harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\varphi\left(\begin{array}{c c}a b 0 d \end{array}\right) =\left(\begin{array}{c c c}a^{2 \, p^{e_1}}c_1 \, a^{p^{e_1}}\, b^{p^{e_1}}\frac{1}{2}\, \lambda \, c_1^2 \, b^{2 \, p^{e_1}}0 1 \lambda \, c_1 \, b^{p^{e_1}}\, d^{p^{e_1}}0 0 d^{2 \, p^{e_1}}\end{array}\right) .
I_{2}=\left(\int_{X}\left(\int_{B\left(0,2|x|\right)\setminus B\left(0,\frac{|x|}{2}\right)}|G_{\xi, \sigma}(y^{-1}x)u(y)|\,dy\right)^{q}|x|^{-\beta q}dx \right)^{\frac{1}{q}}\leq C\||x|^{\alpha}u\|_{L^{p}(X)}.
\mathcal{E}(t) :=\frac{1}{2}\int_{G}\big\|\mathbf{u}(t, \cdot)\big\|_{\mathbb{R}^{k}}^{2}\mathrm{d}\mathbf{x}+\frac{\tau}{2}\int_{G}\big\|\mathbf{H}(t, \cdot) - \mathbf{F}(\mathbf{0})\big\|_{\mathbb{R}^{(k \times d) \times (k \times d)}}^{2}\mathrm{d}\mathbf{x}.
E=\begin{pmatrix}e_1 \sigma^{-1}(e_1) \cdots \sigma^{-\nu+1}(e_1) e_2 \sigma^{-1}(e_2) \ldots \sigma^{-\nu+1}(e_2) \vdots \vdots \ddots \vdots e_\nu \sigma^{-1}(e_\nu) \cdots \sigma^{-\nu+1}(e_\nu)\end{pmatrix}.
dX= \sqrt{2X}\,dB +\left(1+2\sqrt{X/\beta}W'(X)\right)\, dt, \Phi(X)= -\frac{a^2}{4}\int_{0}^t \frac{du}{X_u}- \frac{a}{\sqrt{\beta}}\int_{0}^1 \frac{L_X(x,t)}{\sqrt{x}}\circ dW(x).
\forall \, p< r \leq q, \;\;\;\; \sharp\{r\leq s \leq q\,|\, D_{\lambda}(s) = \times \}< \sharp\{r\leq s \leq q\,|\, D_{\lambda}(s) = \circ \}\sharp\{p\leq s \leq q\,|\, D_{\lambda}(s) = \times \}= \sharp\{p\leq s \leq q\,|\, D_{\lambda}(s) = \circ \}.
L{^{\pm}}_{C S}= \int_{\cal M}\varepsilon^{i j k}\bigg ({^{\pm}}A_{i}^{A B}\partial_{j}{\^{\pm}}A_{k A B}+{\frac{2}{3}}{^{\pm}}A_{i A}^{B}{^{\pm}}A_{j B}^{C}{^{\pm}}A_{k C}^{A}\bigg ) ,
\|\boldsymbol{\theta}\|^2\left(n-\prod_{l=1}^d2^{J_{n,l}}\right)\lesssim n\|K_{\boldsymbol{J}_n}(f)\|_n^2=\boldsymbol{\theta}^T\boldsymbol{\Psi}^T\boldsymbol{\Psi}\boldsymbol{\theta}\lesssim \|\boldsymbol{\theta}\|^2\left(n+\prod_{l=1}^d2^{J_{n,l}}\right).
(\begin{matrix}n\\k_{1},k_{2},...,k_{m}\end{matrix})=\frac{n!}{k_{1}!k_{2}!\cdot\cdot\cdot k_{m}!}=(\begin{matrix}k_{1}\\k_{1}\end{matrix})(\begin{matrix}k_{1}+k_{2}\\k_{2}\end{matrix})\cdot\cdot\cdot(\begin{matrix}k_{1}+k_{2}+\cdot\cdot\cdot+k_{m}\\k_{m}\end{matrix})
{j \choose i}{4m-j \choose 2m-i}{4m-i \choose 2m}= \frac{j!}{i! (j-i)!}\cdot \frac{(4m-j!}{(2m-i)! (2m+i-j)!}\cdot \frac{(4m-i)!}{(2m)! (2m-i)!}= \left[\frac{B^{B}\cdot (4-B)^{(4-B)}\cdot (4-AB)^{4-AB}}{2^{2}\cdot (AB)^{AB}\cdot ((1-A)B)^{(1-A)B}\cdot (2-AB)^{2(2-AB)}}\right]^m \cdot \left[\frac{1}{(2+AB-B)^{2+AB-B}}\right]^{m}
\begin{array}{r@{}c@{}ll}\partial_t u + \widehat A u{}={}0 \mbox{in}(0,\infty ) \times \Omega , begin{align*}5pt] u = 0 \mbox{on}(0,\infty ) \times \partial\Omega , begin{align*}5pt] u (0,\cdot ) = u_{0}\mbox{in}\Omega ,\end{array}
\begin{array}{rcr}f(Z_t)-f(Z_0) = \int_0^t{\partial_{k^i}(\tilde{f})(Z_{s_-})d\tilde{Z}^i_s}+\frac{1}{2}\int_0^t{\partial_{k^ik^j}(\tilde{f})(Z_{s_-})d[\tilde{Z}^i,\tilde{Z}^j]^c_s}+ & +\sum_{0 \leq s \leq t}(f(Z_s)-f(Z_{s_-})-\Delta{Z}^i_s\partial_{k^i}(\tilde{f})(Z_{s_-}))\end{array}
\int_{\mathbb{R}^2}\int_{\mathbb{R}}|I(\xi,\tau)|^2 d\tau d\xi \leq C \int_{\mathbb{R}^2}\int_{\mathbb{S}^1}\int_{\rho = 0}^{\rho = \infty}|\hat{F}(\xi-\rho \omega)|^2 |\hat{g}(\rho \omega)|^2 \rho^2 d\rho d\omega d\xi = C \|F\|_{L^2}^2 \|g\|_{\dot{H}^{1/2}}^2 = C \|f\|_{\dot{H}^{1/2}}^2 \|g\|_{\dot{H}^{1/2}}^2.
\begin{cases}H_A(x,0,0)>0, H_B(x,0,0)>0, begin{align*}5pt] \partial_x H_A(x,A,B)<0,\partial_x H_B(x,A,B)>0, begin{align*}5pt]\partial_BH_A(x,A,B)<0,\partial_AH_B(x,A,B)<0,\end{cases}
\sum_{n=1}^{\infty}R^{E_n}_{M^D_Y(\cdot, z)}(x_0)<\infty \sum_{n=1}\frac{2^{n(d+4)}\phi'(2^{2n})}{\phi(2^{2n})^2}\, R^{E_n}_v(x_0)<\infty\, .
S(x) =\sum_{n\leq y}\left(\frac{D(n)}{m}\right)\mu^{2}(D(n))= \sum_{n\leq y}\left(\frac{D(n)}{m}\right)\sum_{r^2\mid D(n)}\mu(r) &= \sum_{\substack{r\leq \sqrt{x}(r, m)=1}}\mu(r)\sum_{\substack{n\leq y r^{2}\mid D(n)}}\left(\frac{D(n)}{m}\right).
\theta_{n+k}(\theta_{p,q}(\pi')) =\pi(\lambda_{d},\Psi,\mu,v,(\varepsilon|((-1)^{\frac{p-q}{2}},(-1)^{\frac{p-q}{2}},\dots,(-1)^{\frac{p-q}{2}})), \qquad(\kappa|(1+n-\frac{p+q}{2},2+n-\frac{p+q}{2},\dots,k+n-\frac{p+q}{2})))
= - \mathrm{t r}\left( \Delta \bar{F}_{\mu \nu}\Delta \bar{F}_{\mu \nu}\right) - \mathrm{t r}\left( \Delta \partial_{\mu}\partial_{\sigma}\Delta \bar{F}_{\sigma \tau}\Delta \partial_{\tau}\partial_{\nu}\Delta 2 \bar{F}_{\mu \nu}\right) + \textrm{c u b i c}
\delta_q P(\xi) = -\frac{1}{6}\sum_{\alpha, \beta=1}^3\Phi_q^{\beta, \alpha}\Big( -\epsilon(\xi^0)\, \hat{\vec{\xi}}\,\Big)\:\xi^0\:\frac{e^{\frac{i}{2}q \xi}}{q \xi}\: P(\xi) \Big( 1 + \O \Big( \frac{\varepsilon}{t}\Big) \bigg) + (\deg < 2) \:.
\int_{-\infty}^{+\infty}\langle\hat{x}(t-i\hbar\beta)\hat{x}(0)\rangle e^{i\omega t}dt=e^{-\beta\hbar\omega}\int_{-\infty}^{+\infty}\langle\hat{x}(t)\hat{x}(0)\rangle e^{i\omega t}dt
-G \left(A_t + \frac{1-\tilde{Q}_t(A_t, \dot{\varphi}_0)}{q_{\sigma, t}\varphi_\sigma(0)}\right) + \tilde{T}_t(\dot{\varphi}_y) = (\ln_2 t)^{\mu-2}h_{t,\aleph}(\varphi_\sigma(y)),
a_l^0(f^{(d)}) =-\beta_{l,0}\,a_l^1(f^{(d-1)})\qquad(n\geq3), a_l^0(f^{(d)}) =-2\beta_{l,0}\,a_l^1(f^{(d-1)})\qquad(n=2), a_l^{2j}(f^{(d)}) =\beta_{l,2j-1}\,a_l^{2j-1}(f^{(d-1)})-\beta_{l,2j}\,a_l^{2j+1}(f^{(d-1)}), a_l^{2j+1}(f^{(d)}) =0
S_{\omega,\mathbf{c}}(U) :=E(U)+\omega Q(U)+\mathbf{c}\cdot \mathbf{P}(U), K_{\omega,\mathbf{c}}(U) :=\partial_{\lambda}S_{\omega,\mathbf{c}}(\lambda U)|_{\lambda =1}=2L(U)+3N(U)+2\omega Q(U)+2 \mathbf{c}\cdot \mathbf{P}(U), L_{\omega,\mathbf{c}}(U) :=K_{\omega, \mathbf{c}}(U)-3N(U)=2L(U)+2\omega Q(U)+2\mathbf{c}\cdot \mathbf{P}(U).
\ddot{\vec{\phi}}=-[(\vec{A_{1}^{2}}+\vec{A_{2}^{2}})\vec{\phi}-(\vec{A_{1}}\cdot\vec{\phi}\vec{A_{1}}+\vec{A_{2}}\cdot\vec{\phi}\vec{A_{2}}]-\frac{\kappa}{2}\vec{\phi}(\vec{\phi}^{2}-1).
\Delta_{F}^{\prime}( x^{2}) ={\frac{i \left( m^{2}\right)^{2}}{4 \pi^{2}}}\int_{0}^{\infty}\kappa d \kappa e^{- i \kappa ( m^{2}x^{2}- i \delta )}e^{- i ( 1 - i \epsilon ) / ( 4 \kappa )}.
a_{3}^{1234}=a_{3}^{1243}=a_{3}^{1324}=a_{3}^{1342}=a_{3}^{1423}=a_{3}^{1432}= a_{3}^{2134}=a_{3}^{2143}=a_{3}^{2314}=a_{3}^{2341}=a_{3}^{2413}=a_{3}^{2431}= a_{3}^{3124}=a_{3}^{3142}=a_{3}^{3214}=a_{3}^{3241}=a_{3}^{3412}=a_{3}^{3421}= a_{3}^{4123}=a_{3}^{4132}=a_{3}^{4213}=a_{3}^{4231}=a_{3}^{4312}=a_{3}^{4321}
\tilde{S}_+=\sum_{x\in \Lambda}\gamma_xc_{x\uparrow}c_{x\downarrow},\ \ \ \tilde{S}_-=\sum_{x\in \Lambda}\gamma_x c_{x\downarrow}^* c_{x\uparrow}^*,\ \ \tilde{S}^{(z)}=\frac{1}{2}|\Lambda|-\frac{1}{2}(N_{\mathrm{e}\uparrow}+N_{\mathrm{e}\downarrow}),
f_m(r)=\alpha_m r^m+\beta_m r^{-m}+ \dfrac{h_m r^m}{2m}\int_{r_e}^r s^{1-m}1_{(a,b)}(s)ds-\dfrac{h_m r^{-m}}{2m}\int_{r_e}^r s^{1+m}1_{(a,b)}(s) ds, \alpha_m=-\dfrac{h_m}{2m r_e^m}\dfrac{1}{\Big( \dfrac{R}{r_e}\Big)^m-\Big( \dfrac{r_e}{R}\Big)^m}\int_a^b \bigg(\Big( \dfrac{R}{s}\Big)^m-\Big( \dfrac{s}{R}\Big)^m\bigg) s\,ds,\beta_m=-r_e^{2m}\alpha_m,
\langle x(t, 0, u), p_{k}\rangle_2^2 \leq \lambda_{k}\int_0^t 2 x^T(s) P^{-1}B u(s)-x^T(s) P^{-1}BB^T P^{-1}x(s)ds &=\lambda_{k}\int_0^t \left\|u(s)\right\|_2^2 ds - \left\|B^T P^{-1}x(s)-u(s)\right\|_2^2ds.
a^{(k)}_{ij}= \begin{cases}\displaystyle\partial_{u_i}\partial^j_{\theta}R+\frac{\theta^{i-j}}{(i-j)!}j\ne k, \displaystyle-\frac{\Psi_{d_1-1,j+1}}{\Psi_{d_1-1,d_1}}\left(\frac{\theta^{i-d_1+1}}{(i-d_1+1)!}+\partial_{u_i}\partial^{d_1-1}_{\theta}R\right) j= k. \end{cases}
\psi(t)= \; p_0(t)\,\lambda(1-{\rm e}^{-\mu}) \bigg[1+ \sum_{i=0}^{k-1}\left(\lambda{\rm e}^{-\mu}t\right)^i C(i;k-1)\bigg] -p_0(t)\, \lambda{\rm e}^{-\mu}\sum_{i=1}^{k-1}i \left(\lambda{\rm e}^{-\mu}t\right)^{i-1}C(i;k-1)
\mathbb{E}\left( \mathbb{E}_{n}^{(N-1)}(f|A)-\mathbb{E}(f|A)\right)=\mathbb{E}\left( \mathbb{P}_{n}^{(N-1)}(f1_{A})\left( \frac{1}{\mathbb{P}_{n}^{(N-1)}(A)}-\frac{1}{P(A)}\right) \right) \neq0,
A_{p^{i}}=\{r_{i}p^{i}\,|\,r_{i}= 1,2,3,....,p^{i}-1\, \, r_{i}\nmid p^{i}\}A_{q^{j}}=\{s_{j}q^{j}\,|\,s_{j}= 1,2,3,....,q^{j}-1\, \, s_{j}\nmid q^{j}\}A_{p^{i}q^{j}}=\{t_{ij}p^iq^{j}\,|\,t_{ij}= 1,2,3,....,p^{i}q^{j}-1\, \, t_{ij}\nmid p^i\, \, t_{ij}\nmid q^j \}.
\mathcal{D}_1 ={\partial}_{1}+ i{\partial}_{2}+ [A_1 + i A_2, \cdot\; ], \quad\mathcal{D}_2 ={\partial}_{3}+ [A_3 - i \phi_0, \cdot\; ], \quad\mathcal{D}_3 = [\phi_1 - i \phi_2, \cdot\; ].
E((X^{(m)}(t+h)-X^{(m)}(t))^2| (V^{(m)}(t),X^{(m)}(t))=(v,x)) &=h \frac{1}{m}\big(\lambda\delta+(1-q)\delta^2(1-e^{-\beta v})\,(1-e^{-\alpha q(N-x)})\big).
(n+1)\sum_{k=0}^m\tbinom{m}{k}(-1)^ka_{n+k}= \sum_{k=0}^m\tbinom{m}{k}(-1)^k(n+k+1)a_{n+k}& + m\sum_{k=0}^{m-1}\tbinom{m-1}{k}(-1)^ka_{n+k+1}.
({\bf p}; l_{L}, l_{R}) \to ({\bf p}^{\prime}; l_{L}^{\prime}, l_{R}^{\prime}) = ({\bf p}+ w x{\bf A}; u_{L}-{\bf p}\cdot{\bf A}-{\frac{w x}{2}}{\bf A}\cdot{\bf A}, u_{R}-{\bf p}\cdot{\bf A}-{\frac{w x}{2}}{\bf A}\cdot{\bf A}) \quad .
(m(a_1), m(a_2)) \in \Phi_b \iff m(a_1) \wedge b = m(a_2) \wedge b \implies m_*\circ m(a_1) \wedge m_*(b) = m_* \circ m(a_2) \wedge m_*(b) \iff (m_* \circ m(a_1), m_* \circ m(a_2)) \in \Phi_{m_*(b)}.
(A-2I)y_{1}=(\begin{matrix}-1&0&0&0&0\\ 3&-1&0&0&0\\ 6&3&0&0&0\\ 10&6&3&0&0\\ 15&10&6&3&0\end{matrix})(\begin{matrix}0\\ 0\\ 0\\ 0\\ 9\end{matrix})=(\begin{matrix}0\\ 0\\ 0\\ 0\\ 0\end{matrix})=0,
\gamma_k = \frac{1}{\frac{\sin^2\theta_k}{|\hat{r}_{k,k}|^2\cos^2\theta_k}\|\tilde{\mathbf{h}}_k \hat{\mathbf{Q}}^H \|^2+\frac{\kappa}{P}\frac{1}{\rho_k^2|\hat{r}_{k,k}|^2 \cos^2\theta_k}}&= \frac{\frac{P}{\kappa}\rho_k^2|\hat{r}_{k,k}|^2 \cos^2\theta_k}{\frac{P}{\kappa}\rho_k^2 \| \tilde{\mathbf{h}}_k \hat{\mathbf{Q}}^H\|^2 \sin^2\theta_k+1}.
\left( \mathcal{R}^{\left( e\right)}\mathcal{T}^{\left( e\right)}\right) \mathcal{V}^{\left( e\right)}=\mathcal{V}^{\left( e\right)},\quad\mbox{versus}\quad\mathcal{R}^{\left( e\right)}\left(\mathcal{T}^{\left( e\right)}\mathcal{V}^{\left( e\right)}\right) =0.
\frac{\delta W}{\delta J}=\overline{\phi},\frac{\delta W}{\delta J}|_{J=0}=\langle\phi\rangle,\frac{\delta\Gamma[\overline{\phi}]}{\delta\overline{\phi}}|_{J}=-J,\frac{\delta\Gamma[\overline{\phi}]}{\delta\overline{\phi}}|_{\overline{\phi}=\langle\phi\rangle}=0.
\begin{array}{| c | c c c c c c c |}\hline{}\gamma_{\Omega}\gamma_{\Omega R_1}\gamma_{\Omega R_2}\gamma_{\Omega R_1R_2}\gamma_{R_1}\gamma_{R_2}\gamma_{R_1R_2}\hline9 \mbox{\bf 1}M N -D M N -D 5_1 M D \mbox{\bf 1}N N M -D 5_2 M \mbox{\bf 1}D N M N -D 1 \mbox{\bf 1}M N -D M N -D \hline\end{array}
\| D^N_\varepsilon\|_{\mathfrak{D}_{\infty}(X)}= \sup_{t \in \mathbb{R}}\sup_{\delta >0}{\Big\| \sum_{n=1}^{N}{\frac{a_{n}}{n^\varepsilon}\frac{1}{n^{\delta+it}}}\Big\|_X}&= \sup_{t \in \mathbb{R}}{\Big\| \sum_{n=1}^{N}{\frac{a_{n}}{n^\varepsilon}\frac{1}{n^{it}}}\Big\|_X}= \|D^N_\varepsilon\|_{\mathcal{H}_{\infty}(X)}\,.
\hat{R}_{1 2}( q^{- 1}) K_{1}^{\prime}\hat{R}_{1 2}( q^{- 1}) K_{1}^{\prime}= K_{1}^{\prime}\hat{R}_{1 2}( q^{- 1}) K_{1}^{\prime}\hat{R}_{1 2}( q^{- 1}) \quad ,
r\widetilde{F}\left(\beta \right)\approx -\frac{\pi^{4}}{45}\frac{r^{4}}{\beta^{4}}-\frac{1}{240}-\frac{\pi^{2}\left|\mu_{eff}^{2}\right|}{12}\frac{r^{2}}{\beta^{2}}-\frac{3\left|\mu_{eff}^{2}\right|}{16},
\begin{array}{c}{N_{A}= \frac 1 4 \varepsilon_{A B C D E}p^{B}\overline{{p}}^{C}q^{D}\overline{{q}}^{E}\ ,}\\{p^{A}N_{A}= 0 \quad , \qquad N^{A}N_{A}= 1 .}\\ \end{array}
| \mathcal{T}( T ) \rangle = \, U_{h ( z ) \circ z ( v )}^{\dagger}( h ( z ) \circ z ( \rho ) \circ \frac{1}{2 \pi i}\int d \rho \, b ( \rho ) ) \left( h ( z ) \circ z ( w ) \circ \mathcal{B}\right) | 0 \rangle ,
\displaystyle \frac{dx^{i}}{dt}= \rho^{i}_{\alpha}y^{\alpha}, \; \; \; \frac{d}{dt}\left(\frac{\partial L}{\partial y^{\alpha}}\right) = \rho^{i}_{\alpha}\frac{\partial L}{\partial x^{i}}- C_{\alpha \beta}^{\gamma}y^{\beta}\frac{\partial L}{\partial y^{\gamma}},
\sigma_{1}= \left( \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right) ; \sigma_{2}= \left( \begin{array}{c c}{0}&{- i}\\{i}&{0}\\ \end{array}\right) ; \sigma_{3}= \left( \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right)
G_\zeta =\tau G\tau\delta_\zeta\tau+\tau A_\ell\tau\delta_\zeta\tau+\tau R_\ell\tau\delta_\zeta\tau &=\tau\hat N\delta_\zeta\tau+\tau K\delta_\zeta\tau+\tau A_\ell\tau\delta_\zeta\tau+\tau R_\ell\tau\delta_\zeta\tau,
\mathfrak d^\nu(\mathfrak w, V) :P_+^\nu\mathfrak D\big(|D^\nu|^{1/2}\big)\to \mathbb R, \mathfrak d^\nu(\mathfrak w, V)[\varphi] :=\big\||D^\nu|^{1/2}\varphi\big\|^2 +\mathfrak w[\varphi] - \int\limits_{\mathbb{R}^2}\big\langle \varphi(\mathbf x), V(\mathbf x)\varphi (\mathbf x)\big\rangle \mathrm{d}\mathbf{x}
\int_{-\infty}^{\infty}f(a_0t)f(a_1t)\cdot\ldots\cdot f(a_nt)\dfrac{\sin(bt)}{t}\, dt = \int_{-\infty}^{\infty}f(\dfrac{\pi a_0}{b}t)f(\dfrac{\pi a_1}{b}t)\cdot\ldots\cdot f(\dfrac{\pi a_n}{b}t)\dfrac{\sin(\pi t)}{t}\, dt = 1
\begin{cases}\frac{\partial H}{\partial t}=- \frac{\partial L}{\partial t}\frac{\partial H}{\partial q_{i}}=- D_{t}^{\alpha}(\frac{\partial L}{\partial(D_{t}^{\alpha}q_{i})})=-D_{t}^{\alpha}(p_{i}^{\alpha}) \frac{\partial H}{\partial p_{i}^{\alpha}}= D_{t}^{\alpha}q_{i}\end{cases}
\overline{\mbox{\Large$v$}}_{D}(x)\equiv{2(4\pi)^{D/2}\over{\mathrm{tr}}{\bf1}\,\Lambda^{D}}{\mbox{\Large$v$}}_{D}(m)={(4\pi)^{D/2}\over{\mathrm{tr}}{\bf1}\,g^{2}\Lambda^{D-2}}\left(x-{\epsilon\over\Lambda^{2}}\right)+\int_{1}^{\infty}\!\!d\tau\,\tau^{-{D\over2}-1}\left\{{\mathrm e}^{-\tau x}G_{D}(\tau{\cal F})-1\right\}\ .
F_{\mathrm{d i v}}^{C}( \beta , \mu ) = -{\frac{1}{4 8 \kappa \beta^{2}}}\int_{\Sigma}\left[ 2 c \mu^{2}+ \ln{\frac{\mu^{2}}{m^{2}}}\left({\frac{1}{6 0}}{\frac{4 \pi^{2}}{\kappa^{2}\beta^{2}}}{\cal P}+ \left( \frac 1 6 - \xi \right) R - m^{2}\right) \right] .
\ell y\equiv -y'' + q(x)y=\mu y,x\in (0, \pi),\; \mu\in \mathbb{C}, y(0) \cos \alpha + y'(0) \sin \alpha = 0, \alpha\in (0, \pi], y(\pi) \cos \beta + y'(\pi) \sin \beta = 0,\beta\in[0, \pi),
|\mathcal{E}_{q}(ix)|^{2}=\mathcal{C}_{q}^{2}(x)+q^{1/2}\mathcal{S}_{q}^{2}(x)={}_{2}\phi_{2}\left[\begin{matrix}0, q^{1/2}-q^{1/2}, -q \end{matrix}\ ;q, -x^{2}q^{1/2}\right], \mbox{for}x\in\mathbb{R}.
\Delta_g(\Phi_{\Lambda_p}(u)-\Phi_{\Lambda_t}(u))+(m_1^2+q^2u_{\Lambda_p}^2)(\Phi_{\Lambda_p}(u)-\Phi_{\Lambda_t}(u)) &= q(u_{\Lambda_p}^2-u_{\Lambda_t}^2)(1-q\Phi_{\Lambda_t}(u))~.
A_{0}(z) =\frac{1}{z-a_{0}^{(0)}-\sum_{j=1}^{p}a_{0}^{(j)}\,A_{j-1}^{(1)}(z)}A_{j}(z) =A_{0}(z)\,A^{(1)}_{j-1}(z)1\leq j\leq p.
s_{\lambda/\mu}=(-1)^q\det \begin{pmatrix}\Bigl( s_{(\alpha_i|\beta_j)}\Bigr)_{1 \le i, j \le p}\Bigl( h_{\alpha_i - \gamma_j}\Bigr)_{\substack{1 \le i \le p 1 \le j \le q}}\Bigl( e_{\beta_j - \delta_i}\Bigr)_{\substack{1 \le i \le q 1 \le j \le p}}O\end{pmatrix},
W^{(\eta)}_{N}(t) := \log\bigg{|}\det\left(\mathcal{H}_{\mathrm{GUE}}-E-\frac{\tau+i\eta}{d_{N}}\right)\bigg{|}-\log\bigg{|}\det\left(\mathcal{H}_{\mathrm{GUE}}-E-\frac{i\eta}{d_{N}}\right)\bigg{|}
\P\Bigl(\bigl(B^{0}\cap B^{2}\bigr)^{\complement}\Bigr) \leq C'h+\P\Bigl(\bigl(B^{2}\bigr)^{\complement}\Bigr)\leq C'h+h^{-1}\frac{1}{N}\sum_{i=1}^{N}\P\bigl(B^{i,2}\bigr)\leq C'h,
\overline{J}_{p}( u,v) :=\inf\{\underset{n\rightarrow +\infty}{\lim\inf}J(u_{n},v_{n}) :u_{n}\in W^{1,1}(\Omega;\mathbb{R}^d),~v_{n}\in L^{p}(\Omega;\mathbb R^m),~u_{n}\rightarrow uL^{1},~v_{n}\rightharpoonup vL^{p}\},
c^{2}=a^{2}+b^{2}-\frac{K}{3}a^{2}b^{2}-\frac{K^{2}}{45}a^{2}b^{2}(a^{2}+b^{2})-\frac{2K^{3}}{945}a^{2}b^{2}(a^{2}-b^{2})^{2}+O(K^{4}c^{10}).
(b\theta)(a_0\otimes a_1\otimes a_2) =\theta(a_0a_1\otimes a_2)-\theta(a_0\otimes a_1a_2)+\theta(a_2a_0\otimes a_1) &={\rm Tr}(\Gamma a_0a_1[F,a_2]T-\Gamma a_0[F,a_1a_2]T)+{\rm Tr}(\Gamma a_2a_0[F,a_1]T) &=-{\rm Tr}(\Gamma a_0[F,a_1]a_2T)+{\rm Tr}(\Gamma a_0[F,a_1]Ta_2)={\rm Tr}(\Gamma a_0[F,a_1][T,a_2]).
\left(\int_{\Omega\cap\{|\eta|\geq 2|\gamma|\}}\left| |\eta\gamma|^{\alpha-Q}-|\eta|^{\alpha-Q}\right|^{Q'}d\eta\right)^{1/Q'}\leq C \left(\int_{\{|\eta|\geq 2|\gamma|\}}\left(|\gamma||\eta|^{\alpha-Q-1}\right)^{Q'}d\eta\right)^{1/Q'}\leq C|\gamma|\cdot|\gamma|^{\alpha-Q-1+\frac Q{Q'}}=C|\gamma|^{\alpha'},
\mathbf{Q}_\alpha ( d(\chi_{\sigma_L}^{-1}\circ \chi_{\sigma_R})_p ) = e^{2\pi i(k_L/3-k_R/3+1/2)}, ~~ ~~ \alpha =e^{2\pi i\left( \frac{k_R}{3}+ \frac{5}{12}\right)}\ .
X_{L}^{\mu}( \sigma_{+}) = \frac{1}{2}X^{\mu}+ \frac{1}{2}P_{L}^{\mu}\sigma_{+}+ \frac{i}{2}\sum_{k \neq 0}\frac{1}{k}\alpha_{k}^{\mu}\mathrm{e}^{- 2 i k \sigma_{+}}.
\begin{aligned}\bigg| \left( D^\gamma (v\partial_y u)\cdot D^\alpha u\right)|_{y=0}\bigg| \leq C \sum_{\tilde{\beta}\leq \beta, 0\leq j\leq k-2}\bigg(E_u^2 \Vert D^\alpha u\Vert_{L^2_y}+E_u^2\Vert \partial_y D^\alpha u\Vert_{L^2_y}\bigg) \leq \frac{\nu}{14}\Vert \partial_y D^\alpha u\Vert_{L^2_y}^2+CE_u^4+C E_u^2,\end{aligned}
H(r_{1},r_{2})=\sum_{i=1,2}(-\frac{\hbar^{2}}{2\mu}\nabla_{r_{i}}^{2}-\frac{Ze^{2}}{4\pi\epsilon_{0}r_{i}})-\frac{\hbar^{2}}{M}\nabla_{r_{1}}\cdot\nabla_{r_{2}}+\frac{e^{2}}{4\pi\epsilon_{0}r_{12}}
h_a(x):=\left\lbrace\begin{aligned}\mathbf{E}_x(T_a),x\in [a,r]\cap E, &\frac{h_{b}(b_k)-h_{b}(a_k)}{b_k-a_k}\cdot (x-a_k)+h_{b}(a_k),x\in (a_k,b_k)\subset (a,r), k\geq 1, &\mathbf{E}_rT_a,x\geq r. \end{aligned}\right.
\ln(F_m(x)) =-\sum_{k=1}^\infty \big(\ln(1-x^{km})+\ln(1-x^{km-(m-1)})+\ln(1-x^{km-1}) \big) = \sum_{k=1}^\infty \big((n-1)!\cdot \sigma'_m(n) \big) \frac{x^n}{n!}.
S_n\left( \lambda \right)=\left( p_0-\lambda q_0 \right)\prod_{k=1}^n\frac{\lambda q_k-p_k}{\lambda q_k}=\left( p_0-\lambda q_0 \right)\prod_{k=1}^n\left(1-\frac{c_k}{\lambda}\right), \mbox{where}c_k=\frac{p_k}{q_k}\in \left( 0,1 \right).
\begin{aligned}b_{k,N}(L) =\int_0^1R_N^2(L,\alpha,\Delta)e(-k\alpha)\,d\alpha =\frac{L}{N^2}\sum_{n\in\mathbb{Z}}\widehat{\Delta}\left(\frac{Ln}{N}\right)\sum_{1\leq i\not=j\leq N}\int_0^1e((na_i-na_j-k)\alpha)\,d\alpha =\frac{L}{N^2}\sum_{n\not=0}\sum_{\substack{1\leq i\not=j\leq N n(a_i-a_j)=k}}\widehat{\Delta}\left(\frac{Ln}{N}\right),\end{aligned}
\mathcal{E}(\mathcal{B}_{d, k}) = \sum_{j = 0}^{k-1}(d-1)^{k-1-j}\Psi(E_{j+1}(x, d-1)) - \sum_{j = 1}^{k-1}(d-1)^{k-1-j}\Psi(E_j(x, d-1)) .
\begin{array}{r c l}{{\hat{\phi}_{( I )}( t , \vec{x}\, )}}&{{=}}&{{e^{i ( t - t_{0}) \hat{H}_{0}}\, \hat{\phi}( t_{0}, \vec{x}\, ) \, e^{- i ( t - t_{0}) \hat{H}_{0}}\ ,}}\\{{}}&{{}}&{{}}\\{{\hat{\pi}_{( I )}( t , \vec{x}\, )}}&{{=}}&{{e^{i ( t - t_{0}) \hat{H}_{0}}\, \hat{\pi}( t_{0}, \vec{x}\, ) \, e^{- i ( t - t_{0}) \hat{H}_{0}}\ ,}}\\ \end{array}
I_2^{(d)}=-\frac{(d-4)}{(d-3)}I_1^{(d)}+\frac{m^2}{(d-3)}\left(1-\frac{q^2}{4m^2}\right)\left(I_1^{(d)}\right)' +\frac{m^{2d-10}}{(d-3)(d-4)}H_1^{(d)},
\tilde{\varphi_t}^{\star}F_0(x,\lambda y) ={F_{0}}(\tilde{\varphi_t}(x,\lambda y))={F_0}(\varphi_t (x),\lambda y^{j}\frac{\partial \varphi_t}{\partial x^{j}}) &=\lambda F_{0}(\varphi_t(x),y^{j}\frac{\partial \varphi_t}{\partial x^{j}})=\lambda F_{0}(\tilde{\varphi_t}(x,y))=\lambda \tilde{\varphi_t}^{\star}(F_0)(x,y).
\sum_{a=1}^{p_j^{2\beta_j+1}}\left(\frac{f(a)}{p_j}\right)^{2\beta_j+1}= \sum_{b=0}^{p_j^{2\beta_j}-1}\sum_{c=1}^{p_j}\left(\frac{f(bp_j+c)}{p_j}\right)=p_j^{2\beta_j}\sum_{c=1}^{p_j}\left(\frac{f(c)}{p_j}\right).
\beta+o(1) =I^\gamma_g(\hat{u}_\alpha) &=\frac{1}{2}\int_X\rho^{1-2\gamma}|\nabla\hat{u}_\alpha|_g^2\,dv_g-\frac{1}{2^*}\int_M|\hat{u}_\alpha|^{2^*}d\sigma_{\hat{h}}&=\frac{\gamma}{n}\int_X\rho^{1-2\gamma}|\nabla\hat{u}_\alpha|_g^2\,dv_g+o(1) &=\frac{\gamma}{n}\int_M|\hat{u}_\alpha|^{2^*}d\sigma_{\hat{h}}+o(1).
\footnotesize\left(\bar{\xi}\circ \tilde{\alpha}_*\right)\left([\varphi] \right)=\bar{\xi}\left( \tilde{\alpha}_*\left([\varphi] \right) \right)=\bar{\xi}\left( \alpha \circ \varphi, 0 \right)=\left(\varphi \right)+\bar{B}_n.
\left(D_{-}^{\alpha,\alpha',\beta,\beta',\gamma}t^{-\rho}\right)(x)=\frac{\Gamma\left(\beta'+\rho\right)\Gamma\left(-\alpha-\alpha'+\gamma+\rho\right)\Gamma\left(-\alpha'-\beta+\gamma+\rho\right)}{\Gamma\left(\rho\right)\Gamma\left(-\alpha'+\beta'+\rho\right)\Gamma\left(-\alpha-\alpha'-\beta+\gamma+\rho\right)}x^{\alpha+\alpha'-\gamma-\rho}.
\begin{cases}\widetilde{W}^{(u)}= W (K^{(u)})^{-1}=W+W(H^{(u)})^{-1}W, \widetilde\eta^{(u)}= \widetilde{W}^{(u)}T(u)\eta + \eta, \widetilde\theta_i^{(u)}= e^{\rho_i(u)}i\in V\end{cases}
W_{q}^{2}= q^{2}+ V^{\prime \prime}+{\frac{5}{1 6}}\left[ \left({\frac{V^{\prime \prime \prime}}{q^{2}+ V^{\prime \prime}}}\right)^{2}-{\frac{V^{\prime \prime \prime \prime}}{4 ( q^{2}+ V^{\prime \prime})}}\right] \dot{\phi}^{2}-{\frac{V^{\prime \prime \prime}}{4 ( q^{2}+ V^{\prime \prime})}}\ddot{\phi}+ \ldots ,
\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}K_{t}(x-y)u(x)u(y)dxdy\le\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}K_{t}(x-y)u^{*}(x)u^{*}(y)dxdy
f(x,y,n,z)^{-\alpha}=\sum_{m=0}^\infty n^{-2m}\sum_{k+j = m}\frac{F'_{k,j}(x,y)}{(x^2+y^2+z^2)^{\alpha+j}}&= \sum_{m=0}^{N-1}n^{-2m}\sum_{j=0}^m \frac{F'_{m-j,j}(x,y)}{(x^2+y^2+z^2)^{\alpha+j}}+ n^{-2N}\frac{G \cdot (x^2+y^2)^N}{(x^2+y^2+z^2)^{\alpha}},
P_{\min}^{\rm{O}}\left({\overline S,\overline \varepsilon ,\overline R |{W_s}}\right) = \frac{{\left({W -{W_s}}\right){N_0}}}{{{{\left|{{h_b}}\right|}^2}}}\left({{2^{\frac{{\overline R}}{{\left({W -{W_s}}\right)}}}}- 1}\right) +{p_K}\left({\overline S,{W_s}}\right),
\langle T_{i}^{k}\rangle^{(b)}_N\left( \xi_{1},\xi_{2},\xi\right) =\langle T_{i}^{k}\rangle^{(1b)}_N(\xi_{1},\xi)-A_d\delta_{i}^{k}\int_{0}^{\infty}dk\,k^{d}\int_{0}^{\infty}d\omega \frac{K'_{\omega}(k\xi_{2})}{K'_{\omega}(k\xi_{1})}\frac{F^{(i)}[N_{\omega}(k\xi ,k\xi_{1})]}{N'_{\omega}(k\xi_{1},k\xi_{2})},
\begin{array}{rcl}x_0 = \displaystyle{\frac{a+c+bd-\theta}{2b}}y_0 = \displaystyle{\frac{a+c-bd+\theta}{2d}}\end{array}\mbox{where}\theta := \sqrt{(a+c+bd)^2 - 4abd}= \sqrt{(a+c-bd)^2 + 4bcd}\,.
M_{n}^{\left( \alpha\right)}\left( \upsilon\right) =\frac{1}{\overline{\mathbf{F}}_{n}\left( X_{n-\upsilon:n}\right)}\int_{X_{n-\upsilon:n}}^{\infty}\log^{\alpha}\left( x/X_{n-\upsilon:n}\right)d\mathbf{F}_{n}\left( x\right) .
L_{0}[ a_{n}, a_{0}] = - i \sum_{n > 0}a_{n}\dot{a}_{n}^{*}- m^{2}L a_{0}^{2}- \sum_{n > 0}\frac{m^{2}L}{4 \pi n}a_{n}^{*}a_{n}\equiv - i \sum_{n > 0}a_{n}\dot{a}_{n}^{*}- H ,
\begin{array}{r@{\: = \:}l@{\qquad\ \}r@{\: = \:}l}\Delta(z^{\,i}{}_j) z^{\,i}{}_k \otimes z^{\,k}{}_j \Delta({\bf 1}){\bf 1}\otimes{\bf 1}[1ex]\varepsilon(z^{\,i}{}_j) \delta^{\,i}{}_j \varepsilon({\bf 1}) 1 begin{align*}1ex]S(z^{\,i}{}_j) (S(Z))^{\,i}{}_j S({\bf 1}){\bf 1}\end{array}
M_{1}(\mathcal{NC}(G)) =(8n-4)(8n-5)^{2}-4(8n-5)((2n-2)(4n-5)+6n) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+(4n-4)(4n-5)^{2}+36n = 320n^{3}-576n^{2}+256n = 8n(40n^{2}-72n+32)
\begin{aligned}P_{N,r}=\frac{N}{r^{N/r}(N/r)!}\int_0^1\bigl[t^r+(-1)^{r+1}(1-t)^r\bigr]^{N/r}\,dt &=\frac{N}{(N+1)r^{N/r}(N/r)!}\sum_{0\le j\le N\atop j\equiv 0(r)}\!\!\!\!(-1)^{j(r+1)/r}\frac{\binom{N/r}{j/r}}{\binom{N}{j}}\end{aligned}
\Upsilon_{\ell j}^0:=\Upsilon^0(z_\ell, z_j)=\frac{1}{|\mathbf{B}_\ell|}\frac{1}{|\mathbf{B}_j|}\int_{\mathbf{B}_\ell}\int_{\mathbf{B}_j}\Upsilon^0(x, z)\,dx\,dz,
H_{X}(R_{X}) = \sum_{[P]}\sum_{n \geq 2}\frac{1}{n}R_{X}^{n \ell(P)}\leq \sum_{[P]}\sum_{n \geq 2}R_{X}^{n \ell(P)}&= \sum_{[P]}\frac{R_{X}^{2 \ell(P)}}{1- R_{X}^{\ell(P)}}\leq \frac{1}{1-R_{X}}\sum_{[P]}R_{X}^{2 \ell(P)}= \frac{P_{X}(R_{X}^{2})}{1-R_{X}}< \infty.
[a_{m},a^{\dagger}_{m}] - [\psi_{\dot \alpha}, \; \psi^{\dagger}_{\dot \alpha}]_{+}= [\tilde{a}_{m},\tilde{a}^{\dagger}_{m}] - [\tilde{\psi}_{\dot \alpha}, \; \tilde{\psi}^{\dagger}_{\dot \alpha}]_{+}= B-F-1+\mid 0 \rangle \langle 0 \mid .
\delta_{i}= \int_{a^-_i}^{a^+_i}\int_{a^-_{i+1}}^{a^+_{i+1}}P(y_{B},y_{C}) dy_B dy_C + \int_{a^-_{i+1}}^{a^+_{i+1}}\int_{a^-_i}^{a^+_i}P(y_{B},y_{C}) dy_B dy_C \leq \frac{\eta}{2^{b}- 1}
\Vert \L[\nabla\phi, \L^{-\alpha}]\L^\alpha \psi\Vert_{L^2}\le C\Vert \nabla\phi\Vert_{W^{1, \infty}}\Vert \L^\alpha \psi\Vert_{L^2}= C\Vert \nabla\phi\Vert_{W^{1, \infty}}\Vert \psi\Vert_{D(\L^\alpha)}.
\lim_{n\to\infty}n^\alpha \rho_n^\pm( \omega ) =\varkappa(\alpha) |{b}^{\pm}|, \lim_{n\to\infty}n^\alpha \rho_n( \omega) =\varkappa(\alpha)\bigl(|{b}^{+}|^{1/\alpha}+ |{b}^{-}|^{1/\alpha}\bigr)^{\alpha}.
|A'_r(n,\epsilon)| \leq 2^{F(n)}(m(r)^2-2)^n(1+ 2^{-\beta_{\ref{boundA}}(n-2)^2}+2^{N^2-\beta (n-2)})|C_r(n-2)| &= 2^{F(n)}(m(r)^2-2)^n m(r)^{-2n+3}(1+ 2^{-\beta_{\ref{boundA}}(n-2)^2}+2^{N^2-\beta (n-2)})|C_r(n)| &= 2^{F'(n) - 2\beta n}(1+ 2^{-\beta_{\ref{boundA}}(n-2)^2}+2^{N^2-\beta (n-2)}) |C_r(n)|.
i_{\omega}\circ \widehat{h}_R= \widehat{h}_R \circ i_{\scriptstyle \widehat{(h^t)}_R(\omega)},\,{\rm and}\,\, i_{\omega}\circ \widehat{h}_L= \widehat{h}_L \circ i_{\scriptstyle \widehat{(h^t)}_L(\omega)}.
S_{q f}^{( \mathrm{e x})}\sim N_{e f f}H_{m a x}^{3}V \sim e^{- \phi_{e x i t}}l_{s}^{- 2}H_{m a x}V \sim l_{P}^{- 2}H_{m a x}V \sim S_{H B}^{( \mathrm{e x})}.