harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\log \frac{\Pr(X = k + \ell)}{\Pr(X = k)}= \ell \log \frac{p}{1-p}+ \sum_{i=1}^\ell (\log(m-k-i+1) - \log(k+i)) \le \ell \log \frac{p}{1-p}+ \ell \log(m-k) - \sum_{i=1}^\ell \log(k+i).
S = S^{0}+ \phi_{i}^{*}R^{i}{}_{a}c^{a}+ ( - )^{b}\frac{1}{2}c_{a}^{*}T_{b c}^{a}c^{c}c^{b}+ ( - )^{i + a}\frac{1}{4}\phi_{i}^{*}\phi_{j}^{*}E_{a b}^{j i}c^{b}c^{a}+ . . . \ .
\left\langle \mathsf{D}_{j}\left( s\right) \psi,\mathsf{L}_{j}\left(s\right) \overline{v}\right\rangle_{\mathbb{R}^{3}}=\left\langle \psi,\gamma_{\operatorname*{N};j}^{\operatorname*{ext}}\left( s\right)\overline{v}\right\rangle_{\Gamma_{j}}\qquad\forall v\in H^{1}\left(\mathbb{R}^{3},\mathbb{A}_{j}^{\operatorname*{ext}}\right) .
\begin{aligned}\left|T_1\right| \lesssim \int \left<\xi\right>^{\gamma}\left|\nabla_{\xi}h\right|\left|D^2_\xi h\right| \varrho\, m_0 & \lesssim \varepsilon \left\|\left<\xi \right>^{\frac{\gamma}{2}}\left|D^2_\xi h\right| \right\|_{L^2(\varrho\,m_0)}+C(\varepsilon) \left\|\left<\xi \right>^{\frac{\gamma}{2}}\left|\nabla_\xi h\right| \right\|_{L^2(\varrho\,m_0)}.\end{aligned}
\tilde{D}= \left( \begin{array}{cccc|ccccc}0 1 1 1 2 \cdots \cdots \cdots 2 1 0 1 1 1 \cdots \cdots \cdots 1 1 1 0 2 2 \cdots \cdots 2 1 1 1 2 0 1 2 \cdots \cdots 2\end{array}\right)
\left| B_{N}\right\rangle_{B = 0}= \left( \frac{- \det g_{\mu \nu}}{( 2 \alpha^{\prime})^{p + 1}}\right)^{\frac{1}{4}}\prod_{n = 1}^{\infty}\exp \left( - \frac{1}{n}\alpha_{- n}^{\mu}g_{\mu \nu}\tilde{\alpha}_{- n}^{\nu}\right) \left| \mathbf{0}\right\rangle ~ .
(\Delta - D_t)(uP)\ =\ m_n (m_{n}+1)x^{m-\overline{n}}t^\ell + \sum_{i \neq n}m_i (m_i - 1) x^{m - 2 \overline{i}+ \overline{n}}t^\ell - \ell x^{m+ \overline{n}}t^{\ell-1}+\ \sum_{|m| + 2\ell \leq |q| + 2\kappa \leq k-1}c^{m ,\ell}_{q,\kappa}x^q t^\kappa + w_{m,\ell}(x,t),
\begin{cases}(\xi_G \cdot e_{4g-2}) =-1+(e_n \cdot e_{4g-2})-(f_1 \cdot e_{4g-2}), (\xi_G \cdot f_1) =(f_1\cdot e_{4g-2})-1, (\xi_G \cdot f_2) =(f_2\cdot e_{4g-2})-1.\end{cases}
e^{- 2 U ( r )}= 1 +{\frac{( p^{1})^{2}}{8 \pi r^{2}}}\log{\frac{c ( p^{1})^{2}}{r^{2}}}\pm . . \quad , \quad z^{1}= i{\frac{\sqrt{c}p^{1}}{r ( 1 \mp . . )}},
V_i=\left[v^0_{Li}(v^1_{Li}v^2_{Li}v^3_{Li})(v^4_{Li}v^5_{Li}v^6_{Li})(v^7_{Li}v^8_{Li}v^9_{Li}) || v^0_{Ri}(v^1_{Ri}v^2_{Ri}v^3_{Ri})(v^4_{Ri}v^5_{Ri}v^6_{Ri}) (v^7_{Ri}v^8_{Ri}v^9_{Ri})\right]~.
\varepsilon^{(1)}(\vec{x}) = \frac{g_\varepsilon \lambda^2}{4\pi c \tau_S}\, \frac{\cos (|\vec{x}-\vec{x}_S| /\lambda)}{|\vec{x}-\vec{x}_S|}\sum_{l\, m_l}c_{l\, m_l}Y_{l\,m_l}(\theta,\phi)\,.
f \mathcal{M}^\nu(f_1,\ldots,f_{n-1}) = f \langle\widetilde{\mathcal{M}}_\Pi^\nu,d_{A}f_1\wedge\cdots\wedge d_{A}f_{n-1}\rangle &= f\iota_{d_{A}f_1\wedge\cdots\wedge d_{A}f_{n-1}}\widetilde{\mathcal{M}}_\Pi^\nu &= \iota_{fd_{A}f_1\wedge\cdots\wedge d_{A}f_{n-1}}\widetilde{\mathcal{M}}_\Pi^\nu &= \iota_\alpha \widetilde{\mathcal{M}}_\Pi^\nu,
C_{i_{1}. . . i_{s}}^{g}= \int_{M_{g , s + 1}}\langle \oint_{C_{z_{1}}}\! \! G^{-}{\bar{G}}^{-}\phi_{i_{1}}. . . \oint_{C_{z_{s}}}\! \! G^{-}{\bar{G}}^{-}\phi_{i_{s}}\prod_{j ,{\bar{j}}= 1}^{3 g - 3}G^{-}( \chi_{j}){\bar{G}}^{-}({\bar{\chi}}_{\bar{j}}) \rangle
-(\hat f \circ \gamma)^-(t) = \cos(\measuredangle(-\dot \gamma(t_k),\Gamma_k)) \geq \cos(\pi - \measuredangle (\dot \gamma(t_k),\Gamma_k)) = -\cos(\measuredangle(\dot \gamma(t_k),\Gamma_k) = (\hat f \circ \gamma)^+(t).
g^{i j}= \frac{e^{2}L N}{4 \pi T}\left( \begin{matrix}{{N - 1}}&{{- 1}}&{{\ldots}}&{{- 1}}\\{{- 1}}&{{N - 1}}&{{\ldots}}&{{- 1}}\\{{\vdots}}&{{\vdots}}&{{\ddots}}&{{\vdots}}\\{{- 1}}&{{- 1}}&{{\ldots}}&{{N - 1}}\\ \end{matrix}\right) \ \ .
\int_{\xi}^{\alpha_{2}}{e^{-ut}G_{n}\left( t\right) dt}=\frac{1}{u}e^{-u\xi}G_{n}\left( \xi \right) +\frac{1}{u}\int_{\xi}^{\alpha_{2}}{e^{-ut}{G}_{n}^{\prime}\left( t\right) dt.}
\int_{\mathbb{R}^{m}}a(p)\left\langle \nabla v,\nabla\varphi\right\rangle\,dx=\int_{\mathbb{R}^{m}}c(p)|v|^{2^{\ast}-2}v\varphi\,dx\qquad\forall\varphi\in C_{c}^{\infty}(\mathbb{R}^{m}).
\frac{\partial}{\partial\tau}[\Delta(\tau,\tau_2)-\Delta(\tau,\tau_1)]={1\over2}\frac{\partial}{\partial\tau}[G_B(\tau,\tau_1)- G_B(\tau,\tau_2)],\quad\frac{\partial^2}{\partial\tau\partial\tau'}\Delta(\tau,\tau')={1\over 2}\ddot{G}_B(\tau,\tau'),
z_{0}^{2}\varepsilon^{2}<{\frac{1}{3 2}}\left( - 3 2 \omega z_{0}^{2}+ 9 l^{2}- 3 \sqrt{- 6 4 \omega z_{0}^{2}l^{2}+ 9 l^{4}- 6 4 l^{2}z_{0}^{2}\omega^{2}}\right) / ( \omega^{2}l^{2}) \, .
F_0 = x^{3}+x^{2}y-x y^{2}+y^{3}+x^{2}z+x y z+y^{2}z-x z^{2}+z^{3}F_1 = x^{3}+x^{2}y-x^{2}z-x y z+y^{2}z+z^{3}F_2 = x^{3}-x^{2}y+x y^{2}+y^{3}+x^{2}z+x y z+y^{2}z-y z^{2}
|\partial_t\log (B(t)^n)|=n|\partial_t\log B(t)|\geq\varepsilon_0^{-21/40}\cdot\bigg(\varepsilon_0^{101/100}+\frac{1}{t}\bigg)\geq D\bigg(\varepsilon_0^{1/2}+\frac{1}{t}\bigg),
N_\infty(\pi_{\infty,j}^*\omega) =\lim_{k\to\infty}\sup_{i\ge k}N_i(\pi_{k,j}^*\omega) =\lim_{k\to\infty}\sup_{i\ge k}(N_i(\pi_{i,j}^*\omega)+O(m^{-k})) =\lim_{k\to\infty}(N_j(\omega)+O(m^{-k})) =N_j(\omega).
{\cal L}= -{( S_{o}{\bar{S}}_{o})}^{3 / 2}( \frac{\hat{L}}{2})^{- \frac{1}{2}}e^{- \frac{G^{( o )}}{2}}+ ( \frac{\hat{L}}{2}) G^{( 1 )}+{( S_{o}^{3}w )_{F}},
\frac{(p + t)!(i_1 + \alpha_1t)!\cdots(i_{n-1}+ \alpha_{n-1}t)!}{(i_1 + 1 + t)!(i_2 + \alpha_1t)!\cdots(i_{n-1}+ \alpha_{n-2}t)!(\alpha_{n}t)!((\alpha_{n-1}- \alpha_n)t-\beta - 1)!}= c\frac{(i_{n-1}+ \alpha_{n-1}t)!}{(\alpha_nt)!((\alpha_{n-1}- \alpha_n)t - \beta - 1)!}
\mathbb{E}_{n,m}' [\alpha \gamma] =\mathbb{E}_{n,m}' [ \chi_{n,m}(w,x_1) \mathbb{1}_{X_1(M)}(x_1) \chi_{n,m}(w, x_2) \mathbb{1}_{X_1(M)}(x_2) ] = \mathbb{E}_{\mathcal{M}}[\mathbb{1}_{X_1(M)}(x_1) \mathbb{1}_{X_1(M)}(x_2) \mathbb{E}_{n,m}[\chi_{n,m}(w, x_1) \chi_{n,m}(w, x_2) | M ]]
\vec{M}\equiv ( M ,{\cal Q}_{d},{\cal Q}_{e}^{1},{\cal Q}_{e}^{2}) , \vec{N}\equiv ( N ,{\cal Q}_{a},{\cal Q}_{m}^{1},{\cal Q}_{m}^{2}) , \vec{J}\equiv ( J ,{\cal F},{\cal P}_{m}^{1},{\cal P}_{m}^{2}) ,
\begin{aligned}{d\lambda_k \over dt}= -4\nu_0 \left(e^{q_{k+1}-q_k}(\vec{n}_{k+1}^\dag ,\vec{n}_{k}) -e^{q_k -q_{k-1}}(\vec{n}_{k}^\dag ,\vec{n}_{k-1}) \right) + M_k +i N_k ,{d q_k \over dt}= - 4\nu_0 \lambda_k + 2i(\mu_0 +i\nu_0) \Xi_k -iX_k ,{d\vec{n}_k \over dt}= \mathcal{O}(\epsilon),\end{aligned}
q\cdot N_{d}^{\prime}=q^2+q\phi(b)+\frac{\phi(-a)}{q-1}\sum_{n=0}^{q-2}G_{-(d-1)n}G_{nd}G_{-n+\frac{q-1}{2}}G_{\frac{q-1}{2}}T^n\left(\frac{b}{a^d}\right)\phi(ab) &=q^2+q\phi(b)+\frac{\phi(-b)}{q-1}\sum_{n=0}^{q-2}G_{-(d-1)n}G_{nd}G_{-n+\frac{q-1}{2}}G_{\frac{q-1}{2}}T^n\left(\frac{b}{a^d}\right).
24 \mathcal{F}(u^2) = 24 \mathcal{F}((1+|x|^2)^{-4}) &= 8\pi^{\frac{3}{2}}\left( \frac{|\xi|}{2}\right)^{\frac{5}{2}}K_{\frac{5}{2}}(|\xi|) &= (|\xi|^2+3|\xi|+3) \mathcal{F}((1+|x|^2)^{-2}) &= \mathcal{F}(-\Delta u +3 (-\Delta)^{1/2}u+3u).
E_{q^{-1}\cdot X}L_g E_{p^{-1}\cdot Y}L_h = E_{q^{-1}\cdot X}(L_gE_{p^{-1}\cdot Y}L_g^*) L_gL_h & = E_{q^{-1}\cdot X}E_{(gp^{-1})\cdot Y}L_{gh}= E_{(q^{-1}\cdot X)\cap((gp^{-1})\cdot Y)}L_{gh}.
\Big(|\langle v \rangle_h(t_1)|+|\langle v \rangle_h(t_2)|\Big)^{\frac{p-2}{2}}\big|\langle v \rangle_h(t_1)-\langle v \rangle_h(t_2)\big| \leq c \Big||\langle v \rangle_h|^{\frac{p-2}{2}}\langle v \rangle_h(t_1)-|\langle v \rangle_h|^{\frac{p-2}{2}}\langle v \rangle_h(t_2)\Big| begin{align*}3mm] =c\big|[w]_h(t_1)-[w]_h(t_2)\big|
\mathcal{B}^{\rm nre}_{h,r}(\sigma)= \Big\{F\in \mathcal{B}_{h,r}:F(x)=\mathcal{T}_{N}F(x)\Big\}, &\mathcal{B}^{\rm re}_{h,r}(\sigma) = \Big\{F\in \mathcal{B}_{h,r}:F(x)=\mathcal{R}_{N}F(x)\Big\}.
d b^{\frac{2}{10 \cdot 3^{k-4}}}\le d (1 + k C_k)^{\frac{2}{10 \cdot 3^{k-4}}}N^{\frac{2}{10 \cdot 3^{k-4}}}&\le C_k (1 + k C_k)^{\frac{2}{10 \cdot 3^{k-4}}}N^{1 - \frac{1}{10 \cdot 3^{k-5}}+ \frac{2}{10 \cdot 3^{k-4}}}= C_k (1 + k C_k)^{\frac{2}{10 \cdot 3^{k-4}}}N^{1 - \frac{1}{10 \cdot 3^{k-4}}}.
Q_u=\begin{pmatrix}1 . -1 -1 -1 . . -1 -1 -1 . 1 \multicolumn{2}{c}{?}\end{pmatrix}, Q_v=\begin{pmatrix}1 . -1 -1 . -1 . -1 -1 -1 1 . \multicolumn{2}{c}{?}\end{pmatrix}.
x_s = \bigoplus_i \sqrt{\lambda_i}\begin{pmatrix}\Re(X_s \psi_i) \Im(X_s \psi_i) \end{pmatrix}y_t = \bigoplus_i \sqrt{\lambda_i}\begin{pmatrix}\Re(Y_t \psi_i) \Im(Y_t \psi_i) \end{pmatrix}
h_{2g-L}^{\mathcal{L}}(t)=\sum_{l=2}^{L}\frac{(-1)^{l}}{l}\sum_{\sum_{s=1}^lg_s=g}\sum_{\cup_{s=1}^l\mathbf{I}_s=\mathbf{L}}\prod_{s=1}^lh_{2g_s-I_s}^{\mathcal{L}_{\mathbf{I}_s}}(t).
0 \leq \varphi_1(n) - \lfloor \varphi_1(n) \rfloor= \varphi_1(n) -1 - \lfloor \varphi_1(n) - \psi(n) \rfloor & < \varphi_1(n) - 1 + 1 + \psi(n) - \varphi_1(n) = \psi(n).
x_2=f(x_1)+l\chi_2\ge f(x_1)-l\ge f(x_1)+\tilde \Delta_l(x_1)-f(x_1)+x_1\ge x_0+\tilde \Delta_l(x_0)+\tilde \Delta_l(x_1)\ge x_0+2\tilde \Delta_l(x_0).
\begin{aligned}H_{00}^{\frac{1}{2}}\Lambda^{0}(\Gamma_{1}):= \{\mu\in H^{\frac{1}{2}}\Lambda^{0}(\Gamma_{1})\mid\rho^{-\frac{1}{2}}\mu\in L^{2}\Lambda^{0}(\Gamma_{1}),\rho=\mathrm{dist}(x,\partial\Omega)\}= \{\mu\in H^{\frac{1}{2}}\Lambda^{0}(\Gamma_{1})\mid\mu_{0}\in H^{\frac{1}{2}}\Lambda^{0}(\partial\Omega)\}, \end{aligned}
\frac 13\varepsilon^{ij}\left( \begin{array}{l}2\bar{\varepsilon}\gamma_{\mu \rho}\partial_i\theta \bar{\theta}\gamma^{\mu \nu}\partial_j\theta -\bar{\varepsilon}\gamma_{\mu \rho}\theta \partial_i\bar{\theta}\gamma^{\mu \nu}\partial_j\theta\end{array}\right) \lambda_\nu \lambda^\rho ,
\gamma_{{5}}= \left( \begin{array}{c c}{\sigma_{3}}&{0}\\{\noalign{\medskip}0}&{\sigma_{3}}\\ \end{array}\right) \ , \quad \Lambda^{\pm}= \frac{1}{2}( 1 \pm \gamma^{0}\gamma^{3}) \ , \quad \gamma^{0}\gamma^{3}= \left( \begin{array}{r r}{{\bf 1}}&{0}\\{\noalign{\medskip}0}&{{\bf - 1}}\\ \end{array}\right) \ .
\bigl ( \partial_{\mu}-{\frac{1}{4}}\Omega_{\mu +}{}^{a b}\gamma_{a b}\bigr ) \epsilon_{+}= 0 \ , \quad \bigl ( \partial_{\mu}-{\frac{1}{4}}\Omega_{\mu -}{}^{a b}\gamma_{a b}\bigr ) \epsilon_{-}= 0 \ .
H_{0,1}= H_{1,0}= \dfrac{1}{p}\left( - p \log p - (1-p) \log (1-p)\right), H^c_{0,1}= \dfrac{2}{p}\left( - p \log p - (1-p) \log (1-p)\right), (t^{rel}+ 1) H(X) = H^{av}= \left(\dfrac{1}{p}+1\right) \left( - p \log p - (1-p) \log (1-p)\right) \leq (t^{rel}+ 1) \log 2.
\begin{cases}\dot{u}^\varepsilon (t,x) = \Delta u^\varepsilon (t,x)+\displaystyle{\frac{1}{\varepsilon}}f(u^\varepsilon (t,x) ) + \varepsilon^\gamma a(x) \dot{W}_t(x),\ \ \t> 0,\x\in \mathbb{R}, u^\varepsilon (0,x) = u^\varepsilon_0 (x),\x\in \mathbb{R}, \ \ \u^\varepsilon (t,\pm \infty) = \pm 1,\ \ \t\geq 0,\end{cases}
a_1 =1 , a_{3k}=-(4k - 1)^2(2k - 1)x^3, a_{3k+1}=-4k^2x^2 , a_{3k+2}=-(4k + 1)^2(2k + 1)x^3 ; b_0 =0 , b_{3k}= -(6k - 1)x + 1 , b_{3k+1}= -(6k + 1)x + 1, b_{3k+2}= -4(2k + 1)^2x^2 - 2(2k + 1)x + 1 .
g^1=r\displaystyle\bar{g}^1 =r(\displaystyle\frac{t}{2}+a_1v+c_1u+d_1rw^2+\cdots) g^2=r\displaystyle\bar{g}^2 =r(b_2t+a_2v+c_2u+d_2rw^2+\cdots)
\psi_{Q}= \left ( \begin{array}{c}\frac{\psi_{(0)}}{\sqrt{N_{f}}}I_{N_{f}}+\psi_{(1)}\psi_{(2)}\end{array}\right )\; , \qquad \quad \psi_{\bar Q}= \left ( \begin{array}{cc}\frac{\psi_{(\bar 0)}}{\sqrt{N_{f}}}I_{N_{f}}+ \psi_{(\bar 1)}\psi_{(\bar 2)}\end{array}\right )
E_{\infty}(\hat{x})=- i \, k \, \eta \, \sum_{m=1}^{\aleph}\, e^{i \, k \, \hat{x}\cdot z_{m}}\left[ \hat{x}\times \int_{D_{m}}F_{m}(y) \, dy \right]+\mathcal{O}\left(a^h\right)=-i \, k \, \eta \, \sum_{m=1}^{\aleph}\, e^{i \, k \, \hat{x}\cdot z_{m}}\hat{x}\times Q_m+\mathcal{O}\left(a^h\right),
z_{n-1}' =\lambda(d(z)a^{-1}b^{-1}, d(z)a^{-1}b^{-1}a) z_n' =\lambda(d(z)a^{-1}b^{-1}a, d(z)a^{-1}b^{-1}ab) z_{n+1}' =\lambda(d(z)a^{-1}b^{-1}ab, d(z)a^{-1}b^{-1}ab^2) z' =z_1...z_{n-2}z_{n-1}'z_n'z_{n+1}'
\begin{pmatrix}1\!\!1_{k}0 0 t1\!\!1_{n-k}\end{pmatrix}\begin{pmatrix}A 0 0 \alpha\end{pmatrix}-\begin{pmatrix}A 0 0 \alpha\end{pmatrix}^{\top}\begin{pmatrix}1\!\!1_{k}0 0 t1\!\!1_{n-k}\end{pmatrix}=
{\bf h_\varphi}= \left(\begin{array}{cc}e^{\varphi/4}{1\over 2}\,\varphi_w e^{-\varphi/4}0 e^{-\varphi/4}\end{array}\right),\qquad{\bf \bar{h}_\varphi}= \left(\begin{array}{cc}e^{\varphi/4}0 -{1\over 2}\,\varphi_{\bar{w}}e^{-\varphi/4}e^{-\varphi/4}\end{array}\right).
{\delta\over\delta\psi_{\alpha}^{a}(x)}=\int dz\int dy{\delta\sigma_{\beta\rho}(y,z)\over\delta\psi_{\alpha}^{a}(x)}{\delta\over\delta\sigma_{\beta\rho}(y,z)}=-\int dy\bar{\psi}_{\beta}^{a}(y){\delta\over\delta\sigma_{\beta\alpha}(y,x)}
W_{\pi}(g_{t,l,v}) = \chi_2(v^{-1})\epsilon(\frac{1}{2}, \chi_1^{-1}) q^{\frac{l}{2}-1}\chi_2(y_0)\chi_1^{-1}(1+\varpi^{\frac{a_1-a_2}{2}}y_0) \psi(\frac{y_0}{v}\varpi^{-\frac{a_1+a_2}{2}}) S(f_{y_0},a_2-1).
C_1(a,b,\lambda) = \ \lambda^{-1}\sum_{k=1}^N \frac{a_k}{1+\alpha b_{k+1}}\,E_{k,k+1}, C_2(a,b,\lambda) = \ \lambda^{-1}\sum_{k=1}^N \frac{a_k}{1+\alpha b_k}\,E_{k,k+1}
-\widehat{H}'_{nH}(t)=-e^{-\alpha(t)}\widehat{H}_1+\left(\frac{d\alpha}{dt}(t)+2\beta(t)\right)\widehat{H}_2+e^{\alpha(t)}\left(\frac{d\beta}{dt}(t)-\beta^2(t)-\omega^2(t)\right)\widehat{H}_3-e^{\frac{k \alpha(t)}2}c(t)\widehat{H}_4,
\frac{d}{dt}\int_M | \rho (u(x ,t)) |^2 = 2 \int_M \sum_{a, b}(\rho (u))^a \rho^a_b (u) \frac{\partial u^b}{\partial t}= 2 \int_M \sum_a (\rho (u))^a \Delta_H (\rho (u))^a = - 2 \int_M \sum_a | \nabla_H (\rho (u))^a |^2 \leq 0
\limsup_{\lambda\to0^+}\lambda\left\|\left[\int_{\mathbb{R}}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]\cap[(-2\beta,2\beta)^\complement\times\mathbb{R}]}(\cdot,y)\left|\cdot-y\right|^{\gamma-1}\,dy\right]^\frac{1}{q}\right\|_X=0,
\omega^+_n \cdot e_1 - \frac{p-1}{2}\cdot \widetilde{\omega}^+_n \cdot e_2 = ( \omega^+_n \cdot \frac{p-1}{2}, \omega^+_n) - (\frac{p-1}{2}\cdot \widetilde{\omega}^+_n \cdot X, 0) & = ( 0 , \omega^+_n) & = (0,0) \in \Lambda_n/ \omega^-_n \oplus \Lambda_n/ \omega^+_n .
f_t \left( \int_{\Omega}^{}\alpha (\omega) d \omega \right) = \int_{\Omega}^{}\alpha (\omega) d \omega - t = \int_{\Omega}^{}( \alpha (\omega) - T ( \omega ) ) d \omega \leq \int_{\Omega}^{}f_{T (\omega)}\circ \alpha (\omega) d \omega.
h_{T_0}^1(t) \ = \t(v_1+v_2+v_3) h_{T_1}^1(t) \ = \tv_1+t^2(v_2+v_3) h_{T_2}^1(t) \ = \t^2((v_1+v_2)+(v_1+v_3)-2v_1)+t^3v_1
\sigma_{n,-1}(t) = (-1)^{n}\frac{H_{n+1}^{(-n-1)}(t)}{H_{n}^{(-n)}(t)}, \sigma_{n,n}(t) = \frac{H_{n+1}^{(-n)}(t)}{H_{n}^{(-n)}(t)}, n \geq 1.
{\cal F}= ( R^{A B}+{\frac{k}{l^{2}}}e^{A A^{\prime}}\wedge e^{B}{}_{A^{\prime}}, ~ ~{\frac{1}{l^{2}}}D ( e^{A A^{\prime}}\wedge e^{B}{}_{A^{\prime}}) ) M_{A B}+ c . c .
\mathbf{d}_{p}+(-a_{0}^{(p)}A_{0}^{(p)},-\sum_{j=p-1}^{p}a_{1}^{(j)}A_{j-p+1}^{(p)},-\sum_{j=p-2}^{p}a_{2}^{(j)}A_{j-p+2}^{(p)},\ldots,-\sum_{j=1}^{p}a_{p-1}^{(j)}A_{j-1}^{(p)}).
- i \hat{\nabla}\left( \begin{array}{c}{\alpha_{\lambda}( \theta , \phi )}\\{\beta_{\lambda}( \theta , \phi )}\\ \end{array}\right) = \lambda \left( \begin{array}{c}{\alpha_{\lambda}( \theta , \phi )}\\{\beta_{\lambda}( \theta , \phi )}\\ \end{array}\right) .
h_q(x) =\frac{1}{\pi}\int^\infty_0 \Re\left(\frac{1-e^{iux}}{q-\eta(u)}\right)du &=\frac{1}{\pi}\int^\infty_0 \Re\left(\frac{1-\cos(ux)}{q-\eta(u)}\right)du+\frac{1}{\pi}\int^\infty_0 \Im\left(\frac{\sin(ux)}{q-\eta(u)}\right)du.
\begin{aligned}z-y = 2^{pd}p^{pe-1}r'^p, \phi_p(z, y) = pr_1^p, x = 2^{d}p^{e}r' r_1, z-x = s^p, \phi_p(z, x) = s_1^p, y = s s_1, x+y = t^p, \phi_p (x, -y) = t_1^p, z = t t_1,\end{aligned}
\rho (xy,\psi\varphi) = T_{xy}M_{\psi \varphi}&= T_xT_yM_\psi M_\varphi &= \overline{\psi (y)}T_xM_\psi T_y M_\varphi &=\sigma ((x,\psi ),(y,\varphi )) \rho (x,\psi )\rho(y,\varphi)\, .
- i \widehat{p}\Psi_{p}^{( 1 / 2 )}= \varepsilon m \Psi_{p}^{( 1 / 2 )}\hspace{0 . 5 i n}\sigma_{p}^{( 1 / 2 )}\Psi_{p}^{( 1 / 2 )}= s_{p}\Psi_{p}^{( 1 / 2 )}
\Phi_{1}^{m}\equiv \frac{\sigma_{1}+ i \sigma_{2}}{2}\otimes \psi_{- 1 / 2}^{m}| 0 , k \rangle , \Phi_{2}^{m}\equiv \frac{\sigma_{1}- i \sigma_{2}}{2}\otimes \psi_{- 1 / 2}^{m}| 0 , k \rangle
\lim_{N \to \infty}e^{-\lambda(\tau_N-t_N)}\cdot N^{-1}S_j(\tau_N) = Y^{-1}\lim_{N \to \infty}e^{-\lambda \tau_N}S_j(\tau_N) = \nu \int_0^\infty e^{-\lambda s}p_j(s) ds,
E(\lambda n_s^{-1})E(\mu) = q_\lambda^{1/2}q_\mu^{1/2}q_s^{-1}\theta(\lambda)T_{n_s^{-1}}\theta(\mu) & = q_\lambda^{1/2}q_\mu^{1/2}q_s^{-1}\theta(\lambda n_s^{-1}(\mu))T_{n_s^{-1}}&- \sum_{k = 1}^{\langle \nu(\mu),\alpha\rangle}q_\lambda^{1/2}q_\mu^{1/2}q_s^{-1}\theta(\lambda)c_{s,k}\theta(\mu_{n_s^{-1}}(-k)\mu).
{\sigma}^{\prime \prime}+ [ 2 / r + ( 1 / 2 ) ({\nu}^{\prime}-{\lambda}^{\prime}) ]{\sigma}^{\prime}+ e^{\lambda}{( \omega + e A_{0})}^{2}e^{- \nu}\sigma - e^{\lambda}\frac{d U}{d{\sigma}^{2}}\sigma = 0 ,
\begin{array}{rl}{\rm E}\Big[|\sum_{j\in B}\alpha_{ji}(c_j-{\rm E}[c_j]) |^{\gamma_i}\Big]^{\frac1{\gamma_i}}={\rm E}\Big[|\sum_{j\in B}\alpha_{ji}(c_j-{\rm E}[c_j^\prime]) |^{\gamma_i}\Big]^{\frac1{\gamma_i}}& \leq{\rm E}\Big[|\sum_{j\in B}\alpha_{ji}(c_j-c_j^\prime) |^{\gamma_i}\Big]^{\frac1{\gamma_i}}\end{array}
W_2(\bar{\mu}_t,\bar{\mu}_\infty)^2 = (\bar{m}(t)-\bar{m}_\infty)^2+(\sqrt{\bar{\sigma}(t)}-\sqrt{\bar{\sigma}_\infty})^2 &\leq \bar{m}(t)^2+|\overline{\sigma}(t)-\overline{\sigma}_\infty| &=e^{-2\lambda_2 t}\Big[x_0^2+\frac{1}{\beta\omega^2}\Big].
-T_0^3T_2 - 2T_0^2T_1T_2 + 14T_0^2T_2^2 - 2T_0T_1^3 + 9T_0T_1^2T_2 + 4T_0T_1T_2^2 - 7T_0T_2^3 + T_1^4 - 2T_1^3T_2{}- 7T_1^2T_2^2 + 3T_2^4 = 0
\phi(t)=(-1)^{\frac{m(m-1)p}{2}}\prod\limits_{l=0}^{m-1}{t+pl \choose(m+1)p-1}\frac{((m+1)p-1)!((l+1)p)!}{((m+l+1)p-1)!p!}, \sigma=\big[\tfrac{1}{2}(\begin{smallmatrix}1+i -1+i 1+i 1-i\end{smallmatrix})\big]\in{\rm PSL}(2,\mathbb{C}),
\langle \frac{(u_3)_k^i-2(u_3)_k^{i-1}+(u_3)_k^{i-2}}{\delta_k^2}, \varphi_3 \rangle +\langle \sigma_k^i, E \varphi \rangle=\langle \mathcal L_k^i,\varphi \rangle.
u(t)=\cos\left( t\sqrt{-\Delta}\right)u_0+\frac{\sin\left( t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}u_1+\int_0^t\frac{\sin\left( (t-s)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}|u|^{\frac{4}{N-2}}u(s)\,ds.
\mathcal{B}_{\Sigma}\; = \int d^{D}x \, \left( \frac{\delta \Sigma}{\delta \varphi^{i}}\frac \delta{\delta \varphi^{i *}}\; + \; \frac{\delta \Sigma}{\delta \varphi^{i *}}\frac \delta{\delta \varphi^{i}}\right) \; \; ,
L^t_k := \frac{1}{mk}[-t^{mk+1}\partial_t, L_0^t] = \frac{1}{mk(c+\check{h})}\left(\sum_{n>0}n \sum_{a\in A_{-\underline{n}}}\left(u_{a}[t^{-n+mk}] u^a[t^n] -u_a[t^{-n}]u^a[t^{n+mk}]\right)\right), \,\,\,,
[X_{c(t)}^*]^{\perp}= \left\{\tilde{u}\in L^2_{\mu}(\mathbb{R}\times \mathbb{T}) : \langle u_{c(t)}, \tilde{u}\rangle_{L^2(\mathbb{R}\times \mathbb{T})}=\langle \partial_{\xi}^{-1}\partial_c u_{c(t)}, \tilde{u}\rangle_{L^2(\mathbb{R}\times \mathbb{T})}= 0\right\}.
\omega{f^a}{e^b}{\omega^c}{\omega^{'}}^d ={q^{b - a}}{f^a}{e^b}{\omega^{c + 1}}{\omega^{'}}^d,{\omega^{'}}{f^a}{e^b}{\omega^c}{\omega^{'}}^d ={q^{a - b}}{f^a}{e^b}{\omega^c}{\omega^{'}}^{d + 1}.
I_{\Psi}= -\int_{{\cal M}_b}d^5x\sqrt{-g}\left[\frac{1}{2}g^{MN}\partial_M\Psi\partial_N\Psi +U(\Psi)\right] - \sum_{\sigma=\pm}\int_{\Sigma_{\sigma}}d^4y_{\sigma}\sqrt{-q_{\sigma}}V_{\sigma}(\Psi_{\sigma})
{\frac{1}{\sum_{n^{\prime}, k^{\prime}}\exp \! \left[{-{\frac{( E_{n^{\prime}}- E )^{2}- ( E_{n}- E )^{2}}{2 \varepsilon^{2}}}}\right]}}< \exp \! \left[{-{\frac{( E_{n}- E )^{2}- ( E_{n_{0}}- E )^{2}}{2 \varepsilon_{0}^{2}}}}\right]
E_\pi(T_1, \ldots, T_n) = E\left( T'_{q_1}\cdots T'_{q_{k_1}}X_m T'_{q_{k_1+1}}\cdots T'_{q_{k_{m-p+1}}}X_{p}T'_{q_{k_{m-p+1}+1}}\cdots T'_{q_{k_{m-p+2}}}R_{E_{\pi|_{Y}}((T_1, \ldots, T_n)|_{Y})}\right)
-\frac{\partial^2f}{\partial s^2}+ \frac{\partial}{\partial u}\cdot(\mathcal{G}f)-\epsilon f\kappa : u\otimes u+ \epsilon \frac{\partial}{\partial s}\Big( f\kappa:\lambda(f)\Big) &+\dfrac{\epsilon}{4\pi}\int_{S_2}\kappa:v \otimes vf(s,v )d\mu(v)= \dfrac{3+\epsilon}{4\pi}\kappa:u \otimes u \textrm{on}Q f(s=0) =f(s=1) = 0\phantom{UYTUY}
S_{r,s}\, =\,\sum_{\substack{\rho(\underline{i})= r \sigma(\underline{i}) = s}}\left| \mathbb{E}\left[ X_N(\underline{i}) \right]\right| \intertext{and}S_{r}\, =\,\sum_{\rho(\underline{i})= r}\left| \mathbb{E}\left[ X_N(\underline{i}) \right]\right|
(x+y)^{m}P_{m}^{(\alpha,\beta)}\left(\frac{x-y}{x+y}\right)=\frac{(\alpha+1)_{m}}{m!}\,x^{m}\,{}_2F_{1}\left( \genfrac{}{}{0pt}{}{-m,-m-\beta}{\alpha+1}\;\Big \rvert \; -\frac{y}{x}\right),
\int \frac{1}{(-x^2;q^2)_\infty}\left( qx[n-1]_q+c[n]_q\right)\widetilde{h}_n(x;q) d_qx= & \frac{1-q}{(-x^2;q^2)_\infty}\left( \widetilde{h}_n(x;q)- q^{1-n}[n]_q (c+x) \widetilde{h}_{n-1}(x;q)\right),
u_i =\left(1_n\backslash\big\{\{i,i'\},\{i+1,(i+1)'\}\big\}\right)\cup\big\{\{i,i+1\},\{i',(i+1)'\}\big\}\sigma_i =\left(1_n\backslash\big\{\{i,i'\},\{i+1,(i+1)'\}\big\}\right)\cup\big\{\{i,(i+1)'\},\{i+1,i'\}\big\}.
f_{n}=z^{n}\sum_{k=0}^{\infty}\left(q^{-k}z\alpha;q\right)_{-n}\left(q^{k+1}z^{-1}\alpha^{-1};q\right)_{\!\infty}\frac{q^{\frac{1}{2}k(k+1)}}{(q;q)_{k}}(-1)^{k}\alpha^{-k}z^{k}.
2{\displaystyle\sum\limits_{i=1}^{s}}(\mu_{n-i+1}^{2}+\overline{\mu}_{n-i+1}^{2}) \leq2sn+n\left(\mu+\overline{\mu}\right) -\mu^{2}-\overline{\mu}^{2}& \leq2sn+n\left( \mu+\overline{\mu}\right) -\left( \mu+\overline{\mu}\right)^{2}/2 & \leq2sn+n^{2}/2,
\mathcal{R}_{t}^c(\alpha)=\left\{t_{np}\ge \frac{p(p-1)}{2(n-1)}(\sqrt{\frac{n-2}{n+1}}-1)+\chi^2_{\alpha}(p(p-1)/2)\sqrt{\frac{(n-2)}{(n-1)^2(n+1)}}\right\}
\varepsilon_1^U := C_{s,p}\left[ \frac{\log(1+d/\log d)}{\log d}\right]^{s/p}, \varepsilon_2^U := C_{s,p}d^{-s/p}, \varepsilon_3^U := C_{s,p}2^{-s}d^{-s(1/p+1/\gamma)},
\lim_{\epsilon\to 0}\frac{D_\epsilon \tilde{L}_{0,1}D_\epsilon^{-1}- D_\epsilon \tilde{L}_{0,0}D_\epsilon^{-1}}{\epsilon^3}= \lim_{\epsilon \to 0}\frac{\epsilon(U_\epsilon - U) + \O(\epsilon^4)}{\epsilon^3}= \lim_{\epsilon \to 0}\frac{U_\epsilon - U}{\epsilon^2}= \frac{d}{dt}U.
\tilde{\cal L}_{\kappa = 0}= - 8 ( D - 2 )^{2}\int d r \left({\frac{\partial W}{\partial \phi}}\right)^{2}\left( 1 +{\frac{1}{2}}\partial_{\mu}\tilde{r}\partial^{\mu}\tilde{r}\ -{\frac{1}{2}}\bar{\tilde{\eta}}_{- i}\tilde{\Gamma}^{\mu}\partial_{\mu}\tilde{\eta}_{-}^{i}\right) \ .
B(x|t)B(y|t) = B(y|t)B(x|t) &D(x|t)B(y|t) = D(y|t)B(x|t) & B(y|t)D(x|t)=(y\ominus x)D(x|t)B(y|t)+(1+\beta y\ominus x)B(x|t)D(y|t) &D(x|t)D(y|t) = D(y|t)D(x|t)
\det(F^*+pF-sI) = \det\begin{bmatrix}\overline{w_{11}}+ w_{11}p-s p(w_{12}+\overline{w_{21}}) (\overline{w_{12}}+w_{21}) w_{22}+ \overline{w_{22}}p -s\end{bmatrix}&=(\overline{w_{11}}+ w_{11}p-s)(w_{22}+ \overline{w_{22}}p -s)-p|\overline{w_{12}}+w_{21}|^2.
\lim_{\varepsilon \to 0}\frac{1}{\sigma}\int_{\mathbb{R}^d \times (0,\infty)}-\lambda^\varepsilon \sqrt{2W(\varphi^\varepsilon)}\nabla\varphi^\varepsilon \cdot \Phi \, dxdt = \int_{\mathbb{R}^d \times (0,\infty)}g \cdot \Phi \, d\mu
\frac{\partial^2 h}{\partial t^2}-c_1c_2\frac{\partial^2 h}{\partial x^2}+ (c_1-c_2)\frac{\partial^2 h}{\partial x\partial t}=-\frac{1}{t}\Biggl[(m+n+2)\frac{\partial h}{\partial t}+\Bigl[ (c_1-c_2)(m+n+1)-(c_1m-c_2n) \Bigr]\frac{\partial h}{\partial x}\Biggr]
\Big( \underset{i=1}{\overset{b}{\sum}}Z_i^2 \leq \frac{24{b}m k}{C_\alpha(m-1)}N_\alpha \Big) + \left( \underset{1 \leq i \leq b}{\max}Z_i^2 \geq C_\alpha^{-1}{m \sqrt{b}k}\right) \qquad\leq \left( \underset{i=1}{\overset{b}{\sum}}Z_i^2 \leq \frac{24{b}m k}{C_\alpha(m-1)}N_\alpha \right) + \alpha/8.
\gamma_{k}=\frac{\overline{\lambda}_{k-1}-\overline{\lambda}_{k}}{\overline{\lambda}_{k-1}}\leq \frac{\overline{\lambda}_{k-1}-\rho (A^{-1})}{\rho (A^{-1})}=\frac{\varepsilon_{k-1}}{\rho (A^{-1})}.
J_{r}^{( n )}( t ; \{q \}; L ) \approx \frac{\hat{S}_{r}^{( n )}( \{q \}; L ) + \hat{\Sigma}_{r}^{( n )}( \{q \}; L ) \vartheta}{z - z_{0}}\equiv \frac{\hat{J}_{r}^{( n )}( \{q \}; L ; \vartheta )}{z - z_{0}}.
\sum_{k=1}^{p-1}a_kw^k=\begin{cases}-\sqrt{q}(q-1)=\displaystyle\sum_{k =1}^{p-1}-\sqrt{q/p}\ (q-1)\left(\frac k p\right)w^{k}q\equiv 1 \pmod 4, begin{align*}20pt]\;i\sqrt{q}(q-1)=\displaystyle\sum_{k =1}^{p-1}\sqrt{q/p}\ (q-1)\left(\frac k p\right)w^{k}q\equiv 3 \pmod 4. \end{cases}
\aligned2\,a_3+b_4+(\rho-3)\,a_2=0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\,b_2+\frac{\rho-1}{2}a_1=0, 2\,b_2+a_1+(\rho-3)\,b_3=0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\,a_3+\frac{\rho-1}{2}b_4=0.\endaligned
=2\binom{n+2}{4}+ 2\sum_{i=2}^{n+1}\binom{i}{2}= 2 \left ( \binom{n+2}{4}+ \sum_{i=2}^{n+1}\binom{i}{2}\right ) = 2 \left ( \binom{n+2}{4}+ \binom{n+2}{3}\right ) =2 \binom{n+3}{4}.
B = \frac{\pi}{2}+ \gamma + \log{8\pi}- \log \operatorname{rd}(K) - \int_0^\infty \frac{1-F(x)}{2}\left( \frac{1}{\sinh{\frac{x}{2}}}+ \frac{1}{\cosh{\frac{x}{2}}}\right) \, dx + 2 \sum_{p \in S}\sum_{m=1}^\infty \frac{\log p}{p^{m/2}}F(m \log p).