harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
M =\begin{pmatrix}1 0 \dots 0 m_{21}1 \dots 0 \vdots \vdots \ddots \vdots m_{n1}m_{n2}\dots 1\end{pmatrix},\begin{cases}m_{ij}= (-1)^{i+j}e_{i-j}(S_i) S_{i}= \{x_1,\dots,x_{i-1}\}.\end{cases}
\sum\limits_{k=0}^{m}a_{nk}x_{k}=\sum\limits_{k=0}^{m}\left( \sum\limits_{j=k}^{m}\frac{1}{r}\left(-\frac{s}{r}\right)^{j-k}\frac{f_{j+1}^{2}}{f_{k}f_{k+1}}a_{nj}\right)y_{k}=\sum\limits_{k=0}^{m}d_{nk}^{(m)}y_{k}=D_{n}^{(m)}(y),
h(B,\nu_{B})=h\left(Y,\nu\right)+\int_{Y}{\rm I}(\mathbb{P}_{\mu,1}^{y},\mathcal{T}_{y})d\nu(y)=h\left(Y,\nu\right)+\int_{Y}\inf_{n\in\mathbb{N}}{\rm I}(\mathbb{P}_{\mu,1}^{y},\mathbb{P}_{\mu,n}^{y})d\nu(y).
\bar{r}_{k-1,k}=r_{k-1,k}- \lfloor r_{k-1,k}/r_{k-1,k-1}\rceil r_{k-1,k-1}, \bar{r}_{k-1,k-1}=\sqrt{\bar{r}^2_{k-1,k}+r_{kk}^2}, |\bar{r}_{kk}| = |r_{k-1,k-1}r_{kk}/\bar{r}_{k-1,k-1}|.
\left<\mathcal W_\Psi f,\mathcal W_\Psi h\right> &=\int_{\mathcal S^n}\!\int_{\mathcal S^n}\!\int_0^\infty\!\!\! \underbrace{\int_{SO(n+1)}\Psi_\rho(g^{-1}x)\,\overline{\Psi_\rho(g^{-1}y)}\,d\nu(g)}_{\Psi_\rho\hat\ast\overline{\Psi_\rho}(x\cdot y)}\alpha(\rho)\,d\rho\,\overline{f(x)}\,d\sigma(x)\,g(y)\,d\sigma(y).
y =((A\circ I)^{[\dagger]})^{[*]}\circ ((A\circ I)^{[*]}\circ A\circ I\circ u)+w &=((A\circ I)\circ (A\circ I)^{[\dagger]})^{[*]}\circ (A\circ I)\circ u+w &= (A\circ I)\circ (A\circ I)^{[\dagger]}\circ (A\circ I)\circ u+w &=(A\circ I)\circ u+w,
\Sigma_{n,i}=\begin{cases}2q^{2i+2}\mu_{n-i-1}(q^{2})+4\frac{\mu_{n-i-2}(q^{2})(q^{n-i-1}-1)}{q^{n-3i-5}(q^{2}-1)}n-i\;, 2q^{2i+2}\left[\mu_{n-i-1}(q^{2})+2\frac{q^{4}\mu_{n-i-3}(q^{2})+q^{2}-1}{q^{2}-1}\right] n-i\;. \end{cases}\;.
\begin{aligned}\chi_{2k}(z) = z^k\varphi_{2k}^{L,\dag}(z^{-1}), x_{2k}(z) = z^{-k}\varphi_{2k}^R(z), & \chi_{2k+1}(z) = z^{-k}\varphi_{2k+1}^R(z), x_{2k+1}(z) = z^k\varphi_{2k+1}^{L,\dag}(z^{-1}). \end{aligned}
\begin{cases}-\operatorname{div}(y^a\varphi(x)\nabla_{x,y}v)=0, \hbox{in}~\mathcal{C}_{\Omega}^{\displaystyle\star}, v=0, \hbox{on}~\partial_{L}\mathcal{C}_{\Omega}^{\displaystyle\star}, -\underset{y\rightarrow0^{+}}{\lim}y^av_y=f^{\displaystyle\star}, \hbox{on}~\Omega^{\displaystyle\star},\end{cases}
I =c_n\int^\pi_0 z^{-n}\frac{\phi_{2n+1}(z)}{z-1}\, x^j(1-\cos\theta)f_3(\theta)d\theta &+c_n\int^\pi_0 z^n\frac{\phi_{2n+1}(1/z)}{1/z-1}\, x^j(1-\cos\theta)f_3(\theta)d\theta,
\lim_{(0,\infty)\ni t\to 0}\frac{\int_E f(x,u t) \rho(dx)}{\int_E f(x, t) \rho(dx)}= \lim_{(0,\infty)\ni t\to 0}\frac{\int_E f(x,t) \rho(dx)}{\int_E f(x, u^{-1}t) \rho(dx)}= \big(( u^{-1})^{\alpha_0}\big)^{-1}= u^{\alpha_0}.
R(z) = \sum_{s=1}^\infty z^s \biggl(1-\sum_{k=0}^s p(k)\biggr) &= \frac{z}{1-z}- \sum_{s=1}^\infty \sum_{k=0}^s p(k) z^k z^{s-k}&= \frac{z}{1-z}- \frac{1}{1-z}P(z) = -1 + \frac{1}{\sqrt{1-z}}.
-(\partial_x^2+\partial_y^2) a +\sqrt{2}\,g^{4}\bar\mu_D\,\left(\tilde{q}q+\frac i2\,\eta +\frac{\mu_{D}}{\sqrt{2}}a\right)+ 2g^2 a\left(\bar{q}q+\bar{\tilde{q}}\tilde{q}\right)=0
V_{l,k}^{m,n}(z)=:\prod_{i=-l/ 2}^{l/2}V_{1,0}^{m_i,0}\left(x^{2i(\xi+1)}z\right)\prod_{j=-k/2}^{k/2}V_{0,1}^{0,n_j}\left(x^{2j\xi}z\right):
\pi_{i}^{( 3 ) a}= - \, \frac{1}{3 m^{2}}\, \eta^{1 b}\, G_{1 i}^{( 2 ) b a}+ \frac{g}{6 m^{2}}\, \bar{\eta}^{b c}\, ( D_{j}\pi^{j})^{b}\, G_{1 i}^{( 2 ) c a}- \frac{1}{3}\, \eta^{2 b}\, G_{2 i}^{( 2 ) b a}\, ,
\begin{array}{l c l}{{\cal C}_{c o n}}&{\equiv}&{\{8 6 4 \psi^{6}+ \phi = \pm 1 \},}\\{{\cal C}_{1}}&{\equiv}&{\{\phi = \pm 1 \},}\\{{\cal C}_{\infty}}&{\equiv}&{\{\phi , \psi = \infty \},}\\{{\cal C}_{0}}&{\equiv}&{\{\psi = 0 \}.}\\ \end{array}
\left(\frac{\pi}{\sqrt{c}}\right)^{-s}\Gamma(s)\,\eta_{p}(2s)=c^{1/4}\left[\frac{1}{1+\frac{1}{\pi p}}\cdot\frac{1}{2s-1}+C_{p}^{(2)}-\frac{2e^{2\pi p}}{1+\frac{1}{\pi p}}Q_{2\pi p}(0)-\frac{1}{2}\left(\gamma+\log\left(\frac{4\pi}{\sqrt{c}}\right)\right)+O\left(s-\frac{1}{2}\right)\right].
i \Pi_{\mu \nu}^{( a b c d )}( p ) = \frac{i e^{2}}{( 4 \pi )^{2}\epsilon}\left( \frac{2 0}{3}- \frac{8 N_{f}}{3}\right) \left( \eta_{\mu \nu}p^{2}- p_{\mu}p_{\nu}\right) + f i n i t e
a= a_1 \mathbf{e}_1 + a_2 \mathbf{e}_2, c=\cos\theta\cos \phi \mathbf{e}_1+ \cos\theta \sin \phi \mathbf{e}_2+\sin\theta \mathbf{e}_3,(a-c)\times b = -(a_2-\cos\theta \sin \phi) \mathbf{e}_3 + \sin\theta \mathbf{e}_2
\Delta_F\left( x,x^{\prime}\right) =<IN\left| T(\Phi (x)\Phi (x^{\prime}))\right| IN>=(-i)\int \frac{d^4k}{\left( 2\pi \right)^4}e^{ik(x-x^{\prime})}\frac 1{k^2+M^2-i\varepsilon}
\int{\partial X \bar \partial X^{\ast}+ \psi \bar \partial \psi^{\ast}+\bar{\psi}\partial\bar{\psi}^{\ast}+ FF^{\ast}+ W'(X)F +\psi\bar{\psi}W''(X) + W'(X^{\ast})F^{\ast}+\psi^{\ast}\bar{\psi}^{\ast}W''(X^{\ast})}
I_2\leq 6\sum_{i=1}^2\big\|u^i\big\|_{L^\infty(L^\infty)}\|g\|_{L^1(L^1)}+ \frac{2}{\delta^2}\|u^2\|_{L^\infty(L^\infty)}\|\sigma^1 -\sigma^2\|_{L^2(L^2)}^2.
e^{\int_0^t\alpha(s)ds}\Bigg[\int_0^t\beta(s)e^{\int_0^s-\alpha(r)dr}ds+C\Bigg] &=e^{\int_0^{t+T}\alpha(s)ds}\Bigg[\int_0^{t+T}\beta(s)e^{\int_0^s-\alpha(r)dr}ds+C\Bigg] =e^{\int_0^{t}\alpha(s)ds}e^{\int_t^{t+T}\alpha(s)ds}\Bigg[\int_0^{t}\beta(s)e^{\int_0^s-\alpha(r)dr}ds &+\int_t^{t+T}\beta(s)e^{\int_0^s-\alpha(r)dr}ds+C \Bigg],
\cos(\phi_k -\psi_k) = \frac{|b_{13}^{(k)}+b_{24}^{(k)}|}{|b_{14}^{(k-1)}-b_{23}^{(k-1)}|}\geq \frac{1.41421\delta_k S(B)}{\sqrt{2}\sqrt{(b_{14}^{(k-1)})^2+(b_{23}^{(k-1)})^2}}& > \frac{1.41421\delta_k S(B)}{\sqrt{2}S(B)}> 0.999997\delta_k.
| S_{1 2}\rangle = \prod_{m = 1}^{\infty}\exp{\left( c_{- m + \frac{1}{2}}^{( 1 )}b_{- m + \frac{1}{2}}^{( 2 )}+ c_{- m + \frac{1}{2}}^{( 2 )}b_{- m + \frac{1}{2}}^{( 1 )}\right)}| 0 \rangle_{1}\otimes | 0 \rangle_{2}\; ,
\sum_{g \in G}{f_2}_g(x){\tau_2}_g =\sum_{g \in G}(f_1\pi_{g^{-1}})(x)\pi_g\tau_1=\sum_{g \in G}(fU^{-1})(\pi_{g^{-1}}x)\pi_g U\tau &=\sum_{g \in G}(f\pi_{g^{-1}})(U^{-1}x)U\pi_g \tau=U\left(\sum_{g \in G}(f\pi_{g^{-1}})(U^{-1}x)\pi_g \tau\right) &=U\left(\sum_{g \in G}f_g(U^{-1}x)\tau_g\right)=UU^{-1}x=x, \forall x\in \mathcal{X}.
\hat{T}(F)=\frac{i}{\hbar}\frac{\delta F}{\delta\phi^A}\frac{\delta}{\delta\bar{\phi}_A}-\frac{1}{2}\varepsilon_{ab}\frac{\delta}{\delta\phi^*_{Aa}}\frac{\delta^2F}{\delta\phi^A \delta\phi^B}\frac{\delta}{\delta\phi^*_{Bb}}\;,
M_{2}(\mathcal{C}(G)) = \dfrac{q(q+1)}{2}(q^{2}-3q+2)\dfrac{(q^{2}-3q+1)^{3}}{2}+ \dfrac{q(q-1)}{2}(q^{2}-q)\dfrac{(q^{2}-q-1)^{3}}{2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + (q+1)(q^{2}-2q+1)\dfrac{(q^{2}-2q)^{3}}{2}= \dfrac{q(q-1)}{2}(q^{8}-6q^{7}+14q^{6}-15q^{5}+3q^{4}+12q^{3}-16q^{2}+9q-1).
\tilde{f}_A (\tau_1, \xi_1 , \eta_1) = \tilde{f}( \tau_1 , \xi_1, A^{\frac{1}{2}}\eta_1), \tilde{g}_A (\tau_2, \xi_2, \eta_2) = \tilde{g}( \tau_2, \xi_2, A^{\frac{1}{2}}\eta_2), \tilde{h}_A (\tau, \xi, \eta) = \tilde{h}( \tau, \xi, A^{\frac{1}{2}}\eta).
\partial_\theta(V^{(\alpha,a)}+H^{(\alpha,a)}\cdot\omega) =\widetilde{\Omega}V^{(\alpha,a)}+\widetilde{\Omega}H^{(\alpha,a)}\cdot\omega &=\widetilde{S}^\alpha\widetilde{\Omega}\Gamma^aV+\widetilde{S}^\alpha\widetilde{\Omega}\Gamma^aH\cdot\omega
p_{1,j}- q_{1,j}= \frac{\varphi(x_1+\xi-x_j) - \varphi(y_1+\xi-y_j)}{N\varrho_N^\varphi(x_1+\xi)}+ \varphi(y_1+\xi-y_j) \frac{\sum_k [ \varphi(y_1+\xi-y_k) - \varphi(x_1+\xi-x_k)]}{N^2\varrho_N^\varphi(x_1+\xi) \tilde\varrho_N^\varphi(y_1+\xi)}\,,
S(I_1,I_2) = \left\lbrace p \in \mathbb{P}: p\nmid 4N_1N_2, \frac{a(tp^2)}{2p^{k_1-\frac{1}{2}}}\in I_1, \frac{b(tp^2)}{2p^{k_2-\frac{1}{2}}}\in I_2 \right\rbrace S(I_1,I_2)(x) = \left\lbrace p\leq x: p\in S(I_1,I_2) \right\rbrace.
W_{2r,0}(Z)= (q-1)(q^{2r-1}-q^{r-1})Z^{q^m-q^{m-1}-q^{m-r-1}(q-1)-1}&+(q^{2r-1}+q^{r-1}(q-1))Z^{q^m-q^{m-1}-q^{m-r-1}(q-1)}&+(q-1)(q^m-q^{2r})Z^{q^m-q^{m-1}-1}+(q^m-q^{2r})Z^{q^m-q^{m-1}}&+(q-1)((q-1)q^{2r-1}+q^{r-1})Z^{q^m-q^{m-1}+q^{m-r-1}-1}&+(q-1)(q^{2r-1}-q^{r-1})Z^{q^m-q^{m-1}+q^{m-r-1}}.
R=\frac{\Gamma (1/2+i(\rho + \frac{(2mV_0)^{1/2}}{\omega}) )\Gamma (1/2+i(\rho - \frac{(2mV_0)^{1/2}}{\omega}) )\Gamma (-i\rho)}{\Gamma (1/2+i \frac{(2mV_0)^{1/2}}{\omega})\Gamma (1/2-i\frac{(2mV_0)^{1/2}}{\omega})\Gamma (i\rho)}
\delta \varepsilon (i |\omega|) \, \delta ({\bf r}_2 -{\bf r}) \,\omega^2 D_{jk}(\varepsilon,{\bf r},{\bf r}^\prime, i \omega) +L_{jm}( \varepsilon,{\bf r}, i \omega )\delta D_{mk}({\bf r},{\bf r}^\prime, i \omega ) = 0 \, .
\begin{aligned}A_p(z, y) = \sum_{i=0}^{k-1}(zy)^{i}(z^{2(k-i)}+ y^{2(k-i)}), D_p(z, y) = \sum_{i=0}^{k-1}(zy)^{i}(z^{k-i}- y^{k-i})^2\end{aligned}
\mathbf{s}_{m}^{a}C^{\alpha_{0}b}= Z_{\alpha_{1}}^{\alpha_{0}}C^{\alpha_{1}a b}+ \epsilon^{a b}B^{\alpha_{0}}- \mathrm{~ \frac{1}{2}~}F_{\beta_{0}\gamma_{0}}^{\alpha_{0}}C^{\beta_{0}a}C^{\gamma_{0}b},
c(m\otimes A)=(A\otimes m)(c\otimes A)(A\otimes c), c(A\otimes m)=(m\otimes A)\left( A\otimes c\right) (c\otimes A), c(u\otimes A)l_{A}^{-1}=\left( A\otimes u\right) r_{A}^{-1},c(A\otimes u)r_{A}^{-1}=\left( u\otimes A\right) l_{A}^{-1}.
|<z^{*}_{j},x_{n_{j}}>| =|<e^{*}_{p_{j}},x_{n_{j}}>-<e^{*}_{q_{j}},x_{n_{j}}>| &\geq |<e^{*}_{p_{j}},x_{n_{j}}>|-|<e^{*}_{q_{j}},x_{n_{j}}>| &>\delta-\frac{\delta}{2}=\frac{\delta}{2}.
\frac{1}{s}=\sum_{k=2}^{\infty}{[\frac{(k-1)arctan(s(k-1))+s^2(k-1)^3arctan(s(k-1))+s(k-1)^2}{(1+s^2(k-1)^2)(s(k-1)arctan(s(k-1))+\alpha)}}
p(x_{1},x_{2}|\mu,\sigma)=\frac{\exp\left\{-\frac{1}{2\sigma^{2}(1-\rho^{2})}\left[ (x_{1}-\mu)^{2}-2\rho(x_{1}-\mu)(x_{2}-\mu)+(x_{2}-\mu)^{2}\right] \right\}}{2\pi\sigma^{2}\sqrt{1-\rho^{2}}}
e^{{\frac{i \hbar}{2}}( \stackrel{\leftarrow}{\partial}_{x}\stackrel{\rightarrow}{\partial}_{p}- \stackrel{\leftarrow}{\partial}_{p}\stackrel{\rightarrow}{\partial}_{x}) + \hbar \stackrel{\leftarrow}{\partial}_{\theta}\stackrel{\rightarrow}{\partial}_{\theta}}~ .
\left<{\cal O}\left[\vec \xi{\,}\right]\right>_{\vec\xi}=\frac{\int\limits_{\vec\xi(0)=\vec\xi(1)}^{}{\cal D}\vec\xi e^{-S}{\cal O}\left[\vec\xi{\,}\right]}{\int\limits_{\vec\xi(0)=\vec\xi(1)}^{}{\cal D}\vec\xi e^{-S}}.
t^{1}= S , \qquad d_{A B C}= \left( \begin{matrix}{{d_{1 j k}= \eta_{j k}}}\\{{0 \; \mathrm{o t h e r w i s e}}}\\ \end{matrix}\right) , \hspace{2 c m}A , B , C = 1 , 2 , \dots , n + 1 \ ,
{\rm Det}^{\star}_2\begin{pmatrix}a+\lambda\star c b+\lambda\star d c d \end{pmatrix}={\rm Det}^{\star}_2\begin{pmatrix}a b c+ \lambda\star a d+\lambda\star b \end{pmatrix}={\rm Det}^{\star}_2 \begin{pmatrix}a b c d \end{pmatrix}
\displaystyle w(x,\tau) = (\psi w)(x,\tau) \leq (\psi w)(x_1,t_1) \displaystyle \leq w(x_1,t_1) \displaystyle \leq \frac{C}{\tau-(t_0-T)}+ (N-1)K + C(\delta) D^{\alpha-1}|\nabla q|+ \alpha D^{\alpha-1}(q^+)
{\cal L}_{fermion}~=~ \bar{\psi}~\left[i\gamma^{\mu}~\left(\partial_{\mu}~-~ \sigma^{\rho \beta}v^{\nu}_{\rho}g_{\lambda\nu}\partial_{\mu}v^{\lambda}_{\beta}~-~ g_{\alpha \delta}\sigma^{a b}v^{\beta}_a v^{\delta}_b \tilde{\Gamma}^{\alpha}\,_{\mu \beta}\right) \right]~\psi
A_{n,1}(x) =(-1)^n\sum_{j=0}^n \binom{n}{j}\frac{\Gamma(\mu+\nu+1+n)}{\Gamma(\mu+\nu+1+j)}\left(\frac{a^2-b^2}{a^2}\right)^jr_{j,\mu}(a^2 x), A_{n,2}(x) =(-1)^n a \sum_{j=0}^n \binom{n}{j}\frac{\Gamma(\mu+\nu+1+n)}{\Gamma(\mu+\nu+1+j)}\left(\frac{a^2-b^2}{a^2}\right)^js_{j,\mu}(a^2 x).
\mathrm{~ \frac{1}{2}~}\epsilon_{a b}\mathbf{s}_{m}^{b}\mathbf{s}_{m}^{a}A^{i}= R_{\alpha_{0}}^{i}B^{\alpha_{0}}+ \mathrm{~ \frac{1}{2}~}\epsilon_{a b}R_{\alpha_{0}, j}^{i}R_{\beta_{0}}^{j}C^{\beta_{0}b}C^{\alpha_{0}a},
G_{f,\tau}\coloneqq [f_j(\tau_k)]_{1\leq j,k \leq n}=\begin{pmatrix}f_1(\tau_1) f_1(\tau_2) \cdots f_1(\tau_n) f_2(\tau_1) f_2(\tau_2) \cdots f_2(\tau_n) \vdots \vdots \vdots f_n(\tau_1) f_n(\tau_2) \cdots f_n(\tau_n) \end{pmatrix}_{n\times n}\in \mathbb{M}_n(\mathbb{K}).
n =\sum_{x \in \mathbb{F}_q^*}\left( \frac{1}{p}\sum_{y \in \mathbb{F}_p}\zeta_p^{y^m_1(x^d)}\right) &=\frac{1}{p}\sum_{y \in \mathbb{F}_p}\sum_{x \in \mathbb{F}_q^*}\chi_m(yx^d)
(w_{k}^{-})^{-1}= (\underbrace{2k,1},\ldots,\underbrace{2k+1-i,i},\ldots,\underbrace{k+1,k}), (w_{k}^{+})^{-1}= (\underbrace{2k+1,1},\ldots,\underbrace{2k+2-j,j},\ldots, \underbrace{k+2,k},k+1)
\frac{1}{2}W_{ijkl,tts}+W_{ijkl,stt}+W_{vjkl, t}R_{vist}+W_{ivkl, t}R_{vjst}+W_{ijvl, t}R_{vkst}+W_{ijkv, t}R_{vlst}+\frac{R}{4}W_{ijkl, s}=0\,.
u_{( 1 )}( y ) = - \frac{1}{5 4 \hat{T}^{1 / 3}}\frac{5 ( \hat{z}_{+}( y )^{2 / 3}+ \hat{z}_{-}( y )^{2 / 3}) + 1 1 \hat{T}^{1 / 3}}{( \hat{z}_{+}( y )^{2 / 3}+ \hat{z}_{-}( y )^{2 / 3}+ \hat{T}^{1 / 3})^{5}}.
\hat\mu_{n,j+R_1}(\theta'_n) =\min\left\{\max\left(0,\frac{\frac{\bar m_{n,j}(\theta'_n)}{\hat\sigma_{n,j}(\theta'_n)}}{\frac{\bar m_{n,j+R_1}(\theta'_n)}{\hat\sigma_{n,j+R_1}(\theta'_n)}+\frac{\bar m_{n,j}(\theta'_n)}{\hat\sigma_{n,j}(\theta'_n)}}\right),1\right\}, \hat\mu_{n,j}(\theta'_n) =1-\hat\mu_{n,j+R_1}(\theta'_n).
(\Gamma^{0})^{\underline\alpha}_{~~\underline\beta}= \left(\begin{array}{cc}0 -1 1 0\end{array}\right)\,, \quad(\Gamma^{1})^{\underline\alpha}_{~~\underline\beta}= \left(\begin{array}{cc}-1 0 0 1\end{array}\right)\, , \quad \Gamma^2=\Gamma^0\Gamma^1=\left(\begin{array}{cc}0 -1 -1 0\end{array}\right) \,.
\lim_{n\rightarrow\infty}\frac1n\log\mathbb{P}\left[ X^n\in \mathcal{T}^{(n)}_{q_X}, Y^n\in \mathcal{T}^{(n)}_{q_Y}\right]=-\min_{r_{XY}:\: r_X=q_X, r_Y=q_Y}D(r_{XY}\| p_{XY}).
{\left|{\hat h_{\max}^{{\rm{rough}}}}\right|^{\rm{2}}}={\sum\limits_{{{\bf{r}}_n}\in{I_1}}^{}{\max \left({{{\left|{r_{n,1}^{}}\right|}^{\rm{2}}},{{\left|{r_{n,2}^{}}\right|}^{\rm{2}}}}\right)}}/{\left|{{I_1}}\right|}-{N_0}
m_X^{N_0,N_1}(x,\xi):= N_1\left(2-m_X\left(x,\frac{\xi}{\|\xi\|_x}\right)-\tilde{m}_X\left(x,\frac{\xi}{\|\xi\|_x}\right)\right)-2N_0\tilde{m}_X\left(x,\frac{\xi}{\|\xi\|_x}\right),
\alpha-2^{-m}\leq \int_{(\delta_{m},\infty)}\log(t)\,d\mu_{|x_{i_{m}}|}(t) &=\int_{(\delta_{n},\infty)}\log(t)\,d\mu_{|x_{i_{m}}|}(t)+\int_{(\delta_{m},\delta_{n}]}\log(t)\,d\mu_{|x_{i_{m}}|}(t) &\leq \int_{(\delta_{n},\infty)}\log(t)\,d\mu_{|x_{i_{m}}|}(t).
\|\tilde{h}^{(m)}\|_{\bar{{\cal H}}([0,a_{m+1}])}^{2}=C_{H}\cdot\int_{0}^{a_{m+1}}\left|t^{H-\frac{1}{2}}\left(t^{1-2H}\dot{\tilde{h}}^{(m)}(t)\right.\right. \ \ \ \left.\left.+\left(H-\frac{1}{2}\right)\int_{0}^{t}\frac{t^{\frac{1}{2}-H}\dot{\tilde{h}}^{(m)}(t)-s^{\frac{1}{2}-H}\dot{\tilde{h}}^{(m)}(s)}{(t-s)^{H+\frac{1}{2}}}ds\right)\right|^{2}dt.
e_{[k,k+m]}\qquad={\tau}^{-m(m-1)/2}(e_{k+m+1}e_{k+m}...e_{k+2})(e_{k+m+2}e_{k+m+1}...e_{k+3}) &\qquad\qquad...(e_{k+2m}e_{k+2m-1}....e_{k+m+1}).
U_{j}=\left[\frac{a-1}{a-b}A^{a-2}-\frac{b-1}{a-b}A^{b-2}\right]p_{j}q^{\beta-2}_{j}&V_{j}=\left[\frac{a-1}{a-b}B^{a-2}-\frac{b-1}{a-b}B^{b-2}\right]q^{\beta-1}_{j}
a = 2\alpha(-Y, s) = \rho, b = 2\beta(-Y, s) + 1 = \rho + 1, c = 2\gamma(-Y, s) + 2 = \begin{cases}2\delta(-Y, s) + 2 \quad 2\delta(-Y, s) = \rho \mod 4 2\delta(-Y, s) \quad 2\delta(-Y, s) = \rho + 2 \mod 4. \ \end{cases}
\frac{V}{N}=-\frac{\sigma}{g_0}+\frac{1}{4\pi\rho^2}\sum_{l=0}^{I}\int_{|k|<\Lambda_1}\frac{dk}{2\pi}(2l+1) \ln(1+\frac{\sigma}{k^2+\frac{l(l+1)}{\rho^2}+\frac{2\xi}{\rho^2}+i\epsilon}).
2\int_{\tilde{Z}_\lambda^\alpha}|\nabla^T\triangle e_\lambda|^2 ds \leq 2\int_{\tilde{Z}_\lambda^\alpha}|\nabla^T\phi_\lambda|\sqrt{1+|\nabla^T\phi_\lambda|^2}ds &\leq\int_{\partial M}\sqrt{1+|\nabla^T\phi_\lambda|^2}|\triangle^T\phi_\lambda|+|\langle\nabla^T \sqrt{1+|\nabla^T\phi_\lambda|^2}, \nabla^T\phi_\lambda\rangle| dV.
\frac{d \ln Z}{d \ln k}= \frac{1}{Z}\frac{d Z}{d \ln k}= \frac{1}Z \frac{d Z}{d \phi}\frac{d \phi}{d \ln k}= \frac{1}Z \frac{d Z}{d \phi}\frac{\dot{\phi}}{H}= \frac{1}Z \frac{d Z}{d \phi}\left( - \frac{M_{P}}{Z}\sqrt{2 \epsilon}\right) = - \sqrt{2 \epsilon}\left( M_{P}\frac{Z^{\prime}}{Z^{2}}\right)
V_0^T(g) = \sum_{k=1}^M V_{t_{k-1}}^{t_k}(g) = \sum_{k=1}^M \big| g(t_k) - g(t_{k-1}) \big| &= \sum_{k=1}^M \big| f(t_k + h) - f(t_k) - f(t_{k-1}+ h) + f(t_{k-1}) \big| \leq \sum_{k=1}^M \Big(\big| f(t_k + h) - f(t_k) \big| + \big| f(t_{k-1}+ h) - f(t_{k-1}) \big| \Big) &\leq 2MC |h|^\mu \leq 2LC |h|^\mu,
K|_{x_h}=9 (N^{'}|_{x_h})^2 (\frac{\sigma^{'}}{\sigma}|_{x_h})^2+6N^{'}|_{x_h}N^{''}|_{x_h}\frac{\sigma^{'}}{\sigma}|_{x_h}+(N^{''}|_{x_h})^2+6\frac{(N^{'}|_{x_h})^2}{x_h^2}+\frac{12}{x_h^4}\ .
1- \int_{-\infty}^{0}\int_{-\infty}^{0}p(y_n^{\rm R},y_n^{\rm I})\mathrm{d}y_n^{\rm R}\mathrm{d}y_n^{\rm I}= 1- \left(\frac{1}{2}-\frac{1}{2}{\rm{erf}}\big(z_n^{\rm R}+\sqrt{P/2}\big)\right)\left(\frac{1}{2}-\frac{1}{2}{\rm{erf}}\big(z_n^{\rm I}+\sqrt{P/2}\big)\right).
1+\sum_{n=1}^{\infty}\frac{P(n)}{(q^{n}-1)(q^{n}-q)\cdot\cdot\cdot(q^{n}-q^{n-1})}z^{n}=\prod_{i=1}^{\infty}\prod_{j=0}^{\infty}\frac{1}{1-q^{1-j}z^{i}}.
-\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\psi\equiv\sum_{\mu=0}^{3}\sum_{\nu=0}^{3}-\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\psi=\frac{1}{c^{2}}\partial_{0}^{2}\psi-\sum_{\nu=1}^{3}\partial_{\nu}\partial_{\nu}\psi=\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}\psi-\nabla^{2}\psi.
X^1_{\tau}- X^2_{\tau}= e^{\tau A}(x_1 - x_2) + e^{\tau A}[v^{(0)}(0,x_1) - v^{(0)}(0, x_2)] - [v^{(0)}(\tau,X_\tau^{1}) - v^{(0)}(\tau,X_\tau^{2})]+{\int_0^\tau e^{(\tau-s)A}[\widetilde Z^{x_1}_s- \widetilde Z^{x_2}_s]\;dW_s,}\;\; \tau \in [0,T_0],
|\langle T(f_1, \ldots, f_n ), f_{n+1}\rangle | \lesssim \left\|\mathrm{M}\left(|f_1|_{X_1},\ldots,|f_n|_{X_n}, |f_{n+1}|_{Y_{n+1}^*}\right)\right\|_{1}, \mathrm{M}(g_1,\ldots, g_{n+1})(x) \coloneqq \sup_{x\in Q}\prod_{j=1}^{n+1}\langle |g_j| \rangle_Q, \langle g \rangle_Q{\coloneqq}\frac{1}{|Q|}\int_Q g.
: a ( \vec{k}\, ) a^{\dagger}( \vec{\ell}\, ) : \ = a^{\dagger}( \vec{\ell}\, ) a ( \vec{k}\, ) \ \ \ , \ \ \ : a^{\dagger}( \vec{k}\, ) a ( \vec{\ell}\, ) : \ = a^{\dagger}( \vec{k}\, ) a ( \vec{\ell}\, ) \ .
\frac{d\sigma(s,t)}{dt}= \frac{1}{16\pi s^2}\{|A_1(s, t)|^2 - i[A_1(s, t) A_2^\ast(s, t) - A_1^\ast(s, t) A_2(s, t)] &+ |A_2(s, t)|^2 - [A_1(s, t) A_3^\ast(s, t) + A_1^\ast(s, t) A_3(s,t] \}\;.
a ( k^{+}) = \alpha ( k^{+}) \cdot \mathrm{{\bf 1}_{N \times N}}+{\tilde{a}}( k^{+}) \mathrm{a n d}b ( k^{+}) = \beta ( k^{+}) \cdot \mathrm{{\bf 1}_{N \times N}}+{\tilde{b}}( k^{+}) ,
\lim_{\gamma_{+}\rightarrow 0^{+}}\mathrm{P}_{\Phi +\Phi^{\mathfrak{b}_{+},\gamma_{+}}+\Phi^{-\mathfrak{b}_{-},\gamma_{-}}}=\mathrm{P}_{(\Phi+\Phi^{-\mathfrak{b}_{-},\gamma_{-}},\mathfrak{a}_{\mathfrak{b}_{+}})}^{\sharp}
\tilde F_{E,\omega}(p,z)e^{-tz}\tilde F_{E,\omega}(p+p',z')e^{-(t+t')z'}-\tilde F_{E,\omega}(p',z')e^{-t'z'}\tilde F_{E,\omega}(p,z+z')e^{-t(z+z')}+\tilde F_{E,\omega}(p+p',z+z')e^{-(t+t')(z+z')}\tilde F_{E,\omega}(p',-z)e^{t'z}=0
\delta(S) = \lim_{x\to\infty}\frac{1}{\log x}\sum_{n\in S, n\le x}\frac{1}{n},\qquad \Delta(S) = \lim_{x\to\infty}\frac{1}{\log\log x}\sum_{n\in S, 1<n\le x}\frac{1}{n\log n},
- \frac{1}{2}\phi_x^2 + \frac{(G(h)\phi+h_x\phi_x)^2}{2(1+h_x^2)}= \frac{1}{2}\big[B^2(1+h_x^2)-(V+Bh_x)^2\big] = \frac{1}{2}\big[B^2-2VBh_x-V^2\big].
g(x) =\int_{\ell}^r\left(s(x)-s(y)\right)^+ g(y)\mu_A(dy), \mbox{such that}g(c)=a; g(x) =\int_{\ell}^r\left(s( y)-s(x)\right)^+g(y)\mu_A(dy), \mbox{such that}g(c)=a
\ll X^{1-\frac{2}{k}}\sum_{\substack{m,n \le X^{\theta}m \ne n}}\frac{|a_m a_n| (\max(m,n))^{1-\frac1k}}{|m - n|}\ll X^{1-\frac{2}{k}+\theta(1-\frac1k)+\varepsilon}\sum_{n \le X^{\theta}}|a_n|^2 \ll X^{1-\frac1k-\varepsilon}.
\frac{d w}{d r}= \frac{4 R}{3 \sqrt{M}}\left[ \frac{1}{1 6 R^{4}}( w^{3}- 3 w + 4 \kappa ) ( 1 - w^{2}) + \frac{1}{2 R^{2}}( 2 \kappa - w^{3}) - w \right] ,
\hat{A}_p = Y_p(Q_p^*Q_p)^{-1}Q_p^* + M_p(M_p^*M_p)^{-1}X_p - M_p(M_p^*M_p)^{-1}M_p^*Y_p(Q_p^*Q_p)^{-1}Q_p^* + \mathsf{P}_{M_p}^{\perp}\hat{A}_p \mathsf{P}_{Q_p}^{\perp}.
\lim_{s\downarrow1}\zeta((\alpha+1)s - \alpha)(s-1) =\lim_{t\downarrow1}\zeta(t)\left(\frac{t+\alpha}{1+\alpha}-1\right) &=\lim_{t\downarrow1}\zeta(t)\frac{t-1}{1+\alpha}&=\frac{1}{1+\alpha}.
\left. \bar{q}_{\mu \nu}^{( 0 )}\right|_{x = Z ( y )}= q_{\mu \nu}^{( 0 )}( y ) + \delta Z^{M}( y ) \left. \partial_{M}\bar{q}_{\mu \nu}^{( 0 )}\right|_{x = Z^{( 0 )}( y )},
\mathbf{d}_{k}(z):=\begin{cases}(0,\ldots,0,-a_{0}^{(k-1)},-a_{1}^{(k-2)},\ldots,-a_{k-2}^{(1)},z-a_{k-1}^{(0)}), 1\leq k\leq p, begin{align*}0.5em](-a_{k-p}^{(p-1)},-a_{k-p+1}^{(p-2)},-a_{k-p+2}^{(p-3)},\ldots,-a_{k-2}^{(1)},z-a_{k-1}^{(0)}), k\geq p+1,\end{cases}
P_{\infty}^{({\pm}N ) ( J )}( g ) = \lim_{l{\rightarrow}{\infty}}P^{({\pm}N ) ( J )}( u ) = \left( \begin{array}{c c}{{\frac{1 +{\gamma}}{2}{\cal P}^{({\pm}N )}( g )}}&{{0}}\\{{0}}&{{\frac{1 -{\gamma}}{2}{\cal P}^{({\pm}N )}( g )}}\\ \end{array}\right) .
\bigg(|(-\Delta_x)^{1/6}u_h - ((-\Delta_x)^{1/6}u_h)_{Q_r (z_0)}|^p\bigg)^{1/p}_{Q_r (z_0)}\leq N \nu^{-1}(|(-\Delta_x)^{1/6}u_h|^p)^{1/p}_{Q_{\nu r}(z_0)}\le N \nu^{-1}(|(-\Delta_x)^{1/6}u|^p)^{1/p}_{Q_{\nu r}(z_0)}+ N \nu^{-1}\sum_{k = 0}^{\infty}2^{-k}F_k (1).
S_{eff}^{2+1}=-2\pi \int d\tau \sqrt{g_{\tau\tau}}\;\;\;,\;\;\; g_{\tau\tau}=(\frac{d\xi^{0}}{d\tau})^{2}-h_{\alpha\beta}(\xi) \frac{d\xi^{\alpha}}{d\tau}\frac{d\xi^{\beta}}{d\tau}\;\;,
\quad+\sum_{n\geq0}\left(\sum_{n-1>i_1>i_2>\cdots>i_{m}\geq1}a(n;i_1i_2\cdots i_m(n-1)n)\right)x^n\qquad &=B_{m+1}(x)+x^{m+3}\sum_{i=2}^mC^i(x)+xB_m(x)+x^{m+2}\frac{1-x}{1-2x},
\left\| \partial_{t}^{\ell}\nabla_{x}^{k}\tilde{K}_{1}(t)g\right\|_{2}+\left\| \partial_{t}^{\ell}\nabla_{x}^{k}\tilde{K}_{2}(t)g\right\|_{2}\le C(1+t)^{-\frac{n}{2}-(\ell+ k)}\| g \|_{1}+ C e^{-ct}\| \nabla_{x}^{\ell+k}g\|_{2},
- \partial_{0}\left( \begin{array}{c}{-{\frac{\kappa}{2 e}}{\frac{\delta A_{+}}{\phi^{*}}}}{\delta \phi}\end{array}\right) = D_{( \infty )}\left( \begin{array}{c}{-{\frac{\kappa}{2 e}}{\frac{\delta A_{+}}{\phi^{*}}}}{\delta \phi}\end{array}\right) .
|I|^{-1}\int_I \left|\log w-(\log w)_I\right|\,dx =2 |I|^{-1}\int_I \left[\log w-(\log w)_I\right]_+\,dx &\le 2\log\left(|I|^{-1}\int_I \exp\left[\log w-(\log w)_I\right]_+\,dx\right) &\le 2\log \left(|I|^{-1}\int_I \exp\left[\log w-(\log w)_I\right]\,dx+1\right) &\le 2\log ([w]_{A_\infty(\mathbb R)}+1),
\sum_{j=0}^a(-1)^j\binom{a}{j}\binom{2j}{b}=(-1)^a\binom{a}{b-a}2^{2a-b}, \sum_{j=0}^a(-1)^j\binom{a}{j}\binom{2j+1}{b}=(-1)^a\left[\binom{a}{b-a}2^{2a-b}+\binom{a}{b-a-1}2^{2a+1-b}\right],
\Big( a_{\chi, \mathfrak{\overline{p}}}= \frac{\varOmega^{j-k}}{\varOmega_p^{j-k}}\left( \frac{\sqrt{d_E}}{2\pi}\right)^j G(\chi) \left(1 - \frac{\chi(\mathfrak{\overline{p}})}{p}\right) L_{\infty, \mathfrak{g}{\mathfrak{p}}}(\chi^{-1}, 0) \Big).
T_k(u_k)(x)=\begin{cases}\overline{u}_k(x) u_k(x)>\overline{u}_k(x), u_k(x) \underline{u}_k(x)\leq u_k(x)\leq \overline{u}_k(x), \underline{u}_k(x) u_k(x)<\underline{u}_k(x).\end{cases}
\begin{cases}\partial_t\tilde{w}+ (u^s + \tilde{u}) \partial_x\tilde{w}+ \tilde{v}\partial_y(u^s_y+\tilde{w})= \partial^2_y\tilde{w}, & \partial_y \tilde{w}|_{y=0}= 0, & \tilde{w}|_{t=0}= \tilde{w}_0,\end{cases}
\begin{aligned}|J_1^h| \leq C \big( 1+ |\nabla v^h|\big), |J_2^h| \leq C \big(1+ |\nabla w^h| + |\nabla v^h|^2 + |\nabla^2 v^h| + |\nabla \vec r^{\,h}|\big), |J_3^h| \leq C h^3 \big(1+ |\nabla w^h| + |\nabla^2 w^h| + |\nabla v^h|^2 + |\nabla^2 v^h|+ |\nabla v^h|\cdot|\nabla^2 v^h|+|\nabla \vec r^{\,h}|\big) + o(h^2).\end{aligned}
K_{N-n}^{\rm qtm}(q^{-x};p^{-1}q^{-N-1},N;q) =\frac{K_x^{\rm Aff}(q^{-n};p,N;q)}{K_x^{\rm Aff}(q^{-N};p,N;q)}&=\frac{K_n^{\rm Aff}(q^{-x};p,N;q)}{K_N^{\rm Aff}(q^{-x};p,N;q)}\,.
{\beta^r}(\xi)=\frac{\sqrt{2\pi}e^{\frac{i\pi}{4}}e^{-\frac{\pi \nu(-k_0)}{2}}}{{q}^r_{1}(-k_0)\Gamma(-i\nu(-k_0))},{\gamma^r}(\xi)=\frac{\sqrt{2\pi}e^{-\frac{i\pi}{4}}e^{-\frac{\pi \nu(-k_0)}{2}}}{{q}^r_{2}(-k_0)\Gamma(i\nu(-k_0))}
\frac{\mathbb{P}(S_{(\eta)}>\mathit{xy})}{\mathbb{P}(S_{(\eta)}>x)}=\frac{\sum_{n=1}^{K}\mathbb{P}(S_{(n)}>\mathit{xy})\mathbb{P}(\eta=n)}{\mathbb{P}(S_{(\eta)}>x)}&+\frac{\sum_{n=K+1}^{\infty}\mathbb{P}(S_{(n)}>\mathit{xy})\mathbb{P}(\eta=n)}{\mathbb{P}(S_{(\eta)}>x)}&=:\mathcal{I}_{1}+\mathcal{I}_{2},
\Lambda_{+}(X) = \sum_{i\in\mathcal{I}}\left( (A_i-X)^T \otimes(\Phi^{+}_{i,T}(X)) \right) + \sum_{i\in\mathcal{I}^-}\left( (A_i-\overline{X}_i)^T\otimes( \Phi^{+}_{i,T}(X)) \right) (\overline{M}^+ -I_4),
F ( \lambda ) = x_{+}^{2 \ell - \mu + \lambda}\int_{0}^{\infty}( \rho + 1 )^{2 \ell - \mu}\rho^{\lambda - 1}d \rho = \frac{\Gamma ( \lambda ) \Gamma ( - 2 \ell + \mu - \lambda )}{\Gamma ( \mu - 2 \ell )}x_{+}^{2 \ell - \mu + \lambda},
C_{\epsilon}^{\textrm{np}}= \gamma\overline{\lambda}_{\epsilon}^{\textrm{np}}\sum_{i=1}^{N}\eta_i\left( 1- q_i^{\textrm{np}}\left( \overline{\lambda}_{\epsilon}^{\textrm{np}}\right)\right) &= \frac{-\gamma\log{\left(1-\epsilon\right)}}{\beta^{\frac{2}{\alpha}}\kappa_{\alpha}r_N^2}\sum_{i=1}^{N}\eta_i(1-\epsilon)^{\frac{r_i^2}{r_N^2}}.
\prod_{k=0}^{n-1}(1-q^k) =(-1)^{n}q^{-\frac{n}{2}}(q)_{n}\sum_{\substack{r,s\geq 0 r+s=n}}\frac{1}{(q)_{r}(q^{-1})_{s}}q^{\frac{r^{2}-s^{2}}{2}}.