harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{h^{(\alpha)}_{k}}{h_{k-1}^{(\alpha+1)}}+\frac{h_{k}^{(\alpha+1)}}{h^{(\alpha)}_{k}}=\frac{h_{k+1}^{(\alpha-1)}}{h_{k}^{(\alpha)}}+\frac{h_{k}^{(\alpha)}}{h_{k}^{(\alpha-1)}}.
\|\left(\Phi(X^*AY)\Phi(Y^*AY)^{-1}\Phi(Y^*AX)\right)^{\frac{p}{2}}\Phi(X^*AX)^{-\frac{p}{2}}\| &\le \frac{(M-m)^{p}(M^2+m^2)^{\frac{p}{2}}}{2^{2+\frac{p}{2}}M^{\frac{3p}{4}}m^{\frac{3p}{4}}(M+m)^{\frac{p}{2}}},~~~p\ge 4.
{\frac{1}{5 \cdot 2^{7 / 3}\alpha ( h )}}\;{\epsilon^{4 / 3}\Phi \left({z / \alpha ( h )}, 5 t \right)}= \int \mathrm{d}l \int \mathrm{d}a \; \; e^{- z l - t a}\tilde{\phi}( l , a ) \ .
f \left( m_0 + \sum_{j \in J'}x_j e_j \right) = \f(m_0) +\sum_{\emptyset \neq J \subseteq J'}\Psi_J^{(m_0)}\prod_{j \in J}x_j, \Psi_J^{(m_0)}:= \ \sum_{K \subseteq J}(-1)^{|J|-|K|}f(m_0 + \sum_{k \in K}e_k).
\langle \tilde{M}_{g^{*}}\, \phi, \psi \rangle_{\mathcal{P}}= \langle \psi, \tilde{M}_{g^{*}}\, \phi \rangle_{\mathcal{P}}^{*}&=\big( \langle g^{*},\phi^{(2)}\rangle_{\mathcal{C}}\, \langle \psi, Q \phi^{(1)}\rangle_{\mathcal{P}}\big)^{*}&=\langle \phi^{(2)}, g^{*}\rangle_{\mathcal{C}}\, \langle Q \phi^{(1)}, \psi \rangle_{\mathcal{P}}
\frac{\overline{x}+\overline{y}}{2}=\frac{\alpha x+\beta y}{2}=\alpha\frac{x+\gamma y}{2}=\alpha(\gamma\frac{x+y}{2}+(1-\gamma)\frac{x}{2})=\alpha\gamma\frac{x+y}{2}+\alpha(1-\gamma)\frac{x}{2}
\bigwedge_{i\in I_1}(f_i(x,y)=0) \wedge \bigwedge_{i\in I_2}(0<f_i(x,y))\wedge \bigwedge_{i\in I_3}(v(f_i(x,y))=1) \wedge \bigwedge_{i\in I_4, j\in I_5}v(f_i(x,y))<v(f_j(x,y))
\prod_{j_1=1}^\infty\left(1+c_{H_1}(\max_{1\leq j\leq n}|s_j-s_{j-1}|^{H_1}|\xi_j|)^2\widetilde{\mu}_{j_1}^4\right)^{-\frac14}\leq c\,\exp(-c_{H_1}\max_{1\leq j\leq n}|s_j-s_{j-1}||\xi_j|^{\frac1{H_1}}).
\sum_{k\geq 1}P \Bigg( \sum_{i=1}^k Y_i > \frac{\gamma k \log k}{\log 1/p}\Bigg) \leq \sum_{k\geq 1}\sum_{i=1}^k P \left( Y_i > \frac{\gamma \log k}{\log 1/p}\right) \leq \sum_{k \geq 1}k A p^{\gamma \log k / \log 1/p}=A \sum_{k\geq 1}k^{1 - \gamma}< \infty, \gamma > 2 .
n_i\cdot q_j + n_j\cdot q_i = n_i\cdot(v_j - n_j) + n_j\cdot (v_i - n_i) = n_i\cdot v_j - n_j \cdot v_i - 2n_i\cdot n_j = a_{ii}^{-1/2}A^{1/2}e_i\cdot a_{jj}^{1/2}A^{-1/2}Re_j + a_{jj}^{-1/2}A^{1/2}e_j\cdot a_{ii}^{1/2}A^{-1/2}Re_i - 2a_{ii}^{-1/2}a_{jj}^{-1/2}A^{1/2}e_i\cdot A^{1/2}e_j.
I'_{X,2}: = \langle\,\alpha_X^*\ (\iota_{X,1}^*(I_{X,1}))\ \rangle = \langle\,_rT_{ij}T_\infty\ -\_rf_{ij};~r=1,\ldots,\mathbf r,~0\le i<j\le n_r+2,\i\le n_r\,\rangle, I'_{Y,2}: = \langle\,\alpha_Y^*\ (\iota_{Y,1}^*(I_{Y,1}))\ \rangle = \langle\,S_{ij}S_{n+1,n+2}\ -\g_{ij};~1\le i<j\le n+2,\i\le n\,\rangle.
R_{S_{0}}( e_{\otimes}^{S_{\mathrm{i n t}}}, \sum_{n \geq 1}\lambda^{n}\sigma \pi ( K_{\nu}^{( n )}) f ) = R_{S_{0}}^{N}( e_{\otimes}^{S_{1}}, \sigma \pi ( K_{\nu}^{( 1 )}) f ) \ ,
\frac{dF_j\left(\frac 1{w_1},\frac 1{w_2},\ldots,\frac 1{w_n}, t\right)}{F_j\left(\frac 1{w_1},\frac 1{w_2},\ldots,\frac 1{w_n},t\right)}=\frac{d{\widetilde F_j}(w,t)}{{\widetilde F_j}(w,t)}-\sum_{k=1}^n m_{jk}\cdot\frac{dw_k}{w_k}.
\tilde{\lambda}_{n,j}= \frac{U_{n}(z_{n,j})}{(1-z_{n,j})V_{n}(z_{n,j})}= \frac{2^{2n-1}d_1 d_2 \cdots d_{n}}{z_{n,j}^{-(n-2)}(z_{n,j}-1)^{-1}\,V_{n}(z_{n,j})}\frac{z_{n,j}}{(z_{n,j}- 1)(1 - z_{n,j})}> 0,
E_{2}^{p , q}\: = \: H^{p}\left( S , \underline{{{\mathrm{E x t}}}}_{{\cal O}_{S}}^{q}({\cal S}_{1},{\cal S}_{2}) \right) \Longrightarrow \mathrm{E x t}_{S}^{p + q}\left({\cal S}_{1},{\cal S}_{2}\right)
\begin{array}{r c l}{{{\hat{A}}_{i}^{( 1 )}}}&{{=}}&{{\left( 1 + e^{2 \phi}k^{- 2}( i_{k}C^{( 1 )})^{2}\right)^{- 1}\left( A_{i}^{( 1 )}- e^{2 \phi}k^{- 2}C_{i}^{( 1 )}( i_{k}C^{( 1 )}) \right) \, ,}}\\{{}}&{{}}&{{}}\\{{{\hat{A}}_{6}^{( 1 )}}}&{{=}}&{{- \left( 1 + e^{2 \phi}k^{- 2}( i_{k}C^{( 1 )})^{2}\right)^{- 1}e^{2 \phi}k^{- 2}( i_{k}C^{( 1 )}) \, ,}}\\ \end{array}
\det_{1\leq j,k\leq m}\left(\sum_{l=0}^NA_{jl}B_{lk}\right)=\sum_{0\leq l_1<\dots<l_m\leq N}\det_{1\leq j,k\leq m}(A_{j,l_k})\det_{1\leq j,k\leq m}(B_{l_k,j})
\sum_{k=1}^m\int_{B_i^{\varepsilon}}\frac{(U_2^{(k)}(|x-a_i|))^2}{|x-a_i|^{2(m-k)}}dx\leq C_{m,\Omega,j}\sum_{i=0}^{m-2}\varepsilon^{-2m-2i}\int_0^{\varepsilon}t^{2i+N-1}\leq C_{m,\Omega,j}.
e_B(w) =\min\{\ell_B(\sigma) + \ell_B(\tau) - \ell_B(w): \sigma, \tau \in W, w = \sigma\tau, \sigma^2 = \tau^2 = 1\}; e_D(w) =\min\{\ell_D(\sigma) + \ell_D(\tau) - \ell_D(w): \sigma, \tau \in \hat W, w = \sigma\tau, \sigma^2 = \tau^2 = 1\}.
\hat{\psi}:=\mathcal{D}_{\omega}^{-1}\{g_1+M_1(\varphi)M_{\varphi}[\hat{\eta}]+M_2(\varphi) g_2+M_3(\varphi) g_3-M_2(\varphi)[\partial_{\psi}\theta_0]^T M_{\varphi}[ g_2 ]\}.
h_j(x,t) = \sum_k \sum_{(q,m)\in\Gamma_j}\big[b^j_{(q,m),k}\,Hg(t - (q-k)T) \hskip-.25in e^{-2\pi i m(q-k)/L}\,\Phi^j_{(q,m)}(t,x-t+(q-k)T)\big]
\left(\le N\right)= \sum_{x=2}^{N}\left(x; \right)=\sum_{x=2}^{N}\mathcal{B}\left(\sum_{j=2}^{\left \lfloor x/2 \right \rfloor}\mathcal{B}\left(\mathbf{d}_j(0,x), \frac{1}{2}\right), \frac{1}{2}\right)
{\lim_{\left| k_{2 \mu}\right| \rightarrow \infty}}T_{\mu \nu}^{( 3 ) a b c d}T_{\mu^{\prime}\nu^{\prime}}^{( 3 ) a^{\prime}b^{\prime}c d *}g^{\mu \mu^{\prime}}Q^{\nu \nu^{\prime}}( k_{2}) = \widetilde{S}
\frac{d}{dt}\sum_{i=1}^{N}\|v_i\|_4^4 + C\sum_{i=1}^{4}\|v_i^2\|_{H^1(\Omega)}^2 \leq C\sum_{i=1}^{N}\|v_i^2\|_{H^1(\Omega)}^2(\|v_i\|_2^2 - 1) + C.
\sigma ({\bf q}) = \left\{N_{0}\left( 1 - e^{- q^{2}\overline{{{u_{t}^{2}}}}}\right) + e^{- q^{2}\overline{{{u_{t}^{2}}}}}\left| \sum_{n}e^{i{\bf q r}_{n 0}}\right|^{2}\right\}\sigma_{0}({\bf q}) ,
\begin{matrix}3\\ 4&4&4\\ 5&5&5&5&5&5\\ 6&6&6&6&6&6&6&6&6&6\\ 7&7&7&7&7&7&7&7&7&7&7&7\\ 8&8&8&8&8&8&8&8&8&8&8&8\\ 9&9&9&9&9&9&9&9&9&9\\ 10&10&10&10&10&10\\ 11&11&11\\ 12\end{matrix}
\mathcal{A}(a, b) = \sum_k \vartheta(k, a)\vartheta(k + 1, b) + \sum_\ell \vartheta(\ell + 1, a)\vartheta(\ell, b) +{}\phantom{{}={}}\vartheta(0, a)\vartheta(0, b) + 2\sum_{i = 1}^\infty \vartheta(i, a)\vartheta(i, b).
x = \left( \begin{array}{c c}{{\bar{u}_{2}}}&{{u_{1}}}\\{{- \bar{u}_{1}}}&{{u_{2}}}\\ \end{array}\right) , \bar{x}= \left( \begin{array}{c c}{{u_{2}}}&{{- u_{1}}}\\{{\bar{u}_{1}}}&{{\bar{u}_{2}}}\\ \end{array}\right)
\sum_{k=s+1}^{rn}a_{k,s}\psi_{{i,0}}\circ{\mathcal{T}}_{\bold{c}}\circ B(\theta_t) \bigcirc_{{{w}}=r+1}^{r-s-1+k}(\theta_t+\gamma_{{w}}) \circ B(\theta_t)^{-1}(t^nH_{\ell}(t)) &=\sum_{k=s+1}^{rn}a_{k,s}[\alpha_i]\circ{\rm{Eval}}_{\alpha_i}\bigcirc_{{{w}}=r+1}^{r-s-1+k}(\theta_t+\gamma_{{{w}}}) (t^nH_{\ell}(t))=0\enspace.
\!\!\!{\cal E}_n^{(1)}\propto\!\!\!\int_{\infty}^{\infty}\Pi_{j=1}^{n}dz_j (\chi_n^{(0)})^{*}(\sum_{i<j}^{n}\delta(z_i-z_j)\partial_{z_i})\chi_n^{(0)}\propto\!\!\!\int_{\infty}^{\infty}\Pi_{j=1}^{n}dz_j (\chi_n^{(0)})^{*}\chi_n^{(0)}(\sum_{i\neq k}^{n}{\rm sign}(z_i-z_k))=0,
u_n(t)-u_n(s)=\int_{s}^{t}\Delta u_n(r)dr+\int_{s}^{t}P_n\Big[u_n(r)\log\big(u_n(r)\big)\Big]dr+\int_{s}^{t}P_n\Big[\sigma\big(u_n(r)\big)\Big]h(r)dr.
b^-= \big( \prod_{j=n+1}^{K}F_{i_{j}}H_{\omega^{\vee}_{i_{j}}}(X_{j}^{-1}) \big) F_n\dots F_3 F_2 F_1 =F_{i_{n+1}}H_{\omega^{\vee}_{i_{n+1}}}(X_{n+1}^{-1}) F_{i_{n+2}}H_{\omega^{\vee}_{i_{n+2}}}(X_{n+2}^{-1}) \dots F_{i_{K-1}}H_{\omega^{\vee}_{i_{K-1}}}(X_{K-1}^{-1})F_{i_K}H_{\omega^{\vee}_{i_K}}(X_{K}^{-1}) F_n\dots F_3 F_2 F_1.
\prod_{i = 1}^{m_{a}^{\prime}}\psi_{2}^{( a )}({y^{\prime}}_{i}^{( a )}) \prod_{i = 1}^{n_{a}}\overline{{{\psi}}}_{1}^{( a )}( x_{i}^{( a )}\! + \hat{\tau}) \prod_{i = 1}^{m_{a}}\psi_{2}^{( a )}( y_{i}^{( a )}\! + \hat{\tau}) \prod_{i = 1}^{n_{a}^{\prime}}\overline{{{\psi}}}_{1}^{( a )}({x^{\prime}}_{i}^{( a )}) \Big \rangle \; .
\gamma_k^\mathrm{B}[\iota] = \frac{M\sigma_{\mathrm{s}k}^2[\iota]}{\sum_{i=1}^{K}\beta_{\mathrm{s}i}+\left(\rho_\mathrm{p}\sum_{i=1}^{K}\beta_{\mathrm{d}i}+1\right)/\rho_\mathrm{s}}
\int\prod_{j=1}^gd\nu_jd\bar\nu_jd\hat\nu_jd\hat{\bar \nu}_j\,\,{\rm e}^{\int_{\Sigma_g}{\cal L}_5}=\det\left({1\over \Omega-\bar\Omega}\bar\partial\bar\Omega{1\over \Omega-\bar \Omega}\partial\Omega\right),
p(t,x) =\mathbb{E}[p(T,x)-\frac{1}{2}\int_t^T \frac{\partial p}{\partial s}(s,x)ds|\mathcal{F}^{v,w}_t] &=\mathbb{E}[p(T,x)-\frac{1}{2}[p(s,x)]_t^T|\mathcal{F}^{v,w}_t] &=\mathbb{E}[p(T,x)-\frac{1}{2}p(T,x)+\frac{1}{2}p(t,x)|\mathcal{F}^{v,w}_t].
2\left(D_x^\dagger D_x\right)^{-1}(z,z')=(\sigma_0+i\sigma_3)e^{-\phi(z)}K_+(z,z';x)e^{-\phi(z')}+(\sigma_0-i\sigma_3)e^{\phi(z)}K_-(z,z';x)e^{\phi(z')},
\delta :=e^{-\frac{K}{5.4}}+e^{-\frac{\eta n}{1.8}}+e^{-\frac{K}{C_{\alpha_1}}}+e^{-\frac{K}{C_{\alpha_2}}}+e^{-\frac{K}{C_{\alpha_3}}}+e^{-\frac{\rho n}{1.8}}+e^{-\frac{K}{C_{\alpha_4}}}+e^{-\frac{\bar\rho n}{1.8}}+e^{-\frac{K}{C_{\bar\alpha_4}}}&+T_1T_2e^{-\frac{S_2}{669}}+T_1e^{-\frac{S_1}{353}}.
S = S_{a_{f}}+ S_{\Phi}= \int L_{a_{f}}( \zeta_{f}) \, d \zeta^{( 1 , 1 , 1 )}\wedge \cdots \wedge d \zeta^{( 2 , 2 , 3 )}+ \int \sqrt{g}L_{\Phi}( \zeta ) \, d \zeta^{( 1 1 1 )}\wedge \cdots \wedge d \zeta^{( 2 2 3 )},
\begin{aligned}\|V(\cdot,0,\cdot)\|_{\widetilde H^{(k+1)/3,k+1}(B_\delta)}&\leq c(T,k,b,L) \left(\|J(\cdot,R,\cdot;\mu)\|_{\widetilde H^{(k+1)/3,k+1}(B_\delta)}+\|\partial_x J(\cdot,R,\cdot;\mu)\|_{\widetilde H^{k/3,k}(B_\delta)}\right).\end{aligned}
u_n \Phi(x) u_n^{-1}- \Phi (x) = \Phi\left(v_n x v_n^{-1}-x\right)= -2\Phi(y_n) +\sum\limits_{k=1}^{n-1}\left(u_n \Phi(y_k) u_n^{-1}-\Phi(y_k)\right).
Q/Q(2) \downarrow L' = (1,1,0,01) + (0,1,1,00), Q(2)/Q(3) \downarrow L' = (1,0,1,01) + (0,0,0,10), Q(3)/Q(4) \downarrow L' = (0,1,1,10), Q(4)/Q(5) \downarrow L' = (0,0,0,10) + (1,0,1,00), Q(5)/Q(6) \downarrow L' = (1,1,0,00), Q(6) \downarrow L' = (0,0,0,01).
a^{^{\dag}}( \vec{p}) = \int{\frac{d^{\nu}x}{( 2 \pi )^{\frac{\nu}{2}}}}{\frac{e^{i \vec{p}\cdot \vec{x}}}{\sqrt{2 g ( \vec{p})}}}\left( \int d^{\nu}y{\frac{1}{2}}G ( x , y ) \eta ( y ) -{\frac{\delta}{\delta \eta ( \vec{x})}}\right) .
S_{\mathrm{o d d}}( a ; b_{1}, \ldots , b_{k - 1}) = \sum_{n = 1}^{\infty}\frac{1}{( 2 n - 1 )^{a}}\prod_{i = 1}^{k - 1}\sum_{m_{i}= 1}^{n - 1}\frac{1}{( 2 m_{i}- 1 )^{b_{i}}},
{\cal T}={f_\pi^2\over 2}{\cal R}_0\cdot{\cal R}_0 -{1\over{2 e^2}}\biggl [{\cal R}_0\cdot{\cal R}_{\mathrm{i}}\;{\cal R}_0\cdot{\cal R}_{\mathrm{i}}-{\cal R}_0\cdot{\cal R}_0\;{\cal R}_{\mathrm{i}}\cdot{\cal R}_{\mathrm{i}}\biggr ]
{}\mathcal H(\tilde{x},\tilde{x}^{(0)},\tilde{\lambda})=\psi(\tilde{x})- \tilde{\lambda}\psi(\tilde{x}^{(0)})=\left[\begin{array}{c}\psi_1(\tilde{x})-{\lambda_1}\psi_1(\tilde{x}^{(0)}) \psi_2(\tilde{x})-{\lambda_2}\psi_2(\tilde{x}^{(0)}) \vdots \psi_n(\tilde{x})-{\lambda_n}\psi_n(\tilde{x}^{(0)})\end{array}\right]
\frac{d^{2}x}{dt^{2}}=\frac{\epsilon}{\mu}\frac{1}{\beta^{3}}X,\frac{d^{2}y}{dt^{2}}=\frac{\epsilon}{\mu}\frac{1}{\beta}(Y-\frac{v}{V}N),\frac{d^{2}z}{dt^{2}}=\frac{\epsilon}{\mu}\frac{1}{\beta}(Z+\frac{v}{V}M)
\sum_{a=1}^n\langle\omega^a\wedge (\nabla_{V_a}H^{1,0})\lrcorner\,\phi,\phi\rangle =\sum_{a,b=1}^n g_Q(\nabla_{V_a}H^{1,0},\bar V_b)\langle V_b\lrcorner\,\phi,V_a\lrcorner\,\phi\rangle &=\sum_{a=1}^n g_Q(\nabla_{V_a}H^{1,0},\bar V_a) |V_a\lrcorner\,\phi|^2.
\sum_{m}( X_{m_{1}}^{J_{1}})_{l m}( X_{m_{2}}^{J_{2}})_{m n}= \sum_{J , M}\left( \begin{array}{c c c}{{J_{1}}}&{{J_{2}}}&{{J}}\\{{m_{1}}}&{{m_{2}}}&{{M}}\\ \end{array}\right) \left\{\begin{array}{c c c}{{J_{1}}}&{{J_{2}}}&{{J}}\\{{S}}&{{S}}&{{S}}\\ \end{array}\right\}( X_{M}^{J})_{l n},
E_1 = \sum_{j=1}^{\infty}\lambda_{j}^{(2)}a_j^{(2)}= \sum_{j=1}^{\infty}\lambda_{j}^{(2)}\, \Pi_l(G_j^{(2)},\tilde G_j^{(2)})+\sum_{j=1}^{\infty}\lambda_{j}^{(2)}\left(a_j^{(2)}-\Pi_l(G_j^{(2)},\tilde G_j^{(2)})\right) &:= M_2+E_2.
\sum_{|k_{24}|\sim 2^q}\left(\sum_{\substack{k_2: |k_2|\leq|k_4| k_2+k_4=k_{24}}}+ \sum_{\substack{k_2: |k_4|\leq|k_2| k_2+k_4=k_{24}}}\right)\frac{1}{|k_2|^4 |k_4|^2}&\lesssim \sum_{|k_{24}|\sim 2^q}\frac{1}{|k_{24}|^2}\sum_{k_2}\frac{1}{|k_2|^4}+\sum_{|k_{24}|\sim 2^q}\frac{1}{|k_{24}|^{3-\delta}}\sum_{k_2}\frac{1}{|k_2|^{3+\delta}}\lesssim2^{\frac{q}{2}}
\begin{aligned}\qquad\nabla^2 V=0 s.t. V(0,t) =0 V(1/2,t) =\int_{0}^{\infty}\left[\frac{1}{x}-1\right]\frac{\sin\left(t\log x\right)}{\left(e^{x}+1\right)}dx V(\theta,0) =0 V(\theta,M) =0\end{aligned}
\frac{h_{11}-h_{11t}}{h-h_{t}}-\frac{(h_{1}-h_{1t})^{2}}{(h-h_{t})^{2}}&=-\sum_{i}b^{ii}\nabla_{11}b_{ii}+\sum_{i,k}b^{ii}b^{kk}(\nabla_{1}b_{ik})^{2}+\nabla_{11}\psi.
0= -\div_x \bigg(\int_{\mathcal{Y}^d}\widetilde{m}_0 (x )\widetilde{m}_1 (x,y ) (P+\nabla\widetilde{u}_0 (x )+\nabla_y\widetilde{u}_1 (x,y ) )dy\bigg ) = -\div_x \bigg(\int_{\mathcal{Y}^d}\widetilde{m}(x,y ) (P+\nabla\widetilde{u}_0 (x )+\nabla_y\widetilde{u}_1 (x,y ) )dy\bigg ).
\epsilon_{\mathrm{C a s}}( x ) = \frac{1}{2}\omega_{B}\phi_{B}^{2}( x ) - m \int_{0}^{\Lambda}\frac{d k}{2 \pi}\left( \frac{\omega}{k^{2}+ \frac{1}{4}m^{2}}- \frac{1}{\omega}\right) \phi_{B}^{2}( x ) - \frac{m}{2 \pi}\left( \phi_{B}^{2}( x ) + 2 \phi_{0}^{2}( x ) \right) \ \ .
\frac{d}{ds}\int\mathbb{E}[U|Z_{st}=z]f(z)\mu_{st}(z)dz =\frac{d}{ds}\mathbb{E}[Uf(Z_{st})] &=\mathbb{E}[Uf'(Z_{st})U] &=\int \mathbb{E}[U^2|Z_{st}=z]f'(z)\mu_{st}(z)dz
J_{a,b}(T) = \int_{T}^{\infty}(\log y)^a y^{-b}dy = \int_{(\log T)(b-1)}^{\infty}t^a (b-1)^{-a}e^{-t}\frac{dt}{b-1}=\frac{1}{(b-1)^{a+1}}\Gamma(a+1,(b-1)\log T),
S(a,b,\alpha,\beta)=\left\{\begin{array}{c|c}\!\!\! n\in{\mathbb N}\! \begin{array}{c}\!\!\! a_1x_1 +\cdots+ a_px_p + \alpha \leq n \leq b_1x_1 +\cdots+ b_px_p - \beta begin{align*}3pt] \mbox{for some}(x_1,\ldots x_p) \in{\mathbb N}^p\end{array}\end{array}\!\!\!\!\! \right\}.
{\mathbb E}\left\{R_{\rm PL}\right\}\le \sum\limits_{k = 1}^{{N_{\rm{R}}}}{{{\log}_2}\left({1 + \frac{{\mathbb E}{\left\{{{{\left|{\xi_k^\star}\right|}^2}}{p_k^\star}\right\}}}{\sigma^2}}\right)}\buildrel \Delta \over ={R_{{\rm{PL,upper}}}}.
L=P_\mu\Pi^\mu_\tau +Z_{\alpha\beta}\Pi^{\alpha\beta}_\tau +\bar Z_{\dot\alpha\dot\beta}\bar\Pi^{\dot\alpha\dot\beta}_\tau -\lambda_v h_v - \bar\lambda_v \bar h_v -\lambda_u h_u - \bar\lambda_u \bar h_u \, .
D_{T_{x}}\approx[0.3\pm0.1][\frac{\langle u^{\prime}u^{\prime}\rangle+\langle u\rangle^{2}}{\langle u^{\prime}u^{\prime}\rangle}]\int_{\tau}^{\infty}u^{\prime}(t_{0})u^{\prime}(t_{0}+\tau)d\tau.
{\cal O}\equiv \int d^{4}x \, \mathrm{t r}\, \phi^{2}, \qquad \bar{Q}_{\mu}{\cal O}, \qquad \bar{Q}_{\mu}\bar{Q}_{\nu}{\cal O}, \qquad \bar{Q}_{\mu}\bar{Q}_{\nu}\bar{Q}_{\rho}{\cal O}, \qquad \bar{Q}_{\mu}\bar{Q}_{\nu}\bar{Q}_{\rho}\bar{Q}_{\sigma}{\cal O}.
\frac{\partial M_{pq}}{\partial x_{q}}=\mathcal{T}_{p}+\frac{\partial\mathcal{L}_{p}}{\partial t},\qquad\overrightarrow{\mathcal{L}}=\mathbf{r\times}\overrightarrow{\mathcal{G}},\quad\overrightarrow{\mathcal{T}}=\mathbf{r\times}\overrightarrow{\mathcal{F}},
U^{n}(t)\leq I_1(t)=\frac{1-\sqrt{1-4C\max(t^{\frac{1}{2}-\frac{1}{q}},t^{\frac{1}{p}})U_{0}}}{2\max(t^{\frac{1}{2}-\frac{1}{q}},t^{\frac{1}{p}})}\leq2CU_{0},
y_1=x_1, \, y_2=\frac{1}{\sqrt{\nu}}x_2, \, y_3=\frac{1}{\sqrt{\mu}}x_3; &[y_1,y_2]=0, \,[y_2,y_3]=-\frac{1}{\sqrt{\mu\nu}}y_1 - \frac{1}{\sqrt{\mu}}y_2, \,[y_3,y_1]=\frac{1}{\sqrt{\mu}}y_1.
t P_n^{( q)}(t) = P_{n+1}^{( q)}(t) +[n]_q(1 + q^{n-1})P_{n-1}^{( q)}(t), n=2,3,4,\dots, &P_0^{( q)}(t) =1, P_1^{( q)}(t) =t, P_{2}^{( q)}(t) =t^2-1,
\frac{\partial\xi}{\partial\zeta}=\frac{1}{\pi}\begin{pmatrix}\frac12 1 \cdots 1 1 \frac12 \cdots 1 \cdots \cdots \cdots \cdots 1 1 \cdots \frac12\end{pmatrix}\mbox{diag}(j_b:1\leq b\leq n)
\Sigma ( Z_{1}, Z_{2}, T ) = \sigma_{c}+{\frac{\left( e^{2 h}- 1 \right)^{2}\left( 3 + ( 2 + \sqrt{7}) e^{2 h}\right)^{2}}{2 0 \cdot 2^{1 / 3}e^{4 h}\left( 1 + e^{2 h}\right)^{2}}}\left[{\frac{\Phi ( Z_{1}, T ) - \Phi ( Z_{2}, T )}{Z_{1}- Z_{2}}}\right] \epsilon^{1 / 3}+{\cal O}( \epsilon^{2 / 3})
{\partial_t}= \sum\limits_{n = 1}^{+ \infty}{{\varepsilon^n}\partial{t_n}}, \; \;\;\nabla = \varepsilon{\nabla_1}, \; \;\;{g_i}= \sum\limits_{n = 1}^{+ \infty}{{\varepsilon^n}g_i^{\left( n \right)}}, \; \;\; \bar F = \varepsilon{\bar F^{\left( 1 \right)}},
\int_{0}^{\infty}d x x^{2}\log \left[ 1 - e^{- \left( x^{2}+ m_{e f f}^{2}\beta^{2}\right)^{1 / 2}}\right] = - \left( m_{e f f}\beta \right)^{2}\sum_{n = 1}^{\infty}\frac{1}{n^{2}}K_{2}\left( m_{e f f}\beta n \right) .
\left \|\sum\limits_{j=1}^{\infty}\mu_{j}b_{j}\right \|_{h_{\omega}^{p}}\leq C\left \|\sum\limits_{j=1}^{\infty}\mu_{j}g(b_{j})\right \|_{L_{\omega}^{p}}\leq C\left \|\sum\limits_{j=1}^{\infty}\frac{\mu_{j}\chi_{P_{j}}}{\omega(P_{j})^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}.
p_\beta(y \mid x)= \prod_{i=1}^d \left\{\frac{\Gamma(x_i + y_i + \beta_i)}{\Gamma(x_i + \beta_i) y_i !}\frac{r_i^{x_i+\beta_i}s_i^{y_i}}{(r_i + s_i)^{x_i + y_i + \beta_i}}\right\},
\left[ T_{(A,i+n)},T_{(B,j+n)}\right] =K_{i+n,j+n}^{\gamma}C_{AB}{}^{C}T_{(C,\gamma)}, & =K_{i+n,j+n}^{k}C_{AB}{}^{C}T_{(C,k)}+K_{i+n,j+n}^{k+n}C_{AB}{}^{C}T_{(C,k+n)}, & =-K_{i+n,j+n}^{k}C_{AB}{}^{C}T_{(C,k+n)}+K_{i+n,j+n}^{k+n}C_{AB}{}^{C}T_{(C,k+n)}, & =-\left( K_{ij}^{k}-K_{ij}^{k+n}\right) C_{AB}{}^{C}T_{(C,k+n)}.
\Delta_{2}(n,m,n',m') &\qquad=\begin{cases}1, \mbox{if either $m'=0$ or $2^{m'}\langle n'\rangle^{2}\le 2^{m+4}\langle n\rangle^{2}$;}\max\{2^{m}\langle n\rangle^{2},2^{m'}\langle n'\rangle^{2}\}, \mbox{otherwise}.\end{cases}
\lim_{m \to \infty}\frac{1}{m}\sum_{j = -m/2+1}^{m/2}\frac{e^{2 \pi i \frac{j + \theta_1/2}{m}\lfloor sm \rfloor}}{1 - \exp \left\{- 2 \pi i \frac{j + \theta_1/2}{m}\right\}+ \frac{z}{m}}= e^{\pi i \theta_1 s}F(z + \pi i \theta_1 , s)
\beta^{\ss_{1i}}=\sum_{j: j\ne i}^n\sigma_{\xi_{12}}(x_{ij}) \sigma_{\eta+\nu}(x_{ij}^+)\prod_{l\ne i, j}\sigma_\mu(x_{jl})\sigma_\mu(x_{li})\,, \beta^{\ss_{1i}^+}=\sum_{j: j\ne i}^n \sigma_{\xi_{12}}(-x_{ij}^+) \sigma_{\eta+\nu}(x_{ji})\prod_{l\ne i, j}\sigma_\mu(x_{jl})\sigma_\mu(x_{li}^+)\,.
E_M(n,t) \leq |p_M(t,\hat x_n)-G_M(t,\hat x_n)|+|G_M(t,\hat x_n)-G_M(t,\check x_n)|+|G_M(t,\check x_n)- p_M(t,\check x_n)| &=:A_1(n)+A_2(n)+A_3(n)=A_1+A_2+A_3,
\dfrac{(f^{\tilde{\phi}^{-n}}\circ \rho_{\eta}(X))'}{f^{\tilde{\phi}^{-n}}\circ \rho_{\eta}(X)}= \sum_{w \in W_{\rho}^{m_0}}\dfrac{(f^{\tilde{\phi}^{-(n+1)}}(X+w))'}{f^{\tilde{\phi}^{-(n+1)}}(X+w)}= \sum_{w \in W_{\rho}^{m_0}}\dfrac{(f^{\tilde{\phi}^{-(n+1)}})'(X+w)}{f^{\tilde{\phi}^{-(n+1)}}(X+w)}.
\begin{aligned}\forall l\in I^{0+}(\bar x)\cap I^{\varphi^t_\textup{FB}}(x)\colon\quad \alpha_l\neq 0\quad \beta_l\approx 0 &\forall l\in I^{+0}(\bar x)\cap I^{\varphi^t_\textup{FB}}(x)\colon\quad \alpha_l\approx 0\quad \beta_l\neq 0\end{aligned}
\tilde{h}^{\mu \nu}= - h^{\mu \nu}+ \frac 1 2 h_{\; \; \lambda}^{\lambda}\, \eta^{\mu \nu}- \frac 1 2 h_{\; \; \lambda}^{\lambda}\, h^{\mu \nu}+ h^{\mu \lambda}\, h_{\; \; \lambda}^{\nu}+ \frac 1 8 \left( h_{\; \; \lambda}^{\lambda}\right)^{2}\, \eta^{\mu \nu}- \frac 1 4 h_{\alpha \beta}h^{\alpha \beta}\, \eta^{\mu \nu}+ \cdots \; \; .
\stackrel{\mathrm{G}}{\tau}_{\alpha \beta}{}^{\! i}= \frac{1}{4 m i}\left[ \left( D^{i}\overline{{\Psi}}\right) \sigma_{\alpha \beta}\Psi - \overline{{\Psi}}\sigma_{\alpha \beta}\left( D^{i}\Psi \right) \right] ,
\sum\limits_{r=-\infty}^{k}PZ(0,t_{r})\int_{t_{r}}^{\zeta_{r}}\Phi(t_{r},s)\,ds, \sum\limits_{r=-\infty}^{k}PZ(0,t_{r+1})\int_{\zeta_{r}}^{t_{r+1}}\Phi(t_{r+1},s)\,ds,
\sum_{X\in L(M)}\frac{g(X)}{||X||_F^t}< K_1 t>s, & \sum_{X\in L(M)}\frac{g(X)}{||X||_F^t}< K_2 \log(M) t=s, & \sum_{X\in L(M)}\frac{g(X)}{||X||_F^t}< K_3 M^{s-t}t<s,
I_{{B_3\{1,2,3\}}}= \{\theta_y^1,\, \theta_y^2,\, \theta^1_z, \, \theta^2_z,\, \theta^3_z,\, \theta^4_z\,\}&= \{dy^1 - p^1 dx,\, dy^2 - p^2 dx, dz^1 - p^1p^2 dx,\, dz^2 - \frac12(p^2)^2 dx,\, dz^3 - \frac12 p^1(p^2)^2 dx,\, & dz^4- \frac12p^2(xp^1p^2 - y^1p^2 - 2y^2 p^1)\, dx \}
q^{n,[a,b]}(t)=\begin{cases}\int_a^b \left[\prod_{\substack{i\in\{1,\ldots,n\}, c_i^n\neq \frac{2t-(a+b)}{b-a}}}\tfrac{2x-(b-a)c_i^n-(a+b)}{2t-(b-a)c_i^n-(a+b)}\right]\,dx \colon (a<b) \big(\frac{2t-(a+b)}{b-a}\in\{c_1^n,\ldots,c_n^n\}\big) 0 \colon \end{cases}
\epsilon_i a_i <\lambda_i < \epsilon_{i-1}a_{i-1}~~ i=2, \cdots m, & \min_{i}\{\epsilon_i a_i \}+ \sum_{i=1}^m a_i \leq \lambda_1 \leq \max_{i}\{\epsilon_ia_i \}+ \sum_{i=1}^m a_i.
\frac{1-\tau^k}{(\tau^k)^2}\|\breve{x}^{k}-\breve{x}^{k-1}\|^2+\frac{1}{\tau^k}\|\breve{x}^{k}-x\|^2=\|\widetilde{x}^{k}-x\|^2+\frac{1-\tau^k}{\tau^k}\|\breve{x}^{k-1}-x\|^2,~ &2(\widetilde{x}^k-x)^T\frac{1}{\tau^k}(\breve{x}^{k}-\breve{x}^{k-1})=\|\widetilde{x}^k-x\|^2+\frac{1}{(\tau^k)^2}\|\breve{x}^{k}-\breve{x}^{k-1}\|^2-\|\breve{x}^{k-1}-x\|^2.
\widetilde{\mathbf{x}}(t)\leq \exp\Bigg(\int_{t_0}^t\widetilde{A}_\nu(\tau)d\tau\Bigg)\widetilde{\mathbf{x}}(t_0)+\int_{t_0}^t\exp\Bigg(\int_u^t\widetilde{A}_\nu(\tau)d\tau\Bigg)\widetilde{\mathbf{f}}(u)du,t\geq t_0.
(\rho_{i_1}\cdots \rho_{i_p})\rho_0 (\rho_{i_1}\cdots \rho_{i_p})^{-1}\cdot (\rho_{j_1}\cdots \rho_{j_q})\rho_0 (\rho_{j_1}\cdots \rho_{j_q})^{-1}= (\rho_{j_1}\cdots \rho_{j_q})\rho_0 (\rho_{j_1}\cdots \rho_{j_q})^{-1}\cdot (\rho_{i_1}\cdots \rho_{i_p})\rho_0 (\rho_{i_1}\cdots \rho_{i_p})^{-1},
\sum\limits_{i=0}^{2}\varepsilon^{\frac{i}{2}}\left\vert \lambda\right\vert^{1-\frac{i}{2}}\left\Vert u^{\left( i\right)}\right\Vert_{X}+\left\Vert Au\right\Vert_{X}\leq C\sum\limits_{i=0}^{2}\varepsilon^{\frac{i}{2}}\left\vert \lambda \right\vert^{1-\frac{i}{2}}
\frac{\ell+1}{\sin^8 \phi}\ell^3 \sum_{v=0}^4{_4}\omega_{3,v}(\cos \phi) p_{2,\ell+v}(\phi) = \frac{\ell^4}{\sin^8 \phi}\Big[ \sin^4 \phi f_b(\phi) s_{2,0}(\ell,\phi) \Big] + \phi^{-8-1/2}O(\ell^{-1/2})+\phi^{-5}O(\ell),
AB=[\begin{matrix}x_{1}&x_{1}a_{12}&x_{1}a_{13}&\cdot\cdot\cdot&x_{1}a_{1n}\\ 0&x_{2}&x_{2}a_{23}&\cdot\cdot\cdot&x_{2}a_{2n}\\ \vdots&\vdots&\vdots&&\vdots\\ 0&0&0&\cdot\cdot\cdot&x_{n}\end{matrix}].
\mathcal{I}^{\theta^0}= \{I \mid \lvert I \rvert > n \}, \mathcal{I}^{\theta^+}= \mathcal{I}^{\theta^0}\sqcup \{I \mid \lvert I \rvert = n 1 \in I \}, \mathcal{I}^{\theta^-}= \mathcal{I}^{\theta^0}\sqcup \{I \mid \lvert I \rvert = n 1 \notin I \}.
D_{2N}(\lambda) =\sum_{k=0}^N a_k^{(N)}(-\lambda) (\Delta^\prime)^k \iota^* \partial_n^{2N-2k}, D_{2N+1}(\lambda) =\sum_{k=0}^N b_k^{(N)}(-\lambda) (\Delta^\prime)^k \iota^* \partial_n^{2N-2k+1}
4\pi T=\int d^2x\left[ iJ^2 \epsilon_{ij}\partial_i\left(\frac{\psi^*\partial_j\psi-\psi\partial_j\psi^*}{1-\vert\psi\vert^2}\right)-J\epsilon_{ij}\partial_i\left((Q^3-v^{''})A_j\right)\right],
C_Q=-2\pi r_+^2\left[1-\frac{r_-}{r_+}\right]^{2q^2/(2q^2+3)}\left[1-\frac{3}{2q^2+3}\frac{r_-}{r_+}\right] \left[1-\frac{2q^2+9}{2q^2+3}\frac{r_-}{r_+}\right]^{-1}.
\arraycolsep=2ptX^{i_1,\ldots,i_r}_{j_1,\ldots,j_s}=\left[\,\begin{matrix}x_{i_1j_1}\ldots x_{i_1j_s}begin{align*}4pt]\vdots \ddots \vdots x_{i_rj_1}\ldots x_{i_rj_s}\end{matrix}\,\right]=\big(x_{i_aj_b}\big)_{1\le a\le r,1\le b\le s}
\eqref{va3}= \psi(0)\sqrt{\tfrac{it}{2\pi}}\int_- e^{-\frac{it}{2}(x-\xi)^2}\,d\xi & \quad+ [\psi(x)-\psi(0)]\sqrt{\tfrac{it}{2\pi}}\int_- e^{-\frac{it}{2}(x-\xi)^2}\,d\xi & + \sqrt{\tfrac{it}{2\pi}}\int_- e^{-\frac{it}{2}(x-\xi)^2}[\psi(\xi)-\psi(x)]\,d\xi.
\begin{aligned}\partial_i\partial_j h_1\!=\!-r^{-n}\,E_+(r)\!\left(\delta_{ij}-(n+R(r))\frac{x_ix_j}{r^2}\right)\!\int_0^r\rho^{n-1}E_-(\rho)B[v](\rho)\,d\rho\!-\!\frac{x_ix_j}{r^2}B[v](r),\end{aligned}
G_{k,N}^\chi(Z_1, Z_2, r; Q_\tau) := \pi^{2r+4 -2k}\times (\mathfrak{p}^\circ_{\ell, m}\otimes \mathfrak{p}^\circ_{\ell, m}) (E_{k,N}^\chi(Z_1, Z_2, 1 - \frac{k-r}2; Q_\tau)).
\rho_1 = \Gamma_{11}\psi, \quad \rho_3 ={1\over 6}\psi^3 +{\cal F}\Gamma_{11}\psi, \quad \rho_5 ={1\over 120}\Gamma_{11}\psi^5+{1 \over 6}{\cal F}\psi^3 +{1 \over 2}{\cal F}^2 \Gamma_{11}\psi, \quad{\rm etc.},
\frac{d}{dt}\Bigg|_{t=0}j_{0,r_i-1}\left( f(u_t(z)) \right) = j_{0,r_i-1}\left(\frac{d}{dt}\Bigg|_{t=0}m^i_t(z) g_i(0,z)\right) = j_{0,r_i-1}\left(\frac{d}{dt}\Bigg|_{t=0}m^i_t(z)\right)M(g_i),
\mathbb{E}\int_{Q_T}|\tilde P|^2{\rm d}x{\rm d}y{\rm d}t= \mathbb{E}\int_{Q_T}\left|(\tau-1) l_{xx}\tilde z- l_t\tilde z-(x+\varepsilon)^{2\gamma}l_{yy}\tilde z\right|^2{\rm d}x{\rm d}y{\rm d}t \leq C(\lambda)\mathbb{E}\int_{Q_T}s^2\widehat\phi^2\xi^3 |\tilde z|^2{\rm d}x{\rm d}y{\rm d}t.
I ( s , t ) = \frac{3^{9}}{2^{1 2}}\int_{0}^{\infty}d T e^{T}\frac{\det ( 1 + S \tilde{X}( T ) )}{\det ( 1 - S \tilde{N}( T ) )^{1 3}}\exp \left( - \frac{1}{2}p_{i}Q_{i j}p_{j}\right)
\lim_{\frac{A}{\sigma_{DE}}\to 0}\frac{C_k^{UB}}{\frac{A^2}{2 \sigma_D^2}}=\lim_{\frac{A}{\sigma_{DE}}\to 0}\frac{\log(1+ \frac{A^2}{\sigma_D^2 (1+\frac{A^2}{\sigma_E^2})})}{\frac{A^2}{\sigma_D^2}}=1.
(\Omega_3+\Omega_2 + M_{\Omega_1})x^{(k+1)}= (\Omega_3+N_{\Omega_1})[\theta x^{(k)}+(1-\theta)x^{(k-1)}] &+(\Omega_2 - M_{\Omega_2})\vert x^{(k)}\vert + N_{\Omega_2}\vert x^{(k-1)}\vert - q.