harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{H_{m,n}(z_1,z_2\mid q)}{(q;q)_m(q;q)_n}=\sum_{k=0}^{m\wedge n}\frac{z_1^{m-k}z_2^{n-k}}{(q;q)_{m-k}(q;q)_{n-k}(q;q)_k}.
\Gamma_a:= \left(I+a v_n^* R_n^*(a)H_{1,n}^{-1}u_{1,n}\right)^{-1}v_n^* R_n^*(a)H_{1,n}^{-1}v_n, \Gamma_b:= \left(I+b v_n^* R_n^*(b)H_{1,n}^{-1}u_{1,n}\right)^{-1}v_n^* R_n^*(b)H_{1,n}^{-1}v_n
G\left(\sum_{i=1}^{n}x_{i}e_{i}\right)=\sum_{i=1}^{n}g_{i}(x_{i})e_{i}:=\sum_{i=1}^{n}\frac{f(x_{i}+\alpha_{i})-\alpha_{i}}{f(1+\alpha_{i})-\alpha_{i}}e_{i}.
d_{\chi_i \varphi_{j}^{(\pi)}}^{\pi}= \varepsilon_i d_{\chi'_{i}\varphi_{j}'^{(\pi)}}^{\pi}, \ \varepsilon_i = \pm 1 \ (1 \leq i \leq k(b), \ 1 \leq j \leq l(b_{\pi}))
e^{- 2 \hat{\phi}}= e^{- 2 \hat{\phi}_{0}}+ \sum_{1}^{s}{\frac{2 \tilde{m}_{k}}{r_{k}}}\ , \qquad A_{u}= ( A_{u})_{0}+ \sum_{1}^{s}{\frac{2 \hat{m}_{k}}{r_{k}}}\ , \qquad A_{i}= ( A_{i})_{0}+ \sum_{1}^{s}{\frac{2 ( q_{k})_{i}}{r_{k}}}\ , \qquad i = 4 , \dots 9 .
DS_{N}(u, v) = \Big(\int_{\widetilde{\mathcal{C}}}dz \int_{\widetilde{\Sigma}}dw- \int_{\widetilde{\mathcal{C}}_{+}}dz \int_{\widetilde{\Sigma}}dw - \int_{\widetilde{\mathcal{C}}_{-}}dz \int_{\widetilde{\Sigma}}dw\Big)\Big(\cdot\Big) &=:I_{1}(u, v) - I_{2}(u, v) - I_{3}(u, v).
g_1\cdots g_k = u_1^{-1}h_1u_1\cdots u_k^{-1}h_ku_k, g_1^{n_1}\cdots g_k^{n_k}= u_1^{-1}h_1^{n_1}u_1 \cdots u_k^{-1}h_k^{n_k}u_k.
\Theta^I_a|_{loops}=-\frac{i}{2}\,(\omega+\beta^I_a)\,\langle t^I+\bar{t}^{\bar{I}}\rangle^{-1}~;{}~~~~~~~~\omega=-\frac{C}{8\pi^2}~,~~~\beta^I_a=\frac{b'^I_a}{8\pi^2}.
\iint_Q{h_\epsilon \varphi}dx dt =\iint{(u^\epsilon)_+^n \partial_x I(u^\epsilon) \varphi}dx dt & = - \iint{n (u^\epsilon)_+^{n-1}\partial_x u^\epsilon I(u^\epsilon) \varphi}dx dt - \iint{(u^\epsilon)_+^n I(u^\epsilon) \partial_x \varphi}dx dt .
V= (\nabla \phi_3)^2+ \phi_3^2\left[(\partial_i{\mathbf{n}})^2+ (\mathbf{n}\cdot{\rm rot}{\ \mathbf{n}})^2\right] -(\mathbf{n}\cdot \nabla \phi_3)^2+ ([\mathbf{n}\times \mbox{rot\}\mathbf{n}] \cdot \nabla \phi_3^2)~,
D^p_{{\mu}{\nu}}(q)=\frac{i}{q^2+i{\epsilon}}\left( -g_{{\mu}{\nu}}+\frac{n_{\mu}q_{\nu}+n_{\nu}q_{\mu}}{q_-}-n^2\frac{q_{\mu}q_{\nu}}{q_-^2}\right)-{\delta}_{{\mu}+}{\delta}_{{\nu}+}\frac{i}{q_-^2},
I_{\tilde{u}_l}^{(l)}(1)\leq 2^{2\alpha}I_{\tilde{u}_l}^{(l)}(\frac{1}{2}) \ , I_{\tilde{u}_l}^{(l)}(\frac{1}{2})> 2^{2\alpha}I_{\tilde{u}_l}^{(l)}(\frac{1}{4})\ , \tilde{u}_l(p)= 0
D_{q}(z_{0},r;p):=\frac{1}{r^2}\int\limits_{t_0-r^2}^{t_0}\|p(\cdot, t)-[p]_{B(x_{0},r)}(t)\|_{L^{\frac{3}{2},\frac{q}{2}}(B(x_{0},r))}^{\frac{3}{2}}dt
J_5 =\int_0^r z^{2}(\bar\phi+e_\delta-\delta)(1-\delta+\bar\phi+e_\delta)\cdot (e''_\delta \phi+e''_\sigma\psi){\rm{d}}z \leq\Vert z(\phi,\psi)\Vert_{L^2}\cdot \Vert z (e''_\delta,e''_\sigma)\Vert_{L^2}\cdot\Vert (\bar\phi+e_\delta-\delta)(1-\delta+\bar\phi+e_\delta)\Vert_{L^\infty},
[a + b, a^\prime + b^\prime] := ([a, a^\prime]_A + \rho_B(b)a^\prime - \rho_B(b^\prime) a ) + ([b, b^\prime]_B + \rho_A(a)b^\prime - \rho_A(a^\prime)b ), (a + b) \cdot (a^\prime + b^\prime) := (a \cdot_A a^\prime + \mu_B(b)a^\prime + \mu_B(b^\prime)a) + (b \cdot_B b^\prime + \mu_A(a) b^\prime + \mu_A(a^\prime)b ),
\mathcal{R}' := \bigg \{(\mathbf{x}',\mathbf{y}') \in \mathbb{R}^{M-M_2}\times \mathbb{R}^{M-M_2}: - C (j+M_2+1) + 3 \log M_2 \le \sum_{m'=1}^j x_{m'}', \sum_{m'=1}^j y_{m'}' \le C(M_2+1) - 3 \log(j+M_2) \; \forall \; 1 \le j \le M - M_2 \bigg \}.
\min_{X \succ 0, \, Z\succ 0}\, -\log \det (Z) \mbox{\rm s.t.}\ \ \ \ \H (X) \succeq 0, & \left[ \begin{array}{@{}c@{\,\,}c@{}}X - Z X K^{1/2}K^{1/2}X I + K^{1/2}X K^{1/2}\end{array}\right] \succeq 0 \,,
p_{n}(t) =\sum_{r=0}^{\infty}\frac{(-1)^{r+n}}{r!}\lambda^{\alpha r}t^{r}\binom{\alpha r}{n}& =\sum_{r=0}^{\infty}\frac{(-1)^{r+n}}{r!}\lambda^{\alpha r}t^{r}\frac{\Gamma (\alpha r+1)}{n!\Gamma (\alpha r-n+1)},
LLR(u_{km})=\log\left ( \frac{\sum_{i_{\bar{m}}}\sum_{s}\sum_{s'}\lambda_{k}^{i_m=1,i_{\bar{m}}}(\mathbf{s},\mathbf{s'})}{\sum_{i_{\bar{m}}}\sum_{s}\sum_{s'}\lambda_{k}^{i_m=-1,i_{\bar{m}}}(\mathbf{s},\mathbf{s'})}\right ),
(E, h^E)\cong (E_{x_0}, h^{E_{x_0}})\,, \: (L, h^L)\cong (L_{x_0}, h^{L_{x_0}})\,,\:(\Lambda (T^{*(0,1)}M), h^{\Lambda^{0,\bullet}})\cong(\Lambda (T^{*(0,1)}_{x_0}M), h^{\Lambda^{0,\bullet}_{x_0}}) &(E_p,h_p)\cong(E_{p,x_0}, h^{E_{p,x_0}})\,,
\frac{1}{(1-\alpha_{\xi}\delta^{1/n})(1+\beta_{\xi}\delta')}=\frac{1}{(1-\alpha_{\xi}\delta^{1/n})(1+\beta_{\xi}c_0\delta^{1/n})}= 1+(\alpha_{\xi}-c_0\beta)\delta^{1/n}+o(\delta^{1/n}) \leq 1+G_{c_0}(P)\delta^{1/n}+o(\delta^{1/n})\leq 1+G(P)\delta^{1/n}+o(\delta^{1/n}).
|\Phi (\xi_1', \eta_1', \xi_2 ,\eta_2) | = |\xi_1' \xi_2(\xi_1' + \xi_2) + \eta_1' \eta_2 (\eta_1' + \eta_2)|\leq 2^4 A^{-1}N_1^3 , |F (\xi_1', \eta_1', \xi_2 ,\eta_2)| = |\xi_1' \eta_2 + \xi_2 \eta_1' + 2 (\xi_1' \eta_1' + \xi_2 \eta_2)|\leq 2^4 A^{-1}N_1^2,
\begin{aligned}\int_{\Omega}\left|\left|u_n\right|^{p-2}u_n-|u|^{p-2}u\right|^{p'}dx \leq C\int_{\Omega}\left|u_n- u\right|^{p'(p-1)}dx = C\left\|u_n- u\right\|_{p}^{p}. \end{aligned}
(G_i)_{i,j}= \left(\frac{E_{2 (i-j+1)}}{4^{2-j}}+ \frac{\zeta (2 j-2 i-3)}{2 j-2 i-3}(1-4^{i-j+2})\right)\frac{4}{1-4^i}\begin{cases}\binom{2i-1}{2j-3}j\neq1 1/(2i) \end{cases}
O_A^{(gb)}(p,s)=\left(M(p,s)\right)^{-1}O_A^{(g)}(p,s/p)M(p,s),O_B^{(gb)}(p,s)=\left(M(p,s)\right)^{-1}O_B^{(g)}(p,s/p)M(p,s),
Q_{n}(x; t) = \frac{1}{H_{n}^{(-n)}(t)}\left|\begin{array}{cccc}\nu_{-n}^{(t)}\nu_{-n+1}^{(t)}\cdots \nu_{0}^{(t)}\vdots \vdots \vdots \nu_{-1}^{(t)}\nu_{0}^{(t)}\cdots \nu_{n-1}^{(t)}1 x \cdots x^n \end{array}\right|, n \geq 1,
\left[\mathrm{Td\,}(\nabla^{T^{1,0}U},T^{1,0}U)\wedge\mathrm{ch\,}(\nabla^{E\otimes L^m},E\otimes L^m)\right]_{2n}(z)\wedge d\theta &=\left[\mathrm{Td_b\,}(\nabla^{T^{1,0}X},T^{1,0}X)\wedge\mathrm{ch_b\,}(\nabla^{E},E)\wedge e^{-m\frac{d\omega_0}{2\pi}}\wedge\omega_0\right]_{2n+1}(z,\theta).
D_{F}= \left( \begin{array}{c c}{D_{+},}&{- \sqrt{2}e \phi}\\{- \sqrt{2}e \phi^{*},}&{\partial_{-}}\\ \end{array}\right) , \qquad - D_{F}^{\dag}= \left( \begin{array}{c c}{D_{-},}&{\sqrt{2}e \phi}\\{\sqrt{2}e \phi^{*},}&{\partial_{+}}\\ \end{array}\right) .
w(t,x)= \psi(\phi(T;t,x)) \exp \Bigl[ -\int_{t}^{T}\frac{\lambda(s,\phi(s;t,x))}{s}ds \Bigr] - \int_t^T \exp \Bigl[ -\int_t^{\tau}\frac{\lambda(s,\phi(s;t,x))}{s}ds \Bigr] \frac{g(\tau,\phi(\tau;t,x))}{\tau}d\tau.
\| \varphi \|^p_{2, p, \lambda}\ = \ \int | \varphi(x)|^p e(\lambda)(x) dx + \sum_i \int \Big| \frac{\partial \varphi(x)}{\partial x_i}\Big|^p e(\lambda)(x) dx + \sum_{i,j}\int \Big| \frac{\partial^2 \varphi(x)}{\partial x_i x_j}\Big|^pe(\lambda)(x)| dx.
\hat{V}\left( x \right) = V \left( x \right) \left( \begin{array}{c c c c}{0}&{1}&{\cdots}&{1}\\{1}&{0}&{\cdots}&{\vdots}\\{\vdots}&{\vdots}&{\ddots}&{1}\\{1}&{\cdots}&{1}&{0}\\ \end{array}\right)
\Vert \bar X_k - \hat X_k^{\Gamma_k}\Vert_{_2}\le K_{2, \eta}\sum_{k'=1}^{k}e^{(k-k') \Delta C_{b,\sigma}}a_{k'}\big(b, \sigma,d,k,\Delta,x_0 ,L,2+\eta\big) N_{k'}^{-1/d}
\int_{- \infty}^{\infty}d \theta \gamma_{i}( \theta ) \, e^{i t \theta}= 2 \pi \sum_{m = 0}^{\infty}( - i )^{m}\nu_{i}^{( m )}t^{m}= \pi \sum_{m = 0}^{\infty}\frac{E_{2 m}}{( 2 m ) !}( \pi \eta_{i})^{2 m}\, t^{2 m}\, \, .
(D_{1})_{j1}= \left\langle\Psi_{j}^{0}, \mathcal{A}_{0}[\phi]\phi_{k}+ \mathcal{A}_{1}[\phi]\Phi_{1}^{0}\right\rangle + \left\langle\frac{1}{ik}\partial_{\tau}\Psi_{j}^{\tau}\bigg|_{\tau = 0},\mathcal{A}_{0}[\phi]\Phi_{1}^{0}\right\rangle &= \left\langle\Psi_{j}^{0}, -kc_{k}\Phi_{1}^{0}\right\rangle &= -kc_{k}\delta_{j1},
\int_0^\infty \int_{\mathbb R^n}u(t,x)\overline{(i \partial_t + (- \Delta)^{1/2}) \phi (t,x)}dx \thinspace dt &= i \int_{\mathbb R^n}u_0(x) \overline{\phi (0,x)}dx + \int_0^\infty \int_{\mathbb R^n}|u(t,x)|^p \overline{\phi(t,x)}dx \thinspace dt
\lambda_{\imath, \jmath}=(\varpi_{\imath}, \phi_{\mathcal{Q}}^{\mathrm{tw}}\varpi_{\jmath})-(\phi_{\mathcal{Q}}^{\mathrm{tw}}\varpi_{\imath}, \varpi_{\jmath}) &=(\varpi_{\imath}, \phi_{\mathcal{Q}}^{\mathrm{tw}}\varpi_{\jmath}-\varpi_{\jmath})-(\varpi_{\imath}, (\phi_{\mathcal{Q}}^{\mathrm{tw}})^{-1}\varpi_{\jmath}-\varpi_{\jmath}).
E=[\begin{matrix}\frac{\partial v_{1}}{\partial x_{1}}&0&0\\ \frac{\partial v_{2}}{\partial x_{1}}&\frac{\partial v_{2}}{\partial x_{2}}&0\\ 0&\frac{\partial v_{3}}{\partial x_{2}}&\frac{\partial v_{3}}{\partial x_{3}}\\ 0&0&\frac{\partial v_{4}}{\partial x_{3}}\\ \end{matrix}].
\begin{aligned}\mathcal{B}_s := \|\underline{\beta}_{s}, \underline{y}\|^2_{L^2(N_{s}\cap B_{4\rho_s})}\mathcal{F}_s := \|\underline{f}_s\|^2_{L^{2}}(2\rho_s,\tau_0) \mathcal{V}_{s}:={\rm VolEx}_{N_{s}}(2\rho_s)\end{aligned}
\Gamma^{A_1,A_2, \ldots, A_n}(f_1\otimes\cdots\otimes f_n) (X_1,\ldots, X_{n-1})= & f_1(A_1)X_1f_2(A_2) \cdots f_{n-1}(A_{n-1})X_{n-1}f_n(A_n).
A =(1,1,1,-,1,1,-,1), \ \ \ \ \ \ \ \ \B=(1,1,1,-,-,-,1,-), C =(1,i,-,1,-,i,\overline{i},-,i,i,1), \ \D=(1,1,\overline{i},\overline{i},\overline{i},1,1,i,-,1,-).
g((T\times T^2)^n(a,b,a,b)) =(n\alpha,n^2\alpha+2na,2n\alpha,4n^2\alpha+4na) =(S_{2a}\times S_{2a}^2)^n(0,0,0,0) =(S_{2a}\times S_{2a}^2)^n(g(a,b,a,b)),
\nabla^{2}(\frac{\partial\varphi_{1}}{\partial x_{2}}-\frac{\partial\varphi_{2}}{\partial x_{1}})=\frac{2\kappa Gh}{D(1-\nu)}(\frac{\partial\varphi_{1}}{\partial x_{2}}-\frac{\partial\varphi_{2}}{\partial x_{1}}).
c_0(x,\xi;t) = -\ 2 \xi \cdot \partial_{x}q_{in}(x,\xi) - \{(g^{ij}-\delta^{ij}) \xi_i \xi_j, q_{in}(x,\xi)\}- 2 \Im b(t,x,\xi) q_{in}(x,\xi) := \c_0^{main}+ c_0^{err}.
S_{a,1}:\,(r_1,y_1,z,\epsilon_1) = (r_1,\left(\frac{2(\vert \beta \vert+\mathcal O(z))}{\phi^{[k]}(\vert b\vert + \mathcal O(z))(-z)}\right)^{1/k}+ \mathcal O(r_1),z,0).
{\cal L}_{g a u g e}^{e f f}= \Pi \partial_{+}A_{-}+ \partial_{-}A_{i}\partial_{+}A_{i}- \frac 1 2 \Pi^{2}- \frac 1 2 ( \partial_{i}A_{j})^{2}- \Pi \partial_{i}A_{i}+ A_{i}j^{i}+ A_{-}j^{-},
|\operatorname{Ker}(\rho_n)| = \prod_{i=0}^l \prod_{D_{n-1}^{(i)}}(d^{(i)}-1)! \prod_{j \neq i}d^{(j)}!= \left(\prod_{i=0}^{l}\frac{d^{(i)}!^{(d+1)}}{d^{(i)^{d^{(i)}}}}\right)^{d^{n-2}}.
\begin{array}{l}{{R^{( 1 )}T_{1}T_{2}= T_{2}T_{1}R^{( 1 )}\quad ,}}\\{{\,}}\\{{T_{1}^{\dagger}R^{( 2 )}T_{2}= T_{2}R^{( 2 )}T_{1}^{\dagger}\quad ,}}\\{{\,}}\\{{T_{2}^{\dagger}R^{( 3 )}T_{1}= T_{1}R^{( 3 )}T_{2}^{\dagger}\quad ,}}\\{{\,}}\\{{R^{( 4 )}T_{1}^{\dagger}T_{2}^{\dagger}= T_{2}^{\dagger}T_{1}^{\dagger}R^{( 4 )}\quad .}}\\ \end{array}
SINR^j_{1,k}= \frac{|\mathbf{v}_{1,k}^H\mathbf{H}_{1,k}\mathbf{p}_1 |^2 \alpha_{1,j}^2}{\sum^{j-1}_{l=1}|\mathbf{v}_{1,k}^H\mathbf{H}_{1,k}\mathbf{p}_1 |^2 \alpha_{1,l}^2 +\sum_{m=2}^{M}|\mathbf{v}_{1,k}^H\mathbf{H}_{1,k}\mathbf{p}_{m}|^2 +|\mathbf{v}_{1,k}|^2\frac{1}{\rho}}.
\phi^{\pm}=\sum_{k\in \mathbb{Z}}\phi_k^{\pm}(r)e^{i k\theta},\quad\mbox{or}\phi=\sum_{k\in \mathbb{Z}}\phi_k(r) e^{i k\theta}\ \mbox{with}\phi_k(r)=\big(\phi_k^+(r),\phi_k^-(r)\big),
U_{j+1}^{(k)}(a_{j+1}) \geq U_{j+1}^{(k)}(x_1^*)=U_{j+1}(x_1^*)\geq W_{j+1}(x_1^*) &=W_{j+1}^{(j+1)}(x_1^*)\geq W_{j+1}^{(k)}(x_1^*)\geq W_{j+1}^{(k)}(a_{j+1}),
X_{\ell_i^{(k)}}=\begin{pmatrix}\left\{\zeta_j, \zeta_i\right\}_t \left\{\varphi_j, \zeta_i\right\}_t begin{align*}0.3cm]\left\{\zeta_j, \varphi_i\right\}_t \left\{\varphi_j, \varphi_i\right\}_t \end{pmatrix}_{i,j}\begin{pmatrix}\dfrac{\partial\ell_i^{(k)}}{\partial \zeta_j}begin{align*}0.5cm]\dfrac{\partial\ell_i^{(k)}}{\partial \varphi_j}\end{pmatrix}_{j}
I_1(v) \sim (1-\theta^2)w_2^2\int_1^\infty \frac{x^3v^2}{\pi\bigl( w_2^2x^4 +(1-\theta^2)v^2x^2\bigr)^{3/2}}\,dx = w_2 \int_{w_2/(v(1-\theta^2)^{1/2})}^\infty \frac{1}{\pi(x^2+1)^{3/2}}\,dx \to w_2 \int_0^\infty \frac{1}{\pi (x^2+1)^{3/2}}\,dx = w_2/\pi.
\int_{0}^{\infty}\int_{0}^{t}F_u^{(C_s)}1_{\{C_{s}< u \}}dX_u d\phi(s) =\int_{0}^{t}\int_{[\sqrt{[X]}_{0-},\sqrt{[X]}_{u-})}F^{(C_s)}_u d\phi(s) dX_u .
f(x+iy) = \lim_{k\to\infty}\frac{1}{\pi}\langle F_k(\cdot, 0), P(x-\cdot, y) \rangle = \lim_{k\to\infty}\frac{1}{\pi}\int_{-\infty}^\infty F_k(t,0) P(x-t,y)\,dt &= \lim_{k\to\infty}\frac{1}{\pi}\int_{-\infty}^\infty f(t+iy_k) P(x-t,y)\,dt= \frac{1}{\pi}\langle f_b, P(x-\cdot, y)\rangle.
\|\mathcal{T}_{1,a}^{(a_1,a_2)}-\mathcal{T}_{2,a}^{(a_1,a_2)}\|_{\rm F}^2 =O(p^{a-a_1-a_2-c_1+1+2(a_1-c_2)})=O(p^{a+a_1-a_2-c_1-2c_2+1}) &=O(p^{a+a_1-a_2-l+1})=O(p^{a+a_1-a_2-2r+2}),
\Bigl ( X^{a A , b B}( x , y ) \Bigr ) = \left( \begin{array}{c c}{\frac{g}{2 m^{2}}f^{a b c}\bigl ( D_{i}\pi^{i}\bigr )^{c}}&{\delta^{a b}}\\{- \frac{1}{m^{2}}\delta^{a b}}&{0}\\ \end{array}\right) \delta ( x - y ) .
L_{3}= \dot{x}p_{x}+ \dot{y}p_{y}+ \dot{z}p_{z}- \frac{1}{2}p_{x}^{2}- \frac{1}{2}p_{y}^{2}- \frac{1}{2}p_{z}^{2}- \frac{1}{2}\omega^{2}\left( x^{2}+ y^{2}\right) - \xi \left[ p_{z}+ g ( x p_{y}- y p_{x}) \right] \ \ \ .
S_{0}^{+ Q}L = \{\left( \begin{matrix}{{0}}&{{- i}}\\{{- i}}&{{0}}\\ \end{matrix}\right) , \left( \begin{matrix}{{i}}&{{0}}\\{{0}}&{{i}}\\ \end{matrix}\right) , \left( \begin{matrix}{{- 1}}&{{0}}\\{{0}}&{{1}}\\ \end{matrix}\right) , \left( \begin{matrix}{{0}}&{{- i}}\\{{i}}&{{0}}\\ \end{matrix}\right) \}
A_{1}(s)=\frac{\zeta(2s)}{\pi}\frac{R^{2s-1}}{c_2^{2s}\,\Gamma(1-s)}\!\int\limits_{0}^{1}d\eta\,(1-\eta^2)^{-s}\!\int\limits_{0}^{\infty}\! dk\, k^{-2s+1}\left[\frac{\partial}{\partial k}-\frac{\eta}{k}\frac{\partial}{\partial \eta}\right]D_{1}(k,\eta),
\psi^{-1}(x,y)=\phi\Bigl(\frac{x+y}{2}+\frac{x-y}{2}S\Bigr)=\frac{x+y}{2}+\frac{x-y}{2}\cdot \frac{T}{\varpi'}=\frac{x+y}{2}+\frac{x-y}{2}\cdot \frac{T^{1/p^m}}{\varpi'^{1/p^m}}.
(p^v, X^v):=(D_x\phi(\hat x, \hat y), X-\delta D^2d(\hat x) + 2\delta I) \in \bar J^{2,+}_{\bar \Omega}v(\hat x), &(p^{\tilde u}, Y^{\tilde u}):=(- D_y\phi(\hat x, \hat y), Y+\delta D^2d(\hat y)) \in \bar J^{2,-}_{\bar \Omega}\tilde u(\hat y),
\mathbb{E}\left(\int\int K(x,y)\,d\nu_t(x)\,d\nu_t(y)\right) = \int_X\int_X\frac{\mathbb{P}(x,y\in E_t)p(x,d(x,y)))d(x,y)^{-s}}{p(x,t) p(y,t)}\,d\mu(x)\,d\mu(y) &\le C\int\int d(x,y)^{-s}d\mu(x)d\mu(y)<\infty\,.
\Omega_{2}^{( 1 )}( x ) = \int d^{3}y \ \Sigma_{i_{1}}^{( 1 )}( x , y ) L_{i_{1}}^{( 1 )}( y ) = \partial^{i}\alpha_{A^{i}}^{( 1 )}+ \alpha_{5}^{( 1 )}+ \alpha_{6}^{( 1 )}= 0 .
-\int_{\Omega}h(v_\rho)dx\le \sum_{i=0}^{n}\overline \sigma_i\|v_\rho\|^{q_i}_{q_n+1}, k_n \|v_\rho\|_{q_n+1}^{q_n+1}\le \sum_{i=0}^{n-1}\sigma_i\|v_\rho\|^{q_i+1}_{q_n+1}+\sum_{i=0}^{n}\overline \sigma_i\|v_\rho\|^{q_i}_{q_n+1},
\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]\frac{(-1)^{n-1}q^{n(n+1)/2}}{(1-q^n)}&=\sum_{n=1}^{N}\sum_{k=1}^{\infty}\left[\begin{matrix}N-1 n-1\end{matrix}\right]k(-1)^{n-1}q^{nk+n(n-1)/2}-q^N\sum_{n=1}^{N}\sum_{k=1}^{\infty}\left[\begin{matrix}N-1 n-1\end{matrix}\right]k(-1)^{n-1}q^{nk+n(n-1)/2}.
\left | \int_{B^+_\rho}\frac{\partial \widetilde{\varphi}_r}{\partial y_2}\frac{\partial \widetilde{\psi}}{\partial y_2}\right | \leq C \| \widetilde{\psi}\|_{L^2(B^+_\rho)}, \hbox{for every}\widetilde{\psi}\in C^\infty_0(B^+_\rho), \r=1,2,
\sum\limits_{n \in S_2^* \cup S_3^*}\left( \frac{1}{p_n}+ \frac{1}{p_{-n}}\right)^2 \leq \sum_{n = \lceil \gamma R \rceil + 1}^{\infty}\left(\frac{\gamma^2-1}{n}\right)^2 \leq (\gamma^2-1)^2 \int_{\gamma R}^{\infty}\frac{dx}{x^2}\leq \frac{(\gamma^2-1)^2}{R}\leq \frac{3(\gamma^2-1)}{R}.
f(y)-f(x) =\frac{1}{S'(y)}-\mu(y)y m((0,y)) - \left(\frac{1}{S'(x)}-\mu(x)x m((0,x))\right) &=\int_x^y \mu(t)tm'(t)dt-\mu(y)y m((0,y))+\mu(x)x m((0,x)) &=\int_x^y (\mu(t)t-\mu(y)y)m'(t)dt + (\mu(x)x-\mu(y)y) m((0,x)).
\frac{1}{[v\theta/\pi]_{H_{\scriptscriptstyle 0}}^{\frac{1}{2}}}\frac{1}{[\det(\theta/\pi)]_{H^{\mu}\phi_{\scriptscriptstyle 0}}^{\frac{1}{2}}}\frac{1}{[\det(\theta/4\pi)]_{H_{\scriptscriptstyle 0}B^{\mu}}^{\frac{1}{2}}},
\begin{aligned}\tilde{\mathbf{y}}_k = \mathbf{M}_k^H \tilde{\mathbf{H}}_k \mathbf{B}_k \mathbf{s}_k + \underbrace{\sum_{i=1, i\neq k}^{K}\mathbf{M}_k^H \tilde{\mathbf{H}}_k \mathbf{B}_i \mathbf{s}_i}_{\mathrm{interference}}+ \underbrace{\mathbf{M}_k^H\mathbf{W}_k^H\mathbf{n}_k}_{\mathrm{noise}},~k=1,2,\cdots,K,\end{aligned}
\frac{A_+ e^{5i\alpha_+}+ A_-e^{5i\alpha_-}}{A_+ e^{i\alpha_+}+ A_-e^{i\alpha_-}}= \frac{e^{6i\alpha_+}- e^{6i\alpha_-}}{e^{2i\alpha_+}-e^{2i\alpha_-}}= -\frac{\sin 3\pi j/2}{\sin \pi j/2}= -(1 + 2 \cos \pi j).
\sigma_{\varepsilon}(-2i(TS^{\ast}+S^{\ast}T)) =\sigma_{\varepsilon}(T\diamond S\circ_{\ast}iI)=\sigma_{\varepsilon}(\psi(T)\diamond \psi(S)\circ_{\ast}\psi(iI)) &=\sigma_{\varepsilon}(-2i(\psi(T)\psi(S)^{\ast}+\psi(S)^{\ast}\psi(T))).
\theta_{k_1,a}(q,u,q^{-2a-1}v) =q^{-a^2-a}T^*_{a\tau_{12}}\vartheta_{k_1}+\cdots, \theta_{k_1,a+1}(q,u,q^{-2a-1}v) =q^{-a^2-a}T^*_{(a+1)\tau_{12}}\vartheta_{k_1}+\cdots, \theta_{k_1,k_2}(q,u,q^{-2a-1}v) \equiv 0\ \q^{-a^2-a+1},\ (k_2\neq a,a+1),
\mathbf{z}(\alpha)=(1-\alpha)(\mathbf{w}_{xs}-\mu \mathbf{e}) + \alpha(\mathbf{W}_{xs}-\mathbf{R}_{xs})\widehat{\Delta \mathbf{x}}+\alpha^2 \widehat{\Delta \mathbf{x}}\circ \widehat{\Delta \mathbf{s}},
b_A(\{8',4\}) = b_2^{(0,01,01,1)}= a_2^1 + a_2^2 + a_2^{2'}+a_2^3= 9 + 12 + 11 + 14 =46, b_B(\{8',4\}) = b_1^{(0,1,1,0)}= a_1^{2''}= 4, b_C(\{8',4\}) = b_2^{(0,01,0,0)}= a_2^0 + a_2^1 = 6+9=15, b_D(\{8',4\}) = b_4^{(1,0,0,0)}= a_4^1 = 89,
p_k = \int_0^{\infty}\int\limits_0^{\frac{1}{\theta y}}\int\limits_0^{\frac{1}{\theta (y+x_k^{-1})}}\int\limits_0^{\frac{1}{\theta (y+\sum_{i=k-1}^k x_i^{-1})}}\cdots\int\limits_0^{\frac{1}{\theta (y+\sum_{i=2}^k x_i^{-1})}}f_{\xi_1,\xi_2,\cdots,\xi_k, I_k}(x_1, x_2, \cdots, x_k, y)\d x_1 \d x_2 \cdots \d x_k \d y.
\begin{cases}a \cdot ( b \cdot u) - ( b \cdot u ) \cdot a + H (a, T (b \cdot u)) - H ( T (b \cdot u), a) = (ab - ba) \cdot u + b \cdot ( a \cdot u - u \cdot a + H (a, Tu) - H (Tu, a)), (ab-ba) \cdot ( a \cdot u - u \cdot a + H (a, Tu) - H (Tu, a)) = 0, ~ b \in A, u \in M.\end{cases}
\Bigr\lvert\sum_{\substack{i\in M'_x 1\leq i\leq N}}\mu(i)\Bigr\rvert \leq \epsilon N +\Bigr\lvert\displaystyle\sum_{\substack{i\in E 1\leq i\leq N}}\mu(i)\Bigr\rvert \leq \epsilon N+m\epsilon N+\epsilon N=(m+2)\epsilon N.
f(t) \le\{M\int_0^te^{-Lu}du\}e^{Lt}=Me^{Lt}\int_0^te^{-Lu}du &=Me^{Lt}\frac{-1}{L}[e^{-Lt-1}]=\frac{M}{L}e^{Lt}[1-e^{-Lt}]=\frac{M}{L}[e^{Lt}-1]
\mu(S_{N+1}\in \cdot \ | \S_N) =\frac{\nu_{N+1}(S_N)}{\nu_{N}(S_N)}\delta_{S_N}(\cdot) + \left(1-\frac{\nu_{N+1}(S_N)}{\nu_{N}(S_N)}\right)\delta_{N+1}(\cdot)
\nabla_{M}\nabla^{M}\phi + 2 \alpha_{1}\Lambda e^{\alpha_{1}\phi}+ \frac{\alpha_{2}T_{3}}{2}\frac{1}{\sqrt{| g |}}\int d^{4}\xi \sqrt{| \gamma |}\delta^{5}( x - X ) \gamma^{\alpha \beta}\partial_{\alpha}X^{A}\partial_{\beta}X^{B}g_{A B}e^{\alpha_{2}\phi}= 0 ,
q_{\infty}^*=\max_{\mathcal F_1^c, \mathcal F_2^c}q_{1,\infty}(\mathcal F_1^c, \mathcal F_2^c)+q_{2,\infty}^*\left(\mathcal F_2^c\right) s.t. \eqref{eqn:cache-constr}.
\frac{\partial K( \delta,r)}{\partial \delta}= \frac{2}{(1+r)^2}\left( \frac{1-r^2}{2}-\frac{k_{\phi}(r)}{k'_{\phi}(r)}\right) &= \frac{2}{(1+r)^2}\left( \frac{1-r^2}{2}-\frac{r}{\psi(-r)}\right) &<0
u^{\Delta x}(x,t) = \begin{cases}u^{n-1/2}(x) + (t-t_{n-1/2}) D^{+}_{t}u^{n-1/2}(x), t \in [t_{n-1/2}, t_{n+1/2}), n \ge 1, u^{0}(x) + 2t \frac{u^{1/2}(x)-u^{0}(x)}{\Delta t}, t \in [0, t_{1/2}).\end{cases}
P_{n}\left( g \cdot v, \ell_{f,v}(g) \right)= \frac{a_2 - a_1}{\sigma \sqrt{2\pi (n-1)}}e^{- \frac{n-1}{2}\ell^2_{f,v}(g)}\big[ \nu(\varphi) + \| \varphi \|_{\gamma}o(1) \big].
|\{f_1,\ldots,f_{k}\}|= |i_{x_{\{f_1,\ldots,f_{k-1}\}}}f_{k}|= &|x_{\{f_1,\ldots,f_{k-1}\}}|+|f_{k}|= &|x_1|+\cdots+ |x_{k-1}| + |f_{k}|-1= &|f_1|+\cdots+|f_{k}|-1+(k-1)n\;.
S = \int \! \! d^{4}x \int \! \! d y \Bigl [{\cal L}_{\mathrm{5 D}}^{S U ( 5 )}+ \delta ( y ){\cal L}_{\mathrm{4 D}}^{S U ( 5 )}+ \delta ( y - \pi R ){\cal L}_{\mathrm{4 D}}^{3 - 2 - 1}\Bigr ] .
\Delta(K_i) =K_i\otimes K_i \varepsilon(K_i) = 1 S(K_i) = K_i^{-1}, \Delta(E_i) =1\otimes E_i + E_i \otimes K_i \varepsilon(E_i) = 0 S(E_i) = -E_i K_i^{-1}.
[\underline{\tau}_*]\left[x,{r^0},s;e^u\right]= \left[\underline{\tau}(x,{r^0},s);te^u \right] = \left[\underline{\tau}(x,{r^0},s);e^vt\right], = \left[\underline{\tau}(x,r^0,s);e^v\right] &.
\frac{c_{2 r - 2}^{- \nu}}{c_{2 r}^{- \nu}}= \frac{1}{\displaystyle h^{- 2}[ s^{2}- ( - \nu + 2 r - 2 )^{2}] - \frac{1}{\displaystyle h^{- 2}[ s^{2}- ( - \nu + 2 r - 4 )^{2}] - \frac{1}{\displaystyle h^{- 2}[ s^{2}- ( - \nu + 2 r - 6 )^{2}] \cdot \cdot \cdot}}}
\exp(x_\rho\otimes\partial^\rho)(1\otimes x_\mu) \exp(-x_\sigma\otimes\partial^\sigma) = 1\otimes x_\mu + x_\mu \otimes 1, \exp(x_\rho\otimes\partial^\rho)(x_\mu\otimes 1)\exp(-x_\sigma\otimes\partial^\sigma) = x_\mu \otimes 1.
\exp \!\left (\!\hat{q}\!\sum_{a>b(\ge 1)}\! x^a x^b \!\right )\!=\!\int \!{\rm D}z \exp \left (\sum_{a=1}^n \left (-\frac{\hat{q}}{2}(x^a)^2 \!+\! \sqrt{\hat{q}}z x^a \!\right ) \!\right ) \!
s^{(2)}_{4}(x) ={Q'}^{(2)}_{4}(x;-1), s^{(2)}_{31}(x) ={Q'}^{(2)}_{31}(x;-1)+{Q'}^{(2)}_{4}(x;-1), s^{(2)}_{22}(x) ={Q'}^{(2)}_{31}(x;-1), s^{(2)}_{211}(x) ={Q'}^{(2)}_{31}(x;-1)+{Q'}^{(2)}_{4}(x;-1), s^{(2)}_{1111}(x) ={Q'}^{(2)}_{4}(x;-1).
\|dE(u_n)\|_*=\sup_{\psi \in T_{u_n}S(c), \|\psi\|\leq 1}|dE(u_n)[\psi]|=\sup_{\psi \in T_{u_n}S(c), \|\psi\|\leq 1}|dE(u_n)[(\psi_{\frac{1}{t_n}})_{t_n}]|.
A(u)|\phi_{\{l;k\}}\rangle = (u+\omega+\eta\sum_{i=1}^{n-1}l_i)(u-\omega+\eta\sum_{i=1}^{m-1}k_i) |\phi_{\{l;k\}}\rangle B(u)|\phi_{\{l;k\}}\rangle = 0 C(u)|\phi_{\{l;k\}}\rangle \neq 0 D(u)|\phi_{\{l;k\}}\rangle = \eta^{-2}|\phi_{\{l;k\}}\rangle.
A_i = \begin{cases}I_{M_i - M_{i-1}}^{\circ}(p^{-1}; x_{M_{i-1}+ 1}, \dots, x_{M_i}) i\in[r-1], I_{n - M_{r-1}}(p^{-1}; x_{M_{r-1}+ 1}, \dots, x_n) i=r. \end{cases}
(-1)^{\frac{k(k-1)}{2}}\frac{G^2(k+1)}{G(2k+1)}x^{\frac{k^2-k-1}{2}}\sum_{i=0}^\infty \left( \sum_{q=0}^{\min(i, \lfloor \frac{k+1}{2}\rfloor)}\sum_{m=0}^{k+1-2q}\Big( \prod_{s=0}^{m-1}(i-q+\frac{k^2}{2}-s) \Big) b_{q,m}^{(k+1)}a_{i-q}\right)x^i=0.
f_{n-4}^{(0)}= f_{n-4}^{(1)}+A_{2-n}A_{4-n}A_{6-n}\psi_{n-4}^{(0)}+K_{6-n}\psi_{n-4}^{(0)}= O(r^{n-3})\log r+O(r^{n-3})+O(r^{n-2})\log^2 r.
{}_2 F_1 (\alpha,\beta ;\gamma ;z)=1+ \frac{\alpha \beta}{\gamma}z + \frac{\alpha (\alpha +1) \beta (\beta +1 )}{2! \gamma (\gamma +1)}z^2 + \dots +\frac{(\alpha )_n (\beta )_n}{n!(\gamma )_n}z^n + \cdots,
\begin{array}{rcl}T_3=\displaystyle\dfrac{t^4}{120}\sum_{i=1}^{k-1}f_{iiii}, \hbox{and}T_4=\displaystyle\dfrac{t^4}{36}\,\sum_{\substack{i,j=1 j>i}}^{k-1}{f}_{iijj}.\end{array}
{\rm Tr}( |D|^{-p-1}(1-f(s|D|)))\leq{\rm Tr}(|D_0|^{-p-1}(1-f(s|D_0|)))=\sum_{k=1}^{\infty}(k^{1/(p+\varepsilon)})^{-p-1}(1-e^{-(sk^{1/(p+\varepsilon)})^{p+1}}) &\leq\int_0^{\infty}u^{-(p+1)/(p+\varepsilon)}(1-e^{-(su^{1/(p+\varepsilon)})^{p+1}})du=s^{1-\varepsilon}\int_0^{\infty}v^{-(p+1)/(p+\varepsilon)}(1-e^{-v^{(p+1)/(p+\varepsilon)}})dv.
\xi^{(k,\ell)}_i(t):=\int_\Sigma \left\langle \mathcal{L}_{\Sigma}^{\ell}\partial^{k}_t u,\varphi_{i}\right\rangle\, d\mu,\ \mathcal{E}^{(k,\ell)}_i(t):=\int_\Sigma \left\langle \mathcal{L}_{\Sigma}^{\ell}\partial^{k}_t E_2(u),\varphi_{i}\right\rangle\, d\mu
B_j^{m, K}(x)=\begin{cases}\frac{(x-a_j)B_j^{m-1, K}(x) + (a_{j+m+1}-x)B_{j+1}^{m-1, K}(x)}{a_{j+m+1}-a_j},\a_j< a_{j+m+1},a_j\leq x < a_{j+m+1}, 0,\ \,.\end{cases}
(A(t) + \zeta I)^{-1}A(t) F(t) = t^2 \Xi(t,\zeta) SP + t^3 Z \Xi(t,\zeta) SP+ t^3 \Xi(t,\zeta) K - t^5 \Xi(t,\zeta) N \Xi(t,\zeta) SP + Y(t,\zeta),
\Phi(x) := \prod_{i \in I}l_i(x)^{l_i(x)}\prod_{j \in J}m_j(x)^{-m_j(x)}, u(x; n) := (2 \pi)^{\frac{|I|-|J|}{2}}\prod_{i \in I}l_i(x)^{\alpha_i(n) - \frac{1}{2}}\prod_{j \in J}m_j(x)^{\frac{1}{2}- \beta_j(n)},
\sum_{m_1,m_2,m_3=0}^{p-1}\binom{m_1+m_2+m_3}{m_1,m_2,m_3}=CT\frac1{(xyz)^{p-1}}\prod_{cyc}\frac{(x+y+z)^p-x^p}{y+z}&=CT\frac1{(xyz)^{p-1}}\prod_{cyc}\left((y+z)^{p-1}+\sum_{i=1}^{p-1}\binom{p}ix^{p-1}(y+z)^{i-1}\right).
g ( x , y ) = - \frac{1}{2}\left( 2 \frac{y}{1 - \beta x}- \frac{1}{\tilde{A}}\right) \pm \sqrt{\frac{1}{4}\left( 2 \frac{y}{1 - \beta x}- \frac{1}{\tilde{A}}\right)^{2}- \frac{y^{2}}{( 1 - \beta x ) ( 1 - \beta y )}}.
B_{n}(u): =\sum_{t=1}^{n}m_{t}\left( b(x_{t}^{T}\beta^\prime + x_t^T u/\sqrt{n})- b(x_{t}^{T}\beta^{\prime}) - \dot{b}(x_{t}^{T}\beta^{\prime})x_{t}^{T}u/\sqrt{n}\right)
| I_{2j}- \int_{K_j}\frac{e^{-itx}f_t (x) G(0)}{x- \lambda - i\delta}dx | = | \int_{K_j}\frac{e^{-itx}f_t (x) G(x - \Re \lambda)}{x-\lambda - i\delta}dx - \int_{K_j}\frac{e^{-itx}f_t (x)G(0)}{x- \lambda - i\delta}dx | \leq \frac{1}{G(0)}\sup_{K_j}\frac{|f_t (x)|}{|x- \lambda - i\delta|}\int_{K_j}| G(x - \Re \lambda) - G(0) | dx.
\begin{array}{lllll}\mbox{for any $\psi$,}\mbox{at}\phi=+1 ~:~ (i^k,1,\pm\sqrt{\psi},z_4,z_5,1,0) ,(k=0,...,3)~, & \mbox{or}\phi=-1 ~:~ (i^k e^{i\pi/4},1,\pm \sqrt{\psi},z_4,z_5,1,0) ,(k=0,...,3)~, \mbox{and for any $\phi$,}\mbox{at}\psi=\pm \sqrt{\phi^2-1}~:~ (\phi^{1/4}i^k,1,0,z_4,z_5,1,0) ,(k=0,...,3)~,\end{array}