harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
k(t,s):=\frac{a}{\Gamma(b/a)}(t^a-s^a)^{\frac{b}{a}-1}t^{a-\nu}s^{\nu-1}F_3\left(\frac{\nu}{a}-1,\frac{b}{a},1,\mu, \frac{b}{a}, 1-\left(\frac{s}{t}\right)^a,1-\left(\frac{t}{s}\right)^a\right),
y \coloneqq \psi_1^{g - 2 - A}\prod_{i=2}^n\psi_i^{a_i}\kappa_1, x_\ell \coloneqq \psi_1^{g-2-A}\prod_{i=2}^n\psi_i^{a_i+\delta_{i\ell}}, \ell=2, \dots, n.
{\mathrm{Tr}}\bar{K}^{\beta}(s)= \theta_3 \Big(0,{\mathrm{e}}^{-\frac{\beta^2}{4s}}\Big) \frac{\beta}{(4\pi s)^{1/2}}\int{\mathrm{d}}^3{\mbox{\boldmath $x$}}\, \bar{K}^{(3)}(s|\mbox{\boldmath $x$}, \mbox{\boldmath $x$}).
d s^{2}={\frac{1}{\rho}}\left\{d y_{i}d y^{i}-{\frac{1}{\rho}}y_{i}d y^{i}y_{\bar{j}}d y^{\bar{j}}\right\}+ \left\{\rho^{n}d w d{\bar{w}}+ n \rho^{n - 1}\left({\bar{w}}d w y_{\bar{i}}d y^{\bar{i}}+ w d{\bar{w}}y_{i}d y^{i}\right) \right\}
\tfrac{1}{2}k\left(k+1\right)\left(1-\beta\right) \leq \sum_{j=1}^{k}a_{2j}\leq \tfrac{1}{2}k\left(k+1\right)\left(1-\beta\right) + k \qquad\tfrac{1}{2}k\left(k+1\right)\beta \leq \sum_{j=1}^{k}a_{2j-1}\leq \tfrac{1}{2}k\left(k+1\right)\beta + k.
\tilde\lambda=\sup_{t\in[0,T]}\sum_{i,j=1}^d\sup_{x\in\varOmega}\sup_{\substack{|\xi-u(x,t)|\le r *[2pt] |\vec\eta-\nabla u(x,t)|\le r}}\bigg| \frac{\partial g_i(\xi,\vec\eta,x,t)}{\partial\eta_j}\bigg| \, .
m_{\Gamma_2}(\Gamma_2q_0B) =\mu(\Gamma_1\backslash G\times \Gamma_2q_0B) &=m_{\Gamma_0}(\psi^{-1}(\Gamma_1\backslash G\times \Gamma_2q_0B)) &=\sum_{\alpha}m_{\Gamma}(\Gamma \gamma_{\alpha}B),
\bigg(2\frac{\partial E}{\partial b}(t,x;b,y;\mu,\nu^2)\mp 2\frac{\partial E}{\partial y}(t,x;b,y;\mu,\nu^2)-\frac{\mu}{1+b}E(t,x;b,y;\mu,\nu^2)\bigg)_{y=x\pm (t-b)}& = 2^{\sqrt{\delta}-1}(1+t)^{-\frac{\mu}{2}-1}(1+b)^{\frac{\mu}{2}-1}\Big[ 2\big(\tfrac{\mu}{2}+\tfrac{1-\sqrt{\delta}}{2}\big)(1+t)+2^{-1}(\sqrt{\delta}-1)(2t+2)-\mu (1+t)\Big]=0.
a ( k^{+}) = \alpha ( k^{+}) \cdot \mathrm{~{\bf ~ 1}_{N \times ~ N}~}+{\tilde{a}}( k^{+}) \hspace{4 m m}\mathrm{a n d}\hspace{4 m m}b ( k^{+}) = \beta ( k^{+}) \cdot \mathrm{~{\bf ~ 1}_{N \times ~ N}~}+{\tilde{b}}( k^{+}) ,
\int_{\mathbb{R}^{n}}|\nabla u|^{2}=lim_{t\rightarrow0}\frac{1}{t}(\int_{\mathbb{R}^{n}}|u|^{2}-\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}K_{t}(x-y)u(x)u(y)dxdy),
\int_{\tau_0}^s \frac{dr}{u_r^1}> \int_{\tau_0}^s \frac{dr}{u_r^2}, \;\; \hbox{and}\;\; \int_{\tau_0}^s \frac{dr}{u_r^i}= \int_{\tau_0}^s \frac{dv}{\dot{t}^i[(t^1)^{-1}_v]}=(t^i)^{-1}_s - (t^i)^{-1}_{\tau_0}.
\frac{V(X_N,\phi_N)}{N\sigma(C(\cdot,\phi_N))}= \sum_{n=1}^\infty Z(d,n)\frac{a_n(\phi_N)^2}{\sigma(C(\cdot,\phi_N))}\frac1N\sum_{i,j=1}^N P_n^{(d)}(\langle\mathbf{x}_i,\mathbf{x}_j\rangle)\to0.
K_{ijl}^{m}=K_{ij}{}^{k}K_{kl}{}^{m}+K_{ij}{}^{k+n}K_{k+n,l}{}^{m}K_{ijl}^{m+n}=K_{ij}{}^{k}K_{kl}{}^{m+n}+K_{ij}{}^{k+n}K_{k+n,l}{}^{m+n}
\tau^{(2)}\bigl(\Lambda \circ \eta_{W,[B_W]}(f_*) \bigr) + \tau^{(2)}\bigl(\Lambda(T_u(C_*))\bigr) = \tau^{(2)}\bigl(\Lambda(T_u(D_*))\bigr) + \tau^{(2)}\bigl(\Lambda \circ \eta_{V,[B_V]}(f_*)\bigr).
f_{a,c,\gamma}(z) = a \left( \frac{4!}{4-\sum_l k_l-1!}\sum_{j_0, j_1 = 0}^{1, 1}\left[ \prod_{i = 0}^{1}(-\gamma_i)^{k_i-j_i}\binom{k_i}{j_i}\right] \frac{z^{4+\sum_{l=0}^{1}j_l - \sum_{l=0}^1 k_l}}{4 + \sum_{l = 0}^{1}j_l - \sum_{l = 0}^1 k_l}\right) + c.
E_u(\rho_j) =\left\lbrace \frac{f}{D}H_u\left( u_j^H\right) \;\middle|\; f\in \mathbb{C}_{m-1}[z]\right\rbrace , F_u(\rho_j) = \left\lbrace \frac{g}{D}H_u\left( u_j^H\right) \;\middle|\; g\in \mathbb{C}_{m-2}[z]\right\rbrace ,
f_{m}(r+c_{m},k+1)=\sum_{j=1}^{K}\frac{\partial C}{\partial f_{j}(r,k)}|_{ss}f_{j}(r,k)+\sum_{j=1}^{K}\sum_{p=1}^{K}\frac{\partial C}{\partial f_{j}(r+c_{p},k)}|_{ss}f_{j}(r+c_{p},k)
\frac{1}{2}\frac{d}{dt}\|\nabla\Psi\|_{L^{2}}^{2}+\int_{\mathbb{R}^{3}}(v+w)|\nabla\Psi|^{2}dx+\int_{\mathbb{R}^{3}}|\Delta\Psi|^{2}dx +\int_{\mathbb{R}^{3}}(v-w)u\cdot\nabla\Psi dx =0.
K_\omega(u_n,v_n) =H_\omega(u_n,v_n)-3P(u_n,v_n) &=\sum_{j=1}^l H_\omega(U^j,V^j) + H_\omega(u^l_n,v^l_n) - 3P(u_n,v_n)+ o_n(1) &=\sum_{j=1}^l K_\omega(U^j,V^j) + 3\sum_{j=1}^l P(U^j,V^j)- 3 P(u_n,v_n) + H_\omega(u^l_n,v^l_n)+o_n(1).
\left\langle T_{11}(\vec{x})\right\rangle =-\frac{1}{32\pi^{2}}F_{0}(\vec{x})-\frac{1}{16\pi^{2}}\left( D_{1}(\vec{x})-D_{2}(\vec{x})\right) +\frac{1}{64\pi^{2}}\left( F_{1}(\vec{x})-F_{2}(\vec{x})\right)
\frac{H_2(0,-2\delta+2it,\delta,t)}{\zeta(1-2\delta)\zeta(1-2\delta-2it)\zeta(1-2\delta+2it)}\Psi_{\Upsilon,M}(-2\delta-2it) \Psi_{\Upsilon,M}(-2\delta+2it) \hskip 240pt + \mathcal{O}\left( M^{-2\delta(1-\Upsilon)}e^{-c'\log^{1/4}q}\right),
\theta_{s,a^1}^1= \big(\bar{r}^1(s,a^1)-\bar{r}^1(s,a_s^1)\big)-\beta\left(\sum_{s'\in S}p_{s'}\bar{r}^1(s',a_{s'}^1)-\sum_{s'\in S}p^1(s'|s,a^1)\bar{r}^1(s',a_{s'}^1)\right),
|x_{k+1}| < h_k |x_k|^3< h_k |x_0|^{3^{k+1}}\prod_{i=0}^{k-1}h_{k-1-i}^{{3}^{i+1}}= |x_0|^{3^{k+1}}\prod_{i=-1}^{k-1}h_{k-1-i}^{{3}^{i+1}},
\beta^{n-k,k}_D(v,w)=\begin{cases}\sum_{i=2}^{k-1}(-1)^{i+1}\left(\xi_i\lambda_{k-i+1}+\mu_i\nu_{k-i+1}\right) k=m, \sum_{i=2}^{n-k-1}(-1)^{i+1}\lambda_i\nu_{n-k+1-i}+\sum_{i=2}^{k-1}(-1)^i\mu_i\xi_{k+1-i}k<m.\end{cases}
\Big(\frac{1}{2}I_{d}+ \mathcal{K}_{\alpha_{\mathrm{p}}}\Big)\Big[\gamma^{\textbf{int}}_{0}\mathrm{U}_{\mathrm{i}}\Big](\mathrm{x},t) = \mathcal{S}_{\alpha_{\mathrm{p}}}\Big[\gamma^{\textbf{int}}_{1}\mathrm{U}_{\mathrm{i}}\Big](\mathrm{x},t) + \gamma^{\textbf{int}}_{0}\mathcal{V}\Big[ f \Big](\mathrm{x},t), \ \ \mathrm{x}\in \partial\Omega
\frac{\big(F(x,y)\big)^4}{4x^2}=\frac14(z^2-2xy^2)\sum_{k\ge 0}\frac{(2k+2)!}{(k+2)!(k+1)!}\frac{x^ky^{2k}}{z^{2k+2}}&=\frac14\sum_{k\ge 0}\frac{(2k+2)!}{(k+2)!(k+1)!}\frac{x^ky^{2k}}{z^{2k}}-\frac12\sum_{k\ge 0}\frac{(2k+2)!}{(k+2)!(k+1)!}\frac{x^{k+1}y^{2k+2}}{z^{2k+2}}&=\frac14+\frac12\sum_{k\ge2}\frac{(2k)!(k-1)}{k!(k+2)!}\frac{x^ky^{2k}}{z^{2k}},
a = a_{k(n+1)+l}= a_{kn+l}\prod_{i=1}^k f(kn+l+i) + \sum_{j=1}^k h_1(kn+l+j)\prod_{i=j+1}^k f(kn+l+i) & = a\prod_{i=1}^k f(kn+l+i) + \sum_{j=1}^k h_1(kn+l+j)\prod_{i=j+1}^k f(kn+l+i).
\begin{array}{c c c}{{( \gamma^{0})^{\dagger}= \gamma^{0},}}&{{( \gamma^{0})^{T}= - \gamma^{0},}}&{{( \gamma^{0})^{*}= - \gamma^{0},}}\\{{( \gamma^{1})^{\dagger}= - \gamma^{1},}}&{{( \gamma^{1})^{T}= \gamma^{1},}}&{{( \gamma^{1})^{*}= - \gamma^{1},}}\\{{( \gamma^{2})^{\dagger}= - \gamma^{2},}}&{{( \gamma^{2})^{T}= \gamma^{2},}}&{{( \gamma^{2})^{*}= - \gamma^{2}.}}\\ \end{array}
\frac{d}{dt}[\begin{matrix}x\\y\\z\end{matrix}]=[\begin{matrix}10(y-x)\\ 28x-y-xz\\ -\frac{8}{3}z+xy\end{matrix}],[\begin{matrix}x\\y\\z\end{matrix}]=[\begin{matrix}-10\\ 10\\ 25\end{matrix}],t=0...30
( d s )^{2}= \left( - 1 + \frac{2 G m_{1}}{r}\right) d t^{2}+ \left( 1 + \frac{2 G m_{1}}{r}\right) \bigl ( d r^{2}+ r^{2}\, d \theta^{2}+ r^{2}\sin^{2}\theta \, d \phi^{2}\bigr )
w_{\mathrm{s p i n o r}}= 2 \mathrm{I m}{\cal L}_{\mathrm{s p i n o r}}^{( 1 )}= - \Bigl ( \frac{q E}{2 \pi}\Bigr ) \Bigl ( \frac{q B}{2 \pi}\Bigr ) \sum_{j = 0}^{\infty}\sum_{\sigma = \pm 1}\ln \Biggl \{1 - \exp \Bigl ( - \pi \frac{m^{2}+ q B ( 2 j + 1 - \sigma )}{q E}\Bigr ) \Biggr \}.
E_{ij}\rhd_{\epsilon}E_{kl}=\begin{cases}E_{kl}k< j=i+1\leq l, -E_{kl}l< j=i+1\leq k, 0 .\end{cases}E_{kl}\rhd_{\epsilon}E_{ij}=\begin{cases}E_{ij}i< l=k+1\leq j, -E_{ij}j< l=k+1\leq i, 0 .\end{cases}
-{\frac{g_{\Lambda}^{2}C_{F}\Lambda^{2}}{2 \pi^{2}P^{+}}}\ln \Biggl ({\frac{1}{\epsilon}}\Biggr ) \int{\frac{d Y d^{2}Q}{1 6 \pi^{3}}}\theta ( 1 - Y ) \theta ( Y ) | \phi ( Q , Y ) |^{2}\; .
T^{\,m-(j-b_l)}e_{b_{2^{k-1}+l+1}-m+j-b_{l}}=v^{(k)}\, \Bigl(\prod_{i=\Delta^{(k)}-m+j-b_l+1}^{\Delta^{(k)}-1}w_i^{(k)}\Bigr)\, e_{b_l}&-\Biggl(\; \prod_{i=1}^{\Delta^{(k)}-m+j-b_l}w_i^{(k)}\Biggr)^{-1}e_{b_{2^{k-1}+l}},
D(i) = \sum_{j=0}^{r-1}\sum_{k \ne j}p^i_{jk^*}= \sum_{j=0}^{r-1}\sum_{k \ne j}\frac{n_j p^j_{ik}}{n_i}= \frac{1}{n_i}\sum_{j=0}^{r-1}n_j \sum_{k \ne j}p^j_{ik}= \frac{1}{n_i}\sum_{j=0}^{r-1}n_j(n_i - p^j_{ij}).
(\tilde{u}_R(t),\varphi(t))_{\mathcal{H}}-(\tilde{u}_{R,0},\varphi(0))_{\mathcal{H}}= \int_0^t \Big[ (\tilde{u}_R,\partial_t \varphi)_{\mathcal{H}}+2\nu a(\tilde{u}_R,\varphi)+2\nu a(H,\varphi) -b_R(\tilde{u}_R,\varphi,\tilde{u}_R)-b(H,\varphi,\tilde{u}_R)-b(\tilde{u}_R,\varphi, H) \Big]dt.
W^l_1(x,t)=\sum_{i=1}^{\lfloor l/2\rfloor}\bar{w}_{1,i}+\sum_{i=1}^{\lceil l/2\rceil}{w}_{2,i},\ \W_2^l(x,t)=\sum_{i=1}^{\lceil l/2\rceil}{w}_{1,i}+\sum_{i=1}^{\lfloor l/2\rfloor}\bar{w}_{2,i},
0 = EF + \kappa^2 K'^2 = -\frac{1}{4}(1+b^2)(x_2^2 + x_3^2) + 4 \kappa^2 b^2 x_1^2 = -\frac{1}{4}(1+b^2)(x_2^2 + x_3^2) - \frac{(1+b^2)^2}{4}x_1^2 = -\frac{1}{4}(1+b^2) \big(x_2^2 + x_3^2 + (1+b^2)x_1^2 \big).
\mbox{Prod}\left(\mathbf{A}^{(1)},\mathbf{A}^{(2)},\cdots,\mathbf{A}^{(m)}\right)^{\top}=\mbox{Prod}\left(\left(\mathbf{A}^{(2)}\right)^{\top},\cdots,\left(\mathbf{A}^{(m)}\right)^{\top},\left(\mathbf{A}^{(1)}\right)^{\top}\right),
u_{m + k -2}(s, 0) = F_{m + k-2}(s) + \int_{0}^s{\cal G}_{m+ k-2, m + k}(\xi) u_{m + k}(\xi, 0) \, d \xi begin{align*}6pt] + \int_{0}^s{\cal G}_{m + k-2, m + k-1}(\xi) u_{m + k - 1}(\xi, 0) \, d \xi \mbox{for}0 \le s \le t_{k-2},
\phi(j,i')-\phi(j,i) =2(l_{i+1}-l_{i'+1})-\Delta(i+1,i'+1)+\Delta(i+1-\alpha(j,i),i+1) +\Delta(i'+1,i'+1-\alpha(j,i')) =2(l_{i+1}-l_{i'+1}-\Delta(i+1,i'+1))+\Delta(i+1-\alpha(j,i),i'+1-\alpha(j,i')) \geq 2\alpha(i+1,i'+1)+ \Delta(i+1-\alpha(j,i),i'+1-\alpha(j,i'))\,\cdot
- i{\frac{\gamma^{+}( m - \gamma \cdot p ) \gamma^{+}}{m^{2}+ p^{2}}}\to{\frac{- i p^{+}\gamma^{+}\gamma^{-}\gamma^{+}}{m^{2}- 2 p^{+}p^{-}}}={\frac{i \gamma^{+}}{p^{--}m^{2}/ 2 p^{+}}},
\int \vert \eta, z_1 , z_2 \rangle d\mu (\eta ,\bar \eta, z_1, \bar z_1, z_2 , \bar z_2)\langle \eta, z_1 , z_2 \vert = \sum_{l = 0}^{k}\sum_{n = 0}^{k-l}\vert n,l \rangle \langle n,l \vert
u\,=\,F(x,y,z,w)\ \ \ \ \ \ \ \{\scriptstyle{(F(0,0,0,0)\,=\,0)}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \v\,=\,G(r,s,t,p)\ \ \ \ \ \ \ \{\scriptstyle{(0\,=\,G(0,0,0,0))}},
\partial_t \left(\begin{array}{c}\hat{\xi}(k,t) \hat{v}(k,t) \hat{\phi}(k,t) \end{array}\right) =\left(\begin{array}{ccc}0 -ik_2\eta^{\top}0 -ik_2\eta -a\mathbf{I}_3 0 0 0 0 \end{array}\right)\left(\begin{array}{c}\hat{\xi}(k,t) \hat{v}(k,t) \hat{\phi}(k,t) \end{array}\right),
H\tbinom{1}{x_1}H\tbinom{1}{x_2}= \sum_{0<d_1,d_2}e^{d_1x_1}\frac{q^{d_1}}{1-q^{d_1}}e^{d_2x_2}\frac{q^{d_2}}{1-q^{d_2}}&= \bigg( \sum_{0<d_1<d_2}+\sum_{0<d_2<d_1}+\sum_{0<d_1=d_2}\bigg)e^{d_1x_1}\frac{q^{d_1}}{1-q^{d_1}}e^{d_2x_2}\frac{q^{d_2}}{1-q^{d_2}}&= H\tbinom{1,1}{x_1,x_2}+H\tbinom{1,1}{x_2,x_1}+H\tbinom{2}{x_1+x_2}.
A=\begin{pmatrix}2 1 0 0 0 0 0 0 0 \cdots 0 2 1 1 0 0 0 0 0 \cdots 0 0 2 0 1 1 1 1 0 \cdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \ddots\end{pmatrix}.
\frac{Z_{\scriptscriptstyle MCS}(J)}{Z_{\scriptscriptstyle SD}(\hat{J})}=e^{i \alpha}\tau_{\scriptscriptstyle RS}(M)^{\scriptscriptstyle 1/2}e^{i \langle J, \hskip 3pt \frac{1}{\ast d_{\scriptscriptstyle 1}}\hskip4pt J \rangle}=e^{i \alpha}Z_{\scriptscriptstyle CS}(J)
H_{z}=\frac{k_{o}^{2}\epsilon_{r}-k_{z}^{2}}{j\omega\mu}(Ce^{jk_{x\epsilon}w}+De^{-jk_{x\epsilon}w})e^{jk_{xo}(x+w)}cos(\frac{m\pi}{a}y)e^{-jk_{z}z}(50)
t_{(\sigma,b)}(z):=\textbf{t}_{(\sigma,b)}(z)+\sum\limits_{\sigma^{\prime}:\sigma \dagger \sigma^\prime}\sum\limits_{\substack{c\in\mathbb{Q}:c>0, \langle c\rangle=\hat{b}_{j}}}q^{d_{c,\sigma,j}}\frac{Rec(c)^{(\sigma^{\prime},b^{\prime})}_{(\sigma,b)}}{(-z+U_{j}(\sigma)/c)}[f_{(\sigma^{\prime},b^{\prime})}]_{z=U_{j}(\sigma)/c},
0=\frac{\partial}{\partial \lambda}\L(\widetilde{\phi}_{\lambda})|_{\lambda=1}=a_{1}\int_{\mathbb{G}}|\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}}\phi(x)|^{p}dx+a_{2}\int_{\mathbb{G}}|\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}}\phi(x)|^{p}dx-\frac{Q(q-p)}{pq}\int_{\mathbb{G}}|\phi(x)|^{q}dx.
x_1^2 -mx_1y_1+{y_1}^2 -(a^2 -m ab + b^2 ) = mj(mjv^4 - m^2jv^mw + mjv^2w^2 + mv^2 - 2vw)a^2 & + mj(mjv^2w^2 - m^2jvw^m + mjw^4 + 2vw - mw^2)b^2 &- 2mj(mjv^mw - m^2jv^2w^2 + mjvw^m + v^2 - w^2)ab .
\begin{aligned}X_i =a_i-(2n_i-1)d, X_{i+3}= b_i+(2n_i+1)d,\;i=1,\,2,\,3, Y_i =a_i+(2n_i+1)d, Y_{i+3}=b_i-(2n_i-1)d,\;i=1,\,2,\,3,\end{aligned}
\partial_s (\gamma_n(s)(R_n(\lambda)f)(s)) =\partial_s (c_n\gamma_n(s)H_n(s)+\gamma_n(s)G_n(s)H_n(s)) & =-(\mu_n(s)+\lambda)(c_n+G_n(s))H_n(s) + f(s) & =-(\mu_n(s)+\lambda)(R_n(\lambda)f)(s) + f(s),
\delta A[f_{0},f_{1}]=f_{1}(0)[n_{(-)}\frac{f_{0}^{\prime}(0^{-})}{\sqrt{1+f_{0}^{\prime}(0^{-})^{2}}}-n_{(+)}\frac{f_{0}^{\prime}(0^{+})}{\sqrt{1+f_{0}^{\prime}(0^{+})^{2}}}].
X^{\prime}=[\begin{matrix}ct^{\prime}\\r^{\prime}\end{matrix}]B(v)=[\begin{matrix}\gamma_{v}&-\frac{\gamma_{v}}{c}v^{T}\\ -\frac{\gamma_{v}}{c}v&I+\frac{\gamma_{v}^{2}}{\gamma_{v}+1}\frac{vv^{T}}{c^{2}}\\ \end{matrix}]X=[\begin{matrix}ct\\r\end{matrix}]
v_j(t)=c_j e^{\mu_j^{\infty}t}+\tilde{v}_j(t), \tilde{v}_j(t):=\sum_{l\in\mathbb{Z}^{\nu}}\frac{f_{j l}}{\mathrm{i}\omega\cdot l+\mu_j^{\infty}}e^{\mathrm{i}\omega\cdot l t}.
-\partial_{t}U[v_{j}\left( t,\beta\right) ] =-2\operatorname{Im}\zeta_{j}\left( \beta\right) U[v_{j}\left( t,\beta\right) ],\text{\ \}1\leq j\leq2N, U[v_{j}\left( t,\beta\right) ] =\frac{1}{2}\left( w_{j}\left(\beta\right) ,w_{j}\left( \beta\right) \right) e^{2\operatorname{Im}\zeta_{j}\left( \beta\right) t}
c_{s_{\alpha}}(\lambda)=c_{0}\frac{2^{-i(\lambda,\alpha_{0})}\Gamma(i(\lambda,\alpha_{0}))}{\Gamma(\frac{1}{2}(\frac{1}{2}m_{\alpha}+1+i(\lambda,\alpha_{0})))\Gamma(\frac{1}{2}(\frac{1}{2}m_{\alpha}+m_{2\alpha}+i(\lambda,\alpha_{0})))},
\mu_t(-tf) = \mu_t \left(\sum_{i \in \mathbb{N}}-tf|_{[i]}\right) &\leq \sum_{i \in \mathbb{N}}\sup(-tf|_{[i]}) \mu_t[i] &\leq C_t \sum_{i \in \mathbb{N}}\sup(-tf|_{[i]}) \exp(\sup(tf|_{[i]})) < \infty \,.
\begin{array}{l l}ch(L_1)=(u+\frac{1}{2}u^2)v_{2n-4}ch(L_2)=u^2v_{2n-4}begin{align*}15pt]ch(L_3)=(u+\frac{1}{2}u^2)v_{2n-2}ch(L_4)=u^2v_{2n-2}.\end{array}
\tau_{J'}= \Bigg( \bigcap_{\substack{\hat{s}\neq s \in S Z_s \in J'}}\mathrm{H}_{Z_s - Z_{\hat{s}}}\Bigg) \cap \Bigg( \bigcap_{\substack{s \in S Z_s \notin J'}}\mathrm{H}_{Z_s - Z_{\hat{s}}, \ge 0}\Bigg).
\sum_{n=1}^{N}\frac{r^{n}w^{n}}{n}=\sum_{n=1}^{N}\int_{0}^{1}r^{n}w^{n}t^{n-1}dt=\int_{0}^{1}\frac{rw(1-r^{N}w^{N}t^{N})}{1-rwt}dt.
{\cal H}_{j,j+1}=\sigma^{+}_{j}\sigma^{-}_{j+1}+\sigma^{-}_{j}\sigma^{+}_{j+1}+\Gamma\left(\sigma^{+}_{j}\sigma^{+}_{j}+\sigma^{-}_{j}\sigma^{-}_{j+1}\right)-\frac{h}{2}\left(\sigma^{z}_{j}+\sigma_{j+1}^{z}\right)\;.
{\tau}_1(\bar{a}) = \{(i_1,\dots,i_n)\mid i_1<\dots <i_n\in\{1,\dots ,m\}(a_{i_1},\dots,a_{i_n})\in\varrho\},{\tau}_2(\bar{a}) = \{(i,j)\mid i<j\in \{1,\dots ,m\}, \; a_i=a_j\}.
[x_j, \frac{\nabla_k}{\sqrt{-\Delta+m^2}}] =[x_j,\nabla_k] \frac{1}{\sqrt{-\Delta+m^2}}+\nabla_k [x_j,\frac{1}{\sqrt{-\Delta+m^2}}] &=-\frac{\delta_{jk}}{\sqrt{-\Delta+m^2}}-\frac{\nabla_k\nabla_j}{({-\Delta+m^2})^{\frac{3}{2}}}.
\sum E_{q}^{h}+(\sum E_{q}^{h}-\sum\pi_{P}^{f})\frac{dp}{dt}=\sum\frac{dI^{h}}{dt}+\{\sum\pi_{z}^{f}\frac{dz^{f}}{dt}-\sum E_{z}^{h}\frac{dz^{h}}{dt}\}
\bigl(X^+f\bigr)\circ\pi_q(g) =\bigl(X^+f\bigr)(g\cdot q)=X^+_{g\cdot q}(f) &=X_e(f\circ\pi_{g\cdot q})=X_e(f\circ\pi_q\circ R_g) &=\hat X_g(f\circ\pi_q)=\bigl(\hat X(f\circ\pi_q)\bigr)(g).
\Omega^{ij}\;=\; \left( \begin{array}{cccc}k_1 0 0 -k_2 0 0 k_2 0 0 k_2 0 0 -k_2 0 0 0 \end{array}\right) \, , \, \Omega_{ij}\;=\; \left( \begin{array}{cccc}0 0 0 -q_2 0 0 q_2 0 0 q_2 0 0 -q_2 0 0 q_1 \end{array}\right) \, ,
\begin{aligned}\left\| u_d(x) - \frac{(x-1)p(x)}{q(x)}\right\| =\left\| (x-1)\left( h_d(x) -\frac{p(x)}{q(x)}\right)\right\| &=\left\| h_d(x) -\frac{p(x)}{q(x)}\right\|+1 &= -2 \|q(x)\| - c + 1.\end{aligned}
\frac{1}{3}\frac{2\omega-4}{2\omega-3}\oint d\tau_1\,d\tau_2\,d\tau_3\,\epsilon(\tau_1\,\tau_2\,\tau_3)\frac{\sin\tau_{12}+\sin\tau_{23}+\sin\tau_{31}}{\alpha\beta(1-\cos\tau_{12})+\beta\gamma(1-\cos\tau_{23})+\gamma\alpha(1-\cos\tau_{13})}.
d [ \sigma ] = \prod_{j = 1}^{2}\prod_{p , q}d ( \mathrm{R e}\sigma_{p q}^{j j}) d ( \mathrm{I m}\sigma_{p q}^{j j}) \prod_{j \neq l}\prod_{p , q}d \sigma_{p q}^{j l *}d \sigma_{p q}^{j l}
\begin{array}{lcrcl}\epsilon = 1\hspace{10mm}\delta\varepsilon^{1}_{1}(\frac{\pi}{5}) = \delta\varepsilon^{1}_{3}(\frac{\pi}{5})\;, \delta\varepsilon^{1}_{1}(\frac{\pi}{4}) = \delta\varepsilon^{1}_{2}(\frac{\pi}{4})\;, \delta\varepsilon^{1}_{0}(\frac{\pi}{4}) = \delta\varepsilon^{1}_{3}(\frac{\pi}{4})\;, \end{array}
m_{\cdot}: h^A(R)\times h^A(R) \to h^A(R) (f_1,f_2) \mapsto f_1\cdot f_2 := (f_1,f_2)\circ \Delta_\cdot, m_{\circ}: h^A(R)\times h^A(R) \to h^A(R) (f_1,f_2) \mapsto f_1\cdot f_2 := (f_1,f_2)\circ \Delta_\circ.
\sum_{m \in \Delta(n)}\frac{2}{m^{d_k}(1 - m^2)}\sim\frac{1}{n^{d_k}(n^2 - 1)}- \sum_{j=0}^{\infty}\sum_{\ell=0}^{\infty}\frac{4^{\ell}B_{2\ell}\Gamma(2\ell+2j+d_k+1)}{(2\ell)! \Gamma(d_k+2j+2)n^{2\ell+2j+d_k+1}}.
{\cal I}= \frac{1}{4 \pi}\int \left( \frac{( | R_{z}|^{2}- | R_{{\bar{z}}}|^{2}) ( 1 + | z |^{2})^{2}}{( 1 + | R |^{2})^{2}}\right)^{2}\frac{2 i ~ \mathrm{d}z \mathrm{d}{\bar{z}}}{( 1 + | z |^{2})^{2}}.
(q^{(1-n)/2}uv^{-1};q)_{n}=\begin{cases}(-1)^{k}u^{k}v^{-k}q^{-k^{2}/2}(u^{-1}vq^{1/2};q)_{k}(uv^{-1}q^{1/2};q)_{k}, \mbox{if}n=2k, (-1)^{k}u^{k}v^{-k}q^{-k(k+1)/2}(u^{-1}vq;q)_{k}(uv^{-1};q)_{k+1}, \mbox{if}n=2k+1. \end{cases}
\Phi_{v^o_n}(z^o_n, x^o_n) = \Phi_{v^\iota_n}(z^\iota_n, x^\iota_n) + \sum_{j=n^2+k_n}^{n^2+2n-k_n}\left(\frac{\sqrt{w_j}}{2}+o\left(\frac{1}{n^2}\right)\right).
(-\Delta)^m_{y}{|y-x|^{-N-2\sigma}}=-(N+2\sigma)(2\sigma+2) (-\Delta)_y^{m-1}{|y-x|^{-N-2\sigma-2}}&=(-1)^{m}\prod\limits_{i=0}^{m-1}(N+2\sigma+2i)(2\sigma+2(i+1)){|y-x|^{-N-2\sigma-2m}}.
A_k =\frac{m^2 V_k^2 \left(c^2+(2G^2+\tilde M V_k)\delta^2\right)}{(2G^2+\tilde M V_k)^2}B_k =\frac{(2G^2 V_k+(\tilde M-m^2) V_k^2)((2G^2+\tilde M V_k)(1-\delta^2)-c^2)}{(2G^2+\tilde M V_k)^2}Z_k =\frac{\bigl(2 G^2+\tilde M V_k\bigr)(\delta^2+\zeta_k )+c^2}{(2G^2+\tilde M V_k)(1-\delta^2)-c^2}
\varepsilon^{\alpha -1}\, w_{\alpha -1}^{(i)}(x_i,t) + \varepsilon^{\alpha}\, w_{\alpha}^{(i)}(x_i,t) + \sum\limits_{k=2}^{-\lfloor\alpha\rfloor}\varepsilon^{\alpha +k -1}\, \Big(w_{\alpha +k -1}^{(i)}(x_i) + u_{\alpha +k -1}^{(i)}\big( x_i, \tfrac{\overline{x}_i}{\varepsilon}, t\big)\Big).
\mathcal{T}^{(\gamma)}={\rm tr}^{(\gamma)}\gamma^{(2n+1)}\left(\frac{u_1-a}{|u_1-a|}-\frac{u_2-a}{|u_2-a|}\right)\cdot\gamma \left(\frac{u_2-a}{|u_2-a|}-\frac{u_3-a}{|u_3-a|}\right)\cdot\gamma \qquad\qquad\qquad\cdots \left(\frac{u_{2n}-a}{|u_{2n}-a|}-\frac{u_{2n+1}-a}{|u_{2n+1}-a|}\right)\cdot\gamma
\left( \begin{array}{c c c}{{- 1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{- 1}}\\{{0}}&{{- 1}}&{{0}}\\ \end{array}\right) , \quad \left( \begin{array}{c c c}{{0}}&{{- 1}}&{{0}}\\{{- 1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{- 1}}\\ \end{array}\right) , \quad \left( \begin{array}{c c c}{{0}}&{{0}}&{{- 1}}\\{{0}}&{{- 1}}&{{0}}\\{{- 1}}&{{0}}&{{0}}\\ \end{array}\right)
\partial_{\xi}^{2}\left(\xi^{2}\frac{K_{1}(\xi\lambda(t))}{\log^{b-1}(\frac{1}{\xi})}\right)= \partial_{\xi}^{2}\left(\frac{\xi}{\lambda(t)\log^{b-1}(\frac{1}{\xi})}\right) + F_{v_{2}}(\xi,\lambda(t)), \xi \leq \frac{1}{4}
\frac{1}{\pi R^2}\int_{B(R)}|F_{\mu}|^2 = \alpha \frac{1}{\pi R^2}\int_{B(R)}|F_{\mu_{A}}|^2 + \beta \frac{1}{\pi R^2}\int_{B(R)}|F_{\mu_B}|^2 +2\sqrt{\alpha\beta}\frac{1}{\pi R^2}\int_{B(R)}(F_{\mu_{A}}\cdot F_{\mu_B}).
K_{p}(z) \stackrel{(z \rightarrow 0)}{\sim}\frac{1}{2}\,\left[\Gamma (p)\left( \frac{z}{2}\right)^{-p}+\Gamma (-p)\left( \frac{z}{2}\right)^{p}\right]\,\left[ 1 + O(z^{2}) \right]\; .
u(x',t)=\frac{2t}{\omega_{n-1}\sqrt{{\rm det}\,(A_{\rm sym})}}\int_{{\mathbb{R}}^{n-1}}\frac{f(y')}{\big\langle(A_{\rm sym})^{-1}(x'-y',t),(x'-y',t)\big\rangle^{\frac{n}{2}}}\,dy'.
-4 \int_{t}^{\infty}\frac{\lambda''(x) dx}{1+x-t}+ \frac{4b}{t^{2}\log^{b}(t)}+ 4 \alpha \log(\lambda(t)) \lambda''(t) - 4 \int_{t}^{\infty}\frac{\lambda''(s) ds}{(\lambda(t)^{1-\alpha}+s-t)(1+s-t)^{3}}= G(t,\lambda(t))
2 1 \hat{d}^{- 1}( p^{2}) = \frac{N g^{2}}{4 \pi}[ - \alpha p \operatorname{a r c t a n}( \frac{p}{2 m}) - \beta m + \gamma \frac{m^{2}}{p}\operatorname{a r c t a n}( \frac{p}{2 m}) ] .
\prod_{i=0}^{j}\big(\chi(g_{k_i})-1\big)-\prod_{i=0}^{j}\big(\varphi(g_{k_i})-1\big) &=\big(\chi(g_{k_0})-\varphi(g_{k_0})\big)\prod_{i=1}^{j}\big(\chi(g_{k_i})-1\big)+\big(\varphi(g_{k_0})-1\big)\Bigg(\prod_{i=1}^{j}\big(\chi(g_{k_i})-1\big) -\prod_{i=1}^{j}\big(\varphi(g_{k_i})-1\big)\Bigg)
R = \begin{bmatrix}1 -1/2 0 0 \ldots 0 0 -1/2 1 -1/2 0 \ldots 0 0 0 -1/2 1 0 \ldots 0 0 \vdots \vdots \vdots \vdots \ddots \ddots \ddots 0 0 0 0 \ldots 1 -1/2 0 0 0 0 \ldots -1/2 1\end{bmatrix},
(\Gamma_2^{(\kappa)}(y)^{*}W^{(\kappa)}(y))''-(\Gamma_1^{(\kappa)}(y)^{*}W^{(\kappa)}(y))'=0, \,\, \lim_{y\to0,1}\left(\Gamma_2^{(\kappa)}(y)^{*}W^{(\kappa)}(y)\right)'-\Gamma_1^{(\kappa)}(y)^* W^{(\kappa)}(y)=0,
Pr\left(\underset{v\in\mathcal{F_{\varepsilon}}}{\sup}B_{n,1}(v)\geq B\sqrt{\frac{\mu}{2n}}\right) =Pr\left(\underset{v\in\mathcal{F_{\varepsilon}}}{\sup}\left|R_{L,P}(v)-R_{L^n,D,g}(v)\right|\geq B\sqrt{\frac{\mu}{2n}}\right) &\leq\underset{v\in\mathcal{F_{\varepsilon}}}{\sum}Pr\left(|R_{L,P}(v)-R_{L^n,D,g}(v)|\geq B\sqrt{\frac{\mu}{2n}}\right).
D_{0}= \frac{1}{2}\left( \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right) , D_{+}= \left( \begin{array}{c c}{0}&{\sigma_{1}}\\{0}&{0}\\ \end{array}\right) , D_{-}= \left( \begin{array}{c c}{0}&{0}\\{\sigma_{1}}&{0}\\ \end{array}\right) ;
H_{\mathrm{f r e e}}= \int \frac{1}{2}\left\{| \pi ( \stackrel{\rightharpoonup}{x}) |^{2}+ | \bigtriangledown \varphi ( \stackrel{\rightharpoonup}{x}) |^{2}+ m | \varphi ( \stackrel{\rightharpoonup}{x}) |^{2}\right\}d^{3}\stackrel{\rightharpoonup}{x}\ .
\sum_{i,j=1}^3 \sigma^{ij}\big(u^0\hat{p}_i-u_i\big)\big(u^0\hat{p}_j-u_j\big)\geq \sum_{i,j=1}^3\frac12\big(u^0\big)^2\sigma^{ij}\hat{p}_i\hat{p}_j-\sum_{i,j=1}^3\sigma^{ij}u_iu_j.
\; = \frac{1}{Z}\sum_{n = 0}^{\infty}\frac{1}{n !}\Bigg \langle \left( \sum_{b = 1}^{N}m^{( b )}\int_{\Lambda}d^{2}x \; t ( x ) \Big [ \overline{{{\psi}}}^{( b )}( x ) P_{+}\psi^{( b )}( x ) + \overline{{{\psi}}}^{( b )}( x ) P_{-}\psi^{( b )}( x ) \Big ] \right)^{n}
a=\frac12\cdot\#\{i \mid \varsigma(i)\neq i, \quad\dim\phi_{\pi_i}=1\}, b=\frac12\cdot\#\{i \mid \varsigma(i)\neq i, \dim\phi_{\pi_i}=2\}, c=\#\{i \mid \varsigma(i)=i \}.
\begin{array}{c}{{< j_{1}- m_{1}\j_{2}- m_{2}| J - M >_{q}= < j_{2}m_{2}\j_{1}m_{1}| J M >_{q}=}}\\{{= ( - )^{j_{1}+ j_{2}- J}< j_{1}m_{1}\j_{2}m_{2}| J M >_{q^{- 1}}}}\\ \end{array}
-2 \Phi(N) \rho(N-1)^{2}\Delta A(N-1) +2 \Phi(N+1) \rho(N)^{2}\Delta A(N) = \zeta +\sum_{i=1}^{M}\lambda_{i}A(N)^{i}-\alpha_{1}B_0(N) -2 \alpha_2 B_0(N)A(N) -\beta [B_0(N)^2 + \rho(N-1)^2 \Phi(N) +\rho(N)^2 \Phi(N+1)],
\left(\begin{array}{ccccc}-z_1 z_2 0 0 0 1 1 0 0 0 0 z_2 -z_3-z_4 z_5 0 0 0 z_3 -z_5 0 0 0 z_3 z_5 -z_6\end{array}\right)\left(\begin{array}{c}x_1 x_2 x_3 x_4 x_5 \end{array}\right) +\left(\begin{array}{c}0 -z_7 z_8 0 z_9 \end{array}\right)= \left(\begin{array}{c}0 0 0 0 0\end{array}\right).
B_{k+1}= B_k - B_kDP(P^TD^TB_kDP)^{-1}P^TD^TB_k + G DP(P^TD^TGDP)^{-1}P^TD^TG &= B_k - B_kD(D^TB_kD)^{-1}D^TB_k + G D(D^TGD)^{-1}D^TG.
a\cdot(b\times c)=det[\begin{matrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\ \end{matrix}]=det[\begin{matrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{matrix}]=det[\begin{matrix}a&b&c\end{matrix}].
f(g_{t_1},\ldots, g_{t_{n-1}}) =\sum_{g_0\in G}g_0^{-1}\theta(g_0, g_{t_1},\ldots, g_{t_{n-1}}) &\quad+\sum_{i=1}^{n-1}(-1)^i\sum_{\substack{g, \bar{g}\in G g\bar{g}=g_{t_i}}}\theta(g_{t_1},\ldots, g_{t_{i-1}}, g, \bar{g}, g_{t_{i+1}},\ldots, g_{t_{n-1}}) &\quad+(-1)^n\sum_{g_n\in G}\theta(g_{t_1},\ldots, g_{t_{n-1}}, g_n).
\Gamma_{\mu ab}(p,q) = \gamma_{\mu ab}+ \int \frac{d^3\ell}{(2\pi)^3}\left[S(p + \ell)\Gamma_\mu(p + \ell,q + \ell)S(q + \ell)\right]_{dc}K_{cd,ba}(q + \ell,p + \ell,\ell),
\alpha(s,\mathcal{I})^{x(t,\mathcal{I})}=(\alpha(s,\rho_1),\ldots,\alpha(s,\rho_r))^{(x_{t\rho_1},\ldots,x_{t\rho_r})}&=(\alpha(s,\rho_1)^{x_{t\rho_1}},\ldots,\alpha(s,\rho_r)^{x_{t\rho_r}}) &=(\alpha(st,\rho_1),\ldots,\alpha(st,\rho_r)) \textnormal{(by Lemma \ref{start})}&=\alpha(st,\mathcal{I}).
f_0(N;x,p)={1\over \pi}\! \int^{\infty}_0\! \! \! \! dy ~{\cos(yp) \over (\cosh(2x/\hbar )+\cosh(y))^N}={1\over (N-1)!}~ \left(\frac{-\hbar}{2 \sinh(2x/\hbar )}\partial_x\right)^{N-1}f_0(1;x,p),
\mathrm{P}_{0}e_{2}\left(\eta\right) =g^{-1}\cdot\mathrm{P}_{0}\psi_{2}\left(\left|\eta\right|\right) & =\mathtt{a}_{2,1}^{0}(\left|\eta\right|^{2})\chi_{0}+i\mathtt{a}_{2,2}^{1}(\left|\eta\right|^{2})\sum_{j=1}^{3}\eta_{j}\chi_{j}+\mathtt{a}_{2,1}^{4}(\left|\eta\right|^{2})\chi_{4},