harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{l}\lambda_{1}=-2(\frac{A^{2}-3B}{3})^{\frac{1}{2}}\cos\frac{1}{3}\alpha-\frac{A}{3}, \lambda_{2}=-2(\frac{A^{2}-3B}{3})^{\frac{1}{2}}\cos\frac{1}{3}(\alpha+2\pi)-\frac{A}{3}, \lambda_{3}=-2(\frac{A^{2}-3B}{3})^{\frac{1}{2}}\cos\frac{1}{3}(\alpha+4\pi)-\frac{A}{3}, \end{array}
h(\mu)+\frac{s}{d}\int \log|\det A_{x_1}|d\mu(x) = h(\mu)+\lim_{n\to\infty}\frac{s}{nd}\int \log |\det A_{x_n}\cdots A_{x_1}|d\mu(x) & = h(\mu)+\lim_{n\to\infty}\frac{1}{n}\int \frac{s}{d}\sum_{i=1}^d \log \alpha_i( A_{x_n}\cdots A_{x_1})d\mu(x) &=h(\mu)+\frac{s}{d}\sum_{i=1}^d \lambda_i(\mathsf{A},\mu)=h(\mu)+s\lambda_1(\mathsf{A},\mu)
\Bigl|\frac{T^*_N}{N}- T^* \Bigr| \le \sup_{t \in [z^{-1}(1) - \delta, z^{-1}(1) + \delta]}z^{-1}(t) = \sup_{t \in [T^*-\delta, T^*+\delta]}z^{-1}(t) \le A \delta.
Z^{( 1 )}[ \eta ] = \mathrm{S d e t}^{- \frac 1 2}{\sf Y}[ \eta ] = \mathrm{S d e t}^{- \frac 1 2}\left( \begin{array}{c c}{p^{2}}&{\lambda \bar{\eta}}\\{\lambda \eta}&{i p \llap /}\\ \end{array}\right) .
\left[ -\frac{g^2_{{\rm YM}}N \Gamma(\omega-1)}{8\pi^\omega (2-\omega)(2\omega-3)}\right] \cdot[x^2]^{(2-\omega)}2^{-J}\Delta^J(x)\frac{J^2}{2}\langle{\rm Tr}Z^{J-1}\bar Z^{J-1}-\frac{1}{N}{\rm Tr}Z^{J-1}{\rm Tr}\bar Z^{J-1}\rangle
I(T_2;x) =\!\begin{multlined}[t] x^{14}+ 48x^{13}+ 2372x^{12}+ 15498x^{11}+ 48086x^{10}+ 90178x^9 + 112870x^8 + 98968x^7 + 62183x^6 + 28147x^5 + 9089x^4 + 2037x^3 + 300x^2 + 26x + 1, \end{multlined}
q(p^{-1}(U_2 \cdot [\phi_6])) = (U_2 \cdot [xy]) \cup (U_2 \cdot [x^2 - y^2]) \cup (U_2 \cdot [x^2 + y^2]) = &= \{x(sx + y)\}\cup \{x^2 - (sx + y)^2\}\cup \{x^2 + (sx + y)^2\},
h^{k}_{pp\bar{m}\bar{m}}= M^2_{mp}, h^{k}_{pp\bar{m}\bar{n}}= 2M_{mp}M_{nq},m<n, h^{k}_{pq\bar{m}\bar{m}}= 2M_{mp}M_{mq},p<q, h^{k}_{pq\bar{m}\bar{n}}= 2(M_{mp}M_{nq}+M_{mq}M_{np}),p<q,m<n.
f_2^p+{\bar f_2}f_1^p = f_2^p+\left(\tfrac{2c_1}{c_0}X^{p^s-2p^{s-1}}f_2 -f_1^2) \right)f_1^p = f_2^p -f_1^pf_1^2 +\tfrac{2c_1}{c_0}X^{p^s-2p^{s-1}}f_2 f_1^p.
\alpha_n(t,q) = \Big{(}(n-2)tq + (n-1)t + 2q + 1\Big{)}\alpha_{n-1}(t,q) + (t-t^2)(q+1)\frac{\partial}{\partial t}\alpha_{n-1}(t,q) + (1-t)(q^2+q)\frac{\partial}{\partial q}\alpha_{n-1}(t,q)
\begin{aligned}\mathcal K_{\ell}:={\rm Span}\left\{\i^*(Z_j^0),\j=1,\dots,\ell\right\}\mathcal K^\bot_{\ell}:=\left\{\phi\in H^1_g(M)\,:\, \hbox{$\phi$ is symmetric and}\ \langle \phi, \i^*(Z_j^0)\rangle =0,\j=1,\dots,\ell\right\}. \end{aligned}
-\Delta f_{n}^{+}=(k^{2}-4)f^{+}_{n}+(4\widetilde\chi_{n}'(y)-\widetilde\chi_{n}''(y))\chi_{n}(x)e^{ikx-2y}&-\big(2ik\chi'_{n}(x)+\chi''_{n}(x)\big)\widetilde{\chi}_{n}(y)e^{ikx-2y}.
\|w_{0}+l_{0}w_{\epsilon,\eta}\|^{2}= \|w_{0}\|^{2}+ (l_{0})^{2}\|w_{\epsilon,\eta}\|^{2}+2l_{0}\langle w_{0}, w_{\epsilon,\eta}\rangle >\|w_{0}\|^{2}+ |c^{2}-\|w_{0}\|^{2}| + 2l_{0}\langle w_{0}, w_{\epsilon,\eta}\rangle >c^{2}>\left(t^{-}\left(\frac{w_{0}+l_{0}w_{\epsilon,\eta}}{\|w_{0}+l_{0}w_{\epsilon,\eta}\|}\right)\right)^{2}
f_{\varphi}\:=\: |(ac+bc)-(c-ab)| & \:=\: \left(\left(ac+bc\right)-\left(a^{\prime}c+b^{\prime}c\right)\right)+\left(\left(a^{\prime}c+b^{\prime}c\right) -\left(ac+bc\right)\right) & \:=\: abc+a^{\prime}b^{\prime}c
\langle{\cal O}\rangle_\sigma =G^{(n)}_\sigma(\overbrace{x^{-1}\zeta,\cdots,x^{-1}\zeta}^{n}, \overbrace{\zeta,\cdots,\zeta}^{n})^{-\varepsilon_{n}\cdots -\varepsilon_{1}\varepsilon'_1 \cdots \varepsilon'_{n}}.
\Gamma(a)\, |U(a,b,z) | = \, \int_0^1 e^{-zt}\, t^{a-1}\, (t+1)^{b-a-1}\, dt + \int_1^\infty e^{-zt}\, t^{a-1}\, (t+1)^{b-a-1}\, dt & \leq \frac 1a\, 2^{b-a-1}+ \int_1^\infty e^{-zt}\, \Big(\frac{1+t}{t}\Big)^{1-a}\, (t+1)^{b-2}\, dt & \leq \frac 1a\, 2^{b-a-1}+2^{1-a}\e^z\, z^{1-b}\, \Gamma(b-1).
( 1 - \gamma ) ( \partial_{y}+ 2 \ell^{- 1}) h |_{0_{+}}- \frac{2}{3}\gamma \ell \big ( \partial_{\lambda}\partial_{\rho}h^{\lambda \rho}- \partial_{\lambda}^{2}h \big ) |_{0_{+}}= \frac{1}{6 M_{*}^{3}}S_{\mu}^{\mu}.
C_w(r)\geq \tau^r ((\tau-1)r+1)^{-1}\sup_{0<\varepsilon<1}\varepsilon^{r-1}[1-(1-\varepsilon)^{(\tau-1)r+1}] =\frac{1}{(1/\tau)^{r-1}}\frac{r'-1}{r'-\tau^{-1}}
f_{1}\star_{\vec{c}}^{n}f_{2}\ (\mathbf{n})=\int_{S^{2}}\left\{\int_{S^{2}}\mathbb{T}_{\vec{c}}^{j}(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n})\star_{\vec{c}}^{n}f_{1}(\mathbf{n}_{1})d\mathbf{n}_{1}\right\}\star_{\vec{c}}^{n}f_{2}(\mathbf{n}_{2})d\mathbf{n}_{2},\quad\quad
\frac{p(x+\bar{h})\cdot q - p(x)\cdot q}{h}\le\frac{\left(1-\frac{1}{1-\partial_{x_{1}}\varphi(x,p(x+\bar{h}))h+o(h)}\right)p(x+\bar{h})\cdot q}{h}& = \frac{1}{1-\partial_{x_{1}}\varphi(x,p(x+\bar{h}))h+o(h)}\left(-\partial_{x_{1}}\varphi(x,p(x+\bar{h}))+\frac{o(h)}{h}\right) p(x+\bar{h})\cdot q
\int f\otimes\bigl(\otimes_{i=1}^n f_i\bigr)\otimes g\,d\rho_n=\int f\cdot \bigl(\prod_{i=1}^n Vf_i\bigr)\cdot Jg\,d\mu=\int f\cdot \bigl(\prod_{i=1}^n Vf_i\bigr)\otimes g\,d\eta.
=\sum_{n\neq a^{*}}\beta \delta(n)\mathbb{E}[\theta_{n,K}] & \leq \beta \left[8\sum_{n\neq a^{*}}\frac{\ln K}{\delta_n}+ \left(1+ \frac{\pi^2}{3}\right)\sum_{n\neq a^{*}}\delta_n \right].
\left( 1-\frac{x_{j}}{x_{i}}\right) \sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(j,i)}x^{\gamma}=-\frac{x_{j}}{x_{i}}\left(1-\frac{x_{i}}{x_{j}}\right) \sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(j,i)}x^{\gamma}=-\frac{x_{j}}{x_{i}}\sum_{\gamma\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\gamma}x^{\gamma}.
Y^{t,\Upsilon;v}(s)=\Phi(X_T^{t,\Upsilon;v})+\int_{s}^{T}g(r,X_r^{t,\Upsilon;v},Y^{t,\Upsilon;v}(r),Z^{t,\Upsilon;v}(r),v(r))dr -\int_{s}^{T}Z^{t,\Upsilon;v}(r)dW(r).
M_1[n+1] := \mathrm{diag}\Big(\{(\upsilon_N\ast|\phi_{n+1}|^2)(x) + \frac{1}{N}(\upsilon_N \ast\rho^\mathrm{pair})(x)\}\delta(x-y) &+\frac{1}{N}(\upsilon_N \Big(\frac{k_{n+1}\circ\overline{k_{n+1}}}{\delta - k_{n+1}\circ \overline{k_{n+1}}}\Big))(x,y)\Big),
\int \left[ G_{h/2}\ast,\nabla \xi \right] u \, \nabla^2 G_{h/2}\ast u\,dx = h\sum_{i,j,k}\int \partial_i\partial_j \xi_k \colon \nabla_i G_{h/2}\ast u \, \partial_j\partial_k G_{h/2}\ast u \,dx &+ O\Big(\|\nabla^3\xi\|_\infty h \int \left|\nabla^2 G_{h/2}\ast u \right|dx\Big).
F'(y^*,u^*,p^*,\mu^*) = \begin{bmatrix}M 0 L^T \alpha_y I 0 \nu M -M \alpha_u I L - M 0 0 c\alpha_y \Pi_{\mathcal{A}_*}c \alpha_u \Pi_{\mathcal{A}_*}0 \Pi_{\mathcal{I}_*}\end{bmatrix},
\mathcal{L}^{(\alpha)}_n(x)=\frac{(q^{\alpha+1};q)_n}{(q;q)_n}\sum_{k=0}^n \begin{bmatrix}n k \end{bmatrix}\frac{q^{k^2+k \alpha}}{(q^{\alpha+1};q)_k}(-x)^k.
F_a \to{\cal F}_a = \partial V_a - i \frac e{\sin \theta_W}\Bigl((\hat \Phi + \zeta \mbox{v})^\dagger \frac{\tau^T_{a}(\hat G)}2( \Phi + \mbox{v}) -( \Phi + \mbox{v})^\dagger \frac{\tau^T_{a}(\hat G)}2( \hat \Phi + \zeta \mbox{v}) \Bigr) .
(g\rtimes \sigma)\cdot \mathcal{R}_h(x_1,\ldots, x_k):=\mathcal{R}_g\left((-1)^{h_{\sigma^{-1}(1)}}x_{\sigma^{-1}(1)}, \ldots, (-1)^{h_{\sigma^{-1}(k)}}x_{\sigma^{-1}(k)}\right).
\begin{array}{lll}g([W, X_1+X_2+X_3]_{\frak{m}}, X_1+X_2+X_3)_o\ & \ \ \ +g(W, [X,X_1+X_2+X_3]_{\frak{m}}+[Y, Z]_{\frak{m}})_o=0. \end{array}
X := \{(\vec{g},\vec{z})\in G^d\times\Delta^L\mid \forall i\not=j: d_{\vec{g}}^{(\alpha_1,\ldots,\alpha_l)}(z_i,z_j)>2l+2 &\forall i: |\{z_i,\nu_1(z_i),\ldots,\nu_l(z_i)\}|\leq l\},
F = i \frac{d g_{1}\wedge d \overline{{g}}_{1}}{( 1 - g_{1}\overline{{g}}_{1})^{2}}- i \frac{d g_{2}\wedge d \overline{{g}}_{2}}{( 1 + g_{2}\overline{{g}}_{2})^{2}}= - \frac{1}{2}( R_{2}^{1}+ R_{4}^{3}) .
\begin{aligned}\int_0^1 (au'')' v' dx = (au''v')(x_0- \delta) - (au''v')(0)-\int_0^{x_0-\delta}au''v''dx & + \int_{x_0-\delta}^{x_0+\delta}(au'')'v' dx + (au''v')(1) - (au''v')(x_0+ \delta) &- \int_{x_0+\delta}^1 au''v'' dx.\end{aligned}
\mathcal I_v^{\sharp}(\mathbf h, \mathbf g, \pmb{\phi}) = \frac{L(1,\pi_v,\mathrm{ad})L(1,\tau_v,\mathrm{ad})}{L(1/2,\pi_v\times \mathrm{ad}\tau_v)}\alpha_v^{\sharp}(\mathbf h, \mathbf g, \pmb{\phi}),
S_{l}Ad(O_{<k}) \ = \ 2^{-l}S_{l}\partial_{x,t}(Ad(O)_{<k}) = 2^{-l}S_{l}(ad(O_{;(x,t)}Ad(O)_{<k}) \ = \ \ldots \ = \ 2^{-5l}\prod_{j=1}^5 (S^{(j)}_{l}ad(O_{;(x,t)}) \cdot Ad(O)_{<k}\ ,
\begin{aligned}p(y) ={}\left(\frac{2}{\pi}\right)^{1/2}\biggl[1+\tau^{1/2}\left(-\frac{8}{3}y^{3}+2y\right) & +\tau\left(\frac{32}{9}y^{6}-\frac{31}{3}y^{4}+\frac{15}{2}y^{2}-\frac{37}{48}\right)\biggr]e^{-2y^{2}}+O(\tau^{3/2})\end{aligned}
\frac{\alpha(C_{r_j}(x))}{r_j^N}\geq \frac{\alpha_{k(j)}(C_{r_j}(x))}{r_j^N}- r_j &= \frac{(1-s_{k(j)}) \,r_j^{N-s_{k(j)}\,p}\,r_{j}^p\,\Big[(u_{k(j)})_{r_j,x}\Big]^p_{W^{s_{k(j)},p}(Q)}}{r_j^N}- r_j \geq \left(1- \frac{1}{j}\right)\,(1-s_{k(j)})\,\Big[(u_{k(j)})_{r_j,x}\Big]^p_{W^{s_{k(j)},p}(Q)}- r_j.
{\cal L}_{\mathrm{W Z}}^{( p )}= \epsilon^{i_{1}\ldots i_{p + 1}}\sum_{k = 0}^{( p - 1 ) / 2}a_{p , k}C_{i_{1}\ldots i_{p - 2 k}}( F^{k})_{i_{p - 2 k + 1}\ldots i_{p}}\partial_{i_{p + 1}}T ,
\sum_{n=1}^{\infty}\frac{1}{\left(n^{2}+x^{2}\right)^{s}}=\frac{\sqrt{\pi}\,x^{1-2s}}{2\Gamma(s)}\Gamma\left(s-\frac{1}{2}\right)-\frac{x^{-2s}}{2}+\frac{2\,\pi^{s}x^{\frac{1}{2}-s}}{\Gamma(s)}\,\sum_{n=1}^{\infty}n^{s-\frac{1}{2}}\,K_{s-\frac{1}{2}}(2\pi nx).
I \geq \frak{H}(\omega; f(\lambda T_{1}+I),...,f(\lambda T_{n}+I)) & =\left[\sum_{i=1}^{n}w_{i}f(\lambda T_{i}+I)^{-1}\right]^{-1}\geq \left[\sum_{i=1}^{n}w_{i}f(\lambda T_{i}+I)^{\frac{-p}{\lambda}}\right]^{\frac{-\lambda}{p}}
\Vert u_{\alpha, \beta}\Vert^{2}_{L_{g_{0}}^{\frac{n}{n-2}r}}\leq C_{n,p}(\Vert v_{\alpha, \beta}\Vert^{r}_{L_{g_{0}}^{\frac{nr}{n+r}}}+\Vert u_{\alpha, \beta}\Vert^{r}_{L_{g_{0}}^{r}}).
P_{v}= \exp \left\{- \mu \tilde{n}^{c r}\int d{\bf x}\right\}, \; \; \mu = \sum_{n = 0}^{\infty}\frac{\beta ( n + 1 )}{( n + 1 )^{3 / 2}\beta ( 1 )}\exp \left\{- n \pi \frac{( a M )^{2}}\rho \right\}\; .
f(x^{k+1})-f(x) \ = f(x^{k+1})-f(x^k)+f(x^k)-f(x) \ \leq \langle \nabla f(x^k),x^{k+1}-x^k \rangle +\frac{L}{2}\|x^k-x^{k+1}\|_2^2+\langle \nabla f(x^k),x^{k}-x \rangle =\langle \nabla f(x^k),x^{k+1}-x\rangle +\frac{L}{2}\|x^k-x^{k+1}\|_2^2 \ ,
0\rightarrow\mathbb{Z}_{2}^{\infty}\oplus(\begin{matrix}n\\ 2\end{matrix})\mathbb{Z}_{2}\oplus\sum_{i=0}^{n}(\begin{matrix}n\\i\end{matrix})\Gamma_{i+1}\rightarrow MCG(T^{n})\rightarrow GL(n,\mathbb{Z})\rightarrow0
MSE=\frac{1}{N}+\frac{3(N-1)}{N(N+1)}+f_{3}^{2}\frac{5(N-1)(N-2)}{N((N+1)(N+2)}+\rho_{3}^{2}[\frac{(N-1)(N-2)}{6}]^{2}(1-f_{3})^{2}
C_{2}= \frac{1}{2}\tilde{A}_{\mu}^{\nu}\tilde{A}_{\nu}^{\mu}= \frac{P ( N + P ) ( N - 1 )}{2 N}, \; \tilde{A}_{\mu}^{\nu}- \frac{\delta_{\mu}^{\nu}}{N}\sum_{\lambda}A_{\lambda}^{\lambda}\; .
d(\epsilon) = \sum_{v\in I_1\setminus M}p_v\lg{x^\epsilon_v\over x^*_v}+ \sum_{v\in M\setminus I_1}p_v\lg{x^\epsilon_v\over x^*_v}= \lg (1-\epsilon\chi_f(G))\sum_{v\in I_1\setminus M}p_v + \lg(1 + \epsilon\chi_f(G))\sum_{v\in M\setminus I_1}p_v.
M_{t}^{n,f}={\frac{\sigma_{n}}{n^{3/2}}\sum_{m=1}^{n}\int_{0}^{t}f^{\prime}(\lambda_{m}^{(n)}(X_{s-}^{(n)}))dW_{s}^{m}}+ \frac{1}{n}\int_{0}^{t}\int_{\mathbb{S}(\mathbb{H}_{n}^{1})}\int_{\mathbb{R}}\mathrm{tr}\Big [f(X_{s-}^{(n)}+ry)-f(X_{s-}^{(n)})\Big ]\widetilde{J}_{X}(ds,dr,dy),
S^{(1)}_h(x) =\frac{1}{x^{h+1}}{}_2F_{1}\left(\begin{array}{c}\frac{h}{2}+\frac{1}{2},\frac{h}{2}+1 begin{align*}3pt] h+1\end{array};\,\frac{4}{x^2}\right), T^{(1)}_h(u) =u^{h+1}\sum^{\infty}_{n=0}P^{(h,0)}_{1,k}(-1)(-u^2)^{k}=\frac{u^{h+1}}{1+u^2}.
\sum_{\substack{a,b\in \mathbb{Z}^+ ab=2n b> 3a a\not\equiv b \pmod{2}}}b= 3\sigma(n)-4\sigma\left(\frac{n}{2}\right)-\sum_{\substack{a,b\in\mathbb{Z}^+ ab=2n 1\leq b< 3a a\not\equiv b \pmod{2}}}b.
\sum_{a+b=n}\sum_{0\le i\le a-1}(-1)^{\deg x_1+\dots+\deg x_i+i}\overline{m}_a(x_1,\dots,x_i,\overline{m}_b(x_{i+1},\dots,x_{i+b}),x_{i+b+1},\dots,x_{a+b-1})=0
\begin{gathered}\Psi^{(1)}(g,h)\circ\partial_{sh}^{(0)}=\lambda^{(0)}(g)\circ\lambda^{(0)}(h)-\lambda^{(0)}(gh) \partial_{tg}^{(0)}\circ\Psi^{(1)}(g,h)=\lambda^{(1)}(g)\circ\lambda^{(1)}(h)-\lambda^{(1)}(gh) \end{gathered}
\mu_{1}\ell_{\bf i}+\mu_{2}\ell_{\bf j}+\mu_{3}\ell_{\bf k}+c_{0}u_{0}+c_{1}u_{\bf i}+c_{2}u_{\bf j}++c_{3}u_{\bf k}+d_{1}F^{\prime}_{\bf i}+d_{2}F_{\bf j}+d_{3}F_{\bf k}=0.
(\left. G_{\rho ,1}(\omega_{1})\right\vert_{\Omega},\omega_{1})_{\Omega}=(\left. R_{\rho}P_{\rho}^{-1}\Delta^{2}\omega_{1}\right\vert_{\Omega},\omega_{1})_{\Omega}& =\left( P_{\rho}^{-1}\Delta^{2}\omega_{1},\left[ R_{\rho}\omega_{1}\right]_{\Omega}\right)_{\Omega}.
F_{1,p,\nu}(b_{1},b_{2},b_{3};c_{1};x,y)=\sum_{n,m=0}^{\infty}\frac{(b_{2})_{m}(b_{3})_{n}~B_{p,\nu}(b_{1}+m+n,c_{1}-b_{1})}{~B(b_{1},c_{1}-b_{1})}\frac{x^{m}}{m!}\frac{y^{n}}{n!},
\begin{array}{l@{{}\iff{}}l}g\left( r_1 \right) >f\left( r_1 \right) r_1^{L}-\sum_{i=1}^{L}c_{i}r_1^{L-i}+r_1^{m}-1>r_1^{L}-\sum_{i=1}^{L}c_{i}r_1^{L-1}r_1^{m}-1 >0 r_1>1. \end{array}
\sup_{z,w\in Y_{N,\varepsilon}}\bigg|g_{X_{N},\mathrm{can}}(z,w)-\sum_{\gamma\in S_{\Gamma_{X}}(\delta;z,w)}g_{\mathbb{H}}(z,\gamma w)\bigg|=O_{X_{q_{\mathcal{N}}},\varepsilon,\delta}(1).
\begin{array}{l}{w_{2 i + 1}= x_{i}^{- 1}x_{i + 1}, i = 0 , 1 , 2 , \cdots , N - 1 ,}\\{w_{2 j}= y_{j}, j = 1 , 2 , \cdots , N - 1 ,}\\{w_{2 N}= \prod_{l = 1}^{N - 1}y_{l}^{- 1},}\\ \end{array}
P=\frac{2}{3}[\begin{matrix}cos(\theta)&cos(\theta-\frac{2\pi}{3})&cos(\theta+\frac{2\pi}{3})\\sin(\theta)&sin(\theta-\frac{2\pi}{3})&sin(\theta+\frac{2\pi}{3})\\ \frac{1}{2}&\frac{1}{2}&\frac{1}{2}\end{matrix}]
\Big [ \frac{t_{n}^{a}t_{m}^{a}}{z_{n}- z_{m}}, \frac{t_{m}^{a}t_{\ell}^{a}}{z_{m}- z_{\ell}}\Big ] + \mathrm{p e r m u t a t i o n s c y c l i q u e s}= 0 .
F(x,t) := \sum_k \int d\eta \, \frac{|c(k,\eta)|}{\left(1+|\eta+k^2|\right)^{1-b}}e^{ikx+it\eta}\,, G(x,t) := \sum_k\int d\eta \, |\tilde{v}_\delta(k,\eta)|e^{2\pi i k x + 2 \pi it\eta}\,.
E \Big\{\mathbf{h}_k^H \mathbf{h}_j\Big\}= \sum_{m=0}^{N-1}{\rm sinc}^2\left(am\right){\rm var}\Big\{\mathbf{h}_k^H \mathbf{h}_j\Big\}= \sum_{m_1=0}^{N-1}\sum_{m_2=0}^{N-1}{\rm sinc}^2\left(a(m_1-m_2)\right) &-\left(\sum_{m=0}^{N-1}{\rm sinc}^2\left(am\right)\right)^2
\Psi_{\delta_{i}}^{R}\equiv \Psi_{l_{i}}^{R}, \Psi_{q_{i}}^{R}\qquad \Psi_{l_{i}}^{R}= \left( \begin{array}{c}{{\psi_{\nu_{i}}^{R}}}\\{{\psi_{e_{i}}^{R}}}\\ \end{array}\right) \qquad \Psi_{q_{i}}^{R}= \left( \begin{array}{c}{{\psi_{u_{i}}^{R}}}\\{{\psi_{d_{i}}^{R}}}\\ \end{array}\right)
\tilde{I}\{a_{A i}\}= - \frac{1}{N_{c}- 1}\sum_{i , j = 1}^{N_{+}}\sum_{A < B}^{N_{c}}\ln ( a_{A i}- a_{B j})^{2}+ \frac{N_{c}}{2}N_{+}( \ln N_{c}- 1 ) .
f\left(u; t\right) = \begin{pmatrix}t \mu_1( 1 - u_1) u_1 + (1-t) \frac{\mu_1}{\alpha^2}\left(\alpha - (\alpha u_{1}- d u_2)^+\right)(\alpha u_{1}- d u_2)^+ - k \omega u_1 u_2 \frac{1}{d}\left(t \mu_2( 1 - u_2) u_2 + (1-t) \frac{\mu_2}{d^2}\left(d - (\alpha u_{1}- d u_2)^-\right)(\alpha u_{1}- d u_2)^- - \alpha k \omega u_1 u_2\right) \end{pmatrix}.
\rho_{XX}(t_{1},t_{2})=\frac{K_{XX}(t_{1},t_{2})}{\sigma_{t_{1}}\sigma_{t_{2}}}=\frac{E[(X_{t_{1}}-\mu_{t_{1}})(X_{t_{2}}-\mu_{t_{2}})]}{\sigma_{t_{1}}\sigma_{t_{2}}}
L_{SS}=-{1\over 8\pi}\int d^2 \sigma\sqrt{-h}\bigg(h^{ab}\partial_aX^\mu\partial_bX_\mu +2i \bar{\psi}^{\mu}\gamma^{a}\partial_{a}\psi_{\mu}-i \bar{\chi}_a\gamma^b\gamma^a\psi^{\mu}\big(\partial_bX_\mu-{i\over 4}\bar{\chi}_b\psi_\mu \big)\bigg),
T^{F}( z ) ={\frac{2 \sum_{i , j = 1}^{h}\omega_{i}^{\prime \prime}A_{i j}\overline{\omega}_{j}- 3 \left( \sum_{i , j = 1}^{h}\omega_{i}^{\prime}A_{i j}\overline{\omega}_{j}\right)^{2}}{2 \left( \sum_{i , j = 1}^{h}\omega_{i}A_{i j}\overline{\omega}_{j}\right)^{2}}}.
\varphi (e_k) =e_i,\varphi(W_{<k>}\cap U_+^\perp) \subset U_+^\perp, \varphi(W_{<k>}\cap U_-) \subset U_-, & \varphi(W_{<k>}\cap U_-^\perp)\subset U_-^\perp \mbox{and that}\varphi(W_{<k>}\cap V^\perp)\subset V^\perp.
G_{k_{1}, k_{2}}( z_{1 2}) = ( z_{1 2})^{- h_{1}- h_{2}}\left( D_{( h_{1}; k_{1}) ( h_{2}; k_{2})}- \int^{z_{1 2}}\! \frac{\mathrm{d}\zeta}{\zeta^{1 - h_{1}- h_{2}}}[ G_{k_{1}- 1 , k_{2}}( \zeta ) + G_{k_{1}, k_{2}- 1}( \zeta ) ] \right) \, .
\lim_{\ell \longrightarrow \infty}\frac{\mathcal{N}_{q,P_{V_\omega}(b)}\big( \vert \omega \vert \nu_\ell,\vert \omega \vert r_0 \big)}{\textup{Tr}\,\textbf{\textup{1}}_{[\nu_{\ell},\infty)}\Big( \widetilde{\bf P}_q \vert W \vert \widetilde{\bf P}_q \Big)}= 1.
E(\xi,v)=\frac{v^2}2+\frac{\sigma\xi^2}2, \lambda_2(\xi,v)=\frac{\sigma(a+b)-v^2-\sigma\xi^2+\sqrt{(\sigma(a+b)-v^2-\sigma\xi^2)^2-4\sigma p(0)}}{2\sigma}, p(0)=(v_2\xi_1-v_1\xi_2)^2-(bv_1^2+av_2^2)-\sigma(b\xi_1^2+a\xi_2^2)+ab\sigma.
{\it L}_{g h}= \overline{{c}}^{\mu}\biggl ( g_{\mu \nu}\nabla^{2}+ R_{\mu \nu}- \frac{2 \xi \varphi}{\alpha ( \varphi )}( \nabla_{\nu}\varphi ) \nabla_{\mu}- \frac{2 \xi \varphi}{\alpha ( \varphi )}( \nabla_{\mu}\nabla_{\nu}\varphi ) \biggr ) c^{\nu}
\mbox{minimize}\sum_{l =1}^\mathcal{L}\left( c_{il}(g_{il}) - p_l(\bar s_l)s_{il}\right)\cr\mbox{subject to}\sum_{l=1}^\mathcal{L}g_{il}{\geq}\sum_{l=1}^\mathcal{L}s_{il}, & g_{il}, s_{il}\geq 0,g_{il}\leq \mathrm{cap}_{il}, l = 1,\hdots, \mathcal{L}.
\begin{pmatrix}x_1+x_2 -x_1-x_3 x_2-x_3\end{pmatrix}=-\begin{pmatrix}0 x_1 0\end{pmatrix}+\begin{pmatrix}x_1 0 0\end{pmatrix}+\begin{pmatrix}0 0 x_2\end{pmatrix}+\begin{pmatrix}x_2 0 0\end{pmatrix}-\begin{pmatrix}0 0 x_3\end{pmatrix}-\begin{pmatrix}0 x_3 0\end{pmatrix}.
\int_{\Theta}v^{2}(w)\|(\chi_{w}+\xi_{w})\pi_{F(w)}(L+G)^{\ast}\pi_{(L+G)F(w)}f\|^{2}d\mu(w) \geq \int_{\Theta}v^{2}(w)\|\chi_{w}\pi_{F(w)}(L+G)^{\ast}f\|^{2}d\mu(w) &\geq A_{1}\|K^{\ast}(L+G)^{\ast}f\|^{2}&=A_{1}\|(L+G)^{\ast}K^{\ast}f\|^{2}&\geq A_{1}\|(L+G)^{-1}\|^{-2}\|K^{\ast}f\|^{2}.
=\frac{\frac{1}{2}+\sigma}{2\sigma\tau}\left(-\boldsymbol{z}\cdot\nabla\hat{\boldsymbol{\psi}}_{k}\left(\boldsymbol{z}\right)+\hat{\boldsymbol{\psi}}_{k}\left(\boldsymbol{z}\right)\right)\,\equiv\,\boldsymbol{f}\left(\boldsymbol{z},\,\tau\right)
\textsf{Var}(|F_1|^2) = \mathbb{E}[|F_1|^4] - (\mathbb{E}[|F_1|^2])^2 &= \frac{711 - 640 a - 72 a^2 + a^4 + 396 \log{a}+ 384 a \log{a}+ 72 (\log{a})^2}{18 (\log{a})^4}
\beta_n^{(2)}(\Sigma,\{\Sigma_i\}_{i\geq1},G) = \dim_{L(G)}H_n(L(G) \otimes_{\mathbb{C}[G]}C_\bullet^b(\Sigma))= \dim_{L(G)}H_n(L(G) \otimes_{\mathbb{C}[G]}C_\bullet(\Sigma))
\begin{array}{ll}R_{j;\,m+1}\left(-\dfrac{1}{\tau}, \dfrac{v}{\tau}\right)- e^{\frac{2\pi i (m+1)}{\tau}(2v+1)}\; R_{j;\,m+1}\left(-\dfrac{1}{\tau}, \dfrac{v+1}{\tau}\right) = -2 e^{\frac{\pi i j^2}{2(m+1)\tau}}e^{\frac{2\pi i j v}{\tau}}.\end{array}
0=f\left(\pi(\lambda)^{-1}\left(\sum_{\mu \in \Lambda^0}c_\mu \pi(\mu)^{-1}\tau\right)\right)=\sum_{\mu \in \Lambda^0}c_\mu f(\pi(\lambda)^{-1}\pi(\mu)^{-1}\tau)=c_\lambda \frac{o(G)}{o(\Lambda)}.
\oint_C\mu ds =\;\oint_C\left(\kappa^{4/3}-\frac{5}{9}\kappa^{-8/3}\kappa^2_s+\frac{1}{3}\kappa^{-5/3}\kappa_{ss}\right) ds =\;\oint_C\Big(\kappa^{4/3}-\frac{5}{9}\kappa^{-8/3}\kappa^2_s\Big) ds-\oint_C\Big(\frac{1}{3}\kappa_{s}\Big) d\left(\kappa^{-5/3}\right) =\; \oint_C\kappa^{4/3}ds>0.
\begin{aligned}{\bf{y}}={\bf{W}}_{{\rm{DS}}}^H{\bf{r}}={\bf{W}}_{{\rm{DS}}}^H{{\bf{H}}_{{\rm{DS}}}}{{\bf{F}}_{{\rm{DS}}}}{\bf{s}}+{\bf{W}}_{{\rm{DS}}}^H{\bf{n}}\approx \sqrt{\frac{E}{{{N_{\rm{R}}}}}}\sum\limits_{m = 1}^{{M_{\rm{R}}}}{{{\bf{\Xi}}_{{\rm{DS}},m}}}{\bf{s}}+{\bf{W}}_{{\rm{DS}}}^H{\bf{n}}.\end{aligned}
Z_{m}( A | J ) = \mathrm{e x p}\Bigr \{\! - ( i / \hbar ) z_{Q}^{- 1}\bigr ( J_{i}A^{i}- \mathrm{~ \frac{1}{2}~}m^{2}g_{i j}A^{i}A^{j}\bigr ) \Bigr \}\, \mathrm{e x p}\Bigr \{- m^{2}g_{i j}A^{i}\frac{\delta}{\delta J_{j}}\Bigr \}Z_{m}( 0 | J ) .
\Vert (B_\varepsilon (\vartheta)+\lambda_0 Q_0^\varepsilon)^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}\leqslant \Vert f\Vert^2_{L_\infty}\Vert (\widetilde{B}_\varepsilon (\vartheta)+\lambda_0 I)^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}&\leqslant \lambda_0^{-1}\Vert f\Vert^2_{L_\infty}.
U(1) = \frac{1}{a}U(0) \rightarrow U(1)=0, 2U(2) = \frac{1}{a}U(1) +2 \rightarrow U(2)=1, 3U(3) = \frac{1}{a}U(2) - \frac{1}{a}\rightarrow U(3)=0, \vdots \vdots
\log D_j \geqslant (pq)^{\frac{j-1}{2}}\left(\log D_1+\frac{\log(pq)}{2(pq-1)^2}\left(1-7pq-4p^2q\right)+\frac{\log E_0}{pq-1}\right) &=(pq)^{\frac{j-1}{2}}\log \left(D_1(pq)^{-(4p^2q+7pq-1)/(2(pq-1)^2)}E_0^{1/(pq-1)}\right) &=(pq)^{\frac{j-1}{2}}\log \left(E_1\varepsilon^p\right)
{{\left( \mathbf{x}+\Delta \mathbf{x}\right)}^{T}}\left( \mathbf{s}+\Delta \mathbf{s}\right) +\Delta{{\mathbf{y}}^{T}}\mathbf{Ax}-\Delta{{\mathbf{x}}^{T}}\left({{\mathbf{A}}^{T}}\mathbf{y}+\mathbf{C}\mathbf{x}+\mathbf{s}\right) =\nu{\mathbf{x}}^{T}\mathbf{s}+\Delta \mathbf{x}^T \Delta \mathbf{s}.
F_1(x) = \ln[x+ \alpha(x)] -2\biggl(\ln x+\frac1{2x}-\frac1{12x^2}+\frac1{240x^4}-\frac1{252x^6}\biggr) &- \ln\biggl(\frac1x-\frac1{2x^2}+\frac1{6x^3}-\frac1{30x^5}+\frac1{42x^7}-\frac1{30x^9}\biggr) -\frac1{120x^4}
(T_wf)(x) =(T_aT_{w'}f)(x)=(T_ag)(x)=g(a^{-1}x) &=(T_{w'}f)(a^{-1}x)=I_{w'S}(a^{-1}x)f((w'^{-1})_Ga^{-1}x) &=I_{aw'S}(x)f(((aw')^{-1})_Gx) = I_{wS}(x)\cdot f((w^{-1})_Gx).
\int \prod_{j = 1}^{g}d \nu_{j}d \bar{\nu}_{j}d \hat{\nu}_{j}d \hat{\bar{\nu}}_{j}\mathrm{e}^{\int_{\Sigma_{g}}{\cal L}_{5}}= \det \left({\frac{1}{\Omega - \bar{\Omega}}}\bar{\partial}\bar{\Omega}{\frac{1}{\Omega - \bar{\Omega}}}\partial \Omega \right) ,
\left(\int_{\mathbb{R}^N}\Psi_\varepsilon(U)^{p^*}\,dx\right)^\frac{p}{p^*}\le \frac{1}{\alpha}\,\|U\|^{p^*+\alpha-1}_{\infty}\,|\{U>K_0\}| &+\frac{1}{\alpha}\,\left(\int_{\{U\le K_0\}}U^{p^*}\,dx\right)^\frac{p^*-p}{p^*}\,\left(\int_{\mathbb{R}^N}\Psi_\varepsilon(U)^{p^*}\,dx\right)^\frac{p}{p^*},
\int_{0}^{\infty}d x x^{\nu^{1}}\exp \left[ -{\frac{\beta}{x}}- \gamma x \right] = 2 ( \beta \gamma^{- 1})^{\nu / 2}K_{\nu}( 2 \sqrt{\beta \gamma}) \quad ( \Re \beta > 0 , \Re \gamma > 0 ) ) ,
E_{\xi_{i}}^{*}E_{i}=(E_{\xi_{i}}F_{i})^{*}K_{i}+ (-q^{-1}K_{i}^{-1}E_{\xi_{i + 1}})^{*}K_{i}& =(K_{i}E_{\xi_{i}}F_{i})^{*}- q^{-1}E_{\xi_{i + 1}}^{*}.
U e_{2n}= \begin{cases}e_{2n+1}n \ \textrm{odd}e_{2n-3}n \ \textrm{even}\end{cases}Ue_{2^k n - (2^{k-1}+1)}= \begin{cases}e_{2^k n + (2^k - 2^{k-1})}n \ \textrm{odd}e_{2^k n - (2^k + 2^{k-1})}n \ \textrm{even}\end{cases}
(t_\lambda,L_\lambda,S_\lambda) = \left(\frac{1}{\sqrt{\gamma_0}\lambda^\frac{n}{4}}f(x), \frac{1}{\sqrt{\gamma_1}\lambda^\frac{n+2}{4}}d_xf, \frac{1}{\sqrt{\gamma_2}\lambda^\frac{n+4}{4}}\nabla^2_xf\right)
[D_{w\lambda,\lambda}] =\mathcal{I}_{w}([c_{w\lambda,\lambda}^{\lambda}][c_{\lambda,\lambda}^{\lambda}]^{-1}) D_{w\lambda,\lambda}=(\varphi\circ\mathcal{I}_{w}^{+})([c_{w\lambda,\lambda}^{\lambda}]^{-1}[c_{\lambda,\lambda}^{\lambda}])
g = t^{-2}\cdot a_0 \cdot a_4t^{-3}\cdot t^2 \cdot t^2 \cdot a_0 \cdot t^{2}\cdot a_0 \cdot t^{2}= s_2 \, s_4 \, s_1 \, s_5 \, s_5 \, s_4 \, s_5 \, s_4 \, s_5
I = \frac{1}{1 6 \pi G_{N}}\int d^{4}x \sqrt{- g}\left[ R + a_{1}R^{2}+ a_{2}( 3 R_{\nu}^{\mu}R_{\mu}^{\nu}- R^{2}) + a_{3}( R_{\ \ \rho \sigma}^{\mu \nu}R_{\ \ \mu \nu}^{\rho \sigma}- 4 R_{\nu}^{\mu}R_{\mu}^{\nu}+ R^{2}) \right] ,
\begin{aligned}\Vert \nabla_\epsilon u^\epsilon\Vert_{L^2}+\Vert \nabla_\epsilon v^\epsilon\Vert_{L^2}+\Vert \nabla_\epsilon h^\epsilon\Vert_{L^2}+\Vert \nabla_\epsilon g^\epsilon\Vert_{L^2}&\lesssim \Vert f_1\Vert_{L^2}+\Vert f_3\Vert_{L^2}+\sqrt{\epsilon}(\Vert f_2\Vert_{L^2}+\Vert f_4\Vert_{L^2}).\end{aligned}
h_{k_ii}(x)v_{(m_i-1)i}(x) + h_{(k_i-1)i}(x)v_{m_ii}(x) + m_ih'_{k_ii}(x)v_{m_ii}(x) = a_{t_ii}(x)^{n_i}h_{(k_i-1)i}(x) + (n_ia_{t_ii}(x)^{n_i-1}a_{(t_i-1)i}(x) + (t_in_i-t_i+m_in_i)a_{t_ii}(x)^{n_i-1}a_{t_ii}'(x))h_{k_ii}(x).
\nabla_{\mu}\Sigma_{AA^{\prime}}^{\lambda}=\widetilde{\nabla}_{\mu}\Sigma_{AA^{\prime}}^{\lambda}+T_{\mu \nu}{}^{\lambda}\Sigma_{AA^{\prime}}^{\nu}-(\Gamma_{\mu A}^{{\footnotesize (T)}}{}^{B}\Sigma_{BA^{\prime}}^{\lambda}+).
e^{2 \pi i B_{C}(m+a, b)}f_{-\frac{1}{N\tau}}(m+a) = P_{k-1}(m+a) e^{2 \pi i \left(B_{C}(m+a,b) - \frac{Q_{C}(m+a)}{N\tau}\right)}&= e^{2 \pi i \left(B_{C}(a,b) - \frac{Q_{C}(a)}{N\tau}\right)}P_{k-1}(m+a) e^{2 \pi i \left(B_{C}\left(m, \frac{bN\tau -a}{N\tau}\right) - \frac{Q_{C}(m)}{N\tau}\right)}.
\sigma^{ab}n_{ab}= \frac{1}{r}+ \frac{r}{2}[(\rho +2 \kappa^2+ (div_{\sigma}\eta)^{(0)}] + \frac{r^2}{3}[ D \rho+ (div_{\sigma}\eta)^{(1)}]+ \frac{r^3}{4}\Big [ \frac{1}{2}(D^2 \rho -\frac{2}{3}|\beta|^2) &+ (div_{\sigma}\eta)^{(2)}-\frac{1}{9}|\beta|^2+ \frac{1}{15}|\alpha|^2 \Big ] + O(r^4).
\begin{dcases}\rho_t+\rho_x +u_{x}=0 (0,T)\times (0,1), u_{t}-u_{xx}+u_x +\rho_x =0 (0,T)\times (0,1), \rho(t,0)= h(t) t\in (0,T) , u(t,0)=0, \u(t,1)=0 t\in (0,T) , \rho(0,x)=\rho_0(x),\u(0,x)=u_0(x) x\in(0,1),\end{dcases}
\bar{f}(p) = f(p) + \sum_{i>n}\frac{\xi^{i-1}}{p^i}\delta(x-y) = \sum_{i>0}\frac{\xi^{i-1}}{p^i}\delta(x-y) + a_0(x) + p\,a_{-1}(x) = \frac{1}{p-\xi}\delta(x-y) + a_0(x) + p\,a_{-1}(x),
[\begin{matrix}A&B\\C&D\end{matrix}]^{-1}=[\begin{matrix}A^{-1}+A^{-1}B(D-CA^{-1}B)^{-1}CA^{-1}&-A^{-1}B(D-CA^{-1}B)^{-1}\\ -(D-CA^{-1}B)^{-1}CA^{-1}&(D-CA^{-1}B)^{-1}\end{matrix}]