harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{k=1}^4\log\frac{1-|h(\omega_k)|^2}{|h'(\omega_k)|}+\sum_{k=1}^4\sum_{{\substack{l=1 l\neq k}}}^4\delta_k\delta_l\log\left|\frac{1-\overline{h(\omega_k)}h(\omega_l)}{h(\omega_k)-h(\omega_l)}\right|\le 2\log \frac{4\rho^2}{1-\rho^2}.
\mathcal P^6_{\geq 3}=_{\mathsf{def}}\, \bar{\mathcal P}^6_{(3111)}- \bar{\mathcal P}^6_{(1311)}+ \bar{\mathcal P}^6_{(1131)}- \bar{\mathcal P}^6_{(1113)}+ \bar{\mathcal P}^6_{(31)}- \bar{\mathcal P}^6_{(13)}
W^C_{106}= 1+(35245 + 2 d )y^{18}+(416262 + 128 c - 2 d )y^{20}&+(6586310 + 8192 b - 896 c - 34 d )y^{22}&+(86626645 + 524288 a - 106496 b + 1024 c + 34 d )y^{24}+ \cdots, W^S_{106}= a y^5+(- 24 a - b )y^9+( 276 a + 22 b + c )y^{13}&+(- 2024 a - 231 b - 20 c - d )y^{17}+ \cdots,
\sum_{i=0}^{n-1}u(i) =\int_0^n u(x) dx+\frac{1}{2}\big(u(0)-u(n)\big)+\sum_{j=1}^{M}\frac{B_{2j}}{(2j)!}\big(u^{(2j-1)}(n)-u^{(2j-1)}(0)\big) &+\frac{1}{(2M+1)!}\int_0^n B_{2M+1}(x-[x]) u^{(2M+1)}(x)dx.
B_{8}= \left( \begin{array}{c c}{P_{1}}&{0}\\{0}&{0}\\ \end{array}\right) , B_{9}= \left( \begin{array}{c c}{Q_{1}}&{0}\\{0}&{0}\\ \end{array}\right) , B_{7}= \left( \begin{array}{c c}{0}&{0}\\{0}&{b I}\\ \end{array}\right) .
\hat f(t,x,\omega)~=~\begin{cases}f(t,x_1,\omega) \hbox{if}~~x<x_1\,, f(t,x,\omega) \hbox{if}~~x\in[x_1, x_2]\,, f(t,x_2,\omega) \hbox{if}~~x>x_2\,,\end{cases}
K = \begin{bmatrix}0 0 0 K_{12,1}0 K_{1,22}^- K_{1,23}^- K_{12,2}0 K_{1,23}^{-\top}K_{1,33}^+ K_{12,3}K_{12,1}^{\top}K_{12,2}^{\top}K_{12,3}^{\top}K_2\end{bmatrix}\,.
{\rm Tr}(e^{-(nV)^{-\alpha}})) \leq{\rm Tr}(e^{-(nW)^{-\alpha}}))=\sum_{k=1}^{\infty}\exp(-(n\|V\|_{1,\infty})^{-\alpha}k^{\alpha}) &\leq\int_0^{\infty}\exp(-(n\|V\|_{1,\infty})^{-\alpha}s^{\alpha})ds=n\|V\|_{1,\infty}\int_0^{\infty}\exp(-u^{\alpha})du.
P=(\begin{matrix}\frac{1}{2}&\frac{1}{2}&0&0\\ \frac{1}{2}&\frac{1}{2}&0&0\\ 0&0&\frac{1}{2}&\frac{1}{2}\\ 0&0&\frac{1}{2}&\frac{1}{2}\\ \end{matrix})+\epsilon(\begin{matrix}-\frac{1}{2}&0&\frac{1}{2}&0\\ 0&-\frac{1}{2}&0&\frac{1}{2}\\ \frac{1}{2}&0&-\frac{1}{2}&0\\ 0&\frac{1}{2}&0&-\frac{1}{2}\\ \end{matrix})
\rho=\frac{\sum_{t=i+1}^{T}(\hat{\epsilon}_{t}^{2}-\hat{\sigma}_{t}^{2})(\hat{\epsilon}_{t-1}^{2}-\hat{\sigma}_{t-1}^{2})}{\sum_{t=1}^{T}(\hat{\epsilon}_{t}^{2}-\hat{\sigma}_{t}^{2})^{2}}
\sigma(B') = \sigma(B) + \tfrac{\sigma(b)}{n}\cdot I_n =ABA^{-1}+\delta(A)A^{-1}-\tfrac{1}{n}\delta(\mathrm{det}(A))\mathrm{det}(A)^{-1}\cdot I_n +\tfrac{\sigma(b)}{n}\cdot I_n =A(B+\tfrac{b}{n}\cdot I_n)A^{-1}+ \delta(A)A^{-1}-\tfrac{1}{n}\delta(\mathrm{det}(A))\mathrm{det}(A)^{-1}\cdot I_n +\tfrac{\sigma(b)-b}{n}\cdot I_n = AB'A^{-1}+\delta(A)A^{-1},
\widetilde{T}_{n,i}(x)\ =\x^{2}\,\sum_{j=0}^{i-1}T_{n-1,j}(x)+x\sum_{j=i}^{2n-i-3}T_{n-1,j}(x)+\sum_{j=2n-i-2}^{2n-3}T_{n-1,j}(x)\,,
\lim_{\epsilon \to 0}\lim_{N \to \infty}\frac{1}{N}\log \left( \rho_{N}^{\otimes N}\left( \overline{\mathbf{P}}_{N}\in B( \overline{\mathbf{P}},\epsilon) \right)\right) = -\overline{\rm Ent}[ \overline{\mathbf{P}}| \overline{\mathbf{\Pi}}^{1}]- c_{\omega, \Sigma}.
\frac{d\mathbb{Q}_{n,\sigma}}{d\mathbb{P}_{n}}:= \exp \left\{\sum_{i<j}\left(\frac{2\beta}{\sqrt{n}}\sigma_{i}\sigma_{j}A_{i,j}- \frac{2\beta^2}{n}\right) + \frac{\beta J}{n}\left( \sum_{i=1}^{n}\sigma_{i}\right)^2 \right\}.
[a_{(1\,l)}, a_{(3\,l-1)}]_c=-\lambda_{l-1l-1l}(1-\xi)\chi_{l-1}(g_{(3\,l-2)})[a_{(1\,l-2)},a_{(3\,l-2)}]_cg_{l-1l-1l}&- \lambda_{l-1l-1l}(1-\xi^2)^2\chi_{(3\,l-2)}(g_{(1\,l)})a_{(3\,l-2)}a_{(1\,l-2)}g_{l-1l-1l}.
l_3 \coloneq \left[\left(1 - \dfrac{g}{2}\right)l_1 + \dfrac{g}{2}l_2\right] + \underbrace{\left[(g - 1)l_1 - (g + 1)l_2\right]}_{}&= \dfrac{g}{2}l_1 - \dfrac{g + 2}{2}l_2,
T_{0}^{1}\; = \; \alpha^{- \frac{L + 1}{2}}\; \sum_{j = 1}^{L}\alpha^{j}\, \tau_{j}^{1}\, , \hspace{1 . 5 c m}T_{0}^{2}\; = \; \beta^{- \frac{L + 1}{2}}\; \sum_{j = 1}^{L}\beta^{j}\, \tau_{j}^{2}\, , \hspace{1 . 5 c m}E \; = \; L \, ,
\begin{aligned}D_x^{\beta}( |x|^{\alpha}\,W(t) u_0)(x,y)= W(t)(D_x^{\beta}|x|^{r_1}u_0)(x,y) + W(t) (D_x^{\beta}\{\Phi_{1,t,r_1}(\widehat u_0)(\xi,\eta)\}^{\lor})(x,y). \end{aligned}
f_1 =u_1^{n_1}-u_0^{l_0^{(1)}}=0 f_2 =u_2^{n_2}-u_0^{l_0^{(2)}}u_1^{l_1^{(2)}}=0 \;\;\vdots f_g =u_g^{n_g}-u_0^{l_0^{(g)}}\dots u_{g-1}^{l_{g-1}^{(g)}}=0
||w^s_{pp}(x)||^{2}_{H^{-1/2}(\partial B_R)}= \sum_{n \in \mathbb{Z}}(1+n^2)^{-1/2}|D_n H^{(1)}_n(t_p)|^2 =\sum_{n \in \mathbb{Z}}|D_{n}|^{2}\frac{2^{2n}(n-1)! (n-2)!}{\pi^2 t_p^{2n}}\left(1+\mathcal{O}(\frac{1}{n})\right).
S(A|M)_{(\mathcal{N}(t)\otimes\mathbb{I}_M)(\hat{\rho}_{AM})}\ge \lim_{\nu\to\infty}S(A|A')_{(\mathcal{N}(t)\otimes\mathbb{I}_{A'})(\hat{\omega}_{AA'}(\nu))}= n\ln t + n\;.
Z(s,f,\chi,\Delta)=\sum\limits_{m=1}^{\infty}\int\limits_{\pi^{am}O_v^\times \times \pi^{bm}O_v^\times}\chi(ac(f(x,y))\ |f(x,y)|^s\ |dxdy| =\sum\limits_{m=1}^{\infty}q^{-(a+b)m-d_0ms}\int\limits_{O_v^{\times 2}}\chi(ac \ (F^{(m)}(x,y)))\ |F^{(m)}(x,y)|^s\ |dxdy|,
- ({\frac{e}{c}}) \int d \vec{y}\psi^{( 3 ) +}( y ) ) K_{i}( y , x ) ( i \hbar \vec{\partial}_{y^{i}}- ({\frac{e}{c}}) A_{i}^{[ 3 ]}( y ) ) \psi^{( 3 )}( y ) \left. \right] \psi^{( 3 )}( x ) \, ,
A'(t)\geq \int_0^{p}\frac{ye^{3G}}{(e^{-(k-1)A}+1)^2}dy=\frac{e^{3A}}{(e^{-(k-1)A}+1)^2 (3B)^2}(1-3Bp e^{-3A/2}-e^{-3A/2}).
L^{(2)}(\phi, \chi)=\frac{1}{2}\int d^3x\left[\sum_{i=1}^{3}\dot{\phi}_i^2+\sum_{i,j =1}^{3}\dot{\chi}_iG^{-1}_{ij}\dot{\chi}_j - V(\phi, \chi)\right] - \theta\int d^3x\,\,Q^{(2)}(\phi, \chi)\,,
\left(\prod_{j=1}^r \prod_{s=1}^n e^{d_{j,s}\varpi_s}(b_{k(j,s), m(j,s)}) \right)\left( \prod_{s=1}^n \prod_{j=1}^r e^{d_{j,s}\varpi_s}(\zeta(j,s))^{-1}\right).
\bar{A}{}_{\,\,m}^{l\,\,\,\,n}=\frac{\partial\bar{x}{}^{l}}{\partial x^{i}}\frac{\partial x^{j}}{\partial\bar{x}{}^{m}}\frac{\partial\bar{x}{}^{n}}{\partial x^{k}}A{}_{\,\,\,j}^{i\,\,\,k}
R^\#Rf(u) =\int\limits_{S^{n-1}}Rf(\omega,u \cdot \omega) d\omega =|S^{n-2}| \int\limits_{S^{n-1}}\int\limits_{0}^\infty f(u+r\omega)r^{n-2}drd\omega =|S^{n-2}| \int\limits_{\mathbb{R}^n}f(u+x)|x|^{-1}dx= |S^{n-2}| (|x|^{-1}\ast f)(u).
{{\mathsf{R}}^{\lambda}_{\mathcal{N}_{[1:N]}}\leq{\mathsf{R}}^{\lambda}_{\mathcal{N}_{\mathcal{M}_{1}}}+{\mathsf{R}}^{\lambda}_{\mathcal{N}_{\mathcal{M}_{2}}}={\mathsf{R}}^{\lambda}_{\mathcal{N}_{\mathcal{K}}}+{\mathsf{R}}^{\lambda}_{\mathcal{N}_{[1:N]\backslash \mathcal{K}}}}.
\beta_r(q_{1 3}^0, |\vec{q}_{1 3}|, q_{2 4}^0, |\vec{q}_{2 4}|) =\frac{- (q_{1 3}^0 - q_{2 4}^0)^2 + \vec{q}_{1 3}^2 + \vec{q}_{24}^2 + m_r^2 - i \varepsilon}{2 |\vec{q}_{1 3}| |\vec{q}_{2 4}|}\,.
P \left(S_n(\theta)/\sqrt{n}\in K \right) \geq 1- \sum_{k=1}^{\infty}l_k \sum_{j>N_k}E\big\vert\langle S_{n}(\theta)/\sqrt{n},v_j \rangle\big\vert^2 &= 1- \sum_{k=1}^{\infty}2\pi l_k \sum_{j>N_k}\langle \mathcal{F}_{n;\theta}(v_j),v_j \rangle,
D_{\tau}^{\alpha,\sigma,\gamma,m,n}u = \tau^{-\alpha}\sum_{k=0}^n\omega^{(\alpha,\sigma)}_{n-k}(u(t_k)-u_0) + \tau^{-\alpha}\sum_{k=1}^mw^{(\alpha,\sigma)}_{n,k}(u(t_k)-u_0) ,
|{\Psi_\alpha}_2 (p) \rangle_{tree}=\Big\{[U_2^{(2)}(\alpha) + U_2^{(11)}(\alpha)]|\Psi_0(p) \rangle+ U_1(\alpha)|\Psi_1(p) \rangle+ U_0(\alpha) |\Psi_2 (p) \rangle \Big\}_{tree}=|\Psi_2(p_\alpha) \rangle\; .
\displaystyle E(\xi)-\frac{c}{24}=\frac{1}{\pi}\lim_{R\rightarrow 0}RE(R) =\displaystyle \frac{1}{\pi}\left[ \sum^{m_{1}}_{j=1}2e^{-y^{(1)}_{j}}-\int^{\infty}_{-\infty}\! \! \frac{dx}{\pi}e^{-x}\log (1-e^{-\epsilon_{2}(x)}) \right]
f_k(x)= x(x-1)^{k-1}=\sum_{j>0}(-1)^{k-j}{k-1\choose j-1}x^j, f_{-k}=\frac{1}{(x-1)^{k}}=\sum_{j=1}^{\infty}{{j-1}\choose{k-1}}\frac{1}{x^j}, f_{0}=0.
B(x_1x_4+x_4x_1) = A(x_2x_3+x_3x_2), B(x_2x_0+x_0x_2) = A(x_3x_4+x_4x_3), B(x_3x_1+x_1x_3) = A(x_4x_0+x_0x_4), B(x_4x_2+x_2x_4) = A(x_0x_1+x_1x_0), B(x_0x_3+x_3x_0) = A(x_1x_2+x_2x_1).
\langle \xi, \mathcal{G}\zeta\rangle = \frac{1}{2}(\|\xi\|_\mathcal{G}^2 + \|\zeta\|_\mathcal{G}^2 - \|\xi-\zeta\|_\mathcal{G}^2) = \frac{1}{2}(\|\xi+\zeta\|_\mathcal{G}^2 - \|\xi\|_\mathcal{G}^2 - \|\zeta\|_\mathcal{G}^2).
d_{\mathbf{p}S_i}=\left(\begin{smallmatrix}d_{P_i[3]}0 0 0 \rho d_{P_{s(\rho)}[2]}0 0 -\tau^\ast -\partial_{\rho\tau}w d_{P_{t(\tau)}[1]}0 e_i^\ast \rho^\ast \tau d_{P_i}\end{smallmatrix}\right)
L_{fk}=\frac{e^{\frac{1}{2}\alpha\phi /M_{p}}}{\zeta +k}\left[(\zeta +h_{E}) f_{E}\overline{L_{L}}\,H\,e_{R}+(\zeta +h_{U}) f_{U}\overline{Q_{L}}\,\tilde{H}\,U_{R}+(\zeta +h_{D}) f_{D}\overline{Q_{L}}\,H\,D_{R}+H.c.\right]
\sum_{k=0}^{\tilde n}T_k \leq C \frac{{(\tilde n +1)}^{d+ \alpha +1}}{|i-j|^{d+\alpha}}& \leq C \frac{(\log |i-j|^{d + \alpha}+1 )^{d+\alpha+1}}{|i-j|^{d+\alpha}}\leq C \frac{1}{|i-j|^{d+\frac{\alpha}{2}}}.
\frac{d}{d\epsilon}J(\varphi^{\epsilon})|_{\epsilon=0}=\int_{0}^{1}\int_{X}Av_{t}\cdot\delta v_{t}dxdt=\int_{0}^{1}\int_{X}Av_{t}\cdot(\frac{d}{dt}w_{t}-((Dv_{t})w-(Dw)v_{t}))dxdt.
\left(\mathcal{R}\cdot\psi_{0}\right)\left(-\left|\eta\right|\right) =\psi_{1}\left(\left|\eta\right|\right), \left(\mathcal{R}\cdot\psi_{j}\right)\left(-\left|\eta\right|\right) =\psi_{j}\left(\left|\eta\right|\right),\mbox{for}j=2,3,4.
(-\Delta_{p,q})^{s}u(x)=2\lim_{\delta\searrow 0}\int_{\mathbb{G}\setminus B_{q}(x,\delta)}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{q^{Q+sp}(y^{-1}\circ x)}dy, \,\,\,x\in \mathbb{G},
\nabla^2 V_k(\zeta) = 0 ~~~~~~~~~~~~~~\, \mbox{if}\zeta\in G_m, V_k(\zeta) = 1-\Psi_k(\zeta) \mbox{if}\zeta\in C_j, j=1,\ldots,k-1, V_k(\zeta) = 0 ~~~~~~~~~~~~~~\, \mbox{if}\zeta\in C_k, V_k(\zeta) = -\Psi_k(\zeta) ~~~~~~ \mbox{if}\zeta\in C_j, j=k+1,\ldots,m.
w_\mu(q^\mu t^{\delta(n)}; q, t) = q^{-|\mu|}\, t^{(n-1)|\mu|-2n(\mu)}\, ( qt^{n-1})_{\mu}\!\!\prod_{1\leq i < j\leq n}\!\!\frac{(qt^{j-i-1})_{\mu_i-\mu_j}}{(qt^{j-i})_{\mu_i-\mu_j}}
\begin{cases}C_p^T\Psi''-C_p^T\Delta \Psi+A^TC_p^T\Psi=0 \hbox{in}(0, +\infty)\times \Omega, C_p^T\Psi=0 \hbox{on}(0, +\infty)\times \Gamma_0, C_p^T\partial_\nu\Psi + B^TC_p^T\Psi=0 \hbox{on}(0, +\infty)\times \Gamma_1.\end{cases}
0 =2\, \langle w_0\cdot\delta_G-\delta_K,\theta'-w_{0}\cdot\theta_{i}\rangle+\|\theta'-w_0\cdot\theta_i\|^2\,, &= 2\,\langle w_0\cdot\delta_G,\theta'\rangle-\|\theta_i\|^2+\|\theta'-w_0\cdot\theta_i\|^2\,, &= 2\,\langle \delta_G,w_{0}^{-1}\cdot\theta'\rangle-\|\theta_i\|^2+\|\theta'-w_0\cdot\theta_i\|^2\,,
({\cal N}^c(f_{\ell})) = \frac{[{\cal I}_2- 2^3 \cdot 5 \;{\cal I}_0]^2}{2^{10}}\ell^3 + \frac{[2^6 \cdot 3 \cdot 17 \;{\cal I}_0-2^4 \cdot 11\;{\cal I}_2+{\cal I}_4]^2}{2^{18}\, \pi^2}\ell^2 \log \ell +O(\ell^2).
\left|\Delta_1n(x,y)\right| = \lim \limits_{k\rightarrow \infty}\left|\frac{1}{19683}\right|^{pk}\left|\Delta_1n\left(\frac{x}{3^{pk}},\frac{y}{3^{pk}}\right)\right| \leq \lim \limits_{k\rightarrow \infty}\left|\frac{1}{19683}\right|^{pk}\zeta\left(\frac{x}{3^{pk}},\frac{y}{3^{pk}}\right)=0.
\lim_{a\to0}\frac{\mathbb{P}_y(\tau^{(-a,a)}<\infty)}{\mathbb{P}_x(\tau^{(-a,a)}<\infty)}= \lim_{a\to\infty}\frac{\mathbf{P}_{0,{\rm sign}(x)}(T^-_{\log (a/|y|)}<\infty)}{\mathbf{P}_{0,{\rm sign}(x)}(T^-_{\log (a/|x|)}<\infty)}= \frac{v_{{\rm sign}(y)}(\theta)}{v_{{\rm sign}(x)}(\theta)}\left|\frac{y}{x}\right|^\theta, x,y\in\mathbb{R}.
\begin{array}{c}S=-\Sigma_nm_n\int\sqrt{-g_{\mu\nu}U^{\mu}_nU^{\nu}_n}d\tau_n+\frac{1}{2}\Sigma_n\int\lambda_n(\tau_n)[g_{\mu\nu}U^{\mu}_nU^{\nu}_n+1]d\tau_n +\alpha\Sigma_nm_n\int\phi[x_n(\tau_n)]d\tau_n-\frac{1}{2\beta}\int[\nabla^{\mu}\phi\nabla_{\mu}\phi-\Lambda^2e^{2\phi}]\sqrt{g}d^4x \end{array}
\underbrace{f\circ\pi_J(a)}_{=\rho(a)}=f\circ\pi_{J,k}(x) \xrightarrow[x\to\pi_k(c)]{}f\circ\pi_{J,k}\big(\pi_k(c)\big)=\underbrace{f\circ\pi_J(c)}_{=\rho(c)=c}.
K K^\top =K^2=\frac{1}{4a^2}(H_1^\top H_1-H_2^\top H_2)^2=\frac{1}{4a^2}(H_1^\top (H_1H_1^\top) H_1+H_2^\top (H_2H_2^\top) H_2) &=\frac{n}{4a^2}( H_1^\top H_1+H_2^\top H_2)=\frac{n^2}{4a^2}I_n.
U_{ijk}= \vec{\omega_{j}}\times \vec{\omega_{k}}U_{ikl}= \vec{\omega_{k}}\times \vec{\omega_{l}}U_{ijl}= \vec{\omega_{j}}\times \vec{\omega_{l}}U_{jkl}= (\vec{\omega_{k}}- \vec{\omega_{j}}) \times (\vec{\omega_{l}}- \vec{\omega_{j}})
4 \left( \begin{array}{c}{{{\bf E}^{\prime}}}\\{{{\bf B}^{\prime}}}\\ \end{array}\right) = R \left( \begin{array}{c}{{{\bf E}}}\\{{{\bf B}}}\\ \end{array}\right) \equiv \left( \begin{array}{l l}{{\cos \theta}}&{{\sin \theta}}\\{{- \sin \theta}}&{{\cos \theta}}\\ \end{array}\right) \left( \begin{array}{c}{{{\bf E}}}\\{{{\bf B}}}\\ \end{array}\right)
\varepsilon_{n}^2 :\,= \min_{\lambda\in\Lambda}\, \Big\{\min_{\varepsilon_\lambda>0}\Big\{(\varepsilon^2_{\lambda}+ \Big(-\frac{1}{n}\log \Pi_\lambda(B_n(\theta^\ast,\,\varepsilon_\lambda))\Big)\Big\}+ \Big(-\frac{1}{n}\log \pi_\lambda\Big)\Big\},
P^{(\infty)}_{+}(z) = P^{(\infty)}_{-}(z) \begin{pmatrix}0 1 -1 0\end{pmatrix}, z\in (-\infty,-1), &P^{(\infty)}_{+}(z) = P^{(\infty)}_{-}(z) e^{\pi i \alpha \sigma_3}, z \in (-1,0)\setminus \{-a^{-1}\},
\frac{1}{2\pi i}\int_{(1 + \epsilon)}D_G^{\mathrm{cm}}(s)\widehat{\phi_{Y}}(s)X^sds = \widehat{\phi_{Y}}(1)r_d(G)X + \frac{1}{2\pi i}\int_{(\alpha')}D_G^{\mathrm{cm}}(s)\widehat{\phi_{Y}}(s)X^sds.
\frac{\delta_{D}(z)r_{D}(x,y)}{r_{D}(y,z)r_{D}(x,z)}\le\frac{\delta_{D}(z)(r_{D}(x,z)+r_{D}(y,z))}{r_{D}(y,z)r_{D}(x,z)}=\frac{\delta_{D}(z)}{r_{D}(y,z)}+\frac{\delta_{D}(z)}{r_{D}(x,z)}\le2,
\sigma_{11}-\sigma_{33}=2C_{1}(\lambda_{1}^{2}-\lambda_{3}^{2})-2C_{2}(\frac{1}{\lambda_{1}^{2}}-\frac{1}{\lambda_{3}^{2}});\sigma_{22}-\sigma_{33}=2C_{1}(\lambda_{2}^{2}-\lambda_{3}^{2})-2C_{2}(\frac{1}{\lambda_{2}^{2}}-\frac{1}{\lambda_{3}^{2}})
0 = 3 x (254 x^6-714 x^5+681 x^4-250 x^3-6 x^2-28 x-1) f & + x(x-1) (1397 x^5-2482 x^4+1094 x^3-28 x^2+197 x+14) f_{x}& + 6 x (x-1)^3 (127 x^3+36 x^2+33 x+4) f_{xx}& + (x-1)^3 (127 x^4+48 x^3+66 x^2+16 x-1) f_{xxx}
x_1 = r\cos\theta_1, x_2 = r\sin\theta_1\cos\theta_2, x_3 = r\sin\theta_1\sin\theta_2\cos\theta_3, & \vdots x_{d-2}= r\sin\theta_1\sin\theta_2\ldots\sin\theta_{d-3}\cos\theta_{d-2}, x_{d-1}= r\sin\theta_1\sin\theta_2\ldots\sin\theta_{d-2}\cos\varphi, x_{d}= r\sin\theta_1\sin\theta_2\ldots\sin\theta_{d-2}\sin\varphi,
w=\sum_{p\in J}S(p)^{d}w \geq \sum_{p\in J}\prod_{j=1}^d \frac{\tilde S(p, \pi_j(p),\alpha, \lambda)}{{\lambda+k_j\choose k_j}}\geq \frac{1}{\prod_{j=1}^d{\lambda+k_j\choose k_j}}{\lambda + n \choose n}\gtrsim_{n}1,
\widehat\alpha(mtu) = \alpha_Z(t)\in \alpha(t) \cdot E= \alpha(m)^{-1}\cdot \alpha(m) \alpha(t) \alpha(u) \cdot \alpha(u^{-1}) E &\subseteq K_M \cdot \alpha(mtu) \cdot E= \alpha(mtu) \cdot K_M E.
SNR_{ins}= \frac{E[{\boldsymbol s}^\emph{H}{\boldsymbol s}]Tr({\boldsymbol W}^\emph{H}[i]{\boldsymbol H}_D[i]{\boldsymbol H}^\emph{H}_D[i]{\boldsymbol W}[i])}{E[{\boldsymbol n}^\emph{H}_D[i]{\boldsymbol n}_D[i]]},
|-4b \int_{t}^{t+\lambda_{0,0}(t)^{1-\alpha}}\frac{dx}{x^{2}\log^{b+1}(x) (\lambda_{0,0}(t)^{1-\alpha}+x-t)(1+x-t)^{3}}| &\leq C \int_{t}^{t+\lambda_{0,0}(t)^{1-\alpha}}\frac{dx}{x^{2}\log^{b+1}(x)}\frac{1}{\lambda_{0,0}(t)^{1-\alpha}}&\leq \frac{C}{t^{2}\log^{b+1}(t)}
\left[\hat{Q}^k,\;\hat{Q}^l\right] = 0\;,\quad\left[\hat{P}^k,\;\hat{P}^l\right] = 0\;,\quad\left[\hat{Q}^k,\;\hat{P}^l\right] = i\,\delta^{kl}\,\hat{\mathbb{I}}\;,k,\,l=1,\,\ldots,\,n\;.
{\cal L}_{\rm c}= -{3\over 2}\left[S_0\overline S_0\,e^{-K/3}\right]_D+\left[S_0^3(ih +\alpha M^3)\right]_F+{1\over 4}\sum_{a=1}^{2}{\left[(S+\beta^aT){\cal W}^a{\cal W}^a\right]_F},
g_t(1) =\gamma_+(u^++tu^-)=I'(u^+)[u^+]+t^2\lambda\int_{\mathbb R^3}\phi_{u^-}(u^+)^2dx \leq I'(u^+)[u^+]+\lambda\int_{\mathbb R^3}\phi_{u^-}(u^+)^2dx=\gamma_+(u)<0
\frac{H_{\mu}^{( 1 )}( \lambda z )}{H_{\nu}^{( 1 )}( z )}F ( z ) = \frac{H_{\mu}^{( 2 )}( \lambda z_{1})}{H_{\nu}^{( 2 )}( z_{1})}F ( z_{1}) , z_{1}= e^{- \pi i},
\sum_{n \leq x}\mu(n+a)e^{i 2 \pi u n /q}=\frac{1}{p}\sum_{n \leq x,}\sum_{0 \leq t\leq p-1,}\sum_{1 \leq s\leq p-1}\mu(s+a) \omega^{t(n-s)}e^{i2\pi u s/q}.
\star \equiv \exp \left({\frac{i \hbar}{2}}\int d x \left( \stackrel{\leftarrow}{\frac{\delta}{\delta \phi ( x )}}\stackrel{\rightarrow}{\frac{\delta}{\delta \pi ( x )}}- \stackrel{\leftarrow}{\frac{\delta}{\delta \pi ( x )}}\stackrel{\rightarrow}{\frac{\delta}{\delta \phi ( x )}}\right) \right) ,
\int_0^T DG(B^H_r)d\mathbf{B}^H_r = \lim\limits_{|\mathcal{P}| \to 0}\sum_{[s,t]\in \mathcal{P}}DG(B^H_s)B^H_{s,t}+ D^2G(B^H_{s})\mathbb{B}^H_{s,t}= \lim\limits_{|\mathcal{P}| \to 0}\sum_{[s,t]\in \mathcal{P}}DG(B^H_s)B^H_{s,t}+ \frac{1}{2}D^2G(B^H_{s}){B^H_{s,t}}^{\otimes 2}+ \frac{1}{2}D^2G(B^H_{s})Anti(\mathbb{B}^H_{s,t}).
B_0 =M_0^\dagger +Q, B_1 =-(M_0^\dagger +Q) M_1(M_0^\dagger +Q), B_2 =-(M_0^\dagger +Q) M_2(M_0^\dagger +Q) &\quad+(M_0^\dagger +Q) M_1(M_0^\dagger +Q) M_1(M_0^\dagger +Q).
F_{1/4}(t) := \frac{2^{3n}}{n!}\left( t+\frac{1}{4}\right)_n, F_{3/4}(t) := \frac{2^{3n}}{n!}\left( t+\frac{3}{4}\right)_n, G(t) := \frac{n!}{(t)_{n+1}}.
R^j_\alpha \frac{\delta R^i_\beta}{\delta \phi^j}- R^j_\beta \frac{\delta R^i_\alpha}{\delta \phi^j}= C^\gamma_{\alpha \beta}R^i_\gamma + M^{ij}_{\alpha \beta}\frac{\delta I}{\delta \phi^i}, \; \; M^{ij}_{\alpha \beta}= - M^{ji}_{\alpha \beta}
P_{Y_{i-1},{\bf S}}(1y^{i-2},{\bf s}) \leq \frac{B_{a,i-1}}{| \mathcal S(y^{i-2})|}P_{Y_{i-2},{\bf S}}(y^{i-2},{\bf s}), & P_{Y_{i-1},{\bf S}}(0y^{i-2},{\bf s}) = 1 - P_{Y_{i-1},{\bf S}}(1y^{i-2},{\bf s}).
\frac{a_{\mathrm{I R}}}{a_{\mathrm{U V}}}= \frac{c_{\mathrm{I R}}}{c_{\mathrm{U V}}}= \frac{2 7}{3 2}, a_{\mathrm{U V}}= c_{\mathrm{U V}}, a_{\mathrm{I R}}= c_{\mathrm{I R}}
-\pi^{\frac{p^k-1}{2}}\left(\alpha x_1 x_2(x_1^{p^k-1}+\pi^{\frac{p^k-1}{2}}x_2^{p^k-1})\right)^{p^k}=\alpha x_1 x_2(x_1^{p^k-1}+\pi^{\frac{p^k-1}{2}}x_2^{p^k-1}),
\Psi_{E}= \frac{1}{\sqrt{r}}\left( \begin{array}{c}{{e^{- i \frac{\theta}{2}}\varphi_{1 E}\left( r , \theta \right)}}\\{{e^{i \frac{\theta}{2}}\chi_{1 E}\left( r , \theta \right)}}\\ \end{array}\right)
\max h_{\ell+1}(I) \le \max \big\{\phi_{2}(h_{\ell}(I)), \phi_{1}(h_{\ell}(I))\big\}= \max \phi_{1}(h_{\ell}(I)) \leq \phi_{1}(\phi_{1}^{\ell}(1)) = \phi_{1}^{\ell+1}(1).
e^{\frac{-r\frac{y^2}{4}+ \frac{1}{2}(r)^{\frac{1}{2}}(yx)-r\frac{x^2}{4}}{1-r}}e^{-\frac{x^2}{4}}\leq e^{\frac{1}{2}(r)^{\frac{1}{2}}(yx)}e^{-\frac{x^2}{4}}=e^{-\frac{x}{2}(\frac{x}{2}-\sqrt{r}y)}
\tau_1 = ( 1, 1; 0, 0; 0, 0 ), \tau_2 = ( 0, 1; 0, 1; 0, 0 ), \tau_3 = ( 1, 2; 0, 1; 0, 1 ), \tau_4 = ( 1, 1; 1, 1; 0, 1 ), \tau_5 = ( 1, 2; 1, 2; 1, 1 ), \tau_6 = ( 2, 3; 1, 2; 1, 2 ).
(z_0+z_2)^{k+\frac{j_2}{T}}Y_{M^3}(u, z_0+z_2)I^{\circ}(w_1, z_2)w_2 = (z_2+z_0)^{k+\frac{j_2}{T}}I^{\circ}(Y_{M^1}(u, z_0)w_1, z_2)w_2
\Big\Vert u \Big\Vert_{\mathrm{H}^{1,\frac{1}{2}}\Big(\partial\Omega \times \mathbb{R}\Big)}^2 = \int_\mathbb{R}\int_{\partial\Omega}\Big[|\nabla_{\textbf{tan}}u(\mathrm{x},t)|^2 + u^2(\mathrm{x},t) + \Big(\partial_{t}^{\frac{1}{2}}u(\mathrm{x},t)\Big)^2\Big] d\sigma_{\mathrm{x}}d\mathrm{t},
\Phi_N : = \max\left\{\frac{1+m +\frac{Q_{N-1}}{Q_N}}{k + \ell m +\ell \frac{Q_{N-1}}{Q_N}}\ : \ \left( \begin{smallmatrix}\ell k \end{smallmatrix}\right) \in \mathcal{S}(N,m) \ \ 0\leq m < a_{N+1}\right\}\,.
\chi_H \left(ev_1^* \alpha_1\otimes \dots \otimes ev_r^* \alpha_r \otimes [\O_{\overline{{\mathcal{M}}_{0,r}}(X, \d)}] \right) = \sum_{e_1, \dots, e_r \in I}\chi_H\left(a_{1,e_1}(ev_1^* \phi_{e_1}) \otimes \dots \otimes a_{r,e_r}(ev_r^* \phi_{e_r})\right)
- 2 i \int \frac{d^{\, 4}P}{( 2 \pi )^{4}}\, e^{- i P \cdot ( x - y )}\sum_{\tau = \pm}\theta ( \tau p_{0}) \left[ P \cdot \partial_{X}f^{( \tau )}( X ;{\bf p}) \right] \hat{\Delta}^{( \tau )}( X ; P ) = 0 ,
r^{4}\left\{\left[T(f^{2})-2fT(f)-2A(f)^{2}\right]-2\left[A(f^{2})^{2}-4f^{2}A(f)^{2}\right] \right\}+ r^{2}\left\{T(f^{2})-2fT(f)-2A(f)^{2}\right\}=0.
E_3 =\sum_{x^{\epsilon}\leq d<x}\frac1d \sum_{\substack{{p\leq x}{p\equiv 1 (d)}}}\sum_{\substack{{d'|d}{d'\geq p^{\frac{C}{\log\log p}}}}}\left|\mu\left(\frac{d}{d'}\right)\right| d'e^{-\log^{C_2}p}.
H_{\lambda = q^{- 1}}( q = e^{i \pi / N}) = - \sum_{j = 1}^{L}\sum_{n = 1}^{N - 1}\left( \frac{1}{s i n ( \frac{\pi n}{N})}\left[ \left( \Sigma_{j}^{+}\Sigma_{j + 1}^{-}\right)^{n}+ \left( \Sigma_{j}^{-}\Sigma_{j + 1}^{+}\right)^{n}\right] + ( 1 - \frac{2 n}{N}) e^{i \pi n / N}Z_{j}^{n}\right)
S_k(\chi_{K^{(l)}})= \left(\frac{1}{[K:K^{(l)}]}\right)^2+\sum_{j=1}^lq^{jk}\left[\left(\frac{\lambda(K^{(j-1)}-K^{(j)})}{[K^{(j-1)}:K^{(l)}]}\right)^2+\left(\frac{\lambda(K^{(j)})}{[K^{(j)}:K^{(l)}]}\right)^2\right] \asymp q^{(k-\textrm{dim}G)l}.
\lim_{a\downarrow -\infty}\frac{W_{\alpha,\beta}^a(x)}{W_{\alpha+\beta}(-a)}= Z_{\alpha}(x,\Phi_{\alpha+\beta}) \textrm{and}\lim_{x\uparrow \infty}\frac{W_{\alpha,\beta}^a(x)}{W_{\alpha}(x)}=Z_{\alpha+\beta}(-a,\Phi_{\alpha}).
\Delta n\sum_{l=1}^\infty (2l+1)A_l=\left . y\Delta n \left(\frac{\partial^2}{\partial r^2}-2\frac{\partial^2}{\partial r\partial \rho}+\frac{\partial^2}{\partial \rho^2}\right )\overline{\cal D}(r,\rho,y)\right |_{r=\rho=1}+1\,{.}
\begin{aligned}\Phi_2(z,s,a) =\sum_{m_1,m_2=0}^\infty\frac{z^{m_1+m_2}}{(m_1+m_2+a)^{s}}&=\sum_{m_2=1}^\infty\sum_{m_3=0}^\infty\frac{z^{m_2+m_3}}{(m_2+m_3+a)^{s}}+\sum_{m_3=0}^\infty\frac{z^{m_3}}{(m_3+a)^{s}}.\end{aligned}
A ( p + \frac{q}{2}, p - \frac{q}{2}) \: = \: - \sum_{n = 0}^{\infty}\frac{( - 1 )^{n}}{n !}\left( \frac{q^{2}}{4}\right)^{n}\sum_{k = 0}^{\infty}\frac{1}{( 2 k ) !}\sum_{l = 0}^{n}\frac{( - 1 )^{l}}{2 k + 2 l + 1}\: \left( \! \! \begin{array}{c}{{n}}\\{{l}}\\ \end{array}\! \! \right) \: \left( \frac{q^{j}}{2}\frac{\partial}{\partial p^{j}}\right)^{2 k}T_{a}^{( n + 1 )}( p ) \; \; .
\omega_{x+1}^2 - \omega_x^2 = \nabla_{x,x+1}\left(\omega_x^2\right) \omega_x\omega_{x+k}= -\frac{1}{2}\nabla_x\big(\omega_x\omega_{x+1}\big)+\sum_{\ell=1}^{k-1}\nabla_{x+\ell, x+\ell+1}\big(\omega_x\omega_{x+\ell}\big).
A =205\xi+4300\xi^2+34000\xi^3+120000\xi^4+160000\xi^5, B =215\xi+4475\xi^2+35000\xi^3+122000\xi^4+160000\xi^5, C =85\xi+1750\xi^2+13525\xi^3+46500\xi^4+60000\xi^5, D =15\xi+305\xi^2+2325\xi^3+7875\xi^4+10000\xi^5, E =\xi+20\xi^2+150\xi^3+500\xi^4+625\xi^5.
\begin{array}{l}{\varphi_{1 E}\left( r , \theta \right) = \sum_{n = - \infty}^{\infty}f_{n}\left( r \right) e^{i \left( n + \frac{1}{2}\right) \theta}}\\{\chi_{1 E}\left( r , \theta \right) = \sum_{n = - \infty}^{\infty}g_{n}\left( r \right) e^{i \left( n + \frac{1}{2}\right) \theta}}\\ \end{array}
\langle\boldsymbol{\phi}_2(f)\rangle=\phi_0\int dp\f_R(p)\lambda(p)\left[\frac{9}{7!\pi^4}[(8\lambda^{(7)}\ast \vartheta\ln)(p)-(\lambda^{(8)}\ast S\vartheta\ln)(p)]+\sum_{\alpha=0}^6(-1)^\alpha a_\alpha\lambda^{(\alpha)}(p)\right]
\|F^{*}dy\| =\sqrt{\langle dy^{1}\wedge\cdots\wedge dy^{n},dy^{1}\wedge\cdots\wedge dy^{n}\rangle}=\sqrt{\mathrm{det}\left(\left(\langle dy^{i},dy^{j}\rangle\right)_{1\leqslant i,j\leqslant n}\right)}=\sqrt{\mathrm{det}\left(\frac{dy}{dx}\left(\frac{dy}{dx}\right)^{*}\right)}.
\begin{array}{c}{{\left\{L_{1}^{n}, L_{2}^{n}\right\}= \left( a - \alpha \right) L_{1}^{n}L_{2}^{n}- L_{1}^{n}L_{2}^{n}\left( a + \alpha \right) ,}}\\{{\left\{L_{1}^{n}, L_{2}^{n + 1}\right\}= L_{1}^{n}\left( \alpha - s \right) L_{2}^{n + 1},}}\\{{\left\{L_{1}^{n}, L_{2}^{m}\right\}= 0 , \left| n - m \right| \geq 2 .}}\\ \end{array}
\begin{aligned}\sum_{\alpha\in\{\alpha=(\beta,k):|\alpha|\leq m, \beta\leq m-1\}}s(x)+25\vartheta_0^{-4}s_m(x) &\leq \left(F_0+\int_0^x F\right)\left\{1-2C\vartheta_0^{-8}\left(F_0+\int_0^x F\right)^2x\right\}^{-1/2},\end{aligned}
- g^{2}T^{2}\sum_{n , l}\int \frac{d^{5}p}{( 2 \pi )^{5}}\, \frac{d^{5}k}{( 2 \pi )^{5}}\, \frac{e^{i ( \theta_{1 2}( p_{1}\, k_{2}- p_{2}\, k_{1}) + \theta_{3 4}( p_{3}\, k_{4}- p_{4}\, k_{3}) )}}{( p^{2}+ \frac{4 \pi^{2}n^{2}}{\beta^{2}}) ( k^{2}+ \frac{4 \pi^{2}l^{2}}{\beta^{2}}) ( ( p + k )^{2}+ \frac{4 \pi^{2}( n + l )^{2}}{\beta^{2}})}.
\mathrm{T r}K^{( j )}( s ) = \mathrm{T r}\left[ e^{- s \bigtriangleup^{( j )}}\right] ={\frac{1}{( 4 \pi s )^{d / 2}}}\left( A_{0}^{( j )}+ s A_{1}^{( j )}+ s^{2}A_{2}^{( j )}+ . . . . \right) ~ ~ ~
\binom{n}{k}=\sum_{i=1}^{k-1}\left(\frac{j-1}{2}\right)^i\binom{n}{k-i}+\left(\frac{j-1}{2}\right)^{k-1}\binom{n}{1}+\sum_{i=1}^{k-1}\left(\frac{j-1}{2}\right)^{i-1}\frac{r+1+j (i-1)}{k-(i-1)}\binom{n}{k-i}.
R(f) = \frac{1}{2}\log_2\left(1+\frac{\left|H_{\rm SD}(f) + \frac{H_{\rm RD}(f)H_{\rm SR}(f)\Theta(f)}{1-\hat\alpha(f)\Theta(f)}\right|^2P(f)}{\left(\left|\frac{H_{\rm RD}(f)\Theta(f)}{1-\hat\alpha(f)\Theta(f)}\right|^2 + 1\right)N_0}\right).
\mathcal{V}_{i}^{1}= \left\{\mathbf{z}\in \mathbb{R}^2 | \Vert \mathbf{z}- \mathbf{z}_{i}\Vert \leq \Vert \mathbf{z}- \mathbf{z}_k\Vert, ~ \forall ~\mathbf{z}_k \in \Phi_{\mathrm b}\! \setminus \! \mathbf{z}_{i}\right\}.
v_x(\tau,x)~=~u_x\bigl(t,x+y_2(t)\bigr),v_{\tau}(\tau,x)~=~{u_t\bigl(t,x+y_2(t)\bigr)+\dot{y}_2(t)\cdot u_x\bigl(t,x+y_2(t)\bigr)\over \dot{y}_1(t)-\dot{y}_2(t)}.