harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\partial \mathbb{P}(t,\rho)}{\partial t}=\Pi(\rho)^{\frac{1}{2}}\nabla_\rho\cdot(L(\rho)\big(\mathbb{P}(t,\rho)d_\rho\mathcal{F}(\rho)+\beta d_\rho\mathbb{P}(t,\rho)\big) \Pi(\rho)^{-\frac{1}{2}}).
(\alpha^{p^{e'}+1}(v_1')^{p^{e'}+1}f^{p^{e'}}(a_1),\dots,\alpha^{p^{e'}+1}(v_s')^{p^{e'}+1}f^{p^{e'}}(a_s),(v_{s+1}')^{p^{e'}+1}f^{p^{e'}}(a_{s+1}), \dots, (v_n')^{p^{e'}+1}f^{p^{e'}}(a_n),0) = (u_1g(a_1),\dots,u_sg(a_s),u_{s+1}g(a_{s+1}),\dots,u_ng(a_n),0).
\mathcal{M}^T_n=\left(\begin{array}{ccc}{-\mathcal{A}^T_{1,n}}{\mathcal{B}^T_n}{\mathcal{C}^T_n}{\left(\mathcal{B}^T_n\right)^*}{\mathcal{A}^T_{2,n}}{-\mathcal{D}^T_n}{\left(\mathcal{C}^T_n\right)^*}{-\left(\mathcal{D}^T_n\right)^*}{-P\left(T^{-2}-l^T\right)}\end{array}\right)
(-1)^{\ell(w_1)}(-1)^{\ell(w_2)}\chi(T_{w_1})\chi(T_{w_2})\pi(T_{w_1})\pi(T_{w_2}) & = (-1)^{\ell(w_1)}\chi(T_{w_1})\pi(T_{w_1})(-1)^{\ell(w_2)}\chi(T_{w_2})\pi(T_{w_2}) & = \pi'(T_{w_1})\pi'(T_{w_2}).
\sum_{k = 0}^{\frac{r + s}{2}- \frac{\mid r - s \mid}{2}}\frac{( C_{k + \mid \frac{r - s}{2}\mid}^{\frac{r + s}{2}}) ( C_{k}^{\frac{r + s}{2}}) \mid 2 k + \mid \frac{r - s}{2}\mid - \frac{r + s}{2}\mid}{C_{\frac{r + s}{2}}^{r + s}}\, .
\|\mathcal{H}_1\| = \|\widehat{\Theta}_1-\Theta_1\| \left\|I - \frac{2\eta_1}{|\widehat{\mathcal{C}}_1|}ZZ^{\top}\right\|+ \frac{2 \eta_1}{|\widehat{\mathcal{C}}_1|}\sum_{i \in \widehat{\mathcal{C}}_1}\| WZ^{\top}\|,
C_{\star}:= \sup_{k \in \mathbb{N}}\bigg\{\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\bigg(\frac{|u_{k}(x)-u_k(y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}+ \frac{|v_{k}(x)-v_k(y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}\bigg)\,dy\,dx +\int_{\mathbb{R}^{N}}\big(|u_k|^{\overline{p}}+ |v_k|^{\overline{p}}\big)\,dx \bigg\}< \infty.
D_{\alpha}{F}^{\alpha \beta , a}+ \frac{\kappa}{2}\epsilon^{\beta \alpha \lambda}{F}_{\alpha \lambda}^{a}= - i \bar{\psi}\gamma^{\beta}{T}^{a}\psi - i \frac{u}{2}\epsilon^{\beta \alpha \lambda}D_{\alpha}( \bar{\psi}\gamma_{\lambda}{T}^{a}\psi ) .
\frac{\delta X V_2}{N}\gg |\mathbf{x}\cdot \mathbf{w}_2|=|x_3|\,\|\mathbf{w}_2\|_2, \frac{\delta X V_1}{N}\gg |\mathbf{x}\cdot \mathbf{w}_1|=\frac{|x_2|\,\|\mathbf{w}_1\times \mathbf{w}_2\|_2}{\|\mathbf{w}_2\|_2}+O\Bigl(|x_3|\, \|\mathbf{w}_1\|_2\Bigr).
f_{|\Delta_n|^2}(x) = \frac{\left({\alpha_n}/{b_{0,n}}\right)^{\alpha_n}}{2\Gamma(\alpha_n)}& \times \int_0^{\infty}b_n^{\alpha_n-2}\exp\left(-\frac{\alpha_n b_n}{b_{0,n}}-\frac{x+\tilde{A}_n^2}{2b_n}\right)I_0\left(\frac{\tilde{A}_n\sqrt{x}}{b_n}\right)\mathrm{d}b_n.
\delta_2(z) = - \frac{1}{m}\sum_{k=1}^{m}\frac{\epsilon_k}{(z+ \mathbb{E}[s_{n,m}^{1}(z)])^2}- \frac{1}{m}\sum_{k=1}^{m}\frac{\epsilon_k^2}{(z+ \mathbb{E}[s_{n,m}^{1}(z)])^3}- \frac{1}{m}\sum_{k=1}^{m}\frac{\epsilon_k^3}{\beta_k (z+ \mathbb{E}[s_{n,m}^{1}(z)])^3}= S_1 + S_2 + S_3.
w = \sum_{x \in A^G_{}}\vert (x)\tau \vert_k = \sum_{y \in A^S}N_y ( 1 - \delta_{(y)\mu, k}) & = n q^{n-s}\sum_{y \in A^S_{}}( 1 - \delta_{(y)\mu, k}) + n ( q^{n-s}- 1 ) \sum_{y \in A^S_{}}( 1 - \delta_{(y)\mu, k}).
k_1 = \frac{\left(\mu^{(2)}_{X_1}\right)^2}{\mu_{X_1}^{(4)}- \left(\mu^{(2)}_{X_1}\right)^2}, \theta_1 = \frac{\mu_{X_1}^{(4)}- \left(\mu^{(2)}_{X_1}\right)^2}{\mu^{(2)}_{X_1}},
\Vert \sum_{i}n_i x b_i \Vert_2^2 =\sum_{i, j}\tau( b_j^\ast x^\ast n_j^\ast n_i x b_i )= \sum_{i,j}\tau(n_j^\ast n_i) \tau( b_j^\ast b_i ) = \Vert \sum_i n_i \otimes b_i \Vert_2^2,
\left\langle u,v\right\rangle \geq\cos\angle\left(v,V^{\bot}\right)\,\left\Vert u\right\Vert \left\Vert v\right\Vert \quad\mbox{and}\quad\left\langle Pu,Pv\right\rangle =\varepsilon\,\left\Vert Pu\right\Vert \left\Vert Pv\right\Vert \mbox{.}
\lim\limits_{x\to 0-0}h'(x) = \lim\limits_{x\to 0-0}f'(x) - \lim\limits_{x\to 0-0}g'(x) = L-L = 0 \lim\limits_{x\to 0+0}h'(x) = \lim\limits_{x\to 0+0}f'(x) - \lim\limits_{x\to 0+0}g'(x) = R-R = 0,
X^{k}(\alpha^{n+1}) = \sum_{j=0}^{k}X^{j}(\alpha^{n}) \otimes X^{k-j}(\beta^{n}) = \sum_{j=0}^{k}X^{j}(\beta^{n}) \otimes X^{k-j}(\alpha^{n}) = X^{k}(\beta^{n+1}),
\int_X |\Gamma(f_m)-\Gamma(f_n)|dm =\int_X |\Gamma(f_m)^{1/2}-\Gamma(f_n)^{1/2}|\left(\Gamma(f_m)^{1/2}+\Gamma(f_n)^{1/2}\right)dm &\leq 2\left(\int_X |\Gamma(f_m)^{1/2}-\Gamma(f_n)^{1/2}|^2dm\right)^{1/2}\:\sup_n\mathcal{E}(f_n)^{1/2},
\varphi\left( a\wedge^{{\mathbf{L}}}b\right) =\varphi\left( a\right)=f\left( a^{\prime{\mathbf{L}}}\right) =f\left( a^{\prime{\mathbf{L}}}\right) \wedge^{\mathcal{T}_{i}\left({\mathbf{L}}\right)}b=\varphi\left( a\right) \wedge^{\mathcal{T}_{i}\left( P\left({\mathbf{L}}\right)\right)}\varphi\left( b\right)
A_I(N)= A_{N-k+2,\hdots,N}(N)\cap A_{N-k+1,N-k+3,\hdots,N}(N)= = \hat{A}_{N-k+2,\hdots,N}(N)\cap \hat{A}_{N-k+1,N-k+3,\hdots,N}(N)+ + (A_{N-k+2,\hdots,N}(N)\cap A_{N-k+1,N-k+3,\hdots,N}(N))\setminus &(\hat{A}_{N-k+2,\hdots,N}(N)\cap \hat{A}_{N-k+1,N-k+3,\hdots,N}(N)).
\tau_{1}^{(n)}= \frac{\tau_{4}^{(n+1)}}{2}(t_{2}- \lambda ) + (\tau_{1}^{(n)})_{0}\tau_{3}^{(n)}= \frac{\tau_{4}^{(n+1)}}{2}(t_{2}+\lambda ) + (\tau_{3}^{(n)})_{0},
\partial_- (\partial_+ \Gamma^{(1)}\, \Gamma^{(1)-1}) = -\Gamma^{(2)}\, \Gamma^{(1)-1}, \qquad \partial_-(\partial_+\Gamma^{(2)}\, \Gamma^{(2)-1}) = \Gamma^{(2)}\,\Gamma^{(1)-1}.
0 <\psi(j',i') = l_{j'}-l_{i'}-\Delta(j',i') = l_{j'}-l_{i'+1}-\Delta(i',i'+1)-\Delta(j',Q(j'))-\Delta(Q(j'),Q(i')+1)-\Delta(Q(i')+1,i') = (l_{j'}+\Delta(Q(j'),j'))-(l_{i'+1}+\Delta(Q(i')+1,i'+1))-\Delta(Q(j'),Q(i')+1)\,,
u_1 = \ \frac{1}{2}( d_{111}-1,d_{211}), u_2 = \u_1 \ + \ \left(\frac{1}{2},0\right), u_3 = \ \frac{1}{2}( -1,0 ), u_4 = \u_3 \ + \ \left(\frac{5}{6},0\right), u_5 = \ \frac{1}{3}( -2d_{111}-1,-2d_{211}), u_6 = \u_5 \ + \ \left(\frac{2}{3},0\right).
a\left(\sum_{n=1}^{m}|c_n|^2 \right)^\frac{1}{2}\leq \left\|\sum_{n=1}^{m}c_n\tau_n\right\|\leq b\left(\sum_{n=1}^{m}|c_n|^2 \right)^\frac{1}{2}, \forall c_1, \dots, c_m \in\mathbb{R}\mathbb{C}
\frac{\tau^{N(N-1)/2}}{(2\pi \mathrm{i})^n}\int_{\mathcal{C}^N}\prod_{1\leq A<B\leq N}\frac{z_A-z_B}{z_A-\tau z_B}\prod_{i=1}^{N}\exp\big[ -\frac{z_i(p-q)^2}{(z_i+1)(p+qz_i)}t \big]\big(\frac{1+z_i}{1+z_i/\tau}\big)^{j_i}\frac{dz_i}{z_i+\tau}\,,
\frac{1}{X}\int_{X}^{2X}\left|\frac{1}{h}\sum_{\substack{x \leq n \leq x+h n \in \mathcal{S}}}f(n) \right|^2 dx \ll \int_{1}^{1+iX/h_1}\left|F(s)\right|^2 |ds| + \max_{T \geq X/h_1}\frac{X/h_1}{T}\int_{1+iT}^{1+i2T}\left|F(s)\right|^2 |ds|.
\lim_{n\to+\infty}\P_{\theta}\left[f\left(\left(\mathrm{e}^{r_{n}}X_{\mathrm{e}^{-\alpha r_{n}}t}\right)_{t\ge 0}\right)\right]=\P_{\theta}\left[f\left(\left(\mathrm{e}^{r}X_{\mathrm{e}^{-\alpha r}t}\right)_{t\ge 0}\right)\right].
Q(x)e^{\lambda x}=1+\frac{\beta}{\alpha}\left[\; \; \int_0^x e^{\lambda y}G(Q(y))\mathrm{d}y \right. &+e^{2\alpha x}\int_x^{+\infty}e^{-(\alpha-\mu )y}G(Q(y))\mathrm{d}y &-\left.\int_0^{+\infty}e^{-(\alpha-\mu ) y}G(Q(y))\mathrm{d}y \;\right].
\delta(\tau(x_i)) = \delta(\tau'(x_i)+\epsilon_i)=\delta(\tau'(x_i)) + \delta(\epsilon_i) + \sum_{i=1}^{p-1}p^{-1}\binom{p}{i}\tau'(x_i)^{i}\epsilon_i^{p-i}, \tau(\delta(x_i)) = \sum_{I}\tau'(\delta(x_i)^{(I)}) \cdot \epsilon_1^{I_1}\cdots \epsilon_n^{I_n},
F(n^{-1},\tau) = \begin{cases}1-\mathcal{O}\big(e^{-j^{3/2}N_{f}}\big) n^{-1}>2\sqrt{1+2\tau}\mathcal{O}\big(e^{-|j|^3N_{f}^{2}}) n^{-1}<2\sqrt{1+2\tau}\end{cases}.
\frac{\rm d}{{\rm d}x}L_n^{(\alpha)}(x)=-L_{n-1}^{(\alpha+1)}(x), begin{align*}.5ex]L_n^{(\alpha)}(x)=L_{n}^{(\alpha+1)}(x)-L_{n-1}^{(\alpha+1)}(x), xL_{n}^{(\alpha+1)}(x)=(n+\alpha+1)L_n^{(\alpha)}(x)-(n+1)L_{n+1}^{(\alpha)}(x).
\frac{1+t}{2}\int_0^{|u|}\frac{\xi''(s)}{{b}-{d}(s)}ds+ \frac{1-t}{2}\int_0^{|u|}\frac{\xi''(s)}{{b}-{d}(s)}ds=\int_{0}^{|u|}\frac{\xi''(s)}{{b}-{d}(s)}ds
f_N^{(\alpha)}(|p-q|) = \sum_{l=0}^{N-1}\frac{e^{i\kappa_l(p-q)}}{N}\omega_{\alpha}^2(\kappa_l) = \frac{\Omega_{\alpha}^2}{N}\sum_{l=0}^{N-1}e^{i\kappa_l(p-q)}\left(4\sin^2{(\frac{\kappa_l}{2}})\right)^{\frac{\alpha}{2}}
f_1(v) =e^{vQ_1}\left\{1+vS_1S_2e^{v(Q_1+Q_2)}\int_0^ve^{-S_1S_2\int_{v'}^vv''e^{v''(Q_1+Q_2)}dv''}dv'\right\}, f_2(v) =vS_1e^{vQ_2}f_1(v)
V_i^*V_i =\frac{(1-T_iT_i^*)\otimes 1+1\otimes(1-T_iT_i^*)}{2}e_+ +\frac{(1-T_iT_i^*)\otimes 1+1\otimes(1-T_iT_i^*)}{2}e_- +\frac{(1-T_iT_i^*)\otimes 1-1\otimes(1-T_iT_i^*)}{2}e_+ +\frac{(1-T_iT_i^*)\otimes 1-1\otimes(1-T_iT_i^*)}{2}e_- =(1-T_iT_i^*)\otimes 1.
\alpha^{*}_n(aq,b, q)=\frac{(1-aq^{2n+1})(aq/b;q)_n(-b)^nq^{n(n-1)/2}}{(1-aq)(bq)_n}\sum_{n\ge j\ge0}\frac{(b)_j}{(aq/b)_j}(-b)^{-j}q^{-j(j-1)/2}\alpha_j(a,q)
p(N,\vec\ell;2)= \frac{(-1)^{N+t}\prod_j\ell_j!}{(N)_{\ell}}\sum_{r=\ell-t}^{N-1}(-1)^{r}\binom{N-1}{r}^{-1}\!\!K(N,\ell,t;r), K(N,\ell,t;r):= \,[\xi^{r-\ell+t}z^{N-\ell}]\,(1-\xi)^{t-1}(1-z)^{-t+1}(1-\xi z)^{-\ell-1}.
\mathcal{N}(t)(\hat{\rho}) := \int_{\mathbb{R}^{2n}}\hat{D}(\mathbf{x})\,\hat{\rho}\,{\hat{D}(\mathbf{x})}^\dag\,\mathrm{e}^{-\frac{|\mathbf{x}|^2}{2t}}\frac{\mathrm{d}^{2n}x}{(2\pi\,t)^n}\;.
\varepsilon\mu\frac{\partial p_i\psi_i}{\partial z}(z,\mu) + \Sigma_{ti}p_i\psi_i (z,\mu) = \frac{\Sigma_{ti}-\varepsilon^2\sigma_{ai}}{2}\int_{-1}^1p_i\psi_i(z,\mu')d\mu' &+ \eta|\mu|\left(\frac{p_j\psi_j(z,\mu)}{\lambda_j}-\frac{p_i\psi_i(z,\mu)}{\lambda_i}\right) +\varepsilon^2\frac{p_iq_i}{2}.
du(t,x,z)=u(t,x,z)b(t,x,z) d G(t)+k(t,x,z)dF(t,x,z)+xu(t,x,z)dG(t)+u(t,x,z) b(t,x,z)x dt &+k(t,x,z) b(t,x,z)y(t,x,z)q(t,x,z)dt &=u(t,x,z)[\{b(t,x,z)+x\}d G(t)+b(t,x,z)x dt]+k(t,x,z)[dF(t,x,z)+b(t,x,z)y(t,x,z)q(t,x,z)dt]
u_i^{\mathrm{s}}(\hat{\boldsymbol{\tau}}_i)=\sum_{k=0}^\infty{\delta^k}\pi_i(\boldsymbol{d}_i^*; \hat{\boldsymbol{\tau}}_i\!)= \frac{\pi_i(\boldsymbol{d}_i^*; \hat{\boldsymbol{\tau}}_i\!)}{1-\delta}\forall\;\hat{\boldsymbol{\tau}}_i \in \mathcal{A}_i^{\mathrm{s}}
\bar{L}_{2N}^{(p)}: =(\tfrac{\beta}{2}-N+1)_{2N-1}L_{2N}^{(p)}=(\tfrac{2J}{n})^N\Big[(\tfrac{\beta}{2}-N+1)_{2N}R_N(y^{(+)};0)+(\tfrac{\beta}{2}-N)_{2N}R^{(1)}_N(y^{(1)})\Big]
\lVert k\cdot B\psi\lVert^2 =\sum_{i,j=1}^{\nu}\langle k_iB_i\psi,k_jB_j\psi \rangle\leq \sum_{i,j=1}^{\nu}\lvert k_i\lvert \lvert k_j\lvert \lVert B_i\psi \lVert\lVert B_j\psi \lVert &\leq \sum_{i,j=1}^{\nu}\frac{1}{2}\lvert k_i\lvert^2 \lVert B_j\psi \lVert^2 +\frac{1}{2}\lvert k_j\lvert\lVert B_i\psi \lVert^2=\lvert k\lvert^2\lVert B\psi\lVert^2
\sum_{n = 1}^{N}\left[\begin{matrix}N n \end{matrix}\right] \frac{q^{n(n + 1)}}{(q)_n (1 - q^n)}F\left(q^N, q^n; -q^n\right)=\frac{1}{2(-q)_N}\sum_{k=1}^{N}\left[\begin{matrix}N k \end{matrix}\right] \frac{q^{k(k+1)/2}}{(1-q^k)}\left(\frac{(-q)_k}{(q)_k}- 1 \right).
h(s,y) = \frac{e^{-s}}{b(t(s))^2}\left( -2 \frac{d\alpha}{ds}(s) \psi_0(y)+\alpha(s)\left(\frac{y}{2}\cdot\nabla_y\psi_0(y)+\left(\frac{n}{2}+1\right)\psi_0(y) \right) \right) &+ r(s,y) - \left(\int_{\mathbb{R}^n}r(s,y) dy \right) \varphi_0(y).
\sigma_{n,-1}(t) = (-1)^n \frac{\alpha_{n+1}(t) \alpha_{n}(t)\cdots \alpha_{2}(t) \sigma_{0,0}(t)}{\beta_{n+1}(t)\beta_{n}(t) \cdots \beta_{2}(t)\beta_{1}(t)}= \frac{\sigma_{n,n}(t)}{\beta_{n+1}(t)a_{n,0}(t)}.
h_{\underline{k},\ell}=\frac{\tau_{k+1,\ell}}{\tau_{k,\ell}},h_{{k},\underline{\ell}}=\frac{\tau_{k,\ell+1}}{\tau_{k,\ell}},h_{\underline{k},\underline{\ell}}=\frac{\tau_{k+1,\ell+1}}{\tau_{k,\ell}}.
\left|\frac{\delta F}{\delta \nu}(\nu_t, \mu_t, x) + \frac{\sigma^2}{2}\log \left( \frac{\nu_t(x)}{\pi(x)}\right) - \frac{\sigma^2}{2}\operatorname{D_{KL}}(\nu_t|\pi)\right| &\leq 3C_{\nu}+ \frac{\sigma^2}{2}\left( \max\{|\log r_{1, \nu}|, \log R_{1, \nu}\}+ 2 \log R_{\nu}\right) =: C_{V, \nu},
T = \frac{1}{d}\int_0^{R_0}\frac{z}{g\left( \frac{m}{\alpha_d}z^{-d}\right)}\, \mathrm{d}z = C(m,d,s) \int_0^{R_0}z^{sd+1}\, \mathrm{d}z= \frac{C(m,d,s)}{sd+2}\, z^{sd+2}\Big|_0^{R_0}
\begin{array}{ll}\|u^s -1\|_{L^2(\mathbb{R_{+}})}^2 + 2\int\limits_0^t \int\limits_0^{+\infty}|u_y^s|^2 \,\mathrm{d}y\mathrm{d}t+ 2\beta \int\limits_0^t \big|u^s|_{y=0}\big|^2 \,\mathrm{d}t begin{align*}10pt]\leq \|u_0^s -1\|_{L^2(\mathbb{R_{+}})}^2 + 2\int\limits_0^t\big|\partial_y u^s|_{y=0}\big|_{\infty}\,\mathrm{d}t \leq C(T).\end{array}
\lambda^{a , \alpha \beta}\lambda_{a ~ \delta}^{~ \gamma}- \lambda^{a , \gamma \alpha}\lambda_{a ~ \delta}^{~ \beta}= \frac{2}{\sigma_{0}}\left( \Omega^{\alpha \beta}\delta_{~ \delta}^{\gamma}- 2 \Omega^{\beta \gamma}\delta_{~ \delta}^{\alpha}+ \Omega^{\gamma \alpha}\delta_{~ \delta}^{\beta}\right) ,
\sum_{k\geq M(\tau)}f(k) \geq \int_{M(\tau)}^\infty f(y)\, dy= \int_{h(\tau)}^\infty f(y)\, dy - \int_{h(\tau)}^{M(\tau)}f(y)\, dy \geq \int_{h(\tau)}^\infty f(y)\, dy - \tau\, , begin{align*}5pt] \sum_{k\geq M(\tau)}f^2(y) \leq \int_{M(\tau)}^\infty f^2(y)\, dy + f^{2}(M(\tau)) \leq \int_{h(\tau)}^\infty f^2(y)\, dy + \tau^2\, .
\mathcal{A}=\begin{pmatrix}A_{11}I_{d\times d}-A_{11}A_{13}A_{13}A_{21}-A_{21}I_{d\times d}-A_{11}^T -A_{11}^T 0_{d\times d}0_{d\times d}I_{d\times d}I_{d\times d}-I_{d\times d}I_{d\times d}0_{d\times d}0_{d\times d}\end{pmatrix},
\chi^{\rm (TM)}={i\hbar A\over \pi^2}\int d\omega_{\rm in}\int_{0}^{|\omega_{\rm in}|}dk_{\|}\, k_{\|}\,\frac{\left[k_{\|}^2-\omega_{\rm in}(\omega+\omega_{\rm in}) \right]^2}{\biggl[(\omega+\omega_{\rm in}+ i\varepsilon)^2-k_{\|}^2\biggr]^{1/2}\, \sqrt{{\omega_{\rm in}}^2-k_{\|}^2}}\,\left[{1\over 2}+{\overline n}(|\omega_{\rm in}|)\right].
\sum_{r\in[s,T]}q^{n,[s,T]}(r)\,\psi(r) \leq \sum_{i=1}^{n}q^{n,[t,T]}(\tfrac{T-t}{2}c_i^n+\tfrac{T+t}{2})\,\psi(\tfrac{T-t}{2}c_i^n+\tfrac{T+t}{2}) = \sum_{r\in[t,T]}q^{n,[t,T]}(r)\,\psi(r).
A = \left( \begin{array}{c c c}{- 2 M - 2 \Sigma / \sqrt{3}}&{- 2 Q}&{2 N}\\{}&{}&{}\\{2 Q}&{4 \Sigma / \sqrt{3}}&{2 P}\\{}&{}&{}\\{2 N}&{- 2 P}&{2 M - 2 \Sigma / \sqrt{3}}\\ \end{array}\right) .
L_{\psi^{\prime}\psi^{\prime}\psi \psi}= g^{2}\sum_{i , j = L , R}\sum_{q = u , d}\frac{\epsilon_{i j}}{( \Lambda_{i j}^{\epsilon})^{2}}( \bar{\psi}_{i}^{\prime}\gamma_{\mu}\psi_{i}^{\prime}) ( \bar{\psi}_{j}\gamma^{\mu}\psi_{j})
{\cal S}_{W Z W}\, = \, - \frac{k}{4 \pi}\int_{\Sigma}d^{2}\sigma A_{\alpha}^{I}A^{J \alpha}\Omega_{I J}\, - \, \frac{k}{1 2 \pi}\int_{B}d^{3}\sigma \, \epsilon^{\alpha \beta \gamma}A_{\alpha}^{I}A_{\beta}^{J}A_{\gamma}^{L}{f_{I J}}^{N}\Omega_{N L}\, ,
f_\lambda(w) =\frac{1}{2}\int_\Omega{\langle\nabla u(x),\nabla u(x)\rangle\,dx}-\frac{1}{2}\int_\Omega{\langle \nabla v(x),\nabla v(x)\rangle\,dx}+\int_\Omega{F(\lambda,x,u(x),v(x))\, dx},
R(M_s,M_p) \big|_{\textrm{Scheme2}}= \frac{\sum_{n=1}^{\Lambda-t_s}\binom{\Lambda-n}{t_s}\binom{\mathcal{L}_1}{t_p+1}}{\binom{\Lambda}{t_s}\binom{\mathcal{L}_1}{t_p}}= \frac{\binom{\Lambda}{t_s+1}\binom{\mathcal{L}_1}{t_p+1}}{\binom{\Lambda}{t_s}\binom{\mathcal{L}_1}{t_p+1}}=\frac{(\Lambda-t_s)(\mathcal{L}_1-t_p)}{(t_s+1)(t_p+1)}
A_{77,\ell}= \int_{C/\ell}^{\pi -C/\ell}\frac{a^2_{7,\ell}(\phi)}{(1-4 \alpha_2^2/\lambda_{\ell}^2)^{1/2}(1-4 \alpha_1^2/\lambda_{\ell}^2)^{1/2}}\sin \phi d \phi
K_{1}(w,\lambda(t)) = \int_{0}^{\infty}\frac{R dR}{(1+R^{2})^{3}}\int_{0}^{w}\frac{\rho d\rho}{w}\left(1+\frac{R^{2}\lambda(t)^{2}-1-\rho^{2}}{\sqrt{(R^{2}\lambda(t)^{2}-1-\rho^{2})^{2}+4R^{2}\lambda(t)^{2}}}\right)
\sum_{i=2}^{n-1}\big[\ell(x_{i+1},\ldots, x_n),[r(x_1,\ldots,x_{i-1}),x_i]\big] - \big[[x_i,\ell(x_{i+1},\ldots, x_n)], r(x_1,\ldots,x_{i-1}) \big] = \sum_{i=2}^{n-1}\big[\ell(x_{i+1},\ldots, x_n),r(x_1,\ldots,x_{i-1},x_i)\big] - \big[\ell(x_i,x_{i+1},\ldots, x_n),r(x_1,\ldots,x_{i-1})\big].
x =\|\phi\|_{L^2}^{\frac{4\sigma-1}{\sigma}}\|\mathcal{P}_{\lambda_n}^{1/2}\phi\|_{L^2}^{\frac{1}{\sigma}}, y= \alpha\|\phi\|_{L^2}^3\|\mathcal{P}_{\lambda_n}^{1/2}\phi\|_{L^2}, X =\|v_n\|_{L^2}^{\frac{4\sigma-1}{\sigma}}\|\mathcal{P}_{\lambda_n}^{1/2}v_n\|_{L^2}^{\frac{1}{\sigma}}, Y=\alpha\|v_n\|_{L^2}^3\|\mathcal{P}_{\lambda_n}^{1/2}v_n\|_{L^2}.
\dot{z}_i =A_i^0z_i+f_i(x_i), \dot{\hat x}_{i}=S\hat x_i+dv+dK\sum\nolimits_{j=0}^{N}a_{ij}(\hat x_{i}-\hat x_j)+\bar g_i(\bar z_i,\,z_i,\,x_i) \dot{\xi}_i =A_i^0\xi_i+f_i(x_i) \dot{w}=Sw+dv
\psi(h_5) =\psi(h_6)=\psi(h_7)=\psi(h_8)=\frac{n+1}{4}e n\equiv 3 \pmod{4}\,, \psi(h_5) =\psi(h_6)=\psi(h_7)=\psi(h_8)=\frac{3n+1}{4}e n\equiv 1 \pmod{4}\,.
\frac{(q^2;q^2)_\infty^2}{(-q^2;q^2)_\infty^2}\sum_{n=-\infty}^\infty\frac{(-1)^nq^{3n^2+3n}}{1-q^{2n+1}}=-3\sum_{n=-\infty}^\infty\frac{(2n-1)q^{n(3n+1)-2}}{1+q^{6n-3}}-\sum_{n=-\infty}^\infty\frac{(6n-1)q^{n(3n+1)-1}}{1+q^{6n-1}}.
2J_l(g) \leq \int_{|\tau| \leq \psi_{l+1}}\left|\sum_{n \geq 1}\frac{g(n)\Lambda(n)}{n^s}\right|^2 d\tau \leq 3B^2\int_{|\tau| \leq \psi_{l+1}}\left|\sum_{n \geq 1}\frac{\Lambda(n)}{n^s}\right|^2 d\tau = 3B^2 \int_{|\tau| \leq \psi_{l+1}}\left|\frac{\zeta'(s)}{\zeta(s)}\right|^2 d\tau =: 3B^2\iota_l
\partial_1(a) \cdot \partial_2(b) + \eth_1(a \cdot \partial_2(b)) - \partial_2(a) \cdot \partial_1(b) - \eth_2(a \cdot \partial_1(b)) \qquad{}\overset{\eqref{eq:qadm2}}{=}a \cdot \eth_1(\partial_2(b)) - a \cdot \eth_2(\partial_1(b)) \overset{\eqref{eq:vip1}}{=}0.
{\cal W}( \theta ) = \left({\cal P}\int d \theta^{\prime}\rho ( \theta^{\prime}) \cot \frac{\theta - \theta^{\prime}}{2}\right)^{2}- 2{\cal P}\int d \theta^{\prime}\rho ( \theta^{\prime}) d \theta^{\prime \prime}\rho ( \theta^{\prime \prime}) \cot \frac{\theta - \theta^{\prime}}{2}\cot \frac{\theta^{\prime}- \theta^{\prime \prime}}{2}\; .
\int_{0}^{\infty}e^{-s x}F_1(x)dx=\frac{1}{s}\left(\frac{\mathbb E\left[e^{-s \overline{X}_{e(q)}}\right]}{\mathbb E\left[e^{-s \overline{X}_{e(p+q)}}\right]}-1\right), \ \s > 0.
\begin{array}{rcccl}G_{i\mu}\to G^{\prime}_{i\mu}= U_i\star G_{i\mu}\star U_i^{-1}+ig_i^{-1}U_i\star\partial_{\mu}U_i^{-1}, \Phi \to \Phi^{\prime}= U_1\star\Phi\star U_2^{-1},\end{array}
H = \int d x \Biggl [ \frac{1}{2 R}\Pi^{2}( x , t ) + \frac{m^{2}R}{2}\Phi^{2}( x , t ) + \frac{1}{2 R}( \frac{d}{d x}\Phi )^{2}+ \frac{\lambda R}{4}\Phi^{4}( x , t ) \Biggr ] .
\eta(\eta(\tau_i, \tau_j), \eta(\tau_i, \tau_\ell)) =\eta((\tau_n^q\tau_1)^{n-i}\tau_n^{q-1+D(n-i+j)}\tau_{n-i+j+1},(\tau_n^q\tau_1)^{n-i}\tau_n^{q-1+D(n-i+\ell)}\tau_{n-i+\ell+1}) =\eta(\tau_n^{D(n-i+j)}\tau_{n-i+j+1},\tau_n^{D(n-i+\ell)}\tau_{n-i+\ell+1}).
\begin{aligned}U_R(x) =\left[a+b\left\{\frac{1}{|x|^{\frac{n-p}{p-1}}}-\frac{1}{R^{\frac{n-p}{p-1}}}\right\}^{-\frac{p}{n-p}}\right]^{1-\frac{n}{p}}&=\left[a+bR^{\frac{p}{p-1}}\left\{-\frac{n-p}{p-1}\log_{\frac{n-1}{p-1}}\frac{|x|}{R}\right\}^{-\frac{p}{n-p}}\right]^{1-\frac{n}{p}}\end{aligned}
A_{s}=\int_{0}^{\pi}A_{T}=\int_{0}^{\pi}\frac{(2H-z_{0})z_{0}}{2H^{2}}r\sqrt{r^{2}+H^{2}}d\theta=\frac{kr\sqrt{r^{2}+H^{2}}}{2H^{2}}\int_{0}^{\pi}(2H-z_{0})y_{0}d\theta
\cap_{i=1}^m\mathfrak{m}^{\mathbf{a}+1-\mathbf{c}_i}=\cap_{i=1}^m\mathfrak{m}^{\mathbf{a}\backslash\mathbf{c}_i}+\mathfrak{m}^{\mathbf{a}+2}=I^{[\mathbf{a}]}+\mathfrak{m}^{\mathbf{a}+2}.
K ( n , k^{2}) \equiv \frac{1}{2 n}\int_{0}^{2 n \pi}d \theta_{2}\int_{0}^{2 n \pi}d \theta_{1}~{\cal F}( \theta_{1})^{k^{2}- 1}{\cal F}( \theta_{2})^{k^{2}- 1}\left| 2 n \sin \frac{\theta_{2}- \theta_{1}}{2 n}\right|^{- 2 k^{2}},
\sum_{\ell=0}^{N-1}\big|\mathcal{E}_{N,\ell}^{h,k,2,1}\big|\le \frac{C(1+|x|_{L^p})^K}{\tau^{2\kappa}}\bigl(t_{N-1}^{-\frac12+\kappa}\Delta t |h|_{L^{2q}}|k|_{L^{2q}}+t_{N-1}^{\frac12+\kappa-\beta-\gamma}|(-A)^{-\beta}h|_{L^{2q}}|(-A)^{-\gamma}k|_{L^{2q}}\bigr).
y_1^2 = x_1\big(x_1^2 + x_1(t_1^4/16 + t_1^3(b-1)(c-a)/2 + t_1^2(3b^2c^2-abc^2-4bc^2+2ac^2-4ab^2c-b^2c+2a^2bc +3abc+2bc-4a^2c+2ac+2ab^2+2a^2b-4ab)/2 + 2t_1(b-1)b(b-a)(c-1)c(c-a) + b^2(b-a)^2c^2(c-1)^2 ) - (a-1)a(b-1)(c-a)(c-b)t_1^3(t_1+2bc-2b)(t_1+2bc-2ac)/2 \big)
|A(t,k,r)|=\left |\sum_{j=0}^{\min (k,t)}(-1)^j\binom{k}{j}\binom{j+t}{j}\binom{r-j-1}{r-t-1}\right|\leq 4\left(\frac{q}{r-1-q}\right)^{q-t}\binom{r-1}{t}.
b_{1}(\sqrt{x}) = \lim_{z \to -1}i \sqrt{x}(z+1) \left[ \partial_{z}\Phi(xz;x,a)\Phi^{-1}(xz;x,a) \right]_{12}, & b_{2}(\sqrt{x}) = \lim_{z \to -a}i \sqrt{x}(z+a) \left[ \partial_{z}\Phi(xz;x,a)\Phi^{-1}(xz;x,a) \right]_{12}.
M_0 = \{6 a: 0 \leq a \leq 18 k\}, M_1 = \{6 a + 1 : 12 k \leq a \leq 24 k-1\}, M_3 = \{6 a + 3 : 0 \leq a \leq 54 k -2\}, M_5 = \{6 a - 1 : 12 k \leq a \leq 24 k-1\}.
M_{\pm}^2 = \frac{(\kappa + \Pi_{\rm odd})^2 + M_1^2 + M_2^2 \pm ((M_1^2-M_2^2)^2 + 2 (M_1^2 + M_2^2)(\kappa + \Pi_{\rm odd})^2 + (\kappa +\Pi_{\rm odd})^4)^{1/2}}{2}.
{\cal S}\,=\,{1 \over 2}\int\,dx\,dt\,\left[\frac{\dot\phi^a\dot\phi^a\,+\,\phi^a\partial_i\partial_i\phi^a}{\phi^a\,\phi^a}\,-\,\frac{(\,\dot\phi^a\dot\phi^a\,)^2}{(\,\phi^a\,\phi^a\,)^2}\right]\;\;,
\sum_{n = - \infty}^{\infty}\int d \tau d^{d - 2}x^{i}\left( \epsilon^{2}\partial_{\tau}\phi_{n}^{*}\partial_{\tau}\phi_{n}^{\vphantom x}+ \frac{2 i n}{R}\phi_{n}^{*}\partial_{\tau}\phi_{n}^{\vphantom x}- \partial_{i}\phi_{n}^{*}\partial_{i}\phi_{n}^{\vphantom x}- M^{2}\phi^{*}\phi \right) \ .
\frac{{d}^j}{{d}\alpha^j}\tan \alpha = \sum_{n=1+[j/2]}^\infty \frac{2^{2n}\,(2^{2n}-1)\,\left|B_{2n}\right|}{(2n)!}\,\frac{(2n-1)!}{(2n-1-j)!}\,\alpha^{2n-1-j}\,, |\alpha|<\pi/2\,,
\begin{aligned}\|SWu\|_{L_t^{\theta_q}(L^{q,2}_D)}= \lim_{n\rightarrow\infty}\left|\left<SWu.v_n\right>_{L_t^{\theta_q}(L^{q,2}_D),L_t^{\theta_q'}(L^{q',2}_D)}\right|\lesssim_{p,q,W}T^{1/\theta_{\alpha,\beta}}\|u\|_X.\end{aligned}
h_{n,d}(u) \, = (d-1)! \sum_{j=0}^d (-1)^j \binom{d}{j}C_{n-2j}^d(u) = (d-1)! \sum_{j=0}^{d-1}(-1)^j \binom{d-1}{j}Z_{n-2j}^{d-1}(u).
\mathrm{D}f_\infty^{-1}(2)=\frac{n}{2\sqrt{\mathrm{disc(f)}}}\omega_ny^{\frac{n}{2}-1}\le \sqrt{2^{n-5}\cdot\frac{n}{\pi}\cdot\left(\frac{2\pi e}{n}\right)^{n}}\le \frac{1}{50}.
\|\mathcal{L}(\partial_{n-1}\tilde{\varphi})\|_{L^q(B_1)}\leq \Lambda\|D^3\tilde{\varphi}\|_{L^q(B_1)}= \Lambda\|s_k\varphi(s_kx)\|_{W^{3,q}(B_1)}&= \Lambda s_k^{1-n/q}\|\varphi\|_{W^{3,q}(B_{s_k})}\rightarrow 0.
\tilde F_k= \sum_{s=1}^k F_k=(1-t)\sum_i (x_i+\dots +x_i^k)\prod_{j\ne i}\frac{x_i-tx_j}{x_i-x_j}=(1-t)\sum_i \frac{x_i(1-x_i^{k})}{1-x_i}\prod_{j\ne i}\frac{x_i-tx_j}{x_i-x_j}.
\mathcal{F}(\mathbf{h}_*,\mathbf{g}_*) \leq \liminf_{N \to \infty}\mathcal{F}_N (\mathbf{h}_N,\mathbf{g}_N) \leq \limsup_{N \to \infty}\mathcal{F}_N (\mathbf{h}_N,\mathbf{g}_N) \leq \lim_{N \to \infty}\mathcal{F}_N (\mathbf{h},\mathbf{g}) = \mathcal{F}(\mathbf{h},\mathbf{g}).
\left( ( D^{2})^{a b}+ g \epsilon^{a d c}A^{d}D^{c b}\right) f_{b}= j_{\mathrm{t o t}, 0}^{b}; j_{\mathrm{t o t}, 0}^{b}= g \epsilon^{a b c}A_{i}^{a}E_{i}^{c}+ j_{0}^{b},
\sigma^{-\tau - n}e^{n\sigma}\int_{n\sigma}^\infty \ell^{\tau +n -1}e^{-\ell}\,d\ell & < \sigma^{-\tau - n}e^{n\sigma}\int_0^\infty \ell^{\tau +n -1}e^{-\ell}\,d\ell = \sigma^{-\tau - n}e^{n\sigma}\Gamma (\tau + n).
\big\| \int_0^t e^{i(t-s) \Delta}[F(u+w)(s) - F(u(s))] ds \big\|_{X^s([0,T])}&\lesssim \| w \|_{X^s([0,T])}\big( \| u \|_{X^s([0,T])}+ \| w \|_{X^s([0,T])}\big)^{\frac{4}{d-2}}.
c \frac{d}{d t}\left( ( 1 - \frac{1}{c^{2}}| \frac{d{\bf x}_{1}}{d t}|^{2})^{- 1 / 2}\right) = \frac{1}{c}\sum_{i = 1}^{3}\frac{d x_{1}^{i}}{d t}\frac{d}{d t}\left( ( 1 - \frac{1}{c^{2}}| \frac{d{\bf x}_{1}}{d t}|^{2})^{- 1 / 2}\frac{d x_{1}^{i}}{d t}\right) .
|\Phi_{K\to L}^n(v^n_K)|^2=-2\int_{v^n_K}^{v^n_L}\Phi_{K\to L}^n(\xi)\partial_\xi\Phi_{K\to L}^n(\xi)d\xi=2\int_{v^n_K}^{v^n_L}\Phi_{K\to L}^n(\xi)a_{K\to L}^n(\xi)d\xi,
\int_{X^{[2]}}s_4\left( \left(E \otimes H\right)^{[2]}\right) = \alpha_{0, 0}+ \alpha_{1,0}g + \alpha_{0, 1}d + \alpha_{1, 1}g d + \alpha_{2, 0}g^2 + \frac{1}{2}d^2 >0,
A =G\vee C(\ddot3)\vee G, A' =G'\vee C(\ddot3)\vee G', B =G\vee C(\dot1,2,\dot1)\vee G, B' =G'\vee C(\dot1,2,\dot1)\vee G', C =G\vee C(\dot1,1,1,\dot1)\vee G, C' =G'\vee C(\dot1,1,1,\dot1)\vee G'.
1+ W^{(2)}_{u_1 u_1}{u_1}_x+ W^{(2)}_{u_1 u_2}{u_2}_x=0\, , W^{(2)}_{u_2 u_1}{u_1}_x+W^{(2)}_{u_2 u_2}{u_2}_x=0\, , & u_1 + W^{(2)}_{u_1 u_1}{u_1}_t+W^{(2)}_{u_1 u_2}{u_2}_t=0\, , 1 +W^{(2)}_{u_2 u_1}{u_1}_t+W^{(2)}_{u_2 u_2}{u_2}_t=0\, .
T_{{\bar{m}}{\bar{n}}{\bar{s}}}\rightarrow T_{{\bar{m}}{\bar{n}}{\bar{s}}}+ \nabla_{\bar{m}}\nabla_{\bar{n}}\Lambda_{\bar{s}}H_{\alpha \nu}\rightarrow H_{\alpha \nu}+ W_{\alpha \nu s}G^{s{\bar{s}}}\Lambda_{\bar{s}}\ ,
|I_{1}(x)| \leq\sum_{k=-\infty}^{-1}\int_{2^{k}R\leq |xy^{-1}|<2^{k+1}R}\frac{|xy^{-1}|^{\alpha-Q}|f(y)|}{(1+|xy^{-1}|)^{\gamma}}dy &\leq \sum_{k=-\infty}^{-1}\frac{(2^{k}R)^{\alpha-Q}}{(1+2^{k}R)^{\gamma}}\int_{2^{k}R\leq |xy^{-1}|<2^{k+1}R}|f(y)|dy &\leq CMf(x)\sum_{k=-\infty}^{-1}\frac{(2^{k}R)^{\alpha-Q+Q/p_{2}}(2^{k}R)^{Q/p_{2}^{'}}}{(1+2^{k}R)^{\gamma}}.
\textsf{E}(\Xi^{2})^{p}= \int_{0}^{\infty}\textsf{P}\{(\Xi^{2})^{p}>t\}\,dt=\int_{0}^{\infty}\textsf{P}\{\Xi^{2}>u\}pu^{p-1}\,du = (C\log n\sum_{i\le N}\Vert B_{i}\Vert_{2}^{2})^{p}\int_{0}^{\infty}\textsf{P}\{\Xi^{2}>Cz\log n\sum_{i\le N}\Vert B_{i}\Vert_{2}^{2}\}pz^{p-1}\, dz.
S\left(\sum_{i\in\Gamma}(\operatorname{id}\otimes\varphi)\bigl(\Delta(c^*p_i)(1\otimes q_i^*d)\bigr)\right)=\sum_{i\in\Gamma}(\operatorname{id}\otimes\varphi)\bigl((1\otimes c^*p_i)\Delta(q_i^*d)\bigr).