harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
C_{\lambda}: y^2 = (x^2+1)^2 - \lambda (x^2 - 1)^2 = (1 - \lambda) \left( x^2 + \frac{1+\sqrt{\lambda}}{1-\sqrt{\lambda}}\right) \left( x^2 + \frac{1-\sqrt{\lambda}}{1+\sqrt{\lambda}}\right)
\begin{array}{r c l}{L_{3}u_{\lambda , l , m}}&{=}&{m u_{\lambda , l , m}, \nonumber}\\{L_{\pm}u_{\lambda , l , m}}&{=}&{\sqrt{l ( l + 1 ) - m ( m \pm 1 )}\u_{\lambda , l , m \pm 1},}\\ \end{array}
{\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}.
\sum_{i,j,k,\ell=1}^n \Big[ \big( \rho(i-k) + \rho(j-\ell) \big)^{2q+1}- \rho(i-k)^{2q+1}- \rho(j-\ell)^{2q+1}\Big] = O\left\{n^2 \left( \sum_{\vert i\vert < n}\vert\widetilde{\rho}(i)\vert \right) \left( \sum_{\vert i\vert < n}\vert\widetilde{\rho}(i)\vert^2 \right) \right\}\,,
P_{\mu}^5(k)= -\frac{m}{\pi}\epsilon_{\mu\nu}k^{\nu}\frac{1}{k^2\sqrt{1-\frac{4m^2}{k^2}}}\ln \left( \frac{\sqrt{1-\frac{4m^2}{k^2}}+1}{\sqrt{1-\frac{4m^2}{k^2}}-1}\right) .
\lim_{h\to0}\lim_{\alpha\to 0}\frac{i}h\int_{\Omega_\Gamma}[-h^2\Delta-1, \chi_\alpha(x_n)hD_n] u_h \overline{u_h}dx=\int_{S^*_\Gamma M \backslash S^* \Gamma}|\xi_n| d\mu^\perp,
T^{5}= \left( \begin{matrix}{{\gamma^{\mu \nu}\omega_{1 \mu}^{5 b}e_{\nu}^{b}}}&{{g^{\mu \nu}\gamma^{5}V_{\nu}( e_{\mu}^{b}l^{5 b}- \partial_{\mu}( \frac{1}{\sqrt{V^{2}}}) )}}\\{{- g^{\mu \nu}\gamma^{5}V_{\nu}( e_{\mu}^{b}\tilde{l}^{5 b}- \partial_{\mu}( \frac{1}{\sqrt{V^{2}}}) )}}&{{\gamma^{\mu \nu}\omega_{2 \mu}^{5 b}e_{\nu}^{b}}}\\ \end{matrix}\right)
\begin{array}{ll}\left. \partial_s^2{\rm Log}\left( e^{\int a_1(t,s) ds}e^{\int a_2(t,s) ds}\right) \right|_{s=0}- \left. \partial_s^2{\rm Log}\left( e^{\int a_2(t,s) ds}e^{- \int a_1(t,s) ds}\right) \right|_{s=0}= 0\end{array}
D^i_sD^j_tp\vert_{\tilde R}(w_1)=D^i_sD^j_tp(w_1), i=0,...,n_1-r_1-2\,j=0,...,r_2, &D^i_sD^j_tp\vert_{\tilde R}(w_5)=D^i_sD^j_tp(w_5), i=0,...,r_1\,j=0,...,r_2,
c_{n}= \frac{( - 1 )^{n ( n + 1 ) / 2}}{x^{n ( n - 1 ) / 2}( x^{2}; x^{2})_{\infty}^{n ( n - 1 ) / 2}}\frac{( x^{2}; x^{4}, x^{4})_{\infty}^{2 n}}{( x^{8}; x^{4}, x^{4})_{\infty}^{2 n}}.
\omega^{I \pm}= \omega_{( 0 ) \alpha \beta}^{I \pm}d X^{\alpha}\wedge d X^{\beta}+ \omega_{( 0 ) a b}^{I \pm}d \Phi^{a}\wedge d \Phi^{b}+ \mathrm{s m a l l c o r r e c t i o n s}
\theta_{\mu \nu}- \theta_{\nu \mu}= - \frac{\mathrm{i}}4 \partial_{\alpha}( \psi^{+}\{\gamma_{\alpha}, \sigma_{\mu \nu}\}\psi ) = \frac 1 2 \varepsilon_{\mu \nu \alpha \lambda}\partial_{\alpha}( \psi^{+}\gamma_{\lambda}\gamma_{5}\psi ) .
[ h_{A_2}(J),k_{A_2}(J) ] \ =\D_1 + D_2\tau + D_3\mu + D_4 \tau^2 + D_5 \tau\mu + D_6 \mu^2 + D_7 \tau^2\mu\ + & D_8 \tau\mu^2 + D_9 \mu^3 + D_{10}\tau^3\mu+ D_{11}\tau^2\mu^2 + D_{12}\tau\mu^3 \ ,
d(c', b_1) =(1-\frac{d(a,c)}{\delta})(r-d(p,c)) \geq (1-\frac{d(a,c)}{\delta})\frac{\delta}{2}=\frac{\delta}{2}-\frac{d(a,c)}{2}\tag*{Then}d(a, b_1) =d(a, c')+d(c', b_1) >(d(a, c)-\epsilon)+d(c', b_1) &>d(a,c)-\frac{d(a,c)}{2}+\frac{\delta}{2}-\frac{d(a,c)}{2}=\frac{\delta}{2}
\Omega_{2 1}= Q \bar{K}_{1}Q \bar{K}_{1}Q = \left( \begin{array}{c c}{- d_{1}d_{2}^{2}}&{u_{2}d_{1}d_{2}}\\{- d_{2}^{3}}&{u_{2}d_{2}^{2}}\\ \end{array}\right)
\bar \Phi =c_0(\sin \frac \vartheta 2)^m(\cos \frac \vartheta2)^me^{i(m\varphi -Et)}\left(\begin{array}{c}c(\cos \frac \vartheta 2)P_n^{(m-1/2,m+1/2)}A(r) c(\sin \frac \vartheta 2)P_n^{(m+1/2,m-1/2)}A(r) (\cos \frac \vartheta 2)P_n^{(m-1/2,m+1/2)}B(r) -(\sin \frac \vartheta 2)P_n^{(m+1/2,m-1/2)}B(r)\end{array}\right)
\psi_k(\xi,m)= \,(\log 16-2)\sum_{j=1}^km_j^2+\sum_{j=1}^km_j^2\log m_j^2-4\pi\sum_{j=1}^km_j^2H(\xi_j,\xi_j) &\,-4\pi\sum_{i=1}^k\sum_{j=1,j\ne i}^km_i m_jG(\xi_i,\xi_j),
\min_{\boldsymbol{y},\boldsymbol{u}}\mathcal{J}(\boldsymbol{y},\boldsymbol{u}) = \min_{\boldsymbol{y},\boldsymbol{u}}\frac{1}{2}\| \boldsymbol{y}- \boldsymbol{y_d}\|_{L^2(Q)}^2 + \frac{\alpha}{2}\| \boldsymbol{u}\|_{L^2(Q)}^2
\vec{v}_{i}(U,X,Y,\Delta)_{C}= \begin{cases}(-1)^{|\tau|}\det(U,C)_{B}\mbox{if $C = X_{\tau}\cup (Y \setminus Y_{\tau}) \cup \Delta$ for $\tau \subseteq \{1, \ldots, i\}$}0 \mbox{otherwise}. \end{cases}
\sum_{k=0}^a \binom{a}{k}\binom{b}{m+k}\binom{m+k}{d}H_k H_{b-m-k}={}\binom{a+b-d}{m+a-d}\binom{b}{d}[H_{b-m}^{(2)}-H_{a+b-d}^{(2)}+ (H_{b-m}-H_{a+b-d}+H_{b-d})(H_{b-m}-H_{a+b-d}+H_a)]
c_Ey_1 =\omega(1-\omega^2)\varphi_2,c_Ey_2=(1-\omega^2)\varphi_1+(1-\omega^2)\varphi_2, c_Ey_3 =(1-\omega)\varphi_3,c_Ey_4=(1-\omega^2)\varphi_4-\omega^2(1-\omega^2)\varphi_2.
u(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i\lambda x-\lambda^{2}t}Q(\lambda,t)d\lambda-\frac{1}{2\pi}\int_{\partial D^{+}}e^{i\lambda x-\lambda^{2}t}[\tilde{g_{1}}(\lambda^{2},t)+i\lambda\tilde{g_{0}}(\lambda^{2},t)]d\lambda,
D_{\mu \nu}^{ab}(p,p_3) = \frac{\delta^{ab}\delta^{a3}}{p^2-p_3^2+i\epsilon}\left[-g_{\mu \nu}+ p_{\mu}p_{\nu}\left((1-\delta_{p_3,0})\frac{1}{p_3^2}+\delta_{p_3,0}(1-\xi)\frac{1}{p^2+i\epsilon}\right) \right]
\lambda^i ={2(D-2)\over \kappa^{D-1}}{1\over W}{\partial W\over \partial\phi}e^{-{1\over2}A}\tilde \eta^i_-\ ,\qquad\psi^i_r =-{2(D-2)\over \kappa^{D-1}}\left({1\over W}{\partial W\over \partial\phi}\right)^2e^{-{1\over2}A}\tilde\eta^i_-\ ,\qquad \psi^i_\mu=0\ .
h_{\mu\nu}^{\rm even}= \left (\begin{array}{cccc}H_0\left ( 1 -{2M\over r}\right ) H_1 0 0 H_1 H_2\left ( 1 -{2M\over r}\right )^{-1}0 0 0 0 r^2 K(r) 0 0 0 0 r^2\sin^2\theta K(r)\end{array}\right ) e^{-i\omega t}P_l(\theta).
N_\lambda = \bigcup_{0\leq i< j\leq r-1}[\tilde{\lambda}_i+1,\tilde{\lambda}_{i+1}]\times [\tilde{\lambda}_j+1,\tilde{\lambda}_{j+1}], I_\lambda = N_\lambda\cup\left(\bigcup_i [\tilde{\lambda}_i+1,\tilde{\lambda}_{i+1}]^2\right)\setminus \left\{(i,i):1\leq i\leq n \right\}.
\sum_{A, B}\int\limits_{x}^{\infty}d \theta \frac{L_A^{>}(\theta)}{d \theta}(\Phi_{AB}* L_{A}^{>}) (\theta) \simeq \sum_{A}\int\limits_{\epsilon_A^{>}(x)}^{\epsilon_A^{>}(\infty)}d \epsilon_A^{>}\frac{\epsilon_A^{>}}{(1+ e^{\epsilon_A^{>}})}- \sum_A \int\limits_{x}^{\infty}d \theta M_A e^\theta L_A^{>}(\theta).
F_*\big((x_0,t_0),\phi(x_0,t_0),D\phi(x_0,t_0),D^2\phi(x_0,t_0)\big) &=F^*\big((x_0,t_0),\phi(x_0,t_0),D\phi(x_0,t_0),D^2\phi(x_0,t_0)\big)
\left[\omega_{WP}^{(n)}\right]^{n-4}\cap\left[\overline V^{(k+2)}\times \overline V^{(n-k)}\right]=\left[\omega_{WP}^{(k+2)}+\omega_{WP}^{(n-k)}\right]^{n-4}\cap\left[\overline V^{(k+2)}\times \overline V^{(n-k)}\right],
S_n^{\delta_3}(W,t)\le \frac{3 \delta_0}{\delta_3}\frac{n}{m_1}\theta_0^{t n}+ \frac{3{\delta_{t_*}}}{\delta_3}\sum_{k=1}^{\ell-1}\sum_{j=1}^{km_1}{Km_1}\theta_0^{t (n -j)}L_j^{\delta_{t_*}}(W,t) \, .
u_{2 \epsilon \kappa}^{\prime \prime}- \left[ \frac{\kappa \left( \kappa - 1 \right)}{r^{2}}+ \frac{\kappa}{r}\frac{V^{\prime}}{\epsilon - m - V}- V^{2}+ 2 \epsilon V - k^{2}\right] u_{2 \epsilon \kappa}+ \frac{V^{\prime}}{\epsilon - m - V}\, u_{2 \epsilon \kappa}^{\prime}= 0 \, .
\frac{d}{dz}\left\lbrace{}^{\Psi}\Phi_{p}^{(\alpha,\beta)}(b;c;z)\right\rbrace =\frac{d}{dz}\left\lbrace \sum_{n=0}^{\infty}\frac{{}^{\Psi}B_{p}^{(\alpha,\beta)}(b+n,c-b)}{B(b,c-b)}\frac{z^{n}}{n!}\right\rbrace &=\sum_{n=1}^{\infty}\frac{{}^{\Psi}B_{p}^{(\alpha,\beta)}(b+n,c-b)}{B(b,c-b)}\frac{z^{n-1}}{(n-1)!}.
0 = a_n \left[f(x_0, \dots , x_{n-1})^p - f(x_0, \dots , x_{n-1})\right] - a_0(x_0^p - x_0) &= a_n \left[g(x_0)^p - g(x_0) + (\beta_0^p x_0^p - \beta_0 x_0)+ \sum_{j=1}^{n-1}\beta_j( x_j^p- x_j)\right] - a_0(x_0^p - x_0) .
\sum_{\substack{a = 1 \gcd(a,m) = 1}}^{m}\zeta^{-ar}_m = \sum_{a = 1}^{m}\sum_{b \mid \gcd(a,m)}\mu(b)\zeta^{-ar}_m = \sum_{b \mid m}\mu(b)\sum_{a = 1}^{m/b}\zeta^{-abr}_m = m\sum_{\substack{b\mid m m\mid br}}\frac{\mu(b)}{b}
cn[\frac{2}{5}K(k);k]=\{\frac{\sqrt{5}\vartheta_{00}[q(k)^{5}]}{\vartheta_{00}[q(k)]}+1\}\{\frac{5\vartheta_{01}[q(k)^{10}]^{2}}{\vartheta_{01}[q(k)^{2}]^{2}}-1\}^{-1}
-\frac{uv_y}{v}+u_y+2uu_x-vv_x-\frac{u^2v_x}{v}&=\varepsilon u_{xx}-\frac{\varepsilon uv_{xx}}{v}-\frac{2\varepsilon u_xv_x}{v}+\frac{2\varepsilon uv_x^2}{v^2}-\tilde{\Gamma}^1_{22}-2\tilde{\Gamma}^1_{12}u-\tilde{\Gamma}^1_{11}(u^2-v^2).
\begin{aligned}\lim_{n\to\infty}k\sum_{j=1}^{n-1}\sum_{l=1}^k \frac{\binom{M_n}l(k)_l(k(j-1))_{M_n-l}}{(kn)_{M_n}}\frac 1{k(n-j+1)-l}= &k\sum_{l=1}^k\binom kl(\frac c{1-c})^l\sum_{s=1}^\infty(1-c)^{k(s+1)}\frac 1{k(s+1)-l},\ \M_n\sim cn, \c\in(0,1).\end{aligned}
W [ f(t); x_f, t_f; x_i, t_i ]\equiv \int_{q(t_i) = x_i}^{q(t_f) = x_f}{\cal D}q(t) \exp \left\{i \int_{t_i}^{t_f}d t \ [ \{\cal L}[ q(t), \dot q (t)] - f(t) q(t) \ ] \right\}.
D_ux_t^n =\int^t_u \exp\Big(\int_s^t\tilde{f}'(r,x_r^n)dr\Big)\tilde{f}'_n(s,x_s^n)K_H(s,u)ds+K_H(t,u) &= J_1^n(t,u)+J_2^n(t,u)+K_H(t,u), u<t,\lambda\times,
\begin{array}{rcl}Q = S_+ C^+ + S_- C^- +A_1 C^3 + A_2 \bar{C}_- C^- C^3+A_3 \bar{C}_+ C^+ C^3 +A_4 \bar{C}_3 C^+ C^- begin{align*}3mm] + A_5 \bar{C}_+ \bar{C}_- C^+ C^- C^3 ~,\end{array}
\left( \begin{matrix}{{a}}&{{b}}\\{{c}}&{{d}}\\ \end{matrix}\right) = \left( \begin{matrix}{{I - e_{i}}}&{{e_{i}}}\\{{e_{i}}}&{{I - e_{i}}}\\ \end{matrix}\right) \ \ \mathrm{s . t .}\ \e_{i}= \mathrm{d i a g}( \underbrace{\overbrace{0 , . . . , 0}^{i - 1}, 1 , \overbrace{0 , . . . , 0}^{d - i - 2}}_{d}) .
\left( \begin{array}{c c}{{u}}&{{i \alpha}}\\{{i \beta}}&{{\bar{u}}}\\ \end{array}\right) = \frac{1}{\sqrt{r_{+}^{2}+ r_{-}^{2}}}\left( \begin{array}{c c}{{e^{i ( r_{+}\phi + r_{-}t )}\sqrt{r^{2}+ r_{-}^{2}}}}&{{i e^{r_{-}\phi - r_{+}t}\sqrt{r_{+}^{2}- r^{2}}}}\\{{i e^{- r_{-}\phi + r_{+}t}\sqrt{r_{+}^{2}- r^{2}}}}&{{e^{- i ( r_{+}\phi + r_{-}t )}\sqrt{r^{2}+ r_{-}^{2}}}}\\ \end{array}\right) .
\left\langle (x-\alpha)_{j+1}p(x),q(x)\right\rangle_{\lambda,j,\ell}= \left\langle (x-\alpha)_{j+1}{\bf u}^{M}, p(x)q(x)\right\rangle begin{align*}2mm]= \left\langle p(x), (x-\alpha)_{j+1}q(x)\right\rangle_{\lambda,j,\ell},
\begin{matrix}\hat{N}\left| n \right> = n \left| n \right>, \hat{N}(a^\dagger \left| n \right> ) = a^\dagger (\hat{N}+ \hat{I}) \left| n \right> = (n + 1) (a^\dagger \left| n \right> ), \hat{N}(a \left| n \right> ) = a(\hat{N}- \hat{I}) \left| n \right> = (n - 1) (a \left| n \right> ), a \left| 0 \right> = 0 = \hat{N}\left| 0 \right>,\end{matrix}
\Phi_{\nu}(\phi) = N_{\nu}\cos^{\lambda}\phi\, C^{(\lambda)}_{\nu}(\sin\phi), \lambda = \tfrac{1}{2}(1+2\alpha), N_{\nu}= 2^{\lambda-\frac{1}{2}}\Gamma(\lambda) \left(\frac{\nu! (\nu+\lambda)}{\pi \Gamma(\nu +2\lambda)}\right)^{1/2},
\int_0^{\infty}\exp(-s) R (\eta(\cdot \mid s)\| \eta_{(1)}(\cdot \mid s) \otimes P^o(\cdot,\cdot)) ds &\ge R\left (\int_0^{\infty}\exp(-s) \eta(\cdot \mid s) ds\| \int_0^{\infty}\exp(-s) \eta_{(1)}(\cdot \mid s) \otimes P^o(\cdot,\cdot) ds\right) &= R(\gamma \| m \otimes P^o)
K_{i\Theta_{l}}(z) \stackrel{(z \rightarrow 0)}{\sim}-\sqrt{\frac{\pi}{\Theta_{l}\sinh \left( \pi \Theta_{l}\right)}}\,\sin\left[\Theta_{l}\ln \left( \frac{z}{2}\right) - \delta_{\Theta_{l}}\right]\,\left[ 1 + O \left( z^{2}\right) \right] \; ,
\delta A[f_{0},f_{1}]=\int_{x_{0}}^{x_{1}}[\frac{n(x,f_{0})f_{0}^{\prime}(x)f_{1}^{\prime}(x)}{\sqrt{1+f_{0}^{\prime}(x)^{2}}}+n_{y}(x,f_{0})f_{1}\sqrt{1+f_{0}^{\prime}(x)^{2}}]dx.
\int_{G_{\infty}/K_{\infty}}\phi_{\infty}(\tilde{\tau}_m(g_{\infty}\times h_{\infty}),s)\mathbf{f}(h_{\mathbf{h}}\cdot h_{\infty})\mathbf{d}h_{\infty}=c_k(s)\mathbf{f}(h_{\mathbf{h}}\cdot g_{\infty}),
\psi_{\Lambda_1}(a_\Lambda) = \sum_{i\in I}\prod_{x\in\Lambda}\hbox{Tr}(h_{x,i}h_{x,i}^{*}a_x )\beta_{ii}^{|\Lambda_1\setminus \Lambda|}+ \sum_{i\ne j}\prod_{x\in\Lambda}\hbox{Tr}(h_{x,i}h_{x,j}^{*}a_x )c^{|\Lambda_1\setminus \Lambda|},
| \mathcal E_\lambda^\gamma f(x) | = \Big|\int_I e^{i \lambda x \cdot (\gamma(ht + \tau) - \gamma(\tau))}a(x) f_h(t) dt \Big| = \Big|\int_I e^{i \lambda (M_\tau^{\gamma, k}D_h^k)^t x \cdot \gamma_\tau^h(t)}a(x) f_h(t) dt \Big|.
B_{n m}= \left( \begin{array}{c c}{0}&{- q^{- 1 / 2}}\\{q^{1 / 2}}&{0}\\ \end{array}\right) , \quad B^{n m}= \left( \begin{array}{c c}{0}&{q^{- 1 / 2}}\\{- q^{1 / 2}}&{0}\\ \end{array}\right) , \,
I_{1 1 d}= -{\frac{1}{2 \kappa_{( 1 1 )}^{\ 2}}}\int d^{1 1}x \ \left[ \sqrt{- G}\left( R +{\frac{1}{4 8}}( d A )^{2}\right) +{\frac{1}{6}}A \wedge d A \wedge d A \right] \ ,
q \, F_{\ldots , z_{i}, z_{i+1}, \ldots}^{\ldots , \mu_{i}, \mu_{i+1}, \ldots}=f(z_{i}, z_{i+1})F_{\ldots , z_{i+1}, z_{i}, \ldots}^{\ldots , \mu_{i+1}, \mu_{i}, \ldots}-g(z_{i+1}, z_{i})F_{\ldots , z_{i}, z_{i+1}, \ldots}^{\ldots , \mu_{i+1}, \mu_{i}, \ldots},
\Omega_{C_2}[1/\tau] = \left(\Omega_{C_2}\to \Sigma^{\sigma - |\sigma|}\Omega_{C_2}\to \Sigma^{2\sigma - |2\sigma|}\Omega_{C_2}\to \dots\right) & \simeq \left(MU_{C_2}^{\mathbf{C}^\infty}\to MU_{C_2}^{\mathbf{C}^{\infty,1}}\to M_{C_2}U^{\mathbf{C}^{\infty,2}}\to \dots\right) & = MU_{C_2}
t_{\mu \nu}= \frac{1}{8 \pi{\cal G}}\left[ R_{\mu \nu}- \frac{1}{2}g_{\mu \nu}R_{\alpha}^{\alpha}- R_{\mu \nu}^{( 1 )}+ \frac{1}{2}\eta_{\mu \nu}R_{\alpha}^{( 1 ) \alpha}\right] .
\sum_{d\leqslant \varepsilon_{1}^\frac{1}{2}\Xi(\alpha )^{-\frac{1}{2}}B^{1-\frac{1}{2r}}}\mu(d)\frac{\varepsilon_1-\varepsilon_2}{2(\alpha^\prime)^2 d^2}B^{2-\frac{1}{r}}=\frac{6(\varepsilon_1-\varepsilon_2)}{\pi^2 (\alpha^\prime)^2}B^{2-\frac{1}{r}}+O_{\varepsilon_i}(\Xi(\alpha)^\frac{1}{2}B^{1-\frac{1}{2r}}),
\aligned \sum_{k_1+k_2=k+1}\sum_{\beta_1+\beta_2=\beta}\sum_{i=1}^{k-k_2+1}&(-1)^*{\frak m}^{\epsilon,\rho,[0,1]^{\boxplus\tau_0}}_{k_1,\beta_1}(h_1,\ldots,{\frak m}^{\epsilon,\rho,[0,1]^{\boxplus\tau_0}}_{k_2,\beta_2}(h_i,\ldots,h_{i+k_2-1}),\ldots,h_{k}) = 0,\endaligned
+\sum_{\substack{{r_1,r_2,r_3=1}r_1\ne r_2\ne r_3}}^{n}\frac{a_{r_1}b_{r_2}c_{r_3}}{(r_3-r_2)(r_3-r_1)}\left(\frac{1}{r_3+k+1}-\frac{1}{r_1+k+1}\right).
\Omega_{a_{1}a_{2}\ldots a_{2 m + 1}}\psi_{+}^{a_{1}}\psi_{+}^{a_{2}}\ldots \psi_{+}^{a_{2 m + 1}}, \qquad \Omega_{a_{1}\ldots a_{2 m}a_{2 m + 1}}\psi_{+}^{a_{1}}\ldots \psi_{+}^{a_{2 m}}j_{+}^{a_{2 m + 1}},
\int_{\partial (G_{-}\setminus B_{-}(r,x))\setminus\Omega}\left( u \frac{\partial v}{\partial n_{-}}- v \frac{\partial u}{\partial n_{-}}\right) d\sigma + \int_{\partial (G_{-}\setminus B_{-}(r,x))\cap \Omega}\left( u \frac{\partial v}{\partial n_{-}}- v \frac{\partial u}{\partial n_{-}}\right) d\sigma= 0.
Z^{I J}= \left( \begin{array}{c c}{{Z_{1}\epsilon}}&{{0}}\\{{0}}&{{Z_{2}\epsilon}}\\ \end{array}\right) \ , \quad \mathrm{w h e r e}\quad \epsilon = \left( \begin{array}{c c}{{0}}&{{1}}\\{{- 1}}&{{0}}\\ \end{array}\right) .
H(p)=4\pi N\int \frac{d^4l}{{(2\pi)}^4}\left\{\frac{(l+p)^2-m^2}{{[(l+p)^2-m^2]}^2+\varepsilon^2}+(p\to -p) \right\}\delta(l^2-m^2)\sin^2\theta(l^0,\mu)
\sum_{y=x-n-1}^{x+n}\bar K^{(n)}_1({y,x})=\frac{1}{n+1}\sum_{j=0}^n \Phi\left(\frac{j}{n+1}, \frac{j}{n+1}\right) \cos\left(\frac{j\ell}{n+1}\right) &= G_{\omega_0}(\ell) + o(1),
\frac{\partial}{\partial t}(x^1+ix^2)=\frac{2}{(1+\xi\bar{\xi})^2}\frac{\partial\eta}{\partial t}-\frac{2\xi^2}{(1+\xi\bar{\xi})^2}\frac{\partial\bar{\eta}}{\partial t}+\frac{2\xi}{1+\xi\bar{\xi}}\frac{\partial r}{\partial t},
F_{1,j}=\Big\{x\in \prod_{i=1}^dB_{i}(y_i, 2r_n^{a_i})\colon \max_{1\le i\le j}\rho_i(x_i,y_i)<4r_n^{a_j}\Big\}, & F_{2,j}=\Big\{x\in \prod_{i=1}^dB_{i}(y_i, 2r_n^{a_i})\colon \max_{1\le i\le j-1}\rho_i(x_i,y_i)\ge r_n^{a_j}\Big\}.
(A)_0 :=(A;q)_0 =1, (A)_n :=(A;q)_n = (1-A)(1-Aq)\cdots(1-Aq^{n-1}),n \geq 1, (A)_{\infty}:=(A;q)_{\i}= \lim_{n\to\i}(A;q)_n, |q|<1, (A_1,A_2,\cdots,A_m;q)_n :=(A_1;q)_n(A_2;q)_n\cdots(A_m;q)_n.
\frac{\partial L_{+}}{\partial \theta}=-\frac{1}{1+\mathrm{e}^{\varepsilon_{+}(\theta )}}\frac{\partial \varepsilon_{+}}{\partial \theta}\sim -\frac{1}{1+\mathrm{e}^{\varepsilon_{0}}}\frac{\partial \varepsilon_{+}}{\partial \theta}\quad \mathrm{for}\quad \theta \rightarrow -\infty
\eta^{\pm}(x_{0}):=\sqrt{\frac{\lambda_{2}(x_{0})-1}{\lambda_{2}(x_{0})-\lambda_{1}(x_{0})}}\xi_{1}(x_{0})\pm\sqrt{\frac{1-\lambda_{1}(x_{0})}{\lambda_{2}(x_{0})-\lambda_{1}(x_{0})}}\xi_{2}(x_{0}),
\Gamma_{0,\left(5\right)}=\sum_{i,k(i<k)}\varphi^{ik\dagger}\gamma^{0}\left(\gamma^{5}\right)\varphi^{ik}+\sum_{p=1}^{N-1}\psi^{p}\gamma^{0}\left(\gamma^{5}\right)\psi^{p}\ ,
[\begin{matrix}M_{11}\\M_{22}\\M_{12}\end{matrix}]=-\{\int_{-h}^{h}x_{3}^{2}[\begin{matrix}C_{11}&C_{12}&C_{13}\\C_{12}&C_{22}&C_{23}\\C_{13}&C_{23}&C_{33}\end{matrix}]dx_{3}\}[\begin{matrix}w_{,11}^{0}\\w_{,22}^{0}\\w_{,12}^{0}\end{matrix}].
\rho_O(\mathcal D_O) = \rho(\mathcal D^a) \cup \lbrace \frac12 \iota(\beta^\vee) \mid \beta \in \Pi \cap \frac12 \Sigma \rbrace \cup \lbrace \iota(\beta^\vee) \mid \beta \in \Pi \setminus (\Gamma^\perp \cup \Sigma \cup \frac12 \Sigma \rbrace.
\begin{array}{r l}{{\displaystyle \varphi ( \sigma , \tau ) =}}&{{\phi ( z ) + \bar{\phi}( \bar{z}) \: ,}}\\{{\phi ( z ) =}}&{{\frac{1}{2}\varphi_{0}- i p_{+}\log z + i \sum_{k \neq 0}\frac{1}{k}a_{k}z^{- k}\: ,}}\\{{\bar{\phi}( \bar{z}) =}}&{{\frac{1}{2}\varphi_{0}- i p_{-}\log \bar{z}+ i \sum_{k \neq 0}\frac{1}{k}\bar{a}_{k}\bar{z}^{- k}\: ,}}\\ \end{array}
J^B_w(Z;X,Y) :=\left\langle A'(Y)B(Z)A(X), w\right\rangle, J^C_w(Z;X,Y) :=\left\langle A'(Y)C(Z)A(X), w\right\rangle, \ \ \mathrm{and}I^D_w(Z;X,Y) :=\left\langle A'(Y)D(Z)A(X), w\right\rangle.
\frac{\mathrm{d^2}}{\mathrm{d}p^2}\ln \pi_p=\frac{(p^2+\pi^2)}{p^4}-\frac{2 \pi}{p^3}\cot\Big(\frac{\pi}{p}\Big)+\frac{\pi^2}{p^4}\cot^2\Big(\frac{\pi}{p}\Big)>0.
\int_{\mathcal O}m_{V}(\xi)d\xi= \int_{K}\sigma_{V}(\xi)d\xi=\sum_{\psi\in \Psi}\sum_{j=1}^{\infty}\int_{K}|\widehat{\psi}(\mathfrak{p}^{-j}\xi|^2 =\sum_{\psi\in \Psi}||\psi||^2 /(q-1) \leq L/(q-1)<\infty.
\begin{aligned}\dot{y}_1 =e^{y_2-y_1}-\frac{1}{2},\dot{y}_2=-e^{y_2-y_1}-e^{y_2-y_3}+\frac{1}{2}, \dot{y}_3 =e^{y_2-y_3}+e^{y_4-y_3}-\frac{1}{2},\dot{y}_4=-e^{y_4-y_3}+\frac{1}{2}.\end{aligned}
\begin{aligned}PD^1_a=\varprojlim_{l\to\infty}PD(-l, a), \ \PD_1^b=\varprojlim_{t\to\infty}PD(b,t) PD= \varprojlim_{l\to\infty,l\to \infty}PD(-l, t) PD_2^b=\varprojlim_{l\to\infty, t=b}PD(-l, t), \ \PD_2^a=\varprojlim_{t\to\infty, -l=a}PD(-l, t) .\end{aligned}
\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}e^{ih^{\top}u}|\widehat g(x,u)|^2du =\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}e^{ih^{\top}u}\widehat g(x,u) \overline{\widehat g(x,u)}du &=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}e^{ih^{\top}u}\widehat g(x,u) \widehat g(x,-u)du &= \int_{\mathbb{R}^d}g(x,u) g(x,u-h)du =\int_{\mathbb{R}^d}g(x,h+s)g(x, s)ds,
\tau^{(a)}_{m}(E) \tau^{(a)}_m (E\Omega^n) = \tau^{(a+1)}_{m}(E) \tau^{(a-1)}_{m}(E \Omega^n) + \tau^{(a)}_{m+1}(E) \tau^{(a)}_{m-1}(E \Omega^n)
0\le I \le \int_0^{2x_1}\frac{1}{x_1}\log\frac{(x_1+y)}{|x_1-y|}~dy+\int_0^{2x_1}\frac{1}{x_2}\log\frac{(x_2+y)}{|x_2-y|}~dy &=\frac{1}{x_1}3x_1\log 3+ \frac{1}{x_2}\left[2x_1 \log\frac{1+\frac{2x_1}{x_2}}{1-\frac{2x_1}{x_2}}+x_2\log\left(1-\frac{2x_1}{x_2}\right)+x_2\log\left(1+\frac{2x_1}{x_2}\right)\right] &\leq 3\log 3 + 2\log 2 .
\|\rho^{(r)}_k(x)\|^2 =\|\rho^{(r)}_k(x)^*\rho^{(r)}_k(x)\| &=\|\rho^{(r)}_n(\rho^{(r)}_{n,k}(x)^*)\rho^{(r)}_n(\rho^{(r)}_{n,k}(x))\| &\leq \|\rho^{(r)}_n(\rho^{(r)}_{n,k}(x)^*\rho^{(r)}_{n,k}(x))\|,
\tilde{L}_{F}( \psi_{R}, \psi_{L}; A_{+}, A_{-}; S ) = \bar{\psi}_{L}\bar{\partial}\psi_{L}+ \bar{\psi}_{R}\partial \psi_{R}+ A_{-}^{a}J_{L}^{a}+ A_{+}^{\bar{a}}J_{R}^{\bar{a}}+ ( S^{- 1})_{a \bar{a}}A_{-}^{a}A_{+}^{\bar{a}}.
\operatorname*{l i m}_{N \to \infty}{\cal W}_{0}= \sqrt{\frac{A_{1}+ A_{2}}{\hat{g}^{2}A_{1}A_{2}}}J_{1}\Bigl ( \sqrt{\frac{4 \hat{g}^{2}A_{1}A_{2}}{A_{1}+ A_{2}}}\Bigr )
\|\mathcal{T}_{1,a}^{(a_1,a_2)}-\mathcal{T}_{2,a}^{(a_1,a_2)}\|_{\rm F}^2=O(p^{a-a_1-a_2+2(a_1-c_2+1)})=O(p^{a+a_1-a_2-2c_2+2})=O(p^{a+a_1-a_2-2r+2}),
v_{2n-2}^* v_{2n-2}+ v_{2n-1}^* v_{2n-1}= v_{2n-2}^* v_{2n-2}+ (1 \otimes f_n -v_{2n-2}^*v_{2n-2})w_n^*w_n(1 \otimes f_n -v_{2n-2}^*v_{2n-2}) = v_{2n-2}^* v_{2n-2}+ 1 \otimes f_n -v_{2n-2}^*v_{2n-2}= 1\otimes f_n.
\alpha(t,x) =\frac{\mathrm{e}^{2\lambda \|\eta^0\|_\infty}- \mathrm{e}^{\lambda \eta^0(x)}}{\gamma(t)}, \xi(t,x) =\frac{\mathrm{e}^{\lambda \eta^0(x)}}{\gamma(t)}, \gamma(t)=(t-t_0)(t_1-t)
\left[ T_{\mu \nu}\right]_{P}( x ) = \frac{C_{P}}{2}\frac{1}{\sqrt{| g ( x ) |}}g_{\mu \rho}( x ) g_{\nu \lambda}( x ) \int_{- \infty}^{+ \infty}d \xi \frac{1}{\beta ( \xi )}\frac{d y^{\rho}( \xi )}{d \xi}\frac{d y^{\lambda}( \xi )}{d \xi}\delta^{( 1 1 )}( x - y ( \xi ) )
\tilde{\nabla}_i\tilde{h}_{jk}=\varphi f^2\nabla_ih_{jk},\tilde{\nabla}_0\tilde{h}_{ij}=\varphi 'f^2h_{ij},\tilde{\nabla}_i\tilde{h}_{j0}=\tilde{\nabla}_i\tilde{h}_{0j}=-\varphi f'f h_{ij},
\prod_{k=1}^r \prod_{l=1}^{N_k}e^{\mathbf{a}_k(l) \varpi_{i_{k,l}}}(b_{k,l}) = \prod_{s=1}^n \prod_{j=1}^r e^{d_{j,s}\varpi_s}(b_{k(j,s), m(j,s)}).
\Delta(\mathbf{F}_{1\bar{4}\,\bar{2}3})=\iota\otimes \mathbf{F}_{1\bar{4}\,\bar{2}3}+\mathbf{F}_{1}\otimes \mathbf{F}_{\bar{3}\,\bar{1}2}+\mathbf{F}_{1\bar{2}}\otimes \mathbf{F}_{\bar{1}2}+\mathbf{F}_{1\bar{3}\,\bar{2}}\otimes \mathbf{F}_{1}+\mathbf{F}_{1\bar{4}\,\bar{2}3}\otimes \iota.
U(C_6)=\begin{pmatrix}0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0\end{pmatrix},U(P_7)=\begin{pmatrix}0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0\end{pmatrix}
Z^{\cal N}\, =\,\left( \begin{array}{clcl}Z_1\,\epsilon 0 0 0 0 Z_2\,\epsilon 0 0 0 0 Z_3\,\epsilon 0 0 0 0 Z_4\,\epsilon\end{array}\right) \qquad,\qquad \epsilon\,=\,\left( \begin{array}{cl}0 1 -1 0\end{array}\right)
u_{n+N}-T_{[n+N]}\cdots T_{[n+1]}(u_n) =\sum_{k=1}^{N-1}\Big(T_{[n+N]}\cdots T_{[n+N-k+1]}(u_{n+N-k}) \qquad\qquad\qquad-T_{[n+N]}\cdots T_{[n+N-k]}(u_{n+N-k-1})\Big) \quad+u_{n+N}-T_{[n+N]}(u_{n+N-1}).
\sup u \leq \left((1-p) \sup u +K\right)\sum_{i=0}^{n-1}p^i+ p^n\sup g &= \left((1-p) \sup u +K\right)\frac{1-p^n}{1-p}+ p^n\sup g &= (1-p^n) \sup u +K\frac{1-p^n}{1-p}+ p^n\sup g.
{\cal F}(a(u))={1\over 2\pi^2}\int^u_1dx\int_{-1}^1dy{4\sqrt{(x-u)(y-u)}-(y-x)\log\left[{2u-x-y+2\sqrt{(u-x)(u-y)}\over x-y}\right]\over \sqrt{(x^2-1)(y^2-1)}}.
S_{B}[ A ] = i \frac{1}{\eta}S_{C S}[ A ] + R [ \widetilde{F}] + \frac{i}{\theta}\int d^{3}x d^{3}y j_{\mu}^{T}( x ){\mathcal K}_{\mu \nu}( x - y ) j_{\nu}^{T}( y ) ,
F_{T,R}(y) = \frac{1}{(\pi i)^2}\int \limits_{-i \infty}^{i \infty}\int \limits_{-i \infty}^{i \infty}F^{\#}_{T,R}(\nu) \overline{W_{\nu}(y)}\frac{d\nu_1 d\nu_2}{\prod \limits_{j=1}^3 \Gamma\left(\frac{3\nu_j}{2}\right) \Gamma\left(-\frac{3\nu_j}{2}\right)}.
\omega_{1 1}= \partial_{z_1}\bar \partial_{z_1}\left( \frac{i}{z_1 - \bar z_1}+ \vert z_2 \vert^2 \right) = \frac{1}{4}\left( \frac{\partial}{\partial x_1}- i \frac{\partial}{\partial y_1}\right) \left( \frac{\partial}{\partial x_1}+ i \frac{\partial}{\partial y_1}\right) \frac{1}{2 y_1}= \frac{1}{4}\frac{1}{y_1^3}
\left< u_{t},\varphi\right> -\left< u_{0},\varphi\right> = \mathbb{E}\left[ \int_{0}^{t}\varphi'\left(\overline{S}^{1}_{z}\right)dz \right] + \mathbb{E}\left[ \int_{0}^{t}R\varphi\left(\overline{S}^{1}_{z-}\right) \overline{N}^{1}(dz) \right].
\tau(xk_\gamma,yk_\delta)=\tau(x,y)q^{-(\gamma,\delta)}(x\in U^+, y\in U^-,\gamma, \delta\in Q), &\tau(U^+_\gamma, U^-_{-\delta})=\{0\}(\gamma, \delta\in Q^+, \gamma\ne\delta), &\tau|_{U^+_\gamma\times U^-_{-\gamma}}\; (\gamma\in Q^+), &\tau(Sx,Sy)=\tau(x,y) (x\in{}^\sharp{U}^{\geqq0}, y\in{}^\sharp{U}^{\leqq0}).
\begin{array}{rcll}Z^m_n (\frac{\pi}{2}-\vartheta_1,\, \phi) = Z^m_n (\frac{\pi}{2}+\vartheta_1,\, \phi), \quad\mbox{if}|m|+n\mbox{is odd}; Z^m_n (\frac{\pi}{2}-\vartheta_1,\, \phi) = -Z^m_n (\frac{\pi}{2}+\vartheta_1,\, \phi), \quad\mbox{if}|m|+n\mbox{is even}.\end{array}
\ell_k^+ = k \ell + \left\lceil \frac{\ell n^+}{n}\right\rceil =k \ell + \frac{\ell n^+}{n}+\left( \frac{-\ell n^+}{n}{\rm ~mod~}1 \right) =\frac{\ell n_k^+}{n}+ \left( \frac{\ell d}{n}{\rm ~mod~}1 \right) \;,
{\frac{( - i )^{d}}{N}}\sum_{m}{}^{\prime}e^{2 \pi i{\frac{2 \mu}{N}}m}{\frac{1}{2^{d}\prod_{j}\sin{\frac{\alpha_{j}m}{2}}}}= \sum_{\nu}w_{\nu}{\frac{( - i )^{d}}{N}}\sum_{m}{}^{\prime}e^{2 \pi i{\frac{\mu - \nu}{N}}m}{\frac{1}{2^{d}\prod_{j}\sin{\frac{\alpha_{j}m}{2}}}}
\mbox{Tr}K^{(j)}(s)=\mbox{Tr}\left[e^{-s\bigtriangleup^{(j)}}\right] ={1 \over (4\pi s)^{d/2}}\left( A_0^{(j)}+s A_1^{(j)}+ s^{2}A_{2}^{(j)}+....\right)~~~
[ P ( \Omega_{p}) ]_{a_{1}\ldots a_{p}}= \Omega_{\mu_{1}\ldots \mu_{p}}\left( \delta_{a_{1}}^{\mu_{1}}I_{N}+ \lambda \frac{\partial \phi^{\mu_{1}}}{\partial \sigma^{a_{1}}}\right) \ldots \left( \delta_{a_{p}}^{\mu_{p}}I_{N}+ \lambda \frac{\partial \phi^{\mu_{p}}}{\partial \sigma^{a_{p}}}\right)
\langle X_i^{(m)}-Y_i^{(m)},Z_{i-1}^{(m)}\rangle =\langle{\bf e}_{n+1}+X_{i+1}^{(m)},X_{i-1}^{(m)}\rangle \langle{\bf e}_{n+1}+X_{i+1}^{(m)},X_i^{(m)}-X_{i-1}^{(m)}\rangle =\langle{\bf n},X_{i-1}^{(m)}\rangle \langle{\bf n},X_i^{(m)}-X_{i-1}^{(m)}\rangle