harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|Df^{m}(x)(v)| =|Df^{s+1}(x)(v)||Df^{m-s-1}(f^{s+1}(x))(v)| &\geq \frac{|\xi_{s+1}(x)-c(\lambda^{s}z_0)|}{QdC_d (s+1)^2 |w_0-c(z_{0}/\lambda)|}\kappa \lambda_0^{m-s-2}d|\xi_{m-1}(x)|^{d-1}&\geq \frac{d\delta \kappa \lambda_0^{m-s-2}|\xi_{m}(x)-c(\lambda^{m-1}z_0)|}{QdC_d (s+1)^2 |w_0-c(z_{0}/\lambda)|}.
\overline{\dim}_B\prod_{i=1}^dE_{S_i}=\limsup_{k\rightarrow\infty}\sum_{i=1}^dd_k(S_i)\mbox{and}\underline{\dim}_B\prod_{i=1}^dE_{S_i}=\liminf_{k\rightarrow\infty}\sum_{i=1}^dd_k(S_i).
(p-q)P_{q+1}^{n-2,p-q}(x)+2s\Big(P_{q+1}^{n-2,p-q}(x)\Big)' =-(n+q)P_{q+1}^{n-2,p-q}(x)+(n+p+q+1)P_{q+1}^{n-1,p-q-1}(x) &-(n+q)P_{q}^{n-1,p-q}(x) &=-(n+q)\Big(P_{q+1}^{n-2,p-q}(x)+P_{q}^{n-1,p-q}(x)\Big)+(n+p+q+1)P_{q+1}^{n-1,p-q-1}(x).
X^{N}_y(t)=\frac{1}{N^{1/2}}X_{\lfloor Ny \rfloor}(t)+ \frac{1}{N^{1/2}}(Ny -\lfloor Ny \rfloor) (X_{\lfloor Ny \rfloor+1}(t) - X_{\lfloor Ny \rfloor}(t)) ,\, t\ge 0,\, y\in [0,1].
\frac{\partial h}{\partial z}(\tilde{z}_{0},\alpha_{0}^{-1})=\frac{\partial^{2}h}{\partial z^{2}}(\tilde{z}_{0},\alpha_{0}^{-1})=0,\qquad\frac{\partial^{3}h}{\partial z^{3}}(\tilde{z}_{0},\alpha_{0}^{-1})\neq0
\phi\Big(\Big(q_{r(\mu)}- \sum_{e \in r(\mu)E^1}t_e t^*_e\Big)\pi_{(\mu,k)}\Big) = \phi\Big(\pi_{(\mu,k)}- \sum_{e \in r(\mu)E^1}t_e t^*_e\pi_{(\mu,k)}\Big) = e^\beta m^\phi(Z(\mu,k)) - (A_\omega m^\phi)(Z(\mu,k)),
\frac{\xi'(a)\zeta'(\bar s)-\xi'(\bar s)\zeta'(a)}{\xi'(a)\zeta(\bar s)-\xi(\bar s)\zeta'(a)}+ \frac{\xi'(b)\zeta'(\bar s)-\xi'(\bar s)\zeta'(b)}{\xi'(b)\zeta(\bar s)-\xi(\bar s)\zeta'(b)}=0.
\left|\frac{d^m}{d\omega^m}\frac{1}{1-\mathcal{L}[\beta(k,\cdot)](2\pi i \omega)}\right|\epsilon^{m+1}&\le\sum_{i=0}^{m-1}\cdot \left[\sum_{j_1+j_2 \cdots +j_{i+1}=m}\left(\begin{array}{c}m j_1,j_2,\ldots ,j_{i+1}\end{array}\right)|g^{(j_1)}(\omega)||g^{(j_2)}(\omega)| \cdots |g^{(j_{i+1})}(\omega)|\right],
\hat{\cal H}= E_{s p h}+{\frac{\hat{\pi}_{0}^{2}}{2}}+{\frac{\hat{\pi}_{- 1}^{2}}{2}}-{\frac{\Omega^{2}\hat{\phi}_{- 1}^{2}}{2}}+{\frac{1}{2}}\sum_{l \ge 1}( \hat{\pi}_{l}^{2}+ \omega_{l}^{2}\hat{\phi}_{l}^{2})
\widehat{\widetilde{\mathbf{H}}}_k^H\widehat{\widetilde{\mathbf{H}}}_k = \widetilde{\mathbf{H}}_k^H\widetilde{\mathbf{H}}_k+ \widetilde{\boldsymbol{\Delta}}_k^H\widetilde{\mathbf{H}}_k + \widetilde{\mathbf{H}}_k^H\widetilde{\boldsymbol{\Delta}}_k + \widetilde{\boldsymbol{\Delta}}_k^H\widetilde{\boldsymbol{\Delta}}_k,
\forall \, h \in H, \:\:\: \mu^r(h?) \varphi = \mu^r\!\left(h'?\right) \varphi\!\left(S^{-1}(h'')\right), &\mu^l = \mu^r(g^2 ?), &\forall\, x,y \in H, \:\:\: \mu^r(xy) = \mu^r(S^2(y)x), \:\:\: \mu^l(xy) = \mu^l(S^{-2}(y)x), &\forall\, x,y \in H, \:\:\: \mu^r(gxy) = \mu^r(gyx), \:\:\: \mu^l(g^{-1}xy) = \mu^l(g^{-1}yx) .
\theta_{\mathcal{C}}(\mathbf{z}) = a_{000}+ a_{100}\exp[z_1] + a_{010}\exp[z_2] + a_{001}\exp[z_3] + a_{110}\exp[z_1 + z_2] + a_{101}\exp[z_1 + z_3]+ a_{011}\exp[z_2 + z_3] + a_{111}\exp[z_1 + z_2 + z_3]
S_{n,M}(x) := \sum_{i=1}^{n}X_{i,M}(x)=\sum_{i=1}^{n}\sum_{j=0}^{M}2^{\frac{j}{2}}c_{j,\lfloor2^{j}P_{i}(x)\rfloor}(-1)^{\epsilon_{j+1}(P_{i}(x))}
d s_{1 0}^{2}= g_{\mu \nu}^{5}d x^{\mu}d x^{\nu}+ l^{2}\sum_{i = 1}^{3}[ d \mu_{i}^{2}+ \mu_{i}^{2}( d \varphi_{i}+ \frac{2}{\sqrt{3}}A_{\mu}d x^{\mu})^{2}] ,
R_{\mathrm{d i l a t o n}}^{\mathrm{b o u n d a r y}}( r ) = A_{\mathrm{o u t}}\left( 1 - i \frac{r_{H}^{2}}{\pi r^{2}}\right) + A_{\mathrm{i n}}\left( 1 + i \frac{r_{H}^{2}}{\pi r^{2}}\right) .
A_\infty=\mathrm{diag}(\sigma_0,\dots,\sigma_{\mu-1})\quad\arraycolsep=5ptA_0=\mu\begin{pmatrix}0 1 1 0 0 0 1 0 0 \vdots \ddots \ddots \ddots \vdots 0 \cdots 0 1 0\end{pmatrix}.
\epsilon_x^1(t_0,t) := \dot{\mathbf P}_{\delta_x}^{(\phi)}[Y^{(t_0,t]}_t(\phi)^{-1}] - \dot{\mathbf P}_\nu^{(\phi)}[Y^{(t_0,t]}_t(\phi)^{-1}]; \epsilon_x^2(t_0,t) := \dot{\mathbf P}_{\delta_x}^{(\phi)}[Y_t(\phi)^{-1}- Y^{(t_0,t]}_t(\phi)^{-1}].
\mathbb{P}\left\{\mathcal{M}_i^{1, j}\right\}\leq c_1 \sum_{t=0}^{T-1}\exp\left(-c_2 N_in_x \left(\frac{\alpha \rho^{(i)}\|\Sigma^{(i)}_t\|}{\rho^{(i)}\|\Sigma^{(i)}_t\| + \sqrt{n_x}}\right)^2\right).
R_0^m(\lambda, r,r') = I_{|\alpha+m|}(\sqrt{|\lambda|}\, r)\, K_{|m+\alpha|}(\sqrt{|\lambda|}\, r') +\frac{\widetilde B_m(\lambda)}{\widetilde A_m(\lambda)}\K_{|\alpha+m|}(\sqrt{|\lambda|}\, r) \, K_{|m+\alpha|}(\sqrt{|\lambda|}\, r') .
g^2\left(\begin{array}{c|c}-\frac{1}{\bar{D}^2}\delta_\sigma\nu+\frac{1}{\bar{D}^2}\bar{D}_\sigma\frac{1}{E}\bar{D}_\nu\frac{1}{\bar{D}^2}-i\frac{1}{\bar{D}^2}\bar{D}_\sigma\frac{1}{E}i \frac{1}{E}\bar{D}_\nu\frac{1}{\bar{D}^2}\frac{1}{E}-1\end{array}\right)
\tilde{\iota}_a((V_X)_{X\in \mathbb{Z}})_{Q',A'}= \begin{cases}V_X\{X\}_{T'}, (Q', A') = \tilde{\varphi}_a(X), 0, , \end{cases}\iota_a((V_X)_{X\in \mathbb{Z}})_{Q',A'}= \begin{cases}V_X\{X\}_{T'}, (Q', A') = \varphi(X), 0, . \end{cases}
\int_0^{\pi}|f^*\mbox{}'(\sin f^*)| \, dr \geqslant \int_0^{r_1}f^*\mbox{}'(\sin f^*) \, dr - \int_{r_1}^{r_2}f^*\mbox{}' (\sin f^*) \, dr + \int_{r_2}^{\pi}f^*\mbox{}' (\sin f^*) \, dr = \left.-\cos f^*(r)\right|_0^{r_1}+ \left.\cos f^*(r)\right|_{r_1}^{r_2}- \left.\cos f^*(r)\right|_{r_2}^{\pi}= 6.
\langle a g, v^{\ell}_{r,\zeta_j}\rangle_{q^r}= \lim_{t \to 1^{-}}\langle a g, v^{\ell}_{r, t \zeta_j}\rangle_{q^r}= \lim_{t\to 1}(a g)^{(\ell)}(t \zeta_j) & = \lim_{t\to 1}\sum_{p = 0}^{\ell}{\ell \choose p}a^{(p)}(t \zeta_j) g^{(\ell - p)}(t \zeta_j).
\frac{1 a + b \vert}{1 + \vert a + b \vert}= 1 - \frac{1}{1 + \vert a + b \vert}\vert \leq 1 - \frac{1}{1 + \vert a \vert + \vert b \vert}= \frac{1 a \vert + \vert b \vert}{1 + \vert a \vert + \vert b \vert}
Z(\gamma) = Z_{mon}(\gamma) Z_{0}(\gamma) ={\cal N}' \sum_{l=-\infty}^{\infty}\exp \left[ - \frac{e^{2}}{2}V \left( \frac{S}{V}\nu + l \right)^{2}- \frac{e^{2}}{2}\nu^2 \frac{S(V-S)}{V}\right].
\xi = \xi - \pi^{*}x_{p}+ \pi^{*}x_{p}, \: \: \: \: \: \upsilon = \upsilon - \pi^{*}y_{p}+ \pi^{*}y_{p}, \: \: \: \: \: \zeta = \zeta - \pi^{*}z_{p}+ \pi^{*}z_{p}
\mu= \frac{v_{2}}{v_{1}}=\frac{(c_{1}c_{4}+c_{2}c_{3})\gamma^{2^k}+(c_{1}c_{3}+c_{2}c_{4})\gamma+(c_{3}+c_{4})^{2}}{(c_{1}c_{3}+c_{2}c_{4})(\gamma+\overline{\gamma})+(c_{1}+c_{2}+c_{3}+c_{4})^{2}}.
\left|\sum_{k=1}^Dk \mu(A^{[k_n]})\mu(V_k^{(x_A)})-\int_A \sum_{k=1}^Dk \mu(V_k^{(x)})d\mu(x)\right| \leq \sum_{k=1}^Dk\left( \mu((\partial A)^{[k_n]})+\mu(A^{[k_n]})\mu(B_A^{[k_n]}) \right) & \leq C\mu(A)n^{-1/20}.
\psi\left(W^1_{n}(p,p_1,\dots,p_{n-1})\right) = \sum_{\{p_1,\dots,p_{n-1}\}}\sum_{i=0}^{n-1}_{i}\leftrightarrow_{i+1}\left(\mathbf{(1)}\ast\mathbf{(1)}\ast\dots\ast\mathbf{(1)}\right)
\begin{array}{r l}\Psi_\alpha(x)= \Big(\min_{1\leq i\leq N_\alpha}d_g(x_{i,\alpha},x)\Big)^n &\times \left(u^q_\alpha(x)+\left\vert\frac{\nabla u_\alpha(x)}{u_\alpha(x)}\right\vert^n+\left\vert\frac{\nabla^2u_\alpha(x)}{u_\alpha(x)}\right\vert^\frac{n}{2}+\vert\mathcal{L}_g W_\alpha\vert_g(x)\right)\end{array}
\chi_{0}^{+}\chi_{1}^{-}-\chi_{0}^{-}\chi_{1}^{+}=q^{1/4}\left(\mathcal{E}_{q}(x)\mathcal{E}_{q}(-q^{1/2}x)+\mathcal{E}_{q}(-x)\mathcal{E}_{q}(q^{1/2}x)\right)=2q^{1/4}.
\sum_{i=1}^4g(N(X,D^g_{Je_i}\theta^\sharp),Je_i) =-\frac{1}{2}\sum_{i,j=1}^4g(D^g_{Je_i}\theta^\sharp,e_j)g(N(e_j,Je_i),X), =\frac{1}{2}\sum_{i,j=1}^4g(D^g_{e_i}\theta^\sharp,e_j)g(N(e_i,e_j),X), =\frac{1}{2}g(d\theta,N_X).
\sum_{s=\lfloor{b}\rfloor}^{n}g(k-1,s)\Big(\frac{s}{n}\Big)^b=\frac{g(k-1)}{n^b}\sum_{s=\lfloor{b}\rfloor}^{n}s^b+\frac{1}{n^b}\sum_{s=\lfloor{b}\rfloor}^{n}O(\varepsilon_s) s^b,
A_{k,T}= \frac{\frac{1}{2}\tanh \left( \frac{\beta_{k,T}E_0}{2}\right)u_1}{{2 E_0 E\sin (E_0 \tau ) \sin (E \tau )-\left(E_0^2+E^2\right) (1-\cos (E_0 \tau ) \cos (E \tau))}}
f ({\bf r}) = \frac{1}{2 \pi}\, e^{i \theta}\, \left( \frac{\partial}{\partial r}+ \frac{i}{r}\, \frac{\partial}{\partial \theta}\right) \, \int_{D}\, d^{2}r^{\prime}\, \ln{|{\bf r - r^{\prime}}|}\, e^{2 \phi ({\bf r^{\prime}})}\, b ( r^{\prime}e^{i \theta^{\prime}}) + C \, ,
\begin{array}{rll}\partial^{\gamma,\mathbf{A}}_b\mathbf{F}{_i}^{ab}= \partial_kp{_i}^{ak}\partial^{\gamma,\mathbf{A}}_bp{_i}^{jb}+ \partial_kp{_i}^{jk}+ \frac{1}{2}c^j_{k\ell}p{_i}^{k\ell}= \frac{1}{2}\|\mathbf{F}\|^2\delta{_i}^j+ \frac{1}{2}\mathbf{F}{_i}^{ab}\mathbf{F}{^j}_{ab}\end{array}
\varphi_{p,q}(\lambda)-(\lambda+1)\phi_{p,q}(\lambda)= \left( 5\,pq-5\,p-2\,q+2 \right){\lambda}^{3}&+ \left( 14\,pq-14\,p-4\,q+4 \right){\lambda}^{2}&+ \left( 18\,pq-18\,p-8\,q+8 \right) \lambda+8\,pq-8\,p-4\,q+4.
\begin{aligned}0 =\nabla f(x_k)+\sum\limits_{i\in I^g(\bar x)}\lambda_i^k\nabla g_i(x_k)+\sum\limits_{j\in\mathcal P}\rho_j^k\nabla h_j(x_k) &+\sum\limits_{l\in \mathcal I(\bar x)}\xi^k_l\left(\alpha^k_l\nabla G_l(x_k)+\beta^k_l\nabla H_l(x_k)\right).\end{aligned}
Z_{t}^{\mu}= \sum_{\mathrm{\tiny f u l l c u t s C o f t}}( - 1 )^{n_{C}}\left[ \prod_{i}B_{t_{i}^{C}}\mu_{t_{i}^{C}}^{- \# ( t_{i}^{C}) x}\right] B_{t_{R}}\mu_{t_{R}}^{- \# ( t_{R}) x},
dT_\xi(\gamma') h-dT_\xi(\gamma) h =(dQ_{\gamma,\xi}h)\gamma_\xi P_{\gamma,\xi}+P_{\gamma,\xi}\gamma_\xi (dQ_{\gamma,\xi}h)+P_{\gamma,\xi}h_\xi P_{\gamma,\xi}&{}-(dQ_{\gamma',\xi}h)\gamma'_\xi P_{\gamma',\xi}-P_{\gamma',\xi}\gamma'_\xi (dQ_{\gamma',\xi}h)-P_{\gamma',\xi}h_\xi P_{\gamma',\xi}.
q_m = \mathrm{B}_{\alpha- (m-1)}^{-1}\mathrm{B}_{\alpha - (m-2)}^{-1}\ldots \mathrm{B}_\alpha^{-1}\widetilde q_m, \ \beta \in [0,1], \q_0 = \widetilde q_0, & p_{0,m}= \mathrm{B}^{-1}_{1-\beta}\mathrm{A}_{1-\beta}^{-1}q_m, \ \beta \in (0,1), & p = p_{0,m}- m \mathrm{B}_{-m-\beta}p_{0,m}, \p_{0,m}= p + m \mathrm{B}_{-\beta}p, \ \beta \in (0,1).
\delta(g^{ij}\Gamma_{ij}^k f_k) = -ug^{ij}g^{kl}h_{ij,l}f_k - g^{ij}g^{kl}u_i h_{jl}f_k - g^{ij}g^{kl}u_j h_{il}f_k + g^{ij}g^{kl}u_l h_{ij}f_k &= -2u \langle \nabla H, \nabla f \rangle - 2 \, h (\nabla{u},\nabla{f}) + 2H \langle \nabla u, \nabla f \rangle,
I_{\omega}= ( - )^{r_{5}}( \omega^{1 / 2})^{r_{3}^{~ 2}+ r_{3}r_{4}- r_{1}r_{3}- r_{2}r_{3}- r_{3}r_{5}- r_{4}r_{5}}\Bigg [ \frac{\Phi ( r_{1}) \Phi ( r_{2})}{\Phi ( r_{3}) \Phi ( r_{4})}\Bigg ]^{1 / 2}
\mu_1(t) =\frac{1}{2}\Big((e^{(a-2k)t}+e^{at})x_1+( e^{at}-e^{(a-2k)t})x_2\Big) &=\frac{1}{2}e^{at}\Big((1+e^{-2kt})x_1+(1-e^{-2kt})x_2\Big) & \leq C e^{at}.
{\bf E}\Bigl[\frac{Z_{\varepsilon}(n\beta)}{Z^n_{\varepsilon}(\beta)}\Bigr] \approx N^{1+n^2\beta^2-(1+\beta^2)n}\prod\limits_{j=0}^{n-1}\frac{\Gamma(1-n\beta^2+(j+1)\beta^2)^2 \,\Gamma(1-\beta^2)}{\Gamma(1-2n\beta^2+(j+1)\beta^2)\,\Gamma(1+j\beta^2)}.
\mathbb{P}\left(\Psi(x,\gamma,Z)=y\right)= \mathbb{P}\left(( \mathbf{1}_{[0, \frac{1}{2}]}- \mathbf{1}_{(\frac{1}{2},1]})(Z)=-1\right)=\mathbb{P}\left(Z \in \left(\frac{1}{2},1\right]\right)=\frac{1}{2};
\left(1+\frac{1}{2}\delta^2 \right)d'\binom{|G-B|}{2}> \left(\max\left\{\frac{r-2}{r-1}k, (h-1)\right\}-2\eta + \frac{1}{2}\delta^2\right)\binom{|G-B|}{2}> \max\left\{k,\frac{r-1}{r-2}(h-1)\right\}t_{r-1}(|G-B|).
I(X^n\rightarrow Y^n) \triangleq \sum_{t=0}^n{\bf E}^{{p}}\Big\{\log \Big( \frac{dq_t(\cdot|Y^{t-1},X^t)}{d\nu_t^{{p}}(\cdot|Y^{t-1})}(Y_t)\Big)\Big\}= \sum_{t=0}^n \int_{{\cal X}^{t}\times{\cal Y}^{t}}\log \Big( \frac{dq_t(\cdot|y^{t-1}, x^t)}{d\nu_t^{{p}}(\cdot|y^{t-1})}(y_t)\Big){\bf P}^{p}( dx^t, dy^t).
(\pi \sigma_{1\dots n})^{\vee}= \delta^{(n)\,\vee}=\begin{pmatrix}1 0 0 0 0 -1 1 0 0 0 0 -1 1 0 0 0 0 -1 \ddots 0 0 & \ddots \ddots 0 0 0 \ddots 1 0 0 0 0 -1 0 0 0 0 0 1\end{pmatrix}.
\int d^{2}x \left| x \right|^{- 2 r}\left| 1 - x \right|^{- 2 s}= \pi \frac{\Gamma ( 1 - r ) \Gamma ( 1 - s ) \Gamma ( r + s - 1 )}{\Gamma ( r ) \Gamma ( s ) \Gamma ( 2 - r - s )}
{\cal X}{\bar H}_{\mbox{\scriptsize I}}=\Xi^{\dagger}{\cal A}^{\dagger}H_{\mbox{\scriptsize I}}+\left(1-{\cal X}\right)H_0-\left(1-\Xi^{\dagger}{\cal A}^{\dagger}\right)H_{0}+{\cal X}{\bar H}\left(1-\Xi^{\dagger}{\cal A}^{\dagger}\right).
\Delta\omega_\epsilon=\widetilde{\tau_\epsilon}:=-|\nabla\omega_\epsilon|^2\omega_\epsilon-2f_\epsilon^{-1}\nabla f_\epsilon\cdot\nabla\omega_\epsilon+(\tau_\epsilon-\langle\tau_\epsilon, \omega_\epsilon\rangle\omega_\epsilon),\{\rm{in}}\B_{\frac{r_1}2}.
ds^2 =\left[ H(r)\right]^{-\frac{\widetilde{d}}{D-2}}\, dx^\mu \otimes dx^\nu \, \eta_{\mu\nu}+ \left[ H(r)\right]^{\frac{d}{D-2}}\, \left( dr^2 +r^2 ds^2_{S^{D-p-2}}\right)
C_i(t_i)=\tfrac{t_i}{t_i-_F t_{i+1}}+\tfrac{t_{i+1}}{t_{i+1}-_F t_i}=\tfrac{t_i-_F t_{i+1}+_F t_{i+1}}{t_i-_F t_{i+1}}+\tfrac{t_{i+1}}{t_{i+1}-_F t_i}=\tfrac{F(\rho_i,t_{i+1})-t_{i+1}}{\rho_i}+t_{i+1}\kappa_i.
\left| \mathbb E_{\mu}\prod_{j=0}^{l-1}f_{jk}\, \mathbb E_{\mu}f_{lk}- \prod_{j=0}^{l-1}\mathbb E_{\mu}f_{jk}\, \mathbb E_{\mu}f_{lk}\right|= \|f\|_{L_1(P)}\left| \mathbb E_{\mu}\prod_{j=0}^{l-1}f_{jk}- \prod_{j=0}^{l-1}\mathbb E_{\mu}f_{jk}\right| = \|f\|_{L_1(P)}D_{l-1}.
\sum_{m_{1}=-j_{1}}^{j_{1}}\sum_{m_{2}=-j_{2}}^{j_{2}}\sum_{m_{3}=-j_{3}}^{j_{3}}|j_{1}m_{1}\rangle|j_{2}m_{2}\rangle|j_{3}m_{3}\rangle(\begin{matrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{matrix})=|00\rangle.
\phi_+(0)=\sqrt{m}\begin{pmatrix}\cos\left(\frac\theta2\right)\mathrm{e}^{-i\frac\varphi2}\sin\left(\frac\theta2\right)\mathrm{e}^{i\frac\varphi2}\end{pmatrix}, \phi_-(0)=\sqrt{m}\begin{pmatrix}\sin\left(\frac\theta2\right)\mathrm{e}^{-i\frac\varphi2}-\cos\left(\frac\theta2\right)\mathrm{e}^{i\frac\varphi2}\end{pmatrix}
B_{n,\mathbf{m}}^{(d)}(\mathbf{z}\mid{\boldsymbol{\omega}})+B_{n,\mathbf{m}}^{(d)}(\mathbf{z}\mid{\boldsymbol{\omega}}^{-}[j])= -\sum_{i=1}^{r}B_{n-1,\mathbf{m}_{i}}^{(d)}(z\mid \widehat{{\boldsymbol{\omega}}}(j)) \left(m_{i}+\frac{d}{2}(r-i)\right) h_{-,i}^{(d)}(\mathbf{m}).
A^S(x)=\frac{1}{x^{3/2}}\begin{pmatrix}0 1 a^{(1)}_{21}0\end{pmatrix}+\frac{1}{x}\begin{pmatrix}a^{(1)}_{11}0 0 a^{(1)}_{22}-\frac12\end{pmatrix}+\cdots.
p_+ = \frac{\hat{m}+ \omega n \sqrt{p_+}}{\binom{n}{2}}= \frac{\hat{m}+ (\sqrt2+o(1))\omega \sqrt{\hat m}}{\binom{n}{2}}\ge \frac{\hat{m}+ \omega \sqrt{\hat m \left(\binom{n}{2}-\hat m \right)/\binom{n}{2}}}{\binom{n}{2}},
\mathbf{d}_\mathbf{F}(r) =\sup_{x\in X}\sup_{F\in[x_\bullet^r]^{<\omega}}\inf_{F\leq^\mathbf{d}y}\mathbf{d}(y,x). \mathbf{d}_\Phi(r) =\sup_{x\in X}\sup_{F\in[x_\bullet^r]^{<\omega}}\sup_{\mathbin{\preceq}\in\Phi^\mathbf{d}}\inf_{F\preceq y}\mathbf{d}(y,x).
\widetilde{I}_{\varepsilon}(\xi) =I_{\varepsilon}\left( W_{\varepsilon,\xi}+\phi_{\varepsilon,\xi}\right) & =I_{\varepsilon}\left( W_{\varepsilon,\xi}\right) +o(1)=J_{\varepsilon}\left( W_{\varepsilon,\xi}\right) +\frac{\omega^{2}}{2}G_{\varepsilon}\left( W_{\varepsilon,\xi}\right) +o\left( 1\right)
f(x_1,x_2)=f_{n_{1}}(x_1,x_2)+\lim_{m\longrightarrow\infty}{\sum_{k=1}^{m}{(f_{n_{k+1}}-f_{n_{k}})}}&=\lim_{m\longrightarrow\infty}{\sum_{k=1}^{m}{f_{n_{m}}(x_1,x_2)}}for every (x_1,x_2)\notin N.
A_n^{(t,\theta_t)}=\left(\left(\mathbf{S}_n^{(t-1)}\right)^{\mathsf{T}}\mathbf{\Xi}_{n,\ell,i}^{(t)}\mathbf{Q}_{n,i}^{(t,\theta_t)}\right)_{\ell\in[\#_r^{(t)}],i\in[K_c]}, n\in\overline{\mathcal{S}}_r^{(t)}.
g(s) =\left(c_{1}\alpha_{1}\lambda_{1}^{\alpha_1-1}+c_{2}\alpha_{2}\lambda_{2}^{\alpha_2-1}\right)s^{-p}+\left(c_{1}(\alpha_{1}-1)\lambda_{1}^{\alpha_1-2}+c_{2}(\alpha_{2}-1)\lambda^{\alpha_2-2}\right)s^{p-1}+.... = \sum_{k=0}^{\infty}{b_{k}s^{k+\gamma+1}},
||\mathbb{B}''((W_1,\psi_1),(W_2,\psi_2))||_{H^s(\Gamma_T)}&\leq C\sum_{i\neq j}\Big(||W_i||_{H^s(\Gamma_T)}||\psi_j||_{W^{1,\infty}(\Gamma_T)}+||W_i||_{L^{\infty}(\Gamma_T)}||\psi_j||_{H^{s+1}(\Gamma_T)}&\quad\quad+ ||W_i||_{H^s(\Gamma_T)}||W_j||_{L^{\infty}(\Gamma_T)}\Big).
\langle D^k\xi,D^k\eta\rangle(t_0,x)=\sum_{i_1,\ldots,i_k=1}^n \langle D^k_{e_{i_1},\ldots,e_{i_k}}\xi,D^k_{e_{i_1},\ldots,e_{i_k}}\eta\rangle(t_0,x),
\sigma_{-1}= J_* - J_\ell \,=\, -\left( f^-_* - f^-_\ell\right) \,=\,-\left( \rho_* - \rho_\ell \right), \sigma_{1}=J_r - J_* \,=\, f^+_r - f^+_* \, = \, \rho_r - \rho_*.
F_2^\mathrm{s}(t,y_1,y_2):=-(1/2N)v_N(y_1-y_2)\Big\{u(t,y_1,y_2)+(\bar p\circ u)(t,y_1,y_2)\Big\}, &F_3^\mathrm{s}(t,y_1,y_2,y_3):=-N^{-1/2}v_N(y_1-y_2)\phi(t,y_2)u(t,y_3,y_1), & F_4^\mathrm{s}(t,y_1,y_2,y_3,y_4):=-(1/2N)v_N(y_1-y_2)u(t,y_3,y_1)u(t,y_2,y_4) & F_l^{\mathrm{r}}:=F_l- F_l^{\mathrm{s}}l=2,3,4.
T = \sum_{a = 1}^{3}T^{a}\otimes T^{a}= \left( \begin{array}{c c c c}{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{- 1}}&{{2}}&{{0}}\\{{0}}&{{2}}&{{- 1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\ \end{array}\right)
\sum_{i=1}^{n}Q[J_i \varphi] \ge - \frac{1}{2}{\rm Tr}A\sum_{i=1}^{n}\int_{\Omega_i}\varphi^2 \, d\mu -\frac{(n-1)c}{r_0^2}\sum_{i=1}^{n}\int_{\Omega_i}J_i^2\varphi^2 \, d\mu
[ j_{+}{}^{n}, ( j_{+}{}^{m})^{\dagger}] = n \delta_{m , n}, \qquad [ j_{-}{}^{n}, ( j_{-}{}^{m})^{\dagger}] = n \delta_{m , n}, \qquad [ j_{+}{}^{n}, j_{-}{}^{m}] = 0 .
\sum_{\sigma' \in \Sigma'}(\rho' \otimes_0 \varphi')_{\sigma'}^{\sigma''}\circ (\rho \otimes_0 \varphi)_{\sigma}^{\sigma'}= \sum_{\sigma' \in \widehat{\Sigma'}}(\rho' \otimes_0 \varphi')_{\sigma'}^{\sigma''}\circ (\rho \otimes_0 \varphi)_{\sigma}^{\sigma'}.
\alpha!=\prod_{k=1}^{\infty}\alpha_{k},\left| \alpha\right| = \sum_{k=1}^{\infty}\alpha_{k}, x^{\alpha}=\prod_{k=1}^{\infty}x_{k}^{\alpha_{k}},\binom{\alpha}{\beta}= \prod_{k=1}^{\infty}\binom{\alpha_{k}}{\beta_{k}}.
{\cal L}= - \frac{1}{2 ( p + 2 ) !}F^{\mu_{0}\mu_{1}\ldots \mu_{p + 1}}F_{\mu_{0}\mu_{1}\ldots \mu_{p + 1}}+ \frac{1}{2 ( p + 1 ) !}\frac{g^{2}}{e^{2}}\psi^{\mu_{0}\mu_{1}\ldots \mu_{p}}\psi_{\mu_{0}\mu_{1}\ldots \mu_{p}}.
\mathbb{E}\left[ \mathcal{I}_{\mathrm{a}}\right] = \mathbb{E}\left[ \sum_{x \in \Phi_{\mathrm{b}}\setminus \{i, \bar{i}\}}\frac{g_{xik}}{\Vert x \Vert^{\alpha}}\right]= \frac{2\pi \lambda s^{-(\alpha - 2)}}{\alpha - 2}.
P\{\max_{0\le s\le t}\mathcal{T}(s)< \beta\ |\V(0) = c,\N(t) = 2k+1\}= \frac{\beta}{ct}\sum_{j=0}^{k}\binom{2j}{j}\Bigl(\frac{\sqrt{c^2t^2-\beta^2}}{2ct}\Bigr)^{2j}
\Delta= \left\{\emptyset, \{F\}, \{S\}, \{FB\}, \{E\}, \{TV\}, \{S, F\}, \{FB, F\}, \{FB, S\},\{FB, S, F\}, \right. &\{FB, E\}, \{FB, TV\}, \{TV, E\}, \{FB, TV, E\},\{FB, G\}, \{TV, G\}, &\left. \{FB, TV, G\},\{E, G\}, \{FB, E, G\}, \{TV, E, G\}\right\}.
[ d \Phi ] [ d \Phi^{\dagger}] [ d \zeta ] [ d \zeta^{\dagger}] = \Pi_{x , a}\frac{d \Phi_{a}( x ) d \Phi_{a}^{\dagger}( x )}{2 \pi i}d \zeta_{a}( x ) d \zeta_{a}^{\dagger}( x ) .
\Gamma_1^h(u) = \langle(I_h -I)z,I_h (z u)\rangle_{H_x^\mathfrak{s}}, \Gamma_2^h(u) = \langle z,(I_h -I)(z u)\rangle_{H_x^\mathfrak{s}}, \Gamma_3^h(u) = \mathfrak{s}\langle|I_h z|^2I_h z -|z|^2z,I_h (z u)\rangle_{L_x^2}, \Gamma_4^h(u) = \mathfrak{s}\langle |z|^2z,(I_h -I)(z u)\rangle_{L_x^2}.
M_{\{\{1,3\},\{2,4\}\}}M_{\{\{1,3,4\},\{2\}\}}= M_{\{\{1,3\},\{2,4\},\{5,7,8\},\{6\}\}}+M_{\{\{1,3,5,7,8\},\{2,4\},\{6\}\}}begin{align*}5pt] +M_{\{\{1,3\},\{2,4,5,7,8\},\{6\}\}}+M_{\{\{1,3,6\},\{2,4\},\{5,7,8\}\}}begin{align*}5pt] +M_{\{\{1,3\},\{2,4,6\},\{5,7,8\}\}}+M_{\{\{1,3,5,7,8\},\{2,4,6\}\}}begin{align*}5pt] +M_{\{\{1,3,6\},\{2,4,5,7,8\}\}}.
A\psi_{j-1}+B\psi_j +C\psi_{j+1}= hK \ \sum_{i=1}^m q_i c_{ij}+ \frac{h^2}{2}\sum_{n=1}^{Q_1}\omega_n \rho_0(x_j+hs_n/2) L(s_n),, 2\leq j\leq N-1,
\Gamma\big(\tfrac14u^2+\tfrac12 b\big)2^{b-2}u^{1-b}e^{-\frac12z^2}z^b U\big(\tfrac14u^2+\tfrac12 b,b,z^2\big) \qquad{}= zK_{b-1}(uz)\left(\sum_{s=0}^{N-1}\frac{a_s(z)}{u^{2s}}+g_2(u,z)\right)-\frac{z}{u}K_b(uz)\left(\sum_{s=0}^{N-1}\frac{b_s(z)}{u^{2s}}+zh_2(u,z)\right),
\Big | \frac{\partial R_i}{\partial \alpha_i}\Big | \le \; Z_0 = \frac{\max\{|y_1|, |y_N|\}(1 + \frac{1}{4}|r|_{\infty}) + \frac{1}{4}|I| \max\{|d_1|,|d_N|\}}{\min\{c_i : i \in J \}}~ \forall~ i \in J,
a=\sum_{j=1}^m S_n \left(a_{1,j},\dotsc,a_{n,j}\right) =\sum_{j=1}^m \left\Vert a_{1,j}\right\Vert \dotsb \left\Vert a_{n,j}\right\Vert S_n \left(b_{1,j},\dotsc, b_{n,j}\right),
{\dot{X}}^{a}= \{H , \X^{a}\}={\cal P}^{a}( 1 +{\frac{1}{m^{2}}}{\frac{\partial g}{\partial m^{( i ) ( k )}}}m_{( i ) ( k )}+{\frac{1}{m^{2}}}{\frac{\partial g}{\partial{\bar{m}}^{( i ) ( k )}}}{\bar{m}}_{( i ) ( k )}) ,
-\kappa_5\frac{(\eta\cdot\zeta)(\eta'\cdot\zeta)}{2P_+}\frac{\partial}{\partial\theta'_\eta}\psi = \frac{\kappa_5}{2P_+}(\eta\cdot\zeta)(\eta'\cdot\zeta)\chi_\eta + \kappa_5\frac{\theta_\eta}{2P_+}(\eta\cdot\zeta)(\eta'\cdot\zeta) B.
\begin{cases}D_{A_0}\psi_0 =0 F_{A_0}^{+}= \frac{r}{2}(q(\psi_0,\psi_0)- i\Omega_X) + i\mu_0 *d*b_0 -2^{-\frac{1}{2}}r^{\frac{1}{2}}(\eta_0^* \psi_I-\psi_I^* \eta_0)=0, \end{cases}
\widehat{F}(\widehat{z}_1,\widehat{z}_2)=\begin{cases}(a-\frac{1}{c})\widehat{z}_1 +\frac{a^2b}{2}\left( \widehat{z}_1^2-2\widehat{z}_1\widehat{z}_2+\widehat{z}_2^2 \right) (a+\frac{1}{c})\widehat{z}_2 +\frac{a^2b}{2}\left( \widehat{z}_1^2-2\widehat{z}_1\widehat{z}_2+\widehat{z}_2^2 \right).\end{cases}
\mu\equiv \frac{\sqrt3h}{\xi}, \quad\chi\equiv \frac{\psi}{\xi}, \quad\nu\equiv \frac{N}{\xi^2}, \quad\zeta\equiv \frac{\pm 3 K}{\xi^2}, \quad\lambda\equiv \frac{\pm \Lambda_{\rm M}}{\xi^2}, \quad\frac{d}{dT}\equiv \xi \frac{d}{d\tau},
\frac{16\, \pi^2}{g_{i,\,{\rm bare}}^2}=\frac{16\, \pi^2}{g_{i}^2(\mu)}-b_i \, (4\pi)^\epsilon\int_0^\infty\frac{dt}{t^{1-\epsilon}}\, e^{-t\frac{\mu^2}{M^2}}\, .
z(t_1) = z(s_1) - \int_{t_1}^{s_1}z'(t) \,dt < w(x_n) -2 \theta_n g_n(s_1) - \int_{t_1}^{s_1}(-2 \theta_n g_n'(t)) \,dt &= w(x_n) -2 \theta_n g_n (t_1) & = w(x_n) + 2 \theta_n | g_n(t_1)|.
\prod_{k=1}^\infty \exp\left[-2k\left((a_k^Ba_{-k}^B+b_k^Bb_{-k}^B)\frac{1+q^{2k}}{1-q^{2k}}-(a_k^Bb_{-k}^B+b_k^Ba_{-k}^B)\frac{2q^k}{1-q^{2k}}\right)\right].
\begin{aligned}0 = \, (A \Psi \pm T_\alpha (u - \varphi))_n = \, - A + 2 A K x_n \pm \left[(u - \varphi)_{\alpha n}+\sum_{\beta<n}B_{\alpha\beta}(x_{\beta}(u-\varphi)_{nn}-x_n(u-\varphi)_{\beta n})-\sum_{\beta<n}B_{\alpha\beta}(u-\varphi)_{\beta}\right] < \, 0,\end{aligned}
\{F , G \}={\frac{1}{2}}\sum \int_{\Omega}\Biggl ({\binom{J}{M}}D_{L + J - M}{\frac{\partial g}{\partial \phi_{B}^{( K )}}}D_{K}\biggl ({\frac{\partial f}{\partial \phi_{A}^{( J )}}}D_{M}I_{A B}^{L}\biggr )
\Lambda^m(1\otimes x) = \sum_{k=1}^p \Big(1\otimes \frac{2D_k^\theta}{(1-\Delta^\theta)^{\frac12}}\Big)\cdot \Lambda^{m-1}(1\otimes [D_k^\theta,x]) -\sum_{k=1}^p\Big(1\otimes\frac1{(1-\Delta^\theta)^{\frac12}}\Big)\cdot \Lambda^{m-1}(1\otimes [D_k^\theta,[D_k^\theta,x]]).
\begin{aligned}D_{ij}^{2}= (x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}+(z_{i}-z_{j})^{2}= 2, \ \ \ \A_{ij}= 1, D_{ij}^{2}= (x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}+(z_{i}-z_{j})^{2}\geq 2, \ \ \ \A_{ij}= 0.\end{aligned}
H_{\ell\ell}\left( \beta\right) =\operatorname*{span}\left\{w_{j}\left( \beta\right) :N_{R}+1\leq j\leq2N\right\}, H_{h\ell}\left( \beta\right) =\operatorname*{span}\left\{w_{j}\left(\beta\right) :1\leq j\leq N_{R}\right\}.
\tilde{\mathbf{w}}_{\mathrm{a},ik}= \frac{1}{\sqrt{\tilde{\zeta}_{\mathrm{a},i}}}\! \left( \rho N \mathbf{I}_N \!+\!\! \sum_{l=1}^K \hat{\mathbf{h}}_{iil}\hat{\mathbf{h}}_{iil}^{\mathrm{H}}\!+\!\! \sum_{l=1}^{K'}\hat{\mathbf{h}}_{i\bar{i}l}\hat{\mathbf{h}}_{i\bar{i}l}^{\mathrm{H}}\right)^{-1}\!\!\!\!\!\! \hat{\mathbf{h}}_{iik}
\vec L_D=(I_n \otimes P_{\Omega})^{-1})(DC) \vec L_C \ \ \DC=(I_n \otimes E^T) (P_{erm})^T (P_{\partial\Omega}\otimes I_n), \vec L_C=((L_C)_1,(L_C)_2,...,(L_C)_N)^T (L_C)_j=(2 (J^- T^{-1})^T \Sigma (\sqrt{|\Lambda^-|}\vec W^--R \sqrt{\Lambda^+}\vec W^+ - S \vec G))_j.
\frac{1}{\inf_{t>\tau,x\in\Omega}v^{q+1-\varepsilon_0/2}(t,x)}=\sup_{t>\tau,x\in\Omega}\frac{1}{v^{q+1-\varepsilon_0/2}(t,x)}\le \sup_{t>\tau}\frac{\big(\int_\Omega u^{-p}\big)^{\frac{q+1-\varepsilon_0/2}{p}}}{\big(\delta_0|\Omega|\big)^{q+1-\varepsilon_0/2}}.
\mu_j \nearrow \bar{\mu}, u_j \rightharpoonup u \mbox{weakly in}W^{2,2}(M^n), u_j \rightarrow u \mbox{strongly in}L^2(M^n), f_j = - u_j + \mu_j P_{g_0}^{-1}(u_j^{\frac{n+4}{n-4}}) \rightarrow 0 \mbox{strongly in}W^{2,2}(M^n).
[ \hat{x}_{i}, \hat{x}_{j}] = i \hbar \, \frac{( 2 \beta - \beta^{\prime}) + ( 2 \beta + \beta^{\prime}) \beta \hat{p}^{2}}{( 1 + \beta \hat{p}^{2})}\left( \hat{p}_{i}\hat{x}_{j}- \hat{p}_{j}\hat{x}_{i}\right) \; .
\frac{1}{N}\sum_{i=1}^{N}\left[J(T,\,x_{i}^{N}(T))-J(0,\,x_{i}^{N}(0))-\int_{0}^{T}\left(J_{t}+\frac{1}{2}J_{xx}\right)(t,\,x_{i}^{N}(t))dt\right]
\Omega_r\left[\frac{\mu}{1+t}u_{t}\right](t,x) =\frac{\mu}{1+t}\bigg(\frac{\partial}{\partial t}\bigg)\, \Omega_r[u](t,x) , \Omega_r\left[\frac{\nu^2}{(1+t)^2}u \right](t,x) =\frac{\nu^2}{(1+t)^2}\, \Omega_r[u](t,x) .
u^{{m ( k )}}\delta u_{{m}}^{{( n )}}( \Gamma_{{( k ) ( n )}})_{{b}}^{{a}}= D ( v^{-{1}}\delta v )_{{b}}^{{a}}+ ( C \Gamma^{{( k )}}( v^{-{1}}\delta v ) \Gamma_{{( k )}}C^{-{1}})_{{b}}^{{a}}\qquad
\Sigma ( p ) ={\frac{i}{\theta}}\int_{0}^{1}d x \int{\frac{d^{3}q}{( 2 \pi )^{3}}}{\frac{\left( q_{\lambda}+ x p_{\lambda}\right) \left[ \left( p_{\rho}( 1 - x ) - q_{\rho}\right) 2 g_{\lambda}^{\rho}- 2 M T^{3}\gamma_{\lambda}\right]}{\left[ q^{2}+ p^{2}x ( 1 - x ) - M^{2}x \right]^{2}}}