harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\log\dfrac{|\det D_1(z_0)|.||f(z_0)||}{\prod_{j=1}^{q}|(f,\tilde a_j)(z_0)|}\le\log^+\biggl ( \Psi (z_0)\cdot \biggl(\dfrac{|\det D_i(z_0)|}{\prod_{j=1,j\ne i}^{q}|(f,\tilde a_j)(z_0)|}\biggl)\biggl ) & \le\log^+\biggl(\dfrac{|\det D_i(z_0)|}{\prod_{j=1,j\ne i}^{q}|(f,\tilde a_j)(z_0)|}\biggl)+\log^+\Psi (z_0).
S_{1,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,k\right)=r!\sum_{t=0}^{r}\frac{1}{t!}\binom{b+k-t}{r-t}S_{1,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p-1,k-t\right) .
\sum_{r=r_1+1}^{\delta - \delta_0 +r_1}S(r_1,r) = \sum_{r=r_1+1}^{\delta - \delta_0 +r_1}(-1)^r r \frac{(\delta - \delta_0 + r_1 -1)!}{r_1! (r-r_1)!(\delta - \delta_0 + r_1 -r)!}\binom{d-d_0-1}{r_1-1}; & = (-1)^{r_1+1}\binom{\delta - \delta_0 + r_1 -1}{r_1-1}\binom{d-d_0-1}{r_1-1}.
\vec{a}= (a \cos \varphi_1, a \sin \varphi_1) \;, \vec{b}= (b \cos \varphi_2, b \sin \varphi_2) \;, \vec{c}= (c \cos \varphi_3, c \sin \varphi_3) \;, \vec{d}= (d \cos \varphi_4, d \sin \varphi_4) \;.
v_{\delta^{1}}g_{1}(x_{1})+u_{\delta^{1,n}}g_{1}(x_{1})g_{n}(x_{n})=\sum_{\delta\in\{0,1\}^{n}}v_{\delta}[g_{1}(x_{1}-x_{n})^{\delta_{1}}-g_{1}(-x_{n})^{\delta_{1}}]f_{n}^{\delta_{n}}(x_{n})\prod_{i=2}^{n-1}g_{i}^{\delta_{i}}(-x_{n})
\hat{H}(\tau )={\rm bdiag}\left( \hat{h}(\tau ),\,-\sigma_{3}\hat{h}(-\tau)\sigma_{3}\right) \,,\;\;\hat{h}(\tau )=\left. \hat{h}(x^{0})\right|_{x^{0}=\tau},\;\;\hat{h}(x^{0})=qA_{0}I+\hat{\omega}\;.
\sum_{T'\in{\bar S}\setminus \{\{i,\bar{i},2n+1\}, i=1,\ldots,n\},i\in{\bar S}}v_{T'}= \sum_{T'\in{\bar S}\setminus \{\{i,\bar{i},2n+1\}, i=1,\ldots,n\},\bar{i}\in{\bar S}}v_{T'}=\alpha_i\in \{\pm 2,\pm 4\}.
G_{k,-m,0}(z) = P_{k,-m}(z,k/2) &= \varphi_{k,-m}(z,k/2) + \sum_{(-1)^{\lambda_k}n \equiv 0,1 (4)}b_{k,-m}(n,k/2)\mathcal{W}_{k,n}(y,k/2)e^{2\pi inx}&= \frac{1}{\Gamma(k)}\biggl\{q^{-m}+ \sum_{(-1)^{\lambda_k}n \equiv 0,1 (4), n>0}n^{k-1}b_{k,-m}(n,k/2)q^n\biggr\}.
( \frac{5 a b^{3}}{- 3 c^{2}})^{2}= \frac{( 5 a b^{3}) \cdot ( 5 a b^{3})}{( - 3 c^{2}) \cdot ( - 3 c^{2})}= \frac{2 5 a^{2}b^{2}}{9 c^{4}}
\varepsilon_{\delta_{1}\delta_{2}\delta_{3}}^{a_{1}a_{2}a_{3}}:=-\mathrm{sign}\left[\prod_{u<w}(a_{u}-a_{w})\cdot\prod_{x<z}(\delta_{x}-\delta_{z})\cdot\prod_{r\neq s}(a_{r}-\delta_{s})\right].
{\frac{\partial \widetilde{A}_{b}^{M i}({\bf x})}{\partial A_{c}^{M i}({\bf x})}}= \delta_{b c}+{\frac{\partial \delta A_{b}^{M i}({\bf x})}{\partial A_{c}^{M i}({\bf x})}}= \delta_{b c}- g f_{b d}^{c}\delta \Lambda_{d}^{M}({\bf x}) .
\xi_n^l\partial_\xi^\alpha p_L(x,\xi)=\xi_n^l\sum_{|\beta|\leq N}\left.\dfrac{1}{\alpha!}\partial_\xi^{\alpha+\beta}D_y^\beta p_\kappa(x,y,\xi)\right|_{y=x}+\xi_n^lr_{N,\alpha}(x,\xi)
0\leq u(x,t) \leq \omega(t) = e^{-\beta t}\|u_0\|_{L^\infty(\Omega)}+ (1-e^{-\beta t})\frac{\alpha \tilde{\tilde C}}{\beta}M(1) \leq e^{-\beta t}\|u_0\|_{L^\infty(\Omega)}+ \frac{\alpha \tilde{\tilde C}}{\beta}M(1),
a_{2k}B^{(+)}_k =B^{(+)}_{k-1}, \tfrac 14 b_{2m}=2(m-1)(\lambda+m-1)+\tfrac{\beta}{2}\lambda, \tfrac{1}{16}c_{2m}a_{2m-2}=-(m-1)(\lambda+m-2)(\lambda-\tfrac{\beta}{2}+m-1)(\tfrac{\beta}{2}+m-2).
\|\exp(\alpha \log A + (1-\alpha) \log B )\|_\infty =\lim_{p/2\ge q\searrow0}\|(P_{t^n h}(A^q,B^q))\|_\infty^{1/q}&\le \|P_{t^n h}(A^p,B^p) \|_\infty^{1/p}<1.
F(h) \leq F(f) \leq c + \epsilon~\!, d(h,f):=\sup_{p \in \overline{B_R}}\|h(p)-f(p) \| \le \epsilon^{1/2}~\!, F(g)>F(h) - \epsilon^{1/2}d(h,g)~~\forall g \in \Phi \ \hbox{with}\g\neq h~\!.
\Delta_{\tau}=\left\{\lambda_{1}a_{1}+\cdots+\lambda_ka_k\ ;\ \lambda_{i}\in\mathbb{R},\lambda_{i}>0\right\}, \quad\overline{\Delta}_{\tau}=\left\{\lambda_{1}a_{1}+\cdots+\lambda_ka_k\ ;\ \lambda_{i}\in\mathbb{R},\lambda_{i}\geq0\right\}.
f_*(f^*y\cdot \mu)(p) = \nu(p)\int_{f^{-1}(p)}(f^*y\cdot \mu)/f^*(\nu) = \nu(p)\int_{f^{-1}(p)}f^*y\cdot (\mu/f^*(\nu)) = y(p)\nu(p)\int_{f^{-1}(p)}(\mu/f^*(\nu)) = (y \cdot f_*\mu)(p).
{\cal L}= \bar{\psi}_{\alpha}( i \partial \! \! \! / - e A \! \! \! / ) \psi_{\alpha}+ \frac{1}{2}( \phi^{*}\bar{\psi}_{\alpha}^{c}\psi_{\alpha}- \phi \bar{\psi}_{\alpha}\psi_{\alpha}^{c}) - \lambda N \phi^{*}\phi .
|\sum_{m=1}^{N}\sum_{n=1}^{N}\alpha_{m}\alpha_{n}log\frac{g(z_{m})-g(z_{n})}{z_{m}-z_{n}}|^{2}\le\sum_{m=1}^{N}\sum_{n=1}^{N}\alpha_{m}\overline{\alpha_{n}}log\frac{1}{1-(z_{m}\overline{z_{n}})^{-1}}.
\vartheta_{3}\left( 0\left| i v\right.\right)=\dfrac{1}{\sqrt{v}}\left\lbrace 1+O(e^{-\pi /v})\right\rbrace \ \ \mathrm{and}\ \ \vartheta_{4}\left( 0\left| i v\right.\right)=\dfrac{2}{\sqrt{v}}e^{-\pi /4v}\left\lbrace 1+O(e^{-2\pi /v})\right\rbrace,
{\dot X}^{a}=\{H, \X^{a}\}={\cal P}^{a}(1+{1 \over m^{2}}{{\partial g}\over{\partial m^{(i)(k)}}}m_{(i)(k)}+{1 \over m^{2}}{{\partial g}\over{\partial{\bar m}^{(i)(k)}}}{\bar m}_{(i)(k)}),
\frac{\partial \widehat{H}(t,x)}{\partial y}=\frac{\partial H}{\partial y}(t,x,\hat{Y}(t,x,z),\widehat{Y}(t,\cdot,z)(x),\hat{u}(t,x,z),\hat{p}(t,x,z),\hat{q}(t,x,z),\hat{r}(t,x,z,.)).
\frac{1}{N}\sum_{\ell=L+10}^{\infty}X_{\ell}(a_N T) \leq C_2 e^{-s(\gamma_{L+10}- a_N T)}+ \frac{s}{N \mu}\leq C_2 \bigg( \frac{s}{\mu}\bigg)^{-1/3k_N}+ \frac{s}{N \mu},
\sum_{k=1}^\infty (-1)^k\frac{\rho_k}{k}Y^k =-\sum_{i=1}^n\sum_{k=1}^\infty (-1)^{k-1}\frac{(X_iY)^k}{k}=-\sum_{i=1}^n\log(1+X_iY)\overset{\eqref{funclog}}{=}-\log\Bigl(\prod_{i=1}^n(1+X_iY)\Bigr) &\overset{\eqref{vieta1}}{=}-\log\Bigl(1+\sum_{i=1}^n\sigma_iY^i\Bigr)=\sum_{l=1}^\infty\frac{(-1)^l}{l}\Bigl(\sum_{i=1}^n\sigma_iY^i\Bigr)^l.
(f_l\circ T^u)(\omega)=\begin{cases}\frac 1{L_l}\left(\frac{n_l}{k_{l,J_l}}\right)^{1/p}, \mbox{if}s-k_{l,J_l}<u\leqslant s; \frac 1{L_l}\left(\frac{n_l}{k_{l,j}}\right)^{1/p}, \mbox{if}s-k_{l,j-1}< u\leqslant s-k_{l,j}, \mbox{and}1\leqslant j\leqslant J_l; 0, \quad\mbox{if}s-n_l \leqslant u< s-k_l. \end{cases}
{\cal S}= \int d^4x \sqrt{-\bar{g}}\ \left[ \frac{1}{2}\bar{R}- \frac{1}{4}\left(\frac{\partial Y}{Y}\right)^2 - \frac{3}{4}\left(1 - \frac{c}{Y}\right) (\partial \sigma)^2- V(Y,\sigma) \right] +{\cal S}_m \hspace{5mm}\ ,
\Lambda_D(r)\equiv\begin{cases}\arcsin\Big(\frac{\sqrt{3}\, (R_{11}(r)+1)}{2 R_{23}(r)}\Big)\ 1\,<\,r\,<\,\sqrt{10}\big/3 \pi - \arcsin\Big(\frac{\sqrt{3}\, (R_{11}(r)+1)}{2 R_{23}(r)}\Big)\ \sqrt{10}\big/3\,<\,r\,<\,\sqrt{2}. \end{cases}
Z [ C ] = \int D \phi \exp \{- \int d x^{3}[ \frac{g^{2}}{3 2 \pi^{2}}\, ( \partial_{\mu}\phi - 2 \pi j_{\mu}( x ) ) ( \partial^{\mu}\phi - 2 \pi j_{\mu}( x ) ) + \frac{M_{\gamma}^{2}g^{2}}{1 6 \pi^{2}}\, ( 1 - \cos \phi ) ] - M_{W}L \}\, \, \, .
c_f+c_g \geq f^{*}_\Phi \left(\phi\right)+g^{*}_\Psi\left(\bar{\psi}\right) =\sup_{x\in X}\left\{\varphi\left(x\right)-\bar{\psi}(Lx)-f\left(x\right) \right\}+g^{*}_\Psi\left(\bar{\psi}\right) = (f+\bar{\psi}\circ L)^*_\Phi (\varphi) + g^*_\Psi (\bar{\psi}) \geq \left(f+g\circ L\right)^*_\Phi (\varphi).
C_0={e}^{\pi i(\nu+1)/4}2^{3iab-1/2}b^{-1/2},~D_0=\frac{\sqrt{2\pi}}{\Gamma(-\nu)}{e}^{\frac{3i\pi}{4}\nu}{e}^{-\frac{\pi i}{2}}2^{-3iab}b^{1/2}.
\Bigl(\phi\bigl(\lambda_{t}(x,y,z,\ldots ,z^{t})\bigr), \phi\bigl(\rho_{t}(x,y,z,\ldots ,z^{t})\bigr),\phi(z^{t+1})\Bigr) =\Bigl(\phi\bigl(\lambda_{r}(x,y,z,\ldots ,z^{r})\bigr), \phi\bigl(\rho_{r}(x,y,z,\ldots ,z^{r})\bigr),\phi(z^{r+1})\Bigr).
\left( \frac{i\epsilon_{\mu\lambda\alpha}p_\lambda}{p^2}\right)\left(\frac{i\alpha\epsilon_{\alpha\sigma\beta}p_\sigma}{\sqrt{p^2}}\right)\left( \frac{i\epsilon_{\beta\kappa\nu}p_\kappa}{p^2}\right)= \frac{i\alpha\epsilon_{\mu\lambda\nu}p_\lambda}{\left(p^2\right)^{3/2}}\, .
f_1(x_0,x_1,x_2,x_3):=x_0^4+x_1^4+\lambda x_0x_1x_2x_3, f_2(x_0,x_1,x_2,x_3):=x_0x_1\big(x_0^2+x_1^2\big)+\lambda x_0x_1x_2x_3 , g_1(x_2,x_3):=x_2^4+x_3^4, g_2(x_2,x_3):=x_2x_3\big(x_2^2+x_3^2\big), g_3(x_2,x_3):=x_2^3x_3+x_3^4, h_1(u,v,x_2,x_3):=u^4-4u^2v+2v^2+\lambda vx_2x_3, h_2(u,v,x_2,x_3):=v\big(u^2-2v\big)+\lambda vx_2x_3.
\Pr(Y > X + \ell) = \Pr(Y \ge X + \ell + 1) \ge \Pr(Y \ge X) e^{\left(-C (\ell+1) \sqrt{\frac{\log m}{mp}}\right)}- 2 m^{-2}\ge \Pr(Y > X) e^{\left(-C (\ell+1) \sqrt{\frac{\log m}{mp}}\right)}- 2 m^{-2}
G^{\mu_{1}\ldots \mu_{D - p - 2}}( z ) = ( - )^{p + 1}g \int_{{\cal M}_{D}}\, \delta^{( D )}\left( z - Z ( \varsigma ) \right) d Z^{\mu_{1}}\wedge \ldots \wedge d Z^{\mu_{D - p - 2}}\, .
\lambda_0(T_a^n)=0,\,\,\, \lambda_{2m-1}(T_a^n)=\lambda_{2m}(T_a^n)=4\pi^2m^2a^{2(n-1)}, \,\,1\leq m\leq \left\lfloor\frac{1}{a^n}\right\rfloor, \lambda_{2\left\lfloor\frac{1}{a^n}\right\rfloor+1}(T_a^n)=\lambda_{2\left\lfloor\frac{1}{a^n}\right\rfloor+2}(T_a^n)=4\pi^2\frac{1}{a^2}.
\int_{\Omega}\rho^{-1}\big|Lu+\nabla P+\nabla |H|^2-H \cdot \nabla H\big|^2x \geq C^{-1}|Lu|^2_2-C(|\nabla P|^2_2+|H|^2_{\infty}|\nabla H^2|_2) \geq C^{-1}|u|^2_{D^2}-C(|\nabla \rho|^2_2+|\nabla \theta|^2_2+|\nabla u|^2_2+|\nabla H|^2_2),
[f_3 (k + k')]^{-1}\cdot [h (f_1 + f_1')] = [(f_3 k)^{-1}+ (f_3 k')^{-1}] \cdot [h f_1 + h f_1'] = [(f_3 k)^{-1}\cdot h f_1] + [(f_3 k')^{-1}\cdot h f_1']
F(t;x,y,1,u,z,v) &=\sum_{k=0}^{\infty}z(\delta_k tuv(1-z)+z)H(t;x,\delta_k,1,u,1,v)\prod_{i=0}^{k-1}\frac{tx(\delta_i-\delta_izr+z)}{(tux+\delta_i^{-1}-tu)(\delta_i-\delta_i r+1)}.
\mathbb{P}_{couple}\left(W>i\right)\leq\sum_{k=\left[\log\left(i\right)\right]}^{\left[\log\left(ic_{t}\right)\right]-1}\left|g\left(2^{k}\right)-g\left(2^{k+1}\right)\right|+L\left(\frac{ic_{t}}{4}\right)\left(\frac{ic_{t}}{4}\right)^{-\alpha}+\left|g\left(\frac{ic_{t}}{4}\right)\right|.
\|(u-d-\textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_p\mathcal{O}_t}u^+-\textrm{esssup}_{(\omega,t,x)\in\Omega\times\mathcal{O}_t}\hat{\xi}^+)^+\|_{\mathcal{V}_2(\mathcal{O}_t)}=0,
|-4 \int_{t}^{\infty}\frac{e'''(s) ds}{(1+s-t)^{3}(\lambda_{0,0}(t)^{1-\alpha}+s-t)}\left(\frac{1}{\log(\lambda_{0,0}(s))}-\frac{1}{\log(\lambda_{0,0}(t))}\right)| &\leq \frac{C \sup_{x \geq t}|e'''(x)|}{t \log(t) (\log(\log(t)))^{2}}
\mathfrak{s}(x_3)=\mathfrak{s}\left(\begin{array}{cc}-1 1 \end{array}\right)\mathfrak{s}\left(\begin{array}{cc}1 z_1 &1\end{array}\right)\mathfrak{s}\left(\begin{array}{cc}1 -1 \end{array}\right)=\kappa(x_3)
W(\gamma) =\frac{1}{2\pi i}\int_{0}^1\frac{d}{ds}\log L_x(z(s))ds &=\frac{1}{2\pi i}\int_{0}^1\frac{-2(1+e^{2\pi is}/2)\pi i e^{2\pi is}ds}{-e^{i2\pi s}-e^{4\pi is}/4}&=\int_{0}^1\frac{1+e^{2\pi is}/2ds}{1+e^{2\pi is}/4}.
T = \left( \begin{matrix}{{- \frac{\Delta_{\phi}}{d}}}&{{0}}&{{\ldots}}&{{0}}\\{{1}}&{{- \frac{\Delta_{\phi}}{d}}}&{{\ldots}}&{{\vdots}}\\{{0}}&{{1}}&{{\ldots}}&{{0}}\\{{\vdots}}&{{\ldots}}&{{\ddots}}&{{0}}\\{{0}}&{{\ldots}}&{{1}}&{{- \frac{\Delta_{\phi}}{d}}}\\ \end{matrix}\right) ~ .
{\rm Re}\langle g(\vec{e}),\,\vec{e}\rangle = \Vert \vec{e}\Vert_{H}^{2}-{\rm Re}\langle \vec{c},\, \vec{e}\rangle \geq \Vert \vec{e}\Vert_{H}\big(\Vert \vec{e}\Vert_{H}- \Vert \vec{c}\Vert_{H}\big), \forall \,\vec{e}\in H,
\int \frac{x}{(-x^2;q^2)_\infty}\widetilde{h}_n(x;q) d_qx= \frac{1-q}{[n-1]_q(-x^2;q^2)_\infty}\left( \frac{1}{q}\widetilde{h}_n(x;q)- q^{-n}[n]_q x \widetilde{h}_{n-1}(x;q)\right),
\operatorname*{Law}\nolimits_{U_{\hbar}\left( \alpha_{0}\right) \psi}\left(\left\{A_{\hbar}\left( t_{i}\right) \right\}_{i=1}^{n}\right)\cong\operatorname*{Law}\nolimits_{\psi}\left( \left\{\alpha\left(t_{i}\right) +\sqrt{\hbar}a\left( t_{i}\right) \right\}_{i=1}^{n}\right)0<\hbar\ll1.
\begin{array}{c}\displaystyle{\frac{e^{\displaystyle{-\frac{i}{2}\,\pi \,\left( 1 + s \right)}}}{{\nu}^{1 + s}\,8\, \pi}}\, \left( \frac{8\,{\nu}^2}{1 - s}+ \frac{4\,\nu}{s}+ \frac{1 - 4\,{\nu}^2}{1 + s}\right)\end{array}
F(u,-\mu)\sum_{s=0}^\infty \frac{A_s(z)}{u^{2s}}= \sum_{s=0}^\infty \frac{a_s(z)}{u^{2s}}, F(u,-\mu)\sum_{s=0}^\infty \frac{B_s(z)}{u^{2s}}= \sum_{s=0}^\infty \frac{b_s(z)}{u^{2s}},
0 \le E \sim_{\mathbb Z}h^{e,*}(1-p^{e-r})K_Z +\gamma_e^*M_e = p^e(B|_U +(1+d_{\delta,t}+\lambda_t)D|_U) +h^{e,*}\big((t(1-p^e)(1-\delta) +1-p^{e-r})K_V +H \big) \le (p^e+1)B|_U +\mu_eD|_U +h^{e,*}\big((t(1-p^e)(1-\delta) +1-p^{e-r})K_V \big),
\varphi(x_1,x_2,x_3) & \;\;\; = \left(2\sqrt{AG(x_1)}\,\cos \frac{x_2}{2\sqrt{A}},\,2\sqrt{AG(x_1)}\,\sin \frac{x_2}{2\sqrt{A}},\,f(x_1) \cos x_3, \,f(x_1) \sin x_3 \right).
\dfrac{(2n+\nu)!}{n!^2}= \dfrac{2^{2n+\nu}n^{\nu-\frac{1}{2}}}{\sqrt{\pi}}\left(1+\dfrac{(-1)^{\nu+1}(2\nu+1)}{8n}+ \dfrac{(-1)^\nu(6\nu+1)}{128n^2}+\cdots\right).
\left\| \nabla v(\cdot,U,\xi') - \nabla v(\cdot,U,\xi) - \sum_{i=1}^d (\xi_i'-\xi_i) \nabla \tilde{w}_{e_i}(\cdot,U,\xi) \right\|_{\underline{L}^2(U)}\leq C \left| \xi-\xi' \right|^{1+\beta}.
( \partial^{\lambda \dot{\mu}}) = \left[ \begin{array}{c c}{\partial^{1 \dot{1}}}&{\partial^{1 \dot{2}}}\\{\partial^{2 \dot{1}}}&{\partial^{2 \dot{2}}}\\ \end{array}\right] = \left[ \begin{array}{c c}{\partial_{0}+ \partial_{3}}&{\partial_{1}+ i \partial_{2}}\\{\partial_{1}- i \partial_{2}}&{\partial_{0}- \partial_{3}}\\ \end{array}\right] ,
\left(1+\frac{b}{2}\right)\Re\left\langle w,Aw\right\rangle \geqslant \frac{|b|}{2}|w||Aw|\delta_A = \frac{|b|}{2}|w||Aw|+\frac{|b|}{2}|w||Aw|\left(\delta_A-1\right) \geqslant \frac{|b|}{2}|w||Aw|+\frac{|b|}{2}\frac{|w|}{|A^{-1}|}\left(\delta_A-1\right) = \frac{|b|}{2}|Aw|+\gamma_A,
\left\langle \varepsilon, \phi_{0, b, \beta}\left( \frac{d+2}{2\beta}- \frac{y^2}{2}\right) \right\rangle_{L^2_{\rho_\beta}}=\frac{\gamma \varepsilon_1}{\beta}\|\phi_{0, b, \beta}\|^2_{L^2_{\rho_\beta}}+O(b^{\frac{\alpha}{2}+\delta})
(\phi_i',\phi_j')+4e_ie_j(\widetilde \phi_n,\widetilde \phi_n) &=(\psi_i,\psi_j)+d_i(\psi_{i_0},\psi_j)+d_j(\psi_{i_0},\psi_i)+d_id_j(\psi_{i_0},\psi_{i_0})
-\sum_{i\in I}\nu_{i}(a_{i}+1) =-\sum_{i\in I}N_{i}\sigma_{i}(a_{i}+1) &\leq-\sum_{i\in I}N_{i}(a_{i}+1)(\sigma_{i}-c_0(f))-mc_0(f) &\leq -mc_0(f).
S ={\frac{1}{8 \pi}}\int d x d y \left[ \sum_{i = 1}^{2}( \partial_{x}\phi_{i})^{2}+ ( \partial_{y}\phi_{i})^{2}- \Lambda^{\prime}\left( e^{i{\frac{\beta}{\sqrt{2}}}( \phi_{1}+ \sqrt{3}\phi_{2})}+ e^{i{\frac{\beta}{\sqrt{2}}}( \phi_{1}- \sqrt{3}\phi_{2})}+ e^{- i \sqrt{2}\beta \phi_{1}}\right) \right]
\| \, \|u(s)\|^2_{L^2(\Sigma_s)}\, \|^2_{L^2(I)}= \int_I \int_{\Sigma_s}|u|^2 dA\, ds\sim\int_I \int_{\Sigma_s}\beta^{1/2}|u|^2 dA\, ds=\int_{t^{-1}(I)}|u|^2 dV \, .
\sum_{i= -\infty}^{k-1}2^{-i\frac{r}{p-r}}V_r(x_{i-1}, x_i)^{\frac{pr}{p-r}}\approx \sum_{i= -\infty}^{k-1}\int_{x_i}^{x_{i+1}}\bigg(\int_0^t w \bigg)^{-\frac{p}{p-r}}w(t) dt V_r(x_{i-1}, x_i)^{\frac{pr}{p-r}}\le \int_{0}^{x_{k}}\bigg(\int_0^t w \bigg)^{-\frac{p}{p-r}}w(t) V_r(0, t)^{\frac{pr}{p-r}}dt.
f(x)=f(x_{0})+f^{\prime}(x_{0})(x-x_{0})+\cdot\cdot\cdot+\frac{f^{(k)}(x_{0})}{k!}(x-x_{0})^{k}+\frac{R_{k+1}(x)}{(k+1)!}(x-x_{0})^{k+1}
\bar{Z}_{j_{1}\ldots j_{p - k}}^{i_{1}\ldots i_{p - k + 1}}= \frac 1{\left( p - k \right) !}\partial_{}^{\left[ i_{1}\right.}\delta_{j_{1}}^{i_{2}}\ldots \delta_{j_{p - k}}^{\left. i_{p - k + 1}\right]}, k = 1 , \ldots , p .
\|B_1 (z)\|^2_{HS}\leq \frac{\delta^2 \alpha^4}{4}\int_0^{2 r_\varepsilon}\int_0^{\pi/2}\frac{\sin^2 \phi \; d\rho \: d\phi}{\rho|\ln \rho|^2}=\frac{\delta^2 \alpha^4 \pi}{16}\int_0^{2 r_\varepsilon}\frac{d\rho}{\rho|\ln \rho|^2}.
\Vert (B(\varepsilon ;\vartheta)+\lambda \varepsilon^2 Q_0)^{-1}-(B^0(\varepsilon ;\vartheta )+\lambda\varepsilon^2 \overline{Q_0})^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}&\leqslant C\varepsilon^{-1},0 < \vartheta\leqslant 1, 0<\varepsilon\leqslant \vartheta^{-1}.
\begin{aligned}\frac{\partial y}{\partial t}= \int_{-\infty}^x \partial_t \frac{1}{\sqrt{g_{[<\lambda^{\sigma}]}}}\, dx' = \ \Re \int_{-\infty}^x h(u_{[<\lambda^{\sigma}]}) \partial_x^2u_{[<\lambda^{\sigma}]}+ h_1(u_{[<\lambda^{\sigma}]}) (\partial_x u_{[<\lambda^{\sigma}]})^2\, dx' .\end{aligned}
S_jK_s ={}s(4\langle X,S_j\rangle|X|^2-\langle J_ZS_j,X\rangle)K_{s-1}, S_j^2K_s ={}s(s-1)(4\langle X,S_j\rangle|X|^2-\langle J_ZS_j,X\rangle)^2K_{s-2}+ s(4|X|^2+8\langle X,S_j\rangle^2+\tfrac{1}{2}\langle J_{[X,S_j]}X,S_j\rangle)K_{s-1}.
t \left(t + \frac{m}{2}\right) \left(t - \frac{m}{2}\right) \tilde{\alpha}''(t) + \left( a(\frac{m}{2})^2 + 2(1 - a)\frac{m}{2}t + (a + 2)t^2 \right) \tilde{\alpha}'(t) = 0.
\sum_i \sum_{p \neq q}\mathrm{Cov}(X_i^2 s_{ii}, X_p s_{pq}X_q) = \sum_i \sum_{p \neq q}\mathbb{E}[X_i^2 X_p X_q] \mathbb{E}[s_{ii}s_{pq}] = \sum_i \sum_{p \neq q}\mathbb{E}[(X_i^2-1) X_p X_q] \mathbb{E}[s_{ii}s_{pq}]
\begin{array}{cc}{\cal Q}^{a}=\left(\begin{array}{c}{\cal Q}^{a}_{+}{\cal Q}^{a}_{-}\end{array}\right) ~~~~ ~~~~ \bar{{\cal Q}}_{a}={{\cal Q}^{a}}^{\dagger}\Gamma^{0}=(\bar{{\cal Q}}_{a}^{-}, \bar{{\cal Q}}_{a}^{+})\end{array}
\int_{-\infty}^\infty dq_-\, e^{i \omega q_-}\delta\epsilon_{+}(q_-)= (2\overline\mu)^{i \omega}\int_{-\infty}^\infty dq_+\, e^{-i\omega q_+}\sum_{k=1}^\infty \frac{\overline\mu^{1-k}}{k!}\frac{\Gamma(1+i\omega)}{\Gamma(2 - k + i\omega)}\delta\epsilon_{+}(q_+)^k.
c^* =\min\left\{\left(\frac{1}{p}-\frac{1}{2\cdot p^\flat}\right) \left(\frac{p^\flat}{b}\right)^{\frac{p}{2\cdot p^\flat-p}}(a S^\flat)^{\frac{2\cdot p^\flat}{2\cdot p^\flat-p}}, \left(\frac{1}{p}-\frac{1}{2\cdot p^\sharp}\right) \left(\frac{p^\sharp}{b}\right)^{\frac{p}{2\cdot p^\sharp-p}}(S^\sharp)^{\frac{2\cdot p^\sharp}{2\cdot p^\sharp-p}}\right\}.
\mathbb{E}[ -\log( q_{V|X_A}(V|X_A) ) ]= \mathbb{E}\left[ \frac{(V-X_A^2 \Delta \tilde{G}- \sigma^2_N)^2}{4 X_A^2 \Delta^2 \sigma^2_N}\right]+ \log{\Delta}+ \frac{1}{2}\log(4 \pi\sigma^2_N)+ \frac{1}{2}\mathbb{E}[ \log(X_A^2) ]
{\cal M}= \frac{i}{\sqrt{{\mathrm{\boldmath ~ \b e t a ~}}^{2}}}\left( \begin{matrix}{{E_{\mathrm{\boldmath ~ \b e t a ~}}}}&{{- E_{-{\mathrm{\boldmath ~ \m u ~}}}}}\\{{E_{\mathrm{\boldmath ~ \m u ~}}}}&{{\, \, \, E_{-{\mathrm{\boldmath ~ \b e t a ~}}}}}\\ \end{matrix}\right) \, \, .
T_{G,H}(xy)=\prod_{j=1}^{n}g_{\pi_{x}(\pi_{y}(j))}^{-1}xg_{\pi_{y}(j)}g_{\pi_{y}(j)}^{-1}yg_{j}\cdot H^{\prime}=\prod_{j=1}^{n}u_{x}(\pi_{y}(j))\cdot u_{y}(j)\cdot H^{\prime}
- \frac{2\pi}{\rm L}\hbar p \sum_{k \in \cal Z}\frac{1}{2}[a_k^{\dagger},a_{k+p}]_{-}\cdot |\lambda\varepsilon_{k,\rm R}|^{-s/2}|\lambda \varepsilon_{k+p,\rm R}|^{-{3s}/2}.
{\cal L}_{e f f}^{n m}= \partial_{+}A_{i}\partial_{-}A_{i}- \frac 1 2 ( \partial_{i}A_{i}+ \frac{1}{\partial_{-}}\ast J_{n m}^{+})^{2}- \frac 1 4 \left( \partial_{i}{A}_{j}- \partial_{j}{A}_{i}\right)^{2}+ A_{i}J_{n m}^{i}\ .
{Q_\gamma}(\lambda ) = P(\lambda ) +{\Delta_\gamma}(\lambda ) = P(\lambda ) +{\Delta_\gamma}= I \lambda - \left({A +{s_\rho}\left({{F_\gamma}\left[{P,\Sigma}\right]}\right) \hat U \left( \gamma \right)\hat V{{\left( \gamma \right)}^\dag}}\right).
[X_{n1},X_{n2},X_{n3}]_c =[X_{n1},[X_{n2},X_{n3}]_c]_c =X_{n1}\cdot[X_{n2},X_{n3}]_c\cdot X_{n1}^{-1}\cdot[X_{n2,X_{n3}}]_c^{-1}=X_{n1}\cdot \bigl(X_{n2}\cdot X_{n3}\cdot X_{n2}^{-1}\cdot X_{n3}^{-1}\bigr)\cdot X_{n1}^{-1}\cdot \bigl(X_{n3}\cdot X_{n2}\cdot X_{n3}^{-1}\cdot X_{n2}^{-1}\bigr).
\Sigma_A(\bar{p}) \simeq \frac{i e^2}{(2 \pi)^3}|eH| \int dq_0 dq_3\int_0^\infty dr^2{\rm e}^{- r^2}\frac{G_1}{q^2}\frac{\Sigma_A(\bar{p"})}{\bar{p"}^2 + \Sigma_A(\bar{p"})}
[ \Delta_{\perp}^{d}]^{- 1}( x_{\perp}) = - \int \frac{d^{d}k_{\perp}}{( 2 \pi )^{d}}\frac{e^{i k_{\perp}\cdot x_{\perp}}}{k_{\perp}^{2}}= - \frac{1}{2^{2 - d / / 2}}\frac{1}{( 2 \pi )^{d / 2}}\frac{1}{( x_{\perp})^{d / 2 - 1}}\Gamma ( d / 2 - 1 )
\langle 0 | e^{k g \varphi ( 0 )}| 0 \rangle = \sum_{j = 0}^{\infty}\frac{k^{j}g^{j}}{j !}\langle 0 | : \! \varphi^{j}( 0 ) \! : | 0 \rangle = 1 + o ( k^{2}) .
\eta_{M, M-1}(q|a, b) = \exp\Bigl( \int\limits_0^\infty \frac{dt}{t}\Bigl[\frac{\bigl(\mathcal{S}_{M-1}e^{-xt}\bigr)(q\,|\,b)-\bigl(\mathcal{S}_{M-1}e^{-xt}\bigr)(0\,|\,b)}{\prod\limits_{i=1}^M (1-e^{-a_i t})}+ + qe^{-t}\frac{\prod\limits_{j=1}^{M-1}b_j}{\prod\limits_{i=1}^M a_i}\Bigr]\Bigl),
{\bf\Phi}_{M_n,n}(f)(x,t)= \dfrac{1}{n}\sum_{k=1}^n f(x_k, t)+\sum_{p=1}^{M_n}e^{(T-t)p^{2\beta}}\Bigg[ \dfrac{\pi}{n}\sum_{k=1}^n f(x_k,t) \phi_p(x_k) \Bigg]\phi_p(x)
\left[{\begin{array}{*{20}c}{u_1 (\gamma_* )^*}{u_2 (\gamma_* )^*}\end{array}}\right]F[P(\mu_1 ,\mu_2 );\gamma_* ] = s_* \left[{\begin{array}{*{20}c}{v_1 (\gamma_* )^*}{v_2 (\gamma_* )^*}\end{array}}\right],
R_L(F)(x,y) =R_L(\sum_{i=1}^{m}f_i\otimes g_i)(x,y)=R(\sum_{i=1}^{m}(f_i\otimes g_i)\restriction_{[0,\alpha]\times{y}})(x) =\sum_{i=1}^{m}R(f_i)(x)\cdot g_i(y)=\sum_{i=1}^{m}(R(f_i)\otimes g_i)(x,y)=G(x,y).
m(0) \cdot s^+=\lambda s_1^+>0 \implies m(x^+)\cdot s^+=m_1(x^+) s_1^+ +m_2(x^+)s_2^+ \geq 0 \implies m_2(x^+)\geq -\frac{m_1(x^+) s^+_1}{s^+_2}\geq -\tilde{C}\alpha^+_1.
\beta \,{\frac{F}{L}}= e_{0}\, \beta - \int_{- \infty}^{\infty}{\frac{d p}{2 \pi}}\ln \left( 1 + e^{- \beta \, \epsilon ( p )}\right) = e_{0}\, \beta +{\frac{m}{\pi}}\sum_{n = 1}^{\infty}{\frac{( - )^{n}}{n}}\, K_{1}( n \, m \, \beta )
\sum_{j \in N_{i'}}\max \left\{a_{i'j}, b - \sum_{k \in M_C - i'}a_{k t_k}\right\}x_{i'j}& + \sum_{i \in M_C - i'}\left(\sum_{j =1}^{t_i}a_{i t_i}\max \left\{1, \frac{a_{ij}}{b - \sum_{k \in M_C -{i, i'}}a_{k t_k}- a_{i' n_{i'}}}\right\}x_{ij}+ \sum_{j=t_i+1}^{n_i}a_{ij}x_{ij}\right) \leq b
H_{m,n}(0,0) = \lim\limits_{\rho \to +\infty}\frac{\mathcal{Z}_{m,n}^{\rho^2}(0,0)}{\rho^{m+n}}= \lim\limits_{\rho \to +\infty}(-1)^m m! \delta_{m,n}\dfrac{\Gamma(\rho^2+2m+1)}{\rho^{2m}\Gamma(\rho^2+m+1)}= (-1)^m m! \delta_{m,n}.
L_n=\Big\{\vec{x}=(x_0,x_1,\ldots,x_n)\in \{\mathbb{Z}^d\}^{n+1}: x_0=O, x_{i+1}\sim x_i0\leq i\leq n-1 &x_i\neq x_j 0\leq i<j\leq n\Big\}
d\mu^{\varepsilon}=Z^{-1}\exp \left[ -\beta \left( \varepsilon \right)\sum_{b\in \left( \varepsilon \mathbb{Z}^{d}\right)^{\ast}}V\left( \eta^{\varepsilon}\left( b\right) \right) \right] \prod_{i\in \varepsilon \mathbb{Z}^{d}}d\theta^{\varepsilon}\left( i\right) ,
\varphi(x,z) =\frac 1{z^3}\left\{\left(\frac{1-z^2x}{x-1}\right)\sin (xz)+\left(-\frac{zx}{x-1}+z^3\right)\cos(xz)\right\}, \psi(x,z) =\frac 1{z^3}\left\{\left(z^2-\frac{1}{x-1}\right)\sin (xz)+\left(\frac{zx}{x-1}\right)\cos(xz)\right\}
{\cal L}_{m a s s}^{\psi}= - 2{\frac{\beta}{\pi^{2}}}\left({\frac{\alpha}{2}}\right)^{1 2}{\frac{1}{m^{3}}}\left\{F^{i j}F^{k l}\psi_{i j}^{\dag}\psi_{k l}^{\dag}+ 4 F^{i j}F^{k l}\psi_{l i}^{\dag}\psi_{j k}^{\dag}\right\},
TZ_{\Lambda}\left( \lambda \ast_{s}\mu \right) :=\{\left( x,d\left(\lambda \right) -d\left( \mu \right) ,y\right) \in \mathcal{TG}_{\Lambda}:x\in TZ_{\Lambda}\left( \lambda \right) ,y\in TZ_{\Lambda}\left( \mu\right) & \sigma^{d\left( \lambda \right)}x=\sigma^{d\left( \mu \right)}y\}
I_n : = \Big|\int_{\Omega}L_{\mu}(u_n-u)\mathcal{A}(x,\nabla u_n)\cdot\nabla\phi_K\,dx\Big| &\leq ||\nabla\phi_K||_{L^\infty(\Omega)}\int_{K}\,w|u_n-u||\nabla u_n|^{p-1}\,dx &\leq ||\nabla\phi_K||_{L^\infty(\Omega)}||u_n-u||_{L^p(\Omega,w)}||u_n||^{p-1}_{W^{1,p}(\Omega,w)}.
\mathrm{e i t h e r}\quad \tilde{\gamma}^{a}: = i \; d x^{0}\; \tilde{\tilde{\vee}}\; d x^{a}\; \tilde{\vee}, \quad \mathrm{o r}\qquad \tilde{\gamma}^{a}= i \; \tilde{\tilde{a}}{}^{0}\; \tilde{a}^{a}
\sigma(|\psi\rangle) = \left[ \frac{1}{2}\langle L^{\dagger}+ L\rangle L - \frac{1}{8}\langle L^{\dagger}+ L\rangle^2 \right] |\psi\rangle, \delta_1(|\psi\rangle) = \left[ L - \frac{1}{2}\langle L^{\dagger}+ L\rangle \right] |\psi\rangle, \delta_2(|\psi\rangle) = \left[\dfrac{L}{\sqrt{\langle L^{\dagger}L\rangle}}- 1\right]|\psi\rangle.
\sum_{n \geq 1}A(n, m) e\Bigl( \frac{n d}{c}\Bigr) \psi(n) &= c \sum_{\pm}\sum_{n_1 \mid c m}\sum_{n_2 \geq 1}\frac{A(n_1, n_2)}{n_1 n_2}S(m \bar d, \pm n_2; m c n_1^{-1}) \Psi^{\pm}\Bigl( \frac{n_2 n_1^2}{c^3 m}\Bigr)
a_{4,4}=ra_{4,4}^{\mbox{\tiny{scal}}}+{1 \over 4!}\int_{\Sigma_r}\bar{h}^{1/2}d^3x\left[{1\over 16}\mbox{Tr}(\bar{\gamma}^i\bar{\gamma}^j\bar{F}_{ij})^4-{r \over 2}(\bar{F}^{ij}\bar{F}_{ij})^2 \right]~~~,
{\cal L}_{e f f}^{p z m}= \frac{1}{2 L}\pi \partial_{+}a_{-}-{\cal H}_{D}^{p z m}= \frac{1}{2 L}\pi \partial_{+}a_{-}+ \frac{1}{2 L}\pi \frac{1}{\Delta_{\perp}}\ast \partial_{i}J_{p z m}^{i}+ a_{-}J_{p z m}^{-}- \frac 1 2 J_{p z m}^{i}\frac{1}{\Delta_{\perp}}\ast J_{p z m}^{i}\; ,
\Re{\left( \int_y^M g(u) e^{2\pi i \left( f(u) -u \right)}du \right)}=\Re{\left( \int_y^M \frac{e^{-2\pi i w(u)}}{u^{\sigma_1}(u+n)^{\sigma_3}}du \right)}&= \int_{w(y)}^{w(M)}\frac{1}{u^{\sigma_1}(u+n)^{\sigma_3}}\frac{\cos(2\pi w) dw}{1-\frac{1}{2\pi}\left( \frac{t_1}{u}+\frac{t_3}{u+n}\right)}.
\bar E_n =\frac{c}{4\pi a^2}\int_{0}^{\infty}y\, dy\,\left \{\ln\left [1-\xi^2\sigma^2_n(y)\right ]+\frac{\xi^2}{4}\frac{y^4}{(n^2+y^2)^3}\right \},\quad n=1,2,\ldots{.}
\frac{1}{A_{1}\cdot\cdot\cdot A_{n}}=\int_{0}^{1}d\alpha_{1}\cdot\cdot\cdot d\alpha_{n-1}\int_{0}^{\infty}d\alpha\alpha^{n-1}exp(-\alpha\{\alpha_{1}A_{1}+\cdot\cdot\cdot+\alpha_{n-1}A_{n-1}+(1-\alpha_{1}-\cdot\cdot\cdot-\alpha_{n-1})A_{n}\}).
dim(K_{1,1})=1, \,\,dim(K_{1,n})=2\,\, \,\, n\ge 2 &dim(K_{2,2})=2, \,\,dim(K_{2,n})=3\,\, \,\, n\ge 3 &dim(K_{m,n})=4, \,\,\,\,\,\, n\ge m\ge 3.