harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{Z}_{2^\infty,3^\infty}=\left\{a\in C\big([0,1], M_{2^\infty}\otimes M_{3^\infty}\big)\colon \begin{aligned}a(0)\in M_{2^\infty}\otimes 1_{M_{3^\infty}}, & a(1)\in 1_{M_{2^\infty}}\otimes M_{3^\infty}\end{aligned}\right\}
\left( x(s),y(s),z(s)\right)=\left(\frac{1}{5}\left(-2\sqrt{\wp( s)}+\frac{1}{4}\frac{\wp^\prime( s)}{\wp( s)}\right),\frac{1}{20}\left(\sqrt{\wp( s)}+\frac{1}{2}\frac{\wp^\prime( s)}{\wp( s)}\right),\frac{\sqrt{-g_3}}{2\sqrt{12}\wp(s)}\right).
{\cal H}_{\epsilon_{3}}={\cal H}_{\theta}\left({\frac{2 r_{5}^{2}}{g}}\epsilon_{3}\right) , ~ ~{\cal H}_{*_{6}\epsilon_{3}}={\cal H}\left( 2 r_{1}^{2}g e^{- 2 \bar{\Phi}}*_{6}\epsilon_{3}\right) .
\Theta(z) =\sum_{j=1}^{p}q_{j}\left( z-\sum_{\nu=k+1}^{\infty}a_{\nu,j}z^{\nu}\right) &=\sum_{j=1}^{p}q_{j}z -\sum_{\nu=k+1}^{\infty}\left( \sum_{j=1}^{p}q_{j}a_{\nu,j}z^{\nu}\right) &=z -\sum_{\nu=k+1}^{\infty}\left( \sum_{j=1}^{p}q_{j}a_{\nu,j}\right)z^{\nu}.
L_{i}=det^{-1}[(\pi_{S^{i}X})_{!}\{\xi\mathcal{O}_{X\times S^{i}X}(-\Delta_{i})\}]\otimes det^{-1}[(\pi_{S^{i}X})_{!}\{\mathcal{O}_{X\times S^{i}X}(\Delta_{i})\}],
U_{2i,j-k}= U_{2i,j+k}= (-1)^i 2^{k-j}\tilde K_{2i}(k;\textstyle{\frac{1}{2}},2j), \; k\in\{1,\ldots,j-1\}; & U_{2i,0}= U_{2i,2j}= (-1)^i 2^{-1/2}\tilde K_{2i}(j;\textstyle{\frac{1}{2}},2j); & U_{2i,j}=(-1)^i 2^{-j+1/2}\tilde K_{2i}(0;\textstyle{\frac{1}{2}},2j).
{\sum}_{i=1}^r{w_i\otimes\bar{w}_i}={\sum}_{j=1}^r{\sum}_{i,h=1}^r{a}_j^{i}\bar{a}_j^{h}v_i\otimes\bar{v}_h ={\sum}_{i,h=1}^r\big({\sum}_{j=1}^r{a}_j^{i}\bar{a}_j^{h}\big) v_i\otimes\bar{v}_h
\int_{\mathcal R_j}(cd,d)_p\chi_{\psi}(d)\chi_{\delta}(d)|d|_p^{-3/2}\mathbf h_p(h) dh = (-1)^j\chi_{\psi}(p^j)p^{3j/2}\int_{\mathcal R_j}(cd,p^j)_p dh.
\left. \mathbb{K}_{N}f_{N}\right\vert_{I_{n}}={\displaystyle\sum\limits_{m=1}^{N}}c_{m}^{2}\left[ \int_{S}g_{n}\left( s\right) g_{m}\left( s\right)\mathrm{d}s\right] f_{m}& =\left( \boldsymbol{K}_{N}\boldsymbol{f}_{N}\right)_{n}
\sigma P = \left( \frac{( a z + b ) ( \bar{c}\bar{z}+ \bar{d}) + a \bar{c}r^{2}}{| c z + d |^{2}+ | c |^{2}r^{2}}, \frac{r}{| c z + d |^{2}+ | c |^{2}r^{2}}\right)
\begin{array}{l}\bar{X}_{0}=x_{0},\quad\bar{K}_{0}=0,\quad\bar{X}_{t}\in\overline{\mathrm{Dom}\left( \varphi\right)},\smallskip \left\updownarrow \bar{K}\right\updownarrow_{t}-\left\updownarrow \bar{K}\right\updownarrow_{s}\leq\bar{V}_{t}-\bar{V}_{s}\quad\quad0=\bar{V}_{0}\leq\bar{V}_{s}\leq\bar{V}_{t}\end{array}
\|\big(u(f,g)(\cdot,T),\u(f,g)\big)\|_{H^2(\Omega)\times{H^2(\Gamma\times(T-\delta_1,T+\delta_1))}}&=\left(\|u(f,g)(\cdot,T)\|_{H^2(\Omega)}^2+\|u(f,g)\|_{H^2(\Gamma\times(T-\delta_1,T+\delta_1))}^2\right)^{\frac{1}{2}}
W(x) = \frac{(x+\epsilon)\sin(\pi z)\sin(\pi z')}{\pi^2}\cdot\frac{\Gamma(x+a+b+1)\Gamma(x+a+1)}{\Gamma(x+b+1)\Gamma(x+1)}\times\frac{\Gamma(x+1-z-N)\Gamma(x+1-z'-N)}{\Gamma(x+z+N+2\epsilon)\Gamma(x+z'+N+2\epsilon)}.
\mu[U]\left(b^{(1)}\otimes \ldots \otimes b^{(m)}\right) = V[v^{(i_0 + 1)}\leadsto x^{(i_0 + 1)}], &\mu[U]\left(c^{(1)}\otimes \ldots \otimes c^{(m)}\right) = W[w^{(i_0 + 1)}\leadsto y^{(i_0 + 1)}].
\begin{gathered}w\left(g(w)+a(w)+wg(w)a(w)\right)=0\Longleftrightarrow a(w)=-(1+wg(w))^{-1}g(w) z\left(b(z)+f(z)+zb(z)f(z)\right)=0\Longleftrightarrow b(z)=-(1+zf(z))^{-1}f(z) \begin{aligned}zw\big(2e(z,w)+f(z)g(w)+ zf(z)e(z,w)+a(w)b(z)+wa(w)e(z,w) + wg(w)e(z,w)+zb(z)e(z,w)+zwe^2(z,w))\big)=0.\end{aligned}\end{gathered}
\langle \mu + \rho, \epsilon_1-\epsilon_{j}\rangle = \langle \mu + \rho, \epsilon_1-\epsilon_{j_1}\rangle + \langle \mu + \rho, \epsilon_{j_1}-\epsilon_{j}\rangle &= p + \langle \nu + \rho, \epsilon_{j_1}-\epsilon_{j}\rangle < (n_{j_1j}+1)p \leq n_{1j}p.
G_{L_{n-1}}(z) = \mathbf{E}[z^{L_{n-1}}] &= \mathbf{E}[\mathbf{E}[z^{L_{n-1}}\lvert L_{n-2}]] &= \mathbf{E}[\left(q_{n-1}+ p_{n-1}z\right)^{(L_{n-2}+1)K-1}] &= \left(q_{n-1}+ p_{n-1}z\right)^{K-1}G_{L_{n-2}}(\left(q_{n-1}+p_{n-1}z\right)^{K}),
\ell_0\ge\frac{2 (M_1+1) r_2+2 (M_2+1) r_1+(r_1-r_2)^2}{4 (M_1+M_2+2)}=\frac{2 (a+1) k (a r_2+r_1)-a (r_1-r_2)^2}{4 (a+1)^2 k}
\epsilon_{( 0 , 4 . 8 )}^{'}= \epsilon \qquad \epsilon_{( 2 , 6 )}^{'}= \gamma_{1 1}\epsilon \qquad \epsilon_{( - 1 , 3 , 7 )}^{'}= \imath \epsilon \qquad \epsilon_{( 1 , 5 )}^{'}= \imath \epsilon^{\ast}
\Gamma_F[\varphi_0, 0]\,=\,- \int d^3x\,\int \frac{d^3p}{(2\pi)^3}\left\{\ln[1 - e^{-L (p + i e \varphi_0)}]+\ln[1 - e^{-L(p - i e \varphi_0}]\right\}\;,
\alpha_{1}(\alpha_{2}\alpha_{3})=[\alpha_{1},\alpha_{2}]_{A^{-}}\alpha_{3}-\varepsilon\alpha_{2}(\alpha_{3}\alpha_{1})-\frac{1}{n}[[\alpha_{1},\alpha_{2}]_{A^{-}},\alpha_{3}]+\frac{2}{n}\langle\alpha_{1},\alpha_{2}\rangle\alpha_{3}.
f(Oz+v,t-2\langle Oz,Jv\rangle) =f(O(z+O^Tv),t-2\langle Oz,JOO^Tv\rangle) &=f(O(z+O^Tv),t-2\langle Oz,OJO^Tv\rangle) &=f(O(z+O^Tv),t-2\langle z,JO^Tv\rangle) &=f(z+O^Tv,t-2\langle z,JO^Tv\rangle) &\ge f(z,t)+\langle z,O^Tv\rangle &=f(Oz,t)+\langle Oz,v\rangle,
\sum_{i=1}^m\|E_{L(\Sigma_2)'\cap L(\Gamma)}(u_{k_igk_i^{-1}})\|_2^2+\sum_{i=m+1}^n\|E_{L(\Omega_1\cap k_i^{-1}\Delta k_i)}(u_gu_{b_i}^*)\|_2^2 \geq d,\;\;\;\;
\mu \otimes \eta([\alpha_\pi](T) A_n \bigtriangleup A_n) = \int_X \eta(\alpha_\pi(T^{-1}x,x) A_n^{T^{-1}x}\bigtriangleup A_n^x)\d\mu(x) & = \frac{1}{\pi}\int_X \theta_{n,T^{-1}x,x}\d\mu(x)
E(d,j,j')=\frac{1}{d}\frac{\prod\limits_{k=0}^{5d}\left(5\alpha_j+k\frac{\alpha_{j'}-\alpha_j}{d}\right)}{\prod\limits_{l=0}^4\prod\limits_{k=0 \atop (l,k)\neq (j,0)}^d\left(\alpha_j-\alpha_l+k\frac{\alpha_{j'}-\alpha_j}{d}\right)},
P\left(\tau_N<s_N(\rho)\right)=P\left(\sup_{t\leq s_N(\rho)}Z_0(t) \geq N\right)\leq \frac{1}{N^2}E\left[Z_0\left(s_N(\rho)\right)^2\right]=O\left(N^{2(\rho-1)}\right).
|N_\mathcal{S}^w(Q,\delta)-\frac{2\hat{w}(0)}{n}\delta Q^n|\ll \delta Q^{n-1}+Q^{n}/J+J^{\frac{n-1}2}Q^{\frac{n-1}2}\log Q+Q^{\frac{n+1}2}G_n(J) \stackrel{J=Q}{\ll}E_n(Q),
S_{\scriptscriptstyle Y M}= \int d^{4}x T r \left\{- \frac{1}{4}F_{\mu \nu}^{a}F^{a \mu \nu}- \frac{1}{2}\bar{\lambda^{a}}\hat{D}\lambda^{a}+ \frac{1}{2}( D^{a})^{2}\right\},
{\bf E}S_n=n^\theta l(\theta,n)+o(\sqrt{{\bf E}S_n}), \ \ \frac{{\bf Var}S_n}{{\bf E}S_n}\to \sigma^2, \ \ \frac{S_n}{{\bf E}S_n}\stackrel{a.s.}{\to}1, \ \ \frac{S_n-{\bf E}S_n}{\sqrt{{\bf Var}S_n}}\Rightarrow{\bf N}_{0,1},
\gamma \left( r \right) = \frac{\operatorname{e x p}\left\{\sqrt{\frac{q}{k}}\operatorname{l o g}\frac{r^{2}}{\rho^{2}\sqrt{q}}\right\}}{\sqrt{q}\operatorname{e x p}\left\{\sqrt{\frac{q}{k}}\operatorname{l o g}\frac{r^{2}}{\rho^{2}\sqrt{q}}\right\}+ 1},
\displaystyle{\int_{\Sigma}\int_{|x_N|<\frac{1}{2(2k+1)}}f(\xi((2k+1)x), \varphi((2k+1)x), (2k+1)\nabla \xi((2k+1)x))dx}&\displaystyle{= \frac{1}{2k+1}\int_Q f(\xi(y), \varphi(y), (2k+1)\nabla \xi(y))dy \underset{k\to +\infty}{\to}\int_{Q}f^{\infty}(\xi(y), \varphi(y), \nabla \xi(y))dy.}
[(z\phi_0z^{-1})^* ,z\phi_0z^{-1}]= &=[\phi_0^*,\phi_0]+\frac{1}{2}[\phi_0^*,[G(k),\phi_0]]+\frac{1}{2}[[G(k),\phi_0]^*,\phi_0]+O(t^{-2}\mathcal L) &=\theta-\Delta G(k)+O(t^{-2}\mathcal L) &=\theta+(P-1)(k)+O(t^{-2}\mathcal L)
\aligned \langle \phi_1, \phi_2 \rangle_{G_g,a}= \sum_{\gamma}\frac{\widehat{\phi_1}(-\gamma)-\widehat{\phi_1}(0)}{\gamma}\cdot \frac{\widehat{\overline{\phi_2}}(\gamma)-\widehat{\overline{\phi_2}}(0)}{\gamma}\endaligned
\|u\|_{X_{|\tau|=|\xi|, \pm}^{s,b}(S_T)}=\inf\{\|v_{X_{|\tau|=|\xi|, \pm}^{s,b}}:v\in X_{|\tau|=|\xi|, \pm}^{s,b}\ \ \mbox{and}\ \v=u \ \ \mbox{on}\S_T\}.
\begin{aligned}P_1(x,y,y) = x, P_n(x,x,y) = y, \begin{aligned}P_i(x,y,x) = x 1\leq i\leq n, P_i(x,x,y) = P_{i+1}(x,y,y) 1\leq i<n. \end{aligned}\end{aligned}
||\xi_{\infty}||_{L^2}= \lim_{t\rightarrow \infty}\left|\left|\frac{\xi(t)}{t}\right|\right|_{L^2}\leq \lim_{t\rightarrow \infty}\frac{1}{t}\int_0^t ||\dot{\gamma}(s)||_{L^2}\, ds = ||*F_{A_{\infty}}+ \mu(u_{\infty})||_{L^2}= m
\beta(v,w)+\beta(v,x)+\beta(w,x)+\beta(x,w) & =\beta_{X}(v,y_{0})+\beta_{X}(v,x)+\beta_{X}(y_{0},x)+1 & =\beta_{X}(v,y_{0})+\beta_{X}(v,x)+\beta_{X}(y_{0},x)+\beta_{X}(x,y_{0})
\begin{cases}\partial_{t}u+(u\cdot\nabla)u-\Delta u+\nabla \Pi=0,\ \ x\in\mathbb{R}^{3},\t>0, \nabla\cdot u=0,\ \ x\in\mathbb{R}^{3},\t>0, u|_{t=0}=u_0,\ \ x\in\mathbb{R}^{3}. \end{cases}
\lambda_{a}\equiv l_{j_{a}}- \sum_{q = 1}^{a - 1}k_{j_{q}j_{a}}; \theta_{a}= \sum_{j}( l_{j}- \sum_{q = 1}^{a}k_{j_{q}j}) x_{j}, \rho_{j_{a}}= \sum_{j}k_{j_{a}j}x_{j}.
\begin{aligned}c_{2,1}(A)\leq \c_{2,1}\left(\bigcap_{i=1}^{d}\left\{\left|B_{s_{1}}^{i}-B_{t_{0}}^{i}\right|<2\eta\right\}\right) +\sum_{i=1}^{d}c_{2,1}\left(\sup_{s\in I}\left|B_{s_{1}}^{i}-B_{s}^{i}\right|>\eta\right) +\sum_{i=1}^{d}c_{2,1}\left(\sup_{t\in J}\left|B_{t}^{i}-B_{t_{0}}^{i}\right|>\eta\right).\end{aligned}
\mathcal{T}^\nu_{2,\mp}(\mathcal{T}^\nu_{1,\pm}f)(\xi) = \mathcal{C}_{{\mp}i{\psi_2}}(\mathcal{C}_{{\pm}i{\psi_1}}f)(\xi) &= \mathcal{C}_{{\pm}i{\psi_1}}f \left(\frac{\mp i \xi}{\sqrt 2}, \frac{{\mp}\xi}{\sqrt 2}\right) &= f(\xi).
\mu_2([z_1,\ldots,z_{d}]) =\mu_2(p_1(z_1,\ldots,z_{d}))=\iota_2^{\ast}(\Phi(z_1,\ldots,z_{d})) &=\iota_2^{\ast}(-\frac{1}{2}(2\lambda,\ldots,2\lambda)+(\lambda,\ldots,\lambda))=\iota_2^{\ast}(0)=0.
e^{\frac{1}{2(n+1)}}\frac{1}{\sqrt2}\left(\sqrt{p_n}+\sqrt{1-p_n}\right)=e^{\frac{1}{2(n+1)}}\sqrt{\frac{1}{2}+\sqrt{p_n(1-p_n)}}=e^{\frac{1}{2(n+1)}}\sqrt{e^{-\frac{1}{n+1}}}=1.
\delta_{n, 1}= \langle s_{(n)}, s_{(1^n)}\rangle=\langle h_n, e_n \rangle = \sum_{\lambda \vdash n}\langle \frac{p_\lambda}{z_\lambda}, \frac{\varepsilon_{\lambda}p_\lambda}{z_\lambda}\rangle= \sum_{\lambda \vdash n}\frac{\varepsilon_{\lambda}}{z_{\lambda}}
\sum_{k=0}^{n}\binom{n}{k}H_k=2^n\left(H_n-\sum_{k=1}^{n}\frac{1}{k2^k}\right)\;, &\sum_{k=0}^{n}\binom{n}{k}^2H_k=\binom{2n}{n}(2H_n-H_{2n})\;.
\begin{array}{rl}\lambda^* = \max \lambda \textnormal{subject to:}\displaystyle p_n - \lambda = v_1^T Q_0 v_1 + \sum_{i=0}^{n-1}v_1^T Q_i^+ v_1(1+X_i) + \sum_{i=0}^{n-1}v_1^T Q_i^- v_1 (1-X_i), & \displaystyle \lambda \in \mathbb{R}, & \displaystyle Q_0,Q_i^+,Q_i^- \succeq 0 \textnormal{for all}i \in \mathbb{Z}_n,\end{array}
A(\omega)|x|^{-\frac{1}{2}+i\rho}= (1- 2^{\frac{1}{2}+i(\rho-\frac{\omega}{2})})\zeta(\frac{1}{2}- i(\rho-\frac{\omega}{2})) |x|^{-\frac{1}{2}+i(\rho-\omega)}= 0, H_{-}|x|^{-\frac{1}{2}+i\rho}= |1-2^{\frac{1}{2}-i(\rho-\frac{\omega}{2})}|^{2}|\zeta(\frac{1}{2}-i(\rho-\frac{\omega}{2}))|^{2}|x|^{-\frac{1}{2}+i\rho}= 0.
Z_{ab}={}- \tfrac{4}{3}{*}F_{[a}{}^{c}Y_{b]c}, \beta_a={}- \tfrac{1}{2}U^{b}Z_{ab}- \tfrac{1}{2}i U^{b}{*}Z_{ab}- \tfrac{1}{2}\nabla_{b}Z_{a}{}^{b}- \tfrac{1}{2}i \nabla_{b}{*}Z_{a}{}^{b}.
\mathcal{D}_{k,\ell}(s_1,s_2) = \sum_{d \mid h}\frac{1}{d^{1+s_1+s_2}}\sum_{q=1}^{\infty}\frac{\mu(q) G_{k}(qd,s_1+1) G_{\ell}(qd,s_2+1)}{q^{2+s_1+s_2}}.
\begin{aligned}\uppercase\expandafter{\romannumeral1}_{2}\leq C\sum\limits_{j=1}^{\infty}\mu_{j}{\omega(P_{j})}^{-\frac{1}{p}}{\left({\left(\frac{l(P_{j})}{\vert x-x_{P_{j}}\vert}\right)}^{n}\right)}^{\gamma}\chi_{(4P_{j})^{c}}(x) &\leq C\sum\limits_{j=1}^{\infty}\mu_{j}{\omega(P_{j})}^{-\frac{1}{p}}(M\chi_{P_{j}})^{\gamma}(x)\end{aligned}
\lim_{n\to+\infty}\mathbb{P}(\tau_k^{(n)}\in I)=\lim_{n\to+\infty}\mathbb{P}(\tau_k^{(n)}\in A_1)-\lim_{n\to+\infty}\mathbb{P}(\tau_k^{(n)}\in A_2) =\varphi_k\left(-f'(a)\right)-\varphi_k\left(-f'(b)\right)=\int_a^{b}\frac{f''(x)(-f'(x))^{k-1}}{(k-1)!}e^{f'(x)}dx.
\begin{gathered}\int_0^x x^{-2\alpha}\rho^{\nu+1}({q^2\rho^2}/{x^2}; q^2)_{-\alpha}\int_0^\infty C_1(u) J_\nu (u\rho; q^2) \,d_q u\,d_q \rho = \int_0^x g_1 (\rho) x^{-2\alpha}\rho^{\nu+1}({q^2\rho^2}/{x^2}; q^2)_{-\alpha}\,d_q\rho.\end{gathered}
\mathcal I_v^{\sharp}(\mathbf h, \mathbf g, \pmb{\phi}) := \frac{\mathcal I_v(\mathbf h, \mathbf g, \pmb{\phi})}{\langle \mathbf h_v, \mathbf h_v \rangle\langle \mathbf g_v, \mathbf g_v \rangle \langle \pmb{\phi}_v, \pmb{\phi}_v \rangle}= \frac{\mathcal I_v(\mathbf h, \mathbf g, \pmb{\phi})}{||\mathbf h_v||^2 ||\mathbf g_v||^2||\pmb{\phi}_v||^2},
F(x,y)=2(x^4+y^4)\cdot\sum_{i=0}^{\lfloor \frac{N-2}{4}\rfloor}e_i(2xy)^{N-2-4i}\left(-\frac 1 4 x^8+\frac 1 2 x^4y^4-\frac 1 4y^8\right)^i.
\langle \Psi_{1}^{\prime I_{1}}( z_{1}) \Psi_{2}^{\prime I_{2}}( z_{2}) \cdots \Psi_{n}^{\prime I_{n}}( z_{n}) \rangle = \langle \Psi_{1}^{I_{1}}( z_{1}) \Psi_{2}^{I_{2}}( z_{2}) \cdots \Psi_{n}^{I_{n}}( z_{n}) \rangle
d(f_X, t) = (2 \sqrt{\pi})^{-1}\Gamma \left(\frac{s+1}{2}\right) \Gamma \left(1 - \frac{s}{2}\right) \int_{0}^{U}\left( P_{s-1}^0 (i v) + P_{s-1}^0 (-i v) \right) dv.
Z={V_{p+1}}{{\left({{\mbox{T}}\over{2\pi}}\right)}^{(p+1)/2}}\int d\tau{{\mbox{det}(\psi,B)}\over{\sqrt{\mbox{det}\hat{(B,B)}}}}{{\sqrt{\mbox{Det}'{\hat{P}^\dagger}\hat{P}}}\over{\mbox{Vol}(\mbox{CKV})}}{{\left({{{\mbox{Det}_N}'\hat{\Delta}}\over{\int{d^2}\xi\sqrt{\hat{g}}}}\right)}^{-(p+1)/2}}
- \Sigma = p ! g^{p - 1}\left( \frac{p - 1}{p^{2}}\right)^{\frac{p - 1}{2}}\frac{2^{\frac{p - 1}{2}}}{\left( \frac{p - 1}{2}\right) !}\prod_{i = 1}^{\frac{p - 1}{2}}\int \frac{d^{D}k_{i}}{( 2 \pi )^{D}}p^{- \frac{1}{2}k_{i}^{2}}
T_{\hat{M}}^{~ \hat{N}}= \mathrm{d i a g}\left( \frac{n ( n + 1 )}{2 \ell^{2}}- \frac{{\cal Q}^{2}}{a^{2 n}}, \frac{n ( n + 1 )}{2 \ell^{2}}+ \frac{{\cal Q}^{2}}{a^{2 n}}, \cdots , \frac{n ( n + 1 )}{2 \ell^{2}}+ \frac{{\cal Q}^{2}}{a^{2 n}}, \frac{n ( n + 1 )}{2 \ell^{2}}- \frac{{\cal Q}^{2}}{a^{2 n}}\right) .
P_k^{(m)}(\kappa^2) = (-1)^m \frac{\kappa}{m!}\partial_z^m[z^{m-1}(1-z^2)^k]_{|z = \kappa}& = \sum_{j=0}^{m-1}\frac{(-1)^m}{(m-j)!}\binom{m-1}{j}\kappa^{m-j}\partial_z^{m-j}[(1-z^2)^k]_{|z=\kappa}& = \sum_{j=1}^{m}\frac{(-1)^m}{j!}\binom{m-1}{m-j}\kappa^{j}\partial_z^{j}[(1-z^2)^k]_{|z=\kappa}.
\begin{array}{l}{{\Psi ( u ) = R ({\bf B}_{f}) \Psi_{f}( u_{f}) , \quad \bar{\Psi}( u ) = \bar{\Psi}_{f}( u_{f}) \widetilde{R}^{+}({\bf B}_{f}) ,}}\\{{g ( u ) \nabla \Psi ( u ) = S ( B_{f}) R ({\bf B}_{f}) \gamma_{f}D \Psi_{f}( u_{f}) ,}}\\{{\left( \nabla \bar{\Psi}( u ) \right) g ( u ) = S ( B_{f}) \left( D \bar{\Psi}_{f}( u_{f}) \right) \gamma_{f}\widetilde{R}^{+}({\bf B}_{f}) .}}\\ \end{array}
xy=\sum_{(x)_2, (y)_2}\tau(x_{(0)},y_{(0)})\tau(x_{(2)},Sy_{(2)})y_{(1)}x_{(1)}, &yx=\sum_{(x)_2, (y)_2}\tau(Sx_{(0)},y_{(0)})\tau(x_{(2)},y_{(2)})x_{(1)}y_{(1)}.
\displaystyle g ( \lambda | \lambda_{j}) = \sum_{k = 1}^{N_{H}}\chi ( \lambda - h_{k}) - 2 \sum_{k = 1}^{N_{S}}\chi ( \lambda - y_{k}) - \sum_{k = 1}^{M_{C}}\chi ( \lambda - c_{k}) - \sum_{k = 1}^{M_{W}}\chi ( \lambda - w_{k})_{\mathrm{I I}}\: ,
\begin{array}{c}{{{\bf J}={\bf p}\times{\bf x}+{\bf p}_{+}\times{\bf e}_{+}+{\bf p}_{1}\times{\bf e}_{1}=}}\\{{={\bf p}\times{\bf x}+ c ( 2 \epsilon - q ){\bf e}_{+}- 2 c{\bf e}_{-}.}}\\ \end{array}
\begin{bmatrix}z^{u_0}\cdots z^{u_N}u_{0,1}\cdot z^{u_0}/z_1 \cdots u_{N,1}\cdot z^{u_N}/z_1 \vdots \vdots u_{0,n}\cdot z^{u_0}/z_n \cdots u_{N,n}\cdot z^{u_N}/z_n \end{bmatrix}.
(2-7\epsilon)n \leq \sum_{uv\in E(S,L)}\mathbf{v}_u+\mathbf{v}_v =\left( \sum_{\substack{uv\in E(S,L) u=x}}\mathbf{v}_u+\mathbf{v}_v\right) + \left(\sum_{\substack{uv\in E(S,L) u\not= x}}\mathbf{v}_u+\mathbf{v}_v\right) \leq \epsilon e(S,L) + d_x + \sum_{\substack{uv\in E(S,L) u\not= x}}\mathbf{v}_u,
\int_{[-1, 1]^n}|\nabla v|^pdx = \int_{[-1, 1]^n}|\lambda(2+f(x_1))^{\lambda-1}f'(x_1)|^pdx \le C\int_{-1}^{1}\alpha^p|x_1|^{(\alpha-1)p}dx_1\le C\alpha^{p}\to 0
E\left[\exp\left(\left(\frac{|Y|}{M\Lambda}\right)^{2/q}\right)\right] =\sum_{k=0}^\infty\frac{1}{k!}E\left[\left(\frac{|Y|}{M\Lambda}\right)^{2k/q}\right]\leq1+\sum_{k=1}^\infty C^{2k/q}\frac{(2k/q)^k}{k!M^{2k/q}}.
{\Lambda_n}= \int_0^\infty \int_0^\infty \cdots \int_0^\infty{{\log}^n}\left({1 + \sum\limits_{i = 1}^M{{\gamma_i}}}\right) &\times{f_\gamma}\left({{\gamma_1},{\gamma_2}, \cdots{\gamma_M}}\right)d{\gamma_1}d{\gamma_2}\cdots d{\gamma_M},
\int_M (\eta+g^{-1})|D^{''*}v|^2_QdV_M+\int_M \eta|D^{''}v|^2_QdV_M \ge \int_M \langle[\eta \sqrt{-1}\Theta_Q-\sqrt{-1}\partial \bar{\partial}\eta-\sqrt{-1}g\partial\eta \wedge\bar{\partial}\eta, \Lambda_{\omega}]v,v\rangle_QdV_M.
V^2 + V^4 |\nabla \tau|^2 +\frac{(div V^2 \nabla \tau)^2}{|H_0|^2}= A^2 +2AB_i \tilde{X}^i r + r^2 \left [ (B_i \tilde{X}^i)^2 +A^2+(C_i \tilde{X}^i)^2 + g_1+ \frac{g_2}{4}- h_0^{(1)}\sum_{ij}C_iC_j \tilde{X}^i \tilde{X}^j \right],
\left| \left| w_{\dot{\alpha}}^{~ u}\right| \right| = \left| \left| \bar{w}_{~ u}^{\dot{\alpha}}\right| \right| = \rho \, T \, \left( \begin{array}{l}{0_{[ N - 2 ] \times [ 2 ]}}{1_{[ 2 ] \times [ 2 ]}}\end{array}\right)
x^2 y^2 = x^2 \left(1-\frac{q\nu_z\nu_x - q^{-1}\nu_x\nu_z}{q-q^{-1}}\right) = x^2 - \frac{qx^2 \nu_z\nu_x - q^{-1}x^2 \nu_x \nu_z}{q-q^{-1}}.
V = e^{- a_{p + 1}\phi}\sqrt{- g_{t t}^{E}g_{w_{1}w_{1}}^{E}. . . g_{w_{p}w_{p}}^{E}g_{y y}^{E}}= H^{\frac{4 ( D + p ) - ( D - 2 )^{2}a a_{p + 1}}{2 ( D - 2 ) \Delta}}.
\frac{d}{dt}\mathcal{M}(L^+_{\gamma}(t)) = KR \cos\gamma\int_{-M}^M\big[B_{-,\omega}(t)+B_{+,\omega}(t)\big] ~ d\omega + \int_{-M}^M\big(\dot{\phi}(t)-\omega\big)\big[B_{+,\omega}(t)-B_{-,\omega}(t)\big] ~ d\omega.
\begin{aligned}\partial_ty(t,\omega)-\Delta_\omega y(t,\omega)-\tfrac1{10}\chi_{\Omega_u}(\omega)u(t)-\tfrac1{10}\chi_{\Omega_v}(\omega)v(t) \,=\,0 \qquad I\times\Omega \vec{\mathbf n}(\omega)\cdot\nabla_\omega y(t,\omega) \,=\,0 I\times\Gamma y(0,\omega) \,=\,0 \Omega\end{aligned}
\begin{gathered}\varrho_{K}^{(N,J)}[u,X]\leq{\varrho}_E^{(N)}[u,X] + L h_N^\alpha \;\; u\in U,{\vartheta}_{K}^{(N,J)}\leq{\vartheta}_E^{(N)}+ L h_N^\alpha \;\; .\end{gathered}
\eta_Y^{-1}\psi_k\eta_{k-1}(H^{2,0}(X_{k-1}))=\eta_Y^{-1}(\psi_k(\ell_{k-1}))=\eta_Y^{-1}(\psi(\ell_0))=\eta_Y^{-1}(P(Y,\eta_Y))=H^{2,0}(Y),
K = \frac{8 \pi^{2 \mu}( 2 \mu - 1 )^{2}( \mu - 2 )^{2}}{( 2 \mu - 3 )^{2}\Gamma ( \mu ) \Gamma ( \mu + 1 )}, P = \frac{1 6 \pi^{2 \mu}( 2 \mu - 1 ) ( 2 - \mu )}{( 2 \mu - 3 ) \Gamma ( \mu ) \Gamma ( \mu + 1 )}
Ts_t(v_m)=T\Phi_{\xi_0(p)^{-1}}\left(T\varphi_t\left(\operatorname{Hor}^{\Gamma_0}_p(v_m)\right)\right)=T\Phi_{\xi_0(p)^{-1}}\left(\operatorname{Hor}^{(\varphi_t)_*\Gamma_0}_{\varphi_t(p)}(v_m)\right).
\begin{array}{c c c c c c}{{}}&{{}}&{{1}}&{{}}&{{}}&{{}}\\{{}}&{{0}}&{{}}&{{0}}&{{}}&{{}}\\{{1}}&{{}}&{{2 0}}&{{}}&{{1}}&{{}}\\{{}}&{{0}}&{{}}&{{0}}&{{}}&{{}}\\{{}}&{{}}&{{1}}&{{}}&{{}}&{{.}}\\ \end{array}
\displaystyle|\mathcal{SC}_{(s,s+d,s+2d)}|=\begin{cases}\displaystyle\sum_{i=0}^{\lfloor s/4 \rfloor}\binom{(s+d-1)/2}{i, d/2+i, (s-1)/2-2i}& &\displaystyle\sum_{i=0}^{\lfloor s/2\rfloor}\binom{\lfloor (s+d-1)/2 \rfloor}{\lfloor i/2 \rfloor, \lfloor (d+i)/2\rfloor, \lfloor s/2 \rfloor -i}\end{cases}
\Psi_t(\rho\oplus \rho(x,y)(\xi\oplus\eta)) = \Psi_t(\rho(x,y)\xi\oplus\rho(x,y)\eta) & = (\rho(x,y)\xi)_t \otimes (\rho(x,y)\eta)_t & = \rho_t(x,y) \xi_t \otimes \rho_t(x,y) \eta_t & = (\rho_t \otimes \rho_t)(x,y) (\xi_t \otimes \eta_t) & = (\rho_t \otimes \rho_t)(x,y) \Psi_t(\xi\oplus \eta).
\||W_1(t, x-x_n)|^\alpha \times |W_2(t+t_n-s_n, x-y_n)|\|_{L^{r'}_x}=\||W_1(t, x)|^\alpha \times |W_2(t+t_n-s_n, x+x_n-y_n)|\|_{L^{r'}_x}\overset{n\rightarrow \infty}\longrightarrow 0
\Pr(\pi'(\lambda)=i,\, \xi(\lambda) = i + j ) &=\Pr( \exists \, t < \lambda : \pi'(t) = i, \, \xi(t) =i+ j )\,\frac{b + c(i+j)}{1 + b + c(i+j)}&=[b + c(i+j)]v_{i,i+j}.
A_{\sigma l}= (l +1)(l+2+2\sigma_n) p_{\sigma, l+1}+ 2s(\sigma_n + 1) p_{\sigma+\bar n, l}+ (\sigma_i +1)(\sigma_i + 2) p_{\sigma + 2\bar \imath, l-1}+ c_{\sigma l}^{\mu m}p_{\mu m}.
\left\{\left\vert |V| \cdot \frac{N}{M}- |V'| \cdot \frac{n'}{m+1}\right\vert > \frac{t}{2}~\right\}\subset \left\{2c(2\ell+3) \cdot n^{1-\alpha}k^{\frac{\ell}{\ell+1}}>2c(\ell+4)\cdot n^\alpha k^{\frac{1}{\ell+1}}\right\}=\emptyset,
|\mathfrak m_N(\xi)-\lambda_N^1(\xi)| \le 17\min\Big\{e^{-\frac{c\kappa(d, N)^2}{400}\sum_{i=1}^d\sin^2(\pi\xi_i)}, \kappa(d, N)^2\sum_{i=1}^d\sin^2(\pi\xi_i)\Big\},
\sum_{{n=-\infty}\atop{n\ne \alpha,\beta}}\frac{|{\widehat f}(n)|^2}{\displaystyle\sum_{j=1}^{4s+1}\left|\cos\left(\left(\frac{\alpha-\beta}{2}\right)x_j\right)-\cos \left(\left(n-\frac{\alpha+\beta}{2}\right)x_j\right)\right|^{2s}}
\theta_{L^1}(\mathcal{C}) =\sum_{\sigma\in\mathcal{C}_{\nu\leq K}}\mu(\sigma)\delta_{\nu}^2(\sigma)+2K\sum_{\sigma\in\mathcal{C}_{\nu>K}}\mu(\sigma)\left|\delta_{\nu}(\sigma)\right| &\leq\sum_{\sigma\in\mathcal{C}_{\nu\leq K}}\mu(\sigma)K^2+2K\sum_{\sigma\in\mathcal{C}}|\nu|(\sigma) &\leq K^2+2K||\nu||_{L^1}.
U \pi(x,\omega,\tau) f = U \Big( \sum_{k=1}^n c_k e^{2 \pi i \tau}\pi(x,\omega,0) \pi(x_k,\omega_k,0) g \Big) & = \sum_{k=1}^n c_k e^{2 \pi i \tau}\rho(x,\omega) \rho(x_k,\omega_k) \varphi & = \rho(x,\omega,\tau) U f .
|\nabla \tilde{w}_\varepsilon(\tilde{z}_\varepsilon)|=\frac{|\nabla u_\varepsilon(z_\varepsilon)|}{|\nabla u_\varepsilon(y_\varepsilon)|}\le \frac{u_\varepsilon(y_\varepsilon)}{u_\varepsilon(z_\varepsilon)}\frac{1}{|\tilde{x}_\varepsilon-\tilde{z}_\varepsilon|}\le \frac{1+o(1)}{|\tilde{x}_\varepsilon-\tilde{z}_\varepsilon|}
F[f](x) - c_{1}^{\prime}\epsilon (x^{0}) \delta ((x^{0})^{2}- |{\bf x}|^{2}) =\sum_{k = 0}^{N}a_{k}\left( \sum_{\nu = 0}^{3}\eta^{\nu \nu}\left( \frac{\partial}{\partial x^{\nu}}\right)^{2}\right)^{k}\delta (x).
det^{S^2}(\otimes_{1\leq i<j\leq 2d}(v_{i,j}))=\sum_{(\Gamma_1,...,\Gamma_d)\in \mathcal{P}^{h,cf}_d(K_{2d})}\varepsilon_d^{S^2}((\Gamma_1,...,\Gamma_d))M_{(\Gamma_1,...,\Gamma_d)}((v_{i,j})_{1\leq i<j\leq 2d}),
p_2(u) = \frac{u(u-1)}{(-\tfrac{1}{2})(-\tfrac{3}{2})}f(a) + \frac{(u+\tfrac{1}{2})(u-1)}{(\tfrac{1}{2})(-1)}f(a+\tfrac{1}{2}h) + \frac{(u+\tfrac{1}{2})u}{(\tfrac{3}{2})(1)}f(a+\tfrac{3}{2}h)
\kappa^{2}_{1,\pm}= \left[ \frac{1}{2}\left( \kappa^{2}_{1,1}+\kappa^{2}_{1,2}\right) - \kappa^{2}_{1,3}\right] \pm i \left(\kappa_{1,1}+ \kappa_{1,2}\right) \sqrt{\kappa^{2}_{1,3}-\left[\frac{1}{2}\left( \kappa_{1,1}- \kappa_{1,2}\right)\right]^{2}}\, .
< D^{\alpha \beta \, 6}> = \sum_{a = 1}^{5}q^{6}{}_{a}< ( h^{* a \alpha}h^{a \beta}+ h^{* a \beta}h^{a \alpha}) > = - 2 \, ( v^{* \alpha}v^{\beta}+ v^{* \beta}v^{\alpha}) ) \neq 0 .
\sum_{n\geqslant2}\sum_{i=2}^{2^n-1}f\bigl(s_{i-1}^{n},s_{i}^{n}\bigr) =\sum_{n\geqslant2}\sum_{i=2}^{2^n-1}\frac{1}{\ln(4/3)}\iint\limits_{A_i^n}\frac{f\bigl(s_{i-1}^{n},s_{i}^{n}\bigr)}{(s-t)^2}\,dt\,ds &\leqslant4\int_0^1\int_s^1\frac{f(s,t)}{(s-t)^2}\,dt\,ds,
\|\varphi_t\|_{H_\Gamma^{3/2}(\partial \mathrm{M})}^2=\sum_{N_t^1}\tau_j^{-2}|a_j|^2\le t^{-2}\sum_j |a_j|^2=t^{-2}\|u\|_{L^2(\mathrm{M}_0)}^2.
|u(x)-u(y)|^{p-2}\, (u(x)-u(y))\, \left[\frac{v(x)^q}{u(x)^{q-1}}-\frac{v(y)^q}{u(y)^{q-1}}\right] &\le|v(x)-v(y)|^q\,|u(x)-u(y)|^{p-q}.
\tau(\langle({\partial}a^*)c, ({\partial}a^*)c\rangle_{\mathcal A})=\tau(c^*\langle{\partial}a^*,{\partial}a^*\rangle_{\mathcal A}c) \leq \|\langle{\partial}a^*,{\partial}a^*\rangle_{\mathcal A}\|_\infty \tau(c^*c).
(J\circ r_2\circ J^{-1})(x,y,z) =J(r_2(x,u,\lambda_{u}^{-1}(v))) =J(x,v,\rho_{\lambda_u^{-1}(v)}(u)) =(x,z,\lambda_x(\lambda_v(\rho_{\lambda_u^{-1}(v)}(u)))) =(x,z,\sigma_z(y)) =s_2(x,y,z).
\sum_{\substack{p\le x p\textrm{splits in}K}}N_{p}^k(E[\ell])=1^k\cdot \pi_E^s(x;\ell,0) +\ell^k\cdot \pi_E^s(x;\ell,1) +\ell^{2k}\cdot \pi_E^s(x;\ell,2).
A_{i}= ( 2 \pi R )^{p - 3}\sqrt{\left( \left( n_{1}^{( i )}\right)^{2}+ \left( m_{1}^{( i )}\right)^{2}\right) \left( \left( n_{2}^{( i )}\right)^{2}+ \left( m_{2}^{( i )}\right)^{2}\right)}.
\lbrace A,B\rbrace{}^{*}=\lbrace A,B\rbrace -i[\lbrace A,\pi_{\theta \, i}+{\frac{i}{2}}\theta^{*}_i\rbrace \lbrace \pi_{\theta^{*}\, i}+{\frac{i}{2}}\theta_i,B\rbrace +\lbrace A,\pi_{\theta^{*}\, i}+{\frac{i}{2}}\theta_i\rbrace \lbrace \pi_{\theta \, i}+{\frac{i}{2}}\theta^{*}_i,B\rbrace ],
\left| \left| \frac{1}{nh}\sum_{k=b+1}^n K\left(\frac{k/n-t_1}{h}\right)\mathbb{E}(x_{k-1}\epsilon_k)^2-\mathbb{E}(\mathbf{U}_1^2(t_1, \mathcal{F}_0)) \right| \right| \leq C\Big( (nh)^{-1/2}+ h^2 +n^{-1}\Big ).
T^{p}S \left( a | n > + b | \widehat{n^{\prime}}> \right) = i \ \mathrm{e x p}( - i \pi / 6 ) V \sum_{m m^{\prime}}\frac{i}{2 \epsilon}a | \widehat{m}> + ( 2 i \epsilon b + p ) | m^{\prime}>
Skew[X]=\frac{2(15\kappa^{6}+6\kappa^{4}+2\kappa^{2}+1)}{(1-9\kappa^{2})(2\kappa^{4}+1)}\sqrt{\frac{1-4\kappa^{2}}{1+2\kappa^{4}}}for0\le\kappa<1/3
\begin{aligned}\int_0^T \int |\Delta d|^2 dxd\tau =\int_0^T \int |\nabla d|^4 dx d\tau+\int_0^T \int \left|\Delta d+|\nabla d|^2 d\right|^2 dxd\tau &\le \int_0^T \int |\nabla d|^4 dx \tau+C.\end{aligned}
0 =\lim_{t_j\to 0}\frac{1}{t_j}\left( S'|_{g_{t_j}}\big(h(t_j), v(t_j) \big)-S'|_{g_{t_j}}\big(\psi_{t_j}^* (h, v)\big) \right) =\lim_{t_j\to 0}\frac{1}{t_j}\left( S'|_{g_{t_j}}\big(h(t_j)-h, v(t_j)-v \big)-S'|_{g_{t_j}}\big( \psi_{t_j}^* (h, v) -(h,v)\big)\right) = S' \big(p-L_X h, z- X(v) \big).
\frac{p^{s_F^\ell(L_1^{m_r};x)}-1}{m_r/(\ell+1)}\le \frac{p^{s_F^\ell(L_2^{m_r+j};x)}-1}{m_r/(\ell+1)}= \frac{p^{s_F^\ell(L_2^{m_r+j};x)}-1}{(m_r+j)/(\ell+1)}\cdot \frac{m_r+j}{m_r}.