harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
| V_{C K M}| = \left( \begin{array}{l l l}{0 . 9 7 5}&{0 . 2 2 0}&{0 . 0 0 3}\\{0 . 2 2 0}&{0 . 9 7 4}&{0 . 0 4}\\{0 . 0 0 8}&{0 . 0 4}&{0 . 9 9 9}\\ \end{array}\right) ,
\Vert \nabla\varphi \Vert^4_{\mathbb{L}^4(\Omega)}= \int_{\Omega}|\nabla\varphi(\mathrm{x})|^4 d\mathrm{x}= \delta^2 \int_{\mathrm{B}}\delta^{-4}|\nabla_{\xi}\varphi(\delta \xi + z)|^4 d\xi = \delta^{-2}\Vert \nabla \hat{\varphi}\Vert^4_{\mathbb{L}^4(\mathrm{B})},
= \frac{1}{2}\times ( \frac{7 - 5}{5 \times 7}+ \frac{9 - 7}{7 \times 9}+ \frac{1 1 - 9}{9 \times 1 1}+ \frac{1 3 - 1 1}{1 1 \times 1 3}+ \frac{1 5 - 1 3}{1 3 \times 1 5})
- \; \frac{m c}{2 \pi}\frac{4 \pi^{2}}{\pi \! + \! g}\; \frac{1}{( 2 \pi )^{2}}\int d^{2}p d^{2}q \; \hat{C}_{m_{d}}( p ) \hat{C}_{m_{d}}( q ) \int_{\Lambda}d^{2}x \; e^{i p ( x - w )}e^{i q ( x - z )}\; 2 \cos ( \theta )
[X,Y]_\star:=\mathcal{L}^\mathcal{F}_XY =[\overline{\mathcal{F}}_1\rhd X,\overline{\mathcal{F}}_2\rhd Y] =X\star Y-(\overline{\mathcal{R}}_1\rhd Y)\star(\overline{\mathcal{R}}_2\rhd X),
L_j u:=a_{11}^j \frac{\partial^2 u}{\partial x^2}-2a_{12}^j \frac{\partial^2 u}{\partial x \partial y}+a_{22}^j \frac{\partial^2 u}{\partial y^2}+b_1^j \frac{\partial u}{\partial x}+b_2^j \frac{\partial u}{\partial y}=0
\widehat{v}(\xi,y_d)=\frac{1}{2\omega_\lambda(\xi)}\int^{\infty}_0 \left(e^{-\omega_\lambda(\xi)|y_d-z_d|}- e^{-\omega_\lambda(\xi)(y_d+z_d)}\right)\widehat{f}(\xi,z_d)dz_d,
\tilde{M}:= \left( \begin{matrix}1 0 0 \mathcal{V}\end{matrix}\right) M \left( \begin{matrix}1 0 0 \mathcal{U}^T \end{matrix}\right) = \left( \begin{matrix}\alpha (U v_1)^T v_2^T V w_1 V A U^T V B w_2 C U^T D \end{matrix}\right) \, .
m\left( \bigcup_{i=1}^s B(p_i;\delta_n) \right) \geq m\left( \bigcup_{i=1}^s B'(p_i;\delta_n)\right) &= \sum_{i=1}^s m(B'(p_i;\delta_n)) &= \sum_{i=1}^s \left(1-\frac{1}{p_i}\right) 2 \delta_n &\geq \left(1-\frac{4}{n}\right) \sum_{i=1}^s m( B(p_i;\delta_n) ).
\big((\mathbb{I}+E_n)\mathcal{W}_n\mathcal{F}_n\varphi\big)_{j}=\begin{pmatrix}\tilde{\chi}_{j}c_{n,T_n j}\tilde{\chi}_{j}\end{pmatrix}&\tilde{\chi}_{j}(p):=\frac{c_{n,j}\Big(p^2+(1+\sqrt{1+p^2})^2\Big)}{(1+\sqrt{1+p^2})(c_{n,j}+p+c_{n,j}\sqrt{1+p^2})}\chi_{j}(p)p\in \mathbb{R}_{+},
\bigcap_{i=1}^m f_i(A) \supset \bigcap_{i=1}^m \bigcup_{p=1}^M f_i \circ g_p(K_p\cap B_M) &\supset \bigcap_{i=1}^m h_i\left(K_{p(i)}\cap B_M\right) &\supset \bigcap_{i=1}^m h_i\left(K_{p(i)}\right) \cap \bigcap_{i=1}^m h_i(B_M) &\supset \bigcap_{i=1}^m \bigcap_{j=1}^\infty h_i\left(F_{(2j-1) 2^{p(i)-1}}\right) \cap B'_M.
\left\|(\frac{\lambda_u}{\lambda})^{s}\ss\frac{1}{\lambda}\sum_{m=s+1}^{t-1}\frac{\lambda_u^{m-s}\cdot{m}^{d_u}}{\lambda^{m-s}}\frac{1}{\lambda_u^{m}\cdot{m}^{d_u}}M^{m}\vec u_{t-m}\right\| \leq \epsilon.
\alpha\leq\alpha_{\kappa}=\bar{\upsilon}_{i}^{p}\dfrac{\frac{1}{\gamma_{i}^{\mathsf{th}}}-\kappa\sum_{j\neq i}^{L}\mathbb{E}\left[\mathsf{\bar{L}}_{ji}^{2}\right]}{1+\sum_{j\neq i}^{L}\mathbb{E}\left[\mathsf{\bar{L}}_{ji}\right]}.
\alpha^{2}= ( 1 - r^{- 1 0 / 3})^{- 1}; \hspace{2 m m}\beta^{2}= \frac{9}{1 0 0}r^{2}( 1 - r^{- 1 0 / 3})^{- 1}; \hspace{2 m m}\gamma^{2}= \frac{9}{2 0}r^{2}.
\chi_{A \, 1}~=~ - \, \frac{(2\alpha-1)}{8(\alpha-1)}\left[y_1 \left( P_1 + \frac{X_1}{2(2\alpha-1)}\right) + z_1 \left( P_2+ \frac{X_2}{2(2\alpha-1)}\right) \right]
g_{n}=\sum_{\pi\in\mathcal{C}\left(n\right)}\left(-1\right)^{\pi+1}x_{\pi}=\sum_{p=1}^n\left(-1\right)^{p+1}\sum_{\underset{k_{i}\ge1}{k_{1}+\dots+k_{p}=n}}x_{k_{1}}\dots x_{k_{p}}
\theta_{1/w}(z) = \theta(N^{\prime\prime}_{1/w}z), |b_{\theta,1/w}(n)|: =\begin{cases}\big(N^{\prime\prime}_{1/w}\big)^{1/4}, n=m^2N^{\prime\prime}_{1/w}m \in \mathbb{Z}, 0, \end{cases}
(\psi_r\psi_{r+1}\psi_r-\psi_{r+1}\psi_r\psi_{r+1})e(i)=\begin{cases}e(i) \textit{if}i_{r+2}=i_r\rightarrow i_{r+1}; -e(i) \textit{if}i_{r+2}=i_r\leftarrow i_{r+1}; (y_r-2y_{r+1}+y_{r+2})e(i) \textit{if}i_{r+2}=i_r\leftrightarrow i_{r+1}; 0 \textit{otherwise}.\end{cases}
q_{1}= \left( \begin{matrix}{1}&{0}\\{0}&{{- 1}}\\ \end{matrix}\right) ; \quad q_{2}= \left( \begin{matrix}{0}&{{- 1}}\\{0}&{0}\\ \end{matrix}\right) ; \quad q_{3}= \left( \begin{matrix}{0}&{0}\\{1}&{0}\\ \end{matrix}\right)
\tau^M(T) = -\log\left(\frac{1+\sum_{i}\rho^{M-1}_{\inf}(T_i)}{1+\sum_{i}\rho^{M-1}_{\sup}(T_i)}\right) = -\log\left(\frac{1+\sum_{i}\rho_i\exp(-\tau^{M-1}(T_i))}{1+\sum_{i}\rho_i}\right).
\mu^{i\rightarrow j}(\vec{\sigma}_{i}||\vec{\sigma}_{j})=\sum_{\{\vec{\sigma}_{k}\}_{k\in\partial i\backslash j}}W_{i}(\vec{\sigma}_{i};\{\vec{\sigma}_{k}\}_{k\in\partial i})\prod_{k\in\partial i\backslash j}\mu^{k\rightarrow i}(\vec{\sigma}_{k}||\vec{\sigma}_{i}),
j \overline{{\Psi}}_{L}\Psi_{R}= j \overline{{\psi}}_{L}j \psi_{L}^{\prime}- j \overline{{\psi}}_{R}j \psi_{R}^{\prime}+ j ( \overline{{\psi}}_{L}\psi_{R}+ \overline{{\psi}}_{R}^{\prime}\psi_{L}^{\prime}) .
\delta_{\xi,\zeta}(V):= \frac{8}{(\xi-1)}+ \sum |\langle{V_\beta},q_1\otimes \cdots\otimes q_r\rangle| \Bigl(\sum_{j=1}^r{\rm deg}_U(q_j)\Bigr)\Bigl[\sum_{\ell=1}^r \xi^{{\rm deg}_U(q_{\ell})}\zeta^{\deg_{A,B}(q_\ell)}\Bigr].
\sup_{\|Y\|_2=1}Y^T\Sigma_t(\varphi)Y=Y^T_0\Sigma_t(\varphi)Y_0 \leq Y^T_0\Sigma_t(0)Y_0 \leq \|\Sigma_t(0)\| = \sup_{\|Y\|_2=1}Y^T \Sigma_t(0)Y .
H(X,Y,Z) =2\langle [\sigma X,\sigma Y]_{H_0},\sigma Z \rangle &=2\langle [X,Y]+L_{X}b(Y)-i_{Y}(db(X))+i_Yi_X H_0, Z+i_Z b \rangle &=H_0(X,Y,Z)+(L_X b(Y)-i_Y(db(X)))(Z)+b(Z,[X,Y]) &=H_0(X,Y,Z)+db(X,Y,Z),
G_0(z,\lambda) = G_0(u,\mu) + DG_0(u,\mu)\left( \begin{matrix}z-u \mu- \lambda \end{matrix}\right) + \left( \begin{matrix}(z - u) (\mu - \lambda) + z(|u|^2 - |z|^2) + 2u ( Re(\bar{u}z) - |u|^2) 0 \end{matrix}\right).
-x/t^2 \cdot (\nabla u) (x/t) = \frac{\lvert x \rvert}{t^3}\chi_{\lvert x \rvert < t}\geq \frac{\chi_{t/2 \leq \lvert x \rvert < t}}{2 t^2}\geq \frac{1}{2 \omega_{+}^2}\chi_{B(x_0, \omega_{-}/4)}\geq 2\chi_{B(x_0, \omega_{-}/4)}
{D}( h_{m_j,k_j})= C^\infty_c(0,+\infty)^2, & h_{m_j,k_j}( f^+, f^-):=\left(\begin{array}{cc}m+\frac{\nu+\mu}{r}-\partial_r+\frac{k_j+\lambda}{r}\partial_r+\frac{k_j+\lambda}{r}-m+\frac{\nu-\mu}{r}\end{array}\right)\begin{pmatrix}f^+ f^-\end{pmatrix};
\begin{aligned}\psi_1(x) = w_0(x) \psi_2(x) = w_0(x) \int_a^{x}w_1(\xi_1) d\xi_1, \psi_{r+1}(x) = w_0(x) \int_a^{x}w_1(\xi_1) \int_a^{\xi_1}\dots \int_a^{\xi_{r-1}}w_{r}(\xi_r) d\xi_r \dots d\xi_2 d\xi_1, r=2,\dots,m-1.\end{aligned}
\widehat{J}_0:=\frac{\sin(|\xi|^2t)}{|\xi|^2}\mathrm{e}^{-\frac{1}{2\sigma}|\xi|^4t}\ \ \mbox{and}\ \ \widehat{J}_1:=\cos(|\xi|^2t)\mathrm{e}^{-\frac{1}{2\sigma}|\xi|^4t}.
M_n''(\tau_n)=\bigg(\frac{b_n-a_n}{e^{(b_n-a_n)\tau_n/2}+e^{-(b_n-a_n)\tau_n/2}}\bigg)^2 = \bigg(\frac{(b_n-a_n)e^{-(b_n-a_n)\tau_n/2}}{e^{(b_n-a_n)\tau_n/2}+e^{-(b_n-a_n)\tau_n/2}}\bigg)^2 =\bigg(h_n'(\tau_n)\bigg)^2.
\Pr_{a \sim_{\rho}b}[a \in A, b \in B] = 2^{-n}\sum_{d=0}^{n}\left(\frac{1+\rho}{2}\right)^{n-d}\left(\frac{1-\rho}{2}\right)^d |W_d| \le 2^{-2n}\sum_{d=0}^{n}(1+\rho)^{n-d}(1-\rho)^d \binom{n}{d}2^{d}&= 2^{-2n}(3-\rho)^n\,,
g(\underline{x}) = \bigl( \prod_{s = 1}^{l}\frac{\partial}{\partial y_{j_{s}}^{0}}\bigr)^{K}\sum_{|\underline{m}| \leq K - 1, m_{j_{1}}^{0}= ... = m_{j_{l}}^{0}= 0}(\frac{\partial}{\partial \underline{y}})^{\underline{m}}\nu_{\underline{m}}(\underline{y}),
\frac{d^{r}}{dx^{r}}[\ln [_{p}\Gamma_{k}(x)]]=\frac{d^{r-1}}{dx^{r-1}}\:_{p}\psi_{k}(x)=(-1)^{r}k(r-1)!\sum_{n=0}^{\infty}\frac{1}{(x+nk)^{r}}, for \: \:r\geq 2
\rho_t^{Z, 0\oplus 1}(S_c) = S_c,\rho_t^{Z, 0\oplus 1}(S_d) = e^{2\pi\sqrt{-1}t}S_d, \rho_t^{Z, 1\oplus 0}(S_c) = e^{2\pi\sqrt{-1}t}S_c,\rho_t^{Z, 1\oplus 0}(S_d) = S_d
E \Big[ \Big( \sum_{u \in N_t}f(X^u_t) \Big) \Big( \sum_{u \in N_t}g(X^u_t) \Big) \Big] = S_{fg}(t) + 2 \int_0^t S_f(t - s) S_g(t - s) \frac{\partial}{\partial s}\Big( 2 \Phi(\beta \sqrt{s}) \mathrm{e}^{\frac{\beta^2}{2}s}\Big) \mathrm{d}s
\sum_{j=1}^{n}e^{2 \pi i k \varepsilon_j}e^{-2 \pi i k \frac{j}{n}}= 2 \pi i k \sum_{j=1}^{n}\varepsilon_j e^{-2 \pi i k \frac{j}{n}}+ \mathcal{O}\left(k^2 \sum_{j=1}^{n}\varepsilon_j^2\right).
R^{i}_{}(\boldsymbol{p_{}}, \boldsymbol{p_{}},\boldsymbol{\alpha}) =(1+\alpha_{i})B\log(1+\gamma^{i}_{}(\boldsymbol{\alpha})), R^{i}_{}(\boldsymbol{p_{}}, \boldsymbol{p_{}},\boldsymbol{\alpha}) =(1+\alpha_{i})B\log(1+\gamma^{i}_{}(\boldsymbol{\alpha})).
ts = \frac{\eta(t) (1+t L_1(\eta(t)))}{1-L_0(\eta(t))}\left[ 1 - L_0(\eta(t)) + k \ln{(1-\mu(t))}\right] s = \frac{1-\mu(t)}{\mu(t)}\left[ 1 - L_0(\eta(t)) + k \ln{(1-\mu(t))}\right] \mu(t) s = (1-\mu(t))\ (1 - L_0(\eta(t))) + k (1-\mu(t)) \ln{(1-\mu(t))}\ ,
B_{P^{g}}(\mathbf{n)}=h(\tilde{Z},P^{g}\tilde{Z})=h(\tilde{Z},\varphi_{j}(g)P\varphi_{j}(g)^{-1}\tilde{Z})=h(\varphi_{j}(g)^{-1}\tilde{Z},P\varphi_{j}(g)^{-1}\tilde{Z})
Z ( \alpha ) = \int{\cal D}A_{\mu}^{\omega}{\cal D}B{\cal D}\bar{c}{\cal D}c e^{- S_{0}( A^{\omega}) - \frac{\alpha}{2}\int B^{2}d x + \int [ - i B \frac{1}{\sqrt{- \partial_{\mu}\partial_{\mu}}}\partial_{\mu}A_{\mu}^{\omega}+ \bar{c}\sqrt{- \partial_{\mu}\partial_{\mu}}c ] d x}
E[F|\mathcal{F}_{t}]=\exp \Big( \Big(-\int_t^T e^{2\tilde b(t)-2\tilde b(u)}\mathrm{d}u\Big) r(t)\Big) (A_0(\tilde b)^d+A_0(\tilde b)^{d-1}A_1(\tilde b)r(t)+\ldots),
0=\int(-2g_{\mu\nu}\frac{d^{2}x^{\nu}}{d\tau^{2}}+\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}\partial_{\mu}g_{\alpha\nu}-2\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}\partial_{\alpha}g_{\mu\nu})\delta x^{\mu}d\tau
c_{Y_{\ell}}\left(y_{\ell}, y_{\ell}^{\prime}\right) = \inf_{x_\ell, x_\ell^\prime \in \mathcal{X}}c_\ell\left( (y_\ell, x_\ell), (y^\prime_\ell, x^\prime_\ell) \right) \leq c_\ell\left( (y_\ell, x_\ell), (y^\prime_\ell, x^\prime_\ell) \right) .
N(\phi)=\left[\frac{\rho}{\int_{\mathbb{S}^2}e^{w_{\lambda}+\phi}}e^{\phi}-\frac{\rho}{\int_{\mathbb{S}^2}e^{w_{\lambda}}}-\left(\phi-\frac{\int_{\mathbb{S}^2}e^{w_{\lambda}}\phi}{\int_{\mathbb{S}^2}e^{w_{\lambda}}}\right)\right]e^{w_{\lambda}}.
\omega_n = \begin{cases}-\frac{1}{2}\pi - \xi_n, \xi_n \in [-\pi, -\frac{1}{2}\pi), begin{align*}-0.5em] 0, \xi_n \in [-\frac{1}{2}\pi, \frac{1}{4}\pi), begin{align*}-0.5em] \frac{1}{2}\pi, . \end{cases}
\chi_{({\cal V}_{m_{1}, n_{1}}\otimes{\cal V}_{m_{2}, n_{2}})_{\mathrm f}}( \tau ) = \sum_{m_{3}= | m_{1}- m_{2}| + 1}^{m_{1}+ m_{2}- 1}\sum_{n_{3}= | n_{1}- n_{2}| + 1}^{n_{1}+ n_{2}- 1}\chi_{m_{3}, n_{3}}( \tau ) \, ,
H = \frac{\sqrt{\alpha^{\prime}}}{2}g^{\mu \nu}( P_{\mu}- Q_{e}A_{\mu}) ( P_{\nu}- Q_{e}A_{\nu}) + ( \frac{m^{2}R^{2}}{2 \sqrt{\alpha^{\prime}}\alpha^{\prime}}+ \frac{\sqrt{\alpha^{\prime}}Q_{e}^{2}}{2 k^{2}}- \frac{Q_{e}\lambda R}{k}) ,
V_N(0,v,k)\ll \begin{cases}\frac{l^{1/2}(lT)^{\epsilon}}{N}\max{\left( \frac{\sqrt{T}}{k},\frac{1}{\sqrt{k}}\right)}4\pi elT\geq Nk, \frac{1}{\sqrt{lT}}\left(2\pi e\frac{lT}{Nk}\right)^k 4\pi elT<Nk. \end{cases}
+ \left[ 2i{\left(-1\right)}^{n-1}{\epsilon}_2 \cdot p_1 P^{n-1}\left(p_1^2,q^2 \right) \right]\frac{{\left(-1\right)}^n i}{q^{2n}-{\left(-m^2 \right)}^n}\left[ 2i{\left(-1\right)}^n{\epsilon}_1 \cdot p_2 P^{n-1}\left(q^2,p_2^2 \right) \right] +
\mathcal{R}=\left\{(x,y)~|~\langle J_Ax-J_Ay,x-y\rangle\geq \left(\tfrac{\delta}{2}+\tfrac{(1-\alpha-\delta/2)^2-\alpha^2+(\delta/2)^2}{2(1-\alpha-\delta/2)}\right)\|x-y\|^2\right\}.
\sigma^2:= \frac{1}{n_i}\sum_{j\in [m_i]}(f(x_{i,j})- \mu/n_i )^2 + \sum_{j\in [n_i]\setminus [m_i]}( 0 -\mu/n_i)^2 \leq \frac{1}{n_i}\left(w q_i( w^{-1}- \frac{\epsilon q_i}{n_i})^2 + ( n_i- w q_i)( \frac{\epsilon q_i}{n_i})^2 \right) &\leq \frac{q_i}{w n_i}- \frac{2\epsilon q_i^2}{n_i^2}+ \frac{\epsilon^2 q_i^2}{n_i^2}\leq \frac{2}{w n_i}.
\begin{array}{c}\mathbf{E}_{n}^{(k_1,k_2,...,k_r)}(a,b)=\sum\limits_{i=0}^{n}r^{n-i}(\ln a +\ln b)^i(\ln a)^{n-i}\binom{n}{i}\mathbf{E}_{i}^{(k_1,k_2,...,k_r)}\end{array}.
\mathcal{A}^{(2)}= \int_{0}^{1}\left[ (1+\gamma)(u_n^{\delta+1}-v_n^{\delta+1})(x)-\gamma(u_n-v_n)(x)-(u_n^{2\delta+1}-v_n^{2\delta+1})(x)\right]\psi_k'(w_n(x))\d x.
\displaystyle\sum_{\substack{U \pmod{D}(U,D)=1}}\psi (U) N_2(H, M, N, D,S,U) = \frac{\varphi(H)^2 \psi(S)}{\varphi(M)\varphi(N)}\sum_{\substack{J\mid M D\mid J}}\frac{\mu(J)}{\varphi(D)}.
(P_k)_j=\begin{bmatrix}U_0(\alpha_r) \overbrace{0 \ldots 0}^{(k-1)}\mu^{\frac{1}{2}}U_1(\alpha_r) \ldots \overbrace{0 \ldots 0}^{(k-1)}\mu^{\frac{n-k}{2k}}U_{\frac{n-k}{k}}(\alpha_r) \overbrace{0 \ldots 0}^{(k-1)}\end{bmatrix}^T,
S_{g a u g e d}= S_{L R}+ \frac{1}{2 \pi}\int d^{2}z \; \left( \frac{1}{2}A \bar{A}( 1 + B ( x ) \sin 2 \alpha ) + \frac{1}{\sqrt{2}}A \bar{L}_{R}+ \frac{1}{\sqrt{2}}L_{L}\bar{A}\right) ,
U\bigl(A_{i+n}^j (b,c)\bigr) = \frac{p^{(j-i)j}\chi(h)^j}{(j)!_{qp}(i+n-j)!_{qp}}U\bigl(U^j(b)\alpha^j\bigl(U^{i+n-j}(c)\bigr)\bigr) & = \frac{p^{(j-i)j}\chi(h)^j}{(j)!_{qp}(i+n-j)!_{qp}}\Bigl(q^j U^j(b)\alpha^j\bigl(U^{j+1+n-j}(c)\bigr) +p^{j-i}\chi(h) U^{j+1}(b)\alpha^{j+1}\bigl(U^{i+n-j}(c)\bigr)\Bigr).
\chi : \{( V_{i_{0}}, E_{i_{1}}, F_{i_{2}}) \, | \, i_{k}= 1 , \cdots , M_{k}( k = 0 , 1 , 2 ) , \, E_{i_{1}}\cap F_{i_{2}}= E_{i_{1}}, \, V_{i_{0}}\cap E_{i_{1}}= V_{i_{0}}\}\longrightarrow X
R_{\mathbf{a},\mathbf{b}}(\tau)=\begin{cases}\sum_{k=0}^{M-1-\tau}a_{k}b_{k+\tau}^{*}, 0\leq \tau\leq M-1, \sum_{k=0}^{M-1+\tau}a_{k-\tau}b_{k}^{*}, 1-M\leq \tau\leq -1, 0, \mid \tau \mid \geq M.\end{cases}
{\cal L}= ( D_{\mu}\phi )^{*}( D^{\mu}\phi ) - m^{2}\phi^{*}\phi - \frac \lambda 4 ( \phi^{*}\phi )^{2}+ \frac \Theta 2 \epsilon_{\sigma \mu \nu}A^{\sigma}\partial^{\mu}A^{\nu}- \frac \xi 2 ( \partial_{i}A^{i})^{2},
t_1=\frac{2\cdot \frac{\frac{p}{2}t_0}{\frac{p}{2}+t_0}}{2-\frac{\frac{p}{2}t_0}{\frac{p}{2}+t_0}}=\frac{p}{p-(\frac{p}{2}-2)t_0}t_0>t_0,
\begin{array}{l}{{\delta \xi^{i}= \left({\frac{d z}{d t}}+ i z \right) C^{i j}\bar{\epsilon}_{j}+ i \overline{{{W^{\prime}}}}( \bar{z}) \varepsilon^{i j}\epsilon_{j}}}\\{{\delta z = i \epsilon_{i}\xi^{i}}}\\ \end{array}
- R_{0}^{2}\cos^{2}\tau f^{i}( \tau , \sigma ) = \sqrt{\frac{\alpha^{\prime}}{2}}\sum_{n = - \infty}^{\infty}f_{n}^{i}( \tau ) e^{- i n \sigma}\; \; ; \; f_{- n}^{i}= f_{n}^{i *}
\int_{0}^{\infty}\frac{\rho d\rho}{(1+2(\rho^{2}+r^{2})\lambda(t)^{2\alpha-2}+(\rho^{2}-r^{2})^{2}\lambda(t)^{4\alpha-4})}\leq \int_{0}^{\lambda(t)^{1-\alpha}}\rho d\rho + \int_{\lambda(t)^{1-\alpha}}^{\infty}\frac{d\rho}{\rho^{3}\lambda(t)^{4\alpha-4}}&\leq C \lambda(t)^{2-2\alpha}
cn[\frac{1}{3}K(\frac{1}{2});\frac{1}{2}]=sech[\frac{1}{2}ln(\sqrt{2\sqrt{(\frac{4}{3})^{4/3}-(\frac{4}{3})^{2/3}+1}-(\frac{4}{3})^{2/3}+2}+\sqrt{(\frac{4}{3})^{2/3}+1})]
[y^{\nu}]\left(\frac{1-x}{1-x\xi}\right)^y =\frac{1}{\nu!}\left(\frac{1}{1-\eta}\right)^{\nu}=\sum_{\ell\ge\nu}\frac{s(\ell,\nu)}{\ell!}\left(\frac{x(\xi-1)}{1-x}\right)^{\ell}.
u\left( t\right) = P_p(T-t) \Phi+ \int_{t}^{T}P_p (r-t)\left( \nabla u\left( r\right) b\left( r\right)\right) dr +\int_{t}^{T}P_p\left( r-t\right) f \left(r, u(r) , \nabla u (r) \right)dr,
|A(\nabla\operatorname{grad}f)|^2-|A(\nabla\operatorname{grad}f)|\big(|A||\Delta f|+|A|^3f+m|H||d\phi||f|\big) &\geq-\frac{1}{4}\big(|A||\Delta f|+|A|^3|f|+m|H||d\phi||f|\big)^2 &=-\frac{3}{4}|A|^2|\Delta f|^2-\frac{3}{4}|A|^6f^2-\frac{3}{4}m^3H^2f^2,
\dim\left(\bigcap_{i=1}^t \Im(H_{[n'-k], A_{i}})\right) =\max_{P_1 \sqcup P_2 \sqcup \dots \sqcup P_s=[t]}\left(\sum_{i=1}^s\Big|\bigcap_{j \in P_i}A_j\Big|-(s-1) (n'-k)\right).
\begin{cases}\partial_t u_m(x,t)+P^-\big(D^2u_m(x,t)\big)+\big\langle \mu,Du_m(x,t)\big\rangle-ru_m(x,t)=0, \partial_t u_m(x,t)+P^+\big(D^2u_m(x,t)\big)+\big\langle \mu,Du_m(x,t)\big\rangle-ru_m(x,t)=0, \end{cases}
\begin{aligned}x_i^2-(\mu_i+K_i)x_i+\underbrace{\mu_iK_i+\frac{a_iK_i}{a_jK_j}\frac{\rho_ir_j}{\rho_jr_i}\frac{(a_j-d_j)}{(a_i-d_i)}(x_j-\mu_j)(x_j-K_j)}_{\phi_i(x_j)}= 0\end{aligned}
\lim_{t\to\infty}\frac{\mathcal{T}^{T(2,2a+1)^{(2,2b+1)}}(\Phi^{\rm{NN}}_{u,\omega_1,\omega_3})}{(t-1)^3\frac{d\,v(u)}{d\,u}}= -2(2b+1) \left( \frac{(2a+1)}{(\omega_1-\omega_1^{-1})}\right)^2.
I_{2}(\mu_{1},\mu_{2},\mu_{3},\mu_{4}) = \int_0^Le(\left\langle \mu_{1}+\mu_{2}+\mu_{3}+\mu_{4},\gamma(t)\right\rangle)\mu_{1}^{t}\dot{\gamma}(t)\mu_{2}^{t}\dot{\gamma}(t)\mu_{3}^{t}\dot{\gamma}(t)\mu_{4}^{t}\dot{\gamma}(t)dt.
L:=\left[\begin{matrix}\xi_1^{h-1}\xi_2^{h-1}\ldots \xi_h^{h-1}\xi_1^{h-2}\xi_2^{h-2}\ldots \xi_h^{h-2}\vdots \vdots \ddots \vdots \xi_1 \xi_2 \ldots \xi_h 1 1 \ldots 1 \end{matrix}\right].
|A_{I}\backslash\bigcup_{i\in[n]\backslash I}A_{I\cup\{i\}}|=\sum_{K\subset[n]\backslash I}(-1)^{|K|}|\bigcap_{i\in K}A_{I\cup\{i\}}|=\sum_{K\subset[n]\backslash I}(-1)^{|K|}|A_{K\cup I}|.
\left[ B_{\vec{k}}^b,\exp \sum\limits_aC_3\left(\left|\vec{p}\right| \right) B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}\right]=-C_3\left(\left| \vec{p}\right| \right) B_{\vec{p}}^{b+}\delta_{\vec{k},\vec{p}}\exp \sum\limits_aC_3\left(\left| \vec{p}\right|\right) B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}.
\psi(t) =-\gamma + (t-1) \sum_{n=0}^{\infty}\frac{1}{(1+n)(n+t)},t>0 \psi_{p,q}(t) = \ln[p]_q + (\ln q)\sum_{n=1}^{p}\frac{q^{nt}}{1-q^{n}}, t>0.
\frac{|x(t)|}{|y(t)|}= \frac{|x(T_{2k+2})+\delta(t-T_{2k+2})y(T_{2k+2})|}{|y(T_{2k+2})|}\ge \frac{|x(T_{2k+2})|}{|y(T_{2k+2})|}= = e^{-(T_{2k+2}-T_{2k+1})}\frac{|x(T_{2k+1})|}{|y(T_{2k+1})|}\stackrel{\eqref{rel1}}{\ge}e^{-(T_{2k+2}-T_{2k+1})}\delta(T_{2k+1}-T_{2k})\stackrel{\eqref{rel2}}{\ge}\frac{\delta}{\varepsilon}\,.
T(\lambda^-,\lambda^+) =\prod_{d^- \in B(\lambda^-)}|C_{d^-}\wr \Sigma_{E(\lambda^-,d^-)}| \prod_{d^- \in B(\lambda^-)}|C_{d^-}\wr \Sigma_{E(\lambda^-,d^-)}| U(\lambda^-,\lambda^+) =\prod_{d^- \in B(\lambda^-)}(q^{d^-}-1)^{E(\lambda^-,d^-)}\prod_{d^+ \in B(\lambda^+)}(q^{d^+}+1)^{E(\lambda^+,d^+)}
\varphi(a,b)\equiv \int_u^v \frac{(\gamma bx,qx/u,qx/v;q)_{\infty}}{(bx,cx,dx;q)_{\infty}}{}_1\phi_2\left(\begin{gathered}\gamma q^{\alpha+1}, \gamma bx \end{gathered};\,q, -q^{\alpha+1}ax \right) d_qx.
\frac{d}{dr}\left(r^{1-n}\cdot \mathcal{F}_S(\{E_j\},B_r) \right) = \frac{d}{dr}\sum^2_{j=0}\alpha_j \int_{B_r\cap\partial^*\!E_j}\frac{(\nu_{E_j}(x)\cdot x)^2}{|x|^{n+1}}\, d\mathcal{H}^{n-1}(x).
T(n_i) =\sum_{t=0}^{k_i-1}\frac{1}{C^t(n_i)}< \sum_{t=0}^{k_i-1}\frac{1}{n_i \cdot \left(\tfrac{3}{2}\right)^t}= \frac{1}{n_i}\cdot \frac{1-\left(\frac{2}{3}\right)^{k_i}}{1-\frac{2}{3}}= \frac{1}{n_i}\cdot \left(3 - 3 \cdot \left(\frac{2}{3}\right)^{k_i}\right)
J(s,x)=e^{\phi'(s) x}\left [\hat{m}(s,x,s+\epsilon,\phi'(s+\epsilon))e^{-\frac{1}{2}\int_{s}^{s+\epsilon}\phi'(u)^2 du}\right.+ \left. \int_{s+\epsilon}^{\infty}e^{-\frac{1}{2}\int_{s}^{u}\phi'(\omega)^2 d\omega}\Upsilon(s,x,u)du \right]
y_j=d\rho(X_{r_{j_1},1})^{k_1-1}d\rho(X_{r_{j_2},2})^{k_2}\cdots d\rho(X_{r_{j_m},m})^{k_m}d\rho(Y_{1})^{l_1}d\rho(Y_2)^{l_2}\cdots d\rho(Y_n)^{l_n}y
C^2 = d^2 - \sum_{p \in S}m_p(\bar{C})^2 = d^2 - \sum_{i=1}^s m_{q_i}(\bar{C})^2 -t &\geq d - \sum_{i=1}^s m_{q_i}(\bar{C}) -t \geq d(1-s) -t \geq -d(s+t) \geq -nd,
\overset{g}{R}(X,Y)AZ = A\overset{g}{R}(X,Y)Z, \overset{g}{R}(X,Y,AZ,AW) = \overset{g}{R}(AX,AY,Z,W), \overset{g}{R}(X,AY,AZ,W) = \overset{g}{R}(AX,Y,Z,AW), \overset{g}{R}(AX,AY,AZ,AW) = \overset{g}{R}(X,Y,Z,W), \overset{g}{R}(X,Y,Z,AW) = - \overset{g}{R}(X,Y,AZ,W).
\mathcal{E}_1=y^{i}\frac{\partial}{\partial y^{i}}\in\Gamma(V_1)\,,\,\mathcal{E}_2=p_i\frac{\partial}{\partial p_i}\in\Gamma(V_2)\,,\,\mathcal{E}=\mathcal{E}_1+\mathcal{E}_2\in\Gamma(V).
\lambda_n= 2(1-\cos\phi_n),\ \ \ \ \n=0,1,2,\cdots,N-1 u_{0m}= \frac{1}{\sqrt{N}}, \ \ \ \ \m=0,1,\cdots N-1 u_{nm}= \sqrt{\frac{2}{N}}\cos(m+1/2\phi_n), \ \ \ \ n=1,2,\cdots,N-1,m=0,1,\cdots N-1
M^a_{0c}M^c_{0b}=(M_0^2)^a_b=\left(\begin{array}{cc}-\frac{\alpha'}{2}[A_{\mu},A_{\nu}]\psi^{\mu}\psi^{\nu}+\overline{T}T -i\sqrt{\alpha'}(\overline{T}A'_{\mu}-A_{\mu}\overline{T})\psi^{\mu}-i\sqrt{\alpha'}(TA_{\mu}-A'_{\mu}T)\psi^{\mu}-\frac{\alpha'}{2}[A_{\mu}',A'_{\nu}]\psi^{\mu}\psi^{\nu}+T\overline{T}\end{array}\right) \ .
\left. \int \, e^{i q A_{b}}\, G(A) \, d \mu_{I_{\beta}}\, (A) = e^{- \tilde{E}_{\beta}(a,q)}\, \int \, e^{i q A_{b}}\, \left[ \prod_{p : b \in \partial p}\, i_{\beta}\, (\epsilon (b,p) a , dA ) \right] \, G(A) \, d \mu_{I_{\beta}}\, (A) \ , \right.
\sum_{\substack{\mathbf{r}\in\mathcal{D}_k r_i/(r_i,p)=u_i\forall i (r_i,W_i)=1}}\frac{S_{p}(\mathbf{r},\mathbf{s},p)}{\phi_\omega(r)}=\frac{\mu((s_m,p))\phi((s_m,p))}{\phi_\omega(u)}\Bigl(1+\frac{\omega(p)-1}{p-\omega(p)}-\frac{p-1}{p-\omega(p)}\Bigr)=0.
\{L_1,L_2\}=L_3-2\mu_3\mathcal{Q}+2\mu_1\mu_2, \{L_2,L_3\}=L_1-2\mu_1 \mathcal{Q}+2\mu_2\mu_3, \{L_3,L_1\}=L_2-2\mu_2 \mathcal{Q}+2\mu_1\mu_3.
H(Y^a, p_a) = \left(p_a\dot{Y}^a - L\right)\vert_{\eqref{momenta_pA},\eqref{momenta_g}}+ v_gp_g &= \frac{1}{2m}\eta^{AB}p_Ap_B + g\eta^{AB}p_Ax_B + \xi g + v_gp_g
\widetilde J_{p,f}(u)=\frac{1}{2}\, \int_{\mathbb{R}^N}|\nabla u|^2\, dx+\frac{c_2^{2-q}}{q}\, \int_{\mathbb{R}^N}|u|^{q}\, dx-\int_{\mathbb{R}^N}f\,u\, dx,
\omega_{i j}+ \tau_{i \, l}^{1}\omega^{l \, k}\tau_{j k}^{1}= 0 \, , \qquad \nabla_{[ i}\tau_{j ] k}^{1}+ 2 \tau_{[ i l k}^{2}\omega^{l m}\tau_{j ] m}^{1}= 0 \, ,
\begin{aligned}0 =\int_{\mathbb{R}^N}\left( \nabla\bar{\phi}(y)\cdot \nabla{\bar\psi}( y)+ \bar{\phi}(y){\bar\psi}( y) - p \left(U_0(y)\right)^{p-1}\bar{\phi}(y){\bar\psi}(y)\right)dy.\end{aligned}
(T^T \tilde F)^T \Lambda^{-1}(T^T \tilde F)= \begin{bmatrix}\tilde F_n \tilde F_{\tau}- \tilde F_n/u_n\end{bmatrix}^T\begin{bmatrix}-1/u_n 0 0 0 -1/u_n 0 0 0 u_n\end{bmatrix}\begin{bmatrix}\tilde F_n \tilde F_{\tau}- \tilde F_n/u_n\end{bmatrix}=-\tilde F_{\tau}^2/u_n
\lim_{n \to \infty}S_n\left( \lambda \right)=0 \Longleftrightarrow \lim_{n \to \infty}\left| S_n\left( \lambda \right) \right|^2=0 \Longleftrightarrow \lim_{n \to \infty}|p_0-\lambda q_0|^2\prod_{k=1}^n \left| 1-\frac{c_k}{\lambda}\right|^2=0.
g_1 = \frac 12 -12q_4^2+12q_4^4-48q_4^6+12q_4^8-72q_4^{10}+48q_4^{12}-96q_4^{14}+\cdots, g_2 = -8q_4^2-32q_4^6-48q_4^{10}-64q_4^{14}-104q_4^{18}-96q_4^{22}-112q_4^{26}+ \cdots,
\textbf{r}_{k}=\textbf{b}-\textit{A}\textbf{x}_{k}\bot\textit{G}_{k}(\textit{A}^{T}, \textbf{y})=\texttt{span}(\textbf{y}, \textit{A}^{T}\textbf{y},\ldots,\textit({A}^{T})^{k-1}\textbf{y}),
\begin{cases}|\phi(x_1-1)|\leq e^{30\varepsilon q_n}\max\{e^{-((\ell-1)q_n-x_1)L}r_{\ell-1}^-, e^{-(x_1-(\ell-2)q_n)L}r_{\ell-2}^+\}|\phi(x_2+1)|\leq e^{30\varepsilon q_n}\max\{e^{-((\ell+1)q_n-x_2)L}r_{\ell+1}^-, e^{-(x_2-\ell q_n)L}r_{\ell}^+\}\end{cases}
L_{s o f t}= \frac{1}{2}\sum_{\alpha}M_{\alpha}{\tilde{\lambda}}{\tilde{\lambda}}- \sum_{i}m_{i}^{2}|{\tilde{\phi}}|^{2}- ( A_{i j k}{\tilde{Y}}_{i j k}{\tilde{\phi}}_{i}{\tilde{\phi}}_{j}{\tilde{\phi}}_{k}+ B_{i j}{\tilde{\mu}}_{i j}{\tilde{\phi}}_{i}{\tilde{\phi}}_{j}+ h . c ) ,
\frac{1}{p_{s}^{2}}{\cal I}_{\mu_{1}\cdots \mu_{s}, \alpha_{1}\cdots \alpha_{s}}^{( s )}( x ){\cal I}_{\alpha_{1}\cdots \alpha_{s}, \nu_{1}\cdots \nu_{s}}^{( s )}( x ) = \Im_{\mu_{1}\cdots \mu_{s}, \nu_{1}\cdots \nu_{s}}^{( s )},
\rho_{\tau}(k)=\frac{E\left[ \psi_{\tau_{1}}(y_{1t}-q_{1,t}(\tau_{1}))\psi_{\tau_{2}}(y_{2,t-k}-q_{2,t-k}(\tau_{2}))\right]}{\sqrt{E\left[\psi_{\tau_{1}}^{2}(y_{1t}-q_{1,t}(\tau_{1}))\right]}\sqrt{E\left[\psi_{\tau_{2}}^{2}(y_{2,t-k}-q_{2,t-k}(\tau_{2}))\right]}},
L_{2,n}h_n=\int_{E_n}\left(L_{1,n}f_n+\tfrac{1}{\theta_n}L_{0,n}f_n\right)(\cdot,y)\pi_n(dy)-\left(L_{1,n}f_n+\tfrac{1}{\theta_n}L_{0,n}f_n\right)\,.
\frac{d^2}{d y^2}\begin{pmatrix}\epsilon_A(y) \epsilon_B(y)\end{pmatrix}\approx\begin{pmatrix}-\partial_AH_A/d_A \partial_BH_A/d_A \partial_AH_B/d_B -\partial_BH_B/d_B\end{pmatrix}\begin{pmatrix}\epsilon_A(y) \epsilon_B(y)\end{pmatrix},
(\L_2-\lambda_2)\, \phi_1 = C_1 \bigl(\lambda_2^{(k+1)}- \lambda_2^{(k)}\bigr) \phi_0^{(k+1)}+ C_2 \bigl(\lambda_2^{(k-1)}- \lambda_2^{(k)}\bigr) \phi_0^{(k-1)}& = c_{11}\, \frac{\sqrt{k}\, \phi_0^{(k+1)}+ \sqrt{k-1}\, \phi_0^{(k-1)}}{(2\gamma)^{\frac12}},
S=\int d^{2}x\left\{{\frac 1 2}{\partial}_{\mu}{\phi}{\partial}^{\mu}{\phi}+\frac{1}{2({\partial}_{\nu}a)({\partial}^{\nu}a)}\left[{\partial}^{\mu}a({\partial}_{\mu}{\phi}-{\epsilon}_{{\mu}{\sigma}}{\partial}^{\sigma}{\phi})\right]^2\right\},
C(x \mid p_{[2..\ell]}) \stackrel{(a)}{\leq}\log(\delta_1 \cdot |B|) + O(\log n) &\stackrel{(b)}{\leq}A(x_1 \mid p_{[2..\ell]})+ 8 \gamma_2 - \log(1/\delta_1) + O(\log n) & \stackrel{(c)}{\leq}C(x_1 \mid p_{[2..\ell]}) + 16 \gamma_2 - \log(1/\delta_1) + O(\log n) & \stackrel{(d)}{<}C(x_1 \mid p_{[2..\ell]}).
\|u_i^{(\tau)}\|_{L^p(Q_T)}^p = \int_0^T\|u_i^{(\tau)}\|_{L^p(\Omega)}^p dx\le C\int_0^T\|u_i^{(\tau)}\|_{H^1(\Omega)}^{\theta p}\|u_i^{(\tau)}\|_{L^1(\Omega)}^{(1-\theta)p}dt &\le C\|u_i^{(\tau)}\|_{L^\infty(0,T;L^1(\Omega)}^{(1-\theta)p}\int_0^T\|u_i^{(\tau)}\|_{H^1(\Omega)}^2 dt \le C(u^0,b,T).
P_n(\boldsymbol\nu)=\frac{N}{\prod_{\ell}\ell^{\nu_{\ell}}\nu_{\ell}!}\int_0^1\prod_{\ell\ge 1}\bigl[t^{\ell}+(-1)^{\ell+1}(1-t)^{\ell}\bigr]^{\nu_{\ell}}\,dt.