harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\mathrm{d}}{\mathrm{d}\operatorname{l n}s}\, \, \operatorname{l n}\left| \frac{F ( s )}{s}\right| \, \, \rightarrow \, \, \frac{2}{\pi}\; \operatorname{a r c t a n}\left( \frac{\mathrm{R e}\, F ( s )}{\mathrm{I m}\, F ( s )}\right) \, \, \, .
d\tilde{\omega}_t= d(\omega_t+ \partial_t \overline{u_t^{min}}+ \bar\partial_t u_t^{min})=\partial_t\omega_t+\bar\partial_t\omega_t+\bar\partial_t\partial_t \overline{u_t^{min}}+\partial_t\bar\partial_t u_t^{min}.
T=(\Phi(x),B(x))=\left(\left(\begin{array}{c}x^1 x^2\end{array}\right),\left(\begin{array}{cc}\frac{x^1}{\sqrt{(x^1)^2+(x^2)^2}}\frac{x^2}{\sqrt{(x^1)^2+(x^2)^2}}\frac{-x^2}{\sqrt{(x^1)^2+(x^2)^2}}\frac{x^1}{\sqrt{(x^1)^2+(x^2)^2}}\end{array}\right)\right)
T^H_{(-i_k)}(t) = \bigl(g_k^H\bigr)^{-1}\Bigl( X^H(t)\Bigr) = \Biggl(\bigl(g_k^H\bigr)_h^{-1}\Bigl( X^H(t)\Bigr)\Biggr)_{\substack{h=0,\dots, H h\not=k}}= \Biggl[v_{i}^H - v_{i_k}^H\Biggr]^{-1}_{\substack{i\in I_H i\not=i_k}}\,\Bigl(X^H(t)-v_{i_k}^Ht\Bigr).
\eta=(1-R_{1})(1-e^{-\alpha d})[\frac{(1+R_{2}e^{-\alpha d})}{1-2\sqrt{R_{1}R_{2}}e^{-\alpha_{c}d}cos(2\beta L+\phi_{1}+\phi_{2})+(R_{1}R_{2})e^{-\alpha_{c}d}}]
T_1 T_2 (D f^{(j)}) (w) =\int_{\Delta^2}\frac{\partial u_2(z)}{\partial \bar z_1}\frac{S_1(w_1, z_1)S_2(w_2, z_2) |w_1-z_1|^2}{|w-z|^2}dV(z) &~~~+ \int_{\Delta^2}\frac{\partial u_1(z)}{\partial \bar z_2}\frac{S_1(w_1, z_1)S_2(w_2, z_2) |w_2-z_2|^2}{|w-z|^2}dV(z) &:=I_1(w) + I_2(w).
\hskip -2mm\sum_{k=0}^{n-1}(2k+1)R_k^2 &=n\sum_{s=0}^{n-1}{n+s\choose 2s}{2s\choose s}\frac{n-s}{s+1}\sum_{m=s}^{2s}\sum_{i=0}^s{m\choose i}{s\choose i}{i\choose m-s}\frac{1}{(2i-1)(2m-2i-1)}.
{\frac{\phi^{\prime \prime}}{A}}+{\frac{1}{A}}\left( -{\frac{1}{2}}{\frac{A^{\prime}}{A}}+ 2{\frac{B^{\prime}}{B}}+{\frac{2}{r}}\right) \phi^{\prime}-{\frac{2}{r^{2}}}\phi - \lambda \phi ( \phi^{2}- \eta^{2}) = 0 .
{\cal F}= \det \left[ \sum_{r = -{\infty}}^{\infty}\sum_{p = -{\infty}}^{\infty}A_{r - N + j + p}A_{p}\left( t_{i}^{r + q}\pm t_{i}^{- ( r + q )}- \frac{1}{2}\delta_{q 0}\delta_{j N}( t_{i}^{r}+ t_{i}^{- r}) \right) \right] .
\log\|\theta_{1,1}^{8}\|^{2}= \log{\|\theta_{1,1}^{8}\|'}^{2}-\frac{4}{N}\frac{\log (u_{\nu}\bar u_{\nu})\log (v_{\nu}\bar v_{\nu})}{\log (u_{\nu}\bar u_{\nu})+\log (v_{\nu}\bar v_{\nu})}.
p(t,x) = \left(\int_{|y-x|\geq r/2}+ \int_{|y-x| < r/2}\right) p_{\nu}(t,x-y) p_a(t,y) dy\leq c_{14}t \left(\frac{\Psi_{\nu}(1/r)}{r^{d}}+ \frac{(1/r^2)}{r^{d}}\right) + t^{-d/2}e^{-\frac{r^2}{16at}}.
\sum_{j=0}^{n-1}\sum_{k=\max\{j,1\}}^{n-1}(-1)^{j}\frac{1}{k!}\binom{k}{j}N_k(x_n)\Big(\sum_{l=1}^{k}l x_l-jt\Big)^k = \sum_{k=1}^{n-1}N_k(x_n) t^k .
\begin{aligned}\binom{k+c}{c+2}= h^0(\O_{\P^{c+2}}(k-2)) = h^0(\O_S(k-2))=\chi(\O_S(k-2))+h^1(\O_S(k-2)) \ge \ge \chi(\O_S)+\frac{d(k-2)}{2}\big(3-k+\frac{4(k-1)}{c+2}\big).\end{aligned}
\phi_{1}= a P - \frac{\zeta_{3}}{b}Q^{- 1}P , \phi_{2}= - b P^{- 1}Q + \frac{\zeta_{1}}{c}Q , \ \ \ \, p h i_{3}= c Q^{- 1}+ \frac{\zeta_{2}}{a}P^{- 1}
\left\langle\rho,\tau^{x_{3}}\alpha_{1},\sigma\alpha_{2}\alpha_{3}^{(1+x_{3})x_{3}^{-1}}\right\rangle\cong C_{p}^{3},\ \left\langle\rho,\tau^{x_{3}}\alpha_{1},\sigma\alpha_{2}\alpha_{3}^{a}\right\rangle\cong M_{1}& \a=0,...,p-1, \x_{3}=1,...,p-1 \ \a- (1+x_{3})x_{3}^{-1}\not\equiv 0 \ \mathrm{mod}\p.
\langle\psi(0,0)\psi^\dagger(x,t)\rangle_T=\frac{\mathop{\rm tr}\nolimits\left(e^{-\frac{H}{T}}\psi(0,0)\psi^\dagger(x,t)\right)}{\mathop{\rm tr}\nolimits\left(e^{-\frac{H}{T}}\right)}.
\begin{cases}\displaystyle B_{i,j}^N (V_N)=-\frac{a^N(v_i-v_j)}{N}\ \ \ \ \i\neq j, \displaystyle B_{i,i}^N (V_N)=\frac{1}{N}\sum_j a^N(v_i-v_j)+\frac{1}{N}.\end{cases}
d_t\,\mathbb{P}^\delta_x\, \stackrel{\mathcal{L}}{=}\,\max\{t,\,\tau_0'\}\,\inf\{s\geq 1:X_s=0\}\,\mathbb{Q}_0^\delta(X)\times\mathbb{Q}_{x^2}^0(X') &\stackrel{\mathcal{L}}{=}\,\max\left\{t,\,\frac{1}{E_x}\right\}\,d_1\,\mathbb{P}_0^\delta.
h_{\beta,\gamma}(r) = r^{2\beta+\gamma-2\zeta}\int_0^\infty \frac{dt}{t}\, t^\beta\int_0^\infty ds\, s^\gamma p_\zeta^{(2)}(t,1,s) = C(\beta,\gamma,\zeta) r^{2\beta+\gamma-2\zeta},
G^t=\mathcal P\exp\!\left(\int_{\frac 1 2}^ta^s_kds\right)=1+\int_{\frac 1 2}^t a^s_kds+\int_{\frac 1 2}^t\left(\int_\frac 1 2^{t_1}a^{s_2}_kds_2\right)a^{s_1}_kds_1+\cdots
\Delta_{\alpha}^{r}f^{-}= \sum_{\ell_{1},\ell_{2},\ell_{3}=0}^{m-1,n-1,k-1}\frac{\ell_{2}!}{(\ell_{2}-r)!}\frac{\ell_{3}!}{(\ell_{3}-r)!}\overline{\beta}^{\ell_{1}}\overline{\alpha}^{l_{2-r}}\alpha^{\ell_{3}-r}g_{\ell_{1}\ell_{2}\ell_{3}}(\beta).
\||u_n|+|D_v u_n|+|D_v^2 u_n|\|_{L_2((-1,0)\times B_{R^3}\times B_R)}&\le N (d) \delta^{-1}\sum_{k=0}^\infty 2^{-k(k-1)/4}R^{-k}\|f_n\|_{L_2(Q_{1,2^{k+1}R/\delta^2})}
x^{-\frac{k^2}{2}-l}f_l(x)=(-1)^{\frac{k(k-1)}{2}}\frac{G^2(k+1)}{G(2k+1)}\sum_{i=0}^\infty \left( \sum_{q=0}^{l}\sum_{m=0}^{l-q}\Big(\prod_{s=0}^{m-1}(i+2l-q+\frac{k^2}{2}-s)\Big) c_{q,m}^{(l)}a_{i+2l-q}\right)x^i.
\varphi(t)/r+|x(t)| \leq \varphi(t)/r+|x_0|+ B(\varphi(t_0)-\varphi(t)) = \varphi(t_0)/r+|x_0|-(1/r-B)(\varphi(t_0)-\varphi(t)) \leq \varphi(t_0)/r+|x_0| <R.
B^{\infty}_r\left(u_*\right) := \left\{u \in{H^{-1/2}_0}: \left\|u-u_*\right\|_{{H^{-1/2}_0}}\leq r\right\}, C^{\infty}_{k,r}\left(z\right) := \left\{u \in{H^{-1/2}_0}: \left|k\right|^{-1/2}\left|\widehat{u}\left(k\right)-z\right| \leq r\right\}.
V_{11}=-\varepsilon\bar{m}\int_{\mathbb{T}^{d}}(\mu^{1}-\mu^{2})\sum_{i=1}^{d}\left[\Theta_{p_{i}q}(t,x,\mu^{1},Dw^{1})\frac{\partial\mu^{1}}{\partial x_{i}}-\Theta_{p_{i}q}(t,x,\mu^{2},Dw^{1})\frac{\partial\mu^{1}}{\partial x_{i}}\right]\dx,
\mathbb{E}_{N}\exp\sum_{i,k=1}^{N}\Biggl[ \int_{0}^{\eta}\frac{2au_{ik}(s)}{N\epsilon}dM_{ik}^{N}(s) -\int_{0}^{\eta}\left(\exp\frac{2au_{ik}(s)}{N\epsilon}-\frac{2au_{ik}(s)}{N\epsilon}-1\right)\lambda NdA_{ik}^{N}(s)\Biggl]=1.
(-\alpha_2)(\mu(\partial_1(a))v) - (-\alpha_1)(\mu(\partial_2(a))v) + \mu(\partial_1(a))\alpha_2(v) - \mu(\partial_2(a))\alpha_1(v) \qquad{}= (\mu(\partial_1(a))\alpha_2(v) - \alpha_2(\mu(\partial_1(a))v) ) - (\mu(\partial_2(a))\alpha_1(v) - \alpha_1(\mu(\partial_2(a))v) ) \qquad{}= - \mu(\partial_2(\partial_1(a)))v + \mu(\partial_1(\partial_2(a)))v = 0.
\begin{aligned}P_N^* =(e^{r^*}-1)N_0\sum_{i=0}^{K-1}\frac{e^{ir^*}}{|h_{K-i}|^2}\leq (e^{r^*}-1)N_0 \frac{1}{K}\sum_{i=0}^{K-1}e^{ir^*}\sum_{i=0}^{K-1}\frac{1}{|h_{K-i}|^2}=(e^{Kr^*}-1)\frac{N_0}{K}\sum_{i=1}^{K}\frac{1}{|h_i|^2}=P_{O1}^*.\end{aligned}
W_{s,t}(x) = \int_s^t [V_{du_1}, V_{d u_2}](x) = \frac{1}{2}\left( \int_s^t V^i_{s,u}(x) \partial_i V_{du}(x) - \int_s^t \partial_i V_{s,u}(x) V^i_{du}(x) \right).
\tilde\eta-\frac{1}{2}\tilde\eta^he^{-|h|}-\frac{1}{2}\tilde\eta^{-h}e^{-|h|}=\tilde\eta(1-e^{-|h|})+e^{-|h|}(\tilde\eta-\frac{1}{2}\tilde\eta^h-\frac{1}{2}\tilde\eta^{-h})
\gamma(AB) = \delta(AB)+AB\tau(I) & = A \delta ( B) + \delta (A) B - A \delta (I) B+ AB\tau(I) & = \delta (A) B+A(\delta ( B)-\delta (I) B)+ AB\tau(I) & = \delta (A) B+A(\tau ( B)- B\tau ( I))+ AB\tau(I) & = \delta (A) B+A\tau ( B)
\mathcal{I}_{\mathrm{D}}(\delta_1, 0) = \sup_{\pi \in \Pi\left(\mu_1, \mu_2\right)}\inf_{\lambda_1 \in \mathbb{R}_{+}}\left[ \lambda_1 \delta_1 + \int_{\mathcal{V}}g_{\lambda, 1}\, d \pi\right].
\mu_{1 , 2}^{2}= m_{3 / 2}^{2}+ V_{o}+{\tilde{\mu}}^{2}, \mu_{3}^{2}= - B m_{3 / 2}{\tilde{\mu}},{\tilde{\mu}}= e^{\frac{1}{2}K_{o}}\mu \frac{W_{o}^{*}}{| W_{o}|},
\mathbb{E}\Big(e^{-\beta\tau_i^{*}}\Big)=e^{-\frac{\lambda_-}{\sigma}\ln\left(\frac{x^*}{x}\right)}=e^{-k_1}\ln\left(\frac{x^*}{x}\right)=\left(\frac{x^*}{x}\right)^{-k_1}.
(\Delta(f) \circ \tilde{\Phi})_0(a \otimes b) = (0,z \varphi_0(a \otimes b) - \varphi_0(a \otimes b) z) , (\Delta(f) \circ \tilde{\Phi})_i = (f_\lambda^{E_{i}}- f_\varrho^{E_i}) \circ \varphi_i .
\sum_{p|c}\frac{\mathcal{S}(i, j, c )}{c}\,J_{2k-1}\left(\frac{4\pi \sqrt{ij}}{c}\right)=O\left(p^{-\frac{1}{2}}\, (i,j)^{\frac{1}{2}}\,(ij)^{\frac{1}{2}}\right).
\left( \frac{d}{dt}-\Delta_{\omega_{\phi}}\right) \nabla_{\phi}{\rm Rm}_{\phi}=\nabla_{\phi}\left(\frac{d}{dt}-\Delta_{\omega_{\phi}}\right){\rm Rm}_{\phi}+{\rm Rm}_{\phi}\ast \nabla_{\phi}({\rm Rm}_{\phi}+\widetilde{\eta}+\nabla_{\phi}X),
\begin{aligned}\bar h_{1122}^{2^{*}}=\bar h_{2211}^{2^{*}}, \ \bar h_{1112}^{1^{*}}=\bar h_{1121}^{1^{*}}, &\bar h_{1112}^{2^{*}}-\bar h_{1121}^{2^{*}}=(\bar \lambda_1-2\bar \lambda_2)\bar K, & \bar h_{2212}^{2^{*}}-\bar h_{2221}^{2^{*}}=3\bar \lambda_2\bar K.\end{aligned}
f(x) \in h(x) \ast (g(x))^{-1}= h(x) \ast r(g)(x) \mbox{implies}f \in h \ast r(g) g(x) \in (f(x))^{-1}\ast h(x) = r(f)(x) \ast h(x) \mbox{implies}g\in r(f) \ast h
\widehat{P}^\omega\left((\log t)^{2/d}\left(\frac{\log N_t}{t}-\beta\right)+c(d,\nu)<-\varepsilon\right)=\widehat{P}^\omega\left(N_t<\exp\left[t\left(\beta-\frac{c(d,\nu)+\varepsilon}{(\log t)^{2/d}}\right)\right]\right) .
G_1 = \left[\begin{array}{cc}I_{n_1-1}0\end{array}\right], F_1 = \left[\begin{array}{cc}0 I_{n_1-1}\end{array}\right], G_2 = \left[\begin{array}{cc}0 I_{n_2-1}\end{array}\right], F_2 = \left[\begin{array}{cc}I_{n_2-1}0 \end{array}\right].
F_{n}^{\mathcal{O}| \ldots \mu_{i}\mu_{j}\ldots}( \ldots \theta_{i}, \theta_{j}\ldots ) = F_{n}^{\mathcal{O}| \ldots \mu_{j}\mu_{i}\ldots}( \ldots \theta_{j}, \theta_{i}\ldots ) S_{\mu_{i}\mu_{j}}( \theta_{i , j}) .
\max\{|S|:S\in\bigcap_{i=1}^n\mathcal{F}_i\}=\min\{r_1(X)+r_2(X_1\backslash X)+\dots+r_{n-1}(X_{n-2}\backslash X_{n-3})+r_n(E\backslash X_{n-2}): X\subseteq X_1\subseteq X_2\subseteq \dots\subseteq X_{n-2}\subseteq E\}
\alpha+\Delta \leq h_{k}(H_{k}) = \sum_{r\in R}\frac{|E^{r}(H_{k})|}{\binom{k}{r}}= \sum_{r\in R}\frac{r!}{k(k-1)...(k-r+1)}|E^{r}(H_{k})| = \sum_{r\in R}\frac{r!}{k^{r}(1-\frac{1}{k})(1-\frac{2}{k})...(1-\frac{r-1}{k})}|E^{r}(H_{k})| \leq \frac{1}{c}\sum_{r\in R}\frac{r!}{k^{r}}|E^{r}(H_{k})|.
\alpha_{1}=\|B^{\dagger}\|_{2}^{2}\big(\|EB^{\dagger}\|_{F}^{2}-\|AA^{\dagger}EB^{\dagger}\|_{F}^{2}\big), \alpha_{2}=\|A^{\dagger}\|_{2}^{2}\big(\|EA^{\dagger}\|_{F}^{2}-\|BB^{\dagger}EA^{\dagger}\|_{F}^{2}\big).
W \cong P\bigl(-\xi_1 z_1 - \xi_2 z_2\bigr) := \bigl\{f \in R \,\big|\, \exp(-\xi_1 z_1 - \xi_2 z_2) f \;\mbox{\rm is}\; (\Pi, \underline{\mu})-\mbox{\rm quasi--invariant}\bigr\}.
L^{+}= \left( \begin{array}{l l}{q^{S^{3}}}&{\Omega S^{-}}\\{0}&{q^{- S^{3}}}\\ \end{array}\right) \ , \L^{-}= \left( \begin{array}{r r}{q^{- S^{3}}}&{0}\\{- \Omega S^{+}}&{q^{S^{3}}}\\ \end{array}\right)
\mathrm{d i v \int \frac{d^{4}q}{( 2 \p i )^{4}}\frac{1}{q^{2}( q + p )^{2}\vec{p}^{2}( \vec{q}+ \vec{p})^{2}}= \frac{- 4 \Gamma ( 2 - \o m e g a - \s i g m a )}{p^{2}\vec{p}^{2}( 4 \p i )^{2}}\left|{\begin{array}{l}{{{\scriptstyle \s i g m a \to ( 1 / 2 )^{+}}}}\\{{{\scriptstyle \o m e g a \to ( 3 / 2 )^{+}}}}\\ \end{array}}~ , \right.}
T_\psi^{a A_0,a A_1,a A_2}(X,Y) =\sum_{i=1}^{n_0}\sum_{k=1}^{n_1}\sum_{j=1}^{n_2}\psi\bigl(a\lambda_i^{(0)}, a\lambda_k^{(1)},a \lambda_j^{(2)}\bigr)E_{i}^{A_0}X E_{k}^{A_1}Y E_{j}^{A_2}&=T^{A_0,A_1,A_2}_{\psi_a}(X,Y).
\left\vert\pi_1(X_n)\left(\frac{1}{\pi_2(X_n)}-\frac{1}{H_n}\right)\right\vert =\left\vert\frac{\pi_1(X_n)}{\pi_2(X_n)}\right\vert \frac{H_n-\pi_2(X_n)}{H_n}\leqslant \frac{n}{H_n/2}\frac{H_n^{1/2}}{H_n}=2n H_n^{-3/2}.
\lim_{k\rightarrow+\infty}(\mathbf{P}_{\mathcal{B}|\mathcal{A}}\mathbf{P}_{\mathcal{A}|\mathcal{B}})^{k}\cdot\mathbb{E}[f|\mathcal{A}]=P_{1}[f],\text{}\lim_{k\rightarrow+\infty}(\mathbf{P}_{\mathcal{A}|\mathcal{B}}\mathbf{P}_{\mathcal{B}|\mathcal{A}})^{k}\cdot\mathbb{E}[f|\mathcal{B}]=P_{2}[f],
\begin{cases}-\partial_{tt}w_n + \Delta w_n =|\nabla F_n^\omega + \nabla w_n|^2 + 2 \nabla u_{n-1}\nabla w_n + 2 \nabla P_{>N^\gamma}u_{n-1}\cdot \nabla F_n^\omega~, w_n|_{t=0}= 0~, ~~ \partial_t w_n|_{t=0}=0~. \end{cases}
S_{\sigma}=-\frac{k}{4\pi}\int_{M}d\tau d\sigma\;\eta^{\alpha\beta}\mbox{Tr}[g^{-1}\partial_\alpha g\;g^{-1}\partial_\beta g]-\frac{k}{6\pi}\int_{B}\mbox{Tr}[g^{-1}dg\wedge g^{-1}dg\wedge g^{-1}dg].
C=\begin{pmatrix}1 0 1 0 1 1 0 0 1 0 0 0 0 0 1 1\end{pmatrix},\hat{C}=\begin{pmatrix}\frac{1}{\sqrt{6}}0 \frac{1}{2}0 begin{align*}3mm]\frac{1}{\sqrt{6}}\frac{1}{\sqrt{2}}0 0 begin{align*}3mm]\frac{1}{\sqrt{3}}0 0 0 begin{align*}3mm]0 0 \frac{1}{2}\frac{1}{\sqrt{2}}\end{pmatrix}.
\displaystyle\sum_{n=1}^{N}\left[\begin{array}{c}N n\end{array}\right] \frac{(q)_n (zc)^n q^{n^2}}{(zq)_n (cq)_n}= z \displaystyle\sum_{n=1}^{N}\left[\begin{array}{c}N n\end{array}\right]\frac{(q)_n (cq)_{N-n}(cq)^n}{(zq)_n (cq)_N}.
\Big ( \tau^{2}\Big )^{\frac{1}{N}\Big [ \big ( \sum_{a}n_{a}\big ) \big ( \sum_{a}m_{a}^{\prime}\big ) + \big ( \sum_{a}m_{a}\big ) \big ( \sum_{a}m_{a}^{\prime}\big ) - \big ( \sum_{a}n_{a}\big ) \big ( \sum_{a}n_{a}^{\prime}\big ) - \big ( \sum_{a}n_{a}\big ) \big ( \sum_{a}m_{a}^{\prime}\big ) \Big ]}\; .
\sum_{n=-\infty}^\infty \frac{(6n+3)q^{3n(n+1)+2n+2}}{1+q^{6n+3}}-\sum_{n=-\infty}^\infty \frac{(6n-1)q^{3n(n+1)-2n}}{1+q^{6n-1}}&=\frac{2(q^{2};q^{2})^2_\infty}{(-q^{2};q^{2})^2_\infty}\sum_{n=-\infty}^\infty \frac{(-1)^nq^{3n(n+1)+2n+2}}{1-q^{6n+3}}+\frac{q[q^{2},-q^3,q^{4};q^{6}]_\infty(q^{6};q^{6})^4_\infty}{[q,-q,-q^2,-q^{2},-q^4;q^{6}]_\infty}.
a\circ(b\circ c) =a\circ(B_1(b_1)cS(B_2(b_2))) &=B_1(a_1)B_1(b_1)cS(B_2(b_2))S(B_2(a_2)) &=B_1(B_1(a_1)b_1S(B_2(a_2)))cS(B_2(B_1(a_3)b_2S(B_2(a_4)))) &=B_1(a_1\circ b_1)cS(B_2(a_2\circ b_2))=(a\circ b)\circ c.
\int_{0}^{\infty}{\frac{d s}{s}}G ( s ) ={\frac{1}{\pi}}\int_{0}^{\infty}d \lambda{\frac{\Im \widetilde E ( \lambda )}{\lambda^{2}}}= c_{1}={\frac{3}{1 6}}( 2 K^{2}+ 2 K + 1 ) ,
g_{k}(\alpha^{-1}q^{m}) =\alpha^{k}q^{-mk}\sum_{j=k-m}^{\infty}(q^{m-k+j+1};q)_{\infty}\frac{(-1)^{j}q^{\frac{1}{2}j(j+1)+2mj}}{(q;q)_{j}}\alpha^{-2j}=(-1)^{m+k}\alpha^{2m-k}q^{-\frac{1}{2}m(3m+1)+\frac{1}{2}k(k+1)}\sum_{j=0}^{\infty}(q^{j+1};q)_{\infty}\frac{(-1)^{j}q^{\frac{1}{2}j(j+1)+(m+k)j}}{(q;q)_{k-m+j}}\alpha^{-2j},
ds^2 ={X^{1/2}\over \rho}( e^{2 A}dx^2 + dr^2) +{X^{1/2}\over \rho^3}( d \theta^2 +{\rho^6 \cos^2 \theta \over X}d \Omega_3^2 +{\sin^2 \theta\over X}d \phi^2 )
\int\limits_{S(r)}\log^+\Psi (z) \eta \le \sum_{i=1}^{n+1}\sum_{v=0}^n \int\limits_{S(r)}\log^+|A_{vi}(z)|\eta+O(1) &\le \sum_{i=1}^{n+1}\sum_{v=0}^n T(r,A_{vi}) +O(1) &= O(\max_{1\le i\le q}T_{a_i}(r))+O(1).
[W', \i A]_{\circ}\approx [W_1', \i A_1]_{\circ}\otimes W_2' \otimes ...\otimes W_d' \ + \W_1' \otimes [W_2', \i A_2]_{\circ}\otimes ...\otimes W_d' & + \ ... \ + \W_1' \otimes ...\otimes W_{d-1}' \otimes [W_d', \i A_d]_{\circ}&:= K_{W'}+ B_{W'}
\frac{df}{dE}=\frac{1}{m_{w}^{*}}\frac{dk}{dE}tan(\frac{kl_{w}}{2})+\frac{k}{m_{w}^{*}}sec^{2}(\frac{kl_{w}}{2})\times\frac{l_{w}}{2}\frac{dk}{dE}-\frac{1}{m_{b}^{*}}\frac{d\kappa}{dE}(9-1)
M :=\begin{bmatrix}m_{0}m_{1}m_2 \cdots m_{1}m_{2}m_3 \cdots m_{2}m_{3}m_4 \cdots \vdots \vdots \vdots \cdots \end{bmatrix}, M_{[k]}:=\begin{bmatrix}m_{0}\cdots m_{k-1}\vdots \vdots m_{k-1}\cdots m_{2k-2}\end{bmatrix}.
\sum_{k,p}\partial_{t^i}\partial_{t^j}\partial_{t^k}\mathbb{F}(t)~ \eta^{kp}~\partial_{t^p}\partial_{t^q}\partial_{t^s}\mathbb{F}(t) = \sum_{k,p}\partial_{t^s}\partial_{t^j}\partial_{t^k}\mathbb{F}(t) ~\eta^{kp}~\partial_{t^p}\partial_{t^q}\partial_{t^i}\mathbb{F}(t),
|\nabla^j (u\rho^{\frac{1}{2}})| \le | \nabla^j u| \rho^{\frac{1}{2}}+ \sum_{i=1}^{j}C_{ij}|\nabla^{j-i}u| \; |\nabla^i (\rho^{\frac{1}{2}})| &\le | \nabla^j u| \rho^{\frac{1}{2}}+ C\sum_{i=1}^{j}N^i |\nabla^{j-i}u | d^{-2i}\rho^{\frac{1}{2}}.
F_1(y,q) =\sum_{n>0}\sum_{d|n}\frac{1}{2}\left(-\frac{n^3}{d^3}+\frac{n^2}{d}-\frac{n}{d}\right)(y^{d/2}-y^{-d/2})^2 q^n F_2(y,q) =\sum_{n>0}\sum_{d|n}\left(\frac{n^2}{d^2}-\frac{n}{2}\right)\frac{y^{d}-y^{-d}}{y-y^{-1}}q^n.
A_1 = 4 \log \left(t^9+6t^6\right) A_2 = 2 \log \left(t^9+5t^6\right) A_3 = 2 \log \left(t^9+7t^6\right) B= m' \frac{\log \left(t^9+7t^6\right)}{\log \left(t^9+5t^6\right)}> \left(1.0007\right)m'
S_{\mathrm{W}_{A}}^{\mathrm{B I}}= - T_{0}\int d \tau ~ \mathrm{S t r}\Bigl \{k^{- 1}\sqrt{- P [ E_{0 0}+ E_{0 i}( Q^{- 1}- \delta )_{k}^{i}E^{k j}E_{j 0}] \ \mathrm{d e t}( Q_{j}^{i})}\ \Bigr \},
\begin{aligned}\alpha =e^2+|\vec{j}|^2-|\vec{p}|^2-|\vec{c}|^2 \beta =(e^2-|\vec{j}|^2-|\vec{p}|^2-|\vec{c}|^2)^2-4|\vec{j}\times\vec{p}|^2-4|\vec{p}\times\vec{c}|^2-4|\vec{c}\times\vec{j}|^2+8e\vec{c}\cdot (\vec{p}\times\vec{j}).\end{aligned}
\biggl[ \frac{1}{n^d}\biggr] (t_+ - t_-) = \biggl[ \frac{1}{n^d}\biggr] \sum_\ell \binom{k}{\ell}(-1)^{k+\ell}\Bigl( U_{i+\ell}- \frac{1}{2}(U_{i+\ell})^2 + \frac{1}{3}(U_{i+\ell})^3 \cdots \Bigr) = \begin{cases}\hfil 0 d < k - 1 \displaystyle\frac{(k-2)!}{r^{k-1}}d = k-1 \end{cases}
d\lambda_i(t)=\sqrt{2(1+\lambda^2_i(t))}d\beta_i(t)+\left(2\Im(s)+\left(2-2N-2\Re(s)\right)\lambda_i(t)+\sum_{j\ne i}^{}\frac{2\left(1+\lambda^2_i(t)\right)}{\lambda_i(t)-\lambda_j(t)}\right)dt.
f^\lambda_{\lvert\vec{x}\rvert}\!\left(\vec{y}+\frac{\vec{x}}{2}\right)=\sqrt{\frac{8}{3\pi}}\,\lambda\,\frac{\mathrm{K}_2\Big(\sqrt{\frac{4\lambda}{3}}\,\textstyle{\sqrt{y^2+x^2+\vec{x}\!\cdot\!\vec{y}}}\,\Big)}{y^2+x^2+\vec{x}\!\cdot\!\vec{y}}.
\frac{u-v+h}{u-v}f_{1}^{\pm}(v)k_{1}^{\pm}(v)k_{1}^{\pm}(u)=\frac{h}{u-v}f_{1}^{\pm}(u)k_{1}^{\pm}(u)k_{1}^{\pm}(v)+k_{1}^{\pm}(u)f_{1}^{\pm}(v)k_{1}^{\pm}(v)
R e f_{\pm}( s , t ) = \frac{( s )^{k}}{\pi}P \! \int_{s_{0}}^{+ \infty}d s^{\prime}\frac{I m f_{\pm}( s^{\prime}, t )}{( s^{\prime})^{k}}\left[ \frac{1}{s^{\prime}- s}\pm \frac{( - 1 )^{k}}{s^{\prime}+ s}\right] .
\widehat{\xi}_{[(\sigma,k,l)]}(s) =\frac{(r_\alpha-s)^{\frac{k-(m_\sigma-2)}{m_\sigma}}}{m_\sigma^2 k!}k\neq m_\sigma-2\ \operatorname{mod}m_\sigma \widehat{\xi}_{[(\sigma,k,l)]}(s) =-\frac{(r_\alpha-s)^{\frac{k-(m_\sigma-2)}{m_\sigma}}\log(r_\alpha-s)}{m_\sigma^2 k!}k= m_\sigma-2\ \operatorname{mod}m_\sigma.
\|\pi_n u-\pi_nv\|_{\L^p}= \bigg\| \frac{n}{\|u\|_{\L^p}}u-\frac{n}{\|v\|_{\L^p}}v\bigg\|_{\L^p}&= \frac{n}{\|u\|_{\L^p}\|v\|_{\L^p}}\|(u-v)\|v\|_{\L^p}+v(\|v\|_{\L^p}-\|u\|_{\L^p})\|_{\L^p}& \leq \|u-v\|_{\L^p}+(\|v\|_{\L^p}-\|u\|_{\L^p}) \leq 2\|u-v\|_{\L^p},
\nu_A^*\theta_i =\sigma_i, \nu_B^*(kh)^*\theta_i =(kh)^*\sigma_i=\sum_j \sigma_i(k e_j)h^*\sigma_j, \nu_A^*\varphi_i =S_A^*\sigma_i, \nu_B^*(kh)^*\varphi_i =(k h S_B)^*\sigma_i=\sum_j \sigma_i(d(kh)\,S_B\,dh^{-1}\,e_j) h^*\sigma_j.
\bigl\langle \Gamma(\phi)(X_1,\ldots,X_{n-1}), X_n\bigr\rangle = \langle\phi,\varphi\rangle & = \int_{\Sigma}\langle \widetilde{\phi}(t) ,\varphi \rangle \,\mu(t) & = \int_{\Sigma}\bigl\langle\Gamma(\widetilde{\phi}(t))(X_1,\ldots,X_{n-1}), X_n\bigr\rangle \,\mu(t)\,.
I_{(X_+,D_+)}(y,z)=ze^{t_+/z}\sum_{d\in\mathbb K_{+}}y^{d}\left(\prod_{i\in M_0}\frac{\prod_{a\leq 0,\langle a\rangle=\langle D_i\cdot d\rangle}(\bar{D}_i+az)}{\prod_{a \leq D_i\cdot d,\langle a\rangle=\langle D_i\cdot d\rangle}(\bar{D}_i+az)}\right)\textbf{1}_{[-d]}I_{D_+,d},
T(n_i)+T(n_{i+1})+T(n_{i+1}) < \left(\frac{297}{81}+\frac{128}{81}\right) \cdot \frac{1}{X_0}&= \frac{425}{81}\cdot \frac{1}{X_0}< 7 \cdot \frac{3}{4}\cdot \frac{1}{X_0}&\leq (k_i+k_{i+1}+k_{i+2}) \cdot \frac{3}{4}\cdot \frac{1}{X_0},
l_{\bar{q}}^{\mu}= \left( k_{g}+ k_{\bar{q}}^{\prime}- k_{\bar{q}}\right)^{\mu}={\frac{\vec{k}_{g_{\! \perp}}^{\, 2}}{2 k_{g}^{+}}}\ \eta^{\mu}\quad \mathrm{a n d}\quad l_{q}^{\mu}= \left( k_{g}+ k_{q}- k_{q}^{\prime}\right)^{\mu}={\frac{\vec{k}_{g_{\! \perp}}^{\, 2}}{2 k_{g}^{+}}}\ \eta^{\mu}\ .
\left( \begin{array}{c c}{n_{1}^{\prime}}{n_{2}^{\prime}}{l_{1}^{\prime}}{l_{2}^{\prime}}\end{array}\right) = \left( \begin{array}{c c}{a}{c / 2}{2 b}{d}\end{array}\right) \left( \begin{array}{c c}{n_{1}}{n_{2}}{l_{1}}{l_{2}}\end{array}\right)
\begin{array}{r c l}{{M_{M 2_{6}}}}&{{=}}&{{{\displaystyle \frac{R_{3}R_{4}( R_{5}\ldots R_{1 0})^{2}}{({}^{-}\! \! \! \! \ell_{\mathrm{P l a n c k}}^{( 1 1 )})^{1 5}}}\, ,}}\\{{}}&{{}}&{{}}\\{{M_{M 5_{3}}}}&{{=}}&{{{\displaystyle \frac{R_{3}\ldots R_{6}( R_{7}R_{8}R_{9})^{2}R_{1 0}}{({}^{-}\! \! \! \! \ell_{\mathrm{P l a n c k}}^{( 1 1 )})^{1 2}}}\, .}}\\ \end{array}
{\mbox{\bf M}}(v)= \left(\begin{array}{cc}\frac{1+v^2}{1-v^2}\frac{+2v}{1-v^2}\frac{+2v}{1-v^2}\frac{1+v^2}{1-v^2}\end{array}\right)=\frac{1}{\gamma^2}\left(\begin{array}{cc}1+v^2 +2v +2v 1+v^2\end{array}\right)\equiv\frac{1}{\gamma^2}{\mbox{\bf L}}(v)\, .
0=N_F(e_1,e_2) = 2(AC-B\mu)e_1+2(AB-\lambda C)e_2-(1+A^2+B^2+C^2-\lambda\mu+\lambda\tau+\mu\tau)e_3, begin{align*}5pt]0=N_F(e_1,e_3) =-2(A\tau+BC)e_1-(1-A^2-B^2+C^2+\lambda\mu-\lambda\tau+\mu\tau)e_2-2(AB-\lambda C)e_3, begin{align*}5pt]0=N_F(e_2,e_3) =(1-A^2+B^2-C^2+\lambda\mu+\lambda\tau-\mu\tau)e_1+2(A\tau+BC)e_2+2(AC-B\mu)e_3.
\|f\|_{C^{0,\mu}(Q_\ell^*;H_\xi^s(Q_\ell\times Q_\ell))}:=\|f\|_{L^\infty H^s}+\sup_{\xi\neq \xi'\in Q_\ell^*}\frac{\|f(\xi,\cdot,\cdot)-f(\xi',\cdot,\cdot)\|_{H^s(Q_\ell\times Q_\ell)}}{|\xi-\xi'|^\mu}.
\widetilde{A}^{AVF}:=\left(\begin{array}{c}\delta_n^+u_{m,n}+\delta_m^{(2)}\mu_nv_{m,n}+\mu_n(v_{m,n}^2+\frac{2}3u_{m,n}^2)\mu_nv_{m,n}+\frac{1}3\mu_n(u_{m,n}^2v_{m,n}) -\delta_n^+v_{m,n}+\delta_m^{(2)}\mu_nu_{m,n}+\mu_n(u_{m,n}^2+\frac{2}3v_{m,n}^2)\mu_nu_{m,n}+\frac{1}3\mu_n(v_{m,n}^2u_{m,n})\end{array}\right)=\mathbf{0}.
\left\|\frac{f'}{r}\right\|_{L_{\beta}^2}^2=\left\|\left(\frac{f}{r}\right)'+\frac{f}{r^2}\right\|_{L_{\beta}^2}^2=\left\|\left(\frac{f}{r}\right)'\right\|_{L_{\beta}^2}^2+2\Re\left\langle\left(\frac{f}{r}\right)',\frac{f}{r^2}\right\rangle+\left\|\frac{f}{r^2}\right\|_{L_{\beta}^2}^2.
h_{1-L}^{\mathcal{L}}(t)=h_{1-L}^{\mathcal{L}_+}(t)-h_{-L}^{\mathcal{L}_{10L+1}}(t)=h_{1-L}^{\mathcal{L}_+}(t)-h_{-1}^{\mathcal{K}_{10L+1}}\prod_{\alpha=2}^Lh_{-1}^{\mathcal{K}_\alpha}(t).
\Bigg(\sum_{j=1}^\infty|c_j|^2|k_{z_j}(z)|z|^{-m}|^2\Bigg)^{q/2}\lesssim \int_0^1\Bigg|\sum_{j=1}^\infty c_jr_jk_{z_j}(z)|z|^{-m}\Bigg|^qdt.
\Delta_{i,i+1}(u^{(n)}w_0^{(n)}) =\prod_{k=1}^{i-1}y_{n-1,n-k}\Delta_i(u^{(n-1)}w_0^{(n-1)}) +y_{n-1,n-i-1}\prod_{k=1}^{i-1}y_{n-1,n-k}\Delta_{i,i+1}(u^{(n-1)}w_0^{(n-1)}).
P_1 =\{(x,y,z) \mid x+y \geq 1, y \geq 0, y+z \geq 0, x+z \geq 0, x \geq 0 \}, P_2 =\{(x,y,z) \mid 2y+z \geq 1, y \geq 0, y+z \geq 0, x+z \geq 0, x \geq 0\}, P_3 =\{(x,y,z) \mid x+y+2z \geq 1, y \geq 0, y+z \geq 0, x+z \geq 0, x \geq 0\}, P_4 =\{(x,y,z) \mid 2x+z \geq 1, y \geq 0, y+z \geq 0, x+z \geq 0, x \geq 0\}.
\left\vert \sum_{k=3}^{+\infty}\frac{1}{k!}\left( i \frac{t}{v_u}f_u (x) \right)^k \right\vert \leq \sum_{k=3}^{+\infty}\frac{\vert t \vert^k K^{k-2}f_u (x)^2}{k! v_u^k}&\leq \frac{\vert t \vert^3 K f_u(x)^2}{v_u^3}\exp \left(\frac{\vert t \vert K}{v_u}\right).
\int_{0}^{1}t^{k-3/2}(1-xt)^{-1}\left(\tfrac{1-t}{1-tx}\right)^{k-3/2}\d t \leq (1-x)^{-1}\int_{0}^{1}t^{k-3/2}\d t=(1-x)^{-1}\tfrac{1}{k-1/2}.
((z_{1g_1}, z_{2g_1}), ..., (z_{1g_n}, z_{2g_n})) a = z_{1g_1}^{i_{g_1}}z_{2g_1}^{j_{g_1}}\ ... \z_{1g_n}^{i_{g_n}}z_{2g_n}^{j_{g_n}}a.
\bigl\{\lambda_{3}\bigl(\psi(\alpha),\psi(\beta),1,\psi(\gamma_1),\psi(\gamma_2)\bigr), \rho_{3}\bigl(\psi(\alpha),\psi(\beta),1,\psi(\gamma_1),\psi(\gamma_2)\bigr)\bigr\}\quad=\left\{\begin{pmatrix}a' a \end{pmatrix}, \begin{pmatrix}b' b \end{pmatrix}\right\},
\int_{\mathbb R^{n}}\ldots \int_{\mathbb R^{n}}\prod_{k=1}^{M}F^{(k)}( x_{1}, \ldots , x_{\ell}) \prod_{i=1}^{N}f_{i}^{\ast}\left( \sum_{j=1}^{\ell}a_{ij}x_{j}| \theta \right) d \mu(x_{\ell}) \ldots d\mu(x_{1}) .
d_{\lambda}^{( 1 )}= d_{( \lambda_{1}+ 1 , \lambda_{2}, \cdots \lambda_{j})}+ d_{( \lambda_{1}, \lambda_{2}+ 1 , \cdots \lambda_{j})}+ \cdots + d_{( \lambda_{1}, \lambda_{2}, \cdots \lambda_{j}+ 1 )}+ ( M - j ) d_{( \lambda_{1}, \lambda_{2}, \cdots \lambda_{j}, 1 )}.
\alpha - \alpha^{p^m}= \left(\frac{1}{2}+ \sqrt{d}{2}\right) - \left(\frac{1}{2}+ \frac{\sqrt{d}}{2}\right)^{p^m}\equiv \left(\frac{1}{2}+ \sqrt{d}{2}\right) - \left(\frac{1}{2}\right)^{p^m}+ \left(\frac{\sqrt{d}}{2}\right)^{p^m}\mod p \equiv 0 \mod p,
Y = \frac{k(k-1)(m-1)(m-2)}{(m+1)(m^2-2)}+ \frac{k(k-1)(m-2)}{(m+1)(m^2-2)}+ \frac{2k(k-1)(m-1)}{(m+1)(m^2-2)}&=\frac{k(k-1)}{(m+1)(m^2-2)}\left( (m-1)(m-2) + (m-2) + 2(m-1) \right) &= \frac{k(k-1)}{(m+1)(m^2-2)}\left( m^2-2\right) = \frac{k(k-1)}{m+1}.
\begin{aligned}\mathbb{E}^{u_1,u_2}[\bar{H}_{i{v_i}}(t,x,y,z,z_1,z_2,u_1,u_2;q_i,k_i,k_{1i},k_{2i},p_i,Q_{1i},Q_{2i})|\mathcal{F}_t^i]=0\quad(i=1,2),\end{aligned}
H_{A s}= - R^{2}/ 2 \, ( \pi / 2 )^{2}\, ( 1 / 4 \, a_{2}^{2}+ 1 1 / 2 4 \, a_{4}^{2}+ 2 9 / 6 0 \, a_{6}^{2}- 1 / 3 \, a_{2}\, a_{4}- 1 / 1 2 \, a_{2}\, a_{6}- 1 / 3 \, a_{4}\, a_{6}) .
\begin{aligned}2\,CY_n^\ast C=C(X- U_2^nXU_1^{-n})^\ast C+C(X-U_2^{-n}XU_1^n)^\ast C = (X-U_2^{-n}XU_1^n)+(X-U_2^nXU_1^{-n})=2\,Y_n\end{aligned}
\int\limits_{S_2}\mathbb{E}_{m^1}[f(\zeta(t),v)] m_0^2(dv) = \int\limits_{S_1}\mathbb{E}_{m^{2,t}}[f(u,\xi^t(t)) \exp \{\int\limits_0^t g(s,\xi^t(s))ds\}] m_0^1(du)