harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
P_{e_{s}}= P(\xi^{c})\cdot c\cdot E_{H\vert\xi^{c}}\left[Q\left(\sqrt{g\frac{\chi_{1}}{N_{0}}}\right)\right]+ P(\xi)\cdot c\cdot E_{\mathcal{H}\vert\xi}\left[Q\left(\sqrt{g\frac{\chi_{1}+\chi_{2}}{N_{0}}}\right)\right],
\begin{cases}\mu^n \to \mu \mbox{weakly-$*$ in $ L^\infty(0, T; H) $,}\mu^n(t) \to \mu(t) \mbox{in $ H $, in the pointwise sense,}\mbox{for a.e. $ t \in (0, T) $,}\end{cases}\mbox{as $ n \to \infty $.}
e^{-\mathcal{A}_{D,\varepsilon}t}-e^{-\mathcal{A}_D^0t}=-\frac{1}{2\pi i}\int_\gamma e^{-\zeta t}\left((\mathcal{A}_{D,\varepsilon}-\zeta I)^{-1}-(\mathcal{A}_D^0 -\zeta I)^{-1}\right)\,d\zeta .
F_1^\delta \left( \begin{array}{c}z x \end{array}\right) = \left( \begin{array}{c}a_1^\delta(x) z + a^\delta_2(x) z^2 + \dots b^\delta_0(x) + b^\delta_1(x) z + b^\delta_2(x) z^2 + \dots \end{array}\right),
G_{p,q}^{\,m,n}\!\left( \,\begin{matrix}a_1, \cdots , a_p b_1, \cdots , b_q \end{matrix}\; \Big| z \right) & = \sum_{k=1}^{m}A_{p,q,k}^{m,n}(z){}_p F_{q-1}\left( \begin{matrix}1+b_k - a_1,\cdots, 1+ b_k - a_p 1+ b_k - b_1, \cdots, *, \cdots, 1 + b_k - b_q \end{matrix}\Big| (-1)^{p-m-n}z \right),
\mathcal{L}(q,p_{r})= \int_{s_{1}}^{s_{2}}\sqrt{\langle \gamma^{\prime}(s),\gamma^{\prime}(s)\rangle}ds \geq \int_{s_{1}}^{s_{2}}\frac{|\langle\gamma^{\prime}(s),\nabla f\rangle|}{|\nabla f|}ds \geq \frac{1}{\sqrt{R_{max}}}|\int_{s_{1}}^{s_{2}}\langle\gamma^{\prime}(s),\nabla f\rangle ds| = \frac{1}{\sqrt{R_{max}}}|f(p_{r})-f(q)|.
p =p^{0}X_{1}+p^{1}X_{2}+p^{2}X_{3}+p^{3}X_{4},\;\;\;\;\;\;\;\;q=q^{0}\partial_{y^{2}}+q^{1}\partial_{y^{3}}+q^{2}\partial_{y^{4}}+q^{3}\partial_{y^{5}}=p^{0}+p^{1}i+p^{2}j+p^{3}k\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=q^{0}+q^{1}i+q^{2}j+q^{3}k.
\begin{cases}\partial_t v = \mathcal{L}_\beta^* v \coloneqq \beta\Delta v_t + x\cdot \nabla v_t + n v_t,\;\;\; (t,x) \in (0,\infty)\times \mathbb{R}^n, v(0,x) = v_0(x),\;\;\; x\in \mathbb{R}^n. \end{cases}
J_n^{p}\ge \sum_{l_1=0}^{n-(p-1)k}\sum_{l_2=l_1+1}^{l_1+k}\sum_{l_3=l_2+1}^{l_2+k}...\sum_{l_p=l_{p-1}+1}^{l_{p-1}+k}1_{\overline{G}_l}(\omega), J_n^{p}\le 2k+\sum_{l_1=k}^{n}\sum_{l_2=l_1+1}^{\infty}...\sum_{l_p=l_{p-1}+1}^{\infty}1_{\overline{G}_{l,k}}(\omega),
A_{n}(x) := \frac{\mathbf{K}(x)}{\mu(\widetilde{c}_{n}(x,t))}, F_{n}(x):= \rho(\widetilde{c}_{n}(x,t))\mathbf{g}, A(x) := \frac{\mathbf{K}(x)}{\mu(c(x,t))}, F(x):= \rho(c(x,t))\mathbf{g}.
E(z)^\top + Q_{\Phi(z)}Q_{E(z)}^\top \frac{\Phi(1)^*}{|\Phi(1)|^2}= \frac{\Phi(1)^*}{|\Phi(1)|^2}- \frac{[\Phi(z) - \Phi(1)] \Phi(1)^*}{|\Phi(1)|^2}E(z)^\top & = \frac{\Phi(1)^*}{|\Phi(1)|^2}- \frac{\sum_{i =1}^\infty [\varphi_i(z) - \varphi_i(1)] \overline{\varphi_i(1)}}{|\Phi(1)|^2}E(z)^\top.
\lim_{a_+ - a_- \rightarrow 0}\frac{e^{ia_- (x_n-s)/h}-e^{ia_+ (x_n-s)/h}}{(1+|K|^2)(a_+ - a_-)}= \frac{i(x_n-s)}{h(1 + |K|^2)}e^{ia_- (x_n-s)/h}.
V_{\beta}^{1 - l o o p}( \phi_{0}) = \int \frac{d^{\nu}k}{( 2 \pi )^{\nu}}{\frac{E}{2}}+{\frac{1}{\beta}}\int \frac{d^{\nu}k}{( 2 \pi )^{\nu}}\ln [ 1 - e^{- \beta E}] \; .
P^\eta\Big(\sum_{k=1}^{t_n-1}Z_k\geq c_{n+1}t_{n+1}+r\Big) \leq P^\eta\Big(\sum_{k=1}^{t_n-1}\big(Z_k-E^\eta[Z_k]\big)\geq (t_{n}-1)\big(c_{n+1}t_n- E^\eta[Z_k]\big)\Big) &\leq P^\eta\Big(\sum_{k=1}^{t_n-1}\big(Z_k-E^\eta[Z_k]\big)\geq t_{n}^{3/4}(t_n-1)\Big)\leq{\rm e}^{-c\,t_n^{1/2}}\,,
{{cdt}\over{dx}}={{\hat{c}d\hat{t}}\over{d\hat{x}}},\ \ \ \ \ \ \{\hbar\over{dEdt}}={\hat{\hbar}\over{d\hat{E}d\hat{t}}},\ \ \ \ \ \ \{{G_4dE}\over{c^4dx}}={{\hat{G}_4d\hat{E}}\over{\hat{c}^4d\hat{x}}},\ \ \ \ \ \ \{e^2\over{dEdx}}={\hat{e}^2\over{d\hat{E}d\hat{x}}},
\begin{aligned}\hat{W}_k\left({\bf 1}\otimes \bar{x}^k\right) =\left(\left(I+\gamma^kW-\frac{{\bf 1}{\bf 1}^T}{m}\right){\bf \times 1}\right)\otimes\left(I_d\times \bar{x}^k\right)= 0\end{aligned}
E[B_n \mathbb{1}_{\tau> n}| \mathfrak{B}_{n-1}] = E[B_n \mathbb{1}_{\tau\ge n}| \mathfrak{B}_{n-1}] + E[0 \mathbb{1}_{\tau= n}| \mathfrak{B}_{n-1}] &= \mathbb{1}_{\tau\ge n}E[B_n | \mathfrak{B}_{n-1}] \le\mathbb{1}_{\tau\ge n}M.
\vec{p}_{a}+ \vec{p}_{b}-{\tilde{L}}_{p}( E_{a}\vec{p}_{b}+ E_{b}\vec{p}_{a}) / c - \vec{p}_{c}- \vec{p}_{d}+{\tilde{L}}_{p}( E_{c}\vec{p}_{d}+ E_{d}\vec{p}_{c}) / c \simeq 0 ~ .
\xi \frac{\partial V^{( 1 )}}{\partial \xi}= \frac 1 2 ( m_{2}^{2}\varphi ) \xi e^{2}\int \frac{d^{d}k}{( 2 \pi )^{d}}\frac 1{D_{N}\, ( k^{2}+ m_{g h}^{2})}\left\{( \varphi - 2 u ) k^{2}- \xi u e^{2}\varphi^{2}) \right\}.
\begin{array}{lll}c = (2\alpha + \frac{\beta}{\delta}(\beta-\gamma + \sqrt{(\beta-\gamma)^2 + 4 \alpha \delta}))^{1/2}d = (2\delta + \frac{\gamma}{\alpha}(\gamma-\beta + \sqrt{(\beta-\gamma)^2 + 4 \alpha \delta}))^{1/2}. \end{array}
G_{r s}= \left( \begin{matrix}{{{\frac{H_{0}}{H_{1}}}}}&{{0}}\\{{0}}&{{{\frac{H_{2}}{H_{3}}}}}\\ \end{matrix}\right) ={\frac{H_{2}}{H_{3}}}\left( \begin{matrix}{{( \mathrm{R e}\, U )^{2}}}&{{0}}\\{{0}}&{{1}}\\ \end{matrix}\right) \; ,
\frac{d}{dt}\|x_\nu^{(\frac{N}{2})}(t)-x_\nu^{(\frac{N}{2}+3)}(t)\|^2 \leq-\alpha\nu\|x_\nu^{(\frac{N}{2})}(t)-x_\nu^{(\frac{N}{2}+3)}(t)\|^2+\nu\|x_\nu^{(\frac{N}{2}-1)}(t)-x_\nu^{(\frac{N}{2}+4)}(t)\|^2 &+\nu\|x_\nu^{(\frac{N}{2}+1)}(t)-x_\nu^{(\frac{N}{2}+2)}(t)\|^2+\frac{1}{\nu}M_{T_1,T_2}^{\frac{N}{2},\frac{N}{2}+3,2-\alpha}(\omega)
\frac{(\beta)_{p+k}}{(1)_{p+k}}= \frac{\Gamma_p(\beta+p+k)\Gamma_p(1)}{\Gamma_p(\beta)\Gamma_p(p+k+1)}\cdot\frac{tp}{p}\cdot\frac{(t+1)p}{p}\equiv(t+1)\cdot\frac{(\beta)_k}{(1)_k}\equiv t\cdot\frac{(\beta)_k}{(1)_k}+tp\cdot\frac{(\beta)_k}{(1)_k}\cdot\bigg(H_{p-1-b}+\sum_{j=0}^{k-1}\frac{1}{j+\beta}-H_k\bigg)\pmod{p^2}.
\deg(\alpha\alpha_1^{-1}\cdots\alpha_n^{-1}) \leq (\deg\alpha)(\deg\alpha)^{2^0}(\deg\alpha)^{2^1}\cdots(\deg\alpha)^{2^{n-1}}& = (\deg\alpha)^{1 + \sum_{i=0}^{n-1}2^i}& = (\deg\alpha)^{2^n}
\mathbf E[ C_{k,\ell}(n+1)-C_{k,\ell}(n)\mid\mathcal F_n] begin{align*}0.5em] = \frac{2(k-\ell)p^{k-1}+\ell p^{2k-\ell-1}}n C_{k,\ell}(n) + \frac{2(\ell+1) p^{k-1}}n C_{k,\ell+1}(n) \intertext{and}\mathbf E[ C_{k,k-1}(n+1)-C_{k,k-1}(n)\mid\mathcal F_n] begin{align*}0.5em] = \frac{2p^{k-1}+(k-1)p^k}n C_{k,k-1}(n) + \frac{kp^{k-1}}nC_k(n).
E_{2}(q) = 1 - 24 \sum_{j=1}^{\infty}\sigma_1(j)q^j\ \E_{4}(q) = 1+240 \sum_{j=1}^{\infty}\sigma_3(j)q^j \ \E_6(q) = 1-504 \sum_{j=1}^{\infty}\sigma_5(j)q^j
A_{m+1}(v) =e^{-K_{m+1}(v,v)}\left(\frac{\rho_{m+1}(v)}{v}+\frac{\kappa}{c_{+0}-c_{-0}}\right)-\frac{1}{v}\left(1-e^{-K_{m+1}(v,v)}\right), \Delta_{m+1}B(v) =\frac{e^{-K_{m+1}(v,v)}}{v^2}\left(1+\rho_{m+1}(v)+\frac{\kappa v}{c_{+0}-c_{-0}}\right)\left(\frac{c_{+0}-c_{-0}}{\kappa}P_{m+1}(v)-\Delta_{m+1}I(v)\right) &\qquad+\frac{\Lambda_{m+1}(v)}{v}.
\left( \delta\tilde{v},\nabla\delta\tilde{Q}\right) =\left( \tilde{v}_{1}-\tilde{v}_{2},\nabla\tilde{Q}_{1}-\nabla\tilde{Q}_{2}\right) , \left( \delta\tilde{u},\nabla\delta\tilde{P}\right) =\left( \tilde{u}_{1}-\tilde{u}_{2},\nabla\tilde{P}_{1}-\nabla\tilde{P}_{2}\right) ,
\P_1: =\{p\equiv 1 \bmod 12 \; :\; p=(6k+2)^2 + (6t+3)^2 \text{for some $k,t\in \mathbb Z$}\}, \P_2: =\{p\equiv 1 \bmod 12 \; :\; p=(6k)^2 + (6t+1)^2 \}.
{\frac{5}{2}}\partial_{z}\partial_{0}b_{z}^{0}-{\frac{5}{2}}{\frac{1}{z}}\partial_{0}b_{z}^{0}-{\frac{5}{4}}{\frac{1}{z}}\partial_{z}b_{i}^{i}-{\frac{1 9}{2}}{\frac{1}{z^{2}}}b_{i}^{i}= 0 .
=\lambda\begin{bmatrix}\tilde{X}_{11}\tilde{X}_{12}\hdots \tilde{X}_{1N}\tilde{X}_{21}\tilde{X}_{22}\hdots \tilde{X}_{2N}\vdots \hdots \ddots \vdots \tilde{X}_{N1}\hdots \hdots \tilde{X}_{NN}\end{bmatrix}.
W_0 =\langle v_1,w_1\rangle^\perp \cap W; W_1 = \bigcap\limits_{W'\in \Lambda}W' = \langle W_0, w_1\rangle; W_2 = \langle W' \mid W'\in\Lambda\rangle = \langle W_0, w_1, v_1,v_2,v_3\rangle
R^{2}=\frac{\sum(\hat{y}_{i}-\overline{y})^{2}}{\sum(y_{i}-\overline{y})^{2}}=\frac{y^{T}P^{T}LPy}{y^{T}Ly}=1-\frac{y^{T}My}{y^{T}Ly}=1-\frac{RSS}{TSS}
\left(\int_{1}^{\hat{X}}\frac{(X-\lambda/3)dX}{2\sqrt{X(X-1)(X-\lambda)}}\right)\frac{1}{2\sqrt{\hat{X}(\hat{X}-1)(\hat{X}-\lambda)}}= \left(\int_{1}^{\hat{X}}\frac{(X-\lambda/3 -\sqrt{X(X-\lambda)})dX}{2\sqrt{X(X-1)(X-\lambda)}}\right)\frac{1}{2\sqrt{\hat{X}(\hat{X}-1)(\hat{X}-\lambda)}}+ \frac1{2\sqrt{\hat{X}(\hat{X}-\lambda)}}.
\mathcal{Q}_0^{(k)}(1\!+\!\varepsilon)= \delta_{k,0}\,,\mathcal{Q}_{m+N}^{(k)}(1\!+\!\varepsilon)=\sum_{j=1}^{m+N}\frac{(-1)^{k+j-1}}{(j\!-\!1)!\,(m\!+\!N\!-\!j)!}\,\frac{1}{(\varepsilon\!+\!j)^{k+1}}\,.
\left[\int_{0}^{a}\left| r j_{l}(k r) \right|^2 dr \right] \left[ \int_{0}^{a}\left| r j_{l}(K r) \right|^2 dr\right] \geq \left| \int_{0}^{a}j_{l}(k r) j_{l}(K r) r^2 dr\right|^2;j=2,
J(t)=t^{-n/2}\int_{T^{\ast}M}\overline{\psi^{\prime}\circ\pi}~\psi\circ\pi\circ\Phi_{i}~e^{-E/t\hbar}\left( \sqrt{d\mathrm{vol}_{g}},\sqrt{\Phi_{i}^{\ast}d\mathrm{vol}_{g}}\right)_{}\frac{\omega^{n}}{(2\pi\hbar)^{n}n!}.
min\{n|S|,n|T|,n^{2}|\frac{e(S,T)}{|S||T|}-\frac{e(X,Y)}{|X||Y|}|\}\ge|\sum_{x\in S,y\in T}a_{xy}|\ge\frac{1}{K_{G}^{\mathbb{R}}}\epsilon^{3}n^{2}\ge\frac{1}{2}\epsilon^{3}n^{2}.
\lim_{t \to 0}\frac{\mathbb{E}\phi(X_t,Y_t) - \phi(x,y)}{t}=\dfrac{1}{2}\dfrac{\partial^2 \phi}{\partial y^2}- (c_0 y + kx) \dfrac{\partial \phi}{\partial y}+ y \dfrac{\partial \phi}{\partial x}, \mbox{if}|x| <1.
\int d^{3}x_{1}K(2,1,R_{+}^{+}u_{+}^{+}) \gamma^{4}\psi(1) = a e^{i \theta_{2}/ 2}u_{+}^{+}= aR_{+}^{+}u_{+}^{+}= \psi(2)_{b=c=d=0},
[\begin{matrix}t_{i}\\A_{i}\end{matrix}]=[\begin{matrix}0,&0.5\pi/\omega_{d},&\pi/\omega_{d},&1.5\pi/\omega_{d},&2\pi/\omega_{d}\\I/(1+m),&I/K^{1/2},&I/K,&I/K^{3/2},&mI/[(1+m)K^{2}]\end{matrix}]
J_{r}={\frac{e n}{4}}( J_{2}+{\frac{r_{1}^{2}}{\sigma^{2}J_{2}}}-{\frac{2 r_{1}}{\sigma}}) ={\frac{e n}{4}}( J_{r}+ J_{1}+{\frac{r_{1}^{2}}{\sigma^{2}J_{2}}}-{\frac{2 r_{1}}{\sigma}}) .
dg_t =g_t\xi_tdt d\xi_t^{\vee}= \mathbb{M}^{-1}(\mathbb{M}\xi_t^{\vee}\times \xi_t^{\vee}+ \mathbb{H}\sigma u_t) dt + \mathbb{M}^{-1}\mathbb{H}\sigma dW_t (g_0,\xi_0) \sim \pi_0
\partial_t\varphi - \Delta\mu = \left( \mathcal P\sigma - \alpha u \right)h(\varphi_1) + (\mathcal P\sigma_2 - a - \alpha u_2)\left( h(\varphi_1) - h(\varphi_2) \right) \mu = - A\Delta \varphi + B(\psi'(\varphi_1) - \psi'(\varphi_2)),
\hbox{either}t^{n}=0, \forall n, \mbox{or}t^{n}\overset{n\rightarrow \pm \infty}\longrightarrow \pm\infty, &\hbox{either}x^{n}=0, \forall n, \mbox{or}x^{n}\overset{n\rightarrow \infty}\longrightarrow \pm\infty.
E ( x ) = \frac{1}{{\partial}_{-}^{2}}{\Delta}( x ; m^{2}) - \frac{x^{-}}{m^{2}- n_{-}{\partial}_{-}^{2}}{\delta}( x^{+}) + \frac{2 n_{+}{\partial}_{+}}{( m^{2}- n_{-}{\partial}_{-}^{2})^{2}}{\delta}( x^{+}) .
\tilde{\Delta}f(u) = \partial_x \langle \nabla f(u), v_x \rangle + \partial_y \langle \nabla f(u), v_y \rangle &= \langle \nabla^A_x \nabla f(u), v_x \rangle + \langle \nabla^A_y \nabla f(u), v_y \rangle + \langle \nabla f(u), \nabla^A_x v_x + \nabla^A_y v_y \rangle
\left( U ( g ) f \right) ( z ) = \epsilon ( g ; z ) ( \alpha + \overline{{{\beta}}}\overline{{{z}}})^{- \frac{1}{2}}( \overline{{{\alpha}}}+ \beta z )^{- \frac{1}{2}}f \left( \frac{\alpha z + \overline{{{\beta}}}}{\beta z + \overline{{{\alpha}}}}\right)
M(x_i|t)=\sum_{a,b=0,1}e_{ab}\otimes M^{(ab)}(x_i|t),(M^{(ab)}(x_i|t))_{a,b=0,1}=\left( \begin{array}{cc}A(x_{i}|t) B(x_{i}|t) C(x_{i}|t) D(x_{i}|t)\end{array}\right)
\sum_{j = 1}^k{( - 1)^{j - 1}G_{2j - 1}^2}= \frac{1}{6}G_0^2 + \frac{1}{2}G_1^2 - \frac{1}{6}G_2^2 - \frac{5}{6}( - 1)^k G_{2k + 1}^2 &\qquad- \frac{{( - 1)^k}}{6}G_{2k + 2}^2 + \frac{{( - 1)^k}}{6}G_{2k + 3}^2\,,
V_\mu^I|_{\widehat{G}^I \rtimes \langle \tau \rangle}= \bigoplus_{\bar{\lambda}\in{\mathcal Wt}(\bar{\mu})^{+, \tau}}V_{\bar{\lambda}, 1}\otimes{\mathbb H}_\mu(\bar{\lambda}) ~\oplus ~ \bigoplus_{\underset{\tau\bar{\lambda}\neq \bar{\lambda}}{\bar{\lambda}\in{\mathcal Wt}(\bar{\mu})^+}}V^I_\mu(\bar{\lambda}).
[v_i,[v_j,v_k]]+[v_j,[v_k,v_i]]+[v_k,[v_i,v_j]] =0 \implies [v_i,z_b]+[v_j,0]+[v_k,0] =0 \mbox{(by Equation \ref{2bracket})}\implies [v_i,z_b] = 0 \mbox{for all $v_i$ not in the closed path of length 3 or 4}.
\begin{alignedat}{2}\#(h_{m+1}) + \#(f_{m}) + \#(f_{m+1}) = k+1, \quad \#(e_{m+1}) + \#(h_{m+1}) + \#(f_{m}) = k+1, &\#(e_{m+1}) + \#(h_{m}) + \#(f_{m}) = k+1,\quad \#(e_{m}) + \#(e_{m+1}) + \#(h_{m}) = k+1,\end{alignedat}
\Tilde{z}_{k,\mathrm{avg}}= \sum_{k'=k-K}^{k+K}\left(\sum_{l=0}^{L-1}z'_{k',l}\right)^{\mu}\varphi_{k,\mathrm{est},\mathrm{mod}}= \frac{1}{\mu}\mathrm{unwrap}\left(\arg\left(\Tilde{z}_{k, \mathrm{avg}}\right)\right).
G_n(k)=\frac{\left|\sum\limits_{i=1}^kR_i-\frac{k}{n}\sum\limits_{i=1}^nR_i\right|}{\left\{\frac{1}{n}\sum\limits_{t=1}^k S_t^2(1,k)+\frac{1}{n}\sum\limits_{t=k+1}^n S_t^2(k+1,n)\right\}^{\frac{1}{2}}},
\{(x_1,y_1,z_1,w_1,\hdots,x_{g+1},y_{g+1},z_{g+1},w_{g+1})\,|\,x_i,z_i\in\mathbb{R}-\{0\}\ \mbox{and}\y_i,w_i\in\mathbb{R}\ \forall\i, &x_{g+1}^2 = 1,\ \mbox{and}\ \sum_{i=1}^g p(x_i,y_i,z_i,w_i) + x_{g+1}y_{g+1}(z_{g+1}^2+1) = 0 \}
\frac{(1 - \frac{c_n}{n})^{2 n}}{(1 - \frac{c_n}{n}) - (1 - \frac{c_n}{n})^{n}}=\frac{e^{-2 c}}{1 - e^{- c}}\Bigl(1 +\frac{c (2 - c e^{-c}- 2 c)}{2 n (1 - e^{- c})}\Bigr) + \ldots
\lambda =\lambda b(u,u_{\mathrm{nc}})+\lambda b(u,u-u_{\mathrm{nc}})= \lambda b(u,u_{\mathrm{nc}}-Ju_{\mathrm{nc}})+a_{\mathrm{pw}}(u, Ju_{\mathrm{nc}})+\lambda/2\ \Vert u-u_{\mathrm{nc}}\Vert^2_{L^2(\Omega)}.
\Psi_{E}^{( - )}= \left( \begin{array}{l}{e^{i \frac{\theta}{2}}\sum f_{n}e^{i \left( n + \frac{1}{2}\right) \theta}}\\{e^{- i \frac{\theta}{2}}\sum g_{n}e^{i \left( n + \frac{1}{2}\right) \theta}}\\ \end{array}\right)
- 2 + \cos d_{PQ}+ \cos d_{RS}= - \frac{1}{2}d_{PQ}^2 -\frac{1}{2}d_{RS}^2 + O( d_{RS}^4 + d_{PQ}^4) \left(\sqrt{2-2\cos d_{QR}}- \sqrt{2-2\cos d_{SP}}\right)^2 =\left( \frac{\sin d_{SP}}{\sqrt{2-2\cos d_{SP}}}(d_{QR}- d_{SP}) + O\left((d_{QR}- d_{SP})^2\right)\right)^2 &= \frac{1+\cos d_{PS}}{2}(d_{QR}- d_{SP})^2 + O\left((d_{QR}- d_{SP})^3\right).
\bigg| \frac{1}{2}x_1+ \frac{\sqrt{3}}{2}x_n \bigg| \leq \frac{1}{2}|x_1|+ \frac{\sqrt{3}}{2}|x_n| = \frac{1}{2}(2-r)+ \frac{\sqrt{3}}{2}|x_n| & \leq \frac{1}{2}(2-r)+ \frac{r}{4}= 1- \frac{r}{4}=: \lambda.
\sum_{i=1}^n g(zn^{1/4}A_{i,n}) \leq -nc\,\frac{1}{n}\sum_{i=1}^n\frac{(zn^{1/4}A_{i,n})^4}{1+(zn^{1/4}A_{i,n})^2}&\leq -nc\,\frac{\left(\frac{1}{n}\sum_{i=1}^n(zn^{1/4}A_{i,n})^2\right)^2}{1+\frac{1}{n}\sum_{i=1}^n(zn^{1/4}A_{i,n})^2}=-\frac{cz^4}{1+z^2/\sqrt{n}},
\P \Big\{\langle \bar{u},X_i\rangle \ge \tau\sqrt{d}\vert b_i\vert i=1,2,\dots ,m\Big\}= \P \Big\{d(0,E)\ge \tau\sqrt{d}\Big\}&= \P \Big\{\sqrt{\sum_{i\le m}\langle \bar{u}, X_i\rangle^2}\ge \tau\sqrt{d}\Big\}&=\P\Big\{\Vert G\bar{u}\Vert\ge \tau\sqrt{d}\Big\}&\ge \P\Big\{s_{\min}(G)\ge \tau\sqrt{d}\Big\}.
\overline{J}_{\dot{\alpha}}^{0}= -{\frac{i}{2}}\overline{\sigma}^{0 \dot{\beta}\beta}( \overline{D}_{\dot{\alpha}}R_{\beta \dot{\beta}}- 2 \varepsilon_{\dot{\alpha}\dot{\beta}}\overline{D}^{\dot{\gamma}}R_{\beta \dot{\gamma}}) \bigr |_{\theta = \overline{\theta}= 0}.
G_{M N}= \left( \begin{array}{c | c}{{\begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}}}&{{0}}\\ \hline{{0}}&{{\mathrm{{\bf ~ 1}}}}\\ \end{array}\right) , \B_{M N}= 0 , \D = X^{1}\quad ( M , N = 0 , 1 , \dots ) .
l(k) = \frac1{2\pi}\int_0^{\infty}\frac{\chi(\xi)}{\xi-k}~d\xi &= \frac{1}{2\pi}\int_0^1\frac{1}{\xi-k}~d\xi + \frac1{2\pi}\int_1^2\frac{\chi(\xi)}{\xi-k}~d\xi &= \frac1{2\pi}\log k + h(k),
\sum_k \mathsf{P}_{k}^{i}\mathsf{P}_{j}^{k}= \sum_{m,n,o,p}c_{n}^{m}c_{p}^{o}\sum_{k}(\mathsf{M}_{m}^{n})_{k}^{i}(\mathsf{M}_{o}^{p})_{j}^{k}= \sum_{m,n,p}c_{n}^{m}c_{p}^{n}(\mathsf{M}_{m}^{p})_{j}^{i}.
\begin{aligned}F^{ij}\eta_{ij}- \eta_t &\geq 2 B \sum_k F^{ij}u_{kj}u_{ki}+ A F^{ij}(\underline{u}-u)_{ij}- A(\underline{u}- u)_t - CK(1 + \sum_i F^{ii}) & \geq B \sum_i F^{ii}U_{ii}^2 + A F^{ij}(\underline{u}-u)_{ij}- A(\underline{u}- u)_t - CK(1 + \sum_i F^{ii}).\end{aligned}
C_{\alpha,\beta}=\ln (1+4(1-\alpha-\beta)^2)>0 \;\; \;\;G_D(C_0,C_1)=\frac{1}{\sigma \sqrt{2\pi}}\int_{C_0b_{D}}^{C_1b_{D}}\exp\left\{-\frac{1}{2\sigma^2}(t-b_{D})^2\right\}dt,
J_{1}= - \left( \begin{array}{c c c}{{D^{- 1}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{\frac{\textstyle{1}}{\textstyle{q_{x x}}}}}\\{{0}}&{{- \frac{\textstyle{1}}{\textstyle{q_{x x}}}}}&{{\frac{\textstyle{1}}{\textstyle{q_{x x}}}D \frac{\textstyle{r_{x}}}{\textstyle{q_{x x}}}+ \frac{\textstyle{r_{x}}}{\textstyle{q_{x x}}}D \frac{\textstyle{1}}{\textstyle{q_{x x}}}}}\\ \end{array}\right)
\begin{aligned}\left|\| WW^{T}\bar{U}\bar{\Sigma}^{-1}\|_F - \|( SS^{T}- WW^{T}) \bar{U}\bar{\Sigma}^{-1}\|_F\right| \leq \| WW^{T}\bar{U}\bar{\Sigma}^{-1}+ ( SS^{T}- WW^{T}) \bar{U}\bar{\Sigma}^{-1}\|_F = \|S \hat{V}\|_F, \end{aligned}
N_{F,k}\left(Z, \frac{1}{2kd}\log Z \right) = A_F \prod_{p \leq \log Z/(2kd)}\left(1 - \frac{\rho_F(p^k)}{p^{2k}}\right) Z^{\frac{2}{d}}+ O_{F,k}\left(Z^{\frac{2}{d}}/ (\log Z)^{6}\right).
T_g=\cap_{m\in{\mathbb N}}\Big\{t\in [\frac32,3]\, : \, V_t^{(m)}\mbox{is a unit density 1-varifold},\ \h(V_t^{(m)},\cdot)\in L_{loc}^2(\|V_t^{(m)}\|) &\mbox{and}\Theta(\|V_t^{(m)}\|,x)=1\mbox{or}\frac32,\, \ \forall x\in{\rm spt}\,\|V_t^{(m)}\|\Big\},
\langle \mathcal{T}_{a^+,a^-}\circ B(\mathfrak{w}),c\rangle=Re(\int_{\mathbb{S}^1}\bar{u}^-w^+- u^+\bar{w}^-)=Re(\int_{M\setminus\Sigma}\langle D\mathfrak{u},\mathfrak{w}\rangle+\langle \mathfrak{u},D\mathfrak{w}\rangle)=0
\Sigma_{\mu \nu}\rightarrow \Sigma_{\mu \nu \rho}={\cal A}_{\mu \nu \rho}^{F}-8{\cal A}_{\mu \nu \rho}^{V},\qquad \Xi_{\mu \nu}\rightarrow \Xi_{\mu \nu \rho}={\cal A}_{\mu \nu \rho}^{F}+{\frac{16}{5}}{\cal A}_{\mu\nu \rho}^{V},
\varphi(x) = |uJ|^{\frac{1}{2}}\varphi_1'(x_1') \varphi_4'(x_4') \int_F \int_F \varphi_2'(x_2') \varphi_3'(x_3') \psi(x_2'y_2' + x_3' y_3') \, dx_2' \, dx_3' = |uJ|^{\frac{1}{2}}\varphi_1'(x_1') \hat{\varphi}_2'(y_2') \hat{\varphi}_3'(y_3') \varphi_4'(x_4').
\lambda(p, q) - \frac\gamma4 = \frac{p(q-q_-)(q-q_+)}{2{(p-1)}^2(q-1)}, q_{\pm}= \frac{\gamma (p-1)}{4p}\left(p - 1 + \frac2\gamma \pm \sqrt{(p-p_-)(p-p_+)}\right), p_\pm = 1+\tfrac2\gamma \pm 2\sqrt{\tfrac2\gamma}.
g_{ij}=\frac{1}{u^2}(\sigma_{ij}+\frac{1}{u^2}\nabla_iu\nabla_j u) & \eta=-\frac{1}{(u^2+|\nabla u|^2)^{1/2}}(\nabla u-u x) & h_{ij}=\frac{1}{u\sqrt{u^2+|\nabla u|^2}}(u\sigma_{ij}+\nabla_{ij}u)
\gamma(v_n^A) = \frac{\pi_A}{\pi_B}v_n^B +\sum_{i=1}^{n-1}\ell_i^B \left( \frac{\pi_A}{\pi_B}(v_{n-i}^B)^{q^i}- \gamma(v_{n-i}^A)^{q^i}\right) .
\xi \cdot \Phi'(p) = \phi'\left(\int_{\Omega}f(p(x)d\nu(x)\right) \int_{\Omega}f'(p(x))\xi(x)d\nu(x), (\xi,\eta)\cdot\Phi''(p)= \phi'\left(\int_{\Omega}f(p(x))d\nu(x)\right) \int_{\Omega}f''(p(x))\xi(x)\eta(x)d\nu(x) + \phi''\left(\int_{\Omega}f(p(x))d\nu(x)\right)\int_{\Omega}f'(p(x))\xi(x)d\nu(x)\int_{\Omega}f'(p(x))\eta(x)d\nu(x).
\left\lbrace\begin{tabular}{l}$u = 2a \widetilde{u}+2a \widetilde{v}$, $v = \left(\left(\frac{1}{c}-1\right)+\lambda\right) \widetilde{u}+ \left(\left(\frac{1}{c}-1\right)-\lambda\right) \widetilde{v}$\end{tabular}\right.
S = \frac{( d - 2 ) !}{2^{d - 2}}\frac{\cosh ( t / 2 )}{\sinh^{d - 1}( t / 2 )}\left[ 1 - \frac{t \eta^{2}( d - 1 )}{2 \sinh ( t )}\left( 1 + 4 \xi \sinh^{2}( t / 2 ) \right) \right] \ .
\widehat{\left(i\ \epsilon\right)}\boxtimes\widehat{\left(j\ \epsilon_{1}\right)}=\widetilde{\left(\frac{i+j}{2}\ \epsilon+\epsilon_{1}\right)}+\widetilde{\left(k+\frac{i+j}{2}\ \epsilon+\epsilon_{1}\right)}+\sum_{1\le r\le2k-1,r\ \mbox{even}}\left(i+r\j-r\right)
u(x_j) \ge \frac{C(m,n,\lambda)\psi(|x_j|)}{|x_j|^{2m-2}}\frac{\psi(|x_j|)^{\frac{\lambda(n-2m)}{n-\lambda(n-2m)}}}{|x_j|^{(1+2b)(n-2m)}}&= \frac{C(m,n,\lambda) \psi(|x_j|)^{\frac{n}{n-\lambda(n-2m)}}}{|x_j|^{(n-2m) + (2m-2) + (n-2m)2b}}&\ge C(m,n,\lambda) \frac{\varphi(|x_j|^{-1})^{1/2}}{|x_j|^{a+n-2m}}
\left\langle \gamma_j(w), \gamma_i(z)\right\rangle &\quad= \big(K_2\partial_i\bar{\partial}_jK_3\big)(z,w)+ \big(K_3\partial_i\bar{\partial}_jK_2\big)(z,w) -\big(\partial_iK_2\bar{\partial}_j K_3\big)(z,w)-\big(\partial_iK_3\bar{\partial}_j K_2\big)(z,w).
S(a,b,q,N) = \sum_{l=0}^{\lfloor N/2 \rfloor}(q^3)^{N-2l \choose 2}\genfrac{[}{]}{0pt}{}{N+1}{2l+1}_{q^3}(-aq^2,-bq^4;q^6)_{l}, T(a,b,q,N) = \sum_{l=0}^{\lfloor N/2 \rfloor}(q^3)^{N-2l \choose 2}\genfrac{[}{]}{0pt}{}{N+1}{2l+1}_{q^3}(-aq^5,-bq;q^6)_{l},
\langle 2 div\h - \nabla \mathcal{H}, \nabla f \rangle = \langle 2 g^{ij}\nabla_i( \frac{1}{2}Rg_{ij}) - \frac{n}{2}\nabla_k R, \nabla f \rangle \displaystyle = \frac{2n-n}{2}\langle \nabla R, \nabla f \rangle
I_2^{(n)}= \int_{1/(\pi n+\pi/2)}^{1/(\pi n-\pi/2)}\frac{t^2\sin^2(t^{-1})dt}{(t-\frac{1}{\pi n})^2}=(\pi n)^2\int_{-\pi /2}^{\pi/2}\frac{\sin^2(x)dx}{(x+\pi n)^2x^2}\leq (\pi n)^2\int_{-\pi /2}^{\pi/2}\frac{x^2dx}{(x+\pi n)^2x^2}&\leq\frac{1}{(1-\frac{1}{2 n})^2}.
|(\mu \times \nu)(\gamma^b B_i \cap B_j) - \mu(\gamma^c C_i \cap C_j)| = \left \vert (\mu \times \nu) \left( \bigsqcup_{s,s' = 1}^t \gamma^b P^b_{\theta_i(s)}\cap P^b_{\theta_j(s')}\right) - \mu\left( \bigsqcup_{s,s' =1}^t \gamma^c Q^c_{\theta_i(s)}\cap Q^c_{\theta_j(s')}\right) \right \vert \leq t^2 \delta \leq (2r)^{2|F|}\delta < \frac{\epsilon}{4},
\lambda_{h}^{R, (n,k)}= i R_0 - h^{\frac 23}a_n^R(\kappa) e^{-i\pi/3}+ h (2k-1) \frac{e^{-i\pi/4}}{\sqrt{2 R_0}}& + h^{\frac 43}\frac{i}{2R_0 (\kappa^2 - a_n^R(\kappa) e^{-\pi i/3})}+ \mathcal O(h^{\frac 53})\, .
\left\Vert y_n (0) - r(x)\right\Vert = \left\Vert \Pi^-_n r( x_n) - r(x)\right\Vert \leq \left\Vert \Pi^-_n \left(r( x_n) - r(x)\right)\right\Vert + \left\Vert \left( 1 - \Pi^-_n \right)r(x)\right\Vert
\sum_{k\leq x}\frac{1}{k}\sum_{j=1}^{k}\tau(\gcd(k,j)) \log j = \sum_{n\leq x}\frac{\sigma(n)}{n}\log \frac{n}{e}+ \frac{1}{2}\sum_{n\leq x}\frac{l(n)}{n}+ \log\sqrt{2\pi}\sum_{n\leq x}\frac{\tau(n)}{n}+{\Theta}\sum_{n\leq x}\frac{\sigma_{-1}(n)}{n}.
\|A\|_{q,r}= \begin{cases}\left[\sum_{i=1}^m (\sum_{j=1}^n |a_{ij}|^q)^{r/q}\right]^{1/r}\mbox{if $q \geq r$}\left[\sum_{j=1}^n (\sum_{i=1}^m|a_{ij}|^r)^{q/r}\right]^{1/q}\mbox{if $r > q$,}\end{cases}
(\begin{matrix}n\\k_{1},k_{2},...,k_{r}\end{matrix})=(\begin{matrix}n-1\\k_{1}-1,k_{2},...,k_{r}\end{matrix})+(\begin{matrix}n-1\\k_{1},k_{2}-1,...,k_{r}\end{matrix})+...+(\begin{matrix}n-1\\k_{1},k_{2},...,k_{r}-1\end{matrix})
q(t)=\left(\begin{array}{c}-\varepsilon(b\bar{v}-c\bar{w}) -\bar{v}^3+(a+1)\bar{v}^2-a\bar{v}-\bar{w}\end{array}\right) &\times\frac{1}{\sqrt{(-\bar{v}^3+(a+1)\bar{v}^2-a\bar{v}-\bar{w})^2+\varepsilon^2(b\bar{v}-c\bar{w})^2}}.
C_{\alpha\beta}=[\begin{matrix}C_{11}&C_{12}&C_{12}&0&0&0\\C_{12}&C_{11}&C_{12}&0&0&0\\C_{12}&C_{12}&C_{11}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{44}&0\\ 0&0&0&0&0&C_{44}\end{matrix}].
\frac{1}{\mu_1}=e^{t}(a^2-b^2+c^2+\sqrt{-N(v)}) &\mu_1=e^{-t}(a^2-b^2+c^2-\sqrt{-N(v)}) &\frac{1}{\mu_2}=e^{t}(a^2-b^2+c^2-\sqrt{-N(v)}) &\mu_2=e^{-t}(a^2-b^2+c^2+\sqrt{-N(v)})~.
\gamma_f = \sum_{x \in E_N}\xi_f(x) (\{g_x \rho, g_x \infty\}+ \{g_x \infty, g_x \rho^2\}) & = \sum_{x \in E_N}\xi_f(x) (\{g_x \rho, g_x \infty\}- \sum_{x \in E_N}\xi_f(x) \{g_x 0, g_x \rho\}) & = 2 \sum_{x \in E_N}\xi_f(x) \{g_x \rho, g_x \infty\}- \sum_{x \in E_N}\xi_f(x) \xi(x).
\b A = \begin{pmatrix}A \tfrac{1}{\ell}e \tfrac{-\epsilon}{\ell}e^t 0 \end{pmatrix}, \b F = d\b A + \b A^2= \begin{pmatrix}F \tfrac{1}{\ell}T \tfrac{-\epsilon}{\ell}T^t 0 \end{pmatrix}=\begin{pmatrix}R-\tfrac{\epsilon}{\ell^2}e e^t \tfrac{1}{\ell}D^A e \tfrac{-\epsilon}{\ell}(D^Ae)^t 0 \end{pmatrix}.
\int_0^\infty u(t,x) \varphi(x)\,dx=\int_0^\infty u_0(x)\varphi(x)\,dx+\tau\int_0^t V(s)\frac{\int_0^\infty xu(s,x)\varphi'(x)\,dx}{1+\omega\int_0^\infty u(s,x)\,dx}\,ds & -\mu\int_0^t\int_0^\infty u(s,x)\varphi(x)\,dxds+ \beta\int_0^t\int_0^\infty x^\gamma u(s,x)\Bigl[2\int_0^1\varphi(zx)\wp(z)\,dz-\varphi(x)\Bigr]\,dxds.
R_1(v)= \sum_{j=1}^k M_{11}(f_j)=\sum_{j=1}^k M_{11}(f_j^{-1})=\sum_{j=1}^k m_j(M_{12}(f_j^{-1})+M_{12}(h_j^{-1}))= = m_k \sum_{j=1}^k (M_{12}(f_j^{-1})+M_{12}(h_j^{-1})) + D,
{\rm Avg}_{\rm prime}\Phi(\log L_q({\rm Sym}_f^{\mu},\sigma)-\log L_q({\rm Sym}_f^{\nu},\sigma)) &={\rm Avg}_{\rm power}\Phi(\log L_q({\rm Sym}_f^{\mu},\sigma)-\log L_q({\rm Sym}_f^{\nu},\sigma)) =\int_{\mathbb{R}}\mathcal{M}_{\sigma}(u)\Phi(u)\frac{du}{\sqrt{2\pi}}
\tilde{\alpha}_t(a,b) =\Phi_t^{-1}\alpha_t(\Phi_ta,\Phi_tb), \tilde{\mu}_t(x,a) =\Phi_t^{-1}\mu_t(\Psi_tx,\Phi_ta), \tilde{\lambda}_t(x,y) =\Psi_t^{-1}\lambda_t(\Psi_tx,\Psi_ty).
\begin{pmatrix}\bar{f}_1 \bar{f}_2 \bar{f}_3\end{pmatrix}= \begin{pmatrix}f^P_1 f^P_2 f^P_3\end{pmatrix}- C\,A^{-1}\begin{pmatrix}f^Q_1 f^Q_2 f^Q_3\end{pmatrix}.
\mathrm{LC}_{s=3/4}^0 \bigg[\frac{b_{1/2,0}(d,s)}{d^{1/2}\Gamma(s+1/4)}\bigg] = \frac{\pi}{\sqrt{d}}\frac{2}{B(1)}\mathrm{Tr}_{d,1}(3/2\pi) = \frac{3}{\sqrt{d}}\mathrm{Tr}_{d,1}(1),