harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{V}< H^{\theta}> = \frac{1}{8 \pi}[ \frac{4}{3}\Lambda^{3}+ \frac{1}{3}m^{3}- \frac{2}{3}( \lambda \eta^{2})^{\frac{3}{2}}+ \lambda \eta^{2}\Lambda ] - \frac{1}{4}\frac{g^{2}}{\pi^{2}}m^{2}
\int_{\mathbb{R}}f(y)^2 G_{t-s}^2(x-y) dy \leq \Big( \int_{\mathbb{R}}G_{t-s}(x-y)^{2q_1}dy \Big)^{1/q_1}\| f\|^2_{L^q(\mathbb{R})}&= k \| f\|^2_{L^q(\mathbb{R})}(t-s)^{-1+1/(2q_1)},
m_{e}^{+}+\delta \ge I_{\varepsilon}(u)=\left(\frac{p-2}{2p}\right)|u^{+}|_{p,\varepsilon}^{p}+\frac{\omega^{2}q^{2}}{2\varepsilon^{n}}\int_{M}u^{2}\psi^{2}(u)d\mu_{g}-\frac{\omega^{2}q}{2\varepsilon^{n}}\int_{M}u^{2}\psi(u)d\mu_{g}\ge \ge\left(\frac{p-2}{2p}\right)|u^{+}|_{p,\varepsilon}^{p}-\frac{\omega^{2}q}{2\varepsilon^{n}}\int_{M}u^{2}\psi(u)d\mu_{g}
(x\cdot y)\cdot\alpha(z)-\alpha(x)\cdot(y\cdot z)=\varepsilon(x,y)\big((y\cdot x)\cdot\alpha(z)-\alpha(y)\cdot(x\cdot z)\big), &(x\cdot y)\cdot\alpha(z)=\varepsilon(y,z)(x\cdot z)\cdot\alpha(y).
v^1(x_2) := \left(\begin{array}{c}\beta - \xi \end{array}\right) e^{\mathrm{i}\beta x_2}; v^3(x_2) := \left(\begin{array}{c}1 0\end{array}\right); begin{align*}8pt] v^2(x_2) := \left(\begin{array}{c}-\beta - \xi \end{array}\right) e^{- \mathrm{i}\beta x_2}; v^4(x_2) := \left(\begin{array}{c}\mathrm{i}\xi x_2 1\end{array}\right).
\langle \vec{r}t \vert\vec{r}_0 0 \rangle = \bigl[{\cal K}[H_0 ,4u^2, 4u^2](\vec{u}t; \vec{u}_0 0) +{\cal K}[H_0 ,4u^2, 4u^2](\vec{u}t;\vec{u}_0 0) \bigr]
\lim_{n\to \infty}\mu (\mathcal{T}_n(\ell)) = Z(\ell)=1-C_2^{-1}\cdot 2^{-(\ell+1)^2}\left[ 1- 2^{-\ell}+O\left( 4^{-\ell}\right) \right]\ \{\mathrm{as}}\ \ \ell\to \infty\,,
Z(C(2a_1,a_2,a_3,\ldots,a_d))+ Z(C(a_3,\ldots,a_d)) &= 2a_1Z(C(a_2,a_3,\ldots,a_d))+ Z(C(a_3,\ldots,a_d))+ Z(C(a_3,\ldots,a_d)) &= 2(a_1Z(C(a_2,a_3,\ldots,a_d))+ Z(C(a_3,\ldots,a_d))) &= 2Z(C(a_1,a_3,\ldots,a_d))=2m,
\frac{(r-1)!!}{(r-2)!!}= 1+\frac{1}{r-2}+\frac{r-1}{(r-2)(r-4)}+\frac{(r-1)(r-3)}{(r-2)(r-4)(r-6)}+ & + \dots + \frac{(r-1)!!}{3(r-2)!!},
\int_{A}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}x^2{\rm d}x= \frac{A}{\sqrt{2\pi}}e^{-\frac{A^2}{2}}+\frac{1}{\sqrt{2\pi}}\int_{A}^{\infty}e^{-\frac{x^2}{2}}{\rm d}x \le \frac{A}{\sqrt{2\pi}}e^{-\frac{A^2}{2}}+\frac{e^{-\frac{A^2}{2}}}{A\sqrt{2\pi}},
:{\bf X}^{\mu}({\bf z}_{1}, \overline{{{\bf z}}}_{1}){\bf X}^{\nu}({\bf z}_{2}, \overline{{{\bf z}}}_{2}) : \, ={\cal R}{\bf X}^{\mu}({\bf z}_{1}, \overline{{{\bf z}}}_{1}){\bf X}^{\nu}({\bf z}_{2}, \overline{{{\bf z}}}_{2}){}-{\bf G}^{\mu \nu}\left({\bf z}_{1}, \overline{{{\bf z}}}_{1}|{\bf z}_{2}, \overline{{{\bf z}}}_{2}\right) ~ ,
\hat{\theta}=\frac{\mathbf{a}^H\mathbf{H}^H(\mathbf{H}\mathbf{D}\mathbf{V}\mathbf{D}^H\mathbf{H}^H\!+\!\sigma_n^2\mathbf{I}_M)^{-1}\mathbf{y}}{\sigma_{\theta}^{-2}\!+\!\mathbf{a}^H\mathbf{H}^H(\mathbf{H}\mathbf{D}\mathbf{V}\mathbf{D}^H\mathbf{H}^H\!+\!\sigma_n^2\mathbf{I}_M)^{-1}\mathbf{H}\mathbf{a}}\;,
f\left( b \right) -{\left.{D_{{0^+}}^\alpha \,f\left( t \right)}\right|_{t = b}}\frac{{{b^\alpha}}}{{\Gamma \left({\alpha + 1}\right)}}+{\left.{I_{{0^+}}^\alpha \,I_{{b^-}}^\alpha \left(\lambda + q \left(t\right) \right)f\left( t \right)}\right|_{t = b}}= 0
\left(\frac{q^{-N}}{c}\right)_n =\frac{(-1)^n(cq^{N-n+1})_n q^{\frac{n(n-1)}{2}}}{c^n q^{Nn}}, \frac{\left(q^{-N}\right)_n}{\left({\frac{q^{-N}}{c}}\right)_n}=\frac{\left(q^{N-n+1}\right)_n c^n}{\left(cq^{N-n+1}\right)_n}=\frac{(q)_N(cq)_{N-n}c^n}{(q)_{N-n}(cq)_N}
\langle\Psi_e([M]), \Phi([{\rm E}_i N])\rangle_{q=1}=\langle[M],[{\rm E}_iN]\rangle = \langle [{\rm E}_iM],[N]\rangle = \langle\Psi_e([{\rm E_i}M]),\Phi([N])\rangle_{q=1}&= \langle2 E_i\Psi_e([M]),\Phi([N])\rangle_{q=1}= \langle\Psi_e([M]),2 E_i\Phi([N])\rangle_{q=1}.
e^{\gamma}=\sqrt{\frac{2}{1}}\cdot\sqrt[3]{\frac{2^{2}}{1\cdot3}}\cdot\sqrt[4]{\frac{2^{3}\cdot4}{1\cdot3^{3}}}\cdot\sqrt[5]{\frac{2^{4}\cdot4^{4}}{1\cdot3^{6}\cdot5}}\cdot\cdot\cdot
|\Psi_{z}\rangle = \frac{1}{\sqrt{3}}\left\{|0,0,0 \rangle|\tilde{\Phi}_{I}\rangle+ z^{2}|1,1, 0 \rangle |\tilde{\Phi}_{II}\rangle+ z |1,0,-1 \rangle |\tilde{\Phi}_{III}\rangle \right\}
\mathbb{E}_{\mathsf{B}_n}\mathbb{P}_{Q^{(\mathsf{B}_n)}}\Big(\mathbf{U}(I,\mathsf{B}_n)\notin\mathcal{T}_\epsilon^n(Q_U)\Big)=\mathbb{P}_{Q_U^n}\Big(\mathbf{U}\notin\mathcal{T}_\epsilon^n(Q_U)\Big)\leq e^{-n\eta(\epsilon)},
\langle A_{\mu}^{a}( x ) A_{\nu}^{b}( y ) \rangle_{\mathrm{\scriptsize ~ F e y n}}= \delta^{a b}\, D_{\mu \nu}^{\mathrm{\scriptsize ~ F e y n}}( x , y ) = -{\frac{\Gamma ( D / 2 - 1 )}{4 \pi^{D / 2}}}\, g_{\mu \nu}\, \delta^{a b}\, [ ( x - y )^{2}]^{1 - D / 2}\, .
\widetilde{H}_{\Lambda_j}=\int_{\mathcal{Y}^d}(\Lambda_i+\widetilde{w}_{y_i})\widetilde{\delta}_{ij}\widetilde{m}dy+\int_{\mathcal{Y}^d}(\Lambda_i+\widetilde{w}_{y_i})\widetilde{w}_{y_i\Lambda_j}\widetilde{m}dy.
\Phi_m(X)(t) := \int_{0}^{t}\int_{|z| \geq 1}\phi_{m}(z) N_{X}(ds dz)+ \int_{0}^{t}\int_{0< |z| < 1}\phi_{m}(z) \widetilde{N_{X}^{0}}( ds dz),\lambda^0{\text-a.s.}, \phi_m(z) :=\ell_m^{-1}(|z|)\frac{z}{|z|},\ell_m(r):= \frac{2^{\frac{d-1}{2}}\Gamma (\frac{d+1}{2})}{m^{\frac{d+1}{2}}\int_{r}^{\infty}u^{\frac{d-3}{2}}K_{\frac{d+1}{2}}(mu)du}.
\begin{array}{l}\displaystyle [e_1,e_1]=e_{2},[e_i,e_1]=e_{i+1},(3\leq i\leq n-1),[e_1,e_3]=e_2-e_4,[e_1,e_j]=-e_{j+1}, \displaystyle (4\leq j\leq n-1,n\geq4).\end{array}
\big((M\circ J_\lambda) f\big)^{\widehat{\}}(\eta) = -i \pi^{\frac{n}{2}}2^{-n+2\lambda}\left(h_{n-2\lambda}(\eta)\frac{\partial \widehat f}{\partial \eta_n}(\eta)+ \frac{n-2\lambda}{n-\lambda-1}\eta_n h_{n-2-2\lambda}(\eta)\widehat f(\eta)\right).\!\!\!\!
\mathcal{C}(r)=\left \{A\in\mathbb{R}^{m_1\times m_2}\,:\, \left\Vert A\right\Vert_{\infty}=1, \left\Vert A\right\Vert_{L_2(\Pi)}^{2}\geq \dfrac{\log(d)}{0.0006\,\log\left (6/5\right )\,p}, \left\Vert A\right\Vert_{*}\leq \sqrt{r}\left\Vert A\right\Vert_{2}\right \}.
I_N = \sum_{d=1}^\infty z^d \sum_{\substack{\lambda \vdash d \ell(\lambda) \leq N}}s_\lambda(a_1,\dots,a_N)s_\lambda(b_1,\dots,b_N) \Omega_{\frac{q}{N}}^{-1}(\lambda) \Omega_{\frac{1}{N}}^{-1}(\lambda).
K\big(\eta^{(1)}\big) -\sigma K\big( \eta^{(2)}\big) \sigma \qquad{}=\alpha_{0}( x(u_{1},0) )^{\ast}A_{1}\alpha_{0}( x(u_{1},0)) -\alpha_{0}(x(u_{2},0))^{\ast}A_{2}\alpha_{0}(x(u_{2},0)) +O\big(\vert z_{1}\vert^{1-2\vert \kappa\vert}\big) .
\frac{du_j}{dt}=-\Phi_j\frac{1}{\Delta x}\Big( F_{j+\frac{1}{2}}^{\mbox{WENO}}-F_{j-\frac{1}{2}}^{\mbox{WENO}}\Big) -(1-\Phi_j)\frac{1}{\Delta x}\Big( F_{j+\frac{1}{2}}^{\mbox{BSQI}}-F_{j-\frac{1}{2}}^{\mbox{BSQI}}\Big).
f_1 =\Phi_1^{-1}\times \mathrm{ch}_{K_{Uv}'}f_2 =\mathrm{meas}(K_{Uv}')^{-2}\mathrm{ch}_{K_{Uv}'}\times \mathrm{ch}_{K_{Uv}'}\Phi =\Phi_1^{-1}\times \mathrm{ch}_{K_{Uv}'}
f_{bulk}(x,y) = f_{bulk}(xE_1 + yE_0) = -\frac{a^2}{2}(x^2 + y^2)+\frac{c^2}{4}\left(x^2+y^2\right)^2 -\frac{b^2}{3 \sqrt{6}}y(y^2 -3x^2).
(\begin{matrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{matrix})\equiv\frac{(-1)^{j_{1}-j_{2}-m_{3}}}{\sqrt{2j_{3}+1}}\langle j_{1}m_{1}j_{2}m_{2}|j_{3}(-m_{3})\rangle.
e^{-B_\tau}\mathcal{Q}_N^{(\beta)}e^{B_\tau}=\frac{1}{2}\sum_{p\in\Lambda^*_+}\left[ -F_p+\sqrt{F_p^2-G_p^2}\right]+ \sum_{p \in \Lambda_+^*}\sqrt{F_p^2-G_p^2}a^*_p a_p+\delta_{N}^{(\beta)}
I(f)=\int_{[0,1]^d}\left|\psi_{\boldsymbol{a},\boldsymbol{c}}(\boldsymbol{x})\frac{\partial^d}{\partial x_1\cdots\partial x_d}\psi_{\boldsymbol{b},\boldsymbol{e}}(\boldsymbol{x})+\psi_{\boldsymbol{b},\boldsymbol{e}}(\boldsymbol{x})\frac{\partial^d}{\partial x_1\cdots\partial x_d}\psi_{\boldsymbol{a},\boldsymbol{c}}(\boldsymbol{x})\right|d\boldsymbol{x}.
\frac{q}{2q -1}\mu \left( \frac{1}{a+ \frac{1}{P +{\bf D}}}\right)= \frac{q}{2q -1}\mu \left( a+ \frac{1}{P +{\bf D}}\right) = \frac{q}{2q -1}\mu \left(\frac{1}{P +{\bf D}}\right).
v = \sum_{i = 1}^{5}( x_{i}^{5}+ \Pi_{i = 1}^{5}x_{i}) .{\bf I}_{{\cal D}\left( G \right)}+ \sum_{i , j , k = 1}^{5}\{\cal F^{\prime}}_{k_{1}k_{2}k_{3}}( \Phi_{1}, \dots , \Phi_{5}) \F_{1}^{k_{1}}. F_{2}^{k_{2}}. F_{3}^{k_{3}}.
\left(\begin{array}{cccc}\cos\omega_{1}T_{0}-1 \sin\omega_{1}T_{0}0 1 -\cos\omega_{2}\left(T-T_{0}\right) 0 -\sin\omega_{2}\left(T-T_{0}\right)\end{array}\right)\left(\begin{array}{c}a_{1}a_{2}b_{1}b_{2}\end{array}\right)=\left(\begin{array}{c}0 0\end{array}\right).
df_\tau = \sum_{\sigma\in \tau^+}(df_\tau)_\sigma= \sum_{\sigma\in \tau_0^+}(df_\tau)_\sigma\ +\ \sum_{\sigma\in \tau_1^+}(df_\tau)_\sigma +\ \sum_{\sigma\in \tau_2^+}(df_\tau)_\sigma.
\max_{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)|\geq \sqrt{\frac{\frac{1}{(G_{f,\tau})}\left(\sum_{j=1}^nf_j(\tau_j)\right)^2-\sum_{j=1}^n|f_j(\tau_j)|^2}{n^2-n}}.
0<\bar{\gamma}_1:=\min_{1\leq i\leq m}\min\left\{\frac{\ell_i^-}{p_i^+},\frac{h_i^-}{q_i^+}\right\}-1\leq\bar{\gamma}_2:=\max_{1\leq i\leq m}\max\left\{\frac{\ell_i^+}{p_i^-},\frac{h_i^+}{p_i^-}\right\}-1.
\tau^{n+1}(c)(0) = \tau(\tau^{n}(c))(0) = \tau(d)(0) = d(1) - d(0)^2 = c(n+1) + \nu_k(c(1), \dots, c(n)) - \mu_n(c(0), \dots, c(k))^2 & = c(n+1) + \nu_{n+1}(c(0), \dots, c(n)),
6D(X,Y,Z) = \sum_{\{i,j,k\}=\{1,2,3\}}s_i t_j u_k + \sum_{\{i,j,k\}=\{1,2,3\}}(x_i y_j z_k) & - \sum_i s_i(y_i\overline{z_i})- \sum_i t_i(x_i\overline{z_i})- \sum_i u_i(x_i\overline{y_i}).
\begin{cases}\partial_t f^n_t = -v \partial_x f^n_t - G(M^{n-1}(t,x)) \partial_v f^n_t + \partial_v(vf^n_t)+ \sigma \partial_{vv}f^n_t \,, f^n_0(x,v)=f_0(x,v)\,, \end{cases}
\lim_{L_1\to0}\tfrac{1}{L_1}\left(L_1-R(L_1,\ell_\nu,2\ell_\mu)-R(L_1,\ell_\nu,2\ell_{\mu'})\right) &=1-\left(\frac{\cosh\frac{\ell_\nu}{2}+e^{-\ell_\mu}}{\cosh\frac{\ell_\nu}{2}+\cosh\ell_\mu}+\frac{\cosh\frac{\ell_\nu}{2}+e^{-\ell_{\mu'}}}{\cosh\frac{\ell_\nu}{2}+\cosh\ell_{\mu'}}\right).
a_{63K}(t)= \, -2(64 t^8 - 1952 t^7 - 4652 t^6 - 10172 t^5 - 28955 t^4 + 35602 t^3 & \mbox{}- 56987 t^2 + 83692 t + 9604) b_{63K}(t)= \, (t - 7)^3 (t + 2)^3 (2 t + 1)^3 (4 t - 7)^3 (2 t^2 - 91 t + 98) (4 t^2 + 13 t + 1)
\frac{1}{A_{1}\cdot\cdot\cdot A_{n}}=(n-1)!\int_{0}^{1}d\alpha_{1}\cdot\cdot\cdot\int_{0}^{1}d\alpha_{n}\frac{\delta(1-\alpha_{1}-\cdot\cdot\cdot-\alpha_{n})}{[\alpha_{1}A_{1}+\cdot\cdot\cdot+\alpha_{n}A_{n}]^{n}}.
\sigma_{(a_{i}^{\dag}a_{k})}^{-}=<{\bf A}_{-}\mid\frac{1}{2}(a_{i}^{\dag}a_{k}+a_{k}a_{i}^{\dag})\mid{\bf A_{-}}>=\alpha_{i}^{*}\alpha_{k}coth\mid{\bf A}\mid^{2}+\frac{1}{2}\delta_{ik}.
(1-2x)H(x,v)-x(1-2x)F_T(x) &=xv(1-x)H(x,v)+\frac{xv(1-x)}{1-v}\big(H(x,1)-H(x,v)\big)-\frac{x^2v}{1-v}\big(H(x,1)-vH(x,v)\big),
E^{P^n}\left[ \left|kM^{f^n}_{t \wedge \tau_{\lambda_{m}}}- k M^{f}_{t \wedge \tau_{\lambda_{m}}}\right| \right] \leq 2c \sup_{m \in \mathbb{N}}P^m (K^c) + c \sup_{\alpha \in K}\big| M^{f^n}_{t \wedge \tau_{\lambda_{m}}(\alpha)}(\alpha) - M^{f}_{t \wedge \lambda_{m}(\alpha)}(\alpha)\big| &\leq \frac{\varepsilon}{2}+ \frac{\varepsilon}{2}= \varepsilon.
|\frac{1}{r^{3/2}\langle \log(r) \rangle}\phi(r,\xi)| \leq \begin{cases}\frac{C}{\langle \log(r) \rangle}\left(\frac{\phi_{0}(r)}{r}+\frac{\log(1+r^{2})}{r^{2}}\right), r^{2}\xi \leq 4 \frac{C |a(\xi)|}{\xi^{1/4}r^{1/2}\cdot r \langle \log(r) \rangle}, r^{2}\xi > 4\end{cases}
\bar{\psi}(\gamma,\alpha,\theta) =E[\psi(W,\gamma,\alpha,\theta)]=E[Z\gamma(X)]-\theta+E[\alpha(X)\{Y-\gamma(X)\}] & =E[\alpha_{0}(X)\gamma(X)]-\theta+E[\alpha(X)\{\gamma_{0}(X)-\gamma(X)\}].
\sum_{m=k}^{n-1}(4m+3){m\choose k}=(4n-1){n\choose k+1}-4{n\choose k+2}, begin{align*}5pt]\sum_{m=k}^{n-1}(8m^2+12m+5){m\choose k}=(8n^2-4n+1){n\choose k+1}-(16n-12){n\choose k+2}+16{n\choose k+3}.
\hat{\psi}( z_{1}, . . . , z_{n}) = \int_{0}^{\infty}\prod_{m = 1}^{n}\mathrm{d}t_{m}\t_{m}^{\Delta_{m}- 1}\mathrm{e}^{- t_{m}}\psi ( t_{1}z_{1}, . . . , t_{n}z_{n}) .
\check{p}:= -p, \check{q}:=-q, \check{r}:=r-p-q; \check{x}:=1-x, begin{align*}2mm] \check{a}:= \dfrac{(r-q)(1-a)-p b}{r-p-q}, \check{b}:= \dfrac{(r-p)(1-b)-q a}{r-p-q}.
\mu\bigl(X(0,H,\alpha,\tfrac1{10}\ell(P),r) \cap F_{mN}(R')\bigr) \lesssim B\,\Theta(R)\,r^s\,\alpha^{2+s-d}\sum_{k=0}^{n_0-1}(1-\alpha)^{sk}& \leq \frac{B\,\Theta(R)\,r^s\,\alpha^{2+s-d}}{1-(1-\alpha)^s}\approx B\,\Theta(R)\alpha^{1+s-d}\,r^s.
E_2^2D_1^2+E_1E_2^2D_1+F_3^2+E_1E_2F_3+E_1^2F_4=0, &E_1D_1^2+E_1^2D_1^2+E_0F_3+E_2F_1+E_1F_2=0, &E_0^2D_1^2+E_0^2E_1D_1+F_1^2+E_0E_1F_1+E_1^2F_0=0.
f_{n,k}(r)=A\rho^{\gamma}e^{-\rho/2}((\gamma-k)\rho L_{n-|k|-1}^{(2\gamma+1)}(\rho)+Z\alpha\frac{\gamma\mu c^{2}-kE}{\hbar cC}L_{n-|k|}^{(2\gamma-1)}(\rho))
e^{- \nu}( \partial_{3}+ i A_{3}+ i \partial_{4}- A_{4}+{\frac{1}{2}}\partial_{3}\nu +{\frac{i}{2}}\partial_{4}\nu ) \psi_{1}+ e^{- \mu}( \partial_{1}+ i A_{1}- i \partial_{2}+ A_{2}+{\frac{1}{2}}\partial_{1}\mu -{\frac{i}{2}}\partial_{2}\mu ) \psi_{2}= 0 ,
J:=\int_{A(\theta)}\frac 1s (2\cos\alpha(s,r))^j ds=2^{j/2}(1-r^2)^{j/2}\int\limits_{\stackrel{\sqrt{\epsilon(r)}\leq x\leq 1/\sqrt{\epsilon(r)}}{x+1/x\leq \frac{2\cos\theta}{\sqrt{1-r^4}}}}\frac 1x(x+\frac 1x)^j dx+\cdots
\frac{\Pr(\max(Z_1,Z_2)>xt,Z_2>t)}{\Pr(\max(Z_1,Z_2)>t,Z_2>t)}\leq \frac{\Pr(Z_1+Z_2>xt,Z_2>t)}{\Pr(Z_2>t)}=\frac{\Pr(Z_1+Z_2>xt,Z_2>t)}{\Pr(Z_1+Z_2>t,Z_2>t)}.
\varepsilon_{n-1}^2 =\Vert y(x,t)-y_{n-1}(x,t)\Vert^2 &=\Vert\sum_{i=n}^{\infty}\langle y(x,t),\psi_i(x,t)\rangle\psi_i(x,t)\Vert^2 &=\sum_{i=n}^{\infty}\langle y(x,t),\psi_i(x,t)\rangle^2M^2_i .
L_5(\lambda) = \begin{bmatrix}0 0 -I_n \lambda I_n 0 -P_4 \lambda P_4-P_3 \lambda P_3 -I_n \lambda P_4-P_3 \lambda P_3-P_2 \lambda P_2 \lambda I_n \lambda P_3 \lambda P_2 \lambda P_1+P_0\end{bmatrix}.
\|Q_n\pi(a)-Q_n\pi(a)Q_n\| =\|V_n^*P_nV_n\pi(a)-V_n^*P_nV_n\pi(a)V_n^*P_nV_n\| &=\|V_n^*P_n\sigma(a)V_n-V_n^*P_n\sigma(a)P_nV_n\| &\leq\|P_n\sigma(a)-P_n\sigma(a)P_n\|\approx 0,
\begin{aligned}\widehat\alpha :=\sup\{\widehat{x}\in [\widehat{l}_0,0]\cap \widehat{I}: \tilde{f}_1(\widehat{x})\neq \tilde{f}_1(0) \}, \widehat\beta :=\inf\{\widehat{x}\in [0,\widehat{r}_0]\cap \widehat{I}: \tilde{f}_1(\widehat{x})\neq \tilde{f}_1(0)\},\end{aligned}
\|u(s,0)-u(s,h)\|_p \leq \sum_{n\geq 1}(p-1)^{n/2}\|I_n({f}_{n,h}(\cdot,0,s))\|_2=\sum_{n\geq 1}(p-1)^{n/2}[J_{n,h}(s)]^{1/2}
\sum_{n=0}^\infty \overline{U}_{2k+1,2a}(n)q^n=\sum_{n=0}^\infty \overline{U}_{2k+1,2a-1}(n)q^n=\frac{(-q^2;q^2)^2_\infty(q^{2a},q^{4k-2a+2},q^{4k+2};q^{4k+2})_\infty}{(q^2;q^2)_\infty}.
T_0 T_1 \cdots T_{N-2}\big(x_{N-1}^+\big) = T_0 T_1 \cdots T_{N-3}\big(\big[x_{N-2}^+, x_{N-1}^+\big]\big) = \big[T_0 T_1 \cdots T_{N-3}\big(x_{N-2}^+\big), T_0\big(x_{N-1}^+\big)\big] \hphantom{T_0 T_1 \cdots T_{N-2}\big(x_{N-1}^+\big)}{}= [E_{N,N-1}(1), -E_{N-1,1}(1)] = -E_{N,1}(2).
\begin{cases}(I) :\hphantom{(II)}m_1+4+2b+n_1+bn_2\equiv_n 0 (II) :\hphantom{(I)}m_2-2b-bn_1-n_2\equiv_n 0 0\leq m_1,n_1,m_1+m_2,n_1+n_2\leq 2n-6 0\leq m_2,n_2\leq n-1 m_0+m_1+m_2=n_0+n_1+n_2=2b-6\end{cases}
\int^r_0s^{n-1}\frac{d^{n-1}}{ds^{n-1}}F_x(s)ds =s^{n-1}\frac{d^{n-2}}{ds^{n-2}}F_x(s)\big|^{r}_0-\int^r_0(n-1)s^{n-2}\frac{d^{n-2}}{ds^{n-2}}F_x(s)ds &=r^{n-1}\frac{d^{n-2}}{dr^{n-2}}F_x(r)-\int^r_0(n-1)s^{n-2}\frac{d^{n-2}}{ds^{n-2}}F_x(s)ds.
\left. \frac{d a}{d T}\right|_{T = T_{c}^{+}}= \left. \frac{d a_{1}}{d T}\right|_{T = T_{c}^{-}}= - \frac{1}{4 ( 1 + c_{1})^{2}}\, ; \qquad \left. \frac{d b}{d T}\right|_{T = T_{c}^{+}}= \left. \frac{d b_{2}}{d T}\right|_{T = T_{c}^{-}}= \frac{1}{4 ( 1 - c_{1})^{2}}\, ,
\ll h_0,j_{0,1}^A\gg_\star = - \ll h_0, J \gg_{\mathbb{1}\star}= - \lim_{z \to 0}\ll h_z, J\gg_{\mathbb{1}\star}= \lim_{z \to 0}\ll h_z, j_{0,1}^A \gg_\star.
\pi_B(\bar{h}(\bar{x})) = \sigma_B^{f_1(x)}(h(x)) \bar{x}\in \bar{X}_A, \pi_A(\bar{h}^{-1}(\bar{y})) = \sigma_A^{f_2(y)}(h^{-1}(y)) \bar{y}\in \bar{X}_B
\xi_1d_i^{(1)}(z_{\ast})-(\xi_1+\xi_2)d_i^{(2)}(z_{\ast})+\cdots +(-1)^N(\xi_{N-2}+\xi_{N-1})d_i^{(N-1)}(z_{\ast})+ +(-1)^{N+1}\xi_{N-1}d_i^{(N)}(z_{\ast}) =0\ \i=1,\hdots m-1 \xi_1d_m^{(1)}(z_{\ast}) =0 (\xi_{i-1}+\xi_i)d_m^{(i)}(z_{\ast}) =0\ \i=2,\hdots,N-1 \xi_{N-1}d_m^{(N)}(z_{\ast}) =0.
P_1 = v_{(1|3,0)}x^3_1 + v_{(1|2,0)}x^2_1 + v_{(1|1,0)}x_1 + v_{(1|0,0)}+ v_{(1|-1,0)}x_1^{-1}+ v_{(1|0,1)}x_2 \,, P_2 = v_{(2|0,1)}x_2 + v_{(2|0,0)}\,,
B = \frac{4 \, \tilde{e}^{2}\, e^{\beta \mu}}{3 ( 2 \pi )^{3 / 2}\tilde{m}^{2}}\, \sqrt{\frac{\tilde{m}}{\beta}}\Bigl ( \operatorname{c o s}\frac{\theta}{3}+ \frac{1}{\sqrt{3}}\operatorname{s i n}\frac{\theta}{3}\Bigr )^{- 1}\, ,
\begin{gathered}H_{0/1}^X H_{0/1}Y=\widetilde{H}_{0/1}Y\;,\widetilde{H}_{0/1}^X \widetilde{H}_{0/1}Y=H_{0/1}Y\;, XG_{0/1}G_{0/1}^X=X\widetilde{G}_{0/1}\;, X\widetilde{G}_{0/1}\widetilde{G}_{0/1}^X=XG_{0/1}\;, \widetilde{S}= H_0^XSG_0^X\;,\end{gathered}
f^{( 5 )}( T ) ={\frac{5 1 2 \pi^{2}}{( T -{\bar{T}})^{3}}}\sum_{\ell = 1}^{6}\int \frac{d^{2}\tau}{\tau_{2}^{\; 3 / 2}}\bar{C}_{\ell}( \bar{\tau}) \, \partial_{\bar{\tau}}\left[ \tau_{2}^{2}\partial_{\bar{\tau}}( \tau_{2}^{\; 3 / 2}\sum_{p_{L}, p_{R}\in \Gamma_{\ell}}p_{L}^{3}{p}_{R}\, e^{\pi i \tau \, | p_{L}|^{2}}\, e^{- \pi i \bar{\tau}\, p_{R}^{2}}) \right] \, .
\left[~\mbox{\begin{minipage}{15cm}The following Riemann curvatures of the metric $G$ vanish on $\omega\times \{0\}$:$$R_{1212}(x',0) = R_{1213}(x', 0) = R_{1223}(x',0) = 0\mbox{for all}x'\in \omega.$$\end{minipage}}\right.
P_{0,s}^V(F_i) = P_{0,s}^V(F_1) = P_{0,s}^V \Big( \Big| \sum_{j = 1}^{j_0}(\mu_s - \Delta_{\tau,j}) \Big| > n^{\frac{1}{2}\epsilon_1 + \epsilon_2}\Big) ~,~\mbox{for all}i \geq 1.
S_{\vec a, \vec b}=T_{\vec a, \vec b}T^*_{\vec a, \vec b}= (P T D_{\vec a}+ (I-P)T D_{\vec b})(D_{\vec a}T^*P + D_{\vec b}T^*(I-P)) &= PT D_{\vec a}^2 T^* P +(I-P) T D_{\vec b}^2 T^* (I-P)+ \{R+R^* \}
\xi_{N+k+1}'(a) = T_a'(\xi_{N+k}(a)) \xi_{N+k}'(a) \biggl(1 + \frac{\partial_a T_a(\xi_{N+k}(a))}{T_a'(\xi_{N+k}(a)) \xi_{N+k}'(a)}\biggr) = (T_a^{k+1})'(\xi_N(a)) \xi_N'(a) \prod_{j=0}^k \biggl(1+\frac{\partial_a T_a(\xi_{N+j}(a))}{T_a'(\xi_{N+j}(a)) \xi_{N+j}'(a)}\biggr).
\lim_{y \to \infty}\frac{\#\{\gamma \mid \ell(\gamma) \leq y, a \leq \dfrac{\psi_{p,q}(\gamma)}{\ell(\gamma)}\leq b\}}{\#\{\gamma \mid \ell(\gamma) \leq y\}}= \frac{\arctan \dfrac{2\pi b}{pq}- \arctan \dfrac{2\pi a}{pq}}{\pi}.
U ( \tau ) = : \sum_{\mathrm{\tiny{\cal N}}}e^{i \beta_{\mathrm{\tiny{\cal N}}}( \tau )}|{\mathrm{\tiny{\cal N}}}; \tau \rangle \langle{\mathrm{\tiny{\cal N}}}; \tau | \: , \beta_{\mathrm{\tiny{\cal N}}}( \tau ) \in [ 0 , 2 \pi ) \: .
U(x,y) =\frac{1}{\Gamma(s)}\int_0^{\infty}e^{-tL}(L^s u)(x)e^{-\frac{y^2}{4t}}\frac{dt}{t^{1-s}}&=\frac{y^{2s}}{4^s\Gamma(s)}\int_0^{\infty}e^{-tL}u(x)e^{\frac{-y^2}{4t}}\frac{dt}{t^{1+s}}
\frac{\phi(n)}{n}= \sum_{d|n}\frac{\mu(d)}{d}= \sum_{d|n,~3 \nmid d}\frac{\mu(d)}{d}- \sum_{3d'=d|n,3 \nmid d'}\frac{\mu(d')}{3d'}= \frac{2}{3}\sum_{d|n,~3 \nmid d}\frac{\mu(d)}{d}.
\Phi_{\ell}^{\tau}= \Phi_{\ell}^{0}+ (ik\tau)\left(\frac{1}{ik}\partial_{\tau}\Phi_{\ell}^{\tau}\bigg|_{\tau = 0}\right) + (ik\tau)^{2}\left(\frac{1}{2(ik)^{2}}\partial_{\tau}^{2}\Phi_{\ell}^{\tau}\bigg|_{\tau = 0}\right) + \mathrm{O}(\tau^{3}),
\frac{1}{r^2-1}2\pi i\frac{2}{i}\frac{1}{2\sqrt{\left(\frac{1+r^2}{1-r^{2}}\right)^{2}-1}}\frac{1}{(k-1)!}\frac{d^{k-1}}{d^{k-1}w}\left.\left(\frac{1}{w-w_{+}}- \frac{1}{w-w_{-}}\right)\right|_{w=0}= -\frac{1}{r^2-1}2\pi\frac{1}{\sqrt{\left(\frac{1+r^2}{1-r^{2}}\right)^{2}-1}}\left(\frac{1}{w_{+}^{\tilde{k}}}- \frac{1}{w_{-}^{\tilde{k}}}\right).
\epsilon_{a , 0}( \beta ) = T \sum_{b = 1}^{N - 1}I_{a b}^{( N )}\int_{- \infty}^{\infty}\frac{d \beta^{\prime}}{2 \pi}\frac{N / 2}{\cosh [ N ( \beta - \beta^{\prime}) / 2 ]}\ln \left( 1 + e^{- \epsilon_{b , 0}( \beta^{\prime}) / T}\right)
\mathcal{A}(t_1,\ldots, t_8,p;q^r):=\frac{\theta\left(\frac{t_1}{pq^{1-r}t_3},t_3t_1,\frac{t_3}{t_1};q^r\right)}{\theta\left(\frac{t_1}{t_2},\frac{t_2}{pq^{1-r}t_1},\frac{t_1t_2}{pq^{1-r}};q^r\right)}\prod_{a=4}^8\frac{\theta\left(\frac{t_2t_a}{pq^{1-r}};q^r\right)}{\theta\left(t_3t_a;q^r\right)}
\kappa ( h ) = \frac{2 \Gamma ( h / 2 )}{\pi \Gamma ( 1 - h / 2 )}\left( \frac{\sqrt{\pi}\Gamma \left( \frac{1}{2 - h}\right)}{2 \Gamma \left( \frac{h}{4 - 2 h}\right)}\right)^{2 - h}.
F(x)-F(x+1)= \sum_{k=1}^{\infty}\frac{1}{(k-1+x+\frac{a}{2})^2}-\sum_{k=1}^{\infty}\frac{1}{(k+x+\frac{a}{2})^2}= \frac{1}{(x+\frac a2)^2}=\frac{1}{x^2+a x+b}.
I_{Vf}(x,t)|_{(x,t)\in \partial\Omega\times(0,T)}=-\frac{(Vf)(x,t)}{2}+\int_{0}^{t}\int_{\partial\Omega}\bigg( \frac{\partial \varepsilon_{n}\big(x-\xi,b(t,\tau)\big)}{\partial\nu(\xi)}a(\tau)(Vf)(\xi,\tau) &-\varepsilon_{n}\big(x-\xi,b(t,\tau)\big)a(\tau)\frac{\partial (Vf)(\xi,\tau)}{\partial\nu(\xi)}\bigg) dS_{\xi}d\tau=0.
2 \pi x \sum_{m = 1}^{\infty}\int_{0}^{1}{\frac{\pi^{2}B_{2}( t )}{( x^{2}+ ( 2 \pi ( t + m ) )^{2})^{{\frac{3}{2}}}}}d t = 2 \pi^{3}x \sum_{k = 0}^{\infty}{\binom{-{\frac{3}{2}}}{k}}{\frac{x^{2 k}}{( 2 \pi )^{3 + 2 k}}}\sum_{m = 1}^{\infty}\int_{0}^{1}{\frac{B_{2}( t )}{( t + m )^{3 + 2 k}}}d t \ .
B_r(x_0)=\{y\in \mathbb{R}^d: |x_0-y|<r\},Q_r(t_0,x_0)=(t_0-r^{2m},t_0)\times B_r(x_0), & B_r^+(x_0)=B_r(x_0)\cap \{x_d>0\},Q^+_r(t_0,x_0)=Q_r(t_0,x_0) \cap\mathcal{O}_\infty^+ .
d e t \left| \begin{array}{c c c}{Q ( \lambda - 3 i / 2 )}&{Q ( \lambda \pm i / 2 )}&{Q ( \lambda + 3 i / 2 )}\\{P ( \lambda - 3 i / 2 )}&{P ( \lambda \pm i / 2 )}&{P ( \lambda + 3 i / 2 )}\\{R ( \lambda - 3 i / 2 )}&{R ( \lambda \pm i / 2 )}&{R ( \lambda + 3 i / 2 )}\\ \end{array}\right| = T^{\pm}( \lambda ) .
|a^0_k - a^0_{k-1}| \leq C r_0^{k(2+{2\alpha})}, |a_k^j - a_{k-1}^j| \leq C r_0^{{2k\alpha}},j=1,\dots, n-1, |b_k - b_{k-1}| \leq C r_0^{{2k\alpha}}.
\sum_{j=0}^r \alpha_j(v) \sum_{u\in N^j_G(v)}\hat{w}_{i+1}(u) = \alpha_0(v)\hat{w}_{i+1}(v) + \sum_{j=1}^r \alpha_j(v) \sum_{u\in N^j_G(v)}\hat{w}_{i+1}(u) &\le \alpha_0(v)\hat{w}_{i+1}(v) + \sum_{j=1}^r \alpha_j(v) \big|N^j_G(v)\big|
\alpha^4-4(2+\beta)\alpha^3 +2 (3 + 5 \beta + \beta^2 - 4 (2 + \beta) s)\alpha^2+4(1 + \beta) (3 + 4 \beta + \beta^2 + 4 n + 4 (3 + \beta) s) \alpha &+(1 + \beta) (3 + \beta) (\beta + 4 s) (2 + \beta + 4 s)=0
f (1 + |\nabla \tau|^2 + \frac{(\Delta \tau)^2}{|H||H_0|}) = a^0 r(h_0^{(1)}- h^{(1)}) + a^0 r^2 (h_0^{(2)}- h^{(2)}) &+ a^0 r^3 \Big [(h_0^{(3)}- h^{(3)}) + \frac{(W_0) ( g_1 + \frac{g_2}{4}-\frac{3}{2}W_0 \sum_{ij}a^ia^j\tilde{X}^i \tilde{X}^j )}{2(a^0)^2}\Big ]+O(r^4).
\mathbb{E}\left[\mathcal{I}_{\mathrm{p}}\right] = \mathbb{E}\left[ \sum_{x \in \Phi_{\mathrm{P}}\cap B^c(0,R_{\mathrm{e}})}\| x \|^{-\alpha}\right] &= \frac{2 \left( \lambda \pi / F \right)^{\frac{\alpha}{2}}}{\alpha-2},
\big|v_s^{(\lambda{+}1)}\big\rangle = \big( (\lambda+1)! p^{\lambda}(\ell+1)^{\lambda}P_{\ell{-}\lambda{-}1}\!- (\ell+2) (\ell+3)\cdots (\ell+\lambda+1) P_{\ell{-}1}^{\lambda{+}1}\big)\! \big|u_0^{(\lambda)}\big\rangle .\!\!\!\!
\int_{- \infty}^{\infty}\Phi_{z^{\prime}}^{\ast \mu}( \eta ) \Phi_{z}^{\mu}( \eta ) d \eta = \delta ( z - z^{\prime}) , \int_{- \infty}^{\infty}\Phi_{z}^{\ast \mu}( \eta^{\prime}) \Phi_{z}^{\mu}( \eta ) d z = \delta ( \eta - \eta^{\prime}) .
B_{4}= - \frac{b_{4}^{\prime \prime}}{2 b_{1}}+ \frac{1}{2 b_{1}^{2}}( 4 b_{1}^{\prime \prime}b_{4}- 4 b_{2}^{\prime}b_{4}+ 6 b_{3}b_{4}+ 5 b_{1}^{\prime}b_{4}^{\prime}- 3 b_{4}^{\prime}b_{2}) + \frac{1}{b_{1}^{3}}( 3 b_{1}^{\prime}b_{2}b_{4}- 4 ( b_{1}^{\prime})^{2}b_{4})
\Psi_{\mu_+}\Big(A_{\ell_i}\,\Big|\,gF_n\setminus\big[R^g_n\cup\bigcup_{j=i}^m A_{\ell_j}\big]\Big) \geq\Psi_\mu\Big(A_{\ell_i}\,\Big|\,gF_n\setminus\big[R^g_n\cup\bigcup_{j=i}^m A_{\ell_j}\big]\Big) + \varepsilon \;.
\mathrm{c^{sm}}(\Omega_i) =\sum_{d=m-i+1}^{m}\binom{i-1}{m-d}h^{m-d}H^d &=H^{m-i+1}\sum_{d=0}^{i-1}\binom{i-1}{d}h^{i-1-d}H^d &=H^{m-i+1}(h+H)^{i-1}.
w(\lambda^o+\Lambda_0)=w(\Lambda_{i_1}+\Lambda_{i_3}-\Lambda_0) = \Lambda_{i_1}+\Lambda_{i_3}-\Lambda_0+\sum_{j=0}^{i_1-1}\alpha_j+\sum_{j=i_3}^n\alpha_j=\mu^o+\Lambda_0+\delta.
\lim\limits_{k\to\infty}\Vert \prod_{n=2}^kp^{-1}C(T_n){\bf 1}\Vert^2= \prod_{n=2}^\infty\Big(\sum_{r\in{\mathbb F}_p}\sqrt{\frac{\alpha_{1n}(r)}{p}}\,\Big)^2=\Big(H(\mu_{\alpha},\mu_{inv})\Big)^2.
\langle k_{1}, k_{2}, k_{3}| V | p \rangle = ( 1 6 \pi^{3}) \delta^{3}( p - k_{1}- k_{2}- k_{3}) \; \Theta \bigl ( p^{--}k_{1}^{--}k_{2}^{--}k_{3}^{-}\bigr ) \; u_{4}( - k_{1}, - k_{2}, - k_{3}, p ) \; .
\frac{q A K^{-1}- q^{-1}K^{-1}A}{q-q^{-1}}= a^{-1}K^{-2}+ aI,\frac{q A B^{-1}- q^{-1}B^{-1}A}{q-q^{-1}}=a B^{-2}+ a^{-1}I.
X_{1}=\left[\begin{array}[c]{ccc}0 1 0 0 0 0 0 0 0\end{array}\right] ,X_{2}=\left[\begin{array}[c]{ccc}0 0 0 0 0 1 0 0 0\end{array}\right] ,X_{3}=\left[\begin{array}[c]{ccc}0 0 1 0 0 0 0 0 0\end{array}\right]
\iota^{\odot n}(U)\pi_n(x_1,\ldots,x_n) =\pi_n(\alpha(\gamma_1)x_1,\ldots,\alpha(\gamma_n)x_n) &=\pi_n(\alpha^n(g)(x_1,\ldots,x_n)) &=T\pi_n(x_1,\ldots,x_n).
\begin{aligned}\widehat Lu = \sum_{i,j=1}^n \sqrt{x_ix_j}a_{ij}(z) u_{x_ix_j}+ \sum_{i=1}^n\sum_{l=1}^m \sqrt{x_i}\left(a_{i,n+l}(z)+a_{n+l,i}(z)\right) u_{x_iy_l}&+\sum_{l,k=1}^m a_{n+l,n+k}(z) u_{y_ly_k}+\sum_{i=1}^n b_i(z)u_{x_i}+\sum_{l=1}^m e_l(z) u_{y_l}.\end{aligned}