harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{R}^{\Sigma}_{A, B, P}(\rho) \{\mathrm{Pair}_{A, B}\}= \rho \{\mathrm{Pair}_{A, B}^{-1}\}\{\mathrm{Pair}_{A, B}\}&= \rho \{\mathrm{Pair}_{A, B}^{-1}\bullet \mathrm{Pair}_{A, B}\}&= \rho \{\mathrm{id}_{\Sigma(\Sigma(\Gamma, A), B)}\}&= \rho.
\iint e^{i\lambda \Phi(y,x,\xi,\eta)}\tilde{\tilde{a_{N}}}(x,\xi)f_{i_{1}....i_{m}}(z){\tilde{b}}^{i_{1}}(x,\xi)\cdots{\tilde{b}}^{i_{m}}(x,\xi)dxd\xi =0.
k!p_{k-1}^2 = k! \sum\limits_{1 \leq u,v \leq k}\rho_V^{u+v-2}\frac{\rho_R^{2k-u-v}}{(k-u)!(k-v)!}&> \sum\limits_{i=0}^{k-2}\rho_V^{i}\rho_R^{2k-i-2}\sum\limits_{j=1}^{i+1}\frac{k!}{(k-2-i+j)!(k-j)!}
\# R_{1}+ \# R_{2}+ \# M+ \alpha = k+1 \# H_{3}+ \# M + \beta + \gamma_{1}+ \gamma_{2}= k+1 \# R_{1}+ \# M + \alpha+ \beta+ \gamma_{1}\leq k \# R_{2}+ \# M + \alpha+ \beta+ \gamma_{2}\leq k
[L_{m},L_{n}]=(m-n)L_{m+n}, &[L_{m},Y_{n}]=-(m+n)Y_{m+n}, &[L_{m},M_{n}]=-(3m+n)M_{m+n}, &[Y_{m},Y_{n}]=(m-n)M_{m+n}, &[Y_{m},M_{n}]=[M_{m},M_{n}]=0,
-C_1(1-\eta)^{\alpha_0}\leq\partial_\xi w+\partial_\tau w\leq \frac{(\delta b)^{\frac{1}{\alpha_0}}}{K}(1-\eta)^{\alpha_0}, & -C_1(1-\eta)^{\alpha_0}\leq\partial_\tau w\leq b\delta(1-\eta)^{\alpha_0}in [0,T^*]\times[0,X]\times[0,1].
C_2(\lambda) = \left[\begin{array}{c|ccc}\lambda P_d+P_{d-1}-I_m P_{d-2}\lambda I_m \ddots \vdots \ddots -I_m P_0 \lambda I_m\end{array}\right]= \left[ \begin{array}{c|c}M_2(\lambda) L_{d-1}(\lambda)^T\otimes I_m\end{array}\right],
b_{n-1}^2 - \tilde{b}_{n-1}^2 = \big(b_{n-1}- \tilde{b}_{n-1}\big) \big(b_{n-1}+ \tilde{b}_{n-1}\big) = \big(b_{n-1}- \tilde{b}_{n-1}\big) \big(2\tilde{b}_{n-1}+ b_{n-1}- \tilde{b}_{n-1}\big).
(\partial_{0,1}^{P})^{2}=0, \partial_{1,0}^{\gamma}\partial_{0,1}^{P}+ \partial_{0,1}^{P}\partial_{1,0}^{\gamma}=0, \partial_{2,-1}^{\sigma}\partial_{0,1}^{P}+ \partial_{0,1}^{P}\partial_{2,-1}^{\sigma}+ (\partial_{1,0}^{\gamma})^{2}=0, \partial_{2,-1}^{\sigma}\partial_{1,0}^{\gamma}+ \partial_{1,0}^{\gamma}\partial_{2,-1}^{\sigma}= 0.
G_{t t}= - \kappa e^{4 \kappa \Psi ( r )}\left[ 2 \Psi^{'^{\prime}}+ 4 \frac{\Psi^{'}}{r}- \kappa ( \Psi^{'})^{2}\right] \ , \ \G_{r r}= - \frac{G_{\theta \theta}}{r^{2}}= \frac{G_{\varphi \varphi}}{r^{2}\sin^{2}\theta}= \kappa^{2}( \Psi^{'})^{2}
Kt^2 n^{2\alpha-3}\Big( \sum_{y=1}^{K-1}y a(y)\Big)^2\sum_x f'\big(\tfrac{x}{n}\big)^2 \int \psi_x^{K}(\eta)^2 \mu_\rho(d\eta) \leq C(f,\rho)\frac{t^2 n^{2\alpha-2}}{K},
{\cal L}_{\rm g}=-\frac{1}{4}\Delta \, F_{\mu\nu}F^{\mu\nu}+\frac{ik}{4}(G^{(1)}_{j}\partial_{\mu}{z^j}-G^{(1)}_{\bar{\jmath}}\partial_{\mu}{\bar{z}^{\bar{\jmath}}})\omega^{\mu}-\frac{k}{4l^2}h_{\mu}\omega^{\mu}+\frac{1}{4}{\rm Im}f F_{\mu\nu}\widetilde{F}^{\mu\nu},
(U^i\cdot\nabla\Theta^{i+1},\phi)-(U\cdot\nabla\Theta,\phi) &=((U^i-U)\cdot\nabla\Theta^{i+1},\phi)-(U\cdot\nabla(\Theta^{i+1}-\Theta),\phi) &=-((U^i-U)\Theta^{i+1},\nabla\phi)+(U(\Theta^{i+1}-\Theta),\nabla\phi).
\sum_{n \leq x}c_K(n) \log n = \left(\rho_Kx+f_1(x,K)\right)\log x - \int_1^x \left(\rho_Ku+f_1(u,K) \right) \frac{du}{u}& = \rho_K(x\log x -x+1)+f_1(x,K)\log x - \int_1^x f_1(u,K) \frac{du}{u}
\begin{array}{l}E_r=\displaystyle\frac{gJ\cos\theta}{4}\frac{2R^2+2r^2+Rr\sin\theta}{(R^2+r^2+2Rr\sin\theta)^{5/2}}~~~ E_\theta=-\displaystyle\frac{gJ\sin\theta}{4}\frac{2R^2+2r^2+Rr\sin\theta}{(R^2+r^2+2Rr\sin\theta)^{5/2}}~ E_\varphi=0 .\end{array}
\liminf_{\lambda \rightarrow \infty}\frac{N_{sc}[\mathfrak{e},\lambda \tilde{V}]}{N_{sc}^{>}[\mathfrak{e},\lambda \tilde{V}]}<\infty,\;\limsup_{\lambda \rightarrow \infty}\frac{N_{sc}[\mathfrak{e},\lambda \tilde{V}]}{N_{sc}^{>}[\mathfrak{e},\lambda \tilde{V}]}=\infty .
E_{k}\approx mc^{2}(1+\frac{1}{2}\frac{v^{2}}{c^{2}}+\frac{3}{8}\frac{v^{4}}{c^{4}})-mc^{2}=\frac{1}{2}mv^{2}+\frac{3}{8}m\frac{v^{4}}{c^{2}}
\sum_{i_n=1}^{i_{n-1}-1}\frac{(i_n + \alpha_nt + \beta)!}{(i_n + \alpha_{n-1}t)!}= \frac{(i_{n-1}+ \alpha_nt + \beta)!}{(i_{n-1}+ \alpha_{n-1}t - 1)!(\beta + 1 - (\alpha_{n-1}- \alpha_n)t)}\qquad\qquad\qquad\qquad\qquad\qquad- \frac{(\alpha_nt + \beta + 1)!\ell(\alpha_{n-1}, \alpha_n)}{(\alpha_{n-1}t)!(\beta + 1 - (\alpha_{n-1}- \alpha_n)t)}
\begin{aligned}(1-\lambda)\sum_i p_{0,i}e^{\varpi \left( \frac{p_{0,i}}{q_{0,i}}\right)}+ \lambda \sum_i p_{1,i}e^{\varpi \left( \frac{p_{1,i}}{q_{1,i}}\right)}\ge \sum_i p_{\lambda,i}e^{\varpi \left( \frac{p_{\lambda,i}}{q_{\lambda,i}}\right)}, \end{aligned}
a_{\mathcal{F}_{k l}}(\xi^i_{\mathcal{F}_{k l}},v)=\lambda_{\mathcal{F}_{k l}}^i b_{\mathcal{F}_{k}}(\xi^i_{\mathcal{F}_{k l}}, v)\forall v\in V^0_h(\mathcal{F}_{k l})
\left( \begin{array}{c c}{E - m}&{2 i \left( \frac{\partial}{\partial_{z}}- i e A_{z}\right)}\\{- 2 i \left( \frac{\partial}{\partial_{\bar{z}}}- i e A_{\bar{z}}\right)}&{- E - m}\\ \end{array}\right) \left( \begin{array}{c}{\chi^{1}}\\{\chi^{2}}\\ \end{array}\right) = 0
\begin{aligned}III' = 4\pi c - c^{-1}\int W(hx) Q_{a, c}(x)^2 \, dx + \frac12 c^{-1}\int \partial_x[ W(hx) (x-a) ] Q_{a, c}(x)^2 \, dx &= 4 \pi c + c^{-1}\int [-\frac12 W(hx) + \frac12 hW'(hx)(x-a)] Q_{a, c}(x)^2 \, dx\end{aligned}
Z_{2}( s ) = \frac{1}{8}\zeta ( s + 1 ) \Gamma \left( \frac{3 - s}{2}\right) \Gamma \left( \frac{1 + s}{2}\right) \left[ - 1 + 3 ( 1 + s ) - \frac{5}{8}( 3 + s ) ( 1 + s ) \right] .
|f\rangle \;=\; \sum_{m,q=-\infty}^{\infty}\int_{-1}^{+1}dx\; |x,m,q\rangle \, f^{m,q}(x) \;=\sum_{m,q=-\infty}^{+\infty}\;\;\;\sum_{j=sup(|m|,|q|)}^\infty |j,m,q\rangle \;f^{m,q}_j
-\frac{9}{2}R =R-\frac{11}{2}R=g^{\mu\nu}\left(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\right) &=\frac{1}{2}g^{\mu\nu}\langle i_{\partial_\mu}G,i_{\partial_\nu}G\rangle-\frac{11}{4}|G|^2 &=2|G|^2-\frac{11}{4}|G|^2 &=-\frac{3}{4}|G|^2.
-{\frac{\lambda^{2}}{4}}( \overline{{{\psi}}}_{+}\Gamma_{\mu}\partial_{\nu}\psi_{+}\overline{{{\psi}}}_{+}\Gamma^{\nu}\partial^{\mu}\psi_{+}+ \overline{{{\psi}}}_{-}\Gamma_{\mu}\partial_{\nu}\psi_{-}\overline{{{\psi}}}_{-}\Gamma^{\nu}\partial^{\mu}\psi_{-})
f-\frac{\partial f}{\partial\kappa_{i}}\kappa_{i}=H_{k}^{\frac{1}{k}}\left(1-\frac{\sigma_{k-1}(\kappa|i)\kappa_{i}}{k\sigma_{k}(\kappa)}\right)\geq \frac{1}{k}H_{k}^{\frac{1}{k}}\left(1-\frac{\sigma_{k-1}(\kappa|i)\kappa_{i}}{\sigma_{k}(\kappa)}\right).
\Gamma(E)=\frac{e^{2}}{2\pi\hbar mc^{3}\epsilon_{0}}\frac{1}{N_{p}}\sum_{i}^{N_{p}}\Delta E_{1,0}(x_{i})^{2}|f_{1,0}(x_{i})|g(E-\Delta E_{1,0}(x_{i}),\delta)
T_n(\tau_1, \tau_2)=\sup\limits_{\lambda\in \left[\tau_1, \tau_2\right]}\frac{\left|W_n(\lambda)\right|}{\left\{\int_0^{\lambda}\left(W_n(r)-\frac{c_n(r)}{c_n(\lambda)}W_n(\lambda)\right)^2dr+\int_{\lambda}^1\left(W_n(r)-\frac{1-c_n(r)}{1-c_n(\lambda)}W_n(\lambda)\right)^2dr\right\}^{\frac{1}{2}}}.
\langle \overline{{\phi}}^{2}( x ) \rangle_{\beta}= \frac{1}{1 2 \beta^{2}}( 1 + \eta^{2}) - \eta^{2}\xi \frac{\sqrt{2}}{8 \pi^{2}r^{2}}S_{1}+ \eta^{2}\frac{1}{3 2 \pi^{2}r^{2}}S_{2},
L_{n}^{(\alpha)}(z)=\frac{z^{-\alpha}e^{z}}{n!}\frac{d^{n}}{dz^{n}}(z^{n+\alpha}e^{-z})=\frac{\Gamma(\alpha+n+1)/\Gamma(\alpha+1)}{n!}_{1}F_{1}(-n,\alpha+1,z).
\tilde q_{k+1}=\tilde q_k+\epsilon\,v_k, v_{k+1}=v_k-\epsilon\,M^{-1}\nabla_qV(\tilde q_{k+1})+\epsilon\,(\lambda_{k+1})_{\alpha}M^{-1}\mu^{\alpha}(\tilde q_{k+1}), \mu^{\alpha}(\tilde q_{k+1})\,v_{k+1}=0,
\varrho_* \Big( \sum \limits_{k = 1}^{n}\, f_k \, d' f_k' \Big) \, = \, \sum \limits_{k = 1}^{n}\, \big( r_{U_{\iota}\cap U}^U (f_k) \, \# \, 1\big) \, d \big( r_{U_{\iota}\cap U}^U (f_k') \, \# \, 1 \big),
c_{(a+1,1^{b-1})(c^k)}^{(a+c,c^{k-1},1^b)}\ge c_{(a+1,1^{b-1})(c)}^{(a+c,1^b)}\ge c_{(a+1)(c)}^{(a+c,1)}\ge c_{(1)(c)}^{(c,1)}=1
\phi_1 := 16 \frac{\eta(2\tau)^4 \eta(8\tau)^4}{\eta(4\tau)^4}\phi_{-2, 1}(\tau, z) = (16 \zeta^{\pm 1}- 32)q + (-32 \zeta^{\pm 2}+ 128 \zeta^{\pm 1}- 192) q^2 + O(q^3)
(p-1)\eta^2 u_k^{p-2}(a\nabla u_k,\nabla u_k)+2u_k^{p-1}\eta (a\nabla u_k,\nabla \eta) = \frac{4(p-1)}{p^2}(a\nabla (\eta u_k^{p/2}),\nabla (\eta u_k^{p/2})) \;\;\;\; - \frac{(2p-4)}{p}u_k^{p-1}\eta(a\nabla u_k,\nabla \eta)-\frac{4(p-1)}{p^2}u_k^p(a\nabla \eta,\nabla \eta).
\frac{2(1-\kappa^{2})(144\kappa^{8}+23\kappa^{6}+27\kappa^{4}-6\kappa^{2}+1)}{\beta^{3}\sigma_{\kappa}^{3}(4\kappa^{2}-1)^{3}(144\kappa^{4}-25\kappa^{2}+1)}
Q_{n}=1+(\begin{matrix}n\\ 2\end{matrix})+\frac{1}{1\times2}(\begin{matrix}n\\ 2\end{matrix})(\begin{matrix}n-2\\ 2\end{matrix})+\frac{1}{1\times2\times3}(\begin{matrix}n\\ 2\end{matrix})(\begin{matrix}n-2\\ 2\end{matrix})(\begin{matrix}n-4\\ 2\end{matrix})+\cdot\cdot\cdot.
L_{1 1}^{( \pm )}= q^{\pm \frac{1}{2}}q^{\pm J}= q^{\pm \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{0}}\\ \end{array}\right)}, \; \; L_{2 2}^{( \pm )}= q^{\pm \frac{1}{2}}q^{\mp J}= q^{\pm \left( \begin{array}{c c}{{0}}&{{0}}\\{{0}}&{{1}}\\ \end{array}\right)}, \; \;
\Delta\left(a\right) = \int_0^{\infty}\frac{\ln\left(\left|a\right| - ix\right)}{\cosh\left(\pi x\right)}\, \mathrm{d}x + \int_0^{\infty}\frac{\ln\left(\left|a\right| + ix\right)}{\cosh\left(\pi x\right)}\, \mathrm{d}x
\Omega_{X}^{1}(logD)=O_{X}\frac{dz_{1}}{z_{1}}\oplus\cdot\cdot\cdot\oplus O_{X}\frac{dz_{k}}{z_{k}}\oplus O_{X}dz_{k+1}\oplus\cdot\cdot\cdot\oplus O_{X}dz_{n}.
{\frac{\delta \phi}{\Delta \phi}}={\frac{H \phi}{M_{\mathrm{p}}^{2}}}( 1 + 6 | \xi | )^{1 / 2}\simeq \left({\frac{2 \pi \lambda}{3}}\right)^{1 / 2}{\frac{\phi^{3}}{M_{\mathrm{p}}^{3}}}> 1 \ .
({\cal E}\xi^{\mu})_{\nu}\equiv [ \bot_{\mu\nu}D^{a}D_{a}+ (n_{\nu i}\widetilde{\nabla}^{a}n^{i}_{\mu}) D_{a}+h^{\alpha\beta}\bot^{\sigma}{_{\nu}}\bot^{\lambda}{_{\mu}}R_{\sigma\alpha\lambda\beta}- 2 \omega^{a}{_{ij}}n_{\nu}{^{i}}\widetilde{\nabla}_{a}n^{j}_{\mu}] \xi^{\mu}= 0;
h_{x y}^{\prime \prime}+{\frac{1 - 3 u^{2}}{u f}}h_{x y}^{\prime}+{\frac{1}{( 2 \pi T )^{2}f^{2}u}}\biggl ( f{\frac{\partial^{2}h_{x y}}{\partial z^{2}}}-{\frac{\partial^{2}h_{x y}}{\partial t^{2}}}\biggr )
D ( P^{\dagger}) = \{F_{\psi}\in L^{2}( 0 , \infty ) : F_{\psi}\mathrm{a b s .}\mathrm{c o n t .}, \int_{0}^{\infty}\mathrm{d}q |{\frac{\mathrm{d}F_{\psi}}{\mathrm{d}q}}|^{2}< \infty \},
\P_x(Z_d(n_1) \leq \xi_1, \ldots, Z_d(n_m) \leq \xi_m) & = \frac{1}{h(x^0)}\int_{I^{\xi_1}}\ldots \int_{I^{\xi_m}}\det(f_{n_1-d+i}(x_j^1 - x_i^0))\det(f_{n_2 - n_1}(x_j^2 - x_i^1)) & \ldots \det(f_{n_m - n_{m-1}}(x_j^m - x_i^{m-1})) \Delta(x^m) \prod_{k=1}^m \prod_{j=1}^d dx_j^k.
\int_{B_{\mathcal{R}}}(|\tilde{u}_k(x)|^{t(x)}+ |\tilde{v}_k(x)|^{t(x)})\bigg(\int_{B_{\mathcal{R}}^{c}}\frac{| \phi (x)- \phi (y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}\,dy\bigg)\,dx \leqslant c_{6}\int_{B_{\mathcal{R}}}(|\tilde{u}_k(x)|^{t(x)}+|\tilde{v}_k(x)|^{t(x)})\,dx.
{\cal T}_{o}\, = \, \quad \left[ \begin{array}{c c c c}{{1}}&{{\qquad 0}}&{{\qquad 0}}&{{\qquad 0}}\\{{}}&{{}}&{{}}&{{}}\\{{1}}&{{\qquad 1}}&{{\qquad 0}}&{{\qquad 0}}\\{{}}&{{}}&{{}}&{{}}\\{{- 2}}&{{\qquad 0}}&{{\qquad 1}}&{{\qquad 0}}\\{{}}&{{}}&{{}}&{{}}\\{{0}}&{{\qquad 0}}&{{\qquad 0}}&{{\qquad q}}\\ \end{array}\right] \hphantom{x x x x x x x x x x x x x x x x x x}
L_{a d d i t i o n a l}= - i l ({\dot{\Theta}}^{\nu}D \Theta_{\mu}E_{\nu}^{\alpha}E_{\alpha}^{\mu}+ D \Theta^{\nu}D \Theta_{\mu}D X^{\underline{{k}}}({\tilde{\Omega}}_{\underline{{k}}})_{\nu}^{\mu})
p(\mathbf{y}|\mathcal{H}_1) =\frac{1}{\pi^M\mathrm{det}(\mathbf{C}_s\!+\!\mathbf{C}_w)}\mathrm{exp}\left(-\mathbf{y}^H(\mathbf{C}_s\!+\!\mathbf{C}_w)^{-1}\mathbf{y}\right) p(\mathbf{y}|\mathcal{H}_0) =\frac{1}{\pi^M\mathrm{det}(\mathbf{C}_w)}\mathrm{exp}\left(-\mathbf{y}^H\mathbf{C}_w^{-1}\mathbf{y}\right)\;,
D_3 := \{x_2 + x_4 = 0, 2x_1^2 + 2x_3^2 + 2x_1 x_3 - 1 = 0 \}D_4 := \{x_2 - x_4 = 0, 2x_1^2 + 2x_3^2 + 2x_1 x_3 - 1 = 0 \}D_5 := \{x_1 + x_4 = 0, 2x_2^2 + 2x_3^2 + 2x_2 x_3 - 1 = 0 \}.
S_{n}\left( y,a,b,c\right) =\left( -1\right)^{n}\sqrt{\frac{\left(a+b\right)_{n}\left( a+c\right)_{n}}{n!\left(b+c\right)_{n}}}_{3}F_{2}\left(\begin{array}{ccc}-n a+it a-it . a+b a+c\end{array}\mid 1\right) ,
\mathrm{d}{x}_t^{i,N}= v_t^{i,N}\, \mathrm{d}t\,, mathrm{d}{v}_t^{i,N}= - v_t^{i,N}\mathrm{d}t + G\left(\frac{\frac 1N \sum_{j=1}^N \varphi (x_t^{i,N}- x_t^{j,N}) v_t^{j,N}}{\frac 1N \sum_{j=1}^N \varphi (x_t^{i,N}- x_t^{j,N})}\right) \mathrm{d}t + \sqrt{2\sigma}\mathrm{d}W_t^{i}\,,
f^{n+1}_\omega(z,w)=\begin{pmatrix}1/2^n \alpha_n 0 1\end{pmatrix}\begin{pmatrix}z w\end{pmatrix},\qquad\textrm{with}\alpha_n=\begin{cases}\frac{\alpha_n}{2}\textrm{if}f_{T^n\omega}=g \frac{\alpha_n}{2}+1 \textrm{if}f_{T^n\omega}=h.\end{cases}
{\cal F}( \Lambda_{2 4}) = \wedge_{i = 1}^{3}d \lambda_{i}\frac{a_{1 2}a_{1 5}^{2}a_{2 5}^{2}a_{3 5}^{2}a_{3 6}a_{4 6}}{a_{4 5}a_{5 6}}\prod_{k < l}a_{k l}^{- 3}\Theta ( \Lambda_{2 4}) ,
\mbox{Fraction of non-zero entries}< \frac{n^m(\displaystyle\sum_{i=1}^{t}\binom{m+e_i-1}{m})}{n^m N_m}& < \frac{\displaystyle\sum_{i=1}^{t}\binom{m+e_i-1}{m}}{\displaystyle\sum_{i=1}^{t}p_i^{m}\binom{m+e_i-1}{m}}.
\widetilde{\phi}=e\Omega^{2}\delta (\omega -\overrightarrow{k}\cdot \overrightarrow{v})\left( \omega^{2}-v_{s}^{2}k^{2}\right) /2\pi\varepsilon k^{2}\left( v_{s}^{2}k^{2}-\omega^{2}-\Omega^{2}\right)
d s^{2}={\frac{1}{4 q^{2}}}\left( q{\frac{d}{d q}}\right)^{2}K \, d q^{2}+ \left( q{\frac{d}{d q}}\right) K \, \left[{\frac{2}{3}}d \hat{z_{i}}d \hat{\bar{z_{i}}}-{\frac{2}{9}}| \hat{z_{i}}d \hat{\bar{z_{i}}}|^{2}\right] +{\frac{4}{9}}\left( q^{2}{\frac{d^{2}}{d q^{2}}}\right) K \, | \hat{z_{i}}d \hat{\bar{z_{i}}}|^{2}.
A\cdot B=(\begin{matrix}A^{0}&A^{1}&A^{2}&A^{3}\end{matrix})(\begin{matrix}1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{matrix})(\begin{matrix}B^{0}\\B^{1}\\B^{2}\\B^{3}\end{matrix})
\begin{vmatrix}\det\left((\lambda I_N-H)_{k|k}\right) \det\left((\lambda I_N-H)_{k|k+1}\right) \det\left((\lambda I_N-H)_{k+1|k}\right) \det\left((\lambda I_N-H)_{k+1|k+1}\right)\end{vmatrix}=f(\lambda)\det\left((\lambda I_N-H)_{kk+1|kk+1}\right),
{\cal H}_3={\mathrm e}^{-{\mathrm i}\frac{\pi}{2}{\cal S}_2}{\mathrm e}^{-{\mathrm i}\frac{\pi}{2}{\cal S}_3}{\cal H}_1{\mathrm e}^{{\mathrm i}\frac{\pi}{2}{\cal S}_3}{\mathrm e}^{{\mathrm i}\frac{\pi}{2}{\cal S}_2}={\cal S}_3^2-{\cal S}_2^2
L_{m a s s}= A \sigma^{\frac{1}{1 - 2 \delta}}( f_{\mu}f^{\mu})^{\frac{\delta}{2 \delta - 1}}- \frac{D \delta}{2 \delta - 1}\epsilon^{\mu \nu \rho}f_{\mu}\partial_{\nu}f_{\rho}+ \frac{1}{2}( \partial_{\mu}\phi )^{2}.
\left[g_{ij}\right]=\left[\begin{array}{ccc}1 0 0 0 \rho^{2}0 0 0 1\end{array}\right]\,\,\,\,\,\,\,\,\,\,\&\,\,\,\,\,\,\,\,\,\,\left[g^{ij}\right]=\left[\begin{array}{ccc}1 0 0 0 \frac{1}{\rho^{2}}0 0 0 1\end{array}\right]
R^{mj}_{pn}\widetilde{R}^{in}_{mq}=\widetilde{R}^{mj}_{pn}R^{in}_{mq}=\delta^i_p\,\delta^j_q\,,\ \ overline{R}{}^{mj}_{pn}\widetilde{\overline{R}}{}^{in}_{mq}=\widetilde{\overline{R}}{}^{mj}_{pn}\overline{R}{}^{in}_{mq}=\delta^i_p\,\delta^j_q\,.
\Phi_{L}\left( p^{\mu}\right) =\Lambda_{L}\left( \dot p^{\mu}\to p^{\mu}\right)\Phi_{L}\left( \dot p^{\mu}\right) =e^{i\vec{J}\cdot (\vec{\vartheta}+i\vec{\varphi})}~\Phi_{L}\left( \dot p^{\mu}\right).
\begin{pmatrix}\phi_{d_1,d_0}-b(d_2/d_0)\phi_{d_1,d_3}-\phi_{d_2,d_0}a(d_1/d_0)\phi_{d_2,d_3}\end{pmatrix}\! \colon\{\cal E}_{d_0}\times{\cal E}_{d_3}\to{\cal E}_{d_1}\times{\cal E}_{d_2}
\sum_{l,m}\sum_{i=1}^n |T_l ||C_{k,l,m}^{(i)}| \geq \sum_{l,m}|T_l ||C_{k,l,m}| - Mn \sum_l |T_l | &\geq \bigg| \bigcup_l T_l C_{k,l}\bigg| - \beta |S_k | &\geq (1-2\beta ) |S_k | .
S_M=-\int d^{10}x\sqrt{-g}[{1\over{60}}M_{m_1...m_5}M^{m_1...m_5}-\L^{(1)}_{m_1...m_5}{M}^{m_1...m_5}+\sum_{n=0}^{\infty}(-1)^n\L^{(n+1)}\L^{(n+2)}]
\|T_{\Lambda}^{(j)}g_i\| =\|\sum_{i\in J}\Lambda_{ij}^*g_i\|=\sup_{\|g\|=1}\left| \left\langle g,\sum_{i\in J}\Lambda_{ij}^*g_i\right\rangle \right| &\le\sup_{\|g\|=1}(\sum_{i\in J}\|\Lambda_{ij}g\|^2)^{1/2}(\sum_{i\in J}\|g_i\|^2)^{1/2}&=\|T_{\Lambda}^{(j)}\|(\sum_{i\in J}\|g_i\|^2)^{1/2}&\le \sqrt{B_j}(\sum_{i\in J}\|g_i\|^2)^{1/2}
\left.\frac{\partial}{\partial t}\rho_{\phi}(t) (x) \right|_{t=0}= \frac{\sum_{i=1}^{n}e^{i}(\nabla_{Z}e_{i}) \rho_{\phi}[\iota]^{2}}{2 \rho_{\phi}[\iota]}+\frac{\sum_{i=1}^{n}f^{i}(\nabla_{Z}(\phi e_{i})) \rho_{\phi}[\iota]^{2}}{2 \rho_{\phi}[\iota]}-\frac{\eta^{*}(\phi (Z)) \rho_{\phi}[\iota]^{2}}{2 \rho_{\phi}[\iota]}.
k' = tkt^{-1}k^{-1}= \left( I + \begin{pmatrix}a+c -a+b-c+d c -c+d \end{pmatrix}p \right) \left( I - \begin{pmatrix}a b c d \end{pmatrix}p \right) &= I + \begin{pmatrix}c -a-c+d 0 -c \end{pmatrix}p.
S = \frac{1}{2 \kappa_{5}^{2}}\int d z \, d^{4}x \, \sqrt{g}\left( \frac{1}{2 e^{2}}g^{i j}g^{k l}F_{i k}^{a}F_{j l}^{a}+ \frac{1}{2}\left({\cal M}^{2}\right)_{a b}\, g^{i j}A_{i}{}^{a}A_{j}{}^{b}\right)
\Psi = \left(\begin{matrix}\Psi_{11}\Psi_{12}\Psi_{21}\Psi_{22}\end{matrix}\right),\Psi_{12}(\zeta) = \frac{i}{\pi}K_0\big(2\zeta^{1/2}\big), \Psi_{22}(\zeta) = -2\zeta^{1/2}K_0'\big(2\zeta^{1/2}\big).
g^{[k](\alpha)}_{ab}(z)=\frac{(-1)^{(a+1)\ell}\epsilon_{ab}^{[k](\alpha)}(z)}{\tau_{k}^{(\alpha)}}\prod_{i=1}^{\ell}c_{i}^{(\alpha)}\left( \det\big(V^{(\ell)}_{\{z_{i}\}}\big)^{2}\prod_{j=1}^\ell(z-{z}_{j})^{1-2a}\right).
\begin{array}{c c c c c c c c c}{{}}&{{}}&{{}}&{{}}&{{\frac{1}{l_{M}^{3}}}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{}}&{{\mid}}&{{}}&{{}}&{{}}&{{}}\\{{\frac{R_{1}}{l_{M}^{3}}}}&{{-}}&{{\frac{R_{1}R_{2}}{l_{M}^{6}}}}&{{-}}&{{\frac{R_{1}R_{2}R_{3}}{l_{M}^{9}}}}&{{-}}&{{\frac{R_{1}R_{2}R_{3}R_{4}}{l_{M}^{9}}}}&{{- \dots -}}&{{\frac{1}{R_{d + 1}}}}\\ \end{array}
X_{l , m}^{( L )}\equiv ( \partial_{A_{1}}\cdots \partial_{A_{l + 2 m}}\omega^{L}) \, K_{[ l , m ]}^{A_{1}\dots A_{l + 2 m}, B_{1}\dots B_{l + 2 m}}\, ( \partial_{B_{1}}\cdots \partial_{B_{l + 2 m}}\omega^{\prime L}) \; .
M_{p}^{2}= M_{5}^{3}\int_{- \infty}^{\infty}e^{2 A_{0}( y )}d y = \frac{4 M_{5}^{3}\sqrt{\pi}}{a ( k^{2}- 1 )}\frac{\Gamma \left( ( k^{2}+ 3 ) / 4 ) \right)}{\Gamma \left( ( k^{2}+ 1 ) / 4 \right)},
I_{\textbf{s}\rightarrow\textbf{i}}(l,\tau) = \frac{1}{K^2}\left|\sum\limits_{k=-K+1}^{K-1}G_{|k|}\int\limits_0^T e^{j2\pi(\frac{k(t+\tau)}{KT}+\frac{l t}{T})}dt\right|^2 = \frac{T}{K^2}\left|\sum\limits_{k=-K+1}^{K-1}G_{|k|}e^{j\pi\frac{k}{K}\tau}(\pi(\frac{k}{K}+l)) \right|^2,
\frac{1}{2}\left(a^2(s,\mu)-a^2(1,\mu)\right) =-2\mu \int_1^s a(u,\mu)\mathrm{d}u-2\beta\int_1^s\left(G(u)-u\right)\mathrm{d}u a^2(s,\mu) =(Q^{\prime})^2(0,\mu)-4\mu \int_1^s a(u,\mu)\mathrm{d}u-4\beta\int_1^s\left(G(u)-u\right)\mathrm{d}u a^2(s,\mu) \geq(Q^{\prime})^2(0,\mu)-4\beta\int_1^{r}\left(G(u)-u\right)\mathrm{d}u.
\tilde{\phi}= \phi -{R_y \over \ell^2}\, a_2 \varphi -{R_y \over \ell^3}\, a_1 \tau, \quad \tilde{\psi}= \psi -{R_y \over \ell^2}\, a_1 \varphi -{R_y \over \ell^3}\, a_2 \tau,
\vert \mathcal{L}(D)^n \cap \bigcap_{j=1}^r U_{P_j}\vert = \vert \mathcal{L}(D) \cap \bigcap_{j=1}^r P_j \vert^n = q^{n \ell(D)}\prod_{j=1}^r q^{-n \deg(P_j)}.
\mathcal S_0^{\frac{1}{2}}(\partial\Omega):=\left\{f\in L^2(\partial\Omega): f=\sum_{j=1}^{\infty}\hat b_j\hat v_{j,0}{\rm\with\}\left(\sqrt{\lambda_j(0)}\hat b_j\right)_{j=1}^{\infty}\in l^2\right\}.
V ( y_{i}) = \frac{- k \, \pi^{\frac{D - 2}{2}}}{\Gamma ( \frac{D - 2}{2}) V_{T}}\int_{0}^{\infty}d x \, \prod_{i = 1}^{D}\theta_{3}\Big ( \frac{y_{i}}{L_{i}}, \frac{i x}{L_{i}^{2}}\Big ) \, ,
\widetilde\Sigma = \mu(f \otimes f) + \frac{1}{N}\sum_{n=1}^{N-1}\sum_{\substack{m \in [n]_K m < n}}\mu(f^{n-m}\otimes f) + \frac{1}{N}\sum_{n=0}^{N-2}\sum_{\substack{m \in [n]_K m > n}}\mu(f \otimes f^{m-n}).
\Tilde{e}_{i}^{\pm P}T=\mathrm{PR}\circ\Tilde{e}_{i}^{P}\circ \mathrm{DPR}(T), \Tilde{f}_{i}^{\pm P}T=\mathrm{PR}\circ\Tilde{f}_{i}^{P}\circ \mathrm{DPR}(T) ,
I_1 = \mathbb{E}[\int_0^T\int_D\{H(t,x)-\widehat{H}(t,x)-\widehat{p}(t,x) [A_uY(t,x)-A_{\hat{u}}\widehat{Y}(t,x) +\widetilde{a}(t,x)] - \widehat{q}(t,x)\widetilde{b}(t,x) -\int_{\mathbb{R}}\hat{r}(t,x,\zeta)\tilde{c}(t,x,\zeta)\nu(d\zeta)\}dx dt].
B ( z ) = i \frac{8 m_{D}}{g_{m}^{2}}\frac{\mathrm{e}^{- m_{D}| z |}}{1 + \mathrm{e}^{- 2 m_{D}| z |}}, \psi ( z ) = 4{}\mathrm{s g n}{}z \cdot \arctan \left( \mathrm{e}^{- m_{D}| z |}\right) .
\lambda_{}=\lambda^{\rm N}\, ,\ \,\,\,\mu_{}=\mu^{\rm N}\, ,\!\mu_{}=\varkappa^{\rm N}\, , \alpha_1 =\frac{2}{L_{\rm c}^2 \, \mu^{\rm N}}\gamma^{\rm N}\, ,\alpha_2=\frac{2}{L_{\rm c}^2 \, \mu^{\rm N}}\beta^{\rm N},\qquad\, \alpha_3 =\frac{2}{L_{\rm c}^2 \, \mu^{\rm N}}\alpha^{\rm N}\, .
2x\cdot\mu_n(x) + (p_0-1)\cdot (\sigma_n(x))^2 & \qquad = \frac{2x\cdot\mu(x)}{1+n^{-1/2}|x|^{\ell_\mu}}+ \frac{(p_0-1)\cdot\sigma^2(x)}{(1+n^{-1/2}|x|^{\ell_\mu})^2}\le \frac{2x\cdot\mu(x)+(p_0-1)\cdot\sigma^2(x)}{1+n^{-1/2}|x|^{\ell_\mu}}
M=(\begin{matrix}m_{11}&m_{12}\\m_{21}&m_{22}\end{matrix})=\prod_{i=0}^{N}(\begin{matrix}q_{i}&1\\ 1&0\end{matrix})=(\begin{matrix}q_{0}&1\\ 1&0\end{matrix})(\begin{matrix}q_{1}&1\\ 1&0\end{matrix})\cdot\cdot\cdot(\begin{matrix}q_{N}&1\\ 1&0\end{matrix}).
\!\begin{multlined}[t]x^{k+3}\cdot x^{k+1}\cdot 11x^3 + x^{k+3}\cdot (2^k+k)x^k \cdot x^4 + (2^{k+2}+k+2)x^{k+2}\cdot x^{k+1}\cdot x^4 = ( 5 \cdot2^k +2k + 13)x^{2k + 7},\end{multlined}
\psi (x) = \frac{84}{\pi^3}x \zeta(3) + \frac{8}{\pi}x^3 \log (\pi x) - \frac{8}{\pi}\left( \frac{11}{6}+ \log 2 \right) x^3 - 3 x (1-x) &{}{}+ 48 \pi x^5 \sum_{n=0}^\infty \frac{| B(2n+2) | \left( 2^{2n+1}- 1 \right)}{(n+1) (2n+5)!}\pi^{2n}x^{2n}- \frac{96}{\pi^4}\sum_{n}\frac{d_n}{n^4}\sin n \pi x .
\begin{aligned}X_{\tau}=x_{n}+ \tau h\varphi_{1}(\tau hM ) v_{n}+h^2 \int_{0}^{1}{\alpha}_{\tau \sigma}(hM)F (X_\sigma)d\sigma, &x_{n+1}=x_{n}+ h\varphi_1(hM)v_{n}+h^2 \int_{0}^{1}\beta_{\tau}(hM)F (X_{\tau})d\tau,\ \v_{n+1}=\varphi_0(hM)v_{n}+h\int_{0}^{1}\gamma_{\tau}(hM)F(X_{\tau})d\tau,\end{aligned}
\mathcal{Z}\left( \binom{an+b}{r}^{p}\prod_{s=2}^{L}\binom{\alpha_{s}n+\beta_{s}}{r_{s}}^{p_{s}}\right) =\frac{1}{\left( r!\right)^{p}\prod_{s=2}^{L}\left( r_{s}!\right)^{p_{s}}}\sum_{k=0}^{rp+\sigma}k!S_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,k\right) \frac{z}{\left( z-1\right)^{k+1}},
d_{i,j}:=\frac{a_{i}}{a_{N}}\prod_{k=i+1}^{N}\left(1-\frac{a_{k}}{a_{j}}\right)e_{i,j}:=\frac{a_{N}}{a_{i}}\overset{N}{\underset{\genfrac{}{}{0pt}{}{k=j}{k\neq i}}{\prod}}\frac{1}{1-\frac{a_{k}}{a_{i}}},
\begin{aligned}\phi(f_1,f_2) = \phi(1, 1) + \bigl(\phi(f_1,1) - \phi(1,1)\bigr) + \bigl(\phi(f_1,f_2) - \phi(f_1,1)\bigr) & \leq \phi(f_1,1) + \bigl(\phi(2,f_2) - \phi(2,1)\bigr).\end{aligned}
dq^\prime_t= \tilde\gamma^{-1}(t^*)\left(-\nabla_qV(t^*,q^\prime_t)+\tilde F(t^*,q_t)\right) dt & +\tilde S(t^*,q^\prime_t)dt+\tilde\gamma^{-1}(t^*)\sigma(t^*,q^\prime_t)\circ dW_t = \tilde\gamma^{-1}(t^*)\left(-\nabla_qV(t^*,q^\prime_t)+\tilde F(t^*,q_t)\right) dt+\tilde\gamma^{-1}(t^*)\sigma(t^*,q^\prime_t) dW_t,
C(\underbrace{1,\cdots,1}_s) =\frac{1}{4^s3^k}\biggl(3^k+(-1)^{k-s}\sum_{i=0}^{s-1}\binom{k-s+i}{i}3^{s-1-i}4^i\biggr) &=\frac{1}{4^s3^k}\biggl(3^k+(-1)^{k-s}\sum_{i=0}^{s-1}\frac{(k-s+1)(k-s+2)\cdots(k-s+i)}{i!}3^{s-1-i}4^i\biggr) &=\frac{1}{4^s3^k}\biggl( 3^k+(-1)^{k-s}\sum_{i=0}^{s-1}\frac{3^{s-1-i}4^i}{i!}(k-s+1)(k-s+2)\cdots(k-s+i)\biggr).
S [ A_{\mu}] = \int_{0}^{\beta}d x_{5}\int d^{4}x \ \frac{1}{2 e^{2}}[ \mathrm{T r}\F_{\mu \nu}F_{\mu \nu}+ \frac{1}{c^{2}}\mathrm{T r}\ \partial_{5}A_{\mu}\partial_{5}A_{\mu}] .
\mathbb{E}[\rho^2_{i,j}] = \mathbb{E}\left[ \frac{|\mathbf{H}_{k}\cdot \mathbf{H}_{l}^H|^2}{\| \mathbf{H}_{k}\|^2 \| \mathbf{H}_{l}\|^2}\right] &\approx \frac{\mathbb{E}\left[ |\mathbf{H}_{k}\cdot \mathbf{H}_l^H|^2 \right]}{\mathbb{E}\left[ \Vert \mathbf{H}_{k}\Vert^2 \right] \mathbb{E}\left[ \Vert \mathbf{H}_{l}\Vert^2 \right]}.
\tilde{f}( \omega , \kappa , \bar{\kappa}, 0 ) = \frac{1}{2 \pi^{2}}\int_{- \infty}^{\infty}d \nu \sum_{n = - \infty}^{\infty}{\frac{\vert \kappa^{2}\vert^{- 1 + i \nu}\left({\frac{\kappa}{\vert \kappa \vert}}\right)^{n}\vert \bar{\kappa}^{2}\vert^{- 1 - i \nu}\left({\frac{\bar{\kappa}}{\vert \bar{\kappa}\vert}}\right)^{- n}}{\omega - \frac{g^{2}C_{2}}{8 \pi^{2}}\Omega ( \omega , \nu , n )}}
\begin{aligned}\|U_{\lambda_n}(t,0)u_{0n}\|_\infty \leq Mt^{-\frac{N}{2p}}e^{\omega t}\|u_{0n}\|_p \|U_{\lambda_n}(t,\tau)f_n(\tau)\|_\infty \leq M(t-\tau)^{-\frac{N}{2p}}e^{\omega(t-s)}\|f_n(\tau)\|_p \end{aligned}
Y^{-k,i}_t[\beta] = \xi^i + \int_0^t\int_Ab(s,Y^{-k,i}_s[\beta],\widehat{\mu}^{-k,x}_s[\beta],a)\beta^i_s(da)dt + \int_0^t\sigma(s,Y^{-k,i}_s[\beta],\widehat{\mu}^{-k,x}_s[\beta])dW^i_s &\quad+ \int_0^t\sigma_0(s,Y^{-k,i}_s[\beta],\widehat{\mu}^{-k,x}_s[\beta])dB_s, \widehat{\mu}^{-k,x}[\beta] := \frac{1}{n-1}\sum_{i \neq k}^n\delta_{Y^{-k,i}[\beta]}.
E\big[|I_3(t)|^2\big] = E\Big[\hat{\textbf{h}}~\!\!_{1,k}^H \big(\phi_{j,k}\hat{\textbf{h}}~\!\!^{H}_{1,k}+ \textbf{w}_{j,k}\big)\Big] & = |\phi_{j,k}|^2 E\big[\|\hat{\textbf{h}}_{1,k}\|^4\big] + \sigma_{w_{j,k}}^2 E\big[\|\hat{\textbf{h}}_{1,k}\|^2\big] & = |\phi_{j,k}|^2 \sigma_{\hat{h}_{1,k}}^4(M^2+M) + \sigma_{w_j}^2 \sigma_{\hat{h}_{1,k}}^2 M.