harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{i=0}^{n-1}\Pi_i =1-\bigg[\frac{(p_0/q)(p/q)^{n-1}-q_0/q}{(p/q)^n-1}\bigg]\bigg[\frac{(p/q)((p/q)^{n-1}-1)}{p/q-1}-(n-1)\bigg] &\qquad{}+\frac{(p_0/q)((p/q)^{n-1}-1)}{p/q-1}&=1-\frac{p_0-q_0}{p-q}+n\,\frac{p_0p^{n-1}-q_0q^{n-1}}{p^n-q^n},
\partial_{t}^{\alpha}u =\left[D_{x^{i}}\left(a^{ij}u_{x^{j}}+b^{i}u+f^{i}(u)\right)+cu+h(u)\right] \qquad+\partial_{t}^{\beta}\int_{0}^{t}[\sigma^{ijk}u_{x^{i}x^{j}}+\mu^{ik}u_{x^{i}}+\nu^{k}u+g^{k}(u)]dw_{s}^{k}.
I_1=\int_0^{\infty}\frac{J_1(r\sqrt{\alpha^2+\beta^2})}{r^2+a^2}{\rm d}r=\frac{1}{a^2(\sqrt{\alpha^2+\beta^2)}}\bigl(1-a\sqrt{\alpha^2+\beta^2}K_1\bigl(a\sqrt{\alpha^2+\beta^2}\bigr) \bigr).
\frac{\log (\left \|XQ^{\frac{1}{2p}}\right \|_{S^{2p}_n}^{2p}) -\log(\left \|XQ^{\frac{1}{2}}\right \|_{HS}^2)}{1-p}\leq \frac{2p \log(\mathrm{tr}(Q)^{-\frac{1}{2p'}})+ (2p-2)\log (\left \|XQ^{\frac{1}{2}}\right \|_{HS})}{1-p}&=\log(\mathrm{tr}(Q))-\log (\left \|XQ^{\frac{1}{2}}\right \|_{HS}^2)
(1-\alpha_t) \sum_{i=1}^2 |w_{t,k_ti}|^2 p_{t,i}+ \alpha_t\tilde{p}_t \le A_{t,k_t}, & (1-\alpha_t)\sum_{i=1}^2 |w_{t,\bar{k}_ti}|^2 p_{t,i}\le A_{t, \bar{k}_t},
\left\lbrace \begin{array}{rcl}\tilde{G}_{(x,u)}(X^{h},Y^{h}) = (a+c)g_{x}(X,Y)+dg_{x}(X,u)g_{x}(Y,u), \tilde{G}_{(x,u)}(X^{h},Y^{t}) = b g_{x}(X,Y), \tilde{G}_{(x,u)}(X^{t},Y^{t}) = ag_{x}(X,Y)-\frac{\phi}{a+c+d}g_{x}(X,u)g_{x}(Y,u), \end{array}\right.
\tilde L(t)=\alpha|M|t +\beta|\Omega|t+\alpha\beta\Vert\varphi\Vert_{1,\Omega\times(0,t)}+{\alpha}^2\Vert\varphi\Vert_{1,M\times(0,t)}+{\alpha}\Vert u_j(x,0)\Vert_{1,\Omega}\cdot\Vert\varphi(x,0)\Vert_{\infty,\Omega}&+\Vert v_i(\zeta,0)\Vert_{1,M}+{\alpha}\Vert v_i(\zeta,0)\Vert_{1,M}\cdot \Vert\varphi(x,0)\Vert_{\infty, M}+\Vert u_j(x,0)\Vert_{1,\Omega}
m_{01}=2(g_{7}+g_{17}), m_{02}=2(g_{18}\!-\!g_{3}), m_{03}=2(g_{19}\!+\!g_{2}), m_{10}=-\!2(g_{26}\!+\!g_{16}), m_{12}= 2(g_{5}\!+\!g_{25}), m_{13}=2(g_{6}\!-\!g_{22}), m_{20}= 2(g_{15}\!-\!g_{30}), m_{21}=2(g_{8}\!+\!g_{28}), m_{23}=2(g_{21}\!+\!2g_{10}), m_{30}=-\!2(g_{31}\!+\!g_{14}), m_{31}=2(g_{11}\!-\!g_{27}), m_{32}=2(g_{23}\!+\!g_{12}).
w(0, \cdot) = w^{0}, \; w_{t}(0, \cdot) = w^{1}, \;v(0, \cdot) = v^{0}, \; v_{t}(0, \cdot) = v^{1}, \;\theta(0, \cdot) = \theta^{0}, \; q(0, \cdot) = q^{0},
\begin{aligned}p_{[1^05^1]}\bigg(5^{2r}m+\frac{5-5^{2r+1}}{24}\bigg)\equiv 0\pmod{5^{2r-1}}, p_{[1^65^{-5}]}\bigg(5^{2r}m+\frac{19\cdot5^{2r}-19}{24}\bigg)\equiv 0\pmod{5^{2r}}.\end{aligned}
\varphi(x)=\sum_{{\vec k}}{1\over \sqrt{(2\pi^3)^3 2\omega({\vec k})}}( a_{{\vec k}}e^{-i\omega({\vec k}) x^0 + i{\vec k}\cdot{\vec x}}+a^{\dag}_{{\vec k}}e^{+i\omega({\vec k}) x^0 - i{\vec k}\cdot{\vec x}})
\dfrac{1}{2}(P(C<N | \mathcal{D}= U)+P(C \geq N|\mathcal{D}= \mathcal{G}_{\sigma})) & = \dfrac{1}{2}(P(C<N | \mathcal{D}= U)+1-P(C < N|\mathcal{D}= \mathcal{G}_{\sigma})) & = \dfrac{1}{2}+ \dfrac{1}{2}(P(C<N | \mathcal{D}= U)-P(C < N|\mathcal{D}= \mathcal{G}_{\sigma})) > \dfrac{1}{2}
d^{k_1}y\cdots d^{k_n}y \mapsto \psi(d^{k_1}y\cdots d^{k_n}y)(z):=\sum_{m_1,\ldots,m_n\geq 0}\frac{B_{m_1}}{m_1!}\cdots \frac{B_{m_n}}{m_n!}\cdot C^{k_1,\ldots,k_n}_{m_1,\ldots,m_n}z^{m_1+\cdots+m_n-n}
\Pi_+^{\phi_{+},T}D_{+,T}(\psi_{+,T}^1) =-\Pi_+^{\phi_{+},T}( D_{+,T}(\phi_{+,T}) + e_{+,T}+\mathcal L_+(\phi_{-,T},\phi_{+,T}) +\mathcal{R}_+(\phi_{+,T})).
Q(z)=\sum_{\alpha\in \Delta_{>0}}x_\alpha(z)\psi_{\alpha}^*(z)+\sum_{\alpha\in \Delta_{>0}}\Phi_{\alpha}(z)\psi_{\alpha}^*(z)-\frac{1}{2}\sum_{\alpha,\beta,\gamma\in \Delta_{>0}}c_{\alpha,\beta}^\gamma\psi_{\alpha}^*(z)\psi_{\beta}^*(z)\psi_{\gamma}(z),
\#E_5(x)\leq\sum_j\sum_k\sum_{p\in\mathcal{P}_{j,k}}\sum_{\substack{n\leq x P(n)|u_n P(n)=p}}1\leq \sum_j\sum_k\sum_{p\in\mathcal{P}_{j,k}}\psi\left(\frac{x}{pz_u(p)},p\right)
\mathbf{d}\in \mbox{Pie}_{i, p}(u) \Longleftrightarrow \ell(u\cdot\tau_\mathbf{d})=\ell(u)-\ell(\tau_\mathbf{d}); w\in \mbox{Per}(\mathbf{d}) \Longleftrightarrow \ell(w\cdot \phi_\mathbf{d})=\ell(w)+\ell(\phi_\mathbf{d}).
g = \frac{4 \pi^{2}R}{e^{2}}\, \left( C_{A B}d \vec{r}_{A}\cdot d \vec{r}_{B}+ ( C^{- 1})_{A B}( d \psi_{A}+ \vec{\omega}_{A C}\cdot d \vec{r}_{C}) ( d \psi_{B}+ \vec{\omega}_{B D}\cdot d \vec{r}_{D}) \right) .
\mathcal D = \mathcal D(n,p,s) = 2 \pi^{\frac{n-1}2}\frac{\Gamma(\frac{1+sp}2)}{\Gamma(\frac{n+sp}2)}\int_0^1 \left| 1 - r^{(sp-1)/p}\right|^p \frac{dr}{(1-r)^{1+sp}}\,,
A_1= b^2(1-\theta_1) -2\theta_2=\dfrac{9ab^3cd-b^4c^2}{6 a c^3 +6b^3 d-4 b^2 c^2}-\dfrac{2c^2 (3ac-b^2)^2}{6 a c^3 + 6 b^3 d-4b^2 c^2}= \dfrac{3c(4 a b^2 c^2 + 3 a b^3 d- 6 a^2 c^3 -b^4 c)}{6 a c^3 +6b^3 d-4 b^2 c^2}= \dfrac{3ab^2c^3}{2c^2(3 a c-b^2)+2b^2(3bd-c^2)}\cdot(4+\dfrac{3bd}{c^2}-\dfrac{6ac}{b^2}-\dfrac{b^2}{ac}).
\mathbb{T}^{N}:=\big\{x\in\mathbb{C}^{N}\colon \vert x_{j}\vert =1,\, 1\leq j\leq N\big\}, \mathrm{d}m(x) =(2\pi)^{-N}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N},x_{j}=\exp( \mathrm{i}\theta_{j}) , -\pi<\theta_{j}\leq\pi, 1\leq j\leq N, K_{\kappa}(x) =\prod_{1\leq i<j\leq N}\vert x_{i}-x_{j}\vert^{2\kappa}, \kappa>-\frac{1}{N};
\left(i\j\right)=V_{\mathbb{Z}\beta+\frac{i+j}{4k}\beta}\otimes V_{\mathbb{Z}\beta+\frac{i-j}{4k}\beta}+V_{\mathbb{Z}\beta+\frac{2k+i+j}{4k}\beta}\otimes V_{\mathbb{Z}\beta+\frac{2k-i+j}{4k}\beta},
\|x^A -x^B + t_m(w^A_m-w^B_m)\| = \Big\|x^A-x^B + t_m \alpha \frac{x^B-x^A}{\|x^B-x^A\|}\Big\| &= \|x^A-x^B\|\left|1-\frac{t_m \alpha}{\|x^B-x^A\|}\right| = \|x^A-x^B\| - t_m \eta \, .
\min s_g =\min\left\{\frac{1}{h}(t-t_0)\right\}+s_0 &=\frac1h\min t-\frac{2}{h[\omega]^2}\int_Mtd\mu+\frac{8\pi c_1[\omega]}{[\omega]^2}&=-\frac{\omega(F)}{4\pi h}-\frac{1}{hT_0}T_1+\frac{4\pi c_1[\omega]}{T_0}.
M_{pl}^2 ={{M_X^3}\over k}e^{2 \pi k \rho -2c}\left(1- e^{-2k y_2}+ e^{-2 k ( \pi \rho - y_4 +y_2)}- e^{-2 k ( 2 \pi \rho - y_4)}\right)~.~\,
\int_{\Lambda^\infty}S_\lambda f^{m,v}\overline{f^{m',v'}}\, dM = \sum_{\mu \in D_v^J}c^{m,v}_\mu \overline{c^{m',v'}_\lambda}\rho(\Lambda)^{d(\lambda)/2}M(Z(\lambda\mu)) &= \overline{c^{m',v'}_\lambda}\rho(\Lambda)^{-d(\lambda)/2}\sum_{\mu \in D_v^J}c^{m,v}_\mu \rho(\Lambda)^{-d(\mu)}x^\Lambda_{s(\mu)}&= 0.
u\in C_{loc}^{0,\alpha}(M\times (0,+\infty )\backslash\{\{x^{1,k}\cdots, x^{l_k,k}\}\times T_k \}_{k=1}^L), & \nabla u\in L_{loc}^{\infty}(M\times (0,+\infty )\backslash\{\{x^{1,k}\cdots, x^{l_k,k}\}\times T_k \}_{k=1}^L) .
\omega_f \big(L_i(I)\big) = ~ \sup_{x,y \in I}\big|f(L_i(x))-f(L_i(y))\big| = ~ \sup_{x,y \in I}\big|\alpha_i (f(x)-f(y))+ q_i(x)-q_i(y)\big| \le ~|\alpha|_\infty \omega_f(I) + Q|I|,
\mathrm{tr}\, (\mathcal{H}(B)-B_0)_-^\sigma \leq L^{\mathrm{cl}}_{\sigma, 1}\ 2^{\, \sigma+\frac 12}\int_{\mathbb{R}}\, \sum_{k=0}^\infty\ \Lambda_k(x_3)^{\sigma+\frac 12}\, \mathrm{d}x_3\,, \sigma \geq \frac 32\,,
\varphi_p^o (x) ={{2a}\over{3(1+\beta^2)}}\left[ - \varphi^h_1(x) \int^x_0 dy \left( h(y) \varphi^h_2 (y) \right) + \varphi^h_2 (x) \int^x_0 dy \left( h(y) \varphi^h_1(y) \right) \right] ,
L^2({\cal A}_\gamma \times{\cal F}_\gamma) =\bigoplus_{\rho_e\in{\rm Rep}(G),\; \rho_v\in S}\quad\bigotimes_{v\in V}\bigl ( \bigotimes_{t(e)=v}\rho_e^* \otimes \bigotimes_{s(e)=v}\rho_e \otimes \rho_v \bigr )
\sum_{j=0}^\infty \left(\begin{array}{c}m j \end{array}\right)(a_{(n+j)}b)_{(m+k-j)}c &=\sum_{j=0}^\infty (-1)^j\left(\begin{array}{c}n j \end{array}\right)a_{(m+n-j)}b_{(k+j)}c-\sum_{j=0}^\infty (-1)^{j+n}\left(\begin{array}{c}n j \end{array}\right)b_{(n+k-j)}a_{(m+j)}c.
c_1(k_{22},k_{12},k_{11})=q^{2n}c_2(k_{22}-1,k_{12},k_{11})+q^{2n-1}c_2(k_{22},k_{12}-1,k_{11})+c_1(k_{22},k_{12},k_{11}-1), c_2(k_{22},k_{12},k_{11})=c_2(k_{22}-1,k_{12},k_{11})+q^{2n-1}c_2(k_{22},k_{12}-1,k_{11})+c_1(k_{22},k_{12},k_{11}-1),
\mathfrak{h}'_n =\underset{\gamma^*\in\Gamma_*}{\sum}\tau_{\gamma^*}\left(g_{1/\delta}\,\sharp^{\epsilon,\kappa}\,\widetilde{\mathfrak{g}}_n\,\sharp^{\epsilon,\kappa}\, \,\big[r^{\epsilon,\kappa}_1\big]^{\,\sharp^{\epsilon,\kappa}\,N}\right)\,,
\lim_{\varepsilon \rightarrow 0}P_{n}\left[ F_{1}, \dots , F_{n}\right] \left( \varepsilon \tau_{i}, \tau_{j}\right) = P_{n - m + 1}\left[ O_{m}\left[ F_{1}, \dots , F_{m}\right] \left( \tau_{i}\right) , F_{m + 1}, \dots , F_{n}\right] \left( 0 , \tau_{j}\right) .
\begin{array}{r c l}{{\nu}}&{{=}}&{{\nu_{o}+ \frac{\displaystyle \left( E_{q - \mathrm{r o t}}( J + 2 ) - E_{q - \mathrm{r o t}}( J ) \right)}{\displaystyle h c}~ ,}}\\{{}}&{{=}}&{{\nu_{0}+ B \left( [ J + 3 ]_{q}[ J + 2 ]_{q}- [ J + 1 ]_{q}[ J ]_{q}\right) ~ ,}}\\ \end{array}
a_h(u,w) = \int_{\Omega}\left(\sum_{i,j=1}^d a_{ij}(x) \frac{\partial u}{\partial x_i}\frac{\partial w}{\partial x_j}+ \sum_{j=1}^d b_j(x) \frac{\partial u}{\partial x_j}w + b_0(x) u w\right) dx
\|S(x^{k-d_k})\|; \left\|\left(\frac{1}{n}\sum_{i=1}^n y_{i,j}^{k-e^{i}_{k}}\right)_{j=1}^m \right\|; r_{ij}^d(y_i^{k-e^{i}_{k}}); r_{ij}^p(x^{k-d_k}); \sum_{\substack{h=0 h\neq \tau_d - e^{i}_{k, j}}}^{\tau_d}\frac{p_{ij}\gamma_{ij}}{\tau_d + 1}r_{ij}^d(y_i^{k - \tau_d +h}),
\binom{pn}{qn}=\frac{(pn)!}{(qn)!((p-q)n)!}<\left(\frac{p}{2\pi q(p-q)n}\right)^{\frac{1}{2}}\cdot \exp\left(f(p,q,n)\right)\cdot \left(\frac{p^{p}}{q^q(p-q)^{p-q}}\right)^n,
B^{z-1+is}[BA^{\frac12},A^{z-\frac12+is}]Y^{-is}= B^{z-1+is}[BA^{\frac12},A^{z-1}A^{\frac12+is}]Y^{-is}= B^{is}B^{z-1}A^{z-1}[BA^{\frac12},A^{\frac12+is}] Y^{-is}\quad+B^{is}B^{z-1}[BA,A^{z-1}] A^{is}Y^{-is}.
\frac{f(m+1,\ell)}{f(m,\ell)}\geq \left(\frac{\ell^{\frac{\ell}{6+\ell}}m}{m+1}\right)^{6+\ell}\geq \left(\frac{21^{\frac{21}{6+21}}2}{2+1}\right)^{6+\ell}> 1^{6+\ell}=1,
R_{vv}^\ast -{\sigma}^{ij}\frac{\partial \beta}{\partial y^i}\frac{\partial\beta}{\partial y^j}- \frac{2\Lambda}{(n-1)}\beta^2 = \frac{1}{\alpha^3}\frac{d^2\alpha}{du^2}-\frac{1}{\alpha^4}\left ( \frac{d\alpha}{du}\right )^2=\frac{1}{\alpha(v)}\frac{d^2\alpha}{dv^2}\quad ,
\varphi_{tt}-\frac{1}{\gamma^2}\frac{\varphi_{ttxx}}{\varphi_x^2}+ \frac{1}{\gamma^2}\frac{\varphi_{ttx}\varphi_{xx}}{\varphi_x^3}= \frac{\gamma-3}{\gamma}\frac{\varphi_t \varphi_{tx}}{\varphi_x}- \frac{1}{\gamma^2}\frac{\varphi_{tx}\varphi_{txx}}{\varphi_x^3}+ \frac{1}{\gamma^2}\frac{\varphi_{tx}^2 \varphi_{xx}}{\varphi_x^4}
U_k^{SR1}=\frac{-1}{\theta_{k-1}(1-\theta_{k-1}) p_{k-1}^Tg_{k-1}}(g_k - (1-\theta_{k-1}) g_{k-1})(g_k - (1-\theta_{k-1}) g_{k-1})^T &= \frac{-(1-\theta_{k-1})}{\theta_{k-1}p_{k-1}^T g_{k-1}}\left(\frac1{1-\theta_{k-1}}g_k - g_{k-1}\right)\left(\frac1{1-\theta_{k-1}}g_k-g_{k-1}\right)^T,
\frac{1}{[ ( k + k_{i})^{2}- m^{2}]}= \sum_{j = 0}^{N}\frac{\left( - 1 \right)^{j}\left( k_{i}^{2}+ 2 k_{i}\cdot k \right)^{j}}{\left( k^{2}- m^{2}\right)^{j + 1}}+ \frac{\left( - 1 \right)^{N + 1}\left( k_{i}^{2}+ 2 k_{i}\cdot k \right)^{N + 1}}{\left( k^{2}- m^{2}\right)^{N + 1}\left[ \left( k + k_{i}\right)^{2}- m^{2}\right]},
\partial_{x}\int\mathbf{1}_{\{k\geq0\}}e^{ikx}\mathfrak{Q}(x,k)e^{ik^{2}t}h(k)\,dk & = \int\mathbf{1}_{\{k\geq0\}}ike^{ikx}\mathfrak{Q}(x,k)e^{ik^{2}t}h(k)\,dk+ \int\mathbf{1}_{\{k\geq0\}}e^{ikx}\partial_{x}\mathfrak{Q}(x,k)e^{ik^{2}t}h(k)\,dk.
D^{BS}_{2N}(\lambda) =(-2)^{N}(\lambda-\frac n2-2N)_{N}(2N-1)!!\Big[A_0(\lambda)\iota^*\partial_n^{2N}+A_N(\lambda)(\Delta^\prime)^N +\sum_{k=1}^{N-1}A_k(\lambda)(\Delta^\prime)^k\iota^*\partial_n^{2N-2k}\Big]
{\hat{\cal C}}^{a}| Q_{i}^{\prime},{\cal C}^{\prime a}, \bar{\cal C}_{a}^{\prime},{\pi}_{a}^{\prime}\rangle = \hat{\bar{\cal C}}_{a}| Q_{i}^{\prime},{\cal C}^{\prime a}, \bar{\cal C}_{a}^{\prime},{\pi}_{a}^{\prime}\rangle ={\hat{\pi}}_{a}| Q_{i}^{\prime},{\cal C}^{\prime a}, \bar{\cal C}_{a}^{\prime},{\pi}_{a}^{\prime}\rangle = 0 ,
&\tilde A_-^{(1)}=\tilde A_-^{(2)}, \tilde A_j^{(1)}=\tilde A_j^{(2)}(s,\tau,\hat y), 1\leq j\leq n-1, &V_1^{(1)}(s,\tau,\beta^{(1)}(s,\tau,\hat y'))=V_1^{(2)}(s,\tau,\beta^{(2)}(s,\tau,\hat y')).
G_{\hat{\mu}\hat{\nu}}^{1 1}= \left( \begin{array}{c c}{{e^{- \frac{2}{3}\phi}G_{\mu \nu}+ e^{- \frac{2}{3}\phi}C_{\mu}C_{\nu}}}&{{e^{\frac{1}{3}\phi}C_{\mu}}}\\{{}}&{{}}\\{{e^{\frac{1}{3}\phi}C_{\nu}}}&{{e^{\frac{4}{3}\phi}}}\\ \end{array}\right) .
\begin{cases}|b_1|\leq C \max(e^{\delta_n q_n}, |\ell |, 1) e^{(\tilde{L}-\beta_n+4\varepsilon) q_n}|b_3|\leq \max(e^{\delta_n q_n}, |\ell |, 1) c_{n,\ell+1}e^{(\tilde{L}-\beta_n+4\varepsilon) q_n}|b_4|\leq c_{n,\ell+1}e^{(\tilde{L}+\varepsilon) q_n}\leq C \max(e^{\delta_n q_n}, |\ell |, 1) e^{(\tilde{L}-\beta_n+\varepsilon) q_n}\end{cases}
{\cal L}= \frac{1}{2}\{[ \dot{X}( t ) + g \xi ( t ) Y ( t ) ]^{2}+ [ \dot{Y}( t ) - g \xi ( t ) X ( t ) ]^{2}+ [ \dot{Z}( t ) - \xi ( t ) ]^{2}\}- U ( X ( t )^{2}+ Y ( t )^{2})
J_{jl}(t, \tau) = J^{(1)}_{jl}(t, \tau) + J^{(2)}_{jl}(t, \tau), |t|\le t^{00}_{jl}, J^{(r)}_{jl}(t, \tau) := t |t| \int_0^{\tau}e^{i \tilde{\tau}A(t)^{1/2}}F^{(r)}_{jl}(t) P_j G_* P_l e^{-i \tilde{\tau}(t^2 S)^{1/2}P}P \, d \tilde{\tau}, r=1,2.
\min(t,|S_{1}|)+\min(t,|S_{2}\backslash S_{1}|)+\ldots+\min(t,|S_{n}\backslash(S_{1}\cup\ldots\cup S_{n-1})|) \geq \frac{1}{2}|S_{1}|(|\Lambda|-|\Lambda^{c}|)+t|\Lambda^{c}| \geq \frac{1}{2}\min(t,|S_{1}|)(|\Lambda|-|\Lambda^{c}|)+\min(t,|S_{1}|)|\Lambda^{c}|=\frac{1}{2}\min(t,|S_{1}|)(|\Lambda|+|\Lambda^{c}|)=\frac{\min(t,|S_{1}|)}{2}r.
\begin{array}{l}{{[ H , X^{\pm}] = X^{\pm}~ ,}}\\{{[ X^{+}, X^{-}] = [ 2 H ] ~ ,}}\\{{\Delta ( X^{\pm}) = X^{\pm}\otimes q^{- H}+ q^{H}\otimes X^{\pm}~ ,}}\\{{\Delta ( H ) = H \otimes 1 + 1 \otimes H ~ ,}}\\{{\epsilon ( X^{\pm}) = 0 = \epsilon ( H ) ~ ,}}\\{{S ( X^{\pm}) = - q^{\mp 1}X^{\pm}~ , ~ S ( H ) = - H ~ .}}\\ \end{array}
K_{p}(z) \stackrel{(z \rightarrow 0)}{=}\frac{1}{2}\,\left[\Gamma (p) \left(\frac{z}{2}\right)^{-p}+\Gamma (-p) \left(\frac{z}{2}\right)^{p}\right]\,\left[1 +O \left(z^{2}\right)\right]\; ,
\textstyle \Psi (\overline{\textstyle (T - a) \tau}) \textstyle = \left( T - \overline{p/a}\sum_{j=1}^t c_j\right) \left[1, X^{p-1-s}\right]= \left( T - \overline{p/a}\cdot \frac{t}{s}\right) \left[1, X^{p-1-s}\right].
r(T,JT) =\sum_{l,k=1}^{6}g\left(JT,N(e_l,e_k) \right)g\left(JT,N(e_l,e_k) \right)+\sum_{l,k=1}^{6}g\left(N(T,e_k),e_l \right)g\left(N(JT,e_k),Je_l \right), &=\sum_{l,k=1}^{6}g\left(JT,N(e_l,e_k) \right)^2 -\sum_{l,k=1}^{6}g\left(N(T,e_k),e_l \right)^2, &=\frac{1}{8}{\textbf{H}^\prime}^2.
Y(s,t) = \big((1-m)s^{\alpha-1}+ mt^{\alpha-1}\big)^{1/(\alpha-1)}, Y_{11}(s,t) = -m(1-m)(2-\alpha)(st)^{\alpha-3}Y(s,t)^{3-2\alpha}t^2, Y_{22}(s,t) = -m(1-m)(2-\alpha)(st)^{\alpha-3}Y(s,t)^{3-2\alpha}s^2, Y_{12}(s,t) = m(1-m)(2-\alpha)(st)^{\alpha-3}Y(s,t)^{3-2\alpha}st,
Z = \beta^{N/2}\Gamma((N-1)/2)^{-1}{\LARGE\int_0^\infty}d\sigma{\LARGE\int_{-\infty}^\infty}d\sigma_1 ( 4 \beta^2 \sigma^2 + (2 \beta \sigma_1 + v )^2 + m^2 )^M \sigma^{N-2}exp[ -\beta (\sigma^2 + \sigma_1^2) ]
U_{E,T}=T;\quad U_{G_{p-1},T}=G_{p-1}T\,{\overline{G_{p-1}T}\,}^{-1}=ST^{p}S^{-1}; U_{G_0,S}=S^2;\quad U_{G_k,S}=G_k S\,{\overline{G_k S}\,}^{-1}=ST^k S\,{\overline{ST^k S}\,}^{-1}\qquad(k=1,2,\ldots,p-1)
\psi_{({y_1}_*,{y_2}_*)}(\xi_1(th, tw))\leq \psi_{({y_1}_*,{y_2}_*)}(1) = J({y_1}_*,{y_2}_*) = \inf_{\mathcal{N}_{\lambda}^{-}}J(y_1, y_2) \leq J(\xi_1(th, tw)({y_1}_* + th,{y_2}_* + tw)).
\begin{minipage}[c]{35pc}For every $x\in M,$ one has $x=x x^* x,$ and, further, every pairof idempotent elements of $M$ commute\cite[p.\,6, Theorem~3]{MVL},\cite[Theorem~1.17(i)]{C+P1},cf.\ \cite[Theorem~II.1.2]{MP}.\end{minipage}
V_+(0, 1) = \min_{\gamma \in \{0,1\}}\Bigg\{c_0 \gamma + \mathbb{E}_+\left[V_+\left(1, (1,\Psi(1,\gamma,\xi_2))\right)\right] \Bigg\}& = \min \Bigg\{\frac{1}{2}\left( c_1 - \frac{5}{16}\beta \right), c_0 + \left( c_1 - \frac{5}{16}\beta \right) \frac{3}{4}- \beta \frac{1}{2}\Bigg\}.
\eta_1= \frac 18(-(1-\alpha)t^{-1-\alpha}\tau+(1-\alpha)t^{-\alpha}\tau_t-t^{1-\alpha}\tau_{tt})x^2-\frac 12t^{1-\alpha}\sigma_{1t}x &+\frac{k+1}{4}\bigg(\tau_t-\frac{1-\alpha}{t}\tau\bigg)+\frac{k-c}{2x}\sigma_1+\frac{1}{mx^k}\eta_{2x}+\frac12\sigma_2.
\begin{aligned}dX^N_t(k) =(\mathcal A^N X^N_t)(k)\,dt+\sqrt{g(X^N_t(k))}\,dB_t(k) &=\frac{1}{N}\sum_{l\in S^N}\big(X^N_t)(k)-X^N_t)(k)\big)\,dt+\sqrt{g(X^N_t(k))}\,dB_t(k),k\in S^N,\, t\geq 0.\end{aligned}
\Delta^{+}_N = \frac{\pi}{\Gamma^2(1-\epsilon)}\left( 2 \int^a_0 \frac{dr}{a}\left(\frac{r}{a}\right)^{2N+1}e^{2 e \varphi}+ \frac{1}{\epsilon - 1}\right) \left(\frac{ka}{2}\right)^{2 - 2\epsilon}+ O(ka)^{4-4\epsilon},
\begin{aligned}[b] \prod_{j\in[t_2]}x_j^{-\tau+\zeta_j}\prod_{i=t_2+1}^{t_1+t_2}\tilde{h}_{p^*_i}(i,\boldsymbol{x}) & \leq \tilde{K}\prod_{j\in[t_2]}x_j^{-\tau+\zeta_j+|Q_j|}\prod_{i=t_2+1}^{t_1+t_2}(1/x_{{j_{p^*_i}}})^{\tau-1-\zeta_i-{(l-p^*_i+1)}},\end{aligned}
\left((\cos \theta\vec{B}+\sin \theta\vec{C})\cdot\vec{D}\right)^2 = \cos^2\theta (\vec{B}\cdot\vec{D})^2+\sin^2\theta (\vec{C}\cdot\vec{D})^2+2\sin\theta\cos\theta(\vec{B}\cdot\vec{D})(\vec{C}\cdot\vec{D})
\frac{d^mG}{dv^m}=v^3\frac{d^m\hat{B}}{dv^m}+c_{2,m}v^2\frac{d^{m-1}\hat{B}}{dv^{m-1}}+c_{1,m}v\frac{d^{m-2}\hat{B}}{dv^{m-2}}+c_{0,m}\frac{d^{m-3}\hat{B}}{dv^{m-3}}.
\mathcal{L}= - \overline{{\Psi}}( x ) \left[ \Gamma_{\mu}\left( \partial_{\mu}- i e A_{\mu}- g A_{\mu}^{B}I^{B}\right) + m \right] \Psi ( x ) - \frac 1 4 \mathcal{F}_{\mu \nu}^{2}- \frac 1 4 \left( F_{\mu \nu}^{B}\right)^{2}
\Phi_{\Delta G}(\Upsilon(u)) = \Phi_{\Delta G}(\Omega(h) * (\hat{0}, \hat{0}) )= (p^{G \times G}\circ \Psi^{G \times G})(h) \cdot \Phi_{\Delta G}(\hat{0}, \hat{0}) &=(h(0), h(1)) \Delta G =\phi^{-1}(h(0)h(1)^{-1}) = \phi^{-1}(\Phi(u)).
L_0 \Big(\left(\chi_{-j_k}\right)^{m_k}\dots \left(\chi_{-j_2}\right)^{m_2}\left(\chi_{-j_1}\right)^{m_1}|0\rangle \Big)= \frac{1}{2}\left(m_k\cdot j_k +\dots m_2\cdot j_2 +m_1\cdot j_1\right)\Big(\left(\chi_{-j_k}\right)^{m_k}\dots \left(\chi_{-j_2}\right)^{m_2}\left(\chi_{-j_1}\right)^{m_1}|0\rangle \Big).
\dot N={}\partial_tN+\partial_qN\cdot\dot q+ \partial_{\dot q}N\cdot\ddot q= ={}\underbrace{\partial_tN+\partial_qN\cdot\dot q+ \partial_{\dot q}N\cdot \Lambda}_{=0}+ \partial_{\dot q}N\cdot(\ddot q-\Lambda)= ={}\partial_{\dot q}N\cdot(\ddot q-\Lambda).
\bar p^*Ric_{jk}=\frac{1}{2}\frac{\partial^2}{\partial\tilde y^i\partial\tilde y^j}\Big(\bar p^*(F^{2})\bar p^*(\mathcal{R}ic)\Big)=\frac{1}{2}\frac{\partial^2}{\partial\tilde y^i\partial\tilde y^j}(\tilde F^{2}\tilde{\mathcal{R}}ic)=\tilde Ric_{jk}.
\alpha_\ell=\frac{[(\ell-1)!]^{2}}{4^{\ell-1}}\sum^{\ell-1}_{k=0}{\ell-1\choose k}^2{2(\ell-1-k)\choose\ell-1-k}{2k\choose k}=\frac{[(\ell-1)!]^{2}}{4^{\ell-1}}D_{\ell-1},
\lambda x_{b_n}=-\frac{1}{\prod_{\nu=b_n+1}^{b_{n+1}-1}w_{\nu}}x_{b_{n+1}-1}+\sum_{l\in\varphi^{-1}(n)}v_lx_{b_{l+1}-1}&=-\frac{1}{\lambda^{\Delta b_{n}-1}}x_{b_{n}}+\sum_{l\in\varphi^{-1}(n)}v_l\, \frac{\prod_{\nu=b_l+1}^{b_{l+1}-1}w_{\nu}}{\lambda^{\Delta b_l-1}}\, x_{b_{l}}.
\langle 0 | a ( k^{\prime}s^{\prime}) a^{*}( q , r ) a ( q^{\prime}, r^{\prime}) a^{*}( k , s ) | 0 \rangle = \frac{2 \kappa^{\prime}g ( s^{\prime})}{( 2 \pi )^{4}\lambda}\delta_{r s^{\prime}}\delta^{4}( q - k^{\prime}) \frac{2 \kappa g ( s )}{( 2 \pi )^{4}\lambda}\delta_{r^{\prime}s}\delta^{4}( q^{\prime}- k )
f(x)=\sum_{k\in\mathbb{Z}^3}\widehat{f}(k)e^{2i\pi k/\ell \cdot x},\quad\textrm{with}\widehat{f}(k)=\frac{1}{\ell^3}\int_{Q_\ell}f(x)e^{-2i\pi k/\ell \cdot x}dx.
\int \widehat f\, d\mathfrak{S}^{n}({\widehat{\mu}})=\sum_{j=1}^{k}\int f_{j}\,d (\mathfrak{S}^{n}({\widehat{\mu}}))_{j},\mathfrak{S}^{n}({\widehat{\mu}})=\big( (\mathfrak{S}^{n}({\widehat{\mu}}))_{1}, \dots, (\mathfrak{S}^{n}({\widehat{\mu}}))_{k}\big).
\sum_{k=2}^{6}(-1)^k \binom{8}{k}\binom{k+1}{m}\frac{B_{k-2}}{k+1}= \frac{28}{3}B_{0}-56 B_{1}+140B_{2}-\frac{560}{3}B_3+ 140B_{4}&= \frac{28}{3}+ 28 + \frac{70}{3}-0 - \frac{14}{3}=56=\binom{8}{3},
\int_0^{r}yN(x,y))dy \geq\int_{y_0}^{r}\int_{y}^{\infty}yf(x,u)dudy &=\int_{y_0}^{r}\int_y^{r}yf(x,u)dudy+\int_{y_0}^{r}\int_r^{\infty}yf(x,u)dudy &\geq\int_{y_0}^{r}\int_y^{r}yf(x,u)dudy &=\frac{r^{3}-3ry_0^{2}+2y_0^{3}}{6}f(x,r) \geq \frac{2y_0^{3}}{3}r^{3}f(x,r).
\begin{array}{c}{{d s^{2}= - \lambda d t^{2}+ \lambda^{- 1}d r^{2}+ R^{2}d \Omega ,}}\\{{e^{2 \phi}= e^{2 \phi_{0}}\frac{r + \Sigma}{r - \Sigma},}}\\{{F = \frac{Q e^{\phi_{0}}}{( r - \Sigma )^{2}}d t \wedge d r , \; G = P e^{\phi_{0}}\sin \theta d \theta \wedge d \varphi ,}}\\ \end{array}
\mathbb{E}h(W_d)-\chi^2_{(d)}h =\mathbb{E}[W_df''(W_d)+\tfrac{1}{2}(d-W_d)f'(W_d)] &=\sum_{j=1}^d\mathbb{E}[W_{(j)}f''(W_d)+\tfrac{1}{2}(1-W_{(j)})f'(W_d)] &=\sum_{j=1}^d\mathbb{E}[\mathbb{E}[W_{(j)}f''(W_d)+\tfrac{1}{2}(1-W_{(j)})f'(W_d) \: | &\quad\: W_{(1)},\ldots,W_{(j-1)},W_{(j+1)},\ldots,W_{(d)})]].
\partial_t v_3=\nu\Delta v_3-\nabla q_3-\frac{1}{2}\textrm{div}[(v_3+v_2)\otimes (v_3+v_2)]-\frac{1}{2}\textrm{div}((v_3+v_2)\otimes z)-\frac{1}{2}\textrm{div}(z\otimes (v_3+v_2)), \textrm{div}v_3=0.
(v\sigma)^{2}q^{\frac{2}{p}}((1+3\lambda^{2})\frac{B(\frac{3}{p},q-\frac{2}{p})}{B(\frac{1}{p},q)}-4\lambda^{2}\frac{B(\frac{2}{p},q-\frac{1}{p})^{2}}{B(\frac{1}{p},q)^{2}})
g ( z ) = \sum_{i j}\frac{\sigma ( z + s_{i j})}{\sigma ( z ) \sigma ( s_{i j})}e^{- \frac{\zeta ( \pi )}{\pi}z s_{i j}}e^{i s_{i j}\frac{z - \bar{z}}{\Delta - \overline{\Delta}}}Y_{i j}E_{i j}= \sum_{i j}W ( z , s_{i j}) Y_{i j}E_{i j}.
\mathbb E_{P_\sigma}\left[\psi\left(\frac{1}{\|f_1\|_\sigma}W^{(\sigma)}(f_1)\right)\left(\psi\left(\frac{1}{\|f_2\|_\sigma}W^{(\sigma)}(f_2)\right)\right)\right]=[\psi]\left(\frac{\langle f_1,f_2\rangle_\sigma}{\|f_1\|_\sigma\|f_2\|_\sigma}\right).
I_2 =\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\frac{(\frac{1}{2})_{l}}{l!}\int_{\mathcal{C}_{\{0\}}}\frac{ds}{2 \pi is}e^{\frac{v}{s}}\big(s(z+1)\big)^k (2s)^l = \sum_{l=0}^{\infty}\big(\frac{2}{z+1}\big)^l \frac{(\frac{1}{2})_{l}}{l!}\sum_{k=l}^{\infty}\frac{1}{k!}\big(v(z+1)\big)^k,
\sum_{(d^{\bullet}; d^{\infty})}\sum_{(d^{1, \bullet}, \cdots , d^{n, \bullet}; d^{\infty, \bullet})}(-1)^{\sum_{i=1}^n (l_i -1) + l_{\infty}}\left( \prod_{i=1}^{\substack{\longrightarrow \infty}}\prod_{j=1}^{\substack{\longrightarrow l_i}}\mathbf{1}_{d^{i,j}}\right) \star \mathbf{1}_{d^{\infty, \infty}}^{\sigma}.
\begin{array}{llr}x_1 x_2 + \mu_{12}x_2 x_1 = x_4^2, x_1 x_4 + \mu_{14}x_4 x_1 = x_2^2 , x_2 x_4 = -\mu_{24}x_4 x_2, begin{align*}3mm]x_1x_3 + \mu_{13}x_3 x_1 = x_3^2, x_2 x_3 + \mu_{23}x_3 x_2= x_1^2, x_3x_4 = -\mu_{34}x_4x_3. begin{align*}3mm]\end{array}
\begin{array}{c}E_{N\, bulk}=E^{+}_{N,\, bulk}+E^{-}_{N,\, bulk}=\displaystyle\frac{N^{2}}{L}\left[ \displaystyle\int \displaystyle\frac{dx}{\pi}\left[ \displaystyle\frac{\phi (x,1/2)}{\cosh (2\Theta -x)}\right] -2\pi \right] =-\displaystyle\frac{N^{2}}{L}(\pi +\gamma ) P_{N\, bulk}=E^{+}_{N\, bulk}-E^{-}_{N\, bulk}=2\omega \end{array}
= \int_{S^1}\left(f_1(\frac12 h_0'+k_1c-k_2d)+\frac1{k_0}f_1'(c'+\frac12 k_0h_0 - \frac12 (k_1h_1+k_2h_2)) + f_2(e'+k_2c+k_1d)\right)dx
\left\{x ( 1 + x)^{2}\frac{d^{2}}{dx^{2}}+ (1 + x) (3x +1) \frac{d}{dx}- \frac{m^{2}(1+x)^{2}}{4 x}+ m (1 + x) + 2 + \lambda^{2}\rho^{2}\right\}\xi^{(m)}= 0
K_{\Psi , w}( t_{1}, t_{2}, \ldots , t_{l}) = \prod_{j = 1}^{l}\Gamma ( - \beta_{j})^{- 1}\times \{\exp ( - t_{1}F_{i_{1}}) \cdots \exp ( - t_{l}F_{i_{l}}) \prod_{j = 1}^{l}t_{j}^{- \beta_{j}- 1}\Psi \}.
\frac{\partial^2}{\partial z_1 \partial z_1^\rho}= \frac{1}{4}\left( \frac{\partial^2}{\partial x_1^2}- u \frac{\partial^2}{\partial x_2^2}\right), \frac{\partial^2}{\partial z_2 \partial z_2^\rho}= \frac{1}{4}\left( \frac{\partial^2}{\partial x_3^2}- u \frac{\partial^2}{\partial x_4^2}\right),
\varphi_{v_0}(e_1,e_2,f_1,f_2) = (a_0,b_0) = (0,0), \varphi_{v_1}(e_1,e_2,f_1,f_2) = (a_1,b_1) = (-e_1,f_1+f_2), \varphi_{v_2}(e_1,e_2,f_1,f_2) = (a_2,b_2) = (e_2,f_1).
\left\vert \sum_{n_{1}=0}^{\infty}\ldots \sum_{n_{d}=0}^{\infty}\frac{a^{n_{1}+\ldots + n_{d}}}{n_{1}! \ldots n_{d}!}\exp \left( \nu R^{c,p}(p,d,\mathbf{n}^{(d)}) - a(c-1) \right) - \frac{(d-p)!}{(c-1-p)!}\right\vert <R e^{-Sa},
\tau_{\infty}=\frac{(-1)^{\mathsf{N}}\kappa_{+}\kappa_{-}e^{\epsilon (\tau_{-}-\tau_{+})}\prod_{b=1}^{\mathsf{N}}\alpha_{b}\beta_{b}}{\left( \zeta_{+}-1/\zeta_{+}\right) \left( \zeta_{-}-1/\zeta_{-}\right)}
\Lambda_C(r)\equiv\begin{cases}\arcsin\big(\frac{17 + 18\, r^2\, (-2 + r^2)}{2\, R_{23}^4(r)}\big)\ \sqrt{3}\,\big/2\,<\, r\, <\, \sqrt{5/6}, -\pi - \arcsin\big(\frac{17 + 18\, r^2\, (-2 + r^2)}{2\, R_{23}^4(r)}\big)\ \sqrt{5/6}\,<\,r\,<\,1, \end{cases}
W_{6}^{j}=\frac{\varepsilon}{2}\sum_{i=1}^{d}\int_{\mathbb{T}^{d}}(\partial_{x_{j}}w^{1}-\partial_{x_{j}}w^{2})^{2}\partial_{x_{i}}\left(\Theta_{p_{i}}(t,x,\mu^{2},Dw^{2})\right)\dx.
\mathbb{I}\equiv \sum_{h_{1},...,h_{\mathsf{N}}=0}^{p-1}\prod_{1\leq b<a\leq \mathsf{N}}(X_{a}^{(h_{a})}-X_{a}^{(h_{a})})|h_{1},...,h_{\mathsf{N}}\rangle\langle h_{1},...,h_{\mathsf{N}}|.
[X_{1},X_{2}]=\sqrt{\frac{2}{3}}X_{3}, &[A,X_{1}]=\frac{2}{\sqrt{33}}X_{1},\,[A,X_{2}]=\frac{2}{\sqrt{33}}X_{2},\,[A,X_{3}]=\frac{4}{\sqrt{33}}X_{3},\,[A,X_{4}]=\frac{3}{\sqrt{33}}X_{4}.