harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\lim_{n \uparrow \infty}\mathbb P_x\left(X^n_{e(q)}> z \right)=\lim_{n \uparrow \infty}q\mathbb E\left[\int_{0}^{\infty}e^{-qt}\textbf{1}_{\{X^n_t>z\}}dt\right]=\mathbb P_x\left(X_{e(q)}> z \right).
\xi_t=\begin{cases}\xi_0+\xi_t^{0,J_0}: t< T_1 \xi_{T_n-}+\Delta^n_{J_{T_n-},J_{T_n}}+\xi^{n,{J_{T_n}}}_{t-T_n}: t\in [T_n,T_{n+1}), t<\mathbf k \xi_t=-\infty :t\geq \mathbf k \end{cases},
r_{j}( w ) = w^{n ( a )}\prod_{a}\frac{E ( - x^{\frac{6 s a}{g}}/ w , x^{1 2 s}) E ( - x^{\frac{6 s ( g - a )}{g}}/ w , x^{1 2 s})}{E ( - x^{\frac{6 s a}{g}}w , x^{1 2 s}) E ( - x^{\frac{6 s ( g - a )}{g}}w , x^{1 2 s})}.
\sum_{l=-k}^k |\nabla u_{k, l}|^2= \sum_{l=-k+1}^{k-1}\Big[ 2\left\{\frac{(k+l)(k+l+1)(2k+1)}{4(2k-1)}+\frac{(k-l)(k-l+1)(2k+1)}{4(2k-1)}\right\}&\quad\quad+\frac{(k+l)(k-l)(2k+1)}{2k-1}\Big]|Y_{k-1, l}|^2 = \sum_{l=-k+1}^{k-1}\frac{k(2k+1)^2}{2k-1}|Y_{k-1, l}|^2 = \frac{k(2k+1)^2}{2k-1}\cdot \frac{2k-1}{4\pi}=\frac{k(2k+1)^2}{4\pi}.
A_k:= \{|\xi(\sqrt{k})|\le s(\sqrt{k})\}= \{-2\sqrt{k}s(\sqrt{k})+s^2(\sqrt{k})-a_k\le S_k+\eta\le2\sqrt{k}s(\sqrt{k})+s^2(\sqrt{k})-a_k\}\supset \{|S_k+\eta|\le \sqrt{k}s(\sqrt{k})\}.
K_{-1}:= \{|x - \frac{X}{t}| \geq \frac{\mathcal{Y}}{M\rho (0)^2 t}\}, &K_j := \{|x - \frac{X}{t}| \sim 2^{-j}\frac{\mathcal{Y}}{M\rho (0)^2 t}\}, 1\leq j \leq k , &K_{0}:= \{|x - \frac{X}{t}| \leq \frac{Y + M\rho (0)}{t}\},
\begin{aligned}\frac{\tilde{M}}{\alpha}\leq \frac{(\alpha+1-\beta')^{2}}{2\alpha}\cdot \Big\{\frac{1}{\alpha}\log d_L + \frac{G_1(\alpha; 0)}{\alpha}\cdot r_1 + \frac{G_2(\alpha; 0)}{\alpha}\cdot 2r_2 + \frac{2}{\alpha^2}+ \frac{2}{\alpha+\alpha^2}\Big\}\end{aligned}
\begin{cases}\rho(x) \varphi_{tt}- \operatorname{div}(K(x) \nabla w) + 2\alpha^2 zw\psi +\alpha^2z^2\varphi=0 ~\hbox{in}~ \Omega \times (0,T), \rho(x) \psi_{tt}- \operatorname{div}(K(x) \nabla \psi) + 2\alpha^2 wz\varphi+\alpha^2w^2\psi=0 ~\hbox{in}~ \Omega\times (0,T), \varphi=\psi=0 ~\hbox{in}~ \omega\times(0,T).\end{cases}
\left | \iint_{\begin{subarray}{l}\xi=\xi_{1}+\xi_{2}\tau=\tau_{1}+\tau_{2}\end{subarray}}| 5\delta \xi^{4}+3\gamma \xi^{2}+2 \xi_1 |^{\frac{1}{2}}\hat{f}(\xi, \tau) \hat{g}(\xi_1, \tau_1)\hat{h}(-\xi_{2}, -\tau_{2}) \right | \lesssim \| f\|_{Y^{0,\frac12,1}}\|g\|_{X^{0,\frac12,1}}\|h\|_{X^{0,0,1}}.
\Vert (B^0(\vartheta)+\lambda_0 \overline{Q_0})(B^0(\vartheta)-\widehat{\zeta}\overline{Q_0})^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}&\leqslant2(1+\lambda_0)\Vert f\Vert_{L_\infty}\Vert f^{-1}\Vert_{L_\infty}c(\phi).
HS_{R_{k}}(t)=\frac{(1-t^{\delta_{1}})\cdot\cdot\cdot(1-t^{\delta_{k}})}{(1-t)^{n}}=\frac{(1+t+\cdot\cdot\cdot+t^{\delta_{1}})\cdot\cdot\cdot(1+t+\cdot\cdot\cdot+t^{\delta_{k}})}{(1-t)^{n-k}}.
B_{p,\nu}(x,y)=\sqrt{\frac{2p}{\pi}}\displaystyle{\int_{0}^{1}t^{x-\frac{3}{2}}(1-t)^{y-\frac{3}{2}}}K_{\nu+\frac{1}{2}}\left(\frac{p}{t(1-t)}\right)dt,
\mathrm{Tr}(\Omega(c)B^zA^z) = \mathrm{Tr}(\Omega(c)(1+D^2)^{-\frac{z}{2}}a^{2z}) = \mathrm{Tr}(a^{2z}\Omega(c)(1+D^2)^{-\frac{z}{2}}) = \mathrm{Tr}(\Omega(c)(1+D^2)^{-\frac{z}{2}}).
i_{s,1}= \ \min\{j\in \{1\ldots,m\}\ | \s_j > N\}, i_{s,2}= \ \min\{j > i_{s,1}\ | \s_j \geq s_{i_{s,1}}\}, \vdots i_{s,l(s)}= \ \min\{j > i_{s,l(s)-1}\ | \s_j \geq s_{i_{s,l(s)-1}}\}.
I\left(\prod_{i=1}^n \Lambda_i'\right) = I\left(\prod_{d=1}^{d_a}\prod_{i=1}^{N_d}e_{1,1}^d\right) = I\left(e_{1,1}^{\sum_{d=1}^{d_a}dN_d}\right) = I(e_{1,1}^N) = N(N+3).
\xi_k(S', N) = \mathrm{d}\left(\!\!\sqrt[N]{\alpha_{k+1}}\right) \underset{E_{k+1}/ F}{\sqcup}\Psi(E_{k+1}, S', N)(\rho_k(S', N)) = \rho_k(S', N)_*(\xi_{k+1}(S', N)).
\tilde{\Gamma}^{abc,r}_{\mu \nu}=(G^{-1})^r_{\nu \alpha}(y)(G^{-1})^r(z)<\tilde{z}_2 \tilde{g}t^{aed}B^e_{\mu}(x)c^d(x) \bar{c}^c(z)B^b_{\nu}(y)>^r
\omega^1 =d \gamma^1, \omega^2 =d \gamma^2-X_1(\gamma^2)d\gamma^1, \omega^3 =d \gamma^3-X_2(\gamma^3)d\gamma^2-(X_1(\gamma^3) -X_2(\gamma^3)X_1(\gamma^2))d\gamma^1, \vdots \omega^{j}\equiv d \gamma^{j}~\mod~d\gamma^1,\ldots, d\gamma^{j-1},
\dfrac{1}{\Phi_1[\alpha_0,\beta_0, L_1(0), N_1(0)]}+\dfrac{u_c}{\Phi_2[\beta_0,\infty,L_2(u_c),N_2(u_c)]}=\dfrac{2l_m\gamma_m(\beta_0)^{\nu+1}}{\theta_b-\theta_m}
\int\int\cdot\cdot\cdot\int|\Psi_{n_{1}n_{2}\cdot\cdot\cdot n_{N}}^{(S/A)}(x_{1},x_{2},...,x_{N})|^{2}d^{3}x_{1}d^{3}x_{2}\cdot\cdot\cdot d^{3}x_{N}=1
\Phi_{0 , 2}^{( 0 , 3 )}= a_{1 2}^{*}V^{( 0 , 3 )}\hspace{1 5 m m}\Phi_{1 , 2}^{( 0 , 3 )}= a_{2 2}^{*}a_{1 2}^{*^{2}}V^{( 0 , 3 )}
\delta:=\begin{cases}1, \textrm{if}\;\; \bar{s}+ s\alpha \leq \frac{N}{2}+1 -\alpha, \frac{N/2 -\bar{s}-\alpha +1}{s\alpha}, \textrm{if}\;\; \bar{s}+ s\alpha > \frac{N}{2}+ 1-\alpha.\end{cases}
\begin{aligned}\|v^k-\widetilde{v}^k\|^2_{G^k}=\tau^k\beta^k\|\bar{x}_2^k-\bar{x}_2^{k+1}\|^2_{D}+(2-\gamma)\tau^k\beta^k\|A\bar{x}^{k+1}-b\|^2 &+2\tau^k\beta^k(\bar{x}_2^k-\bar{x}_2^{k+1})^TA_2^T(A\bar{x}^{k+1}-b)\geq(1-\gamma)\tau^k\beta^k\|A\widetilde{x}^k-b\|^2,\end{aligned}
a_R(u,v)+b(u,\psi)+b(v,\phi)+a_S(\phi,\psi)=\langle f_R,v\rangle+\langle f_S,\psi\rangle, \forall\,v\in V,\ \mbox{and}\ \psi\in S,\ \mbox{such\that}\Bv=\psi.
\left| \begin{array}{c c c c}{{f^{1}}}&{{\cdots}}&{{f^{N}}}&{{f}}\\{{f^{( 1 ) \, 1}}}&{{\cdots}}&{{f^{( 1 ) \, N}}}&{{f^{( 1 )}}}\\{{\vdots}}&{{\cdots}}&{{\vdots}}\\{{f^{( N ) \, 1}}}&{{\cdots}}&{{f^{( N ) \, N}}}&{{f^{( N )}}}\\ \end{array}\right| = 0 .
\left( -{\frac{3}{4}}\omega -{\frac{9}{3 2}}\right)^{2}>{\frac{9}{1 0 2 4}}\left( - 6 4 \omega{\frac{z_{0}^{2}}{l^{2}}}+ 9 - 6 4{\frac{z_{0}^{2}}{l^{2}}}\omega^{2}\right) \, ,
\Psi_\pi(v)=\frac{\Vert v_1\Vert^2}{\Vert v\Vert^2}\Psi_{\pi_1}(v_1)+\cdots+\frac{\Vert v_m\Vert^2}{\Vert v\Vert^2}\Psi_{\pi_m}(v_m)
\mathcal{F}_{h_{f}}(x)=\frac{2\left(\varrho \beta \right)^{\left(\varrho+\beta\right)/2}}{\Gamma(\varrho)\Gamma(\beta)}{x}^{\left(\varrho+\beta\right)/2-1}\mathcal{K}_{\varrho-\beta}\left(2\sqrt{\varrho\beta x}\right),
\int_{D}{h}^{\ast}(\eta -{g}^{\ast}\eta) \wedge (\eta -{h}^{\ast}\eta) = \int_{D}h^{\ast}df_{g}\wedge (\eta - g^{\ast}\eta) = \int_{D}d(h^{\ast}f_{g}(\eta - g^{\ast}\eta)) = \int_{\partial D}h^{\ast}f_{g}(\eta - g^{\ast}\eta) = 0
\mathfrak{n}_\mu (\mathcal{R}(z), \varepsilon) \leq \sum_{k=0}^2 \mathfrak{n}_\mu (\mathcal{R}_{2k}(z), \varepsilon/7) + \sum_{k=1}^2 2\mathfrak{n}_\mu (\mathcal{R}_{2k-1}(z), \varepsilon/7) .
_{3}F_{2}\left( \left.\begin{array}{c}\alpha ,\beta ,\beta \beta +1,\beta +1\end{array}\right\vert 1\right) =\beta^{2}\,\mathrm{B}\left( 1-\alpha ,b\right) \left[\psi \left( 1+\beta -\alpha \right) -\psi \left( \beta \right) \right] .
\epsilon_{l}( H_{G}^{2 l}( M ) ) = < \left( \begin{array}{c c c}{1}\\{\vdots}\\{1}\\ \end{array}\right) , \left( \begin{array}{c c c}{\mu_{1}}\\{\vdots}\\{\mu_{k}}\\ \end{array}\right) , \ldots , \left( \begin{array}{c c c}{\mu_{1}^{l}}\\{\vdots}\\{\mu_{k}^{l}}\\ \end{array}\right) >
\varphi(\tau)=\prod_{j=1}^{\infty}tanh\frac{(2j-1)\pi i}{2\tau}=\sqrt{2}e^{\pi i\tau/8}\prod_{j=1}^{\infty}\frac{1+e^{2j\pi i\tau}}{1+e^{(2j-1)\pi i\tau}},Im\tau>0
x_i=x_0 -\sum_{j=1}^{i-1}\frac{\tilde\gamma_{i,j}}{\tilde\gamma_{i,i}}(x_j-x_0)-\sum_{j=0}^{i-1}\frac{\tilde\beta_{i,j}}{\tilde\gamma_{i,i}}f'(x_j), i=1,\hdots,N,
\Delta\Sigma_\pm = \frac{g^2}6\int \frac{d^4k}{(2\pi)^4}\frac1{k^2+m^2 \pm\frac{\tilde c}{\tilde k^2}}(2+\cos(k\tilde p)) =\frac{g^2}6\int \frac{d^4k}{(2\pi)^4}\frac{\tilde k^2}{k^2\tilde k^2 +m^2\tilde k^2 \pm \tilde c}(2+e^{ik\tilde p}).
\begin{cases}c_0|p|^q \le H^*(p,y) \le C_0|p|^q, |D_y H^*(p,y)| \le C|p|^q, begin{align*}1mm]c_0|p|^{q-1}\le |D_p H^*(p,y)| \le C_0|p|^{q-1}, |D^2_{py}H^*(p,y)| \le C|p|^{q-1}, begin{align*}1mm]c_0|p|^{q-2}\; I_d \le D^2_p H^*(p,y) \le C_0|p|^{q-2}\; I_d, |D^2_{y}H^*(p,y)| \le C|p|^q.\end{cases}
q^{-{{n}\choose{2}}}C_n\left(q,t\right)=\sum_{\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k),\atop \lambda_1+k\leq n}\prod_{i \geq 1}\left[\lambda_i-\lambda_{i+2}+1 \atop \lambda_{i+1}-\lambda_{i+2}\right]_t q^{-{\lambda_1\choose 2}-|\lambda|}.
g(t) := 1 - \frac{b(q - 2)(6 - q)}{a(p - 2)(6 - p)}t +\frac{c(r - 2)(6 - r)}{a(p - 2)(6 - p)}t^{\frac{r - p}{q - p}}h(t) := 1 - \frac{b(q - 2)}{a(p - 2)}t + \frac{c(r - 2)}{a(p - 2)}t^{\frac{r - p}{q - p}}.
{\phi}= \left( \psi^{{}^{\parallel}}\; , \; P_{k}+ i \psi_{k}\right) \; , \; \; T_{0}= \epsilon_{j k l}\pi_{j}\psi^{k \perp}\psi^{l \perp}- \frac{i \alpha}{2}\zeta \omega \; , \; \; T_{k}= - \zeta \omega \epsilon_{k j l}\psi^{j \perp}\psi^{l \perp}+ \frac{i \alpha}{2}\pi_{k}\; .
3\ell_{a}\ell_{b}E_{0}= \ell_{b}(\ell_{a}-\ell_{ab})^2+\ell_{a}(\ell_{b}-\ell_{ab})^2 - 2\ell_{ab}(\ell_{a}-\ell_{ab})(\ell_{b}-\ell_{ab}) &= -2 \ell_{ab}^3 + 3(\ell_{a}+\ell_{b})\ell_{ab}^2 - 6\ell_{a}\ell_{b}\ell_{ab}+\ell_{a}\ell_{b}(\ell_{a}+\ell_{b}).
i_{A}(a)\xi = i_{A}(a)V_{x}^{*}i_{A}(i)V_{y}=(V_{x}i_{A}(a^{*}))^{*}i_{A}(i)V_{y}& =(i_{A}(\alpha_{x}(a^{*}))V_{x})^{*}i_{A}(i)V_{y}= V_{x}^{*}i_{A}(\alpha_{x}(a)i)V_{y};
{}^{\Psi}B_{p}^{(\alpha,\beta)}(x,1-y) =\sum_{n=0}^{\infty}\frac{(y)_{n}}{n!}\int_{0}^{1}t^{x+n-1}{}_{0}\Psi_{1}\left( \alpha,\beta;-\frac{p}{t(1-t)}\right) dt &=\sum_{n=0}^{\infty}\frac{(y)_{n}}{n!}{}^{\Psi}B_{p}^{(\alpha,\beta)}(x+n,1),
R_i^{(2)}(t)=\sum_{|\ell| \geq 1}P_{c}\int_{0}^{t}\frac{D^{\ell}\mathcal{F}_{i}(a)}{\ell !}\mathcal{(}\varepsilon \psi )^{\ell}d\tau =\mathcal{O}(\varepsilon^{1-}).
[(h_r\Theta)^{-1}\mathbf w^r] \bmod \mathbb Z^m = \left[\sum_{l=1}^{L}a_{l}(\mathbf f_l^r - \mathbf{\bar k}_l^r)\right]\bmod \mathbb Z^m = \left[\sum_{l=1}^{L}a_{l}\mathbf f_l^r\right]\bmod \mathbb Z^m.
\left( \begin{array}{c}k_{1}+k_{2}+n-1 n\end{array}\right) =\sum_{v_{1}=0}^{n}\left( \begin{array}{c}k_{1}+v_{1}-1 v_{1}\end{array}\right) \left( \begin{array}{c}k_{2}+n-v_{1}-1 n-v_{1}\end{array}\right) .
\frac{d^2R_{n\ell}(z)}{dz^2}+\frac{(1-z)}{z(1-z)}\frac{dR_{n\ell}(z)}{dz}+\left[\frac{z^2(2a^2[E_{n\ell}-\alpha])+z(2a^2[\beta-\alpha-2E_{n\ell}]-\ell(\ell+1))+2E_{n\ell}a^2}{z^2(1-z)^2}\right]R_{n\ell}(z)=0.
y(t, \Phi) = G(t)\phi_0 + \int^0_{-r}U_t(\theta)\phi_1(\theta)d\theta + \int^t_0 G(t-s)f(s)dB(s) &= G(t)\phi_0 + \int^0_{-r}G(t+\theta)(S\phi_1)(\theta)d\theta + \int^t_0 G(t-s)f(s)dB(s),\hskip 20pt t\in [0, T].
D(P^{\mu}|_{\mathcal{F}^{\mathcal{X}}_{n}\vee \mathcal{F}^{\mathcal{Y}}_{0,n}}\|P^{\nu}|_{\mathcal{F}^{\mathcal{X}}_{n}\vee \mathcal{F}^{\mathcal{Y}}_{0,n}}) &=D(P^{\mu}|_{\mathcal{F}^{\mathcal{X}}_{n,\infty}\vee \mathcal{F}^{\mathcal{Y}}_{0,\infty}}\|P^{\nu}|_{\mathcal{F}^{\mathcal{X}}_{n,\infty}\vee \mathcal{F}^{\mathcal{Y}}_{0,\infty}})
\exp(\phi(z, \bar{z}))=a \tau_2\eta^2\bar{\eta}^2 \tilde{\tau}_2\tilde{\eta}^2\bar{\tilde{\eta}}^2 \left|\prod_{i=1}^{12}(z-z_i)^{-\frac{1}{12}}\right|^2\left|\prod_{j=1}^{12}(z-y_j)^{-\frac{1}{12}}\right|^2 .
\begin{aligned}\tau_{\hat{A}}(C,u+j) = C^{n-u}\sum_{p=0}^n \lfloor C^p \rfloor C^{-p}\tau_A(C,j+n-p) & \leq C^{n-u}\sum_{p=0}^n \tau_A(C,j+n-p) & \leq C^{n-u}\sup_{k \in \mathbb{N}}\sum_{j=k}^{k+n}\tau_A(C,j).\end{aligned}
G_{M}(q)=\frac{\eta_{W}(q^{2})^{2}}{\eta_{W}(q)^{2}}[\frac{\vartheta_{01}(q^{5})}{\vartheta_{01}(q)}]^{1/2}[\frac{5\vartheta_{01}(q^{5})^{2}}{4\vartheta_{01}(q)^{2}}-\frac{1}{4}]^{-1/2}R(q)^{-1/2}
d y^0=0\;, &d y^J=-\tfrac{1}{2}c^J_{KL}(x) y^K\wedge y^L-c^J_{K0}(x)y^K\wedge y^0\;, &d y^n=-c^n_{J0}(x)y^J\wedge y^0-\tfrac{1}{2}c^n_{JK}(x)y^J\wedge y^K-c^n_{n0}(x)y^n\wedge y^0.
W_{1/2}= \int{\cal D}{\mbox{\boldmath $x$}}\exp\left[-\int^1_0 dt\{\left(M + ib({\mbox{\boldmath $x$}})\right)\sqrt{\dot{{\mbox{\boldmath $x$}}}^2}+i{\mbox{\boldmath $a$}}\cdot\dot{\mbox{\boldmath $x$}}+i{\mbox{\boldmath $C$}}\cdot\dot{\mbox{\boldmath $x$}}\}+ \frac{i}{2}\Xi[ \frac{\dot{{\mbox{\boldmath $x$}}}}{|\dot{{\mbox{\boldmath $x$}}}|}]\right]\;.
\rho_x\rho_a =(x+[1]_x\tau+\tau^2)\left(\sum_{k=0}^d \rho_{a,k}\tau^k\right) &=\sum_{k=0}^d\left(x\rho_{a,k}\tau^k+[1]_x\rho_{a,k}^2\tau^{k+1}+\rho_{a,k}^4\tau^{k+2}\right)
\dfrac{h(w;\lambda)}{\varGamma((r-p-q)w+1-a-b)}= C_3 \cdot \tilde{d}^w \cdot \dfrac{\prod_{i=0}^{p-1}\varGamma\left(w+\frac{i+a}{p}\right) \prod_{i=0}^{q-1}\varGamma\left(w+\frac{i+b}{q}\right)}{\prod_{i=1}^{r}\varGamma\left(w+v_i\right)}.
f(t)=\left|\Phi_t(x)-\Phi_t(y)\right| =\left|x-y+\int_0^t X(\Phi_s(x))-X(\Phi_s(y))\,ds\right| &\leq |x-y|+\int_0^t M\left|\Phi_s(x)-\Phi_s(y)\right|\,ds &= |x-y|+\int_0^t Mf(s)\,ds
\mathcal{T}^c = \left \{(m',n') \in \mathbb{Z}^2_{0,n, >}: |\eta| \le \frac{3}{2}~, ~\Big | (2\zeta-1) \eta-(2\zeta-1) g(\zeta) \Big | > k (n) ~,~ |\eta-\zeta| > k (n) \right \}\ ,
\left(\boldsymbol\tau,D \boldsymbol\varphi\right) -\left(\nabla^\star\cdot \boldsymbol\tau-\nabla\cdot \boldsymbol\tau,\boldsymbol\varphi\right)+{\cal R}e\,a_\star\left(\boldsymbol u, \boldsymbol u,\boldsymbol\varphi\right)=(\tfrac{G}{B},\varphi_3)\mbox{for all}\ \boldsymbol\varphi\in \boldsymbol V^p.
1 = -b_2b_1+b_2^2+a_2^2+b_1a_2+b_1^2+a_1^2+a_1a_2+a_1b_2 = \left(\frac{a_1 + a_2}{\sqrt{2}}\right)^2 + \left(\frac{a_2 + b_1}{\sqrt{2}}\right)^2 + \left(\frac{b_1 - b_2}{\sqrt{2}}\right)^2 + \left(\frac{b_2 + a_1}{\sqrt{2}}\right)^2 .
\left|K_{N}^{1}(z,\zeta)\right|\lesssim\frac{\rho(\zeta)^{N-1}\left(\left|\rho(\zeta)\right|+\varepsilon\right)\varepsilon^{n-1}}{\prod_{i=1}^{n-1}\tau_{i}(z,\varepsilon)\left|\frac{1}{K_{0}}S(z,\zeta)+\rho(\zeta)\right|^{N+n-1}}\frac{1}{\left|z-\zeta\right|}.
\frac{\partial}{\partial z}h(z,y,u)=\frac{u(y-zu)}{1-u^2}\, h(z,y,u), \frac{\partial^2}{\partial z^2}h(z,y,u)=\Bigl(\frac{u^2(y-zu)^2}{(1-u^2)^2}-\frac{u^2}{1-u^2}\Bigr)\, h(z,y,u),
|I_2| \leq \frac{\zeta(5/2)^4 p^4 k^{4-A(3/2+\beta)}}{2^9 \pi^9}\cdot \int_{-\infty}^\infty \frac{|1/2+it|^2|3/2+it|^2|1+it|^3|256/225+it|}{|12/5+it||2/5+it|\prod_{r=3}^{10}|r-5/2+it|}dt &\leq \frac{.18047}{10!}p^4k^{4-A(3/2+\beta)}\leq \frac{.18047}{10!}k^{8-A(3/2+\beta)}.
\tilde{Y}_{\mathbf{i}}=\alpha_{Y}(s_{\mathbf{i}})+Y_{\mathbf{i}},\tilde{X}_{\mathbf{i}}=\alpha_{X}(s_{\mathbf{i}})+X_{\mathbf{i}},\tilde{U}_{\mathbf{i}}=\alpha_{U}(s_{\mathbf{i}})+U_{\mathbf{i}},
p=\rho RT+(B_{0}RT-A_{0}-\frac{C_{0}}{T^{2}}+\frac{D_{0}}{T^{3}}-\frac{E_{0}}{T^{4}})\rho^{2}+(bRT-a-\frac{d}{T}+\frac{c}{T^{2}})\rho^{3}+\alpha(a+\frac{d}{T})\rho^{6}
\prod_{k=1}^r \prod_{l=1}^{N_k}e^{\mathbf{a}_k(l) \varpi_{i_{k,l}}}(b_{k,l})= \prod_{s=1}^n ~~~ \prod_{\substack{1 \leq k \leq r, 1 \leq l \leq N_k, i_{k,l}= s}}e^{\mathbf{a}_{k}(l) \varpi_s}(b_{k,l}).
\dot{r}= \eta - \frac{\sqrt{M_1 \pi}}{\lambda e^{\lambda r}}\: \:\: \: ;\: \: \: \:(e^{\lambda r}- \eta \sqrt{M_1 \pi}/\lambda ) =e^{\eta\lambda t}(e^{\lambda r(0)}- \eta \sqrt{M_1 \pi}/\lambda )
\lim_{r\to +\infty}\int_{-r}^r |\eta_2(x_1, t)|\, dx_1 = \lim_{r\to +\infty}\int_0^t \eta_1(-r, x_2) - \eta_1(r, x_2)\, dx_2 = \int_0^t \lim_{r\to +\infty}(\eta_1(-r, x_2) - \eta_1(r, x_2))\, dx_2 = 0\,.
AG \geq \frac{2}{n}\phi G^{2}-\bigg(2K+\frac{12\left\vert b\right\vert}{n}\bigg)\phi G-\frac{2C_{1}\phi^{\frac{1}{2}}G^{\frac{3}{2}}}{R}-\bigg( \frac{(2-\lambda)ap}{\left\vert J \right\vert^{p-1}}+\frac{\lambda ap(p-1)}{\left\vert J \right\vert^{p-2}}& +\frac{2(\lambda-1)a}{n\left\vert J\right\vert^{p}}\bigg)\phi G.
c \psi_n(a_n) \psi(b_n) + c \sum_{i=1}^{n-1}\left(\psi_n(a_i)\psi(b_i) - \frac{1}{2}\left( \psi_{n-1}(a^0_i) \psi(b^0_i) +\psi_{n-1}(a^1_i)\psi(b^1_i) \right) \right) \leq \frac14 a_n b_n.
A_t^{W,Y}(l) = \left( \int_0^t l^*(r, W_r) \mathrm d[W,Y]_r \right)^* = \left(\int_0^t l^*(r, W_r)\nabla \gamma(r, W_r) \mathrm dr \right)^* = \int_0^t \nabla \gamma^*(r, W_r) l(r, W_r) \mathrm dr = A_t^{W,W}(\nabla \gamma^*\, l).
\left[ \widehat{{\mathcal J}}_{\ell / r}^{0}\left( t , \underline{{{x}}}\right) , ~ \widehat{{\mathcal J}}_{\ell / r}^{0}\left( t , \underline{{{y}}}\right) \right] \; = \; \pm ~ i ~ \frac{q^{2}}{4 \pi^{2}}~ \left( \underline{{{B}}}\left( \underline{{{x}}}, t \right) \, \cdot \, \underline{{{\nabla}}}\right) \; \delta \left( \underline{{{x}}}- \underline{{{y}}}\right) ~ ,
S_{(n)}=\int d^4x\Bigl[\frac{1}{6}\,H_{(n)}^{\mu\nu\rho}H_{(n)\mu\nu\rho}^\ast-\frac{2n^2\pi^2}{R^2}\Bigl(iB_{(n)}^{\mu\nu}-\frac{R}{2n\pi}\,F_{(n)}^{\mu\nu}\Bigr)\Bigl(-\,iB_{(n)\mu\nu}^\ast-\frac{R}{2n\pi}\,F_{(n)\mu\nu}^\ast\Bigr)\Bigr]
\epsilon^{\alpha\mu\nu\beta}\gamma_5\gamma_\mu \partial_\nu \Psi^i_\beta-{\kappa \epsilon^{ij}\over{\sqrt 2}}\big[F^{\alpha\beta}+{\scriptstyle{1\over 2}}\gamma_5\epsilon^{\alpha\beta\rho\sigma}F_{\rho\sigma}\big]\Psi^j_\beta = 0 \,.
cov(w_{i},z_{i})+E(w_{i}\Delta z_{i})=\sum_{i}q_{i}w_{i}z_{i}-wz+\sum_{i}q_{i}w_{i}z_{i}^{\prime}-\sum_{i}q_{i}w_{i}z_{i}=\sum_{i}q_{i}w_{i}z_{i}^{\prime}-wz
c_{v,+}(h B^+/B^+) = \sigma_{v,+}(h t^{-1}g^{-1}\dot{v}^{-1}) \dot{v}B^+/B^+ = g_1^{-1}\dot{v}B^+/B^+ = h_1 t_1g^{-1}B^+/B^+ = h_1 B^+/B^+.
\begin{array}{rl}s_{11}=e_1+e_2+e_3,s_{12}=e_4+e_5+e_6,s_{13}=e_7+e_8+e_9, s_{21}=e_1e_2+e_1e_3+e_2e_3,s_{22}=e_4e_5+e_4e_6+e_5e_6, s_{23}=e_7e_8+e_7e_9+e_8e_9,s_{31}=e_1e_2e_3,s_{32}=e_4e_5e_6, s_{33}= e_7e_8e_9,s=-(s_{11}+s_{12}+s_{13}-6)/3.\end{array}
p(x_{[1:N],[1:B-1]},y_{[2:N],[1:B-1]},\hat{y}_{[2:N],[1:B-1]}) \cr =\!\!\prod_{b\in [1:B-1]}\! \!\big( p_{X_1}(x_{1,b})\!\!\prod_{k\in [2:N]}p_{X_k}(x_{k,b})p_{\hat{Y}_k|X_k,Y_k}(\hat{y}_{k,b}|x_{k,b},y_{k,b}) p_{Y_{[2:N]}|X_{[1:N]}}(y_{[2:N],b}|x_{[1:N],b})\big).
\begin{array}{lcr}I = \displaystyle{\frac{1}{2}H^{ij}p_i p_j + V(x_1,x_2)+F_{ij}\theta_1^i \theta_2^j+G_{ij}\theta_1^i \theta_1^j+ J_{ij}\theta_2^i \theta_2^j+}\displaystyle{+ L^i_{jk}p_i\theta_1^j \theta_1^k+ M^i_{jk}p_i \theta_2^j \theta_2^k+N^i_{jk}p_i \theta_1^j \theta_2^k+ S_{ijkl}\theta_1^i \theta_2^k\theta_1^j \theta_2^l}\end{array}
\mathcal{L}H^{p^{\ast}}=\sum_{k}H_{,kk}^{p^{\ast}}-\langle X,\sum_{k}H_{,k}^{p^{\ast}}e_{k}\rangle=H^{p^{\ast}}-\sum_{i,j,q}h_{ij}^{p^{\ast}}h_{ij}^{q^{\ast}}H^{q^{\ast}}.
\frac{\mu_3^2}{t_3}= \frac{\lambda^2 v_2 (y - Z_3)^2}{1 + \lambda v_2}\frac{\mu_4^2}{t_4}= \frac{(1 - \lambda \rho)^2 (y - Z_3)^2 v_1}{1 + (1 - \lambda \rho) v_1}.
U_{n}( \theta , \phi ) = \left( \begin{array}{l l}{\operatorname{c o s}( \theta / 2 )}{- e^{- i n \phi}\operatorname{s i n}( \theta / 2 )}{\operatorname{s i n}( \theta / 2 ) e^{i n \phi}}{\operatorname{c o s}( \theta / 2 )}\end{array}\right)
A_n = \sum_{k=0}^{\infty}\frac{\left(\frac12\right)_k^2}{(1)_k^2}d_n^{2k}\sum_{k=0}^{\infty}\frac{\left(\frac12\right)_k^2}{(1)_k^2}(a_n+b_nk) d_n^{2k}.
\theta_{i,j}= \frac{\frac{m_{i,j}}{j-i}\overline{\partial}(h_i^{-1}\partial^{j-i}h_i)}{(\frac{h_j}{h_i}+|\frac{m_{i,j}}{j-i}|^2(\frac{h_i\partial^{j-i}\overline{\partial}^{j-i}h_i-\partial^{j-i}h_i\overline{\partial}^{j-i}h_i}{h^2_i}) )^{\frac{1}{2}}}.
\left\| \frac{d}{d t}\; \eta_\rho \circ s_\gamma(t) \right\| = \left\| \frac{d}{dt}\; g \circ \eta_\rho(t) \right\| c_\rho \cdot |s_\gamma'(t)| = \left\| D_{g}(\eta_\rho(t)) \cdot \eta_\rho'(t) \right\|
E[X]=\sum_{i}x_{i}p_{i}=1(\frac{c}{2})+2(\frac{c}{8})+3(\frac{c}{24})+\cdot\cdot\cdot=\frac{c}{2}+\frac{c}{4}+\frac{c}{8}+\cdot\cdot\cdot=c=\frac{1}{ln2}.
\widetilde{\mathbf{S}}_{n}(z) =\mathbf{C}_{n}^{\prime}(z-1/4) \widetilde{\mathbf{C}}_{n}(z) =\mathbf{S}_{n}^{\prime}(z+1/4), \widetilde{\mathbf{S}}_{n}^{\prime}(z) =\mathbf{C}_{n}(z-1/4), \widetilde{\mathbf{C}}_{n}^{\prime}(z) =\mathbf{S}_{n}(z+1/4).
\int \frac{u-v}{h}\cdot \frac1h \left[ G_{h}\ast, \phi \right]\Big(\frac{u+v}{2}\Big) dx = \int G_{h/2}\ast\Big(\frac{u-v}{h}\Big) \cdot \frac1h \left[ G_{h/2}\ast, \phi \right]\Big(\frac{u+v}{2}\Big) - \frac1h \left[ G_{h/2}\ast, \phi \right]\Big(\frac{u-v}{h}\Big) \cdot G_{h/2}\ast \Big(\frac{u+v}{2}\Big) \,dx.
1={Q(\infty,s) \over 2 \pi N}\equiv{2 \over (2 \pi)^2 N}\left[\left(1-{(0.125)^2 \over (0.125)^2+s^2}\right){\pi \over 2s}-{0.125\over (0.125)^2 +s^2}\ln \left({s \over 0.125}\right) \right].
H = \pi_{\perp}\left( \sum_{i=1}^{n}\nabla_{e_{i}}e_{i}\right) = \pi_{\phi}\left(\sum_{i=1}^{n}\nabla_{e_{i}}e_{i}\right) + \pi_{\xi}\left(\sum_{i=1}^{n}\nabla_{e_{i}}e_{i}\right), \pi_{\xi}\left(\nabla_{e_{i}}e_{i}\right) =g(\nabla_{e_{i}}e_{i}, \xi) \xi =e_{i}(g(e_{i}, \xi)) \xi + g(e_{i}, \phi e_{i}) \xi = 0,
\det ( L ( \lambda )_{C_{n}}- v \cdot I d ) = \sum_{j = 0}^{2 n}\frac{( \sigma ( \lambda ) )^{^{( j - 1 )}}\sigma ( \lambda + j \gamma )}{( \sigma ( \gamma + \lambda ) )^{j}}( - v )^{2 n - j}( H_{j})_{C_{n}}= 0 ,
y_k:=\frac{\sum_{j=1}^{n-1}w_{j,n-k}b_{j,n-k}}{\sum_{j=1}^{n-1}b_{j,n-k}^2}\,, z_k:=\frac{\sum_{j=1}^{n-1}w_{j,n-k}b_{j,n-k+1}}{\sum_{j=1}^{n-1}b_{j,n-k+1}^2}\,,
\sum_{i=1}^{2n-2}I_\gamma(W_i,W_i) =\sum_{i=1}^{2n-2}\int_0^d[g_T(D_T^TW_i,D_T^TW_i)-g_T(W_i,W_i)\mathbf{K}(T,W_i)]dt &= \sum_{i=1}^{2n-2}\int_0^d[(s'_\lambda(t))^2-(s_\lambda(t))^2\mathbf{K}(T,E_i)]dt &= \int_0^d[(2n-2)(s'_\lambda(t))^2-\mathbf{Ric}^\perp(T)(s_\lambda(t))^2]dt &\leq (2n-2)\int_0^d[(s'_\lambda(t))^2-\lambda(s_\lambda(t))^2]dt=0.
\rho_t(x, y)=\frac{t^{-d/\alpha}\tilde p\big(1, t^{-1/\alpha}x , t^{-1/\alpha}y \big)}{t^{-\delta/\alpha}h(t^{-1/\alpha}x)t^{-\delta/\alpha}h(t^{-1/\alpha}y)}=t^{\frac{2\delta-d}{\alpha}}\rho_1\big(t^{-1/\alpha}x , t^{-1/\alpha}y \big),
|z|\leq t_1\left\{\left(\dfrac{2|a_k|+2t_2|a_{k+1}|}{t_1^{n-k}|a_n|}-1\right)\cos\alpha+\sin\alpha\right\}+2\dfrac{\sin\alpha}{|a_n|}\sum\limits_{j=0}^{n-1}\dfrac{|a_j|}{t_1^{n-j-1}}.
\mathcal{R}^+ =\sum_{m \geq 1}\sum_{n \geq 1}\frac{A(m,n)}{\sqrt{m^2 n}}g\Big(\frac{m^2 n}{N}\Big)\sum_{c > 0}\frac{S(n,1;c)}{c}H_{m,n}^+ \Big( \frac{4\pi \sqrt{n}}{c}\Big);
\frac{-v_y}{v^2}-\frac{uv_x-vu_x}{v^2}=\frac{2\varepsilon v_x^2-\varepsilon vv_{xx}}{v^3}-\tilde{\Gamma}^2_{22}\frac{1}{v}-2\tilde{\Gamma}^2_{12}\frac{u}{v}-\tilde{\Gamma}^2_{11}\frac{u^2-v^2}{v},
| \psi_{+}({\bf a}) \rangle = \frac{1}{\sqrt{2}}\left( \begin{array}{c}{{\sin \theta e^{- i \phi}}}\\{{- \cos \theta}}\\{{0}}\\{{- 1}}\\ \end{array}\right) , | \psi_{+}({\bf a}) \rangle^{\prime}= \frac{1}{\sqrt{2}}\left( \begin{array}{c}{{- \cos \theta}}\\{{- \sin \theta e^{i \phi}}}\\{{1}}\\{{0}}\\ \end{array}\right) ,
Z^{(1)}= \binom{a_1-1,a_2-1,\ldots,a_{k-1}-1,d_{k'+1},\ldots,d_{m'}}{b_1-1,b_2-1,\ldots,b_{l-1}-1,c_{l'+1},\ldots,c_m'}, Z'^{(1)}=\binom{c_1,c_2,\ldots,c_{l'},b_{l+1},\ldots,b_{m}}{d_1,d_2,\ldots,d_{k'},a_{k+1},\ldots,a_{m+1}}.
p_{x}(-1)-1\leq q_{x}(-1)=\left(\frac{2}{1-x}\right)\left(\frac{P_a(-1)}{(1+x)P_a'(x)}\right)^2 \leq C(w)\frac{\lambda(x)}{(1+x)^2}\min\left\{\frac{1}{|a|^2},\frac{1}{n^2}\right\}.
\begin{gathered}I_{03}^d=\iota(u_{yyy}^{m,n})=\frac{u_{yyy}^{m,n}}{(u_{yy}^{m,n})^{3/2}},I_{1,2}^d=\iota(u_{xyy}^{m,n}) = \frac{u_{xyy}^{m,n}+ u_{m,n}\, u_{yyy}^{m,n}+ 2 u_{y}^{m,n}\, u_{yy}^{m,n}}{(u_{yy}^{m,n})^{3/2}}.\end{gathered}
\mu^{2}\equiv \frac{1}{2}( 1 + \sqrt{1 - 4 H^{2}L^{2}}) , \; \; \nu^{2}\equiv \frac{1}{2}( 1 - \sqrt{1 - 4 H^{2}L^{2}}) , \; \; k \equiv \frac{\sqrt{\mu^{2}- \nu^{2}}}{\mu}.
2 \sum_{k=1}^{n_{c}-\tilde{n}_{c}}\frac{\delta_{\ell}- \gamma_{k}}{1 - \omega^{2k-2\ell+1}}= (n_{c}-\tilde{n}_{c}) \delta_{\ell}- 2 \sum_{k=1}^{n_{c}-\tilde{n}_{c}}\frac{\gamma_{k}}{1 - \omega^{2k-2\ell+1}}.
\check{y}_{\mathrm{d},i}[n] =\sum_{k\in\mathcal{K}}\sum_{\ell\in\mathcal{L}_{m_i,k}}h_{i,k,\ell}x_{\mathrm{d},k}[n-d_{m_i,k,\ell}]+z_{\mathrm{d},i}[n]+e_{\mathrm{d},i}[n].
\frac{1}{2\pi i}\int_{-T_2}^{+T_2}dt\; \frac{1}{t-is}\exp[i\Delta E_{n,m}t]e^{\alpha s^2}&=\frac{1}{2\sqrt{\pi\alpha}}\int_{-\infty}^{+\infty}e^{-sw}\exp[-(w-\Delta E_{n,m})^2/(4\alpha)]+R_3.
\mathbf{x}_1=(x_1,x_1,\cdots, x_1, x_2,x_2,\cdots, x_2), &\mathbf{x}_2= (x_2,x_2,\cdots, x_2,x_1,x_1,\cdots, x_1 ), &\mathbf{y}_1=(y_1,y_1,\cdots, y_1,\, y_2,\, y_2,\cdots,\, y_2), &\mathbf{y}_2= (\underbrace{y_2,y_2,\cdots, y_2}_{n/2},\,\underbrace{y_1,\, y_1,\cdots,\, y_1}_{n/2}).
\overline{\zeta}(\theta,\theta^\prime) := - \log \langle v_{\theta,\theta^\prime}| w_{\theta,\theta^\prime}\rangle + \log \max_y v_{\theta,\theta^\prime}(y) + \theta \underline{\xi}(\theta^\prime), \underline{\zeta}(\theta,\theta^\prime) := - \log \langle v_{\theta,\theta^\prime}| w_{\theta,\theta^\prime}\rangle + \theta \overline{\xi}(\theta^\prime)
r_{x_1}=\frac{a_1+b_1-b_2}{2},\ r_{x_2}=\frac{b_1+b_2-a_1}{2},\ r_{x_3}=\frac{a_1+b_2-b_1}{2}, r_{y_1}=\frac{a_2+c_1-c_2}{2},\ r_{y_2}=\frac{c_1+c_2-a_2}{2},\ r_{y_3}=\frac{a_2+c_2-c_1}{2}.
g^{*}h_{\alpha \beta \gamma}^{B}\: = \: \left( h_{\alpha \beta \gamma}^{B}\right) \, \left( \nu_{\alpha \beta \gamma}^{F g}\right)^{- 1}\, \left( \nu_{\alpha \beta \gamma}^{R g}\right) \, \left( \kappa_{\alpha \beta}^{g}\right) \, \left( \kappa_{\beta \gamma}^{g}\right) \, \left( \kappa_{\gamma \alpha}^{g}\right)
a_n\cdot a_{n-2}= (3a_{n-1}-a_{n-2})a_{n-2}= 3a_{n-1}a_{n-2}-a_{n-2}^2 = 3a_{n-1}a_{n-2}-(a_{n-1}a_{n-3}-1) = a_{n-1}(3a_{n-2}-a_{n-3})+1 = a_{n-1}^2+1 \, .