harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F\left( \omega \right) =\mathit{A}_{\mathit{m}}\,{\mathrm{M}\left( \frac{1}{2}-\frac{\,k}{2},\,1,\,2\,\omega \right)}+\mathit{B}_{\mathit{m}}\,{\mathrm{U}\left( \frac{1}{2}-\frac{\,k}{2},\,1,\,2\,\omega \right) ,}
lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=lim_{x\rightarrow\infty}(\frac{x+sin(x)}{x})=lim_{x\rightarrow\infty}(1+\frac{sin(x)}{x})=1+lim_{x\rightarrow\infty}(\frac{sin(x)}{x})=1+0=1.
\begin{array}{rcl}T_a = \displaystyle +[3D_m^2(k_1)D_m(k_2)+D^2(k_1)D(k_2)](2+K_{12}) & \displaystyle +[3D_m^2(k_1)D(k_2)+D^2(k_1)D_m(k_2)](2-K_{12}) & \displaystyle +(N-1)[6D_m^2(k_1)D(k_2)+4D^2(k_1)D(k_2)+2D^2(k_1)D_m(k_2)] & +4(N-1)ND^2(k_1)D(k_2), \end{array}
\sum_j\left(\frac{\partial X^\mu}{\partial x_i}\frac{\partial X^\mu}{\partial x_j}+F_{ij}\right)\hat L_{jk}\,+\,\left(\frac{\partial X^\mu}{\partial x_i}\frac{\partial X^\mu}{\partial x_j}-F_{ij}\right)\hat L_{kj}\,=\, 2L\delta_{ik}
\theta_k(\omega,z+\frac{a+b\omega}{3}) =\zeta_3^{ak}(q^bw)^{-b}\theta_{k+b}(\omega,z), \vartheta_k(\omega,z+\frac{a+b\omega}{3}) =\zeta_3^{ak}(q^{b-3}(-w))^{-b}\vartheta_{k+b}(\omega,z).
\mathcal{I}(\hat u,\delta,|c|,2)=\mathcal{I}_{\!1}(\hat u,\delta):=\begin{cases}\hat u\Bigl[ e^{\mu r}+ \frac{\sigma}{\mu}(e^{\mu r}- 1) \Bigr]+ \frac{\sigma}{\mu}D_1\delta\Bigl[\frac{1}{\mu}(e^{\mu r}- 1) - r e^{\mu r}\Bigr], \mu\ne0, \hat u +\sigma r\hat u -\frac{\sigma D_1 r^2}{2}\delta, \mu=0.\end{cases}
\begin{cases}\partial_t g_n + ( u^s +{u}) \partial_x g_n - \partial^2_y g_n - \epsilon \partial^2_xg_n \qquad\qquad - \epsilon \,2\, (\partial_x \partial_y^{-1}g_n) \partial_y \eta_1 = M_n, (\partial_y g_n)|_{y=0}=0, g_n|_{t=0}= g_{n,0},\end{cases}
\int_{y=-2\sqrt{3}}^{y=\sqrt{6}}\int_{x=y}^{x=\sqrt{24-y^{2}}}dxdy=\cdot\cdot\cdot=[-\frac{y^{2}}{2}+\frac{1}{2}y\sqrt{24-y^{2}}+12arcsin(\frac{y}{\sqrt{24}})]_{-\sqrt{12}}^{\sqrt{6}}
S(\{a_{(1)}, b_{(1)}\})a_{(2)}b_{(2)}= -S(a_{(1)}b_{(1)})\{a_{(2)}, b_{(2)}\}=-(-1)^{|a_{(1)}||b_{(1)}|}S(b_{(1)})S(a_{(1)})\{a_{(2)}, b_{(2)}\}= (-1)^{|a_{(1)}||b_{(1)}|+|b_{(2)}||a_{(2)}|}S(b_{(1)})\{S(a_{(1)}),b_{(2)}\}a_{(2)}=-(-1)^{|b_{(2)}||a_{(2)}|}\{S(a_{(1)}),S(b_{(1)})\}b_{(2)}a_{(2)}= -\{S(a_{(1)}), S(b_{(1)})\}a_{(2)}b_{(2)}.
\begin{array}{l}\displaystyle \binom nkB_{n-k,k}\left(a_1b_1,\dots,\frac1{m+1}\sum_{i=1}^m\binom{m+1}i a_ib_{m+1-i},\dots \right)= \displaystyle\sum_{i=k}^{n-k}\binom niB_{i,k}(a_1,a_2,\dots)B_{i,k}(b_1,b_2,\dots).\end{array}
{\cal T}|{\bf b}+ \frac{1}{2}; n_{1}, n_{2}, \ldots , n_{b}\rangle = \sum_{i = 0}^{2^{b - 1}}( - )^{p_{i}+ 1}|{\bf{p_{i}+ 1 / 2}}; \tau_{i}( n_{b}, n_{b - 1}, \ldots , n_{1}) , \frac{1}{2}\rangle ,
p(x) = \det\left(xI - (M-I)\right) = \det\begin{pmatrix}x+w_1 -w_1 0 \cdots 0 x + w_2 -w_2 \vdots \vdots \ddots \ddots \vdots -w_n \cdots 0 x+w_n \end{pmatrix}= \prod\limits_{i = 1}^n \left(x+w_i\right) + (-1)^{n-1}\prod\limits_{i = 1}^n (-w_i) = \prod\limits_{i = 1}^n \left(x+w_i\right) - \prod\limits_{i = 1}^n w_i
\alpha_{n}(a) := a \ln(n/k) \bigg(\frac{[nx]+1}{n}-x\bigg)^{p}, \beta_{n}(a):= a \ln(n/k) \bigg(\frac{[n(x+\epsilon)]+1}{n}-x\bigg)^{p}, \tau_{n}(a) := a \ln(n/k) \bigg(\frac{[nx]}{n}-x\bigg)^{p}, \mu_{n}(a):= a \ln(n/k) \bigg(\frac{[n(x+\epsilon)]}{n}-x\bigg)^{p}
- \left[ J^{1}( x ) \right]_{- L / 2}^{L / 2}= \left[ ( / \hspace{- 0 . 5 e m}\partial \phi + U ) \gamma^{1}\psi - ( - / \hspace{- 0 . 5 e m}\partial \phi + U ) \gamma^{1}\gamma^{3}\psi \right]_{L / 2}= \left. ( 1 + \gamma_{3}) ( / \hspace{- 0 . 5 e m}\partial \phi + U ) \gamma^{1}\psi \right|_{L / 2}.
\frac{(1-\alpha^2)(1-|z|)^3}{(1+\alpha|z|)^2(1+|z|)^5}\leq J_f(z)\leq\begin{cases}\displaystyle \frac{(1-\alpha^2)(1+|z|)^3}{(1-\alpha|z|)^2(1-|z|)^5}, |z|<\alpha, begin{align*}8pt]\displaystyle \frac{(1+|z|)^2}{(1-|z|)^6}, |z|\geq\alpha,\end{cases}
\|\sigma^\ast(x)\|_q \; =\; \sup\{\,|x_\nu^q|\,|\zeta|^{\nu q}:\;\nu\le m\,\}\;\leq\; \sup\{\,\bigl(|x_\nu|\,|\zeta|^{\nu q}\bigr)^q:\;\nu\le m\,\}\;=\; \|x\|_q^q\,.
\!\!\! \sum_{\kappa_{1,n}\in S_1,\n\ge0}\!\!\!\!\!\!\!\!\frac{(\kappa_{1,n}-n^2\pi^2/\sigma_1^2)_+}{n^2}+ \!\!\!\!\!\!\!\! \sum_{\kappa_{2,n}\in S_2,\n\ge0}\!\!\!\!\!\!\!\frac{(\kappa_{2,n}-\gamma_n^2/\sigma_2^2)_+}{n^2}<\infty,\sigma_1+\sigma_2=2b,
\begin{aligned}\sum_{j=1}^L\frac{1}{n}\sum_{m=0}^{n-1}\frac{1}{S}\bigl|(f,e^{imSH}({\tilde P}^a)^* e^{iS(H^a-E_j)}A^a P^{a,E_j}{\hat P}^{|a|-1}e^{-imSH}f) &\quad\quad\quad\quad\quad-(f,e^{imSH}({\tilde P}^a)^* A^a P^{a,E_j}{\hat P}^{|a|-1}e^{-imSH}f)\bigr|.\end{aligned}
j^{0}= - \frac{J^{0}}{| \psi |^{2}}= \frac{1}{2}i \partial_{\tau}\ln \left( \frac{\psi}{\psi^{*}}\right) + \varphi , \vec{j}= - \frac{\vec{J}}{| \psi |^{2}}= - \frac{1}{2}i \vec{\partial}\ln \left( \frac{\psi}{\psi^{*}}\right) + \vec{a}.
0 >F(x(t_n^\epsilon)) - Mt_n^\epsilon + \frac{2}{\epsilon}\log x(t_n^\epsilon) &= -F(u_n^\epsilon/K(\epsilon)) +F(x(t_n^\epsilon)) + \frac{2}{\epsilon}\log x(t_n^\epsilon) &> -F(u_n^\epsilon/K(\epsilon)) +F(u_n^\epsilon) + \frac{2}{\epsilon}\log u_n^\epsilon>0,
-2\mathrm{tr}\ln\left(\bar{D}^2\right)-\frac{1}{2}\mathrm{tr}\ln E+\frac{1}{2}\mathrm{tr}\left(G_0F\right)+\frac{1}{4}\mathrm{tr}\left(G_0FG_0F\right)+\frac{1}{6}\mathrm{tr}\left(G_0FG_0FG_0F\right)+\ldots
H^\textsc{csm}\equiv \frac{1}{2}\sum_{j=1}^N p_j^2 + g^2 \sum_{i < j}^N \frac{1}{4 \sin^2[(x_i - x_j)/2]}= \frac{1}{2}\sum_{j=1}^N p_j^2 + g^2 \sum_{i < j}^N \frac{z_i \, z_j}{z_{ij}\, z_{ji}}\, ,
\mathbb{E}_{\mathbf{H}}\{\Vert\hat{\mathbf{x}}_{lmmse}-\mathbf{x}\Vert^2\}=\sum_{k=0}^{l}\frac{\left(-1\right)^k}{\sigma_x^{2k}}\mu_{\mathbf{\Lambda}}\left(-k-1\right) +o\left(\sigma_x^{-2l}\right)
\| \prod_{i=1}^k \mathcal E_\lambda^\gamma f_i \|_{L^2(d\mu)}\le C \langle \mu\rangle_\alpha^{\,\,\frac12}L^{-\frac{k^2 -k}{4}}\lambda^{- \frac{\alpha - d +k}{2}}\prod_{i=1}^k \|f_i\|_{L^2}.
\frac{d^nK_{\alpha}(t)}{dt^n}=-\sum_{m_1,\dots,m_n}\frac{n!(m_1+\dots+m_n-1)!}{m_1!m_2!\dots m_n!}\times \prod_{1\leq j\leq n}\left(-\frac{M_{\alpha}^{(j)}(t)}{j!M_{\alpha}(t)}\right)^{m_j},
\mathfrak{g}_1=\big(z_2-z_1+z_0:2z_1-z_0:z_3:z_4:\ldots:z_n:z_1-z_0\big), & \mathfrak{g}_2=\big(z_0+z_2:z_0:z_1:z_3:z_4:\ldots:z_n\big), & \mathfrak{g}_3=\big(-z_1:z_0+z_2-3z_1:z_0:z_3:z_4:\ldots:z_n\big), & \mathfrak{g}_4=\big(z_1-z_n:-2z_n-z_0:2z_n-z_1:-z_2:-z_3:\ldots:-z_{n-1}\big).
\begin{array}{r l}{{d s^{2}= \alpha^{\prime}g_{6}\sqrt{N_{1}N_{5}}[}}&{{u^{2}( d x_{0}^{2}+ d x_{1}^{2}) +{\frac{d u^{2}}{u^{2}}}+ d \Omega_{3}^{2}}}\\{{\nonumber +}}&{{\beta ( N_{1}, N_{5}) ( d x_{2}^{2}+ . . . + d x_{5}^{2}) ] ,}}\\ \end{array}
\hat{W}= i \hat{\zeta}m \sum_{a = 1}^{N}[ \hat{\theta}_{a}^{2}, \hat{\theta}_{a}^{1}] = \hat{\zeta}m \left( \frac{N}{2}- \hat{n}\right) , \hat{n}= \sum_{a = 1}^{N}\hat{n}_{a}.
{\mathcal J}_p(u,v; E) \leq \inf\{{\mathcal J}_p(u,v;A): A \subset \Omega , E\subset A \}&\leq\inf\left\{\int_A f(x,u,v,0)dx + \int_{\partial^\ast E\cap A}K_p(x,0,c,d,\nu)d{\cal H}^{N-1}: A \subset \Omega , E \subset A\right\}&=\int_E K_p(x,0,c,d,\nu)d{\cal H}^{N-1},
r\dot{F}= -F^2r + 4nr \frac{FB}{b}- 2n\frac{B}{b}- 2n(2n-1)r \frac{B^2}{b^2}& \qquad\qquad\qquad\qquad\qquad\qquad+ \frac{n(n+1)}{hb^2}- \frac{n}{2hb^4}\left(r + 2 F r^2 \right) r\dot{h}= \frac{2nr^2}{b^4}- 4nhr \frac{B}{b}+ 2hrF r\dot{b}= Br r\dot{B}= \frac{n+1}{2hb}- \frac{r}{2hb^3}- B - \frac{Br^2n}{hb^4}+ \frac{rB^2}{b}
[X,X]_{\mathcal{G}_1}(a,b,c) = \frac{2}{3}( X(X(a,b),c) - X(a,X(b,c)) + X(X(b,c),a) - X(b,X(c,a))+ X(X(c,a),b) - X(c,X(a,b) ) = \frac{4}{3}(X(X(a,b),c) + X(X(b,a),c)+X(X(c,a),b) )
M_G\left( 1+x^4 + m\:k(x)\right)= 2^4 (1+6m), & M_G\left( (1+x^4)^2 -\;h(-x^2)+m\:k(x)\right) =2^6 (1+3m), M_G\left( x-1 + m\;k(x)\right) = 2^4 3^2 m.
F(x_n, c_n) \ge f(x_n) - \sum_{i = 1}^r \frac{p(x_n)}{2 c_n}\| \lambda_i(x_n) \|^2 - \frac{q(x_n)}{2 c_n}\| \mu(x_n) \|^2 \ge f(x_n) - \frac{\alpha}{c_n}.
\b{13}r^{-1}= q^{-1}r^{-1}\b{13}\; , \b{14}r^{-1}= q^{-1}r^{-1}\b{14}\; , \b{23}r^{-1}= q r^{-1}\b{23}\; , \b{24}r^{-1}= q r^{-1}\b{24}\; ,
\frac{d\widetilde{\mathcal{N}}^{B}}{d\beta}(\beta)=\frac{\tilde{\phi}(\beta)}{\int_{B}\tilde{\phi}(\tilde{\beta})d\tilde{\beta}}\ frac{d\Pi_{\beta}^{B}(\cdot \mid Y,\sigma_{0}^{2})}{d\beta}(\beta)=\frac{\pi(\beta)\tilde{\phi}(\beta)}{\int_{B}\pi(\tilde{\beta})\tilde{\phi}(\tilde{\beta})d\tilde{\beta}}
\lambda_X(i,j) =\lim_{n\to\infty}n\P[ X_i >n, X_j>n ] =\lim_{n\to\infty}n(\P[ X_i >n]+\P[ X_j>n ]-\P[ X_i \vee X_j>n ]) &=\|X_i\|_1+\|X_j\|_1-\|X_i\vee X_j\|_1.
G_{r}(z,v_{I},v_{L}) = \Phi^{r}(T(zv_{I},tv_{L}))|_{t=z}= \frac{1 - u^{r+2}}{(1-u^{r+1})(1+u)}T\Big(\frac{u (1-u^{r+1})^{2}}{(1-u^{r+2})^{2}}v_{I}, \frac{u^{r+1}(1-u)^{2}}{(1-u^{r+2})^{2}}v_{L}\Big).
J_{l}d\boldsymbol{x}^{2}=(-d\boldsymbol{x}^{3},0,d\boldsymbol{x}^{1}),\;\ \ \;J_{l}d\boldsymbol{x}^{3}=(d\boldsymbol{x}^{2},-d\boldsymbol{x}x^{1},0).
D_{s\eta^i}\mathfrak{h}_{i+1,A}=s\mathfrak{f}_{i,A}\mbox{}mod(\ker(D|_{L^2_1})) D_{s\eta^i}\mathfrak{h}_{i+1,B}=\mathfrak{f}_{i,B}\mbox{}mod(\ker(D|_{L^2_1}))
\tau \cdot (v_1 \otimes v_1) = (\tau \cdot v_1) \otimes (\tau \cdot v_1) = (t_1 v_1) \otimes (t_1 v_1) = t_1^2 (v_1 \otimes v_1), \tau \cdot (v_1 \otimes v_2) = (\tau \cdot v_1) \otimes (\tau \cdot v_2) = (t_1 v_1) \otimes (t_2 v_2) = t_1 t_2 (v_1 \otimes v_2),
-\int\limits_{y_n=0}\Big(\frac{\partial}{\partial\tau}+iA_-'\Big)u^f\overline{v^g}d\tau-\int\limits_{y_n=0}\frac{\partial u^f}{\partial s}\overline{v^g}ds=\int_{y_n=0}\Big(\frac{\partial}{\partial y_n}-iA_-'\Big)u^f\overline{v^g}dy_0
<O[A]>=Z=\int D\phi DA_i\exp\left\{-\frac{1}{2}A_i^{\phi}G^{-1}A_i^{\phi}+ i\kappa \partial_i\phi)\epsilon_{ij}A_j\right\}O[A]\exp\left\{-\frac{1}{2}A_iG^{-1}A_i\right\}
\eta_{m \kappa}\left( 0 \right) + \eta_{- m \kappa}\left( 0 \right) = \left( N_{\kappa}^{+}+ N_{\kappa}^{-}\right) \pi + \left( - 1 \right)^{l}\frac{\pi}{2}\left[ \sin^{2}\eta_{m \kappa}\left( 0 \right) + \sin^{2}\eta_{- m \kappa}\left( 0 \right) \right] \, ,
\int_{K}\mathrm{Res}_{s=1/2}Z_{\leq d}(f_{k\rho},k\varphi^{U_k,\psi_k},k{\varphi'}^{U_k,\psi_k^{-1}},s)\, dk &=\int_{K}\mathrm{Res}_{s=1/2}Z(f_{k\rho},k\varphi^{U_k,\psi_k},k{\varphi'}^{U_k,\psi_k^{-1}},s)\, dk.
u_d=u_d^{(0)}+\sum_{n=1}^{\infty}(u_d^{(n+1)}-u_d^{(n)})=u_d^{(0)}+\sum_{n=1}^{\infty}\big(\Phi_d(u_d^{(n)})-\Phi_d(u_d^{(n-1)})\big),
0 =\bar\partial_B Z\lrcorner\,\omega^b + Z\lrcorner\,\bar\partial_B\omega^b &=\sum_a \bar\omega^a\wedge \nabla_{\bar V_a}( Z\lrcorner\,\omega^b) -\sum_a \bar\omega^a\wedge Z\lrcorner\nabla_{\bar V_a}\omega^b &=\sum_a (\nabla_{\bar V_a}Z\lrcorner\,\omega^b)\bar\omega^a.
\begin{aligned}- \int_0^\infty(I_1\cdot \langle y\rangle^{2(l+k)}D^\alpha u+I_2\cdot \langle y\rangle^{2(l+k)}D^\alpha h)\mathrm{d}y &=-\sum_{i=1}^3 \int_0^\infty (I_1^i\cdot \langle y\rangle^{2(l+k)}D^\alpha u+I^i_2\cdot \langle y\rangle^{2(l+k)}D^\alpha h)\mathrm{d}y &=:J_1+J_2+J_3.\end{aligned}
\|A\| = \frac{1}{{d+1 \choose 1}}\frac{|A|}{|V_i|},\; \|B\| = \frac{1}{{d+1 \choose 1}}\frac{|B|}{|V_j|}, \mbox{and}\|E(A,B)\| = \frac{1}{{d+1 \choose 2}}\frac{|E(A,B)|}{|X(1)\cap V_i \times V_j|}
B_{n}(t_{1}+u_{1}, \ldots, t_{n}+u_{n})= B_{n}(t_{1}, \ldots, t_{n})+\sum_{k=1}^{n-1}\binom{n}{k}B_{n-k}(t_{1}, \ldots, t_{n-k})B_{k}(u_{1}, \ldots, u_{k}) +B_{n}(u_{1}, \ldots, u_{n})
27D(e,e,X)D(e,e,Y)D(e,e,Z) = (s_1+s_2+s_3)(t_1+t_2+t_3) (u_1+u_2+u_3) & = \sum_i s_it_iu_i + (s_1t_1u_2+\cdots)+ \sum_{\{i,j,k\}=\{1,2,3\}}s_it_ju_k.
B =\sum_{y,z\in\mathbb{F}_{q}^{\times}}\theta(bz)\theta(-zy^2) &=\frac{1}{(q-1)^2}\sum_{l,m=0}^{q-2}G_{-m}G_{-l}T^{m}(b)T^l(-1)\sum_{y\in\mathbb{F}_{q}^{\times}}T^{2l}(y)\sum_{z\in\mathbb{F}_{q}^{\times}}T^{l+m}(z).
e^{Ns}- \sum_{m = 0}^{N-1}\frac{N^m}{m !}{s}^m = \sum_{m \geq N}\frac{N^m}{m !}{s}^m &= \frac{{( Ns )}^{N}}{N !}\cdot \bigg( 1+\sum_{m \geq 1}\frac{{( Ns )}^{m}}{(N+1) \cdots (N+m)}\bigg) \leq C(\lambda)\cdot \frac{{( Ns )}^{N}}{N !},
Q_v \equiv d + i_v=\int dt\Bigl(c^i \frac{\partial}{\partial a^i}+q_i \frac{\partial}{\partial\bar c_i}+(\partial_t a^i-h^i) \tau_i+(\partial_t \bar c_i+\bar c_j\partial_kh^j) \pi^i\Bigr).
(\partial_t-X\cdot\nabla_Y)w(X,Y,t) =\frac\theta 2 e^{-\lambda (T-t)}\bigl (-2X\cdot Y+\lambda (|Y|^2+A)\bigr )+\eta/(T-t)^2 &\geq \frac\theta 2 e^{-\lambda T}(\lambda (|Y|^2+A)-2R|Y|),
\frac{{f'}_t^T(\theta, x | X_1^{t-1})}{f_t(\theta, x |X_1^{t-1})}=-\frac{g_t'(x-\theta^TX_{t-m}^{t-1})}{g_t(x-\theta^TX_{t-m}^{t-1})}X_{t-m}^{t-1}.
\hat{T}_{MN}^{\varphi}={1 \over 4 \epsilon}( \frac{1}{2}\hat{g}_{MN}\partial_{P}\varphi \partial^{P}\varphi - \partial_{M}\varphi \partial_{N}\varphi ) + \frac{V}{\epsilon}e^{-\varphi}\hat{g}_{MN}\: .
(a+b)^{p-2}= a^{p-2}+\sum^{p-2}_{j=1}C_{p-2}^{j}a^{p-2-j}b^{j}\leq(1+\varepsilon)a^{p-2}+C(p)b^{p-2}\sum^{p-2}_{j=1}\varepsilon^{-\frac{p-2-k}{k}}\leq (1+\varepsilon)a^{p-2}+\frac{C(p)}{\varepsilon^{p-3}}b^{p-2},a,b\geq0.
T^{- 1}= \frac{1}{2 ( 2 \alpha - 1 )}\left( \begin{matrix}{{0}}&{{0}}&{{\alpha - 1}}&{{\alpha}}\\{{0}}&{{0}}&{{- \alpha}}&{{1 - \alpha}}\\{{1 - \alpha}}&{{\alpha}}&{{0}}&{{0}}\\{{- \alpha}}&{{\alpha - 1}}&{{0}}&{{0}}\\ \end{matrix}\right) \ .
\delta A_{0}={\frac{\delta h_{1}^{( 1 )}}{r}}+ O \! \left({\frac{1}{r^{2}}}\right) \, , \qquad \delta A_{4}={\frac{\delta h_{2}^{( 1 )}}{r}}+ O \! \left({\frac{1}{r^{2}}}\right) \, .
H_l(M^k,M^{k-1};\mathbb{Z})=\begin{cases}\bigoplus_{p\in Cr_k(f)}\mathbb{Z}[\sigma_p]\oplus \bigoplus_{\gamma\in Cr_k^+(f_N)}\mathbb{Z}[\sigma_{\gamma}], l=k, 0,{\rm otherwise},\end{cases}
z{\cal H}_k^{(s,t)}(z)={\cal H}_{k+1}^{(s,t)}(z)+(q_{k+1}^{(s,t)}+e_{k}^{(s,t)}){\cal H}_{k}^{(s,t)}(z)+q_{k}^{(s,t)}e_k^{(s,t)}{\cal H}_{k-1}^{(s,t)}(z).
\frac{dE}{dt}=-\alpha_{2}\left\vert \eta( L,t) \right\vert^{2}-\alpha_{1}\left\vert \eta_{x}( L,t) \right\vert^{2}-\alpha_{0}\left\vert \eta_{x}( 0,t) \right\vert^{2}-\frac{1}{3}w^{3}( L,t) -\int_{0}^{L}( \eta w)_{x}\eta dx,
Z_{a}= \prod_{k \neq 0}\left( \det \frac{N^{2}}{i}\Omega k^{2}\right)^{- \frac{1}{2}}= \det \left( \frac{N^{2}}{i}\Omega \right)^{- \zeta ( 0 )}e^{( N - 1 ) \zeta^{\prime}( 0 )}= \sqrt{N}\left( \frac{2 T}{e^{2}L}\right)^{\frac{N - 1}{2}}\ \ .
W(n,k)=\frac{\binom{2 n}{n}\binom{n}{k}\binom{k+n}{2 k}\binom{k+2 n}{n}\binom{2 k+4 n}{k+2 n}}{\binom{2 k}{k}^2}=\frac{k!^3 (2 n)! (2 k+4 n-2)!}{(2 k)!^3 n!(n-1)! (n-k)!^2 (k+2 n-1)!},
\sum_{\substack{s \geq i, t \geq j (s,t) \neq (i,j)}}( A_k )_{is}( X_k )_{st}( B_k )_{tj}= \sum_{t > j}( A_k )_{ii}( X_k )_{it}( B_k )_{tj}+ \sum_{\substack{s > i, t \geq j}}( A_k )_{is}( X_k )_{st}( B_k )_{tj}.
\max_{x\in \Omega_{1i}}\Phi_1(x)=\max_{x\in \Omega'_i}\Phi(x) = \min_{y\in(0;1]}\sum_{\substack{\ell=1, \ell\ne i}}^r y^{\gamma_\ell}< \min_{y\in(0;1]}\sum_{\ell=1}^r y^{\gamma_\ell}=\sum_{j=1}^r \zeta^{\gamma_j}.
\chi_{\nu}(\rho)\sim \left(\begin{array}{c}\sqrt{E+m}\ [\sin\mu J_{\nu}(k\rho) +(-1)^n \cos\mu J_{-\nu}(k\rho)] i\sqrt{E-m}\ [\sin\mu J_{\nu+1}(k\rho) +(-1)^{n+1}\cos\mu J_{-(\nu+1)}(k\rho)] \end{array}\right)
h_{1}= - f_{b c}^{a}\left( \frac{1}{2}A_{j}^{b}A_{k}^{c}B_{a}^{j k}+ \left( \partial^{l}B_{0 l}^{b}\right) \left( B_{a}^{0 i}A_{i}^{c}+ \eta_{a}^{\left( 2 \right) i}{\cal P}_{i}^{\left( 2 \right) c}+ \eta_{a}{\cal P}^{c}\right) \right) ,
A_{j}^{2}\mathcal{L}^{n-2}=[A_{j}]_{A_{j}}\mathcal{L}_{A_{j}}^{n-2}=\sum_{k_{1}+...+k_{r}=n-2}\frac{(n-2)!}{k_{1}!...k_{r}!}a_{1j}^{k_{1}}...a_{jj}^{k_{j}+1}...a_{rj}^{k_{r}}l_{j}
c_{2i+3}=(-1)^{i+1}\frac{2^{s}q+1}{2^{s-1}q+i+2}{2^{s-1}q+i+2 \choose2i+3}2^{2i+2}=(-1)^{i+1}\frac{(2^{s}q+1)(2^{s-1}q+i+1)!}{(2i+3)!(2^{s-1}q-i-1)!}\cdot 2^{2i+2}= -\frac{(2^{s-1}q+i+1)(2^{s-1}q-i)}{(2i+2)(2i+3)}\cdot 2^{2}\cdot c_{2i+1}= -\frac{(2^{s-1}q+i+1)(2^{s-1}q-i)}{(i+1)(2i+3)}\cdot 2\cdot c_{2i+1}.
E_{t o t}^{\bar{M}}( \Sigma ) ={\frac{\lambda^{2}}{4 8 \pi}}\left[ \int_{- \infty}^{v_{1}}\left( 1 - \partial_{v}\bar{v}\right)^{2}+ \int_{v_{1}}^{v_{2}}\left( 1 - \partial_{v}\bar{v}\right)^{2}+ \int_{v_{2}}^{\infty}\left( 1 - \partial_{v}\bar{v}\right)^{2}\right]
|\tilde{\mathcal{L}}^d_{\leq x}| = \frac{1}{2^{d-1}d!}(1+O_{d}(x^{-1+O(1/\log \log x)}))|\mathcal{L}^d_{\leq x}| & = (1+O_{d}(x^{-1+O(1/\log \log x)}))\frac{\prod_{i=2}^{d}\zeta(i)}{2^{d-1}d! d}x^d & = (1+O_{d}(x^{-1+O(1/\log \log x)}))c_d x^d,
\phi ( r , t )_{\mathrm{s t r a d d l e}( r > r_{H})}\approx{\cal N}_{\mathrm{s t r a d d l e}}{\frac{| r - r_{H}|^{\pm i \omega c_{H}/ g_{H}}}{\sqrt{2 \omega}r_{H}}}\exp \left[ \mp i \omega t \right] .
\frac{H_{QM}[\rho,\Pi]}{N^2}~=~\frac{\gamma}{8}\int d\theta \left\{\rho(\theta) \left(\frac{\partial}{\partial\theta}\frac{\delta~S}{\delta\rho(\theta)}\right)^2+\frac{\pi^2}{3}\rho^3(\theta)\right\}+\frac{1}{N^2}V[\rho] ~=~\frac{E}{N^2}\; .
+\Big|\frac{1}{2 \varepsilon n^2}\sum_{x}\Phi_n(s, \tfrac{x}{n}) \sum_{r=1}^{\varepsilon n}\int_{\Omega_1(x)}\overleftarrow{\eta}^{\ell}(x) [\eta(x+1)-\eta(x+1+r) ] [f(\eta)-f(\eta^{x+1,x+1+r})] d\nu_{b}\Big|.
\big(x-q^{\lambda+1}\beta^{-1}\big)\widehat{y}_0(x/q) + q^{-\mu}\beta^{-1}(\alpha x-q )\widehat{y}_0(qx) &{}-\big\{\big(q^{-\mu}\alpha\beta^{-1}+1\big)x- q\beta^{-1}\big(1 +q^{\lambda-\mu}\big)\big\}\widehat{y}_0(x)=0 .
i \left< A_{\mu}^{3}A_{\nu}^{3}A_{\rho}^{a}A_{\sigma}^{b}\right>_{\mathrm{b a r e}}= - i g^{2}\delta^{a b}\left[ 2 g_{\mu \nu}g_{\rho \sigma}- \left( 1 - \frac 1 \alpha \right) ( g_{\mu \rho}g_{\nu \sigma}+ g_{\mu \sigma}g_{\nu \rho}) \right] .
\sum_{i=1}^D\sum_{j=1}^D \alpha_i \bar{\alpha}_j \sum_{f\in \mathcal{B}_{2k}(p)}\omega_f \lambda_f(i) \lambda_f(j)\big|L(f, n+1) \big|^2 =\large{S_{\mathrm{off}}+S_{\mathrm{main}}}.
W(x;z)=\begin{pmatrix}\dfrac{\sin (xz)\! -\! z\cos (xz)}{z(x\!-\!1)}\left(\dfrac 1{z^2}\!-\!(x\!-\!1)\right)\sin (xz)\!-\!\dfrac{x\cos (xz)}z begin{align*}2mm]\dfrac{\sin (xz)}{(x\!-\!1)}\dfrac{\sin (xz)}z \!-\! (x\!-\!1)\cos (xz)\end{pmatrix},
|u^2-v^2D| =v^2\left|\left(\frac{u}{v}+\sqrt{D}\right)\left(\frac{u}{v}-\sqrt{D}\right)\right| \geqslant v^2\left(2\sqrt{D}-\frac{\varepsilon}{B^2}\right)\frac{C(D)}{v^2}=2C(D)\sqrt{D}-\frac{\varepsilon C(D)}{B^2}=2C(D)\sqrt{D}-o(1).
\begin{array}{c c c}A=\begin{pmatrix}1 0 0 0 -2 0 1 0 4m+2 1 -(2m+1) 0 4 -2 -2 1\end{pmatrix}B=\begin{pmatrix}1 -m -(2m+1) 0 0 1 0 1 2\end{pmatrix}.\end{array}
\alpha_1 := 3H_1 + 3H_2 - \textstyle{\sum}_{q=1}^3 E_q + \textstyle{\sum}_{q=4}^{10}E_q + 2(E_{11}+ E_{12}), \alpha_2 := -3H_1 + 2H_2 + \textstyle{\sum}_{q=1}^3 E_q + \textstyle{\sum}_{q=13}^{18}E_q + 2E_{19}\alpha_3 := H_1 + H_2 + E_{12}- E_{13}, \alpha_4 := 2H_1 - H_2 - \textstyle{\sum}_{q=1}^3 E_q - E_{13}+ E_{14}
\|w_\rho \|_{L^2(B_1)}= \rho^{-N/2}\|v\|_{L^2(B_{\rho}(x_0))}\leq \rho^{-N/2}\rho^{-s}\|u-\psi(z_0)d^s\|_{L^2(B_{3\rho}(z_0))}\leq C \rho^{s-\max(\b,\b')}.
\hspace{15mm}\times\prod_{i = 1}^{q}u_{i}^{a^{'}}\prod_{i = 1}^{q - 1}(u_{i}- 1)^{b^{'}- 1}(u_{i}- z)^{c^{'}}\prod_{i = q - k + 1}^{q}(1 - u_{i})^{b^{'}- 1}(z - u_{i})^{c^{'}}
E_{D B I}= \frac{\mu_{5}}{g}\int d^{2}\Omega \sqrt{\det ( G - B + 2 \pi \alpha^{\prime}F )}- \mu_{5}\int_{D^{3}}\tilde{F}_{( 3 )}- \mu_{5}\int ( 2 \pi \alpha^{\prime}F - B ) \wedge C_{4}
(\Delta_{(2,2,2)})^\vee \,\, = \,\,\, \begin{small}{\rm det}\begin{pmatrix}a_0 a_1 a_2 a_3 a_1 a_2 a_3 a_4 a_2 a_3 a_4 a_5 a_3 a_4 a_5 a_6 \end{pmatrix}.\end{small}
\left(\gamma^* (\Phi_-)^\lambda \mathcal{Q}_- \omega\right)_{(x,\xi)}( \eta_1,\dots,\eta_m)=(\Phi_-)^\lambda_{(x,\xi)}\left((T_\gamma)^{\lambda+m}U_\gamma^*\omega\right)_{B_-(x,\xi)}({\tau_-}\eta_1, \dots ,{\tau_-}\eta_m)
\sum_{n=0}^{\infty}h(r_{n})=\frac{\mu(X)}{4\pi}\int_{-\infty}^{\infty}rh(r)tanh(\pi r)dr+\sum_{\{T\}}\frac{logN(T_{0})}{N(T)^{\frac{1}{2}}-N(T)^{-\frac{1}{2}}}g(logN(T)).
\int\limits_0^\pi Cf(u, e_n, \psi)d\psi= \int\limits_0^\pi \int\limits_0^\infty \int\limits_{S^{n-2}}f(u+\rho((\sin\psi) \omega,\cos \psi))(\rho \sin\psi)^{n-2}d\omega d\rho d\psi =\int\limits_{S^{n-1}}\int\limits_0^\infty f(u+\rho\sigma)\rho^{n-2}d\rho d\sigma =\int\limits_{\mathbb{R}^n}f(u+x)|x|^{-1}dx = \frac{1}{|S^{n-2}|}R^\#Rf(u),
\hat{\mathbb{E}}[(X+k)I_{D_k}]=\lim_{n\rightarrow\infty}\hat{\mathbb{E}}[(X_n+k)I_{D_k}]=\lim_{n\rightarrow\infty}\hat{\mathbb{E}}[(\hat{\mathbb{E}}_{\tau+}[X_n]+k)I_{D_k}]=\hat{\mathbb{E}}[(\eta+k) I_{D_k}].
\frac{\partial \bar u^{\varepsilon,N}_t}{\partial x_{\ell}}(\cdot) ={\displaystyle \frac{1}{N \varepsilon}\sum_{i=1}^N \frac{\partial K_{\varepsilon}}{\partial x_{\ell}}( \cdot-\bar \xi^{i}_t) \bar V_t\big (\bar \xi^{i}, \bar u^{\varepsilon,N}(\bar \xi^{i}),\nabla \bar u^{\varepsilon,N}(\bar \xi^{i}) \big )}\ , l = 1, \cdots, d \ ,
\begin{aligned}\sum_{n\in\mathbb N}\|\varphi(T)x_{nt}\|^2 =\sum_{n\in\mathbb N}\|(\varphi\kappa_{nt}\omega_n\varphi_n)(T)x_0\|^2 \leq \Bigl(KC_{{\rm pol}, T}(1+\varepsilon_0)\frac{(2(1+c_1^2))^{1/2}}{1-c_1}\Bigr)^2 \sum_{n\in\mathbb N}\int_{\mathbb T}|\varphi\kappa_{nt}|^2|\psi_{nt}|^2m, \end{aligned}
\mathrm{\left\{R , \Pi_{R}\right\}= \left\{R^{\prime}, \Pi_{R}^{\prime}\right\}= 1 , \qquad \left\{\v a r p h i , \Pi_{\v a r p h i}\right\}= \left\{\v a r p h i^{\prime}, \Pi_{\v a r p h i}^{\prime}\right\}= 1 \, .}
\begin{array}{rl}\displaystyle \min_{\mathbf{D}, \mathbf{W},\mathbf{C},\mathbf{U}}\displaystyle\sum_{i=1}^N\left\{\int_\Omega\| d_i(x)\|dx + \lambda\int_\Omega \left|I(x) - c_i\right|w_i(x) \dx\right\}+\delta_\Delta(\mathbf{W}), \displaystyle \mbox{subject to}\nabla u_i = d_i, u_i=w_i, \forall i=1,\dots,N, \end{array}
={\frac{\pi ( \Gamma ( \ell + 1 ) )^{2}}{2^{2 \ell + 2}( \Gamma ( \ell +{\frac{3}{2}}) )^{2}( \Gamma ( 2 \ell + 1 ) )^{2}}}k^{2 \ell - 1}r_{0}^{2 \ell + 1}e^{- 2 \pi k r_{0}}| \Gamma ( \ell + 1 + 2 i k r_{0}) |^{2}
\mbox{var}\left[ \frac{\left( \sum^K_{i=1}a_i \right)}{p_s}\mathbf{S}(j,j) \right] = \sigma^4 p_s \frac{\left( \sum^K_{i=1}a_i\right)^2}{p_s^2}= \sigma^4 \frac{\left( \sum^K_{i=1}a_i\right)^2}{p_s}.
\sum_{r=0}^{j}(-1)^{r}e_{r}(t_{1},\ldots ,t_{N})h_{j-r}(y_{1},\ldots,y_{n})=0,j=k+1,\ldots ,N-1 \sum_{r=0}^{N}(-1)^{r}e_{r}(t_{1},\ldots ,t_{N})h_{N-r}(y_{1},\ldots,y_{n})=(-1)^{n-1}q
(1-x) \prod_{j=1}^{r-1}\left(1-dxq^{a(j)}\right) G(x) = G(xq^N) &+ \sum_{j=1}^{r-1}\left( \sum_{m=0}^{r-j-1}d^m \sum_{\substack{\alpha < a(r) w(\alpha)=j+m}}xq^{\alpha}\left( (-x)^{m-1}{j+m-1 \brack m-1}_{q^N}+ (-x)^m{j+m \brack m}_{q^N}\right) \right) G\left(xq^{jN}\right).
i\raise.15ex\hbox{$/$}\kern-.53em\hbox{$D$}=e^{g\gamma_5[\phi(x)+f(x)]}U(x)i\raise.15ex\hbox{$/$}\kern-.57em\hbox{$\partial$}U^{\dagger}(x)e^{g\gamma_5[\phi(x) +f(x)]},
\int_{\mathbb{R}}\frac{P_{m}(1,q;x)P_{n}(1,q;x)}{\gamma\big(2q\chi_{1}(x)+x\varphi_{1}'(x)-\varphi_{1}(x)\big)^{2}+\gamma^{-1}\varphi_{1}(x)^{2}}\,\mbox{d}x=\pi\delta_{mn}.
e^{{\frac{1}{2}}\epsilon \varphi_{\alpha}- ( p + 1 ) A}= H_{\alpha}, d s^{2}= \prod_{\alpha = 1}^{N}H_{\alpha}^{-{\frac{\tilde{p}+ 1}{D - 2}}}d x^{\mu}d x^{\nu}\eta_{\mu \nu}+ \prod_{\alpha = 1}^{N}H_{\alpha}^{\frac{p + 1}{D - 2}}d y^{m}d y^{m},
\mathcal{E}\left\{\Delta\right\}=\frac{M}{2}\mathcal{E}\left\{\log \left( 1+\rho x_{k}\right)\right\}+M\mathcal{E}\left\{\log \left(1+\rho x_n\alpha_{1,n}^2 \right) \right\}- M \mathcal{E}\left\{\log\left(1+ \rho x_k \alpha_{1,n}^2\right)\right\}-\frac{M}{2}\mathcal{E}\left\{\log \left(1+\rho x_n \right) \right\}.
z(t)= e^{-\frac{1}{2}\gamma t}\left [ B_1 J_1\left(\frac{2\omega_0}{\gamma}e^{-\frac{1}{2}\gamma t}\right)+ B_2Y_1\left( \frac{2\omega_0}{\gamma}e^{-\frac{1}{2}\gamma t}\right)\right],
\beta(t) =\begin{cases}\alpha(t) t \in [0,t_1) \cup (t_2,1] \Big( x_1 + \frac{r_1^2}{r(t)}\cos \theta(t), y_1 + \frac{r_1^2}{r(t)}\sin \theta(t) \Big) t \in [t_1, t_2] \end{cases}
\mathcal{I}_D =\langle x_1^{a}, x_2(x_1+x_2)^{b-1}, x_3(x_1+x_2+x_3)^{a-1}\rangle, \mathcal{I}_{\widetilde{D}}=\langle y_1^{a}, y_2(y_1+y_2)^{a-1}, y_3(y_1+y_2+y_3)^{b-1}\rangle.
\bar{a}\Big(1-\frac{1}{n}\Big)+ \frac{1-\bar{a}}{\bar{q}}-\frac{1}{\bar{s}}= \bar{a}+ \frac{1-\bar{a}}{\bar{q}}-\frac{1}{\bar{s}}-\frac{\bar{a}}{n}=\frac{s}{\bar{s}(1-a+as/\bar{s})}\left(a\Big(\frac{1}{p}-\frac{1}{n}\Big)+\frac{1-a}{q}-\frac{1}{s}\right).
\phi(|z-e_\varepsilon|)+\phi(|z+e_\varepsilon|)-2\phi(|e_\varepsilon|) = - \alpha(1+\alpha) | e_\varepsilon|^{\alpha - 1}z^2\int_0^1dt\int_{-1}^1\left(1+ \frac{zt\tau}{|e_\varepsilon|}\right)^{\alpha-1}t\,d\tau &\le - \alpha(1+\alpha) 2^{\alpha -1}|e_\varepsilon|^{\alpha-1}.
b_1 = [ \mathit{a}_0\left(b_0,p_1\right)+\mathit{a}_1\left(b_0,p_2\right)](-b_0) b_2 = [\mathit{a}_0\left(b_0,p_0\right)+\mathit{a}_0\left(b_1,p_1\right)+\mathit{a}_1\left(b_0,p_1\right)+\mathit{a}_1\left(b_1,p_2\right)+\mathit{a}_2\left(b_0,p_2\right)] (-b_0).
G=(\begin{matrix}(\begin{matrix}00\\ 11\end{matrix})(\begin{matrix}01\\ 01\end{matrix})(\begin{matrix}11\\ 01\end{matrix})\\ (\begin{matrix}00\\ 11\end{matrix})(\begin{matrix}11\\ 11\end{matrix})(\begin{matrix}00\\ 00\end{matrix})\end{matrix})
2\xi_2 = -\frac{2a(a-1) (8 a^2-a-1)}{3}< \xi_1= - \frac{2a(a-1) (2 a^2+2a-1)}{3}begin{align*}3mm] < \frac{\xi_2}{2}=-\frac{a(a-1) (8 a^2-a-1)}{6}