harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
U_{i}^{(c)}= \int_{0}^{T}b_{c}(t,\,x_{i}^{N}(t))d\beta_{i}(t)-\frac{1}{2}\int_{0}^{T}b_{c}^{2}(t,\,x_{i}^{N}(t))dt V_{ij}^{(c_{1},\,c_{2})}= \int_{0}^{T}\log\left(1+\frac{\gamma_{c_{1},c_{2}}(t,\,x_{i}^{N}(t))}{\lambda N}\right)\left[dM_{ij}^{N}(t)+\lambda NdA_{ij}^{N}(t)\right] -\int_{0}^{T}\gamma_{c_{1},c_{2}}(t,\,x_{i}^{N}(t))dA_{ij}^{N}(t)
E_{_{a}}^{\; \mu}=\left(\begin{array}{ccc}1 0 0 0 \frac{\beta}{\rho}\cos\phi -\frac{1}{\rho}\frac{\alpha}{\sin\alpha\theta}\sin\phi 0 \frac{\beta}{\rho}\sin\phi \frac{1}{\rho}\frac{\alpha}{\sin\alpha\theta}\cos\phi \end{array}\right).
G (1, ..., n; 1', ..., n') = i \frac{\delta^n Z_C [J]}{\delta J^* (1) ... \delta J^* (n) \delta J (1') ... \delta J (n')}\rule[-3mm]{.14mm}{8.5mm}\raisebox{-2.85mm}{\scriptsize{$\; J^* = J = 0$}}.
R_{x,C,S}^{t+1}=\prod_{D\in\mathcal{U}_{x,C}}(1- Q_{D,x,S}^t)\prod_{D\in\mathcal{S}_{x,C}}(1- Q_{D,x,U}^t)-\prod_{D\in\mathcal{U}_{x,C}}(1- Q_{D,x,*}^t)\prod_{D\in\mathcal{S}_{x,C}}(1- Q_{D,x,*}^t),
( \sigma_{0})_{\alpha \beta}= \left( \begin{array}{c c}{1}&{0}\\{0}&{1}\\ \end{array}\right) \quad ( \sigma_{1})_{\alpha \beta}= \left( \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right) \quad ( \sigma_{2})_{\alpha \beta}= \left( \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right)
\prod_{i=1}^m\prod_{j=1}^{k - 1}{\big(x_it^{N-1}- q^{j - \theta}\big)}= (-1)^{m(k - 1)}q^{m\left({k\choose 2}- \theta(k - 1)\right)}\prod_{i=1}^m{\big(x_iq^{\theta N - k+1}; q\big)_{k-1}}.
p(u,w) =\left (\cos u \cos w,\cos u \sin w,\sin u \cos w,\sin u \sin w\right), q(u,v) =\frac{1}{\sqrt{2}}\left (\cos v \left(\sin u+\cos u\right),\sin v \left(\sin u+\cos u\right)\right . , \left . \cos v \left(\sin u-\cos u\right),\sin v \left(\sin u-\cos u\right) \right).
\sum_{i=1}^{n}\int_{\mathbb{H}}\left|\hat{\eta}_{q}^{i}\right\rangle\left\langle \hat{\eta}_{q}^{i}\right|d\mu(q)=UA^{-1/2}A(UA^{-1/2})^{\dagger}=UU^{\dagger}=I_{V_\mathbb{H}^R},
{\cal L}_{\mathrm{W Z}}^{( p )}= \epsilon^{i_{1}\ldots i_{p + 1}}\sum_{k = 0}^{( p - 1 ) / 2}\, \, a_{p , k}C_{i_{1}\ldots i_{p - 2 k}}( F^{k})_{i_{p - 2 k + 1}\ldots i_{p}}\partial_{i_{p + 1}}T \, ,
\alpha ={s(v)\over2}{2e^{w\over2}v+ns(w)\over e^{w\over2}vs(v)-e^{v\over2}ws(w)}\cr\tilde u = -w \cr\tilde v = -v \cr\tilde c = \Big({\alpha\over 1-e^{-v}}+{\beta\over 1-e^{-w}}\Big)(cw+dv) -\alpha{w\over v}c-\beta{v\over w}d + e
\mathcal{L}_L^{\lambda}f:=\sum_{\ell=0}^L\sum_{k=1}^{2\ell+1}\mathcal{S}_{\lambda\mu_{\ell}}\left(\sum_{j=1}^Nw_jf(\mathbf{x}_j)Y_{\ell,k}(\mathbf{x}_j)\right)Y_{\ell,k}.
{\cal D}^{K}= \{{\cal D}, \; \; \}_{\cal A}- t ( \phi^{a}- t I_{a}g^{a b}) \{I_{a}, \; \; \}_{\cal A}+ \frac{t ( 1 - t )}{2}f_{a b}^{c}c^{a}c^{b}\{I_{a}, \; \; \}_{\cal A}-- t c^{a}\{Q_{a}, \; \; \}_{\cal A},
Z_C(s)=\frac{\zeta(s)}{\pi^{1/2+s}}\int_{-\infty}^{\infty}\frac{\sigma_{2ir}(l^2)}{l^{2ir}}\frac{\zeta(s+2ir)\zeta(s-2ir)}{\left|\zeta(1+2ir\right)|^2}h(\omega;s;r)dr,
q_{x1}=\int_{-b/2}^{b/2}q_{0}(\frac{1}{2}-\frac{y}{b})dy=\frac{bq_{0}}{2};q_{x2}=\int_{-b/2}^{b/2}yq_{0}(\frac{1}{2}-\frac{y}{b})dy=-\frac{b^{2}q_{0}}{12}.
\mathcal L_{\mathrm{m a s s}}= \frac{1}{2}\left( \psi_{\mu L}^{1}\ \psi_{\mu L}^{2}\right) \sigma^{\mu \nu}\left( \begin{array}{c c}{{\omega / R}}&{{i \, p_{5}}}\\{{- i \, p_{5}}}&{{\omega / R}}\\ \end{array}\right) \left( \begin{array}{c}{{\psi_{\nu L}^{1}}}\\{{\psi_{\nu L}^{2}}}\\ \end{array}\right) + h . c .
\nu_{n}= \frac{\eta}{\lambda}- ( 2 n + 1 ) \frac{\pi}{2 \lambda}, \qquad w_{k}= \frac{\bar{\eta}}{\lambda}- ( 2 k + 1 ) \frac{\pi}{2 \lambda}, \qquad \bar{\eta}= \pi ( \lambda + 1 ) - \eta \ .
H_{a}(s)=\frac{b_{0}s^{2}+b_{1}s+b_{2}}{a_{0}s^{2}+a_{1}s+a_{2}}=\frac{b_{0}+b_{1}s^{-1}+b_{2}s^{-2}}{a_{0}+a_{1}s^{-1}+a_{2}s^{-2}}.
I=I_{2 \times 2}=\begin{pmatrix}1 0 0 1\end{pmatrix}, X=\begin{pmatrix}0 1 1 0\end{pmatrix}, Y=\begin{pmatrix}0 -i i 0\end{pmatrix}, ~~ ~~Z=\begin{pmatrix}1 0 0 -1\end{pmatrix},
|Kl_p(\psi_p,\psi_p';c,w_{G_n})| \leq D_n \cdot \min(p^{\sigma +a_2+\cdots+ \varrho/2+\frac{n(n-1)}{2}m},p^{\ell/2+ 2 \sigma+(n-3)\varrho +a_2+\cdots+a_{n-2}-\ell +\frac{n(n-1)}{2}m}).
F_X(x)=1 -\frac{(1-\beta^2/\alpha^2)^{\nu+1/2}}{2\sqrt{\pi}\Gamma(\nu+1/2)}\sum_{k=0}^\infty\frac{1}{k!}\bigg(\frac{2\beta}{\alpha}\bigg)^k\Gamma\bigg(\frac{k+1}{2}\bigg)\Gamma\bigg(\nu+\frac{k+1}{2}\bigg)\tilde{G}_{\nu+k,\nu}(\alpha (x-\mu)),
{\frac{d s_{1 1}^{2}}{\ell_{p}^{2}}}= e^{- 2 \phi / 3}{\frac{d s_{I I A}^{2}}{\alpha^{\prime}}}+ e^{4 \phi / 3}({\frac{d x^{1 1}}{R}}-{\frac{C_{1}}{\sqrt{\alpha^{\prime}}}})^{2}\ ,
{\frac{\hbar^{2}}{1 6 m^{2}}}{\frac{| p_{1 , n}|^{2}}{E_{1 , n}- E_{+}}}={\frac{\hbar^{2}}{8 m}}{\frac{E_{1 , n}- U_{1}( 0 )}{E_{1 , n}- E_{+}}}<{\frac{\hbar^{2}}{8 m}},
v_{0}(r) =A\left( r/r_{0}\right)^{1/2}\left[ 1+c\ln\left(r/r_{0}\right) \right] ,r>R, & =\frac{A}{\sin\sqrt{\lambda}}\left( R/r_{0}\right)^{1/2}\left[1+c\ln\left( R/r_{0}\right) \right] \sin(k_{0}r),r<R.
\hat{C}\; = \; C_{( 3 )}- \textstyle{\frac{1}{2}}\vec{A}_{( 1 )}^{T}\varepsilon \vec{A}_{( 2 )}+ \vec{A}_{( 2 )}^{T}\vec{\theta}+ \textstyle{\frac{1}{2}}A_{( 1 )}\vec{\theta}^{T}\varepsilon \vec{\theta}\; ,
1-\alpha\lambda_\mathrm{n_0}^{(3)}= \mathcal{O}(\delta^\mathrm{h}) \iff \varepsilon_\mathrm{p}= \lambda_\mathrm{n_0}^{(3)}- \dfrac{\Big(\lambda_\mathrm{n_0}^{(3)}\Big)^2}{\varepsilon_\mathrm{m}+ \lambda_\mathrm{n_0}^{(3)}}+ \mathcal{O}(\delta^\mathrm{h})
\frac{y(w)}{x(w)}=\frac{sl(\frac{1}{2}\sqrt{2}w)[cl(\frac{1}{2}\sqrt{2}w)^{2}+1]^{1/2}}{cl(\frac{1}{2}\sqrt{2}w)[sl(\frac{1}{2}\sqrt{2}w)^{2}+1]^{1/2}}=slh(w)
F_{ij}(t) = \sum_{k=j}^{i}V_{ik}\exp(Q_{(n-k)\lambda}t)V_{kj}& = \sum_{k=j}^{i}{n-k \choose n-i}(-1)^{i-k}\exp(Q_{(n-k)\lambda}t){n-j \choose n-k}& ={n-j \choose n-i}\sum_{k=j}^{i}(-1)^{i-k}{i-j \choose i-k}\exp(Q_{(n-k)\lambda}t),
\ln D = - \frac{1}{2}\frac{d}{d s}\frac{1}{\Gamma ( s )}\int_{0}^{\infty}d T \T^{s - 1}e^{- m^{2}T}\mathrm{t r}\: e^{- T ( - \nabla_{\mu}\nabla_{\mu}+ \sigma^{\mu \nu}F_{\mu \nu})}\raisebox{- 9 p t}{{\scriptstyle s = 0}}\label 9
\int_{-\infty}^{\infty}A_{q}\left(q^{2\alpha}u\right)A_{q}\left(vq^{2\alpha}\right)q^{2\alpha^{2}}d\alpha=\sqrt{\frac{\pi}{\log q^{-2}}}\frac{\left(q^{1/2}u,q^{1/2}v;q\right)_{\infty}}{\left(uv;q\right)_{\infty}}
I_{3}( p ) \sim \frac{1}{( \sigma \omega )^{( 3 - D ) / 2}}\Gamma \left( \frac{3 - D}{2}\right) \; F \left( \frac{3 - D}{4}, \; \frac{5 - D}{4}; \; \frac{D}{2}; \frac{p^{2}}{\sigma^{2}}\right) \; ,
(P,Q)^{k}=\left.\left|\begin{array}{cc}{\partial\over\partial x_{1}}{\partial\over\partial y_{1}}\{\partial\over\partial x_{2}}{\partial\over\partial y_{2}}\end{array}\right|^{k}P(x_{1},y_{1})Q(x_{2},y_{2})\right|_{x_{1}=x_{2}=x\atop{}y_{1}=y_{2}=y}.
(1-v)a+vb\geq\xi(a,b)^{r_{N}(v)}a^{1-v}b^{v}+\sum_{n=0}^{N-1}r_{n}(v)\sum_{k=1}^{2^{n}}g_{n,k}(a,b)\chi_{(\frac{k-1}{2^{n}},\frac{k}{2^{n}})}(v)
P^{(\infty)}_{+}(z) = P^{(\infty)}_{-}(z) \begin{pmatrix}0 1 -1 0\end{pmatrix}, z\in \mathbb{R}^{-}, &P^{(\infty)}_{+}(z) = P^{(\infty)}_{-}(z) \begin{pmatrix}1 0 e^{\pi i \alpha}1\end{pmatrix}, \arg(z) = \frac{2\pi}{3}, &P^{(\infty)}_{+}(z) = P^{(\infty)}_{-}(z) \begin{pmatrix}1 0 e^{-\pi i \alpha}1\end{pmatrix}, \arg(z) = -\frac{2\pi}{3}.
\delta_{t}(U_C)^{k}_{ij}-\mu\Delta_{h}(U_C)^{k}_{ij}- \mathfrak{w}_{1,1}\Delta_{h}(U_C)^{k}_{ij}=b^{k}_{ij}+\frac{g((U_C)^k_{ij})+g((U_C)^{0}_{ij})}{2},(x_i,y_j)\in\Omega_{h},
\frac{\partial V_{e f f}}{\partial \sigma}= 0 \Longrightarrow - m_{B}^{2}+ \lambda_{B}v^{2}= 2 g_{B}^{2}\int^{\Lambda}\frac{d^{d}q}{( 2 \pi )^{d}}\frac{1}{q^{2}+ g_{B}^{2}v^{2}}.
\widetilde w_s(x,\omega,x^\ast,\omega^\ast) := w_s(x^\ast+\beta(\omega^\ast)x,x^\ast,\omega^\ast+\beta(\omega^\ast)^{-1}\omega,\omega^\ast) & \leq (1+|\omega|)^{\frac{s}{1-\alpha}}(1+\beta(\omega^\ast)|x|)^{t}&\leq (1+|\omega|)^{\frac{s}{1-\alpha}}(1+|x|)^{t}.
{\cal A}( p ) \sim \frac{g^{m}\alpha^{2 m - 1}}{R^{2 m + 2}r_{\mathrm{m i n}}^{m + 2}}\int d r \, r^{3}\biggl ( \frac{r_{\mathrm{m i n}}}{r}\biggr )^{\Delta}F ( \tilde{p}\sqrt{\alpha^{\prime}}) \ ,
X_{0}( [ B_{\infty}, \Gamma ]_{ik}) = \sum_{j=1}^{2}X_{0}( (B_{\infty})_{ij}\Gamma_{jk}- \Gamma_{ij}(B_{\infty})_{jk}) = \sum_{j=1}^{2}( (B_{\infty})_{ij}\delta_{jk}- (B_{\infty})_{jk}\delta_{ij}) = (B_{\infty})_{ik}- (B_{\infty})_{ik}=0
\begin{aligned}\log\det \Delta_\beta=\frac 1 6 \left(2\beta+\frac 4{\beta+1}-\frac 1{2\beta+1}\right)\log 2-\frac 1 6 \left(\frac 2{\beta +1}-\frac 1{2\beta+1}-1\right)\log c_\beta -4\zeta'_B(0;\beta+1,1,1)-2\zeta'_B(0;-2\beta-1,1,1)+2\zeta_R'(-1) -\log(\beta+1)-\frac 1 2 \log(-2\beta-1)-\frac{5}{2}\log 2-\log\pi.\end{aligned}
T ( r , \theta ) = r^{p_{0}}\frac{\sin [ \rho \theta ]}{\rho \sin \theta}\qquad , \qquad \rho^{2}= \frac{1}{1 - \frac{1}{2 k}}( p_{0}^{2}- 2 q p_{0}+ 4 k - \frac{1}{2 k}+ 1 )
\exp(4 \pi i (\eta_{l}+\eta_{m})_1(\eta_k)_2) \exp(4 \pi i (\eta_k)_1 (\eta_l+\eta_m)_2) = \exp(4 \pi i (\eta_{l}+\eta_{m})_1(\eta_k)_2)\exp(-4 \pi i (\eta_k)_1 (\eta_l+\eta_m)_2) = e_2(\eta_l+\eta_m,\eta_k) = e_2(\eta_l,\eta_k)e_2(\eta_m,\eta_k).
\begin{array}{ll}{\big |}H{\big |}^{t_0,\Delta}_{sup,1}:=\sup_{(t,x,v)\in [t_0,t_0+\Delta] \times{\mathbb R}^d\times{\mathbb R}^d}\left({\big |}H(t,x,v){\big |}+{\big |}\nabla_{(x,v)}H(t,x,v){\big |}\right),\end{array}
W(t,\xi) =\sum_{j=1}^N\mu_je_j(\xi)\beta_j(t), t\ge0, \ \xi\in \mathbb{R}^d, \mu(\xi) =\frac{1}{2}\sum_{j=1}^N|\mu_j|^2e_j^2(\xi), \xi \in \mathbb{R}^d.
\begin{cases}t_1 := \min \bigl\{t \in [0,\,1] \mid H_{j}^{\eta}(t) = m_{j}^{\eta}\bigr\}, begin{align*}1.5mm]t_0 := \max \bigl\{t \in [0,\,t_{1}] \mid H_{j}^{\eta}(t) = m_{j}^{\eta}+ 1 \bigr\};\end{cases}
\left(1-\frac{1-\varepsilon}2\left(\frac r{\sqrt n}\right)^2\right)^{n-1}=\left(1-\frac{1-\varepsilon}2\cdot\frac{r^2}n\right)^{\left(-\frac{2n}{(1-\varepsilon)r^2}\right)\cdot\left(-\frac{(1-\varepsilon)r^2}2\right)-1}&\to e^{-\frac{r^2}2\cdot(1-\varepsilon)}
\frac{1}{ps}\sum_{l=1}^{ps-1}\frac{\xi_{s}^{-ltr}}{(1-\xi_{ps}^{l\cdot rq})(1-\xi_{ps}^{l})}=\frac{1}{ps}\sum_{i=1}^{p-1}\frac{1}{(1-\xi_{p}^{irq})(1-\xi_{p}^{i})},
\sum_{k=0}^{\infty}\frac{u^k}{k!}V_k^r(x) =\frac1{2u}(e^{(2X+1)u}-e^{(2X-1)u}) \intertext{and}\sum_{k=0}^{\infty}u^{k+1}V_k^r(x) =\frac12\log \frac{1-(2X-1)u}{1-(2X+1)u}, \intertext{where $\displaystyle{k\sum_{n \le x}J_{k-1}^r(n)}$ is replaced by $X^k$ when $k\ge1$, and $X^0$ are assigned to be the value $0$.}
\begin{array}{l}\displaystyle [e_{n+1},e_1]=-e_3,[e_{n+1},e_{2}]=-e_2,[e_{n+1},e_i]=-e_i-\epsilon e_{i+2}-\sum_{k=i+3}^n{b_{k-i-2}e_k}, \displaystyle (\epsilon=0,\pm1,3\leq i\leq n,5\leq j\leq n);DS=[n+1,n-1,0],LS=[n+1,n-1,n-1,...].\end{array}
M(\lambda) \varphi = \Gamma_1 \gamma(\lambda) \varphi = \Gamma_1 \big( I_4 + \lambda (A_0 - \lambda)^{-1}\big) \gamma \varphi = M \varphi + \bigl(\lambda (A_0 - \lambda)^{-1}\gamma \varphi\bigr)|_\Sigma.
\hat{\Theta}_{\Delta}^{i}[ N ] = - \frac{2}{3 !}\frac{1}{i \hbar}\; N ( v ( \Delta ) ) \; \epsilon^{J K L}{\hat{\Phi}}^{E}( F_{J K}) \, t r \left( \tau^{i}\; h_{s_{L}( \Delta )}\left[ h_{s_{L}( \Delta )}^{- 1}, \; \sqrt{\hat{V}( v ( \Delta ) , \epsilon )}\right] \right)
\|K\tilde{D}^{-1}K^T-K_{\varepsilon}\tilde{D}^{-1}K_{\varepsilon}^T\| \leq \sum_{j=1}^k d_i^{-1}\frac{3\pi}{2\varepsilon^{1/4}\|M\|^{3/4}}(\|M^{1/2}\|+\|M_{\varepsilon}^{1/2}\|)\|M^{1/2}-M_{\varepsilon}^{1/2}\| +n^3\|\tilde{D}^{-1}\|^{2}\|M\|^{3/4}\varepsilon^{1/4}.
|K_0^{(j)}(x)|^\frac{\beta}{\beta-j}= \left| \frac{p_j(x)}{(1-x^2)^{2j}}\right|^{\frac{\beta}{\beta-j}}e^{-\frac{j}{(\beta-j)(1-x^2)}}K_0(x) \leq C'(\beta,j) K_0(x).
|\xi+i\frac{\Im (z)}{\cosh(2t)}-z| =|\xi+i\frac{\Im(z)}{\cosh(2t)}-\left[\Re(z)\left(\frac{1}{\cosh(2t)}-\left(\frac{1}{\cosh(2t)}-1\right)\right)+i\Im z|\right] & \le |\xi-\frac{\Re (z)}{\cosh(2t)}|+(|\Re(z)|+|\Im(z)|) \left(1-\frac{1}{\cosh(2t)}\right) &< \frac{\delta}{2}+2\times \frac{\delta}{4}=\delta.
(\nabla_X\xi_Y)_{\mathfrak{g}}=[\xi_X,\xi_Y]_{\mathfrak{g}}, &(\nabla_X\xi_Y)_{\mathfrak{m}}=\nabla '_X\xi_Y+[\xi_X,\xi_Y]_{\mathfrak{m}}, &R(X,Y)_{\mathfrak{g}}=R'(X,Y)+[\xi_X,\xi_Y]_{\mathfrak{g}}, &R(X,Y)_{\mathfrak{m}}=\nabla '_X\xi_Y-\nabla '_Y\xi_X+[\xi_X,\xi_Y]_{\mathfrak{m}}-\xi_{[X,Y]}.
V = 2D\left. Y_{N+2}\right|_{t_i = x\delta_{i,3}}+ 2D \left. Y_{N-2}\right|_{t_i = x\delta_{i,3}}- 4D \left. Y_{N}\right|_{t_i = x\delta_{i,3}}&= L^{(3)}(x,N+2) + L^{(3)}(x,N-2) - 2L^{(3)}(x,N).
{\mathbf D}_{l}=\frac{1}{P_{l}}\sum_{a}\frac{\partial}{\partial Q^{a}}P_{l}\frac{\partial}{\partial Q^{a}}= \sum_{a}\frac{\partial^{2}}{\partial \left(Q^{a}\right)^{2}}+ \sum_{a}\frac{\partial \ln P_{l}}{\partial Q^{a}}\frac{\partial}{\partial Q^{a}},
|D_k^\# H_j D_{k'}^\#(x,y)| &=\bigg|\int_{S(x,9(A_0)^22^j)}\int D_k^\#(x,u)[k_j(u,v)-k_j(x,v)][D_{k'}^\#(v,y)-D_{k'}^\#(u,y)]d\mu(u)d\mu(v)\bigg| &\lesssim \frac{2^{-|j-k'|\varepsilon}}{V_{-k'}(y)}\int_{S(x,9(A_0)^22^j)}\bigg(\int |D_k^\#(x,u)||k_j(u,v)-k_j(x,v)|d\mu(u)\bigg)d\mu(v)
\phi_{1}= \frac{\lambda}{m^{2}}{\bigg [}\left( \frac{1}{2 \sqrt{3}}+ \frac{3}{4 \pi}\right) \frac{1}{\cosh^{2}( m x / 2 )}- \frac{\sqrt{3}}{4}( m \partial_{m}+ 2 \lambda \partial_{\lambda}){\bigg ]}\phi_{\mathrm{k i n k}}\, .
\mathcal{G}_i:= \sum_{k\ge 1,~ 2\le k+l\le 4}\Theta_i \|\nabla^k g_i\|_{L^\infty(B_3)}^l+\sum_{k\ge 1,~ 6\le k+l\le 8}\|\nabla^k g_i\|_{L^\infty(B_3)}^l ,
\mathrm{T}_t=\boldsymbol{F}\mathbf{Y}^t_0\mathbf{T}_0 \boldsymbol{F}^\star+ \boldsymbol{F}\boldsymbol{\nabla}_t^\star \mathbf{Y}^t_\cdot\mathbf{L}\mathbf{T}_\cdot\boldsymbol{\nabla}_t\boldsymbol{F}^\star,
C_p \int |\nabla u_f - \nabla f|^2d\mu =C_p \int |\nabla u_f |^2d\mu +C_p \int | \nabla f|^2d\mu - 2 C_p \int \nabla u_f \cdot \nabla f d\mu &=C_p \int |\nabla u_f |^2d\mu +C_p \int | \nabla f|^2d\mu - 2 \int | f|^2 d\mu &\leq \epsilon^2 C_p \int | \nabla f|^2d\mu.
\varepsilon^+_u = i \eta_{1}\gamma^{1}\varepsilon^-_l,\ \ \ \ varepsilon^+_u = i \eta_{13}\gamma^{13}\varepsilon^+_l, \ \ \ \ varepsilon^+_u = - \eta_{p}\varepsilon^+_l, \ \ \ \ varepsilon^+_u = \eta_{3}\gamma^{3}\varepsilon^-_l ,
{}^b \mathcal{A}^\epsilon_G (M):= (\mathcal{A}(G)\hat{\otimes}\,{}^b \Psi^{-\infty,\epsilon}(S))^{K\times K}+(\mathcal{A}(G)\hat{\otimes}\,\Psi^{-\infty,\epsilon}(S))^{K\times K}
\frac{1}{2 \pi}\int d \psi k^{a}k^{b}R_{a b}^{0}= - ( k^{r})^{2}\frac{3 R_{1 1}^{2}}{3 2 r^{4}}\Delta^{4}\left( 1 +{\cal O}( \frac{r}{R_{1 1}}) \right) < 0 .
H^*(\mu)(\xi)= \lim_{z \to \xi}\left( \frac{1}{1-b(z)}-\frac{1}{b'(\xi)(1-\bar{\xi}z)}\right)+C = \lim_{z \to \xi}\frac{1}{(1-\bar{\xi}z)}\left( \frac{1-\bar{\xi}z}{1-b(z)}-\frac{1}{b'(z)}\right)+C = -\frac{b''(\xi)}{b'(\xi)^2}+C\ ,
(\frac{\phi^{n+1}_{h}-\phi^{n}_{h}}{\Delta t},v_{h}) = (\frac{\eta^{n+1}-\eta^{n}}{\Delta t},v_{h}) + \nu(\nabla e^{n+1}_{u},\nabla v_{h}) + b(u^{n+1}-u^{n},u^{n+1},v_{h}) + b(e^{n}_{u},u^{n+1},v_{h}) + b(u^{n}_{h},e^{n+1}_{u},v_{h}) - (p^{n+1}-q^{n+1}_{h},\nabla \cdot v_{h}) - (\frac{u^{n+1}-u^{n}}{\Delta t}-u^{n+1}_{t},v_{h}) \; \; \forall v_{h}\in V_{h}.
\begin{aligned}J_3\ge -C s^2 \lambda^2 \iint_{Q}\xi^3 \left|A \nabla \eta \cdot \nabla \eta \right|^2 u^2 dx dydt -C s^2 \lambda^2 \int_{0}^{T}\int_{\omega_0}\xi^3 \left|A \nabla \eta \cdot \nabla \eta \right|^2 u^2 dx dydt,\end{aligned}
\frac{1}{3v^2}\int_0^v\rho_0(v')v'^2dv' =\frac{1}{3v^2}\left\{\int_0^v\tfrac{1}{2}(y(v')+1)v'^2dv'+\int_0^v\left(\rho_0(v')-\tfrac{1}{2}(y(v')+1)\right)v'^2dv'\right\}&=\frac{1}{18v^2}\left\{v^3(y(v)+1)-\int_0^vv'^3\frac{dy}{dv}(v')dv'\right\}+\mathcal{O}(v^2),
{\cal L}( \varphi ) = \frac{1}{2}( \partial_{\mu}\varphi )^{2}- \frac{1}{2}m_{0}^{2}\varphi^{2}- \frac{\lambda_{0}}{4 ! N}\varphi^{4}- \frac{\eta_{0}}{6 ! N^{2}}\varphi^{6},
\delta v^{\ell}( \omega ) = - e \left( \Phi_{p}, \left( L^{\ell}{\frac{1}{\omega - ( H - E )}}L^{k}- L^{k}{\frac{1}{\omega + H - E}}L^{\ell}\right) \Phi_{p}\right){\frac{A_{k}^{e}( \omega )}{Z_{3}^{\, 2}}}~ ~ .
\begin{gathered}\int_0^\infty u^{2\alpha}C(u) J_\nu (u\rho; q^2) \,d_q u- \int_0^\infty \dfrac{w(u)}{1+w(u)}u^{2\alpha}C(u) J_\nu (u\rho; q^2) \,d_q u = f_3 (\rho), \rho\in B_{q,b}\end{gathered}
d_{\lambda \mu}(q) =q^{\delta(\lambda,\mu)}\sum_{\substack{\alpha^0,\dotsc,\alpha^e \beta^0,\dotsc, \beta^{e-1}}}\prod_{0\leq j \leq e-1}c^{\mu^{(j)}}_{\alpha^j \beta^j}c^{\lambda^{(j)}}_{\beta^j (\alpha^{j+1})'},
\bigg(\int_{\mathcal{B}^+_{r}(z)\cap \{y=0\}}w^{p}\,dx\bigg)^{\frac 1p}\leq \frac C{(R-r)^{\frac\sigma\kappa}}\|w\|_{L^1(\mathcal{B}^+_{R}(z),y^{1-2\alpha})}
\frac{2}{N^{2\alpha+1}}\int_0^\infty \d x \,x^{2\alpha}\left. P^{(\alpha)}_{2n}(\frac{x}{N})\right|_{n=Nt/2}\left. P^{(\alpha)}_{2m}(\frac{x}{N})\right|_{m=Nt'/2}= h^{(\alpha)}_{2n}\,\frac2N \delta (t-t')
\Phi_{N,k}\left( \theta\right) =2^{N}\varphi\left( 2^{N}\theta_{1}-k_{1}\right) \varphi\left( 2^{N}\theta_{2}-k_{2}\right) E_{N}=\mathrm{Span}\left\{\Phi_{N,k}\left( \theta\right)\ |\k=\left( k_{1},k_{2}\right) ,\ 0\leq k_{1},k_{2}\leq2^{N}-1\right\}
\tilde{G}^-_{1,\varepsilon}(x)={I}+\sum\limits_{r=1}^{\infty}\varepsilon^r \tilde{N}_{r,\varphi}^{-}(x) (\Lambda^{+}_{0}(x))^{-1}, G^+_{1,\varepsilon}(x)={I}+\sum\limits_{r=1}^{\infty}\varepsilon^r \tilde{N}_{r,\varphi}^{+}(x).
J(Y_m^{M,N})-J(Y_{m-1}^{M,N})=\big<A Y_m^{M,N}, \overline{\partial}Y_m^{M,N}\big>+\big<F(Y_m^{M,N}), \overline{\partial}Y_m^{M,N}\big>+\big<Y_m^{M,N}-Y_{m-1}^{M,N},\overline{\partial}Y_m^{M,N}\big>.
\mathrm{Pr}\{X_{1}+X_{2}+\dots+X_{k}=n\}=\underset{m_j\in\mathbb{N}}{\underset{m_{1}+m_{2}+\dots+m_{k}=n}{\sum}}\prod_{j=1}^{k}\mathrm{Pr}\{X_{j}=m_j\}&=\underset{m_j\in\mathbb{N}}{\underset{m_{1}+m_{2}+\dots+m_{k}=n}{\sum}}\left(\frac{(1-\rho)^{r}}{1-(1-\rho)^{r}}\right)^{k}\prod_{j=1}^{k}\rho^{m_{j}}\binom{r+m_{j}-1}{m_{j}},
\frac{J^w(\textbf{X}_{maxRSSU}^{(n+1)})}{J^w(\textbf{X}_{maxRSSU}^{(n)})}=-2J^w(X_{n+1:n+1}) &=\int_{0}^{+\infty}w(x) f^2_{1:n+1}(x)dx &= (n+1)^2 \int_{0}^{1}w(F^{-1}(u)) u^{2n}f(F^{-1}(u))du & \geq \frac{(n+1)^2}{2n+1}& \geq 1.
K^{\Omega}( t ) = \frac{1}{\left| \Omega \right|}\sum_{x , y \in \Omega}\delta_{x^{y}, x^{- 1}}\prod_{\xi \in \mathcal{O}_{-}( x , y )}K \left( \frac{\lambda_{\xi}^{2}t}{\left| \xi \right|}\right) \prod_{\xi \in \mathcal{O}_{+}( x , y )}Z \left( \frac{\left| \xi \right|}{2 \lambda_{\xi}^{2}i t}+ \frac{\kappa_{\xi}}{\lambda_{\xi}}\right)
\mathbb{P}_\eta\left(\mathcal{G},\mathcal{T}_t(0) \leq \frac{1-p}{4}\delta^d t\right) \leq |D| d e^{-\frac{c(1-p)\delta^{d-1}}{8}t}= (\lfloor 2d\alpha t \rfloor+1)^d d e^{-\frac{c(1-p)\delta^{d-1}}{8}t}
t^{(\pm)}_{\mu\nu}(p,k) \;=\; \sum_{n=-\infty}^{+\infty}\,{\rm tr}\left[\frac{1}{i \not \! p + \omega_n^{(\pm)}}\gamma_\mu \frac{1}{i (\not \! p + \not \! k) + \omega_n^{(\pm)}}\gamma_\nu \right]
y_i =\frac{1}{1+\theta_i}Ty_i+\frac{\theta_i}{1+\theta_i}z =\frac{1}{1+\theta_i}(I-U)(y_i)+\frac{\theta_i}{1+\theta_i}z =\frac{1}{1+\theta_i}y_i-\frac{1}{1+\theta_i}Uy_i+\frac{\theta_i}{1+\theta_i}z,
d(w_{N+1}) = \frac{\#\{1\leq i\leq 2^{N+1}: \lambda_{i}=0\}}{2^{N+1}}&= \frac{2\#\{1\leq i\leq 2^{N}: \lambda_i=0\}- 1}{2^{N+1}}&=d(w_N)-\frac{1}{2^{N+1}}&=-\sum_{i=1}^{N+1}\Big(\frac{-1}{2}\Big)^i
\P( Z_n^* \leq -a) = \P( \forall k\leq n \, :\, Z_k + f(k) \leq -a + f(k) ) & \geq \P( \forall k\leq n \, :\, Z_k + f(k) \leq a ) & \geq \P( Z_n^* \leq a ) \, \exp( - \sqrt{2 \|f\|_{\mathcal H}^2 \log ( 1/\P(Z_n^* \leq a ))}- \|f\|_{\mathcal H}^2/2)\, .
\lim_{\beta \to 0}\sum_{n = 0}\frac{e^{- \beta ( n + 1 / 2 )}}{n !}H_{n}( \xi_{1}x_{1}) H_{n}( \xi_{2}x_{2}) \exp \Bigl [ - \frac{\xi_{1}^{2}x_{1}^{2}+ \xi_{2}^{2}x_{2}^{2}}{4}\Bigr ] = \sqrt{\frac{\pi}{2}}\delta \Bigl ( \frac{\xi_{1}}{2}x_{1}- \frac{\xi_{2}}{2}x_{2}\Bigr ) .
\sum_{n\ge0}\{\begin{matrix}2n\\j\end{matrix}\}\frac{log(q)^{n}}{n!}=\int_{0}^{\infty}\frac{e^{-t^{2}/2}}{\sqrt{2\pi}\cdot j!}[\sum_{b=\pm1}(e^{b\sqrt{2log(q)}\cdot t}-1)^{j}]dt,
[\textstyle{\frac{1}{2}}s^{a}_{~b}t_{1}^{b}{}_{a},\textstyle{\frac{1}{2}}s^{c}_{~d}t_{2}^{d}{}_{c}]=\textstyle{\frac{1}{2}}s^{a}_{~b}[t_{1},t_{2}]^{b}_{~a}\,.
\psi_{\beta,\mu}(T_{b^t})=\begin{cases}(1-e^{-\beta}d)\big(M_t(\mu)+\sum_{k=1}^\infty e^{-\beta k}d^kM_{c^kd^{-k}t}(\mu)\big) (1-e^{-\beta}d)M_t(\mu) \end{cases}
\Pi^q_{M_1}\begin{pmatrix}x y z\end{pmatrix}= \begin{pmatrix}x y 0\end{pmatrix}, \Pi^q_{M_2}\begin{pmatrix}x y z\end{pmatrix}= \begin{pmatrix}x 0 z\end{pmatrix}\qquad\Pi^q_{M_1\cap M_2}\begin{pmatrix}x y z\end{pmatrix}= \begin{pmatrix}x 0 0\end{pmatrix}.
Ad_{\Theta_n(\phi)^{-1}}(d\Theta_n(e_{\phi(\lambda)}))\stackrel{(\ref{eq-differential-of-Theta-n-sends-e-lambda-to})}{=}\Theta_n(\phi)^{-1}e_{\theta(\phi(\lambda))}\Theta_n(\phi)=\frac{t(\lambda,\lambda)}{2}e_{\theta(\lambda)}^\vee.
T_{P}\left( f_{m}\right)=\sum_{j=-1}^{m}\sum_{s_{0}=0}^{2k-1}\sum_{s_{1}=0}^{2k-1}\dots\sum_{s_{j}=0}^{2k-1}T(P_{s_{j}}T(\cdots P_{s_{1}}T(P_{s_{0}}T(f_{m}))\cdots))
{}^{*}( d x^{i_{1}}\cdots d x^{i_{p}}) = \sum_{j_{p + 1}< \ldots < j_{D}}g^{i_{1}j_{1}}\ldots g^{i_{p}j_{p}}\epsilon_{j_{1}\ldots j_{p}j_{p + 1}\ldots j_{D}}d x^{j_{p + 1}}\cdots d x^{j_{D}},
J_{11}= \mathcal{O}\left( a^{2}\; \left\Vert V \right\Vert_{\mathbb{L}^{2}(D_{j_{0}})}\; d^{-4}\; \max_{j}\left\Vert \overset{3}{\mathbb{P}}\left(E^{T}_j\right) \right\Vert_{\mathbb{L}^{2}(D_j)}\right)\overset{(\ref{max-P1P3})}{=}\mathcal{O}\left( \left\Vert V \right\Vert_{\mathbb{L}^{2}(D_{j_{0}})}\; d^{-4}\; a^{\frac{10+h}{2}}\right).
H(X_j | X_{[1,n] \backslash j}) = H(W_j | S_{[1,j-1] \rightarrow j}W_{[j+1,n]}) &= H(W_j | S_{[1,n] \backslash j \rightarrow j}W_{[j+1,n]}) &= H(W_j | S_{[1,n] \backslash j \rightarrow j}) - C^{(3)}_j begin{align*}1ex]C^{(3)}_j = I(W_j; W_{[j+1,n]}| S_{[1,n] \backslash j}) \geq 0
W=W^{{\rm cl}}\frac{H}{H_{0}}\frac{3n\left( 1-\beta^{2}\right)}{10}\left( \frac{1+\zeta}{2}\frac{n+\mu -1}{n+\mu}+\frac{1-\zeta}{2}\right) \left( \frac{5}{6}l_{2}^{2}+\frac{1}{6}l_{3}^{2}\right) \;.
\langle J,\gamma\vert J,\gamma\rangle = \mathcal{N}(J)^{-1}\sum_{n =0}^{+\infty}\frac{J^n}{\rho_n}\equiv 1, \textrm{then}\mathcal{N}(J) \equiv \sum_{n =0}^{+\infty}\frac{J^n}{\rho_n}\,.
\mathbb{P}(L(\sigma) \ge x) = \mathbb{P}\left(\bigcup_{z \ge x}E_z\right) \le \sum_{z \ge x}\mathbb{P}(E_z) < n \max_{z \ge x}\mathbb{P}(E_z) \le(1 + o(1)) e^{-x(\ln x - \ln r - 2) + \ln n},
\lim_{q\to 0}[\theta_{k_1,k_2}(q,u,q^{-2a-1}v)]_{(k_1,k_2)\in H}=[\theta_{k_1,a}u^a,\theta_{k_1,a+1}u^{a+1}]_{k_1}&=[\underset{k_2=a}{\underbrace{T^*_{a\tau_{12}}\vartheta_{k_1}}},\underset{k_2=a+1}{\underbrace{(T^*_{(a+1)\tau_{12}}\vartheta_{k_1})v}}]_{k_1}
\sum_{i=1}^{m}[a_i] \left( \prod_{j \not= i}{{[x_i-x_j+a_j] \over [x_{i}-x_{j}]}}\right){{[x_i+y-b] \over [x_i+y]}}=[b] \prod_{1 \le i \le m}{{[y+x_{i}-a_{i}] \over [y+x_i]}}.
\kappa_{+}\neq 0,\kappa_{-}\neq 0, &\exists i\in \left\{0,1\right\},\exists (\epsilon_{\tau},\epsilon_{\alpha},\epsilon_{\beta})\in \{-1,1\}\times \{-1,1\}\times \{-1,1\}:Y^{(i,2\mathsf{N})}(\epsilon_{\tau}\tau_{\pm},,\epsilon_{\alpha}\alpha_{\pm},\epsilon_{\beta}\beta_{\pm})=0
\zeta_T (t)\chi_{N}(t) =\zeta_T (0)\chi_{N}(t) +\chi_{N}(t)\int_{0}^{t}L_T\bigl(\xi_T(s) \bigr)\chi_{N}(s)\,ds &+\chi_{N}(t)\int_{0}^{t}G_T'\bigl(\xi_T(s) \bigr)\chi_{N}(s)\,dW_T(s).
\gamma= c\mathcal E(\rho_0,u_0|r(0,\cdot),U(0,\cdot))+c(D,T)(\varepsilon^\alpha+R^{-1}+(\varepsilon R)^{-1}+\nu+\varepsilon^2\nu^2(1+R^{-2})+\varepsilon\nu^{-1}), \delta(\tau)= c(D,T)\Bigl(\frac{1}{|\Omega_R|}\int_{\Omega_R}p(\rho)+\varepsilon^\frac{4}{3}\nu^{-1}(1+R^{-4})+(1+\varepsilon^2)(1+R^{-2})+\varepsilon^2\nu+1\Bigr).
U\left(\sum_{j\ge 1}d_{2\alpha-1}(j)X^j\right) =\sum_{j\ge 1}d_{2\alpha-1}(j)U(X^j)=\sum_{j\ge 1}d_{2\alpha-1}(j) \sum_{\ell\ge 1}a(j,\ell)X^\ell &=\sum_{\ell\ge 1}\left(\sum_{j\ge 1}a(j,\ell) d_{2\alpha-1}(j)\right)X^\ell=\sum_{\ell\ge 1}d_{2\alpha}(\ell)X^\ell.
\Delta t \varphi_1(\Delta t L)g_n \approx \Delta t \varphi_1(\Delta t V_mH_mV_m^T)g_n = \Delta t V_m\varphi_1(\Delta t H_m)V_m^Tg_n = ||g_n||\Delta t V_m\varphi_1(\Delta t H_m)e_1.
\begin{aligned}\|a(z)-\hat{a}\|_{H_z^1(\mathbb{R})}\leq \sup_{x\in \mathbb{R}}\| u_x \|_{L^2_x}^2\sup_{x\in \mathbb{R}}\| \Psi^-_{11}-e^{-ic_-}\|_{H_z^1}+\|\bar{u}_x \|_{L^1_x}\sup_{x\in \mathbb{R}}\| \Psi^-_{21}\|_{H_z^1},\end{aligned}
G_{a b , c d}^{( k )}( x_{1}, x_{2}; x_{3}, x_{4}) = e^{i k \theta}\frac{1}{Z [ 0 , 0 , 0 ]}\left. \frac{\delta^{4}Z^{( k )}[ \eta , \overline{{{\eta}}}, J ]}{\delta \overline{{{\eta}}}_{a}( x_{1}) \delta \overline{{{\eta}}}_{b}( x_{2}) \delta \eta_{c}( x_{3}) \delta \eta_{d}( x_{4})}\right|_{\eta , \overline{{{\eta}}}, J = 0}\; .
m^{2}= 2 T \left( 1 + \cosh \frac{\mu}{T}\right) \left\{T_{c}\ln \left[ 2 \left( 1 + \cosh \frac{\mu}{T_{c}}\right) \right] - T \ln \left[ 2 \left( 1 + \cosh \frac{\mu}{T}\right) \right] \right\}.
F_z(z(\alpha,t),t)=\frac{\partial_{\alpha}F(z(\alpha,t),t)}{z_{\alpha}}=\frac{\bar{z}_{t\alpha}}{z_{\alpha}}-\sum_{j=1}^N\frac{\lambda_j i}{2\pi}\frac{1}{(z(\alpha,t)-z_j(t))^2}.