harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\hat{\nabla}_{\hat{\mu}}\hat{F}_{(\hat{d})}{}^{\hat{\mu}\hat{\rho}_{1}\cdots\hat{\rho}_{d}}+{\textstyle\frac{2\chi}{\sqrt{|\hat{g}|}}}\sum_{n}\mu_{n}{\displaystyle\int}d^{d}\xi_{n}\epsilon^{i_{1}\cdots i_{d}}\partial_{i_{1}}\hat{X}^{\hat{\mu}_{1}}\cdots \partial_{i_{d}}\hat{X}^{\hat{\mu}_{d}}\delta^{\hat{d}}(\hat{x}-\hat{X}_{n}) = 0\, ,
M_{0}= ( - \frac{\mathrm{d e t}\, ( c ) \Lambda_{2}^{4}}{m_{0}^{2}}, \, 0 ) , \quad M_{1}= ( \frac{m_{0}m_{2}- m_{0}^{2}\Lambda_{1}^{4}/ \Lambda_{2}^{4}}{\, \mathrm{d e t}\, ( c )}, \, \frac{m_{0}m_{2}- m_{2}^{2}\Lambda_{2}^{4}/ \Lambda_{1}^{4}}{\, \mathrm{d e t}\, ( c )}) ,
{}{}\textrm{maximize :}\;\;\;C_{s}= E \left\{\sum_{i=1}^{M_R}\log_2 \left( 1 + \frac{P}{N_0}l_i \right) \right\}&\textrm{s.t.}\;\;\;{\mathcal U}= E\left\{\eta P \right \}\leq Q, \;\; 0 \leq P \leq P_{\max}
[\sigma_{y,z}] = [\ell_y] + [\gamma_{vy^4,y^4z}]+ [\gamma_{wy^4,y^4z}]+ [\gamma_{xy^4,y^4z}]+ [\gamma_{vy^3z,y^3z^2}]+ [\gamma_{wy^3z,y^3z^2}]- [\gamma_{xy^2z^2,y^3z^2}].
G_{F}^{0}=\frac{G_{F}}{(\hbar c)^{3}}=\frac{\sqrt{2}}{8}\frac{g^{2}}{M_{W}^{2}c^{4}}=1.1663787(6)\times10^{-5}GeV^{-2}\approx4.5437957\times10^{14}J^{-2}.
d_{n,k}^{m,l}=\binom{l}{l-k}((\theta+m-1)_{m-n\downarrow})^{k}(\sum_{j=1}^{m-n}(\alpha + \theta )_{m-j\uparrow}(\theta+m-1)_{j-1\downarrow})^{l-k}&=\binom{l}{k}((\theta+n)_{m-n\uparrow})^{k}(\sum_{j=1}^{m-n}(\alpha + \theta )_{m-j\uparrow}(\theta+m-1)_{j-1\downarrow})^{l-k}.
e^{ia\ln|\frac{\eta+\nu}{\eta-\nu}|}=e^{ia\ln|1+\eta/\nu|}e^{-ia\ln|1-\eta/\nu|}= \left(1 + ia\frac{\eta}{\nu}+ O(\eta^2/\nu^2)\right)\left(1 + ia\frac{\eta}{\nu}+ O(\eta^2/\nu^2)\right) &= 1+2ia\frac{\eta}{\nu}+ O(\eta^2/\nu^2),
A_i(u) \xi_{\lambda}= g_i(u) \left( \prod_{s=1}^{l}\prod_{p=1}^{i}\dfrac{u - \hbar'(p - 1 - \lambda_{i,p}^{(s)}) + a_s}{u - \hbar'(p - 1) + a_s}\right) \xi_{\lambda}.
\left| D\cdot\left(a_{n,0}+\sum_{j=1}^{m}a_{n,j}\alpha_j\right) \right|_p \geqslant \frac{1}{\left| D\cdot\left(a_{n,0}+\sum_{j=1}^{m}a_{n,j}\alpha_j\right) \right|}\geqslant \frac{1}{D\cdot\left(1+ \sum_{j=1}^{m}|\alpha_j|\right)\cdot \max_{0\leqslant j \leqslant m}|a_{n,j}|},
S = S_{c l}+ \int_{- \infty}^{\infty}d t \{-{\frac{1}{2}}\xi^{a}({\frac{d}{d t}}+ \lambda_{a}( t ) ) ({\frac{d}{d t}}- \lambda_{a}( t ) ) \xi^{a}+ \psi^{a}({\frac{d}{d t}}- \lambda_{a}( t ) ) \psi_{a}^{*}\},
\begin{aligned}\Delta_{\frac{1}{d}(1,p)}(k-(r-1)p)+\Delta_{\frac{1}{d}(1,q)}(k-(s-1)q)=\frac{(1-d)(r+s-2)}{2d}+\left\{\frac{-k}{d}\right\}(r+s-1)\end{aligned}
\lim_{T\rightarrow\infty}\frac{1}{T^{nd-2\left(n-1\right)}}\Theta\left(f_{1},f_{2},T\right)=2^{1-n}d_{Q}\int_{\mathbb{R}^{n-1}}f_{2}\left(u'_{2},\dots,u'_{2n-2}\right)du'_{e}\int_{\left[0,\infty\right)^{n+1}}\overline{\overline{\Psi}}_{f_{1}}\left(u'_{1},\dots,u'_{2n-1},u'_{2n}\right)du'_{o}.
[|D_0|,|D_0|^{-\frac13}a^2|D_0|^{-\frac13}] = |D_0|^{-\frac13}\delta_0(a^2)|D_0|^{-\frac13}= |D_0|^{-\frac13}\delta_0(a)a|D_0|^{-\frac13}+|D_0|^{-\frac13}a\delta_0(a)|D_0|^{-\frac13}.
\hat{f}(t,p)=\hat{f}_{0}(p)E_{\beta,1}(-a|p|^{\alpha}t^{\beta}) + \int_{0}^{t}(t-s)^{\beta - 1}E_{\beta, \beta}(-a(t-s)^{\beta}|p|^{\alpha}))\hat{H}(s,y, p)ds,
\forall \psi \in \mathcal C^2(\mathbb{R}^d), A_n \psi(z) \triangleq \frac{1}{2}\partial_{z_1 z_1}\psi(z) + \sum_{i=1}^d F_{n,i}(z) \partial_{z_i}\psi(z), z \in \mathbb{R}^d.
L_{\mathbf{g}}( \psi )^{\mu a}= \psi^{\mu a}\ \Rightarrow \ \psi^{\mu a}( \mathbf{x}, \mathbf{z}\mathbf{g}) = \psi^{\mu a}( \mathbf{x}, \mathbf{e}) \equiv \psi^{\mu a}( \mathbf{x}) .
T_+(x) = T_-(x)\begin{pmatrix}e^{-2n\xi_+}x^\alpha 0 e^{-2n\xi_-}\end{pmatrix}, x\in (0,b), &T_+(x) = T_-(x)\begin{pmatrix}1 x^\alpha e^{n \xi(x)}0 1\end{pmatrix}, x\in (b,\infty).
|T_kx-Tx| \leq |T_kx - T_kx_j| + |T_kx_j - Tx_j| + |Tx_j -Tx| &= |T_k(x-x_j)| + |T_kx_j - Tx_j| + |T(x_j -x)| &= 2|x-x_j| + |T_kx_j - Tx_j|,
m(\chi_d) = e^{-\gamma}\max_{\alpha \in [0, 1)}\Bigg|\sum_{\substack{1\leq n\leq Z P^{+}(n)\leq y}}\frac{\chi_d(n) \sin(2\pi n\alpha)}{n}\Bigg|+O(1) &= e^{-\gamma}\max_{\alpha \in [0, 1)}\Bigg|\sum_{\substack{n\geq 1 P^{+}(n)\leq y}}\frac{\chi_d(n) \sin(2\pi n\alpha)}{n}\Bigg|+O(1).
\frac{A_{\tau(k)+1}}{\overline{B}_{\tau(k)+1}}= \frac{\prod_{j=1}^{\tau(k)+1}{A_j}}{\overline{B}_{\tau(k)+1}\prod_{j=1}^{\tau(k)}{A_j}}\leq \frac{\overline{B}_{\tau(k)+2}\cdot \prod_{j=1}^{\tau(k)+1}{A_j}}{e^{-k}e^x}\leq \frac{e^{M}}{e^{-k}e^x}\leq \frac{\varepsilon e^{x}}{e^{-k}e^x}\leq e^{k}.
S^{({\widehat G}/H)}_{ji}(\theta, q) ~=~ \left(x^{-\rho_H \cdot h}\otimes x^{-\rho_H \cdot h}\right) ~ S^{({\widehat G})}_{ji}~\left(x^{\rho_H \cdot h}\otimes x^{\rho_H \cdot h}\right) \ ,
\Gamma\big(1+\tfrac14u^2-\tfrac12 b\big)2^{-b}u^{b-1}e^{-\frac12z^2}z^b U\big(\tfrac14u^2+\tfrac12 b,b,z^2\big){}= zK_{b-1}(uz)\left(\sum_{s=0}^{N-1}\frac{A_s(z)}{u^{2s}}+g_2(u,z)\right) -\frac{z}{u}K_b(uz)\left(\sum_{s=0}^{N-1}\frac{B_s(z)}{u^{2s}}+zh_2(u,z)\right),
\bf{DC}_{\lambda_1}\bf{DC}_{\lambda_2}\cdots\bf{DC}_{\lambda_r}=\sum_{\delta\in \mathcal{J}}k_{\lambda_1\lambda_2\cdots\lambda_{r-1}}^{\delta}\bf{DC}_{\delta}\bf{DC}_{\lambda_r}&=\sum_{\delta\in \mathcal{J}}\sum_{\rho\in \mathcal{J}}k_{\lambda_1\lambda_2\cdots\lambda_{r-1}}^{\delta}k_{\delta\lambda_r}^\rho\bf{DC}_{\rho},
V(\Gamma_1*\Gamma_2) =V(\Gamma_1)\sqcup V(\Gamma_2), E(\Gamma_1*\Gamma_2) =E(\Gamma_1)\sqcup E(\Gamma_2)\sqcup\{\,\{v_1,v_2\}: v_1\in V(\Gamma_1),~v_2\in V(\Gamma_2)\,\}.
(g_{21}y_1+g_{22}y_2+g_{23}y_3)\{(g_{11}+g_{21})y_1+(g_{12}+g_{22})y_2+(g_{13}+g_{23})y_3\}^{a-1}=g_{23}(g_{13}+g_{23})^{a-1}y_3(y_1+y_2+y_3)^{a-1}.
E^{Q_n}\left ( f_n\right) \ = \ \frac{E_{\tilde \varphi_c}^Q \left ( \frac{f_{j_n}}{\varphi_c (Z_n)}\right)}{E_{\tilde \varphi_c}^Q\left ( \frac{1}{\varphi_c(Z_n)}\right)}+ o(1).
I = \frac{is}{4\pi}\int{\mathrm{Tr}}(A\wedge dA + \frac 23 A\wedge A\wedge A) - \frac{is}{4\pi}\int{\mathrm{Tr}}(\bar{A}\wedge d\bar{A}+ \frac 23 \bar{A}\wedge \bar{A}\wedge \bar{A})\,,
\chi_{\{\xi^{2}_{1}\neq\xi^{2}_{2}\}}=\sum_{Q''\in\mathbb{Q}''}\phi_{Q''}(\xi^{2}_{1},\xi^{2}_{2})= \{\sum_{Q''\in\mathbb{Q}''_{\mathbb{I}}}+\sum_{Q''\in\mathbb{Q}''_{\mathbb{II}}}\}\phi_{Q''}(\xi^{2}_{1},\xi^{2}_{2}) =\chi_{\{\xi^{2}_{1}<\xi^{2}_{2}\}}+\chi_{\{\xi^{2}_{1}>\xi^{2}_{2}\}},
\limsup_{n \rightarrow \infty}\|\sum_{\nu=0}^{n+|\textup{\textbf{m}}|-m_\alpha}b_{\nu,n}^{(\alpha)}\Phi_\nu\|_K^{1/n}\leq \frac{\|\Phi\|_{K}}{\rho_{|\textup{\textbf{m}}|}(\textup{\textbf{F}})}.
\bigg|D_kT_M(f)(x)\bigg|=\bigg|D_k\bigg(-\ln r\sum\limits_{j=-\infty}^\infty\sum\limits_{Q\in Q^j}w(Q) \psi_{Q}(\cdot,x_{Q})q_{Q}f(x_{Q})\bigg)(x)\bigg| \lesssim\sum\limits_{j=-\infty}^\infty\sum\limits_{Q\in Q^j}w(Q)|D_k \psi_{Q}(x,x_{Q})| |q_{Q}f(x_{Q})|.
\Vert (\rho-1,u)^\ell(t)\Vert_{\dot{B}^{\frac{N}{2}+1-\alpha+s\alpha}_{2,1}}+ \Vert \rho^h(t)-1\Vert_{\dot{B}^{\frac{N}{2}}_{2,1}}+\Vert u^h(t)\Vert_{\dot{B}^{\frac{N}{2}+1-\alpha}_{2,1}}\le C(1+t)^{-s},
\langle \tilde{G},G \rangle =G_0\tilde{G}_1^\ast \frac{ic y}{2\pi}\left ( J_0(y,-\lambda y)\left (1+\lambda y\dfrac{c}{2\pi}J_0(y,-\lambda y)\right )-\lambda y\frac{c}{2\pi}\sum_{n\geq 1}\dfrac{J_n(y,-\lambda y)}{n}\right )
\left( \hat{q}_{i}^{a}\right)^{\dagger}= \hat{q}_{i}^{a}, \left( \hat{p}_{i}^{a}\right)^{\dagger}= \hat{p}_{i}^{a}, \left[ \hat{q}_{i}^{a}, \hat{p}_{j}^{b}\right] = i \hbar \delta^{a b}\delta_{i j},
\begin{aligned}-dy_t =\big[g(x_t,y_t,z_t,\tilde{z}_t,\gamma_{(t,e)})-g(x_t,0,0,0,0)+g(x_t,0,0,0,0)\big]dt -z_tdW_t-\tilde{z}_td\xi_t-\int_{\mathcal{E}}\gamma_{(t,e)}\tilde{N}(de,dt) :=\big[G(y_t,z_t,\tilde{z}_t,\gamma_{(t,e)})+\varphi(t)\big]dt-z_tdW_t-\tilde{z}_td\xi_t-\int_{\mathcal{E}}\gamma_{(t,e)}\tilde{N}(de,dt). \end{aligned}
\Delta (e^{\tilde S/\hbar}) = \frac{e^{\tilde S/\hbar}}{\hbar}\left(\Delta\tilde{S}+ \frac{1}{2!}l_2(\tilde S,\tilde S) + \frac{1}{3!}l_3(\tilde S,\tilde S,\tilde S) + \cdots\right),
(\Theta_{j\bar k}({\cdot_\beta\eta}),{\cdot_\alpha\zeta})=\int_{\partial D_t}\theta_{j\bar k}(\rho) \langle{\cdot_\beta\eta},{\cdot_\alpha\zeta}\rangle d \sigma + (c_{j\bar k}(h){\cdot_\beta\eta},{\cdot_\alpha\zeta}) + R,
\left\vert L_{7,5}\right\vert \lesssim k^4 \sum_{\ell = 1}^\aleph\sum_{j\neq\ell}^\aleph|[P_{D_\ell}]Q_j||V_\ell| \lesssim k^4 \, \left\lVert[P_0]\right\rVert^2 \, \aleph \, \sum_{j = 1}^\aleph \lvert Q_j \rvert^2.
\begin{aligned}I^2_{1,2}=\sum_{0 <\tilde{\alpha}\leq \alpha}\begin{pmatrix}\alpha \tilde{\alpha}\end{pmatrix}\bigg\{D^{\tilde{\alpha}}(u_e\phi+u_b(1-\phi))D^{\alpha-\tilde{\alpha}}(\partial_x u)-D^{\tilde{\alpha}}(h_e\phi)D^{\alpha-\tilde{\alpha}}(\partial_x h)\bigg\}.\end{aligned}
A_\alpha(p_\alpha f)(x) = \sum_{k=1}^d \left(R_k^{\alpha}(\partial_k(p_\alpha),f)(x)+R_k^{\alpha}(\partial_k(f),p_\alpha)(x)\right) \quad+ \sum_{k=1}^d \left(\partial_k(p_\alpha)(x) D_k^{\alpha-1}(f)(x) + p_\alpha(x) \partial_kD^{\alpha-1}_k(f)(x)\right).
\Pi_{\varepsilon,\xi}^{\bot}\left\{W_{\varepsilon,\xi}+\phi-i_{\varepsilon}^{\ast}\left( b(x)f(W_{\varepsilon,\xi}+\phi)\right) \right\}=0, \Pi_{\varepsilon,\xi}\left\{W_{\varepsilon,\xi}+\phi-i_{\varepsilon}^{\ast}\left( b(x)f(W_{\varepsilon,\xi}+\phi)\right) \right\}=0,
r_{+}^{2}={\frac{m l^{2}}{2}}( 1 + \sqrt{1 + ({\frac{2 j}{m l}})^{2}}) ~ , ~ ~ | r_{-}|^{2}={\frac{m l^{2}}{2}}( \sqrt{1 + ({\frac{2 j}{m l}})^{2}}- 1 ) ~ ~
\tilde{\mu}\circ (\varphi \otimes \varphi )(a\otimes b)=\varphi(a)*\varphi (b)=\alpha_M\circ \varphi (a)\circ \alpha_M^{-1}\circ \varphi (b)\circ\beta_M^{-1}=\varphi (\alpha_A(a))\circ \varphi (b)\circ \beta_M^{-1}.
L_{D}= \int_{\cal M}\varepsilon^{i j k}\bigg ( \dots +{\frac{k}{8 \pi}}f_{a b c}A_{i}^{a}A_{j}^{b}B_{k}^{c}+{\frac{1}{2 \pi}}\chi_{i}^{a}\partial_{j}B_{k a}+{\frac{1}{4 \pi}}f_{a b c}\chi_{i}^{a}B_{j}^{b}B_{k}^{c}+ \dots \bigg ) .
U(s,x)=U(0,x)-\alpha\int_0^s \int_0^1 A(x,y)U(u,y)\mathrm{d}y \mathrm{d}u+ \alpha \beta\int_0^s \mathrm{d}\xi_2(u,x)+\alpha \zeta \int_0^s \mathrm{d}\xi_3(u,x).
\begin{aligned}\vert m \vert^2 \Lambda(\tilde{m}) \tilde{u}(\tilde{m}) = C(\tilde{m}) \Gamma(\omega)C(\tilde{m})^\top\tilde{u}(\tilde{m}) +\alpha(\tilde{m})C(\tilde{m}) \Gamma(\omega) \tilde{m},\ \ \ \ begin{align*}2.6pt] \Gamma(\omega) C(\tilde{m})^\top\tilde{u}(\tilde{m})\cdot \tilde{m}= - \alpha(\tilde{m})\Gamma(\omega) \tilde m \cdot \tilde m,\end{aligned}
(Y^{M(\mu,t)^\vee}(b,z)f_m)(m') = H_\mu(Y^\mu(e^{zL(\sqrt{-1}\Im(\mu))_1}\prod_{n=1}^{\infty}e^{-\frac{2(t-\sqrt{-1}\Im(\mu))}{n}z^na_n}z^{-2L(t)_0}g(b),z^{-1})m',m) = H_\mu(Y^\mu(e^{zL(\sqrt{-1}\Im(\mu))_1}z^{-2L(t)_0}g\prod_{n=1}^{\infty}e^{\frac{2(-t+\sqrt{-1}\Im(\mu))}{n}(-z)^{-n}a_n}b,z^{-1})m',m)
\lim_{T\rightarrow\infty}\frac{\left|\mathbb{Z}^{nd}\cap P_{Q}^{n}\left(I_{1},\dots,I_{n-1},T\right)\right|}{T^{nd-2\left(n-1\right)}}=C_{Q,n}\prod_{i=1}^{n-1}\left|I_{i}\right|.
\left( q_{e}^{I \ \prime}\, \, \, \, q_{m}^{I \ \prime}\right) = \left( q_{e}^{I}\, \, \, \, q_{m}^{I}\right) \left( \begin{array}{r r}{{a}}&{{- c}}\\{{}}&{{}}\\{{- b}}&{{d}}\\ \end{array}\right) \, ,
\sigma^{2}=\sum_{j=0}^{\infty}\left(1+\sum_{p=0}^{\infty}\Vert h\Vert_{L^{1}}^{p}\right)^{2}\overline{\gamma}_{0}\Vert h\Vert_{L^{1}}^{j}=\frac{\overline{\gamma}_{0}}{(1-\Vert h\Vert_{L^{1}})^{3}},
(u-v-h)X_{i}^{+}(u)X_{i}^{+}(v)=(u-v+h)X_{i}^{+}(v)X_{i}^{+}(u), &(u-v+h)X_{i}^{+}(u)X_{i+1}^{+}(v)=(u-v)X_{i+1}^{+}(v)X_{i}^{+}(u), &X_{i}^{+}(u)X_{j}^{+}(v)=X_{j}^{+}(v)X_{i}^{+}(u)if~ |i-j|>1,
\delta S_g = \frac{i}{48}[ \int (H-{\bar H}) F^{{\mathbf i}{\mathbf j}}F_{{\mathbf i}{\mathbf j}}+ \frac{3}{10}(H-{\bar H})_{{\mathbf i}{\mathbf j}{\mathbf k}{\mathbf l}}F^{{\mathbf i}{\mathbf j}}F^{{\mathbf k}{\mathbf l}}\; ]
A_1 =\{1,3,5,7\}. q(1,1)^2 =d_1^{-2}d^2_{15}, q(1,3)^2 =d_3^{-2}d^2_{13}, q(1,5)^2 =d_5^{-2}d^2_{11}, q(1,7)^2 =d_7^{-2}d^2_9, F_1=\langle q(1,1)^{2}\rangle \times\langle q(1,3)^{2}\rangle\times\langle q(1,5)^{2}\rangle\times\langle q(1,7)^{2}\rangle.
\begin{aligned}\big\|x^{k+1}-x^*\big\|_D^2 \le\big\|x^k - x^*\big\|_D^2 + 2\big(F(x^*) - \frac{1}{\beta}F(x^{k+1}) + \frac{1-\beta}{\beta}F(x^k)\big) &=\big\|x^k-x^*\big\|_D^2 + 2\big(F(x^*) - F(x^k)\big) + \frac{2}{\beta}\big(F(x^k) -F(x^{k+1})\big).\end{aligned}
\int_{a}^{\infty}\int_{\mathbf{R}^{d}}|K(t,r,x)-K(t,s,x)|dxdt & \leq N\int_{a}^{\infty}\frac{|s-r|}{(t-b)^{2}}dt \leq N|s-r|\int_{a-r}^{\infty}\frac{1}{t^{2}}dt =\frac{N|s-r|}{a-r}.
\langle\imath_{\xi}\gamma(x),\eta \rangle =\langle x,[\xi,\eta]\rangle &=\langle [x,\xi],\eta\rangle\qquad &=\langle -ad_x^*\xi+ad_{\xi}^*x,\eta\rangle &=\langle ad_{\xi}^*x,\eta\rangle.
d_{\mathcal{P}(\Omega \times{\rm Config})}( \overline{\mathbf{P}}_{1}, \overline{\mathbf{P}}_{2}) = \sup_{F \in{\rm Lip}_{1}(\Omega \times{\rm Config})}\int F d ( \overline{\mathbf{P}}_{1}- \overline{\mathbf{P}}_{2}),
\begin{array}{c}\displaystyle{\frac{1}{{\rm e}^\lambda +1}}=\displaystyle{\int_C}\displaystyle{\frac{{\rm d}z}{2\pi i}}\zeta_{+}\left(z\right)\Gamma\left(z\right)\,\lambda^{-z}\left( \zeta_{+}(z) \equiv \sum_{k=1}^\infty\displaystyle{\frac{(-1)^{k+1}}{k^z}}=(1 - 2^{1-z})\zeta\left(z\right);\;C{\rm is\;the\;contour\;}\rm{Re}\;z>1-2\epsilon\right)\end{array}
\begin{aligned}\sum_{k=0}^3(-1)^k\begin{bmatrix}3 k\end{bmatrix}_{t_{m+n}}E_{m+n}^kE_{m+n-1}E_{m+n}^{3-k}=\Phi_{m+n}\sum_{k=0}^3(-1)^{\frac{k(k+1)}{2}}\left\{\begin{matrix}3 k\end{matrix}\right\}_{q_{m+n}}e_{m+n}^k e_{m+n-1}e_{m+n}^{3-k}=0,\end{aligned}
\sum_{a\mod{r}}' \chi_*(a) e\left(n\bar{s}\tfrac{a}{r}\right)=\chi_*(s) \sum_{a\mod{r_0}}' \chi_*(a) e\left(n\tfrac{a}{r}\right) \sum_{j\mod{r/r_0}}e\left(n\tfrac{j}{r/r_0}\right)=\delta_{\frac{r}{r_0}\mid n}\frac{r}{r_0}\chi_*(s)c_{\chi_*}(n).
\sum_{\substack{j_1,\ldots,j_k\ge 1 j_1+\cdots +j_k=\nu}}a_{j_1}\cdots a_{j_k}= \sum_{\substack{t_1,\ldots,t_{\nu}\ge 0 t_1+2t_2+\cdots + \nu t_{\nu}=\nu t_1+\cdots +t_{\nu}=k}}\binom{t_1+\cdots +t_{\nu}}{t_1,\ldots,t_{\nu}}a_1^{t_1}\cdots a_{\nu}^{t_{\nu}}.
{\bf E}\Bigl[L^{q/\tau}\bigl(\frac{1}{\tau}\bigr)\Bigr] ={\bf E}\Bigl[e^{(q/\tau)\mathcal{N}(0,\,4\tau\log 2)}\Bigr], & = e^{4q^2\log 2/2\tau}\equiv{\bf E}\bigl[L^{q}(\tau)\bigr].
-\frac{4f_{i}f_{ij}f_{j}}{(1-f)^{3}}-\frac{4f^{2}_{i}f^{2}_{j}}{(1-f)^{4}}+\frac{2g'(\overline{\nabla}{f},\overline{\nabla}\left\|\overline{\nabla}{f}\right\|^{2})}{(1-f)^{2}}&=\frac{-2}{1-f}g'(\overline{\nabla}{f},\overline{\nabla}{\phi})+2g'(\overline{\nabla}{f},\overline{\nabla}{\phi})=\frac{-2}{1-f}g'(\overline{\nabla}{f},\overline{\nabla}{\phi}).
{F}^{\infty}_{\mathrm{ren}}= - \frac1{8\pi^2}\int\!{\mathrm d}^3 x \,{g}^{1/2}{\mathrm tr}\Big\{R_{i j}\gamma_1(-\triangle) R^{i j}+ R \gamma_2(-\triangle) R +{\mathrm{O}}[\Re^3]\Big\}.
|L(a)|^{2^k}\leq L(a^2)^{2^{k-1}}L(1)^{2^{k-1}}\leq L(a^4)^{2^{k-2}}L(1)^{2^{k-2}+2^{k-1}}\leq \dots &\leq L(a^{2^k}) L(1)^{1+\dots+2^{k-1}}=0.
0 \leq \phi'_1(z_1) =w'(z_1)-Kf'(u(z_1))w(z_1) begin{align*}0.2cm] =cw(z_1)-f(u(z_1))p(u(z_1))+f(u(z_1))v(z_1)-Kf'(u(z_1))w(z_1) begin{align*}0.2cm] =\left[cK-p(u(z_1))+v(z_1)-K^2f'(u(z_1))\right]f(u(z_1))<0.
(2p-1)\left(1+\frac{2pK}{f(x)}\right)\frac{x}{f(x)(1-x)}+(2p-1)\frac{\displaystyle \int_x^1\frac{dt}{f(t)}}{1-x}=\frac{2p-1}{2}(1+x)+\frac{B(x)}{1-x}.
\sum_{n=0}^{\Delta-1}X^{n}= \frac{X^{\Delta}-1}{X-1}= \prod_{a=1}^{\Delta-1}\left( X- e^{-2 \pi i a/\Delta}\right), \Delta= \prod_{a=1}^{\Delta-1}\left( 1- e^{-2 \pi i a/\Delta}\right).
|F_\varepsilon|_{\operatorname{Lip}^k(H,H_{-\alpha})}\leq\chi_0 \, |F|_{\operatorname{Lip}^k(H,H_{-\alpha})}\qquad &|B_\varepsilon|_{\operatorname{Lip}^k(H,HS(U,H_{-\beta}))}\leq\chi_0 \, |B|_{\operatorname{Lip}^k(H,HS(U,H_{-\beta}))}.
\begin{array}{rcll}\partial_t v^+_0 - \Delta v^+_0 + k(v^+_0) = f_0 \ \Omega_0\times(0, T), begin{align*}1mm]\partial_{\nu}v^+_0 = 0 \ \big(\partial\Omega_0 \setminus I_0\big) \times(0, T).\end{array}
[\kappa R(\kappa) - 1]\chi_y = R(\kappa) C_+ q \kappa R_0(\kappa) \chi_y + [\kappa R_0(\kappa) - 1]\chi_y &= R (\kappa)q_+ - R(\kappa) C_+(q-q\chi_y) + [R(\kappa) C_+ q + 1][\kappa R_0(\kappa) - 1]\chi_y.
\tilde{J}(u)\overset{\triangle}{=}\int_{0}^{+\infty}\Big(d_{1}\big(\tilde{x}(t)-\tilde{f}_{1}(t)\big)^{2}+ d_{2}\big(\tilde{y}(t)-\tilde{f}_{2}(t)\big)^{2}\Big)dt\rightarrow \min_{u(t)},
Q = \left( \begin{array}{c c c c c c r}{{0}}&{{- 1}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{1}}&{{- 1}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}&{{- 1}}&{{- 1}}\\{{0}}&{{0}}&{{1}}&{{1}}&{{- 1}}&{{- 1}}\\{{0}}&{{0}}&{{0}}&{{1}}&{{- 1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}&{{0}}&{{- 1}}\\ \end{array}\right) .
F(k_0)=\frac{k_0}{2}\int_0^1 du\frac{\sqrt{\sqrt{1+4k_0^2(1-u)}-1}}{\sqrt{u\left(2k_0^2+1-\sqrt{1+4k_0^2(1-u)}~\right)\left(1+4k_0^2(1-u)\right)}}=\frac{1}{2}\ln \frac{R_2}{R_1}.
W = \prod_{a = 1}^{p - 1}\left( \frac{n^{\frac{1}{l}}}{L_{a}}\right) \int_{0}^{\infty}\frac{d s}{s}\left( \frac{\pi}{2 s}\right)^{\frac{p}{2}- 1}e^{- r^{2}s}\frac{4 \sinh^{2}( \omega_{l}s ) - \sinh^{2}( 2 \omega_{l}s )}{~ \sinh ( 2 \omega_{l}s )}\, .
{\sf I}(\alpha)=\frac{1}{s}\vartheta^a\wedge \sum_{a=1}^m\left[\iota_{\frac{\partial}{\partial u^a}}\alpha-\sum_{j=1}^n\frac{\sf d}{{\sf d}x_j}\left(\iota_{\frac{\partial}{\partial u^a_j}}\alpha\right)\right].
s_i^2=1, (s_is_{i+1})^3=1, (s_is_j)^2=1 \mbox{~for~}|i-j|\geq 2, s_ib_i=b_{i+1}s_i, s_ib_j=b_js_i \mbox{~for~}j\neq i, i+1, b_j^2=1, b_ib_j=b_jb_i \mbox{~for~}i\neq j.
\frac{\partial\mathcal{L}}{\partial v}(X_t, \dot X_t, t) = e^{\gamma_t}\left(\nabla h(Z_t) - \nabla h(X_t)\right) \frac{\partial\mathcal{L}}{\partial x}(X_t, \dot X_t, t) = e^{\alpha_t}\frac{\partial\mathcal{L}}{\partial v}(X_t, \dot X_t, t) - e^{\gamma_t}\frac{d}{dt}\nabla h(X_t) - e^{\alpha_t + \beta_t + \gamma_t}\nabla f(X_t),
rsin\frac{x}{r}=x-x\cdot\frac{x^{2}}{(2^{2}+2)r^{2}}+x\cdot\frac{x^{2}}{(2^{2}+2)r^{2}}\cdot\frac{x^{2}}{(4^{2}+4)r^{2}}-\cdot\cdot\cdot
\left[ i \left( \rho^{m}\nabla_{m}\right)_{\, \beta}^{\alpha}- \delta_{\beta}^{\alpha}\rho^{3}\right] \left( \begin{array}{c}{{\theta^{1 \beta \alpha^{\prime}}}}\\{{\theta^{2 \beta \alpha^{\prime}}}}\\ \end{array}\right) = 0 .
M_{1}= \left( \begin{array}{c c c c}{{I}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{A}}&{{- B}}&{{0}}\\{{0}}&{{B}}&{{A}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{I}}\\ \end{array}\right) \: \: , \: \: I = \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{1}}\\ \end{array}\right) \: \: , A = \cos \frac{2 \pi}{n}I \: , \: B = \sin \frac{2 \pi}{n}I .
\begin{array}{ll}\frac{\mathrm{d}}{\mathrm{d}t}\int\limits_{\Omega}|\partial_t^{\ell}\mathcal{D}^{\alpha}\omega|^2 \,\mathrm{d}x+ 2a\int\limits_{\Omega}|\partial_t^{\ell}\mathcal{D}^{\alpha}\omega|^2 \,\mathrm{d}x = I_4 + I_5,\end{array}
S_1\geq \frac{X\log X}{\zeta(2-2\theta_3-0.01)^{35}}\prod_{j=1}^R\Bigg ((1-2^{-\frac{\ell_j}{2}})\exp\Big(\frac{k^2-1}{2}\sum_{p_j\in P_j}(\frac{A(p_j,1)^2}{p_j}+O(\frac{|A(p_j,1)|}{p_j^{2-2\theta_3}}))\Big)\Bigg ).
=(g_{\mu\nu,\lambda}\dot{x}^{\mu}\dot{x}^{\nu})(g_{\alpha\beta}\dot{x}^{\alpha}\dot{x}^{\beta})+\frac{1}{2}(g_{\lambda\nu}\dot{x}^{\nu}+g_{\lambda\mu}\dot{x}^{\mu})\frac{d}{d\tau}(g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu})(6)
\gamma=\gamma^*:=\left(\frac{\rho}{2-\rho}\frac{\hat{M}^{\frac{2}{2-\rho}}}{3\rho}\frac{\sigma_d}{4d(x^*)}\right)^{\frac{2-\rho}{2}}=\hat{M}\left(\frac{\sigma_d}{12(2-\rho)d(x^*)}\right)^{\frac{2-\rho}{2}}
S_{s e l f}= - \frac{e^{2}c}{2}\overline{{{\sum}}}_{n , m , r , s}\int_{- \infty}^{\infty}d t \int_{- \infty}^{\infty}d x_{1}\int_{- \infty}^{\infty}d t^{\prime}\int_{- \infty}^{\infty}d x_{1}^{\prime}[ j_{n m}^{\mu}( x_{1}) D^{c}( x_{1}- x_{1}^{\prime}; t - t^{\prime}) j_{r s ,{\mu}}( x_{1}^{\prime})
\lefteqn{(-\partial_1^2-|\partial_1|^{-1}\partial_2^2)w^\ell}&+P\Big(\sigma^2F^\ell+\sigma v_\ell\partial_2Rw^\ell+\sigma w^\ell\partial_2Rv_\ell+w^\ell\partial_2Rw^\ell &+\partial_2\frac{1}{2}R(w^\ell+\sigma v_\ell)^2-(w^\ell+\sigma v_\ell) \partial_1\frac{1}{2}R(w^\ell+\sigma v_\ell)^2\Big)=0.
H_{0_{B}}=H_{0}+\int d^{3}x\left( \eta_{1a}^{i}\mathcal{P}_{2i}^{a}-\eta_{1a}\mathcal{P}_{2}^{a}-\eta_{2a}\partial^{i}\mathcal{P}_{2i}^{a}+\eta_{2a}^{i}\partial_{i}\mathcal{P}_{2}^{a}\right) \equiv \int d^{3}xh_{0_{B}},
2 D_n^* (\mathbf{x}_1, \dots, \mathbf{x}_n) \geq \left( \textup{vol~}(A) - \frac{\# (A \cap \mathcal{P})}{n}\right) - \left( \frac{\# (A \cap \mathcal{P})}{n}- \textup{vol~}(B) \right) = \textup{vol~}(A \setminus B) \geq \frac{2d}{n},
\eta^+=\Phi_{0}^++\Phi_{2}^+={\frac{U^+}{r}},\qquad\eta^-=\Phi_{0}^-+\Phi_{2}^-={\frac{U^-}{r}}, begin{align*}1mm]\zeta^+=\Phi_{0}^+-\Phi_{2}^+={U^+}',\qquad\zeta^-=\Phi_{0}^--\Phi_{2}^-={U^-}'.
v_{{\mathcal{R}},{\mathcal{T}},p,n}^i(u)=\alpha_{{\mathcal{R}},{\mathcal{T}}}^{\bar{\mathcal{R}}_i}(u)c_p(u) z_{{\mathcal{R}},{\mathcal{T}},n}^{\bar{\mathcal{R}}_i}(u),
\kappa_{11}=\sum_{m\geq 0}\prod_{j=0}^{m-1}q_{0^j11}(0){\rm ~~and~~}\kappa_{10^k1}=1+\sum_{m\geq k+1}\prod_{j=0}^{m-1}q_{0^j10^k1}(0),~\forall k\geq 1.
\frac{\partial F(w)}{\partial w_i}= e^{\widehat{w}}\frac{\partial \widehat{w}}{\partial w_i}J_d-J_d (e^{\widehat{w}}\frac{\partial \widehat{w}}{\partial w_i})^T = e^{\widehat{w}}\frac{\partial \widehat{w}}{\partial w_i}J_d+J_d \frac{\partial \widehat{w}}{\partial w_i}(e^{\widehat{w}})^T
h_{T,S}= \frac{h_{L,S'}[L':K][H^0_T(G,O_{L,S'}^{*})]}{h_{K,S}[\ker(H^0_T(G,L^{*})\to H^0_T(G,I_L))]\prod_{v\in S}[L_w':K_v] \prod_{v\notin S}e_v(L'/K)}.
U(t) = \sum_{\lambda,\mu}e^{it(\lambda+\mu)}\phi_\lambda\otimes \psi_\mu &= \sum_{\lambda,\mu}\left(e^{it\lambda}\phi_\lambda\right)\otimes \left(e^{it\mu}\psi_\mu\right) &= \left(\sum_\lambda e^{it\lambda}\phi_\lambda\right)\otimes\left(\sum_\mu e^{it\mu}\psi_\mu\right) &= U_1(t) \otimes U_2(t)
\begin{cases}x_{1}^{\left(t+1\right)}=\;\gamma_{1}x_{1}^{\left(t\right)}y^{\left(t\right)}\medskip y^{\left(t+1\right)}=\;\left(1-\gamma_{1}\right)x_{1}^{\left(t\right)}y^{\left(t\right)}.\end{cases}
\int_{B_{u_p}^{\ell}}f(x) \lvert g_{u_p}(x)\lvert^2d\lambda_\nu(x)=\frac{1}{\ell!}\int_{A_{u_p}}f(x) \prod_{i=1}^\nu \lvert g_{u_p}^{(i)}(x_i)\lvert^2 d\lambda_\nu(x)
\begin{array}{l}{{\lambda_{1}= - 2 ( \frac{A^{2}- 3 B}{3})^{\frac{1}{2}}\cos \frac{1}{3}\alpha - \frac{A}{3},}}\\{{\lambda_{2}= - 2 ( \frac{A^{2}- 3 B}{3})^{\frac{1}{2}}\cos \frac{1}{3}( \alpha + 2 \pi ) - \frac{A}{3},}}\\{{\lambda_{3}= - 2 ( \frac{A^{2}- 3 B}{3})^{\frac{1}{2}}\cos \frac{1}{3}( \alpha + 4 \pi ) - \frac{A}{3},}}\\ \end{array}
\begin{aligned}p_G(x) \:\:\: = \:\:\: \frac{1}{(G+1)^d}\sum_{\Gamma\in\mathbb{N}_0^d,||\Gamma||_{\infty}\leq G}\psi_{\Gamma}(x) = \:\:\: \sum_{j\in\mathbb{Z}^d,|j_i|\leq G}a'_j\cos(2\pi\langle j,x\rangle)+b'_j\sin(2\pi\langle j,x\rangle) \end{aligned}
\overline{\zeta}^+_0(a)=\frac{-1+i^\frac{p-1}{2}\sqrt{p}}{2}\tau_l=o\bigg{(}\bigg{(}\bigg{(}\sum_{j=1}^{l-1}\beta_j\bigg{)}+1,\dots,\sum_{j=1}^l\beta_j\bigg{)}\bigg{)}l.
\Psi_{\mathcal{A}}([\gamma_{(t,p)}])((t,v),u_m) =[(t,p),\mathcal{A}(t,p)((t,v),a_pv+\gamma_p(u_m))]_G &=\left[p,\mathcal{A}_M(a_p) v+\mathcal{A}_M(\gamma_p(u_m))\right]_G &=\sigma_x^{-1}[a_p]v+\left[p,\mathcal{A}_M(\gamma_p(u_m))\right]_G.
U^{\prime}( \lambda ) = \frac{\lambda J_{0}( x ) + J_{1}( x )}{1 -{\lambda}^{2}}= J_{0}( g + \lambda ) [ ( g - \lambda ) ( g + \lambda ) ]^{- 1}= J_{0}( g - \lambda )^{- 1}= - ( g + \lambda )^{- 1}J_{0}
\left\Vert f\right\Vert_{\mathcal{CMO}^{p}}:=\sup_{\substack{R_{j}\geq1 j=1,...,n}}\left[ \frac{1}{R_{1}...R_{n}}{\displaystyle\int\nolimits_{\left[ -R_{1},R_{1}\right] \times...\times\left[ -R_{n},R_{n}\right]}}\left\vert f\left( x\right) -f_{R_{1}...R_{n}}\right\vert^{p}dx\right]^{1/p}
C(\theta_1,x + iy) := \frac{1}{2}\sum_{k \in \mathbb{Z}}\int_{-1/2}^{1/2}\left\{\log\left( 1 + \left( \frac{y/2\pi + \theta_1/2 + k}{x/2 \pi}\right)^2 \right) - \log\left( 1 + \left(\frac{y/2\pi + \theta_1/2 + k + \phi}{x/2\pi}\right)^2 \mathrm{d}\phi \right) \right\}.
\int_{-\sqrt{1-|x_1|^2}}^{\sqrt{1-|x_1|^2}}\big[1+4\mu -v_{x_2}\big]\,dx_2 &\quad= 2(1+4\mu)\sqrt{1-|x_1|^2}-\big[v(x_1,\sqrt{1-|x_1|^2})-v(x_1,-\sqrt{1-|x_1|^2})\big] &\quad=8\mu \sqrt{1-|x_1|^2}&\quad\quad- \Big\{[v (x_1,\sqrt{1-|x_1|^2})-Q(x_1,\sqrt{1-|x_1|^2})]-\big[v(x_1,-\sqrt{1-|x_1|^2})- Q(x_1,-\sqrt{1-|x_1|^2})\big] \Big\}&\quad\le 8\mu.
\phi\big(M(D_i,D_j)\big) =\prod_{k=1}^K\sum_{\pi\in N_{o}^{(k)}}\kappa_{\pi}[a_{k;i}^{n_1},a_{k;j}^{n_2},\cdots,a_{k;i}^{n_{2t-1}},a_{k;j}^{n_{2t}}].
y_m(t) =\sum_{n=0}^{N-1}\sum_{l=0}^{L-1}s_{n,m}\alpha_l \cos(w_n(t-\tau_l)+\zeta_{n,m,l}+\phi_{n,m}) &=\sum_{n=0}^{N-1}s_{n,m}A_{n,m}\cos(w_n t+\psi_{n,m})
{\left(\rho^{(2)}_{ab}\right)}^{-1}_{ij}=\left(\begin{array}{cccc}0 -\delta_{ab}D_{ij}0 \partial_{i}^x \nabla^{-2}\delta_{ab}D_{ij}0 \partial_i^x\nabla^{-2}0 0 -\partial_j^x \nabla^{-2}0 \nabla^{-2}-\partial_j^x \nabla^{-2}0 -\nabla^{-2}0\end{array}\right)\delta(\vec x-\vec y),
\sum_{|\ell| > 1/(A \Delta)}d_{\ell A_2/D,\Delta}(a) d_{-\ell A_1/D,\Delta}(b) = O\left( \Delta \right), \sum_{|\ell| > 1/(A \Delta)}d_{\ell A_2/D}(a) d_{-\ell A_1/D}(b) = O\left( \Delta \right).