harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\nabla F_{\gamma}^{\rm{DR}}(z) =(2\gamma)^{-1}\big(\nabla R_{\gamma f}(z)z+R_{\gamma f}-R_{\gamma f}-\nabla R_{\gamma f}(z)R_{\gamma g}(R_{\gamma f}(z))\big) &=(2\gamma)^{-1}\nabla R_{\gamma f}(z)(z-R_{\gamma g}R_{\gamma f}(z)).
\sum_{\substack{p\leq x p\in \mathcal{P}_u}}\frac{1}{p}=\int_{x_0}^{x}\frac{1}{t}d \pi_u(t) =\frac{\pi_u(x)}{x}+c_(x_0)+\int_{x_0}^{x}\frac{\pi_u(t)}{t^2}d t,
\sum_{j\ge 3}jn_{T,j}=\sum_{j\ge 3}(j-2)n_{T,j}+2n_{T,j}= n_{T,1}-2+2\sum_{j\ge 3}n_{T,j}&\le n_{T,1}-2+2\sum_{j\ge 3}(j-2)n_{T,j}= 3n_{T,1}-6.
K_{p,\varphi_p}(z)\geq\frac{|K_{2,\varphi_p}(z)|^2}{\|K_{2,\varphi_p}(\cdot,z)\|_{2,\varphi_p}^2}=K_{2,\varphi_p}(z),\ \ \ |z|<\frac{2-pk_p}{pk_p}.
H_0= \int |u|^2 dx, H_1= \frac1i \int u \partial_x \bar u dx , H_2= \int |u_x|^2 + |u|^4 dx, H_3 = i \int \u_x \partial_x \overline{u}_x + 3 |u|^2 u \overline{u}_x dx, H_4 = \int |u_{xx}|^2 + \frac32 |(u^2)_x|^2+ ||u|^2_x|^2 + 2 |u|^6 dx.
t(s)=\gamma_3(s),r(s)=\left(\gamma^2_1(s)+\gamma^2_2(s)\right)^{\frac{1}{2}}=|\xi^H|,\quad\mbox{and}&\begin{cases}\cos{\left(\theta(s)\right)}= \left\langle \nu^H,\frac{\xi^H}{|\xi^H|}\right\rangle, \sin{\left(\theta(s)\right)}= \left\langle \eta,\frac{\xi^H}{|\xi^H|}\right\rangle.\end{cases}
w(t,y) \geq w_0(y) + \left( -g\big(c_0(y)\big) \frac{1}{w_0(y)}- \varepsilon \right) \cdot t = 1 + ( -g_O - \varepsilon ) \cdot t &> 1 + ( -g_I + \varepsilon ) \cdot t= w_0(x) + \left( -g\big(c_0(x)\big) \frac{1}{w_0(x)}+ \varepsilon \right) \cdot t\geq w(t,x)
~f \circ \delta_\alpha g - \delta_\alpha ( f \circ g) + (-1)^{n-1}\delta_\alpha f \circ g = ~ (-1)^{n-1}f \circ ( \mu \circ g) - f \circ (g \circ \mu) - (-1)^{m+n -2}\mu \circ ( f \circ g) + (f \circ g) \circ \mu &~ + (-1)^{n-1}(-1)^{m-1}(\mu \circ f) \circ g - (-1)^{n-1}(f \circ \mu) \circ g .
N \simeq \left( \frac{M_{s}}{M_{P}}\right)^{2}\frac{( M_{s}^{3}A_{T})^{3 / 2}}{( 2 \pi )^{7 / 2}g_{s}^{3 / 2}V_{6}^{1 / 6}F}\frac{\Psi_{*}}{M_{s}}.
[B_1(a),B_1(b)]=B_1([B_1(a),B_1(b)]-[B_2(a),B_2(b)]),\quad\forall a,b\in \mathfrak{g}, &[B_2(b),B_2(a)]=B_2([B_1(a),B_1(b)]-[B_2(a),B_2(b)]),\quad\forall a,b\in \mathfrak{g}.
\sum_{m>0}\int \frac{d^2\rho}{\pi}\e^{-|\rho|^2}(\rho|e^{\sqrt{\alpha'/2}k_\mu \alpha_{-n}^\mu z^n/n}\left[z^{-m-1}\alpha_m^\mu+z^{m-1}\alpha_{-m}^\mu\right]e^{-\sqrt{\alpha'/2}k_\mu \alpha_{n}^\mu z^{-n}/n}|q^n\rho)
p_{L,N+1}^{\ell_2}(x)=\dfrac{\sum\limits_{j=0}^N\left(\omega_j\sum\limits_{\ell=0}^N\dfrac{\tilde{\Phi}_{\ell}(x_j)\tilde{\Phi}_{\ell}(x)}{1+\lambda\mu_{\ell}^2}\right)f(x_j)}{(1+\lambda\mu_0^2)\sum\limits_{j=0}^N\omega_j\sum\limits_{\ell=0}^N\dfrac{\tilde{\Phi}_{\ell}(x_j)\tilde{\Phi}_{\ell}(x)}{1+\lambda\mu_{\ell}^2}}.
\mu\frac{\partial p_i\Psi_i}{\partial x}(x,\mu) + \Sigma_{ti}p_i\Psi_i(x,\mu) = \frac{\Sigma_{si}}{2}\int_{-1}^1p_i\Psi_i(x,\mu')d\mu' & +\left(\frac{p_j\overline\Psi_j(x,\mu)}{\Lambda_j(x,\mu)}-\frac{p_i\overline\Psi_i(x,\mu)}{\Lambda_i(x,\mu)}\right) +\frac{p_iQ_i}{2}, \,\,\,\,-X\leq x\leq X, -1\leq\mu\leq 1,
\begin{cases}\begin{array}{ll}v(.,t)\in SPSH(\bar{\Omega}), \dot{v}=\log\det (v_{\alpha\bar{\beta}})-Av+g(z,t)\;\;\; \mbox{on}\;\Omega\times (0,T), v=\psi \mbox{on}\;\partial\Omega\times [0,T). \end{array}\end{cases}
(F^{f(x)})^{x^i}=\left(F^{a_n x^n}F^{a_{n+1}x^{n+1}}\cdots F^{a_{n+m}x^{n+m}}\right)^{x^i}&=\left(F^{a_n x^n}\right)^{x^i}\left(F^{a_{n+1}x^{n+1}}\right)^{x^i}\cdots \left(F^{a_{n+m}x^{n+m}}\right)^{x^i}&=(F^{a_n})^{x^{n+i}}(F^{a_{n+1}})^{x^{n+1+i}}\cdots (F^{a_{n+m}})^{x^{n+m+i}}&=F^{x^i f(x)}.
P^{rc}_{n-k,\delta,k}=\frac{\operatorname{Pr}((A,B,C)\in\mathbb F^{\delta\times\delta}\times\mathbb F^{\delta\times k}\times\mathbb F^{(n-k)\times\delta}\ )}{\operatorname{Pr}((A,B)\in\mathbb F^{\delta\times\delta}\times\mathbb F^{\delta\times k}\ )}&=1-t^{n-k}+O(t^{n-k+1}).
{\bf A}_k={\bf a}_k{\bf a}^{H}_k,{\bf A}^{'}_j={\bf a}^{'}_j{\bf a}^{'^{H}}_j,{\bf B}_k={\bf b}_k{\bf b}^{H}_k,{\bf B}^{'}_j={\bf b}^{'}_j{\bf b}^{'^{H}}_j,{\bf S}={\bf s}{\bf s}^{H}.
\bar DD\Phi_2=i\mu_f\nu_fe^{\Phi_2-\Phi_1},\qquad\bar DD\Phi_1=e^{2\Phi_1-\Phi_2}\bar\Psi^{+b}\Psi^{-b},\qquad \bar D\Psi^{-b}=0=D\bar\Psi^{+b}.
\partial_{\pm}T = A_{\pm}T ,{}~ A_{\pm}= \pm \left[ \frac{1}{2}\partial_{\pm}\Phi + \exp \left( \mp \frac{1}{2}\mathrm{a d}\Phi \right) \bar{\Psi}_{\pm}+ \exp \left( \mp \frac{1}{2}\mathrm{a d}\Phi \right) \mu_{\pm}\right] ,
e^{-i \psi_{<\ell, \pm}}(t,x,D) a(D) e^{i \psi_{<\ell, \pm}}(D, y, s) - e^{-i \psi_{<\ell, \pm}}_{<k}(t,x,D) a(D) e^{i \psi_{<\ell, \pm}}_{<k}(D, y, s) = e^{-i \psi_{<\ell, \pm}}_{\geq k}(t,x,D) a(D) e^{i \psi_{<\ell, \pm}}(D, y, s) + e^{-i \psi_{<\ell, \pm}}_{< k}(t,x,D) a(D) e^{i \psi_{<\ell, \pm}}_{\geq k}(D, y, s) .
\det(Y,Z_{2,6,7,8}) = \Delta_{13}(X) \Delta_{1,3,2,6,7,8}(Z) + \Delta_{14}(X) \Delta_{1,4,2,6,7,8}(Z) + \Delta_{15}(X) \Delta_{1,5,2,6,7,8}(Z) + \Delta_{35}(X) \Delta_{3,5,2,6,7,8}(Z) + \Delta_{45}(X) \Delta_{4,5,2,6,7,8}(Z).
\alpha_{A}( \varphi_{0}) \simeq \alpha_{3}( \varphi_{0}) +{\frac{\partial \widehat \sigma}{\partial \varphi_{0}}}\left({\frac{B}{M}}\right)_{A}+{\frac{\partial \widehat \delta}{\delta \varphi_{0}}}\left({\frac{D}{M}}\right)_{A}+ a_{3}{\frac{\partial \alpha}{\partial \varphi_{0}}}\left({\frac{E}{M}}\right)_{A}\ ,
z A_{0}(z)-1 = a_{0}^{(0)}A_{0}(z)+\sum_{j=1}^{p}a_{0}^{(j)}A_{j}(z)=a_{0}^{(0)}A_{0}(z)+\sum_{j=1}^{p}a_{0}^{(j)}A_{0}(z) A_{j-1}^{(1)}(z) =A_{0}(z) (a_{0}^{(0)}+\sum_{j=1}^{p}a_{0}^{(j)}A_{j-1}^{(1)}(z))
y_1 =\underbrace{\sum_{j=1}^{J-1}h_{1j}x_j}_{}+\underbrace{h_{1J}x_J+z_1}_{=s_1}, y_2 =\underbrace{h_{2J}x_J}_{}+\underbrace{\sum_{j=1}^{J-1}h_{2j}x_j+{z_2}}_{=s_2},
E[M(t^* \wedge \tau)] = \frac{1}{N}\sum_{j=1}^{\infty}j E[X_j^{\tau}(t^*)] &\leq \frac{1}{N}\sum_{j=1}^{\infty}j \cdot \frac{N \mu^j}{s^j j!}(e^{st^*}- 1)^j &= \frac{\mu}{s}(e^{st^*}- 1) \sum_{j=1}^{\infty}\frac{1}{(j-1)!}\bigg( \frac{\mu}{s}(e^{st^*}- 1) \bigg)^{j-1}&= ye^y,
=\sum_{j}\nabla_{e_{j}}(e_{j}\cdot\langle\star,\bar{s}^{+}\otimes \bar{f}\rangle_{h})-\sum_{j}(\nabla_{e_{j}}e_{j})\cdot\langle\star,\bar{s}^{+}\otimes \bar{f}\rangle_{h}-\sum_{j}e_{j}\cdot\langle\star, \nabla_{e_{j}}(\bar{s}^{+}\otimes \bar{f})\rangle_{h},
\beta_{1,d-3}={{d-1}\choose{2}},\beta_{2,d-3}=2(d-1)(d-3),\quad\beta_{3,d-3}=3(d^2-5d+5) \beta_{4,d-3}=2(d-2)(d-4)\mbox{and}\beta_{5,d-3}={{d-3}\choose{2}}.
C(N) =\sum_{k=1}^{N-2}(-1)^k\binom{N}{k}\sum_{y=2}^{m+1}\binom{N-k}{y}-\sum_{i=N-1}^{\min(m,N-1)}1= &=\sum_{k=1}^{N-2}\sum_{y=2}^{\min(m+1,N-k)}(-1)^k\frac{N!}{k!\cdot y!\cdot (N-k-y)!}-\sum_{i=N-1}^{\min(m,N-1)}1= &=\sum_{y=2}^{\min(m+1,N-1)}\sum_{k=1}^{N-y}(-1)^k\frac{N!}{k!\cdot y!\cdot (N-k-y)!}-\sum_{i=N-1}^{\min(m,N-1)}1
-\int_{\Omega}|\nabla(u_n-v_n)^-|^2 =\int_{\Omega}\nabla (u_n-v_n).\nabla (u_n-v_n)^- & =\int_{\Omega}\bigg(h_n\left(u_n +\frac{1}{n}\right)-h_n\left(v_n +\frac{1}{n}\right) \bigg)f_n\cdot (u_n-v_n)^- & + \int_{\Omega}\mu_n\cdot(u_n-v_n)^- & \geq 0.
\rho_{\mathcal{T}}(\mathcal{Z}_\alpha) = \rho^k_\alpha\partial_k =: \partial_\alpha,\rho_{\mathcal{T}}(\mathcal{V}_\alpha) = \dot{\partial}_\alpha, &\rho_{\mathcal{T}}(\mathcal{Z}_{\bar{\alpha}}) = \rho^{\bar{z}}_{\bar{\alpha}}\partial_{\bar{k}}=:\partial_{\bar{\alpha}},\rho_{\mathcal{T}}(\mathcal{V}_{\bar{\alpha}}) = \dot{\partial}_{\bar{\alpha}}.
A_{2^n}=\sum_{i=0}^{2^n-1}a_{2^n-i}Jq^{2^n-i}Jq^i = a_{2^n}Jq^{2^n}+\sum_{i=1}^{2^n-1}a_{2^n-i}Jq^{2^n-i}Jq^i=0
q^{x}\cdot \Omega ,_{\alpha r}\Omega ,_{\beta s s}+ q^{- x}\cdot \Omega ,_{\beta s}\Omega ,_{\alpha r s}- ( q^{x}\cdot \Omega ,_{\alpha s}\Omega_{, \beta r s}+ q^{- x}\cdot \Omega ,_{\beta r}\Omega_{, \alpha s s}= 1 .
T(x; k, r) = \sum_{u (n r_{1}+ k_{1}) \leq x}a( u( n r_{1}+ k_{1}))= \sum_{\stackrel{n \leq x/u}{n \equiv k_{1}(mod r_{1})}}a(u n) &=\frac{1}{\varphi(r_{1})}\sum_{\chi \, (mod \,r_{1})}\bar{\chi}(k_{1}) \left(\sum_{n\leq x/u}\chi(n) a(u n)\right ),
\lambda^{\left( p\right)}\left( G\right) \leq2\sum_{1\leq i<j\leq n}x_{i}x_{j}<\left( n-1\right) n\left( \frac{1}{n}\sum_{i=1}^{n}x_{i}\right)^{2}\leq\left( n-1\right) n\left( \frac{1}{n}\sum_{i=1}^{n}x_{i}^{p}\right)^{2/p}=\frac{n-1}{n^{2/p-1}}.
T^3_{\overline{1}}=T^3_{\overline{2}}=Q^1_{\overline{1}}=Q^1_{\overline{2}}=Q^2_{\overline{1}}=Q^2_{\overline{2}}=P^1_{0\overline{1}}=P^1_{0\overline{2}}=P^2_{0\overline{1}}=P^2_{0\overline{2}}=0.
Z=\lim_{\delta\to0}\int^{2\pi}_0{d\lambda\over2\pi}e^{-i\lambda Q}\sum^\infty_{l_1=0}\cdots\sum^\infty_{l_{N+1}=0}\exp{\left\{{-iT\sum^{N+1}_{\alpha=1}c_\alpha l_\alpha +i\left(\lambda+i\delta\right)\sum^{N+1}_{\alpha=1}l_\alpha}\right\}}\ ,
||u||^2_{\widetilde{X}_{[0,T]}}=||\langle \xi \rangle^{-1}u||^2_{L^2([0,T]\times \mathbb{R}^d)}+\sum_{k=-\infty}^{\infty}2^k||S_ku||^2_{X_k(T)}, \d\not=2, ||u||^2_{\widetilde{X}_{[0,T]}}=||\langle \xi \rangle^{-1}(\log (2+|\xi|))^{-1}u||^2_{L^2([0,T]\times \mathbb{R}^d)}+\sum_{k=-\infty}^{\infty}2^k||S_ku||^2_{X_k(T)}, \d=2.
\phi_{\alpha_{1}\alpha_{2}}\phi_{\alpha_{2}\alpha_{3}}\phi_{\alpha_{3}\alpha_{1}}= f^{( \alpha_{1})}f^{( \alpha_{2})}{}^{- 1}f^{( \alpha_{2})}f^{( \alpha_{3})}{}^{- 1}f^{( \alpha_{3})}f^{( \alpha_{1})}{}^{- 1}= 1
\begin{array}{r c c c l}{{G_{i \mu}}}&{{\to}}&{{G_{i \mu}^{\prime}}}&{{=}}&{{U_{i}\star G_{i \mu}\star U_{i}^{- 1}+ i g_{i}^{- 1}U_{i}\star \partial_{\mu}U_{i}^{- 1},}}\\{{\Phi}}&{{\to}}&{{\Phi^{\prime}}}&{{=}}&{{U_{1}\star \Phi \star U_{2}^{- 1},}}\\ \end{array}
H^{(n)}:y^{2}=x^3-\frac{1354}{7}x^2+\frac{936}{7}\Bigl(t^{n}+\frac{42989}{819}+\frac{4}{t^{n}}\Bigr)x+\frac{6084}{49}\Bigl(t^{n}-\frac{4}{t^{n}}\Bigr)^2.
\P(\tau > n) = \int_{W^d}\det(q_{n+i-j}(y_j - x_i))_{i, j = 1}^d \prod_{i=1}^d \lambda_i^n e^{-\sum_{i=1}^d \lambda_i(y_i - x_i)}dy_1 \ldots dy_d, \P(\rho > n) = \int_{W^d}\det(q_{n}(y_j - x_i))_{i, j = 1}^d \prod_{i=1}^d \lambda_i^n e^{-\sum_{i=1}^d \lambda_i(y_i - x_i)}dy_1 \ldots dy_d.
H = V_{1}e^{i ( p_{+}+ p_{-})}Y^{- 1}X^{- 1}+ V_{2}e^{i ( p_{+}- p_{-})}X^{- 1}Y + V_{1}e^{- i ( p_{+}+ p_{-})}X Y + V_{2}e^{- i ( p_{+}- p_{-})}Y^{- 1}X .
e^{-\int_{\tau_{\ell+1}}^{a_N T}G_{\ell}(v) \: dv}X_{\ell}^{\tau_{\ell+1}, a_NT}(a_N T) = \int_{\tau_{\ell+1}}^{a_N T}\mu X_{\ell-1}(u) e^{-\int_{\tau_{\ell+1}}^u G_{\ell}(v) \: dv}\: du + Z_{\ell}^{\tau_{\ell+1}, a_N T}(a_N T),
\hat{B}_{(0)}\hat{Q}_{(n)}\vert 0 \rangle_{[0]}= \hat{a}\hat{Q}_{(n)}\vert 0 \rangle_{[0]}= - \sum_{k = 0}^{n-1}\hat{B}_{(n-k)}\hat{Q}_{(k)}\vert 0 \rangle_{[0]}.
-\dfrac{1}{2N^2}\left(\partial^2_t g_{ii}+2\partial_t N\,h_{ii}\right)-\frac{k}{2N^2}\left( g^{ij}\partial^2_t g_{ij}+2\partial_t N\,H\right) \geq -(1-q)\lambda_i^2-(1-q)k\sum_i \lambda_i^2 &=-(1-q)\lambda_i^2-(1-q)k|h|^2\,.
\ln(\beta(t,q_t)/\beta(s,q_s))=\ln(\beta(t,q_t))-\ln(\beta(s,q_s)) = \int_s^t \beta^{-1}(r,q_r)\partial_r\beta(r,q_r)dr+\int_s^t(\beta^{-1}\nabla_q\beta)(r,q_r)\cdot dq_r+\frac{1}{2}\int_s^t \partial_{q^j}(\beta^{-1}\partial_{q^i}\beta)(r,q_r)d[q^i,q^j]_r
\begin{array}{c}{\epsilon_{a b}= \left( \begin{array}{c c}{0}&{q^{\frac{1}{2}}}\\{- q^{- \frac{1}{2}}}&{0}\\ \end{array}\right)}\\{\epsilon_{a b}\epsilon_{b c}= - \delta_{a c}}\\{( \epsilon_{\overline{{a}}\overline{{b}}})_{q}= - ( \epsilon_{a b})_{q^{- 1}}}\\ \end{array}
z\phi_0z =y\left(1+\frac{1}{2}G(y^{-1}ky)\right)\phi_0\left(1-\frac{1}{2}G(y^{-1}ky)\right)y^{-1}+O(t^{-2}\mathcal L) &=\phi_0+\frac{1}{2}\left[G(k),\phi_0\right]+O(t^{-2}\mathcal L)
|D^\nu| \geqslant C_\nu\mathcal A^*\bigg(\bigoplus_{\varkappa\in \mathbb Z +1/2}\mathcal U_{\varkappa}^*R^1\mathcal U_{\varkappa}\bigg)\mathcal A =C_\nu\mathcal T^*\bigg(\bigoplus_{m\in \mathbb Z}R^1\bigg)\mathcal T =C_\nu\sqrt{-\Delta}
C_n(x) \coloneqq \sum_{t=0}^{\infty}\beta_0^t c_{n,t}(x) \leq \frac{L}{1-\beta_0 \alpha}w_n(x) + \frac{L \beta_0}{(1-\beta_0)(1-\beta_0 \alpha)}\gamma &\coloneqq L_1 w_n(x) + L_2.
\begin{aligned}u\left(\frac{a+b}{2}\right)w\left(\frac{a+b}{2}\right) \leq\frac{u\left(\xi a+(1-\xi)b\right)w\left(\xi a+(1-\xi)b\right)}{4}+\frac{u\left((1-\xi)a+\xi b\right)w\left((1-\xi)a+\xi b\right)}{4}& +\frac{t(1-\xi)}{2}\left[u(a)w(a)+u(b)w(b)\right] +\frac{(2\xi^2-2\xi+1)}{4}\left[u(a)w(b)+u(b)w(a)\right].\end{aligned}
V(\mathcal{M}((V_i)_{i \in [k]})) :=[k] \cup \left( \bigcup_{1 \leq i<j \leq k}(V_i \cap V_j) \times \{\star\}\right), E(\mathcal{M}((V_i)_{i \in [k]})) :=\{\{i,(V_i \cap V_j ,\star)\}\, \{j, (V_i \cap V_j, \star)\}\vert \, V_i \cap V_j \neq \emptyset \},
F = \frac{P_T}{\dot{B}}- \frac{\dot{A}_1}{A_1'\dot{B}}P_1 -\frac{\dot{A}_2}{A_2'\dot{B}}P_2 + \frac{1}{\dot{B}}\left[\int\left(\dot{D}_1A_1' -\dot{A}_1D_1'\right)dQ_1 + \int\left(\dot{D}_1A_2' -\dot{A}_2D_2'\right)dQ_2 \right]\;,
E_{\chi_0 , \chi_1}(-q) = - E_{\chi_0 , \chi_1}(q) - 2E_{\chi_0 , \chi_1}(q^2) + 4 E_{\chi_0 , \chi_1}(q^4), &E_{\chi_1 , \chi_0}(-q) = - E_{\chi_1 , \chi_0}(q) + 2E_{\chi_1 , \chi_0}(q^2) + 4 E_{\chi_1 , \chi_0}(q^4) .
i \frac{\partial}{\partial t}\psi ( \vec{x}, t ) \left|_{\vec{x}}\right. = i \left[ \frac{\dot{a}( t )}{a ( t )}\right] \ \vec{r}\cdot \left( \frac{\partial f}{\partial \vec{r}}\left|_t \right. \right) + i \frac{\partial f}{\partial t}\left|_{\vec{r}}\right. .
A_t=f_t(r)\begin{pmatrix}1 0 0 -1\end{pmatrix}\left(\frac{dz}{z}-\frac{d\bar z}{\bar z}\right), \Phi_t=\begin{pmatrix}0 r^{k/2}e^{h_t(r)}z^kr^{-k/2}e^{-h_t(r)}0\end{pmatrix}\,dz
\sup_{\ell \le \gamma_t}\big\{P(S_{t-\ell}^1 \ge \psi(t)-\sqrt{\ell}) \big\}\le C P(A_t) \,, P\big(A_t \cap L_t^c ) \le 4 \zeta_t P(A_t) \,, L_t := \bigcap_{\ell \in (\gamma_t, t]}\Big\{S_\ell^1 \ge \frac{\psi(t) \ell}{(1+\delta) t}\Big\}\,.
\langle f - gG , F \rangle = \int \int g(x) h(x,y) \overline{F(x)}\d (\mu_{4}\times \lambda) - \int \int g(x)G(x)\overline{F(x)}\d (\mu_{4}\times \lambda) &= \int_{C_{4}}g(x)\overline{F(x)}\left(\int_{[0,1]}h(x,y) - G(x) \d \lambda(y) \right) \d \mu_{4}(x) &= \int_{C_{4}}g(x)\overline{F(x)}\left( G(x) - G(x) \right) \d \mu_{4}(x) &= 0.
P_{n+1}(x, x_{n+1}) = \tfrac{1}{6}\sum_{i = 1}^{n}(x_{n+1}^{3}- 3x_{n+1}x_{i}^{2}) + \sqrt{\tfrac{n+2}{n}}P_{n}(x) & = \tfrac{n+3}{6}\left(x_{n+1}^{3}- \tfrac{3}{n+3}x_{n+1}E_{n+1}(x, x_{n+1})\right) + \sqrt{\tfrac{n+2}{n}}P_{n}(x).
\psi_{p-1}(x^{(p)}, \xi^{(p)})=\big\{(\xi_p-h_p(x^{(p)}, \xi^{(p+1)}))^2+g_p(x^{(p)},\xi^{(p+1)})\big\}b_p(x^{(p)}, \xi^{(p)})
r_i(P^{-(r-1)}\mathcal{A}(H)P) =\sum_{\{i,i_2,\ldots,i_r\}\in E(H)}\frac{\sqrt[r]{d_id_{i_2}\cdots d_{i_r}}}{d_i}& \leq\max_{\{i_1,i_2,\ldots,i_r\}\in E(H)}\left\{\sqrt[r]{d_{i_1}d_{i_2}\cdots d_{i_r}}\right\}.
{\mathcal N}: =\{u\in W^{s, p}_0(\Omega)\setminus\{0\}: \langle J'(u), u\rangle =0\},{\mathcal N}_+: =\{u\in{\mathcal N}: u^-=0\},{\mathcal N}_{\rm sc}: =\{u\in{\mathcal N}: u^\pm\neq 0, \langle J'(u), u^\pm\rangle =0\},
d_{\;\;\alpha_{0}}^{\alpha_{1}}=\left( \delta_{\;\;\beta_{1}}^{\alpha_{1}}+m_{\;\;\;\beta_{1}}^{\alpha_{1}\beta_{0}}\chi_{\beta_{0}}\right) \bar{D}_{\;\;\gamma_{1}}^{\beta_{1}}A_{\alpha_{0}}^{\;\;\gamma_{1}},
\Delta(D_{ij}) = \sum_{P \cap (k<l)}D_{ij}^{kl}\otimes D_{kl}^{12}- (a_{i5}a_{j6}+q^{-1}a_{i6}a_{j5}) \otimes D_{56}&- (1+q^{-2})\sum_{5\leq k \leq 6}a_{ik}a_{jk}\otimes D_{kk}\,.
\dot{z}_j(t)=(v-\frac{\lambda_j i}{\overline{2\pi(z-z_j(t)})})\Big|_{z=z_j(t)}=\bar{F}(z_j(t),t)-\sum_{k: k\neq j}\frac{\lambda_k i}{2\pi\overline{z_k(t)-z_j(t)}}
\left({D}^{\alpha}_{N}U^{n-\sigma}, U^{n, \sigma}\right) \, + \, a\big(l(U^{n, \sigma})\big) \, (\nabla U^{n, \sigma}, \nabla U^{n, \sigma}) \, = \, (f^{n-\sigma}, U^{n, \sigma}), \; \, \forall v_h \in V_h.
F_{Y}\Bigl(\sum_{i=1}^{k}\frac{t_i}{10^i}\Bigr) \le \prod_{i=1}^k\sup_{|\gamma|\le 10^{-J-1}}\frac{1}{9}\Bigl|\frac{e(\sum_{j=0}^{J}\frac{t_{i+j}}{10^{j}}+10\gamma)-1}{e(\sum_{j=0}^{J}\frac{t_{i+j}}{10^{j+1}}+\gamma)-1}-e\Bigl(a_0\sum_{j=0}^{J}\frac{t_{i+j}}{10^{j+1}}+a_0\gamma\Bigr)\Bigr| &=\prod_{i=1}^k G(t_i,\dots,t_{i+J}),
\lambda_{1,K^4}= \frac{2}{\pi}\left(\frac{17(n-1)(n-3)}{180}\int_{-\pi/2}^{\pi/2}x^6\cos^2(x)dx-\int_{-\pi/2}^{\pi/2}(y'_{1,2})^2dx +\int_{-\pi/2}^{\pi/2}y_{1,2}^2dx\right).
<\hat{{\cal{J}}}_{\mu_{1}\cdots \mu_{s}}^{(s)}(x)~\hat{{\cal{J}}}_{\nu_{1}\cdots \nu_{s}}^{(s)}(0)>={\prod}_{\mu_{1}\cdots \mu_{s},\nu_{1}\cdots \nu_{s}}^{(s)}\left({\frac{c_{s}(t)}{x^{4}}}\right) .
\textbf{II}\geq C \|f_\lambda\|_{L^{2n/(n+\alpha)}(B_\delta(0)}\|f_\lambda\|_{L^{2n/(n+\alpha)}(\mathbb{R}^n\backslash B_\delta(0))}=O(\frac\delta\lambda)^{-\frac{n+\alpha}2}\quad\lambda\rightarrow 0.
\|\hat{V}\Lambda - \Sigma\hat{V}\|_{\mathrm{F}}= \|VV^T\hat{V}\Lambda + V_1V_1^T \hat{V}\Lambda - V\Lambda V^T\hat{V}- V_1 \Lambda_1 V_1^T\hat{V}\|_{\mathrm{F}}&\geq \|V_1V_1^T\hat{V}\Lambda - V_1\Lambda_1 V_1^T\hat{V}\|_{\mathrm{F}}\geq \|V_1^T\hat{V}\Lambda - \Lambda_1 V_1^T\hat{V}\|_{\mathrm{F}},
\binom{r-j-1}{r-t-1}=\binom{r-j-1}{t-j}=\frac{(r-j-1)(r-j-2)...(r-t)}{(t-j)!}&=(-1)^{t-j}\frac{(-(r-t))(-(r-t+1))...(-(r-j-1))}{(t-j)!}=(-1)^{t-j}\binom{-r+t}{t-j},
\alpha_{\psi}^3 \cdot (\ast \alpha_{\psi}^3) = (\ast \alpha_{\psi}^3) \cdot \alpha_{\psi}^3, \alpha_{\psi}^3 \cdot \psi =_1 \cdot \psi, (\ast \alpha_{\psi}^3) \cdot \psi =_2 \cdot \psi,
\theta^n_K - \frac{n(n-1)}{12}\left(\frac{1+2K}{1-K}\right) \theta_K^{n-1}+\sum_{r = 1}^{n-2}\left(\frac{p_{nr}(K)}{(1-K)^{\rho(n,r)}}+ \sum_{k = r}^{n-2}\sum_{2a+3b = n-k}\frac{c_{ab}p_{kr}(K)}{(1-K)^{a+b+\rho(k,r)}}\right)\theta_K^r + &\quad+\sum_{2a+3b = n}\frac{c_{ab}}{(1-K)^{a+b}}.
\psi_n(t)=\log\left(\prod_{k=0}^{n-1}\frac{\gamma[\gamma_k(t)]}{\gamma_k(t)}\right)=\sum_{k=0}^{n-1}\log\frac{\gamma[\gamma_k(t)]}{\gamma_k(t)}=\sum_{k=0}^{n-1}\psi[\gamma_k(t)].
\langle + \infty | - \infty \rangle_{\lambda + \delta \lambda}= \langle + \infty | - \infty \rangle_{\lambda}- \delta \lambda \frac{1}{\tilde{N}}{\int}d \mu [ B ( t ) X^{\omega}( t ) + i g \bar{c}( t ) Y^{\omega}( t ) c ( t ) ] \e x p \{i{\int}({\cal L}+{\cal L}_{g}) d t \}
\Pi^{\rm one-loop}_{B \cdot P}(p^2) ={Ng^2m \over 8\pi}\left[ \left(-\frac{27}{16}\frac{p^2}{m^2}+ \frac{9}{4}\right) \frac{2m}{p}\tan^{-1}\frac{p}{2m}-\frac{3}{4}\right]
\times \exp \left( \sum_{b , b^{\prime}= 1}^{N}\sum_{j = 1}^{n_{b}}\sum_{j^{\prime}= 1}^{m_{b^{\prime}}}2 \frac{\pi}{\pi + g N}\frac{1}{N}\mathrm{K}_{0}\Big ( e \sqrt{\frac{N}{\pi + g N}}| x_{j}^{( b )}- y_{j^{\prime}}^{( b^{\prime})}| \Big ) \right)
\psi^0(y)=\frac{(n-2)\alpha_n}{2}\frac{|y|^2-1}{(1+|y|^2)^{\frac{n}{2}}},\psi^h(y)= (n-2)\alpha_n\frac{y_h}{(1+|y|^2)^{\frac{n}{2}}}, \quad\mbox{for}\h=1,\cdots,n.
a_{1}=-\frac{1}{2}\left( a^{4}-2a^{3}b-2a^{2}b^{2}+8ab^{3}-6b^{4}\right) , a_{2}=\frac{1}{4}b\left( a-2b\right) \left( a-b\right)^{2}\left(a^{2}-3ab+3b^{2}\right) \left( a^{2}-2ab+2b^{2}\right) , a_{3}=-\frac{1}{8}ab\left( a-2b\right) \left( a-b\right)^{5}\left(a^{2}-3ab+3b^{2}\right) \left( a^{2}-2ab+2b^{2}\right) .
\frac1{\omega(B)}\|(b-b_B)f_1\|^s_{L^s_\omega}=\frac1{\omega(B)}\int_{5B}|b(y)-b_B|^s|f|^sd\omega(y) &\le{\Big(\frac1{\omega(B)}\int_{5B}|b(y)-b_B|^{ss_1'}d\omega(y) \Big)^{\frac1{s_1'}}}\Big(\frac1{\omega(B)}\int_{5B}|f|^{ss_1}d\omega(y)\Big)^{\frac1{s_1}}\lesssim \ell(B)^{\beta s}\|b\|^s_{\Lambda^\beta}M^s_{ss_1}(f)(x)
F_0(z,s) = y^s + \frac{\Lambda(2s-1)}{\Lambda(2s)}y^{1-s}+ 2\sqrt y\sum_{n\neq 0}\frac{|n|^{s-\frac 12}\sigma_{1-2s}(|n|)}{\Lambda(2s)}K_{s-\frac 12}(2\pi|n|y)e(nx),
\bar{\eta}{}^{A}= \frac{i}{\sqrt{2}}\left[ \begin{array}{l}{\cos \left( \frac{\tau}{2}- \frac{\pi}{4}\right) \mathrm{e}^{i \sigma / 2}}\\{\sin \left( \frac{\tau}{2}- \frac{\pi}{4}\right) \mathrm{e}^{- i \sigma / 2}}\\ \end{array}\right] .
\begin{gathered}Q(x_0,x_1,x_2,x_{12},\alpha,\beta)=0\,, Q(x_0,x_1,x_3,x_{13},\alpha,\gamma)=0\,, Q(x_0,x_2,x_3,x_{23},\beta,\gamma)=0\,, Q(x_{13},x_3,x_{123},x_{23},\alpha,\beta)=0\,, Q(x_{12},x_2,x_{123},x_{23},\alpha,\gamma)=0\,, Q(x_{12},x_1,x_{123},x_{13},\beta,\gamma)=0\,, \end{gathered}
\gamma_k=\frac{a_k^2\lvert\mathbf{\hat{h}}_k^{\textrm{T}}\mathbf{p}_k\rvert^2}{\sum\limits_{\substack{i=1 i\neq k}}^K a_i^2\lvert\mathbf{h}_k\mathbf{p}_i\rvert^2+\sigma_w^2}.
Y_s^{t,\eta}\ = \G(\mathbb W_T^{t,\eta}) + \int_s^T F(r,\mathbb W_r^{t,\eta},Y_r^{t,\eta},Z_r^{t,\eta}) dr - \int_s^T Z_r^{t,\eta}dW_r, t \leq s \leq T.
\frac{d\Delta_{m+1}f}{dv}(v)=v\Delta_{m+1}B(v)-\frac{1}{v^2}(1+vA_{m+1}(v))\int_0^ve^{-\int_{v'}^vA_{m+1}(v'')dv''}v'^2\Delta_{m+1}B(v')dv'.
{\cal L}_{S C M}= \frac 1 2 \! \left( J_{\mu}^{j}J^{\mu j}+ i \overline{{\chi}}^{j}/ \kern - . 5 e m \partial \chi^{j}+ i \varepsilon^{j k l}\overline{{\chi}}^{j}/ \kern - . 5 e m J^{k}\chi^{l}+ \frac 1 4 ( \bar{\chi}^{j}\chi^{j})^{2}\right) .
\sum_{l=1}^{2k}\frac{(-1)^{l-1}}{l}\|\partial^{(l-1)}_{c}\phi^\prime\|^{2}_{\alpha^{\otimes l}}\le \langle\mathcal{N}\phi,\phi \rangle\le \sum_{l=1}^{2k-1}\frac{(-1)^{l-1}}{l}\|\partial^{(l-1)}_{c}\phi^\prime\|^{2}_{\alpha^{\otimes l}}.
\mathnormal{H}_{2n+1}^{N_{f}}(x)=(-1)^{n}(\bar{h}_{n}^{N_{f}})^{-\frac{1}{2}}c^{\frac{3+N_{f}}{4}}\x \ \mathnormal{L}_{n}^{\frac{N_{f}+1}{2}}(cx^{2}),\qquad \bar{h}^{N_{f}}_{n}=\frac{\Gamma(\frac{N_{f}}{2}+n+\frac{3}{2})}{n!},
(A_{d}(\delta_{n})) = \int_{\{d(b(\theta),b_{0}) \geq \epsilon^*+ \delta_{n}\}}P_{\theta}\left[d(\eta(\mathbf{z}), \eta (\mathbf{y}) ) \leq \epsilon_n \right] d\Pi(\theta) & \leq \int_\Theta P_\theta \left[d(b(\theta),\eta(\mathbf{z})) \geq \delta_{n}/4 \right] d\Pi(\theta) .
\int_M \langle \nabla f(0), \nabla f(0) \rangle \: d\mu_0 \: = \:\int_M \langle \nabla f(1), \nabla f(1) \rangle \: d\mu_1 \: = \:\int_M \langle \nabla \eta_1, \nabla \eta_1 \rangle \: d\mu_1.
\bigg|\int \limits_{|a_n|}^{|b_n|}\int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}\frac{1}{\sqrt[4s+3]{\sum \limits_{j=1}^{n}x^{4s+3}_j}}dx_1dx_2\cdots dx_n\bigg| \geq \frac{2\pi \times |\log (\langle a,b \rangle)|\bigg|\prod_{j=1}^{n}|b_j|-|a_j|\bigg|}{||\vec{a}||^{4s+4}+||\vec{b}||^{4s+4}}.
a_{1 \mu}= \frac{1}{\sqrt{2}}( v_{\mu}^{( 1 )}+ v_{\mu}^{( 2 )}) , \hspace{5 m m}a_{2 \mu}= \frac{1}{\sqrt{2}}( v_{\mu}^{( 1 )}- v_{\mu}^{( 2 )}) ,
\begin{array}{cc}\mathring{K}^p_4[H_0]\geq \frac{1}{C_1}e^{p-1}[H_0] \textup{in}\ (R_0,\infty), \Big|\mathring{K}^p_4[H_0]\Big|\lesssim e[H_0] \textup{in}\ [R_0-M,R_0], \mathring{K}^p_4[H_0]=0 \textup{in}\ [2M,R_0-M).\end{array}
\begin{array}{ll@{}ll}\displaystyle x_k - x_\ell \displaystyle v_{ij}- x_j \leq t_i \,, i\in[m] j\in[n] \displaystyle x_{1}+ \dots + x_{n}= 0 \displaystyle t_{1}+ \dots + t_{m}= p^*/n \end{array}
\frac{1}{\mu}\left(1-\frac{N_{f}}{N_{c}}\right)m^{\frac{2N_{c}-N_{f}}{N_{c}-N_{f}}}+m_{f}m^{\frac{N_{c}}{N_{c}-N_{f}}}-\Lambda^{\frac{3N_{c}-N_{f}}{N_{c}-N_{f}}}=0.
\begin{aligned}\Delta\log\frac{\rho}{e^{-\Psi}}=\mbox{div}\frac{\nabla\frac{\rho}{e^{-\Psi}}}{\frac{\rho}{e^{-\Psi}}}=-\frac{\bigg|\nabla\frac{\rho}{e^{-\Psi}}\bigg|^{2}}{\bigg(\frac{\rho}{e^{-\Psi}}\bigg)^{2}}+\frac{\Delta\frac{\rho}{e^{-\Psi}}}{\frac{\rho}{e^{-\Psi}}}=:I_1+I_2. \end{aligned}
\dot{D}= \begin{pmatrix}1 1 1\end{pmatrix}A_1 A_2 A_3 + \dfrac{1}{A_1 A_2 A_3}\begin{pmatrix}-A_1^2 A_2^2 A_3^2 A_1^2 -A_2^2 A_3^2 A_1^2 A_2^2 -A_3^2\end{pmatrix}D.
\underline{\underline{A_{1}}}=[\begin{matrix}-1&0&0\\ 0&1&0\\ 0&0&1\end{matrix}];\underline{\underline{A_{2}}}=[\begin{matrix}1&0&0\\ 0&-1&0\\ 0&0&1\end{matrix}];\underline{\underline{A_{3}}}=[\begin{matrix}1&0&0\\ 0&1&0\\ 0&0&-1\end{matrix}]
\mathbf{r}_{\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3}}=\left(\begin{array}{c}1,e^{\int \tau (s)ds}+\int \cosh \left( \int \tau (s)\,ds\right) \kappa(s)ds, e^{\int \tau (s)\,ds}+\int \sinh \left( \int \tau (s)\,ds\right) \kappa (s)ds\end{array}\right).
d_i = (d^v_i p_0, d^v_{i-1}y_1, \dots, d^v_1p_{i-1}, d_i^h p_{i+1}, \dots, d^h_ip_n), s_i = (s^v_i p_0, s^v_{i-1}p_1, \dots, s^v_0p_i,s^h_i p_i, s^h_i p_{i+1}, \dots, s^h_i p_n)
\, - \frac{1}{3 !}\int d^{3}x_{1}d^{3}x_{2}d^{3}x_{3}{\mathcal W}_{\mu_{1}\mu_{2}\mu_{3}}^{( 3 )}( x_{1}, x_{2}, x_{3}) b_{\mu_{1}}( x_{1}) b_{\mu_{2}}( x_{2}) b_{\mu_{3}}( x_{3}) + \ldots
\|\Pi_R y_h-\Pi_R^h\, y_h\|_{L^2(\Omega)}= \|\gamma p_s-\gamma_h p_s^h\|_{L^2(\Omega)}\le |\gamma-\gamma_h|\,\|p_s^h\|_{L^2(\Omega)}+ |\gamma|\,\|p_s-p_s^h\|_{L^2(\Omega)}
\frac{\partial (X.Y)}{\partial q_{j}}=\beta(1-\beta)\left[\left(\frac{\sum_{i}p_{i}q^{\beta-1}_{i}}{\sum_{i}q^{\beta}_{i}}\right)^{\beta}q^{\beta-1}_{j}-\left(\frac{\sum_{i}p_{i}q^{\beta-1}_{i}}{\sum_{i}q^{\beta}_{i}}\right)^{\beta-1}p_{j}q^{\beta-2}_{j}\right]
f^{*}\omega=\sum_{i_{1}<\cdot\cdot\cdot<i_{k}}\sum_{j_{1}<\cdot\cdot\cdot<j_{k}}(\omega_{i_{1}\cdot\cdot\cdot i_{k}}\circ f)\frac{\partial(f_{i_{1}},...,f_{i_{k}})}{\partial(x^{j_{1}},...,x^{j_{k}})}dx^{j_{1}}\wedge\cdot\cdot\cdot\wedge dx^{j_{k}}.
\kappa_m\left(\sum_{i=1}^n c_i Z_{\gamma_1,\gamma_2}(t_i)\right)\sim (m-1)! \left(\sum_{i=1}^n c_i t_i\right)^m \sum_{0\le r\le m/2}{m\choose 2r}U(\gamma_1,\gamma_2;m,r).