harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
V_{\beta}' / (E-B) \subset \bigoplus_{k=0}^{m-1}\psi^k V_{\alpha}/ I_{>k}/ (I_k -B) / (I_{<k}- B) \setminus (I_{<k}\cap B) &= \bigoplus_{k=0}^{m-1}\psi^k V_{\alpha}/ (E -B) / (I_{>k}\cap B) \setminus(I_{<k}\cap B).
\sum_{\ell=2}^{14}V_{\ell}\leq \varepsilon(\mathcal{G}_{1}(K)+\mathcal{G}_{2}(K))(E_{w}+E_{\mu}) + \frac{1}{4}\sum_{j=1}^{d}\int_{\mathbb{T}^{d}}(\partial_{x_{j}}^{2}(Dw^{1}-Dw^{2}))^{2}\dx.
\mathbb{E}[{\bf e}_k(t){\bf e}^H_k(t)]=\beta_k \left(1-\beta_k \boldsymbol{\omega}_k^H \boldsymbol{\Theta}_{\sigma(t)}\boldsymbol{\Sigma}^{-1}\boldsymbol{\Theta}_{\sigma(t)}\boldsymbol{\omega}_k\right){\bf I}_N.
F r a c t i o n \; P \! \! - \! w a v e ={\frac{( \overline{{{p}}}p \rightarrow \pi^{+}\pi^{-})_{P}+ ( \overline{{{p}}}p \rightarrow \pi^{0}\pi^{0})_{P}}{( \overline{{{p}}}p \rightarrow \pi^{+}\pi^{-})_{S \& P}+ ( \overline{{{p}}}p \rightarrow \pi^{0}\pi^{0})_{P}}}.
A_{\alpha}^{\,\,\,\, i}= \xi^{\beta}f_{\beta \alpha}^{\,\,\,\,\,\,\,i},\,\,\,\,\,\,\,\,\,\,\,\,A_i^{\,\,\alpha}= \xi^{\beta}f_{\beta i}^{\,\,\,\,\, \alpha}.
L_{\Sigma+}= \epsilon^{\mu \nu \rho \sigma}\bigg[\dots + \bigg(C -{B^2\over 4A}\bigg) \Sigma^{+ab}_{\mu \nu}\Sigma^{+}_{\rho \sigma ab}-{BE\over 2A}G^{+ab}_{\mu \nu}\Sigma^{+}_{\rho \sigma ab}+\dots \bigg],
\begin{aligned}-\left(\frac{p(x)}{x^{k_1}}+\frac{l_1}{l_2}\right)+\frac{p(x^{*})}{(x^{*})^{k_1}}=\frac{1}{l_2}\frac{(1-\theta)}{\left(\frac{Kk_1}{k_1p(x^{*})-p'(x^{*})x^{*}}+\theta\right)}.\end{aligned}
\omega_\phi\wedge T:=\omega^T\wedge T+dd^c_B(\phi T) &\omega_\phi\wedge T\wedge (\omega^T)^{n-p-1}\wedge \eta= T\wedge (\omega^T)^{n-p}\wedge \eta+dd^c_B(\phi T)\wedge (\omega^T)^{n-p-1}\wedge \eta.
(m,L^{\mu,s/2}_nm')_\mu =(L^{\mu,s/2}_{-n}m,m')_\mu, (m,(J^{\{u\}})^{\mu,s/2}_nm')_\mu =-((J^{\{\phi(u)\}})^{\mu,s/2}_{-n}m,m')_\mu, (m, (G^{\{v \}})^{\mu,s/2}_nm')_\mu =((G^{\{\phi(v) \}})^{\mu,s/2}_{-n}m,m')_\mu.
\sum_{x_i\in T_q}f_i(x_i)^{\sum_{j=1}^r \sum_{k=1}^{L_j}h_{ijk}\beta_{jk}}=qf_i(0)^{\sum_{j=1}^r \sum_{k=1}^{L_j}h_{ijk}\beta_{jk}}+(q-1)B_i,
\Psi ( \omega ; \lambda ) \sim{\frac{2 \omega^{D - 2}}{( 4 \pi )^{( D - 1 ) / 2}}}\sum_{n = 0}^{\infty}{\frac{\tilde{a}_{n}( \lambda )}{\Gamma \left({\frac{D - 1}{2}}- n \right)}}\omega^{- 2 n},
f_{49}|\gamma_7 = \frac{1}{7}\left((-2 \zeta_{7}^{5}- 4 \zeta_{7}^{4}- 6 \zeta_{7}^{3}- 8 \zeta_{7}^{2}- 3 \zeta_{7}- 5)q + (6 \zeta_{7}^{5}- 2 \zeta_{7}^{4}+ 4 \zeta_{7}^{3}+ 3 \zeta_{7}^{2}+ 2 \zeta_{7}+ 1)q^2 + O(q^3)\right)
b_1(\ell,k)=\frac 1{(2\pi i)^4}\int_{\Gamma_{D_2}}dz\int_{\Gamma_{D_1}}dw\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{\tau_2}}d\omega\frac{G_{n_1+1,\mu_1,\xi_1}(z)G_{\Delta n-1,\Delta\mu,\Delta\xi}(w)}{G_{k,\mu_1,\xi_1}(\zeta)G_{n_2+1-\ell,\Delta\mu,\Delta\xi}(\omega)(z-w)(z-\zeta)(w-\omega)}.
{\sf g}_{\ell}({\alpha}) = \frac{1}{\kappa}\left[ \left({\frac{\alpha^{2}}{\kappa}+ 1}\right) \left( \frac{\alpha^{4}}{\kappa^{2}}+ \frac{\alpha^{2}}{\kappa}-{\ell}({\ell}+ 1 ) \right) \right]^{- 1}.
I_{\omega}= ( - )^{r_{5}}( \omega^{1 / 2})^{r_{3}^{2}+ r_{3}r_{4}- r_{1}r_{3}- r_{2}r_{3}- r_{3}r_{5}- r_{4}r_{5}}\Bigg [ \frac{\Phi ( r_{1}) \Phi ( r_{2})}{\Phi ( r_{3}) \Phi ( r_{4})}\Bigg ]^{1 / 2}
W^g(p,p_1,\dots,p_{k-1})=\psi^\ast\left(\sum_{\substack{(t)^g\in (Y^n)^g K=\{p_1,\dots,p_{k-1}\}}}(t)^g\right)= &\sum_{h=0}^g\sum_{\substack{L\cup M=K K}}K_p(q,\bar{q})\left(W^{g-1}_{k+1}(q,\bar{q},K)+ W_{l+1}^h(q,L)W_{m+1}^{g-h}(\bar{q},M)\right)
\varepsilon_R^{\rm K}=E[{\psi}_{\rm K}]+\Delta M_K+o(\hbar^2)=E[{\psi}_{\rm K}]+ \lim_{s \rightarrow-\frac{1}{2}}\left[ \Delta_1 \varepsilon^{\rm K}(s)+\Delta_2\varepsilon^{\rm K}(s) \right]+o(\hbar^2)
\Phi = \tilde{\Phi}, \Psi ~ = ~{\cal F}~ \tilde{\Psi}, ~ ( \tilde{\Psi}~ = ~{\cal G}{\Psi}) , \Phi^{i}= e_{\mu}^{i}f_{a}^{\mu}\tilde{\Phi}^{a}, ~ ( \tilde{\Phi}^{a}= f_{\mu}^{a}e_{i}^{\mu}{\Phi}^{i}) .
\nabla_{v'}^2 g(v_0')=\frac1{\| \mathbf{n}_{x_0}\|_{K^\circ}}P_{\langle e_1,\ldots, e_{n-1}\rangle}^* R_{x_0}\nabla^2_{x_1,\ldots,x_n}\big(\frac12 \| \cdot\|_K^2 \big)(x_0) R_{x_0}^{-1}P_{\langle e_1,\ldots, e_{n-1}\rangle}.
J_{D}( B_{1}, B_{2}) = \mathrm{p . v .}\int_{0}^{\infty}\frac{t^{D}d t}{\frac{( B_{1}t - 1 ) ( B_{2}t - 1 )}{( B_{1}t + 1 ) ( B_{2}t + 1 )}e^{2 t}- 1},
N_\mu(n) = \begin{cases}\left[\Gamma(\frac{n}{2}+\mu+1/2)(\frac{n}{2})!\right]^{-1/2}n \left[\Gamma(\frac{n+1}{2}+\mu+1/2)(\frac{n-1}{2})!\right]^{-1/2}n\end{cases}
G=\left(\begin{array}{ccccc}\pi^0I_{k_0}M_{0,1}M_{0,2}M_{0,3}M_{0,4}\cr0 \pi I_{k_1}\pi M_{1,2}\pi M_{1,3}\pi M_{1,4}\cr0 0 \pi^2 I_{k_2}\pi^2 M_{2,3}\pi^2 M_{2,4}\cr0 0 0 \pi^3 I_{k_3}\pi^3 M_{3,4}\end{array}\right)U
A(r,t)=\frac{\mu_{0}}{4\pi}\iint\frac{qv_{s}(t^{\prime})\delta^{3}(r^{\prime}-r_{s}(t^{\prime}))}{|r-r^{\prime}|}\delta(t^{\prime}-t_{r}^{\prime})dt^{\prime}d^{3}r^{\prime}
\sum_{N\ge n_1\ge \cdots\ge n_k\ge 1}\frac1{n_1^{i_1}\cdots n_k^{i_k}}=\sum_{N-1\ge n_1\ge \cdots\ge n_k\ge 1}\frac1{n_1^{i_1}\cdots n_k^{i_k}}&\quad+ \sum_{N=n_1\ge n_2\ge \cdots\ge n_k\ge 1}\frac1{n_1^{i_1}\cdots n_k^{i_k}}.
\int d^{4}q{\frac{1}{[ q^{2}]^{1 + n \epsilon}( q - p )^{2}}}={\frac{\Gamma ( 1 - \epsilon ) \Gamma ( ( n + 1 ) \epsilon ) \Gamma ( 1 - ( n + 1 ) \epsilon )}{\Gamma ( 1 + n \epsilon ) \Gamma ( 2 - ( n + 2 ) \epsilon )}}{\frac{1}{[ p^{2}]^{( n + 1 ) \epsilon}}}
A_{0}(J_{\varepsilon}\textbf{u}^{\varepsilon})\partial_{t}\textbf{u}^{\varepsilon}+\sum_{j=1}^{d}J_{\varepsilon}A_{0}A_{j}(J_{\varepsilon}\textbf{u}^{\varepsilon})\partial_{x_{j}}J_{\varepsilon}\textbf{u}^{\varepsilon}+\left( \begin{array}{c}0 \nabla P^{\varepsilon}\end{array}\right)=0,
\int_n^{n+1}| \log |t-\gamma_a||^{2k}dt\leq \int_{n-r}^{n+r+2}|\log |t-\gamma_a||^{2k}dt \leq 2 \int_0^{2r+2}|\log x|^{2k}dx = 2 \Gamma(2k+1)+O\Big( (\log (2r+2))^{2k}\Big).
(\Gamma^{\alpha\eta}_{\phi})_{abc}= 2e^3(\gamma^\lambda \gamma^\phi \gamma^\tau)_{\alpha\eta}\int \frac{d^4 S}{(2\pi)^4}(Q+S)_{\tau}(P+S)_\lambda [\tau_3 D(S)]_{ab}[\tau_3 D(P+S) ]_{bc}[\tau_3 D(Q+S)]_{ca}.
\underset{x^\prime \in \mathbb{R}^d}{\arg \max}\left[ \sum_{1\leq \ell \leq 2}\varphi_\ell(x^{\prime}, \lambda_\ell)\right] = \left(\sum_{1 \leq \ell \leq 2}\lambda_{\ell}V_{\ell, X X}^{-1}\right)^{-1}\left(\sum_{1 \leq \ell \leq 2}a_{\ell}V_{\ell, X X}^{-1}V_{\ell, X Y}\right),
H^{(n)}_{\varepsilon}=H_{\varepsilon}^{0,(n)}+\sum_{\ell=2}^{r}\varepsilon^{\ell}\frac{n!}{(n-\ell)!}\mathcal{S}_{n}(\tilde{Q}_{\ell}\otimes\mathrm{Id}_{\bigvee^{n-\ell}\mathcal{Z}})\mathcal{S}_{n}\,,n\geq2r\,,
\left(B^{F}(T_{N,l})\right)^{l}\times A^{(N-l)^{F}}_{\varepsilon_{N,l}}\subset A^{N,F}_{\varepsilon_N,\theta}[Uz_1,\ldots,Uz_l], B^{F}(T):=\{z \in F \ |\ \|z\|<T\}.
[M, v\wedge \partial_v]=2\sum_{1\leq i\leq 2}\big([\Lambda_i , v]\wedge \partial_v\big)\Lambda_i=2\sum_{1\leq i\leq 2}\Lambda_i\big([\Lambda_i , v]\wedge \partial_v\big), \textrm{with}[\Lambda_i , v]=c_i\big ((t-t_0)^{1\over 2}, 0, 0\big ),
\begin{array}{rl}\displaystyle R(3,\mathbb{Z}_n,2) \displaystyle \leq{(n+3)/4 \choose 2}+3{(n-1)/4 \choose 2}&\displaystyle = 3{(n+1)/4 \choose 2}+{(n-3)/4 \choose 2}&\displaystyle = \frac{(n-1)(n-3)}{8}\end{array}
\sup_{|\tau|=1, \tau\cdot \nu=0}u_{\tau\tau}(x_0) \ ({\rm or}\ \sup_{|\tau|=1, \tau\cdot \nu=0}|u_{\tau\tau}(x_0)| ) \le \epsilon M_2(R^\prime) + C_\epsilon(1+\frac{1}{(R^\prime)^2}).
\langle \Psi_{1}( x_{1}) \cdots \Psi_{n}( x_{n}) \rangle = \displaystyle{\frac{F ( \hat{X}_{1 ( i )}{\cdot \hat{X}_{1 ( j )}})}{\displaystyle{\prod_{l < m}( x_{l m}^{2})^{- \frac{1}{( n - 1 ) ( n - 2 )}( \eta_{1}+ \cdots + \eta_{n}) + \frac{1}{n - 2}( \eta_{l}+ \eta_{m})}}}},
\begin{array}{r l}{( b , \lambda , p_{2}) \longleftrightarrow ( c , \mu , p_{3}) ;}&{P \leftrightarrow R , s \leftrightarrow u ;}\\{( b , \lambda , p_{2}) \longleftrightarrow ( d , \nu , p_{4}) ;}&{P \leftrightarrow Q , s \leftrightarrow t .}\\ \end{array}
\lim_{n \to\infty}\frac{\mathcal{L}\{t \in[0,s_{n}(\omega)] \;|\; \pmb\alpha(\omega)(t) = i\}}{s_{n}(\omega)}= \lim_{n\to\infty}\frac{n}{s_{n}(\omega)}\frac{1}{n}\sum_{j=0}^{n-1}\varphi_{i}(\theta^{j}\omega) = \frac{p_{i}\tau_{i}}{m}.
\begin{aligned}P= \chi_{0}+ \chi_{1}w' + \chi_{2}(w')^2,R= \Bigl(\chi_3+\chi_4w'+\chi_5(w')^2+\chi_6(w')^3+\chi_7(w')^4+ + \chi_8(w')^5+\chi_9(w')^6\Bigr)+ \Bigl(\chi_{10}+\chi_{11}w'+\chi_{12}(w')^2+\chi_{13}(w')^3\Bigr)w''+\Bigl(\chi_{14}+\chi_{15}w'\Bigr)(w'')^2.\end{aligned}
\begin{aligned}qcb^{-1}c - ab^{-1}a = q^2y(qy + q^{-1}y^{-1})^{-1}x^2(y-y^{-1}) & -q^2\Lambda\alpha^{-1}(\Lambda)y^{-1}(qy^{-1}+ q^{-1}y)^{-1}x^{-2}(y-y^{-1}) &+ (q^2+1)\Big(\alpha(\Lambda)(qy + q^{-1}y^{-1})^{-1}+ \Lambda(qy^{-1}+ q^{-1}y)^{-1}\Big)\end{aligned}
(\phi_{n}(g)P)(\begin{matrix}z_{1}\\z_{2}\\ \end{matrix})=[\phi_{n}(\begin{matrix}a&b\\c&d\\ \end{matrix})P](\begin{matrix}z_{1}\\z_{2}\\ \end{matrix})=P((\begin{matrix}a&b\\c&d\\ \end{matrix})^{-1}(\begin{matrix}z_{1}\\z_{2}\\ \end{matrix})),P\in P_{n}^{2}.
(\pi_{n}^{*}(g_{i,j}^{n}))(h_{0},h_{1},\cdots,h_{n}) = [T_{i}(1-TT^{*})T_{j}^{*}](h_{0},h_{1},\cdots,h_{n}) & = (0,\cdots,h_{j},\cdots,0), \mbox{where $h_{j}$ is in the $i$-th slot,}
\sum_{l}\int_{0}^{\infty}\, d r r \, \psi_{\scriptscriptstyle{l , L}}^{\ast}( k , r ) \, \psi_{\scriptscriptstyle{l , L^{\prime}}}( k^{\prime}, r ) = \delta_{\scriptscriptstyle{L , L^{\prime}}}\, \delta ( E - E^{\prime}) \, .
\zeta(s)^{-1}Z_{r_2}(f,s) =q^{-As}f(0)+(1-q^{-s})\sum_{j=-A}^{A-1}q^{-js}f(0,0,\varpi^j) =f(0)-s\log q\big(Af(0)-\sum_{j=-A}^{A-1}f(0,0,\varpi^j)\big)+O_{f}(s^2).
j_{\sigma_l}^i(\xi) = \left( \left. \frac{\partial g''^{\eta_k}(g', g)}{\partial g'^{\sigma_l}}\right|_{g' = e}\left. \frac{\partial g''_{\rho_k}(g', g)}{\partial g^{\eta_k}}\right|_{g' = g^{-1}}\right) \Lambda^{\rho_k i}
\mathrm{P}\Big( \sum_{i=1}^k \xi_{[i]}^2 \ge Ck \log\big(\tfrac{en}{k}\big)+c\Big) \le \big(\tfrac{en}{k}\big)^{-(0.4C-2)k}e^{-0.4c}, \mathrm{E}\sum_{i=1}^k \xi_{[i]}^2 \le 6k\log\big(\tfrac{en}{k}\big).
\lim_{\epsilon \to 0}\int_{-\infty}^{0}\frac{e^{r|x|}e^{\pm i \epsilon r}}{r \mp i z}( -ir)^{s_k}dr = \Big( (-i \partial_s)^{s_k}\int_{-\infty}^{0}\frac{e^{rs}}{r \mp i z}dr \Big)\Big|_{s=|x|}
\sigma(XA,1) = \ \sum_I \M_I(X)\ \big( \ \sum_{J\preceq I}\R_J(A) \ \big)=\ \sum_J \ \big(\ \sum_{I\succeq J}\M_I(X)\ \big) \R_J(A)=\ \sum_J \F_J(X)\, R_J(A) \ .
J_{\mu \nu}= 2 i \big \{e_{[ \mu}{}^{\alpha \dot{\alpha}}e_{\nu ]}{}^{\beta}{}_{\dot{\alpha}}\widehat A_{\alpha}^{( 1 )}\star \widehat A_{\beta}^{( 1 )}+ \mathrm{h . c .}\big \}_{z = 0}+ L_{\mu \nu}+ \mathrm{~ W ~ - t e r m s},
Q_{n,\textup{\textbf{m}}}(z)F_\alpha(z)-P_{n,\textup{\textbf{m}},\alpha}(z)=\sum_{k=n+1}^{\infty}[ Q_{n,\textup{\textbf{m}}}F_\alpha]_{k}\,\Phi_k(z), z\in D_{\rho_{0}(F_\alpha)},
\dot{H}= | k | \; \frac{( 2 + | k | ) ( 1 + 2 \operatorname{s i n h}^{2}t ) \operatorname{s i n h}^{2}t - 2}{[ ( 2 + | k | ) \operatorname{s i n h}^{2}t + 2 ]^{2}\operatorname{c o s h}^{2}t}
\frac{1}{Z}= ( \det A )^{1 / 2}= \frac{\det A}{( \det A )^{1 / 2}}= \int \prod_{i}d \phi_{i}d \bar{\theta}_{i}d \theta_{i}e^{\bar{\theta}_{i}A_{i j}\theta_{j}+ \phi_{i}A_{i j}\phi_{j}}
c_{k-2}= - n(n-1) a_k - n b_{k-1}, c_l = - \frac{n(n-1)}{k(k-1)}(l+2)(l+1) a_{l+2}- \frac{n}{k-1}(l+1) b_{l+1}+ \frac{C_{k-l-2,n}}{l!}, l=0, 1, \ldots, k-3.
g = \left( \begin{array}{c c c c c}{{- 1}}&{{0}}&{{0}}&{{\dots}}&{{0}}\\{{0}}&{{( g_{i j})}}&{{\dots}}&{{0}}&{{0}}\\{{0}}&{{\dots}}&{{1}}&{{\dots}}&{{0}}\\{{0}}&{{\dots}}&{{\dots}}&{{\ddots}}&{{0}}\\{{0}}&{{0}}&{{\dots}}&{{\dots}}&{{1}}\\ \end{array}\right)
{\mit \Xi}_{[ 1 / 2 ]}\, = \, \left( \begin{array}{c c}{{e^{i \phi}}}&{{0}}\\{{0}}&{{e^{- i \phi}}}\\ \end{array}\right) \quad , \qquad{\mit \Xi}_{[ 1 ]}\, = \, \left( \begin{array}{c c c}{{e^{i \, 2 \, \phi}}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{e^{- i \, 2 \, \phi}}}\\ \end{array}\right) \quad .
= \int_{\rho B_{1}}\int_{\rho B_{1}}(\tilde{u}(s,x)-\tilde{u}(s,y))(\psi^{2}(y)\tilde{u}^{-q}(s,y) - \psi^{2}(x)\tilde{u}^{-q}(s,x))k(x,y)dxdy,
R_{[c,M,s]}(x)\geq \c(u_1,t)-c(s,t)+\sum_{j=1}^{mk}[c(u_{i+1},v_i)-c(u_i,v_i)]+c(x,v_{mk+1})-c(u_{mk+1},v_{mk+1}) = \c(x_1,t)-c(s,t)+mq+c(x,y_1)-c(x_1,y_1)\underset{m\rightarrow\infty}{\longrightarrow}\infty,
\rho(x) c_{\rho}(x) \partial_{t}\theta(t, x) + \mathrm{div}\, q(t, x) = h(t, x) t \in (0, T), \; x \in \Omega, \tau \partial_{t}q(t, x) + q(t, x) + \lambda(x) \nabla \theta(t, x) = 0 t \in (0, T), \; x \in \Omega.
\tilde{S}_{\lambda}^{(n)}=\det \left( \tau^{j-1}\tilde{H}_{\lambda_{i}-i+j,n}\right)_{1\leq i,j\leq n}=\det \left( \tau^{j-1}\tilde{E}_{\lambda_{i}^{\prime}-i+j,n}\right)_{1\leq i,j\leq k}~,
\left( \begin{matrix}{z_{1}}\\{z_{2}}\\ \end{matrix}\right) \rightarrow U_{L}\left( \begin{matrix}{z_{1}}\\{z_{2}}\\ \end{matrix}\right) ; \left( \begin{matrix}{z_{1}}\\{\bar{z}_{2}}\\ \end{matrix}\right) \rightarrow U_{R}\left( \begin{matrix}{z_{1}}\\{\bar{z}_{2}}\\ \end{matrix}\right)
\int_0^rB(X_t^\epsilon,y)(1_{\{Y_t^\epsilon=y\}}-\pi(y))dt= \int_0^rB(X_t^\epsilon,y)(1_{\{Y_{t/\epsilon}=y\}}-\pi(y))dt ={\epsilon}\int_0^{r/\epsilon}B(X_{\epsilon s}^\epsilon,y)(1_{\{Y_{s}=y\}}-\pi(y))ds.
n(R + \delta_n) \geq H(W) \geq I(X^n,Z^n;W|Y^n) &= \sum_{i=1}^n H(X_i,Z_i|Y_i) -H(X_i,Z_i|W,X^{i-1},Z^{i-1},Y^n) &\overset{(a)}{\geq}\sum_{i=1}^n H(X_i,Z_i|Y_i) -H(X_i,Z_i|U_i,Y_i) &\geq \sum_{i=1}^n I(X_i;U_i|Y_i),
(\frac{\partial^{2}U}{\partial x\partial y})=(\frac{\partial T}{\partial x})_{y}(\frac{\partial S}{\partial y})_{x}+T(\frac{\partial^{2}S}{\partial x\partial y})-(\frac{\partial P}{\partial x})_{y}(\frac{\partial V}{\partial y})_{x}-P(\frac{\partial^{2}V}{\partial x\partial y})
\underline q = (q_0,\ldots,q_r), \underline q^{-}(j) = (q_0, \ldots, q_{j-1},q_{j+1},\ldots,q_r), \underline q [j] = (q_0,\ldots,q_j^{-1},\ldots,q_r ) , \underline q^{-1}= (q_0^{-1},\ldots,q_r^{-1}).
\sum_{\zeta\subset\eta\setminus\xi}(-1)^{|(\eta\setminus\xi)\setminus\zeta|}\prod\limits_{y'\in \zeta}e^{-E^\phi\left( y',\omega\right)}=\Bigl(K_0^{-1}\prod\limits_{y'\in\cdot}e^{-E^\phi\left(y',\omega \right)}\Bigl)(\eta\setminus\xi) &=\prod\limits_{y'\in\eta \setminus \xi}\left( e^{-E^\phi\left( y',\omega \right)}-1\right)
8kd_k = -\,[h^{k+2}]\, \Bigg(\sum_{j=0}^{k-1}d_j h^j \bigg( B(h)(1-h)^{-(j+1/2)}+ C(h)(1-2h)^{-(j+1/2)}+ D(h)(1-3h)^{-(j+1/2)}\bigg)\Bigg),
d(\delta_{\lambda}\Gamma_{t}) =\sum_{\alpha=1}^{d}\left((\delta_{\lambda})_{*}\tilde{W}_{\alpha}\right)(\delta_{\lambda}\Gamma_{t})dB_{t}^{\alpha}=\lambda\sum_{\alpha=1}^{d}\tilde{W}_{\alpha}(\delta_{\lambda}\Gamma_{t})dB_{t}^{\alpha}.
\|\, \mbox{\boldmath $u$}(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R}^{n})}}\;\!=\;\Bigl\{\,\sum_{i\,=\,1}^{n}\int_{\mathbb{R}^{n}}\!|\:u_{i}(x,t)\,|^{q}\,dx\,\Bigr\}^{\!\!\:\!1/q}
\alpha\, \int_0^{+\infty}u\bar{u}_tvdx =\frac{\alpha}{2}\int_0^{+\infty}v \frac{d}{dt}|u|^2 dx &=\frac{\alpha}{2}\frac{d}{dt}\int_0^{+\infty}|u|^2vdx -\frac{\alpha}{2}\int_0^{+\infty}|u|^2v_tdx
h_{r e l}^{( 2 )}( \tau_{2}) = \tau_{2}^{- \alpha}\exp ( \frac{a}{2}\tau_{2}^{2}+ b \tau_{2}) h_{r e l}( \tau_{2}) \tau_{2}^{\alpha}\exp ( - \frac{a}{2}\tau_{2}^{2}- b \tau_{2})
\mu^{*}:=\sup\Big\{\bar{\mu}\in \Big(0\ ,\ \min\{1,\sqrt{\frac{\lambda_1}{a}},\sqrt{\frac{\lambda_2}{a}}\}\Big) \ : \ \mu\sqrt{a}K_{\mu}+\overline{L}_{\mu}\leq b+\chi_2\mu_2-\chi_1\mu_1,\ \forall\ 0<\mu\leq \bar{\mu}\Big\}
T(X,Y) =\nabla_XY-(-1)^{(k-1)(l-1)}\nabla_Y X-[X,Y] &=\widetilde{\nabla}_XY+F^{k,l}(X,Y)-(-1)^{(k-1)(l-1)}\widetilde{\nabla}_Y X-(-1)^{(k-1)(l-1)}F^{l,k}(Y,X) &-[X,Y] &=\widetilde{T}(X,Y)+F^{k,l}(X,Y)-(-1)^{(k-1)(l-1)}F^{l,k}(Y,X),
\tilde{Z}( 0 ) = \lim_{x \rightarrow 0}\prod_{h}\frac{1}{\sqrt{2 \pi x}}e^{- \frac{1}{2 x^{2}}\Omega_{h}^{i j}{\Omega_{i j}}_{h}}({\frac{\delta}{i \delta J}}) \cdot Z ( J ) |_{J = 0}
-\frac{1}{\rho_{L}}\frac{\partial P}{\partial r}=\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial r}-\nu_{L}[\frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}\frac{\partial u}{\partial r})-\frac{2u}{r^{2}}]
(\psi_x\circ \Phi^{-1})(W_u) =\psi_x \big(\overline{b(p)}W_{p}\big) &=\overline{b(p)}\sigma_c\big((y,m+p,x),\eta^{-1}\big)\sigma_c(\eta,(x,p,x))\overline{\sigma_c(\eta^{-1},\eta)}\psi_y(W_{p}) &=\sigma_c\big((y,m+p,x),(x,p,x)\big)\sigma_c\big(\eta,(x,p,x)\big)\overline{\sigma_c(\eta^{-1},\eta)}(\psi_y\circ \Phi^{-1})(W_{u}).
V_{\mathrm{\footnotesize 1 - l o o p}}( H ) = \frac{3 N}{2 \pi^{6}R^{4}}\sum_{k = 0}^{\infty}\frac{\cos \left[ ( 2 k + 1 ) \pi R m_{t}( H ) \right]}{( 2 k + 1 )^{5}}.
\sigma_{x}^{2}\sigma_{p}^{2}=(\langle x^{2}\rangle-\langle x\rangle^{2})(\langle p^{2}\rangle-\langle p\rangle^{2})\ge(\langle xp\rangle-\langle x\rangle\langle p\rangle)^{2}+\frac{\hbar^{2}}{4}.
\delta \theta^{\mu\nu}=-\theta^{\mu\alpha}\epsilon_{\alpha\beta}\theta^{\beta\nu}-\frac{1}{4}\theta^{\mu\alpha}\epsilon_{\alpha\gamma}\theta^{\gamma\rho}\epsilon_{\rho\beta}\theta^{\beta\nu}= -\theta^{\mu\alpha}\delta\theta^{-1}_{\alpha\beta}\theta^{\beta\nu}+O(\epsilon^2).
-v_{i+1}z^{3n^{i+2}+n^{i+1}+\cdots+n^2+n}&= v_{i+1}z^{3n^{i+2}+n^{i+1}+\cdots+n^2-n}(z^t-z^nx^n)-z^{3n^{i+2}+n^{i+1}+\cdots+n^2}f_{i+4}+(v_{i+1}z^{3n^{i+1}+n^{i}+\cdots+n^2+n})^n &-\left(\left(\sum_{j=i+2}^{d-3}v_jz^{3n^{i+1}+n^{i}+\cdots+n^2+n}\right)+(-1)^{d-1}xz^{3n^{i+1}+n^{i}+\cdots+n^2+n}\right)^n \in I_n,
\frac{(k-(t'+d'))(t'-d')}{t'+d'}=\frac{(k-(t'+d'))((t'+d')-2d')}{t'+d'}&=k+2d'-\frac{2kd'}{t'+d'}-(t'+d')\le k+2d'-2\sqrt{2kd'}=(\sqrt{k}-\sqrt{2d'})^2
\Delta_A B_0 = \ [B_j,\partial_0 A_j] + [A_j,\partial_0 B_j] + 2 [B_j,\partial_j A_0] + 2 [B_j,[B_j,A_0]], \Delta \partial_0 B_0 = \ \partial_0 \partial_j( [ B_0,A_j]+[ A_0,B_j]).
\begin{aligned}G(z, \tau) = F(z,\tau)^2 \left( y \frac{d}{dy}\right)^2 \log( F(z,\tau) ) & = F(z,\tau)^2 \cdot \bigg\{\frac{y}{(1+y)^2}- \sum_{d \geq 1}\sum_{m | d}m \big( (-y)^{-m}+ (-y)^m \big) q^d \bigg\}& = 1 + (y^{-2}+ 4 y^{-1}+ 6 + 4 y^{1}+ y^2) q & + (6y^{-2}+ 24y^{-1}+ 36 + 24y + 6y^2)q^2 + \, \dots \ .\end{aligned}
\big| \| h_1-h_2 \|_{\dot{H}^{-1,p}(\mu*\gamma_{\sigma})}- \| \tilde{h}_1-\tilde{h}_2 \|_{\dot{H}^{-1,p}(\mu*\gamma_{\sigma})}\big| \le \sum_{i=1}^2 \| h_i-\tilde{h}_i \|_{\dot{H}^{-1,p}(\mu*\gamma_{\sigma})}< 2\epsilon.
\frac{E_{tot}}{c^{2}}=\frac{E_{em}+E_{p}}{c^{2}}=\frac{E_{em}+\frac{E_{em}}{3}}{c^{2}}=\frac{4}{3}\frac{E_{em}}{c^{2}}=\frac{4}{3}m_{es}=m_{em}
Z[\eta,\overline{\eta}] := \int [d\psi][d\overline{\psi}][dA]e^{-S[\psi,\overline{\psi},A] + \sum_a \Big\{( \overline{\eta}^{(a)},\psi^{(a)})+ (\overline{\psi}^{(a)},\eta^{(a)}) \Big\}}\; .
|(\frak p^{m}W_i+u(k)) \cap (\frak p^{m}W_i+u(k'))|= q^m |(W_i+\frak p^{-m}u(k)) \cap (W_i+\frak p^{-m}u(k'))| = q^m |(W_i+ u(q^m k)) \cap (W_i+ u(q^m k'))| = 0,
f_A(n_1+n_2) = \left| A \cap [u^*_1, u^*_1+ n_1 + n_2 -1] \right| & = \left| A \cap [u^*_1, u^*_1+ n_1 -1] \right| + \left| A \cap [u^*_1+n_1, u^*_1+ n_1 + n_2 -1] \right| & = \left| A \cap [u^*_1, u^*_1+ n_1 -1] \right| + \left| A \cap [u^*_2, u^*_2 + n_2 -1] \right| & \leq f_A(n_1) + f_A(n_2).
[ \delta_{1}, \delta_{2}] \lambda_{i}= 2 i D_{\mu}\lambda_{i}\overline{{\epsilon}}_{1}^{j}\gamma^{\mu}\epsilon_{2 j}+ \frac{a}{2}\gamma^{\mu \nu}\lambda_{i}M_{k j}\overline{{\epsilon}}_{1}^{k}\gamma_{\mu \nu}\epsilon_{2 j}
[\begin{matrix}X&X&X&\cdot&\cdot&\cdot&\cdot&\\X&X&\cdot&X&X&\cdot&\cdot&\\X&\cdot&X&\cdot&X&\cdot&\cdot&\\ \cdot&X&\cdot&X&\cdot&X&\cdot&\\ \cdot&X&X&\cdot&X&X&X&\\ \cdot&\cdot&\cdot&X&X&X&\cdot&\\ \cdot&\cdot&\cdot&\cdot&X&\cdot&X&\\ \end{matrix}]
\prod_{i=1}^k\lambda_i\bigl(\Phi(A^{p_\alpha})^{1/2}\Psi(B^{p_\alpha})\Phi(A^{p_\alpha})^{1/2}\bigr) &\quad\le\prod_{i=1}^k\lambda_i^{1-\alpha}\bigl(\Phi(A^{p_0})^{1/2}\Psi(B^{p_0})\Phi(A^{p_0})^{1/2}\bigr)\lambda_i^\alpha\bigl(\Phi(A^{p_1})^{1/2}\Psi(B^{p_1})\Phi(A^{p_1})^{1/2}\bigr)
\|R_0(z)f\|_{L^2(|V|)}^2 =|z|^{-2}|z|^{-n/2}\|R_0(z/|z|)[f(|z|^{-1/2}\cdot)]\|_{L^2(|V(|z|^{-1/2}\cdot)|)}^2 &\leq C|z|^{-2}|z|^{-n/2}[V(|z|^{-1/2}\cdot)]_{\mathcal{S}_n}^2\|f(|z|^{-1/2}\cdot)\|_{L^2(|V(|z|^{-1/2}\cdot)|^{-1})}^2 &\leq C[V]_{\mathcal{S}_n}^2\|f\|_{L^2(|V|^{-1})}^2.
F ( T ) = - \Delta n^{2}\frac{3}{1 2 8}\frac{T}{R}\left( \alpha + \ln \frac{R T}{\hbar c}+ \frac{8 5 7}{3 4 5 6 0}\frac{c^{2}\hbar^{2}}{T^{2}R^{2}}\right) +{\cal O}( T^{- 2}){,}
\sum\limits_{k = 1}^n{\int\limits_0^\infty{{x^{s - 1}}f\left({kx}\right)dx}= \phi \left({- s}\right)}\Gamma \left( s \right)\left({\zeta \left( s \right) + \frac{{{{\left({n + \frac{1}{2}}\right)}^{1 - s}}}}{{1 - s}}+{\rm O}\left({{n^{- s - 1}}}\right)}\right)
\begin{aligned}\lim_{t\to +\infty}\| ({\eta}(t ) ,{\xi}(t )) -(u_{lin}(u_+,v_+)(t) , \frac{d}{dt}u_{lin}(u_+,v_+)(t)) \|_{H^1\times L^2}=0 , & \lim_{t\to +\infty}|z_J(t)| =\rho_{+J}.\end{aligned}
240\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})\right\}^{(8)}=\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})\right. &\left.\cdot{\rm ch}(\widetilde{TX}+2\wedge^2\widetilde{L_R\otimes C}-\widetilde{L_R\otimes C}\otimes \widetilde{L_R\otimes C}+\widetilde{L_R\otimes C})\right\}^{(8)}.
{\rm Ker\,}{\rm P\,}=\hat{\mathcal{P}}^\perp\oplus(\hat{\mathcal{P}}^\perp\bigcap{\rm Ker\,}{\rm P\,}), &{\rm Ker\,}{\rm P\,}=\hat{\mathcal{P}_0}^\perp\oplus(\hat{\mathcal{P}_0}^\perp\bigcap{\rm Ker\,}{\rm P\,}), &\hat{\mathcal{P}}=\hat{\mathcal{P}_0}\oplus(\hat{\mathcal{P}_0}^\perp\bigcap\hat{\mathcal{P}}).
\begin{aligned}\frac1{\alpha_k}\frac k{k+1}-\frac{1-\alpha_k}{\alpha_k}\le\liminf_{n\to\infty}E_{n,k}(X_{n,k}^{\mathcal{T}}|W_{\mathcal{T}(n,k;R_n)})\le &\limsup_{n\to\infty}E_{n,k}(X_{n,k}^{\mathcal{T}}|W_{\mathcal{T}(n,k;R_n)})\le\frac1{\alpha_k}\frac k{k+1}.\end{aligned}
p(2+p)(4+p)\cdots (j-2+p) &=2^{j/2}(0+\frac{p}{2})(1+\frac{p}{2})(2+\frac{p}{2})\cdots (\frac{j}{2}-1+\frac{p}{2}) &=2^{j/2}\prod_{k=1}^{j/2}\Big( k-1 +\frac{p}{2}\Big)=2^{j/2}\frac{\Gamma(j/2+p/2)}{\Gamma(p/2)}
(11,7) - \frac{1}{2}(16,0) = \frac{1}{2^2}(4,22) + \frac{1}{2^3}(16,0) + \frac{1}{2^3}(0,12), & 2\left((11,7) - \frac{1}{2}(16,0) - \frac{1}{2^2}(4,22)\right) = \frac{1}{2^2}(16,0) + \frac{1}{2^2}(0,12) & 2^2\left((11,7) - \frac{1}{2}(16,0) - \frac{1}{2^2}(4,22) - \frac{1}{2^3}(16,0)\right) = \frac{1}{2}(0,12)
z\cdot(g_1,\ldots,g_n)= (g_1,\ldots,g_n), a_i\cdot(g_1,\ldots,g_n)= (g_1,\ldots,g_n),\ \i=1,\ldots,n \sigma\cdot(g_1,\ldots,g_n)= (g_{\sigma^{-1}(1)},\ldots,g_{\sigma^{-1}(n)}),\sigma\in S_n.
v(y) = L_m^{\ell+1/2}(-y)|_{y=\frac{1}{2}\omega r^2}, \bar{w}(r) =w_3(r) - \omega r\left[ \frac{\partial_yL_m^{(\alpha+1)}(-y)}{L_m^{(\alpha+1)}(-y)}\right]_{y=\frac{1}{2}\omega r^2}.
H_k(\tau_r) =\dfrac{a^{4k+2}}{2^{(2k+1)(r+3)}}\cdot \left(\dfrac{e^{-2^{r+1}(k+1)\pi}}{4^{k+1}}+\sum_{\ell=1}^{k-1}\dfrac{\alpha_k(\ell)}{4^\ell}\cdot e^{M(\ell,r)}+\sum_{\ell=k}^{2k}\dfrac{\beta_k(\ell)}{4^\ell}\cdot e^{M(\ell,r)}\right)\cdot e^{J(k,r)},
\dot{\widetilde{\Pi}}^{\varepsilon}_{\vec{\mathbb{P}}}(f_{1},f_{2})(x)=\int_{0}^{1}\int_{0}^{1}\sum_{\vec{P}\in\vec{\mathbb{P}}}\frac{\widetilde{C}^{\varepsilon}_{Q_{\vec{P}}}}{|I_{\vec{P}}|^{\frac{1}{2}}}\langle f_{1},\Phi^{1,\lambda}_{\vec{P}_{1}}\rangle\langle f_{2},\Phi^{2,\lambda}_{\vec{P}_{2}}\rangle\Phi^{3,\lambda}_{\vec{P}_{3}}(x)d\lambda.
=2\sum_{n\ge0}\sum_{i=0}^n\sum_{0\le m_1\le m_2\le\ldots\le m_r \atop c_1+c_2+\ldots=r}\sum_{j=0}^{m_r}\frac{(rx-j)^{n-i}r!(-1)^{j+c_1+2c_2+\ldots}(c_1+2c_2+\ldots)^i\binom{m_r}{j}\binom{n}{i}}{(c_1!c_2!\ldots)(m_1^{k_1}m_2^{k_2}\ldots m_r^{k_r})}\frac{t^{n}}{n!}
e_{i}={\bf 1}\otimes \cdots \otimes \left( \begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{q^{- 1}}}&{{- 1}}&{{0}}\\{{0}}&{{- 1}}&{{q}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right) \otimes \cdots \otimes{\bf 1}\in{\bf C}^{\otimes 2 l}\; ,
\|\chi\|_Y=\left\{\int_0^\ell\|\chi(r)\|_{V\times H^2(\bar{\Omega},d\sigma)}^2dr+\left(\int_0^\ell\|\chi_t(r)\|_{V^*\times (H^1(\bar{\Omega},d\sigma))^*}\,dr\right)^2\right\}^{\frac{1}{2}}.
S_2 =\frac{1}{c}\sum_{n=1}^N \frac{\left(q/c\right)_{n-1}(c q)_{N-n}(cq)^n}{(q)_n (c q)_N (q)_{N-n}}\left( \frac{1}{c-1}- \frac{1}{(1/c)_n}\sum_{k=1}^n \left[\begin{matrix}n k\end{matrix}\right] \frac{(qc)_k (1/c)_{n-k}c^{-k}}{1- q^k}\right) &=\frac{-1}{(1-c)^2(q)_N}\left( 1- \frac{(q)_N}{(cq)_N}\right) +S_{2}^{*},
\Bigg[\mathrm{H}^{1,\frac{1}{2}}\Big(\partial \Omega\times \mathbb{R}\Big), \mathrm{L}^2\Big((\partial\Omega)_T\Big)\Bigg]_{\theta = \frac{1}{2}}= \mathrm{H}^{\frac{1}{2},\frac{1}{4}}\Big((\partial\Omega)_T\Big).
F_t(p,v) = \frac{\left[ (1-t^2) \langle p, p_0 \rangle^2 + t^2 \right]^{3/2}}{t \left[ (1-t^2) (\langle p, p_0 \rangle^2 + \langle v, p_0 \rangle^2) + t^2 \right] \left[ 1 + (t^2-1) \langle \eta(p), p_0 \rangle^2 \right]^{1/2}}.
\det (\mathbf{I}- \mathbf{B}^{k}) = \det (\mathbf{I}- \mathbf{B}^{k-1})- \frac{\lambda_{k-1}\, \mu_{k-1}}{(\alpha +\lambda_{k-2}+ \mu_{k-1}) \, (\alpha + \lambda_{k-1}+ \mu_k)}\, \det(\mathbf{I}- \mathbf{B}^{k-2})