harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{_2}\phi_{1}\left[ \begin{array}{c}{{\begin{array}{c c}{{a ,}}&{{b}}\\ \end{array}}}\\{{c}}\\ \end{array}; q , u \right] = \sum_{n = 0}^{\infty}\frac{( a ; q )_{n}( b ; q )_{n}}{( q ; q )_{n}( c ; q )_{n}}u^{n},
\begin{aligned}\frac{1}{2}\frac{d}{dt}\int |\nabla \Delta d|^2 dx+\int |\Delta d_t|^2 dx &= \frac{d}{dt}\int \nabla (u \cdot \nabla d-|\nabla d|^2 d)\cdot \nabla \Delta d \dx -\int \frac{\partial}{\partial t}\left[\nabla(u\cdot \nabla d-|\nabla d|^2 d)\right] \cdot \nabla \Delta d \dx.\end{aligned}
\|\rho R_V(\lambda)\rho\|_{L^2(B_R)\rightarrow H^j(B_R)}\leq C|\lambda|^{\frac{-2 + j}{4}}\big(e^{2R(\Re\sqrt[4]{\lambda})_-}+ e^{2R(\Im\sqrt[4]{\lambda})_-}\big),j = 0, 1, 2, 3, 4,
G= g_2^{-1}g_3^{-1}g_1^{}g_3^{-1}\Delta_3\bigsqcup g_3^{-1}g_1g_3^{-1}g_1g_3^{-1}\Delta_1\bigsqcup g_3^{-1}\Delta_4 G= g_2^{-1}g_3^{-1}g_1g_3^{-1}g_1g_3^{-1}\Delta_2\bigsqcup e\Delta_5,
\|\tilde P_r - P_r\|_{\infty}\leq 2\pi\frac{\bar g_r}{2}\frac{1}{2\pi}\left(\frac{2}{\bar g_r}\right)^2\|E\|_{\infty}\sum_{k=0}^{\infty}\left(\frac{2}{\bar g_r}\|E\|_{\infty}\right)^k &\leq \frac{2\|E\|_{\infty}/\bar g_r}{1- 2\|E\|_{\infty}/\bar g_r}.
\dfrac{\pi_{>0}(x,\zeta)}{\pi_\zeta(x)}+\dfrac{\pi_\zeta\left(\frac{1}{\epsilon^2}\right)}{\pi_\zeta(x)}\ge \dfrac{\#\left\{p\leq x : \chi(p)=\zeta, \theta_p\in I_\epsilon\right\}}{\pi_\zeta(x)}.
\begin{aligned}r(\tau) =e^{-b(\tau-t)}r(t)+\frac{a}{b}(1-e^{-b(\tau-t)}) + \sigma\int_{t}^{\tau}e^{-b(\tau-s)}dZ(s) &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(0\le t\le \tau).\end{aligned}
L(uP) = \m_n (m_{n}+1)x^{m-\overline{n}}t^{\ell}+ \sum_{i \neq n}m_i (m_i - 1) x^{m - 2 \overline{i}+ \overline{n}}t^{\ell}- \ell x^{m+ \overline{n}}t^{\ell-1}+ \sum_{|m| + 2 \ell \leq |q|+2\kappa\le k-1}c^{m , \ell}_{q, \kappa}x^q t^\kappa+ w_{m, \ell},
\left(\frac{t}{e^{t}-1}\right)^{r}e^{xt}=\underset{r-\mbox{times}}{\underbrace{\left(\frac{t}{e^{t}-1}\right)\times\cdots\times\left(\frac{t}{e^{t}-1}\right)}}e^{xt}=\sum_{n=0}^{\infty}B_{n}^{\left(r\right)}\left(x\right)\frac{t^{n}}{n!}.
\pi\left((X\times{U})\cap\bigcap_{l\in{J_1}}V_{l,1}\cap\bigcap_{l\in{J_2}}V_{l,2}\right)=\bigcap_{l\in{J_1}}(X\setminus{A(n_l,k_l,i_l)})\cap\pi\left((X\times{U})\cap\bigcap_{l\in{J_2}}V_{l,2}\right).
\rho_{\Phi}^T:=\frac{1}{T^{k/2}}\sum_{j_1 \neq \ldots \neq j_k}\sigma_{j_1}\ldots \sigma_{j_k}\int_{[0,T]^k}\Phi(x^{j_1}+\xi^{j_1}_{s_1},\ldots,x^{j_k}+\xi^{j_k}_{s_k})ds_1 \ldots ds_k,
e^{-1}(V\otimes\O_\alpha)=\frac{1}{\alpha^r}\sum_{0\leq k\leq \dim(X)}\left(-\frac{c_1(V)}{\alpha}-\cdots-\frac{c_{r-1}(V)}{\alpha^{r-1}}-\frac{c_r(V)}{\alpha^r}\right)^k.
125C_{\tau}\eta^4 d^4(t_2-t_1)\big( C_2^4 (t_2-t_1)^2+( C_8'+C_{5}'^2) \big)\sum_{t=t_1}^{t_2-1}C_{d,T}(t) \le C' T^{-4}(t_2-t_1)^3 (t_2-t_1) CC_{s,2}^2 \big(\frac{t_2-1}{T}\big)^2 =O\bigg(\big(\frac{t_2-t_1}{T}\big)^2 \bigg).
K_{J}=\sum_{i=l}^{m}\bigg\{q^{i-l}(T_{i-1}^{-1}\cdots T_{l}^{-1})(T_{l}^{-1}\cdots T_{i-1}^{-1}) +(1-q)\sum_{j=l}^{i-1}q^{j-l}(T_{i-1}\cdots T_{j+1})(T_{j-1}^{-1}\cdots T_{l}^{-1})(T_{l}^{-1}\cdots T_{i-1}^{-1})\bigg\}X_{i}.
a_0\sum_{u=0}^{v}b_{u+2}+a_1\sum_{u=0}^{v}b_{u+1}+a_2\sum_{u=0}^{v}b_{u}+\sum_{k=3}^{v+2}a_{k}\sum_{u=k-2}^{v}b_{u+2-k}&=a_0\bigl({B}(v+2)-b_0-b_1\bigr)+a_1\bigl({B}(v+1)-b_0\bigr)+a_2{B}(v) &+\sum_{k=3}^{v+2}a_{k}{B}(v+2-k) &=\sum_{k=0}^{v+2}a_{k}{B}(v+2-k)-a_0b_0-a_0b_1-a_1b_0.
I_m = - \int d \bar{x}e^{\rho - 2 \phi + \frac{b}{2}f}\left[ 2 \phi^{\prime}\rho^{\prime}+ \frac{1}{4}f^{\prime 2}+ \frac{1}{4}e^{\chi f + 4 \phi}A^{\prime 2}+ \frac{\Lambda}{4}\right] .
L_nL_{-n}a^{\lambda}= \frac{\lambda}{2\rho_0^n}\left(\lambda a^{\lambda-1}\rho_0L_n\left(\frac1{\rho_{\infty}}\right)P_n(u_1,\dots,u_n)+ a^{\lambda}L_n(P_n(u_1,\dots,u_n))\right).
F_r\sum_{j = 0}^k{( - 1)^{rj}\left({\frac{{F_{n + r}}}{{F_n}}}\right)^j G_{m + n + r + rj}}= ( - 1)^{kr}F_n\left(\frac{{F_{n+r}}}{{F_n}}\right)^{k+1}G_{m + (k + 1)r}- ( - 1)^r F_nG_m,n\ne 0\,,
\frac{1}{N}\int\limits_{0}^N \sum\limits_{n\in [N]}u(n) 1_{E_{x,\delta}}(n) \, dx = \frac{1}{N}\sum\limits_{n\in [N]}u(n)\int\limits_{0}^N 1_{E_{x,\delta}}(n)\, dx & = \sum\limits_{n\in [N]}u(n) 2\delta & = 2\delta N^{d-m}T_{F,G,N}^L(1,\dots,1)
\theta_a^{H_u}(x + a(tu +s)){}=\theta^{H_{0,u}^-}_a(a(t+\langle s, u\rangle)u) = \theta_u(t+\langle s,u \rangle) \Theta_u(t;S){}= \{\theta_a^{H_u}(x+a(tu+s))\}_{s\in S}.
\bar{S}_{f r e e}^{( 0 )}[ \phi + \delta \phi ] - \bar{S}_{f r e e}^{( 0 )}[ \phi ] = : \mathrm{T r}\left({\frac{\delta \bar{S}_{f r e e}^{( 0 )}[ \phi ]}{\delta \phi ( \hat{x})}}\delta \phi ( \hat{x}) \right) .
|C(P(G(n),1))| = dis(P(G(n)), 1) & = 2^{n-1}+ (2^{n-1}-1) + (2^{n-1}-1)-1 + (2^{n-1}-1)-2 + \cdots + (2^{n-1}-1) & -((2^{n-1}-1)-1) = 2^{n-1}+ (2^{n-1}-1) + (2^{n-1}-1)(2^{n-1}-2) - \frac{(2^{n-1}-1)(2^{n-1}-2)}{2}& = \frac{2^{n-1}(2^{n-1}+1)}{2}.
\frac{(m+\eta)^2(m-\eta+1)^2}{4 (2 m + 1)^2}=\frac{(m+\frac 12)^4}{4 (2 m + 1)^2}=\frac{(2 m + 1)^2}{2^6}, &\frac{(1 + m)^2 - \eta + \eta^2}{(2 m + 1) (2 m + 3)}=\frac{(1 + m)^2 - (\frac 12)^2}{(2 m + 1) (2 m + 3)}=\frac{(m+\frac 12)(m+\frac 32)}{(2 m + 1) (2 m + 3)}=\frac 14.
\mathbf E\left[\int_0^{\tau_{t,x}}f_n\left(-X^{t,\beta}(s),X^{x,\alpha}(s)\right) ds\right] = \int_0^\infty P_s^{\beta,}P_s^\Omega f_n(t,x)\,ds &\to \int_0^\infty P_s^{\beta,}P_s^\Omega f(t,x)\,ds &= \mathbf E\left[\int_0^{\tau_{t,x}}f\left(-X^{t,\beta}(s),X^{x,\alpha}(s)\right) ds\right],
[m_{1,1}, m_{1}] + [m_{1}, m_{1,1}]+ [m_{1}, m_{1}] = 0, &[m_{2,1}, m_{2}] + [m_{2}, m_{2,1}] + [m_{2}, m_{2}] = 0, &[m_{1,1},m_{2}] + [m_{1}, m_{2,1}] + [m_{1}, m_{2}] = 0.
\tilde r^2_{\varepsilon,\sigma}\geq \left\{\frac{C^2_{\alpha,\beta}}{16}\, \sigma^2\, \max_{1 \leq D \leq M_2}[ b_D^{-2}a_D^{-2}] \right\}\vee \left\{\sup_{D \in \mathbb{N}}\left[c_{\alpha,\beta}\,\varepsilon^2 \sqrt{\sum_{j=1}^D b_j^{-4}}\wedge a_D^{-2}\right] \right\},
\sum_{j=1}^k\log\lambda_j(|G(\theta)|) \leq \sum_{j=1}^k\int_{-\infty}^{+\infty}dt\Big((1-\theta)\beta_{1-\theta}(t)\log\lambda_j\big(|G(it)|\big)+\theta\beta_\theta(t)\log\lambda_j\big(|G(1+it)|\big)\Big).
L_B(F \cup G) \leq L_B(F) + \left(C_1 + C_4 \sqrt{\mbox{\it dim}}\right) |G|^{\frac{\mbox{\it dim}- 1}{\mbox{\it dim}}}& \leq L_B(F) + C_5 |G|^{\frac{\mbox{\it dim}- 1}{\mbox{\it dim}}}
{\cal F}= \frac{H^{2}}{1 + H^{2}r^{2}}\left( \begin{array}{c c c}{0}&{2 E \dot{r}}&{0}\\{0}&{2 E^{2}- m^{2}\alpha^{2}+ H^{2}m^{2}r^{2}}&{0}\\{0}&{0}&{( 1 + H^{2}r^{2}) m^{2}\alpha^{2}}\\ \end{array}\right) .
K_\phi(\lambda)^T\otimes I_n = \begin{bmatrix}-\alpha_{d-2}\lambda-\beta_{d-2}-\alpha_{d-3}-\gamma_{d-2}\lambda-\beta_{d-3}\ddots \ddots \ddots -\alpha_0 -\gamma_1 \lambda-\beta_0\end{bmatrix}\otimes I_n\in\mathbb{F}[\lambda]^{dn\times (d-1)n}.
\big\langle Q^{m-n}v_0,\psi^{+}(w_1)\cdots \psi^{+}(w_m)\psi^{-}(y_1)\cdots \psi^{-}(y_n)v_0\big\rangle \qquad{}= \big\langle\psi_{(m-n)}^{-}\psi_{(m-n+1)}^{-}\cdots \psi_{(-1)}^{-}v_0,\psi^{+}(w_1)\cdots \psi^{+}(w_m)\psi^{-}(y_1)\cdots \psi^{-}(y_n)v_0\big\rangle,
\theta^{(p,q)}(g_1g_2,yH) =\big( \theta((g_1g_2)^p,y^pH),(\theta((g_1g_2)^q,y^qcH)\big) = \big(\theta(g_1^p,y^pH),\theta(g_1^qy^qcH)\big)\cdot \big(\theta(g_2^p,g_1^py^pH),\theta (g_2^q,g_1^qy^qcH)\big) = \theta^{(p,q)}(g_1,yH)\cdot\theta^{(p,q)}(g_2,\tau_{g_1}(yH)).
\langle \Phi^{-1}u, v \rangle_{W^{- 1 , p}_{\sigma},W^{1,p^{\prime}}_{0,\sigma}}= \langle u,v \rangle_{L^p_{\sigma}, L^{p^{\prime}}_{\sigma}}, u\in L^p_{\sigma}(\Omega), \, v\in W^{1,p^{\prime}}_{0,\sigma}(\Omega).
\begin{aligned}\zeta(v;\alpha)= \sum_{\begin{subarray}{c}0{\le}m_1<_{c^{'}_1}\cdots<_{c^{'}_{q-1}}m_q<\infty\end{subarray}}\frac{(m_q+1)!}{(\alpha)_{m_q+1}}\left\{\prod_{i=1}^{q}\frac{1}{(m_i+1)^{k^{'}_i}}\right\}= H^{*}_{(\{0\}_{i=1}^q)}(\tau(v);\alpha)\end{aligned}
\tau_{m,2}^{k}:= \frac{2L_{2,p}c_{h}}{\beta_{2,p}^{2}}( \| F(p_{m,k}^{H}) - P_{k}^{L}[F(p_{m,k}^{H})] \|_{H^{-1}(\Omega)}+ \| P_{k}^{L}[F(p_{m,k}^{H})] \|_{H^{-1}(\Omega)})
\Sigma = \Sigma({\bf b}, x,y, Q) = \sum_{\substack{Q\leq q \leq 2Q, q\,}}\biggr|\sum_{\substack{\chi\,{\rm mod}\, q, \chi \neq \chi_{0}.}}c_{q}(\chi) \sum_{x < n \leq x+y}b_n \chi(n)\biggr|.
\upsilon^{-1}\rho\big( z^{-\kappa},z^{\kappa}\big) \alpha_{n}(y,u) \upsilon=\big\{\upsilon^{-1}\rho\big( z^{-\kappa},z^{\kappa}\big) \upsilon\big\}\big\{\upsilon^{-1}\alpha_{n}(y,u) \upsilon\big\}
\int_0^\infty K_\mu(\alpha x)K_\nu(\beta x) x^{s-1}dx = 2^{s-3}\alpha^{-s-\nu}\beta^\nu \Gamma\left(\frac{1}{2}(s+\mu+\nu) \right)\Gamma\left(\frac{1}{2}(s-\mu-\nu)\right) &\int_0^1 t^{\frac{s-\mu+\nu}{2}-1}(1-t)^{\frac{s+\mu-\nu}{2}-1}\left(1 - \left(1-\frac{\beta^2}{\alpha^2}\right)t\right)^{-\frac{s+\mu+\nu}{2}}dt.
\zeta^{\prime}( z , t ) = \sum_{l = 0}^{\infty}\frac{( - t )^{l}}{l !}\frac{\Gamma ( z + l )}{\Gamma ( z )}\left\{\left[ \psi ( z + l ) - \psi ( z ) \right] \zeta ( z + l ) + \zeta^{\prime}( z + l ) \right\}- t^{z}\mathrm{l n}( z ) .
D(H) = \bigl\{x\in D(A^\star); T \mathbf{S}_{-,P_1/\alpha}{F}_2(\mathbf{S}_{a,b}^{-1}x) = \mathbf{S}_{+,P_1/\alpha}{F}_1(\mathbf{S}_{a,b}^{-1}x) \bigr\}
-q^{-n/2}\left[\#C(\mathbb F_{q^n})-(q^n+1)\right] =\sum_{i=0}^{q(q-1)}\zeta_i^n \textrm{by}\eqref{eqn-roots-of-unity}&= b_0\cdot 1^n+ c_0\cdot(-1)^n+\displaystyle\sum_{k=1}^{p-1}b_k(w^k)^n+\sum_{k=1}^{p-1}c_k(-w^k)^n =\sum_{k=0}^{p-1}a_kw^k \textrm{by the properties of $\sigma$}=-\sqrt{q}(q-1) \textrm{by}\eqref{eqn-sum-with -0}.
- \int_{V}\lambda_{p}\p \delta D^{r i_{2}. . . i_{p}}\wedge \delta A_{r i_{2}. . . i_{p}}= ( - 1 )^{p + 1}\int_{V}\lambda_{p}\ \delta \Pi_{A_{2}. . . A_{p}}^{( 1 )}\wedge \delta Q_{( 1 )}^{A_{2}. . . A_{p}}\ .
v_{x_{0}}\left( x,x_{0}\right) \mid_{x\in \Gamma ,x_{0}\in \left[ 0,1\right]}=\partial_{x_{0}}\widetilde{g}_{0}\left( x,x_{0}\right) ,\partial_{n}v_{x_{0}}\left( x,x_{0}\right) \mid_{x\in \Gamma ,x_{0}\in \left[ 0,1\right]}=\partial_{x_{0}}\widetilde{g}_{1}\left( x,x_{0}\right) ,
< \frac{1}{2}, m_{s}|{\psi}_{-}^{( \frac{1}{2})}( g ) > \equiv D_{m_{s}- \frac{1}{2}}^{( \frac{1}{2})}( g ) = < g , l + \frac{1}{2}| T_{m_{s}-}^{\frac{1}{2}}| g , l > .
||\partial^{\alpha}u||_{L^p(\mathcal{D})}= \sup_{\varphi \in C_c^{\infty}(\mathcal{D})\setminus\{0\}}\Bigg|\int_{\mathcal{D}}u(x)\frac{\partial^{\alpha}\varphi(x)}{||\varphi||_{L^q(\mathcal{D})}}dx \Bigg|
g(x_0) - \underline{g}\geq \sum_{k=0}^K g(x_k) - g(x_{k+1}) &\geq K \min\left( c_1 \tau_1 \min(\underline{\alpha_1}, \underline{t_1}) \varepsilon_1^2, c_2 \tau_2^2\min\left( \min\left(\alpha_{02}, \dfrac{3|2c_2-1| \varepsilon_2}{M_g}\right), \underline{t_2}\right)^2 \varepsilon_2 \right).
\mathbb{E}[(\xi(z,s) +1)^{n(z,s)}] \le [\mathbb{E}((\xi(z,s) +1)^{2n(z,s)})]^{1/2}= \biggl( 1+ \sum_{k=1}^{2n(z,s)}{2n(z,s) \choose k}\mathbb{E}(\xi(z,s)^{k}) \biggr)^{1/2}= 1 + O(N^{-1/2}),
T^{\prime}_{z}C_{\alpha}\otimes T^{\prime}_{z}C_{\beta}\mapsto e^{(\frac{\lambda_{i}-\lambda_{j}}{d_{\alpha}}+\frac{\lambda_{i}-\lambda_{l}}{d_{\beta}})t}T^{\prime}_{z}C_{\alpha}\otimes T^{\prime}_{z}C_{\beta}
\sum_{C_s}\frac{3}{\pi^2}\cdot \frac{m^2}{q_n(q_n+q_{n-1})}(1+O(m^{-\delta/10})) = \frac{3}{\pi^2}m^2 \lambda( \Gamma'_m) (1+O(m^{-\delta/10})) &= \frac{3}{\pi^2}m^2 \left(1+O\left(\frac{1}{\log m}\right)\right) (1+O(m^{-\delta/10})) &= \frac{3}{\pi^2}m^2 + O\left( \frac{m^2}{\log m}\right).
| \Omega_{i_{1}\ldots i_{n}}^{( n ) \, \alpha_{1}\ldots \alpha_{n}}\rangle = \frac{1}{\sqrt{n !}}\left( a_{\alpha_{n}}^{i_{n}}\right)^{\dagger}\ldots \left( a_{\alpha_{1}}^{i_{1}}\right)^{\dagger}| \Omega \rangle \, .
- r \lambda^{\frac{1}{2}}h^{- \frac{1}{2}}\Gamma^{0}\partial_{0}\psi = \left\{r \Gamma^{i}\partial_{i}+ \left( ( h^{\frac{1}{2}}- 1 ) x^{i}\partial_{i}+ C + i \Delta \Gamma^{0}\right) \right\}\Pi \psi
S_{1,\lambda}(k+1,m+1) &=\sum_{l=m}^{k}S_{1,\lambda}(k,l)\bigg\{\binom{l}{m}\langle 1\rangle_{l-m,\lambda}(-1)^{l-m}+l\lambda\binom{l-1}{m}(-1)^{l-m-1}\langle 1\rangle_{l-m-1,\lambda}\bigg\}.
E_{r}=A_{r}^{\prime}+qD_{r}^{\prime}=\sum_{|w|=r}^{\circlearrowright}(\delta_{j_{1}}+\check{t}_{j_{1}})\cdots (\delta_{j_{r}}+\check{t}_{j_{r}})=\sum_{|w|=r}^{\circlearrowright}t_{w}\rho_{t}(\bar{\pi}_{w})~,
\tau_i=\left[\begin{array}{cccccc}1_{m}0 0 0 0 0 0 1_r 0 0 0_r 0 0 0 1_{m}0 0 0 0 0 e_t 1_{m}0 0 0 0_r 0 0 1_r 0 e_t^{\ast}0 0 0 0 1_{m}\end{array}\right],e_t=\left[\begin{array}{cc}1_t 0 0 0\end{array}\right]\in\mathbb{B}^{m}_{m}.
\sqrt{x^{2}+y^{2}}=x+\frac{3}{2}y^{2}-\frac{3}{2}xy^{2}+\frac{1}{2}x^{2}y^{2}+O(y^{3})+O(y^{3})(x-1)+O(y^{3})(x-1)^{2}+O(x-1)^{3}
e_{-1}(m_i) =im_{i-1}1 \leq i \leq 6, e_{-1}(m_0) = 0, e_0(m_i) =(i+4)m_i 0 \leq i \leq 6, e_1(m_i) =(i+1)m_{i+1}0 \leq i \leq 5, e_1(m_6) = 0.
\sum_{p=1}^{r_i}\left(E_{\Lambda^n}\left[\frac{h(\textbf{A}_{p,i,n}\textbf{X}^n+B_{p,i,n}|\Omega^n,\Lambda^n=\lambda^n,W_{i+1},\ldots,W_K)}{r_ir_{i+1}}\right.\right. &\left.\left.-\frac{h(\textbf{C}_{i,n}\textbf{X}^n+D_{i,n}|\Omega^n,\Lambda^n=\lambda_n,W_{i+1},\ldots,W_K)}{r_ir_{i+1}}\right]\right)
\frac{f\left( x\right) -f\left( a\right)}{g\left( x\right) -g\left(a\right)}>\left( <\right) \lim_{x\rightarrow a^{+}}\frac{f\left( x\right)-f\left( a\right)}{g\left( x\right) -g\left( a\right)}=\frac{f^{\prime}\left( a^{+}\right)}{g^{\prime}\left( a^{+}\right)}=\lambda .
\mu_2\circ \sigma_e (Z(\lambda_n)) = \mu_2 (Z(e \lambda_n b_0)) + \mu_2(Z(e \lambda_n b_1) &= 2^{n+1}(1 + (-1)^{m_Q}2 \delta_0) \left( \prod_{i=1}^n (1 + (-1)^{m_i}2 \delta_{2i})\right) \left( 1/2 + \delta_{2n+2}+ 1/2 - \delta_{2n+2}\right) &= 2^{n+1}(1 + (-1)^{m_Q}2 \delta_1) \prod_{i=1}^n (1 + (-1)^{m_i}2 \delta_{2i}).
c^2 + g^2 + h^2 + (j^2 + l^2 + n^2 + q^2) + \sum{\gamma_i^2}= 4 + 4c^2 c^2 + 2gh + jk + lm + np + qr + \sum{\gamma_i\gamma_i^*}= 4c^2 \intertext{Because of duality constraints, the set $\{j, l, n, q\}$ is a permutation of the set $\{k, m, p, r\}$ so, \eqref{squares}implies}c^2 + g^2 + h^2 + (k^2 + m^2 + p^2 + r^2) + \sum{\gamma_i^2}= 4 + 4c^2.
\theta:=1_{\alpha\in(0,1]}\tfrac{\alpha+\bar\alpha}{2}+1_{\alpha\in(1,2)}\tfrac{\alpha+\bar\alpha\vee 1}{2},\ \q>\tfrac{d}{\gamma_0(\gamma_\sigma(\alpha\wedge 1)\wedge\gamma_\nu)}\vee \tfrac{d}{\theta-\bar\alpha}\vee\tfrac{\alpha}{\alpha-\theta}\vee p.
\left\lfloor\frac{t-2i-2j}{1-\frac{\sqrt{2}}{2}}\right\rfloor =\lfloor(t-2i-2j)(2+\sqrt{2})\rfloor begin{align*}5pt] =\lfloor(t-2i-2j)(4-(2-\sqrt{2}))\rfloor begin{align*}5pt] =4t-8i-8j+\left\lfloor-\frac{t-2i-2j}{1+\frac{\sqrt{2}}{2}}\right\rfloor.
\prod \limits_{i=j}^{t_1 - 1}|c(\theta_i)p(x_i)| = \left( \prod \limits_{i=j}^{t - 1}|c(\theta_i)p(x_i)| \right)\cdot \left( \prod \limits_{i=t}^{t_1 - 1}|c(\theta_i)p(x_i)| \right) \leq 4^{4K_{p_0}}\cdot (3/5)^{\left(1 - \frac1{M_0}\right)(t_1 - j)/2}< 1,
\mathbb{E}\left[g_{\psi,x}(\widetilde{R}^{z})-g_{\psi,x}(\widetilde{R}^{u})\right]=\int_0^1 \Psi^{\prime}(t)dt =\frac{1}{2}\sum_{i=b+1}^n \sum_{j=1}^{p}\int_0^1 \mathbb{E}[\partial_jF(Z(t))\dot{Z}_{ij}(t)]dt & =\frac{1}{2}(I_1+I_2+I_3),
m_{T+1}^{i\rightarrow j}(\tau_{i}||\tau_{j})=\sum_{\{\tau_{k}\}_{k\in\partial i\backslash j}}W_{i}(\tau_{i};\{\tau_{k}\}_{k\in\partial i})\prod_{k\in\partial i\backslash j}m_{T}^{k\rightarrow i}(\tau_{k}||\tau_{i}),
u(t)\chi_1 ={S_{1,\alpha}}(\mu_1^{- 1}){1_{D(t_*^j) > \mu_4^{- 3}}}u(t - \mu_1^{- 1}) +{S_{2,\alpha}}(\mu_1^{- 1}){1_{D(t_*^j) > \mu_4^{- 3}}}{\partial_t}u(t - \mu_1^{- 1}) &+ \int_{t - \mu_1^{- 1}}^t{{S_{2,\alpha}}(t - s){{{\left| u \right|}^{p - 1}}u}}{1_{D(t_*^j) > \mu_4^{- 3}}}(s)ds.
\hat{V}^{C}_{\sigma}(C_{1},C_{2}) = \frac{1}{2\pi\sigma^{2}}\frac{1}{N}\sum\limits_{n=1}^N exp \left ( - \frac{|C_1|^2 + |C_2|^2}{2\sigma^2}- \frac{|C_1||C_2| cos(\theta-\phi)}{\sigma^2}\right)
\frac{h\left(\frac{iT}{N}\right)-h\left(\frac{(i-1)T}{N}\right)}{\frac{T}{N}}=-\frac{\pi}{2}\tan\left(\frac{\pi}{2}\left(h\left(\frac{(i-1)T}{N}\right) -W\left(\frac{(i-1)T}{N}\right)\right)\right)
\bigl\|f(e^{i \tilde Z_n}\tilde U_n) -f(\tilde U_n)-\frac{d}{dt}\bigl(f(e^{it \tilde Z_n}\tilde U_n)\bigr)_{\vert t=0}\bigr\|_1 & = N_n \bigl\|f(e^{i Z_n}U_n)-f(U_n)-\frac{d}{dt}\bigl(f(e^{it Z_n}U_n)\bigr)_{\vert t=0}\bigr\|_1 = N_n\beta_n.
\begin{cases}|\phi(x_1-1)|\leq e^{30\varepsilon q_n}\max\{e^{-(x_1-(\ell-1)q_n)L}r_{\ell-1}^+, e^{-(\ell q_n-x_1)L}r_{\ell}^-\}|\phi(x_2+1)|\leq e^{30\varepsilon q_n}\max\{e^{-(x_2-(\ell+1)q_n)L}r_{\ell+1}^+, e^{-((\ell+2)q_n-x_2)L}r_{\ell+2}^-\}\end{cases}
x ( u , \xi ) ={\frac{\sigma ( \xi - u )}{\sigma ( \xi ) \sigma ( u )}}, \quad x_{d}( u , \xi ) ={\frac{\sigma ( 2 \xi - u )}{\sigma ( 2 \xi ) \sigma ( u )}}\quad x_{t}( u , \xi ) ={\frac{\sigma ( 3 \xi - u )}{\sigma ( 3 \xi ) \sigma ( u )}},
\begin{array}{c}p^A=p_0^A x^A\left( \tau \right) =x_0^A+2\left( E_1+E_3\delta_2^2\right)p_0^A-m^{-2}V^A\left( \tau \right) V^A\left( \tau \right) =V_1^A\cos \left( 2m^2E_3\delta_2\right) +V_2^A\sin\left( 2m^2E_3\delta_2\right)\end{array}
I_{Y M , \theta}={\frac{1}{8 \pi}}\int_{X}d^{4}x \sqrt{g}\bigg ({\frac{4 \pi}{g_{Y M}^{2}}}\mathrm{t r}[ F_{\mu \nu}F^{\mu \nu}] +{\frac{i \theta}{4 \pi}}\varepsilon_{\mu \nu \rho \sigma}\mathrm{t r}[ F^{\mu \nu}F^{\rho \sigma}] \bigg ) ,
\lim{F_a(\Sigma^{*}, x_0, \{l_P\}) U^{(\nu)}(C, x_0) \over\langle F_a(\Sigma^{*}, x_0, \{l_P\}) \rangle\langle tr U^{(\nu)}(C, x_0)}\rangle={1 \over n_{\nu}}D^{\nu}\left( a^{k (\Sigma^{*}, C)}\right)
{\widetilde{\cal C}}_{T}^{\prime}= 2 b{\widehat c}\sum_{\alpha , \beta}2 B^{2}\left( q_{\alpha}^{2}+ q_{\beta}^{2}\right) \Delta_{2}( X_{\alpha \beta}^{2}) \Big |_{\sigma}- 8 d^{2}\sum_{\alpha , \beta}2 B^{2}q_{\alpha}q_{\beta}\Delta_{2}( X_{\alpha \beta}^{2}) \Big |_{C_{\mu \nu}}.
\mathbb{P}_n \left( \frac{\sum_{i = 1}^n \left(g_{i,n}^{(j)}- \hat{g}_{i,n}^{(j)}\right)^2}{\sum_{i = 1}^n \mathbb{V}_n \left( g_{i,n}^{(j)}\right)}\geq \varepsilon \right) \to 0.
\begin{array}{l}Y^{(1/2)}= -2 D_2 \Phi; Y^{(n+1/2)}= \partial_- Y^{(n-1/2)}- 2i \partial_- \Phi \cdot \sum_{l=1}^{n-1}D_2 Y^{(l-1/2)}\cdot Y^{(n-l-1/2)}; n=1,2,3, \cdots\end{array}
xu = \sum_{i \leq k}y^{f_i}\gamma_i (\alpha_i + y^{e_i}s_i) = \sum_{i \leq k}y^{f_i}\gamma_i \alpha_i + \sum_{i \leq k}y^{e_k - e_i}y^{e_i}\gamma_i s_i = \sum_{i \leq k}y^{f_i}\gamma_i \alpha_i + y^{e_k}s = \delta + y^{e_k}s,
\frac{\varepsilon}{G(\varepsilon)}=\frac{\frac{m\theta^2}{2\sqrt{ab}}+\delta_1}{G(\frac{m\theta^2}{2\sqrt{ab}})}-\frac{\delta_1}{2G(\frac{m\theta^2}{2\sqrt{ab}})}\leqslant \frac{m\theta^2}{2\sqrt{ab}}-\frac{\delta_1}{2G(\frac{m\theta^2}{2\sqrt{ab}})}\leqslant\frac{m\theta^2}{2\sqrt{ab}}-\frac{\delta_1}{2|\varepsilon_D^*|^2}<\eta,
[ \ell_{\mu}^{+}, \ell_{\nu}^{+}] = [ \ell_{\mu}^{-}, \ell_{\nu}^{-}] = 0 , \quad [ \ell_{\mu}^{-}, \ell_{\nu}^{+}] = 2 \omega \left( \mu \cdot \nu + \sum_{\rho \in \Delta_{+}}g_{| \rho |}\, ( \rho \cdot \mu ) \, ( \rho^{\vee}\! \cdot \nu ) \check{s}_{\rho}\right) .
\mathbf{G}_{ij}^T \triangleq{}\begin{bmatrix}\mathbf{G}_{ij}[1] 0 \ldots 0 0 \mathbf{G}_{ij}[2] \ldots 0 \vdots \vdots \ddots \vdots 0 0 \ldots \mathbf{G}_{ij}[T] \end{bmatrix}.
\beta_{\ell}= t_1 + \frac{(t_1 - t_2)\frac{\beta_1 - t_1}{\beta_1-t_2}p^{\ell-1}}{1 - \frac{\beta_1 - t_1}{\beta_1-t_2}p^{\ell-1}}, &= t_2 + \frac{t_2 - t_1}{\frac{\beta_1 - t_1}{\beta_1-t_2}p^{\ell-1}- 1}.
v^i(x_1^\ast,y_1^\ast) = \sum\limits_{x_2,\dots,x_{k-1}\in X}p(x_{k-1}|x_{k-2},\mu_{k-2}^\ast)\dots p(x_2|x_1^\ast,\mu_1^\ast) v^i(x_{k-1},y_{k-1}^\ast).
a_7 = \sqrt{a_5 a_6}= \sqrt{{\cal A}^{- 1}(a_2){\cal A}^{- 1}(a_3){\cal A}^{- 1}(1 + \alpha_x)}={\cal A}^{- 1}( \sqrt{a_2 a_3}){\cal A}^{- 1}\big( \sqrt{1 + \alpha_x}\big).
\widetilde{c}_1(t; T)= \frac{\prod_{n=1}^t \sin \left( \frac{n}{r}\right) \prod_{n=1}^{2T-t}\sin \left( \frac{n}{r}\right)}{\prod_{n=1}^{2T}\sin \left( \frac{n}{r}\right)}\prod_{n=1}^T\frac{\sin^2 \left( \frac{n}{r}\right)}{\sin\left( \frac{2 \alpha_0+(2n-T-1)}{2r}\right)}.
\Pr\left\{\frac{1}{t}\sum_{\tau=0}^{t-1}p_k(\tau) - c_k > \epsilon_k \right\}\leq u_t \exp\left\{\frac{-2 \epsilon_{t,k}^2 v_t^2}{u_{,k}^2}\right\}+ t \beta_{\texttt{ALG},k}(u_t),
I_3=\int_{\mathbb{R}^{n-1}}g\eta^2-g\mathbf{x}\cdot\nabla\Big(\frac12\eta^2\Big)~d\mathbf{x}=\Big(1+\frac{n-1}{2}\Big)\int_{\mathbb{R}^{n-1}}g\eta^2~d\mathbf{x}.
X_n^{\left( \mathrm{e}^{2i\pi \alpha/n}, \mathrm{e}^{2i\pi\beta/n}\right]}\overset{}{\underset{n\to\infty}{\longrightarrow}}\beta - \alpha - \sum_{j=1}^{+\infty}(\lbrace y_j \beta \rbrace - \lbrace y_j \alpha \rbrace )
\lim_{t \rightarrow 0^{+}}\left\{\langle 0 | \hat{\psi}_{a}^{+}\left( t \right) \hat{\psi}_{b}\left( 0 \right) | 0 \rangle + \langle 0 | \hat{\psi}_{b}\left( 0 \right) \hat{\psi}_{a}^{+}\left( t \right) | 0 \rangle \right\}= \delta_{a b ,}^{}
\big(t_\lambda^\Lambda{t_\mu^\Lambda}^*\big)^*\big(t_{\lambda'}^\Lambda{t_{\mu'}^\Lambda}^*\big)=t_\mu^\Lambda{t_\lambda^\Lambda}^* t_{\lambda'}^\Lambda{t_{\mu'}^\Lambda}^*=\sum_{(\alpha,\beta)\in \Lambda^{\min}(\lambda,\lambda')}t_{\mu\alpha}^\Lambda{t_{\mu'\beta}^\Lambda}^*\in \phi(\mathcal{T}C^*(\Lambda^i)).
\begin{aligned}\zeta'_B(0;a,1,1)=\left(\frac 1{12}-\zeta'_R(-1)\right)\frac 1 a -\frac 1 4 \log (2\pi)+\frac{\gamma a}{12}&+\sum_{k=2}^\infty\frac{B_{2k}\zeta_R(2k-1)}{2k(2k-1)}a^{2k-1},a\to 0^+,\end{aligned}
W^{ac}\sum_{n=0}^\infty(-1)^n\,\left(\Lambda^n\right)^{cb}=\sum_{p=1}^\infty a_p\,\left[(g\bar\theta)^p\right]^{ac}\sum_{n=0}^\infty c_n\,\left[(g\bar\theta)^{2n}\right]^{cb}\,,
t=\frac{1}{2}(x_{1}+x_{2}+x_{3}+x_4),u=\frac{1}{\sqrt{2}}(x_{1}-x_{2}),v=\frac{1}{\sqrt{6}}(x_{1}+x_{2}-2x_{3}),w=\frac{1}{2 \sqrt{3}}(x_1+x_2+x_3-3 x_4).
\begin{aligned}\overline{E_{\rm el}}(w_0) + E_{\rm g}(w_0)+E_{\rm sf}(w_0) \le \liminf_h\overline{E_{\rm el}}(w_h) + E_{\rm g}(w_h)+E_{\rm sf}(w_h) &\le \liminf_h E(w_h)=\inf_WE=\inf_W\overline{E_{\rm el}}+ E_{\rm g}+E_{\rm sf}.\end{aligned}
\bar{b}^S + a_i^S = \begin{cases}\displaystyle \theta_i^1 + \sum_{j \in N}p_{ij}^1 \, a^S_j i \in S \cup N^{\{1\}}\displaystyle \sum_{j \in N}p_{ij}^0 \, a^S_j i \in N^{\{0, 1\}}\setminus S.\end{cases}
\begin{aligned}C = \{v \in M_{\mathbb R}\, | \, \langle v, n_i \rangle \geq 0\,\,\,{}i = 1, \dots, t\}= \bigcap_{i=1}^t \{v \in M_{\mathbb R}\, | \, \langle v, n_i \rangle \geq 0\},\end{aligned}
\ll\frac{(y^{(m)}_{max})^2X}{\varphi(V)\log X}\Bigl(\sum_{\substack{u<R (u,VP_f)=1}}\frac{\mu(u)^2}{g(u)}\Bigr)^{k-1}\Bigl(\sum_s\frac{\mu(s)^2}{g(s)^2}\Bigr)^{k(k-1)-1}\sum_{s_{i,j}>D_0}\frac{\mu(s_{i,j})^2}{g(s_{i,j})^2}&\ll\frac{(y^{(m)}_{max})^2\varphi(VP_f)^{k-1}X(\log X)^{k-2}}{\varphi(V)(VP_f)^{k-1}D_0}.
\boldsymbol{I}_{ij}^{+}\boldsymbol{P}_{i}^{\pm}=\boldsymbol{I}_{ij}^{+}\boldsymbol{P}_{j}^{\pm},\;\;\boldsymbol{OI}_{ij}^{-}\boldsymbol{P}_{i}^{+}=\boldsymbol{I}_{ij}^{-}\boldsymbol{P}_{j}^{\mp}.
x_1x_2+x_2x_1=0,\;\; x_2x_3+x_3x_2=0, x_1x_3+x_3x_1=0,\;\; x_3x_4+x_4x_3=0, x_1x_4+x_4x_1=x_3^2, x_2x_4+x_4x_2=0.
\sum_{i=1}^{2n+2}N(r,\nu_i)\ge \dfrac{2}{n}\sum_{u=1}^3\sum_{i=1}^{2n+2}N^{(n)}_{(f^u,H_i),\leqslant k_i}(r)+\dfrac{1}{3}\sum_{u=1}^3\sum_{i=1}^{2n+2}N^{(1)}_{(f^u,H_i),\leqslant k_i}(r) &-\dfrac{2(n+1)}{n}T(r)+o(T(r)).
[\bar{e}_{0,k+3}, \bar{e}_{0,l}]-(1+d+d^{-1})[\bar{e}_{0,k+2}, \bar{e}_{0,l+1}]+ (1+d+d^{-1})[\bar{e}_{0,k+1}, \bar{e}_{0,l+2}]-[\bar{e}_{0,k}, \bar{e}_{0,l+3}]=0,
{\textstyle\sum_{k=\widehat{k}_1}^{\nu}}\sqrt{\Phi(x^{\ell(k)})-\Phi(x^k)}\le \sqrt{b^2/2}\,{\textstyle\sum_{k=\widehat{k}_1}^{\nu}}\Xi_k+\sqrt{(\rho\!+\!1)/2}\,{\textstyle\sum_{k=\widehat{k}_1}^{\nu}\sum_{j=k-\widehat{k}_1}^{k-1}}\big\|x^{j+1}\!-\!x^{j}\big\|.
\mu_1\colon e_1e_1' =[e_6e_7e_9]+v^{-1}[e_3e_4], \mu_3\colon e_3e_3' =[e_6e_9e_{13}]+v^{-1}[e_1e_5], \mu_4\colon e_4e_4' =v^{-1}[e_{1}^2e_{2}]+v[e_{6}e_{7}^2e_{9}^2e_{12}], \mu_6\colon e_6e_6' =v^{-1}[e_3e_4e_{11}e_{13}]+v[e_1^2e_{5}^2].
= i^{1-k}\frac{N^{-\frac{k}{2}}N_1^sR^{s}\lambda_p(\kappa)^{-1}p^{(2-k)/2}{p}^{s+1}s! G\left({(\chi_1\chi')}^{-1}\right)}{{(2 \pi i )}^{s+1}2^{s}\chi_1\chi'(p)}.
x^2 -mxy+y^2 -(a^2 -m ab + b^2 ) = mj(mju^4 - m^2ju^mv + mju^2v^2 - mu^2 + 2uv)a^2 & + mj(mju^2v^2 - m^2juv^m + mjv^4 - 2uv + mv^2)b^2 &- 2mj(mju^mv - m^2ju^2v^2 + mjuv^m - u^2 + v^2)ab
\frac{\mathrm{det}(h_{ij})}{\mathrm{det}(b_{ij})}= \frac{h(V_1,V_1)h(V_2,V_2)}{b(V_1,V_1)b(V_2,V_2)}= \frac{\lambda_1\lambda_2\langle du^{-1}_{\eta(p)}V_1,V_1\rangle \langle du^{-1}_{\eta(p)}V_2,V_2\rangle \langle\eta,\xi\rangle^{-2}}{\langle du^{-1}_{\eta(p)}V_1,V_1\rangle \langle du^{-1}_{\eta(p)}V_2,V_2\rangle \langle\eta,\xi\rangle^{-2}}= \lambda_1\lambda_2,
f(a)\cdot (h_1\cdots h_{n+1})=\langle(id\otimes f)\circ\Delta_A(a), (h_1\cdots h_{n+1})\otimes id\rangle =\langle(id^{\otimes n+1}\otimes f)\circ\Delta_A^{(n+1)}(a), (h_1\otimes \cdots \otimes h_{n+1})\otimes id\rangle=0,
\tilde T_6(z) = \frac{i}{(2\pi)^{\frac12}}\int_{\xi_1+\xi_2+\xi_3=0}\frac{2 z^2- \frac13 (\xi_1^2+\xi_2^2 +\xi_3^2)}{z(z^2-\xi_1^2)(z^2-\xi_2^2)(z^2-\xi_3^2)}\hat u(\xi_1) \hat u(\xi_2) \hat u(\xi_3) d\xi_1d\xi_2