harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left( \sum_{m=0}^{r}p_{r, m}x^{m}\right) \sum_{m \geq 0}\frac{a_{m}}{x^{m+1}}=\sum_{m \geq 0}\frac{\begin{displaystyle}\sum\nolimits_{k=0}^{r}\end{displaystyle}\p_{r, k}a_{m+k}}{x^{m+1}}+\sum_{m=0}^{r-1}x^{m}\sum_{k=m}^{r-1}p_{r, k+1}a_{k- m}\ ,
v_{m_ii}(x)s_i(x) + v_{(m_i-1)i}(x)r_i(x) + 2\ell_iv_{m_ii}'(x)r_i(x) = (-1)^{m_i}r_i(x)(m_iv_{m_ii}'(x)^*-v_{(m_i-1)i}(x)^*) + (-1)^{m_i}s_i(x)v_{m_ii}(x)^* + (-1)^{m_i}m_ir_i'(x)v_{m_ii}(x)^*.
\sqrt{-h}e^{-2\varphi}h^{\bar{\mu}\bar{\mu}^{\prime}}h^{\bar{\nu}\bar{\nu}^{\prime}}(ML)_{ij}{\cal F}^j_{\bar{\mu}^{\prime}\bar{\nu}^{\prime}}=\epsilon^{\bar{\mu}\bar{\nu}\bar{\rho}}\partial_{\bar{\rho}}\psi^{i},
\begin{array}{ll}u^1 -u^2 = -\partial_y\tilde{u}\int\limits_y^{\infty}w(t,x,\tilde{y}) \,\mathrm{d}\tilde{y}, begin{align*}8pt]\partial_y^m u^1|_{y=0}- \partial_y^m u^2|_{y=0}= -\sum\limits_{m_1+m_2=m}\partial_y^{m_1+1}\tilde{u}|_{y=0}\int\limits_0^{\infty}\partial_y^{m_2}w(t,x,\tilde{y}) \,\mathrm{d}\tilde{y},\end{array}
\begin{aligned}|\langle \lambda^{k+1}, \zeta^* - \zeta^{k+1}\rangle + \langle D_u\theta(u^{k+1}), u^* - u^{k+1}\rangle_\mathcal{U}| & \leq \beta\sum\limits_{i = n}^{k+1}\|r^{i}\|^2 + |\langle \lambda^n, \zeta^* - \zeta^{k+1}\rangle + \langle D_u\theta(u^n), u^* - u^{k+1}\rangle_\mathcal{U}|.\end{aligned}
d_{\; \; \alpha_{0}}^{\alpha_{1}}= \left( \delta_{\; \; \beta_{1}}^{\alpha_{1}}+ m_{\; \; \; \beta_{1}}^{\alpha_{1}\beta_{0}}\chi_{\beta_{0}}\right) \bar{D}_{\; \; \gamma_{1}}^{\beta_{1}}A_{\alpha_{0}}^{\; \; \gamma_{1}},
\lim_{x\downarrow 0}\frac{(w_p-v_q)(x)}{W_q(x)}=\frac{(w_p-v_q)'(0)}{W_q'(0+)}=\frac{\sigma^2}{2}(w_p-v_q)'(0), \lim_{x\uparrow 0}\frac{(w_p-v_q)(x)}{|x|}= -(w_p-v_q)'(0).
\mathcal{W}_{\frac{p}{2}}(g\circ W_{n},g\circ W) =\mathcal{W}_{\frac{p}{2}}(g\circ W_{n}\circ \pi_\Delta,W) &\le \mathcal{W}_{\frac{p}{2}}(g\circ W_{n}\circ \pi_\Delta,\sigma X_n)+\mathcal{W}_{\frac{p}{2}}(\sigma X_n ,\sigma B) &\le Cn^{-\frac{1}{4}+\frac{1}{4(p-1)}}+Cn^{-\frac{1}{4}+\delta}\le Cn^{-\frac{1}{4}+\frac{1}{4(p-1)}},
{\cal L}_{\mathrm{s c a l}}^{( 2 )}[ B ] ={\frac{D - 1}{D}}\times \mathrm{e q .}( \ref{G a m m a 2 s c a l}) +{\frac{1}{D}}\times \mathrm{e q .}( \ref{G a m m a 2 s c a l p I})
\vec{p}_{a}+ \vec{p}_{b}-{\tilde{L}}_{p}( E_{a}\vec{p}_{b}+ E_{b}\vec{p}_{a}) / c - \vec{p}_{c}- \vec{p}_{d}+{\tilde{L}}_{p}( E_{c}\vec{p}_{d}+ E_{d}\vec{p}_{c}) / c \simeq 0 .
I(n,m):=\int_{0}^{1}\frac{artanh^{n}x}{x^{m}}dx=\int_{0}^{1}log^{n/2}(\frac{1+x}{1-x})\frac{dx}{x^{m}}=\int_{0}^{\infty}z^{n}\frac{coth^{m-2}z}{sinh^{2}z}dz.
\mathcal{F}_{D,m}(z,s) := B(s)\Gamma\biggl(\frac{s+1}{2}\biggr)\sum_{n|m}\biggl(\frac{D}{n}\biggr)P_{\frac{1}{2}, \frac{m^2D}{n^2}}\biggl(z,\frac{s}{2}+\frac{1}{4}\biggr),
\begin{array}{lcl}{\cal C}_{con}\equiv \{864 \psi^6+ \phi = \pm 1 \},{\cal C}_{1}\equiv \{\phi = \pm 1 \},{\cal C}_{\infty}\equiv \{\phi, \psi = \infty \},{\cal C}_{0}\equiv \{\psi = 0 \}. \end{array}
\left\langle\; \mathcal{C}^{(g)}(p)\; \right\rangle = \left. \frac{\mathcal{C}^{(g)}\; \left( \frac{1}{i}\, \frac{\delta}{\delta J}\right) \; W_0 (J)}{W_0 (0)}\; \right\vert_{J=0}= \; \bar{p}_3
\left.{\frac{1 6 \, \pi^{2}}{g_{i , \, \mathrm{e f f}}^{2}}}\right\vert_{\mathrm{\f i e l d \t h e o r y}}^{\ \overline{{D R}}}= \frac{1 6 \, \pi^{2}}{g_{i}^{2}( \mu )}+ b_{i}\, ( 2 \gamma + 2 ) \, ;
\sum_{n=0}^{\infty}\frac{\left(b;q\right)_{n}S_{n}\left(x;q\right)z^{n}}{\left(c;q\right)_{n}}=\frac{\left(b;q\right)_{\infty}}{\left(c,z;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(c/b,z;q\right)_{n}b^{n}}{\left(q;q\right)_{n}}A_{q}\left(xzq^{n}\right).
x_u(z) = \frac{1}{N}\sum\limits_{n=1}^N{\frac{{{q}{{\left|{{\lambda_{u,n}}}\right|}^2}}}{{\frac{1}{N}\sum\nolimits_{i=1}^U{\frac{{{q}}}{{1 +{x_i(z)}}}{{\left|{{\lambda_{i,n}}}\right|}^2}}+ \sigma_n^2 -{z}}}}.
\left({{d\tilde{x}}\over{d\tilde{\lambda}}}\right)^2+\left[\tilde{H}^{-{4\over{\Delta_{p+1}}}}_{p+1}-{\tilde{\cal J}^2\over{\tilde{H}^{{8(p+2)}\over{(D-2)\Delta_{p+1}}}_{p+1}\tilde{x}^2}}\right]\tilde{f}-\tilde{E}^2\tilde{H}^{{4(D-2p-6)}\over{(D-2)\Delta_{p+1}}}_{p+1}=0,
\sum_{l}\lambda_{i j l}= 0 \quad ( i < j ) \ , \ \sum_{\tiny l = 1 , 3 , 5}\mu_{i l}= 0 \quad ( i \neq 1 ) \ , \ \sum_{\tiny l = 1 , 3 , 5 , 7}\mu_{1 l}= 0 \quad .
{\Gamma_{\theta}^{i j}}^{- 1}( \nu ) = \frac{\nu \sigma}{\omega^{2}}\left( - \delta_{i j}\cot \left( \pi \frac{\nu}{\omega}\right) + \left( 1 - \delta_{i j}\right) \csc \left( \pi \frac{\nu}{\omega}\right) \right) ,
T^{\mathcal S}_\rho:=(\pi^{\sharp,\mathcal S}_{\rho})^{-1}\circ({\mathcal P}^{\mathcal S}_\rho)^{-1}\circ{\mathcal J}^{\mathcal S}_\rho\circ{\rm Tr}\circ\pi^{\sharp,\mathcal S}_{\rho},
\lim_{\zeta,\xi\to\chi}D_{\zeta}^{r}\left(R(\zeta/\xi)X^{(s)}(\zeta)X^{(p)}(\xi)\right) = \sum_{t=0}^{r}\binom{r}{t}\lim_{\zeta,\xi\to\chi}\left(D_{\zeta}^{t}\left(R(\zeta/\xi)\right)\left(D_{\zeta}^{r-t}X^{(s)}(\zeta)\right)X^{(p)}(\xi)\right).
R_f = \int\limits_{0}^{R}\underset{t>0}\int\left(\underset{j \in \psi}\prod\frac{1}{1+\max\{e^t-1,T\}r^\alpha d_j^{-\alpha}}+\frac{1}{3}\underset{i \in \phi}\prod\frac{P[\eta(r)<T-\Delta]}{1+\max\{e^t-1,T\}r^\alpha d_i^{-\alpha}}\right)\mathrm{d}tf_R(r)\mathrm{d}r.
P(t,x,y)=\varepsilon p^\varepsilon(t,x,y/\varepsilon), Q_{\alpha\beta}(t,x,y)= \begin{cases}Q^\varepsilon_{\alpha\beta}(t,x,y/\varepsilon) \quad\mbox{if}\quad\alpha,\beta= 1\quad\mbox{or}\quad\alpha,\beta=2 \varepsilon Q^\varepsilon_{\alpha\beta}(t,x,y/\varepsilon) \quad\mbox{if}\quad\alpha=1,\beta=2\quad\mbox{or}\quad\alpha=2,\beta=1\end{cases}
\begin{array}{l}(\epsilon_4,\epsilon_5,\epsilon_6) =\begin{cases}(\min\{n-r+k_2,n-r+k_1-r_1+k_4\}+\hat e_2, \mbox{if~}n-r+k_1>r_1, ~~~~~~~n-r+k_3, n-r+k_1-r_1+k_5) (\min\{k_4,r_1-k_1+k_2\}+\hat e_2, r_1-k_1+k_3, k_5) \mbox{if~}n-r+k_1\le r_1,\end{cases}\end{array}
\partial_{+}g_Lg_L^{-1}=\left(\begin{array}{cc}v_L\partial_{+}u_L+b_L\partial_{+}a_L u_L\partial_{+}a_L-a_L\partial_{+}u_L b_L\partial_{+}v_L-v_L\partial_{+}b_L u_L\partial_{+}v_L+a_L\partial_{+}b_L\end{array}\right) ,
\mathcal{N}\le \frac{1}{a_r!}\cdot m^{1-\frac{a_r}{s_{r-1}}}\cdot \left(\sum_{\mathcal{A}}n_{\mathcal{A}}^{s_{r-1}}\right)^{a_r/s_{r-1}}\le \frac{1}{a_r!}\cdot \left(\frac{n^{a_1+\cdots+a_{r-1}}}{\prod_{i=1}^{r-1}a_i!}\right)^{1-\frac{a_r}{s_{r-1}}}\left(\sum_{\mathcal{A}}n_{\mathcal{A}}^{s_{r-1}}\right)^{\frac{a_r}{s_{r-1}}}.
\limsup_{r \rightarrow 0^+}r^{-\alpha Q}\mu(B_r(t_0)) \lesssim C_\mu \left| \left[ \sum_{j_1,\ldots,j_Q=1}^d \left|{\mathcal A}_{t_0}(\partial_{t_{j_1}},\ldots,\partial_{t_{j_Q}}) \right|^2 \right]^{\frac{1}{2}}\right|^{\alpha}
C_1 =\frac{2^{2/p-1}a^{1/2-1/p}\left[(1+\delta_{ak})^{1/p}+(1-\delta_{(a+1)k})^{1/p}\right]}{[(1-\delta_{(a+1)k})-a^{p/2-1}(1+\delta_{ak})\gamma]^{1/p}}, C_2 =\frac{2^{1/p}m^{1/p-1/2}(1+a^{1/2-1/p}\gamma^{1/p})}{[(1-\delta_{(a+1)k})-a^{p/2-1}(1+\delta_{ak})\gamma]^{1/p}}.
I_{YM}^0 = \int_{M}{\rm Tr}F_t^* F_t = \frac{1}{2}\int_{M}d^4 x \{\rm Tr}\left[ \left( F_{\mu \nu}+{\cal A}_{\mu \nu}\right) \left( F^{\mu \nu}+{\cal A}^{\mu \nu}\right) \right]
\gamma_x(w_1,w_2) := \log \sum_{y \in \mathcal{Y}}\sum_{y' \in \mathcal{Y}}\exp ( \, w_1 \tilde{D}(x,y) \ + & w_2 [\tilde{D}(x,y') - \tilde{D}(x,y)] \, ) f(y') W(y|x)
\langle \xi_1, \ldots, \xi_{k-3}, \gamma, \delta, \epsilon \star \phi \rangle = \langle \xi_1, \ldots, \xi_{k-3}, \gamma, \epsilon, \delta \star \phi \rangle + \langle \xi_1, \ldots, \xi_{k-3}, \gamma \star \epsilon, \delta, \phi \rangle &-\langle \xi_1, \ldots, \xi_{k-3}, \gamma \star \delta, \epsilon, \phi \rangle +S
r_{i}\left( k\right) =\sum_{j=1;j\neq i}^{N}\left( 1-a_{ij}\right)(x_{k})_{j}^{2}+\frac{1}{2d_{i}}\sum_{j=1}^{N}a_{ij}\sum_{l=1}^{N}a_{il}\left( (x_{k})_{l}-(x_{k})_{j}\right)^{2}
\frac{2 7 i}{2 0}\int \! \! d^{4}x \eta_{{\mathbf m}{\mathbf i}}[ D^{\mathbf m}{}_{{\mathbf j}}{\bar{D}}_{{\mathbf k}{\mathbf l}}+{\bar{D}}^{\mathbf m}{}_{{\mathbf j}}D_{{\mathbf k}{\mathbf l}}] T^{{\mathbf i}{\mathbf j}{\mathbf k}{\mathbf l}}| .
\intop_\Omega\!\psi_{1,rrz}\psi_{1,zzz}dx+\!\intop_\Omega\!\psi_{1,zzz}^2dx+3\!\intop_\Omega{1\over r}\!\psi_{1,rz}\psi_{1,zzz}dx=-\!\intop_\Omega\omega_{1,z}\psi_{1,zzz}dx.
\phi ({\bf x}, t ) = \int \! \frac{d^{3}p}{( 2 \pi )^{3}}\frac{1}{\sqrt{2 E_{p}}}\left. \left[ a_{\bf p}e^{- i p \cdot x}+ a_{\bf p}^{\dag}e^{i p \cdot x}\right] \right|_{p_{0}= E_{p}}
g^{\ast}\eta \wedge (gh)^{\ast}\eta = (\eta - df_{g}) \wedge h^{\ast}(\eta - df_{g}) = \eta \wedge h^{\ast}\eta - df_{g}\wedge h^{\ast}\eta - \eta \wedge h^{\ast}df_{g}+ df_{g}\wedge h^{\ast}df_{g}= \eta \wedge h^{\ast}\eta - d(f_{g}\wedge h^{\ast}\eta) + f_{g}\omega + d(\eta \wedge h^{\ast}f_{g}) - h^{\ast}f_{g}\omega + d(f_{g}\wedge h^{\ast}df_{g}).
e^{i W}=\int dg_{\mu \nu}~ d\bar{\Gamma}^\sigma_{~\mu \nu}e^{i \left( S_{clas}+ S_{gf}\right)}\left(det \omega_{\mu \nu}\right)^\frac{1}{2}\left(det \omega_{\mu \alpha}\triangle^\alpha_{~\nu}\right)\left(det \zeta_{\mu \nu}\right)^\frac{3}{2}\left(det \varsigma_{\mu \nu}\right)^\frac{3}{2}
H = \sum_{n = 1}^{N - 1}h_{n , n + 1}+ \frac{1}{2}( K_{1}^{-}( 0 )^{- 1}) \dot{K}_{1}^{-}( 0 ) + \frac{t r_{0}[ K_{0}^{+ t}( - \eta ) h_{N 0}]}{t r [ K^{+}( - \eta ) ]},
\sup_{\alpha\in \mathcal A}\big\{-Tr\big(\sigma_{\alpha}(x)\sigma_{\alpha}^T(x)D^2u(x)\big)-I_{\alpha}[x,u]+b_{\alpha}(x)\cdot Du(x)+c_{\alpha}(x)u(x)+f_{\alpha}(x)\big\}=0,\,\,\mathbb R^n,
\|O_1y_1+y_2\| \ge \|O_1y_1+z_2\|-\|z_2-y_2\| &\ge(1-\alpha)\|y_1+z_2\|-\|z_2-y_2\| &\ge(1-\alpha)\|y_1+y_2\|-(2-\alpha)\|z_2-y_2\| &>(1-\alpha)\|y_1+y_2\|-(2-\alpha)\alpha\|y_2\| &\ge(1-\alpha(1+(2-\alpha)A))\|y_1+y_2\|.
\left\lbrace\begin{gathered}\Delta s = 3 |\nabla n|^2 s +\frac{1}{L}(2s\partial_1\varphi + s^2 \partial_2\varphi), s\Delta n + 2 (\nabla s \cdot\nabla)n = -s|\nabla n|^2 n,\end{gathered}\right.
JTx = J\mathcal{Z}_{T}(I-\mathcal{Z}_{T}^{*}\mathcal{Z}_{T})^{-\frac{1}{2}}x =\mathcal{Z}_{T}J (I-\mathcal{Z}_{T}^{*}\mathcal{Z}_{T})^{-\frac{1}{2}}x &= \mathcal{Z}_{T}(I-\mathcal{Z}_{T}^{*}\mathcal{Z}_{T})^{-\frac{1}{2}}Jx &=TJx.
(p+1) (a_1+\dotsb+ a_p) \leq \sum_{i=1}^{p}(p+1 - i) b_i + \sum_{i=2}^{p+1}(i-1) b_{i}\leq p \sum_{i=1}^{p}b_i + \sum_{i=1}^{p}(1 - i) b_i + p b_{p+1}+ \sum_{i=2}^{p}(i-1) b_{i}\leq p (b_1 + \dotsb + b_{p+1}).
\int e^{\frac{\phi_0 y_0+\phi_1 y_1+...+\phi_n y_n}{\hbar}}~\psi ~ d\phi_1\wedge...d\phi_n=(2\pi\hbar)^{\frac{n}{2}}\cdot y_0^{-\frac{n}{2}}~\hat{g}\Bigg(\frac{y_1}{y_0},...,\frac{y_n}{y_0}\Bigg)\cdot e^{\frac{y_0}{\hbar}\hat{f}\big(\frac{y_1}{y_0},...,\frac{y_n}{y_0}\big)}.
\theta_k = \texttt{log}_2\left|\textbf{I}_N+\frac{\beta\textbf{H}_k\textbf{Q}_k\textbf{H}_k^H}{\sigma^2}\right| = \sum_{m=1}^M\texttt{log}_2\left(1+\frac{\left(\widetilde{\textbf{Q}}_k\right)_{m}\left(\mathbf{\Lambda}_k\right)_{m}}{\sigma^2}\right)\stackrel{\Delta}{=}\sum_{m=1}^M \theta_{k,m},
ctlh[\frac{1}{2}aclh(x)]^{2}=\frac{1}{2}\sqrt{2+2\sqrt{2}x\sqrt{\sqrt{x^{4}+1}-x^{2}}}=(2x^{2}+2+2\sqrt{x^{4}+1})^{-1/2}(\sqrt{\sqrt{x^{4}+1}+1}+x)
\langle 0 | a_{p}( 6{\textstyle \sum{}_{4}}){\frac{1}{E - g \sum_{2}}}( 6{\textstyle \sum_{4}}){\frac{1}{E - g \sum_{2}}}( 6{\textstyle \sum_{4}}) a_{p}^{\dagger}| 0 \rangle ={\frac{5 4}{p g^{2}}}I \hbar^{5},
{\frac{M_{C l .}( N )}{M_{C l .}( 1 )}}= \displaystyle{{\frac{\tan N \lambda}{\tan \lambda}}}, \ \mathrm{a n d}\{\frac{M_{Q .}( n )}{M_{Q .}( n )}}= \displaystyle{{\frac{\tan [ n \arctan \lambda ]}{\tan [ \arctan \lambda ]}}},
\frac{1}{\sigma(x)\,e^{2\pi x}-1}=-\frac{1}{2}+\frac{1}{2\pi x}\cdot\frac{1}{1+\frac{1}{\pi p}}+\frac{1}{\pi}\,\intop_{0}^{\infty}\sin(xt)\,\sigma_{p}(t)\,dt.
{\tilde{a}}_{k}=\frac{1}{\sqrt{2\pi}}\int_{-\pi}^{\pi}e^{{\mathrm{i}}kx}\sqrt{f_{b}(x)}dx\,\,Z_{k\ell}^{b}=\sum_{j\in{\mathbb{Z}}}{\tilde{a}}_{j}\xi_{k,\ell-j}\,,
\exp(-\mathcal{P}_1(s)) =\exp\left(-\sum_{k\geq 1}\sum_{p\leq Y}\frac{(p^{-s})^k}{k}\right)=\exp\left(\sum_{p\leq Y}\log(1-p^{-s})\right) &=\prod_{p\leq Y}(1-p^{-s})=M_1(s)+\mathcal{E}_1(s).
\mathbf{T}^{l,1}_{\varepsilon_1,i,j}(\mu_l)(x)=\sup_{\rho\in (0,2R),e\in S^{d-1}}\varepsilon_1^{-d+1}\left|\left(\frac{1}{|.|^{d-1}}\Theta^{\varepsilon_1,e}_{l,\rho}(.)\right)\star \mathbf{K}_j^i\star\mu_l(x)\right|~~\forall~~x\in \mathbb{R}^d.
\begin{array}{rcl}\bar{A}= p \big(1-\frac{\displaystyle 2x}{\displaystyle u}\big)-ay-bz, \bar{B}= q \big(1-\frac{\displaystyle 2y}{\displaystyle v}\big)-cx-ze, \bar{C}= r \big(1-\frac{\displaystyle 2z}{\displaystyle w}\big)-fx-gy. \end{array}
\sum_{m \ge \frac{1}{T_2 - T_1}}\frac{\log^{d-1}(m+1)}{m^2 \prod_{k \neq j}\left\| m \frac{a_j}{a_k}\right\| (T_2 - T_1)}= O \left( (T_2 - T_1)^{1 - \gamma_{d-1}- \varepsilon}\right) .
m(\mathbf{y}| M)=\frac{\sum_{\alpha \in A_{n}}\lambda_{n,\alpha}\int_{p \in \mathcal{P}_{n,\alpha}: p \in B}\prod_{i=1}^{n}p(y_{i})d\Pi_{n,\alpha}(p)}{\Pi_{n}(B | y_{1}, \ldots, y_{n})}.
2\cdot r_{2}\frac{\sqrt{2mE}}{\hbar}\cdot(cos^{-1}(\sqrt{x})-\sqrt{x}\sqrt{1-x})=2\frac{\sqrt{2m}z(Z-z)k_{e}e^{2}}{\hbar\sqrt{E}}\cdot(cos^{-1}(\sqrt{x})-\sqrt{x}\sqrt{1-x})
(LM_{t}-MM_{s})_t+(NM_{s}-MM_t)_s-\frac{2M}{NL}(LM_t^2-2MM_sM_t+NM_s^2) +\left(\widehat A_1-\frac{B_t}{B}M\right)M_s+\left(\widehat A_2-\frac{B_t}{B}L+\frac{B_s}{B}M-BB_tN\right)M_t+\widehat A_0 =0.
\left[h'k-hk'\right]_{bd}=\tfrac{3}{2}\left(\int_{0}^{\frac{1}{2}}\varphi\int_{0}^{\frac{1}{2}}\psi-\int_{0}^{\frac{1}{2}}\varphi\int_{0}^{\frac{1}{2}}\psi\right)=0.
a_n = \big(Q_1^{(n)}\big)_{11}- \big(Q_1^{(n+1)}\big)_{11}, b_{n-1}^2 - \frac{1}{4}= \big(Q_1^{(n)}\big)_{12}\big(\big(Q_1^{(n)}\big)_{21}- \big(Q_1^{(n+1)}\big)_{21}\big).
{\cal V}= \frac{P_{\mathrm{M A X}}- P_{\mathrm{m i n}}}{P_{\mathrm{M A X}}+ P_{\mathrm{m i n}}}=_{N}\! < 0 \vert S^{[ N ]}\vert 0 >_{N}= \left( 1 - q \right)^{N / 2}.
\int_{\Omega}\mathrm{div}\left( (\nabla u_{0})q_{0}u_{0}^{q_{0}-1}\phi_{1}\right) =\int_{\Omega}(\Delta u_{0})q_{0}u_{0}^{q_{0}-1}\phi_{1}& +\int_{\Omega}q_{0}(q_{0}-1)|\nabla u_{0}|^{2}u_{0}^{q_{0}-2}\phi_{1}& +\int_{\Omega}q_{0}(\nabla u_{0}\nabla\phi_{1})u_{0}^{q_{0}-1}.
\begin{aligned}f_{a(k)}(x) = \left(1-dxq^{a(k-1)}\right) f_{a(k-1)}(x) - q^{a(k-1)-N}f_{a(1)}\left(xq^N\right) &+ q^{a(k-1)-N}\left(1-xq^N\right) f_{a(k-1)}\left(xq^N\right).\end{aligned}
g_{i,j}^{-1}g_{k,l}g_{i,j}= \begin{cases}g_{k,l}\ (i-k)(i-l)(j-k)(j-l) > 0, g_{i,l}g_{k,l}g_{i,l}^{-1}\i < k=j < l , g_{i,l}g_{j,l}g_{k,l}g_{i,l}^{-1}g_{j,l}^{-1}\i=k < j < l, g_{i,l}g_{j,l}g_{i,l}^{-1}g_{j,l}^{-1}g_{k,l}g_{j,l}g_{i,l}g_{j,l}^{-1}g_{i,l}^{-1}\i < k < j < l,\end{cases}
\sum_{k=0}^{n-1}\left(4kS_k^{(r)}-S_k^{(r+1)}+3S_k^{(r)}\right) = \sum_{k=0}^{n-1}\sum_{j=0}^k(4k-2j+2)\binom kj^2\binom{2j}j(2j+1)^r = \sum_{j=0}^{n-1}\binom{2j}j(2j+1)^r\sum_{k=j}^{n-1}(4k-2j+2)\binom kj^2=2\sum_{j=0}^{n-1}(2j+1)^{r-1}u_j.
\sup_{\theta \in A \times W_n^{(\eta)}}\ln(n)^q \left| \frac{1}{n}\sum_{i=1}^n f(X_{\frac{i-1}{n}}, \delta_0, \alpha_0) \left( h(n^{1/ \alpha_0}\Delta_i^n L,\alpha) - h(n^{1/ \alpha_0}\Delta_i^n L, \alpha_0) \right) \right| \to 0,
b_1(y,y',p_0)=a_1(p_0)+{1 \over 12}\left(R_{\mu\nu}(p_0)(y-y')^{\mu}(y-y')^{\nu}+R_{\mu\lambda\nu\rho}(p_0)(y-y')^{\mu}(y-y')^{\nu}y^{\lambda}y^{\rho}\right)
I_n^{(1)}(\lambda):=\sum_{j=1}^n \omega_j \left( 4 \tau+\lambda (t_j+1)^2 \right)^{-1}, I_n^{(2)}(\lambda) := \sum_{j=1}^n \omega_j \left( \tau (t_j+1)^2 +4 \lambda \right)^{-1},
Q^1_{i\bar i}=T_{ik\bar m}T_{\bar i\bar k m}\,,Q^2_{i\bar i}=T_{\bar k\bar m i}T_{km \bar i}\,, &Q^3_{i\bar i}=T_{ik\bar k}T_{\bar i\bar m m}\,,Q^4_{i\bar j}=\frac12(T_{mk\bar k}T_{\bar m\bar ii}+T_{\bar m\bar kk}T_{mi\bar i})\,, &q^1=q^2=\|T\|^2\,,q^3=q^4=\|w\|^2\,,
\bigg\|\sum_{j,l=1}^m a_j^*\phi^{-1}\big({s_{\lambda_j}^\Lambda}^*s_{\lambda_l}^\Lambda\big)a_l\bigg\|_{C^*(\Lambda^i)}=\bigg\|\bigg\langle \sum_{j=1}^m s_{\lambda_j}^\Lambda\phi(a_j), \sum_{j=1}^m s_{\lambda_j}^\Lambda\phi(a_j)\bigg\rangle_{C^*(\Lambda^i)}\bigg\|_{C^*(\Lambda^i)}&=\bigg\|\sum_{j=1}^m s_{\lambda_j}^\Lambda\phi(a_j)\bigg\|_X^2.
\mathcal{F}(\rho)=\int_{\mathbb{R}^d}V(x)\rho(x)dx+\frac{1}{2}\int_{\mathbb{R}^d\times \mathbb{R}^d}W(x,y)\rho(x)\rho(y)dxdy+\beta\int_{\mathbb{R}^d}\rho(x)\log\rho(x)dx,
\overrightarrow{0}=\sum\limits_{j=1,\textrm{}j\neq i}^{n}\frac{m_{j}\left(Z-\sigma(q_{i}\odot_{k}q_{j})Z\right)}{(\sigma-\sigma(q_{i}\odot_{k}q_{j})^{2})^{\frac{3}{2}}}-\sigma(\dot{q}_{i}\odot_{k}\dot{q}_{i})Z,\textrm{}i\in\{1,...,n\},
p(n_1, \ldots, n_V; \pmb{\rho}^{(N)}) \triangleq \prod_{v = 1}^V{N^{(N)}_v \choose n_v}\left(\rho^{(N)}_v p^{(N)}_v \right)^{n_v}\left(1 - \rho^{(N)}_v p^{(N)}_v\right)^{N^{(N)}_v - n_v}.
\iota_{\ll \alpha,\beta\gg_\ast}\widetilde{\mathcal{M}}_\Pi^\nu= (a\circ\Pi)(\alpha) \iota_\beta\widetilde{\mathcal{M}}_\Pi^\nu- (a\circ\Pi)(\beta)\iota_\alpha\widetilde{\mathcal{M}}_\Pi^\nu.
\min_{\mathfrak{E}}s_g= \quad32\pi\left(1+\sum a\right)\left(8\sum a-12\sum a^2+8\sum a^3-\sum a^4+12\sum ab+4\sum a^3b-3\sum a^2b^2\right)\Big/ begin{align*}0.1cm] \left(32\sum a^2-32\sum a^3+32\sum a^4-12\sum a^5\right.+\sum a^6+32\sum ab+32\sum a^2b+32\sum a^3b+4\sum a^4b begin{align*}0.1cm] \left.-6\sum a^5b+8\sum a^3b^2+15\sum a^4b^2-10\sum a^3b^3\right)
\frac{1}{2N}\sum_{c=1}^{m}\sum_{i\in I_{N}^{c}}\int_{0}^{T}b_{c}^{2}(t,\,x_{i}^{N}(t))dt +\frac{1}{2\lambda N}\sum_{\substack{c_{1}<c_{2},\,i\in I_{N}^{c_{1}}}}\int_{0}^{T}\gamma_{c_{1},c_{2}}^{2}(t,\,x_{i}^{N}(t))dA_{i,c_{2}}^{N}(t)+O\left(\frac{1}{N}\right).
\sum_{j = 0}^k{2^j G_j^3}= - \frac{5}{{11}}G_0^3 - \frac{{58}}{{11}}G_1^3 - \frac{{20}}{{11}}G_2^3 + \frac{8}{{11}}G_3^3 + \frac{5}{{11}}2^{k + 1}G_{k + 1}^3 &\qquad+ \frac{{29}}{{11}}2^{k + 2}G_{k + 2}^3 + \frac{5}{{11}}2^{k + 3}G_{k + 3}^3 - \frac{1}{{11}}2^{k + 4}G_{k + 4}^3\,.
h_0h_1\cdots h_N \,\pi\big(L(z)\big)\,A^{(N)}_{1\dots N}= \pi\big(L(z)\big)\,h_0h_1\cdots h_N \,A^{(N)}_{1\dots N}& = \pi\big(L(z)\big) \,A^{(N)}_{1\dots N}\,h_0h_1\cdots h_N \,.
\begin{cases}|\mu_1-\mu_2|\leqslant|\lambda_k+1/2|\leqslant N+\mu_1+\mu_2+1, , |\mu_1-\mu_2|\leqslant|\lambda_k+1/2|\leqslant N+\mu_1+\mu_2+2\mu_3+1, , \end{cases}
\sum_{k = 1}^{\infty}\sum_{n = 0}^{\infty}\frac{\sigma_{k}^{n + 1}z^{n}}{n !}\frac{1}{1 - \sigma}= \sum_{m = 1}^{\infty}\sigma_{m}^{1}\frac{1}{( 1 - \sigma )^{2 m}}.
(z-1)^n\frac{(1+\xi_t(z))^{2n}}{\xi_t^n(z)}= \binom{2n}{n}(z-1)^n + \sum_{k \in [-n..n]\setminus \{0\}}\binom{2n}{n+k}e^{kt}\sum_{m \geq 0}L_{m}^{(-k-m)}(-2kt)\frac{(z-1)^{m+n+k}}{2^{m+k}}.
\frac{d(\phi (y_1), \phi (y_2))}{d(\phi (y_1), \phi (y_3))}\leq \limsup_{\substack{x,x' \in \phi (Y),\, \pi (x)=\pi(x') x\to x'}}\, \eta\left( \frac{d(x, \pi^{-1}(y_1))}{d(x', \pi^{-1}(y_1))}\right) < H.
\left\{\begin{array}{c c c}{a}&{b}&{i}\\{c}&{d}&{j}\\ \end{array}\right\}= \frac{T e t \left[ \begin{array}{c c c}{a}&{b}&{i}\\{c}&{d}&{j}\\ \end{array}\right] \Delta_{i}}{\theta ( a , d , i ) \theta ( b , c , i )}.
\sigma h_{11}= - \lambda_2 e^{-(2 \lambda_2 +\lambda_1)z}f_z, \sigma h_{12}=\sigma h_{21}= 0, \sigma h_{22}= [f_{zz}-2\lambda_1 f_z -\lambda_1 f_z^3 e^{-2\lambda_1 z}] e^{-\lambda_1 z}.
\begin{array}{rl}{\rm E}\Big[|\sum_{j\in B}\alpha_{ji}(c_j-{\rm E}[c_j]) |^{\gamma_i}\Big]^{\frac1{\gamma_i}}={\rm E}\Big[|\sum_{j\in B}\alpha_{ji}\epsilon_j (c_j-c_j^\prime) |^{\gamma_i}\Big]^{\frac1{\gamma_i}}& \leq 2{\rm E}\Big[|\sum_{j\in B}\alpha_{ji}\epsilon_j c_j |^{\gamma_i}\Big]^{\frac1{\gamma_i}}\end{array}
\int_X\psi\omega^n = C_H\int_{(-\Delta^t)\cap \bar{C}^-}n! 2^{|\Phi_s^+|-r}\psi(d_{2p}u^*)\prod_{\alpha\in\Phi_{Q^u}}(\chi-p)(\alpha^{\vee}) \prod_{\beta\in\Phi_s^+}(-p)(\beta^{\vee}) dp
\frac{\partial^{n-1}t}{\partial v^{n-1}}(u,v)=e^{-L(u,v)}\left\{\frac{\partial^{n-1}t}{\partial v^{n-1}}(v,v)+\int_v^ue^{L(u',v)}R_{v,n-1}(u',v)du'\right\},
\|y\| \leq \sum_{j=1}^m\alpha_j \|{y_j}\|=\sum_{j=1}^m\frac{\alpha_j}{2}\|m_{a_j,b_j}+y_j-m_{a_j,b_j}+y_j\| &\leq \sum_{j=1}^m\frac{\alpha_j}{2}\bigl(\|m_{a_j,b_j}+y_j\|+\|m_{a_j,b_j}-y_j\|\bigr) &\leq \sum_{j=1}^m\alpha_j\bigl(1+\frac{\varepsilon}{5m\alpha_j}\bigr)< 1+\frac{2\varepsilon}{5},
\tilde{E}_{i}^{2}= E_{i}^{b}R_{2 b}= E_{i}^{b}\hat{E}_{3}^{b}\hat{E}_{3}^{a}\hat{\lambda}^{a}- E_{i}^{b}\hat{\lambda}^{b}\hat{E}_{3}^{a}\hat{E}_{3}^{a}\, .
F_{j}( \omega , \vec{k}) = \int \frac{d \omega^{\prime}}{2 \pi}\int \frac{d \vec{k}^{\prime}}{( 2 \pi )^{3}}\tilde{B}( \omega - \omega^{\prime}, \vec{k}- \vec{k}^{\prime}) \: C_{j - 1}( \omega^{\prime}, \vec{k}^{\prime}) \: F_{j - 1}( \omega^{\prime}, \vec{k}^{\prime})
-\tilde{u}^{-q}\partial_{s}(g_{1-\alpha,m}*\tilde{u}) \geq -\frac{1}{1-q}\partial_{s}(g_{1-\alpha,m}*\tilde{u}^{1-q})+ \left( \frac{\tilde{u}^{1-q}}{1-q}- \tilde{u}^{1-q}\right)g_{1-\alpha,m}& \geq - \frac{1}{1-q}\partial_{s}(g_{1-\alpha,m}*\tilde{u}^{1-q}) + \frac{q}{1-q}\tilde{u}^{1-q}g_{1-\alpha,m}.
p\log \lambda^{(p)}(G) \leq p\log (r^{1-r/p}\alpha^{-1/p}) & = (p-r)\log r - \mu \log \alpha_1 -(1-\mu) \log \alpha_2 & =\mu[(p_1-r)\log r-\log \alpha_1] + (1-\mu)[(p_2-r)\log r-\log \alpha_2] & =\mu p_1\log \lambda^{(p_1)}(G) +(1-\mu) p_2\log \lambda^{(p_2)}(G).
min_{\beta,B}\sum_{i=1}^{n}\|x_{i}-\sum_{x_{j}\in B_{i}}K(x_{i},x_{j})x_{j}\|^{2}+\gamma_{1}\sum_{i=1}^{n}\sum_{x_{j}\in B_{i}}K(x_{i},x_{j})\|x_{i}-x_{j}\|^{2}+\gamma_{2}\sum_{i}|B_{i}|
\mathcal{W}=X_{0,0}+X_{a,b}+ \sum_{i=1}^{a-1}\sum_{j=1}^b X_{i,b}X_{i,b-1}\cdots X_{i,j}+\sum_{j=1}^{b-1}\sum_{i=1}^a X_{a,j}X_{a-1,j}\cdots X_{i,j}.
Z^m(t_0, t_1) \equiv \Bigl\{\psi \in H^j(t_0, t_1; H^{m-j}(\Omega));~j=0,\cdots,m,\ \|\psi\|_{Z^m(t_0, t_1)}< \infty \Bigr\}\ \ \mbox{with}\ \ \|\psi\|_{Z^m(t_0, t_1)}\equiv \biggl\{\sum_{j=0}^m \|\psi\|_{H^j(t_0,t_1; H^{m-j}(\Omega))}^2 \biggr\}^{1/2},
M(X)= \left(\prod_{R \in \mathcal{R}}\left(\left(\left\langle X,R \right\rangle \right)^2-1^2\right) \prod_{R \in \mathcal{R}}\left(\left(\left\langle X,R \right\rangle \right)^2-3^2\right) \ldots\prod_{R \in \mathcal{R}}\left(\left(\left\langle X,R \right\rangle \right)^2-(i-1)^2\right) \right)^2.
\vert p , \sigma , q , \lambda \colon f ( p ) \rangle = e x p \biggl [ - e \int \sum_{\lambda = 1 , 2}{\frac{d k^{+}}{\sqrt{2 k^{+}}}}\int{\frac{d^{2}k_{\perp}}{( 2 \pi )^{\frac{3}{2}}}}\bigl [ f ( k , \lambda \colon p ) a^{\dagger}( k , \lambda ) - f^{*}( k , \lambda \colon p ) a ( k , \lambda ) \bigr ] \biggr ] \vert p , \sigma , q , \lambda \rangle \; .
- i \, \hat{\nabla}\left( \begin{array}{c}{\alpha_{\lambda}( \theta , \, \phi )}{\beta_{\lambda}( \theta , \, \phi )}\end{array}\right) = \lambda \, \left( \begin{array}{c}{\alpha_{\lambda}( \theta , \, \phi )}{\beta_{\lambda}( \theta , \, \phi )}\end{array}\right) .
\textrm{Pr}^{\textrm{OFDMA}}\left\lbrace \textrm{FDF=0}\right\rbrace \to\frac{d_{\textrm{BF}}^{\alpha}\sigma_{\textrm{F}}^{2}\theta}{\lambda_{\textrm{BF}}\rho}\left(2^{\frac{R}{\theta}}-1\right)\frac{1}{P_{\textrm{B}}}.
{\cal L}^{\prime}= \bar{\psi}\gamma^{\mu}\left[ i \partial_{\mu}+ e{\cal A}_{\mu}\right] \psi - \frac{1}{2 G}{\cal A}_{\mu}{\cal A}^{\mu}- \frac{\kappa}{2}\epsilon_{\mu \nu \rho}{\cal A}^{\mu}\partial^{\nu}{\cal A}^{\rho},
\Phi_\lambda = [I -(I- \frac{1}{\sqrt\lambda}q A^{-1})(I + \frac{1}{\sqrt\lambda}\Theta U(q))]^{-1}\times [\frac{1}{\sqrt\lambda}q A^{-1}+(I - \frac{1}{\sqrt\lambda}q A^{-1}) \frac{1}{\sqrt\lambda}\Theta (U - U(q))] + O(1/\lambda) @ = I + (q A^{-1}- \Theta U(q))^{-1}\Theta U + O(1/\sqrt\lambda),
\begin{array}{c}{{P R^{\prime}\left( \sqrt{\frac{z}{w}}\right) \left( q^{S_{3}}\otimes{\frac{E}{z}}+{\frac{E}{w}}\otimes q^{- S_{3}}\right) =}}\\{{= \left( q^{S_{3}}\otimes{\frac{E}{w}}+{\frac{E}{z}}\otimes q^{- S_{3}}\right) P R^{\prime}\left( \sqrt{\frac{z}{w}}\right)}}\\ \end{array}
\mathcal{S}= \mathcal{S}_{c s}+ \int d^{3}x \; \left( b^{a}\partial A^{a}+ \partial^{\mu}\overline{{{c}}}^{a}( D_{\mu}c )^{a}+ \; \hat{A}_{\mu}^{*}( D^{\mu}c )^{a}- \frac 1 2 \; f_{a b c}C^{* a}c^{b}c^{c}\right) \; ,
\begin{array}{c}{A_{\mu}= \left( \frac 1 2 \left( l_{\mu}^{1}+ l_{\mu}^{2}\right) , \frac 1 2 \left( l_{\mu}^{1}+ l_{\mu}^{2}\right) \right) ,}\\{B_{\mu}= \left( \frac 1 2 \left( l_{\mu}^{1}- l_{\mu}^{2}\right) , - \frac 1 2 \left( l_{\mu}^{1}- l_{\mu}^{2}\right) \right) ,}\\ \end{array}
\frac{1}{V(u)}\Pr[ \rho(X) > u ] = \frac{1}{V(u)}\Pr[ \rho(u^{-1}X) > 1 ] \to \mu ( \{x : \rho(x) > 1 \}) = \mu( \{x : \rho(x) \ge 1 \}), u \to \infty.
C_{a b c}= \theta_{0 a}\left( F_{b c}^{0}+ \lambda_{b}A_{c}^{0}- \lambda_{c}A_{b}^{0}\right) + \theta_{0 b}\left( F_{c a}^{0}+ \lambda_{c}A_{a}^{0}- \lambda_{a}A_{c}^{0}\right) - \theta_{0 c}\left( F_{a b}^{0}+ \lambda_{a}A_{b}^{0}- \lambda_{b}A_{a}^{0}\right) ,
\left[\mathcal{H}yp^{ct}_{2,n}\right]=\pi_n^*\left[\mathcal{H}yp^{ct}_{2,n-1}\right] \cdot \rho_n^*\left[\mathcal{H}yp^{ct}_{2,1}\right] - \sum_{i=1}^{n-1}{\sigma_{i}}_\ast\left[ \mathcal{H}yp^{ct}_{2,n-1}\right].