harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\partial_{\theta}\left( \mathfrak{p}_{+}g_{1}f-\mathfrak{p}_{+}g_{2}f\right) \left( re^{i\theta}\right) & =i\sum_{m\geq1,n-m\geq0}\left( n-m\right) r^{n-m}\left(g_{1,n}-g_{2,n}\right) f_{m}e^{i\left( n-m\right) \theta}& =i\sum_{m\geq1}f_{m}\sum_{k\geq0}kr^{k}e^{ik\theta}\left(g_{1,k+m}-g_{2,k+m}\right)
F_i(\phi_i) = \int_M (\phi_i -\phi_0)d\mu_i = \int_M(\bar{\phi}- \phi_0)d\mu + \int_M(\phi_i - \bar{\phi})d\mu_i + \int_M(\bar{\phi}- \phi_0) (d\mu_i - d\mu) = F(\bar{\phi}) + \int_M(\phi_i - \bar{\phi})d\mu_i + \int_M(\bar{\phi}- \phi_0) (d\mu_i - d\mu).
\Lambda^{(4)}_{n,\delta}=\exp \left( i\mu n^{\alpha+\frac 12}\int_0^{t-\delta}(t-s)^\alpha \xi^n_s dW_s +i \lambda n^{\alpha+\frac 12}\int_{t- \delta}^{\eta_n(t)}\psi_{n,1}(s,\eta_n(t)) dW_s \right).
\frac{2v}{\alpha}\leq \begin{cases}\frac{2\sqrt{2}\,v}{\log^{1/2}_+\log^{1/2}_+ v}\mbox{if}v \geq u \log^{1/2}_+\v, 2\,{u \log_+^{1/2}v}\mbox{if}v < u \log^{1/2}_+ v.\end{cases}
\widetilde{E}(n,\mathcal{M}_1,\mathcal{M}_2)=\frac{HH'n^{-\frac{1}{2}}}{\Lambda}\sum_{b\asymp \mathcal{M}_1+\mathcal{M}_2}\sum_{\substack{m_1+m_2=b m_1\asymp\mathcal{M}_1,\m_2\asymp\mathcal{M}_2}}\lambda_{1}(m_1)m_1^{-\frac{1}{4}}\lambda_{2}(m_2)m_2^{-\frac{1}{4}}\sum_{c}\frac{S(b,2n;c)}{c}\Phi\left(\frac{4\pi\sqrt{2nb}}{c}\right),
\left(\begin{array}{l}T_{\alpha{\dot{\alpha}}}begin{align*}3mm]W_\alpha begin{align*}3mm]\tau\end{array}\right)\to\left(\begin{array}{rl}T^{new}_{\alpha{\dot{\alpha}}}={\widetilde T}_{\alpha{\dot{\alpha}}}-\frac{1}{3}[D_\alpha,\bar D_{\dot{\alpha}}]U begin{align*}3mm]W^{new}_\alpha = \frac{1}{2}{\bar D}^2D_\alpha\U begin{align*}3mm] 0\end{array}\right)\ ,
k_1 = \varphi_1 - \frac 13 \sum_n \varphi_n~, ~~~~~ k_2 = \varphi_2 - \frac 13 \sum_n \varphi_n~, ~~~~~ -(k_1 + k_2) = \varphi_3 - \frac 13 \sum_n \varphi_n~.
\chi_Q(x)=1 \operatorname{dist}(x,\partial Q)>\alpha\varepsilon \chi_Q(x)=0 \operatorname{dist}(x,\partial Q)<\frac{\alpha}{2}\varepsilon, \psi_Q(x)=1 \operatorname{dist}(x,\partial Q)>\frac{\alpha}{2}\varepsilon \psi_Q(x)=0 \operatorname{dist}(x,\partial Q)<\frac{\alpha}{4}\varepsilon,
\left(t^{a_1}\frac{\partial}{\partial t}t^{a_2}\right)^{\alpha}\widehat{h}_{\alpha}(\beta,t)= -\beta^2 \widehat{h}_{\alpha}(\beta,t)+\frac{t^{-\alpha(1-a_1-a_2)-a_2}}{(1-a_1-a_2)^{-\alpha}}\frac{1}{\Gamma(1-\alpha)},
[T_{N-1}(t)-T_{M-1}(t)]F_N(x_1,t)F_M(x_2,t) = (t^N/N!)(t^M/M!)e^{(x_1+x_2)t}\frac{A-B}{(e^t-A)(e^{t}-B)}& = \frac{t^{M+N}}{M!N!}e^{(x_1+x_2)t}\left(\frac{1}{e^t-A}-\frac{1}{e^{t}-B}\right) & =\frac{t^M}{M!}F_N(x,t)-\frac{t^N}{N!}F_M(x,t)
\big( S^{\wedge, (l)}\:V\: S^{\wedge, (r)}\big)(x,y) &= \sum_{n=0}^\infty \frac{1}{n!}\int_0^1 \alpha^{l}\:(1-\alpha)^r\:(\alpha - \alpha^2)^n \: (\Box^n V) \big|_{\alpha y + (1-\alpha) x}\:d\alpha\;S^{\wedge, (n+l+r+1)}(x,y) \:.
A ( p ) - 1 = \frac{- e^{2}\xi}{4 \pi^{2}p^{2}}\int_{0}^{\infty}d q \, \frac{1}{A ( q )}\left( \frac{p^{2}A ( p ) - q^{2}A ( q )}{p^{2}- q^{2}}- \frac{p^{2}A ( p ) + q^{2}A ( q )}{2 p q}\ln \left| \frac{p + q}{p - q}\right| \right) ,
\left| u_{\beta}( \phi \psi_{\beta}) \right| \leq C \sup_{\stackrel{{| p | \leq m + n}}{z \in K}}\left| D^{p}( \phi \psi_{\beta}) ( z ) \right| \, \, , \quad \phi \psi_{\beta}\in G_{0}^{\infty}( \widetilde{X}_{\beta}) \, \,
E_s:=\Big\{n\in \mathbb N: 0\leq n<p^{k+\ell},\ n=\lfloor a_{k+\ell-1}\dots a_0\rfloor_p a_{k-m-1}=p-1 a_{k-m-2}\neq 0 a_{k-m-i}=0\forall \, 3\leq i \leq \ell a_{k-m-\ell -1}< b_{k-m-\ell -1}a_j \ge b_j \forall\, j \in \{0, \dots, \, k-m-\ell -2\}\cup \{k-m, \dots, k \}\Big\}.
D_i.D_j = \begin{cases}1 \textnormal{if $v_i$ and $v_j$ are next to each other on an edge.}l^*(m_{ij}^*)+1 \textnormal{if $v_i$ and $v_j$ are vertices that are connected}\textnormal{by an edge $m_{ij}$ whose dual is $m_{ij}^*$.}0 \textnormal{otherwise}. \end{cases}
\chi^{\epsilon,L}_t(\lambda,\mu,m^2) := \max_{x\in\Lambda_{\epsilon,L}}\epsilon^d\sum_{y\in\Lambda_{\epsilon,L}}\big<\varphi_x\varphi_y\big>^{\epsilon,L}_{\lambda,\mu+1/t,m^2}= \epsilon^d\sum_{x\in\Lambda_{\epsilon,L}}\big<\varphi_0\varphi_x\big>^{\epsilon,L}_{\lambda,\mu+1/t,m^2},
R_{bc}^{\bar b' c'}( \beta_2) R_{a c'}^{\bar a' d}(\beta_1) S_{\bar b' \bar a'}^{\bar b \bar a}(\beta_1 - \beta_2) = S_{ab}^{a' b'}(\beta_1 - \beta_2) R_{a' c}^{\bar a c'}(\beta_1) R_{b' c'}^{\bar b d}( \beta_2).
\sum_{n=1}^{\infty}e^{-\beta k E_n}=\frac{1}{\sqrt{\pi}}\int_0^{\infty}dx \; e^{- \frac{x^2}{4}-\frac{\beta^2 k^2m^2}{x^2}}\frac{1}{e^{\frac{2 \beta^2 eB k^2}{x^2}}-1}
\underset{t\rightarrow \infty}{\lim}\dfrac{\mathbb{U}_{\mathbf{G}}\left(tx\right) /\mathbb{U}_{\mathbf{G}}\left( t\right) -x^{\gamma_{2}}}{\mathbf{A}_{\mathbf{G}}\left( t\right)}=x^{\gamma_{2}}\dfrac{x^{\tau_{2}}-1}{\tau_{2}},
{\cal I}_{n}( \kappa ) ={\frac{n}{( n + 1 ) \kappa}}+ \sum_{k = 0}^{K}{\frac{\Gamma ( k - 1 / n )}{\Gamma ( - 1 / n ) \Gamma ( k + 1 )}}I_{k}\, \kappa^{k}+{\cal O}( \kappa^{K + 1})
c b_\alpha^{-1}N_{\theta,\alpha}^{1-\alpha}-2\sum\limits_{i=1}^{d}c_i \Big[ \sum\limits_{j=1}^d \sigma^{ij}\mu_j\Big]+ \sum\limits_{i=1}^d\sum\limits_{j=i}^d c_{ij}\sigma^{ij}=0.
I_1 :{\mathcal C}^\infty({{\mathcal G}}[{\mathcal X}])[1]\to{\mathcal C}^\infty({{\mathcal G}})[1], \ \ \I_1= \sum_{i_1, i_2 \in S}(s^* \chi_{i_1})(t^* \chi_{i_2})\, \hat{\sigma}_{i_1, i_2}^*
S_{s}^{( m )}( \theta ) = \frac{\sinh \theta + i \cos \frac{\pi m p}{2}}{\sinh \theta - i \cos \frac{\pi m p}{2}}\prod_{l = 1}^{m - 1}\frac{\sin^{2}\left( \frac{m - 2 l}{2}\pi p - \frac{p}{4}+ i \frac{\theta}{2}\right)}{\sin^{2}\left( \frac{m - 2 l}{2}\pi p - \frac{p}{4}- i \frac{\theta}{2}\right)}\, .
\mathbb{P}_{\Lambda,\rho,\beta,H}^{|\vec{h}|}(\omega_e=1|\omega_{G_t}=\omega^{\rho}_{G_t})) \leq \mathbb{P}_{\Lambda,w,\beta,H}^{|\vec{h}|}(\omega_e=1|\omega_{G_t}=\omega^{\rho}_{G_t}) &\leq \mathbb{P}_{\Lambda,w,\beta,H}^{|\vec{h}|}(\omega_e=1|\omega_{G_t}=\omega^{w}_{G_t}).
{\cal H}_{B_{n}}={\frac{1}{2}}p^{2}-{\frac{g^{2}}{2}}\sum_{\alpha \in \Delta}x ( \alpha \cdot q ) x ( - \alpha \cdot q ) -{g_{s}^{2}}\sum_{\lambda \in \Delta_{s}}x ( \lambda \cdot q ) x ( - \lambda \cdot q ) .
\limsup_{s\to\infty}\mathbb P(Z_t\xrightarrow{t\to\infty}0\mid\sigma(Z_r;r\leq s)) \geq \tfrac1{1+\lambda}\limsup_{s\to\infty}(1-p)^{Z_s}= \tfrac1{1+\lambda}(1-p)^{\liminf\limits_{s\to\infty}Z_s}.
P=\begin{bmatrix}1 \frac12(q-2)q^{2m-1}\frac12q^{2m}q(q^{2m-2}-1) q-2 1 \frac12(q-2)q^{m-1}\frac12q^{m}-(q-1)(q^{m-1}+1) q-2 1 -\frac12(q-2)q^{m-1}-\frac12q^{m}-(q-1)(q^{m-1}-1) q-2 1 \frac12q^m -\frac12q^m 0 -1 1 -\frac12q^m \frac12q^m 0 -1\end{bmatrix},
(M_r{V})_S = & \sum_{a,b}\left [\binom{a}{2}+\binom{b}{2}\right ]\left [\binom{|S|-1}{a-1}\binom{r-|S|-3}{b}+\binom{|S|-1}{a}\binom{r-|S|-3}{b-1}\right . &\left .\qquad-\binom{|S|-1}{a-1}\binom{r-|S|-3}{b}-\binom{|S|-1}{a}\binom{r-|S|-3}{b-1}\right ]=0.
\det(\xi^{\alpha,\mu\pm i\nu})=\left({\nu}^{2}+{\mu}^{2}\right) \left({\mu}^{2}-2\,\alpha\,\mu+{\alpha}^{2}+{\nu}^{2}\right)=(\nu^{2}+{\mu}^{2})(\lambda-\alpha)(\bar\lambda-\alpha).
n(R + \delta_n) \geq H(W) &\geq I(X^n;W) &= \sum_{i=1}^n H(X_i) -H(X_i|W,X^{i-1}) &\overset{(a)}{=}\sum_{i=1}^n H(X_i) -H(X_i|W,X^{i-1},Y^{i-1},\hat{X}^{i-1}) &\geq \sum_{i=1}^n I(X_i;U_i),
\begin{array}{rll}\left|M^{\perp}\right| =\left|K\right|^{\sum_{i=0}^{r-1}\left(r-i\right)k_{i}\left(M^{\perp}\right)}=\left|K\right|^{rn-\sum_{i=0}^{r-1}\left(r-i\right)k_{i}\left(M\right)}&=\dfrac{\left|K\right|^{rn}}{\left|K\right|^{\sum_{i=0}^{r-1}\left(r-i\right)k_{i}\left(M\right)}}=\dfrac{\left|\mathbb{Z}_{p^{r}}^{n}\right|}{\left|M\right|}.\end{array}
c_\lambda^{+}({\bf p})|0> = 0\;,~~|f> = \int{\tilde f}^{(\lambda)}({\bf p)}c_\lambda^{+}({\bf p)}|0>\;,~~{\tilde f}^{(\lambda)}({\bf p)}= \left.\int\frac{d{\bf p}}{\left(2\pi\right)^{3/2}}e^{-i{\bf px}}e^{(\lambda)}_k({\bf p})f^{k\perp}(x)\right|_{x^0= 0}\;,
W_{k+1}^D=W_{k}^D*\boldsymbol{\mathcal{A}}-\boldsymbol{\alpha}_{k}\times W_{k}^D-\boldsymbol{\beta}_{k}\times W_{k-1}^D, V_{k+1}\times \boldsymbol{\beta}_{k+1}=\boldsymbol{\mathcal{A}}*V_{k}- V_{k}\times \boldsymbol{\alpha}_{k}-V_{k-1},
\begin{array}{c c c}{{\zeta^{a b}\bar{\zeta}_{b c}= \delta_{~ c}^{a}\, , ~}}&{{~ \zeta^{a b}= \zeta^{b a}\, , ~}}&{{~ \bar{\zeta}_{a b}= ( \zeta^{a b})^{\ast}\, .}}\\ \end{array}
x = \left(1+u_0(t)\right) \tilde x \left(1 + \sum_{\substack{i \in K_{N_0}\beta \le i < N_0}}c_i(t) \tilde x^i + O\left(\tilde x^{N_0}\right)\right) \mbox{as}\, \tilde x \searrow 0 \, \mbox{and}\, t > 0,
\left\{e^{\frac{1}{24}A_3}\widehat{L}(TM)[-A_3+2{\rm ch}(\widetilde{T_CM})-8+{\rm ch}(W_i)]\right\}^{(12)}&=\left\{e^{\frac{1}{24}A_3}\widehat{A}(TM){\rm ch}[2116\widetilde{T_CM}+4\wedge^2\widetilde{T_CM}+4W_i-15872]\right\}^{(12)}&-\frac{2}{15}\left\{c_2(W_i)e^{\frac{1}{24}A_3}\widehat{A}(TX)\right\}^{(12)}.
\rho^{0}\partial_{t}u^{0}+\rho^{0}u^{0}\cdot\nabla u^{0}+\nabla p^{0}=0,{\rm in}\ \Omega, &\partial_{t}\rho^{0}+u^{0}\cdot\nabla \rho^{0}=0,{\rm in}\ \Omega, &\nabla\cdot u^{0}=0,{\rm in}\ \Omega, &u^{0}(0,x)=u_0(x),\rho^{0}(0,x)=\rho_0(x),{\rm in}\ \Omega,
\frac{d^{n-1}I}{dv^{n-1}}(v)=\int_0^v\frac{\partial^{n-1}f}{\partial u^{n-1}}(v,v')dv'+\sum_{l=0}^{n-2}\binom{n-1}{l+1}\frac{\partial^{n-2}f}{\partial u^{n-2-l}\partial v^l}(v,v).
\widetilde{\xi}_a^{}=\frac{\tau_2}{8\pi^2}\sum_{\hat{\gamma}\in \Lambda^{+}\cup\{0\}}n_{\hat{\gamma}}q_a\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau+ n|^2}\frac{1+t_{+}^{m, n}t}{t-t_{+}^{m,n}}\,.
\ell_{i,j}^{2}= (s_{j}\ldots s_{i})(s_{i}\ldots s_{j})\ell_{i,j-1}^{2}= (s_{i}\ldots s_{j})(s_{j}\ldots s_{i})\ell_{i+1,j}^{2}\ell_{i,j}^{2}= \ell_{i,j-1}^{2}(s_{j}\ldots s_{i})(s_{i}\ldots s_{j})= \ell_{i+1,j}^{2}(s_{i}\ldots s_{j})(s_{j}\ldots s_{i}).
g_{k, \ell}:= \begin{cases}\max \left(e^{\delta_{n}q_{n}},|\ell|, 1\right) e^{-\left(\beta_{n}-6 \varepsilon\right) q_{n}}\beta_{n}\geq \delta_{n}+200 \varepsilon e^{2 \varepsilon k}\beta_{n}<\delta_{n}+200 \varepsilon\end{cases}.
\Psi_\mathcal{A}\left(\gamma_{(x,g)}\right) =\left[(x,g),\mathcal{A}(x,g)\!\cdot\!\left(id_{T_xX},b_{(x,g)}\right)\right]_G &=\left[(x,g),\operatorname{Ad}_{g^{-1}}\left(\overline{\mathcal{A}}(x)+TR_{g^{-1}}\!\cdot\!b_{(x,g)}\right)\right]_G &=\left[(x,e),\overline{\mathcal{A}}(x)+TR_{g^{-1}}\!\cdot\!b_{(x,g)}\right]_G.
\mathcal L_{p\mu}(\delta p,\delta \mu,\delta \b u):=\mathcal L_{p\mu\rho}(\delta p,\delta\mu,0,\delta \b u) = \begin{pmatrix}\delta p + 2\nabla\cdot(\delta\mu\;\varepsilon(\b u)) + 2\nabla\cdot(\mu\;\varepsilon(\delta\b u))-\rho(\delta\b u)_{tt}\delta \b u\end{pmatrix}.
q^{2}\xi_{0 2}( p , q ) - p . q \xi_{1 1}( p , q ) ={\frac{1}{2}}\left\{- \left[{\frac{1}{2}}+ m^{2}\xi_{0 0}( p , q ) \right] +{\frac{p^{2}}{2}}\xi_{1 0}( p , q ) +{\frac{3 q^{2}}{2}}\xi_{0 1}( p , q ) \right\}
n^{m}(y)= \begin{cases}\displaystyle 2\left (\frac{m}{2\pi}\right)^{\frac{d+1}{2}}\frac{K_{\frac{d+1}{2}}(m|y|)}{|y|^{\frac{d+1}{2}}}, m>0, \displaystyle \frac{\Gamma (\tfrac{d+1}{2})}{\pi^{\frac{d+1}{2}}}\frac{1}{|y|^{d+1}}, m=0,\end{cases}
\mbox{Span}\Bigl \{\begin{pmatrix}1 0 0 0\end{pmatrix}e^{2\pi in \cdot x}, \frac{1}{\sqrt2}\begin{pmatrix}1 0 0 1\end{pmatrix}e^{2\pi in \cdot x},\begin{pmatrix}0 0 0 1\end{pmatrix}e^{2\pi in \cdot x}\Bigr \}_{|n| \leq k}
\begin{array}[c]{ll}K_{2}(s) &= (1-p(s)\sigma_{z}(s))^{-1}\left\{p(s)\sigma_{y}(s)+2\left[\sigma_{x}(s)+\sigma_{y}(s)p(s)+\sigma_{z}(s)K_{1}(s)\right] \right\}P(s) &\ \ +(1-p(s)\sigma_{z}(s))^{-1}\left\{Q(s)+p(s)\left( 1,p(s),K_{1}(s)\right)D^{2}\sigma(s)\left( 1,p(s),K_{1}(s)\right)^{\intercal}\right\},\end{array}
F(p,m,\varepsilon )=i\int \frac{d^{4}k}{(k^{2}_{0}-|\mathbf{k}|^{2}-m^{2}+i\varepsilon )[(k+p)^{2}_{0}-|\mathbf{k}+\mathbf{p}|^{2}-m^{2}+i\varepsilon ]}
I_n =\begin{cases}O(n^{1-2d-\rho}) (1-2d<\rho<1-d), O(n^{-d}\log n) (\rho=1-d), O(n^{-d}) (\rho>1-d),\end{cases}n\to\infty, J_n = O(n^{1-2d-\rho}), n\to\infty.
P_{{\cal S}}(A)= \sum_{j=1,2}P_{{\cal S}_j/{\cal S}}P_{{\cal S}_j}(A) \pm2 \sqrt{P_{{\cal S}_1/{\cal S}}P_{{\cal S}_1}(A)P_{{\cal S}_2/{\cal S}}P_{{\cal S}_2}(A)}\cosh \theta(A;{\cal S},{\cal S}_j)\;.
\left\| \nabla \left(e^{ikx_1}\pi^k\right) \right\|_{H^{-1}(\Omega)}\leq \left\| e^{ikx_1}\begin{bmatrix}\widetilde H_{1}^{k}\widetilde H_{2}^{k}\end{bmatrix}\right\|_{H^{-1}(\Omega)}+\left\| \nabla \left(e^{ikx_1}w^k\right) \right\|_{L^{2}(\Omega)}
\Omega_{1 2}= Q \bar{K}_{2}Q \bar{K}_{2}Q = \left( \begin{array}{c c}{{- u_{1}^{2}d_{1}}}&{{u_{1}^{3}}}\\{{- u_{1}u_{2}d_{1}}}&{{u_{1}^{2}u_{2}}}\\ \end{array}\right)
Ric(\nabla_{\nu}e_i,e_i) =Ric\left(-h_{ii}e_i+\frac{e_i(R)}{2|\nabla f|}\partial_0,e_i\right)=-h_{ii}R_{ii}+\frac{e_i(R)}{2|\nabla f|}R_{0i}&=-h_{ii}R_{ii}+\frac{(e_i(R))^2}{4|\nabla f|^3},
\mu^e(t) =\frac{1}{2^3 \pi}\sqrt{\frac{2}{\pi}}[(-1-18 t^2+19 t^4) e^{-t^2}+1+17 t^2-11 t^4+t^6] e^{-\frac{t^2}{2}}, \mu^s(t) =\frac{1}{2^3 \pi}\sqrt{\frac{2}{\pi}}(-1-18 t^2+19 t^4) e^{-\frac{3 t^2}{2}}.
\mathcal{E}_{V}^{\theta}({\mu}_{\theta}^{\epsilon}) = \mathcal{E}_{V}^{\theta}(\mu_{\theta}) - \epsilon |X| \left( \mathcal{E}_{V}^{\theta}(\mu_{\theta}) \right) + \epsilon \int_{X}h^{\mu_{\theta}}(x) + V(x) \, dx + \frac{1}{\theta}|X| \epsilon \log \epsilon + O(\epsilon^{2}).
g_1(t)^{(-1)}(-1)^{s_r}(t-\theta)^{w-s_r}\Omega^w\alpha_r\mathcal{Li}_{K,(s_1, \ldots, s_{r-1})}(\alpha_1, \ldots, \alpha_{r-1})+g_1(t)^{(-1)}(t-\theta)^w\Omega^w\mathcal{Li}_{K,\mathfrak{s}}({\boldsymbol \alpha}) &+g_2(t)^{(-1)}=0.
\frac{2}{3}f(-1)+\frac{3+\sqrt{-3}}{18}f \left(\frac{-1+\sqrt{-3}}{2}\right)+\frac{3-\sqrt{-3}}{18}f \left(\frac{-1-\sqrt{-3}}{2}\right) = -\frac{1}{4 \pi \sqrt{-1}}\int_{S^1}f(z) e^{-\frac{2}{z}}dz
\begin{array}{r l}{{}}&{{X^{+ +}X^{+ +}= X^{+ -}X^{+ +}= X^{+ +}X^{- +}= 0}}\\{{}}&{{X^{+ -}X^{- +}= X^{- +}X^{+ -}= 0}}\\{{}}&{{X^{+ +}X^{+ -}= -{\frac{r^{2}}{2}}X^{+ +}= - X^{- +}X^{+ +}}}\\{{}}&{{X^{+ -}X^{+ -}= -{\frac{r^{2}}{2}}X^{+ -}\ , \qquad X^{- +}X^{- +}={\frac{r^{2}}{2}}X^{- +}}}\\{{}}&{{X^{+ -}- X^{- +}= -{\frac{r^{2}}{2}}I .}}\\ \end{array}
0=f_{Cayley}(s_1,\ldots,s_{d-1},x_1^{\ast},\ldots,x_{d-1}^{\ast})= \sum_{i_1=0}^{\tau_1}\cdots\sum_{i_{d-1}=0}^{\tau_{d-1}}Y_{i_1,\ldots,i_{d-1}}\prod_{k=1}^{d-1}\phi_{i_k}(s_k).
\langle \, p_{f}\, | \, q_{i}\, \rangle = \int \frac{d P_{1}}{2 \pi \hbar}\, d Q_{N}\, \langle \, p_{f}\, | \, Q_{N}\, \rangle \, e^{i P_{1}Q_{N}/ \hbar}\langle \, P_{1}\, | \, q_{i}\, \rangle \, .
A_n = \begin{pmatrix}\alpha_1^n \cdots \alpha_k^n c_{k_{1}}^n \cdots c_{k_p}^{n}\vdots \ddots \vdots \vdots \ddots \vdots \alpha_1^{n+k-1+p}\cdots \alpha_k^{n+k-1+p}c_{k_{1}}^{n+k-1+p}\cdots c_{k_p}^{n+k-1+p}\end{pmatrix}.
E_{i}\cdot v_r =\delta_{r, i+1}v_{r-1}, F_{i}\cdot v_r =\delta_{r, i}v_{r+1}, K_i\cdot v_r =\begin{cases}qv_{i}\hbox{if}r=i, q^{-1}v_{i+1}\hbox{if}r=i+1, v_{r}\hbox{else}.\end{cases}
\begin{array}{c}{{\omega_{a f t e r}( A \mathcal{) =}\omega_{b e f o r e}( P A P ) + \omega_{b e f o r e}( \left( 1 - P \right) A \left( 1 - P \right) ) \, , \, \,}}\\{{\, \, \, A \mathcal{\in A}, \, \, \mathrm{a l g e b r a ~ o f ~ o b s e r v a b l e s ~}}}\\ \end{array}
\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 .}
\bar\partial_E = \begin{pmatrix}\bar\partial -\beta^t 0 0 0 \bar\partial_W \beta 0 0 0 \bar\partial 0 0 0 0 \bar\partial\end{pmatrix},\Phi=\begin{pmatrix}0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 \end{pmatrix},
\sum q^{\frac{1}{2}p^2}=\sum_n q^{\frac{3}{2}\left(2n+1+\frac{m_1+m_2}{3}\right)^2}=\sum_n q^{6\left(n+\frac{m_1+m_2+3}{6}\right)^2}=\theta_{2m_1+2m_2+6,6}\:(\tau).
S_{\mathrm{D i r a c}}= \mathrm{\frac{i}{2 \p i}}\int d^{2}\! \sigma \, \,{\bar{\Psi}}\gamma^{a}\partial_{a}\Psi = \mathrm{\frac{i}{\p i}}\int d^{2}\! \sigma \, \,{\bar{\Psi}}_{L}{\bar{\partial}}\Psi_{L}- \mathrm{\frac{i}{\p i}}\int d^{2}\! \sigma \, \,{\bar{\Psi}}_{R}{\partial}\Psi_{R}\ ,
\begin{array}{rl}L_{n,s}(\mathcal{E}_{\rho})= \displaystyle\dfrac{1}{2\pi\sqrt{2}}\int_{0}^{2\pi}|K_{n,s}(z)|\sqrt{a_2-\cos{2\theta}}\,d\theta = \displaystyle\dfrac{1}{2\sqrt{2}}\int_{0}^{2\pi}\dfrac{\sqrt{a}}{\rho^{(2s+1)n}2^{s-1/2}c^s}\,d\theta=\dfrac{1}{2^s\rho^{(2s+1)n}}\int_0^{\pi}\dfrac{\sqrt{a}}{c^s}\,d\theta. \end{array}
\label[Ieq]{eq:optimalcouplingjumpparam}\overline{X}{}^i_{T^i_n}= \psi\big(\overline{X}{}^i_{T^{i,-}_n},f_{T^i_n},\theta\big),X^i_{T^i_n}= \psi\Big(X^{i}_{T^{i,-}_n},\mu_{X^i_{T^{i,-}_n}},\theta\Big),
\begin{alignedat}{2}c= (a_1)^{10n-6}(a_2)^{2}b_1b_2 = (a_1)^{10n-5}a_2b_1b_3 = (a_1)^{10n-4}b_2b_3 = (a_1)^{10n+5}= (a_2)^{8n+4}\end{alignedat}
R\int_{A_1}D\Phi(\bar{g}^R, \bar{\pi}^R)\cdot (1,0) \, dx = R \left(B^2_{((g^\theta)^R,(\pi^\theta)^R)}(1,0) - B^1_{(g^R,\pi^R)}(1,0)\right) &= B^{2R}_{(g^\theta, \pi^\theta)}(1,0) - B^R_{(g,\pi)}(1,0),
1\otimes (E(\lambda n_s^{-1}) + dE(\lambda \mu_{n_s^{-1}}(-1))c_{s,-1})E(\mu) &= (sw\widetilde{\chi})(E(\mu))\otimes (E(\lambda n_s^{-1}) + dE(\lambda \mu_{n_s^{-1}}(-1))c_{s,-1})
[a_{(1\,l-1)},a_{(1\,l)}]_c=[[a_{(1\,l-2)},a_{l-1}]_c,a_{(1\,l)}]_c=[a_{(1\,l-2)},[a_{l-1},a_{(1\,l)}]_c]_c &\,+\chi_{(1\,l)}(g_{l-1})\big([a_{(1\,l-2)},a_{(1\,l)}]_ca_{l-1}-\chi_{l-1}(g_{(1\,l)}g_{(1\,l-2)})a_{l-1}[a_{(1\,l-2)}, a_{(1\,l)}]_c\big).
F_{l}^{( N )}= \sum_{j = 0}^{N}( - 1 )^{j + l}\; D_{j l}^{( N - 1 )}\; \frac{m_{l}}{m_{j}}= \frac{\partial}{\partial m_{l}^{2}}\left( m_{l}^{2}D^{( N )}\right) ,
e^{-x}\int_{-\infty}^xe^\eta\big(vv_x^2(\eta)+v^3(\eta)\big)d\eta \geq 2e^{-x}\int_{-\infty}^xe^\eta v^2(\eta)v_x(\eta)d\eta =\dfrac{2}{3}v^3(x)-\dfrac{2e^{-x}}{3}\int_{-\infty}^xe^\eta v^3(\eta)d\eta,
\lambda_i - \left \lfloor \frac{k(m-(2i-1))}{2}\right \rfloor \geq 1+k(m-i) - \left \lfloor \frac{k(m-(2i-1))}{2}\right \rfloor &= 1+km - \left \lfloor \frac{k(m+1))}{2}\right \rfloor &= 1+k(m-1) - \left \lfloor \frac{k(m-1))}{2}\right \rfloor &= 1 + \left \lceil \frac{k(m-1)}{2}\right \rceil.
F \equiv \frac{I_{r e g}}{\beta}= - 2 \pi^{4}L^{2}( N \alpha^{\prime})^{5 / 2}\lim_{r \to \infty}\left( \nu ( R^{3}e^{- 2 \Phi})^{\prime}- \sqrt{\nu_{B P S}}( R^{3}e^{- 2 \Phi})_{B P S}^{\prime}+ \frac{K}{2}\right) \, .
\hspace{-1.5cm}\sum_{\Sigma \rightarrow{\cal M}}\, \int_{{\cal M}}\,[dxd\psi]\, g_s^{-\chi(\Sigma)}\exp\Bigl[ -{1\over 4\pi \alpha'}\int_{\Sigma}d^2\xi \, L(x,\partial_{\xi}x, \psi, \partial_{\xi}\psi,\ldots)\Bigr] \prod_i \int d^2 \xi_i \, V_i(\xi_i)
\mu\left\{x:S^{\sigma}_{\tau_{n}}f\geq t\right\}=\mu\left\{x:S^{\sigma}_{\tau_{n}}f\geq t,\tau_{n}\leq\epsilon t\right\}+\mu\left\{x:S^{\sigma}_{\tau_{n}}f\geq t,\tau_{n}>\epsilon t\right\},
{\bf{E}}\Bigl[\bigl(\int_0^1 s^{\lambda_1}(1-s)^{\lambda_2}\,dM_\mu(s)\bigr)^l\Bigr] = \prod_{k=0}^{l-1}\frac{\Gamma(1-(k+1)\mu/2)\Gamma(1+\lambda_1-k\mu/2)\Gamma(1+\lambda_2-k\mu/2)}{\Gamma(1-\mu/2)\Gamma(2+\lambda_1+\lambda_2-(l+k-1)\mu/2)}.
q^d = \sum_{(a,b)\in K_0}e^{i(\theta_a - \theta_b)t}+ e^{i(\theta_a - \theta_b)t}\sum_{\lambda \ne 0}e^{i 2 \pi \lambda /q}&=\sum_{(a,b) \in K_0}e^{i(\theta_a - \theta_b)t}- \sum_{(a,b) \in K_1}e^{i(\theta_a - \theta_b)t}.
{\rm lim}_{L\rightarrow +\infty}\frac{k_{\gamma,\Sigma}^{L,s}}{\sqrt{L}}=\frac{(-\overline{q}\frac{\dot{\gamma}_1}{\gamma_1}+\overline{p}\dot{\gamma}_3)\frac{d}{dt}(\omega(\dot{\gamma}(t)))}{|\overline{q}\frac{\dot{\gamma}_1}{\gamma_1}-\overline{p}\dot{\gamma}_3|^3},~~if ~~\omega(\dot{\gamma}(t))= 0~~and~~\frac{d}{dt}(\omega(\dot{\gamma}(t)))\neq 0.
((V' (v(i + 1)) - V' (v(i - 1))) \partial_i)^{\ast}= \partial_i^{\ast}(V' (v(i + 1)) - V' (v(i - 1))) = (- \partial_i - (\lambda - V' (v_i))) (V' (v(i + 1)) - V' (v(i - 1))) = - (V' (v(i + 1)) - V' (v(i - 1))) \partial_i - (\lambda - V' (v(i))) (V' (v(i + 1)) - V' (v(i - 1))).
{\cal L}= \mbox{Tr}\left[ \frac{1}{2r}( D_t X_i )^2 - \frac{T_A^2}{4r}[X_i,X_j]^2 - \theta^T D_t \theta - \frac{T_A^2}{4}\theta^T \gamma_i [\theta, X_i]. \right],
\begin{array}{cc}h(r)= 3C_0 \textup{in}\ [2M,R_1], h(r)= 1 \textup{in}\ [2R_1,\infty), |h'(r)|\leq \frac{3C_0}{\log 2}r^{-1}\textup{in}\ [R_1,2R_1].\end{array}
J(u)=\bar{\mathbb{E}}\bigg[\int_0^\infty e^{-\beta t}f\Big(x_t^u,y_t^u,z_t^u,\tilde{z}_t^u,\int_{\mathcal{E}}\gamma_{(t,e)}^u\nu(de),u_t\Big)dt+\phi(y_0^u)\bigg],\ \beta\ ,
\sigma_V(p^2) = \frac{1}{m^2}\frac{1 + \frac{\Sigma_A}{m}}{\epsilon\left(1 + \frac{\Sigma_A}{m}\right)^2 + 2\left(1 + \frac{\Sigma_A + \Sigma_B}{2m}\right) \frac{\Sigma_B - \Sigma_A}{m}},
\begin{array}{c}{a^{A *}= a^{A}b^{A *}= b^{A}\omega_{A B}{}^{*}= \omega_{A B}\lambda^{*}= \lambda T^{\dagger}= - T}\\{{}}\\{\bar{\rho}_{i}= \rho^{i \dagger}\gamma^{0}\bar{\varepsilon}_{i}= \varepsilon^{i \dagger}\tilde{\gamma}^{0}}\\ \end{array}
\Delta=256 \tilde{u}_1^2 \tilde{u}_2^2 + 288 \Lambda_1^2 \Lambda_2^2 \tilde{u}_1 \tilde{u}_2 +256 \Lambda_2^2 \tilde{u}_1^3 -256 \Lambda_1^2 \tilde{u}_2^3 -27 \Lambda_1^4 \Lambda_2^4,
\frac{\mu^s_{p^+_{\ell_0}x}(Q \cdot u^-_\ell)}{\mu^s_{p^+_{\ell_0}x}(N^-_{r_j}\cdot u^-_\ell)}= \frac{\mu^s_{p^+_{\ell}x}(Q )}{\mu^s_{p^+_{\ell}x}(N^-_{r_j})}= \frac{\mu^s_{y_\ell}(Q^\prime)}{\mu^s_{y_\ell}(N^-_{e^{-s_{\rho,\ell}}r_j})}.
Z^X_v\left(-\frac{1}{\tau}\right)=2^{-\frac{b_2(X)}{2}}\left(\frac{\tau}{i}\right)^{-\frac{\chi(X)}{2}}\cdot\sum_{u\in H^2(X,Z_2)}(-1)^{u\cdot v}Z^X_u(\tau).
[A,B] = \left(\begin{smallmatrix}x y 0 1/x \end{smallmatrix}\right)\left(\begin{smallmatrix}z w 0 1/z \end{smallmatrix}\right)\left(\begin{smallmatrix}1/x -y 0 x \end{smallmatrix}\right)\left(\begin{smallmatrix}1/z -w 0 z \end{smallmatrix}\right) = \left(\begin{smallmatrix}1 xy(1-z^2) - zw(1-x^2) 0 1 \end{smallmatrix}\right)
{Ric}_{\rho\rho,\rho}[\bar g](p)=0, &{R}_{\rho i\rho j}[\bar g](p)=0, &{Ric}_{\rho\rho,\rho\rho}[\bar g](p)=0, &{R}_{\rho i\rho j,ij}[\bar g](p)=\frac{R_{,ii}[\hat h](p)}{2(n-1)}=-\frac{|W|^2[\hat h](p)}{12(n-1)}, &{R}_{,\rho\rho}[\bar g](p)=0.
(\nabla_X\theta)Y= \sum^3_{a=1}\{(X(\eta_a(\xi))\theta_a Y+\eta_a(\xi)(\nabla_X\theta_a)Y\}= \sum^3_{a=1}\{2\eta_a(\phi AX)\theta_aY+\eta_a(\xi)\eta_a(\phi Y)AX-g(AX,Y)\eta_a(\xi)\phi_a\xi\}= \sum^3_{a=1}2\eta_a(\phi AX)\theta_aY+\eta^\perp(\phi Y)AX-g(AX,Y)\phi\xi^\perp.
\left| \frac{y_{i+1}x_{i+1}- \zeta y_i}{x_{i+1}- \zeta}- y_{i+1}\right|=\left| \frac{\zeta y_{i+1}- \zeta y_i}{x_{i+1}- \zeta}\right|\le\frac{|\zeta| 2 D}{|x_{i+1}- \zeta|}\le\frac{|\zeta| 8D}{3 x_{i+1}}\le\frac{|\zeta| 8D}{3 \xi}
P = \begin{cases}0 \bar{\eta}\geq \frac{\sum_{i=1}^{M_R}l_i}{\lambda N_0}P_{max}\bar{\eta}\leq \sum_{i=1}^{M_R}\frac{{l_i}}{\lambda\left(N_0 + l_i P_{max}\right)}\rho\left(\sum_{i=1}^{M_R}\frac{l_i}{N_0 + P l_i}- \lambda \bar{\eta}=0 \right) \end{cases}
\begin{aligned}A^1_{Q, j, R_0}= \{\Gamma_{\widehat Q_j, j, 1,k_1(0),k_2(0), R_0, S_0}: 0 \leq S_0 \leq k_2(j)\}; A^2_{Q, j, S_0}= \{\Gamma_{\widehat Q_j, j, 2, k_1(0),k_2(0),R_0, S_0}: 0 \leq R_0 \leq k_1(j)\}.\end{aligned}
f_2(u, v)= \pi^{k+l}u^p+v^r\sum_{i=0}^k(-1)^i\dbinom{k+r}{k-i}\dbinom{i+r-1}{i}t^{k-i}\pi^iv^i = \pi^{k+l}u^p+t^k\dbinom{k+r}{k}v^r+\sum_{i=1}^k(-1)^i\dbinom{k+r}{k-i}\dbinom{i+r-1}{i}t^{k-i}\pi^iv^{r+i}.
A^3B-[3]A^2BA+[3]ABA^2-BA^3=(q^2-q^{-2})^2(BA-AB), B^3A-[3]B^2AB+[3]BAB^2-AB^3=(q^2-q^{-2})^2(AB-BA), B^2A^2-(q^2+q^{-2})BABA+(q^2+q^{-2})ABAB-A^2B^2=(q-q^{-1})^2(AB-BA)\gamma, \gamma A=A\gamma,\gamma B=B\gamma.
L_n\left(\sqrt{\bar{x}\bar{y}},\frac{\bar{x}+\bar{y}}{2}\right)=\sqrt{\bar{x}\bar{y}^{-1}}\sum_{k=0}^n{n \choose k}A_k(\bar{x},\bar{y})\frac{(\bar{y}-\bar{x})^{n-k}}{2^{n-k}}.
\begin{aligned}C^{i}=-\frac{1}{4}G^{i}-\frac{1}{4}, C^{ij}=-\frac{1}{4}K^{ij}, P^{ij}=C^{ij}-C^{i}-C^{j}=-\frac{1}{4}K^{ij}+\frac{1}{4}\left(G^{i}+G^{j}\right)+\frac{1}{2},\end{aligned}
\psi(0) =1-(2-\mathbb{E}S)\lim\limits_{n\rightarrow\infty}\frac{b_{n+1}-b_n}{a_n-a_{n+1}}, 1-\psi(u) = \alpha_u\bigl(1-\psi(0)\bigr)+\beta_u(2-\mathbb{E}S), u\in\mathbb{N},
J\overline \Psi(t)=\left[\begin{array}{ccc}I_s 0 0 \phi_1 \phi_2 \phi_3 \end{array}\right],~~~J\Phi(t)=\left[\begin{array}{ccc}I_s 0 0 0 I_{r-s}0 \phi_1 \phi_2 \phi_3\end{array}\right]
\frac{1}{2}{y^{I , I I}}= \alpha{T^{I , I I}}_{5 z}+ \beta{Y^{I , I I}}_{5}\ , \quad{T^{I , I I}}_{5 z}= \mathrm{d i a g}( - 1 / 3 , - 1 / 3 , - 1 / 3 , 1 / 2 , 1 / 2 ) .
G_{GR}(\tau)=1+\frac{\langle\delta I_{G}(t)\delta I_{R}(t+\tau)\rangle}{\langle I_{G}(t)\rangle\langle I_{R}(t)\rangle}=\frac{\langle I_{G}(t)I_{R}(t+\tau)\rangle}{\langle I_{G}(t)\rangle\langle I_{R}(t)\rangle}
\bigl(\Delta_{a}(f_1\otimes f_2)\bigr)(x_1,x_2) \,=\,f_1(x_1+a) f_2(x_2) - f_1(x_1) f_2(x_2) \,=\,(f_1(x_1+a) - f_1(x_1) )f_2(x_2) \,=\,\bigl((\Delta_{a}f_1)\otimes f_2\bigr)(x_1,x_2)\,.
P^\lambda_n(t)=\frac{(-2)^n\Gamma(n+\lambda)\Gamma(n+2\lambda)}{n!\Gamma(\lambda)\Gamma(2n+2\lambda)}(1-t^2)^{1/2+\lambda}\left(\frac{d}{dt}\right)^n(1-t^2)^{n+\lambda - 1/2},