harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}\frac{2 - h(1+\kappa)}{1-\kappa h}\big(f_{0}-f_{1}\big){}\geq{}\frac{1}{2L(1-\kappa)}\frac{h(1-\kappa)[\kappa h^2 - 2h(1+\kappa) + 3]}{1-\kappa h}\big\|g_{0}\big\|^2 + \frac{h}{2L\big(1-\kappa \big)}\frac{1-\kappa}{1-\kappa h}\|g_1\|^2 .\end{aligned}
\Phi_h\bigg( \bigg( \sum_{i=0}^\ell r_i x_i \bigg)^h \bigg) = \sum_{j_1=0}^\ell \cdots \sum_{j_h=0}^\ell r_{j_1}\cdots r_{j_h}z_{j_1j_2}\cdots z_{j_{h-1}j_h}= \bigg( \sum_{j_1=0}^\ell \sum_{j_2=0}^\ell r_{j_1}r_{j_2}z_{j_1j_2}\bigg)^{h/2},
\Big| \frac{1}{2m \ell n}\sum_{x}\Phi_{n}(s,\tfrac{x}{n}) \sum_{j=1}^{m-1}\int_{\Omega_2^j(x)}\sum_{z = x-\ell}^{x-1}[ \eta(z) - \eta(z - j \ell) ] \overrightarrow{\eta}^{\varepsilon n}(x+1) [f ( \eta_{2,x,j,z}) - f( \eta_{3,x,j,z}) ] d\nu_{b}\Big|,
\delta^{5}{\mathbf q}= 2 i ({\mathbf q}+ ( m , 0 , 0 ) ) \tilde{\omega}( x ) \ , \qquad \delta^{5}{\mathbf{\bar{q}}}= - 2 i ({\mathbf{\bar{q}}}+ ( m , 0 , 0 ) ) \tilde{\omega}( x )
\bigg|\prod_{j=1}^n\sin(\theta_j)^{j-n}\cos(\theta_j)^{1-j}\bigg|=\bigg(\prod_{j=1}^na_j\bigg)^{-\frac{n+1}2}\bigg(\prod_{j=1}^na_j^j\bigg)\bigg(\prod_{j=1}^n\frac1{a_j+a_j^{-1}}\bigg)^{\frac{1-n}2}.
\eta_{j}:=\int_{P_{\rho}}(u_{j}(x)-\widetilde{u}_{j}(x))^{2}dx+\int_{P_{\rho}}\int_{P_{\rho}\setminus B_{\varepsilon_{j}}(y)}J_{\varepsilon_{j}}(x-y)(u_{j}(x)-\widetilde{u}_{j}(x))^{2}dxdy~.
(\begin{matrix}n\\k\end{matrix})=\frac{n^{\underline{k}}}{k!}=\frac{n(n-1)(n-2)\cdot\cdot\cdot(n-(k-1))}{k(k-1)(k-2)\cdot\cdot\cdot1}=\prod_{i=1}^{k}\frac{n+1-i}{i},
T_{K}^{+}= K_{q}^{+}\frac{( K_{q}^{3}- k + 1 )}{[ K_{q}^{3}- k + 1 ]}\hspace{1 . 5 c m}\mathrm{a n d}\hspace{1 . 5 c m}T_{K}^{-}= \frac{( K_{q}^{3}- k + 1 )}{[ K_{q}^{3}- k + 1 ]}K_{q}^{-}\ ,
S ={\frac{1}{2 \kappa_{2}^{2}}}\int d^{2}x \sqrt{- \tilde{g}}\left[ \Phi{\cal R}_{\tilde{g}}+{\frac{1}{2}}\partial_{\mu}\Phi \partial^{\mu}\Phi + \Lambda \Phi^{1 -{\frac{\gamma}{\delta^{2}}}}e^{\frac{\Phi}{2}}\right] .
\frac{\partial}{\partial t}\Phi (t*u)|_{t=1}=\Psi'_{u}(1)+\mu\Psi''_{u}(1)+\mu \Psi^{'}_{u}(1) &=(1+\mu)\Psi'_{u}(1)+\mu \Psi''_{u}(1) &=(1+\mu)P(u)+\mu \Psi''_{u}(1) &=\mu \Psi''_{u}(1).
\begin{multlined}[t][12.5cm]\widetilde{\Pi}^{P}_{\nu_{G},6c+\nu_{1}+3\nu_{2}}\left((z_{0},z_{1},z_{2}),(z_{0},z_{1},z_{2})\right)\sim \frac{9\sqrt{3}c}{2\pi^3}x^{\nu_{1}+\nu_{2}}y^{\nu_{2}}e^{cF(x,y)},\end{multlined}
\begin{aligned}\int_{Q_T}\left(s\Theta (z_x)^2 + s^3 \Theta^3 z^2\right)e^{2s\Phi}dxdt\le C\left(\int_{Q_T}h^{2}e^{2s\Phi}dxdt + \int_0^T\int_\sigma^\gamma z^2e^{2s\Phi}dxdt\right).\end{aligned}
\limsup_{n \rightarrow \infty}|(Q_{n,\textup{\textbf{m}}})^{(k)}(\lambda_j)|^{1/n}\leq \frac{|\Phi(\lambda_j)|}{\rho_{|\textup{\textbf{m}}|}(\textup{\textbf{F}})}, k=0,1,\ldots,\tau_j-1
A_{w,1}f(v_{\ell,0,0})=(q^3+q^2+q)f(v_{\ell,1,0})+f(v_{\ell+1,0,0}) &A_{w,2}f(v_{\ell,0,0})=(q^4+q^3+q^2)f(v_{\ell,1,1})+(q^2+q+1)f(v_{\ell+1,1,0}) &A_{w,3}f(v_{\ell,0,0})=q^3f(v_{\ell-1,0,0})+(q^2+q+1)f(v_{\ell+1,1,1}).
\varepsilon^i =\left(\mathcal B({\mathbf V}_i,\psi_i)_1-\mathcal B({\mathbf V}_{i-1},\psi_{i-1})_1\right)+\mathcal B({\mathbf V}_{i-1},\psi_{i-1})_1 &=\partial_t\delta\psi_{i-1}-\delta u_{1,i-1}+(u_2^a+u_{2,i-1})\partial_2\delta\psi_{i-1}+\delta u_{2,i-1}\partial_2(\varphi^a+\delta\psi_{i-1}+\psi_{i-1}) &\quad+\mathcal B({\mathbf V}_{i-1},\psi_{i-1})_1\,.
H = \int \! \frac{d^{3}p}{( 2 \pi )^{3}}E_{p}\left( a_{\bf p}^{\dag}a_{\bf p}+ a_{\bf p}^{\dag}{\bar{a}}_{\bf p}+{\bar{a}}_{\bf p}^{\dag}a_{\bf p}-{\bar{a}}_{\bf p}^{\dag}{\bar{a}}_{\bf p}\right) ,
[ j_{+}{}^{n}, ( j_{+}{}^{m})^{\dagger}] = \lim_{s \to 0 \atop r \to 0}\sum_{p}\left( \frac{1}{\lambda_{p + m , s}\lambda_{p + n , r}}- \frac{1}{\lambda_{p , s + r}}\right) a_{p + m}^{\dagger}a_{p + n},
V(y) =\frac{1}{\pi}\Re Ae^{i R(\xi_0)}\sqrt{\frac{2\pi}{R''(\xi_0)}}e^{ia\ln \xi_0}\ +\L^2\mbox{-remainder}&= \frac{|A|}{\sqrt{\pi}|3y|^{1/4}}\cos\left(- \frac{2}{3\sqrt 3}|y|^{3/2}+ \frac{a}{2}\ln|y|+\theta_0\right) \ +\L^2,
\displaystyle \sum_{p \mid n}\frac{1}{p}+\frac{1}{n}= \sum_{p \mid n}\frac{1}{p}+\frac{1}{n\pm1}+\left(\frac{1}{n}-\frac{1}{n\pm1}\right)= \sum_{p \mid n(n\pm1)}\frac{1}{p}\pm \frac{1}{n(n\pm1)},
T_a\times L \rightarrow L, (t+i\theta)\circ[(z_1,\ldots,z_{n+2},\lambda)] \rightarrow[(e^{\frac{t+i\theta}{a_1}}z_1,\ldots,e^{\frac{t+i\theta}{a_{n+2}}}z_{n+2},e^{t-i\frac{\theta}{\alpha}}\lambda)],
\begin{array}{l c l}{{\{p^{n}, f \}_{\theta}}}&{{=}}&{{\sum_{s = 0}^{n}\theta^{s - 1}c_{n}^{s}\{\frac{1 - ( - )^{s}}{2}\}f^{s}p^{n - s},}}\\{{\{p^{- n}, f \}_{\theta}}}&{{=}}&{{\sum_{s = 0}^{\infty}\theta^{s - 1}c_{s + n - 1}^{s}\{\frac{( - )^{s}- 1}{2}\}f^{s}p^{- n - s}.}}\\ \end{array}
f_1 = \Delta^{p^{s-1}}+ \tfrac{a_1}{a_0}X^{p^{s-1}}Y^{p^{s-1}}+\tfrac{a_2}{a_0}\Delta^{p^{s-2}}X^{2p^{s-1}-2p^{s-2}}\ldots + \tfrac{a_r}{a_0}YX^{2p^{s-1}-1}
I_4(\lambda) = \frac{1}{(3+22\lambda)(4+22\lambda)}\int\limits_{0<x_2<x_3<1}x_2^{2\lambda}x_3^{4\lambda}(1-x_2)^{4\lambda}(1-x_3)^{2\lambda}(x_3-x_2)^{2\lambda}dx, = \frac{1}{(3+22\lambda)(4+22\lambda)}\int\limits_{[0,1]^2}y^{2\lambda}(1-y)^{2\lambda}z^{8\lambda+1}(1-z)^{2\lambda}(1-yz)^{4\lambda}dydz.
\phi_{ij}(x^{-},0) = \frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}\frac{dk^{+}}{\sqrt{2k^{+}}}\left( a_{ij}(k^{+}){\rm e}^{-ik^{+}x^{-}}+a_{ji}^{\dag}(k^{+}){\rm e}^{ik^{+}x^{-}}\right) ,
\sum\limits_{j=1}^n1_{\left\{X_i\leq X_j\right\}}=\sum\limits_{j=1}^{k^*}1_{\left\{\mu+G(\xi_i)\leq \mu+G(\xi_j)\right\}}+\sum\limits_{j=k^*+1}^n1_{\left\{\mu+G(\xi_i)\leq \mu+G(\xi_j)+\Delta\right\}}&=\sum\limits_{j=1}^n1_{\left\{G(\xi_i)\leq G(\xi_j)\right\}}+\sum\limits_{j=k^*+1}^n1_{\left\{G(\xi_j)<G(\xi_i)\leq G(\xi_j)+\Delta\right\}}.
\Delta(\tau) = \tau \otimes 1 + 1 \otimes \tau, S(\tau) = -\tau, \Delta(\theta) = \theta \otimes 1 + 1 \otimes \theta, S(\theta) = -\theta, \Delta(\sigma_i) = \textstyle \sum_{u+v=i}\sigma_u \otimes \sigma_v + \sum_{u+v+p=i}\sigma_u \tau \otimes \sigma_v \tau, S(\sigma_i) = (-1)^i \sigma_i.
0 \ne \ell (\pi_{v_0}(\varphi) \cdot w_1) = \int_{H(k_{v_0})}\varphi(h) \cdot \ell(h \cdot w_1) \, dh = \int_{N(k_{v_0}) \backslash H(k_{v_0})}\ell (h \cdot w_1) \cdot\left( \int_{N(k_{v_0})}\chi(u)\cdot \varphi(uh) \, du \right) \, dh
G_{T}^{+}( x , y ) = \sum_{m = - \infty}^{\infty}\int_{0}^{\infty}d q \int_{- \infty}^{\infty}d k N_{2}^{2}e^{- i [ \frac{m}{R_{0}}\sinh \Omega R_{0}+ \omega \cosh \Omega R_{0}] \Delta t}J_{m}^{2}( q R_{0}) .
\frac{\partial h_{ij}}{\partial z_{k}}(q)\int_{\mathbb{R}_{+}^{n}}tz_{k}\frac{\partial U}{\partial z_{i}}\frac{\partial U}{\partial z_{j}}dzdt=(2-n)\frac{\partial h_{ij}}{\partial z_{k}}(q)\int_{\mathbb{R}_{+}^{n}}\frac{tz_{k}z_{i}z_{j}dzdt}{\left[(1+t)^{2}+|z|^{2}\right]^{\frac{n}{2}}}=0;
\Lambda_{n_{12}}^{\mu_{12}}F_j^{\mu_1,\mu_2}(\phi(z))(\csc\phi(z))^{n_{12}+j}= \displaystyle\sum\limits_{n=0}^{n_{12}+j}C^{n_1, n_2}_{n_{12}, j}N_{\mu_1}(n)N_{\mu_2}(n_{12}+j-n)z^{n},
d\theta =2e^{1}\wedge e^{2}, \nabla_{b}e_{1}=\sigma_{1}^{2}\otimes e_{2},\nabla_{b}e_{2}=\sigma_{2}^{1}\otimes e_{1}, de^{1}=e^{2}\wedge \sigma_{2}^{1}\theta ;de^{2}=e^{1}\wedge \sigma_{1}^{2}\theta .
\begin{array}{l}{\vec{a}^{\prime}( u ) = c o s ( m u ) \hat{x}- s i n ( m u ) \hat{z},}\\{\vec{b}^{\prime}( v ) = c o s ( n v ) ( c o s \Psi \hat{x}+ s i n \Psi \hat{y}) - s i n ( n v ) \hat{z}.}\\ \end{array}
- D_{4}({\frac{\partial S}{\partial t}})^{2}+ ({\frac{\partial S}{\partial r}})^{2}+{\frac{1}{r^{2}}}({\frac{\partial S}{\partial \theta}})^{2}+ D_{4}({\frac{\partial S}{\partial x^{1 1}}})^{2}= 0 ~
\psi_{_{( \alpha )}}^{2}( x ) = ( \frac{\mu}{2 \pi})^{^{1 / 2}}{\cal K}( \phi_{1}){\cal K}( \phi_{3}) : e^{- i \sqrt{\pi}\{\int_{- \infty}^{x}d z^{1}\pi_{_2}( z ) + \gamma_{_{\alpha \alpha}}^{5}\phi_{_2}( x ) \}}: ,
\lim_{n\rightarrow\infty}\frac{\left\vert \left\{x\in V_{n}\mid x\phi(a)\phi(b)=x\phi(ab)\right\}\right\vert}{\left\vert V_{n}\right\vert}=\lim_{n\rightarrow\infty}\mathcal{P}(w\in K_{n})=1
\begin{aligned}\bar H^6-3\bar H^4S+3\bar H^2S^2-3S^2+2S &\geq \dfrac{(\bar H^2-S)^2}4\bigl[(6+\dfrac34)S-(4+\dfrac3{8})\bar H^2-\dfrac{3S(3S-2)}{8\bar H^2}\bigl ].\end{aligned}
M_{0j}=\frac{1}{2}\big( d\Gamma(X_jP_0)+d\Gamma(P_0X_j)\big) =\frac{1}{2}d\Gamma([X_j,P_0])+d\Gamma(P_0X_j) =\frac{i}{2}d\Gamma(V_j)+d\Gamma(P_0X_j) ,
u_n (t,x) = \frac{\lambda_n}{\tau_n}\, u \left( \frac{t}{\tau_n}, \frac{x}{\lambda_n}\right) \,,\rho_n(t,x) = \gamma_n \, \rho \left( \frac{t}{\tau_n}, \frac{x}{\lambda_n}\right) \,.
\left< \mathrm{T r}U_{\mathrm{a d j}}( u ) \mathrm{T r}U_{\mathrm{a d j}}( v ) \right> = 1 + \left< \mathrm{T r}U^{2}( u ) \mathrm{T r}U^{2}( v ) \right> + \left< \mathrm{T r}U^{2}( u ) \right> + \left< \mathrm{T r}U^{2}( v ) \right> \ .
\frac{3}{8 \pi^{2}}\omega_{i}^{( 4 )}+ \frac{1}{8 \pi^{2}}\sum \Omega_{i j k}^{( 2 )}~ \omega_{i j k}^{( 2 )}+ \frac{1}{4 \pi^{2}}\sum \Omega_{i j k l m}^{( 4 )}= \frac{1}{2},
I(X_1,X_2;Y|U) = I(X_1,X_2;Y) + I(X_1,X_2;U|Y) - I(X_1,X_2;U) = I(X_1,X_2;Y){+}I(X_1,X_2;U|Y){-}I(X_1;U){-}I(X_2;U|X_1) = I(X_1,X_2;Y) + I(X_1,X_2;U|Y) - I(X_1;U) \geq I(X_1 ,X_2;Y) - I(X_1;U),
M^{(0)}= (1), M^{(1)}= \left(\begin{array}{ccc}1 0 0 0 1 0 0 0 q^2\end{array}\right), M^{(2)}= \left(\begin{array}{ccc}1 0 0 0 q^4 0 0 0 q^2\end{array}\right), M^{(3)}= (1).
\begin{aligned}\mathcal{M}^{\oplus_0}_{|\cdot|}= \sqrt{2}\Big(C_F \, \big(\|{\mathcal{R}_1}^c_0 \|_{\Omega}+ \|{\mathcal{R}_3}^c_0 \|_{\Omega}\big) + \|{\mathcal{R}_2}^c_0 \|_{\Omega}+ \|{\mathcal{R}_4}^c_0 \|_{\Omega}\Big) \end{aligned}
\begin{aligned}\#(\{c_{i,j}\}) \leq 2^{\frac{n(n-1)}{2}}\times (\ell+(n-1)m+1)^{\frac{(n-2)(n-1)}{2}}& \times p^{2a_1+a_2+\cdots+a_{k-1}+a_{k+1}+\cdots+a_{n-2}+(n-3)a_{n-1}+(n-3)(n-1)m+\frac{(n-2)(n-1)}{2}m},\end{aligned}
\Theta_{i,h}(x):=\begin{cases}\emptyset, H_i=(h=)0, \neg\theta_{i,1}(x), h=0<H_i, \theta_{i,h}(x)\wedge(\neg\theta_{i,h+1}(x)), 0<h<H_i, \theta_{i,H_i}(x), h=H_i>0.\end{cases}
\rho(k,\omega)=2\pi\sum_{\alpha}[\delta(E_{\alpha}-E_{0}-\omega)|\langle\alpha|\psi_{k}^{\dagger}|0\rangle|^{2}-\zeta\delta(E_{0}-E_{\alpha}-\omega)|\langle0|\psi_{k}^{\dagger}|\alpha\rangle|^{2}]
\epsilon_{i,1}= \frac{{{D_i}+ \sqrt{{{\left({{D_i}}\right)}^2}-{C_i}{D_i}}}}{{{C_i}}}= \frac{{\sqrt{{D_i}}\left({\sqrt{{D_i}}+ \sqrt{{D_i}-{C_i}}}\right)}}{{{D_i}- \left({{D_i}-{C_i}}\right)}}= \frac{{\sqrt{{D_i}}}}{{\sqrt{{D_i}}- \sqrt{{D_i}-{C_i}}}},
\begin{aligned}\left \vert \frac{\tau}{\eta\cdot \omega_j}\right \vert =\frac{\sqrt{1+s^2}}{\vert \mathbf z \cdot \omega_j-s\omega_1\cdot \omega_j+\mathrm i\sqrt{1+s^2}\mathbf d^\perp \cdot \omega_j\vert}=\frac{\sqrt{1+s^2}}{\sqrt{\vert \mathbf z\cdot \omega_j-s\omega_1\omega_j\vert^2+(1+s^2)}}\geq\frac{1}{2},j=2,3. \end{aligned}
\frac{0}{1},\frac{1}{1},\frac{0}{2},\frac{1}{2},\frac{2}{2},\frac{3}{2},\frac{4}{2},\frac{0}{3},\frac{1}{3},\frac{2}{3},\frac{3}{3},\frac{4}{3},\frac{5}{3},\frac{6}{3},\frac{7}{3},\frac{8}{3},\frac{9}{3},\frac{0}{4}\cdot\cdot\cdot
S [ q ] = \lim_{T \rightarrow \infty}\left\{\int_{- T}^{T}d t \; \left[ \frac 1 2 m \dot{q}^{2}- \frac 1 4 m \omega_{0}^{2}q^{2}\right] - \frac 1 8 m \omega_{0}^{2}\ell_{0}^{2}T \sin \left[{\frac{2}{T}}\int_{- T}^{T}d t{\frac{q^{2}}{\ell_{0}^{2}}}\right] \right\}\; .
R_{N}( \beta ) ={\frac{\beta^{N + 1}}{( 1 - \beta )^{\sigma}}}\oint_{\Gamma}{\frac{d \beta^{\prime}}{2 \pi i}}{\frac{\Psi ( \beta^{\prime}, \rho )}{( \beta^{\prime})^{N + 1}( \beta^{\prime}- \beta )}}.
0 \ge{}\Delta Q_A ={}\Delta u_{ee}- A \Delta u ={}- f'(u) u_{ee}- f''(u) u_e^2 + A f(u) \ge{}- A f'(u) u - \sigma Af'(u) x_1 + Af(u) >{}A(- f'(u) u + f(u)) - A f(0) \ge0,
u_1 = \ \frac{1}{l_{01}+1}( d_{101}-l_{01},d_{201}), u_2 = \u_1 \ + \ \left(\frac{l_{01}}{l_{01}+1},0\right), u_3 = \ \frac{1}{2}( -1,0 ), u_4 = \ \frac{1}{2l_{01}+1}( l_{01}^2+d_{101}+l_{01}, d_{201}) u_5 = \ \frac{1}{l_{01}+2}( -d_{101},-d_{201}), u_6 = \u_5 \ + \ \left(\frac{l_{01}+1}{l_{01}+2},0\right).
\int |\chi_R(x-z) \nabla u(t,x)|^2 dx -\frac{3\alpha}{2(\alpha+2)}\int |\chi_R(x-z) u(t,x)|^{\alpha+2}dx &= \|\nabla [\chi_R(\cdot-z) u(t)]\|^2_{L^2}- \frac{3\alpha}{2(\alpha+2)}\|\chi_R(\cdot-z) u(t)\|^{\alpha+2}_{L^\alpha+2}+ O(R^{-2}) &\geq \nu \|\nabla [\chi_R(\cdot-z) u(t)]\|^2_{L^2}+ O(R^{-2})
{\cal L}_{+ +}^{+ +}( \Lambda ) ={\frac{1}{2}}\Lambda_{+}^{i}[ \delta^{i j}D^{+ +}+ ( V^{+ +})^{i j}] \Lambda_{+}^{j}+ \Lambda_{+}^{+ Y^{\prime}}[{\frac{1}{2}}( V )_{Y^{\prime}Z^{\prime}}\Lambda_{+}^{+ Z^{\prime}}+ ( V^{+ +})_{Y^{\prime}}^{- i}\Lambda_{+}^{i}+ m \Phi_{+ Y^{\prime}}^{+}] \; ,
\begin{array}{rcl}p' = P(p) begin{align*}3mm] = p^8+8p^7(1-p)+28p^6(1-p)^2+56p^5(1-p)^3+\frac{525}{8}p^4(1-p)^4+\frac{35}{2}p^3(1-p)^5 begin{align*}3mm] = \frac{105}{8}p^8-55p^7+\frac{315}{4}p^6-\frac{63}{2}p^5-\frac{175}{8}p^4+\frac{35}{2}p^3\,\,\,\,\,.\,\,\,\,\,\end{array}
d^6([(x_6)_my_1])=4(x_6)_{m-1}[y_1^2\gamma_1^3\gamma_2]=0 & d^6([(x_6)_my_2])=4(x_6)_{m-1}[y_1y_2\gamma_1^4]=0 & d^6([(x_6)_m\gamma_1])=4(x_6)_{m-1}[y_1\gamma_1^4]=0 \;\; d^6([(x_6)_m\gamma_2])=4(x_6)_{m-1}[y_1\gamma_1^3\gamma_2]=0.
tan\theta=\frac{u_{y}}{u_{x}}=\frac{\sqrt{1-\frac{v^{2}}{c^{2}}}u_{y}^{\prime}}{u_{x}^{\prime}+v}=\frac{\sqrt{1-\frac{v^{2}}{c^{2}}}u^{\prime}sin\theta^{\prime}}{u^{\prime}cos\theta^{\prime}+v}.
\boldsymbol{\psi}_\alpha(z)\boldsymbol{\psi}_\beta(w)(z-w)^{-\frac{\alpha\beta}{k}}&\qquad=(z-w)^{n(\alpha+\beta)-2}[\delta_{\alpha+\beta,0}I_{\psi_{\alpha}(z)\psi_{\beta}(w)}+\varepsilon_{\alpha,\beta}\boldsymbol{\psi}_{\alpha+\beta}+\mathcal{O}(z-w)].
A_{ai}^kC_{kj}^m=A_{aj}^kC_{ki}^m+C_{ij}^kA_{ak}^m,\,A_{ia}^kC_{kj}^m=C_{ij}^kA_{ka}^m+A_{aj}^kC_{ik}^m,\,C_{ij}^kA_{ka}^m=A_{ia}^kC_{kj}^m+A_{ja}^kC_{ik}^m,
e_{\hat{0}}^{\;\;\mu}e_{\alpha}^{\;\; \nu}\left. \frac{\partial \Gamma_{R}}{\partial g^{\mu \nu}}\right|_{b_{\mu\nu}}= -{\bar{e}}_{\alpha}^{\;\; i}{e}_{\hat{0}}^{\;\;0}\frac{\partial \Gamma_{R}}{\partial v^{i}},
lim_{t\rightarrow\infty,F_{e}\rightarrow0}\frac{1}{t}ln(\frac{p(\beta\overline{J}_{t}=A)}{p(\beta\overline{J}_{t}=-A)})=-lim_{t\rightarrow\infty,F_{e}\rightarrow0}AVF_{e},F_{e}^{2}t=c.
\left( \partial_{\mu}1 \otimes 1 + \frac{1}{4}\epsilon_{\alpha \beta \delta}\omega_{\mu}^{\alpha \beta}1 \otimes \gamma^{\delta}+ \frac{i}{2 \ell}A_{\mu}^{I}( - \frac{1}{\ell}N_{I}^{c}\mp K_{I}^{c}) \sigma_{c}\otimes 1 \pm \frac{1}{2 \ell}e_{\mu \alpha}1 \otimes \gamma^{\alpha}\right) \varepsilon_{r}= 0 \ ,
\dim_H ( \mu ) =\begin{cases}1 + \frac{h_\nu + \int \log \lambda \, d \nu}{\int \log \tau ' \, d \nu}\mbox{if}\dim_H ( \mu ) \geqslant 1 \frac{h_\nu}{- \int \log \lambda \, d \nu}\mbox{if}\dim_H ( \mu ) < 1 \end{cases}.
\vert l_{2j,2N+1}(z)\vert^2+\vert l_{2j+1,2N+1}(z)\vert^2 \le \frac{\vert z^2-e_N\vert^2+2\vert z^2-e_j\vert^2+2\vert e_j-e_N\vert^2}{2\vert e_j-e_N\vert^2}\vert l_{j,N}(z^2)\vert^2 &=\frac{1}{2}\vert l_{j,N+1}(z^2)\vert^2+\vert l_{N,N+1}(z^2)\vert^2\vert l_{j,N}(e_N)\vert^2+\vert l_{j,N}(z^2)\vert^2
Z=\left(\begin{array}{cc}A 0 0 A\end{array}\right),T=\left(\begin{array}{cc}B 0 0 B\end{array}\right),X=\left(\begin{array}{cc}0 C C 0 \end{array}\right),Y=\left(\begin{array}{cc}0 iC -iC 0 \end{array}\right),
\chi ( \tau , \bar{\tau}) = \left[ \sum_{s}\Lambda_{s}( \tau , \bar{\tau}) \right] \theta_{3}^{'}\left( 0 \left| \frac{i \beta^{2}}{2 \pi^{2}\tau_{2}}\right. \right) - \tau_{2}^{- \frac{d - 2}{2}}\frac{2 \bar{\theta}_{2}^{4}}{\bar{\eta}^{1 2}}( z_{s}+ z_{c}) \theta_{2}\left( 0 \left| \frac{2 i \beta^{2}}{\pi^{2}\tau_{2}}\right. \right)
\gamma_{2i}(G_k) = \langle [y,x,\overset{2i-1}{\ldots},x] \rangle \; \gamma_{2i+1}(G_k), \gamma_{2i+1}(G_k) = \langle [y,x,\overset{2i}{\ldots},x], [y,x,\overset{2i-1}{\ldots},x,y] \rangle \; \gamma_{2i+2}(G_k)
T_{\alpha\beta}=-\left(\nabla_\alpha\phi\nabla_\beta\phi-\frac 12 ((\nabla \phi)^2+m^2+\xi\tilde{R}~\phi^2)\gamma_{\alpha\beta}+\xi (\gamma_{\alpha\beta}\nabla^2-\nabla_\alpha\nabla_\beta )\phi^2\right)~,
\int_{0}^{t-\delta}\left[(t-\eta_n(s))^{\alpha}-(t-s)^{\alpha}\right]^2\, ds \leq \int_{0}^{t-\delta}(t-s)^{2\alpha-2}(s-\eta_n(s))^2ds \leq \frac{1}{n^2}\int_{0}^{t-\delta}(t-s)^{2\alpha-2}ds = \frac 1{n^2}\int_\delta^t s^{2\alpha -2}ds \leq \frac{2\delta^{2\alpha-1}}{n^2(1-2\alpha)}= C_2{n^{-2}}\delta^{2\alpha-1}.
w(X) \ge w(f) &= b(n-6)w(V(G))-(3b(n-6)-1)\sum_{i=1}^4w(v_i) &> b(n-6)w(V(G))-\frac{12(3b(n-6)-1)\varepsilon}{n}w(V(G)) &= \left[b(n-6)-\frac{36\varepsilon^2}{n(n-6)}\right]w(V(G)) &= \left[b(n)+\frac{6\varepsilon}{n(n-6)}-\frac{36\varepsilon^2}{n(n-6)}\right]w(V(G)) &\ge b(n)w(V(G))=w(c_{b(n)}).
D^{( 3 )}\equiv \left| \begin{array}{c}{{{\; 1 \; c_{1 2}c_{1 3}\;}}}\\{{{c_{1 2}\; 1 \; c_{2 3}\;}}}\\{{{c_{1 3}c_{2 3}\; 1 \; \;}}}\\ \end{array}\right| = 1 - c_{1 2}^{2}- c_{1 3}^{2}- c_{2 3}^{2}+ 2 c_{1 2}c_{1 3}c_{2 3}.
\frac{d\pi_{n}^{\mu}}{d\pi_{n}^{\nu}}(x) =\frac{E^{\nu}[\frac{d\mu}{d\nu}(X_{0})|Y_{[0,n]},X_{n}=x]}{E^{\nu}[\frac{d\mu}{d\nu}(X_{0})|Y_{[0,n]}]}~~~~~P^{\mu}~a.s.
\rho\left(H^{n_k}R^{m_k}\cdots H^{n_1}R^{m_1}\right)^{\frac{1}{\sum_{i=1}^k (n_i+m_i)}}\geq \min\left\{\rho(R),\rho\left(HR^n\right)^{\frac{1}{n+1}}\right\}.
F_{1}^{\prime\prime\prime}+F_{1}F_{1}^{\prime\prime}-F_{1}^{\prime2}+1=0,\frac{p_{o1}-p_{1}}{\rho_{1}}=\frac{1}{2}k_{1}^{2}x^{2}+k_{1}\nu_{1}F_{1}^{\prime}+\frac{1}{2}k_{1}\nu_{1}F_{1}^{2}
G_2(z;q) := \sum_{n,m \geq 0}p_2(m,n)z^m q^n =1 + \sum_{n \geq 1}\frac{z^n q^n \left(q^6; q^6\right)_{n-1}}{\left(1-q^n\right)\left(q^2; q^2\right)_{n-1}\left(q^3; q^3\right)_{n-1}},
\Gamma_a :=\frac{1}{2k}\sum_{i=1}^k \| g(x_i) + \delta \|_2^2, \Gamma_b:=\frac{1}{2k}\sum_{i=1}^k\| \delta_i - \delta \|_2^2 \Gamma_c := \frac{1}{k}\sum_{i=1}^k (\delta_i - \delta)^T (g(x_i) + \delta) \Gamma_d :=\frac{2}{k}\sum_{i=1}^k\| \epsilon_i \|_2 \| g(x_i) +\delta\|_2 \Gamma_e := \frac{1}{k}\sum_{i=1}^k (\lambda_i-\theta)^T (\delta_i- \delta)
\kappa^2\delta \rho=-6 \left(\dot{\alpha_0}\dot{\xi}^y_0-\dot{\alpha}_0^2\xi^y_0-\dot{\alpha}_0^2 \alpha_1 l^2 \phi_0+\dot{\alpha_0}\ddot{\alpha}_0 e^{\alpha_0}l b_0-\frac{\dot{\alpha}_0 \ddot{\alpha}_0}{\alpha_1}e^{2 \alpha_0}\dot{\hat{E}}_0 -\hat{\Psi}_1\right),
{\mathbb{E}}(X_{i}|{\mathcal{F}}_{0}) =\sum_{k\geq1}e_{-N_{k}+i}\mathbf{1}_{i\leq N_{k}}\mathbf{1}_{T^{-i}(A_{k})}& =\sum_{k\geq1}e_{-N_{k}+i}\mathbf{1}_{i\leq N_{k}}\mathbf{1}_{A_{k}}+\sum_{k\geq1}e_{-N_{k}+i}\mathbf{1}_{i\leq N_{k}}(\mathbf{1}_{T^{-i}(A_{k})\backslash A_{k}}-\mathbf{1}_{A_{k}\backslash T^{-i}(A_{k})}).
ds^2 = \frac{1}{H_1 \tilde{H}_1}dt^2 - \frac{H_1}{\tilde{H}_1}ds^2_1 -\frac{\tilde{H}_1}{H_1}ds^2_1 - H_1 \tilde{H}_1 ds^2_3 \quad , \quad e^{-2 \phi}\sim 1 \ .
D(\gamma\cdot n_1,\ldots, \gamma\cdot n_g)(\gamma \cdot \tau)=\kappa(\gamma)^g\det(c\tau+d)^{g/2+1}\chi_{n_1}(\gamma)\cdot\ldots\cdot\chi_{n_g}(\gamma)D(n_1,\ldots, n_g)(\tau).
\begin{aligned}F_{t}(z;x) \geq (F(z)+\tfrac{\mu + t^{-1}}{2}\|z-x\|^2)-\mu \|z-x\|^2 &\geq F(\hat x)+\tfrac{\mu+t^{-1}}{2}\|\hat x-x\|^2+\tfrac{1}{2t}\|\hat x-z\|^2-\mu\|z-x\|^2,\end{aligned}
\stackrel{.}{a_{1}}=\Gamma^{\rho}\sum_{i,j}T_{i}X_{0,\rho}T_{j}\,e^{\left( m_{i}^{2}-m_{j}^{2}\right)t}-\sum_{i,j}T_{i}X_{1}T_{j}\,e^{\left( m_{i}^{2}-m_{j}^{2}\right) t}
\frac{d^{l+m}}{dx^{l+m}}(x^{2}-1)^{l}=\sum_{i=m}^{l}\frac{(l+m)!}{i!(l+m-i)!}\frac{l!}{(i-m)!}(x+1)^{i-m}\frac{l!}{(l-i)!}(x-1)^{l-i}
n_1+2\sum_{i=1}^{r-1}n_i n_{i+1}=r+2\sum_{i=1}^{r-1}x_i+x_{r}+\sum_{i=1}^{r-1}x_ix_{i+1}&\le r + 2(n-r-1)+x_ox_e &\le r + 2(n-r-1)+\left\lfloor \frac{(n-r-1)^2}4\right\rfloor
\begin{array}{r c l}E^{(1)}_n=\hbar g\omega\left(2n+\frac{9}{2}\right)\ \ \ , \ \ \d^{(1)}_n=\frac{1}{120}(n+1)(n+2)(n+3)(n+4)(n+5)\ \ \ , E^{(2)}_n=\hbar g\omega\left(2n+\frac{15}{2}\right)\ \ \ , \ \ \d^{(2)}_n=\frac{1}{120}(n+1)(n+2)(n+3)(n+4)(n+5)\ \ \ .\end{array}
\sum_{n=1}^{\infty}q^{n^2-1}+\sum_{n=1}^{\infty}q^{2n^2-1}+\sum_{n=1}^{\infty}q^{5n^2-1}+\sum_{n=1}^{\infty}q^{10n^2-1}&\equiv \sum_{n=0}^{\infty}b_{10}(n)q^{n}\sum_{n=0}^{\infty}q^{\frac{5n(n+1)}{2}}\pmod{2}.
{C}^{N_1\ldots N_4\,N}_{\gamma_1\ldots\gamma_4}w_{\gamma_1}\ldots w_{\gamma_4}= \sqrt{{\rm dim}_N}\ \left({C}^{N_1 N_2 N}_{\gamma_1\gamma_2\gamma}w_{\gamma_1}w_{\gamma_2}w_{\gamma}\right) \left({C}^{N_1 N_2 N}_{\gamma_1\gamma_2\gamma'}w_{\gamma_1}w_{\gamma_2}w_{\gamma'}\right) ={1\over \sqrt{{\rm dim}_{N_1}\ldots{\rm dim}_{N_4}}}.
\int^{Q-\sum^{\alpha-1}_{\beta=1}p^\beta+\varepsilon_\alpha}_{-\varepsilon_\alpha}dp^\alpha\to\int^{2\lambda-\sum^{\alpha-1}_{\beta=1}\tilde p^\beta-\tilde\varepsilon_\alpha}_{\tilde\varepsilon_\alpha}d\tilde p^\alpha\ .
\delta_{H}^{2}\simeq \frac{5 1 2 \pi}{7 5}\, \frac{A}{q^{3}{M_{P}}^{6}}\, \frac{\phi^{2 + q}}{\phi_{0}^{q}}\left[ 1 + \xi ( 3 q - \frac{3 \kappa^{2}\phi^{2}}{2}+ \frac{\kappa^{2}\phi^{2}}{q}) \right]^{2}
\begin{array}{l}\lbrace \alpha(A_{00},B)+(\dfrac{1}{q}-\dfrac{1}{q^2})\alpha(A_{10},B)-\dfrac{1}{q^3}\alpha(A_{11},B)\rbrace -\dfrac{1}{q^4}\lbrace \alpha(A_{11},B)+(\dfrac{1}{q}-\dfrac{1}{q^2})\alpha(A_{21},B)-\dfrac{1}{q^3}\alpha(A_{22},B)\rbrace=0\end{array}
\Big\Vert u\Big\Vert_{\mathrm{L}^\mathrm{p}\Big(\mathrm{B}\Big)}\lesssim \Big\Vert u\Big\Vert_{\mathrm{L}^\mathrm{q}\Big(\mathrm{B}\Big)}^{1-\mathrm{r}}\Big\Vert u\Big\Vert_{W^{1,\mathrm{N}}\Big(\mathrm{B}\Big)}^\mathrm{r}\forall u \in W^{1,\mathrm{N}}\Big(\mathrm{B}\Big), \textbf{where}\mathrm{r}= 1-\frac{\mathrm{q}}{\mathrm{p}}
\Omega(a,a,a)=a, \,\,\,\Omega(b,b,b)=b, \Omega(b,a,a)=\Omega(a,b,a)=\Omega(a,a,b)=b, \Omega(a,b,b)=\Omega(b,a,b)=\Omega(b,b,a)=a.
L = \frac{1}{4 \pi \alpha'}\sum_{n}\left[ \epsilon(\dot{X}^{i}_{n})^2 - \frac{1}{\epsilon}\left(X^{i}_{n+1}- X^{i}_{n}\right)^2 + 4\pi \alpha' B_{ij}\dot{X}^{i}_{n}\left(X^{j}_{n+1}- X^{j}_{n}\right)\right]
\sqrt{-g}\approx1+\frac{1}{2}h_{\alpha\beta}\eta^{\beta\alpha}+\frac{1}{8}h_{\alpha\beta}\eta^{\beta\alpha}h_{\gamma\delta}\eta^{\delta\gamma}-\frac{1}{4}h_{\alpha\beta}\eta^{\beta\gamma}h_{\gamma\delta}\eta^{\delta\alpha}.
\begin{aligned}p_0(x)=\sqrt{1/\gamma_0},\sqrt{b_1}\, p_1(x)= (x-a_0)p_0(x) \sqrt{b_{m+1}}\, p_{m+1}(x) = (x-a_m)p_m(x) - \sqrt{b_m}\, p_{m-1}(x), (m\ge1),\end{aligned}
A_1=\begin{pmatrix}2 0 0 0 2 0 0 0 2\end{pmatrix}, A_2=\begin{pmatrix}2 1 0 0 2 0 0 0 2\end{pmatrix},A_3=\begin{pmatrix}2 1 0 0 2 1 0 0 2\end{pmatrix}
f(t) = e^{2\pi i \eta \mu}\Omega_{2\eta}(t; -8\eta, -2\kappa \eta) \frac{\theta_0(t - 4\eta; -8\eta)}{\theta_0(t - 2\eta; -8\eta)}\frac{\theta_0(t + 2 \mu \eta; -2 \kappa \eta)}{\theta_0(t - 2\eta; -2\kappa\eta)}\theta_0(1/2 - 8\mu\eta - 4\kappa\eta + 2t; -16\kappa\eta).
{\cal L}= a^{3}\left[ 6 F ( \varphi ) \left( \frac{\dot{a}}{a}\right)^{2}+ 6 F^{\prime}( \varphi ) \dot{\varphi}\left( \frac{\dot{a}}{a}\right) + \frac{\gamma}{2}\dot{\varphi}^{2}- V ( \varphi ) \right] \ .
G ( x ) ={\frac{2{B_{0}}-{C_{1}}+ 2 ({B_{1}}-{C_{2}}){\omega}+ ( 2{B_{2}}- 3{C_{3}}){{\omega}^{2}}- 4{C_{4}}n{{\omega}^{3}}}{4{\sqrt{{C_{0}}+{C_{1}}{\omega}+{C_{2}}{{\omega}^{2}}+{C_{3}}{{\omega}^{3}}+{C_{4}}{{\omega}^{4}}}}}}.
\int_{v_1>0}F_+(x,v_1)dv_1 =C_{\ell,u}\int_{0}^{x}\int_{v_1>0}\frac{1}{\tau|v_1|}e^{-\frac{a_{\ell}(x-y)}{\tau|v_1|}}\mathcal{M}_1(v_1)dv_1dy\cr \leq C_{\ell,u}\left(\frac{\ln \tau+1}{\tau}\right).
H_{(M_{ij})}(z,y) : = \frac{1}{2}\langle z, (Id+(M_{ij}))z \rangle + \langle z, (M_{ij})NP^* y\rangle + \langle z, s \rangle + \sum_{i=1}^N \delta \psi (z_i + ( NP^* y )_i ),
\lim_{n \to \infty}n^{\frac{m(m+1)}{2}- (m + 1)s + 1}\Big[ q_{\kappa_n}^{(n )}(x_i^{(n)}) - q_{\kappa_n-1}^{(n)}(x_i^{(n)}) \Big] = 2^{(m +1)s}(v_1 \cdots v_m)^{-\frac{s}{2}}z_i^{-\frac{s}{2}}B_{II}^{(s, v)}(\kappa, z_i),
f(b_1,\cdots,b_{d-1},b_d)=\sum_{\alpha\in\{-m,\cdots,-1,0,1,\cdots,m\}^{d-1}}A_{\alpha}(b_d) e^{\alpha_1\frac{2\pi ib_1}{T_1}}\cdots e^{\alpha_{d-1}\frac{2\pi ib_{d-1}}{T_d}}
S_{0}\!=\!\sum_{n=0}^{\infty}\!\int\!dtd\xi_{2}d\xi_{3}\left\{|\partial_{t}\Phi_{\!n}(\xi_{2},\xi_{3},t)|^{2}\!\!-\!|\partial_{\xi_{3}}\!\Phi_{\!n}(\xi_{2},\xi_{3},t)|^{2}\!\!-\!M_{\!n}^{2}|\Phi_{\!n}(\xi_{2},\xi_{3},t)|^{2}\!\right\},
\left\{e^{\frac{1}{24}A_3}\widehat{L}(TM)\right\}^{(12)}=-\frac{1}{2}\left\{e^{\frac{1}{24}A_3}\widehat{A}(TM){\rm ch}[17\widetilde{T_CM}+\wedge^2\widetilde{T_CM}+W_i+128]\right\}^{(12)}&+\frac{1}{60}\left\{c_2(W_i)e^{\frac{1}{24}A_3}\widehat{A}(TX)\right\}^{(12)}.
C_p{}\leq \biggl(\frac{2B}{T^2}\biggr)^\frac{p}{2}\int\limits_0^{A}\biggl( \biggl(T\tau+\frac{T}{2a}\biggr)e^{-2a\tau}+\frac{T}{2a}+\frac{1}{2a^2}e^{-2aT}\biggr)^{p/2}d\tau &{}= (2B )^\frac{p}{2}\frac{T^{p/2}}{T^p}I_5= (2B)^\frac{p}{2}\frac{1}{T^{p/2}}I_5,
E^{j}{\cal P}_{j q}^{i p}= E^{p}\delta_{q}^{i}, t i l d e{\varepsilon}_{i}{\cal P}_{j q}^{i p}= \tilde{\varepsilon}_{q}\delta_{j}^{p}, \ \varepsilon_{p}\tilde{E}^{q}{\cal P}_{j q}^{i p}= \tilde{E}^{i}\varepsilon_{j},