harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S_n(z_1, z_2) = \frac{(2n^2)^{2m-s}}{\frac{h_{n-1}^{(s)}k_{n+2m}^{(s)}}{k_{n-1}^{(s)}}\Big[ \delta_n^{(n)}\Big]^2}\times \times \frac{D_n^{(n)}(1- \frac{z_1}{2n^2}) \cdot D_{n-1}^{(n)}( 1- \frac{z_2}{2n^2}) - D_n^{(n)}(1- \frac{z_2}{2n^2}) \cdot D_{n-1}^{(n)}( 1- \frac{z_1}{2n^2})}{z_2 - z_1}.
\mathcal{E}_N := \frac{T}{2}\sum_{k=N+1}^\infty \|\boldsymbol{y_d}_k\|_{\Omega}^2 = \frac{T}{2}\sum_{k=N+1}^\infty \left(\|{y_d^c}_k\|_{\Omega}^2 + \|{y_d^s}_k\|_{\Omega}^2\right)
\frac{\partial}{\partial y}\frac{\partial^{2}z}{\partial x\partial y}=\frac{\partial^{3}z}{\partial y\partial x\partial y}=\frac{\partial^{3}z}{\partial x\partial y\partial y}=\frac{\partial}{\partial x}\frac{\partial^{2}z}{\partial y^{2}}=0
W(u) =\left[a^+\int_{B_1}|\nabla u_+|^2\,dx+a^-\int_{B_1}|\nabla u_-|^2\,dx\right] &-\left[a^+\int_{\partial B_1}u_+^2\,d\mathcal H^{d-1}+a^-\int_{\partial B_1}u_-^2\,d\mathcal H^{d-1}\right]+|\Omega_u^+\cup\Omega_u^-\cap B_1|.
\dim \{V^{n-\beta}WU^{\beta-1}\}\geq \dim H^{\beta-1}V^{n+1-2\beta}W &\geq 3(n+1-2\beta)+ (\beta-1)\dim H+2 &\geq 3n+5-6\beta+4(\beta-1) &=3n+1-2\beta\,,
|I_{1,\rho}(x)| \leq\int_{|xy^{-1}|<R}\frac{\rho(|xy^{-1}|)}{(1+|xy^{-1}|)^{\gamma}}|f(y)|dy\leq\int_{|xy^{-1}|<R}\frac{\rho(|xy^{-1}|)}{|xy^{-1}|^{\gamma}}|f(y)|dy = \sum_{k=-\infty}^{-1}\int_{2^{k}R\leq|xy^{-1}|<2^{k+1}R}\frac{\rho(|xy^{-1}|)}{|xy^{-1}|^{\gamma}}|f(y)|dy.
\sum_{k=1}^{n-3}\sum_{\ell=k+1}^{n-2}\binom{n-\ell-2}{2}(p-1)(p-2)(p-3)p^{n-4}&= (p-1)(p-2)(p-3)p^{n-4}\sum_{k=1}^{n-3}\sum_{\ell=k+1}^{n-2}\binom{n-\ell-2}{2}&= (p-1)(p-2)(p-3)p^{n-4}\sum_{k=1}^{n-3}\binom{n-k-2}{3}&= (p-1)(p-2)(p-3)p^{n-4}\binom{n-2}{4}.
B:=\langle x\rangle^{\frac{1+\mu}{2}}\langle D\rangle^{-1/2}\langle x\rangle^{-(1+\mu)}\widetilde{A_1}\cdot\nabla \langle D\rangle^{-1/2}\langle x\rangle^{\frac{1+\mu}{2}}\in\mathcal{B}(L^2).
I(K_{\overline{U}}; V_B,V_C|U=W ) \leq I(K_{\overline{U}}; V_B,V_C, J|U=W) = \sum_{j=0,1}Pr[J=j] \,I(K_{\overline{U}}; V_B,V_C| J=j,U=W) + I(K_{\overline{U}}; J|U=W) .
\infty >\int_{-1}^{1}\left\vert f(x)\right\vert^{2}(1-x)^{-2}(1+x)^{\beta}\,dx\geq\int_{x^{\ast}}^{1}\left\vert f(x)\right\vert^{2}(1-x)^{-2}(1+x)^{\beta}\,dx & \geq\left( \frac{c}{2}\right)^{2}\int_{x^{\ast}}^{1}(1-x)^{-2}(1+x)^{\beta}\,dx=\infty\,,
\delta_h*\delta_l:= (\delta_h*_n\delta_l)|_{D_n\setminus\{e_n\}}+ + (\delta_h*_n\delta_l)(\{e_n\})\Bigl[\sum_{k=2}^{n-1}\Bigl( \prod_{m=k+1}^{n-1}\omega_{D_m}(\{e_m\})\Bigr)\cdot \omega_{D_k}|_{D_k\setminus\{e_k\}}+\Bigl( \prod_{m=2}^{n-1}\omega_{D_m}(\{e_m\})\Bigr)\cdot \omega_{D_1}\Bigr].
a_{jk}= M\int_0^M\phi_j(\eta)\,d\eta\int_0^M\phi_k(\eta')\,d\eta' - \int_0^M\phi_j(\eta)\,d\eta\int_0^M\eta'\phi_k(\eta')\,d\eta'- J_{jk}, \mbox{where}J_{jk}= \int_0^M\int_0^M(\eta-\eta')_+\phi_j(\eta)\phi_k(\eta')\,d\eta d\eta',
G_{m,1}\leq \begin{cases}1 \L_1 - 2\L_2 \leq 1 (1/\sqrt{\L_1 - 2\L_2})\left[ 2 - (1/\sqrt{\L_1-2\L_2}) \right] \L_1 - 2\L_2 > 1\end{cases}, G_{m,2}\leq 1 \wedge \frac{1}{\L_1 - 2\L_2}\left[ \frac{1}{4(\L_1 - 2\L_2)}+ \log^+2(\L_1 - 2\L_2) \right].
I_s={1\over4\epsilon^2}{\Gamma(1+2\epsilon)\Gamma(1-\epsilon)^3\over(1-3\epsilon)(1-2\epsilon)\Gamma(1-3\epsilon)}={1\over 2}\left[{1\over 2\epsilon^2}+{1\over2\epsilon}+{1\over\epsilon}(2-C)\right]
e_{\Sigma_3}(B_r) = \, \left(\frac{(q^2-1)^r}2+\frac16 (q-1)^{2r}+ \frac13 (q^2+q+1)^r - (q-1)^r \right) T + \left(- \frac{(q^2-1)^r}2+\frac16 (q-1)^{2r}+ \frac13 (q^2+q+1)^r\right) S +\left( \frac13 (q-1)^{2r}- \frac13 (q^2+q+1)^r -(q-1)^r+3^r\right) V .
\alpha_0^{+}(F_p)= \;r-\#(\mathrm{Supp}(\widetilde{R}_h)\cap \mathrm{Supp}(\widetilde{\Gamma}_p)) = \; \sum_{i}m_i\widetilde{R}_hG_i-\#(\mathrm{Supp}(\widetilde{R}_h)\cap \mathrm{Supp}(\cup_i G_i)) \ge \sum_{i}(m_i-1)\widetilde{R}_hG_i.
\sqrt{F}/F =\langle d_8\rangle F\times\langle1+r_4\rangle F\times\langle1+r_4+r_6\rangle\times\langle1+r_4+r_2\rangle F\times &\times\langle1+r_4+r_6+(r_1+r_3+r_7)\rangle F\times &\times\langle1+r_4+r_2+(r_3+r_5+r_7)\rangle F\times &\times\langle1+r_4+r_2+(r_1+r_3+r_5)\rangle F\times &\times\langle1+r_4+r_6+(r_1+r_5+r_7)\rangle F.
K_W(Z) =\frac{1}{(1-W^{\dagger_3}Z)^2}=e_1\frac{1}{(1-\overline{w}_1z_1)^2}+e_2\frac{1}{(1-\overline{w}_2z_2)^2}= e_1K_{w_1}(z_1)+e_2K_{w_2}(z_2),
P_k^0 =\langle\{x=x_1^{a_1}x_2^{a_2}\ldots x_k^{a_k}\ : \a_1a_2\ldots a_k=0\}\rangle, P_k^+ = \langle\{x=x_1^{a_1}x_2^{a_2}\ldots x_k^{a_k}\ : \a_1a_2\ldots a_k>0\}\rangle.
\pi_i(x_1\otimes\cdots \otimes x_n) = x_1 \otimes \cdots \otimes x_{i-1}\otimes x_{i+1}\otimes x_{i}\otimes x_{i+2}\otimes \cdots \otimes x_{n}, n \geq2, &\widehat{\pi}_{n-1}(x_1\otimes\cdots \otimes x_n)= x_1\otimes x_2\otimes\cdots \otimes x_{n-2}\otimes \overline{x}_{n}\otimes \overline{x}_{n-1}, n \geq 2.
\int_R^{2R1}\frac{v(r) r}{(r^2-R^2)^{1/2}}\,dr \leq \left( \int_R^{2R}v(r)^p r^{p-1}\,dr \right)^{1/p}\left( \int_R^{2R}\left(\frac{r}{\sqrt{r^2-R^2}}\right)^{p'}\frac{dr}{r}\right)^{1/p'}& = c_p \left( \int_R^{2R}v(r)^p r^{p-1}\,dr\right)^{1/p}\,.
\left.\begin{array}{ll}\displaystyle\dfrac{a_2}{m_1}\int_{\mathbb{R}^N}|\nabla \xi_n|^2dx+\dfrac{a_1}{2m_2}\int_{\mathbb{R}^N}|\nabla \eta_n|^2dx-\dfrac{N}{2}a_1a_2\int_{\mathbb{R}^N}\eta_n\xi_n^2dx=0.\end{array}\right.
|\int_{x_{1}}^{x_{2}}\frac{1}{h}(f(x+h)-f(x))dx|\le\int_{x_{1}}^{x_{2}}\frac{1}{h}|f(x+h)-f(x)|dx\le\int_{a}^{b}\frac{1}{h}|f(x+h)-f(x)|dx<\epsilon
S_{2 2}= \left\langle \frac{2 \pi}{H}\right\rangle \left\langle \frac{2 \pi}{3}\right\rangle \left\langle \frac{2 \pi}{H}+ \frac{\pi}{3}\right\rangle \left\langle \frac{\pi}{3}- \frac{4 \pi}{H}\right\rangle \left\langle \frac{\pi}{3}- \frac{6 \pi}{H}\right\rangle \left\langle \frac{4 \pi}{H}- \pi \right\rangle \left\langle \frac{6 \pi}{H}- \pi \right\rangle \ ,
J_{1}=-\frac{1}{2}\int_{F_{ji}}f_{ji}(x)dS_{ji}(x)-\frac{1}{2}\int_{F_{ij}}f_{ij}(x)dS_{ij}(x). =-\frac{1}{2}\left(\int_{F_{ij}}f_{ij}(x)dS_{ij}(x)+f_{ji}(x)dS_{ji}(x)\right).
Z_{\mathrm{1 - l o o p}}^{( c )}( M ) ={\frac{( 2 \pi x )^{{\frac{1}{2}}( \mathrm{d i m}\, H_{A^{( c )}}^{0}- \mathrm{d i m}\, H_{A^{( c )}}^{1})}}{\mathrm{v o l}( H_{c})}}\mathrm{e}^{-{\frac{1}{x}}S_{\mathrm{C S}}( A^{( c )}) -{\frac{i \pi}{4}}\varphi}{\sqrt{| \tau_{R}^{( c )}|}},
V \left( M \right) = - C \left( N_{c}, N_{f}\right) \left[ \frac{\Lambda_{Q C D}^{\frac{1 1}{3}N_{c}- \frac{2}{3}N_{f}}}{\mathrm{d e t}\, M}\right]^{\frac{1 2}{1 1 \left( N_{c}- N_{f}\right)}}\ ,
\prod_{j=1}^m \sigma_{(n-1)(m-j)}(y_j, \ldots, y_m) = \prod_{1 \leq j < k \leq m}\dfrac{y_j^n - y_k^n}{y_j - y_k}= \dfrac{a_{n\delta}}{a_{\delta}}= s_{(n-1)\delta}(y_1, \ldots, y_m).
\frac{P_{X/T}(m)}{m^{2n-k-1}}\leq \frac{P_M(m)}{m^{2n-k-1}}= \frac{P_M(m)}{m^{2(n-k)-1}}\cdot \frac{1}{m^k}\; \overset{m\to \infty}{\longrightarrow}\; 0
\bar{S}_{i j}^{- 1}( \vec{x}, \vec{y}) = \eta_{i j}\, \left( \begin{array}{c c}{{0}}&{{1}}\\{{- 1}}&{{0}}\\ \end{array}\right) \, \frac{1}{\nabla^{2}}\, \delta ( \vec{x}- \vec{y})
O = \left( \begin{array}{c c c c}{{x_{3}}}&{{0}}&{{\sqrt{1 - x_{3}^{2}}}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{0}}\\{{- x_{1}}}&{{0}}&{{\frac{x_{1}x_{3}}{\sqrt{1 - x_{3}^{2}}}}}&{{\frac{x_{2}}{\sqrt{1 - x_{3}^{2}}}}}\\{{x_{2}}}&{{0}}&{{- \frac{x_{2}x_{3}}{\sqrt{1 - x_{3}^{2}}}}}&{{\frac{x_{1}}{\sqrt{1 - x_{3}^{2}}}}}\\ \end{array}\right) .
d s_{\mathrm{s t r}}^{2}= - d t^{2}+ e^{- 4 U}d \vec{x}^{2}\ , \qquad e^{- 4 U}= \chi = e^{4 \phi}= ( 1 +{\frac{2 M}{r}})^{2}\ , \qquad M = P / \sqrt 2
r_n(G)\left(\overline\bigotimes^{s-1}_{i=0}[\langle1,-g_{1,i}\rangle]\overline\otimes...\overline\otimes[\langle1,-g_{n,i}\rangle]\overline\otimes I^{n+1}(G)\right):= \sum^{s-1}_{i=0}l(-1)^{2^{n-1}-n}l(g_{1,i})\otimes...\otimes l(g_{n,i})+\mathcal Q_{2n-1}(G)
wz = \e-e^2-e^2(1-\sigma_i(e))x_i + e(1-\sigma_i(e))x_i - e(1-\sigma_i(e))x_ie - e(1-\sigma_i(e))x_ie(1-\sigma_i(e))x_i = \ -e(1-\sigma_i(e))(\sigma_i(e)x_i + \delta_i(e)) - e(1-\sigma_i(e))[\sigma_i(e(1-\sigma_i(e)))x_i + \delta_i(e(1-\sigma_i(e)))]x_i.
\max_{\mathbf{q},\mathbf{V},\rho_{s,k}}\min_{k,l}R_{k}= R_{s,k}- R_{e,l}^{\textrm{UP}}s.t. ~ \min_{k}E_{s,k}\geq \bar{E}_{s}, ~\min_{l}E_{e,l}\geq \bar{E}_{e},~ \forall k,l, &~ \|\mathbf{q}\|^{2}+ \textrm{tr}(\mathbf{V}) \leq P_{\textrm{total}}, &~ 0 < \rho_{s,k}\leq 1, \mathbf{V}\succeq \mathbf{0}.
u_i^2 = 0 i=1,2,\dotsc,n-1, u_iu_j = u_ju_i i,j = 1,\dotsc,n-1 |i-j|>1, u_iu_{i+1}u_i = u_{i+1}u_iu_{i+1}i=1,2,\dotsc,n-2.
\tau_{em}= \frac{1}{1 + b \chi + \chi^2}\left( \begin{array}{ccc}\chi^2 -2 -\chi -\frac{1}{2}(b+\chi )^2 - \frac{b+ \chi}{2}2 \chi 4(b+ \chi ) -1 + b \chi + \chi^2 \end{array}\right)
e^{-2\psi_2}= \left| \frac{D_2 Q_1 Q_2}{\sqrt{c_1 c_2}}\frac{\sinh \left[ \sqrt{c_2}(I(A) +\tilde{c}_2 ) \right] \sinh \left [\sqrt{c_1}(I(A) + \tilde{c}_1 ) \right ]}{\sinh^2 \left[\sqrt{D_2}(I(A)+\tilde{c}) \right]}\right|^{1/2}e^{-(f_0 I(A)+f_1)/\sqrt{2}},
\mathbb{E}\left[A_{j}^{(N+1)}\mid\mathcal{A}^{(l)}\right]=\left( \frac{P(A_{j}^{(N+1)}\cap A_{1}^{(l)})}{P( A_{1}^{(l)})},...,\frac{P( A_{j}^{(N+1)}\cap A_{m_{l}}^{(l)})}{P(A_{m_{l}}^{(l)})}\right)^{t},
G(x,0) = \frac{{\displaystyle \int}{\cal D}S{\cal D}{\bar \psi}{\cal D}\psi ~{\bar \psi}(x)~e^{-2i\gamma^5 S(x)}~ \psi (x)~{\bar \psi}(0)~{\displaystyle e}^{-2i\gamma^5 S(0)}~ \psi (0) \,\,\, e^{iS_{eff}}}{{\displaystyle \int}{\cal D}S{\cal D}{\bar \psi}{\cal D}\psi \,\,\,{\displaystyle e}^{iS_{eff}}},
{}_3\phi_2\begin{bmatrix}q^{-N}, abt/c, b ;q,q bt, bq^{1-N}/c.\end{bmatrix}=\frac{(c,t;q)_N}{(c/b, bt; q)_N}{}_3\phi_2\begin{bmatrix}q^{-N}, a, b ;q,q c, q^{1-N}/t\end{bmatrix}
g_{\alpha\beta}(\dot{y}^{\alpha}-\Gamma^{\alpha})=\frac{d}{dt}\left( \frac{\partial L}{\partial y^{\beta}}\right)-\rho^{i}_{\beta}\frac{\partial L}{\partial x^{i}}+C^{\gamma}_{\beta\nu}y^{\nu}\frac{\partial L}{\partial y^{\gamma}}
u_{k}=e^{-\overline{\xi}\overline{\omega}t_{k}}(\overline{c}_{1}cos\overline{\omega}_{d}t_{k}+\overline{c}_{2}sin\overline{\omega}_{d}t_{k}),\overline{\omega}_{d}=\overline{\omega}\sqrt{1-\overline{\xi}^{2}}
s = K - \sqrt{\frac{K}{\rho_V}}\sqrt{\mu/\lambda}+ \frac{3+\rho_V-2K}{2\rho_V}\mu/\lambda &+ O (\mu/\lambda)^{3/2}) P_b = 1 - (K-1)\left(\frac{\lambda}{\mu}\right)^{-1}+ \frac{(\rho_V+1)(K-1)}{\sqrt{\rho_V K}}\left(\frac{\lambda}{\mu}\right)^{-3/2}&+ O \left(\left(\frac{\lambda}{\mu}\right)^{-2}\right)
\frac{\ell^n-1}{\sum_{i=0}^{n-1}e_i \ell^i}\geq \frac{\ell^n-1}{\sum_{i=0}^{n-2}\ell^i}= \frac{(\ell^n-1)(\ell-1)}{(\ell^{n-1}-1)}\geq \ell(\ell-1),
S(\lambda) = S_{}(\lambda) S^{-1}_{,0}(\lambda) = I + a_d \mathcal{E}(\lambda) (I + VR_0(\lambda))^{-1}V\mathcal{E^*}(\bar{\lambda}) : L^2(\mathbb{S}^{d-1}) \to L^2(\mathbb{S}^{d-1})
\lbrack f,g]_{\widehat{w}_{\alpha,\beta}}(x):=\frac{(1-x)^{\alpha+1}(1+x)^{\beta+1}}{(x-b)^{2}}\left( f(x)\overline{g}^{\prime}(x)-f^{\prime}(x)\overline{g}(x)\right) \quad(-1<x<1),
|V_\omega(y)-V_\omega(y')| \le \sum_{\ell=0}^{r(a)-1}|v(f^\ell y)-v(f^\ell y')|\le |v|_\eta\sum_{\ell=0}^{r(a)-1}d(f^\ell y,f^\ell y')^\eta \le C^\eta |v|_\eta r(a)d(Fy,Fy')^\eta\le C^\eta |v|_\eta r(a)d_\theta(Fy,Fy')= C^\eta \theta^{-1}|v|_\eta r(a)d_\theta(y,y'),
\begin{array}{l}M = ((m - 2) + \Gamma_{Q(H_2 )}(x - 2))I_{2k}+ (2k - 2)I_{2k}+ J_{2k}+ \Gamma_{Q(H_2 )}(x - 2)(I_k \otimes A(K_2 )) \;\;\;\;\;= \left({m - 2 + \frac{{m - 2}}{{x - 2 - 2r_2}}+ 2k - 2}\right)I_{2k}+ \frac{{m - 2}}{{x - 2 - 2r_2}}(I_k \otimes A(K_2 )) + J_{2k}. \end{array}
\tilde{\omega}_{0}^{3}= \left({\frac{2 m^{2}}{( r + m )^{2}}}-{\frac{2 m \lambda}{( r - m + \lambda ) ( r + m )}}\right) \sigma_{z}, \: \: \: \: \: \tilde{\omega}_{1}^{2}= \left({\frac{2 m^{2}}{( r + m )^{2}}}- 1 \right) \sigma_{z}.
\begin{cases}\begin{aligned}\lim_{m\rightarrow\infty}H^2(p_m)=\sup H^2=\bar H^2,\lim_{m\rightarrow\infty}|\nabla H^2(p_m)|=0, &0\geq\lim_{m\rightarrow\infty}|\nabla^{\perp}\vec H|^2(p_m)-\sum_{i,j,k,p}(\bar h^{p^{\ast}}_{ijk})^2+\dfrac12(\bar H^2-S)(\bar H^2-3S+2).\end{aligned}\end{cases}
\,=\, \left<\,f_{\,1},\, f_{\,2}\,|\, f_{\,3},\, \,\cdots,\, f_{\,n}\,\right>_{1}\,+\, \left<\,g_{\,1},\, g_{\,2}\,|\, g_{\,3},\, \,\cdots,\, g_{\,n}\,\right>_{2},
s_f = \boldsymbol{x}^T(t_f)F(t_f)\boldsymbol{x}(t_f)- \boldsymbol{x}^T(t_f) P(t_f)\boldsymbol{x}(t_f) &=\boldsymbol{x}^T(t_f) \big[F(t_f) - P(t_f) \big] \boldsymbol{x}(t_f)
(N_r V)_s = \sum_t \left[ \binom{s}{2}\binom{r-2}{r-t}+ \binom{r-s}{2}\binom{r-2}{t}\right] \left[ \binom{t}{2}+ \binom{r-t}{2}\right] =\binom{s}{2}P + \binom{r-s}{2}Q,
\bar\nabla^{H}_{\frac{\partial}{\partial t}}D(\pi\circ\Phi)\Big|_0 \! w =\pi^*\bar\nabla^{(\pi\circ\Phi)^{-1}TY}_{\frac{\partial}{\partial t}}D(\pi\circ\Phi) w\Big|_0\!-D(\pi\circ\Phi)\;\bar\nabla^{T(X\times\mathbb R)}_{\frac{\partial}{\partial t}}w
\ln \frac{\det{}^{\prime}D}{\det{}^{\prime}D^{\left(N\right)}}=-\Gamma \left[ a_\mu \right] -\overline{\Gamma}\left[ A_\mu^{\left( N\right)},a_\mu \right] +\ln \det \left| \left\langle \varphi_{0i}^{\left( N\right)^{\dagger}}\exp \left( 2\phi \gamma_5\right)\varphi_{0j}^{\left( N\right)}\right\rangle \right| ,
t_{1}^{-}=0,\,t_{n}^{-}=t_{n-1}^{-}+(\tau+s/\bar{v})\left[1+\left(\frac{C}{Gf^{\mbox{s}}}-1\right)\left(1-\alpha+\alpha\epsilon_{n}\right)\right],\forall n\in\mathcal{N}\backslash\{1\},
1 = P^0(\{S_m\}_{m=1}^{2n}\cap \{0,b_{i}\}= \emptyset) + \sum_{k=1}^{n}\frac{1}{\pi k}P^0(\{S_m\}_{m=1}^{2n-2k}\cap \{0,b_{i}\}= \emptyset)(1+o(1)) + P^{b_i}(\{S_m\}_{m=1}^{2n-1}\cap \{0,b_{i}\}= \emptyset) + \sum_{k=1}^{n-1}\frac{1}{\pi k}P^{b_i}(\{S_m\}_{m=1}^{2n-2k-1}\cap \{0,b_{i}\}= \emptyset)(1+o(1)).
\|v\|_{X}:=\|v\|_{L_t^\infty H_x^1\cap L_t^4 L_x^\infty}+ \|\langle x\rangle^{-\frac32}v\|_{L_x^\infty L_t^2}+ \|\partial_x v\|_{L_x^\infty L_t^2}, \|z\|_{Y}:=\|\dot z+iEz\|_{L_t^1 \cap L_t^2}.
\frac{\partial\,|f(x_1, x')|^2}{\partial x_1}=\frac{\partial}{\partial x_1}\left(f(x_1, x')\overline{f(x_1, x')}\right)=2\Re\left(\overline{f(x_1, x')}\,\frac{\partial f(x_1, x')}{\partial x_1}\right).
u(x) =\sup_{\tau\ge 0}\inf_{\sigma\ge 0}E_x\Big(\int_0^{\tau\wedge\sigma}f(\cdot,u)(X_r)\,dr &\quad+\int_0^{\tau\wedge\sigma}dA^\mu_r+h_1(X_\tau)\mathbf{1}_{\{\tau<\sigma\}}+h_2(X_\sigma)\mathbf{1}_{\{\sigma\le\tau\}}\Big),
A = (C - x_{n,1}) \cap (C-x_{n,2}) \cap \cdots \cap (C-x_{n,n}) \cap (C-z_n), B = \{y_1, \ldots y_\alpha\}\cup \{u_{i,j}: 2 \leq i < n, 1 \leq j \leq i\}\qquad\cup \{v_i : 2 \leq i < n\}\cup \{x_{n,i}- z_n : 1 \leq i \leq n\}, D = A \setminus (B \cup (B-x_{n,1}) \cup \cdots \cup (B-x_{n,n}) \cup (B-z_n) ).
\frac{1}{\pi}\int_{1}^{\infty}\frac{|D(\nu,\alpha)|}{\alpha+\tau}d\alpha\leq \frac{\frac{1}{\pi}\int_{1}^{\infty}\omega(\nu,\alpha)/\alpha d\alpha}{1+\frac{1}{\pi}\int_{1}^{\infty}\frac{\omega(\nu,\beta)}{\beta}d\beta}<1.
\mathcal{W}^{\mathrm{Wick}}:=\frac{1}{2}\int dx\,dy\, \bigl(|\varphi(x)|^2-\mathbb{E}_{\mu}[|\varphi(x)|^2]\bigr)\,w(x-y)\,\bigl(|\varphi(y)|^2-\mathbb{E}_{\mu}[|\varphi(x)|^2]\bigr)\,,
x_{\varepsilon_i-\varepsilon_j}=E_{i,j}, h_{\varepsilon_i-\varepsilon_j}=E_{i,i}-E_{j,j}, \forall\,\, 1\leq i\neq j\leq r+1, & x_{i,j}^+=x_{\alpha_{i,j}}^+=x_{\varepsilon_i-\varepsilon_{j+1}}, x_{i,j}^-=x_{\alpha_{i,j}}^-=x_{\varepsilon_{j+1}-\varepsilon_i}, \forall\,\,1\leq i\leq j\leq r.
\begin{aligned}\mathbb{E}T^{\rm Proac}_j=\tau^{\rm Proac}_{m}\mathbb{E}R_j+\mathbb{E}\tau^{\rm Proac}_{V_j}+\tau_{\rm f}, &\mathbb{E}\left(T^{\rm Proac}_j\right)^2=\mathbb{E}\left(\tau^{\rm Proac}_{m}R_j+\tau^{\rm Proac}_{V_j}+\tau_{\rm f}\right)^2. \end{aligned}
y\left( \alpha\right) +\int K\left( \alpha,s;x,t\right) y\left( s\right)\mathrm{d}\sigma_{t}\left( s\right) =\psi\left( x,t;\mathrm{i}\alpha\right) ,\ \alpha\in\operatorname*{Supp}\sigma, \mathrm{d}\sigma_{t}\left( s\right) :=\mathrm{e}^{8s^{3}t}\mathrm{d}\sigma\left( s\right) ,
0 < \alpha_- < \widehat{A}_+ \left(\frac{T_{-}}{T_{+}}\right)^{\frac{3}{4}}e^{-\frac{T_{\rm i}}{2T_{-}}}= \sqrt{\frac{T_{+}T_{\rm i}^{\frac{3}{2}}}{\kappa p_{+}}}\left(\frac{T_{-}}{T_{\rm i}}\right)^{\frac{3}{4}}e^{-\frac{T_{\rm i}}{2T_{-}}}.
\psi_{(y_1, y_2)}^{\prime}(t) = t^{\kappa_1 + \kappa_2 + 1}\left[F_{(y_1, y_2)}(t) - \lambda(\kappa_1 + \kappa_2 + 2)\int_{\Omega}\vert y_1\vert^{\kappa_1 + 1}\vert y_2\vert^{\kappa_2 + 1}dz\right],
\xi^{i}=\zeta^{i}- \mathrm{i}R(t^{-1}z^i +t\overline{z}^i) \widetilde{\xi}_i =\widetilde{\zeta}_{i}- \mathrm{i}R(t^{-1}F_{i}+t\overline{F}_i) \alpha = \sigma -\mathrm{i}R(t^{-1}\langle \widetilde{Z},\zeta \rangle +t \langle \overline{\widetilde{Z}}, \zeta \rangle)-8\mathrm{i}c_{\ell}\log(t)\,,
\sum_{\substack{i+j+k=n i,j,k\ge 0}}\beta^{i+j}\gamma^k=\gamma^{n}\sum_{k=0}^n (n-k+1) \left(\frac{\beta}{\gamma}\right)^{n-k}=\gamma^{n}\sum_{k=0}^n (k+1) \left(\frac{\beta}{\gamma}\right)^{k}
x=-\tau^{2}\left( \alpha \bar{\nu}^{T}+\beta \bar{\nu}^{T}\Lambda_{\bar{r}}\right) P^{2}\bar{w}_{2^{k}}\left( t\right) +\tau \dot{x}_{0}e_{1}^{T}P\bar{w}_{2^{k}}\left( t\right) +x_{0}
u( x, t) = \sum^{+\infty}_{n=1}2 i n\pi e^{-i (n \pi)^2t}\int^t_0e^{i (n \pi)^2\tau}\Big (h_1 ( \tau ) - (-1)^n h_2 (\tau ) \Big ) d \tau\sin n\pi x &= W_{h}h_1 + (W_{h}h_2)\Big|_{x\rightarrow 1-x}\, .
\left\langle \Delta_{\mathcal{X}}, \nabla_{\mathcal{X}\mathcal{X}}^2L(\overline{\mathcal{X}},\overline{\lambda},\overline{\mathcal{Y}})\Delta_{\mathcal{X}}\right\rangle +\left\langle G'(\overline{\mathcal{X}})\Delta_{\mathcal{X}},\Delta_{\mathcal{Y}}\right\rangle =0.
\left| \frac{{\partial \varphi}_{i}}{{\partial y}_{j}}\right|\le \frac{l_{i}^{x}}{l_{1}^{x}}\frac{M\left( M^{2}-\mu^{2}\right)l_{1}^{y}}{\mu^{3}L},\thinspace i=1,\thinspace \mathellipsis ,\thinspace m;\thinspace j=1,..,n
D_{\infty}(\xi):=\exp\left\{-\frac{1}{2\pi i}\left[\left(\int_{-\eta}^{-C_L}+\int_{C_L}^{\eta}\right)\frac{\log(1-r(s)r^{*}(s))}{X_{L}(s)}ds+\left(\int_{-C_L}^{-C_R}+\int_{C_R}^{C_L}\right)\frac{\log r_{+}(s)}{X_{L+}(s)}ds\right]\right\}.
L=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}[\partial_{\mu}\eta\partial^{\mu}\eta+(\eta+v)^{2}(\partial_{\mu}\xi+eA_{\mu})^{2}]-[\lambda v^{2}\eta^{2}+\lambda v\eta^{3}+\frac{1}{4}\lambda\eta^{4}].
\Gamma_0(t,y)=\exp\{\int_0^t[\ln(1-\frac{b_0(s,y)}{\lambda\gamma_0(s,y)})+\frac{b_0(s,y)}{\lambda\gamma_0(s,y)}]\lambda ds+\int_0^t\ln(1-\frac{b_0(s,y)}{\lambda\gamma_0(s,y)})\tilde{N}(ds)\}.
\begin{matrix}M=\sqrt{\alpha^{2}+\beta^{2}}&cos(\theta)=\frac{\alpha}{M}&sin(\theta)=\frac{\beta}{M}\\C,D=E\mp Fi&&\\G=\sqrt{E^{2}+F^{2}}&cos(\delta)=\frac{E}{G}&sin(\delta)=\frac{F}{G}\end{matrix}
(T_i+q)(T_i-q^{-1}) = 0, \mspace{40mu}T_iT_j =T_jT_i \mspace{10mu}\mspace{10mu}\vert i-j\vert>1, \mspace{40mu}T_iT_{i+1}T_i = T_{i+1}T_iT_{i+1}, \rho\rho^{-1}=1=\rho^{-1}\rho,\mspace{40mu}\rho T_i \rho^{-1}= T_{i+1},
P\bigg\{\bigcap_{j\in I_{H^-}}\{N_{j}(t) = 0\}\bigg\}= P\big\{V(0)\in \{v_{i_0},\dots, v_{i_M}\}\big\}\sum_{n=0}^\infty P\{N(t) = n\}\, \alpha_{I_H}^n & = e^{-\Lambda(t)(1-\alpha_{I_H})}\sum_{i\in I_H}p_i.
W [ f ( t ) ; x_{f}, t_{f}; x_{i}, t_{i}] \equiv \int_{q ( t_{i}) = x_{i}}^{q ( t_{f}) = x_{f}}{\cal D}q ( t ) \exp \left\{i \int_{t_{i}}^{t_{f}}d t \ [ \{\cal L}[ q ( t ) , \dot{q}( t ) ] - f ( t ) q ( t ) \ ] \right\}.
I = ( \sigma^{2}i \partial_{\mu}\gamma^{\mu})_{\alpha_{j}\alpha}( i \partial_{\nu}\gamma^{\nu})_{\alpha \alpha_{i}}\Psi_{\alpha_{i}\alpha_{j}}^{( \pm )}= m^{2}\sigma_{\alpha_{j}\alpha_{i}}^{2}\Psi_{\alpha_{i}\alpha_{j}}^{( \pm )}= - I \; .
\frac{d}{dx}\varphi_A^\varepsilon=-\sqrt{\varepsilon}\;\frac{\frac{d}{dx}A_\varepsilon}{A_\varepsilon}\qquad\qquad\frac{d^2}{dx^2}\varphi_A^\varepsilon=-\sqrt{\varepsilon}\;\Big(\frac{\frac{d^2}{dx^2}A_\varepsilon}{A_\varepsilon}-\frac{|\frac{d}{dx}A_\varepsilon|^2}{A_\varepsilon^2}\Big),
(\nabla{\hskip-0.3cm^{^{^{\pi_i}}}}_{X}U)(p_1,p_2)=\big(\nabla{\hskip-0.2cm^{^{^{i}}}}_{d{\hskip -0.2cm_{_{_{(p_1,p_2)}}}}\hskip -0.6cm\pi_i(X_{(p_1,p_2)})}{\widetilde{U}}\big)(p_i),
\begin{array}{rl}D_\beta [\tilde v_1 - \zeta^2(0)w_1(0) f ]= \!\! \!\!\displaystyle D_\beta (\zeta^2 \tilde w_1) - \zeta^2(0)w_1(0) D_\beta f = \!\! \!\!\displaystyle \zeta^2 D_\beta \tilde w_1- \zeta^2(0)w_1(0) D_\beta f,{\rm on}\B_R\cap \partial\Omega,\end{array}
\int \, \Big( \mathfrak{J}_L \, f_x , \sqrt{\frac{d\mu_{\mathfrak{G}/H}(x{g_0}^{-1})}{d\mu_{\mathfrak{G}/H}(x)}}g_{x{g_0}^{-1}}\, \Big) \, d\mu_{\mathfrak{G}/H}(x) = \int \, (\mathfrak{J}_L \, f_x , k_x ) \, d\mu_{\mathfrak{G}/H}(x) \,\,\, \textrm{for all}\,\, f \in \mathfrak{D}\big(U^{L}_{g_0}\big);
c_{n,\rho,q}^{(k)}(z) =\sum_{m=0}^n S_1(n,m)(-\rho)^{n-m}\sum_{i=0}^m\binom{m}{i}\frac{(-z)^i}{[m-i+1]_q^k}&=\sum_{m=0}^n S_1\left(n,m,\frac{z}{\rho}\right)\frac{(-\rho)^{n-m}}{[m+1]_q^k}\,.
c \bigg( \sum_{j=2}^{d}|\eta_{\lambda,k-1}^{(j)}|^{2}\bigg) e^{- 2 \lambda l_{k}}< \sum_{j=2}^{d}|\eta_{\lambda,k}^{(j)}|^{2}< C \bigg( \sum_{j=2}^{d}|\eta_{\lambda,k-1}^{(j)}|^{2}\bigg) e^{- 2 \lambda l_{k}}
\mathcal{H}: = \Biggl\{\prod_{i = 1}^d\bigl[a_{i\tau(i,t_i-1)},a_{i\tau(i,t_i)}\bigr] \Biggm| t=(t_1, \ldots, t_d ) \in \prod_{i=1}^d \{1, \ldots, l_i \}\Biggr\}.
[-] \circ ( L\otimes [-] ) \big(\overline{a}_{L,L,L}\big( \overline{\gamma}_{L,L}\otimes L\big)\overline{a}_{L,L,L}^{-1}\big( L\otimes \overline{\gamma}_{L,L}\big)\big) ( x\otimes ( y\otimes z ) )=\big[ c^{-1}d^{-2}z, [ dx,cdy ] \big].
P(f^{2^{k+1}}, s\Phi_{2^{k+1}}) = \lim_{\epsilon \rightarrow 0}\limsup_{n \rightarrow \infty}\frac{1}{n}P_n(f^{2^{k+1}}, s\Phi_{2^{k+1}}, \epsilon) &\geq 2\lim_{\delta \rightarrow 0}\limsup_{n \rightarrow \infty}\frac{1}{2n}P_{2n}(f^{2^k}, s\Phi_{2^k}, \delta) &\geq 2P(f^{2^k}, s\Phi_{2^k}).
\int_0^{\widetilde{T}}(|x|^{-s}u^m_t,\omega)dt+\int_0^{\widetilde{T}}(|\nabla u^m|^{p-2}\nabla u^m,\nabla \omega)dt=\int_0^{\widetilde{T}}(|u^m|^{q-2}u^m\ln|u^m|,\omega)dt,
D^{(n)}_{q,t}(d) =\sum_{\sigma\in \{\pm 1\}^n}\prod_{1\le i\le n}\frac{1}{\vartheta(2\sigma_i z_i;q)_d}\prod_{1\le i<j\le n}\frac{\vartheta(t+\sigma_i z_i+\sigma_j z_j;q)_d}{\vartheta(\sigma_i z_i+\sigma_i z_j;q)_d}\prod_{1\le i\le n}T_i^{-d/2}.
(1,1)_{L}^{+}=1 , (e_{i}x,y)_{L}^{+}=\frac{1}{1-q_{i}^{2}}(x,f'_{i}(y))_{L}^{+}, (xe_{i},y)_{L}^{+}=\frac{1}{1-q_{i}^{2}}(x,{_{i}f'}(y))_{L}^{+}.
\frac{1}{q(x)}\int_{-\infty}^{x}\frac{2}{a(y)}q(y) dy = \ \frac{q(-1)}{q(x)}\frac{1}{q(-1)}\int_{-\infty}^{-1}\frac{2}{a(y)}q(y) dy + \frac{1}{q(x)}\int_{-1}^{x}\frac{2}{a(y)}q(y) dy \leq \ \frac{q(-1)}{q(x)}\frac{1}{\mu}+ \frac{1}{q(x)}\int_{-1}^{x}\frac{2}{a(y)}q(y) \mathrm dy.
\langle z_\pm,j | n_{12},j \rangle =\begin{cases}Q_{n_{12}}^{\mu_{12}}(z) \langle z_+,j | 0,j \rangle n_{12}, Q_{n_{12}-1}^{\mu_{12}}(z) \langle z_-,j | 1,j \rangle n_{12},\end{cases}
({\bf A}_1^\frac{1}{2})^{-}= \left[\begin{array}{cc}{\bf I}_r{\bf 0}{\bf 0}{\bf 0}\end{array}\right]{\bf G}^{-1}\mbox{and}({\bf A}_2^\frac{1}{2})^{-}= \left[\begin{array}{cc}{\bf D}_r^\frac{-1}{2}{\bf 0}{\bf 0}{\bf 0}\end{array}\right]{\bf G}^{-1}~_.
(a_1,a_2,a_3) =(\mu_1\sqrt{t/n},\mu_2\sqrt{t/n}, \mu_3\sqrt{t/n}), (b_1,b_2,b_3) =(C_1\mu_1m_1\sqrt{t/n},C_1\mu_2m_2\sqrt{t/n}),C_1\mu_3m_3\sqrt{t/n}).
\int_\Omega \left|Dg_2-Dg_3\right|^p\,d\mu =\int_{\bigcup_{j=0}^{k-1}D(x_j,r)}\left|Dg_2-Dg_3\right|^p\,d\mu &\leq K^p\bigl[g_2-g_3\bigr]^p_{\mathrm{Lip}}\sum_{j=0}^{k-1}\mu\left(D(x_j,r)\right) &=K^pC^p\pi k r^2 \ .
N_{F/E}(a\theta_1^2 + b\theta_1 - 1) = 4a^3 + 2b^3 + 6ab - 1 = 3v, N_{F/E}(a\theta_1^2 + b\theta_1) = 4a^3 + 2b^3 = 4v, N_{F/E}(a\theta_1^2 + b\theta_1 + 1) = 4a^3 + 2b^3 - 6ab + 1 = 5v, N_{F/E}(a\theta_1^2 + (b+1)\theta_1) = 4a^3 + 2b^3 + 6b^2 + 6b + 2 = 6v,
\sqrt{-1}{\bigwedge}_{\omega}F_{H}=\lambda Id &\frac{\alpha_{0}}{2}\left(\frac{\Omega_{h}}{\int_{X}\Omega_{h}}-\frac{\omega^{n}}{\int_{X}\omega^{n}}\right)-\alpha_{1}\frac{2}{(n-2)!}tr(F_{H}\wedge F_{H})\wedge\omega^{n-2}=\tilde{C}vol_{\omega},
F_{2}(r,0) = -r\int_{0}^{2\pi}\log\left((1+r^2)+(r^2-1)\cos(y)\right)(\sin(2x-y))dy & = -r\sin(2x)\int_{0}^{2\pi}\log\left((1+r^2)+(r^2-1)\cos(y)\right)\cos(y)dy & = 2\pi r\sin(2x) \frac{1-r}{1+r}
R^{4}= t^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}t^{\rho_{1}\sigma_{1}\rho_{2}\sigma_{2}\rho_{3}\sigma_{3}\rho_{4}\sigma_{4}}R_{\mu_{1}\nu_{1}\rho_{1}\sigma_{1}}R_{\mu_{2}\nu_{2}\rho_{2}\sigma_{2}}R_{\mu_{3}\nu_{3}\rho_{3}\sigma_{3}}R_{\mu_{4}\nu_{4}\rho_{4}\sigma_{4}}.
{}_3\phi_2\!\left[\begin{matrix}(1-r)^j,1-a,b e,(1-r)^{j+1}(1-a)b/ce\end{matrix};1-r,1-r\right] &=\frac{((1-r)/e,(1-r)(1-a)b/ce;1-r)_j}{((1-r)(1-a)/e,(1-r)b/ce;1-r)_j}\;{}_3\phi_2\!\left[\begin{matrix}(1-r)^j,1-a,c ce/b,(1-r)^{j+1}(1-a)/e\end{matrix};1-r,1-r\right].
\mathbf{P}_{1}^{s_{O}}(\omega)=\begin{pmatrix}p_{1}^{s_{O}}(\omega,1) \mathbf{0}\mathbf{0}\dots \mathbf{0}p_{1}^{s_{O}}(\sigma\omega,1) \mathbf{0}\dots \mathbf{0}\mathbf{0}p_{1}^{s_{O}}(\sigma^{2}\omega,1) \vdots \vdots \ddots\end{pmatrix}.
\det\left[\begin{array}{cccccc}f(x) f(x+h) \cdots f(x+nh) f(x+h) f(x+2h) \cdots f(x+(n+1)h) \vdots \vdots \ \ddots \vdots f(x+nh) f(x+(n+1)h) \cdots f(x+2nh) \end{array}\right] =0 x,h\in\mathbb{R}.