harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{cases}\dot{{\bf x}}= A{\bf x}+BR^{-1}B^T{\bf y}+G{\bf m}+\gamma{\bf p}, \dot{{\bf m}}= (A + G){\bf m}+B\bar u +\gamma{\bf p}, \dot{{\bf p}}= -(A+G)^T{\bf p}-(I-\Gamma)^T Q [{\bf x}- (\Gamma{\bf m}+\eta)] , \dot{{\bf y}}=-A^T{\bf y}+Q[{\bf x}-(\Gamma{\bf m}+\eta)] ,\end{cases}
\bar\alpha_1\colon Tq_f\cdot{}T{L}f\cdot{}q_f= Ti_A\cdot{}q_f \Longrightarrow i_{TA}\cdot{}q_f= Tq_f\cdot{}i_{Kf}\bar\alpha_2\colon T{R}f\cdot{}T{L}f\cdot{}q_f= Tf\cdot{}q_f \Longrightarrow i_B\cdot{}{R}f=T{R}f\cdot{}i_{Kf}
f ( x ) = \int_{- \infty}^{\infty}\frac{d \xi}{2 \pi}\mathrm{e}^{i ( 1 - x ) \xi}\langle \widetilde{H}( p ) | W_{\scriptstyle \Pi}({v \! \cdot \! n}\xi \mu - i 0 ) | \widetilde{H}( p ) \rangle ,
\int_{0}^{1}{\frac{B_{2}( t )}{( x^{2}+ ( 2 \pi t )^{2})^{{\frac{3}{2}}}}}={\frac{1}{6 x^{2}\sqrt{x^{2}+ 4 \pi^{2}}}}-{\frac{1}{4 \pi^{2}| x |}}+{\frac{1}{8 \pi^{3}}}\sinh^{- 1}{\frac{2 \pi}{| x |}}\ .
\Delta J_{s}= \frac{1}{2}\left( \begin{array}{c c c}{{c}}&{{Q_{\nu}- B_{\nu}}}&{{0}}\\{{0}}&{{0}}&{{- Q^{\mu}+ B^{\mu}}}\\{{0}}&{{0}}&{{- c}}\\ \end{array}\right) ( x ) \delta ( x - y ) \Lambda \ .
Z = 4 \pi \cdot \int_0^\infty \cosh(t)^{-k-l-2}\sinh(2t) \, dt = 8 \pi \cdot \int_0^{\infty}\cosh(t)^{-k-l-1}\sinh(t) \, dt = 8 \pi \cdot \int_1^{\infty}t^{-k-l-1}\, dt = \frac{8 \pi}{k+l}.
\frac{P_{\mathrm{1 C G}}}{m_{\mathrm{\scriptscriptstyle{M T}}}^{2}}\; \sim \; \left( \frac{m_{\mathrm{\scriptscriptstyle{M T}}}g_{\mathrm{\scriptscriptstyle{M T}}}}{2 \pi T}\right)^{2}\; \ll \; 1 .
\mathcal{J}^+_1(n_1^2n_2,m,q) = \int_0^\infty U(y) \,y^{-it}\, e\left(\frac{2\sqrt{mNy}}{p_1q \sqrt{p_2}}+\frac{3(N(y+w) n_1^2 n_2)^{1/3}}{p_1qr^{1/3}}\right) dy.
u_t+u\cdot\nabla u +\rho^{\delta-1}Lu = -\frac{2A\gamma}{\delta-1}(\rho^{\frac{\delta-1}{2}})^{\frac{2\gamma-\delta-1}{\delta-1}}\nabla \rho^{\frac{\delta-1}{2}}+\frac{2\delta}{\delta-1}\rho^{\frac{\delta-1}{2}}\nabla \rho^{\frac{\delta-1}{2}}\cdot \mathbb{S}(u).
Lv=-\frac{i(L_\mathrm{prin})_{p_\alpha}}{2\sqrt{\rho}}\,\frac{\partial(\sqrt{\rho}\,v)}{\partial x^\alpha}-\frac{i}{2\sqrt{\rho}}\,\frac{\partial((L_\mathrm{prin})_{p_\alpha}\sqrt{\rho}\,v)}{\partial x^\alpha}+L_\mathrm{sub}\,v\,.
\{j^a_0 (x) , \psi^b_\pm (y) \}= f^{abc}\psi_\pm^c \delta(x{-}y)\, , \qquad\{j^a_1 (x) , \psi^b_\pm (y) \}= \pm{1 \over 2}f^{abc}\psi_\pm^c \delta(x{-}y) \ .
\|S_{\theta}(u)\|_{\beta}\le C\theta^{(\beta-\alpha)_{+}}\|u\|_{\alpha}(\alpha,\beta\ge0) &\|S_{\theta}(u)-u\|_{\beta}\le C\theta^{\beta-\alpha}\|u\|_{\alpha}(\alpha>\beta\ge0) &\Big{\|}\frac{d}{d\theta}S_{\theta}(u)\Big{\|}_{\beta}\le C\theta^{\beta-\alpha}\|u\|_{\alpha}(\alpha,\beta\ge0),
M_{t}:= \exp\left(-\int_0^{t}V(W_s)dW_s - \frac{1}{2}\int_0^{t}V^2(W_s)ds\right) &= \exp\left( F(W_0) - F(W_t) + \frac{1}{2}\int_0^t (V'(W_s)-V^2(W_s))ds\right),
{\beta_{h}}( g ) = -{\frac{g^{3}}{1 6 \pi^{2}}}\left( 3 N_{c}- N_{f}( 1 - \gamma ) \right) \, , \qquad \gamma ( g ) = -{\frac{g^{2}}{8 \pi^{2}}}{\frac{N_{c}^{2}- 1}{N_{c}}}+ O ( g^{4}) ,
\mathbb{E}\big|N^{-H}\bigl(S^{\frac{\lambda}{N}}_{N}(t)-S^{\frac{\lambda}{N}}_{N}(s)\bigr) \big|^2 &\leq e^{2\lambda k}k!\int_{\mathbb{R}^k}^{'}\Biggl[\int_{0}^{t-s}\prod_{j=1}^{k}(y-z_j)^{d-1}_{+}e^{-\lambda(y-z_j)_{+}}\,dy\Biggr]^{2}\,dz_{1}\ldots dz_{k}
V:=H_{0}^1(\Omega),\H:=L^2(\Omega), L := I : H \to H, A_{1}:= - \Delta : D(A_{1}):=H^2(\Omega) \cap H_{0}^{1}(\Omega)\subset H \to H, B_{1}:= I : D(B_{1}):=H \to H, A_{2}:= - \Delta : D(A_{2}):=H^2(\Omega) \cap H_{0}^{1}(\Omega)\subset H \to H, B_{2}:= I : D(B_{2}):=H \to H
\boldsymbol{A}_{\rm par}=\frac{1}{2}\left[\begin{matrix}-2(y+z) (x+y) (x+z) (x+y) -2(x+z) (y+z) (x+z) (y+z) -2(x+y)\end{matrix}\right ] , \boldsymbol{B}_{\rm par}=\frac{1}{2}\left[\begin{matrix}0 p_y-p_x p_z-p_x -(p_y-p_x) 0 p_z-p_y -(p_z-p_x) -(p_z-p_y) 0\end{matrix}\right ] .
p= \min_{l=1, \ldots , N}\frac{1}{2\left(\sum_{m':m'>l}\mathbf{1}_{\{\{s^l,s^{m'}\}\in E \}}+\sum_{m':m'<l}\frac{\mu_k(s^{m'})}{\mu_k(s^l)}\mathbf{1}_{\{\{s^l,s^{m'}\}\in E \}}\right)}.
{\frac{\partial \widetilde{A}_{b}^{M i}({\bf x})}{\partial A_{c}^{M i}({\bf x})}}= \delta_{b c}+{\frac{\partial \, \delta A_{b}^{M i}({\bf x})}{\partial A_{c}^{M i}({\bf x})}}= \delta_{b c}- g f_{b d}^{c}\delta \Lambda_{d}^{M}({\bf x}) .
|\nabla \hat{f_{3}}(re^{i\theta})| \lesssim\int_{0}^{2\pi}\frac{|f_{3}(e^{it})|}{(1-r)^{2}+(\theta-t)^{2}}dt &\lesssim \int_{|t|>2|J|/3}|f(e^{it})-f_{J}|\frac{dt}{t^{2}}.
\begin{aligned}\underset{n \in{1, \ldots , Q}}{}\underset{\{\boldsymbol{\delta}_{k,q}\}_{q = 1}^Q}{}\| \mathbf{h}_{k,m}+ \boldsymbol{\delta}_{k,m}- \mathbf{h}_{k,n}- \boldsymbol{\delta}_{k,n}\|^2 & & n \neq m, & \|\boldsymbol{\delta}_{k,q}\|^2 \leq \gamma_{k,q}. \end{aligned}
F(T)=\frac{\Delta n^2}{\pi a}\left (\frac{23}{384}-\frac{\zeta (3)}{8\pi^2}t^3+\frac{7}{1080}t^4 -\frac{22}{14175}t^6+\frac{46}{55125}t^8+{\cal O}(t^{10})\right )
R^{\mu}{}_{\nu , \lambda \rho}= \partial_{\lambda}\Gamma^{\mu}{}_{\nu , \rho}- \partial_{\rho}\Gamma^{\mu}{}_{\nu , \lambda}+ \Gamma^{\mu}{}_{\sigma , \lambda}\Gamma^{\sigma}{}_{\nu , \rho}- \Gamma^{\mu}{}_{\sigma , \rho}\Gamma^{\sigma}{}_{\nu , \lambda},
c_\ell(p_\ell,q_\ell) = \sum_{T \in \mathcal{T}_\ell}\delta_T\, h_{T}^2 \,\langle \nabla p_\ell, \nabla q_\ell \rangle_T,g_\ell(q_\ell) = -\sum_{T \in \mathcal{T}_\ell}\delta_T\, h_{T}^2 \, \langle f, \nabla q_\ell \rangle_T,
r_n(j) := Corr\left( X_{\frac{j+1}{n}}- X_{\frac{j}{n}},X_{\frac{1}{n}}- X_{0}\right) &= \frac{\gamma_2((j+1)/n;X) -2 \gamma_2(j/n;X) + \gamma_2((j-1)/n;X)}{2\gamma_2(1/n;X)}&\rightarrow \frac{1}{2}\left( |j+1|^{2\alpha+1}-2 |j|^{2\alpha+1}+|j-1|^{2\alpha+1}\right), n\rightarrow \infty,
[\begin{matrix}U_{1}&U_{2}\end{matrix}][\begin{matrix}[\begin{matrix}D^{\frac{1}{2}}&0\\ 0&0\end{matrix}]\\ 0\end{matrix}][\begin{matrix}V_{1}&V_{2}\end{matrix}]^{*}=[\begin{matrix}U_{1}&U_{2}\end{matrix}][\begin{matrix}D^{\frac{1}{2}}V_{1}^{*}\\ 0\end{matrix}]=U_{1}D^{\frac{1}{2}}V_{1}^{*}=M,
{\rm Tutte}(K_{2,2,2},x,y)= x(11,25 ,20,7,1)_{x}+ y (11,46,39,8)_{x}+y^2(32,52,12)_{x}\hphantom{{\rm Tutte}(K_{2,2,2},x,y)=}{}+y^3(40,24)_{x}+ y^4(29,6)_{x}+15y^{5}+5 y^{6}+y^{7}.
\sum_{j = 0}^k{\sum_{s = 0}^j{\binom kj\binom js\left({\frac{{f_3}}{{f_2}}}\right)^j\left({\frac{{f_1}}{{f_3}}}\right)^s X_{m - bk + (b - c)j + (c - a)s}}}= \frac{{X_m}}{{f_2{}^k}}\,,
\int_{(r^n)^{-1}(v)}\xi (e_1 \ldots e_n) \, d\lambda_v^n (e_1 \ldots e_n) &:= \int_{r^{-1}(s(e_2))}\ldots \int_{r^{-1}(s(e_n))}\int_{r^{-1}(v)}\xi (e_1 \ldots e_n) \d\lambda_{s(e_2)}(e_1) \ldots d\lambda_{s(e_n)}(e_{n-1}) d \lambda_{v}(e_n).
F(b) =0, \int_a^b F(x)dx =0, \int_a^b \int_a^{x_{k-1}}\ldots \int_a^{x_1}F(t)dtdx_1\ldots dx_{k-1}=0, for\ \k=2,\ldots ,n, (-1)^{n+1}\int_a^x \int_a^{x_{n-1}}\ldots \int_a^{x_1}F(t)dtdx_1\ldots dx_{n-1}\geqslant 0,for \any \x\in(a,b).
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}
\langle \mu + \rho, \epsilon_1-\epsilon_{j_1-1}\rangle = \langle \mu + \rho, \epsilon_1-\epsilon_{i_1}\rangle + \langle \mu + \rho, \epsilon_{i_1}-\epsilon_{i_1+1}\rangle + \langle \mu + \rho, \epsilon_{i_1+1}-\epsilon_{j_1-1}\rangle &= (i_1-1)a + \left( p-\frac{j_1-i_1-1}{n}\right) + \frac{j_1-i_1 -2}{n}&= (i_1-1)a + p - \frac{1}{n}< p
\frak F ([V \xrightarrow h X;E]) \bullet \frak F([W \xrightarrow k Y;F] ) = [X \xleftarrow h V \xrightarrow{f \circ h}Y; E] \bullet [Y \xleftarrow k W \xrightarrow{g \circ k}Z; F] & = [X \xleftarrow{h \circ{k}''}V' \xrightarrow{(g \circ k) \circ (f' \circ f')}Z;{{k}''}^*E \oplus (f' \circ{h}')^*F ]
\sum_{j=f(n)}^{f(n+1)}\lambda_j =\sum_{j=f(n)}^{f(n+1)}j^{-p}\le\int_{f(n)-1}^{f(n+1)-1}j^{-p}dj =\frac1{1-p}\left(\left\lceil(n+1)^{d/(1-p)}-1\right\rceil^{1-p}-\left\lceil n^{d/(1-p)}-1\right\rceil^{1-p}\right).
\left[ x \prod_{a=1}^{n_{c}}(x-\phi_{a}^{2}) + 2\Lambda m_{1}\cdots m_{n_{f}}\right]^{2}- 4\Lambda^{2}\prod_{i=1}^{n_{f}}(x+m_{i}^{2}) = -4\Lambda^{2}x \prod_{a=1}^{n_{c}}(x-\alpha_{a})^{2}.
S^2(M) = \sum_{x=0}^{\infty}Z_x \le \left( 1 +2\sum_{k=1}^{\infty}\dfrac{1}{(\delta_{2k})^2}\right) \cdot \left( \sum_{x=0}^{\infty}\dfrac{1}{f(0,x)^2}\right)=:\lambda <+\infty
\max \, \left\{\mathbf{x}^1 (\mathbf{R}- \nu \mathbf{1})\colon \begin{bmatrix}\mathbf{x}^{0}\mathbf{x}^{1}\end{bmatrix}\begin{bmatrix}(1-\beta) \mathbf{I}\mathbf{I}- \beta \mathbf{P}\end{bmatrix}= \mathbf{e}_i, \begin{bmatrix}\mathbf{x}^{0}\mathbf{x}^{1}\end{bmatrix}\geq \mathbf{0}\right\}
\|S_\eta^j(A)\| = \sum_{\sigma \in S_\eta^j(A)}w(\sigma) \leq \frac{\eta^{-2^{k-j-1}}}{j+2}\cdot \sum_{\sigma \in S_\eta^j(A)}\|I^\sigma(I_\sigma(S_\eta^{j+1}(A)))\|= \eta^{-2^{k-j-1}}\cdot \|(S_\eta^{j+1}(A))\|
\ll \frac{T_k\phi(a_m W B)^2(\log{R})(\log\log{R})^2}{k^2 a_m^2B^2W^2}\sum_{(A,WB)=1}\log{A}\frac{\omega(A)}{\phi_\omega(A)^2}\sum_{\substack{\mathbf{r}\in\mathcal{D}'_k r_m=1 A|r}}\frac{Y_{\mathbf{r}}^2\phi(r/A)}{\phi_\omega(r/A)^2}.
\min \sum_{p \in P_{CVRP}}\tilde{c}_{p}\lambda_{p}, \sum_{p \in P_{CVRP}}a_{pi}\lambda_{p}= 1, \forall i \in I \setminus \{0 \}, & \sum_{p \in P_{CVRP}}\lambda_{p}= |K|, & \lambda_{p}\in \{0,1\}, \forall p \in P_{CVRP}.
\mathcal{U}^{t \, *}=\mathcal{U}^{-t}\ , \ \ \mathcal{U}^t\circ\mathcal{U}^{t'}=\mathcal{U}^{t+t'}\ , \ \ \mathcal{U}^0={\rm Id}_{\mathfrak{F}^2}\ , \ \t,t'\in\mathbb{R}\ ,
{\mathbb E}\left\{{\left|{\left[{{{\bf{\Xi}}_{{\rm{BF}}}}}\right]_{n,n}^*{{\left[{{{\bf{\Xi}}_{{\rm{BF}}}}}\right]}_{m,m}}}\right|}\right\}= \frac{\pi}{4}\frac{{{N_{\rm{T}}}{N_{\rm{R}}}\kappa}}{{{L_{\rm{R}}}\left({\kappa + 1}\right)}}{N_{{\rm{S}},n}}{N_{{\rm{S}},m}}{\rho_n}{\rho_m}.
\sum_{n=1}^\infty \frac{1}{\left( 2n \atop n \right)}\frac{1}{n^c}\prod_{a,b,i,j}\left[ \sum_{m=1}^{n-1}\frac{1}{m^a}\right]^i\left[ \sum_{k=1}^{2n-1}\frac{1}{k^b}\right]^j
{\rm Tr}e^{iF_{5i}}- 2{\rm Tr}_A e^{iF_{5i}}= m_i - 2{\rm tr}F_{5i}^2 + 2 \Big({\rm tr}F_{5i}\Big)^2 +{2 \over 3}{\rm tr}F_{5i}^4 -{2\over 3}{\rm tr}F_{5i}{\rm tr}F_{5i}^3 + \dots
\big|\big((z-B')^{-1}f_0\big)(\xi_r)\big| =\big|f_0\big((z-B)^{-1}\xi_r\big)\big|=\big|\frac{1}{z}+\frac{z^{m-1}}{r^{m}}\big(e^{\frac{r}{z}}-\sum\limits_{n=0}^{m}\frac{1}{n!}\frac{r^n}{z^n}\big)\big|.
\max_{\{p_{k_j}\}}\;\; \mathcal{U}_B=\sum_{k=1}^K \sum_{j=1}^J x_{k_j}p_{k_j}\mbox{s.t.}\;\; \sum_k x_{k_{j(p_{k_j})}}\leq L_j, \; j=1, 2, \cdots, J.
(C_{\alpha,\frac{\theta}{\alpha}}\boxplus C_\theta) (A_{\bar{\varphi}}^{\alpha,\theta}\oplus A_{\bar{\varphi}}^{\frac{\theta}{\alpha},\theta}\bar{\alpha})(f_1\oplus \alpha f_2) &= C_{\alpha}(P_{\alpha}(\bar{\varphi}f_1))+\alpha C_{\frac{\theta}{\alpha}}(P_{\frac{\theta}{\alpha}}(\bar\alpha\bar{\varphi}f_1))+C_\theta (P_\theta(\bar\varphi f_2)).
\int_{m}^{\infty}d k \ ( k^{2}- m^{2})^{1 - s}\partial_{k}\frac{t^{j}}{l^{n}}= - m^{2 - 2 s}\frac{\Gamma ( 2 - s ) \Gamma \left( s + \frac j 2 - 1 \right) l^{j - n}}{\Gamma \left( \frac j 2 \right) ( R m )^{j}\left( 1 + \left( \frac{l}{m R}\right)^{2}\right)^{s + \frac j 2 - 1}}\ ,
\limsup_a a^{-2}b^{-d+1}|\mathcal{F}(g_a)(|\xi|)|^2 = 4 |\xi|^{-d+1}R^{d-1}\alpha_f^2 >0, &\liminf_a a^{-2}b^{-d+1}|\mathcal{F}(g_a)(|\xi|)|^2 =0.
T^{(2)}_{m^{\varepsilon},(lh,\mathbb{I})}(f_{1},f_{2})(x):=\sum_{\widetilde{Q'}\in\widetilde{\mathbb{Q}'}, \, Q''\in\mathbb{Q}''_{\mathbb{I}}}\int_{\mathbb{R}^{4}}m^{\varepsilon}(\xi)\phi_{\widetilde{Q'}}(\bar{\xi}_{1})\phi_{Q''}(\bar{\xi}_{2})\widehat{f_{1}}(\xi_{1})\widehat{f_{2}}(\xi_{2}) e^{2\pi ix\cdot(\xi_{1}+\xi_{2})}d\xi
\delta{\cal E}= \delta{\cal E}_{0}- \frac{3 \hbar^{2}}{4 m}\int d^{3}\vec{r}\varphi^{\dagger}[ ( \vec{\sigma}\cdot \vec{\epsilon}) ( \vec{\sigma}\cdot \vec{\nabla}) + ( \vec{\sigma}\cdot \vec{\nabla}) ( \vec{\sigma}\cdot \vec{\epsilon}) ] \varphi{,}
\zeta_{AV,2}^{[2]}(s_1,s_2,s_3) \ll \sum_{k=3}^\infty\begin{cases}k^{-2\sigma_1-\sigma_3}(\sigma_2>1) k^{-2\sigma_1-\sigma_3}(\log k)^2 (\sigma_2=1) k^{2-2\sigma_1-2\sigma_2-\sigma_3}(\sigma_2<1), \end{cases}
{H}_{i n t}^{1}= \bar{\psi}e \left( \gamma^{-}\bar{B}_{-}+ \gamma^{i}\bar{B}_{i}\right) \ast \left[ 1 + \frac{e}{2}\gamma^{+}\frac{1}{i \partial_{-}- e \bar{B}_{-}}\left( \gamma^{-}\bar{B}_{-}+ \gamma^{j}\bar{B}_{j}\right) \right] \ast \psi .
f_{\boldsymbol{X}^{\ast\ast}}(\boldsymbol{x})=\frac{c_{n}^{\ast\ast}}{\sqrt{|\boldsymbol{\Sigma}|}}\overline{\mathcal{G}}_{n}\left\{\frac{1}{2}(\boldsymbol{x}-\boldsymbol{\mu})^{T}\mathbf{\Sigma}^{-1}(\boldsymbol{x}-\boldsymbol{\mu})\right\},~\boldsymbol{x}\in\mathbb{R}^{n}.
{D}^{(1/2)}({\bf p},{\bf n})=\frac{pn+m-i{\vec \sigma}\cdot({\bf p}{\times}{\bf n})}{\sqrt{2(p_0+m)(pn)}},\quad{D}^{\dag(1/2)}({\bf p},{\bf n})\,{D}^{(1/2)}({\bf p},{\bf n})=1,
r_{\theta}(t) = \begin{cases}0 b=0, b^{p\theta}t^{p-1}+\sum_{m=1}^{p-1}\binom{p-1}{m}\left( a^{p\theta-m}b^m t^{p\theta+p-1-m}+ a^{-m}b^{p\theta+m}t^{p-1-m}\right). \end{cases}
\langle(C_{\varphi}^{*}- M_{g}C_{\sigma}M^{*}_{h})K_{w},K_z\rangle = -\big(\overline{\varphi(w)}z-1\big)^{N}\Big(\log(\mu(z))+\log(\overline{\eta(w)})\Big) &+ Q(\overline{\varphi(w)}z) - g(z)Q(\bar{w}\sigma(z))\overline{h(w)}.
a = -\frac{\det(Y,Z_1)}{\det(Y,Z_2)}= \frac{-u z_{1,1}- v z_{1, 2}+ z_{1,3}}{u z_{2, 1}+ v z_{2, 2}- z_{2, 3}},b = -\frac{\det(Y,Z_3)}{\det(Y,Z_4)}= \frac{-u z_{3,1}- v z_{3, 2}+ z_{3,3}}{u z_{4, 1}+ v z_{4, 2}- z_{4, 3}}.
\tilde{\cal T}_\mu=\epsilon_{\mu\nu\lambda}p^\nu\gamma^\lambda\otimes\sigma_1=\frac{\sqrt{2}}{m}\epsilon_{\mu\nu\lambda}p^\nu\widehat{\Xi}{}^\lambda,\quad\tilde{\cal K}_\mu=\epsilon_{\mu\nu\lambda}p^\nu\gamma^\lambda\otimes\sigma_2=2i\tilde{\cal T}_\mu\widehat{N},
{\textsl{\footnotesize R}}_j = \textstyle{\left[\sin\left(\frac{2\pi j}{n}\right), \frac{1}{2}\sin\left(\frac{4\pi j}{n}\right), \ldots, \frac{1}{n-1}\sin\left(\frac{2(n-1)\pi j}{n}\right), 0\right]^{\tt T},}
(\sum_{n=0}^{\infty}a_{n}x^{n})(\sum_{n=0}^{\infty}b_{n}x^{n})=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}a_{i}b_{j}x^{i+j}=\sum_{n=0}^{\infty}(\sum_{i=0}^{n}a_{i}b_{n-i})x^{n}=\sum_{n=0}^{\infty}c_{n}x^{n}.
T_j (s>0)=\left(\begin{array}{cc}-i\rho_j/2s+O(1) s^{(k-3)/2}p_j+O(s^{(k-1)/2}) -s^{(k-3)/2}\bar{p}_j^T+O(s^{(k-1)/2}) T_j(0-)+O(s) \end{array}\right) ,
\log |\hat f_L(\lambda)|^2 \leq \frac{1}{\mu (E(\lambda,s))}\int_{E(\lambda,s)}\log | f_L(\xi)|^2 d\mu(\xi) +L\log(1-|z|^2) =\frac{1}{\mu (E(\lambda,s))}\int_{E(\lambda,s)}\log |\hat f_L(\xi)|^2 d\mu(\xi)+ + L\left[\log(1-|\lambda|^2) - \frac{1}{\mu (E(\lambda,s))}\int_{E(\lambda,s)}\log (1-|\xi|^2) d\mu(\xi)\right]\ .
1/\underline{\alpha_2}= \max \left( 1/\alpha_{02}, \dfrac{M_g}{3|2c_2 -1|\varepsilon_2}\right) &\leq \varepsilon_2^{-1}\max\left( 1/\alpha_{02}, \dfrac{M_g}{3|2c_2 -1|}\right) &\leq b_3 \beta \varepsilon_2^{-1}.
M_1(R,T) := \int_{T^{\beta}}^{T\log{T}}\left( \int_{-(\log{t})^2}^{(\log{t})^2}K(u)du \right)|R(t)|^{2}\Phi\left( \frac{t}{T}\right)dt, M_2(R,T) := \int_{T^{\beta}}^{T\log{T}}\left( \int_{-(\log{t})^2}^{(\log{t})^2}e^{-i\theta}\log\zeta(\sigma+i(t+u)) K(u)du \right)|R(t)|^{2}\Phi\left( \frac{t}{T}\right)I(\sigma,t)dt,
{\cal Z}_{\mathrm{G - L}}^{\mathrm{3 D}}= \int{\cal D}{\bf h}{\cal D}\Phi{\cal D}\Phi^{*}\exp \left\{- \int d^{3}x \left[ \frac{1}{4 \eta^{2}}H_{\mu \nu}^{2}+ \frac{q^{2}}{2}{\bf h}^{2}+ \left| \left( \partial_{\mu}+ 2 \pi i h_{\mu}\right) \Phi \right|^{2}+ m_{H}^{2}\left| \Phi \right|^{2}+ \lambda \left| \Phi \right|^{4}\right] \right\}.
(\begin{matrix}r\\r^{\prime}\\ \end{matrix})_{4}=(\begin{matrix}r\\r^{\prime}\\ \end{matrix})_{1}=(M_{1}\cdot M_{2})_{4}\cdot(M_{1}\cdot M_{2})_{3}\cdot(M_{1}\cdot M_{2})_{2}\cdot(M_{1}\cdot M_{2})_{1}\cdot(\begin{matrix}r\\r^{\prime}\\ \end{matrix})_{1}
M_1(\rho) = 1+\rho^2 M_2(\rho) = 3+8\rho^2+3\rho^4 M_3(\rho) = 12 + 54 \rho^2 + 54 \rho^4 + 12 \rho^6 M_4(\rho) = 55 + 352\rho^2 + 616 \rho^4 +352 \rho^6 + 55 \rho^8
|(\tilde\Delta - \lambda_s)v|_2^2 = |(\tilde\Delta - c)v|_2^2 - i\cdot d\langle(\tilde\Delta -c)v,v\rangle_2 + i\cdot d\langle v, (\tilde\Delta -c)\rangle_2 + d^2|v|^2{\geq}d^2|v|^2.
|\psi|^2_{\Omega^p(M;E)}= \frac{1}{p!}\langle\psi_{i_1\ldots i_p}, \psi^{i_1\ldots i_p}\rangle = \frac{1}{p!}g^{i_1j_1}\cdots g^{i_pj_p}\langle \psi_{i_1\ldots i_p}, \psi_{j_1\ldots j_p}\rangle.
B^{m}_{i}(t) = \dfrac{1}{h^{m}}\sum^{m+1}_{j=0}\begin{pmatrix}m+1 j \end{pmatrix}(-1)^{m+1-j}(x_{i-2+j}-t)^{m}_{+}=\dfrac{1}{h^{m}}\nabla^{m+1}(x_{i-2}-t)^{m}_{+}
b_{\sigma_{0}^{( 1 )}}= \left({\frac{d \rho_{0}}{d \sigma_{0}^{( 1 )}}}\right) b_{\rho_{0}}= - \left({\frac{d \rho_{0}}{d \sigma_{0}^{( 2 )}}}\right) b_{\rho_{0}}= - b_{\sigma_{0}^{( 2 )}},
\frac{n^{4d}}{m^4}\sum_{l = 2}^m \mathbb{E}_0\left[(D^S_{nl})^4 \middle| \mathcal{F}_{n, l- 1}\right] , \frac{n^{4d}}{m^4}\sum_{l = 2}^m \mathbb{E}_0\left[(D^T_{nl})^4 \middle| \mathcal{F}_{n, l- 1}\right] \underset{p}{\longrightarrow}0 \quad
\begin{aligned}a\int \left<\nabla u_q, \nabla v\right> = \int R' (1+u_q)^{q-2}\chi^2 w^2 -\int R \chi^2 w^2 &\le \left|\int R \chi^2 w^2\right| &\le \epsilon\|\nabla(\chi w)\|_{2}^2 + C_\epsilon \|\chi w\|_{2}^2.\end{aligned}
| \nu_{e}( t ) \rangle \equiv e^{- i H t}| \nu_{e}( 0 ) \rangle = e^{- i \omega_{1}t}\left( \cos \theta \; | \nu_{1}\rangle \; + \; e^{- i ( \omega_{2}- \omega_{1}) t}\; \sin \theta \; | \nu_{2}\rangle \; \right) ,
c\delta(u)W_3(u,-\mu,z)=zI_{-\mu}(uz)\sum_{s=0}^{N-1}\frac{A_s(z)}{u^{2s}}+\frac{z}{u}I_{-\mu-1}(uz)\sum_{s=0}^{N-1}\frac{B_s(z)}{u^{2s}}+f(u,z),
\sum\limits_{i = 1}^n{F^{ii}}= \frac{\sigma_k}{\sigma_l}\frac{{(n - k + 1)\sigma_{k - 1}\sigma_l - (n - l + 1)\sigma_k \sigma_{l - 1}}}{{\sigma_l^2}}\ge c(n,k)\big(\frac{\sigma_k(D^2u)}{\sigma_l(D^2u)}\big)^{\frac{-1}{k-l}}= c({n,k})f^{\frac{-1}{k-l}}e^{\frac{-1}{k-l}u_t}\ge c_9 >0.
c_{a b}= \left( \begin{array}{c c c c c}{0}&{- \omega_{1}}&{}&{}&{}\\{\omega_{1}}&{0}&{}&{}&{}\\{}&{}&{\ddots}&{}&{}\\{}&{}&{}&{0}&{- \omega_{\frac{p + 1}{2}}}\\{}&{}&{}&{\omega_{\frac{p + 1}{2}}}&{0}\\ \end{array}\right) .
\lambda^{\prime}(t) \mathcal{E}(t) + \mu \alpha^{\prime}(t) \mathcal{E}^r(t)\langle P(u_t), u\rangle=\frac{\lambda^{\prime}(t)}{\lambda(t)}\mathcal{H}(t) + \mu\bigl(\frac{\alpha(t)}{\lambda(t)}\bigr)^{\prime}\lambda(t) \mathcal{E}^r(t)\langle P(u_t), u\rangle.
g_{t}'(t,x) :=\frac{\partial}{\partial t}g(t,x)=ce^{ct}V(x), g_{x}'(t,x) :=\frac{\partial}{\partial x}g(t,x)=\begin{cases}\kappa e^{ct}x^{\kappa-1}, x>1, e^{ct}V'(x), x\in[0,1],\end{cases}g_{x}''(t,x) :=\frac{\partial^{2}}{\partial x^{2}}g(t,x)=\begin{cases}\kappa(\kappa-1)e^{ct}x^{\kappa-2}, x>1, e^{ct}V''(x), x\in[0,1].\end{cases}
\mathcal{E}(f)=\sum_k\sum_l \mathcal{E}(f_{k,n},f_{l,n}) =\sum_k\left(\sum_{l\leq k-1}\mathcal{E}(f_{k,n},f_{l,n})+\sum_{l\geq k+1}\mathcal{E}(f_{k,n},f_{l,n})\right)+\sum_k\mathcal{E}(f_{k,n}) &\geq \sum_k\left(\sum_{l\leq k-1}2^{-n}K(f_{l,n})+\sum_{l\geq k+1}2^{-n}K(f_{k,n})\right) &=\sum_k\sum_{k\neq l}K(f_{k,n}f_{l,n}).
\sum_{k=0}^{\infty}\sum_{\substack{{r_1,r_2=1}r_1\ne r_2}}^{n}\frac{S_{r_1r_1r_2}}{(r_1+k+1)^2(r_2-r_1)}=\sum_{\substack{{r_1,r_2=1}r_1\ne r_2}}^{n}\frac{S_{r_1r_1r_2}}{r_2-r_1}\left(\zeta(2) -H^{(2)}_{r_1}\right).
U({\mathcal F})=\underline{\mathrm{Hom}}(A^*,{\mathcal G}[p^N])\setminus\left( \bigcup_{A'\in{\mathcal F}}\bigcap_{\rho\in(A/A')^*}\underline{\mathrm{Hom}}(\mathrm{ker}(\rho)^*,{\mathcal G}[p^N]) \right).
||f||_{\dot{H}^{1}_{e}}^{2}= \int_{0}^{\infty}\left((\partial_{r}f)^{2}+\frac{f^{2}}{r^{2}}\right) r dr \leq C\left(||Lf||_{L^{2}(r dr)}^{2}+||f||_{L^{2}(r dr)}^{2}\right)
\Lambda_{\b}^{a}={\frac{1}{2}}\left( \begin{array}{c c c c}{{0}}&{{1}}&{{0}}&{{0}}\\{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{i}}\\{{0}}&{{0}}&{{- i}}&{{0}}\\ \end{array}\right) .
\lim_{n=1}B_{m, n-2}= \lim_{n=1}\frac{(n+m-2)!}{m!\,(n-2)!}\,, = \lim_{n=1}(n-1) \frac{(n+m-2)!}{m!\,(n-1)!}\,, \lim_{n=1}B_{m, n-2}= 0\,.
c_{0}= e^{i \phi_{0}}\left[ 1 + a_{- 1}a_{1}+ \frac{1}{2 \sqrt{2}}a_{- 1}^{2}a_{2}+ \frac{1}{2 \sqrt{2}}a_{- 2}a_{1}^{2}+ \frac{1}{2}a_{- 2}a_{2}+ \cdots \right] .
\int_{\Lambda_L^*}| \hat{B}_\#( -u_{s}) | \left|\nu\left( u_{s}+ u_0 + \sum_{l \in \{1,...,s-1 \}\cap I_A}\sigma_{A , s}(l) u_l \right) - E - i \eta \right|^{-1}d u_{s}&=\int_{\Lambda_L^*}| \tilde{B}(v) | \left|\nu\left(v \right) - E - i \eta \right|^{-1}d v &\leq C(E,d,L,\eta,\tilde{B}) ={C}(E,d,L,\eta,\hat{B}_\#) ,
{}_1F^2_0\left(a;X,Y\right) = \int_{\mathcal{S}(r)}\frac{1}{\left(1-x q_r' Y q_r\right)^a}({\rm d}q_r) = \int_0^\infty t^{a-1}e^{-t}\int_{\mathcal{S}(r)}e^{txq_r' Y q_r}({\rm d}q_r) \;{\rm d}t.
L=-\int_{t_{0}}^{t_{1}}\iint\{\int_{-h(x)}^{\eta(x,t)}\rho[\frac{\partial\Phi}{\partial t}+\frac{1}{2}|\nabla\Phi|^{2}+\frac{1}{2}(\frac{\partial\Phi}{\partial z})^{2}]dz+\frac{1}{2}\rho g\eta^{2}\}dxdt.
\mathcal{P}_{n}^{\left( +\right)}\left( E\right) =\sqrt{\frac{b_{0}\mathcal{P}_{1}\left( 0\right)}{b_{n}\mathcal{P}_{n}\left( 0\right) \mathcal{P}_{n+1}\left( 0\right)}}\mathcal{K}_{n}\left( E,0\right)
\mu(A) - \nu(A) = \hat\mu(A \times \Omega) - \hat\mu(\Omega \times A) &= \hat\mu \bigl( (A \times \Omega) \cap \Delta^C \bigr) - \hat\mu \bigl( (\Omega \times A) \cap \Delta^C \bigr) &\le \hat \mu \bigl( (\Omega \times \Omega) \cap \Delta^C \bigr) \le \hat\mu(\Delta^C)
\alpha_{j}^p(x,z)= \sup_{\ell \in \big(\mathbb L^{0,p}_{k}(x)\big)^c}x_{k+1}^{\ell,p,j_{\ell}-}(x,z) \textrm{and}\beta_{j}^p(x,z)= \inf_{\ell \in \big(\mathbb L^{0,p}_{k}(x)\big)^c}x_{k+1}^{\ell,p,j_{\ell}+}(x,z).
|\mathcal W|\le F(n-1,3,0)+6, \ \ \ \ \ \ \ \ &|\mathcal W|\le 8+6 = 14\ \ \ \ \ \ \ \ \n = 5, &|\mathcal W|\le 14+6 <21\ \ \ \ \ \n = 6,\ &|\mathcal W|\le{n-1\choose 3}+6 <{n\choose 3}\ \ \ \ \ \ \ \n \ge 7.
\begin{cases}\ddot{x_1}+\left(\alpha x_1^2 + \beta \dot{x}_1^2 -\gamma \right) \dot{x}_1 + \omega_1^2 x_1 = I(x_1,x_2) \ddot{x_2}+\left(\alpha x_2^2 + \beta \dot{x}_2^2 -\gamma \right) \dot{x}_2 + \omega_2^2 x_2 = I(x_2,x_1) \end{cases}
P\Bigl(\bigcap_{i=0}^\infty G_i\Bigr) =\lim_{m\to\infty}P\Bigl(\bigcap_{i=0}^m G_i\Bigr)=\lim_{m\to\infty}\prod_{i=0}^m\bigl(1-P\bigl(G_i^c|G_0\cap\ldots \cap G_{i-1}\bigr)\bigr) &\geq\lim_{m\to\infty}\bigl(1-\sum_{i=0}^m P\bigl(G_i^c|G_0\cap\ldots \cap G_{i-1}\bigr)\bigr) &=1-\sum_{i=0}^\infty P\bigl(G_i^c|G_0\cap\ldots \cap G_{i-1}\bigr)\geq 1-g(k)\,,
\begin{aligned}f_c(x_i^{a-c,(1)}(k),\varrho(k))=f_c(x_i^{a-c,(2)}(k),\varrho(k)), &f_c(y_i^{a-c,(1)}(k),\varrho(k))=f_c(y_i^{a-c,(2)}(k),\varrho(k)),\end{aligned}
\lim_{m\rightarrow\infty}(\mu^{(m)})^{-2}\int_{B_{\frac58}\setminus B_{\frac{\tau}{2}}}\rho_{(\xi_*^{(m)}(t_0),t_0)}{\rm dist}\,(\cdot,J_{\xi_*^{(m)}(t_0)})^2\, d\|V_t^{(m)}\|= \sum_{j=1}^3 \int_{\frac{\tau}{2}}^{\frac58}\rho_{(0,t_0)}|\tilde f_j -\tilde\xi_j^{\perp}(t_0)|^2\, dx.
\Vert u(t)\Vert_{\infty}\leq \begin{cases}K \, t^{-\frac{N}{2q_0+N(m_2-1)}}\left( t + \| u_0 \|_{q_0}^{1-m_2}\right)^{-\frac{2q_0}{(m_2-1)[2q_0 + N(m_2-1)]}}\forall t \in \left( 0, 1 \right) , K \, t^{-\frac{1}{m_1-1}}\forall t >1 \, ,\end{cases}
H^{\prime}\left( \begin{array}{c}{\psi_{+}}\\{\psi_{-}}\\ \end{array}\right) = \left( \begin{array}{c}{P_{-}P_{+}+ p_{3}^{2}\qquad 0}\\{0 \qquad P_{+}P_{-}+ p_{3}^{2}}\\ \end{array}\right) \left( \begin{array}{c}{\psi_{+}}\\{\psi_{-}}\\ \end{array}\right) .
\left\vert \int(\psi\circ R_{k})\,\left( u-\frac{1}{M}{\textstyle\sum_{j=0}^{M-1}}\widehat{T}^{j}u\right) \,d\mu\right\vert <\frac{\varepsilon}{3}\text{for}k\geq k_{0}u\in\mathcal{U}
H(\hat{P}_B) = -\frac{(S-1)a}{n+Sa}\ln\left(\frac{a}{n+Sa}\right) -\frac{n+a}{n+Sa}\ln\left(\frac{n+a}{n+Sa}\right) \ge \frac{(S-1)a}{n+Sa}\ln\left(\frac{n+Sa}{a}\right),
\Psi_{z}:=\sum_{\sigma \in \mathfrak{S}_{k}}\prod_{1\le i<j \le k}\frac{qz_{\sigma(i)}-z_{\sigma(j)}}{z_{\sigma(i)}-z_{\sigma(j)}}\prod_{i=1}^{k}\left(\frac{1-\nu z_{\sigma(i)}}{1-z_{\sigma(j)}}\right)^{\epsilon_{i}}
L(P) = - \int_0^{\frac{\pi}{2}}P'(x) \left[\int_0^{x}\log\left( \tan u \right)\d u \right] \d x = \sum_{n = 0}^{\infty}\frac{1}{(2n+1)^2}\int_0^{\frac{\pi}{2}}P'(x) \sin\left( 2(2n+1)x\right) \d x;
I_k = B\Big(\frac12, \frac{k-1}{2}\Big) = \sqrt{\pi}\frac{\Gamma\Big(\frac{k-1}{2}\Big)}{\Gamma\Big(\frac{k}{2}\Big)}= \frac{4 \pi}{2^{k}}\frac{\Gamma(k)}{\Gamma(k/2)^{2}}.
\frac{1}{|\xi|}\partial_t\hat{H}\cdotp \frac{\bar{\hat{H}}}{|\hat{H}|}+\mu |\xi||\hat{H}|=\frac{\hat{B}}{|\xi|}\cdot \frac{\bar{\hat{H}}}{|\hat{H}|}+\frac{\hat{G}}{|\xi|}\cdot \frac{\bar{\hat{H}}}{|\hat{H}|},
Te_{n}=\sum_{k=1}^{n}T_{k}=\sum_{k=1}^{n}\frac{k(k+1)}{2}=\sum_{k=1}^{n}(\sum_{i=1}^{k}i)=\frac{n(n+1)(n+2)}{6}=\frac{n^{\overline{3}}}{3!}
\frac{|P+\nabla \widetilde{u}_0 (x )+\nabla_y\widetilde{u}_1 (x,\frac{x}{\epsilon}) |^2}{2}+V\Big (x,\frac{x}{\epsilon}\Big)=\ln \widetilde{m}_0 (x )+\ln \widetilde{m}_1 \Big(x,\frac{x}{\epsilon}\Big)+\overline{H}.