harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
q=\frac{1}{\hbar^{3}}\sum_{p}(E_{e}-E_{F})u_{e}f_{e}^{\prime}=\frac{1}{\hbar^{3}k_{B}T}\sum_{p}u_{e}\tau_{e}(-\frac{\partial f_{e}^{o}}{\partial E_{e}})(E_{e}-E_{F})(u_{e}\cdot F_{te}),
g_{1+\tau+\tau^3}(x)\cr=\, (x-1)\sum_{0\le i_0,i_1,i_3\le 1}(-1)^{i_0+i_1+i_3}\frac{y^{i_0+i_1+i_3}-1}{y-1}\cr \cdot \bigl[a_0s_0(x)+a_1s_1(x)+a_2s_2(x)+a_3s_3(x)\cr -(i_0+1-2k)(i_1+i_3+k+1)-(i_0+1-k)(i_3+1-k)x\bigr]-1,
tilde u(t) = Q^{-1}\tilde B_N'e^{(\tau-t)\tilde A_N'}\nu,\nu = \left(\int_0^\tau e^{s \tilde A_N}\tilde B_N Q^{-1}\tilde B_N' e^{s \tilde A_N'}\,ds\right)^{-1}(\tilde x^1_N-e^{\tau \tilde A_N}\tilde x^0_N),
h_N =\frac{1}{2\pi N i}\int_{|z|=\delta}\frac{d h(z)}{z^{N}}&=\frac{1}{2\pi N i}\int_{|z|=\delta}\frac{d (\log g(z))}{z^{N+r}}-\frac{1}{2\pi N i}\int_{|z|=\delta}\frac{d z}{z^{N+r+1}}= \frac{N+r}{N}b_{N+r}
\Omega_{\alpha,\bar{\alpha}}= \frac{-e^{2\alpha}}{2}(du-2\chi)(\alpha^{\vee}), \Omega_{\alpha_1,\bar{\alpha}_2}= \frac{2\chi([\theta(e_{\alpha_2}),e_{\alpha_1}])}{e^{-2\alpha_1}+e^{-2\alpha_2}},
\mathcal{F}^{-1}(\Phi_{-(B_1^{-1}A_1+D_2B_2^{-1})})=\frac{1}{|\det(B_1^{-1}A_1+D_2B_2^{-1})|}\Phi_{(B_1^{-1}A_1+D_2B_2^{-1})^{-1}}.
\overline{{{Z}}}_{\Psi}= \int \prod D \Phi^{a}\prod D \vartheta^{b}\prod \delta ( d^{\gamma}) e x p \left({\frac{i}{\hbar}}W ( \Phi^{a},{\frac{\partial \overline{{\Psi}}}{\partial \Phi^{a}}}, \vartheta^{b},{\frac{\partial \overline{{\Psi}}}{\partial \vartheta^{b}}}\right)
\sum_{j = 0}^k{\sum_{s = 0}^j{\binom kj\binom js\left({\frac{{f_1}}{{f_3}}}\right)^j\left({-\frac{{1}}{{f_1}}}\right)^s X_{m - (c-b)k + (c - a)j + as}}}= \left(-\frac{f_2}{f_3}\right)^kX_m
\Xi_{[ 1 / 2 ]}= \left( \begin{matrix}{{e^{i \phi}}}&{{0}}\\{{0}}&{{e^{- i \phi}}}\\ \end{matrix}\right) , \quad \Xi_{[ 1 ]}= \left( \begin{matrix}{{e^{i 2 \phi}}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{e^{- i 2 \phi}}}\\ \end{matrix}\right) \quad ,
q^2t_{s_0s_1s_0}-q^{\frac{3}{2}}t_{s_0s_1s_0s_1}+q^{\frac{5}{2}}t_{s_0s_1}-q^2t_{s_0s_1s_0}+q^3t_{s_0}-q^\frac{5}{2}t_{s_0s_1}-q^3t_{s_0}=-q^{\frac{3}{2}}t_{s_0s_1s_0s_1}.
\frac 1{\Phi_{ev}(t^{n-h})}\sum_{m=0}^n \sum_{\substack{g\in \mathcal M_m g(0)\neq 0}}\alpha(t^{n-m})\alpha(g)= \frac{q^{h+1}}{q^{n}}\sum_{f\in \mathcal M_n}\alpha(f)
\det\left(\frac{x_j^{i-1}}{(i-1)!}\right)_{1\le i,j\le n_2}=\frac 1{(2\pi i)^{n_2}}\int_{\gamma_{\tau_1}^{n_2}}d^{n_2}\zeta\prod\limits_{j=1}^{n_2}e^{x_j\zeta_j-\frac 12\mu_1\zeta_j^2}\det\left(\frac 1{\zeta_j^i}\right)_{1\le i,j\le n_2}.
\hat L-\hat L^*=\frac{1}{nh_n}\sum_{i=1}^n [w_n(x_i/h_n)-w_n^*(x_i/h_n)]r(x_i)+\frac{1}{nh_n}\sum_{i=1}^n [w_n(x_i/h_n)-w_n^*(x_i/h_n)]u_i
\langle \varphi_1, \varphi_2 \rangle=\frac{1}{|G|}\sum_{g \in G}\varphi_1(g)\overline{\varphi_2(g)}=\frac{1}{|G|}\sum_{\substack{g=g_{p'}g_p \in G, g_{p}\in{H_1 \cap H_2}}}\varphi_1(g_{p'}) \overline{\varphi_1(g_{p'})},
\mu_{V X^{(n)}}(A) = \mathbb{P}(VX^{(n)}\in A) = \mathbb{P}\Big(\sum_{j=1}^n n^{1/p}\frac{Z_j}{(\lVert Z^{(n)}\rVert_p^p + W_n)^{1/p}}V_{\bullet,j}\in A\Big),
H\big(x;\frac{1}{1-x}\big) =H_1(x)C(x)^2+E\big(x;C(x)\big)-H_{1,0}(x)C(x)^2-\frac{xC(x)^2}{1-x}E\big(x;C(x)\big) &\ +J\big(x;C(x)\big)C(x)-J_1(x)C(x)^2 &-\frac{1}{(1-x)C(x)}\left(E\big(x;C(x)\big)-(1-x)E\big(x;\frac{1}{1-x}\big)C(x)\right) &=x^2C(x)^2F_T(x)-\frac{2x^2-4x+1}{1-x}-\frac{x^5-16x^4+36x^3-28x^2+9x-1}{(1-2x)^2(1-x)}C(x)\,.
{\cal L}= -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+ \frac{1}{2}(\partial_{\mu}\phi)^2 + e(g^{\mu\nu}- \epsilon^{\mu\nu})(\partial_{\mu}\phi)A_{\nu}+ \frac{1}{2}e^2\alpha A_{\mu}^2,
|(R1)^{n}|\leq 2\int\limits^{t_n}_{t_{n-1}}\int\limits^{t_n}_{t_{n-1}}\left| \Delta u_{ss}(\cdot,s)\right|dsdt\leq \bar{C}\tau\int\limits^{t_n}_{t_{n-1}}s^{\alpha-1}ds= \bar{C}\tau(t_{n}^{\alpha}-t_{n-1}^{\alpha}).
\gamma_{ik}= \frac{\vert \mathbf{h}_{iik}^{\mathrm{H}}\mathbf{w}_{ik}\vert^2}{\sum_{l=1,l \neq k}^K \vert \mathbf{h}_{iik}^{\mathrm{H}}\mathbf{w}_{il}\vert^2 + \mathcal{I}},
\ln R(n+N)^2 = \ln R(n)^2-\sum_{l=0}^{N-1}\frac{V(n+l)}{\sin\pi k}\sin2\pi\theta(n+l)+\frac{O(1)}{(n-b)^2}= \ln R(n)^2-K_1\sum_{l=0}^{N-1}\frac{\sin2\pi\theta(n+l)+\sin2\pi\tilde{\theta}(n+l)+100}{(n-b)\sin\pi k}\sin2\pi\theta(n+l) &+\frac{O(1)}{(n-b)^2}.
\int_{\theta}\prod_{j=1}^{M}P(\theta_{j};\alpha)\prod_{t=1}^{N}P(Z_{j,t}|\theta_{j})d\theta=\prod_{j=1}^{M}\int_{\theta_{j}}P(\theta_{j};\alpha)\prod_{t=1}^{N}P(Z_{j,t}|\theta_{j})d\theta_{j}.
D=\frac{1}{(m|g|)^2}\sum_{\substack{\ell=1 (\ell\neq j)}}^n\sum_{i,h,h'=0}^{m-1}[\tilde{v}_i\tilde{w}_i]_{k,\ell}[\tilde{v}_{i-h-1}\tilde{w}_{i-h-1}]_{\ell,j}[\tilde{v}_{i-h'-1}\tilde{w}_{i-h'-1}]_{j,k}\frac{\lambda_\ell^{m-h-1}\lambda_j^{m-h'-1+h}\lambda_k^{h'}}{(\lambda_\ell^m-\lambda_j^m)(\lambda_j^m-\lambda_k^m)}.
\frac{\partial_z F(z)}{F(z)}=12 \frac{\partial_z f(z)}{f(z)}- k \frac{\partial_z \Delta(z)}{\Delta(z)}= 12 \frac{\partial_z f(z)}{f(z)}- k (2 \pi i) E_2(z)=(24 \pi i) f_{\theta}(z).
\sup_{z\in [0,(b_1/a_1-1)\ln n]}e^z\frac{D_n((1+z/\ln n) G_n(x)/2)}{D_n(G_n(x)/2)}\frac{(1+z/\ln n)a_1}{\sqrt{4-(1+z/\ln n)^2a_1^2}}= \sup_{w\in[1,b_1/a_1]}e^{(w-1)\ln n}\frac{D_n(w G_n(x)/2)}{D_n(G_n(x)/2)}\frac{wa_1}{\sqrt{4-w^2a_1^2}}\leq \sup_{w\in[1,b_1/a_1]}e^{(w-1)\ln n}e^{1-(w-1)\ln n}\frac{b_1}{\sqrt{4-b_1^2}}=\frac{eb_1}{\sqrt{4-b_1^2}}
\textstyle M_{r+p-2}\textstyle = \left(-M_{r+p-3}' - \cdots - M_r' -\frac{1}{s}- \cdots - \frac{1}{p-1}\right)p + O(p^2) \textstyle =\left(-\frac{t}{p-2}\cdots - \frac{t}{1}- \frac{1}{1}- \cdots - \frac{1}{p-1}\right)p + O(p^2)\textstyle =\frac{t}{p-1}p + O(p^2).
\begin{aligned}\left|(f^n)'(z)\right| =\left|(f^{n-M_{l+1}}\circ f^{m_{l+1}}\circ f^{2n_{l}}\circ \dots \circ f^{m_2}\circ f^{2n_{1}}\circ f^{m_{1}})(z)\right| \geq \lambda_{l+1}^{(n-M_{l+1})/2}\cdot \prod_{k=1}^{l}\lambda_k^{n_k}\cdot \prod_{k=1}^{l+1}\alpha_k\geq \lambda_{l+1}^{(n-M_{l+1})/2}(1+\beta_{l+1}^2) \lambda_l^{n_l/2}\end{aligned}
N \epsilon_{Y_{j,K}}(A) \lambda_{Y_{j,K}}\left(\pi_{j,K}(P)\right) \le N\left(\epsilon_{j}+\gamma'\right) \lambda_{Y_{j,K}}\left(\pi_{j,K}(P)\right)=N\left(\epsilon_{j}+\gamma'\right) \lambda_{E_{j,K}}(P) & \le \lambda_{\pi_{j}^* F_{j,K}}(P)=\lambda_{F_{j,K}}\left(\pi_{j,K}(P)\right)
III =\int_{\frac{1}{2}x_2}^{2x_2}\left[K\left(\frac{x_1}{y}\right)-K\left(\frac{x_2}{y}\right)\right]\frac{\omega(y)}{y}~dy \geq -\int_{\frac{1}{2}x_2}^{2x_2}K\left(\frac{x_2}{y}\right)\omega(y)\frac{1}{y}~dy &\geq-\int_{\frac{1}{2}}^2\frac{1}{s^2}\log\frac{|s+1|}{|s-1|}~ds \geq -C.
f^{\stackrel{\underbrace{\mbox{\scriptsize 1\ldots 1}}}{m_1}\stackrel{\underbrace{\mbox{\scriptsize 2\ldots 2}}}{m_2}\cdots \stackrel{\underbrace{\mbox{\scriptsize N-1\ldots N-1}}}{m_{N-1}}}(u)\;, \qquad{m_1 \geq m_2\geq \ldots \geq m_{N-1}}\;.
\mathrm{\boldmath ~ S ~}= \kappa^{\frac{1}{2}}~ \left( \begin{array}{c c}{{m_{q}^{- 1}{\mathrm{\boldmath ~ S ~}}_{q}~ I^{( 6 )}}}&{{0}}\\{{0}}&{{m_{D}^{- 1}{\mathrm{\boldmath ~ S ~}}_{D}~ I^{( 2 1 )}}}\\ \end{array}\right)
R ( \theta ) = \left( \begin{array}{c c c c}{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{\frac{\sinh{\gamma}}{\sinh{( \theta + \gamma )}}}}&{{\frac{\sinh{\theta}}{\sinh{( \theta + \gamma )}}}}&{{0}}\\{{0}}&{{\frac{\sinh{\theta}}{\sinh{( \theta + \gamma )}}}}&{{\frac{\sinh{\gamma}}{\sinh{( \theta + \gamma )}}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\ \end{array}\right)
\min \Big( \lambda \varepsilon_K + \frac{1}{K}\sum_{k = 1}^K s_k \Big) \mbox{s.t.}\hat{\mathbf{c}}^{\mbox{\tiny \upshape (\itshape k\upshape)}\top}\mathbf{x}\leq s_k \forall k \in \mathcal{K}& \lambda \geq 1.
(g\cdot\alpha)_p(Y_1,\dots,Y_k) = \alpha_{g^{-1}p}(TL_{g^{-1}}(Y_1),\dots,TL_{g^{-1}}(Y_k)) (g\cdot\eta)_x(X_1,\dots,X_k) = g\cdot\left(\eta_{g^{-1}x}(TL_{g^{-1}}(X_1),\dots,TL_{g^{-1}}(X_k))\right) ,
\int e^{\frac{n^2}{m^2}(\sqrt{\Gamma}f)^\top \sum_{j=1}^{m}\Xi^j_u (\sqrt{\Gamma}g)}dN(0, \epsilon^2 I_{2d}) (f,g) \leq e^{C \frac{n^4 \epsilon^4}{m^4}\left((\sqrt{\bar{\Gamma}}^\top \Xi_u \sqrt{\bar{\Gamma}})^2 \right)},
\tilde{A}_{ij}(\cdot,\tilde{u},D\tilde{u}) = D_i\Upsilon^kD_j\Upsilon^lA_{kl}(\cdot,\tilde{u},[D\Upsilon]^{-1}D\tilde{u})+[D\Upsilon]^{\alpha k}D_\alpha \tilde{u}D_{ij}\Upsilon^k, \tilde{B}(\cdot,\tilde{u},D\tilde{u}) = [\det D\Upsilon]^2 B(\cdot,\tilde{u},[D\Upsilon]^{-1}D\tilde{u}).
{{\cal A}}({\cal T}) \wedge{{\cal A}}({\cal N}) ={{\cal A}}({\widehat{\cal S}}) = 1 +{\frac{( 4 \pi^{2}\alpha^{\prime})^{2}}{1 9 2 \pi^{2}}}\mathrm{t r}({\widehat{\cal R}}\wedge{\widehat{\cal R}}) + \cdots \ ,
\mathcal{A}(u,\phi)+\mathcal{A}(v,\psi)=\int_\Omega \left(\lambda |u|^{q-2}u\phi+ \mu |v|^{q-2}v\psi \right)dx+\frac{2\alpha}{\alpha+\beta}\int_\Omega|u|^{\alpha-2}u|v|^\beta\phi dx+\frac{2\beta}{\alpha+\beta}\int_\Omega|u|^{\alpha}|v|^{\beta-2}v\psi dx
Res_1\zeta^h_N(s_0)= Res_1\left(\frac{1}{(4\pi)^{n/2}\Gamma(s_0)}\sum_{i=0}^\infty(-1)^i\frac{h^{2i}}{i!}\sum_{j=0}^{\infty}\frac{a^N_j}{(s_0+i+j-\frac{n}{2})}\right).
\delta \left({\frac{1}{2}}\int_{M_{1 0}}* F_{2}^{\prime}\wedge F_{2}^{\prime}\right) = \int_{M_{1 0}}* F_{2}^{\prime}\wedge m \delta B_{2}= \mu_{8}\int_{M_{1 0}}d * F_{2}^{\prime}\wedge \Lambda_{1}= \mu_{8}\mu_{0}\int_{W_{1}}\hat{\Lambda}_{1},
\psi(\textbf{c})=\begin{bmatrix}c_0(x) c_1(x) \vdots c_{\ell-1}(x) \end{bmatrix}=\begin{bmatrix}c_{00}+c_{10}x+~~\ldots~~+c_{n-1,0}x^{n-1}c_{01}+c_{11}x+~~\ldots~~+c_{n-1,1}x^{n-1}\vdots c_{0,\ell-1}+c_{1,\ell-1}x+\ldots+c_{n-1,\ell-1}x^{n-1}\end{bmatrix}.
g(d,j) =((d-4)^2+2d-7)\left(\binom{d-4}{j-1}+\binom{d-4}{j-2}\right)-\left(\binom{d-4}{j-3}+\binom{d-4}{j-4}\right) &=g(d-1,j)+g(d-1,j-1)+(2d-7)\binom{d-3}{j-1}.
{\Phi_{5}}({\rho}) = 4 \pi^{2}\, g_{s}\left({\frac{m^{I}d_{I}}{| \Gamma |}}+{\frac{g_{s}}{2 \pi}}{\widehat C_{I J}m^{I}m^{J}}\ln{\frac{\rho}{\rho_{0}}}\right) = 4 \pi^{2}\, g_{s}\,{Q}({\rho}) ~ ,
\left[{dn_d(\omega|D) \over d\omega}\right]_{\tiny\mbox{div}}=r_d\,{\Gamma\left(1-\frac D2\right) \over (4\pi)^{D/2}}\,\,{m^{D-4}\over \kappa}\,\, \int_{\Sigma}\left[2\left(m^2+{R\over 12}-{{\cal Q}\over 8}\right)-{\omega^2\over \kappa^2}{\cal P}\right]~~~.
\mathbb{P}\left[X(t,i)=0|X(t-1,i)=0,t\leq T \right]=\frac{1-\hat{\pi}(t)}{1-\hat{\pi}(t-1)}=1-\frac{\hat{\pi}(t)-\hat{\pi}(t-1)}{1-\hat{\pi}(t-1)}.
\sum_{n=1}^{\infty}\frac{1}{n}\,\intop_{0}^{\infty}e^{-2\pi ny}\,|J_{1}(2\pi xy)|\,dy\leq\frac{D}{\sqrt{2\pi x}}\,\sum_{n=1}^{\infty}\frac{1}{n}\,\intop_{0}^{\infty}\frac{e^{-2\pi ny}}{\sqrt{y}}\,dy\leq\frac{D^{\prime}}{\sqrt{x}}\sum_{n=1}^{\infty}\frac{1}{n^{3/2}},
M_{S_2}(2n,j,0;a) =(-1)^{n-j}\binom{a+n}{a}\binom{a+j}{j}\binom{a}{n-j}, M_{S_2}(2n,j,1;a) =(-1)^{n-j}\binom{2n}{n}\sum_{u=0}^{n-j}(-1)^u\binom{n}{j+u}\binom{j+u}{u}\binom{a+j+u}{j+u}\binom{a+n}{2n-j-u}
\b Q_{i(p-1)}(x\otimes y)\mapsto (-1)^{\tfrac{nm(p-1)}{2}}\sum_{j+k=i}\big(\b Q_{j(p-1)}(x) \otimes Q_{k(p-1)}(y) +(-1)^nQ_{j(p-1)}(x) \otimes \b Q_{k(p-1)}(y)\big).
f_l(z) =\sqrt{|z|}e^{i\left({1\over 2}{\rm arg}\, z +\pi l\right)}\sum_{n=0}^\infty b_n\left(\sqrt{|z|}e^{i\left({1\over 2}{\rm arg}\, z +\pi l\right)}\right)^{-5n}+f^{n.p.}_l(z),
\times \exp \left( 2 \sum_{b,b^\prime=1}^N\sum_{j,j^\prime=1}^{n_b,n_{b^\prime}}\Big( \delta_n(x_j^{(b)})\!-\!\delta_n(y_j^{(b)}),\Big[ (\frac{e \pi}{\pi\!+\!g N})^2 \tilde{Q}+ g \tilde{C}\Big]\Big[ \delta_n(x_{j^\prime}^{(b^\prime)})\!-\!\delta_n(y_{j^\prime}^{(b^\prime)}) \Big] \Big) \right)
{}^* ( dx^{\alpha_1}\wedge dx^{\alpha_2}\wedge ... \wedge dx^{\alpha_m}) =\frac{\sqrt{|g|}}{(n-m)!}\varepsilon^{\alpha_1 ... \alpha_m}_{\ \ \ \ \ \ \ \beta_{m+1}...\beta_n}dx^{\beta_{m+1}}\wedge ... \wedge dx^{\beta_n},
f(x+s+t)+f(x+s-t)-2f(x+s) =t\bigg(\frac{f(x+s+t)-f(x+s)}{t}\bigg)-(f(x+s)-f(x+s-t)) &\leq t\frac{f(x+2s)}{s}\leq Ct\frac{f(x+s)}{s},
\rho_{L\oplus M}(a(x_1+m_1),x_2+m_2)=\rho_{L\oplus M}(ax_1+am_1),x_2+m_2) =\rho(ax_1,x_2)=(-1)^{\bar a\bar x_1}\rho(x_1,ax_2) =(-1)^{\overline a~\overline{x_1+m_1}}\rho_{L\oplus M}(x_1+m_1, ax_2+am_2)=(-1)^{\overline a~\overline{x_1+m_1}}\rho_{L\oplus M}(x_1+m_1, a(x_2+m_2)).
\frac{d}{d t}\langle F ( x ) \rangle_{t}\sim \frac{1}{Z_{t}}\int_{D_{t}}d x \, \frac{\partial}{\partial x}e^{S ( x )}F^{\prime}( x ) = \left. \frac{1}{Z_{t}}e^{S ( x )}F^{\prime}( x ) \right|_{x_{-}( t )}^{x_{+}( t )}.
\Psi_{\varepsilon}(x)=\int_{(a,b)}p_{\varepsilon}(y-x)\,\Psi_{\varepsilon}(y)\,\mathrm{d}y+\int_{-\infty}^{+\infty}p_{\varepsilon}(y-x)\,\varphi(y)\,\mathrm{d}y+(1-\mathrm{e}^{\lambda \varepsilon})\Phi_{\varepsilon}(x)-\varphi(x).
h_{\mu \nu}( z , t ) ={\frac{1 6 \pi}{2}}{\frac{( z - \beta t - \delta / 2 )^{2}}{( 1 - \beta^{2})}}{\frac{k \ell_{P}^{2}}{\delta^{4}}}\left( \eta_{\mu \nu}- 4 n_{\mu}n_{\nu}\right) .
c_dm(\varphi^{0,1}) =\frac{c_d^2}{c_*}\int_0^\infty \varphi^{0,1}(r)U(r)r^{N-1}\,dr=\frac{c_d^2}{c_*}|{\bf S}^{N-1}|^{-1}\int_{{\bf R}^N}\varphi^{0,1}(|x|)U(|x|)\,dx =\frac{c_d^2}{c_*}|{\bf S}^{N-1}|^{-1}\int_{{\bf R}^N}\varphi(x)U(|x|)\,dx=M(\varphi).
\frac{y_{n+1}}{x_{n+1}}=\prod_{k \leq n+1}\frac{p_k^2}{p_k^2-\chi(p_k)}=\left (\frac{p_{n+1}^2}{p_{n+1}^2-1}\right)\left (\frac{p_n^2}{p_n^2-1}\right)\prod_{k \leq n-1}\frac{p_k^2}{p_k^2-1}.
V ( \rho ) = N h \left[ \frac{2 h^{2}}{3 \pi}+{\frac{\rho^{2}}{2 \lambda}}+{\frac{1}{4 \pi}}\rho^{2}\left( \ln{\frac{\rho^{2}}{1 6 h^{2}}}-{\frac{4}{\sqrt{\pi}}}+ 3 \right) \right] + O ( 1 ) ,
f \in I_m(W^c_m)I_{m}(W^c_m) &\subseteq I_t(W^c_m)^{(m-t+1)}I_{t}(W^c_m)^{(m-t+1)}\hbox{\cite[Theorem 3.16 (a)]{conca98}}&\subseteq I_{t}(W^c_m)^{(2m-2t+2)}= I_t(W^c_m)^{(h)}.
\zeta_{AV,2}(s_1,s_2,s_3) = \sum_{m \leq at_3}\sum_{n<m}\frac{1}{m^{s_1}n^{s_2}(m+n)^{s_3}}+ D_1+D_2 &+\sum_{n \leq at_3}\frac{E(s_1,s_3;at_3,n,M)}{n^{s_2}}+\sum_{at_3 < n \leq N}\frac{E(s_1,s_3;n,M)}{n^{s_2}}&-B_2-B_3-B_4-C_2-C_3-C_4
\psi(x)={}_7F_6\bigg[\begin{matrix}-a -a 1-\frac12a -b+x -c -d -e 1 -\frac12a 1-a+b-x 1-a+c 1-a+d 1-a+e\end{matrix}\bigg|\,1\bigg] = \frac{(1-a)_d(1-a+c+b-x)_d}{(1-a+c)_d(1-a+b-x)_d}\cdot{}_4F_3\bigg[\begin{matrix}1+e -b+x -c -d 1 1-a+e a-c-d-b+x\end{matrix}\bigg|\,1\bigg]
\Sigma(Q,N;\mathcal{S}) \ll NZ\cdot \max\limits_{\substack{q_2\in \mathcal{S}Q/2<\mathcal{N}(q_2)\le Q r_2 \bmod q_2 (r_2,q_2)=1}}\sharp\left\{(q,x,y)\ :\q\in \mathcal{S}, \Q/2<\mathcal{N}(q)\le Q, \ \binom{x}{y}\in \mathbb{Z}^2, \right. & \left.x\binom{u'}{v'}+y\binom{v'}{-u'}\in \sqrt{\mathcal{N}(q)}S(k,l)\right\},
S_{p}^{( 0 )}= - \tau_{p}\int d^{p + 1}\sigma \, e^{\Phi ( - 1 - \gamma ( p + 1 ) / 2 )}\sqrt{- d e t [ \tilde{G}_{a b}+ e^{\gamma \Phi}( \tilde{B}_{a b}+ 2 \pi \alpha^{\prime}F_{a b}) ]}~ ,
\frac{d u^i}{dt}=\,{u'}^i =\,\frac{\partial H(u,p)}{\partial p_i}=\,2(1-\kappa(t,\tau))\,\beta^i(u)-\,\frac{\partial\kappa(t,\tau)}{\partial p_i}\,\left(\sum^{8N}_{k=1}\,\beta^k(u)p_k\right) =\,2(1-\kappa(t,\tau))\,\beta^i(u),
(P_k)_i=\begin{bmatrix}\overbrace{0 \ldots 0}^{(k-2)}\eta_r U_0(\alpha_r) \overbrace{0 \ldots 0}^{(k-1)}\ldots \overbrace{0 \ldots 0}^{(k-1)}\eta_r \mu^{-\frac{n-k}{2k}}U_{\frac{n-k}{k}}(\alpha_r) 0\end{bmatrix}^{T},
\lambda_{v_\infty}(D_1,[a:b:c]) - 3 h(P) = \log \frac{\max(|a|,|b|,|c|)^4}{|a| |b| |c| |b+c|}- 3 \log \max(|a|,|b|,|c|) &= \log \frac{\max(|a|,|b|,|c|)}{|a| |b| |c| |b+c|}\le 0,
N ( \theta ) = \frac{m}3 \left( \ln \sin \theta - \frac 1{\sin^{2}\theta}\right) + \frac{C_{1}}4 \left( \ln \frac{1 - \cos \theta}{1 + \cos \theta}- \frac{2 \cos \theta}{\sin^{2}\theta}\right) + C_{2}.
k_{1}= \left( \begin{array}{c}{e}\\{p}\\ \end{array}\right) k_{2}= \left( \begin{array}{c}{e}\\{- p}\\ \end{array}\right) k_{3}= \left( \begin{array}{c}{- e}\\{- p}\\ \end{array}\right) k_{4}= \left( \begin{array}{c}{- e}\\{p}\\ \end{array}\right) .
\sum_{k=0}^{\infty}\frac{q^{k^{2}}\left(-q^{-2n}\right)^{k}}{\left(q,-q;q\right)_{k}}=\frac{\left(-1\right)^{n}\left(q;q^{2}\right)_{n}}{\left(-q;q\right)_{\infty}q^{n^{2}}}.
\frac{\int_{D_\delta^N}\exp\left( -\frac{\beta}{2}\sum_{j=1}^N \lambda^\pm_j (b^\pm_j)^2\right) d\vec b^\pm}{\int_{\mathbb{R}^N}\exp\left( -\frac{\beta}{2}\sum_{j=1}^N \lambda^\pm_j (b^\pm_j)^2\right) d\vec b^\pm}> 1- \hat{\delta}
{\cal E}_{KE}[\tilde{A}_0;R] =\frac{2\pi}{R}\sum_{n \in{\bf Z}}n^2|a_0^{(n)}|^2, \quad{\cal E}_{KE}[\tilde{A}_{\pm};R] =\frac{2\pi}{R}\sum_{n \in{\bf Z}}[(n \pm \alpha)^2-\alpha^2]|a_{\pm}^{(n)}|^2.
\Big\{x\in \Lambda: \textrm{card}\Sigma_{\Lambda}(x)<2^{\aleph_{0}}\Big\}\subseteq \bigcup_{N=0}^{\infty}\bigcup_{(\epsilon_{i})\in\{1,\ldots, n\}^{N}}f_{\epsilon_{1}}\cdots f_{\epsilon_{N}}(U_{\Lambda}),
(z-1)^{-a}P\{\begin{matrix}0&1&\infty&\\ 0&0&a&z(z-1)^{-1}\\ 1-c&b-a&c-b&\end{matrix}\}=P\{\begin{matrix}0&1&\infty&\\ 0&0&a&z\\ 1-c&c-a-b&b&\end{matrix}\}
\Lambda(Z) = \frac{e^{\frac{\sigma_{\mathrm{a}}^2}{\zeta}}\int_{\sigma_{\mathrm{w}}^2+\sigma_{\mathrm{a}}^2}^{\infty}\left ( \frac{1}{v}\right)^{n}e^{-\frac{Z}{v}}e^{-v/\zeta}dv}{\int_{\sigma_{\mathrm{w}}^2}^{\infty}\left ( \frac{1}{v}\right)^{n}e^{-\frac{Z}{v}}e^{-\frac{v}{\zeta}}dv}\mathop{\gtrless}_{H_0}^{H_1}\gamma.
\beta B_{j+1}=B_n\cdots B_{j+2}B_{j+1}B_{j}B_{j-1}\cdots B_{1}B_0 B_{j+1}{}=B_n\cdots B_{j+2}B_{j+1}B_{j}B_{j+1}B_{j-1}\cdots B_{1}B_0{}=B_n\cdots B_{j+2}B_{j}B_{j+1}B_{j}B_{j-1}\cdots B_{1}B_0{}=B_{j}\beta.
\begin{array}{l}\displaystyle u(t,x) = (B(t)+1)^{-n/2}v(\log(B(t)+1), (B(t)+1)^{-1/2}x), begin{align*}5pt]\displaystyle u_t(t,x) = b(t)^{-1}(B(t)+1)^{-n/2-1}w(\log(B(t)+1), (B(t)+1)^{-1/2}x).\end{array}
\Theta_p(x)=(2\epsilon_p)^{-1/2}e^{-i\epsilon_p t}\varphi_p(z),\quad \varphi_p(z)=(2\pi)^{-1/2}e^{ipz},\quad \epsilon_p=\sqrt{p^2+m^2},\quad -\infty<p<\infty,
Y^n_t = n^{\alpha+\frac 12}\int^t_0 [(t-\eta_n(s))^{\alpha}- (t-s)^{\alpha}] \sigma(X^n_{\eta_{n}(s)})\, dW_s +n^{\alpha+\frac 12}\int^t_0 (t-s)^{\alpha}[ \sigma(X^n_{\eta_n(s)}) -\sigma(X_s)]\, dW_s := A^n_t +Y^{n,1}_t.
||U(t)A(t)U^*(t)-B(t)|| &\leq ||U(t)P_{m(n)}A(t)P_{m(n)}U^*(t)-P_{m(n)}B(t)P_{m(n)}||+\frac{18}{n}&=||U_{2m(n)}''(t)P_{m(n)}A(t)P_{m(n)}U_{2m(n)}''^*(t)-P_{m(n)}B(t)P_{m(n)}||+\frac{18}{n}\leq \frac{37}{n}
2\begin{pmatrix}2\mu^2\lambda(s_1^2+\nu+1)s_1-(\mu^2\lambda+\nu)(s_2^2+\mu(1+\lambda))s_1 -(\mu^2\lambda+\nu)(s_1^2+\nu+1)s_2+ 2\nu(s_2^2+\mu(1+\lambda))s_2\end{pmatrix}.
X(z) = \begin{pmatrix}1 0 0 0 1 0 0 0 i\end{pmatrix}Y(z)\begin{pmatrix}1 0 0 0 z^{\frac{1}{4}}0 0 0 z^{-\frac{1}{4}}\end{pmatrix}\begin{pmatrix}1 0 0 0 \frac{1}{\sqrt{2}}\frac{1}{\sqrt{2}}0 \frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}\end{pmatrix}\begin{pmatrix}1 0 0 0 1 0 0 0 i\end{pmatrix},
\frac{d^{n-1}\rho}{dv^{n-1}}=P_0-\frac{\kappa}{c_{+0}-c_{-0}}\frac{u+c_{+0}-\bar{c}_-}{\nu^2}\frac{d^{n-1}\nu}{dv^{n-1}}&=P_0+\frac{1}{2}\hat{I}_{n,0}.
\int_{B_1}|y|^{1-2s}U^2 \le C^{m}\int_{B_{\frac{1}{2^m}}}|y|^{1-2s}U^2 &=C^m2^{-mn}\frac{1}{|B_{\frac{1}{2^m}}|}\int_{B_{\frac{1}{2^m}}}|y|^{1-2s}U^2 &\le C^m(\frac{1}{2})^{m(n+\ell)}(\frac{1}{2})^{-m\ell}\frac{1}{|B_{\frac{1}{2^m}}|}\int_{B_{\frac{1}{2^m}}}|y|^{1-2s}U^2.
e_{_{\mu}}^{\; a}=\left(\begin{array}{ccc}1 0 0 0 \rho\frac{1}{\beta}\cos\phi \rho\frac{1}{\beta}\sin\phi 0 -\rho\frac{\sin\alpha\theta}{\alpha}\sin\phi \rho\frac{\sin\alpha\theta}{\alpha}\cos\phi\end{array}\right).
\sup_{B_r(y)}|u(x)-u(y)-(x-y)\cdot\nabla u(y)| \leq\sup_{B_{r_y}(y)}|u(x)-u(y)-(x-y)\cdot\nabla u(y)| &\leq C r_y^{1+\alpha}= C\frac{r_y^{1+\alpha}}{r^{1+\alpha}}r^{1+\alpha}\leq \frac{C}{r_0^{1+\alpha}}r^{1+\alpha}.
R=[\frac{1+(\frac{b}{t})^{2}}{4-2.521\frac{t}{b}(1-\frac{1.991}{e^{\pi\frac{b}{t}}+1})}][1+\frac{0.00851b^{2}}{L^{2}}]-0.060(\frac{b}{L})^{\frac{3}{2}}(\frac{b}{t}-1)^{2}
\|\{f_j\}_{j\in\mathbb Z}\|_{L\dot B_{pq}(E\times J)}:=\|\{f_j\}_{j\in\mathbb Z}\|_{\ell^qL^p(E\times J)}:=\|\{f_j\}_{j\in\mathbb Z}\|_{\ell^q(J;L^p(E))}:=\left[\sum_{j\in J}\|f_j\|_{L^p(E)}^q\right]^{\frac{1}{q}}
\begin{aligned}M L(2^{2n_1+1}) = \left( \begin{matrix}\displaystyle 1 0 2^{4n_1+2}/c 2^{2n_1+1}/c \end{matrix}\right) \equiv \left( \begin{matrix}\displaystyle 1 0 d 2^{2n_1+1}/c \end{matrix}\right) \equiv L(d) M \equiv N^{-1}\psi\left(L(2^{2n_1+1})\right)NM \pmod{2^{T-3n_1-2}}.\end{aligned}
S = S^{( 1 )}+ S^{( 2 )}+ . . . = - \int_{0}^{\infty}\! \! d l \ \eta ( l ) - \frac{1}{2}\int_{0}^{\infty}\! \! d l \int_{0}^{l}\! \! d l^{\prime}\ [ \eta ( l ) , \eta ( l^{\prime}) ] + . . . \, .
\tau_{j}^{x}\; = \; \Big ( \prod_{i = 1}^{j - 1}\sigma_{i}^{z}\Bigr ) \sigma_{j}^{x}\, , \hspace{2 . 5 c m}\tau_{j}^{y}\; = \; \Big ( \prod_{i = 1}^{j - 1}\sigma_{i}^{z}\Bigr ) \sigma_{j}^{y}
C_0^{U^{\star}}=\begin{cases}\,\,\int_{\eta^{\star}-\beta^{\star}\eta^{\star}}^{+\infty}{p(\gamma)d\gamma}, \eta^{\star}> 0, \,\,\int_{-\infty}^{\eta^{\star}-\beta^{\star}\eta^{\star}}{p(\gamma)d\gamma}, \eta^{\star}< 0;\end{cases}
\lim_{y\rightarrow\pm\infty}iye^{by}\int_{\mathbb{R}}\frac{d\mu(t)}{iy-t}=\lim_{y\rightarrow\pm\infty}iy\int_\mathbb{R}\frac{e^{ibt}}{iy-t}d\mu(t)=\int_\mathbb{R}e^{ibt}d\mu(t)=0.
\Lambda_{\alpha}\bar{\Lambda}_{\dot{\beta}}\bar{\mathrm{\raisebox{0 e x}[ 1 . 7 5 e x ] [ 0 e x ]{D}}}\bar{\Theta}^{\dot{\beta}}= 0 , \L a m b d a_{\alpha}\bar{\Lambda}_{\dot{\beta}}D \Theta^{\alpha}= 0
\sum_{i=1}^{k}\theta(t_{i})(t_{i}-t_{i-1})\geq\int_{a}^{b}\theta(t)\,dt-\frac{2\pi}{\gamma}\,\omega\left(\frac{1}{\lambda}\right)\geq\int_{a}^{b}\theta(t)\,dt-\frac{16\pi\gamma^{2}}{\nu_{2}}.
\frac{\frac{1-\alpha}{2}- \frac{1-\alpha}{2q}}{\frac{1-\alpha}{2}+ \frac{1}{n}- \frac{1}{2}}= \frac{(1-\alpha) \frac{q-1}{q}}{1-\alpha + \frac{2}{n}- 1}= \frac{1-\alpha}{q}= \frac{q - \frac{2}{n}}{q}\in (0, 1),
\frac{\# D_{ab}}{\# \{0,1\}^d}\,=\, \frac{1}{2^d}\sum_{k=a}^{b}\binom{d}{k}\,=\, \P\biggl\{\alpha \leq \frac{1}{\sqrt{d}}\, \sum_{j=1}^d Z_j \leq \beta \biggr\}&\,\geq\, \underbrace{\Phi(\beta) - \Phi(\alpha)}_{=: C_{\alpha\beta}}- \frac{2 \, C_0}{\sqrt{d}}\,=:\, r_0(\alpha,\beta,d) \,.
\sum_{i \ge 0}b_{i,r}+ b_{r,r}\lambda_M - 1 = c_{0,r}+ b_{r,r}\lambda_M - 1 &= - 3r(r + 1)\mu/2 + b_{r,r}\lambda_M &= - 3r(r + 1)\mu/2 + (2^{-r}- 2r\mu)\lambda_M &= 2^{-r}\lambda_M - \mu r(3(r + 1)/2 + 2\lambda_M).
{\cal L}_b = -{1 \over 4}(\partial_\mu b_\nu^R - \partial_\nu b_\mu^R)^2+{1 \over 2}(m_b^R)^2 b_\mu^R b^\mu{}^R -{1 \over 4}\delta_b (\partial_\mu b_\nu^R - \partial_\nu b_\mu^R)^2+{1 \over 2}\delta_m b_\mu^R b^\mu{}^R .
\kappa \cos^{2}{(\frac{\pi}{2}-\delta)}\approx 1+\frac{1}{2\kappa^{2}\cos^{4}{(\frac{\pi}{2}-\delta)}}- \frac{c_{1}\sin{(2\kappa\sin{t})}+c_{2}\cos{(2\kappa\sin{t})}}{\kappa^{2}\cos{(\frac{\pi}{2}-\delta)}}.
\{\, g_{a b}( z , \bar{z}) , \, g_{c d}( y , \bar{y}) \, \}= \frac{\gamma^{2}}{4}\, \, \Theta \, [ 2 g_{a d}( z , \bar{y}) \, g_{c b}( y , \bar{z}) - g_{a b}( z , \bar{z}) \, g_{c d}( y , \bar{y}) ] ,
{\bigl<t_{\eta,v}^{(\rho)};t_{\theta,w}^{(\sigma)}\bigr>}_{L^2}= \int_G\overline{t_{\eta,v}^{(\rho)}(g)}\cdot t_{\theta,w}^{(\sigma)}(g)\,dg = \frac{1}{\dim V_{\rho}}\delta_{\rho\sigma}\overline{\left<\eta;\theta\right>}\left<v;w\right>,
i\hbar\frac{\partial}{\partial t}\Psi_{\alpha}(r,t)=\hat{H}\Psi_{\alpha}(r,t)=(-\frac{\hbar^{2}}{2m}\nabla^{2}+V(r))\Psi_{\alpha}(r,t)=-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi_{\alpha}(r,t)+V(r)\Psi_{\alpha}(r,t)
CS(A)-CS(A_{0}) =\int_{0}^{1}\Gamma_{A(t)}\big{(}\dot{A}(t)\big{)}dt &=-2\int_{0}^{1}\Big{(}\int_{M}Tr\big{(}(F_{A_{0}}+td_{A_{0}}a+t^{2}a\wedge a)\wedge a\big{)}\wedge\ast_{M}P\Big{)}dt &=-\int_{M}Tr\big{(}d_{A_{0}}a\wedge a+\frac{2}{3}a\wedge a\wedge a\big{)}\wedge\ast_{M}P+C,
\lim_{t\to 0^+}\int_{[\frac{1}{m},m]^d}{b_{\kappa_m, p}}(x,t)x^{2\alpha}dx =\int_{[\frac{1}{m},m]^d}{b_{\kappa_m, p}}(x,0)x^{2\alpha}dx &=\int_{[\frac{1}{m},m]^d}B_{\kappa_m,p}(f(x),g(x))x^{2\alpha}dx.
F_{_{\! S}}^{- 1}= \frac{\cosh \frac{1}{2}\sigma \otimes \cosh \frac{1}{2}\sigma + 2 \xi v_{+}e^{- \frac{1}{2}\sigma}\otimes v_{+}e^{- \frac{1}{2}\sigma}}{\cosh \frac{1}{2}\Delta_{_{\! J}}( \sigma )}\,
as_{\lambda, i}(\vec \varphi) = \ \left( c_1^{i}\c_2^{n-i}\right)^{\frac{1 - 2 \lambda}{n}}\ \left( (\det(A_1))^{i}( \det(A_2) )^{n-i}\right)^{\frac{\lambda}{n}}\ \frac{( 2 n \pi)^{\frac{n}{2}}}{\left( \det( i A_1 + (n-i) A_2) \right)^{\frac{1}{2}}}.
\left[ i \left( \rho^{m}\nabla_{m}\right)_{\beta}^{\alpha}- \delta_{\beta}^{\alpha}\rho^{3}\right] \left( \begin{array}{c}{\theta^{1 \beta \alpha^{\prime}}}\\{\theta^{2 \beta \alpha^{\prime}}}\\ \end{array}\right) = 0 .
\begin{array}{cl}\Delta(y_1, \cdots, y_{M-L}| \zeta^{(n)}|\cdots|\zeta^{(1)}|\cdots|\zeta^{(n-1 \cdots 2 1)}) = \det \left( A_{\lambda}(y_{\mu}|\zeta^{(n)}|\cdots |\zeta^{(1)}| \zeta^{(nn-1)}|\cdots |\zeta^{(21)}|\cdots |\zeta^{(n-1 \cdots 1)}\right)_{1\leq \lambda ,\mu \leq M-L},\end{array}
\int_{-\infty}^{\infty}s e^{sq_j [\log(1/sq_j)/sq_j + v]}\cdot sq_j \cdot P(W e^{-sq_jv}> x) \: dv = \int_{-\infty}^{\infty}s e^{sq_jv}e^{-xe^{sq_j v}}\: dv = \frac{1}{q_jx}.
\lim_{\epsilon \rightarrow 0}\psi_k^{\epsilon,\hat{v}}(x) = \lim_{\epsilon \rightarrow 0}\mathbb{E}_{x,k}^{\epsilon,\hat{v}}\Bigl\{\psi_{\zeta^{\epsilon}(\tau_D^{\epsilon,\hat{v}})}^{\epsilon,\hat{v}}(X^{\epsilon,\hat{v}}(\tau_{D}^{\epsilon,\hat{v}})) \Bigr\}&= g_{k_0}(\bar{y}_0)