harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal F}_{\mathrm{F}= 0}= ( 1 - \mathrm{N}^{2}) \frac{e_{+}^{2}}{2{\pi}^{2}{\hbar}^{2}}\sum_{n > 0}\frac{1}{n}\sin ( \frac{2 \pi}{2 \pi}{L}n ( x - y ) ) = ( 1 - \mathrm{N}^{2}) \frac{e_{+}^{2}}{2 \pi{\hbar}^{2}}( \frac{1}{2}\epsilon ( x - y ) - \frac{1}{1}{L}( x - y ) ) .
L = E^{- 1}\left( \begin{array}{c c}{\frac{\sqrt{1 - 3 V_{o o}}}{\sqrt{1 + V_{o o}}}}&{0}\\{0}&{\frac{\sqrt{1 - V_{e e}}}{\sqrt{1 + 3 V_{e e}}}}\\ \end{array}\right) E^{- 1}.
\Sigma(Q,N;\mathcal{S}) \ll NZ\times \max\limits_{\substack{q_2\in \mathcal{S}Q/2<\mathcal{N}(q_2)\le Q r_2 \bmod q_2 (r_2,q_2)=1}}\sharp\left\{q\in \mathcal{S}\ : \Q/2<\mathcal{N}(q)\le Q, \ \binom{f\left((uk+vl)/\mathcal{N}(q_2)\right)}{f\left((-vk+ul)/\mathcal{N}(q_2)\right)}\in D_R({\bf 0})\right\},
\langle a_\alpha \otimes (b_\alpha - b), a_\alpha \otimes (b_\alpha - b) \rangle \ = \ (b_\alpha - b)^* \,\varphi (a_\alpha^* a_\alpha) \,(b_\alpha - b) \ \leq \B^2 \|\varphi\| \,(b_\alpha - b)^* (b_\alpha - b),
\left. S_{n , m}( x_{1}q_{1}, . . . . . . , x_{n}q_{n}; p_{1}. . . . . . p_{m}) = \langle{\cal D}( x_{1}q_{1}, . . . . . . , x_{n}q_{n}) \psi_{p_{1}}. . . . . . . \psi_{p_{m}}\rangle \right.
v=\sqrt{\dot{\varphi}^{2}+(\sin^{2}\varphi)\dot{\theta}^{2}}\dot{v}=\frac{d}{dt}v=\frac{1}{v}[\dot{\varphi}\ddot{\varphi}+(\sin\varphi\cos\varphi)\dot{\varphi}\dot{\theta}^{2}+\sin^{2}(\varphi)\dot{\theta}\ddot{\theta}]
u [a_1 a_4, a_2, a_3] = u \bigl( [a_1, a_2, a_3] a_4 + [a_1, a_2] [a_4, a_3] + [a_1, a_3] [a_4, a_2 ] + a_1 [a_4, a_2, a_3] \bigr) .
E_{i,j}^{(r)}:= [E_{i, j-1}^{(r - s_{j-1,j})}, E_{j-1}^{(s_{j-1, j}+ 1)}] 1 \leq i < j \leq n r > s_{i,j}, \smallskip F_{i,j}^{(r)}:= [F_{j-1}^{(s_{j,j-1}+ 1)}, F_{i,j-1}^{(r-s_{j, j-1})}] 1 \leq i < j \leq n r > s_{j,i}.
\Theta e_0 \;=\; (y_0^2+y_1^2+y_2^2+y_3^2)e_0 & \;=\; (bcy_0-iy_1-iby_2-cy_3)e_1 & \;=\; \big((bc)^2-1+b^2-c^2\big)e_2 & \;=\; (\b-1)(\c+1)e_2
RA_{2}^{+}:=\left\{0,1,\frac{1}{2},\frac{1}{4},\frac{3}{4},\frac{1}{8},\frac{3}{8},\frac{5}{8},\frac{7}{8},\frac{1}{16}\ldots,\frac{k}{2^{n}},\ldots\right\}\;\mbox{where}S\left(n\right) :=\left\{k:k=1,3,\dots,\;\mbox{\ensuremath{\left(odd\right)}}<2^{n}\right\},\; n\geq1.
=-e^{-\lambda t}\sum_{k=0}^\infty I_{2k+2}\Bigl(\frac{2\lambda}{c_1+c_2}\sqrt{(c_1t-\beta)(c_2t+\beta)}\Bigr)\Bigl(\frac{c_2}{c_1}\Bigr)^{2k+1}\Biggl(\sqrt{\frac{c_1t-\beta}{c_2t+\beta}}\Biggr)^{2k+2}
\begin{cases}u''-\Delta u + au +bv=0 \hbox{in}(0, +\infty)\times \Omega, v''-\Delta v + cu+dv =0 \hbox{in}(0, +\infty)\times \Omega, u=v=0 \hbox{on}(0, +\infty)\times \Gamma_0, \partial_\nu u +\alpha u=0 \hbox{on}(0, +\infty)\times \Gamma_1, \partial_\nu v + \beta v =0 \hbox{on}(0, +\infty)\times \Gamma_1.\end{cases}
\hat{\chi}= \sum_{M = 1}^{n}\left( \bar{z}_{+}^{M}\, \, \bar{z}_{-}^{M}\right) \Big ( \begin{array}{c c}{\mathrm{e}^{V}}\end{array}\Big ) \left( \begin{array}{c}{z_{+}^{M}}{z_{-}^{M}}\end{array}\right) \ ,
Q_{(q_1,\ldots, q_k)}\Phi_{T_1}^{p_1}\circ \cdots \circ \Phi_{T_k}^{p_k}({\bf \Delta_{T}}(I))=\sum_{s_1=c_1}^{d_1}\cdots \sum_{s_k=c_k}^{d_k}Q_{(q_1,\ldots, q_k)}\Phi_{T_1}^{p_1}\circ \cdots \circ \Phi_{T_k}^{p_k}(D_{(s_1,\ldots, s_k)})
q_{2i}= p_{2i}+\omega^{a_{h(2i)}(2i)}\left(\frac{z}{2}\right)^{h(2i)}(z^1+z^2+\dots+z^{k-h(2i)}) q_{2i+1}= p_{2i+1}+\omega^{a_{h(2i+1)}(2i+1)}\left(\frac{z}{2}\right)^{h(2i+1)}(z^1+z^2+\dots+z^{k-h(2i+1)})
I_{j}(x)= \int e^{ir^a}\chi_{j}(r)r^{\frac{2a}{q}-1}(r|x|)^{-\frac{d-2}{2}}J_{\frac{d-2}{2}}(r|x|)dr, d\geq 2, I_{j}(x)= \int e^{i|\xi|^a}e^{ix\xi}\chi_{j}(\xi)|\xi|^{\frac{2a}{q}-d}d\xi, d=1.
(z-\mu^{(t)}){\cal H}_{k-1}^{(s,t+1)}(z)=\left(\frac{H_{k-1}^{(s,t)}H_{k}^{(s,t+1)}}{H_{k}^{(s,t)}H_{k-1}^{(s,t+1)}}\right){\cal H}_{k-1}^{(s,t)}(z)+{\cal H}_{k}^{(s,t)}(z),
w-\widetilde{w}=\left(\bar{u}_{xx}+\frac{1}{2 i}|\tilde{u}_x|^2 \bar{u}_x\right)e^{ic_-(x)}-\left(\bar{\tilde{u}}_{xx}+\frac{1}{2i}|\tilde{u}_x|^2 \bar{\tilde{u}}_x\right) e^{ic_-(x)}(\tilde{u}).
\exists \mu>0, \sup_{u\in\mathcal{X},v\in\mathcal{Y}}\frac{B(u,v)}{\|u\|_{\mathcal{X}}\|v\|_{\mathcal{Y}}}= \mu; \exists \beta>0,\inf_{u\in\mathcal{X}}\sup_{v\in\mathcal{Y}}\frac{B(u,v)}{\|u\|_{\mathcal{X}}\|v\|_{\mathcal{Y}}}= \beta ; v\in \mathcal{Y},(\forall u\in\mathcal{X},\B(u,v)=0) \Longrightarrow (v=0).
\int_{\rho}^{2\rho}\frac{\exp \left( -\frac{u^2}{4t^2}\right)}{\sqrt{\cosh u - \cosh \rho}}\d u \leq \exp \left(-\frac{\rho^2}{4t^2}\right)\int_{\rho}^{2\rho}\frac{\d u}{\sqrt{(u-\rho)\rho}}= 2 \exp \left(-\frac{\rho^2}{4t^2}\right).
\gamma_{\mathfrak n}(x,v,w) =B\left(d'(x,v),w \right)=B_{\mathfrak a}(\rho(x)w,v) \gamma_{\mathfrak n}(v,w,x) =B\left(d'(v,w),x \right)=B_{\mathfrak a}\left(\rho(x)v,w \right)
\psi(e^{-\pi}) = a 2^{-5/8}e^{\pi/8}, \psi(e^{-2\pi}) = a 2^{-5/4}e^{\pi/4}, \psi(e^{-\pi/2}) = a 2^{-7/16}(\sqrt{2}+1)^{1/4}e^{\pi/16},
F(\mu, y) = 2\mu\tau_0 + \frac{2i}{\pi}\int_0^\pi \frac{d \theta}{\cos \theta}\, \ln \left[ \frac{\Gamma(1-i\mu \cos \theta)}{\Gamma(1-i\mu y \cos\theta)\Gamma(1-i\mu (1-y) \cos \theta)}\right].
\frac{\zeta(2)}{2}=\frac{\pi^{2}}{12}=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{2}}=\frac{1}{1^{2}}-\frac{1}{2^{2}}+\frac{1}{3^{2}}-\frac{1}{4^{2}}+\cdot\cdot\cdot
\omega_{2n+2|0}= \begin{bmatrix}-P_{2n|0}^{-1}h_{2n+2|0}^{(1)}P_{2n|0}^{-1}\left(P_{2n|1}P_{2n|0}^{-1}h_{2n+2|0}^{(1)}- 2 h_{2n+2|0}^{(2)}\right) P_{2n|0}^{-1}\left(h_{2n+2|0}^{(2)}- 2h_{2n+2|0}^{(3)}\right) M_0\left(-2P_{2n|1}P_{2n|0}^{-1}h_{2n+2|0}^{(1)}+ 3h_{2n+2|0}^{(2)}+ 2h_{2n+2|0}^{(3)}\right) - 4P_{2n|0}^{-1}h_{2n+2|0}^{(4)}\end{bmatrix}.
\det(\mathbf{D}_{i,j})=\begin{cases}\prod\limits_{k=1}^{\frac{m}{2}}(-c_{2k-1}^{2}), m\equiv 0\bmod 2; (-2e)\times \prod\limits_{k_{1}=1}^{\frac{s-1}{2}}(-c_{2k_{1}-1}^{2}) \times \prod\limits_{k_{2}=\frac{s+1}{2}}^{\frac{m-1}{2}}(-c_{2k_{2}}^{2}), m\equiv 1\bmod 2.\end{cases}
\mathbb{E}\Big\lVert \sum_{j=1}^n Z_j V_{\bullet, j}\Big\lVert_2^2= \mathbb{E}\Big[ \sum_{i,j=1}^n Z_iZ_j \langle V_{\bullet, i}, V_{\bullet, j}\rangle_2 \Big]= \mathbb{E}[Z_1^2] \sum_{j=1}^n \langle V_{\bullet, i}, V_{\bullet, j}\rangle_2= k \mathbb{E}[Z_1^2].
\rho_i (z) = 2h_{E_{3-i},\mathfrak{p}}(f_{3-i}(z))-h_{E_i ,\mathfrak{p}}(f_i (z)+Q_i )-h_{E_i ,\mathfrak{p}}(f_i (z)-Q_i ) &-2\alpha_{3-i}h_{E_{3-i}}(f_{3-i}(z))+2\alpha_i (h_{E_i}(f_i (z))+\log_{E_i}(Q_i )^2)\in \Omega_i,
\lim_{n\to\infty}\frac{\tilde{H}(n,n)}{n}= \max_{\kappa\in[0,1]}\left\lbrace \frac{1-\kappa}{\alpha}+ (1+\sqrt{\kappa})^2 \right\rbrace =\begin{cases}4 \mbox{if}\alpha\in(1/2, 1), \frac{1}{\alpha(1-\alpha)}\mbox{if}\alpha \leqslant 1/2.\end{cases}
R_{j,y}^{s}=\sum_{k \in \Psi}\sum_{i \in \Omega_j}\log_2 \left(1+\frac{(|H_i^{pp}| \sqrt{T_i}+|H_{k,i}^{sp}| \sqrt{\alpha_k P_{y,i}})^2}{N_o+|H_{k,i}^{sp}|^2 (1-\alpha_k) P_{y,i}}\right).
\rho_n(E) =-\frac{1}{4\pi^2}\int_{c-\mathrm{i}\infty}^{c+\mathrm{i}\infty}\mathrm{d}\beta\, \mathrm{e}^{\beta\, E}\oint \frac{Z(\beta, x)}{x^{n+1}}\mathrm{d}x\ .
F(n,k)=\frac{16^{k-4 n}\left(120 n^2-84 k n-10 k+34 n+3\right) \binom{2 n}{n}^3\binom{2 k+4 n}{k+2 n}\binom{k+2 n}{n}\binom{k+n}{2 k}\binom{n}{k}}{\binom{2 k}{k}^2}
\kappa_2(x) = x^{4n}- 2x^{3n}- x^{2n+1}- 2x^{2n}- 3x^{2n-1}- 2x^{n}+ 1, \intertext{and}\kappa_1(x) = 4x^{2n}+ 2x^{n+1}+ 4x^{n}+ 2x^{n-1}+ 4.
\beta_{n}^{R^{[1]}}= q+\alpha_{1}+\alpha_{2}-a\beta \; , n \geq 0 \alpha_{n}^{R^{[1]}}= \frac{n(n+3)}{(n+1)(n+2)}\left( \alpha_{1}\alpha_{2}- a\gamma+\beta\gamma \right) \; , & \gamma_{n}^{R^{[1]}}= \frac{n(n+5)}{(n+2)(n+3)}\gamma^{2}\; , \; n \geq 1.
f_3 \, f_1 \, f_4 = \omega^N x_1^{p_1}\cdots x_n^{p_n}x_j= \omega^{N + 2\sum_{i>j}\varepsilon_{ij}p_i}\, x_1^{p_1}\cdots x_j^{p_j+1}\cdots x_n^{p_n}f_3 \, f_2 \, f_4 = \omega^{N'}x_1^{p_1'}\cdots x_n^{p_n'}x_j = \omega^{N' + 2\sum_{i>j}\varepsilon_{ij}p_i'}\, x_1^{p_1'}\cdots x_j^{p_j'+1}\cdots x_n^{p_n'}.
B^{m}(t\wedge\tau_{m}) =B_0^{m}+\int_0^{t\wedge\tau_{m}}p\left(S^{m}(u)\right)du+\int_0^{t\wedge\tau_{m}}r\left(S^{m}(u)\right)dZ(u), V^{m}(t\wedge\tau_{m},x) =V_0^{m}+\int_0^{t\wedge\tau_{m}}\mu\left(S^{m}(u);x\right)du+\sum_{i\in\mathcal{I}_{m}}f_i(x)\sum_{j\leq i}\int_0^{t\wedge\tau_{m}}d_{ij}\left(S^{m}(u)\right)dW^j(u),
P(x,q) = \frac{1+x}{x^2}+ \sum_{n=1}^\infty \frac{nq^n}{1-q^n}\left[ (1+x)^{n}+ (1+x)^{-n}\right] Z(x,q) = \frac{1+x}{x}- \sum_{n=1}^\infty \frac{q^n}{1-q^n}\left[ (1+x)^{n}- (1+x)^{-n}\right].
\sum\limits_{\substack{k=1}}^{n^s}\psi(k;n^s)(k,n^s)_s = \sum\limits_{\substack{k=1}}^{n^s}\psi(k;n^s)\sum\limits_{d^s|(k,n^s)_s}\Phi_s(d^s) =\sum\limits_{\substack{k=1}}^{n^s}\psi(k;n^s)\sum\limits_{\substack{d^s\mid k d^s\mid n^s}}\Phi_s(d^s).
w_{2+f_{3,opt}}=\omega_{2}+f_{3,opt}\nu_{3}=\omega_{2}+\upsilon_{3}T_{3}f_{3,opt}=\omega_{2}+\upsilon_{3}T_{3}\frac{\rho_{3}^{2}}{\rho_{3}^{2}+|\upsilon_{3}|^{2}}
\left|\Delta_C\Delta_B(A)\right| =\left|\big((A+B)\setminus A\big)+C\right|- \left|(A+B)\setminus A\right| &\geq \left|(A+B+C)\setminus (A+C)\right|- \left|A+B\right|+\left|A \right| &= |A+B+C|- |A+C|- |A+B|+ |A|,
B_{(\chi,\chi')}(Q):=\Big(\mbox{vec}(Q_{\chi}^T{\cal J}_{x_1}F(\overline{x})Q_{\chi'}) \,\, \cdots \,\,\mbox{vec}(Q_{\chi}^T{\cal J}_{x_n}F(\overline{x})Q_{\chi'})\Big)\,
\delta \upsilon^{k}= - \frac{\Lambda}{1 2}\epsilon^{k}+ \rho^{\sigma}\omega_{\sigma}^{k}\ , \qquad \delta \theta_{\mu}^{k}= D_{\mu}\epsilon^{k}+ 2 \rho^{\nu}\Sigma_{\nu \mu}^{k}\ , \qquad \delta \xi^{\mu}= - \rho^{\mu}\ ,
h^{p,q}(V) = \begin{cases}1 (p,q) = (0,0) k - 1 (p,q) = (1,0) 0 q \neq 0\end{cases}h^{p,q}_c(V) = \begin{cases}1 (p,q) = (1,1) k - 1 (p,q) = (0,1) 0 q \neq 1.\end{cases}
|P|^{eq^{n}-(q-1)\sum_{k=1}^{n-1}q^{k}\min\{e,\lfloor\frac{k}{d}\rfloor\}}= |P|^{eq^{n}-\sum_{k=1}^{q^{n}-1}\min\{e,\lfloor\frac{k}{q^d}\rfloor+\lfloor\frac{k}{q^{2d}}\rfloor+\cdots\}}.
\langle \:{\hat{\phi}}_{\bar{t}}\prod_{i \in S}\phi_{i}\: \rangle_{g , \:{\Delta_{2}}}= \frac{1}{2}\: \sum_{r = 0}^{g}\: \sum_{X \cup Y = S}{\bar{C}}_{\bar{t}}^{\alpha \beta}\: \langle \phi_{\alpha}\: \prod_{j \in X}\phi_{j}\rangle_{r}\: \langle \phi_{\beta}\: \prod_{k \in Y}\phi_{k}\rangle_{g - r}\,
\frac{\partial}{\partial t}x^3=-\frac{2\bar{\xi}}{(1+\xi\bar{\xi})^2}\frac{\partial\eta}{\partial t}-\frac{2\xi}{(1+\xi\bar{\xi})^2}\frac{\partial\bar{\eta}}{\partial t}+\frac{1-\xi\bar{\xi}}{1+\xi\bar{\xi}}\frac{\partial r}{\partial t}.
2DM\sum_{r=0}^\infty\left(1-\frac{1}{2DM}\right)^r c\left(1+\frac{1}{4DM}\right)^r \geq 2cDM\sum_{r=0}^\infty \left(1-\frac{1}{4DM}\right)^r = 8cD^2 M^2.
G_{\mathcal R}(x,y)= \frac{i}{4\pi}\bigg\{\int^{\pi}_{0}F(\eta,x,y) d\eta+ \int_{+\infty}^{0}F(i\eta, x,y) i d\eta + \int^{-\infty}_{0}F(i\eta+\pi, x,y)id\eta\bigg\}= \frac{i}{4\pi}\int_{\mathcal L}F(\zeta, x, y) d\zeta,
L_{f}=\overline{Q}_{i}iD/Q_{i}+\overline{u}_{i}^{c}iD/u_{i}^{c}+\overline{d}_{i}^{c}iD/d_{i}^{c}+\overline{L}_{i}iD/L_{i}+\overline{e}_{i}^{c}iD/e_{i}^{c}
f(x,t) = \frac{1}{2 \pi}\int_{\mathbb{R}}e^{-i \beta x}E_{\nu, 1+\frac{2H-1}{\nu}, \frac{2H-1}{\nu}}\left(c_k 2H (-i\beta)^k t^{\nu+2H-1}\right) \mathrm d\beta.
H ( z_{2}^{\prime}, z_{3}^{\prime}, P_{2}^{\prime}, P_{3}^{\prime}) = - \int_{z_{0}}^{z_{A}^{\prime}}\frac{\partial \beta_{A}}{\partial \mu}( z_{2}^{\prime}, z_{3}^{\prime}, z_{A}^{\prime \prime}) d z_{A}^{\prime \prime}+ f ( z_{2}^{\prime}, z_{3}^{\prime})
P_\alpha(t) = \ \frac{1}{\alpha}\int_0^t f_\alpha(u^{-1/\alpha})\, u^{-1/\alpha-1}\, du = \int_{t^{-1/\alpha}}^\infty f_\alpha(y)\, dy &= 1- \int_0^{t^{-1/\alpha}}f_\alpha(y)\, dy \equiv 1- F_\alpha(t^{-1/\alpha}) = 1- F_\alpha(1\vert t)
{\cal L}_{\pi \sigma}= \bar{\psi}^{a}\left( i \gamma^{\mu}\partial_{\mu}- \sigma - i \pi \gamma_{5}\right) \psi^{a}+{\frac{N m_{0}}{g_{0}}}\sigma -{\frac{N}{2 g_{0}}}\left( \sigma^{2}+ \pi^{2}\right) .
V_{1}=\sum_{\lambda \in \Sigma^{+}\setminus \Sigma_{H}}\mathfrak{m}_{\lambda}\oplus \sum_{\alpha \in W^{+}\setminus W_{H}}V_{\alpha}^{\perp}(\mathfrak{m}_{1}\cap \mathfrak{k}_{2}), V_{2}=\sum_{\lambda \in \Sigma^{+}}\mathfrak{k}_{\lambda}\oplus \sum_{\alpha \in W^{+}}V_{\alpha}^{\perp}(\mathfrak{k}_{1}\cap \mathfrak{m}_{2}).
\frac{\partial x_r}{\partial z}(z,\tau)=2\pi i x_r\left( -r + n \sum_{m \geq 0}\frac{q^{nm+r}u^{-n}}{1-q^{nm+r}u^{-n}}-n \sum_{m \geq 1}\frac{q^{nm-r}u^{n}}{1-q^{nm-r}u^{n}}\right) ,
\mathbf{N}_E:=\mathbb{Z}^E/ \langle\mathbf{e}_E\rangle, \mathbf{M}_E:=\mathbf{e}_E^\perp \subset \mathbb{Z}^E, \langle -,- \rangle:\mathbf{N}_E \times \mathbf{M}_E \longrightarrow \mathbb{Z}.
{f_{cor}}\left( \gamma \right) = A{\left({\frac{\eta}{{\bar \gamma}}}\right)^U}{\gamma^{U - 1}}{e^{- \frac{\eta \gamma}{{\bar \gamma}}}}&\times \sum\limits_{k = 0}^\infty{{D_k}}{}_1{F_1}\left({mM+k,U;\frac{{\eta \gamma}}{{\bar \gamma \left({1 +{\lambda^{- 1}}}\right)}}}\right),
\Gamma_{\nu}(p) = - \int \frac{d^3q}{(2\pi)^3}D(p-q) \, \gamma_\mu \, S\left[-\left(m+\frac{\delta}{2}\right)iw+q\right] \Gamma_{\nu}(q) \, S\left[\left(m+\frac{\delta}{2}\right)iw+q\right] \gamma_\mu.
{}\displaystyle\sum_{n=1}^{\infty}| A_n-A_{n-1}|^{2}\geq\displaystyle\sum_{n=1}^{\infty}w_n|A_n|^{2}>\frac{1}{4}\displaystyle\sum_{n=1}^{\infty}\frac{|A_n|^{2}}{n^{2}},
B[\widehat \psi, \widehat \eta, \widehat w, \widehat \zeta] = \begin{pmatrix}B_1 [\widehat \psi, \widehat \eta, \widehat w, \widehat \zeta] B_2 [\widehat \psi, \widehat \eta, \widehat w, \widehat \zeta] B_3 [\widehat \psi, \widehat \eta, \widehat w, \widehat \zeta] \end{pmatrix},
(\tau^{s,0})^2 =\sigma_W^2+\sum_{p=1}^P\delta_p^{-1}\mathbb{E}\left[\left(\eta_{s,0}(X+\tau^{s,0}Z)-X\right)^2\right] &=\sigma_W^2+\delta^{-1}\mathbb{E}\left[\left(\eta_{s-1,0}(X+\tau^{s-1,0}Z)-X\right)^2\right],
\mathcal{W}_{p}(\mu,\nu)=W_{p}(\bar{\mu}_{u},\bar{\nu}_{v})\le W_{p}(\bar{\mu}_{u},\bar{\zeta}_{t})+W_{p}(\bar{\zeta}_{t},\bar{\nu}_{v})=\mathcal{W}_{p}(\mu,\zeta)+\mathcal{W}_{p}(\zeta,\nu).
\begin{aligned}2\psi(\mu-g)\omega^2 \leq\Bigg\{\left(\frac{2(g-\ln D)}{\mu-g}\nabla g\cdot\nabla\psi\right)\omega+2\frac{|\nabla\psi|^2}{\psi}\omega-(\Delta_f\psi)\omega &\quad\,\,\,+\psi_t\omega+2(a+(n-1)K)\psi\omega+\frac{2a\,g}{\mu-g}\psi\omega\Bigg\}.\end{aligned}
\left(\begin{array}{c}u_1(t + 2\bar{T}) u_2(t +2 \bar{T}) u_3(t + 2\bar{T}) \end{array}\right)=\left(\begin{array}{ccc}1 0 0 0 \cos (\frac{4\pi p}{n}) -\sin (\frac{4\pi p}{n}) 0 \sin (\frac{4\pi p}{n}) \cos (\frac{4\pi p}{n}) \end{array}\right)\left(\begin{array}{c}u_1(t) u_2(t) u_3(t)\end{array}\right),
\begin{array}{}{{t ]{c}{\displaystyle{\mathop{{\sum}^{\prime}}}}}}\\{{{\scriptstyle 1 \leq{n_{1}}<{n_{2}}< \cdots <{n_{m}}\leq D - 1}}}\\ \end{array}= \begin{array}{}{{t ]{c}{\displaystyle{\mathop{{\sum}}}}}}\\{{{\scriptstyle 1 \leq{n_{1}}<{n_{2}}< \cdots <{n_{m}}\leq D - 1}}}\\{{{\scriptstyle{l_{j}}{\not{|}}{n_{i}}}}}\\ \end{array}\, \, \, .
\hat{\delta}(f)(m \otimes \{a_1 | ... |a_n \}) := d_Nf ( m \otimes \{a_1 | ... |a_n \}) - (-1)^{|f|}f ( b_M ( m \otimes \{a_1 | ... |a_n \})) & + (-1)^{|m|+ |a_1| + ... + |a_{n-1}| - n+1}f( m \otimes \{a_1 | ... | a_{n-1}\}) \cdot a_n.
\big< \theta(t), f_{\lambda}'(t) \big> = - \big< \theta(t), \theta'(t) \big> - \lambda \cdot \coth \rho_{\lambda}(t) \bigg( 2 \big< \theta(t), f_{\lambda}(t) \big> + \big< \theta(t), f_{\lambda}(t) \big>^{2}\bigg).
\mathrm{Lift}(f) = \left\lbrace \prod_{\alpha=1}^d z^{i_{\alpha}}_{j_{\alpha}k_{\alpha}}- \prod_{\alpha=1}^d z^{i_{\alpha}}_{j'_{\alpha}k_{\alpha}}\mid k_{\alpha}\in [t_{i_{\alpha}}] \alpha \in [d]\right\rbrace.
\sum_{i=1}^{k}\left(-\epsilon_{1}^{2}\Delta U_{\epsilon,y_{i}}^{(i)}+V(x)U_{\epsilon,y_{i}}^{(i)}\right)+(-\epsilon_{1}^{2}\Delta\varphi_{\epsilon}+V(x)\varphi_{\epsilon})=\Big(\sum_{i=1}^{k}U_{\epsilon,y_{i}}^{(i)}+\varphi_{\epsilon}\Big)^{p}.
\lim_{n\rightarrow \infty}\int_{\mathbb R^N}\frac{f\left(t_{\varepsilon_n}\Psi_{\varepsilon_n,y_n}\right)}{t_{\varepsilon_n}\Psi_{\varepsilon_n,y_n}}\Psi_{\varepsilon_n,y_n}^2=0,
g(s_t, a_t) = \max_{k_t\!, \alpha_t\!, \tilde{p}_t\!, \mathbf{\emph{p}}_t}\alpha_t\!\log_2\!\Big(\!1\!+\!\frac{\tilde{p}_t|H_{t,\tilde{i}k_t}|^2}{\sigma^2_n}\!\Big) \!+ \! (\!1\!-\!\alpha_t\!)\sum_{i=1}^{2}\log_2\Big(\!1\!+\!\frac{p_{t,i}}{\sigma^2_n}\!\Big),
|\partial_{t}^{2}\left(\langle\left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}}(r))-1}{r^{2}}\right)v_{3}\vert_{r=R\lambda(t)}, \phi_{0}\rangle\right)| \leq \frac{C}{t^{3}\log^{1+2\alpha b-3b}(t)}
(n,m)\in Y^{(\omega)}\Longleftrightarrow \exists \; X \in \mathrm{ARITH}(Y) \; X =\oplus_{i=1}^{n}Y^{(i)}\; \& \; m\in X \Longleftrightarrow \forall \; X \in \mathrm{ARITH}(Y) \; X =\oplus_{i=1}^{n}Y^{(i)}\rightarrow m\in X
\frac{\Delta_{i,i+1}(u^{(n)}w_0^{(n)})}{\Delta_i(u^{(n)}w_0^{(n)})}=\sum_{k=i}^{n-1}\prod_{m=k}^{n-1}\frac{y_{m+1,m-i+1}}{y_{m,m-i+1}},
A_1 = M_4 M_2^{-1}+ M_2^{-1}X^{-1}Y^{-1}(I-X)-M_2^{-1}X^{-1}, A_2 = X M_2, A_3 = M_2^{-1}X^{-1}(Y-I), A_4 = Y^{-1}(I-X)M_2, A_5 = M_2^{-1}(M_1-Y),
\Delta^x_{[x_0:x_1]}= t^2 \Big[ (d_{-1,0}x_1^2 - \frac{1}{t}x_0x_1 + d_{1,0}x_0^2)^2 &- 4(d_{-1,1}x_1^2 + d_{0,1}x_0x_1 + d_{1,1}x_0^2)(d_{-1,-1}x_1^2 + d_{0,-1}x_0x_1 + d_{1,-1}x_0^2) \Big].
{\mathcal E}(V^{\pm}_{n+1}, \Phi^{\pm}_{n+1})-{\mathcal E}(V^{\pm}_n,\Phi^{\pm}_n)={\mathcal E}'_{(V^{\pm}_{n+\frac 12}, S_{\theta_n}\Phi_n^\pm)}(\delta \tilde{V}^{\pm}_n,\delta \Phi^{\pm}_n)+\overline{e}_{\pm,n}
I_0(\tau,p) =J_1(\tau,p)+J_2(\tau,p) &\equiv\int_{\partial D}\left(\frac{\partial}{\partial\nu}+\rho\right)E_{\tau}(x,p)V_{\Omega}(\tau)\varphi(x,\tau)dS_x &\hskip10mm+\int_{\partial D}\left(\frac{\partial}{\partial\nu}+\rho\right)E_{\tau}(x,p)V_{D}(\tau)\psi(x,\tau)dS_x.
\hat{P}^{(\mathcal{C}_n)}(i,j|m,\mathbf{s})=\frac{p^n_{S|U,V}\big(\mathbf{s}\big|\mathbf{u}(i),\mathbf{v}(i,j,m)\big)}{\sum_{(i',j')\in\mathcal{I}_n\times\mathcal{J}_n}p^n_{S|U,V}\big(\mathbf{s}\big|\mathbf{u}(i'),\mathbf{v}(i',j',m)\big)},
K \sim \frac{\tau_{3}}{2}L^{2}\frac{3}{2^{5 / 3}}\left( \frac{u^{2}{\alpha^{\prime}}^{2}}{L^{2}}\right)^{4 / 3}= \frac{1}{8 \pi^{2}g_{Y M}^{2}}\frac{3}{2^{5 / 3}}\left( \frac{{\alpha^{\prime}}^{2}}{L^{2}}\right)^{1 / 3}( u^{2})^{4 / 3}\ ,
\begin{array}{ll}\| \mathbf{1}_{B_1}f \|_{\dot{H}^{s}}= \| \mathbf{1}_{B_1}\bar{\chi}f \|_{\dot{H}^{s}}& \lesssim \| \bar{\chi}f \|_{H^{s}}& \lesssim \| f \|_{\dot{H}^{s}},\end{array}
L_3(u) \equiv \frac{d}{du}\left(u\frac{d}{du}\right)- \left(u\frac{d}{du}+\frac{5}{6}\right)\, \left(u\frac{d}{du}+\frac{1}{6}\right )= u \, \left(1 - u \right) \frac{d^2}{du^2}+\left(1-2 \, u\right)\, \frac{d}{du}- \frac{5}{36}
\widehat{C}(k) =\int_{-1}^1 e^{-2\pi ikx}\log \frac{1}{|x|}dx=2\int_0^1 \cos(2\pi kx)\log \frac{1}{x}dx= \frac{1}{\pi k}\int_0^{2\pi k}\cos y\log \frac{2\pi k}{y}dy =\frac{1}{\pi k}\int_0^{2\pi k}\frac{\sin y}{y}dy,
\left( \begin{array}{c c}{D^{2}e^{2 V}}{\sqrt{2}\bar{\Phi}}{\sqrt{2}\Phi}{\bar{D}^{2}e^{- 2 V}}\end{array}\right) \left( \begin{array}{c}{Q}{\widetilde{Q}^{\dagger}}\end{array}\right) = 0 .
\times \sum_{n + 2 m + l = q}\sum_{k = 0}^{n}\frac{1}{k ! ( n - k ) ! m ! ( l + m ) !}z_{1}^{2 k - n}z_{2}^{n - 2 k}\left. \frac{4 l^{2}}{p^{2}- l^{2}}\right] .
f_n = \sum_{k=1}^{n}\frac{(k-1)!}{n!}B_{n,k}(1!\psi_1,2!\psi_2,\dots), \phi_n = \sum_{k=1}^{n}\frac{k!}{n!}B_{n,k}(1!\psi_1,2!\psi_2,\dots), \theta_n = \sum_{k=1}^{n}\binom{-(\alpha+\beta)n+k}{k-1}\frac{(k-1)!}{n!}B_{n,k}(1!\psi_1,2!\psi_2,\dots).
\mu_{(\boldsymbol{x},\boldsymbol{\phi}(t,\boldsymbol{x}))}^{N,i}:=k_i^N(t,\nu^{N,1}_{\boldsymbol{x}},\ldots,\nu^{N,N}_{\boldsymbol{x}},\lambda^{N,1}_{\boldsymbol{\phi}(t,\boldsymbol{x})},\ldots,\lambda^{N,N}_{\boldsymbol{\phi}(t,\boldsymbol{x})}),\quad\mu^i:=k\big(t,\nu^{N,i}_{\boldsymbol{x}},\Phi(t,\nu^{N,i}_{\boldsymbol{x}})\big),
Z_{k}= i e^{i \pi \mathrm{\boldmath \hat{\scriptstyle \psi}\scriptstyle_{\perp}}\cdot \mathrm{\boldmath \scriptstyle \tau}/ 2}e^{i ( 2 k + 1 ) \pi \mathrm{\boldmath \hat{\scriptstyle \psi}}\cdot \mathrm{\boldmath \scriptstyle \tau}x^{3}/ L}\quad \mathrm{w i t h}\quad \mathrm{\boldmath \hat{\psi}}\cdot \mathrm{\boldmath \hat{\psi}_{\perp}}= 0
\eta_{M, M-1}(q|a, b) \triangleq \exp\Bigl( \bigl(\mathcal{S}_{M-1}\log\Gamma_M\bigr)(q\,|a,\,b) - \bigl(\mathcal{S}_{M-1}\log\Gamma_M\bigr)(0\,|a,\,b) \Bigr).
M_E(x\odot e)= M_E(x) \odot e+ (-1)^{|x|}x \odot M_E(e)+ (-1)^{|e||x_{(2)}|}l(x_{(1)}, e)\odot x_{(2)}+ l (x, e_{(1)})\odot e_{(2)}+ (-1)^{|e_{(1)}||x_{(2)}|}l(x_{(1)}, e_{(1)})\odot x_{(2)}\odot e_{(2)}+ l(x, e)
\left( 1 - \alpha^{i_0-j_0- 3j}\right) \lambda_{i_0-2j,j_0+j,k_0+j}= \alpha^{i_0-3j_0 -3j -3}\left( \sum_{\nu=0}^{j_0 +j}\alpha^{3\nu}\right) \lambda_{i_0-2j -2,j_0+j+1,k_0 + j+1}.
\left\{x_{1}, y_{1}\right\}_{\theta}^{\mu_{1}}\left\{x_{2}, y_{2}\right\}_{\theta}^{\mu_{2}}\ldots \left\{x_{n}, y_{n}\right\}_{\theta}^{\mu_{n}}= : \left\{x_{1}, y_{1}^{\mu_{1}}; x_{2}, y_{2}^{\mu_{2}}; \cdots ; x_{n}, y_{n}^{\mu_{n}}\right\}_{\theta}
I(X_{\pi(s)};Y_1|X_{\pi(1)},\ldots,X_{\pi(s-1)},W_{\sigma(1)},\ldots,W_{\sigma(s-1)}) - I(X;Y_1) = I(W_{\sigma(1)},\ldots,W_{\sigma(s-1)};Y_1|X) - I(X_{\pi(1)},\ldots,X_{\pi(s-1)};Y_1) - I(W_{\sigma(1)},\ldots,W_{\sigma(s-1)};Y_1|X_{\pi(1)},\ldots,X_{\pi(s-1)}) \stackrel{(e)}{=}0,
{\cal E}_{n + 1}-{\cal E}_{n}= \frac{1 6}{\beta^{2}}\sin \left( \frac{\beta^{2}}{8}\right) \cos \frac{\pi}{2}\left( \left( n + 1 \right) \left( \frac{\beta^{2}}{2 \pi}\right) - 2 a + 1 \right) .
{}_3F_2\bigg[\begin{matrix}\alpha \alpha \beta &1 \alpha-\beta+1\end{matrix}\bigg|\,z\bigg]=(1-z)^{-\alpha}{}_3F_2\bigg[\begin{matrix}1-\beta \frac12\alpha \frac12\alpha+\frac12 &1 \alpha-\beta+1\end{matrix}\bigg|\,-\frac{4z}{(1-z)^2}\bigg].
|p^{-\sigma - it}-1 |^2 = p^{-2\sigma}- 2p^{-\sigma}\cos (t \log p) + 1 & \leq p^{-2\sigma}- 2p^{-\sigma}+ 1 + 2p^{-\sigma}t^2 (\log p)^2 & \leq (p^{-\sigma}-1)^2 + 2p^{-\sigma}t^2 (\log p)^2 & \leq \sigma^2 (\log p)^2 + 2t^2 (\log p)^2,
d(\lambda):=\prod_{j=1}^3\frac{(-1)^{\frac{\lambda_j(\lambda_j+1)}{2}}}{\lambda_j!j^{\lambda_j}}\prod_{k=0}^{\lambda_j-1}\prod_{\ell=1}^{j-1}(k-\lambda_\ell-j+\ell+1-[k-\lambda_\ell ]_2).
S = ((\langle x \rangle \rtimes \langle y \rangle) \times \langle z \rangle) \rtimes \langle w \rangle \cong ((C_8 \rtimes C_8) \times C_2) \rtimes C_2, & \x^y = x^{-1}, \x^w = xy^4z, \y^w = y^5, \z^w = z.
[ u_{q_1m+1}, u_{q_{2m+1}}] = (q_1-q_2) \u_{(q_1+q_2)m+1}{[}u_{q_1m}, u_{q_{2m+1}}{]}= -{[}u_{q_{2m+1}}, u_{q_1m}{]}= q_1 \u_{(q_1+q_2)m}
\begin{aligned}\left|\widetilde{u}(z)-\widetilde{u}(w)\right| \leq 2\left\|\widetilde{u}\right\|_{C(X, d)}& =2.3^{\alpha-\frac{s}{p_{0}}}\left\|\widetilde{u}\right\|_{C(X, d)}\frac{1}{3^{\alpha-\frac{s}{p_{0}}}}& \leq 2.3^{\alpha-\frac{s}{p_{0}}}C_3\|u\|_{W_s^{\alpha, G}(X, d, \mu)}d(z, w)^{\alpha-\frac{s}{p_{0}}}.\end{aligned}
u =(c_1\cos (n\theta)+c_2\sin(n\theta))\Big((n-\tanh{t})e^{nt}-(n+\tanh{t})e^{-nt}\Big), \delta = T\frac{\Big(n(n-\tanh{T})-\frac{1}{\cosh^2{T}}\Big) e^{nT}+\Big(n(n+\tanh{T})-\frac{1}{\cosh^2{T}}\Big) e^{-nT}}{(n-\tanh{T})e^{nT}-(n+\tanh{T})e^{-nT}}.
\frac{F}{V}= \frac{1}{4}\left( \frac{1}{4 \pi^{2}\alpha^{\prime}}\right)^{5}\int_{- 1 / 2}^{1 / 2}d \tau_{1}\int_{0}^{\infty}\frac{d \tau_{2}}{\tau_{2}^{6}}\left| \frac{1}{\eta ( \tau )}\right|^{2 4}\sum_{n = - \infty}^{+ \infty}e^{- \beta^{2}n^{2}/ ( 4 \pi \alpha^{\prime}\tau_{2})}C ( n , \tau )
S_{R}= - \int d^{2}x \, \sqrt{g}\, \left\{\frac{1}{2}Z g^{\mu \nu}\partial_{\mu}\Phi \partial_{\nu}\Phi + C R + V + \frac{1}{\epsilon}\left( - \frac{V^{\prime}}{C^{\prime}}+ \frac{Z V}{2{C^{\prime}}^{2}}\right) \right\}.
\left( \sum_{|\alpha |=m}\Vert vc_{\alpha}\Vert_{X}^{\rho}\right)^{\frac{1}{\rho}}& \leq \left( \frac{2}{s_{k}}\right)^{\frac{m-k}{2}}\sqrt{\frac{(m-k)!}{m!}}\frac{m^{m}}{(m-k)^{m-k}}M_{2}(X)\left( C_{2}\left(X\right) \right)^{k-1}\left( \prod_{j=1}^{k-1}K_{\frac{2rj}{2+(j-1)r},2}\right) \pi_{(r,1)}(v)\Vert P\Vert_{\mathbb{D}^n}.
\mathsf{U}^n =\{1,0,g(\xi_2^{\mathsf{U}^n}),\cdots,g(\xi_{n-1}^{\mathsf{U}^n})\}, \mathsf{V}^n =\{0,1,g(\xi_2^{\mathsf{V}^n}),\cdots,g(\xi_{n-1}^{\mathsf{V}^n})\},
D^{a,b}= \left\{\liminf_{n \to \infty}M_n < a \right\}\cap \left\{\limsup_{n \to \infty}M_n > b \right\}= \left\{\liminf_{n \to \infty}M_n \wedge (b+1) < a \right\}\cap \left\{\limsup_{n \to \infty}M_n\wedge (b+1) > b \right\}.
l_{3}= \left( \frac{N_{2}^{( 1 )}\bar{N}_{2}^{( 1 )}N_{5}^{( 2 )}\bar{N}_{5}^{( 2 )}}{N_{2}^{( 2 )}\bar{N}_{2}^{( 2 )}N_{5}^{( 1 )}\bar{N}_{5}^{( 1 )}}\right)^{\frac 1 4}l_{2}
f_{3}( T , A ) = \frac{T^{3}}{8}\left[ \frac{A}{3}\{A , \tau_{3}\}^{2}+ c_{1}^{( 3 )}( T ) A \{A , \tau_{3}\}[ \tau_{3}, A ] + c_{2}^{( 3 )}( T ) ( A^{3}- A \tau_{3}A \tau_{3}A ) \right] ,
DY\overline{u}= [DXv_\rho]^{-1}DY\overline{u}= Y_p[D^2\overline{u}-A(\cdot,\overline{u},D\overline{u})] DXv_\rho = X_q[D^2v_\rho-A^*(\cdot,v_\rho,Dv_\rho)].