harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
b_{r,\alpha}:=\Theta \,a(u_{r,\alpha})b_{r,\alpha}^*:=\Theta^{-1}a^*(u_{r,\alpha})c_{\ell,\alpha}:= \Theta^{-1}a(u_{\ell,\alpha})c^*_{\ell,\alpha}:=\Theta a^*(u_{\ell,\alpha}).
b \left( \phi_{o}^{4}+ \phi^{4}\right){\bf M}^{2}= \frac{1}{6}\left[ 1 + \frac{2{\bf Q}^{2}}{9{\bf M}^{2}}\left( 1 + \frac{2{\bf Q}^{2}}{a^{2}}\right)^{2}\right]^{2}
\begin{aligned}\alpha(t)= C_2t^{a+1}-C_1\frac{1}{a+1}, \beta(t)= C_1\frac{2a+1}{a(a+1)}\mathfrak{c}_{p}^{-1}t^a-C_2\mathfrak{c}_{p}^{-1}t^{2a+1}, \gamma(t)= - C_2\mathfrak{c}_{p}t-C_1\mathfrak{c}_{p}\frac{1}{a(a+1)}t^{-a}.\end{aligned}
{\cal A}_{33}=\frac{3g^2}{4}\sum_{\pi(p_1,\ldots p_6)}\bar{u}(p_{\pi_1})\gamma_{\mu}u(p_{\pi_4})\bar{u}(p_{\pi_2})\gamma^{\mu}\frac{i}{p_{\pi_1}\! \! \!\! \! \! \!/+p_{\pi_2}\!\! \! \!\! \! \!/-p_{\pi_4}\! \!\!\!\!\!\!/-m}\gamma_{\nu}u(p_{\pi_5})\bar{u}(p_{\pi_3})\gamma^{\nu}u(p_{\pi_6})
\Omega_{N}= \Bigg\{\sup_{\substack{t\in [0,T] i\in \{1,\dots,N\}}}\frac{1}{N}\bigg| \sum_{k=1}^{N}(K \ast V^{N})(X_{t}^{i,N,(A)}- X_{t}^{k,N,(A)})\bigg| \leq A\Bigg\}
\#_{C_j,N_2,n}\lesssim \#\{\lambda_1\in \Lambda^+\cap C_j\}\cdot\max\#\{\lambda_2\in\Lambda^+:\ |\lambda_2|_\rho\sim N_2,\ |\lambda_2|_\rho^2=n-|\lambda_1|_\rho^2\}&\lesssim_{\varepsilon}N_2^{r}\cdot N_2^{r-2+\varepsilon}
\varphi \psi = \sum_{i,j}\delta_if_i \delta_jg_j &= \sum_{i}\delta_i^2 f_i g_i + \sum_i \sum_{j\not=i}\delta_i\delta_j f_i g_j &=\sum_i\delta_i f_ig_i - w \sum_{i}\sum_{j\not=i}[\sigma_{ij}\delta_i(z) +\sigma_{ji}\delta_j(z)]f_ig_i &+ w \sum_i\sum_{j\not=i}[\sigma_{ij}\delta_i(z) +\sigma_{ji}\delta_j(z)]f_i g_j.
\hat{H}=\sum_{a}H_{a}^{a}\hat{a}^{\dagger}\hat{a}+\sum_{a,b}H_{ab}^{ab}\hat{a}^{\dagger}\hat{a}\hat{b}^{\dagger}\hat{b}+\sum_{a,b,c}H_{abc}^{abc}\hat{a}^{\dagger}\hat{a}\hat{b}^{\dagger}\hat{b}\hat{c}^{\dagger}\hat{c}+...
\begin{aligned}\|s\|^2\ = \int_{\Omega_{\lambda_1}}|s(x)|^2\, d\mu(x)+\,\int_{M\setminus \Omega_{\lambda_1}}|s(x)|^2\, d\mu(x) &\leq \|\rho(s)\|^2\ +\ \frac{\lambda}{\lambda_1-\lambda}\,\|\rho(s)\|^2= \left(\,\frac{\lambda_1}{\lambda_1-\lambda}\,\right)\, \|\rho(s)\|^2.\end{aligned}
V_{matter}^{fd}={{2N_F^{fd}}\over{\pi^2 L^4}}{\beta^2\over{24}}\Biggl(\sum_{i=1}^{N-1}\Bigl(6\theta_i^2 -12\pi\theta_i +\beta^2+4\pi^2\Bigr)+6\Bigl(\sum_{i=1}^{N-1}\theta_i\Bigr)^2-12\pi\sum_{i=1}^{N-1}\theta_i+\beta^2+4\pi^2\Biggr).
\gamma^{\textbf{ext}}_{1}\mathrm{U}_{\mathrm{e}}= -\mathcal{S}_{\alpha_\mathrm{m}}^{-1}\Big(\frac{1}{2}I_{d}- \mathcal{K}_{\alpha_\mathrm{m}}\Big)\Big[\gamma^{\textbf{ext}}_{0}\mathrm{U}_{\mathrm{e}}\Big].
\bigl\| |\nabla\varrho/\varrho|\bigr\|_{L^2(\mu)}^2=\int_{\mathbb{R}^d}\Bigl|\frac{\nabla\varrho}{\varrho}\Bigr|^2 \varrho\, dx\le\int_{\mathbb{R}^d}|b|^2 \varrho\, dx=\bigl\| |b|\bigr\|_{L^2(\mu)}^2.
\int_{s_{0}}^{\infty}s^{2}\tilde{w}_{s}(\tau, s)^{2}ds + \int_{s_{0}}^{\infty}s^{2}\tilde{w}_{\tau}(\tau, s) ds = \int_{s_{0}}^{\infty}s^{2}\tilde{u}_{s}(\tau, s)^{2}ds + \int_{s_{0}}^{\infty}s^{2}\tilde{u}_{\tau}(\tau, s) ds.
\begin{array}{c}\delta_1^{^{\prime}2}=s_2\left( s_2+1\right) \ ,\quad \delta_2^{^{\prime}2}=m^2s_1\left( s_1+3\right) s_1\geq s_2\qquad ,\qquad s_1,s_2=0,1,2,...\end{array}
P(x) = \det \begin{pmatrix}x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}x_{7}x_{8}x_{9}\end{pmatrix}& = x_{1}x_{5}x_{9}+ x_{2}x_{6}x_{7}+ x_{3}x_{4}x_{8}- x_{1}x_{6}x_{8}- x_{2}x_{4}x_{9}- x_{3}x_{5}x_{7},
\begin{aligned}\frac{\partial u(x,y,t)}{\partial t}-\varepsilon_x\frac{\partial^{\beta_1}u(x,y,t)}{\partial x^{\beta_1}}&\qquad\qquad-\varepsilon_y\frac{\partial^{\beta_2}u(x,y,t)}{\partial y^{\beta_2}}=f(x,y,t),\end{aligned}
\int\limits_{\mathbb{B}^n}\prod\limits_{i,j\leq n}\left( \sin \theta_{i,j}\right)^{2n+1-j}\d{\rm Leb}_{\mathbb{B}^n}= \left(\prod\limits_{j=1}^n \left(\int\limits_0^{\pi}\left( \sin \theta \right)^{2n+1-j}d\theta \right)\right)^n = \left(\pi^{\lceil{\frac{n}{2}\rceil}}2^{\lfloor{\frac{n}{2}}\rfloor}\frac{n!!}{2n!!}\right)^n.
\langle |F_k|^2, \phi \rangle = \frac{3}{\pi}\cdot \langle 1 , \phi \rangle+ O \Big ( (CM)^{3/2+\varepsilon}(\log k)^{\varepsilon}\cdot \Big ( \prod_{p \leq k}\Big ( 1 - \frac{n(p)}{p}\Big ) + \prod_{p \leq k}\Big ( 1 - \frac{\lambda_f(p^2)+1}{p}\Big ) \Big ) \| \phi \|_2 \Big ) + O_{A}( (\log k)^{-A})
\prod_{\substack{n\geq 1 n\not\equiv 0,\pm r(\textrm{mod}2l+s)}}(1-q^n)^{-1}= \sum_{n_1 , n_2 , \ldots , n_{l-1}\geq 0}\frac{q^{N_{1}^{2}+N_{2}^{2}+...+ N^{2}_{l-1}+N_{r}+N_{r+1}+\ldots+N_{l-1}}}{(q)_{n_1}(q)_{n_2}\cdots (q)_{n_{l-2}}(q^{2-s})_{n_{l-1}}},
H ={\frac{1}{1 6 \pi G}}\int d^{2}x \left[ e_{t}^{a}\varepsilon^{i j}\left( R_{a i j}+{\frac{1}{l^{2}}}\varepsilon_{a b c}e_{i}^{b}e_{j}^{c}\right) + \omega_{t}^{a}\left( D_{i}e_{a j}- D_{j}e_{a i}\right) \right] \, ,
A_{T}^{x}:=\sum_{\gamma\in S_{T}:a\rightarrow\alpha\backslash\{a\}}x^{l(\gamma)},B_{T}^{x}:=\sum_{\gamma\in S_{T}:a\rightarrow\beta}x^{l(\gamma)},E_{T}^{x}:=\sum_{\gamma\in S_{T}:a\rightarrow\epsilon\cup\overline{\epsilon}}x^{l(\gamma)}.
\|(z-A)^{-1}\|\leq\frac{1}{|z|}+\sum\limits_{k=1}^{m_0}\frac{\prod\limits_{j=1}^k w_j}{|z|^{k+1}}+\frac{1}{|z|}\big(e^{\frac{1}{|z|}}-\sum\limits_{k=0}^{m_0}\frac{1}{k!|z|^{k}}\big).
S=\int d^{6}x\left[{\frac 1 6}F_{{\mu}{\nu}{\sigma}}F^{{\mu}{\nu}{\sigma}}+{\frac 1 3}{\lambda}_{{\mu}{\nu}{\sigma}}{\cal F}^{{\mu}{\nu}{\sigma}}+{\frac 1 3}G^{{\mu}{\nu}{\sigma}}(F_{{\mu}{\nu}{\sigma}}-{\partial}_{[{\mu}}A_{{\nu}{\sigma}]})\right],
a^{(l, \sigma)}_-(y) = \left( \Lambda \left( \xi^{(l, \sigma)}(y) \right) - \left( \frac{1}{2}- 8\varepsilon \right) \right) \cdot 8L \;, a^{(l, \sigma)}_+(y) = \left( \Lambda \left( \xi^{(l, \sigma)}(y) \right) + \left( \frac{1}{2}- 8\varepsilon \right) \right) \cdot 8L \;.
|f\rangle=\sum_{l=1}^{\infty}\sum_{m=-l}^{m=l}|Y_{m}^{l}\rangle\langle Y_{m}^{l}|f\rangle,f(\theta,\phi)=\sum_{l=1}^{\infty}\sum_{m=-l}^{m=l}f_{lm}Y_{m}^{l}(\theta,\phi),
\sum_{\ell=0}^\infty\binom{2\ell}{\ell}\frac{(-\zeta;\zeta^2)_{d}^{2\ell}}{(-\zeta^2;\zeta^2)_{d}^{2\ell}(-\zeta^4;\zeta^4)_{d}^{2\ell}}=\sum_{\ell=0}^\infty\binom{2\ell}{\ell}\frac{1}{4^\ell}=\infty,
I(f)(0) =\sum_{j=-B}^{A-1}q^{j(\ell-2)}f(0,0,\varpi^j)+\frac{q^{A(\ell-2)}}{1-q^{\ell-2}}f(0)-\frac{q^{(n-A+1)(\ell-2)}}{1-q^{\ell-2}}f(0) +q^{(n-A+1)(\ell-2)}\sum_{j=0}^{B+A-1}q^{j(\ell-2)}f(\varpi^{A-j-1}\xi,0,0).
A(\phi(x_0), \nabla\phi(x_0), \nabla^2 \phi(x_0))\ge 0 \u(x_0)>0 B(\phi(x_0), \nabla\phi(x_0), \nabla^2 \phi(x_0))\ge 0 \u(x_0)<0 -\mathcal{Q}_\infty \phi(x_0)\ge0 \u(x_0)=0
s_{i,l}= \tilde a_i+ l q_i,0\le l\le L_i, L_i = [1/ q_i],q_i = \theta \frac{\Delta(y_i)}{y_i}, \tau_{i,n}= \tau(y_i)+ nq_i, 0\le |n| \le N_i,N_i = [\tau^*(y_i)/q_i].
\frac{\partial A}{\partial q_{j}}=\frac{\partial A(p_{j},q_{j})}{\partial q_{j}}\:\:\:\:;\:\:\:\:\frac{\partial (X+Y)}{\partial q_{j}}=\frac{\partial X(p_{j},q_{j})}{\partial q_{j}}+\frac{\partial Y(p_{j},q_{j})}{\partial q_{j}}
{\mathcal J}^{0}( x ) \; = \; \frac{q}{2 \pi}\; \mathrm{d i v}\, \underline{{{\varphi}}}( x ) ~ , ~ ~ \underline{{{{\mathcal J}}}}( x ) \; = \; - \, \frac{q}{2 \pi}~ \frac{\partial}{\partial t}~ \underline{{{\varphi}}}( x ) ~ ,
\omega\mathbf{\bar{y}}= \gamma\left(h_1\mathbf{\check{c}}_1^2 + h_2\mathbf{\check{c}}_2^2\right) +\gamma\left(h_1\mathbf{\check{c}}_1^1 + h_2\mathbf{\check{c}}_2^1\right)\omega + \omega\mathbf{\bar{z}},
S(x=x_L,y,t)= S(x=x_R,y,t)= S(x,y=y_L,t)= S(x,y=y_R,t)= 0 I(x=x_L,y,t)= I(x=x_R,y,t)= I(x,y=y_L,t)= I(x,y=y_R,t)= 0 V(x=x_L,y,t)= V(x=x_R,y,t)= V(x,y=y_L,t)= V(x,y=y_R,t)= 0.
V_{P}( B ) = \frac{1}{4 \pi^{2}l^{3}}\int_{0}^{1}d \eta \int_{0}^{1}d u u^{2}\ln \left[ \frac{[ r + c ] [ r + c - c \eta^{2}/ 2 ]}{[ r + \pi / 2 u ]^{2}}\right] .
\hat{a}_{B D}( \zeta ) = \frac{e^{\pi \vert \zeta \vert / 2}}{( e^{\pi \vert \zeta \vert}- e^{- \pi \vert \zeta \vert})^{1 / 2}}\hat{a}_{H}( \zeta ) - \frac{e^{- \pi \vert \zeta \vert / 2}}{( e^{\pi \vert \zeta \vert}- e^{- \pi \vert \zeta \vert})^{1 / 2}}\hat{a}_{H}^{\dagger}( \zeta ) \, .
B_{m n}^{( + )}= -{\frac{i^{k}}{2 \pi}}\sqrt{{\frac{n}{m}}}( m + n )^{- k}e^{i n T^{+}( 0 )}\int_{0}^{2 \pi}e^{i ( m + n ) y}{\frac{d^{k + 1}\varphi_{t}}{d y^{k + 1}}}d y ,
L_{\varphi}(a,r,s,A): = \frac{1}{|P_{r,s,A}|K_{P_{r,s,A}, \varphi}(a)}&=\inf \left\{\frac{\int_{a+P_{r,s,A}}|f|^2e^{-\varphi}}{|P_{r,s,A}|e^{-\varphi(a)}}~\middle|~ f\in A^2(a+P_{r,s,A}, \varphi) \And f(a)=1 \right\},
\vec{S}= \left[ \begin{array}{c}{{\mathrm{R e}\left({\mathcal{W}}\right)}}\\{{\mathrm{I m}\left({\mathcal{W}}\right)}}\\ \end{array}\right] , \qquad \vec{a}= - \left[ \begin{array}{c}{{\mathrm{I m}( \phi \partial_{x}\bar{\phi})}}\\{{\mathrm{I m}( \phi \partial_{t}\bar{\phi})}}\\ \end{array}\right] ,
\mathbf{E}_{x}\big[ \Vert h^{(0)}(x) - h^{(0)}_{\le d}(x)\Vert_2^2 \big] \geq \frac{1}{k^2}\cdot \big( \Pr_{x}[x \in \mathsf{Collision}(g)] + \Pr_{x}[x \not \in \mathsf{Collision}(g) \ \wedge \g(x) \not = h(x) ]\big).
Du_\lambda(t,x) = v_{0,\lambda}(t) + \int_0^t\!\!\int_G K_{t-s}(x,y) f'_\lambda(u_\lambda(s,y)) Du_\lambda(s,y)\,dy + \int_0^t\!\!\int_G K_{t-s}(x,y) \sigma'_\lambda(u_\lambda(s,y)) Du_\lambda(s,y)\,\bar{W}(dy,ds),
\langle u , Z_{s'}\rangle = \sum_{k = 0}^{\ell_s + 1}b_{k,\ell_s + 1}\langle u , \zeta(V_{s',k}) \rangle = \sum_{k = 0}^{\ell_s + 1}(\sum_{i = k}^{\ell_s + 1}b_{i,\ell_s + 1})\langle u , \zeta(V_{s',k}) - \zeta(V_{s',k-1}) \rangle,
\sum_{\ell \in I(k,k')}F(k\ell) \overline{F(k'\ell)}= \sum_{Q \leq q,q' < 2Q}\sum_{\substack{\ell \in I(k,k',q,q') k\ell \equiv a_q\pmod{q}k'\ell \equiv a_{q'}\pmod{q'}}}\psi_q((k\ell-a_q)/q) \overline{\psi_{q'}((k'\ell - a_{q'})/q')},
\alpha^{2}= ( 1 - r^{- 1 0 / 3})^{- 1}; \beta^{2}= \frac{9}{1 0 0}r^{2}( 1 - r^{- 1 0 / 3})^{- 1}; \gamma^{2}= \frac{9}{2 0}r^{2}.
\left| e_2^\perp - \left( (1-\dot x_k^2)e_2 - \dot x_k\dot y_ke_3 \right) \right| = O(\varepsilon), \left| e_3^\perp - \left( (1-\dot y_k^2)e_3 - \dot x_k\dot y_ke_2 \right) \right| = O(\varepsilon).
W\left[{\eta \left( t \right)}\right] = \mathop \sum \limits_n \frac{1}{{n!}}\mathop \int{W^{\left( n \right)}}\left({{t_1}-{t_2}, \ldots ,{t_1}-{t_n}}\right)\eta \left({{t_1}}\right)...\eta \left({{t_n}}\right)d{t_1}...d{t_n}
|f(\phi(0))| = |f(-r)| \le |f_0|+|f_+(-r)|+|f_-(-r)| \le |f_0|+ \sum_{n>0}r^n |f_n| + \sum_{n<0}r^n |f_n| & \le |f_0| + r \Big( \sum_{n\ne0}\frac{r^{2(n-1)}}{\omega(n)^2}\Big)^{1/2}\Big(\sum_{n\ne0}|f_n|^2\omega(n)^2\Big)^{1/2}& \le |f_0|+r \,d \,\|f-f_0\|_{2,\omega},
\mathrm{d e t}\left( 1 \! \! 1 \pm \hat{F}\right) = 1 - \frac{1}{2}T r ( \hat{F}^{2}) + \frac{1}{8}\left[ T r ( \hat{F}^{2}) \right]^{2}- \frac{1}{4}T r ( \hat{F}^{4}) + \mathrm{d e t}( \hat{F})
\mathrm{d e t}\left[{{\cal Y}^{\sigma}}_{\alpha}\right] = \frac{( j^{2})^{2}}{({\tilde{j}}^{2})^{2}}\, \mathrm{d e t}\left[{{\cal Y}^{\sigma}}_{\alpha}\right] \, = \frac{({\cal J}^{2})^{2}}{({\tilde{j}}^{2})^{2}}\, \mathrm{d e t}\left[{{\cal Y}^{\sigma}}_{\alpha}\right] \, = a \frac{({\cal X}^{2})^{4}}{({\tilde{j}}^{2})^{2}}.
(1-H^*)^n\ 1 = \sum_{j=0}^n (-1)^j\binom{n}{j}\ (H^*)^j\ 1 &=\sum_{j=0}^n (-1)^j\binom{n}{j}\big((-1)^j \frac{(\log x)^j}{j!}\big) &=\sum_{j=0}^n \binom{n}{j}\frac{(\log x)^j}{j!}.
\left| V_{\mathrm{b o s}}\right> = \exp \left\{\frac{1}{2}\sum_{u , v = 1}^{3}\left( \sum_{m , n}a_{m ( u )}^{I \dagger}{\bar{N}}_{m n}^{u v}a_{n ( v )}^{I \dagger}\right) \right\}\left| 0 \right>_{1 2 3}\, .
\tau_{k}=\|w_{k+1}\|^{2}=\frac{1}{\alpha_{k+1}^{2}}\left(\beta_{k}^{2}\|w_{k}\|^{2}+1\right)=\frac{1}{\alpha_{k+1}^{2}}\left(\beta_{k}^{2}\tau_{k-1}+1\right),
\| \partial^{\rho}\widehat{a}_{j}\|_\infty = \| \partial_1^{\rho_1}\partial_2^{\rho_2}W_{j,0}( A_{j}\cdot) \|_\infty =2^{js \rho_1}2^{js \alpha \rho_2}\| \big( \partial_1^{\rho_1}\partial_2^{\rho_2}W_{j,0}\big) ( A_{j}\cdot) \|_\infty \lesssim 1.
\begin{aligned}\frac{1}{\tau^k}[f_1(x_1)-f_1(\breve{x}_1^{k})]-\frac{1}{\tau^{k-1}}[f_1(x_1)-f_1(\breve{x}_1^{k-1})] + (x_1-\widetilde{x}_1^{k})^T\{-A_1^T\widetilde{\lambda}^k+(1-\gamma)(1-\tau^k)\beta^kA_1^T(A\breve{x}^{k-1}-b)\}\geq0,~\forall x_1.\end{aligned}
b_{s,s+k}\left[Z_s,t;\left\{t_j,v_{s+j},\omega_j,i_j\right\}_{j=1}^k\right]= b_{s,s+k}^0\left[Z_s,t;\left\{t_j,v_{s+j},\omega_j,i_j\right\}_{j=1}^k \right]
\int b(x)|u|^{\tilde{r}(x)}= \int b(x)^\alpha b(x)^\beta |u|^{\tilde{r}(x)}&\leq \int \epsilon_1^{-\frac{r(x)}{r(x)-\tilde{r}(x)}}b(x)^{\alpha \frac{r(x)}{r(x) - \tilde{r}(x)}}+ \int \epsilon_1 b(x)^{\beta \frac{r(x)}{\tilde{r}(x)}}|u|^{r(x)}= \int \epsilon_1^{-\frac{r(x)}{r(x)-\tilde{r}(x)}}b(x) + \int \epsilon_1 b(x) |u|^{r(x)}.
\mbox{with}\;\;\; \omega_{l,m,n}\equiv \sqrt{\Biggl(\frac{l\pi}{2L_x}\Biggr)^2 +\Biggl(\frac{m\pi}{2L_y}\Biggr)^2 +\Biggl(\frac{n\pi}{2L_z}\Biggr)^2}\;\; ,
\lambda ={\frac{2^{2 / 3}( - 1 + 2 \sqrt{7}) ( 1 + ( - 1 + \sqrt{7}) e^{2 h}+ ( 2 + \sqrt{7}) e^{4 h}) \Lambda ( Z , T )}{8 1 g_{c}e^{2 h}( 1 + e^{2 h})}}\epsilon^{- 2 / 3}+{\cal O}( \epsilon^{- 1 / 3}) \ ,
\mu_{B^!}(y_1^*)= (-1)^{d+1}\big((qX+\frac{q}{p}X+\frac{1}{p}W)y_1^*+ (qZ+\frac{q}{p}Z+\frac{1}{p}Y)y_2^*\big), \mu_{B^!}(y_2^*)= (-1)^{d+1}(pXy_1^*+ pZy_2^*).
V' (t) =-\Bigl (\frac{4t}{\alpha}-\frac{8}{\alpha}-2-te^{-t}\Bigr ) v' (t)^2-\Bigl (\frac{2(\alpha +1)}{\alpha^2 t^2}+ \frac{1}{2\alpha}e^{-t}\Bigr )v (t)^2.
0=Q_n\left\{\left(\frac{d}{dt}\right)^n\left[\begin{array}{c}x_1 \vdots x_n\end{array}\right]\right\}+\ldots+Q_1\left\{\frac{d}{dt}\left[\begin{array}{c}x_1 \vdots x_n\end{array}\right]\right\}+Q_0\left\{\left[\begin{array}{c}x_1 \vdots x_n\end{array}\right]\right\}.
\lim_{t\to 0}\int_{-\infty}^{\infty}q_t(x,y)f(y)dy =\lim_{t\to 0}\int_{\infty}^{x}m(z)dz\int_{-\infty}^{\infty}p_t(z,y)\frac{f'(y)}{m(y)}dy &=\int_{-\infty}^{x}m(z)\frac{f'(z)}{m(z)}dz=f(x).
\max \, \mathbf{x}^{1}(\mathbf{R}- \nu \mathbf{1}) & & \begin{bmatrix}\mathbf{x}^{0}\mathbf{x}^{1}\end{bmatrix}\begin{bmatrix}(1-\beta) \mathbf{I}\mathbf{I}- \beta \mathbf{P}\end{bmatrix}= \mathbf{e}_i & \begin{bmatrix}\mathbf{x}^{0}\mathbf{x}^{1}\end{bmatrix}\geq \mathbf{0}.
[\begin{matrix}a&b\\c&d\end{matrix}]=[\begin{matrix}1&\lambda z_{2}\\ 0&1\end{matrix}][\begin{matrix}1&0\\ -1/\lambda f&1\end{matrix}][\begin{matrix}1&\lambda z_{1}\\ 0&1\end{matrix}]=[\begin{matrix}1-z_{2}/f&\lambda(z_{1}+z_{2})-\lambda z_{1}z_{2}/f\\ -1/\lambda f&1-z_{1}/f\end{matrix}].
\langle A_{\{\alpha \}}( x ) \, A_{\{\beta \}}( 0 ) \rangle ={\frac{\Gamma \left({\frac{n}{2}}\right)}{2 ( n - 2 ) \pi^{\frac{n}{2}}}}{\frac{{\cal A}_{\{\alpha \}, \{\beta \}}}{| x |^{n - 2}}}.
\widetilde{f^\chi}_{R'}(z) = f^\chi \mid W_{R'}(z)= \sum_{\ell\mid r_{*0}}\beta_{F_\chi}(\ell) \ell^{-\frac{k}{2}}\left(Q_*\frac{r_{*0}}{\ell}\right)^{\frac{k}{2}}\widetilde{F_\chi}_{\frac{R R_*'}{R_*}}\left(Q_*\frac{r_{*0}}{\ell}z\right).
Q = \tilde{Y}= \left( \begin{array}{c c c}{{0}}&{{0}}&{{0}}\\{{0}}&{{- 1}}&{{0}}\\{{0}}&{{0}}&{{- 1}}\\ \end{array}\right) \: , \: \tilde{T}_{3}= \left( \begin{array}{c c c}{{- 1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{- 1}}\\ \end{array}\right)
g \left( q^{i}, p_{i}, z^{a}, \overline{{{p}}}_{a}\right) = \sum_{k = 1}^{\infty}\frac 1{k ! \lambda^{k}}\left[ C_{a_{1}}^{0}, \left[ C_{a_{2}}^{0}, \ldots , \left[ C_{a_{k}}^{0}, \tilde{H}\right] \ldots \right] \right] z^{a_{1}}\ldots z^{a_{k}}.
u^{*}(x)=[\frac{L_{i}}{n_{i}}\sum_{l=1}^{n_{i}}H(d_{l})+\frac{L_{t}}{n_{t}}\sum_{j=1}^{n_{t}}H(d_{j})+\frac{S}{n_{\Omega}}\sum_{k=1}^{n_{\Omega}}H(d_{k})]e
{\cal C}_{a}(\nu_I)=\alpha^{\prime}\sum_{I,\overline{J}}2 \kappa_{aI}\overline{\kappa}_{a\overline{J}}G^{I\overline{J}}\left\{\gamma + \frac{1}{2}\Big(\mbox{\boldmath${\psi}$}(\nu_{I}) +\mbox{\boldmath${\psi}$}(1-\nu_{I})\Big)\right\}~,
\begin{aligned}0 \leq C_2 \beta \sum_i F^{ii}- (A + \frac{M}{2}) \gamma_1 \sum_i F^{ii}- \frac{\beta}{2n}F^{11}U_{11}+ \gamma_2 \sum_{U_{ii}> 0}F^{ii}U_{ii}&\leq - \frac{\gamma_1 (A + M)}{2}\sum_i F^{ii}- \frac{\beta}{2n}F^{11}U_{11}+ \gamma_2 \sum_{U_{ii}> 0}F^{ii}U_{ii},\end{aligned}
\frac{n(n-1)(n-2)}{j(j-1)}\left[y_j(3)-\frac{1}{j+1}\sum_{i=1}^{j+1}h_{j+1}(i)\right] =n-2-(2n-4)\sum_{i=j+1}^{n}\frac{1}{i-2}&=(n-2)\left[1-2\sum_{i=j+1}^{n}\frac{1}{i-2}\right] &\ge(n-2)\left[1-2\sum_{i=b_n+1}^{n}\frac{1}{i-2}\right] &\ge0\;\;\mbox{(by (\ref{v4e1}))}.
w_{ki}= 1, w_{k+1,i}= 1+\frac{(i-k-l)(i+k+l+\sigma)}{(\beta+2k+1)(k+l-n)}, begin{align*}1ex] w_{hi}= T_{i}(h)w_{h-1,i}+ V(h)w_{h-2,i}(h=k+2,k+3,\ldots,n-l),
F ( x ) \equiv \int_{0}^{x}d x \, \sqrt{x^{2}+ 1}+{\frac{1}{i \alpha}}\ln ( x^{2}+ 1 ) , \quad \alpha \equiv{\frac{2 \beta}{\hbar v g}}={\frac{4}{\hbar v g^{2}[ V_{1}^{\prime}( 0 ) - V_{2}^{\prime}( 0 ) ]}}> 0 .
T(\varphi) = p_+ \circ \varphi{|_{H_+^{(n)}}}; H(\varphi) = p_+ \circ \varphi \circ \iota{|_{H_+^{(n)}}}; \widetilde H(\varphi) = \iota\circ p_- \circ \varphi{|_{H_+^{(n)}}}.
y^{2}\partial_{y}y^{- 2}\partial_{y}H +{\frac{1}{y^{2}}}\left( f^{-{\frac{3}{2}}}( \mu ) \partial_{\mu}f^{{\frac{3}{2}}}( \mu ) \, \partial_{\mu}- m^{2}f \right) H = - \omega^{2}H \ .
\min_{\gamma\in\mathcal{W}}\{\int_{U\times Y\times Z}G(u,y,z)\gamma(du,dy,dz) \}= \min_{p\in\mathcal{\tilde{W}}}\{\int_{U\times Y\times Z}G(u,y,z)\Phi(p)(du,dy,dz) \}\ = \ \tilde{G}^*
\nu^\prime_N\left(Q^\prime_{N,M}(x,y)\right) \geq \mu_N\left(P_{N,M}(x,y)\right) \geq \exp\left\{NM\left(- \frac{\log Z_N}{N}- \frac{\varepsilon\log b}{2}\right)\right\}= \exp\left\{-M\log b \left(\log_b Z_N + \frac{\varepsilon N}{2}\right)\right\}= \left(b^{-M}\right)^{\log_b Z_N + \frac{\varepsilon N}{2}}.
\eta_{ij}= \sum_m b_m \theta_{mi}\theta_{mj}= \frac{1}{\pi^2}\sum_mb_m \frac{\omega_m}{\omega_m^2-i^2}\frac{\omega_m}{\omega_m^2-j^2}\sin^2(\pi \omega_m) (-1)^{i+j}\stackrel{(*)}{\lesssim}(ij)^{-\emph{min}\left(1,\frac{\delta+1}{2}\right)},
B_j^0[\phi_j]:= p\left[U^{p-1}\left(\frac{\mu_{0j}}{\mu_j}y\right)-U^{p-1}(y)\right]\phi_j + p\left[\mu_{0j}^{2s}(u^{*}_{\mu,\xi})^{p-1}-U^{p-1}\left(\frac{\mu_{0j}}{\mu_j}y\right)\right]\phi_j.
|\mathcal{I}'_k| =\sum^k_{i=0}|\mathcal{B}_{i,i}|+|\{(i,i,t,p):(i,i,t,p)\in\mathcal{I}'_{k-1}\}|+|\mathcal{I}'_{k-1}| begin{align*}0.1cm] =\frac{(k+1)(k+2)(k+3)}{6}+{k+3\choose 4}\ \ \ \ \ &={k+4\choose 4},
(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z})=-\frac{1}{\rho}\frac{\partial P}{\partial x}+\nu(\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}v}{\partial y^{2}}+\frac{\partial^{2}w}{\partial z^{2}}),
W_{p , N_{r}}^{( o )}( t ) = ( - 1 )^{e ( N_{r})}\sum_{N_{o}, N_{o}^{\prime}}\left[ E_{p , N_{o}}^{( - 2 )}\left( Y_{b , N_{r}}( t ) \right) + \sum_{N}[ \partial q_{N}Y_{p , N_{r}}( t ) ] B_{N N_{o}}\right] A_{N_{o}N_{o}^{\prime}}^{- 1}\chi_{N_{o}^{\prime}}( t )
G_n^{(t)}(s,q) := (G_n)^{(t)}(s,q) = (1-t)G_n\left(t + s(1-t),\frac{q}{\sqrt{1-t}}\right) &= (1-t)g\left(nt + ns(1-t) - \lfloor nt + ns(1-t)\rfloor, \frac{q}{\sqrt{n(1-t)}}\right).
H_{0,q}\left( x\right) =1,\ H_{1,q}\left( x\right) =x, H_{2,q}\left( x\right) =x^{2}-1, H_{3,q}\left( x\right) =x^{3}-\left[ 3\right]_{q}x, H_{4,q}\left( x\right) =x^{4}-\left( 1+q^{2}\right) \left[ 3\right]_{q}x^{2}+\left[ 3\right]_{q}q^{2}.
d s^{2}= \alpha^{\prime}\left\{\frac{U^{2}}{R^{2}}\left( ( 1 - U_{T}^{4}/ U^{4}) d t^{2}+ \sum_{i}d x_{i}^{2}\right) + \frac{R^{2}}{U^{2}}\frac{d U^{2}}{1 - U_{T}^{4}/ U^{4}}+ R^{2}d \Omega_{5}^{2}\right\}.
\sum_{t=0}^{2}q^{4\binom{t}{2}+3t}j(-q^{4t+2};q^3)m\Big (-q^{7-4t},-1;q^{12}\Big ) &=\overline{J}_{1,3}\Big ( m(q^2,-1;q^3)+\Theta_2(q)\Big ) +\overline{J}_{0,3}m(-q^{3},-1;q^{12}).
\begin{array}{l}h_3=a,h_2=-2 \dot{a}x+b,h_1=g_1+6aU, h_0=-{4\over3}\stackrel{\dots}{a}x^3+2 \ddot{b}x^2-2 \dot{c}x-4 \dot{a}\varphi+4(b-2 \dot{a}x)U+d \end{array}
L_{1}\,=\,-\frac{a}{4}\left( F_{\mu \nu}\right)^{2}+\,b\,\partial_{\mu}F^{\mu \lambda}\,\partial^{\nu}F_{\nu \lambda}\,+\,\theta \,\epsilon^{\mu \nu \rho}\,\partial_{\sigma}\,A_{\mu}\,\partial_{\nu}\partial^{\sigma}\,A_{\rho},
Y_{i_{i'}, 2}= \mathcal{C}_1(\alpha', \nu', \tau^{-1}(i'), I'_b \setminus \lbrace \tau^{-1}(i') \rbrace, I_b) \leq \mu^{\circ}+ \ell - 2{i_{i'}}+ 1 = X_{i_{i'}, 1}+ \ell -2i_{i'}+ 1 = Y_{i_{i'}, 1}.
{\bf \Lambda}_k \stackrel{\bigtriangleup}{=}\left({\Re [{\text C}({\bf v}_k) +{\underline{\text C}}({\bf v}_k)] \atop \Im[{\text C}({\bf v}_k) +{\underline{\text C}}({\bf v}_k)]}{-\Im[{\text C}({\bf v}_k) -{\underline{\text C}}({\bf v}_k) ] \atop \Re[{\text C}({\bf v}_k) -{\underline{\text C}}({\bf v}_k) ]}\right)
\lim_{\delta\to 0}\limsup_{n\to\infty}\,\P\,\bigg( \sup_{\substack{t_1-t_2<\delta 0\leq t_2\leq t_1\leq T}}\big\|U^n(t_1)-U^n(t_2) \big\|\,>\epsilon \bigg) = 0.
\mathbf{1}_\lambda \mathbf{1}_Y = \sum_{w\in W_\lambda}(t^{\frac12})^{\ell(w)-\ell(w_\lambda)}T_w\mathbf{1}_Y= t^{-\frac12\ell(w_\lambda)}\Big(\sum_{w\in W_\lambda}t^{\ell(w)}\Big)\mathbf{1}_Y= t^{-\frac12\ell(w_\lambda)}W_\lambda(t)\mathbf{1}_Y.
\begin{aligned}K(t,x)\triangleq{\textstyle\sum\limits_{n \in Z}}\frac{1}{2}(\delta(x+t+nJ)+\delta(x-t+nJ)), & S(t,x)\triangleq{\textstyle\sum\limits_{n \in Z}}\frac{1}{2}I_{[-t,t]}(x+nJ), \end{aligned}
T^{\prime}_{m}=R_{m}{}^{n}T_{n}\, ,\hspace{1cm}[T^{\prime}_{m},T^{\prime}_{n}] =R_{m}{}^{r}R_{n}^{s}\epsilon_{rsp}T_{p}={\rm det}R\, \epsilon_{mnq}\left(R^{-1}\right)_{p}{}^{q}\left(R^{-1}\right)_{p}{}^{r}T^{\prime}_{r}\, ,
\xi_{s_{\alpha_\tau}\cdot z}:= \bigotimes_{v \in \Sigma_p}\left(\psi_{\sigma_v(3),v}z^{h_{\sigma_v(3),v}-h_{\tau_v(3),v}}\varepsilon^2 \otimes \psi_{\sigma_v(2),v}z^{h_{\sigma_v(2),v}-h_{\tau_v(2),v}}\varepsilon \otimes \psi_{\sigma_v(1)}z^{h_{\sigma_v(1),v}-h_{\tau_v(1),v}}\right).
\epsilon^{\ast}dK_{\widehat{X}}= \epsilon^{\ast}(\psi^{\ast}dK_X + dE) = \epsilon^{\ast}\psi^{\ast}dK_X + \epsilon^{\ast}dE = \epsilon^{\ast}\psi^{\ast}dK_X + \epsilon^{\ast}dE = \phi^{\ast}\pi^{\ast}dK_X + \epsilon^{\ast}dE = \phi^{\ast}(dK_Y - dR)+ \epsilon^{\ast}dE .
\bar{Z}_{+}(z_{2})Z_{+}(z_{1})=F_{+}(z_{12})=\left(\begin{array}{cc}ix_{\bar{2}1}{\cdot\tilde{\sigma}}\,\tau_{+}-2\bar{\theta}_{21b}\,p_{+}-2\theta^{a}_{21}\,\tau_{+}\delta^{a}_{~b}\,p_{+}\end{array}\right)
[w]^p =[v]^p\left(1+\frac{2^{\frac{2s-1}{1-s}}}{p r_0(1-s)}||u||_p^p\right)^{\frac{1-s}{s}}&\leq\left(1-||u||_p^p\right)\left(1+\frac{2^{\frac{2s-1}{1-s}}}{p r_0(1-s)}||u||_p^p\right)^{\frac{1-s}{s}}
\begin{aligned}-\partial_{h}u_k\Delta u_{k}=e^{au_k}\partial_{h}u_k+\left|x\right|^{2N}e^{u_k}\partial_{h}u_k &=\partial_{h}\left[\frac{e^{au_k}}{a}+\left|x\right|^{2N}e^{u_k}\right]-\partial_{h}\left(\log\left|x\right|^{2N}\right)\left|x\right|^{2N}e^{u_k}.\end{aligned}
\Delta{\frac{\partial^{2}\Psi}{\partial t^{2}}}-{\frac{1}{r^{2}}}\left( 1 -{\frac{r_{0}}{r}}\right){\frac{d}{d r}}\left( r ( r - r_{0}){\frac{d \Psi}{d r}}\right) +{\frac{1}{r^{2}}}\left( 1 -{\frac{r_{0}}{r}}\right) L^{2}\Psi = 0
M^1=\{(x_1, x_2)\,:\, x_2\leq 0\}, M^2=\{(x_1, x_2)\,:\, x_1+x_2^2\leq 0\}, &M^3=\{(x_1, x_2)\,:\, -x_1-x_1|x_1|+x_2^2\leq 0\}, &Q^1=\{x^0\}, Q^2=M^1\cap M^3, Q^3=M^1\cap M^2.
p_j(w^{**})=\begin{cases}a_j-1 a_j a_j+1, 1\leq j \leq k-1, b_{2k-1-j}-1 b_{2k-1-j}b_{2k-1-j}+1, k\leq j \leq 2k-2, m, \textnormal{if}j=2k-1. \end{cases}
\big( (\sqrt{\bar{\Gamma}}^\top \Xi_u \sqrt{\bar{\Gamma}})^2 \big) = \left( (\check{V}^\top \Xi \check{V})^2 \right) = \underset{i=\lfloor d/2 \rfloor+1}{\overset{d}{\sum}}\xi_i^2 \leq d \xi_{\lfloor d/2 \rfloor}^2 \leq \frac{4}{d}( \Xi )^2,
(-\mathcal{L}B_{ij,0})'=-a_{ij}\sigma_i\Lambda Q+2p\kappa_0\sigma_j(yQ^{p-1})', &\int B_{ij,0}Q'=0,\quad\lim_{y\rightarrow+\infty}B_{ij,0}(y)=0, &\lim_{y\rightarrow-\infty}B_{ij,0}(y)=-\frac{a_{ij}\sigma_i(p-2)}{p-1}\int Q,
\frac{\kappa}{z}\sum_{n=0}^{\infty}( \sigma\alpha_{n}-\alpha_{n}\sigma ) z^{n}+\sum_{n=1}^{\infty}n\alpha_{n}z^{n-1}=\kappa\sum_{n=0}^{\infty}\alpha_{n}z^{n}\sum_{m=0}^{\infty}B_{m}( x (u,0 ) ) z^{m}.
- \frac{1}{\pi}\int_{\rho}^{\infty}{\frac{\bar{K}_{\nu}( x )}{\bar{I}_{\nu}( x )}\left[ e^{- \nu \pi i}f ( x e^{\pi i / 2}) + e^{\nu \pi i}f ( x e^{- \pi i / 2}) \right] d x}+ \int_{\gamma_{\rho}^{+}}{f ( z ) \frac{\bar{H}_{\nu}^{( 1 )}( z )}{\bar{J}_{\nu}( z )}d z}- \int_{\gamma_{\rho}^{-}}{f ( z ) \frac{\bar{H}_{\nu}^{( 2 )}( z )}{\bar{J}_{\nu}( z )}d z},
F \epsilon^{i j}={\frac{1}{2}}E^{{\alpha}i}\wedge E^{{\beta}j}\epsilon_{{\alpha}{\beta}}+{\frac{1}{2}}\bar{E}^{\dot{\alpha}i}\wedge \bar{E}^{\dot{\beta}j}\epsilon_{\dot{\alpha}\dot{\beta}}+{\frac{1}{2}}E^{a}\wedge E^{b}f_{a b}\epsilon^{i j}\; , \quad d F \equiv 0 \; .
|f(X) - f(X_1)| = |\sum_{j=1}^{r+s}\sum_{i=1}^{\ell}\bigg( |a_j^i|^2(X) \lambda_i (X)- |a_j^i|^2(X_1) \lambda_i(X_1)\bigg) -\sum_{j=1}^{r+s}\sum_{i=\ell+1}^{k}|a_j^i|^2(X) \lambda_i (X)|
\frac{\widehat{W}_{\xi , \eta}( i )}{\widehat{W}_{\xi , \eta}( i - 1 )}= \frac{\mu_{\eta}y_{\xi}- \mu_{\xi}y_{\eta}\omega^{i - 1}}{\mu_{\eta}x_{\eta}- \mu_{\xi}x_{\xi}\omega^{i}}\nonumber