harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left[ - \gamma_{4 4}E^{2}+ 2 i E \gamma_{4 i}\vec{p}_{i}+ \gamma_{i j}\vec{p}_{i}\vec{p}_{j}+ m^{2}\right] u_{1}^{\sigma}( \vec{p}) = 0 \quad ( \mathrm{o r}\, \, v_{1}^{\sigma}( \vec{p}) ) \quad ,
r_2=(\frac{-t_0}{1}\ominus\frac{-1}{0})\bullet\frac{1}{0}-(\frac{1}{0}\ominus\frac{1}{1})\bullet\frac{1}{0}-(\frac{1}{1}\ominus\frac{1}{2})\bullet\frac{1}{0}-\cdots-(\frac{1}{-t_{2}-1}\ominus\frac{1}{-t_{2}})\bullet\frac{1}{0}=t_{2}-1.
\Gamma_0 = \sum_{k=0}^\infty (A')^k A^k = A'\Gamma_0 A + I_2 \Omega_0 = \sum_{k=0}^\infty (B')^{-k}(B')^{-1}T'TB^{-1}B^{-k}= (B')^{-1}\Omega_0 B^{-1}+ (B')^{-1}T'TB^{-1}\Gamma_k = \Gamma_0 A^k = (\Gamma_0 A \Gamma_0^{-1})^k \Gamma_0 \Omega_k = (B')^{-k}\Omega_0
F(x_{1},x_{2},...,x_{n})=\sum_{\underset{\delta_{j}=0}{\delta\in\{0,1\}^{n}}}u_{\delta}(x_{j})\left[\prod_{k\neq i,j}f_{k}^{\delta_{k}}(x_{k})\right]f_{i}^{\delta_{i}}(x_{i}),
\frac{\sqrt{D}}{2a(a^2+c^2)}={}\frac{\sqrt{D}}{2a^3 + (2a)^{-1}(D+b^2)}={}K_E^{-1}\left( \frac{\pi^{2/3}}{K_E^{4/3}A^{2/3}}+ 1 + \frac{(\partial_s K_E)^2}{9K_E^4}\right)^{-1},
( \lambda \overline{{{R}}}_{2 3}^{- 1}- - -- \overline{{{R}}}_{2 3}) x = ( \overline{{{R}}}_{1 2}\overline{{{R}}}_{2 3}- \overline{{{R}}}_{2 3}) x = ( \overline{{{R}}}_{1 2}- 1 ) \overline{{{R}}}_{2 3}x \in \mathrm{I m}( \overline{{{R}}}_{1 2}- 1 )
\phi (p_{r(j_0 \beta j_1)}) = \phi (p_{r(\beta j_1)}) = \phi (p_{r(j_1 \beta^{-1})}) = \phi (p_{r(\beta^{-1})}) = \phi (p_{r(\beta)}) = \phi (p_{r(j_0 \beta)}) = \phi (p_{r(\beta^{-1}j_0)}) = \phi (p_{r(j_1 \beta^{-1}j_0)}).
1+2\sum_{n\geq 1}\frac{(-1)^n q^{n(n+1)/2}}{(1+q^n)(q;q)_n}= \lim_{\rho\rightarrow\infty}{}_2\phi_1 \left( \genfrac{}{}{0pt}{}{-1,\ \rho q}{-q};q, 1/\rho \right) = \frac{(q;q)_\infty}{(-q;q)_\infty},
{\cal L}+{\cal L}_{c t}={\frac{1}{2}}( \partial_{\mu}\phi )^{2}-{\frac{1}{2}}M^{2}\phi^{2}-{\frac{\lambda}{4 !}}\mu^{2 \epsilon}\phi^{4}+{\frac{1}{2}}A ( \partial_{\mu}\phi )^{2}-{\frac{1}{2}}M^{2}B \phi^{2}-{\frac{\lambda}{4 !}}\mu^{2 \epsilon}C \phi^{4}\, -{\frac{1}{2}}( m^{2}- M^{2}) ( 1 + B ) \phi^{2}
E_4 :=K n^{\theta}E\int_{t_0}^{u}\Big|\int^{s}_{\kappa(n,s)}\int_{Z}|x_{\kappa(n,s)}^n|^{p-2}\gamma_r^n(x^n_{\kappa(n,r)},z)\tilde N(dr, dz)\Big| ds & \leq K n^{\theta}\int_{t_0}^{u}E\Big(\int^{s}_{\kappa(n,s)}\int_{Z}|x_{\kappa(n,s)}^n|^{2p-4}|\gamma_r^n(x^n_{\kappa(n,r)},z)|^2 \nu(dz) dr \Big)^\frac{1}{2}ds
\mathrm{G r}_{2 , 1}= \left( \begin{array}{c c c c c}{{\sqrt{Q}}}&{{1}}&{{0}}&{{\alpha_{1}}}\\{{1}}&{{\sqrt{Q}}}&{{1}}&{{z}}\\{{0}}&{{1}}&{{\sqrt{Q}}}&{{1}}\\{{\alpha_{1}^{- 1}}}&{{z}}&{{1}}&{{\sqrt{Q}}}\\ \end{array}\right)
Q_1\sqrt{\frac{2^{\sum_{l=1}^dh_{n,l}(\boldsymbol{\alpha})}\log{n}}{n}}+\sqrt{2Q_2C_{I}}2^{\sum_{l=1}^dh_{n,l}(\boldsymbol{\alpha})/2}\epsilon_n\leq2\sqrt{2Q_2C_I}\epsilon_n^{2\alpha^{*}/(2\alpha^{*}+d)},
\forall\;0\le k<n,\quad\left(y_{k}\right)^{m}=\mbox{Prod}_{\mathbf{P}_{k}}\left(\mathbf{x}^{\top^{\left(m-1\right)}},\mathbf{x}^{\top^{\left(m-2\right)}},\cdots,\mathbf{x}^{\top^{j}},\cdots,\mathbf{x}^{\top^{1}},\mathbf{x}^{\top^{0}}\right).
M =\left(\begin{array}{ccccc}0 0 \cdots 0 0 0 m_{2,2}\cdots m_{2,c}m_{2,c+1}\vdots \vdots \ddots \vdots \vdots 0 0 \cdots m_{c,c}m_{c,c+1}0 0 \cdots 0 0\end{array}\right).
\int{\cal D}V = \int \prod_{n}\left( d V_{n}^{1}d V_{n}^{2}d V_{n}^{3}d \lambda_{n}^{1}d \lambda_{n}^{2}d\bar{\lambda}_{n}^{1}d\bar{\lambda}_{n}^{2}d D_{n}\right) \prod_{i}^{2 n_{0}}d V_{0}^{i}\prod_{i}^{n_{0}}d\lambda_{0}^{i}
\begin{array}{c c l}{J_{n}^{\pm}}&{\rightarrow}&{J_{n \pm w}^{\pm}}\\{J_{n}^{3}}&{\rightarrow}&{J_{n}^{3}-{\frac{k}{2}}w \delta_{n}\qquad \qquad w \in{\bf Z}}\\{L_{n}}&{\rightarrow}&{L_{n}- w J_{n}^{3}-{\frac{k}{4}}w^{2}\delta_{n}.}\\ \end{array}
2(\beta+1) DP_N^{(\alpha,\beta)}(-1)=-\lambda^{(\alpha,\beta)}_N P_N^{(\alpha,\beta)}(-1), \;\;\; 2(\alpha+1) DP_N^{(\alpha,\beta)}(1)=\lambda^{(\alpha,\beta)}_N P_N^{(\alpha,\beta)}(1), & -(1-x^2_j)D^2 P^{(\alpha,\beta)}_{N}(x_j)=\lambda^{(\alpha,\beta)}_N P^{(\alpha,\beta)}_{N}(x_j),1\le j\le N-1.
\frac{d^2\phi}{d\eta^2}= \left[A^{2}+BC-A'+A\frac{C'}{C}+\frac{3}{4}\left(\frac{C'}{C}\right)^{2}-\frac{1}{2}\frac{C''}{C}\right]\phi = \left[\frac{1}{4}+\frac{1}{\eta^2}\left(\frac{v^2(1-y)^2}{y}-vt\right)-\frac{1}{2\eta}-\frac{1}{4\eta^2}+g(\eta,t)\right]\phi= F(\eta,t)\phi.
m_{t}^d \le 8m^d_0+216\eta^4 d^4 \big[ C_2^4 t^3+3\big( C_8'+C_{5}'^2 \big)t \big] \sum_{k=0}^{t-1}m^d_k +216 \big( C_{5}C_1+C_2^2 \big)\eta^4 \sigma^4 t^2 \le 8m^d_0+216 \big(C_2^4+ 3C_8'+3C_{5}'^2 \big)\eta^4 d^4t^3 \sum_{k=0}^{t-1}m^d_k +216 \big( C_{5}C_1+C_2^2 \big)\eta^4 \sigma^4 t^2.
\chi_{(\lambda_1,\lambda_2)}^\pm(s_1s_2)=\begin{cases}\overline{\chi}^\pm_{\lambda_1}(s_1)\chi^\pm_{\lambda_2}(s_2)+\overline{\chi}^\mp_{\lambda_1}(s_1)\chi^\mp_{\lambda_2}(s_2) s_1\in\tilde{A}_{n_1}, 0 \end{cases}
\bigoplus_{m,n=0\atop \max(m,n)=\ell}^{\ell}\mathcal{A}^{[1]}_{m,n}(U) = \ker\left( \frac{\partial^{\ell}}{\partial{Z^*}^{\ell}}\right) \cap \ker\left( \frac{\partial}{\partial Z^\dagger}\right) \cap \ker \left( \frac{\partial}{\partial \widetilde{Z}}\right).
H_{cl}(q,p,Q,P)=\frac{\hbar S\omega_{c}}{2}(q^{2}+p^{2})+\frac{\hbar S\omega_{z}}{2}(Q^{2}+P^{2})+2\hbar S\lambda Qq\sqrt{1-\frac{Q^{2}+P^{2}}{4}}-\hbar S\omega_{z}.
A^k(X,\mathbf{F}):=\left\{\omega\in\Omega^{k,k}(X,\mathbf{C})\,\big{\vert}\,[\omega]\in H^{k,k}(X,\mathbf{F}),\mathcal{F}^{\omega^{2((n-1)-k)},\{U_i\}_{i=0,\dots,K}}\neq\emptyset\right\}.
\begin{matrix}v_{1}&=&\frac{m_{1}-m_{2}}{m_{1}+m_{2}}u_{1}+\frac{2m_{2}}{m_{1}+m_{2}}u_{2}\\v_{2}&=&\frac{2m_{1}}{m_{1}+m_{2}}u_{1}+\frac{m_{2}-m_{1}}{m_{1}+m_{2}}u_{2}.\end{matrix}
e^{-\gamma (a +c)}= \lim_{x\to\infty}\frac{P(x +a+c< X +Y_c)}{P( x < X +Y_c)}\le \liminf_{x\to\infty}\frac{P(x+a < X )}{P( x < X)}\le\limsup_{x\to\infty}\frac{P(x+a < X )}{P( x < X)}\le\lim_{x\to\infty}\frac{P(x+a < X +Y_c)}{P( x+c < X+Y_c)}=e^{-\gamma (a -c)}.
\mathcal{G}(x,x^{\prime})=\sum_{q,\alpha}\int dk_0 \frac{\psi_{q}(\vec{x})e^{-ik_0 x^0}\otimes\psi_{q}^\dag (\vec{x}^{\prime})e^{ik_0 x^{\prime 0}}}{k_0 + \mu +{\alpha}{E_q}}\,,
{E}\Vert \mathcal{P}_0(X_s) \Vert^2 ={E}\Vert \mathcal{P}_0 (X_s-X_s^{(s)}) \Vert^2 \leq 2{E}\left( \Vert{E}[ X_s-X_s^{(s)}\vert \mathcal{G}_0]\Vert^2 + \Vert{E}[ X_s-X_s^{(s)}\vert \mathcal{G}_{-1}]\Vert^2 \right) \leq 4{E}\Vert X_0-X_0^{(s)}\Vert^2=4\nu_2^2\big(X_0-X_0^{(s)}\big).
\mathcal{Z}\bigotimes_S \big(S/(Z_0-1,Z_1,\ldots,Z_g)\big)= 0\rightarrow Z_g\bigotimes_Q Q[Z_0]/(Z_0-1)\rightarrow \cdots \rightarrow Z_2\bigotimes Q[Z_0]/(Z_0-1) \rightarrow \rightarrow Z_1\bigotimes_Q Q[Z_0]/(Z_0-1)\rightarrow Q[Z_0]/(Z_0-1)\rightarrow 0,
\lefteqn{[X(z),X(w)]\;= X(z)X(w)-X(w)X(z)}&=E^+_+(z)E^+_+(w)E^+_-(z)E^+_-(w)\otimes\big(\boldsymbol{:}A_{\alpha}(z)A_\alpha(w)\boldsymbol{:}(z-w)^{-\frac{2}{k}}&\qquad\qquad-\boldsymbol{:}A_{\alpha}(w)A_\alpha(z)\boldsymbol{:}(w-z)^{-\frac{2}{k}}\big)e^{2\alpha}(zw)^{-\frac{\alpha}{k}}&=0
\begin{aligned}\left\|\dfrac{\partial \epsilon(s,t)}{\partial s}\right\| \le \int_{NT}^t \left\|\dfrac{\partial f(s,\tau)}{\partial s}\right\|+\left\|\dfrac{\partial f_a(s)}{\partial s}\right\| \,d\tau \le 2 L_\mathcal{D}T,\end{aligned}
\int_{-1}^{1}\left\vert f(x)\right\vert^{2}(1-x)^{-2}(1+x)^{\beta}dx=\int_{-1}^{1}\left\vert \frac{f(x)}{(1-x)^{2}}\right\vert^{2}(1-x)^{2}(1-x)^{\beta}dx;
\left(v^{lk'}_{i k}\left(W',Z',\overline{W}',\overline{Z}'\right)\right)_{1 \leq k \leq N'}^{1 \leq k'\leq N'}\overline{\left(\left(v^{l'k'}_{jk}\left(W',Z',\overline{W}',\overline{Z}'\right)\right)_{1 \leq k \leq N'}^{1 \leq k'\leq N'}\right)^{t}}=a_{ij}^{ll'}\left(W',Z',\overline{W}',\overline{Z}'\right)I_{N'},\quad\mbox{for all $i,j,l,l'=1,2$,}
-\widetilde{B_s}^{-T}=\left(\begin{smallmatrix}-1 -1 1 1 0 0 1 0 -1 \end{smallmatrix}\right) -\widetilde{B_{s'}}^{-T}=\left(\begin{smallmatrix}-1 1 -1 -1 0 0 -1 0 -1 \end{smallmatrix}\right).
{\rm E}\Bigg[\sum_{\ell=2}^n \sigma^2_{n\ell}\Bigg] =\frac{2}{n^2{\rm E}[u_{n1}^2]}\sum_{\ell=2}^n \sum_{i,j=1}^{\ell-1}{\rm E}[u_{ni}u_{nj}]{\rm E}[\rho_{n,ij}] =\frac{2}{n^2}\sum_{\ell=2}^n\, (\ell-1)=\frac{n-1}{n}.
A_{q^{2}}\left(-c^{2}\right)=\left(c^{2}q;q^{2}\right)_{\infty}\sum_{n=0}^{\infty}\frac{q^{(3n^{2}-n)/2}c^{2n}}{\left(q,cq^{1/2},-cq^{1/2};q\right)_{n}}.
{\frac{\partial}{\partial a}}P_n(a,{\boldsymbol{b}}={\boldsymbol{1}},{\boldsymbol{\beta}}={\boldsymbol{0}},{\boldsymbol{\alpha}}={\boldsymbol{1}},{\boldsymbol{y}}={\boldsymbol{z}}={\boldsymbol{1}})= A005775,
S= 2 \pi (\det )^{1/4}\left\{r_0 \delta_{ik}+ \sqrt{- 2 \Upsilon_{ij}\Upsilon^*_{jk}}\right\}= 2 \pi (\det )^{1/4}\left\{r_0 \delta_{ik}+\sqrt{- 2 \tilde \Upsilon_{ij}\tilde \Upsilon^*_{jk}}\right\}\ .
f^{2}g^{2}{\frac{\partial^{2}}{\partial r^{* 2}}}t + f ( f g^{\prime \prime}+ f^{\prime}g^{\prime}){\frac{\partial}{\partial r^{*}}}t - \{\sqrt 2 f f^{\prime}- 2 f ( 1 - f ) \}t -{\frac{\partial^{2}}{\partial t^{2}}}t = 0 .
\left\{\begin{array}{ll}-\Delta_pu_{\epsilon}(z)=f(z,u_{\epsilon}(z),Du_{\epsilon}(z))+\epsilon e(z)\ \mbox{for almost all}\z\in\Omega, \frac{\partial u_{\epsilon}}{\partial n_p}+\beta(z)u^{p-1}_{\epsilon}=0\ \mbox{on}\ \partial\Omega \end{array}\right\}
_i (1_n^{*3}\otimes1_1) = \tfrac{1}{8}( 1_n^{\ast 3}\otimes 1_1+3\cdot1_n^{\ast 2}i_n^{\ast}\otimes i_1-3\cdot1_n^{\ast}i_n^{\ast 2}\otimes 1_1-i_n^{\ast 3}\otimes i_1).
\Phi_n := N(-n)^{{n+2 \choose 2}}\xrightarrow{\;\; F_n \;\;}N(-n+1)^{{n+1 \choose 2}}\mbox{and}\Psi_n := N(n-1)^{{n+1 \choose 2}}\xrightarrow{\;\; F^t_n \;\;}N(n)^{{n+2 \choose 2}}
\langle \Psi | \pi_{\mathrm{R^{\prime}}}( x ) | \Omega_{\mathrm{R^{\prime}}}\rangle = \frac{1}{\sqrt{2}}\langle \Psi_{+}| \pi_{E_{\mathrm{R}}, +}( x ) | \Omega_{E_{\mathrm{R}}}\rangle + \frac{1}{\sqrt{2}}\langle \Psi_{-}| \pi_{E_{\mathrm{R}}, -}( x ) | \Omega_{E_{\mathrm{R}}}\rangle = 0
e_k\big(\theta_1^{(n)},\ldots, \theta_m^{(n)}\big) = \sum_{r \ge 0}(-h)^{r}N(m-k,2~r) \left\{\sum_{S \subset [1,m] \atop I=\{i_a\},\, J=\{j_a\}}X_{S}u_{i_{1},j_{1}}\cdots u_{i_{|I|},j_{|J|}}\right\},
\begin{array}{c c c}{{O 3 ~ b e f o r e ~ s p l i t t i n g ~ ~}}&{{~ \Delta L_{l e f t}~}}&{{\Delta L_{r i g h t}}}\\{{O 3^{+}}}&{{- N}}&{{N}}\\{{\widetilde{O 3^{+}}}}&{{- N}}&{{N}}\\{{O 3^{-}}}&{{- \frac{1}{2}- N}}&{{\frac{1}{2}+ N}}\\{{\widetilde{O 3^{-}}}}&{{\frac{1}{2}- N}}&{{- \frac{1}{2}+ N}}\\ \end{array}
\begin{aligned}\int_{\Sigma_s}(\partial_t \psi) (\partial_t \phi) + (\partial^i \psi) (\partial_i \phi) d x + \int_0^s \int_{\Sigma_t}(\Box \psi) (\partial_t \phi) + (\Box \phi) (\partial_t \psi) d x d t = \int_{\Sigma_0}(\partial_t \psi) (\partial_t \phi) + (\partial^i \psi) (\partial_i \phi) d x. \end{aligned}
\lim_{m_i\rightarrow 0}\rho_N(p)=2\Im I_N(p)=\frac{(4\pi)^{1+\ell-N\ell}[\Gamma(\ell-1)]^N}{2\Gamma(N\ell-N)\Gamma((N-1)(\ell-1))}(p^2)^{N\ell-\ell-N}\theta(p^2).
m\lfloor\frac{m-1}{k-1}\rfloor-\frac{k-1}{2}\lfloor\frac{m-1}{k-1}\rfloor(\lfloor\frac{m-1}{k-1}\rfloor+1)>\frac{m^{2}}{2(k-1)}+\frac{m-mk-k}{2(k-1)}
u \cdot (v_1, w_1, x_2 + i y_2) = (\sigma_1 (u) v_1, \sigma_1 (u)^{-\frac{1}{2}}w_1, \sigma_2 (u) (x_2 + i y_2)) &= (\sigma_1 (u) v_1, \vert \sigma_2 (u) \vert w_1, \sigma_2 (u) (x_2 + i y_2)).
I_{D_3}= \ (x_1 x_5, x_1 x_7, x_1 x_9, x_5 x_7, x_5 x_8, x_5 x_9, x_6 x_8, x_6 x_9, x_7 x_9, x_1 x_2 x_3, x_1 x_4 x_6, x_1 x_4 x_8, &\x_2 x_3 x_4, x_2 x_3 x_6, x_2 x_3 x_8, x_2 x_4 x_6, x_2 x_4 x_7, x_2 x_4 x_8, x_2 x_4 x_9, x_2 x_5 x_6, x_2 x_7 x_8, \x_3 x_4 x_5, x_3 x_4 x_6, x_3 x_4 x_7, x_3 x_4 x_8, x_3 x_6 x_7, x_3 x_8 x_9) \subseteq k[x_1,\dots,x_9].
(0, -dL_{x}^{-1}(\tau_{H})_{x})^{\ast}_{k_{2}x}= \left. \frac{d}{dt}k_{2}x\exp (tdL_{x}^{-1}(\tau_{H})_{x}) \right|_{t=0}&= dL_{k_{2}}dL_{x}\left. \frac{d}{dt}\exp (tdL_{x}^{-1}(\tau_{H})_{x}) \right|_{t=0}&= dL_{k_{2}}(\tau_{H})_{x}&= (\tau_{H})_{k_{2}x}.
(\alpha+\beta\xi)\in[\hat{\alpha}+\hat{\beta}\xi\pm t_{n-2}^{*}\sqrt{(\frac{1}{n-2}\sum\hat{\epsilon}_{i}^{2})\cdot(\frac{1}{n}+\frac{(\xi-\overline{x})^{2}}{\sum(x_{i}-\overline{x})^{2}})}].
\begin{array}{c}\displaystyle\int_{\partial D}E_{\tau}(x,p)Y_{21}(\tau)\varphi(x,\tau)dS_x \displaystyle=\left(\frac{1}{2\pi}\right)^2\int_{\partial\Omega}dS_y\varphi(y,\tau)\int_{\partial D}\frac{H(x,y,\tau)}{\vert x-p\vert}e^{-\tau\,l_p(x,y)}dS_x.\end{array}
\bigl[4J_n^+ J_n^- - 4 J_n^0 J_n^- - 8b_1 J_n^+ + 2(4a+4b_1+3+2n) J_n^0 - 2(4a-2l-d+3+n) J_n^- - A + 2a(2a+1) + 2b_1(4a-2l-d+3) + n(4a+4b_1+3+2n)\bigr] \phi_n(z) = 0,
\sum_{j=1}^n\sum_{k=1}^n\langle \tau_j,\tau_k\rangle \langle x_k,x_j\rangle =\sum_{j=1}^n\sum_{k=1}^n\langle \tau_j,x_k\rangle \langle \tau_k,x_j\rangle= \operatorname{Trace}(S^2_{x,\tau}) &= \operatorname{Trace}(b^2I_\mathcal{H})=b^2m=\left(\frac{1}{m}\sum_{j=1}^n\langle x_j, \tau_j \rangle\right)^2m .
{{\cal A}}({\cal T})\wedge{{\cal A}}({\cal N})\; = \;{{\cal A}}({\widehat{\cal S}})\; =\; 1 +{(4\pi^2\alpha^\prime)^2\over 192\pi^2}\;{\rm tr}\; ({\widehat{\cal R}}\wedge{\widehat{\cal R}}) + \cdots \ ,
\zeta_\pm^2(z)=N_2\left(\begin{array}{c}z^{\frac{1-M}{2}}(1-z)^{-\frac{M}{2}}F(\frac{3}{2}-M,-\frac{1}{2}-M,\frac{3}{2}-M;z) \mp i(\frac{1}{2}-M)z^{-\frac{M}{2}}(1-z)^{\frac{M}{2}}F(-1,1,\frac{1}{2}-M;z) \end{array}\right)
(F E_{1}^{n}F_{2}^{m}\otimes F_2) \cdot w_s = nm (E_{1}^{n-1}F_{2}^{m-1}H_2 \otimes F_2) \cdot w_s & - mn(m-1)(E_{1}^{n-1}F_{2}^{m-1}\otimes F_2) \cdot w_s & + n (E_{1}^{n-1}F_{2}^{m}E_2 \otimes F_2) \cdot w_s - m (E_{1}^{n}F_{2}^{m-1}F_1 \otimes F_2) \cdot w_s & + (E_{1}^{n}F_{2}^{m}F \otimes F_2) \cdot w_s.
K_{F,1}(\tau)n\log{n}-\frac{1}{20}R^n & <5N^2 K_{F,1}(\tau)(A_F\log{N}+M_F)\log{\left(5N^2(A_F\log{N}+M_F)\right)}-\frac{1}{20}R^N & < 5N^2 K_{F,1}(\tau)(A_F\log{N}+M_F)\cdot & \cdot\left(\frac{5}{2}\log{N}+\log{5}+\log{\left(A_F+\frac{|M_F|}{1.732}\right)}\right)-\frac{1}{20}R^N.
\log|\phi(z)| =\Re e \big(h(z)\big) =\Re e(a_0)+\left(\Re e(a_1)x-\Im m(a_1)y\right) \quad+\left(\Re e(a_2)\left(x^2-y^2\right)-2\Im m(a_2)xy\right)+o\left(x^2+y^2\right)
\phi(0) =\int_{\mathfrak{a}_f}^{a_1}(a_1-x)^{p-1}\Psi(x)dx-\int_{a_1}^{\widetilde r_2}(x-a_1)^{p-1}\Psi(x)dx &=\int_{a_1}^{r_2}(x-a_1)^{p-1}\Psi(x)dx-\int_{a_1}^{\widetilde r_2}(x-a_1)^{p-1}\Psi(x)dx<0,
A ( Z ) ={\frac{1}{3 2 0 \pi^{4}}}\left[{\frac{1}{( Z - 1 )^{4}}}+{\frac{1}{( Z + 1 )^{4}}}\right] ={\frac{1}{1 0}}\left[ ( \Delta_{1}^{\prime})^{2}+ ( \Delta_{2}^{\prime})^{2}\right] \, ,
V_{i}^{(k)}:=\partial_i+\sum_{j=1}^n \sum_{0\leq \left|I\right|<k}u_{I,i}^j\partial_{u^j_I},1\leq i\leq m, V_{j}^{I}:=\partial_{u^j_I},\left|I\right|=k,1\leq j \leq n.
{}\displaystyle\sum_{n=1}^{\infty}|A_n-A_{n-1}|^p \geq \sum_{n=1}^{\infty}w_n|A_n|^p>\Big(\frac{p-1}{p}\Big)^p\sum_{n=1}^{\infty}\frac{1}{n^p}|A_n|^p,
f^{(s)}(t) = \sum_{(i_1,\ldots,i_n,j) \in I}s!\binom{s+2}{i_1}\cdots \binom{s+2}{i_n}\cdot (t+1)^{s+2-i_1}\cdots (t+n)^{s+2-i_n}\cdot \frac{g^{(j)}(t)}{j!}= \sum_{(i_1,\ldots,i_n,j) \in I}f_{(i_1,\ldots,i_n,j)}(t),
C[u^{2}] = \iint_{x,y>0}\min{\{x,y\}}(u(x))^{2}(u(y))^{2}\, dx \, dy + \iint_{x,y<0}\min{\{-x,-y\}}(u(x))^{2}(u(y))^{2}\, dx \, dy &=: C_{+}[u^{2}] + C_{-}[u^{2}],
E_{y}=[-\frac{k_{z}}{\omega\epsilon_{o}\epsilon_{r}}\frac{m\pi}{a}(Ae^{-jk_{x\epsilon}x}+Be^{jk_{x\epsilon}x})-jk_{x\epsilon}(Ce^{-jk_{x\epsilon}x}-De^{jk_{x\epsilon}x})]cos(\frac{m\pi}{a}y)e^{-jk_{z}z}(34)
D_{i j}^{k l}= \frac 1 2 \left( \delta_{i}^{\left[ k \right.}\delta_{j}^{\left. l \right]}- \frac 1{\triangle}\delta_{m}^{\left[ l \right.}\partial_{}^{\left. k \right]}\delta_{\left[ j \right.}^{m}\partial_{\left. i \right]}^{}\right) .
\sum_{m=1}^\infty \lambda_f (m) e\left( \frac{am}{c}\right) F(n) = \frac{1}{c}\frac{\eta_f(D_2)}{\sqrt{D_2}}\sum_{m=1}^\infty \lambda_{f}( m) e\left( \frac{-m\overline{a D_2}}{c}\right) H\left( \frac{m}{D_2 c^2}\right),
\widetilde{\omega}(1)\frac{\zeta(2s)}{\zeta(s)}\sum_{q=1}^{\infty}\frac{1}{q^{1+s}}\sum_{a,b=1}^{q}\delta_q(ab-l^2)=\widetilde{\omega}(1)\sigma_{-s}(l^2)\frac{\zeta(2s)}{\zeta(1+s)}.
\bar{t}_{\mu}\lambda_{\mu}= \sum_{k = 1}^{D}\tilde{\alpha}_{k}\frac{\bar{t}_{\mu}c_{k - 1}^{( \mu )}}{c_{k}}\leq \sum_{k = 1}^{D}\tilde{\alpha}_{k}, \quad \forall \mu = 1 , \ldots , D ,
s\sum^{N-1}_{n=0}\int_\Omega |\hat{A}^{\epsilon,s}_n|^4 d\hat x\leq sN C\lVert \hat{A}^{\epsilon,s}\rVert_{H^{1}(\omega)}^4 = LC\lVert \hat{A}^{\epsilon,s}\rVert_{H^{1}(\omega)}^4.
c_{\psi}(X_3) =\frac{1}{16\pi}+\frac{1}{6\pi}+0.184+\frac{18.626}{\log X_3}+\frac{233.925}{\log^2 X_3}+\frac{725.316}{\log^3 X_3}\qquad+\frac{2.015}{\sqrt{X_3}}+\frac{4.179}{\sqrt{X_3}\log X_3}+\frac{263.886}{\sqrt{X_3}\log^2X_3}.
qa_n'(q) = \sum_{k,\ell}{n \brack k}_q q^{f(n,k,\ell)}\Big({n \brack k}_q \big({n \brack \ell}_q{k \brack \ell}_q{k+\ell \brack n}_q\big)' & + \big({n \brack k}_q f(n,k,\ell) + 2{n \brack k}_q' q\big){n \brack \ell}_q{k \brack \ell}_q{k+\ell \brack n}_q \Big).
\begin{aligned}\zeta(z_1(k_1)z_{c_1}(k_2){\cdots}z_{c_{p-1}}(k_p);\alpha)=\sum_{\begin{subarray}{c}0{\le}m_1<_{c_1}\cdots<_{c_{p-1}}m_p<\infty\end{subarray}}\prod_{i=1}^{p}\frac{1}{(m_i+\alpha)^{k_i}},\end{aligned}
\frac{\partial F}{\partial x_i}(x)=\sum_{k;k \neq i}\frac{1}{x_i-x_k}=:u_{i1}(x), \frac{\partial^2 F}{\partial x_i^2}(x)=-\sum_{k;k \neq i}\frac{1}{(x_i-x_k)^2}=:-u_{i2}(x),
0=\sum_{i=1}^D\sum_{j=1}^D \alpha_i \bar{\alpha}_j \sum_{f\in \mathcal{B}_{2k}(p)}\omega_f \lambda_f(i) \lambda_f(j)\big|\langle (z^n\otimes\lbrace 0, \infty\rbrace), f \rangle \big|^2.
V_{- \textstyle{\frac{1}{2}}; \chi}= e^{- \textstyle{\frac{\phi}{2}}}\prod_{j = 1}^{3}\sigma_{1 - \nu}^{j}e^{i ( \textstyle{\frac{1}{2}}- \nu_{j}) H_{j}}e^{i k_{\mu}X^{\mu}}\, , \ \ \V_{- \textstyle{\frac{1}{2}}; \chi^{*}}= e^{- \textstyle{\frac{\phi}{2}}}\prod_{j = 1}^{3}\sigma_{\nu}^{j}e^{- i ( \textstyle{\frac{1}{2}}- \nu_{j}) H_{j}}e^{i k_{\mu}X^{\mu}}\, .
\left|\lim_{i\to\infty}F_{\Omega}(u_i)-|h'(0)|^{2-\beta}\lim_{i\to\infty}F_{{B_1}}(v_i)\right| = \left|\lim_{i\to\infty}\int_{B_{\delta}}\frac{e^{\alpha v_i^2}-1}{|y|^{\beta}}\left(\chi(y)-\chi(0)\right)\right| \smallskip \leq \epsilon\, F^{sup}_{B_1}\,,
m_j(f) := \sum_{(i_1,\ldots,k_{l})\in I_{j}(f)}2^{(k_1+\ldots+k_{l})r}d_{i_1}^r\ldots d_{i_l}^r \Pr(B_{i_1,k_1,\ldots,i_{l},k_{l}}) & \leq \prod_{s=1}^lg(f(s))=\prod_{\rho=0}^{\infty}g(\rho)^{n_\rho},
\lambda_1 = [-1,0,0,0,1]\, ,\lambda_2 = [-1,0,0,1,0]\, ,\lambda_3 = [0,-1,0,0,1]\, , & \lambda_4 = [0,-1,0,1,0]\, ,\lambda_5 = [0,0,-1,0,1]\, ,\lambda_6 = [0,0,-1,1,0]\, .
\langle \langle \widehat \chi \rangle \rangle ={\varepsilon^{\alpha}}_{\mu \beta}\left({\cal F}_{+ \alpha}\partial^{\mu}{\cal F}_{+}^{\beta}+{\cal F}_{- \alpha}\partial^{\mu}{\cal F}_{-}^{\beta}\right) + m \left({\cal F}_{+ \gamma}{\cal F}_{+}^{\gamma}-{\cal F}_{- \gamma}{\cal F}_{-}^{\gamma}\right) .
\theta^{\mu \nu}= \frac{2 \pi \alpha^{\prime}}{g_{2}g ( 1 - e^{2}) + g^{2}b^{2}}\left( \begin{array}{c c c}{0}&{g_{2}e}&{0}\\{- g_{2}e}&{0}&{- g b}\\{0}&{g b}&{0}\\ \end{array}\right) .
\mu_{X}(\{a\})=lim_{T_{1}\rightarrow\infty}\cdot\cdot\cdot lim_{T_{n}\rightarrow\infty}(\prod_{k=1}^{n}\frac{1}{2T_{k}})\int_{[-T_{1},T_{1}]\times...\times[-T_{n},T_{n}]}e^{-i(t\cdot a)}\varphi_{X}(t)\lambda(dt)
\psi = e^{- i \omega t}\cdot \left\{F_{l \pm 1 , l}\Psi_{\frac{l \pm 1}{2}\frac{l}{2}m m^{\prime}}^{( l )}+ G_{l \pm 1 , l}\Psi_{\frac{l \pm 1}{2}\frac{l}{2}m m^{\prime}}^{( l \pm 1 )}\right\}
\int f(t) h(t/q) \left( -\frac{q(1+q)+t}{q^2 (1-q)t}u(t/q) +\frac{1}{q(1-q)^2t}\right)y(t) d_qt &=f(x/q )h(x/q) \left(y(x/q) u(x/q)-D_{q^{-1}}y(x) \right).
\psi_k(\theta) = (k+1) \theta - 2 \left(\Im \log \Phi_{k}^*(e^{i \theta}) - \Im \log \Phi_{k}^*(1) \right) = (k+1) \theta - 2 \sum_{j=0}^{k-1}\left( \Im \log \left(1 - \gamma_{j}e^{i \psi_j(\theta)}\right) - \Im \log \left(1 - \gamma_{j}\right) \right).
\langle \psi(p_1) \chi(p_1) \cdots \psi(p_n) \chi(p_n) \rangle = \frac{1}{Z_f(X,L)}\int_{\mathcal{H}^+}\int_{\mathcal{H}^-}\psi(p_1) \chi(p_1) \cdots \psi(p_n) \chi(p_n) \exp\left( \frac{1}{2}\langle \psi, D_L \chi \rangle \right) \mathcal{D}\chi
B^n(t,\omega_1,\ldots,\omega_n) = \begin{cases}\omega_1(t) - \omega_1(0) t \in [0,1] \omega_{k+1}(t-k) - \omega_{k+1}(0) +{\sum_{i=1}^k[\omega_i(1)-\omega_i(0)]}t \in [k,k+1], \k \leq n-1.\end{cases}
D(H_1) = \bigl\{x\in X\:;\: ((\mathcal{H}_ex_e)')_{e\in E}\in X \mathcal{H}_ex_e|_{\partial(a(e),b(e))}=0 e\in E\bigr\}D(H_2) = \bigl\{x\in X\:;\: (P_0\mathcal{H}_ex_e)_{e\in E}\in X\bigr\}.
\lfloor y_i^{\textsc{d}}k_- \rfloor + 1 = a_{\pi(i)},\;\;\;\;\;\lceil y_{j+}^{\textsc{g}}k_- \rceil = a_{\pi(n_{{\textsc{d}}}+ j)},\;\;\;\;\;y_- = \frac{a_{\pi(n_{{\textsc{d}}}+ n_{{\textsc{g}}}+ 1)}}{k_-}
(T_x -q) \big[ (1 - q x \alpha_1) (1 - x \alpha_2) T_x^2 &{}- \big\{q ( \beta_1 \beta_2 +\alpha_1 \alpha_2 ) x^2 - (q(\beta_1+\beta_2) + q^{\lambda}(q \alpha_1+\alpha_2) ) x + q^{\lambda}(q + 1) \big\}T_x &{}+ q \big(\beta_1 x-q^{\lambda}\big) \big(\beta_2 x-q^{\lambda}\big) \big] \widehat{y}_{1}( x) =0.
\|u(t)\|_{L^k_yL^\infty_x}\lesssim\frac{1}{|t|^{(d-1)\left(\frac{1}{2}-\frac{1}{k}\right)}}\sum_{j=0}^3\|A_j u\|_{L^\infty_TL^2_{xy}}\lesssim \frac{1}{|t|^{(d-1)\left(\frac{1}{2}-\frac{1}{k}\right)}}\|u\|_{\tilde Z_T}.
{\cal M}_{0}^{( r )}\sim{\textstyle{\frac{1}{2}}}\, ( n_{1}+ n_{2}) \hat{Q}_{o}( 0 ) -{\textstyle{\frac{1}{2}}}\, \left[ 2 \cdot 2^{r / 2}\cdot k_{1}k_{2}+ 2 \right] ( m + \bar{m}) \hat{Q}_{v}( \zeta \tau ) \, .
\omega_{( 2 )}^{r x r}= \left( \begin{array}{l l}{{\widetilde{\omega}_{( 2 )}^{s x s}}}&{{0^{( r - s ) x s}}}\\{{0^{s x ( r - s )}}}&{{0^{( r - s ) x ( r - s )}}}\\ \end{array}\right) \; ,
\begin{array}{l}{{\ell_{3}^{+}= \ell^{+}, \ell^{-}= 0}}\\{{\hat{h}= \frac{k^{-}}{k}\ell^{+}= \frac{\hat{q}}{2}}}\\{{\hat{h}= \hat{h}^{\prime}, \hat{q}= \hat{q}^{\prime}}}\\ \end{array}
E_{L}=(\begin{matrix}0&e_{2,1}&e_{3,1}&e_{4,1}\\ -e_{2,1}&0&e_{3,2}&e_{4,2}\\ -e_{3,1}&-e_{2,3}&0&e_{4,3}\\ -e_{4,1}&-e_{4,2}&-e_{4,3}&0\end{matrix})
\begin{array}{r c l}{{| \psi_{i}>}}&{{=}}&{{| q_{i}> \otimes | p_{\lambda}= 0 > \otimes | - + > = \sum_{n = 0}^{\infty}| n > \otimes | p_{\lambda}= 0 > \otimes | - + > < n | q_{i}> \ \ \ ,}}\\{{| \psi_{f}>}}&{{=}}&{{| q_{f}> \otimes | p_{\lambda}= 0 > \otimes | - + > = \sum_{n = 0}^{\infty}| n > \otimes | p_{\lambda}= 0 > \otimes | - + > < n | q_{f}> \ ,}}\\ \end{array}
\sqrt{a_{i}h_{i}}=\sqrt{\frac{a_{i}+h_{i}}{\frac{a_{i}+h_{i}}{h_{i}a_{i}}}}=\sqrt{\frac{a_{i}+h_{i}}{\frac{1}{a_{i}}+\frac{1}{h_{i}}}}=\sqrt{a_{i+1}h_{i+1}}
\begin{aligned}\int_{B_{s}}(|\nabla u|^{2})^{\mu}dx \le C_{0}\int_{B_{s}-B_{t}}(|\nabla u|^{2})^{\mu}dx +C_{0}\left\{\frac{1}{(s-t)^{n}}\int_{B_{s}}d^{n}(u,p_{0})dx+\int_{B_{R}}gdx\right\}\end{aligned}
w_-^* ( h, \widetilde{h}_0 ) \delta_{0/1}= [\delta_{0,0/1}- J_0 \langle \delta_0, (h - E + \i 0)^{-1}\delta_{0/1}\rangle ] \oplus [ \delta_{1,0/1}- J_0 \langle \delta_1, (h - E + \i 0)^{-1}\delta_{0/1}\rangle ]
\Pi = \mathrm{\O}\int \left[{\frac{e_{1}^{2}}{( p \! \! \! / - m_{1}) ( q \! \! \! / - m_{1})}}-{\frac{e_{2}^{2}}{( p \! \! \! / - m_{2}) ( q \! \! \! / - m_{2})}}\right] d^{4}p
\sigma_{1}=[\begin{matrix}0&0&0\\ 0&10&0\\ 0&0&10\\ \end{matrix}]+\sigma=[\begin{matrix}0&0&0\\ 0&10&0\\ 0&0&10\\ \end{matrix}]+[\begin{matrix}1&0&0\\ 0&0&3.5\\ 0&-1&0\\ \end{matrix}]
\theta_{1}=[\begin{matrix}0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&1&0\end{matrix}]\theta_{2}=[\begin{matrix}0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&-1&0&0\end{matrix}]\theta_{1}\theta_{2}=-\theta_{2}\theta_{1}=[\begin{matrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\end{matrix}].
\bigg| \int_{B^C_{S_1}}\sum^3_{k,\ell=1}\partial y_{\ell}\mathfrak Z_{jk}(x,y) u_k (y) u_{\ell}(y) dy \bigg| &\le C \int_{B^C_{S_1}}\sum^3_{k,\ell=1}|\partial y_{\ell}\mathfrak Z_{jk}(x,y) | |y|^{-2}\nu (y)^{-2}dy &\le C \int_{B^C}[(1+|x-y|)\nu (x-y)]^{-3/2}(1+|y|)^{-2}|(\nu )|^{-2}dy +
V_{i_{1}. . . i_{n}}^{a_{1}. . . a_{n}}\left( 1 ; p_{1}, . . . , p_{n}\right) + V_{i_{1}. . . i_{n}}^{a_{1}. . . a_{n}}\left( 2 a ; p_{1}, . . . , p_{n}\right) + V_{i_{1}. . . i_{n}}^{a_{1}. . . a_{n}}\left( 2 b ; p_{1}, . . . , p_{n}\right) = 0
\mathbb{T}_L =\Delta^{-1}\left(\begin{matrix}([b,d]d^{-1}c+[d,a]+[c,b]) (a-bd^{-1}c)^{-1}\left([d,a]+[c,b]+[b,d]b^{-1}a\right)(c-db^{-1}a)^{-1}[a,c](a-bd^{-1}c)^{-1}[a,c](c-db^{-1}a)^{-1}\end{matrix}\right).
\mathbf{u}^{\mathcal{L}}(x)\mathcal{J}_n \mathbf{u}^{\mathcal{R}}(x)=\sqrt{d_{n}}[(x+i\omega)\{u_{n-1}^{\mathcal{R}}(x)\}'u_{n}^{\mathcal{L}}(x)-(x-i\omega)\{u_{n}^{\mathcal{R}}(x)\}'u_{n-1}^{\mathcal{L}}(x)].
e=\frac{i}{2}\left\{d\tau+\sum_{a=1}^N\frac{S_a}{2i}\left({\tt \frac{1}{z-z_a}}dz-{\tt \frac{1}{\bar{z}-\bar{z}_a}}d\bar{z}\right)\right\}\left(\begin{array}{rr}1 0 0 -1\end{array}\right)+\frac{i}{2}\left(\begin{array}{cc}0 dz d\bar{z}0\end{array}\right)\, ,