harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|e^{e_0 t}\alpha_+ (t) |\lesssim \sum_{n=[t]}^\infty \Bigg| \int_n^{n+1}e^{e_0 s}B(R , \mathcal{Y}_-) ds \Bigg| \le \sum_{n=[t]}^\infty e^{(e_0 - c_1) n}\lesssim e^{(e_0-c_1) t},
I=\sum m_{i}\left\vert \mathbf{a}_{i}\right\vert^{2}T=\frac{1}{2}\sum m_{i}\left\vert \mathbf{\dot{a}}_{i}\right\vert^{2}\text{,\ \ \}\mathbf{\Omega}=\sum m_{i}(\mathbf{a}_{i}\times\mathbf{\dot{a}}_{i})
x^kD^n = x(x^{k-1}D^{n}) = \sum_{j\ge 0}(-1)^{j}{{k-1}\choose{j}}{{n}\choose{j}}j! (xD^{n-j})x^{k-1-j}= \sum_{j\ge 0}(-1)^{j}{{k-1}\choose{j}}{{n}\choose{j}}j! (D^{n-j}x^{k-j}-(n-j)D^{n-1-j}x^{k-1-j}).
\frac{\mathsf{C_{\mathcal{N}_{\mathcal{K}}}}}{{\mathsf{C}}_{\mathcal{N}_{[1:N]}}}\geq \begin{cases}\max \left\{\frac{1}{N}\ , \frac{1}{4}\right \}, k=1 \max \left\{\frac{k}{N}\ , \frac{1}{2}\right \}, N \geq k \geq 2 \end{cases}.
h_k[\boldsymbol{t}] := \langle 1 \mid e^{H[\boldsymbol{t}]}\psi^\ast_{k+1/2}\mid 0 \rangle = \langle 1 \mid e^{H[\boldsymbol{t}]}\psi^\ast(z) \mid 0 \rangle \mid_{z^{k}}& = \left[ e^{\sum_{q \geq 1}t_q z^q}\langle 1 \mid \psi^\ast(z) \mid 0 \rangle \right] \mid_{z^{k}}= \left[ e^{\sum_{q \geq 1}t_q z^q}\right] \mid_{z^{k}}.
\langle u , Z_s \rangle = \sum_{k = 0}^{\ell_s}b_{k,\ell_s}\langle u , \zeta(V_{s,k}) \rangle = \sum_{k = 0}^{\ell_s}(\sum_{i = k}^{\ell_s}b_{i,\ell_s})\langle u , \zeta(V_{s,k}) - \zeta(V_{s,k-1}) \rangle,
\left\|\frac{\delta^2_hu}{|h|^{\beta_\infty}}\right\|_p=\lim_{n\to +\infty}\left\|\frac{\delta^2_hu}{|h|^{\beta_n}}\right\|_p\leq C\, \|f\|_r^{\frac{1}{p-\theta}}\, \|u\|_t^{\frac{1-\theta}{p-\theta}},
D_{x_r,y_l}Q_k = -\frac{1}{g_z}\left(D_{y_l}E_{rk}-E_{rk}\frac{g_{,l,z}}{g_z}\right), D_{x_r,z}Q_k = -\frac{1}{g_z}\left(D_{z}E_{rk}-E_{rk}\frac{g_{,zz}}{g_z}\right).
\left\| Z_{n+1}-m \right\|^{2}= \left\| Z_{n}- m - \gamma_{n}\Phi (Z_{n}) \right\|^{2}+ \gamma_{n}^{2}\left\| \xi_{n+1}\right\|^{2}+2\gamma_{n}\left\langle Z_{n}-m - \gamma_{n}\Phi (Z_{n}) , \xi_{n+1}\right\rangle & \leq \left\| Z_{n}- m - \gamma_{n}\Phi (Z_{n}) \right\|^{2}+ \gamma_{n}^{2}+ 2\gamma_{n}\left\langle Z_{n}-m , \xi_{n+1}\right\rangle .
\frac{\left( e_{j}^{( \alpha )}\cdot e_{l}^{( \alpha )}\right)_{C}}{| e_{j}^{( \alpha )}|_{C}\; | e_{l}^{( \alpha )}|_{C}}= \frac{C_{j l}}{\sqrt{C_{j j}\; C_{l l}}}= c_{j l}\equiv \cos \tau_{j l}.
\gamma_{k}(x_{+}(t)) = k\sum_{r = 1}^{p-1}\binom{p-1}{r}t^{rp^{s}}e_{2r}, \gamma_{k}(x_{-}(t)) = k\sum_{r = 1}^{p-1}\binom{p-1}{r}t^{rp^{s}}e_{-2r}.
P^{\mbox{\scriptsize{p}}}\:P^{\mbox{\scriptsize{sea}}}\;=\; 0 \;=\;P^{\mbox{\scriptsize{sea}}}\:P^{\mbox{\scriptsize{p}}}\;\;\;\;\;{\mbox{and}}\;\;\;\;\;P^{\mbox{\scriptsize{a}}}\:P^{\mbox{\scriptsize{sea}}}\;=\; P^{\mbox{\scriptsize{a}}}\;=\;P^{\mbox{\scriptsize{sea}}}\:P^{\mbox{\scriptsize{a}}}\;.
W_8 \otimes W_8 = \odot^2 W_8 \oplus \Lambda^2 W_8 \cong (W_1 \oplus W_{35}) \oplus (W_7 \oplus W_{21}), W_7 \otimes W_7 = \odot^2 W_7 \oplus \Lambda^2 W_7 \cong (W_1 \oplus W_{27}) \oplus W_{21},
\langle \varphi^{2}(x) \rangle = \int d^{3}p \left[ |\psi_{0}(z_0)|^{2}|\chi_{0}|^{2}+ \int^{\infty}_{-\frac{H}{\Delta + 2}}dm \; |\psi_{m}(z_0)|^{2}|\chi_{m}|^{2}\right].
(\#S_1 + \#S_2 + \#S_3-3) + (\#S_1-1) s_1^{-1}s_2 + (\#S_2-1) s_1 s_2^{-1}+ (\#S_1+\#S_3-1) s_1 s_3^{-1}+ (\#S_2 + \#S_3-1) s_2 s_3^{-1}+ \sum_{i \not\in S_1 \cup S_2 \cup S_3}(s_1 t_i^{-1}+ s_2 t_i^{-1}+ s_3^{-1}t_i )
f^{(n,0)}(r_1...r_{n-2},r_{n-1}+2,r_n)\left(\frac{(r_n+1)^2}{(2r_n+3)(2r_n+1)}-\frac{(r_{n-1}+1)^2}{(2r_{n-1}+3)(2r_{n-1}+1)}\right)=0.
\phi A\phi A+\frac\alpha 2(A-\phi A\phi)-c(\phi^2+\theta)=c\sum^3_{a=1}\{\phi\xi_a\otimes\eta_a\phi-\xi_a\otimes\eta_a\}+c(2\xi\otimes\eta^\perp+2\xi^\perp\otimes\eta)+\left\{\frac{\alpha^2}2-2c||\xi^\perp||^2\right\}\xi\otimes\eta.
\pi=i(i-1)\cdots i'n\gamma^{(1)}(n-1)\gamma^{(2)}\cdots\gamma^{(d-1)}(n-d+1)\gamma^{(d)}(n-d-1)\cdots\gamma^{(e-1)}(n-e) &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\gamma^{(e)}(n-d)(n-e-1)\cdots(i+1),
d \bar{s}^{2}\equiv \mathrm{e}^{2 \phi}d s^{2}= \Gamma \left[ \frac{( d t + 2 N \cos \theta \, d \varphi )^{2}}{( r + 2 M )^{2}}- \frac{d r^{2}}{r^{2}}- d \Omega^{2}\right]
(z\! -\! \zeta)\,\exp \! \left(\sum_{n,m\geq1}\frac{v_{nm}}{nm}z^{-n}\zeta^{-m}\right)\! =\! z\exp \! \left(-\sum_{k\geq1}\frac{v_{0k}}{k}z^{-k}\right)\! -\!\zeta\exp \! \left(-\sum_{k\geq1}\frac{v_{0k}}{k}\zeta^{-k}\right)
I\left( s^{\circ}\right)^{c}=\left( I\left( s\right)^{c}-\left\{\overline{n-j_{1}^{\circ}},\cdots,\overline{n-j_{m_{1}}^{\circ}}\right\}\right) \cup\left\{\overline{n-j_{1}},\cdots,\overline{n-j_{m_{1}}}\right\}.
K_{G\left( x,b;c\right)}^{-}\left( u\right) =\frac{\Gamma\left(x/b+1\right)}{\gamma\left( x/b+1,c/b\right)}K_{G\left( x,b;c\right)}^{L}\left( u\right) K_{G\left( x,b;c\right)}^{+}\left( u\right) =\frac{\Gamma\left(x/b+1\right)}{\Gamma\left( x/b+1,c/b\right)}K_{G\left( x,b;c\right)}^{U}\left( u\right) .
\langle C_\varphi^*f,f\rangle= \langle f_1+\mu f_2+\mu^2f_3,f_1+f_2+f_3\rangle = ||f_1||^2-2||f_2||||f_3||\delta_1\cos{(\theta_1-\frac{\pi}{3})}+\mu\left(||f_2||^2-2||f_1||||f_3||\delta_2\cos{(\theta_2-\frac{\pi}{3})}\right) +\mu^2\left(||f_3||^2-2||f_1||||f_2||\delta_3\cos{(\theta_3-\frac{\pi}{3})}\right).
\Phi=\sum_{\sigma=\pm}\int_0^\infty d\Omega \sum_{\{T,\lambda\}}\sum_A\left(u_{\Omega\{T,\lambda\}A}^{(\sigma)}b_{\Omega\{T,\lambda\}}^{(\sigma)A}+ b_{\Omega\{T,\lambda\}}^{(\sigma)A\dagger}u_{\Omega\{T,\lambda\}A}^{(\sigma)*}\right)\ .
\mathcal L_{\mathrm{Dirac}}(\psi) = \psi^* \left( i \partial_t + i \alpha \partial_x - M \beta \right) \psi , \mathcal L_{\mathrm{meson}}(\phi) = \frac 12 ( \partial_t \phi )^2 - \frac 12 ( \partial_x \phi )^2 - \frac12 m^2 \phi^2,
{}_{2}F_{1}( a , b ; c ; x ) \equiv F ( a , b ; c ; x ) = \sum_{k = 0}^{\infty}{\frac{( a )_{k}( b )_{k}}{( c )_{k}k !}}\, x^{k}, \quad | x | < 1 , \; c \ne 0 , - 1 , - 2 , \ldots \; ,
\big\{c\left((x_1,\ldots,x_{a-1}, x_{a}= u^{\ast}, x_{a+1},\ldots, x_t); ({u},{v})\right)~:~(x_1,\ldots, x_{a-1}, x_{a+1},\ldots, x_t) \in [r]^{t-1}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~({u},{v}) \in [r] \times [s]~~({u},{v}) \neq (u^{\ast}, v^{\ast})\big\},
M=\max\left\{\|u_0\|_{1}\|v_0\|_{1}\ ,\ \frac{(a_0+b_0)^2|\Omega|^2}{4\min\{(a_1-|\Omega|((a_3)_- +(b_3)_-))^2 , (b_2-|\Omega|((b_4)_- +(a_4)_-))^2\}}\right\}.
a_n \sim \frac{2^{\alpha+\beta+1}}{\pi}\bigg\{- \sin(\alpha\pi)\sum_{k=0}^{\infty}\frac{(-1)^k \hat{\psi}^{(2k)}(0)\Gamma(2k+2\alpha+1)}{n^{2k+2\alpha+1}(2k)!}& - (-1)^n \sin(\beta\pi) \sum_{k=0}^{\infty}\frac{(-1)^k\hat{\phi}^{(2k)}(\pi) \Gamma(2k+2\beta+1)}{n^{2k+2\beta+1}(2k)!}\bigg\}, n \rightarrow \infty,
A_{k(2j)}^* = (A_{k(2j-1)}^2 + A_{k(2j)}^2)A_{k(2j)}+ A_{k(2j-1)}[\sum_{t=1}^{j-1}(A_{k(2t-1)}A_{k(2t)}+ A_{k(2t)}A_{k(2t-1)}) \ \ \ + \sum_{t=j+1}^n(A_{k(2t-1)}A_{k(2t)}+ A_{k(2t)}A_{k(2t-1)})] \ \ \forall \k,j
\mathcal{T}_{L1}(t_n,x_i,y_j,\Delta t, \Delta x, \Delta y) = \frac{\Delta t}{2\epsilon}\; \partial_t (y_j \; \partial_x f^{\epsilon}- x_i \; \partial_y f^{\epsilon})- \; \frac{y_j \Delta x}{2}\partial_{xx}q^{\epsilon}- \frac{x_i \Delta y}{2}\; \partial_{yy}q^{\epsilon}+ \mathcal{O}(\Delta t^2) + \mathcal{O}(\Delta x^2) + \mathcal{O}(\Delta y^2).
Q(A,\dots,A) = \frac{\partial^d}{\partial t_1\cdots\partial t_d}\det(\sum_{j=1}^d t_j A)\Big|_{t=0}= \frac{\partial^d}{\partial t_1\cdots\partial t_d}(\sum_j t_j)^d \det(A)\Big|_{t=0}= d!\,\det(A).
g_{ij}=\mathbf{E}_{i}\cdot\mathbf{E}_{j}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,g^{ij}=\mathbf{E}^{i}\cdot\mathbf{E}^{j}
\begin{aligned}h_{ijkln}^{p^{\ast}}-h_{ijknl}^{p^{\ast}}=\sum_{m}h_{mjk}^{p^{\ast}}R_{miln}+ \sum_{m}h_{imk}^{p^{\ast}}R_{mjln}+ \sum_{m}h_{ijm}^{p^{\ast}}R_{mkln}&\ \ +\sum_{m}h_{ijk}^{m^{\ast}}R_{m^{\ast}p^{\ast}ln}.\end{aligned}
S_{e e}( k_{0},{\bf k}) = \cos^{2}\theta \; \frac{\not \! k + m_{1}}{k^{2}- m_{1}^{2}+ i \delta}\; + \; \sin^{2}\theta \; \frac{\not \! k + m_{2}}{k^{2}- m_{2}^{2}+ i \delta}\; ,
S_{\rm ferm}= \int\! d^2x\, \Bigl( i\bar\psi \gamma^{\mu}\partial_{\mu}\psi +{\pi F_e\over\mu}\, \bar\psi\psi +{\pi F_m\over\mu}\, \bar\psi^C \psi -{G\over 2}\, \bar\psi\gamma^{\mu}\psi\, \bar\psi\gamma_{\mu}\psi\Bigr)
\begin{aligned}J^4_{\lambda}(u,v) = \g_{[<\lambda]}^{x_0}J^4_{main}(u_\lambda,v_\lambda)+ ( g_{[<\lambda]}- g_{[<\lambda]}^{x_0})\left(M_\lambda(v) E_\lambda(u)- 2 P_\lambda(u) P_\lambda(v)\right) : = \J^{4,a}_{\lambda}(u,v)+J^{4,b}_{\lambda}(u,v).\end{aligned}
\begin{array}{r l}{{T}}&{{= X - 2 s ,}}\\{{2 S}}&{{= \displaystyle \sum_{i = 1}^{s}( x_{i}- 2 ) ( x_{i}- 3 ) + 2 \sum_{1 \le i < j \le s}( x_{i}- 1 ) ( x_{j}- 1 ) ,}}\\{{}}&{{= X^{2}- ( 2 s + 3 ) X + s ( s + 5 ) ,}}\\{{P}}&{{= X - s .}}\\ \end{array}
\widehat{J}\psi(x) =e^{-in\pi/4}\mathcal{F}\psi(x), \pi^{\operatorname*{Mp}}(\widehat{J})=J \widehat{V}_{-P}\psi(x) =e^{\frac{i}{2\hbar}Px^{2}}\psi(x), \pi^{\operatorname*{Mp}}(\widehat{V}_{-P})=V_{-P}\widehat{M}_{L,m}\psi(x) =i^{m}\sqrt{|\det L|}\psi(Lx), \pi^{\operatorname*{Mp}}(\widehat{M}_{L,m})=M_{L,m}.
r_0 - r_1 \ge \sum_{l=0}^m \dim R_{\lambda(l)^T}= \sum_{l=0}^m \frac{2m - 2l + 1}{2m + 1}\binom{2m + 1}{l}&= \sum_{l=0}^m \binom{2m + 1}{l}- 2 \sum_{l=0}^m \binom{2m}{l - 1}.
Q_{g+2-n}(a_1)-Q_{g+2-n}(a_1-\textstyle{\frac 12}) = \frac{(a_1)(a_1-\frac 12)\cdots (a_1-g-1+n)}{(g+1-n)!}& Q_{g+2-n}(a_1+1)-Q_{g+2-n}(a_1+\textstyle{\frac 32}) = \frac{-(a_1+\frac 32)(a_1+1)\cdots (a_1-g+\frac 12+n)}{(g+1-n)!}
\mathbb{E}^Q_{n,0}\left(L_n^2\right) -1 = d(n)^{-2}\sum_{i = 1}^{d(n)}\sum_{j = 1}^{d(n)}\mathbb{E}^Q_{n,0}\left(e^{\sqrt{n}a_n(z_i+z_j) -n a_n^2}\right) - 1 \leq d(n)^{-1/2}\to 0.
m_0\ge m_1=h+3, \,\, m_2=h+1,\,\,{\rm hence}\,\,\,{\rm ad}_h(C)=(m_0-h+1; m_0-h, 3, 1^{k-1}), m_0\ge m_1=m_2=h+2 ,\,\,{\rm hence}\,\,\,{\rm ad}_h(C)=(m_0-h+1; m_0-h, 2^{k}, 1^{l}).
S ={\frac{1}{2 \kappa_{1 0}^{2}}}\int d x^{1 0}\sqrt{- \hat{G}}e^{- \hat{\Phi}}\left[{\cal R}_{\hat{G}}+ \partial_{M}\hat{\Phi}\partial^{M}\hat{\Phi}-{\frac{1}{2 \cdot 3 !}}\hat{H}_{M N P}\hat{H}^{M N P}\right] ,
\langle{T^{j}_{\k}}\rangle-\langle T^{\,j}_{_{\oslash}k}\rangle \sim{\pi M^2\over 2}\Phi^2\Big({1\over c^2}u^j u_k -{c_{_{\rm S}}^2\over c^2}\gamma^j_{_k}\Big)\,\ln \Big\{{w_{\!_\odot}\over w}\Big\}.
N_{\mathcal L}(aq+r,\ell )= \sum_{s=0}^{n-\ell}2^s\binom{\ell +s}{s}\sum_{t=s}^{a}\binom{t-s+n-\ell -1}{n-\ell -1}\,N_{\mathcal L}^{\mathrm{red}}((a-t)q+r,\ell +s).
Q^-_{\perp}=\left(\frac{ig2^{-1/4}\sqrt{L}}{\pi}\right) \sum_{n,n_\perp}\frac{n_\perp}{\sqrt{n}}\left(A^{\dagger}(n,n_\perp)B(n,n_\perp) -B^{\dagger}(n,n_\perp)A(n,n_\perp)\right)\,.
\mathcal C'^\kappa = \varprojlim_n (\mathcal C^n)^\kappa = \varprojlim_n \varinjlim_i (\mathcal C_i^n)^\kappa = \varinjlim_i \varprojlim_n (\mathcal C_i^n)^\kappa = \varinjlim_i \mathcal C^\kappa = \mathcal C^\kappa,
\Phi^{(2)}(N,H,u) = \prod_{p=1}^{2m}\{H-[2m(N+u)-p][2m(N+u)-p+1]\}\times \prod_{r=1}^{n}\{[2n(N+u)-2r][2n(N+u)-2r+2]-(\alpha+\beta+1)(\alpha+\beta-1)\}\times \prod_{r=1}^n \{[2n (N+u)-2r][2n (N+u)-2r +2]-(\alpha-\beta+1)(\alpha-\beta-1)\},
\Delta_{0}^{( \widetilde{1})}= \frac 1{2 m^{2}}i p^{( \widetilde{1})}\left( i p^{( \widetilde{1})}- \varepsilon m \right) \left( 1 - \sigma_{p}^{( \widetilde{1}) 2}\right) = \Psi_{0}^{( \widetilde{1})}\cdot \overline{{\Psi}}_{0}^{( \widetilde{1})}
RQ_3(a,b;p;z)=q^{(a^2-b^2)/(2p)-(a-b)/2}\frac{\vartheta_3\left(\left(\frac{p}{2}-a\right)\pi z;q^{p/2}\right)}{\vartheta_3\left(\left(\frac{p}{2}-b\right)\pi z;q^{p/2}\right)}.
\left|\mathbb{E}(\widehat{\alpha}_{t,\varepsilon}(x)-\widehat{\alpha}_{t,\varepsilon}(y))^n\right| &\leq\frac1{(2\pi)^{n}}\int_{E^{n}}\int_{\mathbb{R}^{n}}\prod_{j=1}^n\left(e^{-\frac{\varepsilon}{2}|p_j|^2}\left|e^{-\iota p_j x}-e^{-\iota p_jy}\right|\right)\prod_{j=1}^\infty(1+4\lambda_{j}^2)^{-\frac14}dpd\ell.
\xi(s)\vert_{s=0}= (s-1)\pi^{-\frac{s}{2}}\Gamma(1+\frac{s}{2})\zeta(s)\vert_{s=0}\Leftrightarrow \xi(s) = -1\cdot 1\cdot \Gamma(1)\cdot \zeta(0) =\frac{1}{2}
{\cal T}^{( \mathrm{g r a v})}( p_{1}, p_{2}; q )^{\mathrm{o d d}}= i{\cal E}_{i}^{\mu}( p_{1}, p_{1}- q ){\cal E}_{j}^{\nu}( p_{2}, p_{2}+ q ) D_{\mu \nu}^{i j}( q )^{\mathrm{o d d}}\equiv 0
\begin{cases}\lambda_{1}= \frac{-(1 - 3 d D) + \sqrt{(1 - d D )^{2}+ 2 a d^{2}C}}{2 d C}, \lambda_{2}= \frac{-(1 - 3 d D) - \sqrt{(1 - d D )^{2}+ 2 a d^{2}C}}{2 d C}.\end{cases}
\begin{array}{llll}\nabla_{e_{1}}v=a_{1}v+b_{1}w, \nabla_{e_{1}}w=c_{1}v+d_{1}w, &\nabla_{e_{2}}v=a_{2}v+b_{2}w, \nabla_{e_{2}}w=c_{2}v+d_{2}w, &\nabla_{e_{3}}v=a_{3}v+b_{3}w, \nabla_{e_{3}}w=c_{3}v+d_{3}w, &\nabla_{e_{4}}v=a_{4}v+b_{4}w, \nabla_{e_{4}}w=c_{4}v+d_{4}w, \end{array}
\| T_{\mu}x -f\|^{2}= \langle T_{\mu}x -f , x_{2}^{*}\rangle= \mu_{t}\langle T_{t}x -f , x_{2}^{*}\rangle \leq \sup_{t}\|T_{t}x -f\|\|T_{\mu}x -f\| \leq \|x -f\| \|T_{\mu}x -f\|,
\theta(x_\gamma+\gamma(t+\lambda\tau+\mu)) =e^{-\pi \sqrt{-1}(\gamma^2(\lambda^2\tau+2\lambda t)+2\gamma \lambda x_\gamma)}\theta(x+\gamma\,t,\tau), \theta_1(u_\nu+\nu(t+\lambda\tau+\mu)) =e^{-\pi \sqrt{-1}(\nu^2(\lambda^2\tau+2\lambda t)+2 \nu\lambda u_\nu)}\theta_1(u_\nu+\nu\,t,\tau).
\left( \begin{pmatrix}1 1 0 1 \end{pmatrix}+ \begin{pmatrix}a b c d \end{pmatrix}p \right) \left( I + \begin{pmatrix}\alpha 0 \gamma \delta \end{pmatrix}p \right) = \begin{pmatrix}1 1 0 1 \end{pmatrix}+ \begin{pmatrix}a + \alpha + \gamma b + \delta c + \gamma d + \delta \end{pmatrix}p.
1=\lim\limits_{t\rightarrow +\infty}\frac{Vol(\partial B^{g^+}_p(t))}{Vol(\partial B^\mathbb{H}_o(t))}\leq\lim\limits_{t\rightarrow 0^+}\frac{Vol(\partial B^{g^+}_p(t))}{Vol(\partial B^\mathbb{H}_o(t))}=1
\Upsilon(x,t) = \sum_{j=0}^{k_1/2}b_{k_1,k_2,j}\frac{\left(\omega -\frac{\pi}{2}\right)^{k_1-2j}}{|x|^{2j}}+ O\left( \sum_{j=0}^{k_1/2}\frac{\left(\omega -\frac{\pi}{2}\right)^{k_1-2j+1}}{|x|^{2j}}+ \frac{1}{|x|^{k_1+2}}\right);
= \int{\cal D}j_{\mu}{\cal D}\lambda_{\mu}\exp \left[ - \frac{1}{8 \pi^{2}}\int d^{4}x d^{4}x^{\prime}j_{\mu}( x ) \frac{1}{( x - x^{\prime})^{2}}j_{\mu}( x^{\prime}) - i \int d^{4}x \lambda_{\mu}j_{\mu}+ 2 \zeta \int d^{4}x \cos \left( \frac{\left| \lambda_{\mu}\right|}{\Lambda}\right) \right] ,
\kappa\sum_{j\neq i}\sum_{\ell=0}^{N-2}(-1)^{\ell}\sum_{J\subset E_{ij},\, \#J=\ell}\big\{\tau( (i,j)) \widehat{K}_{\alpha+\ell\varepsilon_{i}-\varepsilon_{j}-\varepsilon_{J}}+\widehat{K}_{\alpha+(l+1) \varepsilon_{i}-\varepsilon_{J}}\tau((i,j)) \big\}.
2\kappa \int_0^\infty \frac{dp}{\pi p^2}\left(\frac{p\pi \coth p\pi}{(1+p^2)}\Re \left[\Phi_{p}^{L}(\tau )\Phi_{-p}^{L}(\tau ')\right]W_{\iji'j'}^{L(p)}(\chi)-\Phi_{0}^{L}(\tau)\Phi_{0}^{L}(\tau ')W_{\iji'j'}^{L(0)}(\chi)\right),
\langle 2,m_1 ; 1,m_2 | L,M \rangle \: \{T^{[0]}_{2,m_1}, P^{[1]}_{1,m_2 | 0}\}\qquad ( \langle 2,m_1 ; 1,m_2 | L,M \rangle \: \{S^{[0]}_{2,m_1}, P^{[1]}_{1,m_2 | 0}\})
\begin{aligned}2 \bigg\langle \sum_{j \in \mathbb{Z}}\overline{W}_{j - 1}( \overline{W}_j - \overrightarrow{W}^\ell_j ) \varphi_j , f \bigg\rangle_{L^2 (\nu^n_\rho )}= 2 \sum_{j \in \mathbb{Z}}E_{\nu^n_\rho}[ F_j ( W_j - W_{j + 1}) f ] & = - 2 \sum_{j \in \mathbb{Z}}E_{\nu^n_\rho}[ W_j (\nabla_{j , j + 1}f ) F_j ] . \end{aligned}
{\cal L}={\frac{1}{2}}D_{\mu}\phi{\overline{{D^{\mu}\phi}}}-{\frac{1}{4}}F_{\mu \nu}F^{\mu \nu}-{\frac{1}{8}}\left( | \phi |^{2}- 1 \right)^{2}, \qquad \phi = \phi_{1}+ i \phi_{2},
\Bar{\delta}^{\Phi\Phi}_{p(p-1)qqj}= \frac{\dim \Phi_{pqj}}{\dim \Phi_{(p-1)qj}}\biggl( 1-\frac{\dim \Phi_{p(q+1)j}}{\dim \Phi_{pqj}}\Bar{\delta}^{\Phi\Phi}_{ppq(q+1)j}\biggr).
\tilde{b}^0(t,x)=b^0(t,x)+\frac{1}{2}\int_0^x[a^0(t,z)-a^0_0(t)]_tdz-\frac{1}{4}(a^0(t,x)^2-a^0_0(t)^2)-\frac{1}{2}a^0_x(t,x).
-\frac{1}{2}\left[ \sum_{K,L =1}^{M}(\frac{\partial^2W}{\partial \sigma_K \partial\sigma_L})\chi_{_K}\chi_{_L}+ \sum_{k,l =1}^{N}(\frac{\partial^2 W}{\partial \phi_k\partial \phi_l})\psi_k \psi_l + cc.\right],
\int d\xi_{0}^{(3)}d\xi_{0}^{(2)}d\xi_{0}^{(1)}\mathrm{Tr}(\hat{A}_1(\xi_{0}^{(1)},\xi )\star \hat{A}_2(\xi_{0}^{(2)},\xi )\star \hat{A}_3(\xi_{0}^{(3)},\xi ))~~~\leftrightarrow ~~~\langle \Psi_1|\Psi_2\star^{W}\Psi_3\rangle
\begin{array}{c}{{\sum_{i_{1}. . . i_{N}}d^{- N}\sum_{S}\prod_{j}{}\left( \varphi_{j}, \psi_{S ( j )}\right) \left( e_{i_{P ( j )}}, e_{i_{Q ( j )}}\right) =}}\\{{= \sum_{S}\prod_{j}\left( \varphi_{j}, \psi_{S ( j )}\right) \varphi^{\left( N \right)}( P S Q^{- 1})}}\\ \end{array}
\partial_\lambda \Big (\lambda^{-\nu}\, J_{\nu}(\sqrt{\lambda}\, r) \, J_{\nu}(\sqrt{\lambda}\, r') \Big ) = - \frac{\lambda^{-\nu-\frac 12}}{2}\ \left( r\, J_{\nu+1}(\sqrt{\lambda}\, r) \J_{\nu}(\sqrt{\lambda}\, r') + r'\, J_{\nu+1}(\sqrt{\lambda}\, r') \J_{\nu}(\sqrt{\lambda}\, r) \right).
D_{r,p}(\varphi_{1}) =\{u\in\overline{D(\partial\varphi_{1})}:\varepsilon\longmapsto\varepsilon^{-r}|u-J_{\varepsilon}u|\in L^{p}(0,1,\varepsilon^{-1}\mathrm{d}\varepsilon)\}J_{\varepsilon}=(id+\varepsilon\partial\varphi_{1})^{-1}.
\overline{\Theta}_{i}^{(\partial \Omega )}(\overline{\mathbf{r}})\equiv\overline{\Theta}_{i}^{(\partial \Omega )}(\overline{\mathbf{r}}_{i})=\overline{\Theta}\left( \left\vert \mathbf{r}_{i}-\mathbf{r}_{Wi}\right\vert -\frac{\sigma}{2}\right) ,
\Vert x_n-y_i\Vert^2\le\exp\left(-2\theta_i\sum_{j=i}^{n-1}\lambda_j\right)\Vert x_i-y_i\Vert^2+2M^2(\theta_i-\theta_n)\sum_{j=i}^{n-1}\lambda_j+4M^2\sum_{j=i}^{n-1}\lambda_j^2,\textmd{for all}n\ge i.
J_+(\tilde x)=\frac{\pi}{2\vert \mbox{det}\,\psi''(\tilde x)\vert^\frac 12}(-(2\partial_z\mathcal AU_0, V_0)-(\mathcal AU_0, A_2^*V_0) -(\mathcal B A_1 U_0, V_0)+(\mathcal Q U_0,V_0))(\tilde x),
\begin{aligned}T_{q, k}((F*G)_{q}^{(g_1,g_2;h_1,h_2)})(y) &=T_{q,\mathbf{s}(g_1k,h_1)}(F)\big(\mathcal Z_{g_1}(y,\cdot)\big)T_{q,\mathbf{s}(g_2k,h_2)}(G)\big(\mathcal Z_{g_2}(y,\cdot)\big). \end{aligned}
\int_T^{2T}\zeta(\sigma_0+it)M(\sigma_0+it)\,dt =\int_T^{2T}\left[\sum_{n\leq T}\frac{1}{n^{\sigma_0+it}}+O(T^{-\frac{1}{2}})\right]\sum_m\frac{\mu(m)a(m)}{m^{\sigma_0+it}}\,dt &=\sum_{n\leq T}\sum_m\frac{a(m)\mu(m)}{(mn)^{\sigma_0}}\int_T^{2T}(mn)^{-it}\,dt+O(T^{\frac{1}{2}+\frac{1}{K'}}) &=T+O(T^{\frac{1}{2}+\frac{1}{K'}}),
W^{({D})}({\bf r}) ={\mathcal F}_{({D})}^{- 1}\left\{\widetilde{W}({\bf k}) \right\}= \int \frac{d^{{D}}k}{( 2 \pi )^{{D}}}\; e^{i{\bf k}\cdot{\bf r}}\, \widetilde{W}({\bf k}) \; ,
\begin{array}{c}\left\{\partial(t):=\Big(\alpha_1 t\cdot J \oplus\cdots\oplus\alpha_m t\cdot J\Big) \oplus\beta t\cdot J |\; t \in{\mathbb R}\right\}\hbox{for}\n=2m, \left\{\partial(t):= \Big(0 \oplus \alpha_1 t\cdot J \oplus\cdots\oplus\alpha_m t\cdot J\Big) \oplus\beta t\cdot J |\; t \in{\mathbb R}\right\}\ \hbox{for}\n=2m+1, \end{array}
\Psi_{\textrm{IPLA}}(t;i,R) = \frac{e^{-jt(2^R-1)}}{t}\left[ \frac{2\left( - \frac{jst}{\bar{L}}\right)^{\frac{K}{2}}}{(K-1)!}K_{K}\left( \sqrt{- \frac{4jst}{\bar{L}}}\right) \right]^i.
\begin{array}{l}{c ( p^{4}h + h p^{4}- \frac{1}{2}S p ( p^{2}) ( p^{2}h + h p^{2}) ) +}\\{( p^{3}h p + p h p^{3}- S p ( p^{2}) p h p ) + ( 1 + c ) G_{4}h = 0}\\ \end{array}
x y^{2}= \left[ x^{n_{c}}(x-\phi_{n_{c}}^{2}) \right]^{2}- 4\Lambda^{4}x^{2n_{c}}= x^{2n_{c}}(x-\phi_{n_{c}}^{2}-2\Lambda^{2}) (x-\phi_{n_{c}}^{2}+2\Lambda^{2}).
\hat{G}^{-}_{(5)}={}^{\star}\tilde{\hat{G}}{}^{-}_{(5)}\, ,\hspace{1cm}\tilde{\hat{G}}{}^{-}_{(5)}=5\partial\tilde{\hat{C}}{}^{-}_{(4)}-10\hat{\cal H}\hat{C}^{-}_{(2)}\, ,
I \left( a \right) = \frac{\int_{{\cal F}_{a}}\left[{\cal D}\Psi{\cal D}\overline{{\Psi}}\right] e^{-{\cal A}_{M T M}}}{\int_{{\cal F}_{0}}\left[{\cal D}\Psi{\cal D}\overline{{\Psi}}\right] e^{-{\cal A}_{M T M}}},
\mu\big(\begin{smallmatrix}b_{11}b_{12}b_{21}b_{22}\end{smallmatrix}\big) = \mu_1(b_{21}/b_{11})\, \mathcal{F}^\ast \mu_1(-b_{11}b_{12}) \,\mathcal F \, \mu_2(b_{11}) & \hskip-.35in = \mu_1(a_{21}/a_{11})\, \mathcal{F}^\ast \mu_1(-La_{11}a_{12}) \,\mathcal F \, \mu_2(\sqrt{L}a_{11}).
U_{+}=\begin{pmatrix}0 A 0 0 \end{pmatrix}, U_{-}=\begin{pmatrix}0 0 -\sigma A 0 \end{pmatrix}, V_{+}(k)=\begin{pmatrix}0 4k^2A 0 0 \end{pmatrix}, V_{-}(k)=\begin{pmatrix}0 0 -4\sigma k^2A 0 \end{pmatrix}.
\partial_y\Phi_1(t,y) =-\frac{2\mu_i}{\sqrt{t}}[\phi_i^-]'(y)-\frac{2\mu_{i+1}}{\sqrt{t}}[\phi_{i+1}^+]'(y) &=-\frac{2\mu_i}{\alpha_i-\alpha_{i+1}}\psi'\bigg(\frac{3(y/\sqrt{t}-\alpha_i)}{\alpha_i-\alpha_{i+1}}\bigg)+\frac{2\mu_{i+1}}{\alpha_i-\alpha_{i+1}}\psi'\bigg(\frac{3(y/\sqrt{t}-\alpha_i)}{\alpha_i-\alpha_{i+1}}\bigg)
\frac{m_i}{m_j}=\frac{M_{c}^{i}}{M_{c}^j}=\sqrt{\frac{(\vec{\alpha}_i \cdot \Lambda_{+}) (\vec{\alpha}_i \cdot \Lambda_{-})}{(\vec{\alpha}_j \cdot \Lambda_{-}) (\vec{\alpha}_j \cdot \Lambda_{+})}}.
\hat{\hat{k}}_{(m)}{}^{\hat{\hat{\mu}}}\partial_{\hat{\hat{\mu}}}= \partial_{m}\hspace{.5cm}\rightarrow\hspace{.5cm}\hat{\hat{k}}_{(m)}{}^{\hat{\hat{\mu}}}\hat{\hat{k}}_{(n)}{}^{\hat{\hat{\nu}}}\hat{\hat{g}}_{\hat{\hat{\mu}}\hat{\hat{\nu}}}=G_{nm}\; ,
S_{f i}= \overline{{{u}}}_{R \alpha^{^{\prime}}}^{( 0 )}( p_{1}^{\prime}) \Gamma_{R}^{( 0 ) \mu}( p_{1}^{\prime}, p_{1}) u_{R \alpha}^{( 0 )}( p_{1}) i D_{R \mu \nu}^{( 0 )}( k )
\frac{\mathbb E\gamma_G}{v(G)}=\frac1{v(G)}\frac{\sum km_k(G)}{\sum m_k(G)}=\frac12\left(1-\frac1{v(G)}\frac{|\mu'(G,\sqrt{-1})|}{|\mu(G,\sqrt{-1})|}\right)= =\frac12\left(1-\sqrt{-1}\int\frac{\mathrm d\rho_G(x)}{\sqrt{-1}-x}\right)=\frac12\int\frac{x^2}{1+x^2}\mathrm d\rho_G(x).
{\mathcal P}_j(p)= \sum_{\substack{X^{\alpha_{j-1}}< q \leq X^{\alpha_{j}}q}}\frac{\chi_{8p}(q)\lambda_f(q)}{\sqrt{q}},{\mathcal M}_j(p, \alpha) = E_{e^2\alpha^{-3/4}_j}\Big (\alpha{\mathcal P}_j(p) \Big ),{\mathcal M}(p, \alpha)= (\log X)^{1/2}\prod^{\mathcal{J}}_{j=1}{\mathcal M}_j(p, \alpha).
\eta_{X, Y}^\lambda(\mathsf{P}) = \sum_{i,j}q^{(\lambda,\lambda_{i})}(2c_{j}^{i}-\delta_{j}^{i})\sum_{m,n,o,p}\pi(S(X_{(1)}))_{m}^{j}c_{n}^{m}\pi(X_{(2)}S(Y_{(1)}))_{o}^{n}c_{p}^{o}\pi(Y_{(2)})_{i}^{p}.
\prod_{\rho \in CP_{\underline{r},n}}\prod_{i \geq 1}\rho_{i}=\prod_{j \geq 1}j^{V_{\underline{r},j,n}}=\prod_{j \not\equiv 0\!\!\!\! \pmod{\underline{r},}}j^{V_{\underline{r},,j,n}}.
\begin{aligned}\int_\Omega \left[|\nabla t_\varepsilon|^2+c_1\frac{t_\varepsilon^2}{\varepsilon^2}\right]\leq C\varepsilon\left(1+\int_\Omega \left[|\nabla t_\varepsilon|^2+|\nabla\psi_\varepsilon|^2+\frac{t_\varepsilon^2}{\varepsilon^2}\right]\right)^{1/2}+C\|t_\varepsilon\|_\infty\left(1+\int_\Omega|\nabla\psi_\varepsilon|^2\right).\end{aligned}
h(S_3^n)-h(Y_2^n|X_1^n) =h(X_2^n+\eta N_3^n)-h(X_2^n+\tfrac{1}{\sqrt{b}}Z_2^n)-\tfrac{n}{2}\log b &=h(X_2^n+\eta N_3^n)-h(X_2^n+\eta N_3^n+V^n)-\tfrac{n}{2}\log b &=-I(V^n;X_2^n+\eta N_3^n+V^n)-\tfrac{n}{2}\log b &\leq -nI(V;X_{2,G}+\eta N_3+V)-\tfrac{n}{2}\log b
[H]\hat\otimes_{A}[\lambda_k] \;= \;[H]\hat\otimes_A \big( [\lambda_1]\hat\otimes_{A_e}[\lambda_{k-1}] \big) \; =\; (-1)^{k-1}\big( [H]\hat\otimes_{A}[\lambda_1] \big) \hat\otimes_{A_e}[\lambda_{k-1}]\;,
\chi_{1,0}^{m,0}(x|\rho) =\sum_{i=0}^{\infty}\rho^{i}T_{i+m}(x)\allowbreak=\allowbreak\frac{T_{m}(x)-\rho T_{m-1}(x)}{w_{1}(x|\rho)}, \chi_{0,1}^{0,m}(x|\rho) =\sum_{i=0}^{\infty}\rho^{i}U_{i+m}(x)=\frac{U_{m}(x)-\rho U_{m-1}(x)}{w_{1}(x|\rho)},
\widehat{h}_1^{(n)}= (2n^2)^{-1-s}\int_{0}^{4n^2}\left( y - \frac{\int_{0}^{4n^2}\frac{t^{s + 1}}{( t+ u)^2}dt}{\int_{0}^{4n^2}\frac{t^{s}}{( t + u)^2}dt}\right)^2 dy .
a_n = \sum_{j=0}^n h_1(j)h_2(j)^j \prod_{i=j+1}^n f(i) \equiv \sum_{j=n-kp+1}^n h_1(j)h_2(j)^j \prod_{i=j+1}^n f(i) & = \sum_{j=0}^{kp-1}h_1(n-j)h_2(n-j)^{n-j}\prod_{i=0}^{j-1}f(n-i) \pmod{p^k}.
\begin{array}{rcl}\displaystyle{\frac{dD(t)}{dt}}= - r(\beta) D(t) + I_4(t) \displaystyle{\frac{dE(t)}{dt}}= r(\beta) D(t) - O_5(t) \displaystyle{\frac{dF(t)}{dt}}= r(\beta) D(t) - O_6(t).\end{array}
\xi\equiv\zeta{\rm e}^{\frac{g_m^2}{2}D_{m_H}(0)}=\frac{m_W^{7/2}}{g}\delta\left(\frac{\lambda}{g^2}\right){\rm e}^{\frac{2\pi m_W}{g^2}\left(-2\epsilon+{\rm e}^{-c}\right)}
\begin{cases}\left\{a_1-2\frac{\chi_1}{d_3}k-|\Omega||a_3|\right\}\left\{b_2-2\frac{\chi_2}{d_3}l-|\Omega||b_4| \right\}\cr> \left(|a_2|+|\Omega||a_4|+l\frac{\chi_1}{d_3}\right)\left(|b_1|+|\Omega||b_3|+k\frac{\chi_2}{d_3}\right) .\end{cases}
\int_0^T \|\nabla (T_n(b_n))^{1-\frac{\lambda}{2}}\|_2^2\; dt = C(\lambda)\int_0^T \int_{\Omega}\frac{|\nabla T_n(b^n)|^2}{(T_n(b^n))^{\lambda}}\;dx \; dt &=C(\lambda)\int_0^T \int_{\Omega}\frac{T_n(b^n)}{(b^n)^{1+\lambda}}|\nabla T_n(b^n)|^2\;dx \; dt \le C(\lambda^{-1}).
-1 +2\sum_{m=0}^\infty\left(\frac{q^{6m+1}}{1-q^{6m+1}}-\frac{q^{6(m+1)-1}}{1-q^{6(m+1)-1}}+\frac{q^{6m-2}}{1+q^{6m-2}}-\frac{q^{6(m+1)+2}}{1+q^{6(m+1)+2}}\right) & =-\frac{[q^{10},-q^3,-q^3;q^6]_\infty(q^6;q^6)^2_\infty}{[q,q,-1/q^2,-q^8;q^6]_\infty},
\bigl( t^{l'}_{n'\,\underline{m'}}(Z), t^l_{m\underline{n}}(Z^{-1}) \cdot N(Z)^{-1}\bigr)_R= -\bigl(t^l_{m\underline{n}}(Z^{-1}) \cdot N(Z)^{-1},t^{l'}_{n'\,\underline{m'}}(Z)\bigr)_R= \delta_{ll'}\delta_{mm'}\delta_{nn'},