harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A=[\begin{matrix}1&2\\ 3&4\end{matrix}]=1[\begin{matrix}1&0\\ 0&0\end{matrix}]+2[\begin{matrix}0&1\\ 0&0\end{matrix}]+3[\begin{matrix}0&0\\ 1&0\end{matrix}]+4[\begin{matrix}0&0\\ 0&1\end{matrix}]=1L_{1,1}+2L_{1,2}+3L_{2,1}+4L_{2,2}
| h^{-1}(p) - g^{n_k}\circ h^{-1}(p_0)| = | h^{-1}(p) - h^{-1}(p_{n_k}) | begin{align*}0.2em] \leq \left( \left| \frac{\alpha - \bar{\alpha}}{(p-1)^2}\right| + \delta \right) |p - p_{n_k}|
\partial_{t}\theta^{\varepsilon}+\mathcal{A}_{\kappa}^{\varepsilon}\theta^{\varepsilon}=\tau^{\varepsilon}\sum_{i=1}^{N}\nabla^{\delta}u_{i}^{\varepsilon}\cdot\nabla\theta^{\varepsilon}\quad\mbox{in}\;Q_{T}^{\varepsilon},
J(u)= \int L_{g_{0}}uu = \int L_{g_{0}}u_{\alpha, \beta}u_{\alpha, \beta}+\sum_{i}\alpha_{i}^{2}\int L_{g_{0}}\varphi_{i}\varphi_{i}+o(1) = \alpha^{2}(c_{\omega}+\Vert \beta \Vert^{2}+o(\Vert \beta \Vert^{2}))+c_{0}r_{\infty}^{\frac{2-n}{2}}\sum_{i}K_{i}^{\frac{2-n}{2}}+o(1),
\left|5 \Bigl(\sum_{i=1}^k w_{\mu_i',\zeta_i'}+ \varphi_i\Bigr)^4 \,D_{\zeta_1'}\varphi_1\right| \leq C \left( w_{\mu_1',\zeta_1'}^4 + \varepsilon^4 \right) \varepsilon^2 & \leq C \,\varepsilon^2\, w_{\mu_1',\zeta_1'}^4.
H^\infty_0 Y_1= H^\infty_0 [Y\cap L^2(\mathcal M,\tau)]_2\subseteq [(H^\infty_0Y)\cap L^2(\mathcal M,\tau)]_2\subseteq [\overline{H^\infty_0Y}^{w^*}\cap L^2(\mathcal M,\tau)]_2.
\limsup_{V\setminus\{0\}\ni h \to 0}\bigg[ \frac{\| f^{(n)}(x+h) \mathbf{u}- f^{(n)}(x) \mathbf{u}- g(x)(u_1,u_2,\ldots,u_n,h) \|_W}{\|h\|_V}\bigg] =0.
q\frac{d}{dq}\log s= \sum_{n=0}^{\infty}A_{n}\left ( \frac{r(r^{2}s+8rs-r-s)}{8r^{3}s-3r^{2}s+r-s}\right )^{n}, \frac{1}{s}= r+\frac{1}{r}-2\sqrt{\frac{4}{r}- 4r - 15},
\Sigma_{-}\ : \ \tilde{x}^{3}- \frac{3}{4}\tilde{u}_{0}^{4}\tilde{x}- \frac{1}{4}( \tilde{u}_{0}^{6}+ \tilde{u}_{1}) + \frac{1}{4}( \zeta + \frac{1}{\zeta}) +{\cal O}( \tilde{\epsilon}) = 0 \ ,
\sum_{n \ge 1}\phi_G(n) \, z^n \ = \ \frac{\phi^*_G(z)}{(1-z)^{\xi + 1}}\, , \sum_{n \ge 1}f_G(n) \, z^n \ = \ \frac{f^*_G(z)}{(1-z)^{\xi + 1}}\, , \sum_{n \ge 1}t_G(n) \, z^n \ = \ \frac{t^*_G(z)}{(1-z)^{d - c + 1}}\, ,
f^{\stackrel{\underbrace{\mathrm{\scriptsize ~ 1 \ldots ~ 1}}}{m_{1}}\stackrel{\underbrace{\mathrm{\scriptsize ~ 2 \ldots ~ 2}}}{m_{2}}\cdots \stackrel{\underbrace{\mathrm{\scriptsize ~ N - 1 \ldots ~ N - 1}}}{m_{N - 1}}}( u ) \; , \qquad{m_{1}\geq m_{2}\geq \ldots \geq m_{N - 1}}\; .
G_{l+\nu}\left(\left. r'', r' ; E\right| \lambda; a\right)= - \frac{2M}{\hbar^{2}}\,\frac{\sqrt{r'r''}}{K_{s_{l}}(\kappa a)}\,\left[K_{s_{l}}(\kappa a) I_{s_{l}}(\kappa r_{<})-I_{s_{l}}(\kappa a) K_{s_{l}}(\kappa r_{<})\right]K_{s_{l}}(\kappa r_{>})\; ,
\mathbf{c}^{T}E_{P}\left[ var_{P}\left[ \mathbf{\psi}_{P}\left( \mathbf{G,B};\mathcal{G}\right) |\mathbf{A},Y,\mathbf{G}\right] \right] \mathbf{c=}\sum_{\mathbf{a}\in \mathcal{A}}c_{\mathbf{a}}^{2}E_{P}\left\{\pi_{\mathbf{a}}(\mathbf{G};P)var_{P}(Y\mid \mathbf{A}=\mathbf{a},\mathbf{G})var_{P}\left[ \frac{1}{\pi_{\mathbf{a}}(\mathbf{G,B};P)}\mid \mathbf{A=a},\mathbf{G}\right] \right\}.
h_i^0 = 0, \forall i \in [m] h_i^{j}\leq h_i^{j+1}, \forall j \in [m - 1], i \in [m] \sum_{i \in [m]}h_i^{j}= j, \forall j \in [m] \sum_{k \in [j]}h_i^{k}\geq j - y_i + 1, \forall j \in [m], \forall i \in [m]
Z = \int{\cal D}{^+}G \,{\cal D}{^-}G \, \, d e t \,{\bf G}^{- 1 / 2}\, \, d e t ({^+}M )^{- 1 / 2}d e t ({^-}M )^{- 1 / 2}\, \, e^{- \int d^{4}x \tilde{L}},
\sigma_{y,Dick}^{2}(\tau)=\frac{1}{\tau}\frac{t_{c}^{2}}{t_{i}^{2}}\cdot\sum_{n=1}^{\infty}\frac{sin^{2}(\pi nt_{i}/t_{c})}{\pi^{2}n^{2}}S_{y}^{LO}(\frac{n}{t_{c}})
\int \eta_{k+1}(\d x_{k+1}| x_1,\ldots, x_{k}) =\frac1{N-k}\int \|K_{H_{N-k}}\psi_{k+1}\|^2\mu(\d x_{k+1}) & =\frac1{N-k}\int \|K_{H_{N-k}}(x_{k+1},\cdot)\|^2\mu(\d x_{k+1}) &=\frac1{N-k}\int K_{H_{N-k}}(x_{k+1},x_{k+1})\mu(\d x_{k+1}) &=\frac1{N-k}\mathrm{Tr}(K_{N-k})=1.
(Hf)(x) =T_{w_{x}}^{-1}\sum_{a=1}^{r}\sum_{l=1}^{c_{a}}q^{c_{a}-l}(T_{c_{1}+\cdots +c_{a-1}+l}^{-1}\cdots T_{c_{1}+\cdots +c_{a}-1}^{-1}).f(w_{x}x-v_{c_{1}+\cdots +c_{a}}) -\frac{\alpha}{1-q}\sum_{a=1}^{r}(c_{a}-[c_{a}]_{q})f(x),
\int_0^\infty r f(r) \log f(r) ~ \mathrm{d}r = \int_{|u|\lambda}^\infty \frac{K_{3/2}(\rho)}{|u|^2\rho^{1/2}}\log \left( \frac{K_{3/2}(\rho)}{\rho^{3/2}}\right) \mathrm{d}\rho ,
G^{\mathrm{r e t}}( \omega ,{\bf x}) = \frac{i^{D}\pi}{2 ( 2 \pi )^{( D - 1 ) / 2}}\left( \frac{\omega}{r}\right)^{( D - 3 ) / 2}H_{( D - 3 ) / 2}^{1}( \omega r ) ,
\sum_{j = m_0}^{m/2}a_{j,m}b^j = \sum_{j = m_0}^{m/2}b^j \left( a_{m/2,m}+ \sum_{\ell = j}^{m/2-1}c_{\ell,m}\right) &= a_{m/2,m}\sum_{j = m_0}^{m/2}b^j + \sum_{\ell = m_0}^{m/2}c_{\ell,m}\sum_{j = m_0}^{\ell}b^j .
\hat{\delta}\hat{A}_{\mu}( x ) = \delta ( A_{\mu}( x ) +{\frac{1}{2}}\theta^{\sigma \nu}A_{\nu}( x ) F_{\sigma \mu}( x ) +{\frac{1}{2}}\theta^{\sigma \nu}A_{\nu}( x ) \partial_{\sigma}A_{\mu}( x ) ) ~ ,
C_{4,1}\lesssim A_3^{\frac{p-q}{pq}}+ \sum_{k= -\infty}^{M-1}2^{-k\frac{q}{p-q}}\int_{x_k}^{x_{k+1}}\bigg(\int_t^{x_{k+1}}u\bigg)^{\frac{q}{p-q}}u(t) V_r(x_k,t)^{\frac{pq}{p-q}}dt.
\begin{pmatrix}0 0 \dots 0 0 t+\varepsilon 1 0 \dots 0 0 0 0 1 \dots 0 0 0 \vdots \vdots \ddots \vdots \vdots \vdots 0 0 \dots 1 0 0 0 0 \dots 0 1 0\end{pmatrix}.
X^{\dagger\!\dagger}_{t}\, =\, \int^{t}_{0}e^{t-s}\widetilde{X}^{\dagger\!\dagger}_{s}{\mathrm d}s + \int^{t}_{0}e^{t-s}{\mathrm d}B_{s}\, , \widetilde{X}^{\dagger\!\dagger}_{t}\, =\, \int^{t}_{0}\sum_{k=0}^{\infty}e^{t-s}\cdot \frac{\,(-1)^{k}(t-s)^{k}\,}{\,k!\,}\,{\mathrm d}W_{s, k}\, ; t \ge 0 \,
L^2(S^2(J\times\Omega))=\bigoplus_{i=1}^{\infty}V_{1,i}\oplus\bigoplus_{i=0}^{\infty}V_{2,i}\oplus\bigoplus_{i=1}^{\infty}V_{3,i}\oplus\bigoplus_{i=0}^{\infty}V_{4,i},
0 \le J[u^*,p_\varepsilon](x_\varepsilon,t_\varepsilon)-J[v_*,p_\varepsilon](y_\varepsilon,s_\varepsilon) &\quad+K_{(0,x_\varepsilon)}[u^*,p_\varepsilon](x_\varepsilon,t_\varepsilon)-K_{(0,y_\varepsilon)}[v_*,p_\varepsilon](y_\varepsilon,s_\varepsilon) &\quad+f(x_\varepsilon,t_\varepsilon)-f(y_\varepsilon,s_\varepsilon).
S^r = \begin{pmatrix}s_0\alpha \sigma^{-1}(s_1)\alpha \ldots \sigma^{-r+1}(s_{r-1})\alpha s_1\sigma(\alpha) \sigma^{-1}(s_2)\sigma(\alpha) \ldots \sigma^{-r+1}(s_r)\sigma(\alpha) \vdots \vdots \ddots \vdots s_t\sigma^t(\alpha) \sigma^{-1}(s_{t+1})\sigma^t(\alpha) \ldots \sigma^{-r+1}(s_{t+r-1})\sigma^t(\alpha) \end{pmatrix}_{(t + 1) \times r}.
\forall i < i^{*}(k), \mu((t^n_{i},t^n_{i+1}]\times Z_k) =\mu((t^n_{i-1},t^n_{i}]\times Z_k)= 2^{-n}, & \mu((t^n_{i^{*}(k)},t^n_{i^{*}(k)+1}]\times Z_k) \leq\mu((t^n_{i^{*}(k)-1},t^n_{i^{*}(k)}]\times Z_k) = 2^{-n}.
\lefteqn{\beta_j^K(t_K)-\beta_{i_K}^K(t_K)}& =\widetilde{\beta}^K(t_K,j\delta_K)-\widetilde{\beta}^K(t_K,x_K)-\left(\beta_{i_K}^K(t_K)-\widetilde{\beta}^K(t_K,x_K)\right) \leq\varphi(t_K,j\delta_K)-\varphi(t_K,x_K)+\widetilde{M}^K(t_K,j\delta_K)-\widetilde{M}^K(t_K,x_K)-\left(\beta_{i_K}^K(t_K)-\widetilde{\beta}^K(t_K,x_K)\right)
d_{D'}^\pm(v) =d_D^\pm(v)-d_{D^{\rm abs}}^\pm(v)-d_{\mathcal{SC}\cup\mathcal{SC}'}^\pm(v)-d_{\mathcal{SF}\cap D}^\pm(v)-d_{\mathcal{SF}'\cap D}^\pm(v) &=r-r_0(s-1)-(qr_3+7r^\diamond)-0-0=r'.
\frac{\partial Q}{\partial x}\left(x,t\right)=\intop_{0}^{a\left(x,t\right)}2\pi r\frac{\partial u}{\partial x}\left(x,r,t\right)dr+2\pi a\left(x,t\right)\frac{\partial a}{\partial x}\left(x,t\right)u\left(x,a\left(x,t\right),t\right).
1 = \frac{|G:\Phi(G)|\cdot |\Phi(G)|}{|G:M|\cdot |M:\Phi(M)|\cdot |\Phi(M)|}=\frac{p^{d(G)}|\Phi(G)|}{p^{1+d(M)}|\Phi(M)|}=\frac{|\Phi(G)|}{|\Phi(M)|}.
{\frac{\partial}{\partial r}}\left[(r^2+A^2-2Mr){\frac{\partial G}{\partial r}}\right]+{\frac{1}{sin\theta}}{\frac{\partial}{\partial \theta}}\left[sin\theta{\frac{\partial G}{\partial\theta}}\right] = -\delta(r-r_o) \delta(cos\theta-cos{\theta}_o)\delta(\varphi-{\varphi}_o)
\int_0^{\infty}e^{-\theta x}\sum_{k=1}^{\infty}f_{\xi_k I_k}(x) \d x = \sum_{k=1}^{\infty}\int_0^{\infty}e^{-\theta x}f_{\xi_k I_k}(x) \d x= \sum_{k=1}^{\infty}\L_{\xi_k I_k}(\theta).
{\cal Z}_{\mathrm{m o n .}}= \int D \rho \exp \left\{- \left[ \frac{\pi}{2 e^{2}}\int d^{3}x d^{3}y \rho \left( \vec{x}{}\right) \frac{1}{\left| \vec{x}- \vec{y}{}\right|}\rho \left( \vec{y}{}\right) + V [ \rho ] \right] \right\},
\Big|\Big|Q^{(\mathcal{B}_n)}_{I,\mathbf{U}}Q^{(\mathcal{B}_n)}_{\mathbf{Z}|M=m,I,\mathbf{U}}-Q^{(\mathcal{B}_n)}_{I,\mathbf{U}}Q^n_{Z|U}\Big|\Big|_{\mathsf{TV}}\xrightarrow[n\to\infty]{}0
\delta_{\alpha_{1}\beta_{1}\cdot\cdot\cdot\alpha_{n}\beta_{n}}^{\mu_{1}\nu_{1}...\mu_{n}\nu_{n}}=(2n)!\delta_{[\alpha_{1}}^{\mu_{1}}\delta_{\beta_{1}}^{\nu_{1}}\cdot\cdot\cdot\delta_{\alpha_{n}}^{\mu_{n}}\delta_{\beta_{n}]}^{\nu_{n}}.
\binom{n+j}{k}k^{\underline{m}}(n+j-k)^{\underline{j-m}}= \frac{(n+j)!}{k!(n+j-k)!}\frac{k!}{(k-m)!}\frac{(n+j-k)!}{(n+j-k-(j-m))!}& = \frac{(n+j)!}{n!}\frac{n!}{(k-m)!(n+j-k-(j-m))!}& = \frac{(n+j)!}{n!}\binom{n}{k-m},
\frac{1}{||\beta-\alpha||_g}\sum_{l=1}^n (\Omega_g(\alpha, \beta, l)\omega_g(\alpha+[l], \beta+[l], l) +\Omega_g(\alpha-[l], \beta-[l], l) \omega_g(\alpha, \beta, l))(x^{\alpha}\otimes g) \epsilon_{\beta}^* & = (x^{\alpha}\otimes g) \epsilon_{\beta}^*
\Sigma_1 := \sum_{p^{\nu}\leq y \atop p \in E_j}\prod_{0 \leq l \leq n}\left(1-\frac{E_l(x)-E_l(x/p^{\nu})}{E_l(x)}\right)^{k_l}\left(1-\frac{\log p^{\nu}}{\log x}\right)^{-1}, \Sigma_2 := \sum_{y < p^{\nu}\leq \sqrt{x}}\prod_{0 \leq l \leq n}\left(1-\frac{E_l(x)-E_l(x/p^{\nu})}{E_l(x)}\right)^{k_l}\left(1-\frac{\log p^{\nu}}{\log x}\right)^{-1}.
\frac{\partial A_i(a)}{\partial a_i}=-\sum\limits_{k\neq i}\frac{[A_i(a),A_k(a)]}{a_i-a_k},\frac{\partial A_j(a)}{\partial a_i}=\frac{[A_i(a),A_j(a)]}{a_i-a_j},j\neq i.
{{\tilde{x}_{L}}^{\alpha \beta}}{}^{\prime}={{\tilde{x}_{L}}^{\alpha \beta}}- \frac{i}{4}( \varepsilon^{\alpha}{\tilde{\theta}}^{\beta}+ \xi^{\alpha}\psi^{\beta}({\tilde{x}_{L}},{\tilde{\theta}}) + \alpha \leftrightarrow \beta ) ,
\sum_{k=1}^{n}kb_kz^{k-1}+\sum_{k=n+1}^{\infty}kb_kz^{k-1}=\alpha\left(-2(1-\beta)+\sum_{k=1}^{n-1}(k-2(1-\beta))b_kz^k\right)w(z) & + \alpha\left(\sum_{k=n}^{\infty}(k-2(1-\beta))b_kz^k\right)w(z).
\lim_{m \to \infty, m}(-1)^{\lfloor sm \rfloor}\frac{1}{m}\sum_{j = 0}^{m-1}\frac{e^{- 2 \pi i \frac{j + \theta_1/2}{m}\lfloor sm \rfloor}}{1 + \exp \left\{2 \pi i \frac{j + \theta_1 /2}{m}\right\}+ \frac{z}{m}}= (-1)^{\theta_1 \mathbf{1}_{s < 0}}\frac{e^{- z (s+\mathbf{1}_{s < 0})}}{1 - (-1)^{\theta_1}e^{-z}}.
\|(R_+ - 1)^{j+1}\|_1 = \left\|(R_+ - 1)\sum_{i=0}^j{j \choose i}R_+^i(-1)^{j-i}\right\|_1 \le \sum_{i=0}^j{j \choose i}\left\|(R_+ - 1) R_+^i\right\|_1 \le 2^j \left\|(R_+ - 1) R_+^j\right\|_1 \le e^{-\Omega(n)}.
:\! |g |^{2m}\!: \, = \frac 1{2^m}\sum_{\l = 0}^m \begin{pmatrix}m \l\end{pmatrix}H_{2\l}(h_1 ) H_{2m - 2\l}(h_2) & = \sum_{\l = 0}^m \begin{pmatrix}m \l\end{pmatrix}H_{2\l}(\Re g; \tfrac 12 ) H_{2m - 2\l}(\Im g; \tfrac 12 ),
\Gamma (\sigma_1) \Gamma (\sigma_2) \Phi_2 (\sigma_1,\sigma_2, a, 1,z_2) = \int_0^\infty \int_0^\infty \frac{y^{\sigma_2-1}}{e^y-z_2}\frac{x^{\sigma_1-1}e^{(1-a)(x+y)}}{e^{x+y}-1}dx dy
Z_{5}(t) = \frac{20-504t^{2}+ 3516t^{4}- 6776t^{6}+ 2304t^{8}}{1-30t^{2}+273t^{4}-820t^{6}+576t^{8}}&= 20+96t^{2}+936t^{4}+ \cdots + 2212976684616t^{20}+ \cdots.
G(i,j,x,r,c) := \Pr\left( \sum_{\ell=1}^{j-1}2\lfloor Y'_{i,\ell}(q_\ell^2, c_j, \epsilon)\rfloor + 2\eta''_{i,j,i}+ \sum_{\ell=j+1}^{n}\xi'_{i,\ell}(q_\ell) = r_i - k\right) \ \ \ \times \Pr\left( \sum_{\ell=1}^i 2\xi_{\ell,j}(q_j^2) + \sum_{\ell=i+1}^{m}\xi_{\ell,j}(q_j) = c_j - k\right),
\begin{array}{r c l}{{\left[ M_{a b}, M_{c d}\right]}}&{{=}}&{{- \eta_{a c}M_{b d}- \eta_{b d}M_{a c}+ \eta_{a d}M_{b c}+ \eta_{b c}M_{a d}\, ,}}\\{{}}&{{}}&{{}}\\{{\left[ P_{c}, M_{a b}\right]}}&{{=}}&{{- 2 P_{[ a}\eta_{b ] c}\, , \hspace{1 . 5 c m}\left[ P_{a}, P_{b}\right] = - g^{2}M_{a b}\, .}}\\ \end{array}
\sum_{n=1}^{\infty}(v_{ij}+ (n-1)^2v_{jj})(H_{jj}^{n-1}- H_{jj}^{n}) = v_{ij}+ v_{jj}+ v_{jj}\left(\frac{H_{jj}+1}{(1-H_{jj})^2}\right) -\frac{2v_{jj}}{1-H_{jj}}& = v_{ij}+ v_{jj}\left( \frac{3H_{jj}-1}{(H_{jj}-1)^2}+ 1 \right)
\beta F = - \left({2^{21}3^2 5^7 \pi^{14}\over7^{19}}\right)^{1/5}\!\! N^2 \left({T \over (g_{YM}^2 N)^{1/3}}\right)^{9/5}= - 4.115 ~ N^2 \left({T \over (g_{YM}^2 N)^{1/3}}\right)^{1.8}
h_{1,j}(t)~_{s+1}\mathcal{F}_{s}({\bf a};{\bf b})(\alpha_1)^{q^d}+\cdots+h_{r-j, j}(t)~_{s+1}\mathcal{F}_{s}({\bf a};{\bf b})(\alpha_{r-j})^{q^d}+R_j=0
D^{\alpha\vee \beta}T=(-i)^{|\alpha|}i^{|\beta|}\mathcal L_{P_N}^{\alpha_n}\circ\cdots \circ \mathcal L_{P_1}^{\alpha_1}\circ\mathcal L_{Q_N}^{\beta_N}\circ\cdots\circ\mathcal L_{Q_1}^{\beta_1}(T).
\phi\big(M(D_i,D_j)\big) =\prod_{k=1}^K\sum_{\pi\in N_{o}}\kappa_{\pi}[a_{k;i}^{n_1},a_{k;j}^{n_2},\cdots,a_{k;i}^{n_{2t-1}},a_{k;j}^{n_{2t}}].
F_D^{(3)}\left(a,b_1,b_2,b_3;c;x,y,z\right):=\sum_{m,n,p=0}^\infty\frac{(a)_{m+n+p}\,(b_1)_{m}\,(b_2)_{n}\,(b_3)_{p}}{(c)_{m+n+p}}\,\frac{x^{m}}{m!}\frac{y^{n}}{n!}\frac{z^{p}}{p!}.
\tilde{\Phi}^{J}(x,t,\theta) = \Phi^{J}\Bigl\vert_{\theta^+=\theta^-=\theta}\qquad{\rm and}\qquad\tilde{\Sigma}_{J}(x,t,\theta) =\Sigma_{J}\Bigl\vert_{\theta^+=\theta^-=\theta}\ .
\Psi\big(\mathcal{X},\lambda,\mathcal{Y}\big) :=\left(\begin{matrix}L_{\mathcal{X}}'\big(\mathcal{X},\lambda,\mathcal{Y}\big) h(\mathcal{X}) G(\mathcal{X})-\Pi_{K}(G(\mathcal{X})+\mathcal{Y}) \end{matrix}\right)=0.
(f_r)_{\{J,J^2+I_1,\ldots,J^{2^{a-1}}+I_1,I_1,\ldots,I_k,I_k^2,\ldots,I_k^{2^{b-1}},I_k^{2^b}\}}=(f_r)_{\{J,J^2+I_1\}}\vee \ldots\vee (f_r)_{\{J^{2^{a-1}}+I_1,I_1\}}\vee (f_r)_{\{I_1,\ldots,I_k\}}\vee (f_r)_{\{I_k,I_k^2\}}\vee\ldots\vee(f_r)_{\{I_k^{2^{b-1}},0\}}=(f_r)_{\mathbf{I}}.
|Df^{m_q}(f(x))(v)| =\left(\prod_{i=0}^{q-1}|Df^{m_{i+1}-m_i}(f^{m_i+1}(x))(v)| \right) &\geq \lambda_0^{2n}\frac{|\xi_{m_{q}+1}(x)-c(\lambda^{m_{q}}z_0)|}{|\xi_1(x)-c(z_0)|}.
F^{[1,0]}_{(d)0}\left(p^{2}_{1}, p^{2}_{2}; p^{2}_{3}\right) =x_{1,8}\left( \Pi_{1}- \Pi_{2}\right) + \left[ \left( x_{1,9}+ x'_{1,9}\frac{1}{\Pi_{2}}\right) \Pi_{1}- \left(x_{1,9}+x'_{1,9}\frac{1}{\Pi_{1}}\right) \Pi_{2}\right] \Pi_{3}\, ,
k\left\{\int_{\Omega^{\rho+}}\left(Z+U^2\right) d\mu \right\}^2 \leq \sum_{i=1}^{k}\int_{\Omega^{2^{-i-1}\rho, 2^{-i}\rho}}\left(Z+U^2\right) d\mu =\int_{\Omega^{2^{-k-1}\rho, 2^{-1}\rho}}\left(Z+U^2\right) d\mu \leq \int_{M}\left(Z+U^2\right) d\mu \leq C.
N_d =\#\{(x, y)\in \mathbb{F}_q^2: x^d+ax+b-y^2=0\}&=\sum_{\substack{y\in\mathbb{F}_q y\neq\pm\sqrt{b}}}N(x^d+ax+b-y^2=0)+2\cdot N(x(x^{d-1}+a)=0).
\begin{aligned}(z (t_2)-z(t_1))^{-1}(z(t_1)-z(t))= =\frac{t_1-t}{t_2-t_1}\Bigl(1+(t-t_2)\bigl(\frac12(z')^{-1}z''+\frac16(z')^{-1}z'''(t+t_1+t_2)-\frac14((z')^{-1}z'')^2(t_2+t_1)\bigr)\Bigr)+...\end{aligned}
\widetilde{P}_N^{[n]}(t; \mathbf{x}, \mathbf{y}) = \int_0^t{\prod_{i=1}^N{e^{-sq_{x_i}}}\sum_{\substack{\mathbf{z}\in E_N \mathbf{z}\neq\mathbf{x}}}{S^{(N)}_{\mathbf{x}, \mathbf{z}}\widetilde{P}_N^{[n-1]}(t-s; \mathbf{z}, \mathbf{y})}ds},
\mathcal{L}:=\frac{d}{dz}\left( \sigma(z) \frac{d}{dz}\right) + \frac{nh_0^{n-2}}{\sqrt{1+h_{0z}^2}}, \sigma(z):=\frac{h_0^n}{\left( 1+h_{0z}^2 \right)^{3/2}}.
\int_{\Omega}^{}v_n \eta \,dx + \frac{1}{n}\int_{\Omega}^{}\nabla v_n \cdot \nabla \eta \, dx + \int_{\Gamma}^{}(v_n)_{|_\Gamma}\eta_{|_\Gamma}d\Gamma = \int_{\Omega}^{}u \, \eta\, dx+ \int_{\Gamma}^{}u_{\Gamma}\,\eta_{|_\Gamma}d\Gamma \hbox{for all}\eta \in V .
\left.\frac{sh{\omega}^{'}x_0}{{\omega}^{'}}\Theta\left({\mu}^2-{\vec{p}}^{\,2}\right) \right]=\frac{-1}{\left(2\pi\right)^{\nu /2}}\int d^{\nu-1}p e^{i\vec{p}\cdot\vec{r}}\frac{sin\Omega x_0}{\Omega}
h_{\mu\nu}(x,y)\rightarrow h^\prime_{\mu\nu}(x,y)=h_{\mu\nu}(x,y)-\frac{1}{k}\partial_\mu\partial_\nu \xi^5 (x)+e^{-2ky}[2\partial_{(\mu}\hat{\xi}_{\nu )}(x)-2k\xi^5 (x)\eta_{\mu\nu}].
\hat{Q}( x , l , \alpha , \bar{\beta}) = \int d \mu ( \gamma , \bar{\gamma}) \frac{( \alpha \bar{\gamma})^{2 l + i x}}{\Gamma ( 2 l + i x + 1 )}\prod_{m = 1}^{N}K_{m}( x ) \frac{( \gamma \bar{\beta})^{2 l + i x}}{\Gamma ( 2 l + i x + 1 )}.
\sigma^2 = m_2 - m_1^2 = \sum_{\Sigma\subseteq [\![1,n]\!]}e_{_\Sigma}\frac{\beta_{_\Sigma}^3-\alpha_{_\Sigma}^3}{3}- \left( \sum_{\Sigma\subseteq [\![1,n]\!]}e_{_\Sigma}\frac{\beta_{_\Sigma}^2-\alpha_{_\Sigma}^2}{2}\right)^2 .
N_j^{\frac{s}{2}}\|u_j\|_{L^q_{t,x}}\lesssim N_j^{-\frac{\varepsilon}{2}+}\|u_j\|_{X^{s,b}}, j=1,2 \|u_j\|_{L^p_{t,x}}\lesssim N_j^{-\varepsilon+}\|u_j\|_{X^{s,b}},3\leq j\leq 2k+1, \|v\|_{L^r_{t,x}}\lesssim N^{0+}\|v\|_{X^{0,1-b'}}.
\gamma_{\mathbf{n};j}^{-}\left( \left. \mbox{\boldmath$ \psi$}\right\vert_{\Omega_{j}^{-}}\right) =-\gamma_{\mathbf{n};j}^{+}\left( \left. \mbox{\boldmath$ \psi$}\right\vert_{\Omega_{j}^{+}}\right)
\sum_{n=0}^{\infty}\frac{q^{\binom{n}{2}}}{\left(q;q\right)_{n}}\left(cz\right)^{n}A_{q}\left(cq^{n-1}\right)=\frac{\left(c;q\right)_{\infty}}{\left(cz;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{n^{2}}\prod_{j=0}^{2n-1}\left(z-q^{j}\right)\left(-c^{2}\right)^{n}}{\left(q^{2},c,cq;q^{2}\right)_{n}}
\lim_{\beta \to 0}{\frac{{\cal W}^{( 3 )}\pi^{2 \omega}e^{2 i \pi \omega}}{C_{F}C_{A}( 2 T )^{4 - 4 \omega}( L T )^{2}}}= -{\frac{3}{2 ( \omega - 1 )^{2}}}+{\frac{3 \gamma - 1 1 / 2}{( \omega - 1 )}}-{\frac{3 5}{2}}+ 1 1 \gamma - 3 \gamma^{2}+{\frac{\pi^{2}}{1 2}}+{\cal O}( \omega - 1 ) \ .
\theta^*:{\mathfrak H}_V(\widetilde{G},\widetilde{K})\cong{\mathfrak H}(V)[T_{\lambda}]_{\lambda\in\nabla}={\mathfrak H}(V)\otimes_RR[T_{\lambda}]_{\lambda\in\nabla}\stackrel{\theta\otimes{\rm id}}{\longrightarrow}R[T_{\lambda}]_{\lambda\in\nabla}
P_{x,r}^U(A_k^c | U_{k-1}^- = z)= P_{x,r}^U\Big( |U_k^- - U_{k-1}^-| > \Big(\frac{\epsilon^{2/3}x^{2/3}}{z}\Big) \cdot z \Big| U_{k-1}^- = z \Big) & \leq c_1 \exp\Big( -c_2 \frac{(\epsilon^{2/3}x^{2/3})^2}{2\epsilon x}\Big) = c_1 \exp\Big(-\frac{c_2}{2}\epsilon^{1/3}x^{1/3}\Big)
B(t:s):=b_{12}t^{12}+\sum_{i=5}^{11}b_it^is^{12-i}+(-a_4+\frac{a_1^4}{1728}+\frac{a_3a_1}{6}+\frac{a_2^2}{12}+\frac{a_2a_1^2}{72})t^4s^8+ + (-a_3+\frac{a_2a_1}{6}+\frac{a_1^3}{216})t^3s^9+(-a_2+\frac{a_1^2}{12})t^2s^{10}-a_1t^1s^{11}+2s^{12}.
c_1(n,k)=\sum_{2j_1+1+2j_2+1+\cdots+2j_k+1=n}a^{j_1}\cdot a^{j_2}\cdots a^{j_k}=a^{\frac{n-k}{2}}\sum_{j_1+j_2+\cdots+j_k=\frac{n+k}{2}}1=a^{\frac{n-k}{2}}{\frac{n+k}{2}-1\choose k-1}.
1 - \frac14\omega_0^2r_0^2 + \frac{1}{64}\omega_0^4r_0^4 -\frac{1}{384}\omega_0^6r_0^6 - \frac{7}{64}\kappa^2r_0^2 -\frac{15}{128}\omega_0^2\kappa^2r_0^4 = \frac{c_1}{\alpha_0}K_0(k_0r_0),
{\mathcal C}\approx ( 1 - 1 ) [ - i V_{4}( 8 \pi^{2}\alpha^{\prime})^{- 2}] \frac{1}{2 N}\int_{0}^{\infty}\frac{d t}{t^{2}}\sum_{k = 0}^{N - 1}\prod_{r = 3}^{5}\vert 2 \sin ( \pi k v_{r}) \vert^{- 1}\cdot \vert T r \gamma_{k , 9}+ \prod_{r = 3}^{5}[ 2 \sin ( \pi k v_{r}) ] \sum_{i = 3}^{5}{[ \frac{T r \gamma_{k , 5_{i}}}{2 \sin ( \pi k v_{i})}]}\vert^{2}
B_{h}:=\textrm{cof}\nabla \chi_h=\begin{pmatrix}h -h' z 0 1\end{pmatrix},A_h := \frac{1}{h}(\textrm{cof}\nabla \chi_h)^T\ \textrm{cof}\nabla \chi_h=\begin{pmatrix}h -h'z -h'z \frac{1}{h}+ \frac{(h'z)^2}{h}\end{pmatrix},
\dot v(\theta)=\dot u(t^{**}+\theta), &= f\bigl(t^{**}+\theta,u(t^{**}+\theta),u(t^{**}+\theta-\tau_1(t^{**}+\theta,u(t^{**}+\theta)) ),\dotsc, u(t^{**}+\theta-\tau_N(t^{**}+\theta,u(t^{**}+\theta)))\bigr), &= f\bigl(t^{**}+\theta,v(\theta),\eta_1(\theta),\dotsc,\eta_N(\theta)\bigr).
\begin{aligned}[\alpha]^2 =[1]\,, [\alpha][\sigma_g] =[\sigma_g]\,, [\rho]^2 =[1]+\sum_{g\in G_+}[\sigma_g]\,, [\alpha][\rho] =[\alpha\rho]\,, [\sigma_g][\sigma_{g}] = [1]+[\alpha]+[\sigma_{|2g|}]\,, [\sigma_g][\sigma_h] =[\sigma_{|g+h|}]+[\sigma_{|g-h|}]\,, \end{aligned}
\begin{matrix}&\vdots&\vdots&\vdots&&\vdots&\vdots&\vdots&\\ \cdot\cdot\cdot&0&E_{0,2}^{2}&0&\cdot\cdot\cdot&0&E_{n,2}^{2}&0&\cdot\cdot\cdot\\ \cdot\cdot\cdot&0&E_{0,1}^{2}&0&\cdot\cdot\cdot&0&E_{n,1}^{2}&0&\cdot\cdot\cdot\\ \cdot\cdot\cdot&0&E_{0,0}^{2}&0&\cdot\cdot\cdot&0&E_{n,0}^{2}&0&\cdot\cdot\cdot\\ \end{matrix}
V ( \overline{{{\rho}}}) = - \frac{1}{1 2 \pi}[ | M_{+}( \overline{{{\rho}}}^{2}) |^{3}+ | M_{-}( \overline{{{\rho}}}^{2}) |^{3}] + \frac{\lambda}{4 !}\overline{{{\rho}}}^{4}+ \frac{\eta}{6 !}\overline{{{\rho}}}^{6}.
y_{t}^{n}=\xi +\int_{0}^{t}\beta_{u}^{n}\left( g_{u}\right) dg_{u}\forall t\in \left[ 0,T\right] \left\Vert \beta^{n+1}-\beta^{n}\right\Vert_{\gamma}\leq \frac{C^{n-\left[ p\right]}}{\left( \frac{n-\left[ p\right]}{p}\right) !}\forall n\geq \left[ p\right] +1
u(x,t)=-2\kappa P_{12}(x,t)+2i\lim_{k\rightarrow\infty}k\breve{N}^{r}_{12}(x,t,k), x>0, \t>0, u(x,t)=-2\kappa P_{21}(-x,-t)+2i\lim_{k\rightarrow\infty}k\breve{N}^{r}_{21}(-x,-t,k), x<0, \t<0,
\int_{C_{h}^{-}}{f ( z ) d z}- \int_{C_{h}^{+}}{f ( z ) d z}= 2 \int_{a}^{b}{f ( x ) d x}+ 2 \pi i \left[ \sum_{k}{\mathrm{R e s}}_{{\mathrm{I m}}z_{f , k}< 0}f ( z ) - \sum_{k}{\mathrm{R e s}}_{{\mathrm{I m}}z_{f , k}> 0}f ( z ) \right] .
W_{\varphi_{\lambda}^{(t^*)}}u_0(x-\lambda t\xi + \delta_1(\lambda ), \lambda \xi + \delta_2(\lambda ) ) & = \int \varphi_{\lambda}^{(t^*)}(y - (x-\lambda \xi t +\delta_1(\lambda ))) u_0(y)e^{-iy(\lambda \xi + \delta_2(\lambda ))}dy .
(\xi^{2}-1)\frac{d^{2}R_{mn}(c,\xi)}{d\xi^{2}}+2\xi\frac{dR_{mn}(c,\xi)}{d\xi}-(\lambda_{mn}(c)-c^{2}\xi^{2}+\frac{m^{2}}{\xi^{2}-1})R_{mn}(c,\xi)=0
rank\left(\mathbf{A}_{jk_1}- \mathbf{A}_{jk_2}\right) = L~~~\forall 1 \leq j \leq \omega, 1 \leq k_1 < k_2 \leq d_j, rank\left(\mathbf{I}+ (-1)^{\omega-1}\mathbf{M}_\omega\mathbf{M}_{\omega-1}\cdots\mathbf{M}_1\right) = L ~~~~~~~~~~\forall~\mathbf{M}_j \in \left\{\mathbf{A}_{j1}, \cdots, \mathbf{A}_{jd_j}\right\}, 1 \leq j \leq \omega.
\overrightarrow{\Delta}_\xi\left(\varphi_\alpha^{2-q}\left(\Phi_\alpha\right)_*W_\alpha\right)_i-q\xi^{kl}\partial_k(\ln \varphi_\alpha)\mathcal{L}_\xi\left(\varphi_\alpha^{2-q}(\Phi_\alpha)_* W_\alpha\right)_{il}=\left(\Phi_\alpha\right)_*\left(\overrightarrow{\Delta}_g W_\alpha\right)_i,
- T_{2}({\bf{k}},{\bf{q}})\sum_{{\bf{q}}_{1}}a^{\dagger}_{{\bf{k}}-{\bf{q}}/2 +{\bf{q}}_{1}/2}({\bf{q}}_{1})a_{{\bf{k}}+{\bf{q}}_{1}/2}({\bf{q}}_{1}-{\bf{q}})
\delta^{4}( 1 2 ) \equiv \delta^{4}( \theta_{1}- \theta_{2}) = \delta^{2}( \theta_{1}- \theta_{2}) \delta^{2}( \bar{\theta}_{1}- \bar{\theta}_{2}) \equiv \delta^{2}( 1 2 ) \delta^{2}( \bar{1}\bar{2}) .
b_{a}(t,x)\lesssim \begin{cases}\ (t+R)^{-a}, \ \ 0<a<\frac{n-1}{2}, \ (t+R)^{-\frac{n-1}{2}}(t+R+1-|x|)^{\frac{n-1}{2}-a}, \ \a>\frac{n-1}{2}.\end{cases}
\frac{1}{2}\frac{d}{dt}\|\rho(t)\|_3^2 =-(\sum_{0\leq\alpha_2\leq 2 \atop \alpha_1+\alpha_2=3}D^{\alpha_1}u\cdot\nabla D^{\alpha_2}\rho,D^{3}\rho) &\leq C\| u(t)\|_{3}\|\rho(t)\|_3^2.
\int_{B}|\nabla v|^{p-2}\Big(\frac{\partial v^i}{\partial x_\alpha}\frac{\partial \Psi^j}{\partial x_\beta}\delta_{ij}-\Gamma_{ij}^l(v)\frac{\partial v^i}{\partial x_\alpha}\frac{\partial v^j}{\partial x_\beta}\Psi^l \Big)\delta^{\alpha\beta}dx=0,
\mathcal{E}_f(V) \le G_{p,f}(v_0)+\frac{1}{2}\left[\int_\Omega V\,v_0^2\,dx-\left(\int_\Omega|v_0|^{2p/(p-1)}\,dx\right)^{(p-1)/p}\right] &=\mathcal{E}_f(V_0)+\frac{1}{2}\left[\int_\Omega V\,v_0^2\,dx-\left(\int_\Omega|v_0|^{2p/(p-1)}\,dx\right)^{(p-1)/p}\right].
B_2 \le \sum_{\substack{0 < |h| < y (m,n) \mid h}}(m,n) \min \Bigl(\frac{2s}{Nn}, \frac s N \Bigl( \frac1n + \frac1m \Bigr) - \frac{|h|}{mn}\Bigr) = \sum_{0 < |t| < y/(m,n)}(m,n) \min \Bigl(\frac{2s}{Nn}, \frac s N \Bigl( \frac1n + \frac1m \Bigr) - \frac{(m,n)|t|}{mn}\Bigr).
\sigma_{1}= \left\| \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right\| \quad \sigma_{2}= \left\| \begin{array}{c c}{{0}}&{{- i}}\\{{i}}&{{0}}\\ \end{array}\right\| \quad \sigma_{3}= \left\| \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right\|
|F(z)|> |a_n||z|^{n+1}\Bigg\{\left|z+\dfrac{a_{n-1}}{a_n}-(t_1-t_2)\right| &-\left(t_2+\left|\dfrac{a_{n-1}}{a_n}\right|\right)(\sin\alpha-\cos\alpha) &\qquad-\dfrac{2t_2|a_{k+1}|+2|a_k|}{|a_n|t_1^{n-k-1}}\cos\alpha-2\sin\alpha\sum\limits_{\nu=0}^{n-2}\left|\dfrac{a_{\nu}}{a_n}\right|\dfrac{1}{t_1^{n-\nu-1}}\Bigg\}>0,
D_{AB}q_{C}= - u_{0}^{*}( (q_{A}^{2}q_{B}^{3}+ q_{B}^{2}q_{A}^{3}) q_{C}^{1}+ (q_{A}^{3}q_{B}^{1}+ q_{B}^{3}q_{A}^{1}) q_{C}^{2}+ (q_{A}^{1}q_{B}^{2}+ q_{B}^{1}q_{A}^{2}) q_{C}^{3}) ,
\P\left(\left\|\hat{U}_{[(r+1): p, :]}P_{\hat{U}_{[1:r, :]}}\right\| > C(d/n)^{1\over 2}\right) = \P\left(\left\|R^{\intercal}\hat{U}_{[(r+1): n, :]}P_{\hat{U}_{[1:r, :]}}\right\| > C(d/n)^{1\over 2}\right) \leq C\exp(-cd).
\langle \Psi | \textit{\textbf{G}}_{2}| \Psi \rangle = \frac{i}{2 \pi}\sum_{n \neq 0}\frac{q_{0}+ q_{\pi}}{2 n}\langle \Psi | \mathrm{\boldmath ~ \a l p h a_{n}\times ~ \a l p h a_{- n}~}| \Psi \rangle = \frac{q}{2 \pi}\langle \Psi | \textit{\textbf{S}}| \Psi \rangle .