harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}\frac{2\lambda^k}{m}\sum_{i=1}^m\left\langle g_i^k, \bar x^k -\theta^* \right\rangle\geq -\frac{\gamma^k}{m}\sum_{i=1}^m\|x_i^k-\bar{x}^k\|^2 &\qquad- \frac{L^2(\lambda^k)^2}{\gamma^k}\|\bar x^k -\theta^*\|^2+2\lambda^k (F(\bar{x}^k)-F(\theta^{\ast})) \end{aligned}
\sum_{\mathbf{v}\in\mathcal{D}_k}\frac{\prod_{p'|uv}|S_{p'}(\mathbf{u},\mathbf{v})|}{\phi_\omega(v)}=\prod_{p'|u}\Bigl(\sum_{\substack{\mathbf{w}\in\mathcal{D}_k w_i|p'\,\forall i}}\frac{|S_{p'}(\mathbf{u},\mathbf{w})|}{\phi_\omega(w)}\Bigr)=\prod_{p'|u}\Bigl(\frac{p-1}{p-\omega(p)}+\frac{\omega(p)-1}{p-\omega(p)}\Bigr).
B_{k+1}^{-1}=B_{k}^{-1}+\frac{(s_{k}^{T}y_{k}+y_{k}^{T}B_{k}^{-1}y_{k})(s_{k}s_{k}^{T})}{(s_{k}^{T}y_{k})^{2}}-\frac{B_{k}^{-1}y_{k}s_{k}^{T}+s_{k}y_{k}^{T}B_{k}^{-1}}{s_{k}^{T}y_{k}}.
\widehat{L}=\frac{1}{2}\partial_{\mu}\phi (x;\alpha )\partial^{\mu}\phi(x;\alpha )-\frac{m^{2}}{2}\phi (x;\alpha )^{2}-\frac{1}{2}\partial_{\mu}\widetilde{\phi}(x;\alpha )\partial^{\mu}\widetilde{\phi}(x;\alpha )+\frac{m^{2}}{2}\widetilde{\phi}(x;\alpha )^{2},
M_{n}[\mathfrak{m}_{D(\lambda)}]= \frac{1}{n}M_{n}[\tau_{D(\lambda)}] +\sum_{\begin{subarray}{c}\nu =(\nu_1,\nu_2,\dots)\vdash n \nu_1 < n \end{subarray}}z_\nu^{-1}M_{\nu_1}[\tau_{D(\lambda)}] M_{\nu_2}[\tau_{D(\lambda)}] \cdots.
\begin{array}{c}g_1=z_0g(z_0, z_1, z_2, z_3)+(z_0-z_4)\alpha(z_0, z_1, z_2, z_3) g_2=g(z_4, z_5, z_2, z_3) g_2=g(z_4, z_5, z_6, z_3)\end{array}
\begin{bmatrix}u_y v_y \end{bmatrix}+\begin{bmatrix}u -v -v u \end{bmatrix}\begin{bmatrix}u_x v_x\end{bmatrix}=\begin{bmatrix}f(u,v) g(u,v) \end{bmatrix}+\begin{bmatrix}\varepsilon u_{xx}-\frac{2\varepsilon u_xv_x}{v}\varepsilon v_{xx}-\frac{2\varepsilon v_x^2}{v}\end{bmatrix}.
\Delta^{p, \, q}_{BC}:=\partial^{\star}\partial + \bar\partial^{\star}\bar\partial + (\partial\bar\partial)^{\star}(\partial\bar\partial) + (\partial\bar\partial)(\partial\bar\partial)^{\star}+ (\partial^{\star}\bar\partial)^{\star}(\partial^{\star}\bar\partial) + (\partial^{\star}\bar\partial)(\partial^{\star}\bar\partial)^{\star},
v_t(r, t)=C \zeta'(t) G^{\frac 1{m-1}}-\frac{C}{m-1}\zeta(t) \frac{\eta'(t)}{\eta(t)}G^{\frac 1{m-1}-1}+ \frac{C}{m-1}\zeta(t)\frac{\eta'(t)}{\eta(t)}G^{\frac 1{m-1}}\,;
\begin{array}{c}x=p^{-1}_1q_1\dots p^{-1}_nq_n r_{n+1}, \mbox{where}r_{n+1}\in S \mbox{and}r_k=p^{-1}_kq_k\dots p^{-1}_nq_n r_{n+1}\in S \mbox{for all}k=1,\dots,n.\end{array}
\mathcal{A}_\ell := \{T \in \mathcal{A}\, \mid \, T\varepsilon(1_B \otimes b) = \varepsilon(1_B \otimes b) T \mbox{for all}b \in B\}\mathcal{A}_r := \{T \in \mathcal{A}\, \mid \, T\varepsilon(b \otimes 1_B) = \varepsilon(b \otimes 1_B) T \mbox{for all}b \in B\}.
R \leq \frac{1}{n}\sum_{i=1}^{\ell}I({\bf X}_{i,1}^{L}; Y_{i,1}^{L}|{\bf S}_{i,1}^{L}) + \epsilon_n & = \frac{\ell}{n}I({\bf X}_{T,1}^{L}; Y_{T,1}^{L}|{\bf S}_{T,1}^{L}, T ) + \epsilon_n & \leq \frac{1}{L}I({\bf X}_{T,1}^{L}; Y_{T,1}^{L}|{\bf S}_{T,1}^{L}) + \epsilon_n
\partial_\kappa A_\mu^n(x) = -\sum_{\nu>\{\mu, \kappa\}}\int_0^{x^\nu}\partial_\kappa B_{\mu\nu}(x^1, \ldots, x^{\nu-1}, t, 0, \ldots, 0) dt & + \delta_{\mu>\kappa}B_{\mu\kappa}(x^1, \ldots, x^{\kappa}, 0, \ldots, 0)
- \frac{S_{E}}{\hbar}- W = - \frac{\pi q}{\hbar B}- \frac{2 \pi}{q B}\, \biggl \{\sum \frac{a_{m}}{( q{\cal M})^{2 m}}\biggr \}+ \cdots = - \frac{\pi q}{\hbar B}\biggl ( 1 + \sigma \frac{\hbar}{q^{2}}\biggr ) + \cdots .
\begin{array}{c}e^{-B}Ae^B =\sum\limits_{n=0}^\infty\displaystyle\frac 1{n!}[A,B]_{(n)}, begin{align*}2ex]{}[A,B]_{(0)}\equiv A, \qquad [A,B]_{(n+1)}=[[A,B]_{(n)},B].\end{array}
\xi(x) = \frac{\eta^{\prime\prime}(x)+\mu\eta(x)}{\mu}= \Big(\frac{m_1^2 +\mu}{\mu}\Big) (e^{m_2}-e^{m_3}) e^{m_1 x}+ \Big(\frac{m^2_2 + \mu}{\mu}\Big) (e^{m_3}-e^{m_1}) e^{m_2 x}+ \Big(\frac{m_3^2 + \mu}{\mu}\Big) ( e^{m_1}- e^{m_2}) e^{m_3 x}.
\lim_{n\rightarrow+\infty}n^{p\frac{\alpha}{2}-1}\sum_{j=1}^{n}\left|B_{\alpha,\beta}\left(\frac{j}{n}\right)-B_{\alpha,\beta}\left(\frac{j-1}{n}\right)\right|^{p}=\mathbb{E}\left(\left|B_{\alpha,\beta}(1)\right|^{p}\right).
x \sqrt{\beta - x}( 1 6 x - \beta ) = \frac{5 \beta - 6 x}{\sqrt{\beta - x}}\left( 4 x^{2}- \frac{\beta}{2}x + \frac{\beta^{2}}{4 8}+ \frac{8}{3}\frac{x^{3}( 5 \beta - 6 x )^{2}}{\beta - x ) ( 1 6 x - \beta )^{2}}\right)
\left( \mathbb E_{\nu,\alpha\vphantom{\beta}}^{U}\mathbb E_{\nu,\beta}^{V}-\mathbb E_{-\nu-1,\alpha\vphantom{\beta}}^{U}\mathbb E_{-\nu-1,\beta}^{V}\right)\log\frac{1-\beta UV}{1-\beta V}={}0.
R(x)=\frac{\sum_{j=0}^{m}a_{j}x^{j}}{1+\sum_{k=1}^{n}b_{k}x^{k}}=\frac{a_{0}+a_{1}x+a_{2}x^{2}+...+a_{m}x^{m}}{1+b_{1}x+b_{2}x^{2}+...+b_{n}x^{n}},
\left | \left \{\Theta^4 \left[ \begin{array}{ll}0 0 0 0\end{array}\right] + \Theta^4 \left[\begin{array}{ll}0 0 0 1/2\end{array}\right] + \Theta^4 \left[\begin{array}{ll}0 0 1/2 0\end{array}\right] + \Theta^4 \left[\begin{array}{ll}0 0 1/2 1/2\end{array}\right] \right \}(0 \vert \Omega ) \right |^2
F_N\left( \left[ \frac1n\sum_{i=1}^{n}\frac{\frac1N\Vert z_i\Vert^2}{\check{d}_i}\right]^{-1}\right) \leq F_N\left( \check{d}_i \left[ \frac1n\sum_{i=1}^n \frac1N\Vert z_i\Vert^2 \right]^{-1}\right)
\quad\left((2j-1)^{2}-(2n+5)^{2}\right)\frac{(n+2)^{2}}{4(4n+9)}\frac{(2n+3+2j)(2n+5-2j)}{(n+2+j)(n+3-j)}=\left(\frac{(2n+4)^{2}((2j-1)^{2}-(2n+4)^{2})}{4(4n+9)}+\frac{(2n+3)^{2}((2j-1)^{2}-(2n+3)^{2})}{4(4n+5)}\right) \quad+\frac{(2n+3)^{2}(n+1+j)(n+2-j)}{(4n+5)},
\mu^{p*,F}1_{\{p(F,l)\neq 0\}}= 1_{\{\xi_1\in F,\xi_T\neq\xi_1\}}\frac{p(l)}{p(F,l)}\mu^{p*}\mu^{p*,F}1_{\{p(F,l)=0\}}= 1_{\{\xi_1\in F,p(F,l)=0\}}\frac{p(l)}{q(F,l)}\mu^{p*}.
M = \left(\begin{array}{c|c|c|c}A_{1}^{\top}A_{2}^{\top}\dots A_{n}^{\top}\end{array}\right) \in (W^{*}\o V_{1})\oplus \dots\oplus (W^{*}\o V_{n}) = W^{*}\o (\textstyle \bigoplus_{j}V_{j}).
\begin{aligned}(g^1_i) =(f_{Pos(1)},f_{pos(1)},f_{pos(k+1)},f_{pos(2k+1)}, f_{pos(3k+1)},\ldots ), (g^2_i) =(f_{Pos(2)},f_{pos(2)}, f_{pos(k+2)}, f_{pos(2k+2)}, f_{pos(3k+2)},\ldots), \ \ \vdots (g^k_i) =( f_{Pos(k)}, f_{pos(k)}, f_{pos(2k)}, f_{pos(3k)},f_{pos(4k)}, \ldots).\end{aligned}
\prod_{1\leq i \leq j \leq n}\dfrac{r+i+j-1}{i+j-1}= \prod_{j=1}^n \dfrac{(r+2j-1) \cdots (r+j)}{(2j-1) \cdots j}= \prod_{j=1}^n\dfrac{(r+2j-1)! (j-1)!}{(r+j-1)! (2j-1)!}= \dfrac{\Phi(r+2n-1)F(r-1) F(n-1)}{\Phi(r-1)F(r+n-1)\Phi(2n-1)}= \dfrac{\Phi(r+2n-1) \Phi(r-2) F(n-1)}{F(r+n-1)\Phi(2n-1)},
\widetilde\psi_{u,v}'(\tau) = - \frac{\sigma^2}{2}\widetilde\psi_{u,v}(\tau)^2 - \frac{\delta^\alpha}{\alpha}(-\widetilde\psi_{u,v}(\tau))^\alpha + b \widetilde\psi_{u,v}(\tau) , \tau \leq 0 ,
\theta_{\varphi}\theta_{\varphi^{\vee}}f = \lambda f \ \Rightarrow \ \langle \theta_{\varphi}\theta_{\varphi^{\vee}}f , f \rangle = \lambda \langle f , f \rangle \ \Rightarrow \ \frac{\langle \theta_{\varphi^{\vee}}f , \theta_{\varphi^{\vee}}f \rangle}{\langle f , f \rangle}= \lambda \ .
\Phi(\Psi(\varphi))(X) =\varphi(Xj^+_{P'}(E^{P'}_{o_{+,P'}}(n\lambda n^{-1}))) &=\varphi(X)(n\sigma)(E^{P'}_{o_{+,P'}}(n\lambda n^{-1})) &=\varphi(X)\sigma(E^{P}_{o_{+,P}}(\lambda)).
\int_I \frac 12(f')^2+W(f\sigma)= \mathfrak{E}_{W}(f\sigma,I)-\frac 12\int_I (\sigma'f)^2\leq \mathfrak{E}_{W}(E\sigma,I)-\frac 12\int_I |\sigma'|^2 E^2= \int_{s_0}^{+\infty}\frac 12(E')^2+W(E\sigma).
\big(z_0z_n+z_1^2:z_1z_n:z_2z_n:\ldots:z_{n-1}z_n:z_n^2\big)=\mathfrak{g}_1\sigma_n \mathfrak{g}_2\sigma_n \mathfrak{g}_3\sigma_n \mathfrak{g}_2\sigma_n\mathfrak{g}_4
M_{\mathrm{CsCl}}:=\dfrac{1}{2^{n-2}a^{n-2}}\ \underset{\epsilon \rightarrow 0+}{\lim}\left( \int_{\epsilon}^{+\infty}\left\lbrace\vartheta_{3}(0\vert i v)^{n}-\vartheta_{4}(0\vert i v)^{n}\right\rbrace dv -\dfrac{1}{\left( n/2 -1\right) \epsilon^{n/2-1}}\right)
\begin{array}{lcl}-a_1\mu_1k^{-1}= s^{-1}\log_1(\pi_2) (v_1-v_2), - (\lambda_1+k^{-1}\mu_1 a_1) = v_1 , -a_2\mu_2k^{-1}= s^{-1}\log_1(\pi_2) (v_2-v_1), -(\lambda_2 +k^{-1}\mu_2 a_2) = v_2.\end{array}
\begin{aligned}f_r(T_{s,t}) = f_r(T_{s,0}\ast T_{0,t}) = \sum_{i=-1}^{r}f_{r-i-1}(T_{s,0})f_i(T_{0,t}) = \sum_{i=-1}^{r}\sum_{j=0}^s\sum_{k=0}^t \tbinom sj \tbinom j{2j+i-r}\tbinom tk \tbinom k{2k-i-1}n^jm^k\end{aligned}
\eta_k^{\star}= \frac{a_{k}{E}_{k}\left|\left(\boldsymbol{v}_{k}\right)^{\mathrm{H}}\!\boldsymbol{g}_{k}\right|^{2}}{\sum_{l \neq k}a_{l}{E}_{l}\left|\left(\boldsymbol{v}_{k}\right)^{\mathrm{H}}\boldsymbol{g}_{l}\right|^{2}\!+\!L\sigma^{2}\left\|\boldsymbol{v}_{k}\right\|^{2}}, \forall k \in \mathcal{K}.
\theta^\ell_{i_1,\ldots,i_{\ell-1},1,i_{\ell+1},\ldots,i_n}(f_{i_1,\ldots,i_{\ell-1},1,i_{\ell+1},\ldots,i_n})=\theta^\ell_{i_1,\ldots,i_{\ell-1},2,i_{\ell+1},\ldots,i_n}(f_{i_1,\ldots,i_{\ell-1},2,i_{\ell+1},\ldots,i_n})
J^{(N)}(N-1; 1, \ldots, 1)= -\frac{1}{2\; D^{(N)}}\sum\limits_{i=1}^N \frac{1}{m_i^2}\; F_i^{(N)}\;\left.J^{(N-1)}(N-1;1,\ldots, 1)\right|_{\mbox{\scriptsize without}\; i}.
g_{n M N}= \sum_{k = 0}^{\infty}C_{\xi}^{* k}C_{\eta}^{k}e_{k M}\otimes e_{k - n , N}( \frac{q^{- n}}{1 - q^{2}})^{k}q^{k^{2}}\sqrt{\frac{[ - n ]_{q^{2}}!}{[ - n + k ]_{q^{2}}! [ k ]_{q^{2}}!}}
Pr(x|x_{0},\alpha_{0},\alpha)=x_{0}\frac{\Gamma(\sum_{i=0}^{m}x_{i})\Gamma(\sum_{i=0}^{m}\alpha_{i})}{\Gamma(\sum_{i=0}^{m}(x_{i}+\alpha_{i}))}\prod_{i=0}^{m}\frac{\Gamma(x_{i}+\alpha_{i})}{\Gamma(x_{i}+1)\Gamma(\alpha_{i})}.
d_{p+1}(f)(m) = d_{p+1}(f)(k\wedge\langle x_n\rangle) = (\tilde d(\tilde f)(k))\wedge\langle x_n\rangle + (-1)^{p+1}L_{p+1}(k) &= (-1)^{p+1}L_{p+1}(k) \in (E\otimes\wedge^p L_{n-1}),
R_{L,P}(f)\equiv E_{Z}E_{T|Z}L(T,f(Z))= E_{Z}\left[\int_{0}^{\tau}L(t,f(Z))dF(t|Z)\right] = E_{Z}\left[\int_{0}^{\tau}\ell(t,f(Z))(1-F(t|Z))dt-\left.L(t,f(Z))(1-F(t|Z))\right|_{0}^{\tau}\right] = E_{Z}\left[\int_{0}^{\tau}\ell(t,f(Z))(1-F(t|Z))dt\right]+E[L(0,f(Z))]\,,
\gamma_F(\frac{1}{2}\psi \circ q_2) = \gamma_F(\frac{1}{2}\psi)^4 \cdot (- \frac{\nu_1 J_1}{b_1^2 - d_1^2 J_1}, J_2)_F \cdot (-1, -1)_F = (-(b_1^2 - d_1^2 J_1) \nu_1 J_1, J_2)_F.
=\frac{(\begin{matrix}3\\ 3\end{matrix})}{2^{4}}+\frac{(\begin{matrix}4\\ 3\end{matrix})}{2^{5}}+\frac{(\begin{matrix}5\\ 3\end{matrix})}{2^{6}}=\frac{1}{16}+\frac{4}{32}+\frac{10}{64}=\frac{11}{32}
\|\hat{Q}_r(\Sigma-\lambda_r I)\hat{Q}_r\|_1 \leq \sum_{s\geq 1}\|\hat{Q}_rQ_s(\Sigma-\mu_rI)\hat{Q}_r\|_1 &=\sum_{s\neq r}\|(\mu_s-\mu_r)\hat{Q}_rQ_s\hat{Q}_r\|_1 &=\sum_{s\neq r}|\mu_r-\mu_s|\|\hat{Q}_rQ_s\|_2^2 &\leq \sum_{\substack{s<r_0: s\neq r}}|\mu_r-\mu_s|\|\hat{Q}_rQ_s\|_2^2+2|\mu_r-\mu_{r_0}|\|\hat{Q}_rQ_{\geq r_0}\|_2^2.
\partial_{t}u(x,t)=a\left[ \mathcal{\partial}_{x}^{2}+\lambda t(I-e^{\mathcal{\partial}_{x}^{2}})\right] u(x,t)+\frac{\lambda}{2}(I-e^{\mathcal{\partial}_{x}^{2}})\int_{-\infty}^{x}zu(z,t)dz,
\tilde{T}_{\cdot \nu}^{\mu \cdot}= \frac{i}{2}\left[ \overline{{\psi}}\gamma^{\hat{\alpha}}e_{\hat{\alpha}}^{\mu}\partial_{\nu}\psi - ( \overline{{\partial_{\nu}\psi}}) \gamma^{\hat{\alpha}}e_{\hat{\alpha}}^{\mu}\psi \right]
\Delta R \approx \sum\limits_{k = 1}^{{N_{\rm{R}}}}{{{\log}_2}\left({{{1 +{{{\bar C}_k}}}}}\right)}- \sum\limits_{k = 1}^{{N_{\rm{R}}}}{{{\log}_2}\left({{{1 +{{\bar C}_k}\left({1 -{2^{- 1 -{x_k}}}}\right)}}}\right)}.
\rho(|G^k(|P|)|) = \lim_{r \to \infty}\frac{\lambda(|G^k(|P|)| \cap r C)}{r^n}\leq \lim_{r \to \infty}\frac{1}{r^n}\sum_{B \in P : B \cap rC \neq \emptyset}\lambda(|G^k(B)|).
\Delta^x \circ R - \Delta^x = c \, \Delta^y , \Delta^y \circ R - \Delta^y = p\circ(K^x + \Delta^x) - p\circ K^x + K^y \cdot (q\circ(K^x + \Delta^x) - q\circ K^x) & \quad+ \, \Delta^y \cdot q\circ (K^x + \Delta^x) + u \circ (K + \Delta) - u \circ K+ g\circ (K + \Delta).
\begin{array}{l}{{D_{\mu}^{i j}[ A ] ( x ) \langle T [ j^{\mu , j}( x ) \frac{\delta j^{\beta , l}}{\delta A_{\lambda}^{k}}( y ) ] \rangle}}\\{{+ i g \delta^{4}( x - y ) ( f^{b i l}\langle \frac{\delta j^{\beta , k}}{\delta A_{\lambda}^{b}}( x ) \rangle + f^{k i a}\langle \frac{\delta j^{\beta , a}}{\delta A_{\lambda}^{l}}( x ) \rangle ) = 0 .}}\\ \end{array}
\iint_\Omega \begin{vmatrix}\varphi(z) \end{vmatrix}^p \mathrm{d}x\mathrm{d}y=\int_0^1\int_{\|\varphi\|=r}\begin{vmatrix}\varphi(z) \end{vmatrix}^{p-1}\mathrm{d}s\mathrm{d}r
\beta_{s,a^1}^1=\frac{\left[r^1(s,a^1,a_s^2)-r^1(s,a_s^1,a_s^2)\right]}{\left[r^1(s,a^1,a_{s}^2)-\sum_{s'\in S}p(s'|s,a^1,a_s^2)r^1(s',a_{s'}^1,a_{s'}^2)\right]}.
\varphi(u) =\sum_{k=0}^{u+2}s_k\ \varphi(u+3-k)-a_{u+1}b_0c_0\varphi (2) &-a_{u+1}b_0c_1\varphi(1)-c_0\varphi(1)\sum_{k=0}^{u+2}a_k b_{u+2-k}
k_{\gamma,\Sigma_1}^{\infty}=\frac{\sqrt{\frac{1}{2}\overline{q}^2\left(-e^{-\gamma_3}\dot{\gamma_1}+e^{\gamma_3}\dot{\gamma_2}\right)^2+\overline{p}^2\dot{\gamma_3}^2}}{|\omega(\dot{\gamma}(t))|},~~if ~~\omega(\dot{\gamma}(t))\neq 0,
2^d\lceil 2^{dn}/n \rceil-\lceil 2^{d(n+1)}/(n+1) \rceil\leq 2^{d(n+1)}\left(\frac{1}{n}-\frac{1}{n+1}\right)+1=\frac{2^{d(n+1)}}{n(n+1)}+1\leq \frac{1}{2}2^d \lceil 2^{dn}/n \rceil,
\begin{array}{c}{{\Psi_{i}[{\bf r}( \beta^{\prime}) ; \beta_{n}, \beta_{1}] = \psi_{i}({\bf r}_{n}, \beta_{n}) \psi_{i}({\bf r}_{n - 1}, \beta_{n - 1}) . . . \psi_{i}({\bf r}_{1}, \beta_{1})}}\\{{\int D{\bf r}( \beta^{\prime}) \cdot \Psi_{i}^{*}[{\bf r}( \beta^{\prime}) ; \beta , \beta_{0}] \cdot \Psi_{j}[{\bf r}( \beta^{\prime}) ; \beta , \beta_{0}] = \delta_{i j}}}\\ \end{array}
\int_{0}^{2}(1-t)P_{k}(t)dt=0 \implies \frac{\alpha_{0}(k+1)!}{2C_{k}}-\frac{2\alpha_{1}k(k+1)}{(2\pi)^{2}}+\tilde{C_{k}}=-\frac{2\alpha_{1}k(k-1)}{(2\pi)^{2}}R_{k},
\langle f \rangle_{1 + \alpha,\mathcal{G}}:= \sup_{(x,t) \in \mathcal{G}}\sup_{~(x,s) \in \mathcal{G}\setminus \{(x,t)\}}\frac{|f(x,t) - f(x,s)|}{|t - s|^\frac{1 + \alpha}{2}}.
\varphi_{ij}^{ab}(\theta )=\int\limits_{-\infty}^{\infty}dt\left[ \delta_{ab}\delta_{ij}-\left( 2\cosh \frac{\pi t}{h}-\tilde{I}\right)_{ab}\left(2\cosh \frac{\pi t}{h}-I^{su(k)}\right)_{ij}^{-1}\,\right]\,\,e^{-it(\theta +\sigma_{ab})}\;,
\begin{matrix}F_{R}(q;k,\nu)=\frac{\sqrt{2\pi}k\nu^{\nu/2}}{\Gamma(\nu/2)2^{(\nu/2-1)}}\int_{0}^{\infty}s^{\nu-1}\varphi(\sqrt{\nu}s)\times\\ [\int_{-\infty}^{\infty}\varphi(z)[\Phi(z+qs)-\Phi(z)]^{k-1}dz]ds\end{matrix}
\frac{\partial MPK}{\partial L}=\frac{\partial^{2}Y}{\partial K\partial L}=\frac{\partial}{\partial L}(AL^{\beta}\alpha K^{\alpha-1})=A\beta L^{\beta-1}\alpha K^{\alpha-1}=A\alpha\beta\frac{L^{\beta}K^{\alpha}}{LK}=\alpha\beta\frac{Y}{LK}
B =\{\{1\}, \{2\}, \{3\}, \{4\}, \{5\}, \{6\}, \{2, 5\}, \{2, 3, 4\}, \{3, 4, 5\}, \{1, 2, 3, 4\}, &\{2, 3, 4, 5\}, \{3, 4, 5, 6\}, \{1, 2, 3 ,4, 5\}, \{2, 3, 4, 5, 6\}, \{1, 2, 3, 4, 5, 6\}\},
(-v\partial_x)^*h(x,v) = v\partial_xh(x,v) -v\frac{\partial_x g(x,v)}{g(x,v)}h(x,v) =v\partial_xh(x,v) - g(x,v)^{-1}\rho_g(x)(\alpha \mathcal{M}_{T_g(x)}(v) + (1-\alpha) \mathcal{M}_{\tau(x)}(v))h(x,v) -h(x,v),
\lim_{t_0\to-\infty}\overline{U}(t,t_0)=\frac{1}{b+\chi_2\mu_2-\chi_1\mu_1}\Big\{a+\frac{1}{\lambda_2}\big(\left[(\chi_2\mu_2\lambda_2-\chi_1\mu_1\lambda_1)_{+}+\chi_1\mu_1(\lambda_1-\lambda_2)_{+}\right]\overline{u}_{0}-\left[(\chi_2\mu_2\lambda_2-\chi_1\mu_1\lambda_1)_{-}+\chi_1\mu_1(\lambda_1-\chi_2)_{-}\right]\underline{u}_0\big)\Big\}
\Theta(\nu)=1-\sum_{n\geq1}\frac{1}{j_{\nu,n}^4-1}-\frac{\sum\limits_{n\geq1}\frac{j_{\nu,n}^4}{(j_{\nu,n}^4-1)^2}}{1-\sum\limits_{n\geq1}\frac{1}{j_{\nu,n}^4-1}},
+C^{*k}_{\xi}C^{k}_{\eta}e_{n+1+k,M}\otimes e_{kN}(\frac{q^{n+1}}{1-q^{2}})^{k}q^{k^{2}}\sqrt{\frac{[n+1]_{q^{2}}!}{[n+1+k]_{q^{2}}![k]_{q^{2}}!}}(1-q^{2(n+k+1)}))
V^{[n]}_{z_1z_1z_1}- V^{[n]}_{z_1}-\frac{U}{2n}+\frac{\hat A_1(z_2)}{a_3}U_{z_1}=0, & V^{[j]}_{z_1z_1}-V^{[j]}+U\,\Omega^{(j+1)}=0, j=1\dots n-1, & \left(\Omega^{[1]}\,U\right)_{z_1}+U_{z_2}=0, & \Omega^{[j]}_{z_1}=UV^{[j]}_{z_1}, j=1\dots n.
(1-2\rho\cos(\alpha+\beta)+\rho^{2})((1-2\rho\cos(\alpha-\beta)+\rho^{2})\allowbreak=\allowbreak (1+\rho^{2})^{2}-2\rho(1+\rho^{2})(\cos(\alpha+\beta)+\cos(\alpha-\beta))+4\rho^{2}\cos(\alpha+\beta)\cos(\alpha-\beta).
M_0(X)= X^{10}\sum\limits_{k_2|2}\sum\limits_{k_3|3}\sum\limits_{\substack{h\le \xi/(k_2k_3) (h,6)=1}}\sum\limits_{\substack{e\le \xi/(k_2k_3h) (e,6h)=1}}\frac{\mu(k_2)}{k_2}\cdot \frac{\mu(k_3)}{k_3}\cdot \frac{\mu(h)}{h^2}\cdot \frac{\mu(e)\varphi(e)}{e^3}
\frac{\partial\rho\langle v_{\phi}\rangle}{\partial t}+\frac{\partial\rho\langle v_{r}v_{\phi}\rangle}{\partial r}+\frac{\rho}{r}[3\langle v_{r}v_{\phi}\rangle+2\langle v_{\theta}v_{\phi}\rangle cot(\theta)]=0
\begin{aligned}B\geq\mathbb{E}\int_0^T\Big[- p_1(t)\Big(f^{v_1}(t)-f(t)-\sum_{j=1}^2(z_j^{v_1}(t)-z_j(t))h_j(t)\Big)-\big(y^{v_1}(t)-y(t)\big)H_{1y}(t)-\big(z^{v_1}(t)-z(t)\big)H_{1z}(t) - \sum_{j=1}^2\big(z_j^{v_1}(t)-z_j(t)\big)H_{1z_j}(t)\Big]dt-p_1(T)\Big(g(x^{v_1}(T)-g(x(T))\Big).\end{aligned}
\frac{\Delta_0}{\surd{(\Delta_{1})}}=\frac{\|\mu_1-\mu_2\|_2^2}{\surd{\frac{2}{n(n-1)}(\Sigma_{1}^{2})+\frac{2}{m(m-1)}(\Sigma_{2}^{2})+\frac{4}{nm}(\Sigma_{1}\Sigma_{2})}}\{1+o(1)\},
\frac{1}{\Gamma(\alpha)}\int_{T_0}\mathcal{D}_\lambda\gamma_{\alpha,\beta',\infty}(\tau)={\beta'}_\lambda^{-\alpha}=\frac{1}{\Gamma(\alpha)}\int_{T_0}\mathcal{D}_\lambda\Gamma_{\alpha,\beta',0}(\tau)\;.
\frac{d}{d\xi}F_z(1,(0,0)) =p\|(u,v)\|^p-q\int_{\Omega}\left(\lambda|u|^{q}+\mu |v|^{q}\right)dx- 2(\alpha+\beta)\int_{\Omega}|u|^{\alpha}|v|^{\beta}dx = (p-q)\|(u,v)\|^p-2(\alpha+\beta-q)\int_{\Omega}|u|^{\alpha}|u|^{\beta}dx\not=0.
\sum_{i_q=i_{q-1}+1}^{\min(i_{q-1}+t, i_{q-1}+x/2)}u(m - i_{q}) u(i_q-i_{q-1})^2 \leq c_{7}u(m-i_{q-1}) \sum_{i_q=i_{q-1}+1}^{i_{q-1}+t}u(i_q-i_{q-1})^2 \leq c_{7}u(m-i_{q-1}) D(t) .
\langle s | P | s^{\prime}\rangle \sim \langle s | s^{\prime}\rangle + \int [ d N ] \, \left( \langle s | \hat{H}[ N ] | s^{\prime}\rangle \ + \langle s | \hat{H}[ N ] \hat{H}[ N ] | s^{\prime}\rangle \ + \ldots \right) \; .
\widetilde{\mathcal{R}}_{}\! =\! \bigcup_{P_{U|\widetilde{X}}}\!\Big\{\left(R_s,R_\ell,R_w\right)\!\colon\! &0\leq R_s\leq I(U;Y), &R_\ell\geq I(U;X) - I(U;Y), &R_w\geq I(U;\widetilde{X})\Big\}
ds^2 ={1 \over 4q^2}\left(q{d \over dq}\right)^2 K\, dq^2 + \left(q{d \over dq}\right) K\, \left[{2 \over 3}d\hat{z_i}d\hat{\bar{z_i}}-{2 \over 9}| \hat{z_i}d\hat{\bar{z_i}}|^2 \right] +{4 \over 9}\left(q^2{d^2 \over dq^2}\right) K \, | \hat{z_i}d\hat{\bar{z_i}}|^2 .
\begin{aligned}\qquad\sum_{\mu \in P}m(\pi_\lambda, \sigma_\mu) \delta_{\mu}= &\sum_{j=1}^s m(\pi_{\lambda +\rho_n}^K, \pi_{\mu_j +\rho_\mathfrak z}^{K_\mathfrak z})\sum_{n\geq 0}\sum_{h=0}^{[\frac{n}{2}]}\binom{n-2h+c-1}{c-1}\binom{h+d-1}{d-1}\delta_{[(\mu_j +\rho_\mathfrak z)(Z_0)+ n-1] \varphi}. \end{aligned}
\omega_{z}z-\omega_{y}y =(y'(x)z''(x)-z'(x)y''(x))[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}\omega_{y}x-\omega_{x}z = -z''(x)[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}\omega_{x}y-\omega_{z}x = y''(x)[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}
K = \sum_{\Lambda}z_\Lambda^{-1}(-1)^{\sum_{i =0}^{2^{\mathcal{N}}-1}\binom{M_i}{2}}\prod_{i=0}^{2^{\mathcal{N}}-1}p_{\Lambda^{(i)}}^{\{T^{(i)}\}}(x,\theta) p_{\Lambda^{(i)}}^{\{T^{(i)}\}}(y,\phi),
\mathbb{E}e^{-\langle\tilde{Z}_{\lambda}^{S_{n},M,m},f_{m,\delta}\rangle}\leq\mathbb{E}\left[\exp\left(-e^{-\varepsilon}\sum_{j=1}^{16^{m}}\mathbb{E}[\tilde{X}_{j}1_{\{X_{j}\leq\varepsilon\}}|\Phi^{S_{n},\tilde{S}_{n,m}}]\right)\right].
\theta_{\bot \parallel}= \frac{1}{y^{2}}\left( \partial_{\tau}\vec{x}\, \partial_{\sigma}\vec{x}+ \partial_{\tau}y \, \partial_{\sigma}y \right) = \frac{1}{a^{2}\tau}\left[ \vec{f}{\vec{c}\,}^{\prime}+ a a^{\prime}\right] + \frac{1}{a^{2}}\left( \vec{g}{\vec{c}\,}^{\prime}\right) + \ldots
{\frac{d n ( \omega )}{d \omega}}\simeq{\frac{2 \omega^{D - 2}}{( 4 \pi )^{( D - 1 ) / 2}}}\sum_{n = 0}^{\infty}\left[{\frac{c_{n}}{\Gamma \left({\frac{D - 1}{2}}- n \right)}}\omega^{- 2 n}+{\frac{d_{n}}{\Gamma \left({\frac{D - 2}{2}}- n \right)}}\omega^{- ( 2 n + 1 )}\right] ,
F_{~ \mu \nu ~ \alpha \beta}^{- 1 \sigma ~ ~ \rho}F_{\rho ~ ~ \omega}^{~ \alpha \beta ~ \eta \kappa}= \frac{1}{2}\delta_{\omega}^{\sigma}\left( \delta_{\mu}^{\eta}\delta_{\nu}^{\kappa}- \delta_{\mu}^{\kappa}\delta_{\nu}^{\eta}\right)
Y^N_0 = \xi, Y^N_{n + 1}= \left( \operatorname{Id}_H - \tfrac{T}{N}A \right)^{- 1}\Big( Y_n + F( Y_n ) \tfrac{T}{N}+ \smallint\nolimits_{\frac{n T}{N}}^{\frac{( n + 1 ) T}{N}}B( Y_n ) \, dW_s \Big) .
P_{z,\omega}[S_0=z] =1, P_{z,\omega}[S_{n+1}=S_n+1|(S_m)_{1\leq m\leq n}] = \omega(S_n,\#\{m\leq n:\: S_m=S_n\}), P_{z,\omega}[S_{n+1}=S_n-1|(S_m)_{1\leq m\leq n}] = 1-\omega(S_n,\#\{m\leq n:\: S_m=S_n\})
\textrm{BR}(t)=-t\cdot{}_4F_3\left[\left\{\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5}\right\};\left\{\frac{1}{2},\frac{3}{4},\frac{5}{4}\right\};-\frac{3125 t^4}{256}\right].
\Gamma_{g.f.}+ \Gamma_{\phi\pi}\longrightarrow \Gamma_{g.f.}+ \Gamma_{\phi\pi}+ \chi \int\left\{\frac 12 \bar c B + (\partial_{\xi}\xi_A) m \bar c \varphi_2 -(\partial_{\xi}\hat \xi_A) m \bar c\hat \varphi_2 \right\}\; .
( e^{- 2 \eta})_{\mathrm{f i x}}= \left| \frac{p_{2}p_{4}}{q_{1}q_{3}}\right| \ , \qquad ( e^{- 2 \sigma})_{\mathrm{f i x}}= \left| \frac{p_{2}q_{3}}{q_{1}p_{4}}\right| \ , \qquad ( e^{- 2 \rho})_{\mathrm{f i x}}= \left| \frac{p_{4}q_{3}}{q_{1}p_{2}}\right| \ .
\Gamma^0:=\sum_{\ell = 1}^\aleph\left\langle [P_{D_\ell}]\cdot\sum_{j=1 \atop j \neq \ell}^\aleph\Upsilon_{\ell j}^0\cdot Q_j, \ [P_{D_\ell}] \cdot Q_\ell\right\rangle.
\eta^{12}b_1 =(\eta^{12},0) + u(bE_4^2-aE_6)e_1+u(bE_6-aE_4)e_2, \eta^{12}b_2 =(0,\eta^{12}) + u(dE_4^2-cE_6)e_1+u(dE_6-cE_4)e_2.
\delta x^{\mu}={\dot{x}}^{\mu}\xi \; , \; \delta e = \frac{d}{d \tau}\left( e \xi \right) \; , \; \delta f_{a b}= \frac{d}{d \tau}\left( f_{a b}\xi \right) \; , \; \; \delta \psi_{a}^{n}= \dot{\psi}_{a}^{n}\xi \; , \; \; \delta{\chi}_{a}= \frac{d}{d \tau}\left( \chi_{a}\xi \right) \; ,
\frac{1}{N}\int_{0<s_1<s_2<1}d\iota_N(X^\star)(s_1)\otimes d\iota_N(X^\star)(s_2) = \frac{1}{2N}\sum_{k=0}^{N-1}F^\star_k\otimes F^\star_k &\phantom{=}+ \frac{1}{N}\sum_{0\leq k<l<N}F^\star_k\otimes F^\star_l
\begin{aligned}\sum_{s_1\ge s_2\ge\cdots\ge s_{k}\ge 0}\frac{q^{s_{1}^2+\cdots +s_k^2+s_{k-r+1}+\cdots +s_k}}{(q)_{s_1-s_2}\cdots (q)_{s_{k-1}-s_{k}}(q)_{s_{k}}}&=\frac{(q^{k+1-r},q^{k+2+r},q^{2k+3};q^{2k+3})_\infty}{(q;q)_\infty},0\le r\le k.\end{aligned}
S_{\mathrm{d i v}, 1}^{( j )}= ( - 1 )^{2 j}{\frac{c^{( j )}}{8 \delta^{2}}}\int_{\Sigma}~ ~ , ~ ~ \mathrm{f o r}~ ~ j = 0 , ~ 1 / 2 , ~ 1 , ~ 3 / 2 ,
\begin{aligned}{\|f(y+\tilde y,u)-2f(y,u)+f(y-\tilde y,u)\|}_2 \le C_f\,{\|\tilde y\|}^2_2, \big|g(y+\tilde y,u)-2g(y,u)+g(y-\tilde y,u)\big| \le C_g\,{\|\tilde y\|}^2_2\end{aligned}
K_n(x) =A_{n}(x)+ k x\,A_{n-2}(x) begin{align*}5pt] =\sum_{j=0}^{n-2}\big((n-j-1)x+j+1\big)A_{n-1,j}(x)+ k x\, A_{n-2}(x) begin{align*}5pt] =\big((n-1)x+1\big)A_{n-1,0}(x) +\sum_{j=1}^{n-3}\big((n-j-1)x+j+1\big)A_{n-1,j}(x) begin{align*}5pt] + \big(x+n-1\big)A_{n-1,n-2}(x)+ k x\, A_{n-2}(x).
\min_{w \in \Sigma_{+}^{4}}\Re{f(w)}- \Re{f(z_{0})}\geq \Re{f(w_{5})}- \Re{f(z_{0})}= 4 \eta_{-}(w_5-z_{0})+\log\big(1 - \frac{1}{w_5-\tau}\big( \frac{1}{\tau}-\tau\big)\big) >4-\log 3.
\psi_2\rightarrow \exp\left((i\gamma_i\partial^i+e\gamma_iA^i)/(2m_2)\right) \psi_2, \ \ \ \ \ \overline \psi_2 \rightarrow \overline \psi_2 \exp\left((-i\gamma_i\stackrel{\leftarrow}{\partial^i}+e\gamma_iA^i)/(2m_2)\right).
\langle \tilde{\pi}_p(\beta_m)\mathbf h_p, \mathbf h_p\rangle = \int_{\mathcal B_1(m)}\mathbf h_p(h\beta_m)\overline{\mathbf h_p(h)}dh + \int_{\mathcal B_2(m)}\mathbf h_p(h\beta_m)\overline{\mathbf h_p(h)}dh =: B_1(m) + B_2(m).
\frac{d}{d\psi_0}\int\limits_{\psi_0}^{\pi-\psi_0}Cf(0,e_n,\psi)h(\psi_0,\psi)d\psi &= (2-n)\cos \psi_0 \sin\psi_0 \int\limits_{\psi_0}^{\pi-\psi_0}Cf(0,e_n,\psi)g(\psi_0,\psi)d\psi &=\frac{(2-n)sin\psi_0(\cos\psi_0)^{n-2}}{|S^{n-3}|}\int\limits_{S^{n-2}}Rf(((\cos\psi_0)\omega,\sin \psi_0),0)d\omega.
(\lambda_1, \dots, \lambda_n) \propto |\Delta(\lambda)|^\beta \prod_{i = 1}^n \lambda_i^{\frac{\beta}{2}(m_1 - n + 1) - 1}(1 - \lambda_i)^{\frac{\beta}{2}(m_2 - n + 1) - 1}, &= |\Delta(\lambda)|^\beta \prod_{i = 1}^n \lambda_i^{a}(1 - \lambda_i)^{b}, \lambda_i \in [0,1].
Z_{V}\; : = \; \int{\cal D}A{\cal D}\overline{{{q}}}{\cal D}q \; \exp \left( i S_{V}\; + \; i \frac{\theta g^{2}}{1 6 \pi^{2}}\int_{V}d^{4}x \mathrm{T r}( F \tilde{F}) \right) \; .
(-1)^{k-1}\left( \Big(\sum_{n=0}^\infty u_m(n)\, q^n \Big) \Big( \sum_{n=1-k}^k (-1)^n q^{n(3n-1)/2}\Big) - \sum_{n=0}^\infty C_m(n)\, q^{n}\right) = \left( \sum_{n=0}^\infty C_m(n)\, q^{n}\right) \left( \sum_{n=0}^\infty M_k(n)\, q^n\right).
|2\sqrt[\circ ]{e_2e_3}|\leq \, \, \Big|\sqrt[\circ ]{e_2e_3+h_{2n}^+}+ \sqrt[\circ ]{e_2e_3+h_{2n}^-}\Big| \, \, +\Big|\sqrt[\circ ]{e_2e_3+h_{2n}^+}- \sqrt[\circ ]{e_2e_3}\Big|+\Big|\sqrt[\circ ]{e_2e_3+h_{2n}^-}- \sqrt[\circ ]{e_2e_3}\Big| =\, \, \frac{\ell^2_n}{n^{N+1}}+ |h_{2n}^+|^{1/2}+|h_{2n}^-|^{1/2}= \frac{\ell^2_n}{n^{N+\frac{1}{2}}}.