harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\dfrac{1+\psi(1-\lambda)-\psi(\lambda)}{2}q(t) =\dfrac{1}{\int_a^b \nu(t)\Delta t}\int_a^b K(s,t)q^{\Delta}(s)\Delta s + \dfrac{1}{\int_a^b \nu(t)\Delta t}\int_a^b \nu(s)q(\sigma(s))\Delta s & - \dfrac{\psi(\lambda)q(a)+\left(1-\psi(1-\lambda)\right)q(b)}{2}.
P(F_r)(\eta) = = \int_{|1-r| \le \delta_\eta}\frac{F_r - F_1}{1-r}(\eta) \,dr +\int_{\delta_\eta <|1-r|<\delta}\frac{F_r(\eta)}{1-r}dr + \int_{\delta \le |1-r|}\frac{F_r(\eta)}{1-r}\,dr := \, P_{A}(\eta) + P_{B}(\eta) + P_{C}(\eta),
\begin{aligned}\|x_i^{k+1}-{x'_i}^{k+1}\|_1 \leq (1 -\gamma^k_1|R_{ii}|)\|x_i^k-{x'_i}^k\|_1 + \lambda^k\|y_i^k-{y'_i}^k\|_1. \end{aligned}
W_m^*(r)-\frac{1-r}{1-\lambda}W_m^*(\lambda) =\int J(x)dF(x)\left\{\left(Z(r)-rZ(1)\right)-\frac{1-r}{1-\lambda}\left(Z(\lambda)-\lambda Z(1)\right)\right\}&=\int J(x)dF(x)\left\{Z(r)-Z(\lambda)-\frac{r-\lambda}{1-\lambda}\left(Z(1)-Z(\lambda)\right)\right\}.
E_{\mathrm{b o u n d}}= \frac{1}{2 e^{2}}\left( \frac{m_{W}m_{H}}{m_{Z}}\right)^{2}\left[ 2 g^{\prime}\Phi_{B}- \left( \frac{m_{W}m_{H}}{m_{Z}}\right)^{2}A \right] ~ ~ .
\eta\Pi=\begin{pmatrix}P P P \cdots P 1 P L'_{1,1}L'_{1,2}\cdots L'_{1,n-a}1 \vdots \vdots \vdots \ddots \vdots \vdots P L'_{a,1}L'_{a,2}\cdots L'_{a,n-a}1 1 1 1 \cdots 1 1\end{pmatrix}.
\frac{1}{2}\left(-R_2+R_3+2R_4+R_6\right)= (0, 1, 1, 1, 1, 1, 1)\in R(A(D_6)) \frac{1}{2}\left(R_1-R_2+R_5+2R_6\right)= (1, 1, 1, 1, 1, 1)\in R(A(D_{14})) \frac{1}{2}\left(R_4+R_5+R_6\right)= (1, 1, 1, 1, 1, 1) \in R(A(D_{15})) \frac{1}{2}\left(-R_1+R_2+2R_3+R_5\right)= (1, 1, 1, 1, 1, 1, 1)\in R(A(D_{17})).
Z^{\ast}(\theta_{2})Z^{\ast}(\theta_{1})\Omega =\frac{1}{\sqrt{2}}(\chi_{21}a^{\ast}(\theta_{2})a^{\ast}(\theta_{1})\Omega +\chi_{12}S(\theta_{2}-\theta_{1})a^{\ast}(\theta_{1})a^{\ast}(\theta_{2})\Omega )
\sigma_{k}\left( D^{2}u\right) =\binom{n-2}{k-2}e^{kt}2^{k-1}\left(e^{-t}h^{\prime\prime}-r^{2}\right) & +\binom{n-2}{k-1}e^{kt}2^{k-1}\left( r^{2}+e^{-t}h^{\prime\prime}\right) & +\binom{n-1}{k}e^{kt}2^{k}.
\pi_{i}^{( 1 ) a}= - \frac{1}{m^{2}}\eta^{1 b}G_{1 i}^{( 0 ) b a}+ \frac{g}{2 m^{2}}f^{b c d}\eta^{2 b}( D_{i}\pi^{i})^{c}G_{1 i}^{( 0 ) d a}- \eta^{2 b}G_{2 i}^{( 0 ) b a},
\left\| R_{21}\right\|_2 \leq C\left\| Au_0\right\| + C\int_0^t \left\|E(s) \big( v_1'(t-s) + v_2'(t-s) \big)\right\| ds \leq C\left( \left\| u_0\right\|_2 + \sum\limits_{j=1}^{m}t^{\alpha_j}+ \int_0^t \left\| v_2'(s) \right\|ds \right),
Z_{r_1}(d_2(f), s+1) =\int_{F^\times}\int_{F}f(0,0,0,y,0,0)\psi_v(ay)d(y,\psi_v)|a|^{s+1}d^\times (a,\psi_v) =q_v^{e(\psi_v)(s+\frac{1}{2})}\int_{F^\times}\int_{F}f(0,0,0,y,0,0)\psi'(ay)d(y,\psi_v)|a|^{s+1}d^\times (a,\psi').
\frac{(\frac{tn+1}{2})!(n+1)!(n+2)!}{(tn+2)!(2n+3)!(n+\frac{3}{2}-\frac{tn}{2})!}&=e^n(4n)^{-n}\left(1-\frac{t}{2}\right)^{-n(1-\frac{t}{2})-2}\frac{(2t)^{-\frac{nt}{2}}}{16\sqrt{2}t^{3/2}n^3}\left(1-\frac{104-12t+27t^2}{24(2-t)nt}+O\left(\frac{1}{n^2}\right)\right).
e_s^\perp k_m^{(r)}= \sum_{i=0}^m \binom{r+i-1}{i}(h_{m-i}e_s^\perp + h_{m-i-1}e_{s-1}^\perp) = k_m^{(r)}e_s^\perp + k_{m-1}^{(r)}e_{s-1}^\perp\,.
\int_{\mathbb{T}}H_{n}(\rho_{n}(t)) - H_{n}(\rho_{n}^{0})~dx + \| \sqrt{\rho_{n}}\partial_{x}p_{n}(\rho_{n})\|^{2}_{L^{2}_{t}L^{2}_{x}}= \int_{0}^{t}\int_{\mathbb{T}}(\partial_{x}p_{n}(\rho_{n})) \rho_{n}w_{n}~dxds.
\mathcal S_A^{\ne 0}(D;\ell) =\frac{D}{2\pi i}\int_{(\frac 12)}\sum_{(n,2)=1}n^{-w}\frac{\tau_A(n)}{\sqrt{n}}\frac{1}{2n\ell}\sum_{(c,2n\ell)=1\atop c\le Y}\frac{\mu(c)}{c^2}\sum_{k\ne 0}(-1)^k G_k(n\ell) \int_0^\infty W\left(\frac t N\right)\tilde \Psi\left(\frac{kD}{2c^2t\ell}\right)t^{w-1}~dt ~dw
\int_X\mathrm{Tr\,}e^{-t\widetilde{\Box}^{\pm}_{b,m}}(x,x)dv_X(x)=\mathrm{dim\,}\widetilde{\mathcal{E}}^{\pm}_{m,0}(X,E)+\sum_{\nu\in\mathrm{Spec\,}\widetilde{\Box}^{\pm}_{b,m},\nu>0}e^{-\nu t}\mathrm{dim\,}\widetilde{\mathcal{E}}^{\pm}_{m,\nu}(X,E).
\lim_{\delta \to 0}~~\Big|~~y_0+\int_{0}^{\cdot}b_{j^{\delta}(s)}(s,y^{\delta}(s))ds+\int_{0}^{\cdot}\sigma_{j^{\delta}(s)}(s,y^{\delta}(s))dB^{\delta}(s) ~~ -~~\Big(y_0+\int_{0}^{\cdot}b_{j(s)}(s,y(s))ds+\int_{0}^{\cdot}\sigma_{j(s)}(s,y(s))dB(s)~~\Big)~~\Big|_{(0,T)}~~=~~0.
\nu\left( B\right)=\int_0^{\infty}\int_{\left\vert w\right\vert=1}1_B\left( rw\right)a\left( r,w\right)j\left(r\right)r^{d-1}\Sigma\left( dw\right)dr, \forall B\in\mathcal{B}\left( \mathbf{R}^d_0\right),
F_{\lambda}\hat{O}_{\lambda}= \int [ 1 . . n \, n + 1 \, . . m ] \tilde{\delta}\, f_{\lambda}\, v ( 1 , . . , m ) \, a_{\lambda 1}^{\dagger}\cdots a_{\lambda n}^{\dagger}a_{\lambda \, n + 1}\cdots a_{\lambda m}\; ,
E(y)=\Bigl(\sum_{i=1}^k\bigl[w_{\mu_i^{\prime},\zeta_i^{\prime}}(y)+\varepsilon^{\frac{1}{2}}\pi_i(\varepsilon\, y)\bigr]\Bigr)^5- \sum_{i=1}^kw_{\mu_i^{\prime},\zeta_i^{\prime}}^5(y),y\in \Omega_{\varepsilon}.
\intertext{Hence we obtain following equation}V_k^r(x) =\sum_{i=0}^{k-1}\binom ki 2^{k-i}\left(\sum_{n \le x}\sum_{j=0}^{k-i-1}(-1)^{k-i-1-j}\binom{k-i}jJ_j(n)\right). \intertext{Applying the binomial theorem and changing the order of summation of it, we show}V_k^r(x) =\frac1{2(k+1)}\{(2X+1)^{k+1}-(2X-1)^{k+1}\}.\intertext{This proves the theorem.}
\times D^{-2}({\bf y}-{\bf z})\cos\left(\phi_{\bf x}-\phi_{\bf y}+\phi_{\bf z}\right)+D^2({\bf x}-{\bf y})D^{-2}({\bf x}-{\bf z})D^{-2}({\bf y}-{\bf z})\cos\left(\phi_{\bf x}+\phi_{\bf y}-\phi_{\bf z}\right)\Bigr].
\mathbb{E}(X_iX_{i+1}X_jX_{j+1})= f_k(i)f_k(i+1)f_k(j)f_k(j+1)+O_{\alpha}\Big(\frac{\tau_3(i)\tau_3(i+1)\tau_3(j)\tau_3(j+1)}{\sqrt{i}}\Big) &+O_{\alpha}\Big(\frac{\tau_3(j)\tau_3(j+1)}{\sqrt{j-i-1}}\Big),
Q\log |f_t'(z)| = (h_0' - h_t)(z) + b_t &= (h' - h\circ f_t)(z) + \frac{2}{\sqrt \kappa}\log|z| - \frac{2}{\sqrt \kappa}\log |f_t(z)| + b_t &= (h' - h\circ f_t)(z) + \frac{2}{\sqrt \kappa}\log \frac{|z|}{|f_t(z)|}+ b_t.
(a,b,c) = \pm\left( x_{(2m-1)(2n-1)}^{(d)},x_{(2m-1)(2n+1)}^{(d)},\frac{y_{(2m-1)\cdot 2n}^{(d)}}{y_{2m-1}^{(d)}}\right), &\pm (x_{2n-1}^{(2)},-x_{2n+1}^{(2)},x_{2n}^{(2)})
\varDelta_0 = \underset{u \in\mathcal{D}^*}{\max}L_0^*(u), L_0^*(u) = \sum_{l=1}^{k}\bigl(u_l\zeta_l +u'_l\zeta'_l \bigr), u = \bigl(u_1,\ldots,u_k,u'_1,\ldots,u'_k\bigr), \mathcal{D}^* = \Biggl\{u\geq0: \sum_{l=1}^{k}\bigl(u_l - u'_l\bigr)v_{li}= 0, \, \sum_{l=1}^{k}\bigl(u_l+ u'_l\bigr) = 1 ,\i = \overline{1,q}\Biggr\},
h^{2}= \frac{p^{2}}{( n + 1 )^{2}{\cal G}}, \; \; \kappa^{2}= \frac{1}{{\cal G}}\left( \frac{- ( n + 2 )}{( n + 1 )^{2}}p^{2}+ p^{i j A}p_{i j A}\right)
I_{1}= ( \Pi_{2 3})^{2}\int d^{4}x_{4 , 5}d^{4}u_{4 , 5}d^{4}\theta_{4 , 5}^{+}\; \Pi_{1 4}\Pi_{4 2}\Pi_{3 5}\Pi_{5 1}\; ( D_{4}^{+})^{4}\left({\frac{\delta^{8}( \theta_{4}- \theta_{5})}{x_{4 5}^{2}}}\right) \delta^{( - 2 , 2 )}( u_{4}, u_{5}) \ .
\hat{a}( \varphi_j) = \varphi_j (a) = \sum_{x \in G}\varphi_j ( f (x)) &= \int_G \varphi_j \circ f(x) dx &= \int_G \overline{I(x)}\varphi_j \circ f (x) dx &= I \otimes \varphi_j(f)
A_{\mu}= \mp{\frac{2 i}{g}}( H_{1}{\frac{v_{\mu}^{1}}{\rho_{1}}}+ g \kappa E_{1 3}^{\pm}R_{\mu}^{1}) \mp{\frac{2 i}{g}}( H_{2}{\frac{v_{\mu}^{2}}{\rho_{2}}}+ g \kappa E_{2 3}^{\pm}R_{\mu}^{2}) .
-\Phi_r(\{X_I \}) = \frac{\sum_{I \in \mathcal{P}[r]}(-1)^{|I|+1}X_I \prod_{J \in \mathcal{P}[r] \atop J \neq I}(1 - X_J)}{\prod_{J \in \mathcal{P}[r]}(1 - X_J)}.
\frac{d}{dt}\log F_{6}\big(3^{-2/3}t\big)=\frac{1}{12}t^2-\frac{\sqrt{2}}{3}(-t)^{1/2}+\frac{1}{24\,t}+O\big(|t|^{-\frac{5}{2}}\big)\mbox{as \ \ $t\to-\infty$},
{\cal M}(a,b) =\int_{{\mathrm i}\,\eta}^{\infty +{\mathrm i}\,\eta}{\mathrm d}t\, \exp(-t) \,\left( \frac{1}{1 -{\mathrm i}\, a \,t}+ \frac{1}{1 +{\mathrm i}\, a \,t}+\frac{1}{1 - b \,t}+ \frac{1}{1 + b \,t}\right),
Z_{\Delta}( U ) = \sum_{n}e^{- \frac{n \lambda A}{2}}\sum_{i}\frac{( \lambda A )^{i}}{i !}\sum_{\sigma}\left[ \frac{( - 1 )^{i}}{C_{\sigma}}\phi_{n , i}^{\sigma}\Upsilon_{\sigma}( U ) N^{n - i}+{\cal O}( N^{n - i - 1}) \right] ,
\frac{-c_{b}r}{2\pi}\int_{0}^{\pi}\sin^{2}(\theta) d\theta \left((b-1) \int_{0}^{\frac{t_{+}}{2}}\frac{du \sin(u)}{u \log^{b}(\frac{t_{+}}{u}) t_{+}^{2}}\right)= \frac{-b r}{2 t^{2}\log^{b}(t)}+ E_{v_{2},2}(t,r), r \leq \frac{t}{2}
\int_{[N]}\sum_{(\xi_1',\xi_2')/\sim}r_i(n g)\mathcal{F}_2(f)\left(\xi,\xi'_1,\xi'_2\right) dn &=\int_{[N]}r_i(n g)\mathcal{F}_2(f)\left(\xi,0,1\right) dn+\int_{[N]}\sum_{\alpha \in F}r_i(ng)\mathcal{F}_2(f)\left( \xi,1,\alpha\right) dn.
\sum_{j , j^{\prime}; \mu , \mu^{\prime}}z_{j}^{( \mu )}\; \{J_{( \mu )}^{j , - j}, J_{( \mu^{\prime})}^{j^{\prime}, - j^{\prime}}\}^{*}( y ) \; t_{j^{\prime}, - j^{\prime}}^{( \mu^{\prime})}= [ R ( \vec{z}; y ) , y ]
t\cdot \sum_{i=1}^m T(n_i) = \sum_{i=1}^{t\cdot m}T(n_i) < \frac{97 (t\cdot m) +73}{54}\cdot \frac{1}{X_0}, \sum_{i=1}^m T(n_i) < \frac{97}{54}\cdot m \cdot \frac{1}{X_0}+ \frac{1}{t}\cdot \frac{73}{54}\cdot \frac{1}{X_0}.
\left( a(1 \otimes s) \right) \otimes w = (a(1 \otimes sp')) \otimes w = (a(1 \otimes s)(1 \otimes p')) \otimes w & = (a (1 \otimes s)(1 \otimes p')) \otimes w - (a(1 \otimes s)) \otimes \underbrace{(1 \otimes p')w}_{0}\in Z.
\left\{j_\alpha^0,j_\beta^0\right\}\approx \left[ 2\Gamma_{\alpha \beta}^m\cdot j_m^0\;+\left( emb^{*}S_{\alpha \beta}\left( \Delta\right) \right)_r\cdot \Lambda^r\right] \cdot \delta \left( \sigma -\sigma^{\prime}\right) \quad ,
p(\nu,\psi)=\frac{1}{Z}\sum_{h}exp(\sum_{ij}W_{ij}^{(1)}\nu_{i}h_{j}^{1}+\sum_{jl}W_{jl}^{(2)}h_{j}^{1}h_{l}^{2}+\sum_{lm}W_{lm}^{(3)}h_{l}^{2}h_{m}^{3}),
S_{k}(a,b,-\gamma) := \int_{[0,1]^{k}}\prod_{j=1}^{m}x_{j}^{a}(1-x_{j})^{b}\prod_{1 \leq i < j \leq k}|x_{i}-x_{j}|^{-2\gamma}\,dx_{1}\ldots dx_{k}&= \prod_{j=1}^{k}\frac{\Gamma(a+1-(j-1)\gamma)\Gamma(b+1-(j-1)\gamma)\Gamma(1-j\gamma)}{\Gamma(a+b+2-(k+j-2)\gamma)\Gamma(1-\gamma)}&=\frac{1}{\Gamma^{k}(1-\gamma)}\tilde{S}_{k}(a,b,-\gamma)
V(r,\theta,\varphi)=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}C_{l}^{m}(r)Y_{l}^{m}(\theta,\varphi)=\sum_{j=1}^{\infty}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\frac{D_{l,j}^{m}}{r^{j}}Y_{l}^{m}(\theta,\varphi).
\big|(\pi(g)w_{k_1}\xi_\varphi|v_g w_{k_2}\xi_\varphi)_{L^2(M)}\big| &= \big|(\pi_i(g)a_{k_1}\xi_{\varphi_i}|v_g a_{k_2}\xi_{\varphi_i})_{L^2(M_i)}\big|\cdot\big|(\pi(g)w'_{k_1}\xi_\varphi|w'_{k_2}\xi_\varphi)_{L^2(M)}\big| &\leq \big(\max_{1 \leq k \leq K}\Vert a_k \Vert\big)^2\,\big|(\pi(g)w'_{k_1}\xi_\varphi|w'_{k_2}\xi_\varphi)_{L^2(M)}\big|
U (b) =\begin{pmatrix}1 0 0 0 0 1 b -b^2/2 0 0 1 -b 0 0 0 1\end{pmatrix}, U (c) =\begin{pmatrix}1 0 0 0 0 1 c -c^2/2 0 0 1 -c 0 0 0 1\end{pmatrix},
\begin{aligned}W_{1,a}-\sum_{b\in A}{{H}^{\sharp}_{ab}W_{1,b}{{H}^{\sharp}}_{ab}^{\ast}}= &\sum_{b\in A}{\left[ E_{ab}P_b Q_b^{\ast}{{H}^{\sharp}}_{ab}^{\ast}+{{H}^{\sharp}}_{ab}Q_b P_b{E}_{ab}^{\ast}+ E_{ab}P_b{E}_{ab}^{\ast}\right]}:=T_a.\end{aligned}
\begin{array}{r c l}{{\tilde{e}_{n_{+}, n_{-}}^{\alpha}( z )}}&{{=}}&{{( P_{+})^{n_{+}}( P_{-})^{n_{-}}V_{\alpha}( z )}}\\{{}}&{{=}}&{{\displaystyle \int_{\infty}^{z}d t_{1}J_{+}( t_{1}) \cdots \int_{\infty}^{z}d t_{n_{+}}J_{+}( t_{n_{+}}) \int_{\infty}^{z}d t_{1}^{\prime}J_{-}( t_{1}^{\prime}) \cdots \int_{\infty}^{z}d t_{n_{-}}^{\prime}J_{-}( t_{n_{-}}^{\prime}) V_{\alpha}( z ) ,}}\\ \end{array}
Z_{\mathrm{o r b .}}= \mathrm{T r}_{\mathrm{u n t w^{\prime}d}}\left({\frac{( 1 + R_{2 5})}{2}}q^{L_{0}-{\frac{1}{2 4}}}{\bar{q}}^{{\bar{L}}_{0}-{\frac{1}{2 4}}}\right) + \mathrm{T r}_{\mathrm{t w^{\prime}d}}\left({\frac{( 1 + R_{2 5})}{2}}q^{L_{0}-{\frac{1}{2 4}}}{\bar{q}}^{{\bar{L}}_{0}-{\frac{1}{2 4}}}\right)
w^l\pi_k(e_k) = (\phi_{k+1}\circ\cdots\circ\phi_{k+l})(\pi_{k+l}(e_{k+l})) =\pi_k(e_{k+l}) \quad (k+l<r) w^{r-k+l}\pi_k(e_k) = (\phi_{k-1}\circ\cdots\circ\phi_{r-1}\circ\phi_r \circ(\mathrm{id}\otimes\phi_1)\circ\cdots(\mathrm{id}\otimes\phi_l)) (z\otimes\pi_l(e_l)) = \pi_k(ze_l) \quad (l<k).
\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^T |L(\sigma+it)|^2dt = \sum_{n=1}^\infty\frac{|a_L(n)|^2}{n^{2\sigma}}<\infty,\qquad\sigma>\max\left\{\frac{1}{2},1-\frac{1}{d_L}\right\},
q^2 \nu_x \left(\frac{q\nu_x\nu_y - q^{-1}\nu_y\nu_x}{q-q^{-1}}\right) - q^{-2}\left(\frac{q\nu_x\nu_y - q^{-1}\nu_y\nu_x}{q-q^{-1}}\right)\nu_x = (q^2-q^{-2})\nu_x.
(L_{-1}^{+}V)_0=\sum_{n\ge 1}L_{-1}^{(n)}V_{-n}=\sum_{n=1}^{p^t}L_{-1}^{(n)}V_{-n}\subset L_{-1}^{(p^t)}V_{-p^t}+(L_{1}^{+}V)_0 =(L_{1}^{+}V)_0,
\frac{p_{\alpha}( p + k_{1})_{\beta}( p + k )_{\gamma}}{p^{2}( p + k_{1})^{2}( p + k )^{2}}= \frac{p_{\alpha}( p + k_{1})_{\beta}}{p^{4}( p + k_{1})^{2}}\left[ ( p + k )_{\gamma}- \frac{2 ( k p ) p_{\gamma}}{p^{2}}\right] + \ldots \; .
\begin{aligned}\Vert \Lambda^{\alpha}(uv)- u\Lambda^{\alpha}v \Vert_{\dot{B}^{s}_{2,1}}\lesssim_{\alpha,N,s,s_1,s_2}\Vert u \Vert_{\dot{B}^{\frac{N}{2}+1-s_1}_{2,1}}\Vert v \Vert_{\dot{B}^{s+s_1+\alpha-1}_{2,1}}+\Vert u \Vert_{\dot{B}^{s+s_2+\alpha}_{2,1}}\Vert v \Vert_{\dot{B}^{\frac{N}{2}-s_2}_{2,1}}.\end{aligned}
\sup_{\{p(x_t|s_{t-1},y^{t-1})\}_{t=1}^N}\sum_{i=1}^N I(X_i,S_{i-1};Y_i|Q_{i-1}) =\sup_{\{p(x_t|s_{t-1},q_{t-1})\}_{t=1}^N}\sum_{i=1}^N I(X_i,S_{i-1};Y_i|Q_{i-1}).
\ast_{4,5}=(4k+15)b_3 \mathrel{\phantom{}}\bigg( (32k^3+504k^2+2648k+4641) a_1a_2b_3 - (4k^2+52k+168)a_2^2b_2 &\mathrel{\phantom{}}\mathrel{\phantom{=}}- \, (64k^3+1008k^2+5276k+9177)b_3^2 \bigg)
\frac{1}{2\pi}\int_0^\pi R (\xi) \rho_\epsilon (\xi) \xi = \sum_{0\leq{l}\leq m}R\bigl( \xi_{{l}}^{(m+1)}\bigr) \rho_\epsilon \bigl(\xi_{l}^{(m+1)}\bigr){\Delta}^{(m+1)}_{{l}},
f_{t}(x)=\sum\limits_{k=-\infty}^{\infty}f(x+k)e^{ikt},\int_{-\infty}^{\infty}\left\vert f(x)\right\vert^{2}dx=\frac{1}{2\pi}\int\limits_{0}^{2\pi}\int\limits_{0}^{1}\left\vert f_{t}(x)\right\vert^{2}dxdt
Q(B_2,B_3) \ = \Q(B_3, B_4) \ = \Q(B_4,B_2) \ = Q(B_3,B_1) \ = \Q( B_1,B_4) \ = \ 1, Q(B_1^-,B_2^-) \ = \ 0, Q(B_0, B_1^-) \ = \ Q(B_0, B_2^-) \ = \ 2.
{\chi}_U(\mathcal{F})= \frac{[H^0_{et}(U,\mathcal{F})_{tor}][Ext^1_U(\mathcal{F},\mathbb{G}_m)]R(\mathcal{F})}{[Hom_U(\mathcal{F},\mathbb{G}_m)^D_{tor}][\ker(\Psi^1(\mathcal{F}))][\mathrm{cok}(\delta(\mathcal{F})_{tor})]\prod_{v\in S}[H^0(K_v,\mathcal{F}_v)_{tor}]}
M = \frac{1}{\sqrt{1+{\bf \bar{w}}\cdot{\bf w}-{1\over 4}({\bf w}\times{\bf{\bar w}})^2}}\left(\begin{array}{cc}1-{i\over 2}({\bf w}\times{\bf \bar{w}})\cdot{\bf \sigma}-{\bf w}\cdot{\bf \sigma}begin{align*}2ex]{\bf \bar{w}}\cdot{\bf \sigma}1+{i\over 2}({\bf w}\times{\bf\bar{w}})\cdot{\bf \sigma}\end{array}\right)
P\{T_1 < \cdots < T_g\}= & K \int_0^{\infty}\cdots \int_0^{t_3}e^{-(p_g t_g + \cdots + p_2 t_2)}\left(1 - e^{-p_g t_g}\right)^{M_g - 1}\cdots\left(1 - e^{-p_2 t_2}\right)^{M_2 - 1}\left(1 - e^{-p_1 t_2}\right)^{M_1}\, dt_2 \cdots dt_g,
m_{\mathrm{r e n}}= - \left\langle{\cal W}^{\prime \prime}\right\rangle_{0}= -{\cal W}_{0}^{\prime \prime}- \frac{1}{2}\,{\cal W}_{0}^{\prime \prime \prime \prime}\, \left\langle \chi^{2}( x ) \right\rangle_{\, 0}\, ,
\begin{aligned}15^n_{23106}\to/<->/<1em> 15^n_{56014}15^n_{23432}\to/<->/<1em> 15^n_{56014}15^n_{44028}\to/<->/<1em> \overline{15}^n_{50224}15^n_{73047}\to/<->/<1em> \overline{15}^n_{91280}\end{aligned}
\overline{v_L}=\left| \begin{array}{ccc}X_1, X_2, \widetilde{X_3}\frac{(f_1)_{u_1}}{f_1}, (f_3)_{u_1}, \sqrt{L}\left[\frac{(f_2)_{u_1}}{f_1}-(f_3)_{u_1}\right] \frac{(f_1)_{u_2}}{f_1}, (f_3)_{u_2}, \sqrt{L}\left[\frac{(f_2)_{u_2}}{f_1}-(f_3)_{u_2}\right] \end{array}\right|.
0 = y'' - \dfrac{((2n-3)t^6+t^4(-4\beta n + 2n-5)+t^2(4\beta-4\beta n+2n+3)+2n+1)}{t(t^2+1)(t^4-2\beta t^2+1)}y' - \dfrac{2(2nt^5+nt^3(\beta + (\beta+1)n+5)+nt(-\beta+(\beta+1)n+1))}{t(t^2+1)(t^4-2\beta t^2+1)}y
\begin{pmatrix}c_1 \vdots c_{n}\end{pmatrix}= \begin{pmatrix}a_{1,1}a_{1,2}\cdots a_{1,{k_0}}a_{2,1}a_{2,2}\cdots a_{2,{k_0}}\vdots \vdots \ddots \vdots a_{n,1}a_{n,2}\cdots a_{n,{k_0}}\end{pmatrix}\begin{pmatrix}s_1 \vdots s_{k_0}\end{pmatrix}= A \bf{s}.
= \int_{Q}(\partial_{t}u_h v_h + \theta h \partial_{t}u_h \partial_{t}v_h + \nabla_x u_h \cdot \nabla_x v_h - \theta h \partial_{t}\nabla_x u_h \cdot \nabla_x v_h)\,dxdt + \int_{\partial Q}n_t \nabla_x u_h\cdot \nabla_x v_h\, ds - \int_{\partial Q}n_x \cdot \nabla_x u_h \,(v_h + \theta h \partial_{t}v_h)\, ds.
\langle \widetilde{S}_{ij}(\mathbf{x},t) \rangle = 0, &\langle \widetilde{S}_{ik}(\mathbf{x},t) \widetilde{S}_{lm}(\mathbf{x^{'}},t^{'})\rangle = 2k_{B}T\mu(\delta_{il}\delta_{km}+ \delta_{im}\delta_{kl})\delta(\mathbf{x}- \mathbf{x^{'}})\delta(t - t^{'}),
d\,G_r(\xi,\sigma_1,\ldots ,\sigma_r,\alpha_1,\ldots ,\alpha_r,\beta_1,\ldots ,\beta_r)\,=\, &\textbf{e}((\xi-\sigma_r)\beta_r)\,F(\sigma_r-\xi,\alpha_r-\tau\beta_r)\,G_{r-1}(\xi,\sigma_1,\ldots ,\sigma_{r-1},\alpha_1,\ldots ,\alpha_{r-1},\beta_1,\ldots ,\beta_{r-1})\,d\xi,
\Sigma :=\begin{cases}\{\sigma_{11}^{\pm}, \sigma_{12}^{\pm}, \sigma_{21}^{\pm}, \sigma_{22}^{\pm}, \theta^{\pm}, \iota' \}, , \{\sigma_1^{\pm}, \sigma_2^{\pm}, \theta^{\pm}, \iota' \}, , \{\sigma_1^{\pm}, \sigma_2^{\pm}, \iota' \}, .\end{cases}
X_{N_{2}k_{1}+k_{2}}=\sum_{n_{1}=0}^{N_{1}-1}\sum_{n_{2}=0}^{N_{2}-1}x_{N_{1}n_{2}+n_{1}}e^{-\frac{2\pi i}{N_{1}N_{2}}\cdot(N_{1}n_{2}+n_{1})\cdot(N_{2}k_{1}+k_{2})}
\frac{\alpha_l\psi_l^{\frac{1}{q}}}{\alpha_{l-1}\psi_{l-1}^{\frac{1}{q}}}= 2^{\frac{1}{q}}\left(1-\frac{2^{l-1}K}{1+2^lK}\right)^{2\gamma}\leq 2^{\frac{1}{q}}\exp\left(-2\gamma \frac{2^{l-1}K}{1+2^lK}\right) \leq 2^{\frac{1}{q}}e^{-\frac{2\gamma}{3}}= 2^{\frac{1}{q}-\frac{2\gamma}{3\log 2}}.
\left|A_t^{-1}(x) - \left(x + \gamma(x,t) + \frac{1}{N}\sigma(t)\right)\right| = \left|\gamma(A_t^{-1}(x),t) - \gamma(x,t)\right| \leq \|\gamma_x(t)\|_{L^\infty}\left|A_t^{-1}(x) - x\right| \leq \|\gamma_x(t)\|_{L^\infty}\left(\left|\gamma\left(A_t^{-1}(x),t\right)\right| + \frac{1}{N}|\sigma(t)|\right),
{\cal C}_{0}= i \int d{\bf{r}}{\textstyle{\cal X}^{\alpha}}({\bf r}) j_{0}^{\alpha}({\bf r}) \mathrm{a n d}{\bar{\cal C}}= i \int d{\bf{r}}\overline{{{\cal Y}^{\alpha}}}({\bf r}) j_{0}^{\alpha}({\bf r})
\lim_{x\rightarrow \infty}\frac{1}{\pi(x)}\sum_{\substack{p\leq x p\textrm{is inert or ramifies in}K}}N_p^k(E[\ell])= \frac{1}{2}M_k(\ell)= \frac{\ell-2}{2(\ell-1)}+\ell^{k}\frac{1}{2(\ell-1)}.
\boldsymbol{\delta}_h = -A_{\Delta}\mathbf{P}_R= \sum\limits_{k=1}^R \Delta_1{\bf p}^{(1)}_k \otimes{\bf p}^{(2)}_k \otimes{\bf p}^{(3)}_k + \sum\limits_{k=1}^R{\bf p}^{(1)}_k \otimes \Delta_2{\bf p}^{(2)}_k \otimes{\bf p}^{(3)}_k + \sum\limits_{k=1}^R{\bf p}^{(1)}_k \otimes{\bf p}^{(2)}_k \otimes \Delta_3{\bf p}^{(3)}_k.
\begin{aligned}L_{7}(q) = \sum_{\substack{n \geq 0 -n-1 \leq j \leq n}}q^{6n^2 + 16n + 10 - 2j^2 - 2j}+ q^{6n^2 + 8n +2 - 2j^2 - 2j}& + \sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{6n^2 + 2n - 2j^2}+ q^{6n^2 + 10n + 4 - 2j^2}\end{aligned}
\lim_{R_{P}\rightarrow 0}f_{\mu \nu}( x ) = \left( \begin{array}{c c c c}{{0}}&{{- B_{z}^{2}}}&{{- B_{y}^{2}}}&{{- E_{x}^{2}}}\\{{- B_{z}^{2}}}&{{0}}&{{- B_{x}^{2}}}&{{- E_{y}^{2}}}\\{{- B_{y}^{2}}}&{{- B_{x}^{2}}}&{{0}}&{{- E_{z}^{2}}}\\{{- E_{x}^{2}}}&{{- E_{y}^{2}}}&{{- E_{z}^{2}}}&{{0}}\\ \end{array}\right) ,
\mathrm{T r}\, \left[ \left( P e^{- i g \int_{0}^{\beta}d t^{\prime}\, A_{0}( t^{\prime}, \vec{x})}\right) B ( 0 , \vec{x}) \right] = \mathrm{T r}\, \left[ \left( P e^{- i g \int_{x_{0}}^{x_{0}+ \beta}d t^{\prime}\, A_{0}( t^{\prime}, \vec{x})}\right) B ( x_{0}, \vec{x}) \right]
{\underbrace{h_{\beta\alpha}~h_{\delta\gamma}}}= \left[\frac{1}{2}\left(\eta_{\beta\delta}\eta_{\alpha\gamma}+ \eta_{\alpha\delta}\eta_{\beta\gamma}\right) - \frac{1}{8}\,\eta_{\alpha\beta}\eta_{\gamma\delta}\right]\,\frac{1}{k_\bot^2}
\phi ( r , t )_{\mathrm{s t r a d d l e}( r > r_{H})}\approx{\cal N}_{\mathrm{s t r a d d l e}}\;{\frac{| r - r_{H}|^{\pm i \omega c_{H}/ g_{H}}}{\sqrt{2 \omega}\; r_{H}}}\; \exp \left[ \mp i \omega t \right] .
\boldsymbol{\xi}^{(m,n)}_{\texttt{b};\lambda}= \left( \xi^{(m+n)}_{\lambda_1+n-1}, \xi^{(m+n)}_{\lambda_2+n-2},\ldots , \xi^{(m+n)}_{\lambda_{n-1}+1}, \xi^{(m+n)}_{\lambda_n}\right)
\frac{M_{H}^{2}}{2 M^{2}}\simeq \frac{2 ( A - 1 )}{2 A - 1}\left( \ln M^{2}/ \Lambda_{Q C D}^{2}\right)^{A}\frac{\left( \ln M^{2}/ \Lambda_{Q C D}^{2}\right)^{- 2 A + 1}- \left( \ln \Lambda^{2}/ \Lambda_{Q C D}^{2}\right)^{- 2 A + 1}}{\left( \ln M^{2}/ \Lambda_{Q C D}^{2}\right)^{- A + 1}- \left( \ln \Lambda^{2}/ \Lambda_{Q C D}^{2}\right)^{- A + 1}}.
P_{T}=\prod_{i=1}^{n}(\frac{p_{i,t}}{p_{i,0}})^{\frac{1}{2}[\frac{p_{i,0}\cdot q_{i,0}}{\sum(p_{0}\cdot q_{0})}+\frac{p_{i,t}\cdot q_{i,t}}{\sum(p_{t}\cdot q_{t})}]}
(T_{\rho}\cdot(T_{\mu}\cdot X))(A,w) &=R(\rho)(T_{\mu}\cdot X)(R(\rho_{w})^{-1}A,\rho(w))R(\rho)^{-1}&=R(\rho)R(\mu)X(R(\mu_{\rho(w)})^{-1}R(\rho_{w})^{-1}A,\mu(\rho(w)))R(\mu)^{-1}R(\rho)^{-1}.
f^{p^t-p}= x_1^{p^ta_1-pa_1}\cdots x_d^{p^ta_d - pa_d}= (x_1^{a_1-j_1}\cdots x_d^{a_d-j_d})^{p^t}\cdot x_1^{j_1p^t-pa_1}\cdots x_d^{j_dp^t-pa_d}.
-\beta+\sum_{i=0}^{s-2}\delta_3^{(s-i)}= -\beta\left(1-\gamma\beta\sum_{i=0}^{s-2}2^{1+2(s-i)-4s^2}\right) &< -\beta\left(1-2^{-4s^2+2s+2}\right) &\leq -\beta\left(1-2^{-10}\right),
ad_{z_{2}}^{l_{2}-1}ad_{z_{1}}^{l_{1}}(z_{2})=q^{l_{1}+l_{2}-1}\prod_{i=1}^{l_{1}}(-(i-1)\mu_{0}+i-2)\prod_{j=1}^{l_{2}-1}(-(l_{1}+j-1)\mu_{0}+l_{1})L_{\alpha,0}.
-\varepsilon u_{\varepsilon}^{\prime\prime}+u_{\varepsilon}^{\prime}+\partial\phi(u_{\varepsilon}) \ni f(u_{\varepsilon})\text{\ \ \ \ \a.e.in}(0,T), u_{\varepsilon}(0) =u_{0}u_{\varepsilon}^{\prime}(T)=0
\begin{aligned}\dot q = -\frac{1}{4}(q+p)^3 \left(q+ \O((q+p)^3)\right), \dot p = \frac{1}{4}(q+p)^3 \left(p+ \O((q+p)^3)\right), \dot z = \O((q+p)^6,\rho^6), \dot \rho = \O((q+p)^{12},\rho^{12}), \dot \phi = \omega^0 + \O((q+p)^6,\tilde \rho).\end{aligned}
\sum_{\alpha = 1}^A \zeta_\alpha^\gamma \le \zeta_1^\gamma + \frac{R^\gamma}{R^\gamma-1}\sum_{\alpha=2}^A (\zeta_\alpha^\gamma - \zeta_{\alpha-1}^\gamma ) = \zeta_1^\gamma + \frac{R^\gamma}{R^\gamma-1}(\zeta_A^\gamma - \zeta_1^\gamma) \le \frac{R^\gamma}{R^\gamma-1}\zeta_A^\gamma
w(\mbox{twig}_j) := \tilde w (\mbox{twig}_j) - \frac{1}{n-1}\cdot \sum_{k=1}^{N}w(e_k) = \frac{1}{n-1}\cdot \left(\left( \sum_{\ell=1 , \ell \neq j}^n D_{\hat \ell}\right) - nD_{\hat j}- \sum_{k=1}^{N}w(e_k) \right), \; \forall j\in [n]
\int\limits_{\mathbb{R}^{N}}\frac{f\left( D_{N,d,p}u\left( x\right) \right)}{C\left( x\right)^{t\left( dq+1\right) -Nd}}dx=\frac{1}{1+dq}\int\limits_{\mathbb{R}^{N}}\frac{f\left( \left( \frac{1}{1+dq}\right)^{\frac{p-1}{p}}u\left(y\right) \right)}{C\left( y\right)^{t}}dy.
\sum_{\pi}q^{|\pi|}=\prod_{i=1}^{n}\frac{1-q^{3i-1}}{1-q^{3i-2}}\prod_{1\leq i<j\leq n}\frac{1-q^{3(2i+j-1)}}{1-q^{3(2i+j-2)}}\prod_{1\leq i<j,k\leq n}\frac{1-q^{3(i+j+k-1)}}{1-q^{3(i+j+k-2)}},
\gamma_{D,2}^{M,r}= \frac{{{P_s}}}{{{\sigma^2}}}\sum\nolimits_{k = 1}^M{{{\left|{h_{SD}^k}\right|}^2}}+ \frac{{{P_s}}}{{{\sigma^2}}}\sum\nolimits_{l = r + 1}^M{{{\left|{h_{SR}^l}\right|}^2}}
\sum_{\substack{n\le M n \in S}}A_s(r_{f(n)}) = O \left( \sum_{\substack{n\le M n \in S}}L(r_{f(n)}) \right) = O\left( \max_{n\le M}L(r_{f(n)}) \cdot \sum_{\substack{n\le M n \in S}}1\right) &= o \left( M \cdot \max_{n\le M}L(r_{f(n)})\right) = o(N).
\check{Q}_{i b}( \tau ) \check{\vec{A}}_{a \perp}( \tau ,{\vec{\eta}}_{i}( \tau ) ) = Q_{i a}( \tau ) [ P e^{\Omega_{s a}^{( \hat{\gamma})}( \eta^{( A )}( \tau , -{\vec{\eta}}_{i}( \tau ) ) ){\hat{T}}^{a}}]_{a b}\check{\vec{A}}_{b \perp}( \tau ,{\vec{\eta}}_{i}( \tau ) ) ,
P_n(T\widetilde{f}||f^{\star})- P_n(f||f^{\star}) + D^{\prime}_n(Tf,T\widetilde{f}) \leq\frac{2v v_f}{m_1}+ \frac{L_2v^2_fv^2_0}{m_0}+ \frac{L^2_2v^2_fv^4_0}{4m_0}+ \left(\frac{1}{n}\sum_{i=1}^n|Y_i|\right)\frac{v_fL_2 v^2_0}{m_0}+ \frac{T_n}{n}.
\left(F_{_{1;0,\dots,0}}^{\left(1\right)}(z),\dots,F_{_{1;0,\dots,0}}^{\left(N\right)}(z),F_{_{1;0,\dots,0}}^{\left(N+1\right)}(z),\dots,F_{_{1;0,\dots,0}}^{\left(N'\right)}(z)\right)=\left(z_{1},\dots,z_{N},0,\dots,0\right),
\mathrm{Mant}_{G,b, \mu}= \sum\limits_{(M_b, \mu') \in \mathcal{I}^{G, \mu}_{M_b, b'}}\mathrm{Ind}^G_{P_b}(\otimes^k_{i=1}\mathrm{Mant}_{M_{b'_i}, b'_i, \mu'_i}),
{\frac{\phi^{\prime \prime}}{A}}+{\frac{1}{A}}\left( -{\frac{1}{2}}{\frac{A^{\prime}}{A}}+ 2{\frac{B^{\prime}}{B}}+{\frac{2}{r}}\right) \phi^{\prime}-{\frac{2}{r^{2}}}\phi - \lambda \phi ( \phi^{2}- \eta^{2}) = 0 \, .
\tilde\nu_v\cdot \psi = \left(\frac{g_u\times\psi}{|g_u\times\psi|}\right)_v \cdot \psi = \left(\frac{g_{uv}\times\psi+g_u\times\psi_v}{|g_u\times\psi|}\right) \cdot\psi + \bigl((g_u\times\psi)\cdot\psi\bigr)\left(\frac{1}{|g_u\times\psi|}\right)_v =-\frac{\det(g_u,\psi,\psi_v)}{|g_u\times\psi|}\neq 0.
\hat{H}_{coul}=-\sum_{i=1}^{D}\frac{\partial^2}{\partial x_i^2}-\frac{\eta}{r}+\frac{\alpha_1}{x_1^2+\cdots+x^2_{n_1}}+\cdots+\frac{\alpha_{N-1}}{x^2_{n_{N-2}+1}+\cdots+x^2_{n_{N-1}}},
F_0 =1 F_1 =t^{-a_1}\hat c_0^{-1}\hat c_1 a_1 F_2 =t^{-a_2}\hat c_0^{-1}\hat c_2 a_2(a_2-a_1) F_3 =t^{-a_3}\hat c_0^{-1}\hat c_3 a_3(a_3-a_1)(a_3- a_2)