harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\displaystyle \int_{N_n(F)\backslash P_n(F)}\int_{N_{n}(F)\backslash G_{n}(F)}W_f(p,h)dhdp= \int_{P_{n}(F)\backslash G_{n}(F)}\int_{P_{n-1}(F)U_n(F)\backslash P_n(F)}\int_{N_{n-1}(F)\backslash P_{n-1}(F)}\int_{N_{n-1}(F)\backslash G_{n-1}(F)}W_f(p'p,h'h)\lvert \det(p'h')\rvert^{-1}dh'dp'dpdh
\nu_k(j_k) = \frac{c_{k}}{\mu_{k}}\, \left[\left(\frac{2j_{k}+3}{(1-\rho_k)^2}-\frac{2}{(1-\rho_{k})^3}\right) \rho_{k}^{-j_k-1}\right. & \left. - \frac{(j_{k}+1)^{2}}{1-\rho_k}-\frac{1}{(1-\rho_k)^2}+\frac{2}{(1-\rho_{k})^3}\right] - r_k - s_k.
M_{\ell+8,\ell}(\theta) = 2\cos(4 \theta) +2\cos(4 \theta + \tfrac{\ell \theta}{2})(2i\sin(\tfrac{\theta}{2}))^{-\ell}+ 2\cos(\tfrac{\ell \theta}{2})(2i\sin(\tfrac{\theta}{2}))^{-\ell - 8}.
\left.\mathrm{F}^\ast (g)\right|_{G_\lambda}:=\left.{\mathrm{F}(g^{-1})}^\ast\Delta(g^{-1})\right|_{G_\lambda}:={f(g_\lambda^{-1})}^\ast\Delta(g_\lambda^{-1})\;\;\;\;\;\;\;\;\forall\lambda\in\Lambda\;;
\widetilde{\mathcal{P}}^{-1}(U) = (M_3+d_u\tau B_1)^{-1}U(M+d_u\tau(A+B_2))^{-1}; &\widehat{\mathcal{P}}^{-1}(U) = (M_0D_1+d_u\tau A_1)^{-1}U(M_0+d_u\tau(A+A_2))^{-1};
Z_{b}= \int \prod_{p}d C_{p}\int \prod_{p}d D_{p}\exp \left\{- \frac{1}{2}( \begin{array}{l l}{C_{p}}&{D_{p}}\\ \end{array}) G ( p ) \left( \begin{array}{l}{C_{p}}\\{D_{p}}\\ \end{array}\right) \right\},
M(3+L)\sqrt{N_T T}\leq M(3+L)\sqrt{\Big(1+ 4(3dT)^{d/(d+2)}\Big)T}& \leq M(3+L) \Bigl(\sqrt{T}+ \sqrt{4(3dT)^{d/(d+2)}T}\Bigr) = M(3+L) \Bigl(\sqrt{T}+ 2(3d)^{\frac{d}{2(d+2)}}T^{\frac{d+1}{d+2}}\Bigr) \,.
\frac{\mathrm{d}}{\mathrm{d}\mu}\ell= -\frac{\alpha-1}{\sigma}\frac{\phi_s\left(\frac{x-\mu}{\sigma}\right)}{\Phi_s\left(\frac{x-\mu}{\sigma}\right)}+\frac{\beta-1}{\sigma}\frac{\phi_s\left(\frac{x-\mu}{\sigma}\right)}{1-\Phi_s\left(\frac{x-\mu}{\sigma}\right)}+\frac{s}{\sigma}\left|\frac{x-\mu}{\sigma}\right|^{s-1}\mathrm{sign}(x-\mu)
|h(x)-h(y)| \leq |h(x)-h(\bar{x})|+|h(y)-h(\bar{y})| +|h(\bar{x})-h(\bar{y})| \leq Ar^\alpha + A r^\alpha + A|\bar{x}- \bar{y}|^\alpha \leq 2Ar^\alpha + A|2r + d|^\alpha \leq C d^\alpha = C|x-y|^\alpha
\big[\mathrm{T}\psi\big](\vartheta^\bullet)=\sum_{\vartheta^\bullet_+\sqcup\vartheta^\bullet_\bullet=\vartheta^\bullet}\int T(\vartheta^\bullet_+,\vartheta^\bullet_\bullet,{\vartheta}^-_\bullet)\psi(\vartheta_\bullet)\mathrm{d}\vartheta^-_\bullet,
\left(hbh\otimes e_{11}\right)^{\sigma}=\left((h\otimes e_{11})(b\otimes e_{11})(h\otimes e_{11})\right)^{\sigma}& =(h\otimes e_{12})(b^{\tau}\otimes e_{22})(h\otimes e_{21}) & =hb^{\tau}h\otimes e_{11}.
\begin{array}{ccc}f_0= X_0^{d-3}X_1(X_0^2-X_1^2), f_2=X_0^{d-3}X_2(X_1^2-X_2^2), f_1=X_0^{d-3}X_2(X_0^2-X_1^2), f_3=X_1^{d-3}X_2(X_1^2-X_2^2).\end{array}
P I = \int_{- \infty}^{\infty}d N ( \tau^{\prime \prime}- \tau^{\prime}) \int{\cal D}p{\cal D}x \exp \left( i \int_{\tau^{\prime}}^{\tau^{\prime \prime}}d \tau ( p_{\mu}\dot{x}^{\mu}- N \tilde{\cal H}_{0}) \right) .
U'(x(T,y))\mathbb{E}[\delta_Y(y)|\mathcal{F}_T]=p(0,y) \exp(-\int_0^T \frac{b_0(s,y)}{\sigma_0(s,y)}dB(s) - \frac{1}{2}\int_0^T (\frac{b_0(s,y)}{\sigma_0(s,y)})^{2}ds).
L_{2}(h,u,T) : = |T\cap E_{h}(u)| L_{1}(h,u,T) : = \frac{|\partial (T\cap E_{h}(u))|_{1}}{2}, L_{0}(h,u,T) : =\sharp \mbox{connected components in}T\cap E_{h}(u) - \sharp \mbox{holes in}T\cap E_{h}(u) .
C({\bf H}) = \int_{\mathbb{R}}\log \left( 1+Px\right) F^{\mathbf{H}\Phi_0\mathbf{H}^{\ast}}(x) &= \lim_{N\rightarrow \infty}\frac{I(x^{\Phi_0,N};(y^{\Phi_0,N},H_{dN}))}{dN}.
\alpha_{1,\,2,\,3}=\alpha_{3,\,4,\,5}=\alpha_{1,\,5,\,6}=\sqrt{\tfrac{3}{2}},\,\alpha_{2,\,4,\,6}=-\sqrt{\tfrac{3}{2}},\,\alpha_{1,\,3,\,5}=-2\sqrt{\tfrac{3}{2}}
{\cal L}_{\rm eff}/ N ={1 \over 2}\tilde A_{\mu}^{(1)}\left\{( k^2 -{e^2 \over \pi}) \left( -g^{\mu \nu}+{k^\mu k^\nu \over k^2}\right) + 2 e^2 \xi_0^2 U(k^2){k^\mu k^\nu \over k^2}\right\}\tilde A_{\nu}^{(1)}~.
L^{d=4}=\frac{1}{4e^2}\int\left( W^2 d^2\theta + \bar W^2 d^2\bar \theta\right) +\frac{1}{e^2}\sum_{r,s,t=0}^\infty a_{rst}\int d^4\theta \left(W^2\bar W^2\right)^r X^s Y^t
a_{1}^{+},...,a_{p_{1}}^{+},a_{1}^{-},...,a_{q_{1}}^{-},a_{p_{1}+1}^{+},...,a_{p_{2}}^{+},a_{q_{1}+1}^{-},...,a_{q_{2}}^{-},a_{p_{2}+1}^{+},....
f_{T T T}= -{\frac{2 i}{\pi}}\frac{j_{T}( T )}{j ( T ) - j ( U )}\left\{\frac{j ( U )}{j ( T )}\right\}\left\{\frac{j_{T}( T )}{j_{U}( U )}\right\}\left\{\frac{j ( U ) - j ( i )}{j ( T ) - j ( i )}\right\}\equiv 2 W ( T , U ) .
\Big| \int_{I}\int_{|\mathbf v|\lesssim \lambda^{-1}}e^{i \lambda x \cdot (\gamma(t) + (0,\mathbf{v}))}G_{\eta'}(t,\mathbf{v})d\mathbf{v}dt \Big|= \lambda^{-d}\Big| \int_{\lambda\gamma(I) + O(1)}e^{i x \cdot \xi}g_{\eta'}(\lambda^{-1}\xi) d\xi \Big|
G(a/q) = \prod_{\gamma = (\gamma', 0) \in \Gamma}\bigg(\frac{1}{q}\sum_{x = 1}^q e^{2\pi i a_\gamma x /q}\bigg) \prod_{\gamma = (0, \gamma'') \in \Gamma}\bigg(\frac{1}{\varphi(q)}\sum_{x \in A_q}e^{2\pi i a_\gamma x /q}\bigg).
I_{n}(\Pi;q)=I_{n-1}(\Pi;q)+\sum_{k=1}^{\lfloor{\frac{n}{2}}\rfloor}\sum_{0 < a_1 < \cdots < a_k < n}q^{k(n-2k-1) + \frac{3k(k+1)}{2}- (a_1 + a_2 + \cdots + a_k)}I_{n-2k-1}(\Pi;q)
R_V(\lambda) =R_0(\lambda)\,\bigl(I+K(\lambda)\rho\bigr)^{-1}\,\bigl(I-K(\lambda)(1-\rho)\bigr) &=\Bigl( R_0(\lambda)-R_0(\lambda)\gamma^*(I+VG(\lambda))^{-1}V\gamma R_0(\lambda)\rho \Bigr)\,\bigl(I-K(\lambda)(1-\rho)\bigr)\,.
\breve\S_{k\Delta}:=\max_{1\leq j\leq k}\prod_{i=j}^{k}\frac{dQ_{1}^{(\Delta)}}{dQ_{0}^{(\Delta)}}(\Delta_i{X})=\max(\breve\S_{(k-1)\Delta},1)\frac{dQ_{1}^{(\Delta)}}{dQ_{0}^{(\Delta)}}(\Delta_k{X}),k\geq 1,
z_1 = \frac{e^s\cos\theta+ i (z\cos\theta-p_\theta\sin\theta)}{\sqrt{e^{2s}+z^2+p_\theta^2}}, z_2 = \frac{e^s\sin\theta+ i (z\sin\theta+p_\theta\cos\theta)}{\sqrt{e^{2s}+z^2+p_\theta^2}}, e^u = e^{2s}+z^2+p_\theta^2.
\textrm{Pr}^{\textrm{NOMA}}\left\lbrace \textrm{FDF=1,FDN=0}\right\rbrace \to\frac{d_{\textrm{BF}}^{\alpha}\sigma_{\textrm{F}}^{2}\left(2^{R}-1\right)(1+k)\left(k-2^{\textrm{R}}\right)}{\lambda_{\textrm{BF}}\left(k-\left(2^{\textrm{R}}-1\right)\right)}\frac{1}{P_{\textrm{B}}}.
X_\tau^{x}={e^{\tau A}x+e^{\tau A}v(0,x)-v(\tau,X_\tau^{x})+\int_0^\tau e^{(\tau-s)A}\widetilde Z_s^x\;dW_s +\int_0^\tau e^{(\tau-s)A}GdW_s}= e^{\tau A}x+e^{\tau A}v(0,x)-v(\tau,X_\tau^{x})+\int_0^\tau e^{(\tau-s)A}\nabla^G v(s,X_s^x)\;dW_s +\int_0^\tau e^{(\tau-s)A}GdW_s
(f_k)_1(r) =e^{\frac{r^2}{4}}r^{2l_k}y(-\frac{r^2}{4}) &=c_k\Big(r^{2l_k}+\cdots+(\frac{1}{4})^{2i}\frac{r^{2\lambda}}{(l_k+\frac{m}{2}+\lambda-i)\cdots(l_k+\frac{m}{2}+\lambda-1)}\Big) &=c_k q_k(r).
\mathbb{E}(Q_i) = \mathbb{E}\left(T_{i-1}Y_{i-1}\right)+\mathbb{E}\left(\frac{Y_{i-1}^2}{2}\right) = \mathbb{E}\left(T_{i-1}\right)\mathbb{E}\left(Y_{i-1}\right)+\mathbb{E}\left(\frac{Y_{i-1}^2}{2}\right).
x\lambda_{1}\left( a\mathbf{t}\left( a\right) b\mathbf{t}\left( b\right) \right) \lambda_{2}\left( \mathbf{n}\left( a\mathbf{t}\left( a\right) \right) \left( a\mathbf{t}\left( a\right) \right) \left( b\mathbf{t}\left( b\right)\right) \right) ,\lambda_{1},\lambda_{2}\in K
\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|\Psi_a(x)-\Psi_a(y)|^{p-2}\,(\Psi_a(x) - \Psi_a(y))}{|x-y|^{N+s\,p}}\,(\varphi(x)-\varphi(y))\,dx\,dy=0.
L = -{\frac{1}{g^{2}}}\! \int \! d^{2}x \Bigl ({\frac{1}{4}}F_{\mu \nu}* F^{\mu \nu}+ D_{\mu}\phi * ( D^{\mu}\phi )^{\dagger}+{\frac{\lambda}{2}}( \phi * \phi^{\dagger}\! - \! v^{2})^{2}\Bigr )
\frac{d^{2}F ( 0 )}{d^{2}t}= - K_{L}( \mathcal{I}) \star [ Q , K ]_{L}( \mathcal{I}) + [ Q , K ]_{L}( \mathcal{I}) \star K_{L}( \mathcal{I}) = - [ [ Q , K ] , K ]_{L}( \mathcal{I}) = E_{L}( \mathcal{I})
\frac{4}{K(K-1)}\log \left( \prod_{k=1}^{K-1}k! \right) \geq 2 \log K - 3 + \frac{2 \log \left( 2 \pi K/e^{3/2}\right)}{K-1}- \frac{2+ 6 \pi^{-2}+ \log K}{3 K (K-1)}.
\partial_{x}^\beta\widetilde{q}(x,\xi)=\partial_{x}^\delta\partial_{x_j}\widetilde{q}(x,\xi) =\partial_{x}^\delta\left(\left.\partial_{z_j}q_{x}(x,\xi)\right|_{z=x}\right)+\partial_{x}^\delta\left(\left.\partial_{x_j}{q_{z}}(x,\xi)\right|_{z=x}\right) .
p \in D ( q ) \equiv \{0 0 0 \circ p^{\prime}\mid U ( p^{\prime}, q ) \mathrm{~ i s ~ d e f i n e d}\}\cup \{0 0 1 \circ p^{\prime}\mid U ( p^{\prime}, q^{\prime}) \mathrm{~ i s ~ d e f i n e d}\}\; ,
\hat{D}(x,y,z)=\frac{2}{x}\left(1+\frac{2\sinh\frac{x}{2}}{e^{\frac{x}{2}}+e^{\frac{y+z}{2}}}\right)\leq\frac{4\sinh\frac{x}{2}}{x(e^{\frac{x}{2}}+e^{\frac{y+z}{2}})}.
\lim_{R\rightarrow\infty}\lim_{i\rightarrow\infty}\int_{B_{R}^{c}(0)}\frac{u_{i}^{p-1}(x)}{|x-y|^{n+\sigma p}}dy= u^{p-1}(x)\lim_{R\rightarrow\infty}\int_{B_{R}^{c}(0)}\frac{dy}{|x-y|^{n+\sigma p}}\leq u^{p-1}(x)\lim_{R\rightarrow\infty}\int_{B_{R-|x|}^{c}(x)}\frac{dy}{|x-y|^{n+\sigma p}}=0.
\sum_{k=1}^{\infty}|\lambda_k|^{p_2}(1-|a_k|^2)^{n+1+\alpha_2}|T_{\mu}^{\beta}f_k(a_k)|^{p_2}\lesssim \|T_{\mu}^{\beta}\|^{p_2}\cdot \big \|\{\lambda_k\}\big \|_{\ell^{p_1}}^{p_2}.
{\bf M}^{RL}_\Lambda=\frac{1}{2}\theta_\Lambda\Lambda-2\,,\quad{\bf M}^{Ri}_\Lambda=-\frac{1}{2\sqrt{2}}\hat{\beta}\theta_\Lambda\gamma^i\theta_\Lambda\,,\quad{\bf M}^{Li}_\Lambda=\frac{1}{2\sqrt{2}\hat{\beta}}\Lambda\gamma^i\Lambda\,,
C_2 \ll \sum_{n=N+1}^\infty \frac{1}{n^{\sigma_2}}\int_n^\infty \frac{1}{u^{\sigma_1+1}(u+n)^{\sigma_3}}du &\ll \sum_{n=N+1}^\infty \frac{1}{n^{\sigma_2}}\int_n^\infty \frac{1}{u^{\sigma_1+\sigma_3+1}}du.
\zeta^{(d)}_{M}(s) =\Vert M\Vert\sum^{\infty}_{h=0}\sum^{\infty}_{k=0}\bigl(-(2d-1)\bigr)^k \left\{\sum_{z\in R^{(d)}_{M}(h)}m^{(d)}_{M}(z)C_z P^{(z,0)}_{d,k}\Bigl(\frac{2d-3}{2d-1}\Bigr)\right\}u^{h+2k+1}_s
{\cal V}_{1 , 1}^{( i )}\geq 0 ,{\cal V}_{2 , 2}^{( i )}\geq 0 ,{\cal V}_{1 , 1}^{( i )}\geq |{\cal V}_{1 , 2}^{( i )}| ,{\cal V}_{2 , 2}^{( i )}\geq |{\cal V}_{2 , 1}^{( i )}| .
q_b(a,r)\approx \frac{(1-2\delta_{B0})\tilde \nu^{3/2}\exp\left[ -2 \tilde \nu \ln(a/r)\right]}{(8\pi )^{1/2}S_D\sigma^{D-1}r^D\sqrt{a^2-r^2}}f_1^{(q)},
b_{j\sigma ks}\ge0,\sum\limits_{k=1}^{N_\nu}\sum\limits_{s=1}^{S_{j\sigma k}}b_{j\sigma ks}\le 1,\sum\limits_{k=1}^{N_\nu}\Bigg(\sum\limits_{s=1}^{S_{j1k}}b_{j1ks}+\sum\limits_{s=1}^{S_{j2k}}b_{j2 ks}\Bigg)<2.
\begin{aligned}H(x;v) =H_1(x)v+E(x;v)-H_{1,0}(x)v+\frac{xv}{1-x}\big(H(x;v)-E(x;v)\big)+J(x;v)-J_1(x)v &+\frac{vx^2\left(-H(x;v)+v(1-x)H\big(x;\frac{1}{1-x}\big)+E(x;v)-v(1-x)E\big(x;\frac{1}{1-x}\big)\right)}{(1-x)(1-v+xv)}\,.\end{aligned}
\begin{array}{cccccccc}u_1(v) =\tau/4, \qquad \qquad u_2(v)=3\tau/4, |v|>z_{\infty}u_1(v) =-{M\over 2\pi i}\log v/\Lambda,\qquad u_2(v)=\tau+{M\over 2\pi i}\log v/\Lambda, \Lambda<|v|<z_{\infty}u_1(v) =1/4,\qquad \qquad u_2(v)=\tau+3/4, |v|<\Lambda. \end{array}
L^{\alpha}_{n}(x)=\frac{1}{\pi}e^{\frac{x}{2}}x^{-(\frac{2\alpha-1}{4})}n^{(\frac{2\alpha-1}{4})}\cos\big[2(nx)^{\frac{1}{2}}-\frac{\alpha \pi}{2}-\frac{\pi}{4}\big]+\mathcal{O}( n^{\frac{2\alpha-3}{4}}).
\sum_{j=0}^\infty f_j \left(2-\frac{1}{x}, 2- \frac{1}{x}\right) = \frac{1-x}{2-x}+ \frac{(1-x)^2}{(1-2x)(2-x)}\sum_{j=0}^\infty f_j \left(3-\frac{1}{x}, 2- \frac{1}{x}\right).
\ll \frac{A}{H'}\sum_{1 \leq h_1 \leq H'}\sum_{1 \leq h_2,\dots,h_k \leq H}\left|\sum_{1 \leq n \leq X}1_{\mathcal S}g_1(a_1 n + b_1 + h_1) \prod_{j=2}^k 1_{\mathcal S}g_j(a_j n + b_j + h_j)\right| + + H^{k-1}H'+ k H^{k-2}H' X.
(a)_{2\times 2}=\frac{1}{\rho^2}\begin{pmatrix}\rho + (w_1)^2 w_1w_2 w_1w_2 \rho + (w_2)^2 \end{pmatrix}=\frac{1}{\rho^2}\begin{pmatrix}1 - (w_2)^2 w_1w_2 w_1w_2 1 - (w_1)^2 \end{pmatrix}.
\left\lfloor\frac{m+n-2}{d}\right\rfloor-\left\lfloor\frac{m-1}{d}\right\rfloor-\left\lfloor\frac{n-1}{d}\right\rfloor=\frac{m+n-d}{d}-\frac{m-d}{d}-\frac{n-d}{d}=1.
R(N, \Sigma) := \{\alpha \in M \, | \, \langle \alpha, n(\alpha) \rangle = 1 \textrm{for some}n(\alpha) \in \Sigma[1] & \textrm{and}\langle \alpha, n_{j}\rangle \leq 0 \textrm{for}n_j \in \Sigma[1]\setminus \{n(\alpha) \}\}
\min_{x}\ \sum_{(i,j,r,s) \in [n]^4}c_{i,j}d_{r,s}x_{i,r}x_{j,s}\mathrm{s.t.}\x \in \mathcal{X}= \{x \in \{0,1\}^{[n]^2}: \sum_{i \in [n]}x_{i,r}= 1 \ \forall r \in [n], \ \sum_{r \in [n]}x_{i,r}= 1 \ \forall i \in [n]\}
W(z)-\Theta =-\frac{(1-b-c_0)^2+(1-b)^2}2\Theta+\frac{1}{d+2}\int_{\partial B_1}\left(|\nabla_\theta \phi|^2-2d\phi^2\right)-\frac{c_0}{d+2}\int_{\partial B_1^+}\phi &\le -\frac{c_0^2}4\Theta+\frac{1}{d+2}\int_{\partial B_1}\left(|\nabla_\theta \phi|^2-2d\phi^2\right)-\frac{c_0}{d+2}\int_{\partial B_1^+}\phi,
\left\{\prod_{i = 0}^{n}( \gamma_{\mu_{i}})_{2 1}\frac{1}{\tilde{x}_{0}- \tilde{x}_{\pi ( 1 )}}\frac{1}{\tilde{x}_{\pi ( 1 )}- \tilde{x}_{\pi ( 2 )}}. . . . . \frac{1}{\tilde{x}_{\pi ( n )}- \tilde{x}_{0}}+ c . c . \right\},
A_{U(\beta)f_j}= a_{\kappa}\nu_{d^2 \beta}\beta^{1-\kappa}Q_\beta(f_j) \left\langle f_j, f_j \right \rangle_{d^2 \beta}= a_{\kappa}\nu_{d^2}\beta^{1-\kappa}Q_\beta(f_j) \left\langle f_j, f_j \right \rangle_{d^2}= \beta^{1-\kappa}Q_\beta(f_j) \cdot A_{f_j}.
\textrm{If}\;\,J\geq n^{-\alpha}I,\;\; \textrm{where}\;\; \alpha >0,\;\;\textrm{then}\int_{|J^{1/2}\xi|\geq\log n}e^{-\frac{1}{4}\xi^{\rm T}J\xi}d\xi =\mathcal{O}(n^{-1/2}).
H(\tau+1) = \begin{pmatrix}\zeta_{24}^{-1}0 0 0 0 \zeta_{3}0 \zeta_{3}0 \end{pmatrix}H(\tau), H\left(-\frac{1}{\tau}\right) = \sqrt{-i\tau}\begin{pmatrix}0 1 0 1 0 0 0 0 -1 \end{pmatrix}H(\tau).
\int \prod_{i > j}{\cal D}c^{i j}{\cal D}\bar{c}^{i j}{\cal D}A_{\mu}^{i j}e^{- S_{\mathrm{o f f - d i a g}}}= \exp \left\{{\frac{1}{8 \pi}}\int_{\cal M}d \mu R \sum_{i > j}\log | \lambda_{i}- \lambda_{j}| \right\}.
H=L^2(\Omega), (\cdot,\cdot)=(\cdot,\cdot)_H=(\cdot,\cdot)_{L^2(\Omega)}, \|\cdot\|_H=\|\cdot\|_{L^2(\Omega)}, V=H^1_0(\Omega), (\cdot,\cdot)_V=(\nabla \cdot,\nabla \cdot)_{L^2(\Omega)}, \|\cdot\|_V=\|\nabla \cdot\|_{L^2(\Omega)}
\varepsilon(\mathcal{A}^T-\mathrm{I})\alpha_{k, 1}(\varphi, y)=\varepsilon (\alpha_{k, 1}(\varphi, y+\tilde{\beta}(\varphi, y))-\alpha_{k, 1}(\varphi, y))=\varepsilon \partial_y (\alpha_{k, 1})(\varphi, y)\,\tilde{\beta}(\varphi, y)+\mathtt{R}_{\tilde{\beta}},
{\binom{x}{t}}\sim{\binom{x}{t}}+{\binom{\sqrt{{\frac{R^{2}}{2}}+ R_{s}^{2}}}{-{\frac{R}{\sqrt{2}}}}}\approx{\binom{x}{t}}+{\binom{{\frac{R}{\sqrt{2}}}+{\frac{R_{s}^{2}}{\sqrt{2}R}}}{-{\frac{R}{\sqrt{2}}}}}
F_M \leq e^{-N\log(2)}+ e^{-\frac{\mu}{2N}|a-b|}\sum_{l=1}^{N}\frac{(Bl^d)^{l-1}}{2^{l+1}}&\leq e^{-N \log(2)}+ e^{-\frac{\mu}{2N}|a-b|}N B^N N^{dN}.
=\Omega_d\left({m_B^2 \over 2}\varphi_B^2 +{\lambda_B \over 24}\varphi_B^4\right)+{\hbar \over 2}\log\det (\triangle +M^2)+O(\hbar^2)+\Sigma_{d-2}\left(\mu_B+\sigma_B \varphi_B^2 \right)
P\left( g_1 -\frac{\operatorname{tr}g_1}{2}\operatorname{Id}\right) \equiv \psi(g_1)P - \frac{\operatorname{tr}g_1}{2}P \equiv \left( \psi(g_1) - \frac{\operatorname{tr}\psi(g_1)}{2}\operatorname{Id}\right)P \pmod{2^{T-7n_1-3}},
\frac{\partial W_{n,1}\left({u,\xi}\right)}{\partial \xi}=u\exp \left\{{u\xi +S_{n}\left({u,\xi}\right) +\ln \left\{{1+u^{-2}T_{n}\left({u,\xi}\right)}\right\}+\eta_{n,1}\left({u,\xi}\right)}\right\},
\frac{\partial f(z)}{\partial z}= \frac{1}{\mathrm{B}(\alpha,\beta)}\phi(z)[\Phi(z)]^{\alpha-2}[1-\Phi(z)]^{\beta-2}\times &\Big\{(\alpha-1)\phi(z)[1-\Phi(z)]-(\beta-1)\phi(z)\Phi(z)-z \Phi(z) [1-\Phi(z)]\Big\}.
\frac{1}{t}\int_{0}^{t}\widetilde{\mathfrak{S}}_{s}\varphi{\rm d}s-\langle \varphi,\rho\rangle =\frac{1}{t}\int_{0}^{t_{0}}\widetilde{\mathfrak{S}}_{s}\varphi{\rm d}s-\frac{t_{0}}{t}\langle \varphi,\rho\rangle &\quad+\frac{1}{t}\int_{t_{0}}^{t}{\rm d}s\int_{E}\left(\widetilde{p}(s,x,y)-1\right)\varphi(y)\rho({\rm d}y).
f(|\vec{v}|)=\frac{\Gamma(5N/2)}{\Gamma(3/2)\Gamma(5N/2-3/2)}\times\frac{m^{3/2}|\vec{v}|^{2}}{2^{1/2}E^{3/2}}\times(1-\frac{m|\vec{v}|^{2}}{2E})^{\frac{5(N-1)}{2}}
\mathrm{T r}e^{i F_{9}}- \mathrm{t r}e^{i F_{9}}\mathrm{t r}e^{i F_{\bar{9}}}+ \mathrm{T r}_{S}e^{i F_{\bar{9}}}={\frac{1}{2}}( \mathrm{S t r}e^{i F_{9}})^{2}-{\frac{1}{2}}\mathrm{S t r}e^{2 i F_{9}}.
|u(x,t)-u(y,s)| \le |u(x,t)-u(x,-1)|+|u(x,-1)-u(y,s)| \le \tilde\rho(|t+1|)+\tilde\rho(|x-y|\vee |1+s|) \le \tilde\rho(c|x-y|)+\tilde\rho((|x-y|)\vee |1+t|) \le \tilde\rho(c(|x-y|\vee|t-s|))+\tilde\rho(c|x-y|\vee|t-s|) = 2\tilde\rho(c(|x-y|\vee|t-s|)) \le \rho_2(|x-y|\vee|t-s|).
\bigvee_{i=-n}^0 T_f^i \xi_{-\ell}^{\ell}\subseteq \xi^{\ell + (c p^{k-1}+ 1) j_p + d (k-1)(r - j_p)}_{-\ell}, \bigvee_{i=N}^{N+n}T_f^i \xi_{-\ell}^{\ell}\subseteq \xi^{\ell - N j_p}_{-\ell - (c p^{k-1}+ 1) j_p - d (k-1)(r - j_p)},
\left \|f\right \|_{h_{\omega,atom}^{p,q,s}}=\inf\left\{\left \|\sum\limits_{j=1}^{\infty}\frac{\lambda_j \chi_{Q_j}}{\omega(Q_j)^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}+\left \|\sum\limits_{j=1}^{\infty}\frac{\mu_j \chi_{P_j}}{\omega(P_j)^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}\right\}
\delta_{\vec{u}}(t) <W(\gamma)> = -{4 \pi i\over k}\int_0^1 ds u^a(s) ({\eta'}^b(s)+t{u'}^b(s)) \epsilon_{abc}\oint_\gamma dy^c \delta^3(\eta(s,t)-y) \left<{\rm Tr}\left(\tau^lH(\eta(t)_x^o)H(\gamma_o^y)\tau^lH(\gamma_y^o)H(\eta(t)_o^x)\right)\right>.
\mathrm{I}_{1}^{s}( m ) \equiv - \frac{1}{2}\sum_{k \in \cal Z}\mathrm{s i g n}( \varepsilon_{k}) | \lambda \varepsilon_{k , \mathrm{R}}|^{- s}( | \lambda \varepsilon_{k + m , \mathrm{R}}|^{- s}- | \lambda \varepsilon_{k - m , \mathrm{R}}|^{- s}) .
\sum_{i=1}^N \int_{\Omega_i}\varepsilon \nabla u^* \cdot \nabla \varphi\, dx-\sum_{i=1}^N \int_{\partial \Omega_i}\varepsilon \nabla u^* \cdot \nu \varphi\, dx=\sum_{i=1}^N \int_{\Omega_i}f \varphi\, dx.
\tilde{\varphi}(r) =\frac{1}{2}P(r)(\hat{X}(r))^{2}; \tilde{\nu}(r) =P(r)\hat{X}(r)(\hat{\Theta}(r)D\sigma(r)+\varepsilon_{2}(r))+\frac{1}{2}Q(r)\left( \hat{X}(r)\right)^{2}.
c_2 = 2\pi \int_0^\pi (\pi^q \lambda^{-q}-1)^2 \sin \lambda{\;\rm d}\lambda = \frac{(q)_3}{6}\sum_{j=0}^\infty \frac{(-1)^j \pi^{2j+3}}{(2j+1)!}\frac{1}{(1-q+j)_3},
\mathrm{LC}_{s=1}^r[(\ref{trpart})]^{\mathrm{hol}}= \sum_{\substack{0 < d \equiv 0, 1(4) dD \neq \square}}\frac{1}{\sqrt{d}}\mathrm{LC}_{s=1}^r \biggl[\mathrm{Tr}_{d,D}(G_m(z,s))\biggr] q^d &= \sum_{\substack{0 < d \equiv 0, 1(4) dD \neq \square}}\frac{1}{\sqrt{d}}\mathrm{Tr}_{d,D}(F_{0,m,r}) q^d.
H_{12}(x,t,\tau ,r,p):=\begin{cases}H_1(x,\frac{t}{1+\frac{k_2}{k_1}},p) \mbox{if}\ (i-1) < \tau \leq (i-1) + \frac{k_1 \Delta t}{\varepsilon}, H_2(r,p) \mbox{if}\ (i-1)+ \frac{k_1 \Delta t}{\varepsilon}< \tau \leq i\end{cases}
\begin{aligned}\binom{n-t_1}{3}-\binom{n-t_1-t_2}{3}=\binom{t_2}{3}+\binom{t_2}{2}(n-t_1-t_2)+t_2\binom{n-t_1-t_2}{2}&=\binom{t_2}{3}+\binom{t_2}{2}(|W|+3t_2)+t_2\binom{|W|+3t_2}{2}&=\binom{t_2}{3}+\binom{t_2}{2}|W|+3\binom{t_2}{2}t_2+t_2\binom{|W|}{2}+t_2\binom{3t_2}{2}+3t_2^2|W| &\ge37\binom{t_2}{3}+8\binom{t_2}{2}|W|, \end{aligned}
\pi_n^*\left(\overline{\mathcal{H}yp}_{2,n-1}\right) \cdot \rho_n^*\left(\overline{\mathcal{H}yp}_{2,1}\right) \equiv \overline{\mathcal{H}yp}_{2,n}+ \sum_{i=1}^{n-1}{\sigma_{i}}_\ast\left( \overline{\mathcal{H}yp}_{2,n-1}\right)+ \sum_{i=1}^{n-1}\Phi_i+ \sum_{1\leq i<j\leq n-1}\Gamma_{i,j}.
\begin{aligned}\sigma(x^{(\beta,s_{1})}_{\ell,1})=x^{(\gamma,r)}_{\ell,1}\ 1\leq \ell\leq j-1, \sigma(x^{(\beta,s_{1})}_{\ell,t_{\ell,1}})=x^{(\gamma,r)}_{\ell,1}, \sigma(x^{(\beta,s_{1})}_{\ell,t_{\ell,2}})=x^{(\gamma,r)}_{\ell,2}\j\leq \ell\leq n.\end{aligned}
\left[ T_{(A,i)},T_{(B,j)}\right] =K_{ij}{}^{\gamma}C_{AB}{}^{C}T_{(C,\gamma)}, & =K_{ij}{}^{k}C_{AB}{}^{C}T_{(C,k)}+K_{ij}{}^{k+n}C_{AB}{}^{C}T_{(C,k+n)}& =K_{ij}{}^{k}C_{AB}{}^{C}T_{(C,k)}-K_{ij}{}^{k+n}C_{AB}{}^{C}T_{(C,k)}& =\left( K_{ij}{}^{k}-K_{ij}{}^{k+n}\right) C_{AB}{}^{C}T_{(C,k)},
F_{\eta}=\prod_{k=0}^{\eta-1}7\sqrt{\frac{2}{\pi c}}\frac{(\pi c)^k}{k!}e^{-\frac{\pi}{2}c}= e^{-\frac{\pi}{2}c \eta}\left(7\sqrt{\frac{2}{\pi c}}\right)^{\eta}A_{\eta}, A_{\eta}= \prod_{k=0}^{\eta-1}\frac{(\pi c)^k}{k!}.
d^{2p-3}(\bar{\xi}_1)\dot{=}\alpha_1, d^{2p-3}(\sigma \bar{\xi})\dot{=}\sigma \alpha_1, d^{2p-3}(\tilde{\tau}_1)\dot{=}v_1, d^{2p-3}(\sigma \bar{\tau}_1 )\dot{=}\sigma v_1, d^{2p-2}(\tilde{\tau}_1v_1^k)\dot{=}v_1^{k+1}d^{2p-2}(\bar{\xi}_1v_1^k)\dot{=}\alpha_1v_1^{k}d^{2p-2}(\tilde{\tau}_1\alpha_1v_1^{k-1})\dot{=}\alpha_1v_1^{k}
\begin{array}{rl}\mbox{Bekenstein-Verlinde bound}: S_{BV}=\frac{2\pi}{n}E{\cal R}, \mbox{Bekenstein-Hawking bound}: S_{BH}=(n-1)\frac{V}{4G_{n+1}{\cal R}}, \mbox{Hubble bound}: S_{H}=(n-1)\frac{HV}{4G_{n+1}}.\end{array}
|\theta_L(z,w)|^2=\prod_{j=1}^n \bigl(1-\bigl|\frac{z_j-w_j}{1-z_j\bar w_j}\bigr|^2\bigr)^{L_j}=\prod_{j=1}^n \bigl(1-\rho^2(z_j,w_j)\bigr)^{L_j}\ ,
\left|\frac{1}{N}\sum_{n=0}^{N-1}\left(e^{2\pi i p(n)t}-\int_{h=0}^{1}e^{2\pi i p(n+h)t}dh \right)\right| \leq \frac{4\pi \lvert t\rvert}{N}\sum_{n=0}^{N-1}\max_{h\in \left[0,1\right]}\lvert p'(n+h)\rvert = 4\pi \lvert t \rvert \cdot O_{p}\left(N^{d-1}\right).
\sum_{i=1}^{\infty}\frac{3^i \big( \sum_{j=1}^i \frac{1}{j^3}-\frac{\sum_{j=1}^i \frac{1}{j^2}}{i}\big)}{i \binom{2 i}{i}}= -\frac{4}{3}\pi^2 c_1+18 c_8-\frac{4 \pi \zeta_3}{9 \sqrt{3}}+18-\frac{4 \pi^2}{3}-\frac{2 \pi^4}{243}
\begin{array}{c l r}{\sum_{i = 0}^{r + 4}q_{i}^{a}s_{i}}&{=}&{0}\\{\sum_{i = 0}^{r + 4}q_{i}^{a}n_{i}}&{=}&{0}\\{\sum_{i = 0}^{r + 4}q_{i}^{a}m_{i}}&{=}&{0 .}\\ \end{array}
\Phi_{\varepsilon,\delta,\sigma}(t,x,s,y):=u^{\varepsilon,\delta}(t,x)-v_{\varepsilon,\delta}(s,y)-\varphi_{\delta,\sigma}(t,x,s,y)\quad &\varphi_{\delta,\sigma}(t,x,s,y):=\varphi_\delta(t,x,s,y)+a_\sigma t+\zeta_\sigma\cdot x-b_\sigma s-\xi_\sigma\cdot y
\|\partial_x^2 v_0^+(x,\cdot)\|^2_{H^{-\frac{1}{3}}(0,T)}\leq C \int_{0}^{\infty}\rho^4\left|\hat{h}_0^+(\rho)\right|^2d\rho =C \int_{0}^{\infty}\rho^4\left|\hat{h}_0(ia\rho^3L^3)\right|^2d\rho &= C \int_{0}^{\infty}\rho^4\left| \int_0^{\infty}e^{-ia\rho^3 L^3 t}h_0(t)dt\right|^2d\rho.
\frac{1}{\frac{1}{\sigma_{0}^{2}}+\frac{n}{\sigma^{2}}}(\frac{\mu_{0}}{\sigma_{0}^{2}}+\frac{\sum_{i=1}^{n}x_{i}}{\sigma^{2}}),(\frac{1}{\sigma_{0}^{2}}+\frac{n}{\sigma^{2}})^{-1}
\Bigg\{\left(\begin{array}{c}z^a\bar{z}^b 0\end{array}\right),\left(\begin{array}{c}0 z^a\bar{z}^b\end{array}\right)\Bigg|(a,b)\in (\mathbb{Z}+\frac{1}{2})\times\mathbb{Z}\mbox{or}(a,b)\in \mathbb{Z}\times(\mathbb{Z}+\frac{1}{2}), a+b=\frac{1}{2}\Bigg\}.
{\cal L}_{gauge}^{eff}= \Pi \partial_{+}A_{-}+ \partial_{-}A_i\partial_{+}A_i - \frac 1 2 \Pi^2 - \frac 1 2 (\partial_iA_j)^2 - \Pi \partial_i A_i + A_i j^i + A_{-}j^{-},
2\frac{d}{ds}\ell\left( s,a,\chi\right) |_{s=0}=2\ell^{\prime}\left(0,a,\chi\right) & =-E_{0,\overline{\chi}}\left( 0\right) \log a-\frac{1}{a}\overline{E}_{1,\overline{\chi}}\left( 0\right) +\chi\left( -1\right)\int\limits_{0}^{\infty}\frac{\overline{E}_{1,\overline{\chi}}\left(x\right)}{\left( x+a\right)^{2}}dx.
{\cal F}^{( N_{F})}( \Psi ) = i \frac{( 4 - N_{F})}{8 \pi}\Psi^{2}\log{\left( \frac{C \, \Psi^{2}}{\Lambda_{N_{F}}^{2}}\right)}- \frac{i}{\pi}\sum_{k = 1}^{\infty}{\cal F}_{2 k}^{( N_{F})}\left( \frac{\Lambda_{N_{F}}}{\Psi}\right)^{2 k ( 4 - N_{F})}\Psi^{2}
e_1^\prime =e_1+\sum\limits_{i=1}^{k}\sum\limits_{j=1}^{i}(-1)^{i+j-1}b_{j}f_{i}+\sum\limits_{i=1}^{k}\sum\limits_{j=1}^{i}(-1)^{i+j-1}b_{k+j}f_{k+i}, x^\prime =x+\sum\limits_{i=1}^{k}\sum\limits_{j=1}^{i}(-1)^{i+j-1}\theta_{k+j}f_{k+i},
\rho = \frac{\alpha (1-\alpha)|\mathbf{v}_I^{\dag}\mathbf{T}\mathbf{V}_N\mathbf{w}_I|^2}{4 \sigma_n^2 \sigma_d^2}, \sigma_n = \sqrt{\frac{\alpha}{2}\mathbf{v}_I^{\dag}\mathbf{T}\mathbf{v}_I}, \sigma_d = \sqrt{\frac{1-\alpha}{2}\mathbf{w}_I^{\dag}\mathbf{V}_N^{\dag}\mathbf{T}\mathbf{V}_N\mathbf{w}_I}.
\mathbf{y}_k= a_c s_c \mathbf{H}_k\mathbf{p}_c+ \mathbf{H}_k\sum_{i\in \mathcal{M}_k}a_i s_i\mathbf{p}_i+\mathbf{H}_k\sum\limits_{\substack{l=1 l \neq k}}^{K}\sum\limits_{j\in \mathcal{M}_l}a_j s_j\mathbf{p}_j+ \mathbf{n}_k.
\begin{aligned}f_{m,k}e_{m,j-k}= q^{N \frac{k(k+1)}{2}+k a(r)}{m-1 \brack k}_{q^N}&\times \left( d^{m-1}\sum_{\substack{\alpha < a(r) w(\alpha)=j+m-k-1}}q^{\alpha}+d^m \sum_{\substack{\alpha < a(r) w(\alpha)=j+m-k}}q^{\alpha}\right){j+m-k-1 \brack m-1}_{q^N},\end{aligned}
\delta_H (c_H)(\psi_g(x_1), \ldots , \psi_g(x_{n+1})) &= \sum_{1\leq i < j \leq n+1}(-1)^j c_H(gx_1, \ldots, gx_{i-1}, [gx_i, gx_j], \ldots , \widehat{gx_j}, \ldots , gx_{n+1}) &= \sum_{1\leq i < j \leq n+1}(-1)^j A(\hat{g})c_K(x_1, \ldots ,x_{i-1}, [x_i, x_j], x_{i+1}, \ldots , \hat{x}_j, \ldots , x_{n+1}) &= A(\hat{g})\delta_K(c_K)( x_1, \ldots, x_{n+1}).
\psi_{0}(x)=\frac{1}{2\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty}(-\frac{\zeta^{\prime}(s)}{\zeta(s)})\frac{x^{s}}{s}ds=x-\sum_{\rho}\frac{x^{\rho}}{\rho}-log(2\pi)-\frac{1}{2}log(1-x^{-2})
|\Phi(\xi_1, \eta_1, \xi, \eta)| = |\xi_1 \xi (\xi_1 +\xi) + \eta_1 \eta (\eta_1+ \eta)| & \geq 2^{-1}r_1 r |r_1-r| - 2^{10}A^{-1}K^{-1}N_1^3.
\delta(L_\ell,q^n) = \frac{1}{2}\Big(\ell^{-2a_{n}^+}+ \ell^{-2a_{n}^-}\Big) + \frac{(-1)^{n/2}}{q^{n/2}+ q^{-n/2}}\Big(\ell^{-2a_n^+}- \ell^{-2a_n^-}\Big),