.
\lim\limits_{r\to+\infty}\frac{\bar u(r,\theta)}{u_0(r)}=\lim\limits_{r\to+\infty}\sum_{k=0}^{\infty}\frac{\bar f_k}{u_0}\varphi_k=\sum_{k=0}^{\infty}\lim\limits_{r\to + \infty}\frac{\bar f_k}{u_0}\varphi_k=\sum_{k=0}^{\infty}\bar C_k\varphi_k=g(\theta).
af(\chi)^2 +\tilde{b}f^{\alpha}(\mathcal{Y}_1,\ldots, \mathcal{Y}_n)bf(\chi)+f(\chi)cf(\chi) +f(\chi)b^{\prime}f^{\alpha}(\mathcal{Y}_1,\ldots, \mathcal{Y}_n)d -cf(\chi)^2 -b^{\prime}f^{\alpha}(\mathcal{Y}_1,\ldots, \mathcal{Y}_n)^2d=0.
{\cal L}_{gK}= \frac{1}{2}\left[i\overline{\Psi}_L\left(\partial\!\!\!/ +\frac{1}{2}(i{\cal A}\!\!\!/\,\hat{C}+{\cal A}\!\!\!/\:\hat{C}i)\right) \Psi_L+ \overline{\Psi}_L\left(\partial\!\!\!/ + \frac{1}{2}(i{\cal A}\!\!\!/\,\hat{C}+{\cal A}\!\!\!/\:\hat{C}i)\right)\Psi_L i\right]
V^{(1)}( \Phi^{(1)}) :=\frac{1}{2}\Big( \Phi^{(1)}\Big)^2 \; \frac{e^2 N}{\pi+gN}\; - \;\frac{1}{\pi}m c N\cos \left(2 \sqrt{\frac{\pi}{N}}\sqrt{\frac{\pi}{\pi\!+\!gN}}\Phi^{(1)}- \frac{\theta}{N}\right)\; .
(|k|^2+m^2)\,\tilde{u_2}(k,t)=\int_0^t\frac{\sin((t-s)\sqrt{|k|^2+m^2})}{\sqrt{|k|^2+m^2}}h(s)ds,\sqrt{|k|^2+m^2}\,\tilde{\dot u_2}(k,t)=\int_0^t\frac{\cos((t-s)\sqrt{|k|^2+m^2})}{\sqrt{|k|^2+m^2}}h(s)ds.
ds^{2}=c^{2}d\tau^{2}=(1-\frac{r_{s}}{r}+\frac{r_{Q}^{2}}{r^{2}})c^{2}dt^{2}-(1-\frac{r_{s}}{r}+\frac{r_{Q}^{2}}{r^{2}})^{-1}dr^{2}-r^{2}d\theta^{2}-r^{2}sin^{2}\theta d\varphi^{2},
q_i=\sum_{j=1}^{n}\rho_j^{(i)}=\sum_{j=i}^{n}(-1)^{j-i}\prod\limits_{k=i}^{j}{1 \over a_k}={1\over a_i}-{1\over a_i a_{i+1}}+ \ldots +(-1)^{n-i}\prod\limits_{k=i}^{n}{1 \over a_k}>0.
\hat{\hat{S}}= \int d^{1 1}x s q r t{| \hat{\hat{g}}|}\ \left[ \hat{\hat{R}}-{\textstyle \frac{1}{2 \cdot 4 !}}\hat{\hat{G}}{}^{2}-{\textstyle \frac{1}{6^{4}}}\frac{1}{\sqrt{| \hat{\hat{g}}}|}\hat{\hat{\epsilon}}\partial \hat{\hat{C}}\partial \hat{\hat{C}}\hat{\hat{C}}\right] .
\exp\bigl(\phi\bigl( \ln x \bigr)\bigr) \ \ +\infty \Leftrightarrow \phi'(u) = o(1) \ \ \ \u \to +\infty, \exp\bigl(\phi\bigl( \ln_2 x \bigr)\bigr) \ \ +\infty \Leftrightarrow \phi'(u) = o\bigl(\exp(u)\bigr) \ \ \ \u \to +\infty.
\delta_{t}(e_F)^{1}_{ij}-\mu\Delta_{h}(e_F)^{1}_{ij}- \mathfrak{w}_{1,1}\Delta_{h}(e_F)^{1}_{ij}=\frac{1}{2}\left[g(u^0_{ij})-g((U_F)^0_{ij})\right] +\frac{1}{2}\left[g(u_{ij}^1)-g((U_C)^1_{ij})-g'((U_C)^1_{ij}) \left((U_F)_{ij}^1-(U_C)_{ij}^1\right)\right]+(R)_{ij}^1,(x_i,y_j)\in\Omega_{h},
\int \dfrac{x \cos(x;q)}{( \frac{-x^2}{q}(1-q)^2;q^2)_\infty}\,_2\phi_1(\frac{-x^2}{q}(1-q)^2,q^2;0;q^2,q)d_qx &= \frac{-\sqrt{q}}{(\frac{-x^2}{q}(1-q)^2;q^2)_\infty}\left(\dfrac{\sqrt{q}\cos(\frac{x}{q};q)}{(1-q)}+ x \sin(q^{\frac{-1}{2}}x;q) \,_2\phi_1(\frac{-x^2}{q}(1-q)^2,q^2;0;q^2,q)\right),
\eqref{eq:test function applied sum}=-\int_0^{\frac{\lfloor sT \rfloor}{T}}\Theta( u ,x) f'(u) du+ f(s){\Theta}(\frac{\lfloor sT\rfloor}{T},x)-f(0){\Theta}(0,x) +o(1), \eqref{eq:test function applied sum}=-(i+1-dx) (N_{i,1}^s+N_{i,2}^s+N_{i,3}^s)-(dx-i)(N_{i+1,1}^s+N_{i+1,2}^s+N_{i+1,3}^s) ,
\phi_s(i)=\left\lbrace \begin{array}{lcl}i+1 i\leq (s-3),\, i i-1 i\leq (s-3),\, i i+1 i=s-2 i-2 i=s-1,\, s i-1 i=s-1,\, s i s\leq i \leq x-1 \end{array}\right.
|(g)_{y,1/2^k}-(g)_{y,1/2^j}|\le \sum_{i=j+1}^k|(g)_{y,1/2^i}-(g)_{y,1/2^{i-1}}|\le C\sum_{i=j+1}^k(2^{1-i})^{\frac{s-(n-2\sigma)}{2}}.
\frac{\chi_2(-a)}{(\frac{1}{2}+ \frac{\delta}{k}+ iq_2 y)^k}= \frac{\chi_2(-a)}{(\frac{1}{2}+ iq_2 y)^k\left(1+\frac{\delta}{k\left(\frac{1}{2}+iq_2 y\right)}\right)^k}=\frac{\chi_2(-a)}{(\frac{1}{2}+ iq_2 y)^k}\left(1+O\left(\frac{\epsilon}{\left|\frac{1}{2}+iq_2 y\right|}\right)\right).
B_{n,k}(x_1, x_2, \ldots) = \sum_{\substack{j_1+j_2+j_3+\cdots = k j_1+2j_2+3j_3+\cdots = n j_i\geq 0}}\frac{n!}{j_1!j_2!j_3!\cdots}\left(\frac{x_1}{1!}\right)^{j_1}\left(\frac{x_2}{2!}\right)^{j_2}\left(\frac{x_3}{3!}\right)^{j_3}\cdots
H_{\mu \nu \rho}= \epsilon_{\mu \nu \rho}\frac{r r_{1}^{2}}{\left( r^{2}+ r_{1}^{2}\right)^{2}}\left( f_{1}f_{5}\right)^{1 / 4}; H_{a b c}= \epsilon_{a b c}\frac{r_{5}^{2}}{r^{3}}\left( f_{1}f_{5}\right)^{- 3 / 4}
\Phi_F^{S}(g) = \begin{cases}\Theta_{S}(n_0) \mu(m_\infty)^k\,j(g_\infty, i1_2)^{-k}\,a(S') e^{2 \pi i\,{\rm Tr}(S'Z_0)}S' \in \mathcal P_2, 0 , \end{cases}
\begin{pmatrix}0 0 0 0 0 0 \epsilon^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \epsilon 0 0 0 \end{pmatrix},\begin{pmatrix}0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \epsilon \frac{\epsilon^2}{4}0 0 0 4 \epsilon 0 0 0 \end{pmatrix}.
- i \widehat{p}\Psi_{p}^{( 1 / 2 )}= \varepsilon m \Psi_{p}^{( 1 / 2 )}\sigma_{p}^{( 1 / 2 )}\Psi_{p}^{( 1 / 2 )}= s_{p}\Psi_{p}^{( 1 / 2 )}
\begin{pmatrix}1 -1 -1 0 0 0 0 \cdots 0 0 1 -1 -1 0 0 \cdots 0 0 0 0 1 -1 -1 \cdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \ddots\end{pmatrix}\begin{pmatrix}x_0 y_0 x_1 y_1 \vdots\end{pmatrix}=\vec 0.
\sup_{x\in K}\langle x, y_0^* \rangle - F(x) = \langle x_0, y_0^* \rangle - F(x_0), \leq F^*(y_0^*) - D(y_0^*, x_0^*), \leq F^*(y_0^*) - \inf_{x^*\in (F^*)}D(y_0^*, x^*), = \sup_{x\in K}\langle x, y_0^* \rangle - F(x).
\sum_{i'=0}^{l_3(x)-1}f(\sigma_C^{i'}(gh(x))) - \{\sum_{i'=l_2^{l_1(x)}(h(x))}^{l_3(x)-1}f(\sigma_C^{i'}(gh(x))) + \sum_{j'=0}^{k_2^{l_1(x)}(h(x))-1}f(\sigma_C^{j'}(g(\sigma_B^{l_1(x)}(h(x))))) \}.