harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\frac{1}{2}}( 1 - \gamma )^{- 3}\left( \sum_{k = 0}^{\infty}( - 1 )^{k}{\frac{( k + 1 ) ( k + 2 )}{2}}\beta^{k}\right) \left( - \sum_{r = 0}^{3}\sigma_{3 , 2 r}\right) = 0 \mathrm{f r o m}W_{\infty}^{5 , D},
\operatorname{minmod}(v_1,\dots,v_n)=\begin{dcases}\max\{v_1,\dots,v_n\}, \textnormal{if}~ v_k<0,~k=1,\dots,n, \min\{v_1,\dots,v_n\}, \textnormal{if}~ v_k>0,~k=1,\dots,n, 0, \textnormal{otherwise}.\end{dcases}
\int{\mathcal{D}_{\hat{g}}}(\bar{\Psi},{\Psi_0})\exp\{-{S_c^{00}}[\bar{\Psi},{\Psi_0},\hat{g}]\}={e^{\mathcal{F}_B^{00}}}{{\left({{Det'{\tilde{K}_D}}\over{\oint{d}\tilde{s}}}\right)}^{-d/2}}\int{d}{\Psi_0}{e^{-{Q_0}{\chi_o}{\Psi_0}/2}}
|D_4(S)| \geq 1 + |D_{\{2,4 \}}(S)| = 1+ |D_{\{4\}}(S) \cup [ D_{\{2 \}}(S) \setminus D_{\{4\}}(S)]| \geq 1 + n-4 +\frac{n}{4}+2 = \frac{5n}{4}-1.
R_{\mu \nu}= 2 \kappa^{2}\lbrack 2 \nabla_{\mu}\phi \nabla_{\nu}\phi + 2 b e^{- 2 a \phi}( F_{\mu \rho}F_{\nu}^{~ \rho}- \frac{1}{4}g_{\mu \nu}F^{2}) \rbrack + \kappa^{2}( T_{\mu \nu}^{m}- \frac{1}{2}g_{\mu \nu}T^{m}) ~ ~ ,
u \star v = \exp \Big ({\frac{\kappa}{2}}\, \omega^{i j}{\frac{\partial \phantom{x^{i}}}{\partial x^{i}}}{\frac{\partial \phantom{x^{j}}}{\partial{\tilde{x}}^{j}}}\Big ) u ({\bf x}) v ({\bf{\tilde{x}}}) \Big \vert_{{\bf x}={\bf{\tilde{x}}}}
\Gamma^t_{xy}(\mathcal{I}_0+\mathcal{J}_{t,y}^0)=(\mathcal{I}_0+\mathcal{J}_{t,x}^0)\Gamma^t_{xy}, \Sigma_x^{st}(\mathcal{I}_0+\mathcal{J}_{t,x}^0)=(\mathcal{I}_0+\mathcal{J}_{s,x}^0)\Sigma_x^{st},
\hat{E}^{01}= (\breve{e}_1, 0),\ \ \hat{E}^{02}= (\breve{e}_2, 0), \ \ \hat{E}^{03}= (\breve{e}_3, 0), \hat{E}^{23}= (0, \breve{e}_1),\ \ \hat{E}^{31}= ( 0, \breve{e}_2), \ \ \hat{E}^{12}= (0, \breve{e}_3),
\begin{array}{lcl}c_{ij}(m) = \textrm{reorder}(m_1,\ldots,m_{i-1},m_{i}+m_{j}, m_{i+1},\ldots,m_{j-1},m_{j+1},\ldots), f_{i\theta}(m) = \textrm{reorder}(m_1,\ldots, m_{i-1},\theta\cdot m_i, m_{i+1},\ldots), \end{array}
\left\{b_{\underline{{n}}\sigma}, b_{\underline{{n}}\sigma}^{\dagger}\right\}= \delta_{\underline{{n}}^{\prime}\underline{{n}}}\delta_{\sigma^{\prime}\sigma}\, , \; \; \left[ a_{\underline{{n}}}, a_{\underline{{n}}}^{\dagger}\right] = \delta_{\underline{{n}}^{\prime}\underline{{n}}}\, .
T_i f_{\ldots,\lambda_i,\lambda_{i+1},\ldots}= T_i^2 f_{\ldots,\lambda_{i+1},\lambda_{i},\ldots}= \left(t+(t-1)T_i \right) f_{\ldots,\lambda_{i+1},\lambda_{i},\ldots}&= t f_{\ldots,\lambda_{i+1},\lambda_{i},\ldots}+(t-1) f_{\ldots,\lambda_{i},\lambda_{i+1},\ldots}.
v_{(-1)}\gamma_g(w)_{(-1)}\otimes gh \otimes v_{(0)}\otimes \gamma_g(w)_{(0)}\otimes gh = v_{(-1)}\alpha_g(w_{(-1)})\otimes gh \otimes v_{(0)}\otimes \gamma_g(w_{(0)}) \otimes gh &= \left((v_{(-1)}\otimes g)(w_{(-1)}\otimes h)\right) \otimes F_2(V,W)(v_{(0)}\otimes g \otimes w_{(0)}\otimes h).
B = \frac{2}{\pi}\int f^{\prime}\sin^2 f~{\rm d}r~\frac{1}{4 \pi}\int \frac{(|R_z|^2 - |R_{{\bar z}}|^2)(1+|z|^2)^2}{(1+|R|^2)^2}\frac{2 i~{\rm d}z{\rm d}{\bar z}}{(1 + |z|^2)^2}.
\frac{\delta}{\delta \omega^{a b}}~ - ~ \,{^*{\hspace{- 1 . 5 p t}}}\, ( e^{c}\wedge{\cal D}e^{d}) \epsilon_{a b c d}{}~ = ~ 0 \qquad \mathrm{a n d}\qquad \frac{\delta}{\delta e^{a}}~ + ~ \,{^*{\hspace{- 1 . 5 p t}}}\, ( e^{b}\wedge{}F^{c d}) \epsilon_{a b c d}~ = ~ 0 \ \ ,
\left|F_n^\ast(k) + F_n^\ast(\ell) - 1 - f^\ast(k)\right| = \left|F_n^\ast(k) - F^\ast(k) + F_n^\ast(\ell) - 1 + F^\ast(k - 1)\right| \le \left|F_n^\ast(k) - F^\ast(k)\right| + \left|F_n^\ast(\ell) - F^\ast(\ell)\right|,
\varphi_{\alpha}( \lambda , \omega ) \sim{\frac{1}{( 4 \pi )^{d / 2}}}\sum_{n = 0}^{\infty}\left[ a_{n}^{\prime}( \omega ) ~{\frac{\lambda^{d / 2 - n + \alpha - 1}}{\Gamma \left( \frac d 2 - n + \alpha \right)}}+ b_{n}^{\prime}( \omega ) ~{\frac{\lambda^{( d - 1 ) / 2 - n + \alpha - 1}}{\Gamma \left({\frac{d - 1}{2}}- n + \alpha \right)}}\right] ,
\{f,g\}=\omega_Q^{T}(X_f,X_g)=\frac{\partial f}{\partial \dot{p}_i}\frac{\partial g}{\partial q^i}-\frac{\partial g}{\partial \dot{p}_i}\frac{\partial f}{\partial q^i}+\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial \dot{q}^i}-\frac{\partial g}{\partial p_i}\frac{\partial f}{\partial \dot{q}^i}.
R_{i\overline{i}k \overline{k}}= \frac{-1}{2}\big( \frac{F_{s_i}}{|p_1|^{2m}}-1 \big)\Big(\frac{\partial^2}{\partial z_k \partial\overline{z}_k}\big( \frac{X^{2m-1}}{F_s^2}\big) \Big)
E_i E_{i+1}^2 = q \sum_{t \lhd r, t \rhd s}w_{i+1}^r w_i^t w_{i+1}^s + q^{-1}\sum_{t \rhd r, t \rhd s}w_{i+1}^r w_i^t w_{i+1}^s &+ q \sum_{t \lhd r, t \lhd s}w_{i+1}^r w_i^t w_{i+1}^s + q^{-1}\sum_{t \rhd r, t \lhd s}w_{i+1}^r w_i^t w_{i+1}^s.
\overline{y}_{1}(t)=\dots=\overline{y}_{p}(t)\ ,\dot{\overline{y}}_{1}(t)=\dots=\dot{\overline{y}}_{p}(t)=\sigma(u^{L},u^{R})\ ,\overline{y}_{1}(r)\leq \dots\leq \overline{y}_{p}(r)
\langle f_0,\beta^R(\lambda)\beta^R(\nu)f_1\rangle =\iint\langle f_0(\omega_0),\beta(x)(\beta^R(\nu)f_1(\omega))\rangle d\lambda^\omega(\omega_0,x)d\omega & =\iiint\langle f_0(\omega_0),\beta(x)\beta(y)f_1(\omega_1))\rangle d\nu_\omega(y,\omega_1)d\lambda^\omega(\omega_0,x)d\omega & =\langle f_0,\beta^R(\lambda\nu)f_1\rangle.
\sum_{k=n^2}^{\infty}\frac{e_k(B_1^n, B_2^n)}{\sqrt{k}}\leq \sum_{k=n^2}^{\infty}\frac{e_k(B_2^n, B_2^n)}{\sqrt{k}}\leq \sum_{k=n^2}^{\infty}\frac{n}{k^{3/2}}\leq 2,
\begin{array}{lcr}D^{++}=u^{+ \alpha}{\partial\over{\partial u^{-\alpha}}};\qquad D^{--}=u^{- \alpha}{\partial\over{\partial u^{+\alpha}}}2D^{++}=[D^0,D^{++}];\qquad -2D^{--}=[D^0,D^{--}] D^0=[D^{++},D^{--}]=u^{+ \alpha}{\partial\over{\partial u^{+\alpha}}}-u^{- \alpha}{\partial\over{\partial u^{-\alpha}}}\end{array}
E\left[\sum_{i=1}^N p^i\tilde{Z}_2^i\right] = \sum_{i=1}^N p^i\sum_{j=1}^N \left(\frac{D_{i,j}}{p^i}\right)Z_2^j = \sum_{j=1}^N \left(\sum_{i=1}^N D_{i,j}\right)Z_2^j = \sum_{j=1}^N q^j Z_2^j,
0= m\langle \tau_{H}, \alpha_{1}\rangle\{(1- (\cot \langle \alpha_{1}, H\rangle )^{2}) & +2(1- (\cot \langle 2\alpha_{1}, H\rangle)^{2}) +9(1- (\cot \langle 3\alpha_{1}, H\rangle)^{2}) \}(2\alpha_{1}+\alpha_{2}).
\prod_{t=0}^{p-1}(\zeta^{tq}x;\zeta)_{e}= \prod_{i=0}^{e-1}\prod_{t=0}^{p-1}(1-\zeta^{tq+i}x) = \prod_{t=0}^{pe-1}(1-\zeta^{tq}x) = (x; \zeta^q)_{pe}.
\|S_{T\Gamma\Lambda U}f\| =\|S_{T\Gamma\Lambda U}f-S_{T\Lambda T}f+S_{T\Lambda T}f\| & \ge\|S_{T\Lambda T}f\|-\|S_{T\Gamma\Lambda U}f-S_{T\Lambda T}f\| &\ge(A-\lambda)\|f\|.
h\left(c(x,\xi),\xi\right) - h\left(c(x,\xi)+\nabla c(x,\xi)(y-x) ,\xi\right) = L_1(\xi)\|\nabla c(x,\xi)(y-x)\|_2 &\leq L_1(\xi)L_3(\xi) \cdot \sqrt{q(\|x\|_2)}\| y-x\|_2 &\leq \sqrt{2}L_1(\xi)L_3(\xi) \cdot \sqrt{D_{\Phi}(y,x)},
-g_{11}\kappa_g=\left(\frac{\partial \Gamma_{12}^2}{\partial s_1}-\frac{\partial \Gamma_{11}^2}{\partial s_2}+\Gamma_{12}^1\Gamma_{11}^2+\Gamma_{12}^2\Gamma_{12}^2-\Gamma_{11}^1\Gamma_{12}^2-\Gamma_{11}^2\Gamma_{22}^2\right),
\left. \begin{array}{l l}{{\partial_{+}q_{1}={\displaystyle \frac{2}{\lambda}}( 1 - q_{1}q_{2}) e^{\phi}, \, \,}}&{{\partial_{+}q_{2}= - 2 q_{1}e^{- 2 \phi}-{\displaystyle \frac{2}{\lambda}}q_{2}^{2}e^{\phi},}}\\{{\partial_{-}q_{1}= - ( q_{2}+ \lambda q_{1}^{2}) + q_{1}\partial_{-}\phi , \, \,}}&{{\partial_{-}q_{2}= \lambda ( 1 - q_{1}q_{2}) - q_{2}\partial_{-}\phi ,}}\\ \end{array}\right.
\Phi^{-1}(0,0) =\Big\{(x,\lambda,v)\ |\v=\nabla g(x)\lambda,\,g(x)={\rm min}(0,g(x)+\lambda)\Big\}=\Big\{(x,0,0)\ |\g(x)<0\big\}\cup\big\{(x,\lambda, v)\ |\g(x)=0,\lambda\geq 0, v=\nabla g(x)\lambda\Big\}.
(t+1)^s\Big(\Vert (\sigma,u)^\ell(t)\Vert_{\dot{B}^{\bar{s}+s\alpha}_{2,1}}+\Vert \sigma^h(t)\Vert_{\dot{B}^{\frac{N}{2}}_{2,1}}+\Vert u^h(t)\Vert_{\dot{B}^{\frac{N}{2}+1-\alpha}_{2,1}}\Big) \le 2^s Z_{s,\bar{s}}(t)\le C.
\left\langle \Bigl [ J [ \xi_{1}^{\prime}] , J [ \xi_{2}^{\prime}] \Bigr ] \right\rangle = \left\langle i J \Bigl [ [ \xi_{1}^{\prime}, \xi_{2}^{\prime}] \Bigr ] + i K [ \xi_{1}, \xi_{2}] \right\rangle ,
-\beta \nu \|x_\nu^{(k)}(t)-x_\nu^{(l)}(t)\|^2+y_\nu^{k,l}(t) \leq \frac 1\nu \Bigg(\frac 4\beta e^{-2O_t^{(k)}}\|f^{(k)}(e^{O_t^{(k)}}x^{(k)}(t))\|^2+\frac 4\beta |O_t^{(k)}|^2\|x^{(k)}(t)\|^2\Bigg) & +\frac 1\nu \Bigg(\frac 4\beta e^{-2O_t^{(l)}}\|f^{(l)}(e^{O_t^{(l)}}x^{(l)}(t))\|^2+\frac 4\beta|O_t^{(l)}|^2\|x^{(l)}(t)\|^2\Bigg).
A_{1,2+p}(r)^2 = \bigg(\sum_{j\ge 1}\sum_{k\ge 1}j^{(p+2)/2}k\sqrt{c_kr^{jk}}\cdot j^{p/2}\sqrt{c_kr^{jk}}\bigg)^2 \le A_{2,3+p}(r)A_{0,1+p}(r).
\langle[A_2]_N({\bf x})\rangle =\frac{1}{\sqrt{2k\omega}}\left\{\frac{1}{2}(1 -\frac{m}{\omega})J_{N+1}(kr)e^{i(N+1)\varphi}- \frac{1}{2}(1 +\frac{m}{\omega})J_{N-1}(kr)e^{i(N-1)\varphi}\right\}.
\aligned\frac{d}{dt}E(Iu(t)) = \langle Iu_{t}, |Iu|^{2k}(Iu) - I(|u|^{2k}u) \rangle = -\langle i \nabla Iu, \nabla (|Iu|^{2k}(Iu) - I(|u|^{2k}u)) \rangle - \langle i I(|u|^{2k}u), (|Iu|^{2k}(Iu) - I(|u|^{2k}u)) \rangle.\endaligned
-{\cal D}_{f}({\cal D}_{f}- 1 ) \Psi = s ( 1 - s ) \Psi = s{\bar{s}}\Psi = E_{n}\Psi \Rightarrow s ={\frac{1}{2}}+ i p_{n}={\frac{1}{2}}+ i \sqrt{E_{n}-{\frac{1}{4}}}.
\begin{cases}\Delta^2v-\tau\Delta v=0,{\rm in}\ \Omega, \frac{\partial^2v}{\partial\nu^2}=0,{\rm on}\ \partial\Omega, \rho\ddot v+\tau\frac{\partial v}{\partial\nu}-{\rm div}_{\partial\Omega}\left(D^2v.\nu\right)-\frac{\partial\Delta v}{\partial\nu}=0,{\rm on}\ \partial\Omega \end{cases}
\pi^b_{0a(n-a)}= \pi^{b+1}_{0a(n-a)}+c_b &=\frac{\sum_{i=b}^{n-b}z_i}{n-1-2b}+\left(\frac{\sum_{i=b}^{n-b}z_i}{n+1-2b}- \frac{\sum_{i=b+1}^{n-b-1}z_i}{n-1-2b}\right) & = \frac{\sum_{i=b}^{n-b}z_i}{n+1-2b},
\mathbf T_{\partial_j}\,(\mathbf R+n)^{-1/2}\,v_\alpha=\sqrt{\alpha_j}\,v_{\alpha-1_j}, \quad(\mathbf R+n)^{1/2}\mathbf T_{z_j}\,v_\alpha=(\alpha_j+1)^{1/2}\,v_{\alpha+1_j}(\mathbf T_{z_j})^* \,v_\alpha= \big(\tfrac{\alpha_j}{\vert \alpha\vert+n}\big)^{1/2}\,v_{\alpha-1_j}\,,
\int_{0}^{1}\left[ - \frac{1}{2 e}\left( \dot{x}^{\mu}- i{\psi}_{a}^{\mu}{\chi}_{a}\right)^{2}+ \frac{i}{2}f_{a b}\left[ \psi_{a \mu}, \psi_{b}^{\mu}\right]_{-}- i \psi_{a \mu}\dot{\psi}_{a}^{\mu}\right] d \tau .
C = \max\limits_{P_{\mathrm{su}_1}\ge 0}\mathcal{E}_{g, u, v}\left\{\log_2\left(1+\frac{P_{\mathrm{su}_1}l^{-\epsilon}|g|^2}{P\left(q^{-\epsilon}|u|^2 + r^{-\epsilon}|v|^2\right)}\right)\right\}
-i\bar{\kappa}_{(\alpha)}\gamma^{\mu}\kappa_{(\beta)}\partial_{\mu}= -i\tilde{\mathcal{C}}_{\alpha\gamma}\Gamma_{s}(T^{I})^{\gamma}{}_{\beta}\Gamma_{\rm Adj}(u^{-1})^{a}{}_{I}e_{a}= -i\tilde{\mathcal{C}}_{\alpha\gamma}\Gamma_{s}(T^{I})^{\gamma}{}_{\beta}k_{(I)}\, ,
\dot{\alpha}_0^2 = \frac{1}{l^2}\left( \frac{f(t/l)'-g(t/l)'}{f(t/l)-g(t/l)}\right)^2 = \alpha_1^2-1/l^2 =\frac{\kappa^2}{3 l}\rho+\frac{\kappa^4 \rho^2}{36}\ .
\begin{cases}s_{i_0}\ge p_{i_0}-q_{i_0}=\bar{p}_{i_0}\ge \bar{p}_{j_1}=s^\sigma_{j_1}\ge 0, i_0\in S^+(\vec x^k), s_{i_0}\le p_{i_0}+q_{i_0}=\bar{p}_{i_0}\le \bar{p}_{j_2}=s^\sigma_{j_2}\le 0, i_0\in S^-(\vec x^k),\end{cases}
W [ \eta ] = \inf_{\phi}: - \frac{N}{2}\mathrm{T r}\ln \left( \gamma \cdot \partial + * \phi * \right) + \int \left( \frac{N}{2 \lambda}\phi^{2}- \frac{1}{2}\bar{\eta}\frac{1}{\gamma \cdot \partial + * \phi *}\eta \right)
\int_{SM}\beta\wedge(r_B)_*(j_B^*p_*q^*dg) =\int_{SM\times N(B)}r_B^*\beta\wedge j_B^*p_*q^*dg &=(-1)^{n}\int_{N(B)\times G}(r_B\circ p)^*\beta\wedge q^*dg.
W[\Phi]={1\over{4\pi}}\int d^2x\:\Big{\{}\Big{(}\partial_+\Phi\partial_-\Phi + 2\,e\, A_+\partial_-\Phi - 2\,e\, A_-\partial_+\Phi\Big{)}+(a_+ + a_- -2)\,e^2\,A_+\,A_-\Big{\}}
\langle |F_k|^2 , P_{h,m}(z) \rangle = \frac{2\pi^2}{(k-1) L(1,^2 f)}\sum_{r \ge 1}\lambda_f(r) \lambda_f(r + m) h \Big ( \frac{k-1}{4\pi(r + m/2)}\Big ) &+ O\Big(\frac{(|m|+CM)^B}{k^{1-\varepsilon}}\Big),
f(\xi) =\int\limits_{\mathbb T^n}\check f(x)\,e(-x\cdot\xi)\,\mathrm dx &=\int\limits_{\mathbb T^n}\check f(x-iy)\,e\bigl(-(x-iy)\cdot\xi)\bigr)\,\mathrm dx &\ll e^{2\pi y\cdot\xi}\int\limits_{\mathbb T^n}\bigl|\check f(x-iy)\bigr|\,\mathrm dx\ll e^{2\pi y\cdot\xi}\,C_\gamma.
\left\|u-u_0\right\|_{L^\frac{2\,p}{p+1}(\Omega)}\le \left|\|u\|_{L^\frac{2\,p}{p+1}(\Omega)}-\|u_0\|_{L^\frac{2\,p}{p+1}(\Omega)}\right| &+c_2\,\left\|\frac{u}{\|u\|_{L^\frac{2\,p}{p+1}(\Omega)}}-\frac{u_0}{\|u_0\|_{L^\frac{2\,p}{p+1}(\Omega)}}\right\|_{L^\frac{2\,p}{p+1}(\Omega)}.
\prod_{i=1}^k |V_i\setminus X_i|-\frac{m^k}{4}\geq m^k\left(\left(1-\frac{1}{2k^2}\right)^k-\frac{1}{4}\right)\geq m^k\left(1-\frac{1}{2k}-\frac{1}{4}\right)\geq \frac{m^k}{2}
\mathbf{e}_{1}=\gamma^{\prime}(x)=(1,y^{^{\prime}}(x),z^{^{\prime}}(x)), \mathbf{e}_{2}=\frac{1}{\kappa (x)}\gamma^{^{\prime \prime}}(x)=\frac{1}{\kappa (x)}(0,y^{^{\prime \prime}}(x),z^{^{\prime \prime}}(x)), \mathbf{e}_{3}=\frac{1}{\kappa (x)}(0,\epsilon z^{^{\prime \prime}}(x),\epsilon y^{^{\prime \prime}}(x)),
\Gamma(U)\left(W(\eta)\right)\star \Gamma(U)\left(W(\eta')\right) =W(U(\eta))\star W(U(\eta')) &= \exp\left(-\frac{1}{2}i\Omega(U(\eta),U(\eta'))\right)W(U(\eta)+U(\eta')) &= \exp\left(-\frac{1}{2}i\Omega(\eta,\eta')\right)W(U(\eta+\eta')) &= \Gamma(U)\left(W(\eta)\star W(\eta')\right).
\sum_{i=1}^{n}\big(\deg_{\widetilde{X}}(v_{i})-2\big) =\sum_{i=1}^{n}\deg_{X}(v_{i})-2n\leq N\cdot \deg_{Y}(a)\cdot \deg_{Z}(b)-2N \cdot \deg_{Y}(a) = N \deg_{Y}(a)\cdot\big(\deg_{Z}(b)-2\big) = N\big(\deg_{\widetilde{Y}}(a) -2\big)\big(\deg_{\widetilde{Z}}(b)-2\big).
\left( \partial_{t}-{\frac{l}{2 \pi \alpha^{\prime}}}\partial_{\theta}\right)^{2}X_{4}- \left({\frac{l \rho}{2 \pi \alpha^{\prime}}}\right)^{2}\left({\frac{1}{\rho^{2}}}\partial_{\theta}^{2}+ \partial_{z}^{2}\right) X_{4}+ O ( l^{3}) = 0 ,
{\cal J}_{i}^{0}\equiv - \frac \partial{\partial x}\Phi^{i}- 2 \varepsilon^{i j k}\Phi^{j}\Pi_{k}= - \sqrt{1 -{\varphi}^{2}}\varpi_{i}- \varepsilon^{i j k}\varphi^{j}\varpi_{k}\equiv J_{i}^{0}.
\lambda_{k, l}(a_1; q)_k (b_1; q)_{l+1}\frac{(q; q)_{k+l}}{(q; q)_{k-1}(q; q)_{l}}=\lambda_{k-1, l+1}(a_1; q)_{k}(b_1; q)_{l+1}\frac{(q; q)_{k+l}}{(q; q)_{k-1}(q; q)_{l}}.
\overrightarrow{W}^{R}= \frac{\vec{V}^{R}+ \vec{n}[ ( \vec{n}\vec{u}) ( \operatorname{c o s h}\alpha - 1 ) - c \operatorname{s i n h}\alpha ]}{\operatorname{c o s h}\alpha [ 1 - \frac{1}{c}( \vec{n}\vec{u}) \operatorname{t a n h}\alpha ]}\, .
\beta(V_0) V_0 +\beta(V_1) V_1 = \beta(V) V+V_0(\beta(V_0)-\beta(V))+ V_1(\beta(V_1)-\beta(V)) - \beta(V)V_{lo}\ge \beta(V) V + V_0(\beta(V/2) - \beta(V)) - V_{lo}
\lambda = \frac{\mathrm{d}x}{y}={R}\frac{\mathrm{d}a}{\mathrm{d}u}{\cal W}={R}\left( \frac{\mathrm{d}a}{\mathrm{d}u}\mathrm{d}B - \frac{\mathrm{d}a_{D}}{\mathrm{d}u}\mathrm{d}A \right) \ ,
\begin{array}{ll}D + \alpha \tilde D + \frac{e}{2}E_I |\sigma_I|^2 + k =0, \tilde D + \alpha D + \frac{q}{2}Q_i |\phi_i|^2 + \eta =0, \bar G_i =- \frac{\partial W}{\partial \bar \phi_i}= -m \phi_i, \bar F_I = - \frac{\partial W}{\partial \bar \sigma_I}= -\tilde m \sigma_I.\end{array}
P'_{j}= \sum\limits_{t=1}^{p}\frac{\mathcal{D}_{C_{i_{t},:}}(j)}{p}= \frac{1}{\|C\|_F^2}\sum\limits_{t=1}^{p}\frac{|C_{i_t,j}|^2}{p\frac{\|C_{i_t,:}\|^2_{2}}{\|C\|_F^2}}= \frac{1}{\|C\|_F^2}\sum\limits_{t=1}^{p}\frac{|C_{i_t,j}|^2}{pP_{i_t}}= \frac{\|S_{:,j}\|^2_{2}}{\|S\|_F^2}.
\frac{1}{n}\max_{x_2^n}C(\mathcal G_{x_2^n}) \geq \log\max_{x_2^n}\mathsf{LP}(\mathcal G_{x_2^n})^{1/n}&= \log\max_{x_2^n}\prod_{i=1}^{n}\mathsf{LP}(\mathcal G_{x_{2i}})^{1/n}&= \log \max_{x_2}\mathsf{LP}(\mathcal G_{x_{2}}) &=\max_{x_2}\log\mathsf{LP}(\mathcal G_{x_{2}}) &=\max_{x_2}\bar{C}(\mathcal G_{x_{2}}).
[Q_P(\theta_1,\tilde\theta_1)]_{j,k}=E_P\Big[\frac{m_j(X_i,\theta_1)}{\sigma_{P,j}(\theta_1)}\frac{m_k(X_i,\tilde \theta_1)}{\sigma_{P,k}(\tilde \theta_1)}\Big]-E_P\Big[\frac{m_j(X_i,\theta_1)}{\sigma_{P,j}(\theta_1)}\Big]E_P\Big[\frac{m_k(X_i,\tilde\theta_1)}{\sigma_{P,k}(\tilde \theta_1)}\Big].
(8+\epsilon) \sum_{d<x^{1/2+\epsilon}}\frac{\mu^2(d)}{\phi(d)}\frac{\pi(x)}{\phi(d)}< (8+\epsilon) \pi(x) \sum_{d \geq 1}\frac{\mu^2(d)}{\phi(d)^2}= (8+\epsilon) \pi(x) \prod_p (1+1/(p-1)^2) < 23 \pi(x).
|X_{t_i}- X_{\underline{t_i}}-\widetilde X_{t_i}+\widetilde X_{\underline{t_i}}| =\Bigl|\int_{\underline{t_i}}^{t_i}(\mu(X_s)-\mu(\widetilde X_s))\, ds\Bigr|\leq \frac{4 \|\mu\|_{\infty}}{n}.
\Delta = 1 -{\frac{a^{4}\cos^{2}\theta}{u^{4}}}\ , \tilde{\Delta}= 1 -{\frac{a^{4}}{u^{4}}}\ , A \equiv{\frac{u_{0}^{4}}{u_{H}^{4}- \frac{1}{3}a^{4}}}\ , u_{H}^{6}- a^{4}u_{H}^{2}- u_{0}^{6}= 0 \ .
p_{\mu}= q^{- 3 / 2}\Bigl ( \frac{\partial L_{0}^{m}}{\partial \dot{x}^{\mu}}\Bigr ) \equiv ( \dot{x}_{\mu}+ q^{1 / 2}i \chi \psi_{\mu}) [ ( \dot{x}+ q^{1 / 2}i \chi \psi )^{2}]^{- 1 / 2}m ,
0 =\ddot{\rho}+\frac{\dot{\rho}^{2}}{\rho}-\frac{1}{\rho}(\frac{1}{\rho}U^{\ast}+2h) 0 =\ddot{\gamma}^{\ast}+P\dot{\gamma}^{\ast}+Q\mathbf{\nu}^{\ast}+R\nabla U^{\ast}
|\tilde v(t)|^2 = |h(0)|^2+ 2\sum_{i=1}^m \int_{(0,t]}\langle v_i^{\ast}(s), v(s) \rangle\, dA(s) & + 2 \int_{(0,t]}(\tilde v(s-), dh(s)) - \int_{(0,t]}\left|\sum_{i=1}^m v_i^{\ast}(s)\right|^2 \Delta A(s)\,dA(s)+[h]_t
c_2|_{\mu=0}=3\nu^7+1330\nu^5+65177\nu^2+93100\nu+(120544-123\nu^6-1918\nu^4-28897\nu^3) >0, c_2|_{\mu=1}=(1-\nu)(161652-147\nu^6+672\nu^5+3842\nu^4+16060\nu^3+(264652-195259\nu)\nu)\ge 0,
\frac{\partial C(w_{n})}{\partial w_{n}(k)}=\sum_{i=0}^{n}2\lambda^{n-i}e(i)\cdot\frac{\partial e(i)}{\partial w_{n}(k)}=-\sum_{i=0}^{n}2\lambda^{n-i}e(i)x(i-k)=0k=0,1,...,p
T_\ell\chi_{\mu,\nu}(g_1)=m_\mathcal{G}(X^\ell(g_1)\cap X_{\mu,\nu})(\chi_{\mu,\nu}(g_1)-1) &+m_\mathcal{G}(X^\ell(g_1)\setminus (X^\ell(g_1)\cap X_{\mu,\nu}))\chi_{\mu,\nu}(g_1).
S_{a_{1},b_{1}}^{a_{2},b_{2},q_{1}+q_{2}}\left( p_{2},l\right)=\sum_{m=0}^{p_{2}+q_{2}}S_{a_{1},b_{1}}^{a_{2},b_{2},q_{2}}\left(p_{2},m\right) S_{a_{2},a_{2}m+b_{2}}\left( q_{1},l-m\right) .
\rho\notin \Gamma(\alpha_1,\ldots,\alpha_n):=\left\{8\pi k_0 + 8\pi \sum_{i=1}^m k_i (1+\alpha_i)\;:\; k_0\in \mathbb{N}, k_i \in \{0,1\},\sum_{i=0}^m k_i>0\right\}.
\begin{aligned}\int_{\Omega}\left|\left|u_n\right|^{p-2}u_n-|u|^{p-2}u\right|^{p'}dx \leq C\int_{\Omega}\left|u_n- u\right|^{p'}\left(\left|u_n\right|+|u|\right)^{p'(p-2)}dx \leq C \left\|u_n- u\right\|_{p}^{p'}\left(\left\|u_n\right\|_{p}+\|u\|_{p}\right)^{p'(p-2)}. \end{aligned}
- 2\sqrt{\frac{2x}{n}}e^{v^2\sigma^2/2}- \frac{v^2M_{2,1/v}^{\theta}}{2n}- \frac{6N_{1/v}^{\theta}}{n}\leq \log\big(\sum_{i=1}^n \frac{\cos(vY_i)}{n}\big) + \frac{v^2\sigma^2}{2}\leq 2\sqrt{\frac{2x}{n}}e^{v^2\sigma^2/2}\ .
A=u_1\prod_{l\ne 1}^n a_{1l}a_{1l}^+\,,B=\tau^{2n-2}\tau_n-A-\sum_{i\ne 1}(C_i+D_i)\,, C_i=u_ib_{1i}a_{1i}^+\prod_{l\ne 1, i}a_{il}a_{il}^+\,,D_i=u_i^-b_{1i}^+a_{1i}\prod_{l\ne 1, i}a_{li}^-a_{li}\,.
G(\varphi(x+u)-\varphi(x))=\sum_{\beta\in\mathbb{N}^{n}}\frac{D^{\beta}G(0)}{\beta!}\left(\sum_{\alpha\in\mathbb{N}^{n}\setminus \{0\}}\frac{D^{\alpha}\varphi(x)}{\alpha!}u^{\alpha}\right)^{\beta}=\sum_{\alpha\in\mathbb{N}^{n}}\frac{H_{\alpha}(x)}{\alpha!}u^{\alpha}.
u_t= Au_{b+4}+Bu_{b+3}u_{b+1}+Cu_{b+2}^2+Eu_{b+2}u_{b+1}^2 +Gu_{b+1}^4 &+ Hu_{b+3}+Iu_{b+2}u_{b+1}+Ju_{b+1}^3+Ku_{b+2}+Lu_{b+1}^2 &+Mu_{b+1}+N.
\mathcal{S}_{n}(z) =2\int_{0}^{z}\mathcal{C}_{n}(\zeta)d\zeta, \mathcal{C}_{n+1}(z) =-2\int_{0}^{z}\mathcal{S}_{n}(\zeta)d\zeta-\frac{(2^{2n+1}-1)}{(2n+2)!}|B_{2n+2}|,
E_{a,b}^{(M)}E_{d,c}^{(N)}=\begin{cases}(-1)^{\bar E_{a,b}\bar E_{d,c}}E_{d,c}E_{a,b}-(-1)^{\bar E_{a,b}\bar E_{d,c}}K_{a,b}E_{d,b}, M=N=1; \displaystyle \sum_{t=0}^{\min(M,N)}(-1)^tq_{b}^{-t(M-1-t)}E_{d,b}^{(t)}E_{d,c}^{(N-t)}K_{a,b}^{t}E_{a,b}^{(M-t)}, .\end{cases}
E_n(f) =\{(x,y)\in G^{SC}_n\times G^{SC}_n \mid |x-y|=3^{-n}\}, R^{SC}_n =R_{(G^{SC}_n,E^{SC}_n)}(G^{SC}_n\cap I_l, G^{SC}_n\cap I_r), R^{SC}_{n,pt}=R_{(G^{SC}_n,E_n^{SC})}(p_1,p_5).
X_{n k \pm}^{( 1 ) l}( t ) = \mp v_{k - , n -}^{-}E_{\pm \omega_{l k}^{( 0 )}+ \Omega + \omega_{l n}^{( 0 )}}^{\mp \omega_{l k}^{( 0 )}}\mp v_{k - , n -}^{+}E_{\pm \omega_{l k}^{( 0 )}- \Omega + \omega_{l n}^{( 0 )}}^{\mp \omega_{l k}^{( 0 )}}
\sum_{k = 1}^\infty{\frac{{\zeta (2k)}}{{k(k + 1)}}}= - \frac{1}{2}+ \ln (2\pi), \sum_{k = 1}^\infty{\frac{\zeta (2k)}{k(k + 2)}}= - \frac{1}{8}+ \frac{\ln (2\pi )}{2}+ \frac{3\,\zeta (3)}{2\pi^2},
B_{jr}=\begin{cases}\frac{f(\|Q_{jr}-Q_{2r}\|)-f(\|Q_{1r}-Q_{jr}\|)}{\|Q_{jr}-Q_{2r}\|-\|Q_{1r}-Q_{jr}\|}\cdot\frac{\|Q_{jr}-Q_{2r}\|-\|Q_{1r}-Q_{jr}\|}{\|Q_{1r}-Q_{2r}\|}\textrm{if}\|Q_{jr}-Q_{2r}\|\neq\|Q_{1r}-Q_{jr}\| 0\textrm{if}\|Q_{jr}-Q_{2r}\|=\|Q_{1r}-Q_{jr}\| \end{cases}
K^{\mathrm{top}}_0(\mathbb{S}^{\mathrm{gr}}_A)\cong \bigotimes_{i=1}^k K^{\mathrm{top}}_0\left(\mathbb{S}^{\mathrm{gr}}(d_i)_{w_i}\right), \K^{\mathrm{top}}_1(\mathbb{S}^{\mathrm{gr}}_A)=0.
g^*= \begin{cases}g_1 \p>\frac{3\beta+1}{7+5\beta}g_2 \p<\frac{3\beta+1}{7+5\beta}\big\{(q,1-q): 0\leq q\leq 1\big\}\p=\frac{3\beta+1}{7+5\beta}.\end{cases}
s = \frac{\alpha^{\prime}}{2}( p D \tilde{p}) = \frac{\alpha^{\prime}}{4}( p + D \tilde{p})^{2}= \frac{\alpha^{\prime}}{4}( V ( p + \tilde{p}) )^{2}= \alpha^{\prime}k_{0}^{2}g^{0 0}.
(r-d)\binom{r-1}{2}- \left( \binom{d}{2}+ (r-1-d)\left(\binom{r}{2}- k\right)\right) &= -\frac12\left((r-d)(r-1)(2)+d^2-d-r^2+r-2kr+2dk+2k\right) &= -\frac12\left(d^2 - 2dr + (1+2k)d + r^2 - (2k+1)r + 2k\right) &= -\frac12\left(d -(r-1)\right)\left(d-(r-2k)\right) &\ge 0,
\Omega_{\hat{g}_1}^1(X_1) = \left\{\ \hat{a}_1 \ \left| \begin{array}{l}\hat{a}_1 \in \sqrt{-1}\ker ( \hat{d}^* : \Omega^1(X_1) \rightarrow \Omega^0(X_1) ), \hat{d}^*( i^* \hat{a}_1 ) = 0, \int_{Y_j}\hat{a}_1(\nu) = 0 \ ( j = 1, \dots, r) \end{array}\right. \right\}.
S_{\chi}\, = \, \int_{t^{\prime}}^{t^{\prime \prime}}\, d t \left( p_{i}\dot{q}^{i}+ \mu_{a}\dot{\lambda}^{a}+ i \zeta_{a}\dot{\eta}^{a}+ i \zeta^{\times}{\!}_{a}\dot{\eta}^{\times a}\, - H \, - i \{\Omega , \phi \}\right) .
W(\psi_+ (z, \cdot \,), \psi_- (z, \cdot \,)) = [m_-(z,x_0) - m_+(z,x_0)] \psi_+ (z, x_0) \psi_- (z, x_0), m_{\pm}(z,x_0) = [p(x_0) \psi_{\pm}^{\prime}(z, x_0)] / \psi_{\pm}(z, x_0).
\omega_0(x)=2\int_0^\infty{x\rho(\lambda)d\lambda\over{x^2-\lambda^2}}=\int_0^\infty\rho(\lambda)d\lambda\left({1\over{x-\lambda}}+{1\over{x+\lambda}}\right)=\int_{-\infty}^\infty{\rho(\lambda)d\lambda\over{x-\lambda}},
\Phi_{\varepsilon,\delta}(t,x,s,y)=u^{\varepsilon,\delta}(t,x)-v_{\varepsilon,\delta}(s,y)-\varphi_\delta(t,x,y)\quad &\varphi_\delta(t,x,s,y)=\varphi(t,x,y)+|t-\bar{t}|^2+|x-\bar{x}|^4+|y-\bar{y}|^4+\frac{|t-s|^2}{\delta}.
E(t)\stackrel{\rm{def}}{=}\frac{1}{2}\int_{\mathbb{R}^d}|u|^2(t,x)\,dx+\int_{\mathbb{R}^d}\frac{L\lambda}{2}|\nabla Q|^2(t,x)+\lambda(\frac{a}{2}\textrm{tr}(Q^2(t,x))-\frac{b}{3}\textrm{tr}(Q^3(t,x))+\frac{c}{4}\textrm{tr}^2(Q^2(t,x)))\,dx
A_{63}(t)= \, -2 (26009437 + 18059772 t + 5057576 t^2 + 813612 t^3 + 89370 t^4 & + 7860 t^5 + 608 t^6 + 36 t^7 + t^8), B_{63}(t)= \, (5 + t)^3 (11 + t)^3 (79 + 8 t + t^2)^3 (8737 + 1248 t - 2 t^2 + t^4).
N(k) =\frac{q^2-1}{q^k-1}\frac{1}{q^{k-2}}q^{2(k-2)}q^2(q-1)\times &\times\frac{(q^k-1)(q^{k-1}-1)}{(q^2-1)(q-1)}N(k-2)= &=q^k(q^{k-1}-1)N(k-2).
S_{\mathrm{G F}}\equiv - \int d^{4}x i \mathrm{\boldmath \delta}_{\mathrm{B}}\biggl \{\bar{C}^{A}\biggl [ \partial^{\mu}A_{\mu}^{A}+ \frac \alpha 2 B^{A}- \frac \zeta 2 \epsilon^{A B C}\bar{C}^{B}C^{C}\biggr ] \biggr \}.
[\Gamma(e_1),\Gamma(e_2)]= [A(e_1),A(e_2)] + 2 \beta(e_2)A(e_1)\omega -2 \beta(e_1)A(e_2)\omega = 2(\det A)\omega-2\beta(e_2)J(A(e_1))+2\beta(e_1)J(A(e_2)).
\Delta S_{b}^{\prime}= \int d^{D}x ~ h \Phi_{b}^{\dagger}\left( T + \frac{1}{2}\stackrel{\sim}{V}+ \frac{1}{2}V + \frac{1}{2}b \Phi_{b}^{\dagger}\Phi_{b}- \mu \right) \Psi_{s}+ c o n j .
\left(\left\lfloor \frac{n}{ep^l}\right\rfloor - \left\lfloor \frac{n}{ep^{l+1}}\right\rfloor\right) = a_l + a_{l+1}p + \dots + a_r p^{r-l}- (a_{l+1}+ a_{l+2}p + \dots + a_r p^{r-l-1}) = (a_l - a_{l+1}) + (a_{l+1}- a_{l+2})p + \dots (a_{r-1}- a_r)p^{r-l-1}+ a_rp^{r-l}.
\||D|^{-m_1}x|D|^{-m_2}e^{-s^2D^2}\|_1 \leq \Big(\sum_{l=0}^{\infty}2^{l(p-m_1-m_2)}\cdot e^{-2^{2l}s^2}\Big)\cdot O(1) = O(s^{m_1+m_2-p}).
\mathbf{R}^{(g)}_{code}=\sum_{\left\{m \; \vert \beta_m > 0\right\}}\beta_m \boldsymbol{\phi}_m \boldsymbol{\phi}_m^H \mathbf{SNR}^{total,(g)}_{mimo}=\boldsymbol{\Gamma}\operatorname{diag}\left[\left\{\lambda_n\right\}_{n=1}^{D}\right]\boldsymbol{\Gamma}^{-1}
0\,\longrightarrow\,{\mathcal K}\, \stackrel{h}{\longrightarrow}\,{\mathcal K}(H_1) \, \longrightarrow\, ({\mathcal K}(H_1)_{x_1}/H^1)\oplus \bigoplus_{j=2}^{d-1}{\mathcal K}(H_1)_{x_j}/\widetilde{H}_j\,\longrightarrow\, 0\, .
_{m}=\frac{1}{(\kappa^{2}_{R}m+N_{0}/p)}&\times \mathbf{h}_{m}^{\mathcal{H}}\left(\mathbf{h}_{m}\mathbf{h}_{m}^{\mathcal{H}}\frac{(\kappa^{2}_{T}(\omega+1)+\omega)}{(\kappa^{2}_{R}m+N_{0}/p)}+\mathbf{I}_{n}\right)^{-1}\mathbf{h}_{m},
W^{(1)}=\frac{1}{2!}\int_{1\ldots 6}V_{123}V_{456}(-2)\frac{\delta W_{A}^{(0)}}{\delta D^{-1}_{36}}\left(\frac{\delta^2 W_{\psi}^{(0)}}{\delta S^{-1}_{12}\delta S^{-1}_{45}}+\frac{\delta W_{\psi}^{(0)}}{\delta S^{-1}_{12}}\frac{\delta W_{\psi}^{(0)}}{\delta S^{-1}_{45}}\right).