harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{rcl}\left(\hat{A}_{3,p}\left(\check{A}_{p}\right)_{S}^{\left(+\right)}\hat{A}_{3,p}^{\top}\right) = \left(\hat{A}_{3,p}P_{S}\left[\begin{array}{c|c}\hat{A}_{3,p}^{\top}\hat{A}_{4,p}^{\top}\end{array}\right]\right) = \left(\hat{A}_{3,p}P_{S}\right),\end{array}
lim_{x\rightarrow\infty}\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=lim_{y\rightarrow\infty}\frac{y+y^{-1}}{y-y^{-1}}=lim_{y\rightarrow\infty}\frac{1-y^{-2}}{1+y^{-2}}=\frac{1}{1}=1.
{}^{c_i}({}^{c_1^{\delta_1}\ldots c_{i-1}^{\delta_{i-1}}c_{i+1}^{\delta_{i+1}}\ldots c_r^{\delta_r}}H)={}^{c_1^{\delta_1}\ldots c_{i-1}^{\delta_{i-1}}c_ic_{i+1}^{\delta_{i+1}}\ldots c_r^{\delta_r}}H={}^{c_1^{\delta_1}\ldots c_{i-1}^{\delta_{i-1}}c_{i+1}^{\delta_{i+1}}\ldots c_r^{\delta_r}c_i}H.
S_1 =\frac{X}{V}\sum_{u_1,\ldots,u_k}\left(\prod_{i=1}^k\frac{\mu(u_i)^2}{\varphi(u_i)}\right)\sideset{}{^*}\sum_{s_{1,2},\ldots,s_{k,k-1}}\biggl(\prod_{\substack{1\leq i,j\leq k i\neq j}}\frac{\mu(s_{i,j})}{\varphi(s_{i,j})^2}\biggr)y_{a_1,\ldots,a_k}y_{b_1,\ldots,b_k}&+O\left(y_{max}^2R^2(\log X)^{4k}\right).
\mathcal{J}_{\delta}u_{\varepsilon}=e^{i\mathcal{J}_{\varepsilon}\partial_{x}^{2}t}\mathcal{J}_{\delta}u_{0}-\int_{0}^{t}e^{i\mathcal{J}_{\varepsilon}\partial_{x}^{2}(t-s)}\mathcal{J}_{\delta}\mathcal{J}_{\varepsilon}\left(|\mathcal{J}_{\varepsilon}u_{\varepsilon}|^{2\sigma}\mathcal{J}_{\varepsilon}u_{\varepsilon,x}\right)\ds.
P_h(z,s) = \sum_j \langle P_h(\cdot, s), \mu_j \rangle \mu_j(z) + \sum_\mathfrak{a}\frac{1}{4\pi}\int_\mathbb{R}\langle P_h(\cdot, s), E_\mathfrak{a}(\cdot, \tfrac{1}{2}+ it)\rangle E_\mathfrak{a}(z, \tfrac{1}{2}+ it) \dt.
F(k,t,n) = \frac{1}{n^t}\sum_{j=0}^{k}(-1)^{k-j}\binom{n-j-1}{n-k}\binom{n}{j}j^t = \frac{(n)_{k+1}^{}}{n^t}\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}\frac{j^t}{n-j}.
(-\frac{\hbar^{2}}{2m_{1}}\frac{\partial^{2}}{\partial x_{1}^{2}}-\frac{\hbar^{2}}{2m_{2}}\frac{\partial^{2}}{\partial x_{2}^{2}}+v(x_{1},x_{2}))\Psi(x_{1},x_{2},t)=i\hbar\frac{\partial}{\partial t}\Psi(x_{1},x_{2},t)
e^{- S_{b}^{f r e e}( m ; \Lambda , \lambda )}= \int{\cal D}A{\cal D}\psi{\cal D}\overline{\psi}\quad J ( A ) \quad e^{\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi + m \overline{\psi}e^{- 2 i \varphi \gamma_{5}}\psi + ( A , d \lambda + i \ast d \Lambda )}.
S_2\leq \frac{Q}{2}\sum_{k= 1}^I \Delta x_k |n_k - \hat{n}_k| \sum_{i=k+1}^{I}(N_{\alpha_{i,k}-1}+\hat{N}_{\alpha_{i,k}-1}) & + \frac{Q}{2}\sum_{k= 1}^I (N_k+\hat{N}_k) \sum_{i=k+1}^{I}\Delta x_{\alpha_{i,k}-1}|n_{\alpha_{i,k}-1}-\hat{n}_{\alpha_{i,k}-1}|.
\int_0^T |u_{n+j}|^2 \,dt \le C \int_0^T e_1(U_{n+j})(t)\,dt &\le D(T) \Big(\sum_{i=1}^n e_{1+i-n}(U_i)(0) + \sum_{l=1}^j e_1(U_{n+l})(0)\Big)\,.
\mathcal{S}_{\alpha\beta}:= \inf_{\substack{(u,v)\in \mathit{W}, u\neq 0, v\ne 0}}\frac{\|(u,v)\|}{\bigg\|u^{\frac{\alpha(\cdot)}{q(\cdot)}}v^{\frac{\beta(\cdot)}{q(\cdot)}}\bigg\|}_{L^{q(\cdot)}(\mathbb{R}^{N})}.\, \,
\mathcal{C}_{q}\sim \sum_{i = 1}^N q^{-i}K_{\omega_N}\mathcal{E}_i \mathcal{E}_i^* K_{\omega_N}= \sum_{i = 1}^N q^{-i}\mathcal{E}_i \mathcal{E}_i^* K_{\omega_N}^2.
\Vert \eta \Vert_3 := \sqrt{\Vert\zeta_1\Vert_2^2+\Vert\zeta_2\Vert_2^2}=\sqrt{\sum_{i=1}^2|z_i|^2+\sum_{i=3}^4|z_i|^2}=\sqrt{\sum_{i=0}^7x_i^2}.
h_\rho(r,x) :=\begin{cases}1+x+x^2+\cdots +x^{r-1}k \mid r 1-\rho^r x^r k \nmid r \end{cases}G_\rho(r,i) := \frac{-1}{h_\rho(r,1) \cdot i!}\left. \frac{d^i}{dx^i}h_\rho(r,x) \right|_{x=1}
\sum_{p=\tau}^{k}\frac{1}{n_p}\log\frac{b\mp\lambda-d_{p+1}}{b\mp\lambda-d_{p}}=\sum_{p=\tau}^{k}\frac{1}{n_p}\bigl(\log (b\mp\lambda-d(q_{p+1}))-\log(b\mp\lambda -d(q_{p}))\bigr) &=\sum_{p=\tau}^{k}\frac{1}{m_{p}}\int_{q_{p}}^{q_{p+1}}\frac{\xi''(s)x(s)}{b\mp\lambda-d(s)}ds &=\int_{q_{\tau}}^1\frac{\xi''(s)}{b\mp\lambda-d(s)}ds.
L_{n}\left( \xi \right) ={\sup_{w\in \mathcal{L}\left( \xi \right)}}\left\vert{G_{n}\left( w\right)}\right\vert +\frac{1}{2}\int_{\mathcal{L}\left( \xi \right)}{\left\vert{{G}_{n}^{\prime}\left( t\right) dt}\right\vert .}
I_1 =\int_0^r \left[\frac{z}{2}(\bar\phi+e_\delta-\delta)(\bar\phi',\bar\psi')\cdot (\phi,\psi)\right]{\rm{d}}z \leq\Vert (\phi,\psi)\Vert_{L^2}\cdot \Vert (\bar\phi',\bar\psi')\Vert_{L^2}\cdot\frac{1}{2}\left[\Vert z\bar\phi\Vert_{L^\infty}+\Vert z(e_\delta-\delta)\Vert_{L^\infty}\right],
\epsilon_{( 1 , 2 )}= \epsilon_{1}\epsilon_{2}^{\prime}- \epsilon_{1}^{\prime}\epsilon_{2}, \quad ( \epsilon \lambda )_{( 1 , 2 )}= \epsilon_{1}\lambda_{2}^{\prime}- 2 \epsilon_{1}^{\prime}\lambda_{2}, \quad \lambda_{( 1 , 2 )}= \lambda_{1}\lambda_{2}^{\prime}- \lambda_{1}^{\prime}\lambda_{2}.
\bar{V}_{i}= ( \bar{a}_{4}^{i}- \bar{a}_{5}^{i}) \bar{Q}^{1 / 2}\Big [ \begin{array}{c c}{0}&{0}\\{1}&{0}\\ \end{array}\Big ] \bar{Q}^{1 / 2}\Big [ \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\Big ] ,
\Delta C = \mathbb{E}_{P_S}[C(S,q_0^*)] - \mathbb{E}_{P_Y}\left[ \mathbb{E}_{P_{S|Y}}[C(S,q_y^*)|Y=y]\right] & = \mathbb{E}_{P_Y}\left[ \mathbb{E}_{P_{S|Y}}[C(S,q_0^*) - C(S,q_y^*)|Y=y ] \right] & \le 2 \sqrt{2}L \mathbb{E}_{P_Y}\left[ D(P_{S|Y=y}|| P_S) \right] \le 2 \sqrt{2}L \sqrt{I(S; Y)},
\log \frac{\Gamma\left(\frac{L^2}{2}+1\right)^2}{\Gamma(L^2+1)}=\log L-(L^2+1)\log 2+\cdots, \log \frac{\Gamma\left(K+\frac{L^2}{2}\right)}{\Gamma\left(K-\frac{L^2}{2}-1\right)}= (L^2+1)\log K+\cdots,
\begin{aligned}F(\pi-\Theta(\theta,\phi,\psi)+\theta,\phi,\psi) =-F(\theta,\phi,\psi), G(\pi-\Theta(\theta,\phi,\psi)+\theta,\phi,\psi) =-H(\theta,\phi,\psi), H(\pi-\Theta(\theta,\phi,\psi)+\theta,\phi,\psi) =G(\theta,\phi,\psi).\end{aligned}
{\frac{m}{2 \pi}}\delta_{j , 0}\, \cosh \theta = \sigma_{j}( \theta ) + \widetilde{\sigma}_{j}( \theta ) + s_{j}\, \ \sum_{k = 0}^{2}\int_{- \infty}^{\infty}\Phi_{j , k}( \theta - \theta^{\prime}) \, \, \sigma_{k}( \theta^{\prime}) \, \, d \theta^{\prime}, \qquad j = 0 , 1 , 2 .
D(p) = \sum_{n\ge 0}p^{n\choose 2}[z^n] X((p/q)z) e^{-z/q}= \sum_{n\ge 0}p^{n\choose 2}[z^n] \prod_{j\ge 0}\frac{e^{(p-1)p^j z}- e^{-p^j z}}{p^{j+1}z}.
\prod_{k=1}^\infty (1-X^{5k})(1-X^{5k-2})(1-X^{5k-3})=\sum_{k=-\infty}^\infty (-1)^kX^{\frac{5k^2+k}{2}}, \prod_{k=1}^\infty (1-X^{5k})(1-X^{5k-1})(1-X^{5k-4})=\sum_{k=-\infty}^\infty (-1)^kX^{\frac{5k^2+3k}{2}}.
(f_{m},f_{m})_{\sigma}=\prod_{i=1}^{|m|}\frac{i-1/2-\sigma}{i-1/2+\sigma}=\frac{\Gamma(1/2+\sigma)\Gamma(|m|+1/2-\sigma)}{\Gamma(1/2-\sigma)\Gamma(m+1/2+\sigma)}.
\begin{array}{|c|c|c|c|}\hline\mbox{order}N_c=2 N_c=3 N_c\longrightarrow\infty \hline\epsilon 0.539 0.505 0.480 \hline\epsilon^2 0.620 0.601 0.585 \hline\epsilon^3 1.03 0.933 0.880 \hline\end{array}
\sum_{i=0}^{n-1}|R_{2,i}|\lesssim \sum_{i=0}^{n-1}(t_{i+1}-t_i)^{2H_0}\leq \max_{0\le i\le n-1}(t_{i+1}-t_i)^{2H_0-1}\sum_{i=0}^{n-1}(t_{i+1}-t_i) \to 0,\,\n\to \infty.
{\mathcal N}_1 u = (u \cdot \nabla) u \,b + \frac{(1-b)\, \nabla |u|^2 + (\mathrm{div}u) u}{2}= \star (\star d_1 u \wedge u) \,b + \frac{d_0 |u|^2-(d_0^* u) u}{2}
\int_0^1 |DF(y+t\, (x-y))-DF(y)|\, dt \le C\, |DF(y)|\, \int_0^1\eta_H\left(t\, \frac{|x-y|}{|y|}\right)\, dx &\le C\, |DF(y)|\, \eta_H\left(\frac{|x-y|}{|y|}\right)\, \int_0^1\eta_H(t)\, dt &\le \frac{CH}{H+1}\, |DF(y)|\, \eta_H\left(\frac{|x-y|}{|y|}\right).
\mathrm{tr}(B_q) = \sum_{m=1}^q S(m,m;q)= \sum_{m=1}^q \sum_{\substack{k=1 \gcd(k,q)=1}}^q \exp (\frac{2 \pi i}{q}(mk+mk^*)) = \sum_{\substack{k=1 \gcd(k,q)=1}}^q \sum_{m=1}^q \exp (\frac{2 \pi i}{q}(k+k^*)m) .
\delta L = \frac{\partial L}{\partial q^\alpha}\delta q^\alpha + \frac{\partial L}{\partial{v}^\alpha}\delta{v}^\alpha ={Z^{\rm C}_\alpha}(L) w^\alpha +{Z^{\rm V}_\alpha}(L) (\delta{\rm v}^\alpha + R^\alpha_{\beta\gamma}{\rm v}^\gamma w^\beta).
\left\Vert u\right\Vert_{W_{p_{1}}^{1}\left( R_{+};D\left( O_{t}\right),X\right)}=\left\Vert O_{t}u\right\Vert_{L_{p}\left( R_{+};X\right)}+\left\Vert u^{\prime}\right\Vert_{L_{p}\left( R_{+};X\right)}=
\begin{array}{l}{\cal D}_{-k}{\cal X}^{-k}=1+q^{-2}{\cal X}^{-k}{\cal D}_{-k}~,~~~~~{\rm for}~k>0~,{\cal D}_{+k}{\cal X}^{+k}=1+q^{2}{\cal X}^{+k}{\cal D}_{+k}~, begin{align*}2mm][{\cal D}_i,{\cal D}_j]=0~,~~~~~[{\cal X}^i,{\cal X}^j]=0~, begin{align*}2mm]{\cal D}_i{\cal X}^j={\cal X}^j{\cal D}_i~,~~~~~{\rm for}~ i\not=j~.\end{array}
W\left\{\psi\left( \mathrm{i}\alpha\right) ,\psi\left( \mathrm{i}s\right) \right\}=-\left( s^{2}-\alpha^{2}\right) \int_{x}^{\infty}\psi\left( z,\mathrm{i}\alpha\right) \psi\left( z,\mathrm{i}s\right)\mathrm{d}z & =-\left( s^{2}-\alpha^{2}\right) K\left( \alpha,s\right) ,
\begin{aligned}-y^*_0-y^*_1-\ldots-y^*_k=\big( -x^*_{0k}-x^*_{00},-x^*_{11},\ldots,-x^*_{k-1,k-1},0,-u^*_{0k}-u^*_{00},\ldots,-u^*_{k-1,k-1},0, &-a^*_{0k}-a^*_{00},-a^*_{11},\ldots,-a^*_{k-1,k-1},0,-X^*_{00},\ldots,-X^*_{k-1,k-1},0,\ldots,0\big).\end{aligned}
[ \tau_{\Gamma_{\min}}(P) , P ) = [\tau_{\Gamma_{\min}}(P) ,\tau_{\Gamma}(P)] \cup[ \tau_{\Gamma}(P) , P ), & [\tau_{\Gamma_{\min}}(P) ,\tau_{\Gamma}(P)] \cap[ \tau_{\Gamma}(P) , P ) = \{\tau_{\Gamma}(P) \}.
\int_{\omega_+}U_{\delta,-e_n}^{(p+1)(1+\gamma)}= \int_{\omega_+}\left( \frac{\alpha_n^{p+1}\delta^n}{(\delta^2+|x+e_1|^2)^n}\right)^{1+\gamma}\, dx \leq \alpha_n^{(p+1)(1+\gamma)}|\omega_+| \delta^{n(1+\gamma)}= O(\delta^{n(1+\gamma)}).
H^{( 0 )}( q ,{\bf n}) - m{\quad}\rightarrow{\quad}- \frac{1}{{2 m}{q}^{2}}\frac{\partial}{{\partial}{q}}{q}^{2}\frac{\partial}{{\partial}{q}}+ \frac{\bf L^{2}}{2 m q^{2}},{\quad}{\bf P}^{( 0 )}( q ,{\bf n}) \rightarrow{\quad}- i \nabla_{\bf q}.
\det \left(M'_{\Xi,\{x_j\}}\right) |_{\Pi^*}= x_{i_1}^{l_1}x_{i_2}^{l_2}, \det \left(M_{\Xi, \{x_{j_1},x_{j_2}\}}\right) |_{\Pi^*}= x_{i_1}^{l_1}x_{i_2}^{l_2}.
lim_{x\rightarrow3}\frac{x^{3}-27}{x-3}=lim_{x\rightarrow3}\frac{d(x^{3}-27)}{d(x-3)}=lim_{x\rightarrow3}\frac{3x^{2}dx}{dx}=lim_{x\rightarrow3}3x^{2}=3\cdot3^{2}=27
\int_{0}^{T}\bigg[\big( \mathbf{A},\frac{\partial^{2}\widetilde{\mathbf{A}}}{\partial t^{2}})+\big(\nabla\times \mathbf{A},\nabla\times \widetilde{\mathbf{A}}\big) +\big(\frac{\mathrm{i}}{2}(\Psi^{*}\nabla{\Psi}-\Psi\nabla{\Psi}^{*}), \widetilde{\mathbf{A}}\big) + \big( |\Psi|^{2}\mathbf{A}, \widetilde{\mathbf{A}}\big)\bigg]\mathrm{d}t= 0,
\Omega_n'(a+b) = P_n'(a+b) &=\lim_{\theta \to 0}\frac{e^{in\theta}a^n+e^{-in\theta}b^n - (a^n+b^n)}{e^{i\theta}a+e^{-i\theta}b - (a+b)}&=\lim_{\theta \to 0}\frac{ine^{in\theta}a^n-ine^{in\theta}b^n}{ie^{i\theta}a-ie^{i\theta}b}&= n\frac{a^n-b^n}{a-b}.
\begin{aligned}\tilde{f}^{(z)}(t) = \sum_{0\leq k<N/2}G_k e^{i 2 \pi k t / T}+ \sum_{-N/2<k<0}G_{N+k}e^{i 2 \pi k t / T}&+ \left( \frac{\pi N}{T}\right)^{z}F_{N/2}\cos\left(\frac{\pi N t}{T}+ z \frac{\pi}{2}\right),\end{aligned}
(- \partial_\tau^2 + i \hat v_1 (s) \, \tau -\mu_0(s)) b_1(s,\tau) + 2 \theta_0'(s) \partial_s b_0 (s,\tau) + \theta_0''(s) b_0 (s,\tau) - \mu_1 b_0 (s,\tau) =0 \,.
\tilde{\mathbf r}_m= \mathbf F\mathbf r_m = \sum_{n\in\mathcal N}\alpha_n g_n^{\frac{1}{2}}\mathbf F\mathbf H_{n,m}\mathbf F^H\tilde{\mathbf s}_n + \tilde{\mathbf n}_m = \sum_{n\in\mathcal N}\alpha_ng_n^{\frac{1}{2}}{\rm diag}(\tilde{\mathbf{s}}_n)\mathbf{F}(\mathbf H_{n,m})_{:,1}+\tilde{\mathbf n}_m,m \in \mathcal{M},
\int_0^1 \frac{1}{t}\left(\frac{1}{e^t-1}-\frac{1}{t}+\frac{1}{2}\right) e^{-xt}\;dt = \int_0^1 \frac{1}{t}\left(\frac{1}{e^t-1}-\frac{1}{t}+\frac{1}{2}\right)\;dt + \mathcal{O}(x)=C_1+\mathcal{O}(x)
H_1^k := -2\tau \varphi_{0}'\partial_{s}w_1^k +\tau^{2}(\varphi_{0}')^{2}w_1^k -\tau \varphi_{0}''w_1^k+h_{1}^{k}, H_2^k := -2\tau \varphi_{0}'\partial_{s}w_2^k +\tau^{2}(\varphi_{0}')^{2}w_2^k -\tau \varphi_{0}''w_2^k+h_{2}^{k}, H_3^k := -2\tau \varphi_{0}'\partial_{s}w_2^k +\tau^{2}(\varphi_{0}')^{2}w_2^k -\tau \varphi_{0}''w_2^k++h_{3}^{k}.
f_{(U,V)}(u,v) =\left|J_{\psi^{-1}}(u,v)\right|\,f_X(x(u,v))\,f_Y(y(u,v))\propto \tfrac{xy}{uv}\,\tfrac{x^{a-1}}{(1+\alpha x)^b}e^{-cx}\,\tfrac{y^{b-1}}{(1+\beta y)^a}e^{-cy}I_{(0,\infty)^2}(u,v) &=\tfrac{1}{uv}\,\left(\tfrac{x}{1+\beta y}\right)^a\,\left(\tfrac{y}{1+\alpha x}\right)^b\,e^{-c(x+y)}\,I_{(0,\infty)^2}(u,v).
u_1(t) = M(t) D(t) e^{-3i\lambda_1 |\alpha|^2 \log t}W_1 + O(t^{-3/4 + C_1 \varepsilon^2_1}) &= M(t) D(t) e^{-3i\lambda_1 |W_1|^2 \log t}W_1 + O(t^{-3/4 + C_1 \varepsilon^2_1}),
|GHZ\rangle=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|i\rangle\otimes\cdot\cdot\cdot\otimes|i\rangle=\frac{1}{\sqrt{d}}(|0\rangle\otimes\cdot\cdot\cdot\otimes|0\rangle+\cdot\cdot\cdot+|d-1\rangle\otimes\cdot\cdot\cdot\otimes|d-1\rangle)
d'_{n-1}=\begin{pmatrix}\beta_{n-2}-\psi_{n-1}\alpha_{n-2}^{-1}\delta_{n-1}\phi_{n-2}\end{pmatrix}, \d'_{n-2}=\begin{pmatrix}\alpha_{n-2}-\delta_{n-2}-\psi_{n-2}\beta_{n-3}0 \phi_{n-3}\end{pmatrix},
K(xw) =\tau(w)^{-1}L^{\ast}(x)M(w,x)^{-1}HM(w,x) L(x) \tau(w) \hphantom{K(xw)}{}=\tau(w)^{-1}L^{\ast}(x) \upsilon^{m}H\upsilon^{-m}L(x) \tau(w)
F_t(f)(v)=\sum_{u\in J(H_{\Delta,0}/tH_{\Delta,0}t^{-1})}(utf)(v)=\sum_{u\in J(H_{\Delta,0}/tH_{\Delta,0}t^{-1})}(tf)(vu)= =\sum_{u\in J(H_{\Delta,0}/tH_{\Delta,0}t^{-1})}\sum_{u'\in J(N_{L_P,0}/tN_{L_P,0}t^{-1})}u't\cdot f(t^{-1}u'^{-1}vut)
\mu_1(2\varepsilon-\varepsilon^2)\alpha_1^2 = \frac{1-c}{t}(2\varepsilon-\varepsilon^2)\frac{1}{2t}\bigg(\sum_{v \in V(K)}\delta_v \bigg)^2 &< \varepsilon t^{-2}\bigg(\sum_{v\in V(K)}\delta_v \bigg)^2 &\leq \min\left\{\varepsilon^{-1}t^{-2}\bigg(\sum_{v\in S}\delta_v \bigg)^2, 2\varepsilon t^{-1}\sum_{v \in V(K)}\delta_v^2\right\}
\begin{cases}(\chi^i_j)^*\geq 0, 1\leq i\leq n, 1\leq j\leq d_i, \sum_{j=1}^{d_i}(\chi^i_j)^*\leq s_i, 1\leq i\leq n, \sum_{\substack{1\leq i\leq n 1\leq j\leq d_i}}(\chi^i_j)^*\geq 2. \end{cases}
\big(\overline{\varphi(w)}z-1\big)^{N}\Big(-\log(1-\overline{\varphi(w)}z)+\log(1-\bar{w}\sigma(z))\Big) &\qquad=-\big(\overline{\varphi(w)}z-1\big)^{N}\Big(\log(\mu(z))+\log(\overline{\eta(w)})\Big),
\begin{array}{ccl}\mathcal{Q}_T^k(x_{1:T-1}^k) \stackrel{Lemma \; \ref{eqqeqthetlemma}}{=}\theta_T^k \stackrel{\eqref{formulathetak}}{=}\rho_T(\mathfrak{Q}_T^{k-1}(x_{1:T-1}^k, \xi_T)) &= \rho_T(\mathfrak{Q}_T(x_{1:T-1}^k, \xi_T)) \stackrel{\eqref{definitionQt}}{=}\mathcal{Q}_T(x_{1:T-1}^k).\end{array}
\frac{1}{m^k k!}e^{rt}(e^{mt}-1)^k = \frac{1}{m^k}\left( \sum_{l=0}^\infty r^l \frac{t^l}{l!}\right) \left( \sum_{i=k}^\infty S_2(i,k) \frac{m^i t^i}{i!}\right) &= \frac{1}{m^k}\sum_{n=k}^\infty \left( \sum_{i=k}^n S_2(i,k){n \choose i}r^{n-i}m^i \right) \frac{t^n}{n!}.
S_{\rm eff}[h]={1 \over 2}\int \biggl( ({\rm e}^{-\phi}Q_A{\rm e}^{\phi})({\rm e}^{-\phi}\eta_0{\rm e}^{\phi})-\int_0^1 dt ({\rm e}^{-t\phi}\partial_t{\rm e}^{t\phi})\{({\rm e}^{-t\phi}Q_A{\rm e}^{t\phi}),({\rm e}^{-t\phi}\eta_0{\rm e}^{t\phi}) \}\biggr),
( C^{- 1})^{\alpha \beta}( C^{- 1})^{\gamma \delta}- ( \Gamma_{\dot{a}}C^{- 1})^{\alpha \beta}( \Gamma^{\dot{a}}C^{- 1})^{\gamma \delta}, C_{\alpha \beta}C_{\gamma \delta}- ( C \Gamma_{\dot{a}})_{\alpha \beta}( C \Gamma^{\dot{a}})_{\gamma \delta}
\cdot \left[ \frac{\Gamma ( n_{1}+ | m | + \delta_{1}+ 1 ) \Gamma ( n_{2}+ | m | + \delta_{2}+ 1 ) \Gamma ( n + l + \delta_{1}+ \delta_{2}+ 1 )}{( n_{1}) ! ( n_{2}) ! ( n - l - 1 ) ! \Gamma ( l + \delta_{2}+ 1 ) \Gamma ( l + | m | + \delta_{1}+ \delta_{2}+ 1 )}\right]^{1 / 2}
L=\frac{1}{2}\begin{pmatrix}x^2-t+x^2q_2-xq_1+q_0 2(x^3-x^2e_1+xe_2-e_3) (x+e_1)\dfrac{1-q_2^2}{2}+q_1q_2 x^2-t-x^2q_2+xq_1-q_0\end{pmatrix},
L(\rho,\phi,m, v, \xi) := \frac{1}{2\tau}\sum_{i=1}^s \int_{\Omega}F(\rho_i, m_i)D^{-1}_i dx +E(\rho,\phi)+\int_{\partial \Omega}\xi (\alpha \phi +\beta \partial_n \phi -\phi^b)ds +\sum_{i=1}^s\int_{\Omega}v_i(\rho_i-\rho_i^n+\nabla \cdot m_i)dx+\int_{\Omega}v_{s+1}(f+\sum_{i=1}^sz_i\rho_i+\nabla \cdot (\epsilon(x)\nabla \phi)) dx.
a ( a^{\dagger})^{2 n}= 2 n ( a^{\dagger})^{2 n - 1}+ ( a^{\dagger})^{2 n}a , \; b ( b^{\dagger})^{2 n}= 2 n ( b^{\dagger})^{2 n - 1}+ ( b^{\dagger})^{2 n}b
s B_{\mu \nu}^{a}= \varepsilon_{\mu \nu \lambda \rho}\left( D^{\lambda}\right)_{\; \; b}^{a}\eta^{\rho b}, \; s A_{\mu}^{a}= 0 , \; s \varphi^{( 1 ) a}= \left( D_{\mu}\right)_{\; \; b}^{a}\eta^{\mu b},
h(x) \le \Bigl(g_3(x)+\frac{2}{6^4}(3^4x-34)(7-2^4x)\Bigr)\Bigr|_{x=\frac{8}{19}}+ \frac1{10\cdot 6^4}- \eta &=-\frac{638551936297453983963801188142263}{3940960181626780198766689107612132000}<0,
F_{\beta_j}F_{\beta_i}-q^{-(\beta_i,\beta_j)}F_{\beta_i}F_{\beta_j}=\sum_{n_{i+1},\ldots,n_{j-1}\geq 0}c(n_{i+1},\ldots,n_{j-1})F_{\beta_{i+1}}^{n_{i+1}}\ldots F_{\beta_{j-1}}^{n_{j-1}},
v_{\Lambda \Omega}= \frac{1}{2}{\rm{sgn}}(\sigma_{R_{ij}^{(\ell)}\Lambda}^a) \, v_{\Lambda_{ij}\left(R_{ij}^{(\ell)}\Lambda \right)_{ij}}n_{\omega^s}(\Lambda_i-\ell)\bigl(n_{\omega^s}(\Lambda_i-\ell)-1 \bigr) \, ,
I_{\varepsilon}\left(t_{\varepsilon}(W_{\varepsilon,\xi})W_{\varepsilon,\xi}\right)\rightarrow\left(\frac{1}{2}-\frac{1}{p}\right)\|U\|_{H^{1}(\mathbb{R}^{3})}^{2}+\omega\left(\frac{1}{4}-\frac{1}{p}\right)G(U)=m_{\infty}
\alpha := \frac{\alpha_1 + \alpha_2 - 2\alpha_1 \alpha_2}{1 - \alpha_1 \alpha_2}= \frac{\frac{1}{2}}{1 - \frac{L\max_j\{\gamma_j\}}{4(1-\sqrt{\delta})}}= \frac{2}{4 - \frac{L\max_j\{\gamma_j\}}{1-\sqrt{\delta}}}
\begin{array}{c}{a^{\dagger}a = | \varphi ( N ) | ,}\\{a a^{\dagger}= | \varphi ( N + 1 ) | ,}\\{a a^{\dagger}- a^{\dagger}a = | \varphi ( N + 1 ) | - | \varphi ( N ) | = G_{1}( N ) ,}\\{c ( N ) = e^{i \arg \varphi ( N )}= \frac{\varphi ( N )}{| \varphi ( N ) |}.}\\ \end{array}
= \int_{0}^{\infty}d \omega \int d^{2}k \left[ n_{\omega}\phi_{\omega , k}^{*}( x ) \phi_{\omega , k}( x^{\prime}) + ( n_{\omega}+ 1 ) \phi_{\omega , k}( x ) \phi_{\omega , k}^{*}( x^{\prime}) \right] .
{\frac{1}{2 \pi}}\int_{0}^{2 \pi}d \varphi \left[ R ( r , \varphi ) K_{1}( R ( r , \varphi ) ) \right]^{2}= \sum_{n = - \infty}^{\infty}\left[ r ( I_{n}K_{n})^{\prime}( r ) \right]^{2}\equiv F ( r ) ,
=\int_{P^t_n\backslash G(\mathbb{A})/K_1(\mathfrak{n})K_{\infty}}\sum_{\xi\in G_{2t+r}}\sum_{\gamma\in P^t_n\backslash G_n}\phi(\tilde{\tau}_t((\xi\times 1_{2n-2t}) g\times\gamma h),s)\mathbf{f}(h)\mathbf{d}h
\left(\frac{r+1}{r+1-r\alpha_p}\right)^{-1}= 1-(1-\frac{1}{r+1})\alpha_p = \frac{1}{r+1}+ (1-\alpha_p) - \frac{1-\alpha_p}{r+1}\asymp (p-p_c) \vee \frac{1}{r}
x_1 =\int_{S^1}\frac{z-z_2}{z_1-z_2}w(z) dz=\frac{11+\sqrt{-11}}{22}, x_2 =\int_{S^1}\frac{z-z_1}{z_2-z_1}w(z) dz=\frac{11-\sqrt{-11}}{22}.
2x^2-2\big(1+n_{{i_0}-1,{i_0}}+n_{{i_0}+1,{i_0}}\big)x+n_{{i_0}-1,{i_0}}^2+n_{{i_0}+1,{i_0}}^2+\sum\limits_{i\in I\setminus\{{i_0}-1,{i_0}\}}\big(n_{i,{i_0}}-n_{i+1,{i_0}}\big)^{2}=0.
|A| \# GL_n(\mathbb{Z}/ q\mathbb{Z}) = q^{\frac{n(n-1)}{2}}\prod_{i = 1}^n (q^i - 1) \implies |A| d:= \gcd_{q \geq 3}\left( q^{\frac{n(n-1)}{2}}\prod_{i = 1}^n (q^i - 1) \right).
P_1(x)= 2^{r-4}(2|S|-r)((|S|-S(x))(|S|-S(x)-1+4S(x)(|S|-S(x)) P_2(x)= 2^{r-4}(2|S|-r)(r-|S|+S(x)-1)(r-|S|+S(x)-2) P_4(x)= -2^{r-4}(r-|S|+S(x)-1)(r-|S|+S(x)-2)(|S|+S(x)) P_5(x)= 2^{r-4}\left [ (|S|-S(x))(|S|-S(x)-1) \right . \left . +4(S(x))(|S|-S(x)) \right ](r-|S|+S(x)-1) P_6(x)= 2^{r-4}(r-|S|+S(x)-1)(r-|S|+S(x)-2)(r-|S|+S(x)+1).
Z^{W}= \int{\cal D}\gamma_{+ +}\,{\cal D}c \,{\cal D}b^{--}~ \mathrm{e x p}\left[ i \Gamma^{W}( \gamma ) + i \int \! d^{2}\xi ~ b^{--}\hat{\nabla}_{-}( g_{- \alpha}\epsilon^{\alpha \beta}\partial_{\beta}c ) \right] ~ .
\Hat{r}(X) = \min_{A\subseteq X}\bigl(|X\backslash A|+\sum_{i=1}^n r_i^*(A)\bigr) = \min_{A\subseteq X}\bigl(|X\backslash A|+\sum_{i=1}^n(|A|+r_i(E\backslash A)-r_i(E)\bigr) = \min_{A\subseteq X}\biggl((n-1)|A|+|X|+\sum_{i=1}^n\bigl(r_i(E\backslash A)-r_i(E)\bigr)\biggr)
F(s) = \frac{1}{2}C_{1}(n,\Lambda,\delta)\vartheta(q)(q-1)(\rho-\rho')^{-2\beta}\int_{\rho B_{1}}\phi w^{2}dx & + q g_{1-\alpha}(s)\phi(s)\int_{\rho B_{1}}\psi^{1+q}w^{2}dx+ \dot{\phi}(s)\left(g_{1-\alpha}*\int_{\rho B_{1}}\psi^{1+q}w^{2}dx\right)(s)
\left|\mathbb{E}\left(\widetilde{\alpha}_{t,\varepsilon}\right)^n\right| \leq c_n|x-y|^{\lambda}\left(\sum_{S_2}\int_{\{0<r_1<\cdots<r_n<t\}}\prod_{j=1}^n(r_j-r_{j-1})^{-\frac{H_1H_2}{H_1+H_2}(1+\lambda\alpha_j)}dr\right)^2.
H \widetilde{t}_1 \,\cap\, H \widetilde{t}_2 \, = \, H (k_1, g_1) \,\cap\, H (k_2, g_2) begin{align*}1.5mm] = (k_1, V(k_1)(H) g_1) \,\cap\, (k_2, V(k_2)(H) g_2) \,\neq \,\emptyset
\min_{0{}\leq{}i{}\leq{}N}\|\nabla f(x_i)\|^2{}\leq{}\frac{2L\,\big(f(x_0)-f_*\big)}{1 + \sum\limits_{i=0}^{N-1}\left(2 h_i - h_i^2 \frac{-\kappa}{1-\kappa}\right)}
\kappa_{010}=\sum_{p\geq 1}\prod_{k=1}^{p-1}q_{0^k10}(0){\rm ~~and~~}\kappa_{110}=\sum_{p\geq 1}\prod_{j=0}^{p-1}q_{0^j1^q0}(0)\prod_{k=2}^{q-1}q_{0^k1^q0}(1).
\varphi_\alpha(\lambda,\omega) \sim{1\over (4\pi)^{d/2}}\sum_{n=0}^\infty\left[a_{n}(\omega)~{\lambda^{d/2-n+\alpha-1}\over \Gamma\left(\frac d2 -n +\alpha\right)}+b_{n}(\omega)~{\lambda^{(d-1)/2-n+\alpha-1}\over \Gamma\left({d-1 \over 2}-n +\alpha\right)}\right].
\Phi(\xi x)(\Omega_\psi y) =L(\eta)^\ast (\xi x\otimes \Omega_\psi y) &=L(\eta)^\ast(\xi\otimes x\Omega_\psi y) &=L(\eta)^\ast(\xi\otimes \Omega_\psi \sigma^\psi_{i/2}(x)y) &=L(\eta)^\ast L(\xi\otimes \Omega_\psi)\Omega_\psi \sigma^\psi_{i/2}(x) y &=\Phi(\xi)(x\Omega_\psi y)
q(t,x)=\pi^{-\frac{d}{2}}|x|^{-d}t^{\alpha-1}\textnormal{H}_{12}^{20}\left[\frac{1}{4}t^{-\alpha}|x|^{2}\ \Big|\begin{array}{cc}(\alpha,\alpha) (\frac{d}{2},1) (1,1)\end{array}\right]
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\begin{aligned}\iint |\omega_t*f(x)|^2\,t^{d-1}dt\,d\lambda(x)= \iint |\widehat{\omega_r}*f(x)|^2\,r^{d-1}dr\,d\lambda(x) = c_d\iint\left|D_x^{-\frac{d-1}{2}}e^{-2\pi i t\sqrt{-\Delta}}f(x)\right|^2\,dt\,d\lambda(x) = c_d'\iint\left|D_x^{-\frac{d-2}{2}}e^{2\pi i t\Delta}f(x)\right|^2\,dt\,d\lambda(x).\end{aligned}
( A , B , C ) \mapsto [ A , B , C ]_{\bullet_{\hbar}}\equiv \sum_{\sigma \in S_{3}}\epsilon ( \sigma ) \Delta_{\sigma_{1}}A \bullet_{\hbar}\Delta_{\sigma_{2}}B \bullet_{\hbar}\Delta_{\sigma_{3}}C \; , \quad \forall A , B , C \in{\cal A}_{\hbar}.
\mathcal{R}=\textrm{cl conv}\left\{\left(\underset{\alpha_{1}\in\left[0,1\right]}{\bigcup}\mathcal{R}_{1}\left(\alpha_{1}\right)\right)\bigcup\left(\underset{\alpha_{2}\in\left[0,1\right]}{\bigcup}\mathcal{R}_{2}\left(\alpha_{2}\right)\right)\right\},
s_0(B) :=\sum_{k=1}^\infty\frac{1}{k!}\sum_{\substack{x_1,\ldots, x_k\in B, d(x_1,\ldots,x_k)<m_{i-1}}}(\rho_+(x_1,\ldots,x_k)^2+\rho_-(x_1,\ldots, x_k)^2),
E_{\mu}\left(w,m,s\right) =\frac{\Gamma\left(1-s+w\right)}{\Gamma\left(1-s\right)}\left(m+\frac{i\mu}{2\pi N}\right)^{s-1-w}\left(\frac{i\mu}{2\pi N}\right)^{w}-\delta_{w0}m^{s-1}
||\mathbf{y}- \mathbf{A}(\rho^k+\mathbf{S}^k) ||_2^2 \leq \varepsilon_1||\mathbf{y}||_2, &||\mathbf{\rho}^k-\mathbf{\rho}^{k-1}||_2\leq \varepsilon_2, \ \ \ ||\mathbf{S}^k-\mathbf{S}^{k-1}||_2\leq \varepsilon_3.
\sum_{j \in [r] \setminus \{1 \}}a_{1,j}= \sum_{j \in [r] \setminus \{1 \}}\left( k+1 - \sum_{l \in [r] \setminus \{1,j \}}a_{l,j}\right) = \sum_{j \in [r] \setminus \{1 \}}(k+1) - \sum_{j \in [r] \setminus \{1 \}}\sum_{l \in [r] \setminus \{1,j \}}a_{l,j}.
w = \underbrace{s_{9}s_{10}s_{11}| s_9 s_{10}| s_9}| \underbrace{s_6s_7s_8 | s_6s_7}| \underbrace{s_5s_6 | s_5}| \underbrace{s_4}|\underbrace{s_3}| \underbrace{s_1s_2 |s_1}.
\tilde Z_n^N(i) := \Delta_n^{-\frac 12}\left( (p_N\tilde{\Delta}_i^nY)^{\otimes 2}-\int_{t_{i-1}}^{t_i}p_N\mathcal S(t_i-s)\Sigma_s\mathcal S(t_i-s)^* p_N ds\right).
t_{l}^{( q )}= \underbrace{\sum_{q_{0}= 0}^{l}\sum_{q_{1}= 0}^{l}\cdots \sum_{q_{l}= 0}^{l}}_{q_{0}+ \dots + q_{l}= q \atop 1 \cdot q_{1}+ 2 \cdot q_{2}+ \dots + l \cdot q_{l}= l + 1 - q}{\binom{n_{0}+ q_{0}- 1}{q_{0}}}{\binom{n_{1}+ q_{1}- 1}{q_{1}}}\cdots{\binom{n_{l}+ q_{l}- 1}{q_{l}}}.
\tilde C_1 = \left(2+2\left\lceil\log_2 \bigg( \frac{2 \omega(1/6)}{\epsilon}\bigg) \right\rceil + \log_2\left(\frac{L}{\mu_0}\right) \right) \left(1+\frac{1}{\sqrt{n}}\cdot\frac{GD}{4V_0+2G^2/\lambda}\right),
\theta \varphi\theta^{-1}= (\tau,T) (\sigma, R) (\tau, T)^{-1}&= (\tau, T) (\sigma \tau^{-1}, R^{\tau^{-1}}(T^{-1})^{\tau^{-1}}) &= (\tau \sigma \tau^{-1}, T^{\sigma \tau^{-1}}R^{\tau^{-1}}(T^{-1})^{\tau^{-1}}).
0=\frac{\partial}{\partial x^{i}}\left[\frac{C^{i}(x)}{\chi(x)}\right]=\left[\frac{\partial}{\partial x^{i}}C^{i}(x)\right]/\chi(x)-\frac{\partial}{\partial x^{i}}\chi(x) C^{i}(x)/\chi(x)^{2}.
q^{3m/2}B(h(l,m-l))=\sum_{i=0}^lq^{i/2}\bigg(\frac{\alpha^{m-i+1}-\alpha^{-(m-i+1)}}{\alpha-\alpha^{-1}}-\Big(\frac{L}{F}\Big) q^{-1/2}\,\frac{\alpha^{m-i}-\alpha^{-(m-i)}}{\alpha-\alpha^{-1}}\bigg)
\phi_v(y) = \phi_v (y)-\phi_v(\theta_{\lambda}v) \leq \max_{z\geq \varepsilon_0 v}\left\{\phi_v''(z)\right\}\frac{(y - \theta_{\lambda}v)^2}{2}\leq - c_2 \frac{(y-\theta_{\lambda}v)^2}{v}.