harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{gathered}\min_{\mathbf{a}}\mathbb{E}\left[\lVert\mathbf{s}^{\left(\right)}-\mathbf{y}'\rVert^2\right] ~~\left(\mathbf{P}^{\left(\right)}\left(\mathbf{a}^{\left(\right)}\odot \mathbf{a}^{\left(\right)}\right)\mathbf{P}^{\left(\right)^{H}}\right)=E_{tr},\end{gathered}
t = \frac{z+1}{z}t_1t_2 = \frac{z}{z+1}\left( \frac{z+1}{z}t_1\right) \left( \frac{z+1}{z}t_2\right) &= \frac{z}{z+1}S_{a_1}(z)S_{a_2}(z) = \frac{z}{z+1}S_{a_1a_2}(z) ={\mathcal X}_{a_1a_2}(z).
\Psi_{\kappa}^{( R )}= \frac 1{2^{3 / 2}\pi}\, e^{\pi \kappa / 2}\left\{\Gamma ( i \kappa ) \left( \frac{m x_{+}}2 \right)^{- i \kappa}+ \Gamma ( - i \kappa ) \left( - \frac{m x_{-}}2 \right)^{i \kappa}+ . . . \right\}.
\Theta^{4}\Big [ \begin{array}{c}{{0}}\\{{0}}\\ \end{array}\Big ] ( \tau ) - \Theta^{4}\Big [ \begin{array}{c}{{0}}\\{{1}}\\ \end{array}\Big ] ( \tau ) - \Theta^{4}\Big [ \begin{array}{c}{{1}}\\{{0}}\\ \end{array}\Big ] ( \tau ) = 0 ,
T^{\,2(b_{n+1}-b_n)}\,e_{b_n}=\smash{v_n\,\Big(\prod_{j=b_n+1}^{b_{n+1}-1}w_j\Big)e_{b_{\varphi(n)}}-T^{\,b_{n+1}-b_{n}}\,e_{b_{n}}=e_{b_{n}}.}
L(X,Y)=\frac{1}{n-2}\{\frac{1}{n}[(n-1)Ric(X,Y)+\stackrel{*}{Ric}(X,Y)]-\frac{\sigma}{2(n-1)}g(X,Y)\}\stackrel{*}{L}(X,Y)=\frac{1}{n-2}\{\frac{1}{n}[(n-1)\stackrel{*}{Ric}(X,Y)+Ric(X,Y)]-\frac{\stackrel{*}\sigma}{2(n-1)}g(X,Y)\}g(\stackrel{*}{L}^{\sharp}(X),Y)=\stackrel{*}{L}(X,Y)
\Phi^\dagger_{vp}(v,p)[v_1,p_1] = Q^\dagger F^{q,\dagger}_{vv}(v+\psi^\dagger(v,p),p)[v_1 + \psi_v^\dagger(v,p)v_1]\psi^\dagger_p(v,p)p_1 &+ Q^\dagger F^{q,\dagger}_{vp}(v+\psi^\dagger(v,p),p)[v_1 + \psi_v^\dagger(v,p)v_1, p_1] &+ Q^\dagger F^{q,\dagger}_{v}(v+\psi^\dagger(v,p),p)\psi^\dagger_{vp}(v,p)[v_1,p_1]
w = e^{2 \phi}- v^{2}f^{- 1}\equiv e^{2 \phi_{0}}- v_{0}^{2}f_{0}^{- 1}, \quad 1 - \beta = f^{- 1}| \Phi |^{2}e^{2 \phi}\equiv f_{0}^{- 1}| \Phi_{0}|^{2}e^{2 \phi_{0}},
\Big\| \sum_{n \leq x}{a_{n}n^{-s}}\Big\|_{\mathcal{H}^{+}_p(X)}= \Big\| \sum_{n \leq x}(a_n n^{-s}) n^{-z}\Big\|_{\mathfrak{D}_{\infty}(\mathcal{H}_{p}(X))}&\leq C \log{x}\| \sum_{n}(a_n n^{-s}) n^{-z}\|_{\mathfrak{D}_{\infty}(\mathcal{H}_{p}(X))}&= C \log{x}\| D\|_{\mathcal{H}^{+}_{p}(X)}\,.
b_{n k}= \frac{1}{n + k}\left[ \frac{( k + 1 ) ( k + 2 )}{2}b_{n - 1 , k + 2}- \frac{\lambda ( \lambda - 1 )}{2}\sum_{m = 0}^{k}\frac{f^{( m )}}{m !}b_{n - 1 , k - m}\right] ,
e^{\int_0^t\eta(s)ds}|\Delta Y_t|^2+\int_t^T e^{\int_0^s\eta(\tau)d\tau}\left(\eta(s)|\Delta Y_s|^2 +|\Delta Z_s|^2+ \|\Delta U_s\|^2\right)ds &=e^{\int_0^T\eta(s)ds}|\Delta\xi|^2+M(t) & \quad+\int_t^T 2e^{\int_0^s\eta(\tau)d\tau}\Delta Y_s \,(f(s,Y_s,Z_s,U_s)-f'(s,Y'_s,Z'_s,U'_s))ds,
\mathcal{K}_m'' (x) (v_1, v_2) (s) = \tilde{h}\sum_{j=1}^m \sum_{i=1}^\rho w_i \; \frac{\partial^2 \kappa}{\partial u^2}(s, \zeta_i^j, x (\zeta_i^j)) v_1 (\zeta_i^j) \; v_2 (\zeta_i^j), \;\;\; s \in [a, b].
\| D_{S,x}[f']\|_2^{q-2}= \| \left( \frac{1}{\sqrt{2}}\right)^{|S|}T_{1/\sqrt{2}}D_{S, x}[f] \|_2^{q-2}= \left( \frac{1}{\sqrt{2}}\right)^{|S|(q-2)}\|T_{1/\sqrt{2}}D_{S,x}f\|_2^{q-2}\le \left( \frac{1}{\sqrt{2}}\right)^{|S|(q-2)}\|D_{S,x}f\|_2^{q-2}\le (r^{|S|}\gamma)^{q-2},
D_{\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}}( x_{1}, x_{2}, x_{3}, x_{4}) = \int_{z_{0}> 0}\frac{d^{5}z}{z_{0}^{5}}\prod_{i = 1}^{4}\frac{z_{0}^{\Delta_{i}}}{( z_{0}^{2}+ ( \vec{z}- \vec{x_{i}})^{2})^{\Delta_{i}}}.
\lvert a_t - \zeta_i^l \vert = \frac{\lvert a_t^i - 1 \rvert}{\left\lvert \sum_{j = 0}^{i-1}a_t^{i-j-1}\zeta_i^{lj}\right\rvert}\ge \frac{1}{i^{D+1 + \varepsilon}}
h(Y_2^n|W_1) = h(X_1^n+\tfrac{1}{\sqrt{a}}(\sqrt{b}X_2^n+Z_2^n)|W_1)+\frac{n}{2}\log a &\leq h(X_1^n+\tilde{Z}^n|W_1)+\frac{n}{2}\log a + 0.5n,
\left\Vert z\right\Vert_{Y^{\alpha/2}}=\left\Vert z\right\Vert_{L^{2}\left(\mathbb{R}^{n}\right)}+\left( \int\limits_{Q}\frac{\left\vert z\left(x\right) -z\left( y\right) \right\vert^{2}}{\left\vert x-y\right\vert^{n+\alpha}}dxdy\right)^{1/2}.
\displaystyle F_n(x)= \sum_{m=-\infty}^{\infty}\frac{e^{2\pi i m x}}{m^n}= \begin{cases}\displaystyle \sum_{m=-\infty}^{\infty}\frac{e^{-2\pi i m x}}{m^n}\displaystyle -\sum_{m=-\infty}^{\infty}\frac{e^{-2\pi i m x}}{m^n}\end{cases}
\begin{cases}[(i\xi)^3+\lambda]\hat{\varphi}(\xi)+\dfrac{ab}{c}(i\xi)^3\hat{\psi}(\xi) =\varphi''(0)+\dfrac{ab}{c}\psi''(0)- \left(\varphi''(L)+\dfrac{ab}{c}\psi''(L)\right)e^{-iL\xi}, \dfrac{1}{c}[(i\xi)^3+r(i\xi)+c\lambda]\hat{\psi}(\xi) + a(i\xi)^3\hat{\varphi}(\xi)=0.\end{cases}
P(L_{n+m}\ge y+z-c(nm)^{\frac{1}{d+1}}) \ge P(S_{n+p}-S_n=\lambda)P(L_{n}\ge y,\hat{L}_m\ge z, N_{n,m}(\lambda)\le \frac{1}{4d}(c-d-2d^2)(nm)^{\frac{1}{d+1}}) \ge \zeta^{dq}P(L_{n}\ge y,\hat{L}_m\ge z, N_{n,m}(\lambda)\le \frac{1}{4d}(c-d-2d^2)(nm)^{\frac{1}{d+1}}).
q_{2}^{( i )}- q_{1}^{( i )}= \frac{1}{\pi}\int_{0}^{\infty}d \omega \sinh \pi \omega s^{( i )}- \int_{0}^{\infty}d \omega \frac{I_{\omega}( k \xi_{1}) I_{- \omega}( k \xi_{2})}{D_{\omega}( k \xi_{1}, k \xi_{2})}S^{( i )},
D_2\pi^{\alpha_1}=\frac 12C_{\;\;\beta_1\gamma_1}^{\alpha_1}\pi^{\beta_1}\eta_2^{\gamma_1}+\frac 12C_{\;\;\beta_1\gamma_3}^{\alpha_1}\left( \pi^{\beta_1}\eta_2^{\gamma_3}-\eta_2^{\beta_1}\pi^{\gamma_3}\right) ,
d s_{4}^{2}= - f ( R ) \left( 1 - \frac{2 m}{R}\right) d t^{2}+ f^{- 1}( R ) \left[ \left( 1 - \frac{2 m}{R}\right)^{- 1}d R^{2}+ R^{2}d \Omega_{2}^{2}\right]
I = \int_{C}d \tau \; \eta_{a b}\; [ p^{a}\partial_{\tau}q^{b}+ t^{\mu}( p^{a}e_{\; \mu}^{b}+ j^{a}\omega_{\; \mu}^{b}) ] + \lambda_{1}( p^{2}- m^{2}) + \lambda_{2}( W^{2}- m^{2}s^{2})
( G_{O M}^{- 1})_{a b}={\frac{\left( ( 2 K^{2}- 1 ) - 2 K^{2}\sqrt{1 - K^{- 2}}\right)^{\frac{1}{6}}}{K}}\left[ \hat{\eta}_{\hat{a}\hat{b}}+{\frac{1}{4}}( \hat{{\cal H}}^{2})_{\hat{a}\hat{b}}\right]
M_{s c a l a r \, \pm}^{2}~ ~ = ~ ~ M_{\pm}^{2}~ - ~{\frac{3}{4}}\, \alpha \gamma \, \mu \, \widetilde{m}_{\lambda}\Bigg ( 1 ~ \pm ~ \sqrt{1 + x}~ \Bigg ) \Bigg ( 2 ~ \pm ~{\frac{1}{\sqrt{1 + x}}}~ \Bigg ) ,
\sum_{k=1}^{n-1}\binom{n-1}{k}S_{k+1}x^{k}(x-1)^{n-1-k}(x-2)^{n-1-k}= \sum_{k=0}^{n-1}\frac{1}{n}\binom{n}{k}\binom{n}{k+1}x^{2k}2^{n-k}= \sum_{k=0}^{n}\binom{n+k}{2k}C_k x^{2k}(2-x^2)^{n-k}.
\mu(K,L) =\mu(L,L)= 2\pi (2(n-1)c_{0}+c_{1}), \mu(K,K) = 2\pi \left( 2(n-1)c_{0}+c_{1}+\frac{((4n+1)c_{1}-2(3n+1)c_{0})\varepsilon^2}{2n\omega_{2n}}\int f^2\; du\right).
(-q;q^2)_\infty (1-q^{2a}) = (1-q^{2a})\sum_{n=0}^\infty \frac{q^{n^2}}{(q^2;q^2)_n}&= (1-q^{2a}) + \frac{(1-q^{2a})}{(1-q^2)}\left( \sum_{n=1}^\infty \frac{q^{n^2}}{(q^4;q^2)_{n-1}}\right) &= 1-q^{2a}+ \left(\sum_{t=0}^{a-1}q^{2t}\right)\left(\sum_{n=1}^\infty \frac{q^{n^2}}{(q^4;q^2)_{n-1}}\right).
\Sigma_1 =\sum_{a\le q}\sup_{|\gamma|\le E/Y}F_{U}\Bigl(\frac{a}{q}+\gamma\Bigr), \Sigma_2 =\sup_{\beta\in\mathbb{R}}\sum_{\substack{|\eta|\le E/Y Y(\eta+\beta)\in\mathbb{Z}}}F_{Y/UV}\Bigl(U V\beta+UV\eta\Bigr) &\le \sup_{\beta'\in\mathbb{R}}\sum_{a\le 2E}F_{Y/UV}\Bigl(\beta'+\frac{U V a}{Y}\Bigr).
\begin{array}{ll}\left[\sum_{j=i}^{k-1}\hat\rho_j \hat\gamma_{i,j}c,\sum_{j=i}^{k}\hat\rho_j \hat\gamma_{i,j}c\right] k=i+1,\dots,i+N-1, begin{align*}2ex]\left[\hat\rho_i \hat\gamma_{i,i}c,\hat\gamma_{i}c\right] k=i.\end{array}
\left\{\mu \frac{\partial}{\partial \mu}+ \left( \beta_{p}- d_{p}p \right) \, \frac{\partial}{\partial p}+ \gamma_{i}\int d^{4}x \sqrt{- g}\; \Phi_{i}\, \frac{\delta}{\delta \Phi_{i}}\right\}\, \Gamma \left[ g_{\alpha \beta}, p , \Psi_{i}, \mu \right] = 0 \, .
\bar{G}_{2 n}= \sum_{l = 1}^{n}\bar{G}_{2 l - 2}^{( 0 )}\bar{G}_{2 n - 2 l}- g \bar{G}_{2 n}\bar{G}_{2}- g \bar{G}_{2 n + 2}\bar{G}_{0}+ 3 g \bar{G}_{2 n},
\lim_{s\to \infty}\frac{M(b,c;s)}{s^{b-c}e^s}= \lim_{s\to \infty}\frac{\frac{d^m}{ds^m}M(b,c;s)}{\frac{d^m}{ds^m}(s^{b-c}e^s)}= \frac{(b)_m}{(c)_m}\lim_{s\to \infty}\frac{M(b+m,c+m;s)}{s^{b-c}e^s+o(s^{b-c}e^s)}= \frac{(b)_m \Gamma(c+m)}{(c)_m \Gamma(b+m)}= \frac{\Gamma(c)}{\Gamma(b)}.
E(f) = (-1)^{j-t}E(f_2) b_2,F(g) = F(g_2) b_2^-, & \varphi(E(f),F(g)) = (-1)^{j-t + |F(g_2)|}\varphi(E(f_2),F(g_2)) \varphi(b_2,b_2^-).
\Phi_1(\lambda_{m}) = 2-\sqrt{2+2\sqrt{1-p\lambda_{m}+p^2\lambda_{m}}}\Phi_2(\lambda_{m}) = 2-\sqrt{2-2\sqrt{1-p\lambda_{m}+p^2\lambda_{m}}}\Phi_3(\lambda_{m}) = 2+\sqrt{2-2\sqrt{1-p\lambda_{m}+p^2\lambda_{m}}}\Phi_4(\lambda_{m}) = 2+\sqrt{2+2\sqrt{1-p\lambda_{m}+p^2\lambda_{m}}}
\begin{array}{l}{Q_{a}= \int d x^{-}d^{2}x_{\bot}( f^{a b c}A_{b}^{i}\partial^{+}A_{c}^{i}+ 2 \psi_{+}^{\dagger}T^{a}\psi_{+})}{~ ~ ~ ~ ~ = \int d x^{-}d^{2}x_{\bot}( \rho_{a}^{g}( x^{-}, x_{\bot}) + \rho_{a}^{q}( x^{-}, x_{\bot}) )}\end{array}
\sum_{k\geq 2}\phi^e_k\Delta_LU^{e}_{k}= - (L^e)^2\,[\phi^e_2\,(U^e_2-U^e_1)+(\phi^e_3-\phi^e_2)\,U^e_2]+\sum_{j\geq 3}U^e_j \Delta_L\phi^{e}_{j}
S_{2}= \left( \begin{matrix}{{\cosh ( \lambda / 2 ) - \sin ( \theta / 2 ) \sinh ( \lambda / 2 )}}&{{\cos ( \theta / 2 ) \sinh ( \lambda / 2 )}}\\{{\cos ( \theta / 2 ) \sinh ( \lambda / 2 )}}&{{\cosh ( \lambda / 2 ) + \sin ( \theta / 2 ) \sinh ( \lambda / 2 )}}\\ \end{matrix}\right) .
\mathbb{E}|\tilde{\mathcal{F}}_{1,2}|^{2}= N^{-4}|s(z_{2})|^{2}\sum_{\substack{j_1,p_1,q_1 j_2,p_2,q_2}}(|W_{p_1j_1}|^{2}-1)(|W_{p_2j_2}|^{2}-1)\mathbb{E}_{k}(G^{(1)}_{kp_1j_1})_{j_1q_1}\mathbb{E}_{k}(G^{(2)}_{kp_1j_1})_{q_1j_1}&\times \overline{\mathbb{E}_{k}(G^{(1)}_{kp_2j_2})_{j_2q_2}\mathbb{E}_{k}(G^{(2)}_{kp_2j_2})_{q_2j_2}}
\mathcal P_a^i:= \big\{P\in{[2,n]\choose k-1}\ :\P\cap [i] = S_i\cap [2,i]\big\}, \mathcal P_b^i:= \big\{P\in{[2,n]\choose k}\ :\P\cap [i] = [i]\setminus S_i\big\},
= \; \sum_{\pi ( n )}\mathrm{s i g n}( \pi ) G_{1 2}^{o}( w_{1}- z_{\pi ( 1 )}) G_{1 2}^{o}( w_{2}- z_{\pi ( 2 )}) . . . G_{1 2}^{o}( w_{n}- z_{\pi ( n )})
\mu(\tau) = (\det A(\tau))^\frac13, \ \ \Lambda(\tau) =\det A(\tau)^{\frac23}A(\tau)^{-1}A(\tau)^{-\top}, \mu_0 : = \frac{3\gamma-3}2\mu_1, \ \ \mu_1:=\lim_{\tau\to\infty}\frac{\mu_\tau(\tau)}{\mu(\tau)}.
C_{\Omega}= 2^{1 0 / 3}3^{- 1}\pi^{1 3 / 6}\kappa^{- 5 / 3}N^{1 / 2}m^{1 / 2}r_{+}\frac{1 5 r_{+}^{4}+ 3 l^{2}r_{+}^{2}- 2 l^{4}}{3 r_{+}^{4}+ 4 l^{2}r_{+}^{2}+ l^{4}}\frac{3 r_{+}^{2}+ l^{2}}{3 r_{+}^{2}- l^{2}}.
\Big( 2 D_4 u(x_1,x_2) - (1+z_1-z_2) D_1u(x_1,x_2) - (z_3-z_1) D_2 u(x_1,x_2) - (1+z_2-z_3) D_3 u(x_1,x_2) \Big) \Big|_{x_1 = 0}= 0.
S_{i_1}:= \bigcup_{u_2=0}^{s_2-1}\bigcup_{q_2=0}^{p_{i_2}-1}\Big( \beta^{u_2 s_1}\alpha_{i_2}^{q_2}W_{i_1}\Big), S_{i_2}:= \bigcup_{u_1=0}^{s_1-1}\bigcup_{q_1=0}^{p_{i_1}-1}\Big( \beta^{u_1}\alpha_{i_1}^{q_1}W_{i_2}\Big),
f(x,v,t) =\, f_0(x-vt)\exp\left( -\int_0^t \rho_g(x+v(s-t),s)ds\right) + \chi_1(v) \int_0^t \exp\left( -\int_s^t \rho_g(x+v(\tau-t),\tau)d\tau\right)ds \,,
\frac{\|F\|_{W^{-s,p'}(\Omega)}}{(1-s)^\frac{1}{p}}=p^\frac{1}{p}\,\left(-p'\,\min_{u\in \widetilde W^{s,p}_0(\Omega)}\left\{(1-s)\,[u]^p_{W^{s,p}(\mathbb{R}^N)}-\langle F,u\rangle\right\}\right)^\frac{1}{p'}.
~10 \sum_{h=0}^{k-5}\alpha^{h-k}\,(k-h-1) + 3 \alpha^{-4}(3) + \alpha^{-3}(2) = ~10 \sum_{j=5}^k \alpha^{-j}\, (j-1) + 9 \alpha^{-4}+ 2 \alpha^{-3}\approx ~10 \left(\alpha^{-3}+ 3 (\alpha-1) \alpha^{-4}\,/\,(\alpha-1)^2 \right) + 9 \alpha^{-4}+ 2 \alpha^{-3}\approx ~~~3.383 ~+~ 0.591 ~+~ 0.259 ~~~= ~4.233.
I_{j}=\int_{-\infty}^{\infty}e^{\frac{1}{2}\lambda\mu_{j}y^{2}}dy=2\int_{0}^{\infty}e^{-\frac{1}{2}\lambda(\sqrt{-\mu_{j}}y)^{2}}dy=2\int_{0}^{\infty}e^{-\frac{1}{2}\lambda|\sqrt{-\mu_{j}}|^{2}y^{2}exp(2iarg\sqrt{-\mu_{j}})}dy.
\sum_{\substack{k=0 n - k \equiv 0 \,(\mathrm{mod}\,2)}}^n \binom{n}{k}2^k F_{k}(\sqrt{5})^{n-k}\frac{B_{n-k+2}}{n-k+2}+ \frac{2}{\sqrt{5}}\sum_{\substack{k=0 n - k \equiv 1 \,(\mathrm{mod}\,2)}}^{n-1}\binom{n}{k}(\sqrt{5})^{n-k}\frac{B_{n-k+1}}{n-k+1}& = \frac{2}{5(n+1)}\Big ( \frac{2^{n+1}F_{n+2}}{n+2}- 1 \Big ).
\begin{array}{l}0=k_{1}+3k_{3}{}0=k_{2}+k_{4}+k_{5}+4k_{6}+4k_{7}{}0=k_{2}+k_{3}+3k_{4}+k_{5}+4k_{6}+12k_{7}{}0=4c_{2}-6c_{1}-k_{2}+k_{3}+k_{4}-k_{5}-4k_{6}+4k_{7}\end{array}
\Phi_{S O ( 6 n + 7 )}~ = ~ \left( \begin{array}{c c c}{{0}}&{{\bar{Q}^{T}}}&{{- i \bar{Q}^{T}}}\\{{- \bar{Q}}}&{{A}}&{{i A}}\\{{i \bar{Q}}}&{{i A}}&{{- A}}\\ \end{array}\right) ~ .
x\sum_{k=0}^4 \frac{k!}{\log^{k+1}x}+ \frac{m_a(x)x}{\log^6x}\leq \pi(x) \leq x\sum_{k=0}^4 \frac{k!}{\log^{k+1}x}+ \frac{M_a(x)x}{\log^6x}
\alpha A^{\lambda}_{dr,2}f(\bar{\eta})=\alpha f(\bar{\eta})\int_{{\Bbb R}^d\times [0,1]\times [0,\infty )}\Bigg(\frac{\sum_{S\subset\bar{\eta}_{y,w}}\bar{H}(\bar{g},\hat{g},S,y,\zeta ,w)}{\prod_{(x,\kappa )\in\bar{\eta}_{y,w}}\bar{g}(x,\kappa )}-1\Bigg)dy\nu^1(w,d\zeta )\nu^2(dw)
\left\Vert f\chi_{C_{j_{1},j_{2},...,j_{n}}}\right\Vert_{p}^{p}={\displaystyle\int\nolimits_{{\prod\nolimits_{i=1}^{n}}\left[ -2^{j_{i}},2^{j_{i}}\right]}}\left\vert f\left( x\right) \right\vert^{p}dx & \leq C\left\Vert f\right\Vert_{\mathcal{B}^{p}}^{p}2^{j_{1}+j_{2}+...+j_{n}}
f(n) = f(n_0)\prod_{k=n_0}^{n-1}\Big(1+\frac ak\Big) + \sum_{k=n_0}^{n-1}\frac{g(k)}k\prod_{\ell=k+1}^{n-1}\Big(1+\frac a\ell\Big) =\prod_{k=n_0}^{n-1}\frac{k+a}k\cdot \Bigg( f(n_0) + \sum_{k=n_0}^{n-1}\frac{g(k)}k\prod_{m=n_0}^{k}\frac m{m+a}\Bigg).
\int_{0}^{\infty}\frac{d \kappa^{2}}{\lambda}\rho_{i i}\left( \frac{\kappa^{2}}{\lambda}\right) H ( \kappa^{2}) \leq C \int_{0}^{\infty}\frac{d \kappa^{2}}{\lambda}\rho_{i i}\left( \frac{\kappa^{2}}{\lambda}\right) G ( \kappa^{2}) .
\int_{{\cal M}_{k}^{\prime}}e^{- d_{x}\omega}= \sum_{\sb{k_{1}, k_{2}}\atop k_{1}+ k_{2}= 1}\int_{{\cal M}_{k_{1}, k_{2}}}\frac{1}{\det^{1 / 2}\left({\cal L}_{\cal N}+{\cal R}_{\cal N}\right)},
\boldsymbol{E}[\sum_{j=1}^{T-2}\sum_{k=j+1}^{T-1}\sum_{l= k+1}^T X_j X_k X_l] \leq \boldsymbol{E}[\sum_{j=1}^{T-2}\sum_{k=j+1}^{T-1}\sum_{l= k+1}^T X_j X_k] &\leq \left([\boldsymbol{E}[\sum_{j=1}^{T-2}\sum_{k=j+1}^{T-1}X_j^2 X_k^2]\right)^{\frac{1}{2}}\left([\boldsymbol{E}[\sum_{j=1}^{T-2}\sum_{k=j+1}^{T-1}(T -k)^2] \right)^\frac{1}{2}.
B = - \frac{1}{8}\bar{D}( 1 - \gamma_{5}) D ( W_{a}^{*})^{2}= ( D^{a})^{2}- \frac{1}{2}( F_{\mu \nu}^{a})^{2}- \frac{i}{2}F_{\mu \nu}^{a}\tilde{F}_{\mu \nu}^{a}+ O ( \theta ) ,
\epsilon_{R P A}({\bf{q}}, \omega ) = 1 + \frac{v_{{\bf{q}}}}{V}\sum_{{\bf{k}}}\frac{{\bar{n}}_{{\bf{k}}+{\bf{q}}/ 2}-{\bar{n}}_{{\bf{k}}-{\bf{q}}/ 2}}{\omega - \frac{{\bf{k . q}}}{m}}
p^* (X_{a,k}) = p^* (X_{n-k}) = \lambda_{b-k}\frac{\Delta_{\{b-k,\dots, n-k-1\}}\Delta_{\{b-k+2,\dots, n-k,n\}}}{\Delta_{\{b-k+1,\dots, n-k\}}\Delta_{\{b-k+1,\dots, n-k-1,n\}}}.
e(H_{2}^{-1}H_{3}^{-1}H_{1}H_{2})eH_{2}^{-1}H_{3}^{-1}H_{1}H_{2}=e(H_{2}^{-1}H_{3}^{-1}H_{1}H_{2})eH_{2}^{-1}H_{1}^{-1}H_{3}H_{2}.
\psi ( x ,{\bf p}; s_{1}, s_{2}) ={\frac{1}{2 P^{+}}}\int{\frac{d p^{-}}{2 \pi}}\bar{u}( x P^{+},{\bf p}; s_{1}) \gamma^{+}\Phi ( p ) \gamma^{+}v ( ( 1 - x ) P^{+}, -{\bf p}; s_{2}) ,
\Omega = \sqrt{\frac{Q^{2}}{4 \pi \epsilon_{0}m r_{c}^{2}}}={\frac{e \pi Q^{2}}{4 \pi \epsilon_{0}J_{1}^{3}}}{\frac{1}{{\bar{\sigma}}^{2}}}={\frac{\pi m Q^{4}}{2 \epsilon_{0}^{2}J_{1}^{3}}}~ .
L'=\sum_{n>0}\left(\frac12|\partial\phi|^2-\frac{M^2n^2}{2}|\phi_n |^2-2\lambda \phi_n \sum_{n>0}|\phi_n|^2 \right) + \frac{\lambda}{3!}\sum_{n,n'}{}' \phi_n \phi_{n'}\phi_{n+n'}^*,
\begin{array}{l l}{\begin{array}{l l}{\frac{d}{d r}\sigma ( r ) = 0}&{r = 0}\\{\sigma ( r ) = 0}&{r \rightarrow \infty}\\ \end{array}}&{\begin{array}{l l}{A ( r ) \neq 0}&{r \rightarrow \infty}\\{A ( r ) = 1}&{r = 0 .}\\ \end{array}}\\ \end{array}
\sum_{i=1}^{\infty}rank\;\pi_i\left( Imm\left(M,\mathbb{R}^{m+k};f\right)\right)x^i = \begin{cases}\begin{aligned}\frac{x^{2k-m-1}\left(1-x^{2m+2}\right)}{1-x^4}P_M\left(x\right) \frac{x^{3k-m-2}\left(1-x^{2m+2}\right)}{1-x^4}\left(P_M\left(x\right)\right)^2 \end{aligned}\end{cases}
\delta (a) = \sum_{i}\delta (a_{i}^{1})\otimes a_{i}^{2}-a_{i}^{1}\otimes \delta (a_{i}^{2}) =\sum_{i}\mu (a_{i}^{1})\otimes a_{i}^{2}- a_{i}^1\otimes \mu (a_{i}^{2})=(\mu \otimes id) (a) - (id \otimes \mu )(a).
L =\int_X-\chi(u_1-u_2)(\theta_{\psi_1}^n-\theta_{u_1}^n) -\int_X-\chi(u_1-u_2)(\theta_{\psi_2}^n-\theta_{u_1}^n) =\int_X-\chi(u_1-u_2)dd^c(\psi_1-u_1)\wedge T_1-\int_X-\chi(u_1-u_2)dd^c(\psi_2-u_1)\wedge T_2 =L_1-L_2,
H_{n}^{(m)}(t) = \left|\begin{array}{cccc}\nu_{m}^{(t)}\nu_{m+1}^{(t)}\cdots \nu_{m+n-1}^{(t)}\nu_{m+1}^{(t)}\nu_{m+2}^{(t)}\cdots \nu_{m+n}^{(t)}\vdots \vdots \vdots \nu_{m+n-1}^{(t)}\nu_{m+n}^{(t)}\cdots \nu_{m+2n-2}^{(t)}\end{array}\right|, n,m \in \mathbb{Z}, n \geq 1.
\| \check{\mathbf{r}}_k \|_2 = \min_{\genfrac{}{}{0pt}{2}{p \in \mathbb{P}_k}{p(0) = 1}}\left\| p(\mathbf{A}\mathbf{C}^{(\ell)}) \check{\mathbf{r}}_0 \right\|_2 \leq \left( \min_{\genfrac{}{}{0pt}{2}{p \in \mathbb{P}_k}{p(0) = 1}}\max_{\lambda \in \sigma(\mathbf{A}\mathbf{C}^{(\ell)})}| p(\lambda) | \right) \| \check{\mathbf{r}}_0 \|_2.
\frac{\delta |G|^2}{\delta g^{\mu\nu}}=\frac{k}{k!}\sum_{\mu_2,\ldots,\mu_k,\nu_2,\ldots,\nu_k}G_{\mu\mu_2\ldots\mu_k}G_{\nu\nu_2\ldots\nu_k}g^{\mu_2\nu_2}\cdots g^{\mu_k\nu_k}&=\frac{1}{(k-1)!}\sum_{\mu_2,\ldots,\mu_k}G_{\mu\mu_2\ldots\mu_k}G_{\nu}^{\,\,\,\mu_2\ldots\mu_k}&=\langle i_{\partial_\mu}G,i_{\partial_\nu}G\rangle.
V:=\left(\begin{array}{cc}V_{{\beta_{n,0}}}P_{{\beta_{n,0}}}(l_{{\beta_{n,0}}}h^T s_{{\beta_{n,0}}}) P_{{\beta_{n,0}}}(l_{{\beta_{n,0}}}h^T s_{{\beta_{n,0}}}) h^TI({{\beta_{n,0}}})h\end{array}\right)
\tfrac{1}{m_1}V_{m_1}(\epsilon_{S/m_1}m_1V_{p^s}(\alpha)) + \tfrac{1}{m_1}V_{m_1}(\epsilon_{S/m_1}(-1)^q d V_{p^s}(\beta) ) = V_{m_1p^s}(F_{p^s}(\epsilon_{S/m_1})\alpha) +(-1)^q d V_{m_1p^s}(F_{p^s}(\epsilon_{S/m_1})\beta) = V_m(\alpha)+(-1)^q dV_m(\beta),
\displaystyle{\int_{B_{\delta}(z_i)}\partial_{1}\left(\log\left|x\right|^{2N}\right)\left|x\right|^{2N}e^{u_k}\to\left[8\pi\partial_{1}\log\left(\left|x\right|^{2N}\right)\right]_{x=z_i},\quad\mbox{as}k\to+\infty}.
x^{2}x^{2}=8\sum_{l=1}^{n}\Bigl(\sum_{k,r=1}^{n,\nu}\gamma_{ipk}\widetilde{\gamma}_{ipr}\gamma_{krl}\Bigr)e_{l}+8\sum_{s=1}^{\nu}\Bigl(\sum_{k,r=1}^{n,\nu}\gamma_{ipk}\widetilde{\gamma}_{ipr}\widetilde{\gamma}_{krs}\Bigr)\widetilde{e}_{s}.
\tau_{n}= \frac{t}{2}(1-\sigma 2^{-n}) \,, h_{n}= k(1-\sigma 2^{-n}) \,, \bar h_{n}= \frac{h_{n}+h_{n+1}}{2}= k(1-3\sigma 2^{-n-2}) \,, n\ge 0 \,,
{1\choose -1}{1\choose 1}{1 \choose -1},\,\,\,\,\,\,\,\,\,\,\,{1\choose 1}{1\choose -1}{1 \choose -1},\,\,\,\,\,\,\,\,\,\,\,{1\choose -1}{1\choose -1}{1 \choose 1},
\alpha_{-1}= -\frac{3t}{2^2}a_1 - \frac{1}{2^2}v_{(2,3)}+ a_1 \cdot v_{(2,3)}\in M_0^{(a_1)}\beta_{-1}= -ta_1 + a_1 \cdot v_{(2,3)}\in M_{\frac{1}{2^2}}^{(a_1)}
0 = - \frac{\partial J ( 0 , \hat{T}_{c})}{\partial \hat{T}}\hat{T}_{c}\left( 1 - \frac{\hat{T}}{\hat{T}_{c}}\right) + \frac{\partial J ( 0 , \hat{T}_{c})}{\partial \hat{m}_{T}^{2}}~ \hat{m}_{T}^{2},
P \bigl\{r_\psi(Y) < r_\psi(y); \psi(\theta)=\psi \bigr\}= \Phi \bigl\{r^*_\psi(y) \bigr\}\bigl[ 1+\mathrm{O}_p \bigl\{(\hat\psi-\psi) n^{-1/2}\bigr\}\bigr] ,
\sum_{R_i \in R}\int_{R_i}\Delta P(w)dm(w) = \sum_{k=1}^7 \oint\limits_{\gamma_k}P_ydx-P_xdy & \leq \sum_{k=1}^7 \Lambda(\gamma_k(0),\L(\gamma_k)) \leq 3f(2\delta) + 4f\left(\frac{3}{2}\delta\right) \leq 7f(2\delta).
\psi_{1}=\int d^{D}x\left( -\mu_{3a}^{(\varphi )}\partial^{\alpha}\mu_{2\alpha}^{a}+\left( \partial^{\alpha}\mu_{3\alpha}^{a}\right) \mu_{2a}^{(\varphi )}+\left( \partial^{\alpha}A_{\alpha}^{a}\right) \nu_{1a}^{(\varphi )}-\left( \partial^{\alpha}\varphi_{a}\right) \nu_{1\alpha}^{a}\right) \,,
\ell^p(\mathbb{N},w)=\left\{f=\{f(n)\}_{n\ge 0}: \|f\|_{\ell^{p}(\mathbb{N},w)}:=\Bigg(\sum_{m=0}^{\infty}|f(m)|^p w(m)\Bigg)^{1/p}<\infty\right\},
\gamma_{2a}\equiv -\left( \left( \mathrm{D}_{i}\right)_{a}^{\;\;b}\pi_{b}^{i}-\mathrm{ig}\bar{\psi}_{\alpha}^{\;B}\left( \gamma^{0}\right)_{\;\;\beta}^{\alpha}T_{a\;\;B}^{A}\psi_{\;A}^{\beta}\right) \approx 0,
\left( k^{2}+ m^{2}\right)^{- z}= \frac 1{\Gamma ( z )}\int_{0}^{\infty}d y y^{z - 1}\mathrm{e x p}\left[ - y \left( k^{2}+ m^{2}\right) \right] , \qquad \mathrm{R e}z > 0 ,
K_{\nu}(z)=\frac{\sqrt{\pi}z^{\nu}}{2^{\nu}\Gamma\left(\nu+\frac{1}{2}\right)}\,\intop_{1}^{\infty}e^{-zt}\left(t^{2}-1\right)^{\nu-\frac{1}{2}}dt,\,\,\,\,\,(\nu)>-\frac{1}{2},\,\,\,|\arg(z)|<\frac{\pi}{2}.
\begin{matrix}\displaystyle \min_{P_{\cal F},P_{\cal R}}~ P_{\text out}= G (\mathcal{A}_F P_{\cal F}^{-a}+ \mathcal{A}_R P_{\cal R}^{-a}) \textrm{s.t.}P_{\cal F}+P_{\cal R}\leq P_{tot}& -P_{\cal R}\leq 0, ~ P_{\cal F}\leq \mathcal{S}\end{matrix}
\begin{array}{c}{\bf H}_{11}{\bf V}_{r_{2}t_{13}1}=-{\bf H}_{13}{\bf V}_{r_{2}t_{13}3}{\bf H}_{12}{\bf V}_{r_{3}t_{23}2}=-{\bf H}_{13}{\bf V}_{r_{3}t_{23}3}\end{array}
\delta_{n}-\delta_{n-1}=\psi_{\beta}\left( \delta_{n-1}\right)-\psi_{\beta}\left( \delta_{n-2}\right) =\left( 1-3\sqrt{\beta}\right)\left( \delta_{n-1}-\delta_{n-2}\right) & =\left( 1-3\sqrt{\beta}\right)^{n-1}\left( \delta_{1}-\delta_{0}\right) =\frac{\beta}{2}\left( 1-3\sqrt{\beta}\right)^{n-1},
\int_{(0,r]}y^2\mu(dy) =\int_{(0,r]}\int_0^y 2z\,dz\mu(dy)=\int_0^r 2z\mu((z,r])\,dz =\int_0^r 2z(\mu(z,\infty)-\mu(r,\infty))\,dz=2\int_0^r z\mu(z,\infty)\,dz-r^2\mu(r,\infty)\,.
\hat{W}^{\hat{a}\hat{b}\hat{c}}\equiv \hat{\cal H}^{\hat{a}\hat{d}\hat{e}}\hat{C}_{\hat{d}}^{\hat{b}}\hat{C}_{\hat{e}}^{\hat{c}}= \hat{P}^{\hat{a}\hat{b}\hat{d}}\hat{C}_{\hat{d}}^{\hat{c}}.
\left( a \, x^{(l)}\, \mu_{(l+1)p}(\widetilde{g}) \right) (s) = a \, \mu_{(l+1)p}(\widetilde{g}) (s) \cdot x^{(l)}= e^{2 \pi i (l+1)p s}\widetilde{g}(s) a x^{(l)}\; ,
||\xi_1||_{\mathfrak{Y}}= \sup \bigg\{||S(\xi_1, \widehat{D})||~:~\widehat{D}\in \mathfrak{Y}\bigg\}& =\sup \bigg\{||\sum_{i=1}^{n}\xi_1(d_i)\overline{\mu}(D_i)||~:~\widehat{D}=(D_i, d_i),~i=1,2,..\bigg\}& = \sum_{i=1}^{n}\xi_1(d_i)\overline{\mu}(D_i) & =0
D \! \! \! \! /^{~2}=\left[{1 \over{\sqrt g}}\left({\sqrt g}g^{\nu \lambda}\right)_{, \lambda}+ (1-{\cal D}) \phi^\nu - 2 \sigma^{q p}e^\mu_{~q}e^\nu_{~p}\phi_\mu + \left( g^{\mu \nu}+ 2 \sigma^{\mu \nu}\right) D_\mu\right] D_\nu .
x =n^{1/r}-\frac{1}{r}-\sum_{N=2}^{r}\frac{\Gamma(N+(N-1)/r)}{(N-1)\Gamma(N+1)\Gamma((N-1)/r)}n^{-(N-1)/r}&\quad\ + \frac{1}{r}n^{-1+1/r}+O\left(\frac{1}{n}\right).
I(u, \vec{x})=\begin{cases}H(u, \vec{x}) u \leq t(\vec{x}) B(\vec{x}) u= t(\vec{x})+1 H'(u-t(\vec{x})-2, \vec{x}) t(\vec{x})+1<u \leq t(\vec{x})+t'(\vec{x})+1\end{cases}
\eta_{n+j}\int_0^T \| \mathcal{\mathbf{B^{\ast}_{n+j}}}(U_{n+j}) \|_{G_{n+j}}^2 dt &\ge\int_0^T e_1(U_{n+j})\,dt - C\delta_{n+j}\int_0^T (|u_{n-1}|^2 +|u_n|^2+\sum_{k=1}^{j-1}|u_{n+k}|^2)\,dt \,.
M_{2k}= \sqrt{\frac{(k+\alpha+1)(k+\alpha+\beta+1)(k+\alpha+\beta+2+N)}{(2k+\alpha+\beta+1)(2k+\alpha+\beta+2)}}, M_{2k+1}= \sqrt{\frac{(k+\beta+1)(k+1)(N-k)}{(2k+\alpha+\beta+2)(2k+\alpha+\beta+3)}},
\Theta = \frac{\kappa}{2 \pi}\int_{D}d^{2}x \epsilon^{i j}\left< \partial_{i}g^{- 1}\partial_{j}g g^{- 1}d g \right> + \frac{\kappa}{2 \pi}\oint_{\partial{D}}d \varphi \left< g^{- 1}\partial_{\varphi}g g^{- 1}d g \right> ,
M_R= \left(\begin{array}{ccc}1 r_1 r_3 0 1 r_2 0 0 1 \end{array}\right),\;\;M_D= \left(\begin{array}{ccc}d_1 0 0 0 d_2 0 0 0 (d_1 d_2)^{-1}\end{array}\right),\;\;M_L= \left(\begin{array}{ccc}1 0 0 l_1 1 0 l_3 l_2 1 \end{array}\right).
R^{\prime \prime}+ \frac{{w^{\prime}}^{2}- 4{R^{\prime}}^{2}}{R}+ \frac{4 ( R^{2}+ 1 )}{\nu R}+ \frac{2 \nu^{\prime}}{\nu}( R \Phi^{\prime}- R^{\prime}) - 4 R{\Phi^{\prime}}^{2}+ 1 0 R^{\prime}\Phi^{\prime}- \frac{( w^{2}- 1 )^{2}}{4 \nu R^{3}}= 0 \, ,
\left. \mathcal{O}_{\widetilde{\mathcal{C}^{2}}}\left( \widetilde{C_{j}\times Z_{i}}\right) \right\vert_{\widehat{C}}=\mathcal{O}_{\widehat{C}}(\cdots+a_{l}\widehat{C_{l}}+a_{k}E+a_{k}\widehat{C_{k}}+\cdots).
\begin{array}{r c l}{{}}&{{}}&{{{\cal R}= \bar{\cal C}{\cal K}~ ,}}\\{{}}&{{}}&{{\bar{\cal C}= \displaystyle \sum_{m = 0}^{p - 1}\frac{q^{- m ( m - 1 ) / 2}}{[ m ] !}\left( - ( q - q^{- 1}) \right)^{m}e^{m}\otimes f^{m}~ ,}}\\{{}}&{{}}&{{{\cal K}= \displaystyle \frac{1}{p}\sum_{n , n^{\prime}= 0}^{p - 1}q^{2 n n^{\prime}}t^{n}\otimes \bar{t}^{n^{\prime}}~ .}}\\ \end{array}
\begin{array}{l@{\extracolsep{1cm}}l}\eta(x)=0 \Longleftrightarrow (1-\xi(x))(1-\omega(x))=1\, , \eta(x)=1 \Longleftrightarrow \xi(x)(1-\omega(x))=1\, , \eta(x)=2 \Longleftrightarrow (1-\xi(x))\omega(x)=1\, , \eta(x)=3 \Longleftrightarrow \xi(x)\omega(x)=1\, . \end{array}
\|(H_{q}[\omega,0]\upharpoonright\Lambda'- E_{i,\Lambda}(\omega))\widetilde\psi'\| \leq \|( H_{q}[\omega,0]\upharpoonright\Lambda' -E')\widetilde\psi'\| + |E'- E_{i,\Lambda}(\omega)|\|\widetilde{\psi}'\| &< 4\delta_{0}+|E'-E|+|E-E_{i,\Lambda}(\omega)| < 9\delta_{0}.
S(x)=[x]-\sum_i\left[\frac{x}{n_i}\right] +\sum_{i<j}\left[\frac{x}{n_in_j}\right] -\sum_{i<j<k}\left[\frac{x}{n_in_jn_k}\right]+\cdots +(-1)^r\left[\frac{x}{n_1n_2\cdots n_r}\right],
\begin{array}{crcll}(i) \rho(ab) = \rho (ba) a,b \in \mathcal{E}_n(q) (ii) \rho(1) = 1 (iii) \rho ( a b_n) = z \rho (a) a \in \mathcal{E}_n(q) (iv) \rho (a \epsilon_n) = E \rho (a) a \in \mathcal{E}_n(q) (v) \rho (a \epsilon_n b_n) = z \rho( a) a \in \mathcal{E}_n(q)\end{array}