harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left( \frac{N}{D e}\right)_{\beta_{H}\gg \beta_{S}}^{\mathrm{o p e n}}\: \simeq \frac{1}{m_{0}^{2}}\: \simeq \: \left( \frac{N}{D e}\right)_{\beta_{H}\gg \beta_{S}}^{\mathrm{c l o s e d}}
S_k=\Phi_{B_m}^{\dot A_n}\left( i\partial_{\beta_{m+1}\dot\alpha_n}\Lambda^{B_{m+1}}_{\dot A_{n-1}}+i\partial^{\beta_1}_{\dot\alpha_1}\cdots i\partial^{\beta_n}_{\dot \alpha_n}\Gamma^{\beta_{n+1}\cdots \beta_m}\right)\ ,
|G(x+iy)|=|G(x)|exp(y^{2}\frac{1+\gamma}{2})\sqrt{1+\frac{y^{2}}{x^{2}}}\sqrt{\prod_{k=1}^{\infty}(1+\frac{y^{2}}{(x+k)^{2}})^{k+1}exp(-\frac{y^{2}}{k})}.
\left[{\bf I}_{N_{\rm T}}\,\,{\bf 0}\right]\bar{\bf H}_{\rm b}^H ={\bf I}_{N_{\rm T}}\left[{\bf I}_{N_{\rm T}}\,\,{\bf 0}\right]{\boldsymbol\Lambda}_{\rm b}\left(\mu_{\rm b}\right) = \mu_{\rm b}\left[{\bf I}_{N_{\rm T}}\,\,{\bf 0}\right] = \mu_{\rm b}\left(\bar{\bf H}_{\rm b}- \left[{\bf 0}_{N_{\rm T}}\,\, \hat{\bf h}_{\rm b}\right]\right).
K^{tL(\frac{\cdot}{t})}(tH'(\rho_{-}),tH'(\rho_{+}))=\frac{\rho_{+}-\rho_{-}}{\sqrt{2\pi t^3(H'(\rho_{+})-H'(\rho_{-}))^3}}\times \left (-\frac{t}{2}\int_{\rho_{-}}^{\rho_{+}}(\rho_{*})^2 H''(\rho_{*})d\rho_{*}\right)\mathbb{E}\left[ \left(-\int_{\rho_{-}}^{\rho_{+}}\textbf{e}(tH'(\rho_{*}))d\rho_{*}\right)\right]
\begin{aligned}\qquad\nabla^2 S_-=0 s.t. S_-(0,t) =0 S_-(1/2,t) =\int_{0}^{\infty}\left[\frac{1}{x}-1\right]\frac{\sin\left(t\log x\right)}{\left(e^{x}+1\right)}dx S_-(\theta,0) =0 S_-(\theta,M) =\int_{0}^{\infty}\left[\frac{1}{x^{1/2+\theta}}-\frac{1}{x^{1/2-\theta}}\right]\frac{\sin\left(M\log x\right)}{\left(e^{x}+1\right)}dx \end{aligned}
S(\nu_a, \nu_b) =\int_{\partial \Sigma}\nu_a D_\eta \nu_b-\nu_a\nu_b &=\int_{\partial \Sigma}(k-1)X_a X_b-\int_{\partial \Sigma}(\left\langle{a^T,b^T}\right\rangle+\nu_a\nu_b)=\int_{\partial \Sigma}kX_a X_b-\left\langle{a,b}\right\rangle.
K_{p,q}(r) = \frac{{\pi_{p,q}}}{2}\sum\limits_{n = 0}^\infty{\frac{1}{p}- 1 \choose n}\frac{1}{{(1 - \lambda )^{n + 1 - \frac{1}{p}}}}\sum\limits_{j = 0}^n (-1)^{j}{n \choose j}{- \frac{1}{q}\choose j}{\frac{1}{p}- 1 - \frac{1}{q}\choose j}^{-1}\lambda^{n - j}r^{qj}.
T_{\rho \sigma \mu \nu}^{( 4 ) a b c d}( p_{1}, p_{2}; k_{1}, k_{2}) = C_{1}\gamma_{\rho \sigma \mu \nu}^{( 1 )}+ C_{2}\gamma_{\rho \sigma \mu \nu}^{( 2 )}+ C_{3}\gamma_{\rho \sigma \mu \nu}^{( 3 )}
\sum_{i=1}^{m}c_i^2 =\mathbb{P}_{\pi}[T_A>0]=\pi(B) c_1^2/\lambda_1 \le \mathbb{E}_{\pi}[T_A]=\sum_{i=1}^{m}c_i^2/\lambda_i \le \pi(B)/\lambda_1, c_1^2 =1-\frac{\|\alpha/\pi \|_2^2-1}{\|\alpha/\pi \|_2^2}p:=\sum_{j=2}^m c_j^2=\pi(B)-c_1^2= \frac{\|\alpha/\pi \|_2^2-1}{\|\alpha/\pi \|_2^2}-\pi(A).
4\delta^{ab}(C_1C_3+{1\over 4}C_1C_2){g^4 \over p^2}\Lambda^{3-\omega}\int{d^\omega k \over (2\pi)^\omega}\int{d^3 q\over (2\pi)^3}{k\cdot p \over k^2 q^2 (k+p+q)^2}.
\mathbf{x}=\mathbf{P}_{\mathcal{K}}\mathbf{s}_{\mathcal{K}}+\sum_{g=1}^{G}\mathbf{P}_{\mathcal{K}_g}\mathbf{s}_{\mathcal{K}_g}+\sum_{k=1}^{K}\mathbf{P}_{k}\mathbf{s}_{k}.
U_m(x,y)\bigl(\xi_\delta(u_m(x))-\xi_\delta(u_m(y))\bigr) &=|u_m(x)-u_m(y)|^{p-2}\bigl(u_m(x)-u_m(y)\bigr)\bigl(\xi_\delta(u_m(x))-\xi_\delta(u_m(y))\bigr) \geq 0
T^{ijk}_{P,Q}(\lambda,\mu):=\{u^i_\lambda \{u^j_\mu u^k\}_P\}_Q, J^{ijk}(A,B):=T^{ijk}_{A,B}(\lambda,\mu)-T^{jik}_{A,B}(\mu,\lambda)+T^{kij}_{A,B}(-\lambda-\mu-\partial,\lambda),
\left\langle A_{\nu}^{n}( y ) \left[ g \epsilon^{3 m n}a_{\mu}( x ) \partial_{x}^{\mu}A^{m \nu}( x ) \right] \right\rangle = 8 i g a^{\mu}( x ) \partial_{\mu x}\mathcal P^{*}( y , x ) \Bigr |_{y = x}
u^\varepsilon(t',\hat{x})-\bar{u}^\varepsilon(t'-\tau,\hat{x}) =u^{\varepsilon,\delta}(\hat{t},\hat{x})-(-\delta^{-1}|\hat{t}-t'|^2+u^\varepsilon(t'-\tau,\hat{x})) &\ge u^{\varepsilon,\delta}(\hat{t},\hat{x})-u^{\varepsilon,\delta}(\hat{t}-\tau,\hat{x}).
{\bf 7 0}\, \stackrel{U s p ( 4 ) \, \times \, S U ( 4 ) \, \times \, U ( 1 )}{\longrightarrow}\, \left({\bf 1}+{\bar{\bf 1}},{\bf 1},{\bf 1}\right) \, \oplus \, \left({\bf 1},{\bf 5},{\bf 6}\right) \, \oplus \, \left({\bf 1},{\bf 1},{\bf 6}\right)
\bigl||u+v|^{p_0-2}- |u|^{p_0-2}\bigr| \le \frac{\bigl||u+v|^{\alpha(p_0-2)}- |u|^{\alpha(p_0-2)}\bigr|}{|u+v|^{(\alpha-1)(p_0-2)}+|u|^{(\alpha-1)(p_0-2)}}& \le \frac{\bigl||u+v|^{\alpha(p_0-2)}- |u|^{\alpha(p_0-2)}\bigr|}{|u|^{(\alpha-1)(p_0-2)}}& = \frac{\bigl||u+v|^{2}- |u|^{2}\bigr|}{|u|^{4-p_0}}\le \frac{2|v|(|u| + |v|)}{|u|^{4-p_0}}.
I'_{1,3}=\left[3M_{2}\sum_{p=0}^{M_{2}-1}\sum_{q=0}^{M_{1}}\left(-1\right)^{q}C_{p}^{M_{2}-1}C_{q}^{M_{1}}e^{-3qa}\int_{\mu}^{\infty}e^{-3(\left(p+1\right)y-qb\frac{1}{y})}dy\right]\left[1-\left(1-e^{-3\mu}\right)^{M_{3}}\right],
K_{ik}\, K_{kj}~=~ K_{ij}~, ~~~~t_0^{kl}\, K_{ij}~=~ e^{\,{i\over 2}\, ( \, \alpha_{ik}+\alpha_{kj}+ \alpha_{jl}+ \alpha_{li}\, ) \,}~ K_{ij}\, t_0^{kl}~.
E \big\vert N_t^{\frac{\beta}{2}t + y}\big\vert \leq \mathrm{e}^{- \beta y}E \big\vert N_{t-s}^{\frac{\beta}{2}t + y}\big\vert = E \big\vert N_{t-s}^{\frac{\beta}{2}(t - s) + \frac{\beta}{2}s + y}\big\vert \leq \mathrm{e}^{- \beta (\frac{\beta}{2}s + y)}
\nu_{\eta_{m}^{r}\rightarrow s_{m}^{r}}(s_{m}^{r})\propto \underset{s_{m}}{\sum}\eta_{m}^{r}(s_{m}^{r},s_{m})\times\nu_{s_{m}\rightarrow\eta_{m}^{r}}(s_{m}) = \pi_{s^{r},m}^{out}\delta(s_{m}^{r}-1)+(1-\pi_{s^{r},m}^{out})\delta(s_{m}^{r}),
\mathrm{C h}_{0 , 0}^{N S , ( 2 8 )}+ \mathrm{C h}_{1 0 , 0}^{N S , ( 2 8 )}+ \mathrm{C h}_{1 8 , 0}^{N S , ( 2 8 )}+ \mathrm{C h}_{2 8 , 0}^{N S , ( 2 8 )}= A_{1}A_{3}.
\frac{\int_0^\infty e^{-\lambda s}p_j(s) ds}{\int_0^\infty e^{-\lambda s}(1-p_0(s)) ds}= \begin{cases}\dfrac1{j(j+1)}, p=0, \displaystyle -\frac{p}{\log(q)}\int_0^{1}(1-p y)^{-1}(1-y) y^{j-1}dy, 0<p<1, \end{cases}
\int \! \frac{\mathrm{d^{3}}k}{( 2 \pi )^{3}}\, \frac{1}{8 ( \omega_{k}^{0})^{3}}= \frac{1}{2}\int \frac{\mathrm{d}^{4}k}{( 2 \pi )^{4}}\frac{- i}{( k^{2}- m^{2}+ i o )^{2}}\; .
D(s, S_f) = \sum_{n \geq 1}\frac{S_f(n)}{n^{s + \frac{k-1}{2}}}D(s, S_f \times S_g) = \sum_{n \geq 1}\frac{S_f(n) S_g(n)}{n^{s + k-1}}D(s, S_f \times \overline{S_g}) = \sum_{n \geq 1}\frac{S_f(n) \overline{S_g(n)}}{n^{s + k-1}}.
\nabla_t^\mathcal{F}:=\mathrm{pr}_{t\star}\circ\nabla^\mathcal{F}|_{\Xi_{t\star}\otimes_{\mathbb{K}[[\nu]]}\Xi_{t\star}}\colon\Xi_{t\star}\otimes_{\mathbb{K}[[\nu]]}\Xi_{t\star}\rightarrow\Xi_{t\star}
\langle [\xi,\eta]_{\ast},[X,Y]\rangle+\langle [{\mathrm{ad}_{X}^{\ast}}^{(l)}\xi,\eta]_{\ast},Y\rangle-\langle[{\mathrm{ad}_{Y}^{\ast}}^{(r)}\xi,\eta]_{\ast},X\rangle-\langle[{\mathrm{ad}_{Y}^{\ast}}^{(l)}\xi,\eta]_{\ast},X\rangle=0.
\{A,B\}=\int_N \left(\frac{\delta_r A}{\delta \Phi^a}\frac{\delta_l B}{\delta \bar{\Phi}_a}-(-1)^{\epsilon(\Phi_a)}\frac{\delta_r A}{\delta \bar{\Phi}_a}\frac{\delta_l B}{\delta \Phi^a}\right)d^{n-1}x\,.
\int_{X_{\epsilon}}\partial_{z}\phi \partial_{\bar{z}}\phi \frac{i d z \wedge d \bar{z}}{2}= \int ( \partial_{z}\phi_{M}\partial_{\bar{z}}\phi_{M}- 2 \alpha_{1}\phi_{M}\partial_{z}\partial_{\bar{z}}\phi_{B}) \frac{i d z \wedge d \bar{z}}{2}+
{\frac{d}{d t}}( p \cdot \hat{H}) = \sum_{\rho \in \Delta}{\frac{g_{| \rho |}^{2}}{| \rho |^{2}}}( \rho \cdot \hat{H}) \left[ y_{| \rho |}( \rho \cdot q , w ) x_{| \rho |}( - \rho \cdot q , w ) - x_{| \rho |}( \rho \cdot q , w ) y_{| \rho |}( - \rho \cdot q , w ) \right] .
e^{\Delta S^{D}+ \Delta S^{C}}= \int d\alpha_C d\alpha_D e^{-\frac{\alpha_c^2G^{CC}}{\Gamma_C^2}}e^{-\frac{\alpha_D^2G^{DD}}{\Gamma_D^2}}e^{-\alpha_C \int d^2 zC_\epsilon -\alpha_D \int d^2z D_\epsilon}
u^\varepsilon(t',\hat{x})-\bar{u}^\varepsilon(t'-\tau,\hat{x}) =u^{\varepsilon,\delta}(\hat{t},\hat{x})+\delta^{-1}|\hat{t}-t'|^2-u^\varepsilon(0,\hat{x}) &\ge u^{\varepsilon,\delta}(\hat{t},\hat{x})-u^{\varepsilon,\delta}(0,\hat{x}).
\mathbb{E}\left(f_{x_l}\mid\mathcal{B}_{\bar{1}^k,k-1}\cup\{f_{x_i}: i < l\}\right) =\mathbb{E}\left(f_{x_l}\mid\mathcal{B}\cup\{f_{x_i}: i < l\}\right) = \mathbb{E}\left(f_{x_l}\mid\mathcal{B}\right),
\left. \begin{array}{l}{f ( \phi ) = e^{2 \sqrt \alpha \phi}, \}\\{w i d e h a t{V}= 0 ,}\\{V_{0}= e^{K_{M}}\left[ | D_{T}W |^{2}K^{T T^{*}}- ( 3 - \alpha ) | W |^{2}\right] .}\\ \end{array}\right.
\left[ t_j \vert M^{(_c)}\geq j+1, \tau^{(_c)}_j = s_j \right] = \int_0^{s_j}r \frac{\lambda \exp\left( \lambda r\right)}{\exp\left( \lambda s_j\right) -1}\textrm{d}r &= \frac{\lambda s_j - \left( 1-\exp\left( -\lambda s_j\right)\right)}{\lambda \left( 1-\exp\left( -\lambda s_j\right)\right)}.
\sum_{l=1}^{c}q^{l-1}(T_{p+l-1}^{-1}\cdots T_{p+2}^{-1}T_{p+1}^{-1}).f(x-v_{p+1}) =\sum_{l=1}^{c}q^{c-l}(T_{p+l}^{-1}T_{p+l+1}^{-1}\cdots T_{p+c-1}^{-1}).f(x-v_{p+c}).
F(\mathbf{x}) = \sum\limits_{\substack{\mathbf{k}\in \mathbb{Z}^m \Vert \mathbf{k}\Vert_\infty \leqslant X}}c_X(\mathbf{k}) e( \mathbf{k}\cdot \mathbf{x}) + O\left(M \frac{\log X}{X}\right)
tr_L(f) = (ev_V \otimes id_W) \circ (id_{V^{\vee}}\otimes f) \circ (coev'_{V}\circ id_w) \in End_T(W) tr_R(f) = (id_V \otimes ev'_W ) \circ (f \otimes id_{W^{\vee}}) \circ (id_V \otimes coev_{W}) \in End_T(V)
f(\theta) =S^{-1}(\theta)=\mathbf{a}^H(\theta)\mathbf{U}_V\mathbf{U}^H_V\mathbf{a}(\theta) &=\mathbf{a}^T(\frac{1}{z})\mathbf{U}_V\mathbf{U}^H_V\mathbf{a}(z)\triangleq f(z)
V_{0}( \sigma ) = \frac{\sigma^{2}}{2 \lambda_{0}}- \int_{0}^{\sigma}d s \int_{- \infty}^{\infty}\frac{d \omega}{2 \pi}\frac{s}{( 4 \pi )^{\frac{D - 1}{2}}}( \omega^{2}+ s^{2})^{\frac{D - 3}{2}}\Gamma ( - \frac{D - 3}{2}) \mathrm{t r}\mathrm{\boldmath ~ 1 ~}.
{\frac{3}{b^{2}}}{\frac{a^{\prime}}{a}}\left({\frac{a^{\prime}}{a}}+{\frac{n^{\prime}}{n}}\right) -{\frac{3}{c^{2}n^{2}}}\left[{\frac{\dot{a}}{a}}\left({\frac{\dot{a}}{a}}-{\frac{\dot{n}}{n}}\right) +{\frac{\ddot{a}}{a}}\right] -{\frac{3 k}{a^{2}}}= -{\frac{8 \pi G_{5}}{c^{4}}}\Lambda .
\mathcal{L}_{1}^{\mu}[\beta b] = - \frac{1}{\beta}\sinh{(\beta b(X) \sqrt{\mu}|\mathrm{D}|)}\mathrm{sech}(\sqrt{\mu}|\mathrm{D}|)\dfrac{1}{\sqrt{\mu}|\mathrm{D}|}\mathcal{L}_{2}^{\mu}[\beta b] = -(\mathcal{L}_{1}^{\mu}[\beta b] + b) \frac{1}{\mu |\mathrm{D}|^2}.
f(r)\; =\;o(r^{1/2})\,, (S_D^{-1}\Phi)(r)\; =\; \begin{pmatrix}p^+\! p^-\!\end{pmatrix}r^B+o(r^{1/2})\,, r^B\Phi(r)\; =\;\begin{pmatrix}1 -\frac{1+\nu-B}{1+\nu+B}\end{pmatrix}\gamma+\begin{pmatrix}q^+\! q^-\! \end{pmatrix}r^{2B}+o(r^{1/2+B})
\widehat{\ddot M}=\widehat\Sigma_{ee}^{-1}-\widehat\Sigma_{ee}^{-1}\widehat\Lambda \bigl(\widehat\Lambda'\widehat\Sigma_{ee}^{-1}\widehat\Lambda\bigr)^{-1}\widehat\Lambda '\widehat\Sigma_{ee}^{-1};
\left( \sum_{i,j=1}^{\infty}\left\vert U\left( e_{i},e_{j},e_{\sigma_{1}(i,j)},\ldots ,e_{\sigma_{n-2}(i,j)}\right) \right\vert^{\frac{4}{3}}\right)^{\frac{3}{4}}\leq \sqrt{2}\Vert U\Vert
E[{\rm ess}\inf L_j+L_{j+1}]-\frac{E[\int_{L_j}^{L_j+L_{j+1}}t dt]}{E[L_{j+1}]}= -\log_2{\delta M}-\frac{E[L^2]}{2E[L]}> -\log_2{\delta M}-\frac{1}{2}h(X)+\frac{1}{2}\log_2{\delta}-\epsilon = -\frac{1}{2}\log_2{\delta}-\log_2{M}-\frac{1}{2}h(X)-\epsilon > 0.
A =\left[\begin{matrix}a_0 a_{k-1}a_{k-2}\ldots a_{2}a_{1}a_1 a_0 a_{k-1}\ldots a_{3}a_{2}a_2 a_1 a_0 \ldots a_{4}a_{3}\vdots \vdots \vdots \vdots \vdots a_{k-1}a_{k-2}a_{k-3}\ldots a_{1}a_{0}\end{matrix}\right]
|E_d(T)| \leq \prod_{j=1}^d \frac 1{r_j}\frac{(d+1)^d (\log T)^d}{d!}\leq \prod_{j=1}^d \frac 1{r_j}\frac{e^{d}}{\sqrt{2 \pi d}}\frac{(d+1)^d}{d^d}(\log T)^d \leq \prod_{j=1}^d \frac 1{r_j}\frac{e^{d+1}}{\sqrt{2 \pi d}}(\log T)^d.
\vec{\mathbf{y}}_{i}^T ={}\mathbf{G}_{i1}^T\left(\mathbf{V}_{11}^T\vec{\mathbf{u}}_{11}+ \mathbf{V}_{21}^T\vec{\mathbf{u}}_{21}\right) + \mathbf{G}_{i2}^T\left(\mathbf{V}_{12}^T\vec{\mathbf{u}}_{12}+ \mathbf{V}_{22}^T\vec{\mathbf{u}}_{22}\right) + \vec{\mathbf{z}}_i^T,
\dot{{x}}_a =-\gamma \int_{0}^{1}\left ( y_\delta({x_a},\tau)-\bar{y}_a \right ) \, u(\tau)\,d\tau \dot{\bar{y}}_a = -\gamma\,\bar{y}_a + \gamma\int_{0}^{1}y_\delta ({x}_a,\tau)\,d\tau.
\mathcal{E}_{\log,\Lambda}^{\rm cp}(m^d)\leq E_{\log,\Lambda}^{\rm cp}(\omega^m) =2\left.\frac{\mathrm{d}}{\mathrm{d}s}E_{s,\Lambda}^{\rm cp}(\omega_N)\right|_{s=0}&=2\left. m^d\left(m^s\log m\cdot \zeta_\Lambda(s)+(m^s-1)\zeta'(s)\right)\right|_{s=0}&=2m^d\log m\cdot\zeta_\Lambda(0)=-2m^d\log m=-\frac{2}{d}m^d\log m^d
\frac{\partial}{\partial s}(b^{(1)}-b^{(2)})+\sum_{j=1}^{n-1}\frac{\partial}{\partial \hat y_j}(b^{(1)}-b^{(2)})\alpha_{js}^{(1)}(s,\tau,\beta^{(1)}(s,\tau,\hat y'))=0.
V(x)=\sup_{\theta\in\mathcal{A}}K(\theta) = \sup_{\theta \in\mathcal{A}}E_x \left[\int_{0}^{\infty}C\left(\theta_t^{(1-\alpha)}e^{\alpha X_t}\right)^{\beta}e^{-rt}\,dt -\int_{0}^{\infty}e^{-rt}\,d\theta_{t}\right]
d \epsilon_{a b}= \left( \begin{array}{c c c c}{{0}}&{{- d v_{1}}}&{{- d v_{2}}}&{{- d v_{3}}}\\{{d v_{1}}}&{{0}}&{{- d \theta_{1 2}}}&{{d \theta_{3 1}}}\\{{d v_{2}}}&{{d \theta_{1 2}}}&{{0}}&{{- d \theta_{2 3}}}\\{{d v_{3}}}&{{- d \theta_{3 1}}}&{{d \theta_{2 3}}}&{{0}}\\ \end{array}\right)
\int_D H(|u(x) -b|) \, dx \le \liminf_{j \to \infty}\int_D \chi_{D_{i_j}}H(|u(x) - u_{i_j}|)\, dx &= \liminf_{j \to \infty}\int_{D_{i_j}}H(|u(x) - u_{i_j}|)\le \liminf_{j \to \infty}C= C
(\pi^+, \pi^-)(s) :=\begin{cases}(\pi^{(x,t)}_+(t+s) - x, \pi^{(x,t)}_-(t+s) - x) 0 \leq s \leq \gamma(\pi^{(x,t)}_+, \pi^{(x,t)}_-) - t (\pi^{(x,t)}_+(t+s) - x, \pi^{(x,t)}_+(t+s) - x) s \geq \gamma(\pi^{(x,t)}_+, \pi^{(x,t)}_-) - t .\end{cases}
f^{n}(x)=x+\frac{(x-a)^{2}}{2}(nf^{\prime\prime}(a))+\frac{(x-a)^{3}}{6}(\frac{3}{2}n(n-1)f^{\prime\prime}(a)^{2}+nf^{\prime\prime\prime}(a))+\cdot\cdot\cdot
Z [ J ] = \lim_{\hbar \to 0}\int \exp \left( \hbar^{- 1}\sum_{k}\left( \Gamma_{0}[ \phi_{k}] + J \phi_{k}\right) + \delta S \big [ \hbar^{- 1}\sum_{k}\phi_{k}\big ] \right) \prod_{k}d \phi_{k}\, .
\begin{array}{rcl}A(y,z) = \displaystyle\sum_{k\geq 0}\frac1{k!}z^{k}y^{k+1}\left(\sum_{n\geq 0}4^{n}y^{n}\right)\left(\sum_{n\geq 0}{\binom{2n+1}{n+1}}^{\star k}y^{n}\right) = \displaystyle\sum_{0\leq k<n}\frac1{k!}\left(\sum_{i+j=n-k-1}{4^{i}\binom{2j+1}{j+1}}^{\star k}\right)z^{k}y^{n}. \end{array}
\biggl ( \sum_{l : l \neq j}{\frac{\theta^{\prime}( t_{j}- t_{l})}{\theta ( t_{j}- t_{l})}}\alpha_{c ( l )}-{\frac{\theta^{\prime}( t_{j})}{\theta ( t_{j})}}\Lambda + 2 \pi i \xi , \alpha_{c ( j )}\biggr ) = 0 , \qquad j = 1 , \dots , n .
i\widehat{u}_t = \left\{\mp\frac{p+1}{4\pi}\int_\mathbb{T}|u|^{p-1}(x,t)\,dx\right\}\frac{\widehat{v}}{L_t[u]}+ i\frac{\widehat{v}_t}{L_t[u]}= \mp\widehat{\mathcal{R}^1}[v, \cdots, v]\frac{1}{L_t[u]}+ i\frac{\widehat{v}_t}{L_t[u]},
\varphi\in C([0,\infty); L^2_\sigma(\Omega))\cap L^\infty_{loc}([0,\infty); L^{3,\infty}(\Omega)), &\nabla\varphi\in L^2_{loc}([0,\infty); L^2(\Omega)), \partial_t\varphi\in L^2_{loc}([0,\infty); L^2_\sigma(\Omega)).
[H(t)e^{-\int_0^tg(s)ds}]' =H'(t)e^{-\int_0^tg(s)ds}+ H(t)[-g(t)]e^{-\int_0^tg(s)ds}&=H'(t)e^{-\int_0^tg(s)ds}-H(t)g(t)e^{-\int_0^tg(s)ds}
\sigma_r (i) =\begin{cases}i + p^r \mbox{if}i \equiv 1, \ldots, (p-1)p^r \qquad\quad\mod p^{r+1}, i - (p-1)p^r \mbox{if}i \equiv (p-1)p^r+1, \ldots, p^{r+1}\mod p^{r+1}.\end{cases}
\begin{array}{l}\displaystyle [e_1,e_{n+1}]=e_1+(a-2)e_3,[e_2,e_{n+1}]=ae_2,[e_{i},e_{n+1}]=\left(a+i-4\right)e_{i}, \displaystyle (3\leq i\leq n),[e_{n+1},e_1]=-e_1-(a-2)e_3,[e_{n+1},e_3]=(1-a)e_3, \displaystyle [e_{n+1},e_4]=e_2-ae_4, [e_{n+1},e_j]=\left(4-j-a\right)e_j,(5\leq j\leq n,n\geq4). \end{array}
\sigma_{e_c}^2\left[t+1\right]= \left\{1-2\mu\left(2\sum_{j=1}^{M}\lvert\phi^{\left(j,c\right)}\rvert^2-f^{\left(c\right)}\right)\right\}\sigma_{e_c}^2\left[t\right] -4\left(\frac{_{}\left(a_c^{\left(o\right)}\right)}{a_c^{\left(o\right)}}+a_c^{\left(o\right)}\sum_{j=1}^{M}\Re\left\{\phi^{\left(j,c\right)}\right\}\right)
J ({\hat{k}}; M_{1}, M_{2}) = \int \frac{d^{2}{\hat{p}}}{( 2 \pi )^{2}}\frac{- i \not \!{\hat{p}}}{[ ({\hat{k}}-{\hat{p}})^{2}+ M_{1}^{2}] ({\hat{p}}^{2}+ M_{2}^{2})}.
\big| \big| S^{1/2}G_{\epsilon}(z) B^* f \big| \big|_{L^2}^2 = Q_{[H,iA]}(G_{\epsilon}(z)B^* f , G_{\epsilon}(z) B^* f) \le \frac{1}{\epsilon}\Big( \epsilon Q_{[H,iA]}(G_{\epsilon}(z) B^* f , G_{\epsilon}(z) B^* f ) + \mbox{Im}(z) || G_{\epsilon}(z) B^* f ||^2 \Big) ,
\varPhi^{ij}\equiv \frac{\partial \varPhi}{\partial (-\tilde{u}_{ij})}= F^{ij}\Theta \frac{\sinh \Theta}{\cosh \Theta}, \varPhi^{ij,kl}= F^{ij,kl}v^{-1}\Theta^2 \frac{\sinh \Theta}{\cosh \Theta}.
= \frac{i \beta}{2 \pi}4 i m \varepsilon^{\mu \nu \alpha}\frac{k_{\alpha}}{\sqrt{k^{2}}}\log \left( \frac{1 - \sqrt{\frac{4 m^{2}}{k^{2}}}}{1 + \sqrt{\frac{4 m^{2}}{k^{2}}}}\right) \, \, ,
t\left( 2n,2l\right) =2\times2^{2l}\sum_{m=1}^{j+1-n}\left(V^{-1}\left[ j\right] \right)_{2l+1,m}\frac{\left( j+1-m\right) ~\left(j+n-m\right) !}{\left( j+1-n-m\right) !}& \equiv\left( 2n\right) !~2^{2l}\sum_{m=1}^{j+1-n}\left( V^{-1}\left[j\right] \right)_{2l+1,m}\frac{\left( j+1-m\right)}{n}\binom{j+n-m}{2n-1}\ .
\dot{{x}}_a = \, -\gamma c_1 \delta \left.\dfrac{\partial h}{\partial x}\right|_{{x}_a}-\gamma \sum_{k=2}^{\infty}c_k \,{\delta^{2k-1}}\,\left.\dfrac{\partial^{2k-1}h}{\partial x^{2k-1}}\right|_{{x}_a}
j_{B,n}:= \lfloor \log_2 \lfloor B/\log_2 n\rfloor\rfloor j_{\max}:=\lceil (2+2s_{\min})^{-1}\log_2 (nB) \rceil\wedge \lceil (1+2s_{\min})^{-1}\log_2 n \rceil.
\langle V_{i_{1}}\dots V_{i_{n}}\rangle_{g}= \int{\cal D}\rho{\cal D}X e^{- S^{*}- g^{i}\int_{\Sigma}d^{2}\sigma V_{i}+ Q^{2}\partial \rho{\overline{{\partial}}}\rho - Q^{2}\rho \int_{\Sigma}d^{2}\sigma \rho R^{( 2 )}}V_{i_{1}}\dots V_{i_{n}}
\mathfrak{h}_{s,p}(\Omega):=\inf_{u\in C^\infty_0(\Omega)}\left\{\iint_{\mathbb{R}^N\times\mathbb{R}^N}\frac{|u(x)-u(y)|^p}{|x-y|^{N+s\,p}}\,dx\,dy\, :\, \int_\Omega \frac{|u|^p}{d_\Omega^{s\,p}}\,dx=1\right\}.
\sum_{j}q_{a}^{j}\varphi_{j}^{+}\overline{{\varphi}}_{j}^{+}= \zeta_{a}^{+}\overline{{\zeta}}_{a}^{+}= u_{( \alpha}^{+}u_{\beta )}^{+}\zeta_{a}^{( \alpha}\overline{{\zeta}}_{a}^{\beta )}.
a_r^-(p,q^d,2N_1/d) \stackrel{\eqref{eq:arminus}}{=}-\frac{1}{2}a_r^-(p,q^d,N_1/d) + \frac{1}{2}a_r^+(p,q^{2d},N_1/d) a_r^+(p,q^d,2N_1/d) \stackrel{\eqref{eq:arplus}}{=}-\frac{1}{2}a_r^+(p,q^d,N_1/d) + \frac{1}{2}a_r^+(p,q^{2d},N_1/d)
\prod_{\alpha > 0}\frac{[ \langle w_{i}( \mu ) + w_{i}( \rho ) , \alpha \rangle ]}{[ \langle \rho , \alpha \rangle ]}= - \prod_{\alpha > 0}\frac{[ \langle \mu + \rho , \alpha \rangle ]}{[ \langle \rho , \alpha \rangle ]}~ ,
C_{m_{d}}( \xi ) \; = \; \frac{1}{2 \pi}\int d^{2}p \; \hat{C}_{m_{d}}( p ) \; e^{i p \xi}\; = \; \frac{1}{( 2 \pi )^{2}}\int d^{2}p \; \frac{1}{p^{2}+ m_{d}^{2}}\; e^{i p \xi}\; ,
E(\bar{Y}_{N}) = E(E(\bar{Y}_{N}|\delta_{1}^{N})) = \sum\limits_{n=1}^{N}\frac{1}{N}\Bigg[ \Big(\frac{1}{2}+ \theta\Big) u_{n}+ \frac{1}{2}(1 - u_{n}) \Bigg] = \theta \frac{U_{N}}{N}+ \frac{1}{2}.
\kappa_t(u) = \, \frac{-a^3\lambda_v ER ( 2 \lambda_v Q + 2F_vD +E\det(\nu, \nu_v , \nu_{vv}))}{a^4\lambda_v^2 E^2 R^2}(u,0) & + \frac{(a^3\lambda_v E^2 R D)(a^2EF_vR)}{|a^2E^2R|\,|a^4\lambda_v^2 E^2 R^2|}(u,0) = \,\frac{-2\lambda_v Q - F_vD - E\det(\nu, \nu_v ,\nu_{vv})}{a \lambda_v E R}(u,0).
\begin{aligned}Ua = \frac{b V\alpha}{\alpha-\beta}, Va = \frac{b U\beta}{\alpha-\beta}, Aa = -\frac{b Ab}{a}, Ub = -\frac{a V\alpha}{\alpha-\beta}, Vb =-\frac{a U\beta}{\alpha-\beta}, V\gamma =\frac{a(\gamma-\beta)U\gamma}{b(\alpha-\gamma)},\end{aligned}
\sum_{\xi,\eta}\langle X_\alpha ,[X_\xi,X_\eta]\rangle\langle X_\alpha,[\tilde RX_\xi,\tilde TX\eta]\rangle= \sum_{\xi\triangle\eta=\alpha}r_\xi t_\eta\langle X_\alpha,[X_\xi,X_\eta]\rangle^2= 4\sum_{\xi\triangle\eta=\alpha}r_\xi t_\eta.
\left|\partial_x^\alpha\partial_y^\beta\mathcal K(x,y)-\partial_x^\alpha\partial_y^\beta\mathcal K(x,y+v)\right| \leq C|v|^\rho|x-y|^{-n-|\alpha|-|\beta|-\rho}&\quad\quad\begin{cases}|\alpha|\leq\lfloor s\rfloor, |\beta|=\lfloor J-n\rfloor-|\alpha|\end{cases}
- \frac{1}{4}e^{c \phi - \psi}\delta^{( d_{t}) i j}F_{i}^{( d_{l})}\cdot F_{j}^{( d_{l})}- \frac{1}{4}e^{c \phi - 2 \psi}\delta^{( d_{t}) i j}\delta^{( d_{t}) k l}D_{\alpha}B_{i k}D^{\alpha}B_{j l}) .
\Pr\{T_{c,n}(p) > k\}= p_0 \sum_{i=0}^{c-1}(-1)^{c-1-i}{n-i-1 \choose n-c}\sum_{J \in S_{i,n}}(p_0 + P_J)^{k-1}& \;\;\; + \sum_{\ell = 1}^n p_\ell \sum_{i=0}^{c-2}(-1)^{c-2-i}{n-i-2 \choose n-c}\sum_{J \in S_{i,n}(\ell)}(p_0 + p_\ell + P_J)^{k-1}.
sin^{n}\theta=\frac{1}{2^{n}}(\begin{matrix}n\\ \frac{n}{2}\end{matrix})+\frac{2}{2^{n}}\sum_{k=0}^{\frac{n}{2}-1}(-1)^{(\frac{n}{2}-k)}(\begin{matrix}n\\k\end{matrix})cos((n-2k)\theta)
4\partial_+ \partial_- \rho + 2 \partial_+ \psi \partial_-\psi+\frac{1}{2}\partial_+ \psi_1 \partial_-\psi_1 + 2 e^{-\psi_1-2\rho}\Omega F^2_{2+-}+2e^{\psi_1-2\rho}\Omega F^2_{+-}=0
\begin{aligned}Ric(X,Y) =(2n+2)T^0(X,Y)+(4n+10)U(X,Y)+2(n+2)Sg(X,Y), T(\xi_{i},\xi_{j})=-S\xi_{k}-[\xi_{i},\xi_{j}]_{|H}, S =-g(T(\xi_1,\xi_2),\xi_3).\end{aligned}
\mathfrak{G}_{w^{({\boldsymbol{n}})}}^{(\beta)}=\mathfrak{G}_{w_{0}^{(n_1)}}^{(\beta)}\times\mathfrak{G}_{1^{n_1}\times w_{0}^{(n_2)}}^{(\beta)}\times\mathfrak{G}_{1^{n_1+n_2}\times w_{0}^{(n_3)}}^{(\beta)}\times \cdots \times\mathfrak{G}_{1^{n_{1}+ \ldots n_{p-1}}\times w_{0}^{(n_{p})}}^{(\beta)}.
H_{t,n}(\chi)=\begin{cases}\chi\bigl(\frac{t+p^s\omega}{2}\bigr)\bigl(2+\alpha\frac{p^e+p^{e-1}-2}{p-1}\bigr) r\geq 2e, \chi\bigl(\frac{t+\sqrt{d}\ell}{2}\bigr) +\chi\bigl(\frac{t-\sqrt{d}\ell}{2}\bigr) r<2e\psi_d(p)=1, 0 \end{cases}
\frac{1}{N}\frac{1}{b^j}\tilde{A}-b^{-2j-1}\geq \frac{1}{N}\frac{1}{b^j}\left(\frac{N-1}{b^{j+1}}-\frac{\varphi(m)}{b}-1\right)-b^{-2j-1}= -\frac{1}{N}\frac{1}{b^{2j+1}}-\frac{1}{N}\frac{\varphi(m)}{b^{j+1}}-\frac{1}{N}\frac{1}{b^j}\geq -\frac{1}{N}\frac{b+2}{b^{j+1}}\geq -\frac{1}{N}\frac{2}{b^j},
\Pi_{M M^{\prime}}^{A A^{\prime}}( m^{2}) = \sum_{i = 1}^{3}\Bigl ( \Pi_{M M^{\prime}}^{( i ) + A A^{\prime}}( m^{2}) + \alpha^{M^{\prime}}\eta^{A^{\prime}}\Pi_{M M^{\prime}}^{( i ) - A A^{\prime}}( m^{2}) \Bigr ) .
\mathfrak{L}=\frac{1}{2}|Q^{\frac{1}{2}}D_x|^2+\frac{1}{8}|Q^{\frac{1}{2}}Q_{\infty}^{-1}x|^2+i\Big\langle \Big(\frac{1}{2}QQ_{\infty}^{-1}+B\Big)x,D_x\Big\rangle.
L(\phi) := \frac{1}{M_1}\sum_{j\in J_1}\sum_{i:X_i \in I_j}\phi(X_i) - \sum_{j \in J_1}\int_{I_j}e^\phi = \sum_{j \in J_1}\frac{N_j}{M_1}\biggl\{\frac{1}{N_j}\sum_{i:X_i \in I_j}\phi(X_i) - \int_{I_j}e^{\phi + \log(M_1/N_j)}\biggr\}
\lim_{n\to+\infty}S_{m\sigma,n\sigma}-\widetilde{S}_{m\sigma,n\sigma}=\lim_{n\to+\infty}\mathbb{E}_{x}[S_{m\sigma,n\sigma}\,\big|\, \mathcal{F}_{n\sigma}]-\mathbb{E}_{x}[\widetilde{S}_{m\sigma,n\sigma}\,\big|\, \mathcal{F}_{n\sigma}]=0\mathbb{P}_{x}\mbox{-a.s.}
\mathcal P_a'(i):=\ \Big\{P\ :\P\in{[2,n]\choose k-1},\P\cap [2,i]=\{i\}\Big\}, \mathcal P_b'(i):=\ \Big\{P\ :\P\in{[2,n]\choose k},\P\cap [2,i]=[2,i-1]\Big\},
B_{\alpha}([x,y]_{\alpha},z)= B(\alpha([x,y]_{\alpha}),z)=B([x,y]_{\alpha},\alpha(z)) = B(\alpha(x)\alpha(y),\alpha(z))-(-1)^{|x||y|}B(\alpha(y)\alpha(x),\alpha(z)) = B(\alpha(x),\alpha(y)\alpha(z))-(-1)^{|x||y|}(-1)^{|x||y|+|y||z|}B(\alpha(x),\alpha(z)\alpha(y)) = B(\alpha(x),[y,z]_{\alpha})=B_{\alpha}(x,[y,z]_{\alpha}),
h_{k,l,c,i,j}^* fh_{k,l,c,i,j}= \sum_{q=1}^Q \sum_{t\in B_{k,l,c,q}}\frac{q^2}{Q^2}\alpha_{tc\Lambda_{k,i,j}(c)^{-1}t^{-1}}(f) \alpha_{tc}(h_k^2 )
\psi_{n+1}(g^{p^n})= \big( ((a^{\alpha}b)^{ki}w_1)^{p^{n-1}}, ((a^{\alpha}b)^{ki}w_2)^{p^{n-1}}, \dots ((a^{\alpha}b)^{ki}w_p)^{p^{n-1}}\big),
X_{\pi_1,t}:\begin{cases}\mathcal{W}_{\pi_1}\rightarrow \mathcal{X}_{\pi_1},0\leq t\leq n_{\pi_1}X_{\pi_1,t}=\psi_{\pi_1,t},n_{\pi_1}< t \leq n \end{cases},X_{\pi_2,t}: \mathcal{W}_{\pi_2}\rightarrow \mathcal{X}_{\pi_2}
\kappa(S) =\|S\|_{\infty}\|S^{-1}\|_{\infty}= \|M^{-1/2}K\tilde{D}^{1/2}\|_{\infty}\|\tilde{D}^{-1/2}K^TM^{1/2}\|_{\infty}&\leq \|M^{-1/2}\|_{\infty}\|\tilde{D}^{1/2}\|_{\infty}\|\tilde{D}^{-1/2}\|_{\infty}\|M^{1/2}\|_{\infty}= \kappa(M^{1/2})\kappa(\tilde{D}^{1/2})
0\geq\left[\frac{n-2}{4(n-1)}\mu([g])-C_2(n)\left(\int_M |W|^\frac{n}{2}\right)^\frac{2}{n}-2\sqrt{\frac{n-1}{n}}\left(\int_M|\mathring{Ric}|^{\frac{n}{2}}\right)^{\frac 2n}\right]\left(\int_M |W|^{\frac{2n}{n-2}}\right)^{\frac{n-2}{n}}+\frac{10n-8-n^2}{4n(n-1)}R\int_M |W|^{2}.
\Phi = \phi_{0}- i \tau_{a}\phi_{a}= \left( \begin{array}{r r}{{\phi_{0}- i \phi_{3}\, ,}}&{{- \phi_{2}- i \phi_{1}}}\\{{\phi_{2}- i \phi_{1}\, ,}}&{{\phi_{0}+ i \phi_{3}}}\\ \end{array}\right) \, .
\tilde{L_{1}}( t ) ( \alpha^{*}, \alpha ) = \frac{g^{2}}{2}\left( \frac{1}{3 !}V_{a b c}W^{a b c}+ \alpha_{a}^{*}( V_{\; \; c d}^{a}W_{b}^{c d}+ V_{b c d}W^{c d a}) \alpha^{b}\right)
\mathbb{P}(n\le n^{*}|n_{b}\le n^{*},s+b)=\frac{\mathbb{P}(n\le n^{*},n_{b}\le n^{*}|s+b)}{\mathbb{P}(n_{b}\le n^{*}|s+b)}=\frac{\mathbb{P}(n\le n^{*}|s+b)}{\mathbb{P}(n\le n^{*}|b)}.
M ( p ; q_{1}, q_{2}) = \int \frac{d x \; d y}{( 1 + x + y )^{3}}e x p [ \frac{x y}{1 + x + y}\frac{p^{2}}{\Lambda^{2}}+ \frac{x}{1 + x + y}\frac{q_{1}^{2}}{\Lambda^{2}}+ \frac{y}{1 + x + y}\frac{q_{2}^{2}}{\Lambda^{2}}] ,