harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Omega = -{\cal H}\mathrm{d}z \wedge \mathrm{d}\bar{z}+ \bar{p}\mathrm{d}z \wedge \mathrm{d}\varphi + p \mathrm{d}\varphi \wedge \mathrm{d}\bar{z}\quad \mathrm{w i t h}\quad{\cal H}: = p v + \bar{p}\bar{v}-{\cal L}.
c_xc(d_{x^*}) \equiv1+2((\delta_{\gamma}+x)p+1)+2((\delta_{\gamma}+x^*)p+1)\mod 4 &\equiv 1+2x+2x^*\mod 4 &\equiv 1+2(x+x^*)\mod 4 &=1+2(x-x^*)\mod 4 &=1+2w_{x,x^*}\mod 4.
\frac{a^{2}}{2 ( \cot B + \cot C )}= \frac{a^{2}}{2 ( \frac{\cos B}{\sin B}+ \frac{\cos C}{\sin C})}= \frac{a^{2}}{\frac{2 \cos B \sin C + \cos C \sin B}{\sin B \sin C}}= \frac{\frac{a^{2}b \sin C}{2 R}}{\frac{a}{R}}= \frac{1}{2}a b \sin
\mu_{tf_k}(K^c) = \mu_{tf_k}(\bigcup_{m \in \mathbb{N}}\{x \in \Sigma : x_m > n_m\}) \leq \sum_{m \in \mathbb{N}}\sum_{i > n_m}\mu_{tf_k}(\{x \in \Sigma : x_m = i\}) = \sum_{m \in \mathbb{N}}\sum_{i > n_m}\mu_{tf_k}([i]) \,.
\hat{a}(n-1) \bigl\lbrack a(n-1) y_{n-1}+ b(n-1) y_{n}\bigr\rbrack + \hat{b}(n-1) \bigl\lbrack a(n) y_{n}+ b(n) y_{n+1}\bigr\rbrack ={\hat{d}}(x) d(x) y_{n}.
g_1^1(x_1,x_2) :=\frac{\chi_{2R}(x_1,x_2)}{|2R|}\iint_{R}f_1(y_1,y_2)dy_1dy_2, f_1^1(x_1,x_2) := f_1(x_1,x_2)- g_1^1(x_1,x_2), \alpha_1^1 :=\|f_1^1\|_\infty |2R|.
\left\{{\frac{D e t^{\prime}\left[ - ( d^{2}/ d \tau^{2}) + V^{\prime \prime}[ x_{c 1}( \tau ) ] \right]}{D e t \left[ - ( d^{2}/ d \tau^{2}) + \nu^{2}\right]}}\right\}^{- 1 / 2}s q r t{{\frac{S_{1}}{2 \pi}}}\d \tau_{c}
\lim_{\delta\downarrow0}\limsup_{n\to\infty}\biggl|{\lambda}_{n}^{-(d-1)}\sum_{\mathbf{i}\in[\partial R_n]^{\delta{\lambda}_{n}}\cap\mathbb{Z}^d}\bigl|\theta_n(\mathbf{i})\bigr|^2 - \sigma_{\mathrm{EE}}^2\biggr| =0
II+III \rightarrow 0; IV = \int_{d(x,p)\geq \rho}(\frac{\rho^2}{4\epsilon}-n)e^{-\frac{\rho^2}{4\epsilon}-C}\rightarrow 0; I = e^{-C}\int_{|y|\leq \frac{\rho}{\sqrt{\epsilon}}}(\frac{|y|^2}{2}-n) (2\pi)^{-n/2}e^{-|y|^2/4}(1+O(\epsilon |y|^2) dy &\rightarrow \int_{\mathbb{R}^n}(\frac{|y|^2}{2}-n) (2\pi)^{-n/2}e^{-|y|^2/4}dy=0.
{\cal S}_{D ( - 1 )}= \frac{1}{2 \pi \alpha^{\prime}g_{s}}\left( \int B_{( 2 )}+ \mathrm{i}\int C_{( 2 )}\right) = - \, \frac{\mathrm{i}}{2 \pi \alpha^{\prime}g_{s}}\gamma = - \, 2 \pi \mathrm{i}\, \tau
(\frac{B_{t}}{Y_{t}}-\frac{B_{t-1}}{Y_{t-1}})+(\frac{M_{t}}{Y_{t}}-\frac{M_{t-1}}{Y_{t-1}})=(r-g)(\frac{B_{t-1}}{Y_{t-1}})+(\frac{G_{t}-T_{t}}{Y_{t}})
F_{x + \sigma}(t) = -t \arctan \left( \tfrac{t}{x+\sigma}\right) + \tfrac12 (x+ \sigma) \log \left( 1 + \tfrac{t^2}{(x + \sigma)^2}\right) & = -\frac{t^2}{2 x}\left( 1 + O(x^{-1/2}+ t^2 x^{-2}) \right).
I_{W}=(I_{CSG})_{|M\approx R\times\Sigma}=\frac{1}{\sqrt{\Lambda}}\int dt\int_{\Sigma}d^{2}x (-2\epsilon^{ij}E_{ia}\dot{\omega}_{j}^{a}+E_{t}^{a}\Psi_{a}+ \omega_{ta}{\cal G}^{a}),
\gamma_{1}\mid D^{n+m_{1}+m_{2}}\left\{\left( n+i,n+m_{1}+j\right)\mid1\leq i\leq m_{1},1\leq j\leq m_{2}\right\}& =\gamma_{2}\mid D^{n+m_{1}+m_{2}}\left\{\left( n+i,n+m_{1}+j\right)\mid1\leq i\leq m_{1},1\leq j\leq m_{2}\right\}
\sum\limits_{i=1}^k\sum\limits_{l=0}^n c^{il}_m\langle P\psi^l_{\mu_m^i,\xi_m^i}, P\psi^h_{\mu_m^j,\xi_m^j}\rangle= \int_\Omega f_\epsilon^{'}(V_m)\phi_m P\psi^h_{\mu_m^j,\xi_m^j}dx.
F (\delta_\lambda (x), d_\lambda(y,\eta) )= (\delta_\lambda(x),0 )^{-1}\star d_\lambda(y,\eta) \stackrel{\eqref{sec.two_2:eq_DstardeltaX}}{=}(d_\lambda(x,0) )^{-1}\star d_\lambda(y,\eta) = d_\lambda ((x,0)^{-1}\star(y,\eta) ) = d_\lambda (F (x, (y,\eta) ) );
h_{t^*}({\bf r}_1, \cdots,{\bf r}_4) = \frac{1}{4!}\sum_{\pi \in \mathfrak{S}_4}\phi\left(r^{(1)}_{\pi_1}, \dots, r^{(1)}_{\pi_4}\right)\phi\left(r^{(2)}_{\pi_1}, \dots, r^{(2)}_{\pi_4}\right),
\Delta_K\beta_i^K(t\wedge\theta_K) =g_i^K(t) - \frac{\|p\|_{}}{\underline{p}}A_{i+1}^K(t\wedge\theta_K)+\Delta_KM_i^K(t\wedge\theta_K) \leq \max_{0\leq j\leq 1/\delta_K}g^K_j(0)+2 C(K,L) t+\frac{\|p\|_{}}{\underline{p}}\left(\max_{0\leq j\leq 1/\delta_K-1}\beta^K_i(0)+C_1 t\right)+\varepsilon_K.
\min\{\beta_J(t), \beta_J(u), \beta_J(t'+u)\}= 0, \min\{\beta_J(t), \beta_J(t-u), \beta_J(t - t' - u)\}= 0, &\min\{\beta_J(u), \beta_J(t-u), \beta_J(t')\}= 0, \min\{\beta_J(t'+u), \beta_J(t - t' - u), \beta_J(t')\}= 0.
S^{k \; \ell}_{i \; j}\left(\begin{array}{cc}a d b c \end{array}\right) :{\cal V}_{a,b}( \theta_1 ) \otimes{\cal V}_{b,c}(\theta_2) \rightarrow{\cal V}_{a,d}( \theta_2 ) \otimes{\cal V}_{d,c}(\theta_1)
\frac{1}{2}\varepsilon_{\mu \nu \rho \sigma}\gamma_{5}( \{( \eta^{2}+ \Phi^{2}) , \hat{F}_{\rho \sigma}\}+ \hat{D}_{[ \rho}\Phi \: \: \hat{D}_{\sigma ]}\Phi ) - ( \{( \eta^{2}+ \Phi^{2}) , \hat{F}_{\mu \nu}\}+ \hat{D}_{[ \mu}\Phi \: \: \hat{D}_{\nu ]}\Phi ) = 0 .
\left(1+2^{-j}\frac{1}{m+1}\eta\sqrt{\eta^2+2m+2}+2^{-2j}\frac{\eta^2+2m+2}{2m+2}\right)^{-\frac{1}{2}}=1-2^{-j}\frac{\eta\sqrt{\eta^2+2m+2}}{2m+2}+r_1,
\sigma_{t}(\mathcal{A}(\mathcal{O})) \equiv\Delta^{it}\mathcal{A}(\mathcal{O})\Delta^{-it}=\mathcal{A}(\mathcal{O}), J\mathcal{A}(\mathcal{O})J =\mathcal{A}(\mathcal{O})^{\prime}=\mathcal{A}(\mathcal{O}^{\prime}).
A = (2a+2)(2a+3) + 2b_1(4a-2l-d+7-4z_1) - 16b_2z_1(z_1-1), B_1 = 2b_1(4a+2b_1+7) - 4b_2(4a-2l-d+9-4z_1), B_2 = - 4b_1^2 + 4b_2(4a+4b_1+9).
\mathcal L_{\mathcal H}\bar\Omega(\cdot)_l\subset\bar\Omega(\cdot)_l\ (\emph{all}l),\bar\Omega(\cdot)_l+\mathcal L_{\mathcal H}\bar\Omega(\cdot)_l=\bar\Omega(\cdot)_{l+1}\ (l\mbox{large enough})
\left\| \nabla^{k}_{x}\left( K_{1L}(t)g-\mathcal{F}^{-1}\left[e^{-\frac{\nu t |\xi|^{2 \sigma}}{2}}\cos (t |\xi|) \chi_{L}\right] \ast g\right)\right\|_{2}& \le C (1+t)^{-\frac{n}{2 \sigma}(\frac{1}{r}-\frac{1}{2})-\frac{k-\tilde{k}}{2 \sigma}-1+\frac{1}{2 \sigma}}\| \nabla^{\tilde{k}}_{x}g \|_{r},
\frac{d E(Y(u))}{du}=-2n_1 q_1^2 Y_1^2 F_1(Y) - 2n_2 q_2^2 Y_2^2 F_2(Y) \sim -2n_1q_1^2 Y_1^2 \cdot (2p_1Y_1)-2n_2q_2^2Y_2^2 \cdot (-n_1q_1^2Y_1^2) < 0,
\partial_{t_{i}}\partial_{t_{j}}\lambda_{SW}=\sum_{k=1}^{r}{C_{ij}}^{k}(t)\partial_{t_{k}}\partial_{t_{r}}\lambda_{SW}+{1\over 2\pi i}\mu\partial_{\mu}\left({x^{-2}Q_{ij}\over (\widetilde{W}_B^2-4\mu^2)^{1/2}}\right)dx+\partial_{x}(*)\,dx,
\mu_{\mathbf{R}}^{1}= \mu_{\mathbf{C}}^{1}= \mu_{\mathbf{C}}^{2}= \mu_{\mathbf{C}}^{2 \, \, \, \prime}= \mu_{\mathbf{R}}^{3}= \mu_{\mathbf{C}}^{3}= 0 \, .
\langle\frac{\partial^{2}\vec{\Psi}}{\partial x^{i}\partial x^{j}},\frac{\partial\vec{\Psi}}{\partial x^{l}}\rangle=\langle{\Gamma^{k}}_{ij}\frac{\partial\vec{\Psi}}{\partial x^{k}}+\vec{n},\frac{\partial\vec{\Psi}}{\partial x^{l}}\rangle={\Gamma^{k}}_{ij}\langle\frac{\partial\vec{\Psi}}{\partial x^{k}},\frac{\partial\vec{\Psi}}{\partial x^{l}}\rangle={\Gamma^{k}}_{ij}g_{kl}.
s=E_{\theta}xwhereE_{\theta}=(\begin{matrix}E_{\theta_{1}}\\E_{\theta_{2}}\\ .\\ .\\ .\\E_{\theta_{Nsh}}\end{matrix})andE_{\theta_{i}}=U_{i}FCT_{xy,i}T_{z,i}R_{i}forshoti=1,2,...,N_{sh}
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}+K_{c,d}E_{a,c}, M=N=1; \displaystyle \sum_{t=0}^{\min(M,N)}q_{b}^{-t(N-t)}E_{d,c}^{(N-t)}K_{c,d}^{t}E_{a,b}^{(M-t)}E_{a,c}^{(t)}, .\end{cases}
=\left( \Gamma_{0}^{+}\Gamma_{0}\right)^{-1}\Gamma_{0}^{+}\left\{-A\left( I-P \right)+A(I-P)\left( \tilde{A}-z \right)^{-1}(I-P)\left( A-zI \right)-\left( A-zI \right)P \right\}
{\cal Z}_{\mathrm{g r a n d}}^{\mathrm{4 D}}\left[ j_{\mu}\right] \stackrel{{\Lambda^{\prime}\to \infty}}{\longrightarrow}\exp \left[ - \frac{1}{8 \pi^{2}Q^{2}}\int d^{4}x d^{4}y j_{\mu}( x ) \frac{1}{( x - y )^{2}}j_{\mu}( y ) \right] .
\int d U \Upsilon_{\sigma}( U ) \Upsilon_{\tau}( U^{\dagger}) = \delta_{T_{\sigma}, T_{\tau}}C_{\sigma}= \prod_{l}\delta_{\sigma^{( l )}, \tau^{( l )}}C_{\sigma^{( l )}}= \prod_{l}\int d U_{l}\Upsilon_{\sigma^{( l )}}( U_{l}) \Upsilon_{\tau^{( l )}}( U_{l}^{\dagger}) .
\sum_{\alpha=1}^{d}{\sum_{i\in I_N(i_0)}{f(\theta_{i+\frac{1}{2}}^{\alpha,n+1})}}\leq \sum_{\alpha=1}^{d}{\sum_{i\in I_N(i_0)}{f(\theta_{i+\frac{1}{2}}^{\alpha,n})}}-\frac{\Delta t}{\Delta x}\sum_{\alpha=1}^{d}\left(F^{\alpha,n}_{i_0+N+1}-F^{\alpha,n}_{i_0-N}\right)
\mu^{\prime \prime}+ \biggl \{\frac{4 \pi^{2}}{l_{\mathrm{C}}^{2}}a^{2}\tanh^{2 / p}\biggl [ \biggl ( \frac{l_{\mathrm{C}}}{\lambda ( \eta )}\biggr )^{p}\biggl ] - \frac{a^{\prime \prime}}{a}\biggr \}\mu = 0 .
h'(z)=-\frac{y'}{y}(z) \sum_{j=1}^r \frac{1}{y^j}=-\frac{y'}{y}(z) \frac{1}{z^{r}}= \frac{g'}{g}(z) \frac{1}{z^{r}}-\frac{1}{z^{r+1}}, z\not=0.
\Phi_{p_k}(\nu^{m/p_k})=\dfrac{\nu^{m}-1}{\nu^{m/p_k}-1}=\dfrac{\frac{\nu^{m}-1}{\nu-1}}{\frac{\nu^{m/p_k}-1}{\nu-1}}=\dfrac{[m]_\nu}{[m/p_k]_\nu}.
\Omega ( q ) = \frac 1{N !}\sum_{P P^{\prime}}\int d x^{+}e^{i q_{P P^{\prime}}^{-}x^{+}}\rho ( x^{+}) l_{c}^{3}( x^{+}) e^{- 4 ( q_{P P^{\prime}}^{+})^{2}l_{c}^{2}( x^{+})}
\partial_t V^{\sigma}(m_t) = - \int_{\mathbb{R}^d}\left| \frac{\delta F}{\delta m}(m_t,x) + \frac{\sigma^2}{2}\log \left( \frac{m_t(x)}{\pi(x)}\right) - \frac{\sigma^2}{2}\operatorname{KL}(m_t|\pi) \right|^2 m_t(x) dx \,.
h=-\frac{1}{k}\int_{\Sigma_{\tau}}d\bar{\eta}d\eta~~{\cal E}(-\eta)\sum_{i}q_i E_{ii}+\frac{1}{2k}\sum_{i\neq j}\int_{\Sigma_{\tau}}d\bar{\eta}d\eta~~\left(\frac{u_j}{u_i}w_{ij}v_{ij}+\frac{u_i}{u_j}w_{ji}v_{ji}\right)\left(E_{ii}-\frac{1}{n}I\right).
\sum_{t=0}^{s_1 - 1}\beta^t \alpha_{i_1}^{u_1+p_{i_1}-1}= \Big( \sum_{t=0}^{s_1 - 1}\beta^t \alpha_{i_1}^{p_{i_1}-1}\Big) \alpha_{i_1}^{u_1}\in W_{i_1}^{(2)}\alpha_{i_1}^{u_1}\subseteq K.
\frac{\partial^n}{\partial t_i^n}g_i(t) \ll E_i^{-n}, n \geq 1, \frac{\partial^m}{\partial t_l^m}\frac{\partial^n}{\partial t_i^n}g_i(t) \asymp B_3^{-n-m}, n \geq 0, m \geq 1, i \not= l.
{\psi}_{odd}^{+}= e^{-\frac{1}{2}z_{+}^2}\left( \begin{array}{c}\sqrt{\frac{\Delta}{2}}e^{i\frac{\pi}{4}}F(-\frac{\nu}{2};\frac{1}{2};z_{+}^2) + \bar{G}z_{+}F(\frac{1-\nu}{2};\frac{3}{2};z_{+}^2) z_{+}F(1-\frac{\nu}{2};\frac{3}{2};z_{+}^2) - \bar{G}\sqrt{\frac{\Delta}{2}}e^{i\frac{\pi}{4}}F(\frac{1-\nu}{2};\frac{3}{2};z_{+}^2)\end{array}\right)
t_{2k}+t_{2k+1}=\frac{(\pi r_1/N)^{4k}}{(4k+1)!}-\frac{(\pi r_1/N)^{4k+2}}{(4k+3)!}&=\frac{(\pi r_1/N)^{4k}}{(4k+1)!}\left\{1-\frac{(\pi r_1/N)^{2}}{(4k+2)(4k+3)}\right\}&\ge\frac{(\pi r_1/N)^{4k}}{(4k+1)!}\left\{1-\frac{48}{6\cdot 8}\right\}=0
\frac{N_h^{(d)}(n)}{N^{(d)}(n)}\sim 2c\beta^4\sqrt{\pi/n}\sum_{m \geq 1}(m\pi)^2 \left(2(\pi m\beta)^2-3 \right) e^{-(\pi m\beta)^2}&= 2c\beta^{-1}n^{-1/2}\sum_{m \geq 1}m^2(2(m/\beta)^2 - 3) e^{-(m/\beta)^2},
\widehat{y_j}=\sum_{i=1}^q \widehat{\theta_i}x_{ji}, \widehat{\epsilon_j}= y_j - \widehat{y_j},j=\overline{1,N}; &Z_N = \max_{1\leq j \leq N}\epsilon_j, \widehat{Z_N}= \max_{1\leq j \leq N}\widehat{\epsilon_j}, &Z_N^* = \max_{1\leq j \leq N}|\epsilon_j|, \widehat{Z_N}^* = \max_{1\leq j \leq N}|\widehat{\epsilon_j}|.
\langle \:{\hat{\phi}}_{\bar{t}}\prod_{i \in S}\phi_{i}\: \rangle_{g , \:{\Delta_{2}}}\; = \; \frac{1}{2}\: \sum_{r = 0}^{g}\: \sum_{X \cup Y = S}{\bar{C}}_{\bar{t}}^{\alpha \beta}\: \langle \phi_{\alpha}\: \prod_{j \in X}\phi_{j}\rangle_{r}\: \langle \phi_{\beta}\: \prod_{k \in Y}\phi_{k}\rangle_{g - r}\,
\sqrt{2\omega}n\frac{d}{dz}B(z) = -z^{N}\exp (-z) L_{n}^{N}(z) \left( z\frac{d^{2}}{dz^{2}}L_{n}^{N}( z ) + ( N+1-z ) \frac{d}{dz}L_{n}^{N}(z) \right) \hphantom{\sqrt{2\omega}n\frac{d}{dz}B(z) =}{}-z^{N+1}\exp (-z) \big( L_{n-1}^{N+1}(z) \big)^{2}.
\mu_{k}^{2}M =M \oplus Me_{k}M \oplus M^{\ast}e_{k}(^{\ast}M) \mu_{k}^{2}P =\rho(P)+\displaystyle \sum_{bt,sa}\left( [btsa][(sa)^{\ast}(^{\ast}bt)]+[(sa)^{\ast}(^{\ast}(bt))](bt)(sa)\right)
{\cal L}={\rm i}\varphi^{\dagger}\partial_{+}\varphi+{\rm i}\chi^{\dagger}\partial_{-}\chi+\frac{{\rm i}}{\sqrt{2}}\left(\varphi^{\dagger}\tilde{\partial}_{m}\chi+\chi^{\dagger}\partial_{m}\varphi\right)+{\cal L}_{\rm int}(\varphi,\chi)
{\cal F}_{\epsilon}^{i}( \vec{x}, t ) = \frac{1}{2 \pi}\int d^{2}\vec{p}\; \mathrm{e}^{i \vec{p}\vec{x}}\, \tilde{\cal F}_{\epsilon}^{i}( \vec{p}, t ) , \qquad \overline{{{\tilde{\cal F}}}}{}_{\epsilon}^{i}( \vec{p}, t ) = \tilde{\cal F}_{\epsilon}^{i}( - \vec{p}, t ) ,
\Gamma_p\left(\left\langle\frac{1}{2}-\frac{2j}{p-1}\right\rangle\right)=\frac{\Gamma_p\left(\frac{1}{2}\left\langle\frac{1}{2}-\frac{2j}{p-1}\right\rangle\right)\Gamma_p\left(\frac{1}{2}\left\langle\frac{1}{2}-\frac{2j}{p-1}\right\rangle+\frac{1}{2}\right)}{\Gamma_p(\frac{1}{2})~\omega(2^{(1-p)(1-\langle\frac{1}{2}-\frac{2j}{p-1}\rangle)})}.
h ( 1 1 ; 1 2 ) = \frac 4{3 \alpha^{2}}- 1 \; \; , \; q ( 1 1 ; 1 2 ) = \frac{( 5 - 3 \alpha^{2}) ( 4 - 3 \alpha^{2})}{2 7 \alpha^{3}}\, .
v_p(f_{\alpha, \beta}(x)) = v_p(\alpha(\mathcal{B}_{d,k}(x)) + \beta) = v_p(\alpha(b_0x^d + b_1 x^{d-1}+ \ldots + b_k x^{d-k})+\beta) \geq \min \{v_p(\alpha b_0 x^d), v_p(\alpha b_1 x^{d-1}), \ldots , v_p(\alpha b_k x^{d-k}), v_p(\beta) \}= \min \{v_p(\alpha) + v_p(b_0) + d v_p(x), \ldots, v_p(\alpha) + v_p(b_k) + (d-k) v_p(x), v_p(\beta) \},
W^{1,2}_{0,rad}(B_1) =\left\{u\in W^{1,2}_0(B_1)\,\big|\, u\right\}\smallskip \mathcal{B}_1(\Omega) =\left\{u\in W^{1,2}_0(\Omega)\,\big|\,\|\nabla u\|_{L^2}\leq 1\right\}.
\mathrm{V}({\rho}_{\mathrm{N}};{\rho}_{\mathrm{N}}) \equiv \frac{e_{+}^{2}\mathrm{L}}{8{\pi}^{2}}\sum_{\stackrel{p \in \cal Z}{p \neq 0}}\frac{1}{p^{2}}{\rho}_{\mathrm{N}}( - p ){\rho}_{\mathrm{N}}( p )
S_{\mathrm{g a u g e f i e l d}}= \int d x^{0}\left\{-{\frac{i n}{2 r}}\overline{{\pi}}_{a}\pi_{a}\sum_{i}( \phi_{i}\widetilde{D}_{0}\overline{{\phi}}_{i}- \overline{{\phi}}_{i}\widetilde{D}_{0}\phi_{i}) \right\}
n_{uv}=(-\kappa t+\tau b)_{v}=(-\kappa_{v}+\lambda_{u}\tau)t+(-\kappa\lambda\tau+\tau\gamma)n+(-\lambda_{u}\kappa+\tau_{v})b, n_{vu}=(-\lambda\tau t+\gamma b)_{u}=(-\lambda_{u}\tau-\lambda\tau_{u})t+(-\kappa\lambda\tau+\tau\gamma)n+(\gamma_{u})b.
(a_{t-2}p^{t-2}+\dots+a_1 p+a_0) +(b_{t-2}t^{t-2}+\dots+b_1+b_0) \geq p^{t-1}&=(2m-2k-1-p^t-(p-1)p^{t-1})+(k-\frac{p-1}{2}p^{t-1}) &=3 p^t-k-p^t-(p-1) p^{t-1}-\frac{p-1}{2}p^{t-1}&=\frac{p^t+3 p^{t-1}}{2}-k &\geq \frac{p^t+3 p^{t-1}}{2}-(\frac{p^t+p^{t-1}}{2}-1) &=p^{t-1}+1.
k_1\geq k_2\geq\cdots\geq k_{a-j_1-j_2\cdots -j_l}>k_{a-j_1-j_2\cdots-j_l+1}=\cdots=k_{a-j_1-j_2\cdots-j_{l-1}}=l >\cdots >k_{a-j_1-j_2+1}=\cdots =k_{a-j_1}=2 >k_{a-j_1+1}=\cdots =k_{a}=1.
F_{k,m}|_{\alpha_1^{-1}}(z) = F_{k,m}(z) = q^m + \sum_{n=\ell}^\infty A_k^{(1)}(m,n)q^n, F_{k,m}|_{\alpha_2^{-1}}(z) = z^{-k}F_{k,m}\left(-\frac{1}{z}\right) = z^{-k}\sum_{n=1}^\infty A_k^{(2)}(m,n)q^{n/2}, F_{k,m}|_{\alpha_3^{-1}}(z) = (-z+1)^{-k}F_{k,m}\left(\frac{z}{-z+1}\right) = (-z+1)^{-k}\sum_{n=1}^\infty A_k^{(3)}(m,n)q^{n/2}.
V_{i}\cdot V_{j}={\vec{U}}_{i}\cdot{\vec{U}}_{j}- \sum_{r}V_{i}^{r}V_{j}^{r}+ \sum_{\ell : r i g h t}T_{i}^{\ell}T_{j}^{\ell}- \sum_{\ell : l e f t}T_{i}^{\ell}T_{j}^{\ell}.
\widetilde \phi_j=\begin{cases}\frac{\varphi^-_{|j|}}{\|\varphi^-_{|j|}\|_{L^2(0,2\pi)}}, j<0, \frac{\varphi^+_{|j|}}{\|\varphi^+_{|j|}\|_{L^2(0,2\pi)}}, j>0,\end{cases}
R_{n}:= \frac{1 \mp c_0 a^{h}}{4 \pi \,\lambda_{n_{0}}^{(1)}\, \left(1 - k^{2}\, \eta \, a^{2}\, \lambda_{n}^{(1)}\right)}\sum_{\ell \geq 1}(-ika)^{\ell+1}\boldsymbol{T}_{- \, k \, a}^{-1}\int_{B}\; \frac{\left\Vert \cdot - y \right\Vert^{\ell}}{(\ell+1)!}e_{n}^{(1)}(y) \, dy.
\omega_{0k_l}^\mathrm{h}= \left. \frac{2\pi \left(n_{0k_l}^\mathrm{h}-1 \right)}{N_\mathrm{F}^{\mathrm{h}}}\,\, \right| \,\, n_{0k_l}^\mathrm{h}= \mathop{\arg\max}\limits_{n \in \left[1,N_\mathrm{F}^{\mathrm{h}}\right]}\Re \left[ \mathcal{F}\left\lbrace \boldsymbol{\rho}_{k_l}^\mathrm{h}\right\rbrace^{N_\mathrm{F}^{\mathrm{h}}}_n \right],
M^*_\lambda (0) = (\lambda I + q I - Q)^{-1}+ P_{\lambda,1}\Psi_\lambda(q) M^*_\lambda (0) = (I - P_{\lambda,1}\Psi_\lambda (q))^{-1}(\lambda I + q I - Q)^{-1}= \frac{1}{\sqrt\lambda}(- U(q))^{-1}\Theta^{-1}+ O(1/\lambda)
\mathbb{E}_\pi(N_t)= \lfloor t/2\rfloor \pi(A_{k^*})= \lfloor t/2\rfloor \frac{|A_{k^*}|}{|V|}= \lfloor t/2\rfloor \frac{|A_{k^*}|}{2|A|}\geq \lfloor t/2\rfloor \frac{\delta}{2(d-1)}\geq \frac{\delta(t-2)}{4(d-1)}=\frac{(1-\lambda)^2}{12(d-1)}(t-2).
\psi_{{i,s}}=\psi_{i,0}\circ (\theta_t+\gamma_1)\circ \cdots \circ (\theta_t+\gamma_s)\enspace, &\psi_{i,0}\circ \mathcal{T}_{\bold{c}}=[\alpha_i]\circ{\rm{Eval}}_{\alpha_i}\ \ 1\le i \le m\enspace.
F(\eta) = \frac{\Gamma((d-2)/2)}{2\, \pi^{(d+2)/2}}\, \frac{1}{\sin \eta}\,\sec^{d-3}\bigg(\frac{\eta}{2}\bigg) \, \frac{d}{d\eta}\int_\eta^\alpha \frac{g(\zeta) \sin \zeta \, d\zeta}{\sqrt{\cos \eta - \cos \zeta}}, 0 \leq \eta \leq \alpha.
\tau_i(\alpha_0,\ldots,\alpha_n,\beta) = i + \alpha_i (\alpha_0 + \ldots + \alpha_{i-1}+ \beta) \sigma_{i,j}(\alpha_0,\ldots,\alpha_n) = i + j + \alpha_i \alpha_j + \alpha_i(\alpha_0 + \ldots + \alpha_{i-1}) + \alpha_j (\alpha_0 + \ldots + \alpha_{j-1})
(\kappa_{\tau}\frac{\partial}{\partial \kappa_{\tau}}+ \beta_{\tau}\frac{\partial}{\partial u_{\tau}}- \frac{1}{2}N \gamma_{\phi (\tau)}(u_{\tau}) + L \gamma_{\phi^2 (\tau)}(u_{\tau})) \Gamma_{R (\tau)}^{(N,L)}= \delta_{N,0}\delta_{L,2}(\kappa_{\tau}^{-2 \tau})^\frac{\epsilon_{L}}{2}B_{\tau}(u_{\tau}) ,
H_{3}^{NS}=\frac{2\sin{\theta}\cos{\theta}}{g^2\Omega(z)^2}\left(\sin{\theta}\cos{\theta}\dot{x}(z)dz-d\theta\right)\wedge\left(d\phi_1+\widetilde{A}\right)\wedge d\phi_2+\frac{e^{-x(z)}\sin^2{\theta}}{g^2\Omega(z)}d\widetilde{A}\wedge d\phi_2.
\begin{array}{lcl}\alpha_{3,0}= 36 E + 18 a E c - 8 a^3, \quad \alpha_{2,1}= 18 E c^2 - 6 a^2 c + 36 a E, \alpha_{2,1}= -18 c E + 24 a^2 + 3 a c^2, \quad \alpha_{0,3}= c^3 + 6 a c + 12 E.\end{array}
D_{i}F^{i k}+ \frac{1}{{m_{e}}^{2}}D_{i}[ 4 a R F^{i k}+ 2 b ({R^{i}}_{l}F^{l k}-{R^{k}}_{l}F^{l i}) + 4 c{R^{i k}}_{l r}F^{l r}] = 0
\sqrt{\kappa}d^{2}\leq O_{p}(n^{\frac{1}{2}-\frac{1}{p}}d^{2})\leq O_{p}(n+(d^{2})^{\frac{1}{\frac{1}{2}+\frac{1}{p}}})=O_{p}(n+d^{\frac{4}{1+\frac{2}{p}}}).
\beta - \beta_{c}\; \approx \;{\frac{k^{2}}{4}}\left[ \operatorname{l n}{\frac{4}{k^{\prime}}}\operatorname{l n}{\frac{2}{k^{\prime}}}+{\frac{1}{2}}\operatorname{l n}{\frac{2}{k^{\prime}}}+{\frac{1}{2}}\right] \; ,
\hat{\mathbb A}_{\phi}:=p\left[\begin{array}{cc}1/2 1/2 1/2 1/2 \end{array}\right]+(1-p)\left[\begin{array}{cc}0 0 1/2 1/2 \end{array}\right]=\left[\begin{array}{cc}p/2 p/2 1/2 1/2 \end{array}\right].
|v|d|v|=(v\cdot v)^{1/2}d(v\cdot v)^{1/2}=(v\cdot v)^{1/2}\frac{1}{2}(v\cdot v)^{-1/2}d(v\cdot v)=\frac{1}{2}(v\cdot dv+dv\cdot v)=v\cdot dv
\gamma_{\tau^*}(0) =x^+,\gamma_{\tau^*}(-\tau^*)=\eta(-\tau^*)+h=\bar{y}+h, x^*:=\gamma_{\tau^*}(-2\tau^*) =2\eta(-\tau^*)-\eta(0)+h=2\bar{y}-(\bar{x}-h)=2\bar{y}-x^-.
S ( D ) = \sum_{n = 0}^{\infty}{\frac{1}{( n + 1 )^{2}}}{\frac{1}{n !}}{\frac{\Gamma ( n + D - 3 )}{\Gamma ( D - 3 )}}{\frac{\Gamma ( n + 4 - D / 2 )}{\Gamma ( 4 - D / 2 )}}{\frac{\Gamma ( D / 2 )}{\Gamma ( n + D / 2 )}}.
\frac{\partial^n V}{\partial \Phi^n}=\sum_{k=1}^n 2{n-1 \choose k-1}\frac{\partial^k W}{\partial\Phi^k}\left( \frac{27}{4}\frac{\partial^{n-k+2}W}{\partial\Phi^{n-k+2}}- 12 \frac{\partial^{n-k}W}{\partial \Phi^{n-k}}\right) .
e^{-2\phi}= \left| \frac{Q_1 \sqrt{c_2}}{Q_2 \sqrt{c_1}}\frac{\sinh \left[ \sqrt{c_1}(I(A) + \tilde{c}_1 ) \right]}{\sinh \left[ \sqrt{c_2}(I(A) + \tilde{c}_2 ) \right]}\right|e^{\sqrt{2}(f_0 I(A)+f_1)}.
\widetilde{H}_{\Lambda_j\Lambda_l}=\int_{\mathcal{Y}^d}(\widetilde{\delta}_{il}+\widetilde{w}_{y_i\Lambda_l}) (\widetilde{\delta}_{ij}+\widetilde{w}_{y_i\Lambda_j})\widetilde{m}dy+\int_{\mathcal{Y}^d}\frac{\widetilde{m}_{\Lambda_l}\widetilde{m}_{\Lambda_j}}{\widetilde{m}}dy.
|IV| =\left|\int_{2x_2}^1\left[K\left(\frac{x_1}{y}\right)-K\left(\frac{x_2}{y}\right)\right]\frac{\omega(y)}{y}~dy \right| \leq \int_{2x_2}^1\left[K\left(\frac{x_2}{y}\right)-K\left(\frac{x_1}{y}\right)\right]\frac{1}{y}~dy &\leq \int_{2x_2}^1\frac{1}{x_2}\log\frac{y+x_2}{y-x_2}~dy-\int_{2x_2}^1\frac{1}{x_1}\log\frac{y+x_1}{y-x_1}&=(i)-(ii).
\sqrt{\frac{b_{0}\mathcal{P}_{1}\left( 0\right)}{b_{n}\mathcal{P}_{n}\left(0\right) \mathcal{P}_{n+1}\left( 0\right)}}=\frac{\left( -1\right)^{n}\left( \gamma +\frac{1}{2}-D\right) \sqrt{n!\left( 2\gamma +1\right)_{n}}}{\left( \gamma +\frac{1}{2}-D\right)_{n}\left( \gamma +\frac{1}{2}-D+n\right)}
A_k(x):= \sum_{i=1}^3 \sum_{t=1}^{3k-2}\sum_{s=1}^t x^{-2ki+2t+2s+6k-4}= \ \frac{(1-x^{6k-4})(1-x^{6k-2})(1-x^{6k})}{(1-x^2)(1-x^4)(1-x^{2k})}
\nu_{a b}~{\mathrm{i s ~ i n d e p e n d e n t ~ o f ~ a , b ~ , ~ w h e r e}}~ \nu_{a b}\equiv{\frac{{\widetilde b}_{a}-{\widetilde b}_{b}}{b_{a}^{*}- b_{b}^{*}}}~{\mathrm{f o r}}~ a \not = b .
\begin{array}{l}\theta_t^k = \rho_t \left( \mathfrak{Q}_{t}^{k-1}(x_{[n]}^{k}, \xi_t) \right)= \displaystyle \sum_{m \in C(n)}p_{k , m}\Phi_{m}\mathfrak{Q}_t^{k-1}(x_{[n]}^k, \xi_{m}) \mbox{and}\beta_t^k = \displaystyle \sum_{m \in C(n)}p_{k , m}\Phi_{m}\pi_{k , m}, \end{array}
\mathcal{D}^{(2)}_1= \{(1:t:u:v)\in\mathbb{CP}^3:|t|^2+|u|^2-|v|^2>1,t^2 +u^2 -v^2=1,Im(u(\overline{t}+\overline{v})>0)\}&\cup \{(0:t:u:v)\in\mathbb{CP}^3: t^2 +u^2 -v^2=0,Im(u(\overline{t}+\overline{v}))>0\}.
M = - \frac{\gamma}{2}\int r^{2}{\mathcal{V}}d^{2}x = - \frac{\gamma}{4}\int_{{\bf{R}}_{0}^{2}}\partial_{i}q_{i}d^{2}x = \frac{\gamma}{4}\sum_{s = 1}^{N}\int_{C^{s}}q_{i}n_{i}^{s}d l .
( \bar{\nabla}_{\mu}+ 2 \partial_{\mu}\bar{\phi}) ({\cal G}^{\mu \nu}- \bar{G}_{\alpha}^{\nu}h^{\alpha \mu}- \bar{G}_{\beta}^{\mu}h^{\beta \nu}) + \bar{G}^{\mu \nu}( \delta \Gamma_{\sigma \mu}^{\sigma}( h ) + 2 ( \partial_{\mu}\varphi ) ) = 0 ,
J_\gamma(t)=\sum_{\|K\|\geqslant 0}(-t)^{\|K\|+n}\sum_J \frac{(-1)^{s(J)}}{\beta(K, J)!}\cdot \frac{\partial^{||\beta(K, J)||}}{\partial w^\beta(K, J)}\left[ \widetilde \Delta \cdot w_1^{\gamma_1+1}\cdots w_n^{\gamma_n+1}\cdot \frac{\widetilde Q^K}{\widetilde q^{K+I}(J)}\right]_{w=a_J}.
(M_u)_{i,j}=\pi_{b\otimes a}^i\circ \sigma_{a,b}\circ \iota_{a\otimes b}^j = (\pi_{b}^{i_1}\otimes \pi_{a}^{i_2}) \circ \sigma_{a,b}\circ (\iota_{a}^{j_1}\otimes \iota_{b}^{j_2}) &= \begin{cases}\sigma_{a^{j_1},b^{j_2}}, j_1=i_2, \; j_2=i_1, 0_{a^{j_1}\otimes b^{j_2}, b^{i_1}\otimes a^{i_2}}, .\end{cases}
\sum_{n\ge0}n^{k}z^{n}=\sum_{j=0}^{k}\{\begin{matrix}k\\j\end{matrix}\}\frac{z^{j}\cdot j!}{(1-z)^{j+1}}=\frac{1}{(1-z)^{k+1}}\times\sum_{0\le m<k}\langle\begin{matrix}k\\m\end{matrix}\rangle z^{m+1}.
<u_1(t)>^{\widehat{}}(k)=e^{-\nu (\frac{\pi k}{h})^2(t-t_0)}<u_1(t_0)>^{\widehat{}}(k)+\sqrt{\frac{2}{h}}\frac{h}{\Pi_1\pi k}\bigg((-1)^k-1\bigg) \int_{t_0}^t e^{-\nu (\frac{\pi k}{h})^2(t-\tau)}p_1(\tau)d\tau.
\xi_V(z) := \int_z^1 \tilde{q}(t) G_V(t) \, \mathrm dt |(z)|<1, \eta_V(z) := \begin{cases}\int_1^z q(t) G_V(t) \, \mathrm dt , (z)>1, \int_{-1}^z q(t) G_V(t) \, \mathrm dt , (z)<-1.\end{cases}
\rho g_{ij}^{\prime\prime}-\rho g^{kl}g_{ik}^{\prime}g_{jl}+\frac{1}{2}\rho g^{kl}g_{kl}^{\prime}g_{ij}^{\prime}+\frac{2-n}{2}g_{ij}^{\prime}-\frac{1}{2}g^{kl}g_{kl}^{\prime}g_{ij}+Ric(g)_{ij}=0.
y - y_{0}^{I I}= - \frac{2 m^{2}p_{y}}{p_{v}}\int_{u_{0}^{I I}}^{u}\frac{d u}{\cos^{2}u}= - \frac{2 m^{2}p_{y}}{p_{v}}[ \tan u - \tan u_{0}^{I I}]
\big| det\big(\begin{bmatrix}\sin \theta \cos\phi - \sin \tilde{\theta}\cos\tilde{\phi}\sin \theta \sin\phi - \sin \tilde{\theta}\sin \tilde{\phi}\cos \theta - \cos \tilde{\theta}-s \sin \theta \sin\phi s \sin \theta \cos\phi 0 s\cos\theta\cos\phi s\cos\theta \sin \phi -s\sin\theta \end{bmatrix}\big)\big|
\sum_{i'=0}^{k_3(x)-1}f(\sigma_C^{i'}(gh(\sigma_A(x)))) - \{\sum_{i'=l_2^{k_1(x)}(h(\sigma_A(x)))}^{k_3(x)-1}f(\sigma_C^{i'}(gh(\sigma_A(x)))) +\sum_{j'=0}^{k_2^{k_1(x)}(h(\sigma_A(x)))-1}f(\sigma_C^{j'}(g(\sigma_B^{k_1(x)}(h(\sigma_A(x)))))) \}.
(\widehat m_t^m)_{t=0}^T=(m_t)_{t=0}^T, \rho_m=\mathbb{P}_m\circ (\Phi^m, \mu_m)^{-1}= \mathbb Q^m \circ (\Phi^*, m)^{-1}=\rho_1(\cdot|m) \otimes \delta_m.
{}_3F_2\bigg[\begin{matrix}\alpha \beta \gamma &\alpha-\beta+1 \alpha-\gamma+1\end{matrix}\bigg|\,1\bigg]=\frac{\Gamma(\frac12\alpha+1)\Gamma(\alpha-\beta+1)\Gamma(\alpha-\gamma+1)\Gamma(\frac12\alpha-\beta-\gamma+1)}{\Gamma(\alpha+1)\Gamma(\frac12\alpha-\beta+1)\Gamma(\frac12\alpha-\gamma+1)\Gamma(\alpha-\beta-\gamma+1)}.
R_2(t) := \{(x_3) \mid 0 \le x_3 \le t\}, R_3(t) := \{(x_3) \in R_2(t) \mid \{(x_1, x_2) \in \mathbb{R}^2 \mid (x_1, x_2, x_3) \in R_1(t)\}= \varnothing\}.
\alpha(v_n) =\left(e^{2\pi i r_n^{(1)}/m_1}u_1^{r_n^{(1)}}\right)\cdots \left(e^{2\pi i r_n^{(d)}/m_d}u_d^{r_n^{(d)}}\right) &=\left(e^{2\pi i /m_1}u_1^{r_n^{(1)}}\right)\cdots \left(e^{2\pi i /m_d}u_d^{r_n^{(d)}}\right) &=e^{2\pi i \left(\frac{1}{m_1}+\cdots+\frac{1}{m_d}\right)}v_n.
\frac{1}{| \vec{x}_{1 2}|^{\lambda}}\sim \frac{1}{d + 2 n - \lambda}\frac{1}{2^{2 n}n !}\frac{2 \pi^{d / 2}}{\Gamma ( d / 2 + n )}\, ( \partial^{2})^{n}\delta^{d}( \vec{x}_{1 2}) \, ,
{P ( \Delta s ) V [ \phi ] P ( \Delta s )}={g}{\frac{\mu^{\epsilon}}{4 !}}\sum_{n = 0}^{4}\left({4 \atop n}\right) \int \! d^{d - 1}x \langle 0 | \phi_{>}( x )^{4 - n}| 0 \rangle \phi_{<}( x )^{n}\quad .