harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S(p^t)=\begin{cases}0, \beta< t-1, -\frac{\prod_{i=1}^3 p^{\min(\frac{\alpha_i+t}{2},t)}}{p^{tk}}p^{t-1}, \beta=t-1, \phi(p^t)\frac{\prod_{i=1}^3 p^{\min(\frac{\alpha_i+t}{2},t)}}{p^{3t}}, \beta\geq t.\end{cases}
\frac{1}{n!}\int_a^{b}\cdots\int_a^b f (\mathbf{x}) \emph{W}^{(n)}(\mathbf{x}) \emph{d}x_1\cdots\emph{d}x_n =\sum_{{\lambda}\in \Lambda^{(m,n)}}f \bigl(\mathbf{x}^{(m,n)}_{{\lambda}}\bigr) \emph{W}^{(m,n)}_{{\lambda}},
\mathcal{F}(\tilde{\mathcal{L}};q,t) =\sum_{l=1}^L\frac{(-1)^{l-1}}{l}\sum_{\cup_{s=1}^l\mathbf{I}_s=\mathbf{L}}\prod_{s=1}^l\mathcal{H}(\mathcal{L}_{\mathbf{I}_s};q,t) &=\left(\sum_{l=1}^L\frac{(-1)^{l-1}}{l}\sum_{\cup_{s=1}^l\mathbf{I}_s=\mathbf{L}}1\right)\prod_{\alpha=1}^L\mathcal{H}(\mathcal{K}_\alpha;q,t) &=0,
\mu=\left(\frac{1}{2}-\frac{1}{p},\frac{1}{2}-\frac{1}{p},\frac{1}{2}-\frac{1}{p},\frac{1}{4}+\frac{3}{2p}-\frac{t}{2},\frac{1}{4}+\frac{3}{2p}+\frac{t}{2}\right),
[\hat{G}_{\mu\nu}^{(n)},\hat{G}_{\lambda\rho}^{(n)}]=2\left(\delta_{\nu\lambda}\hat{G}_{\mu\rho}^{(n)}+\delta_{\mu\rho}\hat{G}_{\nu\lambda}^{(n)}-\delta_{\mu\lambda}\hat{G}_{\nu\rho}^{(n)}-\delta_{\nu\rho}\hat{G}_{\mu\lambda}^{(n)}\right).
(\mathcal M\wr\mathbb E_{\mathcal G})\times_{\mathbb E_{\mathcal G}}\mathbb G_{\mathcal G}= \mathcal M\wr\mathbb G_{\mathcal G}\xrightarrow[\cong]{\Phi}(\mathcal M\rtimes\mathcal G)\wr\mathbb E_{\mathcal G}\to \mathcal M\wr\mathbb E_{\mathcal G}\ .
[\,A_{{\sf GI}\,i}^{{\gamma}}({\bf{r}})\,{\textstyle\frac{{\lambda}^{\gamma}}{2}}\,]=V_{\cal{C}}({\bf{r}})\,[\,A_{i}^{\gamma}({\bf{r}})\,{\textstyle\frac{\lambda^\gamma}{2}}\,]\,V_{\cal{C}}^{-1}({\bf{r}})+{\textstyle\frac{i}{g}}\,V_{\cal{C}}({\bf{r}})\,\partial_{i}V_{\cal{C}}^{-1}({\bf{r}})\,,
V_n(x) = - \frac{16\alpha x_1 \sin^2 (2x_1)}{4 |x_1|^2+|x'|^4}+ \frac{16\alpha(\alpha+1)|x'|^6}{(4 |x_1|^2+|x'|^4)^2}- \frac{4\alpha ( 4x_1 \cos(2x_1) + (\nu+1)|x'|^2)}{4 |x_1|^2+|x'|^4}& + O((|x_1|+|x'|^2)^{-2})
\hat{f}(n)=\int_0^1 [D_\mu(x)-x]e(-nx)\,dx &=\frac{1}{-2\pi in}[D_\mu(x)-x]e(-nx)|_0^1+\frac{1}{2\pi in}\int_0^1 e(-nx)\,d [D_\mu(x)-x] &=\frac{\hat{\mu}(n)}{2\pi in},
x^{(k+1)}= x^{(k)}- \left[ I + \frac{1}{2}F'(x^{(k)})^{-1}F''(x^{(k)}) F'(x^{(k)})^{-1}F(x^{(k)}) \right] F'(x^{(k)})^{-1}F(x^{(k)})
H_{C}= \frac{g^{2}}{2}\, \pi_{a}^{-}\pi_{a}^{-}- \, \epsilon^{a b c}\, A_{+}^{b}\left( A_{-}^{c}\, \pi_{a}^{-}+ A_{k}^{c}\, \pi_{a}^{k}\right) + V ( A_{k})
M(t) = \frac{1}{N}\sum_{j=0}^{\infty}j X_j(t) \leq \frac{1}{N}\sum_{j=0}^{\infty}k^* X_j(t) + \frac{1}{N}\sum_{\ell = 1}^{\infty}\ell X_{k^* + \ell}(t) = k^* + \frac{1}{N}\sum_{\ell = 1}^{\infty}\ell X_{k^* + \ell}(t).
\left\langle L_Fe_{k},e_{j}\right\rangle =\begin{cases}\sum_{l=1}^{n}\, \alpha_l(j) \, a_{l,\alpha(l)}\textrm{if}j=k \alpha_{l}(k) \, a_{l,\alpha(r)}\textrm{if}\alpha(j)=(\alpha_1(k),\cdots, \alpha_{l}(k)-1, \cdots, & \alpha_{r}(k)+1, \cdots,\alpha_n(k)), 0 \textrm{otherwise}.\end{cases}
\bar{\partial}\frac{\delta Z}{\delta h_i(x,\bar{x})}=\frac{1}{\pi}\sum_{j,n}^{}\frac{(-1)^{n-1}}{(n-1)!}\partial^{n-1}\left(h_j\left<\left[\Phi^i\Phi^j\right]_ne^{-\frac{1}{\pi}\int{\bf h \cdot\Phi}}\right>\right).
d S^{2}={H_{7}}^{- 1 / 2}[ d u d v - \mu^{2}\sum_{i = 1}^{4}{z^{i}}^{2}d u^{2}+ \sum_{i = 1}^{4}d z^{i}d z^{i}+ d \tilde{x}^{2}+ d \tilde{y}^{2}] +{H_{7}}^{1 / 2}( d r^{2}+ r^{2}d{\theta}^{2}) .
\sup_{B(p_{r_i},d_{0};{g_{r_i}})}| \nabla_{(g_{r_i})}X_{(i)}|_{g_{r_i}}= \sup_{B(p_{r_i},d_{0};{g_{r_i}})}\frac{|{\rm Ric}|}{\sqrt{R(p_{r_i})}}&\le C(d_0)\sqrt{R(p_{r_i})}\to 0.
{\mathbf Y}_t(\theta-\frac{\pi}{2}i ,\ell){\mathbf Y}_t(\theta+\frac{\pi}{2}i,\ell) =1+{\mathbf Y}_1(\theta,\ell), &{\mathbf Y}_1(\theta-\frac{\pi}{2}i ,\ell){\mathbf Y}_1(\theta+\frac{\pi}{2}i,\ell) =(\xi^3+{\mathbf Y}_t(\theta,\ell)) (\xi^{-3}+{\mathbf Y}_t(\theta,\ell)).
V_{1,+}(\ell_1-1,0) \oplus V_{2,+}(\ell_1, 1) \oplus V_{1,-}(\ell_1-1,1) \oplus V_{2,-}(\ell_1-1, 1) \cong V_{2,+}(\ell_1-1,1) \oplus V_{1,+}(\ell_1-1,1) \oplus V_{1,-}(\ell_1-1,0) \oplus V_{2,-}(\ell_1, 1),
R_{\gamma A}=\frac{2}{1+2\gamma \sigma+(\gamma \beta)^2}\begin{bmatrix}\tfrac{1}{2}(1-(\gamma\beta)^2) \gamma\sqrt{\beta^2-\sigma^2}-\gamma\sqrt{\beta^2-\sigma^2}\tfrac{1}{2}(1-(\gamma\beta)^2)\end{bmatrix}.
\sum_{j=1}^2 \frac{\lambda_j i}{2\pi}\frac{(\dot{z}_j)^2}{(\zeta(\alpha,t)-z_j(t))^3}= \frac{\lambda i (\dot{z}_1)^2}{2\pi}(\frac{1}{(\zeta(\alpha,t)-z_1(t))^3}-\frac{1}{(\zeta(\alpha,t)-z_2(t))^3}) &+\frac{\lambda i ((\dot{z}_1)^2-(\dot{z}_2)^2)}{2\pi}\frac{1}{(\zeta(\alpha,t)-z_2(t))^3}:=I+\it{II}.
\overline{uP(p|_{\phi_2(r,s)})}, \ \overline{P(up|_{\phi_2(r,s)})}, \ \overline{P(d(u)p|_{\phi_2(r,s)})}<_n \overline{\phi_1(u,p|_{\phi_2(r,s)})}=\overline{\phi_1(u,v)}=w
\varphi:\mathcal{O}_{K}\rtimes(\mathcal{O}_{K}^{\times,+})^{2}\longrightarrow\operatorname*{SL}\nolimits_{2}\left( \mathcal{O}_{K}\right) (u,v) \longmapsto\begin{pmatrix}\sqrt{v}\frac{u}{\sqrt{v}}& \frac{1}{\sqrt{v}}\end{pmatrix}
\lim_{n \rightarrow \infty}\ln L = \lim_{n \rightarrow \infty}g(n) \ln f(n) = \lim_{n \rightarrow \infty}\frac{2n}{an^{\alpha}+ \sqrt{an^{\alpha}}}\left( \sqrt{an^{\alpha}}\ln\left(1+ \frac{1}{\sqrt{an^{\alpha}}}\right) - 1 \right)
\tilde{V}=\left[\begin{array}{cc}\tilde{Q}0 0 \tilde{F}\end{array}\right]=\left[\begin{array}{cc}|F_2|^{-2}Q_2/2-|F_1|^{-2}Q_1/2 0 0 -|F_2|^{2}+|F_1|^{2}\end{array}\right]=\tilde V_2-\tilde V_1.
{\bf{\mathcal B}}_i^{AF}\left( \pmb \rho \right) = \frac{{\sqrt{{\left({{X_i}+{Y_i}+ 1}\right)\left({{W_i}+ 1}\right)}}}}{{\sqrt{\left({{X_i}+{Y_i}+ 1}\right)\left({{W_i}+ 1}\right)}+ \sqrt{Z_i +W_i+1}}}
\Omega_{i_{1}i_{2}\cdot\cdot\cdot i_{2m-1}}^{(2m-1)}=f_{i_{1}[i_{2}}^{j_{1}}\cdot\cdot\cdot f_{i_{2m-3}i_{2m-2}]}^{j_{m-1}}k_{j_{1}\cdot\cdot\cdot j_{m-1}i_{2m-1}}^{(m)}
\int_0^{\tau}x^n e^{-px}dx = p^{-n-1}\gamma(n+1, p \tau), [n \geq 0], &\int_{\tau}^{\infty}\frac{e^{-px}dx}{x^{n+1}}= \frac{p^{n}\textbf{Ei}(-p\tau)}{n! (-1)^{n+1}}\!\!-\!\! \frac{e^{\!-\!p\tau}}{\tau^{n}}\sum_{k=0}^{n-1}\frac{(-1)^kp^k \tau^k}{n(n\!-\!1)\dots (n\!-\!k)}, &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~[n > 0].
\Gamma_{R ( \tau )}^{( 0 , 2 )}) = ( \kappa_{\tau}^{- 2 \tau})^{\frac{\epsilon_{L}}{2}}( C_{1 \tau}( \frac{t}{\kappa_{\tau}^{2 \tau}})^{- \alpha_{\tau}}+ \frac{\nu_{\tau}}{\tau \nu_{\tau}( d - \frac{m}{2}) - 2}B_{\tau}( u_{\tau}^{*}) ) .
{\cal L}^{\mathrm{g a u g e}}\ = \ -{\textstyle{\frac{i}{8}}}\left({\cal N}_{I J}F_{\mu \nu}^{+ I}F^{+ \mu \nu J}\-{\bar{\cal N}}_{I J}F_{\mu \nu}^{- I}F^{- \mu \nu J}\right) ,
R_{n\ell}(p) = \sqrt{\frac{2 (2n'+a+b+1)\, n'!\, \Gamma(n'+a+b+1)}{\Gamma(n'+a+1) \Gamma(n'+b+1)}}\,(\beta+\beta')^{D/4}\, c^{\lambda+\delta}\, s^\ell\, P_{n'}^{(a,b)}(z)\;,
C_n(q,t) = \sum_{\mu \vdash n}\dfrac{t^{2\sum l}q^{2\sum a}(1-t)(1-q) \prod_{}^{0,0}(1-q^{a'}t^{l'}) \sum q^{a'}t^{l'}}{\prod (q^{a}- t^{l +1})(t^{l}-q^{a+1})}
d_1= \begin{pmatrix}0 i\mu(g^{\alpha\beta}\xi_\beta) 0 i\lambda (\xi_\beta) 0 0 0 0 0 \end{pmatrix}, d_0= \begin{pmatrix}0 0 0 \lambda (\Gamma^\alpha_{\alpha\beta}) \lambda \Gamma^\alpha_{\alpha n}-\beta 0 0 0 \end{pmatrix}.
{\cal T}( y ) = \left( \begin{array}{c c c c}{{\mathrm{e}^{i \varphi_{1}}}}&{{}}&{{}}&{{}}\\{{}}&{{\ddots}}&{{}}&{{}}\\{{}}&{{}}&{{\ddots}}&{{}}\\{{}}&{{}}&{{}}&{{\mathrm{e}^{i \varphi_{N}}}}\\ \end{array}\right) , \quad \mathrm{w i t h}\quad \varphi_{i}\equiv \sum_{b = 1}^{n}{\frac{h_{b , i}}{L^{b}}}y^{b}, \quad \sum_{i = 1}^{N}h_{b , i}= 0 .
+ \left[ - \frac{1}{| \mu | \varrho}K_{1}( | \mu | \varrho ) + \frac{1}{( \mu \varrho )^{2}}- \frac{\pi}{\mu^{2}}\delta^{2}( \vec{\varrho}) \right] \frac{( \vec{a}\cdot \vec{b})}{2 \pi}~ ~ ,
\int d^4q{1\over [q^2]^{1+n\epsilon}(q-p)^2}={\Gamma(1-\epsilon)\Gamma((n+1)\epsilon)\Gamma(1-(n+1)\epsilon)\over\Gamma(1+n\epsilon)\Gamma(2-(n+2)\epsilon)}{1\over [p^2]^{(n+1)\epsilon}}
\tilde{a}_{\tilde{n}}^{\alpha}= \int_{0}^{\pi}\frac{d \sigma_{-}}{\pi}\exp \left[ 4 i \tilde{n}\frac{e_{\mu}X_{R}^{\mu}( \sigma_{-})}{e_{\mu}P_{R}^{\mu}}\right] \xi_{i}^{\alpha}\partial_{-}X_{R}^{i}( \sigma_{-})
\Gamma(S_i) \le \left(\frac{5e^2\Delta(S_i)}{b^2}\right)^{\kappa}b_1 (3\log_2|S_i|)^{1+\kappa}= b_2 \Delta(S_i)^\kappa (\log_2 |S_i|)^{1+\kappa}\le b_2 |S_i|^\kappa (\log_2 |S_i|)^{1+\kappa},
\bar{\zeta}( s ) = \int_{- \infty}^{\infty}\frac{d k}{2 \pi}\int_{0}^{\infty}\frac{\lambda \, d \lambda}{( k^{2}+ \lambda^{2})^{s}}= \frac{1}{2 \sqrt{\pi}}\frac{\Gamma ( s - 1 / 2 )}{\Gamma ( s )}\int_{0}^{\infty}\lambda^{2 - 2 s}d \lambda \,{.}
s=\sum_{n=0}^{N-1}(-1)^{n}e^{i\Delta k\Lambda n}=1-e^{i\Delta k\Lambda}+e^{i2\Delta k\Lambda}-e^{i3\Delta k\Lambda}+...+(-1)^{N}e^{i\Delta k\Lambda(N-2)}-(-1)^{N}e^{i\Delta k\Lambda(N-1)}.
g_1=\begin{bmatrix}T_1 \cdots T_n\end{bmatrix}\varphi_1=\begin{bmatrix}T_1 \cdots T_n\end{bmatrix}\rho^i\lambda=\begin{bmatrix}q^i_1 q^i_2\end{bmatrix}\lambda=c(p^i_1q^i_2-p^i_2q^i_1).
\Delta (C_0,C_0) = \frac{q-1}{2}\{0\}+(0,0)C_0+(1,1)C_1 & =\frac{q-1}{2}\{0\}+\frac{q-5}{4}C_0+\frac{q-1}{4}(\mathbb{F}_q-C_0-\{0\})
{_{0}\mathcal{F}_0}^{(d)}\left( \begin{matrix}\end{matrix};\mathbf{z},\mathbf{u}\right)\prod_{i=1}^{n}\frac{|\mathbf{u}|}{e^{\omega_{i}|\mathbf{u}|}-1}= \sum_{\mathbf{m}\in \mathcal{P}}B_{n, \mathbf{m}}^{(d)}\left(\mathbf{z}\mid{\boldsymbol{\omega}}\right) \Psi_{\mathbf{m}}^{(d)}(\mathbf{u})
{}I_4 \le \frac{c}{(\prod_{l = 1}^d h_l^*)^2}\frac{1}{T_n^2}\int_0^{T_n}\int_{D}^{T_n}e^{- \rho s'}e^{\rho \Delta_n}dt ds'\le \frac{c}{T_n(\prod_{l = 1}^d h_l^*)^2}e^{- \rho D}
\begin{aligned}\omega_n^{(\alpha,\sigma)}\approx \widehat{\omega}_n^{(\alpha,\sigma)}={\tau^{1+\alpha}e^{-n\sigma\tau}}\Delta{x}\sum_{j=-\infty}^{\infty}(1+e^{x_j}\tau)^{-1-n}\phi(x_j) ={\tau^{1+\alpha}e^{-n\sigma\tau}}\sum_{j=-\infty}^{\infty}w_j(1+\lambda_j\tau)^{-1-n},\end{aligned}
q^{1}=\frac{1}{4}(u^{1}+u^{2}+u^{3})^{2}-(u^{1}u^{2}+u^{1}u^{3}+u^{2}u^{3})-\frac{1}{4}(u^{6})^{2}-u^{4}u^{5}, &q^{2}=\frac{2u^{1}u^{2}u^{3}+(u^{1}+u^{2}+u^{3}-u^{6})q^{1}}{2u^{5}}, &q^{3}=\frac{1}{2}(u^{1}+u^{2}+u^{3}+u^{6}), &q^{4}=u^{4}, &q^{5}=u^{5}, &q^{6}=\frac{1}{2}(u^{1}+u^{2}+u^{3}-u^{6}).
\left( \mu_{3}^{A}, \bar{\Phi}_{B}^{( 2 )}\right)_{1}= \delta_{B}^{A}, \; \left( \bar{\Phi}_{A}^{( 3 )}, \mu_{2}^{B}\right)_{1}= \delta_{A}^{B}, \; \left( \nu_{1}^{A}, \tilde{\Phi}_{B}\right)_{1}= \delta_{B}^{A}.
I = \int_{t_{0}}^{t_{1}}\int_{0}^{1}\Big [ x ( y^{\prime}\dot{z}- \dot{y}z^{\prime}) - H_{1}\Big ( \frac{\partial H_{2}}{\partial x}x^{\prime}+ \frac{\partial H_{2}}{\partial y}y^{\prime}+ \frac{\partial H_{2}}{\partial z}z^{\prime}\Big ) \Big ] d s \, d t .
-8\pi\nabla\left(\log\left|x\right|^{2N}\right)(z_i)=-2\pi\left[4\beta_0\frac{z_{i}}{\left|z_i\right|^{2}}+16\sum^{n}_{i\neq j=1}\frac{z_{i}-z_{j}}{\left|z_i-z_j\right|^{2}}\right],
G_3 =\{f_1,f_2,f_3\}G_4 =\{f_1,f_2,f_3,f_4,x_0x_2^2\}G_5 =\{f_1,f_2,f_3,f_4,f_5,x_0x_2x_3\}G_6 =\{f_1,\ldots, f_6, x_0x_3^2+2 x_0x_2x_4, 2 x_1x_3^2+3 x_0x_3x_4-x_0x_2x_5\}.
\begin{multlined}q(\varphi^y,\varphi^\ast(y)x\varphi^\ast(y)^{-1}) = q(\mathrm{id},y)q(\varphi,x)q(\mathrm{id},\varphi^\ast(y)^{-1})= q(\mathrm{id},y) q(\varphi,\varphi^\ast(y)^{-1})= \varphi^y\ ,\end{multlined}
a_{11}=a_{12},a_{ii}=\max\{a_{i-1,i},a_{i,i+1}\}i=2,\ldots, n-1,a_{nn}=a_{n-1,n}, a_{ik}=\min\{a_{i,k-1},a_{i+1,k}\}i,k1\leq i < k-1 \leq n-1, a_{ki}=a_{ik}i,k i>k.
\left\Vert f\right\Vert^{2}=\left\vert I^{\ast}f(o)\right\vert^{2}+\sum_{z>o}\omega(z)\left\vert I^{\ast}(c_{x})(z)\right\vert^{2},\text{and}\left\Vert f\right\Vert^{2}=\left\vert I^{\ast}f(o)\right\vert^{2}+\sum_{z>o}\omega(z)^{-1}\left\vert Df(z)\right\vert^{2}.
\sum_{j = 1}^\aleph\left\langle Q_j, V_j\right\rangle-\eta \frac{36}{\pi^2 d^6}\int_{\Omega}\int_{\Omega}\left\langle [\mathbf{P}]\cdot\Upsilon^0(x, z)\cdot \Lambda(x), \Pi(z)\right\rangle\,dx\,dz-\frac{2\eta}{\pi d^3}\sum_{m=1}^\aleph|V_m|^2-\eta \mathcal{R}^k= \sum_{j = 1}^\aleph\left\langle J_j, V_j\right\rangle.
B = \overline{\operatorname{span}}\, \left\{\tau_{i(1)}(\xi_1) \cdots \tau_{i(k)}(\xi_k) \pi_j(a) \tau_{i'(1)}(\eta_l)^* \cdots \tau_{i'(l)}(\eta_l)^* u^m : \right. \, \left. \xi_r, \eta_s \in H, a \in A, i(r), i'(s) \in \mathbb{Z}, 1 \leq r \leq k, 1 \leq s \leq l, j, k, l, m \in \mathbb{Z}\right\}.
\exp\bigg[\epsilon^{-2}(t_{k}-t_{k-1})(\Lambda(\bar \alpha_{k,k})-\delta)\bigg] \le \mathbb E\left(e^{\Delta_k}\Big|\mathcal{F}_{t_{k-1}}\right) \le \exp \bigg[\epsilon^{-2}(t_{k}-t_{k-1})(\Lambda(\bar \alpha_{k,k})+\delta)\bigg].
\Big(1+ 2^{p-1}\displaystyle\sum_{n= 3}^{\infty}\frac{1}{n^p}\Big)\langle f(A)\hat{k}_\lambda, \hat{k}_\lambda \rangle^p< \Big(\frac{p}{p-1}\Big)^{p}\langle f(A)^p\hat{k}_\lambda, \hat{k}_\lambda \rangle.
B_{n+1}(v,w,t,q^{\frac{1}{2}})=B_{n}(v,w,t,q^{\frac{1}{2}})\cdot (1-x y^{n})^{-1}, D_{n+1}(v,w,t,q^{\frac{1}{2}})=D_{n}(v,w,t,q^{\frac{1}{2}}) \cdot \prod_{k=0}^{n-1}(1-(q^{\frac{1}{2}})^{1-n+2k}x y^{n})^{-1}, & x=\exp(-2\pi iv/ t), y=\exp(- 2\pi i w/t).
=b_{0}(1+\frac{\alpha\beta}{1!\times(-2)}x+\frac{\alpha(\alpha+1)\beta(\beta+1)}{2!\times(-2)\times(-1)}x^{2})=b_{0}\sum_{r=0}^{3-1}\frac{(\alpha)_{r}(\beta)_{r}}{r!(1-3)_{r}}x^{r}.
\begin{matrix}{G_{\infty}}&{}&{\rightarrow}&{L_{\infty}}\\{}&{d}&{[ \cdot , \cdot ]}&{[ \cdot , \cdot , \cdot ]}&{\cdots}\\{\downarrow}&{\cdot}\\{\mathrm{c o m m .}A_{\infty}}&{\{\cdot , \cdot , \cdot \}}\\{}&{\vdots}\\ \end{matrix}.
V_{e f}\left( \phi_{c}\right) = V_{c l}\left( \phi_{c}\right) + \frac{1}{2}\frac{\hbar}{\Omega}\ln \det \left[ \frac{\delta^{2}S \left[ \phi_{c}\right]}{\delta \phi \left( x \right) \delta \phi \left( y \right)}\right] = V_{c l}\left( \phi_{c}\right) + V_{e f}^{\left( 1 \right)}\left( \phi_{c}\right) ,
\partial^{2}_{\vec{z}}H_{F}+H_{3}\partial^{2}_{\vec{x}}H_{F}=q_F\delta({\vec{x}-\vec{x}_0})\delta(\vec{z}-\vec{z}_0), \ \ \ \partial^2_{\vec{z}}H_3={{q_3}\over{V_y}}\delta({\vec{z}-\vec{z}_0}),
d^Cf-\iota_\xi\omega_1 = -xX_1f dx + (\partial_xf-\psi)x^{-1}\theta^1 - X_3(f-x\psi)\theta^2 + X_2(f-x\psi)\theta^3, \imath_\xi\omega^c = - x(X_2+iX_3)\psi (dx + i x^{-1}\theta^1) + \psi (\theta^2+i\theta^3) ,
d_{\mathcal M}(\mu_1,\mu_2) = \sup \left\{\left| \int_{Y}\phi \,d\mu_1 - \int_{Y}\phi \,d\mu_2 \right| : \phi\in\mathrm{Lip}_1(Y), \,\sup_{x\in Y}|\phi(x)|\le 1 \right\}, d_{\mathcal M}^*(\mu_1,\mu_2) = \sup \left\{\left| \int_{Y}\phi \,d\mu_1 - \int_{Y}\phi \,d\mu_2 \right| : \phi\in\mathrm{Lip}_1(Y) \right\},
L_{p}(\mathbf{f}_x,\alpha(x),\epsilon\omega^{k/2-1},s-1) =L_{p}(\mathbf{f}_x,\alpha(x),\epsilon\omega^{k/2-1}\chi_{s-1}) &=L_{p}(\mathbf{f}_x,\alpha(x),\epsilon\omega^{k/2-s}\chi^{s-1})
\frac{1}{2}\frac{W_{0}}{R^{\beta}}= \frac{W_{0}/2}{(\delta a_{\beta})^{\beta}}= \delta^{-\beta}\left(\frac{a_{\beta}}{\Lambda_{\beta}}\right)^{\beta-2}a_{\beta}^{-\beta}\ge \frac{a_{\beta}}{|Q|}\frac{3^{-3/2}}{\Lambda_{\beta}}\left(\delta\Lambda_{\beta}\right)^{3-\beta}.
\begin{aligned}p(P) =\frac{2}{\pi R^{2}}\int_{0}^{R}\int_{0}^{\pi}p(r',\theta')r'dr'd\theta' &+\frac{4}{\pi R^{2}}\int_{0}^{R}\int_{0}^{r'}\int_{0}^{\pi}\frac{(\tilde{u}-u)v_{\theta'}}{\rho}r'd\theta'd\rho dr' \doteq I_{1}+I_{2}.\end{aligned}
\begin{array}{l}u_1 = \frac{{\cos \psi}}{{k_1}}\left({\frac{{k_4 k_2}}{{k_3}}+ f'}\right) u_2 = - \cos \psi f u_3 = \cos \psi \frac{{k_4}}{{k_3}}u_5 = \cos \psi = cons\tan t. \end{array}
\min_{0{}\leq{}i{}\leq{}N}\|\nabla f(x_i)\|^2{}\leq{}\frac{2L\,\big(f(x_0)-f_*\big)}{1 + \sum\limits_{i=0}^{N-1}\left(2 h_i - h_i^2 \right)}\, , \, \forall h_i \in (0,2)
V_{e f f}= \left[ e^{2}a^{i}a_{i}- \frac{1}{2 \alpha^{\prime}}+ \left( e \phi_{a}- \frac{\delta^{a}}{2 \pi \alpha^{\prime}}\right)^{2}\right] T \overline{{{T}}}+ \frac{e^{2}}{2}\left( T \overline{{{T}}}\right)^{2}
\frac{\sqrt{\frac{u}{f(u)}}}{\phi(u)}=\frac{\frac{1}{2}\sqrt{\frac{f(t)}{t}}\cdot\frac{f(t)-tf'(t)}{f(t)^2}}{\phi'(t)}\lessgtr \frac{f(t)-tf'(t)}{-f(t)}= \frac{tf'(t)}{f(t)}-1
\mathfrak{B}_{\mathcal{E}_n(q)}= \left \{\epsilon_I (b_{i_1}\ldots b_{k_1}) \ldots ( b_{i_p}\ldots b_{k_p}) \, \vert \, I\in P(n), \ 1 \leq i_1 < \ldots < i_p \leq n-1, \k_j \leq i_j , \ 1 \leq j \leq n-1\right \}.
{\cal H}^{(2)}= \frac{2}{l}\gamma_{\mu \nu}\frac{\delta I^{(2)}}{\delta \gamma_{\mu \nu}}-\frac{\sqrt{-\gamma}}{2 \kappa}R + \frac{1}{2}\sqrt{-\gamma}\gamma^{\mu \nu}\partial_{\nu}\phi \partial_{\mu}\phi=0.
|v_j(t)| \leq 1 - \frac{\sup K - \inf K}{2\|K'\|_{\infty}}+ \frac{1}{2\kappa_j \|K'\|_\infty}\Big|\int_{\kappa_j \inf K}^{\kappa_j \sup K}e^{itu}du \Big| &=1 - \frac{\sup K - \inf K}{2\|K'\|_{\infty}}+ \Big|\frac{\sin(t\kappa_j(\sup K - \inf K)/2)}{t\kappa_j \|K'\|_\infty}\Big|.
\|v\|_{L^2(\Omega)}\geq \left(\int_{B_{\delta/2}(p)}v^2\right)^{1/2}= \left(\int_{B_{\delta/2}}v(p+x)^2\right)^{1/2}&\geq \left(\int_{B_{\delta/2}}v(p)^2\chi_E/4\right)^{1/2}\geq \min\{|B_{\delta/2}|,|E|\}^{1/2}v(p)/2.
{\overline{{Q}}}_{a}={\overline{\lambda}}_{a}^{\mu}P_{\mu}- 2 i G_{\mu \nu}{\overline{\lambda}}_{a}^{\mu}m^{\nu}+ i \Gamma_{\mu \nu \rho}{\overline{\lambda}}_{a}^{\mu}{\overline{\lambda}}^{\nu}\lambda^{\rho},
A(p+\frac{q}{2}, p-\frac{q}{2}) \:=\: -\sum_{n=0}^\infty \frac{(-1)^n}{n!}\left(\frac{q^2}{4}\right)^n \sum_{k=0}^\infty \frac{1}{(2k)!}\sum_{l=0}^n \frac{(-1)^l}{2k+2l+1}\: \left(\!\! \begin{array}{c}n l \end{array}\!\!\right) \: \left(\frac{q^j}{2}\frac{\partial}{\partial p^j}\right)^{2k}T_a^{(n+1)}(p) \;\; .
[\begin{matrix}e_{1}^{\prime}(t)\\e_{2}^{\prime}(t)\\ \end{matrix}]=\|\gamma^{\prime}(t)\|[\begin{matrix}0&\kappa(t)\\ -\kappa(t)&0\\ \end{matrix}][\begin{matrix}e_{1}(t)\\e_{2}(t)\\ \end{matrix}]
\sup_{Q_{y,x}}N(\epsilon\lVert F\circ\phi_{y}\rVert_{L_{2}(Q_{y,x})}, \mathcal{F}\circ\phi_{y}, L_{2}(Q_{y,x}))= &=\sup_{R_{y,x}}N(\epsilon\lVert (F\circ\phi_{y})/(w_{\theta}\circ\phi_{x})\rVert_{L_{2}(R_{y,x})}, \mathcal{F}/w_{\theta}, L_{2}(R_{y,x}))
\bar{I}^{( 2 l - 2 \varepsilon )}= \sum_{j = 0}^{l}\left( \frac{m^{2}}{4 m^{2}- q^{2}}\right)^{j - \varepsilon}\frac{\Gamma ( 2 j - 2 \varepsilon - 3 ) \Gamma ( 2 - j + \varepsilon )}{i^{2 j - 2 \varepsilon}\Gamma ( j - 1 - \varepsilon )}+ C_{\varepsilon}^{l},
\begin{array}{r c l}{{\alpha^{\prime}}}&{{\sim}}&{{\varepsilon^{1 / 2}\to 0 ~ ,}}\\{{g}}&{{\sim}}&{{\varepsilon \to 0 ~ ,}}\\{{| b_{I}|}}&{{\sim}}&{{\varepsilon^{- 1 / 2}\to \infty ~ .}}\\ \end{array}
\sum_{k = 0}^{s}\big\|\partial_{t}^{k}z(t, \cdot)\big\|_{H^{s - k}(\Omega)}^{2}+\sum_{k = 0}^{s-2}\big\|\partial_{t}^{k}\theta (t, \cdot)\big\|_{H^{s + 1 - k}(\Omega)}^{2}+\big\|\partial_{t}^{s-1}\theta(t, \cdot)\big\|_{H^{1}(\Omega)}^{2}\to \infty t\nearrow T_{\mathrm{max}}.
(\partial_{2}^{2}\phi(r,\xi))_{0}= 2 \left(a''(\xi) \psi^{+}(r,\xi) + 2 a'(\xi) \partial_{2}\psi^{+}(r,\xi)\right) &+2 \left(a(\xi)\left(\partial_{\xi}^{2}\left(\frac{e^{i r \sqrt{\xi}}}{\xi^{1/4}}\sigma(r\sqrt{\xi},r)\right)+\frac{r^{2}}{4 \xi^{5/4}}e^{ir\sqrt{\xi}}\sigma(r\sqrt{\xi},r)\right)\right)
s_{1B}(k)=\sum_{j=0}^{k}(\begin{matrix}2j\\j\end{matrix})(\begin{matrix}3j\\j\end{matrix})(\begin{matrix}6j\\ 3j\end{matrix})(\begin{matrix}k+j\\k-j\end{matrix})(-432)^{k-j}=1,-312,114264,-44196288,...
| f_{L}, f_{R}\rangle = e^{f_{L}s_{L}^{\dag}}| I \rangle \star | \Xi \rangle \star e^{f_{R}s_{R}^{\dag}}| I \rangle = e^{f_{L}A^{\dag}}| \Xi \rangle \langle \Xi | e^{f_{R}A}\equiv | f_{L}\rangle \langle f_{R}| .
rsin\frac{x}{r}=x-x\cdot\frac{x^{2}}{(2^{2}+2)r^{2}}+x\cdot\frac{x^{2}}{(2^{2}+2)r^{2}}\cdot\frac{x^{2}}{(4^{2}+4)r^{2}}-\cdot
\begin{aligned}\limsup_{k\to\infty}\Bigg\{\left| ||b_{n_k}||_{L^{2}(B(z_ke_z,R_0))}^{2}-\alpha \right|+\left| ||b_{n_k}||_{L^{2}(\mathbb{R}^{3}\backslash B(z_ke_z,R_k))}^{2}-(l-\alpha)\right| &+||b_{n_k}||_{L^{2}( B(z_ke_z,R_k)\backslash B(z_ke_z,R_0))}^{2}\Bigg\}\leq \varepsilon. \end{aligned}
-r^2 \, dt^2 +{dr^2 \over r^2}+ \cos^2\theta \, d\psi^2+ d\theta^2 + \sin^2\theta \, d\Omega_3^2 + r^2 \, dy_1^2 +{r^2 \over 1 + a^4 r^4}\, dy_2^2
\big[{\hat{R}_\mu},{\hat{R}_\nu}\big]=i\big(a_\mu{\hat{R}_\nu}-a_\nu{\hat{R}_\mu}\big)=iC_{\mu\nu\alpha}{\hat{R}_\alpha}, \big[\hat{x}_\mu\otimes1,{\hat{R}_\nu}\big]=i\big(a_\mu{\hat{R}_\nu}-a_\nu{\hat{R}_\mu}\big), \big[O_{\mu\alpha}\otimes\hat{x}^\alpha,{\hat{R}_\nu}\big]=0, \big[\bigtriangleup p^{\rm L}_\mu,{\hat{R}_\nu}\big]=0
= - \, \delta_{\epsilon , \bar{\epsilon}}\langle X \rangle = \frac{1}{2 \pi i}\oint_{C}d z \, \epsilon ( z ) \langle T ( z ) X \rangle - \frac{1}{2 \pi i}\oint_{C}d \bar{z}\, \bar{\epsilon}( \bar{z}) \langle \bar{T}( \bar{z}) X \rangle
\hat{U}^{+}= \lim_{\tau \to - \infty}e^{\frac{i}{\hbar}\, \hat{H}_{0}\tau}e^{- \frac{i}{\hbar}\, \hat{H}\tau}= \int_{0}^{+ \infty}d k \, | k \rangle \, \langle \Psi_{k}^{-}| .
B^{(1)}= \Omega\bigg( \ln \frac{\mathsf{Num}\cdot d}{\xi}\bigg)^{d/2}; \ \delta^{(1)}= \bigg( \frac{\xi}{d \cdot B^{(1)}\cdot \sqrt{\mathsf{Num}}\cdot \mathsf{Coeff}^{1/d}}\bigg)^d; \c_{(1)}= \frac{\mathsf{Num}}{\delta^{(1)}\cdot \sqrt{\xi}},
\times \exp \bigg ( \; - \frac{i}{2}l^{2}e^{2}\; 2 \mathrm{R e}\Big ( \varepsilon_{\mu \nu}\partial_{\nu}j_{\mu}^{( L )}\; , \; \tilde{Q}\; [ \delta ( - L , 0 ) - \delta ( L , 0 ) ] \; \Big ) \; \bigg ) \; .
T^{*}\geq \frac{1}{10(q-1)}\bigg(\frac{b_{1}N_{\alpha}}{M_{0}^{q-1}\,C\, |\Gamma_1|^{\alpha}\,t_{*}^{N_{\alpha}}}-9q\bigg)\,t_{*}&=\frac{9q}{10(q-1)}\bigg(\frac{C_{1}N_{\alpha}}{q\,M_{0}^{q-1}\,|\Gamma_{1}|^{\alpha}}t_{*}^{1-N_{\alpha}}-t_{*}\bigg),
\ddot{v}(t) - \div (A^{(4)}\nabla v (t) ) +{p}(t) \cdot \nabla v(t) - 2{q}(t) \cdot \nabla \dot{v}(t) ={g}(t) \hbox{in}\Omega^{(4)}\setminus \Gamma^{(4)}(t_0)\,,
G(\theta)= \mathbb{E}_{\Phi|\Phi(\mathcal{A})>0}\left\{\left(\sum_{m=1}^{M}C_{M}^m(-1)^{m+1}\right.\right. \left.\left.\prod\limits_{x_{i}\in\Phi\backslash\{x_1\}\cap{\mathcal{A}}}\frac{1}{\left(1+\frac{m\eta\theta r_1^{\alpha}}{Mr_i^{\alpha}}\right)^M}\right)^b\right\},
\bigvee_{j = 1}^{i-1}(-u_i + z'_j) \vee (-u_{i+1}+ 1 + z'_i) \leq \bigvee_{j = i+1}^{n-1}(-u_j + z'_j) \vee (-u_n) \enspace i = 1, \dots, n-2 \, .
\theta'(e_{12}')=\frac1d\left(\frac{f_o\mathrm{d}h_o-h_o\mathrm{d}f_o}{\mathrm{d}\log \frac{t-\lambda}{t-1}}+f_o^2 \left(\frac{t-a}{\lambda-1}\right)^p\right)\cdot \left(e'_{11}\otimes \mathrm{d}\log \frac{t-\lambda}{t-1}\right)
p=m_{0}v+\frac{1}{2}\frac{m_{0}v^{2}v}{c^{2}}+\frac{3}{8}\frac{m_{0}v^{4}v}{c^{4}}+\frac{5}{16}\frac{m_{0}v^{6}v}{c^{6}}+\cdot\cdot\cdot.
0\preceq\Phi\left(\begin{pmatrix}1 \alpha c \beta d \alpha c \alpha^2c^2 \alpha \beta c d \beta d \alpha \beta c d \beta^2d^2\end{pmatrix}\right)=\Phi\left(\sum_{0\leq i+j\le 2}\alpha^i\beta^jB_{i,j}\right)=\sum_{0\leq i+j\le 2}\alpha^i\beta^j\Phi(B_{i,j})=\hat{Q}_{\Phi}(\alpha,\beta).
\binom{z}{\mu}_{\!\!\!q,t}:= \dfrac{q^{|\mu|}t^{2n(\mu)+(1-n)|\mu|}}{(qt^{n-1})_\mu}\prod_{1\leq i<j \leq n}\left\{\dfrac{(qt^{j-i})_{\mu_i-\mu_j}}{(qt^{j-i-1})_{\mu_i-\mu_j}}\right\}w_\mu(q^z t^{\delta(n)}; q, t)
\Psi_{2 2}( r , \theta^{\prime}\pm{\frac{\pi}{\alpha}}, t ) = -{\frac{1}{2}}\sqrt{{\frac{i}{r}}}\, \,{\frac{1}{\sqrt{2 \pi k_{0}}}}\, \, \cot{\frac{\pi}{\alpha}}\, \, \Psi_{\mathrm{{f r e e}}}( r , \alpha^{2}t ) ,
\widehat{u}_{m}\, :=\, \Big( \int^{T}_{0}X_{t}^{2}{\mathrm d}t \Big)^{-1}\cdot \Big( \int^{T}_{0}X_{t}^{2}{\mathrm d}t + \int^{T}_{0}X_{t}{\mathrm d}X_{t}\Big) \, =\, 1 - \Big( 2 \int^{T}_{0}X_{t}^{2}{\mathrm d}t \Big)^{-1}\big( T - X_{T}^{2}\big) \, .
f(\eta) = \frac{F_Q\, \Gamma((d-1)/2)}{2\, \pi^{(d+1)/2}}\, \bigg(\frac{1+\cos\alpha}{1+\cos\eta}\bigg)^{\frac{d-1}{2}}\bigg(\frac{1+\cos\alpha}{\cos\eta-\cos\alpha}\bigg)^{\frac{1}{2}}\,{}_2 F_1\bigg (1, \frac{d-1}{2};\frac{1}{2}; \frac{\cos\eta-\cos\alpha}{1+\cos\eta}\bigg) + F(\eta),
\begin{array}{l}\Phi_{2}=(\phi_{2}^{+})^{-1}\{D_{2}\}_{p,q}, \bar{\Phi}_{2}=\phi_{2}^{+}, \Phi_{1}=(\phi_{1}^{+})^{-1}\{D_{1}\}_{p,q}p^{D_{2}}, \bar{\Phi}_{1}=\phi_{1}^{+}q^{D_{2}}.\end{array}
\Gamma_f(u/\widetilde{q})={\bf P}_-\left(\frac{pu}{q\widetilde{q}}\right)={\bf P}_-\left(\frac{pua}{q}+\frac{pub}{\widetilde{q}}\right)={\bf P}_-\left(\frac{pua}{q}\right)=\frac{R_q(pua)}{q},
p = p_1 \wedge p_2 \wedge \dots \wedge p_n \in \tau_1(q_1) \vee \tau_2(q_2) \vee \dots \vee \tau_n(q_n) \subseteq \tau_1(q) \vee \tau_2(q) \vee \dots \vee \tau_n(q) \subseteq \bigl(\bigvee T\bigr)(q)
(\hat{\mu}_0,\hat{c}_0)=(\Lambda[0])^{(-1)}({d_0})\, ,{(\hat{\mu}_j,\hat{c}_j)=\sum_{\substack{j_1,j_2 \in \mathbb{N}j_1+j_2=j}}L_{j_1}(d_{j_2})}\forall j \in \mathbb{N}\setminus \{0\}\, ,
d\left(\overline{\xi}_{1,0}\overline{\xi}_{1,1}\overline{\xi}_{2,0}\right) = \Delta\left( \overline{\xi}_{1,0}\overline{\xi}_{1,1}\overline{\xi}_{2,0}\right) - \tau\left( \Delta\left( \overline{\xi}_{1,0}\overline{\xi}_{1,1}\overline{\xi}_{2,0}\right)\right) = 0,
-\mathrm{f}''\mathrm{h}- 3\mathrm{f}'\mathrm{h}' = \big(c_2c_3 e^{-c_3x}\big) \Big[c_0c_3+ c_3c_4x- 3c_4- (c_3c_5+ 3c_5c_6)e^{-c_6x}\Big] &\geq \big(c_2c_3 e^{-c_3x}\big) \Big[c_0c_3- 3c_4- (c_3c_5+ 3c_5c_6)\Big] &= \big(c_2c_3 e^{-c_3x}\big) \Big[c_0c_3- 3c_2c_3- c_5c_3\Big]\geq 0