harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}\Big\| \Lambda^{\alpha}(uv)- \sum_{|k|\le \alpha_1}\frac{1}{k!}\partial^k u\, \Lambda^{\alpha,k}v - \sum_{|m| \le \alpha_2}\frac{1}{m!}\partial^{m}v \,\Lambda^{\alpha,m}u \Big\|_{L^p}\lesssim_{\alpha,\alpha_1,\alpha_2,p,p_1,p_2,N}\| \Lambda^{\alpha_1}u \|_{L^{p_1}}\| \Lambda^{\alpha_2}v \|_{L^{p_2}},\end{aligned}
\Delta Y^{m}_{l_1,\ldots,l_{n-2}}(\varphi_1,\ldots,\varphi_{n-2},\phi) = -l_{n-2}(l_{n-2}+n-2)Y^{m}_{l_1,\ldots,l_{n-2}}(\varphi_1,\ldots,\varphi_{n-2},\phi) .
\begin{array}{ll}g_{_E}(\widehat{\mathbf{f}}_\mu(x), \widehat{\mathbf{f}}_\nu(x))=g(\mathbf{f}_\mu(x), \mathbf{f}_\nu(x))=g_{\mu\nu}(x) g_{_E}(\widehat{\mathbf{f}}_\mu(x), \mathbf{e}_k(x))=0 g_{_E}(\mathbf{e}_k(z), \mathbf{e}_l(z))=g_{0 \, kl}(x) \, . \end{array}
X(t) = X(0) + \sum_{k=1}^K \zeta_k \int_{[0,t]\times[0,\infty)}1_{\left[ \overline q_{k-1},\ \overline q_{k-1}+ \lambda_k^X(s-,X(s-) \right)}(x) N(ds\times dx) Z(t)= Z(0) + \sum_{k=1}^K \zeta_k \int_{[0,t]\times[0,\infty)}1_{\left[ \overline q_{k-1},\ \overline q_{k-1}+ \lambda_k^Z(s-,Z(s-) \right)}(x) N(ds\times dx),
S ={\frac{N}{g}}t r \left({\frac{1}{2 \cosh ( \beta ) ( 1 + c^{*})}}S^{2}+{\frac{1}{2 \cosh ( \beta ) ( 1 - c^{*})}}D^{2}+{\frac{1}{4}}S^{4}+{\frac{1}{4}}D^{4}+{\frac{1}{2}}( S D S D + 2 S^{2}D^{2}) \right) ,
R_{n}^{(x_0, y_0), \Pi^{\mu, \nu}}(f) \lesssim \sum_{j=0}^n O\bigg( |x-y|^{j+1}\cdot \int |v-u|^{n-j}d\Pi^{\mu, \nu}(u, v) \bigg), &+ \sum_{j=0}^n O\bigg( |x-y|^{j}\cdot \int |v-u|^{n-j+1}d\Pi^{\mu, \nu}(u, v) \bigg).
C^{(\ell)}\boldsymbol{r}_0 = \sum_{i=0}^{\ell-1}(\mathrm{I}- M^{-1}\! A)^i M^{-1}\boldsymbol{r}_0 = \sum_{i=0}^{\ell-1}\sum_{j=0}^{i}\binom{i}{j}(- \hat{A})^j \hat{\boldsymbol{r}}_0.
\overline{M}^{(n)}\overline{M}_2\overline{R}= \overline{M}^{(n)}\overline{M}_2 (\mathfrak{M}^{(n)}\overline{R})= \overline{M}^{(n)}\overline{M}_2 (\mathfrak{M}^{(n)}\overline{R}^{(n)})= \overline{M}^{(n)}\overline{M}_2\overline{R}^{(n)}.
T_{\psi}At= \sum_{\xi\in\mathcal{X}}(At)_{\xi}\psi_{\xi}=\sum_{\xi\in\mathcal{X}}\sum_{\eta\in\mathcal{X}}a_{\xi\eta}t_{\eta}\psi_{\xi}&=\sum_{\eta\in\mathcal{X}}\Big(\sum_{\xi\in\mathcal{X}}a_{\xi\eta}\psi_{\xi}\Big)t_{\eta}=\sum_{\eta\in\mathcal{X}}m_{\eta}t_{\eta}=:f.
\mathfrak{C}(n;q_1,q_2,q_2')\ll\begin{cases}0 \;\;n=0\;\;\;\;q_2\neq q_2'; Q^3Q_2 \;\;n=0\;\;\;\;q_2=q_2'; Q^{5/2}\sqrt{Q_2}(q_1,n) \;\;n\neq 0\;\;\;\;q_2\neq q_2'; Q^{5/2}Q_2^{3/2}(q_1,n) \;\;n\neq 0\;\;\;\;q_2=q_2'.\end{cases}
(m_B \otimes 1_X)(1_B \otimes l^*) = l^* l = (1_B \otimes l)(m_B^* \otimes 1_X), (1_X \otimes m_A)(r^* \otimes 1_A) = r^* r = (r \otimes 1_A) (1_X \otimes m_A^*).
[ \Gamma_{\mu}, \Gamma_{\nu}{]_{\lambda}}^{\kappa}\equiv ( \Gamma_{\mu}\Gamma_{\nu}- \Gamma_{\nu}\Gamma_{\mu}{)_{\lambda}}^{\kappa}={\Gamma_{\mu \lambda}}^{\sigma}{\Gamma_{\nu \sigma}}^{\kappa}-{\Gamma_{\nu \lambda}}^{\sigma}{\Gamma_{\mu \sigma}}^{\kappa}.
b_{n}^{i}= A_{n}a_{n}^{i}- B_{n}^{*}\tilde{a}_{n}^{i^{\dagger}}, \qquad \tilde{b}_{n}^{i}= A_{n}\tilde{a}_{n}^{i}- B_{n}^{*}a_{n}^{i^{\dagger}}.
g_{\mathrm{sc},h}(x\lambda \partial_x + \omega\partial_y) = x\lambda \frac{dx}{h^2(x_0-x)^4}+\frac{g_2(\omega\partial_y)}{h(x_0-x)^2}& = h^{-1/2}(x_0-x)^{-1}(\tilde{\lambda}\frac{dx}{h(x_0-x)^2}+ \frac{g_1(\omega\partial_y)}{h^{1/2}(x_0-x)}).
V_1^*V_2 =(R_{\overline q}\otimes U^*P^\perp + R_{\overline q}M_z^*\otimes U^*P)(R_{\overline q}\otimes U^*P+R_{\overline q}M_z\otimes U^*P^\perp) &=R_{\overline q}^2\otimes U^*P^\perp U^*P +R_{\overline q}^2M_z\otimes U^*P^\perp U^*P^\perp +R_{\overline q}M_z^*R_{\overline q}\otimes U^*PU^*P &+R_{\overline q}M_z^*R_{\overline q}M_z\otimes U^*PU^*P^\perp.
(x_1^2 - q^{2i-3})(x_1^2 - q^{2i-1}) z_i & \qquad = (q^i-1)(x_1^2 - q^{i-2}) \big( q^d (q^i+1)(x_1^2 + q^{i-2}) - \tau x_1 q^{i-1}(q^{d+1}+1) \big).
\begin{aligned}\int_{B(0, \nu^{-2b})}\chi((\cdots)_{2})\, dx = \int_{B(0, \nu^{-2b})}\chi(\sum_{i = 1}^{3}(s_{i}- s_{i + 3})x_1 + (s_{i}^{2}- s_{i + 3}^{2})x_2)\, dx &= \nu^{-4b}1_{|\sum_{i = 1}^{3}s_{i}- s_{i + 3}| \leq \nu^{2b}, |\sum_{i = 1}^{3}s_{i}^{2}- s_{i + 3}^{2}| \leq \nu^{2b}}\end{aligned}
\left|\sum_{j=0}^{\lfloor n/2 \rfloor}E_{\mu}^Q \tilde g (Z_j) E_{Z_j}h (Z_{n-j}) \right| \ \le \ \|\tilde g\|_\infty \sum_{j=0}^{\lfloor n/2 \rfloor}\sup_{z}|E_{z}h (Z_{n-j}) | \ \le \c_1\|\tilde g\|_\infty \|h\|_\infty \rho^{n/2}n/2\to 0,
D_{A}(q\Vert p) =\frac{1}{\int A\left( q\right) d\mu}\int \left( \log_{A}q-\log_{A}p\right) A\left( q\right) \d\mu &\geq \frac{1}{\int A\left( q\right) d\mu}\int \left( q-p\right) \d\mu =0.
\lim_{a \to 0, \; b/\sqrt{a}\to k}Q_n\left(\frac{x}{a}\right)=(-1)^n\frac{(\mu+\nu+1)_n}{\Gamma(\mu+1)}{\;}_1 F_2 \left({-n \atop \mu+\nu+1, \mu+1}\Big{|}k^2x \right)x^\mu,
P^1_k(x_0, \cdots, x_k) = \widehat P_{k-1}(x_0, \cdots, x_{k-1}) + x_k, Q^1_k(x_0, \cdots, x_k) = \widehat Q_{k-1}(x_0, \cdots, x_{k-1}) + 2 x_k.
a =Tr\left(\frac{r_1 r_2'\varepsilon}{\delta(\mathfrak{a})}\right),~b=Tr\left(\frac{r_1 r_1'\varepsilon'}{\delta(\mathfrak{a})}\right),~c=Tr\left(\frac{r_2 r_2'\varepsilon}{\delta(\mathfrak{a})}\right) d =Tr\left(\frac{r_1 r_2'\varepsilon'}{\delta(\mathfrak{a})}\right).
\bar{\mathbf{u}}_t-\mu\Delta \bar{\mathbf{u}}+(\bar{\mathbf{u}}\cdot\nabla)\bar{\mathbf{u}}+\nabla\bar{p}-\rho\int_0^t(t-s)^{-\beta}e^{-\delta(t-s)}\Delta \bar{\mathbf{u}}ds &\qquad\qquad=\bar{\mathbf{f}}+\frac{\rho}{\delta^{1-\beta}}\int_{\delta t}^\infty s^{-\beta}e^{-s}\Delta \bar{\mathbf{u}}ds.
\frac{1}{3\cdot 4! \cdot 2!}(-18)(\frac{2}{9})\int d\tau_1 \dots d\tau_4 F(\tau_1-\tau_2)F(\tau_3-\tau_4)[\Theta(\tau_1 \geq \tau_3 \geq \tau_2 \geq \tau_4) + \dots ] .
\frac{K_{N,L}}{k!\left((N-k)/2\right)!}\left|\Delta\left(\{\lambda_l\}_{l=1}^k\cup\{x_j\pm\mathrm{i}y_j\}_{j=1}^{(N-k)/2}\right)\right|\prod_{j=1}^kw(\lambda_j;L)\prod_{j=1}^{(N-k)/2}2\left(w\left((x_j,y_j);L\right)\right)^2,
\mathcal{M}_{2,w}^{|\cdot|,u}(K,z) = \mathcal{M}_{2,w}^{\Re, +,u}(K,z) + \mathcal{M}_{2,w}^{\Re, +,u}(K,-z) + \mathcal{M}_{2,w}^{\Re, +,u}(K,iz) + \mathcal{M}_{2,w}^{\Re, +,u}(K,-iz),
\left \|\max_{1 \leq k \leq n}|S_k(t)| \right \|_2^2 \leq 16 \sum_{k=1}^n \left \| Y_k(t) \sum_{i=k}^n{\mathbb E}_k(Y_i(t)) \right \|_1 \leq 16 n L(t) \, .
\int_{\Lambda}\left[ \prod_{i = 1}^{n_{a}}d^{2}x_{i}^{( a )}d^{2}y_{i}^{( a )}\prod_{j = 1}^{n_{b}}d^{2}x_{j}^{( b )}d^{2}y_{j}^{( b )}\right] \Big | B ( a , b ) \Big |^{\frac{g}{\pi + g N}q ( N - 1 )}\;
[\begin{matrix}1&3&2\\ 2&3&1\end{matrix}]\oplus[\begin{matrix}1&6\\ 0&1\end{matrix}]=[\begin{matrix}1&3&2&0&0\\ 2&3&1&0&0\\ 0&0&0&1&6\\ 0&0&0&0&1\end{matrix}]
\sum_{m=-\infty}^{\infty}\sum_{n=0}^{\infty}\hat{N}_{t-1}(m,n)z^{m}q^{n}=\dfrac{(-zq,-z^{-1}q,\cdots,-z^{s}q,-z^{-s}q;q^{2})_{\infty}}{(zq^{2},z^{-1}q^{2},\cdots,z^{s}q^{2},z^{-s}q^{2};q^{2})_{\infty}},
e^{ik\eta_c}c_{-}\simeq \frac{1}{2\sqrt{2}}[(\frac{1}{2}-\nu) \frac{\Gamma(\nu)}{\pi}( \frac{k\eta_{c}}{2})^{-\nu-\frac{1}{2}}- i (\frac{k\eta_c}{2})^{\nu -\frac{1}{2}}(\nu +\frac{1}{2}) \frac{1}{\nu\Gamma(\nu)}],
\frac{d}{dt}\mathcal{E}(t;t_0,x_0) = \int_{B_{t_0-t}(x_0)\cap \Omega}\left( \partial_t^2 u - \Delta u + u \right)\partial_t u \,dx -\frac{1}{2}\int_{S_{t_0-t}\cup S_{b,t_0-t}}(|\partial_t u|^2 + |\nabla u|^2 + |u|^2 - 2 (\mathbf{n}\cdot \nabla u) \partial_t u ) \,dS,
S\left( \begin{array}{cc}a b d c \end{array}\right.\left| \begin{array}{cc}z_1 z_2 z_2 ' z_1 ' \end{array}\right)=\frac{\overline{W}_{ab}(z_1 ,z_2 )\overline{W}_{bc}(z_2 ,z_1 ')\overline{W}_{cd}(z_1 ',z_2 ')}{\overline{W}_{ad}(z_1 ,z_2 ')}.
\begin{aligned}\||D_{\xi,x}|\widetilde{\gamma}_{n,\xi}(\cdot,\cdot)\|_{L^2(Q_\ell\times Q_\ell)}=\||D_{\xi}|\widetilde{\gamma}_{n,\xi}\|_{\mathfrak{S}_2(\xi)}\leq \|\widetilde{\gamma}_{n,\xi}\|_{X^2(\xi)}\leq (q+M)(1-\kappa)^{-2}e^2, \end{aligned}
\int_{0}^{\infty}k(t) \Big|\sum_{n=n_0}^{\infty}R_{n}e^{r_n t}\Big|^2\d t\le \pi T \varepsilon'\sum_{n= n_0}^\infty\frac{|d_nD_{n}|^2}{\pi^2+T^2r_{n}^2}\le \pi T \varepsilon'\sum_{n= n_0}^\infty\frac{|d_nD_{n}|^2}{\pi^2+4T^2(\Im\zeta_{n})^2}\,.
A\left(R_j\right) \leq \frac{1}{2}\left(L_0 \cos \beta_j + L_0 \sin\beta_j\tan(\alpha_j + \beta_j)\right)L_0\sin\beta_j = \frac{1}{2}L_0^2\sin\beta_j \left(\cos \beta_j + \sin\beta_j\tan(\alpha_j + \beta_j)\right).
={\frac{\alpha}{\pi}}\sum_{k = 0}^{\infty}\sum_{n = 0}^{\infty}{\frac{\Gamma ( z + k ) \Gamma ( 2 z + 2 k + 2 n - 1 )}{2^{2 k}k ! ~ \Gamma ( z ) \Gamma ( 2 z + 2 k )}}~ \zeta_{R}( 2 z + 2 k + 2 n - 1 , 1 / 2 ) ~{\frac{B_{2 n}}{( 2 n ) !}}\left({\frac{2 \pi}{\alpha}}\right)^{2 n}
\mathcal{D}_{\mu}\phi\equiv \partial_{\mu}\phi - \vartheta_{\mu}{}^{i}\Gamma_{r}(M_{i})\phi\, ,\hspace{1cm}\mathcal{D}_{\mu}\psi\equiv \partial_{\mu}\psi +\psi\vartheta_{\mu}{}^{i}\Gamma_{r}(M_{i})\,
{\vec f}_i(x)=\sum_{j=1}^N\, \sum_{k=1}^n\,\left(1 + x\Lambda +\frac{x^2}{2!}\Lambda^2 + \cdots + \frac{x^{n-1}}{(n-1)!}\Lambda^{n-1}\right)_{ik}C_{kj}{\vec e}_j.
\tilde{A}^0(x) =A^0(x)+ \frac{1}{(2 \pi)^3}\int \frac{d^3 k}{2 k^0}[ q_0(\vec{k}) e^{i \vec{k}\vec{x}}+q_0^*(\vec{k}) e^{-i \vec{k}\vec{x}}] = A^0(x) + q^0(\vec{x}) \, .
\begin{array}{l}\displaystyle \dot{\beta}_{n}= \beta_{n}\left( \frac{\alpha_{n+1}}{\beta_{n+1}\beta_{n}}- \frac{\alpha_{n}}{\beta_{n}\beta_{n-1}}\right), \displaystyle \dot{\alpha}_{n}= \alpha_{n}\left( \frac{1}{\beta_{n-1}}- \frac{1}{\beta_{n}}\right), \end{array}
\sum_{j=1}^n\left( \sum_{i=j}^n \epsilon_i \right)^2\ge \sum_{i=1}^n i\epsilon^2_i +2\sum_{i=2}^n \epsilon_i^2\left( \sum_{j=1}^{i-1}j \right)=\sum_{i=1}^n (i \epsilon_i)^2,
\Lambda_c =\Bigg(\frac{1}{e^2}\Bigg)^{3/11}m_\lambda^{2/11}\Bigg(C e^{-8\pi^2/e^2}M^{3N_c-N_f}\Big(\mbox{det}(m_q)_i^j\Big)^{2/3}\Big(\mbox{det}(m_{sq})_i^j\Big)^{1/3}\Bigg)^{3/(11 N_c)},
_{t^{m-1}}\prod_{i\in I}\frac{t^{N_{i}}p^{-\nu_{i}}}{1-t^{N_{i}}p^{-\nu_{i}}}=\sum_{(a_{i})_{i\in I}\in J_{I,m}}p^{-\sum_{i\in I}\nu_{i}(a_{i}+1)},
\int_{-1}^1 h^n_{n-j}(t)\frac{t^{n+2+2r}}{(n+2+2r)!}dt \qquad =\frac{(-1)^{\frac{j+1}{2}}2^{j+1}(\frac{j+1}{2})!(n+1)!(n+2)!}{(j+2)!(2n+3)!(n+\frac{3}{2}-\frac{j}{2})!}\sqrt{2n-2j+1}\frac{(\frac{n+3}{2})_r(\frac{n+4}{2})_{r}}{r!(\frac{4+j}{2})_r(n+\frac{5}{2})_r(n+\frac{5}{2}-\frac{j}{2})_r}\frac{1}{4^r}.
(\mu-\phi(\pi))\phi^\prime(x) \geq (\mu-\phi(x))\phi^\prime(x) &=\int_{-\pi}^{0}\left(K_r(x-y)-K_r(x+y)\right)\phi^\prime(y)\,dy &\geq \int_{-\frac{3}{4}\pi}^{-\frac{1}{4}\pi}\left(K_r(x-y)-K_r(x+y)\right)\phi^\prime(y)\,dy,
\tau_U(\hat{\mu_R},1/N_T) \le\frac{N_T-1+\frac{N_R}{N_R\hat{\mu_R}+1}}{N_T}(1-\hat{\mu_R)}&\le\frac{N_T+\frac{N_R}{N_R\hat{\mu_R}+1}}{N_T}&\le\frac{N_T+\frac{N_R}{N_R\mu_R}}{N_T}=1+\frac{1}{N_T\mu_R},
\theta(n_s^{-1}(\lambda))T_{n_s^{-1}}- T_{n_s^{-1}}\theta(\lambda) = -\sum_{k = 0}^{-\langle\nu(\lambda),\alpha\rangle-1}c_{n_s,-k}\theta(\mu_{n_s}(k)n_s^{-2}\lambda) & = -\sum_{k = 1}^{-\langle\nu(\lambda),\alpha\rangle}\theta(n_s(\lambda)\mu_{n_s}(-k)n_s^{-2})c_{n_s,-k}.
(A/I,C_x^+,v)=(A/I,C_y^+,v) \implies S_v((A/I,C_x^+))=S_v((A/I,C_y^+)) &\implies S_u(A,A_x^+)=S_u(A,A_y^+) &\implies (A,A_x^+,u)=(A,A_y^+,u).
\frac{1}{2\pi}\sum^N_{j=1}\left\vert \int_{u\in I'}e^{-\frac{\hat h_{j,+}(x_0,e^{-iu}\circ x_0)}{t}}\Bigr(\mathrm{Tr\,}\hat K_{j,+}(t,x_0,e^{-iu}\circ x_0)-\mathrm{Tr\,}\hat K_{j,-}(t,x_0,e^{-iu}\circ x_0)\Bigr)du\right\vert.
\sum_{j,k=1}^{d+1}\partial_{jk}\psi (\xi_j \xi_k + \partial_j \psi \partial_k \psi) = -\frac{1}{2}\sum_{i=1}^d \xi_i^2 + \xi_{d+1}^2 - \frac{1}{8}\lvert x' \rvert^2 + (x_{d+1}- 1)^2 &= \frac{3}{2}\xi_{d+1}^2 - \frac{1}{4}\lvert x' \rvert^2 + \frac{1}{2}(x_{d+1}-1)^2 .
\beta_{\ell}^{i}=\frac{1}{2}(\beta_{\ell-1}^{i-1}+\beta_{\ell-1}^{i+1}), \gamma_{\ell}^{i}=\frac{1}{2}(\lambda_{\epsilon}\gamma_{\ell-1}^{i-1}+\gamma_{\ell-1}^{i+1}),
\vec{e}_{0}=\partial_{t},\vec{e}_{1}=\partial_{z},\vec{e}_{2}=\partial_{r},\vec{e}_{3}=\frac{\sqrt{2}\omega}{sinh(\sqrt{2}\omega r)}\partial_{\varphi}-\frac{\sqrt{2}sinh(\sqrt{2}\omega r)}{1+cosh(\sqrt{2}\omega r)}\partial_{t}
\sum_{j = 0}^k{\frac{{G_j^2}}{{2^{4j}}}}= \frac{{3552}}{{3553}}G_0^2 + \frac{{224}}{{3553}}G_1^2 + \frac{{16}}{{3553}}G_2^2 &\quad- \frac{{222}}{{3553}}\frac{{G_{k + 1}^2}}{{2^{4k}}}- \frac{7}{{3553}}\frac{{G_{k + 2}^2}}{{2^{4k - 1}}}- \frac{1}{{3553}}\frac{{G_{k + 3}^2}}{{2^{4k}}}\,,
\begin{array}{l}{{[}U_{a}, U^{b}{]}= S_{a}^{b}\in g^{0}}\\{{[}U_{a}, U_{b}{]}= K_{a b}\in g^{+ 2}}\\{{[}U^{a}, U^{b}{]}= K^{a b}\in g^{- 2}}\\{{[}S_{a}^{b}, U_{c}{]}= U_{( a b c )}\in g^{+ 1}}\\ \end{array}
Y_t = \hat \Phi( W_T ) + \int_t^T \hat f\left( r,W_r , Y_r , \frac{[Y,W]_r}{\mathrm dr}\right) \mathrm dr & + A_{T}^{W ,Y}( \hat b) -A_{t}^{W ,Y}( \hat b) -( M_T -M_t ),
(K+1)\boldsymbol{I}_{ij}^{+}\boldsymbol{P}_{i}^{\pm}=\frac{1}{2}(1+d\ldots\pm d\ldots \pm d\ldots )-\frac{1}{2}=2\boldsymbol{I}_{ij}^{\pm}\boldsymbol{P}_{i}^{\pm}-\frac{1}{2}.
\cdot \left[ \hat{K}^{S}F_{S}y_{i j k}+ \hat{K}^{T}F_{T}\left( \left( 1 + \frac{\sum n_{i}}{3}\right) y_{i j k}( T ) - \frac{( T + T^{*})}{3}y_{i j k}^{\prime}( T ) \right) \right] .
g^{-1}= \left( \begin{array}{ll}- 2 \mu_2 - 2 \mu^2_2 + \frac 23 \mu_3, - \mu_3(6 + \frac{16}{3}\mu_2) - \mu_3(6 + \frac{16}{3}\mu_2), \frac 83 \mu^2_2 \mu_3 - 8 \mu^2_3 \end{array}\right)
1-D_N =\nu^{N,n_N,F}(X_{N,n_N}^{F})-D_N &\leq \gamma^{N^{F}n_N}(B_{D_N}(X_{N,n_N}^{F})) \leq \left( \gamma^{N^{F}}( B_{D_N}(X_{N,1}^{F})) \right)^{n_N}\leq \left\{\frac{e^{-\rho(N^F/2)}}{\sqrt{\pi}}e^{1/2}\left(1-\frac1{N^F}\right)^{(N^F-1)/2}\cdot 2D_N \right\}^{n_N},
h_1\cdots h_{m-\ell}\otimes y_1\wedge \cdots \wedge y_\ell \mapsto \frac{1}{(2\pi \i)^\ell}\left(\prod_{i=1}^{m-\ell}h_i\right)\cdot \frac{dy_1}{y_1}\wedge\cdots \wedge \frac{dy_\ell}{y_\ell}
S=\int d^{4}x\Phi e^{\alpha\phi/M_{p}}\left[-\frac{1}{\kappa}R(\Gamma ,g)+\frac{1}{2}g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}\right]+\int d^{4}x\sqrt{-g}e^{\alpha\phi/M_{p}}\left[-\frac{b_{g}}{\kappa}R(\Gamma ,g)+\frac{b_{k}}{2}g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}\right]
H_{1}(t_{1})=\frac{1}{2}\sum_{\lambda\lambda^{\prime}\mu\mu^{\prime}}\int d^{4}x_{1}d^{4}x_{1}^{\prime}\hat{\psi}_{\lambda}^{\dagger}(x_{1})\hat{\psi}_{\mu}^{\dagger}(x_{1}^{\prime})U(x_{1},x_{1}^{\prime})_{\lambda\lambda^{\prime}\mu\mu^{\prime}}\hat{\psi}_{\mu^{\prime}}(x_{1})\hat{\psi}_{\lambda^{\prime}}(x_{1}^{\prime}),
v_{d-j}= (A-\theta_{j+1}I)(A-\theta_{j+2}I) \cdots (A-\theta_{d}I)u^* & = \sum^d_{i=0}E_i(A-\theta_{j+1}I)(A-\theta_{j+2}I) \cdots (A-\theta_{d}I)u^* & = \sum^d_{i=0}(\theta_i-\theta_{j+1})(\theta_i-\theta_{j+2}) \cdots (\theta_i-\theta_{d})E_iu^* & = \sum^d_{i=0}P_{ij}E_iu^*.
[\Gamma : \Pi] \beta_{v_1}\wedge \dots \wedge \beta_{v_d}= \frac{[\Gamma : \Pi ]}{n_1\dots n_d}\beta_{n_1v_1}\wedge \dots \wedge \beta_{n_dv_d}= \frac{[\Gamma : \Pi ]}{n_1\dots n_d}\iota_*(\beta_{\Sigma \cap \Pi})
e^{-\frac{\langle p\vert A^{-1}\vert p\rangle}{2}}=\frac{\det(A)^{\frac{1}{2}}}{\left(2\pi\right)^{\frac{N}{2}}}\int_{\mathbb{R}^N}e^{i\langle c,p\rangle}e^{-\frac{\langle c\vert A\vert c\rangle}{2}}d^Nc,
2t^{4}\left[ S-\left( \frac{u_{0}(z)}{t}+\frac{u_{1}(z)}{t^{3}}\right)\right] +t^{2}\left( t\partial_{t}S+\frac{u_{0}(z)}{t}\right) -2z^{2}t^{2}\left( S-\frac{u_{0}(z)}{t}\right) -z^{2}t\partial_{t}S=0,
{\check{Q}}_{i}= e_{i}\check{\theta}_{i}^{\ast}( \tau ) \check{\theta}_{i}( \tau ) = Q_{i}= e_{i}\theta_{i}^{\ast}( \tau ) \theta_{i}( \tau ) \big [ \dot{Q}_{i}( \tau ) = 0 \Rightarrow Q_{i}( \tau ) \equiv Q_{i}\big ] .
\langle \Psi | \textit{\textbf{G}}_{2}| \Psi \rangle = \frac{i}{2 \pi}\sum_{n \neq 0}\frac{q_{0}+ q_{\pi}}{2 n}\langle \Psi | \mathrm{\boldmath \alpha_n \times \alpha_{- n}}| \Psi \rangle = \frac{q}{2 \pi}\langle \Psi | \textit{\textbf{S}}| \Psi \rangle .
G_{j}(\mu^{(c)}_{a}) \xrightarrow{L^{2}}0 ~, c \in \{2,\ldots,d \}, j\geq c , \frac{G_{j}(\mu^{(c)}_{a})}{a^{j}}\xrightarrow{L^{2}}\frac{I_{j}}{d^{j}}\binom{c-1}{j}~, c \in \{2,\ldots,d \}, j < c ,
\begin{aligned}\sum_{i=1}^n \xi_i^2+ \sum_{i,j=1}^n a_{ij}(z)\xi_i\xi_j+\sum_{i=1}^n \sum_{l=1}^m c_{il}(z)\xi_i\eta_l +\sum_{l,k=1}^m d_{lk}(z)\eta_k\eta_l\geq \Lambda\left(|\xi|^2+|\eta|^2\right).\end{aligned}
2\pi(u+\pi) =\int_{u}^{u+2\pi}\theta(x)dx=\int_{u}^{z}\theta(x)dx+\int_{z}^{u+2\pi}\theta(x)dx \ge \int_{u}^{z}udx+\int_{z}^{u+2\pi}(z+\pi)dx=u(z-u)+(z+\pi)(u+2\pi-z)
Y_{\nu}(x)=G_{1,3}^{2,0}(\begin{matrix}\frac{-\nu-1}{2}\\ \frac{\nu}{2},\frac{-\nu}{2},\frac{-\nu-1}{2}\end{matrix}|\frac{x^{2}}{4}),\frac{-\pi}{2}<argx\le\frac{\pi}{2}
\left[ e^{- \bar{H}_{i}^{2}t}\right]_{\mathrm{d i a g}}\simeq{\frac{1}{( 4 \pi t )^{( D - 1 ) / 2}}}e^{- m^{2}\kappa^{2}\rho^{2}t}\left( r_{i}+ \bar{b}_{i , 1}t + \bar{b}_{i , 2}t^{2}+ . . . \right) ,
C_{01\ell}(N)=\frac{(-1)^{N}}{N!}\sum_{m=1}^{N-\ell}(-1)^m\sum_{\substack{i_1, i_2, \dots ,i_m \geqslant 1 i_1+i_2+ \cdots +i_m=N-\ell 1\leqslant r_1 \leqslant r_2 \leqslant \cdots \leqslant r_m \leqslant N}}\frac{\binom{r_1}{i_1+1}\binom{r_2}{i_2+1}\cdots \binom{r_m}{i_m+1}}{r_1 \cdot r_2 \cdots r_m}
Z_{t_{i}}= \begin{pmatrix}\tilde x_{t_{i}}\tilde \eta_{t_{i}}\end{pmatrix}\tilde A = \begin{pmatrix}A 0 0 A \end{pmatrix}\tilde B(Z_{t_i}, W) = \begin{pmatrix}0 S V(\tilde x_{t_{i}}) W\end{pmatrix}
j^0=-\frac{J^0}{|\psi|^2}=\frac{1}{2}i\partial_\tau\ln\left(\frac{\psi}{\psi^*}\right)+\varphi\ \ ,\ \ \vec{j}=-\frac{\vec{J}}{|\psi|^2}=-\frac{1}{2}i\vec{\partial}\ln\left(\frac{\psi}{\psi^*}\right)+\vec{a}.
M^{\prime}={\cal U}^{T}\left( \begin{matrix}{{U_{5}}}&{{0}}\\{{0}}&{{U_{2 1}}}\\ \end{matrix}\right){\cal U}M{\cal U}^{T}\left( \begin{matrix}{{U_{5}^{T}}}&{{0}}\\{{0}}&{{U_{2 1}^{T}}}\\ \end{matrix}\right){\cal U},
M_t =\begin{cases}0, \;\; 0\leq t\leq1; M_{n}+ M^n_{t-n}\phi_{n}(\sigma_1,\ldots,\sigma_{n-1}), \;\; t\in(n,n+1], n\in \{1\ldots,N\}, M_{N+1}, \;\; t>N+1,\end{cases}
p_{0}( x , \xi ) = \sum_{j , k = 1}^{d}\big( \widehat{q}_{0 ; j , k}( 0 , 0 ) + \nu_{j , k}( x , \xi ) \big) \big( \xi_{j}+ i \partial_{j}\phi (x) \big) \big( \xi_{k}- i \partial_{k}\phi (x) \big).
\int_0^t \!\int_0^r \!\big\langle U_Y(r,v)F(v),B^*\zeta\big\rangle_V\d v\d B^H_r=\int_0^t \!\int_v^t \!\big\langle U_Y(r,v)F(v),B^*\zeta\big\rangle_V\d B^H_r\!\d v
M_{1}(\mathcal{NC}(G)) =(2^{3k}-2^{k}-1)(2^{3k}-2^{k}-2)^{2}~~~~~~~~~~~~~-4 \cdot (2^{3k}-2^{k}-2) \dfrac{(2^{4k}-2\cdot2^{3k}-2\cdot2^{2k}+3\cdot2^{k}+2)}{2}~~~~~~~~~~~~~~+(2^{5k}-4 \cdot 2^{4k}+ 4 \cdot 2^{3k}+4 \cdot 2^{2k}-5 \cdot 2^{k}-4) = 2^{9k}-5\cdot2^{7k}-2^{6k}+9\cdot2^{5k}-5\cdot2^{3k}-3\cdot2^{2k}+3\cdot2^{k}
\xi= (\cos(\theta_l),\sin(\theta_l),0)+h(\sin(\theta_l),-\cos(\theta_l),1)=(\cos(\theta_l)+h\sin(\theta_l),\sin(\theta_l)-h\cos(\theta_l),h) = (\sqrt{1+h^2}\cos(\theta_l-w),\sqrt{1+h^2}\sin(\theta_l-w),h)\,.
k_1 = \left( \begin{array}{c}e p \end{array}\right) \;\;\;k_2 = \left( \begin{array}{c}e -p \end{array}\right) \;\;\;k_3 = \left( \begin{array}{c}-e -p \end{array}\right) \;\;\;k_4 = \left( \begin{array}{c}-e p \end{array}\right).
(\begin{matrix}X^{0}&X^{1}&X^{2}&X^{3}\end{matrix})(\begin{matrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{matrix})(\begin{matrix}Y^{0}\\Y^{1}\\Y^{2}\\Y^{3}\end{matrix})
u=\left(\begin{pmatrix}0 a_{12}0 0\end{pmatrix},\begin{pmatrix}b_{11}b_{12}0 0\end{pmatrix}, \begin{pmatrix}i_1 i_2\end{pmatrix}, 0, 0, b_{11}+f_2, \begin{pmatrix}f_1 f_2\end{pmatrix}\right)
\mathbb{P}\left(\sum_{i \in{\cal J}}w_i Z_i > \frac{1}{2}\right) =\mathbb{P}\left(\Xi >\frac{1}{2}\right) &\geq \mathbb{P}\left(\Xi >\frac{\mathbb{E}(\Xi)}{2}\right) & \geq 1 - e^{-\frac{\mathbb{E}(\Xi)}{8}}\geq 1 - e^{-\frac{1}{8}}.
\left( v_{A}^{-1}v_{B}^{-1}v_{A}^{-1}\right)^2 \triangleright \varphi = \frac{\mu^l(v^{-1})}{\mu^l(v)}\varphi\!\left( S^{-1}(a_i) g^{-1}v^{-1}S(?) b_i \right) \left( v_{A}^{-1}v_{B}^{-1}v_{A}^{-1}\right)^{-2}\triangleright \varphi = \frac{\mu^l(v)}{\mu^l(v^{-1})}\varphi\!\left( b_j S^{-1}(?) a_j g^{-1}v \right)
\epsilon''_e = \Delta \psi_{e}' \, \epsilon'_e = \Delta \psi'_e \, \Delta \psi_e \, \epsilon_e = \Delta \psi''_e \, \epsilon_e, \varphi''_f = \psi'^{(\epsilon')}_{\partial f}\, \varphi'_f = \psi'^{(\epsilon')}_{\partial f}\, \psi^{(\epsilon)}_{\partial f}\, \varphi_f.
\vec{E}_{r}^{\prime}= - \frac{e_{0}}{4 \pi \varepsilon_{0}}\frac{\vec{R}_{\vartheta \varphi}^{\prime}}{r^{2}\sin \vartheta}= - \frac{e_{0}}{4 \pi \varepsilon_{0}}\frac{\vec{e}_{3}}{r^{2}}.
\left(\partial_{t}-\partial_{xx}^{2}-\frac{2\left(n-1\right)}{x}\partial_{x}-\frac{2\left(n-1\right)}{x^{2}}\right)\left(k\left(-t\right)^{\lambda_{2}+\frac{1}{2}}\varphi_{2}\left(\frac{x}{\sqrt{-t}}\right)\right)=0
\mathrm{DTS}_{(p,q)}(X) =\mathrm{DTV}_{(p,q)}^{-3/2}(X)\bigg\{\sum_{k=2}^{3}\binom{3}{k}(-\mathrm{DTE}_{(p,q)}(X))^{3-k}[\mu^{k}+k\mu^{k-1}\sigma\mathrm{DTE}_{(p,q)}(Y)] +2\mathrm{DTE}_{(p,q)}^{3}(X)+3[\mu-\mathrm{DTE}_{(p,q)}(X)]\sigma^{2}\left(L_{1}+2L_{2}\right)+\sigma^{3}\left(L_{1}^{\ast}+2L_{2}^{\ast}\right)\bigg\},
\Sigma_{s}^{(j+1)}=\frac{\sum_{t=1}^{N}h_{s}^{(j)}(t)[x^{(t)}-\mu_{s}^{(j+1)}][x^{(t)}-\mu_{s}^{(j+1)}]^{\top}}{\sum_{t=1}^{N}h_{s}^{(j)}(t)}
\left(\prod_{i=1}^3[\mathrm{cok}(\delta_{i,tor})]^{(-1)^{i+1}}\right) = \frac{\nu(\mathcal{H}^0)_{\mathbb{R}}\nu(\mathcal{H}_B/\mathcal{H}^0)}{\nu(\mathcal{H}_B)_{\mathbb{R}}}.
\int d^{2}\mathrm{\bf x}\Big ( \frac{1}{4}f_{i j}f^{i j}+ \frac{1}{8}( V + A f )_{i j}^{\alpha}( V + A f )_{\alpha}^{i j}- \frac{1}{4}g^{i j}D_{i}A^{\alpha}\cdot D_{j}A_{\alpha}
\begin{aligned}Q_{i,j}^t=F_{i,j}^t+F_{i-1,j}^{t+1}-F_{i-1,j}^t-F_{i,j}^{t+1}, &W_{i,j}^t=F_{i,j}^t+F_{i,j+1}^{t}-F_{i-1,j}^t-F_{i+1,j+1}^{t},\end{aligned}
1={32n^2\sqrt{1-{\rho_0^2\over c^2}}\tilde \beta(1+\sigma+{1\over c})(1+{1\over{\rho_0\over 2^{3\over 2}}-r})\over \tilde \alpha({\rho_0\over 2^{3\over 2}}-r)r(1-r)^{\tilde \alpha+2}},\ \rho_0\in(2^{3\over 2}r,c).
\begin{aligned}f_0 = w_{n+1}= x_1, f_k = w_{k}w_{n+k+1}= \frac{u_{k,k}}{v_{k,k}}\frac{v_{k+1,k}}{u_{k+1,k}},(0<k<n) f_n =w_nw_{n+1}= \frac{u_{nn}}{v_{nn}}\cdot t_{n+1}.\end{aligned}
a_{m-1}- (2z) a_{m}+ a_{m+1}= (-2z)a_{k+1}+ a_{k+2}& = (2z)\left(\sum_{j=1}^{\infty}\tilde{c}_{j+k}\frac{1}{\lambda_{+}^{j}}\right) - \sum_{j=2}^{\infty}\tilde{c}_{j+k}\frac{2z}{\lambda_{+}^{j}}- \tilde{c}_{k+1}\frac{1}{\lambda_{+}^{2}}& = \tilde{c}_{k+1}\left(\frac{2z}{\lambda_{+}}- \frac{1}{\lambda_{+}^{2}}\right) = \tilde{c}_{k+1}
\tilde{f}_S(b,a,b) = \frac{W^{(q+r)}(b)}{W^{(q+r)\prime}(b+)}+ \tilde{f}_S(0,a,b) \Big( I_a^{(q,r)}(b) - \frac{W^{(q+r)}(b)}{W^{(q+r) \prime}(b+)}(I_a^{(q,r)})'(b +) \Big) .
E_{2x}= a^2 u^p_{xxt}=a^2 \big(u^p_x\big)_{xt}=a^2 \big[r_p(u^p) \rho_x\big]_{xt}= a^2 \big[r_p(u^p) \rho_{xx}\big]_t=a^2 r_p(u^p) \rho_{xxt},
\lambda_i^*\left[\sum_{j=1}^iE_s^{j*}-\sum_{j=1}^i(1-\alpha_j^*)\eta P_p\right] =0 \gamma_i^*[h_{sp}^iE_s^{i*}-\alpha_i^*P_{int}] =0 \mu_i^*[\alpha_i^*-1] =0
|\eta_\varepsilon*g(x')-g(x')|= \Bigl|\int_{{\mathbb{R}}^{n-1}}\eta(y')\bigl(g(x'-\varepsilon y')-g(x')\bigr)\,{\rm d}y'\Bigr| \le \int_{{\mathbb{R}}^{n-1}}\eta(y')\,M\,\varepsilon|y'|\,{\rm d}y' \le \varepsilon M,
\frac{1}{t}d_K(S_t x + tv) = \frac{1}{t}d_K \bigg( x + t(Ax + v) - t \bigg( Ax - \frac{S_t x - x}{t}\bigg) \bigg) \leq \frac{1}{t}d_K \big( x + t (Ax + v) \big) + \bigg\| Ax - \frac{S_t x - x}{t}\bigg\|,
\|g_n\|_{L^1}= \int_I (|f_n^{(k+1)}|\circ f_n^{-1})\cdot ((f_n')^{-1}\circ f_n^{-1}) &= \int_I |f_n^{(k+1)}| &= \|f_n^{(k+1)}\|_{L^1}.
\begin{aligned}J_1(x,x,y) = x, \begin{aligned}J_{2i+1}(x,y,y) = J_{2i+2}(x,y,y) 0\leq i\leq k-1, J_{2i}(x,x,y) = J_{2i+1}(x,x,y) 1\leq i\leq k. \end{aligned}\end{aligned}
|\frac{-1}{2\omega}\int_{0}^{\frac{2}{\sqrt{\omega}\lambda(t)}}r \phi_{0}(r) F_{4}(t,r\lambda(t)) dr| = |\frac{1}{2\omega}\int_{\frac{2}{\sqrt{\omega}\lambda(t)}}^{\infty}\phi_{0}(r) F_{4}(t,r\lambda(t)) r dr|=0
g_1(0;x) =(N_1+\nu_1N_1^{2/3}+xN^{1/3})\log(1+d_1N_1^{-1/3}) &+\frac 12(N_1-\nu_1N_1^{2/3})(1+d_1N_1^{-1/3})^2-(2N_1+\lambda_1N^{1/3})(1+d_1N_1^{-1/3})
\begin{aligned}(u_i)_t-\mu_i(|x|,t)\Delta u_i =f_i(t,|x|,u_i)-\alpha_i(|x|,t) u_1u_2, \; \partial_\nu u_i =0, \; u_i(x,0) =u_{0,i}(x) \ge 0 \; \end{aligned}
\vert f ( z ) \vert < \varepsilon ( x ) e^{c \vert y \vert}\quad \textrm{o r}\quad \vert f ( z ) \vert < \frac{M e^{2 \vert y \vert}}{\vert z \vert^{\alpha}}, \quad z = x + i y , \quad \vert z \vert \to \infty ,
\frac{(\eta_k)^j(-1)^{j-1}}{(j-1)!}\frac{\partial^{j-1}}{\partial \eta^{j-1}}\left(\frac{1}{\eta}e^{\eta(b-y)}\Big(\sum_{i=1}^{M}\frac{H_i\eta}{\beta_{i,q}-\eta}+\sum_{i=1}^{N}\frac{\eta Q_i}{\eta+\gamma_{i,q}}+1\Big)\right)_{\eta=\eta_k}=0,
\rho_i:= \rho^{F\mkern.3mu\cap\mkern.3mu \partial_i \mkern-1.2mu\hat{M}}_{\mkern-2.3mu\tilde{f}_i}: \pi_1(\partial_i \hat{M}) \to \widetilde{\mathrm{Homeo}_+}S^1\mkern-2mu,\;\;\;\;[\tilde{f}_i] \to \mathrm{sh}(1),\;\;\;\;[\tilde{h}_i] \to \gamma_{F,\tilde{f}_i, -\tilde{h}_i},
P_{r}f(e^{i\theta})=\sum_{n\in Z}a_{n}r^{|n|}e^{in\theta}=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{(1-r^{2})f(e^{i\theta})}{1-2rcos\theta+r^{2}}d\theta=K_{r}*f(e^{i\theta}),