harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\langle \tilde{\Theta}_\pm \tilde{\Theta}_\pm D_a\varphi, \tilde{\Theta}_\pm S\varphi\rangle =\pm \langle D_a\varphi, \tilde{\Theta}_\pm S\varphi\rangle &=\pm \langle D_a\varphi, S\tilde{\Theta}_\pm \varphi\rangle &=\pm \langle S\varphi, D_a\tilde{\Theta}_\pm \varphi\rangle=\langle S\varphi, \tilde{\Theta}_\pm D_a\varphi\rangle,
d_{\ell}= \frac{(-1)^{\ell - 1}}{2\pi}\sum_{j=1}^{m}\log s_{j}\int_{x_{j-1}}^{x_{j}}u^{\ell - \frac{3}{2}}du = \frac{(-1)^{\ell -1}}{\pi (2\ell -1)}\sum_{j=1}^{m}\log s_{j}\Big( x_{j}^{\ell -\frac{1}{2}}-x_{j-1}^{\ell-\frac{1}{2}}\Big) .
p(x_i | y_i') = \frac{1}{\sqrt{2\pi\frac{\sigma_x^2\sigma_n^2}{\sigma_n^2+\sigma_x^2r^2}}}\exp\left({\frac{-\left (x_i-\frac{r\sigma_x^2 y_i'}{\sigma_n^2+\sigma_x^2 r^2}\right)^2}{2\frac{\sigma_x^2\sigma_n^2}{\sigma_n^2+\sigma_x^2r^2}}}\right).
\psi(x) = [\psi([b_{m-1},a]), \psi(\rho^{-1}(b))] &=([a\rho(b_{m-1}), 1], [\rho(b_{m-1})a, b]) &=(1, a\rho(b_{m-1})b\rho(b_{m-1})ab) &= (1, [a,b]).
\sqrt{p_{1}^{2e_{1}+1}\cdot\cdot\cdot p_{k}^{2e_{k}+1}p_{k+1}^{2e_{k+1}}...p_{n}^{2e_{n}}}=p_{1}^{e_{1}}...p_{n}^{e_{n}}\sqrt{p_{1}...p_{k}}.
\chi=|\mathfrak{S}_+\times\mathfrak{S}_-\times W_1|^{-1}\!\!\!\!\!\!\!\!\sum_{\substack{w=(w_+,w_-,w_1) \in\mathfrak{S}_+\times\mathfrak{S}_-\times W_1}}\!\!\!\!\!\!\!\!\varphi_+(w_+)\varphi_-(w_-)\varphi(w_1)R^{G}_{T_{w}}\theta_{w}.
\min f(x), f(x)=\sum_{i=1}^n f^i(x_i), x_i\in\Omega_i=\{x_i\in\mathbb R^q:g_i(x_i)\leq 0\}, A_i x_i=b_i,i\in\{1,\ldots,n\}, Lx = 0_{nq},
A(1,2,3,4):=\tilde D^\alpha_{4,-p_4}\tilde D_{\alpha,2,-p_2}\tilde{\bar D}_{\dot\alpha,2,-p_2}\tilde{\bar D}^{\dot\alpha}_{1,-p_1}\tilde\delta_V(1,2)\tilde\delta_V(1,3)\tilde\delta_V(1,4).
\left|z+\dfrac{a_{n-1}}{a_n}-t\right|\leq \left|\dfrac{a_{n-1}}{a_n}\right|(\sin\alpha-\cos\alpha) &+\dfrac{2|a_k|}{|a_n|t_1^{n-k-1}}\cos\alpha+2\dfrac{\sin\alpha}{|a_n|}\sum\limits_{\nu=0}^{n-2}\dfrac{|a_{\nu}|}{t_1^{n-\nu-1}}.
\begin{array}{lll}p_{n+1}= a_{n+1}p_n+p_{n-1}\hat{p}_n = a_{n+1}\hat{p}_{n-1}+\hat{p}_{n-2}q_{n+1}= a_{n+1}q_n+q_{n-1}\hat{q}_n = a_{n+1}\hat{q}_{n-1}+\hat{q}_{n-2}\end{array}
\mathbb{P}\left(Q(t) = M\right) = \frac{\Big(\prod_{i}x_i!\Big)\left(\prod_{j}y_j!\right)}{n!\left(\prod_{i,j}m_{ij}!\right)}\sum_{\substack{M'\in\mathcal{E}': M'\ge M}}\prod_{i,j}\frac{\left(1 - (1 - \pi_{ij})^t\right)^{m_{ij}}\left((1 - \pi_{ij})^t\right)^{m'_{ij}-m_{ij}}}{(m'_{ij}- m_{ij})!}
\sup \{\|({\partial}a)b\|_\tau^2; \|b\|_\tau \leq 1\}=\sup \{\tau(b^*\langle{\partial}a,{\partial}a\rangle_{\mathcal A}b): \|b\|_\tau \leq 1\}=\|\langle{\partial}a,{\partial}a\rangle_{\mathcal A}\|_\infty.
\Gamma^{\mathrm{hori}}(f,g) :=\frac{1}{2}(L (fg) -fLg -g Lf) = X(f) X(g) +Y(f) Y(g), \Gamma^{\mathrm{vert}}(f,g) :=Z(f)Z(g), \Gamma^{\mathrm{elli}}(f,g) := \Gamma^{\mathrm{hori}}(f,g) + \nu \Gamma^{\mathrm{vert}}(f,g).
R^{ef}_{ab}(\vartheta -\vartheta ')\left[ t_{ec}(\vartheta )\right]_{\alpha \gamma}\left[ t_{fd}(\vartheta ')\right]_{\gamma \beta}=\left[ t_{ae}(\vartheta ')\right]_{\alpha \gamma}\left[ t_{bf}(\vartheta )\right]_{\gamma \beta}R^{cd}_{ef}(\vartheta -\vartheta ')
\begin{tabular}{ll}$B_{1}=\left( a-h_{n},a\right) \cap E_{n},$ $B_{2}=\left(b,b+h_{n}\right) \cap E_{n},$ $B_{3}=\left( a,a+h_{n}\right) \cap B,$ $B_{4}=\left( b-h_{n},b\right)\cap B.$\end{tabular}
\frac{1}{T_{n}-T_{n-k}}\ =\ \frac{1}{T_{n}}+\frac{1}{T_{n+k}-T_{n}}-\frac{T_{n}^{2}-T_{n+k}T_{n-k}}{T_{n}\left(T_{n}-T_{n-k}\right)\left(T_{n+k}-T_{n}\right)}.
F^{}(x_{i+1/2}) - F^{}(x_{i-1/2}) = \Bigg( \sum_{j=1}^{i}\int_{\Lambda_j}u \left(\mathcal{L}_{x_{i+ 1/2}}(x_j)+ (u-x_j)\mathcal{L}_{x_{i+ 1/2}}^{'}(x_j)\right) du - \sum_{j=1}^{i-1}\int_{\Lambda_j}u \left(\mathcal{L}_{x_{i- 1/2}}(x_j)+ (u-x_j)\mathcal{L}_{x_{i- 1/2}}^{'}(x_j)\right) du \Bigg)+{\cal O}(\Delta x^3).
[ T_{\vec{m}_{1}}, \cdots , T_{\vec{m}_{M}}] \; = \; \frac{- i}{2 \pi \Lambda}\; \sin \; \bigl ( 2 \pi \Lambda \, ( \vec{m}_{1}, \cdots , \vec{m}_{M}) \big ) \; T_{\vec{m}_{1}\, + \cdots + \, \vec{m}_{M}}
\int_0^t f\left (r, W_r, Y^i_r, \frac{\mathrm d[Y^i,W]_r}{\mathrm dr}\right) \mathrm dr &= \int_0^t f\left (r, W_r, \gamma^i(r,W_r), \nabla \gamma^i(r,W_r) \right) \mathrm dr &= \int_0^t \tilde f^i\left (r, W_r \right) \mathrm dr ,
\rho[\frac{\partial(\overline{u_{i}}+u_{i}^{\prime})}{\partial t}+(\overline{u_{j}}+u_{j}^{\prime})\frac{\partial(\overline{u_{i}}+u_{i}^{\prime})}{\partial x_{j}}]=-\frac{\partial(\overline{p}+p^{\prime})}{\partial x_{i}}+\mu[\frac{\partial^{2}(\overline{u_{i}}+u_{i}^{\prime})}{\partial x_{j}\partial x_{j}}].
p\left({{d_{pF}}\leq \delta}\right) = \frac{{M{\delta^{2kn - 2{k^2}}}}}{{k!\Gamma \left({nk -{k^2}+ 1}\right)}}\prod\limits_{i = 0}^{k - 1}{\Gamma \left({n - 2k + 1 + i}\right)\Gamma \left({2 + i}\right)}
\begin{aligned}2A_+{U^+}^2|\tilde\psi_1^+|^2\,+\,4BU^+U^-\tilde\psi_1^+\tilde\psi_1^-\,+\,2A_-{U^-}^2|\tilde\psi_1^-|^2\,\geq\, \Lambda \big(|\tilde\psi_1^+|^2+|\tilde\psi_1^-|^2\big).\end{aligned}
\|\Phi( t)\|_3^2 + \|\Phi_t( t)\|_2^2+ \int_0^t (\|\Phi_x\|_2^2 + \|\Phi_\tau\|_2^2) d\tau \leqslant O(1)\delta^{-1/4}( \|\Phi(0)\|_3^2 + \|\Phi_t(0)\|_2^2)+ C_3 \delta_0^{{1}/{4}}.
\left[q^{d}\right]\frac{(-1)^{(s-1)d}}{\prod_{j=1}^{N}\,\,\alpha_j^{\ell+1}}\,\,\sum_{\vec{d}, I}\prod_{j\not \in I}(z_j+y)^{\ell+1}\,\,\prod_{i\in I}\frac{z_i^{d+\ell+N-s+1}}{P'(z_i)}\prod_{i, j\in I, \,i\ne j}(z_i-z_j)\bigg{|}_{\epsilon=0}.
F_{m}^{\n}= \left[ \begin{array}{c c c}{{0}}&{{\chi{\vec{\partial}}\phi^{9}}}&{{0}}\\{{\chi{\vec{\partial}}\phi^{9}}}&{{0}}&{{-{\vec{\partial}}\phi^{9}}}\\{{0}}&{{+{\vec{\partial}}\phi^{9}}}&{{0}}\\ \end{array}\right] \ ,
J_K\begin{bmatrix}I S\end{bmatrix}= J_K\begin{bmatrix}I_{k \times k}0 0 I_{(n-k) \times (n-k)}S_1 S_2 S_3 S_4\end{bmatrix}= \begin{bmatrix}-S_1 -S_2 0 I I 0 S_3 S_4 \end{bmatrix}.
\hat{\phi}(x^{0}\,,\,x^{1})\,=\, \frac{1}{\sqrt{2\pi}}\left( \frac{\pi}{R}\right) \sum_{n \geq 0}\frac{1}{\sqrt{\omega_{n+\frac{1}{2}}}}\left[ \hat{\Lambda}_{n}e^{-i\omega_{n+\frac{1}{2}}(x^0 + x^{1})}\,+\,\hat{\Lambda}_{n}^{\dagger}e^{i\omega_{n+\frac{1}{2}}(x^0+x^{1})}\right],
\|v\|_2^2 = \frac{\|w - \alpha e_1\|_2^2}{t}= \frac{\|w\|_2^2 - 2w_1\alpha + \alpha^2}{\|w\|_2^2 + \epsilon_{FP}- w_1\alpha}=\frac{2\|w\|_2^2 - 2w_1\alpha + 2\epsilon_{FP}}{\|w\|_2^2 + \epsilon_{FP}- w_1\alpha}= 2
\tau_{W}\left( r_{\zeta}\right) =\left( 1+\varphi_{W}\left( \zeta\right)\right) \left( 1+\varphi_{W}\left( -\zeta\right) \right) \dfrac{m_{W}\left(\zeta\right) -m_{W}\left( -\zeta\right)}{2\zeta}=\det\mathit{\Pi}_{W}\left(\zeta^{2}\right)
A_p(X_1^\top,X_2^\top) = D^2\Pi(p)(X_1^\top,X_2^\top) X_1 D^2\Pi(p)(X_2,X_3) =\sum_{\sigma \in \mathcal{P}_3}X_{\sigma(1)}^\bot D^2\Pi(p)( X_{\sigma(2)}^\top, X_{\sigma(3)}^\top)
b(m,a_1, \ldots , a_r,n) = (ma_1,a_2, \ldots, a_r,n) + \sum_{i=1}^{r-1}(-1)^i (m,a_1, \ldots, a_{i-1}, a_ia_{i+1}, a_{i+2}, \ldots, a_r,n) &+ (-1)^r (m,a_1, \ldots, a_{r-1},a_rn).
S_r(N) = \,\frac{1}{KQp_1}\displaystyle\int_{\mathbb{R}}W(u)\,\int_{\mathbb{R}}\, V_1\left(\frac{\nu}{K}\right)\, \displaystyle\sum_{1\leq q\leq Q}\frac{g(q,u)}{q}\sideset{}{^\star}{\displaystyle\sum}_{a \, \mathrm{mod}\, q}\,\,\sum_{b \, \mathrm{mod}\, p_1}\,S_1(...)S_2(...) \mathrm{d}u\,\mathrm{d}\nu,
\Gamma_{n l o c}= -{\frac{1}{8 \pi}}\int d^{2}x \sqrt{- G}\big \{{\frac{1}{1 2}}R{\frac{1}{\nabla^{2}}}R -{\cal Q}{\frac{1}{\nabla^{2}}}R +{\cal Q}\beta^{( 1 )}{\cal Q}\big \},
\nu [ A ] = \lim_{V_{4}\rightarrow \infty}\frac{g^{2}}{8 \pi^{2}}\int_{V_{4}}d^{4}x \partial_{\mu}K_{\mu}= \lim_{V_{4}\rightarrow \infty}\frac{g^{2}}{8 \pi^{2}}\int_{\partial V_{4}}d^{3}\sigma \hat{n}_{\mu}K_{\mu}.
{\frac{d \zeta}{d \varphi}}= A [ 1 - \varphi^{2}-{\frac{\epsilon}{2}}{\frac{1 - \zeta^{2}}{1 - \varphi^{2}}}+{\frac{\epsilon^{2}}{8}}{\frac{( 1 - \zeta^{2})^{2}}{( 1 - \varphi^{2})^{3}}}+ \cdots ] .
\left(\rho_{\mathbf z}+ \rho_{\mathbf z'}\right)^2 =\left(\frac{RC}{C + 2 R h}+ \frac{RC}{C + 2 R h'}\right)^2 &= \left(\frac{2 R^2 h}{C + 2 R h}\right)^2 + \left(\frac{2 R^2 h'}{C + 2 R h'}\right)^2 - \frac{8 R^2 h h'(R^2 - C^2/(2hh') )}{(C + 2 R h)(C + 2 R h')}.
\|H(f)\|_{L_{|x|_h}^pL_{\theta}^{\bar{p}_1}(\mathbb H^n)\rightarrow L_{|x|_h}^pL_{\theta}^{\bar{p}_2}(\mathbb H^n)}=G\|f\|_{L_{|x|_h}^pL_{\theta}^{\bar{p}_2}(\mathbb H^n)}
\!\! \prod_a \beta_a^{-{\rm i}M_a^{z\bar{z}}}f({\bf X})=\prod_a\beta_a^{-\lambda_a}f({\bf X}(\beta))\,,\quad\!\!e^{-q(\zeta^2\bar{\zeta}+b\zeta)}f(\zeta)=(1+q\zeta)^{-b}f\Bigl(\frac{\zeta}{1+q\zeta}\Bigr)\,,
\begin{aligned}\int \frac{x+1}{(\log(x+1))^{\beta-1}}dx= \sum_{\ell=0}^k\frac{(x+1)^2}{2B_\ell (\log(x+1))^{\beta+\ell-1}}\qquad+\int\frac{x+1}{B_{k+1}(\log(x+1))^{\beta+k}}dx, \end{aligned}
R = V^{a \mu}V^{b \nu}R_{\mu \nu a b}( \omega ) , R_{\mu \nu a b}( \omega ) = \partial_{\mu}\omega_{\nu a b}- \partial_{\nu}\omega_{\mu a b}+ ( \omega_{\mu a}^{c}\omega_{\nu c b}- \omega_{\nu a}^{c}\omega_{\mu c b}) .
D_{r,s,t}\big(y^{-1}z^{-1}\big) =(r-1)!(s-1)!(t-1)!\sum_{\substack{j+h=k j,h\ge1}}\binom{j-1}{r-1}\binom{h-1}{t-1}(-1)^{j+s}y^{-j}z^{-h}, D_{r,s,t}\big(z^{-1}x^{-1}\big) = (r-1)!(s-1)!(t-1)!\sum_{\substack{h+i=k h,i\ge1}}\binom{h-1}{s-1}\binom{i-1}{r-1}(-1)^{h+t}z^{-h}x^{-i}.
L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{n_{f}g^{2}\theta}{32\pi^{2}}F_{\mu\nu}\tilde{F}^{\mu\nu}+\overline{\psi}(i\gamma^{\mu}D_{\mu}-me^{i\theta^{\prime}\gamma_{5}})\psi
[ P_{0}, M_{0 i}] = i x_{0}\int d^{3}x \: \partial_{i}A_{0}( x ) G ( x ) + i \int d^{3}x \: A_{i}( x ) G ( x ) + i P_{i}, \hspace{0 . 2 5 i n}[ P_{0}, M_{k j}] = 0 ,
[X \xleftarrow p V \xrightarrow s pt; E] \circ_{\oplus}[pt \xleftarrow q W \xrightarrow t pt; F]= & [(X \xleftarrow p V \xrightarrow s pt) \circ (pt \xleftarrow q W \xrightarrow s pt);(pr_1)^*E \oplus (pr_2)^*F ],
M_2(\kappa; \boldsymbol{u}):=-\frac{1}{\pi^2}\int_{\mathbb{R}- i u_2}\int_{\mathbb{R}-iu_1}\frac{e^{-\pi w_1^2-\pi w_2^2-2 \pi i (u_1 w_1+u_2 w_2)}}{w_2(w_1-\kappa w_2)}\boldsymbol{dw}.
f(M)=\sum_{n = 0}^\infty\left\lfloor \frac{M}{2^n}\right\rfloor = \sum_{n = 0}^{\lfloor \log_2 M \rfloor}\left\lfloor \frac{M}{2^n}\right\rfloor \leq \sum_{n = 0}^{\lfloor \log_2M \rfloor}\frac{M}{2^n}\leq M \cdot \frac{1-\frac{1}{2M}}{1-\frac{1}{2}}=2M-1.
\delta C ( \theta ) = \epsilon ( \theta ) C^{\prime}( \theta ) \, ; \; \delta ( C , C ) = \oint \frac{d \theta}{2 \pi}\epsilon ( \theta ) 2 C_{\alpha}( \theta ) C_{\alpha}^{\prime}( \theta ) = - \oint \frac{d \theta}{2 \pi}\epsilon^{\prime}( \theta ) C_{\alpha}^{2}( \theta )
\frac{3 L(1, f)}{\pi}\Bigl( \frac{A_f(p, 1)}{p^{\frac{1}{2}}}- \frac{1}{p^{\frac{3}{2}}}\Bigr) T M \log T = \frac{3 c L(1, g)}{\pi}\Bigl( \frac{A_g(p, 1)}{p^{\frac{1}{2}}}- \frac{1}{p^{\frac{3}{2}}}\Bigr) T M \log T + O_p(T M).
\sum\limits_{\alpha =1}^{m}\Phi_{i,\alpha}\left( \left\vert \xi_{i}^{\alpha}\right\vert \right) \leq F_{\alpha}\left( x,\xi^{\alpha}\right) \leq L\left[ \bar{B}_{\alpha}^{\beta_{\alpha}}\left( \left\vert\xi^{\alpha}\right\vert \right) +a\left( x\right) \right]
G_r^C ( z | \tau_0, \ldots, \tau_r) = \prod_{n \in C^\circ \cap \mathbb{Z}^{r+1}}\left (1-e^{2\pi i (- z + n_0\tau_0 + \ldots +n_r \tau_r )}\right ) \prod_{n \in C \cap \mathbb{Z}^{r+1}}\left ( 1 - e^{2\pi i (z+n_0\tau_0 + \ldots + n_r \tau_r )}\right )^{(-1)^r},
\begin{cases}\Delta u_1+\lambda e^{u_2}(1-e^{u_1})=\displaystyle \sum_{i=1}^{N_1}\frac{4\kappa}{(\kappa+|x-p_{1i}|^2)^2}, \Delta u_2+\lambda e^{u_1}(1-e^{u_2})=\displaystyle \sum_{i=1}^{N_2}\frac{4\kappa}{(\kappa+|x-p_{2i}|^2)^2},\end{cases}
{1 \over{\cal A}_c}\int_0^1 d\epsilon\, a_{5/2}(f_\epsilon,P_\epsilon)=\frac{1}{4 \pi^2 z_{+}^{4}}\left[-\frac{27}{128}(\ln z_{+}+\lambda^4 \ln z_{-}) +\frac{53}{5120}(1+\lambda^4)\right].
\begin{array}{c}{u_{i}^{\mu}v_{\mu}^{j}= G_{i}^{j}\equiv [ \exp ( - W ) ]_{i}^{j},}\\{{G^{- 1}}_{i}^{j}( u_{j}^{\mu}v_{\nu}^{i}+ u_{\nu j}v^{\mu i}) = \delta_{\nu}^{\mu}.}\\ \end{array}
b_i(\alpha\tau)=\alpha_1\tau_1\alpha_2\tau_2\cdots\alpha_{k_i}\tau_{k_i}=\alpha_1\left(\alpha_2^{\tau_1^{-1}}\alpha_3^{(\tau_1\tau_2)^{-1}}\cdots\alpha_{k_i}^{(\tau_1\cdots\tau_{k_i})^{-1}}\right)\tau_1\tau_2\cdots\tau_{k_i},
\frac{f(x+y)}{f(x)}= \frac{\inf_{t \in [x_0, x+y]}f (t)}{\sup_{s\ge x}f(s)}\frac{f(x+y)}{\inf_{t \in [x_0, x+y]}f(t)}\frac{\sup_{s\ge x}f(s)}{f(x)}& \le \frac{F(x+y-c+\Delta)}{F(x+\Delta)}\frac{f(x+y)}{\inf_{t \in [x_0, x+y]}f(t)}\frac{\sup_{s\ge x}f(s)}{f(x)}.
{\frac{\d S}{\d e^{a}}}\Rightarrow ( \Pi^{\underline{{m}}}u_{\underline{{m}}}^{a_{0}}- e^{a_{0}}) e^{a_{1}}. . . e^{a_{p - 1}}\varepsilon_{a_{0}a_{1}. . . a_{p - 1}a}= 0 ,
W ( x , x^{\prime}) = W ( x ) - \frac{1}{2}W_{; \alpha}( x ) \sigma^{; \alpha}+ \frac{1}{2}W_{\alpha \beta}( x ) \sigma^{; \alpha}\sigma^{; \beta}+ \frac{1}{4}\{\frac{1}{6}W_{; \alpha \beta \gamma}( x ) - W_{\alpha \beta ; \gamma}( x ) \}\sigma^{; \alpha}\sigma^{; \beta}\sigma^{; \gamma}+ 0 ( \sigma^{2})
XX+Y^TY = XX-XC(-C^TX) = XI_{n-1}X+XCC^TX = X(CC^T+I_{n-1})X = X(CC^T+I_{n-1})(CC^T+I_{n-1})^{-1}(J_{n-1}-nI_{n-1}) = X(J_{n-1}-nI_{n-1}) = -J_{n-1}-nX
\int_{0}^{\infty}d t t^{\frac{4 - p}{2}}e^{-{\frac{t b^{2}}{2 \pi \alpha^{\prime}}}}= \Gamma ({\frac{6 - p}{2}}) ({\frac{2 \pi \alpha^{\prime}}{b^{2}}})^{\frac{6 - p}{2}}
E^{[1]}(f,G \rtimes S)(t, \overline{t}) = \sum \widehat{m}_{\underline{\lambda},\widetilde{\underline{\lambda}}}(-1)^{n_{\underline{\lambda}}}(t \overline{t})^{{\rm age}(\underline{\lambda}) - \frac{n-n_{\underline{\lambda}}}{2}}\left( \frac{\overline{t}}{t}\right)^{{\rm age}(\widetilde{\underline{\lambda}}) - \frac{n-n_{\widetilde{\underline{\lambda}}}}{2}},
\Psi_R(x)=N(k^0) \left(\begin{array}{r}1 i 0 \end{array}\right) e^{i k^3 x^3 -i k^0 x^0}=N(k^0) (\hat{e}_1+ i \hat{e}_2)e^{i k^3 x^3 - i k^0 x^0}\, \, , k^0=k^3 > 0
{\cal D}_{x^{2\xi}}^2V_{1,0}(w)={1\over (x^\xi-x^{-\xi})^2}\left(\oint_w z^{-3}f_{1,0}\left({w\over z}\right)T(z)V_{1,0}(w)dz-(x^\xi-x^{-\xi})w^{-2}V_{1,0}(w)\right).
\langle b-a_2,a_1-a_2\rangle =\langle u,a_1-a_2\rangle+\langle b-a_2-u,a_1-a_2\rangle &\le\gamma\|u\|\,\|a_1-a_2\|+(\sigma+\varepsilon)\|b-a_2\|\,\|a_1-a_2\| &\le(\gamma\sqrt{1-(\sigma+\varepsilon)^2}+(\sigma+\varepsilon))\|b-a_2\|\,\|a_1-a_2\|,
\begin{array}{c}{{f_{-}= f ( - \infty + i \epsilon ) = ( - 1 )^{\delta}e^{i Z_{+}( - \infty )}= e^{i{\cal Q}_{+}( - \infty )}}}\\{{f_{+}= f ( + \infty + i \epsilon ) = ( - 1 )^{\delta}e^{i Z_{+}( + \infty + i \epsilon )}= 0}}\\ \end{array}
B_1(m) = \begin{cases}p^{1-3m/2}(-1)^{m+1}\chi_{\psi}(p^m)(1-p^{-1})(1 + p - p^m) m > 0, p^{3m/2}(-1)^{m+1}\chi_{\psi}(p^m)(1-p^{-1})p^{-m}m \leq 0. \end{cases}
\int_{0}^{T}d \tau \psi_{\mu}\psi_{\nu}\omega_{\mu \nu}= - T \int_{0}^{T}d \tau ( \psi \cdot \dot{x})^{\cdot}( \psi \cdot \dot{x}) + T \int_{0}^{T}d \tau ( \psi \cdot \dot{x}) ( \dot{\psi}\cdot \dot{x}) +{\frac{T}{4}}\int_{0}^{T}d \tau \dot{x}\cdot \ddot{x}
\alpha(u) = - \frac{c'(u)}{24}&\beta_{0}(u) = - p'(u) + \frac{c'(u)^{2}- c(u) c''(u)}{384}&\beta_{1}(u) = -\frac{p''(u)}{2}+ \frac{5 c'(u) c''(u) - 6 c(u) c'''(u)}{1152}&\beta_{2}(u) = \frac{3 c''(u)^{2}+ 2 c'(u) c'''(u) - 4 c (u) c^{(IV)}(u)}{3456}.
E_\pm(0)u[x] =J_a \tilde P_\pm u[x] &=(2\pi)^{-d}\int_{\mathbb{T}^d}\sum_{y\in\mathbb{Z}^d}e^{i(\varphi_a(x,\xi)-\varphi_a(y,\xi))}p_\pm(y,\xi)u\left[y\right]d\xi &=(2\pi)^{-d}\int_{\mathbb{T}^d}\sum_{y\in\mathbb{Z}^d}e^{i(x-y)\cdot\eta}p_\pm(y,\xi(\eta))\left|\det\left(\frac{d\xi}{d\eta}\right)\right|u\left[y\right]d\eta.
\mathbf{L}_{\nu -1}(x)- \mathbf{L}_{\nu +1}(x) = \frac{2\nu}{x}\mathbf{L}_{\nu}(x)+\frac{\big(\frac{1}{2}x\big)^\nu}{\sqrt{\pi}\Gamma(\nu+\frac{3}{2})}, \frac{\mathrm{d}}{\mathrm{d}x}\big(x^{\nu}\mathbf{L}_{\nu}(x) \big) = x^{\nu}\mathbf{L}_{\nu -1}(x).
J^{( N )}( N - 1 ; 1 , \ldots , 1 ) = - \frac{1}{2 \; D^{( N )}}\sum_{i = 1}^{N}\frac{1}{m_{i}^{2}}\; F_{i}^{( N )}\; \left. J^{( N - 1 )}( N - 1 ; 1 , \ldots , 1 ) \right|_{\mathrm{\scriptsize ~ w i t h o u t}\; i}.
\left( \begin{array}{c}{t}\\{c}\\{u}\\ \end{array}\right) ; \left( \begin{array}{c}{b}\\{s}\\{d}\\ \end{array}\right) ; \left( \begin{array}{c}{\tau}\\{\mu}\\{e}\\ \end{array}\right) ; \left( \begin{array}{c}{\nu_{1}}\\{\nu_{2}}\\{\nu_{3}}\\ \end{array}\right) .
\| f_t \circ ( \phi_1 * \phi_2 ) \| = \lim_{n \to \infty}\left\| f_t \circ \left( 1_{A_1^{(n)}}* 1_{A_2^{(n)}}\right) \right\| \leq \lim_{n \to \infty}\left\| f_t \circ \left( 1_{A_1^{(n)}}^* * 1_{A_2^{(n)}}^* \right) \right\| = \| f_t \circ ( \phi_1^* * \phi_2^* ) \|.
R_{B,opt}(N,t) \geq 1-\frac{\log \left[\prod\limits_{k=0}^{2t}(N-k)\right]}{\log N!}&>1-\frac{(2t+1)\log N}{(N+\frac{1}{2})\log N -(\log e)N}&>1-\frac{(2t+1)\log N}{N(\log N -\log e)}&>1-\frac{2t+1}{N}\left(1+\frac{2\log e}{\log N}\right),
\frac{1}{N}\sum_{mp \leq N}f(m) f(p) \Lambda(p) F(g(mp) \Gamma) = \frac{1}{N}\sum_{m \leq X}\sum_{p \leq N/m}f(m) f(p) \Lambda(p) F(g(mp)\Gamma) &+ \frac{1}{N}\sum_{m > X}\sum_{p \leq N/m}f(m) f(p) \Lambda(p) F(g(mp)\Gamma).
\dim(U_{0}\cap\mathcal{K}\left( 0\right)^{\perp}) = \dim U_{0}+\dim\mathcal{K}\left( 0\right)^{\perp}-\dim( U_{0}+\mathcal{K}\left(0\right)^{\perp}) & \geq\dim U_{0}+n-1-\dim\mathcal{K}(0) -\left( n-1\right) & \geq\dim U_{0}-\dim\mathcal{K}& >\ell-\left( \ell-k+1\right) & =k-1.
\pi(\langle x,y\rangle)T=\langle \Pi(x)\Pi(y)\rangle T=\Pi(x)^{*}\Pi(y)T= \Pi(x)^{*}N\Pi(y)=T \Pi(x)^{*}\Pi(y)=T\langle \Pi(x)\Pi(y)\rangle=T\pi(\langle x,y\rangle)
\frac{1}{\sqrt{-g}}{\cal H}=\frac{1}{2}g^{ij}\partial_i\phi\cdot\partial_j\phi+\frac{1}{4}\exp(\sqrt{2\kappa}\phi)(V_{ij}{}^k +\sqrt{2\kappa}A^k f_{ij})(V^{ij}{}_k+\sqrt{2\kappa}A_k f^{ij})
\{F,G\}_1 = \int \mathrm{res}\left([L,X_+] Y-[L,X]_+Y\right) \; dx , &\{F,G\}_2 = \int \mathrm{res}\left( (LY)_+ LX - (YL)_+ XL + \frac{1}{n+1}X[L,K_Y] \right) \; dx.
\int d x \phi_{\star}^{n}= \int d p_{1}d p_{2}\ldots d p_{n}\exp ( - \frac{i}{2}\sum_{i < j}( p_{i})_{\mu}\theta^{\mu \nu}( p_{j})_{\nu}) \phi ( p_{1}) \phi ( p_{2}) \ldots \phi ( p_{n}) \delta ( p_{1}+ p_{2}+ \ldots + p_{n})
\Delta(k_h)=k_h\otimes k_h, &\Delta(e_i)=e_i\otimes1+k_i\otimes e_i,\Delta(f_i)=f_i\otimes k_i^{-1}+1\otimes f_i &\varepsilon(k_h)=1,\varepsilon(e_i)=\varepsilon(f_i)=0, &S(k_h)=k_h^{-1},S(e_i)=-k_i^{-1}e_i,S(f_i)=-f_ik_i
\mathcal E_{0,j}(x)= 4 \alpha_j a_{0,j}\frac{\langle \nabla u_j, \nabla \phi \rangle}{\phi}\mathcal E_{1,j}(x) =2\left(\mathcal Q_j-\frac{h_j'(u_j)}{h_j(u_j)}z_j\right) a_{1,j}.
k_{\pm}=\frac{1}{\sqrt{4\beta \hslash^{2}/3}}\left( \sqrt{\frac{\sqrt{(1+|a|)^{2}+|b|^{2}}+(1+|a|)}{2}}\pm i \sqrt{\frac{\sqrt{(1+|a|)^{2}+|b|^{2}}-(1+|a|)}{2}}\right)
\begin{array}{rcl}z_1^{a_{11}}z_2^{a_{21}}\ldots z_d^{a_{d1}}= x_1 z_1^{a_{12}}z_2^{a_{22}}\ldots z_d^{a_{d2}}= x_2 \multicolumn{3}{c}{\vdots}z_1^{a_{1d}}z_2^{a_{2d}}\ldots z_d^{a_{dd}}= x_d.\end{array}
\max\{R,\underline{\gamma}\}\sum_{(\boldsymbol{j},\boldsymbol{k})\in\mathcal{I}_n(\boldsymbol{\alpha})^c}\prod_{l=1}^dU_{j_l,n}|D^{\boldsymbol{r}}\psi_{\boldsymbol{j},\boldsymbol{k}}(\boldsymbol{x})|.
\ddot{\Phi}- \frac{2}{a}\, \dot{\Phi}_{,r}+ \frac{1}{a^2}\, \Phi_{,rr}+ H\dot{\Phi}+ \frac{2}{a^2}\left(\frac{1}{r}- \dot{a}\right) \Phi_{,r}= 8\pi G\, \dot{\varphi}\left( \delta\dot{\varphi}- \frac{1}{a}\, \delta\varphi_{,r}\right) \ .
| B_{\mathrm{g h}}\rangle = \exp \left\{\sum_{n = 1}^{\infty}\left( c_{- n}{\tilde{b}}_{- n}- b_{- n}{\tilde{c}}_{- n}\right) \right\}{\frac{c_{0}+{\tilde{c}}_{0}}{2}}| \downarrow \rangle \otimes \widetilde{| \downarrow \rangle}
\mathbb P\begin{bmatrix}\begin{Vmatrix}{\frac 1{(2k)^l}}\sum_{s\in{\hat S}^l}{\tilde H}_{t,M}(xs)\end{Vmatrix}_{L^2}\leq{\frac 1{2^l}}||{\tilde H}_{t,M}||_{L^2}\end{bmatrix}\geq 1-{\frac{2^{\frac{n+4}2}}{n!\eta}}\exp\begin{pmatrix}{\frac{-k}{16\ln 2}}\end{pmatrix}
{}_{N}\! \langle \psi_{n}^{\prime}, \bar{\psi}_{n}^{\prime}| \psi_{n}, \bar{\psi}_{n}\rangle_{N}= \prod_{\mu = 0}^{p}\delta \left( \mathrm{R e}\psi_{n \mu}^{\prime}- \mathrm{R e}\psi_{n \mu}\right) \delta \left( \mathrm{I m}\psi_{n \mu}^{\prime}- \mathrm{I m}\psi_{n \mu}\right) .
\alpha_0=1, \alpha_1=-\frac{1}{y_0}, &\alpha_n=\frac{1}{s_0}\Biggl(\alpha_{n-2}-\sum\limits_{i=1}^{n-1}s_i\alpha_{n-i}-x_{n-1}\Biggr), n\geqslant2, &\beta_0=0, \beta_1=\frac{1}{y_0}, &\beta_n=\frac{1}{s_0}\Biggl(\beta_{n-2}-\sum\limits_{i=1}^{n-1}s_i\beta_{n-i}+x_{n-1}\Biggr),n\geqslant2.
I_P:= \int_{\mathcal{O}_K^n}\chi(ac(f_P(x)))|f_P(x)|^s|dx| = \int_{\mathcal{O}_K^n}\chi(ac(f(\pi x_1+p_1,\cdots, \pi x_n+p_n)))|f(\pi x_1+p_1,\cdots, \pi x_n+p_n)|^s|dx_1\cdots dx_n|.
<0|T_{\mu\nu}(x)|0> =\mathop{{\sum}'}_\gamma{\Delta_\gamma(x,x)^{1/2}\over4\pi^2\sigma_\gamma(x,x)^2}\; \left( t_{\mu\nu}(x;\gamma) + s_{\mu\nu}(x;\gamma) \right) + O(\sigma_\gamma(x,x)^{-3/2}).
\frac{d}{dt}\big\|z(t)\big\|^2\leq\Big(-L+\frac{2}{N}\sum\limits_{j=1}^{N}O_t^{(j)}\Big)\big\|z(t)\big\|^2+\frac{1}{N}\sum\limits_{j=1}^{N}\frac{e^{-2O_t^{(j)}}}{L}\big\|f^{(j)}(0)\big\|^2.
\phi_{k,l}(u):\! =\sum_{p\leq u}a_{L_k}(p)\overline{a_{L_l}(p)}=\sum_{j=2}^{2m+1}\frac{c^{(k,l)}_j u}{(\log u)^j}+ O\left(\frac{u}{(\log u)^{2m+2}}\right).
T = t_0 x^{\nu_2(t_0)}= \exp(\theta\sqrt{\log x}) \cdot \exp\left(\frac{\log x}{3.359\theta \sqrt{\log x}}\left(1 - \frac{8.02\log (\theta \sqrt{\log x})}{\theta \sqrt{\log x}}\right)\right) = \exp\left(\sqrt{\log x}\left(\theta + \frac{1}{3.359\theta}\left(1 - \frac{8.02\log (\theta \sqrt{\log x})}{\theta \sqrt{\log x}}\right)\right)\right).
\frac{1}{T_{n}}-\frac{1}{T_{n}-T_{n-k}}+\frac{1}{T_{n+k}-T_{n}}\ =\ \frac{1}{T_{n+k}-T_{n}}-\frac{T_{n-k}}{T_{n}\left(T_{n}-T_{n-k}\right)}\ =\ \frac{T_{n}^{2}-T_{n+k}T_{n-k}}{T_{n}\left(T_{n}-T_{n-k}\right)\left(T_{n+k}-T_{n}\right)},
\frac{\psi^{\prime\prime}(r)}{\psi(r)}+\frac{(D-1)}{r}\frac{\psi^{\prime}(r)}{\psi(r)}=-\frac{(\alpha^2+2\alpha n-\alpha+2)}{2\alpha}\frac{g^{\prime2}}{g}+\frac{(\alpha+1)(\alpha-1)}{4}\frac{g^{\prime2}}{g^2}+\frac{1}{4}g^{\prime2}+\frac{1}{\alpha}\frac{g^{\prime2}}{(g+\alpha)}+2\frac{g^{\prime2}}{(g+\alpha)^2}-\frac{1}{2}\{g,r\}-\frac{(D-1)(D-3)}{4r^2}
0 \ < \g(x,\tau,t) \ \le \g(0,\tau,t) = \frac{1}{\pi\sqrt{\tau(T_1-t)}}\left\{\left[1-\frac{\tau}{T_1-t}\right]^{1/2}-\left(\frac{T_1-t}{T_2-t}\right)^{1/2}\left[1-\frac{\tau}{T_2-t}\right]^{1/2}\ \right\}\ .
\left( A^{\prime}e^{n A}\right)^{\prime}={\frac{( W^{n})^{'^{\prime}}}{n}}={\frac{1 - n}{1 2}}W^{n}\left(^{( 5 )}R_{\mu}{}^{\mu}- R_{g}W^{- 2}\right) +{\frac{n - 4}{1 2}}W^{n}~^{( 5 )}R_{5}{}^{5}~ ,
\mathcal{B}_{m_k}( \hat{f}(t) \star \hat{g}(t) )(\tau) = \tau^{k}\int_{0}^{\tau^{k}}\mathcal{B}_{m_k}(\hat{f}(t))((\tau^{k}-s)^{1/k}) \star \mathcal{B}_{m_k}(\hat{g}(t))(s^{1/k}) \frac{1}{(\tau^{k}-s)s}ds
\begin{aligned}\sum_{s\in A_{}:r_s=r}L(s)Q^N(s)\le e^{-c_1(r-2)N}K_l^{r-2}\sum_{s\in A_{}:r_{s}=2}L(s)Q^N(s)+ &2(l-1)(r-2)e^{-c_1(r-2)N}K_l^{r-2}\sum_{s\in A_{}:r_{s}=2}Q^N(s).\end{aligned}
f_{\alpha,\beta}(\theta,\Sigma_{\alpha})=\frac{(2\pi)^{-\frac{n}{2}}}{\sqrt{\det\Sigma_{\alpha}}}\int_{0}^{\infty}\tau^{-\frac{n}{2}}e^{-\frac{1}{2\tau}\theta^{\top}\Sigma_{\alpha}^{-1}\theta}M_{\beta}(\tau)\,d\tau.
|E_n(Tx ) - E_n(x)| = |\varepsilon_n (x) -\varepsilon_n (Tx) + \langle s_n (T x) , \eta_n \rangle - \langle s_n (x) , \eta_n \rangle | \leq \frac 14 + |\langle e , \eta_n \rangle | \leq \frac 14 + \Vert\eta_n\Vert < \frac 12 .
\hat{\cal H}= -{\frac{1}{2}}{\frac{d^{2}}{d q^{2}}}- a g \rho \cot ( a \rho q ){\frac{d}{d q}}= -{\frac{1}{2}}a^{2}| \rho |^{2}\left\{( 1 - z^{2}){\frac{d^{2}}{d z^{2}}}- ( 1 + 2 g ) z{\frac{d}{d z}}\right\}.
P_{L}^{\prime}= ({\rho}\frac{\tilde{m}}{2}+ ( G_{\bot}- B_{\bot}- \frac{1}{4}A_{\bot}^{t}C_{\bot}A_{\bot}){\tilde{n}}- \frac{1}{2}A_{\bot}^{t}C_{\bot}{\tilde{l}}, \;{\tilde{l}}+ A_{\bot}{\tilde{n}})
\psi_{\vec{\mu}}^{a}( \theta + 2 \pi ) = e x p ( i \vec{a}. \vec{\mu}) \psi_{\vec{\mu}}^{a}( \theta ) \zeta_{\vec{\mu}}^{a}( \theta + 2 \pi ) = e x p ( i \vec{a}. \vec{\mu}) \zeta_{\vec{\mu}}^{a}( \theta )
{\widetilde T}_{\mu \nu}( p ) = T_{\mu \nu}( p ) -{\widehat M}_{P}^{D - 4}\biggl [ p^{2}H_{\mu \nu}- p_{\mu}p_{\nu}( \chi + g ) +{\frac{p^{2}}{2}}\eta_{\mu \nu}\chi \biggr ] .
\sigma\cdot T_{p}^{4}=\frac{\sigma\cdot T_{*}^{4}}{4\pi\cdot d^{2}}\cdot\frac{4\pi\cdot R_{*}^{2}}{4\pi\cdot R_{p}^{2}}\cdot\pi\cdot R_{p}^{2}(1-A_{p})
\tilde{S}(r)= \frac 18\sum_{m=0}^{k_1-1}\frac{2m+1}{\left(m^2+m+\frac{1+r^2}{4}\right)^2}-\frac 18\sum_{m=0}^{k_2-1}\frac{2m}{\left(m^2+(\frac r2)^2\right)^2}&+\frac 18 H_1\left(1,\frac{1+r^2}{4},\frac{1+r^2}{4};k_1\right)-\frac 18 H_1\left(0,\frac{r^2}{4},\frac{r^2}{4};k_2\right).
\sum\limits_{n = 1}^N{\frac{{{{\left|{{\lambda_{u,n}}}\right|}^2}}}{{\frac{1}{N}\sum\nolimits_{i = 1}^U{\frac{q}{{1 +{x_i(z)}}}{{\left|{{\lambda_{i,n}}}\right|}^2}}+ \sigma_n^2 -{z}}}}.
\Psi^\pm(x) = x \int_0^\infty \psi(y) \sum_{j = 1}^K \frac{c_j^\pm e(3 x^\frac{1}{3}y^{\frac{1}{3}}) + d_j^\pm e(-3 x^{\frac{1}{3}}y^{\frac{1}{3}})}{(x y)^{\frac{j}{3}}}\, d y + O\bigl( (x X)^{\frac{-K + 2}{3}}\bigr)