harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{C}_j(n)=\frac{n^a}{2}\log\left(1+\frac{\frac{\gamma}{n^a}}{\sigma_{n,\lceil j/n^{\bar{a}}\rceil n^{\bar{a}}}^2 + \frac{(\lceil j/n^{\bar{a}}\rceil-1)\gamma}{n^a}}\right).
\alpha_{1}= \tau_{1}= s \sigma_{1}\hspace{. 4 c m}, \hspace{. 4 c m}\alpha_{2}= \tau_{2}- \tau_{1}= s ( \sigma_{2}- \sigma_{1}) \hspace{. 4 c m}, \hspace{. 4 c m}\alpha_{3}= s - \tau_{2}= s ( 1 - \sigma_{2})
\{\phi_r(x_{I}) \;|\; \ell(I)-r+1=0 \}, &\{\phi_r(x_{I})\;|\; \ell(I)-r+1>0 \mbox{and $\ell(I)-r+1$ is even}\}, &\{\phi_r(x_{I})\;|\; \ell(I)-r+1>0 \mbox{and $\ell(I)-r+1$ is odd}\},
\sum_{m=2}^{q+1}a_{k+m-1-q}\sum_{p=1}^{[(m-1)/2]}m_{(m-1-p,p,\dot{0})}= \sum_{t=0}^{q-1}a_{k-t}\sum_{s=1}^{[(q-t)/2]}m_{(q-t-s,s,\dot{0})}= \sum_{s=1}^{[q/2]}\sum_{t=0}^{q-2s}a_{k-t}m_{(q-t-s,s,\dot{0})}.
\sup_{c>1}\int_{\mathbb{R}^n}|u_c|^p dx =\sup_{c>1}\int_{\mathbb{R}^n}c \left( \sqrt{|\xi|^2 + m^2 c^2}- mc\right) |\hat{u}_c (\xi)|^2 + \mu |\hat{u}_{c}(\xi)|^2 d \xi &=\sup_{c>1}\left( \frac{1}{2}-\frac{1}{p}\right)^{-1}I_{e,c}(u_c)< \infty.
\begin{gathered}bs-b j_{\delta}\alpha+b-1 >bs - b j_{\delta}\cdot \frac{r-1}{2}+ b - 1 =b(s+1)-1 - b j_{\delta}\cdot \frac{r-1}{2}\geq s+1 -1 - b j_{\delta}\cdot \frac{r-1}{2}\geq s - (j_{\delta}+2) \cdot j_{\delta}\cdot \frac{r-1}{2}.\end{gathered}
\left(-\partial_r^2+r^{-2}\left(\lambda+\frac{n}{2}\left(\frac{n}{2}-1\right)\right)+z^2\right)^{-1}(r_1,r_2) =(v_1'v_2-v_1v_2')^{-1}(r_1,z)v_1(r_1,z)v_2(r_2,z),
T\partial^\alpha(\rho_\epsilon*\tilde\theta) - \rho_\epsilon*T\partial^\alpha\tilde\theta = \lambda\frac{\partial (\rho_\epsilon*\partial^\alpha\tilde\theta)}{\partial x}- \rho_\epsilon*\lambda\frac{\partial(\partial^\alpha\tilde\theta)}{\partial x}\rightarrow 0,\epsilon\rightarrow 0,
T \sum_{n = - \infty}^{\infty}f ( p_{3}) = \frac{1}{2}\int_{- \infty}^{\infty}\frac{d z}{2 \pi}[ f ( z ) + f ( - z ) ] + \int_{- \infty + i \epsilon}^{\infty + i \epsilon}\frac{d z}{2 \pi}\left( \frac{f ( z )}{e^{- i \beta z + \beta \mu}- 1}+ \frac{f ( - z )}{e^{- i \beta z - \beta \mu}- 1}\right) \; ,
e^{A ( r )}= ( r^{2}- r^{\ast 2}) \left( \frac{\sqrt{\bar{\Lambda}}}{2 r^{\ast}}\left( 1 + \frac{1}{4 r^{\ast 2}}( r^{2}- r^{\ast 2}) \right) + c ( r^{2}- r^{\ast 2})^{2}\right) ,
L_{1j}= (-1)^j c_{1(j-1)}^{-1}\prod_{k=1}^{j-1}p^{-m_{1k}}, (2\le j\le n), L_{1(n+1)}= c_{11}^{-1}c_{2n}^{-1}\prod_{k=1}^n p^{-m_{kn}},
\mathbf{r}_{\parallel}^{\prime}=\frac{\mathbf{r}_{\parallel}-\mathbf{v}t}{\sqrt{1-v^{2}/c^{2}}},\qquad\mathbf{r}_{\perp}^{\prime}=\mathbf{r}_{\perp},t^{\prime}=\frac{t-\left( \mathbf{v}\cdot\mathbf{r}\right) /c^{2}}{\sqrt{1-v^{2}/c^{2}}},
\begin{aligned}\int_{\Omega}|\nabla c|^{2}\le \int_{\Omega}|\nabla c|^{2}+\int_{\Omega}nc^{2}+\frac{1}{2}\int_{\partial\Omega}c^{2}&=\int_{\partial\Omega}\gamma c-\frac{1}{2}\int_{\partial\Omega}c^{2}&\le \frac{1}{2}\gamma^{2}|\partial \Omega|.\end{aligned}
f_{r}(n,k,s)\le \begin{cases}2^{(5/2)ks\log n+4s^2\log D}\r=3, k, 2^{(c_2+2r)s^{r-2}\log n+s^{r-1}(\log(c_2+r)+(r-2)\log s)}\r>3. \end{cases}
(v_1,\dots, v_i, v_{i+1},\dots, v_n) \sim (v_1,\dots, v_i, v_{i+2}, \dots, v_n) v_i = v_{i+1}(v_1,\dots, v_i, v_{i+1},\dots, v_n) \sim (v_1,\dots, v_{i+1}, v_i,\dots, v_n) (i,i+1) \in E.
\begin{array}{ll}\int_D f \, \eta \, dx = \int_D \Delta^2 u \, \eta \, dx = \int_{\partial D}\nabla \Delta u \cdot n \, \eta \, dS_x - \int_{\partial D}\Delta u \, \nabla \eta \cdot n \, dS_x [5pt] + \int_D \Delta u \, \Delta \eta \, dx [5pt] = \int_{\partial D}h \, \eta \, dS_x - \int_{\partial D}g \, \nabla \eta \cdot n \, dS_x \end{array}
PD[F_z] \smile i_1^{*}[\Omega] = \langle [F_z], i_1^{*}[\Omega] \rangle &= \langle [F_z], p'^{*}[\omega|_{Z}] \rangle &= \langle [p'(F_z)], [\omega|_{Z}] \rangle &= PD[p'(F_z)] \smile [\omega|_{Z}] &= \langle p'(F_z), [\omega|_{Z}] \rangle
\int_{\mathbb R^{n}}\ldots \int_{\mathbb R^{n}}\prod_{k=1}^{M}F^{(k)}( x_{1}, \ldots , x_{\ell}) \prod_{i=1}^{N}f_{i}^{\ast}\left( \sum_{j=1}^{\ell}a_{ij}x_{j}\right) d \mu(x_{\ell}) \ldots d\mu(x_{1}) .
\mathcal L_n^{-1}(z_1) \mathcal L_n(z_2) = \mathbb{I}+ \mathcal L_n^{-1}(z_1) (\mathcal L_n(z_2)-\mathcal L_n(z_1)) = \mathbb{I}+ \mathcal{O}\left(n^\frac{3}{2}(z_1-z_2)\right).
R_{n l}( r ) = - \left\{\left( \frac{2}{n a_{0}}\right)^{3}\frac{( n - l - 1 ) !}{2 n [ ( n + l ) ! ]^{3}}\right\}^{1 / 2}e^{- \rho / 2}\rho^{l}L_{n + l}^{2 l + 1}( \rho )
\prod_{i=1}^r (a_i + b(1-a_i)) \ge \biggl((1-b^{r-1}) \prod_{i=1}^{r-1}a_i + b^{r-1}\biggr)((1-b)a_r + b) &= (1-b^{r-1})(1-b) \prod_{i=1}^{r}a_i + b^{r-1}(1-b)a_r + (1-b^{r-1})b \prod_{i=1}^{r-1}a_i+b^r &\ge ((1-b^{r-1})(1-b) + b^{r-1}(1-b) + (1-b^{r-1})b)\prod_{i=1}^r a_i + b^r &= (1-b^r)\prod_{i=1}^r a_i + b^r\,.
S[X,g]\ =\ \int^1_0 d^2\sigma \frac{1}{2\pi}\left\{\sqrt{g}g^{\alpha \beta}\partial_\alpha X^I \partial_\beta X^I\ -\iB^{IJ}\varepsilon^{\alpha \beta}\partial_\alpha X^I \partial_\beta X^J\ \right\}\ ,
C^{0}\simeq \bigoplus_{\alpha\in Q_{1},\,s(\alpha)=j}\mathbb{S}_{2}^{-\deg(\alpha)}(M^{t(\alpha)}), C^{1}\simeq \bigoplus_{\alpha\in Q_{1},\,t(\alpha)=j}\mathbb{S}_{2}^{\deg(\alpha)-1}(M^{s(\alpha)}).
\tilde{I}^{\mathrm{L R}}( p_{1}^{\prime}, p_{2}^{\prime}; p_{1}, p_{2}) ={\frac{1}{( 2 \pi )^{3}}}\int d^{3}{\bf r}e^{i ({\bf k}^{\prime}-{\bf k}) \cdot{\bf r}}J ({\bf r},{\frac{{p}_{1}^{'}+ p_{1}}{2}},{\frac{p_{2}^{\prime}+ p_{2}}{2}})
0 \geq \|x-f(w)\|^2-\|x-w\|^2 = 2\langle x, w-f(w)\rangle +\|f(w)\|^2-\|w\|^2 > 2\langle w, w-f(w)\rangle +\|f(w)\|^2-\|w\|^2 = \|w-f(w)\|^2,
\lambda ( p^{\mu}) \, \equiv \left( \begin{array}{c}{{\left( \zeta_{\lambda}\, \Theta_{[ j ]}\right) \, \phi_{_L}^{\ast}( p^{\mu})}}\\{{\phi_{_L}( p^{\mu})}}\\ \end{array}\right) \, \, , \quad \rho ( p^{\mu}) \, \equiv \left( \begin{array}{c}{{\phi_{_R}( p^{\mu})}}\\{{\left( \zeta_{\rho}\, \Theta_{[ j ]}\right)^{\ast}\, \phi_{_R}^{\ast}( p^{\mu})}}\\ \end{array}\right) \, \, \quad ,
\left( \begin{array}{c c}{l_{1 1}}{l_{1 2}}{l_{2 1}}{l_{2 2}}\end{array}\right) = \left( \begin{array}{c c}{1}{0}{1}{0}\end{array}\right) , \hspace{3 m m}\left( \begin{array}{c c}{1}{1}{0}{0}\end{array}\right) \hspace{3 m m}\mathrm{e t c .}
[V_\lambda] = q^{-\frac{c}{24}}q^{\frac{(\lambda| \lambda + 2\rho)}{2(k+h^\vee)}-\lambda(x+y)}\ \prod_{i=1}^L X_i\ \prod_{1\leq i <j \leq L}X_{i,j}\ \Big\vert_{y=1}
\bigcup_{n=N}^{\infty}A_{n}\supseteq\Biggl\{\Delta X_{s}(y)\geq c_{(\ref{100})}\left((t-s)y\log\bigl((t-s)^{-1}\bigr)\right)^{1/(1+\beta)}&s\in(t-2^{-2N},t)y\in\Big(\frac{c}{2}\,(t-s)^{1/2},\,\frac{3c}{2}\,(t-s)^{1/2}\Big)\Bigg\}
\ell_{j}\left( s\right) \left( v,w\right) =\left\langle -\left[ v\right]_{\operatorname*{N};j}^{\operatorname*{ext}}\left( s\right) ,\gamma_{\operatorname*{D};j}\left( s\right) \overline{w}\right\rangle_{\Gamma_{j}},\qquad\forall w\in H^{1}\left( \mathbb{R}^{3}\right) .
\hat{\mathbb{E}}[\varphi(2B_{t_1\wedge\tau_k}-B_{t_1},\cdots,2B_{t_n\wedge\tau_k}-B_{t_n}-(2B_{t_{n-1}\wedge\tau_k}-B_{t_{n-1}}))] =\hat{\mathbb{E}}[\varphi(B_{t_1},\cdots,B_{t_n}-B_{t_{n-1}})].
\sum_{j=1,2}\mathcal{R}_j^2(\chi_{8R}\nu)(x) =\sum_{j=1,2}\chi_R(x)\mathcal{R}_j^2(\chi_{|x|/8}\nu)(x)+\sum_{j=1,2}\chi_R(x)\mathcal{R}_j^2((\chi_{8R}-\chi_{|x|})\nu)(x) &+\sum_{j=1,2}\mathcal{R}_j^2((\chi_{|x|}-\chi_{|x|/2})\nu)(x)~~\mathcal{D}'(B_{R}),
\hspace{1 0 m m}\times \prod_{i = 1}^{k}\frac{\Gamma ( i \rho )}{( \rho )}\prod_{i = 0}^{k - 1}\frac{\Gamma ( 1 + a + i \rho ) \Gamma ( 1 + c + i \rho )}{\Gamma ( 2 + a + c + ( k - 2 + i ) \rho )}
G_1=\{1,x_1,x_1^2,\dots,x_1^{m-1}\}\approx F_1=\{0=t_{1,0}<t_{1,1}<\dots <t_{1,m-1}=1\}\subset I, &G_2=\{1,x_2,x_2^2,\dots,x_2^{n-1}\}\approx F_2=\{0=t_{2,0}<t_{2,1}<\dots <t_{2,n-1}=1\}\subset I,
\begin{array}{rcl}\delta_{\sigma}L_{m}{}^{i}= -\sigma L_{n}{}^{i}\mathsf{m}^{n}{}_{m}\, , & \delta_{\sigma}A^{m}{}_{\mu}= \sigma \mathsf{m}^{m}{}_{n}A^{n}{}_{\mu}\, , & \delta_{\sigma}B_{\mu\nu\, m}= -\sigma B_{m\, \mu\nu}\mathsf{m}^{n}{}_{m}+2\eta_{mn}\partial_{[\mu}\sigma A^{n}{}_{\nu]}\, , \end{array}
R_{\theta,0;0}(u) =\tfrac12\Gamma(1+\theta)(\log u)^{1+\theta}+O(1), R_{\theta,1;0}(u) =\tfrac12(\log 2)\Gamma(1+\theta)(\log u)^{1+\theta}+O(1), S_0(u) =\tfrac12\log u-\tfrac12\log\log u+O(1).
\sum_{i+j=n}\big( m_{2,i}(x, m_{1 ,j}(y,z)) + m_{1,i}(x, m_{2 ,j}(y,z))\big) = \sum_{i+j=n}\big( m_{2,i}(m_{1,j}(x, y),z) + m_{1,i}(m_{2,j}(x, y),z)- m_{2,i}(m_{1,j}(x, z),y)- m_{1,i}(m_{2,j}(x, z),y)\big).
g_{n-1}(y)=\sum_{k=1}^{n-1}\frac{y^k}{m^{n-k-1}}\prod_{h=0}^{k-1}\left(1-\frac{h}{m}\right)a_{n-1,k}&g'_{n-1}(y)=\sum_{k=1}^{n-1}\frac{y^{k-1}}{m^{n-k-1}}k\prod_{h=0}^{k-1}\left(1-\frac{h}{m}\right)a_{n-1,k}
K\tilde{D}^{-1}K^T =\sum_{i=1}^n d_i^{-1}v_i^{\Re}v_i^{\Re\,T}+\sum_{i=1}^n d_i^{-1}v_i^{\Im}v_i^{\Im\,T}=\sum_{i=1}^n d_i^{-1}\qquad~~~ \sum_{\mathclap{j\in \{k|d_k=d_i,k=1,\ldots,n\}}}(v_j^{\Re}v_j^{\Re\,T}+v_j^{\Im}v_j^{\Im\,T}).
|\alpha_\pm|=|B(h, \mathcal{Y}_\mp) | \lesssim \Big| \frac{1}{2}\int (L_+ h_1) \mathcal{Y}_1 dx \Big| + \Big| \frac{1}{2}\int (L_- h_2) \mathcal{Y}_2 dx \Big| \lesssim \| h(t) \|_{L_x^2}\lesssim e^{-c_0 t}.
h (3)=\frac{\pi}{9\sin(\frac{\pi}{3})^2}\left[2+\frac{\pi^2}{2}\right] \left[\left(1-\frac{1}{2^{\frac32}}\right)\zeta\left(\frac32\right)-1\right] &>\frac{\pi}{9\sin(\frac{\pi}{3})^2}\left[2+\frac{\pi^2}{2}\right] \left[\left(1-\frac{1}{2^{\frac32}}\right)\left( \frac{4+\sqrt{2}}{4}+\sqrt{3}\left(\frac{\pi^2}{6}-\frac54\right) \right)-1\right] >1.
=\frac{1}{(x^{4}+1)^{5/4}}[1-(\frac{x}{\sqrt[4]{x^{4}+1}})^{4}]^{-3/4}=\frac{1}{(x^{4}+1)^{5/4}}(\frac{1}{x^{4}+1})^{-3/4}=\frac{1}{\sqrt{x^{4}+1}}
\delta\left(\sigma^{*}_{2}(t)I\right)=\int^{t_{2}}_{t_{1}}\left[-\dot{P}_{1}-Q^{1}\right]\delta Q^{1}dt+\left[\dot{Q}^{1}-P_{1}\right]\delta P_{1}dt+\left[P_{1}\delta Q^{1}\right]^{t_{2}}_{t_{1}}.
H_{}^{(l)}= \nabla^2 F(x^{(l)}) + \sum_{i = 1}^m \lambda_i^{(l)}\nabla^2 g_i(x^{(l)})+ \sum_{i=1}^m \frac{\lambda_i^{(l)}}{s_i^{(l)}}\nabla g_i(x^{(l)})\nabla g_i(x^{(l)})^T,
\begin{cases}(e^{-s\varphi}w)_t + a(e^{-s\varphi}w)_{xx}+ \lambda \displaystyle\frac{e^{-s\varphi}w}{b}=h, (t,x) \in Q_T, w(t,0)=w(t,1)=0, t \in (0,T), w(T,x)= w(0, x)= 0, x \in(0,1).\end{cases}
\left(\alpha_{\l}- \sigma_{\l+1}\lambda_{\l}\left(1+\frac{1}{\l+d-2}\right)^2\right) \gamma_{\l+1}^2 = (2(1-\delta) + O(\l^{-1})) \mu^{\l+1}, \left(\beta_{\l}+ \sigma_{\l-1}\lambda_{\l}\left(1-\frac{1}{\l}\right)^2\right) \gamma_{\l-1}^2 = (2(1+\delta) + O(\l^{-1})) \mu^{\l-1}
{\cal V}^{H}( \mathrm{\bf r}_{1}, \mathrm{\bf r}_{2}) ={\frac{1}{2}}\left\{{\cal V}( \mathrm{\bf r}_{1}, \mathrm{\bf r}_{2}) +{\cal V}^{\dag}( \mathrm{\bf r}_{1}, \mathrm{\bf r}_{2}) \right\}.
p_\alpha^i(\zeta) = p_\alpha^i([\zeta]) , p_\alpha^I(\zeta) = p_\alpha^I([\zeta]) p(\zeta) = L(\pi^{3}_{2}(j^3_x\phi)) - p_\alpha^i([\omega])u_{i}^\alpha(j^3_x\phi) - p_\alpha^I([\omega])u_I^\alpha(j^3_x\phi) \, .
\begin{array}{c c c c}{\mathrm{l o n g r o o t s :}}&{\pm 2 \! e_{k}}&{}&{( 1 \leq k \leq 3 )}\\{\mathrm{s h o r t r o o t s :}}&{\pm e_{k}\pm e_{l}}&{}&{( 1 \leq k < l \leq 3 )}\\ \end{array}.
\times \exp \Biggl [ - \bar{\alpha}\int d^{2}\xi \sqrt{h}h^{a b}( \partial_{a}w_{\mu}) ( \partial_{b}w_{\mu}) - \frac{1}{\alpha_{0}}\int d^{2}\xi \sqrt{h}h^{a b}( \partial_{a}t_{\mu \nu}) ( \partial_{b}t_{\mu \nu}) \Biggr ] ,
W_1 = \int_{\mathcal{X}\times \mathcal{X}}f(x)f(y) \int_{([l_x,v_x]\times [l_y,v_y])^c}\! \! h(u,v) \, d(F_{n,x,y}-F_{n,x}F_{n,y})(u,v) \,dx \, dy &= o(n^{-(9/2-\epsilon)}).
\{\{F , G \}, H \}= \sum \int_{\Omega}D_{I + J}\Biggl ( E_{A}^{I}\biggl ( D_{K + L}\bigl ( E_{C}^{K}( f ) I_{C D}E_{D}^{L}( g ) \bigr ) \biggr ) I_{A B}E_{B}^{J}( h ) \Biggr )
g_\infty = e^{2(\widehat{\eta}- \widehat{U})}(- \: d\widehat{R}^2 + \widehat{a}^{-2}d\widehat{\theta}^2) + e^{2\widehat{U}}(d\widehat{x}+ \widehat{G}d\widehat{\theta})^2 + e^{-2\widehat{U}}\widehat{R}^2 (d\widehat{y}+ \widehat{H}d\widehat{\theta})^2,
\|f\|_{L^2(\Sigma_r;\delta)}= \Big(\int_{\Sigma_r}|f(x)|^2\delta(x)dx\Big)^{1/2}, \|f\|_{L^2(\Sigma_r;\delta^{-1})}= \Big(\int_{\Sigma_r}|f(x)|^2\delta^{-1}(x)dx\Big)^{1/2}.
\begin{array}{l}\overline{Scal}=e^{-2u}\cos{2v}\Scal-e^{-2u}\sin{2v}\Scal^*+\left(e^{-2w}-e^{-2u}\cos{2v}\right)Ric(\xi,\xi), begin{align*}4pt]\overline{Scal}^*=e^{-2u}\sin{2v}\Scal+e^{-2u}\cos{2v}\Scal^*-e^{-2u}\sin{2v}\Ric(\xi,\xi).\end{array}
d=4 \begin{cases}u \approx (n^{\prime})^2\big(\frac{1}{2}p_1 + (k-1)p_2 + \frac{k-1}{2}p_3 + \binom{k-1}{2}p_4\big) v \approx (n^{\prime})^2\big( p_2 + p_3 + \frac{5(k-2)}{2}p_4 + \binom{k-2}{2}p_5\big) \end{cases}
\frac{d^2\bar x^i}{d\tau^2}= \left({\displaystyle\sum_{j=1}^n}\frac{1}{f}\frac{\partial f}{\partial \bar x^{j}}\frac{d\bar x^j}{d\tau}\right) \frac{d\bar x^i}{d\tau}+f^{2}X^{i}\left(\bar x,\frac 1f \frac{d\bar x}{d\tau}\right) .
p_D(t, x, y)\le c_1 \left(1\wedge \frac{\delta_D(x)}{1\wedge t^{1/\alpha}}\right)^{\alpha/2}\left(1\wedge \frac{\delta_D(y)}{1\wedge t^{1/\alpha}}\right)^{\alpha/2}\cdot\begin{cases}\Psi^1_{c_2^{-1},\gamma^{-1}, T}(t, |x-y|/6) \hbox{if}t\in(0,T], \Psi^2_{c_2^{-1}, T}(t, |x-y|) \hbox{if}t\in[T,\infty),\end{cases}
\rho^* = \int{d \rho \over f^2}={R^2 \over 2 (\rho_+^2 - \rho_-^2)}\left [ \rho_+ \ln \left ({{\rho -\rho_+}\over{\rho +\rho_-}}\right ) - \rho_- \ln \left ({{\rho -\rho_-}\over{\rho+ \rho_-}}\right ) \right ].
\eta^M(T) \le \left(\frac{13}{7 \cdot 3}+ \frac{8}{7}\right)\Sigma_0 + \left(\frac{13}{7 \cdot 3}+ \frac{1}{3}\right)\Sigma_* \le \frac{20\cdot 13}{21 \cdot 7}\Sigma_0 + \frac{20}{21}\Sigma_* = \frac{20}{21}\sum_i \eta^{M-1}(T_i).
2 E\left(\omega_{n}\right) =\int_{\Pi}\int_{\Pi}\omega_{n}(x) G_{\Pi}(x, y) \omega_{n}(y) \mathrm{d}x \mathrm{d}y &\geq \int_{\Pi \cap B_{L}(0)}\int_{\Pi \cap B_{L}(0)}\omega_{n}(x) G_{\Pi}(x, y) \omega_{n}(y) \mathrm{d}x \mathrm{d}y,
z^n_i(\theta) = n^{1/\alpha}\left( \frac{X_{\frac{i}{n}}- X_{\frac{i-1}{n}}- \frac{a}{n}+ \frac{b}{n}X_{\frac{i-1}{n}}}{\delta X_{\frac{i-1}{n}}^{1/ \alpha}}\right).
C''_1 = C''_{1,0}+ C''_{1,1}+ C''_{1,2}, C''_2 = C''_{2,0}+ C''_{2,1}+ C''_{2,2}, C''_3 = C''_{3,0}+ C''_{3,1}+ C''_{3,2}, C''_4 = C''_{4,0}+ C''_{4,1}+ C''_{4,2}.
\frac{z}{\xi e^z - 1}= \frac{z}{e^{k z}-1}\sum_{j=0}^{k-1}\xi^{j}e^{j z}= \frac{1}k \sum_{j=0}^{k-1}\xi^{j}\frac{k z \cdot e^{(j/k) \cdot kz}}{e^{k z}-1}= \frac{1}k \sum_{j=0}^{k-1}\xi^{j}\sum_{m=0}^\infty B_m(j/k)k^m \frac{z^m}{m!}
\left|{\mathfrak{u}^I_{k n}\over |\bar{u}_{k n}|}-{1 \over 1+\rho_*}\right| = \left|{1 \over |\bar{u}_{k n}|}\sum_{i=0}^{|\bar{u}_{k n}|-1}\chi_{I}(T_*^i(p)) -{1 \over 1+\rho_*}\int_{-1}^{\rho_*}\chi_{I}(x) d x \right| \le D_{{k n}},
V^{i j}( z , w ) = e^{u^{i}/ w + u^{j}/ z}\left[{\frac{\delta^{i j}}{z + w}}+ \sum_{k , l = 0}^{\infty}( - 1 )^{k + l}\, V_{k l}^{i j}\, w^{k}z^{l}\right] \, ,
G ( \vartheta ) = \frac{1}{2 \pi}\int_{- \infty}^{+ \infty}d k \, e^{i k \vartheta}\frac{\sinh \frac{\pi ( p - 1 ) k}{2}}{2 \sinh \frac{\pi p k}{2}\, \cosh \frac{\pi k}{2}}\, , \, p = \frac{\pi}{\gamma}- 1
\Delta n \sum_{l = 1}^{\infty}( 2 l + 1 ) A_{l}= \left. \Delta n \lim_{\varepsilon \to 0}\left( \overline{{{\cal D}}}_{r r}- 2 \overline{{{\cal D}}}_{r \rho}+ \overline{{{\cal D}}}_{\rho \rho}\right) \right|_{r = 1 +{\varepsilon}/{2}\atop \rho = 1 -{\varepsilon}/{2}}+ 1 \,{,}
M_\mu (x)=\frac 1{\sqrt{2}}\left( \varphi_\mu(x)-i\widetilde{\varphi}_\mu (x)\right) , \hspace{0.3in}M_{\mu \nu}(x)=\frac 1{\sqrt{2}}\left( \varphi_{\mu \nu}(x)-i\widetilde{\varphi}_{\mu \nu}(x)\right) ,
0\neq 4F\left(\frac{2}{3},-\frac{5}{3},\frac{4}{3};1-\lambda\right)(\lambda^2-\lambda+1)-\frac{1}{3}F\left(\frac{2}{3},-\frac{2}{3},\frac{4}{3};1-\lambda\right)\left(\lambda+1 \right)(8\lambda^2-11\lambda+8)+ &F\left(\frac{2}{3},\frac{1}{3},\frac{4}{3};1-\lambda\right)\lambda(1-\lambda)^2,
\|x_k-x_{k+1}\|\leq \frac{1+\theta}{1-\theta}\left[ \frac{1+\theta}{2}\frac{D^{-}f'(\lambda)}{|f'(\lambda)|}\|x_{k-1}-x_k\| +\theta\,\frac{2+f'(\lambda)}{|f'(\lambda)|}\right]\|x_{k-1}-x_k\|, k= 0,1, \ldots .
f_{D_4}=\frac{4\pi}{\sqrt{y^2+x_0^2}}\, K_0 \left( \frac{\sqrt{(y^2+x_0^2+x_1^2)(y^2+x_0^2+x_2^2)(y^2+x_0^2+x_3^2)}}{y^2+x_0^2}\right)e^{- i\, \frac{x_0 x_1 x_2 x_3}{y (y^2 + x_0^2)}}\, .
\int_{0}^{\infty}dn(t)e^{-tz}dt=\frac{1}{z+}\frac{1^{2}k^{2}}{z+}\frac{2^{2}}{z+}\frac{3^{2}k^{2}}{z+}\frac{4^{2}}{z+}\frac{5^{2}k^{2}}{z+}\cdot\cdot\cdot
{\cal O}_{M}= \frac{Z_{M}}{2 \Lambda^{2}}\left({g^{\prime}}^{2}( D_{\mu}H^{\dagger}H ) ( H^{\dagger}D^{\mu}H ) + g^{2}H^{\dagger}H ( D_{\mu}H^{\dagger}D^{\mu}H ) \right) .
\,\, \int_0^\infty \frac{1}{2 \pi i}\int_C e^{-\lambda}\mathbf 1_{j l}. b_0^2.(\nabla k)_l.b_0^2.(\nabla k)_j.b_0d\lambda (r^{m-1}dr) = \,\,k^{-(m/2+3)}H_{2,2,1}(\mathbf y_1, \mathbf y_2 ;m)(\nabla k \nabla k) \cdot g^{-1}.
\mathbf{u}_{k+1}= R\mathbf{u}_k +a_{k+1}\mathbf{v}_k+a_{k+1}^2\mathbf{u}_{k-1}\mathbf{v}_{k+1}= (2R+1)\mathbf{u}_k +a_{k+1}\mathbf{v}_k +a_{k+1}\mathbf{w}_k \mathbf{w}_{k+1}= -\mathbf{u}_k-a_{k+1}\mathbf{w}_k,
\frac{{e^{4}}}{( 2 \pi )^{3}}\int_{0}^{\infty}\frac{{q}^{2}}{{2 Q^{0}}}\mathrm{d}q N ({Q^{0}}) \int d \Omega \sum_{i j k l}{B}_{( i j k l )}^{\mu \nu \lambda \sigma}({k_{1}, k_{2}, k_{3}}; Q ) .
\begin{array}{l}\displaystyle [e_1,e_{n+1}]=e_1,[e_2,e_{n+1}]=2e_2,[e_{i},e_{n+1}]=\left(i-2\right)e_{i},(3\leq i\leq n),[e_{n+1},e_1]=-e_1, \displaystyle [e_{n+1},e_3]=-e_3,[e_{n+1},e_4]=e_2-2e_4,[e_{n+1},e_j]=\left(2-j\right)e_j,(5\leq j\leq n),\end{array}
\begin{cases}D^{1-\epsilon}_0\tilde{y}(t) = \tilde{z}(t), \tilde{y}(0) = y_0, D^{\alpha_2-1}_0\tilde{z}(t) = f(t,\tilde{y}(t),\tilde{z}(t)). \tilde{z}(0) = s.\end{cases}
\chi(h^{\beta}\xi)\tilde{\theta}_h(x)\left[\frac{-d^2\varphi(x)p''(\xi) - 1}{\xi -d\varphi(x)}\right] = \frac{\chi(h^{\beta}\xi)\tilde{\theta}_h(x)}{p''(d\varphi(x))}\int_0^1 p'''(t\xi + (1-t)d\varphi(x)) dt \in h^{-\sigma}S_{\delta,\beta}(1)\,.
\begin{aligned}(g^1_i) =( f_{Pos(1)},f^1,\ldots,f^{minS(2)},f^{minS(3)+1},\ldots,f^{minS(4)},f^{minS(7)+1}, \ldots, f^{minS(8)},\ldots), (g^3_i) =(f_{Pos(2)},f^{minS(2)+1},\ldots,f^{minS(3)},f^{minS(5)+1},\ldots,f^{minS(6)},f^{minS(11)+1},\ldots), (g^5_i) =(f_{Pos(3)},f^{minS(4)+1},\ldots,f^{minS(5)},f^{minS(9)+1},\ldots,f^{minS(10)},f^{minS(19)+1},\ldots),\end{aligned}
[x+p]_{p+1}=\sum_{k=0}^{p+1}(-1)^{p+1-k}\begin{bmatrix}p+1 k \end{bmatrix}(x+p)^{k}& = \sum_{k=0}^{p+1}(-1)^{p+1-k}\begin{bmatrix}p+1 k \end{bmatrix}\sum_{t=0}^{k}\binom{k}{t}p^{k-t}x^t & =\sum_{t=0}^{p+1}\sum_{k=t}^{p+1}(-1)^{p+1-k}p^{k-t}\binom{k}{t}\begin{bmatrix}p+1 k\end{bmatrix}x^t.
[{\cal D}\mu ] ={\cal D}B^{\mu}{\cal D}\lambda{\cal D}\xi \delta ( F_{0 1}+ \dot{\lambda}) \prod_{\beta = 0}^{3}\{\delta ( \Gamma_{\beta}[ B^{0}+ \xi , B^{i}, \lambda ] ) \}d e t \mid \{\tilde{\Omega}_{\alpha}, \Gamma_{\beta}\}\mid ,
-\theta(\partial_c\ln |\theta|+\widetilde{\Gamma}^d_{dc})\Theta^{ab}\widetilde{\Gamma}^c_{ab}= \left(\Theta^{cd}\partial_c \phi\partial_d\phi+V(\phi)\right)+2\int_{\mathbb{R}^n}\textbf{f}\;\frac{\theta\sqrt{|\Theta|}(\textbf{m}^2+ \Theta_{ab}\widetilde{p}^a\widetilde{p}^b)}{\widetilde{p}^1}\;d\widetilde{p}\;';
\psi_{\rho_{N}}\left( x,t,k\right) =\psi\left( x,t,k\right) -{\displaystyle\sum\limits_{n=1}^{N}}\rho\left( I_{n}\right) e^{8\kappa_{n}^{3}t}y_{n}\left( x,t\right)\int_{x}^{\infty}\psi\left( s,t,k\right) \psi\left( s,t,\mathrm{i}\kappa_{n}\right) \mathrm{d}s,
\begin{array}{l}\partial_ix^j=\delta^j_i+q\hat{\cal R}^{jk}_{il}x^l\partial_k~, \tilde{\partial}_kx^v=\delta^v_k+q^{-1}\left.\hat{\cal R}^{-1}\right.^{vi}_{kj}x^j\tilde{\partial}_i~.\end{array}
H(r,z_i)= 1\,+\, f_0(N_I)\, \frac{Q}{r^4}+ \frac{K^2}{4\, r^4}\sum_if_i(N_I)^2 \left[ \log \left( \frac{r^4}{\epsilon^2(r^2-\rho^2_i)}\right) -1 + \frac{\rho^2_i}{r^2-\rho^2_i}\right]
\rho(r)=\frac{2}{\pi^{3/2}(8+5\sqrt{\pi})}e^{-(1/2)r^{2}}((\frac{\pi}{2})^{1/2}(\frac{7}{4}+\frac{1}{4}r^{2}+(r+\frac{1}{r})erf(\frac{r}{\sqrt{2}}))+e^{-(1/2)r^{2}}).
\begin{array}{l}\mathbb{E}\left[\int \left(h_s - \mu(s, \theta_s) \right)\alpha_s \frac{\partial}{\partial x}\mu(s,\theta^H_s) d[B]_s \right] = \mathbb{E}\left[ \int \left(h_s - \mu(s,\theta^H_s) \right)\frac{\partial}{\partial x}\mu(s,\theta^H_s) dB_s \int \alpha_s dB_s\right] = 0\end{array}
\beta_{\tau}( W )=(1-\tau)\varphi_{1}\big((1-\tau)W\big),\gamma_{\tau}( W )=\varphi_{0}\big((1-\tau)W\big), \bar{\alpha}_{\tau \sigma}(W)-\alpha_{\sigma\tau}(W)=(\tau-\sigma)\varphi_{1}\big(-(\tau-\sigma) W\big),
P_t(\xi,\eta)= \begin{cases}t^{-1},\ \{\rm when}\t\geq d_0(\xi,\eta)^{q_0}, \dfrac{t}{d_0(\xi,\eta)^{2q_0}},\ \{\rm when}\ 0<t\leq d_0(\xi,\eta)^{q_0}, \end{cases}
E_{a_{N}'}^{\rm H}- E_{a_{N}}^{\rm H}= \big((a_{*}-a_{N}')^{\frac{q}{q+1}}- (a_{*}-a_{N})^{\frac{q}{q+1}}\big) \Big(\frac{q+1}{q}\cdot\frac{\Lambda}{a_{*}}+ o(1)_{N\to\infty}\Big) & \geq -(a_{N}'-a_{N})^{\frac{q}{q+1}}\Big(\frac{q+1}{q}\cdot\frac{\Lambda}{a_{*}}+ o(1)_{N\to\infty}\Big).
\int_{M_4}dvol_{\eta_4}(x')\,\langle x_k,z_k|\psi(x')\rangle =\,\frac{1}{\sqrt{N}}\, e^{\imath\,\vartheta_{\Psi k}(\varphi^{-1}_{k}(x),z_k)}\,n_{\Psi k}(\varphi^{-1}_{k}(x),z_k).
E\rho\bigl(Y_{\mathbf{i}}-\Psi(U_{\mathbf{i}},X_{\mathbf{i}})\bigr)=E\rho\bigl(Y_{\mathbf{i}}-X_{\mathbf{i}1}\beta_{1}(U_{\mathbf{i}})-\cdots-X_{\mathbf{i}d}\beta_{d}(U_{\mathbf{i}})\bigr),
\partial_\theta D(p||p_{\theta,\varphi}) = -\kappa( \cos(\theta)\cos(\varphi) \mathbb{E}_p[x_1] + \cos(\theta) \sin(\varphi) \mathbb{E}_p[x_2] - \sin(\theta)\mathbb{E}_p[x_3]), \partial_\varphi D(p||p_{\theta,\varphi}) = -\kappa( -\sin(\theta)\sin(\varphi) \mathbb{E}_p[x_1] + \sin(\theta)\cos(\varphi) \mathbb{E}_p[x_2]).
\mathcal{E}_{n}^{\left( \alpha+1\right)}\left( 1;\lambda\right)=2\lambda^{-1}\binom{\alpha}{1}^{-1}\mathcal{E}_{n+1}^{\left( \alpha\right)}\left( \lambda\right) =\genfrac{[}{]}{0pt}{}{1}{1}2\left( -\lambda\right)^{n-1}\binom{\alpha}{1}^{-1}\mathcal{E}_{n+1}^{\left( \alpha\right)}\left( \lambda\right) .
\zeta_{i}=-w_{i}^{[bern]}+(\partial_{i}\ln |\varpi^{-1}\Omega_{\lbrack0]}|)~\left( \ln |\Omega_{\lbrack 1]}|\right)^{\ast}+\left( \Omega_{\lbrack 1]}^{\ast}\right)^{-1}\partial_{i}\Omega_{\lbrack 1]},
L_{m+1}\left( \frac{k+j}{n+j}\right) = \prod_{l=0 \atop l \neq m}^{j-1}\frac{\frac{k+j}{n+j}- \frac{k+l}{n+j}}{\frac{k+m}{n+j}- \frac{k+l}{n+j}}& = \prod_{l=0 \atop l \neq m}^{j-1}\frac{j-l}{m-l}& = \frac{j!}{j-m}\prod_{l=0}^{m-1}\frac{1}{m-l}\prod_{l=m+1}^{j-1}\frac{1}{m-l}& = j! \frac{(-1)^{j-m+1}}{m!(j-m)!}
\hat{H}_0=\int_{(\infty)}\frac{d^3\vec{k}}{(2\pi)^32\omega(\vec{k}\,)}\frac{1}{2}\omega(\vec{k}\,)\left[a^\dagger(\vec{k}\,)a(\vec{k}\,)+a(\vec{k}\,)a^\dagger(\vec{k}\,)\right]\ ,
\begin{array}{ll}\pmb{V}_i^{(\kappa+1)}= c_0 \pmb{V}_i^{(\kappa)}+ c_1r_{1,i}^{(\kappa)}({\pmb{X}_{pbest,i}^{(\kappa)}}- \pmb{X}_i^{(\kappa)}) + c_2r_{2,i}^{(\kappa)}(\pmb{X}_{gbest}^{(\kappa)}- \pmb{X}_i^{(\kappa)}),\end{array}
n^{-1}g \left| \sum_{p=1}^n u^{p\bar p}u_{p\bar p \bar j}\right|^2- g \sum_{p,q=1}^nu^{p\bar p}u^{q\bar q}|u_{p\bar q j}|^2 + g \sum_{p=1}^nu^{p\bar p}u_{p \bar p j \bar j}=n(2 Re( g_{vj}v_{\bar j}) + g_v v_{j\bar j}+ g_{vv}|v_j|^2+g_{j\bar j}).
\begin{cases}\partial u_t(x)= -{\nu}(-\Delta)^{\frac{\alpha}{2}}u_t(x)-a^\beta(-\Delta)^{\frac{\beta}{2}}u_t(x)+\xi\sigma(u_t(x))\dot F(t,x), \ \ \x\in B_R(0), \ \t>0 u_t(x)= 0, \ \ \x\in{B_R(0)^c}. \end{cases}
\frac{A^{2 n + 2}}{V^{2 n + 1}}= \frac{2 \pi}{( n + 1 )}{\mathrm{V o l{( M )}}}\, 2^{2 n + 1}\, ( n + 1 )^{2 n + 1}\frac{1}{\left( 1 - \frac{a^{2 n + 2}}{r^{2 n + 2}}\right)^{n}}
\psi = \psi \left( \varepsilon_{0}^{a b}\right) = \left( 1 , 0 , \sqrt{{\frac{1}{2}}}\varepsilon_{0}^{a b}, 0 , \sqrt{{\frac{3 . 1}{4 . 2}}}\varepsilon_{0}^{\left( a b \right.}\varepsilon_{0}^{\left. c d \right)}, 0 , . . . \right) \; .
2 \pi^{3}x{\binom{-{\frac{3}{2}}}{0}}{\frac{1}{( 2 \pi )^{3}}}\int_{1}^{\infty}{\frac{B_{2}( t - [ t ] )}{t^{3}}}d t ={\frac{x}{4}}\left({\frac{7}{1 2}}- \gamma \right) \ ,
\underbrace{\begin{bmatrix}\tilde{F}_1 0 \rho A^T 0 0 \rho B^T \rho A \rho B 0 \end{bmatrix}}_{A_{}}\begin{bmatrix}\Delta W \Delta x \Delta \bar{u}\end{bmatrix}=\underbrace{\begin{bmatrix}-\tilde{f}_1 -\tilde{f}_2 \rho c\end{bmatrix}}_{b_{}},
\sum_{n=0}^{\infty}\frac{2^n}{\|M_1M_2\|}\int\limits_{B_{3\cdot 2^{-n}\|M_1M_2\|}(M_2)}\frac{\omega(d(M))}{d^2(M)}\, dm_3(M)\leq C\sum_{n=0}^{\infty}\frac{2^n}{\|M_1M_2\|}\cdot 2^{-n}\cdot\|M_1M_2\|\omega(2^{-n}\cdot\|M_1M_2\|)= &=C\sum_{n=0}^{\infty}\omega(2^{-n}\cdot\|M_1M_2\|)\leq C\omega(\|M_1M_2\|),
\begin{array}{rcl}\omega(\tau,t,x(\cdot),s) = \displaystyle\int_\tau^t H(\xi,x(\xi),x(\xi-h),s) \mathrm{d}\xi begin{align*}0.4cm] - \big\langle \Big(x(t) - g(t,x(t-h))\Big) - \Big(x(\tau) - g(\tau,x(\tau-h))\Big), s \big\rangle.\end{array}
\dim_{\mathbb{C}}\mathcal{A}_{m}^{\nu}:=\int_{\mathbb{C}^n}K_{\nu,m}(z,z)d\mu_{n}(z)=\frac{(2m+n+2\nu )\Gamma(m+n)\Gamma (m+n+2\nu )}{n\left( \Gamma (n)\right)^{2}\Gamma (m+1)\Gamma(m+2\nu +1)}.