harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}H^1(G,U_N) \simeq\mathcal{P}_{N/K}/\mathcal{P}_K & \simeq(\mathcal{P}_{L/\mathbb{Q}}/\mathcal{P}_\mathbb{Q}) \times\left((\mathcal{P}_{M/K^+}/\mathcal{P}_{K^+})\cap\ker(N_{M/L})\right) \times\left((\mathcal{P}_{N/K}/\mathcal{P}_K)\cap\ker(N_{N/M})\right)\end{aligned}
\mathcal{V}_2\left( \frac{m}{p_2 (p_1q)^2}\right) = 2\pi i^k \frac{N}{N^{i(t+\nu)}}\int_0^\infty V_2(y)\, y^{-i(t+\nu)}\, e\left( -\frac{N uy}{p_1qQ}\right) J_{k-1}\left( \frac{4\pi}{p_1q}\sqrt{\frac{mNy}{p_2}}\right) \{d}y,
\left(2\lambda K+\frac{4(\lambda-1)}{n}\left(2\left\vert b\right\vert-b\right)\right)\omega^{2}+(2(2-\lambda)\left\vert b\right\vert-\lambda b)p\omega-2\lambda \left\vert b\right\vert p(p-1) \geq -(4-\lambda)p\left\vert b\right\vert \left\vert L\right\vert-2\lambda \left\vert b\right\vert p(p-1).
\Big \langle J_{\mu_{1}}^{( 1 )}( x_{1}) J_{\mu_{2}}^{( 1 )}( x_{2}) . . . . . J_{\mu_{n}}^{( 1 )}( x_{n}) \Big \rangle \; \; \Big \langle J_{\nu_{1}}^{( l_{1})}( y_{1}) J_{\nu_{2}}^{( l_{2})}( y_{2}) . . . . . J_{\nu_{k}}^{( l_{k})}( y_{k}) \Big \rangle \; .
\int_{0}^{1}\left\vert \sum_{j=1}^{n}a_{j}r_{j}\left( t\right) \right\vert^{p}dt=\int_{0}^{1}\left\vert \sum_{j=1}^{n}\delta_{j}a_{j}r_{j}\left(t\right) \right\vert^{p}dt,
\begin{array}{lll}I_2 = \int_{\mathcal{C}^d}|u_{n}(s,X_t(\omega)) - u_{n}(s,X_s(\omega))|^{4}dm^{n}(\omega) \leq L_K \exp(M_{\Lambda}T) \int_{\mathcal{C}^d}|X_t(\omega)-X_s(\omega)|^{4}dm^{n}(\omega) \leq L_K \exp(M_{\Lambda}T)C'|t-s|^{2}\ , \end{array}
\begin{aligned}N_1(T \pm \epsilon,\chi) = \frac{T \pm \epsilon}{\pi}\log{\frac{q(T \pm \epsilon)}{2m\pi}}- \frac{T \pm \epsilon}{\pi}+ O\left( \frac{m^{1/2}\log{(qT)}}{(\log{\log{(qT)}})^{1/2}}\right) &\quad\quad+ O(m^{1/2}\log{q}).\end{aligned}
f_j^{(k)}(\ell,\zeta,\varphi) = \sum_{\substack{|I|=j I \subseteq \{1,\ldots k\}}}e^{t L_I(\ell)}- \sum_{\substack{|I|=|J|=j I,J\subseteq \{n-k+1,\ldots,n\}}}P_{I,J}\left(\zeta,\varphi\right),
I_{14}= \int_{t_q}^T \log{|L(1-b-it,\overline{\chi})|}dt = \int_{t_q}^T \log{|L(1-b+it,\chi)|}dt &= \int_{1-b}^{a_q}\arg{L(\sigma+it_q,\chi)}d\sigma - \int_{1-b}^{a_q}\arg{L(\sigma+iT,\chi)}d\sigma +O(1).
\sin\!\left(\frac{\pi\ln\left(1+q^{m}\epsilon_{m}\right)}{\ln(q)}\right)=\frac{q^{(m+1)m/2}w^{m+1}}{A(q^{-m}+\epsilon_{m})}\frac{(q+q^{m+1}\epsilon_{m};q)_{\infty}}{(q;q)_{\infty}}\, C(w,q^{-m}+\epsilon_{m}).
Z(\lambda, \nu) = \frac1{(2\pi)^{\nu-1}}\int_{-\pi}^\pi \cdots \int_{-\pi}^\pi \exp\Bigg\{\sum_{j=1}^{\nu-1}\exp \big\{{\rm i}x_j\big\}+ \lambda \exp\Big\{-{\rm i}\sum_{j=1}^{\nu-1}x_j\Big\}\Bigg\}\,{\rm d}x_1 \cdots{\rm d}x_{\nu-1}\, .
\sum_{j=1}^{m_1}G\hskip -0.1cm{_{_{_{f_1f_2}}}}\!\!\!(\nabla_{e_j^h}grad(f_1^h),e_j^h)=\!\!\frac{1}{(1-c^2b_1^hb_2^v)}\left\{\left(\triangle_1(f_1)\right)^h\!\!+\!\!\frac{cb_1^hb_2^v}{f_1}(cb_1^h-m_1)\right\}.
y_1 = p_{1}(n)x_2+\dots+p_{d-1}(n)x_s; \vdots = \vdots y_i =p_{1}(n)x_{i+1}+\dots+p_{d-i}(n)x_s; \vdots =\vdots y_{s-2}= p_{1}(n)x_{s-1}+p_{2}(n)x_s; y_{s-1}= p_{1}(n)x_s.
\begin{aligned}\sum_{x \in T_{N}^{-1}(x')}x = \cos\left(\pi u_{0}\right) + 2\cos\left(\pi u_{0}\right)\sum_{j=1}^{\frac{N}{2}-\frac{1}{2}}\cos\left(\frac{2\pi}{N}j\right) = \cos\left(\pi u_{0}\right)\left(1+2\left(\frac{-1}{2}\right)\right) = 0 \ \ \ \square\end{aligned}
\Delta S ( \beta ;{\bf m},{\bf n}) = \frac{\pi}{2}N T_{I I}\beta^{2}[ m_{\ell}\Omega_{\ell i}- n_{i}] ( ( \Im \Omega )^{- 1})_{i j}[ \bar{\Omega}_{j k}m_{k}- n_{j}] = \mathrm{g}^{\mu \nu}( \Omega ){\cal N}_{\mu}{\cal N}_{\nu},
\sum_{i=0}^{n-1}(|\tilde{w}''_i (t)| + |\tilde{w}_i'(t)| +1) \leq K_n\ \textrm{whenever $|t - x_i | \geq \sigma_n$ for all $0 \leq i \leq n-1$}; \shortintertext{and}K_n \geq 1+ K_{n-1}.
\left( \frac{1}{5}, \frac{1}{4}\right] = I(5) = \bigcup_{a_2 = 21}^{\infty}J(5,a_2) = \bigcup_{a_2 = 21}^{\infty}\left( \frac{1}{5}+ \frac{1}{a_2}, \frac{1}{5}+ \frac{1}{a_2-1}\right].
{I}:= [\underline{B},\infty) \times (0,\overline{\rho}] \times(-\infty, \overline{B}] \times (0, \overline{c}] \times (0,\overline{C}] \times [\underline{C'},\infty) \times (0, \overline{C}_X] \times (0, \overline{C}_f].
\begin{aligned}\int_{\Omega}\frac{\delta\mathcal{F}}{\delta v}\wedge\partial v=\lim_{\epsilon\to 0}\frac{1}{\epsilon}\big(\mathcal{F}(v+\epsilon\partial v )-\mathcal{F}(v)\big),\ \forall\partial v\in P^{\ast}\Lambda^{1}(\Omega). \end{aligned}
u_\tau^{k,1}=u_\tau^{k,2}\ \ \Rightarrow\ frac{\tau}{\tau{+}\chi_1}v_\tau^{k,1}+\frac{\chi_1}{\tau{+}\chi_1}u_\tau^{k-1,1}=\frac{\tau}{\tau{+}\chi_2}v_\tau^{k,2}+\frac{\chi_2}{\tau{+}\chi_2}u_\tau^{k-1,2},
r^2 =\lim_{t\rightarrow 0^+}\left( -(s(r,t)-t)^2+\frac{s(r,t)-t}{h(s(r,t))}\right) =\lim_{s\rightarrow 0^+}\frac{s}{h(s)}\lim_{t\rightarrow 0^+}\left(1-\frac{t}{s(r,t)}\right).
{\rm tr}\left[ P\,\Gamma^{,\rho \sigma}\right] =2i\delta^{\rho \sigma}t\,({\rm tr}P)-i\frac{1}{k^{2}}P^{\rho \sigma}\left[ e^{k^{2}(1-\xi^{-1})t}-1\right] +i\frac{1}{k^{2}}P^{\rho \sigma}\left[ e^{k^{2}(-1+\xi^{-1})t}-1\right]
m_{\alpha,\beta}(x|\mu) \equiv f_{\alpha\beta}(x|\mu) = \int_0^\infty f_\alpha(x|y) f_\beta(y|\mu) dy r_{\alpha,\beta}(x) \equiv \rho_{\alpha\beta}(x) = x \int_0^\infty f_\alpha(x|y) y^{-1}\rho_\beta(y) dy
u^{s'}(P)= u^{s}(X)+\nabla u^{s}(X)\cdot (P-X)+ \left(\left( M^{s}(\widetilde{Y}_i)-I \right)\nabla u^{s}(X)\right) \cdot (P-\widetilde{Y}_i) & + R_i(P,X), ~ \; P\in T^{s'}\cap b_i, \; X \in T^{s}_{non},
{}[ \kern - 0 . 4 5 e m \llap [ J_{1}( x ) , \phi^{a b}( y ) ] \kern - 0 . 4 5 e m \llap ] = - \delta ( x - y ) M^{a b}= - [ \kern - 0 . 4 5 e m \llap [ \phi ( x ) , J_{1}^{a b}( y ) ] \kern - 0 . 4 5 e m \llap ~
W^{s,m}_{0}(\Omega) = \{u \in W^{s,m}(\mathbb{R}^N); \,\, u =0 \ \ \mathbb{R}^N\backslash\Omega\}, W^{-s,m'}(\Omega) = \left(W^{s,m}(\Omega) \right)^*,\m'=\frac{m}{m-1}\ (\mbox{dual space}).
B_{\lambda}(x^{\mu}, x^{\nu}) =\dfrac{x^{\mu}y^{\nu}-x^{\nu}y^{\mu}}{x-y}&=\sum_{k=1}^{\mu-\nu}x^{\mu-k}y^{\nu+k-1}=\sum_{k=1}^{\mu-\nu}x^{\lambda-(\lambda-\mu+k)}y^{\lambda-(\lambda-\nu-k+1)},
\lim_{x \uparrow 0}\frac{\partial}{\partial x}\hat{m}(s,x,s+\epsilon,\phi'(s+\epsilon))=-\int_{0}^{\infty}e^{\phi'(s+\epsilon)y-\frac{y^2}{2\epsilon}}\frac{2y}{\sqrt{2\pi \epsilon^3}}\times \mathbb{E}\left[ \left (-\int_{s}^{s+\epsilon}\phi''(u)B(u)du \right ) \vert B(s)=0,B(s+\epsilon)=y\right]dy
u_1(x)= q(2q^3-6q^2+7q-4)x^2-4(q-1)^2(q^{2m}-2q^2+3q-2)x &-4(2q^2-4q+1)q^{2m}+4(2q^4-8q^3+14q^2-12q+3), u_2(x)= q^{2m}x^2-4q^{2m}+4.
\tau_t :=\sup\{r\in(0,\varepsilon_t): g_t(\hat{x}_1,\hat{x}_1+t^{-1/d}(\hat{x}_2-\hat{x}_1)+y_2,\hdots,\hat{x}_1+t^{-1/d}(\hat{x}_k-\hat{x}_1)+y_k)>c/2 y_2,\hdots,y_k\in B^d_r\}& =t^{-1/d}\sup\{r\in(0,\varepsilon): g_1(\hat{x}_1,\hat{x}_2+y_2,\hdots,\hat{x}_k+y_k)>c/2 y_2,\hdots,y_k\in B^d_r\}& =t^{-1/d}\tau_1.
\left[\begin{array}{l}\dot{x}^c \dot{x}^{ss}\dot{x}^u\end{array}\right]=A\left[\begin{array}{l}x^c x^{ss}x^u\end{array}\right]=\left[\begin{array}{lll}B_1 0 0 0 B_2 0 0 0 C\end{array}\right]\left[\begin{array}{l}x^c x^{ss}x^u\end{array}\right],
\zeta^{X_1}[\psi](y)+\zeta^{X_2}[\psi](y)= &\quad=\int_{X_1}G^{X_1}(x,y)\psi(y)\, d_{x\in X_1}+\int_{X_2}G^{X_2}(x,y)\psi(y)\, d_{x\in X_2}= &\quad=\int_{X}E(x,y)\psi(y)\, d_{x\in X}-\int_{X}H^{X,x}(y)\psi(y)\, d_{x\in X_1},
\left(u-u^*\right)(p(u)-v)=\left(u-u^*\right)(p(u)-p(u^*))+\left(u-u^*\right)(v^*-v), begin{align*}0.2cm] \left(v-v^*\right)\left(q(u)-v\right)=\left(v-v^*\right)\left(q(u)-q(u^*)\right)-\left(v-v^*\right)^2.
{f(t_{1},t_{2})t_{1}^{n_{1}-1}t_{2}^{n_{2}-1}dt_{1}dt_{2}=f(t_{1},t_{2})t_{1}^{\frac{\gamma_{1}}{p-1}}t_{1}^{n_{1}-1-\frac{\gamma_{1}}{p-1}}dt_{1}t_{2}^{\frac{\gamma_{2}}{p-1}}t_{2}^{n_{2}-1-\frac{\gamma_{2}}{p-1}}dt_{2}&=g(r_{1},r_{2})r_{1}^{n_{1}-1}r_{2}^{n_{2}-1}dr_{1}dr_{2}}
I(j,\mathbf{k};q)=\frac{w(\mathbf{x}_{j,\mathbf{k}})}{\sqrt{|\det\nabla^2f(\mathbf{x}_{j,\mathbf{k}})|}}(qj)^{-\frac{n-1}2}e\left(-qjf^*\left(\frac{\mathbf{k}}{j}\right)+\frac{\sigma}8\right)+O\left((qj)^{-\frac{n+1}2}\right).
B'\begin{pmatrix}a 0 0 b \end{pmatrix}= V_{2m+2n}W_{2m+2n}B \left( \begin{matrix}a 0 0 b \end{matrix}\right) W_{2m+2n}^* V_{2m+2n}^* = \begin{pmatrix}V_{2m}W_{2m}B(a) W_{2m}^* V_{2m}^* 0 0 V_{2n}W_{2n}B(b) W_{2n}^* V_{2n}^* \end{pmatrix}= \begin{pmatrix}B'(a) 0 0 B'(b) \end{pmatrix}\, .
\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2}){\rm ch}(\widetilde{T_CM}-\widetilde{L_C}+256+W_i)\right\}^{(14)}&=A_1\left\{e^{\frac{1}{24}A_1}\widehat{A}(TX){\rm exp}(\frac{c}{2})-\frac{e^{\frac{1}{24}A_1}-1}{A_1}\widehat{A}(TX){\rm exp}(\frac{c}{2}){\rm ch}(\widetilde{T_CM}-\widetilde{L_C}+256+W_i)\right\}^{(10)}.
\mathcal{M}_{0,\Gamma}(\hat\mu) =(|\hat{\mu}(v_{j})|^{2})_{j=1}^{n}, &\mathcal{M}_{1,\Gamma}(\hat\mu)=(|\hat{\mu}(v_{j})-\hat{\mu}(v_{k})|^{2})_{\{v_j,v_k\}\in E,j<k}, \quad \mathcal{M}_{2,\Gamma}(\hat\mu) =(|\hat{\mu}(v_{j})-i\hat{\mu}(v_{k})|^{2})_{\{v_j,v_k\}\in E,j<k}.
F_{m}(z)=\alpha+\frac{e^{2m\pi i/3}}{(c_{3})^{1/3}}(z-c_{0})^{1/3}+e^{2m\pi i/3}\sum_{n=2}^{\infty}d_{n,m}(z-c_{0})^{n/3},m=0,1,2,
\left(\begin{array}{ccc}\widetilde{v}_{1}\cdots \widetilde{v}_n\end{array}\right)\left(\begin{array}{c}v_{1}^c \vdots v_n^c\end{array}\right)=\widetilde{v}_{1}v_1^c+\cdots+\widetilde{v}_nv_n^c,
\int_0^t P_{t,s}r^m_{1,3}(s)ds = \int_0^t \frac{d}{ds}\big[ P_{t,s}R^m_{1,3}(s)\big] + P_{t,s}(L_s+c\partial_x)R^m_{1,3}(s) ds & = R^m_{1,3}(t) + \int_0^t P_{t,s}(L_s+c\partial_x )R^m_{1,3}(s) ds .
{_2}\phi_{1}\left[ \begin{array}{c}{\begin{array}{c c}{a ,}&{b}\\ \end{array}}\\{c}\\ \end{array}; q , u \right] = \sum_{n = 0}^{\infty}\frac{( a ; q )_{n}( b ; q )_{n}}{( q ; q )_{n}( c ; q )_{n}}u^{n},
\Big\|\sup_{N\ge 1}\big|\sum_{n=1}^N \alpha_n D(\lambda_n t)\big|\, \Big\|_{\S^2}\le C \Big(\sum_{n\ge 1}|\alpha_n|\Big) \Big(\sum_{n\ge 1}\big(\sum_{k\,:\, n\le \mu_k< n+1}|\beta_k|\big)^2\Big)^{1/2}\, .
\det\mathcal{P}(k) =\det\begin{pmatrix}I_1(k+9) I_2(k+9){}{}I_3(k+9) I_4(k+9){}{}{}{}I_5(k+9){}{}{}{}I_6(k+9) \end{pmatrix}&=(k+1)(k+3)(k+4)(k+5)(k+9)^2(k+10)(k+11) &\neq 0
U_{1 .}= \left( \begin{array}{c c c c c c c}{\sqrt{Q}}\\{2}&{0}\\{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}\\{0}&{0}&{0}&{0}&{\sqrt{Q}}\\{Q - \alpha}&{0}&{0}&{0}&{0}&{0}\\ \end{array}\right)
\partial_sM =-\frac{L}{N}\partial_tN+\frac{2M}{N}\partial_sN-\frac{B_t}{B}L+\frac{B_s}{B}M &\qquad-BB_tN-\frac{2B_t}{B}\frac{M^2}{N}+\frac{1}{N}(B^2K)_t, \partial_tM =\partial_sN-\frac{B_t}{B}M.
I_2(w) = \frac{i}{2}\int_{\Delta^2}u_1(z)S_1(w_1, z_1) K_2(w_2, z_2) \frac{|w_2-z_2|^2}{|w-z|^2}dV(z) &~~~+ \int_{\Delta^2}u_1(z)S_1(w_1, z_1) S_2(w_2, z_2) \frac{(w_2-z_2) |w_1-z_1|^2}{|w-z|^4}dV(z) &:=II_1(w) + II_2(w),
\begin{aligned}A^i_t \to A^i_t+A^{N_t+1}_t \mbox{with rate}\quad m\left(X_t^i,X_t^{N_1+1}\right)\gamma\left(X_t^{N_1+1}\right), A^i_t+A^j_t \to A^j_t \mbox{with rate}n^{-1}c\left(X^i_t,X^j_t\right).\end{aligned}
Y^T_{t+1}- Y^{T'}_{t+1}= (1 + \alpha)\big(Y^T_t - Y^{T'}_t\big) - \Big(f\big(\omega,t,Z^T_t\big) - f\big(\omega,t,Z^{T'}_t\big)\Big) &\ \ \ + \big(Z^T_t - Z^{T'}_t\big)^{\ast}M_{t+1}.
-\lim\limits_{\theta_{n}\rightarrow 0}\theta_{n}^{1-2s}\nu \cdot \tilde{\nabla}_{S^n}\theta_n^{\frac{2s-1}{2}}v = \lim\limits_{\theta_{n}\rightarrow 0}\lambda e^{2st}\theta_n^{\frac{2s-1}{2}}v,
\alpha(n,p) - \frac{1}{n-p_+}\left(C_2\delta \right)^{\frac{1}{p+}}\le C_1^{\frac{1}{p+}}\max\{\|\nabla p\|_s^{p_-}, \|\nabla p\|_s^{p_+}\}^{\frac{1}{p+}}+ \frac{p_+}{n}\|\nabla f\|_p.
\vert \mathcal J_{1,1}+\mathcal J_{1,3}\vert \le C_3\left\{\Vert{\mathbf V}\Vert^2_{L^2(\Omega_t)}+\Vert\partial_s{\mathbf V}\Vert^2_{L^2(\Omega_t)}+\Vert\partial_1{\mathbf V}_n\Vert^2_{L^2(\Omega_t)}\right\}&\le C_3\int_0^t(I_{1,\ast}+I_{1,n})(s)ds\,.
K^{(d)}_M(x,t)=\sum_{z\in\prod^{d}_{j=1}m_j\mathbb{Z}}K_{\mathbb{Z}^d}(x+z,t)=\sum_{(z_1,\ldots,z_d)\in\prod^{d}_{j=1}m_j\mathbb{Z}}\prod^{d}_{j=1}K_{\mathbb{Z}}(x_j+z_j,t).
\int_0^\infty{\rm d}r\,\left[u_{1\epsilon\kappa}^2\left(r\right)+u_{2\epsilon\kappa}^2\left(r\right)-v_{1\epsilon\kappa}^2\left(r\right)-v_{2\epsilon\kappa}^2\left(r\right)\right]=2\eta'_{\epsilon\kappa}\left(k\right)+\left(-1\right)^l2\pi\delta\left(k\right)\sin^2\eta_{\epsilon\kappa}\left(k\right)\,,
h_c(G)-h_c(G \backslash v) = f_c(b)-f_c(0)+\sum_{i=1}^b \Delta_c(d_i) \le f_c(b)-f_c(0)+b \cdot \Delta_c\left( \frac{m+\binom{b}{2}}{b}\right) :=LHS(m,b,c).
\mathcal{F}_e(Q)=\int_{\mathbb{R}^d}\frac{L}{2}|\nabla Q|^2+\big(\frac{a}{2}\textrm{tr}(Q^2)-\frac{b}{3}\textrm{tr}(Q^3)+\frac{c}{4}\textrm{tr}^2(Q^2))\,dx
u^+ =\Big(\sum_l\frac{(a_l-sign(l)b_l)}{2}e^{ilt},\sum_l\frac{(b_l-sign(l)a_l)}{2}e^{ilt}\Big)\in Exp^+; u^- =\Big(\sum_l\frac{(a_l+sign(l)b_l)}{2}e^{ilt},\sum_l\frac{(b_l+sign(l)a_l)}{2}e^{ilt}\Big)\in Exp^-.
\omega = \frac{i}{2\pi}\partial \overline{\partial}\log |\mathbf{z}|^2 = \frac{i}{2\pi}\left(\frac{\sum_{j=0}^ndz_j\wedge d\overline{z}_j}{|\mathbf{z}|^2}- \frac{\sum_{j,k=0}^n\overline{z}_jz_kdz_j\wedge d\overline{z}_k}{|\mathbf{z}|^4}\right)
x^{k+1}={\rm Mirr}_{x^k}\left({h\nabla_x f\left({x^k,\xi^k}\right)}\right),{\rm Mirr}_{x^k}\left( v \right):=\arg \mathop{\min}\limits_{x\in Q}\left\{{\left\langle{v,x-x^k}\right\rangle +V_{x^k}\left({x}\right)}\right\},
0 = \delta_{i_1 i_2}[ \sum_{\lambda = 1, \pi \in S_{n}}^{n!}a \delta_{k \pi(i_1}D_{i_2 \cdots i_{n})}+ \sum_{\lambda = 1, \pi \in S_{n}}^{n!}b \delta_{\pi(i_1 i_2}D_{i_3 \cdots i_n) k}].
\frac{\mbox{d}^2}{\mbox{d}t^2}\left[|y+tk|^p-\frac{(2-p)\gamma^{p-1}+\gamma^{p-2}}{p-1}|x+th|^p\right]\Bigg|_{t=0}=p|y|^{p-2}|k|^2+p(p-2)|y|^{p-4}\langle y,k\rangle^2 & -\frac{(2-p)\gamma^{p-1}+\gamma^{p-2}}{p-1}\left(p(p-2)|x|^{p-4}\langle x,h\rangle^2+p|x|^{p-2}|k|^2\right).
\left|r'_\lambda(y',y_d,z_d)\right|+\left|r_{d,\lambda}(y',y_d,z_d)\right| & \leq \frac{C y_d}{(y_d + z_d +|y'|)^{d-1}}\frac{e^{-c |\lambda|^{\frac 12}z_d}}{\big (1+|\lambda|^{\frac 12}(y_d+z_d + |y'|)\big ) \big (1+|\lambda|^{\frac 12}(y_d+z_d)\big )}.
L_t h_{ij}= \Delta_{g_0(t)}h_{ij}+ 2Rm(g_0(t))_{ijkl}h_{kl}-Ric(g_0(t))_{ik}h_{kj}-Ric(g_0(t))_{jk}h_{ki}, R_0[h]= Rm(g_0(t))*h*h + O(h^3)*Rm(g_0(t))+ g^{-1}*g^{-1}*\nabla^{g_0(t)}h * \nabla^{g_0(t)}h, \nabla R_1[h]= \nabla_p^{g_0(t)}\left( \left( ( g_0(t)+h(t))^{pq}- (g_{0}(t))^{pq}\right) \nabla_q^{g_0(t)}h\right),
B_1\theta_1-8B_2B_3\theta_2=0 \Rightarrow\theta_2=\frac{B_1}{8B_2B_3}\theta_1, B_2\theta_1-8B_3B_1\theta_2=0 \Rightarrow\theta_1=\frac{8B_3B_1}{B_2}\theta_2, B_3\theta_1-8B_1B_2\theta_2=0 \Rightarrow\theta_1=\frac{8B_1B_2}{B_3}\theta_2
\sum_{m=n+1}^{\infty}\lambda_m j(l,n,m) = \sum_{m=n+1}^{\infty}\lambda_m h(l,n) + \sum_{m=n+1}^{\infty}\lambda_m g(l,n,m) & = h(l,n) + \sum_{m=n+1}^{\infty}(p(l,i) - p(l, i+1))w(l,i) = l .
ds^2_{TN}(m_1,x_{1}) = (1+ \frac{4m_{1}}{r_{1}})^{-1}(d\psi_{1}+ \cos\theta_{1})^2 + (1+ \frac{4m_{1}}{r_{1}})(dr_{1}^2 + r_{1}^2 d\Omega_{2}^2),
\ker(\varphi^{+}) = \ker \left( \begin{bmatrix}i 0 0 id \end{bmatrix}^{+}\right) begin{align*}1ex] = \ker \left( \begin{bmatrix}i^{+}0 0 id \end{bmatrix}\right) begin{align*}1ex] \cong \ker(i^{+})
\xi^2(x^5)=\frac{2\alpha}{\alpha+\beta}\; e^{-\sqrt{\frac32}\, (\beta-\alpha)\,x^5}\,,\qquad A(x^5)=-\frac{\alpha \,x^5}{{\sqrt{6}}}+ \frac{1}{6}\log \frac{\alpha+\beta-2\alpha e^{- \sqrt{\frac{3}{2}}\, ( \beta-\alpha) \,x^5}}{\beta -\alpha}\,.
\Vert \nabla y - \bar{R}_j\Vert^p_{L^p(U)}\le Cm^{-2}C_m \big( \delta_p + m^{-2}\vartheta_p (\gamma+ \delta_4 + \epsilon \Vert W \Vert_*)\big) \le Cm C^2_m\big( \delta_p + \vartheta_p (\gamma+ \delta_4 + \epsilon \Vert W \Vert_*)\big)
\int_{B_{2}}\left( \kappa_{2}\, u_{\mu \nu}^{0}( x ) T^{\mu \nu}+ \kappa_{2}\sum_{n = 1}^{\infty}\frac{\psi_{n}( R ) e^{2 k R}}{N_{0}}\, u_{\mu \nu}^{n}( x ) T^{\mu \nu}- \frac{\kappa_{1}}{\sqrt{3}}\varphi \, T_{\mu}^{\mu}\right) d z .
\|\nabla u\|_{L^{2}(\Omega)}^{2}= \|\mathrm{div}\, u\|_{L^{2}(\Omega)}^{2}+ \tfrac{1}{2}\|\nabla u - (\nabla u)'\|_{L^{2}(\Omega)}^{2}+ (n - 1) \int_{\partial \Omega}|u|^{2}H_{n}\mathrm{d}S,
\nabla^{2}V(\overline{r})=\frac{-e\delta(\overline{r})}{\epsilon\epsilon_{r}}=-\sum_{\overline{q}}\overline{q}^{2}V_{\overline{q}}e^{i\overline{q}\overline{r}}=\frac{-e}{V\epsilon\epsilon_{r}}\sum_{\overline{q}}e^{i\overline{q}\overline{r}}(8)
\sum_{n=1}^{N}\frac{q^n(q^{n+1})_{N-n}}{(q^{2n};q^2)_{N-n+1}}= \frac{1}{(q)_N}\sum_{n = 1}^{N}\left[\begin{matrix}N n \end{matrix}\right]\frac{(-1)^{n - 1}nq^{n(n + 1)/2}}{1 + q^n}.
\frac{\rho^{r+s}}{1-\rho}\sim\frac{\exp\left(-\log\tfrac{n}{c^2hw}\right)}{1-e^{-cn^{-1/2}}}=O\bigl(n^{-1/2}hw\bigr)=O\bigl(n^{-1/2}(h^2+w^2)\bigr);
\Sigma_1 = \frac{1}{2}\frac{1}{\left(-2 \pi i \tau\right)^r}g_r(\tau) + \frac{1}{2}\sum_{0 <j <r \atop{r \equiv j \pmod{2}}}\frac{\alpha_j}{\left(-2 \pi i \tau\right)^j}g_j(\tau) + \sum_{n \geq 1}(-1)^{n+1}q^{n^2 + \rho_R}\sum_{0 < j \leq r}\frac{\alpha_j}{(-2 \pi i)^j}\sum_{k \geq 1}\frac{1}{(n \tau - k)^j}+ \frac{1}{(n \tau + k)^j}.
J_n (k_1, \cdots, k_n) = \int_{0}^{1}d \tau_1 \cdots \int_{0}^{1}d \tau_n \; \exp \left[ - \frac{i}{2}\sum_{i<j}^n(k_i \times k_j) (2 \tau_{ij}- \epsilon (\tau_{ij})) \right]
M_{L}=\begin{bmatrix}\pi_{1}^{{-1}}\begin{bmatrix}1 1 1\end{bmatrix}\pi_{2}^{{-1}}\begin{bmatrix}1 1 1\end{bmatrix}\pi_{3}^{{-1}}\begin{bmatrix}1 1 1\end{bmatrix}\end{bmatrix}= e [ \pi_{1}^{{-1}}, \pi_{2}^{{-1}}, \pi_{3}^{{-1}}]
\sum_{i=1}^n \frac{\alpha_i}{a\alpha_i+c_i\alpha_{\sigma(i)}}=\frac{1}{a+c_{1}\frac{\alpha_{\sigma(1)}}{\alpha_1}}+\frac{1}{a+c_{2}\frac{\alpha_{\sigma(2)}}{\alpha_2}}+\cdots+\frac{1}{a+c_{n}\frac{\alpha_{\sigma(n)}}{\alpha_n}}&=\frac{1}{a+c_{1}\lambda^{n-1}}+\frac{1}{a+\frac{c_{2}}{\lambda}}+\cdots+\frac{1}{a+\frac{c_{n}}{\lambda}}\leq 1.
\widehat{f}_{0}:= \zeta +\beta , \widehat{f}_{n}:=\zeta +\beta +\sum_{i\in\mathbb N_{\geq 1}:\,\alpha_i\leq\beta_{n}}\mathrm{e}^{-\alpha_i \zeta}P_{i}\left(\zeta \right), n\in \mathbb{N}_{\geq 1},
T_\pi T_0T^{-1}_\pi = T_\pi (t^{-\frac12}q^{-1}x_1\partial_0-t^{\frac12}\partial_0 q^{-1}x_1)T^{-1}_\pi= t^{-\frac12}q^{-1}x_2q\partial_1 -t^{\frac12}q\partial_1 q^{-1}x_2= T_1.
\frac{\mathbb{E}(z_{\beta}^{k})}{\mathbb{E}(z_{\beta}^{k-1})}= \frac{\Gamma(a+1-(k-1)\gamma)\Gamma(b+1-(k-1)\gamma)\Gamma(1-k\gamma)\Gamma(2+a+b-(k-2)\gamma)}{\Gamma(2+a+b-(2k-2)\gamma)\Gamma(2+a+b-(2k-3)\gamma)}
C_{2}\; : = \; \Big \langle \prod_{b = 1}^{N}\prod_{i = 1}^{n_{b}}\overline{{{\psi}}}^{( b )}( x_{i}^{( b )}) P_{+}\psi^{( b )}( x_{i}^{( b )}) \prod_{i = 1}^{m_{b}}\overline{{{\psi}}}^{( b )}( y_{i}^{( b )}) P_{-}\psi^{( b )}( y_{i}^{( b )}) \Big \rangle_{0}
\mathrm{CL}_{(1,2), \ell}(0,0) = \lim_{g \to \infty}\mathrm{Haar}_g(F: \mathrm{coker}(1 - F) = 0, \mathrm{coker}(1 - F^2) = 0 ) &= \lim_{g \to \infty}\mathrm{Haar}_g(F: 1 - F^2 \mod \ell).
\langle W ( C ) \rangle_{\mathrm{l a d d e r s}}= \exp \biggl [ \biggl ( \frac{g^{2}N}{4 \pi}- \frac{g^{4}N^{2}}{8 \pi^{3}}\ln \frac{1}{g^{2}N}+ \cdots \biggr ) \frac{T}{L}\biggr ] , \qquad g^{2}N \ll 1 ;
\sum \beta_0(2-\delta_1,-\delta_2+\delta_3,\delta_4) &\qquad= \beta_0(2,1,1) + \beta(2,-1,1) + \beta(1,1,0) + \beta(1,-1,0) + 2\beta(1,0,0) &\qquad= 2\bigl(\beta(2,1,1) + \beta(1,1,0) + \beta(1,0,0)\bigr).
\partial_a \log Z^{a,b}_{m,n}= E^{a,b}_{m,n}\left[\sum_{i=1}^{t_1}L_{1}(R^1_{0,i})\right] \ge 0 \partial_a \partial_b \log Z^{a,b}_{m,n}= \mathrm{Cov}^{a,b}_{m,n}\left(\sum_{i=1}^{t_1}L_{1}(R^1_{0,i}),\sum_{j=1}^{t_2}L_{2}(R^2_{j,0})\right) \le 0
T=\frac{i\gamma}{2}\sum_{j=1}^{2}\int_{U_{j}}\left(2i(\partial_{\bar{z}}\zeta_{z}-\partial_{z}\zeta_{\bar{z}})+\partial_{t}(\partial_{z}h \partial_{\bar{z}}\chi + \partial_{\bar{z}}h \partial_{z}\chi)-\dot{\chi}\Omega^{2}\right) dz\wedge d\bar{z}
B_{\mu\nu}=A_0(x)\eta_{\mu\nu}+A_1(x)\frac{\eta_{\mu\alpha}\eta_{\nu\beta}(x^\alpha x^\beta)^{d_1}}{l_1^{2d_1}}+A_2(x)\frac{\eta_{\mu\alpha}\eta_{\nu\beta}(x^\alpha x^\beta)^{d_2}}{l_2^{2d_2}}+\cdots +A_n(x)\frac{\eta_{\mu\alpha}\eta_{\nu\beta}(x^\alpha x^\beta)^{d_n}}{l_n^{2d_n}},
u_i v_j =( 3c_i+4 s_{i}+4z_{i})( c_j-4 s_{j}-4z_{j}) &= 3c_i c_j -12{c_i(s_{j}+ z_{j})+4c_j( s_{i}+ z_{i})}-16 s_{i}s_{j}-16s_iz_{j}&\phantom{{}={}}-16s_jz_{i}-16z_{i}z_{j}&= 6t_{i,j}-3c_{i,j}+ 3c_i c_j-16 (s_iz_{j}+ s_jz_{i}+ z_{i}z_{j})
J_{\lambda ,\gamma}\left( W\right) =e^{-2\lambda \left( d+c\right)}\displaystyle\int\limits_{\Omega}\left[ L\left( x,p,W\right) \right]^{2}\varphi_{\lambda}^{2}\left( x\right) dx+\gamma \left\Vert W\right\Vert_{H^{s}\left( \Omega \right)}^{2},
\int_{\mathbb{R}^{n-1}}fd\mu_{\tau,T}=\sum_{v\in\mathbb{Z}^{nd}}f\left(Q\left(v_{1}\right)-Q\left(v_{2}\right),\dots,Q\left(v_{n-1}\right)-Q\left(v_{n}\right)\right)w_{\pm\tau,T}\left(v\right).
s = \bigg ( \begin{array}{c}{r}\\{0}\\ \end{array}\bigg ) \textrm{w i t h s t a b i l i z e r}G = \bigg \{\bigg ( \begin{array}{c c}{1}&{0}\\{0}&{q}\\ \end{array}\bigg ) \bigg | q \in S p ( 1 ) \bigg \}\simeq S U ( 2 )
\begin{array}{rclcl}M+|L| = \sum_{j=0}^{\infty}|q|^{j(j+1)/2-j(k+1/2)}= |q|^{-k^2/2}\sum_{j=0}^{\infty}|q|^{(j-k)^2/2}= |q|^{-k^2/2}(1+2\sum_{\nu =1}^k|q|^{\nu^2/2}+\sum_{\nu =k+1}^{\infty}|q|^{\nu^2/2}) < |q|^{-k^2/2}(1+\tau (|q|))~.\end{array}
W ( x ) = \exp \left( i \frac{x}{\Sigma}\right) \left( \begin{array}{c c c c c c}{\mathrm{}^{\ddots}}&{e^{- 2 \pi i / N}}&{}&{}&{\phi}\\{}&{}&{1}&{e^{2 \pi i / N}}&{}&{}\\{}&{\phi}&{}&{}&{e^{4 \pi i / N}}&{\mathrm{}_{\ddots}}\\ \end{array}\right)
I_1=\pi^{-1}(-r)^{n+m+1}(X;z^{\prime},\bar{z}^{\prime})c_{\alpha}L_{\Omega}(X;z^{\prime},\bar{z}^{\prime}), &I_2=\pi^{-1}(-r)^{n+m+1}(X;z^{\prime},\bar{z}^{\prime})D\big(L_{\Omega}(X;z^{\prime},\bar{z}^{\prime})\big).
\mathcal{Q}_2^{1}\Psi =\frac{1}{\vert \Lambda\vert}\sum_{k\neq 0}\widehat{g}(k)\sum_{i\neq j}e_{k}(x_{j})e_{0}(x_{i})a_{0}a_{k}\Psi &=\frac{1}{\vert \Lambda\vert}\sum_{k\neq 0}\widehat{g}(k)a_{k}^{\dagger}a_{0}^{\dagger}a_{0}a_{k}\Psi.
\mathcal{B}_{3,M}= \ker(\varphi_{3,M}) = \langle e_1+7e_3, e_2 + 5e_3, 13e_3 \rangle, \mathcal{W}_{3,M}= \{-3e_1+\mathcal{B}_{3,M}, 4e_1+\mathcal{B}_{3,M}, -2e_1+\mathcal{B}_{3,M}\}.
\begin{array}{lll}\ast(E_i)=E_i, \ast(F_i)=F_i, \ast(K_{\alpha_i^{\vee}})=K_{-\alpha_i^{\vee}}, \varphi(E_i)=F_i, \varphi(F_i)=E_i, \varphi(K_{\alpha_i^{\vee}})=K_{\alpha_i^{\vee}}, \omega'(E_i)=K_iF_i, \omega'(F_i)=E_iK_{-i}, \omega'(K_{\alpha_i^{\vee}})=K_{\alpha_i^{\vee}},\end{array}
\int_{\mathbb{R}}\frac{P_{m}(a,q;x)P_{n}(a,q;x)}{\gamma\big(a\psi_{a}(x)-\varphi_{a}(x)\big)^{2}+\gamma^{-1}(a-1)^{2}a^{2}\psi_{a}(x)^{2}}\,\mbox{d}x=\frac{\pi}{(a-1)^{2}}\,\delta_{mn},
u_t =\Delta u -\chi \nabla u\cdot \nabla V(u)-\chi u (V(u)-u)+u(a_0(t,x)-a_1(t,x)u-a_2(t,x)\int_{\Omega}u) &\ge \Delta u -\chi \nabla u\cdot \nabla V(u)+u(a_0(t,x)-\epsilon_0\chi -a_1(t,x)u-a_2(t,x)\int_{\Omega}u) &\ge \Delta u-\chi \nabla u\cdot\nabla V(u)+u(a_{0,\inf}-\epsilon_0(\chi+|\Omega|\cdot |a_{2,\sup}|)-a_{1,\sup}u).
n \widehat{\sigma}_s^2 = \| \Pi^{\perp}[\widehat{S}_s] y \|_2^2 \le \| \Pi^{\perp}[\widehat{S}_s] X \beta^* \|_2^2 + \| \Pi^{\perp}[\widehat{S}_s] w \|_2^2 &+ 2 | (\Pi^{\perp}[\widehat{S}_s] X \beta^*)^T w | \,.
\Omega(\mathcal{R}_{1,P})=\begin{cases}E,F,G,H,I, P I(U_2;Y_3)\leq I(U_1;Y_1)+I(X_1,U_2;Y_2|U_1) E,F,G,J, P I(U_2;Y_3)> I(U_1;Y_1)+I(X_1,U_2;Y_2|U_1) \end{cases},
\frac{x \prod_{i=1}^{m}(x-\hat{x}_i)}{\prod_{i=1}^{n}(x-\tilde{x}_i)}=\sum_{i=1}^{n}\frac{\prod_{k=1}^{m}(\tilde{x}_i-\hat{x}_k)}{\prod_{k=1,k\neq i}^{n}(\tilde{x}_i-\tilde{x}_k)}\frac{x}{x-\tilde{x}_i}.
\sum_{r=1}^{m-1}\zeta(s,\frac{r}{m})sin\frac{2\pi rk}{m}=\frac{m\Gamma(1-s)}{(2\pi m)^{1-s}}cos\frac{\pi s}{2}\cdot\{\zeta(1-s,\frac{k}{m})-\zeta(1-s,1-\frac{k}{m})\}
I_K(f;q,t,T;t_0,t_1,t_2,t_3) =I_K\big(\tau_{t_0,t_1,t_2,t_3;t}f;q,t,T t_0t_1t_2t_3/t^2;t/t_0,t/t_1,t/t_2,t/t_3\big).
\int \sqrt{p_{<J}q_{<J}}H^2(p_{=J},q_{=J|<J}) \lesssim \sum_{i=1}^m\sum_{k=0}^{2^J-1}\big(EZ_{i,J,k}^{*}- [EN_{i,J,k}]^{1/2}(2p_{i,J,k}-1)\big)^2 &+ \sum_{i=1}^m\sum_{k=0}^{2^J-1}(p_{i,J,k}-\tfrac 12)^2 +\sum_{i=1}^m\sum_{k=0}^{2^J-1}\sqrt{EN_{i,J,k}^2}\big(p_{i,J,k}-\tfrac 12\big)^4 & =: (i) +(ii) +(iii).
y(q) = \psi(q^3)\left(\frac{\varphi_6\varphi_9^2}{\varphi_3\varphi_{18}}-2q\psi(q^9)\right) = \frac{\varphi_6^2}{\varphi_3}\frac{\varphi_6\varphi_9^2}{\varphi_3\varphi_{18}}-2q\psi(q^3)\psi(q^9) = z(q^3)-2q\psi(q^3)\psi(q^9)
\sum_{n\ge 0}(\sigma_n(x)-\sigma_n(0))w^{n+1}= -\int_0^x V(t)\, dt + \sum_{j,k\ge 0}w^{j+k+2}\int_0^x \sigma_j(t)\sigma_k(t)\, dt & \quad\quad- 2 \sum_{n\ge 0}w^n \int_0^x \sigma_n(t)\, dt .
\left|p_t(y)-p(y)-\frac1{2}tm_2p''(y)-\frac1{6}t^{3/2}m_3p'''(y)\right| &=\frac1{24}t^2\left|\int u^4p''''(\xi_{y,u})q(u)du\right|. &\le \frac{t^2}{24}p(y)\int u^4\left(\sup_{v\colon |v-y|\le \sqrt{t}|u|}\frac{|p''''(v)|}{p(y)}\right)q(u)du &\le Ct^2p(y)\int u^4e^{C\sqrt{t}|u|}q(u)du &\le C^2t^2p(y)\int u^4e^{C|u|-cu^2}du &\le c_1t^2p(y)