harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
w^{m}, \triangle w^{m}\in H^{s - m}(\Omega) \cap H^{1}_{0}(\Omega) m = 2, \dots, s - 1 w^{s}\in H^{2}(\Omega) \cap H^{1}_{0}(\Omega), \theta^{k}\in H^{s + 1 - k}(\Omega) \cap H^{1}_{0}(\Omega) k = 1, \dots, s - 2 \theta^{s - 1}\in H^{1}_{0}(\Omega).
\Delta\left(h(f_0);f\right) =\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|\sum\limits_{j=0}^{\infty}((\bold{B}^0)^{-1}\bold{a})_je^{ij\theta}\right|^2 f_0^{-2}(\theta)f(\theta)d\theta.
d_{12}= \begin{cases}N, \frac{N}{M}\in\left(0,1\right] M, \frac{N}{M}\in\left(1,\frac{3}{2}\right] \frac{2N}{3}, \frac{N}{M}\in\left(\frac{3}{2},3\right] 2M, \frac{N}{M}\in\left(3,\infty\right) \end{cases}
\int\limits_{S_2}\mathbb{E}_{m^{1,t}}[f(\zeta^t(t),v)] m_0^2(dv) =\int\limits_{S_1}\mathbb{E}_{m^{2,t}}[f(u,\xi^t(t)) \exp \{\int\limits_0^t g(s,\xi^t(s))ds\}] m_0^1(du).
u_n = \begin{cases}\frac{n(N+1-n)(N-n+a+c)(n-1+a+c)\left((n-j-1)^2-(a-c)^2\right)}{4(N-2n)(N-2n+2)}begin{align*}1em]\alpha(1-\alpha)(a-c)^2(1+j)^2(a+c+j)^2 begin{align*}1em]\end{cases}
|f_{\nu,L}(X_1,\dots, X_n)-f_{\nu,L}(X_1',\dots, X_n')|\leq &4\|B_1\|_{\infty}\dots \|B_{k+1}\|_{\infty}\|P_r\|_1 (k+2)(3\delta)^{k-1}\frac{\|\Sigma\|_{\infty}^{1/2}+\sqrt{2\delta}}{\sqrt{n}}\biggl(\sum_{j=1}^n \|X_j-X_j'\|^2\biggr)^{1/2}.
\sum_{i=0}^\infty \frac{1-\alpha_i\lambda_i}{\alpha_i\lambda_i}\|z^{i+1}- z^i\|^2 \leq \frac{1}{\rho}\left( 1+ \eta_{\mathrm{p}}\eta_{\mathrm{s}}\right)\|z^0 - z^\ast\|^2_{U_0}.
I_{b}=2\Im[a_{jk}\partial^{b}_{k}v (\partial_{j}b_{\ell}-\partial_{\ell}b_{j}) a_{\ell m}\partial_{m}\psi\ \overline{v}] = 2\Im [(a\nabla^{b}v)\cdot (db \a\nabla \psi)\overline{v}]
D_{p^l+3,k}(1,x)=D_{p^l+2,k}(1,x)-xD_{p^l+1,k}(1,x)\cr = \frac{1}{2}\,(1-4x)^{\frac{p^l+1}{2}}+\frac{k}{2}\,x\,(1-4x)^{\frac{p^l-1}{2}}-\Big(1-\frac{k}{2}\Big)x+\frac{1}{2}\cr -\Big(\frac{1}{2}-\frac{k}{4}\Big)\,x\,(1-4x)^{\frac{p^l+1}{2}}-\frac{k}{4}\,x\,(1-4x)^{\frac{p^l-1}{2}}-\frac{1}{2}\,x
\prod_{i = 1}^{L}\int d^{d}k_{i}\frac{\partial}{\partial k_{r \mu}}\left\{\left( \sum_{l}x_{l}\overline{{k}}_{l \mu}\right) \prod_{j = 1}^{N}P_{\overline{{k}}_{j}, m_{j}}^{\nu_{j}}\right\}\equiv 0 ,
\sum_{i=m+1}^{n}f(i)=\int_{m}^{n}f(x)dx+\frac{f(n)-f(m)}{2}+\sum_{k=1}^{\lfloor\frac{p}{2}\rfloor}\frac{B_{2k}}{(2k)!}(f^{(2k-1)}(n)-f^{(2k-1)}(m))+R_{p}.
E\{(\|X_{1}-X_{2}\|_{2}^{2}-p\tau)^{s}\}+E\{(\|Y_{1}-Y_{2}\|_{2}^{2}-p\tau)^{s}\}-2E\{(\|X_{1}-Y_{1}\|_{2}^{2}-p\tau)^{s}\}&=\sum^{s}_{a=0}\binom{s}{a}(-p\tau)^{s-a}\left(E\|X_1- X_2\|^{2a}_2+E\|Y_1-Y_2\|^{2a}_2-2E\|X_1- Y_1\|^{2a}_2\right) &=\sum^{s}_{a=l}\binom{s}{a}(-p\tau)^{s-a}\left(E\|X_1- X_2\|^{2a}_2+E\|Y_1-Y_2\|^{2a}_2-2E\|X_1- Y_1\|^{2a}_2\right),
\xi_{n,1}^{()}(1) & =\pi^{-n}\cdot 2^{-1-2n}\cdot 3^{-1}\frac{\left(\tfrac{n-5}{4}\right)!\left(\tfrac{n+3}{2}\right)!(n+1)!(n-1)!}{\left(\tfrac{n-1}{4}\right)!\left(\tfrac{n-3}{2}\right)! \left\{\left(\tfrac{n-1}{2}\right)!\right\}^2}\frac{B_{\frac{n-1}{2}}}{B_{\frac{n+3}{2}}B_{n+1}B_{n-1}}.
\sum_{n=0}^{\infty}\frac{\left(q;q^{2}\right)_{n}q^{2n^{2}}c^{2n}}{\left(q^{2},cq,cq^{2};q^{2}\right)_{n}}=\frac{1}{\left(cq;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{n^{2}}c^{n}}{\left(q;q\right)_{n}}A_{q}^{2}\left(cq^{n}\right).
Num_1(A,B,A',t) ={1\over 3}t^3(3-(2A^2+B^2+A^{\prime 2}+2AB+BA'+2AA')t^2 -A(A^3+2A^2B+AB^2+2A^2A'+AA^{\prime 2}+3ABA'+B^2A'+BA^{\prime 2})t^4),
G=\sum_{n=0}^{\infty}(-1)^{n}\prod_{k=1}^{n}\frac{(2k-1)^{2}}{(2k)^{2}}=1-\frac{1^{2}}{2^{2}}+\frac{1^{2}\cdot3^{2}}{2^{2}\cdot4^{2}}-\frac{1^{2}\cdot3^{2}\cdot5^{2}}{2^{2}\cdot4^{2}\cdot6^{2}}+\cdot\cdot\cdot
\psi_k(x) \rightarrow C\left(e^{ikx}\frac{\Gamma(ik)\Gamma(2\lambda +2)}{\Gamma(\lambda+1+ik)\Gamma(\lambda+1)}+e^{-ikx}\frac{\Gamma(-ik)\Gamma(2\lambda +2)}{\Gamma(\lambda+1-ik)\Gamma(\lambda+1)}\right)
\sum_{k=0}^{b-1}F_{bn+k}^{\left(b\right)}=\sum_{k=0}^{b-1}F_{k}F_{n}^{\left(b\right)}=F_{n}^{\left(b\right)}\left(F_{b+1}-1\right) =F_{n}^{\left(b\right)}\left(F_{b}+F_{b-1}-1\right)=F_{n}^{\left(b\right)}\left(F_{b-2}+2F_{b-1}-1\right) =F_{bn+b-2}^{\left(b\right)}+2F_{bn+b-1}^{\left(b\right)}-F_{n}^{\left(b\right)}.
\mathbb{E}(X)=\sum_{\substack{0\leq i<j \leq h j - i \leq t}}\mathbb{P}(s_i=s_j)+\sum_{\substack{0\leq i\leq h<j\leq k j - i \leq t}}\mathbb{P}(s_i=s_j)+\sum_{\substack{h< i<j \leq k j - i \leq t}}\mathbb{P}(s_i=s_j).
A_{0}(x, t)= A(r_0 x, r_{0}^{2}t),\b_{0}(x, t)= r_0 b(r_{0}x, r_{0}^{2}t),\c_0=r_{0}^2 c(r_{0}x, r_{0}^{2}t),\g_0(x,t)=r_0 g(r_{0}x, r_{0}^{2}t).
\begin{array}{cccc}w(t)~=~1/\,t \quad \mbox{with}k = 0~, begin{align*}2mm] w(t)~=~1/\sin t \quad \mbox{with}k = +1~, begin{align*}2mm] w(t)~=~1/\sinh t \quad \mbox{with}k = -1~, \end{array}
\tilde{Q}_n(s)=\begin{bmatrix}\frac{1}{f_{g((s-1)\ell_w^*+1)}-\alpha_n}e_M(\theta)1_{\{1\in J_w^{[s]}\}}+\hat{Z}_{s,1}\vdots \frac{1}{f_{g(s\ell_w^*)}-\alpha_n}e_M(\theta)1_{\{\ell_w^*\in J_w^{[s]}\}}+\hat{Z}_{s,\ell_w^*}\end{bmatrix},
\sum_{r}S_{r}( \omega , \psi , \chi , \varphi ; z , \zeta ) \lim_{\bar{z}\rightarrow 0}\; \bar{z}^{\Delta_{\bar{v}}+ \Delta_{\bar{r}}- \Delta_{\bar{x}}}\; \lim_{\bar{\zeta}\rightarrow 0}\; \bar{\zeta}^{\Delta_{\bar{w}}+ \Delta_{\bar{u}}- \Delta_{\bar{r}}}\; \widehat{W}_{\bar{x}, \bar{r}}^{2}( \bar{z}, \bar{\zeta}, 0 ) ,
g([X,Y], Z) =-g(h(X,tY), \varphi Z)+g((\widetilde\nabla_{X}\varphi) Z+\varphi\widetilde\nabla_{X}Z, nY) -g(\widetilde\nabla_{Y}(tX+nX), \varphi Z)-2g(\widetilde\nabla_{Y}X, Z).
S_{int}= \frac{i}{2\pi\alpha^\prime}\int_{z^2 =0}dz^1\left( \frac{1}{2}\partial_\nu F_{\mu\lambda}\xi^\nu\xi^\lambda\partial_1 \bar{X}^\mu +\frac{1}{3}\partial_\nu F_{\mu\lambda}\xi^\nu\xi^\lambda \partial_1 \xi^\mu\right) .
I(q\,|\,a, \tau) =\log\frac{G(1+a+\tau\,|\,\tau)}{G(1-q+a+\tau\,|\,\tau)}-q\log\Bigl[\Gamma\bigl(1+\frac{a}{\tau}\bigr)\Bigr]+\frac{(q^2-q)}{2\tau}\psi\bigl(1+\frac{a}{\tau}\bigr),
\begin{cases}u(t_i)=\displaystyle\sum_{j=0}^{k}\frac{(\Delta t_i)^j}{j!}\frac{d^ju}{dt^j}(t_0)+\mathcal{O}(\Delta t_i)^{k+1}u(t_{i+1})=\displaystyle\sum_{j=0}^{k}\frac{(\Delta t_{i+1})^j}{j!}\frac{d^ju}{dt^j}(t_0)+\mathcal{O}(\Delta t_{i+1})^{k+1},\end{cases}
x_{1}= \frac{z_7z_2}{z_2+z_1},x_{2}= \frac{z_1z_7}{z_2+z_1},x_{3}= \frac{z_1z_2z_7+(z_1+z_2)z_8}{z_4 \left( z_2+z_1 \right)}, x_{4}= \frac{z_3 \left( z_1z_2z_7+(z_1+z_2)z_8 \right)}{z_4 \left( z_2+z_1 \right) z_5},x_{5}= \frac{2\,z_1z_2z_3z_7+(z_1+z_2)(2\,z_3z_8+z_4z_9)}{z_6z_4 \left( z_2+z_1 \right)}.
\zeta(s_{1},...,s_{k})=\sum_{n_{1}>n_{2}>\cdot\cdot\cdot>n_{k}>0}\frac{1}{n_{1}^{s_{1}}\cdot\cdot\cdot n_{k}^{s_{k}}}=\sum_{n_{1}>n_{2}>\cdot\cdot\cdot>n_{k}>0}\prod_{i=1}^{k}\frac{1}{n_{i}^{s_{i}}},
\begin{array}{l}{{E_{r}= \displaystyle \frac{g J \cos \theta}{4}\frac{2 R^{2}+ 2 r^{2}+ R r \sin \theta}{( R^{2}+ r^{2}+ 2 R r \sin \theta )^{5 / 2}}\ ~ ~}}\\{{E_{\theta}= - \displaystyle \frac{g J \sin \theta}{4}\frac{2 R^{2}+ 2 r^{2}+ R r \sin \theta}{( R^{2}+ r^{2}+ 2 R r \sin \theta )^{5 / 2}}\}}\\{{E_{\varphi}= 0 .}}\\ \end{array}
J_n^{(p)}\ge \sum_{l_1=0}^{n-(p-1)k}\sum_{l_2=l_1+1}^{l_1+k}\sum_{l_3=l_2+1}^{l_2+k}...\sum_{l_p=l_{p-1}+1}^{l_{p-1}+k}1_{G_l}(\omega).
\bar{\lambda}_{\theta}=-\frac{2\pi|\alpha|}{m}\frac{\Gamma(1+|\alpha|)\tan\theta-\Gamma(1-|\alpha|)\left(\frac{\mu r_0}{2}\right)^{2|\alpha|}}{\Gamma(1+|\alpha|)\tan\theta+\Gamma(1-|\alpha|)\left(\frac{\mu r_0}{2}\right)^{2|\alpha|}}.
N(\lambda)\lesssim \iint_{|\xi|^{2k}+|x|^{2\ell}<\lambda}dx d\xi= \lambda^{n(\frac{1}{2k}+\frac{1}{2\ell})}\iint_{|\xi'|^{2k}+|x'|^{2\ell}<1}dx' d\xi' \lesssim \lambda^{n(\frac{1}{2k}+\frac{1}{2\ell})}.
\frac{dy}{dx}=\frac{d}{dx}(x^{\frac{1}{q}})^{p}=p(x^{\frac{1}{q}})^{p-1}\cdot\frac{1}{q}x^{\frac{1}{q}-1}=\frac{p}{q}x^{p/q-1}=rx^{r-1}
\lim_{\epsilon\to 0^+}\int_{|\mu|>\epsilon/(\varrho \sqrt{3})}\big(\varphi(x,y,z)-\varphi(x^\ast,y^\ast,z^\ast)\big) \frac{d\mu}{2\mu}& = \int_{\mathbb{R}}\big(\varphi(x,y,z)-\varphi(x^\ast,y^\ast,z^\ast)\big) \frac{d\mu}{2\mu}
|\mathcal{A}_m| = |\cup_{i=1}^7 \mathcal{B}_i|+|\mathcal{B}_8|+|\mathcal{B}_9|+|\mathcal{B}_{10}| = 7\textstyle \binom{4k}{2k}+\binom{7}{2}\binom{4k}{2k+1}+\binom{7}{1}\binom{4k}{2k+2}+\binom{4k}{2k+3}.
\langle \sqrt{-1}\partial \bar{\partial}\log W, \frac{1}{\sqrt{-1}}v\wedge \bar{v}\rangle = R^M_{1\bar{1}v\bar{v}}-R^N_{1\bar{1}\partial f(v) \overline{\partial f(v)}}+\frac{\sum_{i\ne 1}|f^i_{1v}|^2}{W}.
{\rm{div}}\,\tilde{w}=(u-\tilde{b}_n K_n)\cdot\nabla\zeta\quad\mbox{in}\,\,B_{l_2}^+\setminus B_{l_1}, \qquad\tilde{w}=0\quad\mbox{on}\,\,\partial (B_{l_2}^+\setminus B_{l_1}).
\Big(J_p(a)-J_p(b)\Big)\,(a-b) =(p-1)\,\left(\int_b^a |t|^{p-2}\,dt\right)\,(a-b) &\ge (p-1)\, \left(\int_a^b |t|^\frac{p-2}{2}\,dt\right)^2=(p-1)\,\left(\frac{2}{p}\right)^2\,|V_{p}(a)-V_{p}(b)|^2,
g_{i\bar{j}}=\frac{1}{2\pi i}\oint_{|z|=1}\bigg\{\partial_i\partial_{\bar{j}}\Big( \log{h(z;\boldsymbol{\xi})}\log{\bar{h}(\bar{\xi};\bar{\boldsymbol{\xi}})}\Big)-\partial_i\Big( \partial_{\bar{j}}\log{h(z;\boldsymbol{\xi})}\log{\bar{h}(\bar{\xi};\bar{\boldsymbol{\xi}})}\Big)\bigg\}\frac{dz}{z}
J'(c)= (1-2\alpha)c^{2\alpha-2}\left( \frac c{(1-2\alpha)(c^2+1)^\alpha}-f(c) - f(1) + \frac{2^{-1-\alpha}}{1-2\alpha}- 2^{-1-\alpha}\mu_\alpha \right) =: (1-2\alpha)c^{2\alpha-2}g(c).
\int_{B^{(4)}}\partial_\tau[u]_h\cdot \eta\chi_\delta[\tilde{v}]_h(\cdot,\tau)+ \langle [A(z,Du)]_h,\chi_\delta D\left([\tilde{v}]_h\eta\right)\rangle(\cdot,\tau) dx &=-\int_{B^{(4)}}[B(z,Du)]_h\cdot \eta\chi_\delta[\tilde{v}]_h (\cdot,\tau) dx
F_{\vec{x}}(u_{1}, \ldots , u_{k})=\frac{(1-q^2)^{k}}{\prod_{i=1}^{k}(1-su_{i})}\sum_{\sigma \in \mathcal{S}_{k}}\prod_{1\le i<j \le k}\frac{u_{\sigma(i)}-q^{2}u_{\sigma(j)}}{u_{\sigma(i)}-u_{\sigma(j)}}\prod_{i=1}^{k}\left(\frac{u_{i}-s}{1-su_{i}}\right)^{x_{i}}
{\frac{e^{2}}{6}}\, c_{A}\, \sum_{k}{\frac{1}{\omega_{k}^{3}}}\int d^{3}x \; E^{2}\, ={\frac{e^{2}}{6}}{\frac{c_{A}}{2 \pi^{2}}}\, l o g \left({\frac{\Lambda}{\mu}}\right) \, \int d^{3}x \; E^{2}.
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!((1-t)\times_{p,\bar{\alpha}}f+_{p,\bar{\alpha}}t\times_{p,\bar{\alpha}}g) (z_1,\cdots, z_n) =\sup_{0<\lambda<1}\left( \sup_{z_i=M_{p,\alpha_i}^{(t,\lambda)}(x_i,y_i), 1\leq i\leq n}M_{p,\alpha_{n+1}}^{(t,\lambda)}\left(f(x_1,\cdots, x_n), g(y_1,\cdots, y_n)\right)\right),
S = \int d^{3}x \left[ - \left( \frac{\partial \phi^{A}}{\partial x^{i}}G_{A B}[ \phi ] \frac{\partial \phi^{B}}{\partial x^{j}}\right) + \frac{i}{2 \kappa}\left| \frac{\partial \phi^{A}}{\partial x^{i}}\right| \sqrt{G [ \phi ]}R [ \phi ] \right] ,
\begin{aligned}\pm \ \omega_M\times 1 = f^*(\omega_M\times 1) = f^*(x_M^n\times 1)\cup f^*(x_M^{m-n}\times 1) = ((y_M^{n}\times1)+\xi\cdot(1\times\omega_N))\cup((y_M^{m-n}\times1)+(y_M^{m-2n}\times\omega_{N}).\end{aligned}
\begin{cases}\frac{d}{dt}{\mathbb E}x_i = A{\mathbb E}x_i +BR^{-1}B^T{\mathbb E}y_i +G \bar m_i+\gamma \bar p_i , \frac{d}{dt}{\mathbb E}y_i=-A^T{\mathbb E}y_i +Q[{\mathbb E}x_i-(\Gamma\bar m_i+\eta)] ,\end{cases}
\sup_{\xi\in\mathbb{R}^N}\langle\xi,\phi\rangle-H^*(\xi)\, m=m\, \left[\sup_{\xi\in\mathbb{R}^N}\, \left\langle\xi,\frac{\phi}{m}\right\rangle-H^*(\xi)\right]=m\, H^{**}\left(\frac{\phi}{m}\right),
P(x, \omega) = \sum_{\gamma_1 \leq \alpha}\sum_{\gamma_2 \leq \beta}\begin{pmatrix}\alpha \gamma_1\end{pmatrix}\begin{pmatrix}\beta \gamma_2\end{pmatrix}|x^{\gamma_2}| \, |(2 \pi i \omega)^{\gamma_1}| & = \prod_{k=1}^d (1+|x_k|)^{\beta_k}(1 + 2 \pi |\omega_k|)^{\alpha_k}.
\Lambda_{\rho}(F,G) := \sum_{j=1}^m \int_{\mathbb{R}^4}F(x+u,y)G(x,y+u)F(x+v,y)G(x,y+v) begin{align*}-1ex](\phi\ast\rho)_{2^{k_j}}(u)(\phi_{2^{k_j}}-\phi_{2^{k_{j-1}}})(v) \,dx dy du dv.
\theta_k:GL(V)\times \Lambda^k(V)\to \Lambda^k(V), ~~ \theta_{k,g}\left(\sum_iv_1^i\wedge ...\wedge v_k^i\right)=\sum_i(gv_1^i)\wedge ...\wedge (gv_k^i) \theta^k:GL(V^*)\times \Lambda^k(V^*)\to \Lambda^k(V^*), ~~ \theta^k_g(\alpha)=\alpha\circ \theta_{k,g}.
U = \left( \begin{matrix}{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}\\{\vdots}&{\vdots}&{\vdots}&{\vdots}&{\ddots}\\{1}&{0}&{0}&{0}&{\cdots}\\ \end{matrix}\right) ,
L(fg)-L(f)L(g) =\int\limits_X{f(x)g(x)d\mu(x)}-\int\limits_X{f(x)d\mu(x)}\cdot \int\limits_X{g(y)d\mu(y)}&=\iint\limits_{X^2}{f(x)g(x)d(\mu\otimes\mu)(x,y)}-\iint\limits_{X^2}{f(x)g(y)d(\mu\otimes\mu)(x,y)}&=\iint\limits_{X^2}{f(x)(g(x)-g(y))d(\mu\otimes\mu)(x,y)}.
\alpha_{\kappa}=\dfrac{\frac{1}{\gamma_{i}^{\mathsf{th}}}-\sum_{j\neq i}^{L}\mathbb{E}\left[\mathsf{\bar{L}}_{ji}^{2}\right]}{\left(1+\sum_{j\neq i}^{L}\mathbb{E}\left[\mathsf{\bar{L}}_{ji}\right]\right)\left(1+{\sum_{j\neq i}^{L}}\mathbb{E}\left[\mathsf{\bar{L}}_{ji}\right]+\frac{\sigma^{2}}{P_u}\right)}.
Z_{a}\equiv \nabla_{a}Z = -{\frac{1}{2}}\oint_{S^{2}}g_{a b^{*}}{\cal F}^{+ b^{*}}= ( Q_{\Lambda}\nabla_{a}L^{\Lambda}- P^{\Sigma}\nabla_{a}M_{\Sigma}) = \langle U_{a}|{\cal Q}\rangle .
i \hbar \frac{\partial}{\partial t}\left[ \begin{array}{c c}{{\Psi_{e}( t )}}\\{{\Psi_{\mu}( t )}}\\ \end{array}\right] = H_{\mathrm{f l a v o r}}\left[ \begin{array}{c c}{{\Psi_{e}( t )}}\\{{\Psi_{\mu}( t )}}\\ \end{array}\right] \, ,
R_{s-1}(\lambda_1,...,\lambda_{s-1})=\frac{(-1)^s n!\,C}{n-s+1}\prod_{l=1}^{s-1}w(\lambda_l)\int d\lambda_s w(\lambda_s)\begin{vmatrix}[0]_{\substack{j=1,...,s k=1,...,s}}[\lambda_j^{k-1}]_{\substack{j=1,...,s k=1,...,n}}[f_j(\lambda_k)]_{\substack{j=1,...,n k=1,...,s}}[h_{j,k}]_{\substack{j=1,...,n k=1,...,n}}\end{vmatrix}.
\aligned0 \,=\,-\,\tfrac{1}{10}\,F_{5,0}+\tfrac{1}{120}\,F_{6,0}+\tfrac{1}{36}\,F_{5,1}, 0 \,=\,-\tfrac{7}{120}\,F_{5,0}+\tfrac{1}{120}\,F_{5,1}+\tfrac{2}{27},\endaligned
- f^{\prime \prime}( u ) + \frac{m^{2}+ \frac{d^{2}- 1}4}{u^{2}}f ( u ) = - f^{\prime \prime}( u ) + \frac{( \nu + 1 / 2 ) ( \nu - 1 / 2 )}{u^{2}}f ( u ) = \lambda f ( u ) \ ,
\begin{array}{ll}\ e^{-2\Phi}={M\over\lambda}-\lambda^2x^+x^-={M\over\lambda}+e^{2\lambda\sigma}\Rightarrow e^{-2\left(\Phi+\lambda\sigma\right)}=1+{M\over\lambda}e^{-2\lambda\sigma}\Rightarrow \Phi\sim-\lambda\sigma-{M\over{2\lambda}}e^{-2\lambda\sigma}\;.\end{array}
\tilde{\nabla}_ET_{\theta}e_3=\omega_{34}(E)T_{\theta}e_4+\frac{\kappa_2}{\kappa_1}T_{\theta}e_5-\frac{i\mu_2}{\kappa_1}T_{\theta}e_6-\kappa_1 e^{i\theta}dg_{\theta}(\overline{E}),
=\frac{1}{(q^3;q^3)_{\infty}}\left(\frac{a^2(q)-1}{12}\right)-\frac{2}{(q^3;q^3)_{\infty}}\sum_{n\ge1}\frac{(-1)^n q^{3n(3n+1)/2}(1+q^{3n})}{(1-q^{3n})^2}.
V ( \sigma ; \lambda , \Lambda ; M ) = | \lambda \Lambda^{2}|^{2}\left[ 1 + 2 \frac{N \lambda^{2}}{1 6 \pi^{2}}\left( \ln \frac{| \lambda \sigma |^{2}}{M^{2}}+ 4 \right) +{\cal O}( \lambda^{4}) \right] .
\begin{matrix}\bar{\tau}_1 = [a_0, \hat{j}] + a_0, \bar{\tau}_{-1}= -[a_0, \hat{j}] + a_0, \bar{\tau}^\dagger_1 = [\hat{j}, a_0^\dagger] + a_0^\dagger, \bar{\tau}_{-1}= -[\hat{j}, a_0^\dagger] + a_0^\dagger, \end{matrix}
r_{U^{\alpha}}(z,z^{\prime},w)=r\Big(\frac{|z_1|^2}{(1-|w|^2)^{\alpha_1}},\dots,\frac{|z_n|^2}{(1-|w|^2)^{\alpha_n}};z^{\prime},\bar{z}^{\prime}\Big), &r_{V^{\gamma}}(z,z^{\prime},w)=r\Big(e^{\gamma_1|w|^2}|z_1|^2,\dots,e^{\gamma_n|w|^2}|z_n|^2;z^{\prime},\bar{z}^{\prime}\Big).
\mathcal{L}x_i =\Delta x_i-\left<\nabla f, \nabla x_i\right> &=\sum_{j=1}^n\bar{\nabla}^2x_i(e_j,e_j)+\left<{\bf H}, \bar{\nabla}x_i\right>-\left<\nabla f, \nabla x_i\right> &=\left<{\bf H}+(\bar{\nabla}f)^{\perp}, \bar{\nabla}x_i\right>-\left<\bar{\nabla}f, \bar{\nabla}x_i\right> &=\left<{\bf H}_f, \bar{\nabla}x_i\right>+\dfrac{x_i}{2}.
x_{1}\times x_{2}\times x_{3}=\left\vert\begin{array}{cccc}-e_{1}e_{2}e_{3}e_{4}x_{1}^{1}x_{1}^{2}x_{1}^{3}x_{1}^{4}x_{2}^{1}x_{2}^{2}x_{2}^{3}x_{2}^{4}x_{3}^{1}x_{3}^{2}x_{3}^{3}x_{3}^{4}\end{array}\right\vert ,
K(x):=|x|^{n\beta}e^{n\left(p(x)+q\left(\frac{x}{|x|^2}\right)-\frac{\Lambda}{\Lambda_1}w_0\right)}, c_{\tilde{\psi}}:=\frac{1}{n}\ln \frac{\Lambda}{\int_{\mathbb{R}^n}Ke^{n\tilde{\psi}}dx}.
\frac{1}{T_2 - T_1}\int_{T_1}^{T_2}E_N (t) \, \mathrm{d}t = O \left( \sum_{j=1}^d \sum_{m=1}^{\infty}\frac{\log^{d-1}(m+1)}{m \prod_{k \neq j}\left\| m \frac{a_j}{a_k}\right\|}\min \left( 1 , \frac{1}{(T_2 - T_1) m}\right) \right) .
h'(\tilde{x},\tilde{t})=\frac{h(Z(\tilde{x},\tilde{t}),Y^{-1}(\tilde{t}))X_x^2(Z(\tilde{x},\tilde{t}),Y^{-1}(\tilde{t}))}{\tilde{t}^{\alpha-1}(\mathcal{T}_t^\alpha Y)(Y^{-1}(\tilde{t}))},
{\cal L}_{p}^{\hbar}\left[ H \right] = - \sum_{i = 0}^{p}\frac{2}{\hbar}H \left( \phi_{i}\right) \sin \left[ \frac{\hbar}{2}\stackrel{\leftarrow}{\partial}_{a}^{( i )}\omega^{a b}\stackrel{\rightarrow}{\partial}_{b}^{( i )}\right]
\tilde{C}^{[0, M]}(u; s) =(-su)^{-1}(-s(1-su))^{-(M+1)}\prod_{1\le i \le k}K^{(i)}((-s)^{-(M+1-i)}) \times C^{[0, M]}(-su; s)\prod_{1\le i \le k}K^{(i)}((-s)^{M+1-i}),
\nu_{P_i}(z) \ge \min\{\nu_{(f,a_i)},\nu_{(g,a_i)}\}+\min\{\nu_{(f,a_{\sigma (i)})},\nu_{(g,a_{\sigma (i)})}\}+\sum_{\underset{v\ne i,\sigma (i)}{v=1}}^{q}\nu_{(f,a_v)}^{[1]}(z)
\begin{aligned}A^{0}_{1}(u,v)\partial_{t}\hat{u}+A^{i}_{11}(u,v)\partial_{i}\hat{u}=f_{1}(U,D_{x}v), A^{0}_{2}(u,v)\partial_{t}\hat{v}-B^{ij}_{22}(u,v)\partial_{i}\partial_{j}\hat{v}=f_{2}(U,D_{x}U),\end{aligned}
{\cal R}=(2 S_{cl})^{3 k}\, \prod_{i=1}^3 \, \left( S_i \right)^{-k}\left( \frac{( S_i^2+S_{cl}^2)^{1/2}+ S_{cl}}{( S_i^2+S_{cl}^2)^{1/2}- S_{cl}}\right)^{-k/2}
( a \nabla_{\nu}b)^{2}\geqslant ( a \sharp_{\nu}b )^{2}+ r_{0}^{2}(a-b)^{2}+ \sum_{k=1}^{\infty}r_{k}\big[ a^{1-\frac{m_k}{2^k}}b^{\frac{m_k}{2^k}}- a^{1-\frac{m_k+1}{2^k}}b^{\frac{m_k+1}{2^k}}\big]^{2}.
D_{\varsigma}:=\{\eta:\ \ \varsigma_{i}\eta_{i}\geq 0, \ \i=1,2, \cdots, n\}; D_{\mu}:=\{\xi:\ \ \ \mu_{i}\xi_{i}\geq 0, \ \i=1,2, \cdots, n\}; D_{\nu}:=\{\xi+\eta: \ \ \nu_{i}(\xi_{i}+\eta_{i})\geq 0, \ \i=1,2, \cdots, n\}.
K_i E_j = q^{\langle\alpha_i,\alpha_j\rangle}E_j K_i, K_i F_j = q^{-\langle\alpha_i,\alpha_j\rangle}F_j K_i, L_i E_j = q^{\langle\alpha_i,\alpha_j\rangle}E_j L_i, L_i F_j =q^{-\langle\alpha_i,\alpha_j\rangle}F_j L_i,
F = V_{p + 1}( 8 \pi^{2}\alpha^{\prime})^{- \frac{p + 1}{2}}\int_{0}^{\infty}\frac{d \tau}{\tau}\tau^{- \frac{p + 1}{2}}e^{- \frac{y^{2}\tau}{2 \pi \alpha^{\prime}}}\left( f_{1}( e^{- \pi \tau}) \right)^{- 2 4}=
2 (y-a)\left[a^k+k(y-a)a^{k-1}+\binom{k}{2}(y-a)^2a^{k-2}-y^k\right] + (b-y)\left[a^k+k(y-a)y^{k-1}+k(k-1)(y-a)^2y^{k-2}-y^k\right]>0\,.
\delta^{-\frac{n-4}{2}}\int_{B}\varphi_0\Big(\frac{e_n-x}{\delta}\Big)U_{\delta,-e_n}^p=\delta^{-\frac{n-4}{2}}\int_{B}\varphi_0\Big(\frac{e_n+x}{\delta}\Big)U_{\delta,e_n}^p=O(\delta^{n-2})\delta\to 0.
\sum_{i=1}^2 \frac{\partial^2 Z_\kappa}{\partial x_i^2}(\bar{x},\bar{y}) \leq \sqrt{2}\, \kappa \, |A(\bar{x})| - \frac{\kappa^2-1}{\sqrt{2}\, \kappa}\, \frac{|A(\bar{x})|}{1-\langle F(\bar{x}),F(\bar{y}) \rangle}\sum_{i=1}^2 \Big \langle \frac{\partial F}{\partial x_i}(\bar{x}),F(\bar{y}) \Big \rangle^2.
{\bf r}(\lambda)=\frac{\lambda^{\frac{3}{2}}+\lambda^{-\frac{3}{2}}}{\lambda^{\frac{3}{2}}-\lambda^{-\frac{3}{2}}}\sum_{i=1}^{3}H_i\otimes H_i+\frac{2}{\lambda^{\frac{3}{2}}-\lambda^{-\frac{3}{2}}}\sum_{(ij)}\left(\lambda^{-\frac{1}{2}}E_{ij}\otimes{E}_{ji}+\lambda^{\frac{1}{2}}E_{ji}\otimes{E}_{ij}\right),
x_1x_4+x_4x_1 = ax_0^2, x_2x_3+x_3x_2 = bx_0^2, x_2x_0+x_0x_2 = ax_1^2, x_3x_4+x_4x_3 = bx_1^2, x_3x_1+x_1x_3 = ax_2^2, x_4x_0+x_0x_4 = bx_2^2, x_4x_2+x_2x_4 = ax_3^2, x_0x_1+x_1x_0 = bx_3^2, x_0x_3+x_3x_0 = ax_4^2, x_1x_2+x_2x_1 = bx_4^2.
\sum\limits_{i = 1}^k{({\lambda_{k + 1}}-{\lambda_i}}{)^\alpha}&\le \frac{{2\sqrt{n + 1}}}{n}{\left[{\sum\limits_{i = 1}^k{{{({\lambda_{k + 1}}-{\lambda_i})}^\beta}}}\right]^{\frac{1}{2}}}{\left[{\sum\limits_{i = 1}^k{{{({\lambda_{k + 1}}-{\lambda_i})}^{2\alpha - \beta - 1}}{\lambda_i}}}\right]^{\frac{1}{2}}}.
{1\over B^{2}}\left({1\over A^{2}}-{\ell^{2}\over \tilde{B}^{2}r^{2}}\right)=0,\qquad{1\over B^{2}}\left({(A^{2})'\over A^{4}}-{(\tilde{B}^{2}r^{2})' \ell^{2}\over\tilde{B}^{4}r^{4}}\right) =0.
\bigg\|\bigotimes_{i=1}^n P_f^g-\bigotimes_{i=1}^n P_{\hat f_m}^g\bigg\|_{TV}\leq H\bigg(\bigotimes_{i=1}^n P_f^g,\bigotimes_{i=1}^n P_{\hat f_m}^g\bigg) \leq\sqrt{n H^2\big(P_{f}^g,P_{\hat f_m}^g\big)}.
\left ( \int_{-\mathbb{R}+i\epsilon}+ \int_{\mathbb{R}-i\epsilon}\right ) ( \phi^C (t) + \phi^C ( - t ) ) \log(-t) dt = - i \pi \int_{-\mathbb{R}+i\epsilon}\phi^C ( t) dt + i\pi \int_{\mathbb{R}-i\epsilon}\phi^C ( t) dt
e_{k}^{s,N}= 2\big|t_1 \big(u_1^{s,N}(C_{0,k}) - u_1^{s,N}(B_{0,k})\big) + t_2 \big(u_2^{s,N}(C_{0,k}) - u_2^{s,N}(B_{0,k})\big)\big|^2
f = - \sqrt{2 \pi}\left[ f_{0}( r ) \delta ( \omega ) \tilde{w_{1}}( k ) + \frac{1}{2}f_{+}( r ) \delta ( \omega - 2 \omega_{0}) \tilde{w_{2}}( k ) + \frac{1}{2}f_{-}( r ) \delta ( \omega + 2 \omega_{0}) \tilde{w_{2}}( k ) \right] .
\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1 1 1 -1\end{array}\right)\left(\frac{1}{\sqrt{5}}\left(\begin{array}{cc}2 1 1 -2\end{array}\right)\right)^\top=\frac{1}{\sqrt{10}}\left(\begin{array}{cc}3 -1 1 3\end{array}\right).
(\xi^j,\eta^j) = \begin{cases}(|x_1|, |x_2|) j=0 (\cos (\theta_j) \xi^{j-1}+ \sin (\theta_j) \eta^{j-1}, |-\sin (\theta_j) \xi^{j-1}+ \cos (\theta_j) \eta^{j-1}|) j =1,\dots,\nu.\end{cases}
M_2 = \frac{1}{\#h_{(36)}}\sum_{\substack{r_1,r_2 \geq 0 a}}\frac{1}{2^{r_1}3^{r_2/2}N(a)^{1/2}}V \left(\frac{\pi 2^{2r_1+1}3^{r_2}N(a)}{z}\right)\sum_{\psi \bmod{36}}H(a, \psi, y),
\int_0^1 \{nx\}\log x \mathrm{d}x \underset{n\to \infty}{=}-\frac{1}{2}+ \frac{1}{4n}\log n + \mathcal{O}\left( \frac{1}{n}\right) - \frac{1}{6n}\log n + \mathcal{O}\left( \frac{1}{n}\right) &= -\frac{1}{2}+ \frac{1}{12n}\log n + \mathcal{O}\left( \frac{1}{n}\right).
\dfrac{dH}{dI}\geq \dfrac{F(S,I)-F(S,I^{*})}{F(S,I)F(S,I^{*})}F_{1}(S,I)-\dfrac{I-I^{*}}{II^{*}}& = \dfrac{1}{F(S,I^{*})}\left[ F_{1}(S,I)-F_{1}(S,I^{*})\right]\geq 0.
\frac1T \sum \limits_{k=0}^{T-1}\langle g_{k+1}, x_k -x^* \rangle \leq \frac{\kappa}{\kappa +1}\frac{R_{0}^{\frac{1+\kappa}{\kappa}}}{\nu T}+ \frac{\nu^{\kappa}}{1+\kappa}\frac{1}{T}\sum \limits_{k=0}^{T-1}\|g_{k+1}\|^{1+\kappa}_q.
\frac{1}{N}\mathbb{E}B_1(\boldsymbol{h}+\boldsymbol{y}_0^1,\boldsymbol{h}+\boldsymbol{y}_0^2,\lambda) & = \log \sqrt{\frac{b^2}{b^2-\lambda^2}}+\mathbb{E}J_{1}^1(h+y_0^1,h+y_0^2,\lambda)+ \mathbb{E}J_{1}^2(h+y_0^1,h+y_0^2,\lambda).
Z_{\mu\nu}\equiv-(k^2+i\epsilon)\biggl[g_{\mu\rho}+\biggl({1\over\lambda}(1-\kappa)^2-1\biggr){k_\mu k_\rho\over{k^2+i\epsilon}}-i{{\kappa(1-\kappa)}\over\lambda}{k_{[\mu}\eta_{\rho]}\over{k^2+i\epsilon}}+{\kappa^2\over\lambda}{\eta_\mu\eta_\rho\over{k^2+i\epsilon}}\biggr],
( W^{1 2}( \theta_{3 , 4}, \bar{\theta}^{1 , 2}) )^{p + q + r}( W^{1 3}( \theta_{2 , 4}, \bar{\theta}^{1 , 3}) )^{q + r}( W^{2 3}( \theta_{1 , 4}, \bar{\theta}^{2 , 3}) )^{r}\; .
\alpha^2 \|V_0^{1-3,l}\|_{1,r,r'}\le c\beta c_0^2\alpha^2r',\sum_{m=2}^Nc_0^{\frac{m}{2}}\alpha^m\|V_m^{1-3,l}\|_{1,r,r'}\le c\beta c_0^2\alpha^4 r'.
p_t(s,t)+(\gamma(s,P(t),Q(t))p(s,t))_s=-\mu(s,P(t),Q(t))p(s,t), & \gamma(0,P(t),Q(t))p(0,t)=\int_0^m\beta(s,P(t),Q(t))p(s,t)\,d s, & \frac{d Q(t)}{d t}=Q(t)f(Q(t))-\int_0^m F(s,P(t),Q(t))p(s,t)\,d s,
F_{\mu \nu}^{( H )}= \sum_{\jmath = 1}^{[ d / 2 ] - 1}H_{\jmath}\left( \delta_{\mu}^{2 \jmath}\delta_{\nu}^{2 \jmath - 1}- \delta_{\nu}^{2 \jmath}\delta_{\mu}^{2 \jmath - 1}\right) \; .
F=\frac{dp}{dt}=\frac{d}{dt}(\frac{mv}{\sqrt{1-\frac{v^{2}}{c^{2}}}})=\frac{ma}{\sqrt{1-\frac{v^{2}}{c^{2}}}}+\frac{mvav}{c^{2}(\sqrt{1-\frac{v^{2}}{c^{2}}})^{3}}
w_{kl}^+{''}-\left[\frac{l\left(l+1\right)}{r^2}-\frac{\kappa}{r}\frac{V'}{2m-V}+\frac{1}{2}\frac{V''}{2m-V}+\frac{3}{4}\left(\frac{V'}{2m-V}\right)^2-V^2+2mV-k^2\right]w_{kl}^+=0\,,
\frac{1}{2\pi}\int_{-\pi}^\pi f_{\alpha,\beta}(x) e^{-ikx}\,\mathrm{d}x = \frac{1}{\pi}\int_0^\pi f_{\alpha,\beta}(x) \cos(kx) &= \frac{1}{i4\pi}\int_0^\pi x^\alpha \big( e^{ix^{-\beta}}- e^{-ix^{-\beta}}\big) \big( e^{ikx}+ e^{-ikx}\big) \,\mathrm{d}x ,
(\textrm{ellipticity ratio of $\nabla^{2}E^{\ast}(z_{0})$}) =\frac{(\textrm{the largest eigenvalue of $\nabla^{2}E^{\ast}(z_{0})$})}{(\textrm{the lowest eigenvalue of $\nabla^{2}E^{\ast}(z_{0})$})}&\le (p-1)\left(1+(\delta-b)^{-1}\right)
I_1 := \int_{\frac{|x-y|^2}{|x|^2}}^\infty t^{\frac{d-s}2}e^{-ct}\tfrac{dt}t, I_2 := \tfrac{|x-y|^{\sigma}}{|x|^{\sigma}}\int_{\frac{|x-y|^2}{|y|^2}}^{\frac{|x-y|^2}{|x|^2}}t^{\frac{d-s-\sigma}2}e^{-ct}\tfrac{dt}t, I_3:= \tfrac{|x-y|^{2\sigma}}{|x|^{\sigma}|y|^{\sigma}}\int_{0}^{\frac{|x-y|^2}{|y|^2}}t^{\frac{d-s-2\sigma}2}e^{-ct}\tfrac{dt}t.
|f^{\prime\prime}(z)|=|\frac{1}{2\pi i}\oint_{\gamma}\frac{f^{\prime}(w)}{(w-z)^{2}}dw|\le\frac{1}{2\pi}\cdot2\pi rsup_{w=\gamma(t)}\frac{|f^{\prime}(w)|}{|w-z|^{2}}\le\frac{2}{r},
B_{0}( M_{*}^{2}) = \frac{\Gamma ( \frac{d}{2}) \, \Gamma ( d - 2 + 2 k ) \, \Gamma ( k + \frac{1}{2})}{\sqrt{\pi}\, \Gamma ( d - 2 ) \, \Gamma ( 2 k + 1 ) \, \Gamma ( k + \frac{d}{2})}M_{\ast}^{2 k}\ \cdot