harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
((\phi \otimes f_{n})*\psi)(X,s) =\phi(X)\int f_{n}(t)\alpha_{t}(\psi(X.t,t^{-1}s))1_{X}(t^{-1})dt = \phi(X)\int_{t \in U_{n}}f_{n}(t)\alpha_{t}(\psi(X.t,t^{-1}s))dt =\phi(X) \int_{t \in U_{n}}f_{n}(t)R_{t}(\psi(X,s))dt .
\left\lbrace\begin{array}{ll}g_{ii}= \left\|\mathbf{h}_i \right\|^2_2, i = 1, \ldots, K, g_{ij}= \mathbf{h}_i^{\mathrm{H}}\mathbf{h}_j, j = 1, \ldots, K, \,\, j \neq i.\end{array}\right.
[ x_{i}, p_{j}] = i \delta_{i j}, \quad [ \tilde{p}_{i}, \tilde{p}_{j}] = \frac{i \kappa}{4}\, \epsilon_{i j}, \quad [ x_{i}, x_{j}] = [ p_{i}, p_{j}] = [ x_{i}, \tilde{p}_{j}] = [ p_{i}, \tilde{p}_{j}] = 0 .
\sigma_{\Delta \textbf{h}_{1,k}}^2 = \frac{E\big[\|\Delta \textbf{h}_{1,k}\|^2 \big]}{E\big[\|\textbf{h}_{1,k}\|^2\big]}= \frac{1}{\tau}\bigg( \frac{\alpha(1-\lambda)}{\lambda}\frac{\sum_{j=1}^L\sum_{k=1}^{K}\beta_{j,k}}{\beta_{1,k}}+ \frac{\sigma_n^2}{\lambda \beta_{1,k}}\bigg).
\tau(s) = 1 + T(s) L_0'(s) + k\ \frac{L_1'(s)}{L_1(s)}\ \left[ \frac{s}{k}- T(s) \right] = L_0'(s)\ \left( T(s) - \frac{s}{1-L_0(s)}\right) - k\ \frac{L_1'(s)}{L_1(s)}\T(s) > 0\ ,
\underbrace{\zeta_1 , \ldots , \zeta_{q_1}}_{:= \xi_1},\ \underbrace{\zeta_{q_1 +1}, \ldots , \zeta_{q_2}}_{:= \xi_2},\ \ldots,\ \underbrace{\zeta_{q_{s-2}+ 1}, \ldots , \zeta_{q_{s-1}}}_{:= \xi_{s-1}},\ \underbrace{\zeta_{q_{s-1}+ 1}, \ldots , \zeta_p}_{:= \xi_s = 0};
\forall\,x:n\forall\,y:n(h_n(x+_ny)=h_n(x)+_{n+1}h_n(y)) \forall\,x:n\forall\,a:0(h_n(a\cdot_n x)=a\cdot_n h_n(x)) h_n(\top_n)=\top_{n+1}
\tilde{Z}_\downarrow (w, z; b) =\sum_{k=0}^\infty C(w, z p^{-k})\prod_{i=0}^{k-1}D(w, z p^{-i})+\lim_{k\to\infty}\tilde{Z}_\downarrow (w, z p^{-k};b)\prod_{i=0}^{k-1}D(w, z p^{-i}).
\begin{aligned}\beta_1 = \bar{a}_{21}\bar{z}_1 + \bar{a}_{22}\bar{z}_2+ 2 \bar{b}_{12}z_1 + 2 \bar{b}_{22}z_2 , & \beta_2 = - \bar{a}_{11}\bar{z}_1 - \bar{a}_{12}\bar{z}_2- 2 \bar{b}_{11}z_1 - 2 \bar{b}_{12}z_2 , & \alpha = a_{11}\bar{z}_1 \beta_1 + a_{21}\bar{z}_2 \beta_1+ a_{12}\bar{z}_1 \beta_2 + a_{22}\bar{z}_2 \beta_2 .\end{aligned}
\frac{1}{(q)_{\infty}}\sum_{n=1}^{\infty}\frac{n(-d)^{n-1}(q/d)_{n-1}q^{n(n+1)/2}}{(q)_{n}^2}=\frac{1}{(q)_{\infty}}\sum_{n=1}^{\infty}\frac{nq^n(dq)_{n-1}}{(q)_n}&\quad-\frac{(dq)_{\infty}}{(q)_{\infty}}\sum_{j=1}^{\infty}\frac{q^{j^2}}{(q)_{j}^{2}}\sum_{n=1}^{j}\frac{q^n}{(1-dq^n)(1-q^n)}.
\mathbb{P}\left(\{\#{\cal E}_1 \geq \epsilon n\}\cap \{\tau = k\}\right) \leq \mathbb{P}\left(\bigcup_{j=1}^{k}\{\#{\cal N}_j \geq \epsilon n t^{-1}\}\right)\leq \sum_{j=1}^{k}\mathbb{P}\left(\#{\cal N}_j \geq \epsilon n t^{-1}\right).
\zeta_{1,\varepsilon}\mathcal{T}_{a^+,a^-}(c^k)=\mathcal{T}_{a^+,a^-}(\zeta_{1,\varepsilon}c^k)+(\zeta_{1,\varepsilon}\mathcal{T}_{a^+,a^-}(c^k)-\mathcal{T}_{a^+,a^-}(\zeta_{1,\varepsilon}c^k)).
\lim_{n\to\infty}\tau_{\Lambda^{(n)}\times\Delta^{(n)}}\big( \tilde A(\Lambda^{(n)}\times\Delta^{(n)};f_1)\dotsm \tilde A(\Lambda^{(n)}\times\Delta^{(n)};f_k)\big) =\tau(\tilde A(f_1)\dotsm \tilde A(f_k)).
R_{n,H,f}(V)= T^{H,H+V,H,\ldots,H}_{f^{[n]}}(V,\ldots,V) = \sum_{p=0}^n\sum_{\substack{j_1,\ldots,j_{p}\geq1,j_{p+1}\geq0 j_1+\ldots+j_{p+1}=n}}(-1)^{n-p}\,T^{H,H_{j_1},H,\ldots,H}_{(fu^p)^{[p]}}(\tilde{V}^{j_1},\ldots,\tilde{V}^{j_{p}})\tilde{V}^{j_{p+1}},
( a ( \phi^{\zeta}) , b ( \phi^{\zeta}) ) + \delta_{\epsilon}( a ( \phi^{\zeta}) , b ( \phi^{\zeta}) ) = ( a ( \phi^{\zeta}) + \delta_{\epsilon}a ( \phi^{\zeta}) , b ( \phi^{\zeta}) + \delta_{\epsilon}b ( \phi^{\zeta}) )
\sum_{a,b}\eta_{ab}\vartheta \left[ \begin{array}{c}a b \end{array}\right ]\vartheta \left[\begin{array}{c}a b+v_5 \end{array}\right] \prod_{r=3}^{4}\vartheta \left[ \begin{array}{c}a+ \frac{1}{2}b+v_r \end{array}\right] = 0
\exp \left \{\int_0^T dt \> \rho^{\mu}(t)\gamma_{\mu}\right \}\> = \> \exp \left \{\gamma_{\mu}\frac{\partial}{\partial \Gamma_{\mu}}\right \}\> \> \exp \left \{\int_0^Tdt \> \rho^{\mu}(t)\Gamma_{\mu}\right \}\Biggr|_{\Gamma_{\mu}= 0}
C(N_1,N_2,N_3) := \begin{cases}N_{\min}^{\varepsilon}(d=3), N_{\min}^{\varepsilon}N_3^{\frac{1}{4}}(d=4), N_{\min}^{\frac{d}{2}-\frac{2 d}{d-1}+ \varepsilon}N_3^{\frac{2}{d-1}}(d \geq 5).\end{cases}
\gamma^{N^F n}(X_{N,n,\varepsilon,\theta}^{F})=\int_{z_1 \in A^{N^F}_{\varepsilon}}\int_{z_{2}\in A^{N,F}_{\varepsilon,\theta}[z_1]}\cdots \int_{z_n \in A^{N,{F}}_{\varepsilon,\theta}[z_1,\ldots,z_{n-1}]}d\gamma^{N^F}(z_{n}) \cdots d\gamma^{N^F}(z_2) d\gamma^{N^F}(z_1).
\frac{d}{dt}\Big[(6+t)E(U_1;t)\Big] =\|U_1'\|^2+\|A^{1/2}U_1\|^2+2(6+t)(U_1',-U_1'+Ae^{-tA}(u_0+u_1)) &\leq\|U_1'\|^2+\|A^{1/2}U_1\|^2-(6+t)\|U_1'\|^2+(6+t)\|Ae^{-tA}(u_0+u_1)\|^2
( \prod_{i = 1}^{r - 2}T_{a_{i}}^{x_{i , 1}\ldots x_{i , s}}) ( \prod_{j = 1}^{s}T_{x_{1 , j}\ldots x_{r - 2 , j}x_{w , j}}^{b_{j}}) T_{a_{r - 1}a_{r}}^{x_{w , 1}\ldots x_{w , s}}
III=\frac{1}{|B|}\int_B|L^{-\alpha/2}(\prod\limits_{j=1}^m(b_j-(b_j)_B)f_1)(y)|dy &\ \ \ \ \leq \frac{C}{|B|^{1-\alpha/n}}\|L^{-\alpha/2}(\prod\limits_{j=1}^m(b_j-(b_j)_B)f_1)\|_{L^{\frac{n}{n-\alpha},\infty}}&\ \ \ \ \leq \frac{C}{|B|^{1-\alpha/n}}\int_{2B}|\prod\limits_{j=1}^m(b_j(y)-(b_j)_B)f(y)|dy
\partial_su(s,t)+J_1(u(s,t))(\partial_tu(s,t)-X_{H^{(k)}}(u(s,t))) =(\phi_L^t)_*(\partial_sv(s,t))+J_1(u(s,t))((\phi_L^t)_*(\partial_tv(s,t))-(\phi_L^t)_*X_{K^{(k)}}(v(s,t)))
{\cal L}= -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{\theta}{4}\epsilon^{\mu\nu\alpha}F_{\mu\nu}A_{\alpha}+\frac{i}{2}\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-\frac{i}{2}(\partial_{\mu}\bar{\psi})\gamma^{\mu}\psi-m\bar{\psi}\psi+e\bar{\psi}\gamma^{\mu}A_{\mu}\psi ,
f(t)=\frac{\nu}{\alpha\beta\sqrt(2\pi)}e^{-{\frac{1}{2\alpha^2}[(\frac{t}{\beta})^{2\nu}+(\frac{t}{\beta})^{-2\nu}-2][(\frac{t}{\beta})^{\nu-1}+(\frac{t}{\beta})^{-(\nu+1)}]}}.
A_{6k+3,k}= [(6k+3)a, (6k+4)a - 2]_2, B_{6k+3,k}= \left[ (6k+3)a + \frac{a}{2}+ 3, (6k+4) a + \frac{a}{2}-1 \right]_2, A_{6k+4,k}= [(6k+4) a, (6k+5)a]_2.
e^{-r2^{j-1}l}\geq \Pr(B_{i_1,k_1,\ldots,i_l,k_l})=\prod_{s=1}^{l}\frac{\Pr(B_{i_1,k_1,\ldots,i_s,k_s})}{\Pr(B_{i_1,k_1,\ldots,i_{s-1},k_{s-1}})}\geq e^{-lr}\prod_{s\colon f(s)\geq 1}e^{-r2^{f(s)}}.
\begin{array}{ll}\sum\limits_{k^{'}=k+N}^{K}\int_{B(x_{i_{l_{k^{'}}}},R_{i_{l_{k'}}})}|u(\bar{t})|^{2}\, dx \leq\frac{1}{2}\int_{B(x_{i_{l_{k}}},R_{i_{l_k}})}|u(\bar{t})|^{2}\, dx \cdot\end{array}
\left(\left( Q_{1,\vec{v}}\right)_i \right)_{j,k}= \begin{cases}1 i \leq j , k = j-i+1 \alpha_{n-i+1+j-k}2 \leq i \leq j, k \in \{j-i+2,\dots, j\}- \alpha_{n-i+1+j-k}j<i , j+1\leq k \leq n+j-i+1 0 \end{cases}.
\int_{\mathbb{R}^{d}}\mathrm{d}y\p_{s}(y-x)\,1_{B}(y)\,p_{s}(y-x) & =\int_{\mathbb{R}^{d}}\mathrm{d}y\p_{s}(y-x)\,p_{s}(y-x)-\int_{B^{\mathrm{c}}}\mathrm{d}y\p_{s}(y-x)\,p_{s}(y-x).
\psi = \left( \begin{matrix}{{{\psi_{1}}( \rho , z ){e}^{i ( m_{z}-{\frac{1}{2}}) \phi}}}\\{{{\psi_{2}}( \rho , z ){e}^{i ( m_{z}+{\frac{1}{2}}) \phi}}}\\{{- i{\psi_{3}}( \rho , z ){e}^{i ( m_{z}-{\frac{1}{2}}) \phi}}}\\{{- i{\psi_{4}}( \rho , z ){e}^{i ( m_{z}+{\frac{1}{2}}) \phi}}}\\ \end{matrix}\right)
\frac{d}{dt}\left[ \frac{1}{2}\left( \| \partial_t u^{(j)}(t) \|_{L^2}^2 + \| \nabla u^{(j)}(t) \|_{L^2}^2 \right) + \frac{1}{p+1}\| u^{(j)}(t) \|_{L^{p+1}}^{p+1}\right] = - \| \partial_t u^{(j)}(t) \|_{L^2}^2.
G_{1}^{( m , k )}= \Phi ( \vec{x}_{r}, \vec{x}_{s}) \left( \frac{1}{\mathcal{S}_{+}}- \frac{1}{\mathcal{S}_{-}}\right) = \Phi ( \vec{x}_{r}, \vec{x}_{s}) \left( \frac{\sinh \mathcal{U}}{\cosh \mathcal{U}- \cos \mathcal{T}}\right) \; .
{\rm d}s^2 = G_{\mu\nu}(\gamma , R ){\rm d}X^\mu{\rm d}X^\nu ={\rm d}\rho^2 + \rho^2 \left({\rm d}\phi +{\gamma \over R}{\rm d}y \right)^2 +{\rm d}y^2 +{\rm d}x_i{\rm d}x^i\,,
\operatorname{Re}\psi_3(i\rho,i\tau,\xi+i\eta)=-\alpha\rho\tau-\beta\rho\eta\leqslant-\frac{\alpha}{2}(1+3\rho^2) \leqslant-\frac{\alpha}{2},\shortintertext{and}\operatorname{Re}\psi_4(i\rho,i\tau, \xi+i\eta)=-\alpha\rho\tau-\beta\tau\eta \leqslant-\frac{\alpha}{2}(1+3\rho^2)\leqslant -\frac{\alpha}{2},
\begin{aligned}P(\frac{S^{(i)}_{\tau^{N,i},N+\tau^{N,i}}}N< r_i)\approx e^{-N\big(I_i(r_i)-I_i(r_{i+1})\big)^+}; &P(\frac{S^{(i)}_{\tau^{N,i},N+\tau^{N,i}}}N\ge r_{i+1})\approx e^{-N\big(I_i(r_{i+1})-I_i(r_i)\big)^+}.\end{aligned}
\sum_{n=0}^{\infty}\frac{q^{n^2+n}}{(zq;q^2)_{n+1}}=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}b^3_\nu(m,n)z^mq^n=\sum_{n=0}^{\infty}(-q/z;q^2)_n(zq)^n.
\Gamma_{3}= \frac{\lambda^{3}}{8 \left( 2 \pi \right)^{5}}\int \frac{\mathrm{d}^{5}Q}{|{\bf Q}|}N \left( |{\bf Q}| \right) \left\{\frac{k_{1}^{2}k_{2}\cdot k_{3}}{\left( Q \cdot k_{1}\right)^{2}Q \cdot k_{2}Q \cdot k_{3}}+ p e r m u t a t i o n s \right\},
X^{i}( \tau , \sigma ) = x^{i}+ \frac{p^{i}}{2 \pi}\tau + \frac{i}{2 \sqrt{\pi}}\sum_{n \ne 0}\left( \frac{1}{n}\alpha_{n}^{i}e^{- i n ( \tau - \sigma )}+ \frac{1}{n}\tilde{\alpha}_{n}^{i}e^{- i n ( \tau + \sigma )}\right) ,
F_{0}( \lambda , \mu ) = -{\frac{R}{2}}\mu^{2}\log \mu + R \mu^{2}\sum_{n = 1}^{\infty}{\frac{1}{n !}}\left[ \mu^{R - 2}\tilde{\lambda}^{2}\right]^{n}{\frac{\Gamma ( n ( 2 - R ) - 2 )}{\ \Gamma ( n ( 1 - R ) + 1 )}}.
\begin{cases}\mathcal{M}(E_{B^{q}}^+) \;=\; \{(i,j) \in E_{B^{q}}\mid \Delta t_{i,j}= \Delta t_q, \; \mbox{and}\; i \triangleleft j \}, \mathcal{M}(E_{B^{q}}^-) \;=\; \{(i,j) \in E_{B^{q}}\mid \Delta t_{i,j}= \Delta t_q, \; \mbox{and}\; j \triangleleft i \}, \end{cases}
\int_{0}^\infty \int_{\mathbb{R}^d_+}\partial_{x_i}P^\alpha_t (f)(x)\partial_t \mathbb{P}^{\alpha,i}_t (g)(x)x^{2\alpha}dx t dt &=\int_0^\infty \int_{\mathbb R_+^d}|y| e^{- 2|y|t}h_\alpha (f)(y)h_{\alpha+e_i}\left(\frac{g(\cdot)}{\cdot_i}\right)(y) y^{2(\alpha+e_i)}dy\;t\, dt.
\begin{array}{ll}L_1:=2\xi-e_0+e_1; L_1':=4e_0-2e_1-e_2-e_3-e_4-e_5-e_6; L_2:=2\xi-2e_0+e_1+e_2+e_3+e_4 L_2':=5e_0-2e_1-2e_2-2e_3-2e_4-e_5-e_6 L_3:=2\xi-3e_0+2e_1+e_2+e_3+e_4+e_5+e_6 L_3':=6e_0-3e_1-2e_2-2e_3-2e_4-2e_5-2e_6. \end{array}
{\rm d}\mu_{\,{\rm SU}(N)}= \frac{\pi^{2N-6}\,e^{-8\pi^2/g^2}}{2^{8}\,(N-1)!\,(N-2)!}\, (M_{\rm PV})^{3N}\left( \frac{8\pi^2}{g^2}\right)^{N}\! \!{\rm d}\rho^2\,{\rm d}^4 x_0\,{\rm d}^2\theta_0 \,{\rm d}^2\bar\beta\, \prod_{i=1}^{2N-4}\rho\,{\rm d}\bar\xi_i \, .
E=\langle d_1^8\rangle \times\langle d_3^{-1}d_{13}\rangle\times\langle d_5^{-1}d_{11}\rangle\times\langle d_7^{-1}d_9\rangle\times &\times\langle d_1^{-2}d_7^2\rangle\times\langle d_3^{-2}d_5^2\rangle\times\langle d_1^{-4}d_3^4\rangle.
\widehat{(|\psi_N^n|^2)}_l'(0)=2\widehat{\left(\mathrm{Re}\{\overline{\psi_N^n}\partial_s\psi_N^n\}\right)}_l(0),\quad\widehat{(z_N^n\psi^n_N)}_l'(0)=\widehat{(\partial_sz_N^n\psi^n_N)}_l(0)+\widehat{(z_N^n\partial_s\psi^n_N)}_l(0),
\int_{-1}^1 s^k\big(u_{1-}(s) + \hat{g}_2(s)\phi_+^{-n}\big){\rm d}s = 0 , \int_{-1}^1 P_k(s)\big(p_{n-1}(s) + C_{[-1,1]}^-(\hat{g}_1 \phi_+^{n}) + \hat{g}_2(s)\phi_+^{-n}\big){\rm d}s = 0.
\langle x_{\sigma(UV)}, x_{\sigma(UW)}\rangle =\langle \pi({\sigma(UV)})x_I, \pi({\sigma(UW)})x_I\rangle = \langle \overline{f(UV)}\pi(U)\pi(V)x_I, \overline{f(UW)}\pi(U)\pi(W)x_I \rangle = \overline{f(UV)}f(UW)\langle\pi(V)x_I, \pi(W)x_I \rangle =\overline{f(UV)}f(UW)\langle x_V, x_W \rangle, \forall U, V, W \in \mathcal{U}.
\lim_{n\to \infty}\sup_{m\geq n}E(\sup_{0\leq t\leq T}| X^{n}_{t\wedge \tau_N^{n,m}}- X^{m}_{t\wedge \tau_N^{n,m}}|^2)\leq \lim_{n\to \infty}\sup_{m\geq n}e^{C(N,T)T}C(N,T,M,\tilde M) \epsilon_{n,m}= 0.
{\cal L}={\frac{k}{2}}\varepsilon^{\alpha \beta \gamma}A_{\alpha}\partial_{\beta}A_{\gamma}+ i \Psi^{\ast}( \partial_{t}+ i A_{0}) \Psi -{\frac{1}{2}}\left| ( \nabla - i{\bf A}) \Psi \right|^{2}+{\frac{g}{2}}\left| \Psi \right|^{4}.
={\frac{1}{\Gamma ( 2 z + 2 k - 2 l )}}\int_{0}^{\infty}d y ~{\frac{y^{2 z + 2 k - l - 1}}{1 - e^{- y}}}( - 1 )^{p - l}\left({\frac{d}{d y}}\right)^{p - l}\left[{\frac{\exp \left( -{\frac{d - 1}{2}}y \right)}{e^{\gamma y}- 1}}\right] .
Q_x\xi(r) =\sum_{j\in\S,j\neq r}q_{rj}(x)(\xi_j-\xi_r)=\sum_{j\in F_i,j\in \S}q_{rj}(x)(\xi_j-\xi_r) &=\sum_{k< i}\big(\sum_{j\in F_k}q_{rj}(x)\big)(\xi_k^F-\xi_i^F)+\sum_{k> i}\big(\sum_{j\in F_k}q_{rj}(x)\big)(\xi_k^F-\xi_i^F) &\leq \sum_{k\neq i}q_{ik}^F(\xi_k^F-\xi_i^F)=\big(Q^F\xi^F\big)(i).
\widehat{G_2(h,\phi)}(\xi)=\frac{1}{2\pi}\int_{\mathbb{R}}\widehat{h}(\eta)\widehat{\phi}(\xi-\eta)\big[1-\chi(\xi-\eta,\eta)\big]\big[\xi(\xi-\eta) -|\xi||\xi-\eta|\big]\,d\eta,
\mathcal{H}_{\mathrm{b2u}}= \left[\sqrt{a_{\mathrm{b2u},1}}\mathbf{H}^\mathrm{T}_{\mathrm{b2u},1}\, \cdots \, \sqrt{a_{\mathrm{b2u},K}}\mathbf{H}^\mathrm{T}_{\mathrm{b2u},K}\right]^{\mathrm{T}}
(\zeta_{n,j}^*)'(r)=\frac{o\Big(\frac{1}{\lambda_{n,j}^{(1)}}\Big)}{r}+O\left(\frac{{e^{-\lambda_{n,j}^{(1)}}}}{r^3}\right) \ \ \textrm{for all}\ \r\in (\Lambda_{n,j,R}^{-},{\delta}).
\langle \psi^{\dagger}({\bf{0}}, 0 ) \psi ({\bf{x}}, t ) \rangle = e x p \mathrm{}[ \mathrm{}( \frac{1}{4 N_{0}}) \sum_{{\bf{q}}}f_{{\bf{q}}}({\bf{x}}, t ) \mathrm{}] \langle \psi^{\dagger}({\bf{0}}, 0 ) \psi ({\bf{x}}, t ) \rangle_{f r e e}
X^\mu (\tau ,\sigma ) = q^\mu + a_0^\mu \tau + (\frac{\pi}{2}-\sigma )B^\mu_{\ \nu}a_0^\nu + \sum_{n\neq 0}\frac{1}{n}e^{-in\tau}(ia_n^\mu \cos n\sigma - B^\mu_{\ \nu}a_n^\nu \sin n\sigma ).
x_1^2x_2^2x_3^2 \geq \max\left\{\begin{aligned}\max \{x_1^2 + x_2^2 - 1, 4x_1^2 + 4x_2^2 - 16 \}+ x_3^2 -1 4 \max \bigl\{x_1^2 + x_2^2 - 1, 4x_1^2 + 4x_2^2 - 16 \bigr\}+ 16 x_3^2 -64\end{aligned}\right\}.
\int_{a_{k}}^{b_{k}}G_{n,k}(\tau)\tau^{s+1}d\mu_{k,j}(\tau) =0,0\leq s\leq \lambda-1,j\in[k:\ell(n)], \int_{a_{k}}^{b_{k}}G_{n,k}(\tau)\tau^{s}d\mu_{k,j}(\tau) =0,0\leq s\leq \lambda,j\in[\ell(n)+1:p-1].
\frac{\mathcal{L}^n}{n!}\mathrm{Mab}^l_{\Theta}(\phi) = \sum_Y \Lambda_Y \int_{\tilde{\Delta}^+_Y}(nu^*(p)-u^*(p)\sum \frac{\chi(\alpha^{\vee})}{q(\alpha^{\vee})}+d_pu^*(p))P_{DH}dq & +\int_{\Delta^+}u^*(p)(\sum \frac{\chi^{ac}(\alpha^{\vee})}{q(\alpha^{\vee})}-\bar{S}_{\Theta})P_{DH}dq +\int_{\Delta^+}4\rho_H(a)P_{DH}(q)dq
\phantom{=}C_3\|u_n\|^p-C_4\varepsilon_W(\|u_n\|^{q_1}+ \|u_n\|^{q_2})-C_2 &\geq C_3\|u_n\|^p-C_4\varepsilon (\|u_n\|^{q_1}+ \|u_n\|^{q_2})-C_2.
Q^U_{1,l}\equiv E_U(r^{n-1}\rho f(J_{1,l}))= 2 r^{n-1}\rho U f_{J_{1,l}}^{(l)}, & Q^\rho_{1,l}\equiv E_\rho(r^{n-1}\rho f(J_{1,l})) = r^{n-1}\big( U^2 f_{J_{1,l}}^{(l)}+ f - \sum_{0\leq i\leq l}J_{1,l-i}f_{J_{1,l}}^{(i)}\big) , & Q^S_{1,l}\equiv E_S(r^{n-1}\rho f(J_{1,l})) = -\tfrac{2}{n}p' D_r ( r^n f_{J_{1,l}}^{(l)}) .
d((u_1, w_1 , a_1), (u_0, w_0, a_0)) &:= M_{u^*_1 - u^*_0}+ [w_1 - w_0]_{2\alpha}+ \| u^*_1 - u^*_0\|_{\alpha}+ \|w_1 - w_0 \|_{\alpha}+ (N_0 + N_0^{int}) \| a_1 - a_0\|_{\alpha}
\sigma - p_1 = \gamma + \gamma p_1, \frac{3}{2}\partial_x \sigma - \frac{3}{2}\partial_x p_1 + p_0 = \frac{3}{2}\gamma \partial_x p_1 -\gamma p_0 +\frac{3}{4}\partial_x \gamma +\frac{3}{4}\partial_x \gamma p_1 .
u_{ij}^{(\pi)}D^{\frac{1}{p}}=D^{\frac{1}{p}}\sigma_{\frac{\mathrm{i}}{p}}(u_{ij}^{(\pi)})=D^{\frac{1}{p}}(Q_\pi^{-\frac{1}{p}})_{ii}u_{ij}^{(\pi)}(Q_\pi^{-\frac{1}{p}})_{jj},
\Phi_{xx}+\Phi_{qq}=-\frac{1}{\Phi_{xq}}(\Phi_{xy}\Phi_{yq}+\Phi_{xp}\Phi_{pq}), \Phi_{xx}-\Phi_{qq}=-\frac{1}{\Phi_{xp}}(\Phi_{xy}\Phi_{yp}+\Phi_{xq}\Phi_{qp}).
\delta(x_1^{a_1}\cdots x_{i-1}^{a_{i-1}}x_{i+1}^{a_{i+1}}\cdots x_n^{a_n}dx_1 \wedge \cdots \wedge \widehat{dx_i}\wedge \cdots \wedge dx_n) = (-1)^{i-1}\hbar^{-1}r_i x_1^{a_1}\cdots x_{i-1}^{a_{i-1}}x_i^{r_i-1}x_{i+1}^{a_{i+1}}\cdots x_n^{a_n}\Omega.
\partial_{t}\Vert a \Vert^{2}\leq -c\frac{\Vert a \Vert^{4}}{\lambda^{2}}+O(\frac{\Vert a \Vert\vert \delta J(u)\vert^{2}}{\lambda})=\Vert a \vert^{2}(-\frac{\Vert a \Vert^{2}}{\lambda^{2}}+O(\frac{\vert \delta J(u)\vert^{2}}{\Vert a \Vert\lambda})).
\mid{P}_{0}\rangle \: = \:{\mathbf O^{- 1}}\mid \tilde{P}_{0}\rangle \: = \: \exp \left({\cal W}_{\nu \rho}^{\mu \sigma}{\mathbf a}^{\nu}{\mathbf b}_{\mu}{\mathbf a}^{\rho}{\mathbf b}_{\sigma}\right) \prod_{\alpha = 0}^{3}\,{\mathbf a}_{\alpha}^{\dagger}\prod_{\beta = 0}^{3}\,{\mathbf b}^{\dagger \beta}\, \mid 0 \rangle .
Y_t^{b}= \xi^b-\int_{t}^{T}g^b(s, Y_s^{b}, Z_s^{b})+\lambda_s\frac{1-e^{\alpha(\xi^a_s-Y_s^{b})}}{\alpha}ds-\int_{t}^{T}Z_s^{b}\cdot dW_s, ~t\in[0, T],
\begin{gathered}(e'_0 \alpha'^{(2)}_{\eta}(\phi,\phi')- m (\phi, \phi')h) \wedge (e'_0 \alpha'^{(2)}_{\eta}(\phi,\phi'')- m (\phi, \phi'')h) = e'_0 \phi \Big(m (\phi, \phi')\alpha'_{\eta}(\phi'') -m (\phi, \phi'')\alpha'_{\eta}(\phi')+m (\phi', \phi'')\alpha'_{\eta}(\phi) \Big),\end{gathered}
\frac{dU_{i}}{dt}=-\frac{1}{\rho}\sum_{j=1}^{N}(\frac{\partial p}{\partial x})j\int\Omega\phi_{j}\frac{\partial\phi_{i}}{\partial x}d\Omega+\nu\sum_{j=1}^{N}\int_{\Omega}(\frac{\partial^{2}u}{\partial x^{2}})\phi_{j}\frac{\partial^{2}\phi_{i}}{\partial x^{2}}d\Omega+\int\Omega f_{x}\phi_{i}d\Omega
\hat{x}_{i}=\frac{l_{i}^{x}}{l_{1}^{x}}\frac{\sum\limits_{j=1}^n \sum\limits_{k=1}^p{e_{1jk}y_{j}z_{k}}}{\sum\limits_{j=1}^n \sum\limits_{k=1}^p{e_{ijk}y_{j}z_{k}}},
\begin{aligned}\int_{\Sigma}\frac{\tau k}{\rho}\wedge\mathrm{tr}(\frac{\delta\mathcal{F}}{\delta v})=\int_{\partial\Omega}\mathrm{tr}(\frac{\tilde{p}}{\rho})\wedge\mathrm{tr}(\frac{\delta\mathcal{F}}{\delta v})=(-1)^{n-1}\int_{\Omega}\frac{\delta\mathcal{F}}{\delta v}\wedge d\big(\frac{\tilde{p}}{\rho}\big). \end{aligned}
1 = G I , \; \; I = 2 N \int \frac{d^{4}l}{( 2 \pi )^{4}}\left[ \frac{i}{l^{2}- m^{2}+ i \varepsilon}- 2 \pi \delta ( l^{2}- m^{2}) \sin^{2}\theta ( l^{0}, \mu ) \right] ,
\mathbb{P}(S_t>0\forall t\in [k]) =\mathbb{P}(I_d+\sum_{j=1}^{d-1}I_j+\sum_{i=2}^{t}(X_i-1)>0\forall t\in [k]) &=\mathbb{P}(1+I_d+\left(\sum_{j=1}^{d-1}I_j-1\right)+\sum_{i=2}^{t}(X_i-1)>0\forall t\in [k]) &=\mathbb{P}(1+I_d+\sum_{i=1}^{t}(X'_i-1)>0\forall t\in [k]),
T_{\underline{{{A}}}\ \ \underline{{{B}}}\dot{A}}^{\ \ \dot{A}\ \ \ \dot{D}}H_{\dot{D}\underline{{{C}}}}= - \frac{i}{1 2}\left( \nabla_{\underline{{{A}}}}G_{\underline{{{B}}}}^{\ \ \dot{D}}\right) \overline{{{R}}}\chi_{\underline{{{C}}}\dot{D}}
(x,F_iy) = (E_i,F_i)(r_i(x),y), (x,yF_i) =(E_i,F_i)(r_i'(x),y), (E_ix,y) = (E_i,F_i)(x,\rho_i'(y)), (xE_i,y) = (E_i,F_i)(x,\rho_i(y)).
\left( \widehat{\mathbf{\mu}}\mathbf{,}\widetilde{\mathbf{\Sigma}}\right)=\arg\min S\left( \mathbf{d}\left( \mathbf{\mu,\Sigma}\right) \right),\ \mathbf{\mu\in}R^{p},\ \mathbf{\Sigma\in}R^{p},\ |\mathbf{\Sigma}|=1,
\begin{array}{l}{{x^{-}x^{0}= q x^{0}x^{-}~ , ~ ~ x^{0}x^{+}= q x^{+}x^{0}~ ,}}\\{{x^{+}x^{-}- x^{-}x^{+}= ( q^{1 / 2}- q^{- 1 / 2}) x^{0}x^{0}~ .}}\\ \end{array}
\Big(\mathbb{E}\Big\|\sum_{k=1}^{n}r_{k}T_{k}x_{k}\Big\|_{Y}^{2}\Big)^{1/2}\leq C\Big(\mathbb{E}\Big\|\sum_{k=1}^{n}r_{k}x_{k}\Big\|_{X}^{2}\Big)^{1/2}.
\hat{T}( F ) = \frac{i}{\hbar}\frac{\delta F}{\delta \phi^{A}}\frac{\delta}{\delta \bar{\phi}_{A}}- \frac{1}{2}\varepsilon_{a b}\frac{\delta}{\delta \phi_{A a}^{*}}\frac{\delta^{2}F}{\delta \phi^{A}\delta \phi^{B}}\frac{\delta}{\delta \phi_{B b}^{*}}\; ,
\log G(z\,|\,\omega_1,\widetilde{\omega}_1,\omega_2) = \int_C \frac{-e^{(z+\overline\omega_1)s}}{(e^{\omega_1 s}-1) (e^{\widetilde\omega_1 s}-1)(e^{\omega_2 s}-1)}\cdot \frac{ds}{s},
V(\alpha) \triangleq \left\{t >0 \colon \tau_{t}(\alpha) < \tau_{t+}(\alpha) \right\}, V'(\alpha) \triangleq \left\{t > 0 \colon \alpha (\tau_t (\alpha)) \not = \alpha(\tau_t(\alpha)-) \|\alpha(\tau_t(\alpha) - )\| = t\right\}.
\begin{array}{r c c c l}{\dot{\varphi}_{1}}&{=}&{[ \varphi_{1}, H ]}&{=}&{\varphi_{2},}\\{\dot{\varphi}_{2}}&{=}&{[ \varphi_{2}, H ]}&{=}&{\partial_{i}^{\makebox [ 1 m m ]{}2}\varphi_{1}.}\\ \end{array}
\Re \sum_{n=1}^{\infty}z_n^{j}\leq \frac{1}{(\sigma_0-1)^{2j}}- \frac{1}{(\sigma_0-\omega_0)^{2j}}+\Re \left[\frac{1}{\{(\sigma_0-1)+it_0\}^{2j}}-\frac{1}{\{(\sigma_0-\omega_0)+it_0\}^{2j}}\right].
\frac{P(X)}{Q(X)}= \frac{a_l\left(\frac{G(X)}{H(X)}\right)^l+ \ldots +a_s\left(\frac{G(X)}{H(X)}\right)^s}{b_m\left(\frac{G(X)}{H(X)}\right)^m+ \ldots +b_t\left(\frac{G(X)}{H(X)}\right)^t}& = H(X)^{m-l}G(X)^{s-t}\frac{q(X)}{r(X)},
{\mathop{{\rm{max}}}\limits_{\mathcal R \in \left\{\mathcal R \in \mathbb R_+:{p_{out,K}}\le \epsilon \right\}}}{\bar{\mathcal T}= \frac{{\mathcal R\left({1 -{p_{out,K}}}\right)}}{{\sum\nolimits_{k = 0}^{K - 1}{{p_{out,k}}}}}},
\begin{array}{r c l c r c l}\Phi(uxyzv) = \Phi(u)+\Phi(v)+4, \Phi(ux^3v) = \begin{cases}\Phi(u)+\Phi(v) + 3 \Phi(u)+\Phi(v) , \end{cases}\strut \Phi(uy^3v) = \Phi(u)+\Phi(v)+3, \Phi(uz^3v) = \begin{cases}\Phi(u)+\Phi(v) + 3 \Phi(u)+\Phi(v) . \end{cases}\end{array}
\left(2\pi\right)^{3/2}e^{\frac{1}{2}t^{2}}m_{\Sigma}\left(t/\kappa\right) ={\displaystyle\int\limits_{0}^{+\infty}}e^{-tx}\left [e^{-\frac{1}{2}x^{2}}{\displaystyle\iint\limits_{xT_{N}}}e^{-\frac{1}{2}\left( y^{2}+z^{2}\right)}\, d y \, d z \right]\, dx = \mathcal{L}\left(h\right)(t).
\Psi_1(a,b,c)=\frac{\rho((a.b)c)\psi\left(\frac{|a-(a.b)b|}{4(a.b)(1-(a.b)c)}\right)}{4^{d-1}(a.b)^{d}(1-(a.b)c)^{d-1}},~~ &\Psi_2(a,b,c)=\frac{\rho((a.b)c)\psi\left(\frac{|a-(a.b)b|}{4(a.b)(1-(a.b)c)}\right)}{4^{d-1}(a.b)^{d-1}(1-(a.b)c)^{d}}c (a-(a.b)b).
a(\tilde g)-\Pi_{j}( G, \tilde G)(\tilde g) =a(\tilde g)- \Big( G(\tilde g) R_j(\tilde G)(\tilde g) - \tilde G(\tilde g) R_j^t( G)(\tilde g) \Big) &=\bigg( a(\tilde g) -{\displaystyle a(\tilde g)\over R_j(\tilde G)( g)}\, R_j(\tilde G)(\tilde g) \bigg) +{\tilde G(\tilde g) R_j^t(a)(\tilde g) \over R_j(\tilde G)(g)}&=: W_1(\tilde g)+W_2(\tilde g).
M =\lim_{n\rightarrow \infty}(S_0+S_n) &\ge\lim_{n\rightarrow \infty}M\left((\lambda_0)^{\frac{3-2s}{3}}+(\lambda_n)^{\frac{3-2s}{3}}\right) &= M\left((\lambda_0)^{\frac{3-2s}{3}}+(1-\lambda_0)^{\frac{3-2s}{3}}\right) &> M(\lambda_0+1-\lambda_0)=M,
[ \hat{x}_{i}, \hat{x}_{j}] = i \hbar \frac{( 2 \beta - \beta^{\prime}) + ( 2 \beta + \beta^{\prime}) \beta \hat{p}^{2}}{( 1 + \beta \hat{p}^{2})}\left( \hat{p}_{i}\hat{x}_{j}- \hat{p}_{j}\hat{x}_{i}\right) .
P_0U(\vec{a},1)B_s^n 1= P_0\psi^{\beta_1,n}(g_1)_{U(\vec{a})}\psi^{\beta_2,n}(g_2)_{U(\vec{a})}\cdots \psi^{\beta_{s-1},n}(g_{s-1})_{U(\vec{a})}U(\vec{a},1)\psi^{\beta_s,n}(g_s)1 = e^{i[a^0 \hat{H}- a^1\hat{P}]}P_0B_s^{n,\vec{a}}1= e^{i[a^0 \hat{H}- a^1\hat{P}]}\left[B_s^{n,\vec{a}}1 - \left\langle B_s^n 1, 1 \right\rangle 1 \right],
F_{3}^{(1)}=\sum_{s=0}^{1}\sum_{s'=0}^{1}\sum_{1\leq j<k\leq m}e_{a_{1}}D_{\alpha,\beta}(U(b_{1}+\cdots+b_{j}+s-1),\iota^{c_{j}}(a_{s+1}),U(b_{j+1}+\cdots+b_{k}+s'-1))g_{k,s+s'},
w_2(q_{\infty})+ w_1(q_{\infty})\cdot w_1(q_{\infty})+ w_1(q_{\infty})\cdot w_1(q_{dR,\mathbb{R}})+w_2(q_{dR,\mathbb{R}})=sw_2(\rho_{\infty\mid G_{\mathbb{R}}})+ sp_2(\rho_{\infty\mid G_{\mathbb{R}}})
\delta(p+q)=(2\pi)^{\,4}\,\tilde C_{\alpha}^{\,\alpha_{0}}(q) delta_{\Phi_{c}(p)}\delta_{\Phi_{c}(q)}\,\Gamma_{0}^{\,\alpha,\,\alpha_{0}}(\Phi_{c})\qquad\mbox{for}\r=0\quad ,
F_{_{( 1 )}}= \left( \begin{array}{c c c}{{f}}&{{0}}&{{M_{_{( 1 )}}}}\\{{0}}&{{0}}&{{R_{_{( 1 )}}}}\\{{- M_{_{( 1 )}}^{^T}}}&{{- R_{_{( 1 )}}^{^T}}}&{{0}}\\ \end{array}\right) \; .
R=\frac{1}{2}\rho\frac{fLS}{A^{3}}\equiv\frac{1}{2}\rho\frac{fL}{R_{h}A^{2}}\equiv\frac{1}{2}\rho\frac{4fL}{D_{h}A^{2}}\equiv\frac{1}{2}\rho\frac{\lambda L}{D_{h}A^{2}}
T_{k,h}(s) = \begin{cases}0, \ \mbox{if}0 \le |s| < h, (|s| - h) \, \mathrm{sign}(s) , \ \mbox{if}h \le |s| \le k + h, k \, \mathrm{sign}(s), \ \mbox{if}|s| > k + h. \end{cases}
M_{W^{+}}^{2}={\frac{v^{2}}{4}}g^{2}~ , ~ M_{W^{-}}^{2}={\frac{v^{2}}{4}}g^{2}~ , ~ M_{Z}^{2}={\frac{v^{2}}{4}}~ ( g^{\prime 2}+ g^{2}) ~ , ~ M_{A}^{2}= 0 \ .
\sqrt{\left( 4+4\frac{a}{b}\right)u^{2}+\left( 4+4\frac{a}{b}\right)u+1}= \left( \sqrt{\left( 4+4\frac{a}{b}\right)x^{2}+\left( 4+4\frac{a}{b}\right)x+1}-\left(2x+1\right)\right) -2u-1.
\frac{r( q_{k+1}+q_k -2,{\mathbf x})}{q_{k+1}+q_k -2}\le \frac{|W_{k+1}|+q_{k+1}+2q_k-2}{q_{k+1}+q_k-2}= 1+ \frac{|W_{k+1}|+q_k}{q_{k+1}+q_k-2}\le 1+\frac{|W_k|+q_k}{2q_k+q_{k-1}-2}= 1 + \frac{t_k + 1}{2 + \eta_k - \varepsilon_k}< 1 + \frac{t_k + 1}{2 + \eta_k}+ \varepsilon_k.
<BR>{\cal=20L}_{MCSP}=3D-{1\over 4}A_{\mu\nu}A^{\mu\nu}<BR>+{{\mu}\over=204}\epsilon_{\mu\nu\lambda}A^{\mu\nu}A^\lambda<BR>+{{m^2}\over 2}A_\mu =A^\mu=20,~~A_{\mu \nu}=3D\partial_\mu A_\nu -\partial_\nu A_\mu=20.<BR><BR>
p_{S|Y}(s|y)=\begin{cases}\epsilon\left(1-\epsilon\right)^{s}s\in[0,k-1],y=1 \left(1-\epsilon\right)^{s}s=k,y=1 \frac{\epsilon\left(1-\epsilon\right)^{s}}{1-\left(1-\epsilon\right)^{k}}s\in[0,k-1],y\neq1, 0 ,\end{cases}
\begin{aligned}{}|\nabla\psi_{\tau}(x',y)-\nabla\psi_{\tau}(\bar{x'},\bar{y})| &\leq (1+\tau+\tau^{2}+\cdots+\tau^{k-1})\omega_{\nabla\psi}(\tau|x'-\bar{x'}|+\cdots+\tau^{k}|y_{k}-\bar{y_{k}}|)\rightarrow0,\end{aligned}
\sum_j f_{\le j-2}D^s g_j = f D^s g - \sum_j f_{>j-2}D^s g_j & = f D^s g - \sum_j f_j D^s g_{<j+2}& = f D^s g - \sum_j f_j D^s g_{j-2\le \cdot <j+2}- \sum_j f_j D^s g_{<j-2}.
|\frac{-c_{b}r}{2\pi}\int_{0}^{\pi}\sin^{2}(\theta) d\theta (b-1)(t,r,\theta)| \leq C r \int_{0}^{\pi}\frac{d\theta}{t^{2}\log^{b+1}(t)}&\leq \frac{C r}{t^{2}\log^{b+1}(t)}, r \leq \frac{t}{2}