harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
z ( \sigma , \bar{\sigma}, \varphi ) = - \frac{\sqrt{2}\bar{\sigma}}{c_{0}}+ \frac{4}{c_{0}}\cos ( \varphi / 2 ) \, \, , \quad \bar{z}( \sigma , \bar{\sigma}, \varphi ) = \frac{\sigma}{\sqrt{2}}+ \frac{4}{c_{0}}\cos ( \varphi / 2 )
\int \dfrac{x^{\alpha+1}}{(- x^2 (1-q)^2;q^2)_\infty}\left(q+\frac{1-q^{1-\nu}[\nu]^2}{qx^2}\right) J_\nu^{(2)}( x |q^2) d_qx = & \frac{x^{\alpha+1}}{q^{\alpha}(- x^2 (1-q)^2;q^2)_\infty}\left( \frac{q^{1-\nu}(1-q)[\nu]^2-1}{x}J_\nu^{(2)}( x |q^2)- D_{q^{-1}}J_\nu^{(2)}( x |q^2) \right).
\int_{0}^{\infty}e^{-s x}\hat{F}_2(x)dx =\frac{1}{s}\left(\frac{\mathbb E\left[e^{s \underline{X}_{e(p+q)}}\right]}{\mathbb E\left[e^{s \underline{X}_{e(q)}}\right]}-1\right),\ \s> 0.
\left[\mbox{\boldmath$ \beta$}\right]_{j,k}:=\left( \left. \beta_{\operatorname*{D};j}\right\vert_{\Gamma_{j,k}}-\left. \beta_{\operatorname*{D};k}\right\vert_{\Gamma_{j,k}},-\left. \beta_{\operatorname*{N};j}\right\vert_{\Gamma_{j,k}}-\left.\beta_{\operatorname*{N};k}\right\vert_{\Gamma_{j,k}}\right)
U_{t}^{H_{1}\circ S_{t}}\psi(x)=\left( \frac{1}{2\pi i\hbar}\right)^{1/2}\frac{i^{m}}{\sqrt{|\sin t-t\cos t|}}\int_{-\infty}^{\infty}e^{\frac{i}{\hbar}W(x,x^{\prime})}\psi(x^{\prime})dx^{\prime}
|\Omega|=(\begin{matrix}p^{k}m\\p^{k}\end{matrix})=\prod_{j=0}^{p^{k}-1}\frac{p^{k}m-j}{p^{k}-j}=m\prod_{j=1}^{p^{k}-1}\frac{p^{k-\nu_{p}(j)}m-j/p^{\nu_{p}(j)}}{p^{k-\nu_{p}(j)}-j/p^{\nu_{p}(j)}}
\lim_{n \to \infty}n^{3-d(r,s)}E(S_{\lfloor nt \rfloor}(r)\overline{S_{\lfloor nu \rfloor}(s)}) = \frac{\sigma(r,s)}{(3-d(r,s))(2-d(r,s))}\left[\overline{c(s,r)}t^{3-d(r,s)}+c(r,s)u^{3-d(r,s)}+C(r,s;t-u)|t-u|^{3-d(r,s)}\right].
\|\hat{p}^{(k)}(\cdot|Z^n)-\mu_k^{(b)}\|_1 \leq{1\over k+g}\sum_{i=1}^{k+g}\|\hat{p}^{(1)}(\cdot|S^{(i),t})-\mu_k^{(b)}(\cdot)\|_1+1-{t(k+g)\over n-k}+{k+g\over n-k}&={1\over k+g}\sum_{i=1}^{k+g}\|\hat{p}^{(1)}(\cdot|S^{(i),t})-\mu_k^{(b)}(\cdot)\|_1+{n-t(k+g)+g\over n-k}.
\deg_w(r_{j,1}) \leq \deg_w\Big(\sum_{i=1}^k q_{ji}\cdot g_i\Big), \deg_w(r_{j,2}) \leq \deg_w(a_j \cdot g_j'') \leq \deg_w\Big(a_j \cdot \sum_{i=1}^k q_{ji}\cdot g_i\Big).
\begin{cases}\partial_{t}u-\Delta u=-\nabla\cdot(u\nabla \psi) \mbox{in}\ \ \mathbb{R}^n\times(0,\infty), \partial_{t}\psi-\Delta \psi=u-\psi \mbox{in}\ \ \mathbb{R}^n\times(0,\infty), u(x,0)=u_0(x), \ \ \psi(x,0) =\psi_{0}(x) \mbox{in}\ \ \mathbb{R}^n.\end{cases}
T_i^{1}|_\mathcal{P}= \mathsf{AddDots}_{\gamma_{i+1}}(T_i^{0})|_\mathcal{P}= \mathsf{AddDots}_{\theta_{i+1}}(T_i^{0})|_\mathcal{P}&= \mathsf{AddDots}_{\theta_{i+1}}(T_i^{0}|_\mathcal{P})= \mathsf{AddDots}_{\theta_{i+1}}(C_i^{0}|_\mathcal{P}) &= C_i^1|_\mathcal{P},
\left. \begin{array}{llll}Tr\,[\,Q^{3}_{0}\,]\,= 4 \, \,Tr\,[\,Q^{3}_{1}\,]\,= 4 \, \, Tr\,[\,Q^{3}_{2}\,]\, = -8 \, \,Tr\,[\,Q^{3}_{3}\,]\, = 0\end{array}\right.
P_{{\underline X}_1}( 11000) = P_{{\underline X}_1}( 00010) = \frac{2}{10}, \P_{{\underline X}_1}( 00001) = \frac{3}{10}, P_{{\underline X}_1}( 10100) = P_{{\underline X}_1}( 01100) = P_{{\underline X}_1}( 00110) = \frac{1}{10}.
\frac{1}{2}\int \prod_{i=1}^4 dg_i ~ ~~\phi^2(g_1,\ldots,g_4) = \frac{1}{2}\sum_{N_1\ldots N_4}\sum_\Lambda \phi_{\cal A}^{N_1\ldots N_4\,\Lambda}\phi_{\cal A}^{N_1\ldots N_4\, \Lambda}.
\theta (t_1-t_2)\left[ \begin{array}{cc}0 \frac{\theta (-n-\frac 12)t_1^n}{t_2^{n+1}}\frac{\theta (n+\frac 12)t_2^n}{t_1^{n+1}}0\end{array}\right] -\theta (t_2-t_1)\left[ \begin{array}{cc}0 \frac{\theta (n+\frac 12)t_1^n}{t_2^{n+1}}\frac{\theta (-n-\frac 12)t_2^n}{t_1^{n+1}}0\end{array}\right]
\gamma_{[i,i]}= -2 (w-2i+1) \beta_{[i-1,i]}+ (i+1) \beta_{[i,i+1]}& -2(w-2i+1) \alpha_{[i,i-1]}+ (i+1) \alpha_{[1+i,i]}= (i+1)( \beta_{[i,i+1]}+ \alpha_{[1+i,i]}) -2 (w-2i+1) ( \beta_{[i-1,i]}+ \alpha_{[i,i-1]}) \;.
p'(\alpha) =p'_1p'_2\cdots p'_{R-k}p'_{R-k+1}=(p_1-1)p_2\cdots p_{R-k}(p_{R-k+1}-1)\hbox{and}q'(\alpha) =q'_0q'_1\cdots,\hbox{where}q'_j=|\{i: \p'_i=j\}|.
[a_{m}, \tilde{a}^{\dagger}_{n}] = [\tilde{a}_{m}, a^{\dagger}_{n}] = \delta_{mn}\mid 0 \rangle \langle 0 \mid , \;\;\;\;\;\; [a_{m}, \tilde{a}_{n}] = [a^{\dagger}_{m}, \tilde{a}^{\dagger}_{n}] = 0
\sum_{(i_1,k_1)\in P_1I_0}2^{k_1r}d_{i_1}^r\Pr(B_{i_1,k_1}) \leq \sum_{i_1=1}^nd_{i_1}^r\sum_{k_1=0}^\infty 2^{k_1r}e^{2-2^{k_1-1}t/e}\leq \sum_{i_1=1}^n d_{i_1}^r2e^{2-t/{(}2e{)}}\leq \left(\frac{Cr\sigma_{r,X}(rl)}{t}\right)^r,
t_{\mu \nu}^{( \pm )}( p , k ) \; = \; \sum_{n = - \infty}^{+ \infty}\, \mathrm{t r}\left[ \frac{1}{i \not \! p + \omega_{n}^{( \pm )}}\gamma_{\mu}\frac{1}{i ( \not \! p + \not \! k ) + \omega_{n}^{( \pm )}}\gamma_{\nu}\right]
\iota \left( D_1 \otimes \cdots \otimes D_n \right) = \iota \left( M_1 \otimes \cdots \otimes M_n \right)= \sum_{i=1}^n \iota(D_i) + \sum_{1 \leq j < k \leq n}m \left( h_k, \partial h_j \right).
B_\Omega(\gamma_n^{-1}z_0, z_0) + B_\Omega(z_0, y_n) - B_\Omega(\gamma_n^{-1}z_0, y_n) & = B_\Omega(\gamma_n^{-1}z_0, z_0) + B_\Omega(z_0, y_n) - B_\Omega(z_0, \gamma_n y_n) & \geq B_\Omega(\gamma_n^{-1}z_0, z_0) - B_\Omega(\gamma_n y_n, y_n) \rightarrow \infty.
g_{0}= 2 \left\{4 \ln{( 1 + r^{2})}- 5 + r^{2}+{\frac{r^{2}- 1}{r^{2}+ 1}}\left[ 2 \mathrm{d i l o g}( 1 + r^{2}) + \ln{r^{2}}\right] \right\},
\sum_{i}^{4}c_{i}|\begin{matrix}p_{v}-p_{u_{1}}&(p_{v}-p_{u_{1}})\cdot\lambda_{i,u_{1}}\\p_{v}-p_{u_{2}}&(p_{v}-p_{u_{2}})\cdot\lambda_{i,u_{2}}\\p_{v}-p_{u_{3}}&(p_{v}-p_{u_{3}})\cdot\lambda_{i,u_{3}}\end{matrix}|=0.
e^{M ( t - t_{i})}= \left( \begin{matrix}{{c o s ( \omega{}( t - t_{i}) )}}&{{s e n ( \omega{}( t - t_{i}) )}}&{{0}}\\{{- s e n ( \omega{}( t - t_{i}) )}}&{{c o s ( \omega{}( t - t_{i}) )}}&{{0}}\\{{0}}&{{0}}&{{1}}\\ \end{matrix}\right) .
{{\bf{Y}}^{[2]}}(n) -{{\bf{Y}}^{[2]}}({t_1}) ={{\bf{H}}^{[21]}}(n){\bf{V}}_2^{[1]}(n){{\bf{v}}^{[1]}}+{{\bf{H}}^{[22]}}(n){\bf{V}}_2^{[2]}(n){{\bf{v}}^{[2]}}.
{Z}_{n}^{F_{m - 2}}= d ( n , m ) \left( \sum_{k = 1}^{n - 3}c ( m ; n , k ){Z}_{n - k}^{F_{m - 2}}{Z}_{k + 2}^{F_{m - 2}}+ \mathrm{h i g h e r ~ o r d e r ~ t e r m s}\right) , \qquad n > m + 1 ,
\left( \begin{array}{c c}{{a^{\prime \prime}}}&{{b^{\prime \prime}}}\\{{c^{\prime \prime}}}&{{d^{\prime \prime}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{a}}&{{b}}\\{{c}}&{{d}}\\ \end{array}\right) \left( \begin{array}{c c}{{a^{\prime}}}&{{b^{\prime}}}\\{{c^{\prime}}}&{{d^{\prime}}}\\ \end{array}\right)
C_{Q}= - 2 \pi r_{+}^{2}\left[ 1 - \frac{r_{-}}{r_{+}}\right]^{2 q^{2}/ ( 2 q^{2}+ 3 )}\left[ 1 - \frac{3}{2 q^{2}+ 3}\frac{r_{-}}{r_{+}}\right] \left[ 1 - \frac{2 q^{2}+ 9}{2 q^{2}+ 3}\frac{r_{-}}{r_{+}}\right]^{- 1}.
\frac{d}{dz}\bigl(z^{n+2-q}\frac{d}{dz}S_{n,q}(z)\bigr) = z^n \frac{P_{n,q}(z)}{(1-z)^{2n+1}}-\binom{n+q-1}{n}z^n =\sum_{k=0}^{n+1-q}a_{n+1,k+1}\binom{n+1-q}{k}\frac{z^{n+1-q-k}}{(1-z)^{2n+1-k}}- \binom{n+q-1}{n}z^n.
\tilde{G}(x)=\tilde{S}(x)-4\pi \alpha \hspace{-1mm}\int \hspace{-1mm}d^4 x_1 d^4 y_1 e^{ix A(x_1)+ix_1 A(y_1)}\tilde{S}(x-x_1)\gamma^{\mu}\tilde{G}(x_1-y_1)\gamma^{\nu}\tilde{G}(y_1){\cal D}_{\mu\nu}(x_1-y_1),
\int_{T_1}\int_{T_2}\int_{T_1}u\,d(m_{T_1}\ast m_{T_2}\ast m_{T_1})=\int_0^{2\pi}\int_0^{2\pi}\int_0^{2\pi}u(k_1(t_1)k_2(t_2)k_1(t_3))\,\frac{dt_3\,dt_2\,dt_1}{8\pi^3}
\begin{aligned}{1\over q}(\lambda h_q+(1-\lambda) k_q)+w_q(x_q, t_q)^{q-1}\mu(w_q(x_q, t_q)) &\geq -\lambda v(y_q, t_q)^{1-{1\over q}}\omega(|X_q|)-(1-\lambda)v(z_q, t_q)^{1-{1\over q}}\omega(|X_q|),\end{aligned}
\begin{aligned}\frac{d}{dt}\textrm{Ent}_\pi^\phi(\rho_t) = -{\mathcal E}(\phi'(\rho_t),\rho_t), \frac{d^2}{dt^2}\textrm{Ent}_\pi^\phi(\rho_t) = \pi[\L\phi'(\rho_t)\L\rho_t + \phi''(\rho_t)(\L\rho_t)^2\big],\end{aligned}
\lim_{s\rightarrow 0^+}\frac{v(s)\cos(\pi P_{v(s)}f(s))}{\sin(\pi P_{v(s)}f(s))}=\lim_{s\rightarrow 0^+}\frac{\cos(\pi P_{v(s)}f(s))}{\frac{1}{s}+\frac{\xi}{s}+\frac{1}{s}O(v(s)/s)}=0.
\mathcal{A}^+ = \left\{\begin{aligned}(\tau,v_{s+k+1},\omega_{k+1}) \in[0,\infty) \times \mathbb{R}^d \times \mathbb{S}^{d-1}\textnormal{such that}& \omega_{k+1}\cdot \left(v_{s+k+1}- v_{i_{k+1}}^\prime (\tau;0) \right) > 0\end{aligned}\right\}
\begin{array}{r c l}{i T}&{=}&{\displaystyle + \lambda v \int \left[ 3 D_{k}^{\sigma}+ D_{k}^{Z}+ 2 D_{k}^{W}\right]}\\{}&{}&{\displaystyle + \frac{n}{2}g^{2}v \int \left[ D_{k}^{W}+ D_{k}^{Z}\right]}\\{}&{}&{\displaystyle - \frac{1}{2}g^{2}v \int \left[ D_{k}^{W}+ D_{k}^{Z}\right] ,}\\ \end{array}
k_{i,j}= (x \otimes t^i) \wedge (x \otimes t^j) \otimes v_{-1}- ( (x \otimes t^i) \wedge (h \otimes t^j) \otimes v_1 + (h \otimes t^i) \wedge (x \otimes t^j) \otimes v_1), 1 \leq i < j.
(\det a)^\frac{1}{n}\geq \int_{0}^1\Big[\det \Big( [\det x(t)]^{\frac{1}{n}-1}\tilde{x}(t)\Big)\Big]^\frac{1}{n}dt &= \int_{0}^1[\det x(t)]^{\frac{1}{n}-1}[\det \tilde{x}(t) ]^\frac{1}{n}dt.
\frac{1}{\tau}\int_0^{\tau/2}|\Pi_{\tilde{V}}e^{|z| J^\perp t}\tilde{Y}_0|dt = \frac{1}{\tau}\int_0^{\tau/2}|\Pi_{\tilde{V}}\tilde{Y}_0 + t|z| |\Pi_V \tilde{Y}_0| dt \gtrsim R |z|\tau^{3/2}.
\displaystyle \mathop{{\mathcal A}}^{\; \, \mathrm{o}}\ = \ \left\{a \in{\mathcal A}\bigm | \; \Big \| \ \frac 1 2 \, \left( \triangle a^{2}+ a^{2}\triangle \right) - \; a \triangle a \, \Big \| < \infty \right\}\ ,
\Theta(r,\zeta;\nu,\varepsilon)>0 \quad\mbox{for}0\leq r <\Xi_1(\zeta), &\Theta(\Xi_1(\zeta),\zeta;\nu,\varepsilon)=0, &\Theta(r,\zeta;\nu,\varepsilon)<0\quad\mbox{for}\Xi_1(\zeta)<r\leq 2\xi_1.
\langle s(u),s(v)\rangle =\langle s_0(u)+\lambda_u^0,s_0(v)+\lambda_v^0\rangle=\langle s_0(u),s_0(v)\rangle+\langle s_0(u),\lambda_v^0\rangle+\langle\lambda_u^0,s_0(v)\rangle &=\delta_u\cdot (\lambda_v+\lambda_v^0)-\delta_v\cdot(\lambda_u+\lambda_u^0)-\delta_u\cdot\lambda_v^0+\delta_v\cdot\lambda_u^0 &=\delta_u\cdot\lambda_v-\delta_v\cdot\lambda_u,
\sigma = \left( \frac 1 2 + \frac 1 2 \sqrt{1 + 4 \left( \frac{4 q^{2}}{3}\right)^{3}}\right)^{\frac 1 3}+ \left( \frac 1 2 - \frac 1 2 \sqrt{1 - 4 \left( \frac{4 q^{2}}{3}\right)^{3}}\right)^{\frac 1 3},
m^{\star}= \max \left\{m_{i}, \i = 1, 2, \ldots, n \right\}, \mu^{\star}= \max \left\{\mu_{i}, \i = 1, 2, \ldots, n \right\}, \mu_\star = \min \left\{\mu_{i}, \i = 1, 2, \ldots, n \right\}.
\begin{bmatrix}\lambda P_5-P_4 \lambda P_4 0 \lambda P_4 \lambda P_3+P_2 P_1 0 P_1 -\lambda P_1+P_0\end{bmatrix}\, \mbox{and}\,\,\begin{bmatrix}\lambda P_5+P_4 P_3 P_2 P_3 -\lambda P_3+P_2 -\lambda P_2+P_1 P_2 -\lambda P_2+P_1 -\lambda P_1+P_0\end{bmatrix}.
\begin{array}{c c}{\displaystyle{\langle \phi^{\alpha}( z_{1}) \phi^{\beta}( z_{2}) \rangle = C_{\phi}\frac{I^{\alpha \beta}( \hat{{\cal X}}_{1 2})}{( \det{\cal X}_{1 2})^{\frac{1}{2}\eta}}}}&{I^{\alpha \beta}( \hat{{\cal X}}_{1 2}) = \hat{{\cal X}}_{1 2}^{\alpha \beta}}\\ \end{array}
\textstyle\varphi_t( \overline{\mathcal G})=\{\varphi_t(m): m \in \overline{\mathcal G}\setminus \overline{\mathcal H}\}\cup\{m \frac{x_p}{x_q}+ tm : m \in \mathcal H\}\cup \{x_p^2+2tx_px_q+t^2x_q^2\}.
\Pi(d)(g)=\sigma_0(\widetilde{g})^{-\dot{h}_{\dot{\theta}}}\prod_{j=1}^r\sigma_j(\widetilde{g})^{\dot{h}_j}=\prod_{j=1}^r\left (\frac{\sigma_j(\widetilde{g})}{\sigma_0(\widetilde{g})^{\check{a}_j}}\right )^{\dot{h}_j},
\mathrm{\Gamma_{\m u}^{1}( p^{\prime}, p ) = \frac{\a l p h a_{s}}{2 4 \p i}\g a m m a_{\m u}\Gamma ( 2 - \o m e g a - \s i g m a ) , \o m e g a \to ( 3 / 2 )^{+}, \s i g m a \to ( 1 / 2 )^{+}.}
\sum_{\lambda = j}^{j + n -1}= ~ (-1)^j ~h ( a^{m+n-1}_{1, i-1}, ~ f^n, ~ a^{m+n-1}_{i+m, j-1},~ a_j^{m+n-2}\cdot g^{m-1}, ~ a_{j+n+1, m+n+p-1}^{m+n-1}) &~ + (-1)^{j+n-1}~ h ( a^{m+n-1}_{1, i-1}, ~ f^n, ~ a^{m+n-1}_{i+m, j-1},~ g^{m-1}\cdot a_{j+n}^{m+n-2}, ~ a_{j+n+1, m+n+p-1}^{m+n-1}).
\Xi_{\imath}=\begin{cases}\xi(\imath) \ 1\leq \imath\leq n-1, \frac{1}{2}(\xi(n-1)+\xi(n))+1 \ \imath=n\ \ \flat=>, \frac{1}{2}(\xi(n-1)+\xi(n)) \ \imath=n\ \ \flat=<, \xi(\imath -1) \n+1\leq \imath\leq 2n-1.\end{cases}
\begin{array}{l}\hat\lambda^{\iota_1}={1\over2}\left(\hat\lambda_{1_w4_w}-\hat\lambda_{2_w3_w}\right) begin{align*}2mm]\hat\lambda^{\iota_2}={1\over2}\left(\hat\lambda_{1_w3_w}+\hat\lambda_{2_w4_w}\right) begin{align*}2mm]\hat\lambda^{\iota_3}={1\over2}\left(\hat\lambda_{1_w2_w}-\hat\lambda_{3_w4_w}\right)\end{array}\hskip5.9cm
W(h,U)\varphi(g)W(h,U)^* =\varphi(Ug)-2\textup{Re}(\langle Ug,h \rangle) W(h,U)a(g)W(h,U)^* =a(Ug)-\langle Ug,h \rangle W(h,U)a^\dagger(g)W(h,U)^* =a^\dagger(Ug)-\langle h,Ug \rangle.
T(f^{+}g^{+}) =T((fg)^{+})\wedge((T(f)+T(f^{3}))^{+}+(T(g)+T(g^{3}))^{+}) & =(T(f)T(g))^{+}\wedge((T(f)+T(f)^{3})^{+}+(T(g)+T(g)^{3})^{+}) & =T(f)^{+}T(g)^{+}.
{\cal I}( \boldsymbol{x})= \sum_{j=1}^{d}\bar{R}^{N_j}_j( \boldsymbol{x}) W_j ( \boldsymbol{x}) = \sum_{j=1}^{d}\left( \sum_{k=1}^{N_j}( \sigma^{(j)}_k)^{-1}(f_{| \Omega_j},{\bar u}^{(j)}_k)_{\ell_2(X_{N_j})}{\bar u}^{(j)}_k (\boldsymbol{x}) \right) W_j(\boldsymbol{x}) , \boldsymbol{x}\in \Omega.
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}}.
\#\left(\{x\in \mathbb{Z}_q\mid s_{h}x=0\}\bigcup\{x_i,-x_i\mid 1\leq i\leq h-1\}\right) = \,\#\{x\in \mathbb{Z}_q\mid s_{h}x=0\}+\#\{x_i,-x_i\mid 1\leq i\leq h-1\}&\, -\#\left(\{x\in \mathbb{Z}_q\mid s_{h}x=0\}\bigcap\{x_i,-x_i\mid 1\leq i\leq h-1\}\right).
(-z\xi)^{n}q^{-\frac{1}{2}n(n+1)}\left(z^{-1}\xi;q\right)_{\infty}\!\left[\sum_{j=1}^{n}\frac{(qz\xi^{-1};q)_{j}}{(q;q)_{n-j}}z^{-2j}+\sum_{k=0}^{\infty}\frac{q^{k(k-1)/2}}{\left(z^{-1}\xi;q\right)_{k}(q;q)_{k+n}}(-z\xi)^{k}\right]\!\!.
\mathcal{K}:=\tilde{\mathcal{A}}\Big(\mathcal{F}-\mathcal{A}_0\partial_t{\mathbf V}-\mathcal{A}_2\partial_2{\mathbf V}-\mathcal{A}_3{\mathbf V}-\mathcal{A}_{(0)}\partial_1{\mathbf V}\Big),
\begin{array}{l}{{< \phi_{q}^{4}> = \frac 1{1 2 8 \pi^{3}C}\log \left[ 1 + \frac{\Lambda^{2}}{( m_{r}^{2}+ ( \xi_{r}- \frac 1 8 ) R - 1 2 \kappa_{r}\phi_{c}^{2}+ 1 5 \lambda_{r}^{2}\phi_{c}^{4}+ \frac Q C )}\right]}}\\ \end{array}
\sum_{k=0}^{(n-2)/2}\frac{2k\pi}{n-1}\sin\left(\frac{2k\pi}{n-1}\cdot b\right)=\frac{\pi\csc\left(\frac{b\pi}{2a+1}\right)\left(-2(a+1)+\sin\left(\frac{2b\pi(a+1)}{2a+1}\right)\csc\left(\frac{b\cdot\pi}{2a+1}\right)\right)}{4a+2}.
\{l\in I^G(\bar x,\bar y,\bar z)\cup I^{GH}(\bar x,\bar y,\bar z)\,|\,\bar y_l=0\}=\{l\in\mathcal Q\,|\,G_l(\bar x)=\bar y_l=0\}=I^{0+}(\bar x)\cup I^{00}(\bar x)
V \equiv \sum_{\mu \in{\mathcal R}}g_{| \rho |}( \rho \cdot \mu ) ( \rho^{\vee}\! \cdot \nu ) \frac{\ell_{\mu}^{j}- \ell_{s_{\rho}( \mu )}^{j}}{\ell_{\mu}- \ell_{s_{\rho}( \mu )}}\ell_{s_{\rho}( \nu )}^{k}\check{s}_{\rho}= 0 .
\gamma_{a_{2 k + 1}}^{( 1 )}\equiv \pi_{a_{2 k + 1}}- \pi_{a_{2 k + 1}}^{( 1 )}\approx 0 , \gamma_{a_{2 k + 1}}^{( 2 )}\equiv \pi_{a_{2 k + 1}}^{( 2 )}\approx 0 , k = 0 , \cdots , \Lambda ,
\mathcal{M}\left\lbrace{}^{\Psi}\Phi_{p}^{(\alpha,\beta)}(b;c;z):s\right\rbrace =\frac{{}^{\Psi}\Gamma^{(\alpha,\beta)}(s)B(b+s,c+s-b)}{B(b,c-b)}\Phi(b+s;c+2s;z).
\Delta \tilde{v}_1 =4\bigg(\dfrac{1}{2}k_{11}e^{u_0'+ \tilde{v}_1}+\dfrac{1}{2}k_{12}e^{u_0''+ \tilde{v}_2}-1\bigg)+g_0' \Delta \tilde{v}_2 =4\bigg(\dfrac{1}{2}k_{21}e^{u_0'+ \tilde{v}_1}+\dfrac{1}{2}k_{22}e^{u_0''+ \tilde{v}_2}-1\bigg)+g_0'',
\langle W ( C ) \rangle_{Y M}= e^{[ \cdots ]}\int d \mu_{C}( \xi ) \Biggr \langle \exp \left[ 2 i \rho^{- 1}K^{1 / 2}J g \int_{S_{C}}d S^{\mu \nu}( x ){\cal D}[ \partial_{x}] h_{\mu \nu}^{\xi}( x ) \right] \Biggr \rangle_{A P E G T},
f_{t}(x+1)=e^{it}f_{t}(x),(f,\chi_{k,t})_{(-\infty,\infty).}=(f_{t},\chi_{k,t})_{(0,1)}=\frac{1}{\alpha_{k}(t)}(f_{t},\Psi_{k,t}^{\ast})=:a_{k}(t).
\sum_{\alpha\in\Gamma}{\overline\chi_0}^\dagger{\gamma_\lambda}^\dagger\chi_{\alpha+\lambda}(\chi_0)e^{\bar\alpha\cdot\lambda\phi(1)/3}e^{\bar\alpha\cdot\phi(a)/\sqrt 3}=\sum_{\alpha\in\Gamma}{\overline\chi_0}^\dagger{\gamma_\lambda}^\dagger\chi_{-\alpha+\lambda}(\chi_0)e^{-\bar\alpha\cdot\phi(a)/\sqrt 3}\,.
f(x)\leq m2^{1-s}f(a)+\bigg(\frac{x-ma}{b-ma}\bigg)^s[f(b)-mf(a)]=p_{1}(x). g(x)\leq m2^{1-s}g(a)+\bigg(\frac{x-ma}{b-ma}\bigg)^s[g(b)-mg(a)]=p_{2}(x).
\sum_{n=0}^{\infty}p_{-(25r+k)}(n)q^{n}=\dfrac{1}{E_{1}^{25r+k}}\equiv\dfrac{1}{E_{1}^{k}E_{5}^{5r}}=\dfrac{1}{E_{5}^{5r}}\sum_{n=0}^{\infty}p_{-k}(n)q^{n}.
\textbf{t}^{\alpha_1}F_{\alpha_1}+\cdots+\textbf{t}^{\alpha_l}F_{\alpha_l}=\sum\limits_{j=1}^{r}G_j\left(x_j\textbf{t}^{\beta_j}g_j-f_j\textbf{z}^{\beta_j}\textbf{t}^{\beta_j}\right)+{\sum\limits_{i=1}^{s}H_i({y_i}^{q}-y_i)}+H(h w-1),
T_{H}^{\mathrm{B T Z}}= ( \rho_{+}^{2}- \rho_{-}^{2}) / 2 \pi R^{2}\rho_{-}= T_{H}, ~ ~{\frac{1}{T_{L / R}^{\mathrm{B T Z}}}}={\frac{1}{T_{H}^{\mathrm{B T Z}}}}\left( 1 \pm{\frac{\rho_{+}}{\rho_{-}}}\right) ={\frac{1}{T_{L / R}}}.
\tilde{X}^{(K)}_{jk}= \mathrm{diag}(\underbrace{\tilde{X}_{jk},\tilde{X}_{jk},\ldots, \tilde{X}_{jk}}_{K}), \tilde{X}^{(K)}_{jk}\in \mathcal{M}_{Kd_j,Kd_k}(\mathbb{C}).
\begin{array}{l}{{0 = k_{1}+ 3 k_{3}}}\\{{{}}}\\{{0 = k_{2}+ k_{4}+ k_{5}+ 4 k_{6}+ 4 k_{7}}}\\{{{}}}\\{{0 = k_{2}+ k_{3}+ 3 k_{4}+ k_{5}+ 4 k_{6}+ 1 2 k_{7}}}\\{{{}}}\\{{0 = 4 c_{2}- 6 c_{1}- k_{2}+ k_{3}+ k_{4}- k_{5}- 4 k_{6}+ 4 k_{7}}}\\ \end{array}
\begin{array}{rcl}Y_2 A_1 = L^+_2 SL^-_2 A_1 = L^+_2 A_1 SL^-_2 R_{21}= A_1 R_{12}L^+_2 SL^-_2 R_{21}= A_1 R_{12}Y_2 R_{21},\end{array}
\frac{8\mathcal{N}\ \delta (k_{j}+1)\beta (d)}{32\mathcal{M}\k_{j}\ \delta(k_{j}+1)-8\alpha (2k_{j}+1)\beta (d)+\mathcal{N}\ \left[ \beta (d)\right]^{2}}\leqq \frac{8\mathcal{N}\ \delta (k+1)\beta (d)}{32\mathcal{M}\k\ \delta(k+1)-8\alpha (2k+1)\beta (d)+\mathcal{N}\ \left[ \beta (d)\right]^{2}}(j=1,\ldots ,m),
\overline{g}_{s,d}(\Lambda) \leq \liminf_{N_0\rightarrow \infty}\frac{\mathcal{E}_{s,\Lambda}^{\rm cp}(N_0)}{N_0^{1+\frac{s}{d}}}=\underline{g}_{s,d}(\Lambda), \overline{g}_{\log,d}(\Lambda) \leq \liminf_{N_0\rightarrow \infty}\frac{\mathcal{E}_{\log,\Lambda}^{\rm cp}(N_0)+\frac{2}{d}N_0\log N_0}{N_0}=\underline{g}_{\log,d}(\Lambda).
\mathcal{M}' \models \Gamma^{\bar{\varphi}}_{\leq \varepsilon, \bar{\beta}}(\bar{b}) \iff \textrm{for every}\bar{c}\in V^{\bar{m}_2}, \sum_{i=1}^{n}\beta_i \cdot \mu_{\bar{n}}\left( \varphi_i(\bar{x}, \bar{b}, \bar{c}) \right) \leq \varepsilon.
R_0:=\sup_{{\bf y}\in \mathcal A}r({\bf y}) \le\big( 1+\alpha_0 + \alpha^{-1}\big)\sup_{{\bf y}\in \mathcal A}\big(|x|+|v|+|x'| +|v'|\big) & \le2\big( 1+ \alpha_0 + \alpha^{-1}\big)\sup_{(x,v)\in \mathcal A_0}\big(|x|+|v|\big ) &\le2R^*\big( 1+\alpha_0 + \alpha^{-1}\big).
4 T_o^\alpha T_o^\beta\frac{\partial^2 r}{\partial z^\alpha\partial z^\beta}= T^iT^j \frac{\partial^2 r}{\partial x^i\partial x^j}-V^iV^j\frac{\partial^2 r}{\partial x^i\partial x^j}-2\sqrt{-1}V^iT^i\frac{\partial^2 r}{\partial x^i\partial x^j}.
\delta_{c_3}\circ \delta_{c_3}= u \delta_{c_1}+ u \delta_{c_2}+ x \delta_{c_3}+ u \delta_{c_4}, &\delta_{c_4}\circ \delta_{c_4}= u \delta_{c_1}+ u \delta_{c_2}+ u \delta_{c_3}+ x \delta_{c_4}.
S_{\left\langle p , \phi \right\rangle}^{\left\langle q , \psi \right\rangle}= \frac{1}{| \Omega_{p}| | \Omega_{q}|}\sum_{z \in \Omega}\sum_{x , y \in \Omega_{p}\cap \Omega_{z q}}\overline{{\phi}}( x , y ) \overline{{\psi}}^{z}( y , x ) \prod_{\xi \in \mathcal{O}( x , y )}\Lambda_{p_{\xi}}^{( z q )_{\xi}}\left( \frac{\kappa_{\xi}}{\lambda_{\xi}}\right)
\mathrm{Im}(w^\xi_\eta(\rho))= \mathrm{Im}\sum_{\{\alpha,\beta;\epsilon\}\in\triangle^+(\rho)}\left(e^{\frac{1}{2}(\ell_{\alpha}(\rho)+\ell_{\beta}(\rho))}+(-1)^{\alpha\cdot\beta}\right)^{-1}=-\mathrm{Im}\sum_{\{\alpha,\beta,\epsilon\}\in\triangle^-(\rho)}\left(e^{\frac{1}{2}(\ell_{\alpha}(\rho)+\ell_{\beta}(\rho))}+(-1)^{\alpha\cdot\beta}\right)^{-1}.
\int_{S^2}(k^{(3)}- h^{(3)}) dS^2 = -\frac{3}{4}\int_{S^2}W_0^2 d S^2 + \frac{1}{60}\int_{S^2}|\alpha|^2 dS^2- \int_{S^2}(\sigma^{ab}n_{ab})^{(3)}dS^2 = -\frac{3}{4}\int_{S^2}W_0^2 d S^2 -\frac{1}{60}\int_{S^2}|\alpha|^2 dS^2 + \frac{11}{45}\int_{S^2}|\beta|^2 dS^2.
(K_{|x|^{2k-2}}f)(z) = \int_{\Sigma}| z - x |^{2k-2}\; f(x) \ \mathrm{d}x , z \in \Sigma (K_{|x|^{2k-2}\ln |x|}f)(z) = \int_{\Sigma}| z - x|^{2k-2}\ln |z -x| \; f(x) \ \mathrm{d}x , z\in \Sigma
M^{p,q}(U):=\Big\{f\in L^p_{\rm{loc}}(U)\ \Big|\ \big\|f\big\|_{M^{p,q}(U)}^q\equiv\sup_{B_r\subset U}\frac1{r^{3-q}}\int_{B_r}|f|^p<+\infty\Big\}.
\frac{1}{2}\frac{d}{dt}\int_{\Omega}\rho |u_t|^2 x+\int_{\Omega}|\nabla u_t|^2x \leq C(\|\nabla u\|^2_1+|D(u)|_\infty+1)(|\sqrt{\rho}u_t|^2_{2}+1)+C|\sqrt{\rho}\theta_t|^2_2.
\Omega_{\rm{m}}(\tau_{i}) = \sum_{l=1}^{\zeta_\tau|_{\tau_{i-1}}^{\tau_{i}}}P_{\rm{h}}\frac{d_\beta(\tau_{i},l)}{v_{\rm{h}}}+ I(\Delta h_\beta(\tau_{i},l)) P_{\rm{a}}\frac{\Delta h_\beta(\tau_{i},l)}{v_{\rm{a}}}- \big(1-I(\Delta h_\beta(\tau_{i},l))\big) P_{\rm{d}}\frac{\Delta h_\beta(\tau_{i},l)}{v_{\rm{d}}},
m(I_k \cap (H \setminus P)) \leq \sum_{n=n_k}^{\infty}m(I_k \cap (H_n \setminus K) \cap (\bigcap_{m=1}^{\infty}(H_n \setminus P_m))) \leq \sum_{n=n_k}^{\infty}m(H_n \setminus P_n) < \sum_{n=n_k}^{\infty}\frac{1}{2^{n+1}}= \frac{1}{2^{n_k}}
(k+1)\left(ka_{3,2}+a_{2,2}\right){\mathcal C}_{k+1}+\left(k(k-1)a_{3,1}+ka_{2,1}-\tau_{1,1}\right)){\mathcal C}_k+\left((k-1)a_{2,0}-\tau_{1,0}\right)){\mathcal C}_{k-1}=0,{\mathcal C}_{-1}=0,{\mathcal C}_{0}=1,
\sum_{k=0}^\infty \| (z_2-\Delta_2) s_k\|_2^2 = \sum_{k=0}^\infty \tau\bigl( \langle s_k,s_k\rangle_A \, (z_2-\Delta_2)^2 \bigr) &= \sum_{k=0}^\infty \tau\bigl( \Delta_2^k (z_2 - \Delta_2)^2 \bigr) = \tau(z_2 -\Delta_2) < \infty \; ,
\hat{I}_{n, k}=2v_n\sum^{[{t}/{(2v_n)}]-1}_{j=0}\biggl(c_j^{k}(u_n)-\frac{(\sinh(u_n^2c^{k}_j(u_n)))}{u_n^2(k_n-1)}\biggr)^2, k=0, 1.
\pi_a^i\circ \pi^1_{a,b}\circ \iota^1_{a,b}\circ \iota^{j_1}_a=\begin{cases}\mathbf{1}_{a^i}, i=j_1, 0_{a^{j_1}, a^i}, ,\end{cases}\quad\pi_a^i\circ \pi^1_{a,b}\circ \iota^2_{a,b}\circ \iota^{j_2}_b=0_{b^{j_2},a^i}.
A =p^2\varphi(-1)\sum_{\chi\in\widehat{\mathbb{F}_p^\times}}{\varphi\overline{\chi}\choose\chi}~\chi\left(\frac{-x}{4}\right) &=p\varphi(-1)\sum_{\substack{\chi\in\widehat{\mathbb{F}_p^\times}y\in\mathbb{F}_p}}\varphi(y)\overline{\chi}(y)\overline{\chi}(1-y)\chi\left(\frac{x}{4}\right).
ds^2=(Ff_1\cdots f_7f_8)^{1/2}(a_1(r)\cdots a_7(r))^{1/2}\times\left({{(\sigma_1)^2}\over{f_1^2a_1^2}}+\cdots{{(\sigma_7)^2}\over{f_7^2a_7^2}}+{{dr^2}\over{f_8^2}}\right)
\displaystyle{\sum_{\stackrel{\iota \leq \iota' \leq \gamma_1<\cdots < \gamma_m = \iota + k + 1}{1 \leq m \leq \iota}}}(-1)^{\deg(\omega')}u_{\iota\iota'}^{(i)}x_{\iota' \gamma_1}u_{\gamma_1\gamma_2}^{(\mu)}\cdots u_{\gamma_{m-1},\iota+k+1}^{(\mu)}= x_{\iota, \iota+k+1},
\begin{cases}\partial_tv + v\cdot\nabla v + \nabla p = \nabla\cdot( \frac{\partial W(F)}{\partial F}F^\top), begin{align*}-4mm] \partial_t F+v\cdot\nabla F=\nabla v F, begin{align*}-4mm] \nabla\cdot v = 0,\nabla\cdot F^\top = 0,\end{cases}
\int_{0}^{1}B_{2n-1}(x)log(tan\frac{\pi}{2}x)dx=\frac{(-1)^{n-1}}{\pi^{2n}}\frac{2^{2n-2}}{(2n-1)!}\sum_{k=1}^{n}(2^{2k+1}-1)\zeta(2k+1)\zeta(2n-2k)
{\textstyle\sum_{\overline{K}_1\ni j=k}^{\nu}}\,\big\|x^{j+1}\!-\!x^j\big\|\le(1/2){\textstyle\sum_{j=k}^{\nu}}\,\Xi_j+ba^{-1}\varphi(\Phi(x^{\ell(k)})\!-\!\Phi^*).
d\mu_{[-r,r]}(x) = \frac{(\sqrt{r^2+1}+1)\,dx}{\pi\sqrt{r^2-x^2}(x^2+(\sqrt{r^2+1}+1-\sqrt{r^2-x^2})^2)}+ \frac{(\sqrt{r^2+1}+1)\,dx}{\pi\sqrt{r^2-x^2}(x^2+(\sqrt{r^2+1}+1+\sqrt{r^2-x^2})^2)},\x\in(-r,r).
C_3\sum_{1\le j\le k-1}C^j_k\ \int^t_0\|\tau^{j}\partial^{j}_\tau f\|_{L^2}\|\tau^{k-j}\partial^{k-j}_\tau f\|_{L^2_A}\|\tau^k\partial^k_\tau f\|_{L^2_A}d\tau \le \frac{1}{8}\|\tau^k\partial^k_\tau f\|^2_{L^2([0,t];L^2_A)}+A_2 B^{2(k-2)}((k-2)!)^2,
\begin{aligned}\langle S_N(\ell)^2\rangle =\sum_{i,j=1}^N\sum_{m,n\in\mathbb{Z}}\int_0^1\chi\left(\frac{x_i-x_0+m}{\ell}\right)\chi\left(\frac{x_j-x_0+n}{\ell}\right)\,dx_0 =\sum_{i,j=1}^N\sum_{m\in\mathbb{Z}}\int_\mathbb{R}\chi\left(\frac{x_i-x_0+m}{\ell}\right)\chi\left(\frac{x_j-x_0}{\ell}\right)\,dx_0.\end{aligned}
X_{rs}= \left( \Gamma_{\mu}C\right)_{rs}X^{\mu}+ \left( \Gamma_{[\mu\nu]}C\right)_{rs}\, X^{[\mu\nu]}+ \left( \Gamma_{[\mu_5 - \mu_s]}C\right)_{rs}\, X^{[\mu_1 \ldots \mu_5]}\, ,
\begin{cases}\eta^\zeta(\rho,\rho u)=\int_\mathbb{R}\chi(\rho;s-u)\zeta(s)ds, q^\zeta(\rho,\rho u)=\int_\mathbb{R}(\vartheta s +(1+\vartheta)u)\chi(\rho;s-u)\zeta(s)ds, \vartheta=\frac{\gamma-1}{2},\end{cases}
e^{i\Gamma_{f}[A_{+},A_{-}]}=\int d\psi^{\dagger}d\psi e^{i\int d^{2}x (\psi^{\dagger}_{L}(i\partial_{+}+A_{+})\psi_{L}+\psi^{\dagger}_{R}(i\partial_{-}+A_{-})\psi_{R})}\,,
f(r^k) =r^k\|\vec x^{k+1}\|_{\infty}-I(\vec x^{k+1}) &=r^k\|\vec x^{k+1}\|_{\infty}-r^{k+1}\|\vec x^{k+1}\|_{\infty}&=\|\vec x^{k+1}\|_{\infty}(r^k-r^{k+1})\to 0 k\to +\infty,
\sum^j_{l=0}([u_{1,j-l}]-[u_{2,j-l}]\partial_2\varphi_l)=0,[p_j]+\sum^{j-1}_{l=0}C_{l,j-1}[(H_l,H_{j-l})]=0,\quad\mbox{on}\,\,\{x_1=0\}\,,