harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Psi (P) \ast f(x)= \int \sum_{\alpha}a_{\alpha}\frac{\partial^{|\alpha|}\delta}{\partial y^{\alpha}}(y) f(x-y) d y = \sum_{\alpha}a_{\alpha}\int \delta(y) (-1)^{|\alpha|}\frac{\partial^{|\alpha|}}{\partial y^{\alpha}}f(x-y) d y = \sum_{\alpha}a_{\alpha}\frac{\partial^{|\alpha|}}{\partial x^{\alpha}}f(x).
\vec{0}= \vec{1}P_{G}^{\uparrow n}= \vec{1}(J_{G}^{\uparrow n}D_{1})^{-1}(J_{G}^{\uparrow n}M_{G}^{\uparrow n}) D_{2}^{-1}\; \; \; \; \mbox{so}\; \; \; \; \vec{0}= \vec{1}(J_{G}^{\uparrow n}D_{1})^{-1}H_{G}^{\uparrow n}.
0 = \phi + \phi \circ (1\,\,3) + \phi\circ(1\,\,4) = \phi \circ (1\,\,3) = \alpha \circ (1\,\,3) + \alpha \circ (3\,\,4) \circ (1\,\,3) = \alpha + \alpha \circ (1\,\,4).
T_{n_0}= O \left( 2^{-4n_0 \beta/(1-\beta)}a_{n_0}\left( 2^{-4n_0(1-2\alpha)/(1-\beta)}n_0^{-2}{a_{n_0}}\right)^{-1}\right)= O \big({n_0^2}2^{-4n_0 (\beta+2\alpha-1)/(1-\beta)}\big) \to 0
\max_{x\in M\smallsetminus I_{P_{\varepsilon}}(\varepsilon\rho)}u_{\varepsilon}(x)=\max_{x\in\partial I_{P_{\varepsilon}}(\varepsilon\rho)}u_{\varepsilon}(x)=\max_{|z|=\rho}U(z)+\sigma(\varepsilon)\le ce^{-\alpha\rho}+\sigma_{1}(\varepsilon)
\Delta \Sigma = \frac{g^{2}}6 \int \frac{d^{4}k}{( 2 \pi )^{4}}\frac 1{k^{2}+ m^{2}}\big ( 2 + \cos ( k \tilde{p}) \big ) , \quad \tilde{p}_{\mu}= \theta_{\mu \nu}k_{\nu},
d(A^Tx, \partial g(\bar{y})) \geq \gamma \cdot d(A^Tx, \partial g(\bar{y})\cap R(A^T)) = \gamma \cdot \min_{A^Tu\in\partial g(\bar{y})}\|A^Tx-A^Tu\| &= \gamma \cdot\min_{y\in A^T X_0}\|A^Tx-y\|\geq \gamma \cdot \min_{y\in V_0}\|A^Tx-y\|= \gamma \cdot\|A^Tx- \hat{y}\|,
K_{(y,s)}(x,t) =\langle K_{(y,s)}(\xi,\eta),K_{(x,t)}(\xi,\eta)\rangle_W &=\langle K_{(x,t)}(\xi,\eta),K_{(y,s)}(\xi,\eta)\rangle_W=K_{(x,t)}(y,s), \Vert K_{(x,t)}(y,s)\Vert_W =\langle K_{(x,t)}(y,s),K_{(x,t)}(y,s)\rangle_W=K_{(x,t)}(x,t)<\gamma .
{\cal L}= - \frac{1}{4}F_{\mu \nu}^{a}F^{a \mu \nu}, F_{\mu \nu}^{a}= \partial_{\mu}A_{\nu}^{a}- \partial_{\nu}A_{\mu}^{a}- g f^{a b c}A_{\mu}^{b}A_{\nu}^{c}\ ,
<M>=2V\int_{}^{}\frac{e^{\beta(\mu-\frac{p^{2}}{2m})}(1+e^{\beta(\mu-\frac{q^{-2}p^{2}}{2m}})d^{3}p}{(2\pi\hbar)^{3}(1+2e^{-\beta (\frac{p^{2}}{2m}-\mu)}+e^{-\beta (\frac{p^{2}}{2m}(q^{-2}+1)-2\mu)})}.
\sigma_{1}= \left( \begin{matrix}{1}&{0}\\{0}&{- 1}\\ \end{matrix}\right) , \quad \sigma_{2}= \left( \begin{matrix}{0}&{1}\\{1}&{0}\\ \end{matrix}\right) , \quad \sigma_{3}= \left( \begin{matrix}{0}&{- i}\\{i}&{0}\\ \end{matrix}\right) .
Z_{i}^{\dagger}( \theta ) = \exp \left[ - i \int_{\theta}^{\infty}d \theta^{\prime}\, \delta_{i l}( \theta - \theta^{\prime}) a_{l}^{\dagger}( \theta^{\prime}) a_{l}( \theta^{\prime}) \right] a_{i}^{\dagger}( \theta ) \, .
x=\frac{X}{H},z=\frac{Z}{H},u=\frac{U}{k_{T}/H},w=\frac{W}{k_{T}/H},S^{*}=\frac{S-S_{B}}{S_{T}-S_{B}},T^{*}=\frac{T-T_{B}}{T_{T}-T_{B}},t^{*}=\frac{t}{H^{2}/k_{T}}.
\begin{aligned}\int_{V_{\min}}^{V_{F}}\left(\partial_tp \phi-hp\partial_v\phi+a\partial_vp\partial_v\phi\right) dv + \left(hp\phi|_{V_{\min}}^{V_R^-}+hp\phi|_{V_R^+}^{V_F}\right)-\left(a\partial_vp\phi|_{V_{\min}}^{V_R^-}+a\partial_vp\phi|_{V_{V_R^+}}^{V_F}\right)=0.\end{aligned}
A^{-1}=\frac{1}{det(A)}\sum_{s=0}^{n-1}A^{s}\sum_{k_{1},k_{2},...,k_{n-1}}\prod_{l=1}^{n-1}\frac{(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}tr(A^{l})^{k_{l}},
\left[ e^{+ 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi + g}}\big ( \delta ( x ) , C_{m_{d}}\alpha \big ) + i \theta}\; + \; e^{- 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi + g}}\big ( \delta ( x ) , C_{m_{d}}\alpha \big ) - i \theta}\right] \; .
\begin{tabular}{|c|c|c|c|}\hline$\nu$ $b$ $\rho$ $U$ \hline $(4,4,1,1)$ $(1,3,2,2,2)$ $(4,3,1)$ $U(2,4)=U(4,1)=5$ \hline $(4,3,1)$ $(1,3,2,2)$ $(3,3)$ $U(1,4)=U(3,1)=4$ \hline $(3,3)$ $(1,3,2)$ $(3,1)$ $U(2,2)=U(2,3)=3$ \hline $(3,1)$ $(1,3)$ $(1)$ $U(1,2)=U(1,3)=U(2,1)=2$ \hline $(1)$ $(1)$ $U(1,1)=1$ \hline\end{tabular}
h^1_1(\phi^1_1)=(1-\sin 3\phi^1_1)^{(A-B)/6}(1+\sin 3\phi^1_1)^{(A+B)/6}\cdot \frac{1}{2A-3-2B\sin 3\phi^1_1}\hat{P}^{(\alpha,\beta)}_{l+1}(\sin 3\phi^1_1)
{\cal J}_{i}^{\mu \nu}[ \xi | s ] = 4 \pi \int d^{4}x \, \epsilon^{\mu \nu \rho \sigma}\left[ \Omega_{\xi}( s , 0 ) T_{i}\Omega_{\xi}^{- 1}( s , 0 ) \right]^{j}\dot{\xi}_{\sigma}( s ) \widetilde{j}_{\rho}^{j}( x ) \delta^{4}( x - \xi ( s ) ) .
R_{ij, k}-R_{ik, j}= -R_{tijk, t}= R_{tikj, t}R_{ij, kt}-R_{ij, tk}=R_{likt}R_{lj}+R_{ljkt}R_{li}R_{ij, ktl}-R_{ij, klt}= R_{sj, k}R_{sitl}+R_{is, k}R_{sjtl}+ R_{ij, s}R_{sktl}.
\epsilon\int_{\{\Psi<-t_2\}}|\tilde{F}_{t_1}-F_{t_0}|^2_he^{-\Psi}\le \int_{\{\Psi<-t_2\}}|\tilde{F}_{t_1}-F_{t_0}|^2_h(e^{-\Psi-t_0}c(t_0)-c(-\Psi)) \le \int_{\{\Psi<-t_0\}}|\tilde{F}_{t_1}-F_{t_0}|^2_h(e^{-\Psi-t_0}c(t_0)-c(-\Psi)) = 0.
0 = \frac{1}{2}(\delta_1 + \delta_2) \oplus (\varepsilon_1 + \dots + \varepsilon_5) = &\frac{1}{4}(\delta_1\oplus(\varepsilon_1 + \varepsilon_2 + \varepsilon_3 + \varepsilon_4) + \delta_1\oplus(\varepsilon_4 + \varepsilon_5) + \delta_2\oplus(\varepsilon_1 + \varepsilon_2) + \delta_2\oplus(\varepsilon_3 + \varepsilon_5)).
\theta h \int_{\Sigma}[ n_t \nabla_x u_h \cdot\nabla_x v_h - n_x \cdot \nabla_x u_h \partial_{t}v_h] \, ds & = \theta h\int_{\Sigma}\nabla_x u_h \cdot [ n_t \nabla_x v_h - n_x \partial_{t}v_h] \, ds = 0,
\ell^{\alpha+\beta}[u]_\alpha[f]_\beta \gtrsim\begin{cases}\|\lceil u, (\cdot)_\ell\rceil f\| \textrm{if}\alpha \in (0,1], \| \big( \lceil u, (\cdot)_\ell\rceil-\partial_1 u \lceil x_1, (\cdot)_\ell\rceil \big) f\| \textrm{if}\alpha \in (1, \frac32),\end{cases}
\left| 2 \sum_{i=1}^m n_i p_i Y_i - \frac{1}{2}E(Y)\right| \leq \left|\sum_{i=1}^m n_i Y_i (2 p_i - q_i^2Y_i)\right| + \frac{1}{2}E(Y) \leq \left ( \sum_{i=1}^m n_i \right )^{1/2}\left (\sum_{i=1}^m n_iY_i^2 (2 p_i - q_i^2Y_i)^2 \right )^{1/2}+ \frac{1}{2}E(Y).
\left|R_N\right| \le \left|R_{0N}\right|+\left|R_{1N}^{\rm (cut)}\right| +2\left|R_{1N}^{(1)}\right|_{\rm (I)}+\left|\widetilde R_{1N}^{(0)}\right| +2\left|\widetilde R_{1N}^{(1)}\right|_{\rm (II)}+\left|\widetilde R_{1N}^{(1)}\right|_{\rm (III)}+\left|\widetilde R_{1N}^{(2)}\right|_{\rm (III)},
T_{p+i}^{-1}\,J =\sum_{l=1}^{i-1}q^{c-l}(T_{p+l}^{-1}\cdots T_{p+c-1}^{-1})T_{p+i-1}^{-1}+\sum_{l=i, i+1}q^{c-l}(T_{p+l}^{-1}\cdots T_{p+c-1}^{-1}) +\sum_{l=i+2}^{c}q^{c-l}(T_{p+l}^{-1}\cdots T_{p+c-1}^{-1})T_{p+i}^{-1}.
r^{-B}(g^+\!(r)-br^{-B})\;=\;r^{-B}\big(f^+\!(r)+a(S_D^{-1}\Phi)^+\!(r)+b\gamma^{-1}\Phi^+\!(r)-br^{-B}\big) =\;a\,p^+ +b\,q^+\gamma^{-1}+ o(r^{1/2-B})\,,
N_{r,i}^{n}= \binom{r}{i}^{-1}\frac{(2r)!}{2^{2r}(r!)^{2}}\sum_{k=\max(0,i+r-n)}^{\min(i,r)}(-1)^{r-k}\binom{n-r}{i-k}\binom{r-\frac{1}{2}}{k}\binom{r-\frac{1}{2}}{r-k}&+\sum_{k=0}^{r}\lambda_{k}\binom{k}{i}^{-1}\frac{(2k)!}{2^{2k}(k!)^{2}}\sum_{j=\max(0,i+k-n)}^{\min(i,k)}(-1)^{k-j}\binom{n-k}{i-j}\binom{k-\frac{1}{2}}{j}\binom{k-\frac{1}{2}}{k-j}.
\Bigl[\Lambda(t), H_{\epsilon}(t)\Bigr] = -\Bigl[\Lambda(t), X_{\epsilon}(t)\Bigr]^{\dagger}H(t)X_{\epsilon}(t) + X_{\epsilon}(t)^{\dagger}\Bigl[\Lambda(t), H(t)\Bigr]X_{\epsilon}(t) + X_{\epsilon}(t)^{\dagger}H(t)\Bigl[\Lambda(t), X_{\epsilon}(t)\Bigr]
\int_{t+2(r+1)}^{\infty}\lambda''(s) \frac{2 r (s-t)}{(1+r^{2}+(s-t)^{2}+\sqrt{(1+(r+s-t)^{2})(1+(r-(s-t))^{2})}}ds &=\int_{t+2(r+1)}^{\infty}\lambda''(s) \frac{2r}{(s-t)}\left(\frac{1}{2}+O\left(\frac{(1+r)^{2}}{(s-t)^{2}}\right)\right) ds &=\int_{t+2(r+1)}^{\infty}\lambda''(s) \frac{r}{(s-t)}ds +E_{1}(t,r)
9\Phi_{1,w}(X,Y) = X\times Y - \frac 12 (e) X\times Y - \frac 12 (X\times Y) e - \frac 12 (X\times Y) e + \frac 12 (e)(X\times Y) e. & = -\frac 12 X\times Y + \frac 12 (X\times Y) e.
\sum_\ell c_{k,\ell}x^{|V(H)|-\ell}=\sum_i\sum_j\binom{|B|-i}k\omega_{i,j}x^{|V(H)|-i-j}=\sum_\ell\bigg(\sum_i\binom{|B|-i}k\omega_{i,\ell-i}\bigg)x^{|V(H)|-\ell}
f\Big(\frac xy,\,y\Big) =\sum_{i=0}^{\infty}\sum_{j=0}^{i}[b+c(i+j)]v_{i, i+j}x^i y^j &= b\, G(x,y) + c \big( x\,\partial_x G(x,y) + y\,\partial_y G(x,y) \big).
\delta_{\mathrm{s h i f t}}\int_{\Sigma_{g}}\sqrt{g}= - \int_{\Sigma_{g}}\frac{1}{2}\sqrt{g}t r ( r_{\mu}^{\gamma}g_{\gamma \nu}+ r_{\nu}^{\gamma}g_{\gamma \mu}) = - \int_{\Sigma_{g}}\sqrt{g}r_{\mu}^{\mu}.
\lambda_1 (H_{\Lambda}^D) \leq ~~ < \varphi_0, H_{\Lambda}^D \varphi_0 > ~ = \lambda_0 + \int_{\Lambda}V \varphi_0^2 ~dx & \leq~ \lambda_0 + \frac{C}{|\Lambda^{'}|}\int_{\Lambda}V (x) ~dx,
\langle T_{\tau\tau}\rangle ={1\over4\pi^2}\sum_{\scriptstyle n=-\infty\atop\scriptstyle n\ne0}^\infty{2\Bigl(\cos(n\alpha) + 2\Bigr)\over3\sigma_n(x,x)^2}-{4n^2\beta^2\Bigl(\cos(n\alpha)+5\Bigr)\over3\sigma_n(x,x)^3}
\mathbb{P}_{\rm FA}+ \mathbb{P}_{\rm MD}= E_U\left [\mathbb{P}_{\rm FA}(U) + \mathbb{P}_{\rm MD}(U) \right] \geq E_U \left [\mathbb{P}_{\rm FA}(U) + \mathbb{P}_{\rm MD}(U)|{\mathcal{A}^c}\right] P({\mathcal{A}^c}) > 1 - \epsilon.
\begin{array}{ll}\partial_y u^s(t,y) = \frac{\beta e^{\beta y}}{2\sqrt{\pi t}}\int\limits_{+\infty}^{y}e^{-\beta \hat{y}}\int\limits_0^{+\infty}\Big( e^{-\frac{(\hat{y}-\tilde{y})^2}{4t}}- e^{-\frac{(\hat{y}+\tilde{y})^2}{4t}}\Big)w_0^1(\tilde{y}) \,\mathrm{d}\tilde{y}\,\mathrm{d}\hat{y}\end{array}
U_{{\rm lin}}=\begin{pmatrix}c_{1,1}c_{1,2}c_{1,3}begin{align*}-1pt] c_{2,1}0 c_{1,2}begin{align*}-1pt] c_{3,1}c_{2,1}c_{1,1}\end{pmatrix},U = \left(\begin{matrix}c_0 c_1+1 c_0 c_0^2 begin{align*}-1pt] c_1 0 c_0 begin{align*}-1pt] c_1^2 c_1 c_0 c_1+1\end{matrix}\right),\begin{pmatrix}* * * begin{align*}-1pt] * * begin{align*}-1pt] * * *\end{pmatrix}.
f_k = [ u^k] F(u)=\frac{2^kz^k}{(1+\sqrt{1-4z})^k}, g_k =[ u^k] G(u)=\frac{2^kz^k(1-2z-\sqrt{1-4z})}{(1+\sqrt{1-4z})^{k+1}}, \mbox{and}h_k =[ u^k] H(u)=\frac{2^{k+1}z^{k+1}}{(1+\sqrt{1-4z})^{k+1}}.
n^{1 - 2 \beta_2}m_{0}^{2 \alpha_2}D^{1 - 2 \alpha_2}\le n^{1 - 2 \beta_2}m^{2 \alpha_2}(m^{\alpha_1}n^{-\beta_1})^{1 - 2 \alpha_2}= m^{1 - \frac{k-1}{4 k -1}}n^{1 - \frac{3}{4k -1}}.
\begin{array}{ll}f_0 =1,\mbox{and}f_k=zf_0+z\sum\limits_{\ell=0}^{k-1}g_\ell +z\sum\limits_{\ell=0}^{k-1}h_\ell, k\geq 1, g_k =z f_{k+1}+zh_{k+1},k\geq 0, h_k =zf_k+zg_k,k\geq 0.\end{array}
\mathrm{I m}K_{n}( \omega ) = \mp \frac{2^{1 - n}\pi^{\frac{3 - n}{2}}}{\Gamma ( \frac{n - 1}{2})}\left( \sqrt{\omega^{2}- m^{2}}\right)^{n - 3}\, \theta ( \omega^{2}- m^{2}) .
\sin(\theta) =(1-(\bar{\mathbf{n}}\cdot\mathbf{n}(Y))^2)^{1/2}&\leqslant (1+(1-2\epsilon^2)^{-3/2}) \kappa h \left( 4 - (1+(1-2\epsilon^2)^{-3/2})^2 \kappa^2 h^2 \right)^{1/2}&\leqslant (1+(1-2\epsilon^2)^{-3/2}) \kappa h (4 - \bar{\kappa}/2)^{1/2}&\leqslant C \kappa h,
p^{\gamma\dot{\alpha}}B_{\gamma\epsilon_{1}\epsilon_{2}\cdot\cdot\cdot\epsilon_{n}}^{\dot{\beta}_{1}\dot{\beta}_{2}\cdot\cdot\cdot\dot{\beta}_{n}}=mcA_{\epsilon_{1}\epsilon_{2}\cdot\cdot\cdot\epsilon_{n}}^{\dot{\alpha}\dot{\beta}_{1}\dot{\beta}_{2}\cdot\cdot\cdot\dot{\beta}_{n}}
\Lambda_{ij}\cdot T_{ij}^{(l)}=T_{ij}^{(l)},\;\;\;\Lambda_{ij}\cdot T_{ii}^{(l)}=T_{jj}^{(l)}\;\;\;\textup{and}\;\;\;\Lambda_{ij}\cdot T_{jj}^{(l)}=T_{ii}^{(l)}.
\left\{j_M^0\left( t,\sigma \right) ,\widehat{U_N^0}\left(t,\sigma^{\prime}\right) \right\}=-\delta_MU_N^0\cdot \delta \left(\sigma^{\prime}-\sigma \right) \;+\frac \partial{\partial \sigma^r}\left[ \delta_M\phi^i\frac{\partial \widehat{U_N^0}}{\partial \partial_r\phi^i}\cdot \delta \left( \sigma^{\prime}-\sigma \right) \right] \quad.
A=\frac{1}{\alpha(\alpha+\beta-1)}\sum_{i}q^{\alpha+\beta-1}_{i}+\frac{1}{\alpha(1-\beta)}\sum_{i}p^{\alpha}_{i}q^{\beta-1}_{i}&B=\frac{1}{(1-\beta)(\alpha+\beta-1)}\sum_{i}p^{\alpha+\beta-1}_{i}
\| x\|_{\mathcal{CG}_q^\omega}\ \ge\ \left(\sum_{k=1}^{n_s}( \omega(k)\vartheta_k(x_s))^q\frac{1}{k}\right)^{1/q}\ \stackrel{\eqref{four}}{\gtrsim}\ (\widetilde{\zeta}(n_s))^{1/q}\|1_{V_s}\|\ \stackrel{\eqref{zeta}}{\gtrsim}\ \omega(n_s)\|1_{V_s}\|.
Q_{3,k}=|I_{k}|^{2(1-H)}\int_{0}^{1}\left|(a_{k}+v_1|I_{k}|)^{H-\frac{1}{2}}\cdot\int_{0}^{v_1}\frac{\varphi_k(v_1)-\varphi_k(v_2)}{\left(v_1-v_2\right)^{H+\frac{1}{2}}}dv_2\right|^{2}dv_1,
b_0 = a_0(a_2 b_1 - a_1 b_2)^q/(c_2 b_1 - c_1 b_2)^q,\;\, c_0 = a_0(a_2 c_1 - a_1 c_2)^q/(b_2 c_1 - b_1 c_2)^q.
[n]_q^{(i)}= ( (q^n-1) \frac{1}{q-1})^{(i)}&= \sum_{j=0}^{i}\binom{i}{j}(q^n-1)^{(i-j)}(\frac{1}{q-1})^{(j)}&= \sum_{j=0}^{i-1}\binom{i}{j}n(n-1)\cdots (n-(i-j-1)) q^{n-(i-j)}\frac{j!(-1)^j}{(q-1)^{j+1}}+\frac{(q^n-1)i!(-1)^i}{(q-1)^{i+1}}.
\mathbb{E}\left[y_l^* y_j|\hat{\mathbf{H}}\right]= a_c^2\hat{\phi}^{\left(l,c\right)^*}\hat{\phi}^{\left(j,c\right)}+\sum_{q=1}^M a_q^2\hat{\phi}^{\left(l,q\right)^*}\hat{\phi}^{\left(j,q\right)}.
\frac{2^s\Gamma(1-s)}{s}\frac{\Gamma\big(\frac{2k+n}{2}+\frac{1+s}{2}\big)}{\Gamma\big(\frac{2k+n}{2}+\frac{1-s}{2}\big)}=c_1 +\int_0^{\infty}\big(1-e^{-(2k+n) t}\big)(\sinh t)^{-s-1}\,dt &=c_1 +|\lambda|\int_0^{\infty}\big(1-e^{-(2k+n) |\lambda|t}\big)(\sinh t|\lambda|)^{-s-1}\,dt.
\begin{array}{rl}c =c-\ell\lfloor\frac{c}{\ell}\rfloor+\lfloor\frac{c}{\ell}\rfloor = m_v+(-1)^{m_v}(-1)^{\ell(n'_v-1)}\ell n_v'= m_v+(-1)^{m_v}\ell n_v.\end{array}
\frac{d(\mbox{sn}\,u)}{du}= \mbox{cn}\,u\,\mbox{dn}\,u,\;\;\;\;\frac{d(\mbox{cn}\,u)}{du}= -\mbox{sn}\,u\,\mbox{dn}\,u,\;\;\;\;\frac{d(\mbox{dn}\,u)}{du}= -k^2 \mbox{sn}\,u\,\mbox{cn}\,u.
\langle I (v) , u \rangle = \lim_{n\rightarrow \infty}{E}\bigg[\bigg\langle \sum_{t=0}^{n-h}\mathcal{P}_{-t}(X_h),u\bigg\rangle \: \bigg\langle v,\sum_{s=0}^{n-h}\mathcal{P}_{-s}(X_0) \bigg\rangle\bigg].
var(\sum_{i=1}^{n}a_{i}X_{i})=\sum_{i=1}^{n}a_{i}^{2}\sigma^{2}(X_{i})+2\sum_{i,j:i<j}a_{i}a_{j}cov(X_{i},X_{j})=\sum_{i,j}a_{i}a_{j}cov(X_{i},X_{j})
p_i \circ H (X_1,...,X_n,0) = p_i \circ \Delta \circ F (c_1,...,c_n) = p_i(F(c_1,\dots,c_n),\dots,F(c_1,\dots, X_n)) = F(c_1,\dots,c_n) = p_j \circ \Delta \circ F (c_1,...,c_n) = p_j \circ H (X_1,...,X_n,0).
\epsilon^{\alpha \mu \nu \xi}\Omega_{\left( \mu \nu \xi \right)}^{\left\{\beta \sigma \right\}}+ \epsilon^{\beta \mu \nu \xi}\Omega_{\left( \mu \nu \xi \right)}^{\left\{\sigma \alpha \right\}}+ \epsilon^{\sigma \mu \nu \xi}\Omega_{\left( \mu \nu \xi \right)}^{\left\{\alpha \beta \right\}}= 0 \, ,
X' = \{x\in \mathbb{T}^{\mathbb{Z}^2}:x_\mathbf{n}+x_{\mathbf{n}+\mathbf{e}_1}+x_{\mathbf{n}+\mathbf{e}_2}=0\enspace \textup{for every}\enspace \mathbf{n}\in \mathbb{Z}^2\}.
\ker D \mu \subset \rho_{L^{*}\setminus \{0\}}(\mu^*\circ F_{\pi}\circ \pi^*(O(1)))= \rho_{L^{*}\setminus \{0\}}(F_{\operatorname{pr}}\circ \operatorname{pr}^* \circ F_{\phi}\circ \phi^*(O(1)) ),
\sum_{|i-i_0| + |j-i_0| \leq N^{\nu}}\mathbb{E}[ O ( \rho (E) N (\lambda_i (T) - E), \rho(E) N (\lambda_j (T) - E )) | \{\lambda_k (0) \}_k] = \sum_{i, j}\mathbb{E}[ O ( \rho_T (E) N (\lambda_i (T) - E), \rho_T (E) N (\lambda_j (T) - E )) | \{\lambda_k (0) \}_k] +\O ( N^{-100})
\begin{gathered}I^u_{k_1,\ldots ,k_d}: L^2(\mathcal{S}^d_+)\to L^2(\mathcal{C}^n,\mathcal{B}^n,\mu^u), I^u_{k_1,\ldots ,k_d}a=\int^\infty_0 \ldots \int^{t_2}_0 a(t)dw_{k_1}(t_1)\ldots dw_{k_d}(t_d).\end{gathered}
\left (\begin{array}{cc}A B C D \end{array}\right)^{-1}=\left (\begin{array}{cc}(A - BD^{-1}C)^{-1}-A^{-1}B (D - CA^{-1}B)^{-1}-D^{-1}C (A - BD^{-1}C)^{-1}(D - CA^{-1}B)^{-1}\end{array}\right).
y_{\mathbf{r}}^{(m)}=\mu(r)\phi_\omega(r)\sum_{\mathbf{r}|\mathbf{d},d_m=1}\frac{\lambda'_{\mathbf{d}}}{\phi_L(d)},\lambda_{\mathbf{d}}'=\mu(d)\phi_L(d)\sum_{\mathbf{d}|\mathbf{r}}\frac{y^{(m)}_{\mathbf{r}}}{\phi_\omega(r)}.
\displaystyle{\left(\delta^{IJ}-\frac{(x_{a}^{I}-x_{b}^{I})(x_{a}^{J}-x_{b}^{J})}{(x_{a}-x_{b})^{2}}\right)}(\textstyle{\frac{d\,}{dt}x_{a}^{J}-\frac{d\,}{dt}x_{b}^{J}})=0\,,
\lim_{j\to\infty}F(t'_j,x'_j,u^{\varepsilon,\delta_j}(t_j,x_j),p_j,X_j) =F(\bar{t},\bar{x}',u^\varepsilon(\bar{t},\bar{x}),\nabla_x\varphi(\bar{t},\bar{x},\bar{y}),X) &\ge F_\varepsilon(\bar{t},\bar{x},u^\varepsilon(\bar{t},\bar{x}),\nabla_x\varphi(\bar{t},\bar{x},\bar{y}),X).
\frac{1}{p}-\frac{1}{q}=\frac{1}{p}-\frac{\lambda}{p_3}=\frac{1}{p}+\frac{(2-\varepsilon)\lambda}{n}-\frac{\lambda\sigma}{p_2}=\frac{1}{p}+\frac{(2-\varepsilon)\lambda}{n}+\frac{(2-\varepsilon)\lambda\sigma}{n}-\frac{\lambda\sigma}{p}=-\frac{\lambda\sigma-1}{p}+\frac{(2-\varepsilon)\lambda\sigma+(2-\varepsilon)\lambda}{n}.
\begin{array}{l}{{{\frac{\delta}{\delta C_{\mu}( \sigma )}}W [ C ] = p^{\mu}( \sigma ) = ( p^{0}, p^{1}, p^{2}, p^{3}) = c ( \sigma ) \lambda^{\mu}( x ( \sigma ) ) =}}\\{{= c ( \sigma ) \left({\frac{1 + x_{-}^{2}+ x_{1}^{2}+ x_{2}^{2}}{2}}, \; \; - x_{2}+ x_{-}x_{1}, \; \; x_{1}+ x_{-}x_{2}, \; \; -{\frac{1 + x_{-}^{2}- x_{1}^{2}- x_{2}^{2}}{2}}\right)}}\\ \end{array}
\sum_jg_j\frac{\nabla (h_j+h_j')\otimes\nabla(h_j+ h_j')-\nabla h_j\otimes\nabla h_j-\nabla h_j'\otimes\nabla h_j'}2=\sum_jg_j\frac{\nabla h_j\otimes\nabla h_j'+\nabla h_j'\otimes\nabla h_j}2,
{\cal L}=\bar{\psi}(i\!\not{\!\partial}+J)\psi +\frac{1}{\sqrt{N}}\bar{\psi}\not{\!\!A}\psi -\frac{1}{\sqrt{N}}\sigma\bar{\psi}\psi -\frac{1}{2G_S}\sigma^2 +\frac{1}{2G_V}A_\mu^2 -\frac{1}{2\xi}(\partial^\mu A_\mu)^2\ ,
\begin{cases}|\alpha_+ (t)| \lesssim e^{-c_1^- t}, c_0 < c_1 \le e_0 e_0 < c_0 < c_1, |\alpha_+(t) - A e^{-e_0 t}| \lesssim e^{-c_1 t}, c_0 \le e_0 < c_1.\end{cases}
\leq \; \omega^{\sum_{a = 1}^{N}2 n_{a}}\; \left\langle \prod_{b = 1}^{N}\left[ \chi_{+}^{( b )}[ Q^{M , S}] \right]^{q_{b}}\left[ \chi_{-}^{( b )}[ Q^{M , S}] \right]^{2 n_{b}- q_{b}}\right\rangle_{Q^{M , S}}\; .
\nu_{q ,{\bf n}}= ( q + n + 1 ) \left( \begin{matrix}{q + n - 1}\\{n - 1}\\ \end{matrix}\right) ={\frac{( q + n + 1 ) ( q + 1 ) \cdots \bigl ( q + ( n - 1 ) \bigr )}{( n - 1 ) !}}.
|Imo(1)|= \left|-Im\left[\omega \left( n\left (r_n,\frac{1}{f}\right)+\frac{k-1}{2k}n\left(r_n,\frac{1}{A}\right)\right)\right]\right| &\geq\frac{k-1}{2k}|Im(\omega)| & \geq \frac{k-1}{2k}\Delta,
\sum_{{\bf t}^1_\nu}\lambda_{\nu,\iota^1_\nu,{\bf t}^1_\nu,1}\prod_{i=1}^{k-1}(\lambda(h_i)+(\delta-\beta_{\bar01})(h_i))^{t_{i,\nu}^1}d^{t_{k,\nu}^1}=0.
U=\begin{cases}0 x+y\le 2r x+y-2r x+y\ge 2r,x\le 2r,y\le 2r x 0\le x\le 2r,y\ge 2r y 0\le y\le 2r,x\ge 2r 2r x\ge 2r,y\ge 2r. \end{cases}
L_s^{(K)}=\binom{K-2}{s-1}+\sum_{i=0}^{K-1-s}\binom{K-1}{s+i}(N-1)^iN, D_s^{(K)}=\sum_{i=0}^{K-1-s}\binom{K}{s+1+i}(N-1)^i N.
\textbf{K}_{C}(\upsilon,b,c,\lambda,y)=\int_{0}^{\infty}x^{\upsilon-1}e^{-b\lambda\sqrt{x}}{}_{r}F_{s}\left(\ \begin{array}{lll}\alpha_{1},...,\alpha_{r}; \beta_{1},...,\beta_{s};~\end{array}e^{-c\sqrt{x}}\right)\cos (xy)dx,
u_{{\rm HiPOD}}^{L, M_L}(\alpha^*)=\displaystyle \sum_{k=1}^{m}\Big[ \sum_{j=1}^{N_h}{u}_{{\rm POD}, k, j}^{\alpha^*}\, \vartheta_j(x) \Big] \varphi_k(\psi_x({\bf y})),
{S}_\infty(\tau;\beta) := \lim_{N, k \to \infty \atop \tau \:{\rm fixed}}{S}_N(k;\beta) \Big |_{\tau = 2 \pi k / N}= \int_{-\infty}^\infty \Big ( \rho_{(2), \infty}^T(x, 0) + \delta(x) \Big ) e^{i\tau x}\, dx.
\mathsf{C}\left( s\right) :=\operatorname*{diag}\left[ \mathsf{C}_{j}\left( s\right) :1\leq j\leq n_{\Omega}\right] \quad\mathsf{C}_{j}\left( s\right) :=\left[\begin{array}[c]{ll}-\mathsf{K}_{j}\left( s\right) \mathsf{V}_{j}\left( s\right) \mathsf{W}_{j}\left( s\right) \mathsf{K}_{j}^{\prime}\left( s\right)\end{array}\right] .
\kappa = \frac{\phi' e^{\phi}(e^{2\phi}+ 2x_y^2) - e^{\phi}x_{yy}}{(e^{2\phi}+ x_y^2)^{\frac{3}{2}}}, \x(y,t), &\kappa = \frac{\phi' e^{\phi}y_x (y_x^2 e^{2\phi}+ 2) + e^{\phi}y_{xx}}{(1 + e^{2\phi}y_x^2)^{\frac{3}{2}}}, \y(x,t).
\int_0^1\bigg(\int_0^xu\bigg)^{r/p}\bigg(\int_0^{1/x}w^{1-p'}\bigg)^{r/p'}u(x)\,dx=\frac qr u(0)^{r/p}\bigg(\int_0^1 w^{1-p'}\bigg)^{r/p'}u(0).
\| f \|_{\dot B^s_{p,q}(\mathcal{H})}:=\begin{cases}\displaystyle \left\{\sum_{j =-\infty}^\infty \left(2^{sj}\|\phi_j(\sqrt{\mathcal{H}}) f \|_{L^p(\Omega)}\right)^q \right\}^\frac{1}{q}, \displaystyle \sup_{j\in\mathbb Z}2^{sj}\|\phi_j(\sqrt{\mathcal{H}}) f \|_{L^p(\Omega)},\end{cases}
\left|\frac{zf_1'(z)}{f_1(z)}\right| =\left|1+\frac{\rho_1}{(1-\rho_1^2)}\left( \frac{u\rho_1^2+4\rho_1+u}{\rho_1^2+u\rho_1+1}+\frac{v\rho_1^2+4\rho_1+v}{\rho_1^2+v\rho_1+1}+\frac{q \rho_1^2+4\rho_1+q}{\rho_1^2+q \rho_1+1}\right)\right| =1+\sin{1}=q_0(1),
Q_i' = \frac12\sum_{j=1}^n \gamma_{ij}\sum_{q=0}^{p-1}(Y_i^{-1}BY_j)(Y_i^{-1}A^{-1}BY_j)^q (Q_j-Q_i)(BY_jY_i^{-1}A^{-1})^{p-q-1}(Y_i^{-1}A^{-1}Y_i) &+\frac12\sum_{j=1}^n \gamma_{ij}\sum_{q=0}^{p-1}(Y_i^{-1}AY_i)(Y_j^{-1}B^{-1}AY_i)^q (Q_j-Q_i)(AY_iY_j^{-1}B^{-1})^{p-q-1}(Y_j^{-1}B^{-1}Y_i).
N_{t}^{\max}= \max_{f\in\mathcal{F}_0}\Big\{\#\{\tilde{f}\in\mathcal{F}_0:\, d(f,\tilde{f})\leq t\}\Big\}, N_{t}^{\min}= \min_{f\in\mathcal{F}_0}\Big\{\#\{\tilde{f}\in\mathcal{F}_0:\, d(f,\tilde{f})\leq t\}\Big\}.
-\frac{1}{4}\exp\!\Big(\sqrt{\frac{2\kappa}{3}}\phi\Big)[V_{\mu\nu}{}^j V^{\mu\nu}{}_j +\overline{\Theta}_{\mu\nu}\overline{\Theta}^{\mu\nu}+2\sqrt{2\kappa}(V_{\mu\nu}{}^j A_j +\overline{\Theta}_{\mu\nu}\Theta) f^{\mu\nu}+2\kappa (A_j A^j +\Theta^2)f_{\mu\nu}f^{\mu\nu}]
n(4(k+1)s)^{2s+3}\left(1+4p \right)^{16s^2}2^{6s}K^{22s}\theta_1^{4sk}\cdot \left( \frac{K}{d}\right)^{2sk}\sum_{g=0}^{\infty}\left( \frac{2K^{12}\left(1+4p\right)^{8s}(4(k+1)s)^{6s}}{n}\right)^g.
\dot{F}^{i}(w) =\delta_{ij}+(w-w_{j})\dot{F}_{w}^{i}(w)+o\left(\left|\frac{w-w_{j}}{\log|w-w_{j}|}\right|\right) \intertext{as $w\rightarrow w_{j}$, $ j\neq n$, and}\dot{F}^{i}(w) =w \dot{F}^{i}_{w}(w) + o\left(\frac{|w|}{\log |w|}\right) \ \w\rightarrow\infty.
\rho{e^{\sigma^2\zeta^v(p)}}(1+p\zeta^v(p)) =1, \rho{e^{\sigma^2\zeta^e(p,\hat{p}_{j1}^v)}}(1+p\zeta^e(p,\hat{p}_{j1}^v))(1+\hat{p}_{j1}^v\zeta^e(p,\hat{p}_{j1}^v)) =1.
\sum_k \frac{1}{m(I_k)}\int_{I_k^{(1)}}\int_{[s_k^{(3)},u]}|f'(v)|^p w_r^p(v)\,|dv|\,dm(u)=\frac{1}{4}\sum_k\int_{[s_k^{(3)},u_k]}|f'(v)|^p w_r^p(v)\,|dv|.
\overline{{{Q}}}_{\dot{\alpha}}e^{-{\frac{P_{0}}{4 \kappa}}}\otimes Q_{\beta}e^{\frac{P_{0}}{4 \kappa}}+ Q_{\beta}e^{-{\frac{P_{0}}{4 \kappa}}}\otimes \overline{{{Q}}}_{\dot{\alpha}}e^{\frac{P_{0}}{4 \kappa}}
\delta \theta^{\mu \nu}= - \theta^{\mu \alpha}\epsilon_{\alpha \beta}\theta^{\beta \nu}- \frac{1}{4}\theta^{\mu \alpha}\epsilon_{\alpha \gamma}\theta^{\gamma \rho}\epsilon_{\rho \beta}\theta^{\beta \nu}= - \theta^{\mu \alpha}\delta \theta_{\alpha \beta}^{- 1}\theta^{\beta \nu}+ O ( \epsilon^{2}) .
\begin{aligned}|\mathcal H(k,d,2)|-(k-d-\frac{1}{2})\binom{n-d}{k-d}=(d+\frac{3}{2}-k)\binom{n-d}{k-d}+d\binom{n-d-1}{k-d}&=(2d+\frac{3}{2}-k)\binom{n-d}{k-d}-d\binom{n-d-1}{k-d-1}.\end{aligned}
\underline{\underline{C}}=[\begin{matrix}C_{11}&C_{11}-2C_{66}&C_{13}&0&0&0\\C_{11}-2C_{66}&C_{11}&C_{13}&0&0&0\\C_{13}&C_{13}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{44}&0\\ 0&0&0&0&0&C_{66}\end{matrix}]
\widehat{R}_{i \: \nu \rho \sigma}^{\mu}= \kappa_{i \: \rho}^{\mu}\kappa_{\nu \sigma}^{i}- \kappa_{i \: \sigma}^{\mu}\kappa_{\nu \rho}^{i}+ h_{i \: \nu}^{\alpha}h_{i \: \rho}^{\beta}h_{i \: \sigma}^{\gamma}R_{\: \: \alpha \beta \gamma}^{\mu}
\langle Ay,y|y|^{p-2}\rangle =-\langle \nabla y,\nabla (y|y|^{p-2})\rangle &=-(p-1)\langle \nabla y,\nabla y |y|^{p-2}\rangle &=-(p-1)||y|^{\frac{p-2}{2}}\nabla y|_{L^2}^2 &=-\frac{4(p-1)}{p^2}|\nabla (|y|^{\frac{p}{2}})|_{L^2}^2 &\le -\lambda_1\frac{4(p-1)}{p^2}||y|^{\frac{p}{2}}|_{L^2}^2 &=-\lambda_1\frac{4(p-1)}{p^2}|y|_{L^p}^p
\begin{pmatrix}\partial_{x_1}\zeta^1 \dots \partial_{x_N}\zeta^1 \partial_u\zeta^1 \vdots \vdots \vdots \partial_{x_1}\zeta^{N-s}\dots \partial_{x_N}\zeta^{N-s}\partial_u\zeta^{N-s}\partial_{x_1}\upsilon \dots \partial_{x_N}\upsilon \partial_u\upsilon\end{pmatrix}
D_{f}(n):=\sum_{j=1}^{n}ds_{2j,f}(n)=\sum_{m=1}^{\lfloor\frac{n}{2}\rfloor}\sum_{i=0}^{2m-1}(\begin{matrix}2m-1\\i\end{matrix})(-1)^{i+1}\tilde{ds}_{i+1,f}(n)
\begin{pmatrix}\mbox{Aux}\Lambda \mbox{O}_{N^{p-1}}\dots \mbox{O}_{N^{p-1}}\mbox{O}_{N^{p-1}}\mbox{Aux}\Lambda \dots \mbox{O}_{N^{p-1}}\vdots \vdots \ddots \vdots \mbox{O}_{N^{p-1}}\mbox{O}_{N^{p-1}}\dots \mbox{Aux}\Lambda \end{pmatrix}\tilde{Z}\left[J \right].
f(x_{\lfloor|\Gamma|/2\rfloor+1})\ll C_p(f)^{\frac{1}{p-1}}\left(\frac{1}{\deg(u)^{\frac{1}{p-1}}}+\sum_{n=1}^{\lfloor\log_2(|\Gamma|/\deg(u))\rfloor}\max_{\substack{A\subset X u\in A A |\Gamma|/2^{n+1}<|A|\le|\Gamma|/2^n}}j_{A,p}\right).
{\rm ord}_{1/2}\left(F_{k,2}\right) =\begin{cases}\frac{1}{2}\mbox{if $k$ is odd,}1 \mbox{if $k$ is even,}\end{cases}{\rm ord}_{1/2}\left(F_{k,4}\right) =\begin{cases}2 \mbox{if $k$ is odd,}1 \mbox{if $k$ is even.}\end{cases}
{\cal F}_{\mu \nu}= \partial_{\mu}( \hat{\phi}^{a}W_{\nu}^{a}) - \partial_{\nu}( \hat{\phi}^{a}W_{\mu}^{a}) -{\frac{1}{g}}\epsilon^{a b c}\hat{\phi}^{a}( \partial_{\mu}\hat{\phi}^{b}) ( \partial_{\nu}\hat{\phi}^{c})
\sum_{\substack{z_1<q\le p\le z_2 q\le (X/p)^{1/2}z_3\le p q<z_5}}S_{p q}(q)=\sum_{\substack{z_1<q\le p\le z_2 z_3\le p q<z_5 q^2 p < z_6}}S_{p q}(q)+\sum_{\substack{z_1<q\le p\le z_2 z_3\le p q<z_5 z_6\le q^2 p\le X}}S_{p q}(q).
\begin{array}{c}{\nabla_{\rho}\nabla^{\rho}\Lambda^{\mu_{2}\ldots \mu_{s}}+ m^{2}\Lambda^{\mu_{2}\ldots \mu_{s}}= 0 ,}{\mathrm{w i t h}\; \; m^{2}= ( s - 1 ) ( s + d - 3 )}\end{array}