harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A_N = \sum_{\sum_i k_i = N}\left[\frac{y_{\#_1}^{k_{\#_1}}...y_{\#_R}^{k_{\#_R}}}{R!}\binom{R}{\#_1, ... \#_R}\sum_{\sigma \in S_R / \mathrm{Stab}(k_{r_1}, ... k_{r_l})}\sigma(k_2(k_2 + k_3)...(k_2 + ... + k_R))\right]
\begin{array}{l}{{{\bf X}_{a , a + 2}^{a + 1}{\bf U}_{a + 1}^{'}+ c_{2 a}( u_{+}){\bf D}_{a + 1}{\bf X}_{a , a + 2}^{' a + 1}= c_{2 a}( u_{-}){\bf X}_{a , a + 2}^{' a + 1}{\bf U}_{a + 1}}}\\{{\mathrm{~}+ c_{2 a}( u_{+}) c_{2 a}( u_{-}){\bf D}_{a + 1}^{'}{\bf X}_{a , a + 2}^{a + 1}+ c_{3 a}( u^{'}, u ){\bf U}_{a - 1}^{'}{\bf X}_{a , a + 2}^{a + 1}}}\\ \end{array}
{\hat{\bar{\Psi}}}\ = \ \sum_{j = | k | - 1 / 2}^{\infty}\sum_{| m | \le j}[{\bar{a}}_{- k m}^{j +}{\hat{D}}_{- k - 1 / 2 , m}^{j}a \ + \{\bar{a}}_{- k m}^{j -}{\hat{D}}_{- k + 1 / 2 , m}^{j}a^{*}] \ \in \{\hat{\cal S}}_{- k}\ ,
\widetilde{A_k}(t,\xi)=\frac{1}{(\xi-kt)^2+Q^2k^2},\,k\neq 0,\widetilde{A_0}(t,\xi)=\frac{1}{\xi^2+Q^2};\widetilde{A_*}(t,\xi)=\sum_{k}e^{-2|k|}A_{k}(t,\xi),
( \mathrm{m a s s})^{2}={\frac{( \frac{1}{2}m + \Delta a )^{2}}{N}}- \frac{1}{4}{\frac{m^{2}}{N}}+ \frac{Q^{2}}{8}={\frac{\Delta a ( \Delta a + m )}{N}}+ \frac{1}{4 N}~ ,
\begin{aligned}A_1U=C_1, A_{2}X =C_{2},\X=X^{\eta^{\ast}}, A_{3}Y =C_{3},\Y=Y^{\eta^{\ast}}, A_{4}Z =C_{4},\Z=Y^{\eta^{\ast}}, E_1U+(E_1U)^{\eta^{\ast}}+E_{2}XE_{2}^{\eta^{\ast}}+E_{3}YE_{3}^{\eta^{\ast}}+E_{4}ZE_{4}^{\eta^{\ast}}=C_c.\end{aligned}
\frac1{|Q|^{1+\beta/n}}\int_Q |b(x)-b_Q|dx =\frac1{|Q|^{1+\beta/n}}\int_{E\cup{F}}|b(x)-b_Q|dx =\frac2{|Q|^{1+\beta/n}}\int_E|b(x)-b_Q|dx \le \frac2{|Q|^{1+\beta/n}}\int_E|b(x)-M_Q(b)(x)|dx \le \frac2{|Q|^{1+\beta/n}}\int_Q|b(x)-M_Q(b)(x)|dx.
R_{\mu \nu}- \nabla_{\mu}\Phi \nabla_{\nu}\Phi -{\frac{1}{4}}e^{4 \Phi}H_{\mu \rho \sigma}H_{\nu}^{\rho \sigma}+{\frac{1}{6}}g_{\mu \nu}e^{4 \Phi}H^{2}+{\frac{4}{k}}g_{\mu \nu}e^{- 2 \Phi}= 0 ,
\mathsf{p}_s = 1- \left( 1- \textrm{Pr}\left\{\mathsf{C}_1 \right\}\cdot \textrm{Pr}\left\{\mathsf{C}_2 \right\}\right)^{|\mathcal{N}_s|}& \gtrsim 1- \left( 1- \textrm{Pr}\left\{\mathsf{C}_1 \right\}\cdot \textrm{Pr}\left\{\mathsf{C}_2\right\}\right)^{N \cdot \alpha^{2(S-1)}}.
r_{\langle\rangle}(\mathcal B_{\forall,\psi}^\beta P) =(\forall_x,x\notin\mathbb L_\beta^u\lor\theta), n(\mathcal B_{\forall,\psi}^\beta P,a) =\textstyle\bigvee_{a\notin\mathbb L_\beta^u,\theta(a)}^1\mathcal B_{\forall,\psi}^\beta n(P,a).
L(t,x)\xi=\int_t^T\int_{H}e^{-(s-t){A}}GB(s,z+e^{(s-t){A}}x)\left\langle Q_{s-t}^{-1/2}e^{(s-t){A}}G\xi,Q_{s-t}^{-1/2}z\right\rangle \mu_{s-t}(dz) +\int_{t}^{T}\int_{H}e^{-(s-t){A}}L(s,z+e^{(s-t){A}}x)B(s,z+e^{(s-t){A}}x) \left\langle Q_{s-t}^{-1/2}e^{(s-t){A}}G\xi,Q_{s-t}^{-1/2}z\right\rangle \mu_{s-t}(dz)\,ds.
{\frac{d}{d t}}S = \mathrm{I m}( P ) + b_{\scriptscriptstyle I}\left( \frac{1}{R}\nabla^{2}R - ( \nabla S )^{2}\right) +{\frac{b_{\scriptscriptstyle R}}{R^{2}}}\nabla \cdot ( R^{2}\nabla S ) ~ . ~ ~
\Omega(s)= k^{s/2}2^{s-1}\Gamma(\frac{s+1}{2}) \pi^{-\frac{s+1}{2}}\int_{\Gamma_{0}(4k) \backslash H}\overline{f(z)}\theta(z) E(\frac{s+1}{2},z) \frac{dx dy}{y^{2}}.
\sum_{t_1,\ldots, t_n}\langle f(h_{t_1},\ldots, h_{t_n}), (g_0^{-1}\tau)|_{[t_1,\ldots, t_n]}\pi \rangle=\sum_{t_0,\ldots, t_n}\langle h_{t_0}f(h_{t_1},\ldots, h_{t_n}), \tau|_{[t_0,\ldots, t_n]}\pi\rangle.
r^*_{i,j}:=\sup_{\substack{0\le l\le L_i,0\le p\le L_j |n|\le N_i,|m|\le N_j}}|\tilde r_{y_i,y_j}(s_{i,l},\tau_{i,n},s_{j,p},\tau_{j,m})|\le K (k-t)^{-\lambda}\le \frac{\lambda}{4},
\theta^{\rm N}_n\big((x, y, z, w)\big) = \theta^{\rm LB}_n \Big(\big( (x, y), (x, \langle x,y,z \rangle ), (x, \langle x,y,\langle y,z,w \rangle \rangle ) \big) \Big) &= \theta^{\rm LB}_n \Big(\big( (x, y), (x, -x+y+z ), (x, -x+z+w )\big) \Big) &=2^{-2}(x-z) \frac{(-y+z+2w)^n +( y+z)^n- 2( z+w)^n}{n},
\leq \e^{\, s \alpha 4 m^{2}- m^{2}\sum_{j = 1}^{s}\alpha_{j}}\, P ( \, \vert \vec{p}\vert \, ) \, Q ( \sqrt \alpha_{1}, \ldots , \sqrt \alpha_{s}) \quad \mathrm{f o r}\quad \vec{k}_{0}\not = 0 \ .
\begin{cases}dx(t,y)=\pi(t,y)x(t,y)[b_0(t,y)dt+\sigma_0(t,y)dB(t)+\int_\mathbb{R}\gamma_0(t,y,\zeta)\tilde{N}(dt,d\zeta)]; t\in[0, T] x(0,y)=x_0(y)>0,\end{cases}
G_\beta^B({\rm\bf r}_2,{\rm\bf r}_1) ={{m \omega_c}\over \pi}e^{-\beta \omega_c}\exp \left [ -{{m \omega_c}\over 2}\left \{r_{21}^2 + 2 i \epsilon \hat{\rm\bf k}\cdot ({\rm\bf r}_2 \times{\rm\bf r}_1 ) \right \}\right ],
\frac{1}{L^n}\int_U \varphi(y) \dn_L(y) - \int_U \varphi(x) \bar{\nu}(x) dx \geq -(\sup_U \varphi) \cdot \int_{Q_1}\big|\nu_{R,L}(x,w) - \bar{\nu}(x) \big|\dx & - \omega_\varphi (\delta) \cdot ||\bar{\nu}||_{L^\infty(Q_1)}\cdot |Q_1| .
\sum_{n=1}^{N}\frac{(-1)^{n-1}q^{\frac{n(n+1)}{2}}}{(1-cq^n) (q)_n (q)_{N-n}}\left(\sum_{k=1}^n \frac{q^k}{1- q^k}\right) = \sum_{n = 1}^{N}\left[\begin{matrix}N n \end{matrix}\right] \frac{q^{n(n + 1)}}{(q)_n (1 - q^n)}F(q^N, q^n; cq^n),
\alpha_{n, M}(t)=\frac{1}{M}\sum_{a\bmod{M}}e^{-2\pi in \frac{a}{M}}L(t, f, \frac{a}{M})= \frac{1}{M}\sum_{d\mid M}\sum_{\substack{a\bmod{d}\gcd(a, d)=1}}e^{-2\pi in \frac{a}{d}}L(t, f, \frac{a}{d}).
\Vert \Delta \psi_{n}\Vert^{2}_{{L}^{2}}= (-\Delta \psi_{n}, \,-\Delta \psi_{n}) = (f,\, -\Delta \psi_{n})\leq \Vert f\Vert_{L^{2}}\Vert \Delta \psi_{n}\Vert_{{L}^{2}},
P_{\vert\boldsymbol{Y}\vert=n}\left(\boldsymbol{Y}= \boldsymbol{y}\right) = \binom{n}{\boldsymbol{y}}\prod_{j=1}^J \left(\frac{\theta_j}{\vert\boldsymbol{\theta}\vert}\right)^{y_j}\cdot \mathbb{1}_{\Delta_{n}}(\boldsymbol{y}),
\bar{X}_{L,t_{k}}=\Big(\frac{1}{N_{0}}\sum^{N_{0}}_{i=1}\tilde{\tilde{X}}_{0,t_{k}}^{i}\Big)+\sum^{L}_{l=1}\Big(\frac{1}{N_{l}}\sum^{N_{l}}_{i=1}\big(\tilde{\tilde{X}}_{l,t_{k}}^{i}-\tilde{\tilde{X}}_{l-1,t_{k}}^{i}\big)\Big),
{\mathcal{R}}_{0}=exp(-\sum_{{\bf{k}},{\bf{q}}, i}f^{*}_{{\bf{k}},{\bf{q}}}({\bf{x}})f_{{\bf{k}},{\bf{q}}}({\bf{x}})[b_{i}({\bf{q}}), a^{\dagger}_{{\bf{k}}}({\bf{q}})][a_{{\bf{k}}}({\bf{q}}), b^{\dagger}_{i}({\bf{q}})])
\det(A_G) =(-1)^{|V|-l-1}\left[(t-1)^2-|W|lt-(t-1)(t-1-|W|l+l)\right]\alpha^{|V|}(t-1)^{(l-1)}|E|^{|W|-1}=(-1)^{|V|-l-1}l[1-(t+w)]\alpha^{|V|}(t-1)^{(l-1)}|E|^{|W|-1}.
R_{\mathcal{BS}(\alpha)}(\mathcal{S}^*_\beta) =\begin{dcases}\min\left\{1, \dfrac{1}{\beta(3-2\alpha)}\right\}, \ 0\leq \alpha\leq \frac{1}{9}, \min\left\{1,\dfrac{2\sqrt{\alpha}}{\beta (1+\alpha)\sqrt{1+16\alpha}}\right\}, \frac{1}{9}\leq\alpha \leq 1.\end{dcases}
Q_p(t):= \left(\int_{\mathbb{R}^n}u_1^{1/p}u_2^{1/p}\right)^2-\int_{\mathbb{R}^n}u_1^{2/p}\int_{\mathbb{R}^n}u_2^{2/p}= \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\big((u_1\otimes u_2)^{1/p}(u_2\otimes u_1)^{1/p}- (u_1\otimes u_2)^{2/p}\big) = -\frac{1}{2}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\big((u_1\otimes u_2)^{1/p}-(u_2\otimes u_1)^{1/p}\big)^2
T r \left({\tilde{L}}( \lambda | x_{N}, \partial_{x_{N}}) \ldots{\tilde{L}}( \lambda | x_{1}, \partial_{x_{1}}) \right) = \prod_{i = 1}^{N}- ( \lambda + \beta_{i}) \frac{w_{i}( \lambda - 1 )}{w_{i}( \lambda )}+ \prod_{i = 1}^{N}\frac{w_{i}( \lambda + 1 )}{w_{i}( \lambda )}~ .
\mathcal T_{\kappa,3}^{1,2}=\bigcup_{i,j\in \mathbb N \colon\atop{2(i+j)\leq \kappa}}\{[n_0,n_1,n_2,n_3,n_4,n_5,n_4,n_7]\in \mathcal S_{\kappa,3}:n_1=n_2=i,n_5=n_6=j\},
u= 6 \lambda_1 - 3 \lambda_2 - 3 \lambda_3 -l, v= \sqrt{- 3(2\lambda_1 +3 \lambda_2 - 5\lambda_3 +l)( 2\lambda_1 - 5\lambda_2 +3 \lambda_3 +l)}, l= \sqrt{-3(2\lambda_1+\lambda_2-3\lambda_3) (2\lambda_1-3\lambda_2 +\lambda_3)}.
\|U_\theta(f)\|_{L_{2}(\mathcal{R}_\theta^d)}^2 =\tau_\theta(U_\theta(f)^*U_\theta(f)) &=\tau_\theta(U_\theta(\overline{\tilde{f}}*_\theta f)) =(\overline{\tilde{f}}*_\theta f)(0) &=\|f\|_{L_2(\mathbb{R}^d)}^2=\|\hat{x}\|_{L_2(\mathbb{R}^d)}.
\mu(Z(e g_1 \cdots eg_n e g_{n+1})) + \mu (Z(e g_1 \cdots eg_n e h_{n+1})) &= \mu(Z(e g_1 \cdots e g_n ))(1/2 + \gamma_{n+1}) + \mu(Z(e g_1 \cdots e g_n ))(1/2 - \gamma_{n+1}) &= \mu (Z(e g_1 \cdots e g_n)).
\|W_{\widetilde{\psi}_0,\widetilde{\varphi}_0}f - W_{\psi_{i, 1, 0},\varphi_{i, 1, 0}}f\|_{n,q}= e^{\frac{\left|(z^0)'_{[j]}\right|^2}{2}}\|W_{\widetilde{\psi},\widetilde{\varphi}}f - W_{\psi_{i, 1},\varphi_{i, 1}}f\|_{n,q},
E[(1+\lambda t)^{\frac{1}{\lambda}(U_1+U_2+\cdots+U_{k-1}+1)}] = (1+\lambda t)^{\frac{1}{\lambda}}E[(1+\lambda t)^{\frac{U_1+\cdots+U_{k-1}}{\lambda}}] &=(1+\lambda t)^{\frac{1}{\lambda}}\Big(\frac{1}{\lambda}\log(1+\lambda t)\Big)^{-(k-1)}\Big((1+\lambda t)^{\frac{1}{\lambda}}-1\Big)^{k-1}.
\chi (M_n (\pi/2))= 2 \sum_{s=1}^m (-1)^{m+s}\left(s- \big\lfloor s/2 \big\rfloor \right){n \choose m-s}= 2(-1)^m \sum_{s=1}^m (-1)^s \big\lfloor (s+1)/2 \big\rfloor{2m+1 \choose m-s}= 2 (-1)^{m+1}\sum_{i=0}^{m+1}(-1)^i\big\lfloor i /2\big\rfloor{2m+1 \choose m+i}.
\frac{\partial}{\partial a}\big( g(s,a;r) p_{k-1}(s,a;r)\big) = g(s,a;r)\frac{\partial}{\partial a}p_{k-1}(s,a;r)+g(s,a;r)p_{k-1}(s,a;r)(s-r\psi_0^f(a)) &=g(s,a;r)p_k(s,a;r).
n^{1}_{\gamma}=|\{i\,\mid\, \gamma_i=\gamma\sigma_i=1\}|+|\{i\,\mid\, \gamma_i=\xi\gamma\sigma_i=c\}|, n^{c}_{\gamma}=|\{i\,\mid\, \gamma_i=\gamma\sigma_i=c\}|+|\{i\,\mid\, \gamma_i=\xi\gamma\sigma_i=1\}|,
\int_{x_{0,1}}^{x_{1}}\int_{x_{0,2}}^{x_{2}}\dots \int_{x_{0,n}}^{x_{n}}f(\mathbf{y})(\mathbf{x}-\mathbf{y})^{\beta-\mathbf{1}}d\mathbf{y}=\frac{\gamma(\mathbf{x}-\mathbf{x}_{0})^{\beta}}{\beta^{\mathbf{1}}}+o(|\mathbf{x}-\mathbf{x}_{0}|^{|\beta|}) \ \ \ \mbox{as}\mathbf{x}\to \mathbf{x}_{0},
\sum_{i=1}^{m_1}T(n_i) = T(n_{m_1}) + \sum_{i=1}^{m_1-1}T(n_i)< T(n_{m_1}) + (m_1-1) \cdot \frac{35}{18}\cdot \frac{1}{X_0}&< \frac{3}{X_0}+ (m_1-1) \cdot \frac{35}{18}\cdot \frac{1}{X_0}=\frac{35m_1+19}{18}\cdot \frac{1}{X_0},
\mathrm{Sub}(A)=\sigma_A^{(1)}+\frac{\mathrm{i}}{2}\cdot\frac{\partial^2\sigma_A^{(0)}}{\partial x^k\partial\xi_k}+\frac{\mathrm{i}}{2}\cdot \frac{\partial\sigma_A^{(0)}}{\partial{\xi_k}}\cdot\frac{\partial (\log \rho)}{\partial x^k},
d_t\,\mathbb{P}^\delta_x\, \stackrel{\mathcal{L}}{=}\,\inf\left\{s\geq t:s X_\frac{1}{s}=0\right\}\,\mathbb{P}^{\delta,x}_0 &=\,\frac{1}{\sup\left\{s\leq\frac{1}{t}:X_s=0\right\}}\,\mathbb{P}^{\delta,x}_0 &=\,\frac{1}{g_\frac{1}{t}}\,\mathbb{P}^{\delta,x}_0.
p_1d\lambda=\left(-\frac{\hbar}{2}w_tdv+\frac{\hbar}{2}v_tdw+\frac{\hbar}{2}wdv_t-\frac{\hbar}{2}vdw_t-\frac{\hbar^2}{2m}v_xdv_x-\frac{\hbar^2}{2m}w_xdw_x\right) \omega_0.
P(n_1, p, q) = \Pr(X_1 \le Y_1) &\ge \Pr(X_1 \le Y_1 \mid X_1 + X_2 \le Y_1 + Y_2) \Pr(X_1 + X_2 \le Y_1 + Y_2) &\ge \frac 12 P(n_1 + n_2, p, q).
{\rm Res}_{z^{\prime}=z}^{\Psi}\partial^{\Psi}\!:\!\psi(z^{\prime})\!:\,\bar{\partial}^{\Psi}\!:\!\phi(z)\!:=-:\!(\phi_{;\mu\eta}-iC_{\eta\nu\mu}\phi_{;\nu})\psi^{;\eta}\bar{\partial}^{\Psi}\! X^{\mu}\!:+O(\epsilon^5)
(**) =\int_{[0,a]^{A}}P_N(\phi+\tilde{\psi}\geq0 \,on\,A^c|\phi=0 \, on \, A)g(\psi)d\psi &=\int_{[0,a]^{A}}P_{A^c}(\phi+\tilde{\psi}\geq0 \,on\,A^c)g(\psi)d\psi &\geq P_{A^c}(\Omega_{A^c}^+)
\pi ( u ) = \frac{2 k - 1}{4 \pi}\int_{\tau = 0}^{\infty}d \tau \; \sinh \tau \int_{0}^{2 \pi}d \phi \left[ \cos \frac{\theta_{0}}{2}- i \sin \frac{\theta_{0}}{2}\hat{n}. \hat{r}\right]^{- 2 k}
d\overline{\zeta}^{\alpha+I}[j]=d\overline{\zeta}_{1}^{\alpha_{1}+1}\wedge\cdot\cdot\cdot\wedge d\overline{\zeta}_{j-1}^{\alpha_{j+1}+1}\wedge d\overline{\zeta}_{j+1}^{\alpha_{j-1}+1}\wedge\cdot\cdot\cdot\wedge d\overline{\zeta}_{n}^{\alpha_{n}+1}
\frac{2}{k^{+}k^{' +}}\left[ \left( m^{2}+ k_{\perp}k_{\perp}^{\prime}e^{+ i \lambda ( \varphi - \varphi^{\prime})}\right) \delta_{\lambda^{\prime}}^{\lambda}- m \lambda \left( k_{\perp}^{\prime}e^{+ i \lambda \varphi^{\prime}}- k_{\perp}e^{+ i \lambda \varphi}\right) \delta_{- \lambda^{\prime}}^{\lambda}\right]
\varphi_\pm^{-1}:\begin{pmatrix}x^1 x^2 x^3 x^4 \end{pmatrix}\mapsto\begin{pmatrix}x^1 x^2 x^3 x^4 \end{pmatrix}-a\begin{pmatrix}\cos(x^3+x^4) \pm\sin(x^3+x^4) 0 0\end{pmatrix}.
{\mathbb E}\tilde m_i^T(T) y(T)-{\mathbb E}\tilde m_i^T(0) y(0)={\mathbb E}\int_0^T\left\{\gamma y^T \tilde p_i - \tilde m_i^T (G^T y_i +\Gamma^T Q[x_i-(\Gamma m_i+\eta)])\right\}dt.
16(2\lambda-\beta+2) (2\lambda-\beta+4)(\lambda-\beta)\omega^{(-)}_6 =2 R^{(0)}_2 \delta^h\omega^{(+)}_0 +4(2\lambda-\beta+4)(\beta+6)(\tfrac{J}{2n}) R^{(0)}_1 \delta^h\omega^{(+)}_0 \quad+2(2\lambda-\beta+2)(2\lambda-\beta+4) (\beta+4)(\beta+6)(\tfrac{J}{2n})^2\delta^h\omega^{(+)}_0
\Phi_T(M_{f_1f_2})= \Phi_T(M_{f_1}M_{f_2})= \Phi_T(M_{f_1})\Phi_T(M_{f_2})= = (M_{g_1}+R_1)(M_{g_2}+R_2)= M_{g_1g_2}+R_1M_{g_2}+M_{g_1}R_2+R_1R_2.
\widehat{\left(\begin{matrix}w_h \chi_h\end{matrix}\right)}={\left(\begin{matrix}\{w_h\}+ \alpha[w_h] + \beta_1 [\chi_h] \{\chi_h\}-\alpha[\chi_h]+\beta_2 [w_h]\end{matrix}\right)}, \beta_1, \beta_2\leq 0.
q\left(\frac{1}{2}+\rho .\exp \left[i.\left(\frac{\pi}{2}+\alpha \right)\right]\right) =q\left(\frac{1}{2}-\rho .\exp \left[i.\left(\frac{\pi}{2}+\alpha \right)\right]\right) =\overline{q}\left(\frac{1}{2}+\rho .\exp \left[i.\left(\frac{\pi}{2}-\alpha \right)\right]\right)
I = \frac{1}{2\pi i}\int_{2-i \infty}^{2+i \infty}F_C (s) k(s)\, ds = \sum_{\mathfrak{p}}\sum_{m=1}^{\infty}\theta ({\mathfrak{p}}^m) (\log N_{K/{\mathbb Q}}{\mathfrak{p}}) \widehat{k}(N_{K/{\mathbb Q}}{\mathfrak{p}}^{m}),
\mathbb{E}_{\sigma^1, \sigma^2}^{\mu_0}(1 - \delta_{\epsilon}) \bigg[\sum_{m=0}^{\infty}\beta_{\epsilon}^m u^1(a^1_m,a^2_m)\bigg] + \epsilon & \geq \liminf_{N \to \infty}{1 \over N}\mathbb{E}_{\sigma^1, \sigma^2}^{\mu_0}\bigg[ \sum_{m=0}^{N-1}u^1(a^1_m,a^2_m)\bigg]
\eta = \alpha'^{-1}(r_3 - r_1) \zeta = \left[\frac{2(m_1 +m_3) m_2}{\hbar^2 (m_1 + m_2 + m_3)}\right]^{\frac 12}\left( r_2 - \frac{m_1}{m_1+m_3}r_1 - \frac{m_3}{m_1+m_3}r_3\right) ,
h^{\mathrm{an}}_{(D, g)}(f^{\mathrm{an}}(\xi)) = h^{\mathrm{an}}_{f^*(D, g)}(\xi) = h^{\mathrm{an}}_{d(D, g) + \widehat{(\varphi)}}(\xi)= d h^{\mathrm{an}}_{(D, g)}(\xi).
\widetilde \Pi^{\prime}= \left( \begin{matrix}{{\widetilde F_{0}^{\prime}}}\\{{\widetilde F_{1}^{\prime}}}\\{{\widetilde X^{0}}}\\{{\widetilde X^{1}}}\\ \end{matrix}\right) = \left( \begin{matrix}{{\widetilde X_{0}}}\\{{\widetilde F_{1}}}\\{{- \widetilde F_{0}}}\\{{\widetilde X^{1}}}\\ \end{matrix}\right) \ .
m_{\varphi{\bar{\varphi}}}^{2}\sim G_{\varphi{\bar{\varphi}}}^{- 1}\left( G^{i{\bar{\jmath}}}R_{i{\bar{\jmath}}\varphi{\bar{\varphi}}}- G_{\varphi{\bar{\varphi}}}\right){\frac{m_{3 / 2}^{4}}{M_{p}^{2}}}{\cal J}\ ,
\big [{\frac{d^{2}}{d \, x^{2}}}+ 1 6 x^{N - 2}( x + B )^{2}+ 8 \lambda_{n + 1}+ \sum_{i = 1}^{n}{\frac{\lambda_{i}}{x + A_{i}}}\big ] \Psi_{j}( x ; \lambda_{1}\ldots \lambda_{n + 1}) = 0
\mathrm{[}\hat{\Lambda}_{n}, \hat{\Lambda}_{m}] = [ \hat{\Lambda}_{n}^{\dagger}, \hat{\Lambda}_{m}^{\dagger}] = 0 ; \, \, \, [ \hat{\Lambda}_{n}\, \, , \, \, \hat{\Lambda}_{m}^{\dagger}] = \frac{R}{\pi}\delta_{n , m}.
B_{R}^{3}[u] =\frac{-2N(p-2)}{p}\int_{\mathbb{R}^{N}}|u|^{p}-\frac{2(p-2)}{p}\int_{|x|\geq R}(\Delta \varphi_{R}-N)|u|^{p}&=\frac{-2N(p-2)}{p}\int_{\mathbb{R}^{N}}|u|^{p}+O(R^{-\frac{(p-2)(N-1)}{2}}\|\Delta u\|_{2}^{\frac{p-2}{4}}).
U_{k_{1}\, \, k_{2}}^{\, \, l_{1}\, \, l_{2}}\left( \begin{array}{c c c}{\lambda}{\mu}{\rho}{k_{1}}{k_{2}}{m}\end{array}\right) \left( \begin{array}{c c c}{\lambda}{\mu}{\rho^{\prime}}{l_{1}}{l_{2}}{m^{\prime}}\end{array}\right)^{\ast}
\sum_{\substack{(n ,\, a_{2}) \, =\, 1 begin{align*}1pt] \omega(n) \, \le \, \lambda P(n)\, \ge \, y_u}}\frac{\rho_{\lambda}(n)}{\varphi\big(\l(n)\big)}\le 2C_u (\lambda+1)\Gamma(\lambda+1,\log w_u ) \le 2C_u(\lambda + 1)!.
\mathcal S^{\frac{3}{2}}(\partial\Omega)=\mathcal S^{\frac{3}{2}}_{\lambda}(\partial\Omega):=\left\{f\in L^2(\partial\Omega): f=\sum_{j=1}^{\infty}\hat a_j\hat u_{j,\lambda}{\rm\with\}\left(\sqrt{|\mu_j(\lambda)|}\hat a_j\right)_{j=1}^{\infty}\in l^2\right\},
(x+y)^7+A_1(x^6+x^5y+x^4y^2+x^3y^3+x^2y^4+xy^5+y^6)+A_2(x^5+x^4y+x^3y^2+x^2y^3+xy^4+y^5) &+A_3(x^4+x^3y+x^2y^2+xy^3+y^4) +A_4(x+y)^3+A_5(x^2+xy+y^2)+A_6(x+y)+A_7=0.
M_{n}= \sum_{k=0}^n(-1)^{n-k}\binom{n}{k}C_{k+1}= \sum_{k=0}^n(-1)^k3^{n-k}\binom{n}{k}C_{k+1}= \frac1{(\sqrt{2})^{n+2}}\sum_{k=0}^n(\sqrt{2}-3)^{n-k}\binom{n}{k}S_{k+1}= \frac1{(\sqrt{2})^{n+2}}\sum_{k=0}^n(-1)^k(3+\sqrt{2})^{n-k}\binom{n}{k}S_{k+1}.
(d\pi_{GM})_{p}\circ(dL_{p})_{Id}(W_{i})=(d\pi_{GM})_{p}([\tilde{Y}^{0},\,\tilde{Y}^{\rho}])=[Y^{0},\,Y^{\rho}]_{q}, \rho \in \big{\{}0,{\bf i},{\bf j},{\bf k}\big{\}}
\frac{\partial h}{\partial t}=\alpha[\frac{\partial^{2}h}{\partial r^{2}}+\frac{1}{r}\frac{\partial h}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}h}{\partial\theta^{2}}+\frac{\partial^{2}h}{\partial z^{2}}]-G.
f({\bf z})=\sum_{{\bf J}=0}^\infty c_{{\bf J}}{\bf z}^{\bf J}\equiv \sum_{j_1=0}^\infty\sum_{j_2=0}^\infty ...\sum_{j_D=0}^\infty c_{j_1,j_2,...,j_D}\prod_{d=1}^D z_{d}^{j_d}.
\sum_{i=1}^{\infty}\frac{-24+\frac{2}{i^4}+\frac{2}{i^2}-\frac{4}{i}-\frac{\sum_{j=1}^i \frac{1}{j}}{2 i^3}+\frac{\sum_{j=1}^i \frac{1}{1+2 j}}{i^3}}{i \binom{2 i}{i}}=\frac{11 \zeta_5}{9}-16
\| q_{k_i -1}\theta \| - \left( \tilde r_{k_i+1}+1 \right) \| q_{k_i}\theta \| - \| q_{k_i+1}\theta \| = \left(a_{k_i+1}-1 - \tilde r_{k_i+1}\right) \| q_{k_i}\theta \| &> \frac{a_{k_i+1}-1 - \tilde r_{k_i+1}}{a_{k_i+1}+ 1}\| q_{k_i-1}\theta \| &\ge \frac{\| q_{k_i-1}\theta \|}{5}> \frac{|U|}{5},
L_{\mathrm{B R S}}^{\prime \prime}= \mathrm{}- \frac{1}{2}\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}- \frac{m^{2}}{2}A_{\mu}A^{\mu}- \frac{1}{2}\partial_{\mu}\theta \partial^{\mu}\theta - \frac{m^{2}}{2}\theta^{2}.
\Theta_j(t)\leq \left(\mathcal{P}_j (|t|)\right)^2 \sum\limits_{n=j+1}^{\infty}\frac{\left|\Delta^{k-j}c_{n-j}\right|^2}{n^{2j}}= \left(\mathcal{P}_j (|t|)\right)^2 \Omega_j.
T_{1/f}(x) = \begin{cases}x^{-1}R_{1/f}(x), \ 0<M<\frac1{2}; x^{-1}R_{1/f}(x) \log x, \M=\frac1{2}; x^{\log M/\log 2}\max(\log x, R_{1/f}(x)), \ \frac1{2}<M<1.\end{cases}
Y_l^{\pm k}(\widetilde\theta_1,\widetilde\varphi)=2^k\Gamma\left(k+\frac{1}{2}\right)\sqrt{\frac{(2l+1)(l-k)!}{\pi\,(l+k)!}}\, C_{l-k}^{k+\frac{1}{2}}(\cos\widetilde\theta_1)\sin^k\widetilde\theta_1\,e^{\pm ik\widetilde\varphi}
L_{\tilde{\xi}}\pi^*\omega = L_{\tilde{\xi}}i\partial\bar{\partial}\phi & =i\partial \bar{\partial}L_{\tilde{\xi}}\phi \intertext{and on the other hand,}L_{\tilde{\xi}}\pi^*\omega = \pi^* L_{\xi}\omega & =\pi^* i\partial \bar{\partial}\theta_{\xi}& = i\partial \bar{\partial}(\theta_{\xi}\circ \pi).
D_\lambda(x,y)=\frac{1}{2\alpha\sqrt{\lambda}}\begin{cases}\overline{\alpha}e^{i\sqrt{\lambda}|x-y|}, x\geq 0,\,y\geq 0, 0, x\geq 0,\,y<0, 0, x<0,\,y\geq 0, -\alpha e^{-\sqrt{\lambda}|x-y|}, x<0,\,y<0,\end{cases}
u_n(x)-u_n(y)= \begin{cases}\varphi_n(x)u_n(x) x\in B_n^c\cap B_{2n}, y\in B_{2n}^c, -\varphi_n(y)u_n(y) y\in B_n^c\cap B_{2n}, x\in B_{2n}^c, \varphi_n(x)u_n(x)-\varphi_n(x)u_n(y) x,y\in B_n^c\cap B_{2n}, 0 x,y\in B_{2n}^c. \end{cases}
\int\limits_t^T\left({\bf w}_s^{(i_1)}-{\bf w}_t^{(i_1)}\right)^{(q)}d{\bf w}_s^{(i_2)}=\sum_{j=0}^{q}\int\limits_t^T\phi_j(\tau)d{\bf w}_{\tau}^{(i_1)}\int\limits_t^T\int\limits_t^s\phi_j(\tau)d\tau d{\bf w}_{s}^{(i_2)}
\alpha_u(\mathcal{S}) = \{t \leq n_u(\mathcal{S}) \ | \S_u(t)^T k(t) = 1 R(t) \leq R_u(t)\}\beta_u(\mathcal{S}) = \{t \leq n_u(\mathcal{S}) \ | \S_u(t)^T k(t) \neq 1 R(t) > R_u(t)\}\gamma_u(\mathcal{S}) = \{t \leq n_u(\mathcal{S}) \ | \ \mathcal{X}_u(t) =1 R(t) \leq R_u(t) \},
_C\langle y\cdot d,y'\rangle=_C\langle y,y'\cdot d^*\rangle,\quad \langle c\cdot y,y'\rangle_D=\langle y,c^*\cdot y'\rangle_D_C\langle y,y'\rangle\cdot y'' =y\cdot \langle y',y''\rangle_D.
\left( L_{0},L_{0}\,|\,L_{+},A,L_{-}\right) =2\hbar^{3}\left[L_{0},A\right] +\hbar\left[ L_{-},\left[ L_{+},\left[ A,L_{0}\right]\right] \right] +\hbar\left[ L_{+},\left[ L_{-},\left[ A,L_{0}\right]\right] \right] \;,
I =I_{1}+I_{2}& =\frac{1}{2}\int_{-\infty}^{+\infty}dq\frac{\left\vert q\right\vert^{\alpha}\e^{i(\frac{n\pi x}{2a}+\frac{n\pi}{2})q}}{\ (q+1)(q-1)}\ +\frac{1}{2}\int_{-\infty}^{+\infty}dq\frac{\left\vert q\right\vert^{\alpha}\e^{i(\frac{n\pi x}{2a}-\frac{n\pi}{2})q}}{\ (q+1)(q-1)},
\tilde{\cal A}_{m}^{n}= \frac{3}{4}\left( \begin{array}{c c c c}{z^{- 2}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{z^{- 2}}\\ \end{array}\right) + \mathrm{m e a n f i e l d t e r m s}
\begin{aligned}|\psi_t|\omega =\psi^{1/2}\omega\frac{|\psi_t|}{\psi^{1/2}}&\leq\frac{1}{10}\left(\psi^{1/2}\omega\right)^2+c\left(\frac{|\psi_t|}{\psi^{1/2}}\right)^2 &\leq\frac{1}{10}\psi\omega^2+\frac{c}{(\tau-t_0+T)^2}.\end{aligned}
(q;q)_\infty (q;q^2)_\infty \sum_{n=0}^\infty p(n)q^n\! =(q;q^2)_\infty \mbox{and}(q;q)_\infty (-q;q^2)_\infty \sum_{n=0}^\infty p(n)q^n\! =(-q;q^2)_\infty.
\Lambda_{\pm}( 1 ) = \Lambda ( 1 ) \pm \frac{( 4 \kappa )^{3 / 2}( 1 - k )^{\kappa - \frac{1}{2}}}{( 2 \pi )^{1 / 2}2^{\kappa}}\bigg [ 1 + ( 1 - k ) \{\kappa ( \frac{1}{2}- \ln 2 ) + \frac{1}{4}\}+ O \bigg ( \frac{1}{\kappa}\bigg ) \bigg ]
\hat{\cal{L}}_{S D}^{S}= \frac{1}{2}( \hat{f}^{\mu}+ \hat{h}^{\mu}- \partial^{\mu}\hat{\phi}) ( \hat{f}_{\mu}+ \hat{h}_{\mu}- \partial_{\mu}\hat{\phi}) - \frac{1}{2 m}\epsilon^{\alpha \beta \gamma}( \hat{f}_{\alpha}\partial_{\beta}\hat{f}_{\gamma}+ 2 \hat{h}_{\alpha}\partial_{\beta}\hat{f}_{\gamma}) .
H_NS_N(i,j) =\frac{1}{J_N}(-J_1J_{N-2}+2J_1(J_{N-2}+J_{N-1})-2J_1J_{N-3}) =\frac{1}{J_N}(-J_{N-2}+2(J_{N-2}+J_{N-1})-2J_{N-3}) =\frac{1}{J_N}(-2J_{N-3}+J_{N-2}+2J_{N-1}) =\frac{1}{J_N}(2J_{N-2}+J_{N-1}) =1.
\Delta\log u=\frac{2\left\Vert \alpha_{2}\right\Vert^2}{(K_1^\perp)^2}(K_2^{\perp}-\hat{K}_2^{\perp})+\frac{2\left\Vert \hat{\alpha}_{4}\right\Vert^2}{4\hat{K}_2^{\perp}}
\overset{3}{R}^{+}(\tau, z)= \frac{\eta(\frac{\tau}{2})\eta(2\tau)\vartheta_{11}(\tau,z)}{\vartheta_{00}(\tau,z)}\,\,\,\hbox{and}\,\,\,\overset{3}{R}^{-}(\tau, z)=\frac{\eta(\tau)^3 \vartheta_{11}(\tau,z)}{\eta(\frac{\tau}{2})\vartheta_{01}(\tau,z)}
\operatorname{Tr}f_{E}(Q_1)=\frac{N}{2 \pi}\int_{\mathbb{R}^2}(i \sigma f^{\prime \prime}_E(e) \chi(\sigma)+if_E(e)\chi^{\prime}(\sigma)-\sigma f_{E}^{\prime}(e) \chi^{\prime}(\sigma))m_2(e+i\sigma)de d\sigma.
\hat{\mathbb{E}}_t[\varphi(X_{t+t_1}^{x},X_{t+t_2}^{x},\cdots,X_{t+t_{m}}^{x})] =\hat{\mathbb{E}}_t[\varphi_{m-1}(X_{t+t_1}^{x},X_{t+t_2}^{x},\cdots,X_{t+t_{m-1}}^{x})] &\ \vdots &=\hat{\mathbb{E}}_t[\varphi_{1}(X_{t+t_1}^{x})] &=\hat{\mathbb{E}}[\varphi_{1}(X_{t_1}^{y})]_{y=X_{t}^x},
DI^{\gamma,\alpha}_g(u_\alpha)\cdot\phi= \int_{X}\rho^{1-2\gamma}\langle\nabla u_\alpha,\nabla \phi\rangle_g\,dv_g+\int_MQ^\gamma_\alpha u_\alpha\phi \, d\sigma_{\hat{h}}-\int_M u_\alpha^{2^*-1}\phi \,d\sigma_{\hat{h}}= \,o(\|\phi\|_{W^{1,2}(X,\rho^{1-2\gamma})}) \ \ \hbox{as}\ \alpha\rightarrow+\infty.
\smash[b]{|v^{(k)}|\prod_{j=\Delta^{(k)}-m}^{\Delta^{(k)}-1}|w^{(k)}_j|}\ge C \intertext{and}\smash[t]{|v^{(k)}| \prod_{j=m'+1}^{\Delta^{(k)}-1}|w^{(k)}_j|}>C\quad\textrm{for every}\ \ 0\le m'\le \alpha \Delta^{(k)}.
J(x, y) := \kappa(x,y) j(|x-y|) = \begin{cases}\kappa(x,y) |x-y|^{-d-\alpha}\psi_1 (|x-y|)^{-1}\beta \in [0, \infty), \kappa(x,y) |x-y|^{-d-\alpha}{\bf 1}_{\{|x-y| \le1\}}\beta = \infty. \end{cases}
\prod_{i=1}^N{\frac{\Gamma_q(\lambda_i + \theta(N - i) - z)}{\Gamma_q(\lambda_i + \theta(N - i + 1) - z)}}\qquad{}= \prod_{i, j}{[\theta (N - i) + j - z]_q}\frac{[\lambda_N - z - 1]_q\cdots [-z]_q}{[\lambda_1 + \theta N - z - 1]_q\cdots [\theta N - z]_q}\qquad\quad{}\times\frac{\Gamma_q(-z)}{\Gamma_q(\theta N - z)},
u_{1}=t^{-1}y+t^{-2}+c_{2}/2+\exp\left[ c_{2}^{-1}t^{-2}(tx+ty+1)+1/2\right] , & u_{2}=t^{-1}x+t^{-2}+c_{2}/2-\exp\left[ c_{2}^{-1}t^{-2}(tx+ty+1)+1/2\right] , & p=-t^{-2}(x-y)^{2}/2+(t^{-3}-c_{2}t^{-1}/2)(x+y),
T^{1}=\frac{w^{2}q_{/2}+vq}{\delta w},\,\, T^{2}=\frac{2\delta\,w^{2}q_{/1}+\delta'\,q(\delta^{2}-v^{2})}{2\delta^{2}\, w}+\frac{T^{1}(\kappa w^{2}+\delta'\,v-\delta^{2}\lambda)}{\delta}.
f[x_0,x_1,\dots,x_k] =\int_{\Sigma_k}f^{(k)}\Big( t_0 x_0 +t_1x_1 +\dots t_k x_k\Big) dt & \ge \frac 1{k! L^k}\int_{-0.9}^{0.9}f^{(k)}(s) ds \ge \frac 1{L^k}.
p^*\left(L'_{i,j}\right)=\frac{A'_{i-1,j-1}}{A'_{i,j}}\prod_{k=i-a}^{b-j}\lambda'_k=\frac{A'_{i-1,j-1}}{A'_{i,j}}\prod_{k=i-a-1}^{b-j-1}\lambda_k.
{\cal L}_{q}( x , x^{\prime}) = \bar{q}_{\alpha \alpha^{\prime}}( x , x^{\prime}) \left( i \hbar c \gamma^{\mu}\partial_{\mu}- m c^{2}\right)_{\alpha \beta}\left( i \hbar{c}^{\prime}\gamma^{\mu^{\prime}}\partial_{\mu^{\prime}}- m{c^{\prime}}^{2}\right)_{\alpha^{\prime}\beta^{\prime}}q_{\beta \beta^{\prime}}( x , x^{\prime})
\sum_{j_{12}}(-1)^{I_{12}(I_{123}+1)}\left\{\begin{array}{ccc}j_1 j_2 j_{12}j_3 j j_{23}\end{array}\right\}^S~ \left\{\begin{array}{ccc}j_1 j_2 j_{12}j_3 j j'_{23}\end{array}\right\}^S=(-1)^{I_{23}(I_{123}+1)}\delta_{j_{23}\,j^{'}_{23}}~.
\Bigl[\varphi\bigl(V(t)\bigr),\varphi(X)\Bigr]^i(x) =\sum_{k=1}^n\Bigl(\varphi\bigl(V(t)\bigr)\Bigr)^k(x) \frac{\partial \bigl(\varphi(X)\bigr)^i(x)}{\partial x^k}-\sum_{k=1}^n\bigl(\varphi(X)\bigr)^k(x) \frac{\partial \Bigl(\varphi\bigl(V(t)\bigr)\Bigr)^i(x)}{\partial x^k}