harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\rho_{t, k}(\varphi) \, :=\, \mathbb E_{0}[ Z_{t}^{-1}\varphi (t, \overline{X}_{t, 1}, \ldots , \overline{X}_{t,k}) \vert \mathcal F_{T}^{X}] \, ; k \, =\, 2, \ldots , n \, , \, \, 0 \le t \le T \, ,
m_i(h)=\left(\frac{1+\rho}{\rho}\right)\widehat{h}(i)+\frac{1}{\rho^{i+1}}\sum_{j=0}^{i-1}\widehat{h}(j)\rho^j=\left(\frac{1+\rho}{\rho}\right)\widehat{h}(i)-\frac{1}{\rho^{i+1}}\sum_{j=i}^{\infty}\widehat{h}(j)\rho^j,
{\cal L}= - \frac{1}{4}F_{\mu \nu}\left({1 + e^{2}a \left[{1 - 1 2 \alpha^{2}g^{2}\partial^{2}}\right]}\right) F^{\mu \nu}\, + \frac{\chi}{2}\varepsilon_{\mu \nu \lambda}A^{\mu}\partial^{\nu}A^{\lambda}\, .
\int\frac{d^{3}k}{(2\pi)^{3}}[1-\hat{k}\hat{k}]\frac{exp(ik\cdot r)}{k^{2}+m^{2}}=\frac{1}{2}\frac{e^{-mr}}{4\pi r}\{\frac{2}{(mr)^{2}}(e^{mr}-1)-\frac{2}{mr}\}[1+\hat{r}\hat{r}]
{\mathcal M}^0_{{\mathcal O}_{\dot{K}}}|_{\mathfrak{U}_{Z}}=\bigoplus_{\mu\in X^{*}(T)}\dot{\pi}^{(d+1)\overline{\mu}(Z)}\underline{M}^0_{\mu,{\mathcal{O}}_{\dot{K}}}|_{\mathfrak{U}_{Z}}.
|f(\cdot ,\overline{u},\nabla \overline{u})| \leq M(1+|\overline{u}|^{\alpha}+|\nabla \overline{u}|^{\beta}) & \leq M(1+(\delta^{-p}(Ld)^{\overline{\omega}})^{\alpha}+(\delta^{-p}\hat{L})^{\beta}& \leq C\delta^{-p\max \{\alpha ,\beta \}}
M^u_{\mu}g(x) := \sup_{\{r > 0, y\in X : \d(x,y) < r \mbox{and}\mu B(y, r)^{cl}> 0\}}\frac{1}{\mu(B(y, r)^{cl})}\int_{B(y, r)^{cl}}|g (u)| d\mu (u).
\frac{\alpha_{\varepsilon}/ (e\sigma)}{2\tilde{F}^\varepsilon(\mathbf{x},\tau)/(2+\log\sigma - \tau)}= \frac{\alpha_{\varepsilon}}{e\sigma\tilde{F}^{\varepsilon}(\mathbf{x}, \log\sigma)}= \frac{\alpha_\varepsilon \sqrt{\varepsilon^2+|D{v^\varepsilon}(\mathbf{x})|^2}}{e\sigma}
c+o(1)=\Phi(u_n) =(\frac{1}{2}-\frac{1}{p+1})a(u_n)+(\frac{1}{p+1}-\frac{1}{2^*(s_1)})\lambda_1b(u_n) +(\frac{1}{p+1}-\frac{1}{2^*(s_2)})\lambda_2c(u_n)+o(1)\|u_n\|.
P = (x+4)I + \frac{10(x+6)J}{(x-22)(x+8)}+ A[\mathcal T] Q = \mathbf{u}+\frac{9x+82}{(x-22)(x+8)}\mathbf{1}R = (x+4)+\frac{10(x+6)}{(x-22)(x+8)}= \frac{x^3-10x^2-222x-644}{(x-22)(x+8)}
g\left((X \times Y)\times (Z \times W)\right) =t(t^{-1}g)\left((X \times Y)\times (Z \times W)\right) &=t\left(t^{-1}g(X) \times t^{-1}g(Y)\right) \times \left(t^{-1}g(Z) \times t^{-1}g(W)\right) &= c(g)^{-1}\left(g(X) \times g(Y)\right) \times \left(g(Z) \times g(W)\right).
\begin{array}{c l}{L_{\mu \nu}= - i g \frac{\tau_{i}}{2}W_{\mu \nu}^{i}+ i \frac{g^{\prime}}{2}B_{\mu \nu}; R_{\mu \nu}= i g^{\prime}B_{\mu \nu}}\\{D_{\mu}\phi = ({\partial}_{\mu}- i g \frac{{\tau}_{i}}{2}W_{\mu}^{i}- i \frac{g^{\prime}}{2}B_{\mu}) \phi .}\\ \end{array}
\left(D_{a+}^\alpha y\right)(x):=\frac{1}{\Gamma(1-\alpha)}\left(\frac{d}{dx}\right)^n\underset{a}{\overset{x}{\int}}\frac{y(t)dt}{(x-t)^{\alpha-n+1}},\,(n=\left[Re(\alpha)\right]+1,\,x>a),
{\cal K}^{L}( \sigma = 2 ) = \left( \begin{array}{c c}{- \frac{d^{2}}{d x^{2}}+ 4 - \frac{3}{\cosh^{2}x}}&{\frac{3}{\cosh^{2}x}}\\{\frac{3}{\cosh^{2}x}}&{- \frac{d^{2}}{d x^{2}}+ 4 - \frac{3}{\cosh^{2}x}}\\ \end{array}\right) \qquad ,
\rho_{1}( s ) = \frac{s}{\pi}I m \frac{\Pi_{e}( s )}{\Pi_{e}( s )^{2}s - \Pi_{o}( s )^{2}M^{2}}\hspace{2 c m}\rho_{2}( s ) = \frac{s}{\pi}I m \frac{\Pi_{o}( s )}{\Pi_{e}( s )^{2}s - \Pi_{o}( s )^{2}M^{2}}
{}_{r + 1}\phi_{r}\biggl(\genfrac..{0pt}{}{a,\, b_{1}, \dotsc, b_{r}}{c_{1}, \dotsc, c_{r}}; q,\, x\biggr) = \frac{(b_{1})_{\infty}\dotsm (b_{r})_{\infty}}{(c_{1})_{\infty}\dotsm (c_{r})_{\infty}}\widetilde{\phi}_{D}\biggl(\genfrac..{0pt}{}{x;\, c_{1}/ b_{1}, \dotsc, c_{r}/ b_{r}}{a x}; b_{1}, \dotsc, b_{r}\biggr),
([\delta_{\varepsilon_1},\delta_{\varepsilon_2}]-\delta_{[\varepsilon_1,\varepsilon_2]_Q})A^a= \varepsilon^b_1\varepsilon^c_2(-\nabla^-_\mu (T_{\nabla^-})^a{}_{bc}+2\rho^\nu_{[b|}(R_{\nabla^-})^a{}_{\nu \mu| c]})D_-X^\mu + \varepsilon^b_1\varepsilon^c_2(-\nabla^+_\mu (T_{\nabla^+})^a{}_{bc}+2\rho^\nu_{[b|}(R_{\nabla^+})^a{}_{\nu \mu |c]})D_+X^\mu,
\int_\Omega f \, \overline \varphi \, \pi_i= \sum_{k=1}^d \int_\Omega w_k \, \overline{\partial_k (\varphi \, \pi_i)}+ \lambda \int_\Omega u \, \overline \varphi \, \pi_i= \sum_{k=1}^d \int_\Omega w_k \, \overline{\partial_k \varphi}\, \pi_i + \int_\Omega w_i \, \overline \varphi + \lambda \int_\Omega u \, \overline \varphi \, \pi_i.
U^-(x,t) = \inf_{S \in \mathcal S}\sup_{A\in \mathcal{AC}}\mathbb E\Big[e^{-r(T-t)}g\big(X(T)\big)\Big], U^+(x,t) =\sup_{S \in \mathcal S}\inf_{A\in \mathcal{AC}}\mathbb E\Big[e^{-r(T-t)}g\big(X(T)\big)\Big]
j_{t}^{r}(q):=(2\pi\hbar)^{-n}t^{-n/2}\int_{B(r)}\psi\circ\pi\circ\Phi_{i}(q,p)~\left( \sqrt{d\mathrm{vol}_{g}},\sqrt{\Phi_{i}^{\ast}d\mathrm{vol}_{g}}\right)_{(q,p)}e^{-\left\vert p\right\vert^{2}/2t\hbar}d^{n}p,
I_{21}(y,a)|_{m=0}=\left(\frac{B_{0}}{8\pi^{2}y^{2}}+\frac{B_{1}}{4\pi ya}+\frac{1}{2\pi}\sum_{n=2}^{\infty}\frac{B_{n}}{n!}(\frac{2\pi y}{a})^{n-1}\right).
\int_{\gamma_2}\prod |z-(x_k,0)|^{-\frac{c}{\pi}a_k}\leq \int_{\gamma_2}t_*^{-\sum \frac{c}{\pi}a_k}= |x'_0-x_0| t_*^{-\sum \frac{c}{\pi}a_k}\lesssim \delta^{1-\frac{c}{\pi}\sum a_k},
I_2 :=\int_{D(R, a_2)\setminus D(R, a_3)}f(y; \gamma, \beta, q, \delta, x) \, dy &\asymp R^{\gamma+\alpha-q}\left(F\left(\frac{a_3}{R};\gamma+\alpha-q-1, \beta\right)-F\left(\frac{a_2}{R};\gamma+\alpha-q-1, \beta\right)\right).
\Psi_{\kappa}^{(R)}=\frac1{2^{3/2}\pi}\,e^{\pi\kappa/2}\left\{\Gamma(i\kappa)\left(\frac{mx_+}2\right)^{-i\kappa}+\Gamma(-i\kappa)\left(-\frac{mx_-}2\right)^{i\kappa}+...\right\}.
\displaystyle J(\sigma, \tau) ={\cal D}\mbox{et}(\bar{P}^{\dag}\bar{P})^{\frac{1}{2}}\left[ \det \left( \bar{k}^{i}, ( 1 - \bar{P}\frac{1}{\bar{P}^{\dag}\bar{P}}\bar{P}^{\dag}) \bar{k}^{j}\right) \right]^{\frac{1}{2}}\: .
\big\| (\beta_q \ast \varphi)(t_n) - Q_{n}^{(q)}(\varphi) \big\| \leq 2 \breve{\mu}_{n-1}^{(q)}\int_{0}^{k}\|\varphi_t(s)\|ds + k\sum\limits_{m=2}^{n}\breve{\mu}_{n-m}^{(q)}\int_{t_{m-1}}^{t_{m}}\|\varphi_{tt}(s)\| ds,
\mathbb{S}(R^{\bigstar})=\frac{2}{d^{n+1}}\Big[\frac{d_1d_2\dots d_n}{n+1}+ \frac{W^{(n)}_1}{n}+\dots+\frac{W^{(n)}_s}{n-s+1}+\dots+\frac{W^{(n)}_{n-1}}{2}\Big].
\mathbb{P}_\varphi(\mu^{(T)}=m, X^{(T)}=x^{(T)})\geq \mathbb{P}_\varphi(\mu^{(T)}=m) m_0(\{x_{0}\}) \prod_{t=0}^{T-1}\mathbb{P}_\varphi(X_{t+1}=x_{t+1}| \mu^{(T)}=m ,X^{(t)}=x^{(t)}) >0.
{\cal A}_{0}\left( t ,{\bf x}\right) = \int_{0}^{1}{d \alpha \frac{1}{{1 + e^{2}a \left({1 - 1 2 \alpha^{2}g^{2}\partial^{2}}\right)}}}\frac{{x^{i}\partial_{i}^{\alpha x}\left({- J^{0}\left({\alpha x}\right)}\right)}}{{\nabla_{\alpha x}^{2}- \chi^{2}}}.
\left \|\sum\limits_{j=1}^{\infty}\lambda_{j}a_{j}\right\|_{L_{\omega}^{p}}\leq C\left \|\sum\limits_{j=1}^{\infty}\frac{\lambda_{j}\chi_{Q_{j}}}{\omega(Q_{j})^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}.
W = \sum_{\alpha , \alpha^{\prime}}H_{\alpha , \alpha^{\prime}}V_{\alpha + 1 , \alpha^{\prime}}D_{\alpha + 1 , \alpha^{\prime}+ 1}- \sum_{\alpha , \alpha^{\prime}}H_{\alpha , \alpha^{\prime}+ 1}V_{\alpha , \alpha^{\prime}}D_{\alpha + 1 , \alpha^{\prime}+ 1}.
d_Z(z_1,z_2) < d_{Z,{\delta_1}}(z_1,z_2) \leq d_Z(z_1,q_1)+ d_Z(z_2,q_2) + d_{Z,\delta_1}(q_1,q_2) \leq d_Z(z_1,q_1)+ d_Z(z_2,q_2) + d_{Z}(q_1,q_2) + \varepsilon = d_Z(z_1,z_2)+ \varepsilon,
^{( 3 )}g_{\mu \nu}d x^{\mu}d x^{\nu}= \frac{l^{2}}{4 y^{2}}d y^{2}+ ( \frac{1}{y}g_{( 0 ) i j}+ g_{( 2 ) i j}) d x^{i}d x^{j}+{\cal O}( y ) ,
\nu_s(u_1^{\epsilon_1}v_1^{(p-1)i_1-\epsilon_1}\cdots u_s^{\epsilon_s}v_s^{(p-1)i_s-\epsilon_s})=(-1)^{i_1+\cdots+i_s+\sum_{\ell<k}\epsilon_\ell\epsilon_k}(\lambda_{i_1-1}^{\epsilon_1}\cdots\lambda_{i_s-1}^{\epsilon_s})^*,
\begin{aligned}\mathbb E \exp\left(\sum_{i=1}^k\alpha_i\int_0^{\epsilon^{-2}t_i}\xi(s)ds\right) =\mathbb E \exp\left(\sum^{k}_{\ell=1}\Delta_\ell\right) &=\mathbb E \left[\exp\left(\sum^{k-1}_{\ell=1}\Delta_\ell\right) \mathbb E\left(e^{\Delta_k}\Big|\mathcal{F}_{\epsilon^{-2}t_{k-1}}\right)\right],\end{aligned}
\begin{array}{rcl}\gamma_+ \psi = \mathcal{R^*}\!\left[\gamma_- \psi\right] begin{align*}5pt] \; \gamma_+ \psi(t,x,v) = \gamma_- \psi(t,x,R_x v) \forall\; (t,x,v)\in\Sigma_{+}^{T}, \end{array}
b_4 = \frac{2}{3}\frac{\omega_t^2}{\omega_y^2}\frac{1}{\epsilon_y^2}+ \frac{3\omega_y^2-\omega_t^2}{2\omega_y^2 + 2\omega_t^2}\,,\qquad c_4 = - 2 \frac{\omega_t^2}{\omega_y^2}\frac{1}{\epsilon_y^2}\,.
\left | \iint_{\begin{subarray}{l}\xi=\xi_{1}+\xi_{2}\tau=\tau_{1}+\tau_{2}\end{subarray}}\hat{f}(\xi, \tau)\hat{g}(\xi_1, \tau_1)\hat{h}(-\xi_2,-\tau_2) \right | \lesssim \| f\|_{Y^{0,0,1}}\|g\|_{X^{0,\frac14,1}}\|h\|_{X^{0,\frac12,1}}.
W[{\Phi}]= \frac{1}{2}\int_{-\infty}^{\infty}dx^{+}\int_0^L dx_1^{-}\int_0^L dx_2^{-}{\Phi}^{\star,++}(x_1^{-}, x^{+}|x_2^{-}, x^{+}) ({\pi}_{(1)}^{-}+{\pi}_{(2)}^{-}
f_{\scriptscriptstyle{0 , L}}( a ) = \frac{1}{2 \pi}\int_{0}^{a}d r \int_{0}^{2 \pi}d \theta \, e^{i \theta}\, e^{2 \phi ( r , \theta )}\, \sum_{l = 0}^{\infty}b_{\scriptscriptstyle{l , L}}\, r^{l}\, e^{i l \theta}\, .
\frac{1}{V}<H^{\theta}> = \frac{1}{8\pi}[\frac{4}{3}\Lambda^3+\frac{1}{3}m^3 - \frac{2}{3}(\lambda \eta^2)^{\frac{3}{2}}+\lambda\eta^2 \Lambda] -\frac{1}{4}\frac{g^2}{\pi^2}m^2
\mathcal{B}_\Theta=\displaystyle\left(\bigcup\limits_{i=1}^{r-1}(\mathcal{B}_{i})_1\cup(\mathcal{B}_{i})_2\right)\cup\left(\bigcup\limits_{1\leq n<m\leq r-1}\mathcal{B}_{mn}\right)\cup\left(\bigcup\limits_{\begin{subarray}{c}1\leq i\leq r-1 l_i>1\end{subarray}}\mathcal{B}_{i}\right)
B_{2}=\displaystyle \sum_{\beta \in F^{\times}}\sum_{\substack{X\in S(\beta) t=2}}\Theta_{\theta}\begin{bmatrix}m_{1}X 0 m_{2}\end{bmatrix}\overline{\psi_{A}({m_{1}}^{-1}X})=q^{8}(1-q).
\begin{cases}A_{1}A_{2}A_{3}B_{1}B_{2}B_{3}A_{1}'A_{2}'A_{3}'B_{1}'B_{2}'B_{3}'=1, A_{1}A_{2}A_{3}B_{1}B_{2}B_{3}=A_{1}'A_{2}'A_{3}'B_{1}'B_{2}'B_{3}'=-1, A_{2}B_{3}A_{2}'B_{3}'=A_{3}B_{1}A_{3}'B_{1}'=A_{1}B_{2}A_{1}'B_{2}'=1, A_{3}B_{2}\overline{A_{3}'B_{2}'}=A_{1}B_{3}\overline{A_{1}'B_{3}'}=A_{2}B_{1}\overline{A_{2}'B_{1}'}=1.\end{cases}
\lambda = \langle \Omega_{0}| Z_{1}| \Omega_{0}\rangle = \omega_{0}( \psi ( h_{+}^{( 1 )}) \psi ( h_{-}^{( 1 )}) ) = \langle \Gamma h_{+}^{( 1 )}, S_{\mathrm{N S}}h_{-}^{( 1 )}\rangle .
S^{(m)}_2=\frac{\varphi(VP_f)^{k+1}X(\log R)^{k+1}}{\varphi(V)(VP_f)^{k+1}\log X}J^{(m)}_k(F)+O\left(\frac{F_{max}^2\varphi(VP_f)^{k+1}X(\log X)^k}{\varphi(V)(VP_f)^{k+1}D_0}\right).
\|P_{\delta}(A_{vt,m})e^{-iH_0t}\psi \|=\|P_{\delta}(A_{vt,m})e^{-iH_0t_n}\chi_{S_R}\psi \|\leq \|\chi_{S_R}e^{iH_0t_n}P_{\delta}(A_{vt,m})\|_\textrm{op}\|\psi\|\xrightarrow{t\rightarrow \infty}0
P(dx_3d[x_1,x_2]) + P(dx_3[dx_2,x_1]) + P(dx_3[x_2,dx_1]) = P(dx_3(d[x_1,x_2] + [dx_2,x_1] + [x_2,dx_1])) = P(dx_2(d[x_1,x_2] + d[x_2,x_1])) = 0.
\begin{array}{lr}d_{n+1}^jd^i_n=d^i_{n+1}d_n^{j-1}i<j begin{align*}1.1ex]\varepsilon_n^j\varepsilon_{n+1}^i=\varepsilon_n^i\varepsilon_{n+1}^{j+1}i\leqslant j\end{array}
d s^{2}= - e^{- 2 \phi ( r )}\left( 1 - b ( r ) / r \right) d t^{2}+{\frac{d r^{2}}{\left( 1 - b ( r ) / r \right)}}+ r^{2}( d \theta^{2}+ \sin^{2}\theta d \varphi^{2}) .
x^T(t)P^{-1}x(t)= \int_0^t x^T(s)P^{-1}dx(s)+\int_0^t dx^T(s) P^{-1}x(s) = \int_0^t x^T(s) P^{-1}A x(s) ds+\int_0^t x^T(s)P^{-1}B u(s) ds +\int_0^t x^T(s) A^T P^{-1}x(s)ds+\int_0^t u^T(s)B^T P^{-1}x(s)ds +\sum_{i=1}^m \left(\int_0^t x^T(s) P^{-1}N_i x(s) u_i(s) ds+\int_0^t x^T(s) u_i(s)N_i^T P^{-1}x(s)ds\right).
x\Big(u\ |\ (1-xH_u^2)^{-1}(u)\Big)=x\Big(1\ |\ (1-xH_u^2)^{-1}H_u^2(1)\Big) &\qquad=\sum\limits_{n\geq0}x^{n+1}\Big(H_u^{2(n+1)}(1)\ |\ 1\Big)=\sum\limits_{n\ge0}x^{n}\Big(H_u^{2n}(1)\ |\ 1\Big)-1=J_x-1\ ,
U\left(\Delta_{{\bf T}_1}h, \Delta_{{\bf T}_2}T_{1,1}^*h, 0,\ldots, \Delta_{{\bf T}_2}T_{1,n_1}^*h, 0\right):=\left(\Delta_{{\bf T}_1'}T_{2,1}^*h, \ldots, \Delta_{{\bf T}_1'}T_{2,n_2}^*h,\Delta_{{\bf T}_2}h, 0\right)
\hphantom{{}={}}D_{(o,o)}(\{f,g\}.\{f',g'\}) * =\begin{pmatrix}Df(o)Df'(o) Dg(o)Df'(o)+Df(o)Dg'(o) Df(o)Df'(o)\end{pmatrix}* =\begin{pmatrix}Df(o) Dg(o) Df(o)\end{pmatrix}\begin{pmatrix}Df'(o) Dg'(o) Df'(o)\end{pmatrix}.
\int_{B_{\rho}}r^{2-n}|\nabla u|^2 \d x \leq C \int_{B_{4\rho}\setminus B_{2\rho}}r^{2-n}|\nabla u|^2 \d x + C \varepsilon(1 + \varepsilon \rho) \int_{B_{8\rho}}r^{3-n}|\nabla u|^2 \d x & \rho \leq 1/8 \varepsilon \leq \varepsilon_0.
{\mathcal{C}}_\beta:=\left\{x\in\Omega{\mbox{s.t.}}\frac{x-x_0}{|x-x_0|}\cdot\bar\nu>c_\beta\right\},\qquad{\mbox{where}}\,c_\beta:=\cos\left(\frac\pi2-\beta\right)>0.
[a^{\dagger}_{m},\;\tilde{\psi}_{\dot \alpha}] = [a^{\dagger}_{m},\;\tilde{\psi}^{\dagger}_{\dot \alpha}] = [\tilde{a}^{\dagger}_{m},\;\psi_{\dot \alpha}] = [\tilde{a}^{\dagger}_{m},\;\psi^{\dagger}_{\dot \alpha}] = 0
\begin{aligned}0 \geq \, S_k^{ij}(A \Psi \pm T_\alpha (u - \varphi))_{ij}\geq A \eta_0 S_{k - 1}- C \Big(S_{k - 1}+ f^{1 - 1/ (k - 1)}\Big) \geq \, \frac{A}{2}\eta_0 S_{k-1}+ \frac{A}{2}\eta_0 c_0 f^{1 - 1/ (k - 1)}- C \Big(S_{k - 1}+ f^{1 - 1/ (k - 1)}\Big) > 0\end{aligned}
\lim_{\tau\uparrow \tau_{c}}\frac{E_{\tau}(1)}{(\tau_c-\tau)^{\frac{s}{s-1}}}= \left(1-\frac{1}{s}\right) \left(sz\int_{\mathbb{R}^3}\frac{Q(x)}{|x|^{s}}{\rm d}x\right)^{\frac{1}{1-s}}
P(\mathsf{A},s):=\lim_{n \to \infty}\frac{1}{n}\log \sup_{\|v\|=1}a_n(v) = \limsup_{n \to \infty}\frac{1}{n}\log \left(\max_{\substack{\alpha_1,\ldots,\alpha_d \in\mathbb{R}\sum_{i=1}^d|\alpha_i|^2=1}}a_n\left(\sum_{i=1}^d\alpha_i e_i\right)\right) &\leq \limsup_{n\to\infty}\frac{1}{n}\log \left(d^s \max_{1 \leq i \leq d}a_n(e_i)\right)<P(\mathsf{A},s)
{\cal A T}_{c l a s s}= \left( \begin{matrix}{1}&{0}&{0}&{0}\\{1}&{1}&{0}&{0}\\{-{\frac{5}{6}}}&{-{\frac{5}{2}}}&{1}&{- 1}\\{{\frac{5}{2}}}&{5}&{0}&{1}\\ \end{matrix}\right)
\mathcal{J}_{p}(x,s)=\intop_{0}^{\infty}\,\frac{y^{s-\frac{1}{2}}J_{s-\frac{1}{2}}(2\pi xy)}{\sigma\left(y\right)e^{2\pi y}-1}\,dy=\frac{2\,\pi^{-s}x^{\frac{1}{2}-s}}{\Gamma\left(1-s\right)}\,\intop_{1}^{\infty}\frac{1}{\left(t^{2}-1\right)^{s}}\,\intop_{0}^{\infty}\frac{\sin(2\pi xyt)}{\sigma\left(y\right)e^{2\pi y}-1}\,dy\,dt,
W(\widehat{f}, \widehat{g})(x, \omega) = 2^d e^{4 \pi i x \cdot \omega}\langle \widehat{f}, M_{2 \omega}T_{2x}\widehat{g}^\vee \rangle & = 2^d e^{4 \pi i x \cdot \omega}\langle f, T_{-2\omega}M_{2 x}g^\vee \rangle & = 2^d e^{-4 \pi i x \cdot \omega}\langle f, M_{2x}T_{-2 \omega}g^\vee \rangle & = W(f,g)(-\omega, x).
0<\beta:= \inf_{x\in H \setminus\{0\}}\sup_{y\in H\setminus\{0\}}\frac{b(x,y)}{\| x\|_{H}\| y\|_H}= \inf_{y\in H \setminus\{0\}}\sup_{x\in H\setminus\{0\}}\frac{b(x,y)}{\| x\|_{H}\| y\|_H}.
\tau_{0}= \frac 1 2 \left( \begin{array}{c c}{0}&{1}\\{- 1}&{0}\\ \end{array}\right) , \quad \tau_{1}= - \frac 1 2 \left( \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right) , \quad \tau_{2}= \frac 1 2 \left( \begin{array}{c c}{- 1}&{0}\\{0}&{1}\\ \end{array}\right) \,
\xi^{*}(\delta_{0}\delta_2) = \pi_1^* (\delta_0) + \pi_2^*(\delta_{0}), \xi^{*}(\delta_{1}\delta_2) = \pi_1^*(\delta_1) + \pi_2^*(\delta_{1}), \xi^{*}(\delta_2^2) ={}- \pi_1^*(\psi) - \pi_2^*(\psi).
\alpha = \begin{pmatrix}L_1'(y_1) L_2'(y_1) L_3'(y_1) L_1'(y_2) L_2'(y_2) L_3'(y_2) \end{pmatrix}\beta = \sqrt{\frac34}\begin{pmatrix}-3 -1 4 1 3 -4 \end{pmatrix}\beta.
((z_{1},\ldots,z_{d}),2b,e_{l}) \rightarrow \{(x_{1},\ldots,x_{d}) \in \mathbb{R}^{d}, x_{l}=z_{l};|x_{i}-z_{i}|\leq b,i \in [d] \setminus \{l \}\}, (z_{1},\ldots,z_{d}) \in [0,b]^{d}, l\in [d], [d]= \{1,\ldots,d \}
\bar{g}(X,Y)=e^{\varphi}g(X,Y) g(\bar{\nabla}_{X}Y,Z)=g(\nabla_{X}Y,Z)-\frac{1+\alpha}{2}d\varphi(Z)g(X,Y)+\frac{1-\alpha}{2}[d\varphi(X)g(Y,Z)+d\varphi(Y)g(X,Z)]
G_{\rm ir}^{(n)}({\bf{r}}_1 , \ldots ,{\bf{r}}_n ) = \left. \frac{\delta^{(n)}{\cal{G}}_{\rm ir}\{\varphi \}}{\delta \varphi ({\bf{r}}_1) \ldots \delta \varphi ({\bf{r}}_n )}\right|_{\varphi = 0}\; .
\int \frac{d^d k_1}{[i\pi^{d/2}]}~(qk_1)~ P_{k_1,m_1}^{\nu_1+1}P_{k_1-q,m_2}^{\nu_2}= \nu_2 q^2 I^{(d+2)}_{\nu_1+1~ \nu_2+1}.
F(\tau,E) = \begin{cases}F_+(\tau) \equiv (1 + \Delta\alpha)(1-S_a \cos(2 \pi \tau)) - (L_m + L_a \cos(2 \pi (\tau-\phi))), E>0, F_-(\tau) \equiv (1 - \Delta\alpha)(1-S_a \cos(2 \pi \tau)) - (L_m + L_a \cos(2 \pi (\tau-\phi))), E<0.\end{cases}
\mathcal{L}\left[\mathcal{F}\left\{p(t,\cdot)\right\}\right](s)=\int_{0}^{\infty}e^{-st}\mathcal{F}\left\{p(t,\cdot)\right\}(\xi)dt=\frac{s^{\alpha-1}}{s^{\alpha}+|\xi|^{2}}.
\psi \left( m , z \right) = N_{m}\sqrt{\left| z \right| + \frac{1}{k}}\left[ Y_{2}\left( m \left| z \right| + \frac{1}{k}\right) + \frac{4 k^{2}}{\pi m^{2}}J_{2}\left( m \left( \left| z \right| + \frac{1}{k}\right) \right) \right] ,
a_{n_1}\equiv \sum_{0\leq j\leq kp-1, n_1-j\in\cal{N}}h_1(n_1-j)h_2(n_1-j)^{n_1-j}\prod_{i=0}^{j-1}f(n_1-i) \equiv & \equiv \sum_{0\leq j\leq kp-1, n_2-j\in\cal{N}}h_1(n_2-j)h_2(n_2-j)^{n_2-j}\prod_{i=0}^{j-1}f(n_2-i) \equiv a_{n_2}\pmod{p^k},
\underset{z_1 = \infty}{}\frac{I_n(z_1,\dots,z_n)}{z_1}= - I_{n-1}(z_2,\dots,z_n) &\underset{z_n = 0}{}\frac{I_n(z_1,\dots,z_n)}{z_n}= \gamma \cdot I_{n-1}(z_1,\dots,z_{n-1})
(I) \geq \sum_{k=1}^{n_1}\int_{(P^{k_{0}(k+1)}\setminus P^{k_{0}k})\cap F_{N_0}(R')}\frac{\alpha}{|x-y|^s}d\mu(y) & \geq c(k_0)^{-1}\sum_{k=1}^{n_1}\frac{\alpha\,\mu\bigl(P^{k_{0}(k+1)}\cap F_{N_0}(R')\setminus P^{k_{0}k}\bigr)}{\ell(P^{k_{0}(k+1)})^s},
\begin{array}{l c c c}{\mathrm{f l o o r}}&{\mathrm{S U ( 1 / 2 )}}&{\mathrm{S U ( 2 )}}&{\mathrm{f i e l d}}\\{}\\{\mid \mathrm{g r o u n d}>}&{( 0 1 )}&{( 1 )}&{\lambda}\\{\mid \mathrm{1 s t}>}&{( - 1 0 )}&{( 0 )}&{\phi}\\ \end{array}
\left\| \lambda^{k+1}- \lambda^k \right\|^2 \leq \frac{3L^2}{\sigma_N}\left\| x_N^k - x_N^{k+1}\right\|^2 + \frac{6L^2}{\sigma_N}\left\| x_N^{k-1}- x_N^k \right\|^2 + \frac{3 L^2}{\sigma_N}\sum_{i=1}^{N-1}\left\| x_i^k - x_i^{k+1}\right\|^2.
Z_{l}^{SU}(g,\theta)\sim\frac{\sqrt{2\pi}}{g}\exp\frac{g^{2}}{8}(\frac{\theta}{\pi})^{2}{\rm det}_{jk}\exp[-\frac{g^{2}}{2N}\{(j-k)^{2}-\frac{1}{4}(\frac{\theta}{\pi})^{2}\}].
w^{S \setminus \{i_1, i_2\}}_j = \frac{\displaystyle \frac{w^S_{i_1}}{w^{S \setminus \{i_2\}}_{i_1}}\, w^{S \setminus \{i_2\}}_j + \frac{w^S_{i_2}}{w^{S \setminus \{i_1\}}_{i_2}}\, w^{S \setminus \{i_1\}}_j - w^S_j}{\displaystyle \frac{w^S_{i_1}}{w^{S \setminus \{i_2\}}_{i_1}}+ \frac{w^S_{i_2}}{w^{S \setminus \{i_1\}}_{i_2}}- 1}.
i^{-1}(i_1,i_2)=\left(\frac{i_1+\sqrt{i_1^2-4i_2}}{2},\frac{i_1-\sqrt{i_1^2-4i_2}}{2}\right)\in{D(\lambda_1,\lambda_2)}\cup \gamma_1, (i_1,i_2)\in{D(i_1,i_2)}\cup \gamma_2
\begin{array}{c c c}{\left( \begin{array}{c}{a}\\{b}\\{\varepsilon^{i}}\\{\rho^{i}}\\{\lambda}\\{\omega_{A B}}\\{T_{i}^{j}}\\ \end{array}\right)}&{\longrightarrow}&{\left( \begin{array}{c}{b}\\{a}\\{\rho^{i}}\\{\varepsilon^{i}}\\{- \lambda}\\{\omega_{A B}}\\{T_{i}^{j}}\\ \end{array}\right)}\\ \end{array}
\|qf^2\|_{L^1}=\int_{\mathbb R}|q(t)| |f(t)|^2\,\mathrm dt=\int_{\mathbb R}\big(q_+(t)+q_-(t)\big) |f(t)|^2\,\mathrm dt\leq 2\|q_-\|_{L^1}\|f\|_\infty^2.
< m > = \frac{e^{- \beta ( \varepsilon - \mu )}( 1 + e^{- \beta ( \varepsilon q^{- 2}- \mu )})}{1 + 2 e^{- \beta ( \varepsilon - \mu )}+ e^{- \beta ( \varepsilon ( q^{- 2}+ 1 ) - 2 \mu )}}
\det K ={\det}^2 (i{\not\!\partial}-e{\not\!\!A}-m) \det \left( 1 - 4 \frac{1}{i{\not\!\partial}+e{\not\!\!A}-m}\chi^* \cdot \gamma \frac{1}{i{\not\!\partial}-e{\not\!\!A}-m}\chi \cdot \gamma \right)~,
f_{k}(x)= \oint_{\Sigma_{-\infty}}\frac{dt}{2 \pi i}\frac{\Gamma(t)}{t +k}e^{-\frac{\gamma}{2}t^{2}+x t}, g_{k}(x)= \int^{1 + i\infty}_{1 - i\infty}\frac{ds}{2\pi i}\frac{1}{(s +k)\Gamma(s)}e^{\frac{\gamma}{2}s^{2}-x s},
\left. \begin{array}{c}Y\left( x,y\right) =\delta \left( x_{1}-y_{1}\right) \overline{Y}\left( \overline{x},\overline{y}\right) , \overline{Y}\in L_{\infty}\left( \Omega_{1}\times \Omega_{1}\right) ,\left\Vert \overline{Y}\right\Vert_{L_{\infty}\left( \Omega_{1}\times \Omega_{1}\right)}\leq N_{1}.\end{array}\right.
f(x,x_0) \equiv - \frac{e^2q}{2{\pi}^2}\hbar \sum_{p>0}\frac{1}{p\varepsilon_{\rm R}(p)}\cos(\frac{2\pi p}{\rm L}(x-x_0)) =- \frac{qm_0}{2}\frac{\cosh(\frac{\rm L m_0}{2}- m_0|x-x_0|)}{\sinh{\frac{\rm L m_0}{2}}}.
H_1 := \frac{1}{r}T_0 - \frac{1}{2}I_r = \frac{1}{2}I_r - \mathrm{diag}(0, \frac{1}{r}, \dots, \frac{r-1}{r}). H_2 := I_r - \frac{1}{r}T_0 = \mathrm{diag}(0, \frac{1}{r}, \dots, \frac{r-1}{r}).
\chi(t,x,s,y) =b_{\varepsilon}\left(\frac{x-y}{2}\right)b_{\varepsilon}\left(\frac{t-s}{2}\right)g\left(\frac{x+y}{2}\right)h_{\delta}\left(\frac{t+s}{2}\right) &=b_{\varepsilon}\left(\overline{y}\right)b_{\varepsilon}\left(\overline{s}\right)g\left(\overline{x}\right)h_{\delta}\left(\overline{t}\right).
\eta_{0 \mu}= \operatorname*{l i m}_{\Delta \omega \to 0}\frac{2 \Omega_{\mu}\sqrt{g \Delta \omega}}{\sqrt{4 ( \Omega_{\mu}^{2}- \tilde{\omega}^{2})^{2}+ \pi^{2}g^{2}\Omega_{\mu}^{2}}}\; ,
V:=\mathbb{R}^{3}\setminus\{\{(x,y,z)\in\mathbb{R}^{3}: y = 0\}\cup\{(M,- M\sqrt{\lambda},0):M\in\mathbb{R}\}&\cup\{(M,M\sqrt{\lambda},0):M\in\mathbb{R}\}\cup \{(0,0,M):M\in\mathbb{R}\}\}.
\frac{F}{V}=-\sum_{n,p,q}\int_{\cal S}d\tau_1 \frac{d\tau_2}{2\tau_2}\frac{\left| \eta\left(\tau\right)\right|^{-48}}{(4\pi^2\alpha'\tau_2)^{13}}e^{-S_{w.m}(n,p,q)}
\sum_{k=0}^{j-h-1}l_k \max \{0,d-j-k\}\geq \sum_{k=0}^{j-h-1}l_k (d-j-k) & = (d-j) \sum_{k=0}^{j-h-1}l_k - \sum_{k=0}^{j-h-1}l_k k & \geq (j-1)(d-j) - (j-h-1).
\frac{F}{V}= - \sum_{\tau \in{\cal F}}\frac{\nu^{- 2}e^{- \frac{\beta^{2}| n \tau - m |^{2}}{4 \pi \alpha^{\prime}\tau_{2}}}}{n^{2}+ \nu^{- 2}r^{2}}\left( \frac{1}{4 \pi \alpha^{\prime}\tau_{2}}\right)^{1 3}\vert \eta ( \tau ) \vert^{- 4 8}
\Re\int_1^\infty\frac{\d x}{\sqrt{x(x-1)(x-z)}}=\Re\int_0^1\frac{\d t}{\sqrt{t(1-t)(1-zt)}}=\pi\,\Re{}_2F_1\biggl(\begin{matrix}\frac12, \, \frac12 1 \end{matrix}\biggm|z\biggr),
\frac{x^i}{\rho}F_{\hat i\hat 0}=\partial_\rho\psi=-\frac{Q_{elec}}{\rho^2}+{\cal O}(\rho^{-3}) \,\qquad\frac{x^i}{\rho}\tilde G_{\hat i\hat0}=\partial_\rho \chi=-\frac{P_{elec}}{\rho^2}+{\cal O}(\rho^{-3})\{}.
\int_G f(x)d\tilde{\mu}=\int_G f(x)Q\big(g(q(x)\big) d\tilde{\mu}(x) &=\int_G f(x) \int_K \int_H g (k^{-1}xh) dh dk d\tilde{\mu}(x) &=\int_K\int_H\int_G f(kxh^{-1})g(x) \Delta_K(k)\Delta_H(h)d\tilde{\mu}(x)dhdk &=\int_G g(x) \int_H \int_K f(k^{-1}xh) dkdhd\tilde{\mu}(x) &=\int_G g(x) Q(f) \big(q(x)\big) d\tilde{\mu}(x)=0.
C_{i,k}:= [(1+2i) a - 2 i, (1+2i) a - 2 i + 6 (i-t-1) - 2] & \cup \left[(1 + 2 i) a - 2 i + 6 (i-t-1), (1 + 2 i) a - 2 i + 6 k \right]_2, D_{i,k}:= [(2+2i) a - 2 (i+1), (2+2i) a - 2 (i+1) + 6 (i-t-1) + 2] & \cup \left[(2 + 2 i) a - 2 (i+1) + 6 (i-t-1) + 4, (2 + 2 i) a - 2 (i+1) + (6k+2) \right]_2,
\frac{\partial^{2}\psi}{\partial r^{2}}+\frac{1}{r^{2}}\frac{\partial^{2}\psi}{\partial\phi^{2}}=\frac{\partial^{2}\psi}{\partial x^{2}}+\frac{\partial^{2}\psi}{\partial y^{2}}-\frac{1}{r}(cos\phi\frac{\partial\psi}{\partial x}+sin\phi\frac{\partial\psi}{\partial y})
\int_{T_{t^{\frac{a+\epsilon}{2}}}}^{T_{t^{\frac{1-\epsilon}{2}}}}\frac{dm}{|W(m)|^2}=_d\int_0^{\tilde{T}_{t^{\frac{1-a-2\epsilon}{2}}}}\frac{dm}{|\tilde{W}(m)|^2}, & H(T_{t^{\frac{1-\epsilon}{2}}})-H( T_{t^{\frac{a+\epsilon}{2}}}) H(T_{t^{\frac{a+\epsilon}{2}}}).
dz_i(s) = -\frac{\partial V}{\partial z_i}\big(z(s), y(s)\big)\,ds + \sqrt{2\beta^{-1}}dw_i(s)\,,1 \le i \le m\,, dy_j(s) = -\frac{1}{\delta}\frac{\partial V}{\partial y_j}\big(z(s), y(s)\big)\,ds + \sqrt{\frac{2\beta^{-1}}{\delta}}\,dw_j(s)\,,m+1 \le j \le n \,.
\frac{1}{\psi_2(N,K,s)}\geq N\sum_{i=0}^{\epsilon K}\frac{(K-1)(K-1-s)(K-2-s) \cdots (K-i-s)}{s(s+1)(s+2)\cdots (s+i)}(N-1)^i &\geq N\sum_{i=0}^{\epsilon K}\frac{(K-1)(K-\epsilon K-s)^{i}}{(s+\epsilon K)^{i+1}}(N-1)^i &= N\sum_{i=0}^{\epsilon K}\frac{(1-\frac{1}{K})((1-(\lambda+\epsilon))^{i}}{(\lambda+\epsilon)^{i+1}}(N-1)^i.
(\frac{tanhx_{a}}{\sqrt{1-tanh^{2}x_{a}-tanh^{2}y_{a}}},\frac{tanhy_{a}}{\sqrt{1-tanh^{2}x_{a}-tanh^{2}y_{a}}},\frac{1}{\sqrt{1-tanh^{2}x_{a}-tanh^{2}y_{a}}})
\mathcal G^\ast_{k,(a_1,a_2;N)}(\tau):= \mathcal G_{k,(a_1,a_2;N)}^\ast(0,\tau)=\lim_{\operatorname{Re}(s)>0\atop s\rightarrow 0}\mathcal G_{k,(a_1,a_2;N)}^\ast(s,\tau).
(1_{Y}\boxtimes u_{X,Z})\circ (u_{X,Y}\boxtimes 1_{Z})(b_{i}\boxtimes c_{j}\boxtimes d_{k}) =c_{j}\boxtimes d_{k}\boxtimes b_{i}&=u^{F,K}_{X, Y\boxtimes Z}(b_{i}\boxtimes c_{j}\boxtimes d_{k}) &=u_{X,Y\boxtimes Z}(b_{i}\boxtimes c_{j}\boxtimes d_{k}),
\frac{\mathbb{E}_x\left[Z_T^{b^\ast}\right]}{T}= \frac{x - \mathbb{E}_x\left[X_T^{Z^{b^\ast}}\right]}{T}+ \frac{1}{T}\mathbb{E}_x\int_0^T \mu\left(X_t^{Z^{b^\ast}}\right)X_t^{Z^{b^\ast}}\,dt.
\begin{aligned}\|x(t)-z(t)\| \le \|x(0)-z(0) \|+ \gamma^2 3 L_\mathcal{D}M_\mathcal{D}T t & + \gamma L_\mathcal{D}\int_0^t (\|x(0)-z(0) \|+ \gamma^2 3 L_\mathcal{D}M_\mathcal{D}T \tau) & \cdot e^{\gamma L_\mathcal{D}(t-\tau)}\,d\tau = \|x(0)-z(0) \| e^{\gamma L_\mathcal{D}t}+ \gamma 3 M_\mathcal{D}T \left(e^{\gamma L_\mathcal{D}t}-1\right)\end{aligned}