harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\langle \xi_x^{(n)}, \xi_y^{(n)}\rangle =\phi((r-tp_x^{(n)})^*(r-tp_y^{(n)})) &= r^2-rt\phi(p_x^{(n)})-rt\phi(p_y^{(n)})+t^2\phi(p_x^{(n)}p_y^{(n)})=0.
X^{t,\xi}_s =\xi+\int_{t}^{s}\mu_u(X^{t,\xi}_u)du + \int_{t}^{s}\sigma_u(X^{t,\xi}_u)dW_u, t\leq s\leq T X_s^{t,\xi}=\xi-\int_s^t \mu_u(X^{t,\xi}_u)du-\int_{s}^t Z_u dW_u, 0\leq s\leq t.
\mathrm{T r}\, \exp ( - H t ) = \mathrm{T r}\, \mathrm{I \! P}\int D q \exp \int_{0}^{t}d \tau \left[ - \frac 1 2 \dot{q}^{2}( \tau ) - \dot{q}( \tau ) \cdot \gamma B - \frac 3 2 \lambda^{2}B^{2}\right] .
||\left(\partial_{r}+\frac{1}{r}\right)v_{4}||_{L^{2}(r dr)}^{2}= \int_{0}^{\infty}\left((\partial_{r}v_{4})^{2}+ \frac{\partial_{r}(v_{4}^{2})}{r}+ \frac{v_{4}^{2}}{r^{2}}\right) r dr &=||\partial_{r}v_{4}||_{L^{2}(r dr)}^{2}+||\frac{v_{4}}{r}||_{L^{2}(r dr)}^{2}
a_n=a_n(q,\alpha,\beta):=\int_0^{\pi}|\varphi_n(x)|^2 dx, n=0, 1, 2, \ldots, b_n=b_n(q,\alpha,\beta):=\int_0^{\pi}|\psi_n(x)|^2 dx, n=0, 1, 2, \ldots ,
\mathbb{E}_{\mathbf{H}, \mathbb{W}}\left \{\rho_k^2 \|\tilde{\mathbf{h}}_k \hat{\mathbf{Q}}^H \|^2 \sin^2\theta_k\right\}= \mathbb{E}_{\mathbf{H}, \mathbb{W}}\left \{\rho_k^2\right\}\mathbb{E}_{\mathbf{H}, \mathbb{W}}\left \{\|\tilde{\mathbf{h}}_k \hat{\mathbf{Q}}^H \|^2 \right\}\mathbb{E}_{\mathbf{H}, \mathbb{W}}\left \{\sin^2\theta_k\right\}.
\widehat W_{+}^\star(\eta,k)=\widehat W_{+}(-\eta,k),\widehat Y_{+}(\eta,k)=\widehat Y_{+}(\eta,-k), &\widehat W_{-}(\eta,k)=\widehat W_{+}(\eta,-k),\widehat Y_{-}(\eta,k)=\widehat Y_{+}^\star(-\eta,k).
\left. + ( e^{G / 2})_{, \bar{A}}\bar{\chi}^{\bar{A}}\right] + \frac{N}{\kappa^{2}}\lambda \left[ ( e^{G / 2})_{, A}\chi^{A}+ ( e^{G / 2})_{, \bar{A}}\bar{\phi}^{\bar{A}}\right]
\hat{a}{\cal P}= \frac{1}{q^{2}}{\cal P}\hat{a}- \frac{\mu}{q^{3}}\hat{a}~ ~ , ~ ~ \hat{b}{\cal P}= \frac{1}{q^{2}}{\cal P}\hat{b}- \frac{\mu}{q^{3}}\hat{d}\hat{c}^{- 1}\hat{a},
f\left( \alpha\right) \leq C\left( J_{\infty}g\left( \alpha\right)+\sum_{k=1}^{N\left( \alpha\right)}\left\{\prod_{k\leq\ell\leq N\left(\alpha\right)}\left( 1-\left| z_{m_{\ell}-1}\right|^{2}\right)^{\sigma}\right\}J_{k}g\left( \alpha\right) \right) ,\;\;\;\;\;\alpha\in\mathcal{Y}.
\frac{(k-1)!k!}{(2k+1)!}{\}_2 F_1 (k, \k+1; \ 2k+2; \u) =\frac{1}{k! (k+1)!}\frac{d^k}{du^k}(1-u)^{k+1}\frac{d^k}{du^k}\frac{\log (1 -u)}{u}
-\frac{d}{dt}\left\{\int \,d\alpha\,P_\alpha\,\ln\,(P_\alpha/c_\alpha)\right\}=\, -\int d\alpha\, \int d\beta\,\left(1+\,\ln (P_\alpha/c_\alpha)\right) &\left( P_\beta\,\frac{d\Gamma(\beta\to \alpha)}{d\alpha}-\,P_\alpha\,\frac{d\Gamma(\alpha\to\beta)}{d\beta}\right).
Z(t)=\int_{0}^{\infty}e^{-\mathcal{E}t}d\mathcal{N}(\mathcal{E})=\frac{1}{(2\pi)^d}|\Omega|\frac{(2\Gamma(1+\frac{1}{2s}))^d}{t^{\frac{d}{2s}}}+o(t^{-\frac{d}{2s}})
A_+ = A_1 \oplus A_0 \oplus A_\lambda \oplus A_{\lambda - \frac{1}{2}}\oplus\bigoplus_{m \in wt(D_+)}A_{\nu^{(m, |\alpha|- m)}_\pm}A_- = \bigoplus_{m \in wt(D) - wt(D_+)}A_{\nu^{(m, |\alpha|-m)}_\pm}
0\rightarrow I_{\mathcal{B}_{i-1}}(E_1,-)\rightarrow\mathcal{C}(E_1,-)\rightarrow \frac{\mathcal{C}(E_i,-)}{I_{\mathcal{B}_{i-1}}(E_i,-)}=\Delta_{E_i}^\circ(i)\rightarrow 0,
F^{1111}= \frac{(12)(34)}{x^2_{12}\, x^2_{34}}\; \alpha(x_0,\ldots,x_4) + \frac{(14)(23)}{x^2_{14}\, x^2_{23}}\; \beta(x_0,\ldots,x_4) + \frac{(13)(24)}{x^2_{13}\, x^2_{24}}\; \gamma(x_0,\ldots,x_4) \;.
\frac{\theta}{\pi}|e^{\sigma_0|x|}J_c^{-1}g(x)| \leqq \int_{\mathbb R}e^{(\sigma_0-\theta) |x-y|}e^{\sigma_0|y|}|g(y)|dy &\leqq sup_{y\in \mathbb R}|e^{\sigma_0|y|}g(y)| \int_{\mathbb R}e^{(\sigma_0-\theta) |p|}dp <\infty.
a=a_{M}\left( \left[ A_{i}^{a}\right] ,\left[ \partial_{i}B_{a}^{0i}\right],\left[ \pi \right] ,\left[ \mathcal{P}_{\Gamma}\right] \right) e^{M}\left(\eta_{2a}^{i},\partial^{\left( j\right.}\eta_{2a}^{\left. i\right)}\right) +\gamma_{I}b,
\displaystyle\int_{0}^1\int_{0}^1F(x,y)dxdy=\displaystyle\frac{1}{(n_1+1)(n_2+1)}\sum_{i_1=0}^{n_1}\sum_{i_2=0}^{n_2}F\left(\frac{i_1}{n_1},\frac{i_2}{n_2}\right)+R_{n_1,n_2}[F],
\sum_{z^n}\!\Big(\mathbb{P}[Z^n=z^n, X^n\!\in\!\mathcal{A}_1,Y^n\!\in\!\mathcal{B}_1]^{\frac{1}{2}}\times \mathbb{P}[Z^n=z^n, X^n\!\in\!\mathcal{A}_2,Y^n\!\in\!\mathcal{B}_2]^{\frac 12}\Big) begin{align*}0.5ex] \;>\mathbb{P}[X^n\!\in\!\mathcal{A}_1,Y^n\!\in\!\mathcal{B}_2]^{\frac 12}\;\mathbb{P}[X^n\!\in\!\mathcal{A}_2,Y^n\!\in\!\mathcal{B}_1]^{\frac 12}.
u_0^r(r,z) = \frac 1{\log N_1}\sum_{M_1-N_1\le j\le M_1}2^{-j}a_3^r(2^j(r,z)),a_3^r(r,z)=\frac 1 r (-\partial_z a_3)(r,z); &u_0^z(r,z) = \frac 1{\log N_1}\sum_{M_1-N_1\le j\le M_1}2^{-j}a_3^z(2^j(r,z)),a_3^z(r,z)=\frac 1 r (\partial_r a_3)(r,z).
I_{\ell(\gamma_\ell),s}^{+,(j_0)}(g_\alpha(x))\,(=I_{\overline X_{p_{\ell(\gamma_\ell)}},s}^{+,(j_0)}(g_\alpha(x)))\,=\frac{1}{2\pi}\int_{\overline X_{p_{\ell(\gamma_\ell)}}}g_\alpha(Y)\mathrm{Tr}\,b_{j_0,s}^+(z(Y), z(Y))dv_{\overline X_{p_{\ell(\gamma_\ell)}}}(Y)
v^{-1}h_{ij}= -v^{-2}u_{;ij}- \dot\vartheta \vartheta \sigma_{ij}& = -v^{-2}\{u_{:ij}- \tfrac{1}{2}\bar{g}^{km}\left( (\vartheta^2)_j \sigma_{mi}+ (\vartheta^2)_i \sigma_{mj}-(\vartheta^2)_m \sigma_{ij}\right) u_k \}- \dot\vartheta \vartheta \sigma_{ij}.
\begin{array}{r c l}{{Y}}&{{\sim}}&{{1 \qquad \mathrm{f o r}\quad d_{2 i - 1}= 0 \nonumber}}\\{{}}&{{\sim}}&{{\exp \left( - \frac{1}{2 \pi \sqrt{3}}\sum_{i}R_{2 i - 1}^{2}\right) \mathrm{f o r}\quad d_{2 i - 1}= \pm 1 , \pm 2}}\\ \end{array}
{J}\pi(a){J}^{-1}= p_-{J}\pi_0(f){J}^{-1}p_- + p_+{J}\pi_0(g){J}^{-1}p_+ &= p_- \pi_0(\bar f) + p_+\pi_0(\bar g) = \pi(\bar g, \bar f) = \pi(\rho(a^*)).
I_{4}\equiv \det \left[ \left( \begin{array}{c}{\vec{\tilde{q}}}\\{\vec{\tilde{p}}}\\ \end{array}\right) \left( \begin{array}{c c}{\vec{\tilde{q}}\}\\{v e c{\tilde{p}}}\\ \end{array}\right) \right] = \left( \vec{\tilde{q}}\cdot \vec{\tilde{q}}\right)^{2}\left( \vec{\tilde{p}}\cdot \vec{\tilde{p}}\right)^{2}- \left( \vec{\tilde{q}}\cdot \vec{\tilde{p}}\right)^{2},
\mathbf{A}_{q}\!\left[N_{-}(w)\cap wG_{0}^{\min}\right] :=\mathbf{A}_{q}\!\left[N_{-}\left(w\right)\right]\left[\mathcal{D}_{w}^{-1}\right], \mathbf{A}_{q}\!\left[N_{-}^{w}\right] :=\mathbf{A}_{q}\!\left[N_{-}\cap X_{w}\right]\left[\left[\mathcal{D}_{w}\right]^{-1}\right].
\mathbf{j}' = (0, 0, \dots, \frac{m-1}{3}, 2) \mathbf{j}'' = (0, 0, \dots, \frac{m-1}{3}- 1, 5) \mathbf{j}''' = (0 , 0, \dots, 1, \frac{m-1}{3}- 2, 1).
\dfrac{\lambda}{\Lambda}=\dfrac{\lambda\,\Phi}{\Lambda\,\Phi}\leq C:=\dfrac{\left| A^{1/2}\dfrac{z}{|z|}\right|^2}{\left| A^{1/2}\dfrac{e}{|e|}\right|^2}\leq\dfrac{\Lambda\,\Phi}{\lambda\,\Phi}=\dfrac{\Lambda}{\lambda}.
t_{*}\ge t_{clear}=2\pi\sqrt{\frac{a^{3}}{GM}}\frac{C^{2}}{a^{2}}(\frac{m}{3M})^{2/3}\frac{a^{2}M^{2}}{100m^{2}}=\frac{2\pi}{100\sqrt{G}}\frac{C^{2}}{3^{2/3}}a^{3/2}M^{5/6}m^{-4/3}
p_j M_j e^{-p_j t_j}\left(1 - e^{-p_j t_j}\right)^{M_j - 1}= -\sum_{k_j=1}^{M_j}(-1)^{k_j}\binom{M_j}{k_j}p_j k_j e^{-k_j p_j t_j},
[i\Theta_{E,h_E}\otimes \mathrm{id}_F,\Lambda_\omega]u=\bigl(i\Theta_{E,h_E}\otimes \mathrm{id}_F\bigr) \wedge\Lambda_\omega u-\Lambda_\omega \wedge \bigl(i\Theta_{E,h_E}\otimes \mathrm{id}_F\bigr) u
\Big(P_{n-s}(z)+ F_k(z) +\sum_{j=k+1}^\infty F_j(z)\Big)^m &\ \ \ =\Big(F_k^1(z) +\sum_{j=k+1}^\infty F_j^1(z)\Big)^{m_1}...\Big(F_k^s(z) +\sum_{j=k+1}^\infty F_j^s(z)\Big)^{m_s}\times \Big(z_{s+1}+F_k^{s+1}(z) +\sum_{j=k+1}^\infty F_j^{s+1}(z)\Big)^{m_{s+1}}...\Big(z_n+F_k^n(z) +\sum_{j=k+1}^\infty F_j^n(z)\Big)^{m_n}
\mbox{\fboxsep=.1in \framebox{$\displaystyle a_\mu^{\xi}(x) := \langle \Lambda |{\cal A}_\mu^\xi(x) |\Lambda \rangle , \quad{\cal A}_\mu^\xi(x) = \xi^\dagger(x){\cal A}_\mu(x) \xi(x) +{i \over g}\xi^\dagger(x) \partial_\mu \xi(x) ,$}}
\sigma = \left. 2 \zeta_{1}a^{\prime}\right|{\mathrm{\raisebox{- 1 . 2 e x}{{\tiny{~ \! 0^{-}~}}}}}- \left. 2 \zeta_{2}a^{\prime}\right|{\mathrm{\raisebox{- 1 . 2 e x}{{\tiny{~ \! 0^{+}~}}}}}
\frac{dE}{dt}=-\sum_{A=1}^{N_{layer}}\tau_{A}\sum_{i,j=1}^{N_{A}}\frac{dx_{j}^{A}}{dt}\frac{\partial^{2}L^{A}}{\partial x_{j}^{A}\partial x_{i}^{A}}\frac{dx_{i}^{A}}{dt}\le0
y \coloneqq \psi_1^{g - n-1}\prod_{i=2}^n\psi_i^{1}\kappa_1 \tilde{x}_\ell \coloneqq -\frac{Q_{3}(5/2)}{Q_{1}(1)Q_2(3/2)}\psi_2^{1}\dotsb \psi_\ell^{2}\dotsb \psi_n^{1}\psi_1^{g-n-1}, \ell=2, \dots, n.
\omega_{m}=\frac{\Gamma^{2}\left( \gamma +\frac{1}{2}+D\right)}{\Gamma\left( 2D\right) \Gamma \left( 2\gamma +1\right)}\frac{\left( -2D\right)_{m}\left( -D+1\right)_{m}\left( \left( -D+\gamma +\frac{1}{2}\right)_{m}\right)^{2}\left( -1\right)^{m}}{\left( -D\right)_{m}\left( \left(-D-\gamma +\frac{1}{2}\right)_{m}\right)^{2}m!}
a_d(n) ={}da_d(n-1) + da_d(n-2) - \binom{d}{2}a_d(n-3)+ da_d(n-4) - d^2a_d(n-5) & + da_d(n-6) - \sum_{k \geq 8}^{n+1}\mu_d(k) a_d(n+1-k) \leq{}(d+1)a_d(n-1) + \left( -d^3\binom{d}{2}+ d(d+1)^2 - d^3 + d + 2d^2\right) a_d(n-6) \leq{}(d+1)a_d(n-1),
m_{b}= \frac{5 r_{b}^{8}+ ( \ell^{2}- 2 8 N^{2}) r_{b}^{6}+ 5 N^{2}( 1 4 N^{2}- \ell^{2}) r_{b}^{4}+ 5 N^{4}( 3 \ell^{2}- 2 8 N^{2}) r_{b}^{2}+ 5 N^{6}( \ell^{2}- 7 N^{2})}{1 0 \ell^{2}r_{b}}
\|\delta_t(t)+\bar u(t)\|^2_{L^2(\Gamma)}= \|\delta_t(t)-\bar\delta_t(t)+\bar\delta_t(t)+\bar u(t)\|^2_{L^2(\Gamma)}& \leq 2\|w_t(t)\|^2_{L^2(\Gamma)}+ 2\|\bar\delta_t(t)+\bar u(t)\|^2_{L^2(\Gamma)}.
- \left( \frac{4}{q^{2}}\right)^{1 + \epsilon}\frac{1}{2 \epsilon}\sum_{j = 0}^{\infty}\frac{\Gamma ( 1 + \epsilon + j )}{1 + 2 \epsilon + 2 j}~ \frac{1}{j !}\left( - \frac{4 y}{q^{2}}\right)^{j}+ \frac{1}{y^{\epsilon + 1 / 2}}\frac{1}{\sqrt{q^{2}}}\frac{\sqrt{\pi}\Gamma ( \epsilon + 1 / 2 )}{2 \epsilon}.
2\sum\limits_{\substack{(l',m',s')\in \mathfrak{T}_3 (l,m,s)\in \mathfrak{T}_3}}\Re\left(\big\langle |\boldsymbol{p}|^{-1}\big(\mathcal{F}_3\zeta\big)_{(l',m',s')}^{+}\Omega_{l',m',s'},(\boldsymbol{\sigma}\cdot \boldsymbol{p}) |\boldsymbol{p}|^{-1}\big(\mathcal{F}_3\zeta\big)_{(l,m,s)}^{-}\Omega_{l,m,s}\big\rangle\right) .
simil(x,y)=cos(\vec{x},\vec{y})=\frac{\vec{x}\cdot\vec{y}}{||\vec{x}||\times||\vec{y}||}=\frac{\sum_{i\in I_{xy}}r_{x,i}r_{y,i}}{\sqrt{\sum_{i\in I_{x}}r_{x,i}^{2}}\sqrt{\sum_{i\in I_{y}}r_{y,i}^{2}}}
\begin{cases}\displaystyle(n=2)\qquad\lim_{t\rightarrow T}\int_{f^{-1}(B_{\rho(t)}(x))}|A|^2 d\mu \ge \varepsilon_1\,, \displaystyle(n=4)\qquad\lim_{t\rightarrow T}\int_{f^{-1}(B_{\rho(t)}(x))}|A|^4+|\nabla A|^2 d\mu \ge \varepsilon_1\,,\end{cases}
\;(B(0,r_0)\setminus \Lambda_{2\alpha}) \leq \sum_{m=N}^{+\infty}|\lambda|^{-2m}\;(B(0,|\lambda|^mr_0)\setminus \Omega_{\alpha,m}) &\leq \sum_{m=N}^{+\infty}|\lambda|^{-2m}Qe^{-\gamma \theta m}\;(B(0,|\lambda|^mr_0) &= \sum_{m=N}^{+\infty}Q\pi|r_0|^2e^{-\gamma \theta m}\leq \varepsilon.
\mathrm{R e}\, \chi_{T}^{\mathrm{( T M )}}( \omega ) = \frac{\hbar A}{4 \pi^{2}}\left( \frac{k_{B}T}{\hbar}\right)^{5}\int_{0}^{\Omega}d v \frac{\left( \frac{\Omega}{2}- v \right)^{2}}{\sqrt{v ( \Omega - v )}}\int_{0}^{\infty}d u \, \frac{u^{2}}{e^{\frac{u}{2}}- 1}\left[ 1 +{\cal O}( \Omega^{2}) \right] .
\mathcal{Q}^{(n)}_i = \frac{(q-1)^2 (q+1)}{2^L q^{L+n}(L-1)!}\frac{\prod_{k=1}^{L}y_k^{-\frac{1}{2}}}{\prod_{\stackrel{j=1}{j \neq i}}^{n}(x_j - x_i)}\left[ \sum_{m=0}^{L}\mathcal{G}^{(n)}_m (x_i ; \bar{X}_i^{1,n}) \sum_{1 \leq j_1 < \dots < j_m \leq L}\prod_{\alpha=1}^{m}y_{j_{\alpha}}\right] \; ,
{\delta}{\lambda}_{4}( 1 2 3 5 ) = \frac{8}{3}g_{4}^{4}\int_{{\Lambda}}\frac{d^{4}\vec{p}_{4}}{( 2{\pi})^{4}}\frac{{\Phi}( 4 ; 1 2 3 5 )}{( \vec{p}_{4}^{2}+ m^{2}) ( ( \vec{p}_{1}+ \vec{p_{2}}+ \vec{p}_{4})^{2}+ m^{2})},
|q_{11}^{(2)}| \lesssim\frac{\varepsilon_0|\xi|}{t^2}e^{-5(|k-l|+|\xi-\eta|)}\frac{|A_k(\xi)-A_l(\eta)|}{\sqrt{A_k(\xi)A_l(\eta)}}&\lesssim\frac{\varepsilon_0|\xi|}{t^2}e^{-4(|k-l|+|\xi-\eta|)}\sqrt{A_k(\xi)A_l(\eta)}\cdot\big|\varepsilon_0^{-1}k^2+(\xi-tk)^2-\varepsilon_0^{-1}l^2-(\eta-tl)^2\big|.
\left(1 - q^{3n}\right) u_\S(n) = q^{3n-2}\left(1 + q + q^2\right) u_\S(n-1) & - q^{6n-6}\left(1 + q + q^2 + q^{3n-3}\right) u_\S(n-2)+ q^{9n-12}u_\S(n-3).
X_k=L^{p_k^S}(M_S,\mu_s; L^{p_k^{S-1}}(M_{S-1},\mu_{S-1};\cdots L^{p_k^{1}}(M_{1},\mu_{1}; L^{p_k^{0}}(\mathcal M))\cdots),k=1,\ldots,n, Y_{n+1}=L^{q_{n+1}^S}(M_S,\mu_S; L^{q_{n+1}^{S-1}}(M_{S-1},\mu_{S-1};\cdots L^{q_{n+1}^{1}}(M_{1},\mu_{1}; L^{q_{n+1}^{0}}(\mathcal M))\cdots).
\gamma\int_{0}^{\sigma}{\left({\frac{\sigma^{2}-t^{2}}{1-t^{2}}}\right)^{1/2}dt}\sim\frac{1}{2}\left({n-m+\frac{1}{2}}\right) \pi+\sum\limits_{s=0}^{\infty}{\frac{\kappa_{s}}{\gamma^{2s+1}}},
\cdot(2)_{t^{4}}(6)_{t^{4}}\cdot(2)_{t^{5}}(6)_{t^{5}}\cdot(2)_{t^{6}}^{2}(3)_{t^{6}}\cdot(6)_{t^{7}}\cdot(6)_{t^{8}}\cdot(6)_{t^{9}}
\Delta \phi_1 = \cos^{-1}\left(\frac{r_s - \Delta T_s(v_{o,max}- v_{c,s})}{r_s}\right) + \sin^{-1}\left(\frac{r_c}{r_s}\right) & \Delta \phi_2 = \pi/2 + \sin^{-1}\left(v_{o,max}/v_{c,s}\right) - \Delta \phi_1 & \tau_{f,1}= \frac{c_3 m\Delta \phi_1 v_{c,s}}{f_{planar}- K_d (v_{c,s}+ v_{air})^2}
\boxed{\aligned d\rho_0 =P\,\rho_0\wedge\zeta_0+\overline{P}\,\rho_0\wedge\overline{\zeta_0}+i\,\zeta_0\wedge\overline{\zeta}_0, d\zeta_0 = 0, d\overline{\zeta_0}= 0.\endaligned}
(M_{A,k}+|g_1|)|\tilde{u}-\hat{u}|+(1+\|g_2\|)\sum_{x^\prime\in S}|\tilde{r}^\prime(x^\prime)-\hat{r}^\prime(x^\prime)| \geq \epsilon(r-\hat{r}),\forall\,\, (\hat{u},\underline{\hat{r}}^\prime,\hat{r})\in (f)
M_{\Phi}=L\begin{pmatrix}\frac{n(n-1)}{2}\frac{n(n-1)}{2}\frac{n(n-1)(n-3)}{8}0 n(n-1) \frac{n(n-1)^2}{4}n(n-1) 0 \frac{n(n-1)^2}{4}0 0 \frac{n(n^2-1)}{2}\end{pmatrix},
\rho_{d^{n+1}}(F_{n+1}, G_{n+1}) = \rho_{d}(F, G)^{d^n}\rho_{d^n}(F_n,G_n)^{d^2}= \rho_{d}(F,G)^{d^n + d^2(d^{n-1}(d^{n-1}+\dots + 1))}= \rho_{d}(F,G)^{d^n (d^{n+1}-1)/(d-1)}
- \Pi ( p^{+}= 0 , p^{-}= 0 , \vec{p}\rightarrow 0 ) = \frac{g^{2}}{2 5 6 \sqrt{2}\pi^{3}T}\int d^{3}k \; \frac{1}{( k^{+})^{2}}\mathrm{c o s e c h}^{2}\frac{\omega_{k}^{2}+ 2 ( k^{+})^{2}}{4 \sqrt{2}k^{+}T}
|2c_{b}(e_{1}(t)-e_{2}(t)) \int_{0}^{\infty}d\xi \frac{\sin(t\xi)}{t^{2}}\psi_{v_{2}}(\xi,\lambda_{1}(t))| &\leq C |e_{1}(t)-e_{2}(t)| |\int_{0}^{\infty}d\xi \frac{\sin(t\xi)}{t^{2}}\psi_{v_{2}}(\xi,\lambda_{1}(t))| &\leq C \frac{|e_{1}(t)-e_{2}(t)|}{t^{3}\lambda_{1}(t)}
\frac{\gamma\left( a+1,a\right)}{\Gamma\left( a+1\right)}=\frac{\gamma\left( a,a\right)}{\Gamma\left( a\right)}-\frac{a^{a}\exp\left( -a\right)}{\Gamma\left( a+1\right)}& =\frac{1}{2}+\frac{1}{\sqrt{2\pi}}\left( -\frac{2}{3}a^{-1/2}+\frac{23}{270}a^{-3/2}\right) +O\left( a^{-5/2}\right) .
n_j=\begin{cases}(n+r)/d-1 (d-r-1)/2\leq j \leq (d+r-1)/2 begin{align*}4pt] (n+r)/d 1 \leq j < (d-r-1)/2 \:\: \:\: (d+r-1)/2 < j \leq d-1\end{cases}
\langle \overline{{\phi}}_{1}, \overline{{\phi}}_{2}\rangle = \overline{{\phi}}_{2}( \phi_{1}^{*}) = \int \d v \ \overline{{\phi}}_{2}( x ) \phi_{1}^{*}( x ) = \int \d v \ \phi_{2}( x ) \overline{{\phi_{1}^{*}}}( x ) .
\mathbb{Q}\subseteq\mathbb{Q}(\sqrt{2})\subseteq\mathbb{Q}(\sqrt{2},\sqrt{3})\subseteq\mathbb{Q}(\sqrt{2},\sqrt{3})(e^{2i\pi/5})\subseteq\mathbb{Q}(\sqrt{2},\sqrt{3})(e^{2i\pi/5},a)
\tilde{V}_{\nu}( r ) = - \omega^{2}( f - 1 ) + h \left\{{\frac{3}{4 r^{2}}}\left( 1 +{\frac{3 r_{0}^{2}}{r^{2}}}\right) +{\frac{8 R^{4}}{r^{2}( r^{2}+ R^{2})^{2}}}\right\},
Z \; = \; \prod_{j = 1}^{M}\Gamma_{\, j}, E \; = \; \prod_{j = 1}^{M}V_{\, j}, V_{\, j}\; = \; \left( \begin{array}{c c c}{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}\\{{0}}&{{1}}&{{0}}\\ \end{array}\right)_{\, j}.
\frac{d}{d t}\lambda(t) \int_M \varphi^2 \d\mu = \lambda(t) \int_M \mathcal{H}\varphi^2 \d\mu - \int_M \mathcal{H}| \nabla \varphi |^2 \d\mu + 2 \int_M \langle h, d \varphi \otimes d \varphi \rangle \d\mu.
\, (n^2-1)^{1/2}\cosh(\rho(n))-n - \tfrac{\sinh(2\rho(n))}{4}>(n^2-1)^{1/2}\Big( 1+ \tfrac{\rho(n))^2}{2}\Big)-n - \tfrac{2\rho(n)+(2\rho(n))^3}{4}= \, (n^2-1)^{1/2}- n +(n^2-1)^{1/2}\tfrac{\rho(n))^2}{2}- \rho(n)-4\rho(n)^3.
\left[{\cal H}- \zeta^{\, 0}\right]_{\mathrm{q u a d}}= \frac{1}{2}\left\{\dot{\chi}^{2}+ [ ( \partial_{\, z}+ F^{\prime}) \chi ]^{2}+ i \psi_{2}( \partial_{\, z}- F^{\prime}) \psi_{1}+ i \psi_{1}( \partial_{\, z}+ F^{\prime}) \psi_{2}\right\}\, ,
\frac{1}{2i\pi}\int_{C_N}R(t) \frac{\pi^3}{\sin(\pi t)^3}e^{u t}dt=\sum_{k=n+1}^N \left(\frac{\pi^2+u^2}{2}R(k)+ uR'(k)+ \frac12R''(k)\right)(-e^u)^k.
\begin{aligned}0 \equiv \xi^1_2\wedge\eta^2\wedge\eta^0+\xi^\varrho_2\wedge\eta^2\wedge\eta^1 \mod\{\eta^{\overline{1}}\}\Rightarrow 0 \equiv \xi^1_2,\xi^\varrho_2 \mod\{\eta^0,\eta^1,\eta^2,\eta^{\overline{1}}\}.\end{aligned}
\int_{\mathbb{T}^{N}}\Big\Vert \sum_{\alpha}x_{\alpha}z^{\alpha}\Big\Vert^{q'}dz & = \int_{\mathbb{T}^{N-1}}\left( \int_{\mathbb T}\Big\Vert \sum_{\alpha}\left( x_{\alpha}z_2^{\alpha_2}\cdots z_N^{\alpha_N}\right) z_1^{\alpha_1+(m+1)\alpha_2 + \cdots + (m+1)^{N-1}\alpha_N}\Big\Vert^{q'}dz_1 \right) dz_2 \cdots dz_N.
2m\sum_{1 \leq l \leq p-1}{M^{2l}\sum_{1 \leq t \leq p-l+1, 0 \leq l_0 \leq \min{(\frac{t}{2},l)}}{\binom{l-l_0+t-1}{l-l_0}\binom{t}{2l_0}}b_{p-l,t}}:=2m \cdot (s(p,M)-M^{2p}),
(\mathcal{F}\otimes_{t_1, t_2}\mathcal{G})_{v}:=\bigoplus_{v_1+v_2=v}(\mathbb{S}_{v_1, v_2})_{*}\Big( (\mathcal{F}_{v_1}\boxtimes \mathcal{G}_{v_1})\otimes \mathcal L_{v_1, v_2}\Big),
\Bigl|\Bigl(\prod(1+\alpha_k)\Bigr)\circ\beta-\prod_{k=1}^n(1+\alpha_k(\beta))\Bigr| =\Bigl|\Bigl(\prod_{k=1}^\infty(1+\alpha_k)-\prod_{k=1}^n(1+\alpha_k)\Bigr)\circ\beta\Bigr| &\le\Bigl|\prod_{k=1}^\infty(1+\alpha_k)-\prod_{k=1}^n(1+\alpha_k)\Bigr|\xrightarrow{n\to\infty}0.
S\,=\, \int d^3x \left[ -\frac{1}{4}(\frac{1}{2\rho}) F_{\mu\nu}(A) F^{\mu\nu}(A) + 2 \pi A^\mu{\tilde J}_\mu +\frac{1}{4 \rho}(\partial \rho)^2 - V(\rho) \right]
(P_k)_j=\begin{bmatrix}0 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-2)}\end{bmatrix}^{T}
\biggl| \widehat{h}^q \Bigl(Y, \sum_{i=1}^{r}m_i D_i + \sum_{j=1}^s p_jE_j\Bigr) - \widehat{h}^q \Bigl(Y, \sum_{i=1}^r m_i D_i' \Bigr) \biggr| \leq n! \, C p (m+p)^{n-1}.
\Psi(p)= \lim_{x\to0}{}_3F_2\bigg[\begin{matrix}-a+p+x 1+a-p-x \beta &1 2\beta\end{matrix}\bigg|\,1\bigg] = \lim_{x\to0}\frac{\Gamma(\frac12)\Gamma(\frac12+\beta)\Gamma(\beta)}{\Gamma(\frac12-\frac{p-1-a+x}2)\Gamma(1+\frac{{p-1-a+x}}2)\Gamma(\beta-\frac{p-1-a+x}2)\Gamma(\frac12+\beta+\frac{p-1-a+x}2)}=0.
\beta_{x_1}(\lambda^c_xs)=\prod_{\ell=1}^{\infty}\frac{\lambda^u_{\psi^{x_1}_{-\ell}(\lambda^c_xs)}}{\lambda^u_{x_{-\ell+1}}}=\prod_{\ell=1}^{\infty}\frac{\lambda^u_{\psi^x_{-\ell+1}(s)}}{\lambda^u_{x_{-\ell+1}}}=\frac{\lambda^u_{\Phi^c_x(s)}}{\lambda^u_x}\beta_x(s),
\sum_{n \leq x}c_K(n)\log n = \sum_{N(\mathfrak{a})\leq x}\log N(\mathfrak{a}) = \sum_{N(\mathfrak{a})\leq x}\sum_{\mathfrak{b}| \mathfrak{a}}\Lambda_K(\mathfrak{b}) & = \sum_{N(\mathfrak{b}) \leq x}\Lambda_K(\mathfrak{b}) \sum_{N(\mathfrak{a})\leq \frac{x}{N(\mathfrak{b})}}1 = \sum_{n \leq x}\Lambda_K^\#(n) \sum_{m \leq \frac{x}{n}}c_K(m).
\mathrm{Tr}(XT_z(s)) = \mathrm{Tr}(Y^{-is}XB^{is}B^{z-1}[BA^{\frac12},A^{z-\frac12+is}]) \quad+ \mathrm{Tr}(Y^{-is}XB^{is}[BA^{\frac12},A^{\frac12+is}]Y^{z-1}).
V(x)=\frac{1}{1-\left( \Psi (x)\right)^{\theta_{1}+\theta_{2}+\theta_{3}}}\sum_{i=0}^{\infty}\sum_{j=0}^{\alpha -1}\sum_{k=0}^{\infty}\vartheta_{i,j}^{(k)}(a\Gamma (\alpha^{\ast},x)+b(\Gamma (\alpha^{\ast}+1,x)).
\left [\begin{matrix}a_{11}1 0 a_{21}+ a_{22}\end{matrix}\right ]*\left [\begin{matrix}a_{11}0 a_{21}a_{22}+1\end{matrix}\right ]= \left [\begin{matrix}a_{11}1 a_{21}a_{22}\end{matrix}\right ]+v^{a_{11}-a_{22}-1}\ \overline{[a_{11}+1]}\left [\begin{matrix}a_{11}+1 0 a_{21}-1 a_{22}+1\end{matrix}\right ].
\langle (L_{c_*}+ 1) \hat{u}_{\pm 1},\hat{u}_{\pm 1}\rangle_{L^2(\mathbb{R})}\geq A \| \hat{u}_{\pm 1}\|_{H^1(\mathbb{R})}^2\mbox{\rm if}\;\; \langle \eta_*, \hat{u}_{\pm 1}\rangle_{L^2(\mathbb{R})}= 0.
Y^i(\Psi(x,z))-Y^j(x)\partial_{x^j}(\Psi^i)(x,z)-\tau(x) H^{\alpha}(z)\partial_{z^{\alpha}}(\Psi^i)(x,z)-C^{\ell}(x) K^{\alpha}_{\ell}(z)\partial_{z^{\alpha}}(\Psi^i)(x,z)=0,
\xi'(r) =\frac1r\int_{\partial D_r}\nabla\Big(e^{\frac{i}{2}(\theta_0-\theta_{\mathbf e})}\Psi_k\Big)\cdot\nu \, \overline{\psi^k_2(\theta_0(x))}\,ds =-\frac{i}{\sqrt\pi}r^{-\frac k2-1}\int_{\partial D_r}e^{-\frac{i}{2}\theta_{\mathbf e}}(i\nabla+A_{\mathbf e})\Psi_k\cdot\nu\, \psi_k\, ds.
\dot{q}_{i}=\frac{\partial R}{\partial\alpha_{i}}=f_{i}(\zeta_{1}(t),...,\zeta_{s}(t),\dot{\zeta}_{1}(t),...,\dot{\zeta}_{s}(t),\alpha_{1},...,\alpha_{n},t),\dot{p}_{i}=-\frac{\partial R}{\partial q_{i}}=0,
\int_{-\pi/2}^{\pi/2}(y_{1,2}')(y_{1,3}') = \frac{(n-1)^2(n-3)^2}{24^2}\frac{1}{160}\left(1710\pi - 300\pi^3+17\pi^5-\frac{83\pi^7}{168}+\frac{4\pi^9}{315}\right) & \approx\frac{(n-1)^2(n-3)^2}{24^2}0.993
x=\frac{\boldsymbol{a}_\perp^{(3)}\cdot\boldsymbol{v}^{(3)}}{\boldsymbol{a}^{(3)}\cdot\boldsymbol{v}^{(3)}},y=\frac{(\boldsymbol{a}^{(3)}\times\boldsymbol{a}_\perp^{(3)})\cdot\boldsymbol{v}^{(3)}}{\boldsymbol{a}^{(3)}\cdot\boldsymbol{v}^{(3)}}.
R^{(k+1)}=\frac{N-\tilde{B}-2(B-\tilde{B})}{N-\tilde{B}}\cdot \frac{1-\frac{T}{N-\tilde{B}-2(B-\tilde{B})}}{1-(\frac{T}{N-\tilde{B}-2(B-\tilde{B})})^M}=\frac{N+\tilde{B}-2B}{N-\tilde{B}}\cdot \frac{1-\frac{T}{N+\tilde{B}-2B}}{1-(\frac{T}{N+\tilde{B}-2B})^M}
\lim_{k\to \infty}\|(Y_{t-r_k}(\omega)-Y^-_t(\omega), Y(t-r_k,\omega)-Y^-(t,\omega))\|_{M^p}^p &= \lim_{k\to \infty}\int_{-r}^0 |Y(t-r_k+\beta, \omega)-Y^-(t+\beta, \omega)|^p d\beta + \lim_{k\to \infty}|Y(t-r_k, \omega)-Y^-(t, \omega)|^p=0,
|\mathcal{E}_{q, p}|\leq \sum_{l=1}^{\lfloor\frac{p}{2}\rfloor}\binom{q}{l}l^{p}\leq \lfloor\frac{p}{2}\rfloor \binom{q}{\lfloor\frac{p}{2}\rfloor}\left(\lfloor\frac{p}{2}\rfloor \right)^{p}.
:\hat{b}\hat{b}^{\dagger}:=:1+\hat{b}^{\dagger}\hat{b}:=:1:+:\hat{b}^{\dagger}\hat{b}:=1+\hat{b}^{\dagger}\hat{b}\ne\hat{b}^{\dagger}\hat{b}=:\hat{b}\hat{b}^{\dagger}:
\Phi_k \coloneqq\begin{bmatrix}\phi_{k-n}\dots \phi_k\end{bmatrix}=\begin{bmatrix}x_{k-n}\dots x_k u_{k-n}\dots u_k\end{bmatrix}, U_k \coloneqq \begin{bmatrix}u_{k-n+1}u_{k-n+2}\dots u_{k+1}u_{k-n+2}u_{k-n+3}\dots u_{k+2}\vdots \vdots \ddots \vdots u_{k-n+L-1}u_{k-n+L}\dots u_{k+L-1}\end{bmatrix}.
E_{0 0}= \frac{1}{a^{4}}\int d t a^{4}\left( \partial_{t}\left( \frac{3}{1 6}\dot{\phi}^{2}+ \frac{3}{8}V \right) + \frac{3}{2}H_{B}\dot{\phi}^{2}+ \frac{\dot{U}_{B}}{4}\rho \right) .
S^{\mathrm{A u x}}= \frac{1}{2}\int \! d x^{-}d x^{+}d y^{-}\sigma_{j i}( x^{-}, y^{-}; x^{+}) \left( s^{i j}( y , x ) - q_{+}^{\dag j}( x ) q_{+}^{i}( y ) \right) .
J_{2,-}(t_1,t_2) = \int_0^{t_2}\int_{-\infty}^0\int_{-\infty}^y G_{t_2-s}(x-z) \psi(s,z) \sigma_s(y) W(ds,dy) &-\int_0^{t_1}\int_{-\infty}^0 \int_{-\infty}^y G_{t_1-s}(x-z) \psi(s,z) \sigma_s(y) W(ds,dy).
(Qm_{\langle 0\rangle})_{\langle 0\rangle}\otimes (Qm_{\langle 0\rangle})_{\langle 1\rangle}S^2(P) \otimes R m_{\langle 1\rangle}\qquad{}= P(Rm)_{\langle 0\rangle}\otimes Q((Rm)_{\langle 1\rangle})^1S^2(P) \otimes R((Rm)_{\langle 1\rangle})^2 S^2(Q).
\sum_{(i_1,\ldots, i_k)\in \widetilde{J}_k\setminus J_{n,k}(v)}\prod_{j=1}^k d_{i_j}=k d_v \sum_{(i_1,\ldots, i_{k-1})\in J_{n,k-1}(v)}\prod_{j=1}^k d_{i_j}
\exp \left[\frac{i}{\hbar}S_{cl}(b,a)\right]=\prod_{k=1}^{n-1}\exp\left[ \frac{im}{2\hbar\epsilon}(r_{k}^{2}+r_{k-1}^2-2r_{k}r_{k-1}\cos(\theta_{k}-\theta_{k-1})\right].
\mathcal{J}_k(\boldsymbol{y}_k(\boldsymbol{v}_k),\boldsymbol{v}_k) = \frac{1}{2}\|\boldsymbol{y}_k-{\boldsymbol{y}_d}_k\|_{\Omega}^2 + \frac{\lambda}{2}\|\boldsymbol{v}_k\|_{\Omega}^2.
\sum_{j}\left( \frac{\partial X^{\mu}}{\partial x_{i}}\frac{\partial X^{\mu}}{\partial x_{j}}+ F_{i j}\right) \hat{L}_{j k}+ \left( \frac{\partial X^{\mu}}{\partial x_{i}}\frac{\partial X^{\mu}}{\partial x_{j}}- F_{i j}\right) \hat{L}_{k j}= 2 L \delta_{i k}
e^{i a \ln (\eta^2 - \mu^2/|\eta|)}= e^{i a\ln |\eta|^2}\exp \left( ia \ln \left( 1 - \frac{\mu^2}{|\eta|^3}\right) \right) = e^{i a\ln |\eta|^2}\left( 1 + \frac{\mu^2}{|\eta|^3}\phi \left( \frac{\mu}{|\eta|^{3/2}}\right) \right),
J_{\alpha}= \frac{\pi}{2e}\frac{\delta S}{\delta\bar{\chi}^{\alpha}}=\rho^{\beta}\rho_{\alpha}\bar{\Psi}^{\mu}\partial_{\beta}X_{\mu}-\frac{1}{4}\bar{\Psi}^{\mu}\Psi_{\mu}\rho_{\alpha}\rho^{\beta}\chi_{\beta}=0
\overline{H}\left(\bigsqcup_{a \in \mathcal{A}}^{P_A}G_a\right) = \sum_{a \in \mathcal{A}}P_A(a) \overline{H}(G_a) \Longleftrightarrow \;\; \overline{H}\left(\bigwedge_{a \in \mathcal{A}}G_a\right) = \sum_{a \in \mathcal{A}}\overline{H}(G_a).
Z_{\mathrm{g r}}[ \varphi , \mu , \bar{\mu}] = e^{K [ \varphi , \mu , \bar{\mu}]}\int{\cal D}T{\cal D}\overline{{T}}| \chi [ \mu , T ] |^{2}e^{-{\frac{1}{2}}\int d^{2}w e^{- \varphi}T \overline{{T}}},
\ln \left( \frac{N(\Phi) - 1}{L}\right) = \ln\left(\frac{1}{L}\sum_{j=0}^{L-1}N_{j}\right) \geq \frac{1}{L}\sum_{j=0}^{L-1}\ln N_{j}= \frac{1}{L}\cdot \ln\left( \prod_{j=0}^{L-1}N_{j}\right),