harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\partial_i(e(T) \otimes 1) := \sum_{j=1}^{i+3}\sum_{\sigma \in H}(-1)^{j-1}e(\sigma(T_j)) \otimes x_{a_j}\in V_{(d-1, d-i+2, 1^{i-1})}\otimes_K R(-d-i) =F_{i-1},
\sum_{k = 1}^{2}\left[ B_{i k}( t ) \dot{B}_{j k}( t ) - \dot{B}_{i k}( t ) B_{j k}( t ) \right] = \sum_{k = 1}^{2}\left[ B_{i k}^{\ast}( t ) \dot{B}_{j k}^{\ast}( t ) - \dot{B}_{i k}^{\ast}( t ) B_{j k}^{\ast}( t ) \right] = 0 \; .
\tau^{M}(T)\leq 2^{-(M - 1)}\sum_{i=1}^{w_{M-1}(T)}\tau^1(T_{v_i})\leq 2^{-(M - 1)}\sum_{i=1}^{w_{M-1}(T)}\deg(T_{v_i}) \le 2^{-(M-1)}w_M(T).
\|f\|_{B^{i,\ell}_{q,\theta}}= \bigg(\int\limits_0^{\infty}\Big(h^{-\ell}\big\| \Delta_i^s(h) f\big\|_{L^q(\mathbb{R}^n)}\Big)^{\theta}\frac{dh}{h}\bigg)^{\frac{1}{\theta}}.
f_{d,e}^{(m),i,b}= f_{\delta_{m-1},\varepsilon_{m-1}}^{T_{d,e}^{(m-1)}(i),b}\cdots f_{\delta_1,\varepsilon_1}^{T_{d,e}^{(1)}(i),b}\cdot f_{\delta_0,\varepsilon_0}^{i,b}.
\begin{aligned}\left\Vert\frac{1}{T}\int_0^T B(s) K_1(I-P^d_{M_{k-1}})e^{-it(s)H_d}e^{it(s)H}{\hat P}^{k-2}_{{\hat M}_d}e^{-isH}E_H(B)ds\right\Vert &+\left\Vert\frac{1}{T}\int_0^T B(s) K_2(I-e^{-it(s)H_d}e^{it(s)H}){\hat P}^{k-2}_{{\hat M}_d}e^{-isH}E_H(B)ds\right\Vert.\end{aligned}
\alpha(t[\pi] + (1-t)[\rho])(a) = \alpha(\sigma_M(\pi \otimes p + \rho \otimes p^{\perp}))(a) & = \tau(\sigma_M(\pi(a) \otimes p)) + \tau(\sigma_M(\rho(a) \otimes p^{\perp})) & = t\tau(\pi(a)) + (1-t)\tau(\rho(a)) & = (t\alpha([\pi]) + (1-t)\alpha([\rho]))(a).
\frac{\partial \tilde{\epsilon}}{\partial \lambda_{jb}}(\lambda) = -s_b \epsilon_0'(\lambda_{jb})F(\lambda_{jb}|\lambda) -\sum_i\int_{\lambda_{iL}}^{\lambda_{iR}}d\nu\, \epsilon_0'(\nu) \frac{\partial F}{\partial \lambda_{jb}}(\nu|\lambda),
\begin{array}{l l}{{I_{( a )}+ I_{( b )}+ I_{( c )}+ I_{( d )}}}&{{\sim \left[ G_{+ +}^{( 0 )}( k + p_{2}) - G_{- +}^{( 0 )}( k + p_{2}) \right] G_{R}( k - p_{1})}}\\{{{}}}&{{= G_{A}( k + p_{2}) G_{R}( k - p_{1}) .}}\\ \end{array}
g_1= t_1 t_3+{1\over 2}t_2^2+{1\over 2}t_1^2 t_2+{1\over 8}t_1^4, g_2= t_2 t_3-{1\over 2}t_1 t_2^2+{1\over 2}t_1^3 t_2+{1\over 8}t_1^5, g_3={1\over 2}t_3^2-{1\over 3}t_2^3+t_1^2 t_2^2+{1\over 4}t_1^4 t_2+{1\over 6}t_1^6.
\nabla_U^{nc}\nabla_Z^{nc}\begin{pmatrix}0 X 0 \end{pmatrix}= \begin{pmatrix}* \nabla_U^g \nabla_Z^g X + K^g(X,Z) \cdot U +K^g(U)^{\sharp}-g(U,\nabla^g_Z X)\end{pmatrix}\in \Gamma(\mathcal{H}).
I(f_R) \leq \begin{cases}\frac{2\pi}{\ln \frac{\tanh(R/2)}{\tanh(a/2)}}(R^2 - a^2) : d=2, \frac{4 \pi \tanh a}{\tanh R - \tanh a}\tanh R (R^2 - a^2) : d=3. \end{cases}
B_j^2=B_j\quad\mbox{for all}0\le j\le n, B_jB_{j'}B_j=B_{j'}B_jB_{j'}\quad\mbox{for all}0\le j',j\le n, B_jB_{j'}=B_{j'}B_j\quad\mbox{for all}0\le j'< j\le n\mbox{with}j-j'\ge 2.
\begin{aligned}\overline{15}^n_{23106}\to/<->/<1em> \overline{15}^n_{56014}\overline{15}^n_{23432}\to/<->/<1em> \overline{15}^n_{56014}\overline{15}^n_{44028}\to/<->/<1em> 15^n_{50224}\overline{15}^n_{73047}\to/<->/<1em> 15^n_{91280}\end{aligned}
S = \frac{1}{4 \pi \alpha^{\prime}}\int_{\Sigma}d^{2}\sigma \sqrt{- h}h^{a b}\partial_{a}X^{\mu}\partial_{b}X^{\nu}- \int_{\partial \Sigma}d s \left( A_{\mu}\frac{\partial}{\partial s}X^{\mu}+ \phi^{i}\frac{\partial}{\partial \sigma}X^{i}\right)
{\bf E}\bigl[\beta_{M, N}(a, b)^{k}\bigr] =\exp\Bigl(-\sum\limits_{l=0}^{k-1}\bigl(\mathcal{S}_N\log\Gamma_{M-1}\bigr)(l \,|\,\hat{a}_i, b)\Bigr),{\bf E}\bigl[\beta_{M, N}(a, b)^{-k}\bigr] =\exp\Bigl(\sum\limits_{l=0}^{k-1}\bigl(\mathcal{S}_N\log\Gamma_{M-1}\bigr)(-(l+1) \,|\,\hat{a}_i, b)\Bigr), \; k<b_0.
D_{\psi_i}=\sum_{N=0}^{+\infty}{\left(\frac{\Gamma_i e^{-{{\hbar\omega_i}\over kT}}}{1-\left(1-\Gamma_i\right)e^{-{{\hbar\omega_i}\over kT}}}\right)}^N\left|\rho_i^N\right>\left<\rho_i^N\right|\;.
M_1^\top M_1 =\begin{pmatrix}q J_q -J_q \cdots -J_q -J_q (q+1)I_q-J_q \cdots (q+1)I_q-J_q \vdots \vdots \ddots \vdots -J_q (q+1)I_q-J_q \cdots (q+1)I_q-J_q \end{pmatrix}.
\mathcal{P}_{x_{1}}M =\left\{\theta\in\mathcal{P}M\mid\theta\left(0\right) =x_{1}\right\}\mathcal{P}^{x_{2}}M =\left\{\theta\in\mathcal{P}M\mid\theta\left(1\right) =x_{2}\right\}\mathcal{P}_{x_{1}}^{x_{2}}M =\left\{\theta\in\mathcal{P}M\mid\theta\left( 0\right) =x_{1}\theta\left( 1\right)=x_{2}\right\}
y \in \mathcal{L}(Y)\;\;\iff\;\;\mathcal{L}^{\circ}\mkern-1.5mu(\hat{Y}[y])\mkern1mu \cup \mkern1mu\varphi_{\mkern-1.7mu n_{{\textsc{g}}}\mkern-.5mu *}^{{\mathbb{P}}}(\mathcal{L}^{\circ}\mkern-1.5mu(Y_{n_{{\textsc{g}}}}))={\mathbb{Q}}\cup \{\infty\}
\wp =\underset{n\rightarrow\infty}{\lim}\operatorname*{Vol}\left(X(K_{n},\mathcal{O}_{K_{n}}^{\times,+})\right) & =\frac{(s+1)}{4^{s}\cdot2^{s^{2}}}\cdot\underset{n\rightarrow\infty}{\lim}\sqrt{\left\vert \triangle_{K_{n}/\mathbf{Q}}\right\vert}\cdot R_{K_{n}}
\left|\mathbb{E}\left\{g(X^{x}_{\tau'})\right\}-\mathbb{E}\left\{g(X^{x}_{\tau})\right\}\right| = \left|\mathbb{E}\left\{\sum_{i=\tau}^{\tau'-1}(P g-g)(X^{x}_i)\right\}\right|\leq \mathbb{E}\left\{\sum_{i=N^{x}_\varepsilon}^{\infty}|P g-g|(X^{x}_i)\right\}.
\varphi^*\left(\begin{array}{c c}a b 0 d \end{array}\right) =\left(\begin{array}{c c c}a^{2 \, p^{e_1}}0 0 0 1 b^{p^{e_1}}\, d^{p^{e_1}}0 0 d^{2 \, p^{e_1}}\end{array}\right) (\, e_1 \geq 0 \, ) .
f_{n,d}(n) = Lag\Big\{n, G_1G_2 m_s h_{sr}h_{rr}h_{rd}, G_1G_2 (m_{br_1}+ n_{sp}) & \times h_{rr}h_{rd}+ G_2 (m_{br_2}+ n_{sp})h_{rd}+ m_{bd}, D \Big\},
\left(\bigsqcup_{a \in \mathcal{A}}\mathcal{V}_a\right) \times \left(\bigsqcup_{b \in \mathcal{B}}\mathcal{V}_b\right) = \bigsqcup_{(a,b) \in \mathcal{A}\times \mathcal{B}}\mathcal{V}_a \times \mathcal{V}_b
H_{k+1}=H_{k}+\frac{(s_{k}^{T}y_{k}+y_{k}^{T}H_{k}y_{k})(s_{k}s_{k}^{T})}{(s_{k}^{T}y_{k})^{2}}-\frac{H_{k}y_{k}s_{k}^{T}+s_{k}y_{k}^{T}H_{k}}{s_{k}^{T}y_{k}}
A(b) = \sum_{k = 0}^{1}\left| Z_{b^{-1}}\phi \left( \tfrac{1}{4}+\tfrac{k}{2}, \tfrac{1}{2}\right) \right|^2 = 2 \left| Z_{b^{-1}}\phi \left( \tfrac{1}{4}, \tfrac{1}{2}\right) \right|^2.
\begin{aligned}\phi(\omega): =\mathrm{i}\omega\cdot l+\mu_j^{\infty}(\omega)-\mu_k^{\infty}(\omega)=\mathrm{i}\omega\cdot l-\mathrm{i}m_3(\omega) (j^3-k^3)+\mathrm{i}m_1 (j-k)+(r_j^{\infty}-r_k^{\infty})(\omega)\end{aligned}
\lim_{t\to\pm\infty}v\big(t,\widetilde{x}(t)+\widetilde{x}_+(t)\big)=\sqrt{\widetilde{c}_\pm}\hbox{or equivalently}\lim_{t\to\mp\infty}u\big(t,x(t)-\widetilde{x}_+(-t)\big)=\sqrt{\widetilde{c}_\pm}.
|B_n(A)-B(A)| = | \int_{[0, \infty)-K_n}\eta (T_tA) d \mu (t)| \leq \int_{[0, \infty)-K_n}| \eta (T_tA)| d \mu (t) \leq \| \eta \| \| A \| \int_{[0, \infty)-K_n}Me^{( \omega - \lambda)t}dt \rightarrow 0
P_{0}(z) =e^{\frac{\lambda}{\xi}z}(1-z)^{-\frac{\gamma}{\xi}}\left[P_{0}(0) -\frac{C_{\lambda, \mu}}{\xi}\int_{s=0}^{z}e^{- \frac{\lambda}{\xi}z}(1-s)^{\frac{\gamma}{\xi}-1}ds \right],
(e^{ic_1}\cdots e^{ic_m})(e^{-id_m}\cdots e^{-id_1}) =e^{i(c_m-d_m)}e^{ic_1}\cdots e^{ic_{m-1}}e^{-id_{m-1}}\cdots e^{-id_1}&=e^{i(c_m-d_m)}\cdots e^{i(c_1-d_1)}.
\bigoplus_{j\in I_{i}}K_{n}(R[H_{j}])\oplus\bigoplus_{j\in I_{i}}K_{n-1}(RH_{j})\rightarrow\bigoplus_{j\in I_{i}}K_{n}(RH_{j})\oplus KR_{n}^{G}(X^{i-1})\rightarrow KR_{n}^{G}(X^{i})
\chi R_0(z_m) \chi = \sum_{j=1}^{\frac{d-1}{2}}B_j(r, z_0 ) + i m \Big( \frac{z_0}{2 \pi}\Big)^{d-1}\mathcal{E}^*_{\chi}( \bar{z_0}) \mathcal{E}_{\chi}( z_0)
f(z_i)^*E_i =M_f^{[*]}(K_S(\cdot,z_i)E_i) \&=M_f^{[*]}((E_i-SS(z_i)^*E_i)k_{\rm Sz}(\cdot,z_i)) &=M_f^{[*]}((E_i-SN_i)k_{\rm Sz}(\cdot,z_i)) \quad\mbox{for}i=1,\ldots,n,
(\lambda\ast_{G/H}\lambda')^{\ast^{G/H}}=T_H\left((\lambda\ast_{G/H}\lambda')_q^{\ast^{G}}\right) &=T_H\left((\lambda_q\ast_{G}\lambda'_q)^{\ast^{G}}\right)=T_H\left({\lambda'}_q^{\ast^{G}}\ast_{G}\lambda_q^{\ast^{G}}\right).
\frac{1}{n!}(x_{1}^{1}\cdot x_{2}^{0}\cdot\cdot\cdot x_{n}^{0}+\cdot\cdot\cdot+x_{1}^{0}\cdot\cdot\cdot x_{n}^{1})(n-1)!\ge\frac{1}{n!}(x_{1}\cdot\cdot\cdot\cdot\cdot x_{n})^{1/n}n!
\Gamma_{+}^{(3)^*}= \left(\begin{array}{ccc}0 0 q^4\end{array}\right),\Gamma_{0}^{(3)^*}= \left(\begin{array}{ccc}0 -q^6 0\end{array}\right),\Gamma_{-}^{(3)^*}= \left(\begin{array}{ccc}q^4 0 0\end{array}\right).
u_{n+1}^j - u_{n+1,r}^j \approx (u_{n+1}- u_{n+1,r})u_n^{j-1}, (u_{n+1}+u_n)^{p+1}- (u_{n+1,r}+u_n)^{p+1}\approx \sum\limits_{j=0}^{p+1}\binom{p+1}{j}(u_{n+1}- u_{n+1,r}) u_n^p, \approx (u_{n+1}- u_{n+1,r}) u_n^p \sum\limits_{j=0}^{p+1}\binom{p+1}{j}, \approx 2^{p+1}(u_{n+1}- u_{n+1,r}) u_n^p.
\beta^{n-k-2,k-2}_D\left(z(v),z(w)\right)=\begin{cases}\sum_{i=2}^{k-1}(-1)^i\left(\xi_i\lambda_{k-i+1}+\mu_i\nu_{k-i+1}\right) k=m, \sum_{i=2}^{n-k-1}(-1)^i\lambda_i\nu_{n-k+1-i}+\sum_{i=2}^{k-1}(-1)^{i-1}\mu_i\xi_{k+1-i}k<m.\end{cases}
\|g_0^{j,t}(x)\|^2_{L^2(C^{j,t}_0, \mu_2)}\leq \frac{C}{\omega(B^g(2\sqrt{t}))}\int_{B^g(x^t_j, 4\sqrt{t})}\int_{B^g(x^t_j, 4\sqrt{t})}|f(x)-f(y)|^2 d\mu_2(x)d\mu_2(y).
|\nabla'' s_{k}|^2_{h^k, \omega}= |\xi^k \overline{\partial}\chi_{k}+k\chi_{k}\xi^{k-1}\overline{\partial}\xi|^2_{\omega}e^{-k\varphi_{0}}\lesssim |\overline{\partial}\chi_{k}|^2_{\omega}e^{-k\varphi}+ k^2|\chi_{k}\overline{\partial}\xi|^2_{\omega}e^{-k\varphi}.
\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\Big(J_p(u_h(x)-u_h(y))-J_p(u(x)-u(y))\Big)\,\Big(\varphi(x)-\varphi(y)\Big)\,d\mu(x,y)=\int_\Omega \delta_h f\,\varphi\,dx,
\psi ([a,b]_T) =~ \psi \big( \rho (T(a))b - \rho (T(b))a + \lambda [a,b]_V \big) =~ \rho (\phi(T (a)))(\psi (b)) - \rho (\phi(T (b)))(\psi (a)) + \lambda [\psi (a), \psi (b)]_V =~ \rho(T' (\psi (a)))(\psi (b)) - \rho(T'(\psi (b)))(\psi (a)) + \lambda [\psi (a), \psi (b)]_V = [ \psi (a), \psi (b) ]_{T'}.
z+ \lim\limits_{t \to \infty}{Q\big(z+(1-z)(1-p_t)\big)-(1-p_t) -zp_t \over p_t\big(1-Q'(1-p_t)\big)}=z+\lim\limits_{x \rightarrow 1-}{Q\big(z+(1-z)x\big)-\big(z+(1-z)x\big) \over (1-x)(1-Q'(x))}&=z+{1 \over 2-L}(1-z)^{2-L},
\widetilde{\gamma}_{F,k^*}=\frac{P_F \beta_{F,k^*}|{\bf h}_{F,k^*}^H{\bf w}_{i^*}|^2}{P_M \beta_{M,k^*}|{\bf h}_{M,k^*}^H{\bf f}_{j}|^2+\sigma_F^2}\geq \Gamma_F,
I(p)=\langle a_{d}b_{d-1}, a_{d}b_{d-2}, a_{d-1}b_{d-2}, \dots , a_{d}b_{j}, \dots , a_{j+2}b_{j}\rangle: \langle a_{j+1}b_j \rangle =\langle a_{j+2}, \dots, a_d\rangle.
\int_0^\infty m^s \int |\nabla f_m|^2 dx dm = \int \left(\frac{\sin \pi s}{\pi}\int_0^\infty \frac{m^s}{(|\xi|^2 +m)^2}dm \right) |\xi|^2 |\hat{f}(\xi)|^2 d\xi = s \|f\|^2_{\dot{H}^s}.
E_{t-1}[y^2_{t-1}w_{m,t-1}I_{|y_{t-1}|\geq M}|z_t]= \int_{S_{\mathcal{I}_{t-2}},y_{t-1}\leq -M}{y^2_{t-1}[w_{m,t-1}]p(\mathcal{I}_{t-1}|z_t)d\mathcal{I}_{t-1}}&+\int_{S_{\mathcal{I}_{t-2}},y_{t-1}\geq M}{y^2_{t-1}[w_{m,t-1}]p(\mathcal{I}_{t-1}|z_t)d\mathcal{I}_{t-1}},
\sum_{(d_1, \ldots, d_N)}q_1^{d_1}\cdots q_N^{d_N}\left[h_1^{d_1}\ldots h_N^{d_N}\right] \left(\Phi_1(h_1)^{d_1+1}\cdots \Phi_N^{d_N+1}(h_N) \cdot \Psi (h_1, \ldots, h_N)\right) = \frac{\Psi}{J}
F_{QR,P}(\Omega)(\zeta_i \otimes c(\sigma(j))\alpha_{\sigma(j)}) =(2g_C+1)\left(\int_{C-\gamma}\log(f) dz_i \wedge dx_j + 2\pi i \int_{\gamma}dz_i dx_j \right) &=-(2g_C+1)\left(\int_{C-\gamma}\log(f) dx_j \wedge dz_i - 2\pi i \int_{\gamma}dz_idx_j \right)
{_{5}F_{4}}\left[\begin{array}{cccccccc}a,1+\frac{a}{2},b,c,d \frac{a}{2},1+a-b,1+a-c,1+a-d\end{array};1\right] begin{align*}1mm] \:\:=\frac{\Gamma(1+a-b)\Gamma(1+a-c)\Gamma(1+a-d)\Gamma(1+a-b-c-d)}{\Gamma(1+a)\Gamma(1+a-b-c)\Gamma(1+a-b-d)\Gamma(1+a-c-d)},
\sigma_{\mathrm{ess}}(A_\Gamma) = \sigma_{\mathrm{ess}}(A_\Gamma^{(\mathrm{s})}) = \sigma_{\mathrm{ess}}(A_\Gamma^{(\mathrm{as})}) = \sigma_{\mathrm{ess}}(A_\Gamma^{(1)}) = [ 0, \infty)
U_j(X,t) = \frac{U(r_jX, r_j^2\, t)}{\bigg(\frac{1}{r_j^{n+1+a}}\int_{\mathbb B_{r_j/2}^+}U^2(X, 0) x_{n+1}^a dX \bigg)^{1/2}},\,\,\,\ \ \ \j\geq 1.
\Phi(u^i,u^i{}_j,u^i{}_{jk}) =(u^i,u^i{}_j,u^i{}_{jl}v^l{}_k), *\Psi(x^i,y^i{}_j,z^i{}_{jk}) =(x^i,y^i{}_j,z^i{}_{jl}y^l{}_k),
K(u_0,v_0)=0, K(u_0,v_i)=-\rho_i^2,K(v_0,v_i)=-\lambda_{\,i}^2, 1\leq i\leq n-1, \noalign{\smallskip}K(v_i,v_j)=-(\rho_i\rho_j+\lambda_{\,i}\lambda_j), 1\leq i\not= j\leq n-1,
{f_{16}}={f_{15}}+ \frac{1}{2}\left({{f_3}-{f_4}}\right) + \frac{1}{2}\left({{f_7}-{f_8}}\right) - \frac{1}{2}\left({{f_9}-{f_{10}}}\right) + \frac{1}{8}\left({2{F_y}+{F_z}}\right) - \frac{1}{6}\rho{u_z},
\frac 1 2{\frac{g_{2}^{2}v^{2}}{4}}( 2 W_{+}^{\mu}W_{\mu -}+ W_{3}^{\mu}W_{\mu 3}) +{\frac{g_{2}g_{1}v^{2}}{4}}W_{3}^{\mu}W_{\mu 4}+ \frac 1 2{\frac{g_{1}^{2}v^{2}}{4}}W_{4}^{\mu}W_{\mu 4}
V_{\cal{C}}^{- 1}({\bf{r}}) = \exp \left( i g{\cal X}^{\alpha}({\bf{r}}){\textstyle \frac{\lambda^{\alpha}}{2}}\right) \, \exp \left( \, i g{\overline{{{{\cal{Y}}^{\alpha}}}}}({\bf{r}}){\textstyle \frac{\lambda^{\alpha}}{2}}\, \right) \; .
p_c(T_\xi, r) \geq 1 - \frac{1}{G_\xi^r(1-t_0)}= \frac{G_\xi^r(1-t_0) - 1}{G_\xi^r(1-t_0)}&=\frac{t_0 \left(1-\frac{1}{r}\right)}{1-t_0}\frac{1-t_0}{1-\frac{1}{r}t_0}& = \frac{t_0 \left(1-\frac{1}{r}\right)}{1-t_0/r}\geq t_0 \left(1- \frac{1}{r}\right) &= \left(1-\frac{1}{r}\right) \left(\frac{(r-1)!}{\mathbb{E}[\xi^r]}\right)^{1/(r-1)}.
A=[\begin{matrix}&\vdots&\vdots&\vdots&\vdots\\ \cdot\cdot\cdot&a_{0}&a_{-1}&a_{-2}&a_{-3}&\cdot\cdot\cdot\\ \cdot\cdot\cdot&a_{1}&a_{0}&a_{-1}&a_{-2}&\cdot\cdot\cdot\\ \cdot\cdot\cdot&a_{2}&a_{1}&a_{0}&a_{-1}&\cdot\cdot\cdot\\ \cdot\cdot\cdot&a_{3}&a_{2}&a_{1}&a_{0}&\cdot\cdot\cdot\\ &\vdots&\vdots&\vdots&\vdots\end{matrix}].
\left<\phi_{2}-\frac{\pi}{2\sqrt{3}}\tilde{\phi}_{1},\phi_{2}-\frac{\pi}{2\sqrt{3}}\tilde{\phi}_{1}\right>_{L^{2}}^{2}= \left<\phi_{2},\phi_{2}\right>-\frac{\pi}{\sqrt{3}}\left<\bar{\phi}_{1},\phi_{2}\right>+\frac{\pi^2}{12}= \frac{\pi^{2}}{3}-\frac{\pi^{2}}{6}+\frac{\pi^2}{12}= \frac{3\pi^{2}}{12}= \frac{\pi^{2}}{4}
[ d_{i}\, , d_{j}] = 0 \, , \, \, d_{i}\, e_{j}= q^{{\delta}_{i j + 1}-{\delta}_{i j}}e_{j}\, d_{i}\, , \, \, d_{i}\, f_{j}= q^{{\delta}_{i j}-{\delta}_{i j + 1}}f_{j}\, d_{i}
{\mathcal{I}}=\frac{1}{2x^{2}}\sum_{n=1}^{\infty}l^{D}\int_{0}^{\infty}dz\,z^{D-2}(1+z^{2}x^{2})^{1/2}e^{-2l[\eta(z)-\eta (zx)]}\left[ b^{(-)}(zx)-b^{(+)}(zx)\cos (2l\phi )\right],
\mathbf{y}_{i}=\sum_{k=1}^{K}\mathbf{H}_{i,k}\mathbf{v}_{i,k}s_{i,k}+\sum_{n=1}^{K}\mathbf{G}_{i',n}\mathbf{v}_{i',n}s_{i',n}+\mathbf{n}_{i},
\left( B_{\ell,\hbar}\right) =\left(\mathcal{B}_{\ell,\hbar}\right) \leq\max\left\{\left(b_{0,\hbar}^{n}\right) ,0\right\}& \leq\max\left\{\left( \mathcal{B}_{0,\hbar}\right),0\right\}=\max\left\{\left( B_{0,\hbar}\right) ,0\right\}.
\mathfrak{S}_{1}= \mathfrak{R}_{1}\mathfrak{S}_{2}= [\mathfrak{R}_{2}, \mathfrak{R}_{3}] &= ((y^{2}- 1)x^{4}, (y^{2}- 1)x^{3}y, (y^{2}- 1)x^{2}y^{2}, (y^{2}- 1)xy^{3}, xz, yz, z^{2}).
||p||_\infty \leq 2(n+1)^2||q||_\infty &=2(n+1)^2\sup_x \left|\int_0^x p(y)\,dy\right| &\leq 2(n+1)^2\sup_x \int_0^x |p(y)|\,dy &= 2(n+1)^2\int_0^1 |p(y)|\,dy &= 2(n+1)^2||p||_1.
\left( \bar{D}_{0}^{- 1}\right)_{\Theta \Phi}^{a b}( x , y ) = - i \delta^{a b}\left( \begin{array}{c c}{{{\cal O}_{\mu \nu}( x )}}&{{x_{\mu}}}\\{{x_{\nu}}}&{{0}}\\ \end{array}\right) \delta ( x - y ) \, .
\langle J(y\otimes_D x), m\otimes n\rangle = \sum_i \langle e_ix\otimes ye_i^*,m\otimes n\rangle=\sum_i\tau(e_ixm)\tau( ye_i^*n) &=\sum_i\tau(e_iE_D(xm))\tau( E_D(ny)e_i^*)=\tau(xm E_D(ny)).
\varphi'(U_{\ell}){R_{\ell}}\partial_j U_{\ell,k}= \partial_j ({R_{\ell}}\varphi'(U_{\ell})U_{\ell,k}) - 2\sqrt{R_{\ell}}U_{\ell,k}\varphi'(U_{\ell}) \partial_j \sqrt{R_{\ell}}- R_{\ell}U_{\ell,k}\varphi''(U_{\ell}) \partial_{j}U_{\ell},
\big\langle D^F_{Z_{1,R}}\mu,\mu \big\rangle_{Z_{1,R}}-\big\langle \mu,D^F_{Z_{1,R}}\mu \big\rangle_{Z_{1,R}}= \big\langle \lambda \mu,\mu \big\rangle_{Z_{1,R}}-\big\langle \mu,\lambda\mu \big\rangle_{Z_{1,R}}= 0 .
I_{\beta \mu \lambda \rho}( q ) = \int d^{d}k \frac{2 k_{\beta}k_{\mu}( q_{\nu}p_{\rho}+ q_{\nu}q_{\rho}) + k_{\beta}( q_{\nu}q_{\rho}p_{\mu}- q_{\nu}q_{\mu}p_{\rho})}{( k + p )^{2}( k - q )^{2}k^{2}}.
P(\bigcap_{n_0 \geq 1}\bigcup_{n \geq n_0}\left\{(t,z,\omega):|f(t,z,\omega)|>n \right\}) \leq \inf_{N \geq 1}(\mathbb{P}\circ\mu)(|f(t,z,\omega)|>n) &\leq \inf_{N \geq 1}(\mathbb{P}\circ\mu)(\cup_{n \geq 1}|f(t,z,\omega)|>n) &\leq\inf_{N \geq 1}\sum\limits_{n \geq N}(\mathbb{P}\circ\mu)(|f(t,z,\omega)|>n)=0
\frac{\partial E}{\partial t}(t,x;b,y;\mu,\nu^2) \Big|_{y=x\pm(t-b)}= 2^{\sqrt{\delta}-1}(1+b)^{\frac{\mu}{2}-1}(1+t)^{-\frac{\mu}{2}-1}\bigg[ 2^{-3}(1-\sqrt{\delta})^2(t-b) + \big(-\tfrac{\mu}{2}+\tfrac{1-\sqrt{\delta}}{2}\big)(1+b) +2^{-2}(\sqrt{\delta}-1) (t+b+2) \bigg].
y_{2}^{[m]}+ (-1)^{n}z_{m}=h\displaystyle\sum_{i=1}^{2}b_{2i}F_{i}- y_{2}^{[m-1]}+ (-1)^{n-1}h\displaystyle\sum_{i=1}^{2}\frac{\partial f}{\partial y}b_{2i}u_{i2}z_{m-1}- (-1)^{n-1}z_{m-1}, \Rightarrow z_{m}=\big(1-h\displaystyle\sum_{i=1}^{2}\frac{\partial f}{\partial y}b_{2i}u_{i2}\big)z_{m-1}.
\phi_{m}( z , \bar{z}) = \frac{\bar{z}^{m}}{\sqrt{\pi \Gamma ( m + 1 )}}\left( \frac{e B}{2}\right)^{( m + 1 ) / 2}\exp \left( - \frac{e B}{4}z \bar{z}\right) \left( \begin{matrix}{{0}}\\{{1}}\\ \end{matrix}\right) ,
H_{SR}=\hbar\sum_{j=1}^{2}(\hat{a}_{sj}\hat{\Gamma}_{Rsj}^{\dagger}+\hat{\Gamma}_{Rsj}\hat{a}_{sj}^{\dagger}+\hat{a}_{pj}\hat{\Gamma}_{Rpj}^{\dagger}+\hat{\Gamma}_{Rpj}\hat{a}_{pj}^{\dagger})+\hbar(\hat{a}_{c}\hat{\Gamma}_{Rc}^{\dagger}+\hat{\Gamma}_{Rc}\hat{a}_{c}^{\dagger})
\sigma^{( 1 )}= \frac{m^{2}}{8 \sqrt{3}\pi}\eta^{- 1 / 2}\int_{0}^{1}d{\tilde{h}}({\tilde{h}}- 1 ) ( 1 7 + 1 8{\tilde{h}}) = - \frac{2 3 m^{2}}{1 6 \sqrt{3}\pi}\eta^{- 1 / 2}.
(x_1,y_1, \dots, x_n,y_n,t) *(x_1',y_1', \dots, x_n',y_n',t') & = \Big(x_1+x_1', y_1+y_1', \dots , x_n + x_n' , y_n+y_n', t+t'+2 \sum_{j=1}^n (x_j' y_j - x_j y_j')\Big)
\frac{1}{\mu(B)}\int_B e^{\gamma \frac{|b(x)-b_B|}{\lambda \|b\|_{{\rm bmo}_{\mathcal{B}}}}}\,d\mu \leq \left(\frac{1}{\mu(B)}\int_B e^{\gamma \frac{|b(x)-b_B|}{\|b\|_{{\rm bmo}_{\mathcal{B}}}}}\,d\mu\right)^{\frac{1}{\lambda}}\leq C^{\frac{1}{\lambda}}=2,
I ( g , A ) = \frac{2 k}{\pi}\int d^{2}z ( 1 - u v ) [ A - \frac{u \partial v - v \partial u}{4 ( 1 - u v )}] [ \overline{{{A}}}+ \frac{u \overline{{{\partial}}}v - v \overline{{{\partial}}}u}{4 ( 1 - u v )}] - \frac{k}{4 \pi}\int d^{2}z \frac{\partial u \overline{{{\partial}}}v - \partial v \overline{{{\partial}}}u}{1 - u v}
\varphi\Bigl(u^{\frac{l}{s+nh+1}}\Bigr) \cdot u = \varphi\Bigl(u^{\frac{l+1}{s+nh+1}}u^{\frac{s+nh}{s+nh+1}}\Bigr) \in \varphi\Bigl(u^{\frac{l+1}{s+nh+1}}\Bigr) \cdot \prod_{i=1}^n I^{(s_i+1)}.
\epsilon^{\mu \nu \rho \lambda \sigma}K_{\lambda \sigma}^{( \epsilon )}= - \delta_{\mu^{\prime}\nu^{\prime}\rho^{\prime}}^{\mu \nu \rho}{\frac{G^{5 \rho^{\prime}}}{G^{5 5}}}\tilde{H}^{\mu^{\prime}\nu^{\prime}}.
\frac{\sqrt{n}}{\sin^{2}\left( n ( y_{0}- | y | ) \right)}{\,}_{2}F_{1}\left( 5 / 4 \mp 3 \delta / 4 , \, 5 / 4 \pm 3 \delta / 4 ; \, 3 ; \, \sin^{- 2}\left( n ( | y | - y_{0}) \right) \right)
(i)(\mathcal{M},g) = (M_1 \times M_2, g_1 \otimes g_2), q_0 \in M_2, T= \Sigma = M_1 \times \{q_0\}, (ii) cd(m, \Sigma)^{2 \gamma}\leq b(m) \leq Cd(m, \Sigma)^{2 \gamma}\forall m \in M_1 \times U.
h(D^t_{X_1}X_2, X_3) = h(\nabla_{X_1}X_2, X_3)+\frac{1}{4}[dF(JX_1, JX_2, JX_3)-dF(JX_1, X_2, X_3)] ~~~~~~~~~~~~~~~~~~~~~~~~~~-\frac{t}{4}[dF(JX_1, JX_2, JX_3)+dF(JX_1, X_2, X_3)],
D_{m m^{\prime}}^{j}( \psi ) = \frac{1}{\sqrt{( j + m^{\prime}) !}}~ \frac{1}{\sqrt{( j - m^{\prime}) !}}~ ( \frac{\partial}{\partial \xi_{1}^{*}})^{j + m}~ ( \frac{\partial}{\partial \xi_{2}^{*}})^{j - m}~ < \xi \mid U ( \psi ) \mid j , m^{\prime}>
\lim_{\Lambda_{1}\uparrow\uparrow V}\psi_{\Lambda_{1}}(b_{\Lambda})=\sum_{i\in I}\prod_{x\in \Lambda}\hbox{Tr}_{\mathcal{H}_{x}}\left(h_{x, i}h_{x, i}^* b_{x}\right)=: \hat{\psi}_{\Lambda}(b_{\Lambda}).
(x^2+x) F_{p+2}'(x) + (2x+1) F_{p+2}(x) + F_p'(x) = 2x \sum_{j=1}^\infty \frac{1}{Z_j^p}-p x^2 \sum_{j=1}^\infty \frac{1}{Z_j^{p+1}}\left( \frac{dZ_j}{dx}\right).
\theta_{0}(q,q^{-3}u) =1+u^3+q^{18}u^{-3}+q^{18}u^{6}+\cdots, \theta_{1}(q,q^{-3}u) =q^{-2}u+q^{10}u^{-2}+q^{4}u^4+\cdots, \theta_{2}(q,q^{-3}u) =q^{-2}u^2+q^4u^{-1}+q^{10}u^{5}+q^{28}u^{-4}+\cdots, \lim_{q\to 0}\ [\theta_0,\theta_1,\theta_2] (q,q^{-3}u)=\lim_{q\to 0}\ [1,q^{-2}u,u^2]=[0,1,\overline{u}]
G^+_s(\gamma_1,\gamma_2) = \frac{2 i \sqrt{J_1J_2}}{E_q(J_1,J_2)}\sum_{n_1,n_2 =0}^\infty\frac{J_1^{n_1}J_2^{n_2}}{[n_1]_q![n_2]_q!}\sin(\gamma_1 q^{2[n_1]_q}+\gamma_2q^{2[n_2]_q})\,;
\frac{\partial_{1}\beta}{\beta}+ \frac{\gamma}{8 \Omega}+ \frac{\beta^{2}}{2}\Omega V ( \Omega ) - \frac{\Omega^{\delta}}{4}\{\frac{\beta^{2}}{\alpha^{2}}( \partial_{0}f )^{2}+ ( \partial_{1}f )^{2}\}= 0 ,
K = (1/A_0) \big(\mathop{\sup}\limits_{m \in \mathbb{R}_+}\big|{m\int\limits_0^{+ \infty}{\beta \left({m,m'}\right)dm'}}\big| + \mathop{\sup}\limits_{m \in \mathbb{R}_+}\big|{\frac{m}{2}\int\limits_0^m{\beta \left({m - m',m'}\right)dm'}}\big| \big).
\hat P^{[1]}_n(x) =\Big(I_2p_{n+1}(x)-p_{n+1}(A)\big(p_n(A)\big)^{-1}p_n(x)\Big)(I_2x-A)^{-1}, \hat P^{[2]}_n(x) =-K_{n}(x,A)\big[P_n^{[1]}(A)\big]^{-1}
\sum_{\chi \in \mathcal F_{\pi,n}}\chi(a) L(1/2+\alpha_1,\chi) \overline{L(1/2+\alpha_2,\chi)}=\sum_{d_1=0}^{N}\sum_{d_2=0}^N \sum_{\chi \in \mathcal F_{\pi,n}}\chi(a)a_{d_1}(\chi) \overline{a_{d_2})(\chi)}q^{- d_1 \alpha_1+ d_2 \alpha_2}.
\sum_{k=0}^{n-1}\frac{(q;q^2)_k(-1;q^2)_k}{(q^2;q^2)_k}q^{(n-1)^2-k^2}\equiv q^{1-n}+\frac{(1-n)q(1-q^n)}{2}\pmod{\Phi_n(q)^2}.
g_{\mu \rho}\hat{\Gamma}_{\nu \lambda}^{\rho}\frac{u^{\lambda}}{\hat{\gamma}}= \hat{\Gamma}_{\mu , \nu \lambda}\frac{u^{\lambda}}{\hat{\gamma}}= F_{[ g ] \mu \nu}= \frac{\partial A_{[ g ] \nu}}{\partial x^{\mu}}- \frac{\partial A_{[ g ] \mu}}{\partial x^{\nu}}
F ( \beta ) = N_{F}f_{S}( \beta ) = - 2 N_{F}\int_{0}^{\infty}d s s^{- 1 - \frac{d}{2}}\theta_{2}\left( 0 \left| \frac{2 i \beta^{2}}{\pi s}\right. \right) e^{- m^{2}s / 2}
\mathcal{L}_{m}^{\Gamma,-}= \left([1-C_{-}m^{-\frac{1}{2}}]\mathcal{L}^{\Gamma}-\frac{\kappa^2}{4}+K-C_{-}m^{-1}\right)_{+}^{\frac{1}{2}}\,, \mathcal{L}_{m}^{\Gamma,+}= \left([1+C_{+}m^{-\frac{1}{2}}]\mathcal{L}^{\Gamma}-\frac{\kappa^2}{4}+K+C_{+}m^{-1}\right)^{\frac{1}{2}}\,.
g_{n+1}(f_{n+1}- f_n ) \phantom{olao}= \displaystyle\frac{1}{2}(n+1) (f_{n+1}+ f_n )- \displaystyle\frac{1}{4}(n+1) (f_{n+1}- f_n ) - \big(c_3 + \displaystyle\frac{1}{16}- \displaystyle\frac{c_2^2}{4}\big) (n+1) \,.
\begin{matrix}if&p_{l}=p_{1}(u)&p_{m}=p_{2}(u)&p_{n}=p_{3}(u)\\then&p_{l}^{\prime}=p_{2}(u-1)&p_{m}^{\prime}=p_{1}(u-1)&p_{n}^{\prime}=p_{3}(u-1)\end{matrix}
M_{m}L^i_{0}=(L_0+m)^iM_m, &M_{m}M^i_{0}=M^i_0M_m, &M_{m}Y^{i}_{0}=Y^i_{0}M_m, &Y_{m}L^i_{0}=(L_0+m)^iY_m, &Y_{m}M^i_{0}=M^i_0Y_m, &L_mL^i_0=(L_0+m)^iL_m, &Y_{m}Y^i_{0}=Y^i_{0}Y_m+miY^{i-1}_{0}M_{m}, &L_mM^i_0=M^i_0L_m-3miM^{i-1}_{0}M_m, &L_mY^i_0=Y^i_0L_m-miY^{i-1}_{0}Y_m-\frac{i(i-1)}{2}m^2Y^{i-2}_{0}M_m.
A=P\frac{i(1+i)^{n}}{(1+i)^{n}-1}=Pi\times\frac{(1+i)^{n}}{(1+i)^{n}-1}\times\frac{(1+i)^{-n}}{(1+i)^{-n}}=\frac{P\times i}{1-(1+i)^{-n}}
\frac{1}{\sqrt{-g^{\prime}}}\partial_{\mu}(\sqrt{-g^{\prime}}g^{\prime\mu\nu}\partial_{\nu}\phi)+ \frac{1}{\chi}\left[\frac{dV_{1}}{d\phi}-\frac{1}{\chi}\frac{dV_{2}}{d\phi}+\frac{\alpha}{M_{p}}P(\varphi)e^{\alpha\phi /M_{p}}\right]=-\frac{h\gamma}{\sqrt{2}M_{p}}\overline{\Psi}^{\prime}\Psi^{\prime}\varphi \frac{e^{\gamma\phi/M_{p}}}{\chi^{3/2}};
\|\sum_{\pi\in\mathrm{Irr}(\mathbb{G}_{1})}c_{\pi}\chi_{\pi}\|_{L^{\infty}(\mathbb{G}_{1})}=\|\sum_{\pi\in\mathrm{Irr}(\mathbb{G}_{1})}c_{\pi}\chi_{\Phi(\pi)}\|_{L^{\infty}(\mathbb{G}_{2})}.
f_{0}= \frac 2{\chi}\ln \left( \frac{\alpha}{\cosh \left({\frac{\chi \alpha}{2}}t_{0}\right)}\right) + \frac 1{\chi}\ln \left({\frac{( d - 2 ) \chi}{( q - 1 ) b^{2}}}\right) -{\frac{a c_{2}}{\chi}}.
B_0 =\{\mu \in A^\prime : (\langle f^\prime \oplus \mu,\lambda(\mu),h^0(\mu),h^1(\mu),(H^2)^\prime(\mu(\kappa))\rangle ,a^\prime)\Vdash b \}. B_1 =\{\mu \in A^\prime : (\langle f^\prime \oplus \mu,\lambda(\mu),h^0(\mu),h^1(\mu),(H^2)^\prime(\mu(\kappa))\rangle ,a^\prime)\Vdash \neg b \}.
\sum_n \frac{\overline{\chi}(n)}{n^{\frac{1}{2}}}V \left( \frac{pn}{q}\right) = L \left( \frac{1}{2},\overline{\chi}\right) -\sum_n \frac{\chi(n)}{n^{\frac{1}{2}}}F \left( \frac{n}{p}\right).
\frac{\partial V_{e f f}}{\partial \chi}= \frac{1}{8 \pi \chi}\left[ \chi - m^{2}- \lambda_{r}\left( \phi^{2}+ \frac{1}{4 \pi}{\ln}[ \frac{m^{2}}{\chi}] \right) - \eta \left( \phi^{2}+ \frac{1}{4 \pi}{\ln}[ \frac{m^{2}}{\chi}] \right)^{2}\right]
{\cal A}_{even}\sim \frac{{\hat \mu}^2}4 \cos 3 \left(\theta_{1}- \theta_{2}\right) =e_1 e_2 + g_1 g_2 \;,\;\;{\cal A}_{odd}\sim \frac{{\hat \mu}^2}4 \sin 3 \left(\theta_{1}- \theta_{2}\right) =e_1 g_2 - g_1 e_2
\left(\begin{array}{ccccc}4(2\Delta^{2}+\Delta)+4(4\Delta+1)\theta 6(\Delta+\theta) 6(\Delta+\theta) 4(\Delta+\theta)+\frac{c}{2}\end{array}\right)\left(\begin{array}{c}b^{(1,1)}b^{(2)}\end{array}\right)=0,