harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
[y_i , y_{(j\,j+1)}]_c = \lambda_{i\,j}(1-\chi_{j+1}(g_ig_j))y_{j+1}+\lambda_{i\,j+1}(\chi_{j}(g_i)-\chi_{j+1}(g_j))y_j &= \lambda_{i\,j}(1-\chi_{j+1}(g_ig_j)\chi_i\chi_j(g_{j+1}))y_{j+1}+\lambda_{i\,j+1}(\chi_{j}(g_i)-\chi_i^{-1}(g_j))y_j &= 2\lambda_{i\,j}y_{j+1}.
\langle \hat{\gamma}^0_{q,(-h,h)}(w^s_q(\tau)), e_x(w^s_q(\tau))\rangle-\langle \hat{\gamma}^0_{q,(-h,h)}(w^s_q(0)), e_x(w^s_q(0))\rangle=\beta(a,\omega,n)\tau+\mathcal{O}_*(h)
h = \sum_{i=0}^{s}\sigma_i\cdot g_i+\sum_{j=1}^{n}\omega_j\cdot (p_j^2\cdot w_{j,1}+ q_j^2\cdot (x_j+r_j)^2) = \underbrace{(\sigma_0+\sum_{j=1}^{n}q_j^2\cdot\omega_j\cdot (x_j+r_j)^2)}_{\sigma_{0}}+\sum_{i=1}^{s}\sigma_i\cdot g_i+\sum_{j=1}^{n}\underbrace{p_j^2\,\omega_j}_{\omega_{j}}\cdot w_{j,1}.
\int_0^\infty d\tau \int_0^\infty d\tau'\, \sigma_t^{(\alpha/2)}(\tau)\, \sigma_{t'}^{(\alpha/2)}(\tau')\, f(\tau+\tau') = \int_0^\infty d\tau\, (\sigma_t^{(\alpha/2)}\ast\sigma_{t'}^{(\alpha/2)})(\tau)f(\tau) = \int_0^\infty d\tau\, \sigma_{t+t'}^{(\alpha/2)}(\tau)f(\tau), f\in L^1.
R_4' = \int_{\mathcal{X}_n}f(x) \int_{\frac{a_n}{n-1}}^1 \log\biggl(\frac{(n-1)s}{e^{\Psi(k)}f(x)}\biggr) \, \mathrm{B}_{k,n-k}(s) \, ds \, dx = o(n^{-(3-\epsilon)}),
\protect \eta = -{\frac{\mu}{\ln \mu}}\left( 1 +{\frac{\ln ( - \ln \mu )}{\ln \mu}}+ \left({\frac{\ln ( - \ln \mu )}{\ln \mu}}\right)^{2}-{\frac{\ln ( - \ln \mu )}{( \ln \mu )^{2}}}+ \cdots \right) .
k = \frac{2{\cal D}( 0 ) - p - 2}{2 ( 1 + p )}\, \, , \, \, \, \, \, \, \, \, \, \, k_{1}= \frac{p}{1 + p}\min \left( \frac{C_{0}}{p}- \frac{\nu}{4}, \frac{1}{2}- \nu \right) \, \mathrm{.}
\frac{\Vert \gamma_{1}^{\prime}(t)\Vert_{g^{M_{1}}}}{\Vert \gamma_{1}\Vert}=\frac{\Vert \gamma_{2}^{\prime}(t)\Vert_{g^{M_{2}}}}{\Vert\gamma_{2}\Vert},\ \ \ \ \a\leq t\leq b.
\big(\sqrt{n}T_n^B,\sqrt{n}T_n\big) = \big(\sqrt{n}(\hat{\mu}_n^B - \hat{\mu}_n)*\gamma_\sigma + \sqrt{n}(\hat{\mu}_n - \mu)*\gamma_\sigma, \sqrt{n}(\hat{\mu}_n - \mu)*\gamma_\sigma \big) &\stackrel{d}{\to}\big(G_{\mu}'+G_{\mu},G_\mu \big) \dot{H}^{-1,p}(\mu*\gamma_\sigma) \times \dot{H}^{-1,p}(\mu*\gamma_\sigma)
I_{c o v}( p ) = ( - i )^{n + 1}\int q . d^{2 \omega}\int_{0}^{\infty}d \alpha_{1}\cdots d \alpha_{n}d \beta \cdot 2 q^{-}. e^{i [ \sum_{1}^{n}\alpha_{i}f_{i}( q , p ) + \beta 2 q^{+}q^{-}]},
Op_{1,h}(a)u(x)=\mathcal{F}_h^{-1}(a(x,\xi;h)\mathcal{F}_h u(\xi))(x), &Op_{0,h}(a)u(x)=\mathcal{F}_h^{-1}(\mathcal{F}_h (a(y,\xi;h)u(y))(\xi))(x),
(m + 1)^2(m + k) G_{m+1,k}(t) = (2m + 2k +1) [m^2 + m + 2km + k - (m + k)(2m+ 2k + 2)t]G_{m,k}(t) - (m +k +1)(m + 2k)^2G_{m - 1,k}(t).
k^{u}_{j+1}(T) \leq k^{u}_{0}(T)+ C \Big(\sup_{0< t< T}e^{-\frac{\omega t}{2}}t^{\frac{3}{2}(\frac{1}{p}-\frac{1}{q})}\int_{0}^{t}(t-s)^{-\frac{1}{2}-\frac{3}{2q}}s^{-3(\frac{1}{p}-\frac{1}{q})}\ \d s \Big) [k_{j}^{u}(T)^2+k_{j}^{\nabla y}(T)^2].
\begin{cases}\begin{array}{ll}\dot{v}=\log\det (v_{\alpha\bar{\beta}})-n\log (At+1)+f(z,\frac{\log (At+1)}{A})\;\; \mbox{on}\; \Omega\times (0,\frac{e^{AT}-1}{A}), v(z,0)=u_0(z) \mbox{on}\;\Omega, v(z,t)=\varphi (z,\frac{\log (At+1)}{A}) \mbox{on}\; \partial\Omega\times [0,T). \end{array}\end{cases}
ds^{2}= -\frac{r^{2}}{\gamma^{4}}\left ( 1 - \frac{2{\sqrt 2}\gamma^{2}M}{Qr}\right ) dt^{2}+ \left ( 1 - \frac{2{\sqrt 2}\gamma^{2}M}{Qr}\right )^{-1}dr^{2}+ r^{2}d\Omega^{2}
\hat{V}\left( x \right) = V \left( x \right) \left( \begin{array}{c c c c}{{0}}&{{1}}&{{\cdots}}&{{1}}\\{{1}}&{{0}}&{{\cdots}}&{{\vdots}}\\{{\vdots}}&{{\vdots}}&{{\ddots}}&{{1}}\\{{1}}&{{\cdots}}&{{1}}&{{0}}\\ \end{array}\right)
A=\sum_{i=1}^\infty\frac{1}{b_i\pi_i},~~~B=\sum_{i=1}^\infty{\pi}_{i},~~~R=\sum_{i=1}^\infty\frac{1}{b_i\pi_i}\sum_{j=1}^{i}\pi_{j},~~~S=\sum_{i=1}^\infty\frac{1}{b_i\pi_i}\sum_{j=i+1}^{\infty}\pi_{j}.
\|D_{X}C_{X}^{-1}\|_{1}=\sum_{j=1}^{\infty}D_{j}C_{j}^{-1}=\sum_{j=1}^{\infty}D_{X}(\phi_{j})(\phi_{j})[C_{X}(\phi_{j})(\phi_{j})]^{-1}=\|\rho \|_{1}.
{\frac{\partial}{\partial t}\psi_{\mu}(t,z)}=-\sigma^{2}\psi_{\mu}(t,z)\frac{\partial}{\partial z}\psi_{\mu}(t,z)-\eta \frac{\partial}{\partial z}\psi_{\mu}(t,z)-\frac{\partial}{\partial z}\psi_{\mu}(t,z)\int_{\mathbb{R}\backslash \{0\}}\frac{\psi_{\mu}(t,z)+r}{1-r\psi_{\mu}(t,z)}\rho(dr).
Z = \sum_{h_{i j}}\prod_{n i}\int_{- \infty}^{\infty}\! d A_{i}( n ) \; e x p \: ( - \frac{1}{4 \kappa^{2}}{\sum_{n i j}\: [ \bigtriangleup_{i}A_{j}( n ) - \bigtriangleup_{j}A_{i}( n ) + h_{i j}( n ) ]^{2}}) .
P ( \sigma ) ={\frac{1}{\sqrt{2}}}\sum_{n = 1}^{\infty}( \eta_{n}+ \tilde{\eta}_{n}) \cos ( 2 n \sigma ) +{\frac{1}{\sqrt{2}}}\sum_{n = 1}^{\infty}n ( \xi_{n}- \tilde{\xi}_{n}) \sin ( 2 n \sigma )
\left(\begin{array}{ccccc}E a \frac{c}{2}\frac{1}{3}0 0 E a \frac{c}{2}\frac{1}{3}a c -1 0 0 0 a c -1 0 0 0 a c -1\end{array}\right)\left(\begin{array}{c}I_0 I_1 I_2 I_3 I_4\end{array}\right)=\left(\begin{array}{c}T M 0 2T 4M\end{array}\right)
H = \int d h_{i a}^{*}\exp \left\{\frac{i}{\hbar}\int d^{4}x \left( - \frac{\alpha}{2}\varepsilon_{a b}G_{i}^{a}G^{i b}+ h_{i a}^{*}\bar{G}^{i a}+ \frac{1}{2 \alpha}h_{i a}^{*}{\cal M}^{a b}h_{b}^{* i}\right) \right\},
\left\| W_{n+1}- W \right\|_{\mathrm{F}}^2 \leq \left\|W_{n}- W \right\|_{\mathrm{F}}^2- \beta_n \left(1-\beta_n \right) \left\| W_n - S_n\left(W_n\right) \right\|_{\mathrm{F}}^2 &\leq \left\|W_{n}- W \right\|_{\mathrm{F}}^2,
\int e^{i(\theta' \cdot A^{-1}(w)-\theta\cdot w)}\Phi(w) dw =\int \left(\mathcal{D}_je^{i(\theta' \cdot A^{-1}(w)-\theta\cdot w)}\right)\Phi(w) dw &= \int e^{i(\theta' \cdot A^{-1}(w)-\theta\cdot w)}\cdot \left({}^t\mathcal{D}_j\right)\Phi(w) dw
Z_{g}^{( 1 )}= \delta_{A C \bar{C}}^{2}- Z_{C}^{( 1 )}- \frac 1 2 Z_{A}^{( 1 )}= - \frac{1 1}6 \frac{( g \mu^{- \epsilon})^{2}}{( 4 \pi )^{2}}\frac{C_{2}( G )}{\epsilon}.
\lim_{N \rightarrow \infty}[\mathbb{E}^V-\mathbb{E}^G] \theta(\mathbf{p}_{\alpha^{\prime}}, N \xi_{\alpha^{\prime}}(i) \xi_{\alpha^{\prime}}(j), N \zeta_{\alpha^{\prime}}(\mu) \zeta_{\alpha^{\prime}}(\nu))=0.
\left(f+\bar{\psi}\circ L\right)^{*}_\Phi\left(\phi\right) =\sup_{x\in X}\left\{\phi\left(x\right)-f\left(x\right)-\bar{\psi}\left(Lx\right)\right\}& \geq\sup_{x\in X}\left\{\phi\left(x\right)-\phi_{\varepsilon_2}\left(x\right)-f\left(x\right)\right\}-\sup_{x\in X}\left\{\bar{\psi}\left(Lx\right)-\phi_{\varepsilon_2}\left(x\right)\right\}
X \mapsto \frac{\alpha X + \beta Y + \gamma}{\alpha'' X + \beta'' Y +\gamma''}, \Y \mapsto \frac{\alpha' X + \beta' Y +\gamma'}{\alpha'' X + \beta'' Y +\gamma''}, \textrm{for}\begin{bmatrix}\alpha \beta \gamma \alpha' \beta' \gamma' \alpha'' \beta'' \gamma''\end{bmatrix}\in PGL_3(k)
c_{jk}= M\int_0^M\phi_{N+j}(\eta)\,d\eta\int_0^M\phi_{N+k}(\eta')\,d\eta'- \int_0^M\phi_{N+j}(\eta)\,d\eta\int_0^M\eta'\phi_{N+k}(\eta')\,d\eta'- \gamma_{jk}, &\mbox{where}\gamma_{jk}= \int_0^M\int_0^M(\eta-\eta')_+\phi_{N+j}(\eta)\phi_{N+k}(\eta')\,d\eta d\eta'.
R_{Pol}=\frac{D[(\frac{\pi}{6\phi})^{\frac{1}{3}}-1](1-\frac{\sigma}{M})}{A\sqrt{2mV_{a}}}(\frac{h}{e})^{2}exp(\frac{4\pi D}{h}[(\frac{\pi}{6\phi})^{\frac{1}{3}}-1](1-\frac{\sigma}{M})\sqrt{2mV_{a}})
\begin{aligned}\sum_{\alpha\in H^1(\mathrm{Gal}(E/F),G(E))}\mathrm{dim}\mathrm{Hom}_{G_{\alpha}(F)}(\pi,\omega_{G_{\alpha}(F),E})=\sum\limits_{\tilde{\phi}}m(\lambda_{\pi},\tilde{\phi}),\end{aligned}
{\bf M}_{V_N}\, \big({\bf M}_{V_N}^2+1\big)^{-1/2}w\ =\ \frac{2}{\pi}\int_0^\infty{\bf M}_{V_N}\, \big({\bf M}_{V_N}^2+1+\lambda^2\big)^{-1}\,w\,d\lambda\,,
\tilde{\mathcal{B}}_{ac}^{bd}-(\tilde{\mathcal{B}}^{-1})_{ac}^{bd}=(q^{\frac{1}{2}}-q^{-\frac{1}{2}})(\mathcal{M}_{a c}\mathcal{M}^{bd}-\delta_{a}^{b}\delta_{c}^{d})\hbox{on}V_{2n-2},
H_{T^{n+1}}^{*}(Pl(k,n);\mathbb{Q})= \Big \{\alpha\in \bigoplus_{i=0}^m \mathbb{Q}[y_1,\dots, y_n,z]/(Y_{\lambda^i}+z)~\big{|}~ (\alpha_{i}-\alpha_{j})\in (Y_{\lambda^i}+z,Y_{\lambda^j}+z) |\lambda^i\cap \lambda^j|=k-1 \Big \}.
I ( n_{b}, n_{b}; a^{( b )}) = \int d \mu_{Q}[ A ] d \mu_{C}[ h^{\prime}] \; \; \exp \left( - 2 \sum_{b = 1}^{N}\sum_{j = 1}^{n_{b}}\Big ( \chi^{( b )}, \delta_{n}( x_{j}^{( b )}) \! - \! \delta_{n}( y_{j}^{( b )}) \Big ) \right)
\langle \:{\hat{\phi}}_{\bar{t}}\prod_{i \in S}\phi_{i}\: \rangle_{g , \:{\Delta_{1}}}\; = \; \frac{1}{2}\: B_{\bar{t}}^{' \alpha \beta}\langle \: \phi_{\alpha}\phi_{\beta}\prod_{i \in S}\phi_{i}\: \rangle_{g - 1}\,
\Omega_{i}^{( 2 ; 4 - 2 \varepsilon )}= \int_{0}^{\tau_{0 i}}\frac{\mathrm{d}\theta}{\cos^{2 - 2 \varepsilon}\theta}= 2 \tan \tau_{0 i}_{2}F_{1}\left( \left. \begin{array}{c}{1 / 2 , \varepsilon}\\{3 / 2}\\ \end{array}\right| - \tan^{2}\tau_{0 i}\right) ,
\psi_{\boldsymbol{i}}:= \psi_{i_1}\circ \dots \circ \psi_{i_n}:= \psi^{F^\emptyset}_{i_1}\circ \psi^{F^{i_1}}_{i_2}\circ \psi^{F^{i_1i_2}}_{i_3}\circ \dots \circ \psi^{F^{i_1\dots i_{n-1}}}_{i_n}
\partial_t w-\partial_x^5 w= \; -b_1(u_1\partial_x^3w+\partial_x^3u_2 w)-b_2(\partial_x u_1\partial_x^2w+\partial_x^2u_2\partial_xw) &- b_3(\partial_xu_2(u_1+u_2)w+u_1^2\partial_x w).
\begin{aligned}\uppercase\expandafter{\romannumeral2}=\sum\limits_{j=1}^{\infty}\mu_{j}\vert g(b_{j})(x)\vert\chi_{4P_{j}}(x)+\sum\limits_{j=1}^{\infty}\mu_{j}\vert g(b_{j})(x)\vert\chi_{(4P_{j})^{c}}(x) &=\uppercase\expandafter{\romannumeral1}_{1}+\uppercase\expandafter{\romannumeral1}_{2}.\end{aligned}
||\Phi'(u)-\Phi'(v)||_{H^{-1}(\Omega)}=||f(u)-f(v)||_{H^{-1}(\Omega)}&\leq C(n,p,\Omega)||f(u)-f(v)||_q &\leq C(n,p,\Omega)\left(||u-v||_{pq}+||u-v||_{pq}^{p-1}\right).
2FS_0 - L_{00}- F^2L_{ww}= 2F [ 2w^1{w^1}_{x_1}+ w^2({w^1}_{x_2}+{w^2}_{x_1})] y^1 &+ 2F[ w^1({w^1}_{x_2}+{w^2}_{x_1}) + 2w^2{w^2}_{x_2}] y^2 &- 2{w^1}_{x_1}(y^1)^2 - 2({w^1}_{x_2}+{w^2}_{x_1}) y^1 y^2 - 2{w^2}_{x_2}(y^2)^2 &- F^2 ( 2 (w^1)^2{w^1}_{x_1}+ 2w^1w^2 ({w^1}_{x_2}+{w^2}_{x_1}) &+ 2(w^2)^2{w^2}_{x_2}).
R_1 \geq H(M)_\sigma &\geq \frac{1}{2}[H(A^n)_\theta + H(R^nE')_\theta - H(LT_B')_\sigma] &= \frac{1}{2}[H(A^n)_\theta + H(R^nE')_\theta - H(A^nR^nE')_\theta] &= \frac{1}{2}I(A^n; R^nE')_\theta,
\widetilde V^D (G^D( \cdot, y))(x) =u(\infty) G^D( x, y) +\int_{(0,\infty)}\int_0^s p^D(t,x,y)dt\widetilde \mu(ds) &=u(\infty) G^D( x, y) +\int_{0}^\infty\widetilde \mu((t, \infty)) p^D(t,x,y)dt &=u(\infty) G^D( x, y) +\int_{0}^\infty (u(t) -u(\infty) ) p^D(t,x,y)dt =U^D(x,y).
{J_{5}^{\mu}}= - \sum_{r = 0 , 2 , . . .}{\tilde{\overline{{\Psi}}}}_{L_{r}}\gamma^{\mu}\tilde{\Psi}_{L_{r}}+ \sum_{s = 1 , 3 , . . .}{\tilde{\overline{{\Phi}}}}_{L_{s}}\sigma^{3}\gamma^{\mu}\tilde{\Phi}_{L_{s}}.
a\frac{(q)_\infty^2[-a,1/a^2]_\infty}{[x,-xq/a,-ax]_\infty}= \frac{[1/a^2]_\infty}{[-1/a]_\infty}\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}q^{n(3n+1)/2}}{1-xq^n}+ \sum_{n=-\infty}^{\infty}\frac{q^{n(3n+1)/2}a^{-3n-1}}{1+xq^{n}/a}& - \sum_{n=-\infty}^{\infty}\frac{q^{n(3n+1)/2}a^{3n}}{1+axq^n}.
J=\frac{\partial(q,r)}{\partial (x,s)}=\left(\begin{array}{cc}A B -(B^{-1})^2A B^{-1}\end{array}\right)\,,J^{-1}=\frac{1}{2}\left(\begin{array}{cc}A^{-1}-A^{-1}B^2 B^{-1}B\end{array}\right)\,,
|\zeta_D-\zeta_B| \geq |\zeta_{p(D)}-\zeta_B|-|\zeta_{p(D)}-\zeta_{D}| \geq |\zeta_{p(D)}-\zeta_B|-c|\zeta_{p(D)}-\zeta_{S'}| \geq |\zeta_{p(D)}-\zeta_B|-c c' |\zeta_{p(D)}-\zeta_{B}| &= |\zeta_{p(D)}-\zeta_B|(1-c c') \geq |\zeta_{C}-\zeta_B|\frac{1-c c'}{c'},
R_N^3 = -\frac{17}{5760}\frac{(b-a)^5 f^{(4)}(\xi)}{(N-2)^4}\xi \in [a-h/2, b+h/2] R_N^5 = \frac{367}{967680}\frac{(b-a)^7 f^{(6)}(\xi)}{(N-4)^6}\xi \in [a-3h/2, b+3h/2]
\begin{aligned}u_{11}=\frac{-\cos\theta}{\alpha^2\cos \theta-\alpha\beta \sin \theta},u_{12}=\frac{-\sin \theta}{\alpha^2\sin\theta+\alpha\beta \cos\theta}, u_{21}=\frac{-\sin \theta}{\beta^2\sin\theta-\alpha\beta \cos\theta},u_{22}=\frac{-\cos\theta}{\beta^2\cos\theta+\alpha\beta \sin \theta}.\end{aligned}
<0|a_1\left(\vec{p}_1\right){\phi}_1(x)a_1^{+}\left(\vec{q}_1\right) a_1^{+}\left(\vec{q}_1^{\,'}\right)|0><0|a_2\left(\vec{p}_2\right){\phi}_2(x)a_2^{+}\left(\vec{q}_2\right) a_2^{+}\left(\vec{q}_2^{\,'}\right)|0>
\left(\int_{\Omega\cap\{|\eta|\leq 2|\gamma|\}}\left| |\eta\gamma|^{\alpha-Q}-|\eta|^{\alpha-Q}\right|^{Q'}d\eta\right)^{1/Q'}\leq C\left(\int_{\{|\eta|\leq B|\gamma|\}}|\eta|^{(\alpha-Q)Q'}d\eta\right)^{1/Q'}\leq C|\gamma|^{\alpha-Q+\frac Q{Q'}}=C|\gamma|^{\alpha'}.
S_{k}S_{k}[r_{k}(u_{1})r_{k+1}(u_{2})\cdots r_{2k-1}(u_{k})] [r_{k-1}(u_{k+1})\cdots r_{2k-2}(u_{2k})]\cdots[r_{1}(u_{k^{2}-k+1})\cdots r_{k}(u_{k^{2}})]S_{k}S_{k}\;,
L_{\tilde{a}}^{\ast}g_{(\mu_1,\mu_2)}=-e^{-2\mu_1(a^2+x^2)}\{d(a^0+e^{\mu_1a^2}x^0)\}^2+ e^{-2\mu_2(a^2+x^2)}\{d(a^1+e^{\mu_2a^2}x^1)\}^2+\{d(a^2+x^2)\}^2 =-e^{-2\mu_1x^2}(dx^0)^2+e^{-2\mu_2x^2}(dx^1)^2+(dx^2)^2.
G_{\phi, p}(x)= L_p(1+x, \phi)L_p(1 + 3x, \phi) & \times\left(1 +\frac{\rho_\phi(p^2)-\rho_\phi(p)^2}{p^{2 + 4x}}+O\left(\frac{(1 + |\rho_\phi(p)|)^{3}}{p^{3 + 7x}}\right)\right).
\hat{h}_t(p,k) = \hat{h}_0(p,k)e^{-2\pi i \left(\frac{k}{L}\cdot v(p)\right)t}\;\;\;\;\;\;\;\;- \int_0^t \widehat{S_h}(\tau,p, k)e^{-2\pi i \left(\frac{k}{L}\cdot v(p)\right)(t-\tau)}d\tau.
\frac{[ a^{\prime}]}{a}= - \frac{\kappa^{2}}{3}( \lambda_{-}( \phi ) + \rho ) b \mathrm{,}\frac{[ n^{\prime}]}{n}= \frac{\kappa^{2}}{3}( - \lambda_{-}( \phi ) + 3 p + 2 \rho ) b
\int_0^1 \Theta(s,x)U(s,x)\mathrm{d}x-\frac{1}{d}\sum_{i=1}^d \Theta(\frac{\lfloor sT \rfloor}{T},\frac{i}{d})U(\frac{\lfloor sT \rfloor}{T},\frac{i}{d}) =o(1) \frac{2}{d}\sum_{i=1}^d \Theta(\frac{\lfloor sT \rfloor}{T},\frac{i}{d})\big( U(\frac{\lfloor sT \rfloor}{T},\frac{i}{d})-U^{d,T}(\frac{\lfloor sT \rfloor}{T},\frac{i}{d})\big) =o(1).
\tilde{A}(p) = 3 g^{2}N(N^{2}-1)\delta_{ab}\int \frac{d^{4}k}{(2\pi)^{4}}d^{4}\theta \, \left\{\frac{m^{2}\left[ V^{a}(p,\theta,{\overline\theta}) V^{b}(-p,\theta,{\overline\theta})\right]}{(k^{2}+m^{2})[(p-k)^{2}+m^{2}]}\right\}.
\tilde \gamma^i_t = \tilde \gamma^i (t, X_{t-}) =\frac{\theta+\left(1-\frac{1}{n}\right)\eta_{t}}{1+\frac{1}{\lambda}\left(1-\frac{1}{n}\right)^2\eta_{t}}(\bar X_{t-}-X^i_{t-}) , i=1,\ldots,n,
\frac{d^2}{d\theta^2}= g_zE^{n,j}D_{x_j}(g_zE^{m,i}D_{x_i})q_mq_n = g_{z}^2E^{n,j}E^{m,i}q_mq_nD_{x_ix_j}+g_z^2q_mq_nE^{n,j}D_{x_j}(E^{m,i})D_{x_i}\quad\quad+ g_zg_{j,z}E^{n,j}E^{m,i}q_mq_nD_{x_i}.
g_i(s_j)\begin{cases}<0 \frac{j-i}k\in \cup_{m=0}^{\infty}(2m,2m+1); =0 \frac{j-i}k\in\{0,1,2,\dots\}; >0 \frac{j-i}k\in (-\infty,0)\cup\cup_{m=0}^{\infty}(2m+1,2m+2).\end{cases}
\sup_k\|U_k\|_{H^1(B_{R_0})}\le C\sum_{j=0}^\infty|g_j|(j+1) &\le C\Big(\sum_{j=0}^\infty |g_j|^2(j+1)^{2s}\Big)^{1/2}\Big(\sum_{j=0}^\infty(j+1)^{2-2s}\Big)^{1/2}<\infty.
T^{R}(p^{2}) = T(p^{2}) - T(0) -p^{2}\left. \frac{d}{dp^{2}}T(p^{2})\right|_{p^{2}=0}= \frac{p^{4}}{\pi}\int\limits^{\infty}_{4m^{2}}\!\!\frac{dt}{t-p^{2}}\frac{1}{t^{2}}Im T(t)
E_{\pm N}= \Big \lbrace \mu^{2}+ M_{N}^{2}\pm \Big \lbrack 4 \mu^{2}M_{N}^{2}+ | \bar{\Phi}|^{4}\Big ( V_{t r e e}^{\prime \prime}\Big )^{2}\Big \rbrack^{1 / 2}\Big \rbrace^{1 / 2}\; ,
\mathbf{W}_{\rm D}^{*}\!=\!\frac{1}{1\!-\!\eta_b}\left(\mathbf{\Xi}^{\rm H}\mathbf{\Xi}\!+\!\frac{\sigma^2_{\tilde{\mathbf{e}}}}{(1-\eta_b)^2\sigma^2_{\rm h}}\mathbf{I}_{N_{\rm v}}\right)^{-1}\mathbf{\Xi}^{\rm H}.
\delta T^\psi(X,Y)= \left(\nabla_X\delta\psi,Y\cdot\psi\right)+\left(\nabla_X\psi,Y\cdot\delta\psi\right)=\left(\nabla_X(U\cdot\psi),Y\cdot\psi\right)+\left(\nabla_X\psi,Y\cdot U\cdot \psi\right) = \left(\nabla_X(U)\cdot\psi,Y\cdot\psi\right)+\left(U\cdot(\nabla_X\cdot\psi),Y\cdot\psi\right)+\left(\nabla_X\psi,Y\cdot U\cdot \psi\right)
L = \frac{1}{2}\Gamma_{i}\Lambda^{i j}\dot{\Gamma}_{j}- \rho ( \Gamma ) = q_{i}\dot{x}_{i}+ \frac{B}{2}\epsilon_{i j}x_{i}\dot{x}_{j}- \frac{1}{2}\left( \frac{q_{i}^{2}}{m}+ k x_{i}^{2}\right)
\langle A\rangle=\sum_{j}p_{j}\langle\psi_{j}|A|\psi_{j}\rangle=\sum_{j}p_{j}tr(|\psi_{j}\rangle\langle\psi_{j}|A)=tr(\sum_{j}p_{j}|\psi_{j}\rangle\langle\psi_{j}|A)=tr(\rho A),
Lz(x)=-\sum_{i,j=1}^m a_{ij}(x)\frac{\partial^2 z}{\partial x_i \partial x_j}(x)+\sum_{i=1}^m a_{i}(x) \frac{\partial z}{\partial x_i}(x)+a(x)z(x), \mbox{for $x\in \Omega$,}
+\Big| \frac{1}{2m \ell n}\sum_{x}\Phi_{n}(s,\tfrac{x}{n}) \sum_{j=1}^{m-1}\sum_{z = x-\ell}^{x-1}\int_{\Omega_2^j(x)}[ \eta(z) - \eta(z - j \ell) ] \overrightarrow{\eta}^{\varepsilon n}(x+1) [f(\eta) - f(\eta^{z,z- j \ell}) ] d\nu_{b}\Big|.
\left( \begin{matrix}{{\lambda_{-}}}\\{{\lambda_{+}}}\\ \end{matrix}\right) = \left( \begin{matrix}{{\sin k_{2}x_{2}}}\\{{\cos k_{2}x_{2}}}\\ \end{matrix}\right) e^{i ( \omega t - k_{1}x_{1})}\ \mathrm{w i t h}\k_{2}= \left( n + \mathrm{~ \frac{1}{2}~}\right) \mathrm{~ \frac{\p i}{L}~}\, ,
\Lambda = \langle a, a+d, a+2d, \ldots \rangle . \intertext{Since for $j \geq a$ we have that$a + j \cdot d = d \cdot a + a + (j-a) \cdot d$,the semigroup is generated bythe finite arithmetic progression}\Lambda = \langle a, a+d, a+2d, \ldots, a+(a-1)d\, \rangle .
\langle\hat{b}^{\dagger}\hat{b}\rangle=\frac{Tr(e^{-\beta\omega\hat{b}^{\dagger}\hat{b}}\hat{b}^{\dagger}\hat{b})}{Tr(e^{-\beta\omega\hat{b}^{\dagger}\hat{b}})}=\frac{1}{e^{\beta\omega}-1}
\nu \left( \left \{\omega \in \Omega:\left( f_{n}\left( \omega \right),...,f_{n+k}\left( \omega \right) \right) \in B\right \}\right) =\nu \left(\left \{\omega \in \Omega:\left( f_{n+1}\left( \omega \right),...,f_{n+k+1}\left( \omega \right) \right) \in B\right \}\right) .
C^{1}_1=-B^{1}_{1a,a}=-\mu\omega_{2a}(e_a),~~C^{2}_2=-B^{2}_{2a,a}=\mu\omega_{2a}(e_a), C^{1}_2=-B^{1}_{2a,a}=-\mu\omega_{1a}(e_a), ~~C^{2}_1=-B^{2}_{1a,a}=-\mu\omega_{1a}(e_a).
V \approx \frac{m}{4 \pi}g^{2}\{( 1 + \frac{m^{2}}{8 M^{2}}) \phi + \frac{m^{2}}{4 M^{2}}\mathrm{\boldmath ~ \sigma ~}^{( 1 )}\cdot \mathrm{\boldmath ~ \sigma ~}^{( 2 )}\phi \}
P_{\mu\lambda}^{\ast}\frac{\partial Q^{\lambda\nu}}{\partial x^{\nu}}-\frac{\partial P_{\mu\lambda}}{\partial x^{\nu}}\overset{\ast}{\left.Q^{\lambda\nu}\right.}=-\frac{1}{2}\overset{\ast}{\left. Q^{\sigma\tau}\right.}\frac{\partial P_{\tau\sigma}}{\partial x^{\mu}}.
\Big[r^{a_1,a_2}_{a''_1,a''_2}\Big]_{\le0}=-\sum\limits_{(a_1',a_2')>(a_1'',a_2'')}\Big[r^{a_1,a_2}_{a'_1,a'_2}q_{a_1'',a_2''}^{a_1',a_2'}\Big]_{\le0}.
\frac{\partial^2}{\partial\epsilon^2}D_{GE}(p_{\hat{U}},\epsilon,u) &=\sum\limits_{y\in\mathcal{Y},s\in\mathcal{S}}\frac{p_S(s)\delta^2(u,y,s)}{p_{Y|X,S}(y|\psi(u,s),s)+\epsilon\delta(u,y,s)}&>0,\epsilon\in[0,1], u\in\mathcal{U}\backslash\mathcal{G}_{\delta}.
\underline{A}^{\mu}= \frac 1 \ell \underline{\Theta}^{\mu}+{1\over 2}\epsilon^{\mu}\_{\nu\rho}\underline{\Omega}^{\nu \rho}\qquad , \qquad \underline{\tilde A}^{\mu}= - \frac 1 \ell \underline{\Theta}^{\mu}+{1\over 2}\epsilon^{\mu}\_{\nu\rho}\underline{\Omega}^{\nu \rho}\qquad .
\mathcal{R}\;:=\;\big\{A\in\widetilde{\mathcal{A}}_1\textrm{or}A\in\widetilde{\mathcal{A}}_2\:|\: \partial_t A_j\in L^1_{\mathrm{loc}}(\mathbb{R},L^{b_j}(\mathbb{R}^3,\mathbb{R}^3)),\,j=1, 2\big\}\,.
\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\tilde{\nu})r^{3}=\underbrace{\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\nu_e)}_{O_3(r^{-2\delta-3})}r^3 + \underbrace{\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\tilde{\nu}-\nu_e)}_{O_3(r^{-3\delta-3})}r^{3}=O_3(r^{-2\delta}).
|\mathcal{AI}_{2n}(1432)| = |\mathcal{AI}_{2n}(3214)| = M_n, begin{align*}3pt]|\mathcal{AI}_{2n-1}(3214)| = |\mathcal{RAI}_{2n-1}(1432)| = M_n - M_{n-2}, begin{align*}3pt]|\mathcal{RAI}_{2n}(1432)| = |\mathcal{RAI}_{2n}(3214)| = M_n, begin{align*}3pt]|\mathcal{AI}_{2n+1}(1432)| = |\mathcal{RAI}_{2n+1}(3214)| = M_n.
D_{pert,s\chi_0\eta_0}(\psi+s\mathfrak{c}_0^g-s\mathfrak{h}^g_0+s\mathfrak{e}_0)= s^2(\sum_{i=1}^3\mathcal{Q}_i)-D_{pert}(s\mathfrak{k}^b_0) + s^2\mathcal{A}+s^2\mathcal{B}+s\mathcal{C}\mbox{}mod(\ker(D|_{L^2_1}))
\partial_s^n\left.F_{m,r}(s,\alpha)\right|_{s=-1}=(-1)^n n! \cdot \frac{1}{m}\sum_{j=0}^{m-1}\frac{e^{2 \pi i \frac{jr}{m}}}{\left(-1+2\sin^2\left(\pi\frac{(j+\alpha)}{m}\right)\right)^{n+1}}&= -n! \tilde{S}_{m,r}(\alpha,n+1).
F_\lambda =F_\lambda (x_1,\dots, x_n;t)=\frac{(1-t)^n}{\prod_{i=1}^{n-l}(1-t^i)}\sum_{\sigma \in S_n}\sigma\left(x_{1}^{\lambda_1}\dots x_{n}^{\lambda_n}\prod_{i=1}^{n}\prod_{i < j}\frac{x_{i}-tx_{j}}{x_{i}-x_{j}}\right).
\gamma_{j}^{(r)}(s,\rho) =\frac{1}{2\pi i}\int_{\mathcal{F}}\frac{(\chi_f^{(r)}/\chi_f)(1-s-w)}{w(w+1)\cdots(w+j)}\frac{\Gamma(s+w+\frac{k-1}{2})}{\Gamma(s+\frac{k-1}{2})}(\rho e^{-i\frac{\pi}{2}{\rm sgn}(t)})^wdw
a_{i}^{( \mathrm{e f f})}( x ) ={\frac{\theta_{i}}{L_{i}}}- ( Q + 1 ) \tilde{x}_{i}+ \epsilon_{i j}\partial_{x^{j}}\int \! d \vec{y}G ( \vec{x}, \vec{y}) b^{( 1 )}( y ) -{\frac{e}{\mu c}}\epsilon_{i j}\partial_{x^{j}}\int \! d \vec{y}G_{p}( \vec{x}, \vec{y}) J_{0}( y ) \, .
\eta := \big( \| \nu - 1 \|_\infty + \|\bar{u}^h\|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}\big)\exp\Big\{c_r \|\bar{u}^d\|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}^{2r}\Big\}\leq c_0,
F_{\tau^1}^{(2)}(0)=\frac{n-1}{\mathsf{a}(n,\mathbf{d})}F^{(2)}(0)-\sum_{b=2}^{n}\frac{\partial \tau^b}{\partial t^1}F_{\tau^b}^{(2)}(0)=\frac{n-1}{\mathsf{a}(n,\mathbf{d})}F^{(2)}(0).
g ( x_{+}) = T_{n}\left( \begin{array}{c c}{{1}}&{{0}}\\{{\frac{q x}{2 \pi}}}&{{1}}\\ \end{array}\right) \quad ; \qquad J ( x_{+}) = n \left( \begin{array}{c c}{{0}}&{{- 1}}\\{{1}}&{{0}}\\ \end{array}\right) + \frac{q}{2 \pi}\left( \begin{array}{c c}{{\frac{1}{2}\sin ( 2 n x_{+})}}&{{- \sin^{2}( n x_{+})}}\\{{\cos^{2}( n x_{+})}}&{{- \frac{1}{2}\sin ( 2 n x_{+})}}\\ \end{array}\right) \, \, .
\omega = \sum_{i = 1}^{n}\frac{d u_{i}}{u_{i}}\wedge d \beta_{i}- \sum_{i < j}\frac{d \beta_{i}\wedge d \beta_{j}}{\beta_{i}- \beta_{j}}+ \sum_{i = 1}^{m}\frac{d v_{i}}{v_{i}}\wedge d \kappa_{i}- \sum_{i < j}\frac{d \kappa_{i}\wedge d \kappa_{j}}{\kappa_{i}- \kappa_{j}}.
\boldsymbol{H}^{0,1}(Q) := \{\boldsymbol{v}\in \boldsymbol{L^2}(Q): \partial_t \boldsymbol{v}\in \boldsymbol{L^2}(Q) \}, \boldsymbol{H}^{0,1}_{per}(Q) := \{\boldsymbol{v}\in \boldsymbol{H}^{0,1}(Q): \boldsymbol{v}(\boldsymbol{x},0) = \boldsymbol{v}(\boldsymbol{x},T) \ \mbox{for almost all}\boldsymbol{x}\in \Omega \}.
P_1 f(P_1A_1 P_1) P_1J - \JP_2 f(P_2A_2 P_2) P_2 = \P_1\bigl( f(P_1 A_1 P_1)J - J f(P_2 A_2 P_2)\bigr) P_1 + \P_1[ f(P_2A_2 P_2), J]P_2 - P_1[ f(P_2A_2 P_2), J]P_1
\rho_{\nu}( x ) = \rho_{\nu^{\prime}}( x ) \frac{q^{C ( \nu ) / 2}+ \epsilon ( \nu ) \epsilon ( \nu^{\prime}) x q^{C ( \nu^{\prime}) / 2}}{x q^{C ( \nu ) / 2}+ \epsilon ( \nu ) \epsilon ( \nu^{\prime}) q^{C ( \nu^{\prime}) / 2}}\; , ~ \forall \nu \neq \nu^{\prime}.
\sigma_i^2 := \mathrm{Var}[Z_i] = \mathrm{E}[Z_i^2] - \mathrm{E}[Z_i]^2 &= \left( T_{0i}^2 + T_{1i}^2 + T_{2i}^2 + 3 T_{3i}^2 + 3 T_{4i}^2 + T_{5i}^2 + 2 T_{0i}T_{3i}+ 2 T_{0i}T_{4i}+ 2 T_{3i}T_{4i}\right) & - \left( T_{0i}+ T_{3i}+ T_{4i}\right)^2 &= T_{1i}^2 + T_{2i}^2 + 2 T_{3i}^2 + 2 T_{4i}^2 + T_{5i}^2
G_{R}( x , z ; x^{\prime}, z^{\prime}) = - \int \frac{d^{4}p}{( 2 \pi )^{4}}e^{i p_{\mu}( x^{\mu}- x^{\prime}{}^{\mu})}\Biggl [ \frac{e^{- 2 k ( | z | + | z^{\prime}| )}k}{{\bf p}^{2}- ( \omega + i \epsilon )^{2}}+ \int_{0}^{\infty}d m \, \frac{v_{m}( z ) v_{m}( z^{\prime})}{m^{2}+{\bf p}^{2}- ( \omega + i \epsilon )^{2}}\Biggr ] ,
\textstyle P_{B}P_{A}^{-1}= \left[\begin{array}{cc}{f_{11}}{f_{12}}{\frac{\sigma_{2}}{(\sigma_{2}, \beta_{1})}f_{21}}{f_{22}}\end{array}\right] \left[\begin{array}{cc}{l_{11}}{l_{12}}{(q_{1}\cdots q_{k})l_{21}}{l_{22}}\end{array}\right].
\mathcal{R^{\geq}}:=\{(i,j)\geq(i-1,j')\ |\ 1\leq j\leq i\leq n,\ 1\leq j'\leq i-1\}, \mathcal{R^{>}}:=\{(i-1,j')>(i,j)\ |\ 1\leq j\leq i\leq n,\ 1\leq j'\leq i-1\}, \mathcal{R}^{0}:=\{(n,i)\geq(n,j)\ |\ 1\leq i\neq j\leq n\}.
\frac{\theta}{\pi}|x^2 J_c^{-1}f(x)| =|\int_{\mathbb R}e^{-\theta|x-y|}x^2 f(y)dy|\leqq \int_{\mathbb R}e^{-\theta |x-y|}|x-y|^2 |f(y)|dy + \int_{\mathbb R}e^{-\theta |x-y|}y^2 |f(y)|dy &\leqq |f|_{\infty}\int_{\mathbb R}e^{-\theta |p|}p^2 dp + |p^2 f|_{\infty}\int_{\mathbb R}e^{-\theta |p|}dp <\infty.
\mathcal{E}^{\neq}(\mu) = \int_{M \times M \setminus \Delta}g(x-y) d \mu_{x}d\mu_{y}\mathcal{G}(\mu, \nu) = \int_{M \times M}g(x-y) d \mu_{x}d\nu_{y}\mathcal{G}^{\neq}(\mu, \nu) = \int_{M \times M \setminus \Delta}g(x-y) d \mu_{x}d\nu_{y},
S^{(1)}_0 = \begin{pmatrix}1 0 -is_1 1\end{pmatrix},S^{(2)}_0 = \begin{pmatrix}1 is_2 0 1\end{pmatrix}, S^{(3)}_0 = \begin{pmatrix}1 0 -is_3 1\end{pmatrix}, S^{(4)}_0 = \begin{pmatrix}1 -is_1 0 1\end{pmatrix},S^{(5)}_0 = \begin{pmatrix}1 0 is_2 1\end{pmatrix}, S^{(6)}_0 = \begin{pmatrix}1 -is_3 0 1\end{pmatrix}.
\theta_{E,h}(u):=\theta_{E,h}(u,u) =\sum c_{jk\lambda\mu}u_{j\lambda}\overline{u}_{k\mu},~\,\,\mathrm{for}\,\,~ u=\sum u_{j\lambda}\frac{\partial}{\partial z_j}\otimes e_\lambda\in T_{X,x_0}\otimes E_{x_0}, \mathrm{i.e.}~\,\,\, \theta_{E,h}=\sum c_{jk\lambda\mu}(dz_j\otimes e^*_\lambda)\otimes\overline{(dz_k\otimes e^*_\mu)}.