harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H|k,\mu\rangle\equiv H({a^{\dagger}}^{(\mu)}(k)|0\rangle)=\sum_{k^{\prime},\mu^{\prime}}\hbar\omega^{\prime}N^{(\mu^{\prime})}(k^{\prime}){a^{\dagger}}^{(\mu)}(k)|0\rangle=\hbar\omega({a^{\dagger}}^{(\mu)}(k)|0\rangle)=\hbar\omega|k,\mu\rangle.
h(t+e^\lambda ) = \int \rho_{y,t+e^\lambda}(x,t) \, d\mu_{t+e^\lambda}(y)= \int \rho_{y,t+2 e^\lambda}(x,t+e^\lambda ) \, d\mu_{t+e^\lambda}(y) & \leq \int \rho_{y,t+2 e^\lambda}(x,t+e^{\lambda_i}) \, d\mu_{t+e^{\lambda_i}}(y)= \int \rho^R_x \, d\mu_{t+e^{\lambda_i}},
F_{\gamma_{BU}}(x) = \left(1-e^{-xd^\alpha_{BU}}\right)^K = \sum^K_{k=0}C^k_K(-1)^ke^{-kxd^\alpha_{BU}}, &f_{\gamma_{BU}}(x) = \sum_{k=1}^KC^k_K(-1)^{k+1}kd^\alpha_{BU}e^{-kxd^\alpha_{BU}},
- \frac{( N + 2 ) v^{\prime \prime}}{( 2 V^{\prime}+ 2 )^{3 / 2}}+ 3 \frac{( N + 2 ) v^{\prime}V^{\prime \prime}}{( 2 V^{\prime}+ 2 )^{5 / 2}}+ v^{\prime}( \lambda - 2 )
\frac{1}{|\mathcal F_{N,k}|}\sum_{(p_1,p_2,\dots ,p_t)}^{(1)}\sum_{f \in \mathcal F_{N,k}}a_f(p_1^{2m_1}p_2^{2m_2}\dots p_t^{2m_t}) = \pi_N(x) (\pi_N(x) - 1) (\pi_N(x) - 2) \dots (\pi_N(x) - (t-1)).
A_1=1+\frac{1}{2\pi}\int \frac{|D_tZ(\alpha,t)-D_tZ(\beta,t)|^2}{(\alpha-\beta)^2}d\beta-\sum_{j=1}^N \frac{\lambda_j}{\pi}Re\Big\{\frac{D_tZ-\dot{z}_j}{c_0^j(\alpha-\omega_0^j)^2}\Big\}
\hat{\Phi}= \hat{B}^{- 1 / 2}\hat{\varphi}\hat{B}^{- 1 / 2}= \hat{B}^{- 1 / 2}\left( \begin{matrix}{{\hat{\varphi}^{R}}}&{{0}}\\{{0}}&{{\hat{\varphi}^{L}}}\\ \end{matrix}\right) \hat{B}^{- 1 / 2}
I_\ell(x) := \sum_{j=1}^{k}\binom k j (-1)^{k+j}\frac{f^{(\ell)}(u(x))}{\ell !}\left(\Delta_{jh}^1 u(x) \right)^\ell1\le \ell <k, I_k (x) := \sum_{j=1}^k \binom k j (-1)^{k+j}\int_0^1 \frac{f^{(k)}\left(u(x)+t \Delta_{jh}^1 u(x)\right)}{(k-1)!}(1-t)^{k-1}dt \left(\Delta_{jh}^1 u(x))\right)^k.
\begin{array}{r l | r l}{L_{2}L_{- 2}\Psi}&{= \ ( 4 h + c / 2 ) \Psi}&{L_{2}L_{- 1}{}^{2}\Psi}&{= \ 6 h \Psi}\\{L_{1}{}^{2}L_{- 2}\Psi}&{= \ 6 h \Psi}&{L_{1}{}^{2}L_{- 1}{}^{2}\Psi}&{= \ 4 h ( 2 h + 1 ) \Psi ,}\\ \end{array}
[H_{i+1},H_{i+1}]=\Bigl\{(\bold{0},\bold{s}AB^{G}A^{T}\bold{t}^{T};\bold{0},\bold{s}A'B^{G}A'^{T}\bold{t}^{T})\colon\bold{s},\bold{t}\in\mathbb{R}^{r}\Bigr\}.
(a)[g_{i,j},g_{k,l}]=0, [g_{i,l},g_{j,k}]=0, 1 \le i < j < k < l \le n, (b)\left[g_{\overline{j_{1}+m}, \overline{j_{k-1}+m}}, \prod_{a=1}^{k-1}g_{\overline{j_{a}+m}, \overline{j_{k}+m}}\right]=0,
1-\mathrm{e}^{t(1/\ell_{j+1}-1)}\geqslant 1-\mathrm{e}^{-(\ell_{j+1}-1)}\geqslant 1-\left(1-(\ell_{j+1}-1)+\frac{(\ell_{j+1}-1)^2}{2}\right) &=(\ell_{j+1}-1)\left(1-\frac{\ell_{j+1}-1}{2}\right)=(pq)^{-j/2}\left(1-\frac{1}{2(pq)^{j/2}}\right) &=(pq)^{-j}\left((pq)^{j/2}-1/2\right)\geqslant \left((pq)^{1/2}-1/2\right)(pq)^{-j}.
\iint_{Q_T}u_x^6dxdt\leq C\int_0^T\|u_x\|_{L^\infty(\Omega)}^4\int_\Omega u_x^2dx dt\leq C\int_0^T\|u_x\|_{L^\infty(\Omega)}^4 dt \leq C\int_0^T\left(\int_\Omega|u_x||u_{xx}|dx\right)^2 dt\leq C\int_0^T\left(\int_\Omega u_x^2dx\right)\left(\int_\Omega u_{xx}^2dx\right)dt \leq C.
L(v) = \int_U \left( \langle W_u (u^{-1}(y)), \phi (u^{-1}(y)) \rangle + W_\xi (u^{-1}(y)) : [\nabla_x \phi] (u^{-1}(y)) \right) dy &= \int_\Omega \left( \langle W_u(x), \phi(x) \rangle + W_\xi(x) : [\nabla_x \phi](x) \right) dx.
\frac{1}{\sum\limits_{j=1}^n y_{j}}\le \frac{1}{\sum\limits_{j=1}^n{\mathrm{\thinspace}\frac{l_{j}^{y}\mu_{1}^{y}}{l_{1}^{y}M_{j}^{y}}}}=\frac{l_{1}^{y}}{\mu_{1}^{y}\sum\limits_{j=1}^n \frac{l_{j}^{y}}{M_{j}^{y}}}
\|f\|^2=\displaystyle \sum_{k \in \mathfrak{D}}\sum_{j=0}^{q-1}|t_{j,k}|^2= \displaystyle \sum_{m \in \mathfrak{D}}\sum_{p=0}^{q-1}|s_{p,m}|^2 = \displaystyle \sum_{v \in{\mathbb{Z}}^2_N}|w_{v}|^2.
( - 1 )^{\frac{N_{o d d}\left( N_{o d d}- 1 \right)}{2}}= ( - 1 )^{m}= ( - 1 )^{\frac{N_{o d d}- 1}{2}+ \frac{N_{o d d}+ N_{e v e n}- 3}{2}}= ( - 1 )^{N_{o d d}+ \frac{N_{e v e n}}{2}}.
c(x) = (x-1)^{p^s - p^{s-t}}g(x) =(x^{p^{s-t}}-1)^{p^t - 1}g(x) &=\left[\sum_{j=0}^{p^{t}-1}\binom{p^{t}-1}{j}(-1)^{(p^{t}-1-j)}x^{j p^{s-t}}\right] g(x).
J_g[B^*B\alpha+\tfrac{1}{2}\Phi^{-1}(c)\beta,\beta]- \Phi^{-1}(c) =J_f^{B,c}[\alpha,\beta]+\Phi^{-1}(c)\,\mathbf{E}\bigg[\frac{1}{2}\int_0^1 e^{-\frac{1}{2}\int_0^t\|\beta_s\|^2ds}\|\beta_t\|^2\,dt+ e^{-\frac{1}{2}\int_0^1\|\beta_t\|^2dt}- 1\bigg] &= J_f^{B,c}[\alpha,\beta],
E_{tot}=\frac{1}{2}m_{1}\dot{x}_{1}^{2}+\frac{1}{2}m_{2}\dot{x}_{2}^{2}+U(r)=\frac{1}{2}(m_{1}+m_{2})\dot{R}^{2}+\frac{1}{2}\mu\dot{r}^{2}+U(r)
\widehat{\Psi}^\pm_{11}(x):=-\frac{1}{4}e^{-ic_\pm}\int_{\pm \infty}^{x}\left[u_y(y) \bar{u}_{yy}(y)+\frac{1}{2i}|u_y(y)|^4\right] dy, &\widehat{\Psi}^\pm_{21}(x)=\frac{1}{2i}\partial_{x}(\bar{u}_{x}(x)e^{ic_\pm(x)}).
(x\cdot y)\diamond\alpha(z) =\varepsilon(y,z)(x\diamond z)\cdot\alpha(y), (x\diamond y)\cdot \alpha(z) -\alpha(x)\diamond(y\cdot z) =\varepsilon(x,y)\big( (y\diamond x)\cdot\alpha(z)-\alpha(y)\diamond(x\cdot z)\big).
\int_M \langle \nabla_Mf(x,t), \frac{\nabla \phi_{k,k+1}^c(x)}{\tau}\rangle_x d\mu_{k+1}^{\tau}(x) = \int_{M}\langle \nabla_Mf(x,t), \nabla (W \ast \mu_{k+1}^{\tau}) \rangle_x d\mu_{k+1}^{\tau}(x).
\begin{cases}\partial_{t}\tau-\partial_{z}v=0 \partial_{t}v+\partial_{z}P_{\kappa}\left(z,\tau\right)=0, \end{cases}\mbox{where}P_{\kappa}\left(z,\tau\right)= \begin{cases}P_{\kappa}\left(\tau\right) \mbox{for}z\in \mathcal{L}P_{g}\left(\tau\right) \mbox{for}z\in \mathcal{G}, \end{cases}
[r_{31},r_{32}] =[r^{ij}X_{j}\otimes 1\otimes X_{i},r^{lk}1\otimes X_{k}\otimes X_{l}]=-r^{ij}r^{lk}X_{j}\otimes X_{k}\otimes \mathrm{ad}^{(r)}_{X_{i}}X_{l}&=- r^{ij}r^{lk}X_{j}\otimes X_{k}\otimes [X_{l},X_{i}]=-r^{ij}r^{lk}f_{li}\,^{s}X_{j}\otimes X_{k}\otimes X_{s},
\chi e^{-ith^{-1}\tilde{P}_h(0)}q_1 e^{-(t_0-t)X}e^{-ish^{-1}\tilde{P}_h(z)}\chi f=\chi e^{-tX}q_1 e^{-(t_0-t)X}e^{-ish^{-1}\tilde{P}_h(z)}\chi f=0.
\sum_{n=0}^{N}\begin{bmatrix}N n \end{bmatrix}_{q^m}\frac{(q^m,q^m)_{n}(q^r)_{N-n}q^{rn}}{(q^r)_{N}(1-q^r)(1-q^{m+r})\cdots (1-q^{nm+r})}&= (1-q^{N+r})\sum_{n=0}^{N}\begin{bmatrix}N n\end{bmatrix}_{q^m}\frac{(q^m,q^m)_{n}q^{mn^2+2rn}}{[(1-q^r)(1-q^{m+r})\cdots(1-q^{nm+r})]^2}
\langle f_{\underline{i}}, e_{\underline{j}}\rangle = \frac{q^{- k (k - 1)}}{[k]!^2}\sum_{p, p^{\prime}\in S_{k}}(-q)^{\|p\| + \|p^{\prime}\|}\langle f_{p(i_{k})}\otimes \cdots \otimes f_{p(i_{1})}, e_{p^{\prime}(j_{1})}\otimes \cdots \otimes e_{p^{\prime}(j_{k})}\rangle.
\begin{array}{l l l}{H \left| \Omega ( k , k + 1 ) \right\rangle}&{=}&{\left( Q + Q^{- 1}\right) \left| \Omega ( k , k + 1 ) \right\rangle + \left| \Omega ( k - 1 , k + 1 ) \right\rangle}\\{}&{}&{}\\{}&{+}&{\left| \Omega ( k , k + 2 ) \right\rangle}\\ \end{array}
{\cal L_{B C S}}= \psi_{\uparrow}^{\dagger}( i \partial_{t}- \epsilon ( p ) ) \psi_{\uparrow}+ \psi_{\downarrow}^{\dagger}( i \partial_{t}- \epsilon ( p ) ) \psi_{\downarrow}- \lambda \psi_{\uparrow}^{\dagger}\psi_{\downarrow}^{\dagger}\psi_{\downarrow}\psi_{\uparrow},
u_{3}(x,z_{1},z_{2},z_{3})=-\frac{\partial V_{2}}{\partial x_{2}}g_{2}(x_{2})-k_{3}(z_{3}-u_{2}(x_{2}))+\frac{\partial u_{2}}{\partial x_{2}}(f_{2}(x_{2})+g_{2}(x_{2})z_{3})
\varphi_k = 1 \quad\hbox{on}\{w\in P: \hbox{dist}(w, H_i\cap H_j) \leq 1/k\}\cap B_{r/2}(q), \varphi_k = 0 \quad\hbox{on}\{w\in P: \hbox{dist}(w, H_i\cap H_j) \geq 2/k\}\cup B^c_{r/2+1/k}(q),
\tilde{\partial}{}_{\alpha \beta}= \frac{\partial}{\partial{\tilde{x}^{\alpha \beta}}}, \quad D_{\alpha}^{( \tilde{\tau})}= \frac{\partial}{\partial \tilde{\tau}^{\alpha}}+ \frac{i}{2}\tilde{\tau}^{\beta}\frac{\partial}{\partial{\tilde{x}^{\alpha \beta}}},
\left(\textstyle\sum_i(\theta_t^i)^{\frac{1}{\gamma}}\right)^\gamma\left(\hat{\phi}+\tfrac{\eta}{\gamma}\,\hat{x}_t\right)^{-\gamma}\!= \left(\textstyle\sum_i \theta_t^i\delta^i\right)\left(\textstyle\sum_i(\theta_{t+1}^i)^{\frac{1}{\gamma}}\right)^\gamma\left(\hat{\phi}+\tfrac{\eta}{\gamma}\,\hat{x}_{t+1}\right)^{-\gamma}f'(k_{t+1}).
\prod_{i}\frac{1-ty_i z}{1-y_i z}\prod_{i,j}\frac{1-t x_i y_j}{1-x_i y_j}=\sum_{k,l}(-t)^lz^{k+l}h_k(y)e_l(y)\sum_{\nu}S_{\nu}(x;t)s_{\nu}(y).
(i^{k+1}z^{2^{k+1}-1})\begin{pmatrix}P_k(z^2)/\sqrt{2^{k+1}}Q_k(z^2)/\sqrt{2^{k+1}}\end{pmatrix}= g(z^{2^k})\cdots g(z) \begin{pmatrix}1 0 \end{pmatrix}.
\begin{aligned}\frac{\|A(x)|{E(x)}\|}{\|A(x)|{F(x)}\|}=\frac{\|A(x)v\|\cdot\|A(x)u\|}{\|A(x)u\|\sp2}&=\frac{|\det(A(x))|}{\|A(x)u\|\sp2\sin(\angle(A(x)v,A(x)u))}&<1.\end{aligned}
S(z,z)=\frac{2\mathrm{Im}(z) L(L-1)}{\pi}\frac{|1-z^2|^{L-2}}{(1-|z|^2)^{L+1}}\int_{\frac{2|\mathrm{Im}(z)|}{|1-z^2|}}^1(1-t^2)^{(L-3)/2}\mathrm{d}t\left(1-I_{|z|^2}(N-1,L+1)\right),
\inf_{\substack{\varphi(t) \in C_{0T}([0,T], \mathbb{R}^d) \varphi(0)=x_0}}\biggl \{S_{0T}^{\mathcal{K}}(\varphi(t))\, \Bigl \vert \,\varphi(t) \in D_{-\delta}\cup \partial D_{-\delta}, \,\, \forall t \in [0, T] \biggr\},
\begin{array}{r c l}{{\overline{{{K}}}( z + 1 )}}&{{=}}&{{- g \overline{{{K}}}( z ) g ,}}\\{{\overline{{{K}}}( z + \tau )}}&{{=}}&{{- h \overline{{{K}}}( z ) h \times \exp \left\{- 2 \pi \sqrt{- 1}\left( z + \frac{\tau}{2}+ c \right) \right\},}}\\ \end{array}
\|w\|^2 -\lambda \int_{-1}^1f(x)|w_1(x, 0)|^p|w_2(x,0)|^q dx -\int_{-1}^1 g_1(w(x, 0))w_1(x, 0) dx &-\int_{-1}^1 g_2(w(x, 0))w_2(x, 0) dx=0.
(d_0\circ \frak m^{\epsilon}_{1;\alpha_+,\alpha_-}+ \frak m^{\epsilon}_{1;\alpha_+,\alpha_-}\circ d_0)(h)={\rm ev}_{+}!({\rm ev}_{-}^* h;\partial\widehat{\frak S^{+ \epsilon}}(\alpha_-,\alpha_+)).
\int_\Omega (f(t,x)-f^{in}(x)) \varphi(x)\ \mathrm{d}x +\int_0^t \int_\Omega f(s,x) \nabla[ a f + b g ](s,x) \cdot \nabla\varphi(x)\, \mathrm{d}x\mathrm{d}s =0
B=\left (\begin{array}{cc}\mathcal V_0(h_1, \mathbf t^1, d) \mathcal V_{1}(h_2, \mathbf t^1, d) \mathcal V_{1}(h_1, \mathbf t^2, d) \mathcal V_{0}(h_2, \mathbf t^2, d) \end{array}\right)
\rho :=\left[\Gamma^2 -\frac{\nu^2}{\kappa^2}\Big(\sum_{i\in I\smallsetminus J}\frac{\lambda_i}{2-\lambda_i}\Big)^{-1}\Big(\frac{1+\varepsilon}{1 -\varepsilon}\Big)^{|J|}\right]_+^{\frac{1}{2m}}<1.
|\tilde{\Phi}(\xi_2' ,\eta_2')| := \Bigl| \xi_1'{\xi_2'}^2 + \eta_1'{\eta_2'}^2 - \frac{{\xi_1'}^3 +{\eta_1'}^3}{4}\Bigr|\leq 2^4 A^{-1}N_1^3 , |\tilde{F}(\xi_2' ,\eta_2')| := \Bigl| \frac{3}{2}\, \xi_1' \, \eta_1' + 2 \, \xi_2' \, \eta_2' \Bigr|\leq 2^4 A^{-1}N_1^2,
\overline{R}_{ijkl}=\kappa_2(\delta_{ik}\delta_{jl}- \delta_{il}\delta_{jk})+(\kappa_1-\kappa_2)\left( T_iT_k\delta_{jl}+T_jT_l\delta_{ik}-T_iT_l\delta_{jk}- T_jT_k\delta_{il}\right).
\delta=\frac{1}{4\pi}\iint_{R_{st}}(x^{2}+y^{2}+z^{2})^{-\frac{3}{2}}|\begin{matrix}x&y&z\\ \frac{\partial x}{\partial s}&\frac{\partial y}{\partial s}&\frac{\partial z}{\partial s}\\ \frac{\partial x}{\partial t}&\frac{\partial y}{\partial t}&\frac{\partial z}{\partial t}\end{matrix}|dsdt.
I_{m, \alpha, \beta}(a, b) = - \frac{1}{\alpha}J_{m - 1, \alpha, \beta}(a, b) + \frac{\beta}{\alpha}I_{m - 1, \alpha, \beta}(a, b) + \frac{m - 1}{\alpha}I_{m - 2, \alpha, \beta}(a, b).
J^{(N)}(n; 1, \ldots, 1) = 2\mbox{i}^{1-2N}\pi^{n/2}\;\Gamma\left(N-{\textstyle{n\over2}}\right) \;\frac{m_0^{n-N-1}}{\sqrt{\Lambda^{(N)}}}\;\begin{array}{c}{}\int\limits \ldots \int\limits{}^{\Omega^{(N)}}\end{array}\frac{\prod \mbox{d}\gamma_i}{\gamma_N^{n-N}}\;\delta\left( \sum\gamma_i^2 -1\right) ,
R_{1212}=R_{1414}=R_{2323}=R_{3434}=R_{1223}=R_{1241}= R_{4134}=R_{1234}=R_{2334}= R_{2341}, R_{1313}=R_{2424}=R_{1324}=R_{1213}=R_{1224}=R_{1431}= R_{2441}=R_{2423}=R_{2331}=R_{1334}=R_{2434}=0.
a(x, y, s)\in S^{1}_{{\rm cl\,}}\left(M\times M\times(0, \infty)\right), &\mbox{$a(x, y, s)\sim\sum\limits^\infty_{j=0}a_j(x, y)s^{1-j}$ in $S^1_{1,0}\left(M\times M\times(0,\infty)\right)$}, &a_j(x,y)\in C^\infty(M\times M) \quad, &a_0(x,x)=\frac{1}{2}\pi^{-n}\quad,
\hat{\rm H}_{\rm F}^{\rm phys}= \hat{\rm H}_{0,+}+ \hat{\rm H}_{0,-}-\frac{1}{2}(\xi_{+}+ \xi_{-})- \frac{\pi}{\rm L}{\hbar}(1+{\rm N}^2)([\frac{e_{+}b{\rm L}}{2{\pi}{\hbar}}])^2
f_\infty(\mu) = \int_X \bigwedge_{j=1}^k d(x,\mu_j) P( x) + \lambda r(\mu) = \int_X \bigwedge_{j=1}^k |z-\mu_j(t) |^2 P( x) + \lambda \sum_{j=1}^k \| \partial^2\mu_j\|_{L^2}^2.
\phi^g=\frac{1}{2}({\cal P}^2+m^2-iqN_\theta\xi_\mu\xi_\nu{\cal F}^{\mu\nu})\approx 0,\quad\varrho^g=i\epsilon_{\mu\nu\lambda}{\cal P}^\mu\xi^\nu\xi^\lambda-2mN_\theta\approx 0,
(\mathcal P_1{\mbox{\bf a}},\mathcal P_2{\mbox{\bf b}})(x)=(\tilde\Psi/\Phi')(z)=\tilde\Xi(z),(\mathcal C_1 \bar{\mbox{\bf a}},\mathcal C_2\bar{\mbox{\bf b}})(x)=(\Psi/\bar\Phi')(\bar z)=\Xi (\bar z)\mbox{on}\,\,\tilde \Gamma.
\mathbf{x}^{\mathrm{RF}}_l = \mathbf{W}^{\mathrm{RF}}\mathbf{x}^{\mathrm{BB}}_l = \mathbf{W}^{\mathrm{RF}}\mathbf{W}^{\mathrm{BB}}_l \mathbf{s}_l = \mathbf{W}_l \mathbf{s}_l,
\varphi\left(\begin{array}{c c}a b 0 d \end{array}\right) = \left(\begin{array}{c c c}a^{4 \, p^{e_1 - 1}}c_1 \, a^{4 \, p^{e_1 - 1}}\, b^{p^{e_1}}\, d^{p^{e_1}}c_2 \, a^{4 \, p^{e_1 - 1}}\, b^{p^{e_1}}\, d^{p^{e_1}}0 d^{2 \, p^{e_1- 1}}0 0 0 d^{2 \, p^{e_1- 1}}\end{array}\right) .
\int_0^T \langle C_{n+1 n}u_{n}, u_n \rangle \,dt= \Big[\langle A^{-1/2}u_n^{\prime}, A^{1/2}u_{n+1}\rangle - \langle u_n, u_{n+1}^{\prime}\rangle &\qquad+ \langle C_{n n-1}w_{n-1}, u_{n+1}\rangle\Big]_0^T - \int_0^T \langle C_{n n-1}w_{n-1}, u_{n+1}^{\prime}\rangle \,.
T = \tilde{O}\left( \left[ \frac{2^\frac{r^2 + 1}{r}\sigma_q \beta}{\mu_r^{1/r}}\cdot \frac{1}{\varepsilon^{\frac{(r-1)}{r}}}\right]^\frac{1+\kappa}{\kappa}\right), T_k = \tilde{O}\left( \left[ \frac{\sigma_q \beta 2^{(1+r)}}{\mu_r R_0^{r-1}}2^{k(r-1)}\right]^\frac{1+\kappa}{\kappa}\right).
R_{\bar D}(\theta) = \begin{cases}(0 \leq \theta \leq \frac{\pi}{4}) \theta ( \frac{\pi}{4}< \theta < \frac{3}{4}\pi) ( \frac{3}{4}\pi < \theta < \frac{5}{4}\pi) \theta (\frac{5}{4}\pi < \theta < \frac{7}{4}\pi ) ( \frac{7}{4}\pi \leq \theta <2 \pi)\end{cases}
d S^{2}= \frac{- \rho^{2}g_{\infty}^{2 \gamma}d t^{2}}{[ \rho^{2}+ 2 \rho \mu_{0}\cosh \alpha + \mu_{0}^{2}]}+ g_{\infty}^{2 \gamma}\bigl [ d r^{2}+ r^{2}d \Omega_{D - 2}^{2}\bigr ]
I_{1}+\frac{I_{2}}{e^{\zeta\omega_{n}t_{3}}}+\frac{I_{3}}{e^{\zeta\omega_{n}t_{3}}}=I+\frac{2I}{K}+\frac{I}{K^{2}}=1,K=e^{\zeta\pi/\sqrt{1-\zeta^{2}}}
\begin{array}{ll}T'_{i, \epsilon}=(T''_{i, -\epsilon})^{-1}\omega\circ T'_{i, \epsilon}\circ \omega=T''_{i, \epsilon}, \ast\circ T'_{i, \epsilon}\circ \ast=T''_{i, -\epsilon}, \overline{(\cdot )}\circ T^{\#}_{i, \epsilon}\circ\overline{(\cdot )}=T^{\#}_{i, -\epsilon}\end{array}
\int_{0}^{R_{0}}\mu\xi(\mu,t_{0})\d\mu= \int_{0}^{R_{0}}\mu\xi^{\mathrm{in}}(\mu)\d\mu-\int_{0}^{t_{0}}\int_{R_{0}}^{\infty}\int_{0}^{\mu}\nu\Lambda(\mu,\nu)\xi(\mu,s)\xi(\nu,s)\d\nu d\mu ds.
\int_{C}\int_{C}|g(t)| |K(t,s)| |f(s)| ds dt\leq \int_{C}\frac{|g(t)|}{2\alpha}\left(\int_{a}^{t}\frac{ds}{1-s+t}+ \frac{1}{\lambda_{0}(-t)^{1-\alpha}}\int_{-\infty}^{t}\frac{ds}{(1-s+t)^{3}}\right)dt \leq\frac{\log(1+d-a)+\frac{1}{2\lambda_{0}(-a)^{1-\alpha}}}{2\alpha}
x\otimes E_{o_+}(\lambda n^{-1})j_{P'}^+(E^{P'}_{o_{+,P'}}(w)) = x\otimes j_P^-(E_{o_{+,P}}^P(n^{-1}wn))E_{o_+}(\lambda n^{-1}) & = x\sigma(E_{o_{+,P}}^P(n^{-1}wn))\otimes E_{o_+}(\lambda n^{-1}) & = x(n\sigma)(E_{o_{+,P'}}^{P'}(w))\otimes E_{o_+}(\lambda n^{-1}).
1=\frac{1}{|I|}\int_I \omega(x)^{\frac1p}\omega(x)^{-\frac1p}dx\leq \left(\frac{1}{|I|}\int_I \omega(x) dx\right)^{\frac1p}\left(\frac{1}{|I|}\int_I \omega(x)^{-\frac{1}{p-1}}dx\right)^{\frac{1}{p'}}\leq C
- 4 m^{d + 1}\frac{q ( \vec{x})}{| \vec{v}( \vec{x}) |}\sum_{m = 1 , 3 , 5 , . .}^{d / 2}{\binom{\frac{d}{2}}{m}}( \partial_{i}\hat{n}^{a}\partial_{i}\hat{n}^{a})^{( \frac{d}{2}- m )}| \vec{v}( \vec{x}) |^{m}
J_{\nu-1}^{2}(ix)-e^{2\theta}J_{\nu}^{2}(ix)\sim{(ix)^{2(\nu-1)}\over 2\pi}\alpha_{\nu}e^{2\alpha_{\nu}}e^{-2\nu \log(\nu+\alpha_{\nu})}\Bigr[\Sigma_{1}^{2}A_{\theta}(t)+\Sigma_{2}^{2}+2t \Sigma_{1}\Sigma_{2}\Bigr],
h_R(x_1^{i_1}x_2^{i_2}\cdots x_n^{i_n})=\sum_{k=1}^n(-1)^{|x_1^{i_1}\cdots x_{k-1}^{i_{k-1}}||x_k|}\bigtriangleup(i_k)y_kx_1^{i_1}\cdots x_{k-1}^{i_{k-1}}x_k^{i_k-1}x_{k+1}^{i_{k+1}}\cdots x_n^{i_n}.
\limsup_{n\to\infty}\left( [R^n]_{i,j}\right)^{\frac{1}{n}}= \limsup_{n\to\infty}\left( [R_{11}^n]_{i,j}\right)^{\frac{1}{n}}= e^{-\bar{\theta}}, &\limsup_{n\to\infty}\left( [R^n]_{i,k}\right)^{\frac{1}{n}}\le e^{-\bar{\theta}}.
f_n^{(1\bmod{8})}(\sqrt{2}) = P_{2,1}(n) = \Phi_{8n,1}(\sqrt{2}), f_n^{(3\bmod{8})}(\sqrt{2}) = P_{2,3}(n) = \Phi_{8n,3}(\sqrt{2}), f_n^{(1\bmod{12})}(\sqrt{3}) = P_{3,1}(n) = \Phi_{12n,1}(\sqrt{3}), f_n^{(5\bmod{12})}(\sqrt{3}) = P_{3,5}(n) = \Phi_{12n,5}(\sqrt{3})
H(Y_j | Y^{j-1}, \underline S) = & \sum_{\underline s}\sum_{y^{j-1}}P_{Y^{j-1}, \underline S}( y^{j-1}, \underline s) H( P_{Y_{j}| Y^{j-1}, \underline S}( 1 | y^{j-1}, \underline s) ).
\sum_n \frac{\overline{\chi}(n)}{n^{\frac{1}{2}}}V \left( \frac{Tn}{q}\right) = L \left( \frac{1}{2},\overline{\chi}\right) - \epsilon(\overline{\chi}) \sum_n \frac{\chi(n)}{n^{\frac{1}{2}}}F \left( \frac{n}{T}\right),
R_{m n}= ( \Gamma_{+}^{- 1})_{m n}- \frac{1}{2}\frac{m}{\sqrt{m^{2}+ \mu^{2}}}\delta_{m n}\approx \frac{1}{\pi}m n ( - 1 )^{m + n}\frac{\sin ( \pi m y ) \sin ( \pi n y )}{\omega_{n}\omega_{m}( \omega_{m}+ \omega_{n}) \sqrt{U_{m}U_{n}}},
\frac{1}{r^2-1}\frac{2}{i}\int_{|w|=1}\frac{w^{k}}{w^{2}-2aw+1}dw = \frac{1}{r^2-1}2\pi i \frac{2}{i}\frac{w_{-}^{k}}{w_{-}- w_{+}}= \frac{1}{r^2-1}4\pi \left(\frac{1-r}{1+r}\right)^{k}\frac{r^2-1}{4r}= \frac{\pi}{r}\left(\frac{1-r}{1+r}\right)^{k}
ds^{2}=\epsilon_{AB}\epsilon_{A^{\prime}B^{\prime}}\mathbf{e}^{AA^{\prime}}\otimes\mathbf{e}^{BB^{\prime}}=\epsilon_{AB}\epsilon_{A^{\prime}B^{\prime}}\theta^{AA^{\prime}}\otimes\theta^{BB^{\prime}}=\epsilon_{AB}\epsilon_{A^{\prime}B^{\prime}}\mathbf{dx}^{AA^{\prime}}\otimes\mathbf{dx}^{BB^{\prime}}.
V'_{k_{\ell}x}(M)=\frac{1}{4\pi r_\ell}\int\limits_{\partial B_{r_\ell}({O})}\, V'_{k_{\ell}x}(P)\frac{r^2_{\ell}-\|{O}M\|^2}{\|M P\|^3}\, dm_{2}(P),
\lim_{q\to 0}[\theta_{k_1,k_2}(q,u,q^{-1}v)]_{(k_1,k_2)\in H}=[\theta_{k_1,0},\theta_{k_1,1}]=[\underset{k_2=0}{\underbrace{\vartheta_{k_1}}},\underset{k_2=1}{\underbrace{(T^*_{\tau_{12}}\vartheta_{k_1})v}}].
\mathfrak{g}^{(l)}\ni v\mapsto\left(\left.\frac{\partial U_{t}^{u}}{\partial u}\right|_{u=0}\right)^{-1}\cdot v=\left.\frac{d}{d\eta}\right|_{\eta=0}\log\left({\rm e}^{U_{t}^{0}+\eta v}\otimes{\rm e}^{-U_{t}^{0}}\right)\in\mathfrak{g}^{(l)}.
g_{0}^{o}\left( v\right) =1,g_{n+1}^{o}\left( v\right) =\left(v+3n+1\right) g_{n}^{o}\left( v\right) -n\left( 2n-1\right) g_{n-1}^{o}\left( v\right) , g_{0}^{e}\left( v\right) =1,g_{n+1}^{e}\left( v\right) =\left(v+3n+2\right) g_{n}^{e}\left( v\right) -n\left( 2n+1\right) g_{n-1}^{e}\left( v\right) .
d_{S(T,t)}(u^t,u'u^{i-2}xx_{i+1}\cdots x_t) >d_{S(T,t)}(u^{t},u(u')^{t-1})+d_{S(T,t)}(u'u^{t-1},u'u^{i-2}xx_{i+1}\cdots x_t) &>d_{S(T,t)}(u'u^{t-1},u'u^{i-2}xx_{i+1}\cdots x_t) &=d_{S(T,t)}(u^t,w),
{\cal L}= -{\frac{1}{4}}F_{\mu \nu}F^{\mu \nu}\, +{\frac{\kappa}{4}}\epsilon^{\mu \nu \alpha}A_{\mu}F_{\nu \alpha}\, +{\frac{1}{2}}| D_{\mu}\Phi |^{2}- \frac{m^{2}}{2}| \Phi |^{2}\, ,
1 + \sum_{v^*}\!\Bigg(\prod_{s\ge 1}\frac{q^{\langle v^s\!,\,v^s\rangle}}{q^{\langle \beta^s\!,\, \beta^s \rangle}}\frac{c(v^s, q)}{\mathit{gl}(v^s, q)}X^{sv^s}\Bigg) = \,\mathrm{Exp}\Bigg(\sum_{v\in\mathbb{N}^n\backslash\{0\}}\! \frac{a(v,q)}{q-1}X^v\Bigg),
|E(G)| \,>\, \frac{k^2}{6}\left(\gamma^2+4\gamma-2\right) \,=\,{\gamma k \choose 2}- \frac{1}{3}((\gamma k-k)^2 - 1) +\frac{\gamma k}{2}- \frac{1}{3}\,>\,{\gamma k \choose 2}- \frac{1}{3}((\gamma k-k)^2 - 1),
(\vec U, \vec V)_{\Omega}= \vec U^TP_{\Omega}\vec V \approx \int\limits_{\Omega}U^T V d \Omega, (\vec U, \vec U)_{\Omega}= \| \vec U\|^2_{\Omega}= \vec U^TP_{\Omega}\vec U \approx \int\limits_{\Omega}U^T U d \Omega = \| U\|^2_{\Omega}.
|k|x\left|\int_2^x\frac{e^{i\xi}-1}{\xi}\frac{1}{\xi-kx}~d\xi\right| \le C|k|x\left|\int_2^x\frac{1}{\xi(\xi-|k|x)}~d\xi\right| &=C\left|\log\left(\frac{2(1-|k|)}{2-|k|x}\right)\right| &\le C\log[(1+|k|)(1+|k|x)] &\le C|k|^{\epsilon}(1+|x|)^{\epsilon}.
\Phi_{k}( \sigma_{1}, \sigma_{2}) = \sum_{p \in{\bf Z}}e^{2 \pi i ( \sigma_{2}/ 2 \pi + k ) p / n}e^{2 \pi i ( \sigma_{1}/ 2 \pi - p / m ) m / n}\widehat{\psi}_{p}( \sigma_{1}/ 2 \pi - p / m ) ,
\begin{array}{l c l}{{{\mathcal L}_{3}^{( 0 , 3 )}}}&{{=}}&{{p^{3}+ u_{2}p + ( u_{3}- \theta u_{2}^{\prime})}}\\{{{\tilde{\mathcal L}}_{- 3}^{( - 3 , - 1 )}}}&{{=}}&{{{v_{0}\star p}^{- 3}+ v_{- 1}\star p^{- 2}+ v_{- 2}\star p^{- 1}}}\\ \end{array}
\begin{aligned}z=2\mathrm{Re}\W_Y(z) +\frac{q}{2}\int_{z_-}^{z^+}\frac{\mathrm{d}z'}{\sqrt{z'-\delta_U}}\frac{\rho_Y(z')}{\sqrt{z-\delta_U}+\sqrt{z'-\delta_U}}+q\frac{\sqrt{z-\delta_U}}{\sqrt{t_3}}-q\frac{t_2}{2t_3}\ ,z\in[z_-,z_+]\ ,\end{aligned}
S_{D B I}= - \frac{T_{p}}{\kappa}\int_{V_{p + 1}}d^{p + 1}\xi ~ \mathrm{e}^{- \frac{\kappa \, ( 3 - p )}{2 \sqrt{2}}\, \phi}\sqrt{- \det \left[ \, g + \sqrt{2}\, \kappa \, A +{\hat{F}}\, \mathrm{e}^{- \frac{\kappa}{\sqrt{2}}\, \phi}\, \right]}~ ~ .
t_{uv}=(-\kappa n)_{v}=(\lambda\kappa\tau)t+(-\kappa_{v})n+(-\kappa\gamma)b, t_{vu}=(-\lambda\tau n+\lambda_{u}b)_{u}=(\lambda\kappa\tau)t+(-\lambda\tau_{u}-2\lambda_{u}\tau)n+(-\kappa\gamma)b.
\lim_{n\to\infty}\,\left(\frac{\lambda^j_n}{\lambda_n^{j'}}+\frac{\lambda^{j'}_n}{\lambda^j_n}+\frac{\left|c^j_n-c^{j'}_n\right|}{\lambda^j_n}+\frac{\left|t^j_n-t^{j'}_n\right|}{\lambda^j_n}\right)=\infty.
\begin{aligned}2g-2 = -2 d + \sum_{i=1}^{r_0}\left(\lambda_{0,i}- 1\right) + \sum_{i=1}^{r_1}\left(\lambda_{1,i}- 1\right)+ \sum_{i=1}^{r_\infty}\left(\lambda_{\infty,i}- 1\right) &= d-r_0 - r_1 - r_\infty.\end{aligned}
\frac{{R^{-1}+ (R^{-1})}^T}{2}= \omega^{-\frac{1}{2}}\left ( E - \sigma^2 \right )^{-1}\omega^{-\frac{1}{2}}, \frac{{R^{-1}- (R^{-1})}^T}{2}= \omega^{-\frac{1}{2}}(-\sigma) \left ( E - \sigma^2 \right )^{-1}\omega^{-\frac{1}{2}}.
F_{ext,X}= \left( \begin{array}{rcrcr}F_X F_X^X \hdots \hdots F_X^X F_X^X F_X F_X^X \hdots \vdots \vdots F_X^X \ddots \hdots \vdots \vdots \vdots \vdots F_X F_X^X F_X^X \hdots \hdots F_X^X F_X \end{array}\right).
\delta A_{m}^{\alpha}={\frac{\varphi ( x )}{u^{2}}}{\cal L}^{\alpha \beta}{\cal F}_{m n}^{\beta}u^{n}, \qquad \delta \Lambda_{m n}={\frac{\varphi ( x )}{( u^{2})^{2}}}{\cal F}_{m}^{\alpha r}u_{r}{\cal F}_{n}^{\beta s}u_{s}{\cal L}^{\alpha \beta}.
A*B=\left( \begin{array}{cc}0 1 0 0 \end{array}\right) * \left( \begin{array}{cc}0 1 1 0 \end{array}\right) \neq \left( \begin{array}{cc}0 1 1 0 \end{array}\right) * \left( \begin{array}{cc}0 1 0 0 \end{array}\right)=B*A
W_0= \tilde X^i \tilde X^j \bar W_{0i0j}=\rho W_i= \tilde X^j \tilde X^k \bar W_{0kij}=\frac{1}{2}(\beta^b-2\underline{\beta}^b)\tilde{\nabla}_b \tilde{X}^i P_k= \frac{1}{15}\bar W_{0i0k}\tilde X^i-\frac{1}{6}W_0\tilde X^k=-\frac{1}{30}(\beta^a+2\underline{\beta}^a)\tilde{\nabla}_a \tilde{X}^k-\frac{1}{10}\rho\tilde{X}^k
(1-\varepsilon-\delta) \|x\|\leq \|V\theta_gx\|, \forall x \in \mathcal{X}\implies (1-\varepsilon-\delta) \|y\|=(1-\varepsilon-\delta) \|\theta_g^{-1}y\|\leq \|Vy\|, \forall y \in \ell^p([n]),
\mbox{\sf Prob}\big(\min\limits_{x \in X}\; \|T(p) - T(x)\| > \tau\big) = 1 - \mbox{\sf Prob}\big(\min\limits_{x \in X}\; \|T(p) - T(x)\| \le \tau \big) \ge 1 - \sum\limits_{x \in X}\mbox{\sf Prob}\big(\|T(p) - T(x)\| \le \tau \big) > 1 - |X| \left(\frac{\tau}{d}\right)^k.
\frac{\lambda_1}{q}a_{\ell+1,m+1,0}=(q^3+q^2)a_{\ell+1,m+1,1}+qa_{\ell+1,m+2,0}+a_{\ell+2,m+1,0}\frac{\lambda_3}{q}a_{\ell,m,0}=qa_{\ell-1,m,}+a_{\ell,m-1,0}+(q^3+q^2)a_{\ell+1,m+1,1}.
G(k,\omega_{n})=\frac{1}{Z}\sum_{\alpha,\alpha^{\prime}}e^{-\beta E_{\alpha^{\prime}}}\frac{1-\zeta e^{\beta(E_{\alpha^{\prime}}-E_{\alpha})}}{-i\omega_{n}+E_{\alpha}-E_{\alpha^{\prime}}}\int_{k^{\prime}}dk^{\prime}\langle\alpha|\psi(k)|\alpha^{\prime}\rangle\langle\alpha^{\prime}|\psi^{\dagger}(k^{\prime})|\alpha\rangle.
R_{\alpha \beta}^{\gamma \delta}= ( \delta_{\alpha}^{\gamma}- \delta_{\alpha}^{1}\delta_{3}^{\gamma}h_{3}^{1}- \delta_{\alpha}^{2}\delta_{4}^{\gamma}h_{3}^{1}) ( \delta_{\beta}^{\delta}+ \delta_{\beta}^{1}\delta_{2}^{\delta}g_{3}+ \delta_{\beta}^{3}\delta_{4}^{\delta}g_{3})
\int D\Phi \frac{\delta}{\delta \Phi(p)}\left( \frac{1}{2}\frac{\delta}{\delta \Phi(-p)}+ P_{\Lambda}^{-1}(p) \Phi(p) + Q_{\Lambda}^{-1}(p) J(p) \right) e^{-S_{\Lambda}(\Phi,J)}= 0.
\int\int |\hat{\mu}(\omega)|^2|\hat{\mu}(g\omega)|^2 \hat{\phi^D_\delta}(\omega)\psi(2^{-j}\omega)d\omega dg=C(n) \int \left(\int_{S^{n-1}}|\hat{\mu}(t\sigma)|^2d\sigma\right)^2 \psi(2^{-j}t)\hat{\phi}^D_\delta(t) t^{n-1}dt,
\texttt{> p0 := 80352000*n*(5*n - 1)*(5*n - 2)*(5*n - 4):}&\texttt{> p1 := 25*(2592000*n\^{}4 - 16588800*n\^{}3 + 39118320*n\^{}2}\texttt{- 39189168*n + 14092603):}&\texttt{> p2 := 20*(4500*n\^{}2 - 18900*n + 19739):}