harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\big| F(q)-a_n\big(q-q_0\big)^n\big|_{C_{q_0}}=\left|\sum_{\ell=0}^{n-1}a_{n-\ell}\big(q-q_0\big)^\ell\right|_{C_{q_0}}&\leq\sum_{\ell=0}^{n-1}\big|a_{n-\ell}\big| \left|\big(q-q_0\big)^\ell\right|_{C_{q_0}}& <R^n =a^n\big|a_n(q-q_0)^n\big|_{C_{q_0}}
\underset{m\rightarrow \infty}{\lim}P(K_{n}=k|K_{m}=\omega_{m})=\underset{m\rightarrow \infty}{\lim}\frac{d_{n,k}^{m,\omega_{m}}}{d^{m,\omega_{m}}}d^{n,k}= \bar{V}_{n,k}d^{n,k}
u^s(\mathbf x)=\frac{e^{ik\vert \mathbf x\vert}}{\vert \mathbf x\vert^{(n-1)/2}}\left(u^\infty(\hat{\mathbf x})+\mathcal O\left(\frac{1}{\vert \mathbf x\vert^{(n-1)/2}}\right)\right), \vert \mathbf x\vert \to \infty,
E\{C_{i_1}\cdots C_{i_k}\}= \frac{m_{i_1}\cdots m_{i_k}}{q_{\phi}(n)}\sum_{h_1,\dots,h_k \ge 1}(-1)^{h_1 + \cdots + h_k+k}\phi^{h_1 + \cdots + h_k}q_{\phi}(n-h_1i_1 - \cdots - h_ki_k).
d s^{2}= \bigl ( 1 - \frac{2 G M}{\rho}\bigr ) d t^{2}- \bigl ( 1 - \frac{2 G M}{\rho}\bigr )^{- 1}d \rho^{2}- \rho \bigl ( \rho - \frac{Q^{2}}{G M}\bigr ) d \Omega
h =(0,\frac{\Lambda_2^\vee}{4}+\frac{\Lambda_{10}^\vee}{2}), 2[h] =(0,\frac{\Lambda_2^\vee}{2}-\Lambda_3^\vee+\Lambda_4^\vee-\Lambda_5^\vee+\Lambda_6^\vee-\Lambda_7^\vee+\Lambda_8^\vee-\Lambda_9^\vee+\Lambda_{10}^\vee).
\begin{matrix}K_L = K_\ell^{4+2\varepsilon}, K_H = K_\ell^{4+3\varepsilon}, \frac{\mathcal M}{N}= K_\ell^{-12 -10\varepsilon}, \varepsilon_{\rm{gap}}= K_\ell^{-2}, \varepsilon_K = K_\ell^{-18 +2d +(d-16)\varepsilon}, \end{matrix}
\Gamma ({\bf p},{\bf q}; E ) = K ({\bf p},{\bf q}) + \int \frac{d^{2}{\bf k}}{( 2 \pi )^{2}}K ({\bf p},{\bf k}) \frac{i}{E - \frac{{\bf k}^{2}}{m}+ i \epsilon}\Gamma ({\bf k},{\bf q}; E ) .
C_\theta(\tau) = \langle{\cal A}(\tau){\cal B}\rangle_0 \; - \;\lim_{\tau^\prime \rightarrow \infty}\langle{\cal U}_{\tau^\prime}({\scriptstyle{\cal A}}){\cal A}\rangle_0\lim_{\tau^{\prime \prime}\rightarrow \infty}\langle{\cal U}_{\tau^{\prime \prime}}({\scriptstyle{\cal B}}){\cal B}\rangle_0\stackrel{\tau \rightarrow \infty}{\longrightarrow}
\tilde \varphi_{n,i}(x) = \varphi_{n,i}(x) + \sum_{k=1}^\infty (\tilde Q_{n,i; k,0}(x)\varphi_{k,0}(x) - \tilde Q_{n,i; k,1}(x)\varphi_{k,1}(x)), \tilde \varphi_{n,i}(x) = \varphi^K_{n,i}(x) + \sum_{k=1}^K (\tilde Q_{n,i; k,0}(x)\varphi^K_{k,0}(x) - \tilde Q_{n,i; k,1}(x)\varphi^K_{k,1}(x)).
z\cot(z)=\frac{z}{2^n}\cot(z/2^n)-\frac{z}{2^n}\tan(z/2^n)+\sum_{j=1}^{2^{n-1}-1}\frac{z}{2^n}\left(\cot\left(\frac{z+j\pi}{2^n}\right)+\cot\left(\frac{z-j\pi}{2^n}\right)\right).
\int_{\Gamma\backslash G}\Lambda^TK_(x,x)dx= g(1)\log T-\frac{1}{4\pi}\int^\infty_{-\infty}\frac{m'}{m}(it)\hat{g}(it)dt +\frac{1}{4\pi i}\int^\infty_{-\infty}m(it)\hat{g}(it)\frac{T^{it}}{t}dt.
\fbox{$\begin{array}{ll}(xy)(ax)=x(ay)x; (xy)_{L}x_{R}= x_{L}x_{R}y_{L}; (xa)(yx)=x(ya)x; (yx)_{R}x_{L}= x_{L}x_{R}y_{R}; (xax)y=x(a(xy)); y_{R}x_{L}x_{R}= x_{L}(xy)_{R}; y(xax)=((yx)a)x; y_{L}x_{L}x_{R}= x_{R}(yx)_{L}. \end{array}$}
\sigma_{V}(\xi)=1-\sum_{j\geq 0}\sigma_{D^j_{\frak p}(W)}(\xi)=\sum_{l=1}^L\sum_{j \in \mathbb Z}|\hat{\psi}_l(\frak{p}^j \xi)|^2-\sum_{l=1}^L\sum_{j\geq 0}|\hat{\psi}_l(\frak{p}^j \xi)|^2=\sum_{l=1}^{L}\sum_{j=1}^{\infty}|\widehat{\psi}_l(\mathfrak{p}^{-j}\xi)|^2,
R_{\mu \nu}-{\frac{1}{2}}g_{\mu \nu}^{( 4 )}R = - \ell_{P}^{2}\left( \frac{\delta \stackrel{\sim}{\cal L}_{m}^{\prime}}{\delta g^{\mu \nu}}-{\frac{1}{2}}g_{\mu \nu}\stackrel{\sim}{\cal L}_{m}^{\prime}\right) - \ell_{P}^{2}e^{- 1}\frac{\delta{\cal L}_{g}}{\delta g^{\mu \nu}}.
\sum_{\nu = 0}^{N}\left[ \omega_{\nu}^{2}\eta_{\nu \rho}+ \sum_{k = 1}^{N}\lambda_{k}\left( \eta_{k \rho}\delta_{0 \nu}+ \eta_{0 \rho}\delta_{k \nu}\right) \right] \eta_{\nu \sigma}= \delta_{\rho \sigma}\Omega_{\sigma}^{2}.
^{\dagger}f(x,y,z)=\dfrac{y(ax+by+cz)^3}{x^2z^2},{f(x,y,z)=\dfrac{y^2(ax+by+cz)^2}{xz^{3}}},^{\dagger}f(x,y,z)=\dfrac{y(ax+by+cz)^4}{x^2z^{3}},^{\dagger}f(x,y,z)=\dfrac{y^2(ax+by+cz)^3}{xz^{4}},^{\dagger}f(x,y,z)=\dfrac{y(ax+by+cz)^6}{x^3z^{4}},^{\dagger}f(x,y,z)=\dfrac{y^3(ax+by+cz)^4}{xz^{6}},
\limsup_{N\to\infty}\frac{1}{\#(\Lambda_N^{\vec{v}}(1))}\sum_{(m,n)\in\Lambda_N^{\vec{v}}(1)}\min_{1\leq i<j\leq k}d(T^{(m,n)}x_i,T^{(m,n)}x_j) \geq \limsup_{N\to\infty}\frac{1}{2N}\sum_{n=0}^{N-1}\min_{1\leq i<j\leq k}{d}(T^{(n,[n\beta+t])}x_i,T^{(n,[n\beta+t])}x_j)\geq \eta/2>0.
\left |\begin{array}{cccc}Q_0(x_0) Q_0(x_1) \cdots Q_0(x_n) Q_1(x_0) Q_1(x_1) \cdots Q_1(x_n) \vdots \vdots \vdots Q_n(x_0) Q_n(x_1) \cdots Q_n(x_n)\end{array}\right |= \left (\prod^n_{i=0}a_{ii}\right )V(x_0, x_1,\ldots,x_n),
[\begin{matrix}y_{1}\\y_{2}\\y_{3}\\y_{4}\\y_{5}\\y_{6}\\y_{7}\end{matrix}]=[\begin{matrix}1&x_{1}\\ 1&x_{2}\\ 1&x_{3}\\ 1&x_{4}\\ 1&x_{5}\\ 1&x_{6}\\ 1&x_{7}\end{matrix}][\begin{matrix}\beta_{0}\\ \beta_{1}\end{matrix}]+[\begin{matrix}\epsilon_{1}\\ \epsilon_{2}\\ \epsilon_{3}\\ \epsilon_{4}\\ \epsilon_{5}\\ \epsilon_{6}\\ \epsilon_{7}\end{matrix}]
q_T^{(1)}(x) =G_T'(x)\hat{a}_T(x)+\frac{1}{2}G_T''(x)-a_0\bigl(G_T(x) \bigr), q_T^{(2)}(x) = \bigl[G_T'(x)\bigr]^2- \sigma_0^2 \bigl(G_T(x)\bigr),
\omega= \underbracket{\mathcal Q^{(m)}(\boldsymbol{\Lambda})}_{}&+\underbracket{(T\beta)\mathcal Q^{(m-1)}(\boldsymbol{\Lambda})-\mathcal Q^{(m-1)}(\boldsymbol{\Lambda})\beta}_{}&\shortvdotswithin{+}+ \underbracket{(TH)H^{-1}}_{}
\int \! d x d p ~ f \star g \star h ={\frac{1}{\hbar^{2}\pi^{2}}}\! \int \! d{\bf r}_{1}d{\bf r}_{2}d{\bf r}_{3}f ({\bf r}_{1}) g ({\bf r}_{2}) h ({\bf r}_{3}) ~ \exp \! \left({\frac{- 4 i}{\hbar}}A ( r_{3}, r_{2}, r_{1}) \right) .
\tilde{\kappa}'\colon F_*(N/IN \otimes \omega_R / I\omega_R) \to (N/IN \otimes \omega_R / I\omega_R) \overline{n}\otimes \overline{m}\mapsto \sum_{i=1}^n\overline{\phi_i(1)}\cdot \overline{\gamma_N(n)}\otimes \overline{\kappa_R(r_if^{p-1}m)}&= \sum_{i=1}^t\phi_i(1)\gamma_N(n) \otimes \kappa_R(r_if^{p-1}m) + I(N \otimes \omega_R)
\begin{array}{ccc}\nabla_{X_{1}}X_{1}= \displaystyle\frac{-c\nu b_{2}}{(\lambda_{2}-\lambda_{1})}X_{2}, \nabla_{X_{2}}X_{2}= \displaystyle\frac{c\nu b_{1}}{(\lambda_{2}-\lambda_{1})}X_{1}. \end{array}
\prod_{l=a}^{b}\int_{0}^{\pi}\left(\sin\theta\right)^l d\theta = \prod_{l=a}^{b}\pi^{\frac{1}{2}}\frac{\Gamma\left(\frac{l+1}{2}\right)}{\Gamma\left(\frac{l+2}{2}\right)}= \pi^{\frac{b-a+1}{2}}\frac{\Gamma\left(\frac{a+1}{2}\right)}{\Gamma\left(\frac{b+2}{2}\right)}.
\begin{array}{lll}G_W(t) = g((TX)_{\frak{m}}+(TY)_{\frak{m}}+Z_{\frak{m}},[W, TX+TY+Z]_{\frak{m}})_o\ &+ g(W,[TX, TY+Z]_{\frak{m}}+[TY, Z]_{\frak{m}})_o.\end{array}
H^{( 2 )}= \int d^{2}x \left[ - \frac{1}{2}\partial_{i}\Phi^{0}\partial^{i}\Phi^{0}+ \sqrt{m}\Phi^{0}\epsilon_{i j}\partial^{i}\Phi^{j}+ \frac{1}{2}m \Phi^{i}\Phi^{i}- ( \frac{1}{\sqrt{m}}\partial_{i}\Phi^{i}+ \frac{1}{2}\Phi^{3}) \Phi^{3}\right] .
\langle \phi , \psi \rangle_{W^{2,2}(g)}= \int (P_g \phi) \psi \dv_g & \hskip-.5in = \int \big[ (\Delta_g \phi)(\Delta_g \psi) - 4 A_g(\nabla \phi,\nabla \psi) + (n-2) \sigma_1(A_g) g(\nabla \phi,\nabla \psi) + \frac{n-4}{2}Q_g \phi \psi \big] dv_g,
\sup \limits_{t\in [0,T]}\sum \limits_{k=1}^{\infty}\langle T_th_k,h_k\rangle <\infty\qquad\lim \limits_{N \rightarrow \infty}\sup \limits_{t\in [0,T]}\sum \limits_{k=N}^{\infty}\langle T_th_k,h_k\rangle =0.
\|u\|_{\ell^{\infty}(S)}\equiv \max_{n=0,\cdots, N_T}\|u^n\|_{S}, \|u\|_{\ell^{2}_m(S)}\equiv \biggl\{\Delta t\sum_{n=1}^{m}\|u^n\|_S^2 \biggr\}^{1/2}, \|u\|_{\ell^{2}(S)}\equiv \|u\|_{\ell^{2}_{N_T}(S)}, |p|_h \equiv \mathcal{C}_h (p, p)^{1/2},
\nu\big(B_{\infty}(x, r)\big) = \lim_{i\rightarrow \infty}\frac{\mu_i\Big(B_i(x_i, r)\Big)}{\mu_i\Big(B_i(p, 1)\Big)}= \lim_{i\rightarrow \infty}\frac{\mu\Big(B(x_i, r_ir)\Big)}{\mu\Big(B(p, r_i)\Big)}&=r^{\kappa}\lim_{i\rightarrow \infty}\frac{\mu\Big(B(x_i, r_ir)\Big)}{a_0 (r_ir)^{\kappa}}= r^{\kappa}
u_{z-1}\geq u_z ,\,\, u1_{\{u_z>0\}}=u_z+\lambda(1-2^{-z})1_{\{u_z>0\}}1_{\{u_z>0\}}\leq \left(\frac{2^zu_{z-1}}{\lambda}\right)^{q},\quad\forall \, q>0.
V_{\bar{\lambda}, \xi}|_{\widehat{T}^I \rtimes \langle \tau \rangle}= \bigoplus_{\bar{\nu}, \xi'}m_{\bar{\lambda}, \xi}(\bar{\nu}, \xi') \, \bar{\nu}\boxtimes \xi' ~ \oplus ~ \bigoplus_{\underset{d > 1}{\bar{\nu},\,d,\,\xi'_{n/d}}}m_{\bar{\lambda}, \xi}(\bar{\nu}, d, \xi'_{n/d}) \, V_0(\bar{\nu}, d,\xi'_{n/d}).
\lambda_{\imath, \jmath}=(\varpi_{\imath}, \phi_{\mathcal{Q}}^{\mathrm{tw}}\varpi_{\jmath}-\varpi_{\jmath})-(\varpi_{\imath}, (\phi_{\mathcal{Q}}^{\mathrm{tw}})^{-1}\varpi_{\jmath}-\varpi_{\jmath}).
\min\{(x H' M N^{-1})^{\frac{1}{2}}, H'^{-\frac{1}{2}}M^{\frac{1}{2}}N\}\le ((x H' M N^{-1})^{\frac{1}{2}})^{\frac{1}{3}}(H'^{-\frac{1}{2}}M^{\frac{1}{2}}N)^{\frac{2}{3}}& = (x H'^{-1}M^3 N^3)^{\frac{1}{6}}& \le (x D^3)^{1/6}
-\frac{1}{2}\int_{\mathbb{R}}|\varphi|^2~dx + \frac{1}{2\pi}\sum_n\int_0^{\infty}\left(\frac{\psi_n^2}{2}\right)' \widehat{u\varphi}\bar{\hat\varphi}~d\xi+\frac{1}{2\pi}\int_{\real}\widehat{u\varphi}'\bar{\hat\varphi}~d\xi=0.
\widehat{w}_{n}(\zeta)\leq \mathcal{D}_{n}(\widetilde{\Psi}):=\frac{2\widetilde{\Psi}-n+1+\sqrt{4\widetilde{\Psi}^{2}+17n^{2}-16\widetilde{\Psi}n+8\widetilde{\Psi}-18n+5}}{2}.
S = \frac{1}{2 \kappa_{5}^{2}}\int d z d^{4}x \sqrt{g}\left( \frac{1}{2 e^{2}}g^{i j}g^{k l}F_{i k}^{a}F_{j l}^{a}+ \frac{1}{2}\left({\cal M}^{2}\right)_{a b}g^{i j}A_{i}{}^{a}A_{j}{}^{b}\right)
\left(p^{\theta}(\Delta_n,X_{t_k},y)\right)^2\geq q_{(0)}^{\theta}(\Delta_n,X_{t_k},y)e^{-\lambda \Delta_n}\int_{I}q_{(1)}^{\theta}(\Delta_n,X_{t_k},y;a)\mu(da)e^{-\lambda \Delta_n}\lambda \Delta_n.
\begin{array}{l}{( \partial b )^{*}= - q a b^{- 1}( \partial a ) ,}\\{( \partial c )^{*}= - \frac{1}{q^{2}}c^{- 1}( a \partial a ) ,}\\{( \partial d )^{*}= q ( a^{2}- 1 ) ( b c )^{- 1}( \partial a ) .}\\ \end{array}
h_{11}^2 =0, \omega_{21}^2 =-\frac{h_{13}^3 (\sin (2 c)-\sin (c-2 \Lambda ))}{\cos (2 c)-\cos (c-2 \Lambda )}\omega_{21}^3 = -h_{12}^3 \cot \left(\Lambda +\frac{3 c}{2}\right)-\frac{\sqrt{3}}{6}.
\mathbb{E}_0 \left[ h^{(13)}_{{\bf i}^1, \tilde{{\bf i}}^1}\cdot h^{(23)}_{{\bf i}^2, \tilde{{\bf i}}^2}\cdot h^{(14)}_{{\bf i}^3, \tilde{{\bf i}}^3}\cdot h^{(24)}_{{\bf i}^4, \tilde{{\bf i}}^4}\right] = 0
\langle \alpha_\vartheta' | \alpha_{\vartheta'}\rangle = \delta_{\vartheta,\vartheta'}\langle \alpha_x'| \alpha_\vartheta \rangle =\langle \alpha_\vartheta | \alpha_x\rangle = 0 \langle \alpha_x| \alpha_{x'}\rangle = \delta(x-x')
c(z) := \left( 1- |\theta(0)|^2 \right)^{-1/2}\left( \begin{array}{c}1-\overline{\theta(0)}\theta (z) -\overline{\theta(0)}\Delta(z) \end{array}\right), c_1(z) := \left( 1- |\theta(0)|^2 \right)^{-1/2}\left( \begin{array}{c}z^{-1}\left(\theta(z)- \theta(0)\right) z^{-1}\Delta (z)\end{array}\right),
\Omega_2(N, q)=\begin{cases}\frac{1}{2}-\frac{3}{4}e_2 0\leq e_2 \leq 2, -2 e_2\geq 3, s_2=0, -2s_2 e_2> s_2 \geq 3, -e_2 e_2=s_2\geq 3.\end{cases}
M =a^{\vartheta_n}_{c_n}\cdots a^{\vartheta_1}_{c_1}M_2 =a^{\vartheta_n}_{c_n}\cdots a^{\vartheta_k}_{c_k}M_1 =a^{\vartheta_{k 1}}_{c_{k-1}}\cdots a^{\vartheta_1}_{c_1}
D(v_0,v_1,\ldots,v_k)=\begin{vmatrix}v_0 v_1 \cdots v_k u_{0,0}u_{0,1}\cdots u_{0,k}u_{1,0}u_{1,1}\cdots u_{1,k}\vdots \vdots \vdots u_{k-1,0}u_{k-1,1}\cdots u_{k-1,k}\end{vmatrix},
E\max_{1\leq k\leq n}\left\vert \sum\nolimits_{j=1}^{k}a_{j}E^{j}X\right\vert^{2}\ll\sum\nolimits_{k=1}^{n}k^{-1}\left( s_{4k}^{\ast}\right)^{2}E|E^{k}X|^{2}& +\sum\nolimits_{k=1}^{n-1}\left( s_{k}^{2}-s_{k-1}^{2}\right)E|E^{k}X|^{2}& \leq\sum\nolimits_{k=1}^{n}b_{k}E|E^{k}X|^{2}.
\mathrm{( 4 a )}+ \mathrm{( 4 b )}= \frac{N}{4 8 \pi m}\int_{x}\left[ \mathrm{t r}\left\{\partial_{[ \mu}\tilde{A}_{\nu ]}[ \tilde{A}^{\mu}, \tilde{A}^{\nu}] \right\}+ O ( \partial \frac{\partial^{2}}{m^{2}}) \right] .
B(t+s,u,v,w)=\beta([f_t(f_s(u)),f_t(f_s(v))],f_t(f_s(w)))-\beta([u,v],w) =B(t,f_s(u),f_s(v),f_s(w))+\beta([f_s(u),f_s(v)],f_s(w))-\beta([u,v],w)
\begin{aligned}\frac{(1+t)^{2g}}{(1-t)(1-t)}\,= \, \frac{(1+t)^{2g}}{(1-t)^2}\,=\, (1+t)^{2g}(1-t)^{-2}\,=\, (1+2gt+g(2g-1)t^2+\frac{2g(2g-2)(2g-3)}{3!}t^3+\ldots)(1+2t+3t^2+4t^3+\ldots)\, . \end{aligned}
\zeta_{P{\cal K}}( s ) = \left( \frac{\sqrt{2}}{m}\right)^{2 s}\zeta_{P \bar{\cal K}}( s ) \hspace{1 c m}, \hspace{1 c m}\zeta_{\cal V}( s ) = \left( \frac{\sqrt{2}}{m}\right)^{2 s}\zeta_{\bar{\cal V}}( s ) \, .
u_{0}( \theta^{\prime}) = -{\frac{c_{1}l^{d - 1}}{2^{d}}}( 2 k_{+}+ d ){\frac{1}{k_{+}}}\int_{S^{d}}( \sin^{2}{\frac{\gamma}{2}})^{k_{+}}u_{1}( \theta ) d \mu ( \theta ) ~
d(R, S) = \dim Coker(p\vert_{U}) = \dim W^{\perp}- \dim Im(p\vert_{U}) = \dim W^{\perp}- \dim U +\dim Ker(p\vert_{U}) = = \dim W^{\perp}- \dim U +\dim U\cap W = \dim W^{\perp}-\dim U^{\perp}\cap W = \dim U^{\perp}\cap W^{\perp}= e(S^{\perp}, R^{\perp}).
(S_{n N}F)(z_1,\cdots ,z_N)=\prod_{\mu =1}^{n}\oint_{C}dx_{\mu}F(x_1 , \cdots , x_n | z_1 , \cdots , z_N )\Psi (x_1 , \cdots , x_n | z_1 , \cdots , z_N ).
\mathbb{P}\left\{\left\vert \left(\sum_{i=1}^na_iX_i-\mathbb{M}\sum_{i=1}^na_iX_i\right)-\left(\sum_{i=1}^na_iX_i'-\mathbb{M}\sum_{i=1}^na_iX_i'\right)\right\vert >C_q\left(t\left\vert a \right\vert+e^{t^2/(2q)}\left\vert a \right\vert_q\right)\right\}
Q_\infty[w f, w g] - \lambda \int_\Omega U\, f g\, w^2 dx\, \ \leq Q_\infty[w f, w g] - \lambda \int_\Omega\frac{1}{4 |x|^2}\left( \log \frac{|x|}{\rho}+\beta \right)^{-2}f g\, w^2 dx &\ = \widehat Q_\infty[f,g] .
\overline{G}_{\sigma}^{(n)}(\zeta', x^{2}\zeta_{2n})^{\varepsilon'\,\varepsilon_{2n}}=\sigma\overline{G}_{\sigma}^{(n)}(\zeta_{2n},\zeta')^{\varepsilon_{2n}\,\varepsilon'}\prod_{j=1}^{2n-1}\frac{\zeta_{2n}}{\zeta_j}.
f*(g*h)(G) =\sum_{H\leq G}f(H)(g*h)(G/H) &=\sum_{H\leq G}\sum_{K\leq G/H}f(H)g(K)h(G/H/K) &= \sum_{H\leq G}\sum_{H\leq L\leq G}f(H)g(L/H)h\left(\frac{G/H}{L/H}\right) &= \sum_{H\leq G}\sum_{H\leq L\leq G}f(H)g(L/H)h(G/L)
\begin{array}[c]{rll}\operatorname*{curl}\operatorname*{curl}\mathbf{z}-k^{2}\mathbf{z}=k^{2}\mathbf{v}\Omega, \gamma_{T}\operatorname*{curl}\mathbf{z}+\operatorname*{i}kT_{-k}\mathbf{z}_{T}=-\operatorname*{i}kT_{-k}\left( \mathbf{g}^{\nabla}-\mathbf{h}^{\operatorname*{curl}}\right) \Gamma.\end{array}
|f(x) - p(x)| = |R (x)| \leq \frac{1}{(n-1)!}\cdot \sum_{I \in \underline{d}^n}|(x - x_0)^I| \cdot \int_0^1 (1-t)^{n-1}\cdot B \cdot |t (x - x_0)|^\sigma dt \leq \frac{d^n}{n!}|x-x_0|^n \cdot B \cdot |x - x_0|^\sigma \leq C \cdot B \cdot |x-x_0|^\beta \, ,
c_{a b}\equiv \prod_{\substack{c=1 c\neq a}}^{\mathsf{N}}\frac{\cosh2\zeta_{b}^{(1)}-\cosh 2\zeta_{c}^{(0)}}{\cosh 2\zeta_{a}^{(0)}-\cosh2\zeta_{c}^{(0)}}-\delta_{a b}\frac{\tau (\zeta_{b}^{(0)})}{\mathbf{A}(-\zeta_{b}^{(0)})}\forall a,b\in \{1,...,\mathsf{N}\}
\left[ \left({\frac{H_{1}^{\prime}}{H_{1}}}+{\frac{H_{2}^{\prime}}{H_{2}}}\right) +{\frac{2}{r}}\right] ( h_{1}^{\prime}- h_{2}^{\prime}) = - 2 \left[ \left({\frac{H_{1}^{2}}{H_{1}^{2}}}+{\frac{H_{2}^{2}}{H_{2}^{2}}}\right) -{\frac{2}{r^{2}}}\right] h_{2}- 4 \left({\frac{H_{1}^{2}}{H_{1}^{2}}}-{\frac{H_{2}^{2}}{H_{2}^{2}}}\right) \varphi .
f_w(t) =\int_0^t f'_w(\xi)d\xi= \int_0^t 2g_w(\xi)d\xi \geq\displaystyle 2\inf_{w\in\mathbb{R}^n}g_w(s) t &= \displaystyle 2\inf_{w\in \mathbb{R}^n}g_w(s) |y|.
\left[ \widetilde{I}_{\mu}, I_{\nu}\right] = 2 \widetilde{I}\delta_{\mu \nu}, \left[ I_{\mu}, I_{\nu}\right] = 4 I_{\mu \nu}, \left[ \widetilde{I}_{\mu}, \widetilde{I}_{\nu}\right] = - 4 I_{\mu \nu},
\begin{array}{ll}\frac{u_{yy}}{\tilde{u}_y}= (\frac{u_y}{\tilde{u}_y})_y + \frac{u_y\tilde{u}_{yy}}{(\tilde{u}_y)^2}= (\frac{u}{\tilde{u}_y})_{yy}+ \frac{2 u_y\tilde{u}_{yy}}{(\tilde{u}_y)^2}+ u\Big(\frac{\tilde{u}_{yy}}{(\tilde{u}_y)^2}\Big)_y ,\end{array}
\lim_{L_1\to0}\tfrac{1}{L_1}(L_1-R(L_1,\ell_\nu,2\ell_\mu)-R(L_1,\ell_\nu,2\ell_{\mu'})) &=\frac{e^{-\frac{\ell_{\mu}}{2}}+e^{-\frac{\ell_{\mu'}}{2}}}{\sinh\frac{\ell_\mu}{2}+\sinh\frac{\ell_{\mu'}}{2}}=2(e^{\frac{1}{2}(\ell_\mu+\ell_{\mu'})}-1)^{-1}.
Tr f K(t,x,x)=(4\pi t)^{-3/2}tr (\int_{B^{3}}dx\sqrt{g}\sum_{k=0}^{\infty}\frac{(-t)^{k}}{k!}(fa_{k})+\int_{S^{2}}d\theta \sqrt{\gamma}\sum_{k=0}^{\infty}t^{\frac{k+1`}{2}}c_{\frac{k+1}{2}}(f))
\mathbb{E}\sum_{x \in V_{\tilde{\sigma}}(\mathbf{X}_1)}\tilde{f}(\mathbf{X}_1, x) = \int_0^\infty \dots \int_0^\infty \mathbb{E}\tilde{f}( \tilde{\mathbf{X}}^{z_2,\dots,z_l}, \tilde{x}^{z_2,\dots,z_l})\,dz_2\dots\,dz_l,
e^{- 2 \psi_{2}}= \left| \frac{D_{2}Q_{1}Q_{2}}{\sqrt{c_{1}c_{2}}}\frac{\sinh \left[ \sqrt{c_{2}}( I ( A ) + \tilde{c}_{2}) \right] \sinh \left[ \sqrt{c_{1}}( I ( A ) + \tilde{c}_{1}) \right]}{\sinh^{2}\left[ \sqrt{D_{2}}( I ( A ) + \tilde{c}) \right]}\right|^{1 / 2}e^{- ( f_{0}I ( A ) + f_{1}) / \sqrt{2}},
J = \frac{\partial ( \ldots , \bar{A}_{- 1}^{a}, \bar{A}_{0}^{a}, \bar{A}_{1}^{a}, \ldots )}{\partial ( \ldots , \bar{\varphi}_{- 1}^{a}, a_{1}^{a}, \bar{\varphi}_{1}^{a}, \ldots )}.
\begin{aligned}\|\partial_t^j D_x^l D_v^{m}u\|_{L_p (Q_{1/2})}\le N (d, \delta, R, j, l, m) (\||D_x u| + |D_v u| + \lambda^{1/2}|u|\|_{L_p (Q_{R})}), R \in (1/2, 1],\end{aligned}
G((I-P_h)\widetilde{A}{1\!\!\,{\rm I}}_{[u_k;v_1]}, \ldots,(I-P_h)\widetilde{A}{1\!\!\,{\rm I}}_{[u_k;v_p]})=G\Big(\widetilde{A}{1\!\!\,{\rm I}}_{[u_k;v_1]}, \ldots,\widetilde{A}{1\!\!\,{\rm I}}_{[u_k;v_p]}, \frac{h}{\|h\|}\Big).
\Gamma^{\pm}= \frac{1}{\sqrt{2}}( \Gamma^{0}\pm \Gamma^{3}) \hspace{7 m m}\Gamma^{I}= \left( \begin{array}{c c}{{\mathrm{i}\beta_{I}}}&{{0}}\\{{0}}&{{\mathrm{i}\beta_{I}}}\\ \end{array}\right) ,
\left( \begin{matrix}{{a^{\prime}}}\\{{b^{\prime}}}\\ \end{matrix}\right) = M \left( \begin{matrix}{{a}}\\{{b}}\\ \end{matrix}\right) = \left( \begin{matrix}{{D}}&{{C}}\\{{B}}&{{A}}\\ \end{matrix}\right) \left( \begin{matrix}{{a}}\\{{b}}\\ \end{matrix}\right) ,
u^{1}=\eta^{1}(q^{1},q^{2},q^{3},\omega^{1},\omega^{2},\omega^{3},y_{1,[0,4]}^{1,d},y_{2,[0,4]}^{1,d}) u^{2}=\eta^{2}(q^{1},q^{2},q^{3},\omega^{1},\omega^{2},\omega^{3},y_{1,[0,4]}^{1,d},y_{2,[0,4]}^{1,d})\,.
\begin{aligned}(\chi_{\alpha}\circ \mathrm{Nm}_{K/E_2})|_{E_{1}^{\times}}=\chi_{\alpha}|_{F^{\times}}\circ \mathrm{Nm}_{E_1/F}&=\omega_{E_2/F}(\mathrm{Nm}_{E_1/F})=\omega_{K/E_1},\end{aligned}
\left( \prod_{k=1}^r \prod_{l=1}^{N_k}e^{\mathbf{a}_{k}(l) \varpi_{i_{k,l}}}(b_{k,l}) \right)\left( \prod_{s=1}^n \prod_{j=1}^r e^{d_{j,s}\varpi_s}( \zeta(j,s))^{-1}\right) & = \left(\prod_{j=1}^r \prod_{s=1}^n e^{d_{j,s}\varpi_s}(\zeta(j,s)')^{-1}\right)\left(\prod_{k=1}^r e^{\lambda_k}(b_{k,N_k})\right).
\begin{aligned}\ell\theta_{0,\ell+3}= \left(\frac{1}{2}(\ell+1)\ell-a_1\right)\theta_{0,\ell+2}&+\sum\limits^{\ell+1}_{k=3}\left\{(-1)^{\ell-k}\binom{\ell+1}{k-3}-\left(b_1+2^{\ell-k+2}b_2+3^{\ell-k+2}a_3\right)\binom{\ell+1}{k-1}\right\}\theta_{0,k},\ell\geq1.\end{aligned}
\begin{aligned}\| \{\Phi_{2,t,r_2}(\widehat u_0)(\xi,\eta)\}^{\lor}\|_2 \leq c_{r_2}(1+|t|) (\|u_0\|_2 + \|\,D_x^{s}u_0\|_2 + \|\,D_y^{s}u_0\|_2), \end{aligned}
\begin{cases}\partial_t\tilde{w}_\epsilon + (u^s + \tilde{u}_\epsilon) \partial_x\tilde{w}_\epsilon +{v}_\epsilon (u^s_{yy}+ \partial_y \tilde{w}_{\epsilon})= \partial^2_{y}\tilde{w}_\epsilon + \epsilon \partial^2_{x}\tilde{w}_\epsilon, & \partial_y \tilde{w}_{\epsilon}|_{y=0}=0 &\tilde{w}_\epsilon|_{t=0}=\tilde{w}_{0, \epsilon},\end{cases}
\eta^{b_2}_{b_1b_1}= i\sqrt{2 \pi \left|R^{b_2}_{b_1b_1}\right|}, \eta^{b_1}_{b_1 b_2}= i\sqrt{2 \pi \left|R^{b_1}_{b_1 b_2}\right|}, \eta^{b_1}_{b_2 b_1}= i\sqrt{2 \pi \left|R^{b_1}_{b_2 b_1}\right|}.
I \, m ({\cal R}e s{\cal A}) = \lim_{p^{2}\rightarrow \mu^{2}}{\ - \frac 1{2 a_{1}}( \left| \tau_{\mu \nu}\right|^{2}- \frac 1 2 \left| \tau_{\mu}^{\, \, \, \mu}\right|^{2}) .}
(IMP)_{\epsilon}\,\,\, f^{\epsilon,n+1}_{i,j}+ \frac{1}{\epsilon}\Big[\big(r_x (y_j^{+}-y_j^{-})+r_y(x_i^{+}-x_i^{-}\big)f_{i,j}^{\epsilon,n+1}- r_x (y_j^{+}f^{\epsilon,n+1}_{i-1,j}- y_j^{-}f^{\epsilon,n+1}_{i+1,j}) - r_y(x_i^{+}f^{\epsilon,n+1}_{i,j+1}- x_i^{-}f^{\epsilon,n+1}_{i,j-1})\Big] = f^{\epsilon,n}_{i,j}, \forall (n,i,j) \in Q_h,
\begin{aligned}|N_{G}(h)\backslash V(B\cup T')| = d_{G}(h)-|N_{G}(h)\cap V(B)| &-|N_{G}(h)\cap (V(T')\backslash \{u_{l+1},\cdots,u_{r}\})| &\geq t+1-(t-\lfloor\frac{r-l}{2}\rfloor) &\geq 1+\lfloor\frac{r-l}{2}\rfloor,\end{aligned}
\sum_{i=1}^n\C((x_i,y_i),(y_i,x_i))-C((x_{i+1},y_{i+1}),(y_i,x_i)) = \sum_{i=1}^n\c(x_i,y_i)-c(x_{i+1},y_i)-c(x_i,y_{i+1})+c(x_i,y_i) = \sum_{i=1}^n\c(x_i,y_i)-c(x_{i+1},y_i)-c(x_i,y_{i+1})+c(x_{i+1},y_{i+1}).
+ q^x \cdot \Phi_{\alpha, s}\Phi_{\bar \alpha, rs}) + (q^{-x}\cdot \Phi_{\beta, r}\Phi_{\bar \beta, ss}+ q^x \cdot \Phi_{\bar \beta, s}\Phi_{\beta, rs}) - (q^{-x}\cdot \Phi_{\beta, s}\Phi_{\bar \beta, rs}+ q^x \cdot \Phi_{\bar \beta, r}\Phi_{\beta, ss})]
\left. \frac{d}{dt}F_{\Omega}(\gamma\lbrack t])\right\vert_{t=0}=\int_{\Omega}\frac{1}{2}\sqrt{g[t]}g^{ij}[t]\left. \frac{d}{dt}g_{ij}[t]\right\vert_{t=0}dx & =\frac{1}{2}\int_{\Omega}\sqrt{g}g^{ij}\left( u_{ik}\delta^{kl}\eta_{lj}+\eta_{ik}\delta^{kl}u_{lj}\right) dx & =\int_{\Omega}\sqrt{g}g^{ij}u_{ik}\delta^{kl}\eta_{lj}dx.
Z_N(s,\mathbb{X}_\alpha)(\omega)= \frac{1}{2\pi i}\int_{-\delta-i \infty}^{-\delta+i \infty}\zeta(s+w,\mathbb{X}_\alpha)(\omega) \widehat{\phi}(w) N^w \,dw+ \zeta(s,\mathbb{X}_\alpha)(\omega)
\frac{\mu^{2}}{2^{7}( \pi )^{\frac 5 2}( s - 1 )}\frac{\Gamma ( \frac 3 2 - s )}{\Gamma ( 2 - s )}\int d^{4}k \ \tilde{S}( k ) \tilde{S}( - k ) \left( \frac{k^{2}}{\mu^{2}}\right)^{- s + 1}\, .
\partial_{r}\varphi ={\frac{1}{3 ( r - r_{H})}}[ \eta_{P}e^{3 \varphi}{\bf P}_{i j \infty}+ \eta_{Q}e^{-{3 \varphi}}{\bf Q}_{i k \infty}]^{2}{\frac{1}{\eta_{P}{\bf P}_{i j \infty}+ \eta_{Q}{\bf Q}_{i k \infty}}}.
\sum_{m=0}^{r}\frac{(2d)^m}{m!}\left|\overline{q}\right|\leq \sum_{m=0}^{\infty}\frac{(2d)^m\left(d^{a_1}m^{a_1}+a_2\right)}{m!}=(d^{a_1}b(2d)+a_2)e^{2d}.
A_{\alpha}={\frac{g_{s}^{2}}{M_{s}^{1 0}}}\int_{0}^{\infty}{\frac{d \tau_{2}}{\tau_{2}^{2}}}\int_{R_{\alpha}}d w \ \prod_{i > j}\exp \left({\frac{p_{i}. p_{j}}{M_{s}^{2}}}G ( w_{i}- w_{j}) \right) ,
y+\xi_{\sigma\tau}=\sigma\tau(y)=\sigma(y+\xi_{\tau})=\sigma(y)+\xi_{\tau}=y+\xi_{\sigma}+\xi_{\tau}, \intertext{so that}\theta(\sigma\tau)=\xi_{\sigma\tau}=\xi_{\sigma}+\xi_{\tau}=\theta(\sigma)+\theta(\tau).
T_{\xi_{0}}\circ\cdots\circ T_{\xi_{N-1}}(I)\cap T_{\omega_{0}}\circ\cdots\circ T_{\omega_{N-1}}(I)=\emptyset\mbox{and}& T_{\xi_{0}}\circ\cdots\circ T_{\xi_{N-1}}(I)\cup T_{\omega_{0}}\circ\cdots\circ T_{\omega_{N-1}}(I)\subset J.
I_0(X;Y) =\lim_{\alpha \to 0}\frac{\alpha}{\alpha-1}\log \bigg(\max_{y}\Big[\sum_{x}p(x)p(y|x)^\alpha\Big]^{1/\alpha}\bigg) &=\lim_{\alpha \to 0}\frac{1}{\alpha-1}\log \Big(\max_{y}\big[\sum_{x}p(x)p(y|x)^\alpha\big]\Big) &=-\log(\max_{y}\sum_{p(y|x)>0}p(x)).
V = \int_{0}^{\frac{4 \pi \, C}{n - 2}}d t \int_{C}^{r}s^{n - 1}\, d s \, \, \, \mathrm{V o l ( M )}= \mathrm{V o l ( M )}\, \frac{4 \pi \, C}{n ( n - 2 )}\left( r^{n}- C^{n}\right)
\sum_{\substack{n \leq X}}g_{\mathcal{J}}(n) f(n)n^{-it}= \frac{X^{i(t-t_1)}}{1+ i(t-t_1)}\sum_{n \leq X}g_{\mathcal{J}}(n) f(n)n^{-it_1}+ O\left(\frac{X}{(\log X)^{1/10}}\right).
\partial_{\hat{x}}[(\hat{x}^2 -1)(\hat{x}^2-\alpha^2)\partial_{\hat{x}}\hat{n}(\hat{x})] -2m\zeta \partial_{\hat{x}}\hat{n}(\hat{x}) + 2(\hat{x}^2 - \beta_D^2) \hat{n}(\hat{x}) = 0,
\mathrm{YB}(\alpha, \beta, \gamma) := b_\alpha\,b_\beta\,a_\gamma +b_\alpha\,a_\beta\,b_\gamma+a_\alpha\,b_\beta\,b_\gamma+n\,b_\alpha\,b_\beta\,b_\gamma-\,a_\alpha\,a_\beta\,a_\gamma
(Ds)_{\beta \gamma}= \delta_{\beta,\gamma}(Ds)_{\beta},\quad (Ds)_{\beta}=-i\sum_{\lambda\in\Delta_s,\ \beta\cdot\lambda=1}\left[ g_{s_1}z_{d}(\lambda\cdot q,\xi) +g_{s_2}z_{d}^{(1/2)}(\lambda\cdot q,\xi) \right].
\min_{\pmb{\overline{\theta}}^{'}}\pmb{\mathcal{L}}(\pmb{\overline{\theta}}^{'}) = \min_{\pmb{\overline{\theta}}}[\hat{\mathcal{L}}^{1}(\pmb{\overline{\theta}}), \hat{\mathcal{L}}^{2}(\pmb{\overline{\theta}}), \cdots, \hat{\mathcal{L}}^{M}(\pmb{\overline{\theta}})]^T.
\delta = \underset{k\to \infty}{\limsup}\D_{n_k+1}\leq (\underset{k\to \infty}{\limsup}\ \Psi(d_{n_k})) (\underset{k\to \infty}{\limsup}\D_{n_k}) \leq \underset{d_{n_k}\to d^+}{\limsup}\ \Psi(d_{n_k}) \delta .
\left\{q^{-A}_{(u)},q^-_{(u)B}\right\}=2\delta^A{}_B\delta_{uv}H_{2(u)}+ i\mu\delta_{uv}\left((\gamma^{ij}\Pi){}^A{}_B J^{ij}_{(u)}+(\gamma^{i^\prime j^\prime}\Pi^\prime){}^A{}_B J^{i^\prime j^\prime}_{(u)}-{\tilde\Pi}^A{}_BJ^{LR}_{(u)}\right)\,,
f_2(u) > 0 \ \ \ \u > 0, f_2(0)=0, f_2(u) \to f_2^+ \ \ \ \u \to +\infty; f_1(u) > 0 \ \ \ \u > c_0, f_1(c_0)=0, f_1(u) \to f_1^+ \ \ \ \u \to +\infty,
|\widehat{u}(t_0,\xi)|\geqslant\int_{- \frac{\delta}{2}}^{\frac{\delta}{2}}e^{\xi \psi(r_0 + h)}dh \geqslant \int_{- \frac{\delta}{2}}^{\frac{\delta}{2}}e^{-\xi Ah^2}dh= \int_{- \frac{\delta\sqrt{A}}{2}}^{\frac{\delta\sqrt{A}}{2}}e^{-\xi w^2}dw \geqslant \dfrac{C}{\sqrt{\xi}},