harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
D\left(\frac{g(\lambda)}{H_{{\lambda}}}\right)=\sum\limits_{{\ell(\lambda^+)>\ell(\lambda)}}\frac{g(\lambda^+)-g(\lambda)}{H_{{\lambda^{+}}}}+2\sum\limits_{{\ell(\lambda^+)=\ell(\lambda)}}\frac{g(\lambda^+)-g(\lambda)}{H_{{\lambda^+}}}
|T^+-{\rm Id}|_{\mathcal{V}_+^*,0;s_+}\leq C_1 r_+\delta_{+0},|\partial_\xi^\beta T^+|_{\mathcal{V}_+^*,0;s_+}\leq C_1 r_+\delta_{+\mu},1\leq |\beta|_1=\mu \leq \alpha.
p\left({{d_{pF}}\leq \delta}\right) = M\int_{\mathbf{\Omega}}{\prod\limits_{i = 1}^k{{{\left({{{\sin}^2}{\theta_i}}\right)}^{n - 2k}}}\prod\limits_{1 \leq i < j}^k{{{\left({{{\sin}^2}{\theta_i}-{{\sin}^2}{\theta_j}}\right)}^2}}d{{\sin}^2}{\theta_1}\cdots d{{\sin}^2}{\theta_k}}.
J^k_{1,1,1}=\int_{t_{k-1}}^{t_k}\left\{\int_{t_{k-1}}^u(p-2)|X^N(r)-X^N(s)|^{p-4}\left(X^N(r)-X^N(s),\sigma(X^N(r))\frac{\Delta B_k}{\Delta}\right)dr\right\}&\times\left(X^N(t_{k-1})-X^N(s), \sigma(X^N(t_{k-1}))\frac{\Delta B_k}{\Delta}\right)du,
\log_2\left(\frac{|\pi_{\theta}(D_{a_1\cdots a_{n+1}})|}{|\pi_{\theta}(D_{a_1\cdots a_n})|}\right) =\log_2\left(\frac{|\pi_{\phi_{a_n\cdots a_1}}(D_{a_{n+1}})|/2}{|\pi_{\phi_{a_n\cdots a_1}}(D)|/2}\right) &= \log_2\left(\frac{|\pi_{\phi_{a_n\cdots a_1}}(D_{a_{n+1}})|}{2}\right)
u(t,x) =\mathbf E\left[ q(Y^{x}(t))\mathbf 1_{\{W(t)>1\}}\right] &=\mathbf E\left[ q(Y^{x}(t)) | Y^x(t)\right] \Big( =\mathbf E\left[ q(Y^{x}(t)) | Y^x(t)\right]\Big).
v^{n-\frac{1}{2}}-v(t_{n-\frac{1}{2}}) = \frac{1}{2}\left[ \int_{t_{n-1}}^{t_{n-\frac{1}{2}}}(\zeta-t_{n-1})v''(\zeta)d\zeta - \int_{t_{n-\frac{1}{2}}}^{t_{n}}(\zeta-t_{n})v''(\zeta)d\zeta \right].
\left\{\int_{\Omega^{\rho+}}\left(Z+U^2\right) d\mu \right\}^2 =\left\{\int_{\Omega^{r_1+}}\left(Z+U^2\right) d\mu \right\}^2 \leq \left\{\int_{\Omega^{r_i+}}\left(Z+U^2\right) d\mu \right\}^2 \leq \int_{\Omega^{0.25r_i,0.5r_i}}\left(Z+U^2\right) d\mu=\int_{\Omega^{2^{-i-1}\rho, 2^{-i}\rho}}\left(Z+U^2\right) d\mu.
0 = - \frac{\partial J ( 0 , \hat{T}_{c})}{\partial \hat{T}}\hat{T}_{c}\left( 1 - \frac{\hat{T}}{\hat{T}_{c}}\right) + \frac{\partial J ( 0 , \hat{T}_{c})}{\partial \hat{m}_{T}^{2}}\hat{m}_{T}^{2},
\frac{d}{dt}=\frac 1f\frac{d}{d\tau}, \frac{d^2}{dt^2}=\frac 1f\frac{d}{d\tau}\left(\left(\frac 1f\right)\frac{d}{d\tau}\right)=\frac 1{f^2}\frac{d^2}{d\tau^2}-\frac 1{f^3}\frac{df}{d\tau}\frac{d}{d\tau},
d\tilde{\mathbf{A}}_t = \mathbf{S}_td\mathbf{A}_t + \mathbf{L}_t dt, d\tilde{\mathbf{\Lambda}}_t = \mathbf{S}_t^{*}d\mathbf{\Lambda}\mathbf{S}_t^{\top}+ \mathbf{S}_t^{*}d\mathbf{A}_t^{\ast}\mathbf{L}_t^{\top}+ \mathbf{L}_t^{\ast}d\mathbf{A}_t^{\top}\mathbf{S}_t^{\top}+ \mathbf{L}^{\ast}\mathbf{L}^{\top}dt.
\begin{array}{r c l}{m_{0}^{2}}&{=}&{\displaystyle \frac{1}{2 a^{2}}L^{\alpha \beta}L_{\alpha \beta}}\\{}&{=}&{E^{2}-{\bf P}^{2}- \displaystyle \frac{{\bf L}^{2}}{a^{2}},}\\ \end{array}
ds^2 = \left({{r_0^2}\over{r^2}}-1 + \Delta \right)^{-1}du \left( dv + \left({p \over{r^2}}+ k \right) du+ \left( 2 r_0^2{{\dot{f}_i}\over{r^2}}+ 2 a_i \right) dy^i \right)+ dy^2 + H_5 dx^2.
\sum_{k=0}^5 p_kz^k \sum_{i=1}^n \frac{1}{z-z_i}\sum_{\substack{j=1 j\ne i}}^n \frac{2}{z_i-z_j}+ \sum_{k=0}^4 q_kz^k \sum_{i=1}^n \frac{1}{z-z_i}+ \sum_{k=1}^3 w_kz^k = - w_0.
\sum_{k=0}^N\frac{(1-aq^{2k})(a;q)_k(b;q)_k(c;q)_k(q^{-N};q)_k}{(1-a)(q;q)_k(aq/b;q)_k(aq/c;q)_k(aq^{N+1};q)_k}\biggl(\frac{aq^{N+1}}{bc}\biggr)^k=\frac{(aq;q)_N (aq/bc;q)_N}{(aq/b;q)_N(aq/c;q)_N}
f_{\boldsymbol{Y}}(\boldsymbol{y})=\frac{2c_{n}}{\sqrt{|\boldsymbol{\Sigma}|}}g_{n}\left\{\frac{1}{2}(\boldsymbol{y}-\boldsymbol{\mu})^{T}\mathbf{\Sigma}^{-1}(\boldsymbol{y}-\boldsymbol{\mu})\right\}H\left(\mathbf{\Sigma}^{-\frac{1}{2}}(\boldsymbol{y}-\boldsymbol{\mu})\right),~\boldsymbol{y}\in\mathbb{R}^{n},
\begin{array}{c}\mathcal L_ZD_i=[Z,D_i]=-D_iz_i\cdot D_i,\ \mathcal L_Z\mathcal L_{D_i}-\mathcal L_{D_i}\mathcal L_Z=-D_iz_i\cdot\mathcal L_{D_i}, \mathcal L_Z\omega^j_{Ii}=\mathcal L_Z\mathcal L_{D_i}\omega^j_I=\mathcal L_{D_i}\mathcal L_Z\omega^j_I-D_iz_i\omega^j_{Ii}\end{array}
x_{i_{1},i_{2}}y_{i_{3},i_{4}}=x_{i_{\sigma_{1}},i_{\sigma_{2}}}y_{i_{\sigma_{3}},i_{{4}}}, x_{i_{1},i_{2}}z_{i_{3}}=x_{i_{\sigma_{1}},i_{\sigma_{2}}}z_{i_{\sigma_{3}}}& y_{i_{1},i_{3}}z_{i_{2}}=y_{i_{2},i_{3}}z_{i_{1}}
h(w)=-\frac{2v}{\delta_b R{\rm cosh}^2\frac{w}{\delta_b}}+\frac{1}{2\pi R^4}\sum_i\sum_{KM}\frac{KG_i}{{\cal N}_i}\int_w^{\infty}{\rm d}w'~\frac{w'}{{\rm cosh}^{2\gamma_i}\frac{w'}{\delta_b}}~~,
E^{\gamma_-}_{\tau}(x, z') = \frac{\tau}{2(2\pi)^2\gamma_-^{3/2}}\int_{\Bbb R}I_{\tilde\tau, 0}(x-\tilde{z}', \zeta_2)d\zeta_2, \partial_{x_k}E^{\gamma_-}_{\tau}(x, z') = \frac{\tau^2}{2(2\pi)^2\gamma_-^{2}}\int_{\Bbb R}I_{\tilde\tau, k}(x-\tilde{z}', \zeta_2)d\zeta_2\Theta_k(x, z')\qquad(k = 1, 2, 3),
\prod_{i=1}^n (a_{i1}X_1+\ldots+a_{in}X_n)^{k_i}&=\sum_{k_{11}+\ldots+k_{1n}=k_1}\ldots\sum_{k_{n1}+\ldots+k_{nn}=k_n}\frac{k_1!\ldots k_n!}{\prod_{i,j}k_{ij}!}a_{11}^{k_{11}}a_{12}^{k_{12}}\ldots a_{nn}^{k_{nn}}X_1^{k_{11}+\ldots+k_{n1}}\ldots X_n^{k_{1n}+\ldots+k_{nn}}.
\int \phi\cdot (\psi\circ X^t) \,d\mu=\int (\phi\circ\Phi) \,\cdot\, (\psi\circ X^t\circ\Phi) \,d\widehat{\eta}=\int (\phi\circ\Phi) \,\cdot\, (\psi\circ\Phi) \circ{F}_t\,d\widehat{\eta}
\left\|\sin\Theta(V, W)\right\|_F^2 \overset{\eqref{eq:def_sin_Theta_Frobenius}}{=}\|W^{\intercal}V_{\perp}\|_F^2 = \|[\alpha^{(r+1)}~\cdots~\alpha^{(p_2)}]\|_F^2 \leq \frac{\|\tilde{U}^{\intercal}AW_{\perp}\|_F^2\sigma_{r}^2(AW)}{\left(\sigma_{r}^2(AW) - \sigma^2_{r+1}(A)\right)^2}.
\lVert P_B \rVert_{L^p_{\alpha'}}\le 2 \sum^{\infty}_{j=0}2^{j}\lVert{P^{2^{-j}}}\rVert_{L^p_{\alpha'}}\le 2\delta^{-\varepsilon/s}M \sum^{\infty}_{j=0}2^{-j \varepsilon/s}&\le C(\delta,\alpha,\alpha',\tau,p) M.
\Phi (s) = \Phi_e(s) ={\prod\limits_{1\leq i < j\leq M}\left(1 -{s_i\over s_j}\right) \prod\limits_{M+1 \leq i < j\leq M+N}\left(1 -{s_i\over s_j}\right) \over\prod\limits_{i\leq M < j}\left(1 -{s_i\over s_j}\right)},
W_{\hbar}\left( t,s\right) \psi-W_{\hbar}^{M}\left( t,s\right) \psi=i\int_{s}^{t}W_{\hbar}\left( t,\tau\right) \left[ L_{\hbar}^{M}\left(\tau\right) -\overline{L_{\hbar}\left( \tau\right)}\right] W_{\hbar}^{M}\left( \tau,s\right) \psi d\tau
d^{D}\mathbf{x}^{\prime}\, d^{D}\mathbf{p}^{\prime}= d^{D}\mathbf{x}\, d^{D}\mathbf{p}\left[ 1 - \left\{2 ( D - 1 ) \beta \left( \frac{1 + ( \beta + \beta^{\prime}) p^{2}}{1 + \beta p^{2}}\right) + ( 2 \beta + \beta^{\prime}) \right\}p_{j}\frac{\partial H}{\partial x_{j}}\delta t \right] \; .
\int_{{\cal W}_0 (h)}\exp ( c_1 \langle q(h), g_2 (w;h)\rangle_{{\cal V}}) \pi_{*}^h \mu (dw)=\int_{{\cal W}}\exp ( c_1 \langle q(h), g_2 (\pi^h w;h)\rangle_{{\cal V}}) \mu (dw)\le M
\alpha =b1^da \beta=(b-1)1^d(a+1) \alpha =ab1^d \beta=(b-1)(a+1)1^d \alpha =1^{d+1}a(b-1) \beta=1^{d+1}(b-1)a \alpha =1a1^d(b-1) \beta=1(b-1)1^da \alpha =(b-1)1^db(a-b+1) \beta=b1^d(b-1)(a-b+1)
p_{X_{\mathcal{I},Q_n}, Y_{T^c, Q_n}|Q_n}^{(\lambda, T)}(x_{\mathcal{I}}, y_{T^c}|k) = p_{X_{\mathcal{I},k}, Y_{T^c, k}}^{(\lambda, T)}(x_{\mathcal{I}}, y_{T^c}),
\begin{array}{cclcccl}\{b_k,a_k\}_3 = -a_k (b_k^2+a_k), \; \{a_k,b_{k+1}\}_3 = -a_k (b_{k+1}^2+a_k), \{b_k,b_{k+1}\}_3 = -a_k (b_k+b_{k+1}), \; \{a_k,a_{k+1}\}_3 = -2a_k a_{k+1}b_{k+1}, \{b_k,a_{k+1}\}_3 = -a_k a_{k+1}, \; \{a_k,b_{k+2}\}_3 = -a_k a_{k+1}.\end{array}
\Phi_{1}=\Phi_{1}^{mol}+\Phi_{1}^{eddy}=-Kn\frac{df_{1}}{dz}-D_{12}n_{1}(\frac{n^{2}}{n_{1}n_{2}}\frac{df_{1}}{dz}+\frac{(m_{2}-m_{1})g}{kT}+\frac{\alpha_{T}}{T}\frac{dT}{dz})
\widehat{G}(t) = e^{-t^2/2}\Big[p_1 \lambda e^{-\sigma_1^2t^2/2}+ p_1 (1-\lambda) e^{-\kappa|t|}+ (1-p_1) &- (1 - p_0) e^{-\sigma_0^2t^2/2}- p_0\lfloor a + b(t-t^*)\rfloor_+ \Big] &:= e^{-t^2/2}V(t) = \sqrt{2\pi}\phi(t) V(t),
\ldots +\sum_{t=0}^{r}\binom{r}{t}\left(D_{\zeta}^{t}R\right)(1)j_{1}^{r-t}X^{(s)}(j_1) X^{(p)}(r-j_1) &+\sum_{t=0}^{r}\binom{r}{t}\left(D_{\zeta}^{t}R\right)(1)j_{2}^{r-t}X^{(s)}(j_2) X^{(p)}(r-j_2)+\ldots \ldots +\sum_{t=0}^{r}\binom{r}{t}\left(D_{\zeta}^{t}R\right)(1)j_{2s}^{r-t}X^{(s)}(j_{2s}) X^{(p)}(r-j_{2s})+\ldots .
\epsilon_{a}(f)\approx\sum_{i=1}^{N}|\frac{\partial f}{\partial x_{i}}|\epsilon_{a}(x_{i})=|\frac{\partial f}{\partial x_{1}}|\epsilon_{a}(x_{1})+|\frac{\partial f}{\partial x_{2}}|\epsilon_{a}(x_{2})+...+|\frac{\partial f}{\partial x_{N}}|\epsilon_{a}(x_{N})
\begin{aligned}\sum_{i \in H}\sum_{j \in C}y_{i,j}\sum_{j\in C}y_{i,j}\cdot U_{i,j}\geq h_i i\in H & \sum_{i\in H}y_{i,j}\cdot U_{i,j}\geq c_j j\in C & y_{i,j}\in\{0,1\}i\in H, j\in C\end{aligned}
\sup_{\|x \|=1}|\langle Ax, x \rangle|^{p}\leq \sup_{\|x \|=1}\langle A^*Ax, x \rangle^{p \over 2}= \|A^*A \|^{p \over 2}= \|A \|^p = \||A|\|^p= \||A|^p\|,
\Pi\Bigl( \sum_{i=1}^\infty(\kappa_if_i-\kappa_if_{0,i})^2\leq \epsilon^2\Bigr) \geq \Pi\Bigl(\sum_{i=1}^N(\kappa_if_i-\kappa_if_{0,i})^2\leq \epsilon^2/2\Bigr) \Pi\Bigl(\sum_{i=N+1}^\infty(\kappa_if_i-\kappa_if_{0,i})^2\leq \epsilon^2/2\Bigr).
\frac{d\tilde{n}}{dt}=\frac{D_{R}}{D_{o}}(1-\tilde{n})^{\alpha}-s\tilde{n}^{\beta}exp(\frac{-(E_{t}-E_{b})}{kT})-s\tilde{n}^{\beta}exp(-p^{\prime-\frac{1}{3}}r^{\prime})
\sum_{i=1}^N \Lambda_X\big(\tau \omega_i(N))\big) - N\int_0^1 \Lambda_X\big(\tau \overline F(x)\big){\rm d}x=\sum_{i=1}^N \Lambda_X\left(\tau{N}\int_{(i-1)/N}^{i/N}\overline{F}(x){\rm d}x\right) - N\int_0^1 \Lambda_X\big(\tau \overline F(x)\big){\rm d}x.
\mathcal{E}(S',h\circ \tilde{f}^{-1}) -\mathcal{E}(S,h) = -4\textrm{Re}\int_S \psi\cdot \frac{\mu}{1-|\mu|^2}+ 4\int_S |\psi|\cdot \frac{|\mu|^2}{1-|\mu|^2}.
\Phi^{( I )}= -{\frac{e^{2}}{8 \pi^{2}m}}\operatorname{a r c t a n}\biggl \{\tan \biggl [{\frac{\Theta}{2}}+{\frac{\pi}{4}}\bigl ( 1 - \mathrm{s g n}( m ) \bigr ) \biggr ] \biggr \}\qquad \left( F = \frac{1}{2}\right) .
\begin{aligned}t(u): = -\frac{1}{4\pi}(u_1\wedge u_1^\perp + u_2\wedge u_2^\perp)-\frac{1}{4\pi}u_1\wedge u_2^\perp + \frac{1}{4\pi}u_1^\perp\wedge u_2 & =-\frac{1}{4\pi}(u_1+u_2)\wedge(u_1^\perp + u_2^\perp),\end{aligned}
A_{m+1}= \prod_{j=1}^{|G_0|}g_j^{\mathsf{v}_{g_j}(A_{m+1})}= \prod_{j=1}^{|G_0|}\prod_{i=1}^m g_j^{\alpha_i \mathsf{v}_{g_j}(A_i)}= \prod_{i=1}^m \bigg(\prod_{j=1}^{|G_0|}g_j^{\mathsf{v}_{g_j}(A_i)}\bigg)^{\alpha_i}= \prod_{i=1}^m A_i^{\alpha_i}.
(V\times W\times\{0\})\cdot \Gamma_{t}=(\{0\}\times W)\cdot_{V\times W\times\{0\}}((V\times W\times\{0\})\cdot \Gamma_t)= &=(\{0\}\times W)\cdot Z_{f_t|_{A\cap (V\times W)}}= &=(\{0\}\times W)\cdot Z_{f_t}.
\sum_{n=0}^{\infty}\frac{\left(b;q\right)_{n}q^{n^{2}/2}S_{n}\left(aq^{-n-1/2};q\right)}{\left(c;q\right)_{n}}\left(\frac{c}{ab}\right)^{n}=\frac{\left(c/b;q\right)_{\infty}}{\left(c;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{n^{2}/2}}{\left(q;q\right)_{n}}\left(\frac{c}{ab}\right)^{n}A_{q}\left(\frac{cq^{n-1/2}}{a}\right).
\eta_{\alpha}^{\beta}[\tau](t):= \begin{cases}\mu(\frac{t-\beta}{\tau}), \;\; \forall t\in (\beta, +\infty), 1, \;\; \forall t\in [\alpha, \beta], \mu(\frac{t-\alpha}{\tau}), \;\; \forall t\in (-\infty, \alpha), \end{cases}
\int d^{3}\vec{y}\, \, S ( \vec{x}, \vec{y}) S^{- 1}( \vec{y}- \vec{z}) = \left( \begin{array}{c c c}{{1}}&{{0}}&{{0}}\\{{0}}&{{\delta_{\mu}^{\rho}}}&{{0}}\\{{0}}&{{0}}&{{\delta_{j}^{k}}}\\ \end{array}\right) \, \delta ( \vec{x}- \vec{z}) \, .
|\hat{H}_k([X^n]_b)-H([X_{k+1}]_b|[X^k]_b)| =|H(U^{k+1})-H(U^k)-H([X^{k+1}]_b)+H([X^k]_b)| &\leq |H(U^{k+1})-H([X^{k+1}]_b)|+|H(U^k)-H([X^k]_b)|
\Theta(r,\zeta;\nu,\varepsilon)>0 \quad\mbox{for}0\leq r <\Xi_1(\zeta), &\Theta(\Xi_1(\zeta),\zeta;\nu,\varepsilon)=0, &\Theta(r,\zeta;\nu,\varepsilon)<0\quad\mbox{for}\Xi_1(\zeta)<r\leq 2\xi_1
\dim \mathbf{T}=\binom{c-b+3}{3}+\binom{c-a+3}{3}-\binom{b-a+3}{3}+\binom{c+a-e+3}{3}&-\binom{a+b-e+3}{3}-\binom{2a-e+3}{3}+\delta(e,a,b,c)-3.
\{I_{\alpha}, I_{\beta}\}=I_{\gamma}(K^{-1})_{\epsilon}\hspace{0.1 mm}^{\gamma}C^{\epsilon}\hspace{0.1 mm}_{\zeta \iota}K_{\alpha}\hspace{0.1 mm}^{\zeta}K_{\beta}\hspace{0.1 mm}^{\iota}
T_{(\omega^{-1}\cdot e^*)(\gamma)}((T_se)_{\underline{f}})\stackrel{(i)}{=}T_{e^*(\gamma)}T_{\alpha_1^{\vee}(\gamma^{-1})}((T_se)_{\underline{f}}) \stackrel{(ii)}{=}\gamma^{f-\epsilon_1}(T_se)_{\underline{f}}=\gamma^{f-\epsilon_1-a}(T_s(\gamma^a e))_{\underline{f}}=\gamma^{f-\epsilon_1-a}(T_sT_{e^*(\gamma)}e)_{\underline{f}}.
\begin{aligned}|h(z)-h'(b)(z-b)| =\left|\int^z_b(h'(\zeta)-h'(b))d\zeta\right| \leq\frac{16^2}{R}\int^{|z-b|}_0 t\;dt=2^{7}\frac{|z-b|^2}{R}\quad|z-b|\leq \frac{1}{16}R.\end{aligned}
R(r)=De^{-\eta{r}}r^{(\frac{1-\tau}{2})}\Big[e^{2\eta{r}}r^{-(1-\tau+\mathrm{d})}\Big]_{-(1+\mathrm{d}E^{-1})}\quad\Bigg(\mathrm{d}=\Big(\frac{\beta-\eta(1-\tau)}{2\eta}\Big)\Bigg).
\frac{1}{| \vec{x}_{1 2}|^{\lambda}}\sim \frac{1}{d + 2 n - \lambda}\frac{1}{2^{2 n}n !}\frac{2 \pi^{d / 2}}{\Gamma ( d / 2 + n )}( \partial^{2})^{n}\delta^{d}( \vec{x}_{1 2}) ,
\delta=-\frac{V_p}{(2\pi)^p}\omega_1\prod_{i=1}^{l}\frac{1}{\omega_i^2}\int_0^{\infty}\frac{ds}{s}\left ( \frac{\pi}{2s}\right )^{\frac{p}{2}}e^{-b^2s}\frac{(\cos(4\omega_1s\sinh\epsilon) - 4\cos(2\omega_1s\sinh\epsilon) + 3)}{\cosh\epsilon~\sin(2\omega_1s\sinh\epsilon)}
I(x,z,h_1,h_2,h_3,k,y):=\big( \Delta_{-h_1,-2h_2,2k}B(y+2z)\Delta_{-h_2,-h_3,k}B(y+z) \Delta_{-h_1,-h_2,2k}C(x+y+2z)\Delta_{-h_3,k}C(x+y+z) \Delta_{-h_1,2k}D(2x+y+2z)\Delta_{h_2,-h_3,k}D(2x+y+z)\big),
\tilde{\zeta}_{\infty,p^{\prime}}(s,c):=\lim_{p\rightarrow\infty}\tilde{\zeta}_{p,p^{\prime}}(s,c)=\sum_{m,n\neq0}\frac{p^{\prime2}+\lambda_{n}^{\prime2}}{p^{\prime}\left(p^{\prime}+\frac{1}{\pi}\right)+\lambda_{n}^{\prime2}}\frac{1}{\left(m^{2}+c\lambda_{n}^{\prime2}\right)^{s}}=\zeta_{\infty,p^{\prime}}(s,c),\,\,\,\,\,(s)>1.
\,\, v_2 := \frac{1}{u_2}\,\, ,\,\, w_2 := \frac{t_2}{u_2}\,\, ,\,\, v_3 := \frac{1}{u_3}\,\, ,\,\, w_3 := \frac{t_3}{u_3}\,\, ,\,\, v_4 := \frac{1}{u_4}\,\, ,\,\, w_4 := \frac{t_4}{u_4}\,\, .
\Pi_{0}= \{s \in\mathcal{AR}_{n}: p(s_{0}) < p(s_{1})\ \& \p(s_{1})=p(s_{2})\}, \Pi_{1}= \{s \in\mathcal{AR}_{n}: p(s_{0}) < p(s_{1}) \ \& \p(s_{1})\not=p(s_{2})\}\mbox{and}\Pi_{2}= \{s \in\mathcal{AR}_{n}:p(s_{0})\ge p(s_{1})\}.
A_{\pi_{1}}^{}=d\pi_{1}=\left(\theta^{8}+\theta^{7}\right)\otimes e_{1}+\left(\theta^{9}-\theta^{6}\right)\otimes e_{2}+\left(\theta^{10}+\theta^{5}\right)\otimes e_{3}=A_{}^{}+\left(\theta^{5}\otimes e_{3}-\theta^{6}\otimes e_{2}+\theta^{7}\otimes e_{1}\right),
1 \lesssim \sum_{i=0}^kr_i^{p(-\kappa+\delta-n/p )}\int_{B_i}g(y)^pdy\lesssim (r_k)^{s(-\kappa p-n+p\delta)}\int_{B_i}g(y)^pdy &+\sum_{j=0}^{|\log r_k|}(2^jr_k)^{s(-\kappa p-n+p\delta)}\int_{(2^{j+1}B_k\backslash 2^{j}B_k)\cap\Omega}g(y)^pdy &\lesssim \sum_{l=0}^{|\log r_k|+1}(2^lr_k)^{s(-\kappa p-n+p\delta)}\int_{2^lB_k\cap\Omega}g(y)^pdy.
\frac{\textup{MMD}_{n,m}^{2}}{\surd{\textup{var}(\Delta_{1})}}=\frac{\Delta_0+T_1+\Delta_1+\{\widetilde{\Delta}_{2}-E(\widetilde{\Delta}_{2})\}+\{E(\widetilde{\Delta}_{2})-T_1\}}{\surd{(\Delta_1)}}.
\Psi_D(a+bt^{-1}\sqrt{D})\gamma_{D,2}=\gamma_{D,2}\left[\begin{smallmatrix}t^{-1}0 0 1 \end{smallmatrix}\right]\left[\begin{smallmatrix}a b 5b a \end{smallmatrix}\right] \left[\begin{smallmatrix}t 0 0 1 \end{smallmatrix}\right]
F(p) := \begin{cases}\min_{v}\sum_{i=1}^n cp_i\max(-x_i-y_i,-d_i) + \sum_{i\neq j}\alpha_{ij}v_{ij}s.t.\quad \begin{cases}-v\leq 0 \sum_{j\neq i}v_{ij}- x_{0i}\leq 0,\forall i\in I\end{cases}\end{cases}
\int_0^T \Big(\| f(t, X^k_t)\|^p_{L^p(\Omega;\mathbb{R}^d)}+ \| g(t, X^k_t)\|^p_{L^p(\Omega;\mathbb{R}^{d\times n})}&+\| h(t, X^k_t)\|^p_{L^p(\Omega, L^p(\nu))}+ \| h(t, X_t)\|^p_{L^p(\Omega,L^2(\nu))}\Big) dt &\leq \int_0^T K(1+\| X^k_t \|^p_{S^p(\Omega;\mathcal{D})}) dt \leq KT(1+\| X^k\|^p_{S^p_{ad}(\Omega; \mathcal{D}_T)})<\infty.
\omega_{l}:=\frac{\Gamma \left( b-a\right) \Gamma \left( c-a\right)}{\Gamma \left( -2a\right) \Gamma \left( b+c\right)}\frac{\left( 2a\right)_{l}\left( a+1\right)_{l}\left( a+b\right)_{l}\left( a+c\right)_{l}\left(-1\right)^{l}}{\left( a\right)_{l}\left( a-b+1\right)_{l}\left(a-c+1\right)_{l}l!}
\Theta_{\scriptscriptstyle{A}}= (\mathbf{W}^H \mathbf{R}_{\scriptscriptstyle{A}}\mathbf{W}) (\mathbf{W}^H \mathbf{R}_{\scriptscriptstyle{B}}\mathbf{W}) - (\mathbf{W}^H \mathbf{R}_{\scriptscriptstyle{A}}\mathbf{W}\mathbf{W}^H \mathbf{R}_{\scriptscriptstyle{B}}\mathbf{W})
X_{n+1}= \bigg\{\psi(t),\t\in [0,n+1]\bigg|\ \psi(t) = \psi_n^*(t)\t \in [0,n],\ \|\psi(t)\|_2 \leq 2C_1(1 + t)^{-\frac{d}{4}}\|\psi_0\|_1 & \{\rm and}\ \sqrt{Q(\psi(t))}\leq 2C_1C_Q(1 + t)^{-\frac{d}{4}-\frac{\eta}{2}}\|\psi_0\|_1, \n \leq t \leq n+1 \bigg\}.
\widehat{I}_{p}={\det}^\ast\big(\frac{2-S^p_1\circ S^p_2-S^p_2\circ S^p_1}{4}\big)^{\frac{1}{4}}, I_p ={\det}^\ast\big(\frac{2-C^p_{12}-(C^p_{12})^{-1}}{4}\big)^{\frac{1}{4}}.
-\tilde{H}_{ef}= \left( M_{e_3e_2}M_{f_1e_2}+ M_{e_3f_2}M_{f_1f_2}+ \mathbf{1}\{f_1=e_3\}\sum_{u\in V_2}M_{e_3u}^2\right)A_{f_1f_3,f_2}:=M^{(1)}_{ef}+M^{(2)}_{ef}+M^{(3)}_{ef}.
\Phi (\sigma,a,z) = \sum_{n=0}^{\infty}\frac{z^n}{(n+a)^{\sigma}}= \sum_{n=0}^{\infty}\biggl( \frac{z^{2n}}{(2n+a)^{\sigma}}-\frac{z^{2n+1}}{(2n+1+a)^{\sigma}}\biggr) >0
\sin^2 a\left(z+\frac{1}{2}\right) =\alpha^2\sin^2az+2\alpha\beta\sin az\cos az+\beta^2 \cos^2az &=\sin az\left(\sqrt{(\alpha^2-\beta^2)^2+(2\alpha\beta)^2}\sin (az+\theta)\right)+\beta^2 &=\sin az \sin a\left(z+\frac{\theta}{a}\right)+\beta^2,
W^{\mu \nu}\equiv \frac{\partial^{2}L}{\partial \dot{x}_{\mu}\partial \dot{x}_{\nu}}= - \frac{m}{\sqrt{\dot{x}^{2}}}\left( g^{\mu \nu}- \frac{\dot{x}^{\mu}\dot{x}^{\nu}}{\dot{x}^{2}}\right) = - m \left( g^{\mu \nu}- \dot{x}^{\mu}\dot{x}^{\nu}\right) \; ,
\mathfrak{\widehat D}_{k,m}=(m-k)a^{k-2}X^d+\frac{(k-3)(k-2m)}{2}a^{k-4}X^{2d}-\frac{(k-4)(k-5)(k-3m)}{3!}a^{k-6}X^{3d}+\cdots.
A_{n-1}\tilde{u}=P_{n+1,n}+\frac{\kappa}{2}v^{n+1}\sum_{j=1}^n\sum_{m=0}^{j+1}\frac{(-1)^{j+m-1}(n-m)(n+1-m)}{m!(n-j)!(j+1-m)!}\hat{I}_{n,m}.
W(U)=(\rho u, \rho u^2+p, \rho uv, \rho u(\frac{u^2+v^2}{2}+\frac{\gamma p}{(\gamma -1)\rho}), \rho uZ)^\top, &H(U)=(\rho v, \rho uv, \rho v^2+p, \rho v(\frac{u^2+v^2}{2}+\frac{\gamma p}{(\gamma -1)\rho}), \rho vZ)^\top, &G(U)=(0, 0, 0, q_0\rho Z\phi(T), -\rho\phi(T)Z)^\top.
P^{(\alpha,\beta)}_{n}\Big(1-\frac{2x}{\beta}\Big) =\frac{(\alpha+1)_{n}}{n!}\,_2F_1\left(-n,\alpha +\beta +n+1;\alpha +1;\frac{x}{\beta}\right) &=\sum_{k=0}^{n}\frac{R_{n,k}^{(\alpha)}(x)}{\beta^{k}},
\|f\mid W^{1,p}(\Omega)\|=\biggr(\iint\limits_{\Omega}|f(x,y)|^{p}\, dxdy\biggr)^{\frac{1}{p}}+\biggr(\iint\limits_{\Omega}|\nabla f(x,y)|^{p}\, dxdy\biggr)^{\frac{1}{p}}.
-\alpha(n) - 1 = - \frac{j_n - (1 + j_1 + \dots + j_{n-1})}{2}- 1 = - \frac{1 - \frac{m-1}{3}}{2}- 1 = \frac{\frac{m-1}{3}- 1}{2}- 1 = \frac{\frac{m-1}{3}-3}{2}= \frac{m-10}{6}
p(29)=\frac{29\cdot28\cdot27\cdot26\cdot25\cdot24\cdot23\cdot22\cdot21\cdot20}{30\cdot29\cdot28\cdot27\cdot26\cdot25\cdot24\cdot23\cdot22\cdot21}=\frac{20}{30}=\frac{2}{3}>\frac{1}{2}
\mathrm{e}^{- N \mathrm{t r}_{N}\sum_{a = 1}^{P}\ln ( 1 - g M_{a})}= \int d \psi d \psi^{\dagger}\mathrm{e}^{- \sum_{a = 1}^{P}\sum_{i , j = 1}^{Q}\mathrm{t r}_{N}\psi_{i}^{\dagger a}( 1 - g M_{a})_{i j}\psi_{j}^{a}}.
H^*(X_{D}) =\mathbb{Z}[x_1,x_2,x_3]/\langle x_1^{c}, x_2(x_1+x_2)^{a-1}, x_3(x_1+x_2+x_3)^{b-1}\rangle, H^*(X_{\widetilde{D}}) =\mathbb{Z}[y_1,y_2,y_3]/\langle y_1^{c}, y_2(y_1+y_2)^{b-1}, y_3(y_1+y_2+y_3)^{a-1}\rangle.
\|M\|_{S_p}^2 =\|MM^*\|_{S_{p/2}}=\sup_{\|Z\|_{S_{p/(p-2)}}\le 1}\mathrm{Tr}[ZMM^*] \ge \sup_{\|v\|_{p/(p-2)}\le 1}\mathrm{Tr}[\mathrm{diag}(v)MM^*] = \Bigg(\sum_i (MM^*)_{ii}^{p/2}\Bigg)^{2/p}.
\begin{aligned}d\left(\gamma\left(a+b\over 2\right), \gamma\left(a'+b'\over 2\right)\right) \leq \left|{a+b\over 2}-{a'+b'\over 2}\right| &\leq{1\over 2}|a-a'|+{1\over 2}|b-b'|={1\over 2}d(x, x')+{1\over 2}d(y, y').\end{aligned}
d(t)\approx\frac{\sqrt{c^{4}+c^{2}g^{2}t^{2}}-c^{2}}{g}\approx\frac{gt^{2}}{2}-\frac{g^{3}t^{4}}{8c^{2}}+\frac{g^{5}t^{6}}{16c^{4}}+...
F(x^-,x^+)=\sum_i\left|{L \over \pi}\langle 0 | T^{++}(K) |i \rangle \right|^2\left({x^+ \over x^-}\right)^2{M_i^4 \over 8 \pi^2 K^3}K_4\left(M_i\sqrt{2 x^+ x^-}\right)\,,
( E + m ) \psi ( x ) = \sigma \left( \begin{array}{c}{{( E + m ) \, c o s \, k ( x - \xi )}}\\{{k \, s i n \, k ( x - \xi )}}\\ \end{array}\right) + \tau \left( \begin{array}{c}{{( E + m ) \, s i n \, k ( x - \xi )}}\\{{- k \, c o s \, k ( x - \xi )}}\\ \end{array}\right)
\sum_{\substack{p_1\le \dots\le p_\ell p_j\le p_{\ell+1}\le \dots \le p_r p_1\cdots p_r\ge X^{1-\delta}}}^* w_{p_1\cdots p_r}=o_{\mathcal{L},\eta}\Bigl(\frac{\#\mathcal{A}}{\log{X}}\Bigr).
\mathbb{E}\left[e^{cZ_{t}^{1}}\right]=\exp\left\lbrace \int_{0}^{t}\int_{0}^{1}\left(e^{z\psi(s,c)}-1\right)\nu_{1}(\mathrm{d}z)\mathrm{d}s\right\rbrace<\infty ,c\in\mathbb{R},
|U_{n,q}(f_{k},z)-U_{n,q}(f,z)| & \leq|f_{k}(0)-f(0)|\cdot|(1-z)^{n}|+|f_{k}(1)-f(1)|\cdot|z^{n}| & +[n+1]_{q^{-1}}\sum_{j=1}^{n-1}|p_{n,j}(q;z)|q^{j-1}\int_{0}^{1}p_{n-2,j-1}(q^{-1},q^{-1}t)|f_{k}(t)-f(t)|d_{q^{-1}}t & \leq C_{r,n}||f_{k}-f||_{r},
M=|\begin{matrix}e_{x}&e_{y}&e_{z}\\x_{A}-x&0&0\\ 0&-F&0\end{matrix}|+|\begin{matrix}e_{x}&e_{y}&e_{z}\\ -x&0&0\\ 0&R_{0}&0\end{matrix}|=F(x-x_{A})e_{z}-R_{0}xe_{z}=-\frac{Fx_{A}}{L}(L-x)e_{z}.
P(v_1\cdots v_{i_j}v_{i_{j+1}}\cdots v_{i_n}) = P(v_1\cdots v_{i_{j+1}}v_{i_{j}}\cdots v_{i_n}) + P(v_1\cdots dv_{i_{j+1}}dv_{i_{j}}\cdots v_{i_n}) + P(v_1\cdots [v_{i_{j}},v_{i_{j+1}}]\cdots v_{i_n}).
\frac{\partial^2q}{\partial u\partial v}+\frac{-(1-r^2)+2qr}{(1-q^2)r+q(1-r^2)}\frac{\partial q}{\partial u}\frac{\partial q}{\partial v}=0, \frac{\partial^2 r}{\partial v\partial u}+\frac{-(1-q^2)+2qr}{(1-q^2)r+q(1-r^2)}\frac{\partial r}{\partial u}\frac{\partial r}{\partial v}=0.
\frac{\delta S_{0}^{L}}{\delta B_{\mu \nu}^{a}}\equiv - \frac 1 2 F_{a}^{\mu \nu}= 0 , \; \frac{\delta S_{0}^{L}}{\delta A_{a}^{\mu}}\equiv A_{\mu}^{a}+ \left( D^{\lambda}\right)_{\; \; b}^{a}B_{\lambda \mu}^{b}= 0 .
f \left( t \right) = \alpha_{i-1}\, \forall t \in \left[ t_{i-1}+ \tfrac{\epsilon}{2}\tfrac{1}{k}\Delta_{g}, t_{i}- \tfrac{\epsilon}{2}\tfrac{1}{k}\Delta_{g}\right) \forall i \in \left\lbrace 1 , \ldots , k \right\rbrace, f \left( t \right) = \alpha_{k}\, \forall t \in \left[ t_{k}, t_{k+1}\right],
E\left[\exp\left(\max_{1\leq k\leq d}\left(\frac{|Y_k|}{M_k}\right)^{2/q}\right)\right] = E\left[\max_{1\leq k\leq d}\exp\left(\left(\frac{|Y_k|}{M_k}\right)^{2/q}\right)\right]\leq\sum_{k=1}^dE\left[\exp\left(\left(\frac{|Y_k|}{M_k}\right)^{2/q}\right)\right]\leq2d,
\gamma_t\leq~ (\max_{M_0}\gamma_0)e^{-\delta_2t}\exp\left(\frac{2(\alpha+1)}{\delta_1}(\max_{M_0}\gamma_0)(\max H^{1-\alpha})+2^{\alpha-3}(1-\alpha)(\max_{M_0}|H^2-4|)(\max H^{1-\alpha})\right) =:~ C_3(\max_{M_0}\gamma_0)e^{-\delta_2t},
R=\sum_{j=1}^{\infty}\lfloor\frac{2n}{p^{j}}\rfloor-2\sum_{j=1}^{\infty}\lfloor\frac{n}{p^{j}}\rfloor=\sum_{j=1}^{\infty}(\lfloor\frac{2n}{p^{j}}\rfloor-2\lfloor\frac{n}{p^{j}}\rfloor).
E_{0,l}= \lambda[8ab_1 + 2a(2l+d) - 2b_1 - l(l+d-1)], \psi_{0,l}(r) \propto r^l (1+\lambda r^2)^{-a}\exp[-\lambda (b_1+2b_2+3b_3) r^2 - \lambda^2(b_2+3b_3) r^4 - \lambda^3 b_3 r^6],
\mathbb{E}\,d_M(f(\varepsilon),f(-\varepsilon))^p\leq C^p\sum_{i=1}^n\mathbb{E}\,d_M(f(\varepsilon),f(\varepsilon_1,\dots,\varepsilon_{i-1},-\varepsilon_i,\varepsilon_{i+1},\dots,\varepsilon_n))^p,
\begin{aligned}\|\tilde{f}_P\|_p = \frac{\|g\|_p}{h^d}\left(h^d\sum_{i=1}^Sp_i^p\right)^{\frac{1}{p}}&\le \frac{\|g\|_pR^{\frac{d}{p}}}{2h^d}\left(\frac{1}{S}\sum_{i=1}^Sp_i^p\right)^{\frac{1}{p}}&= \frac{c_0^{\frac{d}{p}}h^s\|g\|_p}{2}\cdot n\ln n\left(\frac{1}{S}\sum_{i=1}^Sp_i^p\right)^{\frac{1}{p}},\end{aligned}
{\cal L}= \frac{1}{\kappa^{d - 2}}\sqrt{g}\left[ \ - R + \Lambda + \lambda \kappa^{2}\widehat{\mathrm{G}}+ \sum_{n = 1}^{\infty}\lambda_{n}\kappa^{2 n + 2}\Im_{n}[ \widehat{R}, \nabla , \Lambda ] \right]
h_{( 1 )}^{( C )}={\frac{\operatorname{s i n h}^{2}{({\frac{3 \gamma}{2}})}}{4 \operatorname{c o s h}{( \gamma )}}}\operatorname{s i n h}{( 2 \gamma )}\operatorname{c o t h}{( \epsilon_{-})}\sigma_{z}^{( 1 )}
y([n]P) = \frac{\Psi^\prime_{n+2}(x,y)\left ( \Psi^\prime_{n-1}(x,y) \right)^2 - \Psi^\prime_{n-2}(x,y)\left (\Psi^\prime_{n+1}(x,y)\right)^2}{4y\left ( \Psi^\prime_{n}(x,y) \right)^3},
\psi_A = \rho^{1/2}_E \cdot [ C_2 K_{i_p}(2 \mu \rho_E) + C_1 I_{i_p}(2\mu\rho_E)] \longrightarrow_{\rho_0 \gg 0}\sqrt{\frac{\pi}{4\mu}}[C_2 e^{\omega\rho_0}+ C_1 e^{- \omega\rho_0}]
b_{s, \varphi}(f, v): = \lim_{n \to \infty}\mathbb{E}_{\mathbb{Q}_{s,f}^v}\left[ \varphi(G_n \cdot v) \left( \log \frac{\langle f, G_n v \rangle}{\langle f, v \rangle}- n \Lambda'(s) \right) \right],\v \in \mathcal{S}_{\epsilon}.
G=\frac{\pi}{4\delta(0)}\biggl(\sum_{n=1}^{L/2}\frac{1}{\sinh\bigl(\kappa_{n-\frac12}\pi/2\bigr)}-\sum_{n=1}^{L/2}\frac{1}{\sinh\bigl(\kappa_n\pi/2\bigr)}\biggr) .