harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H_0=\int dp f(p) A^\dagger_f(p)A_f(p)=\int_x [{\frac{1}{2}}\Pi^{2}_x +{\frac{1}{2}}(\partial_x \phi_x)^2 +{\frac{1}{2}}\mu^2 (\phi_x-\varphi)^2 -{\frac{1}{2}}I_0(\mu)] \;.
\liminf_{N\to\infty}P\big(\cap_{n=1}^{2[\frac1{b_N}]}\big\{Z_n^{N,i}\in\{0,-1\}\big\}\big|\cup_{n=1}^{[\frac1{b_N}]}\{Z_{2n}^{N,i}=-1\}\big)\ge e^{-2}.
{\cal{L}}_{D J}^{S}= \frac{1}{2}( f^{\mu}- \partial^{\mu}\phi ) ( f_{\mu}- \partial_{\mu}\phi ) - \epsilon^{\alpha \beta \gamma}f_{\alpha}\partial_{\beta}A_{\gamma}+ \frac{m}{2}\epsilon^{\alpha \beta \gamma}A_{\alpha}\partial_{\beta}A_{\gamma}.
d(A\cup B,C)=\frac{n_{A}+n_{C}}{n_{A}+n_{B}+n_{C}}d(A,C)+\frac{n_{B}+n_{C}}{n_{A}+n_{B}+n_{C}}d(B,C)-\frac{n_{C}}{n_{A}+n_{B}+n_{C}}d(A,B).
a_0(u,v) = \sum_{i=1}^N \int_{\Omega}\biggl( \sum_{j=1}^N a_{ij}\partial_j u + a_i u \biggr) \partial_i v + \int_{\Omega}\biggl( \sum_{i=1}^N b_i \partial_i u + c_0 u \biggr) v,
\mathbf{\Phi}_{0}\nabla_{S}^{\underline{\mathbf{D}}}(L)\underline{\mathbf{Y}}_{\tau}-\underset{i=1}{\overset{\mathbf{P}}{\sum}}\mathbf{\Phi}_{i}\nabla_{S}^{\underline{\mathbf{D}}}(L)\underline{\mathbf{Y}}_{\tau -i}=\underline{\mathbf{u}}_{\tau},
1 \geq \frac1{1-\rho_n}F_N\left( \left[ \frac1n\sum_{i=1}^{n}\frac{\frac1N\Vert z_i\Vert^2}{\check{d}_i}\right]^{-1}\right) &\times\frac1Nz_1^* \left( \frac1n\sum_{i=1}^{n-1}z_iz_i^* + \check{d}_1 \frac{\rho_n}{1-\rho_n}F_N\left( \check{d}_1 \left[ \frac1n\sum_{i=1}^{n}\frac1N\Vert z_i\Vert^2 \right]^{-1}\right) I_N \right)^{-1}z_1.
\left\{\int_0^{\lambda}\left(f(r)-\frac{r}{\lambda}f(\lambda)\right)^2dr+\int_{\lambda}^1\left(f(r)-f(\lambda)-\frac{r-\lambda}{1-\lambda}\left(f(1)-f(\lambda)\right)\right)^2dr\right\}^{\frac{1}{2}}=0
\begin{array}{l}\|\tilde{\mathcal{G}}_{\tilde{e}_N}^M(s)\tilde\Phi_M-\tilde{\mathcal{G}}_{\tilde{e}}^M(s)\tilde\Phi_M\|_M\leq \sum\limits_{i,j=1}^N\|\tilde{e}_N^i(s)-\tilde{e}_i(s)\|_\infty\|f\circ\sigma_{ij}-f\|_{\infty}\leq 2(N-1)\sum\limits_{i=1}^N\|\tilde{e}_N^i(s)-\tilde{e}_i(s)\|_\infty\|f\|_{\infty},\end{array}
+{\cal C}^{1 \mu}S_{\mu \nu}\bar{\cal P}_{2}^{\nu}+ \frac{1}{M^{2}}{\cal C}^{2 \mu}P_{\mu}P^{\nu}\bar{\cal P}_{1 \nu}+ \frac{1}{M^{2}}{\cal C}^{2 \mu}P_{\mu}S^{\nu \sigma}P_{\sigma}\bar{\cal P}_{2 \nu}-{\cal C}^{\prime}P^{\prime}\bar{\cal P}^{\prime}.
C_{J}( \nu_{1}, \nu_{2}) = ( 2 J + \nu_{1}+ \nu_{2}+ 1 ) \frac{{\mit \Gamma}( J + 1 ){\mit \Gamma}( J + \nu_{1}+ \nu_{2}+ 1 )}{{\mit \Gamma}( J + \nu_{1}+ 1 ){\mit \Gamma}( J + \nu_{2}+ 1 )}\, .
\sum_{j=0}^{n-1}w^{ij}\wp_j\beta_r^{(j)}(s) +\alpha_r(s)\sum_{j=0}^{n-1}w^{ij}\wp_j C(j) \beta_r^{(j+1)}(t) &= \sum_{j=0}^{n-1}w^{ij}\wp_j\beta_r^{(j)}(t)+\alpha_r(t)\sum_{j=0}^{n-1}w^{ij}\wp_j C(j) \beta_r^{(j+1)}(s),
[\Lambda,\Lambda]=[\sum_{\alpha,\beta=1}^n g_{\alpha \beta}^{j}(z_j,t)\frac{\partial}{\partial z_{j}^{\alpha}}\wedge \frac{\partial}{\partial z_{j}^{\beta}},\sum_{\alpha,\beta=1}^n g_{\alpha \beta}^{j}(z_j,t)\frac{\partial}{\partial z_{j}^{\alpha}}\wedge \frac{\partial}{\partial z_{j}^{\beta}}]=0
(\begin{matrix}-\frac{1}{2}\\n\end{matrix})=\frac{(-1)^{n}(2n-1)!!}{2^{n}n!}=\frac{(-1)^{n}(2n)!}{(2^{n}n!)^{2}}=\frac{(-1)^{n}}{4^{n}}(\begin{matrix}2n\\n\end{matrix})
F(z) = (-1)^N z^{\beta(N-1)}p^{-1}\int_{0}^{\infty}\frac{t^{N+q/p-1}\, e^{-t^{1/p}}}{t+z^{-\beta}}\; dt+ \frac{2 \pi i}{p}\; z^{-\beta q/p}& \times \;\; \sum_{j=0}^{M-1}\,e^{(2j+1)i q \pi /p}\, \exp \!\left( -z^{-\beta/p}e^{(2j+1)i \pi/p}\right) \;\;,
n_{{\bf{k}}}= n^{\beta}({\bf{k}}) \frac{N}{\langle N \rangle}+ \sum_{{\bf{q}}}a^{\dagger}_{{\bf{k}}-{\bf{q}}/2}({\bf{q}})a_{{\bf{k}}-{\bf{q}}/2}({\bf{q}}) - \sum_{{\bf{q}}}a^{\dagger}_{{\bf{k}}+{\bf{q}}/2}({\bf{q}})a_{{\bf{k}}+{\bf{q}}/2}({\bf{q}})
G(a) =\left[ \frac{mk\Gamma \left( \frac{1-x}{h}\right) \Gamma\left( \frac{x}{h}\right) x}{2\Gamma \left( \frac{1}{h}\right) h}\right]^{-a^{2}}\exp \left( \int\limits_{0}^{\infty}\frac{dt}{t}[a^{2}e^{-2t}-F(a,t)]\right)
R:=\limsup_{N\to\infty}\,\left( \sup_{\genfrac{}{}{0pt}{1}{n\ge 0}{b_{n+1}-b_{n}>N}}\,\sup_{b_{n}\le k<b_{n+1}-N}\,(w_{k+1}w_{k+2}\cdots w_{k+N})\right)^{1/N}.
\sum_{l=0}^p (-1)^l{a \choose{p-l}}={{a-1}\choose p}\sum_{l=0}^p (-1)^l{a \choose{p-l}}l = -{{a-2}\choose{p-1}}\sum_{l=0}^p (-1)^l{a \choose{p-l}}l(l-1) = 2{{a-3}\choose{p-2}}.
\chi\int_{\Omega}w_{+}^2 (u- A^{-1}u) &=(\chi)_+\int_{\Omega}w_{+}^2 (u- A^{-1}u)-(\chi)_-\int_{\Omega}w_{+}^2 (u- A^{-1}u) &\leq (\chi)_+\left((r_1+\epsilon)-(r_2-\epsilon)\right)\int_{\Omega}w_{+}^2+(\chi)_-\left(-(r_2-\epsilon)+(r_1+\epsilon)\right)\int_{\Omega}w_{+}^2=\chi\left((r_1+\epsilon)-(r_2-\epsilon)\right)\int_{\Omega}w_{+}^2
\begin{pmatrix}\eta_{\lambda}^{(1)}\sinh \rho_{\lambda}\eta_{\lambda}^{(2)}\sinh \rho_{\lambda}\cosh \rho_{\lambda}\end{pmatrix}= \begin{pmatrix}\sinh \lambda L_{1}\sinh \lambda L_{2}\cosh \lambda L_{1}\cosh \lambda L_{1}\cosh \lambda L_{2}\end{pmatrix},
\left( T^{2}\frac{\partial}{\partial T}+ \int d x \frac{e^{2}}{2}\sum_{a = 1}^{N^{2}}\left( \frac{\delta}{\delta A^{a}( x )}\right)^{2}\right) \left< A \right| e^{- H / T}\left| A^{g}\right> \; = \; 0 \; ,
P' =\bigoplus_{i\in I_{13}'}SP_{13}\oplus\bigoplus_{i\in I_{14}'}SP_{14}\oplus\bigoplus_{i\in I_{1234}'}SP_{1234}\oplus\bigoplus_{i\in I_{23}'}SP_{23}\oplus\bigoplus_{i\in I_{24}'}SP_{24}&\quad\oplus\bigoplus_{i\in I_{123}'}SP_{123}\oplus\bigoplus_{i\in I_{124}'}SP_{124}\oplus\bigoplus_{i\in I_{1}'}SP_{1}\oplus\bigoplus_{i\in I_{2}'}SP_{2}
\mu(r,\infty)+r^{-2}\int_{(0,r]}y^2\mu(dy)=2r^{-2}\int_0^ry\mu(y,\infty)\,dy &\int_0^ry\mu(y,\infty)\,dy\leq er^2 H(r^{-1}) &\int_{(0,r]}y\mu(dy)\leq er\phi(r^{-1})\,.
S_{\mathrm{c t}}= - \frac{1}{8 \pi G_{p + 2}}\int d^{p + 1}x \sqrt{- h}\frac{c_{0}}{l_{\mathrm{e f f}}}, \ \ \c_{0}= \sqrt{p ( p + 1 )}, \ \ \ \frac{1}{l_{\mathrm{e f f}}}= b e^{- a \phi / 2}\sqrt{\frac{1}{2 p ( p + 1 )}},
\frac{1}{2}\Delta_0 =\frac{1}{2}(dd^{*_0}+d^{*_0}d) &=\Delta_{\partial_0}\coloneqq(\partial_{J_0}\partial_{J_0}^{*_0}+\partial_{J_0}^{*_0}\partial_{J_0}) &=\Delta_{\bar{\partial}_{J_0}}\coloneqq(\bar{\partial}_{J_0}\bar{\partial}_{J_0}^{*_0}+\bar{\partial}_{J_0}^{*_0}\bar{\partial}_{J_0})
{\hat{\bar \Psi}}\ =\ \sum_{j=|k|-1/2}^{\infty}\sum_{|m|\le j}[{\bar a}^{j+}_{-km}{\hat D}^j_{-k-1/2,m}a\ +\{\bar a}^{j-}_{-km}{\hat D}^j_{-k+1/2,m}a^* ]\ \in \{\hat{\cal S}}_{-k}\ ,
|\langle{T_{Am}\varphi_p, F_{v}}\rangle| = |\sum_{n \in{\mathbb{Z}}^2_N}T_{Am}\varphi_p(n)\overline{F_{v}(n)}| = |\sum_{n \in{\mathbb{Z}}^2_N}\varphi_p(n-Am)\frac{1}{N}e^{-2\pi i \langle{n,v}\rangle/N}| \leq \frac{1}{N}\sum_{t \in{\mathbb{Z}}^2_N}|\varphi_p(t)|,
t_\mathrm{s,2}= \Delta t_{\mathrm{s}_{2,1}}+ \Delta t_{\mathrm{s}_{2,2}}+ \Delta t_{\mathrm{s}_{2,3}}+ \Delta t_{\mathrm{s}_{2,4}}=g_\mathrm{DL}^{-1}l_\mathrm{r}+ t_\mathrm{w}+ g_\mathrm{UL}^{-1}l_\mathrm{r}+ g_\mathrm{DL}^{-1}l_\mathrm{h}.
V ( \phi_{0}) = - \frac{1}{5 4}\frac{1}{g^{2}\bar{\kappa}^{2}}= - \frac{2^{1 1}}{3^{1 0}}\frac{1}{g^{2}}\approx ( 0 . 6 8 ) \left( - \frac{1}{2 \pi^{2}g^{2}}\right)
B = \Phi^{2}, \qquad B_{_{0 0}}= -{\frac{c^{4}}{c_{_\mathrm{P}}^{2}}}\Phi^{2}+{\cal O}\{\Theta^{4}\}\ , \qquad B_{_{1 1}}= \Phi^{2}+{\cal O}\{\Theta^{4}\}\ .
\left(c,z;q\right)_{\infty}\sum_{n=0}^{\infty}\frac{S_{n}\left(x;q\right)z^{n}}{\left(c;q\right)_{n}}=\sum_{n=0}^{\infty}\frac{\left(z;q\right)_{n}q^{\binom{n}{2}}\left(-c\right)^{n}}{\left(q;q\right)_{n}}A_{q}\left(xzq^{n}\right).
\Phi_1(Q,I) = \frac{3}{\pi^2}\, Q^2 \int\limits_I \frac{dx}{\max(1,x^2)}+ O\!\left(Q (\ln Q)^{2/3}(\ln\ln Q)^{4/3}\int\limits_I \frac{dx}{\max(1,x^2)}+ Q\right).
S_{\Gamma}= - \frac{1}{\kappa}\int \sqrt{- g}e^{\alpha \phi / M_{p}}( \zeta + b_{g}) g^{\mu \nu}R_{\mu \nu}( \Gamma ) = - \frac{1}{\kappa}\int \sqrt{- \tilde{g}}\tilde{g}^{\mu \nu}R_{\mu \nu}( \Gamma ) ,
G_{n,k}(z)=A\psi_{n,k}(z)+B\psi_{n+1,k}(z)=\int_{a_{k-1}}^{b_{k-1}}\frac{A\psi_{n,k-1}(\tau)+B\psi_{n+1,k-1}(\tau)}{z-\tau}\,\tau\,d\sigma_{k-1}^{*}(\tau).
U^\pm(\cdot , 0) = \pm (J_0(w_1) - J_0(w_2))(\cdot, 0) - | w |_{0,\mathcal{M}}&= \pm ( w_1 - w_2 ) - |w|_{0,\mathcal{M}}&= \pm w - |w|_{0,\mathcal{M}}\leq 0 \textnormal{in}\mathcal{M}.
\overline{\mathrm{mo}}\,(Y) = \limsup_{\varepsilon\,\to\, 0^+}\,\frac{\log\log S_Y(\varepsilon)}{-\log \varepsilon}\qquad\underline{\mathrm{mo}}\,(Y) = \liminf_{\varepsilon\,\to\, 0^+}\,\frac{\log\log S_Y(\varepsilon)}{-\log \varepsilon}
{\bf P}_l = \sum_{\nu=1}^{N}{z_\nu}\,{\cal W}_\nu (\widetilde{\mathbf{P}}_{R_l})= \sum_{\nu=1}^{N}{z_\nu}\,{\cal W}_\nu (\sum\limits_{k\in{\cal K}_l}\widetilde{\bf p}^{(1)}_k \otimes \widetilde{\bf p}^{(2)}_k \otimes \widetilde{\bf p}^{(3)}_k).
Q_g = \frac{2}{n-4}u^{\frac{n+4}{n-4}}P_{g_0}u, \overline{Q}_g = \frac{2}{n-4}\frac{\int u P_{g_0}u \dv_0}{\int u^{\frac{2n}{n-4}}\dv_0}= \frac{2}{n-4}\mu.
K_{\beta_H}\equiv\beta_H\left(\frac{\partial Q}{\partial \varphi}\right)_{\beta_H}=\frac{\tilde{d}\Omega_{\tilde{d}+1}(r_+r_-)^{\tilde{d}/2}}{\pi}\frac{[1-(\tilde{d}-1)(r_-/r_+)^{\tilde{d}}]}{[1-(r_-/r_+)^{\tilde{d}}]}.
x_{true}(s) = \begin{cases}1 0 \leq s \leq \frac{1}{2}0 \frac{1}{2}< s \leq 1 \end{cases},x_{true}(s) = \begin{cases}- 1 0 \leq s \leq \frac{1}{2}0 \frac{1}{2}< s \leq 1 \end{cases}
{\cal L}_{1 / m^{3}}^{( 2 + 1 ) s c a l}= - \frac{\alpha}{7 2 0 m^{3}}\left[ 6 F^{\nu \lambda}F_{\nu \lambda , \mu}^{~ \mu}+ 4 F^{\nu \lambda , \mu}F_{\nu \lambda , \mu}+ F_{\lambda}^{\nu \lambda ,}F_{\nu \mu ,}^{\mu}\right] ,
\beta_{n,q}\left(x\right) =\sum_{l=0}^{n}\dbinom{n}{l}q^{lx}\beta_{l,q}\left[x\right]_{q}^{n-l}=\left(q^{x}\beta_{q}+\left[x\right]_{q}\right)^{n},\quad\left(n\ge0\right),\quad\mbox{(see \cite{key-2,key-3,key-4,key-12}}).
\Vert B_\varepsilon (\vartheta )^{1/2}{\mathcal J}_3(\vartheta; \varepsilon; \widehat{\zeta}) \Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}\leqslant \mathfrak{C}_6\varepsilon c(\phi )^2, & 0 < \vartheta \leqslant 1,0< \varepsilon \leqslant \vartheta^{-1},\widehat{\zeta}=e^{i\phi},0<\phi <2\pi,
\mathcal{L}_{n+1}(x) = x\mathcal{L}_n(x)-\dfrac{1}{4}(x^2+\omega^2)\mathcal{L}_{n-1}(x), n\geq 1, & \mathcal{L}_{1}(x)=x-\kappa, \mathcal{L}_{0}(x)=1.
\|D^\nu_\varkappa\varphi\|^2= \|\mathcal MD^\nu_\varkappa\varphi\|^2= \int\limits_{-\infty}^\infty\big\langle (R^{-1}\mathcal M\varphi)(s), A_\varkappa^\nu(1, s)(R^{-1}\mathcal M\varphi)(s)\big\rangle\,\mathrm ds.
4v ''+\Bigl ( \frac{4}{t}-e^{-t}\Bigr ) v '-\Bigl (\frac{1}{t^2}+e^{-t}\Bigl (\frac{1}{2 t}-\frac{1}{\alpha}- ( t^{\frac{1}{2}}|v |)^\alpha \Bigr)\Bigr ) v =0.
\chi_{\nu}( \rho ) \sim \left( \begin{array}{c}{\sqrt{E + m}\ [ \sin \mu J_{\nu}( k \rho ) + ( - 1 )^{n}\cos \mu J_{- \nu}( k \rho ) ]}\\{i \sqrt{E - m}\ [ \sin \mu J_{\nu + 1}( k \rho ) + ( - 1 )^{n + 1}\cos \mu J_{- ( \nu + 1 )}( k \rho ) ]}\\ \end{array}\right)
\begin{aligned}E^{*-}(f) =E(f)_{[x_j,x^*]}+C^-+\frac{\alpha^2}{4h}(h-\alpha)^2 [f'']+O(h^4)=-\frac{1}{12}\alpha^3 f^-_{xx}(\eta^-),\textrm{with}\eta^-\in [x_j, x^*], C^- ={\frac{{\alpha}^{2}}{2h}}[f]+{\frac{\alpha^2(h-\alpha)}{2h}}[f'].\end{aligned}
k_{R}= \frac{1}{N n}\sum_{\alpha = 1}^{n}\sum_{\bf i}\frac{\partial^{2}V}{\partial ( \varphi_{\bf i}^{\alpha})^{2}}= 2 J d - m^{2}+ \lambda \frac{n + 2}{N n}\sum_{\alpha = 1}^{n}\sum_{\bf i}( \varphi_{\bf i}^{\alpha})^{2}.
\dot{H}= \frac{{\dot{\varphi}}^{2}}{4 F}- \frac{1}{2}\left( H^{2}+ \frac{{\rho}_{\varphi}}{6 F}\right) - \frac{\ddot{F}}{2 F}+ \frac{p_{m}}{4 F}+ \frac{\rho_{m}}{6 F}\, ,
d\rho_i=\sum_{j\in N(i)}\omega_{ij}(S_i-S_j)\theta_{ij}(\rho) dt; & d S_i+(\sum_{j\in N(i)}\frac 12 \omega_{ij}(S_i-S_j)^2 \frac{\partial \theta_{ij}}{\partial \rho_i}+\frac 18 \frac{\partial}{\partial \rho_i}I(\rho)+\mathbb V_i+\sum_{j\in N(i)}\mathbb W_{ij}\rho_j)dt+\sigma_idW_t=0.
=\sqrt{-g}\Big(\frac{1}{8|\omega_{\cal BD}|}\phi^2R-\mbox{sgn}(\omega_{\cal BD})\frac{1}{2}g^{\mu\nu}\partial_\mu\phi\cdot\partial_\nu\phi+\Lambda(\frac{\phi^2}{4|\omega_{\cal BD}|})\frac{\phi^2}{4|\omega_{\cal BD}|}\Big).
\begin{aligned}p_a^{\varepsilon}(t,x(t),\xi(t))-p_a^{\varepsilon}(s,y,\eta) &=\int_s^t\frac{d}{d\tau}p_a^{\varepsilon}(\tau,x(\tau),\xi(\tau))d\tau &=\int_s^t\left|\frac{1}{\sqrt{\tau}}q_a^{\varepsilon}(\tau,x(\tau),\xi(\tau))\right|^2d\tau-\int_s^t\partial_\xi p_a^{\varepsilon}(\tau,x(\tau),\xi(\tau))\cdot\partial_xV(x(\tau))d\tau.\end{aligned}
L_\omega = \sum_{\omega'''}[ \sum_{\omega'\leq \omega''}\sqrt{|\omega' \omega''|}C_{\omega' \omega''}^{\omega'''}\:C_{\omega \omega'''}^0 \:\tilde{\alpha}_{\omega'}\:\tilde{\alpha}_{\omega''}+ \sum_{\omega''< \omega'}\sqrt{|\omega' \omega''|}C_{\omega' \omega''}^{\omega'''}\:C_{\omega \omega'''}^0 \:\tilde{\alpha}_{\omega''}\:\tilde{\alpha}_{\omega'}]
\vec{\phi}_{\rm TK2L}(x)= \frac{(-1)^\alpha}{4}\left(1-(-1)^\beta\tanh \frac{x+a}{2}\right)\vec{e}_1+ \frac{(-1)^\gamma}{\sqrt{2}}\sqrt{1 +(-1)^\beta \tanh \frac{x+a}{2}}\vec{e}_2
w_{\rm dou.}(x) := \int_{\Gamma_0}d\mathcal{H}^{n-1}(y) \frac 1 2 \left( (\dot \eta^0\circ \pi_1)^2 \frac{N\cdot\nu^0}{J}\, \big(J \Delta h^0)\right)(y) \, \partial_N P(x-y),
\Pi_{\mu \nu}^{( I )}={\frac{e^{2}C^{( I )}}{2}}\int{\frac{d^{n}k}{( 2 \pi )^{n}}}{\frac{( 1 - \cos p \times k )}{k^{2}( k + p )^{2}}}N_{\mu \nu}^{( I )}; I = a , b , c
g\left(x\right)\leq \widetilde{V}_{\varepsilon ,\widehat{z}}\left(0,x\right),x\in \overline{\Omega}_{0}\inf_{\varepsilon}\widetilde{V}_{\varepsilon ,\widehat{z}}\left(0,\widehat{z}\right)=g\left(\widehat{z}\right).
M^l(u^*)= \sum_Y \frac{1-c_Y}{v_{\mathcal{L}}(\mu_Y)}\int_{\Delta'_Y}p(\nu)u^*(p)P_{DH}'(p)d\sigma -\int_{\Delta^+}d_pP_{DH}'(4\rho_H)u^*(p)dp & +\int_{\Delta^+}u^*(p)(\sum \frac{\chi^{ac}(\alpha^{\vee})}{q(\alpha^{\vee})}-\bar{S}_{\Theta})P_{DH}dq.
\int_{-\infty}^{\infty}e^{-\phi(x-y)}d \mathbb P_x\left(X_{e(q)}> y\right)=\mathbb E\left[e^{\phi X_{e(q)}}\right] &=\mathbb E\left[e^{\phi \overline{X}_{e(q)}}\right]\mathbb E\left[e^{\phi \underline{X}_{e(q)}}\right]=\psi_q^+(-\phi)\psi_q^-(\phi),
\partial_{\mu}\pi_{i}^{j}\to \partial_{\mu}\pi_{i}^{j}- \left( \begin{array}{c c}{{O}}&{{- m_{1}A_{\mu}+ m_{2}\gamma_{5}B_{\mu}}}\\{{m_{1}A_{\mu}+ m_{2}\gamma_{5}B_{\mu}}}&{{O}}\\ \end{array}\right)_{i}^{j}
\eta_{J^{2}E}=\begin{bmatrix}\pi'x^{q}+a_{1}'x^{q^{2}}+a_{2}'x^{q^{3}}\Delta(\pi)x^{q^{2}}+\Delta(a_{1})x^{q^{3}}+\Delta(a_{2})x^{q^{4}}\end{bmatrix}.
P_\pm(w, \eta) \cdot P_\pm(w', \eta') = (w^{-1}, \pm iw^{-2}\eta) \cdot (w'^{-1}, \pm iw'^{-2}\eta') &=(w^{-1}w'^{-1}-k(ww')^{-2}\eta \eta', \pm i(w^{-1}w'^{-2}\eta' + w^{-2}w'^{-1}\eta))
R_{z}(90^{\circ})[\begin{matrix}1\\ 0\\ 0\\ \end{matrix}]=[\begin{matrix}cos90^{\circ}&-sin90^{\circ}&0\\sin90^{\circ}&cos90^{\circ}&0\\ 0&0&1\\ \end{matrix}][\begin{matrix}1\\ 0\\ 0\\ \end{matrix}]=[\begin{matrix}0&-1&0\\ 1&0&0\\ 0&0&1\\ \end{matrix}][\begin{matrix}1\\ 0\\ 0\\ \end{matrix}]=[\begin{matrix}0\\ 1\\ 0\\ \end{matrix}]
\left (1-\frac{1}{a}\right )\Vert M-\hat M\Vert^{2}_{2}+\mathrm{pen}(\hat M) \leq \left (1+\frac{1}{a}\right )\Vert M-A\Vert^{2}_{2}+2aV^{2}(\hat M-A)+\mathrm{pen}(A).
\mathcal{S}_{p,s}\,\|g_t(U)\|_{p^*}^p \leq [g_t(U)]_{s, p}^p\leq \int_{\mathbb{R}^{2N}}\frac{J_p(U(x)-U(y))\, \big(g_t(U(x))-g_t(U(y))\big)}{|x-y|^{N+s\,p}}\,dx\,dy &\leq \mathcal{S}_{p,s}\,\int_{\mathbb{R}^N}U^{p^*-1}\,g_t(U)\,dx.
\bar{\omega}_{1}\equiv \sum_{n=-\infty}^{\infty}\breve{\omega}_{1}(t,x^1,x^2,x^3,x^4,x^5-n) \simeq -{2 \pi m_0 a\over 3\bar{R}^3}x^2\, , \qquad\bar{\omega}_2 \simeq 0 \, ,
\eqref{va5}= -\sqrt{\tfrac{it}{2\pi}}e^{-\frac{itx^2}{2}}\tfrac{x}{1-itx^2}[T(x)\psi(x)-T(0)\psi(0)] & -\sqrt{\tfrac{it}{2\pi}}\int_+ e^{-\frac{it}{2}(\xi-x)^2}\tfrac{\xi-x}{1-it(\xi-x)^2}\partial_\xi[T(\xi)\psi(\xi)]\,d\xi & -\sqrt{\tfrac{it}{2\pi}}\int_+ e^{-\frac{it}{2}(\xi- x)^2}\tfrac{2it(\xi-x)}{[1-it(\xi-x)^2]^2}[T(\xi)\psi(\xi)-T(x)\psi(x)]\,d\xi.
(\nabla_X A)Y = B S_{JX}Y - Jh(X,BY) +\frac{1}{2}\left(JG(X,AY)-AJG(X,Y)\right), (\nabla_X B)Y = Jh(X,AY) -A S_{JX}Y +\frac{1}{2}\left(JG(X,BY)-BJG(X,Y)\right),
[\ell^j_\mu,\,q\cdot\!\nu]=-i\left[j\,\mu\cdot\nu \,\ell_\mu^{j-1}+\sum_{\rho\in\Delta_+}g_{|\rho|}(\rho\cdot\mu)(\rho^\vee\!\cdot\nu) \frac{\ell_\mu^j-\ell^j_{s_\rho(\mu)}}{\ell_\mu-\ell_{s_{\rho}(\mu)}}\check{s}_\rho\right],
*\circ\mu_{\omega}(\varphi) =*(\omega \varphi)=i_{\omega \varphi}(\frac{g^n}{n!})=(-1)^{pr+qs}i_{\varphi\omega}(\frac{g^n}{n!}) &=(-1)^{pr+qs}i_{\omega}\circ i_{\varphi}(\frac{g^n}{n!})=(-1)^{pr+qs}i_{\omega}\circ*\varphi.
G(d,kz) = \frac{2(k-1)}{k-2+\sqrt{k^2-4(k-1)(kz)^2}}\left(\frac{k-\sqrt{k^2-4(k-1)(kz)^2}}{2(k-1)kz}\right)^d= &= \underbrace{\frac{2(k-1)}{k-2+k\sqrt{1-4(k-1)z^2}}}_{A(z)}\Biggl(\underbrace{\frac{1-\sqrt{1-4(k-1)z^2}}{2(k-1)z}}_{f(z)}\Biggr)^d,
\iint_{\Omega\times\mathbb{R}^3}f_0(x,v)\psi(0,x,v) dxdv \leq \;\; \|f_0\|_{L^\infty}\iint_{\Omega\times\mathbb{R}^3}\big|\psi(0,x,v)\big| dxdv \;\leq \|f_0\|_{L^\infty}.
\sum_{n \geq 0}|p_n - e^{1/\ell}q_n| = \sum_{n \geq 0}\frac{2^{n+1}}{\ell^{n+1}\, (1 \times 3 \times 5 \cdots \times (2n+1))}= \sum_{n \geq 0}\frac{2^{2n+1}\, n!}{\ell^{n+1}\, (2n+1)!}\cdot
\langle X_{(q)}, \xi \rangle = \frac{1}{2}(2q-1)! \frac{\langle Df^{\alpha}, \xi \rangle}{1 + |Df|^2}\left( T_{(2q-1)\alpha}\right)^1_1 = - \frac{1}{2}(2q-1)!\frac{\langle Df^{\alpha}, \xi \rangle^2}{1 + |Df|^2}\left( \frac{|Df|^2}{1 + |Df|^2}\right)^{q-1}H_{(2q-1)}= - \frac{1}{2}(2q-1)!\left( \frac{|Df|^2}{1 + |Df|^2}\right)^{q}H_{(2q-1)},
\beta_{n,q}^{\left(r\right)}\left(x\right)=\frac{1}{\left(1-q\right)^{n}}\sum_{l=0}^{n}\dbinom{n}{l}\left(-1\right)^{l}q^{lx}\left(\frac{l+1}{\left[l+1\right]_{q}}\right)^{r},
\tilde{R}_{1 2}( z_{1}q / z_{2}) L_{1}^{( + )}( z_{1}) L_{2}^{( - )}( z_{2}) = L_{2}^{( - )}( z_{2}) L_{1}^{( + )}( z_{1}) \tilde{R}_{1 2}( z_{1}q^{- 1}/ z_{2}) .
\int_{0}^{x}f(t) h(t/q) \left( -\frac{q(1+q)+t}{q^2 (1-q)t}u(t/q) +\frac{1}{q(1-q)^2t}\right)y(t)\, d_qt &= \frac{f(x)}{q (1-q)}\sum_{n=0}^{\infty}q^{\frac{n(n-3)}{2}}x^{n-1}\left(1-q+q^{n+2}x\right) y(q^n x)= G(x).
\prod_{j=1}^{kd_1}\prod_{i=1}^{d_1-2}\prod_{t=1}^{i}\prod_{s=1}^{t}\begin{bmatrix}3 1\end{bmatrix}_{1,\frac{-2k d_1+3k+1-2j+2ki+2kt+2ks}{kd_1}}=\prod_{i=1}^{d_1-2}\prod_{t=1}^{i}\prod_{s=1}^{t}\begin{bmatrix}3 1\end{bmatrix}_{\frac{1}{kd_1},\frac{-3d_1+3+2i+2t+2s}{d_1}}.
w(t,x)=\frac{1}{2\pi t}\int\limits_{\Omega_{\delta}}\frac{w^e_0(y)+(y-x)\cdot\nabla w^e_0(y)}{\sqrt{t^2-|y-x|^2}}dy+\frac{1}{2\pi}\int\limits_{\Omega_{\delta}}\frac{w^e_1(y)dy}{\sqrt{t^2-|y-x|^2}}.
S_{BH}= 2\pi \left(\sqrt{N_2^{(1)}}+\sqrt{\bar{N}_2^{(1)}}\right)\left(\sqrt{N_2^{(2)}}+\sqrt{\bar{N}_2^{(2)}}\right)\left(\sqrt{N_5^{(1)}}+\sqrt{\bar{N}_5^{(1)}}\right)\left(\sqrt{N_5^{(2)}}+\sqrt{\bar{N}_5^{(2)}}\right)
Y_{j}\theta_{j i}^{a}+ Y_{i}t_{R}^{a}- t_{L}^{a}Y_{i}= 0 ~ ~ ; ~ ~ Y_{j}^{\dagger}\theta_{j i}^{a}+ Y_{i}^{\dagger}t_{L}^{a}- t_{R}^{a}Y_{i}^{\dagger}= 0
W_{\infty}^{3 , D}={\frac{1}{1 6}}( 1 - \gamma )^{- 1}\sum_{k = 0}^{\infty}( - 1 )^{k}\beta^{k}( k + 1 ) ( k + 3 ) ( k + 5 ) \sum_{m = 0}^{\infty}( m^{2 + k}- m^{1 + k}) \alpha_{m}^{- k - 7}.
E_\pi(T_1, \ldots, T_n) = E\left( T'_{q_1}\cdots T'_{q_{j-1}}R_{b}L_{b'}\left(T_{\theta}X_1 X_2 \cdots X_m\right) R_{b''}T'_{q_{j+1}}\cdots T'_{q_{k_{m+1}}}\right)
\lim_{n \to \infty}\Bigg(\frac{1}{(\sum_{i=1}^{n}\sum_{p = 1}^{\lfloor (n-i)/2 \rfloor}\binom{n}{i}\binom{n-i}{2}^p (2^i - 1)^n 2^{Q(n-i+1,n-i+1-2p)-Q(n,n)})/n}\Bigg) \ge 1.
(t_\mu^j(\gamma_{1}),\, t_\mu^j({\beta}),\, t_\mu^j({\gamma_{-1}}), \mu)= \; (\mu, t_\mu^{j-1}(\gamma_{1}),\, t_\mu^{j-1}({\beta}),\, t_\mu^{j-1}({\gamma_{-1}})) = \; (t_\mu^{j-1}(\gamma_{1}),\, t_\mu^{j-1}({\beta}),\, t_\mu^{j-1}({\gamma_{-1}}), V(\mu)) = \; (t_\mu^{j-1}(\gamma_{1}),\, t_\mu^{j-1}({\beta}),\, t_\mu^{j-1}({\gamma_{-1}}), \mu).
f\cdot g(x, y)= \underbrace{\sum_{k,l}\left( f \ast \varphi_k \otimes \varphi_l \cdot g \ast \psi_k \otimes \psi_l \right)\ast \psi_k \otimes \psi_l(x, y) +\ldots + \left( f \ast \psi_k \otimes \psi_l \cdot g \ast \psi_k \otimes \psi_l \right)\ast \varphi_k \otimes \varphi_l(x, y).}_{9}
P(R_s) = \Pr\left(\gamma_E \!>\! \frac{\gamma_B \!+\! 1}{2^{R_s}}\!-\! 1\right) = 1 \!-\! F_{\gamma_E}\left(\frac{\gamma_B \!+\! 1}{2^{R_s}}\!-\! 1\right),
H^{( \mathrm{d}\pm )}= - \frac{1}{2}\left( \sum_{j = 1}^{N - 1}\sigma_{j}^{x}\sigma_{j + 1}^{x}+ \sum_{j = 1}^{N - 1}\sigma_{j}^{z}\pm \sigma_{N}^{y}\sigma_{1}^{x}\right) .
\vert \beta_{\omega^{\prime}\omega}^{B}\vert^{2}= - \mathrm{R e}\int \! \! \! \int_{- \infty}^{\infty}d u d v \mathrm{e x p}[ i \omega u + i \omega^{\prime}f ( u ) - i \omega^{\prime}v - i \omega g ( v ) ] ,
w(C(X),\tau_{\mathcal{B}}^{w})=w_X(\mathcal{B})\cdot ib(C(X),\tau_{\mathcal{B}}^{w})=w_X(\mathcal{B})\cdot c(C(X),\tau_{\mathcal{B}}^{w}) =w_X(\mathcal{B})\cdot L(C(X),\tau_{\mathcal{B}}^{w})=w_X(\mathcal{B})\cdot d(C(X),\tau_{\mathcal{B}}^{w})=w_X(\mathcal{B})\cdot nw(C(X),\tau_{\mathcal{B}}^{w}).
\tilde{A}_{1}= C_{1},\ \tilde{A}_{2}= C_{2}+ z (\frac{\lambda}{c_{0}}t_{2}+ 1)^{-2-\frac{1}{\lambda}}E & \tilde{B}= (-t_{1}+ c +\alpha z) C_{1}+ (\lambda t_{2}+ c_{0}) C_{2}-\frac{z}{2}(\lambda +1) D + z c_{0}( \frac{\lambda}{c_{0}}t_{2}+1)^{-1-\frac{1}{\lambda}}E,
\left\langle L_{\Omega}\mathbf{r},\mathbf{n}\right\rangle =\sum_{\ell=0}^{\infty}\sum_{m\in\iota_{\ell}}\kappa_{\ell}^{m}Y_{\ell}^{m}\quad\mathbf{r}_{T}=\sum_{\ell=1}^{\infty}\sum_{m\in\iota_{\ell}}\left( r_{\ell}^{m}\mathbf{T}_{\ell}^{m.}+R_{\ell}^{m}\nabla_{\Gamma}Y_{\ell}^{m}\right)
\left[\begin{array}{cc}1 0 0 1\end{array}\right],\hspace{2mm}\left[\begin{array}{cc}0 1 1 0\end{array}\right],\hspace{2mm}\left[\begin{array}{cc}0 i -i 0\end{array}\right],\hspace{2mm}\left[\begin{array}{cc}1 0 0 -1\end{array}\right].
g=\kappa M^{2-2\Delta_{V}}\quad ,\quad \kappa =\frac{2\Gamma (\Delta_{V})}{\pi \Gamma (1-\Delta_{V})}\left( \frac{\sqrt{\pi}}{2\Gamma \left( \frac{1}{2-2\Delta_{V}}\right) \Gamma \left( \frac{h}{4-4\Delta_{V}}\right)}\right)^{2-2\Delta_{V}}.
\sum_{k=0}^N \left ( \dfrac{p}{q}\right )^k Li_{-n}^{\star k}(q)=\sum_{m=0}^n \binom{N+1}{m+1}m!\, S_{Y+1}(n,m) =\sum_{k=0}^n c_{n,N}(k) \left ( \dfrac{p}{q}\right )^k Li_{-n}^{\star k}(q).
d_{TV}(X,\pi)\leq\sup_{h\in\mathcal{H}}\sup_{l\in\mathbb{Z}^+}|m_l(h)|\sum_{j=0}^\infty\left|\mathbb{P}(X>j)-\sum_{k=j+1}^\infty\sum_{i=0}^\infty\mathbb{P}(X=i)P_{i,k}\right|.
\int_{-\infty}^{\infty}dq_-\frac{1}{q_-^2}{\rm e}^{iq_-x^-}=2\int_0^{\infty}dq_-\frac{{\rm cos}q_-x^-}{q_-^2}=2\int_0^{\infty}dq_-\frac{1}{q_-^2}-{\pi}|x^-|
Q/Q(2) \downarrow X = 3^{[r]}\otimes 1^{[r+1]}+ 3^{[r]}\otimes 1^{[s]}+ 0, Q(2)/Q(3) \downarrow X = 3^{[r]}\otimes 1^{[r+1]}+ 3^{[r]}\otimes 1^{[s]}, Q(3) \downarrow X = 4^{[r]}+ 1^{[r+1]}\otimes 1^{[s]}+ 0.
\delta K=\int_{0}^{T}\int_{\Omega^{0}}[J_{1}(\dot{u}_{\alpha}^{0}\delta\dot{u}_{\alpha}^{0}+\dot{w}^{0}\delta\dot{w}^{0})+J_{3}\dot{w}_{,\alpha}^{0}\delta\dot{w}_{,\alpha}^{0}]dAdt
\begin{array}{ccc}(0~0~0~0~-\frac{7}{2})~ \rightarrow (0~0~0~-\frac{7}{2})(\frac{7}{2}) \oplus(0~0~0~-\frac{5}{2})(-\frac{1}{2}), (0~0~0~0~-\frac{1}{2})~ \rightarrow (0~0~0~-\frac{1}{2})(\frac{1}{2}) \oplus(0~0~0~+\frac{1}{2})(-\frac{7}{2}). \end{array}
{\hat A}_{T,t}^{(i_1 i_2)q}=\frac{T-t}{2\pi}\sum_{r=1}^{q}\frac{1}{r}\left(\zeta_{2r}^{(i_1)}\zeta_{2r-1}^{(i_2)}-\zeta_{2r-1}^{(i_1)}\zeta_{2r}^{(i_2)}+\sqrt{2}\left(\zeta_{2r-1}^{(i_1)}\zeta_{0}^{(i_2)}-\zeta_{0}^{(i_1)}\zeta_{2r-1}^{(i_2)}\right)\right).
\iota: \mathrm{SO}_+(m,1) \rightarrow \mathrm{SU}(n,1) G= \left(\begin{matrix}A \mathbf{b}\mathbf{c}d\end{matrix}\right)\mapsto \left(\begin{matrix}A 0 \mathbf{b}0 I 0 \mathbf{c}0 d\end{matrix}\right)
\frac{i}{2}\int_{X}F_{\mu\nu}^{D}(*_{\Omega}F)^{\mu\nu}=-\frac{3i}{2}\int_{X}F_{\mu\nu}^{D+}F^{+\mu\nu}+\frac{i}{2}\int_{X}F_{\mu\nu}^{D-}F^{-\mu\nu}.
N(0,5,5n+4)=N(1,5,5n+4)=\cdots=N(4,5,5n+4)=\dfrac{p(5n+4)}{5}, N(0,7,7n+5)=N(1,7,7n+5)=\cdots=N(6,7,7n+5)=\dfrac{p(7n+5)}{7},
\widetilde{A}_{\mu}( x ) = 4 \pi \int \delta \xi d s \, \epsilon_{\mu \nu \rho \sigma}{\Omega}_{\xi}( s , 0 ) T^{i}{\Omega}_{\xi}^{- 1}( s , 0 ) L_{i}^{\rho \sigma}[ \xi | s ] \dot{\xi}^{\nu}( s ) \delta^{4}( x - \xi ( s ) ) .
\Psi_{1}[ R , \Theta ] \, = \,{\frac{k_{1}}{\pi}}^{1 / 2}\int d y e^{- i k_{1}y}\Theta ( y ) \Psi_{0}[ \Theta ] \exp \left[ \int d z{\frac{1}{R ( z )}}\partial_{i}\partial_{i}R ( z ) \right] \, .
\begin{array}[c]{ll}-d{p}^n(t)= \{b_x(\bar{X}^n{(t)},\bar{u}^n(t))^{}p^n(t)-\beta^{n,0}f_x(\bar{X}^n{(t)},\bar{u}^n(t))-\displaystyle\frac{\sum_{j=i}^{n}\beta^{n,j}}{n}b_x(\bar{X}^n{(t)},\bar{u}^n(t))\}dt, &p^n(t_{i})= -\beta^0\Psi_{x}(\bar{X}^n(t_n)1_{i=n}(i)+p^n(t_{i}^{+}),t\in(t_{i-1},t_{i}),\i=1,2,\ldots,n,\end{array}
{\rm Hilb}\big(3T_n^{(0)}(\Gamma_n)^{ab},t\big)=(t+1)^{n-1},{\rm Hilb}\big(3T^{(0)}\big(\Gamma_n^{\rm af\/f}\big)^{ab},t\big)=t^{-1}\big((t+1)^{n}-t-1\big),
U_{\varepsilon_{n}}^{j,k}(x_{j}+r e^{i\vartheta_{j}})=z(r)e^{i\vartheta_{j}}\,\,\,\, \, B_{\frac{T_{\varepsilon_{n}}}{20d_{k}},\frac{T_{\varepsilon_{n}}}{10d_{k}}}(x_{j})